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Wikipedia:Jos Baeten#0
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Josephus C.M. Baeten (born 15 June 1954) is a Dutch computer scientist and mathematician, who has published on process calculus, concurrency theory, formal methods, model-based software engineering, model-based systems engineering and theory of computation. == Early life and education == Jos Baeten was born in Tilburg. He received his Ph.D. from the University of Minnesota in 1985, advised by Wayne Richter. == Career == He was a researcher at Centrum Wiskunde & Informatica (CWI) and the University of Amsterdam until 1991, when he was appointed as a full professor at the Eindhoven University of Technology. There, he was professor at the Department of Mathematics and Computer Science from 1991 until 2015 (in two periods, he was dean of the department), and professor at the Department of Mechanical Engineering from 2010 until 2012. In 2011, he returned to CWI as its director, and in 2015, he returned to the University of Amsterdam as professor of theory of computing at the Institute of Logic, Language and Computation. He retired from both positions in 2020, and at that time became a CWI Fellow. Baeten chaired the steering committee of the CONCUR conferences 1991-2018 and was president of ERCIM 2018-2019. Since 2010, he is a member of the Koninklijke Hollandsche Maatschappij der Wetenschappen. During the 75th anniversary of CWI, he received a royal decoration of Officer in the Order of Orange-Nassau. == References ==
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Wikipedia:Josef Anton Gmeiner#0
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Josef Anton Gmeiner (1862-1926) was an Austrian mathematician working in number theory and mathematical analysis. Gmeiner studied physics and mathematics at the University of Innsbruck from 1885. In 1890 he passed the examination qualifying him to teach at Gymnasien. After two years as an assistant at the University of Innsbruck's physical institute, he worked as an auxiliary teacher at secondary schools in various locations, including Graz, Fiume, Klagenfurt and Vienna. He earned his doctorate at the University of Innsbruck in 1895, under the joint supervision of Leopold Gegenbauer and Otto Stolz. He then found employment at the German-language Gymnasium in Pula. He became in 1899 a teacher at a Realschule in Vienna and in 1900 a docent in mathematics at the University of Vienna and at TH Wien. At the German University in Prague he was appointed professor extraordinarius in 1901 and promoted to professor ordinarius in 1904. In 1906 he returned to the University of Innsbruck in the professorial chair vacated by the death of Stolz in 1905. == Selected publications == Gmeiner, J. A. (1890). Die Ergänzungssätze zum bicubischen Reciprocitäts-Gesetze. SBer. Kais. Akad. Wissensch. Wien, 100, 1330-1361. Theoretische Arithmetik (2 volumes: vol. 1, 1900, vol. 2, 1902) by Otto Stolz & J. A. Gmeiner == References ==
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Wikipedia:Josef Teichmann#0
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Josef Teichmann (* 27 August 1972 in Lienz) is an Austrian mathematician and professor at ETH Zürich working on mathematical finance, stochastic analysis and machine learning. After studying mathematics at the University of Graz, he pursued his PhD at the University of Vienna. The title of his dissertation in 1999 under the supervision of Peter W. Michor was "The Theory of Infinite-Dimensional Lie Groups from the Point of View of Functional Analysis". After working at the Vienna University of Technology, he obtained the Habilitation there in 2002. Since June 2009 he has been a professor at the Department of Mathematics at ETH Zürich. Since August 2023 he has been chair of the Department of Mathematics at ETH Zürich. In 2005 he was awarded the Prize of the Austrian Mathematical Society and in 2006 the Start-Preis of the FWF. In 2014 he was awarded the Louis Bachelier Prize by the French Academy of Sciences. == External links == Teichmann's personal website Portrait on the website of FWF
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Wikipedia:Joseph Bernstein#0
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Joseph Bernstein (sometimes spelled I. N. Bernshtein; Hebrew: יוס(י)ף נאומוביץ ברנשטיין; Russian: Иосиф Наумович Бернштейн; born 18 April 1945) is a Soviet-born Israeli mathematician working at Tel Aviv University. He works in algebraic geometry, representation theory, and number theory. == Biography == Bernstein received his Ph.D. in 1972 under Israel Gelfand at Moscow State University. In 1981, he emigrated to the United States due to growing antisemitism in the Soviet Union. Bernstein was a professor at Harvard during 1983-1993. He was a visiting scholar at the Institute for Advanced Study in 1985-86 and again in 1997-98. In 1993, he moved to Israel to take a professorship at Tel Aviv University (emeritus since 2014). == Awards and honors == Bernstein received a gold medal at the 1962 International Mathematical Olympiad. He was elected to the Israel Academy of Sciences and Humanities in 2002 and was elected to the United States National Academy of Sciences in 2004. In 2004, Bernstein was awarded the Israel Prize for mathematics. In 1998, he was an Invited Speaker of the International Congress of Mathematicians in Berlin. In 2012, he became a fellow of the American Mathematical Society. == Publications == Publication list Some pdf files of papers by Bernstein including Algebraic theory of D-modules and his notes on Meromorphic continuation of Eisenstein series Beilinson, A.A.; Bernstein, J.; Deligne, P. (2018) [1982], "Faisceaux pervers", Astérisque, 100, Société Mathématique de France, Paris, ISBN 978-2-85629-878-7, MR 0751966 == See also == Bernstein–Sato polynomial Bernstein–Gelfand–Gelfand resolution == References == == External links == Bernstein's home page Joseph Bernstein at the Mathematics Genealogy Project
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Wikipedia:Joseph Delboeuf#0
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Joseph Rémi Léopold Delbœuf (30 September 1831, Liège, Belgium – 14 August 1896, Bonn, Germany) was a Belgian experimental psychologist who studied visual illusions including his work on the Delboeuf illusion. He studied and taught philosophy, mathematics, and psychophysics. He published works across a diverse range of subjects including the curative effects of hypnotism. == Early life and education == Joseph Delboeuf was born in the French speaking town of Liège, Belgium. His father, a pewter craftsman, died early in Delboeuf’s life. He studied at the University of Liège, earning his PhD in philosophy in 1855, followed by physics and mathematics in 1858. He received a scholarship from the University of Bonn and continued his postdoctoral research under philosopher and mathematician Friedrich Ueberweg. In 1860, he began teaching Greek at l’Ecole Normale des Humanités de Liège. In 1863, he was given the post of Maitre de Conférences, which he only held for a few months before being appointed Chair of Philosophy at University of Ghent, leaving his research in mathematics. There, he met Joseph Plateau, who helped him publish his first two notes on optical illusions in the Bulletin of the Royal Academy of Belgium. He then began his research in psychophysics with Gustav Fechner in 1865. == Work == Delboeuf began his psychophysical experimentation on brightness in 1865 with Gustav Fechner. Delboeuf introduced the concept of sense distance (contraste sensible). Delboeuf is known for his description of the Delboeuf illusion in 1893. Many experiments have been performed on this illusion since that time. === Hypnotism === After completing work on sleep and dreams, Delboeuf started researching magnetism and hypnotism. At the First International Congress on scientific and experimental hypnotism (1889), a motion was put forward to ban non-medical practitioners from using hypnosis. However, Delboeuf argued that a medical degree was not required to practice hypnotism; rather, it should be used freely, yet with caution. Along with a group of magnetizers in Verviers, he argued that hypnotists had specific personal skills that could not be acquired by all doctors. Thus, the clinical technique of hypnotism was proposed as an alternative to traditional medical therapies. ==== Curative effects ==== Most of the work Delboeuf published was on the curative effects of hypnotism. However, his hypotheses were often difficult to test and falsify. To test his hypotheses, he suggested that he would need two identical patients to be treated with different approaches. Instead, he used the symmetry of the opposite sides of the body. He would make two identical lesions on two parts of the body (e.g., arms) and would apply hypnotism to one area while leaving the other alone for nature to act upon. He found that not only can the consequences of pain be avoided from hypnotic suggestion (i.e., no pain experienced), but that over the course of the injury or disease, the actual effects of the operation could be halted over time. Hypnotism's mode of action, concluded Delboeuf, is like pain, in that it "hypnotizes" by calling attention to itself. Thus, hypnotic suggestion prevents the effects of pain by withdrawing attention from it. ==== Post hypnotic suggestion ==== Delboeuf, with Hippolyte Bernheim in 1885, introduced a new solution to the problem of post hypnotic suggestion. They argued that subjects drifted into a dream-like state in which they became aware of the suggestion and time remaining in this state. Their idea was very similar to a double consciousness theory suggested by Pierre Janet. They highlighted that subjects are unable to remember the suggestion when they wake because the memory is only available in the dream-like or hypnotic state. In other words, they describe the memory being in a “dormant” state that returns during the hypnotic state. Delboeuf argued that the amnesia from waking from hypnotic sleep was similar to that from waking up from regular sleep. We tend not to remember our dreams because we are unable to establish a connection between the waking and dream state. We do, however, tend to remember dreams when we wake up in the middle of a dream because our waking and dreaming state bridge together. Therefore, Delboeuf predicted that subjects should be able to recall their hypnotic sleep if they were awakened in the middle of it. He tested his predictions in an experiment with Marie Wittman, who was awakened in the middle of a hypnotic hallucination and remembered everything. Delboeuf repeated his experiment with other subjects with similar results. His experiments convinced him of the similarity between normal dreams and hypnotic states. === Mathematics === During his time at the University of Bonn, Delboeuf published Prolégomènes philosophiques à la géométrie (1860), disputing his mentor Ueberweg’s concept of Euclidean space and earning the praise of Bertrand Russell. He argued that, in order to use geometry to find the fundamental qualities of “determinations of space,” we must first understand the concepts of both “determination” and “space.” In this paper, he independently discovered Euclidean postulate 5. Postulate 5 states that, if a line intersects two straight lines that together form two interior angles on the same side that sum to less than 180 degrees, then the two straight lines must meet on that side. ==== Psychophysics ==== Delboeuf first conducted psychophysics experiments at the University of Gand (1855-1866). He primarily worked on Weber's Law (proposed by Fechner): S = K log I {\displaystyle S=K\log I} Where S stands for the strength of a sensation, I stands for the intensity of an external stimulus, and K is a constant. Delboeuf proposed two main changes to Fechner's formula. The first change was related to mathematical aspects. Delboeuf argued that, in some cases, Fechner's formula would result in negative values for S. For example if I = 0 {\displaystyle I=0} , then S < 0 {\displaystyle S<0} . Delboeuf proposed that, because a negative value of S did not make sense, the formula could not be applicable to all cases. For example, in cases where the strength of a sensation may lead to inability to view an external stimulus (for instance due to glare), the formula would not be applicable. To address this issue, Delbeouf added a term to the end of Fechner's equation: S = K log I + c c {\displaystyle S=K\log {\frac {I+c}{c}}} Where c stands for the baseline physiological level of excitation within the senses. Delboeuf's second change to Fechner's formula was the addition of a supplementary equation: f = log m m − I ′ {\displaystyle f=\log {\frac {m}{m-I'}}} Where f stands for the amount of fatigue from effort of the sense organ given a stimulus, m stands for the amount of available sensitivity of the organ, and I' stands for the intensity of the stimulus. This supplementary equation would account for the change that a sense organ experiences due to the magnitude of excitation from an external stimulus. By adding this equation, Delboeuf accounted for fatiguing effects that sensations have on sense organs. ==== General Theory of Sensitivity ==== Delboeuf investigated the laws of nature and sensation in his paper "General Theory of Sensitivity." Delboeuf proposed that a sensation consists of two aspects: (1) formation of the sensation and (2) how sense organs function. Delboeuf further proposed three laws that can be applied to determine sensation strength: 1. Law of Degradation: as soon as a sensation occurs, the strength of a sensation begins to degrade. 2. Law of Intensity: in order for a sensation to increase in strength, it must be stimulated by excitations that are also increasing in magnitude. 3. Law of Tension: as the strength of a sensation increases, the level of tension also increases. Tension is associated with unpleasantness, pain, fatigue and even the destruction of the sensation itself. === Visual illusions === Delboeuf started researching perception after meeting Joseph Plateau (1801-1883), a physicist known for his work on vision. After this encounter, Delboeuf changed his focus from Fechnerian psychophysics to the psychology of perception. His first work on illusions was a discussion of Zollner's Pseudoscopy (1865) in which he proposes a theory that all optical illusions could be explained by the muscular strength and structure of the eye. He went on to publish a work on Optico-Geometric Illusions, in which he reinforces his proposed theory. These works led Delboeuf to develop an experiment on an optico-geometric illusion that later became known as the Delboeuf illusion (1865). This illusion is based on the idea that stimuli can be perceived as larger or smaller based on their illumination and surroundings. Delboeuf is well known for his work on this illusion. Later in his career, he published an explanation of the Muller-Lyer Illusion in terms of eye movements instead of the mis-estimation of angles theory Brentano had published. Many studies have been conducted on the Delboeuf illusion (1865) since the publication of his original study, and many explanations for this illusion have been put forth. == Works == === Articles in Popular Science Monthly === "Dwarfs and Giants" . Popular Science Monthly. Vol. 22. April 1883. "What May Animals Be Taught?" . Popular Science Monthly. Vol. 29. June 1886. "The Psychology of Lizards" . Popular Science Monthly. Vol. 43. September 1893. "Affections and Jealousies of Lizards" . Popular Science Monthly. Vol. 50. January 1897. "In a World Half as Large" . Popular Science Monthly. Vol. 52. March 1898. == See also == The Salpêtrière School of Hypnosis == References == == Further reading == François Duyckaerts, (1992). Joseph Delbœuf philosophe et hypnotiseur, LeBlanc, A. (2004). Thirteen Days: Joseph Delboeuf versus Pierre Janet on the Nature of Hypnotic Suggestion, Journal of the History of the Behavioral Sciences, 40, 123-147. == External links == Works by or about Joseph Delboeuf at the Internet Archive
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Wikipedia:Joseph Diestel#0
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Joseph Diestel (January 27, 1943 – August 17, 2017) was an American mathematician and Professor of Mathematics at Kent State University. In addition to his contribution to functional analysis, particularly Banach space theory and the theory of vector measures, Diestel was known for a number of highly influential textbooks: in 1975 he published "Lecture Notes Geometry of Banach Spaces—Selected Topics"; in 1977, he published "Vector Measures" with J. Jerry Uhl; in 1984, published "Sequences and series in Banach spaces" and in 1995 he published "Absolutely summing operators" with H. Jarchow and A. Tonge; as well as a number of other books. Diestel received his Ph.D. degree in 1969 from Catholic University of America under Victor Michael Bogdan. He had 29 graduate students and over 46 mathematical descendants. == References ==
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Wikipedia:Joseph Dzhun#0
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Joseph Dzhun Volodymyrovych is a Ukrainian scientist, astronomer, and academician who was born in Cherniakhiv, Zhytomyr, Ukraine. He graduated from Lviv Polytechnical Institute. He is a scientist-astronomer, mathematician, representative of an academician E.P. Fedorov's scientific school, doctor of physical and mathematical sciences, professor of the mathematical modelling department of the cybernetics faculty of International University of Economics and Humanities (IUEH), full member of the International Pedagogical Academy (Moscow, 1999), and European Safety Association (2002). == Scientific work == Dzhun specialises in mathematical astronomy, space, and statistical information processing of high volumes, especially in connection with Hempel's Paradox. Dzhun's scientific works are devoted to the development of astrometry. He carried out the fundamental control of the Cambridge professor H. Jeffreys' conclusions about the nonclassical form of the errors distribution law for the sample of the volume of more than 500 observations, which confirmed Jeffreys' conception about the correspondence of the VII type Pearson's distribution real errors. Dzhun performed large scale research on the histograms of the errors of astronomic, gravimetrical, geophysical and economic data in total volume more than 170,000 observations, using the most high-quality series, including the F.V. Bessel historical series. The basic ideas and approaches are stated in Dzhun's doctoral thesis "Mathematical Processing of Astronomic and Space Information in case of the Non-Gauss Observation Errors," presented in 1992. His scientific works are devoted mainly to different aspects of the nonclassical theory of errors development, studying of its axiomatic foundations and creation of the adapted procedures at mathematical modelling and data analysis. He developed the analytical theory of the weight functions and the method of the mathematical models diagnostics on its base. Dzhun is the first to theorize that the relative random vibrations of the major index series in global economy submitted to the VII type Pearson's distribution. He created the theory of informative methods of evaluation of astronomical observations accuracy in case of the non-Gauss errors distribution. Dzhun proposed to substitute the fundamental principle of the Gauss maximal weight by the Fisher principle of a maximum of information with the purpose of evolution and generalization of the procedures of the least-squares classic method. == References ==
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Wikipedia:Joseph Fourier#0
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Jean-Baptiste Joseph Fourier (; French: [ʒɑ̃ batist ʒozɛf fuʁje]; 21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre, Burgundy and best known for initiating the investigation of Fourier series, which eventually developed into Fourier analysis and harmonic analysis, and their applications to problems of heat transfer and vibrations. The Fourier transform and Fourier's law of conduction are also named in his honour. Fourier is also generally credited with the discovery of the greenhouse effect. == Biography == Fourier was born in Auxerre (now in the Yonne département of France), the son of a tailor. He was orphaned at the age of nine. Fourier was recommended to the Bishop of Auxerre and, through this introduction, he was educated by the Benedictine Order of the Convent of St. Mark. The commissions in the scientific corps of the army were reserved for those of good birth, and being thus ineligible, he accepted a military lectureship on mathematics. He took a prominent part in his own district in promoting the French Revolution, serving on the local Revolutionary Committee. He was imprisoned briefly during the Terror but, in 1795, was appointed to the École Normale and subsequently succeeded Joseph-Louis Lagrange at the École Polytechnique. Fourier accompanied Napoleon Bonaparte on his Egyptian expedition in 1798, as scientific adviser, and was appointed secretary of the Institut d'Égypte. Cut off from France by the British fleet, he organized the workshops on which the French army had to rely for their munitions of war. He also contributed several mathematical papers to the Egyptian Institute (also called the Cairo Institute) which Napoleon founded at Cairo, with a view of weakening British influence in the East. After the British victories and the capitulation of the French under General Menou in 1801, Fourier returned to France. In 1801, Napoleon appointed Fourier Prefect (Governor) of the Department of Isère in Grenoble, where he oversaw road construction and other projects. However, Fourier had previously returned home from the Napoleon expedition to Egypt to resume his academic post as professor at École Polytechnique when Napoleon decided otherwise in his remark ... the Prefect of the Department of Isère having recently died, I would like to express my confidence in citizen Fourier by appointing him to this place. Hence being faithful to Napoleon, he took the office of Prefect. It was while at Grenoble that he began to experiment on the propagation of heat. He presented his paper On the Propagation of Heat in Solid Bodies to the Paris Institute on 21 December 1807. He also contributed to the monumental Description de l'Égypte. In 1822, Fourier succeeded Jean Baptiste Joseph Delambre as Permanent Secretary of the French Academy of Sciences. In 1830, he was elected a foreign member of the Royal Swedish Academy of Sciences. Fourier never married. In 1830, his diminished health began to take its toll: Fourier had already experienced, in Egypt and Grenoble, some attacks of aneurysm of the heart. At Paris, it was impossible to be mistaken with respect to the primary cause of the frequent suffocations which he experienced. A fall, however, which he sustained on the 4th of May 1830, while descending a flight of stairs, aggravated the malady to an extent beyond what could have been ever feared. Shortly after this event, he died in his bed on 16 May 1830. Fourier was buried in the Père Lachaise Cemetery in Paris, a tomb decorated with an Egyptian motif to reflect his position as secretary of the Cairo Institute, and his collation of Description de l'Égypte. His name is one of the 72 names inscribed on the Eiffel Tower. A bronze statue was erected in Auxerre in 1849, but it was melted down for armaments during World War II. Joseph Fourier University in Grenoble was named after him. == The Analytic Theory of Heat == In 1822, Fourier published his treatise on heat flow in Théorie analytique de la chaleur (The Analytical Theory of Heat), in which he based his reasoning on Newton's law of cooling, namely, that the flow of heat between two adjacent particles is proportional to the extremely small difference of their temperatures. This treatise was translated, with editorial 'corrections', into English 56 years later by Freeman (1878). The treatise was also edited, with many editorial corrections, by mathematician Jean Gaston Darboux and republished in French in 1888. There were three important contributions in this publication, one purely mathematical, two essentially physical. In mathematics, Fourier claimed that any function of a variable, whether continuous or discontinuous, can be expanded in a series of sines of multiples of the variable. Though this result is not correct without additional conditions, Fourier's observation that some discontinuous functions are the sum of infinite series was a breakthrough. The question of determining when a Fourier series converges has been fundamental for centuries. Joseph-Louis Lagrange had given particular cases of this (false) theorem, and had implied that the method was general, but he had not pursued the subject. Peter Gustav Lejeune Dirichlet was the first to give a satisfactory demonstration of it with some restrictive conditions. This work provides the foundation for what is today known as the Fourier transform. One important physical contribution in the book was the concept of dimensional homogeneity in equations; i.e. an equation can be formally correct only if the dimensions match on either side of the equality; Fourier made important contributions to dimensional analysis. The other physical contribution was Fourier's proposal of his partial differential equation for conductive diffusion of heat. This equation is now taught to every student of mathematical physics. == Real roots of polynomials == Fourier left an unfinished work on determining and locating real roots of polynomials, which was edited by Claude-Louis Navier and published in 1831. This work contains much original matter—in particular, Fourier's theorem on polynomial real roots, published in 1820. François Budan, in 1807 and 1811, had published independently his theorem (also known by the name of Fourier), which is very close to Fourier's theorem (each theorem is a corollary of the other). Fourier's proof is the one that was usually given, during 19th century, in textbooks on the theory of equations. A complete solution of the problem was given in 1829 by Jacques Charles François Sturm. == Discovery of the greenhouse effect == In the 1820s, Fourier calculated that an object the size of the Earth, and at its distance from the Sun, should be considerably colder than the planet actually is if warmed by only the effects of incoming solar radiation. He examined various possible sources of the additional observed heat in articles published in 1824 and 1827. However, in the end, because of the large 33-degree difference between his calculations and observations, Fourier mistakenly believed that there is a significant contribution of radiation from interstellar space. Still, Fourier's consideration of the possibility that the Earth's atmosphere might act as an insulator of some kind is widely recognized as the first proposal of what is now known as the greenhouse effect, although Fourier never called it that. In his articles, Fourier referred to an experiment by Horace Bénédict de Saussure, who lined a vase with blackened cork. Into the cork, he inserted several panes of transparent glass, separated by intervals of air. Midday sunlight was allowed to enter at the top of the vase through the glass panes. The temperature became more elevated in the more interior compartments of this device. Fourier noted that if gases in the atmosphere could form a stable barrier like the glass panes they would have a similar effect on planetary temperatures. This conclusion may have contributed to the later use of the metaphor of the "greenhouse effect" to refer to the processes that determine atmospheric temperatures. Fourier noted that the actual mechanisms that determine the temperatures of the atmosphere included convection, which was not present in de Saussure's experimental device. == Works == == See also == Fourier analysis Fourier–Deligne transform Heat equation Least-squares spectral analysis List of things named after Joseph Fourier == References == == Further reading == == External links == Media related to Joseph Fourier at Wikimedia Commons O'Connor, John J.; Robertson, Edmund F., "Joseph Fourier", MacTutor History of Mathematics Archive, University of St Andrews Fourier, J. B. J., 1824, Remarques Générales Sur Les Températures Du Globe Terrestre Et Des Espaces Planétaires., in Annales de Chimie et de Physique, Vol. 27, pp. 136–167 – translation by Burgess (1837). Université Joseph Fourier, Grenoble, France Archived 22 June 2006 at the Wayback Machine Joseph Fourier and the Vuvuzela on MathsBank.co.uk Archived 28 April 2012 at the Wayback Machine Joseph Fourier at the Mathematics Genealogy Project Joseph Fourier – Œuvres complètes, tome 2 Gallican-Math "Episode 2 - Joseph Fourier". YouTube. École polytechnique. 16 January 2019. Archived from the original on 15 December 2021.
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Wikipedia:Joseph H. Sampson#0
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Joseph Harold Sampson Jr. (1926–2003) was an American mathematician known for his work in mathematical analysis, geometry and topology, especially his work about harmonic maps in collaboration with James Eells. He obtained his Ph.D. in mathematics from Princeton University in 1951 under the supervision of Salomon Bochner. == Mathematical work == In 1964, Sampson and James Eells introduced harmonic maps, which are mappings between Riemannian manifolds which solve a geometrically-defined system of partial differential equations. They can also be defined via the calculus of variations. Generalizing Bochner's work on harmonic functions, Eells and Sampson derived the Bochner identity, and used it to prove the triviality of harmonic maps under certain curvature conditions. Eells and Sampson established the existence of harmonic maps whenever the domain manifold is closed and the target has non-positive sectional curvature. Their proof analyzed the harmonic map heat flow, which is a geometrically-defined heat equation. By establishing a priori estimates for the flow, they were able to prove its convergence under the indicated curvature assumption. The use of the Bochner identity in deriving estimates is where the assumption on sectional curvature plays a crucial role. As a result of Eells and Sampson's (subsequential) convergence theorem, they were able to prove the existence of harmonic maps in any homotopy class. As such, harmonic maps may be regarded as canonically-defined representatives of topological spaces of mappings. This perspective has enabled the application of harmonic maps to many problems in geometry and topology. Eells and Sampson's work is one of the most prominent papers in the field of differential geometry, and was a direct inspiration for Richard Hamilton's epochal work on the Ricci flow. In addition to Eells and Sampson's heat flow, their main results on existence of harmonic maps can also be derived via the calculus of variations, using the regularity theory developed in the 1980s by Richard Schoen and Karen Uhlenbeck. Later in 1978, Sampson developed unique continuation, maximum principles, further rigidity theorems, and deformability results for harmonic maps. He also proved that a harmonic map of degree one between compact hyperbolic Riemann surfaces must be a diffeomorphism. The same result was obtained at the same time by Schoen and Shing-Tung Yau. == Major publications == Over the course of forty years, Sampson published around twenty research articles. Eells, James Jr.; Sampson, J. H. (1964). "Harmonic mappings of Riemannian manifolds". American Journal of Mathematics. 86 (1): 109–160. doi:10.2307/2373037. JSTOR 2373037. MR 0164306. Zbl 0122.40102. Sampson, J. H. (1978). "Some properties and applications of harmonic mappings". Annales Scientifiques de l'École Normale Supérieure. Quatrième Série. 11 (2): 211–228. doi:10.24033/asens.1344. MR 0510549. Zbl 0392.31009. == References ==
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Wikipedia:Joseph J. Rotman#0
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Joseph Jonah Rotman (May 26, 1934 – October 16, 2016) was a Professor of Mathematics at the University of Illinois at Urbana–Champaign and also a published author of 10 textbooks. Rotman was born in Chicago. He did his undergraduate and graduate work at the University of Chicago, where he received his doctorate in 1959 with a thesis in abelian groups written under the direction of Irving Kaplansky. In 1959 he moved to the University of Illinois at Urbana–Champaign, where he spent the rest of his mathematical career. Rotman retired from UIUC in 2004. His research interests lay in the area of algebra, involving abelian groups, modules, homological algebra, and combinatorics. Rotman was the Managing Editor of the Proceedings of the American Mathematical Society in 1972–1973. In 1985 he was the Annual Visiting Lecturer of the South African Mathematical Society. A partial list of Rotman's publications includes: An Introduction to Homological Algebra (1979), Pure and Applied Mathematics, vol. 85, Academic Press; ISBN 0-12-599250-5 An Introduction to Algebraic Topology (1988), Springer-Verlag; ISBN 0-387-96678-1 An Introduction to the Theory of Groups (1995), Springer-Verlag; ISBN 0-387-94285-8 A First Course in Abstract Algebra (2000), Prentice Hall; ISBN 0-13-011584-3 Advanced Modern Algebra (2002), Prentice Hall; ISBN 0-13-087868-5 Journey into Mathematics: an introduction to proofs (2006), Dover Publications; ISBN 0-486-45306-5 == References ==
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Wikipedia:Joseph Jean Baptiste Neuberg#0
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Joseph Jean Baptiste Neuberg (30 October 1840 – 22 March 1926) was a Luxembourgish mathematician who worked primarily in geometry. == Biography == Neuberg was born on 30 October 1840 in Luxembourg City, Luxembourg. He first studied at a local school, the Athénée de Luxembourg, then progressed to Ghent University, studying at the École normale des Sciences of the science faculty. After graduation, Neuberg taught at several institutions. Between 1862 and 1865, he taught at the École Normale de Nivelle. For the next sixteen years, he taught at the Athénée Royal d'Arlon, though he also taught at the École Normale at Bruges from 1868 onwards. Neuberg switched from his previous two schools to the Athénée Royal de Liège in 1878. He became an extraordinary professor in the university in the same city in 1884, and was promoted to ordinary professor in 1887. He held this latter position until his retirement in 1910. A year after his retirement, he was elected president of the Belgian Royal Academy, which he had joined earlier, in 1866, after taking Belgian nationality despite his origins. The professor died on 22 March 1926 in Liège, Belgium, and was commemorated in the Bulletin of the American Mathematical Society. == Contributions == Neuberg worked mainly in geometry, particularly the geometry of the triangle, The Neuberg cubic, a curve defined from a triangle, is named after him, and passes through the isodynamic points of a triangle which he discovered and published in 1885. Neuberg was also involved in a number of mathematical journals. With Eugène Catalan and Paul Mansion, he founded the journal Nouvelle correspondance mathématique. This journal was founded to honour the earlier journal Correspondance mathématique et physique, which had been edited by Lambert Quetelet and Jean Garnier. Correspondance was published until 1880; after this, Catalan advised Mansion and Neuberg to continue publication of a new journal. They followed his advice, creating Mathesis in 1881, which is perhaps Neuberg's best-known journal. Several mathematical societies included Neuberg: the Institute of Science of Luxembourg, the Royal Society of Science of Liège, Mathematical Society of Amsterdam, and the Belgian Royal Academy noted in the biography above. == References == == External links == Mathesis. Gauthier-Villars. 1936. Weisstein, Eric W. "Joseph Neuberg". Wolfram. A generalization of Neuberg's theorem and the Simson-Wallace line at Dynamic Geometry Sketches, an interactive dynamic geometry sketch
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Wikipedia:Joseph Kampé de Fériet#0
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In mathematics, the Kampé de Fériet function is a two-variable generalization of the generalized hypergeometric series, introduced by Joseph Kampé de Fériet. The Kampé de Fériet function is given by p + q F r + s ( a 1 , ⋯ , a p : b 1 , b 1 ′ ; ⋯ ; b q , b q ′ ; c 1 , ⋯ , c r : d 1 , d 1 ′ ; ⋯ ; d s , d s ′ ; x , y ) = ∑ m = 0 ∞ ∑ n = 0 ∞ ( a 1 ) m + n ⋯ ( a p ) m + n ( c 1 ) m + n ⋯ ( c r ) m + n ( b 1 ) m ( b 1 ′ ) n ⋯ ( b q ) m ( b q ′ ) n ( d 1 ) m ( d 1 ′ ) n ⋯ ( d s ) m ( d s ′ ) n ⋅ x m y n m ! n ! . {\displaystyle {}^{p+q}F_{r+s}\left({\begin{matrix}a_{1},\cdots ,a_{p}\colon b_{1},b_{1}{}';\cdots ;b_{q},b_{q}{}';\\c_{1},\cdots ,c_{r}\colon d_{1},d_{1}{}';\cdots ;d_{s},d_{s}{}';\end{matrix}}x,y\right)=\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(a_{1})_{m+n}\cdots (a_{p})_{m+n}}{(c_{1})_{m+n}\cdots (c_{r})_{m+n}}}{\frac {(b_{1})_{m}(b_{1}{}')_{n}\cdots (b_{q})_{m}(b_{q}{}')_{n}}{(d_{1})_{m}(d_{1}{}')_{n}\cdots (d_{s})_{m}(d_{s}{}')_{n}}}\cdot {\frac {x^{m}y^{n}}{m!n!}}.} == Applications == The general sextic equation can be solved in terms of Kampé de Fériet functions. == See also == Appell series Humbert series Lauricella series (three-variable) == References == Exton, Harold (1978), Handbook of hypergeometric integrals, Mathematics and its Applications, Chichester: Ellis Horwood Ltd., ISBN 978-0-85312-122-0, MR 0474684 Kampé de Fériet, M. J. (1937), La fonction hypergéométrique., Mémorial des sciences mathématiques (in French), vol. 85, Paris: Gauthier-Villars, JFM 63.0996.03 Ragab, F. J. (1963). "Expansions of Kampe de Feriet's double hypergeometric function of higher order". J. reine angew. Math. 212 (212): 113–119. doi:10.1515/crll.1963.212.113. S2CID 118329382. == External links == Weisstein, Eric W. "Kampé de Fériet function". MathWorld.
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Wikipedia:Joseph Kouneiher#0
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Joseph Kouneiher is a French mathematical physicist. He is a professor of mathematical physics and engineering sciences at Nice SA University, France. He works primarily on the foundations of science, and his work in the domains of quantum field theory, quantum gravity, string theory and conformal field theory is widely cited and is well known. He holds three PhDs in mathematical physics and Epistemology and history of sciences. == Research work == He developed and generalized (with his colleague Frédéric Hélein) Hermann Weyl's Hamiltonian formalism for quantum fields theories, what we call today a Covariant Hamiltonian formalism for the calculus of variations with several variables. The main purpose is to build a hamiltonian theory of fields which is consistent with the principles of relativity. It's a finite dimension formalism for a quantum theories. He also clarified the topological (or cohomological) aspect of certain approach to quantum gravity and the role of the integrability in the foundations of such theories. In addition to his contributions in mathematical physics, he introduced the cohomological aspect of the mathematical logic theories or what we call now cohomological logic. The aim of the program of "cohomological logic" is to generalize the foundations of the usual theories of logic and connect logic with homotopy theory by introducing a Hopf structure into the generalized logic theories through a geometric approach. This formalism has a great impact on the foundations of some representations of quantum theories His collaborations with Michael Atiyah to figure out a geometric model for matter. Starting from the idea of an ultimate granular structure of the space-time, which generalizes the continuum and the discontinuum aspects of the space-time, he developed a formalism which generalizes the differential equations and difference equations theories to treat such spaces that appear continuous at low energies and exhibit a dual continuum and discontinuum aspects at high energy. Apart from his deep-routed interests in foundational sciences, he is also an aficionado of classical music. He composes musical pieces and plays piano among many other instruments. == Archimedes S.I.E.E. == Along with several leading figures, Joseph is the co-founder of the Foundation Archimedes to promote sciences, innovation, education and environment. The scientific committee includes Sir Roger Penrose, Alain Connes, Simon Donaldson, Mathilde Marcolli, Hugo Duminil-Copin, Misha Gromov, Abhay Ashtekar, Carlo Rovelli, Jeremy Butterfield, Jean-Pierre Luminet, Abhay Ashtekar, Alicia Dickenstein, Jean-Pierre Bourguignon and Sir Michael Atiyah. Located in Saint-Raphaël, Var, provides interdisciplinary research residencies for both French and international researchers, entrepreneurs, and industrialists. Its facilities are designed to promote focus and creativity, offering opportunities for both individual and small group study stays. The Foundation supports work across a broad range of disciplines, including Industrial Sciences, Technologies, and Environmental Studies. A key mission of the Foundation is to host prominent scientists, entrepreneurs, and industrialists whose contributions have advanced understanding of the physical world and humanity, as well as driven technological and cultural innovations. The goal is to foster high-level research and innovation across various fields. == Books authored/co-authored/edited == J. Kouneiher ed., Foundations of Mathematics and Physics, one century after Hilbert, in collaboration with : John Stachel, Michael Atiyah, Alain Connes, Misha Gromov, Roger Penrose, Edward Witten, Ali Chamsddine, Colin Maclarty, Jeremy Butterfield, Abhay Ashtekar, Lee Smolin, Leo Corry, Thierry Masson, Sebastian de Haro, Matilde Marcolli, Springer, 2018. J. Kouneiher, Géométrie au XXe siècle : Histoire et horizons, with Dominique Flament, Philippe Nabonand, Jean Jacques Szczeciniarz, éditions Hermann (pour l’Europe) et Presses internationales Polytechnique (USA, Canada), 2005. J. Kouneiher, C. Barbachoux, and F. Helein, Geometry, Topology, Quantum Fields Theory and Cosmology, Hermann editions. H. Cartan and J. Kouneiher (préface & dernier chapitre), Cours de calcul différentiel, éditions Hermann, 2006. J. Kouneiher & al, Fundamental Frontier of Physics, AIP USA, 2012. P. Baird, J. Kouneiher & al, Systèmes intégrables & théorie des champs quantiques, Hermann editions, 2007. J. Kouneiher, Vers une nouvelle Philosophie de la Nature : Actualités Mathématiques, Physiques et Biologiques, Hermann eds., 2010. == References ==
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Wikipedia:Joseph L. Ullman#0
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Joseph Leonard Ullman (30 January 1923, in Buffalo, New York – 11 September 1995, in Chelsea, Michigan) was a mathematician who worked on classical analysis with a focus on approximation theory. Ullman received his A.B. from the University of Buffalo and his graduate studies were interrupted by service in the U.S. Army in World War II. He was injured, received a Purple Heart, and spent the rest of the war as a mathematics instructor. He received a Ph.D. in 1949 from Stanford University with thesis Studies on Faber Polynomials under the direction of Gábor Szegő. Ullman became an instructor at the University of Michigan in 1949, an assistant professor in 1954, an associate professor in 1962, and a professor in 1966. He wrote forty-three research papers. During his career at the University of Michigan he supervised eleven doctoral theses. == Selected works == Ullman, Joseph L. (1960). "Studies in Faber polynomials. I". Trans. Amer. Math. Soc. 94 (3): 515–528. doi:10.1090/s0002-9947-1960-0112955-2. MR 0112955. Ullman, J. L. (1963). "Tchebycheff quadrature on the infinite interval". Trans. Amer. Math. Soc. 107 (2): 291–299. doi:10.1090/s0002-9947-1963-0147817-0. MR 0147817. Ullman, J. L. (1966). "A class of weight functions for which Tchebycheff quadrature is possible". Bull. Amer. Math. Soc. 72 (6): 1073–1075. doi:10.1090/s0002-9904-1966-11651-8. MR 0204931. with R. C. Lyndon: Lyndon, R. C.; Ullman, J. L. (1967). "Groups of elliptic linear fractional transformations". Proc. Amer. Math. Soc. 18 (6): 1119–1124. doi:10.1090/s0002-9939-1967-0222182-8. MR 0222182. Ullman, J. L. (1972). "The location of the zeros of the derivatives of Faber polynomials". Proc. Amer. Math. Soc. 34 (2): 422–424. doi:10.1090/s0002-9939-1972-0299761-7. MR 0299761. Ullman, J. L. (1972). "On the regular behaviour of orthogonal polynomials" (PDF). Proc. London Math. Soc. 24 (3): 119–148. doi:10.1112/plms/s3-24.1.119. hdl:2027.42/135623. Ullman, J. L. (1980). "Orthogonal polynomials associated with an infinite interval". Michigan Math. J. 27 (3): 353–363. doi:10.1307/mmj/1029002408. MR 0584699. with Matthew F. Wyneken: Ullman, J. L.; Wyneken, M. F. (1986). "Weak limits of zeros of orthogonal polynomials" (PDF). Constr. Approx. 2 (4): 339–347. doi:10.1007/bf01893436. hdl:2027.42/41341. MR 0892160. S2CID 39657045. with Vilmos Totik: Totik, Vilmos; Ullman, Joseph L. (1994). "Local asymptotic distribution of zeros of orthogonal polynomials". Trans. Amer. Math. Soc. 341 (2): 881–894. doi:10.1090/s0002-9947-1994-1150019-2. MR 1150019. == References == == External links == Joseph L. Ullman at the Mathematics Genealogy Project
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Wikipedia:Joseph Lipman#0
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Joseph Lipman (born June 15, 1938) is a Canadian-American mathematician, working in algebraic geometry. Lipman graduated from the University of Toronto with a bachelor's degree in 1960 and then went to Harvard University, receiving his master's degree in 1961. He then earned a Ph.D. there in 1965 under the supervision of Oscar Zariski. In 1965 he was an assistant professor at Queen's University in Kingston and in 1966 was an assistant professor at the Purdue University, where he became professor in 1971. From 1987 to 1992, there, he was head of the mathematics department. He was a member of the MSRI and visiting scholar at the University of Cambridge and the University of Nice and a visiting professor at the Columbia University and Harvard University. He specializes in singularity theory in algebraic geometry. In 1958, while studying at the University of Toronto, he became a Putnam Fellow both in the spring and fall William Lowell Putnam Mathematical Competitions. In 1982 he received the Jeffery–Williams Prize. He is a fellow of the American Mathematical Society. == Writings == Collected Papers of Joseph Lipman. Queen's Papers in Pure and Applied Mathematics, vol. 117, Queen's University Press, Kingston, Ontario, 2000. Editor with Herwig Hauser, Frans Oort, Adolfo Quirós: Resolution of singularities. A research textbook in tribute to Oscar Zariski. Birkhäuser, Basel 2000, ISBN 3-7643-6178-6. (Progress in Mathematics. volume 181.) Mumford, David, ed. (1975). "Introduction to the Resolution of Singularities". Algebraic Geometry (Arcata 1974). Proc. Symp. Pure Math. Vol. 29. American Mathematical Society. Lipman, Joseph (1978). "Desingularization of Two-Dimensional Schemes". Annals of Mathematics. 107 (2): 151–207. doi:10.2307/1971141. JSTOR 1971141. == References == == External links == Homepage Joseph Lipman publications indexed by Google Scholar
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Wikipedia:Joseph Ludwig Raabe#0
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Joseph Ludwig Raabe (15 May 1801 in Brody, Galicia – 22 January 1859 in Zürich, Switzerland) was a Swiss mathematician. == Life == As his parents were quite poor, Raabe was forced to earn his living from a very early age by giving private lessons. He began to study mathematics in 1820 at the Polytechnicum in Vienna, Austria. In the autumn of 1831, he moved to Zürich, where he became professor of mathematics in 1833. In 1855, he became professor at the newly founded Swiss Polytechnicum. He is best known for Raabe's ratio test, an extension of d'Alembert's ratio test. Raabe's test serves to determine the convergence or divergence of an infinite series, in some cases. He is also known for the Raabe integral of the gamma function: ∫ a a + 1 log Γ ( t ) d t = 1 2 log 2 π + a log a − a , a ≥ 0. {\displaystyle \int _{a}^{a+1}\log \Gamma (t)\,dt={\tfrac {1}{2}}\log 2\pi +a\log a-a,\quad a\geq 0.} == Publications == Differential- und Integralrechnung (3 volumes) (Zürich, 1839–1847) Mathematische Mitteilungen (2 volumes) (1857-1858) == References ==
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Wikipedia:Joseph Mugisha#0
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Joseph Y.T. Mugisha is a Ugandan mathematician (specialising in biomathematics), academician and academic administrator. Currently he is a professor of mathematics and principal of the College of Natural Sciences, a constituent college of Makerere University. Makerere University is the oldest university in Uganda. == Background and education == He was born in 1960 in Bushenyi district, Western Uganda. Professor Mugisha received his PhD in Bio-mathematics in 2000 from Makerere university. Prior to that he was awarded a Masters of Science(Bio-mathematics) in 1992 and a Bachelors of Science in Education from Makerere University. He attended Mbarara High School for his Secondary Education; and Masheruka Primary School for foundation Education. == Career == Before his appointment as the principal of the College of Natural Sciences (CONAS), he was the dean of the Faculty of Science from 2009 to 2010, in which position he led the process of conversion of the faculty to a college. He held the principal post initially in acting capacity (2010 to 2011) before substantive appointment in 2012. He has also served as the acting deputy vice chancellor (academic affairs) at Makerere University. Mugisha joined Makerere University as a teaching assistant in 1987, rising through the ranks to professorship in 2008. He has also served the university in various leadership and management capacities; acting director Institute of Computer Science - Makerere University from August to December 2003; deputy director Institute of Computer Science - Makerere University from 2003 to August 2005. He is also a member of the Makerere University Senate, which is the highest academic decision-making body of the university; he has been appointed to several boards and committees within the university. Mugisha has taught courses at undergraduate and graduate level. He has supervised and mentored over 40 students at graduate level (both PhD and MSc) in the region. He is an international researcher and examiner with strong links to several universities in Eastern, Central and Southern Africa. He has served as a reviewer of several international journals like Mathematical Biosciences, Southern Journal of Sciences, Mathematical Biosciences and Engineering, Mathematical Modelling and Analysis, Ecological Modelling, Computers and Mathematics with Application, Computational and Applied Mathematics, Mathematical and Computer Modelling, among others. Mugisha is a founder member and the current president of the African Society for Bio-mathematics since 2009. He has been a member of the American Mathematical Society, Ugandan Mathematical Society, Ugandan Biometric Society and is a Fellow of the Ugandan National Academy of Sciences [UNAS]. Mugisha is widely published in over 50 articles in International Journals. His major research interest is in the application of mathematics in biology and biomedical processes with special emphasis on epidemiological and ecological modeling. He has co - organized, attended and facilitated at various conferences, workshops and seminars nationally and internationally. He is a member Kyambogo University Council, and board member of the Uganda National Council for Science and Technology council. == References ==
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Wikipedia:Joseph Pérès#0
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Joseph Pérès (31 October 1890 – 12 February 1962) was a French mathematician. == Early life and education == Pérès was born in Clermont-Ferrand on 31 October 1890. Former student of the Ecole Normale Superieure, he worked in Rome with Vito Volterra and defended his doctoral thesis in 1915. == Career == In 1920, he became a lecturer at the Faculty of Sciences of Strasbourg and in 1921 held the mechanics chair of the faculty of sciences of Marseille. In 1932, he was appointed lecturer at the Faculty of Paris. He was elected member of the Academy of Sciences in 1942. He held the chair of mechanics in 1950 and Dean of the Faculty of Science in 1954, succeeding Albert Châtelet. During his deanship, he undertakes the creation of the Orsay campus. He was also one of the founders of the Institut des Hautes Études Scientifiques and its first président until his death. == Selected publications == === Articles === Pérès, Joseph (1919), "Le parallélisme de Mr. Levi-Civita et la courbure riemannienne", Rendiconti della Reale Accademia dei Lincei, (Serie 5) (in Italian), 28: 425–428 "Sur les transformations qui conservent la composition." Bull. Soc. Math. France 47 (1919): 16–37. "Choc en tenant compte du frottement." Nouvelles annales de mathématiques: journal des candidats aux écoles polytechnique et normale 2 (1923): 98–107. "Contribution à l'étude des jets fluides." Journal de Mathématiques Pures et Appliquées 11 (1932): 57–66. === Books === Volterra, Vito; Pérès, Joseph (1936). Théorie générale des fonctionnelles. Paris: Gauthier-Villars. Joseph Pérès (1936). Mécanique des Fluides. Gauthier-Villars; published with the collaboration of Lucien Malavard{{cite book}}: CS1 maint: postscript (link) == References == == External links == Joseph Pérès at the Mathematics Genealogy Project A French short biography of Joseph Pérès in BnF
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Wikipedia:Joseph Ser#0
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Joseph Ser (1875–1954) was a French mathematician, of whom little was known till now. He published 45 papers between 1900 and 1954, among which four monographs, edited in Paris by Henry Gauthier-Villars. In the main, he worked on number theory and infinite series. He got important results in the domain of factorial series. His representation of Euler's constant as a series of rational terms is well known. It was used in 1926 by Paul Appell (1855–1930), in an unsuccessful attempt to prove the irrationality of Euler's constant. == References == Ser, Joseph : Sur une expression de la fonction ζ(s) de Riemann (Upon an expression for Riemann's ζ function). CRAS (Paris) vol.182(1926),1075-1077 Ayoub, Raymond G.: Partial triumph or total failure ? The mathematical Intelligencer, vol.7, No 2(1985),55-58. This paper explains exactly Appell's mistake (4 - Appell and the irrationality of Euler's constant).
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Wikipedia:Joseph Tilly#0
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Joseph Marie de Tilly (16 August 1837 – 4 August 1906) was a Belgian military man and mathematician. He was born in Ypres, Belgium. In 1858, he became a teacher in mathematics at the regimental school. He began with studying geometry, particularly Euclid's fifth postulate and non-Euclidean geometry. He found similar results as Lobachevsky in 1860, but the Russian mathematician was already dead at that time. Tilly is more known for his work on non-Euclidean mechanics, as he was the one who invented it. He worked thus alone on this topic until a French mathematician, Jules Hoüel, showed interest in that field. Tilly also wrote on military science and history of mathematics. He died in München, Germany. == References == O'Connor, John J.; Robertson, Edmund F., "Joseph Tilly", MacTutor History of Mathematics Archive, University of St Andrews
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Wikipedia:Joseph Wedderburn#0
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Joseph Henry Maclagan Wedderburn FRSE FRS (2 February 1882 – 9 October 1948) was a Scottish mathematician, who taught at Princeton University for most of his career. A significant algebraist, he proved that a finite division algebra is a field (Wedderburn's little theorem), and part of the Artin–Wedderburn theorem on simple algebras. He also worked on group theory and matrix algebra. His younger brother was the lawyer Ernest Wedderburn. == Life == Joseph Wedderburn was the tenth of fourteen children of Alexander Wedderburn of Pearsie, a physician, and Anne Ogilvie. He was educated at Forfar Academy then in 1895 his parents sent Joseph and his younger brother Ernest to live in Edinburgh with their paternal uncle, J. R. Maclagan Wedderburn, allowing them to attend George Watson's College. This house was at 3 Glencairn Crescent in the West End of the city. In 1898 Joseph entered the University of Edinburgh. In 1903, he published his first three papers, worked as an assistant in the Physical Laboratory of the University, obtained an MA degree with first class honours in mathematics, and was elected a Fellow of the Royal Society of Edinburgh, upon the proposal of George Chrystal, James Gordon MacGregor, Cargill Gilston Knott and William Peddie. Aged 21 on election he remains one of the youngest Fellows ever. He then studied briefly at the University of Leipzig and the University of Berlin, where he met the algebraists Frobenius and Schur. A Carnegie Scholarship allowed him to spend the 1904–1905 academic year at the University of Chicago where he worked with Oswald Veblen, E. H. Moore, and most importantly, Leonard Dickson, who was to become the most important American algebraist of his day. Returning to Scotland in 1905, Wedderburn worked for four years at the University of Edinburgh as an assistant to George Chrystal, who supervised his D.Sc, awarded in 1908 for a thesis titled On Hypercomplex Numbers. He gained a PhD in algebra from the University of Edinburgh in 1908. From 1906 to 1908, Wedderburn edited the Proceedings of the Edinburgh Mathematical Society. In 1909, he returned to the United States to become a Preceptor in Mathematics at Princeton University; his colleagues included Luther P. Eisenhart, Oswald Veblen, Gilbert Ames Bliss, and George Birkhoff. Upon the outbreak of the First World War, Wedderburn enlisted in the British Army as a private. He was the first person at Princeton to volunteer for that war, and had the longest war service of anyone on the staff. He served with the Seaforth Highlanders in France, as Lieutenant (1914), then as Captain of the 10th Battalion (1915–18). While a Captain in the Fourth Field Survey Battalion of the Royal Engineers in France, he devised sound-ranging equipment to locate enemy artillery. He returned to Princeton after the war, becoming Associate Professor in 1921 and editing the Annals of Mathematics until 1928. While at Princeton, he supervised only three PhDs, one of them being Nathan Jacobson. In his later years, Wedderburn became an increasingly solitary figure and may even have suffered from depression. His isolation after his 1945 early retirement was such that his death from a heart attack was not noticed for several days. His Nachlass was destroyed, as per his instructions. Wedderburn received the MacDougall-Brisbane Gold Medal and Prize from the Royal Society of Edinburgh in 1921, and was elected to the Royal Society of London in 1933. == Work == In all, Wedderburn published about 40 books and papers, making important advances in the theory of rings, algebras and matrix theory. In 1905, Wedderburn published a paper that included three claimed proofs of a theorem stating that a noncommutative finite division ring could not exist. The proofs all made clever use of the interplay between the additive group of a finite division algebra A, and the multiplicative group A* = A-{0}. Parshall (1983) notes that the first of these three proofs had a gap not noticed at the time. Meanwhile, Wedderburn's Chicago colleague Dickson also found a proof of this result but, believing Wedderburn's first proof to be correct, Dickson acknowledged Wedderburn's priority. But Dickson also noted that Wedderburn constructed his second and third proofs only after having seen Dickson's proof. Parshall concludes that Dickson should be credited with the first correct proof. This theorem yields insights into the structure of finite projective geometries. In their paper on "Non-Desarguesian and non-Pascalian geometries" in the 1907 Transactions of the American Mathematical Society, Wedderburn and Veblen showed that in these geometries, Pascal's theorem is a consequence of Desargues' theorem. They also constructed finite projective geometries which are neither "Desarguesian" nor "Pascalian" (the terminology is Hilbert's). Wedderburn's best-known paper was his sole-authored "On hypercomplex numbers," published in the 1907 Proceedings of the London Mathematical Society, and for which he was awarded the D.Sc. the following year. This paper gives a complete classification of simple and semisimple algebras. He then showed that every finite-dimensional semisimple algebra can be constructed as a direct sum of simple algebras and that every simple algebra is isomorphic to a matrix algebra for some division ring. The Artin–Wedderburn theorem generalises these results to algebras with the descending chain condition. His best known book is his Lectures on Matrices (1934), which Jacobson praised as follows: That this was the result of a number of years of painstaking labour is evidenced by the bibliography of 661 items (in the revised printing) covering the period 1853 to 1936. The work is, however, not a compilation of the literature, but a synthesis that is Wedderburn's own. It contains a number of original contributions to the subject. About Wedderburn's teaching: He was apparently a very shy man and much preferred looking at the blackboard to looking at the students. He had the galley proofs from his book "Lectures on Matrices" pasted to cardboard for durability, and his "lecturing" consisted of reading this out loud while simultaneously copying it onto the blackboard. == See also == Hypercomplex numbers Wedderburn–Etherington number == References == == Further reading == Artin, Emil (1950). "The influence of J. H. M. Wedderburn on the development of modern algebra". Bull. Amer. Math. Soc. 56 (1, Part 1): 65–72. doi:10.1090/s0002-9904-1950-09346-x. Robert Hooke (1984) Recollections of Princeton, 1939–1941 Karen Parshall (1983) "In pursuit of the finite division algebra theorem and beyond: Joseph H M Wedderburn, Leonard Dickson, and Oswald Veblen," Archives of International History of Science 33: 274–99. Karen Parshall (1985) "Joseph H. M. Wedderburn and the structure theory of algebras," Archive for History of Exact Sciences 32: 223–349. Karen Parshall (1992) "New Light on the Life and Work of Joseph Henry Maclagan Wedderburn (1882–1948)," in Menso Folkerts et al. (eds.): Amphora: Festschrift für Hans Wußing zu seinem 65. Geburtstag, Birkhäuser Verlag, 523–537.
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Wikipedia:Joseph-Louis Lagrange#0
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Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia or Giuseppe Ludovico De la Grange Tournier; 25 January 1736 – 10 April 1813), also reported as Giuseppe Luigi Lagrange or Lagrangia, was an Italian mathematician, physicist and astronomer, later naturalized French. He made significant contributions to the fields of analysis, number theory, and both classical and celestial mechanics. In 1766, on the recommendation of Leonhard Euler and d'Alembert, Lagrange succeeded Euler as the director of mathematics at the Prussian Academy of Sciences in Berlin, Prussia, where he stayed for over twenty years, producing many volumes of work and winning several prizes of the French Academy of Sciences. Lagrange's treatise on analytical mechanics (Mécanique analytique, 4. ed., 2 vols. Paris: Gauthier-Villars et fils, 1788–89), which was written in Berlin and first published in 1788, offered the most comprehensive treatment of classical mechanics since Isaac Newton and formed a basis for the development of mathematical physics in the nineteenth century. In 1787, at age 51, he moved from Berlin to Paris and became a member of the French Academy of Sciences. He remained in France until the end of his life. He was instrumental in the decimalisation process in Revolutionary France, became the first professor of analysis at the École Polytechnique upon its opening in 1794, was a founding member of the Bureau des Longitudes, and became Senator in 1799. == Scientific contribution == Lagrange was one of the creators of the calculus of variations, deriving the Euler–Lagrange equations for extrema of functionals. He extended the method to include possible constraints, arriving at the method of Lagrange multipliers. Lagrange invented the method of solving differential equations known as variation of parameters, applied differential calculus to the theory of probabilities and worked on solutions for algebraic equations. He proved that every natural number is a sum of four squares. His treatise Theorie des fonctions analytiques laid some of the foundations of group theory, anticipating Galois. In calculus, Lagrange developed a novel approach to interpolation and Taylor's theorem. He studied the three-body problem for the Earth, Sun and Moon (1764) and the movement of Jupiter's satellites (1766), and in 1772 found the special-case solutions to this problem that yield what are now known as Lagrangian points. Lagrange is best known for transforming Newtonian mechanics into a branch of analysis, Lagrangian mechanics. He presented the mechanical "principles" as simple results of the variational calculus. == Biography == === Early years === Firstborn of eleven children as Giuseppe Lodovico Lagrangia, Lagrange was of Italian and French descent. His paternal great-grandfather was a French captain of cavalry, whose family originated from the French region of Tours. After serving under Louis XIV, he had entered the service of Charles Emmanuel II, Duke of Savoy, and married a Conti from the noble Roman family. Lagrange's father, Giuseppe Francesco Lodovico, was a doctor in Law at the University of Torino, while his mother was the only child of a rich doctor of Cambiano, in the countryside of Turin. He was raised as a Roman Catholic (but later on became an agnostic). His father, who had charge of the King's military chest and was Treasurer of the Office of Public Works and Fortifications in Turin, should have maintained a good social position and wealth, but before his son grew up he had lost most of his property in speculations. A career as a lawyer was planned out for Lagrange by his father, and certainly Lagrange seems to have accepted this willingly. He studied at the University of Turin and his favourite subject was classical Latin. At first, he had no great enthusiasm for mathematics, finding Greek geometry rather dull. It was not until he was seventeen that he showed any taste for mathematics – his interest in the subject being first excited by a paper by Edmond Halley from 1693 which he came across by accident. Alone and unaided he threw himself into mathematical studies; at the end of a year's incessant toil he was already an accomplished mathematician. Charles Emmanuel III appointed Lagrange to serve as the "Sostituto del Maestro di Matematica" (mathematics assistant professor) at the Royal Military Academy of the Theory and Practice of Artillery in 1755, where he taught courses in calculus and mechanics to support the Piedmontese army's early adoption of the ballistics theories of Benjamin Robins and Leonhard Euler. In that capacity, Lagrange was the first to teach calculus in an engineering school. According to Alessandro Papacino D'Antoni, the academy's military commander and famous artillery theorist, Lagrange unfortunately proved to be a problematic professor with his oblivious teaching style, abstract reasoning, and impatience with artillery and fortification-engineering applications. In this academy one of his students was François Daviet. ==== Variational calculus ==== Lagrange is one of the founders of the calculus of variations. Starting in 1754, he worked on the problem of the tautochrone, discovering a method of maximizing and minimizing functionals in a way similar to finding extrema of functions. Lagrange wrote several letters to Leonhard Euler between 1754 and 1756 describing his results. He outlined his "δ-algorithm", leading to the Euler–Lagrange equations of variational calculus and considerably simplifying Euler's earlier analysis. Lagrange also applied his ideas to problems of classical mechanics, generalising the results of Euler and Maupertuis. Euler was very impressed with Lagrange's results. It has been stated that "with characteristic courtesy he withheld a paper he had previously written, which covered some of the same ground, in order that the young Italian might have time to complete his work, and claim the undisputed invention of the new calculus"; however, this chivalric view has been disputed. Lagrange published his method in two memoirs of the Turin Society in 1762 and 1773. ==== Miscellanea Taurinensia ==== In 1758, with the aid of his pupils (mainly with Daviet), Lagrange established a society, which was subsequently incorporated as the Turin Academy of Sciences, and most of his early writings are to be found in the five volumes of its transactions, usually known as the Miscellanea Taurinensia. Many of these are elaborate papers. The first volume contains a paper on the theory of the propagation of sound; in this he indicates a mistake made by Newton, obtains the general differential equation for the motion, and integrates it for motion in a straight line. This volume also contains the complete solution of the problem of a string vibrating transversely; in this paper, he points out a lack of generality in the solutions previously given by Brook Taylor, D'Alembert, and Euler, and arrives at the conclusion that the form of the curve at any time t is given by the equation y = a sin ( m x ) sin ( n t ) {\displaystyle y=a\sin(mx)\sin(nt)\,} . The article concludes with a masterly discussion of echoes, beats, and compound sounds. Other articles in this volume are on recurring series, probabilities, and the calculus of variations. The second volume contains a long paper embodying the results of several papers in the first volume on the theory and notation of the calculus of variations, and he illustrates its use by deducing the principle of least action, and by solutions of various problems in dynamics. The third volume includes the solution of several dynamical problems by means of the calculus of variations; some papers on the integral calculus; a solution of a Fermat's problem: given an integer n which is not a perfect square, to find a number x such that nx2 + 1 is a perfect square; and the general differential equations of motion for three bodies moving under their mutual attractions. The next work he produced was in 1764 on the libration of the Moon, and an explanation as to why the same face was always turned to the earth, a problem which he treated by the aid of virtual work. His solution is especially interesting as containing the germ of the idea of generalised equations of motion, equations which he first formally proved in 1780. === Berlin === Already by 1756, Euler and Maupertuis, seeing Lagrange's mathematical talent, tried to persuade Lagrange to come to Berlin, but he shyly refused the offer. In 1765, d'Alembert interceded on Lagrange's behalf with Frederick of Prussia and by letter, asked him to leave Turin for a considerably more prestigious position in Berlin. He again turned down the offer, responding that: 361 It seems to me that Berlin would not be at all suitable for me while M.Euler is there. In 1766, after Euler left Berlin for Saint Petersburg, Frederick himself wrote to Lagrange expressing the wish of "the greatest king in Europe" to have "the greatest mathematician in Europe" resident at his court. Lagrange was finally persuaded. He spent the next twenty years in Prussia, where he produced a long series of papers published in the Berlin and Turin transactions, and composed his monumental work, the Mécanique analytique. In 1767, he married his cousin Vittoria Conti. Lagrange was a favourite of the king, who frequently lectured him on the advantages of perfect regularity of life. The lesson was accepted, and Lagrange studied his mind and body as though they were machines, and experimented to find the exact amount of work which he could do before exhaustion. Every night he set himself a definite task for the next day, and on completing any branch of a subject he wrote a short analysis to see what points in the demonstrations or the subject-matter were capable of improvement. He carefully planned his papers before writing them, usually without a single erasure or correction. Nonetheless, during his years in Berlin, Lagrange's health was rather poor, and that of his wife Vittoria was even worse. She died in 1783 after years of illness and Lagrange was very depressed. In 1786, Frederick II died, and the climate of Berlin became difficult for Lagrange. === Paris === In 1786, following Frederick's death, Lagrange received similar invitations from states including Spain and Naples, and he accepted the offer of Louis XVI to move to Paris. In France he was received with every mark of distinction and special apartments in the Louvre were prepared for his reception, and he became a member of the French Academy of Sciences, which later became part of the Institut de France (1795). At the beginning of his residence in Paris, he was seized with an attack of melancholy, and even the printed copy of his Mécanique on which he had worked for a quarter of a century lay for more than two years unopened on his desk. Curiosity as to the results of the French Revolution first stirred him out of his lethargy, a curiosity which soon turned to alarm as the revolution developed. It was about the same time, 1792, that the unaccountable sadness of his life and his timidity moved the compassion of 24-year-old Renée-Françoise-Adélaïde Le Monnier, daughter of his friend, the astronomer Pierre Charles Le Monnier. She insisted on marrying him and proved a devoted wife to whom he became warmly attached. In September 1793, the Reign of Terror began. Under the intervention of Antoine Lavoisier, who himself was by then already thrown out of the academy along with many other scholars, Lagrange was specifically exempted by name in the decree of October 1793 that ordered all foreigners to leave France. On 4 May 1794, Lavoisier and 27 other tax farmers were arrested and sentenced to death and guillotined on the afternoon after the trial. Lagrange said on the death of Lavoisier: It took only a moment to cause this head to fall and a hundred years will not suffice to produce its like. Though Lagrange had been preparing to escape from France while there was yet time, he was never in any danger; different revolutionary governments (and at a later time, Napoleon) gave him honours and distinctions. This luckiness or safety may to some extent be due to his life attitude he expressed many years before: "I believe that, in general, one of the first principles of every wise man is to conform strictly to the laws of the country in which he is living, even when they are unreasonable". A striking testimony to the respect in which he was held was shown in 1796 when the French commissary in Italy was ordered to attend in the full state on Lagrange's father and tender the congratulations of the republic on the achievements of his son, who "had done honour to all mankind by his genius, and whom it was the special glory of Piedmont to have produced". It may be added that Napoleon, when he attained power, warmly encouraged scientific studies in France, and was a liberal benefactor of them. Appointed senator in 1799, he was the first signer of the Sénatus-consulte which in 1802 annexed his fatherland Piedmont to France. He acquired French citizenship in consequence. The French claimed he was a French mathematician, but the Italians continued to claim him as Italian. ==== Units of measurement ==== Lagrange was involved in the development of the metric system of measurement in the 1790s. He was offered the presidency of the Commission for the reform of weights and measures (la Commission des Poids et Mesures) when he was preparing to escape. After Lavoisier's death in 1794, it was largely Lagrange who influenced the choice of the metre and kilogram units with decimal subdivision, by the commission of 1799. Lagrange was also one of the founding members of the Bureau des Longitudes in 1795. ==== École Normale ==== In 1795, Lagrange was appointed to a mathematical chair at the newly established École Normale, which enjoyed only a short existence of four months. His lectures there were elementary; they contain nothing of any mathematical importance, though they do provide a brief historical insight into his reason for proposing undecimal or Base 11 as the base number for the reformed system of weights and measures.: 23 The lectures were published because the professors had to "pledge themselves to the representatives of the people and to each other neither to read nor to repeat from memory" ["Les professeurs aux Écoles Normales ont pris, avec les Représentants du Peuple, et entr'eux l'engagement de ne point lire ou débiter de mémoire des discours écrits": iii ]. The discourses were ordered and taken down in shorthand to enable the deputies to see how the professors acquitted themselves. It was also thought the published lectures would interest a significant portion of the citizenry ["Quoique des feuilles sténographiques soient essentiellement destinées aux élèves de l'École Normale, on doit prévoir quיelles seront lues par une grande partie de la Nation": v ]. ==== École Polytechnique ==== In 1794, Lagrange was appointed professor of the École Polytechnique; and his lectures there, described by mathematicians who had the good fortune to be able to attend them, were almost perfect both in form and matter. Beginning with the merest elements, he led his hearers on until, almost unknown to themselves, they were themselves extending the bounds of the subject: above all he impressed on his pupils the advantage of always using general methods expressed in a symmetrical notation. However, Lagrange does not seem to have been a successful teacher. Fourier, who attended his lectures in 1795, wrote: his voice is very feeble, at least in that he does not become heated; he has a very marked Italian accent and pronounces the s like z [...] The students, of whom the majority are incapable of appreciating him, give him little welcome, but the professeurs make amends for it. ==== Late years ==== In 1810, Lagrange started a thorough revision of the Mécanique analytique, but he was able to complete only about two-thirds of it before his death in Paris in 1813, in 128 rue du Faubourg Saint-Honoré. Napoleon honoured him with the Grand Croix of the Ordre Impérial de la Réunion just two days before he died. He was buried that same year in the Panthéon in Paris. The inscription on his tomb reads in translation:JOSEPH LOUIS LAGRANGE. Senator. Count of the Empire. Grand Officer of the Legion of Honour. Grand Cross of the Imperial Order of the Reunion. Member of the Institute and the Bureau of Longitude. Born in Turin on 25 January 1736. Died in Paris on 10 April 1813. == Work in Berlin == Lagrange was extremely active scientifically during the twenty years he spent in Berlin. Not only did he produce his Mécanique analytique, but he contributed between one and two hundred papers to the Academy of Turin, the Berlin Academy, and the French Academy. Some of these are really treatises, and all without exception are of a high order of excellence. Except for a short time when he was ill he produced on average about one paper a month. Of these, note the following as amongst the most important. First, his contributions to the fourth and fifth volumes, 1766–1773, of the Miscellanea Taurinensia; of which the most important was the one in 1771, in which he discussed how numerous astronomical observations should be combined so as to give the most probable result. And later, his contributions to the first two volumes, 1784–1785, of the transactions of the Turin Academy; to the first of which he contributed a paper on the pressure exerted by fluids in motion, and to the second an article on integration by infinite series, and the kind of problems for which it is suitable. Most of the papers sent to Paris were on astronomical questions, and among these, including his paper on the Jovian system in 1766, his essay on the problem of three bodies in 1772, his work on the secular equation of the Moon in 1773, and his treatise on cometary perturbations in 1778. These were all written on subjects proposed by the Académie française, and in each case, the prize was awarded to him. === Lagrangian mechanics === Between 1772 and 1788, Lagrange re-formulated Classical/Newtonian mechanics to simplify formulas and ease calculations. These mechanics are called Lagrangian mechanics. === Algebra === The greater number of his papers during this time were, however, contributed to the Prussian Academy of Sciences. Several of them deal with questions in algebra. His discussion of representations of integers by quadratic forms (1769) and by more general algebraic forms (1770). His tract on the Theory of Elimination, 1770. Lagrange's theorem that the order of a subgroup H of a group G must divide the order of G. His papers of 1770 and 1771 on the general process for solving an algebraic equation of any degree via the Lagrange resolvents. This method fails to give a general formula for solutions of an equation of degree five and higher because the auxiliary equation involved has a higher degree than the original one. The significance of this method is that it exhibits the already known formulas for solving equations of second, third, and fourth degrees as manifestations of a single principle, and was foundational in Galois theory. The complete solution of a binomial equation (namely an equation of the form a x n {\displaystyle ax^{n}} ± b = 0 {\displaystyle b=0} ) is also treated in these papers. In 1773, Lagrange considered a functional determinant of order 3, a special case of a Jacobian. He also proved the expression for the volume of a tetrahedron with one of the vertices at the origin as the one-sixth of the absolute value of the determinant formed by the coordinates of the other three vertices. === Number theory === Several of his early papers also deal with questions of number theory. Lagrange (1766–1769) was the first European to prove that Pell's equation x2 − ny2 = 1 has a nontrivial solution in the integers for any non-square natural number n. He proved the theorem, stated by Bachet without justification, that every positive integer is the sum of four squares, 1770. He proved Wilson's theorem that (for any integer n > 1): n is a prime if and only if (n − 1)! + 1 is a multiple of n, 1771. His papers of 1773, 1775, and 1777 gave demonstrations of several results enunciated by Fermat, and not previously proved. His Recherches d'Arithmétique of 1775 developed a general theory of binary quadratic forms to handle the general problem of when an integer is representable by the form ax2 + by2 + cxy. He made contributions to the theory of continued fractions. === Other mathematical work === There are also numerous articles on various points of analytical geometry. In two of them, written rather later, in 1792 and 1793, he reduced the equations of the quadrics (or conicoids) to their canonical forms. During the years from 1772 to 1785, he contributed a long series of papers which created the science of partial differential equations. A large part of these results was collected in the second edition of Euler's integral calculus which was published in 1794. === Astronomy === Lastly, there are numerous papers on problems in astronomy. Of these the most important are the following: Attempting to solve the general three-body problem, with the consequent discovery of the two constant-pattern solutions, collinear and equilateral, 1772. Those solutions were later seen to explain what are now known as the Lagrangian points. On the attraction of ellipsoids, 1773: this is founded on Maclaurin's work. On the secular equation of the Moon, 1773; also noticeable for the earliest introduction of the idea of the potential. The potential of a body at any point is the sum of the mass of every element of the body when divided by its distance from the point. Lagrange showed that if the potential of a body at an external point were known, the attraction in any direction could be at once found. The theory of the potential was elaborated in a paper sent to Berlin in 1777. On the motion of the nodes of a planet's orbit, 1774. On the stability of the planetary orbits, 1776. Two papers in which the method of determining the orbit of a comet from three observations is completely worked out, 1778 and 1783: this has not indeed proved practically available, but his system of calculating the perturbations by means of mechanical quadratures has formed the basis of most subsequent researches on the subject. His determination of the secular and periodic variations of the elements of the planets, 1781–1784: the upper limits assigned for these agree closely with those obtained later by Le Verrier, and Lagrange proceeded as far as the knowledge then possessed of the masses of the planets permitted. Three papers on the method of interpolation, 1783, 1792 and 1793: the part of finite differences dealing therewith is now in the same stage as that in which Lagrange left it. === Fundamental treatise === Over and above these various papers he composed his fundamental treatise, the Mécanique analytique. In this book, he lays down the law of virtual work, and from that one fundamental principle, by the aid of the calculus of variations, deduces the whole of mechanics, both of solids and fluids. The object of the book is to show that the subject is implicitly included in a single principle, and to give general formulae from which any particular result can be obtained. The method of generalised co-ordinates by which he obtained this result is perhaps the most brilliant result of his analysis. Instead of following the motion of each individual part of a material system, as D'Alembert and Euler had done, he showed that, if we determine its configuration by a sufficient number of variables x, called generalized coordinates, whose number is the same as that of the degrees of freedom possessed by the system, then the kinetic and potential energies of the system can be expressed in terms of those variables, and the differential equations of motion thence deduced by simple differentiation. For example, in dynamics of a rigid system he replaces the consideration of the particular problem by the general equation, which is now usually written in the form d d t ∂ T ∂ x ˙ − ∂ T ∂ x + ∂ V ∂ x = 0 , {\displaystyle {\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {x}}}}-{\frac {\partial T}{\partial x}}+{\frac {\partial V}{\partial x}}=0,} where T represents the kinetic energy and V represents the potential energy of the system. He then presented what we now know as the method of Lagrange multipliers—though this is not the first time that method was published—as a means to solve this equation. Amongst other minor theorems here given it may suffice to mention the proposition that the kinetic energy imparted by the given impulses to a material system under given constraints is a maximum, and the principle of least action. All the analysis is so elegant that Sir William Rowan Hamilton said the work could be described only as a scientific poem. Lagrange remarked that mechanics was really a branch of pure mathematics analogous to a geometry of four dimensions, namely, the time and the three coordinates of the point in space; and it is said that he prided himself that from the beginning to the end of the work there was not a single diagram. At first no printer could be found who would publish the book; but Legendre at last persuaded a Paris firm to undertake it, and it was issued under the supervision of Laplace, Cousin, Legendre (editor) and Condorcet in 1788. == Work in France == === Differential calculus and calculus of variations === Lagrange's lectures on the differential calculus at École Polytechnique form the basis of his treatise Théorie des fonctions analytiques, which was published in 1797. This work is the extension of an idea contained in a paper he had sent to the Berlin papers in 1772, and its object is to substitute for the differential calculus a group of theorems based on the development of algebraic functions in series, relying in particular on the principle of the generality of algebra. A somewhat similar method had been previously used by John Landen in the Residual Analysis, published in London in 1758. Lagrange believed that he could thus get rid of those difficulties, connected with the use of infinitely large and infinitely small quantities, to which philosophers objected in the usual treatment of the differential calculus. The book is divided into three parts: of these, the first treats of the general theory of functions, and gives an algebraic proof of Taylor's theorem, the validity of which is, however, open to question; the second deals with applications to geometry; and the third with applications to mechanics. Another treatise on the same lines was his Leçons sur le calcul des fonctions, issued in 1804, with the second edition in 1806. It is in this book that Lagrange formulated his celebrated method of Lagrange multipliers, in the context of problems of variational calculus with integral constraints. These works devoted to differential calculus and calculus of variations may be considered as the starting point for the researches of Cauchy, Jacobi, and Weierstrass. === Infinitesimals === At a later period Lagrange fully embraced the use of infinitesimals in preference to founding the differential calculus on the study of algebraic forms; and in the preface to the second edition of the Mécanique Analytique, which was issued in 1811, he justifies the employment of infinitesimals, and concludes by saying that: When we have grasped the spirit of the infinitesimal method, and have verified the exactness of its results either by the geometrical method of prime and ultimate ratios, or by the analytical method of derived functions, we may employ infinitely small quantities as a sure and valuable means of shortening and simplifying our proofs. === Number theory === His Résolution des équations numériques, published in 1798, was also the fruit of his lectures at École Polytechnique. There he gives the method of approximating the real roots of an equation by means of continued fractions, and enunciates several other theorems. In a note at the end, he shows how Fermat's little theorem, that is a p − 1 − 1 ≡ 0 ( mod p ) {\displaystyle a^{p-1}-1\equiv 0{\pmod {p}}} where p is a prime and a is prime to p, may be applied to give the complete algebraic solution of any binomial equation. He also here explains how the equation whose roots are the squares of the differences of the roots of the original equation may be used so as to give considerable information as to the position and nature of those roots. === Celestial mechanics === A theory of the planetary motions had formed the subject of some of the most remarkable of Lagrange's Berlin papers. In 1806 the subject was reopened by Poisson, who, in a paper read before the French Academy, showed that Lagrange's formulae led to certain limits for the stability of the orbits. Lagrange, who was present, now discussed the whole subject afresh, and in a letter communicated to the academy in 1808 explained how, by the variation of arbitrary constants, the periodical and secular inequalities of any system of mutually interacting bodies could be determined. == Prizes and distinctions == Euler proposed Lagrange for election to the Berlin Academy and he was elected on 2 September 1756. He was elected a Fellow of the Royal Society of Edinburgh in 1790, a Fellow of the Royal Society and a foreign member of the Royal Swedish Academy of Sciences in 1806. In 1808, Napoleon made Lagrange a Grand Officer of the Legion of Honour and a Count of the Empire. He was awarded the Grand Croix of the Ordre Impérial de la Réunion in 1813, a week before his death in Paris, and was buried in the Panthéon, a mausoleum dedicated to the most honoured French people. Lagrange was awarded the 1764 prize of the French Academy of Sciences for his memoir on the libration of the Moon. In 1766 the academy proposed a problem of the motion of the satellites of Jupiter, and the prize again was awarded to Lagrange. He also shared or won the prizes of 1772, 1774, and 1778. Lagrange is one of the 72 prominent French scientists who were commemorated on plaques at the first stage of the Eiffel Tower when it first opened. Rue Lagrange in the 5th Arrondissement in Paris is named after him. In Turin, the street where the house of his birth still stands is named via Lagrange. The lunar crater Lagrange and the asteroid 1006 Lagrangea also bear his name. == See also == List of things named after Joseph-Louis Lagrange Four-dimensional space Gauss's law History of the metre Lagrange's role in measurement reform Seconds pendulum == Notes == == References == === Citations === === Sources === == External links == O'Connor, John J.; Robertson, Edmund F., "Joseph-Louis Lagrange", MacTutor History of Mathematics Archive, University of St Andrews Weisstein, Eric Wolfgang (ed.). "Lagrange, Joseph (1736–1813)". ScienceWorld. Lagrange, Joseph Louis de: The Encyclopedia of Astrobiology, Astronomy and Space Flight Clerke, Agnes Mary (1911). "Lagrange, Joseph Louis" . Encyclopædia Britannica. Vol. 16 (11th ed.). pp. 75–78. Joseph-Louis Lagrange at the Mathematics Genealogy Project The Founders of Classical Mechanics: Joseph Louis Lagrange The Lagrange Points Derivation of Lagrange's result (not Lagrange's method) Lagrange's works (in French) Oeuvres de Lagrange, edited by Joseph Alfred Serret, Paris 1867, digitized by Göttinger Digitalisierungszentrum (Mécanique analytique is in volumes 11 and 12.) Joseph Louis de Lagrange – Œuvres complètes Gallica-Math Inventaire chronologique de l'œuvre de Lagrange Persee Works by Joseph-Louis Lagrange at Project Gutenberg Works by or about Joseph-Louis Lagrange at the Internet Archive Mécanique analytique (Paris, 1811-15)
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Wikipedia:Josephine D. Edwards#0
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Josephine Dianne Edwards (18 August 1942 – 25 May 1985) was an Australian mathematician and mathematics educator who founded the Australian Mathematics Competition. == Education and career == Edwards was born in Oxford and was educated at the Ursuline School in Brentwood. She went on to study mathematics at the University of Cambridge. In 1964, Edwards moved to Canberra. She taught mathematics at secondary schools in the Australian Capital Territory. In 1979, she joined the faculty at the College of Advanced Education in Canberra, later the University of Canberra. == Service == For eighteen years, Edwards was a member of the Canberra Mathematical Association, also serving as its vice-president, president and secretary. She helped establish and run the Australian Mathematics Competition, serving as chair of its founding committee, as a member of its board of governors from 1977 to 1985 and as editor for its publications from 1979. She was also an associate editor for the American publication The College Mathematics Journal. Her articles on teaching mathematics appeared in journals in Australia, Canada and France. == Personal life == She was married to Paul Frost whom she met whilst studying mathematics at the University of Cambridge and the couple migrated to Australia and had two children. After Paul died she married Robert A. Edwards and had a daughter. Later she married John Pulley who at the time had three children from his first marriage. Edwards died in Canberra at the age of 42. == Recognition == In 1996, Edwards was posthumously awarded a BH Neumann Award. == References ==
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Wikipedia:Josephine Janina Mehlberg#0
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Josephine Janina Mehlberg (1905-1969) was a Polish-American Jewish mathematician. Using the fictional identity of Countess Janina Suchodolska, she helped thousands of people avoid death at the Majdanek concentration camp during World War II. == Life == Mehlberg was born Pepi Spinner in Zhuravno, now part of Ukraine. Her father was a wealthy landowner who often socialized with Polish nobles. She had a happy childhood until World War I, when her father was abducted by Russian forces; he died in 1918. Mehlberg earned her D.Phil. under the supervision of Kazimierz Twardowski from the University of Lwów (now Lviv) in 1928. She married fellow Jewish philosopher Henry Mehlberg in 1933, and at some point afterwards went by the name Janina Spinner Mehlberg. The two lived in Lwów, where Henry worked as a professor and Janina taught at a girls' school. In 1939, the Nazis invaded Poland, but the city was under Soviet control until 1941, at which point Nazi officials began moving the city's Jews into a ghetto. In response, the Mehlberg's fled to Lublin. There, the two received forged identity documents from Count Andrzej Skrzynski, a friend of Josephine's father. Under her new pseudonym of Countess Janina Suchodolska, Mehlberg began working with the Polish Main Welfare Council (known by the Polish acronym RGO); through them, she began working to bring supplies into the Majdanek concentration camp and to gather information for the Polish resistance. Mehlberg would argue with camp officials, convincing them to accept food, medicine, and clothing for prisoners under the supposition that it would allow for more prisoners to labor for the Nazis. She also negotiated the release of prisoners unable to work, who were transferred to the care of the RGO; biographers White and Sliwa say that Mehlberg negotiated the release of 9,707 Polish prisoners, 4,431 of whom were from Majdanek. Mehlberg was not able to aid Jewish prisoners due to the restraints of the RGO. Within the RGO, she also kept her Jewish identity secret for the duration of the war. After World War II, the Mehlberg's immigrated to the United States, settling in Chicago. She began teaching mathematics as a professor at the Illinois Institute of Technology, where she had two PhD students. Although not religious, both Janina and Henry participated in Jewish community events in Chicago. == Legacy == Late in her life, she wrote a memoir of her life during World War II; it was not published before her death in 1969. Her husband translated the manuscript into English and unsuccessfully sought to publish it. The manuscript was passed on to history professor Arthur Funk, and then to historian Elizabeth B. White, who used the manuscript as the foundation of her biography of Mehlberg, co-authored with historian Joanna Sliwa. The biography was published in 2024 under the title The Counterfeit Countess. == References == == Further reading == White, Elizabeth B.; Sliwa, Joanna (January 2024). The Counterfeit Countess: The Jewish Woman Who Rescued Thousands of Poles During the Holocaust. S. & S. ISBN 9781982189129.
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Wikipedia:Josephine Robinson Roe#0
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Josephine Robinson Roe (May 5, 1858 – April 29, 1946) was an American mathematician and university professor. In 1918, she was the first woman to receive a doctorate in mathematics from Syracuse University, and one of the few American women to earn a PhD in mathematics before World War II. == Biography == Josephine Alberta Robinson was born in Meredith, New Hampshire, to Frances Eliza Weld and Joseph Wadleigh Robinson, who was a farmer. Josephine graduated from the New Hampton Literary Institution in 1880 and taught in New Hampshire schools for more than a decade before entering college. Her early teaching assignments included teaching public district schools in New Hampshire for about 15 months, high school principal in Laconia, New Hampshire from 1880 to 1882, and teaching at her alma mater New Hampton Literary Institution. In 1890 she began studying mathematics at Oberlin College in Ohio, at age 32, and received a bachelor's degree in mathematics in 1894. She then taught Latin at Kimball Union Academy in Meriden, New Hampshire. In 1897 she was appointed to Berea College in Kentucky, where she taught Latin, English literature and mathematics. At Berea she held a series of positions including director of the women's department and acting professor of mathematics from 1897 to 1901, dean of women from 1901 to 1907 and professor of mathematics from 1901 to 1911. During the summer months of 1907 to 1911, she studied at Dartmouth College and it was there that she earned her master's degree in mathematics in 1911. In that same year, she married the widower Edward Drake Roe and left Berea College to study mathematics at Syracuse University where her husband was a math professor. In 1918, at the age of 60, she became the first female PhD student at Syracuse to receive her doctorate in mathematics with the dissertation: Interfunctional Expressibility Problems Symmetric Functions. Her research was directed by her husband, E.D. Roe. She taught as an assistant professor at Syracuse University and was a member of various mathematical organizations in addition to the American Astronomical Society, in acknowledgment of her research assistance to her husband at a private observatory he built behind their home. Josephine Robinson Roe died April 29, 1946 in Newton, Massachusetts, just a few days before she would have turned 88. == Memberships == According to Green, Roe belonged to several professional organizations. American Mathematical Society American Astronomical Society American Association for the Advancement of Science Phi Beta Kappa Sigma Xi == References ==
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Wikipedia:Josip Pečarić#0
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Josip Pečarić (born 2 September 1948) is a Croatian mathematician. He is a professor of mathematics in the Faculty of Textile Technology at the University of Zagreb, Croatia, and is a full member of the Croatian Academy of Sciences and Arts. He has written and co-authored over 1,200 mathematical publications. He has also published a number of works on history and politics that have been described as comprising historical negationism or Holocaust denial. == Education == Pečarić was born in Kotor, Montenegro (at the time part of Yugoslavia) on 2 September 1948, where he remained to attend elementary and high school. He studied at the University of Belgrade's Faculty of Electrical Engineering for his undergraduate and master's degrees, which he completed respectively in 1972 and 1975. The supervisor of his master's degree, mathematics professor, Dobrilo Tošić, inspired him to switch fields to mathematics. Pečarić remained at the University of Belgrade, working on his PhD in mathematics from 1975 to 1982. He received it under the supervision of Petar Vasić. His dissertation was on Jensen's and related inequalities. He began working at the University of Zagreb in 1987. == Mathematics career == Pečarić is known for his work in the theory of inequalities. He has founded several journals, all published by Element in Zagreb: he is currently Editor-in-Chief at Mathematical Inequalities and Applications and at the Journal of Mathematical Inequalities, and also founded Operators and Matrices. Pečarić has written and co-authored over 1,200 articles on mathematics in journals, books, and conference proceedings. He has also coauthored over 20 mathematical books, including 6 that are written in English. == Political views and historical negationism == In addition to his mathematical work, Pečarić has published more than 20 books and 40 articles on history and politics. This work is from a far-right point of view, and has been criticized as comprising historical negationism or Holocaust denial. For example, Pečarić has advocated for the return of the World War II-era fascist salute Za dom spremni. This salute has been called the Croatian equivalent of the German Sieg Heil. His 2017 book General Praljak reinvents the war criminal Slobodan Praljak as a humanist and war hero. His books Serbian Myths about Jasenovac and The Jasenovac Lie Revealed, the latter coauthored with Stjepan Razum, argued that the Jasenovac concentration camp was a labor camp with much lower casualties than the commonly accepted figure, and that the bulk of its victims were Croats killed by Yugoslav communist authorities after the war. This last prompted the Simon Wiesenthal Centre to advocate the Croatian government to ban publications denying the war crimes of the Ustaša. == Honors and awards == Pečarić has received a number of honors and awards. He was awarded the Croatian National Science Award in 1996, and received the Order of Danica Hrvatska in 1999. Pečarić was appointed to full membership of the Croatian Academy of Sciences and Arts in 2000. In 2008, a conference was held in honor of his 60th birthday and an issue (volume 2 no. 2) of the Banach Journal of Mathematical Analysis was dedicated to him. Another conference was held in 2014 in Pečarić's honor on the occasion of the publication of his 1000th mathematical paper. == Selected bibliography == === Textbooks === Recent Advances in Geometric Inequalities, co-authored with Dragoslav Mitrinović and Veno Volenec. Dordecht : Springer Science & Business Media (1989). ISBN 978-94-015-7842-4 Inequalities Involving Functions and Their Integrals and Derivatives, co-authored with Dragoslav Mitrinović and A.M. Fink. Dordecht : Springer; Kluwer Academic Publishers (1991). ISBN 978-94-011-3562-7 Convex Functions, Partial Orderings, and Statistical Applications, co-authored with Frank Proschan and Y.L. Tong. Boston : Academic Press (1992). ISBN 9780080925226 Classical and New Inequalities in Analysis, co-authored with Dragoslav Mitrinović and A.M. Fink. Dordecht : Springer Science & Business Media (1993). ISBN 978-94-017-1043-5 Mond-Pečarić Method in Operator Inequalities, co-authored with Takayuki Furuta, Jadranka Mićić Hot and Yuki Seo. Zagreb : Element (2005). ISBN 9789531975711 Multiplicative Inequalities of Carlson Type and Interpolation, co-authored with Leo Larsson, Lech Maligranda and Lars-Erik Persson. World Scientific Publishing Co. (2006). ISBN 978-981-4478-36-6 === Journal articles === Inequalities for Differentiable Mappings with Application to Special Means and Quadrature Formulæ, with Charles E. M. Pearce in Applied Mathematics Letters (2000). doi:10.1016/S0893-9659(99)00164-0 Hadamard-type Inequalities for S-convex Functions, with Ugur S Kirmaci, Milica Klaričić Bakula and Mehmet Emin Özdemir in Applied Mathematics and Computation (2007). doi:10.1016/j.amc.2007.03.030 New Means of Cauchy's Type, with Matloob Anwar in Journal of Inequalities and Applications (2008). doi:10.1155/2008/163202 === Non-fiction books === Srpski mit o Jasenovcu: Skrivanje istine o beogradskim konc-logorima. Zagreb : Croatian information centre (1998). Srpski mit o Jasenovcu II: O Bulajićevoj ideologijigenocida hrvatskih autora. Zagreb : Element (2000). Serbian myth about Jasenovac. Zagreb : Stih (2001). ISBN 9789536959006 Književnik Mile Budak sada i ovdje [Writer Mile Budak here and now]. Zagreb : Vlastita naklada (2005). ISBN 9789539888747 General Praljak, co-authored with Miroslav Međimorec. Zagreb : Vlastita naklada (2017). ISBN 9789537575212 Razotkrivena jasenovačka laž [The Jasenovac Lie Revealed], co-authored with Stjepan Razum. Zagreb : Društvo za istraživanje trostrukog logora Jasenovac (2018). ISBN 9789535856511 == References == == Sources == Byford, Jovan (2011). Staro sajmište: mesto sećanja, zaborava i sporenja. Beogradski centar za ljudska prava. ISBN 978-86-7202-131-8.
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Wikipedia:José Antonio Villaseñor y Sánchez#0
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José Antonio Villaseñor y Sánchez was an 18th-century geographer, historian, and mathematician in New Spain. He was born in San Luis Potosí, Mexico and studied at San Ildefonso in Mexico City. He became an accountant, and later official cosmographer (geographer) of New Spain. Following royal instructions, Viceroy Pedro Cebrián, 3rd Count of Fuenclara ordered him to prepare an official estimate of the population of New Spain, to be transmitted to the Court. The estimate was completed in April 1744, and the population was found to be 3,865,000. He also produced a plan of Mexico City (1750). == Works == Pantómetra matemática combinatoria de las leyes de la plata de toda ley (1733). Theatro Americano, descripción general de los reinos y provincias de la Nueva España y sus jurisdicciones (2 vols., 1746–48; 2nd ed. 1952). This work is a valuable source for colonial historians. Matemático cómputo de los astros (1756). == References == "Villaseñor y Sánchez, José Antonio," Enciclopedia de México, v. 14. Mexico City, 1988.
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Wikipedia:José Bonet Solves#0
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José Bonet Solves (Valencia, June 18, 1955) is a Spanish mathematician specialist in functional analysis and its applications to complex analysis and linear partial differential equations. == Academic biography == José Bonet graduated in Mathematics at the University of Valencia in 1977. In 1980 he defended his Ph.D. thesis in that University under the supervision of Professor Manuel Valdivia Ureña. Bonet was assistant in the University of Valencia from 1977 to 1983; between 1983 and 1987 he was associate professor, in the Polytechnic University of Valencia. Since 1987, Bonet is full professor in the Applied Mathematics Department at the Polytechnic University of Valencia. He was also Visiting Professor in the University of Paderborn, Germany in 1989 and in 2002. He had an Alexander Von Humboldt fellowship in Düsseldorf (1994), Paderborn (1995, 2008) and Eichstätt (2008, 2013, 2017); Germany. He has been the Director of the institute of mathematics Instituto Universitario de Matemática Pura y Aplicada IUMPA at the Polytechnic University of Valencia from 2004 until November 2016. José Bonet has been Principal Investigator of several research projects of the Spanish Ministry of Education since 1988, integrated actions between Spain and Germany and between Spain and Italy, and of the project of excellence in research PROMETEO of the Generalitat Valenciana (2008–2012, 2013–2016, and 2017–2021). He also organized several international meetings on functional analysis. He supervised fifteen Ph.D. thesis. == Publications == Coauthor of the book “Barrelled Locally Convex Spaces” published by North-Holland in 1987. Coeditor of the books “Progress in Functional Analysis” and “Recent Progress in Functional Analysis”, published by North-Holland in 1992 and 2002, respectively. *Coeditor of the book “Topics in Complex Analysis and Operator Theory” published by the American Mathematical Society in 2012. Author of more than 200 research papers published in international mathematics journals since 1980, like Advances in Mathematics, Journal of Functional Analysis, Transactions of the American Mathematical Society, Journal of the London Mathematical Society, Mathematische Zeitschrift, Studia Mathematica, etc. Most of this papers have been written in collaboration with researchers from Germany, the United States, Italy, Finland, Poland, etc. The publications received several citations. Editor of the journals RACSAM, Journal of Mathematical Analysis and Applications, Mediterranean Journal of Mathematics, Banach Journal of Mathematical Analysis y Functiones et Approximatio Commentarii Mathematici. == Honours == Corresponding member of the Société Royale des Sciences de Liège Belgium since 1992. Corresponding member of the Spanish Royal Academy of Sciences since 1994. Full member of the Spanish Royal Academy of Sciences, elected since 2005. Entrance lecture presented on April 23, 2008. Medal of the Real Sociedad Matemática Española in 2016. President of the Mathematics Section of the Spanish Royal Academy of Sciences since 2018. === Awards === First National Prize for Termination of University Studies in Mathematics, Primer Premio Nacional de Terminación de Estudios en Matemáticas of the Spanish Ministry of Education in 1977. Premio Extraordinario de Licenciatura (Prize for the Graduation) in Mathematics of the University of Valencia in 1978. Premio Extraordinario de Doctorado (Prize for the Doctor Thesis) in Mathematics of the University of Valencia in 1980. == References ==
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Wikipedia:José Felipe Voloch#0
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José Felipe Voloch (born 13 February 1963, in Rio de Janeiro) is a Brazilian mathematician who works on number theory and algebraic geometry and is a professor at Canterbury University. == Career == Voloch earned his Ph.D. from the University of Cambridge in 1985 under the supervision of John William Scott Cassels. He was a professor at the University of Texas, Austin. == Awards == He is a member of the Brazilian Academy of Sciences. == Selected publications == Coleman, Robert F.; Voloch, José Felipe (1992), "Companion forms and Kodaira-Spencer theory", Invent. Math., 110: 263–281, Bibcode:1992InMat.110..263C, doi:10.1007/bf01231333, MR 1185584, S2CID 121416817 == References == == External links == https://scholar.google.com/citations?user=r0Jun08AAAAJ
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Wikipedia:José Luis Massera#0
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José Luis Massera (Genoa, Italy, June 8, 1915 – Montevideo, September 9, 2002) was a Uruguayan dissident and mathematician who researched the stability of differential equations. Massera's lemma is named after him. He published over 40 papers during 1940–1970. A militant Communist, he was a political prisoner during 1975–1984. In the 1930s, Julio Rey Pastor gave regular weekend lectures on topology in Montevideo to a group that included Massera. Stimulated by contact with Argentine mathematics, the 1950s saw Uruguay develop a fine school in mathematics, of which Massera was very much a part. Massera developed new notions of stability, and published several foundational papers and an influential textbook. His results in (Massera 1950) on periodic differential equations have been heavily cited and are referred to as Massera's theorem. His work in (Massera 1949) and (Massera 1956) on the converse to Lyapunov's criterion is also influential, and contain the well known Massera's lemma. His textbook (Massera & Schäffer 1966) is also heavily cited. After military intervention in Uruguay in 1973, Massera was arrested on October 22, 1975 in Montevideo and was held in solitary confinement for nearly a year. During this time he was subjected to repeated torture resulting in injuries including a fractured pelvis. In October 1976 he was taken from solitary confinement, tried and convicted for "subversive association", and given a 24-year prison sentence. On June 22, 1979, as a consequence of a proposal put forward by Gaetano Fichera and unanimously approved by the whole Mathematics Faculty Council of the Sapienza University of Rome, he was awarded the laurea honoris causa while still being under conviction. He was released in 1984. == Honors == The outer main-belt asteroid 10690 Massera, discovered by American astronomer Schelte Bus at the Australian Siding Spring Observatory in 1981, was named in his memory on 13 April 2017 (M.P.C. 103976). == Selected works == Massera, José Luis (1949), "On Liapounoff's conditions of stability", Annals of Mathematics, Second Series, 50 (3), The Annals of Mathematics, Vol. 50, No. 3: 705–721, doi:10.2307/1969558, JSTOR 1969558, MR 0035354, Zbl 0038.25003. Massera, José Luis (1950), "The existence of periodic solutions of systems of differential equations", Duke Mathematical Journal, 17 (4): 457–475, doi:10.1215/S0012-7094-50-01741-8, MR 0040512, Zbl 0038.25002. Massera, José Luis (1956), "Contributions to stability theory", Annals of Mathematics, Second Series, 64 (1), The Annals of Mathematics, Vol. 64, No. 1: 182–206, doi:10.2307/1969955, JSTOR 1969955, MR 0079179, Zbl 0070.31003. Massera, José Luis; Schäffer, Juan Jorge (1966), Linear differential equations and function spaces, Pure and Applied Mathematics, Vol. 21, Boston, MA: Academic Press, MR 0212324, Zbl 0243.34107 == References == == Biographical references == Vernacchia-Galli, Jole (1986), "José Luis Massera", Regesto delle lauree honoris causa dal 1944 al 1985, Studi e Fonti per la storia dell'Università di Roma (in Italian and Spanish), vol. 10, Roma: Edizioni Dell'Ateneo, pp. 559–605. The "regest of honoris causa degrees from 1944 to 1985" (English translation of the title) is a detailed and carefully commented regest of all the documents of the official archive of the Sapienza University of Rome pertaining to the honoris causa degrees, awarded or not. It includes all the awarding proposals submitted during the considered period, detailed presentations of the work of the candidate, if available, and precise references to related articles published on Italian newspapers and magazines, if the laurea was awarded.
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Wikipedia:José Sebastião e Silva#0
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José Sebastião e Silva (12 December 1914 in Mértola – 25 May 1972 in Lisbon) was a Portuguese mathematician who made contributions to functional analysis, distribution theory, and mathematical education. After graduating from the University of Lisbon in 1937 and earning his doctorate in 1949, he taught at the Instituto Superior de Agronomia before becoming Director of the Centre for Mathematical Studies at the University of Lisbon. Silva is particularly remembered for his influential approach to teaching calculus, which combined intuitive understanding with mathematical rigour, and for his co-authored textbook Compêndio de Álgebra that shaped mathematics education in Portugal for decades. His pedagogical methods emphasised introducing students to mathematical concepts through concrete examples before progressing to formal definitions. == Early life and education == Born in Mértola, Silva graduated in mathematics from the University of Lisbon in 1937. In 1942 he received a grant from the Instituto de Alta Cultura to study in Rome, where he worked with members of the Italian school of algebraic geometry. His first doctoral thesis on geometric transformations was rejected by Federigo Enriques, so he composed a second on functional analysis, earning his doctorate in 1949 from the University of Lisbon. == Academic career == From 1951 to 1961, Silva was professor of mathematics at the Instituto Superior de Agronomia. He then returned to the University of Lisbon as Director of the Centre for Mathematical Studies, a post he held for twenty years. His research spanned analytic functionals, the theory of distributions (including vector‑valued distributions and ultradistributions), the operational calculus, and differential calculus in locally convex spaces. == Educational contributions == In 1951, Silva argued in the journal Gazeta de Matemática that infinitesimal calculus ought to be reintroduced into secondary education, warning that "when nothing is sown, what can be harvested?" and stressing that first exposure to the ideas of function and limit should occur in lyceum classrooms. Two years later, in partnership with José da Silva Paulo, he co‑authored the Portuguese: Compêndio de Álgebra (first ed. 1956; second ed. 1957), which was selected by national competition as the official algebra textbook for Portugal’s third‑cycle lyceums until 1968. Silva's pedagogical design unfolds as follows: Introduction of the concept of infinitesimal as a more intuitive precursor to limits. Successive treatment of limits of sequences followed by limits of functions. Continuity of a function. Definition of the derivative via the limit: f ′ ( x 0 ) = lim x → x 0 f ( x ) − f ( x 0 ) x − x 0 = lim h → 0 f ( x 0 + h ) − f ( x 0 ) h {\displaystyle f'(x_{0})=\lim _{x\to x_{0}}{\frac {f(x)-f(x_{0})}{x-x_{0}}}=\lim _{h\to 0}{\frac {f(x_{0}+h)-f(x_{0})}{h}}} where f ′ ( x 0 ) {\displaystyle f'(x_{0})} denotes the rate of change of f {\displaystyle f} at x 0 {\displaystyle x_{0}} . To avoid abstraction, Silva began with the mechanical notion of velocity—interpreting the slope of a distance–time graph as speed—and the geometric notion of slope (coefficient of a secant line approaching the tangent on a curve). Only after these concrete and symbolic motivations did he present the formal ε–δ definition, reserving rigorous mathematical proofs for higher education. This sequence closely mirrors David Tall's "three worlds" of mathematics—corporeal (sensory intuition), symbolic (manipulation of algebraic expressions) and formal (axiomatic proof)—by weaving together concrete examples, algebraic computation and formal definitions in a single narrative. Analyses of lyceum students' notebooks from 1978 confirm that Silva's examples—such as computing the derivative of x 2 {\displaystyle x^{2}} at a given point—were reproduced verbatim in class, and that subsequent textbooks built on his approach by adding lateral derivative concepts and further exercises while preserving his intuitive framework. == Legacy == Silva's integration of intuition and rigour in teaching the derivative shaped Portuguese mathematics education for decades. The Compêndio de Álgebra remained a pedagogical benchmark long after its official adoption, and successor texts continued to draw on his method of motivating abstract definitions through concrete phenomena. == References ==
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Wikipedia:José de Jesús Martínez#0
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José de Jesús Martínez (also known as "Chuchú Martínez" or "Sergeant Chuchú") (June 8, 1929, in Managua – January 27, 1991, in Panama) was a poet, playwright, philosopher, pilot and mathematician and a former aide to General Omar Torrijos Herrera, ruler of Panama from 1968 to 1981. Through his association with Torrijos, Martínez became a major figure in Graham Greene's 1984 book Getting to Know the General: The Story of an Involvement. == Early life == Martínez was born in Managua in 1929. == Career == Martínez became a Panamanian citizen and an aide to General Omar Torrijos Herrera, ruler of Panama from 1968 to 1981. Martínez was known as "Chuchú Martínez" in the intellectual circles, and later as "Sergeant Chuchú" among the military. Martínez was professor of Marxist philosophy at Panama University and also professor of mathematics. In literature, Martínez was awarded the National Theater Prize in Madrid in 1952 for his play, La Perrera. In 1969 and 1971 he won Panama's national literary prize Premio Ricardo Miró for his plays and philosophical essays. For his book, Mi General Torrijos (1987) he won Cuba's Casa de Las Americas prize. Through his association with Torrijos, Martínez became a major figure in Graham Greene's 1984 book Getting to Know the General: The Story of an Involvement. == Death == Martínez died, aged 61, in 1991. == Selected publications == La perrera. c. 1952. Enemigos. 1962. Aquí, ahora. 1963. Poemas a ella. 1963. La retreta. 1964. Amor no a ti, contigo. 1965. Poemas a mí. 1966. Amanecer de Ulises. 1967. One Way. 1967. 0 y van 3. 1970. Teatro. San José, Costa Rica, 1971. El caso Dios. 1975. Mi General Torrijos. 1987. == References == == External links == http://www.epdlp.com/escritor.php?id=3117 http://elpais.com/diario/1991/01/30/agenda/665190001_850215.html http://panamapoesia.com/pt68.php https://journals.ku.edu/index.php/latr/article/download/887/862
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Wikipedia:Joséphine Guidy Wandja#0
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Joséphine Guidy Wandja (born 1945, also Guidy-Wandja) is an Ivorian mathematician. She is the first African woman with a PhD in mathematics. == Early life == She moved to France aged 14. She attended the Lycée Jules-Ferry in Paris, and later the Pierre and Marie Curie University. Her master's degree thesis was entitled Sous les courbes fermées convexes du plan et le théorème des quatre sommets (Under closed convex curves in the plane and the theorem of four peaks). Whilst working in Paris in the late 1960s she was advised by René Thom, Henri Cartan and Paulette Liberman. She studied for a PhD at the University of Abidjan, becoming the first African woman to get a PhD in mathematics. == Career == In 1969, she worked at the Lycée Jacques Amyot in Melun, before working for a year at the Paris Diderot University. In 1971, she joined the University of Abidjan, as a mathematics lecturer. In doing so, she became the first African female university mathematics professor. In 1983, she was appointed the president of the International Committee on Mathematics in Developing Countries (ICOMIDC). The organisation was set up during the International Mathematical Union (IMU) conference in Warsaw, Poland, but without the IMU's knowledge. In 1986, she wrote a humorous 24 page mathematical comic book Yao crack en maths. In 1985, she organised an ICOMIDC conference in Yamoussoukro, Ivory Coast. She is an officer of the Ivorian Order of Merit of National Education, and the French Ordre des Palmes Académiques. == Publications == Guidy Wandja, Joséphine, Yao crack en maths (in French), Nouvelles Éditions africaines, 1985. ISBN 2723607356 == References ==
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Wikipedia:Jouanolou's trick#0
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In algebraic geometry, Jouanolou's trick is a theorem that asserts, for an algebraic variety X, the existence of a surjection with affine space fibers from an affine variety W to X. Moreover, the variety W is homotopy-equivalent to X, and W has the technically advantageous property of being affine. Jouanolou's original statement of the theorem required that X be quasi-projective over an affine scheme, but this has since been considerably weakened. == Jouanolou's construction == Jouanolou's original statement was: If X is a scheme quasi-projective over an affine scheme, then there exists a vector bundle E over X and an affine E-torsor W. By the definition of a torsor, W comes with a surjective map to X and is Zariski-locally on X an affine space bundle. Jouanolou's proof used an explicit construction. Let S be an affine scheme and X = P S r {\displaystyle X=\mathbf {P} _{S}^{r}} . Interpret the affine space A S ( r + 1 ) 2 {\displaystyle \mathbf {A} _{S}^{(r+1)^{2}}} as the space of (r + 1) × (r + 1) matrices over S. Within this affine space, there is a subvariety W consisting of idempotent matrices of rank one. The image of such a matrix is therefore a point in X, and the map W → X {\displaystyle W\to X} that sends a matrix to the point corresponding to its image is the map claimed in the statement of the theorem. To show that this map has the desired properties, Jouanolou notes that there is a short exact sequence of vector bundles: 0 → O X ( − 1 ) → O X ⊕ r + 1 → F → 0 , {\displaystyle 0\to {\mathcal {O}}_{X}(-1)\to {\mathcal {O}}_{X}^{\oplus r+1}\to {\mathcal {F}}\to 0,} where the first map is defined by multiplication by a basis of sections of O X ( 1 ) {\displaystyle {\mathcal {O}}_{X}(1)} and the second map is the cokernel. Jouanolou then asserts that W is a torsor for E = Hom ( F , O X ( − 1 ) ) {\displaystyle {\mathcal {E}}=\operatorname {Hom} ({\mathcal {F}},{\mathcal {O}}_{X}(-1))} . Jouanolou deduces the theorem in general by reducing to the above case. If X is projective over an affine scheme S, then it admits a closed immersion into some projective space P S r {\displaystyle \mathbf {P} _{S}^{r}} . Pulling back the variety W constructed above for P S r {\displaystyle \mathbf {P} _{S}^{r}} along this immersion yields the desired variety W for X. Finally, if X is quasi-projective, then it may be realized as an open subscheme of a projective S-scheme. Blow up the complement of X to get X ¯ {\displaystyle {\bar {X}}} , and let i : X → X ¯ {\displaystyle i\colon X\to {\bar {X}}} denote the inclusion morphism. The complement of X in X ¯ {\displaystyle {\bar {X}}} is a Cartier divisor, and therefore i is an affine morphism. Now perform the previous construction for X ¯ {\displaystyle {\bar {X}}} and pull back along i. == Thomason's construction == Robert Thomason observed that, by making a less explicit construction, it was possible to obtain the same conclusion under significantly weaker hypotheses. Thomason's construction first appeared in a paper of Weibel. Thomason's theorem asserts: Let X be a quasicompact and quasiseparated scheme with an ample family of line bundles. Then an affine vector bundle torsor over X exists. Having an ample family of line bundles was first defined in SGA 6 Exposé II Définition 2.2.4. Any quasi-projective scheme over an affine scheme has an ample family of line bundles, as does any separated locally factorial Noetherian scheme. Thomason's proof abstracts the key features of Jouanolou's. By hypothesis, X admits a set of line bundles L0, ..., LN and sections s0, ..., sN whose non-vanishing loci are affine and cover X. Define Xi to be the non-vanishing locus of si, and define E {\displaystyle {\mathcal {E}}} to be the direct sum of L0, ..., LN. The sections define a morphism of vector bundles s = ( s 0 , … , s N ) : O X → E {\displaystyle s=(s_{0},\ldots ,s_{N})\colon {\mathcal {O}}_{X}\to {\mathcal {E}}} . Define F {\displaystyle {\mathcal {F}}} to be the cokernel of s. On Xi, s is a split monomorphism since it is inverted by the inverse of si. Therefore F {\displaystyle {\mathcal {F}}} is a vector bundle over Xi, and because these open sets cover X, F {\displaystyle {\mathcal {F}}} is a vector bundle. Define P ( E ) = Proj Sym ∗ E {\displaystyle \mathbf {P} ({\mathcal {E}})=\operatorname {Proj} \operatorname {Sym} ^{*}{\mathcal {E}}} and similarly for P ( F ) {\displaystyle \mathbf {P} ({\mathcal {F}})} . Let W be the complement of P ( F ) {\displaystyle \mathbf {P} ({\mathcal {F}})} in P ( E ) {\displaystyle \mathbf {P} ({\mathcal {E}})} . There is an equivalent description of W as Spec ( Sym ∗ E / ( s − 1 ) ) {\displaystyle \operatorname {Spec} (\operatorname {Sym} ^{*}{\mathcal {E}}/(s-1))} , and from this description, it is easy to check that it is a torsor for F {\displaystyle {\mathcal {F}}} . Therefore the projection π : W → X {\displaystyle \pi \colon W\to X} is affine. To see that W is itself affine, apply a criterion of Serre (EGA II 5.2.1(b), EGA IV1 1.7.17). Each si determines a global section fi of W. The non-vanishing locus Wi of fi is contained in π − 1 ( X i ) {\displaystyle \pi ^{-1}(X_{i})} , which is affine, and hence Wi is affine. The sum of the sections f0, ..., fN is 1, so the ideal they generate is the ring of global sections. Serre's criterion now implies that W is affine. == References == Jouanolou, Jean-Pierre, Une Suite exact de Mayer–Vietoris en K-Theorie Algebrique. In Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 293–316. Lecture Notes in Math., Vol. 341. Springer, Berlin, 1973. Weibel, Charles A, Homotopy algebraic K-theory. In Algebraic K-theory and algebraic number theory (Honolulu, HI, 1987), volume 83 of Contemp. Math., pp. 461–488. Amer. Math. Soc., Providence, RI, 1989. == Further reading == https://dornsife.usc.edu/assets/sites/1176/docs/PDF/Jouanolou.pdf
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Wikipedia:Joubert's theorem#0
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In polynomial algebra and field theory, Joubert's theorem states that if K {\displaystyle K} and L {\displaystyle L} are fields, L {\displaystyle L} is a separable field extension of K {\displaystyle K} of degree 6, and the characteristic of K {\displaystyle K} is not equal to 2, then L {\displaystyle L} is generated over K {\displaystyle K} by some element λ in L {\displaystyle L} , such that the minimal polynomial p {\displaystyle p} of λ has the form p ( t ) {\displaystyle p(t)} = t 6 + c 4 t 4 + c 2 t 2 + c 1 t + c 0 {\displaystyle t^{6}+c_{4}t^{4}+c_{2}t^{2}+c_{1}t+c_{0}} , for some constants c 4 , c 2 , c 1 , c 0 {\displaystyle c_{4},c_{2},c_{1},c_{0}} in K {\displaystyle K} . The theorem is named in honor of Charles Joubert, a French mathematician, lycée professor, and Jesuit priest. In 1867 Joubert published his theorem in his paper Sur l'équation du sixième degré in tome 64 of Comptes rendus hebdomadaires des séances de l'Académie des sciences. He seems to have made the assumption that the fields involved in the theorem are subfields of the complex field. Using arithmetic properties of hypersurfaces, Daniel F. Coray gave, in 1987, a proof of Joubert's theorem (with the assumption that the characteristic of K {\displaystyle K} is neither 2 nor 3). In 2006 Hanspeter Kraft gave a proof of Joubert's theorem "based on an enhanced version of Joubert’s argument". In 2014 Zinovy Reichstein proved that the condition characteristic( K {\displaystyle K} ) ≠ 2 is necessary in general to prove the theorem, but the theorem's conclusion can be proved in the characteristic 2 case with some additional assumptions on K {\displaystyle K} and L {\displaystyle L} . == References ==
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Wikipedia:Jouko Väänänen#0
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Jouko Antero Väänänen (born September 3, 1950 in Rovaniemi, Lapland) is a Finnish mathematical logician known for his contributions to set theory, model theory, logic and foundations of mathematics. He served as the vice-rector at the University of Helsinki, and a professor of mathematics at the University of Helsinki, as well as a professor of mathematical logic and foundations of mathematics at the University of Amsterdam. He completed his PhD at the University of Manchester under the supervision of Peter Aczel in 1977 with the PhD thesis entitled "Applications of set theory to generalized quantifiers". He was elected to the Finnish Academy of Science and Letters in 2002. He served as a member of the Senate of the University of Helsinki from 2004 to 2006 and the Treasurer of the European Mathematical Society from 2007 to 2014, as well as the Treasurer of the European Set Theory Society since 2012. Jouko Väänänen received the Magnus Ehrnrooth Foundation Prize in mathematics on April 29, 2024. == Publications == === Books === Dependence Logic, Cambridge University Press, 2007. Models and Games, Cambridge University Press, 2011. == See also == Dependence logic == References == == External links == Jouko Väänänen's home page Jouko Väänänen in mathematics genealogy
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Wikipedia:Journal of Algebra#0
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Journal of Algebra (ISSN 0021-8693) is an international mathematical research journal in algebra. An imprint of Academic Press, it is published by Elsevier. Journal of Algebra was founded by Graham Higman, who was its editor from 1964 to 1984. From 1985 until 2000, Walter Feit served as its editor-in-chief. In 2004, Journal of Algebra announced (vol. 276, no. 1 and 2) the creation of a new section on computational algebra, with a separate editorial board. The first issue completely devoted to computational algebra was vol. 292, no. 1 (October 2005). The Editor-in-Chief of the Journal of Algebra is Michel Broué, Université Paris Diderot, and Gerhard Hiß, Rheinisch-Westfälische Technische Hochschule Aachen (RWTH) is Editor of the computational algebra section. == See also == Susan Montgomery, an editor of the journal == External links == Journal of Algebra at ScienceDirect
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Wikipedia:Journal of Algebra and Its Applications#0
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The Journal of Algebra and Its Applications covers both theoretical and applied algebra, with a focus on practical applications. It is published by World Scientific. According to the Journal Citation Reports, the journal has a 2020 impact factor of 0.736. == Abstracting and indexing == The journal is abstracted and indexed in: Mathematical Reviews Zentralblatt MATH Science Citation Index Expanded Current Contents/Physical Chemical and Earth Sciences Journal Citation Reports/Science Edition INSPEC == References ==
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Wikipedia:Journal of Commutative Algebra#0
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The Journal of Commutative Algebra is a peer-reviewed academic journal of mathematical research that specializes in commutative algebra and closely related fields. It has been published by the Rocky Mountain Mathematics Consortium (RMMC) since its establishment in 2009. It is currently published four times per year. Historically, the Journal of Commutative Algebra filled a niche for the Rocky Mountain Mathematics Consortium when the Canadian Applied Mathematics Quarterly, formerly published by the RMMC, was acquired by the Applied Mathematics Institute of the University of Alberta. Founding editors Jim Coykendall (currently at Clemson University) and Hal Schenck (currently at Auburn University) began the journal with the goal of creating a top-tier journal in commutative algebra. == Abstracting and indexing == The journal is abstracted and indexed in Current Contents/Physical, Chemical & Earth Sciences, Science Citation Index Expanded, Scopus, MathSciNet, and zbMATH. == References == == External links == Official website
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Wikipedia:Journal of Group Theory#0
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The Journal of Group Theory is a bimonthly peer-reviewed mathematical journal covering all aspects of group theory. It was established in 1998 and is published by Walter de Gruyter. The editor-in-chief is Chris Parker (University of Birmingham). == Abstracting and indexing == The journal is abstracted and indexed in: Its 2018 MCQ is 0.48. According to the Journal Citation Reports, the journal has a 2018 impact factor of 0.47, and the 5-year impact factor is 0.52. == References == == External links == Official website
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Wikipedia:Journal of Mathematical Analysis and Applications#0
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The Journal of Mathematical Analysis and Applications is an academic journal in mathematics, specializing in mathematical analysis and related topics in applied mathematics. It was founded in 1960 by Richard Bellman, as part of a series of new journals on areas of mathematics published by Academic Press, and is now published by Elsevier. For most years since 2003 it has been ranked by SCImago Journal Rank as among the top 25% of journals in its topic areas. == References ==
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Wikipedia:Journal of Mathematical Physics, Analysis, Geometry#0
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The Journal of Mathematical Physics, Analysis, Geometry is a quarterly peer-reviewed scientific journal covering mathematics as applied to physics. It is published by the Verkin Institute for Low Temperature Physics and Engineering and was established in 1994 as Mathematical Physics, Analysis, Geometry. Papers are published in English, Ukrainian, and Russian. The journal is abstracted and indexed by Scopus. According to the Journal Citation Reports, the journal has a 2017 impact factor of 0.531. == Editors-in-chief == The following persons are or have been editors-in-chief: Vladimir Marchenko: 1994—1999 Iossif Ostrovskii: 2000—2004 Leonid Pastur: 2005—present == History == The Kharkov Mathematical Society was founded in 1879 and, starting in 1880, the society published the journal named Communications of the Kharkov Mathematical Society (Russian Сообщения и протоколы заседаний математического общества при Императорском Харьковском университете). Publication was suspended in 1960, but in 1965 due to the efforts of Naum Akhiezer the journals Theory of functions, functional analysis and their applications, and Ukrainian Geometric Collection» were established. In 1994, these journals were merged by the Mathematical Division of the Verkin Institute to establish the current journal. The first editor was Vladimir Marchenko. == References == == External links == Official website L. Pastur (2010). "Mathematical publications in ILTPE" (PDF). 50 years of ILTPE (in Russian). Naukova dumka. Retrieved 2018-07-10.
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Wikipedia:Journal of Pure and Applied Algebra#0
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The Journal of Pure and Applied Algebra is a monthly peer-reviewed scientific journal covering that part of algebra likely to be of general mathematical interest: algebraic results with immediate applications, and the development of algebraic theories of sufficiently general relevance to allow for future applications. Its founding editors-in-chief were Peter J. Freyd (University of Pennsylvania) and Alex Heller (City University of New York). The current managing editors are Srikanth Iyengar (University of Utah), Charles Weibel (Rutgers University), and Aldo Conca (Università di Genova). == Abstracting and indexing == The journal is abstracted and indexed in Current Contents/Physics, Chemical, & Earth Sciences, Mathematical Reviews, PASCAL, Science Citation Index, Zentralblatt MATH, and Scopus. According to the Journal Citation Reports, the journal has a 2016 impact factor of 0.652. == References == == External links == Official website
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Wikipedia:Journal of Symbolic Computation#0
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The Journal of Symbolic Computation is a peer-reviewed monthly scientific journal covering all aspects of symbolic computation published by Academic Press and then by Elsevier. It is targeted to both mathematicians and computer scientists. It was established in 1985 by Bruno Buchberger, who served as its editor until 1994. The journal covers a wide variety of topics, including: Computer algebra, for which it is considered the top journal Computational geometry Automated theorem proving Applications of symbolic computation in education, science, and industry According to the Journal Citation Reports, its 2020 impact factor is 0.847. The journal is abstracted and indexed by Scopus and the Science Citation Index. == See also == Higher-Order and Symbolic Computation International Symposium on Symbolic and Algebraic Computation == References == == External links == Official website
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Wikipedia:Jovan Karamata#0
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Jovan Karamata (Serbian Cyrillic: Јован Карамата; February 1, 1902 – August 14, 1967) was a Serbian mathematician and university professor. He is remembered for contributions to analysis, in particular, the Tauberian theory and the theory of slowly varying functions. Considered to be among the most influential Serbian mathematicians of the 20th century, Karamata was one of the founders of the Mathematical Institute of the Serbian Academy of Sciences and Arts, established in 1946. == Life == Jovan Karamata was born in Zagreb on February 1, 1902, into a family descended from merchants based in the city of Zemun, which was then in Austria-Hungary, and now in Serbia. Being of Aromanian origin, the family traced its roots back to Pyrgoi, Eordaia, West Macedonia (his father Ioannis Karamatas was the president of the "Greek Community of Zemun"); Aromanians mainly lived and still live in the area of modern Greece. Its business affairs on the borders of the Austro-Hungarian and Ottoman empires were very well known. In 1914, he finished most of his primary school in Zemun but because of constant warfare on the borderlands, Karamata's father sent him, together with his brothers and his sister, to Switzerland for their own safety. In Lausanne, 1920, he finished primary school oriented towards mathematics and sciences. In the same year he enrolled at the Engineering faculty of Belgrade University and, after several years moved to the Philosophy and Mathematicians sector, where he graduated in 1925. He spent the years 1927–1928 in Paris, as a fellow of the Rockefeller Foundation, and in 1928 he became Assistant for Mathematics at the Faculty of Philosophy of Belgrade University. In 1930 he became Assistant Professor, in 1937 Associate Professor and, after the end of World War II, in 1950 he became Full Professor. In 1951 he was elected Full Professor at the University of Geneva. In 1933 he became a member of Yugoslav Academy of Sciences and Arts, Czech Royal Society in 1936, and Serbian Royal Academy in 1939 as well as a fellow of Serbian Academy of Sciences in 1948. He was one of the founders of the Mathematical Institute of the Serbian Academy of Sciences and Arts in 1946. Karamata was member of the Swiss, French and German mathematical societies, the French Association for the Development of Science, and the primary editor of the journal L’Enseignement Mathématique in Geneva. He also taught at the University of Novi Sad. In 1931 he married Emilija Nikolajevic, who gave birth to their two sons and two twin daughters. His wife died in 1959. After a long illness, Karamata died on August 14, 1967, in Geneva. His ashes rest in his native town of Zemun. == Legacy == Karamata published 122 scientific papers, 15 monographs and text-books as well as 7 professional-pedagogical papers. Karamata is best known for his work on mathematical analysis. He introduced the notion of regularly varying function, and discovered a new class of theorems of Tauberian type, today known as Karamata's tauberian theorems. He also worked on Mercer's theorems, Frullani integral, and other topics in analysis. In 1935 he introduced the brackets and braces notation for Stirling numbers (analogous to the binomial coefficients notation), which is now known as Karamata notation. He is also cited for Karamata's inequality. In Serbia, Karamata founded the "Karamata's (Yugoslav) school of mathematics”. Today, Karamata is the most cited Serbian mathematician. He is the developer and co-developer of dozens of mathematical theorems and has had a lasting influence in 20th-century mathematics. == See also == Mihailo Petrović Alas Bogdan Gavrilović == References == == Further reading == N.H. Bingham, C.M. Goldie, J.L. Teugels, Regular Variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge Univ. Press, 1987. J.L. Geluk, L. de Haan, Regular Variation Extensions and Tauberian Theorems, CWI Tract 40, Amsterdam, 1987. Maric V, Radasin Z, Regularly Varying Functions in Asymptotic Analysis Nikolic A, About two famous results of Jovan Karamata, Archives Internationales d’Histoire des Sciences Nikolic A, Jovan Karamata (1902–1967), Lives and work of the Serbian scientists, SANU, Biographies and bibliographies, Book 5 Tomic M, Academician Jovan Karamata, on occasion of his death, SANU, Vol CDXXIII, t. 37, Belgrade, 1968 (in Serbian) Tomic M, Jovan Karamata (1902–1967), L’Enseignement Mathématique Tomic M, Aljancic S, Remembering Karamata, Publications de l’Institut Mathématique == External links == Tổng quát về Bất đẳng thức Karamata (tiếng Việt)
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Wikipedia:Joy Morris#0
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Karen Joy Morris (Chinese: 莫文蔚; pinyin: Mò Wénwèi; born 2 June 1970), better known as Karen Mok, is a Hong Kong actress and singer. She is considered one of the leading East Asian pop singers and actresses with a career spanning three decades. Mok is the first female Hong Kong singer to win the Golden Melody Award, and to date has won it three times. Mok has released 18 solo studio albums, starred in over 40 movies, and has over 15 million followers on leading Chinese social media site Weibo. The Lhasa concert during The Ultimate Karen Mok Show (2018–2021) set the Guinness World Record for the Highest Altitude Mass-Attended Music Concert. In 2024, Mok was the first female singer to do a solo concert in Beijing National Stadium, the Bird's Nest. == Early life == Karen Mok was born on 2 June 1970 as Karen Joy Morris in Hong Kong. She is of mixed ancestry: her mother is half Chinese, quarter German and quarter Persian, while her father is half Welsh and half Chinese. Her grandfather was Alfred Morris, the first principal of King's College, Hong Kong. She speaks English, Cantonese, Mandarin, Italian, German and French. Mok attended Diocesan Girls' School from primary to secondary grade in Hong Kong. When she was a F.4 student, she received the 1st Hong Kong Outstanding Students Awards. In 1987, she won a scholarship for the United World College of the Adriatic in Duino (Trieste, Italy) from which she graduated in 1989 with the International Baccalaureate. She subsequently studied Italian Literature at Royal Holloway, University of London. == Music career == === 1993–2004: Career beginnings and musical breakthrough === While studying in London, Mok auditioned for the West End Musical Miss Saigon. At the same time, she recorded music demo tapes together with fellow students which landed her first recording contract with Star Records. She decided to put her theatre aspirations on hold, headed back to Hong Kong and released in 1993 her first Cantonese album Karen. She achieved her musical breakthrough with the launch of her first Mandarin album To Be in 1997. In 2000, she gave her debut solo concert, The Very Karen Mok Show, in Taipei in front of 20,000 spectators. === 2005–2014: International efforts === In 2005, she started her international concert tours with the Extremely Karen Mok Show. She also assumed the role of creative director and producer in her concert tours. In 2013, she launched her English album Somewhere I belong, an East-meets-West reinterpretation of jazz classics. The album is recorded in China with East Asian musicians and adding the guzheng on several tracks. She showcased this album at Ronnie Scott's Jazz Club, London, on 30 May 2013. Mok said about the title of the album: “It’s about what I do, when I feel comfortable, that’s when I’m doing what I love most, that’s singing, that’s when I’m performing on stage, and just singing my heart out, sharing my emotions,” she said. “So that’s where I belong.” The Age of Moknificence Tour (2014–2015) celebrated her 20th anniversary in show business. === 2015–2020: Tours and world records === The Regardez Tour (2015–2016) started in Taipei and led Mok to 27 cities in Asia, North America, Europe, Australia, and New Zealand. As part of this tour, she was the first Chinese pop artist to give solo concerts in Madrid and Milan. The Ultimate Karen Mok Show started on 23 June 2018 with the concert in the Hongkou Football Stadium in Shanghai 23 June 2018 in front of more than 20,000 spectators and totaled 48 concerts in 40 cities in Asia, Europe and Australia. The tour was largely conducted in outdoor stadiums, with the maximum attendance of 40,000 in the Beijing Workers' Stadium, Beijing, on 22 September 2019. On 12 October 2019, she performed in Lhasa at an altitude of 3650m, setting the Guinness World Record for the Highest Altitude Mass-Attended Music Concert and being the first solo stadium concert in Tibet. Her concerts in the Taipei Arena on 7 and 8 December 2019 had the special feature that she performed entirely different song lists on each night. The European leg of the tour brought her in London to the Palladium and in Paris to the Folies Bergère, making her the first Chinese pop singer to perform in this venue. === 2021–present: Continued success === With the Ultimate tour, Mok celebrated her 25th anniversary in show business. Interrupted by the events around the global Covid pandemic, the tour finished with three concerts in Hong Kong in June 2021. Her 2021 single "Empty World" had over 3bn streams on Chinese music platforms and became an anthem during the Covid pandemic in mainland China. In 2024, Mok ran a series of large-scale stadium concerts titled "The Big Big Show", commencing in Beijing on 15 June 2024, followed by concerts in Shanghai, Chengdu, Shenzhen, Hangzhou and Wuhan. In the course of this, she was the first female singer to perform a solo concert in Beijing's Bird's Nest stadium. === Notable collaborations === Mok has collaborated on stage and in the studio with numerous leading global artists. Amongst others, she performed together with The Black Eyed Peas, Andrea Bocelli, Far East Movement, Lang Lang, John Legend, Sergio Mendes, Mika, Keanu Reeves and Pharrell Williams. == Acting career == === Films === Mok gave her film debut in 1993 with a cameo role in the movie The Tigers – The Legend of Canton. Her first starring role was in 1995 together with Stephen Chow in A Chinese Odyssey. In the same year, she acted in Wong Kar Wai’s movie Fallen Angels, for which she received the award for Best Supporting Actress at the Hong Kong Film Award and the Golden Bauhinia Awards. In total, she starred in more than 40 movies. Outside Greater China, she acted in the 2004 Hollywood production Around the World in 80 Days with Jackie Chan (credited as Karen Joy Morris, her birth name) and in the Thai horror movie The Coffin. She also played the female lead role in Keanu Reeves’ directorial debut Man of Tai Chi. === Theatre and TV === Throughout her career, Mok has frequently branched into adjacent areas of performances. In 2005/2006, she played the female lead Mimi in the 10th Anniversary Asian Tour of the Broadway musical Rent. In 2001, she supplied the voice of Princess Kida for the Cantonese dub of Walt Disney's Atlantis: The Lost Empire. In 2020, Mok sang the theme song of the Hong Kong TVB drama Flying Tiger II. The theme song, "呼吸有害, Breathing Is Hazardous", topped all radio, TV and digital platforms in Hong Kong, a first in Cantopop. In 2016 and 2017, she starred in the two reality TV productions Up Idol (我们来了), of Hunan Television and The Next (天籁之战), season 1 and season 2 (天籁之战 and 天籁之战 (第二季)), of Dragon Television. On 20 May 2021, Mok won the 星耀年度金曲 award with Breathing is Hazardous at Weibo Starlight Awards 2020. == Public image == === Special events === She was a torch bearer for the 2008 Summer Olympics held in Beijing and performed at the opening and closing ceremonies. In 2016, she was awarded the title of Cultural Ambassador of the Italian city of Bergamo, being the first Chinese to be awarded this title. In 2023, Karen Mok celebrated the 30th anniversary of her career in show business with dedicated exhibitions in Hong Kong, Shanghai, Beijing, Taipei and Chengdu, titled "Reign of Moknificence", featuring materials from her careers across music, film and charity. === Brands === In October 2008, she launched her own line of perfume. Starting in 2015, she collaborated with the Italian brand Rucoline, also developing her own designs. In the following year, she initiated a crossover collection with Replay, leading up to a launch event and pop-up store at Harvey Nichols in London in 2018. Mok has been the face of leading global brands such as Cadillac, Canon, Cartier, Chow Tai Fook, Clear, Kappa, Lux, Mandarin Oriental, Schwarzkopf, Schweppes and Solvil et Titus. === Activism and charity === Mok is a strong advocate for animals and has been involved in numerous campaigns for this purpose. Amongst others, she has joined efforts with Animals Asia to help Asiatic black bears which are exploited for the extraction of bile, and has been on an expedition with Society for the Prevention of Cruelty to Animals (SPCA) to demonstrate against the culling of baby seals. She has also been involved with People for the Ethical Treatment of Animals (PETA). In 2007, Mok was involved with MTV EXIT, a campaign against human trafficking in Asia, presenting Traffic: An MTV EXIT Special, a documentary on trafficking. In 2013 she championed in the campaign Roll Back Malaria (RBM). Mok currently serves as an ambassador for UNICEF, SPCA, Animals Asia Foundation and Care for Children. In 2017, Mok created the Morris Charity Initiative, providing support in animal welfare, education and the environment. As a first initiative, the charity raised money for a scholarship at the United World College Changshu. Since then the charity has provided numerous donations to charities across the target areas. == Personal life == Mok married her boyfriend, German-born Johannes Natterer, at a church near Florence, Italy, on 1 October 2011. Mok has three adult step-children and spends her time between London and her international engagements. In 2017, they celebrated their sixth wedding anniversary with an elaborate party at Kensington Palace. == Filmography == == Discography == Cantonese Karen (1993) Karen Mok in Totality (1996) Golden Flower (2001) The Voyage (2021) Mandarin To Be (1997) I Say (1998) You Can (1999) This is Karen Mok (1999) Karen Mok on the Twelfth Floor (2000) [i] (2002) X (2003) Without You (2006) Live is… Karen Mok (2007) Hui Wei (2009) Precious (2010) Departures (2014) Half Time (2018) English Somewhere I Belong (2013) == Concert tours == Very Karen Mok Live Tour (2000–2001) Karen Mok China Tour (2003) The Extremely Karen Mok Show (2005–2006) The Original Karen Mok Show (2009–2011) The Age of Moknificence Tour (2013–2015) Regardez World Tour (2015–2016) The Ultimate Karen Mok Show (2018–2021) A Big Show (2024–2025) == Awards and nominations == === Film-related === === Baidu Entertainment Hot Point Awards === === Golden Bauhinia Awards === === Golden Horse Awards === === Golden Rooster Awards === === Hong Kong Film Awards === === Hong Kong Film Critics Society Award === === Huading Awards === === Macau International Movie Festival === === Hundred Flowers Awards === === Music-related === === Asia Song Festival === === Asian Pop Music Awards === === Beijing Pop Music Awards === === China Central Television – MTV Music Awards === === China Music Awards === === Chinese Music Awards === === Chinese Top Ten Music Awards === === CMIC Music Awards === === Ultimate Song Chart Awards Presentation === === CASH Golden Sail Music Awards === === Global Chinese Golden Chart Awards === === Global Chinese Music Awards === === Global Chinese Pop Chart === === Golden Melody Awards === === Hito Music Awards === === Jade Solid Gold Best Ten Music Awards Presentation === === Joox Top Music Awards === === KKBox Music Awards === === Ku Music Asian Music Awards === === MAMA Awards === === Metro Radio Hits Music Awards === === Migu Music Awards === === MTV Asia Awards === === MTV Video Music Awards === === Music Pioneer Awards === === Music Radio China Top Chart Awards === === Southeast Music Chart Awards === === Top Chinese Music Awards === === Top Ten Chinese Gold Songs Award === === TVB8 Mandarin Music On Demand Awards Presentation === == References == == External links == Official website (in English) Karen Mok at IMDb Karen Mok at the Hong Kong Movie Database
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Wikipedia:Jozef Dravecký#0
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Jozef Dravecký (25 July 1947 – 23 January 2023) was a Slovak mathematician and diplomat. Between 1991 and 2013, he served as the ambassador of Slovakia to Bulgaria (1993–1998), the Baltic States (2000 – 2005) and the Holy See (2007–2013). Dravecký was born in Spišská Nová Ves. He studied Sciences at the Comenius University graduating in 1971. In 1973 – 1990 he was a professor of Mathematical Analysis at the university. In 2010, Dravecký was awarded the Knights Grand Cross of the Order of Pope Pius IX by the Pope Benedict XVI. His death, on 23 January 2023 at the age of 75, was announced by the Chairman of the Christian Union of Pensioners Peter Mach in a Facebook post. == References ==
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Wikipedia:Jozef Teugels#0
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Jozef Lodewijk Maria Teugels, or Jozef L. Teugels, Jef Teugels (born February 20, 1939) is a Belgian mathematical statistician and actuary. His main contributions are in extreme value theory, stochastic processes, and reinsurance theory. == Education and career == Teugels was born in Londerzeel, Belgium and studied at the Catholic University of Louvain, where he received his licentiate in mathematics in 1963. He moved to the US to pursue further study and obtained his MSc in 1966 and PhD in 1967, both at Purdue University, where he was supervised by Marcel F. Neuts. He returned to Belgium in the same year and took up a faculty position at KU Leuven, where he was promoted to full professorship in 1973. He remained in Louvain until his retirement in 2004. During this period, Teugels was the chair of the Department of Mathematics from 1970 to 1977 and from 1982 to 1989. Teugels was named a distinguished alumnus of Purdue University in 2004. He was president of the International Statistical Institute between 2009 and 2011. == Bibliography == Bingham, N. H.; Goldie, C. M.; Teugels, J. L. (1987). Regular Variation (1 ed.). Cambridge University Press. doi:10.1017/cbo9780511721434. ISBN 978-0-521-30787-1. Beirlant, Jan; Teugels, Jozef L.; Vynckier, Petra (1996). Practical analysis of extreme values. Leuven, Belgium: Leuven University Press. ISBN 90-6186-768-1. OCLC 36890147. Beirlant, Jan; Goegebeur, Yuri; Segers, Johan; Teugels, Jozef L. (2004). Statistics of extremes : theory and applications. Chichester, England: Wiley. ISBN 978-0-470-01237-6. OCLC 85820811. Teugels, Jef L.; Sundt, Bjørn, eds. (2004). Encyclopedia of actuarial science. Hoboken, NJ: Wiley. ISBN 0-470-84676-3. OCLC 55845505. Albrecher, Hansjörg; Beirlant, Jan; Teugels, Jozef L. (2008). Reinsurance : Actuarial and Statistical Aspects. Hoboken: John Wiley & Sons, Incorporated. ISBN 978-1-119-41993-8. OCLC 1003266191. Rolski, Tomasz; Schmidli, Hanspeter; Schmidt, V.; Teugels, Jozef L. (2009). Stochastic processes for insurance and finance. Chicester: Wiley. ISBN 9786612307928. OCLC 1303448068. == References ==
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Wikipedia:João Lucas Marques Barbosa#0
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João Lucas Marques Barbosa (born April 9, 1943) is a Brazilian physicist, researcher, and university professor. Commander and Grand Officer of the National Order of Scientific Merit (Ordem Nacional do Mérito Científico) and member of the Brazilian Academy of Sciences (Academia Brasileira de Ciências - ABC), João Lucas is professor emeritus of the Department of Mathematics of the Federal University of Ceará (UFC), and author of several books on mathematics. He is a member of the Science and Technology Council of the Ceará state government. == Biography == Barbosa was born in Fortaleza in 1943. Coming from a poor family, he lost his father, a grain merchant at the Central Market of Fortaleza, in 1946, when he was only 3 years old. His mother was an elementary school teacher and principal, from the age of 17 until 70. João Lucas was taught literacy at home, and studied his early years at the Domingos Brasileiro Reunited Schools, then at the Castelo Branco School, and, finally, at the Ceará's liceu, in an evening course, where he completed the scientific course. In high school he studied at night because he had entered the Banking Apprenticeship Course, created by the Bank of Northeast (Banco do Nordeste do Brasil - BNB) with the objective of training professionals for a career in banking. Upon graduating, he became an employee of the bank. The university course only came in 1962, when he entered the Federal University of Ceará (UFC), in the first class of the bachelor's degree in Mathematics of the then recently created Faculty of Philosophy, Sciences and Letters. He passed in first place in the entrance exam of the course. In his first year he had classes with the mathematician Elon Lages Lima, from whom he learned calculus, and, in the third year, with Manfredo do Carmo, learning differential geometry, an area in which he became interested almost immediately. He received a scholarship for scientific initiation in his first year, which made him leave the Bank of Northeast. He managed to keep his scholarship until the second semester of 1964, but ended up losing it, as well as his job at the bank. So he started teaching in prep courses and colleges in order to earn money. He was a student leader, academic directory president, member of the direction team of the Catholic University Youth and even candidate for president of the Central Students Directory (Diretório Central dos Estudantes - DCE). He got his bachelor's degree in December 1965 and, through a public contest, he joined the UFC as a teaching assistant. At the same time, he began his master's degree in mathematics, defending his dissertation in July 1967. == Career == Already married in 1969, he received a scholarship from CNPq and left for his doctorate at the University of California, Berkeley, at the time the best center for the study of differential geometry. His thesis, defended in 1972 under the guidance of Shiing-Shen Chern, was on minimal immersions, a trendy subject at that time. Back in Brazil, he taught mathematics at UFC and in elementary and medical schools in the capital city of Ceará. He served as a graduate course coordinator and, lastly, as President of the Ceará Foundation for Research Support. He is the author of several books on Euclidean plane geometry and hyperbolic geometry. His line of research focuses on immersions with prescribed r-curvature and stability issues. == See also == National Order of Scientific Merit Brazilian Academy of Sciences (ABC) Federal University of Ceará (UFC) == References ==
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Wikipedia:Juan Manuel Cajigal y Odoardo#0
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Juan Manuel Cajigal y Odoardo (1803 in Barcelona, Anzoátegui – 1856 in Yaguaraparo, Sucre) was a Venezuelan mathematician, engineer and statesman. Orphaned at age 7, he was raised in Spain by his cousin-once-removed, Field Marshal Juan Manuel Cajigal, former captain general of Venezuela and Cuba. He studied in the University of Alcalá de Henares and later in France, finishing his studies in 1828. He returned to Venezuela that year. He helped found the Sociedad Económica de Amigos del País the following year, and in 1830 the government appointed him to create and direct the new Military Academy of Mathematics. He served in Congress twice, once in 1833 as representative of Caracas, and in 1835 as senator of Barcelona Province. With José Hermenegildo García and Fermín Toro he started the newspaper Correo de Caracas, which ran from 1838 to 1841. His publications include Tratado de mecánica elemental ("Treatise on Fundamental Mechanics") and Curso de astronomía y memorias sobre integrales entre límites ("Course on Astronomy and Report on Integrals between Limits"). The Juan Manuel Cajigal Naval Observatory in the 23 de Enero district of Caracas (Metro Station: Caño Amarillo), Juan Manuel Cajigal Municipality in Anzoátegui, and asteroid (minor planet) 12359 Cajigal are named after him. == Sources == Nieschulz de Stockhausen, Elke. "Juan Manuel Cajigal y Odoardo," Diccionario de Historia de Venezuela. Caracas: Fundacíon Polar, 1997. ISBN 980-6397-37-1
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Wikipedia:Jucys–Murphy element#0
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In mathematics, the Jucys–Murphy elements in the group algebra C [ S n ] {\displaystyle \mathbb {C} [S_{n}]} of the symmetric group, named after Algimantas Adolfas Jucys and G. E. Murphy, are defined as a sum of transpositions by the formula: X 1 = 0 , X k = ( 1 k ) + ( 2 k ) + ⋯ + ( k − 1 k ) , k = 2 , … , n . {\displaystyle X_{1}=0,~~~X_{k}=(1\;k)+(2\;k)+\cdots +(k-1\;k),~~~k=2,\dots ,n.} They play an important role in the representation theory of the symmetric group. == Properties == They generate a commutative subalgebra of C [ S n ] {\displaystyle \mathbb {C} [S_{n}]} . Moreover, Xn commutes with all elements of C [ S n − 1 ] {\displaystyle \mathbb {C} [S_{n-1}]} . The vectors constituting the basis of Young's "seminormal representation" are eigenvectors for the action of Xn. For any standard Young tableau U we have: X k v U = c k ( U ) v U , k = 1 , … , n , {\displaystyle X_{k}v_{U}=c_{k}(U)v_{U},~~~k=1,\dots ,n,} where ck(U) is the content b − a of the cell (a, b) occupied by k in the standard Young tableau U. Theorem (Jucys): The center Z ( C [ S n ] ) {\displaystyle Z(\mathbb {C} [S_{n}])} of the group algebra C [ S n ] {\displaystyle \mathbb {C} [S_{n}]} of the symmetric group is generated by the symmetric polynomials in the elements Xk. Theorem (Jucys): Let t be a formal variable commuting with everything, then the following identity for polynomials in variable t with values in the group algebra C [ S n ] {\displaystyle \mathbb {C} [S_{n}]} holds true: ( t + X 1 ) ( t + X 2 ) ⋯ ( t + X n ) = ∑ σ ∈ S n σ t number of cycles of σ . {\displaystyle (t+X_{1})(t+X_{2})\cdots (t+X_{n})=\sum _{\sigma \in S_{n}}\sigma t^{{\text{number of cycles of }}\sigma }.} Theorem (Okounkov–Vershik): The subalgebra of C [ S n ] {\displaystyle \mathbb {C} [S_{n}]} generated by the centers Z ( C [ S 1 ] ) , Z ( C [ S 2 ] ) , … , Z ( C [ S n − 1 ] ) , Z ( C [ S n ] ) {\displaystyle Z(\mathbb {C} [S_{1}]),Z(\mathbb {C} [S_{2}]),\ldots ,Z(\mathbb {C} [S_{n-1}]),Z(\mathbb {C} [S_{n}])} is exactly the subalgebra generated by the Jucys–Murphy elements Xk. == See also == Representation theory of the symmetric group Young symmetrizer == References == Okounkov, Andrei; Vershik, Anatoly (2004), "A New Approach to the Representation Theory of the Symmetric Groups. 2", Zapiski Seminarov POMI, 307, arXiv:math.RT/0503040(revised English version).{{citation}}: CS1 maint: postscript (link) Jucys, Algimantas Adolfas (1974), "Symmetric polynomials and the center of the symmetric group ring", Rep. Mathematical Phys., 5 (1): 107–112, Bibcode:1974RpMP....5..107J, doi:10.1016/0034-4877(74)90019-6 Jucys, Algimantas Adolfas (1966), "On the Young operators of the symmetric group" (PDF), Lietuvos Fizikos Rinkinys, 6: 163–180 Jucys, Algimantas Adolfas (1971), "Factorization of Young projection operators for the symmetric group" (PDF), Lietuvos Fizikos Rinkinys, 11: 5–10 Murphy, G. E. (1981), "A new construction of Young's seminormal representation of the symmetric group", J. Algebra, 69 (2): 287–297, doi:10.1016/0021-8693(81)90205-2
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Wikipedia:Juha Heinonen#0
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Juha Heinonen (23 July 1960, Toivakka – 30 October 2007) was a Finnish mathematician, known for his research on geometric function theory. == Biography == Heinonen, whose father was a lumberjack and local politician, grew up in a small town in central Finland. He studied mathematics at the University of Jyväskylä and received his doctorate there in 1987 with a thesis on nonlinear potential theory. His thesis advisor was Olli Martio. During the academic year 1987–1988 Heinonen was a visiting researcher at the Deutsche Forschunsgemeinschaft in Bonn and then at the Centre de Recerca Matemática in Barcelona. He first came to the University of Michigan as a visiting graduate student in 1985, and then came back as a three-year postdoctoral assistant professor from 1988 to 1991. In 1992 he was hired there as a tenure-track assistant professor, and spent the rest of his career there until his death from kidney cancer in 2007. He was promoted to full professor in 2000. He was the author or coauthor of three books (one of which was published posthumously) and over 60 research articles. He was a leading contributor to the development of nonsmooth calculus in geometric analysis on metric spaces. His 2007 article Nonsmooth calculus is an important survey of the subject. The objects of interest in nonsmooth calculus as described by Heinonen are spaces of homogeneous type, or metric measured spaces where a generalization of Poincaré inequality is true. In such spaces the differential calculus goes a long way: Sobolev spaces, differentiation theorems, Hardy spaces. It is noticeable that in such a general situation we don't have enough structure to define differentials, but only various constructions corresponding to the norm of a differential of a function. In 1992 Heinonen was a Sloan Research Fellow. In 2002 he was an Invited Speaker with talk The branch set of a quasireglar mapping at the International Congress of Mathematicians in Beijing. In 2004 he was elected a member of the Finnish Academy of Science and Letters. A gifted athlete, Juha is still revered as a local sports celebrity. Many Finns remember his achievements in cross-county skiing, including Finland's 1976 gold medal in the 5 km race in his class. More recently, he traveled around Finland and North America to compete in orienteering, placing in nearly every major US competition he entered. He is widely remembered in US orienteering circles for winning both the US and the North American gold medal in his class in the year 2000 ... In 1991 he married the mathematician Karen E. Smith. They had three children. == Selected publications == === Articles === Heinonen, Juha; Kilpeläinen, Tero (1988). "On the Wiener criterion and quasilinear obstacle problems". Transactions of the American Mathematical Society. 310: 239. doi:10.1090/S0002-9947-1988-0965751-8. ——; Koskela, Pekka (1994). " A {\displaystyle A} ∞-condition for the Jacobian of a quasiconformal mapping". Proceedings of the American Mathematical Society. 120 (2): 535. doi:10.1090/S0002-9939-1994-1169029-X. —— (1995). "The diameter conjecture for quasiconformal maps is true in space". Proceedings of the American Mathematical Society. 123 (6): 1709. doi:10.1090/S0002-9939-1995-1234626-0. ——; Koskela, Pekka (1998). "Quasiconformal maps in metric spaces with controlled geometry". Acta Mathematica. 181: 1–61. doi:10.1007/BF02392747. S2CID 59366197. Hanson, Bruce; —— (2000). "An n {\displaystyle n} -dimensional space that admits a Poincaré inequality but has no manifold points". Proceedings of the American Mathematical Society. 128 (11): 3379–3390. doi:10.1090/S0002-9939-00-05453-8. —— (2006). "What is a quasiconformal mapping?" (PDF). Notices of the AMS. 53 (11): 1134–1135. —— (2007). "Nonsmooth calculus". Bulletin of the American Mathematical Society. 44 (2): 163–233. doi:10.1090/S0273-0979-07-01140-8. ISSN 0273-0979. === Books and monographs === Heinonen, Juha; Kilpeläinen, Tero; Martio, Olli (19 September 2012). Nonlinear Potential Theory of Degenerate Elliptic Equations. Mineola, New York: Dover Publications. ISBN 9780486149257. (originally published by Oxford University Press in 1993) Heinonen, Juha (2001). Lectures on Analysis on Metric Spaces. Springer. doi:10.1007/978-1-4613-0131-8. ISBN 9780387951041. —— (2005). Lectures on Lipschitz analysis (PDF). Vol. 100. University of Jyväskylä. ——; Koskela, Pekka; Shanmugalingam, Nageswari; Tyson, Jeremy T. (5 February 2015). Sobolev Spaces on Metric Measure Spaces. Cambridge University Press. ISBN 978-1-107-09234-1. == References == == External links == "In Memory of Professor Juha Heinonen (1960–2007)". Mathematics, University of Michigan.
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Wikipedia:Juhani Karhumäki#0
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Eero Urho Juhani Karhumäki (born 1949) is a Finnish mathematician and theoretical computer scientist known for his contributions to automata theory. He is a professor at the University of Turku in southwestern Finland. == Biography == Karhumäki earned his doctorate from the University of Turku in 1976. In 1980–1985, he was a junior researcher of Academy of Finland. Since 1986, he has held teaching positions at the University of Turku, attaining full professorship in 1998. In 1998–2015, Karhumäki was the head of the mathematics department at the University of Turku. He has authored altogether around 200 research papers. Karhumäki is a member of the Finnish Academy of Science and Letters since 2000 and of Academia Europaea since 2006. A festschrift in his honour was published in 2009 as a special issue of Theoretical Computer Science. == Research contributions == Karhumäki has been a member of the Lothaire group of mathematicians that developed the foundations of combinatorics of words. In 1991, jointly with Tero Harju, he solved the long-standing equivalence problem for multitape finite automata in automata theory. Karhumäki contributed to different areas of formal language theory, such as word equations, language equations and descriptional complexity of finite automata. == References == == External links == Juhani Karhumäki at DBLP Bibliography Server Juhani Karhumäki at the Mathematics Genealogy Project
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Wikipedia:Jules Molk#0
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Jules Molk (8 December 1857 in Strasbourg, France – 7 May 1914 in Nancy) was a French mathematician who worked on elliptic functions. The French Academy of Sciences awarded him the Prix Binoux for 1913. He was appointed to the chair of applied mathematics at the University of Nancy upon the death of Émile Léonard Mathieu in 1890. From 1902 until his death in 1914, Molk was the leader and editor-in-chief of the publication of a French encyclopedia of pure and applied mathematical sciences based upon Klein's encyclopedia. It was a translation of the volumes in German and required the collaboration of many mathematicians and theoretical physicists from France, Germany, and several other European countries. Among the noteworthy contributors are: Paul Appell, Felix Klein, Jacques Hadamard, David Hilbert, Émile Borel, Paul Montel, Maurice Fréchet, Édouard Goursat, Ernst Zermelo, Ernst Steinitz, Arthur Schoenflies, Philipp Furtwängler, Carl Runge, Vilfredo Pareto, Ernest Vessiot, Gino Fano, George Darwin, Paul Langevin, Jean Perrin, Karl Schwarzschild, Pierre Boutroux, Edmond Bauer, Max Abraham, Arnold Sommerfeld, Ernest Esclangon, Paul Ehrenfest, and Tatyana Pavlovna Ehrenfest. The French edition of the Encyclopaedia of the Pure Mathematical Sciences edited and published after the German edition is neither a straight translation nor a simple adaptation for the French readership. It was intended as an erudite and sometimes controversial re-reading of the German edition, by a group of scholars brought together by Jules Molk, professor of rational mechanics at the University of Nancy. He was familiar with the scientific literature in Germany having lived in Berlin between 1880 and 1884 and having written a thesis focused on an overview of the German breakthroughs in Mathematics. The first volumes of this French edition appeared in 1908, with the help of ... Teubner and Gauthier Villars, and the last in 1916, two years after the death of Jules Molk. Only half of the volumes released from Teubner in 1908 were published, and that with some difficulty. Nevertheless, such a Franco-German scientific collaboration was remarkable and totally new. In 1906 Molk was elected a member of the Academy of Sciences Leopoldina. == Publications == Tannery, Jules; Molk, Jules (1972) [1893], Éléments de la théorie des fonctions elliptiques. Tomes I, II, III, IV. Calcul différentiel, New York: Chelsea Publishing Co., ISBN 978-0-8284-0257-6, MR 0392471 Éléments de la théorie des fonctions elliptiques. Tome 1, sur Gallica. Éléments de la théorie des fonctions elliptiques. Tome 2, sur Gallica. Éléments de la théorie des fonctions elliptiques. Tome 4, sur Gallica. Éléments de la théorie des fonctions elliptiques Volumes 1 et 2, sur Archive.org. Éléments de la théorie des fonctions elliptiques Tome 1, sur Archive.org. === Encyclopédie des sciences mathématiques pures et appliquées === Site des Editions Jacques Gabay Linum, livres numériques mathématiques Encyclopédie des sciences mathématiques pures et appliquées Volume 1, sur Archive.org. Encyclopédie des sciences mathématiques pures et appliquées Tome 4 Volume 2 Archived 2018-01-07 at the Wayback Machine, sur IRIS. Encyclopédie des sciences mathématiques pures et appliquées Tome 2, Premier volume, sur Gallica. Encyclopédie des sciences mathématiques pures et appliquées Tome 2, Deuxième volume, sur Gallica. Encyclopédie des sciences mathématiques pures et appliquées Tome 4, Cinquième volume, sur Gallica. Encyclopédie des sciences mathématiques pures et appliquées Tome 4, Sixième volume, sur Gallica. == References == Hélène Gispert (1999) "Les débuts de l'histoire des mathématiques sur les scènes internationales et le cas de l'entreprise encyclopédique de Felix Klein et Jules Molk", Historia Mathematica 26(4):344–60. Jules Molk at the Mathematics Genealogy Project
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Wikipedia:Julia Böttcher#0
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Julia Böttcher is a German discrete mathematician and a professor of mathematics at the London School of Economics. Her research involves graph theory, including graph and hypergraph packing problems, random graphs and random subgraphs, and the relations between graph parameters including graph bandwidth, degree, and chromatic number. == Education and career == After secondary school in Erfurt, Böttcher studied computer science at the Humboldt University of Berlin, with an exchange year at the University of Toronto. She earned a diploma in 2005. She completed a Ph.D. in mathematics at the Technical University of Munich in 2009, with the dissertation Embedding Large Graphs – The Bollobás-Komlós Conjecture and Beyond supervised by Anusch Taraz. After postdoctoral research at Technical University of Munich and at the University of São Paulo in Brazil, Böttcher became a lecturer at the London School of Economics in 2012. She was named as an associate professor in 2016 and promoted to professor in 2020. == Recognition == Böttcher was a recipient of the 2018 Fulkerson Prize for her research with Robert Morris, Yoshiharu Kohayakawa, Simon Griffiths, an Peter Allen on "the chromatic thresholds of graphs" relating the degree and the chromatic number of graphs with a forbidden induced subgraph. She was an invited speaker at the 2022 (virtual) International Congress of Mathematicians, speaking on hypergraph packing. == References == == External links == Home page Julia Böttcher publications indexed by Google Scholar
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Wikipedia:Julia Chuzhoy#0
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Julia Chuzhoy (Hebrew: ג'וליה צ'וז'וי) is an Israeli mathematician and computer scientist at the Toyota Technological Institute at Chicago, known for her research on approximation algorithms and graph theory. == Education and career == Chuzhoy earned bachelor's, master's, and doctoral degrees from the Technion – Israel Institute of Technology in 1998, 2000, and 2004 respectively. Her dissertation, on approximation algorithms, was supervised by Seffi Naor. She has been at the Toyota Technological Institute since 2007, and also holds a position in the Computer Science Department of the University of Chicago. == Contributions and recognition == Chuzhoy won the best paper award at the 2012 Symposium on Foundations of Computer Science for her paper with Shi Li on approximating the problem of connecting many given pairs of vertices in a graph by edge-disjoint paths.[CL12] She is also known for her work showing a polynomial relation between the size of a grid graph minor of a graph and its treewidth.[CC16] This connection between these two graph properties is a key component of the Robertson–Seymour theorem, is closely related to Halin's grid theorem for infinite graphs, and underlies the theory of bidimensionality for graph approximation algorithms. She was an Invited Speaker at the 2014 International Congress of Mathematicians, in Seoul. == Selected publications == == References == == External links == Home page Julia Chuzhoy publications indexed by Google Scholar
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Wikipedia:Julia Gordon#0
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Julia Gordon is a Canadian mathematician at the University of British Columbia whose research concerns algebraic geometry, including representation theory, p-adic groups, motivic integration, and the Langlands program. Gordon earned her PhD at the University of Michigan in 2003 under the supervision of Thomas C. Hales and Robert Griess. Her dissertation was Some Applications of Motivic Integration to the Representation Theory of P-adic Groups. After postdoctoral research at the University of Toronto and the Fields Institute, she joined the University of British Columbia faculty in 2006. As of 2021, she is an associate professor there. == Awards == In 2017, Gordon won the Ruth I. Michler Memorial Prize of the Association for Women in Mathematics. She is the 2019 winner of the Krieger–Nelson Prize of the Canadian Mathematical Society. == References == == External links == Home page Julia Gordon publications indexed by Google Scholar
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Wikipedia:Julia Pevtsova#0
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Julia Pevtsova is a Russian-American mathematician who works as a professor of mathematics at the University of Washington. Her research concerns representation theory and in particular modular representation theory. Pevstova competed for Russia in the 1992 International Mathematical Olympiad, earning a silver medal. She earned a bachelor's degree in 1997 from Saint Petersburg State University, and completed her doctorate in 2002 at Northwestern University, under the supervision of Eric Friedlander. After postdoctoral studies at the University of Oregon, she joined the University of Washington in 2005. In 2017, she became a fellow of the American Mathematical Society "for contributions to modular representation theory". In 2018 she won the distinguished teaching award of the Pacific Northwest Section of the Mathematical Association of America. The award cited her work teaching problem-solving to undergraduates in preparation for the William Lowell Putnam Mathematical Competition and her leadership of math circles and other activities for local secondary-school students. == References == == External links == Home page Julia Pevtsova at the Oberwolfach Photo Collection
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Wikipedia:Julia Wolf#0
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Julia Wolf is a British mathematician specialising in arithmetic combinatorics who was the 2016 winner of the Anne Bennett Prize of the London Mathematical Society. She is currently a professor in the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge. == Education and career == Wolf writes that her childhood ambition was to become a carpenter, and that she became attracted to science only after subscribing to Scientific American as a teenager. She read mathematics at Clare College, Cambridge, completing the Mathematical Tripos in 2003. She remained at Cambridge for graduate study, and completed her PhD there in 2008. Her dissertation, Arithmetic Structure in Sets of Integers, was supervised by Timothy Gowers. She was also mentored in her doctoral studies by Ben Green, whom she met when he was a postdoctoral researcher at Cambridge from 2001 to 2005. Since earning her doctorate she has been a postdoctoral fellow at the Mathematical Sciences Research Institute in Berkeley, California, Triennial assistant professor at Rutgers University in New Jersey, Hadamard associate professor at the École Polytechnique in Paris (earning a habilitation at the University of Paris-Sud in 2012), and Heilbronn reader in combinatorics and number theory at the University of Bristol. She returned to Cambridge as a university lecturer in 2018, and was a Fellow of Clare College from 2018 to 2022. == Recognition == In 2016 the London Mathematical Society gave Wolf their Anne Bennett Prize "in recognition of her outstanding contributions to additive number theory, combinatorics and harmonic analysis and to the mathematical community." The award citation particularly cited her work with Gowers on counting solutions to systems of linear equations over abelian groups, and her work on quadratic analogues of the Goldreich–Levin theorem. == References == == External links == Home page
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Wikipedia:Julia set#0
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In complex dynamics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly under repeated iteration of the function, and the Julia set consists of values such that an arbitrarily small perturbation can cause drastic changes in the sequence of iterated function values. Thus the behavior of the function on the Fatou set is "regular", while on the Julia set its behavior is "chaotic". The Julia set of a function f is commonly denoted J ( f ) , {\displaystyle \operatorname {J} (f),} and the Fatou set is denoted F ( f ) . {\displaystyle \operatorname {F} (f).} These sets are named after the French mathematicians Gaston Julia and Pierre Fatou whose work began the study of complex dynamics during the early 20th century. == Formal definition == Let f ( z ) {\displaystyle f(z)} be a non-constant meromorphic function from the Riemann sphere onto itself. Such functions f ( z ) {\displaystyle f(z)} are precisely the non-constant complex rational functions, that is, f ( z ) = p ( z ) / q ( z ) {\displaystyle f(z)=p(z)/q(z)} where p ( z ) {\displaystyle p(z)} and q ( z ) {\displaystyle q(z)} are complex polynomials. Assume that p and q have no common roots, and at least one has degree larger than 1. Then there is a finite number of open sets F 1 , . . . , F r {\displaystyle F_{1},...,F_{r}} that are left invariant by f ( z ) {\displaystyle f(z)} and are such that: The union of the sets F i {\displaystyle F_{i}} is dense in the plane and f ( z ) {\displaystyle f(z)} behaves in a regular and equal way on each of the sets F i {\displaystyle F_{i}} . The last statement means that the termini of the sequences of iterations generated by the points of F i {\displaystyle F_{i}} are either precisely the same set, which is then a finite cycle, or they are finite cycles of circular or annular shaped sets that are lying concentrically. In the first case the cycle is attracting, in the second case it is neutral. These sets F i {\displaystyle F_{i}} are the Fatou domains of f ( z ) {\displaystyle f(z)} , and their union is the Fatou set F ( f ) {\displaystyle \operatorname {F} (f)} of f ( z ) {\displaystyle f(z)} . Each of the Fatou domains contains at least one critical point of f ( z ) {\displaystyle f(z)} , that is, a (finite) point z satisfying f ′ ( z ) = 0 {\displaystyle f'(z)=0} , or f ( z ) = ∞ {\displaystyle f(z)=\infty } if the degree of the numerator p ( z ) {\displaystyle p(z)} is at least two larger than the degree of the denominator q ( z ) {\displaystyle q(z)} , or if f ( z ) = 1 / g ( z ) + c {\displaystyle f(z)=1/g(z)+c} for some c and a rational function g ( z ) {\displaystyle g(z)} satisfying this condition. The complement of F ( f ) {\displaystyle \operatorname {F} (f)} is the Julia set J ( f ) {\displaystyle \operatorname {J} (f)} of f ( z ) {\displaystyle f(z)} . If all the critical points are preperiodic, that is they are not periodic but eventually land on a periodic cycle, then J ( f ) {\displaystyle \operatorname {J} (f)} is all the sphere. Otherwise, J ( f ) {\displaystyle \operatorname {J} (f)} is a nowhere dense set (it is without interior points) and an uncountable set (of the same cardinality as the real numbers). Like F ( f ) {\displaystyle \operatorname {F} (f)} , J ( f ) {\displaystyle \operatorname {J} (f)} is left invariant by f ( z ) {\displaystyle f(z)} , and on this set the iteration is repelling, meaning that | f ( z ) − f ( w ) | > | z − w | {\displaystyle |f(z)-f(w)|>|z-w|} for all w in a neighbourhood of z (within J ( f ) {\displaystyle \operatorname {J} (f)} ). This means that f ( z ) {\displaystyle f(z)} behaves chaotically on the Julia set. Although there are points in the Julia set whose sequence of iterations is finite, there are only a countable number of such points (and they make up an infinitesimal part of the Julia set). The sequences generated by points outside this set behave chaotically, a phenomenon called deterministic chaos. There has been extensive research on the Fatou set and Julia set of iterated rational functions, known as rational maps. For example, it is known that the Fatou set of a rational map has either 0, 1, 2 or infinitely many components. Each component of the Fatou set of a rational map can be classified into one of four different classes. == Equivalent descriptions of the Julia set == J ( f ) {\displaystyle \operatorname {J} (f)} is the smallest closed set containing at least three points which is completely invariant under f. J ( f ) {\displaystyle \operatorname {J} (f)} is the closure of the set of repelling periodic points. For all but at most two points z ∈ X , {\displaystyle \;z\in X\;,} the Julia set is the set of limit points of the full backwards orbit ⋃ n f − n ( z ) . {\displaystyle \bigcup _{n}f^{-n}(z).} (This suggests a simple algorithm for plotting Julia sets, see below.) If f is an entire function, then J ( f ) {\displaystyle \operatorname {J} (f)} is the boundary of the set of points which converge to infinity under iteration. If f is a polynomial, then J ( f ) {\displaystyle \operatorname {J} (f)} is the boundary of the filled Julia set; that is, those points whose orbits under iterations of f remain bounded. == Properties of the Julia set and Fatou set == The Julia set and the Fatou set of f are both completely invariant under iterations of the holomorphic function f: f − 1 ( J ( f ) ) = f ( J ( f ) ) = J ( f ) , {\displaystyle f^{-1}(\operatorname {J} (f))=f(\operatorname {J} (f))=\operatorname {J} (f),} f − 1 ( F ( f ) ) = f ( F ( f ) ) = F ( f ) . {\displaystyle f^{-1}(\operatorname {F} (f))=f(\operatorname {F} (f))=\operatorname {F} (f).} == Examples == For f ( z ) = z 2 {\displaystyle f(z)=z^{2}} the Julia set is the unit circle and on this the iteration is given by doubling of angles (an operation that is chaotic on the points whose argument is not a rational fraction of 2 π {\displaystyle 2\pi } ). There are two Fatou domains: the interior and the exterior of the circle, with iteration towards 0 and ∞, respectively. For g ( z ) = z 2 − 2 {\displaystyle g(z)=z^{2}-2} the Julia set is the line segment between −2 and 2. There is one Fatou domain: the points not on the line segment iterate towards ∞. (Apart from a shift and scaling of the domain, this iteration is equivalent to x → 4 ( x − 1 2 ) 2 {\displaystyle x\to 4(x-{\tfrac {1}{2}})^{2}} on the unit interval, which is commonly used as an example of chaotic system.) The functions f and g are of the form z 2 + c {\displaystyle z^{2}+c} , where c is a complex number. For such an iteration the Julia set is not in general a simple curve, but is a fractal, and for some values of c it can take surprising shapes. See the pictures below. For some functions f(z) we can say beforehand that the Julia set is a fractal and not a simple curve. This is because of the following result on the iterations of a rational function: This means that each point of the Julia set is a point of accumulation for each of the Fatou domains. Therefore, if there are more than two Fatou domains, each point of the Julia set must have points of more than two different open sets infinitely close, and this means that the Julia set cannot be a simple curve. This phenomenon happens, for instance, when f(z) is the Newton iteration for solving the equation P ( z ) := z n − 1 = 0 : n > 2 {\displaystyle \;P(z):=z^{n}-1=0~:~n>2\;} : f ( z ) = z − P ( z ) P ′ ( z ) = 1 + ( n − 1 ) z n n z n − 1 . {\displaystyle f(z)=z-{\frac {P(z)}{P'(z)}}={\frac {\;1+(n-1)z^{n}\;}{nz^{n-1}}}~.} The image on the right shows the case n = 3. == Quadratic polynomials == A very popular complex dynamical system is given by the family of complex quadratic polynomials, a special case of rational maps. Such quadratic polynomials can be expressed as f c ( z ) = z 2 + c , {\displaystyle f_{c}(z)=z^{2}+c~,} where c is a complex parameter. Fix some R > 0 {\displaystyle R>0} large enough that R 2 − R ≥ | c | . {\displaystyle R^{2}-R\geq |c|.} (For example, if c is in the Mandelbrot set, then | c | ≤ 2 , {\displaystyle |c|\leq 2,} so we may simply let R = 2 . {\displaystyle R=2~.} ) Then the filled Julia set for this system is the subset of the complex plane given by K ( f c ) = { z ∈ C : ∀ n ∈ N , | f c n ( z ) | ≤ R } , {\displaystyle K(f_{c})=\left\{z\in \mathbb {C} :\forall n\in \mathbb {N} ,|f_{c}^{n}(z)|\leq R\right\}~,} where f c n ( z ) {\displaystyle f_{c}^{n}(z)} is the nth iterate of f c ( z ) . {\displaystyle f_{c}(z).} The Julia set J ( f c ) {\displaystyle J(f_{c})} of this function is the boundary of K ( f c ) {\displaystyle K(f_{c})} . The parameter plane of quadratic polynomials – that is, the plane of possible c values – gives rise to the famous Mandelbrot set. Indeed, the Mandelbrot set is defined as the set of all c such that J ( f c ) {\displaystyle J(f_{c})} is connected. For parameters outside the Mandelbrot set, the Julia set is a Cantor space: in this case it is sometimes referred to as Fatou dust. In many cases, the Julia set of c looks like the Mandelbrot set in sufficiently small neighborhoods of c. This is true, in particular, for so-called Misiurewicz parameters, i.e. parameters c for which the critical point is pre-periodic. For instance: At c = i, the shorter, front toe of the forefoot, the Julia set looks like a branched lightning bolt. At c = −2, the tip of the long spiky tail, the Julia set is a straight line segment. In other words, the Julia sets J ( f c ) {\displaystyle J(f_{c})} are locally similar around Misiurewicz points. == Generalizations == The definition of Julia and Fatou sets easily carries over to the case of certain maps whose image contains their domain; most notably transcendental meromorphic functions and Adam Epstein's finite-type maps. Julia sets are also commonly defined in the study of dynamics in several complex variables. == Pseudocode == The below pseudocode implementations hard code the functions for each fractal. Consider implementing complex number operations to allow for more dynamic and reusable code. === Pseudocode for normal Julia sets === f ( z ) = z 2 + c {\displaystyle f(z)=z^{2}+c} === Pseudocode for multi-Julia sets === f ( z ) = z n + c {\displaystyle f(z)=z^{n}+c} Another recommended option is to reduce color banding between iterations by using a renormalization formula for the iteration. Such formula is given to be, m u = k + 1 − log log | z k | log n {\displaystyle mu=k+1-{\frac {\log {\log {|z_{k}|}}}{\log {n}}}} ∀ f c , n ( z ) = z n + lower power terms + c {\displaystyle \forall f_{c,n}(z)=z^{n}+{\text{lower power terms}}+c} where k {\displaystyle k} is the escaping iteration, bounded by some K {\displaystyle K} such that 0 ≤ k < K {\displaystyle 0\leq k<K} and K ∈ N {\displaystyle K\in \mathbb {N} } , and | z k | {\displaystyle |z_{k}|} is the magnitude of the last iterate before escaping. This can be implemented, very simply, like so: The difference is shown below with a Julia set defined as f c , 2 ( z ) {\displaystyle f_{c,2}(z)} where c = − 0.835 − 0.321 i {\displaystyle c=-0.835-0.321i} . == The potential function and the real iteration number == The Julia set for f ( z ) = z 2 {\displaystyle f(z)=z^{2}} is the unit circle, and on the outer Fatou domain, the potential function φ(z) is defined by φ(z) = log|z|. The equipotential lines for this function are concentric circles. As | f ( z ) | = | z | 2 {\displaystyle |f(z)|=|z|^{2}} we have φ ( z ) = lim k → ∞ log | z k | 2 k , {\displaystyle \varphi (z)=\lim _{k\to \infty }{\frac {\log |z_{k}|}{2^{k}}},} where z k {\displaystyle z_{k}} is the sequence of iteration generated by z. For the more general iteration f ( z ) = z 2 + c {\displaystyle f(z)=z^{2}+c} , it has been proved that if the Julia set is connected (that is, if c belongs to the (usual) Mandelbrot set), then there exist a biholomorphic map ψ between the outer Fatou domain and the outer of the unit circle such that | ψ ( f ( z ) ) | = | ψ ( z ) | 2 {\displaystyle |\psi (f(z))|=|\psi (z)|^{2}} . This means that the potential function on the outer Fatou domain defined by this correspondence is given by: φ ( z ) = lim k → ∞ log | z k | 2 k . {\displaystyle \varphi (z)=\lim _{k\to \infty }{\frac {\log |z_{k}|}{2^{k}}}.} This formula has meaning also if the Julia set is not connected, so that we for all c can define the potential function on the Fatou domain containing ∞ by this formula. For a general rational function f(z) such that ∞ is a critical point and a fixed point, that is, such that the degree m of the numerator is at least two larger than the degree n of the denominator, we define the potential function on the Fatou domain containing ∞ by: φ ( z ) = lim k → ∞ log | z k | d k , {\displaystyle \varphi (z)=\lim _{k\to \infty }{\frac {\log |z_{k}|}{d^{k}}},} where d = m − n is the degree of the rational function. If N is a very large number (e.g. 10100), and if k is the first iteration number such that | z k | > N {\displaystyle |z_{k}|>N} , we have that log | z k | d k = log ( N ) d ν ( z ) , {\displaystyle {\frac {\log |z_{k}|}{d^{k}}}={\frac {\log(N)}{d^{\nu (z)}}},} for some real number ν ( z ) {\displaystyle \nu (z)} , which should be regarded as the real iteration number, and we have that: ν ( z ) = k − log ( log | z k | / log ( N ) ) log ( d ) , {\displaystyle \nu (z)=k-{\frac {\log(\log |z_{k}|/\log(N))}{\log(d)}},} where the last number is in the interval [0, 1). For iteration towards a finite attracting cycle of order r, we have that if z ∗ {\displaystyle z^{*}} is a point of the cycle, then f ( f ( . . . f ( z ∗ ) ) ) = z ∗ {\displaystyle f(f(...f(z^{*})))=z^{*}} (the r-fold composition), and the number α = 1 | ( d ( f ( f ( ⋯ f ( z ) ) ) ) / d z ) z = z ∗ | ( > 1 ) {\displaystyle \alpha ={\frac {1}{\left|(d(f(f(\cdots f(z))))/dz)_{z=z^{*}}\right|}}\qquad (>1)} is the attraction of the cycle. If w is a point very near z ∗ {\displaystyle z^{*}} and w′ is w iterated r times, we have that α = lim k → ∞ | w − z ∗ | | w ′ − z ∗ | . {\displaystyle \alpha =\lim _{k\to \infty }{\frac {|w-z^{*}|}{|w'-z^{*}|}}.} Therefore, the number | z k r − z ∗ | α k {\displaystyle |z_{kr}-z^{*}|\alpha ^{k}} is almost independent of k. We define the potential function on the Fatou domain by: φ ( z ) = lim k → ∞ 1 ( | z k r − z ∗ | α k ) . {\displaystyle \varphi (z)=\lim _{k\to \infty }{\frac {1}{(|z_{kr}-z^{*}|\alpha ^{k})}}.} If ε is a very small number and k is the first iteration number such that | z k − z ∗ | < ϵ {\displaystyle |z_{k}-z^{*}|<\epsilon } , we have that φ ( z ) = 1 ( ε α ν ( z ) ) {\displaystyle \varphi (z)={\frac {1}{(\varepsilon \alpha ^{\nu (z)})}}} for some real number ν ( z ) {\displaystyle \nu (z)} , which should be regarded as the real iteration number, and we have that: ν ( z ) = k − log ( ε / | z k − z ∗ | ) log ( α ) . {\displaystyle \nu (z)=k-{\frac {\log(\varepsilon /|z_{k}-z^{*}|)}{\log(\alpha )}}.} If the attraction is ∞, meaning that the cycle is super-attracting, meaning again that one of the points of the cycle is a critical point, we must replace α by α = lim k → ∞ log | w ′ − z ∗ | log | w − z ∗ | , {\displaystyle \alpha =\lim _{k\to \infty }{\frac {\log |w'-z^{*}|}{\log |w-z^{*}|}},} where w′ is w iterated r times and the formula for φ(z) by: φ ( z ) = lim k → ∞ log ( 1 / | z k r − z ∗ | ) α k . {\displaystyle \varphi (z)=\lim _{k\to \infty }{\frac {\log(1/|z_{kr}-z^{*}|)}{\alpha ^{k}}}.} And now the real iteration number is given by: ν ( z ) = k − log ( log | z k − z ∗ | / log ( ε ) ) log ( α ) . {\displaystyle \nu (z)=k-{\frac {\log(\log |z_{k}-z^{*}|/\log(\varepsilon ))}{\log(\alpha )}}.} For the colouring we must have a cyclic scale of colours (constructed mathematically, for instance) and containing H colours numbered from 0 to H−1 (H = 500, for instance). We multiply the real number ν ( z ) {\displaystyle \nu (z)} by a fixed real number determining the density of the colours in the picture, and take the integral part of this number modulo H. The definition of the potential function and our way of colouring presuppose that the cycle is attracting, that is, not neutral. If the cycle is neutral, we cannot colour the Fatou domain in a natural way. As the terminus of the iteration is a revolving movement, we can, for instance, colour by the minimum distance from the cycle left fixed by the iteration. == Field lines == In each Fatou domain (that is not neutral) there are two systems of lines orthogonal to each other: the equipotential lines (for the potential function or the real iteration number) and the field lines. If we colour the Fatou domain according to the iteration number (and not the real iteration number ν ( z ) {\displaystyle \nu (z)} , as defined in the previous section), the bands of iteration show the course of the equipotential lines. If the iteration is towards ∞ (as is the case with the outer Fatou domain for the usual iteration z 2 + c {\displaystyle z^{2}+c} ), we can easily show the course of the field lines, namely by altering the colour according as the last point in the sequence of iteration is above or below the x-axis (first picture), but in this case (more precisely: when the Fatou domain is super-attracting) we cannot draw the field lines coherently - at least not by the method we describe here. In this case a field line is also called an external ray. Let z be a point in the attracting Fatou domain. If we iterate z a large number of times, the terminus of the sequence of iteration is a finite cycle C, and the Fatou domain is (by definition) the set of points whose sequence of iteration converges towards C. The field lines issue from the points of C and from the (infinite number of) points that iterate into a point of C. And they end on the Julia set in points that are non-chaotic (that is, generating a finite cycle). Let r be the order of the cycle C (its number of points) and let z ∗ {\displaystyle z^{*}} be a point in C. We have f ( f ( … f ( z ∗ ) ) ) = z ∗ {\displaystyle f(f(\dots f(z^{*})))=z^{*}} (the r-fold composition), and we define the complex number α by α = ( d ( f ( f ( … f ( z ) ) ) ) / d z ) z = z ∗ . {\displaystyle \alpha =(d(f(f(\dots f(z))))/dz)_{z=z^{*}}.} If the points of C are z i , i = 1 , … , r ( z 1 = z ∗ ) {\displaystyle z_{i},i=1,\dots ,r(z_{1}=z^{*})} , α is the product of the r numbers f ′ ( z i ) {\displaystyle f'(z_{i})} . The real number 1/|α| is the attraction of the cycle, and our assumption that the cycle is neither neutral nor super-attracting, means that 1 < 1/|α| < ∞. The point z ∗ {\displaystyle z^{*}} is a fixed point for f ( f ( … f ( z ) ) ) {\displaystyle f(f(\dots f(z)))} , and near this point the map f ( f ( … f ( z ) ) ) {\displaystyle f(f(\dots f(z)))} has (in connection with field lines) character of a rotation with the argument β of α (that is, α = | α | e β i {\displaystyle \alpha =|\alpha |e^{\beta i}} ). In order to colour the Fatou domain, we have chosen a small number ε and set the sequences of iteration z k ( k = 0 , 1 , 2 , … , z 0 = z ) {\displaystyle z_{k}(k=0,1,2,\dots ,z_{0}=z)} to stop when | z k − z ∗ | < ϵ {\displaystyle |z_{k}-z^{*}|<\epsilon } , and we colour the point z according to the number k (or the real iteration number, if we prefer a smooth colouring). If we choose a direction from z ∗ {\displaystyle z^{*}} given by an angle θ, the field line issuing from z ∗ {\displaystyle z^{*}} in this direction consists of the points z such that the argument ψ of the number z k − z ∗ {\displaystyle z_{k}-z^{*}} satisfies the condition that ψ − k β = θ mod π . {\displaystyle \psi -k\beta =\theta \mod \pi .\,} For if we pass an iteration band in the direction of the field lines (and away from the cycle), the iteration number k is increased by 1 and the number ψ is increased by β, therefore the number ψ − k β mod π {\displaystyle \psi -k\beta \mod \pi } is constant along the field line. A colouring of the field lines of the Fatou domain means that we colour the spaces between pairs of field lines: we choose a number of regularly situated directions issuing from z ∗ {\displaystyle z^{*}} , and in each of these directions we choose two directions around this direction. As it can happen that the two field lines of a pair do not end in the same point of the Julia set, our coloured field lines can ramify (endlessly) in their way towards the Julia set. We can colour on the basis of the distance to the center line of the field line, and we can mix this colouring with the usual colouring. Such pictures can be very decorative (second picture). A coloured field line (the domain between two field lines) is divided up by the iteration bands, and such a part can be put into a one-to-one correspondence with the unit square: the one coordinate is (calculated from) the distance from one of the bounding field lines, the other is (calculated from) the distance from the inner of the bounding iteration bands (this number is the non-integral part of the real iteration number). Therefore, we can put pictures into the field lines (third picture). == Plotting the Julia set == Methods : Distance Estimation Method for Julia set (DEM/J) Inverse Iteration Method (IIM) === Using backwards (inverse) iteration (IIM) === As mentioned above, the Julia set can be found as the set of limit points of the set of pre-images of (essentially) any given point. So we can try to plot the Julia set of a given function as follows. Start with any point z we know to be in the Julia set, such as a repelling periodic point, and compute all pre-images of z under some high iterate f n {\displaystyle f^{n}} of f. Unfortunately, as the number of iterated pre-images grows exponentially, this is not feasible computationally. However, we can adjust this method, in a similar way as the "random game" method for iterated function systems. That is, in each step, we choose at random one of the inverse images of f. For example, for the quadratic polynomial fc, the backwards iteration is described by z n − 1 = z n − c . {\displaystyle z_{n-1}={\sqrt {z_{n}-c}}.} At each step, one of the two square roots is selected at random. Note that certain parts of the Julia set are quite difficult to access with the reverse Julia algorithm. For this reason, one must modify IIM/J ( it is called MIIM/J) or use other methods to produce better images. === Using DEM/J === Images of Julia sets for f c ( z ) = z 2 + c {\displaystyle f_{c}(z)=z^{2}+c} As a Julia set is infinitely thin we cannot draw it effectively by backwards iteration from the pixels. It will appear fragmented because of the impracticality of examining infinitely many startpoints. Since the iteration count changes vigorously near the Julia set, a partial solution is to imply the outline of the set from the nearest color contours, but the set will tend to look muddy. A better way to draw the Julia set in black and white is to estimate the distance of pixels (DEM) from the set and to color every pixel whose center is close to the set. The formula for the distance estimation is derived from the formula for the potential function φ(z). When the equipotential lines for φ(z) lie close, the number | φ ′ ( z ) | {\displaystyle |\varphi '(z)|} is large, and conversely, therefore the equipotential lines for the function δ ( z ) = φ ( z ) / | φ ′ ( z ) | {\displaystyle \delta (z)=\varphi (z)/|\varphi '(z)|} should lie approximately regularly. It has been proven that the value found by this formula (up to a constant factor) converges towards the true distance for z converging towards the Julia set. We assume that f(z) is rational, that is, f ( z ) = p ( z ) / q ( z ) {\displaystyle f(z)=p(z)/q(z)} where p(z) and q(z) are complex polynomials of degrees m and n, respectively, and we have to find the derivative of the above expressions for φ(z). And as it is only z k {\displaystyle z_{k}} that varies, we must calculate the derivative z k ′ {\displaystyle z'_{k}} of z k {\displaystyle z_{k}} with respect to z. But as z k = f ( f ( ⋯ f ( z ) ) ) {\displaystyle z_{k}=f(f(\cdots f(z)))} (the k-fold composition), z k ′ {\displaystyle z'_{k}} is the product of the numbers f ′ ( z k ) {\displaystyle f'(z_{k})} , and this sequence can be calculated recursively by z k + 1 ′ = f ′ ( z k ) z k ′ {\displaystyle z'_{k+1}=f'(z_{k})z'_{k}} , starting with z 0 ′ = 1 {\displaystyle z'_{0}=1} (before the calculation of the next iteration z k + 1 = f ( z k ) {\displaystyle z_{k+1}=f(z_{k})} ). For iteration towards ∞ (more precisely when m ≥ n + 2, so that ∞ is a super-attracting fixed point), we have | φ ′ ( z ) | = lim k → ∞ | z k ′ | | z k | d k , {\displaystyle |\varphi '(z)|=\lim _{k\to \infty }{\frac {|z'_{k}|}{|z_{k}|d^{k}}},} (d = m − n) and consequently: δ ( z ) = φ ( z ) / | φ ′ ( z ) | = lim k → ∞ log | z k | | z k | / | z k ′ | . {\displaystyle \delta (z)=\varphi (z)/|\varphi '(z)|=\lim _{k\to \infty }\log |z_{k}||z_{k}|/|z'_{k}|.\,} For iteration towards a finite attracting cycle (that is not super-attracting) containing the point z ∗ {\displaystyle z^{*}} and having order r, we have | φ ′ ( z ) | = lim k → ∞ | z k r ′ | / ( | z k r − z ∗ | 2 α k ) , {\displaystyle |\varphi '(z)|=\lim _{k\to \infty }|z'_{kr}|/(|z_{kr}-z^{*}|^{2}\alpha ^{k}),\,} and consequently: δ ( z ) = φ ( z ) / | φ ′ ( z ) | = lim k → ∞ | z k r − z ∗ | / | z k r ′ | . {\displaystyle \delta (z)=\varphi (z)/|\varphi '(z)|=\lim _{k\to \infty }|z_{kr}-z^{*}|/|z'_{kr}|.\,} For a super-attracting cycle, the formula is: δ ( z ) = lim k → ∞ log | z k r − z ∗ | 2 / | z k r ′ | . {\displaystyle \delta (z)=\lim _{k\to \infty }\log |z_{kr}-z^{*}|^{2}/|z'_{kr}|.\,} We calculate this number when the iteration stops. Note that the distance estimation is independent of the attraction of the cycle. This means that it has meaning for transcendental functions of "degree infinity" (e.g. sin(z) and tan(z)). Besides drawing of the boundary, the distance function can be introduced as a 3rd dimension to create a solid fractal landscape. == See also == Douady rabbit Limit set Stable and unstable sets No wandering domain theorem Chaos theory == Notes == == References == == Bibliography == Carleson, Lennart; Gamelin, Theodore W. (1993). Complex Dynamics. Springer. Douady, Adrien; Hubbard, John H. (1984). "Etude dynamique des polynômes complexes". Prépublications mathémathiques d'Orsay. 2; "[op.cit.]". Prépublications mathémathiques d'Orsay. 4. 1985. Milnor, J.W. (2006) [1990]. Dynamics in One Complex Variable. Annals of Mathematics Studies. Vol. 160 (Third ed.). Princeton University Press;First appeared in as a "Stony Brook IMS Preprint". Archived from the original on 2006-04-24. available as Milnor, John W. (1990). "Dynamics in one complex variable: Introductory lectures". arXiv:math.DS/9201272. Bogomolny, Alexander. "Mandelbrot Set and Indexing of Julia Sets". cut-the-knot. Algebra curriculum. Demidov, Evgeny (2003). "The Mandelbrot and Julia sets' anatomy". Beardon, Alan F. (1991). Iteration of Rational Functions. Springer. ISBN 0-387-95151-2. == External links == "Julia set", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Weisstein, Eric W. "Julia Set". MathWorld. Bourke, Paul. "Julia set fractal (2D)" (personal site). Sawyer, Jamie (6 April 2007). "Julia sets" (blog). McGoodwin, Michael. "Julia jewels: An exploration of Julia sets" (personal site). Pringle, Lucy. "Crop circle Julia Set" (personal site). Greig, Josh. "Interactive Julia Set Applet". Archived from the original on 2012-03-26. Joyce, David E. "Julia and Mandelbrot set explorer" (academic personal site). Clark University. "A simple program to generate Julia sets". liazardie.com. Archived from the original on 2011-03-17. – Windows, 370 kB "A collection of applets". SourceForge. – one of the applets can render Julia sets, via Iterated Function Systems. "Julia meets HTML5". Google Labs. Archived from the original on 2011-02-18. HTML5 Fractal generator for your browser "Julia". r-project.org. GNU R Package. 25 November 2014. generate Julia or Mandelbrot set at a given region and resolution "Julia sets". Archived from the original on 2017-06-16. Retrieved 2011-08-24. – A visual explanation of Julia Sets. "FractalTS". github.io. – Mandelbrot, Burning ship and corresponding Julia set generator. "Julia set images, online rendering". finengin.net. Archived from the original on 2019-06-22. Retrieved 2017-01-24. "Understanding Julia and Mandelbrot Sets". - A visual explanation.
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Wikipedia:Julian Sochocki#0
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Julian Karol Sochocki (Polish: Julian Karol Sochocki; Russian: Юлиан Васильевич Сохоцкий; February 2, 1842, in Warsaw, Congress Poland, Russian Empire – December 14, 1927, in Leningrad, Soviet Union) was a Polish-Russian mathematician. His name is sometimes transliterated from Russian in several different ways (e.g. Sokhotski or Sochotski). == Life and work == Sochocki was born in Warsaw under the Russian domination to a Polish family, where he attended state gymnasium. In 1860 he registered at the physico-mathematical department of St Petersburg University. His study there was interrupted for the period 1860–1865 because of his involvement with Polish patriotic movement: he had to return to Warsaw to escape prosecution. In 1866 he graduated from the Department of Physics and Mathematics at the University of Saint Petersburg. In 1868 he received his master's degree and in 1873 his doctorate. His master's dissertation, practically the first text in Russian mathematical literature on Cauchy method of residues, was published in 1868. The dissertation itself contains many original grasps, which have been also ascribed to other mathematicians. His doctoral thesis contains the famous Sokhotski–Plemelj theorem. From 1868 Sochotcki lectured at the St Petersburg university, first as the "privat-docent", from 1882 as an ordinary professor, and from 1893 as a merited professor. In 1894 he was elected corresponding member of the Polish Academy of Arts and Sciences. Sochocki died on December 14, 1927, in a nursing home in Leningrad. Sochocki is mainly remembered for the Casorati–Sokhotski–Weierstrass theorem and for the Sokhotski–Plemelj theorem. == Selected publications == Теорiя интегральныхъ вычетовъ с нѣкоторыми приложенiями (A Theory of Integral Residues with Some Applications) (1868) Объ определенныхъ интегралахъ и функцiяхъ употребляемыхъ при разложенiяхъ въ ряды (On Definite Integrals and Functions Used in Series Expansions) (1873) О суммахъ Гаусса и о законе взаимности символа Лежандра (On Gauss Sums and the Reciprocity Law of the Legendre Symbol) (1877) Высшая алгебра (Higher Algebra) (St. Petersburg, 1882) Теорiя чиселъ (Number Theory) (St. Petersburg, 1888) Начало общего наибольшего делителя въ применении к теорiи делимости алгебраическихъ чиселъ (The Principle of the Greatest Common Divisor Applied to Divisibility Theory of Algebraic Numbers) (1893), JFM 25.0300.01, JFM 25.0297.04 == Notes == == External links == Yulian Vasilievich Sokhotski (in Russian) Youschkevitch, A. P. (1970–1980). "Yulian-Karl Vasilievich Sokhotski". Dictionary of Scientific Biography. New York: Charles Scribner's Sons. ISBN 978-0-684-10114-9. O'Connor, John J.; Robertson, Edmund F., "Julian Sochocki", MacTutor History of Mathematics Archive, University of St Andrews Julian Karol Sochocki Zentralblatt profile
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Wikipedia:Julie Rowlett#0
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Julie Marie Rowlett (born 1978) is an American mathematician. She is a professor of mathematics at the Chalmers University of Technology. Her primary research interest is in geometric analysis with a particular focus on geometric analysis on singular spaces, dynamics, mathematical physics, and spectral theory. == Biography == Rowlett earned her Bachelor of Science from the University of Washington in 2001, and her Ph.D. from Stanford University in 2006. Her dissertation, Spectral Geometry and Asymptotically Conic Convergence, was supervised by Rafe Mazzeo. Rowlett was a postdoctoral researcher at the Centre de Recherches Mathématiques and McGill University in Montreal. After a short period as an instructor in the Education Program for Gifted Youth at Stanford University, and a position as visiting assistant professor at the University of California, Santa Barbara, Rowlett left the US in 2009 for Germany. There, she held a series of temporary positions at the Hausdorff Center for Mathematics from 2009 to 2010, the Max Planck Institute for Mathematics from 2011 to 2012, the University of Göttingen from 2012 to 2013, and Leibniz University Hannover from 2013 to 2014. She received her habilitation at the University of Göttingen in 2013 and took a permanent position at the University of Ingolstadt in 2014. In 2015 Rowlett moved to Sweden for her current position at Chalmers University of Technology. == Awards == In 2016, she received the Paul R. Halmos – Lester R. Ford Award together with Zhiqin Lu of the University of California, Irvine, for their joint work on hearing the shape of a drum. Although there exist pairs of drum shapes that sound the same, Rowlett and Lu showed that certain shapes of drums, such as parallelograms and acute trapezoids, can be distinguished from others by their sounds. == References == == External links == Julie Rowlett publications indexed by Google Scholar
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Wikipedia:Juliusz Schauder#0
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Juliusz Paweł Schauder ([ˈjulʲjuʂ ˈpavɛw ˈʂau̯dɛr]; 21 September 1899 – September 1943) was a Polish mathematician known for his work in functional analysis, partial differential equations and mathematical physics. == Life and career == Born on 21 September 1899 in Lwów to a lawyer father of Jewish descent, he was drafted into the Austro-Hungarian Army right after his graduation from school and saw action on the Italian front. He was captured and imprisoned in Italy. He entered the university in Lwów in 1919 and received his doctorate in 1923. He got no appointment at the university and continued his research while working as teacher at a secondary school. Due to his outstanding results, he obtained a scholarship in 1932 that allowed him to spend several years in Leipzig and, especially, Paris. In Paris he started a very successful collaboration with Jean Leray. Around 1935 Schauder obtained the position of a senior assistant in the University of Lwów. Schauder, along with Stanisław Mazur, was an Invited Speaker of the International Congress of Mathematicians in 1936 in Oslo. Schauder was Jewish, and after the invasion of German troops in Lwów 1941 it was impossible for him to continue his work. Even before the Lwów ghetto was established he wrote to Ludwig Bieberbach pleading for his support. Instead, Bieberbach passed his letter to the Gestapo and Schauder was arrested. In his letters to Swiss mathematicians, he wrote that he had important new results, but no paper to write them down. He was executed by the Gestapo, probably in October 1943. Most of his mathematical work is in the field of functional analysis, being part of a large Polish group of mathematicians, i.e. the Lwów School of Mathematics. They were pioneers in this area with wide applications in all parts of modern analysis. Schauder is best known for the Schauder fixed-point theorem, which is a major tool to prove the existence of solutions in various problems, the Schauder bases (a generalization of an orthonormal basis from Hilbert spaces to Banach spaces), and the Leray−Schauder principle, a way to establish solutions of partial differential equations from a priori estimates. == In memoriam == The Schauder Medal is awarded by the J.P. Schauder Center for Nonlinear Studies at the Nicolaus Copernicus University in Toruń, Poland, to individuals for their significant achievements related to topological methods in nonlinear analysis. == See also == Banach–Schauder theorem Schauder basis Schauder estimates Schauder fixed point theorem List of Polish mathematicians == References == == External links == O'Connor, John J.; Robertson, Edmund F., "Juliusz Schauder", MacTutor History of Mathematics Archive, University of St Andrews Ingarden, Roman (1993), "Juliusz Schauder - personal reminiscences", Topological Methods in Nonlinear Analysis, 2 (1): 1–14, doi:10.12775/TMNA.1993.026, Zbl 0795.01027 Schaerf, H. M. (1993), "My memories of Juliusz Schauder", Topological Methods in Nonlinear Analysis, 2 (1): 15–19, doi:10.12775/TMNA.1993.027, Zbl 0795.01028 Juliusz P. Schauder Center for Nonlinear Studies
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Wikipedia:Junior Mathematical Challenge#0
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The United Kingdom Mathematics Trust (UKMT) is a charity founded in 1996 to help with the education of children in mathematics within the UK. == History == The national mathematics competitions had existed prior to the formation of the trust, but the foundation of the UKMT in the summer of 1996 enabled them to be run collectively. The Senior Mathematical Challenge was formerly called the National Mathematics Contest. Founded in 1961, it was run by the Mathematical Association from 1975 until its adoption by the UKMT in 1996. The Junior and Intermediate Mathematical Challenges were the initiative of Tony Gardiner in 1987, and were run by him under the name of the United Kingdom Mathematics Foundation until 1996. In 1995, Gardiner advertised for the formation of a committee and for a host institution that would lead to the establishment of the UKMT, enabling the challenges to be run effectively together under one organization. == Mathematical Challenges == The UKMT runs a series of mathematics challenges to encourage children's interest in mathematics and to develop their skills. The three main challenges are: Junior Mathematical Challenge (UK year 8/S2 and below) Intermediate Mathematical Challenge (UK year 11/S4 and below) Senior Mathematical Challenge (UK year 13/S6 and below) == Certificates == In the Junior and Intermediate Challenges the top scoring 50% of the entrants receive bronze, silver or gold certificates based on their mark in the paper. In the Senior Mathematical Challenge these certificates are awarded to top scoring 66% of the entries. In each case bronze, silver and gold certificates are awarded in the ratio 3 : 2 : 1. So in the Junior and Intermediate Challenges The Gold award is achieved by the top 8-9% of the entrants. The Silver award is achieved by 16-17% of the entrants. The Bronze award is achieved by 25% of the entrants. In the past, only the top 40% of participants received a certificate in the Junior and Intermediate Challenges, and only the top 60% of participants received a certificate in the Senior Challenge. The ratio of bronze, silver, and gold have not changed, still being 3 : 2 : 1. == Junior Mathematical Challenge == The Junior Mathematical Challenge (JMC) is an introductory challenge for pupils in Years 8 or below (aged 13) or below, taking place in spring each year. This takes the form of twenty-five multiple choice questions to be sat in exam conditions, to be completed within one hour. The first fifteen questions are designed to be easier, and a pupil will gain 5 marks for getting a question in this section correct. Questions 16-20 are more difficult and are worth 6 marks. The last five questions are intended to be the most challenging and so are also 6 marks. Questions to which no answer is entered will gain (and lose) 0 marks. However, in recent years there has been no negative marking so wrong questions will be given 0 marks. Previously, the top 40% of students (50% since the 2022 JMC) get a certificate of varying levels (Gold, Silver or Bronze) based on their score. === Junior Kangaroo === Over 10,000 participants from the JMC are invited to participate in the Junior Kangaroo. Most of the Junior Kangaroo participants are those who performed well in the JMC, however the Junior Kangaroo is open to discretionary entries for a fee. Similar to the JMC, the Junior Kangaroo is a 60 minute challenge consisting of 25 multiple-choice problems. Correct answers for Questions 1-15 earn 5 marks, and for Questions 16-25 earn 6 marks. Blank or incorrect answers are marked 0; there is no penalty for wrong answers. The top 25% of participants in the Junior Kangaroo receive a Certificate of Merit. === Junior Mathematical Olympiad === The highest 1200 scorers are also invited to take part in the Junior Mathematical Olympiad (JMO). Like the JMC, the JMO is sat in schools. Students are given 120 minutes to complete the JMO. This is also divided into two sections. Part A is composed of 10 questions in which the candidate gives just the answer (not multiple choice), worth 10 marks (1 mark each). Part B consists of 6 questions and encourages students to write out full solutions. Each question in section B is worth 10 marks and students are encouraged to write complete answers to 2-4 questions rather than hurry through incomplete answers to all 6. If the solution is judged to be incomplete, it is marked on a 0+ basis, maximum 3 marks. If it has an evident logical strategy, it is marked on a 10- basis. The total mark for the whole paper is 70. Everyone who participates in this challenge will gain a certificate (Participation 75%, Distinction 25%); the top 200 or so gaining medals (Gold, Silver, Bronze); with the top fifty winning a book prize. From 2025, this has changed as Part A has been omitted. Section B has stayed the same, though it is no longer called Section B (it is now the only section). This changes the total number of questions to 10 and the marks to 60. However the time given for the JMO, has stayed at 120 minutes. == Intermediate Mathematical Challenge == The Intermediate Mathematical Challenge (IMC) is aimed at school years equivalent to English Years 9-11, taking place in winter each year. Following the same structure as the JMC, this paper presents the student with twenty-five multiple choice questions to be done under exam conditions in one hour. The first fifteen questions are designed to be easier, and a pupil will gain 5 marks for getting a question in this section correct. Questions 16-20 are more difficult and are worth 6 marks, with a penalty of 1 point for a wrong answer which tries to stop pupils guessing. The last five questions are intended to be the most challenging and so are also 6 marks, but with a 2 point penalty for an incorrectly answered question. Questions to which no answer is entered will gain (and lose) 0 marks. Again, the top 40% of students taking this challenge get a certificate. There are two follow-on rounds to this competition: The European Kangaroo and the Intermediate Mathematical Olympiad. Additionally, top performers can be selected for the National Mathematics Summer Schools. === Intermediate Mathematical Olympiad === To prevent this getting confused with the International Mathematical Olympiad, this is often abbreviated to the IMOK Olympiad (IMOK = Intermediate Mathematical Olympiad and Kangaroo). The IMOK is sat by the top 500 scorers from each school year in the Intermediate Maths Challenge and consists of three papers, 'Cayley', 'Hamilton' and 'Maclaurin' named after famous mathematicians. The paper the student will undertake depends on the year group that student is in (Cayley for those in year 9 and below, Hamilton for year 10 and Maclaurin for year 11). Each paper contains six questions. Each solution is marked out of 10 on a 0+ and 10- scale; that is to say, if an answer is judged incomplete or unfinished, it is awarded a few marks for progress and relevant observations, whereas if it is presented as complete and correct, marks are deducted for faults, poor reasoning, or unproven assumptions. As a result, it is quite uncommon for an answer to score a middling mark (e.g. 4–6). This makes the maximum mark out of 60. For a student to get two questions fully correct is considered "very good". All people taking part in this challenge will get a certificate (participation for the bottom 50%, merit for the next 25% and distinction for the top 25%). The mark boundaries for these certificates change every year, but normally around 30 marks will gain a Distinction. Those scoring highly (the top 50) will gain a book prize; again, this changes every year, with 44 marks required in the Maclaurin paper in 2006. Also, the top 100 candidates will receive a medal; bronze for Cayley, silver for Hamilton and gold for Maclaurin. === European Kangaroo === The European Kangaroo is a competition which follows the same structure as the AMC (Australian Mathematics Competition). There are twenty-five multiple-choice questions and no penalty marking. This paper is taken throughout Europe by over 3 million pupils from more than 37 countries. Two different Kangaroo papers follow on from the Intermediate Maths Challenge and the next 5500 highest scorers below the Olympiad threshold are invited to take part (both papers are by invitation only). The Grey Kangaroo is sat by students in year 9 and below and the Pink Kangaroo is sat by those in years 10 and 11. The top 25% of scorers in each paper receive a certificate of merit and the rest receive a certificate of participation. All those who sit either Kangaroo also receive a keyfob containing a different mathematical puzzle each year. (The puzzles along with solutions) === National Mathematics Summer Schools === Selected by lottery, 48 of the top 1.5% of scorers in the IMC are invited to participate in one of three week-long National Mathematics Summer Schools in July. Each from a different school across the UK, the 24 boys and 24 girls are facilitated with a range of activities, including daily lectures, designed to go beyond the GCSE syllabus and explore wider and more challenging areas of mathematics. The UKMT aims to "promote mathematical thinking" and "provide an opportunity for participants to meet other students and adults who enjoy mathematics". They were delivered virtually during the COVID-19 pandemic but had reverted to in-person events by 2022. == Senior Mathematical Challenge == The Senior Mathematical Challenge (SMC) takes place in late-autumn each year, and is open to students who are aged 19 or below and are not registered to attend a university. SMC consists of twenty-five multiple choice questions to be answered in 90 minutes. All candidates start with 25 marks, each correct answer is awarded 4 marks and 1 mark is deducted for each incorrect answer. This gives a score between 0 and 125 marks. Unlike the JMC and IMC, the top 66% get one of the three certificates. Further, the top 1000 highest scorers who are eligible to represent the UK at the International Mathematical Olympiad, together with any discretionary and international candidates, are invited to compete in the British Mathematical Olympiad and the next around 6000 highest scorers are invited to sit the Senior Kangaroo. Discretionary candidates are those students who are entered by their mathematics teachers, on payment of a fee, who did not score quite well enough in the SMC, but who might cope well in the next round. === British Mathematical Olympiad === Round 1 of the Olympiad is a three-and-a-half hour examination including six more difficult, long answer questions, which serve to test entrants' problem-solving skills. As of 2005, a more accessible first question was added to the paper; before this, it only consisted of 5 questions. Approximately 100 highest scoring candidates from BMO1 are invited to sit the BMO2, which is the follow-up round that has the same time limit as BMO1, but in which 4 harder questions are posed. The top 24 scoring students from the second round are subsequently invited to a training camp at Trinity College, Cambridge or Oundle School for the first stage of the International Mathematical Olympiad UK team selection. === Senior Kangaroo === The Senior Kangaroo is a one-hour examination to which the next around 6000 highest scorers below the Olympiad threshold are invited. The paper consists of twenty questions, each of which require three digit answers (leading zeros are used if the answer is less than 100, since the paper is marked by machine). The top 25% of candidates receive a certificate of merit and the rest receive a certificate of participation. == Team Challenge == The UKMT Team Maths Challenge is an annual event. One team from each participating school, comprising four pupils selected from year 8 and 9 (ages 12–14), competes in a regional round. No more than 2 pupils on a team may be from Year 9. There are over 60 regional competitions in the UK, held between February and May. The winning team in each regional round, as well as a few high-scoring runners-up from throughout the country, are then invited to the National Final in London, usually in late June. There are 4 rounds: Group Questions Cross-Numbers Shuttle (NB: The previous Head-to-Head Round has been replaced with another, similar to the Mini-Relay used in the 2007 and 2008 National Finals.) Relay In the National Final however an additional 'Poster Round' is added at the beginning. The poster round is a separate competition, however, since 2018 it is worth up to six marks towards the main event. Four schools have won the Junior Maths Team competition at least twice: Queen Mary's Grammar School in Walsall, City of London School, St Olave's Grammar School, and Westminster Under School. == Senior Team Challenge == A pilot event for a competition similar to the Team Challenge, aimed at 16- to 18-year-olds, was launched in the Autumn of 2007 and has been running ever since. The format is much the same, with a limit of two year 13 (Upper Sixth-Form) pupils per team. Regional finals take place between October and December, with the National Final in early February the following year. Previous winners are below: == British Mathematical Olympiad Subtrust == For more information see British Mathematical Olympiad Subtrust. The British Mathematical Olympiad Subtrust is run by UKMT, which runs the British Mathematical Olympiad as well as the UK Mathematical Olympiad for Girls, several training camps throughout the year such as a winter camp in Hungary, an Easter camp at Trinity College, Cambridge, and other training and selection of the IMO team. == See also == European Kangaroo British Mathematical Olympiad International Mathematical Olympiad International Mathematics Competition for University Students == References == == External links == United Kingdom Mathematics Trust website British Mathematical Olympiad Committee site International Mathematics Competition for University Students (IMC) site Junior Mathematical Challenge Sample Paper Intermediate Mathematical Challenge Sample Paper Senior Mathematical Challenge Sample Paper
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Wikipedia:Jur Hronec#0
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Gymnázium Jura Hronca (GJH) is a gymnasium (grammar school) in Bratislava, Slovakia. The school focuses on the study of natural sciences, mathematics, and computer sciences. However, its affiliation with the International Baccalaureate, an active bi-lingual (English – Slovak) programme and the option to study several foreign languages such as French and German, the school has a strong reputation for the study of foreign languages. In the school year 2005/2006, GJH launched lower programs of IB Primary Years Programme and IB Middle Years Programme. The school has been recently known as the "Spojená škola Gymnázium Jura Hronca a ZŠ Košická" (United school of the Gymnázium Jura Hronca and the Košická Primary School) after a merge with the primary school Základná škola a osemročné gymnázium Košická sharing the same building. == History == The school was founded on January 9, 1959, as an 11-year secondary school. In the school year 1969/70 the school was granted the status of a Gymnasium, named after the Slovak mathematician Jur Hronec. In 1997 the school donated the head of a mummified Egyptian man to the Slovak National Museum. == References == == External links == Homepage Official BratMUN Homepage An article about the 2010 BratMUN in the SME daily.
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Wikipedia:Justinian Rweyemamu#0
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Justinian F. Rweyemamu (28 September 1942 – 30 March 1982) was Tanzania’s first major economics scholar. Considered by many as the outstanding representative of the post-independence African scholars, he was also a pan-Africanist, political strategist, and international civil servant. The first Tanzanian to get PhD in Harvard University. == Early life and education == Rweyemamu was born on 28 September 1942 in Katoma, a small village in the outskirts of Bukoba town located in Kagera Region, Tanzania. In 1958 he joined St. Thomas More College Ihungo, a Catholic secondary school in Bukoba, and in 1961 graduated top of his class. He then went to the USA on a scholarship to pursue undergraduate education on the eve of his country’s independence from the British. He enrolled at Fordham University, where he majored and graduated with a Bachelor of Science degree in Economics, Applied Mathematics and Philosophy (1965). At Fordham he was an active member of the university's Economics club and the Philosophy club. He went on to Harvard University for his graduate and doctoral studies in Economics (Ph.D 1971) under a fellowship from the Rockefeller Foundation, studying under economists Albert Hirschman and Thomas Weisskopf. While at Harvard, he was a contemporary of the political scientist Mahmood Mamdani and documentary film-maker James Ault, with whom they formed the informal Harvard “Africa Group”. His Ph.D thesis, entitled “An Industrial Strategy for Tanzania”, was a seminal work in the development economics scene, and its revised version was later published by Oxford University Press as Underdevelopment and Industrialization in Tanzania; a study of perverse capitalist industrial development (1973). == Career == On completion of graduate studies, he returned to his native land and took up a faculty position at the Department of Economics of the University of Dar es Salaam, and was later the Dean of its Faculty of Social Sciences. He then took up position in the Government to become more of an actor, than a privileged spectator in academia. In the government he was appointed (1975) Permanent Secretary of the Planning Ministry and subsequently Personal Assistant (Economic affairs) to the then President of the Republic Julius Nyerere. In a span of just a few years he became internationally recognized, due to his thought provoking economic analysis and recommendations of the economic plight of the poor nations. He was thus appointed Chairman of the Council for the Development of Social Science Research In Africa (CODESRIA), member of the Committee of the Third World Forum and a founding member of The International Foundation for Development Alternatives (IFDA). In 1977 he left the country for a high-profile appointment in the UN, first in Switzerland and then later New York, USA. During his time in the UN he was a member of the UN Committee for Development Planning, worked for the Brandt Commission and worked for the UN Director General for Development and International Cooperation till his untimely death caused by cancer on 30 March 1982. Rweyemamu is remembered as the father of Tanzanian economics and made his greatest impact through the remarkable concentration of his students in top echelons of government and academia. These include: Jakaya Kikwete, the fourth President of the United Republic of Tanzania; Benno Ndulu, the Governor of the Bank of Tanzania. Ibrahim Lipumba, Tanzania’s academician and politician; Delphin Rwegasira, Tanzanian economist. In 1982, his friends and colleagues established the Justinian F. Rweyemamu Prize in order to perpetuate the academic spirit of J F Rweyemamu and to stimulate young Africans to follow his example in placing their talent at the service of their people. It was established by four of the institutions with which Justinian Rweyemamu was affiliated: CODESRIA, the Dag Hammarskjöld Foundation, IFDS, and the Third World Forum. Among the recipients of this prize is Calestous Juma, a prominent Kenyan scientist based at Harvard University. == Selected writings == Underdevelopment and Industrialization in Tanzania: A Study of Perverse Capitalist Development (Nairobi: Oxford University Press, 1973) Towards Socialist Planning The Teaching of Economics in Africa Industrialization and Income Distribution in Africa Pugwash on Self-reliance North-South: A Programme for Survival (The Brandt report) Dialogue for a New Order Third World Options: Power, security and hope for another development (Tanzania Publishing House, 1992). == References ==
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Wikipedia:Jyeṣṭhadeva#0
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Jyeṣṭhadeva (c. 1500 – c. 1575) was an astronomer-mathematician of the Kerala school of astronomy and mathematics founded by Madhava of Sangamagrama (c. 1350 – c. 1425). He is best known as the author of Yuktibhāṣā, a commentary in Malayalam of Tantrasamgraha by Nilakantha Somayaji (1444–1544). In Yuktibhāṣā, Jyeṣṭhadeva had given complete proofs and rationale of the statements in Tantrasamgraha. This was unusual for traditional Indian mathematicians of the time. The Yuktibhāṣā is now believed to contain some elements of calculus like Taylor and infinity series of some trigonometric functions.However they did not combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the powerful problem-solving tool we have today.Jyeṣṭhadeva also authored Drk-karana, a treatise on astronomical observations. According to K. V. Sarma, the name "Jyeṣṭhadeva" is most probably the Sanskritised form of his personal name in the local language Malayalam. == Life period of Jyeṣṭhadeva == There are a few references to Jyeṣṭhadeva scattered across several old manuscripts. From these manuscripts, one can deduce a few bare facts about the life of Jyeṣṭhadeva. He was a Nambudiri belonging to the Parangngottu family (Sanskrtised as Parakroda) born about the year 1500 CE. He was a pupil of Damodara and a younger contemporary of Nilakantha Somayaji. Achyuta Pisharati was a pupil of Jyeṣṭhadeva. In the concluding verse of his work titled Uparagakriyakrama, completed in 1592, Achyuta Pisharati has referred to Jyeṣṭhadeva as his aged benign teacher. From a few references in Drkkarana, a work believed to be of Jyeṣṭhadeva, one may conclude that Jyeṣṭhadeva lived up to about 1610 CE. Parangngottu, the family house of Jyeṣṭhadeva, still exists in the vicinity of Trikkandiyur and Alathiyur. There are also several legends connected with members of Parangngottu family. == Mathematical lineage == Little is known about the mathematical traditions in Kerala prior to Madhava of Sangamagrama. Vatasseri Paramesvara was a direct disciple of Madhava. Damodara was a son of Paramesvara. Nilakantha Somayaji and Jyeshthadeva were pupils of Damodara. Jyeṣṭhadeva's pupil was Achyuta Pisharati and Melpathur Narayana Bhattathiri was Achyuta Pisharati's student. == Jyeshthadeva's works == Jyeṣṭhadeva is known to have composed only two works, namely, Yuktibhāṣā and Drkkarana. The former is commentary with rationales of Tantrasamgraha of Nilakantha Somayaji and the latter is a treatise on astronomical computations. Three factors make Yuktibhāṣā unique in the history of the development of mathematical thinking in the Indian subcontinent: It is composed in the spoken language of the local people, namely, the Malayalam language. This is in contrast to the centuries-old Indian tradition of composing scholarly works in the Sanskrit language which was the language of the learned. The work is in prose, again in contrast to the prevailing style of writing even technical manuals in verse. All the other notable works of the Kerala school are in verse. Most importantly, Yuktibhāṣā was composed intentionally as a manual of proofs. The very purpose of writing the book was to record in full detail the rationales of the various results discovered by mathematicians-astronomers of the Kerala school, especially of Nilakantha Somayaji. This book is proof enough to establish that the concept of proof was not unknown to Indian mathematical traditions. == See also == Kerala School Indian mathematics Indian mathematicians List of astronomers and mathematicians of the Kerala school == References == == Sources == Bressoud, David (2002), "Was Calculus Invented in India?", The College Mathematics Journal, 33 (1): 2–13, doi:10.2307/1558972, JSTOR 1558972 == Further references == Details on the English translation of Yuktibhāṣā by K. V. Sarma: Sarma, K.V., Ramasubramanian, K., Srinivas, M.D., Sriram, M.S. (2008). Ganita-Yukti-Bhasa (Rationales in Mathematical Astronomy) of Jyeṣṭhadeva: Volume I: Mathematics, Volume II: Astronomy. Sources and Studies in the History of Mathematics and Physical Sciences. Springer jointly with Hindustan Book Agency, New Delhi, India. ISBN 978-1-84882-072-2.{{cite book}}: CS1 maint: multiple names: authors list (link) (This is a critical translation of the original Malayalam text by K.V. Sarma with explanatory notes by K. Ramasubramanian, M.D. Srinivas and M.S. Sriram.) For a review of the English translation of Yuktibhāṣā: Homer S. White (17 July 2009). "Ganita-Yukti-Bhāsā (Rationales in Mathematical Astronomy) of Jyesthadeva". MAA Reviews. The Mathematical Association of America. Retrieved 30 January 2010. R.C. Gupta (1973). "Addition and subtraction theorems for the sine and the cosine functions in medieval india" (PDF). Indian Journal of History of Science. 9 (2): 164–177. Archived from the original (PDF) on 29 November 2014. K. V. Sarma (1972). A history of the Kerala school of Hindu astronomy (in perspective). Vishveshvaranand Indological series. Vol. 55. Vishveshvaranand Institute of Sanskrit & Indological Studies, Hoshiarpur, Panjab University. Bibcode:1972hksh.book.....S. K.V. Sarma. "Tradition of Aryabhatiya in Kerala : Revision of planetary parameters" (PDF). Indian Journal of History of Science. 12 (2): 194–199. Archived from the original (PDF) on 29 November 2014. Retrieved 30 January 2010. George Gheverghese Joseph (2000). The Crest of the Peacock: The Non-European Roots of Mathematics. Princeton University Press. pp. 416. ISBN 978-0-691-00659-8. Plofker, Kim (2009). "7 The school of Madhava in Kerala". Mathematics in India. Princeton University Press. pp. 219–254. ISBN 9780691120676. For a modern explanation of Jyeṣṭhadeva's proof of the power series expansion of the arctangent function: Victor J. Katz (2009). "12". A history of mathematics: An introduction (3 ed.). Addison Wesley. pp. 450–455. ISBN 978-0-321-38700-4.
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Wikipedia:Jyotirmimamsa#0
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In astronomy, Jyotirmimamsa (analysis of astronomy) is a treatise on the methodology of astronomical studies authored by Nilakantha Somayaji (1444–1544) in around 1504 CE. Nilakantha somayaji was an important astronomer-mathematician of the Kerala school of astronomy and mathematics and was the author of the much celebrated astronomical work titled Tantrasamgraha. This book stresses the necessity and importance of astronomical observations to obtain correct parameters for computations and to develop more and more accurate theories. It even discounts the role of revealed wisdom and divine intuitions in studying astronomical phenomena. Jyotirmimamsa is sometimes cited as proof to establish that modern methodologies of scientific investigations were known to ancient and medieval Indians. Neelkantha Somayaji insisted that computational results should tally with that of observations and astronomical parameters and constants should be revised periodically. To come to more precise conclusions, Neelkantha Somayaji have discussions with the astronomer and mathematicians of other schools. The nature of the astronomical and mathematical work, the divine intuition, the experimental details of the science, corrections to the planetary parameters, reasons for the corrections for the planetary revolutions, Vedic authority for inference in astronomy, relative accuracy of different systems, and correction through eclipses, true motion, position, etc., of planets are some of the topics discussed in Jyotirmimamsa. == Synopsis == The following is an outline of the various topics discussed in Jyotirmimamsa. The necessity of revising the astronomical constants at regular intervals for correcting the parameters connected with astronomy The meaning of devatha prasada, which is manifested in the intuition of the astronomers, as a prerequisite for obtaining the accurate values and the correct approach in the astronomy The experimental determinations of the astronomical constants and the tools used for these determinations The need and the importance of conducting experiments in astronomical studies Significance of the astronomical books and the base of collecting data from those books Application of corrections, called bija correction, for the astronomical figures The corrections known as the Bhatta correction Justification of the changes made by the astronomer Lallacharya in his book Sishyaddhi vruddhi tantra Reasons for difference in the mean planets though the revolutions are identical Vedic authority for inference as a means to derive the number of planetary revolutions Different systems of astronomy The numbers of planetary revolutions enunciated by Sripati and Brahmagupta The numbers of revolutions of planets, apogees and nodes, number of days, in a Kalpa (in astronomy, a period of 14×72×4320000 years) Zero points of the planets at the commencement of Kali Yuga, corrections to planetary revolutions Relative accuracy of the different astronomical systems Mean planets according to the Sunrise system of Aryabhatiya, mean planets, Moon's apogee and Moon's node according to Siddhantasekhara Application of the values in the astronomical calculations, astronomical corrections given on the bases of the eclipses Eclipses observed by Parameswaracharya Method of corrections given by other astronomers Demonstration of the validity of those corrections through eclipses, precision of equinoxes, calculation and the correction, correction of the periphery of the manda epicycle, discussion on the precision of the equinoxes, corrections due to the precession of the equinoxes Sine table for praanakalaantara, sine table for the ascensional differences Derivation of the 36 Rsines Graphic proof for the relation of the sides and hypotenuse Reason for the reduction of the minutes of arc of the planetary orbital to the visible celestial sphere == See also == Indian astronomy Indian mathematics Indian mathematicians History of mathematics == References ==
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Wikipedia:Jyā, koti-jyā and utkrama-jyā#0
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Jyā, koṭi-jyā and utkrama-jyā are three trigonometric functions introduced by Indian mathematicians and astronomers. The earliest known Indian treatise containing references to these functions is Surya Siddhanta. These are functions of arcs of circles and not functions of angles. Jyā and koti-jyā are closely related to the modern trigonometric functions of sine and cosine. In fact, the origins of the modern terms of "sine" and "cosine" have been traced back to the Sanskrit words jyā and koti-jyā. == Definition == Let 'arc AB' denote an arc whose two extremities are A and B of a circle with center 'O'. If a perpendicular BM is dropped from B to OA, then: jyā of arc AB = BM koti-jyā of arc AB = OM utkrama-jyā of arc AB = MA If the radius of the circle is R and the length of arc AB is s, the angle subtended by arc AB at O measured in radians is θ = s / R. The three Indian functions are related to modern trigonometric functions as follows: jyā ( arc AB ) = R sin ( s / R ) koti-jyā ( arc AB ) = R cos ( s / R ) utkrama-jyā ( arc AB ) = R ( 1 - cos ( s / R ) ) = R versin ( s / R ) == Terminology == An arc of a circle is like a bow and so is called a dhanu or chāpa which in Sanskrit means "a bow". The straight line joining the two extremities of an arc of a circle is like the string of a bow and this line is a chord of the circle. This chord is called a jyā which in Sanskrit means "a bow-string", presumably translating Hipparchus's χορδή with the same meaning. The word jīvá is also used as a synonym for jyā in geometrical literature. At some point, Indian astronomers and mathematicians realised that computations would be more convenient if one used the halves of the chords instead of the full chords and associated the half-chords with the halves of the arcs. The half-chords were called ardha-jyās or jyā-ardhas. These terms were again shortened to jyā by omitting the qualifier ardha which meant "half of". The Sanskrit word koṭi has the meaning of "point, cusp", and specifically "the curved end of a bow". In trigonometry, it came to denote "the complement of an arc to 90°". Thus koṭi-jyā is "the jyā of the complementary arc". In Indian treatises, especially in commentaries, koṭi-jyā is often abbreviated as kojyā. The term koṭi also denotes "the side of a right angled triangle". Thus koṭi-jyā could also mean the other cathetus of a right triangle, the first cathetus being the jyā. Utkrama means "inverted", thus utkrama-jyā means "inverted chord". The tabular values of utkrama-jyā are derived from the tabular values of jyā by subtracting the elements from the radius in the reversed order. This is really the arrow between the bow and the bow-string and hence it has also been called bāṇa, iṣu or śara all meaning "arrow". An arc of a circle which subtends an angle of 90° at the center is called a vritta-pāda (a quadrat of a circle). Each zodiacal sign defines an arc of 30° and three consecutive zodiacal signs defines a vritta-pāda. The jyā of a vritta-pāda is the radius of the circle. The Indian astronomers coined the term tri-jyā to denote the radius of the base circle, the term tri-jyā being indicative of "the jyā of three signs". The radius is also called vyāsārdha, viṣkambhārdha, vistarārdha, etc., all meaning "semi-diameter". According to one convention, the functions jyā and koti-jyā are respectively denoted by "Rsin" and "Rcos" treated as single words. Others denote jyā and koti-jyā respectively by "Sin" and "Cos" (the first letters being capital letters in contradistinction to the first letters being small letters in ordinary sine and cosine functions). == From jyā to sine == The origins of the modern term sine have been traced to the Sanskrit word jyā, or more specifically to its synonym jīvá. This term was adopted in medieval Islamic mathematics, transliterated in Arabic as jība (جيب). Since Arabic is written without short vowels – and as a borrowing the long vowel is here denoted with yāʾ – this was interpreted as the homograph jaib, jayb (جيب), which means "bosom". The text's 12th-century Latin translator used the Latin equivalent for "bosom", sinus. When jyā became sinus, it has been suggested that by analogy kojyā became co-sinus. However, in early medieval texts, the cosine is called the complementi sinus "sine of the complement", suggesting the similarity to kojyā is coincidental. == See also == Versine == References ==
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Wikipedia:Ján Mináč#0
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Ján Mináč (born 15 June 1953) is a Slovak-Canadian mathematician who is a professor of mathematics at The University of Western Ontario. His research interests include Galois groups, Galois cohomology, quadratic forms, and nonlinear dynamics. == Early life and education == Mináč received his bachelor's degree and his master's level RNDr. degree from Comenius University, Czechoslovakia in 1976 and 1977 respectively. He then earned his Ph.D. in 1986 from Queen’s University in Canada under the supervision of Paulo Ribenboim. The title of his thesis is "Galois Groups, Order Spaces, and Valuations". His brother Matej Mináč is a film director. == Career == Mináč was a member of Mathematical Sciences Research Institute at Berkeley from 1986 to 1987 and then an NSF Postdoctoral Fellow at the University of California at Berkeley from 1987 to 1989. Afterward, he joined the University of Western Ontario as an assistant professor in 1989. He became an associate professor in 1991 and a full professor in 2003. == Research == Mináč and Nguyễn Duy Tân formulated the Mináč-Tân conjectures on the vanishing of Massey products over fields and the kernel unipotent conjecture. He has also worked on Galois theory and quadratic forms, Galois Demushkin groups, mild pro-2-groups, Galois modules, small quotients of Absolute Galois groups, ghosts in group cohomology, Koszulity properties of Galois cohomology, and Zassenhaus filtrations. Mináč has also worked on non-linear dynamics in networks and its applications to computational neuroscience. == Awards == Mináč received the Distinguished Research Professor Award at Western University during the years 2004-2005 and 2020-2021. In 2019, he became a Fellow of the Canadian Mathematical Society. During the year 2022-2023, he was a fellow at the Western Academy for Advanced Research. In 2013 he received an Excellence in Teaching Award from the Canadian Mathematical Society. Mináč also received multiple teaching awards at the University of Western Ontario. == References == == External links == Website at UWO
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Wikipedia:János Aczél (mathematician)#0
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János Dezső Aczél (Hungarian: [ˈɒt͡seːl]; 26 December 1924 – 1 January 2020), also known as John Aczel, was a Hungarian-Canadian mathematician, who specialized in functional equations and information theory. == Professional career == Aczél earned a doctorate in mathematical analysis from the University of Budapest, and held positions at the University of Cologne, Kossuth University, University of Miskolc, and University of Szeged. He joined the University of Waterloo faculty in 1965, eventually becoming Distinguished Professor in the Department of Pure Mathematics. He was the founder of the journal Aequationes Mathematicae, first published in 1968, and remained its honorary editor-in-chief. == Awards and honors == Aczél held honorary degrees from the University of Karlsruhe, the University of Graz, and the University of Silesia in Katowice. In 1971, he was elected a fellow of the Royal Society of Canada. He was the 1988 winner of the Santiago Ramón y Cajal Medal. In 1990, he became an external member of the Hungarian Academy of Sciences, and, in 2004, he was honored by the academy as one of the "big five" - five distinguished Hungarian mathematicians born in 1924 and graduating/obtaining a doctorate in 1947. The other honorees were John Horvath, Steven Gaal, Ákos Császár and László Fuchs. In 2008, he became an honorary member of the Hamburg Mathematical Society, the oldest active mathematical society in the world. In 2004, he won the Kampé de Fériet Award of the annual Information Processing and Management of Uncertainty conference, "for his pioneering work on the theory of functional equations, with applications in many fields, such as information measures, index numbers, group decision making, aggregation, production functions, laws of science, theory of measurement and utility theory." Issues of the journal Aequationes Mathematicae were dedicated to Aczél in 1999, 2005, and 2010, in honor of his 75th, 80th, and 85th birthdays. He died at the age of 95 on 1 January 2020. == Selected publications == === Articles === Aczél, J. (1965). "Quasigroups, nets, and nomograms" (PDF). Advances in Mathematics. 1 (3): 383–450. doi:10.1016/0001-8708(65)90042-3. Aczél, J. (1989). "The state of the second part of Hilbert's fifth problem". Bulletin of the American Mathematical Society. 20 (2): 153–163. doi:10.1090/S0273-0979-1989-15749-2. Aczél, János; Maksa, Gyula; Páles, Zsolt (2000). "Solution of a functional equation arising in an axiomatization of the utility of binary gambles". Proceedings of the American Mathematical Society. 129 (2): 483–493. doi:10.1090/S0002-9939-00-05545-3. S2CID 31566241. Aczél, János (2005). "Extension of a generalized Pexider equation". Proceedings of the American Mathematical Society. 133 (11): 3227–3233. doi:10.1090/S0002-9939-05-08039-1. Abbas, Ali E.; Aczél, János (2010). "The Role of Some Functional Equations in Decision Analysis". Decision Analysis. 7 (2): 215–228. doi:10.1287/deca.1100.0177. === Books === Aczél was the author or co-author of: Aczél, J.; Gołąb, S. (1960), Funktionalgleichungen der Theorie der geometrischen Objekte, Polska Akademia Nauk. Monografie Matematyczne, Tom 39, Państwowe Wydawnictwo Naukowe, Warsaw, MR 0133763. Aczél, J. (1966), Lectures on functional equations and their applications, Mathematics in Science and Engineering, vol. 19, New York: Academic Press, MR 0208210. Previously published in German as Vorlesungen über Funktionalgleichungen und ihre Anwendungen (Birkhäuser, 1961). Aczél, J.; Daróczy, Z. (1975), On measures of information and their characterizations, Mathematics in Science and Engineering, vol. 115, New York: Academic Press [Harcourt Brace Jovanovich Publishers], MR 0689178. Aczél, J. (1987), A short course on functional equations, Theory and Decision Library. Series B: Mathematical and Statistical Methods, Dordrecht: D. Reidel Publishing Co., doi:10.1007/978-94-009-3749-9, ISBN 90-277-2376-1. Aczél, J.; Dhombres, J. (1989), Functional equations in several variables, Encyclopedia of Mathematics and its Applications, vol. 31, Cambridge: Cambridge University Press, doi:10.1017/CBO9781139086578, ISBN 0-521-35276-2. He was the editor of: Aczél, J., ed. (30 November 2001). Functional Equations: History, Applications and Theory. Springer. ISBN 9781402003295. (1st edition 1984) Aczél, J., ed. (6 December 2012). Aggregating clones, colors, equations, iterates, numbers, and tiles. Birkhäuser. ISBN 9783034890960. (1st edition 1995) == References == == External links == Aczél's homepage at the Hungarian Academy of Sciences. János Aczél at the Mathematics Genealogy Project Author profile in the database zbMATH
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Wikipedia:János Halász (politician)#0
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János Halász (born 11 May 1963) is a Hungarian politician, member of the National Assembly (MP) for Debrecen (Hajdú-Bihar County Constituency II) between 1998 and 2014. He was elected MP from his party, Fidesz's national list in 2014. He served as Parliamentary State Secretary in the Ministry of Human (formerly National) Resources between 2010 and 2013. He served as State Secretary for Culture between 28 February 2013 and 5 June 2014. == Biography == He finished Móricz Zsigmond Secondary School of Kisújszállás in 1981 after four years of studies. He graduated as a mathematician and cybernetics expert from the Faculty of Natural Sciences of Kossuth Lajos University of Debrecen. He received a certificate for community development from the Hungarian Institute for Culture. From 1990 to 1993 he worked for and became the head of the Újkert Community House. He was assistant lecturer teaching non-profit management in the Department of Cultural Sciences of the Faculty of Humanities of Kossuth Lajos University of Debrecen from 1993. He was the chairman of the Derecske Town Welfare Service Foundation from 1993 to 2002. In the meantime, he acquired qualification as cultural and adult education manager from Kossuth Lajos University in 1996. He established the first amateur group of the university called Beauty Club in 1986. He was the chairman of the first community radio foundation from 1992 to 1996. === Political career === He was the member of the Ethics Committee of the National Alliance of Non-profit Human Services from 1997 to 1998. In the 1995 local by-elections he was elected member of the General Assembly of Debrecen. He joined to Fidesz in 1998, he was the chairman of the Ethics Committee of the parliamentary group. After the transformation into a people's party in 2003 he was awarded the chairmanship of the Debrecen constituency. In the 1998 parliamentary elections, he was elected MP representing Constituency II, Debrecen, Hajdú-Bihar County. He was spokesperson and deputy parliamentary faction leader from February 2000 to January 2002. He was the chairman of the North Plains Regional Youth Council from 2000. He served as Parliamentary State Secretary for national cultural heritage from 1 January to 26 May 2002. He was elected incumbent individual MP for his Debrecen constituency in April 2002. He had been active in the Cultural and Press Committee, of which he was deputy chairman, and in the Committee on Civil Organisations. In the October 2002 local elections he was elected incumbent local representative again in Debrecen, and he was elected deputy mayor responsible for culture and education. Following the party elections of 23 June 2003 he was re-elected deputy parliamentary faction leader, head of the cultural cabinet and parliamentary faction spokesman. He secured a seat in the Parliament in the 2006 parliamentary elections from 2nd Constituency of Hajdú-Bihar County. He was a member and vice chairman of the Committee on Culture and the Media. Four years later he also won a mandate. He was appointed Parliamentary State Secretary of the Ministry of National Resources in the Second Cabinet of Viktor Orbán on 2 June 2010. He also held the position when Zoltán Balog became minister and restructured and renamed the Ministry. In 2013 he was appointed State Secretary for Culture, replacing László L. Simon. == Personal life == He is married. His wife is Edit Mojzes. They have two sons, Dániel and Barnabás and a daughter, Magdolna. == References ==
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Wikipedia:Jónsson function#0
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In mathematical set theory, an ω-Jónsson function for a set x of ordinals is a function f : [ x ] ω → x {\displaystyle f:[x]^{\omega }\to x} with the property that, for any subset y of x with the same cardinality as x, the restriction of f {\displaystyle f} to [ y ] ω {\displaystyle [y]^{\omega }} is surjective on x {\displaystyle x} . Here [ x ] ω {\displaystyle [x]^{\omega }} denotes the set of strictly increasing sequences of members of x {\displaystyle x} , or equivalently the family of subsets of x {\displaystyle x} with order type ω {\displaystyle \omega } , using a standard notation for the family of subsets with a given order type. Jónsson functions are named for Bjarni Jónsson. Erdős and Hajnal (1966) showed that for every ordinal λ there is an ω-Jónsson function for λ. Kunen's proof of Kunen's inconsistency theorem uses a Jónsson function for cardinals λ such that 2λ = λℵ0, and Kunen observed that for this special case there is a simpler proof of the existence of Jónsson functions. Galvin and Prikry (1976) gave a simple proof for the general case. The existence of Jónsson functions shows that for any cardinal there is an algebra with an infinitary operation that has no proper subalgebras of the same cardinality. In particular if infinitary operations are allowed then an analogue of Jónsson algebras exists in any cardinality, so there are no infinitary analogues of Jónsson cardinals. == References == Erdős, P.; Hajnal, András (1966), "On a problem of B. Jónsson", Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, 14: 19–23, ISSN 0001-4117, MR 0209161 Galvin, Fred; Prikry, Karel (1976), "Infinitary Jonsson algebras and partition relations", Algebra Universalis, 6 (3): 367–376, doi:10.1007/BF02485843, ISSN 0002-5240, MR 0434822 Jónsson, Bjarni (1972), Topics in Universal Algebra, Lecture Notes in Mathematics, vol. 250, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0058648, ISBN 978-3-540-05722-2, MR 0345895 Kanamori, Akihiro (2003), The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.), Berlin, New York: Springer-Verlag, p. 319, ISBN 978-3-540-00384-7
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Wikipedia:Józef Maria Hoene-Wroński#0
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Józef Maria Hoene-Wroński (; Polish: [ˈjuzɛf ˈxɛnɛ ˈvrɔj̃skʲi]; French: Josef Hoëné-Wronski [ʒozɛf ɔɛne vʁɔ̃ski]; 23 August 1776 – 9 August 1853) was a Polish messianist philosopher, mathematician, physicist, inventor, lawyer, occultist and economist. In mathematics, he is known for introducing a novel series expansion for a function in response to Joseph Louis Lagrange's use of infinite series. The coefficients in Wroński's new series form the Wronskian, a determinant Thomas Muir named in 1882. As an inventor, he is credited with designing some of the first caterpillar vehicles. == Life and work == He was born as Hoëné in 1776 but changed his name in 1815 to Józef Wroński. Later in life he changed his name to Józef Maria Hoene-Wroński, without using his family's original French spelling Hoëné. At no point in his life, neither in Polish or French, was he known as Hoëné-Wroński; nor was the common French transliteration, Josef Hoëné-Wronski, ever his official name in his native Poland (though it might have served as his chosen French nom de plume on some work). His father, Antoni Höhne (pl, de), was the municipal architect of Poznań. Antoni originally came from the small Bohemian village of Leukersdorf (present-day Čermná which is now a part of Libouchec). In later life, he settled in western Poland marrying Elżbieta Pernicka in Wolsztyn in 1773. In the same place and a few years later on, in 1776, their son Józef Maria was born. Józef was educated in Poznań and Warsaw. In 1794 he served in Poland's Kościuszko Uprising as a second lieutenant of artillery, was taken prisoner, and remained until 1797 in the Russian Army. After resigning in the rank of lieutenant colonel in 1798, he studied in the Holy Roman Empire until 1800, when he enlisted in the Polish Legion at Marseille. There he began his scientific and scholarly work and conceived the idea of a great philosophical system. Ten years later he moved to Paris where he would spend most of his life working unremittingly to the last in the most difficult material circumstances. He wrote exclusively in French, in the desire that his ideas, of whose immortality he was convinced, be accessible to all; he worked, he said, "through France for Poland." He published over a hundred works, and left many more in manuscript; at 75 years of age and nearing death, he exclaimed: "God Almighty, there's still so much more I wanted to say!" In science, Hoene-Wroński set himself an extraordinary task: the complete reform of philosophy as well as that of mathematics, astronomy and technology. He elaborated not only a system of philosophy, but also applications to politics, history, economics, law, psychology, music and pedagogy. It was his aspiration to reform human knowledge in an "absolute, that is, ultimate" manner. In 1803, Wroński joined the Marseille Observatory, and began developing an enormously complex theory of the structure and origin of the universe. During this period, he took up a correspondence with nearly all of the major scientists and mathematicians of his day, and was well respected at the observatory. In 1803 Wroński "experienced a mystical illumination, which he regarded as the discovery of the Absolute." In 1810, he published the results of his scientific research in a massive tome, which he advocated as a new foundation for all of science and mathematics. His theories were strongly Pythagorean, holding numbers and their properties to be the fundamental underpinning of essentially everything in the universe. His claims were met with little acceptance, and his research and theories were generally dismissed as grandiose rubbish. His earlier correspondence with major figures meant that his writings garnered more attention than a typical crackpot theory, even earning a review from the great mathematician Joseph Louis Lagrange (which turned out to be categorically unfavorable). In the ensuing controversy, he was forced to leave the observatory. He immediately turned his focus towards applying philosophy to mathematics (his critics believed that this meant dispensing with mathematical rigor in favor of generalities). In 1812, he published a paper purporting to show that every equation has an algebraic solution, directly contradicting results which had been recently published by Paolo Ruffini; Ruffini turned out to be correct. He later turned his attention to disparate and largely unsuccessful pursuits such as a fantastical design for caterpillar-like vehicles which he intended to replace railroad transportation, but did not manage to persuade anyone to give the design serious attention. In 1819, he travelled to England in an attempt to obtain financial backing from the Board of Longitude to build a device to determine longitude at sea. After initial difficulties, he was given an opportunity to address the Board, but his pretentious address, On the Longitude, contained much philosophizing and generalities, but no concrete plans for a working device, and thus failed to gain any support from the Board. He remained for several years in England and, in 1821, published an introductory text on mathematics in London, which moderately improved his financial situation. In 1822, he returned to France, and again took up a combination of mathematics and far-fetched ideas, despite being in poverty and scorned by intellectual society. Along with his continuing Pythagorean obsession, he spent much time working on several notoriously futile endeavors, including attempts to build a perpetual motion machine, to square the circle and to build a machine to predict the future (which he dubbed the prognometre). In 1852, shortly before his death, he did find a willing audience for his ideas: the occultist Eliphas Levi who met Wroński and was greatly impressed and "attracted by his religious and scientific utopianism." Wroński was "a powerful catalyst" for Levi's occultism. Wroński died in 1853 in Neuilly-sur-Seine, France, on the outskirts of Paris. == Legacy == During his lifetime, nearly all his work was dismissed as nonsense. However, some of it came to be regarded in a more favourable light in later years. Although most of his inflated claims were groundless, his mathematical work contains flashes of deep insight and many important intermediary results, the most significant of which was his work on series. He had strongly criticized Lagrange for his use of infinite series, introducing instead a novel series expansion for a function. His criticisms of Lagrange were for the most part unfounded but the coefficients in Wroński's new series proved important after his death, forming a determinant now known as the Wronskian (the name which Thomas Muir had given them in 1882). The level of Wroński's scientific and scholarly accomplishments and the amplitude of his objectives placed Wroński in the first rank of European metaphysicians in the early 19th century. But the abstract formalism and obscurity of his thought, the difficulty of his language, his boundless self-assurance and his uncompromising judgments of others alienated him from most of the scientific community. He was perhaps the most original of the Polish metaphysicians, but others were more representative of the Polish outlook. == Works == Books 1801 - Mémoires sur l'aberration des astres mobiles, et sur l'inégalité dans l'apparence de leur mouvement, par J. Hoehné 1803 - Philosophie critique découverte par Kant, fondée sur le dernier principe du savoir, par J. Hoehne 1811 - Introduction à la philosophie des mathématiques, et technie de l'algorithmie, par M. Hoëné de Wronski 1811 - Programme du cours de philosophie transcendantale, par M. Hoëné Wronski 1812 - Résolution générale des équations de tous les degrés], par Hoëné Wronski 1812 - Réfutation de la théorie des fonctions analytiques de Lagrange, par Hoëné Wronski 1814 - Philosophie de l'Infini, par Hoëné Wronski 1818 - Introduction à un ouvrage intitulé Le Sphinx, ou la Nomothétique séhélienne, par Hoëné Wronski 1821 - A Course of mathematics, by Höené Wronski. Translated from the original French under the inspection of the author. London, 1821. 1827 - Canons de logarithmes de H. W. 1828 - Loi téléologique du hasard. Deuxième aperçu. Signé : Hoëné Wronski. Paris, le 13 avril 1828. 1831 - Messianisme, union finale de la philosophie et de la religion constituant la philosophie absolue (1831-1839) 1831 - Prodrome du Messianisme; Révélation des destinées de l’humanité 1832 - Réflexions philosophiques sur un miroir parabolique 1833 - Loi téléologique du Hasard: réimpression de trois pièces rarissimes (1833); précédée d'une autobiographie et d'un inventaire de l'œuvre Hoëné Wronski 1833 - Resolution of equation polynomials of tous les degries (in anglishe) 1837 - Rails mobiles, ou chemins de fer mouvans, de Hoëné Wroński 1840 - Secret politique de Napoléon comme base de l'avenir moral du monde, par Hoëné Wronski 1844 - Urgente réforme des Chemins de fer et de toute la locomotion terrestre, par Hoëné Wronski 1853 - Secret politique de Napoléon, par Hoëné Wronski, comme introduction à sa récente "Philosophie de l'histoire". Nouvelle édition Letters 1851 - Épître à Sa Majesté l'Empereur de Russie, pour compléter les "Cent pages décisives", et pour accomplir la réforme de la mécanique céleste". Signé : Hoëné Wroński. Metz, 1851 1851 - Épitre secrète a Son Altesse le prince Louis-Napoléon président de la République Française sur les destinées de la France == See also == List of Poles Timeline of Polish science and technology == References == == Sources == Władysław Tatarkiewicz, Historia filozofii (History of Philosophy), 3 vols., Warsaw, Państwowe Wydawnictwo Naukowe, 1978. == Further reading == Lazare Augé: Notice sur Hoené Wronski; suivi du portr. de Wronski par Mme Wronski, née s. de Montferrier de plus. acad. Paris : Libr. philos. de Ladrange, 1865. Samuel Dickstein: Hoene Wroński : jego życie i prace (Hoene Wronski: His Life And Works), Jagiellonian University Press Cracow, 1896. Francis Warrain: L'œuvre philosophique de Hoené Wronski, Textes, commentaires et critique. (Three volumes), Paris, Éditions Véga, 1933, 1936, 1938. == External links == Works by Józef Maria Hoene-Wroński at Open Library O'Connor, John J.; Robertson, Edmund F., "Józef Maria Hoene-Wroński", MacTutor History of Mathematics Archive, University of St Andrews Piotr Pragacz, Notes on the life and work of Jozef Maria Hoene-Wronski, preprint (March 2007) J. Hoëné de Wronski, Introduction à la philosophie des mathématiques, et technie de l'algorithmie, 1811 Roman Murawski, "The Philosophy of Hoene-Wronski" in: Organon 35, 2006, pp. 143–150
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Wikipedia:József Solymosi#0
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József Solymosi is a Hungarian-Canadian mathematician and a professor of mathematics at the University of British Columbia. His main research interests are arithmetic combinatorics, discrete geometry, graph theory, and combinatorial number theory. == Education and career == Solymosi earned his master's degree in 1999 under the supervision of László Székely from the Eötvös Loránd University and his Ph.D. in 2001 at ETH Zürich under the supervision of Emo Welzl. His doctoral dissertation was Ramsey-Type Results on Planar Geometric Objects. From 2001 to 2003 he was S. E. Warschawski Assistant Professor of Mathematics at the University of California, San Diego. He joined the faculty of the University of British Columbia in 2002. He was editor in chief of the Electronic Journal of Combinatorics from 2013 to 2015. == Contributions == Solymosi was the first online contributor to the first Polymath Project, set by Timothy Gowers to find improvements to the Hales–Jewett theorem. One of his theorems states that if a finite set of points in the Euclidean plane has every pair of points at an integer distance from each other, then the set must have a diameter (largest distance) that is linear in the number of points. This result is connected to the Erdős–Anning theorem, according to which an infinite set of points with integer distances must lie on one line.[ID] In connection with the related Erdős–Ulam problem, on the existence of dense subsets of the plane for which all distances are rational numbers, Solymosi and de Zeeuw proved that every infinite rational-distance set must either be dense in the Zariski topology or it must have all but finitely many of its points on a single line or circle.[EU] With Terence Tao, Solymosi proved a bound of ( m n ) 2 / 3 + ε {\displaystyle (mn)^{2/3+\varepsilon }} on the number of incidences between n {\displaystyle n} points and m {\displaystyle m} affine subspaces of any finite-dimensional Euclidean space, whenever each pair of subspaces has at most one point of intersection. This generalizes the Szemerédi–Trotter theorem on points and lines in the Euclidean plane, and because of this the exponent of 2 / 3 {\displaystyle 2/3} cannot be improved. Their theorem solves (up to the ε {\displaystyle \varepsilon } in the exponent) a conjecture of Toth, and was inspired by an analogue of the Szemerédi–Trotter theorem for lines in the complex plane.[HD] He has also contributed improved bounds for the Erdős–Szemerédi theorem, showing that every set of real numbers has either a large set of pairwise sums or a large set of pairwise products,[ME] and for the Erdős distinct distances problem, showing that every set of points in the plane has many different pairwise distances.[DD] == Recognition == In 2006, Solymosi received a Sloan Research Fellowship and in 2008 he was awarded the André Aisenstadt Mathematics Prize. In 2012 he was named a doctor of the Hungarian Academy of Science. == Selected publications == == References == == External links == Home page József Solymosi publications indexed by Google Scholar
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Wikipedia:Jörn Dunkel#0
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Jörn Dunkel is a German-American mathematician and physicist. He is the MathWorks Professor of Mathematics at the Massachusetts Institute of Technology (MIT). His research combines applied mathematics, physics, and biology to understand collective behavior in complex systems. == Education and career == Dunkel earned diplomas in physics and mathematics from Humboldt University of Berlin, and a Ph.D. in statistical physics from the University of Augsburg in 2008. He conducted postdoctoral work at the University of Oxford and the University of Cambridge. He joined MIT in 2013 and became a full professor in 2022. == Research == Dunkel studies active fluids, soft matter, and the mechanics of living systems. His work has addressed bacterial turbulence, pattern formation in cell monolayers, and morphogenesis. In 2023, Dunkel was named a Schmidt Science Polymath for his transdisciplinary research spanning mathematics and biology. == Awards == Schmidt Science Polymath Award (2023) Sloan Research Fellowship in Physics (2015) Gustav Hertz Prize, German Physical Society (2011) == Selected publications == Wensink, H. H., Dunkel, J., et al. (2012). "Meso-scale turbulence in living fluids." PNAS, 109(36), 14308–14313. Dunkel, J., et al. (2013). "Fluid dynamics of bacterial turbulence." Physical Review Letters, 110(22), 228102. Wioland, H., Woodhouse, F. G., Dunkel, J., et al. (2013). "Confinement stabilizes a bacterial suspension into a spiral vortex." Physical Review Letters, 110(26), 268102. == References == == External links == Homepage
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Wikipedia:Jørgen Dybvad#0
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Jørgen Dybvad (died 1612), was a Danish theologian and mathematician of the sixteenth and seventeenth century. Jørgen was born to prosperous Yeoman stock in Dybvad in Gosmer Parish, Aarhus, and took the name of his place of birth. Dybvad studied under Caspar Peucer and Sebastian Theodoricus at the University of Wittenberg. He later became a prominent scholar who wrote on astronomy, meteorology, and mathematics. He tended toward new, radically anti-Aristotelian ways of thinking. He was the first Dane to publish a commentary on Copernicus, and he had no trouble accepting the new star of 1572 as evidence of celestial mutability. However he was not a systematic astronomer. In astrology he merged the Ptolemaic approach with a contemporary tradition of historical chronology. He was also a Hebrew scholar. == Appointment as professor == In 1575 Dybvad was sent by Augustus I of Saxony to Denmark with a letter for Frederick II of Denmark which concerned the Crypto-Calvinists who had recently been imprisoned in Saxony. They had pleaded that they were following a precedent set in Denmark. In particular, the Dane, Niels Hemmingsen had recently published Syntagma institutionum christianarum which offered a Calvinist interpretation of the Eucharist. King Frederick responded by organising a hearing in Copenhagen Castle on 15 June. Despite resolute interrogation by Jørgen Rosenkrantz, Hemmingsen was defended by Peder Oxe. However, after Oxe's death in October and a further recommendation from Augustus I, Dybvad was appointed professor of theology at the University of Copenhagen. However Frederick II secured a retraction from Hemmingsen and forbade further discussion of the Eucharist despite pressure from Augustus I. Nevertheless, Hemmingsen lobbied for Tycho Brahe to be appointed as Rector of the University in the hope that Brahe could curb Dybvad's influence. However Brahe declined. == Augury at Sorø Abbey == At the Feast of St Martin, held in Sorø Abbey, on 11 November 1577, Dybvad was amongst the royal entourage gathered there around Frederick II. A comet was seen which led Dybvad to write En nyttig Vnderuissning Om den COMET, som dette Aar 1577. in Nouembrj først haffuer ladet sig see (Copenhagen: Laurentz Benedicht, 1578). This contained all sorts of apocalyptical prophecies, many of a political nature: "Hungary may well fear highly of the Turk. Hispania will feel a hard rod. Cologne on the Rhine will not be left out. Saxony, Thuiringen, Hesse, Steiermark, the Brandenburg lands, Augsburg, Kostnitz, Cleve, Berg, Ghent, Mecklenburg, Lithuania, must make ready for the effects of this comet, and especially for pestilence. Poland dare not be proud, for it must also drink of the cup, and it appears in particular, that the Muscovite or the Tartar will bring a sour visitation upon them.... Denmark, uplift thine eyes, shake off the sleep ... look about thee, and mark, that this comet does also threaten thee with pestilence and dear times.... The Muscovite, Sweden, Walachia, Westphalia, Trent, Hamburg, Bremen, Salzburg, Calabria, Portugal, Alexandria, and many other realms and lands will also receive something of this comet's effects." Tycho Brahe also observed the comet and wrote a report for Fredrick II. A German language text has been identified as being this report and includes a passage which may be taken as referring to Dybvad: "Not alone they, but many others, who seek their own honor and gain in the guise of the true religion, and as pseudo-prophets, not born of the Divine light from heavens and stars, but set up in the vineyard unsummoned and on their own volition, will be punished and tamed by these pseudo-planets, for the comet has let itself be seen as a pseudo-planet so that the children of the planets, both clerical and secular, who have mounted too high in their arrogance, and have not wandered in divine wisdom, will be punished." His son was Christoffer Dybvad. == References ==
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Wikipedia:Júlia Volaufová#0
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Júlia Volaufová is a Slovak biostatistician whose research has applied statistics to questions involving food intake, dietary supplements, calorie restriction, body weight, and diabetes. Her more theoretical interests include mixed linear models, regression analysis, and statistical hypothesis testing. She is a professor emerita of biostatistics at the LSU Health Sciences Center New Orleans. == Education and career == Volaufová studied probability and mathematical statistics at Comenius University in Bratislava, earning the equivalent of a master's degree in 1974, a rerum naturalis doctor degree in 1975, and a candidate degree (equivalent to a PhD) in 1984, under the supervision of Lubomír Kubáček. She became a researcher in the Institute of Measurement Science of the Slovak Academy of Sciences in 1974, one of "the first generation of scholars in statistics" in Czechoslovakia, and known for her organization of seminars and meetings in the country. Eventually, she became head of the Department of Theoretical Methods in the institute. With Kubáček and Ludmila Kubáčková, she coauthored the book Statistical Models with Linear Structures (Slovak Academy of Sciences, 1995). After visiting Louisiana State University beginning in 1994, she took a position as associate research professor in the Pennington Biomedical Research Center in 1997, and was promoted to full research professor in 2002. In 2004 she moved to the LSU Health Sciences Center New Orleans as professor of biostatistics, maintaining an adjunct affiliation with the Pennington Biomedical Research Center. == Recognition == Volaufová was named a Fellow of the American Statistical Association in 2011, recognizing "her excellence in the application of statistics and collaboration on research in biomedical fields; solicitous and challenging training of graduate students and professionals; organization of multiple international conferences; and persistent, effective efforts to preserve and promote the statistics community in the former Czechoslovakia and Central Europe". == References ==
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Wikipedia:Júlio César de Mello e Souza#0
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Júlio César de Mello e Souza (Rio de Janeiro, May 6, 1895 – Recife, June 18, 1974), was a Brazilian writer and mathematics teacher. He was well known in Brazil and abroad for his books on recreational mathematics, most of them published under the pen names of Malba Tahan and Breno de Alencar Bianco. He wrote 69 novels and 51 books of mathematics and other subjects, with over than two million books sold by 1995. His most famous work, The Man Who Counted, saw its 54th printing in 2001. Júlio César's most popular books, including The Man Who Counted, are collections of mathematical problems, puzzles, curiosities, and embedded in tales inspired by the Arabian Nights. He thoroughly researched his subject matters — not only the mathematics, but also the history, geography, and culture of the Islamic Empire which was the backdrop and connecting thread of his books. Yet Júlio César's travels outside Brazil were limited to short visits to Buenos Aires, Montevideo, and Lisbon: he never set foot in the deserts and cities which he so vividly described in his books. Júlio César was very critical of the educational methods used in Brazilian classrooms, especially for mathematics. "The mathematics teacher is a sadist," he claimed, "who loves to make everything as complicated as possible." In education, he was decades ahead of his time, and his proposals are still more praised than implemented today. For his books, Júlio César received a prize by the prestigious Brazilian Literary Academy and was made a member of the Pernambuco Literary Academy. The Malba Tahan Institute was founded in 2004 at Queluz to preserve his legacy. The State Legislature of Rio de Janeiro determined his birthday, May 6, to be commemorated as the Mathematician's Day. == Early life == Júlio César was born in Rio de Janeiro but spent most of his childhood in Queluz, a small rural town in the State of São Paulo. His father, João de Deus de Mello e Souza, was a civil servant with limited salary and eight (some reports say nine) children to support. In 1905 he was sent with his older brother, João Batista, to Rio de Janeiro to attend preparatory classes for admission to the prestigious Colégio Militar do Rio de Janeiro, where he studied from 1906 to 1909, and later at Colégio Pedro II. As a student, Júlio César was not academically successful. In a 1905 letter to their parents, João Batista tells that little Júlio "is bad at writing, and a failure in mathematics". His grade reports at Colégio Pedro II show that he once failed an Algebra exam, and barely passed one on Arithmetic. He later attributed these results to the teaching practices of the time, based on "the detestable method of salivation". However, he did show signs of his originality and non-conventional approaches in other ways. As a child in Queluz, he used to keep frogs as pets, and at one point he had some 50 animals in his yard. One of them, nicknamed "Monsignor", would follow him through the town. As an adult, he kept up with this hobby by assembling a large collection of frog statuettes. His career as a writer began while he was still in high school, when one of his classmates offered him a brand-new pen and a postage stamp from Chile in exchange for an essay on the theme of "Hope", the homework for the next day. According to his memoirs, Júlio was called late at night by other anxious students, and by the next morning he had provided four different essays on "Hope", at 400 réis a piece. He kept on this activity for the rest of the year, writing on "Hate", "Nostalgia", and whatever else the teacher demanded. Many years later he met his teacher, Silva Ramos, and told him of those dubious activities. When Silva Ramos introduced him jokingly to Raul Pederneiras as a "merchant of Hope and Hate", he got from the man prophetic advice: "Forget Hate and go on selling Hope. Take up this poetic profession, Merchant of Hope: since that business is profitable for the buyer, and even more so for the seller." == Career == === Writing === Júlio began to write tales on his own while still in his teens but did not impress the critics in his family. His brother João Batista recalls that Júlio's tales were full of superfluous characters with bizarre names like "Mardukbarian" or "Protocholóski". In 1918, at the age of 23, Júlio César presented five of his tales to the editor of the newspaper O Imparcial, where he worked, but his boss did not even look at them. Undaunted, Júlio picked up the manuscripts and brought them back a few days later, this time pretending that they were translations of the work of a certain "R. S. Slade," supposedly the rage in New York City. The first of those tales, The Jew's Revenge, was published in the front page of the next issue of the newspaper; and the rest followed suit. This experience convinced Júlio to assume a "foreign" pen name. He chose an Arabian identity — because, as he declared in an interview, the Arabs were unsurpassed in the art of storytelling. For the next seven years he prepared himself by studying Arabic and reading all he could on Islamic culture. In 1925, he sold the idea of a series of tales on Oriental themes to Irineu Marinho, editor of the newspaper A Noite (which would later become a huge Brazilian media conglomerate, the Organizações Globo). His stories, published in the column Contos de Malba Tahan ("Tales of Malba Tahan"), were attributed to a fictitious Arabian scholar of that name, and ostensibly translated by an equally fictitious "Professor Breno Alencar Bianco". Whether for the catchy pseudonym, or (more likely) for the author's lively style and imagination, his books were a resounding success, and he became a national celebrity. Even though his identity soon was known to everybody, he continued to use the name of Malba Tahan in his public life. He had a rubber stamp made with that name in Arabic script, which he used when grading his student's homework; and, in 1952 — by special permission of Brazilian President Getúlio Vargas — he added "Malba Tahan" to his own legal name. === Teaching === Before becoming a teacher, he worked for a time as general assistant at the National Library. Júlio César graduated as an elementary schoolteacher at the Escola Normal do Distrito Federal in Rio de Janeiro, and as a civil engineer at the Escola Politécnica in 1913. He started lecturing as a substitute teacher at the Colégio Pedro II, and later became a teacher at the Escola Normal. He began teaching history, geography and physics, and only later moved to mathematics. In time he became Chair at the Colégio Pedro II, at the Instituto de Educação, at the teacher's school of the Universidade do Brasil (which would become the Federal University of Rio de Janeiro) and at the National School of Education, where he got the title of Professor Emeritus. Besides his classes at the teacher's school, he delivered over 2000 lectures on the teaching of mathematics and wrote many books on the subject. In all his works Júlio defended the use of games as teaching aids, and the replacement of chalk-and-blackboard lectures by "mathematics laboratories" where students could engage in creative activities, self-study, and object manipulation — a proposal that was seen as heretical at the time. In the Brazilian 0-to-10 grading system, Júlio would never give a zero grade. "Why give a zero, when there are so many numbers to choose from?" he used to say. He would give the brightest students the task of teaching the weaker ones: "by the end of the first semester, they would all be above the pass line." he claimed. While his methods and style charmed all his students, he had the opposition of many of his colleagues, who found his approach of connecting mathematics to everyday life as demeaning. Julio César also spread his message through radio programmes of several stations in Rio de Janeiro, including the Rádio Nacional, Radio Clube, and Rádio Mayrink Veiga, as well as in television, at the TV Tupi of Rio and the TV Cultura of São Paulo. Júlio César's last public lecture was delivered in Recife, at the age of 79, to an audience of future teachers. It was about the art of storytelling. Back to his hotel room he apparently suffered a heart attack and expired. He had left instructions for his funeral. He did not want people to wear black: quoting a song by Noel Rosa, he explained that "Black clothes are vanities/of those who enjoy fancy dress;/I only wish for your memories/and memories are colorless". === Other activities === Júlio was an energetic campaigner for the cause of the Hanseniacs (lepers), who had historically been banned and confined in leper colonies. For over 10 years he edited the magazine Damião, which preached the end of the prejudice and re-incorporation of former inmates into the society. In his testament, he left a message to the Hanseniacs, to be read at his funeral. == Books == Aventuras do Rei Baribê, "Adventures of King Baribê" A Caixa do Futuro, "The Box of the Future." Céu de Alá, "Allah's Heaven" A Sombra do Arco-Íris, "The Rainbow's Shadow" (the author's favorite) O Homem que Calculava, "The Man Who Counted", 224p. (1938) Lendas do Céu e da Terra, "Legends of Heaven and Earth" Lendas do Deserto, "Legends of the Desert" Lendas do Oásis, "Legends of the Oasis" Lendas do Povo de Deus, "Legends of God's People" Maktub!, "It is Written!" Matemática Divertida e Curiosa, "Enjoyable and Curious Mathematics", 158p., ISBN 85-01-03375-8. Os Melhores Contos, "The Best Tales" Meu Anel de Sete Pedras, "My Ring of Seven Stones" Mil Histórias Sem Fim, "A Thousand Unending Tales" (2 volumes) Minha Vida Querida, "My Dear Life" Novas Lendas Orientais, "New Oriental Legends" Salim, o Mágico, "Salim, the Magician" Acordaram-me de Madrugada, "They Woke Me Up In the Middle of the Night" (memoirs). == References == Luiza Villamea, article in Revista Nova Escola, September 1995. João Batista de Mello e Souza, Os Meninos de Queluz – "The Boys from Queluz". == External links == A Biography in English by Andréa Estevão, at the Brazil-Arab News Agency. A biography based on article by Luiza Villamea at the Wayback Machine (archived October 27, 2009) (in Portuguese). Another biography (in Portuguese). And another Archived 2012-02-04 at the Wayback Machine (in Portuguese).
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Wikipedia:Jürg Fröhlich#0
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Jürg Martin Fröhlich (born 4 July 1946 in Schaffhausen) is a Swiss mathematician and theoretical physicist. He is best known for introducing rigorous techniques for the analysis of statistical mechanics models, in particular continuous symmetry breaking (infrared bounds), and for pioneering the study of topological phases of matter using low-energy effective field theories. == Biography == In 1965 Fröhlich began to study mathematics and physics at Eidgenössischen Technischen Hochschule Zürich. In 1969, under Klaus Hepp and Robert Schrader, he attained the Diplom (“Dressing Transformations in Quantum Field Theory”), and in 1972 he earned a PhD from the same institution under Klaus Hepp. After postdoctoral visits to the University of Geneva and Harvard University (with Arthur Jaffe), he took an assistant professorship in 1974 in the mathematics department of Princeton University. From 1978 until 1982 he was a professor at Institut des Hautes Études Scientifiques in Bures-sur-Yvette in Paris, and since 1982 he has been a professor for theoretical physics at ETH, where he founded the Center for Theoretical Studies. Over the course of his career, Fröhlich has worked on quantum field theory (including axiomatic quantum field theory, conformal field theory, and topological quantum field theory), on the precise mathematical treatment of models of statistical mechanics, on theories of phase transition, on the fractional quantum Hall effect, and on non-commutative geometry. == Honors and awards == In 1991 he received with Thomas Spencer the Dannie Heineman prize, in 1997 he received the Marcel Benoist Prize, in 2001 he won the Max Planck Medal of the Deutschen Physikalischen Gesellschaft, and in 2009 he was awarded the Henri Poincaré Prize. He is a member of the Academia Europaea and the Berlin-Brandenburg Academy of Sciences and Humanities. In 2012 he became a fellow of the American Mathematical Society. In 1978, Fröhlich gave an invited address to the International Congress of Mathematicians in Helsinki (“On the mathematics of phase transitions”) and in 1994 at the plenary talk of the ICM in Zurich (“The FQHE, Chern–Simons Theory and Integral Lattices”). He also co-authored a book on quantum triviality. In 2020, he was elected international member of the National Academy of Sciences. == Selected works == == See also == == References == == External links == Jürg Martin Fröhlich at the Mathematics Genealogy Project. Literature by and about Jürg Fröhlich in the German National Library catalogue. Jürg Fröhlich at zbMATH. "Personal webpage at ETH". phys.ethz.ch. Retrieved August 8, 2021. Jaffe, Arthur (2007). "Constructive Jürg - A Personal Overview of Constructive Quantum Field Theory" (PDF). arthurjaffe.com. Retrieved August 8, 2021.
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Wikipedia:Jürg Peter Buser#0
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Jürg Peter Buser, known as Peter Buser, (born 27 February 1946 in Basel) is a Swiss mathematician, specializing in differential geometry and global analysis. == Education and career == Buser received his doctorate in 1976 from the University of Basel with advisor Heinz Huber and thesis Untersuchungen über den ersten Eigenwert des Laplaceoperators auf kompakten Flächen (Studies on the first eigenvalue of the Laplace operator on compact surfaces). As a post-doctoral student he was at the University of Bonn, the University of Minnesota. and the State University of New York at Stony Brook, before he habilitated at the University of Bonn with a thesis on the length spectrum of Riemann surfaces. Buser is known for his construction of curved isospectral surfaces (published in 1986 and 1988). His 1988 construction led to a negative solution to Mark Kac's famous 1966 problem Can one hear the shape of a drum?. The negative solution was published in 1992 by Scott Wolpert, David Webb and Carolyn S. Gordon. The Cheeger-Buser inequality is named after him and Jeff Cheeger. He has been a professor at the École Polytechnique Fédérale de Lausanne (EPFL) since 1982. From 2004 to 2005 he was president of the Swiss Mathematical Society. In 2003 he was made an honorary doctor of the University of Helsinki. == Selected publications == Buser, Peter (1978). "Über eine Ungleichung von Cheeger". Mathematische Zeitschrift. 158 (3): 245–252. doi:10.1007/BF01214795. "On Cheeger's inequality λ 1 ≥ h 2 / 4 {\displaystyle \lambda _{1}\geq h^{2}/4} ". Geometry of the Laplace Operator. Proceedings of Symposia in Pure Mathematics. Vol. 36. American Mathematical Society. 1980. pp. 29–78. doi:10.1090/pspum/036. ISBN 9780821814390. with Hermann Karcher: Buser, Peter; Karcher, Hermann (1981). "The bieberbach case in gromov's almost flat manifold theorem". Global Differential Geometry and Global Analysis. Lecture Notes in Mathematics. Vol. 838. pp. 82–93. doi:10.1007/BFb0088844. ISBN 978-3-540-10285-4. ISSN 0075-8434. with Hermann Karcher: Gromov`s almost flat manifolds, Astérisque 1981, Nr. 81, p. 148 "A note on the isoperimetric constant." In Annales scientifiques de l'École Normale Supérieure, vol. 15, no. 2, 1982, pp. 213-230. "On the bipartition of graphs." Discrete Applied Mathematics 9, no. 1 (1984): 105–109. Isospectral Riemann Surfaces, Annales Institut Fourier (Grenoble), vol. 36, 1986, pp. 167–192 Cayley graphs and planar isospectral domains, in Toshikazu Sunada (ed.), Geometry and Analysis on Manifolds, Springer Verlag, Lecture Notes in Mathematics, vol. 1339, 1988, pp. 64–77 doi:10.1007/BFb0083047 Geometry and Spectra of Compact Riemann Surfaces, Birkhäuser 1992; 2010 pbk reprint with John Horton Conway, Peter Doyle, and Klaus-Dieter Semmler: Buser, Peter; Conway, John; Doyle, Peter; Semmler, Klaus-Dieter (1994). "Some planar isospectral domains" (PDF). International Mathematics Research Notices. 1994 (9): 391–400. doi:10.1155/S1073792894000437. with Peter Sarnak: Buser, P.; Sarnak, P. (1994). "On the period matrix of a Riemann surface of large genus (with an Appendix by J.H. Conway and N.J.A. Sloane)". Inventiones Mathematicae. 117 (1): 27–56. Bibcode:1994InMat.117...27B. doi:10.1007/BF01232233. ISSN 0020-9910. S2CID 116904696. with Mika Seppälä: Buser, Peter; Seppälä, Mika (2003). "Triangulations and homology of Riemann surfaces". Proceedings of the American Mathematical Society. 131 (2): 425–432. doi:10.1090/S0002-9939-02-06470-5. ISSN 0002-9939. == References == == External links == Homepage at EPFL
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Wikipedia:Jürgen Moser#0
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Jürgen Kurt Moser (July 4, 1928 – December 17, 1999) was a German-American mathematician, honored for work spanning over four decades, including Hamiltonian dynamical systems and partial differential equations. == Life == Moser's mother Ilse Strehlke was a niece of the violinist and composer Louis Spohr. His father was the neurologist Kurt E. Moser (July 21, 1895 – June 25, 1982), who was born to the merchant Max Maync (1870–1911) and Clara Moser (1860–1934). The latter descended from 17th century French Huguenot immigrants to Prussia. Jürgen Moser's parents lived in Königsberg, German empire and resettled in Stralsund, East Germany as a result of the Second World War. Moser attended the Wilhelmsgymnasium (Königsberg) in his hometown, a high school specializing in mathematics and natural sciences education, from which David Hilbert had graduated in 1880. His older brother Friedrich Robert Ernst (Friedel) Moser (August 31, 1925 – January 14, 1945) served in the German Army and died in Schloßberg during the East Prussian offensive. Moser married the biologist Dr. Gertrude C. Courant (Richard Courant's daughter, Carl Runge's granddaughter and great-granddaughter of Emil DuBois-Reymond) on September 10, 1955 and took up permanent residence in New Rochelle, New York in 1960, commuting to work in New York City. In 1980 he moved to Switzerland, where he lived in Schwerzenbach near Zürich. He was a member of the Akademisches Orchester Zürich. He was survived by his younger brother, the photographic printer and processor Klaus T. Moser-Maync from Northport, New York, his wife, Gertrude Moser from Seattle, their daughters, the theater designer Nina Moser from Seattle and the mathematician Lucy I. Moser-Jauslin from Dijon, and his stepson, the lawyer Richard D. Emery from New York City. Moser played the piano and the cello, performing chamber music since his childhood in the tradition of a musical family, where his father played the violin and his mother the piano. He was a lifelong amateur astronomer and took up paragliding in 1988 during a visit at IMPA in Rio de Janeiro. == Work == Moser completed his undergraduate education at and received his Dr. rer. nat. from the University of Göttingen in 1952, studying under Franz Rellich. After his thesis, he came under the influence of Carl Ludwig Siegel, with whom he coauthored the second and considerably expanded English language edition of a monography on celestial mechanics. Having spent the year 1953 at the Courant Institute of New York University as a Fulbright scholar, he emigrated to the United States in 1955 becoming a citizen in 1959. He became a professor at MIT and later at New York University. He served as director of the Courant Institute of New York University in the period of 1967–1970. In 1970 he declined the offer of a chair at the Institute for Advanced Study in Princeton. After 1980 he was at ETH Zürich, becoming professor emeritus in 1995. He was director (sharing office with Armand Borel in the first two years) of the Forschungsinstitut für Mathematik at ETH Zürich in 1984–1995, where he succeeded Beno Eckmann. He led a rebuilding of the ETH Zürich mathematics faculty. Moser was president of the International Mathematical Union in 1983–1986. == Research == In 1967, Neil Trudinger identified a new function space embedding which could be viewed as a borderline case of the Sobolev embedding theorem. Moser found the sharp constant in Trudinger's inequality, with the corresponding result often known as the Moser–Trudinger inequality. === Elliptic and parabolic partial differential equations === In the late 1950s, Ennio De Giorgi and John Nash independently discovered the fundamental elliptic regularity theory for general second-order elliptic and parabolic partial differential equations, in which (unlike the Schauder estimates) no differentiability or continuity is assumed of the coefficients. In the 1960s, Moser identified a new approach to their basic regularity theory, introducing the technique of Moser iteration. He developed it for both elliptic and parabolic problems, and beyond recovering De Giorgi and Nash's results, he was able to use it to prove a new Harnack inequality. In his original work, a key role was played by an extension of the John–Nirenberg lemma. Enrico Bombieri later found an argument avoiding this lemma in the elliptic case, which Moser was able to adapt to the parabolic case. The collection of these regularity results are often known as De Giorgi–Nash–Moser theory, although the original results were due solely to De Giorgi and Nash. === Differential geometry === In 1965, Moser found new results showing that any two volume forms on a closed manifold are related to one another by scaling and pullback by a diffeomorphism, so that geometrically the total volume is the only invariant of a volume form. He was able to apply the same techniques to symplectic forms, thereby proving that a cohomologous family of symplectic forms are related to one another by diffeomorphisms: this is also known as Moser's stability theorem. Moser also analyzed the case of manifolds with boundary, although his argument was mistaken. Later, with Bernard Dacorogna, Moser fully carried out the analysis of the boundary case. Moser also made an early contribution to the prescribed scalar curvature problem, showing that in any conformal class of Riemannian metrics on the projective plane, every function except for those which are nonpositive arises as a scalar curvature. Moser's prior analysis of the Moser–Trudinger inequality was important for this work, highlighting the geometric significance of optimal constants in functional inequalities. Research of Henri Poincaré and Élie Cartan in the early twentieth century had clarified the two-dimensional CR geometry, dealing with three-dimensional hypersurfaces of smooth four-dimensional manifolds which are also equipped with a complex structure. They had identified local invariants distinguishing two such structures, analogous to prior work identifying the Riemann curvature tensor and its covariant derivatives as fundamental invariants of a Riemannian metric. With Shiing-Shen Chern, Moser extended Poincaré and Cartan's work to arbitrary dimensions. Their work has had a significant influence on CR geometry. == Students == Among Moser's students were Mark Adler of Brandeis University, Ed Belbruno, Charles Conley (1933–1984), Howard Jacobowitz of Rutgers University, and Paul Rabinowitz of University of Wisconsin. == Awards and honours == Moser won the first George David Birkhoff Prize in 1968 for contributions to the theory of Hamiltonian dynamical systems, the James Craig Watson Medal in 1969 for his contributions to dynamical astronomy, the Brouwer Medal of the Royal Dutch Mathematical Society in 1984, the Cantor Medal of the Deutsche Mathematiker-Vereinigung in 1992 and the Wolf Prize in 1995 for his work on stability in Hamiltonian systems and on nonlinear differential equations. He was elected to membership of the National Academy of Sciences in 1973 and was corresponding member of numerous foreign academies such as the London Mathematical Society and the Akademie der Wissenschaften und Literatur, Mainz. At three occasions he was an invited speaker at the quadrennial International Congress of Mathematicians, namely in Stockholm (1962) in the section on applied mathematics, in Helsinki (1978) in the section on Complex Analysis, and a plenary speaker in Berlin (1998). In 1990 he was awarded honorary doctorates from University of Bochum and from Pierre and Marie Curie University in Paris. The Society for Industrial and Applied Mathematics established a lecture prize in his honor in 2000. == Major publications == == Notes == == References == Mather, John N.; McKean, Henry P.; Nirenberg, Louis; Rabinowitz, Paul H. (December 2000). "Jürgen K. Moser" (PDF). Notices of the AMS. 4 (11): 1392–1405. Retrieved 2007-08-20. J.J. O'Connor; E. F. Robertson. "Jürgen Kurt Moser". Retrieved 2008-07-04. Sylvia Nasar (December 21, 1999). "Obituary, New York Times". The New York Times. Retrieved 2010-09-14. American Institute of Physics. "Professional biography Jürgen Moser". Archived from the original on 2012-10-05. Retrieved 2010-12-05. Vladimir Arnold (2000). "Déclin des Mathématiques (après la mort de Jürgen Moser)" (PDF). La Gazette des mathématiciens (in French). 84: 92–94. Archived from the original (PDF) on 2014-08-08. ETH (20 March 2002). "Biography of Jürgen Moser, by ETH". ETH. Retrieved 2013-04-02. Guardian (20 March 2000). "Obituary of Moser, by Guardian". The Guardian. Retrieved 2013-05-27. SIAM (20 April 2001). "Moser Lecture, by SIAM". Retrieved 2013-11-16. Max Planck Institut Leipzig (31 May 2001). "In memoriam Jürgen Moser". Moser Symposium, by MPI Leipzig. Retrieved 2013-11-16. == External links == Paul H. Rabinowitz, "Jürgen Moser", Biographical Memoirs of the National Academy of Sciences (2015)
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Wikipedia:K-Poincaré algebra#0
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The Poincaré group, named after Henri Poincaré (1905), was first defined by Hermann Minkowski (1908) as the isometry group of Minkowski spacetime. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our understanding of the most basic fundamentals of physics. == Overview == The Poincaré group consists of all coordinate transformations of Minkowski space that do not change the spacetime interval between events. For example, if everything were postponed by two hours, including the two events and the path you took to go from one to the other, then the time interval between the events recorded by a stopwatch that you carried with you would be the same. Or if everything were shifted five kilometres to the west, or turned 60 degrees to the right, you would also see no change in the interval. It turns out that the proper length of an object is also unaffected by such a shift. In total, there are ten degrees of freedom for such transformations. They may be thought of as translation through time or space (four degrees, one per dimension); reflection through a plane (three degrees, the freedom in orientation of this plane); or a "boost" in any of the three spatial directions (three degrees). Composition of transformations is the operation of the Poincaré group, with rotations being produced as the composition of an even number of reflections. In classical physics, the Galilean group is a comparable ten-parameter group that acts on absolute time and space. Instead of boosts, it features shear mappings to relate co-moving frames of reference. In general relativity, i.e. under the effects of gravity, Poincaré symmetry applies only locally. A treatment of symmetries in general relativity is not in the scope of this article. == Poincaré symmetry == Poincaré symmetry is the full symmetry of special relativity. It includes: translations (displacements) in time and space, forming the abelian Lie group of spacetime translations (P); rotations in space, forming the non-abelian Lie group of three-dimensional rotations (J); boosts, transformations connecting two uniformly moving bodies (K). The last two symmetries, J and K, together make the Lorentz group (see also Lorentz invariance); the semi-direct product of the spacetime translations group and the Lorentz group then produce the Poincaré group. Objects that are invariant under this group are then said to possess Poincaré invariance or relativistic invariance. 10 generators (in four spacetime dimensions) associated with the Poincaré symmetry, by Noether's theorem, imply 10 conservation laws: 1 for the energy – associated with translations through time 3 for the momentum – associated with translations through spatial dimensions 3 for the angular momentum – associated with rotations between spatial dimensions 3 for a quantity involving the velocity of the center of mass – associated with hyperbolic rotations between each spatial dimension and time == Poincaré group == The Poincaré group is the group of Minkowski spacetime isometries. It is a ten-dimensional noncompact Lie group. The four-dimensional abelian group of spacetime translations is a normal subgroup, while the six-dimensional Lorentz group is also a subgroup, the stabilizer of the origin. The Poincaré group itself is the minimal subgroup of the affine group which includes all translations and Lorentz transformations. More precisely, it is a semidirect product of the spacetime translations group and the Lorentz group, R 1 , 3 ⋊ O ( 1 , 3 ) , {\displaystyle \mathbf {R} ^{1,3}\rtimes \operatorname {O} (1,3)\,,} with group multiplication ( α , f ) ⋅ ( β , g ) = ( α + f ⋅ β , f ⋅ g ) {\displaystyle (\alpha ,f)\cdot (\beta ,g)=(\alpha +f\cdot \beta ,\;f\cdot g)} . Another way of putting this is that the Poincaré group is a group extension of the Lorentz group by a vector representation of it; it is sometimes dubbed, informally, as the inhomogeneous Lorentz group. In turn, it can also be obtained as a group contraction of the de Sitter group SO(4, 1) ~ Sp(2, 2), as the de Sitter radius goes to infinity. Its positive energy unitary irreducible representations are indexed by mass (nonnegative number) and spin (integer or half integer) and are associated with particles in quantum mechanics (see Wigner's classification). In accordance with the Erlangen program, the geometry of Minkowski space is defined by the Poincaré group: Minkowski space is considered as a homogeneous space for the group. In quantum field theory, the universal cover of the Poincaré group R 1 , 3 ⋊ SL ( 2 , C ) , {\displaystyle \mathbf {R} ^{1,3}\rtimes \operatorname {SL} (2,\mathbf {C} ),} which may be identified with the double cover R 1 , 3 ⋊ Spin ( 1 , 3 ) , {\displaystyle \mathbf {R} ^{1,3}\rtimes \operatorname {Spin} (1,3),} is more important, because representations of SO ( 1 , 3 ) {\displaystyle \operatorname {SO} (1,3)} are not able to describe fields with spin 1/2; i.e. fermions. Here SL ( 2 , C ) {\displaystyle \operatorname {SL} (2,\mathbf {C} )} is the group of complex 2 × 2 {\displaystyle 2\times 2} matrices with unit determinant, isomorphic to the Lorentz-signature spin group Spin ( 1 , 3 ) {\displaystyle \operatorname {Spin} (1,3)} . == Poincaré algebra == The Poincaré algebra is the Lie algebra of the Poincaré group. It is a Lie algebra extension of the Lie algebra of the Lorentz group. More specifically, the proper ( det Λ = 1 {\textstyle \det \Lambda =1} ), orthochronous ( Λ 0 0 ≥ 1 {\textstyle {\Lambda ^{0}}_{0}\geq 1} ) part of the Lorentz subgroup (its identity component), S O ( 1 , 3 ) + ↑ {\textstyle \mathrm {SO} (1,3)_{+}^{\uparrow }} , is connected to the identity and is thus provided by the exponentiation exp ( i a μ P μ ) exp ( i 2 ω μ ν M μ ν ) {\textstyle \exp \left(ia_{\mu }P^{\mu }\right)\exp \left({\frac {i}{2}}\omega _{\mu \nu }M^{\mu \nu }\right)} of this Lie algebra. In component form, the Poincaré algebra is given by the commutation relations: where P {\displaystyle P} is the generator of translations, M {\displaystyle M} is the generator of Lorentz transformations, and η {\displaystyle \eta } is the ( + , − , − , − ) {\displaystyle (+,-,-,-)} Minkowski metric (see Sign convention). The bottom commutation relation is the ("homogeneous") Lorentz group, consisting of rotations, J i = 1 2 ϵ i m n M m n {\textstyle J_{i}={\frac {1}{2}}\epsilon _{imn}M^{mn}} , and boosts, K i = M i 0 {\textstyle K_{i}=M_{i0}} . In this notation, the entire Poincaré algebra is expressible in noncovariant (but more practical) language as [ J m , P n ] = i ϵ m n k P k , [ J i , P 0 ] = 0 , [ K i , P k ] = i η i k P 0 , [ K i , P 0 ] = − i P i , [ J m , J n ] = i ϵ m n k J k , [ J m , K n ] = i ϵ m n k K k , [ K m , K n ] = − i ϵ m n k J k , {\displaystyle {\begin{aligned}[][J_{m},P_{n}]&=i\epsilon _{mnk}P_{k}~,\\[][J_{i},P_{0}]&=0~,\\[][K_{i},P_{k}]&=i\eta _{ik}P_{0}~,\\[][K_{i},P_{0}]&=-iP_{i}~,\\[][J_{m},J_{n}]&=i\epsilon _{mnk}J_{k}~,\\[][J_{m},K_{n}]&=i\epsilon _{mnk}K_{k}~,\\[][K_{m},K_{n}]&=-i\epsilon _{mnk}J_{k}~,\end{aligned}}} where the bottom line commutator of two boosts is often referred to as a "Wigner rotation". The simplification [ J m + i K m , J n − i K n ] = 0 {\textstyle [J_{m}+iK_{m},\,J_{n}-iK_{n}]=0} permits reduction of the Lorentz subalgebra to s u ( 2 ) ⊕ s u ( 2 ) {\textstyle {\mathfrak {su}}(2)\oplus {\mathfrak {su}}(2)} and efficient treatment of its associated representations. In terms of the physical parameters, we have [ H , p i ] = 0 [ H , L i ] = 0 [ H , K i ] = i ℏ c p i [ p i , p j ] = 0 [ p i , L j ] = i ℏ ϵ i j k p k [ p i , K j ] = i ℏ c H δ i j [ L i , L j ] = i ℏ ϵ i j k L k [ L i , K j ] = i ℏ ϵ i j k K k [ K i , K j ] = − i ℏ ϵ i j k L k {\displaystyle {\begin{aligned}\left[{\mathcal {H}},p_{i}\right]&=0\\\left[{\mathcal {H}},L_{i}\right]&=0\\\left[{\mathcal {H}},K_{i}\right]&=i\hbar cp_{i}\\\left[p_{i},p_{j}\right]&=0\\\left[p_{i},L_{j}\right]&=i\hbar \epsilon _{ijk}p_{k}\\\left[p_{i},K_{j}\right]&={\frac {i\hbar }{c}}{\mathcal {H}}\delta _{ij}\\\left[L_{i},L_{j}\right]&=i\hbar \epsilon _{ijk}L_{k}\\\left[L_{i},K_{j}\right]&=i\hbar \epsilon _{ijk}K_{k}\\\left[K_{i},K_{j}\right]&=-i\hbar \epsilon _{ijk}L_{k}\end{aligned}}} The Casimir invariants of this algebra are P μ P μ {\textstyle P_{\mu }P^{\mu }} and W μ W μ {\textstyle W_{\mu }W^{\mu }} where W μ {\textstyle W_{\mu }} is the Pauli–Lubanski pseudovector; they serve as labels for the representations of the group. The Poincaré group is the full symmetry group of any relativistic field theory. As a result, all elementary particles fall in representations of this group. These are usually specified by the four-momentum squared of each particle (i.e. its mass squared) and the intrinsic quantum numbers J P C {\textstyle J^{PC}} , where J {\displaystyle J} is the spin quantum number, P {\displaystyle P} is the parity and C {\displaystyle C} is the charge-conjugation quantum number. In practice, charge conjugation and parity are violated by many quantum field theories; where this occurs, P {\displaystyle P} and C {\displaystyle C} are forfeited. Since CPT symmetry is invariant in quantum field theory, a time-reversal quantum number may be constructed from those given. As a topological space, the group has four connected components: the component of the identity; the time reversed component; the spatial inversion component; and the component which is both time-reversed and spatially inverted. == Other dimensions == The definitions above can be generalized to arbitrary dimensions in a straightforward manner. The d-dimensional Poincaré group is analogously defined by the semi-direct product IO ( 1 , d − 1 ) := R 1 , d − 1 ⋊ O ( 1 , d − 1 ) {\displaystyle \operatorname {IO} (1,d-1):=\mathbf {R} ^{1,d-1}\rtimes \operatorname {O} (1,d-1)} with the analogous multiplication ( α , f ) ⋅ ( β , g ) = ( α + f ⋅ β , f ⋅ g ) {\displaystyle (\alpha ,f)\cdot (\beta ,g)=(\alpha +f\cdot \beta ,\;f\cdot g)} . The Lie algebra retains its form, with indices µ and ν now taking values between 0 and d − 1. The alternative representation in terms of Ji and Ki has no analogue in higher dimensions. == See also == Euclidean group Galilean group Representation theory of the Poincaré group Wigner's classification Symmetry in quantum mechanics Pauli–Lubanski pseudovector Particle physics and representation theory Continuous spin particle super-Poincaré algebra == Notes == == References == Wu-Ki Tung (1985). Group Theory in Physics. World Scientific Publishing. ISBN 9971-966-57-3. Weinberg, Steven (1995). The Quantum Theory of Fields. Vol. 1. Cambridge: Cambridge University press. ISBN 978-0-521-55001-7. L.H. Ryder (1996). Quantum Field Theory (2nd ed.). Cambridge University Press. p. 62. ISBN 0-52147-8146.
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Wikipedia:K-Poincaré group#0
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The Poincaré group, named after Henri Poincaré (1905), was first defined by Hermann Minkowski (1908) as the isometry group of Minkowski spacetime. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our understanding of the most basic fundamentals of physics. == Overview == The Poincaré group consists of all coordinate transformations of Minkowski space that do not change the spacetime interval between events. For example, if everything were postponed by two hours, including the two events and the path you took to go from one to the other, then the time interval between the events recorded by a stopwatch that you carried with you would be the same. Or if everything were shifted five kilometres to the west, or turned 60 degrees to the right, you would also see no change in the interval. It turns out that the proper length of an object is also unaffected by such a shift. In total, there are ten degrees of freedom for such transformations. They may be thought of as translation through time or space (four degrees, one per dimension); reflection through a plane (three degrees, the freedom in orientation of this plane); or a "boost" in any of the three spatial directions (three degrees). Composition of transformations is the operation of the Poincaré group, with rotations being produced as the composition of an even number of reflections. In classical physics, the Galilean group is a comparable ten-parameter group that acts on absolute time and space. Instead of boosts, it features shear mappings to relate co-moving frames of reference. In general relativity, i.e. under the effects of gravity, Poincaré symmetry applies only locally. A treatment of symmetries in general relativity is not in the scope of this article. == Poincaré symmetry == Poincaré symmetry is the full symmetry of special relativity. It includes: translations (displacements) in time and space, forming the abelian Lie group of spacetime translations (P); rotations in space, forming the non-abelian Lie group of three-dimensional rotations (J); boosts, transformations connecting two uniformly moving bodies (K). The last two symmetries, J and K, together make the Lorentz group (see also Lorentz invariance); the semi-direct product of the spacetime translations group and the Lorentz group then produce the Poincaré group. Objects that are invariant under this group are then said to possess Poincaré invariance or relativistic invariance. 10 generators (in four spacetime dimensions) associated with the Poincaré symmetry, by Noether's theorem, imply 10 conservation laws: 1 for the energy – associated with translations through time 3 for the momentum – associated with translations through spatial dimensions 3 for the angular momentum – associated with rotations between spatial dimensions 3 for a quantity involving the velocity of the center of mass – associated with hyperbolic rotations between each spatial dimension and time == Poincaré group == The Poincaré group is the group of Minkowski spacetime isometries. It is a ten-dimensional noncompact Lie group. The four-dimensional abelian group of spacetime translations is a normal subgroup, while the six-dimensional Lorentz group is also a subgroup, the stabilizer of the origin. The Poincaré group itself is the minimal subgroup of the affine group which includes all translations and Lorentz transformations. More precisely, it is a semidirect product of the spacetime translations group and the Lorentz group, R 1 , 3 ⋊ O ( 1 , 3 ) , {\displaystyle \mathbf {R} ^{1,3}\rtimes \operatorname {O} (1,3)\,,} with group multiplication ( α , f ) ⋅ ( β , g ) = ( α + f ⋅ β , f ⋅ g ) {\displaystyle (\alpha ,f)\cdot (\beta ,g)=(\alpha +f\cdot \beta ,\;f\cdot g)} . Another way of putting this is that the Poincaré group is a group extension of the Lorentz group by a vector representation of it; it is sometimes dubbed, informally, as the inhomogeneous Lorentz group. In turn, it can also be obtained as a group contraction of the de Sitter group SO(4, 1) ~ Sp(2, 2), as the de Sitter radius goes to infinity. Its positive energy unitary irreducible representations are indexed by mass (nonnegative number) and spin (integer or half integer) and are associated with particles in quantum mechanics (see Wigner's classification). In accordance with the Erlangen program, the geometry of Minkowski space is defined by the Poincaré group: Minkowski space is considered as a homogeneous space for the group. In quantum field theory, the universal cover of the Poincaré group R 1 , 3 ⋊ SL ( 2 , C ) , {\displaystyle \mathbf {R} ^{1,3}\rtimes \operatorname {SL} (2,\mathbf {C} ),} which may be identified with the double cover R 1 , 3 ⋊ Spin ( 1 , 3 ) , {\displaystyle \mathbf {R} ^{1,3}\rtimes \operatorname {Spin} (1,3),} is more important, because representations of SO ( 1 , 3 ) {\displaystyle \operatorname {SO} (1,3)} are not able to describe fields with spin 1/2; i.e. fermions. Here SL ( 2 , C ) {\displaystyle \operatorname {SL} (2,\mathbf {C} )} is the group of complex 2 × 2 {\displaystyle 2\times 2} matrices with unit determinant, isomorphic to the Lorentz-signature spin group Spin ( 1 , 3 ) {\displaystyle \operatorname {Spin} (1,3)} . == Poincaré algebra == The Poincaré algebra is the Lie algebra of the Poincaré group. It is a Lie algebra extension of the Lie algebra of the Lorentz group. More specifically, the proper ( det Λ = 1 {\textstyle \det \Lambda =1} ), orthochronous ( Λ 0 0 ≥ 1 {\textstyle {\Lambda ^{0}}_{0}\geq 1} ) part of the Lorentz subgroup (its identity component), S O ( 1 , 3 ) + ↑ {\textstyle \mathrm {SO} (1,3)_{+}^{\uparrow }} , is connected to the identity and is thus provided by the exponentiation exp ( i a μ P μ ) exp ( i 2 ω μ ν M μ ν ) {\textstyle \exp \left(ia_{\mu }P^{\mu }\right)\exp \left({\frac {i}{2}}\omega _{\mu \nu }M^{\mu \nu }\right)} of this Lie algebra. In component form, the Poincaré algebra is given by the commutation relations: where P {\displaystyle P} is the generator of translations, M {\displaystyle M} is the generator of Lorentz transformations, and η {\displaystyle \eta } is the ( + , − , − , − ) {\displaystyle (+,-,-,-)} Minkowski metric (see Sign convention). The bottom commutation relation is the ("homogeneous") Lorentz group, consisting of rotations, J i = 1 2 ϵ i m n M m n {\textstyle J_{i}={\frac {1}{2}}\epsilon _{imn}M^{mn}} , and boosts, K i = M i 0 {\textstyle K_{i}=M_{i0}} . In this notation, the entire Poincaré algebra is expressible in noncovariant (but more practical) language as [ J m , P n ] = i ϵ m n k P k , [ J i , P 0 ] = 0 , [ K i , P k ] = i η i k P 0 , [ K i , P 0 ] = − i P i , [ J m , J n ] = i ϵ m n k J k , [ J m , K n ] = i ϵ m n k K k , [ K m , K n ] = − i ϵ m n k J k , {\displaystyle {\begin{aligned}[][J_{m},P_{n}]&=i\epsilon _{mnk}P_{k}~,\\[][J_{i},P_{0}]&=0~,\\[][K_{i},P_{k}]&=i\eta _{ik}P_{0}~,\\[][K_{i},P_{0}]&=-iP_{i}~,\\[][J_{m},J_{n}]&=i\epsilon _{mnk}J_{k}~,\\[][J_{m},K_{n}]&=i\epsilon _{mnk}K_{k}~,\\[][K_{m},K_{n}]&=-i\epsilon _{mnk}J_{k}~,\end{aligned}}} where the bottom line commutator of two boosts is often referred to as a "Wigner rotation". The simplification [ J m + i K m , J n − i K n ] = 0 {\textstyle [J_{m}+iK_{m},\,J_{n}-iK_{n}]=0} permits reduction of the Lorentz subalgebra to s u ( 2 ) ⊕ s u ( 2 ) {\textstyle {\mathfrak {su}}(2)\oplus {\mathfrak {su}}(2)} and efficient treatment of its associated representations. In terms of the physical parameters, we have [ H , p i ] = 0 [ H , L i ] = 0 [ H , K i ] = i ℏ c p i [ p i , p j ] = 0 [ p i , L j ] = i ℏ ϵ i j k p k [ p i , K j ] = i ℏ c H δ i j [ L i , L j ] = i ℏ ϵ i j k L k [ L i , K j ] = i ℏ ϵ i j k K k [ K i , K j ] = − i ℏ ϵ i j k L k {\displaystyle {\begin{aligned}\left[{\mathcal {H}},p_{i}\right]&=0\\\left[{\mathcal {H}},L_{i}\right]&=0\\\left[{\mathcal {H}},K_{i}\right]&=i\hbar cp_{i}\\\left[p_{i},p_{j}\right]&=0\\\left[p_{i},L_{j}\right]&=i\hbar \epsilon _{ijk}p_{k}\\\left[p_{i},K_{j}\right]&={\frac {i\hbar }{c}}{\mathcal {H}}\delta _{ij}\\\left[L_{i},L_{j}\right]&=i\hbar \epsilon _{ijk}L_{k}\\\left[L_{i},K_{j}\right]&=i\hbar \epsilon _{ijk}K_{k}\\\left[K_{i},K_{j}\right]&=-i\hbar \epsilon _{ijk}L_{k}\end{aligned}}} The Casimir invariants of this algebra are P μ P μ {\textstyle P_{\mu }P^{\mu }} and W μ W μ {\textstyle W_{\mu }W^{\mu }} where W μ {\textstyle W_{\mu }} is the Pauli–Lubanski pseudovector; they serve as labels for the representations of the group. The Poincaré group is the full symmetry group of any relativistic field theory. As a result, all elementary particles fall in representations of this group. These are usually specified by the four-momentum squared of each particle (i.e. its mass squared) and the intrinsic quantum numbers J P C {\textstyle J^{PC}} , where J {\displaystyle J} is the spin quantum number, P {\displaystyle P} is the parity and C {\displaystyle C} is the charge-conjugation quantum number. In practice, charge conjugation and parity are violated by many quantum field theories; where this occurs, P {\displaystyle P} and C {\displaystyle C} are forfeited. Since CPT symmetry is invariant in quantum field theory, a time-reversal quantum number may be constructed from those given. As a topological space, the group has four connected components: the component of the identity; the time reversed component; the spatial inversion component; and the component which is both time-reversed and spatially inverted. == Other dimensions == The definitions above can be generalized to arbitrary dimensions in a straightforward manner. The d-dimensional Poincaré group is analogously defined by the semi-direct product IO ( 1 , d − 1 ) := R 1 , d − 1 ⋊ O ( 1 , d − 1 ) {\displaystyle \operatorname {IO} (1,d-1):=\mathbf {R} ^{1,d-1}\rtimes \operatorname {O} (1,d-1)} with the analogous multiplication ( α , f ) ⋅ ( β , g ) = ( α + f ⋅ β , f ⋅ g ) {\displaystyle (\alpha ,f)\cdot (\beta ,g)=(\alpha +f\cdot \beta ,\;f\cdot g)} . The Lie algebra retains its form, with indices µ and ν now taking values between 0 and d − 1. The alternative representation in terms of Ji and Ki has no analogue in higher dimensions. == See also == Euclidean group Galilean group Representation theory of the Poincaré group Wigner's classification Symmetry in quantum mechanics Pauli–Lubanski pseudovector Particle physics and representation theory Continuous spin particle super-Poincaré algebra == Notes == == References == Wu-Ki Tung (1985). Group Theory in Physics. World Scientific Publishing. ISBN 9971-966-57-3. Weinberg, Steven (1995). The Quantum Theory of Fields. Vol. 1. Cambridge: Cambridge University press. ISBN 978-0-521-55001-7. L.H. Ryder (1996). Quantum Field Theory (2nd ed.). Cambridge University Press. p. 62. ISBN 0-52147-8146.
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Wikipedia:K-SVD#0
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In applied mathematics, k-SVD is a dictionary learning algorithm for creating a dictionary for sparse representations, via a singular value decomposition approach. k-SVD is a generalization of the k-means clustering method, and it works by iteratively alternating between sparse coding the input data based on the current dictionary, and updating the atoms in the dictionary to better fit the data. It is structurally related to the expectation–maximization (EM) algorithm. k-SVD can be found widely in use in applications such as image processing, audio processing, biology, and document analysis. == k-SVD algorithm == k-SVD is a kind of generalization of k-means, as follows. The k-means clustering can be also regarded as a method of sparse representation. That is, finding the best possible codebook to represent the data samples { y i } i = 1 M {\displaystyle \{y_{i}\}_{i=1}^{M}} by nearest neighbor, by solving min D , X { ‖ Y − D X ‖ F 2 } subject to ∀ i , x i = e k for some k . {\displaystyle \quad \min \limits _{D,X}\{\|Y-DX\|_{F}^{2}\}\qquad {\text{subject to }}\forall i,x_{i}=e_{k}{\text{ for some }}k.} which is nearly equivalent to min D , X { ‖ Y − D X ‖ F 2 } subject to ∀ i , ‖ x i ‖ 0 = 1 {\displaystyle \quad \min \limits _{D,X}\{\|Y-DX\|_{F}^{2}\}\qquad {\text{subject to }}\quad \forall i,\|x_{i}\|_{0}=1} which is k-means that allows "weights". The letter F denotes the Frobenius norm. The sparse representation term x i = e k {\displaystyle x_{i}=e_{k}} enforces k-means algorithm to use only one atom (column) in dictionary D {\displaystyle D} . To relax this constraint, the target of the k-SVD algorithm is to represent the signal as a linear combination of atoms in D {\displaystyle D} . The k-SVD algorithm follows the construction flow of the k-means algorithm. However, in contrast to k-means, in order to achieve a linear combination of atoms in D {\displaystyle D} , the sparsity term of the constraint is relaxed so that the number of nonzero entries of each column x i {\displaystyle x_{i}} can be more than 1, but less than a number T 0 {\displaystyle T_{0}} . So, the objective function becomes min D , X { ‖ Y − D X ‖ F 2 } subject to ∀ i , ‖ x i ‖ 0 ≤ T 0 . {\displaystyle \quad \min \limits _{D,X}\{\|Y-DX\|_{F}^{2}\}\qquad {\text{subject to }}\quad \forall i\;,\|x_{i}\|_{0}\leq T_{0}.} or in another objective form min D , X ∑ i ‖ x i ‖ 0 subject to ∀ i , ‖ Y − D X ‖ F 2 ≤ ϵ . {\displaystyle \quad \min \limits _{D,X}\sum _{i}\|x_{i}\|_{0}\qquad {\text{subject to }}\quad \forall i\;,\|Y-DX\|_{F}^{2}\leq \epsilon .} In the k-SVD algorithm, the D {\displaystyle D} is first fixed and the best coefficient matrix X {\displaystyle X} is found. As finding the truly optimal X {\displaystyle X} is hard, we use an approximation pursuit method. Any algorithm such as OMP, the orthogonal matching pursuit can be used for the calculation of the coefficients, as long as it can supply a solution with a fixed and predetermined number of nonzero entries T 0 {\displaystyle T_{0}} . After the sparse coding task, the next is to search for a better dictionary D {\displaystyle D} . However, finding the whole dictionary all at a time is impossible, so the process is to update only one column of the dictionary D {\displaystyle D} each time, while fixing X {\displaystyle X} . The update of the k {\displaystyle k} -th column is done by rewriting the penalty term as ‖ Y − D X ‖ F 2 = ‖ Y − ∑ j = 1 K d j x j T ‖ F 2 = ‖ ( Y − ∑ j ≠ k d j x j T ) − d k x k T ‖ F 2 = ‖ E k − d k x k T ‖ F 2 {\displaystyle \|Y-DX\|_{F}^{2}=\left\|Y-\sum _{j=1}^{K}d_{j}x_{j}^{\text{T}}\right\|_{F}^{2}=\left\|\left(Y-\sum _{j\neq k}d_{j}x_{j}^{\text{T}}\right)-d_{k}x_{k}^{\text{T}}\right\|_{F}^{2}=\|E_{k}-d_{k}x_{k}^{\text{T}}\|_{F}^{2}} where x k T {\displaystyle x_{k}^{\text{T}}} denotes the k-th row of X. By decomposing the multiplication D X {\displaystyle DX} into sum of K {\displaystyle K} rank 1 matrices, we can assume the other K − 1 {\displaystyle K-1} terms are assumed fixed, and the k {\displaystyle k} -th remains unknown. After this step, we can solve the minimization problem by approximate the E k {\displaystyle E_{k}} term with a r a n k − 1 {\displaystyle rank-1} matrix using singular value decomposition, then update d k {\displaystyle d_{k}} with it. However, the new solution for the vector x k T {\displaystyle x_{k}^{\text{T}}} is not guaranteed to be sparse. To cure this problem, define ω k {\displaystyle \omega _{k}} as ω k = { i ∣ 1 ≤ i ≤ N , x k T ( i ) ≠ 0 } , {\displaystyle \omega _{k}=\{i\mid 1\leq i\leq N,x_{k}^{\text{T}}(i)\neq 0\},} which points to examples { y i } i = 1 N {\displaystyle \{y_{i}\}_{i=1}^{N}} that use atom d k {\displaystyle d_{k}} (also the entries of x i {\displaystyle x_{i}} that is nonzero). Then, define Ω k {\displaystyle \Omega _{k}} as a matrix of size N × | ω k | {\displaystyle N\times |\omega _{k}|} , with ones on the ( i , ω k ( i ) ) th {\displaystyle (i,\omega _{k}(i)){\text{th}}} entries and zeros otherwise. When multiplying x ~ k T = x k T Ω k {\displaystyle {\tilde {x}}_{k}^{\text{T}}=x_{k}^{\text{T}}\Omega _{k}} , this shrinks the row vector x k T {\displaystyle x_{k}^{\text{T}}} by discarding the zero entries. Similarly, the multiplication Y ~ k = Y Ω k {\displaystyle {\tilde {Y}}_{k}=Y\Omega _{k}} is the subset of the examples that are current using the d k {\displaystyle d_{k}} atom. The same effect can be seen on E ~ k = E k Ω k {\displaystyle {\tilde {E}}_{k}=E_{k}\Omega _{k}} . So the minimization problem as mentioned before becomes ‖ E k Ω k − d k x k T Ω k ‖ F 2 = ‖ E ~ k − d k x ~ k T ‖ F 2 {\displaystyle \|E_{k}\Omega _{k}-d_{k}x_{k}^{\text{T}}\Omega _{k}\|_{F}^{2}=\|{\tilde {E}}_{k}-d_{k}{\tilde {x}}_{k}^{\text{T}}\|_{F}^{2}} and can be done by directly using SVD. SVD decomposes E ~ k {\displaystyle {\tilde {E}}_{k}} into U Δ V T {\displaystyle U\Delta V^{\text{T}}} . The solution for d k {\displaystyle d_{k}} is the first column of U, the coefficient vector x ~ k T {\displaystyle {\tilde {x}}_{k}^{\text{T}}} as the first column of V × Δ ( 1 , 1 ) {\displaystyle V\times \Delta (1,1)} . After updating the whole dictionary, the process then turns to iteratively solve X, then iteratively solve D. == Limitations == Choosing an appropriate "dictionary" for a dataset is a non-convex problem, and k-SVD operates by an iterative update which does not guarantee to find the global optimum. However, this is common to other algorithms for this purpose, and k-SVD works fairly well in practice. == See also == Sparse approximation Singular value decomposition Matrix norm k-means clustering Low-rank approximation == References ==
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Wikipedia:K-equivalence#0
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In mathematics, K {\displaystyle {\mathcal {K}}} -equivalence, or contact equivalence, is an equivalence relation between map germs. It was introduced by John Mather in his seminal work in Singularity theory in the 1960s as a technical tool for studying stable maps. Since then it has proved important in its own right. Roughly speaking, two map germs ƒ, g are K {\displaystyle \scriptstyle {\mathcal {K}}} -equivalent if ƒ−1(0) and g−1(0) are diffeomorphic. == Definition == Two map germs f , g : X → ( Y , 0 ) {\displaystyle f,g:X\to (Y,0)} are K {\displaystyle \scriptstyle {\mathcal {K}}} -equivalent if there is a diffeomorphism Ψ : X × Y → X × Y {\displaystyle \Psi :X\times Y\to X\times Y} of the form Ψ(x,y) = (φ(x),ψ(x,y)), satisfying, Ψ ( x , 0 ) = ( φ ( x ) , 0 ) {\displaystyle \Psi (x,0)=(\varphi (x),0)} , and Ψ ( x , f ( x ) ) = ( φ ( x ) , g ( φ ( x ) ) ) {\displaystyle \Psi (x,f(x))=(\varphi (x),g(\varphi (x)))} . In other words, Ψ maps the graph of f to the graph of g, as well as the graph of the zero map to itself. In particular, the diffeomorphism φ maps f−1(0) to g−1(0). The name contact is explained by the fact that this equivalence is measuring the contact between the graph of f and the graph of the zero map. Contact equivalence is the appropriate equivalence relation for studying the sets of solution of equations, and finds many applications in dynamical systems and bifurcation theory, for example. It is easy to see that this equivalence relation is weaker than A-equivalence, in that any pair of A {\displaystyle \scriptstyle {\mathcal {A}}} -equivalent map germs are necessarily K {\displaystyle \scriptstyle {\mathcal {K}}} -equivalent. == KV-equivalence == This modification of K {\displaystyle \scriptstyle {\mathcal {K}}} -equivalence was introduced by James Damon in the 1980s. Here V is a subset (or subvariety) of Y, and the diffeomorphism Ψ above is required to preserve not X × { 0 } {\displaystyle X\times \{0\}} but X × V {\displaystyle X\times V} (that is, y ∈ V ⇒ ψ ( x , y ) ∈ V {\displaystyle y\in V\Rightarrow \psi (x,y)\in V} ). In particular, Ψ maps f−1(V) to g−1(V). == See also == A-equivalence == References == J. Martinet, Singularities of Smooth Functions and Maps, Volume 58 of LMS Lecture Note Series. Cambridge University Press, 1982. J. Damon, The Unfolding and Determinacy Theorems for Subgroups of A {\displaystyle \scriptstyle {\mathcal {A}}} and K {\displaystyle \scriptstyle {\mathcal {K}}} . Memoirs Amer. Math. Soc. 50, no. 306 (1984).
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Wikipedia:K. S. Chandrasekharan#0
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Komaravolu Chandrasekharan (21 November 1920 – 13 April 2017) was a professor at ETH Zurich and a founding faculty member of School of Mathematics, Tata Institute of Fundamental Research (TIFR). He is known for his work in number theory and summability. He received the Padma Shri, the Shanti Swarup Bhatnagar Award, and the Ramanujan Medal, and he was an honorary fellow of TIFR. He was president of the International Mathematical Union (IMU) from 1971 to 1974. == Biography == Chandrasekharan was born on 21 November 1920 in Machilipatnam, Andhra Pradesh. Chandrasekharan completed his high school from Bapatla village in Guntur from Andhra Pradesh. He completed M.A. in mathematics from the Presidency College, Chennai and a PhD from the Department of Mathematics, University of Madras in 1942, under the supervision of K. Ananda Rau. When Chandrasekharan was with the Institute for Advanced Study, Princeton, US, Homi Bhabha invited Chandrashekharan to join the School of Mathematics of the Tata Institute of Fundamental Research (TIFR). Chandrashekharan persuaded mathematicians L. Schwarz, C. L. Siegel and others from all over the world to visit TIFR and deliver lectures. In 1965, Chandrasekharan left the Tata Institute of Fundamental Research to join the ETH Zurich, where he retired in 1988. He was a fellow of the American Mathematical Society. == Selected works == with Salomon Bochner: Fourier Transforms. Princeton University Press. 1949. with S. Minakshisundaram: Typical means. Oxford University Press. 1952. LCCN 53002509. Introduction to analytic number theory. Springer. 1968. LCCN 68021990. reprinting 2012 Arithmetical Functions. Grundlehren der Mathematischen Wissenschaften. Springer. 1970. LCCN 49010722. Elliptic Functions. Springer. 1985. ISBN 9780387152950. LCCN 85009802. Classical Fourier transforms. Springer-Verlag. 1989. ISBN 0387502483. LCCN 88038192. Course on topological groups. Hindustan Book Agency. 2011. ISBN 9789380250205. Course on integration theory. Hindustan Book Agency. 2011. ISBN 9789380250199. LCCN 2012472670. == Notes == == References == – India's who is who == External links == K. S. Chandrasekharan at the Mathematics Genealogy Project Komaravolu Chandrasekharan in Historical Dictionary of Switzerland (German)
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Wikipedia:KPZ fixed point#0
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In probability theory, the KPZ fixed point is a Markov field and conjectured to be a universal limit of a wide range of stochastic models forming the universality class of a non-linear stochastic partial differential equation called the KPZ equation. Even though the universality class was already introduced in 1986 with the KPZ equation itself, the KPZ fixed point was not concretely specified until 2021 when mathematicians Konstantin Matetski, Jeremy Quastel and Daniel Remenik gave an explicit description of the transition probabilities in terms of Fredholm determinants. == Introduction == All models in the KPZ class have in common, that they have a fluctuating height function or some analogue function, that can be thought of as a function, that models the growth of the model by time. The KPZ equation itself is also a member of this class and the canonical model of modelling random interface growth. The strong KPZ universality conjecture conjectures that all models in the KPZ universality class converge under a specific scaling of the height function to the KPZ fixed point and only depend on the initial condition. Matetski-Quastel-Remenik constructed the KPZ fixed point for the ( 1 + 1 ) {\displaystyle (1+1)} -dimensional KPZ universality class (i.e. one space and one time dimension) on the polish space of upper semicontinous functions (UC) with the topology of local UC convergence. They did this by studying a particular model of the KPZ universality class the TASEP („Totally Asymmetric Simple Exclusion Process“) with general initial conditions and the random walk of its associated height function. They achieved this by rewriting the biorthogonal function of the correlation kernel, that appears in the Fredholm determinant formula for the multi-point distribution of the particles in the Weyl chamber. Then they showed convergence to the fixed point. == KPZ fixed point == Let h ( t , x → ) {\displaystyle h(t,{\vec {x}})} denote a height function of some probabilistic model with ( t , x → ) ∈ R × R d {\displaystyle (t,{\vec {x}})\in \mathbb {R} \times \mathbb {R} ^{d}} denoting space-time. So far only the case for d = 1 {\displaystyle d=1} , also noted as ( 1 + 1 ) {\displaystyle (1+1)} , was deeply studied, therefore we fix this dimension for the rest of the article. In the KPZ universality class exist two equilibrium points or fixed points, the trivial Edwards-Wilkinson (EW) fixed point and the non-trivial KPZ fixed point. The KPZ equation connects them together. The KPZ fixed point is rather defined as a height function h ( t , x → ) {\displaystyle {\mathfrak {h}}(t,{\vec {x}})} and not as a particular model with a height function. === KPZ fixed point === The KPZ fixed point ( h ( t , x ) ) t ≥ 0 , x ∈ R {\displaystyle ({\mathfrak {h}}(t,x))_{t\geq 0,x\in \mathbb {R} }} is a Markov process, such that the n-point distribution for x 1 < x 2 < ⋯ < x n ∈ R {\displaystyle x_{1}<x_{2}<\cdots <x_{n}\in \mathbb {R} } and t > 0 {\displaystyle t>0} can be represented as P h ( 0 , ⋅ ) ( h ( t , x 1 ) ≤ a 1 , h ( t , x 2 ) ≤ a 2 , … , h ( t , x n ) ≤ a n ) = det ( I − K ) L 2 ( { x 1 , x 2 , … , x n } × R ) {\displaystyle \mathbb {P} _{{\mathfrak {h}}(0,\cdot )}({\mathfrak {h}}(t,x_{1})\leq a_{1},{\mathfrak {h}}(t,x_{2})\leq a_{2},\dots ,{\mathfrak {h}}(t,x_{n})\leq a_{n})=\det(I-K)_{L^{2}(\{x_{1},x_{2},\dots ,x_{n}\}\times \mathbb {R} )}} where a 1 , … , a n ∈ R {\displaystyle a_{1},\dots ,a_{n}\in \mathbb {R} } and K {\displaystyle K} is a trace class operator called the extended Brownian scattering operator and the subscript means that the process in h ( 0 , ⋅ ) {\displaystyle {\mathfrak {h}}(0,\cdot )} starts. === KPZ universality conjectures === The KPZ conjecture conjectures that the height function h ( t , x → ) {\displaystyle h(t,{\vec {x}})} of all models in the KPZ universality at time t {\displaystyle t} fluctuate around the mean with an order of t 1 / 3 {\displaystyle t^{1/3}} and the spacial correlation of the fluctuation is of order t 2 / 3 {\displaystyle t^{2/3}} . This motivates the so-called 1:2:3 scaling which is the characteristic scaling for the KPZ fixed point. The EW fixed point has also a scaling the 1:2:4 scaling. The fixed points are invariant under their associated scaling. ==== 1:2:3 scaling ==== The 1:2:3 scaling of a height function is for ε > 0 {\displaystyle \varepsilon >0} ε 1 / 2 h ( ε − 3 / 2 t , ε − 1 x ) − C ε t , {\displaystyle \varepsilon ^{1/2}h(\varepsilon ^{-3/2}t,\varepsilon ^{-1}x)-C_{\varepsilon }t,} where 1:3 and 2:3 stand for the proportions of the exponents and C ε {\displaystyle C_{\varepsilon }} is just a constant. ==== Strong conjecture ==== The strong conjecture says, that all models in the KPZ universality class converge under 1:2:3 scaling of the height function if their initial conditions also converge, i.e. lim ε → 0 ε 1 / 2 ( h ( c 1 ε − 3 / 2 t , c 2 ε − 1 x ) − c 3 ε − 3 / 2 t ) = ( d ) h ( t , x ) {\displaystyle \lim \limits _{\varepsilon \to 0}\varepsilon ^{1/2}(h(c_{1}\varepsilon ^{-3/2}t,c_{2}\varepsilon ^{-1}x)-c_{3}\varepsilon ^{-3/2}t)\;{\stackrel {(d)}{=}}\;{\mathfrak {h}}(t,x)} with initial condition h ( 0 , x ) := lim ε → 0 ε 1 / 2 h ( 0 , c 2 ε − 1 x ) , {\displaystyle {\mathfrak {h}}(0,x):=\lim \limits _{\varepsilon \to 0}\varepsilon ^{1/2}h(0,c_{2}\varepsilon ^{-1}x),} where c 1 , c 2 , c 3 {\displaystyle c_{1},c_{2},c_{3}} are constants depending on the model. ==== Weak conjecture ==== If we remove the growth term in the KPZ equation, we get ∂ t h ( t , x ) = ν ∂ x 2 h + σ ξ , {\displaystyle \partial _{t}h(t,x)=\nu \partial _{x}^{2}h+\sigma \xi ,} which converges under the 1:2:4 scaling lim ε → 0 ε 1 / 2 ( h ( c 1 ε − 2 t , c 2 ε − 1 x ) − c 3 ε − 3 / 2 t ) = ( d ) h ( t , x ) {\displaystyle \lim \limits _{\varepsilon \to 0}\varepsilon ^{1/2}(h(c_{1}\varepsilon ^{-2}t,c_{2}\varepsilon ^{-1}x)-c_{3}\varepsilon ^{-3/2}t)\;{\stackrel {(d)}{=}}\;{\mathfrak {h}}(t,x)} to the EW fixed point. The weak conjecture says now, that the KPZ equation is the only Heteroclinic orbit between the KPZ and EW fixed point. === Airy process === If one fixes the time dimension and looks at the limit lim t → ∞ t − 1 / 3 ( h ( c 1 t , c 2 t 2 / 3 x ) − c 3 t ) = ( d ) A ( x ) , {\displaystyle \lim \limits _{t\to \infty }t^{-1/3}(h(c_{1}t,c_{2}t^{2/3}x)-c_{3}t){\stackrel {(d)}{=}}\;{\mathcal {A}}(x),} then one gets the Airy process ( A ( x ) ) x ∈ R {\displaystyle ({\mathcal {A}}(x))_{x\in \mathbb {R} }} which also occurs in the theory of random matrices. == References ==
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Wikipedia:Kaisa Matomäki#0
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Kaisa Sofia Matomäki (born April 30, 1985) is a Finnish mathematician specializing in number theory. Since April 2023, she is a full professor in the Department of Mathematics and Statistics, University of Turku, Turku, Finland. Her research includes results on the distribution of multiplicative functions over short intervals of numbers; for instance, she showed that the values of the Möbius function are evenly divided between +1 and −1 over short intervals. These results, in turn, were among the tools used by Terence Tao to prove the Erdős discrepancy problem. == Awards and honors == Kaisa Matomäki, along with Maksym Radziwill of McGill University, Canada, was awarded the SASTRA Ramanujan Prize for 2016. The Prize was established in 2005 and is awarded annually for outstanding contributions by young mathematicians to areas influenced by Srinivasa Ramanujan. The citation for the 2016 SASTRA Ramanujan Prize is as follows: "Kaisa Matomäki and Maksym Radziwill are jointly awarded the 2016 SASTRA Ramanujan Prize for their deep and far reaching contributions to several important problems in diverse areas of number theory and especially for their spectacular collaboration which is revolutionizing the subject. The prize recognizes that in making significant improvements over the works of earlier stalwarts on long standing problems, they have introduced a number of innovative techniques. The prize especially recognizes their collaboration starting with their 2015 joint paper in Geometric and Functional Analysis which led to their 2016 paper in the Annals of Mathematics in which they obtain amazing results on multiplicative functions in short intervals, and in particular a stunning result on the parity of the Liouville lambda function on almost all short intervals - a paper that is expected to change the subject of multiplicative functions in a major way. The prize notes also the very recent joint paper of Matomäki, Radziwill and Tao announcing a significant advance in the case k = 3 towards a conjecture of Chowla on the values of the lambda function on sets of k consecutive integers. Finally the prize notes, that Matomäki and Radziwill, through their impressive array of deep results and the powerful new techniques they have introduced, will strongly influence the development of analytic number theory in the future." With Radziwill, she is one of five winners of the 2019 New Horizons Prize for Early-Career Achievement in Mathematics, associated with the Breakthrough Prize in Mathematics. She is one of the 2020 winners of the EMS Prize. She was awarded the 2021 Ruth Lyttle Satter Prize by the American Mathematical Society "for her work (much of it joint with Maksym Radziwiłł) opening up the field of multiplicative functions in short intervals in a completely unexpected and very fruitful way, and in particular in their breakthrough paper, 'Multiplicative Functions in Short Intervals' (Annals of Mathematics 183 2016, 1015–1056)." For 2023 she received the Cole Prize in Number Theory of the AMS. She was elected to the Academia Europaea in 2021. == Education and career == Kaisa Matomäki was born in Nakkila, Finland, on 30 April 1985. She attended high school in Valkeakoski, Finland and won the First Prize in the national mathematics competition for Finnish high school students. She did her Masters at the University of Turku and received the Ernst Lindelof Award for the best Masters Thesis in mathematics in Finland in 2005. After completing her PhD at the Royal Holloway College of the University of London in 2009 under the direction of Professor Glyn Harman, she returned to Turku where she worked as an associate professor and as Academy Research Fellow. She was made a full professor in April 2023. == Personal life == Kaisa Matomäki is married to Pekka Matomäki, who is also a mathematician specializing in applied mathematics. They have three children. Currently they live in Lieto, close to Turku. == References == == External links == Homepage of Kaisa Matomäki "Video on Kaisa Matomäki created by Academy of Finland". 25 November 2016 – via YouTube.
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Wikipedia:Kaisa Nyberg#0
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Kaisa Nyberg is a Finnish cryptographer and computer security researcher. == Contributions == Nyberg's research includes the theory of perfect nonlinear S-boxes[N] (now known as Nyberg S-boxes), provably secure block cipher design (resulting in KN-Cipher,[NK] and the cryptanalysis of the stream ciphers E0[HN] and SNOW[B+]). == Education and career == Nyberg received her Ph.D. in mathematics in 1980 from the University of Helsinki. Her dissertation, On Subspaces of Products of Nuclear Fréchet Spaces, was in topology, and was supervised by Edward Leonard Dubinsky. Nyberg began doing cryptography research for the Finnish Defence Forces in 1987, and moved to Nokia in 1998. She became professor of cryptology at Aalto University School of Science in 2005, and retired as a professor emerita in 2016. == Recognition == Nyberg was the 2015 recipient of the Magnus Ehrnrooths Foundation Prize of the Finnish Society of Arts and Letters. == Selected publications == == References == == Further reading == "Billions of individuals depend on security protocols crafted by Professor Emerita Kaisa Nyberg | Aalto University". www.aalto.fi. 2023-11-10. Retrieved 2023-11-10. == External links == Kaisa Nyberg's page at TKK Kaisa Nyberg publications indexed by Google Scholar
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Wikipedia:Kaj Nyström#0
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Kaj Nyström is a Swedish mathematician currently at Uppsala University and was awarded the Göran Gustafsson Prize by the Royal Swedish Academy of Sciences. == References ==
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Wikipedia:Kamel Bidari#0
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Kamel Bidari (born 1 February 1960) is the Algerian Minister of Higher Education and Scientific Research. He was appointed as minister on 9 September 2022. He was born in 1960 in Biskra, Algeria. He is a physicist and mathematician occupying the position of rector of the University of M'Sila, Algeria . In 2016 he was the president-representative of the Algerian Minister of Higher Education and Scientific Research, of the national steering and monitoring committee for the implementation of the pedagogical accompaniment program for the benefit of the teacher-researcher in Algeria. He has been a university professor since 1998, he was considered the youngest professor in Algeria in 1998 and the youngest holder of State Doctorate in 1987. From 1997 to 2001, he was designated associate professor at University of Oklahoma (United States), responsible for the supervision and training in mathematical model simulation of engineers in field development of SONTRACH company at the Algerian Petroleum Institute (Boumerdes-Algeria). He developed a complex parameter, combining several physical fields recorded during the rupture of large blocks of rock, precursor of the different stages of deformation of these solid bodies under high pressure and temperatures conditions. == Education == Bidari holds a Bachelor of Science (1978) from the Lycée Ben Mhidi, a Master of Geophysics (1983) from the University of Boumerdès and a PhD of Physics and Mathematics (1987) from the Academy of Sciences of the Soviet Union. In 1978, he obtained the scientific (Secondary School) Degree from Larbi Ben M’Hidi — Biskra high school. He has been graduated from the National Institute of Hydrocarbons and Chemistry (INH) in Boumerdes-Algeria (first of the student promotion of 1983), when he obtained «State Engineer» Degree in Geophysics. From 1983 to 1987, he prepared a Doctorate of Philosophy (Ph.D) in physics and mathematics at the Russian geological prospecting university and the Institute of Physics of the Earth- Academy of Sciences of USSR in Moscow. In 1988, he obtained the equivalent degree of the Algerian State Doctorate (Ministry of Higher Education and Scientific Research) — Algeria. In 1991, he received the Doctorate Nauk diploma in Physics and Mathematics (doktor physico mathematicheskikh nauk recognized for Soviet talented Scientist) for a thesis realized in Russian geological prospecting university and the Institute of Physics of the Earth - Academy of Sciences of the USSR Moscow about the complex studies of physical precursors of failure of rocks in relation to earthquake prediction, the fractal structure of geophysical field, and the simulation of seismic process. He obtained this highest scientific title for his contribution to the development of science. == Career == From 1990 until 1993, Bidari worked as head of department of Geophysics of Random Phenomena at the University of Boumerdès. Between July 1998 and September 2000, he was the Vice-Rector of the University of Boumerdès. His transversal knowledge led him to contribute to strengthening the higher education. He was the head of several institutions: from 2017 to now: Rector of Mohamed Boudiaf university — M’Sila (Algeria); from 2016 to now: Chairman of the national commission for the training of higher education teachers (Algeria); 08/2016 - 06/2017: Director of Studies, the Minister's advisor of Higher Education (Algeria); 2012 - 2016: Rector of Mohand Akli Oulhadj University — Bouira (Algeria); 2000 - 2012: Dean of the Faculty of Science at University of Boumerdes (Algeria); 07/1998 - 09/2000: Vice-Rector of Mhamed Bougara University - Boumerdes (Algeria); 1990 - 1993: Head of department of geophysics and random phenomena (National Institute of Hydrocarbons and Chemistry INH)— Boumerdes (Algeria); 1987 - 1990: Head of department of physics and mathematics - INH (Algeria). In 1988 he founded the laboratory of Physics of the Earth - INH (Algeria). == Publications == Baddari has been author of numerous scientific publications in field of physics and mathematics, such as: === Scientific papers === Baddari K, Sobolev Guennadi A, Frolov Anatoly D (1988) Complex failure precursors of large scale rock bloks (In Russian). Dokladi Academy of Sciences USSR, 299, 5, 1087-1092. Baddari K, Sobolev Guennadi A, Frolov Anatoly D (1996) Similarity in seismic precursors at different scales. C.R Acad. Sc. Paris, t323, série IIa ? 755-762 Baddari K, Djarfour N, Aïfa T, Ferahtia J (2010) Acoustic impedance inversion by feedback artificial neural network. Journal of Petroleum Science and Engineering 71 106–111. Baddari K, Sobolev Guennadi A, Frolov Anatoli D, Ponomarev Alexander V (1999) An integrated study of physical precursors of failure in relation with earthquake prediction, using large scale rock blocks. Annali Di Geofisica Vol. 42, N°5, 771-787 Djarfour N, Aïfa T, Baddari K, Mihoubi A, Ferahtia J (2008) Application of feedback connection artificial neural network to seismic data filtering. C. R. Geoscience 340 335–344. Ferahtia J, Djarfour N, Baddari K, Guérin R (2009) Application of signal dependent rank-order mean filter to the removal of noise spikes from 2D electrical resistivity imaging data. Near Surface Geophysics 159-169. Saad Saoud L, Rahmoune F, Tourtchine V, Baddari K (2013) Complex-valued forecasting of the global solar irradiation. J. Renewable and Sustainable Energy 5, 043124. Baddari, Kamel; Frolov, Anatoly D.; Tourtchine, Victor; Rahmoune, Fayçal; Makdeche, Said (2015). "Effect of Stress-Strain Conditions on Physical Precursors and Failure Stages Development in Rock Samples". Acta Geophysica. 63 (1): 62–102. Bibcode:2015AcGeo..63...62B. doi:10.2478/s11600-014-0206-9. S2CID 129228813. Baddari, Kamel; Frolov, Anatoly D.; Tourtchine, Victor; Makdeche, Said; Rahmoune, Fayçal (2012). "Effect of temperature on the physical precursors of rock block failure". Acta Geophysica. 60 (4): 1007–1029. Bibcode:2012AcGeo..60.1007B. doi:10.2478/s11600-012-0038-4. S2CID 140588697. Baddari K, Frolov Anatoly D (1997) Modeling of the fractal structure of a geophysical field. C.R Acad Sc Paris, Earth and Planetary Sciences, 325, 925-930. Baddari, Kamel; Makdeche, Said; Bellalem, Fouzi (2013). "Probabilistic model to forecast earthquakes in the Zemmouri (Algeria) seismoactive area on the basis of moment magnitude scale distribution functions". Acta Geophysica. 61 (1): 60–83. Bibcode:2013AcGeo..61...60B. doi:10.2478/s11600-012-0064-2. S2CID 129934223. Kamel Baddari; Anatoly d. Frolov (2011). "Regularities in discrete hierarchy seismo-acoustic mode in a geophysical field". Annals of Geophysics. 53 (5–6). doi:10.4401/ag-4725. Baddari K, Ferahtia J, Aifa T, Djarfour N (2011) Seismic noise attenuation by means of ananisotropic non-linear diffusion filter. Computers & Geosciences 37 456–463. Djarfour N, Ferahtia J, Babaia F, Baddari K, El-adj S, Farfour M (2014) Seismic noise filtering basedon Generalized Regression Neural Networks. Computers & Geosciences 69 1–9 Bouchelaghem H.E, Hamadouche M, Soltani F, Baddari K (2016) Adaptive clutter-map CFAR detection in distributed sensor networks,” AEU-Int. J. Electronics Commun. 70, No. 9, 1288. Baouche R, Aïfa T, Baddari K (2017) Intelligent methods for predicting nuclear magnetic resonance of porosity and permeability by conventional well-logs: a case study of Saharan field. Arabian Journal of Geosciences 10:545. Baouche R, Aïfa T, Baddari K (2017) Intelligent methods for predicting nuclear magnetic resonance of porosity and permeability by conventional well-logs: a case study of Saharan field. Arabian Journal of Geosciences 10:545. Baouche R, Baddari K (2017) Prediction of permeability and porosity from well log data using the nonparametric regression with multivariate analysis and neural network, Hassi R’Mel Field, Algeria. Egyptian Journal of Petroleum 26, 7763-778. Bouchelaghem H. E, Hamadouche M, Soltani F, Baddari K (2019) Distributed Clutter-Map Constant False Alarm Rate Detection Using Fuzzy Fusion Rules (2019) Radioelectronics and Communications Systems, Vol. 62, No. 1, 1–5. © Allerton Press, Inc. I.Ali Zerrouki A, Geraud Y, Diraison M, Baddari K (2019) A Preliminary study of relationships between thermal conductivity and petrophysical parameters in Hamra Quartzites reservoir, Hassi Messaoud field (Algeria). J. African Earth Sciences, Vol 151, 461-471. Baddari Kamel, Aifa T., Djarfour N., Ferahtia J. (2009): Application of a radial basis function artificial neural network to seismic data inversion . Computer and Geosciences, 35, pages 2338–2344. Baddari Kamel, Aifa T., Djarfour N., Ferahtia J. : Acoustic impedance inversion by feedback artificial neural network – Petrol J.—Zci. Eng. 76. Baddari Kamel, Guerin R., Djarfour N., Ferahtia J. : Application of signal depen- dant rank-order mean filter to the removal of noise spikes from 2D electrical resistivity imaging data. Near surface geophysics, 7 (3), pages 159 – 169. Baddari, K., Frolov, A. D., Tourtchine, V., & Rahmoune, F. (2011). An integrated study of the dynamics of electromagnetic and acoustic regimes during failure of complex macrosystems using rock blocks. Rock Mechanics and Rock Engineering, 44(3), 269-280. Zerrouki, A.A, Geraud, Y, Diraison, M Baddari, K Preliminary study of relationships between thermal conductivity and petrophysical parameters in Hamra "A Preliminary study of relationships between thermal conductivity and petrophysical parameters in Hamra Quartzites reservoir, Hassi Messaoud field (Algeria)". Journal of African Earth Sciences. === Books === Kamel is the author numerous books: Physique de la terre. OPU. 388 pages Equations de la physique mathématiques appliquées. OPU – 404 pages. Théorie et pratique des fonctions d’une variable complexe. OPU- 350 pages. Filtrage Analogique-Numérique. Dar El Djazairia Alger, les Seismes Et Leur Prévision. Opu Alger 2002, 355 P, assurance Qualité Dans L'enseignement Supérieur - Conduire Et Réussir L'autoévaluation. Opu Alger 2013, 146 P. Comprendre et pratiquer le LMD, OPU, 140p. Conduire et réussir une au- toévaluation institutionnelle – OPU – 198p. Indicateurs de qualité dans l’E.S. - OPU (2012) Bien enseigner avec le LMD – OPU – (2014) La recherché d’information – OPU – (2016) == References == == External links == The minister of higher education in ALGERIA
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Wikipedia:Kanakkusaram#0
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Kaṇakkusāraṃ is a comprehensive treatise in Manipravalam (a hybrid language which combines Sanskrit lexicon and Malayalam morpho-syntax), composed in the 16th-17th century CE, dealing with elementary arithmetic and methods for solving arithmetical problems arising in the everyday life of members of an agricultural community. The book was composed by one Nīlakṇṭha; but other than the name, nothing more definite is known about the author. The date of composition of the book has also not been determined. The significance of the work is that it was composed in a variant of the local vernacular language addressing the needs of ordinary people and it is one of the earliest such work in Malayalam. In comparison, Yuktibhasha, composed in pure Malayalam and published in the middle of the sixteenth century, is an advanced text on mathematics and astronomy addressed to the advanced students and scholars of mathematics. The book is composed in verses and it also contains a commentary both in Manipravalam language. In the book, the author has claimed that the work is based on Sanskrit texts like Bhaskara II's Līlāvatī and older Malayalam texts on mathematics like Kaṇakkadhikāraṃ. There are two versions of the work: one of them is titled simply Kaṇakkusāraṃ while the other is titled Kaṇakkusāraṃ (Bālaprabodhaṃ). Both have been published by Government Oriental Manuscript Library, Madras (Chennai) in 1950 critically edited with introductions by C. Achyutha Menon. There are some differences in content between the two versions. According to Achyutha Menon the versions are likely to be of different authorship though both might have borrowed material from a common source. == Contents == The original manuscripts of both versions of Kaṇakkusāraṃ had no division into chapters. It was one continuous text from beginning to end. However, at the time of publication of the work by Government Oriental Manuscript Library, the editor of the book C. Achyutha Menon divided it into chapters based on the nature of contents. Chapter 1: Names for numbers less than one and greater than one; measurements of volume, weight, time Chapter 2: Rule of three (mathematics) Chapter 3: Rule of five Chapter 4: Rules of seven and nine Chapter 5: Calculations in respect of gold Chapter 6: Calculations in respect of timber, wages for spade work, grain transactions, house building, land tenure, masonry, ground measurement == External links == Both versions of Kaṇakkusāraṃ are available for free download from Internet Archive. Kanakkusaram. Madras (Chennai): Government Oriental Manuscripts Library. 1950. Retrieved 12 December 2023. (critically edited with introduction by C. Achyutha Menon) Kanakkusaram (Balaprabodham). Madras (Chennai): Government Manuscripts Library. 1950. Retrieved 12 December 2023. (critically edited with introduction by C. Achyutha Menon) == See also == Asthana Kolahalam Kaṇita Tīpikai Kaṇakkatikāram == References ==
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Wikipedia:Kanta Gupta#0
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Nolini Kanta Gupta (13 January 1889 – 7 February 1984) was a revolutionary, linguist, scholar, critic, poet, philosopher and yogi, and the most senior of Sri Aurobindo's disciples. He was born in Faridpur, East Bengal, to a cultured and prosperous Vaidya-Brahmin family. While in his teens, he came under the influence of Sri Aurobindo, then a well known revolutionary fighting for independence against the British. When in his fourth year at Presidency College, Calcutta, he left a promising academic career and rejected a lucrative government job to join a small revolutionary group under Sri Aurobindo. In May 1908 he was among those arrested for conspiracy in the Alipore bomb case. Acquitted a year later, after having spent a year in jail, he worked as a sub-editor for the Dharma and the Karmayogin, two of Sri Aurobindo's Nationalist newspapers, in 1909 and 1910. He was taught Greek, Latin, French and Italian by Sri Aurobindo himself and was among the four disciples who were with Aurobindo in 1910 at Pondicherry. When the Sri Aurobindo Ashram was founded in 1926, he settled permanently in Pondicherry, serving the Mother and Sri Aurobindo as secretary of the ashram and later as one of its trustees. A prolific writer on a wide range of topics, he has about 60 books to his credit of which about 16 are in English and 44 in Bengali, as well as many articles and poems in English, Bengali and French. Nolini Kanta Gupta died at the Sri Aurobindo Ashram on 7 February 1984. == Bibliography == Collected Works of Nolini Kanta Gupta (8 volumes), Sri Aurobindo Ashram, Pondicherry 1. The Coming Race 2. Essays on Poetry and Mysticism 3. The Yoga of Sri Aurobindo 4-5. Light of Lights (poems) 6-7. Sweet Mother 8. Vedic Hymns Reminiscences (with K. Amrita) Evolution and the Earthly Destiny About Woman (a compilation, ed. by Sacar) Tributes to Nolini Kanta Gupta ed. by Nirodbaran Nolini: Arjuna of our Age by Dr. V.M. Reddy Lights from Nolini Kanta Gupta == References == == Citations == === Bibliography === Mohanty, Sachidananda (2008). Sri Aurobindo: A Contemporary Reade (1 ed.). New Delhi: Routledge. ISBN 978-0-415-46093-4. == External links == A Commentary on the First Six Suktas of Rigveda An Introduction to the Vedas Yama A commentary on the Katha Upanishad Nolini Kanta Gupta including a brief life-sketch Nolini page NOLINI KANTA GUPTA PHOTO COLLECTION
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Wikipedia:Kantorovich inequality#0
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In mathematics, the Kantorovich inequality is a particular case of the Cauchy–Schwarz inequality, which is itself a generalization of the triangle inequality. The triangle inequality states that the length of two sides of any triangle, added together, will be equal to or greater than the length of the third side. In simplest terms, the Kantorovich inequality translates the basic idea of the triangle inequality into the terms and notational conventions of linear programming. (See vector space, inner product, and normed vector space for other examples of how the basic ideas inherent in the triangle inequality—line segment and distance—can be generalized into a broader context.) More formally, the Kantorovich inequality can be expressed this way: Let p i ≥ 0 , 0 < a ≤ x i ≤ b for i = 1 , … , n . {\displaystyle p_{i}\geq 0,\quad 0<a\leq x_{i}\leq b{\text{ for }}i=1,\dots ,n.} Let A n = { 1 , 2 , … , n } . {\displaystyle A_{n}=\{1,2,\dots ,n\}.} Then ( ∑ i = 1 n p i x i ) ( ∑ i = 1 n p i x i ) ≤ ( a + b ) 2 4 a b ( ∑ i = 1 n p i ) 2 − ( a − b ) 2 4 a b ⋅ min { ( ∑ i ∈ X p i − ∑ j ∈ Y p j ) 2 : X ∪ Y = A n , X ∩ Y = ∅ } . {\displaystyle {\begin{aligned}&{}\qquad \left(\sum _{i=1}^{n}p_{i}x_{i}\right)\left(\sum _{i=1}^{n}{\frac {p_{i}}{x_{i}}}\right)\\&\leq {\frac {(a+b)^{2}}{4ab}}\left(\sum _{i=1}^{n}p_{i}\right)^{2}-{\frac {(a-b)^{2}}{4ab}}\cdot \min \left\{\left(\sum _{i\in X}p_{i}-\sum _{j\in Y}p_{j}\right)^{2}\,:\,{X\cup Y=A_{n}},{X\cap Y=\varnothing }\right\}.\end{aligned}}} The Kantorovich inequality is used in convergence analysis; it bounds the convergence rate of Cauchy's steepest descent. Equivalents of the Kantorovich inequality have arisen in a number of different fields. For instance, the Cauchy–Schwarz–Bunyakovsky inequality and the Wielandt inequality are equivalent to the Kantorovich inequality and all of these are, in turn, special cases of the Hölder inequality. The Kantorovich inequality is named after Soviet economist, mathematician, and Nobel Prize winner Leonid Kantorovich, a pioneer in the field of linear programming. There is also Matrix version of the Kantorovich inequality due to Marshall and Olkin (1990). Its extensions and their applications to statistics are available; see e.g. Liu and Neudecker (1999) and Liu et al. (2022). == See also == Cauchy–Schwarz inequality – Mathematical inequality relating inner products and norms == References == Weisstein, Eric W. "Kantorovich Inequality". MathWorld. Cauchy-Schwarz inequality at PlanetMath. Mathematical Programming Glossary entry on "Kantorovich inequality" Marshall, A. W. and Olkin, I., Matrix versions of the Cauchy and Kantorovich inequalities. Aequationes Mathematicae 40 (1990) 89–93. Liu, Shuangzhe and Neudecker, Heinz, A survey of Cauchy-Schwarz and Kantorovich-type matrix inequalities. Statistical Papers 40 (1999) 55-73. Liu, Shuangzhe, Leiva, Víctor, Zhuang, Dan, Ma, Tiefeng and Figueroa-Zúñiga, Jorge I., Matrix differential calculus with applications in the multivariate linear model and its diagnostics. Journal of Multivariate Analysis 188 (2022) 104849. https://doi.org/10.1016/j.jmva.2021.104849 == External links == Biography of Leonid Vitalyevich Kantorovich
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Wikipedia:Kantorovich theorem#0
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The Kantorovich theorem, or Newton–Kantorovich theorem, is a mathematical statement on the semi-local convergence of Newton's method. It was first stated by Leonid Kantorovich in 1948. It is similar to the form of the Banach fixed-point theorem, although it states existence and uniqueness of a zero rather than a fixed point. Newton's method constructs a sequence of points that under certain conditions will converge to a solution x {\displaystyle x} of an equation f ( x ) = 0 {\displaystyle f(x)=0} or a vector solution of a system of equation F ( x ) = 0 {\displaystyle F(x)=0} . The Kantorovich theorem gives conditions on the initial point of this sequence. If those conditions are satisfied then a solution exists close to the initial point and the sequence converges to that point. == Assumptions == Let X ⊂ R n {\displaystyle X\subset \mathbb {R} ^{n}} be an open subset and F : X ⊂ R n → R n {\displaystyle F:X\subset \mathbb {R} ^{n}\to \mathbb {R} ^{n}} a differentiable function with a Jacobian F ′ ( x ) {\displaystyle F^{\prime }(\mathbf {x} )} that is locally Lipschitz continuous (for instance if F {\displaystyle F} is twice differentiable). That is, it is assumed that for any x ∈ X {\displaystyle x\in X} there is an open subset U ⊂ X {\displaystyle U\subset X} such that x ∈ U {\displaystyle x\in U} and there exists a constant L > 0 {\displaystyle L>0} such that for any x , y ∈ U {\displaystyle \mathbf {x} ,\mathbf {y} \in U} ‖ F ′ ( x ) − F ′ ( y ) ‖ ≤ L ‖ x − y ‖ {\displaystyle \|F'(\mathbf {x} )-F'(\mathbf {y} )\|\leq L\;\|\mathbf {x} -\mathbf {y} \|} holds. The norm on the left is the operator norm. In other words, for any vector v ∈ R n {\displaystyle \mathbf {v} \in \mathbb {R} ^{n}} the inequality ‖ F ′ ( x ) ( v ) − F ′ ( y ) ( v ) ‖ ≤ L ‖ x − y ‖ ‖ v ‖ {\displaystyle \|F'(\mathbf {x} )(\mathbf {v} )-F'(\mathbf {y} )(\mathbf {v} )\|\leq L\;\|\mathbf {x} -\mathbf {y} \|\,\|\mathbf {v} \|} must hold. Now choose any initial point x 0 ∈ X {\displaystyle \mathbf {x} _{0}\in X} . Assume that F ′ ( x 0 ) {\displaystyle F'(\mathbf {x} _{0})} is invertible and construct the Newton step h 0 = − F ′ ( x 0 ) − 1 F ( x 0 ) . {\displaystyle \mathbf {h} _{0}=-F'(\mathbf {x} _{0})^{-1}F(\mathbf {x} _{0}).} The next assumption is that not only the next point x 1 = x 0 + h 0 {\displaystyle \mathbf {x} _{1}=\mathbf {x} _{0}+\mathbf {h} _{0}} but the entire ball B ( x 1 , ‖ h 0 ‖ ) {\displaystyle B(\mathbf {x} _{1},\|\mathbf {h} _{0}\|)} is contained inside the set X {\displaystyle X} . Let M {\displaystyle M} be the Lipschitz constant for the Jacobian over this ball (assuming it exists). As a last preparation, construct recursively, as long as it is possible, the sequences ( x k ) k {\displaystyle (\mathbf {x} _{k})_{k}} , ( h k ) k {\displaystyle (\mathbf {h} _{k})_{k}} , ( α k ) k {\displaystyle (\alpha _{k})_{k}} according to h k = − F ′ ( x k ) − 1 F ( x k ) α k = M ‖ F ′ ( x k ) − 1 ‖ ‖ h k ‖ x k + 1 = x k + h k . {\displaystyle {\begin{alignedat}{2}\mathbf {h} _{k}&=-F'(\mathbf {x} _{k})^{-1}F(\mathbf {x} _{k})\\[0.4em]\alpha _{k}&=M\,\|F'(\mathbf {x} _{k})^{-1}\|\,\|\mathbf {h} _{k}\|\\[0.4em]\mathbf {x} _{k+1}&=\mathbf {x} _{k}+\mathbf {h} _{k}.\end{alignedat}}} == Statement == Now if α 0 ≤ 1 2 {\displaystyle \alpha _{0}\leq {\tfrac {1}{2}}} then a solution x ∗ {\displaystyle \mathbf {x} ^{*}} of F ( x ∗ ) = 0 {\displaystyle F(\mathbf {x} ^{*})=0} exists inside the closed ball B ¯ ( x 1 , ‖ h 0 ‖ ) {\displaystyle {\bar {B}}(\mathbf {x} _{1},\|\mathbf {h} _{0}\|)} and the Newton iteration starting in x 0 {\displaystyle \mathbf {x} _{0}} converges to x ∗ {\displaystyle \mathbf {x} ^{*}} with at least linear order of convergence. A statement that is more precise but slightly more difficult to prove uses the roots t ∗ ≤ t ∗ ∗ {\displaystyle t^{\ast }\leq t^{**}} of the quadratic polynomial p ( t ) = ( 1 2 L ‖ F ′ ( x 0 ) − 1 ‖ − 1 ) t 2 − t + ‖ h 0 ‖ {\displaystyle p(t)=\left({\tfrac {1}{2}}L\|F'(\mathbf {x} _{0})^{-1}\|^{-1}\right)t^{2}-t+\|\mathbf {h} _{0}\|} , t ∗ / ∗ ∗ = 2 ‖ h 0 ‖ 1 ± 1 − 2 α 0 {\displaystyle t^{\ast /**}={\frac {2\|\mathbf {h} _{0}\|}{1\pm {\sqrt {1-2\alpha _{0}}}}}} and their ratio θ = t ∗ t ∗ ∗ = 1 − 1 − 2 α 0 1 + 1 − 2 α 0 . {\displaystyle \theta ={\frac {t^{*}}{t^{**}}}={\frac {1-{\sqrt {1-2\alpha _{0}}}}{1+{\sqrt {1-2\alpha _{0}}}}}.} Then a solution x ∗ {\displaystyle \mathbf {x} ^{*}} exists inside the closed ball B ¯ ( x 1 , θ ‖ h 0 ‖ ) ⊂ B ¯ ( x 0 , t ∗ ) {\displaystyle {\bar {B}}(\mathbf {x} _{1},\theta \|\mathbf {h} _{0}\|)\subset {\bar {B}}(\mathbf {x} _{0},t^{*})} it is unique inside the bigger ball B ( x 0 , t ∗ ∗ ) {\displaystyle B(\mathbf {x} _{0},t^{*\ast })} and the convergence to the solution of F {\displaystyle F} is dominated by the convergence of the Newton iteration of the quadratic polynomial p ( t ) {\displaystyle p(t)} towards its smallest root t ∗ {\displaystyle t^{\ast }} , if t 0 = 0 , t k + 1 = t k − p ( t k ) p ′ ( t k ) {\displaystyle t_{0}=0,\,t_{k+1}=t_{k}-{\tfrac {p(t_{k})}{p'(t_{k})}}} , then ‖ x k + p − x k ‖ ≤ t k + p − t k . {\displaystyle \|\mathbf {x} _{k+p}-\mathbf {x} _{k}\|\leq t_{k+p}-t_{k}.} The quadratic convergence is obtained from the error estimate ‖ x n + 1 − x ∗ ‖ ≤ θ 2 n ‖ x n + 1 − x n ‖ ≤ θ 2 n 2 n ‖ h 0 ‖ . {\displaystyle \|\mathbf {x} _{n+1}-\mathbf {x} ^{*}\|\leq \theta ^{2^{n}}\|\mathbf {x} _{n+1}-\mathbf {x} _{n}\|\leq {\frac {\theta ^{2^{n}}}{2^{n}}}\|\mathbf {h} _{0}\|.} == Corollary == In 1986, Yamamoto proved that the error evaluations of the Newton method such as Doring (1969), Ostrowski (1971, 1973), Gragg-Tapia (1974), Potra-Ptak (1980), Miel (1981), Potra (1984), can be derived from the Kantorovich theorem. == Generalizations == There is a q-analog for the Kantorovich theorem. For other generalizations/variations, see Ortega & Rheinboldt (1970). == Applications == Oishi and Tanabe claimed that the Kantorovich theorem can be applied to obtain reliable solutions of linear programming. == References == == Further reading == John H. Hubbard and Barbara Burke Hubbard: Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach, Matrix Editions, ISBN 978-0-9715766-3-6 (preview of 3. edition and sample material including Kant.-thm.) Yamamoto, Tetsuro (2001). "Historical Developments in Convergence Analysis for Newton's and Newton-like Methods". In Brezinski, C.; Wuytack, L. (eds.). Numerical Analysis : Historical Developments in the 20th Century. North-Holland. pp. 241–263. ISBN 0-444-50617-9.
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Wikipedia:Kapteyn series#0
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Kapteyn series is a series expansion of analytic functions on a domain in terms of the Bessel function of the first kind. Kapteyn series are named after Willem Kapteyn, who first studied such series in 1893. Let f {\displaystyle f} be a function analytic on the domain D a = { z ∈ C : Ω ( z ) = | z exp 1 − z 2 1 + 1 − z 2 | ≤ a } {\displaystyle D_{a}=\left\{z\in \mathbb {C} :\Omega (z)=\left|{\frac {z\exp {\sqrt {1-z^{2}}}}{1+{\sqrt {1-z^{2}}}}}\right|\leq a\right\}} with a < 1 {\displaystyle a<1} . Then f {\displaystyle f} can be expanded in the form f ( z ) = α 0 + 2 ∑ n = 1 ∞ α n J n ( n z ) ( z ∈ D a ) , {\displaystyle f(z)=\alpha _{0}+2\sum _{n=1}^{\infty }\alpha _{n}J_{n}(nz)\quad (z\in D_{a}),} where α n = 1 2 π i ∮ Θ n ( z ) f ( z ) d z . {\displaystyle \alpha _{n}={\frac {1}{2\pi i}}\oint \Theta _{n}(z)f(z)dz.} The path of the integration is the boundary of D a {\displaystyle D_{a}} . Here Θ 0 ( z ) = 1 / z {\displaystyle \Theta _{0}(z)=1/z} , and for n > 0 {\displaystyle n>0} , Θ n ( z ) {\displaystyle \Theta _{n}(z)} is defined by Θ n ( z ) = 1 4 ∑ k = 0 [ n 2 ] ( n − 2 k ) 2 ( n − k − 1 ) ! k ! ( n z 2 ) 2 k − n {\displaystyle \Theta _{n}(z)={\frac {1}{4}}\sum _{k=0}^{\left[{\frac {n}{2}}\right]}{\frac {(n-2k)^{2}(n-k-1)!}{k!}}\left({\frac {nz}{2}}\right)^{2k-n}} Kapteyn's series are important in physical problems. Among other applications, the solution E {\displaystyle E} of Kepler's equation M = E − e sin E {\displaystyle M=E-e\sin E} can be expressed via a Kapteyn series: E = M + 2 ∑ n = 1 ∞ sin ( n M ) n J n ( n e ) . {\displaystyle E=M+2\sum _{n=1}^{\infty }{\frac {\sin(nM)}{n}}J_{n}(ne).} == Relation between the Taylor coefficients and the αn coefficients of a function == Let us suppose that the Taylor series of f {\displaystyle f} reads as f ( z ) = ∑ n = 0 ∞ a n z n . {\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}z^{n}.} Then the α n {\displaystyle \alpha _{n}} coefficients in the Kapteyn expansion of f {\displaystyle f} can be determined as follows.: 571 α 0 = a 0 , α n = 1 4 ∑ k = 0 ⌊ n 2 ⌋ ( n − 2 k ) 2 ( n − k − 1 ) ! k ! ( n / 2 ) ( n − 2 k + 1 ) a n − 2 k ( n ≥ 1 ) . {\displaystyle {\begin{aligned}\alpha _{0}&=a_{0},\\\alpha _{n}&={\frac {1}{4}}\sum _{k=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }{\frac {(n-2k)^{2}(n-k-1)!}{k!(n/2)^{(n-2k+1)}}}a_{n-2k}\quad (n\geq 1).\end{aligned}}} == Examples == The Kapteyn series of the powers of z {\displaystyle z} are found by Kapteyn himself:: 103, : 565 ( z 2 ) n = n 2 ∑ m = 0 ∞ ( n + m − 1 ) ! ( n + 2 m ) n + 1 m ! J n + 2 m { ( n + 2 m ) z } ( z ∈ D 1 ) . {\displaystyle \left({\frac {z}{2}}\right)^{n}=n^{2}\sum _{m=0}^{\infty }{\frac {(n+m-1)!}{(n+2m)^{n+1}\,m!}}J_{n+2m}\{(n+2m)z\}\quad (z\in D_{1}).} For n = 1 {\displaystyle n=1} it follows (see also : 567 ) z = 2 ∑ k = 0 ∞ J 2 k + 1 ( ( 2 k + 1 ) z ) ( 2 k + 1 ) 2 , {\displaystyle z=2\sum _{k=0}^{\infty }{\frac {J_{2k+1}((2k+1)z)}{(2k+1)^{2}}},} and for n = 2 {\displaystyle n=2} : 566 z 2 = 2 ∑ k = 1 ∞ J 2 k ( 2 k z ) k 2 . {\displaystyle z^{2}=2\sum _{k=1}^{\infty }{\frac {J_{2k}(2kz)}{k^{2}}}.} Furthermore, inside the region D 1 {\displaystyle D_{1}} ,: 559 1 1 − z = 1 + 2 ∑ k = 1 ∞ J k ( k z ) . {\displaystyle {\frac {1}{1-z}}=1+2\sum _{k=1}^{\infty }J_{k}(kz).} == See also == Schlömilch's series == References ==
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Wikipedia:Karanapaddhati#0
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Karanapaddhati is an astronomical treatise in Sanskrit attributed to Puthumana Somayaji, an astronomer-mathematician of the Kerala school of astronomy and mathematics. The period of composition of the work is uncertain. C.M. Whish, a civil servant of the East India Company, brought this work to the attention of European scholars for the first time in a paper published in 1834. The book is divided into ten chapters and is in the form of verses in Sanskrit. The sixth chapter contains series expansions for the value of the mathematical constant π, and expansions for the trigonometric sine, cosine and inverse tangent functions. == Author and date of Karanapaddhati == Nothing definite is known about the author of Karanapaddhati. The last verse of the tenth chapter of Karanapaddhati describes the author as a Brahamin residing in a village named Sivapura. Sivapura is an area surrounding the present day Thrissur in Kerala, India. The period in which Somayaji lived is also uncertain. There are several theories in this regard. C.M. Whish, the first westerner to write about Karanapaddhati, based on his interpretation that certain words appearing in the final verse of Karanapaddhati denote in katapayadi system the number of days in the Kali Yuga, concluded that the book was completed in 1733 CE. Whish had also claimed that the grandson of the author of the Karanapaddhati was alive and was in his seventieth year at the time of writing his paper. Based on reference to Puthumana Somayaji in a verse in Ganita Sucika Grantha by Govindabhatta, Raja Raja Varma placed the author of Karanapaddhati between 1375 and 1475 CE. An internal study of Karanapaddhati suggests that the work is contemporaneous with or even antedates the Tantrasangraha of Nilakantha Somayaji (1465–1545 CE). == Synopsis of the book == A brief account of the contents of the various chapters of the book is presented below. Chapter 1 : Rotation and revolutions of the planets in one mahayuga; the number of civil days in a mahayuga; the solar months, lunar months, intercalary months; kalpa and the four yugas and their durations, the details of Kali Yuga, calculation of the Kali era from the Malayalam Era, calculation of Kali days; the true and mean position of planets; simple methods for numerical calculations; computation of the true and mean positions of planets; the details of the orbits of planets; constants to be used for the calculation of various parameters of the different planets. Chapter 2 : Parameters connected with Kali era, the positions of the planets, their angular motions, various parameters connected with Moon. Chapter 3 : Mean center of Moon and various parameters of Moon based on its latitude and longitude, the constants connected with Moon. Chapter 4 : Perigee and apogee of the Mars, corrections to be given at different occasions for the Mars, constants for Mars, Mercury, Jupiter, Venus, Saturn in the respective order, the perigee and apogee of all these planets, their conjunction, their conjunctions possibilities. Chapter 5 : Division of the kalpa based on the revolution of the planets, the number of revolutions during the course of this kalpa, the number of civil and solar days of earth since the beginning of this kalpa, the number and other details of the manvantaras for this kalpa, further details on the four yugas. Chapter 6 : Calculation of the circumference of a circle using variety of methods; the division of the circumference and diameters; calculation of various parameters of a circle and their relations; a circle, the arc, the chord, the arrow, the angles, their relations among a variety of parameters; methods to memorize all these factors using the katapayadi system. Chapter 7 : Epicycles of the Moon and the Sun, the apogee and perigee of the planets; sign calculation based on the zodiacal sign in which the planets are present; the chord connected with rising, setting, the apogee and the perigee; the method for determining the end-time of a month; the chords of the epicycles and apogee for all the planets, their hypotenuse. Chapter 8 : Methods for the determination of the latitude and longitude for various places on the earth; the R-sine and R-cosine of the latitude and longitude, their arc, chord and variety of constants. Chapter 9 : Details of the Alpha aeries sign; calculation of the positions of the planets in correct angular values; calculation of the position of the stars, the parallax connected with latitude and longitude for various planets, Sun, Moon and others stars. Chapter 10 : Shadows of the planets and calculation of various parameters connected with the shadows; calculation of the precision of the planetary positions. == Infinite series expressions == The sixth chapter of Karanapaddhati is mathematically very interesting. It contains infinite series expressions for the constant π and infinite series expansions for the trigonometric functions. These series also appear in Tantrasangraha and their proofs are found in Yuktibhāṣā. === Series expressions for π === Series 1 Series 2 Series 3 === Series expansions of trigonometric functions === == References == Venketeswara Pai R, K Ramasubramanian, M S Sriram and M D Srinivas, Karanapaddhati of Putumana Somayaji, Translation with detailed Mathematical notes, Jointly Published by HBA (2017) and Springer (2018). == Further references == Open Library reference to Karana-paddhati with two commentaries.[1] Bag, Amulya Kumar (1976). "Madhava's sine and cosine series" (PDF). Indian Journal of History of Science. 11 (1). Indian National Academy of Science: 54–57. Archived from the original (PDF) on 14 February 2010. Retrieved 17 December 2009. Bag, Amulya Kumar (1975). "The method of integral solutions of indeterminate equations of the type BY=AX ± C in ancient and medieval India" (PDF). Indian Journal of History of Science. 12 (1). Indian National Academy of Science: 1–16. Retrieved 12 January 2010. P.K. Koru, ed. (1953). Karanapaddhati of Puthumana Somayaji. Cherpu, Kerala, India: Astro Printing and Publishing Company. Indian National Science Academy has started a project in 2007–08 titled "A Critical Study of Karana-paddhati of Putumana Somayaji and Preparation of English Translation with Mathematical Notes" by Dr. K Ramasubramanian, Assistant Professor, Dept. of History, Indian Institute of Technology, Powai, Mumbai 400076.[2] (Retrieved on 13 January 2010)
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Wikipedia:Karel Slavíček#0
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Karel Slavíček (simplified Chinese: 严嘉乐; traditional Chinese: 嚴嘉樂; pinyin: Yán Jiālè; 12 December 1678 – 24 September 1735) was a Jesuit missionary and scientist, the first Czech sinologist and author of the first precise map of Beijing. == Biography == === Early life and studies in the Czech lands === Karel Slavíček was born in a house called Czech: U Slunce ("By the Sun") on the main square in Jimramov, a small town straddling the historical border between Bohemia and Moravia. His father, Václav Vojtěch Slavíček, was a teacher and town councillor writer. After finishing Gymnasium (school) in Brno, Slavíček entered the Jesuit order as a novice in 1694. Later he studied at Faculty of Philosophy and Faculty of Theology of Olomouc University. After finishing his studies, he stayed at the University as a teacher of mathematics and Hebrew language and history. Later he went to Prague, where he studied Theology at Charles University. He was ordained as a priest in 1707. Astronomy soon became his hobby. During his studies in Prague Slavíček took part in making a precise map of the capital. After finishing studies in Prague he went to teach in Jičín, subsequently lecturing mathematics at the University of Wrocław, finally also returning to the University of Olomouc, where he obtained Doctorate of Philosophy, became professor of mathematics and, in 1712, a fully-fledged member of the Jesuit order. In 1714 he went to Brno, where his work involved assisting his former Olomouc professor Jakub Kresa. Soon after this, he decided to become missionary. Because of his knowledge of mathematics as well as music, it was decided that he should be sent to China. In 1715 he left Brno for Prague, where he joined Ignaz Kögler. Together they went to Lisbon, where they joined with a Jesuit from Portugal. On 13 March 1716, they left for China. It took them 170 days to get there, surviving a major storm, during which many of their belongings were damaged. === In China === After reaching Macao they acquired local clothes and adopted local haircuts. On 3 February 1717, Slavíček was introduced to the Kangxi Emperor in Beijing, who employed him as a court musician. (Slavíček was a Spinet player). In 1717 Slavíček made the first precise map of Beijing, putting to use know-how obtained previously while working on the map of Prague. At the same time he clarified the latitude of Beijing. Slavíček soon became ill and left Beijing in order to recover away from the city. Meanwhile, the Kangxi Emperor was succeeded in 1722 by the Yongzheng Emperor, who was much less friendly towards Christians. Yongzheng ordered all the Jesuits in China to be accommodated in Beijing: consequently, Slavíček had to return there. Slavíček learned the Chinese language rather fast. He worked on astronomy and mathematics and compiled a Treatise on Chinese music: however, this treatise has not survived. Although forced to move back to Beijing, Slavíček did not become isolated. Indeed, he was in frequent contact by letter with European scientists, such as Stéphane Souciet and Guillaume de L'Isle. Many of the letters have been lost, but those that survive provide insight into Slavíček's life in China, and are now housed in the library of Strahov Monastery. Slavíček died after 18 years in China; he was succeeded by another Czech Jesuit, Jan Walter of Bílina. == References ==
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Wikipedia:Karen Aardal#0
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Karen I. Aardal (born 1961) is a Norwegian and Dutch applied mathematician, theoretical computer scientist, and operations researcher. Her research involves combinatorial optimization, integer programming, approximation algorithms, and facility location, with applications such as positioning emergency vehicles to optimize their response time. She is a professor in the Delft Institute of Applied Mathematics at the Delft University of Technology, and the chair of the Mathematical Optimization Society for the 2016–2019 term. == Education and career == Aardal is originally from Norway. She earned her Ph.D. in 1992 at the Université catholique de Louvain in Belgium. Her dissertation, On the Solution of One and Two-Level Capacitated Facility Location Problems by the Cutting Plane Approach, was supervised by Laurence Wolsey. Her dissertation won the second-place SOLA Dissertation Award of the Institute for Operations Research and the Management Sciences Section on Location Analysis. Aardal was formerly a researcher at the Dutch Centrum Wiskunde & Informatica, and additionally affiliated with Eindhoven University of Technology since 2005. She moved to Delft in 2008. She was elected to the 2019 class of Fellows of the Institute for Operations Research and the Management Sciences. == References == == External links == Official website Karen Aardal publications indexed by Google Scholar
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