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Wikipedia:Georg Alexander Pick#0
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Georg Alexander Pick (10 August 1859 – 26 July 1942) was an Austrian Jewish mathematician who was murdered during The Holocaust. He was born in Vienna to Josefa Schleisinger and Adolf Josef Pick and died at Theresienstadt concentration camp. Today he is best known for Pick's theorem for determining the area of lattice polygons. He published it in an article in 1899; it was popularized when Hugo Dyonizy Steinhaus included it in the 1969 edition of Mathematical Snapshots. == Education and career == Pick studied at the University of Vienna and defended his Ph.D. in 1880 under Leo Königsberger and Emil Weyr. After receiving his doctorate he was appointed an assistant to Ernst Mach at the Charles-Ferdinand University in Prague. He became a lecturer there in 1881. He took a leave from the university in 1884 during which he worked with Felix Klein at the University of Leipzig. Other than that year, he remained in Prague until his retirement in 1927 at which time he returned to Vienna. Pick headed the committee at the (then) German university of Prague, which appointed Albert Einstein to a chair of mathematical physics in 1911. Pick introduced Einstein to the work of Italian mathematicians Gregorio Ricci-Curbastro and Tullio Levi-Civita in the field of absolute differential calculus, which later in 1915 helped Einstein to successfully formulate general relativity. Charles Loewner was one of his students in Prague. He also directed the doctoral theses of Josef Grünwald, Walter Fröhlich, and Saly Struik. Pick was elected a member of the Czech Academy of Sciences and Arts, but was expelled after Nazis took over Prague. After retiring in 1927, Pick returned to Vienna, the city where he was born. After the Anschluss when the Nazis marched into Austria on 12 March 1938, Pick returned to Prague. In March 1939 the Nazis invaded Czechoslovakia. Pick was sent to Theresienstadt concentration camp on 13 July 1942. He died there two weeks later. == See also == Nevanlinna–Pick interpolation Pick matrix Pick's theorem == References == == External links == Georg Alexander Pick at the Mathematics Genealogy Project
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Wikipedia:Georg Frederik Ursin#0
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Georg Frederik Krüger Ursin (22 June 1797 – 4 December 1849) was a Danish mathematician and astronomer. == Early life == His father, Georg Jacob Krüger, was a first lieutenant in the Royal Danish Navy, however, in 1798, his was stripped of his functions where was taken to Munkholmen, an islet north of Trondheim, Norway. In the same year, Ursin was given royal license to carry the maternal family name from his mother, Jacobine Ursin (1772–1819), daughter of rear admiral Ursin, married to a wealthy shipbuilder and shipyard owner Lars Larsen. == Education == In 1814, Ursin passed an exam in land surveying before graduating cum laude from Metropolitanskolen in 1815. Having won a prize assignment involving regular polyhedron, he passed a second exam cum laude. == Death and legacy == Ursin died on December 4, 1849, in Copenhagen. A DSB Class EA locomotive no. 3020 is named after Ursin. == References ==
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Wikipedia:Georg Ignaz von Metzburg#0
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Georg Ignaz von Metzburg (June 24, 1735 – May 3, 1798) was an Austrian Jesuit priest, mathematician and cartographer. He taught mathematics at the University of Vienna. == Life and work == Von Metzburg was born in Styria in a family from Upper Austria. His father Christoph Augustin Freiherr von Metzburg was a land rights secretary married to Maria Katharina von Hitzelberg. Von Metzburg joined the Society of Jesus in 1751 and was ordained into priesthood in 1764. He became an assistant at the Vienna Observatory. He then received a doctorate in philosophy in 1772. He used his knowledge of geometry and cartography along with Joseph Liesganig to produce a map of Galicia. In 1774 he took up the chair of mathematics at the University of Vienna, succeeding Joseph Walcher. He taught for more than twenty years. Athanasios Psallidas produced a Greek translation of his book on arithmetic. His brother Franz Leopold (November 15, 1746 – October 6, 1789) worked as a consul in Iași, Moldavia and inherited the library of Von Metzburg. His writings included: Helshami Physica experimentalis Newtoniana ex anglico in latinum versa, Vindobonae 1769 Elementa Arithmeticae regularis seu vulgaris, ibid. 1769 (Vienna 1772, Augsburg 1773) Praxis geometrica ex principiis Geometriae deducta, ibid. 1777, Institutiones mathematicae. Tomus I–VII, ibid. 1775–1790 Institutiones mathematicae ad usum tironum, Editio 4, ibid. 1807 (posthumous) == References ==
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Wikipedia:Georg Limnaeus#0
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Georg Limnaeus (born Georg Wirn, also known as Georgius Lymneus, Limnæus or Limnäus; 24 October 1554 – 14 September 1611) was a German mathematician, astronomer and librarian, who provided noteworthy encouragement to Johannes Kepler shortly after his first heliocentric astronomical work was published. == Early life == Georg Limnaeus' father Antonius Wirn originated from Switzerland and served in the military forces of Frederick I, Elector of Saxony, who had been tutored by George Spalatin and closely followed and supported the works of Martin Luther. Around the time of the Capitulation of Wittenberg, Frederick conceived of the founding of the University of Jena, which was established in 1558 and became the university where Limnaeus was to spend all of his academic years. Upon completion of his military service, Antonius moved to Jena, where Georg Wirn was born and lived, and where, in 1571, he enrolled at the university. In accordance with its tradition, upon enrollment he assumed the name Georgius Lymneus. At Jena, Limnaeus studied under Jacob Flach (1537–1611), who was a graduate of the University of Wittenberg and had exposure to Philip Melanchthon (1497–1569) and frequented the lectures of Erasmus Reinhold (1511–1553.) In 1581, Limnaeus received the "Magisters der Philosophie" degree at the University of Jena. == Career and Kepler connections == Limnaeus issued a prognostication in 1585 in Erfurt and, in 1588, became the professor of mathematics at Jena, a position which he held until his death; concurrently, he also assumed the position of head librarian. He lectured on the Celestial sphere, astronomical and scientific calculations, the theory of planets and the use of astronomical tables, and in the areas of geography, geodesy and cosmography. Although he was not known to have produced any memorable manuscripts, he is known to have engaged in professional correspondence with peers, from time to time, including Tycho Brahe, Galileo Galilei, and Johannes Kepler, and to have maintained a respectable reputation as an academic prognosticator. In 1596, he founded the first observatory in Jena. In 1597, Limnaeus (along with Galileo, Brahe and Ursus) received a draft copy from Kepler of his first major work, Mysterium Cosmographicum. On April 24, 1598, Limnaeus wrote to Kepler, expressing his firm belief that heliocentric considerations should not be dismissed from the studies of astronomy by declaring, "Most illustrious Sir, never was I estranged from the most ancient philosophy of the Platonists – nor have I thought, as have several petty philosophers in our time, that it ought to be shunted outside the borders of the territory of the republic of letters." These words have been used to illustrate that it was not uncommon for traditionalist academicians, such as Limnaeus, to covertly honor heliocentric views of the ancients, while at the same time skillfully avoiding any explicit reference to the more controversial views of Copernicus. Limnaeus added, however, the statement that for any serious student of astronomy, Kepler's work represents "a new path to knowledge of the stars." In light of the disconcerting imprisonment of Giordano Bruno in 1593 (who was executed in 1600), this open expression of both support to young Kepler, and delight in his mathematical astronomical approaches, provided him with some of the earliest, forceful words of encouragement, which he must have welcomed in contrast to the many strong criticisms his work quickly evoked. In addition, Limnaeus provided information to Kepler on Tycho Brahe which may have promoted his final decision to go to Prague and study under him, thereby ensuring access to Brahe's data and the furtherance of his own work. However, beyond serving as an encourager to Kepler, and a facilitator to his decision to assist Brahe, there is no record that Limnaeus ever dove into specific details of Kepler's work or adopted it for his lessons. Kepler assisted Brahe from 1599 until Brahe's sudden death in 1601. By 1609, Kepler would develop and introduce his laws of planetary motion, which would subsequently play a major role in the development of Isaac Newton's law of universal gravitation, as has been noted by Newton. Limnaeus and his wife fell victim to the 1611 plague in Jena. == References ==
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Wikipedia:George A. Elliott#0
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George Arthur Elliott (born 1945) is a Canadian mathematician specializing in operator algebras, K-theory, and non-commutative geometry. He is a professor at the University of Toronto Department of Mathematics, and holds a Canada Research Chair. He is best known for his work on classifying C*-algebras, both for initiating their classification and highlighting the importance of K-theory in this respect. He was an invited speaker at the International Congress of Mathematicians, Zurich–1994. == Awards and honours == 1982 Elected as Fellow of Royal Society of Canada. 1996 CRM/Fields Institute Prize 1996-1998 Killam Research Fellow. 1998 Jeffery–Williams Prize. 1999 John L. Synge Award. 2012 Fellow of the American Mathematical Society. 2019 Fellow of the Canadian Mathematical Society == References == == External links == George A. Elliott at the Mathematics Genealogy Project
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Wikipedia:George Adams Kaufmann#0
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George Adams Kaufmann, also George Adams and George von Kaufmann, (8 February 1894, Maryampol, Galicia, then part of the Austro-Hungarian Empire – 30 March 1963, Edgbaston, UK) was a British mathematician, translator, and anthroposophist. He travelled widely, spoke several languages and translated many of Rudolf Steiner’s lectures into English. Through his studies in theoretical physics, he contributed to the expansion and development of the natural sciences as extended by the concepts of anthroposophy. == Youth == His father, Georg von Kaufmann, a British subject of German descent, was a pioneer of the oil industry. His mother was born Kate Adams in England. Shortly after George's birth, the family moved to Solotwina in the foothills of the Carpathians. In 1897, when he was three years old, his parents divorced. His father retained custody of the children and it was only a short while before her death in 1935, that Adams saw his mother again. The father married again – a young German woman, who created for George and his siblings a happy childhood. Educated by English governesses, he was raised to fluency in several languages, primarily English, German and Polish. From 1905, Adams attended Mill Hill School in England, travelling home alone to his family in Galicia. In 1912 he entered Christ's College, Cambridge to read Chemistry, completing his BA in 1915. He was president of Cambridge University Socialist Society in 1915. Preoccupied with problems of social reform, he rejected all manner of violence, remaining a conscientious objector throughout the First World War – a "militant revolutionary" as he described himself. He was imprisoned after refusing to serve with other conscientious objectors in the Non-Combatant Corps and was only released in 1919, after a hunger strike. The atomistic and materialistic thoughts of his time failed to satisfy him, causing him to seek for alternatives in the work of Alfred North Whitehead and Bertrand Russell. On questioning Russell on how to reach satisfactory conclusions in theoretical physics without the hypothesis of the atom, Russell encouraged him to study projective geometry. Following this advice, Adams began to concern himself primarily with mathematics and theoretical physics. He heard lectures by G. H. Hardy and began to research projective, non-Euclidean geometry. == Encounter with anthroposophy == In 1914 he had encountered Rudolf Steiner’s "Occult Science" and become a member of the Emerson Group in London in 1916. During his time as conscientious objector he had come to know Mary Fox, a Quaker and in 1920 they married. His interest in Steiner's ideas on social reform and his intention to translate the book The Threefold Social Order (GA 23) caused him to visit Steiner together with Ethel Bowen Wedgwood in Dornach, Switzerland. Steiner advised him to become involved in some form of social work, something Adams could readily accept amid the social collapse in Central and Eastern Europe following the war. He went on several journeys to Poland as part of the English and American Quaker organisation. In 1920 he took part in the inauguration of the first Goetheanum building. On his return to England, he cooperated with some friends on spreading the ideas of Social threefolding as well as the anthroposophical ideas of Steiner. His wife Mary Adams began her work as librarian and translator for the Anthroposophical Society in London that she was to carry for many years. In addition, Adams was the free verbal translator of around 110 lectures of Steiner into English. He went on to translate many of Steiner's written works, often with Mary Adams. He was often in Dornach during these times and experienced the burning of the first Goetheanum on New Year's Eve 1922/23 and was part of the Christmas Foundation meeting of the General Anthroposophical Society in 1923/24. In 1924 he became one of the Goetheanum-Speakers authorised by Steiner. == Research and work == While working as a free co-worker of the Anthroposophical Society in Britain as lecturer and workshop holder after 1925, Adams turned again to the study of the natural sciences and mathematics, concentrating particularly on projective geometry and working with Elisabeth Vreede, leader of the Section for Mathematics and Astronomy at the Goetheanum. At the beginning of the 1930s, Adams published a series of articles and essays about projective, synthetic geometry and its relationship to physics, to Goethe's theory of metamorphosis and to anthroposophical spiritual science, particularly the pioneering work "Of Etheric Space" in the magazine Natura of the Goetheanum's Medical Section. Here for the first time is mentioned the concept of "counter-space", as Steiner indicated in the third of his courses on the Natural Sciences (GA 323), explained by means of non-Euclidean geometry. Some years later Louis Locher was to discover the same thing independently of Adams. From that time on the conceptual development of the idea of counter-space in its relation to normal spatial thinking became the focal point of Adams' further scientific research. In 1933 the comprehensive work Space and the Light of the Creation – Synthetic Geometry in the light of Spiritual Science appeared, which was an overview of the spiritual scientific meaning of synthetic geometry. When Elisabeth Vreede and Ita Wegman were dismissed from the executive of the General Anthroposophical Society in Dornach, a number of other prominent members of the German, Dutch and British Societies were expelled, including Adams. This brought to an end his cooperation with the Mathematical/Astronomical Section. When the Chairman of the Anthroposophical Society in Great Britain, D. N. Dunlop, died in May 1935, Adams took over as general secretary. In this task, Olive Whicher became his closest co-worker, and he introduced her to projective geometry. == Wartime years == At the outbreak of the Second World War, Adams volunteered as interpreter in a prisoner of war camp. His close ties to Germany were soon the subject of investigation and he was dismissed after six months. He changed his name at this point from George Kaufmann to George Adams, taking the maiden name of his mother. In the following years Adams was one of the monitors of the Polish broadcasting corporation in the service of the BBC and he learnt several additional Slavic languages. Much of his free time was spent in the British Library studying the development of modern mathematical sciences to augment them with his thoughts about counter-space. After the war, he was given a scholarship by the British Anthroposophical Society at Rudolf Steiner House to investigate with Whicher the geometric principles underlying the world of plants. It had been unclear where the corresponding projective counterpart of the infinite plane of Euclidean space, the infinite midpoint of the non-Euclidean space (called by Locher the "absolute midpoint") was to be found in the plant world. In 1947, Adams expressed the idea that such a midpoint did not just exist, but that there was one to be found in every bud. This idea was connected with that of the lemniscatory correlation between space and counter-space and further researched by Whicher and himself. In 1949 and 1952, two books appeared with the titles: The Living Plant and the Science of Physical and Ethereal Space and The Plant between Sun and Earth. == The Goethean Science Institute == In 1947, at the request of his friends Fried Geuter and Michael Wilson of Sunfield Homes in Clent near Birmingham, Whicher and he moved to Clent, where they founded the Goethean science Foundation with Wilson to undertake scientific research. The peaceful countryside and a secure financial base provided an ideal environment for the work that followed. Shortly before this, in 1946, Adams had made contact with the Goetheanum and its Mathematical-Astronomical Section under the provisional guidance of Louis Locher. He wished, despite their differences, to work together on common issues. He again participated actively in many conferences and discussions at the Goetheanum and in Germany. He also took up contact again with Georg Unger, who went on to found the Mathematisch-Physicalisches Institut in 1956, where Adams reported regularly on his work. Unger visited Clent for common research gatherings. == Collaboration with Unger, Leroi and Schwenk == Adams began his research on rigid body statics and dynamics in mechanics from the point of view of space and counter-space on the basis of projective linear geometry. The results were published in the late 1950s in the magazine Mathematisch-Physikalische Korrespondenz edited by Unger and appeared in book form in 1996. Based on the groundwork done by a number of mathematicians like Felix Klein, Arthur Cayley, William K. Clifford and Eduard Study, Adams pursued the idea of counter-space into the concrete details of rigid body statics and dynamics (in non-Euclidean spaces). Peter Gschwind was to build on this research and lead it into the sphere of Quantum physics. Adams and Unger also began to collaborate with Alexandre Leroi and Theodor Schwenk, leading to the establishment of the Instituts für Strömungsforschung in Herrischried in the Black Forest. During the last years of his life, Adams expended a lot of his energy on this project, desiring to test the efficacy of his ideas on space and counter-space into the realm of technology. He worked out specific planes (W-planes) and with John Wilkes built models for the self-purification and re-enlivening of water through special forms. He often worked in Herrischried with Whicher, following with interest the research of Leroi and Schwenk. After a light stroke in August 1959, Adams divided his time between the Black Forest and England devoting much time to the translation of works of Steiner. He is reported to have combined several polarities in character: on the one hand multilingual and worldly, he tended on the other to be shy and strangely innocent; a clear thinker schooled in mathematics, chemistry and physics, his relationship to nature was rather more introspective and religious; he allowed himself to be incarcerated for his convictions, while in his relationships he showed extreme modesty, humour and never allowed his knowledge to become overbearing. == Books == Fruits of Anthroposophy, London 1922 Christ and the Earth, London 1927 Synthetische Geometrie, Goethesche Metamorphosenlehre und Mathematische Physik, in: Mathesis, Stuttgart 1931 The Anthroposophical Movement, London 1932 Von dem ätherischen Raume, in: Na 1932–33, Nr. 5/6 Space and the Light of the Creation, London 1933 Christ in the Power of Memory and the Power of Love, East Grinstead 1938 The Mysteries of the Rose-Cross, East Grinstead 1955, London Physical and Ethereal Spaces, London 1965, Universal Forces in Mechanics, London 1977 Grundfragen der Naturwissenschaft, Stuttgart 1979 Lemniscatory Ruled Surface in Space and Counterspace, Rudolf Steiner Press (March 1979) ISBN 978-0854403486 === With Olive Whicher === The Living Plant and the Science of Physical and Ethereal Space, Stourbridge 1949; The Plant between Sun and Earth, Clent 1952, London 21980; == Further reading == Unger, G.: George Adams, in: MaK 1963, Nr. 40; Unger, G.: George Adams, in: N 1963, Nr. 15; Götte, F.: Ein Gedenkblatt für George Adams, in: MaD 1963, Nr. 66; Whicher, O.: The Life and Work of George Adams: an Introduction, in: GBl 1964; Whicher, O.: George Adams. Ein Geistsucher in unserer Zeit, Dornach 1973; Gschwind, P.: Zum Nachlass von G. Adams, in: MaK 1981, Nr. 121; Unger, G.: Zum 100. Geburtstag von George Adams, in: N 1994, Nr. 28; Ziegler, R.: Bibliographie von G. Adams, in: Adams, Dornach 31996; Ziegler, R.: Biographien und Bibliographien, Dornach 2001. === Locations of estate and archives === Adams House, 20 Freshfild Bank, Forest Row, Sussex RH18 5HG, England; Peter Gschwind, Benedikt- Hugi-Weg 18, CH–4143 Dornach; Archiv am Goetheanum, Dornach; Archiv der Rudolf Steiner-Nachlassverwaltung, Dornach; Freie Hochschule für Waldorfpädagogik, Mannheim. == References ==
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Wikipedia:George F. C. Griss#0
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George François Cornelis Griss (30 January 1898, Amsterdam – 2 August 1953, Blaricum), usually cited as G. F. C. Griss, was a Dutch mathematician and philosopher, who was occupied with Hegelian idealism and Brouwers intuitionism and stated a negationless mathematics. Griss was a student of L. E. J. Brouwer and formulated an intuitionism based on a Hegelian idealism. He obtained his Ph.D. with Roland Weitzenböck at the University of Amsterdam in July 1925. He was largely influenced by L. E. J. Brouwer, Gerrit Mannoury, Carry van Bruggen and Gerard Bolland, who brought Hegelian thought to the Netherlands. He published a number of articles about a negationless mathematics and one small book about idealistic philosophy, called Idealistische Filosofie (17 February 1946, Gouda), in which he lays down a typically Hegelian idealism, and incorporates elements from Bergson's Creative Evolution (L'Evolution créatrice). == Publications == Het volledige invariantensysteem van 2 covariante antisymmetrische tensoren van den 2den trap en een willekeurig aantal vectoren, K.A.W., Amsterdam, Verslag 34, 1925. Differentialinvarianten von Systemen von Vektoren (Ph.D. thesis) Groningen: Noordhoff, 1925. Differentialinvarianten von zwei kovarianten Vektoren in vier Veränderlichen, Proc. K.A.W. (Amsterdam), vol.33, 1930, pp. 176–179. Der Existenzsatz für ein wesentliches System bei Invarianten von Differentialformen, Proc. K.A.W. (Amsterdam), vol.33, 1930, pp. 491–494. Problemen der Invariantentheorie (public lecture) Groningen: Noordhoff, 1934. Die Differentialinvarianten eines Systems von n relativen kovarianten Vektoren in Rn, Proc. K.A.W. (Amsterdam), vol.37, 1934, pp. 82–87. Die Differentialinvarianten eines kovarianten symmetrischen Tensors vierter Stufe im binären Gebiet, Comp. Math. vol.1, 1934, pp. 238–247. Differentialvarianenten von relativen Vektoren, Comp. Math. vol.1, 1935, pp. 420–428. Die konformen Differentialinvarianten eines kovarianten symmetrischen Tensors vierter Stufe im binären Gebiet, Proc. K.A.W. (Amsterdam) vol. 39, 1936, pp. 947–955. Negatieloze intuïtionistisch wiskunde. Proceedings ("Verslagen") Nederlandse Akademie van Wetenschappen, Afdeling Natuurkunde, Vol.LIII, no.5, 1944, pp. 261–268 (includes German, English, and French summary). Idealistisch Filosofie, Arnhem: Van Loghum Slaterus, 1946. Negationless intuitionistic mathematics I, Proc. K.A.W. (Amsterdam) vo.49, 1946, pp. 1127–1133. Over de negatie, festive collection ("feestbundel") Prof. Dr. H.J. Pos, Amsterdam: Noord-Hollandse uitgeversmaatschappij, 1948, pp. 96–106. Mathématiques, Mystique et Philosophie, Mélanges philosophiques, Libr. 10th Int. Congr. Phil. II (1948), pp. 156–175. Logique des mathématiques intuitionistes sans négation, C. R. [Comptes Rendues] Ac. Sci. Paris, vol.227, 1948, pp. 946–948. Sur la Négation dans les Mathématiques et la Logique, Synthese, vol.7, 1948/1949, no.1-2 pp. 71–74. Negationless intuitionistic mathematics II, Proceedings [Koninklijke Nederlandse Akademie van Wetenschappen] [series A] Vol.LIII, no.4, 1950, pp. 456–463 or Indagationes Mathematicae, Vol.XII, Fasc.2, 1950. Logic of negationless intuitionistic mathematics, Proceedings [Koninklijke Nederlandse Akademie van Wetenschappen] Series A, Vol.LIV, no.1, 1951, pp. 41–49. Negationless intuitionistic mathematics III, Proceedings [Koninklijke Nederlandse Akademie van Wetenschappen] series A, Vol.LIV, no.2, 1951, pp. 193–199. Negationless intuitionistic mathematics IV, Proceedings [Koninklijke Nederlandse Akademie van Wetenschappen] series A, Vol.LIV, no.5, 1951, pp. 452–471 or Indagationes Mathematicae, Vol.XIII, no.5, 1951. == Secondary literature == H.J. Pos: G.F.C. Griss' Idealistische Filosofie, Algemeen Nederlands Tijdschrift voor Wijsbegeerte en Psychologie waarin opgenomen de Annalen van het Genootschap voor Wetenschappelijke Philosophie, 46e Jaargang, aflevering 1, october 1953. pp. 1–7. A. Heyting: Over de betekenis van het Wiskundige werk van G.F.C. Griss, ibidem, pp. 8–12. A. Heyting: G.F.C. Griss and his negationless intuitionistic mathematics, Synthese, Vol. IX, Issue 2, no.2, pp. 91–96 H.J. Pos: G.F.C. Griss als wijsgerig humanist en als mens, De Nieuw Stem [1953], pp. 654–663. B. van Rootselaar, In memoriam Dr. G.F.C. Griss, Euclides [1953?] Tijdschrift voor de Didactiek der Exacte Vakken, pp. 42–45. (Contains bibliography.) G.F.C. Griss, 1898–1953 in the Album Academicum of the University of Amsterdam == See also == L. E. J. Brouwer Gerard Bolland Gerrit Mannoury Arend Heyting Philosophy of mind Philosophy of mathematics == External links == George F. C. Griss at the Mathematics Genealogy Project
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Wikipedia:George F. R. Ellis#0
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George Francis Rayner Ellis, FRS, Hon. FRSSAf (born 11 August 1939), is the emeritus distinguished professor of complex systems in the Department of Mathematics and Applied Mathematics at the University of Cape Town in South Africa. He co-authored The Large Scale Structure of Space-Time with University of Cambridge physicist Stephen Hawking, published in 1973, and is considered one of the world's leading theorists in cosmology. From 1989 to 1992 he served as president of the International Society on General Relativity and Gravitation. He is a past president of the International Society for Science and Religion. He is an A-rated researcher with the NRF. Ellis, an active Quaker, was a vocal opponent of apartheid during the National Party reign in the 1970s and 1980s, and it is during this period that Ellis's research focused on the more philosophical aspects of cosmology, for which he won the Templeton Prize in 2004. He was also awarded the Order of the Star of South Africa by Nelson Mandela in 1999. On 18 May 2007, he was elected a fellow of the British Royal Society. == Life == Born in 1939 to George Rayner Ellis, a newspaper editor, and Gwendoline Hilda MacRobert Ellis in Johannesburg, George Francis Rayner Ellis attended the University of Cape Town, where he graduated with honours in 1960 with a Bachelor of Science degree in physics with distinction. He represented the university in fencing, rowing and flying. While a student at St John's College, Cambridge, where he received a PhD in applied maths and theoretical physics in 1964, he was on college rowing teams. At Cambridge, Ellis served as a research fellow from 1965 to 1967, was assistant lecturer in the Department of Applied Mathematics and Theoretical Physics until 1970, and was then appointed university lecturer, serving until 1974. Ellis became a visiting professor at the Enrico Fermi Institute at the University of Chicago in 1970, a lecturer at the Cargese Summer School in Corsica in 1971 and the Erice Summer School in Sicily in 1972, and a visiting H3 professor at the University of Hamburg, also in 1972. The following year, Ellis co-wrote The Large Scale Structure of Space-Time with Stephen Hawking, debuting at a strategic moment in the development of General Relativity Theory. In the following year, Ellis returned to South Africa to accept an appointment as professor of applied mathematics at the University of Cape Town, a position he held until his retirement in 2005. In 2005 Ellis appeared as a guest speaker at the Nobel Conference in St. Peter, Minnesota. == Work == George Ellis has worked for many decades on anisotropic cosmologies (Bianchi models) and inhomogeneous universes, and on the philosophy of cosmology. He is currently writing on the emergence of complexity, and the way this is enabled by top-down causation in the hierarchy of complexity. Recently Ellis has also collaborated with Teppo Felin, Denis Noble, and Jan Koenderink on a set of articles published in the journal Genome Biology. In terms of philosophy of science, Ellis is a Platonist. == Publications == === Books === === Papers === Ellis has over 500 published articles; including 17 in Nature. Notable papers include: == Honours == In 2019 Rhodes University in Grahamstown announced it would award Ellis an honorary doctorate in laws (LLD, hc) == See also == List of science and religion scholars == Notes and references == == External links == Partial list of Ellis' published papers George Ellis's web page Archived 3 February 2019 at the Wayback Machine Professor George Ellis: a man of many parts, Cape Argus, 18 March 2004 George Ellis's scientific work as listed at SPIRES Interview with George Ellis (Recorded June 2004) at the Wayback Machine (archived 1 April 2013) on Speaking of Faith with Krista Tippett (transcript Archived 30 December 2016 at the Wayback Machine) George Ellis extended interview with transcript for the 'Why Are We Here?' documentary series.
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Wikipedia:George Gheverghese Joseph#0
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George Gheverghese Joseph, also known as G. G. Joseph is an Indian-born African mathematician who is a specialist in the history of mathematics. His works are mainly focused on the achievements of Kerala school of astronomy and mathematics and the transmission of mathematics from India to Europe. == Early life and works == George Gheverghese Joseph was born in Kerala, India. At the age of 9, his family moved to Mombasa, Kenya and he pursued his schooling in Kenya. He completed his degree in mathematics at the University of Leicester. After completing his degree, he worked as a school teacher for six years in Kenya and, then he did a master's degree at the University of Manchester, England. He qualified in Law in 2000. G. G. Joseph studied and conducted researches in applied mathematics and statistics, including multivariate analysis, mathematical programming, and demography. He is conducting three-month research on the history of mathematics in his native place every year. Through his research and the best selling book The Crest of the Peacock, he argued that the infinite series was invented by Kerala mathematicians in 1350, before Europeans. This work garnered publicity and generated discussions in the history of maths on the discoveries made by Indians that predated the Europeans. == Bibliography == The Crest of the Peacock: Non-European Roots of Mathematics, Princeton University Press, 1991. A Passage to Infinity: Medieval Indian Mathematics from Kerala and its Impact, 2009. Kerala Mathematics: History and Its Possible Transmission to Europe, 2009. Indian Mathematics: Engaging with the World from Ancient to Modern Times, 2016 Multicultural Mathematics: Teaching Mathematics from a Global Perspective (with David Nelson and Julian Williams), 1993. Women at Work: The British Experience, 1983. == References == "An Interview with George Gherverghese Joseph | Mathematical Association of America". A Passage to Infinity The Crest of the Peacock Joseph, George Gheverghese (24 October 2010). The Crest of the Peacock. ISBN 9780691135267. "Indians predated Newton 'discovery' by 250 years". The University of Manchester News. 13 August 2007. Retrieved 2020-07-04. Krishnamachari, Suganthy (29 October 2015). "A passage to Europe?". The Hindu. ISSN 0971-751X. Retrieved 2020-07-04 – via www.thehindu.com.
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Wikipedia:George Green (mathematician)#0
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George Green (14 July 1793 – 31 May 1841) was a British mathematical physicist who wrote An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism in 1828. The essay introduced several important concepts, among them a theorem similar to the modern Green's theorem, the idea of potential functions as currently used in physics, and the concept of what are now called Green's functions. Green was the first person to create a mathematical theory of electricity and magnetism and his theory formed the foundation for the work of other scientists such as James Clerk Maxwell, William Thomson, and others. His work on potential theory ran parallel to that of Carl Friedrich Gauss. Green's life story is remarkable in that he was almost entirely self-taught. He received only about one year of formal schooling as a child, between the ages of 8 and 9. == Early life == Green was born and lived for most of his life in the English town of Sneinton, Nottinghamshire, now part of the city of Nottingham. His father, also named George, was a baker who had built and owned a brick windmill used to grind grain. In his youth, Green was described as having a frail constitution and a dislike for doing work in his father's bakery. He had no choice in the matter, however, and as was common for the time he likely began working daily to earn his living at the age of five. === Robert Goodacre's Academy === During this era it was common for only 25–50% of children in Nottingham to receive any schooling. The majority of schools were Sunday schools, run by the Church, and children would typically attend for one or two years only. Recognizing the young Green's above average intellect, and being in a strong financial situation due to his successful bakery, his father enrolled him in March 1801 at Robert Goodacre's Academy in Upper Parliament Street. Robert Goodacre was a well-known science populariser and educator of the time. He published Essay on the Education of Youth, in which he wrote that he did not "study the interest of the boy but the embryo Man". To a non-specialist, he would have seemed deeply knowledgeable in science and mathematics, but a close inspection of his essay and curriculum revealed that the extent of his mathematical teachings was limited to algebra, trigonometry and logarithms. Thus, Green's later mathematical contributions, which exhibited knowledge of very modern developments in mathematics, could not have resulted from his tenure at the Robert Goodacre Academy. He stayed for only four terms (one school year), and it was speculated by his contemporaries that he had exhausted all they had to teach him. === Move from Nottingham to Sneinton === In 1773 George's father moved to Nottingham, which at the time had a reputation for being a pleasant town with open spaces and wide roads. By 1831, however, the population had increased nearly five times, in part due to the budding Industrial Revolution, and the city became known as one of the worst slums in England. There were frequent riots by starving workers, often associated with special hostility towards bakers and millers on the suspicion that they were hiding grain to drive up food prices. For these reasons, in 1807, George Green senior bought a plot of land in Sneinton. On this plot of land he built a "brick wind corn mill", now referred to as Green's Windmill. It was technologically impressive for its time, but required nearly twenty-four-hour maintenance, which was to become Green's burden for the next twenty years. == Adult life == === Miller === Just as with baking, Green found the responsibilities of operating the mill annoying and tedious. Grain from the fields was arriving continuously at the mill's doorstep, and the sails of the windmill had to be constantly adjusted to the windspeed, both to prevent damage in high winds, and to maximise rotational speed in low winds. The millstones that would continuously grind against each other, could wear down or cause a fire if they ran out of grain to grind. Every month the stones, which weighed over a ton, would have to be replaced or repaired. === Family life === In 1823 Green formed a relationship with Jane Smith, the daughter of William Smith, hired by Green Senior as mill manager. Although Green and Jane Smith never married, Jane eventually became known as Jane Green and the couple had seven children together; all but the first had Green as a baptismal name. The youngest child was born 13 months before Green's death. Green provided for his (so-called) common-law wife and children in his will. === Nottingham Subscription Library === When Green was thirty, he became a member of the Nottingham Subscription Library. This library exists today, and was likely the main source of Green's advanced mathematical knowledge. Unlike more conventional libraries, the subscription library was exclusive to a hundred or so subscribers, and the first on the list of subscribers was the Duke of Newcastle. This library catered to requests for specialised books and journals that satisfied the particular interests of their subscribers. === 1828 essay === In 1828, Green published An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, which is the essay he is most famous for today. It was published privately at the author's expense, because he thought it would be presumptuous for a person like himself, with no formal education in mathematics, to submit the paper to an established journal. When Green published his Essay, it was sold on a subscription basis to 51 people, most of whom were friends who probably could not understand it. The wealthy landowner and mathematician Edward Bromhead bought a copy and encouraged Green to do further work in mathematics. Not believing the offer was sincere, Green did not contact Bromhead for two years. == Mathematician == By the time Green's father died in 1829, the senior Green had become one of the gentry due to his considerable accumulated wealth and land owned, roughly half of which he left to his son and the other half to his daughter. The young Green, now thirty-six years old, consequently was able to use this wealth to abandon his miller duties and pursue mathematical studies. === Cambridge === Members of the Nottingham Subscription Library who knew Green repeatedly insisted that he obtain a proper University education. In particular, one of the library's most prestigious subscribers was Sir Edward Bromhead, with whom Green shared many correspondences; he insisted that Green go to Cambridge. In 1832, aged nearly forty, Green was admitted as an undergraduate at Gonville and Caius College, Cambridge. He was particularly insecure about his lack of knowledge of Greek and Latin, which were prerequisites, but it turned out not to be as hard for him to learn these as he had envisaged, as the degree of mastery required was not as high as he had expected. In the mathematics examinations, he won the first-year mathematical prize. He graduated with a BA in 1838 as a 4th Wrangler (the 4th highest scoring student in his graduating class, coming after James Joseph Sylvester who scored 2nd). === College fellow === Following his graduation, Green was elected a fellow of the Cambridge Philosophical Society. Even without his stellar academic standing, the Society had already read and made note of his Essay and three other publications, so Green was welcomed. The next two years provided an unparalleled opportunity for Green to read, write, and discuss his scientific ideas. In this short time he published an additional six publications with applications to hydrodynamics, sound, and optics. == Final years and posthumous fame == In his final years at Cambridge, Green became rather ill, and in 1840 he returned to Sneinton, only to die a year later. There are rumours that at Cambridge, Green had "succumbed to alcohol", and some of his earlier supporters, such as Sir Edward Bromhead, tried to distance themselves from him. Green's work was not well known in the mathematical community during his lifetime. Besides Green himself, the first mathematician to quote his 1828 work was the Briton Robert Murphy (1806–1843) in his 1833 work. In 1845, four years after Green's death, Green's work was rediscovered by the young William Thomson (then aged 21), later known as Lord Kelvin, who popularised it for future mathematicians. According to the book "George Green" by D.M. Cannell, William Thomson noticed Murphy's citation of Green's 1828 essay but found it difficult to locate Green's 1828 work; he finally got some copies of Green's 1828 work from William Hopkins in 1845. In 1871 Norman Macleod Ferrers assembled The Mathematical Papers of the late George Green for publication. Green's work on the motion of waves in a canal (resulting in what is known as Green's law) anticipates the WKB approximation of quantum mechanics, while his research on light-waves and the properties of the aether produced what is now known as the Cauchy–Green tensor. Green's theorem and functions were important tools in classical mechanics, and were revised by Julian Schwinger's 1948 work on electrodynamics that led to his 1965 Nobel Prize in Physics (shared with Richard Feynman and Shin'ichirō Tomonaga). Green's functions later also proved useful in analysing superconductivity. On a visit to Nottingham in 1930, Albert Einstein commented that Green had been 20 years ahead of his time. The theoretical physicist Julian Schwinger, who used Green's functions in his ground-breaking works, published a tribute entitled "The Greening of Quantum Field Theory: George and I" in 1993. The George Green Library at the University of Nottingham is named after him, and houses the majority of the university's science and engineering Collection. The George Green Institute for Electromagnetics Research, a research group in the University of Nottingham engineering department, is also named after him. In 1986, Green's Mill, Sneinton in Nottingham was restored to working order. It now serves both as a working example of a 19th-century windmill and as a museum and science centre dedicated to Green. Westminster Abbey has a memorial stone for Green in the nave adjoining the graves of Isaac Newton and Lord Kelvin. His work and influence on 19th-century applied physics had been largely forgotten until the publication of his biography by Mary Cannell in 1993. == Source of knowledge == Recent historical research suggests that the pivotal figure in Green's mathematical education was John Toplis (c1774-1857), who graduated in mathematics from Cambridge as 11th Wrangler before becoming headmaster of the forerunner of Nottingham High School 1806–1819, and lived in the same neighbourhood as Green and his family. Toplis was an advocate of the continental school of mathematics, and fluent in French, having translated Pierre-Simon Laplace's celebrated work on celestial mechanics. The possibility that Toplis played a role in Green's mathematical education would resolve several long-standing questions about the sources of Green's mathematical knowledge. For example, Green made use of "the Mathematical Analysis", a form of calculus derived from Gottfried Wilhelm Leibniz which was virtually unheard of, or even actively discouraged, in England at the time (due to Leibniz being a contemporary of Newton, who had his own methods that were championed in England). This form of calculus, and the developments of mathematicians such as the French mathematicians Laplace, Sylvestre François Lacroix and Siméon Denis Poisson, were not taught even at Cambridge, let alone Nottingham, and yet Green not only had heard of these developments, but improved upon them. == List of publications == An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. By George Green, Nottingham. Printed for the Author by T. Wheelhouse, Nottingham. 1828. (Quarto, vii + 72 pages.) Green, George (1835). "Mathematical investigations concerning the laws of the equilibrium of fluids analogous to the electric fluid, with other similar researches". Transactions of the Cambridge Philosophical Society. 5 (part i): 1–63. Presented 12 November 1832. Green, George (1835). "On the determination of the exterior and interior attractions of ellipsoids of variable densities". Transactions of the Cambridge Philosophical Society. 5 (part iii): 395–429. Bibcode:1835TCaPS...5..395G. Presented 6 May 1833. Green, George (1836). "Researches on the vibration of pendulums in fluid media". Transactions of the Royal Society of Edinburgh. 13 (1): 54–62. doi:10.1017/S0080456800022183. S2CID 124762445. Presented 16 December 1833. Green, George (1838). "On the reflexion and refraction of sound". Transactions of the Cambridge Philosophical Society. 6 (part iii): 403–413. Bibcode:1838TCaPS...6..403G. Presented 11 December 1837. Green, George (1838). "On the motion of waves in a variable canal of small depth and width". Transactions of the Cambridge Philosophical Society. 6 (part iii): 457–462. Bibcode:1838TCaPS...6..457G. Presented 15 May 1837. Green, George (1842). "On the laws of the reflexion and refraction of light at the common surface of two non-crystallized media". Transactions of the Cambridge Philosophical Society. 7 (part i): 1–24. Presented 11 December 1837. Green, George (1842). "Note on the motion of waves in canals". Transactions of the Cambridge Philosophical Society. 7 (part i): 87–95. Presented 18 February 1839. Green, George (1842). "Supplement to a memoir on the reflection and refraction of light". Transactions of the Cambridge Philosophical Society. 7 (part i): 113–120. Presented 6 May 1839. Green, George (1842). "On the propagation of light in crystallized media". Transactions of the Cambridge Philosophical Society. 7 (part ii): 121–140. Presented 20 May 1839. == Notes == == References == Ivor Grattan-Guinness, 'Green, George (1793–1841)', Oxford Dictionary of National Biography, Oxford University Press, 2004 accessed 26 May 2009 D. M. Cannell, "George Green mathematician and physicist 1793–1841", The Athlone Press, London, 1993. Murphy, Robert (1833). "On the inverse method of definite integrals". Transactions of the Cambridge Philosophical Society. 4: 353–408. (Note: This was the first quotation of Green's 1828 work by somebody other than Green himself.) O'Connor, John J.; Robertson, Edmund F., "George Green (mathematician)", MacTutor History of Mathematics Archive, University of St Andrews "George Green". Archived from the original on 26 December 2010. – An excellent on-line source of George Green information Green, George (1828). "An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism". arXiv:0807.0088 [physics.hist-ph]. Cannel, D. M. and Lord, N. J.; Lord, N. J. (March 1993). "George Green, mathematician and physicist 1793–1841". The Mathematical Gazette. 77 (478). The Mathematical Gazette, Vol. 77, No. 478: 26–51. doi:10.2307/3619259. JSTOR 3619259. S2CID 238490315.{{cite journal}}: CS1 maint: multiple names: authors list (link) Challis, L. and Sheard, F.; Sheard, Fred (December 2003). "The Green of Green Functions". Physics Today. 56 (12): 41–46. Bibcode:2003PhT....56l..41C. doi:10.1063/1.1650227. S2CID 17977976.{{cite journal}}: CS1 maint: multiple names: authors list (link) "Green's Mill and Science Centre" (Web page). Retrieved 22 November 2005. == External links == List of References for George Green Quotations related to George Green at Wikiquote Bowley, Roger. "George Green & Green's Functions". Sixty Symbols. Brady Haran for the University of Nottingham.
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Wikipedia:George Grätzer#0
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George A. Grätzer (Hungarian: Grätzer György; born 2 August 1936, in Budapest) is a Hungarian-Canadian mathematician, specializing in lattice theory and universal algebra. He is known for his books on LaTeX and his proof with E. Tamás Schmidt of the Grätzer–Schmidt theorem. == Biography == His father József Grätzer was famous in Hungary as the "Puzzle King" ("rejtvénykirály"). George Grätzer received his PhD from Eötvös Loránd University in 1960 under the supervision of László Fuchs. In 1963 Grätzer and Schmidt published their theorem on the characterization of congruence lattices of algebras. In 1963 Grätzer left Hungary and became a professor at Pennsylvania State University. In 1966 he became a professor at the University of Manitoba and later a Canadian citizen. In 1970 Grätzer became the founder and editor-in-chief of the journal Algebra Universalis. His mathematical articles—over 260, all listed on Research Gate—are widely cited, and he has written several influential books. Grätzer has received several awards and honours. He is married and has two children (Tom Gratzer and David Gratzer) and five grandchildren. == Awards and honours == Grünwald Memorial Prize (1967) Steacie Prize (1971) Fellow of the Royal Society of Canada (1973) Jeffery–Williams Prize (1978) Zubek Prize (1987) Elected Foreign Member of Magyar Tudományos Akadémia (1997) == Publications == More than 270 research articles in mathematics, and 36 books including Elmesport egy esztendőre 1959 (2008-as kiadása: ISBN 9789639725362); trans. into English as Train your brain: A year's worth of puzzles 2011 Universal Algebra 1960 Lattice Theory 1971 [1]VP-Info Database Language 1986 First Steps in LaTeX 1999 More Math into LaTeX, sixth edition 2007 Lattice Theory: Foundation 2011 Practical LaTeX 2014 The Congruences of a Finite Lattice: A Proof-by-Picture Approach, third edition 2023 Math into LaTeX, sixth edition 2024 Math into English, 2024 Write Better, 2024 == References == == External links == George Grätzer at U. of Manitoba Celebrating Professor George A. Grätzer, by Gábor Czédli, Categories and General Algebraic Structures with Applications, Volume 11, Special Issue Dedicated to Prof. George A. Grätzer.
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Wikipedia:George Jerrard#0
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George Birch Jerrard (25 November 1804 – 23 November 1863) was a British mathematician. He studied at Trinity College, Dublin from 1821 to 1827. His main work was on the theory of equations, where he was reluctant to accept the validity of the work of Niels Henrik Abel on the insolubility of the quintic equation by radicals. He found a way of using Tschirnhaus transformations to eliminate three of the terms in an equation, which generalised work of Erland Bring (1736–1798), and is now called Bring–Jerrard normal form. == Works == An essay on the resolution of equations, part 1, London 1858, (online). == References == Lee, Sidney, ed. (1901). "Jerrard, George Birch" . Dictionary of National Biography (1st supplement). London: Smith, Elder & Co. == External links == O'Connor, John J.; Robertson, Edmund F., "George Jerrard", MacTutor History of Mathematics Archive, University of St Andrews
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Wikipedia:George Karniadakis#0
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George Em Karniadakis (Γιώργος Εμμανουήλ Καρνιαδάκης) is a professor of applied mathematics at Brown University. He is a Greek-American researcher who is known for his wide-spectrum work on high-dimensional stochastic modeling and multiscale simulations of physical and biological systems, and is a pioneer of spectral/hp-element methods for fluids in complex geometries, general polynomial chaos for uncertainty quantification, and the Sturm-Liouville theory for partial differential equations and fractional calculus. == Biography == George Em Karniadakis obtained his diploma of engineering in Mechanical Engineering and Naval Architecture from the National Technical University of Athens in 1982. Subsequently, he received his Scientiæ Magister in 1984 and his Ph.D. in Mechanical Engineering and Applied Mathematics in 1987 from the Massachusetts Institute of Technology (MIT) under the advice of Anthony T. Patera and Borivoje B. Mikic. He then joined the Center for Turbulence Research at Stanford University, NASA Ames Laboratory, as a postdoctoral research associate under the mentorship of Parviz Moin and John Kim. In 1988, Karniadakis joined Princeton University as a tenure-track assistant professor in the Department of Mechanical and Aerospace Engineering, and as an associate faculty in the Program of Applied and Computational Mathematics. In 1993, he held a visiting professor appointment in the Aeronautics Department at the California Institute of Technology, before joining the Division of Applied Mathematics at Brown University as a tenured associate professor in 1994. He became a full professor of Applied Mathematics in 1996. Since 2000, he has been a visiting professor and senior lecturer of Ocean/Mechanical Engineering at MIT. He was entitled the Charles Pitts Robinson and John Palmer Barstow Professor of Applied Mathematics in 2014. He is the lead principal investigator (PI) of an OSD/ARO/MURI on fractional PDEs, and the lead PI of an OSD/AFOSR MURI on Machine Learning for PDEs. He is the Director of the DOE center PhILMS on Physics-Informed Learning Machines and was previously the Director of the DOE Center of Mathematics for Mesoscale Modeling of Materials (CM4). == Honors and awards == Ralph E. Kleinman Prize, Society for Industrial and Applied Mathematics, 2015 MCS Wiederhielm Award of the Microcirculatory Society "for the most highly cited original article in Microcirculation over the previous five year period for the paper", 2015 US Association for Computational Mechanics, 2013, The J Tinsley Oden (inaugural) Medal. US Association for Computational Mechanics, 2007 Computational Fluid Dynamics award. Fellow of the Society for Applied and Industrial Mathematics (SIAM), 2010. Fellow of the American Physical Society (APS), 2004. Fellow of the American Society of Mechanical Engineers (ASME), 2003. Associate Fellow of the American Institute of Aeronautics and Astronautics (AIAA), 2006. == Books == Z. Zhang and G.E. Karniadakis, “Numerical Methods for Stochastic PDEs with White Noise”, Springer, Applied Mathematics Series, 2017. G.E. Karniadakis, A. Beskok, and N. Aluru, “Microflows and Nanoflows: Fundamentals and Simulation, Springer 2005. G.E. Karniadakis and R.M. Kirby, “Parallel Scientific Computing in C++ and MPI”, Cambridge University Press, March 2003. G.E. Karniadakis and A. Beskok, “Microflows: Fundamentals and Simulation”, Springer, 2001. (first textbook/monograph in this field). G.E. Karniadakis & S.J. Sherwin, “Spectral/hp Element Methods for CFD,” Oxford University Press, New York, 1999. (first monograph in this field); second edition, Oxford, 2005; third edition, 2013. == Patents == S. Suresh, L. Lu, M. Dao, and G.E. Karniadakis, “Solving inverse indentation Problems via Deep Learning with Applications to 3D printing and Other Engineering Projects, (NTU Ref: 2019-140) - June 24, 2019. M. Raissi, P. Perdikaris, and G.E. Karniadakis, Physics Informed Learning Machines U.S. Provisional Patent Application 6248319, March 29, 2017. G.E. Karniadakis and Y. Du, “Method and Apparatus for Reducing Turbulent Drag”, Patent No. 6,333,593 B1, Dec 25, 2001. G.E. Karniadakis, K. Breuer and V. Symeonidis, “Method and Apparatus for Reducing Turbulent Drag (continuing part)”, Patent No. 6,520,455 B2, Feb. 18, 2003. C. Chryssostomidis, D. Sura, G.E. Karniadakis, C. Jaskolski, R. Kimbal, “Lorentz Acoustic Transmitter for Underwater Communications”, Patent No. 7,505,365, March 17, 2009. == References == == External links == George Karniadakis publications indexed by Google Scholar
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Wikipedia:George Peacock#0
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George Peacock FRS (9 April 1791 – 8 November 1858) was an English mathematician and Anglican cleric. He founded what has been called the British algebra of logic. == Early life == Peacock was born on 9 April 1791 at Thornton Hall, Denton, near Darlington, County Durham. His father, Thomas Peacock, was a priest of the Church of England, incumbent and for 50 years curate of the parish of Denton, where he also kept a school. In early life, Peacock did not show any precocity of genius. He was more remarkable for daring feats of climbing than for any special attachment to study. Initially, he received his elementary education from his father and then at Sedbergh School, and at 17 years of age, he was sent to Richmond School under James Tate, a graduate of Cambridge University. At this school he distinguished himself greatly both in classics and in the rather elementary mathematics then required for entrance at Cambridge. In 1809 he became a student of Trinity College, Cambridge. In 1812 Peacock took the rank of Second Wrangler, and the second Smith's prize, the senior wrangler being John Herschel. Two years later, he became a candidate for a fellowship in his college and won it immediately, partly by means of his extensive and accurate knowledge of the classics. A fellowship then meant about £200 a year, tenable for seven years provided the Fellow did not marry meanwhile, and capable of being extended after the seven years provided the Fellow took clerical orders, which Peacock did in 1819. == Mathematical career == The year after taking a Fellowship, Peacock was appointed a tutor and lecturer of his college, which position he continued to hold for many years. Peacock, in common with many other students of his own standing, was profoundly impressed with the need of reforming Cambridge's position ignoring the differential notation for calculus, and while still an undergraduate formed a league with Babbage and Herschel to adopt measures to bring it about. In 1815 they formed what they called the Analytical Society, the object of which was stated to be to advocate the d 'ism of the Continent versus the dot-age of the university. The first movement on the part of the Analytical Society was to translate from the French the smaller work of Lacroix on the differential and integral calculus; it was published in 1816. At that time the French language had the best manuals, as well as the greatest works on mathematics. Peacock followed up the translation with a volume containing a copious Collection of Examples of the Application of the Differential and Integral Calculus, which was published in 1820. The sale of both books was rapid, and contributed materially to further the object of the Society. In that time, high wranglers of one year became the examiners of the mathematical tripos three or four years afterwards. Peacock was appointed an examiner in 1817, and he did not fail to make use of the position as a powerful lever to advance the cause of reform. In his questions set for the examination the differential notation was for the first time officially employed in Cambridge. The innovation did not escape censure, but he wrote to a friend as follows: "I assure you that I shall never cease to exert myself to the utmost in the cause of reform, and that I will never decline any office which may increase my power to effect it. I am nearly certain of being nominated to the office of Moderator in the year 1818-1819, and as I am an examiner in virtue of my office, for the next year I shall pursue a course even more decided than hitherto, since I shall feel that men have been prepared for the change, and will then be enabled to have acquired a better system by the publication of improved elementary books. I have considerable influence as a lecturer, and I will not neglect it. It is by silent perseverance only, that we can hope to reduce the many-headed monster of prejudice and make the University answer her character as the loving mother of good learning and science." These few sentences give an insight into the character of Peacock: he was an ardent reformer and a few years brought success to the cause of the Analytical Society. Another reform at which Peacock labored was the teaching of algebra. In 1830 he published A Treatise on Algebra which had for its object the placing of algebra on a true scientific basis, adequate for the development which it had received at the hands of the Continental mathematicians. To elevate astronomical science the Astronomical Society of London was founded, and the three reformers Peacock, Babbage and Herschel were again prime movers in the undertaking. Peacock was one of the most zealous promoters of an astronomical observatory at Cambridge, and one of the founders of the Philosophical Society of Cambridge. In 1831 the British Association for the Advancement of Science (prototype of the American, French and Australasian Associations) held its first meeting in the ancient city of York. One of the first resolutions adopted was to procure reports on the state and progress of particular sciences, to be drawn up from time to time by competent persons for the information of the annual meetings, and the first to be placed on the list was a report on the progress of mathematical science. Whewell, the mathematician and philosopher, was a vice-president of the meeting: he was instructed to select the reporter. He first asked William Rowan Hamilton, who declined; he then asked Peacock, who accepted. Peacock had his report ready for the third meeting of the Association, which was held in Cambridge in 1833; although limited to Algebra, Trigonometry, and the Arithmetic of Sines, it is one of the best of the long series of valuable reports which have been prepared for and printed by the Association. In 1837 Peacock was appointed Lowndean Professor of Astronomy in the University of Cambridge, the chair afterwards occupied by Adams, the co-discoverer of Neptune, and later occupied by Robert Ball, celebrated for his Theory of Screws. An object of reform was the statutes of the university; he worked hard at it and was made a member of a commission appointed by the Government for the purpose. He was elected a Fellow of the Royal Society in January 1818. In 1842, Peacock was elected as a member of the American Philosophical Society. == Clerical career == He was ordained as a deacon in 1819, a priest in 1822 and appointed vicar of Wymeswold in Leicestershire in 1826 (until 1835). In 1839 he was appointed Dean of Ely cathedral, Cambridgeshire, a position he held for the rest of his life, some 20 years. Together with the architect George Gilbert Scott he undertook a major restoration of the cathedral building. This included the installation of the boarded ceiling. While holding this position he wrote a text book on algebra, A Treatise on Algebra (1830). Later, a second edition appeared in two volumes, the one called Arithmetical Algebra (1842) and the other On Symbolical Algebra and its Applications to the Geometry of Position (1845). == Symbolical algebra == Peacock's main contribution to mathematical analysis is his attempt to place algebra on a strictly logical basis. He founded what has been called the British algebra of logic; to which Gregory, De Morgan and Boole belonged. His answer to Maseres and Frend was that the science of algebra consisted of two parts—arithmetical algebra and symbolical algebra—and that they erred in restricting the science to the arithmetical part. His view of arithmetical algebra is as follows: "In arithmetical algebra we consider symbols as representing numbers, and the operations to which they are submitted as included in the same definitions as in common arithmetic; the signs + {\displaystyle +} and − {\displaystyle -} denote the operations of addition and subtraction in their ordinary meaning only, and those operations are considered as impossible in all cases where the symbols subjected to them possess values which would render them so in case they were replaced by digital numbers; thus in expressions such as a + b {\displaystyle a+b} we must suppose a {\displaystyle a} and b {\displaystyle b} to be quantities of the same kind; in others, like a − b {\displaystyle a-b} , we must suppose a {\displaystyle a} greater than b {\displaystyle b} and therefore homogeneous with it; in products and quotients, like a b {\displaystyle ab} and a b {\displaystyle {\frac {a}{b}}} we must suppose the multiplier and divisor to be abstract numbers; all results whatsoever, including negative quantities, which are not strictly deducible as legitimate conclusions from the definitions of the several operations must be rejected as impossible, or as foreign to the science." Peacock's principle may be stated thus: the elementary symbol of arithmetical algebra denotes a digital, i.e., an integer number; and every combination of elementary symbols must reduce to a digital number, otherwise it is impossible or foreign to the science. If a {\displaystyle a} and b {\displaystyle b} are numbers, then a + b {\displaystyle a+b} is always a number; but a − b {\displaystyle a-b} is a number only when b {\displaystyle b} is less than a {\displaystyle a} . Again, under the same conditions, a b {\displaystyle ab} is always a number, but a b {\displaystyle {\frac {a}{b}}} is really a number only when b {\displaystyle b} is an exact divisor of a {\displaystyle a} . Hence the following dilemma: Either a b {\displaystyle {\frac {a}{b}}} must be held to be an impossible expression in general, or else the meaning of the fundamental symbol of algebra must be extended so as to include rational fractions. If the former horn of the dilemma is chosen, arithmetical algebra becomes a mere shadow; if the latter horn is chosen, the operations of algebra cannot be defined on the supposition that the elementary symbol is an integer number. Peacock attempts to get out of the difficulty by supposing that a symbol which is used as a multiplier is always an integer number, but that a symbol in the place of the multiplicand may be a fraction. For instance, in a b {\displaystyle ab} , a {\displaystyle a} can denote only an integer number, but b {\displaystyle b} may denote a rational fraction. Now there is no more fundamental principle in arithmetical algebra than that a b = b a {\displaystyle ab=ba} ; which would be illegitimate on Peacock's principle. One of the earliest English writers on arithmetic is Robert Recorde, who dedicated his work to King Edward VI. The author gives his treatise the form of a dialogue between master and scholar. The scholar battles long over this difficulty—that multiplying a thing could make it less. The master attempts to explain the anomaly by reference to proportion; that the product due to a fraction bears the same proportion to the thing multiplied that the fraction bears to unity. But the scholar is not satisfied and the master goes on to say: "If I multiply by more than one, the thing is increased; if I take it but once, it is not changed, and if I take it less than once, it cannot be so much as it was before. Then seeing that a fraction is less than one, if I multiply by a fraction, it follows that I do take it less than once." Whereupon the scholar replies, "Sir, I do thank you much for this reason, – and I trust that I do perceive the thing." The fact is that even in arithmetic the two processes of multiplication and division are generalized into a common multiplication; and the difficulty consists in passing from the original idea of multiplication to the generalized idea of a tensor, which idea includes compressing the magnitude as well as stretching it. Let m {\displaystyle m} denote an integer number; the next step is to gain the idea of the reciprocal of m {\displaystyle m} , not as 1 m {\displaystyle {\frac {1}{m}}} but simply as / m {\displaystyle /m} . When m {\displaystyle m} and / n {\displaystyle /n} are compounded we get the idea of a rational fraction; for in general m / n {\displaystyle m/n} will not reduce to a number nor to the reciprocal of a number. Suppose, however, that we pass over this objection; how does Peacock lay the foundation for general algebra? He calls it symbolical algebra, and he passes from arithmetical algebra to symbolical algebra in the following manner: "Symbolical algebra adopts the rules of arithmetical algebra but removes altogether their restrictions; thus symbolical subtraction differs from the same operation in arithmetical algebra in being possible for all relations of value of the symbols or expressions employed. All the results of arithmetical algebra which are deduced by the application of its rules, and which are general in form though particular in value, are results likewise of symbolical algebra where they are general in value as well as in form; thus the product of a m {\displaystyle a^{m}} and a n {\displaystyle a^{n}} which is a m + n {\displaystyle a^{m+n}} when m {\displaystyle m} and n {\displaystyle n} are whole numbers and therefore general in form though particular in value, will be their product likewise when m {\displaystyle m} and n {\displaystyle n} are general in value as well as in form; the series for ( a + b ) n {\displaystyle (a+b)^{n}} determined by the principles of arithmetical algebra when n {\displaystyle n} is any whole number, if it be exhibited in a general form, without reference to a final term, may be shown upon the same principle to the equivalent series for ( a + b ) n {\displaystyle (a+b)^{n}} when n {\displaystyle n} is general both in form and value." The principle here indicated by means of examples was named by Peacock the "principle of the permanence of equivalent forms," and at page 59 of the Symbolical Algebra it is thus enunciated: "Whatever algebraic forms are equivalent when the symbols are general in form, but specific in value, will be equivalent likewise when the symbols are general in value as well as in form." For example, let a {\displaystyle a} , b {\displaystyle b} , c {\displaystyle c} , d {\displaystyle d} denote any integer numbers, but subject to the restrictions that b {\displaystyle b} is less than a {\displaystyle a} , and d {\displaystyle d} less than c {\displaystyle c} ; it may then be shown arithmetically that ( a − b ) ( c − d ) = a c + b d − a d − b c {\displaystyle (a-b)(c-d)=ac+bd-ad-bc} . Peacock's principle says that the form on the left side is equivalent to the form on the right side, not only when the said restrictions of being less are removed, but when a {\displaystyle a} , b {\displaystyle b} , c {\displaystyle c} , d {\displaystyle d} denote the most general algebraic symbol. It means that a {\displaystyle a} , b {\displaystyle b} , c {\displaystyle c} , d {\displaystyle d} may be rational fractions, or surds, or imaginary quantities, or indeed operators such as d d x {\displaystyle {\frac {d}{dx}}} . The equivalence is not established by means of the nature of the quantity denoted; the equivalence is assumed to be true, and then it is attempted to find the different interpretations which may be put on the symbol. It is not difficult to see that the problem before us involves the fundamental problem of a rational logic or theory of knowledge; namely, how are we able to ascend from particular truths to more general truths. If a {\displaystyle a} , b {\displaystyle b} , c {\displaystyle c} , d {\displaystyle d} denote integer numbers, of which b {\displaystyle b} is less than a {\displaystyle a} and d {\displaystyle d} less than c {\displaystyle c} , then ( a − b ) ( c − d ) = a c + b d − a d − b c {\displaystyle (a-b)(c-d)=ac+bd-ad-bc} . It is first seen that the above restrictions may be removed, and still the above equation holds. But the antecedent is still too narrow; the true scientific problem consists in specifying the meaning of the symbols, which, and only which, will admit of the forms being equal. It is not to find "some meanings", but the "most general meaning", which allows the equivalence to be true. Let us examine some other cases; we shall find that Peacock's principle is not a solution of the difficulty; the great logical process of generalization cannot be reduced to any such easy and arbitrary procedure. When a {\displaystyle a} , m {\displaystyle m} , n {\displaystyle n} denote integer numbers, it can be shown that a m a n = a m + n {\displaystyle a^{m}a^{n}=a^{m+n}} . According to Peacock the form on the left is always to be equal to the form on the right, and the meanings of a {\displaystyle a} , m {\displaystyle m} , n {\displaystyle n} are to be found by interpretation. Suppose that a {\displaystyle a} takes the form of the incommensurate quantity e {\displaystyle e} , the base of the natural system of logarithms. A number is a degraded form of a complex quantity p + q − 1 {\displaystyle p+q^{\sqrt {-1}}} and a complex quantity is a degraded form of a quaternion; consequently one meaning which may be assigned to m {\displaystyle m} and n {\displaystyle n} is that of quaternion. Peacock's principle would lead us to suppose that e m e n = e m + n {\displaystyle e^{m}e^{n}=e^{m+n}} , m {\displaystyle m} and n {\displaystyle n} denoting quaternions; but that is just what William Rowan Hamilton, the inventor of the quaternion generalization, denies. There are reasons for believing that he was mistaken, and that the forms remain equivalent even under that extreme generalization of m {\displaystyle m} and n {\displaystyle n} ; but the point is this: it is not a question of conventional definition and formal truth; it is a question of objective definition and real truth. Let the symbols have the prescribed meaning, does or does not the equivalence still hold? And if it does not hold, what is the higher or more complex form which the equivalence assumes? Or does such equivalence form even exist? == Private life == Politically, George Peacock was a Whig. He married Frances Elizabeth, the daughter of William Selwyn. They had no children. His last public act was to attend a meeting of the university reform commission. He died in Ely on 8 November 1858, in the 68th year of his age, and was buried in Ely cemetery. == Bibliography == A Treatise on Algebra (J. & J. J. Deighton, 1830). A Treatise on Algebra (2nd ed., Scripta Mathematica, 1842–1845). Vol. 1: Arithmetical Algebra (1842). Vol. 2: On Symbolical Algebra and its Applications to the Geometry of Position (1845) Life of Thomas Young: M.D., F.R.S., &c.; and One of the Eight Foreign Associates of the National Institute of France (John Murray, 1855). == References == == Sources == Macfarlane, Alexander (2009) [1916]. Lectures on Ten British Mathematicians of the Nineteenth Century. Mathematical monographs. Vol. 17. Cornell University Library. ISBN 978-1-112-28306-2. (complete text Archived 29 July 2017 at the Wayback Machine at Project Gutenberg) == External links == O'Connor, John J.; Robertson, Edmund F., "George Peacock", MacTutor History of Mathematics Archive, University of St Andrews Biography of Peacock
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Wikipedia:George Pólya#0
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George Pólya (; Hungarian: Pólya György [ˈpoːjɒ ˈɟørɟ]; December 13, 1887 – September 7, 1985) was a Hungarian-American mathematician. He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University. He made fundamental contributions to combinatorics, number theory, numerical analysis and probability theory. He is also noted for his work in heuristics and mathematics education. He has been described as one of The Martians, an informal category which included one of his most famous students at ETH Zurich, John von Neumann. == Life and works == Pólya was born in Budapest, Austria-Hungary, to Anna Deutsch and Jakab Pólya, Hungarian Jews who had converted to Christianity in 1886. Although his parents were religious and he was baptized into the Catholic Church upon birth, George eventually grew up to be an agnostic. He received a PhD under Lipót Fejér in 1912, at Eötvös Loránd University. He was a professor of mathematics from 1914 to 1940 at ETH Zürich in Switzerland and from 1940 to 1953 at Stanford University. He remained a professor emeritus at Stanford for the rest of his career, working on a range of mathematical topics, including series, number theory, mathematical analysis, geometry, algebra, combinatorics, and probability. He was invited to speak at the ICM at Bologna in 1928, at Oslo in 1936 and at Cambridge, Massachusetts, in 1950. On September 7, 1985, Pólya died in Palo Alto, California due to complications of a stroke he suffered during that summer. == Heuristics == Early in his career, Pólya wrote with Gábor Szegő two influential problem books, Problems and Theorems in Analysis (I: Series, Integral Calculus, Theory of Functions and II: Theory of Functions. Zeros. Polynomials. Determinants. Number Theory. Geometry). Later in his career, he spent considerable effort to identify systematic methods of problem-solving to further discovery and invention in mathematics for students, teachers, and researchers. He wrote five books on the subject: How to Solve It, Mathematics and Plausible Reasoning (Volume I: Induction and Analogy in Mathematics, and Volume II: Patterns of Plausible Inference), and Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving (volumes 1 and 2). In How to Solve It, Pólya provides general heuristics for solving a gamut of problems, including both mathematical and non-mathematical problems. The book includes advice for teaching students of mathematics and a mini-encyclopedia of heuristic terms. It was translated into several languages and has sold over a million copies. The book is still used in mathematical education. Douglas Lenat's Automated Mathematician and Eurisko artificial intelligence programs were inspired by Pólya's work. In addition to his works directly addressing problem solving, Pólya wrote another short book called Mathematical Methods in Science, based on a 1963 work supported by the National Science Foundation edited by Leon Bowden and published by the Mathematical Association of America (MAA) in 1977. As Pólya notes in the preface, Bowden carefully followed a tape recording of a course Pólya gave several times at Stanford in order to put the book together. Pólya notes in the preface "that the following pages will be useful, yet they should not be regarded as a finished expression." == Legacy == There are three prizes named after Pólya, causing occasional confusion of one for another. In 1969 the Society for Industrial and Applied Mathematics (SIAM) established the George Pólya Prize, given alternately in two categories for "a notable application of combinatorial theory" and for "a notable contribution in another area of interest to George Pólya." In 1976 the Mathematical Association of America (MAA) established the George Pólya Award "for articles of expository excellence" published in the College Mathematics Journal. In 1987 the London Mathematical Society (LMS) established the Pólya Prize for "outstanding creativity in, imaginative exposition of, or distinguished contribution to, mathematics within the United Kingdom." In 1991, the MAA established the George Pólya Lectureship series. Stanford University has a Polya Hall named in his honor. == Selected publications == === Books === Aufgaben und Lehrsätze aus der Analysis, 1st edn. 1925. ("Problems and theorems in analysis“). Springer, Berlin 1975 (with Gábor Szegő). Reihen. 1975, 4th edn., ISBN 3-540-04874-X. Funktionentheorie, Nullstellen, Polynome, Determinanten, Zahlentheorie. 1975, 4th edn., ISBN 3-540-05456-1. Mathematik und plausibles Schliessen. Birkhäuser, Basel 1988, Induktion und Analogie in der Mathematik, 3rd edn., ISBN 3-7643-1986-0 (Wissenschaft und Kultur; 14). Typen und Strukturen plausibler Folgerung, 2nd edn., ISBN 3-7643-0715-3 (Wissenschaft und Kultur; 15). – English translation: Mathematics and Plausible Reasoning, Princeton University Press 1954, 2 volumes (Vol. 1: Induction and Analogy in Mathematics, Vol. 2: Patterns of Plausible Inference) Schule des Denkens. Vom Lösen mathematischer Probleme ("How to solve it"). 4th edn. Francke Verlag, Tübingen 1995, ISBN 3-7720-0608-6 (Sammlung Dalp). – English translation: How to Solve It, Princeton University Press 2004 (with foreword by John Horton Conway and added exercises) Vom Lösen mathematischer Aufgaben. 2nd edn. Birkhäuser, Basel 1983, ISBN 3-7643-0298-4 (Wissenschaft und Kultur; 21). – English translation: Mathematical Discovery: On Understanding, Learning and Teaching Problem Solving, 2 volumes, Wiley 1962 (published in one vol. 1981) Collected Papers, 4 volumes, MIT Press 1974 (ed. Ralph P. Boas). Vol. 1: Singularities of Analytic Functions, Vol. 2: Location of Zeros, Vol. 3: Analysis, Vol. 4: Probability, Combinatorics with R. C. Read: Combinatorial enumeration of groups, graphs, and chemical compounds, Springer Verlag 1987 (English translation of Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen, Acta Mathematica, vol. 68, 1937, pp. 145–254) ISBN 978-0387964133 with Godfrey Harold Hardy: John Edensor Littlewood Inequalities, Cambridge University Press 1934 Mathematical Methods in Science, MAA, Washington D. C. 1977 (ed. Leon Bowden) ISBN 0-88385-626-3 with Gordon Latta: Complex Variables, Wiley 1974 with Robert E. Tarjan, Donald R. Woods: Notes on introductory combinatorics, Birkhäuser 1983 ISBN 978-3-7643-3170-2 with Jeremy Kilpatrick: The Stanford mathematics problem book: with hints and solutions, New York: Teachers College Press 1974 with several co-authors: Applied combinatorial mathematics, Wiley 1964 (ed. Edwin F. Beckenbach) with Gábor Szegő: Isoperimetric inequalities in mathematical physics, Princeton, Annals of Mathematical Studies 27, 1951 === Articles === Pólya, G. (1922). "On the mean-value theorem corresponding to a given linear homogeneous differential equation". Trans. Amer. Math. Soc. 24 (4): 312–324. doi:10.1090/s0002-9947-1922-1501228-5. MR 1501228. Pólya, G. (1933). "Untersuchungen Über Lücken Und Singularitäten Von Potenzreihen". Ann. Math. 34 (4): 731–77. doi:10.2307/1968698. JSTOR 1968698. Polya, G. (1941). "On Functions Whose Derivatives Do Not Vanish in a Given Interval". Proc Natl Acad Sci U S A. 27 (4): 216–218. Bibcode:1941PNAS...27..216P. doi:10.1073/pnas.27.4.216. PMC 1078308. PMID 16578010. Polya, G. (1941). "Sur l'existence de fonctions entières satisfaisant à certaines conditions linéaires". Trans. Amer. Math. Soc. 50 (1): 129–139. doi:10.2307/1989913. JSTOR 1989913. MR 0004304. with Ralph P. Boas, Jr.: Boas, R. P.; Polya, G. (1941). "Generalizations of Completely Convex Functions". Proc Natl Acad Sci U S A. 27 (6): 323–325. Bibcode:1941PNAS...27..323B. doi:10.1073/pnas.27.6.323. PMC 1078330. PMID 16588467. Pólya, George (1942). "On converse gap theorems". Trans. Amer. Math. Soc. 52 (1): 65–71. doi:10.1090/s0002-9947-1942-0006577-0. MR 0006577. with Norbert Wiener: Pólya, George; Wiener, Norbert (1942). "On the oscillation of the derivatives of a periodic function". Trans. Amer. Math. Soc. 52 (2): 249–256. doi:10.1090/s0002-9947-1942-0007169-x. MR 0007169. "On the zeros of a derivative of a function and its analytic character" (PDF). Bull. Amer. Math. Soc. 49, Part 1 (3): 178–191. 1943. doi:10.1090/s0002-9904-1943-07853-6. MR 0007781. Polya, G. (1947). "A Minimum Problem About the Motion of a Solid Through a Fluid". Proc Natl Acad Sci U S A. 33 (7): 218–221. Bibcode:1947PNAS...33..218P. doi:10.1073/pnas.33.7.218. PMC 1079030. PMID 16588747. Polya, G. (1950). "Remark on Weyl's Note "Inequalities Between the Two Kinds of Eigenvalues of a Linear Transformation". Proc Natl Acad Sci U S A. 36 (1): 49–51. Bibcode:1950PNAS...36...49P. doi:10.1073/pnas.36.1.49. PMC 1063130. PMID 16588947. == See also == Integer-valued polynomial Laguerre–Pólya class Landau–Kolmogorov inequality Multivariate Pólya distribution Pólya's characterization theorem Pólya class Pólya conjecture Polya distribution Pólya enumeration theorem Pólya–Vinogradov inequality Pólya inequality Pólya urn model Pólya's theorem Pólya's proof that there is no "horse of a different color" Wallpaper group The Martians (scientists) == References == == External links == The George Pólya Award O'Connor, John J.; Robertson, Edmund F., "George Pólya", MacTutor History of Mathematics Archive, University of St Andrews George Pólya at the Mathematics Genealogy Project George Pólya, Gábor Szegö, Problems and theorems in analysis (1998) PolyaPower—an introduction to Polya's Heuristics at the Wayback Machine (archived July 28, 2009) George Pólya on UIUC's WikEd Memorial Resolution George Pólya — Biographical Memoirs of the National Academy of Sciences "Polya Guessing" on Vimeo
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Wikipedia:George Saitoti#0
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George Musengi Saitoti, E.G.H. (3 August 1945 – 10 June 2012) was a Kenyan politician, businessman and American- and British-trained economist, mathematician and development policy thinker. As a mathematician, Saitoti served as Head of the Mathematics Department at the University of Nairobi, pioneered the founding of the African Mathematical Union and served as its vice-president from 1976 to 1979. As an economist, Saitoti served as the Executive Chairman of the World Bank and the International Monetary Fund (IMF) in 1990–91, and as President of the African Caribbean and Pacific (ACP) Group of States in 1999–2000, at the crucial phase of re-negotiating the new development partnership agreement to replace the expired Lomé Convention between the ACP bloc and the European Union (EU). His book The Challenges of Economic and Institutional Reforms in Africa influenced practical policy directions on an array of areas during the turbulent 1980s and 1990s. Saitoti joined politics as a nominated Member of Parliament and Minister for Finance in 1983, rising to become Kenya's longest-serving Vice-President, a proficient Minister for education, Internal Security and Provincial Administration and Foreign Affairs. Few recognise him as a "reformist", but his recommendations as the Chair of the KANU Review Committee, popularly known as the "Saitoti Committee" in 1990–91, opened KANU to internal changes and set the stage for the repeal of Section 2A and Kenya's return to pluralist democracy. Saitoti left KANU and joined the opposition, becoming a kingpin figure in the negotiations that led to the "NARC Revolution" in 2002. As Minister for Internal Security and Provincial Administration, Acting Minister for Foreign Affairs and key member of the National Security Advisory Committee (NSAC), he later worked closely with the national Ministry of Defence to see through the Operation Linda Nchi against the Al-Shabaab insurgent group. In addition, rival factions had for decades invoked the infamous Goldenberg fraud to knock Saitoti out of politics, but the legal courts cleared him of the scandal in July 2006. Saitoti's dual heritage as a Maasai with Kikuyu family members predisposed him to a pan-Kenyan vision, but also denied him a strong ethnic base unlike his competitors. Saitoti was running as a candidate to succeed President Mwai Kibaki when he died in a helicopter crash in 2012. == Early life and education == George Saitoti was born on 3 August 1945 and brought up in Maasailand, where he spent his childhood herding cattle in line with the Masai culture, and attending school. He attended Ololua Primary School, Kajiado where he acquired his basic education in the 1950s. Between 1960 and 1963, he secured a place at Mang'u High School in Thika where he attained his high school education. He joined the ranks of Mang'u High School's highly decorated alumni including Kenya's third President, Mwai Kibaki, former Vice-President Moody Awori, Catholic Archbishop Ndingi Mwana-a-Nzeki, the late Environment Minister John Michuki, the late Trade Unionist and former Minister for Justice and Constitutional Affairs, Tom Mboya, and late Cardinal Maurice Michael Otunga. Saitoti was one of the recipients of the Kennedy Airlift scholarships in 1963 at the age of 18. This was a scholarship program started by Tom Mboya and William X. Scheinmann and supported by the John F. Kennedy foundation and the African American Students Association to address the educational needs of newly independent Kenya. Through this program, hundreds of East Africans studied in the United States of America and Canada. Saitoti studied at Brandeis University between 1963 and 1967 where he was a mathematician. During his time there, he was on the Wien Scholarship, specialising in Mathematics and Economics. His colleagues at the time remember that he enjoyed spending time in Cholmondeleys (the coffeehouse in the Castle) and excelled at high jump, ranking as one of the best in New England. In 1988, Saitoti received the first Brandeis Alumni Achievement Award, the highest honour the university bestows upon its graduates. Saitoti later moved to the United Kingdom where he acquired a Master of Science (MSc) degree in mathematics from the University of Sussex, Brighton. He enrolled for his doctoral studies at the University of Warwick where he acquired his PhD in mathematics in 1972; writing his dissertation under the supervision of professor Luke Hodgkin in the area of algebraic topology under the topic: Mod-2 K-Theory of the Second Iterated Loop Space on a Sphere. == Academic career == Upon his graduation, Saitoti returned to Kenya in 1972, commencing a career as a Mathematics lecturer at the University of Nairobi. One of his contributions was the institutionalisation of Mathematics as a discipline in Africa. During the first Pan-African Conference of Mathematicians held in Rabat, Morocco, in 1976, Saitoti was involved in the creation of the African Mathematical Union (AMU). He was elected the AMU's vice-president, a post which he held on up to 1979. By 1983, Saitoti's academic career was on the rise as associate professor and Head of the Mathematics Department. Outside the academy, Saitoti received several public appointments. On 3 November 1972, the Minister of Labour appointed him as the chairman of the Agricultural Wages Council (AWC). On 4 September 1979, the Minister for Tourism and Wildlife, John Ogutu, also appointed him as a committee member of the Natural Sciences Advisory Research Committee (TNSARC) chaired by professor S. O. Wandiga. In September 1983, he was appointed chairperson of the board of directors for the Rift Valley Institute of Science and Technology. He also served in other public capacities as chairman of Mumias Sugar Company and the Kenya Commercial Bank. === Development thinker === Top decision-makers in government had recognised Saitoti as a policy thinker and technocrat, of whom the KANU desperately needed to fix its institutions, politics and the economy. His seminal book, The Challenges of Economic and Institutional Reforms in Africa was widely praised by leading officials as providing practical policy proposals to deal with the various challenges facing Kenya and Africa. The book drew from Saitoti's experience as a seasoned scholar, consultant and experienced policy-maker/thinker, presenting a rigorous and multidisciplinary analysis of strategies for poverty alleviation, sustainable development, poverty reduction, combating HIV/AIDS and peace diplomacy. Saitoti also emphasised the importance of institutional reforms and sound public policies to sustainable economic growth in Africa. == Political career == Long before joining mainstream politics, Saitoti had a stint in the legislative duties. From 1974 to 1977, he represented Kenya in the defunct (since-revived) East African Community as a member of the East African Legislative Assembly. === Dual ancestry and politics of diversity === In October 1983, President Daniel arap Moi nominated Saitoti as a member of parliament and subsequently appointed him to the Cabinet as Minister for Finance. He held the position until 1989. During the 1988 general elections, Saitoti entered competitive politics and won the Kajiado North parliamentary seat that was previously held by Philip Odupoy. Prior to the tenure of Adupoy and Saitoti, the Kajiado North multi-ethnic constituency was held by the popular politician, John Keen, another half-Maasai who champion a nationalist vision and worked over the years to ensure the advancement of his mother's people. For more than 25 years, professor George Saitoti has represented Kajiado North since 1988, recapturing the seat in consecutive elections in 1992, 1997, 2002 and 2007. Building on John Keen's legacy of a cosmopolitan constituency, Saitoti transformed Kajiado North into Kenya's most ethnically integrated multi-ethnic legislative area that also provided a safe haven to Kenyans, forcibly displaced by the 1991–2008 cycles of ethnic violence in neighbouring areas. The area is also ranked among the top ten wealthiest, economically dynamic and fastest growing regions in Kenya. According to figures released by the Government of Kenya in 2009, Kajiado North has had an average poverty index of 10.66 per cent for the last three years, making it one of the richest constituencies in Kenya (see table 1). === Kenya's sixth Vice-President === After the 1988 General Election, President Moi appointed Saitoti as Kenya's sixth vice-president. Saitoti became Kenya's longest sitting vice-president serving for 13 years under President Daniel arap Moi between May 1989 and January 1998 and again between April 1999 and August 2002 (see table 2). At the same time, he served as Minister for Finance. In 1990–1991, Saitoti was the Executive Chairman of the World Bank and the International Monetary Fund (IMF). In 1999–2000, Saitoti also served as president of the African, Caribbean and Pacific Group of States, becoming instrumental in helping negotiate a new development partnership agreement to succeed the previous Lomè Convention that expired in February 2000 between the ACP and the European Union. The hallmarks of Saitoti tenure as vice-president were efficiency, sobriety and loyalty as President Moi's most trusted lieutenant. Even when President Moi dithered in naming a new deputy after the 1997 elections, Saitoti was still his favoured choice 14 months down the line. The same traits of efficiency, patience and loyalty would make him one of President Mwai Kibaki's trusted Ministers. === Reforming KANU'S one-party system === When Saitoti was appointed vice-president on 1 May 1989, KANU was back-pedaling on re-democratizing the country. At the same time, the party was fragmented over the succession divide between a sit-tight "KANU-A" and a more pro-change "KANU-B" led by Saitoti. The new vice-president was, therefore, compelled to walk the tightrope between being the face of change in the ruling party and remaining loyal to his principal who, after re-election as president in 1988, had amended the constitution to increase his power to dismiss judges and widened police powers. On New Year's Day 1990, the vocal cleric Rev. Timothy Njoya called on all Africans to demand a multiparty system of government. Following the Saba Saba riots on 7 July 1990, President Moi announced the formation of the KANU Review Committee under the chairmanship of Prof George Saitoti, popularly known as the Saitoti Committee. === The Saitoti Review Committee === The Saitoti Review Committee was mandated to investigate the party's internal electoral and disciplinary conduct. The committee traversed the country collecting people's opinions on the party, astounding foe, friend and critics alike and offering a rare forum for direct criticism and outbursts. In January 1991, KANU's executive committee adopted the recommendation by George Saitoti, that critics of the party cease being expelled but suspended for one or two years. The recommendations of the report were open for debate during the National Delegates Conference at Karasani in Nairobi. President Moi backed the adoption and implementation of the report in toto, against what many speakers at the conference had expected. This opened the reforms gates, eventually setting the stage for the repeal of Section 2A in 1991 that returned Kenya to back to a multiparty system of government. The Saitoti Review Committee thrust the party on the reform path, but also widened internal ideological schisms between "KANU-A" conservatives and "KANU-B" pro-reformers over the Moi succession question. === KANU'S war on Saitoti === Saitoti was in the eye of a nasty succession storm that rocked KANU before and after the 1997 elections. Maasai purists led by Minister William Ole Ntimama and senior Maasai elders 're-Kikuyunized' Saitoti's dual ancestry, amplifying his Kikuyu family linkages as a scheme to weaken his political base and to challenge his status as a Maasai elder. Despite his steadfast loyalty to KANU and President Moi, Saitoti was frequently ignored, humiliated and frustrated by the party and its top echelons. Around the same time Foreign Affairs Minister Robert Ouko was murdered in February 1990, Saitoti claims that attempts were made on his life. After the 1997 general elections, he was dropped as vice-president, although no replacement was appointed. Even as President Moi reappointed him in April 1999, on the roadside in Limuru, Kiambu he made a scathing remark to the effect that: "I've given back Prof Saitoti the seat of Vice-President, hopefully now your sufurias (pots) will be full of food." Months before the general elections of 2002, Saitoti's name was deleted from the list of KANU delegates and his ascendancy to the presidency blocked by 'unknown' party members. On 18 March 2002, when KANU held its national delegates conference at the Kasarani sports complex, the move to block Saitoti from the succession game was manifest. The meeting amended the party constitution to allow for the merger between KANU and Raila Odinga's National Development Party (NDP) to create the "New KANU". But it also introduced four new positions of party vice-chairmen primarily to water down Saitoti's position as vice-president and Moi's most likely successor as president. === The National Rainbow Coalition (NARC) === It was clear that Moi did not even want him as one of the four vice-chairmen posts reserved for Uhuru Kenyatta, Kalonzo Musyoka, Katana Ngala and Musalia Mudavadi. Moi told Saitoti to his face that he was not "presidential material". As a "Maasai-Kikuyu," Saitoti lacked the ethnic numbers he needed in the political horse-trading in Moi's power game. Instead, Moi finger-pointed as his heir Uhuru Kenyatta, perceived to have a large ethnic base as a pure-bred Kikuyu with the "Kenyatta" mystique. Saitoti gracefully bowed out of the race, living to fight another day, but not without his famous line: There comes a time when the nation is much more important than an individual. But the KANU-NDP marriage came to a tragic end when Moi named Uhuru rather than Raila Odinga as his successor. In August 2002, Odinga left KANU to defeat Moi's "use and dump game," and joined a group of KANU rebels" coalesced around the "Rainbow Alliance" lobby that later transformed itself into the Liberal Democratic Party (LDP). Saitoti also walking out of KANU and became a key LDP figure. In October 2002, LDP joined the National Alliance of Kenya (NAK) of Mwai Kibaki, Charity Ngilu and Wamalwa Kijana to form the National Rainbow Coalition (NARC). Saitoti became a member of the NARC Summit, the highest organ of the coalition. == The "NARC revolution" == When the NARC flag-bearer, Mwai Kibaki, decisively defeated KANU and Uhuru Kenyatta, Saitoti was appointed to the Ministry of Education. He was the man in charge of implementing NARC's flagship and globally acclaimed free primary education in Kenya. === The Kibaki stalwart === After 2004, as the NARC consensus crumbled, Saitoti left the agitating LDP camp and threw his lot behind President Kibaki. He canvassed for the government-sponsored draft Constitution, which lost to a combined KANU-LDP campaign during the November 2005 referendum. During the 2007 elections, Saitoti defended his Kajiado North parliamentary seat on the Party of National Unity (PNU) ticket, Kibaki's re-election vehicle, launched three months to the election on 16 September 2007. The courts ordered a vote recount in Kajiado North, but Saitoti beat his closest competitor, Moses Ole Sakuda with close to 20,000 votes. Saitoti blamed his re-election glitch on intrigues of power by KANU forces within the PNU campaign which underwrote his rivals to knock him out of politics and potentially out of the 2012 Presidential elections. But he had remained reticent about it. Saitoti's traits of patience, efficiency and loyalty to Kibaki paid off. On 8 January 2008, he was appointed Minister of State for Internal Security and Provincial Administration in the Office of the President, a position previously occupied by a Kibaki confidant, John Michuki. Saitoti retained the Internal Security docket even after President Kibaki and Prime Minister Raila Odinga established the power-sharing government that ended the 2008 post-election crisis. Between October 2010 and August 2011, Saitoti was appointed Minister for Foreign Affairs on an acting capacity after the incumbent, Moses Wetangula, stepped aside to allow investigations on alleged corruption. === Cabinet sub-committee on ICC === In July 2009, Saitoti was appointed to head a special cabinet sub-committee formed to oversee the affairs of the International Criminal Court (ICC) in Kenya. Members of this bi-partisan committee include; George Saitoti, Mutula Kilonzo and Moses Wetangula (PNU) and James Orengo, Otieno Kajwang and Amason Kingi (ODM). (Following a cabinet reshuffle in April 2012, Eugene Wamalwa and Prof. Sam Ongeri have replaced Kilonzo and Wetangula). The role of the sub-Committee as a liaison and co-ordination body between the ICC and the Kenyan government took a center-stage from December 2010 when the ICC Chief Prosecutor, Luis Moreno Ocampo, indicted six prominent Kenyans for alleged crimes against humanity relating to the 2008 post-election violence. As the Minister of Internal Security and the chairman of the Cabinet Sub-committee on ICC on YouTube and security matters, Saitoti is the guarantor of the government's commitment to the ICC process. Arising from this, several analysts have claimed the suspects' fate lie with the sub-committee. Saitoti came out strongly criticising the invocation of President Kibaki in the ICC debate, calling for sobriety from politicians. Saitoti has maintained a legal interpretation on whether the suspects can vie for presidency in the coming elections, stressing that only the constitution can bar or let them free to enter the race. === PNU party politics === On 19 December 2008, President Mwai Kibaki who was unanimously endorsed as Party Leader at the PNU National Delegates Conference (NDC) held at Kasarani sports complex in Nairobi. In accordance with the Political Parties Act (2008), Saitoti was elected PNU chairman, becoming the second-in-command in the party hierarchy since he lost as KANU Vice-President in the battle for the Moi succession in March 2002. His elevation, however, complicated coalition politics and raised the stakes for the Kibaki succession in PNU. Other presidential hopefuls, Uhuru Kenyatta and Kalonzo Musyoka, shunned the party and embarked on consolidating their respective parties. In November 2010, Musyoka, Kenyatta and Saitoti signed a protocol to form and transform the PNU Alliance into a common political vehicle for the 2013 presidential race. But the imperative to comply with the Political Parties Act (2011) forced them to abandon the Alliance and shift attention to their respective parties. == Goldenberg scandal == Saitoti was both Vice-President and Finance Minister at the height of the 1991–1993 Goldenberg scandal. Even though his own culpability in the fraud has never been established, for decades the Goldenberg has become the proverbial Sword of Damocles used against Saitoti in intra-elite power wars. In early 1999, Raila Odinga as a presidential contender to succeed Moi as president, sued Saitoti and others over alleged role in the Goldenberg scandal. Three months after his re-appointment as vice-president on 3 April 1999, Otieno Kajwang', a Raila ally, moved a private member's motion of no confidence in the vice-president for his alleged role in the Goldenberg fraud. Saitoti survived the onslaught. The Goldenberg spectre returned to haunt Saitoti in the wake of the fierce political infighting between the LDP/KANU faction and Kibaki supporters in NARC that followed the 2005 referendum. On 3 February 2006 a report by the Goldenberg Commission of Inquiry, chaired by Justice Samuel Bosire, recommended that George Saitoti should face criminal charges for his involvement in the Goldenberg scandal. On 13 February 2006, Saitoti voluntarily stepped aside from his ministerial docket to pave way for investigations into the allegations. However, on 31 July 2006, a three-judge bench headed by Justice Joseph Nyamu issued a certiorari order clearing Prof Saitoti of any wrongdoing, expunging his name from the Bosire Commission Report and issuing an order on permanent stay of prosecution against Saitoti. In dismissing the 23 paragraphs of the report, the High Court (Kenya) bench cited three inter-related errors of commission and omission by the Bosire Commission: The inquiry into the Goldenberg fraud had created a pyramid of noticeable bias, discriminatory treatment of evidence, submissions and factual errors that undermined the pursuit of justice and fairness. The factual flaws, biased and unprofessional handling of evidence by Inquiry led to wrong findings. The Inquiry's claim that Prof Saitoti illegally approved the 15 per cent ex-gratia payments as additional payment over and above the 20 per cent export compensation allowed at the time under the law were factually wrong. Indeed, the customs refunds, which Saitoti was accused of approving, were actually passed by Parliament. Long delay and wrong findings by the Goldenberg inquiry denied Prof. Saitoti any conceivable chance fair trial and justice. On 15 November 2006, President Kibaki reappointed Saitoti back to Cabinet. In April 2012, the vetting board found Justice Samuel Bosire unfit to serve in the judiciary citing fails as the Chairman of the Goldenberg Commission of Inquiry. He ignored a High Court Order to summon retired President Daniel arap Moi, Musalia Mudavadi and Nicholas Biwott as witnesses. The vetting board also accused Justice Nyamu of undermining public confidence in the courts for issuing a permanent stay of prosecution against Saitoti. == Linda Nchi == Starting October 2011, Saitoti worked closely with national Minister of Defence Mohamed Yusuf Haji to see through Linda Nchi, a coordinated operation in southern Somalia between the Somali military and the Kenyan military against the Al-Shabaab group of insurgents. The mission was officially led by the Somali army, with the Kenyan forces providing a support role. In early June 2012, Haji signed another agreement re-hatting Kenya's deployed military forces in Somalia under the AMISOM general command. == The Kibaki succession race == In November 2011, Saitoti confirmed that he was in the race to succeed President Kibaki, who is set to retire after the next general election. Saitoti reiterated his candidature in January 2012, continuing to tour Kenya, with meet-the-people excursions to the Rift Valley, Eastern and Central provinces. It appeared to be history repeating itself in the battle for the soul of the Kikuyu between, Saitoti, a Maasai with Kikuyu kith and kin, and Uhuru Kenyatta, a thorough-bred Kikuyu. Uhuru is widely thought as the presumptive successor to President Kibaki, but Saitoti was emerging also, as a likely candidate. In the event that Uhuru's run for the presidency is thwarted by the confirmed charges by the ICC, it remains a too-up as to whether Saitoti would have benefited from the spin-off. == Private life == Saitoti was a businessman who had interests in agriculture, horticulture, real estates, hospitality and pastoralism. Saitoti's family life received little publicity. His wife, Margaret Saitoti, was with him when the High Court dropped charges in the 16-year Goldenberg case. His brother, Ronald Musengi, has been a banking executive with the Kenya Commercial Bank. In February 2012, Ronald Musengi applied to be a member of the National Police Service Commission. == Death == Saitoti died on Sunday 10 June 2012 at around 9:00 am when a Eurocopter AS350 helicopter belonging to the Kenya Police Air Wing registration 5Y-CDT, carrying him and the Assistant Minister for Internal Security, Joshua Orwa Ojode, crashed in the Kibiku area of Ngong forest, killing them and four others. He was buried on 16 June in Kajiado North constituency. After the Maasai elders agreed to abandon the traditional burial rites and embrace the Catholic way, fifty bulls were slaughtered at the funeral in accordance with Maasai tradition. Al Shabaab militias and drug traffickers are the main suspects for the disaster. Investigations have not yet found anyone guilty.[1] Saitoti was to table a ministerial statement in Parliament. == List of publications == Saitoti, G. (2005). "Keynote address given during the official opening of the sub-regional seminar for TIVET policy makers and UNESCO UNEVOC Center Coordinators". Nairobi, Kenya. ____________(2004). "Education in Kenya: Challenges and policy responses". Paper presented at the Council on Foreign Relations, Washington D.C. ____________(2003) "National conference on education and training, Meeting the challenges of education and training during 21st century". Nairobi. ____________(2003). "Reflections on Africa Development", Journal of Third World Studies. Saitoti, G. and KANU Review Committee(2002), Report of the KANU Review Committee, 1990. The Committee, Nairobi. ____________(2002).The Challenges of Economic and Institutional Reforms in Africa. Ashgate Publishers Limited. ____________(1985). i mathematica, Politechnika Warszawska Technical. ____________ "A remark on Mod 2 K-Theory fundamental classes". Ann. Fac. Sci. Univ. Nat. Zaïre (Kinshasa)Sect. Math.-Phys. 3 (1977), no. 1, 61–63. ____________"Homology of a differential algebra". Publ. Math. Debrecen 23 (1976), no. 3-4, 235—237. ____________"K-Theory fundamental classes". Demonstration Math. 8 (1975), No. 4, 365–377. ____________A note on the homology of a differential graded algebra. Nigerian Journal of Science. 8 (1974), no. 1-2,127–130. ____________Loop spaces and K-theory. Journal of London Mathematics Society.(2) 9 (1974/75), 423–428. == Positions == Member, National Security Committee (NSAC), Kenya. (2008 – death) Chairman, Cabinet Sub-committee on ICC, Kenya. (2009 – death) Chairman, Party of National Unity (PNU) (2008 – death) Acting Minister for Foreign Affairs (October 2010 – August 2011) Minister of State for Provincial Administration and Internal Security (January 2008 – death) Minister of Education (November 2006 – January 2008) Minister of Education (7 December 2005 – 13 February 2006) Minister of Education (January 2003 – November 2005) Vice-President (3 April 1999 – 30 August 2002) Minister for Planning and National Development (December 1997 – April 1999) Vice-President (1 May 1989 – 8 January 1998) Minister of Finance (1983–1988) Vice-President, African Mathematical Union (1976–1979) == References == == External links == https://web.archive.org/web/20070927182239/http://www.parliament.go.ke/MPs/members_saitoti_prof_g.php Research
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Wikipedia:George Szekeres#0
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George Szekeres AM FAA (Hungarian: [ˈsɛkɛrɛʃ]; 29 May 1911 – 28 August 2005) was a Hungarian–Australian mathematician. == Early years == Szekeres was born in Budapest, Hungary, as Szekeres György and received his degree in chemistry at the Technical University of Budapest. He worked six years in Budapest as an analytical chemist. He married Esther Klein in 1937. Being Jewish, the family had to escape from the Nazi persecution so Szekeres took a job in Shanghai, China. There they lived through World War II, the Japanese occupation and the beginnings of the Communist revolution. == Career == In 1948, he was offered a position at the University of Adelaide, Australia, that he gladly accepted. After all the troubles he had had, he began flourishing as a mathematician. In 1963, the family moved to Sydney, where Szekeres took a position at the University of New South Wales, and taught there until his retirement in 1975. He also devised problems for secondary school mathematical olympiads run by the university where he taught, and for a yearly undergraduate competition run by the Sydney University Mathematics Society. Szekeres worked closely with many prominent mathematicians throughout his life, including Paul Erdős, his wife Esther, Pál Turán, Béla Bollobás, Ronald Graham, Alf van der Poorten, Miklós Laczkovich, and John Coates. == Honours == In 1968 he was the winner of the Thomas Ranken Lyle Medal of the Australian Academy of Science. In May 2001, a festschrift was held in honour of his ninetieth birthday at the University of New South Wales. In January 2001 he was awarded the Australian Centenary Medal "for service to Australian society and science in pure mathematics". In 2001, the Australian Mathematical Society created the George Szekeres Medal in his honour. In June 2002, he was made a Member of the Order of Australia (AM) 'for service to mathematics and science, particularly as a contributor to education and research, to the support and development of the University of New South Wales Mathematics Competition and the Australian Mathematical Olympiad Team.' == Personal life == The so-called Happy Ending problem is an example of how mathematics pervaded George's life. During 1933, George and several other students met frequently in Budapest to discuss mathematics. At one of these meetings, Esther Klein proposed the following problem: Given five points in the plane in general position, prove that four of them form a convex quadrilateral. After allowing George, Paul Erdős, and the other students to scratch their heads for some time, Esther explained her proof. Subsequently, George and Paul wrote a paper (1935) that generalises this result; it is regarded as one of the foundational works in the field of combinatorial geometry. Erdős dubbed the original problem the "Happy Ending" problem because it resulted in George and Esther's marriage in 1937. George and Esther died within an hour of each other on 28 August 2005 in Adelaide, South Australia. == See also == Powerful number Szekeres snark Generalized continued fraction Kruskal–Szekeres coordinates Szekeres–Wilf number Schröder's equation Erdős–Szekeres theorem == Notes == == References == Giles, J. R., Wallis, J. S., "George Szekeres. With affection and respect", Journal of the Australian Mathematical Society, Series A, Vol 21 (1976), No 4, pp. 385–392. Cowling, M., "Obituary George and Esther Szekeres", Gazette of the Australian Mathematical Society, Vol 32 (2005), No 4, pp. 221–224. Erdős, Paul; Szekeres, George (1935). "A combinatorial problem in geometry". Compositio Mathematica. 2: 463–470. == External links == MacTutor Biography The work of George Szekeres on functional equations Keith Briggs
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Wikipedia:Georges Darmois#0
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Georges Darmois (24 June 1888 – 3 January 1960) was a French mathematician and statistician. He pioneered in the theory of sufficiency, in stellar statistics, and in factor analysis. He was also one of the first French mathematicians to teach British mathematical statistics. He is one of the eponyms of the Koopman–Pitman–Darmois theorem and sufficient statistics and exponential families. == Biography == Darmois was born on 24 June 1888 in Éply. He was admitted to École normale supérieure in 1906 and passed subsequently the agrégation de mathematiques in 1909. From 1911 to 1914, he was a qualified assistant (agrégé préparateur) at the École normale supérieure, where his scientific activities were directed by Émile Borel who rapidly appreciated his talent. Darmois earned his doctorate from the University of Paris in 1921. He defended a thesis on algebraic curves and partial differential equations before the jury consisting of Émile Picard and Édouard Goursat. In 1949, he succeeded Maurice René Fréchet as the Chair of Calculus of Probabilities and Mathematical Physics at the University of Paris, who had himself succeeded Émile Borel. His research spanned several fields of pure and applied mathematics, including geometry, general relativity, physics, statistics, time series, and econometrics. He was elected fellow of the Econometric Society in 1952. In 1955 he was elected as a Fellow of the American Statistical Association. He was also the president of International Statistical Institute from 1953 until 1960. == Contributions == His scientific contributions include: Giving the first rigorous proof of Fréchet–Darmois–Cramér–Rao inequality, also known as Cramér–Rao bound, in 1945 independently from Rao and Cramér. Developing the notion of Koopman–Darmois family of distributions also known as the exponential family of distributions. Establishing Koopman–Pitman–Darmois theorem Characterizing Gaussian distributions with Darmois–Skitovich theorem == References == == External links == O'Connor, John J.; Robertson, Edmund F., "Georges Darmois", MacTutor History of Mathematics Archive, University of St Andrews Georges Darmois at the Mathematics Genealogy Project Correspondence between Ronald Fisher and Georges Darmois Laurent Mazliak (2010) Borel, Fréchet, Darmois: La découverte des statistiques par les probabilistes français. Journal Electronique d'Histoire des Probabilités et de la Statistique December. John Aldrich (2010)Tales of Two Societies: London and Paris 1860-1940 Journal Electronique d'Histoire des Probabilités et de la Statistique December.
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Wikipedia:Georges Fournier (Jesuit)#0
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Georges Fournier (31 August 1595 – 13 April 1652) was a French Jesuit priest, geographer and mathematician. == Biography == Fournier served as a naval military chaplain on a ship of the line, and acquired a strong knowledge of technical and naval matters. In 1642, he published the treaty Hydrographie, where he attempted to provide a scientific foundation to the design of ships. At the time, results like Couronne or Sovereign of the Seas were obtained by empirical trials and errors. He also authored a Treaty of fortifications or military architecture, drawn from the most estimated places of our times, for fortifications, whose original edition was published in Paris in 1649 by Jean Hénault at the Salle Dauphine of l'ange gardien. Another edition was published in 1668 in Mayence by Louis Bourgeat. Georges Fournier taught René Descartes. == Works == In 1642, Fournier published his Commentaires géographiques. The following year he published in Paris the work which will give him the essential part of his celebrity: the Hydrographie contenant la théorie et la pratique de toutes les parties de la navigation. Resolutely scientific, it is the first French maritime encyclopedia. Dedicated to Louis XIII, it was reprinted many times (1667, 1679, 1973). Geographer, astronomer, hydrographer, mathematician, Fournier took an interest in everything: winds and tides, ports, fishing, maritime trade, shipbuilding, the conduct of officers and the art of command. The Hydrographie was therefore a first-rate source for the history of the navy during the first half of the seventeenth century, which explains the "magnificent re-edition" of this work in 1973 (Etienne Taillemite). Georges Fournier also published in 1643 a Traité de la sphere and in 1644 a Traité de géométrie. His Traité des fortifications ou Architecture militaire (published in Paris in 1649 ) was translated into Dutch, Spanish and German. His works on Euclidean geometry (Paris, 1644 and 1654) were translated into English, and gave rise to three successive editions. In 1656, posthumously appeared his Asiae nova descriptio (New Description of Asia). The book was made in collaboration with several jesuit missionaries. == See also == Furnerius (crater) == Notes and references == == Bibliography == Fournier, Georges (1643). Hydrographie, contenant la théorie et la practique de toutes les parties de la navigation... Paris: Soly. BNF 304555507 Gille, Bertrand (1978). Histoire des techniques. Paris: Gallimard. ISBN 978-2-07-010881-7. Taillemite, Étienne (2002). Dictionnaire des Marins français. Tallandier. ISBN 2-84734-008-4. Vergé-Franceschi, Michel (2002). Dictionnaire d'Histoire maritime. Paris: Robert Laffont. ISBN 2-221-08751-8.
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Wikipedia:Georges Glaeser#0
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Georges Glaeser (8 November 1918 – 1 September 2002) was a French mathematician who was director of the IREM of Strasbourg. He worked in analysis and mathematical education and introduced Glaeser's composition theorem and Glaeser's continuity theorem. Glaeser was a Ph.D. student of Laurent Schwartz. On 3 July 1973, Glaeser filed a complaint against Vichy collaborator Paul Touvier in the Lyon Court, charging him with crimes against humanity. Glaeser accused Touvier of the 1944 massacre at Rillieux-la-Pape, in which Glaeser's father was murdered. Touvier was eventually imprisoned for life on this charge in 1994. == Affiliations == IAS School of Mathematics (9/1961 – 5/1962) == Education == University of Nancy (Class of 1957) == Selected publications == Glaeser, Georges (1963), "Fonctions composées différentiables", Annals of Mathematics, Second Series, 77 (1): 193–209, doi:10.2307/1970204, JSTOR 1970204, MR 0143058 "Etude de quelques algebres tayloriennes" "Racine carrée d'une fonction différentiable", Annales de l'Institut Fourier 13, no. 2 (1963), 203–210 "Une introduction à la didactique expérimentale des mathématiques" == References == == Sources == Pluvinage, François (2002), "In Memoriam — Georges Glaeser (1918–2002)" (PDF), The International Commission on Mathematical Instruction, Bulletin, 51: 63–66 Georges Glaeser at the Mathematics Genealogy Project Bibliography == External links == Glaeser, Georges – The Crisis of geometry teaching
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Wikipedia:Georges Hostelet#0
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Georges Hostelet (1875–1960) was a Belgian chemist, sociologist, mathematician, and philosopher. He was born in the municipality of Chimay in 1875. He attended the Royal Military Academy, and reached the rank of lieutenant. In 1897, he left the academy and enrolled in the University of Liège, where he received his doctorate in 1905. two years later, Hostelet began work with the Solvay & Cie Company as a chemical engineer and worked closely with Ernest Solvay. In 1911, he attended the First Solvay Conference, eventually becoming its last surviving participant. Years later, Hostelet became opposed to the First World War. During the war, he worked alongside English nurse Edith Cavell and was imprisoned by the occupying German forces, later being released in 1917. In 1919, he accepted an offer from Solvay to become co-director of the Solvay Institute of Sociology; a position he held until he left in 1922 when Solvay died. Hostelet left Belgium in 1925 as part of a Franco-Belgian mission to teach social sciences at the University of Cairo. He returned in 1931 and was appointed as a member of the International Statistical Institute in the Hague the following year. He continued teaching at the University of Antwerp until he retired in 1947. He died in 1960. == References == Jean Ghilain (December 30, 1965). "HOSTELET(Georges)" (PDF) (in French).
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Wikipedia:Georgiy L. Stenchikov#0
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Georgiy L. Stenchikov is an applied mathematician and climate scientist focusing on studies of physical processes that govern the Earth's climate. He is a professor in the Department of Earth Science and Engineering at the King Abdullah University of Science and Technology in Saudi Arabia. Stenchikov's research is focused on atmospheric physics, climate modeling, aerosols, radiative transfer, fluid dynamics, climate impacts of volcanic eruptions, and mineral dust. He has authored two books, Mathematical Modeling of Climate (in Russian) and Gotterdämmerung Globale Folgen eines atomaren Konflikts (in German). He has authored over 300 articles in journals, including Science, Journal of Geophysical Research: Atmospheres, and Atmospheric Chemistry and Physics. He contributed to the Nobel Prize-winning report from the Intergovernmental Panel on Climate Change (IPCC-AR4) of 2007 and has been awarded a Prize from the Council of Ministers of the Soviet Union for his work on climate impact modeling. In 2022 he received a Future of Life Award for Nuclear Winter research. == Education and early career == Stenchikov obtained his master's degree in Physics and Mathematics from the Moscow Physical-Technical Institute in 1973 and went on to earn a Ph.D. in Numerical and Analytical Study of Weak Plasma Turbulence from the same university in 1977. He then completed his Habilitation in Modeling the Large-Scale Anthropogenic Impacts on Climate in 1989 from the Computer Center of the USSR Academy of Sciences in Moscow, Russia. From 1976 to 1992, he worked at the Computer Center of the USSR Academy of Sciences as a junior scientist, senior scientist, and Head of the Branch of Mathematical Modeling of Anthropogenic Impacts. During this tenure, his team conducted computational analyses that explored the impact of human activities on Earth's climate. In the 1980s, with Vladimir Aleksandrov, he led the studies of the climatic consequences of a Nuclear War, conducting the first 3-D simulation of climate response to massive emissions of soot in the forest and urban fires fought by nuclear explosions, the effect called Nuclear Winter. == Career == Since 1992 Stenchikov has worked with Alan Robock on the effects of explosive volcanism on Earth's climate in the Department of Meteorology at the University of Maryland, College Park, USA. In 1998, he became a research professor in the Department of Environmental Sciences of Rutgers University and worked there for a decade. From 2009 to 2021, he was a founding Professor and a chair of the Earth Sciences and Engineering Program at King Abdullah University of Science and Technology (KAUST) in Saudi Arabia. == Research == Stenchikov has made contributions to climate modeling. His work, which involved the development of numerical models to simulate the intricate interactions among aerosols, clouds, and radiation, has enhanced the understanding of the role of stratospheric and tropospheric aerosols in shaping regional and global climate, including their influence on temperature, atmospheric and ocean circulation, and extreme weather events. Among several awards, he received the Highlight Award from the American Geophysical Union for his research publication titled "The Impact of Aerosols on simulated ocean temperature and heat content in the 20th Century". === Volcanic impact on climate === Following his early studies of the climate impact of smoke from urban fires ignited by nuclear explosions, Stenchikov's research has focused on understanding the impacts of volcanic explosions on climate. A volcanic cloud in the aftermath of a strong explosive volcanic eruption could cover the entire Earth for several years, reflecting solar radiation and causing a long-lasting cooling effect. Since 1850, volcanic eruptions have offset about 30% of ocean heat uptake forced by greenhouse gas warming. He found that along with global cooling, volcanic impacts can cause large-scale changes in atmospheric and ocean circulation, forcing the positive phase of the Arctic Oscillation, accelerating Atlantic Meridional Overturning Circulation, increasing the polar sea-ice extent and prolonging El Ninos. Despite the global nature of volcanic forcing, the climate responses have pronounced regional signatures, such as winter warming in Siberia and freezing winters in the Middle East. He conducted new simulations of the Toba super-eruption, a particularly devastating event that occurred in Sumatra around 75,000 years ago and emitted in the atmosphere about 2 billion tons of sulfur dioxide. He modified a regional atmospheric chemistry model, WRF-Chem, to capture the initial stage of the volcanic cloud formed after the 1991 Pinatubo eruption. The simulations showed that in the first week after the eruption, the volcanic cloud rose into the stratosphere at a rate of 1 km/day, driven initially by ash's solar absorption and later by sulfate aerosol absorption of solar and terrestrial radiation. This profoundly affected the long-term evolution and radiative impact of the volcanic cloud. === Regional climate modeling === To study climate change and variability, Stenchikov has calculated the future climate projections employing the global high-resolution atmospheric model (HiRAM) with a spatial resolution of 25x25 km2 and performed climate downscaling using nested regional model WRF-Chem with fine grid spacing down to 3 km. To add to physical consistency in modeling the Middle East climate, he coupled the regional atmospheric model (WRF-Chem) with the regional ocean modeling system (ROMS), accounting for aerosol and atmospheric chemistry processes, and conducted the first coupled regional simulations of the impact of tropospheric dust and stratospheric volcanic aerosols on the Middle East and the Red Sea. His research highlighted that a combination of coupled atmospheric and oceanic simulations and ground-based aerosol measurements are the best tools for informing governments when they have to make critical decisions regarding air quality and climate change. His recent research suggested that forestation of the Red Sea Arabian coastal plain was unlikely to increase precipitation in the breezes zone. However, decreasing land-surface albedo by deploying solar panels could potentially generate 1.5 Gt of rainwater annually to meet the needs of 5 million people. === Study the effects of dust on the middle east climate. === In desert regions like the Middle East, dust profoundly impacts the environment, climate, air quality, and human health. In addition, dust affects the efficiency of solar energy devices by reducing the downward solar flux and settling on their optically active surfaces. Stenchikov demonstrated that the desert climate is extremely sensitive to radiative forcing, the change in Earth's energy balance associated with greenhouse gas emissions, desertification, urbanization, and shrinking vegetation cover. This effect is influenced by an abundance of particulates, including sand, dust, and anthropogenic pollutants, suspended in the air, collectively named aerosols. The Middle East and North Africa (MENA) contribute to over half of global dust emissions, and dust significantly influences the regional radiative balance and atmospheric circulation. The prolonged drought and social conflicts led to the growth of dust aerosol optical depth, which characterizes the radiative effect of aerosols. Over the past three decades, observations and computer simulations have shown that the Arabian Peninsula's average temperature has been increasing by about 0.5 °C per decade, exceeding almost twice the trend observed in the northern hemisphere. The analysis shows that atmospheric heating in the middle troposphere caused by Saharan dust alters Hadley and Walker circulation, moving the rain belt north in the summer and increasing precipitation in the Sahel. He highlighted that the radiative surface cooling of the Southern Red Sea due to accumulated dust layers reaches 40 W/m2, affecting the circulation and energy balance of the sea - an effect that had not been previously recognized. In recent research, he stated that the natural and anthropogenic aerosol mix could severely affect air quality. The interaction between dust and anthropogenic aerosols alters the aerosol's optical properties, radiative forcing, and their interaction with clouds. The direct radiative forcing from aerosols diminishes the pace of regional warming, but its future projections are largely unknown. == Awards and honors == 1985 – Gold Medal Award, National Exhibition of the Economy Achievements of the USSR 1986 – Prize for the development of multi-tasking software, Council of Ministers of the USSR 2003 – Group Achievement Award, The National Aeronautics and Space Administration 2004 – Outstanding Scientific Paper Award, The National Oceanic and Atmospheric Administration 2007 – Co-authored the Nobel Prize-winning IPCC AR4 report, The Noble Foundation 2022 – Award for Nuclear Winter Studies, Future of Life Award Institute, USA == Bibliography == === Books === Mathematical Modeling of Climate (1986) Gotterdämmerung Globale Folgen eines atomaren Konflikts (1990) === Selected articles === Александров, В.В.; Стенчиков, Г.Л. (1975). "О влиянии излучения на течение газа при сильном взрыве для большого значения времени" [Effect of Radiation on Gas Flow in Powerful Explosions for Considerable Time Value]. Прикладная математика и механика (in Russian). 32 (2): 246–252. Александров, В.В.; Стенчиков, Г.Л. (1984). "Об одном вычислительном эксперименте, моделирующем климатические последствия ядерной войны" [Numerical simulation of the climatic consequences of a nuclear war]. Журнал Вычислительной Математики и Математической Физики (in Russian). 24 (1): 140–144. Stenchikov, G. L. (1986). Numerical Modeling of the Nuclear Winter Taking into Account Aerosol Spreading (in Russian). Doklady Akademii Nauk SSSR (translated - Transactions of the USSR Academy of Sciences, Earth Science Sections), 287(3), 598–602. Stenchikov, Georgiy L.; Kirchner, Ingo; Robock, Alan; Graf, Hans-F.; Antuña, Juan Carlos; Grainger, R. G.; Lambert, Alyn; Thomason, Larry (27 June 1998). "Radiative forcing from the 1991 Mount Pinatubo volcanic eruption". Journal of Geophysical Research: Atmospheres. 103 (D12): 13837–13857. Bibcode:1998JGR...10313837S. doi:10.1029/98JD00693. hdl:21.11116/0000-0004-ECBD-E. Yip, C. M. A., Gunturu, U. B., & Stenchikov, G. L. (2017). High-altitude wind resources in the Middle East. Scientific reports, 7(1), 9885. Osipov, S., & Stenchikov, G. (2018). Simulating the regional impact of dust on the Middle East climate and the Red Sea. Journal of Geophysical Research: Oceans, 123(2), 1032–1047. Parajuli, S. P., Stenchikov, G. L., Ukhov, A., & Kim, H. (2019). Dust emission modeling using a new high‐resolution dust source function in WRF‐Chem with implications for air quality. Journal of Geophysical Research: Atmospheres, 124(17-18), 10109–10133. Ukhov, A., Mostamandi, S., Krotkov, N., Flemming, J., Da Silva, A., Li, C., ... & Stenchikov, G. (2020). Study of SO Pollution in the Middle East Using MERRA‐2, CAMS Data Assimilation Products, and High‐Resolution WRF‐Chem Simulations. Journal of Geophysical Research: Atmospheres, 125(6), e2019JD031993. Ukhov, A., Mostamandi, S., Da Silva, A., Flemming, J., Alshehri, Y., Shevchenko, I., & Stenchikov, G. (2020). Assessment of natural and anthropogenic aerosol air pollution in the Middle East using MERRA-2, CAMS data assimilation products, and high-resolution WRF-Chem model simulations. Atmospheric Chemistry and Physics, 20(15), 9281–9310. Stenchikov, G. (2021). The role of volcanic activity in climate and global changes. In Climate change (pp. 607–643). Elsevier. Stenchikov, G., Ukhov, A., Osipov, S., Ahmadov, R., Grell, G., Cady‐Pereira, K., ... & Iacono, M. (2021). How Does a Pinatubo‐Size Volcanic Cloud Reach the Middle Stratosphere? Journal of Geophysical Research: Atmospheres, 126(10), e2020JD033829. == References ==
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Wikipedia:Georgy Adelson-Velsky#0
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Georgy Maximovich Adelson-Velsky (Russian: Гео́ргий Макси́мович Адельсо́н-Ве́льский; name is sometimes transliterated as Georgii Adelson-Velskii) (8 January 1922 – 26 April 2014) was a Soviet mathematician and computer scientist. Born in Samara, Adelson-Velsky was originally educated as a pure mathematician. His first paper, with his fellow student and eventual long-term collaborator Alexander Kronrod in 1945, won a prize from the Moscow Mathematical Society. He and Kronrod were the last students of Nikolai Luzin, and he earned his doctorate in 1949 under the supervision of Israel Gelfand. He began working in artificial intelligence and other applied topics in the late 1950s. Along with Evgenii Landis, he invented the AVL tree in 1962. This was the first known balanced binary search tree data structure. Beginning in 1963, Adelson-Velsky headed the development of a computer chess program at the Institute for Theoretical and Experimental Physics in Moscow. His innovations included the first use of bitboards (a now-common method for representing game positions) in computer chess. The program defeated Kotok-McCarthy in the first chess match between computer programs, also in 1966, and it evolved into Kaissa, the first world computer chess champion. In August 1992, Adelson-Velsky moved to Israel, and he resided in Ashdod. He worked as a professor in the department of Mathematics and Computer Science, Bar Ilan University. Adelson-Velsky died on 26 April 2014, aged 92, in his apartment in Giv'atayim, Israel. == Selected publications == Adel'son-Vel'skiĭ, G. M.; Kronrod, A. S. (1945), "On a direct proof of the analyticity of a monogenic function", Doklady Akademii Nauk SSSR, New Series, 50: 7–9, MR 0051912. Adel'son-Vel'skiĭ, G. M.; Landis, E. M. (1962), "An algorithm for organization of information", Doklady Akademii Nauk SSSR, 146: 263–266, MR 0156719. Adel'son-Vel'skiĭ, G. M.; Arlazarov, V. L.; Bitman, A. R.; Životovskiĭ, A. A.; Uskov, A. V. (1970), "On programming a computer for playing chess", Akademiya Nauk SSSR I Moskovskoe Matematicheskoe Obshchestvo, 25 (2 (152)): 221–260, MR 0261965. Translated as "Programming a computer to play chess", Russian Mathematical Surveys 25: 221–262, 1970, doi:10.1070/RM1970v025n02ABEH003792 == References == == External links == Костинский, Александр; Брауде-Золотарев, Михаил (31 December 2002). Не очень серьёзно о цифровых технологиях (Radio broadcast) (in Russian). Radio Liberty. Archived from the original on 8 August 2007.Near the end of the program, Mikhail Donskoy recounts a trip with Adelson to the University of Waterloo. Dinitz, Yefim (28 April 2014). "G.M. Adelson-Velsky passed away". Theory Announcements. TheoryNet and DMANet. Archived from the original on 15 August 2014. [1] Archived 2012-07-05 at the Wayback Machine from http://chessprogramming.wikispaces.com 1990 Moscow Interview with Adelson-Velsky, Eugene Dynkin Collection of Mathematics Interviews, Cornell University Library (in Russian, English transcript). [2] (In Russian, List of publication translated into English). Author profile in the database zbMATH
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Wikipedia:Georgy Pfeiffer#0
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Georgy Pfeiffer also Yurii or Yury Pfeiffer (Russian: Гео́ргий Васи́льевич Пфе́йффер, German: Georg Ferdinand Pfeiffer, 23 December 1872 – 10 October 1946) was a Russian Empire and Soviet mathematician of German descent. Pfeiffer was known as a specialist in the field of integration of differential equations and systems of partial differential equations. He was also interested in algebraic geometry. He was an Invited Speaker of the International Congress of Mathematicians (ICM) in 1908 at Rome, in 1928 at Bologna, and in 1932 at Zurich. He was a chairman of the Academic Council of the Faculty of Physics and Mathematics at the University of Kiev, Russian Empire. Pfeiffer was also attached to the Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR in Kiev and served as Director during two periods, namely 1934 to 1941 and again from 1944 until his death in 1946. In the three years 1941–44, Pfeiffer was in Ufa, Russia the capital of the republic of Bashkortostan in western Russia. In Ufa, Pfeiffer was Director of the Institute of Mathematics and Physics. == References == O'Connor, John J.; Robertson, Edmund F., "Georgy Pfeiffer", MacTutor History of Mathematics Archive, University of St Andrews == External links == Biography Archived 2014-03-11 at the Wayback Machine
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Wikipedia:Georgy Voronoy#0
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Georgy Feodosevich Voronyi (Russian: Георгий Феодосьевич Вороной; Ukrainian: Георгій Феодосійович Вороний; 28 April 1868 – 20 November 1908) was an Imperial Russian mathematician of Ukrainian descent noted for defining the Voronoi diagram. == Biography == Voronyi was born in the village of Zhuravka, Pyriatyn, in the Poltava Governorate, which was a part of the Russian Empire at that time and is in Varva Raion, Chernihiv Oblast, Ukraine. Beginning in 1889, Voronyi studied at Saint Petersburg University, where he was a student of Andrey Markov. In 1894 he defended his master's thesis On algebraic integers depending on the roots of an equation of third degree. In the same year, Voronyi became a professor at the University of Warsaw, where he worked on continued fractions. In 1895 he had a son with his wife Olga Mytrofanivna, named Yurii Voronyi, who later became a noted surgeon. In 1897 he defended his doctoral thesis On a generalisation of a continuous fraction. He was an Invited Speaker of the ICM in 1904 at Heidelberg. When he was only 40 years of age, Voronyi began having stomach problems. He wrote in his diary: I am making great progress with the question under study [indefinite quadratic forms]; however, at the same time my health is becoming worse and worse. Yesterday I had for the first time a clear idea of the algorithm in the theory of forms I am investigating, but also suffered a strong attack of bilious colic which prevented me from working in the evening and from sleeping the whole night. I am so afraid that the results of my enduring efforts, obtained with such difficulty, will perish along with me. Following a severe gall bladder attack, Voronyi died on 20 November 1908. == Works == Voronyi introduced the concept of what we today call Voronoi diagrams or tessellations. They are used in many areas of science, such as the analysis of spatially distributed data, having become an important topic in geophysics, meteorology, condensed matter physics, and Lie groups. These tessellations are widely used in many areas of computer graphics, from architecture to film making and video games. Blender 3D includes a Voronoi texture generator as one of its main sources of randomly generated images, that can be applied as textures for many different uses. == Legacy == Among his students was Wacław Sierpiński (Ph.D. at Jagiellonian University in 1906). Although he was not formally the doctoral advisor of Boris Delaunay (Ph.D. at Kyiv University), his influence on the latter earns him the right to be considered so. In 2008 Ukraine released two-hryvnia coins commemorating the centenary of Voronyi's death. His son Yuriy Voronyi became a prominent transplant surgeon who performed the world's first human-to-human kidney transplant in 1933, volunteered for the army of Central Council of Ukraine, and fought in the Battle of Kruty. His daughter Mariia Vorona-Vasylenko became a teacher of Ukrainian language. == See also == Bowyer–Watson algorithm Centroidal Voronoi tessellation Delaunay triangulation Fortune's algorithm Laguerre–Voronoi diagram Voronoi deformation density Voronoi formula Voronoi iteration Voronoi pole Weighted Voronoi diagram Wigner–Seitz cell == References == == Further reading == Syta, Halyna; van de Weygaert, Rien (2009). "Life and Times of Georgy Voronoi". arXiv:0912.3269 [math.HO]. == External links == Georgy Voronoy at the Mathematics Genealogy Project O'Connor, John J.; Robertson, Edmund F. "Georgy Voronoy". MacTutor History of Mathematics Archive. University of St Andrews.
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Wikipedia:Gerard Laman#0
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Gerard Laman (August 22, 1924 – September 22, 2009) was a Dutch mathematician who worked on graph theory. == Early life == He completed high school studies at the Stedelijk Gymnasium Leiden in 1942. His study of Mathematics at Leiden University was delayed by a period in hiding to evade enforced labor during the Nazi occupation of the Netherlands. He completed a degree in Mathematics with a minor in Mechanics in 1952. From 1949 onwards, he was a scientific assistant to J. Haantjes. He received private instruction in the combinatorial topology of fiber spaces in Brussels from G. Hirsch of the Agricultural University of Ghent in 1953. During this period he received a stipend from the Dutch-Belgian Cultural Accord. From 1954 to 1957 he taught mathematics at the Delft high school 'Gemeentelijke Hogere Burgerschool HBS'. In 1959 he completed his PhD thesis at Leiden University. W. T. van Est acted as his supervisor, once the original supervisor J. Haantjes was deceased. === Career === From 1957 to 1967 he worked as a lecturer at the 'Technische Hogeschool' of Eindhoven (now Eindhoven University of Technology). From 1967 to his retirement in 1989, he worked as a lecturer at the Mathematical Institute of the University of Amsterdam, teaching discrete mathematics and mathematics for students in Econometrics. Laman regarded himself mostly as a teacher. Clear thinking, as well as brevity in speech and writing, were his forte. Laman is often credited with proving, in 1970, that a particular family of sparse graphs, since named Laman graphs, are precisely those that are minimally generically rigid in the plane. This result, the Geiringer–Laman theorem, had already been proven by Hilda Geiringer back in 1927. Laman's original publication in 1970 went largely unnoticed at first. Only when Branko Grünbaum and G. C. Shephard wrote about Laman's paper in their Lectures on lost mathematics did this work receive more attention. Towards the end of his life, Laman worked to lift the original 'Laman graph' from its original two dimensions to three, inspired by a simple counterexample, the 'double banana graph'. == Notes == == References == Graver, J.; Servatius, B.; Servatius, H. (1993), Combinatorial rigidity, Graduate Studies in Mathematics, vol. 2, American Mathematical Society, Providence, RI, MR 1251062 Grünbaum, B.; Shephard, G.C. (1978), Lectures on lost mathematics, Lecture Notes, Univ. of Washington Laman, G. (1959), On automorphisms of transformationgroups of polynomial algebras, Thesis, Rijksuniversiteit Leiden, MR 0108508{{citation}}: CS1 maint: location missing publisher (link) Laman, G. (1970), "On graphs and rigidity of plane skeletal structures", J. Engrg. Math., 4 (4): 331–340, Bibcode:1970JEnMa...4..331L, doi:10.1007/BF01534980, MR 0269535, S2CID 122631794 Owen, J.C.; Power, S.C. (2007), "The non-solvability by radicals of generic 3-connected planar Laman graphs", Trans. Amer. Math. Soc., 359 (5): 2269–2303, doi:10.1090/S0002-9947-06-04049-9, MR 2276620
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Wikipedia:Gerd Grubb#0
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Gerd Grubb (born 12 February 1939) is a Danish mathematician known for her research on pseudo-differential operators. She is a professor emerita in the Department of Mathematical Sciences at the University of Copenhagen, where she was the first female professor of mathematics. == Education and career == Grubb was born on 12 February 1939 in Copenhagen, the daughter of two chemical engineers. She was a student at the Øregård Gymnasium, and then studied various sciences at the University of Copenhagen from 1956 until 1959. After earning a master's degree in mathematics at Aarhus University in 1963, she went to Stanford University for doctoral study in mathematics, completing a Ph.D. in 1966. Her dissertation, A Characterization of the Non-Local Boundary Value Problems Associated With an Elliptic Operator, was supervised by Ralph S. Phillips. She completed a habilitation (Dr. Phil.) in 1975 at the University of Copenhagen, with the habilitation thesis Semiboundedness and other properties of normal boundary problems for elliptic partial differential operators. She returned to the University of Copenhagen as an assistant professor in 1966, eventually becoming a full professor there in 1994. == Books == Grubb is the author of the books Functional calculus of pseudodifferential boundary problems (Progress in Mathematics 65, Birkhäuser, 1986; 2nd ed., 1996) and Distributions and operators (Graduate Texts in Mathematics 252, Springer, 2009). == Recognition == Grubb is a member of the Danish Academy of Natural Sciences. The University of Reims Champagne-Ardenne gave her an honorary doctorate in 1988. She was promoted to hedersdoktor (an honorary doctorate) at the University of Lund (Sweden) in 2016. In 2020, she received the Gold Medal of the Royal Danish Academy of Sciences and Letters. == References == == External links == Home page
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Wikipedia:Gerda de Vries#0
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Gerda de Vries is a Canadian mathematician whose research interests include dynamical systems and mathematical physiology. She is a professor of mathematical and statistical sciences at the University of Alberta, and the former president of the Society for Mathematical Biology. == Education and career == De Vries graduated from the University of Waterloo in 1989, and completed her doctorate in 1995 at the University of British Columbia. Her dissertation, Analysis of Models of Bursting Electrical Activity in Pancreatic Beta Cells, was supervised by Robert M. Miura. After postdoctoral research with Arthur Sherman at the National Institutes of Health, she joined the University of Alberta faculty in 1998. She was promoted to full professor in 2008. == Publications == De Vries has published highly-cited research on beta cells and beta-actin. With Thomas Hillen, Mark A. Lewis, Johannes Müller, and Birgitt Schönfisch, she is also the author of a 2006 textbook, A Course in Mathematical Biology: Quantitative Modeling with Mathematical and Computational Methods. == Recognition and service == De Vries served as president of the Society for Mathematical Biology for 2011–2013, and became a fellow of the society in 2017. In 2014 the Canadian Mathematical Society gave de Vries their excellence in teaching award. The society listed de Vries in their inaugural class of fellows in 2018. == References == == External links == Home page Archived 2020-01-30 at the Wayback Machine Gerda de Vries publications indexed by Google Scholar
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Wikipedia:Gerhard Hochschild#0
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Gerhard Paul Hochschild (April 29, 1915 in Berlin – July 8, 2010 in El Cerrito, California) was a German-born American mathematician who worked on Lie groups, algebraic groups, homological algebra and algebraic number theory. == Early life == On April 29, 1915, Hochschild was born to a middle-class Jewish family in Berlin, Germany, the son of Lilli and Heinrich Hochschild. Hochschild had an older brother. His father was a patent attorney who had an engineering degree. After the rise of the National Socialist German Workers' Party in 1933, his father sent him to South Africa where he was able to enroll in school with funding from the Hochschild Family Foundation established by Berthold Hochschild, a cousin of his grandfather. == Education == In 1936, Hochschild earned a BS degree in mathematics from University of Cape Town in Union of South Africa. In 1937, Hochschild earned a MS degree in mathematics from University of Cape Town. In 1941, Hochschild earned his PhD in mathematics from Princeton University. Hochschild completed his thesis in 1941 at Princeton University with Claude Chevalley on Semisimple Algebras and Generalized Derivations. == Career == In 1956–7 Hochschild was at the Institute for Advanced Study. Hochschild was a professor at University of Illinois at Urbana-Champaign. In the late 1950s Hochschild was a professor at University of California, Berkeley. Hochschild (1945) introduced Hochschild cohomology, a cohomology theory for algebras, which classifies deformations of algebras. Hochschild & Nakayama (1952) introduced cohomology into class field theory. Along with Bertram Kostant and Alex F. T. W. Rosenberg, the Hochschild–Kostant–Rosenberg theorem is named after him. Among his students were Andrzej Białynicki-Birula and James Ax. In 1955, Hochschild was a Guggenheim Fellow. In 1979 Hochschild was elected to the National Academy of Sciences, and in 1980 he was awarded the Leroy P. Steele Prize of the AMS. In 1982, Hochschild retired but continued teaching part-time until 1985. == Personal life == In July 1950, Hochschild married Ruth Heinsheimer. Ruth was born in Germany and fled with her mother in 1939; the couple met at the University of Illinois where she was earning her M.S. in mathematics and Gerhard was working as an assistant professor. Hochschild's children are Ann Hochschild (b. 1955) and Peter Hochschild (b. 1957). On July 8, 2010, Hochschild died at his home. Hochschild was 95. == See also == Hochschild–Mostow group == Publications == Hochschild, G. (1945), "On the cohomology groups of an associative algebra", Annals of Mathematics, Second Series, 46 (1): 58–67, doi:10.2307/1969145, ISSN 0003-486X, JSTOR 1969145, MR 0011076, S2CID 124400094 Hochschild, G.; Nakayama, Tadasi (1952), "Cohomology in class field theory", Annals of Mathematics, Second Series, 55 (2): 348–366, doi:10.2307/1969783, ISSN 0003-486X, JSTOR 1969783, MR 0047699 Hochschild, G. (1965), The structure of Lie groups, San Francisco, Calif.: Holden–Day Inc., MR 0207883 Hochschild, G. (1971), Introduction to affine algebraic groups, San Francisco, Calif.: Holden–Day Inc., MR 0277535 Hochschild, Gerhard Paul (1981), Basic theory of algebraic groups and Lie algebras, Graduate Texts in Mathematics, vol. 75, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90541-9, MR 0620024 Hochschild, Gerhard Paul (1983), Perspectives of elementary mathematics, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90848-9, MR 0710303 == References == Moore, Calvin C.; Ribet, Kenneth A.; Wolf, Joseph A. (2011), IN MEMORIAM Gerhard Hochschild Professor of Mathematics, Emeritus UC Berkeley 1915 - 2010, archived from the original on 2011-09-15 Ferrer Santos, Walter (2011). "Gerhard Hochschild (1915/2010) A Mathematician of the XXth Century". arXiv:1104.0335 [math.HO]. Moskowitz, Martin; Ferrer Santos, Walter, eds. (2011), "Gerhard Hochschild (1915-2010)" (PDF), Notices of the American Mathematical Society, 58 (8): 1078–1099, ISSN 0002-9920 == External links == Gerhard Hochschild at the Mathematics Genealogy Project Pictures of Gerhard Hochschild from Oberwolfach Finding Aid to the Gerhard P. Hochschild papers, 1941-2004, The Bancroft Library Hochschild Family Foundation at foundation center.org
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Wikipedia:Gerhard J. Woeginger#0
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Gerhard J. Woeginger (31 May 1964 – 1 April 2022) was an Austrian mathematician and computer scientist who worked in Germany as a professor at RWTH Aachen University, where he chaired the algorithms and complexity group in the department of computer science. == Biography == Woeginger was born on 31 May 1964 in Graz, Austria. He obtained a diploma from the Graz University of Technology (TU Graz) in 1987, and completed his Ph.D. at TU Graz 1991 under the supervision of Franz Rendl. He worked on the faculty of TU Graz from 1991 to 2001, where he completed his habilitation in 1995. He then moved to the University of Twente from 2001 to 2004, to TU Eindhoven, from 2004 to 2016, and finally to RWTH Aachen in 2016. He was program chair of the European Symposium on Algorithms in 1997, of the algorithms track of the International Colloquium on Automata, Languages and Programming in 2003, of the European Conference on Operational Research in 2009, and of several other conferences. In 1996, Woeginger won the Start-Preis, the highest Austrian award for scientists under the age of 35. He won a Humboldt Research Award in 2011. In 2014, he was elected to the Academia Europaea. Until September 2016, Woeginger maintained a list of attempts to settle the P versus NP problem on his website. As of its last update, the list tallied 116 attempts dating back to 1986 to settle the question in various ways. Woeginger died on 1 April 2022, at the age of 57. == References == == External links == Home page (archived) Google Scholar profile
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Wikipedia:Gerhard Wanner#0
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Gerhard Wanner (born 1942 in Innsbruck) is an Austrian mathematician. == Education and career == Wanner grew up in Seefeld in Tirol and studied mathematics at the University of Innsbruck, where he received his doctorate in 1965 with advisor Wolfgang Gröbner and dissertation Ein Beitrag zur numerischen Behandlung von Randwertproblemen gewöhnlicher Differentialgleichungen (A contribution to the numerical treatment of boundary value problems of ordinary differential equations). He taught in Innsbruck and from 1973 at the University of Geneva. Wanner's research deals with numerical analysis of ordinary differential equations (about which he wrote a two-volume monograph with Ernst Hairer). Wanner is the co-author of an analysis undergraduate textbook and a geometry undergraduate textbook, both of which give historically oriented explanations of mathematics. In 2003 he was awarded, jointly with Ernst Hairer, the Peter Henrici Prize. In 2015 Wanner received SIAM's George Pólya Prize for Mathematical Exposition. He was president of the Swiss Mathematical Society from 1998 to 1999. == Selected publications == === Articles === Hairer, E.; Wanner, G. (1973). "Multistep-multistage-multiderivative methods for ordinary differential equations". Computing. 11 (3): 287–303. doi:10.1007/BF02252917. ISSN 0010-485X. Hairer, E.; Wanner, G. (1975). "A theory for Nyström methods". Numerische Mathematik. 25 (4): 383–400. doi:10.1007/BF01396335. ISSN 0029-599X. Hairer, E.; Wanner, G. (1981). "Algebraically Stable and Implementable Runge-Kutta Methods of High Order". SIAM Journal on Numerical Analysis. 18 (6): 1098–1108. doi:10.1137/0718074. ISSN 0036-1429. Hairer, Ernst; Wanner, Gerhard (1999). "Stiff differential equations solved by Radau methods". Journal of Computational and Applied Mathematics. 111 (1–2): 93–111. doi:10.1016/S0377-0427(99)00134-X. ISSN 0377-0427. Hairer, E.; Lubich, C.; Wanner, G. (2003). "Geometric numerical integration illustrated by the Stormer-Verlet method". Acta Numerica. 12 (12): 399–450. doi:10.1017/S0962492902000144. ISBN 9780521825238. Gander, Martin J.; Wanner, Gerhard (2012). "From Euler, Ritz, and Galerkin to Modern Computing". SIAM Review. 54 (4): 627–666. CiteSeerX 10.1.1.297.5697. doi:10.1137/100804036. ISSN 0036-1445. === Books === with Ernst Hairer: L'analyse au fil de l'histoire. Springer. 2001. ISBN 978-3-540-67463-4; x+372 pages{{cite book}}: CS1 maint: postscript (link) Hairer, Ernst; Wanner, Gerhard (2008). Analysis by Its History. ISBN 9780387770314. Analysis in der historischen Entwicklung. Berlin/Heidelberg: Springer. 2011. ISBN 978-3-642-13766-2. with Alexander Ostermann: Geometry by Its History. Springer, Berlin/Heidelberg 2012, ISBN 978-3-642-29162-3. with Ernst Hairer and Christian Lubich: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. 2002. 2nd edition. Springer, Berlin/Heidelberg 2010, ISBN 978-3-642-05157-9. pbk reprint with Ernst Hairer and Sylvert Nørsett: Solving Ordinary Differential Equations I. Nonstiff Problems (1st ed.). 1987. Hairer, Ernst; Nørsett, Syvert P.; Wanner, Gerhard (1993). Revised 2nd edition. ISBN 9783540566700. 3rd corrected printing. Springer, Berlin/Heidelberg 2009, ISBN 978-3-642-05163-0. with Ernst Hairer and Sylvert Nørsett: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems (1st ed.). 1991. 2nd edition. Springer, Berlin/Heidelberg 1996, ISBN 978-3-642-05220-0. 2013 pbk reprint Integration gewöhnlicher Differentialgleichungen: Lie-Reihen (mit Programmen), Runge-Kutta-Methoden. BI-Hochschultaschenbücher. Bibliographisches Institut, Mannheim/Zürich 1969. == References == == External links == Homepage
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Wikipedia:Gerrit van Dijk (mathematician)#0
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Van Dijk (Dutch pronunciation: [vɑn ˈdɛik] ) is a Dutch toponymic surname meaning "from (the) dike". With 56,441 people, it was the fifth most common name in the Netherlands in 2007. Abroad, people with this surname usually abandoned the ij digraph, resulting in names like Van Dyke and Van Dyk. People with the original surname include: == Dutch people == Anouk van Dijk (born 1965), choreographer and dancer Ans van Dijk (1905–1948), Nazi collaborator during World War II Arjan van Dijk (born 1987), football player Bill van Dijk (born 1947), singer Bryan van Dijk (born 1981), judoka Cilia van Dijk (1941–2023), film producer Daan van Dijk (1907–1986), track cyclist Dick van Dijk (1946–1997), football striker Dick van Dijk (born 1970), darts player Diederik van Dijk (born 1971), politician Dominique van Dijk (born 1979), football midfielder Edith van Dijk (born 1973), swimmer Ellen van Dijk (born 1987), road and track cyclist Frans Van Dijk (1853–1939), Belgian architect Gé van Dijk (1923–2005), football player and coach Gerrit van Dijk (1938–2012), animator, film maker, and painter Gerrit van Dijk (mathematician) (1939–2022), mathematician Gijs van Dijk (born 1980), politician and union leader Gregoor van Dijk (born 1981), football midfielder Herman K. van Dijk (1946–2025), economist Jan van Dijk (born 1952), sociologist and communication scientist Jan van Dijk (born 1956), football midfielder Jappie van Dijk (born 1944), speed skater Jasper van Dijk (born 1971), politician Joeri van Dijk (born 1983), sailor Johannes van Dijk (1868–1938), rower Joost van Dijk (died 1632), privateer and first settler of the Virgin Islands Kay van Dijk (born 1984), volleyball player Kees van Dijk (1931–2008), politician Ko van Dijk (1916–1978), theater and television actor Leon van Dijk (born 1992), football player Louis van Dijk (1941–2020), pianist Marijn van Dijk (born 1979), developmental psychologist and linguist Nelis van Dijk (1904–1969), boxer Niek van Dijk (born 1951), orthopaedic surgeon Olger van Dijk (born 1979), Dutch politician Otwin van Dijk (born 1975), politician Peter van Dijk (born 1952), politician Philip van Dijk (1683–1753), painter Philip van Dijk (footballer) (1885–1937), midfielder Rob van Dijk (born 1969), football goalkeeper Rudi Martinus van Dijk (1932–2003), composer Ryan van Dijk (born 1990), football midfielder Sam van Dijk (born 1996), basketball player Sergio van Dijk (born 1982), football player Stefan van Dijk (born 1976), racing cyclist Steven van Dijk (born 1969), cricket bowler Teun A. van Dijk (born 1943), scholar in linguistics Virgil van Dijk (born 1991), football player Wendy van Dijk (born 1976), actress and television presenter == Other people == Anne-Mette van Dijk (born 1968), Danish badminton player Germaine van Dijk (born 1983), Surinamese football player Hilbert Van Dijk (1918–2001), Australian fencer Jace Van Dijk (born 1981), Australian rugby league player Jennifer van Dijk-Silos (born 1954), Surinamese Minister of Justice and Police Mijk van Dijk (born 1963), German DJ Nikki Van Dijk (born 1994), Australian professional surfer Ruel (born 2002), Australian singer-songwriter == See also == Van Dijck Van Dyk Van Dyke (disambiguation) Van Dyck (surname) == References ==
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Wikipedia:Gershgorin circle theorem#0
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In mathematics, the Gershgorin circle theorem may be used to bound the spectrum of a square matrix. It was first published by the Soviet mathematician Semyon Aronovich Gershgorin in 1931. Gershgorin's name has been transliterated in several different ways, including Geršgorin, Gerschgorin, Gershgorin, Hershhorn, and Hirschhorn. == Statement and proof == Let A {\displaystyle A} be a complex n × n {\displaystyle n\times n} matrix, with entries a i j {\displaystyle a_{ij}} . For i ∈ { 1 , … , n } {\displaystyle i\in \{1,\dots ,n\}} let R i {\displaystyle R_{i}} be the sum of the absolute values of the non-diagonal entries in the i {\displaystyle i} -th row: R i = ∑ j ≠ i | a i j | . {\displaystyle R_{i}=\sum _{j\neq {i}}\left|a_{ij}\right|.} Let D ( a i i , R i ) ⊆ C {\displaystyle D(a_{ii},R_{i})\subseteq \mathbb {C} } be a closed disc centered at a i i {\displaystyle a_{ii}} with radius R i {\displaystyle R_{i}} . Such a disc is called a Gershgorin disc. Theorem. Every eigenvalue of A {\displaystyle A} lies within at least one of the Gershgorin discs D ( a i i , R i ) . {\displaystyle D(a_{ii},R_{i}).} Proof. Let λ {\displaystyle \lambda } be an eigenvalue of A {\displaystyle A} with corresponding eigenvector x = ( x j ) {\displaystyle x=(x_{j})} . Find i such that the element of x with the largest absolute value is x i {\displaystyle x_{i}} . Since A x = λ x {\displaystyle Ax=\lambda x} , in particular we take the ith component of that equation to get: ∑ j a i j x j = λ x i . {\displaystyle \sum _{j}a_{ij}x_{j}=\lambda x_{i}.} Taking a i i {\displaystyle a_{ii}} to the other side: ∑ j ≠ i a i j x j = ( λ − a i i ) x i . {\displaystyle \sum _{j\neq i}a_{ij}x_{j}=(\lambda -a_{ii})x_{i}.} Therefore, applying the triangle inequality and recalling that | x j | | x i | ≤ 1 {\displaystyle {\frac {\left|x_{j}\right|}{\left|x_{i}\right|}}\leq 1} based on how we picked i, | λ − a i i | = | ∑ j ≠ i a i j x j x i | ≤ ∑ j ≠ i | a i j x j x i | = ∑ j ≠ i | a i j | | x j | | x i | ≤ ∑ j ≠ i | a i j | = R i . {\displaystyle \left|\lambda -a_{ii}\right|=\left|\sum _{j\neq i}{\frac {a_{ij}x_{j}}{x_{i}}}\right|\leq \sum _{j\neq i}\left|{\frac {a_{ij}x_{j}}{x_{i}}}\right|=\sum _{j\neq i}\left|a_{ij}\right|{\frac {\left|x_{j}\right|}{\left|x_{i}\right|}}\leq \sum _{j\neq i}\left|a_{ij}\right|=R_{i}.} Corollary. The eigenvalues of A must also lie within the Gershgorin discs Cj corresponding to the columns of A. Proof. Apply the Theorem to AT while recognizing that the eigenvalues of the transpose are the same as those of the original matrix. Example. For a diagonal matrix, the Gershgorin discs coincide with the spectrum. Conversely, if the Gershgorin discs coincide with the spectrum, the matrix is diagonal. == Discussion == One way to interpret this theorem is that if the off-diagonal entries of a square matrix over the complex numbers have small norms, the eigenvalues of the matrix cannot be "far from" the diagonal entries of the matrix. Therefore, by reducing the norms of off-diagonal entries one can attempt to approximate the eigenvalues of the matrix. Of course, diagonal entries may change in the process of minimizing off-diagonal entries. The theorem does not claim that there is one disc for each eigenvalue; if anything, the discs rather correspond to the axes in C n {\displaystyle \mathbb {C} ^{n}} , and each expresses a bound on precisely those eigenvalues whose eigenspaces are closest to one particular axis. In the matrix ( 3 2 2 1 1 0 1 0 1 ) ( a 0 0 0 b 0 0 0 c ) ( 3 2 2 1 1 0 1 0 1 ) − 1 = ( − 3 a + 2 b + 2 c 6 a − 2 b − 4 c 6 a − 4 b − 2 c b − a a + ( a − b ) 2 ( a − b ) c − a 2 ( a − c ) a + ( a − c ) ) {\displaystyle {\begin{pmatrix}3&2&2\\1&1&0\\1&0&1\end{pmatrix}}{\begin{pmatrix}a&0&0\\0&b&0\\0&0&c\end{pmatrix}}{\begin{pmatrix}3&2&2\\1&1&0\\1&0&1\end{pmatrix}}^{-1}={\begin{pmatrix}-3a+2b+2c&6a-2b-4c&6a-4b-2c\\b-a&a+(a-b)&2(a-b)\\c-a&2(a-c)&a+(a-c)\end{pmatrix}}} — which by construction has eigenvalues a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} with eigenvectors ( 3 1 1 ) {\displaystyle \left({\begin{smallmatrix}3\\1\\1\end{smallmatrix}}\right)} , ( 2 1 0 ) {\displaystyle \left({\begin{smallmatrix}2\\1\\0\end{smallmatrix}}\right)} , and ( 2 0 1 ) {\displaystyle \left({\begin{smallmatrix}2\\0\\1\end{smallmatrix}}\right)} — it is easy to see that the disc for row 2 covers a {\displaystyle a} and b {\displaystyle b} while the disc for row 3 covers a {\displaystyle a} and c {\displaystyle c} . This is however just a happy coincidence; if working through the steps of the proof one finds that it in each eigenvector is the first element that is the largest (every eigenspace is closer to the first axis than to any other axis), so the theorem only promises that the disc for row 1 (whose radius can be twice the sum of the other two radii) covers all three eigenvalues. == Strengthening of the theorem == If one of the discs is disjoint from the others then it contains exactly one eigenvalue. If however it meets another disc it is possible that it contains no eigenvalue (for example, A = ( 0 1 4 0 ) {\displaystyle A=\left({\begin{smallmatrix}0&1\\4&0\end{smallmatrix}}\right)} or A = ( 1 − 2 1 − 1 ) {\displaystyle A=\left({\begin{smallmatrix}1&-2\\1&-1\end{smallmatrix}}\right)} ). In the general case the theorem can be strengthened as follows: Theorem: If the union of k discs is disjoint from the union of the other n − k discs then the former union contains exactly k and the latter n − k eigenvalues of A, when the eigenvalues are counted with their algebraic multiplicities. Proof: Let D be the diagonal matrix with entries equal to the diagonal entries of A and let B ( t ) = ( 1 − t ) D + t A . {\displaystyle B(t)=(1-t)D+tA.} We will use the fact that the eigenvalues are continuous in t {\displaystyle t} , and show that if any eigenvalue moves from one of the unions to the other, then it must be outside all the discs for some t {\displaystyle t} , which is a contradiction. The statement is true for D = B ( 0 ) {\displaystyle D=B(0)} . The diagonal entries of B ( t ) {\displaystyle B(t)} are equal to that of A, thus the centers of the Gershgorin circles are the same, however their radii are t times that of A. Therefore, the union of the corresponding k discs of B ( t ) {\displaystyle B(t)} is disjoint from the union of the remaining n-k for all t ∈ [ 0 , 1 ] {\displaystyle t\in [0,1]} . The discs are closed, so the distance of the two unions for A is d > 0 {\displaystyle d>0} . The distance for B ( t ) {\displaystyle B(t)} is a decreasing function of t, so it is always at least d. Since the eigenvalues of B ( t ) {\displaystyle B(t)} are a continuous function of t, for any eigenvalue λ ( t ) {\displaystyle \lambda (t)} of B ( t ) {\displaystyle B(t)} in the union of the k discs its distance d ( t ) {\displaystyle d(t)} from the union of the other n-k discs is also continuous. Obviously d ( 0 ) ≥ d {\displaystyle d(0)\geq d} , and assume λ ( 1 ) {\displaystyle \lambda (1)} lies in the union of the n-k discs. Then d ( 1 ) = 0 {\displaystyle d(1)=0} , so there exists 0 < t 0 < 1 {\displaystyle 0<t_{0}<1} such that 0 < d ( t 0 ) < d {\displaystyle 0<d(t_{0})<d} . But this means λ ( t 0 ) {\displaystyle \lambda (t_{0})} lies outside the Gershgorin discs, which is impossible. Therefore λ ( 1 ) {\displaystyle \lambda (1)} lies in the union of the k discs, and the theorem is proven. Remarks: It is necessary to count the eigenvalues with respect to their algebraic multiplicities. Here is a counter-example : Consider the matrix, [ 5 1 0 0 0 0 5 1 0 0 0 0 5 0 0 0 0 0 1 1 0 0 0 0 1 ] {\displaystyle {\begin{bmatrix}5&1&0&0&0\\0&5&1&0&0\\0&0&5&0&0\\0&0&0&1&1\\0&0&0&0&1\end{bmatrix}}} The union of the first 3 disks does not intersect the last 2, but the matrix has only 2 eigenvectors, e1,e4, and therefore only 2 eigenvalues, demonstrating that theorem is false in its formulation. The demonstration of the shows only that eigenvalues are distinct, however any affirmation about number of them is something that does not fit, and this is a counterexample. The continuity of λ ( t ) {\displaystyle \lambda (t)} should be understood in the sense of topology. It is sufficient to show that the roots (as a point in space C n {\displaystyle \mathbb {C} ^{n}} ) is continuous function of its coefficients. Note that the inverse map that maps roots to coefficients is described by Vieta's formulas (note for characteristic polynomials that a n ≡ 1 {\displaystyle a_{n}\equiv 1} ), which can be proved an open map. This proves the roots as a whole is a continuous function of its coefficients. Since composition of continuous functions is again continuous, the λ ( t ) {\displaystyle \lambda (t)} as a composition of roots solver and B ( t ) {\displaystyle B(t)} is also continuous. Individual eigenvalue λ ( t ) {\displaystyle \lambda (t)} could merge with other eigenvalue(s) or appeared from a splitting of previous eigenvalue. This may confuse people and questioning the concept of continuous. However, when viewing from the space of eigenvalue set C n {\displaystyle \mathbb {C} ^{n}} , the trajectory is still a continuous curve although not necessarily smooth everywhere. Added Remark: The proof given above is arguably (in)correct...... There are two types of continuity concerning eigenvalues: (1) each individual eigenvalue is a usual continuous function (such a representation does exist on a real interval but may not exist on a complex domain), (2) eigenvalues are continuous as a whole in the topological sense (a mapping from the matrix space with metric induced by a norm to unordered tuples, i.e., the quotient space of C^n under permutation equivalence with induced metric). Whichever continuity is used in a proof of the Gerschgorin disk theorem, it should be justified that the sum of algebraic multiplicities of eigenvalues remains unchanged on each connected region. A proof using the argument principle of complex analysis requires no eigenvalue continuity of any kind. For a brief discussion and clarification, see. == Application == The Gershgorin circle theorem is useful in solving matrix equations of the form Ax = b for x where b is a vector and A is a matrix with a large condition number. In this kind of problem, the error in the final result is usually of the same order of magnitude as the error in the initial data multiplied by the condition number of A. For instance, if b is known to six decimal places and the condition number of A is 1000 then we can only be confident that x is accurate to three decimal places. For very high condition numbers, even very small errors due to rounding can be magnified to such an extent that the result is meaningless. It would be good to reduce the condition number of A. This can be done by preconditioning: A matrix P such that P ≈ A−1 is constructed, and then the equation PAx = Pb is solved for x. Using the exact inverse of A would be nice but finding the inverse of a matrix is something we want to avoid because of the computational expense. Now, since PA ≈ I where I is the identity matrix, the eigenvalues of PA should all be close to 1. By the Gershgorin circle theorem, every eigenvalue of PA lies within a known area and so we can form a rough estimate of how good our choice of P was. == Example == Use the Gershgorin circle theorem to estimate the eigenvalues of: A = [ 10 1 0 1 0.2 8 0.2 0.2 1 1 2 1 − 1 − 1 − 1 − 11 ] . {\displaystyle A={\begin{bmatrix}10&1&0&1\\0.2&8&0.2&0.2\\1&1&2&1\\-1&-1&-1&-11\\\end{bmatrix}}.} Starting with row one, we take the element on the diagonal, aii as the center for the disc. We then take the remaining elements in the row and apply the formula ∑ j ≠ i | a i j | = R i {\displaystyle \sum _{j\neq i}|a_{ij}|=R_{i}} to obtain the following four discs: D ( 10 , 2 ) , D ( 8 , 0.6 ) , D ( 2 , 3 ) , and D ( − 11 , 3 ) . {\displaystyle D(10,2),\;D(8,0.6),\;D(2,3),\;{\text{and}}\;D(-11,3).} Note that we can improve the accuracy of the last two discs by applying the formula to the corresponding columns of the matrix, obtaining D ( 2 , 1.2 ) {\displaystyle D(2,1.2)} and D ( − 11 , 2.2 ) {\displaystyle D(-11,2.2)} . The eigenvalues are -10.870, 1.906, 10.046, 7.918. Note that this is a (column) diagonally dominant matrix: | a i i | > ∑ j ≠ i | a j i | {\textstyle |a_{ii}|>\sum _{j\neq i}|a_{ji}|} . This means that most of the matrix is in the diagonal, which explains why the eigenvalues are so close to the centers of the circles, and the estimates are very good. For a random matrix, we would expect the eigenvalues to be substantially further from the centers of the circles. == Stronger Conclusions and Taussky’s Theorem == While the original Gershgorin Circle Theorem applies to all complex square matrices, stronger conclusions can be drawn when the matrix has additional structure, such as being symmetric or irreducible. For a real symmetric matrix A ∈ R n × n {\displaystyle A\in \mathbb {R} ^{n\times n}} , the Gershgorin disks reduce to intervals on the real line. In this case: Every eigenvalue of A {\displaystyle A} lies within at least one of its Gershgorin intervals. If the intervals I 1 , … , I n {\displaystyle I_{1},\dots ,I_{n}} can be divided into two disjoint groups — one with p {\displaystyle p} intervals and the other with n − p {\displaystyle n-p} — and the union of each group is disjoint from the other, then the group with p {\displaystyle p} intervals contains exactly p {\displaystyle p} eigenvalues (counting algebraic multiplicities), and the other group contains n − p {\displaystyle n-p} . Additionally, a refinement due to Olga Taussky provides further structure for irreducible matrices: If an eigenvalue lies at an endpoint of a Gershgorin interval, and the matrix is irreducible, then that eigenvalue is an endpoint of every Gershgorin interval. This result — known as Taussky’s Theorem — highlights how the geometry of the Gershgorin intervals tightly constrains eigenvalue locations when the matrix exhibits sufficient connectivity and structure. == Example illustrating Taussky’s Theorem == Consider the symmetric and irreducible tridiagonal matrix: A = [ 1 − 1 0 − 1 2 − 1 0 − 1 1 ] {\displaystyle A={\begin{bmatrix}1&-1&0\\-1&2&-1\\0&-1&1\end{bmatrix}}} This matrix is real, symmetric, and irreducible. Each Gershgorin disk reduces to an interval on the real line: Row 1: center = 1, radius = 1 → interval: [0, 2] Row 2: center = 2, radius = 2 → interval: [0, 4] Row 3: center = 1, radius = 1 → interval: [0, 2] The union of these intervals covers [0, 4]. The eigenvalues of this matrix are: λ 1 = 0 , λ 2 = 1 , λ 3 = 3 {\displaystyle \lambda _{1}=0,\quad \lambda _{2}=1,\quad \lambda _{3}=3} Observe that the smallest eigenvalue, λ 1 = 0 {\displaystyle \lambda _{1}=0} , lies exactly at the left endpoint of all three Gershgorin intervals: [0, 2], [0, 4], and [0, 2]. By Taussky’s Theorem, since the matrix is symmetric and irreducible, and one eigenvalue lies at the boundary of a Gershgorin interval, that eigenvalue must lie at the boundary of every Gershgorin interval. This condition is satisfied here. This example illustrates how eigenvalues of symmetric, irreducible matrices can lie exactly on the boundaries of all Gershgorin intervals, as constrained by Taussky’s refinement of the Gershgorin Circle Theorem. == See also == For matrices with non-negative entries, see Perron–Frobenius theorem. Doubly stochastic matrix Hurwitz-stable matrix Joel Lee Brenner Metzler matrix Muirhead's inequality Bendixson's inequality Schur–Horn theorem == References == Gerschgorin, S. (1931), "Über die Abgrenzung der Eigenwerte einer Matrix", Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk (in German), 6: 749–754. Varga, Richard S. (2004), Geršgorin and His Circles, Berlin: Springer-Verlag, ISBN 3-540-21100-4. (Errata). Varga, Richard S. (2002), Matrix Iterative Analysis (2nd ed.), Springer-Verlag. 1st ed., Prentice Hall, 1962. Golub, G. H.; Van Loan, C. F. (1996), Matrix Computations, Baltimore: Johns Hopkins University Press, p. 320, ISBN 0-8018-5413-X. == External links == "Gershgorin's circle theorem". PlanetMath. Eric W. Weisstein. "Gershgorin Circle Theorem." From MathWorld—A Wolfram Web Resource. Semyon Aranovich Gershgorin biography at MacTutor
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Wikipedia:Gerstenhaber algebra#0
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In mathematics and theoretical physics, a Gerstenhaber algebra (sometimes called an antibracket algebra or braid algebra) is an algebraic structure discovered by Murray Gerstenhaber (1963) that combines the structures of a supercommutative ring and a graded Lie superalgebra. It is used in the Batalin–Vilkovisky formalism. It appears also in the generalization of Hamiltonian formalism known as the De Donder–Weyl theory as the algebra of generalized Poisson brackets defined on differential forms. == Definition == A Gerstenhaber algebra is a graded-commutative algebra with a Lie bracket of degree −1 satisfying the Poisson identity. Everything is understood to satisfy the usual superalgebra sign conventions. More precisely, the algebra has two products, one written as ordinary multiplication and one written as [,], and a Z-grading called degree (in theoretical physics sometimes called ghost number). The degree of an element a is denoted by |a|. These satisfy the identities (ab)c = a(bc) (The product is associative) ab = (−1)|a||b|ba (The product is (super) commutative) |ab| = |a| + |b| (The product has degree 0) |[a,b]| = |a| + |b| − 1 (The Lie bracket has degree −1) [a,bc] = [a,b]c + (−1)(|a|−1)|b|b[a,c] (Poisson identity) [a,b] = −(−1)(|a|−1)(|b|−1) [b,a] (Antisymmetry of Lie bracket) [a,[b,c]] = [[a,b],c] + (−1)(|a|−1)(|b|−1)[b,[a,c]] (The Jacobi identity for the Lie bracket) Gerstenhaber algebras differ from Poisson superalgebras in that the Lie bracket has degree −1 rather than degree 0. The Jacobi identity may also be expressed in a symmetrical form ( − 1 ) ( | a | − 1 ) ( | c | − 1 ) [ a , [ b , c ] ] + ( − 1 ) ( | b | − 1 ) ( | a | − 1 ) [ b , [ c , a ] ] + ( − 1 ) ( | c | − 1 ) ( | b | − 1 ) [ c , [ a , b ] ] = 0. {\displaystyle (-1)^{(|a|-1)(|c|-1)}[a,[b,c]]+(-1)^{(|b|-1)(|a|-1)}[b,[c,a]]+(-1)^{(|c|-1)(|b|-1)}[c,[a,b]]=0.\,} == Examples == Gerstenhaber showed that the Hochschild cohomology H*(A,A) of an algebra A is a Gerstenhaber algebra. A Batalin–Vilkovisky algebra has an underlying Gerstenhaber algebra if one forgets its second order Δ operator. The exterior algebra of a Lie algebra is a Gerstenhaber algebra. The differential forms on a Poisson manifold form a Gerstenhaber algebra. The multivector fields on a manifold form a Gerstenhaber algebra using the Schouten–Nijenhuis bracket == References == Gerstenhaber, Murray (1963). "The cohomology structure of an associative ring". Annals of Mathematics. 78 (2): 267–288. doi:10.2307/1970343. JSTOR 1970343. Getzler, Ezra (1994). "Batalin-Vilkovisky algebras and two-dimensional topological field theories". Communications in Mathematical Physics. 159 (2): 265–285. arXiv:hep-th/9212043. Bibcode:1994CMaPh.159..265G. doi:10.1007/BF02102639. Kosmann-Schwarzbach, Yvette (2001) [1994], "Poisson algebra", Encyclopedia of Mathematics, EMS Press Kanatchikov, Igor V. (1997). "On field theoretic generalizations of a Poisson algebra". Reports on Mathematical Physics. 40 (2): 225–234. arXiv:hep-th/9710069. Bibcode:1997RpMP...40..225K. doi:10.1016/S0034-4877(97)85919-8.
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Wikipedia:Gertrude Ehrlich#0
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Gertrude Ehrlich (born January 7, 1923) is an Austrian-American mathematician, specializing in abstract algebra and algebraic number theory. She is a professor emerita of mathematics at the University of Maryland, College Park. == Early life and education == Ehrlich was born on January 7, 1923, in Vienna, the daughter of Jewish lawyer Josef Ehrlich and his wife Charlotte, née Kobak. In the late 1930s, she became a student at the Chajes Gymnasium, a special high school in Vienna for Jewish honor students; her classmates included future Nobel laureate Walter Kohn and mathematicians Rodolfo Permutti and Karl Greger. She was able to escape Nazi-occupied Austria in 1939, traveling with her mother, her older sister Margarete Ehrlich (a philosophy student and later radiographer) and aunt Mathilde Ehrlich (a painter) to the US on the SS Statendam in July 1939; her father rejoined them a year later. They lived for the next several years with her uncle Benedict Kobak in Atlanta. She became a US citizen in 1945. She graduated from the Georgia State College for Women in 1943, and earned a master's degree from the University of North Carolina in 1945. She completed her Ph.D. in 1953 at the University of Tennessee. Her dissertation, The Structure of Continuous Rings, was supervised by Wallace Givens. == Contributions == Ehrlich is the author of the book Fundamental Concepts of Abstract Algebra (PWS-Kent Publishing, 1991; Dover, 2011). She is the coauthor of The Structure of the Real Number System (with Leon Warren Cohen, D. Van Nostrand, 1963) and of Algebra (with Jacob Goldhaber, Macmillan, 1970; Robert E. Krieger Publishing, 1980). In 1964 she became editor of the "Classroom Notes" department of The American Mathematical Monthly. She was the first organizer of the University of Maryland High School Mathematics Competition, held annually for high school students in Maryland and the District of Columbia, starting in 1979. The concept of a morphic group comes from a 1976 research paper of Ehrlich, "Units and one-sided units in regular rings", in the Transactions of the American Mathematical Society, and Ehrlich's theorem on the endomorphisms of morphic groups, from the same paper, is named for her. == References ==
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Wikipedia:Gethyn Hewan#0
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Gethyn Elliot Hewan (23 December 1916 – 1 July 1988) was an English first-class cricketer and schoolmaster. He studied mathematics as the University of Cambridge, during which he played first-class cricket for Cambridge University Cricket Club, before accepting a fellowship to study at Yale University. He served in the Second World War with the Royal Horse Artillery, for which he was mentioned in dispatches. Following the war he took up teaching posts at several educational establishments, most notably as headmaster of Cranbrook School, Sydney. == Early life and first-class cricket == Born at Edinburgh, Hewan was educated at Marlborough College, where he was taught by Brian Hone. From Marlborough he went up to Clare College, Cambridge to study mathematics. He debuted in first-class cricket for Cambridge University in 1938 against the Free Foresters at Fenner's, with Hewan making a total of six first-class appearances for the university in 1938, which was to be his only season of first-class cricket. A bowling all-rounder, Hewan scored 187 runs at an average of 20.77, with a high score of 88. With his off break bowling, he took 20 wickets at a bowling average of 36.25, with best figures of 6/91. These figures, one of two five wicket hauls he took, came against Oxford University. He was awarded a blue in cricket, alongside one in hockey. He was awarded the Andrew Mellon Fellowship to study at Yale University, travelling to the United States in 1938. He graduated with a major in mathematics. == World War II and move to Australia == With the outbreak of the Second World War in 1939, Hewan returned to England and enlisted in the British Army, joining the Royal Horse Artillery. He saw action in North Africa, including at Tobruk. He was mentioned in dispatches in November 1945 in recognition of gallantry and distinguished service in North-West Europe. Following the conclusion of the war he was appointed as the mathematics master at Wellington College, Berkshire. Relocating to Berkshire saw Hewan play minor counties cricket for Berkshire in 1946, making two appearances in the Minor Counties Championship. On the recommendation of Hone, he became the headmaster of Cranbrook School in Sydney. He accepted the position in November 1950, he relinquished his commission in the Royal Horse Artillery in July 1951, upon which he was granted the honorary rank of major. In Australia, he was a founding member of the Australian College of Education, and was a strong supporter of the Outward Bound movement. He was a promoter of the house system at Cranbrook, as well as promoting the benefits of extra-curricular activities to its pupils. He oversaw a number of new developments at the school, including the War Memorial Hall in 1953, Dickins House in Rose Bay in 1959 as the Cranbrook Preparatory School, and the Science Building which was opened in 1962. He was a keen amateur golfer, winning the Royal Sydney Golf Championship eight times. He resigned as the headmaster of Cranbrook in 1963, nominating Mark Bishop as his successor. == Return to England and later life == He returned to England shortly after, where he took up master posts at Winchester College and Charterhouse School. He took up the post of headmaster at Allhallows School in Lyme Regis, a position he held from 1965 to 1973. During his tenure as headmaster he oversaw a period of modernisation at the school. He announced his retirement on the grounds of ill health at the end of the 1973 summer term. He died at Chertsey in July 1988, and was survived by his wife Peggy, who passed away aged 96 in 2010, and their three children. == References == == External links == Gethyn Hewan at ESPNcricinfo
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Wikipedia:Ghulam Dastagir Alam#0
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Ghulam Dastagir Alam Qasmi (Urdu: غلام دستگیر عالم قاسمی; popularly known as G.D. Alam; PhD, HI), was a Pakistani theoretical physicist and professor of mathematics at the Quaid-e-Azam University. Alam is best known for conceiving and embarking on research on the gas centrifuge during Pakistan's integrated atomic bomb project in the 1970s, and he also conceived the research on charge density, nuclear fission, and gamma-ray bursts throughout his career. After the atomic bomb project, Alam joined the Department of Mathematics at the Quaid-e-Azam University (QAU) as well as serving as visiting faculty at the Institute of Physics, and co-authored papers on variation calculus and fission isomer. He was one of the notable theoretical physicists at the Pakistan Atomic Energy Commission (PAEC) and QAU. At one point, his fellow theorist, Munir Ahmad Khan, called Alam "the problem solving brain of the PAEC".: 155 == Biography == Alam was educated at the Government College University in Lahore where he studied in 1951 and graduated with Bachelor of Science (BSc) in Mathematics in 1955 under the supervision of Abdus Salam–a theoretical physicist. He then went to attend the physics program at the Punjab University where he graduated with Master of Science (MSc) in Physics in 1957, supervised under Dr. Rafi Muhammad– a nuclear physicist. His thesis titled: The Emission of Electromagnetic Radiations from metals by high energy particles, had contained investigations on electromagnetic radiation emitted from the heavy metals by bombarding the elementary particles. In 1964, Alam joined the doctoral program in physics at the University College London (UCL) under the Colombo Plan Scholarship, initially joining the doctoral group led by British physicist, John B. Hasted. He learned the course on atomic physics under Harrie Massey and worked on experimental physics under J.B. Hasted.: 137 In 1967, he submitted his doctoral thesis supervised by Dr. J.B. Hasted which he defended successfully and graduated with Doctor of Philosophy in theoretical physics.: 11–142 His doctoral thesis, titled: Electron Capture by Multiply Charged Ions, provided scientific investigations on charge-crossing involving potential curve crossing, a concept in quantum mechanics.: 11–142 In 1967, he published another thesis jointly written by J.B. Hasted and D.K. Bohme on physics of atomic collision and potential energy curves— their work was supported and funded by the United States Department of Defense. While in the United Kingdom, Alam continued publishing and working on the atomic physics and atomic collisions, collaborating with many other of his British colleagues. However, Alam lost interests in atomic physics and became interested in computer programming and mathematics.: 80, 425 In 1970, he published a paper on gamma rays and performed an experiment on isomers, proposing and later proving mathematically that, in the isomer state, the average kinetic energy associated with the decay process of Isomer state is about the same in the ground state of fission. == Pakistan Atomic Energy Commission == === 1971 war, atomic bomb program, and centrifuge technology === In 1958, Alam joined the Pakistan Atomic Energy Commission as a researcher, and was directed to join the Institute of Nuclear Science and Technology (PINSTECH), the national laboratory site in Nilore, in 1970.: 8 During the war with India in 1971, Alam became interested in computing and mathematics, briefly writing a computer program while working in problems relating to logic and the mathematics.: 8 Alam was instrumental in establishing the computer department at the Institute of Nuclear Science and Technology before moving to join the Physics Division.: 53 Alam originally was not part of the team that was investigating the uranium enrichment under Bashiruddin Mahmood in 1974 and had not seen a gas centrifuge albeit of rudimentary knowledge from the work done by Jesse Beams, an American, in 1940s.: 53–54 In 1974–76, Alam directed a separate electromagnetic separation program at the Airport Development Works setup at the Chaklala Air Force Base and begin working on the applications of magnetism and magnetic bearing.: 143 Eventually other physicists such as Fakhr Hashmi, Javed Arshad Mirza, Eqbal Ahmad Khokhar, and Anwar Ali joined his Airport Development Works facility— thus forming the centrifuge program under Alam.: 143–144 In 1976, Alam was asked to review the components and the blueprints of a gas centrifuge brought by Abdul Qadeer Khan for analysis, which he later identified as incorrect and incomplete to the PAEC.: 146–146 During this time, he held discussion with Abdul Qadeer Khan on copper metallurgy which was nascent to his development of centrifuge at the Airport Development Works facility.: 146 Meanwhile, Alam accepted and joined the faculty of mathematics at the Quaid-e-Azam University and met with Tasneem Shah, an Oxford University PhD graduate in mathematics, degelating Shah to join the centrifuge program.: 56–57 Alam and Shah performed mathematical calculations for linearly approximating the reasonable percent error for the actual value of military-grade uranium that proved the Abdul Qadeer Khan's estimation required to enriched the uranium.: 57 In April 1976, Alam succeeded in designing a gas centrifuge and succeeded in rotating the first centrifuge to 30,000 rpm by perfectly balancing and rotating around fixed axis of the centrifuge— Alam and his team was immediately dispatched at the Khan Research Laboratories under Abdul Qadeer Khan.: 57 At KRL, Alam and Shah worked on solving differential equation problems relating the gas centrifuges, and designed a first generation of centrifuges, known as P1, that succeeded in separating the U234, U235, and U238 isotopes from raw natural uranium under supervision of Shah, Alam, and Qadeer Khan.: 58–59 Commenting on the success, Alam famously quoted: "Boys, we have achieved enrichment in Pakistan.": 148–149 Following this success, Alam and other colleagues signed and quickly published a paper on the differential methods of the centrifuges in 1978–79.: 150–151 Alam remained associated with the centrifuge program and was its design director until 1981 when he developed intellectual differences with Abdul Qadeer Khan in 1981.: 431 He was confided by Abdul Qadeer Khan about selling the centrifuge technology to unknown Arab country, which he vehemently opposed the idea on the ground that India (a competitor in that field) might be benefited from that technology.: 431 After this incident, Alam was transferred to PAEC where he remained associated with the electromagnetic separation program under Dr. Shaukat Hameed Khan, and decided to work on the partial differential equations.: 59 Alam went to join the Quaid-i-Azam University and briefly taught courses on calculus, and published a computer model on HIV rate of infection along with American scientists.: 389 At the Department of Mathematics, Alam pioneered a paper in Gamma-ray bursts using mathematical descriptions to analyse the energy released by the gamma rays. Alam died on 5 December 2000 in his residence in Islamabad. == Books == Qadir, Asghar; Faheem Hussain; Riazuddin, M.; Jamil Aslam & Hamid Saleem (June 2007) [1999], "Quantum nonlocality, Black Holes and Quantum Gravity", Mathematical physics Proceedings of the 12th Regional Conference, vol. 1 (1 ed.), National Center for Theoretical Physics, Islamabad: World Scientific Publications Co. Pte. Lmtd, pp. 382–400, ISBN 978-981-270-591-4 Qasmi, Ghulam Dastigar Alam (April 1996) [1996], Calculus: An Approach to Engineering Mathematics, 1, vol. 1 (1 ed.), Department of Mathematics, Quaid-e-Azam University: Punjab Text Board, p. 390 == Science articles == Fission fragment mass distributions and kinetic energies for spontaneous fission isomers, by R.L. Ferguson, F. Plasil, G.D. Alam†, H.W. Schmitt. Gamma Ray Transitions in the de-excitation Californium-253 spontenous fission pragments, by G.D. Alam, Department of Mathematics. == Bibliography == == References ==
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Wikipedia:Giacomo Bellacchi#0
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Giacomo Bellacchi (1838–1924) was an Italian mathematician. After graduating from Scuola Normale Superiore di Pisa, he became a teacher at a military school and at the Tuscan Technical Institute, where one of his pupils was Vito Volterra. Over his career, he carried out research both in geometry and algebra. He wrote many works, among which the most prominent is probably Introduzione storica alla teoria delle funzioni ellittiche (Historical Introduction to Elliptic Function Theory), which became well known and was used worldwide. He also wrote many university textbooks. The library of the Fondazione Scienza e Tecnica in Florence has named after him its precious collection of math books and works. == References == == Bibliography == Bellacchi, Giacomo. Lezioni ed esercizi di algebra. Firenze. Bellacchi, Giacomo; Peri, Giuseppe (1873). I principi della geometria moderna esposti da G. Peri e G. Bellacchi. Pistoia.{{cite book}}: CS1 maint: location missing publisher (link) Bellacchi, Giacomo (1888). Le applicazioni elementari delle matematiche: discorso letto nel R. Istituto tecnico Galilei il dì 17 di ottobre 1888. Firenze.{{cite book}}: CS1 maint: location missing publisher (link) Bellacchi, Giacomo (1891). Galileo e i suoi successori: discorso letto nel R. Istituto tecnico Galilei di Firenze il dì 29 ottobre 1891. Firenze.{{cite book}}: CS1 maint: location missing publisher (link) Bellacchi, Giacomo (1891). Teoria dell'equazioni. Firenze.{{cite book}}: CS1 maint: location missing publisher (link) Bellacchi, Giacomo (1894). Introduzione storica alla teoria delle funzioni ellittiche. Firenze: Tipografia di G. Barbera. Bellacchi, Giacomo (1898). Lezioni ed esercizi di algebra complementare: Fasc. 1. Firenze.{{cite book}}: CS1 maint: location missing publisher (link) Bellacchi, Giacomo (1899). Lezioni ed esercizi di algebra complementare: Fasc. 2. Firenze.{{cite book}}: CS1 maint: location missing publisher (link) Bellacchi, Giacomo (1910). Complementi di geometria e d'algebra. Firenze.{{cite book}}: CS1 maint: location missing publisher (link) == External links == Digitized version of Introduzione storica alla teoria delle funzioni ellittiche (Cornell University – Historical library) Fondazione Scienza e Tecnica di Firenze
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Wikipedia:Giambattista Suardi#0
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Giambattista Suardi (January 9, 1711 – March 2, 1767) was an Italian mathematician. == Life == Born into a noble family in Brescia, he studied mathematics in Padua. Suardi graduated in 1773 under the supervision of Giovanni Poleni. In 1752 he published an essay on drawing and mathematics tools: Nuovi istromenti per la descrizione di diverse curve antiche e moderne e di molte altre che servir possono alla speculazione de' geometri ed all'uso de' pratici: col progetto di due nuove macchine per la nautica ed una per la meccanica, e con alcune osservazioni sopra de' poligoni rettilinei regolari. He married Cecilia Curti, a Venetian woman. == Works == Suardi, Giambattista (1752). Nuovi istromenti per la descrizione di diverse curve antiche e moderne (in Italian). In Brescia: dalle stampe di Gian Maria Rizzardi. Suardi, Giambattista (1764). Trattenimenti matematici (in Italian). In Brescia: Giovanni Battista Bossini. == References ==
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Wikipedia:Giambelli's formula#0
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In mathematics, Giambelli's formula, named after Giovanni Giambelli, expresses Schubert classes as determinants in terms of special Schubert classes. It states σ λ = det ( σ λ i + j − i ) 1 ≤ i , j ≤ r {\displaystyle \displaystyle \sigma _{\lambda }=\det(\sigma _{\lambda _{i}+j-i})_{1\leq i,j\leq r}} where σλ is the Schubert class of a partition λ. Giambelli's formula may be derived as a consequence of Pieri's formula. The Porteous formula is a generalization to morphisms of vector bundles over a variety. In the theory of symmetric functions, the same identity, known as the first Jacobi-Trudi identity expresses Schur functions as determinants in terms of complete symmetric functions. There is also the dual second Jacobi-Trudi identity which expresses Schur functions as determinants in terms of elementary symmetric functions. The corresponding identity also holds for Schubert classes. There is another Giambelli identity, expressing Schur functions as determinants of matrices whose entries are Schur functions corresponding to hook partitions contained within the same Young diagram. This too is valid for Schubert classes, as are all Schur function identities. For instance, hook partition Schur functions can be expressed bilinearly in terms of elementary and complete symmetric functions, and Schubert classes satisfy these same relations. == See also == Schubert calculus - includes examples == References == Fulton, William (1997), Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, ISBN 978-0-521-56144-0, ISBN 978-0-521-56724-4, MR1464693 Sottile, Frank (2001) [1994], "Schubert calculus", Encyclopedia of Mathematics, EMS Press
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Wikipedia:Giang Nguyen#0
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This is a Vietnamese name; The family name is Nguyen. Giang Thu Nguyen (born Nguyễn Thu Giang on 2 October 1985 in Hanoi) is a Vietnamese-Australian chess player and mathematician. She is a senior lecturer in Applied Mathematics at the University of Adelaide. In chess, she is a Woman FIDE Master (WFM), and has represented Australia in many Chess Olympiads. She has won the South Australian women's championship four times, and the South Australian (open) championship once. == Chess == Nguyen began playing chess at the age of nine. She won the gold medal in Vietnamese Junior Girls Under 13 Championship in 1998, and the silver medal in Vietnamese Junior Girls Under 15 Championship in 1999. Nguyen first represented Vietnam in the World Rapid Girls Under 14 Championship (Disneyland, Paris) in 1998. Representing Vietnam, she won a gold medal at the Asian Girls Under 14 Championship in 1999, a silver medal at the 2nd Children of Asia International Children Sports Games in 2000, and a silver medal in the ASEAN Girls Under 16 Championship in 2000. In 2001, Nguyen moved to Adelaide, Australia. She came equal third in the Australian Junior Championship in Sydney in 2002. Since 2008, Nguyen has been playing chess under the Australian flag. She has represented Australia in eight Chess Olympiads: in 2008, 2010, 2012, 2014, 2016, 2018, 2022, and 2024. In 2012, she scored 6/9 and was awarded the Woman FIDE Master (WFM) title for her result. She won the (open) South Australian championship in 2014, and the women's championship in 2003, 2006, 2007 and 2008. == Mathematics == Nguyen completed her PhD in mathematics from the University of South Australia (UniSA) in 2009 at the age of 23. This made her the youngest PhD graduate of UniSA and the second youngest from a South Australian university. Her dissertation, Hamiltonian Cycle Problem, Markov Decision Processes and Graph Spectra, was jointly supervised by Jerzy A. Filar and Vlad Ejov. Nguyen is a senior lecturer in Applied Mathematics at the University of Adelaide, and a 2019 South Australian Tall Poppy Science Award recipient. == References == == External links == Official website Giang Nguyen at the Mathematics Genealogy Project
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Wikipedia:Gianluigi Rozza#0
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Gianluigi Rozza is an aerospace engineer and mathematician best known for his work on reduced-order modeling. He is currently full professor of Numerical Analysis at the International School for Advanced Studies (SISSA) in Trieste, where he serves as head of SISSA Mathematics Area and SISSA Director's Delegate for Research Valorisation, Innovation, and Industrial Cooperation. == Personal life and education == G. Rozza was born and raised in Italy where he earned a bachelor's and master's degree in Aerospace Engineering from the Polytechnic University of Milan in the year 2002. He subsequently moved to Lausanne, Switzerland to join the École Polytechnique Fédérale de Lausanne (EPFL) for postgraduate studies. At EPFL, he received a PhD in Numerical Analysis in 2005. His thesis, titled 'Shape Design by Optimal Flow Control and Reduced basis Techniques: Applications to Bypass Configurations in Haemodynamics', was completed under the supervision of Alfio Quarteroni and Anthony Tyr Patera. == Career == Following his doctoral studies, G. Rozza worked at Ecole Polytechnique Fédérale de Lausanne in the professor Alfio Quarteroni's scientific group for a year. In 2006, he joined the Department of Mechanical Engineering and the Center for Computational Engineering at the MIT as a Postdoctoral Associate Researcher in the professor Anthony Tyr Patera's group. He stayed at the MIT until 2008. After a period as a Senior Researcher and Lecturer at the EPFL, in 2012, Rozza joined the Applied Mathematics group, SISSA mathLab, at the International School for Advanced Studies (SISSA) where, in 2014, he became professor. In 2018 he received an ERC Consolidator Grant (CoG) from the European Research Council (ERC) with the proposal «AROMA-CFD» (2016-2022), and in 2022 he won an ERC Proof of Concept Grant PoC (Proof of Concept). for the project «ARGOS». In December 2021 he was ranked n.1152 in the Top Scientist-Mathematics from research.com, n.35 in Italy. G. Rozza is currently n.20 in the Top Italian Scientists Mathematics ranking. In September 2022 he was listed in the World's Top 2% Scientist ranking, made by Stanford University in collaboration with Elsevier and Scopus. As of January 2023 Rozza co-authored 3 books and edited other 10 (12 volumes) on numerical analysis and model order reduction. In addition to research, G. Rozza is the President of SMACT Innovation Ecosystem's supervisory board, and he is an executive committee member of ECCOMAS (European Community on Computational Methods in Applied Sciences). == Recognitions == In 2004 Rozza received the Bill Morton CFD Prize at the Oxford ICFD conference. Rozza Received the Riconoscenza Civica from his birthplace (Sant'Angelo Lodigiano). In 2006 he received the Phd Award from ECCOMAS Congress (European Community on Computational Methods in Applied Sciences) in Amsterdam. In 2009 Rozza received the Springer Computational Science and Engineering (CSE) Award in Monaco with Phuong Huynh and Cuong Nguyen for the rbMIT software, developed at MIT in Boston. Rozza received the Dardo D'oro from his City of residence (Castiraga Vidardo) in 2010. In 2014 he received the Jacques-Louis Lions Award at ECCOMAS Congress (European Community on Computational Methods in Applied Sciences) in Barcelona. In 2022 Rozza gave the first Solari Lecture at the Polytechnic University of Milan, Italy. == References == == External links == "Official website". Retrieved 1 February 2023. "ORCID profile". Retrieved 1 February 2023. "Google Scholar profile". Retrieved 14 February 2023.
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Wikipedia:Gideon Schechtman#0
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Gideon Schechtman (Hebrew: גדעון שכטמן; born 14 February 1947) is an Israeli mathematician and professor of mathematics at the Weizmann Institute of Science. == Academic career == Schechtman received his Ph.D. in mathematics from the Hebrew University of Jerusalem in 1976 and was a postdoctoral fellow at Ohio State University. Since 1980 he has been affiliated with the Weizmann Institute, where he became emeritus professor in 2017. His research focuses predominantly on functional analysis and the geometry of Banach spaces. Schechtman is an editor of the Israel Journal of Mathematics. == References ==
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Wikipedia:Gigliola Staffilani#0
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Gigliola Staffilani (born March 24, 1966) is an Italian-American mathematician who works as the Abby Rockefeller Mauze Professor of Mathematics at the Massachusetts Institute of Technology. Her research concerns harmonic analysis and partial differential equations, including the Korteweg–de Vries equation and Schrödinger equation. == Education and career == Staffilani grew up on a farm in Martinsicuro in central Italy, speaking only the local dialect, and with no books until her older brother brought some back from his school. Her father died when she was 10, and her mother decided that she did not need to continue on to high school, but her brother helped her change her mother's mind. She came to love mathematics at her school, and was encouraged by her teachers and brother to continue her studies, with the idea that she could return to Martinsicuro as a mathematics teacher. She earned a fellowship to study at the University of Bologna, where she earned a laurea in mathematics in 1989 with an undergraduate thesis on Green's functions for elliptic partial differential equations. At the suggestion of one of her professors at Bologna, she moved to the University of Chicago for her graduate studies, to study with Carlos Kenig. This was a big change in her previous plans, because it would mean that she could not return to Martinsicuro. When she arrived at Chicago, still knowing very little English and not having taken the Test of English as a Foreign Language, she had the wrong type of visa to obtain the teaching fellowship she had been promised. She almost returned home, but remained after Paul Sally intervened and loaned her enough money to get by until the issue could be resolved. At Chicago, she studied dispersive partial differential equations with Kenig, earning a master's degree in 1991 and a Ph.D. in 1995. After postdoctoral studies at the Institute for Advanced Study, Stanford University, and Princeton University, Staffilani took a tenure-track faculty position at Stanford in 1999, and earned tenure there in 2001. While at Stanford, she met her husband, Tomasz Mrowka, a mathematics professor at MIT, and after a year and a half found a faculty position closer to him at Brown University. She moved to MIT in 2002, where, in 2006 she became the second female full professor of mathematics. She served as an American Mathematical Society Council member at large from 2018 to 2020. == Collaboration == Staffilani is a frequent collaborator with James Colliander, Markus Keel, Hideo Takaoka, and Terence Tao, forming a group known as the "I-team". The name of this group has been said to come from the notation for a mollification operator used in the team's method of almost conserved quantities, or as an abbreviation for "interaction", referring both to the teamwork of the group and to the interactions of light waves with each other. The group's work was featured prominently in Fefferman's 2006 Fields Medal citations for group member Tao. == Awards and honors == Staffilani was a Sloan Fellow from 2000 to 2002. In 2009-2010 she was a member of the Radcliffe Institute for Advanced Study. In 2012 she became one of the inaugural fellows of the American Mathematical Society. In 2014 she was inducted into the American Academy of Arts and Sciences. In 2021, she was elected to the National Academy of Sciences. == Major publications == Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Global well-posedness for Schrödinger equations with derivative. SIAM J. Math. Anal. 33 (2001), no. 3, 649–669. Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. A refined global well-posedness result for Schrödinger equations with derivative. SIAM J. Math. Anal. 34 (2002), no. 1, 64–86. Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation. Math. Res. Lett. 9 (2002), no. 5-6, 659–682. Staffilani, Gigliola; Tataru, Daniel. Strichartz estimates for a Schrödinger operator with nonsmooth coefficients. Comm. Partial Differential Equations 27 (2002), no. 7-8, 1337–1372. doi:10.1081/PDE-120005841 MR1924470 Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Sharp global well-posedness for KdV and modified KdV on R {\displaystyle \mathbb {R} } and T {\displaystyle \mathbb {T} } . J. Amer. Math. Soc. 16 (2003), no. 3, 705–749. doi:10.1090/S0894-0347-03-00421-1 MR1969209 arXiv:math/0110045 Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Multilinear estimates for periodic KdV equations, and applications. J. Funct. Anal. 211 (2004), no. 1, 173–218. Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on R 3 {\displaystyle \mathbb {R} ^{3}} . Comm. Pure Appl. Math. 57 (2004), no. 8, 987–1014. Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R 3 {\displaystyle \mathbb {R} ^{3}} . Ann. of Math. (2) 167 (2008), no. 3, 767–865. doi:10.4007/annals.2008.167.767 MR2415387 == References == == External links == Home page
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Wikipedia:Gijs de Leve#0
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Gijsbert "Gijs" de Leve (15 August 1926 – 19 November 2009) was a Dutch mathematician and operations researcher, known for his work on Markov decision process. Gijs de Leve is considered the founder of operations research in the Netherlands. == Biography == Born in Amsterdam, De Leve received his MA in Mathematics and Physics in 1954 at the University of Amsterdam. There he also received his PhD cum laude in 1964 for the thesis Generalized Markovian decision processes, advised by Jan Hemelrijk and Johannes Runnenburg. De Leve was appointed professor in operations research, in particular management scientific applications, at the University of Amsterdam in 1972. Among his doctorate students were Henk Tijms (1972), Alexander Rinnooy Kan (1976), Jan Karel Lenstra (1976), P. Weeda (1978), Awi Federgruen (1978), Antoon Kolen (1982), Roy Jonker (1986), Anton Volgenant (1987), Jeroen de Kort (1992), Erik H.J. van der Sluis (1993), Nanda Piersma (1993) and Cees Duin (1994). He retired from the University of Amsterdam on 1 September 1991. In honor of De Leve, in 1997 the Gijs de Leve prize was initiated for the best PhD thesis in the area of mathematics of operations research. == Publications == Overzicht van een aantal artikelen over "Operations Research". Volume 1. G. de Leve, Jacobus Kriens, 1955. Enige statistische aspecten van de factoranalyse. G. de Leve, 1956 Operations research: Inleidende voorbeelden. Hoofdstuk 1. J. Kriens, G. de Leve, 1959 Oriënterende cursus mathematische besliskunde. G. de Leve, Stichting Mathematisch Centrum. Afdeling Mathematische Statistiek, 1962. Generalized Markovian decision processes – Deel 2. G. de Leve, 1964 Generalized Markovian Decision Processes: Model and method. Gijs de Leve, 1964 Inleiding in de besliskunde. G. de Leve, J.C. van Dalen, Stichting Mathematisch Centrum, 1970 Articles, a selection Leve, Gijsbert. "Wie beslist er eigenlijk*." Statistica Neerlandica 20.2 (1966): 155–169. De Leve, Gijsbert, Awi Federgruen, and Hendrik C. Tijms. "A general Markov decision method I: model and techniques." Advances in Applied Probability (1977): 296–315. Federgruen, A. "Some approximation approaches in large-scale." Twenty-five years of operations research in the Netherlands: papers dedicated to Gijs de Leve 70 (1989): 35. == References == == External links == Prof. dr. G. de Leve, 1926 - 2009 on the University of Amsterdam Album Academicum
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Wikipedia:Gillian Slater#0
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Gillian Lesley Slater (née Filtness) is a retired British mathematician and academic administrator, the former vice chancellor of Bournemouth University. == Education == Slater read mathematics in St Hugh's College, Oxford, where she served as secretary of the Oxford University Liberal Democrats in 1969. She completed a DPhil at the University of Oxford in 1973, with the dissertation Some Topics in Functional–Differential Equations supervised by John Bryce McLeod. == Career == Slater became a mathematics instructor at South Bank Polytechnic (now London South Bank University) and at Sheffield City Polytechnic (now Sheffield Hallam University). She moved into academic administration as dean of science and technology at Manchester Polytechnic, and then after it became Manchester Metropolitan University, as pro-vice-chancellor. Her next step was to become vice chancellor of Bournemouth University, in 1994, succeeding the university's first vice chancellor, Bernard MacManus. She came under pressure from the UK's Labour government in 2004 for taking a stand against the government's push to institute differential fees for different universities, and retired in 2005, replaced as vice chancellor by atmospheric scientist Paul Curran. == References == == Further reading == Slater, Gillian L. (1996), "Oh, Poor You!", in Walton, Karen Doyle (ed.), Against the Tide: Career Paths of Women Leaders in American and British Higher Education, Phi Delta Kappa Educational Foundation
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Wikipedia:Gilman–Griess theorem#0
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In finite group theory, a mathematical discipline, the Gilman–Griess theorem, proved by Robert H. Gilman and Robert L. Griess, classifies the finite simple groups of characteristic 2 type with e(G) ≥ 4 that have a "standard component", which covers one of the three cases of the trichotomy theorem. == References ==
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Wikipedia:Giorgio Bidone#0
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Giovanni Giorgio Bidone (19 January 1781, Casalnoceto, Piedmont – 25 August 1839, Turin, Kingdom of Sardinia) was an Italian engineer, mathematician and an experimenter in the field of hydraulics. Giorgio Bidone's parents were Alessandro Antonio Bidone and Margherita Malaspina. In 1782, when Giorgio was one year old, the family moved to Voghera where he was educated. He then entered the Filipino College in Turin. His plans to become a priest were abandoned because of new laws introduced by the French after Piedmont was annexed by Napoleon Bonaparte. He studied at Collegio delle Province as well as the University of Turin. He graduated in mathematics and hydraulic engineering at Turin at the age of 23 and two years later he graduated in civil engineering. He became a member of the Academy of Sciences of Turin in 1811 and was appointed professor of hydraulics in 1815. His research was in mathematics on the solution of transcendental equations and also on definite integrals. He performed experiments in the field of hydraulics at a laboratory at Parella, which had been established in 1763 by Francesco Domenico Michelotti. His research focused on analysis and hydraulics. In 1820 he published a paper called Experiences sur le remou et sur la propagation des ondes, where he announced the hydrodynamic phenomenon known as the "hydraulic jump". This is how kinetic energy is dissipated when a fast flowing current reaches a slower current. This effect can be seen in both water and air. In his hydraulic research he combined experiments with mathematical models. In his study of overflow he measured the upstream increase in height and the shape caused by an obstruction and then derived the equations for describing this effect. In 1817, together with Ignazio Michelotti, who was supervisor of rivers and chief inspector of canals, he drafted the new laws concerning the waterways of the kingdom. This earned him the Civil Order of Merit of Savoy in 1839. Between 1822 and 1823, he participated in an international collaboration to survey across the Alps to link his region with the rest of Western Europe and Eastern Europe. Bidone also taught mathematics at the University of Turin and he was appointed professor of geometry. He became a member of the National Academy of Sciences of Italy. == Works == Description d'une nouvelle boussole propre a observer les mouvemens de rotation et de translation de l'aiguille aimantée (in French). 1811. Mémoire sur la cause des ricochets que font les pierres et les boulets de canon, lancés obliquement sur la surface de l'eau (in French). Turin. 1811 – via de l'Imprimerie de l'Académie Imperiale des Sciences. Mémoire sur diverses intégrales définies (in French). Turin. 1812 – via chez Felix Galletti imprimeur de l'Academie Imperiale des Sciences. Mémoire sur les transcendantes elliptiques (in French). Vol. 1. Turin. 1817 – via de l'Imprimerie Royale. Mémoire sur les transcendantes elliptiques (in French). Vol. 2. Expériences sur le remou et sur la propagation des ondes (in French). Turin. 1820 – via de l'Imprimerie Royale. Riflessioni sul moto permanente dell'acqua ne' canali orizzontali. Modena. 1824 – via presso la Tipografia camerale. Expériences sur la dépense des reversoirs et sur l'accélération et la courbure qu'ils occasionnent à la surface du courant (in French). Turin. 1824 – via Imprimerie Royale. Osservazioni sopra le macchine in moto. Torino. 1825 – via dalla Stamperia reale. Expériences sur la propagation du remous (in French). Turin. 1825 – via Imprimerie Royale. Expériences sur divers cas de la contraction de la veine fluide, et remarque sur la manière d'avoir égard à la contraction dans le calcul de la dépense des orifices (in French). Turin. c. 1825 – via de l'Imprimerie Royale. Esperienze sulle contrazioni parziali delle vene d'acqua. Modena. 1830 – via dalla Tipografia Camerale. == References == Italian Wikipedia "Idraulici Italiani" by Mario Di Fidio & Claudio Gandolfi Archived 2021-08-04 at the Wayback Machine
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Wikipedia:Giovanni Fagnano#0
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Giovanni Francesco Fagnano dei Toschi (born 31 January 1715 in Senigallia, died 14 May 1797 in Senigallia) was an Italian churchman and mathematician, the son of Giulio Carlo de' Toschi di Fagnano, also a mathematician. == Religious career == Fagnano was ordained as a priest. In 1752 he became canon, and in 1755 he was appointed archdeacon of the cathedral of Senigallia. == Mathematics == Fagnano is known for Fagnano's problem, the problem of inscribing a minimum-perimeter triangle within an acute triangle. As Fagnano showed, the solution is the orthic triangle, whose vertices are the points where the altitudes of the original triangle cross its sides. Another property of the orthic triangle, also proven by Fagnano, is that its angle bisectors are the altitudes of the original triangle. Fagnano also partially solved the problem of finding the geometric median of sets of four points in the Euclidean plane; this is the point minimizing the sum of its distances to the four given points. As Fagnano showed, when the four points form the vertices of a convex quadrilateral, the geometric median is the point where the two diagonals of the quadrilateral cross each other. In the other possible case, not considered by Fagnano, one point lies within the triangle formed by the other three, and this inner point is the geometric median. Thus, in both cases, the geometric median coincides with the Radon point of the four given points. == References ==
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Wikipedia:Giovanni Forni#0
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Giovanni Forni is an Italian mathematician at the University of Maryland known for his research in dynamical systems. After graduating from the University of Bologna in 1989, he obtained his PhD in 1993 from Princeton University, under the supervision of John Mather. He was an invited speaker at the 2002 International Congress of Mathematicians in Beijing. For his work on solutions of cohomological equations for flows on surfaces, and on the Kontsevich–Zorich conjecture concerning deviation of ergodic averages, he was awarded the 2008 Michael Brin Prize in Dynamical Systems. In 2012, he became a fellow of the American Mathematical Society. == References ==
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Wikipedia:Giovanni Gherardi#0
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Giovanni di Gherardo da Prato, or Giovanni Gherardi (Prato, 1360/1367 – before 1446), was an Italian jurist, mathematician, writer and humanist. == Biography == Gherardo's father was a second-hand dealer; his grandfather Bartolo, as shown by the title of "ser", had practiced the profession of notary. He studied law in Padua where he also followed the lessons of Biagio Pelacani, a well-known philosopher and mathematician. Having settled in Florence, he began working as a judge and notary while taking a keen interest in literary studies. From 1414 he was legal consultant and archivist of Orsanmichele. In 1417 he taught courses on Dante's works at the Florentine Studio until the teaching position was discontinued in 1425. In that period he wrote the novel entitled Paradiso degli Alberti, named after a famous villa near Florence. In 1420, he was appointed deputy administrator of the dome of Santa Maria del Fiore in Florence, serving as Lorenzo Ghiberti's assistant and a fierce rival of the other administrator, Filippo Brunelleschi. In a parchment from 1426, preserved in the State Archives of Florence, he criticized several of Filippo's decisions through both drawings and writings. His critiques included issues like the insufficient interior lighting and the sharp curvature of the dome's vaults, referred to as the acute fifth (or sixth). He also opposed the "rippled" inclination of the brick-laying beds, arguing that they should be aligned with the center of the monument, rather than following the acute fifth curvature. According to him, these flaws posed a serious risk of causing the monument's collapse. To dissuade others from supporting what he believed to be a "misguided" project, he pointed out that "the Cathedral of Siena collapsed... for having trusted someone without proper knowledge." Indeed, the unfinished structures of the Siena Cathedral, which were planned to be much larger than the current one, are still visible today as a reminder of that failure. Giovanni's document is particularly important also because it is the only graphic testimony with technical commentary of the great work in progress. The contrast between the two is also demonstrated by an exchange of poetic compositions written in a Burchiellesque tone on the occasion of the dispute over a vessel for the transport of stones by river, patented by Filippo in 1421: on that occasion the "ship" (name which was given to cargo boats operating in the Arno) driven by (mechanical) propellers moved by wind energy, devices and mechanisms specially invented by Brunelleschi. The "ship" took the name of "Badalone". This was short-lived because a few months after its launch, a flood destroyed it and caused it to run aground near Montelupo, losing part of the materials it was transporting in 1428. Brunelleschi himself attempted to recover it in vain. There is a copy of this mechanical ship attributed to Leonardo da Vinci on the basis of which the prof. Massimo Ricci has created a working model. The model is currently located in the premises of the Opera del Duomo of Florence in Piazza San Giovanni together with three Brunelleschi machines for the construction of the Lantern of the Dome of Santa Maria del Fiore. == Literary works == As a disciple of Cino Rinuccini, he was also the author of various poems, including Gioco d'Amore II, the Treatise on an Angelic Thing Shown for a Most Devout Vision (written in the form of a mystical polymeter), a two-book allegorical poem in imitation of Dante, and an unfinished work titled Philomena. The manuscript of Philomena is preserved in the National Central Library of Florence, cataloged as ms. Magliabechiano VII, 702. A prose work consisting of five books has been attributed to him where, in imitation of Boccaccio's Filocolo, short stories, mythological stories, descriptions of Tuscan landscapes, imagined journeys and conversations held in the spring of 1389 by a cultured brigade are narrated at the Villa del Paradiso in the village of Bandino by Antonio Alberti. The work was rediscovered in 1864 by Aleksander Wesselofsky who identified its author and gave him the title Paradiso degli Alberti; the first publication of the work was his (1867–69), accompanied by a vast critical apparatus. The poem L'Acquettino was composed against Giovanni da Prato around 1406. The character of Giovanni da Prato appears with Antonio Alberti in the guesthouse of the Convento del Paradiso in Lorenzo Andreaggi's film trabocchetti del Bandino I, played by Niccolò Biffoli. == References ==
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Wikipedia:Giovanni Maria Tolosani#0
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Giovanni Maria Tolosani (pseudonym Joannes Lucidus Samotheus) (c.1471 – 22 January 1549) was an Italian theologian, writer, a prior of the Dominican order at the convent of St. Mark in Florence a mathematician and an astronomer. He is best known for writing the first notable denunciation of Copernican heliocentric theory in 1545. == Biography == Tolosani came from a Tuscan family based near Florence. He was born in Colle di Val d'Elsa. His father, Leonardo, was a jurist and notary there. His brothers included Nicola, a jurist and Roman senator, and Pietro Antonio, who also became a Dominican friar. Tolosani joined the Dominican friary in Florence on 24 June 1487, officially joining the order on 28 June the following year, remaining in the order until his death. In 1500, he became the first of the SS Annunziata in San Gimignano, the vice-prior of San Marco and was active in other convents. He worked on the chronicle of San Marco and San Maria del Sasso. He contributed to a cosmology in verse by Leonardo Dati, La Sfera (or Spera), in Florence in 1513. Many scholars - including Copernicus - responded to the papal request for specialist help to correct the Roman calendar during the time of the Fifth Council of the Lateran. With such correction in mind, in 1514, Tolosani wrote De computatione annorum Domini in emendationem secundae inquisitionis libri su and Apusculum de correctione calendarii pro vera celebratione Pascatis. The council ended in 1517, just before the Reformation, but further help was requested by the papacy in the following decades. In 1523, he published Lettere di Sant'Antonino arcivescovo di Firenze for the canonization of that saint. In 1537, Tolosani had five documents published in Venice, together titled Opusculum de emendiationibus temporum [Breviloquio de Tempe in Italian]; these formed Tolosani's work on aligning the calendar to the seasons, written under the pseudonym of an invented Frenchman, Joannes Lucidus Samotheus. He also wrote a booklet on the maximum declination of the Sun. In 1546, a further edition of the work was produced containing two additional documents (the original was dated December 1545) under Tolosani's own name; Brevis annotatio emendatoria calendarii romani [short annotation to the Roman calendar corrections] was done at the direction of the vicar general of the Dominican order, including a mention of the right time to celebrate Easter, and was finished on 8 May 1545 for the Council of Trent. The perceived importance of Tolosani's writings was elevated by his lifelong friend, Bartolommeo Spina, a noted philosopher and fellow Dominican from Pisa, who was appointed Master of the Sacred Palace by Pope Paul III in 1542. After Tolosani dedicated his latest work, De purissima veritate divinae Scripturae adversus errores humanos ['On the Very Pure Truth of Divine Scripture, against Human Errors] (divided into 250 inquisitions and completed in 1544) to Pope Paul, Spina received a papal direction to assess it and he praised it fulsomely in a letter of 1546, shortly before his own death. In the fourth appendix to his Divine Scripture text, Tolosani claimed that Copernicus did not believe his own writings. Using his own logic and referencing previous and contemporary scientific opinion, he described Copernicus as an expert in mathematics and astronomy but "very deficient in physics and dialectics. Moreover he seems to be unfamiliar with Holy Scripture since he contradicts some of its principles, not without the risk to himself and to the readers of his book [De revolutionibus orbium coelestium] of straying from the faith." He went on to describe Copernicus as being ignorant of Aristotle and too supportive of Pythagorean ideas. He died in Siena, in the convent of Santo Spirito, on January 22, 1549. == Works == Compendio di sphera et machina del mondo (in Italian). Firenze: Bernardo Zucchetta. 1514. De emendationibus temporum (in Latin). Venezia: Lucantonio Giunta (1.). 1537. == References == == Bibliography == Lavenia, Vincenzo (2019). "TOLOSANI, Giovanni Maria". Dizionario Biografico degli Italiani, Volume 96: Toja–Trivelli (in Italian). Rome: Istituto dell'Enciclopedia Italiana. ISBN 978-8-81200032-6.
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Wikipedia:Giovanni Pagnini#0
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Giovanni Pagnini was an 18th-century Maltese mathematician and hydrographer. He was a member of the Knights Hospitaller. Born in Lucca, Italy, Pagnini came to Malta known as the "Hydrographer of the Order" and the knowledge he brought was instrumental in the foundation of Malta's School of Navigation. Pagnini mainly expanded on theoretical mathematics, but his text Compasso di Proporzioni is focused on detailing new compass-like tools (ex: "La Linea Aritmetica", "La Linea dei Piani[o] Linea Geometrica", "La Linea Cubica [o] Linea de Solidi") for construction and navigation purposes. == Works == Pagnini, Giovanni (1724). Compendio di Trigonometria Piana (in Italian). Pagnini, Giovanni (1728). Trionfo in Mare ossia Scuola di CommandareArmate e Vascelli da Guerra, cosi per la manovra come per Caccia e Combattere il Nemico (in Italian). Pagnini, Giovanni. Dell'Estrazione delle radici quadrati e cube (in Italian). Pagnini, Giovanni (1750). Trattato della sfera, ed introduzione alla navigazione per uso de' piloti (in Italian). Venice: Giovanni Battista Recurti. Pagnini, Giovanni (1753). Costruzione ed uso del compasso di proporzione (in Italian). Naples: Ignazio Russo. == References == == See also == Sector (instrument)
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Wikipedia:Giovanni Paradisi#0
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Giovanni Paradisi (1760, Reggio nell'Emilia – 26 August 1826) was an Italian mathematician, politician and poet. His father was the poet and economist Count Agostino Paradisi. == Biography == He was born in Reggio Emilia, to a poet father, more famous than wealthy. He studied at the University of Modena, and in 1783, he was named professor of geometry at the school. Giovanni and his father's selected works were published in Milan in 1828. With the creation of the Cisalpine Republic under Napoleonic occupation, Giovanni was named one of the directors of the new government. But a dispute with general Brune led to his dismissal. With the return of Hapsburg government, he was imprisoned at the Fort of Cattaro. After the Battle of Marengo, and the defeat of the Austrians, he was recalled to government. He held a variety of positions, including president of the Senate in the years 1809, 1811, and 1812. He supported Napoleon's goals to claim the title of king of Italy, and as a reward was granted a title of count, and awards of the Iron Crown and Legion of Honor. He lobbied unsuccessfully to have Eugène de Beauharnais named as king of the North Italian republic. With Napoleon's fall, his main income was teaching mathematics. == Works == Proposizioni fisico-matematiche, Modena, 1779, in 4; Discorso recitato nella prima adunanza dell instituto italiano, in-4; Ricerche sulla vibrazione delle lamine elastiche, Bologne, 1806; Il vitalizio, commedia, Milano, 1822; == References == == Bibliography == Carlo Capra, ‘La generosa nave’: appunti per una biografia di Giovanni Paradisi (la formazione e l’esordio politico), in Ricerche di storia in onore di F. Della Peruta, Milano, F. Angeli, 1996, vol. I, pp. 65–89 Claudio Chiancone, Il circolo Paradisi e il “Poligrafo”, in Istituzioni e cultura in età napoleonica, a c. di E. Brambilla, C. Capra e A. Scotti, Milano, FrancoAngeli, 2008, pp. 232–250
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Wikipedia:Giovanni Pighizzini#0
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Giovanni Pighizzini is an Italian theoretical computer scientist known for his work in formal language theory and particularly in state complexity of two-way finite automata. He earned his PhD in 1993 from the University of Milan, where he is a full professor since 2001. Pighizzini serves as the Steering Committee Chair of the annual Descriptional Complexity of Formal Systems academic conference since 2006. == Research contributions == Pighizzini obtained optimal state complexity tradeoffs between different types of finite automata over a one-letter alphabet, In particular, in his joint paper with Geffert and Mereghetti he presented the first simulation of two-way nondeterministic finite automata by two-way deterministic finite automata using Savitch's theorem, contributing to the 2DFA vs. 2NFA open question. Jointly with Jirásková, he determined state complexity of self-verifying finite automata. He also contributed to the computational complexity theory by results on sublogarithmic space complexity classes and on the complexity of searching for a lexicographically maximal string. == References == == External links == Official website Giovanni Pighizzini at DBLP Bibliography Server
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Wikipedia:Gisèle Mophou#0
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Gisèle Massengo Mophou (also published as Gisèle Adélie Mophou Loudjom) is a Cameroonian applied mathematician and numerical analyst whose research involves control theory and fractional differential equations. She is a professor at the University of the French Antilles in Pointe-à-Pitre, Guadeloupe, where she directs the Laboratoire de Mathématiques Informatique et Applications (LAMIA). == Education and career == Mophou is originally from Cameroon. She earned a bachelor's degree in 1987, and a diplôme d'études approfondies in numerical analysis in 1997 at Pierre and Marie Curie University in France. She completed her PhD in numerical analysis in 2000, at the University of the French Antilles. She obtained a habilitation in 2010, and became a full professor at the University of the French Antilles in 2013. From 2017 to 2019 she returned to Cameroon, with the support of the Alexander von Humboldt Foundation, as German Research Chair at the African Institute for Mathematical Sciences Cameroon. == Recognition == Mophou was elected to the African Academy of Sciences in 2013. == References == == External links == Laboratoire de Mathématiques Informatique et Applications
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Wikipedia:Giulia Di Nunno#0
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Giulia Di Nunno (born 1973) is an Italian mathematician specializing in stochastic analysis and financial mathematics who works as a professor of mathematics at the University of Oslo, with an adjunct appointment at the Norwegian School of Economics. As well as for her research, Di Nunno is known for promoting mathematics in Africa. == Education and career == Di Nunno earned a degree in mathematics from the University of Milan in 1998, including research on stochastic functions with Yurii Rozanov. She moved to the University of Pavia for doctoral studies, continuing with Rozanov as an informal mentor but under the official supervision of Eugenio Regazzini. She completed her Ph.D. in 2003; her dissertation was On stochastic differentiation with applications to minimal variance hedging. She joined the University of Oslo in 2003, and added her affiliation with the Norwegian School of Economics in 2009. == Activism == Di Nunno is the chair of the European Mathematical Society's Committee for Developing Countries, and has worked to promote young researchers to visit Africa and to establish "Emerging Regional Centres of Excellence" there. The International Council for Industrial and Applied Mathematics gave her their 2019 Su Buchin Prize for this work, citing her "long-lasting record actively and efficiently encouraging top-level mathematical research and education in developing African countries". == Books == With Bernt Karsten Øksendal and Frank Proske, Di Nunno is a co-author of the book Malliavin calculus for Lévy processes with applications to finance (Springer, 2009). She also co-edited Advanced Mathematical Methods for Finance (Springer, 1011) with Øksendal. == References == == External links == Home page Giulia Di Nunno publications indexed by Google Scholar
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Wikipedia:Giulio Ascoli#0
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Giulio Ascoli (20 January 1843, Trieste, Austrian Empire – 12 July 1896, Milan) was a Jewish-Italian mathematician. He was a student of the Scuola Normale di Pisa, where he graduated in 1868. In 1872 he became Professor of Algebra and Calculus of the Politecnico di Milano University. From 1879 he was professor of mathematics at the Reale Istituto Tecnico Superiore, where, in 1901, was affixed a plaque that remembers him. He was also a corresponding member of Istituto Lombardo. He made contributions to the theory of functions of a real variable and to Fourier series. For example, Ascoli introduced equicontinuity in 1884, a topic regarded as one of the fundamental concepts in the theory of real functions. In 1889, Italian mathematician Cesare Arzelà generalized Ascoli's Theorem into the Arzelà–Ascoli theorem, a practical sequential compactness criterion of functions. == See also == Measure (mathematics) Oscillation (mathematics) Riemann Integral == Notes == == Biographical references == Guerraggio, Angelo; Nastasi, Pietro (2005), Italian mathematics between the two world wars, Science Networks. Historical Studies, vol. 29, Basel: Birkhäuser Verlag, pp. x+299, doi:10.1007/3-7643-7512-4, ISBN 3-7643-6555-2, MR 2188015, Zbl 1084.01010. Tricomi, G. F. (1962). "Giulio Ascoli". Matematici italiani del primo secolo dello stato unitario (Italian mathematicians of the first century of the unitary state). Memorie dell'Accademia delle Scienze di Torino. Classe di Scienze fisiche matematiche e naturali, series IV. Vol. I. p. 120. Zbl 0132.24405. Archived from the original on 28 July 2012. Retrieved 31 July 2010.{{cite book}}: CS1 maint: bot: original URL status unknown (link)(in Italian). Available from the website of the. == References == Dshalalow, Jewgeni H. (2001), Real analysis: an introduction to the theory of real functions and integration, Studies in Advanced Mathematics, Boca Raton, Florida: CRC Press, pp. xiv+567, ISBN 1-58488-073-2, MR 1788725, Zbl 0978.28001. Letta, Giorgio (1994) [112°], "Le condizioni di Riemann per l'integrabilità e il loro influsso sulla nascita del concetto di misura" (PDF), Rendiconti della Accademia Nazionale delle Scienze Detta dei XL, Memorie di Matematica e Applicazioni (in Italian), XVIII (1): 143–169, MR 1327463, Zbl 0852.28001, archived from the original (PDF) on 28 February 2014. "Riemann's conditions for integrability and their influence on the birth of the concept of measure" (English translation of title) is an article on the history of measure theory, analyzing deeply and comprehensively every early contribution to the field, starting from Riemann's work and going to the works of Hermann Hankel, Gaston Darboux, Giulio Ascoli, Henry John Stephen Smith, Ulisse Dini, Vito Volterra, Paul David Gustav du Bois-Reymond and Carl Gustav Axel Harnack. == External links == Biography in Italian. Ascoli, Julio in the Jewish Encyclopedia. By Their Fruits Ye Shall Know Them: Some Remarks on the Interaction of General Topology with Other Areas of Mathematics by T. Koetsier, J. Van Mill, an article containing a history of Ascoli's work on the Arzelà-Ascoli theorem.
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Wikipedia:Giulio Vivanti#0
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Giulio Benedetto Isacco Vivanti (24 May 1859 – 19 November 1949) was an Italian mathematician. He was a mentor of Bruno de Finetti and he spent most of his academic career at the University of Pavia and University of Milan. == See also == Vivanti–Pringsheim theorem == References == Janovitz, Alessandro; Mercanti, Fabio. "Sull'apporto Evolutivo Dei Matematici Ebrei Mantovani Nella Nascente Nazione Italiana" (PDF) (in Italian). Archived from the original (PDF) on 2013-06-27. Retrieved 2012-01-09.
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Wikipedia:Giuseppe Ceredi#0
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Giuseppe Ceredi was a 16th‑century Italian hydraulic engineer, physician, pharmacist, and mathematician whose work played a significant role in the evolution of hydraulic science during the Renaissance. Best known for his 1567 treatise, Tre discorsi sopra il modo d'alzar acque da' luoghi bassi, Ceredi advanced the design and application of water‑lifting machines—most notably the Archimedes screw (also known as the cochlea)—and proposed systematic irrigation schemes that integrated theoretical, practical, and economic considerations. == Biography == Very little was known about Ceredi’s life until recent scholarly investigations shed light on his multidisciplinary background. According to a brief entry in the Dizionario biografico piacentino (1899), Ceredi was a skilled pharmacist, physician, and mathematician, who perfected Vitruvius’s cochlea, reflecting his expertise in refining ancient hydraulic technology. He was a student and close friend of Benedetto Labadini, a prominent professor at the time, and is known to have visited Giorgio Valla’s private library, where he encountered rare writings by Hero, Pappus, and Dionysodorus that influenced his understanding of hydraulic science. Ceredi received a patent from Ottavio Farnese for the development of his machines in 1566. == Major work == In 1567, Ceredi published his seminal work, Tre discorsi sopra il modo d'alzar acque da' luoghi bassi. The treatise is richly illustrated with woodcut images—including a title vignette, internal illustrations, and folding plates—which serve not only depict the machinery but also serve as visual proofs of his mechanical innovations. This work represents one of the few detailed Renaissance documents to explain the inner workings of hydraulic machines at a time when most technical knowledge was transmitted only through drawings or fragmentary descriptions. == Contributions to hydraulic engineering == === Improvements to the Archimedes screw === Ceredi is best known for his enhancements to the Archimedes screw—a device originally invented in the 3rd century BC. At a time when detailed construction guidelines for the device were lost, he provided a systematic approach to its design and operation. By refining construction details such as the geometry of the helical surface and the configuration of the drive lever (including experimental evaluations of both straight and curved lever designs), he improved its efficiency and reliability. His work represents one of the earliest documented proposals for using a standardized mechanical device for irrigation and water management. === Development of systematic irrigation schemes === Ceredi’s treatise went beyond mere mechanical design and addressed broader issues of hydraulic science. He discussed the application of hydraulic machines for tasks such as irrigating agricultural land, removing excess spring and rainwater from fields lacking natural drainage, supplying drinking water to urban centers, and for military applications, such as obtaining water for armies. His analysis included economic evaluations, maintenance considerations, and even cost–benefit calculations—underscoring Ceredi's grasp of the practical importance of hydraulic engineering. His experimental approach, which involved repeated testing and empirical verification of the machines’ performance, highlighted his commitment to ensuring that theoretical models were effectively translated into practical applications. === Philosophical and epistemological context === Influenced by the Philosophia Naturalis model and Thales of Miletus, Ceredi viewed hydraulic machines as the tangible manifestation of mathematical and physical laws. He argued that the “beautiful mathematical arguments” underlying natural phenomena must be matched by precise “hands-on” engineering practices. This epistemological stance positioned him as a pioneer among 16th‑century inventors, bridging the gap between ancient technical traditions and emerging scientific thought during the early phases of the Scientific Revolution. == Legacy == Once an obscure figure, recent scholarship has reappraised Ceredi as a key contributor to hydraulic engineering in Renaissance Italy. His integration of multidisciplinary knowledge—from medicine and mathematics to practical engineering—helped lay the groundwork for later developments in water management. After Ceredi developed and patented his improved Archimedean screw design, these pumps spread throughout Southern Europe for drainage and irrigation. Galileo later refined the technology further, securing a patent for a horse-powered water-raising mechanism. Ceredi’s work not only influenced subsequent debates on hydraulic efficiency but also provided a model for how theoretical insights could be applied to solve practical, everyday problems, thereby improving public health, agriculture, and military logistics. Despite this, Ceredi would fall into obscurity over the following centuries, leading Giovanni Aldini to lament how the innovative mechanisms of the 16th century, including those proposed by Ceredi, had largely fallen into disuse. == See also == Benedetto Castelli == References ==
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Wikipedia:Giuseppe Da Prato#0
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Giuseppe Da Prato (23 July 1936 – 6 October 2023) was an Italian academic and mathematician. He taught at the elite Scuola Normale Superiore di Pisa. He mainly researched stochastic calculus, partial differential equations, and control theory. == Biography == Born in La Spezia on 23 July 1936, Da Prato earned his doctorate at the Sapienza University of Rome in 1960 under the direction of Marcello Cini. In 1963, he became an assistant professor at the University of Pisa. He became a professor at the Sapienza in 1968 and won the Bartolozzi Prize the following year. He remained at the Sapienza until 1977, during which time he was also a guest professor at the Université Nice-Sophia-Antipolis in France. He worked at the University of Trento from 1977 to 1979, and subsequently joined the Scuola Normale Superiore, where he stayed for nearly 30 years. He was also a guest professor at the University of Maryland, College Park in the United States from 1981 to 1982. Da Prato served on the editorial board of numerous mathematical journals. He also co-founded the journal Nonlinear Differential Equations and Applications. Giuseppe Da Prato died on 6 October 2023, at the age of 87. == References ==
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Wikipedia:Giuseppe Suzzi#0
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Giuseppe Suzzi (1701 - 5 January 1764) was an Italian mathematician and abbot. == Life == Born in Ragogna, province of Udine, he studied in Udine and later in Murano, Venice, in the seminary of Somaschi Fathers, where he was a relevant student of Giovanni Francesco Crivelli. In 1722 he moved to the University of Padua, when he also took lessons from Jacopo Riccati privately with Lodovico da Riva. These lessons were published in 1761 in a posthumous book by Riccati, with the solutions of many problems by Suzzi. Suzzi, an abbey, published many books on mathematical analysis and in 1744 he became professor of natural history in Padua, where he taught mainly within the tradition of aristotelianism, but also giving space to newer philosophical ideas. In 1750 he became a member of the Accademia dei Ricovrati in Padua (later Accademia galileiana di scienze, lettere ed arti). He retired from University in 1762 and died in Venice 5 January 1764. == Works == Suzzi, Giuseppe (1725). Disquisitiones mathematicae (in Latin). Venetiis: Domenico Lovisa. Suzzi, Giuseppe (1747). Solutio generalis aequationum tertii gradus (in Latin). Patauii: Tipografia del Seminario Padova. == See also == Cubic function == References ==
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Wikipedia:Gjon Gazuli#0
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Gjon Gazuli, OP (Latin: Johannes Gasulus, Croatian: Ivan Gazulić)[a][b] was a Dominican friar, humanist scholar, astronomer and diplomat from the Republic of Ragusa of Albanian origin. == Name == He is known in Albanian as Gjon Gjin Gazuli, or even Gjin Gazulli. In Croatian he is known as Ivan Gazulić, Ivan Gazul and Ivan Gazoli Not to be confused with Dom Gjon Gazulli, an Albanian Catholic Cleric, executed by Ahmet Zogu's regime in 1927. == Life == Gazuli attended schools in Shkodër and Dubrovnik, and in 1430 he graduated from the University of Padua. He lived and worked in Dominican Friary in Dubrovnik. In 1432 he traveled to the Hungarian royal court where he attempted to persuade Sigismund I to support Albanian resistance against the Ottoman Empire. He broke his mission off in 1433, when he was appealed to be a professor of mathematics and astronomy at the University of Padua. Years later he was acting as a diplomat with the Italian principalities' courts, representing the interests of Skanderbeg and of the League of Lezhë. His lasting mathematical and astronomical works were written in Latin. He had a reputation for considerable knowledge in Italy and in Hungary, as well. His brother, Pal Gazuli (1405–1470), was the diplomat of Skanderbeg and of the League of Lezhë in Ragusa. He had a second brother named Andrea Gazuli, also mentioned as prominent. == References == == Sources == Frashëri, Kristo (2002), Gjergj Kastrioti Skënderbeu: jeta dhe vepra, 1405–1468 (in Albanian), Botimet Toena, ISBN 99927-1-627-4 Schmitt, Oliver Jens (2001), Das venezianische Albanien (1392-1479), Oldenbourg Wissenschaftsverlag, ISBN 978-3-486-56569-0 == External links == "Gjin Gazulli" Technical High School in Prishtina Core biographical data (in Albanian) Hrvatska enciklopedija LZMK Gazulli, Gjin (Gazulić Ivan, Joannes) (in Croatian)
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Wikipedia:Glasgow Mathematical Journal#0
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The Glasgow Mathematical Journal is a mathematics journal that publishes original research papers in any branch of pure and applied mathematics. It covers a wide variety of research areas, which in recent issues have included ring theory, group theory, functional analysis, combinatorics, differential equations, differential geometry, number theory, algebraic topology, and the application of such methods in applied mathematics. The editor-in-chief is currently A. Bartel (University of Glasgow). == References ==
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Wikipedia:Glen Van Brummelen#0
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Glen Robert Van Brummelen (born May 20, 1965) is a Canadian historian of mathematics specializing in the history of trigonometry and historical applications of mathematics to astronomy. He is president of the Canadian Society for History and Philosophy of Mathematics, and was a co-editor of Mathematics and the Historian's Craft: The Kenneth O. May Lectures (Springer, 2005). == Life == Van Brummelen earned his PhD degree from Simon Fraser University in 1993, and served as a professor of mathematics at Bennington College from 1999 to 2006. He then transferred to Quest University Canada as a founding faculty member. In 2020, he became the dean of the Faculty of Natural and Applied Sciences at Trinity Western University in Langley, BC. Glen Van Brummelen has published the first major history in English of the origins and early development of trigonometry, The Mathematics of the Heavens and the Earth: The Early History of Trigonometry. His second book, Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry, concerns spherical trigonometry. In 2016 he received a Deborah and Franklin Haimo Award for Distinguished College or University Teaching of Mathematics. == Works == The Mathematics of the Heavens and the Earth: The Early History of Trigonometry Princeton; Oxford: Princeton University Press, 2009. ISBN 9780691129730, OCLC 750691811 Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry Princeton; Oxford: Princeton University Press, 2013. ISBN 9780691175997, OCLC 988234342 Trigonometry: A Very Short Introduction; Oxford: Princeton University Press, 2020 ISBN 9780198814313, OCLC 1101269106 The Doctrine of Triangles: The History of Modern Trigonometry Princeton; Oxford: Princeton University Press, 2021 ISBN 978-0691179414, OCLC 1201300540 == References == == External links == Homepage at Bennington College Glen R. Van Brummelen, Ph.D. homepage at Trinity Western University Publication list Trigonometry Book page
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Wikipedia:Global analysis#0
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In mathematics, global analysis, also called analysis on manifolds, is the study of the global and topological properties of differential equations on manifolds and vector bundles. Global analysis uses techniques in infinite-dimensional manifold theory and topological spaces of mappings to classify behaviors of differential equations, particularly nonlinear differential equations. These spaces can include singularities and hence catastrophe theory is a part of global analysis. Optimization problems, such as finding geodesics on Riemannian manifolds, can be solved using differential equations, so that the calculus of variations overlaps with global analysis. Global analysis finds application in physics in the study of dynamical systems and topological quantum field theory. == Journals == Annals of Global Analysis and Geometry The Journal of Geometric Analysis == See also == Atiyah–Singer index theorem Geometric analysis Lie groupoid Pseudogroup Morse theory Structural stability Harmonic map == References == == Further reading == Mathematics 241A: Introduction to Global Analysis
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Wikipedia:Glossary of areas of mathematics#0
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Mathematics is a broad subject that is commonly divided in many areas or branches that may be defined by their objects of study, by the used methods, or by both. For example, analytic number theory is a subarea of number theory devoted to the use of methods of analysis for the study of natural numbers. This glossary is alphabetically sorted. This hides a large part of the relationships between areas. For the broadest areas of mathematics, see Mathematics § Areas of mathematics. The Mathematics Subject Classification is a hierarchical list of areas and subjects of study that has been elaborated by the community of mathematicians. It is used by most publishers for classifying mathematical articles and books. == A == Absolute differential calculus An older name of Ricci calculus Absolute geometry Also called neutral geometry, a synthetic geometry similar to Euclidean geometry but without the parallel postulate. Abstract algebra The part of algebra devoted to the study of algebraic structures in themselves. Occasionally named modern algebra in course titles. Abstract analytic number theory The study of arithmetic semigroups as a means to extend notions from classical analytic number theory. Abstract differential geometry A form of differential geometry without the notion of smoothness from calculus. Instead it is built using sheaf theory and sheaf cohomology. Abstract harmonic analysis A modern branch of harmonic analysis that extends upon the generalized Fourier transforms that can be defined on locally compact groups. Abstract homotopy theory A part of topology that deals with homotopic functions, i.e. functions from one topological space to another which are homotopic (the functions can be deformed into one another). Actuarial science The discipline that applies mathematical and statistical methods to assess risk in insurance, finance and other industries and professions. More generally, actuaries apply rigorous mathematics to model matters of uncertainty. Additive combinatorics The part of arithmetic combinatorics devoted to the operations of addition and subtraction. Additive number theory A part of number theory that studies subsets of integers and their behaviour under addition. Affine geometry A branch of geometry that deals with properties that are independent from distances and angles, such as alignment and parallelism. Affine geometry of curves The study of curve properties that are invariant under affine transformations. Affine differential geometry A type of differential geometry dedicated to differential invariants under volume-preserving affine transformations. Ahlfors theory A part of complex analysis being the geometric counterpart of Nevanlinna theory. It was invented by Lars Ahlfors. Algebra One of the major areas of mathematics. Roughly speaking, it is the art of manipulating and computing with operations acting on symbols called variables that represent indeterminate numbers or other mathematical objects, such as vectors, matrices, or elements of algebraic structures. Algebraic analysis motivated by systems of linear partial differential equations, it is a branch of algebraic geometry and algebraic topology that uses methods from sheaf theory and complex analysis, to study the properties and generalizations of functions. It was started by Mikio Sato. Algebraic combinatorics an area that employs methods of abstract algebra to problems of combinatorics. It also refers to the application of methods from combinatorics to problems in abstract algebra. Algebraic computation An older name of computer algebra. Algebraic geometry a branch that combines techniques from abstract algebra with the language and problems of geometry. Fundamentally, it studies algebraic varieties. Algebraic graph theory a branch of graph theory in which methods are taken from algebra and employed to problems about graphs. The methods are commonly taken from group theory and linear algebra. Algebraic K-theory an important part of homological algebra concerned with defining and applying a certain sequence of functors from rings to abelian groups. Algebraic number theory The part of number theory devoted to the use of algebraic methods, mainly those of commutative algebra, for the study of number fields and their rings of integers. Algebraic statistics the use of algebra to advance statistics, although the term is sometimes restricted to label the use of algebraic geometry and commutative algebra in statistics. Algebraic topology a branch that uses tools from abstract algebra for topology to study topological spaces. Algorithmic number theory also known as computational number theory, it is the study of algorithms for performing number theoretic computations. Anabelian geometry an area of study based on the theory proposed by Alexander Grothendieck in the 1980s that describes the way a geometric object of an algebraic variety (such as an algebraic fundamental group) can be mapped into another object, without it being an abelian group. Analysis A wide area of mathematics centered on the study of continuous functions and including such topics as differentiation, integration, limits, and series. Analytic combinatorics part of enumerative combinatorics where methods of complex analysis are applied to generating functions. Analytic geometry 1. Also known as Cartesian geometry, the study of Euclidean geometry using Cartesian coordinates. 2. Analogue to differential geometry, where differentiable functions are replaced with analytic functions. It is a subarea of both complex analysis and algebraic geometry. Analytic number theory An area of number theory that applies methods from mathematical analysis to solve problems about integers. Analytic theory of L-functions Applied mathematics a combination of various parts of mathematics that concern a variety of mathematical methods that can be applied to practical and theoretical problems. Typically the methods used are for science, engineering, finance, economics and logistics. Approximation theory part of analysis that studies how well functions can be approximated by simpler ones (such as polynomials or trigonometric polynomials) Arakelov geometry also known as Arakelov theory Arakelov theory an approach to Diophantine geometry used to study Diophantine equations in higher dimensions (using techniques from algebraic geometry). It is named after Suren Arakelov. Arithmetic 1. Also known as elementary arithmetic, the methods and rules for computing with addition, subtraction, multiplication and division of numbers. 2. Also known as higher arithmetic, another name for number theory. Arithmetic algebraic geometry See arithmetic geometry. Arithmetic combinatorics the study of the estimates from combinatorics that are associated with arithmetic operations such as addition, subtraction, multiplication and division. Arithmetic dynamics Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, p-adic, and/or algebraic points under repeated application of a polynomial or rational function. A fundamental goal is to describe arithmetic properties in terms of underlying geometric structures. Arithmetic geometry The use of algebraic geometry and more specially scheme theory for solving problems of number theory. Arithmetic topology a combination of algebraic number theory and topology studying analogies between prime ideals and knots Arithmetical algebraic geometry Another name for arithmetic algebraic geometry Asymptotic combinatorics It uses the internal structure of the objects to derive formulas for their generating functions and then complex analysis techniques to get asymptotics. Asymptotic theory the study of asymptotic expansions Auslander–Reiten theory the study of the representation theory of Artinian rings Axiomatic geometry also known as synthetic geometry: it is a branch of geometry that uses axioms and logical arguments to draw conclusions as opposed to analytic and algebraic methods. Axiomatic set theory the study of systems of axioms in a context relevant to set theory and mathematical logic. == B == Bifurcation theory the study of changes in the qualitative or topological structure of a given family. It is a part of dynamical systems theory Biostatistics the development and application of statistical methods to a wide range of topics in biology. Birational geometry a part of algebraic geometry that deals with the geometry (of an algebraic variety) that is dependent only on its function field. Bolyai–Lobachevskian geometry see hyperbolic geometry == C == C*-algebra theory a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties-(i) A is a topologically closed set in the norm topology of operators.(ii)A is closed under the operation of taking adjoints of operators. Cartesian geometry see analytic geometry Calculus An area of mathematics connected by the fundamental theorem of calculus. Calculus of infinitesimals Also called infinitesimal calculus A foundation of calculus, first developed in the 17th century, that makes use of infinitesimal numbers. Calculus of moving surfaces an extension of the theory of tensor calculus to include deforming manifolds. Calculus of variations the field dedicated to maximizing or minimizing functionals. It used to be called functional calculus. Catastrophe theory a branch of bifurcation theory from dynamical systems theory, and also a special case of the more general singularity theory from geometry. It analyses the germs of the catastrophe geometries. Categorical logic a branch of category theory adjacent to the mathematical logic. It is based on type theory for intuitionistic logics. Category theory the study of the properties of particular mathematical concepts by formalising them as collections of objects and arrows. Chaos theory the study of the behaviour of dynamical systems that are highly sensitive to their initial conditions. Character theory a branch of group theory that studies the characters of group representations or modular representations. Class field theory a branch of algebraic number theory that studies abelian extensions of number fields. Classical differential geometry also known as Euclidean differential geometry. see Euclidean differential geometry. Classical algebraic topology see algebraic topology Classical analysis usually refers to the more traditional topics of analysis such as real analysis and complex analysis. It includes any work that does not use techniques from functional analysis and is sometimes called hard analysis. However it may also refer to mathematical analysis done according to the principles of classical mathematics. Classical analytic number theory Classical differential calculus Classical Diophantine geometry Classical Euclidean geometry see Euclidean geometry Classical geometry may refer to solid geometry or classical Euclidean geometry. See geometry Classical invariant theory the form of invariant theory that deals with describing polynomial functions that are invariant under transformations from a given linear group. Classical mathematics the standard approach to mathematics based on classical logic and ZFC set theory. Classical projective geometry Classical tensor calculus Clifford algebra Clifford analysis the study of Dirac operators and Dirac type operators from geometry and analysis using clifford algebras. Clifford theory is a branch of representation theory spawned from Cliffords theorem. Cobordism theory Coding theory the study of the properties of codes and their respective fitness for specific applications. Cohomology theory Combinatorial analysis Combinatorial commutative algebra a discipline viewed as the intersection between commutative algebra and combinatorics. It frequently employs methods from one to address problems arising in the other. Polyhedral geometry also plays a significant role. Combinatorial design theory a part of combinatorial mathematics that deals with the existence and construction of systems of finite sets whose intersections have certain properties. Combinatorial game theory Combinatorial geometry see discrete geometry Combinatorial group theory the theory of free groups and the presentation of a group. It is closely related to geometric group theory and is applied in geometric topology. Combinatorial mathematics an area primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. Combinatorial number theory Combinatorial optimization Combinatorial set theory also known as Infinitary combinatorics. see infinitary combinatorics Combinatorial theory Combinatorial topology an old name for algebraic topology, when topological invariants of spaces were regarded as derived from combinatorial decompositions. Combinatorics a branch of discrete mathematics concerned with countable structures. Branches of it include enumerative combinatorics, combinatorial design theory, matroid theory, extremal combinatorics and algebraic combinatorics, as well as many more. Commutative algebra a branch of abstract algebra studying commutative rings. Complex algebraic geometry the mainstream of algebraic geometry devoted to the study of the complex points of algebraic varieties. Complex analysis a part of analysis that deals with functions of a complex variable. Complex analytic dynamics a subdivision of complex dynamics being the study of the dynamic systems defined by analytic functions. Complex analytic geometry the application of complex numbers to plane geometry. Complex differential geometry a branch of differential geometry that studies complex manifolds. Complex dynamics the study of dynamical systems defined by iterated functions on complex number spaces. Complex geometry the study of complex manifolds and functions of complex variables. It includes complex algebraic geometry and complex analytic geometry. Complexity theory the study of complex systems with the inclusion of the theory of complex systems. Computable analysis the study of which parts of real analysis and functional analysis can be carried out in a computable manner. It is closely related to constructive analysis. Computable model theory a branch of model theory dealing with the relevant questions computability. Computability theory a branch of mathematical logic originating in the 1930s with the study of computable functions and Turing degrees, but now includes the study of generalized computability and definability. It overlaps with proof theory and effective descriptive set theory. Computational algebraic geometry Computational complexity theory a branch of mathematics and theoretical computer science that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other. Computational geometry a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Computational group theory the study of groups by means of computers. Computational mathematics the mathematical research in areas of science where computing plays an essential role. Computational number theory also known as algorithmic number theory, it is the study of algorithms for performing number theoretic computations. Computational statistics Computational synthetic geometry Computational topology Computer algebra see symbolic computation Conformal geometry the study of conformal transformations on a space. Constructive analysis mathematical analysis done according to the principles of constructive mathematics. This differs from classical analysis. Constructive function theory a branch of analysis that is closely related to approximation theory, studying the connection between the smoothness of a function and its degree of approximation Constructive mathematics mathematics which tends to use intuitionistic logic. Essentially that is classical logic but without the assumption that the law of the excluded middle is an axiom. Constructive quantum field theory a branch of mathematical physics that is devoted to showing that quantum theory is mathematically compatible with special relativity. Constructive set theory an approach to mathematical constructivism following the program of axiomatic set theory, using the usual first-order language of classical set theory. Contact geometry a branch of differential geometry and topology, closely related to and considered the odd-dimensional counterpart of symplectic geometry. It is the study of a geometric structure called a contact structure on a differentiable manifold. Convex analysis the study of properties of convex functions and convex sets. Convex geometry part of geometry devoted to the study of convex sets. Coordinate geometry see analytic geometry CR geometry a branch of differential geometry, being the study of CR manifolds. Cryptography == D == Decision analysis Decision theory Derived noncommutative algebraic geometry Descriptive set theory a part of mathematical logic, more specifically a part of set theory dedicated to the study of Polish spaces. Differential algebraic geometry the adaption of methods and concepts from algebraic geometry to systems of algebraic differential equations. Differential calculus A branch of calculus that's contrasted to integral calculus, and concerned with derivatives. Differential Galois theory the study of the Galois groups of differential fields. Differential geometry a form of geometry that uses techniques from integral and differential calculus as well as linear and multilinear algebra to study problems in geometry. Classically, these were problems of Euclidean geometry, although now it has been expanded. It is generally concerned with geometric structures on differentiable manifolds. It is closely related to differential topology. Differential geometry of curves the study of smooth curves in Euclidean space by using techniques from differential geometry Differential geometry of surfaces the study of smooth surfaces with various additional structures using the techniques of differential geometry. Differential topology a branch of topology that deals with differentiable functions on differentiable manifolds. Diffiety theory Diophantine geometry in general the study of algebraic varieties over fields that are finitely generated over their prime fields. Discrepancy theory Discrete differential geometry Discrete exterior calculus Discrete geometry a branch of geometry that studies combinatorial properties and constructive methods of discrete geometric objects. Discrete mathematics the study of mathematical structures that are fundamentally discrete rather than continuous. Discrete Morse theory a combinatorial adaption of Morse theory. Distance geometry Domain theory a branch that studies special kinds of partially ordered sets (posets) commonly called domains. Donaldson theory the study of smooth 4-manifolds using gauge theory. Dyadic algebra Dynamical systems theory an area used to describe the behavior of the complex dynamical systems, usually by employing differential equations or difference equations. == E == Econometrics the application of mathematical and statistical methods to economic data. Effective descriptive set theory a branch of descriptive set theory dealing with set of real numbers that have lightface definitions. It uses aspects of computability theory. Elementary algebra a fundamental form of algebra extending on elementary arithmetic to include the concept of variables. Elementary arithmetic the simplified portion of arithmetic considered necessary for primary education. It includes the usage addition, subtraction, multiplication and division of the natural numbers. It also includes the concept of fractions and negative numbers. Elementary mathematics parts of mathematics frequently taught at the primary and secondary school levels. This includes elementary arithmetic, geometry, probability and statistics, elementary algebra and trigonometry. (calculus is not usually considered a part) Elementary group theory the study of the basics of group theory Elimination theory the classical name for algorithmic approaches to eliminating between polynomials of several variables. It is a part of commutative algebra and algebraic geometry. Elliptic geometry a type of non-Euclidean geometry (it violates Euclid's parallel postulate) and is based on spherical geometry. It is constructed in elliptic space. Enumerative combinatorics an area of combinatorics that deals with the number of ways that certain patterns can be formed. Enumerative geometry a branch of algebraic geometry concerned with counting the number of solutions to geometric questions. This is usually done by means of intersection theory. Epidemiology Equivariant noncommutative algebraic geometry Ergodic Ramsey theory a branch where problems are motivated by additive combinatorics and solved using ergodic theory. Ergodic theory the study of dynamical systems with an invariant measure, and related problems. Euclidean geometry An area of geometry based on the axiom system and synthetic methods of the ancient Greek mathematician Euclid. Euclidean differential geometry also known as classical differential geometry. See differential geometry. Euler calculus a methodology from applied algebraic topology and integral geometry that integrates constructible functions and more recently definable functions by integrating with respect to the Euler characteristic as a finitely-additive measure. Experimental mathematics an approach to mathematics in which computation is used to investigate mathematical objects and identify properties and patterns. Exterior algebra Exterior calculus Extraordinary cohomology theory Extremal combinatorics a branch of combinatorics, it is the study of the possible sizes of a collection of finite objects given certain restrictions. Extremal graph theory a branch of mathematics that studies how global properties of a graph influence local substructure. == F == Field theory The branch of algebra dedicated to fields, a type of algebraic structure. Finite geometry Finite model theory a restriction of model theory to interpretations on finite structures, which have a finite universe. Finsler geometry a branch of differential geometry whose main object of study is Finsler manifolds, a generalisation of a Riemannian manifolds. First order arithmetic Fourier analysis the study of the way general functions may be represented or approximated by sums of trigonometric functions. Fractal geometry Fractional calculus a branch of analysis that studies the possibility of taking real or complex powers of the differentiation operator. Fractional dynamics investigates the behaviour of objects and systems that are described by differentiation and integration of fractional orders using methods of fractional calculus. Fredholm theory part of spectral theory studying integral equations. Function theory an ambiguous term that generally refers to mathematical analysis. Functional analysis a branch of mathematical analysis, the core of which is formed by the study of function spaces, which are some sort of topological vector spaces. Functional calculus historically the term was used synonymously with calculus of variations, but now refers to a branch of functional analysis connected with spectral theory Fuzzy mathematics a branch of mathematics based on fuzzy set theory and fuzzy logic. Fuzzy measure theory Fuzzy set theory a form of set theory that studies fuzzy sets, that is sets that have degrees of membership. == G == Galois cohomology an application of homological algebra, it is the study of group cohomology of Galois modules. Galois theory named after Évariste Galois, it is a branch of abstract algebra providing a connection between field theory and group theory. Galois geometry a branch of finite geometry concerned with algebraic and analytic geometry over a Galois field. Game theory the study of mathematical models of strategic interaction among rational decision-makers. Gauge theory General topology also known as point-set topology, it is a branch of topology studying the properties of topological spaces and structures defined on them. It differs from other branches of topology as the topological spaces do not have to be similar to manifolds. Generalized trigonometry developments of trigonometric methods from the application to real numbers of Euclidean geometry to any geometry or space. This includes spherical trigonometry, hyperbolic trigonometry, gyrotrigonometry, and universal hyperbolic trigonometry. Geometric algebra an alternative approach to classical, computational and relativistic geometry. It shows a natural correspondence between geometric entities and elements of algebra. Geometric analysis a discipline that uses methods from differential geometry to study partial differential equations as well as the applications to geometry. Geometric calculus extends the geometric algebra to include differentiation and integration. Geometric combinatorics a branch of combinatorics. It includes a number of subareas such as polyhedral combinatorics (the study of faces of convex polyhedra), convex geometry (the study of convex sets, in particular combinatorics of their intersections), and discrete geometry, which in turn has many applications to computational geometry. Geometric function theory the study of geometric properties of analytic functions. Geometric invariant theory a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. Geometric graph theory a large and amorphous subfield of graph theory, concerned with graphs defined by geometric means. Geometric group theory the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces). Geometric measure theory the study of geometric properties of sets (typically in Euclidean space) through measure theory. Geometric number theory Geometric topology a branch of topology studying manifolds and mappings between them; in particular the embedding of one manifold into another. Geometry a branch of mathematics concerned with shape and the properties of space. Classically it arose as what is now known as solid geometry; this was concerning practical knowledge of length, area and volume. It was then put into an axiomatic form by Euclid, giving rise to what is now known as classical Euclidean geometry. The use of coordinates by René Descartes gave rise to Cartesian geometry enabling a more analytical approach to geometric entities. Since then many other branches have appeared including projective geometry, differential geometry, non-Euclidean geometry, Fractal geometry and algebraic geometry. Geometry also gave rise to the modern discipline of topology. Geometry of numbers initiated by Hermann Minkowski, it is a branch of number theory studying convex bodies and integer vectors. Global analysis the study of differential equations on manifolds and the relationship between differential equations and topology. Global arithmetic dynamics Graph theory a branch of discrete mathematics devoted to the study of graphs. It has many applications in physical, biological and social systems. Group-character theory the part of character theory dedicated to the study of characters of group representations. Group representation theory Group theory the study of algebraic structures known as groups. Gyrotrigonometry a form of trigonometry used in gyrovector space for hyperbolic geometry. (An analogy of the vector space in Euclidean geometry.) == H == Hard analysis see classical analysis Harmonic analysis part of analysis concerned with the representations of functions in terms of waves. It generalizes the notions of Fourier series and Fourier transforms from the Fourier analysis. Higher arithmetic Higher category theory the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. Higher-dimensional algebra the study of categorified structures. Hodge theory a method for studying the cohomology groups of a smooth manifold M using partial differential equations. Hodge–Arakelov theory Holomorphic functional calculus a branch of functional calculus starting with holomorphic functions. Homological algebra the study of homology in general algebraic settings. Homology theory Homotopy theory Hyperbolic geometry also known as Lobachevskian geometry or Bolyai-Lobachevskian geometry. It is a non-Euclidean geometry looking at hyperbolic space. hyperbolic trigonometry the study of hyperbolic triangles in hyperbolic geometry, or hyperbolic functions in Euclidean geometry. Other forms include gyrotrigonometry and universal hyperbolic trigonometry. Hypercomplex analysis the extension of real analysis and complex analysis to the study of functions where the argument is a hypercomplex number. Hyperfunction theory == I == Ideal theory once the precursor name for what is now known as commutative algebra; it is the theory of ideals in commutative rings. Idempotent analysis the study of idempotent semirings, such as the tropical semiring. Incidence geometry the study of relations of incidence between various geometric objects, like curves and lines. Inconsistent mathematics see paraconsistent mathematics. Infinitary combinatorics an expansion of ideas in combinatorics to account for infinite sets. Infinitesimal analysis once a synonym for infinitesimal calculus Infinitesimal calculus See calculus of infinitesimals Information geometry an interdisciplinary field that applies the techniques of differential geometry to study probability theory and statistics. It studies statistical manifolds, which are Riemannian manifolds whose points correspond to probability distributions. Integral calculus Integral geometry the theory of measures on a geometrical space invariant under the symmetry group of that space. Intersection theory a branch of algebraic geometry and algebraic topology Intuitionistic type theory a type theory and an alternative foundation of mathematics. Invariant theory studies how group actions on algebraic varieties affect functions. Inventory theory Inversive geometry the study of invariants preserved by a type of transformation known as inversion Inversive plane geometry inversive geometry that is limited to two dimensions Inversive ring geometry Itô calculus extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance and stochastic differential equations. Iwasawa theory the study of objects of arithmetic interest over infinite towers of number fields. Iwasawa-Tate theory == J == Job shop scheduling == K == K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry it is referred to as algebraic K-theory. In physics, K-theory has appeared in type II string theory. (In particular twisted K-theory.) K-homology a homology theory on the category of locally compact Hausdorff spaces. Kähler geometry a branch of differential geometry, more specifically a union of Riemannian geometry, complex differential geometry and symplectic geometry. It is the study of Kähler manifolds. (named after Erich Kähler) KK-theory a common generalization both of K-homology and K-theory as an additive bivariant functor on separable C*-algebras. Klein geometry More specifically, it is a homogeneous space X together with a transitive action on X by a Lie group G, which acts as the symmetry group of the geometry. Knot theory part of topology dealing with knots Kummer theory provides a description of certain types of field extensions involving the adjunction of nth roots of elements of the base field == L == L-theory the K-theory of quadratic forms. Large deviations theory part of probability theory studying events of small probability (tail events). Large sample theory also known as asymptotic theory Lattice theory the study of lattices, being important in order theory and universal algebra Lie algebra theory Lie group theory Lie sphere geometry geometrical theory of planar or spatial geometry in which the fundamental concept is the circle or sphere. Lie theory Line geometry Linear algebra a branch of algebra studying linear spaces and linear maps. It has applications in fields such as abstract algebra and functional analysis; it can be represented in analytic geometry and it is generalized in operator theory and in module theory. Sometimes matrix theory is considered a branch, although linear algebra is restricted to only finite dimensions. Extensions of the methods used belong to multilinear algebra. Linear functional analysis Linear programming a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. List of graphical methods Included are diagram techniques, chart techniques, plot techniques, and other forms of visualization. Local algebra a term sometimes applied to the theory of local rings. Local class field theory the study of abelian extensions of local fields. Low-dimensional topology the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. == M == Malliavin calculus a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to stochastic processes. Mathematical biology the mathematical modeling of biological phenomena. Mathematical chemistry the mathematical modeling of chemical phenomena. Mathematical economics the application of mathematical methods to represent theories and analyze problems in economics. Mathematical finance a field of applied mathematics, concerned with mathematical modeling of financial markets. Mathematical logic a subfield of mathematics exploring the applications of formal logic to mathematics. Mathematical optimization Mathematical physics The development of mathematical methods suitable for application to problems in physics. Mathematical psychology an approach to psychological research that is based on mathematical modeling of perceptual, thought, cognitive and motor processes, and on the establishment of law-like rules that relate quantifiable stimulus characteristics with quantifiable behavior. Mathematical sciences refers to academic disciplines that are mathematical in nature, but are not considered proper subfields of mathematics. Examples include statistics, cryptography, game theory and actuarial science. Mathematical sociology the area of sociology that uses mathematics to construct social theories. Mathematical statistics the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Mathematical system theory Matrix algebra Matrix calculus Matrix theory Matroid theory Measure theory Metric geometry Microlocal analysis Model theory the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic. Modern algebra Occasionally used for abstract algebra. The term was coined by van der Waerden as the title of his book Moderne Algebra, which was renamed Algebra in the latest editions. Modern algebraic geometry the form of algebraic geometry given by Alexander Grothendieck and Jean-Pierre Serre drawing on sheaf theory. Modern invariant theory the form of invariant theory that analyses the decomposition of representations into irreducibles. Modular representation theory a part of representation theory that studies linear representations of finite groups over a field K of positive characteristic p, necessarily a prime number. Module theory Molecular geometry Morse theory a part of differential topology, it analyzes the topological space of a manifold by studying differentiable functions on that manifold. Motivic cohomology Multilinear algebra an extension of linear algebra building upon concepts of p-vectors and multivectors with Grassmann algebra. Multiplicative number theory a subfield of analytic number theory that deals with prime numbers, factorization and divisors. Multivariable calculus the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather than just one. Multiple-scale analysis == N == Neutral geometry See absolute geometry. Nevanlinna theory part of complex analysis studying the value distribution of meromorphic functions. It is named after Rolf Nevanlinna Nielsen theory an area of mathematical research with its origins in fixed point topology, developed by Jakob Nielsen Non-abelian class field theory Non-classical analysis Non-Euclidean geometry Non-standard analysis Non-standard calculus Nonarchimedean dynamics also known as p-adic analysis or local arithmetic dynamics Noncommutative algebra Noncommutative algebraic geometry a direction in noncommutative geometry studying the geometric properties of formal duals of non-commutative algebraic objects. Noncommutative geometry Noncommutative harmonic analysis see representation theory Noncommutative topology Nonlinear analysis Nonlinear functional analysis Number theory a branch of pure mathematics primarily devoted to the study of the integers. Originally it was known as arithmetic or higher arithmetic. Numerical analysis Numerical linear algebra == O == Operad theory a type of abstract algebra concerned with prototypical algebras. Operation research Operator K-theory Operator theory part of functional analysis studying operators. Optimal control theory a generalization of the calculus of variations. Optimal maintenance Orbifold theory Order theory a branch that investigates the intuitive notion of order using binary relations. Ordered geometry a form of geometry omitting the notion of measurement but featuring the concept of intermediacy. It is a fundamental geometry forming a common framework for affine geometry, Euclidean geometry, absolute geometry and hyperbolic geometry. Oscillation theory == P == p-adic analysis a branch of number theory that deals with the analysis of functions of p-adic numbers. p-adic dynamics an application of p-adic analysis looking at p-adic differential equations. p-adic Hodge theory Parabolic geometry Paraconsistent mathematics sometimes called inconsistent mathematics, it is an attempt to develop the classical infrastructure of mathematics based on a foundation of paraconsistent logic instead of classical logic. Partition theory Perturbation theory Picard–Vessiot theory Plane geometry Point-set topology see general topology Pointless topology Poisson geometry Polyhedral combinatorics a branch within combinatorics and discrete geometry that studies the problems of describing convex polytopes. Possibility theory Potential theory Precalculus Predicative mathematics Probability theory Probabilistic combinatorics Probabilistic graph theory Probabilistic number theory Projective geometry a form of geometry that studies geometric properties that are invariant under a projective transformation. Projective differential geometry Proof theory Pseudo-Riemannian geometry generalizes Riemannian geometry to the study of pseudo-Riemannian manifolds. Pure mathematics the part of mathematics that studies entirely abstract concepts. == Q == Quantum calculus a form of calculus without the notion of limits. Quantum geometry the generalization of concepts of geometry used to describe the physical phenomena of quantum physics Quaternionic analysis == R == Ramsey theory the study of the conditions in which order must appear. It is named after Frank P. Ramsey. Rational geometry Real algebra the study of the part of algebra relevant to real algebraic geometry. Real algebraic geometry the part of algebraic geometry that studies real points of the algebraic varieties. Real analysis a branch of mathematical analysis; in particular hard analysis, that is the study of real numbers and functions of Real values. It provides a rigorous formulation of the calculus of real numbers in terms of continuity and smoothness, whilst the theory is extended to the complex numbers in complex analysis. Real Clifford algebra Real K-theory Recreational mathematics the area dedicated to mathematical puzzles and mathematical games. Recursion theory see computability theory Representation theory a subfield of abstract algebra; it studies algebraic structures by representing their elements as linear transformations of vector spaces. It also studies modules over these algebraic structures, providing a way of reducing problems in abstract algebra to problems in linear algebra. Representation theory of groups Representation theory of the Galilean group Representation theory of the Lorentz group Representation theory of the Poincaré group Representation theory of the symmetric group Ribbon theory a branch of topology studying ribbons. Ricci calculus Also called absolute differential calculus. A foundation of tensor calculus, developed by Gregorio Ricci-Curbastro in 1887–1896, and later developed for its applications to general relativity and differential geometry. Ring theory Riemannian geometry a branch of differential geometry that is more specifically, the study of Riemannian manifolds. It is named after Bernhard Riemann and it features many generalizations of concepts from Euclidean geometry, analysis and calculus. Rough set theory the a form of set theory based on rough sets. == S == Sampling theory Scheme theory the study of schemes introduced by Alexander Grothendieck. It allows the use of sheaf theory to study algebraic varieties and is considered the central part of modern algebraic geometry. Secondary calculus Semialgebraic geometry a part of algebraic geometry; more specifically a branch of real algebraic geometry that studies semialgebraic sets. Set-theoretic topology Set theory Sheaf theory The study of sheaves, which connect local and global properties of geometric objects. Sheaf cohomology Sieve theory Single operator theory deals with the properties and classifications of single operators. Singularity theory a branch, notably of geometry; that studies the failure of manifold structure. Smooth infinitesimal analysis a rigorous reformation of infinitesimal calculus employing methods of category theory. As a theory, it is a subset of synthetic differential geometry. Solid geometry Spatial geometry Spectral geometry a field that concerns the relationships between geometric structures of manifolds and spectra of canonically defined differential operators. Spectral graph theory the study of properties of a graph using methods from matrix theory. Spectral theory part of operator theory extending the concepts of eigenvalues and eigenvectors from linear algebra and matrix theory. Spectral theory of ordinary differential equations part of spectral theory concerned with the spectrum and eigenfunction expansion associated with linear ordinary differential equations. Spectrum continuation analysis generalizes the concept of a Fourier series to non-periodic functions. Spherical geometry a branch of non-Euclidean geometry, studying the 2-dimensional surface of a sphere. Spherical trigonometry a branch of spherical geometry that studies polygons on the surface of a sphere. Usually the polygons are triangles. Statistical mechanics Statistical modelling Statistical theory Statistics although the term may refer to the more general study of statistics, the term is used in mathematics to refer to the mathematical study of statistics and related fields. This includes probability theory. Steganography Stochastic calculus Stochastic calculus of variations Stochastic geometry the study of random patterns of points Stochastic process Stratified Morse theory Super linear algebra Surgery theory a part of geometric topology referring to methods used to produce one manifold from another (in a controlled way.) Survey sampling Survey methodology Symbolic computation also known as algebraic computation and computer algebra. It refers to the techniques used to manipulate mathematical expressions and equations in symbolic form as opposed to manipulating them by the numerical quantities represented by them. Symbolic dynamics Symplectic geometry a branch of differential geometry and topology whose main object of study is the symplectic manifold. Symplectic topology Synthetic differential geometry a reformulation of differential geometry in the language of topos theory and in the context of an intuitionistic logic. Synthetic geometry also known as axiomatic geometry, it is a branch of geometry that uses axioms and logical arguments to draw conclusions as opposed to analytic and algebraic methods. Systolic geometry a branch of differential geometry studying systolic invariants of manifolds and polyhedra. Systolic hyperbolic geometry the study of systoles in hyperbolic geometry. == T == Tensor algebra, Tensor analysis, Tensor calculus, Tensor theory the study and use of tensors, which are generalizations of vectors. A tensor algebra is also an algebraic structure that is used in the formal definition of tensors. Tessellation when periodic tiling has a repeating pattern. Theoretical physics a branch primarily of the science physics that uses mathematical models and abstraction of physics to rationalize and predict phenomena. Theory of computation Time-scale calculus Topology Topological combinatorics the application of methods from algebraic topology to solve problems in combinatorics. Topological degree theory Topological graph theory Topological K-theory Topos theory Toric geometry Transcendental number theory a branch of number theory that revolves around the transcendental numbers. Transformation geometry Trigonometry the study of triangles and the relationships between the length of their sides, and the angles between them. It is essential to many parts of applied mathematics. Tropical analysis see idempotent analysis Tropical geometry Twisted K-theory a variation on K-theory, spanning abstract algebra, algebraic topology and operator theory. Type theory == U == Umbral calculus the study of Sheffer sequences Uncertainty theory a new branch of mathematics based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms. Universal algebra a field studying the formalization of algebraic structures itself. Universal hyperbolic trigonometry an approach to hyperbolic trigonometry based on rational geometry. == V == Valuation theory Variational analysis Vector algebra a part of linear algebra concerned with the operations of vector addition and scalar multiplication, although it may also refer to vector operations of vector calculus, including the dot and cross product. In this case it can be contrasted with geometric algebra which generalizes into higher dimensions. Vector analysis also known as vector calculus, see vector calculus. Vector calculus a branch of multivariable calculus concerned with differentiation and integration of vector fields. Primarily it is concerned with 3-dimensional Euclidean space. == W == Wavelets == See also == Lists of mathematics topics Outline of mathematics Category:Glossaries of mathematics == References ==
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Wikipedia:Glossary of linear algebra#0
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This glossary of linear algebra is a list of definitions and terms relevant to the field of linear algebra, the branch of mathematics concerned with linear equations and their representations as vector spaces. For a glossary related to the generalization of vector spaces through modules, see glossary of module theory. == A == affine transformation A composition of functions consisting of a linear transformation between vector spaces followed by a translation. Equivalently, a function between vector spaces that preserves affine combinations. affine combination A linear combination in which the sum of the coefficients is 1. == B == basis In a vector space, a linearly independent set of vectors spanning the whole vector space. basis vector An element of a given basis of a vector space. bilinear form On vector space V over field K, a bilinear form is a function B : V × V → K {\displaystyle B:V\times V\to K} that is linear in each variable. == C == column vector A matrix with only one column. complex number An element of a complex plane complex plane A linear algebra over the real numbers with basis {1, i }, where i is an imaginary unit coordinate vector The tuple of the coordinates of a vector on a basis. covector An element of the dual space of a vector space, (that is a linear form), identified to an element of the vector space through an inner product. == D == determinant The unique scalar function over square matrices which is distributive over matrix multiplication, multilinear in the rows and columns, and takes the value of 1 {\displaystyle 1} for the identity matrix. diagonal matrix A matrix in which only the entries on the main diagonal are non-zero. dimension The number of elements of any basis of a vector space. dot product Given two vectors of the same length, the dot product is the sum of the products of their corresponding indices. dual space The vector space of all linear forms on a given vector space. == E == elementary matrix Square matrix that differs from the identity matrix by at most one entry == H == hyperbolic unit 1. An operator (x, y) → (y, x), reflecting the plane in the 45° diagonal 2. In a linear algebra, a linear map which when composed with itself yields the identity == I == identity matrix A diagonal matrix all of the diagonal elements of which are equal to 1 {\displaystyle 1} . imaginary unit 1. An operator (x, y) → (y, –x), rotating the plane 90° counterclockwise 2. In a linear algebra, a linear map which when composed with itself produces the negative of the identity inverse matrix Of a matrix A {\displaystyle A} , another matrix B {\displaystyle B} such that A {\displaystyle A} multiplied by B {\displaystyle B} and B {\displaystyle B} multiplied by A {\displaystyle A} both equal the identity matrix. isotropic vector In a vector space with a quadratic form, a non-zero vector for which the form is zero. isotropic quadratic form A vector space with a quadratic form which has a null vector. == L == linear algebra 1. The branch of mathematics that deals with vectors, vector spaces, linear transformations and systems of linear equations. 2. A vector space that has a binary operation making it a ring. This linear algebra is also known as an algebra over a field. linear combination A sum, each of whose summands is an appropriate vector times an appropriate scalar (or ring element). linear dependence A linear dependence of a tuple of vectors v → 1 , … , v → n {\textstyle {\vec {v}}_{1},\ldots ,{\vec {v}}_{n}} is a nonzero tuple of scalar coefficients c 1 , … , c n {\textstyle c_{1},\ldots ,c_{n}} for which the linear combination c 1 v → 1 + ⋯ + c n v → n {\textstyle c_{1}{\vec {v}}_{1}+\cdots +c_{n}{\vec {v}}_{n}} equals 0 → {\textstyle {\vec {0}}} . linear equation A polynomial equation of degree one (such as x = 2 y − 7 {\displaystyle x=2y-7} ). linear form A linear map from a vector space to its field of scalars linear independence Property of being not linearly dependent. linear map A function between vector spaces which respects addition and scalar multiplication. linear transformation A linear map whose domain and codomain are equal; it is generally supposed to be invertible. == M == matrix Rectangular arrangement of numbers or other mathematical objects. A matrix is written A = (ai, j), where ai, j is the entry at row i and column j. matrix multiplication If a matrix A has the same number of columns as does matrix B of rows, then a product C = AB may be formed with ci, j equal to the dot product of row i of A with column j of B. == N == null vector 1. Another term for an isotropic vector. 2. Another term for a zero vector. == O == orthogonality Two vectors u and v are orthogonal with respect to a bilinear form B when B(u,v) = 0. orthonormality A set of vectors is orthonormal when they are all unit vectors and are pairwise orthogonal. orthogonal matrix A real square matrix with rows (or columns) that form an orthonormal set. == R == row vector A matrix with only one row. == S == scalar A scalar is an element of a field used in the definition of a vector space. singular-value decomposition a factorization of an m × n {\displaystyle m\times n} complex matrix M as U Σ V ∗ {\displaystyle \mathbf {U\Sigma V^{*}} } , where U is an m × m {\displaystyle m\times m} complex unitary matrix, Σ {\displaystyle \mathbf {\Sigma } } is an m × n {\displaystyle m\times n} rectangular diagonal matrix with non-negative real numbers on the diagonal, and V is an n × n {\displaystyle n\times n} complex unitary matrix. spectrum Set of the eigenvalues of a matrix. split-complex number An element of a split-complex plane split-complex plane A linear algebra over the real numbers with basis {1, j }, where j is a hyperbolic unit square matrix A matrix having the same number of rows as columns. == T == transpose The transpose of an n × m matrix M is an m × n matrix M T obtained by using the rows of M for the columns of M T. == U == unit vector a vector in a normed vector space whose norm is 1, or a Euclidean vector of length one. == V == vector 1. A directed quantity, one with both magnitude and direction. 2. An element of a vector space. vector space A set, whose elements can be added together, and multiplied by elements of a field (this is scalar multiplication); the set must be an abelian group under addition, and the scalar multiplication must be a linear map. == Z == zero vector The additive identity in a vector space. In a normed vector space, it is the unique vector of norm zero. In a Euclidean vector space, it is the unique vector of length zero. == Notes == == References == Curtis, Charles W. (1968) Linear Algebra: an introductory approach, second edition, Allyn & Bacon Dickson, L. E (1914) Linear Algebras via Internet Archive James, Robert C.; James, Glenn (1992). Mathematics Dictionary (5th ed.). Chapman and Hall. ISBN 978-0442007416. Bourbaki, Nicolas (1989). Algebra I. Springer. ISBN 978-3540193739. Williams, Gareth (2014). Linear algebra with applications (8th ed.). Jones & Bartlett Learning.
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Wikipedia:Godofredo García#0
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Godofredo García (born in Lima, Peru, November 8, 1888 - July 16, 1970) was a Peruvian mathematician and engineer. He was the author of more than 80 publications covering mathematics, physics, astronomy, astrophysics, and engineering. == Background == He studied at the Colegio de Lima, under Pedro A. Labarthe. In 1906 he entered the Faculty of Sciences of the National University of San Marcos, where he received a bachelor's degree (1909) and later his doctorate degree in Mathematical Sciences (1912), with his thesis on "Singular points of flat curves" and "Resistance of Columns of reinforced concrete", respectively. Simultaneously, he studied at the School of Engineers of Peru, now called the National University of Engineering (1908-1910), graduating from Civil engineer in 1911. From 1912, he taught at the Chorrillos Military School, where he was in charge of the courses of Flat, Descriptive and Analytical Geometry, Infinitesimal Calculus, Rational Mechanics and Exterior Ballistics. He was also professor of Rational Mechanics in the Faculty of Sciences of the University of San Marcos beginning in 1919 and later served as dean (1928-1940). He became Rector (1941-1943). He was also a professor at the School of Engineers corresponding with Albert Einstein. In the 1920s he worked with the Polish mathematician Alfred Rosenblatt in San Marcos. In 1938, together with Rosenblatt and other San Marcos mathematicians, he founded the National Academy of Exact, Physical and Natural Sciences of Peru, an institution that he presided over from 1960 until he died in 1970. He also directed the publication Actas de la Academia". García was elected to the American Philosophical Society in 1943. He was awarded the national prize for scientific research, in recognition of his contributions in the field of mathematical sciences and his "Exact equations and exact solutions to the movement and stresses of viscous fluids" (1948). He organized conferences in Lima with the participation of Tullio Levi-Civita, Arthur Compton and Garret Birkhoff, among others. In each conference Godofredo García presented a review of the work of these scientists. He married Alicia Rendón (Ecuadorian) and fathered four children. == Bibliography == Lessons of Rational Mechanics, UNMSM; 1937. On a New Cosmogonic Theory, 1940. Algebraic Analysis, Ed. Sanmarti, 1955. == References == Carranza, César. “La Matemática en el Perú”. Discurso pronunciado en XXIII Coloquio Nacional de Matemática, Lambayeque, 2004. Samamé Boggio, Mario: “Godofredo García Díaz”, pg.71-81 de Hacer ciencia en el Perú. Biografías de ocho científicos. Lima, 1990. Tauro del Pino, Alberto (2001): Enciclopedia Ilustrada del Perú. Tercera Edición. Tomo 7. FER/GUZ. Lima, PEISA. ISBN 9972-40-156-1
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Wikipedia:Godunov's theorem#0
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In numerical analysis and computational fluid dynamics, Godunov's theorem — also known as Godunov's order barrier theorem — is a mathematical theorem important in the development of the theory of high-resolution schemes for the numerical solution of partial differential equations. The theorem states that: Professor Sergei Godunov originally proved the theorem as a Ph.D. student at Moscow State University. It is his most influential work in the area of applied and numerical mathematics and has had a major impact on science and engineering, particularly in the development of methods used in computational fluid dynamics (CFD) and other computational fields. One of his major contributions was to prove the theorem (Godunov, 1954; Godunov, 1959), that bears his name. == The theorem == We generally follow Wesseling (2001). Aside Assume a continuum problem described by a PDE is to be computed using a numerical scheme based upon a uniform computational grid and a one-step, constant step-size, M grid point, integration algorithm, either implicit or explicit. Then if x j = j Δ x {\displaystyle x_{j}=j\,\Delta x} and t n = n Δ t {\displaystyle t^{n}=n\,\Delta t} , such a scheme can be described by In other words, the solution φ j n + 1 {\displaystyle \varphi _{j}^{n+1}} at time n + 1 {\displaystyle n+1} and location j {\displaystyle j} is a linear function of the solution at the previous time step n {\displaystyle n} . We assume that β m {\displaystyle \beta _{m}} determines φ j n + 1 {\displaystyle \varphi _{j}^{n+1}} uniquely. Now, since the above equation represents a linear relationship between φ j n {\displaystyle \varphi _{j}^{n}} and φ j n + 1 {\displaystyle \varphi _{j}^{n+1}} we can perform a linear transformation to obtain the following equivalent form, Theorem 1: Monotonicity preserving The above scheme of equation (2) is monotonicity preserving if and only if Proof - Godunov (1959) Case 1: (sufficient condition) Assume (3) applies and that φ j n {\displaystyle \varphi _{j}^{n}} is monotonically increasing with j {\displaystyle j} . Then, because φ j n ≤ φ j + 1 n ≤ ⋯ ≤ φ j + m n {\displaystyle \varphi _{j}^{n}\leq \varphi _{j+1}^{n}\leq \cdots \leq \varphi _{j+m}^{n}} it therefore follows that φ j n + 1 ≤ φ j + 1 n + 1 ≤ ⋯ ≤ φ j + m n + 1 {\displaystyle \varphi _{j}^{n+1}\leq \varphi _{j+1}^{n+1}\leq \cdots \leq \varphi _{j+m}^{n+1}} because This means that monotonicity is preserved for this case. Case 2: (necessary condition) We prove the necessary condition by contradiction. Assume that γ p < 0 {\displaystyle \gamma _{p}^{}<0} for some p {\displaystyle p} and choose the following monotonically increasing φ j n {\displaystyle \varphi _{j}^{n}\,} , Then from equation (2) we get Now choose j = k − p {\displaystyle j=k-p} , to give which implies that φ j n + 1 {\displaystyle \varphi _{j}^{n+1}} is NOT increasing, and we have a contradiction. Thus, monotonicity is NOT preserved for γ p < 0 {\displaystyle \gamma _{p}<0} , which completes the proof. Theorem 2: Godunov’s Order Barrier Theorem Linear one-step second-order accurate numerical schemes for the convection equation cannot be monotonicity preserving unless where σ {\displaystyle \sigma } is the signed Courant–Friedrichs–Lewy condition (CFL) number. Proof - Godunov (1959) Assume a numerical scheme of the form described by equation (2) and choose The exact solution is If we assume the scheme to be at least second-order accurate, it should produce the following solution exactly Substituting into equation (2) gives: Suppose that the scheme IS monotonicity preserving, then according to the theorem 1 above, γ m ≥ 0 {\displaystyle \gamma _{m}\geq 0} . Now, it is clear from equation (15) that Assume σ > 0 , σ ∉ N {\displaystyle \sigma >0,\quad \sigma \notin \mathbb {N} } and choose j {\displaystyle j} such that j > σ > ( j − 1 ) {\displaystyle j>\sigma >\left(j-1\right)} . This implies that ( j − σ ) > 0 {\displaystyle \left({j-\sigma }\right)>0} and ( j − σ − 1 ) < 0 {\displaystyle \left({j-\sigma -1}\right)<0} . It therefore follows that, which contradicts equation (16) and completes the proof. The exceptional situation whereby σ = | c | Δ t Δ x ∈ N {\displaystyle \sigma =\left|c\right|{{\Delta t} \over {\Delta x}}\in \mathbb {N} } is only of theoretical interest, since this cannot be realised with variable coefficients. Also, integer CFL numbers greater than unity would not be feasible for practical problems. == See also == Finite volume method Flux limiter Total variation diminishing == References == Godunov, Sergei K. (1954), Ph.D. Dissertation: Different Methods for Shock Waves, Moscow State University. Godunov, Sergei K. (1959), A Difference Scheme for Numerical Solution of Discontinuous Solution of Hydrodynamic Equations, Mat. Sbornik, 47, 271-306, translated US Joint Publ. Res. Service, JPRS 7226, 1969. Wesseling, Pieter (2001). Principles of Computational Fluid Dynamics. Berlin: Springer-Verlag. ISBN 9783540678533. OCLC 44972030. == Further reading == Hirsch, Ch (1990). Numerical Computation of Internal and External Flows. Vol. 2. Chichester [England]: Wiley. ISBN 0-471-91762-1. OCLC 16523972. Laney, Culbert B. (1998). Computational Gasdynamics. Cambridge: Cambridge University Press. ISBN 978-0-511-77720-2. OCLC 664017316. Toro, Elewterio F. (2009). Riemann Solvers and Numerical Methods for Fluid Dynamics a Practical Introduction (3rd ed.). Berlin. ISBN 978-3-540-25202-3. OCLC 391057413.{{cite book}}: CS1 maint: location missing publisher (link) Anderson, Dale A.; Tannehill, John C.; Pletcher, Richard H.; Munipalli, Ramakanth; Shankar, Vijaya (2020). Computational Fluid Mechanics and Heat Transfer (Fourth ed.). Boca Raton, FL: Taylor & Francis. ISBN 978-1-351-12400-3. OCLC 1237821271.
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Wikipedia:Golagrama#0
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Golagrama was a village or region in India associated with several medieval Indian astronomers, astrologers and mathematicians. Presently there is no place bearing the name Golagrama anywhere in India. It is known that Golagrama was situated in Maharashtra State on the northern banks of river Godavari, was near Partha-puri (Pathari) in Maharashtra and was about 320 km away from Amravati town in Maharashtra. It is sometimes identified as a place named Golgam at latitude 18° N longitude 78° E. The name Golagrama considered as a word in Sanskrit could literally be translated as sphere-village. == Golagrama school of astronomy == Several medieval Indian astronomers and mathematicians are known to have flourished in Golagrama. They were all members of one large extended family. The astronomical and mathematical knowledge of the members of this family was passed from generation to generation through teacher-pupil relationships. In nearly all the cases, the teacher would be an elder member of the family like father being a teacher of his son, elder brother being teacher of a younger brother. The members of this family were also referred to as Daivajnas. One of the most distinguished mathematicians of this family was Kamalakara. Kamalakara was born in Benares about 1616 after some members of the family migrated to Benares. The lineage of astronomers and astrologers begins with one Rāma, a Maharshtian Brahmin residing at Golagrama. Rāma had a son by name Bhaṭṭācārya. He is known to have fathered Divākara Daivjña. Divākara Daivjña had five sons all experts in astrology and astronomy. Divākara Daivjña's sons were Viṣṇu, Kṛṣṇa, Mallāri, Keśava and Viśvanātha. Kṛṣṇa Daivajña had two sons Nṛsiṃha (also known as Nṛsiṃha Gaṇaka) born in 1586, and Śiva. Nṛsiṃha is known to have composed Saurabhāṣya, a commentary on Sūrysidhānta. Nṛsiṃha had composed Sidhāntaśromaṇi-Vāsanāvārttika a commentary on the Gaṇitādhyāya and Golādhyāya of Sidhāntaśiromaṇi-Vāsanābhāṣya of Bhāskara II. Mallari (fl.1575) was a pupil of Ganesa (son of Keśava, born 1507). Mallari was also the uncle and teacher of Nrisimha (born 1586). He wrote a commentary on the Grahalaghava of Ganesa (born 1507) and also another treatise titled Parvadvayasadhana. Gaṇeśa composed in 1520 CE the Grahalāghava or Sidhāntarahasya a compendium of astronomical calculations. Nṛsiṃha (born 1586) had four sons Divākara (born 1606), Kamalākara, Gopinātha and Ranganātha. Kamalākara had composed in 1656 CE a treatise titled Sidhāntatattvaviveka which was course on astronomy founded mainly on Sūryasidhānta. == Chart showing father-son relationships among members of the Golagrama school == == Kamalakara == Kamalakara's father was Nrsimha of Golagrama who was born in 1586. Two of Kamalakara's brothers were also astronomers or mathematicians: Divakara elder than Kamalakara born in 1606 and Ranganatha who was younger than Kamalakara. Kamalakara combined traditional Indian astronomy with Aristotelian physics and Ptolemaic astronomy as presented by Islamic scientists (especially Ulugh Beg). Following his family's tradition he wrote a commentary, Manorama, on Ganesa's Grahalaghava and, like his father, Nrsimha, another commentary on the Suryasiddhanta, called the Vasanabhasya ... Kamalakara's most famous work is Siddhanta-tattva-viveka which was completed in 1658. It deals with standard topics usually dealt with in Indian astronomy texts. The third chapter of the book is mathematically most interesting. In it Kamalakara has used the addition and subtraction theorems for the sine and the cosine to give trigonometric formulae for the sines and cosines of double, triple, quadruple and quintuple angles. In this treatise Kamalakara also makes frequent use of the place-value number system with Sanskrit numerals. == Schools of mathematics in medieval India == Historians of mathematics have identified several schools of mathematics that flourished in different parts of India during the 14th19th– centuries CE. It has also been noted that most of the mathematical activities during this period were concentrated in these schools. The schools were at places identified by the following names: Jambusagaranagara Dadhigrama (Vidarbha) Nandigrama (Maharashtra) Parthapura Golagrama (Maharashtra) Kerala == References ==
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Wikipedia:Goldbach–Euler theorem#0
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In mathematics, the Goldbach–Euler theorem (also known as Goldbach's theorem), states that the sum of 1/(p − 1) over the set of perfect powers p, excluding 1 and omitting repetitions, converges to 1: ∑ p ∞ 1 p − 1 = 1 3 + 1 7 + 1 8 + 1 15 + 1 24 + 1 26 + 1 31 + ⋯ = 1. {\displaystyle \sum _{p}^{\infty }{\frac {1}{p-1}}={{\frac {1}{3}}+{\frac {1}{7}}+{\frac {1}{8}}+{\frac {1}{15}}+{\frac {1}{24}}+{\frac {1}{26}}+{\frac {1}{31}}}+\cdots =1.} This result was first published in Euler's 1737 paper "Variæ observationes circa series infinitas". Euler attributed the result to a letter (now lost) from Goldbach. == Proof == Goldbach's original proof to Euler involved assigning a constant to the harmonic series: x = ∑ n = 1 ∞ 1 n {\displaystyle \textstyle x=\sum _{n=1}^{\infty }{\frac {1}{n}}} , which is divergent. Such a proof is not considered rigorous by modern standards. There is a strong resemblance between the method of sieving out powers employed in his proof and the method of factorization used to derive Euler's product formula for the Riemann zeta function. Let x {\displaystyle x} be given by x = 1 + 1 2 + 1 3 + 1 4 + 1 5 + 1 6 + 1 7 + 1 8 + ⋯ {\displaystyle x=1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{8}}+\cdots } Since the sum of the reciprocal of every power of 2 is 1 = 1 2 + 1 4 + 1 8 + 1 16 + ⋯ {\displaystyle \textstyle 1={\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots } , subtracting the terms with powers of 2 from x {\displaystyle x} gives x − 1 = 1 + 1 3 + 1 5 + 1 6 + 1 7 + 1 9 + 1 10 + 1 11 + ⋯ {\displaystyle x-1=1+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{9}}+{\frac {1}{10}}+{\frac {1}{11}}+\cdots } Repeat the process with the terms with the powers of 3: 1 2 = 1 3 + 1 9 + 1 27 + 1 81 + ⋯ {\displaystyle \textstyle {\frac {1}{2}}={\frac {1}{3}}+{\frac {1}{9}}+{\frac {1}{27}}+{\frac {1}{81}}+\cdots } x − 1 − 1 2 = 1 + 1 5 + 1 6 + 1 7 + 1 10 + 1 11 + 1 12 + ⋯ {\displaystyle x-1-{\frac {1}{2}}=1+{\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{10}}+{\frac {1}{11}}+{\frac {1}{12}}+\cdots } Absent from the above sum are now all terms with powers of 2 and 3. Continue by removing terms with powers of 5, 6 and so on until the right side is exhausted to the value of 1. Eventually, we obtain the equation x − 1 − 1 2 − 1 4 − 1 5 − 1 6 − 1 9 − ⋯ = 1 {\displaystyle x-1-{\frac {1}{2}}-{\frac {1}{4}}-{\frac {1}{5}}-{\frac {1}{6}}-{\frac {1}{9}}-\cdots =1} which we rearrange into x − 1 = 1 + 1 2 + 1 4 + 1 5 + 1 6 + 1 9 + ⋯ {\displaystyle x-1=1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{9}}+\cdots } where the denominators consist of all positive integers that are the non-powers minus 1. By subtracting the previous equation from the definition of x {\displaystyle x} given above, we obtain 1 = 1 3 + 1 7 + 1 8 + 1 15 + 1 24 + 1 26 + 1 31 + ⋯ {\displaystyle 1={\frac {1}{3}}+{\frac {1}{7}}+{\frac {1}{8}}+{\frac {1}{15}}+{\frac {1}{24}}+{\frac {1}{26}}+{\frac {1}{31}}+\cdots } where the denominators now consist only of perfect powers minus 1. While lacking mathematical rigor, Goldbach's proof provides a reasonably intuitive argument for the theorem's truth. Rigorous proofs require proper and more careful treatment of the divergent terms of the harmonic series. Other proofs make use of the fact that the sum of 1/(p − 1) over the set of perfect powers p, excluding 1 but including repetitions, converges to 1 by demonstrating the equivalence: ∑ p ∞ 1 p − 1 = ∑ m = 2 ∞ ∑ n = 2 ∞ 1 m n = 1. {\displaystyle \sum _{p}^{\infty }{\frac {1}{p-1}}=\sum _{m=2}^{\infty }\sum _{n=2}^{\infty }{\frac {1}{m^{n}}}=1.} == See also == Goldbach's conjecture List of sums of reciprocals == References == Viader, Pelegrí; Bibiloni, Lluís; Paradís, Jaume (2006). "On a series of Goldbach and Euler" (PDF). American Mathematical Monthly. 113 (3): 206–220. doi:10.2307/27641889. hdl:10230/382. JSTOR 27641889.. Graham, Ronald; Donald Knuth; Oren Patashnik (1988). Concrete Mathematics. Addison-Wesley. ISBN 0-201-14236-8.
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Wikipedia:Golden–Thompson inequality#0
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In physics and mathematics, the Golden–Thompson inequality is a trace inequality between exponentials of symmetric and Hermitian matrices proved independently by Golden (1965) and Thompson (1965). It has been developed in the context of statistical mechanics, where it has come to have a particular significance. == Statement == The Golden–Thompson inequality states that for (real) symmetric or (complex) Hermitian matrices A and B, the following trace inequality holds: tr e A + B ≤ tr ( e A e B ) . {\displaystyle \operatorname {tr} \,e^{A+B}\leq \operatorname {tr} \left(e^{A}e^{B}\right).} This inequality is well defined, since the quantities on either side are real numbers. For the expression on the right hand side of the inequality, this can be seen by rewriting it as tr ( e A / 2 e B e A / 2 ) {\displaystyle \operatorname {tr} (e^{A/2}e^{B}e^{A/2})} using the cyclic property of the trace. Let ‖ ⋅ ‖ {\displaystyle \|\cdot \|} denote the Frobenius norm, then the Golden–Thompson inequality is equivalently stated as ‖ e A + B ‖ ≤ ‖ e A e B ‖ . {\displaystyle \|e^{A+B}\|\leq \|e^{A}e^{B}\|.} === Motivation === The Golden–Thompson inequality can be viewed as a generalization of a stronger statement for real numbers. If a and b are two real numbers, then the exponential of a+b is the product of the exponential of a with the exponential of b: e a + b = e a e b . {\displaystyle e^{a+b}=e^{a}e^{b}.} If we replace a and b with commuting matrices A and B, then the same inequality e A + B = e A e B {\displaystyle e^{A+B}=e^{A}e^{B}} holds. This relationship is not true if A and B do not commute. In fact, Petz (1994) proved that if A and B are two Hermitian matrices for which the Golden–Thompson inequality is verified as an equality, then the two matrices commute. The Golden–Thompson inequality shows that, even though e A + B {\displaystyle e^{A+B}} and e A e B {\displaystyle e^{A}e^{B}} are not equal, they are still related by an inequality. === Proof === == Generalizations == === Other norms === In general, if A and B are Hermitian matrices and ‖ ⋅ ‖ {\displaystyle \|\cdot \|} is a unitarily invariant norm, then (Bhatia 1997, Theorem IX.3.7) ‖ e A + B ‖ ≤ ‖ e A e B ‖ . {\displaystyle \|e^{A+B}\|\leq \|e^{A}e^{B}\|.} The standard Golden–Thompson inequality is a special case of the above inequality, where the norm is the Frobenius norm. The general case is provable in the same way, since unitarily invariant norms also satisfy the Cauchy-Schwarz inequality. (Bhatia 1997, Exercise IV.2.7) Indeed, for a slightly more general case, essentially the same proof applies. For each p ≥ 1 {\textstyle p\geq 1} , let ‖ A ‖ p p := tr ( A A ∗ ) p / 2 {\textstyle \|A\|_{p}^{p}:=\operatorname {tr} (AA^{*})^{p/2}} be the Schatten norm. === Multiple matrices === The inequality has been generalized to three matrices by Lieb (1973) and furthermore to any arbitrary number of Hermitian matrices by Sutter, Berta & Tomamichel (2016). A naive attempt at generalization does not work: the inequality tr ( e A + B + C ) ≤ | tr ( e A e B e C ) | {\displaystyle \operatorname {tr} (e^{A+B+C})\leq |\operatorname {tr} (e^{A}e^{B}e^{C})|} is false. For three matrices, the correct generalization takes the following form: tr e A + B + C ≤ tr ( e A T e − B e C ) , {\displaystyle \operatorname {tr} \,e^{A+B+C}\leq \operatorname {tr} \left(e^{A}{\mathcal {T}}_{e^{-B}}e^{C}\right),} where the operator T f {\displaystyle {\mathcal {T}}_{f}} is the derivative of the matrix logarithm given by T f ( g ) = ∫ 0 ∞ d t ( f + t ) − 1 g ( f + t ) − 1 {\displaystyle {\mathcal {T}}_{f}(g)=\int _{0}^{\infty }\operatorname {d} t\,(f+t)^{-1}g(f+t)^{-1}} . Note that, if f {\displaystyle f} and g {\displaystyle g} commute, then T f ( g ) = g f − 1 {\displaystyle {\mathcal {T}}_{f}(g)=gf^{-1}} , and the inequality for three matrices reduces to the original from Golden and Thompson. Bertram Kostant (1973) used the Kostant convexity theorem to generalize the Golden–Thompson inequality to all compact Lie groups. == References == Bhatia, Rajendra (1997), Matrix analysis, Graduate Texts in Mathematics, vol. 169, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-0653-8, ISBN 978-0-387-94846-1, MR 1477662 Cohen, J.E.; Friedland, S.; Kato, T.; Kelly, F. (1982), "Eigenvalue inequalities for products of matrix exponentials", Linear Algebra and Its Applications, 45: 55–95, doi:10.1016/0024-3795(82)90211-7 Golden, Sidney (1965), "Lower bounds for the Helmholtz function", Phys. Rev., Series II, 137 (4B): B1127 – B1128, Bibcode:1965PhRv..137.1127G, doi:10.1103/PhysRev.137.B1127, MR 0189691 Kostant, Bertram (1973), "On convexity, the Weyl group and the Iwasawa decomposition", Annales Scientifiques de l'École Normale Supérieure, Série 4, 6 (4): 413–455, doi:10.24033/asens.1254, ISSN 0012-9593, MR 0364552 Lieb, Elliott H (1973), "Convex trace functions and the Wigner-Yanase-Dyson conjecture", Advances in Mathematics, 11 (3): 267–288, doi:10.1016/0001-8708(73)90011-X Petz, D. (1994), A survey of trace inequalities, in Functional Analysis and Operator Theory (PDF), vol. 30, Warszawa: Banach Center Publications, pp. 287–298, archived from the original (PDF) on 2012-02-12, retrieved 2009-01-15 Sutter, David; Berta, Mario; Tomamichel, Marco (2016), "Multivariate Trace Inequalities", Communications in Mathematical Physics, 352 (1): 37–58, arXiv:1604.03023, Bibcode:2017CMaPh.352...37S, doi:10.1007/s00220-016-2778-5, S2CID 12081784 Thompson, Colin J. (1965), "Inequality with applications in statistical mechanics", Journal of Mathematical Physics, 6 (11): 1812–1813, Bibcode:1965JMP.....6.1812T, doi:10.1063/1.1704727, ISSN 0022-2488, MR 0189688 Tao, Terence (July 15, 2010). "The Golden-Thompson inequality". What's new. Retrieved January 19, 2025. Tao, Terence (2012). Topics in random matrix theory. Graduate studies in mathematics. Providence, R.I: American Mathematical Society. ISBN 978-0-8218-7430-1. Forrester, Peter J; Thompson, Colin J (2014). "The Golden-Thompson inequality --- historical aspects and random matrix applications". Journal of Mathematical Physics. 55 (2): 023503. arXiv:1408.2008. Bibcode:2014JMP....55b3503F. doi:10.1063/1.4863477. S2CID 119676709.
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Wikipedia:Gordana Matic#0
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Gordana Matic is a Croatian-American mathematician who works as a professor at the University of Georgia. Her research concerns low-dimensional topology and contact geometry. Matic earned her doctorate from the University of Utah in 1986, under the supervision of Ronald J. Stern, and worked as a C.L.E. Moore instructor at the Massachusetts Institute of Technology before joining the University of Georgia faculty. Matic was the Spring 2012 speaker in the University of Texas Distinguished Women in Mathematics Lecture Series. In 2014, she was elected as a fellow of the American Mathematical Society "for contributions to low-dimensional and contact topology." == References == == External links == Home page
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Wikipedia:Gordana Todorov#0
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Gordana Todorov (born July 24, 1949) is a mathematician working in noncommutative algebra, representation theory, Artin algebras, and cluster algebras. She is a professor of mathematics at Northeastern University. == Biography == Todorov earned her Ph.D. in 1978, at Brandeis University. Her dissertation, Almost Split Sequences in the Representation Theory of Certain Classes of Artin Algebras, was supervised by Maurice Auslander. Todorov is married to mathematician Kiyoshi Igusa. The Igusa–Todorov functions and Igusa–Todorov endomorphism algebras are named for their joint work. Todorov is also the namesake of Todorov's theorem on preprojective partitions, and the Gentle–Todorov theorem on abelian categories. == References == == External links == Home page Archived 2017-06-22 at the Wayback Machine Gordana Todorov publications indexed by Google Scholar
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Wikipedia:Gordon Douglas Slade#0
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Gordon Douglas Slade (born December 14, 1955, in Toronto) is a Canadian mathematician, specializing in probability theory. == Education == Slade received in 1977 his bachelor's degree from the University of Toronto and in 1984 his PhD for research supervised by Joel Feldman and Lon Rosen at the University of British Columbia. == Career and Research == As a postdoc he was a lecturer at the University of Virginia. From 1986 he was at McMaster University and since 1999 he is a professor at the University of British Columbia. He developed the technique of lace expansion (originally introduced by David Brydges and Thomas C. Spencer in 1985) with applications to probability theory and statistical mechanics, such as self-avoiding random walks and their enumeration, random graphs, percolation theory, and branched polymers. In 1989 Slade proved with Takashi Hara that the Aizenman–Newman triangle condition at critical percolation is valid in sufficiently high dimension. The Hara–Slade result has important consequences in mean field theory. In 1991 Slade and Hara used the lace expansion to prove that the average distance covered in self-avoiding random walks in 5 or more dimension grows as the square root of the number of steps and that the scaling limit is Brownian motion. === Honours and awards === Slade was an invited speaker in 1994 at the ICM in Zürich with lecture The critical behaviour of random systems. Slade received in 1995 the Coxeter–James Prize and in the 2010 the CRM-Fields-PIMS Prize. He was elected a Fellow of the Royal Society of Canada (FRSC) in 2000, in 2010 of the Fields Institute, and in 2012 of the American Mathematical Society and of the Institute of Mathematical Statistics. He was elected a Fellow of the Royal Society in 2017. In 2018 Slade was awarded the Jeffery–Williams Prize. === Selected publications === with Neal Madras: Self-avoiding walk, Birkhäuser 1993 The lace expansion and its applications (École d’Eté de Probabilités de Saint-Flour XXXIV, 2004), Springer Verlag 2006 == References ==
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Wikipedia:Gordon Royle#0
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Gordon F. Royle is a professor at the School of Mathematics and Statistics at The University of Western Australia. Royle is the co-author (with Chris Godsil) of the book Algebraic Graph Theory (Springer Verlag, 2001, ISBN 0-387-95220-9). Royle is also known for his research into the mathematics of Sudoku and his search for the Sudoku puzzle with the smallest number of entries that has a unique solution. Royle earned his Ph.D. in 1987 from the University of Western Australia under the supervision of Cheryl Praeger and Brendan McKay. == References ==
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Wikipedia:Gorenstein–Harada theorem#0
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In mathematics, specifically finite group theory, the Gorenstein–Harada theorem, proved by Daniel Gorenstein and Koichiro Harada, classifies the finite simple groups of sectional 2-rank at most 4. It is part of the classification of finite simple groups. Finite simple groups of section 2 with rank at least 5 have Sylow 2-subgroups with a self-centralizing normal subgroup of rank at least 3, which implies that they have to be of either component type or of characteristic 2 type. Therefore, the Gorenstein–Harada theorem splits the problem of classifying finite simple groups into these two sub-cases. == References ==
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Wikipedia:Goro Azumaya#0
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Gorō Azumaya (東屋 五郎, Azumaya Gorō, February 26, 1920– July 8, 2010) was a Japanese mathematician who introduced the notion of Azumaya algebra in 1951. His advisor was Shokichi Iyanaga. At the time of his death he was an emeritus professor at Indiana University. == References == == External links == Goro Azumaya at the Mathematics Genealogy Project Biography of Azumaya by BiRep, Bielefeld University
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Wikipedia:Gosper's algorithm#0
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In mathematics, Gosper's algorithm, due to Bill Gosper, is a procedure for finding sums of hypergeometric terms that are themselves hypergeometric terms. That is: suppose one has a(1) + ... + a(n) = S(n) − S(0), where S(n) is a hypergeometric term (i.e., S(n + 1)/S(n) is a rational function of n); then necessarily a(n) is itself a hypergeometric term, and given the formula for a(n) Gosper's algorithm finds that for S(n). == Outline of the algorithm == Step 1: Find a polynomial p such that, writing b(n) = a(n)/p(n), the ratio b(n)/b(n − 1) has the form q(n)/r(n) where q and r are polynomials and no q(n) has a nontrivial factor with r(n + j) for j = 0, 1, 2, ... . (This is always possible, whether or not the series is summable in closed form.) Step 2: Find a polynomial ƒ such that S(n) = q(n + 1)/p(n) ƒ(n) a(n). If the series is summable in closed form then clearly a rational function ƒ with this property exists; in fact it must always be a polynomial, and an upper bound on its degree can be found. Determining ƒ (or finding that there is no such ƒ) is then a matter of solving a system of linear equations. == Relationship to Wilf–Zeilberger pairs == Gosper's algorithm can be used to discover Wilf–Zeilberger pairs, where they exist. Suppose that F(n + 1, k) − F(n, k) = G(n, k + 1) − G(n, k) where F is known but G is not. Then feed a(k) := F(n + 1, k) − F(n, k) into Gosper's algorithm. (Treat this as a function of k whose coefficients happen to be functions of n rather than numbers; everything in the algorithm works in this setting.) If it successfully finds S(k) with S(k) − S(k − 1) = a(k), then we are done: this is the required G. If not, there is no such G. == Definite versus indefinite summation == Gosper's algorithm finds (where possible) a hypergeometric closed form for the indefinite sum of hypergeometric terms. It can happen that there is no such closed form, but that the sum over all n, or some particular set of values of n, has a closed form. This question is only meaningful when the coefficients are themselves functions of some other variable. So, suppose a(n,k) is a hypergeometric term in both n and k: that is, a(n, k)/a(n − 1,k) and a(n, k)/a(n, k − 1) are rational functions of n and k. Then Zeilberger's algorithm and Petkovšek's algorithm may be used to find closed forms for the sum over k of a(n, k). == History == Bill Gosper discovered this algorithm in the 1970s while working on the Macsyma computer algebra system at SAIL and MIT. == Notes == == References == Gosper, Jr., Ralph William "Bill" (January 1978) [1977-09-26]. "Decision procedure for indefinite hypergeometric summation" (PDF). Proceedings of the National Academy of Sciences of the United States of America. Mathematics. 75 (1). Xerox, Palo Alto Research Center, Palo Alto, California, USA: 40–42. Bibcode:1978PNAS...75...40G. doi:10.1073/pnas.75.1.40. PMC 411178. PMID 16592483. Archived (PDF) from the original on 2019-04-12. Retrieved 2020-01-10. algorithm / binomial coefficient identities / closed form / symbolic computation / linear recurrences
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Wikipedia:Goss zeta function#0
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In the field of mathematics, the Goss zeta function, named after David Goss, is an analogue of the Riemann zeta function for function fields. Sheats (1998) proved that it satisfies an analogue of the Riemann hypothesis. Kapranov (1995) proved results for a higher-dimensional generalization of the Goss zeta function. == References == Goss, David (1996), Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 35, Berlin, New York: Springer-Verlag, ISBN 978-3-540-61087-8, MR 1423131 Kapranov, Mikhail (1995), "A higher-dimensional generalization of the Goss zeta function", Journal of Number Theory, 50 (2): 363–375, doi:10.1006/jnth.1995.1030 Sheats, Jeffrey T. (1998), "The Riemann hypothesis for the Goss zeta function for Fq[T]", Journal of Number Theory, 71 (1): 121–157, arXiv:math/9801158, doi:10.1006/jnth.1998.2232, ISSN 0022-314X, MR 1630979
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Wikipedia:Gottfried Köthe#0
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Gottfried Maria Hugo Köthe (25 December 1905 – 30 April 1989) was an Austrian mathematician working in abstract algebra and functional analysis. == Scientific career == In 1923 Köthe enrolled in the University of Graz. He started studying chemistry, but switched to mathematics a year later after meeting the philosopher Alfred Kastil. In 1927 he submitted his thesis Beiträge zu Finslers Grundlegung der Mengenlehre ("Contributions to Finsler's foundations of set theory") and was awarded a doctorate. After spending a year in Zürich working with Paul Finsler, Köthe received a fellowship to visit the University of Göttingen, where he attended the lectures of Emmy Noether and Bartel van der Waerden on the emerging subject of abstract algebra. He began working in ring theory and in 1930 published the Köthe conjecture stating that a sum of two left nil ideals in an arbitrary ring is a nil ideal. By a recommendation of Emmy Noether, he was appointed an assistant of Otto Toeplitz in Bonn University in 1929–1930. During this time he began transition to functional analysis. He continued scientific collaboration with Toeplitz for several years afterward. Köthe's Habilitationsschrift, Schiefkörper unendlichen Ranges über dem Zentrum ("Skew fields of infinite rank over the center"), was accepted in 1931. He became Privatdozent at University of Münster under Heinrich Behnke. During World War II he was involved in coding work. In 1946 he was appointed the director of the Mathematics Institute at the University of Mainz and he served as a dean (1948–1950) and a rector of the university (1954–1956). In 1957 he became the founding director of the Institute for Applied Mathematics at the University of Heidelberg and served as a rector of the university (1960–1961). Köthe's best known work has been in the theory of topological vector spaces. In 1960, volume 1 of his seminal monograph Topologische lineare Räume was published (the second edition was translated into English in 1969). It was not until 1979 that volume 2 appeared, this time written in English. He also made contributions to the theory of lattices. == Awards and honors == Invited Speaker of the ICM in 1928 in Bologna, in 1932 in Zurich, and in 1936 in Oslo Heidelberg Academy of Sciences (1960) Gauss medal, Brunswick Academy of Sciences (1963) German Academy of Sciences Leopoldina, Halle (1968) Honorary degrees from University of Montpellier (1965), University of Münster (1980), University of Mainz (1981) and Saarland University (1981). == Books == Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704. Köthe, Gottfried (1979). Topological Vector Spaces II. Grundlehren der mathematischen Wissenschaften. Vol. 237. New York: Springer Science & Business Media. ISBN 978-0-387-90400-9. OCLC 180577972. Köthe, Gottfried (1969). Topological vector spaces. Springer Verlag. ISBN 978-0-387-90400-9. MR 0551623. == References == == External links == O'Connor, John J.; Robertson, Edmund F., "Gottfried Köthe", MacTutor History of Mathematics Archive, University of St Andrews Gottfried Köthe, 1905-1989 by Joachim Weidmann, digital edition Univ. Heidelberg Vita (in German) by Heinz Günther Tillmann, digital edition Univ. Heidelberg
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Wikipedia:Gould's sequence#0
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Gould's sequence is an integer sequence named after Henry W. Gould that counts how many odd numbers are in each row of Pascal's triangle. It consists only of powers of two, and begins: 1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, 2, 4, ... (sequence A001316 in the OEIS) For instance, the sixth number in the sequence is 4, because there are four odd numbers in the sixth row of Pascal's triangle (the four bold numbers in the sequence 1, 5, 10, 10, 5, 1). Gould's sequence is also a fractal sequence. == Additional interpretations == The nth value in the sequence (starting from n = 0) gives the highest power of 2 that divides the central binomial coefficient ( 2 n n ) {\displaystyle {\tbinom {2n}{n}}} , and it gives the numerator of 2 n / n ! {\displaystyle 2^{n}/n!} (expressed as a fraction in lowest terms). Gould's sequence also gives the number of live cells in the nth generation of the Rule 90 cellular automaton starting from a single live cell. It has a characteristic growing sawtooth shape that can be used to recognize physical processes that behave similarly to Rule 90. == Related sequences == The binary logarithms (exponents in the powers of two) of Gould's sequence themselves form an integer sequence, 0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, ... (sequence A000120 in the OEIS) in which the nth value gives the number of nonzero bits in the binary representation of the number n, sometimes written in mathematical notation as # 1 ( n ) {\displaystyle \#_{1}(n)} . Equivalently, the nth value in Gould's sequence is 2 # 1 ( n ) . {\displaystyle 2^{\#_{1}(n)}.} Taking the sequence of exponents modulo two gives the Thue–Morse sequence. The partial sums of Gould's sequence, 0, 1, 3, 5, 9, 11, 15, 19, 27, 29, 33, 37, 45, ... (sequence A006046 in the OEIS) count all odd numbers in the first n rows of Pascal's triangle. These numbers grow proportionally to n log 2 3 ≈ n 1.585 {\displaystyle n^{\log _{2}3}\approx n^{1.585}} , but with a constant of proportionality that oscillates between 0.812556... and 1, periodically as a function of log n. == Recursive construction and self-similarity == The first 2i values in Gould's sequence may be constructed by recursively constructing the first 2i − 1 values, and then concatenating the doubles of the first 2i − 1 values. For instance, concatenating the first four values 1, 2, 2, 4 with their doubles 2, 4, 4, 8 produces the first eight values. Because of this doubling construction, the first occurrence of each power of two 2i in this sequence is at position 2i − 1. Gould's sequence, the sequence of its exponents, and the Thue–Morse sequence are all self-similar: they have the property that the subsequence of values at even positions in the whole sequence equals the original sequence, a property they also share with some other sequences such as Stern's diatomic sequence. In Gould's sequence, the values at odd positions are double their predecessors, while in the sequence of exponents, the values at odd positions are one plus their predecessors. == History == The sequence is named after Henry W. Gould, who studied it in the early 1960s. However, the fact that these numbers are powers of two, with the exponent of the nth number equal to the number of ones in the binary representation of n, was already known to J. W. L. Glaisher in 1899. Proving that the numbers in Gould's sequence are powers of two was given as a problem in the 1956 William Lowell Putnam Mathematical Competition. == References ==
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Wikipedia:Govinda Bhattathiri#0
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Govinda Bhaṭṭathiri (also known as Govinda Bhattathiri of Thalakkulam or Thalkkulathur) (c. 1237 – 1295) was an Indian astrologer and astronomer who flourished in Kerala during the thirteenth century CE. Govinda Bhaṭṭatiri was born in the Nambudiri family known by the name Thalakkulathur in the village of Alathiyur, Tirur in Kerala. He was traditionally considered to be the progenitor of the Pazhur Kaniyar family of astrologers. He is an important figure in the Kerala astrological traditions. == Works == Govinda wrote Nauka, a commentary on Brihat Jataka. Earlier scholars also assigned to him the authorship of Daśādhyāyī, another commentary on Brihat Jataka written with same narrative style. Recent research suggests that Nauka was the original commentary written by Govinda and Daśādhyāyī was an abridged version rearranged by another person in the 15th century. The authorship of the Daśādhyāyī was assigned to Govinda Bhattathiri in the Ithihyamala written by Sankunni during late 19th century. Daśādhyāyī is considered to be the most important of the 70 known commentaries on this text. Govinda wrote another important work in astrology titled Muhūrttaratnaṃ. Paramesvara (ca.1380–1460), an astronomer of the Kerala school of astronomy and mathematics known for the introduction of the Dṛggaṇita system of astronomical computations, composed an extensive commentary on this work. In this commentary Paramesvara had indicated that he was a grandson of a disciple of the author of Muhūrttaratnaṃ. == See also == List of astronomers and mathematicians of the Kerala school == References ==
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Wikipedia:Govindasvāmi#0
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Govindasvāmi (or Govindasvāmin, Govindaswami) (c. 800 – c. 860) was an Indian mathematical astronomer most famous for his Bhashya, a commentary on the Mahābhāskarīya of Bhāskara I, written around 830. The commentary contains many examples illustrating the use of a Sanskrit place-value system and the construction of a sine table. His works have been quoted extensively by Sankaranarayana (fl. 869), Udayadivakara (fl. 1073) and Nilakantha Somayaji (c. 1444-1544). Sankaranarayana was the director of the observatory founded in Mahodayapuram, the capital of the Chera kingdom, and is believed to be the student of Govindasvami. In his book, Sankaranarayana gives explanations to the insightful questions of the king Ravi Varma, then ruler of Mahodayapuram and from these references the period of Sankaranarayana is known. His work Govindakriti was a sequel to Āryabhaṭīya and is lost. Other works attributed to Govindasvami includeGovinda-paddhati (on astrology) and Ganita-mukha (on mathematics). Like Govinda-kriti, these are lost, and known only from mentions and quotations by later writers such as Sankaranarayana and Udayadivakara. == See also == List of astronomers and mathematicians of the Kerala school == References == == Further reading == Gupta, R. C. (1971). "Fractional parts of Aryabhata's sines and certain rules found in Govindasvami's Bhasya on the Mahabhaskarya". Indian J. History Sci. 6: 51–59. doi:10.1007/978-981-13-1229-8_33. Jha, S. K.; V N Jha (1991). "Computation of sine-table based on the Mahasiddhanta of Aryabhata II". J. Bihar Math. Soc. 14: 9–17. Meijering, Erik (March 2002). "A Chronology of Interpolation: From Ancient Astronomy to Modern Signal and Image Processing" (PDF). Proceedings of the IEEE. 90 (3): 319–342. doi:10.1109/5.993400. Archived from the original (PDF) on 28 January 2007. Kerala Mathematics and its Possible Transmission to Europe 2009_ by George Gevarghese Joseph == External links == A Chronology of Interpolation
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Wikipedia:Grace Yang#0
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Grace Lo Yang (simplified Chinese: 罗昭容; traditional Chinese: 羅昭容; pinyin: Luó Zhāoróng) is a Taiwanese statistician whose research areas include stochastic processes in the physical sciences, asymptotic theory, and survival analysis. She is a professor of statistics in the department of mathematics at the University of Maryland, College Park. She was president of the International Chinese Statistical Association for 1990–1991 and program director for statistics at the National Science Foundation from 2005 to 2008. == Education and career == Yang is originally from China, but moved to Taiwan in 1949. After doing her undergraduate studies at National Taiwan University, she completed her doctorate in 1966 from the University of California, Berkeley, under the supervision of Lucien Le Cam. Her dissertation was on Contagion in Stochastic Models for Epidemics. With Le Cam, she is the author of Asymptotics in Statistics: Some Basic Concepts (Springer, 1990; 2nd ed., 2002). == Awards and honors == Yang is a member of the International Statistical Institute, and a fellow of the Institute of Mathematical Statistics. == References ==
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Wikipedia:Graciano Ricalde Gamboa#0
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Mauro Graciano Ricalde Gamboa (November 21, 1873 – November 9, 1942) was a Mexican mathematician. Ricalde Gamboa was born in Hoctún, state of Yucatan, Mexico. Son of Don Ambrosio Ricalde Moguel and Isidra Gamboa and studied at Hocabá until 1885, the year he received a scholarship to the Normal School for teachers of the State in the city of Mérida. He graduated as a teacher at the age of 16 years, and because of his young age, was granted special permission by the State Congress. He studied accounting and for several years was a professor of commercial arithmetic books and accounts at the School of Commerce of Mérida. For the purpose of continuing his education, he studied civil engineering at the Institute for Literature of the state of Yucatan, under the wise leadership of Manuel Cepeda Sales. At that time, facing the danger of a collapse of one of the towers of the Cathedral of Yucatan, the authorities asked professor Sales Cepeda to assist in determining the weight of the church bell that had been ignored. The teacher passed the problem on to Ricalde Gamboa who did the calculations exactly. Since then he was appointed a professor of rational arithmetic at the Institute where he studied. For various reasons he could no longer continue his desire to study engineering, but soon after, in 1902, he was appointed professor and later director of his own engineering school, a position he continued until 1905 when he resigned to devote himself to the studies of mathematics and became a correspondent for various specialized institutions abroad. He assembled a large library of over 5000 titles and was a collaborator of the French academic journal L'Intermédiaire des Mathématiciens, published in Paris in the early twentieth century. He was also interested in astronomy. He worked on various research projects in the city of Mérida to his death in 1942 the victim of a throat infection. == Career == He achieved recognition for having calculated in 1910 the orbit dish of Halley's Comet. In 1923 he precisely calculated the solar eclipse that year and became part of the Mexican Geodetic Commission who observed and studied in Champotón in the state of Campeche. He discovered a method for solving the quintic equation by using elliptic functions. He was invited to work as a professor in several foreign schools, which he never accepted, preferring to remain in his homeland dedicating himself to his studies and to organize accounting systems of industrial enterprises seeking professional services == Awards == The municipal library of Hoctún was called Graciano Ricalde Gamboa after the "favorite son of the village". In the town of Ticopó, in the municipality of Acanceh, Yucatan, a high school is named after the mathematician. In 2006, an award for Science and Mathematics was named in his honor. == Resolution of Algebraic Equations: Degree 5 by Elliptic Functions == One of the problems that have troubled most the mathematicians of all times is the resolution of algebraic equations of all grades. Such solutions were found for polynomial equations of degree 3 and 4 by Cardano and Ferrari in the 16th century. For polynomial equations of degree 5 and more, it was proven by Abel and Galois that the solution cannot be expressed in terms of radicals (meaning the roots cannot necessarily be expressed). However, there are other ways one can solve such equations. Hermite instead found a solution involving certain functions in advanced calculus that are called elliptic functions. (They generalize trig functions and are related to elliptic curves and originally to the problem of calculating the perimeter of an ellipse.) His solution was simplified and clarified by Kronecker and Klein. Among Ricalde's greatest accomplishments is that of his contribution to the resolution of the general equation of 5th-degree grade by elliptic functions, a feat that is analyzed and evaluated in the second part of this biography in Enciclopedia Yucanense IV, 1944. Additionally, Ricalde corresponded in a French question-and-answer journal called L'Intermédiaire des mathématiciens. In one entry that was published in circa 1898 (Graciano would have been 25 years of age), he asked about solving the quintic equation. More specifically, his question was about reducing the general quintic equation to the reduced Bring-Jerrard form (to solve the Bring-Jerrard equation requires elliptic functions). However, in modern mathematical literature, Ricalde is rarely cited. A citation in the arXiv (funded by Cornell University) is a review by Lemmermeyer concerning another algebra problem called Pell's equation. (Lemmermeyer includes Ricalde in a list of people who solved special cases of Pell's equation, citing the 1901 issue of L'Intermédiaire des Mathématiciens). Since there are no radical solutions to polynomial equations of degrees greater than 4, roots cannot necessarily be expressed. Also, for applications in science and engineering, “solving an equation” usually means “finding a close approximation of a solution”. For example, if there is a real number which solves this equation, it can be said that it lies somewhere between 1.16730398 and 1.16730399. One can provide more precision, but obtaining the full decimal expansion of the solution remains impossible. The code can still make the right decision, and a bridge will still hold if it is a sufficiently accurate approximation. Today's computers also solve polynomial equations by using iterative approximation techniques. == 1901 == He amazed the Scientist of the Intermediare Mathematicians of Paris, France with his solution to the Pell equation. Ricalde then accepted Professorship of the State College of Civil Engineering. In a moment of stillness and recollection, Graciano Ricalde remembers he was encouraged and led by Lic. and Eng. Capeda Manuel Sales, an eminent scientist Yucatan to begin his serious studies of actual mathematics. == 1902 == Graciano joined his wife and life partner who was then, then a professor Srita Normal, Carmen Manzañilla Camomile. They bore four children, three boys and one girl, Alfonso, Humberto, Enrique and Ofelia who studied under the direction of the father, and mathematics. Enrique moved to New York City. == 1910 == Graciano would be the first to correctly prove by precise calculation that neither the arrival of Halley's Comet or its tail would hit the earth extinguishing life. This was a tremendous concern of its day. His rigorous studies were so serious that he compiled his formulas in a booklet that had great resonance in the National Observatory of Paris. == 1923 == Don Graciano accurately calculated the total eclipse of the sun of that year and joined the Mexican Commission Geodescica happened to observe at Champoton in the state of Campeche, for being there this total eclipse, in which the capital of the republic was seen as partial. But his greatest accomplishment is undoubtedly the resolution of the general equation of 5th-degree grade by elliptic functions, a feat that is analyzed and evaluated in the second part of this biography. == November 9, 1942 == Graciano died in Mérida in the grand family home. == January 18, 1943 == The National Academy of Sciences, Antonio Alzaate organized a mathematical exposition of the merits of Graciano Ricalde during the evening in the Palacio de Bellas Artes of Mexico City. The band played and speakers including Dr. Ruben Moreno Ricalde spoke that evening as chronicled and published in the Daily News. == 1959 == A plaque was placed on the house where he was born in the little town of Hoctun. The Hoctun municipal library is called Ricalde Graciano Gamboa “in honor of the beloved son of the village”. In the town of Ticopo, Yucatan, the high school is named after the mathematician. There is a second school, I believe it is in Hocabá. == 2016 == A plaque was presented and placed on Graciano Ricalde's professional residence by his nephew Arcadio Poveda. The home is now a boutique hotel in Mérida, Yucatán called Hotel Casa San Ángel. == References == This article depends heavily on the Spanish article es:Graciano Ricalde Gamboa
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Wikipedia:Graciela Boente#0
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Graciela Lina Boente Boente is an Argentine mathematical statistician at the University of Buenos Aires. She is known for her research in robust statistics, and particularly for robust methods for principal component analysis and regression analysis. == Education == Boente earned her Ph.D. in 1983 from the University of Buenos Aires. Her dissertation, Robust Principal Components, was supervised by Victor J. Yohai. == Awards and honors == Boente became a Guggenheim Fellow in 2001. In 2008, the Argentine National Academy of Exact, Physical and Natural Sciences gave her their Consecration Prize in recognition of her contributions and teaching. She became an honored fellow of the Institute of Mathematical Statistics in 2013, "for her research in robust statistics and estimation, and for outstanding service to the statistical community". == References ==
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Wikipedia:Graciela Salicrup#0
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Graciela Beatriz Salicrup López (México City, México, April 7, 1935 – July 29, 1982) was a Mexican architect, archaeologist, and mathematician. In the 1970s and 1980s, she was a pioneer in the field of categorical topology. Most of her work was published in Spanish, and her original contributions were not widely recognized until after her premature death. == Personal life == Graciela Beatriz Salicrup López was born in Mexico City on April 7, 1935. As a child, she attended Colegio Alemán, a German language primary school in Mexico City, followed by a religious secondary school for girls. A professor at Colegio Alemán originally encouraged her, inciting an interest in mathematics that her family did not understand or support, even sending her to see a psychiatrist for "extravagance, disorientation, and a bit of madness," according to her friend Claudia Gomez Wulschner. When asked how the story ends, Salicrup López states that she married him. She married the psychiatrist Armando Hinojosa Cavazos. They had three children: Ariel who pursued music, David who became an architect like his mother; and Mariana who studied ballet. Salicrup Lopez had many interests and passions, especially for music and art. She loved the opera and visiting art exhibits. She also enjoyed literature and history. == Education == After completing secondary school, Salicrup Lopez enrolled at the Escuela Nacional Preparatoria where she studied mathematics. Salicrup López attended the National Autonomous University of Mexico (UNAM) to study architecture and the German language. In 1959, Salicrup López graduated from the National Autonomous University of Mexico (UNAM) with a degree in architecture. Ten years later, at the age of 34, she earned her master's degree from the National Autonomous University of Mexico (UNAM). == Career == After graduating from the National Autonomous University of Mexico (UNAM) in 1959, Salicrup Lopez worked with the anthropologist Laurette Séjourné on the restoration of Teotihuacan. She worked in the archaeological zone doing surveys and plans as well as directing excavations in important parts of this zone. Salicrup Lopez still wanted to become a mathematician, and finally enrolled in the Faculty of Sciences in 1964 to study mathematics. Between 1966 and 1968 she taught mathematics at the UNAM Faculty of Architecture. Her thesis, accepted in 1969, was on the Jiang Boju subgroup. After graduating in 1969 Graciela began teaching in the UNAM Faculty of Sciences. In 1970 she was given a position as a researcher in the UNAM Mathematics Institute, where she worked with Dr. Roberto Vázquez, her mentor. That same year she published her first work along with her mentor. Her work was concerned with the structure of the Top category of topological spaces and with continuous functions. Her work related concepts such as reflexivity or coreflexivity to those of connection and coexistence, both in Top and in certain subcategories of Top (and in some more general concrete categories). The publications she co-authored with Vázquez were always in Spanish, so many mathematicians were not aware of her work. During this time, there was a group of important topologists including Horst Herrlich; hence, Salicrup and some fellow mathematicians arranged to take German lessons. She was elected to the Sociedad Matematica Mexicana with reciprocity to the American Mathematical Society in 1973. == Legacy == Shortly before her death, Graciela fell out with her mentor Roberto Vázquez and they stopped collaborating. In the summer of 1982, she was visited by Lamar Bentley and Horst Herrlich, with whom she planned to collaborate. Soon after this Graciela suffered a fall that hurt her badly. She did not recover and died on July 29, 1982. The main hall of the UNAM Institute of Mathematics is named after her. Her research in categorical topology was published in 1986 by Horst Herrlich. == Selected publications == === Architecture === === Mathematics === == References == === Works cited ===
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Wikipedia:Grad operator#0
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In vector calculus, the gradient of a scalar-valued differentiable function f {\displaystyle f} of several variables is the vector field (or vector-valued function) ∇ f {\displaystyle \nabla f} whose value at a point p {\displaystyle p} gives the direction and the rate of fastest increase. The gradient transforms like a vector under change of basis of the space of variables of f {\displaystyle f} . If the gradient of a function is non-zero at a point p {\displaystyle p} , the direction of the gradient is the direction in which the function increases most quickly from p {\displaystyle p} , and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative. Further, a point where the gradient is the zero vector is known as a stationary point. The gradient thus plays a fundamental role in optimization theory, where it is used to minimize a function by gradient descent. In coordinate-free terms, the gradient of a function f ( r ) {\displaystyle f(\mathbf {r} )} may be defined by: d f = ∇ f ⋅ d r {\displaystyle df=\nabla f\cdot d\mathbf {r} } where d f {\displaystyle df} is the total infinitesimal change in f {\displaystyle f} for an infinitesimal displacement d r {\displaystyle d\mathbf {r} } , and is seen to be maximal when d r {\displaystyle d\mathbf {r} } is in the direction of the gradient ∇ f {\displaystyle \nabla f} . The nabla symbol ∇ {\displaystyle \nabla } , written as an upside-down triangle and pronounced "del", denotes the vector differential operator. When a coordinate system is used in which the basis vectors are not functions of position, the gradient is given by the vector whose components are the partial derivatives of f {\displaystyle f} at p {\displaystyle p} . That is, for f : R n → R {\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} } , its gradient ∇ f : R n → R n {\displaystyle \nabla f\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}} is defined at the point p = ( x 1 , … , x n ) {\displaystyle p=(x_{1},\ldots ,x_{n})} in n-dimensional space as the vector ∇ f ( p ) = [ ∂ f ∂ x 1 ( p ) ⋮ ∂ f ∂ x n ( p ) ] . {\displaystyle \nabla f(p)={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(p)\\\vdots \\{\frac {\partial f}{\partial x_{n}}}(p)\end{bmatrix}}.} Note that the above definition for gradient is defined for the function f {\displaystyle f} only if f {\displaystyle f} is differentiable at p {\displaystyle p} . There can be functions for which partial derivatives exist in every direction but fail to be differentiable. Furthermore, this definition as the vector of partial derivatives is only valid when the basis of the coordinate system is orthonormal. For any other basis, the metric tensor at that point needs to be taken into account. For example, the function f ( x , y ) = x 2 y x 2 + y 2 {\displaystyle f(x,y)={\frac {x^{2}y}{x^{2}+y^{2}}}} unless at origin where f ( 0 , 0 ) = 0 {\displaystyle f(0,0)=0} , is not differentiable at the origin as it does not have a well defined tangent plane despite having well defined partial derivatives in every direction at the origin. In this particular example, under rotation of x-y coordinate system, the above formula for gradient fails to transform like a vector (gradient becomes dependent on choice of basis for coordinate system) and also fails to point towards the 'steepest ascent' in some orientations. For differentiable functions where the formula for gradient holds, it can be shown to always transform as a vector under transformation of the basis so as to always point towards the fastest increase. The gradient is dual to the total derivative d f {\displaystyle df} : the value of the gradient at a point is a tangent vector – a vector at each point; while the value of the derivative at a point is a cotangent vector – a linear functional on vectors. They are related in that the dot product of the gradient of f {\displaystyle f} at a point p {\displaystyle p} with another tangent vector v {\displaystyle \mathbf {v} } equals the directional derivative of f {\displaystyle f} at p {\displaystyle p} of the function along v {\displaystyle \mathbf {v} } ; that is, ∇ f ( p ) ⋅ v = ∂ f ∂ v ( p ) = d f p ( v ) {\textstyle \nabla f(p)\cdot \mathbf {v} ={\frac {\partial f}{\partial \mathbf {v} }}(p)=df_{p}(\mathbf {v} )} . The gradient admits multiple generalizations to more general functions on manifolds; see § Generalizations. == Motivation == Consider a room where the temperature is given by a scalar field, T, so at each point (x, y, z) the temperature is T(x, y, z), independent of time. At each point in the room, the gradient of T at that point will show the direction in which the temperature rises most quickly, moving away from (x, y, z). The magnitude of the gradient will determine how fast the temperature rises in that direction. Consider a surface whose height above sea level at point (x, y) is H(x, y). The gradient of H at a point is a plane vector pointing in the direction of the steepest slope or grade at that point. The steepness of the slope at that point is given by the magnitude of the gradient vector. The gradient can also be used to measure how a scalar field changes in other directions, rather than just the direction of greatest change, by taking a dot product. Suppose that the steepest slope on a hill is 40%. A road going directly uphill has slope 40%, but a road going around the hill at an angle will have a shallower slope. For example, if the road is at a 60° angle from the uphill direction (when both directions are projected onto the horizontal plane), then the slope along the road will be the dot product between the gradient vector and a unit vector along the road, as the dot product measures how much the unit vector along the road aligns with the steepest slope, which is 40% times the cosine of 60°, or 20%. More generally, if the hill height function H is differentiable, then the gradient of H dotted with a unit vector gives the slope of the hill in the direction of the vector, the directional derivative of H along the unit vector. == Notation == The gradient of a function f {\displaystyle f} at point a {\displaystyle a} is usually written as ∇ f ( a ) {\displaystyle \nabla f(a)} . It may also be denoted by any of the following: ∇ → f ( a ) {\displaystyle {\vec {\nabla }}f(a)} : to emphasize the vector nature of the result. grad f {\displaystyle \operatorname {grad} f} ∂ i f {\displaystyle \partial _{i}f} and f i {\displaystyle f_{i}} : Written with Einstein notation, where repeated indices (i) are summed over. == Definition == The gradient (or gradient vector field) of a scalar function f(x1, x2, x3, …, xn) is denoted ∇f or ∇→f where ∇ (nabla) denotes the vector differential operator, del. The notation grad f is also commonly used to represent the gradient. The gradient of f is defined as the unique vector field whose dot product with any vector v at each point x is the directional derivative of f along v. That is, ( ∇ f ( x ) ) ⋅ v = D v f ( x ) {\displaystyle {\big (}\nabla f(x){\big )}\cdot \mathbf {v} =D_{\mathbf {v} }f(x)} where the right-hand side is the directional derivative and there are many ways to represent it. Formally, the derivative is dual to the gradient; see relationship with derivative. When a function also depends on a parameter such as time, the gradient often refers simply to the vector of its spatial derivatives only (see Spatial gradient). The magnitude and direction of the gradient vector are independent of the particular coordinate representation. === Cartesian coordinates === In the three-dimensional Cartesian coordinate system with a Euclidean metric, the gradient, if it exists, is given by ∇ f = ∂ f ∂ x i + ∂ f ∂ y j + ∂ f ∂ z k , {\displaystyle \nabla f={\frac {\partial f}{\partial x}}\mathbf {i} +{\frac {\partial f}{\partial y}}\mathbf {j} +{\frac {\partial f}{\partial z}}\mathbf {k} ,} where i, j, k are the standard unit vectors in the directions of the x, y and z coordinates, respectively. For example, the gradient of the function f ( x , y , z ) = 2 x + 3 y 2 − sin ( z ) {\displaystyle f(x,y,z)=2x+3y^{2}-\sin(z)} is ∇ f ( x , y , z ) = 2 i + 6 y j − cos ( z ) k . {\displaystyle \nabla f(x,y,z)=2\mathbf {i} +6y\mathbf {j} -\cos(z)\mathbf {k} .} or ∇ f ( x , y , z ) = [ 2 6 y − cos z ] . {\displaystyle \nabla f(x,y,z)={\begin{bmatrix}2\\6y\\-\cos z\end{bmatrix}}.} In some applications it is customary to represent the gradient as a row vector or column vector of its components in a rectangular coordinate system; this article follows the convention of the gradient being a column vector, while the derivative is a row vector. === Cylindrical and spherical coordinates === In cylindrical coordinates, the gradient is given by: ∇ f ( ρ , φ , z ) = ∂ f ∂ ρ e ρ + 1 ρ ∂ f ∂ φ e φ + ∂ f ∂ z e z , {\displaystyle \nabla f(\rho ,\varphi ,z)={\frac {\partial f}{\partial \rho }}\mathbf {e} _{\rho }+{\frac {1}{\rho }}{\frac {\partial f}{\partial \varphi }}\mathbf {e} _{\varphi }+{\frac {\partial f}{\partial z}}\mathbf {e} _{z},} where ρ is the axial distance, φ is the azimuthal or azimuth angle, z is the axial coordinate, and eρ, eφ and ez are unit vectors pointing along the coordinate directions. In spherical coordinates with a Euclidean metric, the gradient is given by: ∇ f ( r , θ , φ ) = ∂ f ∂ r e r + 1 r ∂ f ∂ θ e θ + 1 r sin θ ∂ f ∂ φ e φ , {\displaystyle \nabla f(r,\theta ,\varphi )={\frac {\partial f}{\partial r}}\mathbf {e} _{r}+{\frac {1}{r}}{\frac {\partial f}{\partial \theta }}\mathbf {e} _{\theta }+{\frac {1}{r\sin \theta }}{\frac {\partial f}{\partial \varphi }}\mathbf {e} _{\varphi },} where r is the radial distance, φ is the azimuthal angle and θ is the polar angle, and er, eθ and eφ are again local unit vectors pointing in the coordinate directions (that is, the normalized covariant basis). For the gradient in other orthogonal coordinate systems, see Orthogonal coordinates (Differential operators in three dimensions). === General coordinates === We consider general coordinates, which we write as x1, …, xi, …, xn, where n is the number of dimensions of the domain. Here, the upper index refers to the position in the list of the coordinate or component, so x2 refers to the second component—not the quantity x squared. The index variable i refers to an arbitrary element xi. Using Einstein notation, the gradient can then be written as: ∇ f = ∂ f ∂ x i g i j e j {\displaystyle \nabla f={\frac {\partial f}{\partial x^{i}}}g^{ij}\mathbf {e} _{j}} (Note that its dual is d f = ∂ f ∂ x i e i {\textstyle \mathrm {d} f={\frac {\partial f}{\partial x^{i}}}\mathbf {e} ^{i}} ), where e i = d x i {\displaystyle \mathbf {e} ^{i}=\mathrm {d} x^{i}} and e i = ∂ x / ∂ x i {\displaystyle \mathbf {e} _{i}=\partial \mathbf {x} /\partial x^{i}} refer to the unnormalized local covariant and contravariant bases respectively, g i j {\displaystyle g^{ij}} is the inverse metric tensor, and the Einstein summation convention implies summation over i and j. If the coordinates are orthogonal we can easily express the gradient (and the differential) in terms of the normalized bases, which we refer to as e ^ i {\displaystyle {\hat {\mathbf {e} }}_{i}} and e ^ i {\displaystyle {\hat {\mathbf {e} }}^{i}} , using the scale factors (also known as Lamé coefficients) h i = ‖ e i ‖ = g i i = 1 / ‖ e i ‖ {\displaystyle h_{i}=\lVert \mathbf {e} _{i}\rVert ={\sqrt {g_{ii}}}=1\,/\lVert \mathbf {e} ^{i}\rVert } : ∇ f = ∂ f ∂ x i g i j e ^ j g j j = ∑ i = 1 n ∂ f ∂ x i 1 h i e ^ i {\displaystyle \nabla f={\frac {\partial f}{\partial x^{i}}}g^{ij}{\hat {\mathbf {e} }}_{j}{\sqrt {g_{jj}}}=\sum _{i=1}^{n}\,{\frac {\partial f}{\partial x^{i}}}{\frac {1}{h_{i}}}\mathbf {\hat {e}} _{i}} (and d f = ∑ i = 1 n ∂ f ∂ x i 1 h i e ^ i {\textstyle \mathrm {d} f=\sum _{i=1}^{n}\,{\frac {\partial f}{\partial x^{i}}}{\frac {1}{h_{i}}}\mathbf {\hat {e}} ^{i}} ), where we cannot use Einstein notation, since it is impossible to avoid the repetition of more than two indices. Despite the use of upper and lower indices, e ^ i {\displaystyle \mathbf {\hat {e}} _{i}} , e ^ i {\displaystyle \mathbf {\hat {e}} ^{i}} , and h i {\displaystyle h_{i}} are neither contravariant nor covariant. The latter expression evaluates to the expressions given above for cylindrical and spherical coordinates. == Relationship with derivative == === Relationship with total derivative === The gradient is closely related to the total derivative (total differential) d f {\displaystyle df} : they are transpose (dual) to each other. Using the convention that vectors in R n {\displaystyle \mathbb {R} ^{n}} are represented by column vectors, and that covectors (linear maps R n → R {\displaystyle \mathbb {R} ^{n}\to \mathbb {R} } ) are represented by row vectors, the gradient ∇ f {\displaystyle \nabla f} and the derivative d f {\displaystyle df} are expressed as a column and row vector, respectively, with the same components, but transpose of each other: ∇ f ( p ) = [ ∂ f ∂ x 1 ( p ) ⋮ ∂ f ∂ x n ( p ) ] ; {\displaystyle \nabla f(p)={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(p)\\\vdots \\{\frac {\partial f}{\partial x_{n}}}(p)\end{bmatrix}};} d f p = [ ∂ f ∂ x 1 ( p ) ⋯ ∂ f ∂ x n ( p ) ] . {\displaystyle df_{p}={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(p)&\cdots &{\frac {\partial f}{\partial x_{n}}}(p)\end{bmatrix}}.} While these both have the same components, they differ in what kind of mathematical object they represent: at each point, the derivative is a cotangent vector, a linear form (or covector) which expresses how much the (scalar) output changes for a given infinitesimal change in (vector) input, while at each point, the gradient is a tangent vector, which represents an infinitesimal change in (vector) input. In symbols, the gradient is an element of the tangent space at a point, ∇ f ( p ) ∈ T p R n {\displaystyle \nabla f(p)\in T_{p}\mathbb {R} ^{n}} , while the derivative is a map from the tangent space to the real numbers, d f p : T p R n → R {\displaystyle df_{p}\colon T_{p}\mathbb {R} ^{n}\to \mathbb {R} } . The tangent spaces at each point of R n {\displaystyle \mathbb {R} ^{n}} can be "naturally" identified with the vector space R n {\displaystyle \mathbb {R} ^{n}} itself, and similarly the cotangent space at each point can be naturally identified with the dual vector space ( R n ) ∗ {\displaystyle (\mathbb {R} ^{n})^{*}} of covectors; thus the value of the gradient at a point can be thought of a vector in the original R n {\displaystyle \mathbb {R} ^{n}} , not just as a tangent vector. Computationally, given a tangent vector, the vector can be multiplied by the derivative (as matrices), which is equal to taking the dot product with the gradient: ( d f p ) ( v ) = [ ∂ f ∂ x 1 ( p ) ⋯ ∂ f ∂ x n ( p ) ] [ v 1 ⋮ v n ] = ∑ i = 1 n ∂ f ∂ x i ( p ) v i = [ ∂ f ∂ x 1 ( p ) ⋮ ∂ f ∂ x n ( p ) ] ⋅ [ v 1 ⋮ v n ] = ∇ f ( p ) ⋅ v {\displaystyle (df_{p})(v)={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(p)&\cdots &{\frac {\partial f}{\partial x_{n}}}(p)\end{bmatrix}}{\begin{bmatrix}v_{1}\\\vdots \\v_{n}\end{bmatrix}}=\sum _{i=1}^{n}{\frac {\partial f}{\partial x_{i}}}(p)v_{i}={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(p)\\\vdots \\{\frac {\partial f}{\partial x_{n}}}(p)\end{bmatrix}}\cdot {\begin{bmatrix}v_{1}\\\vdots \\v_{n}\end{bmatrix}}=\nabla f(p)\cdot v} ==== Differential or (exterior) derivative ==== The best linear approximation to a differentiable function f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } at a point x {\displaystyle x} in R n {\displaystyle \mathbb {R} ^{n}} is a linear map from R n {\displaystyle \mathbb {R} ^{n}} to R {\displaystyle \mathbb {R} } which is often denoted by d f x {\displaystyle df_{x}} or D f ( x ) {\displaystyle Df(x)} and called the differential or total derivative of f {\displaystyle f} at x {\displaystyle x} . The function d f {\displaystyle df} , which maps x {\displaystyle x} to d f x {\displaystyle df_{x}} , is called the total differential or exterior derivative of f {\displaystyle f} and is an example of a differential 1-form. Much as the derivative of a function of a single variable represents the slope of the tangent to the graph of the function, the directional derivative of a function in several variables represents the slope of the tangent hyperplane in the direction of the vector. The gradient is related to the differential by the formula ( ∇ f ) x ⋅ v = d f x ( v ) {\displaystyle (\nabla f)_{x}\cdot v=df_{x}(v)} for any v ∈ R n {\displaystyle v\in \mathbb {R} ^{n}} , where ⋅ {\displaystyle \cdot } is the dot product: taking the dot product of a vector with the gradient is the same as taking the directional derivative along the vector. If R n {\displaystyle \mathbb {R} ^{n}} is viewed as the space of (dimension n {\displaystyle n} ) column vectors (of real numbers), then one can regard d f {\displaystyle df} as the row vector with components ( ∂ f ∂ x 1 , … , ∂ f ∂ x n ) , {\displaystyle \left({\frac {\partial f}{\partial x_{1}}},\dots ,{\frac {\partial f}{\partial x_{n}}}\right),} so that d f x ( v ) {\displaystyle df_{x}(v)} is given by matrix multiplication. Assuming the standard Euclidean metric on R n {\displaystyle \mathbb {R} ^{n}} , the gradient is then the corresponding column vector, that is, ( ∇ f ) i = d f i T . {\displaystyle (\nabla f)_{i}=df_{i}^{\mathsf {T}}.} ==== Linear approximation to a function ==== The best linear approximation to a function can be expressed in terms of the gradient, rather than the derivative. The gradient of a function f {\displaystyle f} from the Euclidean space R n {\displaystyle \mathbb {R} ^{n}} to R {\displaystyle \mathbb {R} } at any particular point x 0 {\displaystyle x_{0}} in R n {\displaystyle \mathbb {R} ^{n}} characterizes the best linear approximation to f {\displaystyle f} at x 0 {\displaystyle x_{0}} . The approximation is as follows: f ( x ) ≈ f ( x 0 ) + ( ∇ f ) x 0 ⋅ ( x − x 0 ) {\displaystyle f(x)\approx f(x_{0})+(\nabla f)_{x_{0}}\cdot (x-x_{0})} for x {\displaystyle x} close to x 0 {\displaystyle x_{0}} , where ( ∇ f ) x 0 {\displaystyle (\nabla f)_{x_{0}}} is the gradient of f {\displaystyle f} computed at x 0 {\displaystyle x_{0}} , and the dot denotes the dot product on R n {\displaystyle \mathbb {R} ^{n}} . This equation is equivalent to the first two terms in the multivariable Taylor series expansion of f {\displaystyle f} at x 0 {\displaystyle x_{0}} . === Relationship with Fréchet derivative === Let U be an open set in Rn. If the function f : U → R is differentiable, then the differential of f is the Fréchet derivative of f. Thus ∇f is a function from U to the space Rn such that lim h → 0 | f ( x + h ) − f ( x ) − ∇ f ( x ) ⋅ h | ‖ h ‖ = 0 , {\displaystyle \lim _{h\to 0}{\frac {|f(x+h)-f(x)-\nabla f(x)\cdot h|}{\|h\|}}=0,} where · is the dot product. As a consequence, the usual properties of the derivative hold for the gradient, though the gradient is not a derivative itself, but rather dual to the derivative: Linearity The gradient is linear in the sense that if f and g are two real-valued functions differentiable at the point a ∈ Rn, and α and β are two constants, then αf + βg is differentiable at a, and moreover ∇ ( α f + β g ) ( a ) = α ∇ f ( a ) + β ∇ g ( a ) . {\displaystyle \nabla \left(\alpha f+\beta g\right)(a)=\alpha \nabla f(a)+\beta \nabla g(a).} Product rule If f and g are real-valued functions differentiable at a point a ∈ Rn, then the product rule asserts that the product fg is differentiable at a, and ∇ ( f g ) ( a ) = f ( a ) ∇ g ( a ) + g ( a ) ∇ f ( a ) . {\displaystyle \nabla (fg)(a)=f(a)\nabla g(a)+g(a)\nabla f(a).} Chain rule Suppose that f : A → R is a real-valued function defined on a subset A of Rn, and that f is differentiable at a point a. There are two forms of the chain rule applying to the gradient. First, suppose that the function g is a parametric curve; that is, a function g : I → Rn maps a subset I ⊂ R into Rn. If g is differentiable at a point c ∈ I such that g(c) = a, then ( f ∘ g ) ′ ( c ) = ∇ f ( a ) ⋅ g ′ ( c ) , {\displaystyle (f\circ g)'(c)=\nabla f(a)\cdot g'(c),} where ∘ is the composition operator: (f ∘ g)(x) = f(g(x)). More generally, if instead I ⊂ Rk, then the following holds: ∇ ( f ∘ g ) ( c ) = ( D g ( c ) ) T ( ∇ f ( a ) ) , {\displaystyle \nabla (f\circ g)(c)={\big (}Dg(c){\big )}^{\mathsf {T}}{\big (}\nabla f(a){\big )},} where (Dg)T denotes the transpose Jacobian matrix. For the second form of the chain rule, suppose that h : I → R is a real valued function on a subset I of R, and that h is differentiable at the point f(a) ∈ I. Then ∇ ( h ∘ f ) ( a ) = h ′ ( f ( a ) ) ∇ f ( a ) . {\displaystyle \nabla (h\circ f)(a)=h'{\big (}f(a){\big )}\nabla f(a).} == Further properties and applications == === Level sets === A level surface, or isosurface, is the set of all points where some function has a given value. If f is differentiable, then the dot product (∇f )x ⋅ v of the gradient at a point x with a vector v gives the directional derivative of f at x in the direction v. It follows that in this case the gradient of f is orthogonal to the level sets of f. For example, a level surface in three-dimensional space is defined by an equation of the form F(x, y, z) = c. The gradient of F is then normal to the surface. More generally, any embedded hypersurface in a Riemannian manifold can be cut out by an equation of the form F(P) = 0 such that dF is nowhere zero. The gradient of F is then normal to the hypersurface. Similarly, an affine algebraic hypersurface may be defined by an equation F(x1, ..., xn) = 0, where F is a polynomial. The gradient of F is zero at a singular point of the hypersurface (this is the definition of a singular point). At a non-singular point, it is a nonzero normal vector. === Conservative vector fields and the gradient theorem === The gradient of a function is called a gradient field. A (continuous) gradient field is always a conservative vector field: its line integral along any path depends only on the endpoints of the path, and can be evaluated by the gradient theorem (the fundamental theorem of calculus for line integrals). Conversely, a (continuous) conservative vector field is always the gradient of a function. === Gradient is direction of steepest ascent === The gradient of a function f : R n → R {\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} } at point x is also the direction of its steepest ascent, i.e. it maximizes its directional derivative: Let v ∈ R n {\displaystyle v\in \mathbb {R} ^{n}} be an arbitrary unit vector. With the directional derivative defined as ∇ v f ( x ) = lim h → 0 f ( x + v h ) − f ( x ) h , {\displaystyle \nabla _{v}f(x)=\lim _{h\rightarrow 0}{\frac {f(x+vh)-f(x)}{h}},} we get, by substituting the function f ( x + v h ) {\displaystyle f(x+vh)} with its Taylor series, ∇ v f ( x ) = lim h → 0 ( f ( x ) + ∇ f ⋅ v h + R ) − f ( x ) h , {\displaystyle \nabla _{v}f(x)=\lim _{h\rightarrow 0}{\frac {(f(x)+\nabla f\cdot vh+R)-f(x)}{h}},} where R {\displaystyle R} denotes higher order terms in v h {\displaystyle vh} . Dividing by h {\displaystyle h} , and taking the limit yields a term which is bounded from above by the Cauchy-Schwarz inequality | ∇ v f ( x ) | = | ∇ f ⋅ v | ≤ | ∇ f | | v | = | ∇ f | . {\displaystyle |\nabla _{v}f(x)|=|\nabla f\cdot v|\leq |\nabla f||v|=|\nabla f|.} Choosing v ∗ = ∇ f / | ∇ f | {\displaystyle v^{*}=\nabla f/|\nabla f|} maximizes the directional derivative, and equals the upper bound | ∇ v ∗ f ( x ) | = | ( ∇ f ) 2 / | ∇ f | | = | ∇ f | . {\displaystyle |\nabla _{v^{*}}f(x)|=|(\nabla f)^{2}/|\nabla f||=|\nabla f|.} == Generalizations == === Jacobian === The Jacobian matrix is the generalization of the gradient for vector-valued functions of several variables and differentiable maps between Euclidean spaces or, more generally, manifolds. A further generalization for a function between Banach spaces is the Fréchet derivative. Suppose f : Rn → Rm is a function such that each of its first-order partial derivatives exist on ℝn. Then the Jacobian matrix of f is defined to be an m×n matrix, denoted by J f ( x ) {\displaystyle \mathbf {J} _{\mathbb {f} }(\mathbb {x} )} or simply J {\displaystyle \mathbf {J} } . The (i,j)th entry is J i j = ∂ f i / ∂ x j {\textstyle \mathbf {J} _{ij}={\partial f_{i}}/{\partial x_{j}}} . Explicitly J = [ ∂ f ∂ x 1 ⋯ ∂ f ∂ x n ] = [ ∇ T f 1 ⋮ ∇ T f m ] = [ ∂ f 1 ∂ x 1 ⋯ ∂ f 1 ∂ x n ⋮ ⋱ ⋮ ∂ f m ∂ x 1 ⋯ ∂ f m ∂ x n ] . {\displaystyle \mathbf {J} ={\begin{bmatrix}{\dfrac {\partial \mathbf {f} }{\partial x_{1}}}&\cdots &{\dfrac {\partial \mathbf {f} }{\partial x_{n}}}\end{bmatrix}}={\begin{bmatrix}\nabla ^{\mathsf {T}}f_{1}\\\vdots \\\nabla ^{\mathsf {T}}f_{m}\end{bmatrix}}={\begin{bmatrix}{\dfrac {\partial f_{1}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\dfrac {\partial f_{m}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{m}}{\partial x_{n}}}\end{bmatrix}}.} === Gradient of a vector field === Since the total derivative of a vector field is a linear mapping from vectors to vectors, it is a tensor quantity. In rectangular coordinates, the gradient of a vector field f = ( f1, f2, f3) is defined by: ∇ f = g j k ∂ f i ∂ x j e i ⊗ e k , {\displaystyle \nabla \mathbf {f} =g^{jk}{\frac {\partial f^{i}}{\partial x^{j}}}\mathbf {e} _{i}\otimes \mathbf {e} _{k},} (where the Einstein summation notation is used and the tensor product of the vectors ei and ek is a dyadic tensor of type (2,0)). Overall, this expression equals the transpose of the Jacobian matrix: ∂ f i ∂ x j = ∂ ( f 1 , f 2 , f 3 ) ∂ ( x 1 , x 2 , x 3 ) . {\displaystyle {\frac {\partial f^{i}}{\partial x^{j}}}={\frac {\partial (f^{1},f^{2},f^{3})}{\partial (x^{1},x^{2},x^{3})}}.} In curvilinear coordinates, or more generally on a curved manifold, the gradient involves Christoffel symbols: ∇ f = g j k ( ∂ f i ∂ x j + Γ i j l f l ) e i ⊗ e k , {\displaystyle \nabla \mathbf {f} =g^{jk}\left({\frac {\partial f^{i}}{\partial x^{j}}}+{\Gamma ^{i}}_{jl}f^{l}\right)\mathbf {e} _{i}\otimes \mathbf {e} _{k},} where gjk are the components of the inverse metric tensor and the ei are the coordinate basis vectors. Expressed more invariantly, the gradient of a vector field f can be defined by the Levi-Civita connection and metric tensor: ∇ a f b = g a c ∇ c f b , {\displaystyle \nabla ^{a}f^{b}=g^{ac}\nabla _{c}f^{b},} where ∇c is the connection. === Riemannian manifolds === For any smooth function f on a Riemannian manifold (M, g), the gradient of f is the vector field ∇f such that for any vector field X, g ( ∇ f , X ) = ∂ X f , {\displaystyle g(\nabla f,X)=\partial _{X}f,} that is, g x ( ( ∇ f ) x , X x ) = ( ∂ X f ) ( x ) , {\displaystyle g_{x}{\big (}(\nabla f)_{x},X_{x}{\big )}=(\partial _{X}f)(x),} where gx( , ) denotes the inner product of tangent vectors at x defined by the metric g and ∂X f is the function that takes any point x ∈ M to the directional derivative of f in the direction X, evaluated at x. In other words, in a coordinate chart φ from an open subset of M to an open subset of Rn, (∂X f )(x) is given by: ∑ j = 1 n X j ( φ ( x ) ) ∂ ∂ x j ( f ∘ φ − 1 ) | φ ( x ) , {\displaystyle \sum _{j=1}^{n}X^{j}{\big (}\varphi (x){\big )}{\frac {\partial }{\partial x_{j}}}(f\circ \varphi ^{-1}){\Bigg |}_{\varphi (x)},} where Xj denotes the jth component of X in this coordinate chart. So, the local form of the gradient takes the form: ∇ f = g i k ∂ f ∂ x k e i . {\displaystyle \nabla f=g^{ik}{\frac {\partial f}{\partial x^{k}}}{\textbf {e}}_{i}.} Generalizing the case M = Rn, the gradient of a function is related to its exterior derivative, since ( ∂ X f ) ( x ) = ( d f ) x ( X x ) . {\displaystyle (\partial _{X}f)(x)=(df)_{x}(X_{x}).} More precisely, the gradient ∇f is the vector field associated to the differential 1-form df using the musical isomorphism ♯ = ♯ g : T ∗ M → T M {\displaystyle \sharp =\sharp ^{g}\colon T^{*}M\to TM} (called "sharp") defined by the metric g. The relation between the exterior derivative and the gradient of a function on Rn is a special case of this in which the metric is the flat metric given by the dot product. == See also == Curl – Circulation density in a vector field Divergence – Vector operator in vector calculus Four-gradient – Four-vector analogue of the gradient operation Hessian matrix – Matrix of second derivatives Skew gradient Spatial gradient – Gradient whose components are spatial derivatives == Notes == == References == Bachman, David (2007), Advanced Calculus Demystified, New York: McGraw-Hill, ISBN 978-0-07-148121-2 Beauregard, Raymond A.; Fraleigh, John B. (1973), A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields, Boston: Houghton Mifflin Company, ISBN 0-395-14017-X Downing, Douglas, Ph.D. (2010), Barron's E-Z Calculus, New York: Barron's, ISBN 978-0-7641-4461-5{{citation}}: CS1 maint: multiple names: authors list (link) Dubrovin, B. A.; Fomenko, A. T.; Novikov, S. P. (1991). Modern Geometry—Methods and Applications: Part I: The Geometry of Surfaces, Transformation Groups, and Fields. Graduate Texts in Mathematics (2nd ed.). Springer. ISBN 978-0-387-97663-1. Harper, Charlie (1976), Introduction to Mathematical Physics, New Jersey: Prentice-Hall, ISBN 0-13-487538-9 Kreyszig, Erwin (1972), Advanced Engineering Mathematics (3rd ed.), New York: Wiley, ISBN 0-471-50728-8 "McGraw Hill Encyclopedia of Science & Technology". McGraw-Hill Encyclopedia of Science & Technology (10th ed.). New York: McGraw-Hill. 2007. ISBN 978-0-07-144143-8. Moise, Edwin E. (1967), Calculus: Complete, Reading: Addison-Wesley Protter, Murray H.; Morrey, Charles B. Jr. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading: Addison-Wesley, LCCN 76087042 Schey, H. M. (1992). Div, Grad, Curl, and All That (2nd ed.). W. W. Norton. ISBN 0-393-96251-2. OCLC 25048561. Stoker, J. J. (1969), Differential Geometry, New York: Wiley, ISBN 0-471-82825-4 Swokowski, Earl W.; Olinick, Michael; Pence, Dennis; Cole, Jeffery A. (1994), Calculus (6th ed.), Boston: PWS Publishing Company, ISBN 0-534-93624-5 Arens, T.; Hettlich, F.; Karpfinger, C.; Kockelkorn, U.; Lichtenegger, K.; Stachel, H. (2022), Mathematik (5th ed.), Springer Spektrum Berlin, doi:10.1007/978-3-662-64389-1, ISBN 978-3-662-64388-4 == Further reading == Korn, Theresa M.; Korn, Granino Arthur (2000). Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. Dover Publications. pp. 157–160. ISBN 0-486-41147-8. OCLC 43864234. == External links == "Gradient". Khan Academy. Kuptsov, L.P. (2001) [1994], "Gradient", Encyclopedia of Mathematics, EMS Press. Weisstein, Eric W. "Gradient". MathWorld.
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Wikipedia:Graded structure#0
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In mathematics, the term "graded" has a number of meanings, mostly related: In abstract algebra, it refers to a family of concepts: An algebraic structure X {\displaystyle X} is said to be I {\displaystyle I} -graded for an index set I {\displaystyle I} if it has a gradation or grading, i.e. a decomposition into a direct sum X = ⨁ i ∈ I X i {\textstyle X=\bigoplus _{i\in I}X_{i}} of structures; the elements of X i {\displaystyle X_{i}} are said to be "homogeneous of degree i". The index set I {\displaystyle I} is most commonly N {\displaystyle \mathbb {N} } or Z {\displaystyle \mathbb {Z} } , and may be required to have extra structure depending on the type of X {\displaystyle X} . Grading by Z 2 {\displaystyle \mathbb {Z} _{2}} (i.e. Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } ) is also important; see e.g. signed set (the Z 2 {\displaystyle \mathbb {Z} _{2}} -graded sets). The trivial ( Z {\displaystyle \mathbb {Z} } - or N {\displaystyle \mathbb {N} } -) gradation has X 0 = X , X i = 0 {\displaystyle X_{0}=X,X_{i}=0} for i ≠ 0 {\displaystyle i\neq 0} and a suitable trivial structure 0 {\displaystyle 0} . An algebraic structure is said to be doubly graded if the index set is a direct product of sets; the pairs may be called "bidegrees" (e.g. see Spectral sequence). A I {\displaystyle I} -graded vector space or graded linear space is thus a vector space with a decomposition into a direct sum V = ⨁ i ∈ I V i {\textstyle V=\bigoplus _{i\in I}V_{i}} of spaces. A graded linear map is a map between graded vector spaces respecting their gradations. A graded ring is a ring that is a direct sum of additive abelian groups R i {\displaystyle R_{i}} such that R i R j ⊆ R i + j {\displaystyle R_{i}R_{j}\subseteq R_{i+j}} , with i {\displaystyle i} taken from some monoid, usually N {\displaystyle \mathbb {N} } or Z {\displaystyle \mathbb {Z} } , or semigroup (for a ring without identity). The associated graded ring of a commutative ring R {\displaystyle R} with respect to a proper ideal I {\displaystyle I} is gr I R = ⨁ n ∈ N I n / I n + 1 {\textstyle \operatorname {gr} _{I}R=\bigoplus _{n\in \mathbb {N} }I^{n}/I^{n+1}} . A graded module is left module M {\displaystyle M} over a graded ring that is a direct sum ⨁ i ∈ I M i {\textstyle \bigoplus _{i\in I}M_{i}} of modules satisfying R i M j ⊆ M i + j {\displaystyle R_{i}M_{j}\subseteq M_{i+j}} . The associated graded module of an R {\displaystyle R} -module M {\displaystyle M} with respect to a proper ideal I {\displaystyle I} is gr I M = ⨁ n ∈ N I n M / I n + 1 M {\textstyle \operatorname {gr} _{I}M=\bigoplus _{n\in \mathbb {N} }I^{n}M/I^{n+1}M} . A differential graded module, differential graded Z {\displaystyle \mathbb {Z} } -module or DG-module is a graded module M {\displaystyle M} with a differential d : M → M : M i → M i + 1 {\displaystyle d\colon M\to M\colon M_{i}\to M_{i+1}} making M {\displaystyle M} a chain complex, i.e. d ∘ d = 0 {\displaystyle d\circ d=0} . A graded algebra is an algebra A {\displaystyle A} over a ring R {\displaystyle R} that is graded as a ring; if R {\displaystyle R} is graded we also require A i R j ⊆ A i + j ⊇ R i A j {\displaystyle A_{i}R_{j}\subseteq A_{i+j}\supseteq R_{i}A_{j}} . The graded Leibniz rule for a map d : A → A {\displaystyle d\colon A\to A} on a graded algebra A {\displaystyle A} specifies that d ( a ⋅ b ) = ( d a ) ⋅ b + ( − 1 ) | a | a ⋅ ( d b ) {\displaystyle d(a\cdot b)=(da)\cdot b+(-1)^{|a|}a\cdot (db)} . A differential graded algebra, DG-algebra or DGAlgebra is a graded algebra that is a differential graded module whose differential obeys the graded Leibniz rule. A homogeneous derivation on a graded algebra A is a homogeneous linear map of grade d = |D| on A such that D ( a b ) = D ( a ) b + ε | a | | D | a D ( b ) , ε = ± 1 {\displaystyle D(ab)=D(a)b+\varepsilon ^{|a||D|}aD(b),\varepsilon =\pm 1} acting on homogeneous elements of A. A graded derivation is a sum of homogeneous derivations with the same ε {\displaystyle \varepsilon } . A DGA is an augmented DG-algebra, or differential graded augmented algebra, (see Differential graded algebra). A superalgebra is a Z 2 {\displaystyle \mathbb {Z} _{2}} -graded algebra. A graded-commutative superalgebra satisfies the "supercommutative" law y x = ( − 1 ) | x | | y | x y . {\displaystyle yx=(-1)^{|x||y|}xy.} for homogeneous x,y, where | a | {\displaystyle |a|} represents the "parity" of a {\displaystyle a} , i.e. 0 or 1 depending on the component in which it lies. CDGA may refer to the category of augmented differential graded commutative algebras. A graded Lie algebra is a Lie algebra that is graded as a vector space by a gradation compatible with its Lie bracket. A graded Lie superalgebra is a graded Lie algebra with the requirement for anticommutativity of its Lie bracket relaxed. A supergraded Lie superalgebra is a graded Lie superalgebra with an additional super Z 2 {\displaystyle \mathbb {Z} _{2}} -gradation. A differential graded Lie algebra is a graded vector space over a field of characteristic zero together with a bilinear map [ , ] : L i ⊗ L j → L i + j {\displaystyle [\ ,]\colon L_{i}\otimes L_{j}\to L_{i+j}} and a differential d : L i → L i − 1 {\displaystyle d\colon L_{i}\to L_{i-1}} satisfying [ x , y ] = ( − 1 ) | x | | y | + 1 [ y , x ] , {\displaystyle [x,y]=(-1)^{|x||y|+1}[y,x],} for any homogeneous elements x, y in L, the "graded Jacobi identity" and the graded Leibniz rule. The Graded Brauer group is a synonym for the Brauer–Wall group B W ( F ) {\displaystyle BW(F)} classifying finite-dimensional graded central division algebras over the field F. An A {\displaystyle {\mathcal {A}}} -graded category for a category A {\displaystyle {\mathcal {A}}} is a category C {\displaystyle {\mathcal {C}}} together with a functor F : C → A {\displaystyle F\colon {\mathcal {C}}\rightarrow {\mathcal {A}}} . A differential graded category or DG category is a category whose morphism sets form differential graded Z {\displaystyle \mathbb {Z} } -modules. Graded manifold – extension of the manifold concept based on ideas coming from supersymmetry and supercommutative algebra, including sections on Graded function Graded vector fields Graded exterior forms Graded differential geometry Graded differential calculus In other areas of mathematics: Functionally graded elements are used in finite element analysis. A graded poset is a poset P {\displaystyle P} with a rank function ρ : P → N {\displaystyle \rho \colon P\to \mathbb {N} } compatible with the ordering (i.e. ρ ( x ) < ρ ( y ) ⟹ x < y {\displaystyle \rho (x)<\rho (y)\implies x<y} ) such that y {\displaystyle y} covers x ⟹ ρ ( y ) = ρ ( x ) + 1 {\displaystyle x\implies \rho (y)=\rho (x)+1} .
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Wikipedia:Graded-commutative ring#0
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In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R i {\displaystyle R_{i}} such that R i R j ⊆ R i + j {\displaystyle R_{i}R_{j}\subseteq R_{i+j}} . The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid. The direct sum decomposition is usually referred to as gradation or grading. A graded module is defined similarly (see below for the precise definition). It generalizes graded vector spaces. A graded module that is also a graded ring is called a graded algebra. A graded ring could also be viewed as a graded Z {\displaystyle \mathbb {Z} } -algebra. The associativity is not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies to non-associative algebras as well; e.g., one can consider a graded Lie algebra. == First properties == Generally, the index set of a graded ring is assumed to be the set of nonnegative integers, unless otherwise explicitly specified. This is the case in this article. A graded ring is a ring that is decomposed into a direct sum R = ⨁ n = 0 ∞ R n = R 0 ⊕ R 1 ⊕ R 2 ⊕ ⋯ {\displaystyle R=\bigoplus _{n=0}^{\infty }R_{n}=R_{0}\oplus R_{1}\oplus R_{2}\oplus \cdots } of additive groups, such that R m R n ⊆ R m + n {\displaystyle R_{m}R_{n}\subseteq R_{m+n}} for all nonnegative integers m {\displaystyle m} and n {\displaystyle n} . A nonzero element of R n {\displaystyle R_{n}} is said to be homogeneous of degree n {\displaystyle n} . By definition of a direct sum, every nonzero element a {\displaystyle a} of R {\displaystyle R} can be uniquely written as a sum a = a 0 + a 1 + ⋯ + a n {\displaystyle a=a_{0}+a_{1}+\cdots +a_{n}} where each a i {\displaystyle a_{i}} is either 0 or homogeneous of degree i {\displaystyle i} . The nonzero a i {\displaystyle a_{i}} are the homogeneous components of a {\displaystyle a} . Some basic properties are: R 0 {\displaystyle R_{0}} is a subring of R {\displaystyle R} ; in particular, the multiplicative identity 1 {\displaystyle 1} is a homogeneous element of degree zero. For any n {\displaystyle n} , R n {\displaystyle R_{n}} is a two-sided R 0 {\displaystyle R_{0}} -module, and the direct sum decomposition is a direct sum of R 0 {\displaystyle R_{0}} -modules. R {\displaystyle R} is an associative R 0 {\displaystyle R_{0}} -algebra. An ideal I ⊆ R {\displaystyle I\subseteq R} is homogeneous, if for every a ∈ I {\displaystyle a\in I} , the homogeneous components of a {\displaystyle a} also belong to I {\displaystyle I} . (Equivalently, if it is a graded submodule of R {\displaystyle R} ; see § Graded module.) The intersection of a homogeneous ideal I {\displaystyle I} with R n {\displaystyle R_{n}} is an R 0 {\displaystyle R_{0}} -submodule of R n {\displaystyle R_{n}} called the homogeneous part of degree n {\displaystyle n} of I {\displaystyle I} . A homogeneous ideal is the direct sum of its homogeneous parts. If I {\displaystyle I} is a two-sided homogeneous ideal in R {\displaystyle R} , then R / I {\displaystyle R/I} is also a graded ring, decomposed as R / I = ⨁ n = 0 ∞ R n / I n , {\displaystyle R/I=\bigoplus _{n=0}^{\infty }R_{n}/I_{n},} where I n {\displaystyle I_{n}} is the homogeneous part of degree n {\displaystyle n} of I {\displaystyle I} . == Basic examples == Any (non-graded) ring R can be given a gradation by letting R 0 = R {\displaystyle R_{0}=R} , and R i = 0 {\displaystyle R_{i}=0} for i ≠ 0. This is called the trivial gradation on R. The polynomial ring R = k [ t 1 , … , t n ] {\displaystyle R=k[t_{1},\ldots ,t_{n}]} is graded by degree: it is a direct sum of R i {\displaystyle R_{i}} consisting of homogeneous polynomials of degree i. Let S be the set of all nonzero homogeneous elements in a graded integral domain R. Then the localization of R with respect to S is a Z {\displaystyle \mathbb {Z} } -graded ring. If I is an ideal in a commutative ring R, then ⨁ n = 0 ∞ I n / I n + 1 {\textstyle \bigoplus _{n=0}^{\infty }I^{n}/I^{n+1}} is a graded ring called the associated graded ring of R along I; geometrically, it is the coordinate ring of the normal cone along the subvariety defined by I. Let X be a topological space, H i(X; R) the ith cohomology group with coefficients in a ring R. Then H *(X; R), the cohomology ring of X with coefficients in R, is a graded ring whose underlying group is ⨁ i = 0 ∞ H i ( X ; R ) {\textstyle \bigoplus _{i=0}^{\infty }H^{i}(X;R)} with the multiplicative structure given by the cup product. == Graded module == The corresponding idea in module theory is that of a graded module, namely a left module M over a graded ring R such that M = ⨁ i ∈ N M i , {\displaystyle M=\bigoplus _{i\in \mathbb {N} }M_{i},} and R i M j ⊆ M i + j {\displaystyle R_{i}M_{j}\subseteq M_{i+j}} for every i and j. Examples: A graded vector space is an example of a graded module over a field (with the field having trivial grading). A graded ring is a graded module over itself. An ideal in a graded ring is homogeneous if and only if it is a graded submodule. The annihilator of a graded module is a homogeneous ideal. Given an ideal I in a commutative ring R and an R-module M, the direct sum ⨁ n = 0 ∞ I n M / I n + 1 M {\displaystyle \bigoplus _{n=0}^{\infty }I^{n}M/I^{n+1}M} is a graded module over the associated graded ring ⨁ 0 ∞ I n / I n + 1 {\textstyle \bigoplus _{0}^{\infty }I^{n}/I^{n+1}} . A morphism f : N → M {\displaystyle f:N\to M} of graded modules, called a graded morphism or graded homomorphism , is a homomorphism of the underlying modules that respects grading; i.e., f ( N i ) ⊆ M i {\displaystyle f(N_{i})\subseteq M_{i}} . A graded submodule is a submodule that is a graded module in own right and such that the set-theoretic inclusion is a morphism of graded modules. Explicitly, a graded module N is a graded submodule of M if and only if it is a submodule of M and satisfies N i = N ∩ M i {\displaystyle N_{i}=N\cap M_{i}} . The kernel and the image of a morphism of graded modules are graded submodules. Remark: To give a graded morphism from a graded ring to another graded ring with the image lying in the center is the same as to give the structure of a graded algebra to the latter ring. Given a graded module M {\displaystyle M} , the ℓ {\displaystyle \ell } -twist of M {\displaystyle M} is a graded module defined by M ( ℓ ) n = M n + ℓ {\displaystyle M(\ell )_{n}=M_{n+\ell }} (cf. Serre's twisting sheaf in algebraic geometry). Let M and N be graded modules. If f : M → N {\displaystyle f\colon M\to N} is a morphism of modules, then f is said to have degree d if f ( M n ) ⊆ N n + d {\displaystyle f(M_{n})\subseteq N_{n+d}} . An exterior derivative of differential forms in differential geometry is an example of such a morphism having degree 1. == Invariants of graded modules == Given a graded module M over a commutative graded ring R, one can associate the formal power series P ( M , t ) ∈ Z [ [ t ] ] {\displaystyle P(M,t)\in \mathbb {Z} [\![t]\!]} : P ( M , t ) = ∑ ℓ ( M n ) t n {\displaystyle P(M,t)=\sum \ell (M_{n})t^{n}} (assuming ℓ ( M n ) {\displaystyle \ell (M_{n})} are finite.) It is called the Hilbert–Poincaré series of M. A graded module is said to be finitely generated if the underlying module is finitely generated. The generators may be taken to be homogeneous (by replacing the generators by their homogeneous parts.) Suppose R is a polynomial ring k [ x 0 , … , x n ] {\displaystyle k[x_{0},\dots ,x_{n}]} , k a field, and M a finitely generated graded module over it. Then the function n ↦ dim k M n {\displaystyle n\mapsto \dim _{k}M_{n}} is called the Hilbert function of M. The function coincides with the integer-valued polynomial for large n called the Hilbert polynomial of M. == Graded algebra == An associative algebra A over a ring R is a graded algebra if it is graded as a ring. In the usual case where the ring R is not graded (in particular if R is a field), it is given the trivial grading (every element of R is of degree 0). Thus, R ⊆ A 0 {\displaystyle R\subseteq A_{0}} and the graded pieces A i {\displaystyle A_{i}} are R-modules. In the case where the ring R is also a graded ring, then one requires that R i A j ⊆ A i + j {\displaystyle R_{i}A_{j}\subseteq A_{i+j}} In other words, we require A to be a graded left module over R. Examples of graded algebras are common in mathematics: Polynomial rings. The homogeneous elements of degree n are exactly the homogeneous polynomials of degree n. The tensor algebra T ∙ V {\displaystyle T^{\bullet }V} of a vector space V. The homogeneous elements of degree n are the tensors of order n, T n V {\displaystyle T^{n}V} . The exterior algebra ⋀ ∙ V {\displaystyle \textstyle \bigwedge \nolimits ^{\bullet }V} and the symmetric algebra S ∙ V {\displaystyle S^{\bullet }V} are also graded algebras. The cohomology ring H ∙ {\displaystyle H^{\bullet }} in any cohomology theory is also graded, being the direct sum of the cohomology groups H n {\displaystyle H^{n}} . Graded algebras are much used in commutative algebra and algebraic geometry, homological algebra, and algebraic topology. One example is the close relationship between homogeneous polynomials and projective varieties (cf. Homogeneous coordinate ring.) == G-graded rings and algebras == The above definitions have been generalized to rings graded using any monoid G as an index set. A G-graded ring R is a ring with a direct sum decomposition R = ⨁ i ∈ G R i {\displaystyle R=\bigoplus _{i\in G}R_{i}} such that R i R j ⊆ R i ⋅ j . {\displaystyle R_{i}R_{j}\subseteq R_{i\cdot j}.} Elements of R that lie inside R i {\displaystyle R_{i}} for some i ∈ G {\displaystyle i\in G} are said to be homogeneous of grade i. The previously defined notion of "graded ring" now becomes the same thing as an N {\displaystyle \mathbb {N} } -graded ring, where N {\displaystyle \mathbb {N} } is the monoid of natural numbers under addition. The definitions for graded modules and algebras can also be extended this way replacing the indexing set N {\displaystyle \mathbb {N} } with any monoid G. Remarks: If we do not require that the ring have an identity element, semigroups may replace monoids. Examples: A group naturally grades the corresponding group ring; similarly, monoid rings are graded by the corresponding monoid. An (associative) superalgebra is another term for a Z 2 {\displaystyle \mathbb {Z} _{2}} -graded algebra. Examples include Clifford algebras. Here the homogeneous elements are either of degree 0 (even) or 1 (odd). === Anticommutativity === Some graded rings (or algebras) are endowed with an anticommutative structure. This notion requires a homomorphism of the monoid of the gradation into the additive monoid of Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } , the field with two elements. Specifically, a signed monoid consists of a pair ( Γ , ε ) {\displaystyle (\Gamma ,\varepsilon )} where Γ {\displaystyle \Gamma } is a monoid and ε : Γ → Z / 2 Z {\displaystyle \varepsilon \colon \Gamma \to \mathbb {Z} /2\mathbb {Z} } is a homomorphism of additive monoids. An anticommutative Γ {\displaystyle \Gamma } -graded ring is a ring A graded with respect to Γ {\displaystyle \Gamma } such that: x y = ( − 1 ) ε ( deg x ) ε ( deg y ) y x , {\displaystyle xy=(-1)^{\varepsilon (\deg x)\varepsilon (\deg y)}yx,} for all homogeneous elements x and y. === Examples === An exterior algebra is an example of an anticommutative algebra, graded with respect to the structure ( Z , ε ) {\displaystyle (\mathbb {Z} ,\varepsilon )} where ε : Z → Z / 2 Z {\displaystyle \varepsilon \colon \mathbb {Z} \to \mathbb {Z} /2\mathbb {Z} } is the quotient map. A supercommutative algebra (sometimes called a skew-commutative associative ring) is the same thing as an anticommutative ( Z , ε ) {\displaystyle (\mathbb {Z} ,\varepsilon )} -graded algebra, where ε {\displaystyle \varepsilon } is the identity map of the additive structure of Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } . == Graded monoid == Intuitively, a graded monoid is the subset of a graded ring, ⨁ n ∈ N 0 R n {\textstyle \bigoplus _{n\in \mathbb {N} _{0}}R_{n}} , generated by the R n {\displaystyle R_{n}} 's, without using the additive part. That is, the set of elements of the graded monoid is ⋃ n ∈ N 0 R n {\displaystyle \bigcup _{n\in \mathbb {N} _{0}}R_{n}} . Formally, a graded monoid is a monoid ( M , ⋅ ) {\displaystyle (M,\cdot )} , with a gradation function ϕ : M → N 0 {\displaystyle \phi :M\to \mathbb {N} _{0}} such that ϕ ( m ⋅ m ′ ) = ϕ ( m ) + ϕ ( m ′ ) {\displaystyle \phi (m\cdot m')=\phi (m)+\phi (m')} . Note that the gradation of 1 M {\displaystyle 1_{M}} is necessarily 0. Some authors request furthermore that ϕ ( m ) ≠ 0 {\displaystyle \phi (m)\neq 0} when m is not the identity. Assuming the gradations of non-identity elements are non-zero, the number of elements of gradation n is at most g n {\displaystyle g^{n}} where g is the cardinality of a generating set G of the monoid. Therefore the number of elements of gradation n or less is at most n + 1 {\displaystyle n+1} (for g = 1 {\displaystyle g=1} ) or g n + 1 − 1 g − 1 {\textstyle {\frac {g^{n+1}-1}{g-1}}} else. Indeed, each such element is the product of at most n elements of G, and only g n + 1 − 1 g − 1 {\textstyle {\frac {g^{n+1}-1}{g-1}}} such products exist. Similarly, the identity element can not be written as the product of two non-identity elements. That is, there is no unit divisor in such a graded monoid. === Power series indexed by a graded monoid === These notions allow us to extend the notion of power series ring. Instead of the indexing family being N {\displaystyle \mathbb {N} } , the indexing family could be any graded monoid, assuming that the number of elements of degree n is finite, for each integer n. More formally, let ( K , + K , × K ) {\displaystyle (K,+_{K},\times _{K})} be an arbitrary semiring and ( R , ⋅ , ϕ ) {\displaystyle (R,\cdot ,\phi )} a graded monoid. Then K ⟨ ⟨ R ⟩ ⟩ {\displaystyle K\langle \langle R\rangle \rangle } denotes the semiring of power series with coefficients in K indexed by R. Its elements are functions from R to K. The sum of two elements s , s ′ ∈ K ⟨ ⟨ R ⟩ ⟩ {\displaystyle s,s'\in K\langle \langle R\rangle \rangle } is defined pointwise, it is the function sending m ∈ R {\displaystyle m\in R} to s ( m ) + K s ′ ( m ) {\displaystyle s(m)+_{K}s'(m)} , and the product is the function sending m ∈ R {\displaystyle m\in R} to the infinite sum ∑ p , q ∈ R p ⋅ q = m s ( p ) × K s ′ ( q ) {\displaystyle \sum _{p,q\in R \atop p\cdot q=m}s(p)\times _{K}s'(q)} . This sum is correctly defined (i.e., finite) because, for each m, there are only a finite number of pairs (p, q) such that pq = m. === Free monoid === In formal language theory, given an alphabet A, the free monoid of words over A can be considered as a graded monoid, where the gradation of a word is its length. == See also == Associated graded ring Differential graded algebra Filtered algebra, a generalization Graded (mathematics) Graded category Graded vector space Tensor algebra Differential graded module == Notes == === Citations === === References ===
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Wikipedia:Graded-symmetric algebra#0
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In mathematics, the symmetric algebra S(V) (also denoted Sym(V)) on a vector space V over a field K is a commutative algebra over K that contains V, and is, in some sense, minimal for this property. Here, "minimal" means that S(V) satisfies the following universal property: for every linear map f from V to a commutative algebra A, there is a unique algebra homomorphism g : S(V) → A such that f = g ∘ i, where i is the inclusion map of V in S(V). If B is a basis of V, the symmetric algebra S(V) can be identified, through a canonical isomorphism, to the polynomial ring K[B], where the elements of B are considered as indeterminates. Therefore, the symmetric algebra over V can be viewed as a "coordinate free" polynomial ring over V. The symmetric algebra S(V) can be built as the quotient of the tensor algebra T(V) by the two-sided ideal generated by the elements of the form x ⊗ y − y ⊗ x. All these definitions and properties extend naturally to the case where V is a module (not necessarily a free one) over a commutative ring. == Construction == === From tensor algebra === It is possible to use the tensor algebra T(V) to describe the symmetric algebra S(V). In fact, S(V) can be defined as the quotient algebra of T(V) by the two-sided ideal generated by the commutators v ⊗ w − w ⊗ v . {\displaystyle v\otimes w-w\otimes v.} It is straightforward to verify that the resulting algebra satisfies the universal property stated in the introduction. Because of the universal property of the tensor algebra, a linear map f from V to a commutative algebra A extends to an algebra homomorphism T ( V ) → A {\displaystyle T(V)\rightarrow A} , which factors through S(V) because A is commutative. The extension of f to an algebra homomorphism S ( V ) → A {\displaystyle S(V)\rightarrow A} is unique because V generates S(V) as a K-algebra. This results also directly from a general result of category theory, which asserts that the composition of two left adjoint functors is also a left adjoint functor. Here, the forgetful functor from commutative algebras to vector spaces or modules (forgetting the multiplication) is the composition of the forgetful functors from commutative algebras to associative algebras (forgetting commutativity), and from associative algebras to vectors or modules (forgetting the multiplication). As the tensor algebra and the quotient by commutators are left adjoint to these forgetful functors, their composition is left adjoint to the forgetful functor from commutative algebra to vectors or modules, and this proves the desired universal property. === From polynomial ring === The symmetric algebra S(V) can also be built from polynomial rings. If V is a K-vector space or a free K-module, with a basis B, let K[B] be the polynomial ring that has the elements of B as indeterminates. The homogeneous polynomials of degree one form a vector space or a free module that can be identified with V. It is straightforward to verify that this makes K[B] a solution to the universal problem stated in the introduction. This implies that K[B] and S(V) are canonically isomorphic, and can therefore be identified. This results also immediately from general considerations of category theory, since free modules and polynomial rings are free objects of their respective categories. If V is a module that is not free, it can be written V = L / M , {\displaystyle V=L/M,} where L is a free module, and M is a submodule of L. In this case, one has S ( V ) = S ( L / M ) = S ( L ) / ⟨ M ⟩ , {\displaystyle S(V)=S(L/M)=S(L)/\langle M\rangle ,} where ⟨ M ⟩ {\displaystyle \langle M\rangle } is the ideal generated by M. (Here, equals signs mean equality up to a canonical isomorphism.) Again this can be proved by showing that one has a solution of the universal property, and this can be done either by a straightforward but boring computation, or by using category theory, and more specifically, the fact that a quotient is the solution of the universal problem for morphisms that map to zero a given subset. (Depending on the case, the kernel is a normal subgroup, a submodule or an ideal, and the usual definition of quotients can be viewed as a proof of the existence of a solution of the universal problem.) == Grading == The symmetric algebra is a graded algebra. That is, it is a direct sum S ( V ) = ⨁ n = 0 ∞ S n ( V ) , {\displaystyle S(V)=\bigoplus _{n=0}^{\infty }S^{n}(V),} where S n ( V ) , {\displaystyle S^{n}(V),} called the nth symmetric power of V, is the vector subspace or submodule generated by the products of n elements of V. (The second symmetric power S 2 ( V ) {\displaystyle S^{2}(V)} is sometimes called the symmetric square of V). This can be proved by various means. One follows from the tensor-algebra construction: since the tensor algebra is graded, and the symmetric algebra is its quotient by a homogeneous ideal: the ideal generated by all x ⊗ y − y ⊗ x , {\displaystyle x\otimes y-y\otimes x,} where x and y are in V, that is, homogeneous of degree one. In the case of a vector space or a free module, the gradation is the gradation of the polynomials by the total degree. A non-free module can be written as L / M, where L is a free module of base B; its symmetric algebra is the quotient of the (graded) symmetric algebra of L (a polynomial ring) by the homogeneous ideal generated by the elements of M, which are homogeneous of degree one. One can also define S n ( V ) {\displaystyle S^{n}(V)} as the solution of the universal problem for n-linear symmetric functions from V into a vector space or a module, and then verify that the direct sum of all S n ( V ) {\displaystyle S^{n}(V)} satisfies the universal problem for the symmetric algebra. == Relationship with symmetric tensors == As the symmetric algebra of a vector space is a quotient of the tensor algebra, an element of the symmetric algebra is not a tensor, and, in particular, is not a symmetric tensor. However, symmetric tensors are strongly related to the symmetric algebra. A symmetric tensor of degree n is an element of Tn(V) that is invariant under the action of the symmetric group S n . {\displaystyle {\mathcal {S}}_{n}.} More precisely, given σ ∈ S n , {\displaystyle \sigma \in {\mathcal {S}}_{n},} the transformation v 1 ⊗ ⋯ ⊗ v n ↦ v σ ( 1 ) ⊗ ⋯ ⊗ v σ ( n ) {\displaystyle v_{1}\otimes \cdots \otimes v_{n}\mapsto v_{\sigma (1)}\otimes \cdots \otimes v_{\sigma (n)}} defines a linear endomorphism of Tn(V). A symmetric tensor is a tensor that is invariant under all these endomorphisms. The symmetric tensors of degree n form a vector subspace (or module) Symn(V) ⊂ Tn(V). The symmetric tensors are the elements of the direct sum ⨁ n = 0 ∞ Sym n ( V ) , {\displaystyle \textstyle \bigoplus _{n=0}^{\infty }\operatorname {Sym} ^{n}(V),} which is a graded vector space (or a graded module). It is not an algebra, as the tensor product of two symmetric tensors is not symmetric in general. Let π n {\displaystyle \pi _{n}} be the restriction to Symn(V) of the canonical surjection T n ( V ) → S n ( V ) . {\displaystyle T^{n}(V)\to S^{n}(V).} If n! is invertible in the ground field (or ring), then π n {\displaystyle \pi _{n}} is an isomorphism. This is always the case with a ground field of characteristic zero. The inverse isomorphism is the linear map defined (on products of n vectors) by the symmetrization v 1 ⋯ v n ↦ 1 n ! ∑ σ ∈ S n v σ ( 1 ) ⊗ ⋯ ⊗ v σ ( n ) . {\displaystyle v_{1}\cdots v_{n}\mapsto {\frac {1}{n!}}\sum _{\sigma \in S_{n}}v_{\sigma (1)}\otimes \cdots \otimes v_{\sigma (n)}.} The map π n {\displaystyle \pi _{n}} is not injective if the characteristic is less than n+1; for example π n ( x ⊗ y + y ⊗ x ) = 2 x y {\displaystyle \pi _{n}(x\otimes y+y\otimes x)=2xy} is zero in characteristic two. Over a ring of characteristic zero, π n {\displaystyle \pi _{n}} can be non surjective; for example, over the integers, if x and y are two linearly independent elements of V = S1(V) that are not in 2V, then x y ∉ π n ( Sym 2 ( V ) ) , {\displaystyle xy\not \in \pi _{n}(\operatorname {Sym} ^{2}(V)),} since 1 2 ( x ⊗ y + y ⊗ x ) ∉ Sym 2 ( V ) . {\displaystyle {\frac {1}{2}}(x\otimes y+y\otimes x)\not \in \operatorname {Sym} ^{2}(V).} In summary, over a field of characteristic zero, the symmetric tensors and the symmetric algebra form two isomorphic graded vector spaces. They can thus be identified as far as only the vector space structure is concerned, but they cannot be identified as soon as products are involved. Moreover, this isomorphism does not extend to the cases of fields of positive characteristic and rings that do not contain the rational numbers. == Categorical properties == Given a module V over a commutative ring K, the symmetric algebra S(V) can be defined by the following universal property: For every K-linear map f from V to a commutative K-algebra A, there is a unique K-algebra homomorphism g : S ( V ) → A {\displaystyle g:S(V)\to A} such that f = g ∘ i , {\displaystyle f=g\circ i,} where i is the inclusion of V in S(V). As for every universal property, as soon as a solution exists, this defines uniquely the symmetric algebra, up to a canonical isomorphism. It follows that all properties of the symmetric algebra can be deduced from the universal property. This section is devoted to the main properties that belong to category theory. The symmetric algebra is a functor from the category of K-modules to the category of K-commutative algebra, since the universal property implies that every module homomorphism f : V → W {\displaystyle f:V\to W} can be uniquely extended to an algebra homomorphism S ( f ) : S ( V ) → S ( W ) . {\displaystyle S(f):S(V)\to S(W).} The universal property can be reformulated by saying that the symmetric algebra is a left adjoint to the forgetful functor that sends a commutative algebra to its underlying module. == Symmetric algebra of an affine space == One can analogously construct the symmetric algebra on an affine space. The key difference is that the symmetric algebra of an affine space is not a graded algebra, but a filtered algebra: one can determine the degree of a polynomial on an affine space, but not its homogeneous parts. For instance, given a linear polynomial on a vector space, one can determine its constant part by evaluating at 0. On an affine space, there is no distinguished point, so one cannot do this (choosing a point turns an affine space into a vector space). == Analogy with exterior algebra == The Sk are functors comparable to the exterior powers; here, though, the dimension grows with k; it is given by dim ( S k ( V ) ) = ( n + k − 1 k ) {\displaystyle \operatorname {dim} (S^{k}(V))={\binom {n+k-1}{k}}} where n is the dimension of V. This binomial coefficient is the number of n-variable monomials of degree k. In fact, the symmetric algebra and the exterior algebra appear as the isotypical components of the trivial and sign representation of the action of S n {\displaystyle S_{n}} acting on the tensor product V ⊗ n {\displaystyle V^{\otimes n}} (for example over the complex field) == As a Hopf algebra == The symmetric algebra can be given the structure of a Hopf algebra. See Tensor algebra for details. == As a universal enveloping algebra == The symmetric algebra S(V) is the universal enveloping algebra of an abelian Lie algebra, i.e. one in which the Lie bracket is identically 0. == See also == exterior algebra, the alternating algebra analog graded-symmetric algebra, a common generalization of a symmetric algebra and an exterior algebra Weyl algebra, a quantum deformation of the symmetric algebra by a symplectic form Clifford algebra, a quantum deformation of the exterior algebra by a quadratic form Proj construction § Proj of a quasi-coherent sheaf, an application of symmetric algebras in algebraic geometry == References == Bourbaki, Nicolas (1989), Elements of mathematics, Algebra I, Springer-Verlag, ISBN 3-540-64243-9
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Wikipedia:Gradient#0
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In vector calculus, the gradient of a scalar-valued differentiable function f {\displaystyle f} of several variables is the vector field (or vector-valued function) ∇ f {\displaystyle \nabla f} whose value at a point p {\displaystyle p} gives the direction and the rate of fastest increase. The gradient transforms like a vector under change of basis of the space of variables of f {\displaystyle f} . If the gradient of a function is non-zero at a point p {\displaystyle p} , the direction of the gradient is the direction in which the function increases most quickly from p {\displaystyle p} , and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative. Further, a point where the gradient is the zero vector is known as a stationary point. The gradient thus plays a fundamental role in optimization theory, where it is used to minimize a function by gradient descent. In coordinate-free terms, the gradient of a function f ( r ) {\displaystyle f(\mathbf {r} )} may be defined by: d f = ∇ f ⋅ d r {\displaystyle df=\nabla f\cdot d\mathbf {r} } where d f {\displaystyle df} is the total infinitesimal change in f {\displaystyle f} for an infinitesimal displacement d r {\displaystyle d\mathbf {r} } , and is seen to be maximal when d r {\displaystyle d\mathbf {r} } is in the direction of the gradient ∇ f {\displaystyle \nabla f} . The nabla symbol ∇ {\displaystyle \nabla } , written as an upside-down triangle and pronounced "del", denotes the vector differential operator. When a coordinate system is used in which the basis vectors are not functions of position, the gradient is given by the vector whose components are the partial derivatives of f {\displaystyle f} at p {\displaystyle p} . That is, for f : R n → R {\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} } , its gradient ∇ f : R n → R n {\displaystyle \nabla f\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}} is defined at the point p = ( x 1 , … , x n ) {\displaystyle p=(x_{1},\ldots ,x_{n})} in n-dimensional space as the vector ∇ f ( p ) = [ ∂ f ∂ x 1 ( p ) ⋮ ∂ f ∂ x n ( p ) ] . {\displaystyle \nabla f(p)={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(p)\\\vdots \\{\frac {\partial f}{\partial x_{n}}}(p)\end{bmatrix}}.} Note that the above definition for gradient is defined for the function f {\displaystyle f} only if f {\displaystyle f} is differentiable at p {\displaystyle p} . There can be functions for which partial derivatives exist in every direction but fail to be differentiable. Furthermore, this definition as the vector of partial derivatives is only valid when the basis of the coordinate system is orthonormal. For any other basis, the metric tensor at that point needs to be taken into account. For example, the function f ( x , y ) = x 2 y x 2 + y 2 {\displaystyle f(x,y)={\frac {x^{2}y}{x^{2}+y^{2}}}} unless at origin where f ( 0 , 0 ) = 0 {\displaystyle f(0,0)=0} , is not differentiable at the origin as it does not have a well defined tangent plane despite having well defined partial derivatives in every direction at the origin. In this particular example, under rotation of x-y coordinate system, the above formula for gradient fails to transform like a vector (gradient becomes dependent on choice of basis for coordinate system) and also fails to point towards the 'steepest ascent' in some orientations. For differentiable functions where the formula for gradient holds, it can be shown to always transform as a vector under transformation of the basis so as to always point towards the fastest increase. The gradient is dual to the total derivative d f {\displaystyle df} : the value of the gradient at a point is a tangent vector – a vector at each point; while the value of the derivative at a point is a cotangent vector – a linear functional on vectors. They are related in that the dot product of the gradient of f {\displaystyle f} at a point p {\displaystyle p} with another tangent vector v {\displaystyle \mathbf {v} } equals the directional derivative of f {\displaystyle f} at p {\displaystyle p} of the function along v {\displaystyle \mathbf {v} } ; that is, ∇ f ( p ) ⋅ v = ∂ f ∂ v ( p ) = d f p ( v ) {\textstyle \nabla f(p)\cdot \mathbf {v} ={\frac {\partial f}{\partial \mathbf {v} }}(p)=df_{p}(\mathbf {v} )} . The gradient admits multiple generalizations to more general functions on manifolds; see § Generalizations. == Motivation == Consider a room where the temperature is given by a scalar field, T, so at each point (x, y, z) the temperature is T(x, y, z), independent of time. At each point in the room, the gradient of T at that point will show the direction in which the temperature rises most quickly, moving away from (x, y, z). The magnitude of the gradient will determine how fast the temperature rises in that direction. Consider a surface whose height above sea level at point (x, y) is H(x, y). The gradient of H at a point is a plane vector pointing in the direction of the steepest slope or grade at that point. The steepness of the slope at that point is given by the magnitude of the gradient vector. The gradient can also be used to measure how a scalar field changes in other directions, rather than just the direction of greatest change, by taking a dot product. Suppose that the steepest slope on a hill is 40%. A road going directly uphill has slope 40%, but a road going around the hill at an angle will have a shallower slope. For example, if the road is at a 60° angle from the uphill direction (when both directions are projected onto the horizontal plane), then the slope along the road will be the dot product between the gradient vector and a unit vector along the road, as the dot product measures how much the unit vector along the road aligns with the steepest slope, which is 40% times the cosine of 60°, or 20%. More generally, if the hill height function H is differentiable, then the gradient of H dotted with a unit vector gives the slope of the hill in the direction of the vector, the directional derivative of H along the unit vector. == Notation == The gradient of a function f {\displaystyle f} at point a {\displaystyle a} is usually written as ∇ f ( a ) {\displaystyle \nabla f(a)} . It may also be denoted by any of the following: ∇ → f ( a ) {\displaystyle {\vec {\nabla }}f(a)} : to emphasize the vector nature of the result. grad f {\displaystyle \operatorname {grad} f} ∂ i f {\displaystyle \partial _{i}f} and f i {\displaystyle f_{i}} : Written with Einstein notation, where repeated indices (i) are summed over. == Definition == The gradient (or gradient vector field) of a scalar function f(x1, x2, x3, …, xn) is denoted ∇f or ∇→f where ∇ (nabla) denotes the vector differential operator, del. The notation grad f is also commonly used to represent the gradient. The gradient of f is defined as the unique vector field whose dot product with any vector v at each point x is the directional derivative of f along v. That is, ( ∇ f ( x ) ) ⋅ v = D v f ( x ) {\displaystyle {\big (}\nabla f(x){\big )}\cdot \mathbf {v} =D_{\mathbf {v} }f(x)} where the right-hand side is the directional derivative and there are many ways to represent it. Formally, the derivative is dual to the gradient; see relationship with derivative. When a function also depends on a parameter such as time, the gradient often refers simply to the vector of its spatial derivatives only (see Spatial gradient). The magnitude and direction of the gradient vector are independent of the particular coordinate representation. === Cartesian coordinates === In the three-dimensional Cartesian coordinate system with a Euclidean metric, the gradient, if it exists, is given by ∇ f = ∂ f ∂ x i + ∂ f ∂ y j + ∂ f ∂ z k , {\displaystyle \nabla f={\frac {\partial f}{\partial x}}\mathbf {i} +{\frac {\partial f}{\partial y}}\mathbf {j} +{\frac {\partial f}{\partial z}}\mathbf {k} ,} where i, j, k are the standard unit vectors in the directions of the x, y and z coordinates, respectively. For example, the gradient of the function f ( x , y , z ) = 2 x + 3 y 2 − sin ( z ) {\displaystyle f(x,y,z)=2x+3y^{2}-\sin(z)} is ∇ f ( x , y , z ) = 2 i + 6 y j − cos ( z ) k . {\displaystyle \nabla f(x,y,z)=2\mathbf {i} +6y\mathbf {j} -\cos(z)\mathbf {k} .} or ∇ f ( x , y , z ) = [ 2 6 y − cos z ] . {\displaystyle \nabla f(x,y,z)={\begin{bmatrix}2\\6y\\-\cos z\end{bmatrix}}.} In some applications it is customary to represent the gradient as a row vector or column vector of its components in a rectangular coordinate system; this article follows the convention of the gradient being a column vector, while the derivative is a row vector. === Cylindrical and spherical coordinates === In cylindrical coordinates, the gradient is given by: ∇ f ( ρ , φ , z ) = ∂ f ∂ ρ e ρ + 1 ρ ∂ f ∂ φ e φ + ∂ f ∂ z e z , {\displaystyle \nabla f(\rho ,\varphi ,z)={\frac {\partial f}{\partial \rho }}\mathbf {e} _{\rho }+{\frac {1}{\rho }}{\frac {\partial f}{\partial \varphi }}\mathbf {e} _{\varphi }+{\frac {\partial f}{\partial z}}\mathbf {e} _{z},} where ρ is the axial distance, φ is the azimuthal or azimuth angle, z is the axial coordinate, and eρ, eφ and ez are unit vectors pointing along the coordinate directions. In spherical coordinates with a Euclidean metric, the gradient is given by: ∇ f ( r , θ , φ ) = ∂ f ∂ r e r + 1 r ∂ f ∂ θ e θ + 1 r sin θ ∂ f ∂ φ e φ , {\displaystyle \nabla f(r,\theta ,\varphi )={\frac {\partial f}{\partial r}}\mathbf {e} _{r}+{\frac {1}{r}}{\frac {\partial f}{\partial \theta }}\mathbf {e} _{\theta }+{\frac {1}{r\sin \theta }}{\frac {\partial f}{\partial \varphi }}\mathbf {e} _{\varphi },} where r is the radial distance, φ is the azimuthal angle and θ is the polar angle, and er, eθ and eφ are again local unit vectors pointing in the coordinate directions (that is, the normalized covariant basis). For the gradient in other orthogonal coordinate systems, see Orthogonal coordinates (Differential operators in three dimensions). === General coordinates === We consider general coordinates, which we write as x1, …, xi, …, xn, where n is the number of dimensions of the domain. Here, the upper index refers to the position in the list of the coordinate or component, so x2 refers to the second component—not the quantity x squared. The index variable i refers to an arbitrary element xi. Using Einstein notation, the gradient can then be written as: ∇ f = ∂ f ∂ x i g i j e j {\displaystyle \nabla f={\frac {\partial f}{\partial x^{i}}}g^{ij}\mathbf {e} _{j}} (Note that its dual is d f = ∂ f ∂ x i e i {\textstyle \mathrm {d} f={\frac {\partial f}{\partial x^{i}}}\mathbf {e} ^{i}} ), where e i = d x i {\displaystyle \mathbf {e} ^{i}=\mathrm {d} x^{i}} and e i = ∂ x / ∂ x i {\displaystyle \mathbf {e} _{i}=\partial \mathbf {x} /\partial x^{i}} refer to the unnormalized local covariant and contravariant bases respectively, g i j {\displaystyle g^{ij}} is the inverse metric tensor, and the Einstein summation convention implies summation over i and j. If the coordinates are orthogonal we can easily express the gradient (and the differential) in terms of the normalized bases, which we refer to as e ^ i {\displaystyle {\hat {\mathbf {e} }}_{i}} and e ^ i {\displaystyle {\hat {\mathbf {e} }}^{i}} , using the scale factors (also known as Lamé coefficients) h i = ‖ e i ‖ = g i i = 1 / ‖ e i ‖ {\displaystyle h_{i}=\lVert \mathbf {e} _{i}\rVert ={\sqrt {g_{ii}}}=1\,/\lVert \mathbf {e} ^{i}\rVert } : ∇ f = ∂ f ∂ x i g i j e ^ j g j j = ∑ i = 1 n ∂ f ∂ x i 1 h i e ^ i {\displaystyle \nabla f={\frac {\partial f}{\partial x^{i}}}g^{ij}{\hat {\mathbf {e} }}_{j}{\sqrt {g_{jj}}}=\sum _{i=1}^{n}\,{\frac {\partial f}{\partial x^{i}}}{\frac {1}{h_{i}}}\mathbf {\hat {e}} _{i}} (and d f = ∑ i = 1 n ∂ f ∂ x i 1 h i e ^ i {\textstyle \mathrm {d} f=\sum _{i=1}^{n}\,{\frac {\partial f}{\partial x^{i}}}{\frac {1}{h_{i}}}\mathbf {\hat {e}} ^{i}} ), where we cannot use Einstein notation, since it is impossible to avoid the repetition of more than two indices. Despite the use of upper and lower indices, e ^ i {\displaystyle \mathbf {\hat {e}} _{i}} , e ^ i {\displaystyle \mathbf {\hat {e}} ^{i}} , and h i {\displaystyle h_{i}} are neither contravariant nor covariant. The latter expression evaluates to the expressions given above for cylindrical and spherical coordinates. == Relationship with derivative == === Relationship with total derivative === The gradient is closely related to the total derivative (total differential) d f {\displaystyle df} : they are transpose (dual) to each other. Using the convention that vectors in R n {\displaystyle \mathbb {R} ^{n}} are represented by column vectors, and that covectors (linear maps R n → R {\displaystyle \mathbb {R} ^{n}\to \mathbb {R} } ) are represented by row vectors, the gradient ∇ f {\displaystyle \nabla f} and the derivative d f {\displaystyle df} are expressed as a column and row vector, respectively, with the same components, but transpose of each other: ∇ f ( p ) = [ ∂ f ∂ x 1 ( p ) ⋮ ∂ f ∂ x n ( p ) ] ; {\displaystyle \nabla f(p)={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(p)\\\vdots \\{\frac {\partial f}{\partial x_{n}}}(p)\end{bmatrix}};} d f p = [ ∂ f ∂ x 1 ( p ) ⋯ ∂ f ∂ x n ( p ) ] . {\displaystyle df_{p}={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(p)&\cdots &{\frac {\partial f}{\partial x_{n}}}(p)\end{bmatrix}}.} While these both have the same components, they differ in what kind of mathematical object they represent: at each point, the derivative is a cotangent vector, a linear form (or covector) which expresses how much the (scalar) output changes for a given infinitesimal change in (vector) input, while at each point, the gradient is a tangent vector, which represents an infinitesimal change in (vector) input. In symbols, the gradient is an element of the tangent space at a point, ∇ f ( p ) ∈ T p R n {\displaystyle \nabla f(p)\in T_{p}\mathbb {R} ^{n}} , while the derivative is a map from the tangent space to the real numbers, d f p : T p R n → R {\displaystyle df_{p}\colon T_{p}\mathbb {R} ^{n}\to \mathbb {R} } . The tangent spaces at each point of R n {\displaystyle \mathbb {R} ^{n}} can be "naturally" identified with the vector space R n {\displaystyle \mathbb {R} ^{n}} itself, and similarly the cotangent space at each point can be naturally identified with the dual vector space ( R n ) ∗ {\displaystyle (\mathbb {R} ^{n})^{*}} of covectors; thus the value of the gradient at a point can be thought of a vector in the original R n {\displaystyle \mathbb {R} ^{n}} , not just as a tangent vector. Computationally, given a tangent vector, the vector can be multiplied by the derivative (as matrices), which is equal to taking the dot product with the gradient: ( d f p ) ( v ) = [ ∂ f ∂ x 1 ( p ) ⋯ ∂ f ∂ x n ( p ) ] [ v 1 ⋮ v n ] = ∑ i = 1 n ∂ f ∂ x i ( p ) v i = [ ∂ f ∂ x 1 ( p ) ⋮ ∂ f ∂ x n ( p ) ] ⋅ [ v 1 ⋮ v n ] = ∇ f ( p ) ⋅ v {\displaystyle (df_{p})(v)={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(p)&\cdots &{\frac {\partial f}{\partial x_{n}}}(p)\end{bmatrix}}{\begin{bmatrix}v_{1}\\\vdots \\v_{n}\end{bmatrix}}=\sum _{i=1}^{n}{\frac {\partial f}{\partial x_{i}}}(p)v_{i}={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(p)\\\vdots \\{\frac {\partial f}{\partial x_{n}}}(p)\end{bmatrix}}\cdot {\begin{bmatrix}v_{1}\\\vdots \\v_{n}\end{bmatrix}}=\nabla f(p)\cdot v} ==== Differential or (exterior) derivative ==== The best linear approximation to a differentiable function f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } at a point x {\displaystyle x} in R n {\displaystyle \mathbb {R} ^{n}} is a linear map from R n {\displaystyle \mathbb {R} ^{n}} to R {\displaystyle \mathbb {R} } which is often denoted by d f x {\displaystyle df_{x}} or D f ( x ) {\displaystyle Df(x)} and called the differential or total derivative of f {\displaystyle f} at x {\displaystyle x} . The function d f {\displaystyle df} , which maps x {\displaystyle x} to d f x {\displaystyle df_{x}} , is called the total differential or exterior derivative of f {\displaystyle f} and is an example of a differential 1-form. Much as the derivative of a function of a single variable represents the slope of the tangent to the graph of the function, the directional derivative of a function in several variables represents the slope of the tangent hyperplane in the direction of the vector. The gradient is related to the differential by the formula ( ∇ f ) x ⋅ v = d f x ( v ) {\displaystyle (\nabla f)_{x}\cdot v=df_{x}(v)} for any v ∈ R n {\displaystyle v\in \mathbb {R} ^{n}} , where ⋅ {\displaystyle \cdot } is the dot product: taking the dot product of a vector with the gradient is the same as taking the directional derivative along the vector. If R n {\displaystyle \mathbb {R} ^{n}} is viewed as the space of (dimension n {\displaystyle n} ) column vectors (of real numbers), then one can regard d f {\displaystyle df} as the row vector with components ( ∂ f ∂ x 1 , … , ∂ f ∂ x n ) , {\displaystyle \left({\frac {\partial f}{\partial x_{1}}},\dots ,{\frac {\partial f}{\partial x_{n}}}\right),} so that d f x ( v ) {\displaystyle df_{x}(v)} is given by matrix multiplication. Assuming the standard Euclidean metric on R n {\displaystyle \mathbb {R} ^{n}} , the gradient is then the corresponding column vector, that is, ( ∇ f ) i = d f i T . {\displaystyle (\nabla f)_{i}=df_{i}^{\mathsf {T}}.} ==== Linear approximation to a function ==== The best linear approximation to a function can be expressed in terms of the gradient, rather than the derivative. The gradient of a function f {\displaystyle f} from the Euclidean space R n {\displaystyle \mathbb {R} ^{n}} to R {\displaystyle \mathbb {R} } at any particular point x 0 {\displaystyle x_{0}} in R n {\displaystyle \mathbb {R} ^{n}} characterizes the best linear approximation to f {\displaystyle f} at x 0 {\displaystyle x_{0}} . The approximation is as follows: f ( x ) ≈ f ( x 0 ) + ( ∇ f ) x 0 ⋅ ( x − x 0 ) {\displaystyle f(x)\approx f(x_{0})+(\nabla f)_{x_{0}}\cdot (x-x_{0})} for x {\displaystyle x} close to x 0 {\displaystyle x_{0}} , where ( ∇ f ) x 0 {\displaystyle (\nabla f)_{x_{0}}} is the gradient of f {\displaystyle f} computed at x 0 {\displaystyle x_{0}} , and the dot denotes the dot product on R n {\displaystyle \mathbb {R} ^{n}} . This equation is equivalent to the first two terms in the multivariable Taylor series expansion of f {\displaystyle f} at x 0 {\displaystyle x_{0}} . === Relationship with Fréchet derivative === Let U be an open set in Rn. If the function f : U → R is differentiable, then the differential of f is the Fréchet derivative of f. Thus ∇f is a function from U to the space Rn such that lim h → 0 | f ( x + h ) − f ( x ) − ∇ f ( x ) ⋅ h | ‖ h ‖ = 0 , {\displaystyle \lim _{h\to 0}{\frac {|f(x+h)-f(x)-\nabla f(x)\cdot h|}{\|h\|}}=0,} where · is the dot product. As a consequence, the usual properties of the derivative hold for the gradient, though the gradient is not a derivative itself, but rather dual to the derivative: Linearity The gradient is linear in the sense that if f and g are two real-valued functions differentiable at the point a ∈ Rn, and α and β are two constants, then αf + βg is differentiable at a, and moreover ∇ ( α f + β g ) ( a ) = α ∇ f ( a ) + β ∇ g ( a ) . {\displaystyle \nabla \left(\alpha f+\beta g\right)(a)=\alpha \nabla f(a)+\beta \nabla g(a).} Product rule If f and g are real-valued functions differentiable at a point a ∈ Rn, then the product rule asserts that the product fg is differentiable at a, and ∇ ( f g ) ( a ) = f ( a ) ∇ g ( a ) + g ( a ) ∇ f ( a ) . {\displaystyle \nabla (fg)(a)=f(a)\nabla g(a)+g(a)\nabla f(a).} Chain rule Suppose that f : A → R is a real-valued function defined on a subset A of Rn, and that f is differentiable at a point a. There are two forms of the chain rule applying to the gradient. First, suppose that the function g is a parametric curve; that is, a function g : I → Rn maps a subset I ⊂ R into Rn. If g is differentiable at a point c ∈ I such that g(c) = a, then ( f ∘ g ) ′ ( c ) = ∇ f ( a ) ⋅ g ′ ( c ) , {\displaystyle (f\circ g)'(c)=\nabla f(a)\cdot g'(c),} where ∘ is the composition operator: (f ∘ g)(x) = f(g(x)). More generally, if instead I ⊂ Rk, then the following holds: ∇ ( f ∘ g ) ( c ) = ( D g ( c ) ) T ( ∇ f ( a ) ) , {\displaystyle \nabla (f\circ g)(c)={\big (}Dg(c){\big )}^{\mathsf {T}}{\big (}\nabla f(a){\big )},} where (Dg)T denotes the transpose Jacobian matrix. For the second form of the chain rule, suppose that h : I → R is a real valued function on a subset I of R, and that h is differentiable at the point f(a) ∈ I. Then ∇ ( h ∘ f ) ( a ) = h ′ ( f ( a ) ) ∇ f ( a ) . {\displaystyle \nabla (h\circ f)(a)=h'{\big (}f(a){\big )}\nabla f(a).} == Further properties and applications == === Level sets === A level surface, or isosurface, is the set of all points where some function has a given value. If f is differentiable, then the dot product (∇f )x ⋅ v of the gradient at a point x with a vector v gives the directional derivative of f at x in the direction v. It follows that in this case the gradient of f is orthogonal to the level sets of f. For example, a level surface in three-dimensional space is defined by an equation of the form F(x, y, z) = c. The gradient of F is then normal to the surface. More generally, any embedded hypersurface in a Riemannian manifold can be cut out by an equation of the form F(P) = 0 such that dF is nowhere zero. The gradient of F is then normal to the hypersurface. Similarly, an affine algebraic hypersurface may be defined by an equation F(x1, ..., xn) = 0, where F is a polynomial. The gradient of F is zero at a singular point of the hypersurface (this is the definition of a singular point). At a non-singular point, it is a nonzero normal vector. === Conservative vector fields and the gradient theorem === The gradient of a function is called a gradient field. A (continuous) gradient field is always a conservative vector field: its line integral along any path depends only on the endpoints of the path, and can be evaluated by the gradient theorem (the fundamental theorem of calculus for line integrals). Conversely, a (continuous) conservative vector field is always the gradient of a function. === Gradient is direction of steepest ascent === The gradient of a function f : R n → R {\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} } at point x is also the direction of its steepest ascent, i.e. it maximizes its directional derivative: Let v ∈ R n {\displaystyle v\in \mathbb {R} ^{n}} be an arbitrary unit vector. With the directional derivative defined as ∇ v f ( x ) = lim h → 0 f ( x + v h ) − f ( x ) h , {\displaystyle \nabla _{v}f(x)=\lim _{h\rightarrow 0}{\frac {f(x+vh)-f(x)}{h}},} we get, by substituting the function f ( x + v h ) {\displaystyle f(x+vh)} with its Taylor series, ∇ v f ( x ) = lim h → 0 ( f ( x ) + ∇ f ⋅ v h + R ) − f ( x ) h , {\displaystyle \nabla _{v}f(x)=\lim _{h\rightarrow 0}{\frac {(f(x)+\nabla f\cdot vh+R)-f(x)}{h}},} where R {\displaystyle R} denotes higher order terms in v h {\displaystyle vh} . Dividing by h {\displaystyle h} , and taking the limit yields a term which is bounded from above by the Cauchy-Schwarz inequality | ∇ v f ( x ) | = | ∇ f ⋅ v | ≤ | ∇ f | | v | = | ∇ f | . {\displaystyle |\nabla _{v}f(x)|=|\nabla f\cdot v|\leq |\nabla f||v|=|\nabla f|.} Choosing v ∗ = ∇ f / | ∇ f | {\displaystyle v^{*}=\nabla f/|\nabla f|} maximizes the directional derivative, and equals the upper bound | ∇ v ∗ f ( x ) | = | ( ∇ f ) 2 / | ∇ f | | = | ∇ f | . {\displaystyle |\nabla _{v^{*}}f(x)|=|(\nabla f)^{2}/|\nabla f||=|\nabla f|.} == Generalizations == === Jacobian === The Jacobian matrix is the generalization of the gradient for vector-valued functions of several variables and differentiable maps between Euclidean spaces or, more generally, manifolds. A further generalization for a function between Banach spaces is the Fréchet derivative. Suppose f : Rn → Rm is a function such that each of its first-order partial derivatives exist on ℝn. Then the Jacobian matrix of f is defined to be an m×n matrix, denoted by J f ( x ) {\displaystyle \mathbf {J} _{\mathbb {f} }(\mathbb {x} )} or simply J {\displaystyle \mathbf {J} } . The (i,j)th entry is J i j = ∂ f i / ∂ x j {\textstyle \mathbf {J} _{ij}={\partial f_{i}}/{\partial x_{j}}} . Explicitly J = [ ∂ f ∂ x 1 ⋯ ∂ f ∂ x n ] = [ ∇ T f 1 ⋮ ∇ T f m ] = [ ∂ f 1 ∂ x 1 ⋯ ∂ f 1 ∂ x n ⋮ ⋱ ⋮ ∂ f m ∂ x 1 ⋯ ∂ f m ∂ x n ] . {\displaystyle \mathbf {J} ={\begin{bmatrix}{\dfrac {\partial \mathbf {f} }{\partial x_{1}}}&\cdots &{\dfrac {\partial \mathbf {f} }{\partial x_{n}}}\end{bmatrix}}={\begin{bmatrix}\nabla ^{\mathsf {T}}f_{1}\\\vdots \\\nabla ^{\mathsf {T}}f_{m}\end{bmatrix}}={\begin{bmatrix}{\dfrac {\partial f_{1}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\dfrac {\partial f_{m}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{m}}{\partial x_{n}}}\end{bmatrix}}.} === Gradient of a vector field === Since the total derivative of a vector field is a linear mapping from vectors to vectors, it is a tensor quantity. In rectangular coordinates, the gradient of a vector field f = ( f1, f2, f3) is defined by: ∇ f = g j k ∂ f i ∂ x j e i ⊗ e k , {\displaystyle \nabla \mathbf {f} =g^{jk}{\frac {\partial f^{i}}{\partial x^{j}}}\mathbf {e} _{i}\otimes \mathbf {e} _{k},} (where the Einstein summation notation is used and the tensor product of the vectors ei and ek is a dyadic tensor of type (2,0)). Overall, this expression equals the transpose of the Jacobian matrix: ∂ f i ∂ x j = ∂ ( f 1 , f 2 , f 3 ) ∂ ( x 1 , x 2 , x 3 ) . {\displaystyle {\frac {\partial f^{i}}{\partial x^{j}}}={\frac {\partial (f^{1},f^{2},f^{3})}{\partial (x^{1},x^{2},x^{3})}}.} In curvilinear coordinates, or more generally on a curved manifold, the gradient involves Christoffel symbols: ∇ f = g j k ( ∂ f i ∂ x j + Γ i j l f l ) e i ⊗ e k , {\displaystyle \nabla \mathbf {f} =g^{jk}\left({\frac {\partial f^{i}}{\partial x^{j}}}+{\Gamma ^{i}}_{jl}f^{l}\right)\mathbf {e} _{i}\otimes \mathbf {e} _{k},} where gjk are the components of the inverse metric tensor and the ei are the coordinate basis vectors. Expressed more invariantly, the gradient of a vector field f can be defined by the Levi-Civita connection and metric tensor: ∇ a f b = g a c ∇ c f b , {\displaystyle \nabla ^{a}f^{b}=g^{ac}\nabla _{c}f^{b},} where ∇c is the connection. === Riemannian manifolds === For any smooth function f on a Riemannian manifold (M, g), the gradient of f is the vector field ∇f such that for any vector field X, g ( ∇ f , X ) = ∂ X f , {\displaystyle g(\nabla f,X)=\partial _{X}f,} that is, g x ( ( ∇ f ) x , X x ) = ( ∂ X f ) ( x ) , {\displaystyle g_{x}{\big (}(\nabla f)_{x},X_{x}{\big )}=(\partial _{X}f)(x),} where gx( , ) denotes the inner product of tangent vectors at x defined by the metric g and ∂X f is the function that takes any point x ∈ M to the directional derivative of f in the direction X, evaluated at x. In other words, in a coordinate chart φ from an open subset of M to an open subset of Rn, (∂X f )(x) is given by: ∑ j = 1 n X j ( φ ( x ) ) ∂ ∂ x j ( f ∘ φ − 1 ) | φ ( x ) , {\displaystyle \sum _{j=1}^{n}X^{j}{\big (}\varphi (x){\big )}{\frac {\partial }{\partial x_{j}}}(f\circ \varphi ^{-1}){\Bigg |}_{\varphi (x)},} where Xj denotes the jth component of X in this coordinate chart. So, the local form of the gradient takes the form: ∇ f = g i k ∂ f ∂ x k e i . {\displaystyle \nabla f=g^{ik}{\frac {\partial f}{\partial x^{k}}}{\textbf {e}}_{i}.} Generalizing the case M = Rn, the gradient of a function is related to its exterior derivative, since ( ∂ X f ) ( x ) = ( d f ) x ( X x ) . {\displaystyle (\partial _{X}f)(x)=(df)_{x}(X_{x}).} More precisely, the gradient ∇f is the vector field associated to the differential 1-form df using the musical isomorphism ♯ = ♯ g : T ∗ M → T M {\displaystyle \sharp =\sharp ^{g}\colon T^{*}M\to TM} (called "sharp") defined by the metric g. The relation between the exterior derivative and the gradient of a function on Rn is a special case of this in which the metric is the flat metric given by the dot product. == See also == Curl – Circulation density in a vector field Divergence – Vector operator in vector calculus Four-gradient – Four-vector analogue of the gradient operation Hessian matrix – Matrix of second derivatives Skew gradient Spatial gradient – Gradient whose components are spatial derivatives == Notes == == References == Bachman, David (2007), Advanced Calculus Demystified, New York: McGraw-Hill, ISBN 978-0-07-148121-2 Beauregard, Raymond A.; Fraleigh, John B. (1973), A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields, Boston: Houghton Mifflin Company, ISBN 0-395-14017-X Downing, Douglas, Ph.D. (2010), Barron's E-Z Calculus, New York: Barron's, ISBN 978-0-7641-4461-5{{citation}}: CS1 maint: multiple names: authors list (link) Dubrovin, B. A.; Fomenko, A. T.; Novikov, S. P. (1991). Modern Geometry—Methods and Applications: Part I: The Geometry of Surfaces, Transformation Groups, and Fields. Graduate Texts in Mathematics (2nd ed.). Springer. ISBN 978-0-387-97663-1. Harper, Charlie (1976), Introduction to Mathematical Physics, New Jersey: Prentice-Hall, ISBN 0-13-487538-9 Kreyszig, Erwin (1972), Advanced Engineering Mathematics (3rd ed.), New York: Wiley, ISBN 0-471-50728-8 "McGraw Hill Encyclopedia of Science & Technology". McGraw-Hill Encyclopedia of Science & Technology (10th ed.). New York: McGraw-Hill. 2007. ISBN 978-0-07-144143-8. Moise, Edwin E. (1967), Calculus: Complete, Reading: Addison-Wesley Protter, Murray H.; Morrey, Charles B. Jr. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading: Addison-Wesley, LCCN 76087042 Schey, H. M. (1992). Div, Grad, Curl, and All That (2nd ed.). W. W. Norton. ISBN 0-393-96251-2. OCLC 25048561. Stoker, J. J. (1969), Differential Geometry, New York: Wiley, ISBN 0-471-82825-4 Swokowski, Earl W.; Olinick, Michael; Pence, Dennis; Cole, Jeffery A. (1994), Calculus (6th ed.), Boston: PWS Publishing Company, ISBN 0-534-93624-5 Arens, T.; Hettlich, F.; Karpfinger, C.; Kockelkorn, U.; Lichtenegger, K.; Stachel, H. (2022), Mathematik (5th ed.), Springer Spektrum Berlin, doi:10.1007/978-3-662-64389-1, ISBN 978-3-662-64388-4 == Further reading == Korn, Theresa M.; Korn, Granino Arthur (2000). Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. Dover Publications. pp. 157–160. ISBN 0-486-41147-8. OCLC 43864234. == External links == "Gradient". Khan Academy. Kuptsov, L.P. (2001) [1994], "Gradient", Encyclopedia of Mathematics, EMS Press. Weisstein, Eric W. "Gradient". MathWorld.
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Wikipedia:Gradient conjecture#0
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In mathematics, the gradient conjecture, due to René Thom (1989), was proved in 2000 by three Polish mathematicians, Krzysztof Kurdyka (University of Savoie, France), Tadeusz Mostowski (Warsaw University, Poland) and Adam Parusiński (University of Angers, France). The conjecture states that given a real-valued analytic function f defined on Rn and a trajectory x(t) of the gradient vector field of f having a limit point x0 ∈ Rn, where f has an isolated critical point at x0, there exists a limit (in the projective space PRn−1) for the secant lines from x(t) to x0, as t tends to zero. The proof depends on a theorem due to Stanisław Łojasiewicz. == References == R. Thom (1989) "Problèmes rencontrés dans mon parcours mathématique: un bilan", Publications Mathématiques de l'IHÉS 70: 200 to 214. (This gradient conjecture due to René Thom was in fact well known among specialists by the early 70's, having been often discussed during that period by Thom during his weekly seminar on singularities at the IHES.) In 2000 the conjecture was proven correct in Annals of Mathematics 152: 763 to 792. The proof is available here.
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Wikipedia:Gradimir Milovanović#0
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Gradimir V. Milovanović (born January 2, 1948) is a Serbian mathematician known for his contributions to approximation theory and numerical analysis. He has published over 280 papers and authored five monographs and more than twenty books in his area. He is a full member of the Serbian Academy of Sciences and Arts and of other Serbian and international scientific societies. == Early life and education == Born in Zorunovac, in the Knjaževac municipality of mideastern Serbia, he studied at University of Niš, obtaining a B.Sc. in electrical engineering and computer science (1971), an M.Sc. in mathematics (1974) and a D.Sc. (1976). His thesis was titled On Some Functional Inequalities advised by Dragoslav Mitrinović. == Career == He served as a member of the faculty of electronic engineering and the department of mathematics at the University of Niš, and was promoted to professor in 1986 before serving as the acting Dean of the Faculty of Electronic Engineering from 2002 to 2004. He served as rector of the University of Niš from 2004 to 2006, as well as dean of the Faculty of Computer Sciences at the Megatrend University from 2008 to 2011 until he joined the Mathematical Institute of the Serbian Academy of Sciences and Arts in Belgrade (2011). He has been a member of the board of the Mathematical Society of Serbia (2003–2006), president of the Scientific Council of the Mathematical Institute at Serbian Academy of Sciences and Arts in Belgrade (1997–2010), vice president of the Scientific Society of Serbia since 2002, president of the National Council for Scientific and Technological Development (2006–2010), and president of the Scientific Committee for Mathematics, Computer Sciences and Mechanics. == References ==
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Wikipedia:Graham Kendall#0
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Graham Kendall (born 21 July 1961) is a professor of computer science at the University of Nottingham, UK. He is currently (2016–present) the provost and CEO of University of Nottingham Malaysia Campus. He is also a pro-vice chancellor of the university. He is a member of the Automated Scheduling, Optimisation and Planning (ASAP) Research Group within the School of Computer Science. He is the editor-in-chief of the IEEE Transactions of Computational Intelligence and AI in Games. == References == == External links == Graham Kendall's home page.
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Wikipedia:Graham–Pollak theorem#0
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In graph theory, the Graham–Pollak theorem states that the edges of an n {\displaystyle n} -vertex complete graph cannot be partitioned into fewer than n − 1 {\displaystyle n-1} complete bipartite graphs. It was first published by Ronald Graham and Henry O. Pollak in two papers in 1971 and 1972 (crediting Hans Witsenhausen for a key lemma), in connection with an application to telephone switching circuitry. The theorem has since become well known and repeatedly studied and generalized in graph theory, in part because of its elegant proof using techniques from algebraic graph theory. More strongly, Aigner & Ziegler (2018) write that all proofs are somehow based on linear algebra: "no combinatorial proof for this result is known". == Construction of an optimal partition == A partition into exactly n − 1 {\displaystyle n-1} complete bipartite graphs is easy to obtain: just order the vertices, and for each vertex except the last, form a star connecting it to all later vertices in the ordering. Other partitions are also possible. == Proof of optimality == The proof of the Graham–Pollak theorem described by Aigner & Ziegler (2018) (following Tverberg 1982) defines a real variable x i {\displaystyle x_{i}} for each vertex v i ∈ V {\displaystyle v_{i}\in V} , where V {\displaystyle V} denotes the set of all vertices in the graph. Let the left sides and right sides of the k {\displaystyle k} th bipartite graph be denoted L k {\displaystyle L_{k}} and R k {\displaystyle R_{k}} , respectively and for any set S {\displaystyle S} of vertices define X ( S ) {\displaystyle X(S)} to be the sum of variables for vertices in S {\displaystyle S} : X ( S ) = ∑ v i ∈ S x i . {\displaystyle X(S)=\sum _{v_{i}\in S}x_{i}.} Then, in terms of this notation, the fact that the bipartite graphs partition the edges of the complete graph can be expressed as the equation ∑ i < j x i x j = ∑ k X ( L k ) X ( R k ) . {\displaystyle \sum _{i<j}x_{i}x_{j}=\sum _{k}X(L_{k})X(R_{k}).} Now consider the system of linear equations that sets X ( V ) = 0 {\displaystyle X(V)=0} and X ( L k ) = 0 {\displaystyle X(L_{k})=0} for each k {\displaystyle k} . Any solution to this system of equations would also obey the nonlinear equations 0 = X ( V ) 2 = ( ∑ i x i ) 2 = ( ∑ i x i 2 ) + ( 2 ∑ i < j x i x j ) = ( ∑ i x i 2 ) + ( 2 ∑ k X ( L k ) X ( R k ) ) = ∑ i x i 2 . {\displaystyle {\begin{aligned}0&=X(V)^{2}={\Bigl (}\sum _{i}x_{i}{\Bigr )}^{2}\\&={\Bigl (}\sum _{i}x_{i}^{2}{\Bigr )}+{\Bigl (}2\sum _{i<j}x_{i}x_{j}{\Bigr )}\\&={\Bigl (}\sum _{i}x_{i}^{2}{\Bigr )}+{\Bigl (}2\sum _{k}X(L_{k})X(R_{k}){\Bigr )}\\&=\sum _{i}x_{i}^{2}.\\\end{aligned}}} But a sum of squares of real variables can only be zero if all the individual variables are zero, the trivial solution to the system of linear equations. If there were fewer than n − 1 {\displaystyle n-1} complete bipartite graphs, the system of equations would have fewer than n {\displaystyle n} equations in n {\displaystyle n} unknowns and would have a nontrivial solution, a contradiction. So the number of complete bipartite graphs must be at least n − 1 {\displaystyle n-1} . == Related problems == === Distance labeling === Graham and Pollak study a more general graph labeling problem, in which the vertices of a graph should be labeled with equal-length strings of the characters "0", "1", and "✶", in such a way that the distance between any two vertices equals the number of string positions where one vertex is labeled with a 0 and the other is labeled with a 1. A labeling like this with no "✶" characters would give an isometric embedding into a hypercube, something that is only possible for graphs that are partial cubes, and in one of their papers Graham and Pollak call a labeling that allows "✶" characters an embedding into a "squashed cube". For each position of the label strings, one can define a complete bipartite graph in which one side of the bipartition consists of the vertices labeled with 0 in that position and the other side consists of the vertices labeled with 1, omitting the vertices labeled "✶". For the complete graph, every two vertices are at distance one from each other, so every edge must belong to exactly one of these complete bipartite graphs. In this way, a labeling of this type for the complete graph corresponds to a partition of its edges into complete bipartite graphs, with the lengths of the labels corresponding to the number of graphs in the partition. === Alon–Saks–Seymour conjecture === Noga Alon, Michael Saks, and Paul Seymour formulated a conjecture in the early 1990s that, if true, would significantly generalize the Graham–Pollak theorem: they conjectured that, whenever a graph of chromatic number k + 1 {\displaystyle k+1} has its edges partitioned into complete bipartite subgraphs, at least k {\displaystyle k} subgraphs are needed. Equivalently, their conjecture states that edge-disjoint unions of k {\displaystyle k} complete bipartite graphs can always be colored with at most k + 1 {\displaystyle k+1} colors. The conjecture was disproved by Huang and Sudakov in 2012, who constructed families of graphs formed as edge-disjoint unions of k {\displaystyle k} complete bipartite graphs that require Ω ( k 6 / 5 ) {\displaystyle \Omega (k^{6/5})} colors. More strongly, the number of colors can be as large as exp log 2 − o ( 1 ) k {\displaystyle \exp \log ^{2-o(1)}k} , tight up to the o ( 1 ) {\displaystyle o(1)} term in the exponent. === Biclique partition === The biclique partition problem takes as input an arbitrary undirected graph, and asks for a partition of its edges into a minimum number of complete bipartite graphs. It is NP-hard, but fixed-parameter tractable. The best approximation algorithm known for the problem has an approximation ratio of O ( n / log n ) {\displaystyle O(n/\log n)} . == References ==
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Wikipedia:Gram–Schmidt process#0
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In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process or Gram-Schmidt algorithm is a way of finding a set of two or more vectors that are perpendicular to each other. By technical definition, it is a method of constructing an orthonormal basis from a set of vectors in an inner product space, most commonly the Euclidean space R n {\displaystyle \mathbb {R} ^{n}} equipped with the standard inner product. The Gram–Schmidt process takes a finite, linearly independent set of vectors S = { v 1 , … , v k } {\displaystyle S=\{\mathbf {v} _{1},\ldots ,\mathbf {v} _{k}\}} for k ≤ n and generates an orthogonal set S ′ = { u 1 , … , u k } {\displaystyle S'=\{\mathbf {u} _{1},\ldots ,\mathbf {u} _{k}\}} that spans the same k {\displaystyle k} -dimensional subspace of R n {\displaystyle \mathbb {R} ^{n}} as S {\displaystyle S} . The method is named after Jørgen Pedersen Gram and Erhard Schmidt, but Pierre-Simon Laplace had been familiar with it before Gram and Schmidt. In the theory of Lie group decompositions, it is generalized by the Iwasawa decomposition. The application of the Gram–Schmidt process to the column vectors of a full column rank matrix yields the QR decomposition (it is decomposed into an orthogonal and a triangular matrix). == The Gram–Schmidt process == The vector projection of a vector v {\displaystyle \mathbf {v} } on a nonzero vector u {\displaystyle \mathbf {u} } is defined as proj u ( v ) = ⟨ v , u ⟩ ⟨ u , u ⟩ u , {\displaystyle \operatorname {proj} _{\mathbf {u} }(\mathbf {v} )={\frac {\langle \mathbf {v} ,\mathbf {u} \rangle }{\langle \mathbf {u} ,\mathbf {u} \rangle }}\,\mathbf {u} ,} where ⟨ v , u ⟩ {\displaystyle \langle \mathbf {v} ,\mathbf {u} \rangle } denotes the inner product of the vectors u {\displaystyle \mathbf {u} } and v {\displaystyle \mathbf {v} } . This means that proj u ( v ) {\displaystyle \operatorname {proj} _{\mathbf {u} }(\mathbf {v} )} is the orthogonal projection of v {\displaystyle \mathbf {v} } onto the line spanned by u {\displaystyle \mathbf {u} } . If u {\displaystyle \mathbf {u} } is the zero vector, then proj u ( v ) {\displaystyle \operatorname {proj} _{\mathbf {u} }(\mathbf {v} )} is defined as the zero vector. Given k {\displaystyle k} nonzero linearly-independent vectors v 1 , … , v k {\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{k}} the Gram–Schmidt process defines the vectors u 1 , … , u k {\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{k}} as follows: u 1 = v 1 , e 1 = u 1 ‖ u 1 ‖ u 2 = v 2 − proj u 1 ( v 2 ) , e 2 = u 2 ‖ u 2 ‖ u 3 = v 3 − proj u 1 ( v 3 ) − proj u 2 ( v 3 ) , e 3 = u 3 ‖ u 3 ‖ u 4 = v 4 − proj u 1 ( v 4 ) − proj u 2 ( v 4 ) − proj u 3 ( v 4 ) , e 4 = u 4 ‖ u 4 ‖ ⋮ ⋮ u k = v k − ∑ j = 1 k − 1 proj u j ( v k ) , e k = u k ‖ u k ‖ . {\displaystyle {\begin{aligned}\mathbf {u} _{1}&=\mathbf {v} _{1},&\!\mathbf {e} _{1}&={\frac {\mathbf {u} _{1}}{\|\mathbf {u} _{1}\|}}\\\mathbf {u} _{2}&=\mathbf {v} _{2}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{2}),&\!\mathbf {e} _{2}&={\frac {\mathbf {u} _{2}}{\|\mathbf {u} _{2}\|}}\\\mathbf {u} _{3}&=\mathbf {v} _{3}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{3})-\operatorname {proj} _{\mathbf {u} _{2}}(\mathbf {v} _{3}),&\!\mathbf {e} _{3}&={\frac {\mathbf {u} _{3}}{\|\mathbf {u} _{3}\|}}\\\mathbf {u} _{4}&=\mathbf {v} _{4}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{4})-\operatorname {proj} _{\mathbf {u} _{2}}(\mathbf {v} _{4})-\operatorname {proj} _{\mathbf {u} _{3}}(\mathbf {v} _{4}),&\!\mathbf {e} _{4}&={\mathbf {u} _{4} \over \|\mathbf {u} _{4}\|}\\&{}\ \ \vdots &&{}\ \ \vdots \\\mathbf {u} _{k}&=\mathbf {v} _{k}-\sum _{j=1}^{k-1}\operatorname {proj} _{\mathbf {u} _{j}}(\mathbf {v} _{k}),&\!\mathbf {e} _{k}&={\frac {\mathbf {u} _{k}}{\|\mathbf {u} _{k}\|}}.\end{aligned}}} The sequence u 1 , … , u k {\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{k}} is the required system of orthogonal vectors, and the normalized vectors e 1 , … , e k {\displaystyle \mathbf {e} _{1},\ldots ,\mathbf {e} _{k}} form an orthonormal set. The calculation of the sequence u 1 , … , u k {\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{k}} is known as Gram–Schmidt orthogonalization, and the calculation of the sequence e 1 , … , e k {\displaystyle \mathbf {e} _{1},\ldots ,\mathbf {e} _{k}} is known as Gram–Schmidt orthonormalization. To check that these formulas yield an orthogonal sequence, first compute ⟨ u 1 , u 2 ⟩ {\displaystyle \langle \mathbf {u} _{1},\mathbf {u} _{2}\rangle } by substituting the above formula for u 2 {\displaystyle \mathbf {u} _{2}} : we get zero. Then use this to compute ⟨ u 1 , u 3 ⟩ {\displaystyle \langle \mathbf {u} _{1},\mathbf {u} _{3}\rangle } again by substituting the formula for u 3 {\displaystyle \mathbf {u} _{3}} : we get zero. For arbitrary k {\displaystyle k} the proof is accomplished by mathematical induction. Geometrically, this method proceeds as follows: to compute u i {\displaystyle \mathbf {u} _{i}} , it projects v i {\displaystyle \mathbf {v} _{i}} orthogonally onto the subspace U {\displaystyle U} generated by u 1 , … , u i − 1 {\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{i-1}} , which is the same as the subspace generated by v 1 , … , v i − 1 {\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{i-1}} . The vector u i {\displaystyle \mathbf {u} _{i}} is then defined to be the difference between v i {\displaystyle \mathbf {v} _{i}} and this projection, guaranteed to be orthogonal to all of the vectors in the subspace U {\displaystyle U} . The Gram–Schmidt process also applies to a linearly independent countably infinite sequence {vi}i. The result is an orthogonal (or orthonormal) sequence {ui}i such that for natural number n: the algebraic span of v 1 , … , v n {\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{n}} is the same as that of u 1 , … , u n {\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{n}} . If the Gram–Schmidt process is applied to a linearly dependent sequence, it outputs the 0 vector on the i {\displaystyle i} th step, assuming that v i {\displaystyle \mathbf {v} _{i}} is a linear combination of v 1 , … , v i − 1 {\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{i-1}} . If an orthonormal basis is to be produced, then the algorithm should test for zero vectors in the output and discard them because no multiple of a zero vector can have a length of 1. The number of vectors output by the algorithm will then be the dimension of the space spanned by the original inputs. A variant of the Gram–Schmidt process using transfinite recursion applied to a (possibly uncountably) infinite sequence of vectors ( v α ) α < λ {\displaystyle (v_{\alpha })_{\alpha <\lambda }} yields a set of orthonormal vectors ( u α ) α < κ {\displaystyle (u_{\alpha })_{\alpha <\kappa }} with κ ≤ λ {\displaystyle \kappa \leq \lambda } such that for any α ≤ λ {\displaystyle \alpha \leq \lambda } , the completion of the span of { u β : β < min ( α , κ ) } {\displaystyle \{u_{\beta }:\beta <\min(\alpha ,\kappa )\}} is the same as that of { v β : β < α } {\displaystyle \{v_{\beta }:\beta <\alpha \}} . In particular, when applied to a (algebraic) basis of a Hilbert space (or, more generally, a basis of any dense subspace), it yields a (functional-analytic) orthonormal basis. Note that in the general case often the strict inequality κ < λ {\displaystyle \kappa <\lambda } holds, even if the starting set was linearly independent, and the span of ( u α ) α < κ {\displaystyle (u_{\alpha })_{\alpha <\kappa }} need not be a subspace of the span of ( v α ) α < λ {\displaystyle (v_{\alpha })_{\alpha <\lambda }} (rather, it's a subspace of its completion). == Example == === Euclidean space === Consider the following set of vectors in R 2 {\displaystyle \mathbb {R} ^{2}} (with the conventional inner product) S = { v 1 = [ 3 1 ] , v 2 = [ 2 2 ] } . {\displaystyle S=\left\{\mathbf {v} _{1}={\begin{bmatrix}3\\1\end{bmatrix}},\mathbf {v} _{2}={\begin{bmatrix}2\\2\end{bmatrix}}\right\}.} Now, perform Gram–Schmidt, to obtain an orthogonal set of vectors: u 1 = v 1 = [ 3 1 ] {\displaystyle \mathbf {u} _{1}=\mathbf {v} _{1}={\begin{bmatrix}3\\1\end{bmatrix}}} u 2 = v 2 − proj u 1 ( v 2 ) = [ 2 2 ] − proj [ 3 1 ] [ 2 2 ] = [ 2 2 ] − 8 10 [ 3 1 ] = [ − 2 / 5 6 / 5 ] . {\displaystyle \mathbf {u} _{2}=\mathbf {v} _{2}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{2})={\begin{bmatrix}2\\2\end{bmatrix}}-\operatorname {proj} _{\left[{\begin{smallmatrix}3\\1\end{smallmatrix}}\right]}{\begin{bmatrix}2\\2\end{bmatrix}}={\begin{bmatrix}2\\2\end{bmatrix}}-{\frac {8}{10}}{\begin{bmatrix}3\\1\end{bmatrix}}={\begin{bmatrix}-2/5\\6/5\end{bmatrix}}.} We check that the vectors u 1 {\displaystyle \mathbf {u} _{1}} and u 2 {\displaystyle \mathbf {u} _{2}} are indeed orthogonal: ⟨ u 1 , u 2 ⟩ = ⟨ [ 3 1 ] , [ − 2 / 5 6 / 5 ] ⟩ = − 6 5 + 6 5 = 0 , {\displaystyle \langle \mathbf {u} _{1},\mathbf {u} _{2}\rangle =\left\langle {\begin{bmatrix}3\\1\end{bmatrix}},{\begin{bmatrix}-2/5\\6/5\end{bmatrix}}\right\rangle =-{\frac {6}{5}}+{\frac {6}{5}}=0,} noting that if the dot product of two vectors is 0 then they are orthogonal. For non-zero vectors, we can then normalize the vectors by dividing out their sizes as shown above: e 1 = 1 10 [ 3 1 ] {\displaystyle \mathbf {e} _{1}={\frac {1}{\sqrt {10}}}{\begin{bmatrix}3\\1\end{bmatrix}}} e 2 = 1 40 25 [ − 2 / 5 6 / 5 ] = 1 10 [ − 1 3 ] . {\displaystyle \mathbf {e} _{2}={\frac {1}{\sqrt {40 \over 25}}}{\begin{bmatrix}-2/5\\6/5\end{bmatrix}}={\frac {1}{\sqrt {10}}}{\begin{bmatrix}-1\\3\end{bmatrix}}.} == Properties == Denote by GS ( v 1 , … , v k ) {\displaystyle \operatorname {GS} (\mathbf {v} _{1},\dots ,\mathbf {v} _{k})} the result of applying the Gram–Schmidt process to a collection of vectors v 1 , … , v k {\displaystyle \mathbf {v} _{1},\dots ,\mathbf {v} _{k}} . This yields a map GS : ( R n ) k → ( R n ) k {\displaystyle \operatorname {GS} \colon (\mathbb {R} ^{n})^{k}\to (\mathbb {R} ^{n})^{k}} . It has the following properties: It is continuous It is orientation preserving in the sense that or ( v 1 , … , v k ) = or ( GS ( v 1 , … , v k ) ) {\displaystyle \operatorname {or} (\mathbf {v} _{1},\dots ,\mathbf {v} _{k})=\operatorname {or} (\operatorname {GS} (\mathbf {v} _{1},\dots ,\mathbf {v} _{k}))} . It commutes with orthogonal maps: Let g : R n → R n {\displaystyle g\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}} be orthogonal (with respect to the given inner product). Then we have GS ( g ( v 1 ) , … , g ( v k ) ) = ( g ( GS ( v 1 , … , v k ) 1 ) , … , g ( GS ( v 1 , … , v k ) k ) ) {\displaystyle \operatorname {GS} (g(\mathbf {v} _{1}),\dots ,g(\mathbf {v} _{k}))=\left(g(\operatorname {GS} (\mathbf {v} _{1},\dots ,\mathbf {v} _{k})_{1}),\dots ,g(\operatorname {GS} (\mathbf {v} _{1},\dots ,\mathbf {v} _{k})_{k})\right)} Further, a parametrized version of the Gram–Schmidt process yields a (strong) deformation retraction of the general linear group G L ( R n ) {\displaystyle \mathrm {GL} (\mathbb {R} ^{n})} onto the orthogonal group O ( R n ) {\displaystyle O(\mathbb {R} ^{n})} . == Numerical stability == When this process is implemented on a computer, the vectors u k {\displaystyle \mathbf {u} _{k}} are often not quite orthogonal, due to rounding errors. For the Gram–Schmidt process as described above (sometimes referred to as "classical Gram–Schmidt") this loss of orthogonality is particularly bad; therefore, it is said that the (classical) Gram–Schmidt process is numerically unstable. The Gram–Schmidt process can be stabilized by a small modification; this version is sometimes referred to as modified Gram-Schmidt or MGS. This approach gives the same result as the original formula in exact arithmetic and introduces smaller errors in finite-precision arithmetic. Instead of computing the vector uk as u k = v k − proj u 1 ( v k ) − proj u 2 ( v k ) − ⋯ − proj u k − 1 ( v k ) , {\displaystyle \mathbf {u} _{k}=\mathbf {v} _{k}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{k})-\operatorname {proj} _{\mathbf {u} _{2}}(\mathbf {v} _{k})-\cdots -\operatorname {proj} _{\mathbf {u} _{k-1}}(\mathbf {v} _{k}),} it is computed as u k ( 1 ) = v k − proj u 1 ( v k ) , u k ( 2 ) = u k ( 1 ) − proj u 2 ( u k ( 1 ) ) , ⋮ u k ( k − 2 ) = u k ( k − 3 ) − proj u k − 2 ( u k ( k − 3 ) ) , u k ( k − 1 ) = u k ( k − 2 ) − proj u k − 1 ( u k ( k − 2 ) ) , e k = u k ( k − 1 ) ‖ u k ( k − 1 ) ‖ {\displaystyle {\begin{aligned}\mathbf {u} _{k}^{(1)}&=\mathbf {v} _{k}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{k}),\\\mathbf {u} _{k}^{(2)}&=\mathbf {u} _{k}^{(1)}-\operatorname {proj} _{\mathbf {u} _{2}}\left(\mathbf {u} _{k}^{(1)}\right),\\&\;\;\vdots \\\mathbf {u} _{k}^{(k-2)}&=\mathbf {u} _{k}^{(k-3)}-\operatorname {proj} _{\mathbf {u} _{k-2}}\left(\mathbf {u} _{k}^{(k-3)}\right),\\\mathbf {u} _{k}^{(k-1)}&=\mathbf {u} _{k}^{(k-2)}-\operatorname {proj} _{\mathbf {u} _{k-1}}\left(\mathbf {u} _{k}^{(k-2)}\right),\\\mathbf {e} _{k}&={\frac {\mathbf {u} _{k}^{(k-1)}}{\left\|\mathbf {u} _{k}^{(k-1)}\right\|}}\end{aligned}}} This method is used in the previous animation, when the intermediate v 3 ′ {\displaystyle \mathbf {v} '_{3}} vector is used when orthogonalizing the blue vector v 3 {\displaystyle \mathbf {v} _{3}} . Here is another description of the modified algorithm. Given the vectors v 1 , v 2 , … , v n {\displaystyle \mathbf {v} _{1},\mathbf {v} _{2},\dots ,\mathbf {v} _{n}} , in our first step we produce vectors v 1 , v 2 ( 1 ) , … , v n ( 1 ) {\displaystyle \mathbf {v} _{1},\mathbf {v} _{2}^{(1)},\dots ,\mathbf {v} _{n}^{(1)}} by removing components along the direction of v 1 {\displaystyle \mathbf {v} _{1}} . In formulas, v k ( 1 ) := v k − ⟨ v k , v 1 ⟩ ⟨ v 1 , v 1 ⟩ v 1 {\displaystyle \mathbf {v} _{k}^{(1)}:=\mathbf {v} _{k}-{\frac {\langle \mathbf {v} _{k},\mathbf {v} _{1}\rangle }{\langle \mathbf {v} _{1},\mathbf {v} _{1}\rangle }}\mathbf {v} _{1}} . After this step we already have two of our desired orthogonal vectors u 1 , … , u n {\displaystyle \mathbf {u} _{1},\dots ,\mathbf {u} _{n}} , namely u 1 = v 1 , u 2 = v 2 ( 1 ) {\displaystyle \mathbf {u} _{1}=\mathbf {v} _{1},\mathbf {u} _{2}=\mathbf {v} _{2}^{(1)}} , but we also made v 3 ( 1 ) , … , v n ( 1 ) {\displaystyle \mathbf {v} _{3}^{(1)},\dots ,\mathbf {v} _{n}^{(1)}} already orthogonal to u 1 {\displaystyle \mathbf {u} _{1}} . Next, we orthogonalize those remaining vectors against u 2 = v 2 ( 1 ) {\displaystyle \mathbf {u} _{2}=\mathbf {v} _{2}^{(1)}} . This means we compute v 3 ( 2 ) , v 4 ( 2 ) , … , v n ( 2 ) {\displaystyle \mathbf {v} _{3}^{(2)},\mathbf {v} _{4}^{(2)},\dots ,\mathbf {v} _{n}^{(2)}} by subtraction v k ( 2 ) := v k ( 1 ) − ⟨ v k ( 1 ) , u 2 ⟩ ⟨ u 2 , u 2 ⟩ u 2 {\displaystyle \mathbf {v} _{k}^{(2)}:=\mathbf {v} _{k}^{(1)}-{\frac {\langle \mathbf {v} _{k}^{(1)},\mathbf {u} _{2}\rangle }{\langle \mathbf {u} _{2},\mathbf {u} _{2}\rangle }}\mathbf {u} _{2}} . Now we have stored the vectors v 1 , v 2 ( 1 ) , v 3 ( 2 ) , v 4 ( 2 ) , … , v n ( 2 ) {\displaystyle \mathbf {v} _{1},\mathbf {v} _{2}^{(1)},\mathbf {v} _{3}^{(2)},\mathbf {v} _{4}^{(2)},\dots ,\mathbf {v} _{n}^{(2)}} where the first three vectors are already u 1 , u 2 , u 3 {\displaystyle \mathbf {u} _{1},\mathbf {u} _{2},\mathbf {u} _{3}} and the remaining vectors are already orthogonal to u 1 , u 2 {\displaystyle \mathbf {u} _{1},\mathbf {u} _{2}} . As should be clear now, the next step orthogonalizes v 4 ( 2 ) , … , v n ( 2 ) {\displaystyle \mathbf {v} _{4}^{(2)},\dots ,\mathbf {v} _{n}^{(2)}} against u 3 = v 3 ( 2 ) {\displaystyle \mathbf {u} _{3}=\mathbf {v} _{3}^{(2)}} . Proceeding in this manner we find the full set of orthogonal vectors u 1 , … , u n {\displaystyle \mathbf {u} _{1},\dots ,\mathbf {u} _{n}} . If orthonormal vectors are desired, then we normalize as we go, so that the denominators in the subtraction formulas turn into ones. == Algorithm == The following MATLAB algorithm implements classical Gram–Schmidt orthonormalization. The vectors v1, ..., vk (columns of matrix V, so that V(:,j) is the j {\displaystyle j} th vector) are replaced by orthonormal vectors (columns of U) which span the same subspace. The cost of this algorithm is asymptotically O(nk2) floating point operations, where n is the dimensionality of the vectors. == Via Gaussian elimination == If the rows {v1, ..., vk} are written as a matrix A {\displaystyle A} , then applying Gaussian elimination to the augmented matrix [ A A T | A ] {\displaystyle \left[AA^{\mathsf {T}}|A\right]} will produce the orthogonalized vectors in place of A {\displaystyle A} . However the matrix A A T {\displaystyle AA^{\mathsf {T}}} must be brought to row echelon form, using only the row operation of adding a scalar multiple of one row to another. For example, taking v 1 = [ 3 1 ] , v 2 = [ 2 2 ] {\displaystyle \mathbf {v} _{1}={\begin{bmatrix}3&1\end{bmatrix}},\mathbf {v} _{2}={\begin{bmatrix}2&2\end{bmatrix}}} as above, we have [ A A T | A ] = [ 10 8 3 1 8 8 2 2 ] {\displaystyle \left[AA^{\mathsf {T}}|A\right]=\left[{\begin{array}{rr|rr}10&8&3&1\\8&8&2&2\end{array}}\right]} And reducing this to row echelon form produces [ 1 .8 .3 .1 0 1 − .25 .75 ] {\displaystyle \left[{\begin{array}{rr|rr}1&.8&.3&.1\\0&1&-.25&.75\end{array}}\right]} The normalized vectors are then e 1 = 1 .3 2 + .1 2 [ .3 .1 ] = 1 10 [ 3 1 ] {\displaystyle \mathbf {e} _{1}={\frac {1}{\sqrt {.3^{2}+.1^{2}}}}{\begin{bmatrix}.3&.1\end{bmatrix}}={\frac {1}{\sqrt {10}}}{\begin{bmatrix}3&1\end{bmatrix}}} e 2 = 1 .25 2 + .75 2 [ − .25 .75 ] = 1 10 [ − 1 3 ] , {\displaystyle \mathbf {e} _{2}={\frac {1}{\sqrt {.25^{2}+.75^{2}}}}{\begin{bmatrix}-.25&.75\end{bmatrix}}={\frac {1}{\sqrt {10}}}{\begin{bmatrix}-1&3\end{bmatrix}},} as in the example above. == Determinant formula == The result of the Gram–Schmidt process may be expressed in a non-recursive formula using determinants. e j = 1 D j − 1 D j | ⟨ v 1 , v 1 ⟩ ⟨ v 2 , v 1 ⟩ ⋯ ⟨ v j , v 1 ⟩ ⟨ v 1 , v 2 ⟩ ⟨ v 2 , v 2 ⟩ ⋯ ⟨ v j , v 2 ⟩ ⋮ ⋮ ⋱ ⋮ ⟨ v 1 , v j − 1 ⟩ ⟨ v 2 , v j − 1 ⟩ ⋯ ⟨ v j , v j − 1 ⟩ v 1 v 2 ⋯ v j | {\displaystyle \mathbf {e} _{j}={\frac {1}{\sqrt {D_{j-1}D_{j}}}}{\begin{vmatrix}\langle \mathbf {v} _{1},\mathbf {v} _{1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{1}\rangle \\\langle \mathbf {v} _{1},\mathbf {v} _{2}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{2}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{2}\rangle \\\vdots &\vdots &\ddots &\vdots \\\langle \mathbf {v} _{1},\mathbf {v} _{j-1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{j-1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{j-1}\rangle \\\mathbf {v} _{1}&\mathbf {v} _{2}&\cdots &\mathbf {v} _{j}\end{vmatrix}}} u j = 1 D j − 1 | ⟨ v 1 , v 1 ⟩ ⟨ v 2 , v 1 ⟩ ⋯ ⟨ v j , v 1 ⟩ ⟨ v 1 , v 2 ⟩ ⟨ v 2 , v 2 ⟩ ⋯ ⟨ v j , v 2 ⟩ ⋮ ⋮ ⋱ ⋮ ⟨ v 1 , v j − 1 ⟩ ⟨ v 2 , v j − 1 ⟩ ⋯ ⟨ v j , v j − 1 ⟩ v 1 v 2 ⋯ v j | {\displaystyle \mathbf {u} _{j}={\frac {1}{D_{j-1}}}{\begin{vmatrix}\langle \mathbf {v} _{1},\mathbf {v} _{1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{1}\rangle \\\langle \mathbf {v} _{1},\mathbf {v} _{2}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{2}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{2}\rangle \\\vdots &\vdots &\ddots &\vdots \\\langle \mathbf {v} _{1},\mathbf {v} _{j-1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{j-1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{j-1}\rangle \\\mathbf {v} _{1}&\mathbf {v} _{2}&\cdots &\mathbf {v} _{j}\end{vmatrix}}} where D 0 = 1 {\displaystyle D_{0}=1} and, for j ≥ 1 {\displaystyle j\geq 1} , D j {\displaystyle D_{j}} is the Gram determinant D j = | ⟨ v 1 , v 1 ⟩ ⟨ v 2 , v 1 ⟩ ⋯ ⟨ v j , v 1 ⟩ ⟨ v 1 , v 2 ⟩ ⟨ v 2 , v 2 ⟩ ⋯ ⟨ v j , v 2 ⟩ ⋮ ⋮ ⋱ ⋮ ⟨ v 1 , v j ⟩ ⟨ v 2 , v j ⟩ ⋯ ⟨ v j , v j ⟩ | . {\displaystyle D_{j}={\begin{vmatrix}\langle \mathbf {v} _{1},\mathbf {v} _{1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{1}\rangle \\\langle \mathbf {v} _{1},\mathbf {v} _{2}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{2}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{2}\rangle \\\vdots &\vdots &\ddots &\vdots \\\langle \mathbf {v} _{1},\mathbf {v} _{j}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{j}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{j}\rangle \end{vmatrix}}.} Note that the expression for u k {\displaystyle \mathbf {u} _{k}} is a "formal" determinant, i.e. the matrix contains both scalars and vectors; the meaning of this expression is defined to be the result of a cofactor expansion along the row of vectors. The determinant formula for the Gram-Schmidt is computationally (exponentially) slower than the recursive algorithms described above; it is mainly of theoretical interest. == Expressed using geometric algebra == Expressed using notation used in geometric algebra, the unnormalized results of the Gram–Schmidt process can be expressed as u k = v k − ∑ j = 1 k − 1 ( v k ⋅ u j ) u j − 1 , {\displaystyle \mathbf {u} _{k}=\mathbf {v} _{k}-\sum _{j=1}^{k-1}(\mathbf {v} _{k}\cdot \mathbf {u} _{j})\mathbf {u} _{j}^{-1}\ ,} which is equivalent to the expression using the proj {\displaystyle \operatorname {proj} } operator defined above. The results can equivalently be expressed as u k = v k ∧ v k − 1 ∧ ⋅ ⋅ ⋅ ∧ v 1 ( v k − 1 ∧ ⋅ ⋅ ⋅ ∧ v 1 ) − 1 , {\displaystyle \mathbf {u} _{k}=\mathbf {v} _{k}\wedge \mathbf {v} _{k-1}\wedge \cdot \cdot \cdot \wedge \mathbf {v} _{1}(\mathbf {v} _{k-1}\wedge \cdot \cdot \cdot \wedge \mathbf {v} _{1})^{-1},} which is closely related to the expression using determinants above. == Alternatives == Other orthogonalization algorithms use Householder transformations or Givens rotations. The algorithms using Householder transformations are more stable than the stabilized Gram–Schmidt process. On the other hand, the Gram–Schmidt process produces the j {\displaystyle j} th orthogonalized vector after the j {\displaystyle j} th iteration, while orthogonalization using Householder reflections produces all the vectors only at the end. This makes only the Gram–Schmidt process applicable for iterative methods like the Arnoldi iteration. Yet another alternative is motivated by the use of Cholesky decomposition for inverting the matrix of the normal equations in linear least squares. Let V {\displaystyle V} be a full column rank matrix, whose columns need to be orthogonalized. The matrix V ∗ V {\displaystyle V^{*}V} is Hermitian and positive definite, so it can be written as V ∗ V = L L ∗ , {\displaystyle V^{*}V=LL^{*},} using the Cholesky decomposition. The lower triangular matrix L {\displaystyle L} with strictly positive diagonal entries is invertible. Then columns of the matrix U = V ( L − 1 ) ∗ {\displaystyle U=V\left(L^{-1}\right)^{*}} are orthonormal and span the same subspace as the columns of the original matrix V {\displaystyle V} . The explicit use of the product V ∗ V {\displaystyle V^{*}V} makes the algorithm unstable, especially if the product's condition number is large. Nevertheless, this algorithm is used in practice and implemented in some software packages because of its high efficiency and simplicity. In quantum mechanics there are several orthogonalization schemes with characteristics better suited for certain applications than original Gram–Schmidt. Nevertheless, it remains a popular and effective algorithm for even the largest electronic structure calculations. == Run-time complexity == Gram-Schmidt orthogonalization can be done in strongly-polynomial time. The run-time analysis is similar to that of Gaussian elimination.: 40 == See also == Linear algebra Recursion Orthogonality (mathematics) == References == == Notes == == Sources == Bau III, David; Trefethen, Lloyd N. (1997), Numerical linear algebra, Philadelphia: Society for Industrial and Applied Mathematics, ISBN 978-0-89871-361-9. Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Johns Hopkins, ISBN 978-0-8018-5414-9. Greub, Werner (1975), Linear Algebra (4th ed.), Springer. Soliverez, C. E.; Gagliano, E. (1985), "Orthonormalization on the plane: a geometric approach" (PDF), Mex. J. Phys., 31 (4): 743–758, archived from the original (PDF) on 2014-03-07, retrieved 2013-06-22. == External links == "Orthogonalization", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Harvey Mudd College Math Tutorial on the Gram-Schmidt algorithm Earliest known uses of some of the words of mathematics: G The entry "Gram-Schmidt orthogonalization" has some information and references on the origins of the method. Demos: Gram Schmidt process in plane and Gram Schmidt process in space Gram-Schmidt orthogonalization applet NAG Gram–Schmidt orthogonalization of n vectors of order m routine Proof: Raymond Puzio, Keenan Kidwell. "proof of Gram-Schmidt orthogonalization algorithm" (version 8). PlanetMath.org.
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Wikipedia:Graph algebra (social sciences)#0
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Graph algebra is systems-centric modeling tool for the social sciences. It was first developed by Sprague, Pzeworski, and Cortes as a hybridized version of engineering plots to describe social phenomena. == Notes and references ==
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