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Wikipedia:Mary Margaret Speer#0 | Mary Margaret Speer, (March 27, 1906 in Midway, Pennsylvania – November 23, 1966 in Pittsburgh) was an American mathematician and university instructor and one of the few women to earn a PhD in math before World War II. == Biography == Mary Margaret Taylor was the only daughter and third of four children born to Hallie Blanche Virginia Criss and Albert Aaron Taylor. Mary completed the three-year curriculum at Midway High School from 1918 to 1921. For her last year, she attended Carnegie High School just west of Pittsburgh where she was chosen to receive a four-year college scholarship based on an annual competitive examination sponsored by nearby Washington County, Pennsylvania. From 1922 to 1926 she undertook a double major in mathematics and Latin offered by the University of Pittsburgh. After receiving her honors BA degree in 1926, she was elected to the Sigma Kappa Phi honor society, a national foreign language honorary fraternity, and the Women's Studies Honor Society in Pittsburgh. From 1926 to 1930, while Taylor was working as a teaching assistant at the University of Pittsburgh, she completed her master's degree in 1928 with a thesis titled: A note on the solution in series of the general homogeneous linear differential equation. She attended summer programs at the University of Chicago during the summer of 1928 and then continued her work toward her PhD at Pittsburgh. From 1930 to 1934 Taylor taught math courses at University's regional campus in Johnstown, Pennsylvania, before moving back to the University's main campus in Pittsburgh as a graduate assistant and she finished her PhD studies. Taylor received her doctorate in 1935 with the dissertation: Reciprocals of Certain Curves and Surfaces with Respect to a Space Cubic Curve directed by Forest Almos Foraker. In 1935, she married the mathematician and lawyer Eugene R. Speer (1903–1978), with whom she had three children, all of whom also went on to study mathematics. For the school year 1936-1937, she is listed in the university catalogue as a graduate assistant using her new name Mary Margaret Speer. From 1937 through 1943, she is listed as an instructor of mathematics but she left the University after being told that men with less experience were receiving promotions to Associate Professor ahead of her because "the men had families to support and needed the money." This occurred even though Mary and her husband were already the parents of one child by that time. Speer was an active member and administrator of the Girl Scouts of Allegheny County, from the early 1950s until her death of cancer in Pittsburgh in 1966 at the age of 60. == Selected publications == === Publication === 1929: [Taylor, M. M.] (Translator from the Latin) On the theory of combinations, by Jacques Bernoulli. In A Source Book in Mathematics, ed. D. E. Smith. New York: McGraw-Hill Book Co. Reprint: 1959. New York: Dover Publications. === Abstracts === 1937: Reciprocals with respect to a space cubic. Amer. Math. Monthly 44:187 #1. Presented to the MAA, Beaver Falls, PA, 26 Oct 1935. 1939: Types of curvature of curves and surfaces. Amer. Math. Monthly 46:536 #4. Presented to the MAA, Greenville, PA, 13 May 1939. == References == |
Wikipedia:Mary Pugh#0 | Mary Claire Pugh is an applied mathematician known for her research on thin films, including the thin-film equation and Hele-Shaw flow. She is a professor of mathematics at the University of Toronto. Pugh completed her Ph.D. in 1993 at the University of Chicago. Her dissertation, Dynamics of Interfaces of Incompressible Fluids: The Hele-Shaw Problem, was supervised by Peter Constantin. Before moving to Toronto, she worked at the Courant Institute of Mathematical Sciences at New York University and then as a faculty member at the University of Pennsylvania, where she won a Sloan Research Fellowship in 1999. == References == == External links == Home page Mary Pugh publications indexed by Google Scholar |
Wikipedia:Mary Sandoval#0 | Mary Ruth Sandoval is an American mathematician, the Seabury Professor of Mathematics and Natural Philosophy at Trinity College (Connecticut). Her research interests include global analysis, the study of differential equations on varying topological spaces, and spectral geometry, the study of these spaces through the systems of fundamental solutions to certain differential equations. Sandoval majored in mathematics at Yale University, graduating in 1989. She worked for the United States Department of Energy on mathematical models related to the Clean Air Act, before returning to the University of Michigan for graduate study in mathematics, earning a master's degree in 1993 and completing her Ph.D. in 1997. Her dissertation, Wave-Trace Asymptotics For Operators Of Dirac Type, was supervised by Alejandro Uribe. After postdoctoral research at Purdue University as a Project NExT Fellow, she took a visiting position at Trinity College in 1999, and remained as an assistant professor beginning in 2001. She became full professor in 2023, and was named as Seabury Professor in 2024. The Seabury Professorship, established in 1830, is the oldest endowed professorship at Trinity College. == References == |
Wikipedia:Mary Schaps#0 | Mary Elizabeth Schaps (Hebrew: מלכה אלישבע שפס; born August 6, 1948), also known as Malka Elisheva Schaps, is an Israeli-American mathematician. She is Professor of Mathematics and Dean of the Faculty of Exact Sciences at Bar Ilan University. She received her Ph.D. from Harvard University, and has published in deformation theory, group theory, and representation theory. She is also a writer, authoring several novels under the pseudonym Rachel Pomerantz. == Early and personal life == Mary Elizabeth Kramer was born on August 6, 1948, in Cleveland, Ohio, United States. From 1965 to 1969, she was educated at Swarthmore College in Pennsylvania, majoring in mathematics, philosophy and history, and graduating summa cum laude in 1969. She then attended Harvard University as a mathematics graduate student: she completed her Master of Arts (MA) degree in 1971 and her Doctor of Philosophy (PhD) degree in 1972. Her doctoral thesis was titled "Non-singular deformations of space curves, using determinantal schemes": her advisors were David Mumford and Heisuke Hironaka. She was brought up a Presbyterian and then a Unitarian Universalism. She became interested in Judaism, and formally converted to Conservative Judaism in college. In 1968, she married David Schaps, a classics professor who was then also studying for a PhD at Harvard. They both gradually felt more and more drawn to Orthodox Judaism and considered themselves Haredi by the time they had completed their doctorates. The couple made aliyah (moved to Israel) in 1972. Together, she and her husband had two children: this was a small family by Haredi standards, and they raised four more foster children. They live in Bnei Brak, a Haredi city near Tel Aviv. == Career == Schaps had some teaching experience at Harvard University: she was teaching assistant from 1971 to 1972, and a lecturer at the 1975 Harvard Summer School. In 1972, having moved to Israel, she was appointed a lecturer in mathematics at Tel Aviv University. In her second year in Israel the Yom Kippur War occurred: the change in language, culture and the threat to her new family were a shock. In 1977, she moved to Bar-Ilan University as a senior lecturer: she was promoted to associate professor in 1991 and full professor in 2006. She was the only female Haredi professor in Israel. In October 2013, she was appointed Dean of Exact Sciences at Bar-Ilan University: having been the only female Haredi professor, she now became the highest ranking Haredi woman in Israeli academia. She stepped down as dean in 2015, and was made Professor Emeritus on retirement in 2016. == Selected publications == Kanovei, Vladimir; Katz, Karin U.; Katz, Mikhail G.; Schaps, Mary (2015), "Proofs and Retributions, Or: Why Sarah Can't Take Limits", Foundations of Science, 20: 1–25, doi:10.1007/s10699-013-9340-0, S2CID 123352636. Katz, Mikhail G.; Schaps, Mary; Vishne, Uzi. Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups. J. Differential Geom. 76 (2007), no. 3, 399–422. Schaps, Mary. Deformations of Cohen–Macaulay schemes of codimension 2 and non-singular deformations of space curves. Amer. J. Math. 99 (1977), no. 4, 669–685. Schaps, Mary; Zakay-Illouz, Evelyne. Combinatorial partial tilting complexes for the Brauer star algebras. Representations of algebras (São Paulo, 1999), 187–207, Lecture Notes in Pure and Appl. Math., 224, Dekker, New York, 2002. == References == == External links == Prof. Malka Schaps Bar Ilan University |
Wikipedia:Mary Tiles#0 | Mary Tiles (born 1946) is a philosopher and historian of mathematics and science. From 2006 until 2009, she served as chair of the philosophy department of the University of Hawaii at Manoa. She retired in 2009. == Life == At Bristol University, Tiles obtained her B.A. in philosophy and mathematics in 1967, and her Ph.D. in philosophy in 1973, followed by a B.Phil. in philosophy in 1974 at Oxford and a M.A. in 1978 at Cambridge. After positions as lecturer and visiting associate professor at different institutions, Tiles became associate professor of philosophy at University of Hawaii at Manoa in 1989, and full professor in 1992. == Work == Tiles' area of work is primarily philosophy and history of logic, mathematics and science, with a special emphasis on French contributions to this area, e.g. by Gaston Bachelard, Georges Canguilhem, Bruno Latour, Michel Foucault, Pierre Bourdieu, Michel Serres, Jean-Claude Martzloff, Karine Chemla, Catherine Jami, and François Jullien. One of her publications is the 1989 book The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise. As the subtitle suggests, it is an example of a book that treats the philosophy of mathematics as inseparable from historical concerns. Despite some criticisms, for its lack of technical detail and correctness, and for pressing the author's philosophical agenda on its readers, it has been recommended as an introductory textbook for undergraduates interested in the philosophy of mathematics. == Bibliography == with Hans Oberdiek, Living in a Technological Culture: Human Tools and Human Values, Routledge 1995. with Jim Tiles, An Introduction to Historical Epistemology: The Authority of Knowledge, Oxford 1993. Mathematics and the Image of Reason, Routledge 1991. The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise, Blackwell 1989; reprinted by Dover 2004. Bachelard: Science and Objectivity, Cambridge University Press 1984. == References == |
Wikipedia:María Emilia Caballero#0 | María Emilia Caballero Acosta is a Mexican mathematician specializing in probability theory, including Lévy processes, branching processes, Markov processes, and Lamperti representations (an exponential relation between Markov processes and Lévy processes). She is a professor in the Faculty of Sciences and Researcher in the Institute of Sciences of the National Autonomous University of Mexico (UNAM). == Education and career == After doing her undergraduate studies at the National Autonomous University of Mexico, Caballero went to Pierre and Marie Curie University in France for graduate study in mathematics. She completed a doctorat de troisième cycle in 1973, with the dissertation Quelques proprietes en theorie du potentiel in potential theory, jointly supervised by Marcel Brelot and Paul Malliavin. Her interest in probability theory developed out of this work and the probabilistic theory of potential. Already in 1964, she had begun working as an adjunct professor at the Escuela Nacional Preparatoria and as an assistant in the Faculty of Sciences at UNAM. On completing her doctorate in 1973, she took her present position at the Institute of Mathematics. == Recognition == Caballero is a member of the Mexican Academy of Sciences. She won UNAM's Juana de Asbaje Medal in 2004. In 2012 she won UNAM's National University Award, the first woman in the Institute of Mathematics to win this award. == References == |
Wikipedia:María J. Carro#0 | María Jesús Carro Rossell (born 1961) is a Spanish mathematician specializing in mathematical analysis, including Fourier analysis, functional analysis, harmonic analysis, operator theory and the analysis of Lorentz spaces. She is a professor at the Complutense University of Madrid, in the Department of Mathematical Analysis and Applied Mathematics. == Education and career == Carro was born in 1961, and motivated to work in mathematics by her father, who was prevented from studying science by the Spanish Civil War. She earned a degree in mathematical sciences in 1984 from the University of Extremadura. Next, she went to the University of Barcelona for doctoral study in mathematics, completing her Ph.D. in 1988 under the supervision of Joan Cerdà, with the dissertation Interpolación compleja de operadores lineales. After postdoctoral study at Washington University in St. Louis with Guido Weiss, she obtained a faculty position at the Autonomous University of Barcelona in 1991, and then returned to the University of Barcelona in 1992. There, she held a professorial chair from 1993 to 2019, when she moved to the Complutense University of Madrid. == Recognition == Carro received the medal of the Royal Spanish Mathematical Society in 2020. She was elected as a corresponding member of the Spanish Royal Academy of Sciences in 2021. == References == == External links == Home page María J. Carro publications indexed by Google Scholar |
Wikipedia:Masatoshi Gündüz Ikeda#0 | Masatoşi Gündüz İkeda (25 February 1926 – 9 February 2003), was a Japanese-born Turkish mathematician known for his contributions to the field of algebraic number theory. == Early years == Ikeda was born on 25 February 1926 in Tokyo, Japan, to Junzo Ikeda, head of the statistics department of an insurance company, and his wife Yaeko Ikeda. He was the youngest child with a brother and two sisters. He grew up reading mathematics books belonging to his father. During his school years, he bought himself used books about mathematics and the life story of mathematicians. He was very impressed by the French mathematician Évariste Galois (1811–1832). == Academic career == Ikeda graduated from the mathematics department of Osaka University in 1948. He received a PhD degree with his thesis "On Absolutely Segregated Algebras", written in 1953 under the direction of Kenjiro Shoda. He was appointed associate professor in 1955. He pursued scientific research at the University of Hamburg in Germany, under the supervision of Helmut Hasse (1898–1979) between 1957 and 1959. On a suggestion from Hasse, he went to Turkey in 1960 and landed at Ege University in İzmir. In 1961, he was appointed a foreigner specialist in the Faculty of Science at the same university. In 1964, Ikeda married Turkish biochemist Emel Ardor, whom he met and followed to Turkey. He was naturalized, converted to Islam and adopted the Turkish name Gündüz. He became associate professor in 1965 and a full professor in 1966. In 1968, with permission of the university, he went to the Middle East Technical University (METU) in Ankara as a visiting professor for one year. However, following the end of his term, he was offered a permanent post as a full professor, which he accepted upon the proposal of the mathematician Cahit Arf, whom he had known since his early years in Turkey. From time to time, Ikeda was invited as a visiting professor to various universities such as the University of Hamburg (1966), San Diego State University, California (1971), and Yarmouk University in Irbid, Jordan (1984, 1985–86). In 1976, Ikeda carried out research work at Princeton University. In 1976, Ikeda went to Hacettepe University in Ankara, where he chaired the mathematics department until 1978, before he returned to METU. He retired in 1992 at METU. His scientific devotion was in Galois theory. Among the research institutions Ikeda served were TÜBİTAK Marmara Research Center and Turkish National Research Institute of Electronics and Cryptology. Finally, he worked at the Feza Gürsey Basic Sciences Research Center in Istanbul. Ikeda was a member of the Basic Sciences Board at the Scientific and Technological Research Council of Turkey (TÜBİTAK), and served as the head of the Mathematic Research Unit at the METU. == Family life and death == Ikeda died on 9 February 2003, in Ankara. Following a religious funeral service held on 12 February at Kocatepe Mosque, he was laid to rest at the Karşıyaka Cemetery. He was the father of two sons, both born in Turkey. == Recognition == In 1979, Ikeda was honored with the TÜBİTAK Science Award. The Mathematics Foundation of Turkey established the "Masatoshi Gündüz İkeda Research Award" in Ikeda's memory. == See also == Anabelian geometry == References == == External links == O'Connor, John J.; Robertson, Edmund F., "Masatoshi Gündüz Ikeda", MacTutor History of Mathematics Archive, University of St Andrews Personal web page |
Wikipedia:Masayoshi Nagata#0 | Masayoshi Nagata (Japanese: 永田 雅宜 Nagata Masayoshi; February 9, 1927 – August 27, 2008) was a Japanese mathematician, known for his work in the field of commutative algebra. == Work == Nagata's compactification theorem shows that algebraic varieties can be embedded in complete varieties. The Chevalley–Iwahori–Nagata theorem describes the quotient of a variety by a group. In 1959, he introduced a counterexample to the general case of Hilbert's fourteenth problem on invariant theory. His 1962 book on local rings contains several other counterexamples he found, such as a commutative Noetherian ring that is not catenary, and a commutative Noetherian ring of infinite dimension. Nagata's conjecture on curves concerns the minimum degree of a plane curve specified to have given multiplicities at given points; see also Seshadri constant. Nagata's conjecture on automorphisms concerns the existence of wild automorphisms of polynomial algebras in three variables. Recent work has solved this latter problem in the affirmative. == Selected works == Nagata, Masayoshi (1960), "On the fourteenth problem of Hilbert", Proc. Internat. Congress Math. 1958, Cambridge University Press, pp. 459–462, MR 0116056, archived from the original on 2011-07-17 Nagata, Masayoshi (1965), Lectures on the fourteenth problem of Hilbert (PDF), Tata Institute of Fundamental Research Lectures on Mathematics, vol. 31, Bombay: Tata Institute of Fundamental Research, MR 0215828 Nagata, Masayoshi (1962), Local rings, Interscience Tracts in Pure and Applied Mathematics, vol. 13, New York-London: Interscience Publishers a division of John Wiley & Sons, ISBN 0-88275-228-6, MR 0155856 {{citation}}: ISBN / Date incompatibility (help) == References == Maruyama, Masaki; Masayoshi Miyanishi; Shigefumi Mori; Tadao Oda (January 2009). "Masayoshi Nagata (1927–2008)" (PDF). Notices of the American Mathematical Society. 56 (1): 58. Retrieved 2008-12-30. O'Connor, John J.; Robertson, Edmund F., "Masayoshi Nagata", MacTutor History of Mathematics Archive, University of St Andrews Masayoshi Nagata at the Mathematics Genealogy Project |
Wikipedia:Master theorem (analysis of algorithms)#0 | In the analysis of algorithms, the master theorem for divide-and-conquer recurrences provides an asymptotic analysis for many recurrence relations that occur in the analysis of divide-and-conquer algorithms. The approach was first presented by Jon Bentley, Dorothea Blostein (née Haken), and James B. Saxe in 1980, where it was described as a "unifying method" for solving such recurrences. The name "master theorem" was popularized by the widely used algorithms textbook Introduction to Algorithms by Cormen, Leiserson, Rivest, and Stein. Not all recurrence relations can be solved by this theorem; its generalizations include the Akra–Bazzi method. == Introduction == Consider a problem that can be solved using a recursive algorithm such as the following: procedure p(input x of size n): if n < some constant k: Solve x directly without recursion else: Create a subproblems of x, each having size n/b Call procedure p recursively on each subproblem Combine the results from the subproblems The above algorithm divides the problem into a number (a) of subproblems recursively, each subproblem being of size n/b. The factor by which the size of subproblems is reduced (b) need not, in general, be the same as the number of subproblems (a). Its solution tree has a node for each recursive call, with the children of that node being the other calls made from that call. The leaves of the tree are the base cases of the recursion, the subproblems (of size less than k) that do not recurse. The above example would have a child nodes at each non-leaf node. Each node does an amount of work that corresponds to the size of the subproblem n passed to that instance of the recursive call and given by f ( n ) {\displaystyle f(n)} . The total amount of work done by the entire algorithm is the sum of the work performed by all the nodes in the tree. The runtime of an algorithm such as the p above on an input of size n, usually denoted T ( n ) {\displaystyle T(n)} , can be expressed by the recurrence relation T ( n ) = a T ( n b ) + f ( n ) , {\displaystyle T(n)=a\;T\left({\frac {n}{b}}\right)+f(n),} where f ( n ) {\displaystyle f(n)} is the time to create the subproblems and combine their results in the above procedure. This equation can be successively substituted into itself and expanded to obtain an expression for the total amount of work done. The master theorem allows many recurrence relations of this form to be converted to Θ-notation directly, without doing an expansion of the recursive relation. == Generic form == The master theorem always yields asymptotically tight bounds to recurrences from divide and conquer algorithms that partition an input into smaller subproblems of equal sizes, solve the subproblems recursively, and then combine the subproblem solutions to give a solution to the original problem. The time for such an algorithm can be expressed by adding the work that they perform at the top level of their recursion (to divide the problems into subproblems and then combine the subproblem solutions) together with the time made in the recursive calls of the algorithm. If T ( n ) {\displaystyle T(n)} denotes the total time for the algorithm on an input of size n {\displaystyle n} , and f ( n ) {\displaystyle f(n)} denotes the amount of time taken at the top level of the recurrence then the time can be expressed by a recurrence relation that takes the form: T ( n ) = a T ( n b ) + f ( n ) {\displaystyle T(n)=a\;T\!\left({\frac {n}{b}}\right)+f(n)} Here n {\displaystyle n} is the size of an input problem, a {\displaystyle a} is the number of subproblems in the recursion, and b {\displaystyle b} is the factor by which the subproblem size is reduced in each recursive call ( b > 1 {\displaystyle b>1} ). Crucially, a {\displaystyle a} and b {\displaystyle b} must not depend on n {\displaystyle n} . The theorem below also assumes that, as a base case for the recurrence, T ( n ) = Θ ( 1 ) {\displaystyle T(n)=\Theta (1)} when n {\displaystyle n} is less than some bound κ > 0 {\displaystyle \kappa >0} , the smallest input size that will lead to a recursive call. Recurrences of this form often satisfy one of the three following regimes, based on how the work to split/recombine the problem f ( n ) {\displaystyle f(n)} relates to the critical exponent c crit = log b a {\displaystyle c_{\operatorname {crit} }=\log _{b}a} . (The table below uses standard big O notation). Throughout, ( log n ) k {\displaystyle (\log n)^{k}} is used for clarity, though in textbooks this is usually rendered log k n {\displaystyle \log ^{k}n} . c crit = log b a = log ( # subproblems ) / log ( relative subproblem size ) {\displaystyle c_{\operatorname {crit} }=\log _{b}a=\log(\#{\text{subproblems}})/\log({\text{relative subproblem size}})} A useful extension of Case 2 handles all values of k {\displaystyle k} : === Examples === ==== Case 1 example ==== T ( n ) = 8 T ( n 2 ) + 1000 n 2 {\displaystyle T(n)=8T\left({\frac {n}{2}}\right)+1000n^{2}} As one can see from the formula above: a = 8 , b = 2 , f ( n ) = 1000 n 2 {\displaystyle a=8,\,b=2,\,f(n)=1000n^{2}} , so f ( n ) = O ( n c ) {\displaystyle f(n)=O\left(n^{c}\right)} , where c = 2 {\displaystyle c=2} Next, we see if we satisfy the case 1 condition: log b a = log 2 8 = 3 > c {\displaystyle \log _{b}a=\log _{2}8=3>c} . It follows from the first case of the master theorem that T ( n ) = Θ ( n log b a ) = Θ ( n 3 ) {\displaystyle T(n)=\Theta \left(n^{\log _{b}a}\right)=\Theta \left(n^{3}\right)} (This result is confirmed by the exact solution of the recurrence relation, which is T ( n ) = 1001 n 3 − 1000 n 2 {\displaystyle T(n)=1001n^{3}-1000n^{2}} , assuming T ( 1 ) = 1 {\displaystyle T(1)=1} ). ==== Case 2 example ==== T ( n ) = 2 T ( n 2 ) + 10 n {\displaystyle T(n)=2T\left({\frac {n}{2}}\right)+10n} As we can see in the formula above the variables get the following values: a = 2 , b = 2 , c = 1 , f ( n ) = 10 n {\displaystyle a=2,\,b=2,\,c=1,\,f(n)=10n} f ( n ) = Θ ( n c ( log n ) k ) {\displaystyle f(n)=\Theta \left(n^{c}(\log n)^{k}\right)} where c = 1 , k = 0 {\displaystyle c=1,k=0} Next, we see if we satisfy the case 2 condition: log b a = log 2 2 = 1 {\displaystyle \log _{b}a=\log _{2}2=1} , and therefore, c and log b a {\displaystyle \log _{b}a} are equal So it follows from the second case of the master theorem: T ( n ) = Θ ( n log b a ( log n ) k + 1 ) = Θ ( n 1 ( log n ) 1 ) = Θ ( n log n ) {\displaystyle T(n)=\Theta \left(n^{\log _{b}a}(\log n)^{k+1}\right)=\Theta \left(n^{1}(\log n)^{1}\right)=\Theta \left(n\log n\right)} Thus the given recurrence relation T ( n ) {\displaystyle T(n)} was in Θ ( n log n ) {\displaystyle \Theta (n\log n)} . (This result is confirmed by the exact solution of the recurrence relation, which is T ( n ) = n + 10 n log 2 n {\displaystyle T(n)=n+10n\log _{2}n} , assuming T ( 1 ) = 1 {\displaystyle T(1)=1} ). ==== Case 3 example ==== T ( n ) = 2 T ( n 2 ) + n 2 {\displaystyle T(n)=2T\left({\frac {n}{2}}\right)+n^{2}} As we can see in the formula above the variables get the following values: a = 2 , b = 2 , f ( n ) = n 2 {\displaystyle a=2,\,b=2,\,f(n)=n^{2}} f ( n ) = Ω ( n c ) {\displaystyle f(n)=\Omega \left(n^{c}\right)} , where c = 2 {\displaystyle c=2} Next, we see if we satisfy the case 3 condition: log b a = log 2 2 = 1 {\displaystyle \log _{b}a=\log _{2}2=1} , and therefore, yes, c > log b a {\displaystyle c>\log _{b}a} The regularity condition also holds: 2 ( n 2 4 ) ≤ k n 2 {\displaystyle 2\left({\frac {n^{2}}{4}}\right)\leq kn^{2}} , choosing k = 1 / 2 {\displaystyle k=1/2} So it follows from the third case of the master theorem: T ( n ) = Θ ( f ( n ) ) = Θ ( n 2 ) . {\displaystyle T\left(n\right)=\Theta \left(f(n)\right)=\Theta \left(n^{2}\right).} Thus the given recurrence relation T ( n ) {\displaystyle T(n)} was in Θ ( n 2 ) {\displaystyle \Theta (n^{2})} , that complies with the f ( n ) {\displaystyle f(n)} of the original formula. (This result is confirmed by the exact solution of the recurrence relation, which is T ( n ) = 2 n 2 − n {\displaystyle T(n)=2n^{2}-n} , assuming T ( 1 ) = 1 {\displaystyle T(1)=1} .) == Inadmissible equations == The following equations cannot be solved using the master theorem: T ( n ) = 2 n T ( n 2 ) + n n {\displaystyle T(n)=2^{n}T\left({\frac {n}{2}}\right)+n^{n}} a is not a constant; the number of subproblems should be fixed T ( n ) = 2 T ( n 2 ) + n log n {\displaystyle T(n)=2T\left({\frac {n}{2}}\right)+{\frac {n}{\log n}}} non-polynomial difference between f ( n ) {\displaystyle f(n)} and n log b a {\displaystyle n^{\log _{b}a}} (see below; extended version applies) T ( n ) = 64 T ( n 8 ) − n 2 log n {\displaystyle T(n)=64T\left({\frac {n}{8}}\right)-n^{2}\log n} f ( n ) {\displaystyle f(n)} , which is the combination time, is not positive T ( n ) = T ( n 2 ) + n ( 2 − cos n ) {\displaystyle T(n)=T\left({\frac {n}{2}}\right)+n(2-\cos n)} case 3 but regularity violation. In the second inadmissible example above, the difference between f ( n ) {\displaystyle f(n)} and n log b a {\displaystyle n^{\log _{b}a}} can be expressed with the ratio f ( n ) n log b a = n / log n n log 2 2 = n n log n = 1 log n {\displaystyle {\frac {f(n)}{n^{\log _{b}a}}}={\frac {n/\log n}{n^{\log _{2}2}}}={\frac {n}{n\log n}}={\frac {1}{\log n}}} . It is clear that 1 log n < n ϵ {\displaystyle {\frac {1}{\log n}}<n^{\epsilon }} for any constant ϵ > 0 {\displaystyle \epsilon >0} . Therefore, the difference is not polynomial and the basic form of the Master Theorem does not apply. The extended form (case 2b) does apply, giving the solution T ( n ) = Θ ( n log log n ) {\displaystyle T(n)=\Theta (n\log \log n)} . == Application to common algorithms == == See also == Akra–Bazzi method Asymptotic complexity == Notes == == References == Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw–Hill, 2001. ISBN 0-262-03293-7. Sections 4.3 (The master method) and 4.4 (Proof of the master theorem), pp. 73–90. Michael T. Goodrich and Roberto Tamassia. Algorithm Design: Foundation, Analysis, and Internet Examples. Wiley, 2002. ISBN 0-471-38365-1. The master theorem (including the version of Case 2 included here, which is stronger than the one from CLRS) is on pp. 268–270. |
Wikipedia:Matching polynomial#0 | In the mathematical fields of graph theory and combinatorics, a matching polynomial (sometimes called an acyclic polynomial) is a generating function of the numbers of matchings of various sizes in a graph. It is one of several graph polynomials studied in algebraic graph theory. == Definition == Several different types of matching polynomials have been defined. Let G be a graph with n vertices and let mk be the number of k-edge matchings. One matching polynomial of G is m G ( x ) := ∑ k ≥ 0 m k x k . {\displaystyle m_{G}(x):=\sum _{k\geq 0}m_{k}x^{k}.} Another definition gives the matching polynomial as M G ( x ) := ∑ k ≥ 0 ( − 1 ) k m k x n − 2 k . {\displaystyle M_{G}(x):=\sum _{k\geq 0}(-1)^{k}m_{k}x^{n-2k}.} A third definition is the polynomial μ G ( x , y ) := ∑ k ≥ 0 m k x k y n − 2 k . {\displaystyle \mu _{G}(x,y):=\sum _{k\geq 0}m_{k}x^{k}y^{n-2k}.} Each type has its uses, and all are equivalent by simple transformations. For instance, M G ( x ) = x n m G ( − x − 2 ) {\displaystyle M_{G}(x)=x^{n}m_{G}(-x^{-2})} and μ G ( x , y ) = y n m G ( x / y 2 ) . {\displaystyle \mu _{G}(x,y)=y^{n}m_{G}(x/y^{2}).} == Connections to other polynomials == The first type of matching polynomial is a direct generalization of the rook polynomial. The second type of matching polynomial has remarkable connections with orthogonal polynomials. For instance, if G = Km,n, the complete bipartite graph, then the second type of matching polynomial is related to the generalized Laguerre polynomial Lnα(x) by the identity: M K m , n ( x ) = n ! L n ( m − n ) ( x 2 ) . {\displaystyle M_{K_{m,n}}(x)=n!L_{n}^{(m-n)}(x^{2}).\,} If G is the complete graph Kn, then MG(x) is an Hermite polynomial: M K n ( x ) = H n ( x ) , {\displaystyle M_{K_{n}}(x)=H_{n}(x),\,} where Hn(x) is the "probabilist's Hermite polynomial" (1) in the definition of Hermite polynomials. These facts were observed by Godsil (1981). If G is a forest, then its matching polynomial is equal to the characteristic polynomial of its adjacency matrix. If G is a path or a cycle, then MG(x) is a Chebyshev polynomial. In this case μG(1,x) is a Fibonacci polynomial or Lucas polynomial respectively. == Complementation == The matching polynomial of a graph G with n vertices is related to that of its complement by a pair of (equivalent) formulas. One of them is a simple combinatorial identity due to Zaslavsky (1981). The other is an integral identity due to Godsil (1981). There is a similar relation for a subgraph G of Km,n and its complement in Km,n. This relation, due to Riordan (1958), was known in the context of non-attacking rook placements and rook polynomials. == Applications in chemical informatics == The Hosoya index of a graph G, its number of matchings, is used in chemoinformatics as a structural descriptor of a molecular graph. It may be evaluated as mG(1) (Gutman 1991). The third type of matching polynomial was introduced by Farrell (1980) as a version of the "acyclic polynomial" used in chemistry. == Computational complexity == On arbitrary graphs, or even planar graphs, computing the matching polynomial is #P-complete (Jerrum 1987). However, it can be computed more efficiently when additional structure about the graph is known. In particular, computing the matching polynomial on n-vertex graphs of treewidth k is fixed-parameter tractable: there exists an algorithm whose running time, for any fixed constant k, is a polynomial in n with an exponent that does not depend on k (Courcelle, Makowsky & Rotics 2001). The matching polynomial of a graph with n vertices and clique-width k may be computed in time nO(k) (Makowsky et al. 2006). == References == Courcelle, B.; Makowsky, J. A.; Rotics, U. (2001), "On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic" (PDF), Discrete Applied Mathematics, 108 (1–2): 23–52, doi:10.1016/S0166-218X(00)00221-3. Farrell, E. J. (1980), "The matching polynomial and its relation to the acyclic polynomial of a graph", Ars Combinatoria, 9: 221–228. Godsil, C.D. (1981), "Hermite polynomials and a duality relation for matchings polynomials", Combinatorica, 1 (3): 257–262, doi:10.1007/BF02579331. Gutman, Ivan (1991), "Polynomials in graph theory", in Bonchev, D.; Rouvray, D. H. (eds.), Chemical Graph Theory: Introduction and Fundamentals, Mathematical Chemistry, vol. 1, Taylor & Francis, pp. 133–176, ISBN 978-0-85626-454-2. Jerrum, Mark (1987), "Two-dimensional monomer-dimer systems are computationally intractable", Journal of Statistical Physics, 48 (1): 121–134, Bibcode:1987JSP....48..121J, doi:10.1007/BF01010403. Makowsky, J. A.; Rotics, Udi; Averbouch, Ilya; Godlin, Benny (2006), "Computing graph polynomials on graphs of bounded clique-width", Proc. 32nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG '06) (PDF), Lecture Notes in Computer Science, vol. 4271, Springer-Verlag, pp. 191–204, doi:10.1007/11917496_18, ISBN 978-3-540-48381-6. Riordan, John (1958), An Introduction to Combinatorial Analysis, New York: Wiley. Zaslavsky, Thomas (1981), "Complementary matching vectors and the uniform matching extension property", European Journal of Combinatorics, 2: 91–103, doi:10.1016/s0195-6698(81)80025-x. |
Wikipedia:Mathemalchemy#0 | Mathemalchemy (French: MathémAlchimie) is a traveling art installation dedicated to a celebration of the intersection of art and mathematics. It is a collaborative work led by Duke University mathematician Ingrid Daubechies and fiber artist Dominique Ehrmann. The cross-disciplinary team of 24 people, who collectively built the installation during the calendar years 2020 and 2021, includes artists, mathematicians, and craftspeople who employed a wide variety of materials to illustrate, amuse, and educate the public on the wonders, mystery, and beauty of mathematics. Including the core team of 24, about 70 people contributed in some way to the realization of Mathemalchemy. == Description == The art installation occupies a footprint approximately 20 by 10.5 feet (6.1 by 3.2 m), which extends up to 9.5 feet (2.9 m) in height (in addition, small custom-fabricated tables are arranged around the periphery to protect the more fragile elements). A map shows the 14 or so different zones or regions within the exhibit, which is filled with hundreds of detailed mathematical artifacts, some smaller than 0.5 inches (13 mm); the entire exhibit comprises more than 1,000 parts which must be packed for shipment. Versions of some of the complex mathematical objects can be purchased through an associated "Mathemalchemy Boutique" website. The art installation contains puns (such as "Pi" in a bakery) and Easter eggs, such as a miniature model of the Antikythera mechanism hidden on the bottom of "Knotilus Bay". Mathematically sophisticated visitors may enjoy puzzling out and decoding the many mathematical allusions symbolized in the exhibit, while viewers of all levels are invited to enjoy the self-guided tours, detailed explanations, and videos available on the accompanying official website [1]. A downloadable comic book was created to explore some of the themes of the exhibition, using an independent narrative set in the world of Mathemalchemy. == Themes == The installation features or illustrates mathematical concepts at many different levels. All of the participants regard "recreational mathematics"—especially when it has a strong visual component—as having an important role in education and in culture in general. Jessica Sklar maintains that "mathematics is, at heart, a human endeavor" and feels compelled to make it accessible to those who don't regard themselves as "math people". Bronna Butler talks about the heritage of JH Conway, whose lectures were "almost magical in quality" because they used what looked like curios and tricks but in the end arrived at answers to "fundamental questions of mathematics". Henry Segerman, who wrote the book Visualizing Mathematics With 3D Printing, contributed 3D pieces that explore stereographic projection and polyhedra. According to Susan Goldstine, "The interplay between mathematics and fiber arts is endlessly fascinating [and] allows for a deeper understanding ways that these crafts can illuminate complex concepts in mathematics". Edmund Harriss says, "You don’t need a background in math to appreciate the installation, just like you can enjoy a concert without being a musician". The creators had the goal of illustrating as much of mathematics as possible. Thus the various exhibits touch on number theory, fractals, tessellations, probability theory, Zeno's paradoxes, Venn diagrams, knot theory, calculus, chaos theory, topology, hyperbolic geometry, symbolic logic—and much else—all in a setting that is beautiful and fun. Mathematicians explicitly mentioned or alluded to include Vladimir Arnold, John H. Conway, Felix Klein, Sofya Kovalevskaya, Henri Lebesgue, Ada Lovelace, Benoit Mandelbrot, Maryam Mirzakhani, August Möbius, Emmy Noether, Marjorie Rice, Bernhard Riemann, Caroline Series, Wacław Sierpiński, Alicia Boole Stott, William Thurston, Helge von Koch, Gladys West, Zeno, and many others. Twenty of the "mathemalchemists" are women, and the facility especially celebrates the contributions of women in mathematics, from amateur Marjorie Rice, who found new kinds of pentagon tilings, to Maryam Mirzakhani, the first woman to ever garner a Fields Medal. == Gallery == == History == Daubechies and Ehrmann presented the project in a special session at the 2020 Joint Mathematics Meetings (JMM) in Denver, Colorado. They soon had a core group of more than a dozen interested mathematicians and artists who in turn suggested other people not at JMM. Eventually the group would grow to 24 people. Originally, the intent was to collectively design and fabricate in a series of workshops to be held at Duke University in Durham, North Carolina, starting in March 2020. The COVID-19 pandemic disrupted these plans. Working instead over Zoom, under the guidance of Dominique Ehrmann and various "team leaders" for different parts of the installation, the 16-by-12-by-10-foot (4.9 by 3.7 by 3.0 m) installation was collectively designed and discussed. In July 2021 the team could finally get together at Duke for the first in-person meeting, where the components that had been fabricated in various locations in the US and Canada were assembled for the first time, leading to the first complete full-scale construction. The 24 members of the team employed ceramics, knitting, crocheting, quilting, beadwork, 3D printing, welding, woodworking, textile embellishment, origami, metal-folding, water-sculpted brick, and temari balls to create the room-sized installation. == Venues == The finished installation was originally displayed at Duke University, then moving to the National Academy of Sciences (NAS) building in Washington DC, where it was on display from December 4, 2021, until June 12, 2022. The installation next showed at Juniata College in Huntingdon, Pennsylvania before moving to Boston University from January to March 2023, partially overlapping with the 2023 Joint Mathematics Meetings in Boston. The exhibit then moved to Beaty Biodiversity Museum in Vancouver, British Columbia and then in November of that year it went to Northern Kentucky University where it remained until February 2024. From May 22 to October 27, 2024 Mathemalchemy was at the National Museum of Mathematics (MoMath) in New York City. From November 6, 2024 to May 2, 2025, the University of Quebec in Montreal (UQAM) hosts the exhibition. As of November 2024, fundraising is underway to mount the exhibition at the Navajo Nation Museum in Window Rock, Arizona. The exhibit is planned to ultimately reside in the Duke University mathematics building, on permanent display. == See also == Mathematica: A World of Numbers... and Beyond – 1961 iconic mathematics exhibition by Ray and Charles Eames Mathematics and art == References == == External links == Official website Mathemalchemy Art Installation on YouTube |
Wikipedia:Mathematical Association#0 | The Mathematical Association is a professional society concerned with mathematics education in the UK. == History == It was founded in 1871 as the Association for the Improvement of Geometrical Teaching and renamed to the Mathematical Association in 1897. It was the first teachers' subject organisation formed in England. In March 1927, it held a three-day meeting in Grantham to commemorate the bicentenary of the death of Sir Isaac Newton, attended by Sir J. J. Thomson (discoverer of the electron), Sir Frank Watson Dyson – the Astronomer Royal, Sir Horace Lamb, and G. H. Hardy. In 1951, Mary Cartwright became the first female president of the Mathematical Association. In the 1960s, when comprehensive education was being introduced, the Association was in favour of the 11-plus system. For maths teachers training at university, a teaching award that was examined was the Diploma of the Mathematical Association, later known as the Diploma in Mathematical Education of the Mathematical Association. == Function == It exists to "bring about improvements in the teaching of mathematics and its applications, and to provide a means of communication among students and teachers of mathematics". Since 1894 it has published The Mathematical Gazette. It is one of the participating bodies in the quadrennial British Congress of Mathematics Education, organised by the Joint Mathematical Council, and it holds its annual general meeting as part of the Congress. == Structure == It is based in the south-east of Leicester on London Road (A6), just south of the Charles Frears campus of De Montfort University. Aside from the council, it has seven other specialist committees. === Regions === Its branches are sometimes shared with the Association of Teachers of Mathematics (ATM): Birmingham Cambridge East Midlands Exeter Gloucester Liverpool London Greater Manchester Meridian Stoke and Staffordshire Sheffield Sussex Yorkshire == Past presidents == Past presidents of The Association for the Improvement of Geometrical Teaching included: 1871 Thomas Archer Hirst 1878 Robert Baldwin Hayward MA, FRS 1889 G M Minchin MA, FRS 1891 James Joseph Sylvester 1892 The Reverend C Taylor DD 1893 R Wormell MA, DSc 1895 Joseph Larmor Past presidents of The Mathematical Association have included: 1897 Alfred Lodge 1899–1900 Robert Stawell Ball 1901 John Fletcher Moulton, Baron Moulton 1903 Andrew Forsyth 1905 George Ballard Mathews 1907 George H. Bryan 1909–1910 Herbert Hall Turner 1911–1912 E. W. Hobson 1913–1914 Alfred George Greenhill 1915–1916 Alfred North Whitehead 1918–1919 Percy Nunn 1920 E. T. Whittaker 1921 James Wilson 1922–1923 Thomas Little Heath 1924–1925 G. H. Hardy 1926–1927 Micaiah John Muller Hill 1928–1929 William Fleetwood Sheppard 1930–1931 Arthur Eddington 1932–1933 G. N. Watson 1934 Eric Harold Neville 1935 A W Siddons 1936 Andrew Forsyth 1937 Louis Napoleon George Filon 1938 W Hope-Jones 1939 W C Fletcher 1944 C O Tuckey MA 1945 Sydney Chapman 1946 Warin Foster Bushell 1947 George Barker Jeffery 1948 Harold Spencer Jones 1949 A Robson MA 1950 Professor H R Hasse MA, DSc 1951 Mary Cartwright 1952 K S Snell MA 1953 Professor T A A Broadbent MA 1954 W. V. D. Hodge 1955 G L Parsons MA 1956 George Frederick James Temple 1957 W J Langford JP, MSc 1958 Max Newman 1959 Louise Doris Adams 1960 Edwin A. Maxwell 1961 J T Combridge MA, MSc 1962 Professor V C A Ferraro PhD, DIC 1963 J B Morgan MA 1964 Ida Busbridge 1965 Elizabeth Williams 1966 F W Kellaway BSc 1967 A.P. Rollett 1968 Charles Coulson 1969 Bertha Swirles 1970 James Lighthill 1971 B T Bellis MA, FRSE, FIMA 1972 C T Daltry BSc, FIMA 1973 William McCrea 1974 Margaret Hayman 1975 Reuben Goodstein 1976 E Kerr BSc, PhD, FIMA, FBCS 1977 Professor G Matthews MA, PhD, FIMA 1978 Alan Tammadge 1979 Clive W. Kilmister 1980 D A Quadling MA, FIMA, later OBE 1981 Michael Atiyah 1982 F J Budden BSc 1983 Rolph Ludwig Edward Schwarzenberger 1984 P B Coaker BSc, ARCS, DIC, FIMA, FBCS 1985 Hilary Shuard 1986 Anita Straker 1987 Margaret Rayner 1988 A.G. Howson 1989 Mr Peter Reynolds MA 1990 Margaret Brown 1991 Alan J. Bishop 1992 Mr John Hersee MA 1993 Dr William Wynne-Wilson BA, PhD 1994 Mary Bradburn 1995 E. Roy Ashley 1996 W. P. Richardson MBE 1997 Tony Gardiner 1998 Professor J Chris Robson 1999 John S Berry 2000 Mr Stephen Abbott BSc, MSc 2001 Dr Sue Sanders Cert.Ed, BA, MEd, PhD 2002 Mr Barry Lewis BSc, BA, FIMA 2003 Christopher Zeeman 2004 Professor Adam McBride OBE 2005 Sue Singer 2006 Mr Doug French 2007 Rob Eastaway 2008 Mr Robert Barbour 2009 Mrs Jane Imrie 2010 David Acheson 2011 Dr Paul Andrews 2012 Professor Marcus Du Sautoy OBE FRS 2013 Mr Peter Ransom MBE 2014 Lynne McClure OBE 2015 Dr Peter M. Neumann OBE 2016 Dr Jennie Golding 2017 Mr Tom Roper 2018 Professor Mike Askew 2019 Dr Ems Lord 2020 Professor Hannah Fry 2021 Dr Chris Pritchard 2022 Dr Colin Foster 2023 Professor Nira Chamberlain OBE 2024 Charlie Stripp 2025 Professor Paul Glaister CBE (President) == Arms == == See also == London Mathematical Society Institute of Mathematics and its Applications == References == Siddons, A. W. (1939). "The Mathematical Association—I". Eureka. 1: 13–15. Siddons, A. W. (1939). "The Mathematical Association—II". Eureka. 2: 18–19. Michael H Price Mathematics of the Multitude? A History of the Mathematical Association (MA, 1994) == External links == The Mathematical Association website Complete list of Presidents of the Association The MA's online shop Annual conference The Mathematical Gazette No. 1, 30, 31, 37–39, 41, 43 (1901–1904) on the Internet Archive digitised by Google from the Harvard University Library === News items === Addressing the downward spiral of UK maths education in February 2004 Proposal to split Maths GCSE into two in August 2003 |
Wikipedia:Mathematical Olympiad Cell#0 | The Mathematical Olympiad Cell (MO Cell) is a body of permanent faculty devoted to organizing and conducting the mathematical Olympiads in India, in particular, the Indian National Mathematical Olympiad and the International Mathematical Olympiad Training Camp. The MO Cell has three main members: B.J. Venkatachala Prithwijit De N.V. Tejaswi (who replaced C.R. Pranesachar after he retired in April, 2013) There are a number of other faculty who work part-time in conjunction with the MO Cell by sending problems for the Olympiads, participating in training activities, going as Leader/Deputy Leader, and participating in regional co-ordination. The members of the MO Cell used to operate solely from the Indian Institute of Science, Bangalore, based on an agreement, and Prof. Venkatachala still does so. The two new members, Dr. De and Dr. Tejaswi, function from the HBCSE, where the IMO Training Camp was held until 3 years ago, and from where co-ordination for the outgoing team is managed. == See also == Indian National Mathematical Olympiad International Mathematical Olympiad Training Camp International Mathematical Olympiad == External links == Official page of the MO Cell Mathematics Olympiad Help Site – India |
Wikipedia:Mathematical Proceedings of the Cambridge Philosophical Society#0 | Mathematical Proceedings of the Cambridge Philosophical Society is a mathematical journal published by Cambridge University Press for the Cambridge Philosophical Society. It aims to publish original research papers from a wide range of pure and applied mathematics. The journal, titled Proceedings of the Cambridge Philosophical Society before 1975, has been published since 1843. == Abstracting and indexing == The journal is abstracted and indexed in MathSciNet Science Citation Index Expanded Scopus ZbMATH Open == See also == Cambridge Philosophical Society == External links == official website == References == |
Wikipedia:Mathematical Sciences Foundation#0 | The Mathematical Sciences are a group of areas of study that includes, in addition to mathematics, those academic disciplines that are primarily mathematical in nature but may not be universally considered subfields of mathematics proper. Statistics, for example, is mathematical in its methods but grew out of bureaucratic and scientific observations, which merged with inverse probability and then grew through applications in some areas of physics, biometrics, and the social sciences to become its own separate, though closely allied, field. Theoretical astronomy, theoretical physics, theoretical and applied mechanics, continuum mechanics, mathematical chemistry, actuarial science, computer science, computational science, data science, operations research, quantitative biology, control theory, econometrics, geophysics and mathematical geosciences are likewise other fields often considered part of the mathematical sciences. Some institutions offer degrees in mathematical sciences (e.g. the United States Military Academy, Stanford University, and University of Khartoum) or applied mathematical sciences (for example, the University of Rhode Island). == See also == Exact sciences – Sciences that admit of absolute precision in their results Formal science – Study of abstract structures described by formal systems Relationship between mathematics and physics == References == == External links == Division of Mathematical Sciences at the National Science Foundation, including a list of disciplinary areas supported Faculty of Mathematical Sciences at University of Khartoum, offers academic degrees in Mathematics, Computer Sciences and Statistics Programs of the Mathematical Sciences Research Institute Research topics studied at the Isaac Newton Institute for Mathematical Sciences Mathematical Sciences in the U.S. FY 2016 Budget; a report from the AAAS |
Wikipedia:Mathematical Treatise in Nine Sections#0 | The Mathematical Treatise in Nine Sections (simplified Chinese: 数书九章; traditional Chinese: 數書九章; pinyin: Shùshū Jiǔzhāng; Wade–Giles: Shushu Chiuchang) is a mathematical text written by Chinese Southern Song dynasty mathematician Qin Jiushao in the year 1247. The mathematical text has a wide range of topics and is taken from all aspects of the society of that time, including agriculture, astronomy, water conservancy, urban layout, construction engineering, surveying, taxation, armament, military and so on. This book contains nine chapters: Da Yan type (Indeterminate equations); Heaven phenomena Area of land and field Surveying Taxation Storage of grains Building construction Military matters Price and interest. Each chapter contains nine problems, a total of 81 problems. Apart from describing Chinese Remainder Theorem for the first time and providing a constructive proof for it, the text investigated: Indeterminate equations "Linglong method" (玲瓏開方; línglóng kāifāng, roughly "method of harmoniously alternating evolution") for numerical solution of algebraic equations, 570 years before Horner's method Areas and volumes of geometric objects and Linear system Like many traditional Chinese mathematical works, the text reflects a Confucian administrator's concern with more practical mathematical problems, like calendrical, mensural, and fiscal problems. The text existed in manuscript form in 1247, it was incorporated into The Yongle Encyclopedia in 1421; in 1787 the book was collected into the Complete Library of the Four Treasuries, in 1842 appeared in woodblock printed edition. The 19th century British Protestant Christian missionary Alexander Wylie in his article Jottings on the Sciences of Chinese Mathematics published in North China Herald 1852, was the first person to introduce Mathematical Treatise in Nine Sections to the West. In 1971 Belgian sinologist Ulrich Libbrecht published his doctorate dissertation, Chinese Mathematics in the Thirteenth Century, which earned him a degree cum laude at Leiden University. == Notes == == References == Guo, Shuchun, "Qin Jiushao". Encyclopedia of China (Mathematics Edition), 1st ed. Complete Library of the Four Treasuries edition of the book - 數學九章 (四庫全書本) |
Wikipedia:Mathematical Tripos#0 | The Mathematical Tripos is the mathematics course that is taught in the Faculty of Mathematics at the University of Cambridge. == Origin == In its classical nineteenth-century form, the tripos was a distinctive written examination of undergraduate students of the University of Cambridge. Prior to 1824, the Mathematical Tripos was formally known as the "Senate House Examination". From about 1780 to 1909, the "Old Tripos" was distinguished by a number of features, including the publication of an order of merit of successful candidates, and the difficulty of the mathematical problems set for solution. By way of example, in 1854, the Tripos consisted of 16 papers spread over eight days, totaling 44.5 hours. The total number of questions was 211. It was divided into two parts, with Part I (the first three days) covering more elementary topics. The actual marks for the exams were never published, but there is reference to an exam in the 1860s where, out of a total possible mark of 17,000, the senior wrangler achieved 7634, the second wrangler 4123, the lowest wrangler around 1500 and the lowest scoring candidate obtaining honours (the wooden spoon) 237; about 100 candidates were awarded honours. The 300-odd candidates below that level did not earn honours and were known as poll men. The questions for the 1841 examination may be found within Cambridge University Magazine (pages 191–208). === Influence === According to the study Masters of Theory: Cambridge and the Rise of Mathematical Physics by Andrew Warwick during this period the style of teaching and study required for the successful preparation of students had a wide influence: on the development of 'mixed mathematics' (a precursor of later applied mathematics, descriptive geometry and mathematical physics, with emphasis on algebraic manipulative mastery) on mathematical education as vocational training for fields such as astronomy in the reception of new physical theories, particularly in electromagnetism as expounded by James Clerk Maxwell Since Cambridge students did a lot of rote learning called "bookwork", it was noted by Augustus De Morgan and repeated by Andrew Warwick: 152 that authors of Cambridge textbooks skipped known material. In consequence, "non-Cambridge readers ... found the arguments impossible to follow." From the 1820s to the 1840s, analytic topics such as elliptical integrals were introduced to the curriculum. Under William Whewell, the Tripos' scope changed to one of 'mixed mathematics', with the inclusion of topics from physics such as electricity, heat and magnetism. Students would have to study intensely to perform routine problems rapidly. === Early history === The early history is of the gradual replacement during the middle of the eighteenth century of a traditional method of oral examination by written papers, with a simultaneous switch in emphasis from Latin disputation to mathematical questions. That is, all degree candidates were expected to show at least competence in mathematics. A long process of development of coaching—tuition usually outside the official University and college courses—went hand-in-hand with a gradual increase in the difficulty of the most testing questions asked. The standard examination pattern of bookwork (mostly memorised theorems) plus rider (problems to solve, testing comprehension of the bookwork) was introduced. === Wranglers and their coaches === The list of wranglers (the candidates awarded a first-class degree) became in time the subject of a great deal of public attention. According to Alexander Macfarlane To obtain high honours in the Mathematical Tripos, a student must put himself in special training under a mathematician, technically called a coach, who is not one of the regular college instructors, nor one of the University professors, but simply makes a private business of training men to pass that particular examination. Skill consists in the rate at which one can solve and more especially write out the solution of problems. It is excellent training of a kind, but there is not time for studying fundamental principles, still less for making any philosophical investigations. Mathematical insight is something higher than skill in solving problems; consequently the senior wrangler has not always turned out the most distinguished mathematician in after life. William Hopkins was the first coach distinguished by his students' performances. When he retired in 1849, one of his students, Edward Routh, became the dominant coach. Another coach, William Henry Besant, published a textbook, Elementary Hydrostatics, containing mathematical exercises and solutions such as would benefit students preparing for Tripos. After Routh retired in 1888, Robert Rumsey Webb coached many of the top wranglers. Warwick notes that college teaching improved toward the end of the 19th century: The expansion of intercollegiate and university lectures at all levels through the 1880s and 1890s meant that, by 1900, it had become unnecessary for coaches either to lecture students or even to provide them with manuscripts covering the mathematical methods they were required to master. The prime job to the coach now was to ensure that students were attending an appropriate range of courses and that they understood what they were being taught. … This curtailment of responsibility made it virtually impossible for a private tutor to dominate undergraduate training the way that Hopkins, Routh, and Webb had done.: 282 A fellow of Trinity College, Robert Alfred Herman, then was associated with several of the top wranglers as their coach; evidently the university was finally providing their students with education. When A. R. Forsyth wrote his retrospective in 1935, he recalled Webb, Percival Frost, Herman, and Besant as the best coaches. Other coaches that produced top wranglers include E. W. Hobson, John Hilton Grace, H. F. Baker, Thomas John I'Anson Bromwich, and A. E. H. Love. === Athletics === Apart from intellectual preparation, the challenge of Tripos was its duration: "The examinations themselves were intended partly as tests of endurance, taking place on consecutive mornings and afternoons for four and five days together.": 186 Brisk walking was taken up by many candidates to build up their stamina. As the nineteenth century progressed walking turned to athletics and other competitive sports including rowing and swimming. The coaches set the example: Routh had a two-hour constitutional walk daily, while "Besant was a mountaineer, Webb a walker, and Frost was extremely proficient in cricket, tennis, running and swimming.": 200 By 1900, there were twenty-three recognized sports contested at Cambridge. === Women === In 1873, Sarah Woodhead became the first woman to take, and to pass, the Mathematical Tripos. In 1880, Charlotte Angas Scott obtained special permission to take the Mathematical Tripos, as women were not normally allowed to sit for that exam. She came eighth on the Tripos of all students taking them, but due to her sex, the title of "eighth wrangler," a high honour, went officially to a male student. At the ceremony, however, after the seventh wrangler had been announced, all the students in the audience shouted her name. Because she could not attend the award ceremony, Scott celebrated her accomplishment at Girton College where there were cheers and clapping at dinner, a special evening ceremony where the students sang "See the Conquering Hero Comes", received an ode written by a staff member, and was crowned with laurels. After this incident women were allowed to formally take the exam and their exam scores listed, although separately from the men's and thus not included in the rankings. Women obtaining the necessary score also received a special certificate instead of the BA degree with honours. In 1890, Philippa Fawcett became the first woman to obtain the top score in the Mathematical Tripos. == 1909 reforms == Reforms were implemented in 1909. The undergraduate course of mathematics at Cambridge still reflects a historically broad approach; and problem-solving skills are tested in examinations, though the setting of excessively taxing questions has been discouraged for many years. Example questions from 1881, before the reforms, are quoted in A Mathematician's Miscellany:(b) A sphere spinning in equilibrium on top of a rough horizontal cylinder is slightly disturbed; prove that the track of the point of contact is initially a helix. (c) If the sphere has a centrally symmetrical law of density such as to make the radius of gyration a certain fraction of the radius then, whatever the spin, the track is a helix so long as contact lasts. (Marked at 200; a second part about further details carried another 105.) == The modern tripos == As of 2018, the Mathematical Tripos course comprises three undergraduate years (Parts IA, IB and II) which qualify a student for a BA degree, and an optional one year masters course (Part III) which qualifies a student for a Master of Mathematics (MMath) degree (with BA) if they are a Cambridge fourth year student or a Master of Advanced Study (MASt) degree if they come from outside just to do Part III. Assessment is mostly by written examination at the end of each academic year, with some coursework elements in the second, third and fourth years. During the undergraduate part of the course, students are expected to attend around 12 one-hour lectures per week on average, together with two supervisions. Supervisions are informal sessions in which a small group of students—normally a pair—goes through previously completed example sheets under the guidance of a faculty member, college fellow or graduate student. During the first year, Part IA, the schedule of courses is quite rigid, providing much of the basic knowledge requisite for mathematics, including algebra, analysis, methods in calculus, and probability. The second year, Part IB, contains no mandatory content but it is recommended that students do particular courses as they are essential prerequisites for further courses. A range of pure courses, such as geometry, complex analysis and a course studying group theory, rings and modules are on offer as well as applied courses on electromagnetism, quantum mechanics and fluid dynamics. In Part II, students are free to choose from a large number of courses over a wide range of mathematical topics; these are separated into more accessible C courses and D courses which are more involved. Some students choose to exchange 25% of the first-year mathematics options in exchange for the Physics option of first-year Natural Sciences Tripos with the possibility of changing to Natural Sciences at the end of the first year. == References == == Further reading == Rouse Ball, A History of the Study of Mathematics at Cambridge Leonard Roth (1971) "Old Cambridge Days", American Mathematical Monthly 78:223–236. The Tripos was an important institution in nineteenth century England and many notable figures were involved with it. It has attracted broad attention from scholars. See for example: Griffin, N.; Lewis, A. C. (1990). "Bertrand Russell's Mathematical Education". Notes and Records of the Royal Society. 44: 51–71. doi:10.1098/rsnr.1990.0004. Stray, C. (2001). "The Shift from Oral to Written Examination: Cambridge and Oxford 1700–1900". Assessment in Education: Principles, Policy & Practice. 8: 33–50. doi:10.1080/09695940120033243. S2CID 144330476. In old age two undergraduates of the 1870s wrote sharply contrasting accounts of the Old Tripos — one negative, one positive. Andrew Forsyth, Senior Wrangler 1881, stayed in Cambridge and was one of the reformers responsible for the New Tripos. Karl Pearson Third Wrangler in 1879 made his career outside Cambridge. Forsyth, A. R. (1935). "Old Tripos Days in Cambridge". Mathematical Gazette. 19 (234): 162–179. doi:10.1017/s0025557200203756. JSTOR 3605871. Pearson, Karl (1936). "Old Tripos Days at Cambridge, as Seen from Another Viewpoint". Mathematical Gazette. 20 (237): 27–36. doi:10.1017/s0025557200205755. JSTOR 3607829. J. J. Thomson, a Second Wrangler in 1880, wrote about his experience in: J. J. Thomson Recollections and Reflections London: G. Bell, 1936. J. E. Littlewood, a Senior Wrangler in the last years of the old Tripos, recalled the experience in: J. E. Littlewood A Mathematician's Miscellany (2nd edition published in 1986), Cambridge University Press. G. H. Hardy, A Mathematician's Apology, Cambridge University Press (1940). 153 pages. ISBN 0-521-42706-1. Kathryn M. Olesko (2004) Review of Masters of Theory from American Scientist magazine. Theodore M. Porter (2003) Review of Masters of Theory from Science. On the importance of the Tripos in the history of mathematics in Britain: search on "tripos" in The MacTutor History of Mathematics archive For statistics on the number of graduates (men and women) between 1882 and 1940 see: Davis archive of female mathematicians: Cambridge For the present-day Tripos see: Cambridge University: Guide to the Mathematical Tripos (pdf) Actual examination papers from 2001 onwards The Cambridge Maths faculty's site explaining Part III Nelson, Graham. "Miss Warren’s Profession" Eureka 51, 1992. Critique of Part III. |
Wikipedia:Mathematical analysis#0 | Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). == History == === Ancient === Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy. (Strictly speaking, the point of the paradox is to deny that the infinite sum exists.) Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids. The explicit use of infinitesimals appears in Archimedes' The Method of Mechanical Theorems, a work rediscovered in the 20th century. In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century CE to find the area of a circle. From Jain literature, it appears that Hindus were in possession of the formulae for the sum of the arithmetic and geometric series as early as the 4th century BCE. Ācārya Bhadrabāhu uses the sum of a geometric series in his Kalpasūtra in 433 BCE. === Medieval === Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere in the 5th century. In the 12th century, the Indian mathematician Bhāskara II used infinitesimal and used what is now known as Rolle's theorem. In the 14th century, Madhava of Sangamagrama developed infinite series expansions, now called Taylor series, of functions such as sine, cosine, tangent and arctangent. Alongside his development of Taylor series of trigonometric functions, he also estimated the magnitude of the error terms resulting of truncating these series, and gave a rational approximation of some infinite series. His followers at the Kerala School of Astronomy and Mathematics further expanded his works, up to the 16th century. === Modern === ==== Foundations ==== The modern foundations of mathematical analysis were established in 17th century Europe. This began when Fermat and Descartes developed analytic geometry, which is the precursor to modern calculus. Fermat's method of adequality allowed him to determine the maxima and minima of functions and the tangents of curves. Descartes's publication of La Géométrie in 1637, which introduced the Cartesian coordinate system, is considered to be the establishment of mathematical analysis. It would be a few decades later that Newton and Leibniz independently developed infinitesimal calculus, which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the calculus of variations, ordinary and partial differential equations, Fourier analysis, and generating functions. During this period, calculus techniques were applied to approximate discrete problems by continuous ones. ==== Modernization ==== In the 18th century, Euler introduced the notion of a mathematical function. Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the modern definition of continuity in 1816, but Bolzano's work did not become widely known until the 1870s. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work, particularly by Euler. Instead, Cauchy formulated calculus in terms of geometric ideas and infinitesimals. Thus, his definition of continuity required an infinitesimal change in x to correspond to an infinitesimal change in y. He also introduced the concept of the Cauchy sequence, and started the formal theory of complex analysis. Poisson, Liouville, Fourier and others studied partial differential equations and harmonic analysis. The contributions of these mathematicians and others, such as Weierstrass, developed the (ε, δ)-definition of limit approach, thus founding the modern field of mathematical analysis. Around the same time, Riemann introduced his theory of integration, and made significant advances in complex analysis. Towards the end of the 19th century, mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof. Dedekind then constructed the real numbers by Dedekind cuts, in which irrational numbers are formally defined, which serve to fill the "gaps" between rational numbers, thereby creating a complete set: the continuum of real numbers, which had already been developed by Simon Stevin in terms of decimal expansions. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the "size" of the set of discontinuities of real functions. Also, various pathological objects, (such as nowhere continuous functions, continuous but nowhere differentiable functions, and space-filling curves), commonly known as "monsters", began to be investigated. In this context, Jordan developed his theory of measure, Cantor developed what is now called naive set theory, and Baire proved the Baire category theorem. In the early 20th century, calculus was formalized using an axiomatic set theory. Lebesgue greatly improved measure theory, and introduced his own theory of integration, now known as Lebesgue integration, which proved to be a big improvement over Riemann's. Hilbert introduced Hilbert spaces to solve integral equations. The idea of normed vector space was in the air, and in the 1920s Banach created functional analysis. == Important concepts == === Metric spaces === In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined. Much of analysis happens in some metric space; the most commonly used are the real line, the complex plane, Euclidean space, other vector spaces, and the integers. Examples of analysis without a metric include measure theory (which describes size rather than distance) and functional analysis (which studies topological vector spaces that need not have any sense of distance). Formally, a metric space is an ordered pair ( M , d ) {\displaystyle (M,d)} where M {\displaystyle M} is a set and d {\displaystyle d} is a metric on M {\displaystyle M} , i.e., a function d : M × M → R {\displaystyle d\colon M\times M\rightarrow \mathbb {R} } such that for any x , y , z ∈ M {\displaystyle x,y,z\in M} , the following holds: d ( x , y ) ≥ 0 {\displaystyle d(x,y)\geq 0} , with equality if and only if x = y {\displaystyle x=y} (identity of indiscernibles), d ( x , y ) = d ( y , x ) {\displaystyle d(x,y)=d(y,x)} (symmetry), and d ( x , z ) ≤ d ( x , y ) + d ( y , z ) {\displaystyle d(x,z)\leq d(x,y)+d(y,z)} (triangle inequality). By taking the third property and letting z = x {\displaystyle z=x} , it can be shown that d ( x , y ) ≥ 0 {\displaystyle d(x,y)\geq 0} (non-negative). === Sequences and limits === A sequence is an ordered list. Like a set, it contains members (also called elements, or terms). Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Most precisely, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the natural numbers. One of the most important properties of a sequence is convergence. Informally, a sequence converges if it has a limit. Continuing informally, a (singly-infinite) sequence has a limit if it approaches some point x, called the limit, as n becomes very large. That is, for an abstract sequence (an) (with n running from 1 to infinity understood) the distance between an and x approaches 0 as n → ∞, denoted lim n → ∞ a n = x . {\displaystyle \lim _{n\to \infty }a_{n}=x.} == Main branches == === Calculus === === Real analysis === Real analysis (traditionally, the "theory of functions of a real variable") is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real numbers, and continuity, smoothness and related properties of real-valued functions. === Complex analysis === Complex analysis (traditionally known as the "theory of functions of a complex variable") is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including algebraic geometry, number theory, applied mathematics; as well as in physics, including hydrodynamics, thermodynamics, mechanical engineering, electrical engineering, and particularly, quantum field theory. Complex analysis is particularly concerned with the analytic functions of complex variables (or, more generally, meromorphic functions). Because the separate real and imaginary parts of any analytic function must satisfy Laplace's equation, complex analysis is widely applicable to two-dimensional problems in physics. === Functional analysis === Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear operators acting upon these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. === Harmonic analysis === Harmonic analysis is a branch of mathematical analysis concerned with the representation of functions and signals as the superposition of basic waves. This includes the study of the notions of Fourier series and Fourier transforms (Fourier analysis), and of their generalizations. Harmonic analysis has applications in areas as diverse as music theory, number theory, representation theory, signal processing, quantum mechanics, tidal analysis, and neuroscience. === Differential equations === A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics, biology, and other disciplines. Differential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space or time (expressed as derivatives) is known or postulated. This is illustrated in classical mechanics, where the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow one (given the position, velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called an equation of motion) may be solved explicitly. === Measure theory === A measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the n {\displaystyle n} -dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} . For instance, the Lebesgue measure of the interval [ 0 , 1 ] {\displaystyle \left[0,1\right]} in the real numbers is its length in the everyday sense of the word – specifically, 1. Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X {\displaystyle X} . It must assign 0 to the empty set and be (countably) additive: the measure of a 'large' subset that can be decomposed into a finite (or countable) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ {\displaystyle \sigma } -algebra. This means that the empty set, countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. Indeed, their existence is a non-trivial consequence of the axiom of choice. === Numerical analysis === Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). Modern numerical analysis does not seek exact answers, because exact answers are often impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors. Numerical analysis naturally finds applications in all fields of engineering and the physical sciences, but in the 21st century, the life sciences and even the arts have adopted elements of scientific computations. Ordinary differential equations appear in celestial mechanics (planets, stars and galaxies); numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology. === Vector analysis === Vector analysis, also called vector calculus, is a branch of mathematical analysis dealing with vector-valued functions. === Scalar analysis === Scalar analysis is a branch of mathematical analysis dealing with values related to scale as opposed to direction. Values such as temperature are scalar because they describe the magnitude of a value without regard to direction, force, or displacement that value may or may not have. === Tensor analysis === == Other topics == Calculus of variations deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. Harmonic analysis deals with the representation of functions or signals as the superposition of basic waves. Geometric analysis involves the use of geometrical methods in the study of partial differential equations and the application of the theory of partial differential equations to geometry. Clifford analysis, the study of Clifford valued functions that are annihilated by Dirac or Dirac-like operators, termed in general as monogenic or Clifford analytic functions. p-adic analysis, the study of analysis within the context of p-adic numbers, which differs in some interesting and surprising ways from its real and complex counterparts. Non-standard analysis, which investigates the hyperreal numbers and their functions and gives a rigorous treatment of infinitesimals and infinitely large numbers. Computable analysis, the study of which parts of analysis can be carried out in a computable manner. Stochastic calculus – analytical notions developed for stochastic processes. Set-valued analysis – applies ideas from analysis and topology to set-valued functions. Convex analysis, the study of convex sets and functions. Idempotent analysis – analysis in the context of an idempotent semiring, where the lack of an additive inverse is compensated somewhat by the idempotent rule A + A = A. Tropical analysis – analysis of the idempotent semiring called the tropical semiring (or max-plus algebra/min-plus algebra). Constructive analysis, which is built upon a foundation of constructive, rather than classical, logic and set theory. Intuitionistic analysis, which is developed from constructive logic like constructive analysis but also incorporates choice sequences. Paraconsistent analysis, which is built upon a foundation of paraconsistent, rather than classical, logic and set theory. Smooth infinitesimal analysis, which is developed in a smooth topos. == Applications == Techniques from analysis are also found in other areas such as: === Physical sciences === The vast majority of classical mechanics, relativity, and quantum mechanics is based on applied analysis, and differential equations in particular. Examples of important differential equations include Newton's second law, the Schrödinger equation, and the Einstein field equations. Functional analysis is also a major factor in quantum mechanics. === Signal processing === When processing signals, such as audio, radio waves, light waves, seismic waves, and even images, Fourier analysis can isolate individual components of a compound waveform, concentrating them for easier detection or removal. A large family of signal processing techniques consist of Fourier-transforming a signal, manipulating the Fourier-transformed data in a simple way, and reversing the transformation. === Other areas of mathematics === Techniques from analysis are used in many areas of mathematics, including: Analytic number theory Analytic combinatorics Continuous probability Differential entropy in information theory Differential games Differential geometry, the application of calculus to specific mathematical spaces known as manifolds that possess a complicated internal structure but behave in a simple manner locally. Differentiable manifolds Differential topology Partial differential equations == Famous Textbooks == Foundation of Analysis: The Arithmetic of Whole Rational, Irrational and Complex Numbers, by Edmund Landau Introductory Real Analysis, by Andrey Kolmogorov, Sergei Fomin Differential and Integral Calculus (3 volumes), by Grigorii Fichtenholz The Fundamentals of Mathematical Analysis (2 volumes), by Grigorii Fichtenholz A Course Of Mathematical Analysis (2 volumes), by Sergey Nikolsky Mathematical Analysis (2 volumes), by Vladimir Zorich A Course of Higher Mathematics (5 volumes, 6 parts), by Vladimir Smirnov Differential And Integral Calculus, by Nikolai Piskunov A Course of Mathematical Analysis, by Aleksandr Khinchin Mathematical Analysis: A Special Course, by Georgiy Shilov Theory of Functions of a Real Variable (2 volumes), by Isidor Natanson Problems in Mathematical Analysis, by Boris Demidovich Problems and Theorems in Analysis (2 volumes), by George Pólya, Gábor Szegő Mathematical Analysis: A Modern Approach to Advanced Calculus, by Tom Apostol Principles of Mathematical Analysis, by Walter Rudin Real Analysis: Measure Theory, Integration, and Hilbert Spaces, by Elias Stein Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, by Lars Ahlfors Complex Analysis, by Elias Stein Functional Analysis: Introduction to Further Topics in Analysis, by Elias Stein Analysis (2 volumes), by Terence Tao Analysis (3 volumes), by Herbert Amann, Joachim Escher Real and Functional Analysis, by Vladimir Bogachev, Oleg Smolyanov Real and Functional Analysis, by Serge Lang == See also == Constructive analysis History of calculus Hypercomplex analysis Multiple rule-based problems Multivariable calculus Paraconsistent logic Smooth infinitesimal analysis Timeline of calculus and mathematical analysis == References == == Further reading == Aleksandrov, A. D.; Kolmogorov, A. N.; Lavrent'ev, M. A., eds. (March 1969). Mathematics: Its Content, Methods, and Meaning. Vol. 1–3. Translated by Gould, S. H. (2nd ed.). Cambridge, Massachusetts: The M.I.T. Press / American Mathematical Society. Apostol, Tom M. (1974). Mathematical Analysis (2nd ed.). Addison–Wesley. ISBN 978-0201002881. Binmore, Kenneth George (1981) [1981]. The foundations of analysis: a straightforward introduction. Cambridge University Press. Johnsonbaugh, Richard; Pfaffenberger, William Elmer (1981). Foundations of mathematical analysis. New York: M. Dekker. Nikol'skiĭ [Нико́льский], Sergey Mikhailovich [Серге́й Миха́йлович] (2002). "Mathematical analysis". In Hazewinkel, Michiel (ed.). Encyclopaedia of Mathematics. Springer-Verlag. ISBN 978-1402006098. Fusco, Nicola; Marcellini, Paolo; Sbordone, Carlo (1996). Analisi Matematica Due (in Italian). Liguori Editore. ISBN 978-8820726751. Rombaldi, Jean-Étienne (2004). Éléments d'analyse réelle : CAPES et agrégation interne de mathématiques (in French). EDP Sciences. ISBN 978-2868836816. Rudin, Walter (1976). Principles of Mathematical Analysis (3rd ed.). New York: McGraw-Hill. ISBN 978-0070542358. Rudin, Walter (1987). Real and Complex Analysis (3rd ed.). New York: McGraw-Hill. ISBN 978-0070542341. Whittaker, Edmund Taylor; Watson, George Neville (1927-01-02). A Course Of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions (4th ed.). Cambridge: at the University Press. ISBN 0521067944. {{cite book}}: ISBN / Date incompatibility (help) (vi+608 pages) (reprinted: 1935, 1940, 1946, 1950, 1952, 1958, 1962, 1963, 1992) "Real Analysis – Course Notes" (PDF). Archived (PDF) from the original on 2007-04-19. == External links == Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis Basic Analysis: Introduction to Real Analysis by Jiri Lebl (Creative Commons BY-NC-SA) Mathematical Analysis – Encyclopædia Britannica Calculus and Analysis |
Wikipedia:Mathematical oncology#0 | Mathematical oncology is the use of modeling and simulations applied to the study of cancer (oncology). == History == Teorell made preliminary efforts to model in a work published 1937 because of the problem of the time a drug injected exists within the body was an unknown. Modelling by epidemiological data originated in 1954. == Modeling == Modeling types: epidemiological data mechanistical: tumor growth conceptualized from conceptualization of the tumor matter as a type of mechanism cancer cell population evolution Models use ordinary differential equations and partial differential equations to represent tumor growth, angiogenesis, metastasis development, and treatment responses. == Simulations == Simulation of cancer behavior potentially reduces the need for early-phase experimental trials. == Treatment/therapy == Researchers develop models that describe tumor dynamics, the effects of treatment, to remedy possible non-optimal treatment responses supporting the development of more effective treatment protocols. Control theory and optimization are applied to treatment planning in cancer therapies, particularly in radiotherapy and chemotherapy. By optimizing dose schedules and timing, mathematical oncology aims to maximize therapeutic efficacy while minimizing adverse effects. == Statistical methods == Statistical methods can be important for understanding cancer progression, analyzing treatment outcomes, and identifying significant trends in large data sets. Advances in artificial intelligence (AI) and machine learning have further impacted the field. AI algorithms can process larger amounts of patient data and identify patterns that may predict individual responses to treatment, personalizing therapeutic strategies. == Computational-AI == AI allows researchers to predict the behavior of individual cells with greater accuracy by integrating diverse types of patient data. AI-driven models can also identify mathematical equations that more precisely reflect tumor growth dynamics, helping researchers uncover relationships between various biological factors more quickly. == Notes == == References == == External links == === Mathematical oncology === Moffitt Cancer Center's Integrated Mathematical Oncology Program mathematical-oncology.org Cook, P. J.; Doll, Richard; Fellingham, S. A. (15 January 1969). "A mathematical model for the age distribution of cancer in man". International Journal of Cancer. 4 (1). John Wiley & Sons: 93–112. doi:10.1002/ijc.2910040113. PMID 5346480. Boeryd, B.; Ganelius, T.; Lundin, P.; Mellgren, J. (15 September 1966). "Counting and sizing of tumor metastases in experimental oncology". International Journal of Cancer. 1 (5). John Wiley & Sons: 497–502. doi:10.1002/ijc.2910010509. PMID 5912537. Iversen, S (December 1954). "Human Cancer and Age". Br J Cancer. 8 (4): 575–584. doi:10.1038/bjc.1954.62. PMC 2007970. PMID 14351598. Arley, Niels; Iversen, Simon (September 1953). "ON THE MECHANISM OF EXPERIMENTAL CARCINOGENESIS 3: IX. Application of the Hit Theory to Tumours Produced by Ultraviolet Radiation". Acta Pathologica Microbiologica Scandinavica. 33 (2): 133–150. doi:10.1111/j.1699-0463.1953.tb01503.x. === Mathematical biology === Society for Mathematical Biology |
Wikipedia:Mathematics#0 | Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of abstract objects that consist of either abstractions from nature or—in modern mathematics—purely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to prove properties of objects, a proof consisting of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of the theory under consideration. Mathematics is essential in the natural sciences, engineering, medicine, finance, computer science, and the social sciences. Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent of any scientific experimentation. Some areas of mathematics, such as statistics and game theory, are developed in close correlation with their applications and are often grouped under applied mathematics. Other areas are developed independently from any application (and are therefore called pure mathematics) but often later find practical applications. Historically, the concept of a proof and its associated mathematical rigour first appeared in Greek mathematics, most notably in Euclid's Elements. Since its beginning, mathematics was primarily divided into geometry and arithmetic (the manipulation of natural numbers and fractions), until the 16th and 17th centuries, when algebra and infinitesimal calculus were introduced as new fields. Since then, the interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method, which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than sixty first-level areas of mathematics. == Areas of mathematics == Before the Renaissance, mathematics was divided into two main areas: arithmetic, regarding the manipulation of numbers, and geometry, regarding the study of shapes. Some types of pseudoscience, such as numerology and astrology, were not then clearly distinguished from mathematics. During the Renaissance, two more areas appeared. Mathematical notation led to algebra which, roughly speaking, consists of the study and the manipulation of formulas. Calculus, consisting of the two subfields differential calculus and integral calculus, is the study of continuous functions, which model the typically nonlinear relationships between varying quantities, as represented by variables. This division into four main areas—arithmetic, geometry, algebra, and calculus—endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then studied by mathematicians, but now are considered as belonging to physics. The subject of combinatorics has been studied for much of recorded history, yet did not become a separate branch of mathematics until the seventeenth century. At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics. The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas. Some of these areas correspond to the older division, as is true regarding number theory (the modern name for higher arithmetic) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations. === Number theory === Number theory began with the manipulation of numbers, that is, natural numbers ( N ) , {\displaystyle (\mathbb {N} ),} and later expanded to integers ( Z ) {\displaystyle (\mathbb {Z} )} and rational numbers ( Q ) . {\displaystyle (\mathbb {Q} ).} Number theory was once called arithmetic, but nowadays this term is mostly used for numerical calculations. Number theory dates back to ancient Babylon and probably China. Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria. The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat and Leonhard Euler. The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss. Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example is Fermat's Last Theorem. This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles, who used tools including scheme theory from algebraic geometry, category theory, and homological algebra. Another example is Goldbach's conjecture, which asserts that every even integer greater than 2 is the sum of two prime numbers. Stated in 1742 by Christian Goldbach, it remains unproven despite considerable effort. Number theory includes several subareas, including analytic number theory, algebraic number theory, geometry of numbers (method oriented), diophantine equations, and transcendence theory (problem oriented). === Geometry === Geometry is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines, angles and circles, which were developed mainly for the needs of surveying and architecture, but has since blossomed out into many other subfields. A fundamental innovation was the ancient Greeks' introduction of the concept of proofs, which require that every assertion must be proved. For example, it is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results (theorems) and a few basic statements. The basic statements are not subject to proof because they are self-evident (postulates), or are part of the definition of the subject of study (axioms). This principle, foundational for all mathematics, was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements. The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane (plane geometry) and the three-dimensional Euclidean space. Euclidean geometry was developed without change of methods or scope until the 17th century, when René Descartes introduced what is now called Cartesian coordinates. This constituted a major change of paradigm: Instead of defining real numbers as lengths of line segments (see number line), it allowed the representation of points using their coordinates, which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry was split into two new subfields: synthetic geometry, which uses purely geometrical methods, and analytic geometry, which uses coordinates systemically. Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functions, the study of which led to differential geometry. They can also be defined as implicit equations, often polynomial equations (which spawned algebraic geometry). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions. In the 19th century, mathematicians discovered non-Euclidean geometries, which do not follow the parallel postulate. By questioning that postulate's truth, this discovery has been viewed as joining Russell's paradox in revealing the foundational crisis of mathematics. This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not a mathematical problem. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space. Today's subareas of geometry include: Projective geometry, introduced in the 16th century by Girard Desargues, extends Euclidean geometry by adding points at infinity at which parallel lines intersect. This simplifies many aspects of classical geometry by unifying the treatments for intersecting and parallel lines. Affine geometry, the study of properties relative to parallelism and independent from the concept of length. Differential geometry, the study of curves, surfaces, and their generalizations, which are defined using differentiable functions. Manifold theory, the study of shapes that are not necessarily embedded in a larger space. Riemannian geometry, the study of distance properties in curved spaces. Algebraic geometry, the study of curves, surfaces, and their generalizations, which are defined using polynomials. Topology, the study of properties that are kept under continuous deformations. Algebraic topology, the use in topology of algebraic methods, mainly homological algebra. Discrete geometry, the study of finite configurations in geometry. Convex geometry, the study of convex sets, which takes its importance from its applications in optimization. Complex geometry, the geometry obtained by replacing real numbers with complex numbers. === Algebra === Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were the two main precursors of algebra. Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in the title of his main treatise. Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers. Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas. Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra), and polynomial equations in a single unknown, which were called algebraic equations (a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices, modular integers, and geometric transformations), on which generalizations of arithmetic operations are often valid. The concept of algebraic structure addresses this, consisting of a set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra, as established by the influence and works of Emmy Noether, and popularized by Van der Waerden's book Moderne Algebra. Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include: group theory field theory vector spaces, whose study is essentially the same as linear algebra ring theory commutative algebra, which is the study of commutative rings, includes the study of polynomials, and is a foundational part of algebraic geometry homological algebra Lie algebra and Lie group theory Boolean algebra, which is widely used for the study of the logical structure of computers The study of types of algebraic structures as mathematical objects is the purpose of universal algebra and category theory. The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces; this particular area of application is called algebraic topology. === Calculus and analysis === Calculus, formerly called infinitesimal calculus, was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz. It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis" is commonly used for advanced parts. Analysis is further subdivided into real analysis, where variables represent real numbers, and complex analysis, where variables represent complex numbers. Analysis includes many subareas shared by other areas of mathematics which include: Multivariable calculus Functional analysis, where variables represent varying functions Integration, measure theory and potential theory, all strongly related with probability theory on a continuum Ordinary differential equations Partial differential equations Numerical analysis, mainly devoted to the computation on computers of solutions of ordinary and partial differential equations that arise in many applications === Discrete mathematics === Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example is the set of all integers. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply. Algorithms—especially their implementation and computational complexity—play a major role in discrete mathematics. The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of the 20th century. The P versus NP problem, which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems. Discrete mathematics includes: Combinatorics, the art of enumerating mathematical objects that satisfy some given constraints. Originally, these objects were elements or subsets of a given set; this has been extended to various objects, which establishes a strong link between combinatorics and other parts of discrete mathematics. For example, discrete geometry includes counting configurations of geometric shapes. Graph theory and hypergraphs Coding theory, including error correcting codes and a part of cryptography Matroid theory Discrete geometry Discrete probability distributions Game theory (although continuous games are also studied, most common games, such as chess and poker are discrete) Discrete optimization, including combinatorial optimization, integer programming, constraint programming === Mathematical logic and set theory === The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century. Before this period, sets were not considered to be mathematical objects, and logic, although used for mathematical proofs, belonged to philosophy and was not specifically studied by mathematicians. Before Cantor's study of infinite sets, mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration. Cantor's work offended many mathematicians not only by considering actually infinite sets but by showing that this implies different sizes of infinity, per Cantor's diagonal argument. This led to the controversy over Cantor's set theory. In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour. This became the foundational crisis of mathematics. It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory. Roughly speaking, each mathematical object is defined by the set of all similar objects and the properties that these objects must have. For example, in Peano arithmetic, the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. This mathematical abstraction from reality is embodied in the modern philosophy of formalism, as founded by David Hilbert around 1910. The "nature" of the objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion—sometimes called "intuition"—to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems assert, roughly speaking that, in every consistent formal system that contains the natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system. This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer, who promoted intuitionistic logic, which explicitly lacks the law of excluded middle. These problems and debates led to a wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory, type theory, computability theory and computational complexity theory. Although these aspects of mathematical logic were introduced before the rise of computers, their use in compiler design, formal verification, program analysis, proof assistants and other aspects of computer science, contributed in turn to the expansion of these logical theories. === Statistics and other decision sciences === The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory. Statisticians generate data with random sampling or randomized experiments. Statistical theory studies decision problems such as minimizing the risk (expected loss) of a statistical action, such as using a procedure in, for example, parameter estimation, hypothesis testing, and selecting the best. In these traditional areas of mathematical statistics, a statistical-decision problem is formulated by minimizing an objective function, like expected loss or cost, under specific constraints. For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence. Because of its use of optimization, the mathematical theory of statistics overlaps with other decision sciences, such as operations research, control theory, and mathematical economics. === Computational mathematics === Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory; numerical analysis broadly includes the study of approximation and discretization with special focus on rounding errors. Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic-matrix-and-graph theory. Other areas of computational mathematics include computer algebra and symbolic computation. == History == === Etymology === The word mathematics comes from the Ancient Greek word máthēma (μάθημα), meaning 'something learned, knowledge, mathematics', and the derived expression mathēmatikḗ tékhnē (μαθηματικὴ τέχνη), meaning 'mathematical science'. It entered the English language during the Late Middle English period through French and Latin. Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely the first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle (384–322 BC) this meaning was fully established. In Latin and English, until around 1700, the term mathematics more commonly meant "astrology" (or sometimes "astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This change has resulted in several mistranslations: For example, Saint Augustine's warning that Christians should beware of mathematici, meaning "astrologers", is sometimes mistranslated as a condemnation of mathematicians. The apparent plural form in English goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural ta mathēmatiká (τὰ μαθηματικά) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, inherited from Greek. In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math. === Ancient === In addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It is in Babylonian mathematics that elementary arithmetic (addition, subtraction, multiplication, and division) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time. In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as the Pythagoreans appeared to have considered it a subject in its own right. Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof. His book, Elements, is widely considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is often held to be Archimedes (c. 287 – c. 212 BC) of Syracuse. He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus. Other notable achievements of Greek mathematics are conic sections (Apollonius of Perga, 3rd century BC), trigonometry (Hipparchus of Nicaea, 2nd century BC), and the beginnings of algebra (Diophantus, 3rd century AD). The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine, and an early form of infinite series. === Medieval and later === During the Golden Age of Islam, especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra. Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarizmi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. The Greek and Arabic mathematical texts were in turn translated to Latin during the Middle Ages and made available in Europe. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe, with innovations that revolutionized mathematics, such as the introduction of variables and symbolic notation by François Viète (1540–1603), the introduction of logarithms by John Napier in 1614, which greatly simplified numerical calculations, especially for astronomy and marine navigation, the introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716). Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and the proof of numerous theorems. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss, who made numerous contributions to fields such as algebra, analysis, differential geometry, matrix theory, number theory, and statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved. Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews (MR) database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs." == Symbolic notation and terminology == Mathematical notation is widely used in science and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. This notation consists of symbols used for representing operations, unspecified numbers, relations and any other mathematical objects, and then assembling them into expressions and formulas. More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally Latin or Greek letters, and often include subscripts. Operation and relations are generally represented by specific symbols or glyphs, such as + (plus), × (multiplication), ∫ {\textstyle \int } (integral), = (equal), and < (less than). All these symbols are generally grouped according to specific rules to form expressions and formulas. Normally, expressions and formulas do not appear alone, but are included in sentences of the current language, where expressions play the role of noun phrases and formulas play the role of clauses. Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous definitions that provide a standard foundation for communication. An axiom or postulate is a mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a conjecture. Through a series of rigorous arguments employing deductive reasoning, a statement that is proven to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a lemma. A proven instance that forms part of a more general finding is termed a corollary. Numerous technical terms used in mathematics are neologisms, such as polynomial and homeomorphism. Other technical terms are words of the common language that are used in an accurate meaning that may differ slightly from their common meaning. For example, in mathematics, "or" means "one, the other or both", while, in common language, it is either ambiguous or means "one or the other but not both" (in mathematics, the latter is called "exclusive or"). Finally, many mathematical terms are common words that are used with a completely different meaning. This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, "every free module is flat" and "a field is always a ring". == Relationship with sciences == Mathematics is used in most sciences for modeling phenomena, which then allows predictions to be made from experimental laws. The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model. Inaccurate predictions, rather than being caused by invalid mathematical concepts, imply the need to change the mathematical model used. For example, the perihelion precession of Mercury could only be explained after the emergence of Einstein's general relativity, which replaced Newton's law of gravitation as a better mathematical model. There is still a philosophical debate whether mathematics is a science. However, in practice, mathematicians are typically grouped with scientists, and mathematics shares much in common with the physical sciences. Like them, it is falsifiable, which means in mathematics that, if a result or a theory is wrong, this can be proved by providing a counterexample. Similarly as in science, theories and results (theorems) are often obtained from experimentation. In mathematics, the experimentation may consist of computation on selected examples or of the study of figures or other representations of mathematical objects (often mind representations without physical support). For example, when asked how he came about his theorems, Gauss once replied "durch planmässiges Tattonieren" (through systematic experimentation). However, some authors emphasize that mathematics differs from the modern notion of science by not relying on empirical evidence. === Pure and applied mathematics === Until the 19th century, the development of mathematics in the West was mainly motivated by the needs of technology and science, and there was no clear distinction between pure and applied mathematics. For example, the natural numbers and arithmetic were introduced for the need of counting, and geometry was motivated by surveying, architecture and astronomy. Later, Isaac Newton introduced infinitesimal calculus for explaining the movement of the planets with his law of gravitation. Moreover, most mathematicians were also scientists, and many scientists were also mathematicians. However, a notable exception occurred with the tradition of pure mathematics in Ancient Greece. The problem of integer factorization, for example, which goes back to Euclid in 300 BC, had no practical application before its use in the RSA cryptosystem, now widely used for the security of computer networks. In the 19th century, mathematicians such as Karl Weierstrass and Richard Dedekind increasingly focused their research on internal problems, that is, pure mathematics. This led to split mathematics into pure mathematics and applied mathematics, the latter being often considered as having a lower value among mathematical purists. However, the lines between the two are frequently blurred. The aftermath of World War II led to a surge in the development of applied mathematics in the US and elsewhere. Many of the theories developed for applications were found interesting from the point of view of pure mathematics, and many results of pure mathematics were shown to have applications outside mathematics; in turn, the study of these applications may give new insights on the "pure theory". An example of the first case is the theory of distributions, introduced by Laurent Schwartz for validating computations done in quantum mechanics, which became immediately an important tool of (pure) mathematical analysis. An example of the second case is the decidability of the first-order theory of the real numbers, a problem of pure mathematics that was proved true by Alfred Tarski, with an algorithm that is impossible to implement because of a computational complexity that is much too high. For getting an algorithm that can be implemented and can solve systems of polynomial equations and inequalities, George Collins introduced the cylindrical algebraic decomposition that became a fundamental tool in real algebraic geometry. In the present day, the distinction between pure and applied mathematics is more a question of personal research aim of mathematicians than a division of mathematics into broad areas. The Mathematics Subject Classification has a section for "general applied mathematics" but does not mention "pure mathematics". However, these terms are still used in names of some university departments, such as at the Faculty of Mathematics at the University of Cambridge. === Unreasonable effectiveness === The unreasonable effectiveness of mathematics is a phenomenon that was named and first made explicit by physicist Eugene Wigner. It is the fact that many mathematical theories (even the "purest") have applications outside their initial object. These applications may be completely outside their initial area of mathematics, and may concern physical phenomena that were completely unknown when the mathematical theory was introduced. Examples of unexpected applications of mathematical theories can be found in many areas of mathematics. A notable example is the prime factorization of natural numbers that was discovered more than 2,000 years before its common use for secure internet communications through the RSA cryptosystem. A second historical example is the theory of ellipses. They were studied by the ancient Greek mathematicians as conic sections (that is, intersections of cones with planes). It was almost 2,000 years later that Johannes Kepler discovered that the trajectories of the planets are ellipses. In the 19th century, the internal development of geometry (pure mathematics) led to definition and study of non-Euclidean geometries, spaces of dimension higher than three and manifolds. At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century, Albert Einstein developed the theory of relativity that uses fundamentally these concepts. In particular, spacetime of special relativity is a non-Euclidean space of dimension four, and spacetime of general relativity is a (curved) manifold of dimension four. A striking aspect of the interaction between mathematics and physics is when mathematics drives research in physics. This is illustrated by the discoveries of the positron and the baryon Ω − . {\displaystyle \Omega ^{-}.} In both cases, the equations of the theories had unexplained solutions, which led to conjecture of the existence of an unknown particle, and the search for these particles. In both cases, these particles were discovered a few years later by specific experiments. === Specific sciences === ==== Physics ==== Mathematics and physics have influenced each other over their modern history. Modern physics uses mathematics abundantly, and is also considered to be the motivation of major mathematical developments. ==== Computing ==== Computing is closely related to mathematics in several ways. Theoretical computer science is considered to be mathematical in nature. Communication technologies apply branches of mathematics that may be very old (e.g., arithmetic), especially with respect to transmission security, in cryptography and coding theory. Discrete mathematics is useful in many areas of computer science, such as complexity theory, information theory, and graph theory. In 1998, the Kepler conjecture on sphere packing seemed to also be partially proven by computer. ==== Biology and chemistry ==== Biology uses probability extensively in fields such as ecology or neurobiology. Most discussion of probability centers on the concept of evolutionary fitness. Ecology heavily uses modeling to simulate population dynamics, study ecosystems such as the predator-prey model, measure pollution diffusion, or to assess climate change. The dynamics of a population can be modeled by coupled differential equations, such as the Lotka–Volterra equations. Statistical hypothesis testing, is run on data from clinical trials to determine whether a new treatment works. Since the start of the 20th century, chemistry has used computing to model molecules in three dimensions. ==== Earth sciences ==== Structural geology and climatology use probabilistic models to predict the risk of natural catastrophes. Similarly, meteorology, oceanography, and planetology also use mathematics due to their heavy use of models. ==== Social sciences ==== Areas of mathematics used in the social sciences include probability/statistics and differential equations. These are used in linguistics, economics, sociology, and psychology. Often the fundamental postulate of mathematical economics is that of the rational individual actor – Homo economicus (lit. 'economic man'). In this model, the individual seeks to maximize their self-interest, and always makes optimal choices using perfect information. This atomistic view of economics allows it to relatively easily mathematize its thinking, because individual calculations are transposed into mathematical calculations. Such mathematical modeling allows one to probe economic mechanisms. Some reject or criticise the concept of Homo economicus. Economists note that real people have limited information, make poor choices and care about fairness, altruism, not just personal gain. Without mathematical modeling, it is hard to go beyond statistical observations or untestable speculation. Mathematical modeling allows economists to create structured frameworks to test hypotheses and analyze complex interactions. Models provide clarity and precision, enabling the translation of theoretical concepts into quantifiable predictions that can be tested against real-world data. At the start of the 20th century, there was a development to express historical movements in formulas. In 1922, Nikolai Kondratiev discerned the ~50-year-long Kondratiev cycle, which explains phases of economic growth or crisis. Towards the end of the 19th century, mathematicians extended their analysis into geopolitics. Peter Turchin developed cliodynamics since the 1990s. Mathematization of the social sciences is not without risk. In the controversial book Fashionable Nonsense (1997), Sokal and Bricmont denounced the unfounded or abusive use of scientific terminology, particularly from mathematics or physics, in the social sciences. The study of complex systems (evolution of unemployment, business capital, demographic evolution of a population, etc.) uses mathematical knowledge. However, the choice of counting criteria, particularly for unemployment, or of models, can be subject to controversy. == Philosophy == === Reality === The connection between mathematics and material reality has led to philosophical debates since at least the time of Pythagoras. The ancient philosopher Plato argued that abstractions that reflect material reality have themselves a reality that exists outside space and time. As a result, the philosophical view that mathematical objects somehow exist on their own in abstraction is often referred to as Platonism. Independently of their possible philosophical opinions, modern mathematicians may be generally considered as Platonists, since they think of and talk of their objects of study as real objects. Armand Borel summarized this view of mathematics reality as follows, and provided quotations of G. H. Hardy, Charles Hermite, Henri Poincaré and Albert Einstein that support his views. Something becomes objective (as opposed to "subjective") as soon as we are convinced that it exists in the minds of others in the same form as it does in ours and that we can think about it and discuss it together. Because the language of mathematics is so precise, it is ideally suited to defining concepts for which such a consensus exists. In my opinion, that is sufficient to provide us with a feeling of an objective existence, of a reality of mathematics ... Nevertheless, Platonism and the concurrent views on abstraction do not explain the unreasonable effectiveness of mathematics (as Platonism assumes mathematics exists independently, but does not explain why it matches reality). === Proposed definitions === There is no general consensus about the definition of mathematics or its epistemological status—that is, its place inside knowledge. A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable. There is not even consensus on whether mathematics is an art or a science. Some just say, "mathematics is what mathematicians do". A common approach is to define mathematics by its object of study. Aristotle defined mathematics as "the science of quantity" and this definition prevailed until the 18th century. However, Aristotle also noted a focus on quantity alone may not distinguish mathematics from sciences like physics; in his view, abstraction and studying quantity as a property "separable in thought" from real instances set mathematics apart. In the 19th century, when mathematicians began to address topics—such as infinite sets—which have no clear-cut relation to physical reality, a variety of new definitions were given. With the large number of new areas of mathematics that have appeared since the beginning of the 20th century, defining mathematics by its object of study has become increasingly difficult. For example, in lieu of a definition, Saunders Mac Lane in Mathematics, form and function summarizes the basics of several areas of mathematics, emphasizing their inter-connectedness, and observes: the development of Mathematics provides a tightly connected network of formal rules, concepts, and systems. Nodes of this network are closely bound to procedures useful in human activities and to questions arising in science. The transition from activities to the formal Mathematical systems is guided by a variety of general insights and ideas. Another approach for defining mathematics is to use its methods. For example, an area of study is often qualified as mathematics as soon as one can prove theorems—assertions whose validity relies on a proof, that is, a purely-logical deduction. === Rigor === Mathematical reasoning requires rigor. This means that the definitions must be absolutely unambiguous and the proofs must be reducible to a succession of applications of inference rules, without any use of empirical evidence and intuition. Rigorous reasoning is not specific to mathematics, but, in mathematics, the standard of rigor is much higher than elsewhere. Despite mathematics' concision, rigorous proofs can require hundreds of pages to express, such as the 255-page Feit–Thompson theorem. The emergence of computer-assisted proofs has allowed proof lengths to further expand. The result of this trend is a philosophy of the quasi-empiricist proof that can not be considered infallible, but has a probability attached to it. The concept of rigor in mathematics dates back to ancient Greece, where their society encouraged logical, deductive reasoning. However, this rigorous approach would tend to discourage exploration of new approaches, such as irrational numbers and concepts of infinity. The method of demonstrating rigorous proof was enhanced in the sixteenth century through the use of symbolic notation. In the 18th century, social transition led to mathematicians earning their keep through teaching, which led to more careful thinking about the underlying concepts of mathematics. This produced more rigorous approaches, while transitioning from geometric methods to algebraic and then arithmetic proofs. At the end of the 19th century, it appeared that the definitions of the basic concepts of mathematics were not accurate enough for avoiding paradoxes (non-Euclidean geometries and Weierstrass function) and contradictions (Russell's paradox). This was solved by the inclusion of axioms with the apodictic inference rules of mathematical theories; the re-introduction of axiomatic method pioneered by the ancient Greeks. It results that "rigor" is no more a relevant concept in mathematics, as a proof is either correct or erroneous, and a "rigorous proof" is simply a pleonasm. Where a special concept of rigor comes into play is in the socialized aspects of a proof, wherein it may be demonstrably refuted by other mathematicians. After a proof has been accepted for many years or even decades, it can then be considered as reliable. Nevertheless, the concept of "rigor" may remain useful for teaching to beginners what is a mathematical proof. == Training and practice == === Education === Mathematics has a remarkable ability to cross cultural boundaries and time periods. As a human activity, the practice of mathematics has a social side, which includes education, careers, recognition, popularization, and so on. In education, mathematics is a core part of the curriculum and forms an important element of the STEM academic disciplines. Prominent careers for professional mathematicians include mathematics teacher or professor, statistician, actuary, financial analyst, economist, accountant, commodity trader, or computer consultant. Archaeological evidence shows that instruction in mathematics occurred as early as the second millennium BCE in ancient Babylonia. Comparable evidence has been unearthed for scribal mathematics training in the ancient Near East and then for the Greco-Roman world starting around 300 BCE. The oldest known mathematics textbook is the Rhind papyrus, dated from c. 1650 BCE in Egypt. Due to a scarcity of books, mathematical teachings in ancient India were communicated using memorized oral tradition since the Vedic period (c. 1500 – c. 500 BCE). In Imperial China during the Tang dynasty (618–907 CE), a mathematics curriculum was adopted for the civil service exam to join the state bureaucracy. Following the Dark Ages, mathematics education in Europe was provided by religious schools as part of the Quadrivium. Formal instruction in pedagogy began with Jesuit schools in the 16th and 17th century. Most mathematical curricula remained at a basic and practical level until the nineteenth century, when it began to flourish in France and Germany. The oldest journal addressing instruction in mathematics was L'Enseignement Mathématique, which began publication in 1899. The Western advancements in science and technology led to the establishment of centralized education systems in many nation-states, with mathematics as a core component—initially for its military applications. While the content of courses varies, in the present day nearly all countries teach mathematics to students for significant amounts of time. During school, mathematical capabilities and positive expectations have a strong association with career interest in the field. Extrinsic factors such as feedback motivation by teachers, parents, and peer groups can influence the level of interest in mathematics. Some students studying mathematics may develop an apprehension or fear about their performance in the subject. This is known as mathematical anxiety, and is considered the most prominent of the disorders impacting academic performance. Mathematical anxiety can develop due to various factors such as parental and teacher attitudes, social stereotypes, and personal traits. Help to counteract the anxiety can come from changes in instructional approaches, by interactions with parents and teachers, and by tailored treatments for the individual. === Psychology (aesthetic, creativity and intuition) === The validity of a mathematical theorem relies only on the rigor of its proof, which could theoretically be done automatically by a computer program. This does not mean that there is no place for creativity in a mathematical work. On the contrary, many important mathematical results (theorems) are solutions of problems that other mathematicians failed to solve, and the invention of a way for solving them may be a fundamental way of the solving process. An extreme example is Apery's theorem: Roger Apery provided only the ideas for a proof, and the formal proof was given only several months later by three other mathematicians. Creativity and rigor are not the only psychological aspects of the activity of mathematicians. Some mathematicians can see their activity as a game, more specifically as solving puzzles. This aspect of mathematical activity is emphasized in recreational mathematics. Mathematicians can find an aesthetic value to mathematics. Like beauty, it is hard to define, it is commonly related to elegance, which involves qualities like simplicity, symmetry, completeness, and generality. G. H. Hardy in A Mathematician's Apology expressed the belief that the aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He also identified other criteria such as significance, unexpectedness, and inevitability, which contribute to mathematical aesthetics. Paul Erdős expressed this sentiment more ironically by speaking of "The Book", a supposed divine collection of the most beautiful proofs. The 1998 book Proofs from THE BOOK, inspired by Erdős, is a collection of particularly succinct and revelatory mathematical arguments. Some examples of particularly elegant results included are Euclid's proof that there are infinitely many prime numbers and the fast Fourier transform for harmonic analysis. Some feel that to consider mathematics a science is to downplay its artistry and history in the seven traditional liberal arts. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematical results are created (as in art) or discovered (as in science). The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions. == Cultural impact == === Artistic expression === Notes that sound well together to a Western ear are sounds whose fundamental frequencies of vibration are in simple ratios. For example, an octave doubles the frequency and a perfect fifth multiplies it by 3 2 {\displaystyle {\frac {3}{2}}} . Humans, as well as some other animals, find symmetric patterns to be more beautiful. Mathematically, the symmetries of an object form a group known as the symmetry group. For example, the group underlying mirror symmetry is the cyclic group of two elements, Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } . A Rorschach test is a figure invariant by this symmetry, as are butterfly and animal bodies more generally (at least on the surface). Waves on the sea surface possess translation symmetry: moving one's viewpoint by the distance between wave crests does not change one's view of the sea. Fractals possess self-similarity. === Popularization === Popular mathematics is the act of presenting mathematics without technical terms. Presenting mathematics may be hard since the general public suffers from mathematical anxiety and mathematical objects are highly abstract. However, popular mathematics writing can overcome this by using applications or cultural links. Despite this, mathematics is rarely the topic of popularization in printed or televised media. === Awards and prize problems === The most prestigious award in mathematics is the Fields Medal, established in 1936 and awarded every four years (except around World War II) to up to four individuals. It is considered the mathematical equivalent of the Nobel Prize. Other prestigious mathematics awards include: The Abel Prize, instituted in 2002 and first awarded in 2003 The Chern Medal for lifetime achievement, introduced in 2009 and first awarded in 2010 The AMS Leroy P. Steele Prize, awarded since 1970 The Wolf Prize in Mathematics, also for lifetime achievement, instituted in 1978 A famous list of 23 open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert. This list has achieved great celebrity among mathematicians, and at least thirteen of the problems (depending how some are interpreted) have been solved. A new list of seven important problems, titled the "Millennium Prize Problems", was published in 2000. Only one of them, the Riemann hypothesis, duplicates one of Hilbert's problems. A solution to any of these problems carries a 1 million dollar reward. To date, only one of these problems, the Poincaré conjecture, has been solved by the Russian mathematician Grigori Perelman. == See also == == Notes == == References == === Citations === === Other sources === == Further reading == |
Wikipedia:Mathematics Subject Classification#0 | The Mathematics Subject Classification (MSC) is an alphanumerical classification scheme that has collaboratively been produced by staff of, and based on the coverage of, the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH. The MSC is used by many mathematics journals, which ask authors of research papers and expository articles to list subject codes from the Mathematics Subject Classification in their papers. The current version is MSC2020. == Structure == The MSC is a hierarchical scheme, with three levels of structure. A classification can be two, three or five digits long, depending on how many levels of the classification scheme are used. The first level is represented by a two-digit number, the second by a letter, and the third by another two-digit number. For example: 53 is the classification for differential geometry 53A is the classification for classical differential geometry 53A45 is the classification for vector and tensor analysis === First level === At the top level, 63 mathematical disciplines are labeled with a unique two-digit number. In addition to the typical areas of mathematical research, there are top-level categories for "History and Biography", "Mathematics Education", and for the overlap with different sciences. Physics (i.e. mathematical physics) is particularly well represented in the classification scheme with a number of different categories including: Fluid mechanics Quantum mechanics Geophysics Optics and electromagnetic theory All valid MSC classification codes must have at least the first-level identifier. === Second level === The second-level codes are a single letter from the Latin alphabet. These represent specific areas covered by the first-level discipline. The second-level codes vary from discipline to discipline. For example, for differential geometry, the top-level code is 53, and the second-level codes are: A for classical differential geometry B for local differential geometry C for global differential geometry D for symplectic geometry and contact geometry In addition, the special second-level code "-" is used for specific kinds of materials. These codes are of the form: 53-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 53-01 Instructional exposition (textbooks, tutorial papers, etc.) 53-02 Research exposition (monographs, survey articles) 53-03 Historical (must also be assigned at least one classification number from Section 01) 53-04 Explicit machine computation and programs (not the theory of computation or programming) 53-06 Proceedings, conferences, collections, etc. The second and third level of these codes are always the same - only the first level changes. For example, it is not valid to use 53- as a classification. Either 53 on its own or, better yet, a more specific code should be used. === Third level === Third-level codes are the most specific, usually corresponding to a specific kind of mathematical object or a well-known problem or research area. The third-level code 99 exists in every category and means none of the above, but in this section. == Using the scheme == The AMS recommends that papers submitted to its journals for publication have one primary classification and one or more optional secondary classifications. A typical MSC subject class line on a research paper looks like MSC Primary 03C90; Secondary 03-02; == History == According to the American Mathematical Society (AMS) help page about MSC, the MSC has been revised a number of times since 1940. Based on a scheme to organize AMS's Mathematical Offprint Service (MOS scheme), the AMS Classification was established for the classification of reviews in Mathematical Reviews in the 1960s. It saw various ad-hoc changes. Despite its shortcomings, Zentralblatt für Mathematik started to use it as well in the 1970s. In the late 1980s, a jointly revised scheme with more formal rules was agreed upon by Mathematical Reviews and Zentralblatt für Mathematik under the new name Mathematics Subject Classification. It saw various revisions as MSC1990, MSC2000 and MSC2010. In July 2016, Mathematical Reviews and zbMATH started collecting input from the mathematical community on the next revision of MSC, which was released as MSC2020 in January 2020. The original classification of older items has not been changed. This can sometimes make it difficult to search for older works dealing with particular topics. Changes at the first level involved the subjects with (present) codes 03, 08, 12-20, 28, 37, 51, 58, 74, 90, 91, 92. == Relation to other classification schemes == For physics papers the Physics and Astronomy Classification Scheme (PACS) is often used. Due to the large overlap between mathematics and physics research it is quite common to see both PACS and MSC codes on research papers, particularly for multidisciplinary journals and repositories such as the arXiv. The ACM Computing Classification System (CCS) is a similar hierarchical classification scheme for computer science. There is some overlap between the AMS and ACM classification schemes, in subjects related to both mathematics and computer science, however the two schemes differ in the details of their organization of those topics. The classification scheme used on the arXiv is chosen to reflect the papers submitted. As arXiv is multidisciplinary its classification scheme does not fit entirely with the MSC, ACM or PACS classification schemes. It is common to see codes from one or more of these schemes on individual papers. == First-level areas == 00: General (Includes topics such as recreational mathematics, philosophy of mathematics and mathematical modeling.) 01: History and biography 03: Mathematical logic and foundations (including model theory, computability theory, set theory, proof theory, and algebraic logic) 05: Combinatorics 06: Order, lattices, ordered algebraic structures 08: General algebraic systems 11: Number theory 12: Field theory and polynomials 13: Commutative algebra (Commutative rings and algebras) 14: Algebraic geometry 15: Linear and multilinear algebra; matrix theory 16: Associative rings and (associative) algebras 17: Non-associative rings and (non-associative) algebras 18: Category theory; homological algebra 19: K-theory 20: Group theory and generalizations 22: Topological groups, Lie groups (and analysis upon them) 26: Real functions (including derivatives and integrals) 28: Measure and integration 30: Functions of a complex variable (including approximation theory in the complex domain) 31: Potential theory 32: Several complex variables and analytic spaces 33: Special functions 34: Ordinary differential equations 35: Partial differential equations 37: Dynamical systems and ergodic theory 39: Difference equations and functional equations 40: Sequences, series, summability 41: Approximations and expansions 42: Harmonic analysis on Euclidean spaces (including Fourier analysis, Fourier transforms, trigonometric approximation, trigonometric interpolation, and orthogonal functions) 43: Abstract harmonic analysis 44: Integral transforms, operational calculus 45: Integral equations 46: Functional analysis (including infinite-dimensional holomorphy, integral transforms in distribution spaces) 47: Operator theory 49: Calculus of variations and optimal control; optimization (including geometric integration theory) 51: Geometry 52: Convex and discrete geometry 53: Differential geometry 54: General topology 55: Algebraic topology 57: Manifolds and cell complexes 58: Global analysis, analysis on manifolds (including infinite-dimensional holomorphy) 60: Probability theory and stochastic processes 62: Statistics 65: Numerical analysis 68: Computer science 70: Mechanics of particles and systems (including particle mechanics) 74: Mechanics of deformable solids 76: Fluid mechanics 78: Optics, electromagnetic theory 80: Classical thermodynamics, heat transfer 81: Quantum theory 82: Statistical mechanics, structure of matter 83: Relativity and gravitational theory (including relativistic mechanics) 85: Astronomy and astrophysics 86: Geophysics 90: Operations research, mathematical programming 91: Game theory, economics, social and behavioral sciences 92: Biology and other natural sciences 93: Systems theory; control (including optimal control) 94: Information and communication, circuits 97: Mathematics education == See also == Areas of mathematics Mathematical knowledge management MathSciNet == References == == External links == MSC2020-Mathematical Sciences Classification System (PDF of MSC2020) The Zentralblatt MATH page on the Mathematics Subject Classification. MSC2020 can be seen here. Mathematics Subject Classification 2010 – the site where the MSC2010 revision was carried out publicly in an MSCwiki. A view of the whole scheme and the changes made from MSC2000, as well as PDF files of the MSC and ancillary documents are there. A personal copy of the MSC in TiddlyWiki form can be had also. The American Mathematical Society page on the Mathematics Subject Classification. Rusin, Dave. "A Gentle Introduction to the Mathematics Subject Classification Scheme". Mathematical Atlas. Archived from the original on 2015-05-16. |
Wikipedia:Mathematics and Computing College#0 | Mathematics and Computing Colleges were introduced in England in 2002 and Northern Ireland in 2006 as part of the Government's Specialist Schools programme which was designed to raise standards in secondary education. Specialist schools focus on their chosen specialism but must also meet the requirements of the National Curriculum and deliver a broad and balanced education to all their pupils. Mathematics and Computing Colleges must focus on mathematics and either computing or ICT. Colleges are expected to disseminate good practice and share resources with other schools and the wider community. They often develop active partnerships with local organisations and their feeder primary schools. They also work with local businesses to promote the use of mathematics and computing outside of school. In 2007 there were 222 schools in England which were designated as specialist Mathematics and Computing Colleges. A further 21 schools were designated in combined specialisms which included mathematics and computing, and 15 had a second specialism in Mathematics and Computing. The Specialist Schools programme ended in 2011. Since then, schools in England have to either become an academy or apply through the Dedicated Schools Grant if they wish to become a Mathematics and Computing College. As of 2021 there are few Mathematics and Computing Colleges left in the United Kingdom. == References == == External links == Vision for Mathematics and Computing Colleges, The Standards Site Specialist Schools Programme |
Wikipedia:Mathematics education in France#0 | Education in France is organized in a highly centralized manner, with many subdivisions. It is divided into the three stages of primary education (enseignement primaire), secondary education (enseignement secondaire), and higher education (enseignement supérieur). Two year olds do not start primary school, they start preschool. Then, by the age of six, a child in France starts primary school and soon moves into higher and higher grade levels until they graduate. In French higher education, the following degrees are recognized by the Bologna Process (EU recognition): Licence and Licence Professionnelle (bachelor's degrees), and the comparably named Master and Doctorat degrees. The Programme for International Student Assessment coordinated by the OECD in 2018 ranked the overall knowledge and skills of French 15-year-olds as 26th in the world in reading literacy, mathematics, and science, below the OECD average of 493. The average OECD performance of French 15-year-olds in science and mathematics has declined, with the share of low performers in reading, mathematics and science developing a sharp upward trend. France's share of top performers in mathematics and science has also declined. France's performance in mathematics and science at the middle school level was ranked 23 in the 1995 Trends in International Math and Science Study. In 2019, France ranked 21 in the TIMSS Science general ranking. == History == Napoleon began the French university and secondary educational systems. Guizot started the elementary system. Intense battles took place over whether the Catholic Church should play a dominant role. The modern era of French education begins at the end of the 19th century. Jules Ferry, the Minister of Public Instruction in 1881, is widely credited for creating the modern school (l'école républicaine) by requiring all children between the ages of 6 and 12, both boys and girls, to attend. He also made public instruction mandatory, free of charge, and secular (laïque). With those laws, known as French Lubbers, Jules Ferry laws, and several others, the Third Republic repealed most of the Falloux Laws of 1850–1851, which gave an important role to the clergy. The French curriculum predominantly emphasized the works of French writers of European descent. Ferry and others considered literature the glue of French identity. The ethnic and cultural demographics of the student body did not factor in to the quest to transmit a "common culture" to the students. Like literature, history education is seen as critical to shaping the identity of young people and the integration of immigrants to French identity. Ferry's views continue to exert influence today. Ministry reports have confirmed that the rule of schools in promoting "common culture" is only made more critical by the rising levels of student diversity. According to the ministry, history education in France has, over the course of one century made possible "the integration of children of Italians, Poles, Africans and Portuguese". == Governance == All educational programmes in France are regulated by the Ministry of National Education and Youth (officially called Ministère de l'Éducation nationale et de la Jeunesse). The head of the ministry is the Minister of National Education. All teachers in public primary and secondary schools are state civil servants, making the ministère the largest employer in the country. Professors and researchers in France's universities are also employed by the state. At the primary and secondary levels, the curriculum is the same for all French students in any given grade, which includes public, semi-public and subsidised institutions. However, there exist specialised sections and a variety of options that students can choose. The reference for all French educators is the Bulletin officiel de l'éducation nationale, de l'enseignement supérieur et de la recherche (B.O.), which lists all current programmes and teaching directives. It is amended multiple times each year. Since 2021, schooling is mandatory and families may only provide teaching outside of a school in exceptional circumstances: Long-term illness, Intensive athletic or artistic activities, Distance from a public school, Or other specific reasons. == School year == In Metropolitan France, the school year runs from early September to early July. The school calendar is standardized throughout the country and is the sole domain of the ministry. In May, schools need time to organize exams (for example, the baccalauréat). Outside Metropolitan France, the school calendar is set by the local recteur. Major holiday breaks are as follows: All Saints (la Toussaint), two weeks (since 2012) around the end of October and the beginning of November; Christmas (Noël), two weeks around Christmas Day and New Year's Day; winter (hiver), two weeks starting in mid-February; spring (printemps) or Easter (Pâques), two weeks starting in mid-April; summer (été), two months starting in early July. (mid-June for high school students). == Primary school == Most parents start sending their children to preschool (maternelle) when they turn 3. Some even start earlier at age 2 in toute petite section ("TPS"). The first two years of preschool (TPS and petite section "PS") are introductions to community living; children learn how to become students and are introduced to their first notions of arithmetic, begin to recognize letters, develop oral language, etc. The last two years of preschool, moyenne section and grande section, are more school-like; pupils are introduced to reading, writing and more mathematics. A preschool can be stand-alone (mostly true in towns and cities) or be affiliated to an elementary school (mostly in villages). As in other educational systems, primary school students in France usually have a single teacher (or two) who teaches the entire curriculum, without specialist teachers. After kindergarten, the young students move on to the école élémentaire (elementary school). In the first 3 years of elementary school, they learn to write, develop their reading skills and get some basics in subjects such as French, mathematics, science and the arts, to name a few. The French word for a teacher at the primary school level is professeur or professeure des écoles (previously called instituteur, or its feminine form institutrice). Children stay in elementary school for 5 years until they are 10–11 years-old. The grades are named: CP (cours préparatoire), CE1 (cours élémentaire 1), CE2 (cours élémentaire 2), CM1 (cours moyen 1) and CM2 (cours moyen 2). == Middle school and high school == The compulsory middle and high school subjects cover French language and literature, history and geography, foreign languages, arts and crafts, musical education, civics, mathematics, physics, chemistry, natural sciences, technology, and PE. The curriculum is set by the Ministry of National Education and applies to most collèges in France and also to AEFE-dependent institutions. Académies and individual schools have little freedom in the State curriculum. Class sizes vary from school to school, but usually range from 20 to 35 pupils. After primary school, two educational stages follow: collège (middle school), for children during their first four years of secondary education from the age of 11 to 15. lycée (high school), which provides a three-year course of further secondary education for children between the ages of 15 and 18. Pupils are prepared for the baccalauréat (baccalaureate, colloquially known as le bac) or the CAP (Certificat d'aptitude professionnelle). The baccalauréat can lead to higher education studies or directly to professional life (there are three main types of baccalauréat: the baccalauréat général, the baccalauréat technologique, and the baccalauréat professionnel). CFA (centre de formation des apprentise, apprentice learning center), which provides vocational degrees: le Certificat d'aptitude professionnelle. == Private schools == Primary and secondary private schools in France are divided into two categories: Private schools which respect the State curriculum (known as "under contract') are private, fee-paying institutions where pupils study the same national curriculum as those in public schools. Teachers in private schools are recruited in the same way and have roughly the same status as their equivalents in public schools. They are also employed directly by the State, but they are not permanently assigned and may not return to a public school position. The great majority of private schools in France are "under contract". Private schools without contract employ their teachers directly and may teach their own curriculum; the State, however, still monitors their educational standards. Most of these schools provide religious instruction in parallel with a broad curriculum. == International education == As of January 2015, the International Schools Consultancy (ISC) listed France as having 105 international schools. ISC defines an 'international school' in the following terms: "ISC includes an international school if the school delivers a curriculum to any combination of pre-school, primary or secondary students, wholly or partly in English outside an English-speaking country, or if a school in a country where English is one of the official languages, offers an English-medium curriculum other than the country's national curriculum and is international in its orientation." That definition is used by publications including The Economist. France has its own international school regulator, the AEFE (Agence pour l'enseignement français à l'étranger). == Higher education == Higher education in France is organized in three levels, which correspond to those of other European countries, facilitating international mobility: the Licence and Licence Professionnelle (bachelor's degrees), and the Master's and Doctorat degrees. The Licence and the Master are organized in semesters: 6 for the Licence and 4 for the Master. Those levels of study include various "parcours" or paths based on UE (Unités d'enseignement or Modules), each worth a defined number of European credits (ECTS). A student accumulates those credits, which are generally transferable between paths. A licence is awarded once 180 ECTS have been obtained; a master is awarded once 120 additional credits have been obtained. Licence and master's degrees are offered within specific domaines and carry a specific mention. Spécialités, which are either research-oriented or professionally oriented during the second year of the Master. There are also professional licences whose objective is immediate job integration. It is possible to return to school later by continuing education or to validate professional experience (through VAE, Validation des Acquis de l’Expérience). Higher education in France is divided between grandes écoles and public universities. The grandes écoles admit the graduates of the level Baccalauréat + 2 years of validated study (or sometimes directly after the Baccalauréat) whereas universities admit all graduates of the Baccalauréat. Higher education in France was reshaped by the student revolts of May 1968. During the 1960s, French public universities responded to a massive explosion in the number of students (280,000 in 1962–63 to 500,000 in 1967–68) by stuffing approximately one-third of their students into hastily developed campus annexes (roughly equivalent to American satellite campuses) which lacked decent amenities, resident professors, academic traditions, or the dignity of university status. With so many students ripe for radicalization after being forced to study in such miserable conditions, change was necessary and inevitable. Rather than expand already-overwhelmed parent campuses, it was decided to split off the annexes as new universities. As a result, French higher education, compared with other countries, is small in size with a multiplicity of establishments, each specialised in a more-or-less broad spectrum of areas. A middle-sized French city, such as Grenoble or Nancy, may have 2 or 3 universities (focused on science, sociological studies, engineering, etc.) as well as a number of other establishments specialised in higher education. In Paris and its suburbs, there are currently 11 universities (there were 13 from 1970 to 2017), none of which is specialised in one area or another, plus many smaller institutions that are highly specialised. It is not uncommon for graduate teaching programmes (master's degrees, the course part of doctorate programmes etc.) to be operated in common by several institutions, allowing the institutions to present a larger variety of courses. In engineering schools and the professional degrees of universities, a large share of the teaching staff is often made up of non-permanent professors; instead, part-time professors are hired to teach one specific subject. Part-time professors are generally hired from neighbouring universities, research institutes or industries. Another original feature of the French higher education system is that a large share of the scientific research is carried out by research establishments such as CNRS or INSERM, which are not formally part of the universities. However, in most cases, the research units of those establishments are located inside universities (or other higher education establishments) and jointly operated by the research establishment and the university. In 2021, 1.65 million students are enrolled in French higher education institutions (61% in licence, 35% in master, 4% in doctorat). === Tuition costs === Higher education is mostly funded by the State which leads to very low tuition fees. For citizens of the EU, EEA, Switzerland or Quebec, the annual fees range from 170 to 380 euros per year depending on the level (licence, master, doctorat). One can therefore get a master's degree (in 5 years) for about €750–3,500. For other international students, these fees range from 2,770 to 3,770 euros. Students from low-income families can apply for scholarships, paying nominal sums for tuition or textbooks, and can receive a monthly stipend of up to €450 per month. The tuition in public engineering schools is comparable to universities but a little higher (around €700). However, it can reach €7,000 a year for private engineering schools. Private business schools typically charge up to €12,000 a year for Bachelor programmes and up to €24,000 for Master programmes, while some elite institutions may charge €40,000 and more. Health insurance for students is free until the age of 20 and so only the costs of living and books must be added. After the age of 20, health insurance for students costs €200 a year and covers most of the medical expenses. Some public schools have other ways of gaining money. Some do not receive funds for class trips and other extra activities and so those schools may ask for a small entrance fee for new students. === Universities === ==== Public universities ==== The public universities in France are named after the major cities near which they are located, followed by a numeral if there are several. Paris, for example, has 13 universities, labelled Paris I to XIII. Some of them are in Paris itself, some in the suburbs. In addition, most of the universities have taken a more informal name that is usually that of a famous person or a particular place. Sometimes, it is also a way to honor a famous alumnus, for example the science university in Strasbourg is known as "Université Louis-Pasteur" while its official name is "Université Strasbourg I" (however, since 2009, the three universities of Strasbourg have been merged). The French system has undergone a reform, the Bologna process, which aims at creating European standards for university studies, most notably a similar time-frame everywhere, with three years devoted to the bachelor's degree ("Licence" in French), two for the Master's degree, and three for the doctorate. French universities have also adopted the ECTS credit system (for example, a licence is worth 180 credits). The traditional curriculum based on end of semester examinations tends to remain in place in some universities. That double standard has added complexity to a system. It is difficult to change a major during undergraduate studies without losing a semester or even a whole year. Students usually also have few course selection options once they enroll in a particular diploma. There are about 4,000 Master programmes offered in the French university system (listed at trouvermonmaster.gouv.fr), and 17,000 undergraduate programmes (offered by Parcoursup). ==== IUTs ==== There are also the IUTs (University Technical Institutes). This is a type of university college, usually in small and medium-sized towns, at satellite campuses of larger universities, that offers post-secondary study programmes designed to provide higher vocational education or the technical skills needed to perform the tasks of a particular and specific job. The degree awarded by IUTs is the Bachelor Universitaire de Technologie, (in English: "Bachelor of Technical Studies") which is very similar to a Bachelor of Applied Science. ==== Other universities ==== France also hosts rare catholic universities recognized by the French Government as "free" private colleges (Facultés Libres), the largest one being the Catholic University of Lille. There are also branch colleges of foreign universities, which include Baruch College, the University of London Institute in Paris, Parsons Paris School of Art and Design and the American University of Paris. === Public and private Grandes écoles === The grandes écoles of France are elite higher-education institutions. They are generally focused on a single subject area (e.g., engineering or business), have a small size (typically between 100 and 300 graduates per year), and are highly selective. They are widely regarded as prestigious, and most of France's scientists and executives have graduated from a grande école. National rankings are published every year by various magazines. While the rankings slightly vary from year to year, the top grandes écoles have been very stable for decades: science and engineering: Écoles Normales Supérieures, École polytechnique, Mines ParisTech, Télécom Paris, ISAE-Supaéro, Ponts Paristech, Ecole Nationale de l'Aviation Civile, École nationale supérieure d'ingénieurs Sud-Alsace and CentraleSupélec; humanities: three Écoles Normales Supérieures, École des Chartes, and CELSA – Sorbonne; business: HEC Paris, NEOMA Business School, ESSEC Business School, ESCP Europe, INSEAD, EMLyon, Audencia, Grenoble École de Management, INSEEC and EDHEC; administration and political sciences: ENA and Institutes of Political Studies (Sciences Po Aix, Sciences Po Bordeaux, Sciences Po Grenoble, Sciences Po Lille, Sciences Po Lyon, Sciences Po Paris, Sciences Po Rennes, Sciences Po Saint-Germain-en-Laye, Sciences Po Strasbourg). === Private higher education === Private higher education in France is made possible by the fundamental principle of freedom of education. The Laboulaye Law of 12 July 1875, repealed in 2000, on the freedom of higher education stipulated that "higher education is free", albeit with a few conditions to be met: declaration to the State; administrators and professors who have not been convicted; annual publication of corporate accounts. In France, there are two main categories of private higher education colleges and universities: "free" private colleges (Facultés Libres): these private higher education colleges generally correspond to free faculties, most of which were created in the 19th century following the 1875 law on the freedom of higher education, and to Catholic Universities – officially "Catholic Institutes" – which may group together several free faculties. Sciences Po was a free private college until its nationalization in 1945 as a public Grande école. These institutions are not-for-profit associations, and have all obtained the French State qualification of "établissement d'enseignement supérieur privé d'intérêt général" (EESPIG); "technical" private colleges (Écoles Techniques): they correspond to the vast majority of private Grandes Écoles and colleges of higher education in France, a small number of which are not-for-profit associations under contract with the State through the EESPIG general interest qualification. The remainder are commercial – for-profit – companies, as well as former public consular colleges that have opted for limited company status (like HEC Paris, EMLyon, or ESSEC Business School); and private high schools offering higher education courses (STS, CPGE). Since 2018, a public university can integrate not only public faculties, but also private colleges (which have EESPIG accreditation, which means of general interest) as part of an experimental public-private partnership. Voices have been raised in the teaching community, fearing the privatization of French public higher education and tuition fee increases. In France, private colleges and universities cannot award national Bachelor's (Licence) and Master's degrees. They can, however, either: set up co-diploma partnerships with public universities to deliver their national diplomas, or have their own diplomas accredited by the French Ministry of Higher Education (the accredited diploma, or "Diplôme Visé"), or have their programs certified (with the Titre) by the public vocational training agency France Compétences, under the authority of the French Ministry of Labor. === Preparatory classes (CPGEs) === Preparatory classes (in French "classes préparatoires aux grandes écoles" or CPGE), widely known as prépas, are courses whose main goal is to prepare students for enrollment in a grande école. Admission to CPGEs is based on academic performance during the last two years of high school, called Première and Terminale. Only 5% of a given cohort is admitted to a prépa. CPGEs are usually located within high schools but are a part of tertiary education, which means that to be admitted, each student must have already successfully passed their Baccalauréat (or equivalent). Most CPGE receive applications from hundreds of applicants every year in April and May, and then selects students based on their own criteria. A few mainly private CPGEs, accounting for 10% of CPGEs, also have an interview process or expect student participation in local community initiatives. The ratio of CPGE students who fail to enter any grande école is lower in scientific and business CPGEs than in humanities CPGEs. Some preparatory classes are widely considered "elite", being extremely selective, and recruiting only the best students from each high school, if not the best student from each high school. These CPGEs practically guarantee their students a place in one of the top "grandes écoles". Among them are the Lycée Louis-Le-Grand, the Lycée Henri-IV, the Lycée Stanislas and the Lycée privé Sainte-Geneviève. ==== Scientific CPGEs ==== The oldest CPGEs are the science ones, which can be accessed only with a bac in science Bacheliers. Science CPGE are called TSI ("Technology and Engineering Science"), MPSI ("Mathematics, Physics and Engineering Science"), PCSI ("Physics, Chemistry, and Engineering Science") or PTSI ("Physics, Technology, and Engineering Science") in the first year, MP ("Mathematics and Physics"), PSI ("Physics and Engineering Science"), PC ("Physics and Chemistry") or PT ("Physics and Technology") in the second year and BCPST ("Biology, Chemistry, Physics, Life and Earth Sciences"). The first year of a CPGE is widely known as "Math Sup", or Hypotaupe, (Sup for "Classe de Mathématiques Supérieures", superior in French, meaning post-high school), and second year as "Math Spé", or Taupe, (Spés standing for "Classe de Mathématiques Spéciales", special in French). The students of these classes are known as Taupins. Both the first and second year programmes include as much as twelve hours of mathematics teaching per week, ten hours of physics, two hours of philosophy, two to four hours of (one or two) foreign languages and four to six hours of options: chemistry, SI (Engineering Industrial Science) or Theoretical Computer Science (including some programming using the Pascal or CaML programming languages, as a practical work). There are also several hours of homework, which can double the class-based workload. A well-known joke among CPGE students is that they become moles for two years, sometimes three, hence the nicknames taupe and taupin (taupe being the French word for a mole). ==== Business CPGEs ==== There are also CPGEs that are focused on economics (who prepare the admission in business schools). They are known as prépa EC (short for Economiques et Commerciales) and are divided into two parts: prépa ECS, which focuses more on mathematics, generally for those who graduated the scientific baccalauréat, and prépa ECE, which focuses more on economics, for those who were in the economics section in high school. ==== Humanities CPGEs (Hypokhâgne and Khâgne) ==== The literary and humanities CPGEs have also their own nicknames, Hypokhâgne for the first year and Khâgne for the second year. The students are called the khâgneux. Those classes prepare for schools such as the three Écoles Normales Supérieures, the École des Chartes, and sometimes Sciences Po. There are two kinds of Khâgnes. The Khâgne de Lettres is the most common, and focuses on philosophy, French literature, history and languages. The Khâgne de Lettres et Sciences Sociales (Literature and Social Sciences), otherwise called Khâgne B/L, also includes mathematics and socio-economic sciences in addition to those literary subjects. The students of Hypokhâgne and Khâgne (the humanities CPGE) are simultaneously enrolled in universities, and can go back to university in case of failure or if they feel unable to pass the highly competitive entrance examinations for the Écoles Normales Supérieures. ==== Colles ==== The amount of work required of the students is high. In addition to class time and homework, students spend several hours each week completing oral exams called colles (sometimes written 'khôlles' to look like a Greek word, that way of writing being initially a khâgneux's joke since khâgneux study Ancient Greek). The colles are unique to French academic education in CPGEs. In scientific and business CPGEs, colles consist of oral examinations twice a week, in French, foreign languages (usually English, German, or Spanish), maths, physics, philosophy, or geopolitics—depending on the type of CPGE. Students, usually in groups of three or four, spend an hour facing a professor alone in a room, answering questions and solving problems. In humanities CPGEs, colles are usually taken every quarter in every subject. Students have one hour to prepare a short presentation that takes the form of a French-style dissertation (a methodologically codified essay, typically structured in 3 parts: thesis, counter-thesis, and synthesis) in history, philosophy, etc. on a given topic, or the form of a commentaire composé (a methodologically codified form of literary criticism) in literature and foreign languages. In Ancient Greek or Latin, they involve a translation and a commentary. The student then has 20 minutes to present his/her work to the teacher, who finally asks some questions on the presentation and on the corresponding topic. Colles are regarded as very stressful, particularly due to the high standards expected by the teachers, and the subsequent harshness that may be directed at students who do not perform adequately. But they are important insofar as they prepare the students, from the very first year, for the oral part of the highly competitive examinations, which are reserved for the happy few who pass the written part. === Recruitment of teachers === Decades ago, primary school teachers were educated in Écoles normales and secondary teachers recruited through the "Agrégation" examination. The situation has been diversified by the introduction in the 1950s of the CAPES examination for secondary teachers and in the 1990s by the institution of "Instituts universitaires de formation des maitres" (IUFM), which have been renamed Écoles supérieures du professorat et de l’éducation (ESPE) in 2013 and then Instituts Nationaux supérieurs du professorat et de l’éducation (INSPE) in 2019. Precisely, school teachers are divided between : Primary school and kindergarten teachers (Professeurs des écoles), educated in an INSPE, have usually a "master" (Bac+5). Their weekly service is about 28 hours a week. Certified teachers (Professeurs certifiés), educated in both a University and an INSPE, have a "master" (Bac+5) and must pass a competitive exam called Certificat d'aptitude au professorat de l'enseignement du second degré (CAPES) in a specific domain. Their rank usually determines their geographic assignment for the first years of their careers. The majority of them are teaching in collège (middle school). Agrégés teachers (Professeurs agrégés) are recruited through a different competitive exam called Agrégation, of much higher level in each domain. They could either be certified teachers or external holders of at least a "master" (Bac+5) in the domain. In the latter case they must attend an additional formation in teaching in an INSPE. Agrégés teachers have a higher salary rate and reduced weekly service. The majority of them are teaching in lycée (high school). University teachers are recruited by special commissions, and are divided between: "teachers-researchers" (enseignants-chercheurs), with at least a doctorate: they teach classes and conduct research in their field of expertise with a full tenure. They are either Maître de conférences (Senior lecturers), or Professeurs (Professors). A Maître de conférence must publish a reviewed work named Habilitation à diriger des recherches (HDR) (professorial thesis) in order to be allowed to become the director of studies for PhD students. The HDR is in turn necessary to be appointed as Professeur. The net pay (all insurances included) is from 2,300 to 8,800 (with extra duties) euros per month. Net salaries of over 4,000 euros per month (2011 level) are however very unusual, and limited to the small minority of teacher-researchers who have held the grade of first class full professor for at least seven years, which is rare. The maximum possible net salary for second-class full professors and chief senior lecturers (maître de conférence hors classe), the end of career status for most full-time teacher-researchers in French universities, is 3,760 euros a month (2011), and only a few of the group ever reach that level. Secondary school teachers who have been permanently assigned away from their original school position to teach in a university. They are not required to conduct any research but teach twice as many hours as the "teachers-researchers". They are called PRAG (professeurs agrégés) and PRCE (professeurs certifiés). Their weekly service is 15 or 18 hours. The net pay is from 1,400 to 3,900 euros per month. CPGE teachers are usually "agrégés" or "chaire sup", assigned by the Inspection générale according to their qualifications and competitive exam rank as well as other factors. Their weekly service is about 9 hours a week, 25 or 33 weeks a year. Net pay : from 2,000 to 7,500 euro (extra hours) France did a great activity of supplying training for their people, via way of means of the 1800s, France had approximately 350 eight-yr faculties and six-yr faculties. Also in the course of the 1800s, they furnished classical training to approximately 50,000 to younger guys from a long time of 10–20. Getting greater specific, the very best increase charge of training in France became in the course of the 1821–1837 whilst towns welcomed new colleges. The increase charge slowed in the course of 1837–1867 because the authorities reached thinly populated rural areas. Then, via way of means of 1867–1906, the emphasis became on deepening the first-rate of colleges and teaching. Camille Sée added secondary colleges for ladies in 1880, which became a vital pass due to the fact he furnished secondary faculty stage of training to ladies. There have been 36 such colleges in France in 1896. == Religion == Religious instruction is not given by public schools (except for 6- to 18-year-old students in Alsace and Moselle under the Concordat of 1801). Laïcité (secularism) is one of the main precepts of the French Republic. In a March 2004 ruling, the French government banned all "conspicuous religious symbols" from schools and other public institutions with the intent of preventing proselytisation and to foster a sense of tolerance among ethnic groups. Some religious groups showed their opposition, saying the law hindered the freedom of religion as protected by the French constitution. == Statistics == The French Republic has 67 million inhabitants, living in the 13 regions of metropolitan France and four overseas departments (2.7 million). Despite the fact that the population is growing (up 0.4% a year), the proportion of young people under 25 is falling. There are now fewer than 19 million young people in Metropolitan France, or 32% of the total population, compared with 40% in the 1970s and 35% at the time of the 1990 census. France is seeing a slow aging of the population, however, that is less marked than in other neighbouring countries (such as Germany and Italy), especially as the annual number of births is currently increasing slightly. Eighteen million pupils and students, a quarter of the population, are in the education system, over 2.9 million of whom are in higher education. In 2000, the French Education Minister reported that only 39 out of 75,000 state schools were "seriously violent" and some 300 were "somewhat violent". == See also == Trouvermonmaster.gouv.fr, a Web portal listing all the masters available in France Parcoursup, a Web portal listing all the undergraduate programmes available in France Academic grading in France Agency for French Teaching Abroad (Agence pour l'enseignement français à l'étranger) Campus France (Agency for the promotion of French Higher Education) Conférence des Grandes Écoles (CGE) Conference of the Directors of French Engineering Schools (Conférence des directeurs des écoles françaises d'ingénieurs (CDEFI)) Homeschooling in France Open access in France Nursery schools of France == References == == Further reading == == External links == Eurydice France Archived 10 October 2017 at the Wayback Machine, Eurydice: Portal for European education systems French Ministry of National Education, Higher Education and Research Archived 18 December 2017 at the Wayback Machine (English) School Education in France, Eduscol: the French portal for Education players (English) Education in France, a webdossier by the German Education Server (English) Understand the French School System in 5 min Archived 22 November 2022 at the Wayback Machine (English & French) |
Wikipedia:Mathematics education in India#0 | In contemporary education, mathematics education—known in Europe as the didactics or pedagogy of mathematics—is the practice of teaching, learning, and carrying out scholarly research into the transfer of mathematical knowledge. Although research into mathematics education is primarily concerned with the tools, methods, and approaches that facilitate practice or the study of practice, it also covers an extensive field of study encompassing a variety of different concepts, theories and methods. National and international organisations regularly hold conferences and publish literature in order to improve mathematics education. == History == === Ancient === Elementary mathematics were a core part of education in many ancient civilisations, including ancient Egypt, ancient Babylonia, ancient Greece, ancient Rome, and Vedic India. In most cases, formal education was only available to male children with sufficiently high status, wealth, or caste. The oldest known mathematics textbook is the Rhind papyrus, dated from circa 1650 BCE. ==== Pythagorean theorem ==== Historians of Mesopotamia have confirmed that use of the Pythagorean rule dates back to the Old Babylonian Empire (20th–16th centuries BC) and that it was being taught in scribal schools over one thousand years before the birth of Pythagoras. In Plato's division of the liberal arts into the trivium and the quadrivium, the quadrivium included the mathematical fields of arithmetic and geometry. This structure was continued in the structure of classical education that was developed in medieval Europe. The teaching of geometry was almost universally based on Euclid's Elements. Apprentices to trades such as masons, merchants, and moneylenders could expect to learn such practical mathematics as was relevant to their profession. === Medieval and early modern === In the Middle Ages, the academic status of mathematics declined, because it was strongly associated with trade and commerce, and considered somewhat un-Christian. Although it continued to be taught in European universities, it was seen as subservient to the study of natural, metaphysical, and moral philosophy. The first modern arithmetic curriculum (starting with addition, then subtraction, multiplication, and division) arose at reckoning schools in Italy in the 1300s. Spreading along trade routes, these methods were designed to be used in commerce. They contrasted with Platonic math taught at universities, which was more philosophical and concerned numbers as concepts rather than calculating methods. They also contrasted with mathematical methods learned by artisan apprentices, which were specific to the tasks and tools at hand. For example, the division of a board into thirds can be accomplished with a piece of string, instead of measuring the length and using the arithmetic operation of division. The first mathematics textbooks to be written in English and French were published by Robert Recorde, beginning with The Grounde of Artes in 1543. However, there are many different writings on mathematics and mathematics methodology that date back to 1800 BCE. These were mostly located in Mesopotamia, where the Sumerians were practicing multiplication and division. There are also artifacts demonstrating their methodology for solving equations like the quadratic equation. After the Sumerians, some of the most famous ancient works on mathematics came from Egypt in the form of the Rhind Mathematical Papyrus and the Moscow Mathematical Papyrus. The more famous Rhind Papyrus has been dated back to approximately 1650 BCE, but it is thought to be a copy of an even older scroll. This papyrus was essentially an early textbook for Egyptian students. The social status of mathematical study was improving by the seventeenth century, with the University of Aberdeen creating a Mathematics Chair in 1613, followed by the Chair in Geometry being set up in University of Oxford in 1619 and the Lucasian Chair of Mathematics being established by the University of Cambridge in 1662. === Modern === In the 18th and 19th centuries, the Industrial Revolution led to an enormous increase in urban populations. Basic numeracy skills, such as the ability to tell the time, count money, and carry out simple arithmetic, became essential in this new urban lifestyle. Within the new public education systems, mathematics became a central part of the curriculum from an early age. By the twentieth century, mathematics was part of the core curriculum in all developed countries. During the twentieth century, mathematics education was established as an independent field of research. Main events in this development include the following: In 1893, a Chair in mathematics education was created at the University of Göttingen, under the administration of Felix Klein. The International Commission on Mathematical Instruction (ICMI) was founded in 1908, and Felix Klein became the first president of the organisation. The professional periodical literature on mathematics education in the United States had generated more than 4,000 articles after 1920, so in 1941 William L. Schaaf published a classified index, sorting them into their various subjects. A renewed interest in mathematics education emerged in the 1960s, and the International Commission was revitalized. In 1968, the Shell Centre for Mathematical Education was established in Nottingham. The first International Congress on Mathematical Education (ICME) was held in Lyon in 1969. The second congress was in Exeter in 1972, and after that, it has been held every four years. Midway through the twentieth century, the cultural impact of the "electronic age" (McLuhan) was also taken up by educational theory and the teaching of mathematics. While previous approach focused on "working with specialized 'problems' in arithmetic", the emerging structural approach to knowledge had "small children meditating about number theory and 'sets'." Since the 1980s, there have been a number of efforts to reform the traditional curriculum, which focuses on continuous mathematics and relegates even some basic discrete concepts to advanced study, to better balance coverage of the continuous and discrete sides of the subject: In the 1980s and early 1990s, there was a push to make discrete mathematics more available at the post-secondary level; From the late 1980s into the new millennium, countries like the US began to identify and standardize sets of discrete mathematics topics for primary and secondary education; Concurrently, academics began compiling practical advice on introducing discrete math topics into the classroom; Researchers continued arguing the urgency of making the transition throughout the 2000s; and In parallel, some textbook authors began working on materials explicitly designed to provide more balance. Similar efforts are also underway to shift more focus to mathematical modeling as well as its relationship to discrete math. == Objectives == At different times and in different cultures and countries, mathematics education has attempted to achieve a variety of different objectives. These objectives have included: The teaching and learning of basic numeracy skills to all students The teaching of practical mathematics (arithmetic, elementary algebra, plane and solid geometry, trigonometry, probability, statistics) to most students, to equip them to follow a trade or craft and to understand mathematics commonly used in news and Internet (such as percentages, charts, probability, and statistics) The teaching of abstract mathematical concepts (such as set and function) at an early age The teaching of selected areas of mathematics (such as Euclidean geometry) as an example of an axiomatic system and a model of deductive reasoning The teaching of selected areas of mathematics (such as calculus) as an example of the intellectual achievements of the modern world The teaching of advanced mathematics to those students who wish to follow a career in science, technology, engineering, and mathematics (STEM) fields The teaching of heuristics and other problem-solving strategies to solve non-routine problems The teaching of mathematics in social sciences and actuarial sciences, as well as in some selected arts under liberal arts education in liberal arts colleges or universities == Methods == The method or methods used in any particular context are largely determined by the objectives that the relevant educational system is trying to achieve. Methods of teaching mathematics include the following: Computer-based math: an approach based on the use of mathematical software as the primary tool of computation. Computer-based mathematics education: involves the use of computers to teach mathematics. Mobile applications have also been developed to help students learn mathematics. Classical education: the teaching of mathematics within the quadrivium, part of the classical education curriculum of the Middle Ages, which was typically based on Euclid's Elements taught as a paradigm of deductive reasoning. Conventional approach: the gradual and systematic guiding through the hierarchy of mathematical notions, ideas and techniques. Starts with arithmetic and is followed by Euclidean geometry and elementary algebra taught concurrently. Requires the instructor to be well informed about elementary mathematics since didactic and curriculum decisions are often dictated by the logic of the subject rather than pedagogical considerations. Other methods emerge by emphasizing some aspects of this approach. Relational approach: uses class topics to solve everyday problems and relates the topic to current events. This approach focuses on the many uses of mathematics and helps students understand why they need to know it as well as helps them to apply mathematics to real-world situations outside of the classroom. Historical method: teaching the development of mathematics within a historical, social, and cultural context. Proponents argue it provides more human interest than the conventional approach. Discovery math: a constructivist method of teaching (discovery learning) mathematics which centres around problem-based or inquiry-based learning, with the use of open-ended questions and manipulative tools. This type of mathematics education was implemented in various parts of Canada beginning in 2005. Discovery-based mathematics is at the forefront of the Canadian "math wars" debate with many criticizing it for declining math scores. New Math: a method of teaching mathematics which focuses on abstract concepts such as set theory, functions, and bases other than ten. Adopted in the US as a response to the challenge of early Soviet technical superiority in space, it began to be challenged in the late 1960s. One of the most influential critiques of the New Math was Morris Kline's 1973 book Why Johnny Can't Add. The New Math method was the topic of one of Tom Lehrer's most popular parody songs, with his introductory remarks to the song: "...in the new approach, as you know, the important thing is to understand what you're doing, rather than to get the right answer." Recreational mathematics: mathematical problems that are fun can motivate students to learn mathematics and can increase their enjoyment of mathematics. Standards-based mathematics: a vision for pre-college mathematics education in the United States and Canada, focused on deepening student understanding of mathematical ideas and procedures, and formalized by the National Council of Teachers of Mathematics which created the Principles and Standards for School Mathematics. Mastery: an approach in which most students are expected to achieve a high level of competence before progressing. Problem solving: the cultivation of mathematical ingenuity, creativity, and heuristic thinking by setting students open-ended, unusual, and sometimes unsolved problems. The problems can range from simple word problems to problems from international mathematics competitions such as the International Mathematical Olympiad. Problem-solving is used as a means to build new mathematical knowledge, typically by building on students' prior understandings. Exercises: the reinforcement of mathematical skills by completing large numbers of exercises of a similar type, such as adding simple fractions or solving quadratic equations. Rote learning: the teaching of mathematical results, definitions and concepts by repetition and memorisation typically without meaning or supported by mathematical reasoning. A derisory term is drill and kill. In traditional education, rote learning is used to teach multiplication tables, definitions, formulas, and other aspects of mathematics. Math walk: a walk where experience of perceived objects and scenes is translated into mathematical language. == Content and age levels == Different levels of mathematics are taught at different ages and in somewhat different sequences in different countries. Sometimes a class may be taught at an earlier age than typical as a special or honors class. Elementary mathematics in most countries is taught similarly, though there are differences. Most countries tend to cover fewer topics in greater depth than in the United States. During the primary school years, children learn about whole numbers and arithmetic, including addition, subtraction, multiplication, and division. Comparisons and measurement are taught, in both numeric and pictorial form, as well as fractions and proportionality, patterns, and various topics related to geometry. At high school level in most of the US, algebra, geometry, and analysis (pre-calculus and calculus) are taught as separate courses in different years. On the other hand, in most other countries (and in a few US states), mathematics is taught as an integrated subject, with topics from all branches of mathematics studied every year; students thus undertake a pre-defined course - entailing several topics - rather than choosing courses à la carte as in the United States. Even in these cases, however, several "mathematics" options may be offered, selected based on the student's intended studies post high school. (In South Africa, for example, the options are Mathematics, Mathematical Literacy and Technical Mathematics.) Thus, a science-oriented curriculum typically overlaps the first year of university mathematics, and includes differential calculus and trigonometry at age 16–17 and integral calculus, complex numbers, analytic geometry, exponential and logarithmic functions, and infinite series in their final year of secondary school; Probability and statistics are similarly often taught. At college and university level, science and engineering students will be required to take multivariable calculus, differential equations, and linear algebra; at several US colleges, the minor or AS in mathematics substantively comprises these courses. Mathematics majors study additional areas of pure mathematics—and often applied mathematics—with the requirement of specified advanced courses in analysis and modern algebra. Other topics in pure mathematics include differential geometry, set theory, and topology. Applied mathematics may be taken as a major subject in its own right, covering partial differential equations, optimization, and numerical analysis among other topics. Courses here are also taught within other programs: for example, civil engineers may be required to study fluid mechanics, and "math for computer science" might include graph theory, permutation, probability, and formal mathematical proofs. Pure and applied math degrees often include modules in probability theory or mathematical statistics, as well as stochastic processes. (Theoretical) physics is mathematics-intensive, often overlapping substantively with the pure or applied math degree. Business mathematics is usually limited to introductory calculus and (sometimes) matrix calculations; economics programs additionally cover optimization, often differential equations and linear algebra, and sometimes analysis. Business and social science students also typically take statistics and probability courses. == Standards == Throughout most of history, standards for mathematics education were set locally, by individual schools or teachers, depending on the levels of achievement that were relevant to, realistic for, and considered socially appropriate for their pupils. In modern times, there has been a move towards regional or national standards, usually under the umbrella of a wider standard school curriculum. In England, for example, standards for mathematics education are set as part of the National Curriculum for England, while Scotland maintains its own educational system. Many other countries have centralized ministries which set national standards or curricula, and sometimes even textbooks. Ma (2000) summarized the research of others who found, based on nationwide data, that students with higher scores on standardized mathematics tests had taken more mathematics courses in high school. This led some states to require three years of mathematics instead of two. But because this requirement was often met by taking another lower-level mathematics course, the additional courses had a “diluted” effect in raising achievement levels. In North America, the National Council of Teachers of Mathematics (NCTM) published the Principles and Standards for School Mathematics in 2000 for the United States and Canada, which boosted the trend towards reform mathematics. In 2006, the NCTM released Curriculum Focal Points, which recommend the most important mathematical topics for each grade level through grade 8. However, these standards were guidelines to implement as American states and Canadian provinces chose. In 2010, the National Governors Association Center for Best Practices and the Council of Chief State School Officers published the Common Core State Standards for US states, which were subsequently adopted by most states. Adoption of the Common Core State Standards in mathematics is at the discretion of each state, and is not mandated by the federal government. "States routinely review their academic standards and may choose to change or add onto the standards to best meet the needs of their students." The NCTM has state affiliates that have different education standards at the state level. For example, Missouri has the Missouri Council of Teachers of Mathematics (MCTM) which has its pillars and standards of education listed on its website. The MCTM also offers membership opportunities to teachers and future teachers so that they can stay up to date on the changes in math educational standards. The Programme for International Student Assessment (PISA), created by the Organisation for the Economic Co-operation and Development (OECD), is a global program studying the reading, science, and mathematics abilities of 15-year-old students. The first assessment was conducted in the year 2000 with 43 countries participating. PISA has repeated this assessment every three years to provide comparable data, helping to guide global education to better prepare youth for future economies. There have been many ramifications following the results of triennial PISA assessments due to implicit and explicit responses of stakeholders, which have led to education reform and policy change. == Research == According to Hiebert and Grouws, "Robust, useful theories of classroom teaching do not yet exist." However, there are useful theories on how children learn mathematics, and much research has been conducted in recent decades to explore how these theories can be applied to teaching. The following results are examples of some of the current findings in the field of mathematics education. === Important results === One of the strongest results in recent research is that the most important feature of effective teaching is giving students "the opportunity to learn". Teachers can set expectations, times, kinds of tasks, questions, acceptable answers, and types of discussions that will influence students' opportunities to learn. This must involve both skill efficiency and conceptual understanding. === Conceptual understanding === Source: Two of the most important features of teaching in the promotion of conceptual understanding times are attending explicitly to concepts and allowing students to struggle with important mathematics. Both of these features have been confirmed through a wide variety of studies. Explicit attention to concepts involves making connections between facts, procedures, and ideas. (This is often seen as one of the strong points in mathematics teaching in East Asian countries, where teachers typically devote about half of their time to making connections. At the other extreme is the US, where essentially no connections are made in school classrooms.) These connections can be made through explanation of the meaning of a procedure, questions comparing strategies and solutions of problems, noticing how one problem is a special case of another, reminding students of the main point, discussing how lessons connect, and so on. Deliberate, productive struggle with mathematical ideas refers to the fact that when students exert effort with important mathematical ideas, even if this struggle initially involves confusion and errors, the result is greater learning. This is true whether the struggle is due to intentionally challenging, well-implemented teaching, or unintentionally confusing, faulty teaching. === Formative assessment === Formative assessment is both the best and cheapest way to boost student achievement, student engagement, and teacher professional satisfaction. Results surpass those of reducing class size or increasing teachers' content knowledge. Effective assessment is based on clarifying what students should know, creating appropriate activities to obtain the evidence needed, giving good feedback, encouraging students to take control of their learning and letting students be resources for one another. === Homework === Homework assignments which lead students to practice past lessons or prepare for future lessons are more effective than those going over the current lesson. Students benefit from feedback. Students with learning disabilities or low motivation may profit from rewards. For younger children, homework helps simple skills, but not broader measures of achievement. === Students with difficulties === Source: Students with genuine difficulties (unrelated to motivation or past instruction) struggle with basic facts, answer impulsively, struggle with mental representations, have poor number sense, and have poor short-term memory. Techniques that have been found productive for helping such students include peer-assisted learning, explicit teaching with visual aids, instruction informed by formative assessment, and encouraging students to think aloud. In particular, research surrounding students with disabilities in a mathematics classroom is mostly done by special education researchers. Some mathematics education researchers have called for more collaboration across disciplines to better understand supports that could be helpful to mathematics students with disabilities. === Algebraic reasoning === Elementary school children need to spend a long time learning to express algebraic properties without symbols before learning algebraic notation. When learning symbols, many students believe letters always represent unknowns and struggle with the concept of variable. They prefer arithmetic reasoning to algebraic equations for solving word problems. It takes time to move from arithmetic to algebraic generalizations to describe patterns. Students often have trouble with the minus sign and understand the equals sign to mean "the answer is...". === Cultural Equity === Despite the popular belief that mathematics is race neutral, some research suggests that effective mathematics teaching of culturally diverse students requires a culturally relevant pedagogy that considers students' cultural backgrounds and experiences. The three criteria for culturally relevant pedagogy are academic success, cultural competence, and critical consciousness. More recent research proposes that culturally sustaining pedagogy explicitly aims to perpetuate and foster cultural and linguistic pluralism within the educational system, ensuring that students can thrive while retaining their cultural identities. === Mathematics Teacher Education === Student teaching is a crucial part of a teacher candidate's path to becoming a teacher. Recommended reform in mathematics teacher education includes a focus on learning to anticipate, elicit, and use students’ mathematical thinking as the primary goal, as opposed to models with an over-emphasis on classroom management and survival. === Methodology === As with other educational research (and the social sciences in general), mathematics education research depends on both quantitative and qualitative studies. Quantitative research includes studies that use inferential statistics to answer specific questions, such as whether a certain teaching method gives significantly better results than the status quo. The best quantitative studies involve randomized trials where students or classes are randomly assigned different methods to test their effects. They depend on large samples to obtain statistically significant results. Qualitative research, such as case studies, action research, discourse analysis, and clinical interviews, depend on small but focused samples in an attempt to understand student learning and to look at how and why a given method gives the results it does. Such studies cannot conclusively establish that one method is better than another, as randomized trials can, but unless it is understood why treatment X is better than treatment Y, application of results of quantitative studies will often lead to "lethal mutations" of the finding in actual classrooms. Exploratory qualitative research is also useful for suggesting new hypotheses, which can eventually be tested by randomized experiments. Both qualitative and quantitative studies, therefore, are considered essential in education—just as in the other social sciences. Many studies are “mixed”, simultaneously combining aspects of both quantitative and qualitative research, as appropriate. ==== Randomized trials ==== There has been some controversy over the relative strengths of different types of research. Because of an opinion that randomized trials provide clear, objective evidence on “what works”, policymakers often consider only those studies. Some scholars have pushed for more random experiments in which teaching methods are randomly assigned to classes. In other disciplines concerned with human subjects—like biomedicine, psychology, and policy evaluation—controlled, randomized experiments remain the preferred method of evaluating treatments. Educational statisticians and some mathematics educators have been working to increase the use of randomized experiments to evaluate teaching methods. On the other hand, many scholars in educational schools have argued against increasing the number of randomized experiments, often because of philosophical objections, such as the ethical difficulty of randomly assigning students to various treatments when the effects of such treatments are not yet known to be effective, or the difficulty of assuring rigid control of the independent variable in fluid, real school settings. In the United States, the National Mathematics Advisory Panel (NMAP) published a report in 2008 based on studies, some of which used randomized assignment of treatments to experimental units, such as classrooms or students. The NMAP report's preference for randomized experiments received criticism from some scholars. In 2010, the What Works Clearinghouse (essentially the research arm for the Department of Education) responded to ongoing controversy by extending its research base to include non-experimental studies, including regression discontinuity designs and single-case studies. == Organizations == Advisory Committee on Mathematics Education American Mathematical Association of Two-Year Colleges Association of Teachers of Mathematics Canadian Mathematical Society C.D. Howe Institute Mathematical Association National Council of Teachers of Mathematics OECD International Association for the Evaluation of Educational Achievement Association of Mathematics Teacher Educators == See also == == References == Voit, Rita (14 February 2020). "Accelerated Math: What Every Parent Should Know". Resources by HEROES Academy. Retrieved 20 September 2023. == Further reading == == External links == History of Mathematical Education A quarter century of US 'math wars' and political partisanship. David Klein. California State University, Northridge, United States |
Wikipedia:Mathematics education in New York#0 | Mathematics education in New York in regard to both content and teaching method can vary depending on the type of school a person attends. Private school math education varies between schools whereas New York has statewide public school requirements where standardized tests are used to determine if the teaching method and educator are effective in transmitting content to the students. While an individual private school can choose the content and educational method to use, New York State mandates content and methods statewide. Some public schools have and continue to use established methods, such as Montessori for teaching such required content. New York State has used various foci of content and methods of teaching math including New Math (1960s), 'back to the basics' (1970s), Whole Math (1990s), Integrated Math, and Everyday Mathematics. How to teach math, what to teach, and its effectiveness has been a topic of debate in New York State and nationally since the "Math Wars" started in the 1960s. Often, current political events influence how and what is taught. The politics in turn influence state legislation. California, New York, and several other states have influenced textbook content produced by publishers. The state of New York has implemented a novel curriculum for high school mathematics. The courses Algebra I, Geometry, and Algebra II/Trigonometry are required courses mandated by the New York State Department of Education for high school graduation. == 2007-present == === Algebra === This is the first course in the new three-year curriculum. It was originally "Math A," but was replaced with "Integrated Algebra." In 2009 when Common Core was adopted, "Algebra I" replaced "Integrated Algebra" and is still in use today. Students learn to how write, solve, and graph equations and inequalities. They will also learn how to solve systems of equations, as well as how to simplify exponents, quadratic equations, exponential functions, polynomials, radicals, and rational expressions. Other topics included are probability and statistics. Some schools divided Algebra 1 into a two-year sequence. The students who receive it begin with Algebra 1A, and will cover the rest of the Algebra 1 topics in Algebra 1B in the next school year. Typically, most students are placed in Algebra 1 which covers all lessons rather than splitting them. At the conclusion of the one-year course, students take the New York State Regents Exam. The first administration of this exam was in June 2008. === Geometry === This is the second course in the new three year curriculum. It replaced part of "Math A" and part of "Math B" Geometric concepts such as right triangles are introduced. The course also covers topics including perpendicular and parallel lines, triangles, quadrilaterals, and transformations. At the conclusion of this one-year course, students take a New York State Regents exam in Geometry. The first administration of this exam was in June 2009. === Algebra II === This is the third and last course of the new three-year curriculum. It replaced the elements of "Math B" not covered in geometry. This course covers concepts of advanced algebra, and as well prepares students for pre-calculus and calculus. In 2016, the Board of Regents removed some of the trigonometry concepts and lessons from the course, and the Regents exam has been renamed from "Algebra 2/Trigonometry" to "Algebra II". At the conclusion of the one-year course, students take the New York State Regents exam for Algebra II. This is the last Regents exam in mathematics students could take. Like the former "Math B" Regents, it is considered one of the hardest High School Regents examinations, along with the Physical Setting/Chemistry regents and the Physical Setting/Physics regents. The Algebra 2/Trigonometry exam was given from June 2010 through January 2017 and the new Algebra II Exam has been given since June 2016. == 2001-2009 == === Math A (former course) === Math A replaced the former "Course 1" curriculum which focused solely on the topic of algebra, while Math A covered a whole range of topics. After algebra, students would take Geometry in the 10th grade and Algebra II in the 11th grade. In Math A, students learned to how write, solve, and graph equations and inequalities. They learned how to solve systems of equations, quadratics, as well as exponents, exponential functions, polynomials, radicals, and rational expressions. Other topics included are probability and statistics. Geometric concepts such as right triangles are also introduced. The course works in conjunction with New York State's Standards for Mathematics.[1] One course lasted three semesters, after which students took the Regents Math A Examination. === Math A/B (former course) === Math A/B took the place of the former "Course 2" curriculum, which focused almost solely on geometry, while Math A/B focused on a whole range of topics. Math A/B served as a bridge between the Math A and Math B courses. Math A/B stayed true to its geometric roots, as the first half of the course covered topics such as perpendicular and parallel lines, triangles, quadrilaterals, and transformations. After their first semester, students took the New York State Math A Regents exam. June 2008 was the last administration of this exam. For the second half of the year, students would begin Math B. They covered logic, geometric figures, and an introduction to trigonometry. === Math B (former course) === Math B was required to receive a High School Regents Diploma with Advanced Designation. The course replaces the former "Course 3" curriculum, which focused almost solely on trigonometry. Math B focused on a whole range of topics. It was taken after the student has completed and passed Math A. A Regents exam was taken at the end of the 1+1⁄2-year course. The Math B Regents was often considered one of the most difficult New York State Regents. Math B covered concepts that can be found in trigonometry and advanced algebra, and prepared students for pre-calculus and calculus and reviewed past topics. During their year of study, students learned different theorems, graphed complex numbers and vectors, as well as reviewed topics such as exponential functions, systems of inequalities, and radicals. As the year progressed, students were expected to relate these functions to the real world, create conjectures through their own research, and begin a classroom discussion about these topics. At the end of their studies, they would take the New York State Math B Regents Examination. The last administration of this exam was in August 2010. === Changes proposed in 2004 === In November 2004, the Mathematics Standards Committee made a report to the Board of Regents[2] about the State's requirements for high school graduation as related to mathematics. The committee recommended that: The curriculum should return to its old format as a one-year course with a Regents exam taken at the end of the year, with the new exam to be administered no earlier than June 2007. Math A would have its name changed to Integrated Algebra. In addition to the current Regents exam at the end of the Math B course, there should be another Regents exam, at the end of the first half of Math B study (currently Math A/B), to be administered no later than June 2006, bringing the total number of Regents exams to three. The course's name should be changed to Geometry for the first half of the course, and Algebra II and Trigonometry for the second half of the course (currently Math B). Integrated is only used in the new Algebra course. The State of New York has announced that these changes will be implemented in the 2009/10 scholastic year. In 2009, Integrated Algebra was switched to Algebra 1, due to the common core adoption. The move was praised by many who thought the changes to the original draft curriculum were unnecessary. == Pre-2002 == === Sequential Mathematics === Prior to the 2001-2002 school year, the New York State Board of Regents offered a three-year Sequential Mathematics series divided into three Courses I, II, and III. == See also == Mathematics education in the United States USA Mathematical Olympiad Math circle == References == Mathematics' Standard Committee's recommendations == External links == Official website of textbook developer New York State Department of Education New York State Department of Education: Curriculum, Instruction and Instructional Technology Math Help and Resources JMAP - Free resources for teachers Regents preparation site for New York |
Wikipedia:Mathematics education in the United Kingdom#0 | Mathematics education in the United Kingdom is largely carried out at ages 5–16 at primary school and secondary school (though basic numeracy is taught at an earlier age). However voluntary Mathematics education in the UK takes place from 16 to 18, in sixth forms and other forms of further education. Whilst adults can study the subject at universities and higher education more widely. Mathematics education is not taught uniformly as exams and the syllabus vary across the countries of the United Kingdom, notably Scotland. == History == The School Certificate was established in 1918, for education up to 16, with the Higher School Certificate for education up to 18; these were both established by the Secondary Schools Examinations Council (SSEC), which had been established in 1917. === 1950s === The Association of Teachers of Mathematics was founded in 1950. === 1960s === The Joint Mathematical Council was formed in 1963 to improve the teaching of mathematics in UK schools. The Ministry of Education had been created in 1944, which became the Department of Education and Science in 1964. The Schools Council was formed in 1964, which regulated the syllabus of exams in the UK, and existed until 1984. The exam body Mathematics in Education and Industry in Trowbridge was formed in 1963, formed by the Mathematical Association; the first exam Additional Mathematics was first set in 1965. The Institute of Mathematics and its Applications was formed in 1964, and is the UK's chartered body for mathematicians, being based in Essex. Before calculators, many calculations would be done by hand with slide rules and log tables. The Nuffield Mathematics Teaching Project started in September 1964, lasting until 1971, to look at primary education, under Edith Biggs, from the Schools Inspectorate. The Nuffield Foundation Primary Mathematics Project began, with the 'Mathematics for the Majority Project', for the years up to 16, for slow learners. === 1970s === Decimal Day, on 15 February 1971, allowed less time on numerical calculations at school. The Metric system has curtailed lengthy calculations as well; the US, conversely, largely does not have the metric system. At Ruskin College on Monday 18 October 1976 Labour Prime Minister Jim Callaghan made a radical speech decrying the lack of numeracy in school leavers, possibly prompted by the William Tyndale affair in 1975. The Prime Minister also questioned why so many girls gave up science before leaving secondary school. But the Labour Party, instead, took curriculum change slowly, merely setting up the Committee of Inquiry into the Teaching of Mathematics in Schools, under Sir Wilfred Cockcroft, with Hilary Shuard and Elizabeth Williams. The subsequent report Mathematics Counts, was published in 1982; it offered few radical changes. In March 1977 the government had a £3.9m scheme to recruit 1,200 more teachers. In England and Wales, there was a shortfall of 1,120 Maths teachers, 424 in physical sciences, and 525 in Design and Technology. It would be paid for by the Training Services Agency, and run by the Local Government Training Board. The scheme was open to people over 28, who had not attended a higher education course in the last five years. === 1980s === Electronic calculators began to be owned at school from the early 1980s, becoming widespread from the mid-1980s. Parents and teachers believed that calculators would diminish abilities of mental arithmetic. Scientific calculators came to the aid for those working out logarithms and trigonometric functions. The BBC2 'Horizon' documentary Twice Five Plus the Wings of a Bird on Monday 28 April 1986, narrated by Peter Jones, looked at why people disliked abstract Maths, notably in the teenage years. The Trends in International Mathematics and Science Study (TIMSS) showed that in some topics, the UK apparently had adequate Mathematics teaching, and from such reports Sir Keith Joseph merely implemented feasibility studies of national attainment standards, but the next education secretary, Kenneth Baker, Baron Baker of Dorking, wanted a lot more than mere feasibility studies. From hearing reports of national industrial failure being caused by insufficient mathematical abilities, he swiftly proposed a national curriculum in January 1987, to start in September 1988. Anita Straker and Hilary Shuard were part of the team that developed the primary national curriculum. Since 1988, exams in Mathematics at age sixteen, except Scotland, have been provided by the GCSE. === 1990s === From the 1990s, mainly the late 1990s, computers became integrated into mathematics education at primary and secondary levels in the UK. On Wednesday 18 November 1992 exam league tables were published for 108 local authorities, in England, under the Education Secretary John Patten, Baron Patten. The tables showed GCSE and A-levels for all 4,400 state secondary schools in England. Independent schools results were shown from 1993, and would include truancy rates. Left-wing parent groups, teachers' unions had opposed the move. Labour said it showed the government's simplistic approach to education standards, adding that raw results cannot reflect the real achievement of schools. The Liberal Democrats were not opposed, but thought that any information being provided was limited. Ofsted would be brought in the next year by Education Minister Emily Blatch, Baroness Blatch. The specialist schools programme was introduced in the mid-1990s in England. Fifteen new City Technology Colleges (CTCs) from the early 1990s often focussed on Maths. In 1996 the United Kingdom Mathematics Trust was formed to run the British Mathematical Olympiad, run by the British Mathematical Olympiad Subtrust. The United Kingdom Mathematics Trust summer school is held at The Queen's Foundation in Birmingham each year. The National Numeracy Strategy, costing £60m, was devised by Anita Straker, for the government's Numeracy Task Force, for primary schools, for implementation in autumn 1999. Prof David Reynolds, of Newcastle University, was the chairman. He had appeared in a Panorama documentary on Maths education on 3 June 1996. The 60-page report in January 1998 recommended that children under the age of 8 should not have calculators. The Millennium Mathematics Project was set up in 1999 at Cambridge by John D. Barrow, with 21 staff, with deputy director Julia Hawkins, now directed by Julia Gog. === 2000s === Mathematics and Computing Colleges were introduced in 2002 as part of the widened specialist schools programme; by 2007 there were 222 of these in England. The Excellence in Cities report was launched in March 1999, which led to the Advanced Extension Award in 2002, replacing the S-level for the top 10% of A-level candidates. Since 2008, the AEA is only available for Maths, provided by Edexcel; the scheme was introduced when the A* grade was introduced; the scheme was provided until 2018. In February 2004, the Smith Report, by the Principal of Queen Mary College, looked at how good exams were. People could pass at grade B at GCSE, but had taken much different type of exams. The report concluded that people could pass such exams, but lacked scant real-life proficiency at Maths. A-level Maths entries dropped from 67,000 in 2000 to 53,000 in 2004. The IGCSE, a more rigorous exam, was introduced in 2004, but the Labour government banned state secondary schools from being allowed to set the exam. It was viewed as 'elitist'. In a 2006 House of Lords report on science education, the Lib Dem chair Baroness Sharp, took an interest in the reduced participation in Maths in schools; she had worked with the Science Policy Research Unit at the University of Sussex. The 2001 report by the Lords Science and Technology Committee led to the National Science Learning Centre (Science Learning Centres) at National STEM Centre, with the University of York in 2006, with a Maths centre at University of Southampton. The National Centre for Excellence in the Teaching of Mathematics was founded 2006, after the Smith Report, being now in Sheffield. The National Higher Education Science, Technology, Engineering and Mathematics (HE STEM) Programme was founded in August 2009 by HEFCE and HEFCW; the scheme had six regions across England and Wales, working with the universities of Bath, Birmingham, Bradford, Manchester Metropolitan, Southampton and Swansea; it was funded by £21m, and developed by the University of Birmingham STEM Education Centre; the scheme finished in July 2012. Also involved was the MSOR centre of the HEA (now Advance HE) Subject Centre, and the Centre for Excellence in University Wide Mathematics and Statistics Support at Loughborough University. === 2010s === The HEA subject centres closed in August 2011. In September 2012 Prof Jeremy Hodgen, the chairman of the British Society for Research into Learning Mathematics, produced a report made by Durham University and KCL, where 7,000 children at secondary school took 1970s Maths exams. Maths exams results over the same time scale had doubled in grades, but the researchers did not find much improvement. Proficiency in routine Maths was better, but proficiency with difficult Maths was not as good. Mathematics free schools were opened in 2014 - the King's College London Mathematics School in Lambeth, and Exeter Mathematics School in Devon; both were selective sixth form colleges; others opened at Liverpool and Lancaster; more selective sixth form maths schools are to open in Cambridge, Surrey, and Durham. A newer curriculum for Maths GCSE (and English) was introduced in September 2015, with a new grading scale of 1–9. In August 2015 the ACME claimed that there was a shortfall of 5,500 secondary school Maths teachers, in England. But this shortfall was hugely uneven. Comprehensive schools in wealthy areas or state grammar schools were not commonly short of Maths teachers, but secondary schools in less-salubrious places were often hideously short of Maths teachers. == Nations == === England === Mathematics education in England up to the age of 19 is provided in the National Curriculum by the Department for Education, which was established in 2010. Early years education is called the Early Years Foundation Stage in England, which includes arithmetic. In England there are 24,300 schools, of which 3,400 are secondary. The National Curriculum for mathematics aims to ensure that all pupils: become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately. reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language. can solve problems by applying their mathematics to various routine and non-routine problems with increasing sophistication, including breaking down problems into a series of more straightforward steps and persevering in seeking solutions. Mathematics is a related subject in which pupils must be able to move fluently between representations of mathematical ideas. It is essential to everyday life, critical to science, technology and engineering, an appreciation of the beauty and power of mathematics, and a sense of and necessary for financial literacy and most forms of employment. A high-quality mathematics education, therefore, provides a foundation for understanding the world, the ability to reason mathematically, and curiosity about the subject. Pupils should build connections across mathematical ideas to develop fluency, mathematical reasoning and competence in solving increasingly sophisticated problems. They should also apply their mathematical knowledge in science, geography, computing and other subjects. === Wales === Wales takes the GCSE and A-level in Mathematics, but has its own Department for Education and Skills. Wales does not produce school league tables. Wales has 1550 schools, of which 180 are secondary. === Scotland === Education Scotland, formed in 2011, regulates education at school in Scotland, with qualifications monitored by the Scottish Qualifications Authority (SQA) and the Mathematics syllabus follows the country's Curriculum for Excellence. Scotland does not produce school league tables. Scotland has 5,050 schools, of which 350 are secondary. === Northern Ireland === Northern Ireland is the only country in the UK to have exclusively selective schools - it has sixty nine grammar schools. Mathematics education is provided by the Department of Education (DENI), with further education provided by the Department for Employment and Learning. Northern Ireland has 1120 schools, of which 190 are secondary. == Relation to other countries == In the 1980s the education researcher Sig Prais looked at mathematics education in Germany and the UK. He found that the teaching of mathematics of an appropriate level, in Germany, worked much better than to bludgeon all levels of mathematics onto all abilities in British comprehensive schools. In preparation for the new national curriculum in 1988, Sig Prais said 'There is an enormous burden on the teacher facing a mixed ability class. At age five, mixed ability classes are not such a problem. You can assume that nobody knows anything, but as the children process through the school, some will not have grasped all that they should, and they never catch up. In some countries, children are not allowed to move into the secondary schools until they are ready for it. They retake the lessons until they are.' == Primary level == The Department of Education and Science set up an Assessment of Performance Unit in 1976 to monitor attainment of children at a national level, with standards of mathematics being monitored from 1978 by the National Foundation for Educational Research (NFER). Before this time, assessment of primary school standards had not been carried out at a national level. Children at primary school are expected to know their times tables. Children are taught about long division, fractions, decimals, averages, ratios, negative numbers, and long multiplication. == Secondary level == Study of Mathematics is compulsory up to the school leaving age. If you do not take Additional Mathematics at GCSE, in some schools, it is likely that you would not be allowed to subsequently take Maths A-level in the sixth form. The Programme for International Student Assessment coordinated by the OECD currently ranks the knowledge and skills of British 15-year-olds in mathematics and science above OECD averages. In 2011, the Trends in International Mathematics and Science Study (TIMSS) rated 13–14-year-old pupils in England and Wales 10th in the world for maths and 9th for science. === Mathematics teachers === Qualifications vary by region; the East Midlands and London have the most degree-qualified Maths teachers and North East England the least. For England about 40% mostly have a maths degree and around 20% have a BSc degree with QTS or a BEd degree. Around 20% have a PGCE, and around 10% have no higher qualification than A level Maths. For schools without sixth forms, only around 30% of Maths teachers have a degree, but for schools with sixth forms and sixth form colleges around 50% have a Maths degree. There are around 27,500 Maths teachers in England, of whom around 21,000 are Maths specialists; there are around 31,000 science teachers in England. == Sixth-form level == You will need at least grade 6 at GCSE to study Maths in the sixth form, and many sixth forms will only accept people with a grade 7 at GCSE. At A-level, participation by gender is broadly mixed; about 60% of A-level entrants are male, and around 40% are female. Further Mathematics is an additional course available at A-level. A greater proportion of females take Further Maths (30%) than take Physics (15%), which at A-level is overwhelmingly a male subject. From the UPMAP project (Understanding Participation rates in post-16 Mathematics and Physics) of the ESRC Targeted Initiative on Science and Mathematics Education (TISME), in conjunction with the Institute of Physics, it was found that uptake of Maths A-level was linked to the grade at GCSE. From 2012 figures, 79% with A*, 48% of A, 15% of B and 1% of grade C chose Maths in the 6th form. For English, History and Geography, 30% with grade B, and 10% with grade C chose the course in the 6th form. The House of Lords July 2012 report Higher Education in STEM Subjects recommended that everyone study some type of Maths after 16. For less-able sixth formers, there was the AS level titled 'Use of Mathematics'. Professor Robert Coe, Director of the Centre for Evaluation and Monitoring (CEM) at Durham University conducted research on grade inflation. By 2007, 25% of Maths A-level grades were an A; he found that an A grade A-level would have been a grade B in 1996 and a grade C in 1988. The Labour government wanted to expand higher education, so required 'proof' that academic standards at A-level appeared to be rising, or at least not falling, so requiring higher education to expand for this wider apparent academic achievement. === Core Maths === People not taking Maths A-level can take the Core Mathematics Level 3 Certificate, developed by Mathematics in Education and Industry in Wiltshire. It was introduced by education minister Liz Truss from September 2015; her father was a university Maths lecturer. From 2014 it had been trialled in 170 schools. It was hoped that 200,000 sixth formers could study the course for three hours per week, but would possibly require 1,000 extra Maths teachers. 20% of sixth formers studied some kind of Mathematics in 2015. The Advisory Committee on Mathematics Education wanted the Core Maths introduction. In August 2016, there were 3,000 entries for the first Core Maths Level 3 exam. Consequently, the Conservative government was looking at making Maths education up to 18 compulsory. == University level == Admission to Mathematics at university in the UK will require three A-levels, often good A-levels. It is prevalently males who study Maths at university, and has been for decades. There are around 42–43,000 Maths undergraduates at British universities, with around 27,000 being male and around 16–17,000 being female. Mathematics at university is also taught for other physical sciences and Engineering, but much fewer women than men are taught on these types of courses. == Broadcasting == === Television === Educational series on television have included Mathematics and Life, BBC TV 18 September 1961 on Mondays at 11am, repeated on Fridays at 2pm, presented by Hugh David, produced by Donald Grattan Pure Mathematics, BBC TV 17 September 1962 on Mondays and Wednesdays at 10am, repeated on Wednesday and Fridays at 9.30am, for the fifth form and sixth-form, presented by Norman Hyland, produced by Donald Grattan; repeated in September 1963 Pure Mathematics Year II, BBC TV 16 September 1963 on Mondays and Wednesdays at 10am, for the age of 14, presented by Alan Tammadge (9 July 1921 - 25 February 2016, educated at Dulwich College and Emmanuel College, Cambridge); the series was the first of its kind on the BBC, to teach in such a formal way, known as 'chalk and talk'; the series was developed in response to an increasing lack of teachers for the sixth-form; in 1965, 38-year old Donald Grattan, a former grammar school Maths teacher, set up a new further education department at the BBC, with twenty producers; a new 'University of the Air' was being planned by Jennie Lee, Baroness Lee of Asheridge; in March 1968 through BBC Education, under Mr Grattan, this proposal turned largely into adult education courses, and from 1974 towards adult illiteracy; in July 1971 44-year-old Mr Grattan took over as Controller of Educational Broadcasting at the BBC, staying until July 1984; he died aged 93 on 21 August 2019. Middle School Mathematics, BBC TV 16 September 1963 on Mondays at 3pm, a 28-part series presented by Dikran Tahta, Alan Tammadge, the President of the Mathematical Association in 1978, David Morris, Stewart Gartside, Gerald Leach (4 January 1933 - 10 January 2004, the husband of child psychologist Penelope Leach), and Maurice Meredith, produced by Donald Grattan, and John Cain; repeated in September 1964, September 1965, September 1966. Mathematics '64, BBC2 on Tuesdays at 7.30pm a 20-part series presented by Alan Tammadge, Raymond Cuninghame-Green, Frank Yates, Stuart Hollingdale, Peter Coaker and Geoffrey Matthews, of 'Tuesday Term', with Wilfred Cockcroft, produced by David Roseveare Mathematics in Action: A Course in Statistics, BBC2 16 September 1965 on Thursdays at 7.30pm, repeated on BBC1 on Mondays at 9.30am and Thursdays at 12pm; a 12-part series presented by Stewart Gartside, Peter Sprent (a statistician and the husband of Janet Sprent and head of the Department of Mathematics at the University of Dundee in the late 1960s), and Bill Coleman, with Peter Armitage (statistician), produced by Edward Goldwyn; repeated September 1966, September 1967, September 1968, September 1969 and September 1970 Mathematics in Action: Logic and the Computer, BBC2 13 January 1966 on Thursdays at 7.30pm, a ten-part series presented by Raymond Cuninghame-Green, Frank Lovis, Philip Woodward and Benedict Nixon, produced by Edward Goldwyn; repeated January 1967, January 1968, January 1969, January 1970 and January 1971 Mathematics in Action: Mathematics Applied, BBC2 21 April 1966 on Thursdays at 7.30pm, presented by Prof John Crank, Kenneth Wigley, Noel Williams, Malcolm Bevan and Sydney Urry (originally from Highcliffe in Dorset, and taught Mechanical Engineering at Wandsworth Technical College, he later helped to set up Brunel University in 1966 and the sandwich course schemes, he had a heart attack aged 74 on 12 June 1999), produced by John Cain (attended Emanuel School and UCL) and Harry Levinson; repeated April 1967, April 1968, April 1969, April 1970, and April 1971 Maths Today, BBC1 21 September 1967 on Thursdays at 10.30am with repeats on Fridays at 10am, Mondays at 10.30am, and Wednesdays at 11.30am, a two-year series for the first and second years at secondary school, with an associated fortnightly series of six programmes for teachers called 'Teaching Maths Today' from Monday 18 September 1967 with Don Mansfield; presented by David Sturgess, Derick Last and Brenda Briggs, the wife of Trevor Jack Cole, produced by John Cain and Peter Weiss; Year 2 was from 23 September 1968; both series were repeated each year until 26 June 1973; Mr Sturgess lived at 8 Cobthorne Drive in Allestree, in Derbyshire, and was a Maths lecturer at the University of Nottingham, previously the head of the Mathematics Department at Bishop Lonsdale College of Education in Mickleover; Derick Last appeared in the 1967 film 'Mathematics and the Village' about the Cambridgeshire school Cottenham Village College, where he was head of Maths from 1963, and attended Eye Grammar School and Loughborough College, who lived at Wickham Skeith and Ditchingham. There was a later radio series on Radio 3. Square Two, BBC2 14 January 1970 on Wednesdays at 7pm, repeated on Saturdays on BBC1 at 9.30am, a 30-part series for people who have left school, presented by Stewart Gartside, Bill Coleman and Alan Tammadge, produced by David Roseveare, written by Hilary Shuard, Douglas Quadling, Ronald Thompson, Leslie Williams and Albert Lawrance; repeated in January 1971 Maths Workshop on BBC1 in the early 1970s with Jim Boucher and Michael Holt (author) Maths Topics, BBC1 17 September 1979 on Mondays at 10.30am, an O-level and CSE course, written by Ian Harris and Colin Banwell (of Bodmin School, and head boy of Sexey's Grammar School in Blackford, Somerset in 1953, who wrote 'Starting Points: For Teaching Mathematics in Middle and Secondary Schools' in 1972 with Ken Saunders ISBN 0906212510), produced by David Roseveare and Peter Bratt, repeated in September 1980, September 1981, September 1982, September 1983, September 1984, September 1985, September 1986, September 1987, and September 1988 Maths Help, BBC1 10 January 1982 on Sundays at 12pm, repeated on Mondays at 2.30pm on BBC2, 12-part series for O-level, but was much more for primary-school level abilities than 16 year olds, presented by Laurie Buxton of ILEA, and partly by Norman Gowar of the Open University, produced by Robert Clamp (attended Coalville Grammar School, and taught Maths at Latymer Upper School); repeated in October 1982, with another series in January 1983, that was more O-level standard Using Mathematics, ITV and C4 1990 for secondary schools Mathsphere, BBC2 7 January 1991 on Mondays at 9.30am repeated on Thursdays at 11.30am, the course was designed for children 'who find Maths difficult' for ages 11-16, produced by David Roseveare and Kevin Newport The Statistics Collection, BBC2 26 April 1995, on Wednesdays at 9.30am, a short course for sixth formers Maths 4 Real, C4 2000, for secondary schools Maths Mansion C4 2001 === Radio === How Mathematicians Think, Third Programme 16 March 1950 on Thursdays at 7.30pm, with George Temple (mathematician) of KCL, Gerald James Whitrow of Imperial College, and Charles Coulson of KCL New Paths in Pure Mathematics, Third Programme 28 November 1950 on Wednesdays at 6.20pm, with parts 1 What Is a Curve? 2 Numbers and Axioms, with Max Newman, Professor of Mathematics at the University of Manchester 3 The Mathematics of Counting, 4 The Mathematics of Measuring, with Richard Rado of KCL 5 The Evolution of Geometrical Ideas, with John Greenlees Semple, Professor of Geometry at KCL Thinking in Numbers, Network Three 4 November 1959 on Wednesdays at 7.30pm, presented and written by statistician Maurice Kendall, with parts 1 Louis Rosenhead of the University of Liverpool, and Alexander Aitken of the University of Edinburgh 2 Staff of the National Physical Laboratory, and Reg Revans of the University of Manchester 3 Richard van der Riet Woolley, Michael Abercrombie of UCL, and Otto Robert Frisch of the University of Cambridge 4 Kenneth Mather Mathematics, part of For Schools, Home Service 6 May 1965 Thursdays at 9.30am, 10-part series presented by James Hawthorne; repeated in April 1966, April 1967, September 1967, and on Radio 4 in September 1968, and September 1969 Developing Maths Today, Radio 3 9 October 1969 on Fridays at 7.30pm, it was a radio version of the BBC2 'Teaching Maths Today', produced by John Turtle == Results by region in England == Of all A-level entrants at Key Stage 5, 23% take Maths A-level, with 16% of all female entrants and 30% of all male entrants; 4% of all entrants take Further Maths, with 2% of female entrants and 6% of male entrants. By number of A-level entries, 11.0% were Maths A-levels with 7.7% female and 15.0% male. In England in 2016 there were 81,533 entries for Maths A-level, with 65,474 from the state sector; there were 14,848 entries for Further Maths with 10,376 from the state sector Entries for Further Maths in 2016 by region - South East 2987 East of England 1270 North West 1111 South West 1070 West Midlands 868 East Midlands 774 Yorkshire and the Humber 749 North East 414 == Results by LEA in England == Results shown are for 2016. In the 1980s, some areas with low Maths participation at A-level lost all sixth forms at the area's comprehensive schools, being replaced with stand-alone sixth form colleges, such as in Manchester and Portsmouth; this course of action may have helped in attracting qualified Maths teachers to those areas. The supply of qualified (QTS in England and Wales) Maths teachers in the UK is largely a postcode lottery. === Lowest number of entries for Maths A-level === The north of England (except Lancashire) has a worse record for Mathematics entries at A-level than other regions. Knowsley 6 Portsmouth 51 Salford 66 (Manchester entered 647 as a comparison) Halton 70 Middlesbrough 79 South Tyneside 85 Barnsley 96 === Highest number of entries for Maths A-level === Hampshire 2573 Hertfordshire 2039 Kent 1775 Surrey 1668 Essex 1499 Lancashire 1492 Birmingham 1403 Buckinghamshire 1284 Barnet 1189 Trafford entered 505, which is high for a small borough and almost the same number as Cumbria. Kirklees entered 661, which is more than Sheffield's 596; Kirklees is a much smaller borough by population than Sheffield. === Lowest number of entries for Further Maths A-level === Knowsley 0 (Knowsley only entered 61 A-level exams in 2016) Sandwell 5 Blackburn with Darwen 6 Salford 7 Portsmouth 8 North East Lincolnshire 9 Middlesbrough 11 Stoke-on-Trent 15 Barnsley 15 Halton 16 Southampton 16 Torbay 16 Bury 18 Merton 18 Rochdale 19 === Highest number of entries for Further Maths A-level === Hampshire and Hertfordshire are the top two for Maths and Further Maths Hampshire 381 Hertfordshire 370 Kent 297 Surrey 276 Essex 260 Buckinghamshire 244 Lancashire 206 == See also == Association of Teachers of Mathematics Education in the United Kingdom National Centre for Excellence in the Teaching of Mathematics Uses of trigonometry == References == == External links == Mathematics 5 to 11, 1979 How to Teach Maths February 2014 with Alex Bellos on BBC Radio 4 Association of Teachers of Mathematics |
Wikipedia:Mathematics education in the United States#0 | Mathematics education in the United States varies considerably from one state to the next, and even within a single state. With the adoption of the Common Core Standards in most states and the District of Columbia beginning in 2010, mathematics content across the country has moved into closer agreement for each grade level. The SAT, a standardized university entrance exam, has been reformed to better reflect the contents of the Common Core. Many students take alternatives to the traditional pathways, including accelerated tracks. As of 2023, twenty-seven states require students to pass three math courses before graduation from high school (grades 9 to 12, for students typically aged 14 to 18), while seventeen states and the District of Columbia require four. A typical sequence of secondary-school (grades 6 to 12) courses in mathematics reads: Pre-Algebra (7th or 8th grade), Algebra I, Geometry, Algebra II, Pre-calculus, and Calculus or Statistics. Some students enroll in integrated programs while many complete high school without passing Calculus or Statistics. Counselors at competitive public or private high schools usually encourage talented and ambitious students to take Calculus regardless of future plans in order to increase their chances of getting admitted to a prestigious university and their parents enroll them in enrichment programs in mathematics. Secondary-school algebra proves to be the turning point of difficulty many students struggle to surmount, and as such, many students are ill-prepared for collegiate programs in the sciences, technology, engineering, and mathematics (STEM), or future high-skilled careers. According to a 1997 report by the U.S. Department of Education, passing rigorous high-school mathematics courses predicts successful completion of university programs regardless of major or family income. Meanwhile, the number of eighth-graders enrolled in Algebra I has fallen between the early 2010s and early 2020s. Across the United States, there is a shortage of qualified mathematics instructors. Despite their best intentions, parents may transmit their mathematical anxiety to their children, who may also have school teachers who fear mathematics, and they overestimate their children's mathematical proficiency. About one in five American adults are functionally innumerate. While an overwhelming majority agree that mathematics is important, many, especially the young, are not confident of their own mathematical ability. On the other hand, high-performing schools may offer their students accelerated tracks (including the possibility of taking collegiate courses after calculus) and nourish them for mathematics competitions. At the tertiary level, student interest in STEM has grown considerably. However, many students find themselves having to take remedial courses for high-school mathematics and many drop out of STEM programs due to deficient mathematical skills. Compared to other developed countries in the Organization for Economic Co-operation and Development (OECD), the average level of mathematical literacy of American students is mediocre. As in many other countries, math scores dropped during the COVID-19 pandemic. However, Asian- and European-American students are above the OECD average. == Curricular content and standards == Each U.S. state sets its own curricular standards, and details are usually set by each local school district. Although there are no federal standards, since 2015 most states have based their curricula on the Common Core State Standards in mathematics. The stated goal of the Common Core mathematics standards is to achieve greater focus and coherence in the curriculum. This is largely in response to the criticism that American mathematics curricula are "a mile wide and an inch deep." The National Council of Teachers of Mathematics published educational recommendations in mathematics education in 1989 and 2000 which have been highly influential, describing mathematical knowledge, skills and pedagogical emphases from kindergarten through high school. The 2006 NCTM Curriculum Focal Points have also been influential for its recommendations of the most important mathematical topics for each grade level through grade 8. However, some states have either abandoned, or never adopted, the Common Core standards, but instead instituted their own. (See Common Core implementation by state.) In fact, there has been considerable disagreement on the style and content of mathematics teaching, including the question of whether or not there should be any national standards at all. In the United States, mathematics curriculum in elementary and middle school is integrated, while in high school it traditionally has been separated by topic, with each topic usually lasting for the whole school year. However, some districts have integrated curricula, or decided to try integrated curricula after Common Core was adopted. Since the days of the Sputnik in the 1950s, the sequence of mathematics courses in secondary school has not changed: Pre-algebra, Algebra I, Geometry, Algebra II, Pre-calculus (or Trigonometry), and Calculus. Trigonometry is usually integrated into the other courses. Calculus is only taken by a select few. Some schools teach Algebra II before Geometry. Success in middle-school mathematics courses is correlated with having an understanding of numbers by the start of first grade. This traditional sequence assumes that students will pursue STEM programs in college, though, in practice, only a minority are willing and able to take this option. Often a course in Statistics is also offered. While a majority of schoolteachers base their lessons on a core curriculum, they do not necessarily follow them to the letter. Many also take advantage of additional resources not provided to them by their school districts. === Primary school === Primary schoolchildren learn counting, arithmetic and properties of operations, geometry, measurement, statistics and probability. They typically begin studying fractions in third grade. === Secondary school === Pre-algebra can be taken in middle school by seventh or eighth graders. Students typically begin by learning about real numbers and basic number theory (prime numbers, prime factorization, fundamental theorem of arithmetic, ratios, and percentages), topics needed for algebra (powers, roots, graphing, order of operations, variables, expressions, and scientific notation) and geometry (quadrilaterals, polygons, areas of plane figures, the Pythagorean theorem, distance formula, equations of a line, simple solids, their surface areas, and volumes), and sometimes introductory trigonometry (definitions of the trigonometric functions). Such courses usually then go into simple algebra with solutions of simple linear equations and inequalities. Algebra I is the first course students take in algebra. Although some students take it as eighth graders, this class is most commonly taken in ninth or tenth grade, after the students have taken Pre-algebra. Students learn about real numbers and the order of operations (PEMDAS), functions, linear equations, graphs, polynomials, the factor theorem, radicals, and quadratic equations (factoring, completing the square, and the quadratic formula), and power functions. This course is considered a gatekeeper for those who want to pursue STEM because taking Algebra I in eighth grade allows students to eventually take Calculus before graduating from high school. As such, tracking students by their aptitude and deciding when they should take Algebra I has become a topic of controversy in California and Massachusetts. Parents of high-performing students are among the most vocal critics of policies discouraging the taking of Algebra I in middle school. Geometry, usually taken in ninth or tenth grade, introduces students to the notion of rigor in mathematics by way of some basic concepts in mainly Euclidean geometry. Students learn the rudiments of propositional logic, methods of proof (direct and by contradiction), parallel lines, triangles (congruence and similarity), circles (secants, tangents, chords, central angles, and inscribed angles), the Pythagorean theorem, elementary trigonometry (angles of elevation and depression, the law of sines), basic analytic geometry (equations of lines, point-slope and slope-intercept forms, perpendicular lines, and vectors), and geometric probability. Students are traditionally taught to demonstrate simple geometric theorems using two-column proofs, a method developed in the early 20th century in the U.S. specifically for this course, though other methods may also be used. Depending on the curriculum and instructor, students may receive orientation towards calculus, for instance with the introduction of the method of exhaustion and Cavalieri's principle. Algebra II has Algebra I as a prerequisite and is traditionally a high-school-level course. Course contents include inequalities, function notation, quadratic equations, power functions, exponential functions, logarithms, systems of linear equations, matrices (including matrix multiplication, 2 × 2 {\displaystyle 2\times 2} matrix determinants, Cramer's rule, and the inverse of a 2 × 2 {\displaystyle 2\times 2} matrix), the radian measure, graphs of trigonometric functions, trigonometric identities (Pythagorean identities, the sum-and-difference, double-angle, and half-angle formulas, the laws of sines and cosines), conic sections, among other topics. Requiring Algebra II for high school graduation gained traction across the United States in the early 2010s. The Common Core mathematical standards recognize both the sequential as well as the integrated approach to teaching high-school mathematics, which resulted in increased adoption of integrated math programs for high school. Accordingly, the organizations providing post-secondary education updated their enrollment requirements. For example, the University of California (UC) system requires three years of "college-preparatory mathematics that include the topics covered in elementary and advanced algebra and two- and three-dimensional geometry" to be admitted. After the California Department of Education adopted the Common Core, the UC system clarified that "approved integrated math courses may be used to fulfill part or all" of this admission requirement. On the other hand, in a controversial decision, the Texas Board of Education voted to remove Algebra II as a required course for high school graduation. In California, suggestions that Algebra II should be de-emphasized in favor of Data Science (a combination of algebra, statistics, and computer science) has faced severe criticism out of concerns that such a pathway would leave students ill-prepared for collegiate education. In 2023, the faculty of the University of California system voted to end an admissions policy that accepts Data Science in lieu of Algebra II. Students interested in taking AP Computer Science A or AP Computer Science Principles must have taken at least one course on algebra in high school. AP Chemistry specifically requires Algebra II.Precalculus follows from the above, and is usually taken by college-bound students. Pre-calculus combines algebra, analytic geometry, and trigonometry. Topics in algebra include the binomial theorem, complex numbers, the Fundamental Theorem of Algebra, root extraction, polynomial long division, partial fraction decomposition, and matrix operations. In chapters on trigonometry, students learn about radian angle measure, are shown the sine and cosine functions as coordinates on the unit circle, relate the six common trigonometric functions and their inverses and plot their graphs, solve equations involving trigonometric functions, and practice manipulating trigonometric identities. In the chapters on analytic geometry, students are introduced to polar coordinates and deepen their knowledge of conic sections. Some courses include the basics of vector geometry, including the dot product and the projection of one vector onto another. If time and aptitude permit, students might learn Heron's formula or the vector cross product. Students are introduced to the use of a graphing calculator to help them visualize the plots of equations and to supplement the traditional techniques for finding the roots of a polynomial, such as the rational root theorem and the Descartes rule of signs. Precalculus ends with an introduction to limits of a function. Some instructors might give lectures on mathematical induction and combinatorics in this course. Precalculus is a prerequisite for AP Physics 1 and AP Physics 2 (formerly AP Physics B). AP Precalculus has only three required chapters. polynomial and rational functions, exponential and logarithmic functions, and trigonometric functions and polar curves. Optional materials include parametric equations, implicit functions, conic sections, vectors, and matrix algebra ( 2 × 2 {\displaystyle 2\times 2} matrix inversion, 2 × 2 {\displaystyle 2\times 2} determinants, and linear transformations). According to the College Board, "AP Precalculus may be the last mathematics course of a student's secondary education, the course is structured to provide a coherent capstone experience and is not exclusively focused on preparation for future courses." Depending on the school district, several courses may be compacted and combined within one school year, either studied sequentially or simultaneously. For example, in California, Algebra II and Precalculus may be taken as a single compressed course. Without such acceleration, it may be not possible to take more advanced classes like calculus in high school. In Oregon, high-school juniors and seniors may choose between three separate tracks, depending on their interests. Those aiming for a career in mathematics, the physical sciences, and engineering can pursue the traditional pathway, taking Algebra II and Precalculus. Those who want to pursue a career in the life sciences, the social sciences, or business can take Statistics and Mathematical Modelling. Students bound for technical training can take Applied Mathematics and Mathematical Modelling. In Florida, students may also receive lessons on mathematical logic and set theory at various grade levels in high school following the new 2020 reforms. The new Floridian standards also promote financial literacy and emphasize how different mathematical topics from different grade levels are connected. In Utah, the final required mathematics course in high school incorporates elements of Algebra II, Trigonometry, Precalculus, and Data Science. However, as of 2023, students may opt out of this class with a signed letter from their parents, and about half do. College algebra is offered at many community colleges as remedial courses for students who did not pass courses before Calculus. It should not be confused with abstract algebra and linear algebra, taken by students who major in mathematics and allied fields (such as computer science) in four-year colleges and universities. Calculus is usually taken by high-school seniors or university freshmen, but can occasionally be taken as early as tenth grade. Unlike many other countries from France to Israel to Singapore, which require high school students aiming for a career in STEM or placed in the track for advanced mathematics to study calculus, the United States generally treats calculus as collegiate mathematics. A successfully completed college-level calculus course like one offered via Advanced Placement program (AP Calculus AB and AP Calculus BC) is a transfer-level course—that is, it can be accepted by a college as a credit towards graduation requirements. Prestigious colleges and universities are believed to require successful completion AP courses, including AP Calculus, for admissions. Calculus is a prerequisite or a corequisite for AP Physics C: Mechanics and AP Physics C: Electricity and Magnetism. Since the 1990s, the role of calculus in the high school curriculum has been a topic of controversy. In this class, students learn about limits and continuity (the intermediate and mean value theorems), differentiation (the product, quotient, and chain rules) and its applications (implicit differentiation, logarithmic differentiation, related rates, optimization, concavity, Newton's method, L'Hôpital's rules), integration and the Fundamental Theorem of Calculus, techniques of integration (u-substitution, by parts, trigonometric and hyperbolic substitution, and by partial fractions decomposition), further applications of integration (calculating accumulated change, various problems in the sciences and engineering, separable ordinary differential equations, arc length of a curve, areas between curves, volumes and surface areas of solids of revolutions), improper integrals, numerical integration (the midpoint rule, the trapezoid rule, Simpson's rule), infinite sequences and series and their convergence (the nth-term, comparison, ratio, root, integral, p-series, and alternating series tests), Taylor's theorem (with the Lagrange remainder), Newton's generalized binomial theorem, Euler's complex identity, polar representation of complex numbers, parametric equations, and curves in polar coordinates. Depending on the course and instructor, special topics in introductory calculus might include the classical differential geometry of curves (arc-length parametrization, curvature, torsion, and the Frenet–Serret formulas), the epsilon-delta definition of the limit, first-order linear ordinary differential equations, Bernoulli differential equations. Some American high schools today also offer multivariable calculus (partial differentiation, the multivariable chain rule and Clairault's theorem; constrained optimization, Lagrange multipliers and the Hessian; multidimensional integration, Fubini's theorem, change of variables, and Jacobian determinants; gradients, directional derivatives, divergences, curls, the fundamental theorem of gradients, Green's theorem, Stokes' theorem, and Gauss' theorem). Other optional mathematics courses may be offered, such as statistics (including AP Statistics) or business math. Students learn to use graphical and numerical techniques to analyze distributions of data (including univariate, bivariate, and categorical data), the various methods of data collection and the sorts of conclusions one can draw therefrom, probability, and statistical inference (point estimation, confidence intervals, and significance tests). High school students of exceptional ability may be selected to join a competition, such as the USA Mathematical Olympiad, or the International Mathematical Olympiad. === Tertiary school === All students in STEM, especially mathematics, physics, chemistry, computer science, and engineering must take single-variable calculus unless they have Advanced Placement credits (or equivalents, such as IB Math HL). Students majoring in mathematics, the physical sciences, and engineering then take multivariable calculus, linear algebra, complex variables, ordinary differential equations, and partial differential equations. Mathematics majors may take a course offering a rigorous introduction to the concepts of modern mathematics before they tackle abstract algebra, number theory, real analysis, advanced calculus, complex analysis, probability theory, statistics, and advanced topics, such as set theory and mathematical logic, stochastic processes, integration and measure theory, Fourier analysis, functional analysis, differential geometry, and topology. They may further choose courses in applied mathematics, such as mathematical modelling, numerical analysis, game theory, or mathematical optimization. The calculus of variations, the history of mathematics, and topics in theoretical or mathematical physics (such as classical mechanics, electrodynamics, nonlinear dynamics, fluid mechanics, quantum mechanics, or general relativity) may be taken as electives. Computer science majors must study discrete mathematics (such as combinatorics and graph theory), information theory, the theory of computation, and cryptography. Students in computer science and economics might have the option of taking algorithmic game theory. Those who study biomedical and social sciences have to study elementary probability and statistics. Students in the physical sciences and engineering need to understand error analysis for their laboratory sessions and courses. Advanced undergraduates and beginning graduate students in physics may take a course on advanced mathematical methods for physics, which may cover contour integration, the theory of distributions (generalized functions), Fourier analysis, Green's functions, special functions (especially Euler's gamma and beta functions; Bessel functions; Legendre polynomials; Hermite polynomials; Laguerre polynomials; and the hypergeometric series), asymptotic series expansions, the calculus of variations, tensors, and group theory. Exact requirements and available courses will depend on the institution in question. At many colleges and universities, confident students may compete in the Integration Bee. Exceptional undergraduates may participate in the annual William Lowell Putnam Mathematical Competition. Many successful competitors have gone on to fruitful research careers in mathematics. Although doing well on the Putnam is not a requirement for becoming a mathematician, it encourages students to develop skills and hone intuitions that could help them become successful researchers. Besides the monetary prize, the winners are virtually guaranteed acceptance to a prestigious graduate school. Such competitions are one way for mathematical talents to stand out. == Attendance and completion rates == For many students, passing algebra is often a Herculean challenge, so much so that many students have dropped out of high school because of it. The greatest obstacle for excelling in algebra is fluency with fractions, something many Americans do not have. Without mastery of high-school algebra—Algebra I and II—students will not be able to pursue collegiate STEM courses. In fact, the lack of adequate preparation in mathematics is part of the reason why the rate of attrition in STEM is so high. From 1986 to 2012, though more students were completing Algebra II, their average performance has fallen. Indeed, students who had passed high-school courses, including those labeled "honors" courses, might still fail collegiate placement exams and had to take remedial courses. As for Algebra I, the number of 13-year-olds enrolled fell from 34% in 2012 to 24% in 2023. Longitudinal analysis shows that the number of students completing high-school courses on calculus and statistics, including AP courses, have declined before 2019. Data taken from students' transcripts ( N = 15 , 188 {\displaystyle N=15,188} ) from the late 2000s to the mid-2010s reveals that majorities of students had completed Algebra I (96%), Geometry (76%), and Algebra II (62%). But not that many took Precalculus (34%), Trigonometry (16%), Calculus (19%), or Statistics (11%) and only an absolute minority took Integrated Mathematics (7%). Overall, female students were more likely to complete all mathematics courses, except Statistics and Calculus. Asian Americans were the most likely to take Precalculus (55%), Statistics (22%), and Calculus (47%) while African Americans were the least likely to complete Calculus (8%) but most likely to take Integrated Mathematics (10%) in high school. Among students identified as mathematically proficient by the PSAT, Asians are much more likely than blacks to attend an honors or Advanced Placement course in mathematics. Asians are also the most likely to have scored at least a 3 on the AP Calculus exams. Students of lower socioeconomic status were less likely to pass Precalculus, Calculus, and Statistics. While boys and girls are equally likely to take AP Statistics and AP Calculus AB, boys are the majority in AP Calculus BC (59%), as well as some other highly mathematical subjects, such as AP Computer Science A (80%), AP Physics C: Mechanics (74%) and AP Physics C: Electricity and Magnetism (77%). Although undergraduate men and women score the same grades in Calculus I (in college) on average, women are more likely than men to drop out because of mathematical anxiety. Perceptions and stereotypes of girls being less mathematically able than boys begin as early as second grade, and they affect how girls actually perform in class or in a competition, such as the International Mathematical Olympiad. Among university students who have taken calculus, engineering disciplines are the most popular among men and biology among women. During the 1970s and 1980s, the number of students taking remedial courses in college rose substantially, partly due to the de-emphasis of calculus in high school, leading to less exposure to pre-calculus topics. In the twenty-first century, American community colleges require 60% of their students to pass at least one course in mathematics, depending on the program. But around 80% fail to meet this requirement, and 60% require remedial courses. Many students at these schools drop out after failing even in remedial courses, such as (the equivalent of) Algebra II. On the other hand, four-year institutions have seen increased student interest in STEM programs, including mathematics and statistics. == Controversies and issues == Mathematics education has been a topic of debate among academics, parents, as well as educators. Majorities agree that mathematics is crucial, but there has been many divergent opinions on what kind of mathematics should be taught and whether relevance to the "real world" or rigor should be emphasized. Another source of contention is the decentralized nature of American education, making it difficult to introduce standard curriculum implemented nationwide, despite the benefits of such a program as seen from the experience of other countries, such as Italy. In the early 2020s, the decision by some educators to include the topics of race and sexuality into the mathematical curriculum has also met with stiff resistance. === Progressive education === During the first half of the twentieth century, there was a movement aimed at systematically reforming American public education along more "progressive" grounds. William Heard Kilpatrick, one of the most vocal proponents of progressive education, advocated for the de-emphasis of intellectual "luxuries" such as algebra, geometry, and trigonometry, calling them "harmful rather than helpful to the kind of thinking necessary for ordinary living." He recommended that more advanced topics in mathematics should only be taught to the select few. Indeed, prior to the Second World War, it was common for educationists to argue against the teaching of academic subjects and in favor of more utilitarian concerns of "home, shop, store, citizenship, and health," presuming that a majority of high school students could not embark on a path towards higher education but were instead, destined to become unskilled laborers or their wives. By the 1940s, however, the deficiency in mathematical skills among military recruits became a public scandal. Admiral Chester Nimitz himself complained about the lack of skills that should have been taught in public schools among officers in training and volunteers. In order to address this issue, the military had to open courses to teach basic skills such as arithmetic for bookkeeping or gunnery. Indeed, many parents opposed the progressive reforms, criticizing the lack of contents. By mid-century, technological marvels, such as radar, nuclear energy, and the jet engine, made progressive education untenable. === New Math === Under the 'New Math' initiative, created after the successful launch of the Soviet satellite Sputnik in 1957, conceptual abstraction rather than calculation gained a central role in mathematics education. The educational status quo was severely criticized as a source of national humiliation and reforms were demanded, prompting Congress to introduce the National Defense Education Act of 1958. The U.S. federal government under President Dwight D. Eisenhower realized it needed thousands of scientists and engineers to match the might of its ideological rival the Soviet Union and started pouring enormous sums of money into research and development as well as education. Conceived in response to the lack of emphasis on content of the progressive education and the technological advances of World War II, New Math was part of an international movement influenced by the Nicholas Bourbaki school in France, attempting to bring the mathematics taught in schools closer to what research mathematicians actually use. Students received lessons in set theory, which is what mathematicians actually use to construct the set of real numbers, normally taught to advanced undergraduates in real analysis (see Dedekind cuts and Cauchy sequences). Arithmetic with bases other than ten was also taught (see binary arithmetic and modular arithmetic). Other topics included number theory, probability theory, and analytic geometry. However, this educational initiative soon faced strong opposition, not just from teachers, who struggled to understand the new material, let alone teach it, but also from parents, who had problems helping their children with homework. It was criticized by experts, too. In a 1965 essay, physicist Richard Feynman argued, "first there must be freedom of thought; second, we do not want to teach just words; and third, subjects should not be introduced without explaining the purpose or reason, or without giving any way in which the material could be really used to discover something interesting. I don't think it is worthwhile teaching such material." In his 1973 book, Why Johnny Can't Add: the Failure of the New Math, mathematician and historian of mathematics Morris Kline observed that it was "practically impossible" to learn new mathematical creations without first understanding the old ones, and that "abstraction is not the first stage, but the last stage, in a mathematical development." Kline criticized the authors of the 'New Math' textbooks, not for their mathematical faculty, but rather their narrow approach to mathematics, and their limited understanding of pedagogy and educational psychology. Mathematician George F. Simmons wrote in the algebra section of his book Precalculus Mathematics in a Nutshell (1981) that the New Math produced students who had "heard of the commutative law, but did not know the multiplication table." By the early 1970s, this movement was defeated. Nevertheless, some of the ideas it promoted still lived on. One of the key contributions of the New Math initiative was the teaching of calculus in high school. === Standards-based reforms and the NCTM === From the late twentieth century to the early twenty-first, there has been a fierce debate over how mathematics should be taught. On one hand, some campaign for a more traditional teacher-led curriculum, featuring algorithms and some memorization. On the other hand, some prefer a conceptual approach, with a focus on problem-solving and the sense of numbers. However, as mathematician Hung-Hsi Wu explained, the apparent dichotomy between basic skills and understanding of mathematical concepts is a delusion. In 1989 the National Council for Teachers of Mathematics (NCTM) produced the Curriculum and Evaluation Standards for School Mathematics. Widespread adoption of the new standards notwithstanding, the pedagogical practice changed little in the United States during the 1990s. In fact, mathematics education became a hotly debated subject in the 1990s and early 2000s. This debate pitted mathematicians (like UC Berkeley mathematician Hung-Hsi Wu) and parents, many of whom with substantial knowledge of mathematics (such as the Institute for Advanced Study physicist Chiara R. Nappi), who opposed the NCTM's reforms against educational professionals, who wanted to emphasized what they called "conceptual understanding." In many cases, however, educational professionals did not understand mathematics as well as their critics. This became apparent with the publication of the book Knowing and Teaching Elementary Mathematics (1999) by Liping Ma. The author gave evidence that even though most Chinese teachers had only 11 or 12 years of formal education, they understood basic mathematics better than did their U.S. counterparts, many of whom were working on their master's degrees. In 1989, the more radical NCTM reforms were eliminated. Instead, greater emphasis was put on substantive mathematics. In some large school districts, this came to mean requiring some algebra of all students by ninth grade, compared to the tradition of tracking only the college-bound and the most advanced junior high school students to take algebra. A challenge with implementing the Curriculum and Evaluation Standards was that no curricular materials at the time were designed to meet the intent of the Standards. In the 1990s, the National Science Foundation funded the development of curricula such as the Core-Plus Mathematics Project. In the late 1990s and early 2000s, the so-called math wars erupted in communities that were opposed to some of the more radical changes to mathematics instruction. Some students complained that their new math courses placed them into remedial math in college. However, data provided by the University of Michigan registrar at this same time indicate that in collegiate mathematics courses at the University of Michigan, graduates of Core-Plus did as well as or better than graduates of a traditional mathematics curriculum, and students taking traditional courses were also placed in remedial mathematics courses. Mathematics instructor Jaime Escalante dismissed the NCTM standards as something written by a PE teacher. In 2001 and 2009, NCTM released the Principles and Standards for School Mathematics (PSSM) and the Curriculum Focal Points which expanded on the work of the previous standards documents. Particularly, the PSSM reiterated the 1989 standards, but in a more balanced way, while the Focal Points suggested three areas of emphasis for each grade level. Refuting reports and editorials that it was repudiating the earlier standards, the NCTM claimed that the Focal Points were largely re-emphasizing the need for instruction that builds skills and deepens student mathematical understanding. These documents repeated the criticism that American mathematics curricula are a "mile wide and an inch deep" in comparison to the mathematics of most other nations, a finding from the Second and Third International Mathematics and Science Studies. === Integrated mathematics === As previously stated, American children usually follow a unique sequence of mathematics courses in secondary school (grades 6 to 12), learning one subject at a time. They take two years of Algebra punctuated by a year of Geometry. Geometry, hitherto a collegiate course, was introduced into high schools in the nineteenth century. In Europe, schools followed Felix Klein's call for Geometry to be integrated with other math subjects. In 1892 the American Committee of Ten recommended the same strategy for the United States, but American teachers had already been developing the habit of teaching Geometry has a separate course. The American high-school geometry curriculum was eventually codified in 1912 and developed a distinctive American style of geometric demonstration for such courses, known as "two-column" proofs. This remains largely true today, with Geometry as a proof-based high-school math class. On the other hand, many countries around the world from Israel to Italy teach mathematics according to what Americans call an integrated curriculum, familiarizing students with various aspects of calculus and prerequisites throughout secondary school. In fact, many topics in Algebra and Geometry that Americans typically learn in high school are taught in middle school in Europe, making it possible for European countries to require and to teach Calculus in high school. In France and Germany, calculus was brought into the secondary-school curriculum thanks to the advocacy of famous mathematicians, such as Henri Poincaré and Felix Klein, respectively. However, as the Singaporean case demonstrates, early exposure to the concepts of calculus does not necessarily translates to actual understanding among high school students. In the U.S., this is reflected in the concerns voiced by many university professors, according to whom their students lack sufficient preparation in pre-calculus mathematics. Proponents of teaching the integrated curriculum believe that students would better understand the connections between the different branches of mathematics. On the other hand, critics—including parents and teachers—prefer the traditional American approach both because of their familiarity with it and because of their concern that certain key topics might be omitted, leaving the student ill-prepared for college. As mentioned above, only 7% of American high school students take Integrated Mathematics. === Preparation for college === Beginning in 2011, most states have adopted the Common Core Standards for mathematics, which were partially based on NCTM's previous work. Controversy still continues as critics point out that Common Core standards do not fully prepare students for college and as some parents continue to complain that they do not understand the mathematics their children are learning. Indeed, even though they may have expressed an interest in pursuing science, technology, engineering, and mathematics (STEM) in high school, many university students find themselves ill-equipped for rigorous STEM education in part because of their inadequate preparation in mathematics. Meanwhile, Chinese, Indian, and Singaporean students are exposed to high-level mathematics and science at a young age. About half of STEM students in the U.S. dropped out of their programs between 2003 and 2009. On top of that, many mathematics schoolteachers were not as well-versed in their subjects as they should be, and might well be uncomfortable with mathematics themselves. An emphasis on speed and rote memorization gives as many as one-third of students aged five and over mathematical anxiety. Parents and high school counselors consider it crucial that students pass Calculus if they aim to be admitted to a competitive university. Private school counselors are especially likely to make this recommendation while admissions officers are generally less inclined to consider it a requirement. Moreover, there has been a movement to de-emphasize the traditional pathway with Calculus as the final mathematics class in high school in favor of Statistics and Data Science for those not planning to major in a STEM subject in college. Nevertheless, Calculus remains the most recommended course for ambitious students. But in the case of Utah, as of 2023, students may skip the final required course for high-school graduation—one that combines elements of Algebra II, Trigonometry, Precalculus, and Statistics—if they submit a letter signed by their parents acknowledging that this decision could jeopardize their chances of university matriculation. By the mid-2010s, only a quarter of American high school seniors are able to do grade-level math, yet about half graduate from high school as A students, prompting concerns of grade inflation. Strong performance in Algebra I, Geometry, and Algebra II predict good grades in university-level Calculus even better than taking Calculus in high school. Another issue with mathematics education has been integration with science education. This is difficult for public schools to do because science and math are taught independently. The value of the integration is that science can provide authentic contexts for the math concepts being taught and further, if mathematics is taught in synchrony with science, then the students benefit from this correlation. === Enrichment programs and accelerated tracks === Growing numbers of parents have opted to send their children to enrichment and accelerated learning after-school or summer programs in mathematics, leading to friction with school officials who are concerned that their primary beneficiaries are affluent white and Asian families, prompting parents to pick private institutions or math circles. Some public schools serving low-income neighborhoods even denied the existence of mathematically gifted students. In fact, American educators tend to focus on poorly performing students rather than those at the top, unlike their Asian counterparts. Parents' proposal for an accelerated track for their children are oftentimes met with hostility by school administrators. Conversely, initiatives aimed at de-emphasizing certain core subjects, such as Algebra I, triggered strong backlash from parents and university faculty members. Students identified by the Study of Mathematically Precocious Youth as top scorers on the mathematics (and later, verbal) sections of the SAT often became highly successful in their fields. By the mid-2010s, some public schools have begun offering enrichment programs to their students. Similarly, while some school districts have proposed to stop separating students by mathematical ability in order to ensure they begin high school at the same level, parents of gifted children have pushed back against this initiative, fearing that it would jeopardize their children's future college admissions prospects, especially in the STEM fields. In San Francisco, for example, such a plan was dropped due to a combination of mixed results and public backlash. === Shortage of instructors === A shortage of qualified mathematics schoolteachers has been a serious problem in the United States for many years. In order to rectify this problem, the amount of instructional hours dedicated towards mathematical contents has been increased in undergraduate programs aimed at training elementary teachers. Teachers oftentimes unknowingly transmit their own negative attitude towards mathematics to their students, damaging the quality of instruction. == Standardized tests == The Program for International Student Assessment (PISA) is held every three years for 15-year-old students worldwide. In 2012, the United States earned average scores in science and reading. It performed better than other progressive nations in mathematics, ranking 36 out of 65 other countries. The PISA assessment examined the students’ understanding of mathematics as well as their approach to this subject and their responses. These indicated three approaches to learning. Some of the students depended mainly on memorization. Others were more reflective on newer concepts. Another group concentrated more on principles that they have not yet studied. The U.S. had a high proportion of memorizers compared to other developed countries. During the 2015 testing, the United States failed to make it to the top 10 in all categories including mathematics. More than 540,000 teens from 72 countries took the exam. American students' average score in mathematics declined by 11 points compared to the previous testing. The 2022 PISA test showed that U.S. national average in mathematics remained behind those of other industrialized nations and remained below the OECD average. Furthermore, one third of American students did not meet the requirements for basic proficiency in mathematics. However, European- and especially Asian-American students perform above the OECD average. See chart below. According to a 2021 report by the National Science Foundation (NSF), American students' mathematical literacy ranks 25th out of 37 nations of the Organization for Economic Cooperation and Development (OECD). During the 2000s and 2010s, as more and more college-bound students take the SAT, scores have gone down. (See chart below.) This is in part because some states have required all high school students to take the SAT, regardless of whether or not they were going to college. In 2015, educational psychologist Jonathan Wai of Duke University analyzed average test scores from the Army General Classification Test in 1946 (10,000 students), the Selective Service College Qualification Test in 1952 (38,420), Project Talent in the early 1970s (400,000), the Graduate Record Examination between 2002 and 2005 (over 1.2 million), and the SAT Math and Verbal in 2014 (1.6 million). Wai identified one consistent pattern: those with the highest test scores tended to pick mathematics and statistics, the natural and social sciences, and engineering as their majors while those with the lowest were more likely to choose healthcare, education, and agriculture. (See the two charts below.) Results from the National Assessment of Educational Progress (NAEP) test show that scores in mathematics have been leveling off in the 2010s, but with a growing gap between the top and bottom students. The COVID-19 pandemic, which forced schools to shut down and lessons to be given online, further widened the divide, as the best students lost fewer points compared to the worst and therefore could catch up more quickly. While students' scores fell for all subjects, mathematics was the hardest hit, with a drop of eight points, the steepest decline in 50 years. Scores dropped for students of all races, sexes, socioeconomic classes, types of schools, and states with very few exceptions. This might be because mathematics education is more dependent upon the classroom experience than reading, as students who were allowed to return to in-person classes generally did better, more so in mathematics than in reading. However, on the topics of statistics and probability, student performance had already declined before the pandemic. As consequence, the entire cohort of college students in the 2022-23 academic year have lower average grades and mathematical standards. A 2023 comparison between parents' views and standardized test scores revealed a significant gap; most parents overestimated their children's academic aptitude. In mathematics, only 26% were proficient, even though 90% of the parents asked thought their children met grade standards. Having a higher NAEP math score in eighth grade is correlated with high academic standing, higher income, lower rates of adolescent parenthood, and lower chances of criminality. == Advanced Placement Mathematics == There was considerable debate about whether or not calculus should be included when the Advanced Placement (AP) Mathematics course was first proposed in the early 1950s. AP Mathematics has eventually developed into AP Calculus thanks to physicists and engineers, who convinced mathematicians of the need to expose students in these subjects to calculus early on in their collegiate programs. In the early 21st century, there has been a demand for the creation of AP Multivariable Calculus and indeed, a number of American high schools have begun to offer this class, giving colleges trouble in placing incoming students. As of 2021, AP Precalculus was under development by the College Board, though there were concerns that universities and colleges would not grant credit for such a course, given that students had previously been expected to know this material prior to matriculation. AP Precalculus launched in Fall 2023. == Conferences == Mathematics education research and practitioner conferences include: NCTM's Regional Conference and Exposition and Annual Meeting and Exposition; The Psychology of Mathematics Education's North American Chapter annual conference; and numerous smaller regional conferences. == See also == Mathematics education Embodied design (mathematics education) Graduate science education in the United States Mathematics education in New York National Museum of Mathematics Stand and Deliver (1988 film) Math 55 at Harvard University Financial literacy curriculum Chicago movement Computer-Based Math Mathematical software Mathethon American Mathematics Competitions == References == == Further reading == Garelick, Barry; Wilson, JR (2022). Traditional Math: An effective strategy that teachers feel guilty using. John Catt Educational. ISBN 978-1-91526-154-0. == External links == Math courses with “Math Is Your Future”; an article about studying math with the use of Internet technologies Math is amazing and we have to start treating it that way, Eugenia Cheng for the PBS Newshour. |
Wikipedia:Mathematics in Education and Industry#0 | MEI (Mathematics in Education and Industry) is an independent educational charity and curriculum development body for mathematics education in the United Kingdom. Income generated through its work is used to support the teaching and learning of mathematics. == History == MEI was founded in 1963 with a grant from the Schools & Industry Committee of the Mathematical Association. In 1965 it produced its first exam, Additional Mathematics, then produced an A level course two years later. MEI's A-level exams were the first to include probability. It was incorporated as a company on 18 October 1996. == Structure == Although independent, MEI works in partnership with many organisations, including the UK Government. MEI is a registered charity with a board of directors and a small professional staff. == Qualifications == GCE AS/A level Mathematics, Further Mathematics and Further Mathematics (Additional) (Published by OCR) AS Level Statistics GCSE Mathematics Foundations of Advanced Mathematics (FAM) – a freestanding course Introduction to Quantitative Methods (in association with OCR) OCR MEI Level 3 Core Maths Qualifications (Level 3 Certificate in Quantitative Reasoning and Level 3 Certificate in Quantitative Problem Solving) == Competitions == MEI organises an annual online competition called Ritangle for teams of students of A level Mathematics, the International Baccalaureate and Scottish Highers. Questions are posted on the Integral website, with correct answers releasing a clue for the final question. == References == == External links == Official website |
Wikipedia:Mathematics in India (book)#0 | Mathematics in India: 500 BCE–1800 CE is a monograph about the history of Indian mathematics. It was written by American historian of mathematics Kim Plofker, and published in 2009 by the Princeton University Press. The Basic Library List Committee of the Mathematical Association of America has classified the book as essential for undergraduate mathematics libraries, their highest rating. == Topics == Plofker has organized Mathematics in India into nine chapters, roughly chronologically, according to the "mainstream narrative" of Indian chronology in a subject where accurate chronology is difficult and disputed. It covers the mathematics of the entire Indian subcontinent, including the modern areas of Afghanistan, India, and Pakistan, but largely restricts itself to Sanskrit-language sources. Unlike many previous works in this area, it views Indian mathematics as a coherent whole, strongly connected to Indian culture and religion, both influencing and being influenced by the other cultures of the world, rather than as a collection of milestones for measuring relative progress against other cultures. Much of the scholarly work on this subject has been contradictory and contentious, and Plofker is careful to provide evidence for the hypotheses she supports, discuss alternative hypotheses, and view the subject neutrally for itself rather than as a way to boost or put down Indian culture. Her book includes some speculative theories, but is well-grounded in recent scholarship, and focused on evidence from the source material. It carefully maintains a balance between the cultural and scientific context needed to understand the mathematics it describes, the major texts and oral traditions through which that mathematics has come down to us, and the cross-cultural transmission of mathematical knowledge with other cultures. The first introductory chapter provides an overview of Indian history of Indian mathematics and its scholarship, and of the religious and linguistic context of early Sanskrit texts, which leads to important differences from Indian mathematics to other ancient mathematical cultures developing from administrative or scientific works. Chapter two discusses the Vedic period from 1500 to 500 BCE, and the Shulba Sutras, religious instructional texts with significant mathematical content that are generally attributed to this period, although (as the book discusses) the absence of concrete astronomical observations within these texts has made it impossible to date them precisely. Topics from this period include its methods for reckoning time, its fascination with large numbers, the beginnings of decimal numbering and integer factorization, geometric constructions using cords or ropes, the Pythagorean theorem, and accurate approximations to pi and the square root of two. This chapter also includes material on speculative links between Vedic India and ancient Mesopotamia, a pet theory of Plofker's advisor David Pingree, but it notes the weakness of evidence for these theories. The third chapter covers the next 500 years, the early classical period of India, including the Bhutasamkhya system for describing numbers in words and the invention of decimal place-value arithmetic (although Plofker suggests that the concept of zero may be an import from China), connections between poetic meter and binary representations, early trigonometry, the works of Pāṇini and Pingala (arguably including the invention of recursion), mathematics in Jainism and Buddhism from this period, and possible Greek influences in trigonometry and astrology, which became one of the driving forces in later mathematics. Chapter four covers roughly the first millennium CE, and focuses mainly on Indian astronomy and geocentrism, including the use of verse forms and interpolation to make memorization of trigonometric tables possible. Chapters five and six concern the medieval period of India. Chapter five overlaps in time with the later parts of chapter four, and concerns the works of Aryabhata, Bhāskara I, and Brahmagupta, and Mahāvīra, and the Bakhshali manuscript, including the invention of negative numbers and algebra, Brahmagupta's formula for the area of cyclic quadrilaterals, and the solution of Pell's equation. Chapter six covers later mathematicians Bhāskara II and Narayana Pandita, Bhāskara's works on geodesy, and the development of ideas related to calculus (although not really calculus itself). It also discusses the position of mathematicians in society, and the nature of mathematical canon, commentary, and proof in those times. The Kerala school of astronomy and mathematics founded by Madhava of Sangamagrama is the topic of the seventh chapter, which includes Madhava's works on series expansions of trigonometric functions and the calculation of pi, and developments by Nilakantha Somayaji in the theory of astronomy. Chapter eight covers the interactions between India and mathematics in medieval Islam, including the transmission of decimal notation to the west and an increased awareness of mathematical rigor in India. Chapter nine concerns colonial and early modern times in India, the influence of European mathematics, and ongoing developments within Indian mathematics from the 16th through 18th centuries. Unfortunately, it stops just before the time of Srinivasa Ramanujan. The book concludes with a collection of still-unresolved major research questions in the area of Indian mathematics. Two appendices cover aspects of Sanskrit grammar and prosody that are important for understanding Indian mathematics, a glossary of technical terms, and a collection of biographies of Indian mathematicians. Throughout, many images of documents and artifacts of mathematical interest are included. == Audience and reception == Mathematics in India does not require that its readers have any background in mathematics or the history of mathematics. It makes scholarship in this area accessible to a general audience, for instance by replacing many Sanskrit technical terms by English phrases, although it is "more of a research monograph than a popular book". Its readers are likely to come from many different audiences, including mathematicians, historians, indologists, philosophers, linguists, and philologists, and it succeeds in navigating the different expectations of these audiences. Reviewer James Rauff recommends Mathematics in India to all students or teachers of the history of mathematics, calling it "meticulously researched, carefully argued, and beautifully written", and Benno van Dalen goes further, calling it required reading for all future students of this topic. Dominik Wujastyk calls it "path-breaking", "a classic work that should be owned and read by any scholar interested in the history of science in South Asia". Although calling it difficult reading for non-specialists, Ward Stewart suggests that it could also be valuable to high school teachers and that some of its material could be incorporated into their lessons, and although A. K. Bag calls it "mainly meant for the foreign audience", B. Ramanujam writes that it deserves to be better known among Indian schoolteachers in particular. Dominik Wujastyk suggests using it as the basis for university-level courses, and Toke Knudsen highlights its value as reference material for researchers in this area. Both van Dalen and Agathe Keller write that the comprehensive English-language history of Indian mathematics in Mathematics in India was long-awaited, and several reviewers point to the History of Hindu Mathematics by Bibhutibhushan Datta and Awadhesh Narayan Singh from the 1930s as the only previous work that filled that role, albeit one organized by topic rather than by time. Reviewers also noted the novelty of the book's focus on mathematical astronomy, with Alexander Jones calling it "the best general introduction to the history of astronomy in India that we currently have". Despite some quibbles, Keller and Clemency Montelle both call the book "destined to be a classic". A rare negative review is given by Satyanad Kichenassamy, who takes issue with the book's consideration of social context rather than purely of the mathematical content of the works it discusses, with its emphasis on astronomy as a force for mathematical development, with its omission of Malayalam-language works, with "a tendency to conflate ancient mathematical concepts with modern ones", and with many details of its conclusions. == References == |
Wikipedia:Mathematics in Nazi Germany#0 | Mathematics in Nazi Germany was heavily affected by Nazi policies. Though Jews had previously faced discrimination in academic institutions, the Civil Service Law of 1933 led to the dismissal of many Jewish mathematics professors and lecturers at German universities. During this time, many Jewish mathematicians left Germany and took positions at American universities. Before the Nazi rise to power, some Jewish mathematicians like Hermann Minkowski and Edmund Landau had achieved success and even were appointed to full professorships with the support of David Hilbert. == University of Göttingen == Göttingen was, along with Berlin, one of Germany's two main centers for mathematical research. Prior to Nazi rule, the University of Göttingen already had an illustrious mathematics tradition that included distinguished mathematicians like Gauss, Riemann, David Hilbert, Dirichlet, Hermann Minkowski and Felix Klein. Abraham Fraenkel has written that Hilbert was "the most significant mathematician in the world" during those years. Fraenkel writes that Hilbert "always remained free of all national and racist prejudices" and had been influenced by two Jewish mathematicians, Adolf Hurwitz and Minkowski. Though prejudice against appointing Jews to academic positions existed before the Nazi era, Hilbert had supported the successful appointments of two Jewish mathematicians to full professorships: Minkowski in 1902 and Edmund Landau in 1909. Like Hilbert himself, Minkowski had first been appointed by Felix Klein. When Klein retired, Hilbert appointed the German Jewish mathematician Richard Courant to replace him. (Courant moved to New York University in 1933 where the Courant Institute of Mathematical Sciences is named after him). Hilbert also supported Emmy Noether, a Jewish woman whose postdoctoral candidacy had been opposed, mostly on account of her gender, even by Jews. In the 1920s, Hilbert became involved in a dispute with L.E.J. Brouwer, a Dutch mathematician whose support for intuitionism had not been widely accepted by Germany's mathematical establishment. Intuition (Anschauung) was contrasted with "modern abstract" mathematics like formalism. There was a rivalry in those years between Berlin and Göttingen, and Berlin sided with Brouwer against Hilbert in the dispute. The dispute took on an ideological dimension as Brouwer presented himself as a "champion of Aryan Germanness". When Brouwer objected to Ostjuden (German Jews of Eastern European descent) writing for the journal Mathematische Annalen, Hilbert removed Brouwer from his position as editor. The Nazis offered Brouwer a position at the University of Berlin in 1933, which he declined. Even so, the Dutch government suspended Brouwer in 1945 because of his connections to the Party; he was, however, eventually reinstated. Though Jewish academics had experienced prejudice prior to 1933, Hilbert had been supportive of Jewish mathematicians and their advancement. When the Civil Service Law of 1933 mandated the dismissal of Jews from the civil service, including university professors and Privatdozent, Landau and Courant were still teaching. Hermann Weyl, who had succeeded Hilbert in 1931, and Gustav Herglotz were not of Jewish descent. Weyl, whose wife was Jewish, chose to accept a position at the Institute for Advanced Study in Princeton in the United States. Other lower ranking professors and lecturers included Paul Bernays, Emmy Noether, Hans Lewy, Otto Neugebauer, Herbert Busemann, Werner Fenchel, Franz Rellich, and Wilhelm Magnus. Paul Bernays was among the scholars who had to leave their positions at Göttingen in 1933. Together with Hilbert, Bernays had co-authored a seminal text on mathematical logic called Grundlagen der Mathematik. The collaboration on the second volume of that work, published in 1939, continued even after 1933; face-to-face collaboration ceased sometime in 1934 when Bernays moved to Zürich. Otto Blumenthal, who had converted to Protestantism when he was 18, lost his position at RWTH Aachen University. == NSDAP == In the mid-1930s, racist Nazi policies that limited the participation of Jewish mathematicians were imposed on the German mathematics journal Zentralblatt für Mathematik. Ivan Niven identified this as a turning point for the journal, saying it began to "deteriorate". Otto Neugebauer, who had been a key figure in founding Zentralblatt, had moved to the United States and taken a position at Brown University. With his expertise a new reviewing journal, Mathematical Reviews, was established in the United States. During the years of Nazi rule, many classes in German universities would begin with a Nazi salute, a practice that Erich Hecke declined to implement in his classroom. Even before Hitler's rise to power, some mathematicians had already emigrated to the United States for various reasons. John von Neumann had taken a position at the California Institute of Technology in 1929 because he felt anti-semitism in Germany was affecting his career. By 1933, von Neumann had a position at Princeton; though he had maintained ties with Germany until then, he canceled his scheduled lectures in Berlin after Hitler became Chancellor. Other early emigrants from Germany included Theodor Estermann, Hans Freudenthal, Eberhard Hopf, Heinz Hopf, Herman Müntz, Wilhelm Meyer, and Abraham Plessner. Some emigrated to the United States, others to European countries; Heinz Hopf spent the years of Nazi rule in Zürich, Switzerland. Hans Rademacher obtained a position at the University of Pennsylvania after he was dismissed from the University of Breslau by the Nazis. In 1933, when Hitler came to power, the following topologists held positions in German universities: Max Dehn, Herbert Seifert, Hans Freudenthal, Hellmuth Kneser, Georg Feigl, Kurt Reidemeister, William Threlfall, Heinrich Tietze, Hermann Künneth, Leopold Vietoris, and Felix Hausdorff. == Deutsche Mathematik == The journal Deutsche Mathematik, published between 1936 and 1943, was edited by Theodor Vahlen and Ludwig Bieberbach. Vahlen had taught in Greifswald before serving in the army in World War I; he was wounded and became a decorated, high-ranking officer. He joined the Nazi Party after the failed Beer Hall Putsch of 1923, visited Hitler at Landsburg prison, and lost his job at the University of Greifswald after taking down the Weimar flag there. He went into voluntary exile in Vienna, returning to Germany only after Hitler became Chancellor in 1933. Bieberbach was a distinguished mathematician who first taught in Königsberg in 1910, moving to Basel in 1913 and then Berlin in 1921. Although he had himself espoused formalistic ideas in the early years of his career, by the 1920s he had become critical of David Hilbert's formalism. During Nazi rule, he became a strong supporter of the "Aryanization" of mathematics. Nazi ideology about mathematics had determined that "Aryan mathematics" would emphasize geometric math and probability theory. Abstract axiomatics were denounced as "Franco-Jewish". Felix Klein was posthumously declared an exemplar of Aryan mathematics. Hilbert and Richard Dedekind were also accepted even though their work did not actually fit the Aryan mold. Bieberbach condemned Cantorian set theory and measure theory as un-Germanic. Even abstract algebra was suspect. == Notes == == References == Dalen, Dirk van (2013). L.E.J. Brouwer – Topologist, Intuitionist, Philosopher: How Mathematics Is Rooted in Life. Springer Science & Business Media. ISBN 978-1-4471-4616-2. Fraenkel, Abraham (2017). "How German Mathematicians Dealt With the Rise of Nazism". Tablet Magazine. Retrieved 2018-08-05. Holfter, Gisela (2015). Voices from Exile: Essays in Memory of Hamish Ritchie. BRILL. ISBN 978-90-04-29639-8. Lane, Saunders Mac (1995). "Mathematics at Gottingen under the Nazis" (PDF). Notices of the AMS. 42 (10): 1134–38. Parsons, Charles (2009). "Paul Bernays' later philosophy of mathematics". In Dimitracopoulos, Costas; Newelski, Ludomir; Normann, Dag; Steel, John R (eds.). Logic Colloquium 2005. Cambridge University Press. pp. 129–150. doi:10.1017/CBO9780511546464.011. ISBN 9780511546464. Segal, Sanford L. (1986). "Mathematics and German politics: The national socialist experience". Historia Mathematica. 13 (2): 118–135. doi:10.1016/0315-0860(86)90025-X. ISSN 0315-0860. Segal, Sanford L. (2003). Mathematicians Under the Nazis. Princeton University Press. ISBN 9780691164632. |
Wikipedia:Mathematics in Nepal#0 | Mathematics in Nepal have been used for measurement since ancient times. Advanced mathematics were used primarily in the field of Astrology to predict position of planets to determine auspicious time for various Hindu rituals. In recent times, mathematics is taught formally in schools from primary level up to doctorate degree. All students must pass mathematics in the SEE exam. == History == The history of mathematics in Nepal is inter-related with the history of mathematics in the Indian sub-continent. However, independent history of mathematics in Nepal also exists. The ancient Licchavi people developed a series of the system for measurement such as Kharika to measure land area and Kosh for measurement of distance. Similarly, Jayasthiti Malla, in 1350 AD standardized Mana and Pathi for volumetric measurement of grains and cereals. The numerals of Ranjana script was developed in 199 BC. It was used until the mid-20th century in Nepal and India. It is still in use in the Newari language. === Formal education === In the Rana Period, Kashi (Banaras) used to be the education hub to learn astrology and mathematics. The mathematics was based mainly on the text of Baskaracharya's Siddhant Siromani. The formal education of mathematics in school started after overthrowing of Rana regime and start of democracy. The M.A./M.Sc. in Mathematics started on July 14, 1959, and Central Department of Mathematics was formed on September 20, 1959, A.D. in Tripureshwor Bengali mathematician Prof. Asutosh Ganguli was the first head of Department in Master level mathematics courses at Tri-Chandra College. ==== Institutions offering graduate courses ==== Central Department of Mathematics TU at Tribhuwan University Department of mathematics at Kathmandu University Department of Mathematics at Pokhara University == Notable books == Siddhānta Shiromani of Bhāskara II Līlāvatī Bijaganita (Algebra) Arithmetic by Yadav Chandra Chakravorty Geometry of Jagannath Wyakta Chandrika by Gopal Pande illustrates the rule of three to the determination of square root and cube roots Nepal Arithmetic by Nepal Bhasa Prakasiri Samiti published in 1834 == Notable figures == Chakra Pani Aryal was a 15th-century astrologer and mathematician. He wrote Uttan Gadit (in Sanskrit) that was used for calculation of solar and lunar eclipses. This book was revised by Padma Nav Keshari Aryal in 1934 A.D. Pd. Gopal Pandey (1883–1914) was the first person to write a book in Nepali about mathematics. He wrote four editions of his book in Nepali. The third edition was also published in Hindi. He was honoured as the Royal Astrologer for successfully predicting the number of lunar eclipses in 1884. He was also responsible to make the plot of Tudikhel. Noor Dutta Pande (second son of Gopal Pande) wrote the Gorkha Bijaganita. He has composed a mathematics book called "Bichitra Ganita". Tikaram Dhananjaya (1909–1936) wrote the first full commentary of Lilavati of Bhaskaracharya in Nepali. He also wrote the book Shishubodha Tarangini along with his wife Chandrakala Dhananjaya. Chandra Kala Dhananjaya was the first woman writer in mathematics. She wrote a book series called Shishubodha Tarangini, in poetic form. In the second part of the series, Shishubodha Tarangini dosro bhaga, more than 100 mathematical poems were written. In 1890 the Rana Prime minister, Chandra Samser was the first Nepali to pass the matriculation examination from the Calcutta University with mathematics. Naya Raj Pant (1913–2002) was the first person to graduate in mathematics from Tribhuwan University. He also used his skill in astronomy to decide the historical events. == See also == Nepal Mathematical Society == References == == External links == Journal of Nepal Mathematical Society (JNMS) Homepage of Nepal Mathematical Society TU central department of mathematics |
Wikipedia:Mathematics in the medieval Islamic world#0 | Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built upon syntheses of Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta). Important developments of the period include extension of the place-value system to include decimal fractions, the systematised study of algebra and advances in geometry and trigonometry. The medieval Islamic world underwent significant developments in mathematics. Muhammad ibn Musa al-Khwārizmī played a key role in this transformation, introducing algebra as a distinct field in the 9th century. Al-Khwārizmī's approach, departing from earlier arithmetical traditions, laid the groundwork for the arithmetization of algebra, influencing mathematical thought for an extended period. Successors like Al-Karaji expanded on his work, contributing to advancements in various mathematical domains. The practicality and broad applicability of these mathematical methods facilitated the dissemination of Arabic mathematics to the West, contributing substantially to the evolution of Western mathematics. Arabic mathematical knowledge spread through various channels during the medieval era, driven by the practical applications of Al-Khwārizmī's methods. This dissemination was influenced not only by economic and political factors but also by cultural exchanges, exemplified by events such as the Crusades and the translation movement. The Islamic Golden Age, spanning from the 8th to the 14th century, marked a period of considerable advancements in various scientific disciplines, attracting scholars from medieval Europe seeking access to this knowledge. Trade routes and cultural interactions played a crucial role in introducing Arabic mathematical ideas to the West. The translation of Arabic mathematical texts, along with Greek and Roman works, during the 14th to 17th century, played a pivotal role in shaping the intellectual landscape of the Renaissance. == Origin and spread of Arab-Islamic mathematics == Arabic mathematics, particularly algebra, developed significantly during the medieval period. Muhammad ibn Musa al-Khwārizmī's (Arabic: محمد بن موسى الخوارزمي; c. 780 – c. 850) work between AD 813 and 833 in Baghdad was a turning point. He introduced the term "algebra" in the title of his book, "Kitab al-jabr wa al-muqabala," marking it as a distinct discipline. He regarded his work as "a short work on Calculation by (the rules of) Completion and Reduction, confining it to what is easiest and most useful in arithmetic". Later, people commented his work was not just a theoretical treatise but also practical, aimed at solving problems in areas like commerce and land measurement. Al-Khwārizmī's approach was groundbreaking in that it did not arise from any previous "arithmetical" tradition, including that of Diophantus. He developed a new vocabulary for algebra, distinguishing between purely algebraic terms and those shared with arithmetic. Al-Khwārizmī noticed that the representation of numbers is crucial in daily life. Thus, he wanted to find or summarize a way to simplify the mathematical operation, so-called later, the algebra. His algebra was initially focused on linear and quadratic equations and the elementary arithmetic of binomials and trinomials. This approach, which involved solving equations using radicals and related algebraic calculations, influenced mathematical thinking long after his death. Al-Khwārizmī's proof of the rule for solving quadratic equations of the form (ax^2 + bx = c), commonly referred to as "squares plus roots equal numbers," was a monumental achievement in the history of algebra. This breakthrough laid the groundwork for the systematic approach to solving quadratic equations, which became a fundamental aspect of algebra as it developed in the Western world. Al-Khwārizmī's method, which involved completing the square, not only provided a practical solution for equations of this type but also introduced an abstract and generalized approach to mathematical problems. His work, encapsulated in his seminal text "Al-Kitab al-Mukhtasar fi Hisab al-Jabr wal-Muqabala" (The Compendious Book on Calculation by Completion and Balancing), was translated into Latin in the 12th century. This translation played a pivotal role in the transmission of algebraic knowledge to Europe, significantly influencing mathematicians during the Renaissance and shaping the evolution of modern mathematics. Al-Khwārizmī's contributions, especially his proof for quadratic equations, are a testament to the rich mathematical heritage of the Islamic world and its enduring impact on Western mathematics. The spread of Arabic mathematics to the West was facilitated by several factors. The practicality and general applicability of al-Khwārizmī's methods were significant. They were designed to convert numerical or geometrical problems into equations in normal form, leading to canonical solution formulae. His work and that of his successors like al-Karaji laid the foundation for advances in various mathematical fields, including number theory, numerical analysis, and rational Diophantine analysis. Al-Khwārizmī's algebra was an autonomous discipline with its historical perspective, eventually leading to the "arithmetization of algebra". His successors expanded on his work, adapting it to new theoretical and technical challenges and reorienting it towards a more arithmetical direction for abstract algebraic calculation. Arabic mathematics, epitomized by al-Khwārizmī's work, was crucial in shaping the mathematical landscape. Its spread to the West was driven by its practical applications, the expansion of mathematical concepts by his successors, and the translation and adaptation of these ideas into the Western context. This spread was a complex process involving economics, politics, and cultural exchange, greatly influencing Western mathematics. The period known as the Islamic Golden Age (8th to 14th century) was characterized by significant advancements in various fields, including mathematics. Scholars in the Islamic world made substantial contributions to mathematics, astronomy, medicine, and other sciences. As a result, the intellectual achievements of Islamic scholars attracted the attention of scholars in medieval Europe who sought to access this wealth of knowledge. Trade routes, such as the Silk Road, facilitated the movement of goods, ideas, and knowledge between the East and West. Cities like Baghdad, Cairo, and Cordoba became centers of learning and attracted scholars from different cultural backgrounds. Therefore, mathematical knowledge from the Islamic world found its way to Europe through various channels. Meanwhile, the Crusades connected Western Europeans with the Islamic world. While the primary purpose of the Crusades was military, there was also cultural exchange and exposure to Islamic knowledge, including mathematics. European scholars who traveled to the Holy Land and other parts of the Islamic world gained access to Arabic manuscripts and mathematical treatises. During the 14th to 17th century, the translation of Arabic mathematical texts, along with Greek and Roman ones, played a crucial role in shaping the intellectual landscape of the Renaissance. Figures like Fibonacci, who studied in North Africa and the Middle East, helped introduce and popularize Arabic numerals and mathematical concepts in Europe. == Concepts == === Algebra === The study of algebra, the name of which is derived from the Arabic word meaning completion or "reunion of broken parts", flourished during the Islamic golden age. Muhammad ibn Musa al-Khwarizmi, a Persian scholar in the House of Wisdom in Baghdad was the founder of algebra, is along with the Greek mathematician Diophantus, known as the father of algebra. In his book The Compendious Book on Calculation by Completion and Balancing, Al-Khwarizmi deals with ways to solve for the positive roots of first and second-degree (linear and quadratic) polynomial equations. He introduces the method of reduction, and unlike Diophantus, also gives general solutions for the equations he deals with. Al-Khwarizmi's algebra was rhetorical, which means that the equations were written out in full sentences. This was unlike the algebraic work of Diophantus, which was syncopated, meaning that some symbolism is used. The transition to symbolic algebra, where only symbols are used, can be seen in the work of Ibn al-Banna' al-Marrakushi and Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī. On the work done by Al-Khwarizmi, J. J. O'Connor and Edmund F. Robertson said: "Perhaps one of the most significant advances made by Arabic mathematics began at this time with the work of al-Khwarizmi, namely the beginnings of algebra. It is important to understand just how significant this new idea was. It was a revolutionary move away from the Greek concept of mathematics which was essentially geometry. Algebra was a unifying theory which allowed rational numbers, irrational numbers, geometrical magnitudes, etc., to all be treated as "algebraic objects". It gave mathematics a whole new development path so much broader in concept to that which had existed before, and provided a vehicle for the future development of the subject. Another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before." Several other mathematicians during this time period expanded on the algebra of Al-Khwarizmi. Abu Kamil Shuja' wrote a book of algebra accompanied with geometrical illustrations and proofs. He also enumerated all the possible solutions to some of his problems. Abu al-Jud, Omar Khayyam, along with Sharaf al-Dīn al-Tūsī, found several solutions of the cubic equation. Omar Khayyam found the general geometric solution of a cubic equation. === Cubic equations === Omar Khayyam (c. 1038/48 in Iran – 1123/24) wrote the Treatise on Demonstration of Problems of Algebra containing the systematic solution of cubic or third-order equations, going beyond the Algebra of al-Khwārizmī. Khayyám obtained the solutions of these equations by finding the intersection points of two conic sections. This method had been used by the Greeks, but they did not generalize the method to cover all equations with positive roots. Sharaf al-Dīn al-Ṭūsī (? in Tus, Iran – 1213/4) developed a novel approach to the investigation of cubic equations—an approach which entailed finding the point at which a cubic polynomial obtains its maximum value. For example, to solve the equation x 3 + a = b x {\displaystyle \ x^{3}+a=bx} , with a and b positive, he would note that the maximum point of the curve y = b x − x 3 {\displaystyle \ y=bx-x^{3}} occurs at x = b 3 {\displaystyle x=\textstyle {\sqrt {\frac {b}{3}}}} , and that the equation would have no solutions, one solution or two solutions, depending on whether the height of the curve at that point was less than, equal to, or greater than a. His surviving works give no indication of how he discovered his formulae for the maxima of these curves. Various conjectures have been proposed to account for his discovery of them. === Induction === The earliest implicit traces of mathematical induction can be found in Euclid's proof that the number of primes is infinite (c. 300 BCE). The first explicit formulation of the principle of induction was given by Pascal in his Traité du triangle arithmétique (1665). In between, implicit proof by induction for arithmetic sequences was introduced by al-Karaji (c. 1000) and continued by al-Samaw'al, who used it for special cases of the binomial theorem and properties of Pascal's triangle. === Irrational numbers === The Greeks had discovered irrational numbers, but were not happy with them and only able to cope by drawing a distinction between magnitude and number. In the Greek view, magnitudes varied continuously and could be used for entities such as line segments, whereas numbers were discrete. Hence, irrationals could only be handled geometrically; and indeed Greek mathematics was mainly geometrical. Islamic mathematicians including Abū Kāmil Shujāʿ ibn Aslam and Ibn Tahir al-Baghdadi slowly removed the distinction between magnitude and number, allowing irrational quantities to appear as coefficients in equations and to be solutions of algebraic equations. They worked freely with irrationals as mathematical objects, but they did not examine closely their nature. In the twelfth century, Latin translations of Al-Khwarizmi's Arithmetic on the Indian numerals introduced the decimal positional number system to the Western world. His Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear and quadratic equations. In Renaissance Europe, he was considered the original inventor of algebra, although it is now known that his work is based on older Indian or Greek sources. He revised Ptolemy's Geography and wrote on astronomy and astrology. However, C.A. Nallino suggests that al-Khwarizmi's original work was not based on Ptolemy but on a derivative world map, presumably in Syriac or Arabic. === Spherical trigonometry === The spherical law of sines was discovered in the 10th century: it has been attributed variously to Abu-Mahmud Khojandi, Nasir al-Din al-Tusi and Abu Nasr Mansur, with Abu al-Wafa' Buzjani as a contributor. Ibn Muʿādh al-Jayyānī's The book of unknown arcs of a sphere in the 11th century introduced the general law of sines. The plane law of sines was described in the 13th century by Nasīr al-Dīn al-Tūsī. In his On the Sector Figure, he stated the law of sines for plane and spherical triangles and provided proofs for this law. === Negative numbers === In the 9th century, Islamic mathematicians were familiar with negative numbers from the works of Indian mathematicians, but the recognition and use of negative numbers during this period remained timid. Al-Khwarizmi did not use negative numbers or negative coefficients. But within fifty years, Abu Kamil illustrated the rules of signs for expanding the multiplication ( a ± b ) ( c ± d ) {\displaystyle (a\pm b)(c\pm d)} . Al-Karaji wrote in his book al-Fakhrī that "negative quantities must be counted as terms". In the 10th century, Abū al-Wafā' al-Būzjānī considered debts as negative numbers in A Book on What Is Necessary from the Science of Arithmetic for Scribes and Businessmen. By the 12th century, al-Karaji's successors were to state the general rules of signs and use them to solve polynomial divisions. As al-Samaw'al writes: the product of a negative number—al-nāqiṣ—by a positive number—al-zāʾid—is negative, and by a negative number is positive. If we subtract a negative number from a higher negative number, the remainder is their negative difference. The difference remains positive if we subtract a negative number from a lower negative number. If we subtract a negative number from a positive number, the remainder is their positive sum. If we subtract a positive number from an empty power (martaba khāliyya), the remainder is the same negative, and if we subtract a negative number from an empty power, the remainder is the same positive number. === Double false position === Between the 9th and 10th centuries, the Egyptian mathematician Abu Kamil wrote a now-lost treatise on the use of double false position, known as the Book of the Two Errors (Kitāb al-khaṭāʾayn). The oldest surviving writing on double false position from the Middle East is that of Qusta ibn Luqa (10th century), an Arab mathematician from Baalbek, Lebanon. He justified the technique by a formal, Euclidean-style geometric proof. Within the tradition of Golden Age Muslim mathematics, double false position was known as hisāb al-khaṭāʾayn ("reckoning by two errors"). It was used for centuries to solve practical problems such as commercial and juridical questions (estate partitions according to rules of Quranic inheritance), as well as purely recreational problems. The algorithm was often memorized with the aid of mnemonics, such as a verse attributed to Ibn al-Yasamin and balance-scale diagrams explained by al-Hassar and Ibn al-Banna, who were each mathematicians of Moroccan origin. == Influences == The influence of medieval Arab-Islamic mathematics to the rest of the world is wide and profound, in both the realm of science and mathematics. The knowledge of the Arabs went into the western world through Spain and Sicily during the translation movement. "The Moors (western Mohammedans from that part of North Africa once known as Mauritania) crossed over into Spain early in the seventh century, bringing with them the cultural resources of the Arab world". In the 13th century, King Alfonso X of Castile established the Toledo School of Translators, in the Kingdom of Castile, where scholars translated numerous scientific and philosophical works from Arabic into Latin. The translations included Islamic contributions to trigonometry, which helps European mathematicians and astronomers in their studies. European scholars such as Gerard of Cremona (1114–1187) played a key role in translating and disseminating these works, thus making them accessible to a wider audience. Cremona is said to have translated into Latin "no fewer than 90 complete Arabic texts." European mathematicians, building on the foundations laid by Islamic scholars, further developed practical trigonometry for applications in navigation, cartography, and celestial navigation, thus pushing forward the age of discovery and scientific revolution. The practical applications of trigonometry for navigation and astronomy became increasingly important during the Age of Exploration. Al-Battānī is one of the islamic mathematicians who made great contributions to the development of trigonometry. He "innovated new trigonometric functions, created a table of cotangents, and made some formulas in spherical trigonometry." These discoveries, together with his astronomical works which are praised for their accuracy, greatly advanced astronomical calculations and instruments. Al-Khayyām (1048–1131) was a Persian mathematician, astronomer, and poet, known for his work on algebra and geometry, particularly his investigations into the solutions of cubic equations. He was "the first in history to elaborate a geometrical theory of equations with degrees ≤ 3", and has great influence on the work of Descartes, a French mathematician who is often regarded as the founder of analytical geometry. Indeed, "to read Descartes' Géométrie is to look upstream towards al-Khayyām and al-Ṭūsī; and downstream towards Newton, Leibniz, Cramer, Bézout and the Bernoulli brothers". Numerous problems that appear in "La Géométrie" (Geometry) have foundations that date back to al-Khayyām. Abū Kāmil (Arabic: أبو كامل شجاع بن أسلم بن محمد بن شجاع, also known as Al-ḥāsib al-miṣrī—lit. "The Egyptian Calculator") (c. 850 – c. 930), was studied algebra following the author of Algebra, al-Khwārizmī. His Book of Algebra (Kitāb fī al-jabr wa al-muqābala) is "essentially a commentary on and elaboration of al-Khwārizmī's work; in part for that reason and in part for its own merit, the book enjoyed widespread popularity in the Muslim world". It contains 69 problems, which is more than al-Khwārizmī who had 40 in his book. Abū Kāmil's Algebra plays a significant role in shaping the trajectory of Western mathematics, particularly in its impact on the works of the Italian mathematician Leonardo of Pisa, widely recognized as Fibonacci. In his Liber Abaci (1202), Fibonacci extensively incorporated ideas from Arabic mathematicians, using approximately 29 problems from Book of Algebra with scarce modification. == Western historians' perception of the contribution of Arab mathematicians == Despite the fundamental works Arabic mathematicians have done on the development of Algebra and algebraic geometry, Western historians in the 18th and early 19th century still regarded it as a fact that Classical science and math were unique phenomena of the West. Even though some math contributions from Arab mathematicians are occasionally acknowledged, they are considered to be "outside history or only integrated in so far as it contributed to science, which is essentially European", and just some technical innovations to the Greek heritage rather than open up a completely new branch of mathematics. In the French philosopher Ernest Renan's work, Arabic math is merely "a reflection of Greece, combined with Persian and Indian influences". And according to Duhem, "Arabic science only reproduced the teachings received from Greek science". Besides being considered as merely some insignificant additions or reflections to the great tradition of Greek classical science, math works from Arabic mathematicians are also blamed for lacking rigor and too focused on practical applications and calculations, and this is why Western historians argued they could never reach the level of Greek mathematicians. As Tannery wrote, Arabic math "in no way superseded the level attained by Diophantus". On the other hand, they perceived that Western mathematicians went into a very different way both in its method employed and ultimate purpose, "the hallmark of Western science in its Greek origins as well as in its modern renaissance, is its conformity to rigorous standards". Thus, the perceived non-rigorous proof in Arabic mathematicians' book authorizes Bourbaki to exclude the Arabic period when he retraced the evolution of algebra. And instead, the history of classical algebra is written as the work of the Renaissance and the origin of algebraic geometry is traced back to Descartes, while Arabic mathematicians' contributions are ignored deliberately. In Rashed's words: "To justify the exclusion of science written in Arabic from the history of science, one invokes its absence of rigor, its calculatory appearance and its practical aims. Furthermore, strictly dependent on Greek science and, lastly, incapable of introducing experimental norms, scientists of that time were relegated to the role of conscientious guardians of the Hellenistic museum." In 18th century Germany and France, the prevailing Orientalist view was "East and West oppose each other not as geographical but as historical positivities", which labeled "Rationalism" as the essence of the West, while the "Call of the Orient" movement emerged in the 19th century was interpreted as "against Rationalism" and a return to a more "spiritual and harmonious" lifestyle. Thus, the prevailing Orientalism in that period was one of the main reasons why Arabic mathematicians were often ignored for their contributions, as people outside the West were considered to be lacking the necessary rationality and scientific spirit to made significant contributions to math and science. == Conclusion == The medieval Arab-Islamic world played a crucial role in shaping the trajectory of mathematics, with al-Khwārizmī's algebraic innovations serving as a cornerstone. The dissemination of Greek mathematics to the West during the Islamic Golden Age, facilitated by cultural exchanges and translations, left a lasting impact on Western mathematical thought. Mathematicians like Al-Battānī, Al-Khayyām, and Abū Kāmil, with their contributions to trigonometry, algebra, and geometry, extended their influence beyond their time. Despite the foundational contributions of Arab mathematicians, Western historians in the 18th and early 19th centuries, influenced by Orientalist views, sometimes marginalized these achievements. The East lacking rationality and scientific spirit perpetuated a biased perspective, hindering the recognition of the significant role played by Arabic mathematics in the development of algebra and other mathematical disciplines. Reevaluating the history of mathematics necessitates acknowledging the interconnectedness of diverse mathematical traditions and dispelling the notion of a uniquely European mathematical heritage. The contributions of Arab mathematicians, marked by practical applications and theoretical innovations, form an integral part of the rich tapestry of mathematical history, and deserves recognition. == Other major figures == 'Abd al-Hamīd ibn Turk (fl. 830) (quadratics) Sind ibn Ali (d. after 864) Thabit ibn Qurra (826–901) Al-Battānī (before 858 – 929) Abū Kāmil (c. 850 – c. 930) Abu'l-Hasan al-Uqlidisi (fl. 952) (arithmetic) 'Abd al-'Aziz al-Qabisi (d. 967) Abū Sahl al-Qūhī (c. 940–1000) (centres of gravity) Ibn al-Haytham (c. 965–1040) Abū al-Rayḥān al-Bīrūnī (973–1048) (trigonometry) Al-Khayyām (1048–1131) Ibn Maḍāʾ (c. 1116–1196) Ismail al-Jazari (1136–1206) Jamshīd al-Kāshī (c. 1380–1429) (decimals and estimation of the circle constant) == Gallery == == See also == Arabic numerals Indian influence on Islamic mathematics in medieval Islam History of calculus History of geometry Science in the medieval Islamic world Timeline of science and engineering in the Muslim world == References == == Sources == == Further reading == == External links == Hogendijk, Jan P. (January 1999). "Bibliography of Mathematics in Medieval Islamic Civilization". O'Connor, John J.; Robertson, Edmund F. (1999), "Arabic mathematics: forgotten brilliance?", MacTutor History of Mathematics Archive, University of St Andrews Richard Covington, Rediscovering Arabic Science, 2007, Saudi Aramco World |
Wikipedia:Mathematics of the Incas#0 | The mathematics of the Incas (or of the Tawantinsuyu) was the set of numerical and geometric knowledge and instruments developed and used in the nation of the Incas before the arrival of the Spaniards. It can be mainly characterized by its usefulness in the economic field. The quipus and yupanas are proof of the importance of arithmetic in Inca state administration. This was embodied in a simple but effective arithmetic, for accounting purposes, based on the decimal numeral system; they too had a concept of zero, and mastered addition, subtraction, multiplication, and division. The mathematics of the Incas had an eminently applicative character to tasks of management, statistics, and measurement that was far from the Euclidean outline of mathematics as a deductive corpus, since it was suitable and useful for the needs of a centralized administration. On the other hand, the construction of roads, canals and monuments, as well as the layout of cities and fortresses, required the development of practical geometry, which was indispensable for the measurement of lengths and surfaces, in addition to architectural design. At the same time, they developed important measurement systems for length and volume, which took parts of the human body as reference. In addition, they used appropriate objects or actions that allowed to appreciate the result in another way, but relevant and effective. == Inca numeral system == The prevailing numeral system was the base-ten. One of the main references confirming this are the chronicles that present a hierarchy of organized authorities, using the decimal numeral system with its arithmometer: Quipu. It is also possible to confirm the use of the decimal system in the Inca system by the interpretation of the quipus, which are organized in such a way that the knots — according to their location — can represent: units, tens, hundreds, etc. However, the main confirmation of the use of this system is expressed in the denomination of the numbers in Quechua, in which the numbers are developed in decimal form. This can be appreciated in the following table: == Accounting systems == === Quipus === The quipus constituted a mnemonic system based on knotted strings used to record all kinds of quantitative or qualitative information; if they were dealing with the results of mathematical operations, only those previously performed on the "Inca abacuss" or yupanas were cancelled. Although one of its functions is related to mathematics — as it was an instrument capable of accounting — it was also used to store information related to census, product amount, and food kept in state warehouses. Quipus are even mentioned as instruments the Incas used to record their traditions and history in a different way than in writing. Several chroniclers also mention the use of quipus to store historical news. However, it has not yet been discovered how this system worked. In the Tahuantinsuyo, it was specialized personnel who handled the strings. They were known as quipucamayoc and they could be in charge of the strings of an entire region or suyu. Although the tradition is being lost, the quipus continue to be used as mnemonic instruments in some indigenous villages where they are used to record the product of the crops and the animals of the communities. According to the Jesuit chronicler Bernabé Cobo, the Incas designated to certain specialists the tasks related to accounting. These specialists were called quipo camayos, in whom the Incas placed all their trust. In his study of the quipu sample VA 42527 (Museum für Völkerkunde, Berlin), Sáez-Rodríguez noted that, in order to close the accounting books of the chacras, certain numbers were ordered according to their value in the agricultural calendar, for which the khipukamayuq — the accountant entrusted with the granary — was directly in charge. === Yupanas === In the case of numerical information, the mathematical operations were previously carried out on the abacuss or yupanas. These could be made of carved stone or clay, had boxes or compartments that corresponded to the decimal units, and were counted or marked with the help of small stones or grains of corn or quinoa. Units, tens, hundreds, etc. could be indicated according to whether they were implicit in each operation. Recent research regarding the yupanas suggests that they allowed to calculate considerable numbers based on a probably non-decimal system, but based in relation to the number 40. If true, it is curious to note the coincidence between the geometric progression achieved in the yupana and the current processing systems; on the other hand, it is also contradictory that they based their accounting system on the number 40. If the investigations continue and this fact is confirmed, it would be necessary to compare its use with the decimal system, which according to the historical tradition and previous investigations, was the one used by the Incas. In October 2010, Peruvian researcher Andrés Chirinos with the support of the Spanish Agency for International Development Cooperation (in Spanish, Agencia Española de Cooperación Internacional para el Desarrollo, AECID), reviewed drawings and ancient descriptions of the indigenous chronicler Guaman Poma de Ayala and finally deciphered the riddle of the yupana — that he calls "pre-Hispanic calculator" — as being capable of adding, subtracting, multiplying, and dividing. This made him hopeful to finally discover how the quipus worked as well. == Units of measurement == There were different units of measurement for magnitudes such as length and volume in pre-Hispanic times. The Andean peoples, as in many other places in the world, took parts of the human body as a reference to establish their units of measurement. There was not a single system of units of obligatory and uniform use throughout the Andean world. Many documents and chronicles have recorded different systems of local origin that remained in use until the 16th century. === Length === Among the units of length measurement, there was the rikra (fathom), which is the distance measured between a man's thumbs with arms extended horizontally. The kukuchu tupu (kukush tupu) was equivalent to the Spanish codo (cubit) and was the distance measured from the elbow to the end of the fingers of the hand. There was also the capa (span), and the smallest was the yuku or jeme, which was the length between the index finger and the thumb, separating one from the other as much as possible. The distance between two villages would have been evaluated by the number of chasquis required to carry an errand from one village to the other. They would have used direct proportionality between the circumference of a sheepfold and the number of chacra partitions. === Surface === The tupu was the unit of measurement of surface area. In general terms it was defined as the plot of land required for the maintenance of a married couple without children. Every hatun runa or "common man" received a plot of land upon marriage and its production had to satisfy the basic needs of food and trade of the spouses. It did not correspond to an exact measurement, since its dimensions varied according to the conditions of each land and from one ethnic group to another. The quality of the soil was taken into consideration and the necessary rest time was calculated accordingly, which had to be considered after a certain number of agricultural campaigns. After that time, the couple could claim a new tupu from their curaca. === Capacity === Among the units of measurement of capacity there is the pokcha, which was equivalent to half a fanega or 27.7 liters. Some crops such as corn were measured in containers; liquids were measured in a variety of pitchers and jars. There were boxes of a variety of cántaros and tinajas, and straw or reed boxes in which objects were kept. These boxes were also used in warehouses to store delicate or exquisite products, such as dried fruits. Coca leaves were measured in runcu or large baskets. Other baskets were known as ysanga. Among these measures of capacity there is the poctoy or purash (almozada), which is equivalent to the portion of grains or flour that can be kept in the concavity formed with the hands together. The ancient inhabitants of the Andes knew the scales of saucers and nets as well as the huipe, an instrument similar to steelyards. Apparently, its presence is associated with the works of jewelry and metallurgy, trades in which it is necessary to know the exact weights to use the right proportions of the alloys. === Volume === Especially the volume of their colcas (trojas) and their tambos (state warehouses, located in key points of the Qhapaq Ñan). They used the runqu (rongos: bales), portable containers or ishanka (baskets) or the capacity of a chacra. They would have handled the proportionality of the volumes of prisms with respect to their heights — without varying the bases. === Time === To measure time, they used the day (workday), which could include a morning, even an afternoon. Time was also useful, indirectly, to appreciate the distance between two cities; for example, 20 days from Cajamarca to Cusco was the accepted time measurement. Months, years, and the phases of the moon — much consulted for the tasks of sowing, aporques and harvests and in navigation — were also measured in days. == See also == Inca Empire History of the Incas History of Peru Mathematics == Notes == == References == == Bibliography == Espinoza Soriano, Waldemar (2003). Los Incas, economía, sociedad y estado en la era del Tahuantinsuyo (in Spanish). Lima: Editorial Sol 90. ISBN 9972-891-79-8. Muxica Editores (2001). Culturas Prehispánicas (in Spanish). Muxica Editores. ISBN 9972-617-10-6. |
Wikipedia:Mathematika#0 | Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allows machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimization, plotting functions and various types of data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other programming languages. It was conceived by Stephen Wolfram, and is developed by Wolfram Research of Champaign, Illinois. The Wolfram Language is the programming language used in Mathematica. Mathematica 1.0 was released on June 23, 1988 in Champaign, Illinois and Santa Clara, California. Mathematica's Wolfram Language is fundamentally based on Lisp; for example, the Mathematica command Most is identically equal to the Lisp command butlast. There is a substantial literature on the development of computer algebra systems (CAS). == Notebook interface == Mathematica is split into two parts: the kernel and the front end. The kernel interprets expressions (Wolfram Language code) and returns result expressions, which can then be displayed by the front end. The original front end, designed by Theodore Gray in 1988, consists of a notebook interface and allows the creation and editing of notebook documents that can contain code, plaintext, images, and graphics. Code development is also supported through support in a range of standard integrated development environment (IDE) including Eclipse, IntelliJ IDEA, Atom, Vim, Visual Studio Code and Git. The Mathematica Kernel also includes a command line front end. Other interfaces include JMath, based on GNU Readline and WolframScript which runs self-contained Mathematica programs (with arguments) from the UNIX command line. == High-performance computing == Capabilities for high-performance computing were extended with the introduction of packed arrays in version 4 (1999) and sparse matrices (version 5, 2003), and by adopting the GNU Multiple Precision Arithmetic Library to evaluate high-precision arithmetic. Version 5.2 (2005) added automatic multi-threading when computations are performed on multi-core computers. This release included CPU-specific optimized libraries. In addition Mathematica is supported by third party specialist acceleration hardware such as ClearSpeed. In 2002, gridMathematica was introduced to allow user level parallel programming on heterogeneous clusters and multiprocessor systems and in 2008 parallel computing technology was included in all Mathematica licenses including support for grid technology such as Windows HPC Server 2008, Microsoft Compute Cluster Server and Sun Grid. Support for CUDA and OpenCL GPU hardware was added in 2010. == Extensions == As of Version 14, there are 6,602 built-in functions and symbols in the Wolfram Language. Stephen Wolfram announced the launch of the Wolfram Function Repository in June 2019 as a way for the public Wolfram community to contribute functionality to the Wolfram Language. At the time of Stephen Wolfram's release announcement for Mathematica 13, there were 2,259 functions contributed as Resource Functions. In addition to the Wolfram Function Repository, there is a Wolfram Data Repository with computable data and the Wolfram Neural Net Repository for machine learning. Wolfram Mathematica is the basis of the Combinatorica package, which adds discrete mathematics functionality in combinatorics and graph theory to the program. == Connections to other applications, programming languages, and services == Communication with other applications can be done using a protocol called Wolfram Symbolic Transfer Protocol (WSTP). It allows communication between the Wolfram Mathematica kernel and the front end and provides a general interface between the kernel and other applications. Wolfram Research freely distributes a developer kit for linking applications written in the programming language C to the Mathematica kernel through WSTP using J/Link., a Java program that can ask Mathematica to perform computations. Similar functionality is achieved with .NET /Link, but with .NET programs instead of Java programs. Other languages that connect to Mathematica include Haskell, AppleScript, Racket, Visual Basic, Python, and Clojure. Mathematica supports the generation and execution of Modelica models for systems modeling and connects with Wolfram System Modeler. Links are also available to many third-party software packages and APIs. Mathematica can also capture real-time data from a variety of sources and can read and write to public blockchains (Bitcoin, Ethereum, and ARK). It supports import and export of over 220 data, image, video, sound, computer-aided design (CAD), geographic information systems (GIS), document, and biomedical formats. In 2019, support was added for compiling Wolfram Language code to LLVM. Version 12.3 of the Wolfram Language added support for Arduino. == Computable data == Mathematica is also integrated with Wolfram Alpha, an online answer engine that provides additional data, some of which is kept updated in real time, for users who use Mathematica with an internet connection. Some of the data sets include astronomical, chemical, geopolitical, language, biomedical, airplane, and weather data, in addition to mathematical data (such as knots and polyhedra). == Reception == BYTE in 1989 listed Mathematica as among the "Distinction" winners of the BYTE Awards, stating that it "is another breakthrough Macintosh application ... it could enable you to absorb the algebra and calculus that seemed impossible to comprehend from a textbook". Mathematica has been criticized for being closed source. Wolfram Research claims keeping Mathematica closed source is central to its business model and the continuity of the software. == See also == == References == == External links == Official website Mathematica Documentation Center A little bit of Mathematica history documenting the growth of code base and number of functions over time |
Wikipedia:Mathethon#0 | A Mathethon is a computational mathematics competition that is primarily focused on computer-based math in contrast to math competitions that use scientific calculators or handwritten only. Mathethons are analogous to hackathons for computer programming competitions. They can very in academic difficulty from elementary competitions, middle school, high school, and college level mathematics. They can be held in person individually, as a group, or hosted virtually online. == Competition categories == Algorithmic problem solving Biomathematics Computational informatics Computational physics Computational mechanics Computational science Computational statistics Cryptography Data mining Discrete mathematics Machine learning - deep learning, mathematics of artificial neural networks. Mathematical modeling - computational model, computer simulation. Numerical analysis Mathematical chemistry Mathematics of paper folding - origami Mathematical optimization Mathematical visualization - computational geometry, geometric modeling, mesh generation, mathematical simulation. Procedural generation - fractal geometry, fractal art, fractal analysis, mathematical sculpture, fourier series. symbolic computational algebra == See also == Comparison of TeX editors Computational education Computer-assisted proof GPU workstations and accelerated computing List of mathematics competitions List of algorithms List of finite element software packages List of graphing software List of interactive geometry software List of mathematical art software Lists of mathematical software List of numerical-analysis software List of numerical libraries Manim Mathematics and art and generative art Mathlete Numerical analysis programming languages Project Euler Parametric surface Parametric equation Recreational mathematics == References == |
Wikipedia:Mathias Lerch#0 | Mathias Lerch or Matyáš Lerch (Czech: [ˈlɛrx]; 20 February 1860, Milínov – 3 August 1922, Sušice) was a Czech mathematician who published about 250 papers, largely on mathematical analysis and number theory. He studied in Prague (Czech Technical University) and Berlin; subsequently held teaching positions at the University of Fribourg in Switzerland, the Brno University of Technology in Brno, and finally at then newly founded (1920) Masaryk University in Brno where he became its first mathematics professor. In 1900, he was awarded the Grand Prize of the French Academy of Sciences for his number-theoretic work. The Lerch zeta function is named after him, as is the Appell–Lerch sum. His doctoral students include Michel Plancherel and Otakar Borůvka. == References == == External links == Mathias Lerch at the Mathematics Genealogy Project |
Wikipedia:Mathieu Lewin#0 | Mathieu Lewin (born 14 November 1977 in Senlis, Oise, France) is a French mathematician and mathematical physicist who deals with partial differential equations, mathematical quantum field theory, and mathematics of quantum mechanical many-body systems. == Biography == Lewin studied mathematics at the École normale supérieure de Cachan, receiving his master's degree in 2000. He then received his PhD in 2004 at the Paris Dauphine University (Dauphine-Paris) PhD under the direction of Éric Séré. His dissertation was titled Some Nonlinear Models in quantum mechanics. From 2004 to 2005 he was a postdoctoral fellow at the University of Copenhagen under Jan Philip Solovej. From 2005, he conducted research for the Centre national de la recherche scientifique (CNRS) at the University of Cergy-Pontoise, then at the Paris-Dauphine university. In July 2012, he was awarded an EMS Prize "for his ground breaking work in rigorous aspects of quantum chemistry, mean field approximations to relativistic quantum field theory and statistical mechanics". == Works == His works concern the mathematical properties of matter at the microscopic scale, and they are mostly based on quantum mechanics. He uses tools from the calculus of variations, nonlinear functional analysis, partial differential equations, and spectral theory. For instance, he studied several nonlinear models for atoms and molecules (e.g. the Multi-configurational self-consistent field and Hartree–Fock methods), or for infinite quantum systems (e.g. in quantum field theory and condensed matter). == Selection of papers == Lewin, Mathieu (2004). "Solutions of the Multiconfiguration Equations in Quantum Chemistry". Archive for Rational Mechanics and Analysis. 171 (1). Springer: 83–114. Bibcode:2004ArRMA.171...83L. doi:10.1007/s00205-003-0281-6. S2CID 16366393. Hainzl, Christian; Lewin, Mathieu; Solovej, Jan Philip (2007). "The mean-field approximation in quantum electrodynamics: the no-photon case". Communications on Pure and Applied Mathematics. 60 (4). Wiley: 150402. arXiv:math-ph/0503075. doi:10.1002/cpa.20145. S2CID 14193609. Esteban, Maria J.; Lewin, Mathieu; Séré, Eric (2008). "Variational methods in relativistic quantum mechanics". Bulletin of the American Mathematical Society. 45 (4). American Mathematical Society (AMS): 535. arXiv:0706.3309. doi:10.1090/s0273-0979-08-01212-3. Lenzmann, Enno; Lewin, Mathieu (2010). "Minimizers for the Hartree-Fock-Bogoliubov theory of neutron stars and white dwarfs". Duke Mathematical Journal. 152 (2). Duke University Press: 257–315. arXiv:0809.2560. doi:10.1215/00127094-2010-013. S2CID 15321236. Lewin, Mathieu (2011). "Geometric methods for nonlinear many-body quantum systems". Journal of Functional Analysis. 260 (12). Elsevier: 3535–3595. arXiv:1009.2836. doi:10.1016/j.jfa.2010.11.017. Frank, Rupert L.; Lewin, Mathieu; Lieb, Elliott H.; Seiringer, Robert (2011). "Energy Cost to Make a Hole in the Fermi Sea". Physical Review Letters. 106 (15). American Physical Society (APS): 150402. arXiv:1102.1414. Bibcode:2011PhRvL.106o0402F. doi:10.1103/physrevlett.106.150402. PMID 21568533. Lewin, Mathieu; Nam, Phan Thành; Solovej, Jan Philip; Serfaty, Sylvia (2014). "Bogoliubov spectrum of interacting Bose gases". Communications on Pure and Applied Mathematics. 68 (3). Wiley: 413–471. arXiv:1211.2778. doi:10.1002/cpa.21519. S2CID 53480317. Lewin, Mathieu; Nam, Phan Thành; Rougerie, Nicolas (2014). "Derivation of Hartreeʼs theory for generic mean-field Bose systems". Advances in Mathematics. 254. Elsevier: 570–621. arXiv:1303.0981. doi:10.1016/j.aim.2013.12.010. Frank, Rupert L.; Lewin, Mathieu; Lieb, Elliott H.; Seiringer, Robert (2014). "Strichartz inequality for orthonormal functions". Journal of the European Mathematical Society. 16 (7). European Mathematical Society (EMS): 1507–1526. arXiv:1306.1309. doi:10.4171/JEMS/467. Lewin, Mathieu; Lieb, Elliott H.; Seiringer, Robert (2018). "Statistical mechanics of the Uniform Electron Gas". Journal de l'École polytechnique — Mathématiques. 5. Ecole Polytechnique: 79–116. arXiv:1705.10676. doi:10.5802/jep.64. Lewin, Mathieu; Nam, Phan Thành; Rougerie, Nicolas (2021). "Classical field theory limit of many-body quantum Gibbs states in 2D and 3D". Inventiones Mathematicae. 224 (2). Springer: 315–444. Bibcode:2021InMat.224..315L. doi:10.1007/s00222-020-01010-4. S2CID 253745623. == References == Web page |
Wikipedia:Maths Mansion#0 | Maths Mansion was a British educational television series for school Years 4 to 6 (nine to eleven year olds) that ran from 19 September 2001 to 26 March 2003. Produced by Channel 4 by Open Mind, It follows the adventures of "Bad Man" taking kids to his mansion, Maths Mansion. There, the kids learn and are tested on maths every week; if they pass the quiz, they get a "Maths Card". The kids are not allowed to leave the mansion until they get enough Maths Cards. They do not always pass the test, and this is shown in various episodes, one of them being Angleman!. Frequently interrupting each programme is another programme, about "Sad Man", who seems to be quite happy. He demonstrates maths with songs, puppets, and games. Sad Man has a puppet called "Decimole", as for him being a mole. Decimole is known for being very greedy around food and attacking people. There were forty episodes in four seasons. Each episode is about ten minutes long and comes with a teacher's guide and activity book and three activity sheets of differing levels for kids to use in class. == Characters == The main characters of Maths Mansion (other than the several kids in each episode) are Bad Man and Sad Man. Bad Man is the game show host who traps the kids and does not let them leave until they acquire Maths Cards that are earned in his game show. One of Bad Man's catchphrases, "No Leaving Without Learning," sums up his attitude towards the kids and his role in the show. Sad Man, a dated, uncharismatic, leather elbow patch wearing alter ego of Bad Man from the seventies, regularly interrupts Bad Man's programming with informational broadcasts that help the kids learn how to leave the mansion and teaches the lesson to the kids watching the show. Sad Man brings with him several minor characters such as Decimole and Snorter the Pig, a collection of puppets. Sad Man also occasionally has other alter egos himself such as Angleman, who is the namesake for one episode 5 of season 3. Other than Bad Man, Sad Man, and the kids, other characters include: Decimole Miss Sniff Angleman Mr. Girhalf Third Bird Twelfth Elf Not So Great Big Hen Snorter Thick Stick == Episodes == === Season 1 === Season 1 is an introduction to the number system and general mathematics. === Season 2 === Season 2 is focused on doing simple mathematical calculations. === Season 3 === Season 3 is focused on numbers and the number system, including natural numbers, integers, and rational numbers. === Season 4 === Season 4 is focused on geometry, specifically shape, space, and problem solving using those ideas. == External links == Channel 4 Learning - Maths Mansion, an archive of the former official website. == References == Discovery Education Search Episode 35, Angleman! |
Wikipedia:Maths school#0 | A maths school is a type of specialist free school sixth form college in England which specialises in the study of mathematics. Each maths school is sponsored by a university and, frequently, also a nearby established sixth form college or multi-academy trust. All students in a maths school must follow a course of study that includes A-Levels in mathematics and further mathematics. Maths schools receive additional funding from central government, above what a standard sixth form college would receive, with the aim of providing an enriched curriculum and student experience, so that students are better prepared for studies in mathematics or related subjects at competitive universities, or for careers requiring high levels of mathematical skill. Maths schools are selective and all students seeking to apply must have, at minimum, a grade 8 in GCSE mathematics. Students must also sit an entry exam before being admitted. == Features of maths schools == Maths free schools are for 16 to 19 year pupils who have a great aptitude for maths. As set out in the government’s Industrial Strategy, maths schools help to encourage highly skilled graduates in sectors that depend on science, technology, engineering and maths (STEM) skills. The aim of maths schools is to prepare the most mathematically able pupils to succeed in mathematics-related disciplines at highly selective maths universities and pursue mathematically intensive careers. Maths schools can also be centres of excellence in raising attainment, supporting and influencing the teaching of mathematics in their surrounding area, and are central to their associated universities’ widening participation commitments. They are free schools, however they are unique in multiple ways. Free schools are legally barred from partaking in academic selection, however all maths schools are selective and therefore exempt from this rule. They are not required to follow both the School Admissions Code and the School Admission Appeal Code. Maths schools are, in most cases, housed in old repurposed buildings that have undergone refurbishment and remodelling. Every maths school is run by an academy trust, sponsored by a university and, sometimes additionally, an existing local sixth form college or multi-academy trust. Each year, they receive an additional £350,000 of funding from central government. The curricula of specialist maths schools are provided through partnerships with sponsor universities. All students at maths schools must study A-levels in mathematics and further mathematics and they usually study physics and/or computer science in addition. They are exclusively for students aged 16 to 19, whereas normal free schools and other academies can serve students from primary education onwards. Maths schools are required, as part of their core business, to deliver significant outreach work – programmes that help establish maths schools as centres of excellence. Outreach plans are developed in collaboration with local schools and colleges and are integral to the widening commitments of a maths school's sponsor university. They prioritise disadvantaged students, primarily girls, and raise awareness of the mathematics curriculum that will be offered by the school to potential students, preparing them for the study of advanced mathematics. Outreach programmes are complemented by maths hubs, regional leadership networks for mathematical improvement. == History == Maths schools were conceptualised by Dominic Cummings during his time as adviser to Education Secretary Michael Gove in 2010. They were largely inspired by the Russian institutions of the same name, established by renowned mathematician Andrey Kolmogorov. They were announced by the Cameron–Clegg coalition a year later, with the aim of 12 being established over a three-year period. The first two maths schools, the King's College London Mathematics School and Exeter Mathematics School, opened in 2014. In January 2017, Prime Minister Theresa May announced her intention for every British city to have a maths schools. A budget of £170 million was allocated for this purpose. In November 2017, Chancellor of the Exchequer Philip Hammond granted an annual fund of £350,000 to every maths school. In March 2017 the Minister for Schools, Nick Gibb, promoted maths schools, asking multiple universities to establish them. In September 2019, Education Secretary Gavin Williamson announced a plan to establish nine more specialist maths schools (for a total of eleven), with four already in development in Cambridge, Lancaster, Liverpool and Surrey. At the 2019 Conservative Party Conference it was announced that the eleven maths school would consist of at least one in every region of England and the establishment of one in Durham was confirmed. These plans were reinforced in March 2020 when Chancellor Rishi Sunak released that year's budget. The budget dedicated £7 million to the establishment of these maths schools. Further developments in 2020 included the announcement of a second maths school in London in July, the opening of the third maths school, University of Liverpool Maths School, in September, and the announcement of a maths school in Leeds in November. More maths schools are opening from 2022. == List of current and approved future maths schools == == See also == University technical college Studio school Academy Free school == References == This article incorporates text published under the British Open Government Licence: "Opening a 16 to 19 specialist maths school" (PDF). assets.publishing.service.gov.uk. Retrieved 26 January 2022. |
Wikipedia:Mathspy#0 | Mathspy is a 1988 BBC Maths Educational programme. == Episodes == Needle and Thread. More Waste, Less Speed. Play Your Cards. Solid Clues. To Make the Pattern Fit. 1 Across, 1 Down. F2 to B4. Locks and Box. Seven Times Able. The Fourth Term. Final challenge. == Cast == == Notes == |
Wikipedia:Mathukumalli V. Subbarao#0 | Mathukumalli (Matukumalli) Venkata Subbarao (May 4, 1921 – February 15, 2006) was an Indo-Canadian mathematician, specialising in number theory. He was a long-time resident of Edmonton, Alberta, Canada. Subbarao was born in the small village of Yazali, Guntur, Andhra Pradesh, India. He received his master's degree from Presidency College, Madras in 1941. He went on to complete a doctorate in functional analysis, advised by Ramaswamy S. Vaidyanathaswamy. He worked at Presidency College, Madras, Sri Venkateswara University, and the University of Missouri, before moving in 1963 to the University of Alberta, where he spent the rest of his professional career. In the 1960s Subbarao began to study the congruence properties of the partition function, p(n), which became one of his favourite problems. For example, he conjectured that if A and B are integers with 0 ≤ B < A, there are infinitely many n for which p(An+B) is even and infinitely many n for which p(An+B) is odd. Ken Ono showed that the even case is always true and that if there is one number n such that p(An+B) is odd, then there are infinitely many such numbers n. The odd case was finally settled by Silviu Radu. A more general variant of the conjecture was formulated by Morris Newman predicting that for any given r and m, there are infinitely many n such that p(n)= r(mod m). At the end of his life, Subbarao co-authored a book on partition theory with A.K. Agarwal and Padmavathamma. Partition theory is ubiquitous in mathematics with connections to the representation theory of the symmetric group and the general linear group, modular forms, and physics. Thus, Subbarao's conjectures, though seemingly simple, will generate fundamental research activity for years to come. He also researched special classes of divisors and the corresponding analogues of divisor functions and perfect numbers, such as those arising from the exponential divisors ("e-divisors") which he defined. Many other mathematicians have published papers building on his work in these subjects. A prolific collaborator, Subbarao had more than 40 joint authors (including Paul Erdős, giving him Erdős number 1). He continued producing mathematics papers into the final years of his life. He died in Edmonton at the age of 84. Subbarao was the father of Prof. Mathukumalli Vidyasagar. == Selected publications == Straus, E. G.; Subbarao, M. V. (1974). "On exponential divisors". Duke Mathematical Journal. 41 (2): 465–471. doi:10.1215/S0012-7094-74-04152-0. == References == == External links == Mathukumalli V. Subbarao at the Mathematics Genealogy Project |
Wikipedia:Matila Ghyka#0 | Prince Matila Costiescu Ghyka (Romanian pronunciation: [maˈtila ˈɟika]; born Matila Costiescu; 13 September 1881 – 14 July 1965), was a Romanian naval officer, novelist, mathematician, historian, philosopher, academic and diplomat. He did not return to Romania after World War II, and was one of the most significant members of the Romanian diaspora. His first name is sometimes written as Matyla. == Life == Ghyka was born in Iași, the former capital of Moldavia, of the Ghica family of boyars. His mother was Maria Ghyljia and his father was Matila Costiecu, a Wallachian officer. Maria's half-brother was Grigoire Ghyka, who adopted Matila when he was a teenager so that he would acquire the title of Prince as Matila was the great-grandson of Grigore Alexandru Ghica, last reigning Prince of Moldavia before the union of the Danubian Principalities. However, much of Ghyka's inherited capital was via his grandmother's Balş family. As a boy he lived in France studying first at the Salesian Order school in Paris, then a Jesuit college in Jersey where he became interested in mathematics. In his early teens he was a cadet at the French Naval Academy in Brest, and of the last generation in the old sailing ship Borda. He became a French Navy midshipman and made a cruise in a frigate to the Caribbean. In later years he attended the École supérieure d'électricité de Paris, and finally took a doctorate in law at the Université libre de Bruxelles. Ghyka entered the Romanian Navy as a junior officer, serving mainly on the Danube. He was also involved in taking newly constructed river gunboats from the Thames Iron Works to Romania via European waterways. During the First World War he was Romanian Navy liaison officer on the Russian cruiser Rostislav, acting as a shore bombardment director along the Black Sea coast. He had joined the diplomatic service in 1909, being stationed at the Romanian Legations in Rome, Berlin, London, Madrid, Paris, Vienna, Stockholm (as Minister Plenipotentiary) and twice again in London between 1936-1938 and between 1939 and 1940. In 1918, at the Brompton Oratory, he married Eileen O'Conor (1897-1963), daughter of the late Sir Nicholas Roderick O'Conor (d. 1908), the former British Ambassador to Istanbul and Saint Petersburg, and Minna Margaret Hope-Scott. Eileen belonged to a junior branch of the Ó Conchobhair Donn, who had anciently been Kings of Connacht. During his first diplomatic assignments in London and Paris, Prince Ghyka was introduced by Paul Morand and Prince Antoine Bibesco to the English and French literary circles. He became a friend of Marcel Proust and a "piéton de Paris" with the poet Léon-Paul Fargue. A frequent visitor of Natalie Clifford Barney's literary salon, he also met most of the American "exiled" writers of the 1920s, but his chief interest was always the synthesis of high mathematics and poetry. After World War II, Ghyka fled Communist Romania, and was visiting professor of aesthetics in the United States, at the University of Southern California and at the Mary Washington College, Virginia. Ghyka published his memoirs in two volumes in French, Escales de ma jeunesse (1955) and Heureux qui, comme Ulysse… (1956) under the collective title Couleur du monde; a shortened and revised version appeared in English in 1961 as The World Mine Oyster. Ghyka died in London and was survived by his son, Prince Roderick Ghyka, and daughter, Princess Maureen Ghyka. He was predeceased by his wife Eileen, who died on 10 February 1963. Both Prince Matila and Princess Eileen are buried in Gunnersbury Cemetery, London. Their funeral monument was restored in 2010 by art historian Dr Radu Varia. == Mathematical aesthetics == In around 1900, Ghyka spent a year studying engineering at the École supérieure d'électricité de Paris, Whilst there he developed his own mathematical ideas on the relationship between thermodynamics and living matter, partly under the influence of Gustave Le Bon. He returned to mathematics around 1920 when Albert Einstein's theories were published, and over the next few years developed ideas on the mathematics of form which he published in 1927 as Esthétique des proportions dans la nature et dans les arts, and revised and expanded in his two volume Le nombre d'or. Rites et rythmes pythagoriciens dans le development de la civilisation occidentale in 1931. Ghyka developed a personal philosophy in which all living things were endowed with an energy and functioned with a rhythm related to that of the golden ratio. Further work was published in French as Essai sur le rythme (1938), Tour d'horizon philosophique (1946) and Philosophie et Mystique du nombre (1952), and in English as The Geometry of Art and Life (1946). Around 1945 Ghyka was offered a visiting Professorship at the University of Southern California in Los Angeles because the President of the university had read Esthétique des proportions, and this was followed in 1947 by a job in the Art Department of Mary Washington College, where he taught his personal aesthetic theories for three years. In 1950 he returned to his wife at their family home in Dublin and his Practical Handbook of Geometry and Design was published in 1952. Salvador Dalí possessed two copy of Ghyka's books which was read by theatre director Peter Brook, who was profoundly influenced by Ghyka's ideas on the mathematical relationships between classical art and the human body. The only monograph on his life and work appeared in Romanian. == Works == Contes marécageux; unpublished juvenilia c1900. Esthétique des proportions dans la nature et dans les arts (1927) (printed in Italian, Russian, Spanish) Le nombre d'or. Rites et rythmes pythagoriciens dans le development de la civilisation occidentale (1931) which ran into many editions and was prefaced by his friend and admirer Paul Valéry (translated into Italian, Czech, Spanish, Polish, English, Romanian) Pluie d'étoiles (1933) (English as Again One Day, 1936) - the only novel Ghyka wrote, printed also in Romanian Essai sur le rythme (1938) Sortilèges du verbe (1949), prefaced by Léon-Paul Fargue A Documented Chronology of Roumanian History from Pre-historic Times to the Present Day (1941), printed also in Romanian The Geometry of Art and Life (1946) (translated into Chinese - 2014 and Japanese - 2021) Tour d'horizon philosophique (1946) A Practical Handbook of Geometry and Design (1952) Philosophie et Mystique du nombre (1952) (translated into Serbian, Spanish, Romanian) Couleur du monde (1: Escales de ma jeunesse (1955), 2: Heureux qui comme Ulysse (1956)) (translated into Romanian) The World Mine Oyster. London, Heinemann, 1961 (English version of "Couleur du monde") == Further reading == Ghyka has been the subject of recent publications in German and Romanian. == References == |
Wikipedia:Matilde Lalín#0 | Matilde Noemí Lalín is an Argentine-Canadian mathematician specializing in number theory and known for her work on L-functions, Mahler measure, and their connections. She is a professor of mathematics at the Université de Montréal. == Education and career == Lalín is originally from Buenos Aires, and is a dual citizen of Argentina and Canada. As a high school student, she represented Argentina twice in the International Mathematical Olympiad, in 1993 and 1995, earning a silver medal in 1995. She earned a licenciatura in 1999 from the University of Buenos Aires. After starting graduate study at Princeton University and spending a term as a visiting student at Harvard University, she completed her doctorate in 2005 at the University of Texas at Austin. Her dissertation, Some Relations of Mahler Measure with Hyperbolic Volumes and Special Values of L-Functions, was supervised there by Fernando Rodriguez-Villegas. She became a postdoctoral researcher at the Institute for Advanced Study, Mathematical Sciences Research Institute, Institut des Hautes Études Scientifiques, Max Planck Institute for Mathematics, and Pacific Institute for the Mathematical Sciences, before obtaining a tenure-track faculty position in 2007 as an assistant professor of mathematics at the University of Alberta. She moved to the Université de Montréal in 2010, earned tenure as an associate professor there in 2012, and was promoted to full professor in 2018. == Recognition == Lalín is the 2022 winner of the Krieger–Nelson Prize of the Canadian Mathematical Society, "for her outstanding contributions to research in Number Theory and related areas". She was named to the 2023 class of Fellows of the American Mathematical Society, "for contributions to number theory, including the study of L-functions, and for service to the mathematical community". In 2024, she will become a fellow of the Association for Women in Mathematics, "For her ongoing contributions to the AWM, most notably her leadership role in the Women in Numbers Network and considerable contributions to its growth; her service to the International Mathematics Union Committee for Women; and for her ardent efforts towards making conferences more welcoming and accessible for researchers by actively advocating for childcare resources." == References == == External links == Home page Matilde Lalín publications indexed by Google Scholar |
Wikipedia:Matilde Marcolli#0 | Matilde Marcolli is an Italian and American mathematical physicist. She has conducted research work in areas of mathematics and theoretical physics; obtained the Heinz Maier-Leibnitz-Preis of the Deutsche Forschungsgemeinschaft, and the Sofia Kovalevskaya Award of the Alexander von Humboldt Foundation. Marcolli has authored and edited numerous books in the field. She is currently the Robert F. Christy Professor of Mathematics and Computing and Mathematical Sciences at the California Institute of Technology. == Career == Marcolli obtained her Laurea in Physics in 1993 summa cum laude from the University of Milan under the supervision of Renzo Piccinini, with a thesis on Classes of self equivalences of fibre bundles. She moved to the USA in 1994, where she obtained a master's degree (1994) and a PhD (1997) in Mathematics from the University of Chicago, under the supervision of Melvin Rothenberg, with a thesis on Three dimensional aspects of Seiberg-Witten Gauge Theory. Between 1997 and 2000 she worked at the Massachusetts Institute of Technology (MIT) as a C.L.E. Moore instructor in the Department of Mathematics. Between 2000 and 2010 she held a C3 position (German equivalent of associate professor) at the Max Planck Institute for Mathematics in Bonn and held an associate professor position (courtesy) at Florida State University in Tallahassee. She also held an honorary professorship at the University of Bonn. From 2008 to 2017 she was a full professor of Mathematics in the Division of Physics, Mathematics and Astronomy of the California Institute of Technology. Between 2018 and 2020 she was a professor in the mathematics department of the University of Toronto and a member of the Perimeter Institute. She is currently the Robert F. Christy Professor of Mathematics and Computing and Mathematical Sciences at the California Institute of Technology. She held visiting positions at the Tata Institute of Fundamental Research in Mumbai, the Kavli Institute for Theoretical Physics in Santa Barbara, the Mittag-Leffler Institute in Stockholm, the Isaac Newton Institute in Cambridge, and the Mathematical Sciences Research Institute in Berkeley, California. == Research == Marcolli's research work has covered different areas of mathematics and theoretical physics: gauge theory and low-dimensional topology, algebraic-geometric structures in quantum field theory, noncommutative geometry with applications to number theory and to physics models, especially related to particle physics, quantum gravity and cosmology, and to the quantum Hall effect. She also worked in linguistics. She has collaborated with several other mathematicians, physicists, and linguists, among them Yuri I. Manin, Alain Connes, Michael Atiyah, Roger Penrose, Noam Chomsky. Twenty six graduate students obtained their PhD under her supervision between 2006 and 2022. == Honors and awards == In 2001 she obtained the Heinz Maier-Leibnitz-Preis of the Deutsche Forschungsgemeinschaft (DFG) and in 2002 the Sofia Kovalevskaya Award of the Alexander von Humboldt Foundation. She was a plenary speaker in the 2008 European Congress of Mathematics in Amsterdam (with a talk on Renormalization, Galois symmetries and motives) and an invited speaker of the 2010 International Congress of Mathematicians in Hyderabad (with a talk on Noncommutative Geometry and Arithmetic). == Books authored == Marcolli, Matilde (1999). Seiberg-Witten gauge theory. New Delhi: Hindustan Book Agency. ISBN 9788185931227. OCLC 468606720. Marcolli, Matilde; Manin, Jurij Ivanovič (2005). Arithmetic Noncommutative Geometry. American Mathematical Society. ISBN 9780821838334. Connes, Alain; Marcolli, Matilde (2008). Noncommutative Geometry, Quantum Fields and Motives. American Mathematical Society. ISBN 9780821874783. Marcolli, Matilde (2010). Feynman Motives. Singapore: World Scientific. ISBN 9789814271202. Matilde, Marcolli (2017). Noncommutative Cosmology. Singapore: World Scientific. ISBN 9789813202863. Marcolli, Matilde (2020). Lumen Naturae: Visions of the Abstract in Art and Mathematics. MIT Press. ISBN 9780262043908. == Books edited == Hertling, Claus; Marcolli, Matilde, eds. (2003). Frobenius Manifolds : Quantum Cohomology and Singularities. Vieweg-Teubner Verlag. ISBN 978-3-322-80238-5. Consani, Caterina; Marcolli, Matilde, eds. (2006). Noncommutative geometry and number theory : where arithmetic meets geometry and physics (1st ed.). Vieweg-Teubner Verlag. ISBN 978-3-8348-0170-8. Albeverio, Sergio; Marcolli, Matilde; Paycha, Sylvie; Plazas, Jorge (2008). Traces in Number Theory, Geometry and Quantum Fields. Vieweg+Teubner Verlag. ISBN 9783834803719. Khalkhali, Masoud; Marcolli, Matilde (2008). An Invitation to Noncommutative Geometry. World Scientific. ISBN 9789812706164. Ceyhan, Özgür; Manin, Yu I.; Marcolli, Matilde (2010). Arithmetic and Geometry Around Quantization. Springer. ISBN 9780817648312. Abbaspour, Hossein; Marcolli, Matilde; Tradler, Thomas (2010). Deformation spaces : perspectives on algebro-geometric moduli. Vieweg+Teubner. ISBN 978-3-8348-1271-1. Marcolli, Matilde; Parashar, Deepak (2011). Quantum groups and noncommutative spaces perspectives on quantum geometry: a publication of the Max-Planck-Institute for Mathematics, Bonn. Vieweg + Teubner. ISBN 9783834814425. OCLC 793144094. == References == Citations Sources == External links == Matilde Marcolli at the Mathematics Genealogy Project Homepage at Caltech Homepage at FSU Papers on the eprints ArXiv |
Wikipedia:Matlis duality#0 | In algebra, Matlis duality is a duality between Artinian and Noetherian modules over a complete Noetherian local ring. In the special case when the local ring has a field mapping to the residue field it is closely related to earlier work by Francis Sowerby Macaulay on polynomial rings and is sometimes called Macaulay duality, and the general case was introduced by Matlis (1958). == Statement == Suppose that R is a Noetherian complete local ring with residue field k, and choose E to be an injective hull of k (sometimes called a Matlis module). The dual DR(M) of a module M is defined to be HomR(M,E). Then Matlis duality states that the duality functor DR gives an anti-equivalence between the categories of Artinian and Noetherian R-modules. In particular the duality functor gives an anti-equivalence from the category of finite-length modules to itself. == Examples == Suppose that the Noetherian complete local ring R has a subfield k that maps onto a subfield of finite index of its residue field R/m. Then the Matlis dual of any R-module is just its dual as a topological vector space over k, if the module is given its m-adic topology. In particular the dual of R as a topological vector space over k is a Matlis module. This case is closely related to work of Macaulay on graded polynomial rings and is sometimes called Macaulay duality. If R is a discrete valuation ring with quotient field K then the Matlis module is K/R. In the special case when R is the ring of p-adic numbers, the Matlis dual of a finitely-generated module is the Pontryagin dual of it considered as a locally compact abelian group. If R is a Cohen–Macaulay local ring of dimension d with dualizing module Ω, then the Matlis module is given by the local cohomology group HdR(Ω). In particular if R is an Artinian local ring then the Matlis module is the same as the dualizing module. == Explanation using adjoint functors == Matlis duality can be conceptually explained using the language of adjoint functors and derived categories: the functor between the derived categories of R- and k-modules induced by regarding a k-module as an R-module, admits a right adjoint (derived internal Hom) D ( k ) ← D ( R ) : R Hom R ( k , − ) . {\displaystyle D(k)\gets D(R):R\operatorname {Hom} _{R}(k,-).} This right adjoint sends the injective hull E ( k ) {\displaystyle E(k)} mentioned above to k, which is a dualizing object in D ( k ) {\displaystyle D(k)} . This abstract fact then gives rise to the above-mentioned equivalence. == See also == Grothendieck local duality == References == Bruns, Winfried; Herzog, Jürgen (1993), Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, ISBN 978-0-521-41068-7, MR 1251956 Matlis, Eben (1958), "Injective modules over Noetherian rings", Pacific Journal of Mathematics, 8: 511–528, doi:10.2140/pjm.1958.8.511, ISSN 0030-8730, MR 0099360, archived from the original on 2014-05-03 |
Wikipedia:Matrix Chernoff bound#0 | In probability theory, a Chernoff bound is an exponentially decreasing upper bound on the tail of a random variable based on its moment generating function. The minimum of all such exponential bounds forms the Chernoff or Chernoff-Cramér bound, which may decay faster than exponential (e.g. sub-Gaussian). It is especially useful for sums of independent random variables, such as sums of Bernoulli random variables. The bound is commonly named after Herman Chernoff who described the method in a 1952 paper, though Chernoff himself attributed it to Herman Rubin. In 1938 Harald Cramér had published an almost identical concept now known as Cramér's theorem. It is a sharper bound than the first- or second-moment-based tail bounds such as Markov's inequality or Chebyshev's inequality, which only yield power-law bounds on tail decay. However, when applied to sums the Chernoff bound requires the random variables to be independent, a condition that is not required by either Markov's inequality or Chebyshev's inequality. The Chernoff bound is related to the Bernstein inequalities. It is also used to prove Hoeffding's inequality, Bennett's inequality, and McDiarmid's inequality. == Generic Chernoff bounds == The generic Chernoff bound for a random variable X {\displaystyle X} is attained by applying Markov's inequality to e t X {\displaystyle e^{tX}} (which is why it is sometimes called the exponential Markov or exponential moments bound). For positive t {\displaystyle t} this gives a bound on the right tail of X {\displaystyle X} in terms of its moment-generating function M ( t ) = E ( e t X ) {\displaystyle M(t)=\operatorname {E} (e^{tX})} : P ( X ≥ a ) = P ( e t X ≥ e t a ) ≤ M ( t ) e − t a ( t > 0 ) {\displaystyle \operatorname {P} \left(X\geq a\right)=\operatorname {P} \left(e^{tX}\geq e^{ta}\right)\leq M(t)e^{-ta}\qquad (t>0)} Since this bound holds for every positive t {\displaystyle t} , we may take the infimum: P ( X ≥ a ) ≤ inf t > 0 M ( t ) e − t a {\displaystyle \operatorname {P} \left(X\geq a\right)\leq \inf _{t>0}M(t)e^{-ta}} Performing the same analysis with negative t {\displaystyle t} we get a similar bound on the left tail: P ( X ≤ a ) = P ( e t X ≥ e t a ) ≤ M ( t ) e − t a ( t < 0 ) {\displaystyle \operatorname {P} \left(X\leq a\right)=\operatorname {P} \left(e^{tX}\geq e^{ta}\right)\leq M(t)e^{-ta}\qquad (t<0)} and P ( X ≤ a ) ≤ inf t < 0 M ( t ) e − t a {\displaystyle \operatorname {P} \left(X\leq a\right)\leq \inf _{t<0}M(t)e^{-ta}} The quantity M ( t ) e − t a {\displaystyle M(t)e^{-ta}} can be expressed as the expected value E ( e t X ) e − t a {\displaystyle \operatorname {E} (e^{tX})e^{-ta}} , or equivalently E ( e t ( X − a ) ) {\displaystyle \operatorname {E} (e^{t(X-a)})} . === Properties === The exponential function is convex, so by Jensen's inequality E ( e t X ) ≥ e t E ( X ) {\displaystyle \operatorname {E} (e^{tX})\geq e^{t\operatorname {E} (X)}} . It follows that the bound on the right tail is greater or equal to one when a ≤ E ( X ) {\displaystyle a\leq \operatorname {E} (X)} , and therefore trivial; similarly, the left bound is trivial for a ≥ E ( X ) {\displaystyle a\geq \operatorname {E} (X)} . We may therefore combine the two infima and define the two-sided Chernoff bound: C ( a ) = inf t M ( t ) e − t a {\displaystyle C(a)=\inf _{t}M(t)e^{-ta}} which provides an upper bound on the folded cumulative distribution function of X {\displaystyle X} (folded at the mean, not the median). The logarithm of the two-sided Chernoff bound is known as the rate function (or Cramér transform) I = − log C {\displaystyle I=-\log C} . It is equivalent to the Legendre–Fenchel transform or convex conjugate of the cumulant generating function K = log M {\displaystyle K=\log M} , defined as: I ( a ) = sup t a t − K ( t ) {\displaystyle I(a)=\sup _{t}at-K(t)} The moment generating function is log-convex, so by a property of the convex conjugate, the Chernoff bound must be log-concave. The Chernoff bound attains its maximum at the mean, C ( E ( X ) ) = 1 {\displaystyle C(\operatorname {E} (X))=1} , and is invariant under translation: C X + k ( a ) = C X ( a − k ) {\textstyle C_{X+k}(a)=C_{X}(a-k)} . The Chernoff bound is exact if and only if X {\displaystyle X} is a single concentrated mass (degenerate distribution). The bound is tight only at or beyond the extremes of a bounded random variable, where the infima are attained for infinite t {\displaystyle t} . For unbounded random variables the bound is nowhere tight, though it is asymptotically tight up to sub-exponential factors ("exponentially tight"). Individual moments can provide tighter bounds, at the cost of greater analytical complexity. In practice, the exact Chernoff bound may be unwieldy or difficult to evaluate analytically, in which case a suitable upper bound on the moment (or cumulant) generating function may be used instead (e.g. a sub-parabolic CGF giving a sub-Gaussian Chernoff bound). === Bounds from below from the MGF === Using only the moment generating function, a bound from below on the tail probabilities can be obtained by applying the Paley-Zygmund inequality to e t X {\displaystyle e^{tX}} , yielding: P ( X > a ) ≥ sup t > 0 ∧ M ( t ) ≥ e t a ( 1 − e t a M ( t ) ) 2 M ( t ) 2 M ( 2 t ) {\displaystyle \operatorname {P} \left(X>a\right)\geq \sup _{t>0\land M(t)\geq e^{ta}}\left(1-{\frac {e^{ta}}{M(t)}}\right)^{2}{\frac {M(t)^{2}}{M(2t)}}} (a bound on the left tail is obtained for negative t {\displaystyle t} ). Unlike the Chernoff bound however, this result is not exponentially tight. Theodosopoulos constructed a tight(er) MGF-based bound from below using an exponential tilting procedure. For particular distributions (such as the binomial) bounds from below of the same exponential order as the Chernoff bound are often available. == Sums of independent random variables == When X is the sum of n independent random variables X1, ..., Xn, the moment generating function of X is the product of the individual moment generating functions, giving that: and: Pr ( X ≤ a ) ≤ inf t < 0 e − t a ∏ i E [ e t X i ] {\displaystyle \Pr(X\leq a)\leq \inf _{t<0}e^{-ta}\prod _{i}\operatorname {E} \left[e^{tX_{i}}\right]} Specific Chernoff bounds are attained by calculating the moment-generating function E [ e − t ⋅ X i ] {\displaystyle \operatorname {E} \left[e^{-t\cdot X_{i}}\right]} for specific instances of the random variables X i {\displaystyle X_{i}} . When the random variables are also identically distributed (iid), the Chernoff bound for the sum reduces to a simple rescaling of the single-variable Chernoff bound. That is, the Chernoff bound for the average of n iid variables is equivalent to the nth power of the Chernoff bound on a single variable (see Cramér's theorem). == Sums of independent bounded random variables == Chernoff bounds may also be applied to general sums of independent, bounded random variables, regardless of their distribution; this is known as Hoeffding's inequality. The proof follows a similar approach to the other Chernoff bounds, but applying Hoeffding's lemma to bound the moment generating functions (see Hoeffding's inequality). Hoeffding's inequality. Suppose X1, ..., Xn are independent random variables taking values in [a,b]. Let X denote their sum and let μ = E[X] denote the sum's expected value. Then for any t > 0 {\displaystyle t>0} , Pr ( X ≤ μ − t ) < e − 2 t 2 / ( n ( b − a ) 2 ) , {\displaystyle \Pr(X\leq \mu -t)<e^{-2t^{2}/(n(b-a)^{2})},} Pr ( X ≥ μ + t ) < e − 2 t 2 / ( n ( b − a ) 2 ) . {\displaystyle \Pr(X\geq \mu +t)<e^{-2t^{2}/(n(b-a)^{2})}.} == Sums of independent Bernoulli random variables == The bounds in the following sections for Bernoulli random variables are derived by using that, for a Bernoulli random variable X i {\displaystyle X_{i}} with probability p of being equal to 1, E [ e t ⋅ X i ] = ( 1 − p ) e 0 + p e t = 1 + p ( e t − 1 ) ≤ e p ( e t − 1 ) . {\displaystyle \operatorname {E} \left[e^{t\cdot X_{i}}\right]=(1-p)e^{0}+pe^{t}=1+p(e^{t}-1)\leq e^{p(e^{t}-1)}.} One can encounter many flavors of Chernoff bounds: the original additive form (which gives a bound on the absolute error) or the more practical multiplicative form (which bounds the error relative to the mean). === Multiplicative form (relative error) === Multiplicative Chernoff bound. Suppose X1, ..., Xn are independent random variables taking values in {0, 1}. Let X denote their sum and let μ = E[X] denote the sum's expected value. Then for any δ > 0, Pr ( X ≥ ( 1 + δ ) μ ) ≤ ( e δ ( 1 + δ ) 1 + δ ) μ . {\displaystyle \Pr(X\geq (1+\delta )\mu )\leq \left({\frac {e^{\delta }}{(1+\delta )^{1+\delta }}}\right)^{\mu }.} A similar proof strategy can be used to show that for 0 < δ < 1 Pr ( X ≤ ( 1 − δ ) μ ) ≤ ( e − δ ( 1 − δ ) 1 − δ ) μ . {\displaystyle \Pr(X\leq (1-\delta )\mu )\leq \left({\frac {e^{-\delta }}{(1-\delta )^{1-\delta }}}\right)^{\mu }.} The above formula is often unwieldy in practice, so the following looser but more convenient bounds are often used, which follow from the inequality 2 δ 2 + δ ≤ log ( 1 + δ ) {\displaystyle \textstyle {\frac {2\delta }{2+\delta }}\leq \log(1+\delta )} from the list of logarithmic inequalities: Pr ( X ≥ ( 1 + δ ) μ ) ≤ e − δ 2 μ / ( 2 + δ ) , 0 ≤ δ , {\displaystyle \Pr(X\geq (1+\delta )\mu )\leq e^{-\delta ^{2}\mu /(2+\delta )},\qquad 0\leq \delta ,} Pr ( X ≤ ( 1 − δ ) μ ) ≤ e − δ 2 μ / 2 , 0 < δ < 1 , {\displaystyle \Pr(X\leq (1-\delta )\mu )\leq e^{-\delta ^{2}\mu /2},\qquad 0<\delta <1,} Pr ( | X − μ | ≥ δ μ ) ≤ 2 e − δ 2 μ / 3 , 0 < δ < 1. {\displaystyle \Pr(|X-\mu |\geq \delta \mu )\leq 2e^{-\delta ^{2}\mu /3},\qquad 0<\delta <1.} Notice that the bounds are trivial for δ = 0 {\displaystyle \delta =0} . In addition, based on the Taylor expansion for the Lambert W function, Pr ( X ≥ R ) ≤ 2 − x R , x > 0 , R ≥ ( 2 x e − 1 ) μ . {\displaystyle \Pr(X\geq R)\leq 2^{-xR},\qquad x>0,\ R\geq (2^{x}e-1)\mu .} === Additive form (absolute error) === The following theorem is due to Wassily Hoeffding and hence is called the Chernoff–Hoeffding theorem. Chernoff–Hoeffding theorem. Suppose X1, ..., Xn are i.i.d. random variables, taking values in {0, 1}. Let p = E[X1] and ε > 0. Pr ( 1 n ∑ X i ≥ p + ε ) ≤ ( ( p p + ε ) p + ε ( 1 − p 1 − p − ε ) 1 − p − ε ) n = e − D ( p + ε ∥ p ) n Pr ( 1 n ∑ X i ≤ p − ε ) ≤ ( ( p p − ε ) p − ε ( 1 − p 1 − p + ε ) 1 − p + ε ) n = e − D ( p − ε ∥ p ) n {\displaystyle {\begin{aligned}\Pr \left({\frac {1}{n}}\sum X_{i}\geq p+\varepsilon \right)\leq \left(\left({\frac {p}{p+\varepsilon }}\right)^{p+\varepsilon }{\left({\frac {1-p}{1-p-\varepsilon }}\right)}^{1-p-\varepsilon }\right)^{n}&=e^{-D(p+\varepsilon \parallel p)n}\\\Pr \left({\frac {1}{n}}\sum X_{i}\leq p-\varepsilon \right)\leq \left(\left({\frac {p}{p-\varepsilon }}\right)^{p-\varepsilon }{\left({\frac {1-p}{1-p+\varepsilon }}\right)}^{1-p+\varepsilon }\right)^{n}&=e^{-D(p-\varepsilon \parallel p)n}\end{aligned}}} where D ( x ∥ y ) = x ln x y + ( 1 − x ) ln ( 1 − x 1 − y ) {\displaystyle D(x\parallel y)=x\ln {\frac {x}{y}}+(1-x)\ln \left({\frac {1-x}{1-y}}\right)} is the Kullback–Leibler divergence between Bernoulli distributed random variables with parameters x and y respectively. If p ≥ 1/2, then D ( p + ε ∥ p ) ≥ ε 2 2 p ( 1 − p ) {\displaystyle D(p+\varepsilon \parallel p)\geq {\tfrac {\varepsilon ^{2}}{2p(1-p)}}} which means Pr ( 1 n ∑ X i > p + x ) ≤ exp ( − x 2 n 2 p ( 1 − p ) ) . {\displaystyle \Pr \left({\frac {1}{n}}\sum X_{i}>p+x\right)\leq \exp \left(-{\frac {x^{2}n}{2p(1-p)}}\right).} A simpler bound follows by relaxing the theorem using D(p + ε || p) ≥ 2ε2, which follows from the convexity of D(p + ε || p) and the fact that d 2 d ε 2 D ( p + ε ∥ p ) = 1 ( p + ε ) ( 1 − p − ε ) ≥ 4 = d 2 d ε 2 ( 2 ε 2 ) . {\displaystyle {\frac {d^{2}}{d\varepsilon ^{2}}}D(p+\varepsilon \parallel p)={\frac {1}{(p+\varepsilon )(1-p-\varepsilon )}}\geq 4={\frac {d^{2}}{d\varepsilon ^{2}}}(2\varepsilon ^{2}).} This result is a special case of Hoeffding's inequality. Sometimes, the bounds D ( ( 1 + x ) p ∥ p ) ≥ 1 4 x 2 p , − 1 2 ≤ x ≤ 1 2 , D ( x ∥ y ) ≥ 3 ( x − y ) 2 2 ( 2 y + x ) , D ( x ∥ y ) ≥ ( x − y ) 2 2 y , x ≤ y , D ( x ∥ y ) ≥ ( x − y ) 2 2 x , x ≥ y {\displaystyle {\begin{aligned}D((1+x)p\parallel p)\geq {\frac {1}{4}}x^{2}p,&&&{-{\tfrac {1}{2}}}\leq x\leq {\tfrac {1}{2}},\\[6pt]D(x\parallel y)\geq {\frac {3(x-y)^{2}}{2(2y+x)}},\\[6pt]D(x\parallel y)\geq {\frac {(x-y)^{2}}{2y}},&&&x\leq y,\\[6pt]D(x\parallel y)\geq {\frac {(x-y)^{2}}{2x}},&&&x\geq y\end{aligned}}} which are stronger for p < 1/8, are also used. == Applications == Chernoff bounds have very useful applications in set balancing and packet routing in sparse networks. The set balancing problem arises while designing statistical experiments. Typically while designing a statistical experiment, given the features of each participant in the experiment, we need to know how to divide the participants into 2 disjoint groups such that each feature is roughly as balanced as possible between the two groups. Chernoff bounds are also used to obtain tight bounds for permutation routing problems which reduce network congestion while routing packets in sparse networks. Chernoff bounds are used in computational learning theory to prove that a learning algorithm is probably approximately correct, i.e. with high probability the algorithm has small error on a sufficiently large training data set. Chernoff bounds can be effectively used to evaluate the "robustness level" of an application/algorithm by exploring its perturbation space with randomization. The use of the Chernoff bound permits one to abandon the strong—and mostly unrealistic—small perturbation hypothesis (the perturbation magnitude is small). The robustness level can be, in turn, used either to validate or reject a specific algorithmic choice, a hardware implementation or the appropriateness of a solution whose structural parameters are affected by uncertainties. A simple and common use of Chernoff bounds is for "boosting" of randomized algorithms. If one has an algorithm that outputs a guess that is the desired answer with probability p > 1/2, then one can get a higher success rate by running the algorithm n = log ( 1 / δ ) 2 p / ( p − 1 / 2 ) 2 {\displaystyle n=\log(1/\delta )2p/(p-1/2)^{2}} times and outputting a guess that is output by more than n/2 runs of the algorithm. (There cannot be more than one such guess.) Assuming that these algorithm runs are independent, the probability that more than n/2 of the guesses is correct is equal to the probability that the sum of independent Bernoulli random variables Xk that are 1 with probability p is more than n/2. This can be shown to be at least 1 − δ {\displaystyle 1-\delta } via the multiplicative Chernoff bound (Corollary 13.3 in Sinclair's class notes, μ = np).: Pr [ X > n 2 ] ≥ 1 − e − n ( p − 1 / 2 ) 2 / ( 2 p ) ≥ 1 − δ {\displaystyle \Pr \left[X>{n \over 2}\right]\geq 1-e^{-n\left(p-1/2\right)^{2}/(2p)}\geq 1-\delta } == Matrix Chernoff bound == Rudolf Ahlswede and Andreas Winter introduced a Chernoff bound for matrix-valued random variables. The following version of the inequality can be found in the work of Tropp. Let M1, ..., Mt be independent matrix valued random variables such that M i ∈ C d 1 × d 2 {\displaystyle M_{i}\in \mathbb {C} ^{d_{1}\times d_{2}}} and E [ M i ] = 0 {\displaystyle \mathbb {E} [M_{i}]=0} . Let us denote by ‖ M ‖ {\displaystyle \lVert M\rVert } the operator norm of the matrix M {\displaystyle M} . If ‖ M i ‖ ≤ γ {\displaystyle \lVert M_{i}\rVert \leq \gamma } holds almost surely for all i ∈ { 1 , … , t } {\displaystyle i\in \{1,\ldots ,t\}} , then for every ε > 0 Pr ( ‖ 1 t ∑ i = 1 t M i ‖ > ε ) ≤ ( d 1 + d 2 ) exp ( − 3 ε 2 t 8 γ 2 ) . {\displaystyle \Pr \left(\left\|{\frac {1}{t}}\sum _{i=1}^{t}M_{i}\right\|>\varepsilon \right)\leq (d_{1}+d_{2})\exp \left(-{\frac {3\varepsilon ^{2}t}{8\gamma ^{2}}}\right).} Notice that in order to conclude that the deviation from 0 is bounded by ε with high probability, we need to choose a number of samples t {\displaystyle t} proportional to the logarithm of d 1 + d 2 {\displaystyle d_{1}+d_{2}} . In general, unfortunately, a dependence on log ( min ( d 1 , d 2 ) ) {\displaystyle \log(\min(d_{1},d_{2}))} is inevitable: take for example a diagonal random sign matrix of dimension d × d {\displaystyle d\times d} . The operator norm of the sum of t independent samples is precisely the maximum deviation among d independent random walks of length t. In order to achieve a fixed bound on the maximum deviation with constant probability, it is easy to see that t should grow logarithmically with d in this scenario. The following theorem can be obtained by assuming M has low rank, in order to avoid the dependency on the dimensions. === Theorem without the dependency on the dimensions === Let 0 < ε < 1 and M be a random symmetric real matrix with ‖ E [ M ] ‖ ≤ 1 {\displaystyle \|\operatorname {E} [M]\|\leq 1} and ‖ M ‖ ≤ γ {\displaystyle \|M\|\leq \gamma } almost surely. Assume that each element on the support of M has at most rank r. Set t = Ω ( γ log ( γ / ε 2 ) ε 2 ) . {\displaystyle t=\Omega \left({\frac {\gamma \log(\gamma /\varepsilon ^{2})}{\varepsilon ^{2}}}\right).} If r ≤ t {\displaystyle r\leq t} holds almost surely, then Pr ( ‖ 1 t ∑ i = 1 t M i − E [ M ] ‖ > ε ) ≤ 1 p o l y ( t ) {\displaystyle \Pr \left(\left\|{\frac {1}{t}}\sum _{i=1}^{t}M_{i}-\operatorname {E} [M]\right\|>\varepsilon \right)\leq {\frac {1}{\mathbf {poly} (t)}}} where M1, ..., Mt are i.i.d. copies of M. == Sampling variant == The following variant of Chernoff's bound can be used to bound the probability that a majority in a population will become a minority in a sample, or vice versa. Suppose there is a general population A and a sub-population B ⊆ A. Mark the relative size of the sub-population (|B|/|A|) by r. Suppose we pick an integer k and a random sample S ⊂ A of size k. Mark the relative size of the sub-population in the sample (|B∩S|/|S|) by rS. Then, for every fraction d ∈ [0,1]: Pr ( r S < ( 1 − d ) ⋅ r ) < exp ( − r ⋅ d 2 ⋅ k 2 ) {\displaystyle \Pr \left(r_{S}<(1-d)\cdot r\right)<\exp \left(-r\cdot d^{2}\cdot {\frac {k}{2}}\right)} In particular, if B is a majority in A (i.e. r > 0.5) we can bound the probability that B will remain majority in S(rS > 0.5) by taking: d = 1 − 1/(2r): Pr ( r S > 0.5 ) > 1 − exp ( − r ⋅ ( 1 − 1 2 r ) 2 ⋅ k 2 ) {\displaystyle \Pr \left(r_{S}>0.5\right)>1-\exp \left(-r\cdot \left(1-{\frac {1}{2r}}\right)^{2}\cdot {\frac {k}{2}}\right)} This bound is of course not tight at all. For example, when r = 0.5 we get a trivial bound Prob > 0. == Proofs == === Multiplicative form === Following the conditions of the multiplicative Chernoff bound, let X1, ..., Xn be independent Bernoulli random variables, whose sum is X, each having probability pi of being equal to 1. For a Bernoulli variable: E [ e t ⋅ X i ] = ( 1 − p i ) e 0 + p i e t = 1 + p i ( e t − 1 ) ≤ e p i ( e t − 1 ) {\displaystyle \operatorname {E} \left[e^{t\cdot X_{i}}\right]=(1-p_{i})e^{0}+p_{i}e^{t}=1+p_{i}(e^{t}-1)\leq e^{p_{i}(e^{t}-1)}} So, using (1) with a = ( 1 + δ ) μ {\displaystyle a=(1+\delta )\mu } for any δ > 0 {\displaystyle \delta >0} and where μ = E [ X ] = ∑ i = 1 n p i {\displaystyle \mu =\operatorname {E} [X]=\textstyle \sum _{i=1}^{n}p_{i}} , Pr ( X > ( 1 + δ ) μ ) ≤ inf t ≥ 0 exp ( − t ( 1 + δ ) μ ) ∏ i = 1 n E [ exp ( t X i ) ] ≤ inf t ≥ 0 exp ( − t ( 1 + δ ) μ + ∑ i = 1 n p i ( e t − 1 ) ) = inf t ≥ 0 exp ( − t ( 1 + δ ) μ + ( e t − 1 ) μ ) . {\displaystyle {\begin{aligned}\Pr(X>(1+\delta )\mu )&\leq \inf _{t\geq 0}\exp(-t(1+\delta )\mu )\prod _{i=1}^{n}\operatorname {E} [\exp(tX_{i})]\\[4pt]&\leq \inf _{t\geq 0}\exp {\Big (}-t(1+\delta )\mu +\sum _{i=1}^{n}p_{i}(e^{t}-1){\Big )}\\[4pt]&=\inf _{t\geq 0}\exp {\Big (}-t(1+\delta )\mu +(e^{t}-1)\mu {\Big )}.\end{aligned}}} If we simply set t = log(1 + δ) so that t > 0 for δ > 0, we can substitute and find exp ( − t ( 1 + δ ) μ + ( e t − 1 ) μ ) = exp ( ( 1 + δ − 1 ) μ ) ( 1 + δ ) ( 1 + δ ) μ = [ e δ ( 1 + δ ) ( 1 + δ ) ] μ . {\displaystyle \exp {\Big (}-t(1+\delta )\mu +(e^{t}-1)\mu {\Big )}={\frac {\exp((1+\delta -1)\mu )}{(1+\delta )^{(1+\delta )\mu }}}=\left[{\frac {e^{\delta }}{(1+\delta )^{(1+\delta )}}}\right]^{\mu }.} This proves the result desired. === Chernoff–Hoeffding theorem (additive form) === Let q = p + ε. Taking a = nq in (1), we obtain: Pr ( 1 n ∑ X i ≥ q ) ≤ inf t > 0 E [ ∏ e t X i ] e t n q = inf t > 0 ( E [ e t X i ] e t q ) n . {\displaystyle \Pr \left({\frac {1}{n}}\sum X_{i}\geq q\right)\leq \inf _{t>0}{\frac {E\left[\prod e^{tX_{i}}\right]}{e^{tnq}}}=\inf _{t>0}\left({\frac {E\left[e^{tX_{i}}\right]}{e^{tq}}}\right)^{n}.} Now, knowing that Pr(Xi = 1) = p, Pr(Xi = 0) = 1 − p, we have ( E [ e t X i ] e t q ) n = ( p e t + ( 1 − p ) e t q ) n = ( p e ( 1 − q ) t + ( 1 − p ) e − q t ) n . {\displaystyle \left({\frac {\operatorname {E} \left[e^{tX_{i}}\right]}{e^{tq}}}\right)^{n}=\left({\frac {pe^{t}+(1-p)}{e^{tq}}}\right)^{n}=\left(pe^{(1-q)t}+(1-p)e^{-qt}\right)^{n}.} Therefore, we can easily compute the infimum, using calculus: d d t ( p e ( 1 − q ) t + ( 1 − p ) e − q t ) = ( 1 − q ) p e ( 1 − q ) t − q ( 1 − p ) e − q t {\displaystyle {\frac {d}{dt}}\left(pe^{(1-q)t}+(1-p)e^{-qt}\right)=(1-q)pe^{(1-q)t}-q(1-p)e^{-qt}} Setting the equation to zero and solving, we have ( 1 − q ) p e ( 1 − q ) t = q ( 1 − p ) e − q t ( 1 − q ) p e t = q ( 1 − p ) {\displaystyle {\begin{aligned}(1-q)pe^{(1-q)t}&=q(1-p)e^{-qt}\\(1-q)pe^{t}&=q(1-p)\end{aligned}}} so that e t = ( 1 − p ) q ( 1 − q ) p . {\displaystyle e^{t}={\frac {(1-p)q}{(1-q)p}}.} Thus, t = log ( ( 1 − p ) q ( 1 − q ) p ) . {\displaystyle t=\log \left({\frac {(1-p)q}{(1-q)p}}\right).} As q = p + ε > p, we see that t > 0, so our bound is satisfied on t. Having solved for t, we can plug back into the equations above to find that log ( p e ( 1 − q ) t + ( 1 − p ) e − q t ) = log ( e − q t ( 1 − p + p e t ) ) = log ( e − q log ( ( 1 − p ) q ( 1 − q ) p ) ) + log ( 1 − p + p e log ( 1 − p 1 − q ) e log q p ) = − q log 1 − p 1 − q − q log q p + log ( 1 − p + p ( 1 − p 1 − q ) q p ) = − q log 1 − p 1 − q − q log q p + log ( ( 1 − p ) ( 1 − q ) 1 − q + ( 1 − p ) q 1 − q ) = − q log q p + ( − q log 1 − p 1 − q + log 1 − p 1 − q ) = − q log q p + ( 1 − q ) log 1 − p 1 − q = − D ( q ∥ p ) . {\displaystyle {\begin{aligned}\log \left(pe^{(1-q)t}+(1-p)e^{-qt}\right)&=\log \left(e^{-qt}(1-p+pe^{t})\right)\\&=\log \left(e^{-q\log \left({\frac {(1-p)q}{(1-q)p}}\right)}\right)+\log \left(1-p+pe^{\log \left({\frac {1-p}{1-q}}\right)}e^{\log {\frac {q}{p}}}\right)\\&=-q\log {\frac {1-p}{1-q}}-q\log {\frac {q}{p}}+\log \left(1-p+p\left({\frac {1-p}{1-q}}\right){\frac {q}{p}}\right)\\&=-q\log {\frac {1-p}{1-q}}-q\log {\frac {q}{p}}+\log \left({\frac {(1-p)(1-q)}{1-q}}+{\frac {(1-p)q}{1-q}}\right)\\&=-q\log {\frac {q}{p}}+\left(-q\log {\frac {1-p}{1-q}}+\log {\frac {1-p}{1-q}}\right)\\&=-q\log {\frac {q}{p}}+(1-q)\log {\frac {1-p}{1-q}}\\&=-D(q\parallel p).\end{aligned}}} We now have our desired result, that Pr ( 1 n ∑ X i ≥ p + ε ) ≤ e − D ( p + ε ∥ p ) n . {\displaystyle \Pr \left({\tfrac {1}{n}}\sum X_{i}\geq p+\varepsilon \right)\leq e^{-D(p+\varepsilon \parallel p)n}.} To complete the proof for the symmetric case, we simply define the random variable Yi = 1 − Xi, apply the same proof, and plug it into our bound. == See also == Bernstein inequalities Concentration inequality − a summary of tail-bounds on random variables. Cramér's theorem Entropic value at risk Hoeffding's inequality Matrix Chernoff bound Moment generating function == References == == Further reading == Chernoff, H. (1952). "A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations". Annals of Mathematical Statistics. 23 (4): 493–507. doi:10.1214/aoms/1177729330. JSTOR 2236576. MR 0057518. Zbl 0048.11804. Chernoff, H. (1981). "A Note on an Inequality Involving the Normal Distribution". Annals of Probability. 9 (3): 533–535. doi:10.1214/aop/1176994428. JSTOR 2243541. MR 0614640. Zbl 0457.60014. Hagerup, T.; Rüb, C. (1990). "A guided tour of Chernoff bounds". Information Processing Letters. 33 (6): 305. doi:10.1016/0020-0190(90)90214-I. Nielsen, F. (2011). "An Information-Geometric Characterization of Chernoff Information". IEEE Signal Processing Letters. 20 (3): 269–272. arXiv:1102.2684. doi:10.1109/LSP.2013.2243726. S2CID 15034953. |
Wikipedia:Matrix addition#0 | In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. For a vector, v → {\displaystyle {\vec {v}}\!} , adding two matrices would have the geometric effect of applying each matrix transformation separately onto v → {\displaystyle {\vec {v}}\!} , then adding the transformed vectors. A v → + B v → = ( A + B ) v → {\displaystyle \mathbf {A} {\vec {v}}+\mathbf {B} {\vec {v}}=(\mathbf {A} +\mathbf {B} ){\vec {v}}\!} However, there are other operations that could also be considered addition for matrices, such as the direct sum and the Kronecker sum. == Entrywise sum == Two matrices must have an equal number of rows and columns to be added. In which case, the sum of two matrices A and B will be a matrix which has the same number of rows and columns as A and B. The sum of A and B, denoted A + B, is computed by adding corresponding elements of A and B: A + B = [ a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋱ ⋮ a m 1 a m 2 ⋯ a m n ] + [ b 11 b 12 ⋯ b 1 n b 21 b 22 ⋯ b 2 n ⋮ ⋮ ⋱ ⋮ b m 1 b m 2 ⋯ b m n ] = [ a 11 + b 11 a 12 + b 12 ⋯ a 1 n + b 1 n a 21 + b 21 a 22 + b 22 ⋯ a 2 n + b 2 n ⋮ ⋮ ⋱ ⋮ a m 1 + b m 1 a m 2 + b m 2 ⋯ a m n + b m n ] {\displaystyle {\begin{aligned}\mathbf {A} +\mathbf {B} &={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\\\end{bmatrix}}+{\begin{bmatrix}b_{11}&b_{12}&\cdots &b_{1n}\\b_{21}&b_{22}&\cdots &b_{2n}\\\vdots &\vdots &\ddots &\vdots \\b_{m1}&b_{m2}&\cdots &b_{mn}\\\end{bmatrix}}\\&={\begin{bmatrix}a_{11}+b_{11}&a_{12}+b_{12}&\cdots &a_{1n}+b_{1n}\\a_{21}+b_{21}&a_{22}+b_{22}&\cdots &a_{2n}+b_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}+b_{m1}&a_{m2}+b_{m2}&\cdots &a_{mn}+b_{mn}\\\end{bmatrix}}\\\end{aligned}}\,\!} Or more concisely (assuming that A + B = C): c i j = a i j + b i j {\displaystyle c_{ij}=a_{ij}+b_{ij}} For example: [ 1 3 1 0 1 2 ] + [ 0 0 7 5 2 1 ] = [ 1 + 0 3 + 0 1 + 7 0 + 5 1 + 2 2 + 1 ] = [ 1 3 8 5 3 3 ] {\displaystyle {\begin{bmatrix}1&3\\1&0\\1&2\end{bmatrix}}+{\begin{bmatrix}0&0\\7&5\\2&1\end{bmatrix}}={\begin{bmatrix}1+0&3+0\\1+7&0+5\\1+2&2+1\end{bmatrix}}={\begin{bmatrix}1&3\\8&5\\3&3\end{bmatrix}}} Similarly, it is also possible to subtract one matrix from another, as long as they have the same dimensions. The difference of A and B, denoted A − B, is computed by subtracting elements of B from corresponding elements of A, and has the same dimensions as A and B. For example: [ 1 3 1 0 1 2 ] − [ 0 0 7 5 2 1 ] = [ 1 − 0 3 − 0 1 − 7 0 − 5 1 − 2 2 − 1 ] = [ 1 3 − 6 − 5 − 1 1 ] {\displaystyle {\begin{bmatrix}1&3\\1&0\\1&2\end{bmatrix}}-{\begin{bmatrix}0&0\\7&5\\2&1\end{bmatrix}}={\begin{bmatrix}1-0&3-0\\1-7&0-5\\1-2&2-1\end{bmatrix}}={\begin{bmatrix}1&3\\-6&-5\\-1&1\end{bmatrix}}} == Direct sum == Another operation, which is used less often, is the direct sum (denoted by ⊕). The Kronecker sum is also denoted ⊕; the context should make the usage clear. The direct sum of any pair of matrices A of size m × n and B of size p × q is a matrix of size (m + p) × (n + q) defined as: A ⊕ B = [ A 0 0 B ] = [ a 11 ⋯ a 1 n 0 ⋯ 0 ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ a m 1 ⋯ a m n 0 ⋯ 0 0 ⋯ 0 b 11 ⋯ b 1 q ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ 0 ⋯ 0 b p 1 ⋯ b p q ] {\displaystyle \mathbf {A} \oplus \mathbf {B} ={\begin{bmatrix}\mathbf {A} &{\boldsymbol {0}}\\{\boldsymbol {0}}&\mathbf {B} \end{bmatrix}}={\begin{bmatrix}a_{11}&\cdots &a_{1n}&0&\cdots &0\\\vdots &\ddots &\vdots &\vdots &\ddots &\vdots \\a_{m1}&\cdots &a_{mn}&0&\cdots &0\\0&\cdots &0&b_{11}&\cdots &b_{1q}\\\vdots &\ddots &\vdots &\vdots &\ddots &\vdots \\0&\cdots &0&b_{p1}&\cdots &b_{pq}\end{bmatrix}}} For instance, [ 1 3 2 2 3 1 ] ⊕ [ 1 6 0 1 ] = [ 1 3 2 0 0 2 3 1 0 0 0 0 0 1 6 0 0 0 0 1 ] {\displaystyle {\begin{bmatrix}1&3&2\\2&3&1\end{bmatrix}}\oplus {\begin{bmatrix}1&6\\0&1\end{bmatrix}}={\begin{bmatrix}1&3&2&0&0\\2&3&1&0&0\\0&0&0&1&6\\0&0&0&0&1\end{bmatrix}}} The direct sum of matrices is a special type of block matrix. In particular, the direct sum of square matrices is a block diagonal matrix. The adjacency matrix of the union of disjoint graphs (or multigraphs) is the direct sum of their adjacency matrices. Any element in the direct sum of two vector spaces of matrices can be represented as a direct sum of two matrices. In general, the direct sum of n matrices is: ⨁ i = 1 n A i = diag ( A 1 , A 2 , A 3 , … , A n ) = [ A 1 0 ⋯ 0 0 A 2 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ A n ] {\displaystyle \bigoplus _{i=1}^{n}\mathbf {A} _{i}=\operatorname {diag} (\mathbf {A} _{1},\mathbf {A} _{2},\mathbf {A} _{3},\ldots ,\mathbf {A} _{n})={\begin{bmatrix}\mathbf {A} _{1}&{\boldsymbol {0}}&\cdots &{\boldsymbol {0}}\\{\boldsymbol {0}}&\mathbf {A} _{2}&\cdots &{\boldsymbol {0}}\\\vdots &\vdots &\ddots &\vdots \\{\boldsymbol {0}}&{\boldsymbol {0}}&\cdots &\mathbf {A} _{n}\\\end{bmatrix}}\,\!} where the zeros are actually blocks of zeros (i.e., zero matrices). == Kronecker sum == The Kronecker sum is different from the direct sum, but is also denoted by ⊕. It is defined using the Kronecker product ⊗ and normal matrix addition. If A is n-by-n, B is m-by-m and I k {\displaystyle \mathbf {I} _{k}} denotes the k-by-k identity matrix then the Kronecker sum is defined by: A ⊕ B = A ⊗ I m + I n ⊗ B . {\displaystyle \mathbf {A} \oplus \mathbf {B} =\mathbf {A} \otimes \mathbf {I} _{m}+\mathbf {I} _{n}\otimes \mathbf {B} .} == See also == Matrix multiplication Vector addition == Notes == == References == Lipschutz, Seymour; Lipson, Marc (2017). Schaum's Outline of Linear Algebra (6 ed.). McGraw-Hill Education. ISBN 9781260011449. Riley, K.F.; Hobson, M.P.; Bence, S.J. (2006). Mathematical methods for physics and engineering (3 ed.). Cambridge University Press. doi:10.1017/CBO9780511810763. ISBN 978-0-521-86153-3. == External links == Direct sum of matrices at PlanetMath. Abstract nonsense: Direct Sum of Linear Transformations and Direct Sum of Matrices Mathematics Source Library: Arithmetic Matrix Operations Matrix Algebra and R |
Wikipedia:Matrix analysis#0 | In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties. Some particular topics out of many include; operations defined on matrices (such as matrix addition, matrix multiplication and operations derived from these), functions of matrices (such as matrix exponentiation and matrix logarithm, and even sines and cosines etc. of matrices), and the eigenvalues of matrices (eigendecomposition of a matrix, eigenvalue perturbation theory). == Matrix spaces == The set of all m × n matrices over a field F denoted in this article Mmn(F) form a vector space. Examples of F include the set of rational numbers Q {\displaystyle \mathbb {Q} } , the real numbers R {\displaystyle \mathbb {R} } , and set of complex numbers C {\displaystyle \mathbb {C} } . The spaces Mmn(F) and Mpq(F) are different spaces if m and p are unequal, and if n and q are unequal; for instance M32(F) ≠ M23(F). Two m × n matrices A and B in Mmn(F) can be added together to form another matrix in the space Mmn(F): A , B ∈ M m n ( F ) , A + B ∈ M m n ( F ) {\displaystyle \mathbf {A} ,\mathbf {B} \in M_{mn}(F)\,,\quad \mathbf {A} +\mathbf {B} \in M_{mn}(F)} and multiplied by a α in F, to obtain another matrix in Mmn(F): α ∈ F , α A ∈ M m n ( F ) {\displaystyle \alpha \in F\,,\quad \alpha \mathbf {A} \in M_{mn}(F)} Combining these two properties, a linear combination of matrices A and B are in Mmn(F) is another matrix in Mmn(F): α A + β B ∈ M m n ( F ) {\displaystyle \alpha \mathbf {A} +\beta \mathbf {B} \in M_{mn}(F)} where α and β are numbers in F. Any matrix can be expressed as a linear combination of basis matrices, which play the role of the basis vectors for the matrix space. For example, for the set of 2 × 2 matrices over the field of real numbers, M 22 ( R ) {\displaystyle M_{22}(\mathbb {R} )} , one legitimate basis set of matrices is: ( 1 0 0 0 ) , ( 0 1 0 0 ) , ( 0 0 1 0 ) , ( 0 0 0 1 ) , {\displaystyle {\begin{pmatrix}1&0\\0&0\end{pmatrix}}\,,\quad {\begin{pmatrix}0&1\\0&0\end{pmatrix}}\,,\quad {\begin{pmatrix}0&0\\1&0\end{pmatrix}}\,,\quad {\begin{pmatrix}0&0\\0&1\end{pmatrix}}\,,} because any 2 × 2 matrix can be expressed as: ( a b c d ) = a ( 1 0 0 0 ) + b ( 0 1 0 0 ) + c ( 0 0 1 0 ) + d ( 0 0 0 1 ) , {\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}=a{\begin{pmatrix}1&0\\0&0\end{pmatrix}}+b{\begin{pmatrix}0&1\\0&0\end{pmatrix}}+c{\begin{pmatrix}0&0\\1&0\end{pmatrix}}+d{\begin{pmatrix}0&0\\0&1\end{pmatrix}}\,,} where a, b, c,d are all real numbers. This idea applies to other fields and matrices of higher dimensions. == Determinants == The determinant of a square matrix is an important property. The determinant indicates if a matrix is invertible (i.e. the inverse of a matrix exists when the determinant is nonzero). Determinants are used for finding eigenvalues of matrices (see below), and for solving a system of linear equations (see Cramer's rule). == Eigenvalues and eigenvectors of matrices == === Definitions === An n × n matrix A has eigenvectors x and eigenvalues λ defined by the relation: A x = λ x {\displaystyle \mathbf {A} \mathbf {x} =\lambda \mathbf {x} } In words, the matrix multiplication of A followed by an eigenvector x (here an n-dimensional column matrix), is the same as multiplying the eigenvector by the eigenvalue. For an n × n matrix, there are n eigenvalues. The eigenvalues are the roots of the characteristic polynomial: p A ( λ ) = det ( A − λ I ) = 0 {\displaystyle p_{\mathbf {A} }(\lambda )=\det(\mathbf {A} -\lambda \mathbf {I} )=0} where I is the n × n identity matrix. Roots of polynomials, in this context the eigenvalues, can all be different, or some may be equal (in which case eigenvalue has multiplicity, the number of times an eigenvalue occurs). After solving for the eigenvalues, the eigenvectors corresponding to the eigenvalues can be found by the defining equation. === Perturbations of eigenvalues === == Matrix similarity == Two n × n matrices A and B are similar if they are related by a similarity transformation: B = P A P − 1 {\displaystyle \mathbf {B} =\mathbf {P} \mathbf {A} \mathbf {P} ^{-1}} The matrix P is called a similarity matrix, and is necessarily invertible. === Unitary similarity === == Canonical forms == === Row echelon form === === Jordan normal form === === Weyr canonical form === === Frobenius normal form === == Triangular factorization == === LU decomposition === LU decomposition splits a matrix into a matrix product of an upper triangular matrix and a lower triangle matrix. == Matrix norms == Since matrices form vector spaces, one can form axioms (analogous to those of vectors) to define a "size" of a particular matrix. The norm of a matrix is a positive real number. === Definition and axioms === For all matrices A and B in Mmn(F), and all numbers α in F, a matrix norm, delimited by double vertical bars || ... ||, fulfills: Nonnegative: ‖ A ‖ ≥ 0 {\displaystyle \|\mathbf {A} \|\geq 0} with equality only for A = 0, the zero matrix. Scalar multiplication: ‖ α A ‖ = | α | ‖ A ‖ {\displaystyle \|\alpha \mathbf {A} \|=|\alpha |\|\mathbf {A} \|} The triangular inequality: ‖ A + B ‖ ≤ ‖ A ‖ + ‖ B ‖ {\displaystyle \|\mathbf {A} +\mathbf {B} \|\leq \|\mathbf {A} \|+\|\mathbf {B} \|} === Frobenius norm === The Frobenius norm is analogous to the dot product of Euclidean vectors; multiply matrix elements entry-wise, add up the results, then take the positive square root: ‖ A ‖ = A : A = ∑ i = 1 m ∑ j = 1 n ( A i j ) 2 {\displaystyle \|\mathbf {A} \|={\sqrt {\mathbf {A} :\mathbf {A} }}={\sqrt {\sum _{i=1}^{m}\sum _{j=1}^{n}(A_{ij})^{2}}}} It is defined for matrices of any dimension (i.e. no restriction to square matrices). == Positive definite and semidefinite matrices == == Functions == Matrix elements are not restricted to constant numbers, they can be mathematical variables. === Functions of matrices === A functions of a matrix takes in a matrix, and return something else (a number, vector, matrix, etc...). === Matrix-valued functions === A matrix valued function takes in something (a number, vector, matrix, etc...) and returns a matrix. == See also == === Other branches of analysis === Mathematical analysis Tensor analysis Matrix calculus Numerical analysis === Other concepts of linear algebra === Tensor product Spectrum of an operator Matrix geometrical series === Types of matrix === Orthogonal matrix, unitary matrix Symmetric matrix, antisymmetric matrix Stochastic matrix === Matrix functions === Matrix polynomial Matrix exponential == Footnotes == == References == === Notes === === Further reading === C. Meyer (2000). Matrix Analysis and Applied Linear Algebra Book and Solutions Manual. Vol. 2. SIAM. ISBN 089-871-454-0. T. S. Shores (2007). Applied Linear Algebra and Matrix Analysis. Undergraduate Texts in Mathematics. Springer. ISBN 978-038-733-195-9. Rajendra Bhatia (1997). Matrix Analysis. Matrix Analysis Series. Vol. 169. Springer. ISBN 038-794-846-5. Alan J. Laub (2012). Computational Matrix Analysis. SIAM. ISBN 978-161-197-221-4. |
Wikipedia:Matrix calculus#0 | In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities. This greatly simplifies operations such as finding the maximum or minimum of a multivariate function and solving systems of differential equations. The notation used here is commonly used in statistics and engineering, while the tensor index notation is preferred in physics. Two competing notational conventions split the field of matrix calculus into two separate groups. The two groups can be distinguished by whether they write the derivative of a scalar with respect to a vector as a column vector or a row vector. Both of these conventions are possible even when the common assumption is made that vectors should be treated as column vectors when combined with matrices (rather than row vectors). A single convention can be somewhat standard throughout a single field that commonly uses matrix calculus (e.g. econometrics, statistics, estimation theory and machine learning). However, even within a given field different authors can be found using competing conventions. Authors of both groups often write as though their specific conventions were standard. Serious mistakes can result when combining results from different authors without carefully verifying that compatible notations have been used. Definitions of these two conventions and comparisons between them are collected in the layout conventions section. == Scope == Matrix calculus refers to a number of different notations that use matrices and vectors to collect the derivative of each component of the dependent variable with respect to each component of the independent variable. In general, the independent variable can be a scalar, a vector, or a matrix while the dependent variable can be any of these as well. Each different situation will lead to a different set of rules, or a separate calculus, using the broader sense of the term. Matrix notation serves as a convenient way to collect the many derivatives in an organized way. As a first example, consider the gradient from vector calculus. For a scalar function of three independent variables, f ( x 1 , x 2 , x 3 ) {\displaystyle f(x_{1},x_{2},x_{3})} , the gradient is given by the vector equation ∇ f = ∂ f ∂ x 1 x ^ 1 + ∂ f ∂ x 2 x ^ 2 + ∂ f ∂ x 3 x ^ 3 , {\displaystyle \nabla f={\frac {\partial f}{\partial x_{1}}}{\hat {x}}_{1}+{\frac {\partial f}{\partial x_{2}}}{\hat {x}}_{2}+{\frac {\partial f}{\partial x_{3}}}{\hat {x}}_{3},} where x ^ i {\displaystyle {\hat {x}}_{i}} represents a unit vector in the x i {\displaystyle x_{i}} direction for 1 ≤ i ≤ 3 {\displaystyle 1\leq i\leq 3} . This type of generalized derivative can be seen as the derivative of a scalar, f, with respect to a vector, x {\displaystyle \mathbf {x} } , and its result can be easily collected in vector form. ∇ f = ( ∂ f ∂ x ) T = [ ∂ f ∂ x 1 ∂ f ∂ x 2 ∂ f ∂ x 3 ] T . {\displaystyle \nabla f=\left({\frac {\partial f}{\partial \mathbf {x} }}\right)^{\mathsf {T}}={\begin{bmatrix}{\dfrac {\partial f}{\partial x_{1}}}&{\dfrac {\partial f}{\partial x_{2}}}&{\dfrac {\partial f}{\partial x_{3}}}\\\end{bmatrix}}^{\textsf {T}}.} More complicated examples include the derivative of a scalar function with respect to a matrix, known as the gradient matrix, which collects the derivative with respect to each matrix element in the corresponding position in the resulting matrix. In that case the scalar must be a function of each of the independent variables in the matrix. As another example, if we have an n-vector of dependent variables, or functions, of m independent variables we might consider the derivative of the dependent vector with respect to the independent vector. The result could be collected in an m×n matrix consisting of all of the possible derivative combinations. There are a total of nine possibilities using scalars, vectors, and matrices. Notice that as we consider higher numbers of components in each of the independent and dependent variables we can be left with a very large number of possibilities. The six kinds of derivatives that can be most neatly organized in matrix form are collected in the following table. Here, we have used the term "matrix" in its most general sense, recognizing that vectors are simply matrices with one column (and scalars are simply vectors with one row). Moreover, we have used bold letters to indicate vectors and bold capital letters for matrices. This notation is used throughout. Notice that we could also talk about the derivative of a vector with respect to a matrix, or any of the other unfilled cells in our table. However, these derivatives are most naturally organized in a tensor of rank higher than 2, so that they do not fit neatly into a matrix. In the following three sections we will define each one of these derivatives and relate them to other branches of mathematics. See the layout conventions section for a more detailed table. === Relation to other derivatives === The matrix derivative is a convenient notation for keeping track of partial derivatives for doing calculations. The Fréchet derivative is the standard way in the setting of functional analysis to take derivatives with respect to vectors. In the case that a matrix function of a matrix is Fréchet differentiable, the two derivatives will agree up to translation of notations. As is the case in general for partial derivatives, some formulae may extend under weaker analytic conditions than the existence of the derivative as approximating linear mapping. === Usages === Matrix calculus is used for deriving optimal stochastic estimators, often involving the use of Lagrange multipliers. This includes the derivation of: Kalman filter Wiener filter Expectation-maximization algorithm for Gaussian mixture Gradient descent == Notation == The vector and matrix derivatives presented in the sections to follow take full advantage of matrix notation, using a single variable to represent a large number of variables. In what follows we will distinguish scalars, vectors and matrices by their typeface. We will let M(n,m) denote the space of real n×m matrices with n rows and m columns. Such matrices will be denoted using bold capital letters: A, X, Y, etc. An element of M(n,1), that is, a column vector, is denoted with a boldface lowercase letter: a, x, y, etc. An element of M(1,1) is a scalar, denoted with lowercase italic typeface: a, t, x, etc. XT denotes matrix transpose, tr(X) is the trace, and det(X) or |X| is the determinant. All functions are assumed to be of differentiability class C1 unless otherwise noted. Generally letters from the first half of the alphabet (a, b, c, ...) will be used to denote constants, and from the second half (t, x, y, ...) to denote variables. NOTE: As mentioned above, there are competing notations for laying out systems of partial derivatives in vectors and matrices, and no standard appears to be emerging yet. The next two introductory sections use the numerator layout convention simply for the purposes of convenience, to avoid overly complicating the discussion. The section after them discusses layout conventions in more detail. It is important to realize the following: Despite the use of the terms "numerator layout" and "denominator layout", there are actually more than two possible notational choices involved. The reason is that the choice of numerator vs. denominator (or in some situations, numerator vs. mixed) can be made independently for scalar-by-vector, vector-by-scalar, vector-by-vector, and scalar-by-matrix derivatives, and a number of authors mix and match their layout choices in various ways. The choice of numerator layout in the introductory sections below does not imply that this is the "correct" or "superior" choice. There are advantages and disadvantages to the various layout types. Serious mistakes can result from carelessly combining formulas written in different layouts, and converting from one layout to another requires care to avoid errors. As a result, when working with existing formulas the best policy is probably to identify whichever layout is used and maintain consistency with it, rather than attempting to use the same layout in all situations. === Alternatives === The tensor index notation with its Einstein summation convention is very similar to the matrix calculus, except one writes only a single component at a time. It has the advantage that one can easily manipulate arbitrarily high rank tensors, whereas tensors of rank higher than two are quite unwieldy with matrix notation. All of the work here can be done in this notation without use of the single-variable matrix notation. However, many problems in estimation theory and other areas of applied mathematics would result in too many indices to properly keep track of, pointing in favor of matrix calculus in those areas. Also, Einstein notation can be very useful in proving the identities presented here (see section on differentiation) as an alternative to typical element notation, which can become cumbersome when the explicit sums are carried around. Note that a matrix can be considered a tensor of rank two. == Derivatives with vectors == Because vectors are matrices with only one column, the simplest matrix derivatives are vector derivatives. The notations developed here can accommodate the usual operations of vector calculus by identifying the space M(n,1) of n-vectors with the Euclidean space Rn, and the scalar M(1,1) is identified with R. The corresponding concept from vector calculus is indicated at the end of each subsection. NOTE: The discussion in this section assumes the numerator layout convention for pedagogical purposes. Some authors use different conventions. The section on layout conventions discusses this issue in greater detail. The identities given further down are presented in forms that can be used in conjunction with all common layout conventions. === Vector-by-scalar === The derivative of a vector y = [ y 1 y 2 ⋯ y m ] T {\displaystyle \mathbf {y} ={\begin{bmatrix}y_{1}&y_{2}&\cdots &y_{m}\end{bmatrix}}^{\mathsf {T}}} , by a scalar x is written (in numerator layout notation) as d y d x = [ d y 1 d x d y 2 d x ⋮ d y m d x ] . {\displaystyle {\frac {d\mathbf {y} }{dx}}={\begin{bmatrix}{\frac {dy_{1}}{dx}}\\{\frac {dy_{2}}{dx}}\\\vdots \\{\frac {dy_{m}}{dx}}\\\end{bmatrix}}.} In vector calculus the derivative of a vector y with respect to a scalar x is known as the tangent vector of the vector y, ∂ y ∂ x {\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}} . Notice here that y: R1 → Rm. Example Simple examples of this include the velocity vector in Euclidean space, which is the tangent vector of the position vector (considered as a function of time). Also, the acceleration is the tangent vector of the velocity. === Scalar-by-vector === The derivative of a scalar y by a vector x = [ x 1 x 2 ⋯ x n ] T {\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}&x_{2}&\cdots &x_{n}\end{bmatrix}}^{\mathsf {T}}} , is written (in numerator layout notation) as ∂ y ∂ x = [ ∂ y ∂ x 1 ∂ y ∂ x 2 ⋯ ∂ y ∂ x n ] . {\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}={\begin{bmatrix}{\dfrac {\partial y}{\partial x_{1}}}&{\dfrac {\partial y}{\partial x_{2}}}&\cdots &{\dfrac {\partial y}{\partial x_{n}}}\end{bmatrix}}.} In vector calculus, the gradient of a scalar field f : Rn → R (whose independent coordinates are the components of x) is the transpose of the derivative of a scalar by a vector. ∇ f = [ ∂ f ∂ x 1 ⋮ ∂ f ∂ x n ] = ( ∂ f ∂ x ) T {\displaystyle \nabla f={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}\\\vdots \\{\frac {\partial f}{\partial x_{n}}}\end{bmatrix}}=\left({\frac {\partial f}{\partial \mathbf {x} }}\right)^{\mathsf {T}}} By example, in physics, the electric field is the negative vector gradient of the electric potential. The directional derivative of a scalar function f(x) of the space vector x in the direction of the unit vector u (represented in this case as a column vector) is defined using the gradient as follows. ∇ u f ( x ) = ∇ f ( x ) ⋅ u {\displaystyle \nabla _{\mathbf {u} }{f}(\mathbf {x} )=\nabla f(\mathbf {x} )\cdot \mathbf {u} } Using the notation just defined for the derivative of a scalar with respect to a vector we can re-write the directional derivative as ∇ u f = ∂ f ∂ x u . {\displaystyle \nabla _{\mathbf {u} }f={\frac {\partial f}{\partial \mathbf {x} }}\mathbf {u} .} This type of notation will be nice when proving product rules and chain rules that come out looking similar to what we are familiar with for the scalar derivative. === Vector-by-vector === Each of the previous two cases can be considered as an application of the derivative of a vector with respect to a vector, using a vector of size one appropriately. Similarly we will find that the derivatives involving matrices will reduce to derivatives involving vectors in a corresponding way. The derivative of a vector function (a vector whose components are functions) y = [ y 1 y 2 ⋯ y m ] T {\displaystyle \mathbf {y} ={\begin{bmatrix}y_{1}&y_{2}&\cdots &y_{m}\end{bmatrix}}^{\mathsf {T}}} , with respect to an input vector, x = [ x 1 x 2 ⋯ x n ] T {\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}&x_{2}&\cdots &x_{n}\end{bmatrix}}^{\mathsf {T}}} , is written (in numerator layout notation) as ∂ y ∂ x = [ ∂ y 1 ∂ x 1 ∂ y 1 ∂ x 2 ⋯ ∂ y 1 ∂ x n ∂ y 2 ∂ x 1 ∂ y 2 ∂ x 2 ⋯ ∂ y 2 ∂ x n ⋮ ⋮ ⋱ ⋮ ∂ y m ∂ x 1 ∂ y m ∂ x 2 ⋯ ∂ y m ∂ x n ] . {\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x_{1}}}&{\frac {\partial y_{1}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{1}}{\partial x_{n}}}\\{\frac {\partial y_{2}}{\partial x_{1}}}&{\frac {\partial y_{2}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{2}}{\partial x_{n}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{m}}{\partial x_{1}}}&{\frac {\partial y_{m}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{n}}}\\\end{bmatrix}}.} In vector calculus, the derivative of a vector function y with respect to a vector x whose components represent a space is known as the pushforward (or differential), or the Jacobian matrix. The pushforward along a vector function f with respect to vector v in Rn is given by d f ( v ) = ∂ f ∂ v d v . {\displaystyle d\mathbf {f} (\mathbf {v} )={\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}d\mathbf {v} .} == Derivatives with matrices == There are two types of derivatives with matrices that can be organized into a matrix of the same size. These are the derivative of a matrix by a scalar and the derivative of a scalar by a matrix. These can be useful in minimization problems found in many areas of applied mathematics and have adopted the names tangent matrix and gradient matrix respectively after their analogs for vectors. Note: The discussion in this section assumes the numerator layout convention for pedagogical purposes. Some authors use different conventions. The section on layout conventions discusses this issue in greater detail. The identities given further down are presented in forms that can be used in conjunction with all common layout conventions. === Matrix-by-scalar === The derivative of a matrix function Y by a scalar x is known as the tangent matrix and is given (in numerator layout notation) by ∂ Y ∂ x = [ ∂ y 11 ∂ x ∂ y 12 ∂ x ⋯ ∂ y 1 n ∂ x ∂ y 21 ∂ x ∂ y 22 ∂ x ⋯ ∂ y 2 n ∂ x ⋮ ⋮ ⋱ ⋮ ∂ y m 1 ∂ x ∂ y m 2 ∂ x ⋯ ∂ y m n ∂ x ] . {\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}={\begin{bmatrix}{\frac {\partial y_{11}}{\partial x}}&{\frac {\partial y_{12}}{\partial x}}&\cdots &{\frac {\partial y_{1n}}{\partial x}}\\{\frac {\partial y_{21}}{\partial x}}&{\frac {\partial y_{22}}{\partial x}}&\cdots &{\frac {\partial y_{2n}}{\partial x}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{m1}}{\partial x}}&{\frac {\partial y_{m2}}{\partial x}}&\cdots &{\frac {\partial y_{mn}}{\partial x}}\\\end{bmatrix}}.} === Scalar-by-matrix === The derivative of a scalar function y, with respect to a p×q matrix X of independent variables, is given (in numerator layout notation) by ∂ y ∂ X = [ ∂ y ∂ x 11 ∂ y ∂ x 21 ⋯ ∂ y ∂ x p 1 ∂ y ∂ x 12 ∂ y ∂ x 22 ⋯ ∂ y ∂ x p 2 ⋮ ⋮ ⋱ ⋮ ∂ y ∂ x 1 q ∂ y ∂ x 2 q ⋯ ∂ y ∂ x p q ] . {\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}={\begin{bmatrix}{\frac {\partial y}{\partial x_{11}}}&{\frac {\partial y}{\partial x_{21}}}&\cdots &{\frac {\partial y}{\partial x_{p1}}}\\{\frac {\partial y}{\partial x_{12}}}&{\frac {\partial y}{\partial x_{22}}}&\cdots &{\frac {\partial y}{\partial x_{p2}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y}{\partial x_{1q}}}&{\frac {\partial y}{\partial x_{2q}}}&\cdots &{\frac {\partial y}{\partial x_{pq}}}\\\end{bmatrix}}.} Important examples of scalar functions of matrices include the trace of a matrix and the determinant. In analog with vector calculus this derivative is often written as the following. ∇ X y ( X ) = ∂ y ( X ) ∂ X {\displaystyle \nabla _{\mathbf {X} }y(\mathbf {X} )={\frac {\partial y(\mathbf {X} )}{\partial \mathbf {X} }}} Also in analog with vector calculus, the directional derivative of a scalar f(X) of a matrix X in the direction of matrix Y is given by ∇ Y f = tr ( ∂ f ∂ X Y ) . {\displaystyle \nabla _{\mathbf {Y} }f=\operatorname {tr} \left({\frac {\partial f}{\partial \mathbf {X} }}\mathbf {Y} \right).} It is the gradient matrix, in particular, that finds many uses in minimization problems in estimation theory, particularly in the derivation of the Kalman filter algorithm, which is of great importance in the field. === Other matrix derivatives === The three types of derivatives that have not been considered are those involving vectors-by-matrices, matrices-by-vectors, and matrices-by-matrices. These are not as widely considered and a notation is not widely agreed upon. == Layout conventions == This section discusses the similarities and differences between notational conventions that are used in the various fields that take advantage of matrix calculus. Although there are largely two consistent conventions, some authors find it convenient to mix the two conventions in forms that are discussed below. After this section, equations will be listed in both competing forms separately. The fundamental issue is that the derivative of a vector with respect to a vector, i.e. ∂ y ∂ x {\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}} , is often written in two competing ways. If the numerator y is of size m and the denominator x of size n, then the result can be laid out as either an m×n matrix or n×m matrix, i.e. the m elements of y laid out in rows and the n elements of x laid out in columns, or vice versa. This leads to the following possibilities: Numerator layout, i.e. lay out according to y and xT (i.e. contrarily to x). This is sometimes known as the Jacobian formulation. This corresponds to the m×n layout in the previous example, which means that the row number of ∂ y ∂ x {\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}} equals to the size of the numerator y {\displaystyle \mathbf {y} } and the column number of ∂ y ∂ x {\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}} equals to the size of xT. Denominator layout, i.e. lay out according to yT and x (i.e. contrarily to y). This is sometimes known as the Hessian formulation. Some authors term this layout the gradient, in distinction to the Jacobian (numerator layout), which is its transpose. (However, gradient more commonly means the derivative ∂ y ∂ x , {\displaystyle {\frac {\partial y}{\partial \mathbf {x} }},} regardless of layout.). This corresponds to the n×m layout in the previous example, which means that the row number of ∂ y ∂ x {\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}} equals to the size of x (the denominator). A third possibility sometimes seen is to insist on writing the derivative as ∂ y ∂ x ′ , {\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} '}},} (i.e. the derivative is taken with respect to the transpose of x) and follow the numerator layout. This makes it possible to claim that the matrix is laid out according to both numerator and denominator. In practice this produces results the same as the numerator layout. When handling the gradient ∂ y ∂ x {\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}} and the opposite case ∂ y ∂ x , {\displaystyle {\frac {\partial \mathbf {y} }{\partial x}},} we have the same issues. To be consistent, we should do one of the following: If we choose numerator layout for ∂ y ∂ x , {\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }},} we should lay out the gradient ∂ y ∂ x {\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}} as a row vector, and ∂ y ∂ x {\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}} as a column vector. If we choose denominator layout for ∂ y ∂ x , {\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }},} we should lay out the gradient ∂ y ∂ x {\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}} as a column vector, and ∂ y ∂ x {\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}} as a row vector. In the third possibility above, we write ∂ y ∂ x ′ {\displaystyle {\frac {\partial y}{\partial \mathbf {x} '}}} and ∂ y ∂ x , {\displaystyle {\frac {\partial \mathbf {y} }{\partial x}},} and use numerator layout. Not all math textbooks and papers are consistent in this respect throughout. That is, sometimes different conventions are used in different contexts within the same book or paper. For example, some choose denominator layout for gradients (laying them out as column vectors), but numerator layout for the vector-by-vector derivative ∂ y ∂ x . {\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}.} Similarly, when it comes to scalar-by-matrix derivatives ∂ y ∂ X {\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}} and matrix-by-scalar derivatives ∂ Y ∂ x , {\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}},} then consistent numerator layout lays out according to Y and XT, while consistent denominator layout lays out according to YT and X. In practice, however, following a denominator layout for ∂ Y ∂ x , {\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}},} and laying the result out according to YT, is rarely seen because it makes for ugly formulas that do not correspond to the scalar formulas. As a result, the following layouts can often be found: Consistent numerator layout, which lays out ∂ Y ∂ x {\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}} according to Y and ∂ y ∂ X {\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}} according to XT. Mixed layout, which lays out ∂ Y ∂ x {\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}} according to Y and ∂ y ∂ X {\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}} according to X. Use the notation ∂ y ∂ X ′ , {\displaystyle {\frac {\partial y}{\partial \mathbf {X} '}},} with results the same as consistent numerator layout. In the following formulas, we handle the five possible combinations ∂ y ∂ x , ∂ y ∂ x , ∂ y ∂ x , ∂ y ∂ X {\displaystyle {\frac {\partial y}{\partial \mathbf {x} }},{\frac {\partial \mathbf {y} }{\partial x}},{\frac {\partial \mathbf {y} }{\partial \mathbf {x} }},{\frac {\partial y}{\partial \mathbf {X} }}} and ∂ Y ∂ x {\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}} separately. We also handle cases of scalar-by-scalar derivatives that involve an intermediate vector or matrix. (This can arise, for example, if a multi-dimensional parametric curve is defined in terms of a scalar variable, and then a derivative of a scalar function of the curve is taken with respect to the scalar that parameterizes the curve.) For each of the various combinations, we give numerator-layout and denominator-layout results, except in the cases above where denominator layout rarely occurs. In cases involving matrices where it makes sense, we give numerator-layout and mixed-layout results. As noted above, cases where vector and matrix denominators are written in transpose notation are equivalent to numerator layout with the denominators written without the transpose. Keep in mind that various authors use different combinations of numerator and denominator layouts for different types of derivatives, and there is no guarantee that an author will consistently use either numerator or denominator layout for all types. Match up the formulas below with those quoted in the source to determine the layout used for that particular type of derivative, but be careful not to assume that derivatives of other types necessarily follow the same kind of layout. When taking derivatives with an aggregate (vector or matrix) denominator in order to find a maximum or minimum of the aggregate, it should be kept in mind that using numerator layout will produce results that are transposed with respect to the aggregate. For example, in attempting to find the maximum likelihood estimate of a multivariate normal distribution using matrix calculus, if the domain is a k×1 column vector, then the result using the numerator layout will be in the form of a 1×k row vector. Thus, either the results should be transposed at the end or the denominator layout (or mixed layout) should be used. The results of operations will be transposed when switching between numerator-layout and denominator-layout notation. === Numerator-layout notation === Using numerator-layout notation, we have: ∂ y ∂ x = [ ∂ y ∂ x 1 ∂ y ∂ x 2 ⋯ ∂ y ∂ x n ] . ∂ y ∂ x = [ ∂ y 1 ∂ x ∂ y 2 ∂ x ⋮ ∂ y m ∂ x ] . ∂ y ∂ x = [ ∂ y 1 ∂ x 1 ∂ y 1 ∂ x 2 ⋯ ∂ y 1 ∂ x n ∂ y 2 ∂ x 1 ∂ y 2 ∂ x 2 ⋯ ∂ y 2 ∂ x n ⋮ ⋮ ⋱ ⋮ ∂ y m ∂ x 1 ∂ y m ∂ x 2 ⋯ ∂ y m ∂ x n ] . ∂ y ∂ X = [ ∂ y ∂ x 11 ∂ y ∂ x 21 ⋯ ∂ y ∂ x p 1 ∂ y ∂ x 12 ∂ y ∂ x 22 ⋯ ∂ y ∂ x p 2 ⋮ ⋮ ⋱ ⋮ ∂ y ∂ x 1 q ∂ y ∂ x 2 q ⋯ ∂ y ∂ x p q ] . {\displaystyle {\begin{aligned}{\frac {\partial y}{\partial \mathbf {x} }}&={\begin{bmatrix}{\frac {\partial y}{\partial x_{1}}}&{\frac {\partial y}{\partial x_{2}}}&\cdots &{\frac {\partial y}{\partial x_{n}}}\end{bmatrix}}.\\{\frac {\partial \mathbf {y} }{\partial x}}&={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x}}\\{\frac {\partial y_{2}}{\partial x}}\\\vdots \\{\frac {\partial y_{m}}{\partial x}}\\\end{bmatrix}}.\\{\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}&={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x_{1}}}&{\frac {\partial y_{1}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{1}}{\partial x_{n}}}\\{\frac {\partial y_{2}}{\partial x_{1}}}&{\frac {\partial y_{2}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{2}}{\partial x_{n}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{m}}{\partial x_{1}}}&{\frac {\partial y_{m}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{n}}}\\\end{bmatrix}}.\\{\frac {\partial y}{\partial \mathbf {X} }}&={\begin{bmatrix}{\frac {\partial y}{\partial x_{11}}}&{\frac {\partial y}{\partial x_{21}}}&\cdots &{\frac {\partial y}{\partial x_{p1}}}\\{\frac {\partial y}{\partial x_{12}}}&{\frac {\partial y}{\partial x_{22}}}&\cdots &{\frac {\partial y}{\partial x_{p2}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y}{\partial x_{1q}}}&{\frac {\partial y}{\partial x_{2q}}}&\cdots &{\frac {\partial y}{\partial x_{pq}}}\\\end{bmatrix}}.\end{aligned}}} The following definitions are only provided in numerator-layout notation: ∂ Y ∂ x = [ ∂ y 11 ∂ x ∂ y 12 ∂ x ⋯ ∂ y 1 n ∂ x ∂ y 21 ∂ x ∂ y 22 ∂ x ⋯ ∂ y 2 n ∂ x ⋮ ⋮ ⋱ ⋮ ∂ y m 1 ∂ x ∂ y m 2 ∂ x ⋯ ∂ y m n ∂ x ] . d X = [ d x 11 d x 12 ⋯ d x 1 n d x 21 d x 22 ⋯ d x 2 n ⋮ ⋮ ⋱ ⋮ d x m 1 d x m 2 ⋯ d x m n ] . {\displaystyle {\begin{aligned}{\frac {\partial \mathbf {Y} }{\partial x}}&={\begin{bmatrix}{\frac {\partial y_{11}}{\partial x}}&{\frac {\partial y_{12}}{\partial x}}&\cdots &{\frac {\partial y_{1n}}{\partial x}}\\{\frac {\partial y_{21}}{\partial x}}&{\frac {\partial y_{22}}{\partial x}}&\cdots &{\frac {\partial y_{2n}}{\partial x}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{m1}}{\partial x}}&{\frac {\partial y_{m2}}{\partial x}}&\cdots &{\frac {\partial y_{mn}}{\partial x}}\\\end{bmatrix}}.\\d\mathbf {X} &={\begin{bmatrix}dx_{11}&dx_{12}&\cdots &dx_{1n}\\dx_{21}&dx_{22}&\cdots &dx_{2n}\\\vdots &\vdots &\ddots &\vdots \\dx_{m1}&dx_{m2}&\cdots &dx_{mn}\\\end{bmatrix}}.\end{aligned}}} === Denominator-layout notation === Using denominator-layout notation, we have: ∂ y ∂ x = [ ∂ y ∂ x 1 ∂ y ∂ x 2 ⋮ ∂ y ∂ x n ] . ∂ y ∂ x = [ ∂ y 1 ∂ x ∂ y 2 ∂ x ⋯ ∂ y m ∂ x ] . ∂ y ∂ x = [ ∂ y 1 ∂ x 1 ∂ y 2 ∂ x 1 ⋯ ∂ y m ∂ x 1 ∂ y 1 ∂ x 2 ∂ y 2 ∂ x 2 ⋯ ∂ y m ∂ x 2 ⋮ ⋮ ⋱ ⋮ ∂ y 1 ∂ x n ∂ y 2 ∂ x n ⋯ ∂ y m ∂ x n ] . ∂ y ∂ X = [ ∂ y ∂ x 11 ∂ y ∂ x 12 ⋯ ∂ y ∂ x 1 q ∂ y ∂ x 21 ∂ y ∂ x 22 ⋯ ∂ y ∂ x 2 q ⋮ ⋮ ⋱ ⋮ ∂ y ∂ x p 1 ∂ y ∂ x p 2 ⋯ ∂ y ∂ x p q ] . {\displaystyle {\begin{aligned}{\frac {\partial y}{\partial \mathbf {x} }}&={\begin{bmatrix}{\frac {\partial y}{\partial x_{1}}}\\{\frac {\partial y}{\partial x_{2}}}\\\vdots \\{\frac {\partial y}{\partial x_{n}}}\\\end{bmatrix}}.\\{\frac {\partial \mathbf {y} }{\partial x}}&={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x}}&{\frac {\partial y_{2}}{\partial x}}&\cdots &{\frac {\partial y_{m}}{\partial x}}\end{bmatrix}}.\\{\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}&={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x_{1}}}&{\frac {\partial y_{2}}{\partial x_{1}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{1}}}\\{\frac {\partial y_{1}}{\partial x_{2}}}&{\frac {\partial y_{2}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{2}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{1}}{\partial x_{n}}}&{\frac {\partial y_{2}}{\partial x_{n}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{n}}}\\\end{bmatrix}}.\\{\frac {\partial y}{\partial \mathbf {X} }}&={\begin{bmatrix}{\frac {\partial y}{\partial x_{11}}}&{\frac {\partial y}{\partial x_{12}}}&\cdots &{\frac {\partial y}{\partial x_{1q}}}\\{\frac {\partial y}{\partial x_{21}}}&{\frac {\partial y}{\partial x_{22}}}&\cdots &{\frac {\partial y}{\partial x_{2q}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y}{\partial x_{p1}}}&{\frac {\partial y}{\partial x_{p2}}}&\cdots &{\frac {\partial y}{\partial x_{pq}}}\\\end{bmatrix}}.\end{aligned}}} == Identities == As noted above, in general, the results of operations will be transposed when switching between numerator-layout and denominator-layout notation. To help make sense of all the identities below, keep in mind the most important rules: the chain rule, product rule and sum rule. The sum rule applies universally, and the product rule applies in most of the cases below, provided that the order of matrix products is maintained, since matrix products are not commutative. The chain rule applies in some of the cases, but unfortunately does not apply in matrix-by-scalar derivatives or scalar-by-matrix derivatives (in the latter case, mostly involving the trace operator applied to matrices). In the latter case, the product rule can't quite be applied directly, either, but the equivalent can be done with a bit more work using the differential identities. The following identities adopt the following conventions: the scalars, a, b, c, d, and e are constant in respect of, and the scalars, u, and v are functions of one of x, x, or X; the vectors, a, b, c, d, and e are constant in respect of, and the vectors, u, and v are functions of one of x, x, or X; the matrices, A, B, C, D, and E are constant in respect of, and the matrices, U and V are functions of one of x, x, or X. === Vector-by-vector identities === This is presented first because all of the operations that apply to vector-by-vector differentiation apply directly to vector-by-scalar or scalar-by-vector differentiation simply by reducing the appropriate vector in the numerator or denominator to a scalar. === Scalar-by-vector identities === The fundamental identities are placed above the thick black line. === Vector-by-scalar identities === NOTE: The formulas involving the vector-by-vector derivatives ∂ g ( u ) ∂ u {\displaystyle {\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}} and ∂ f ( g ) ∂ g {\displaystyle {\frac {\partial \mathbf {f} (\mathbf {g} )}{\partial \mathbf {g} }}} (whose outputs are matrices) assume the matrices are laid out consistent with the vector layout, i.e. numerator-layout matrix when numerator-layout vector and vice versa; otherwise, transpose the vector-by-vector derivatives. === Scalar-by-matrix identities === Note that exact equivalents of the scalar product rule and chain rule do not exist when applied to matrix-valued functions of matrices. However, the product rule of this sort does apply to the differential form (see below), and this is the way to derive many of the identities below involving the trace function, combined with the fact that the trace function allows transposing and cyclic permutation, i.e.: tr ( A ) = tr ( A ⊤ ) tr ( A B C D ) = tr ( B C D A ) = tr ( C D A B ) = tr ( D A B C ) {\displaystyle {\begin{aligned}\operatorname {tr} (\mathbf {A} )&=\operatorname {tr} \left(\mathbf {A^{\top }} \right)\\\operatorname {tr} (\mathbf {ABCD} )&=\operatorname {tr} (\mathbf {BCDA} )=\operatorname {tr} (\mathbf {CDAB} )=\operatorname {tr} (\mathbf {DABC} )\end{aligned}}} For example, to compute ∂ tr ( A X B X ⊤ C ) ∂ X : {\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {AXBX^{\top }C} )}{\partial \mathbf {X} }}:} d tr ( A X B X ⊤ C ) = d tr ( C A X B X ⊤ ) = tr ( d ( C A X B X ⊤ ) ) = tr ( C A X d ( B X ⊤ ) + d ( C A X ) B X ⊤ ) = tr ( C A X d ( B X ⊤ ) ) + tr ( d ( C A X ) B X ⊤ ) = tr ( C A X B d ( X ⊤ ) ) + tr ( C A ( d X ) B X ⊤ ) = tr ( C A X B ( d X ) ⊤ ) + tr ( C A ( d X ) B X ⊤ ) = tr ( ( C A X B ( d X ) ⊤ ) ⊤ ) + tr ( C A ( d X ) B X ⊤ ) = tr ( ( d X ) B ⊤ X ⊤ A ⊤ C ⊤ ) + tr ( C A ( d X ) B X ⊤ ) = tr ( B ⊤ X ⊤ A ⊤ C ⊤ ( d X ) ) + tr ( B X ⊤ C A ( d X ) ) = tr ( ( B ⊤ X ⊤ A ⊤ C ⊤ + B X ⊤ C A ) d X ) = tr ( ( C A X B + A ⊤ C ⊤ X B ⊤ ) ⊤ d X ) {\displaystyle {\begin{aligned}d\operatorname {tr} (\mathbf {AXBX^{\top }C} )&=d\operatorname {tr} \left(\mathbf {CAXBX^{\top }} \right)=\operatorname {tr} \left(d\left(\mathbf {CAXBX^{\top }} \right)\right)\\[1ex]&=\operatorname {tr} \left(\mathbf {CAX} d(\mathbf {BX^{\top }} \right)+d\left(\mathbf {CAX} )\mathbf {BX^{\top }} \right)\\[1ex]&=\operatorname {tr} \left(\mathbf {CAX} d\left(\mathbf {BX^{\top }} \right)\right)+\operatorname {tr} \left(d(\mathbf {CAX} )\mathbf {BX^{\top }} \right)\\[1ex]&=\operatorname {tr} \left(\mathbf {CAXB} d\left(\mathbf {X^{\top }} \right)\right)+\operatorname {tr} \left(\mathbf {CA} (d\mathbf {X} )\mathbf {BX^{\top }} \right)\\[1ex]&=\operatorname {tr} \left(\mathbf {CAXB} (d\mathbf {X} )^{\top }\right)+\operatorname {tr} (\mathbf {CA} \left(d\mathbf {X} )\mathbf {BX^{\top }} \right)\\[1ex]&=\operatorname {tr} \left(\left(\mathbf {CAXB} (d\mathbf {X} )^{\top }\right)^{\top }\right)+\operatorname {tr} \left(\mathbf {CA} (d\mathbf {X} )\mathbf {BX^{\top }} \right)\\[1ex]&=\operatorname {tr} \left((d\mathbf {X} )\mathbf {B^{\top }X^{\top }A^{\top }C^{\top }} \right)+\operatorname {tr} \left(\mathbf {CA} (d\mathbf {X} )\mathbf {BX^{\top }} \right)\\[1ex]&=\operatorname {tr} \left(\mathbf {B^{\top }X^{\top }A^{\top }C^{\top }} (d\mathbf {X} )\right)+\operatorname {tr} \left(\mathbf {BX^{\top }} \mathbf {CA} (d\mathbf {X} )\right)\\[1ex]&=\operatorname {tr} \left(\left(\mathbf {B^{\top }X^{\top }A^{\top }C^{\top }} +\mathbf {BX^{\top }} \mathbf {CA} \right)d\mathbf {X} \right)\\[1ex]&=\operatorname {tr} \left(\left(\mathbf {CAXB} +\mathbf {A^{\top }C^{\top }XB^{\top }} \right)^{\top }d\mathbf {X} \right)\end{aligned}}} Therefore, ∂ tr ( A X B X ⊤ C ) ∂ X = B ⊤ X ⊤ A ⊤ C ⊤ + B X ⊤ C A . {\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {AXBX^{\top }C} \right)}{\partial \mathbf {X} }}=\mathbf {B^{\top }X^{\top }A^{\top }C^{\top }} +\mathbf {BX^{\top }CA} .} (numerator layout) ∂ tr ( A X B X ⊤ C ) ∂ X = C A X B + A ⊤ C ⊤ X B ⊤ . {\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {AXBX^{\top }C} \right)}{\partial \mathbf {X} }}=\mathbf {CAXB} +\mathbf {A^{\top }C^{\top }XB^{\top }} .} (denominator layout) (For the last step, see the Conversion from differential to derivative form section.) === Matrix-by-scalar identities === === Scalar-by-scalar identities === ==== With vectors involved ==== ==== With matrices involved ==== === Identities in differential form === It is often easier to work in differential form and then convert back to normal derivatives. This only works well using the numerator layout. In these rules, a is a scalar. In the last row, δ i j {\displaystyle \delta _{ij}} is the Kronecker delta and ( P k ) i j = ( Q ) i k ( Q − 1 ) k j {\displaystyle (\mathbf {P} _{k})_{ij}=(\mathbf {Q} )_{ik}(\mathbf {Q} ^{-1})_{kj}} is the set of orthogonal projection operators that project onto the k-th eigenvector of X. Q is the matrix of eigenvectors of X = Q Λ Q − 1 {\displaystyle \mathbf {X} =\mathbf {Q} {\boldsymbol {\Lambda }}\mathbf {Q} ^{-1}} , and ( Λ ) i i = λ i {\displaystyle ({\boldsymbol {\Lambda }})_{ii}=\lambda _{i}} are the eigenvalues. The matrix function f ( X ) {\displaystyle f(\mathbf {X} )} is defined in terms of the scalar function f ( x ) {\displaystyle f(x)} for diagonalizable matrices by f ( X ) = ∑ i f ( λ i ) P i {\textstyle f(\mathbf {X} )=\sum _{i}f(\lambda _{i})\mathbf {P} _{i}} where X = ∑ i λ i P i {\textstyle \mathbf {X} =\sum _{i}\lambda _{i}\mathbf {P} _{i}} with P i P j = δ i j P i {\displaystyle \mathbf {P} _{i}\mathbf {P} _{j}=\delta _{ij}\mathbf {P} _{i}} . To convert to normal derivative form, first convert it to one of the following canonical forms, and then use these identities: == Applications == Matrix differential calculus is used in statistics and econometrics, particularly for the statistical analysis of multivariate distributions, especially the multivariate normal distribution and other elliptical distributions. It is used in regression analysis to compute, for example, the ordinary least squares regression formula for the case of multiple explanatory variables. It is also used in random matrices, statistical moments, local sensitivity and statistical diagnostics. == See also == Derivative (generalizations) Product integral Ricci calculus Tensor derivative == Notes == == References == == Further reading == == External links == === Software === MatrixCalculus.org, a website for evaluating matrix calculus expressions symbolically NCAlgebra, an open-source Mathematica package that has some matrix calculus functionality SymPy supports symbolic matrix derivatives in its matrix expression module, as well as symbolic tensor derivatives in its array expression module. Tensorgrad, an open-source python package for matrix calculus. Supports general symbolic tensor derivatives using Penrose graphical notation. === Information === Matrix Reference Manual, Mike Brookes, Imperial College London. Matrix Differentiation (and some other stuff), Randal J. Barnes, Department of Civil Engineering, University of Minnesota. Notes on Matrix Calculus, Paul L. Fackler, North Carolina State University. Matrix Differential Calculus Archived 2012-09-16 at the Wayback Machine (slide presentation), Zhang Le, University of Edinburgh. Introduction to Vector and Matrix Differentiation (notes on matrix differentiation, in the context of Econometrics), Heino Bohn Nielsen. A note on differentiating matrices (notes on matrix differentiation), Pawel Koval, from Munich Personal RePEc Archive. Vector/Matrix Calculus More notes on matrix differentiation. Matrix Identities (notes on matrix differentiation), Sam Roweis. Tensor Cookbook Matrix Calculus using Tensor Diagrams. |
Wikipedia:Matrix congruence#0 | In mathematics, two square matrices A and B over a field are called congruent if there exists an invertible matrix P over the same field such that PTAP = B where "T" denotes the matrix transpose. Matrix congruence is an equivalence relation. Matrix congruence arises when considering the effect of change of basis on the Gram matrix attached to a bilinear form or quadratic form on a finite-dimensional vector space: two matrices are congruent if and only if they represent the same bilinear form with respect to different bases. Note that Halmos defines congruence in terms of conjugate transpose (with respect to a complex inner product space) rather than transpose, but this definition has not been adopted by most other authors. == Congruence over the reals == Sylvester's law of inertia states that two congruent symmetric matrices with real entries have the same numbers of positive, negative, and zero eigenvalues. That is, the number of eigenvalues of each sign is an invariant of the associated quadratic form. == See also == Congruence relation Matrix similarity Matrix equivalence == References == Gruenberg, K.W.; Weir, A.J. (1967). Linear geometry. van Nostrand. p. 80. Hadley, G. (1961). Linear algebra. Addison-Wesley. p. 253. Herstein, I.N. (1975). Topics in algebra. Wiley. p. 352. ISBN 0-471-02371-X. Mirsky, L. (1990). An introduction to linear algebra. Dover Publications. p. 182. ISBN 0-486-66434-1. Marcus, Marvin; Minc, Henryk (1992). A survey of matrix theory and matrix inequalities. Dover Publications. p. 81. ISBN 0-486-67102-X. Norman, C.W. (1986). Undergraduate algebra. Oxford University Press. p. 354. ISBN 0-19-853248-2. |
Wikipedia:Matrix difference equation#0 | A matrix difference equation is a difference equation in which the value of a vector (or sometimes, a matrix) of variables at one point in time is related to its own value at one or more previous points in time, using matrices. The order of the equation is the maximum time gap between any two indicated values of the variable vector. For example, x t = A x t − 1 + B x t − 2 {\displaystyle \mathbf {x} _{t}=\mathbf {Ax} _{t-1}+\mathbf {Bx} _{t-2}} is an example of a second-order matrix difference equation, in which x is an n × 1 vector of variables and A and B are n × n matrices. This equation is homogeneous because there is no vector constant term added to the end of the equation. The same equation might also be written as x t + 2 = A x t + 1 + B x t {\displaystyle \mathbf {x} _{t+2}=\mathbf {Ax} _{t+1}+\mathbf {Bx} _{t}} or as x n = A x n − 1 + B x n − 2 {\displaystyle \mathbf {x} _{n}=\mathbf {Ax} _{n-1}+\mathbf {Bx} _{n-2}} The most commonly encountered matrix difference equations are first-order. == Nonhomogeneous first-order case and the steady state == An example of a nonhomogeneous first-order matrix difference equation is x t = A x t − 1 + b {\displaystyle \mathbf {x} _{t}=\mathbf {Ax} _{t-1}+\mathbf {b} } with additive constant vector b. The steady state of this system is a value x* of the vector x which, if reached, would not be deviated from subsequently. x* is found by setting xt = xt−1 = x* in the difference equation and solving for x* to obtain x ∗ = [ I − A ] − 1 b {\displaystyle \mathbf {x} ^{*}=[\mathbf {I} -\mathbf {A} ]^{-1}\mathbf {b} } where I is the n × n identity matrix, and where it is assumed that [I − A] is invertible. Then the nonhomogeneous equation can be rewritten in homogeneous form in terms of deviations from the steady state: [ x t − x ∗ ] = A [ x t − 1 − x ∗ ] {\displaystyle \left[\mathbf {x} _{t}-\mathbf {x} ^{*}\right]=\mathbf {A} \left[\mathbf {x} _{t-1}-\mathbf {x} ^{*}\right]} == Stability of the first-order case == The first-order matrix difference equation [xt − x*] = A[xt−1 − x*] is stable—that is, xt converges asymptotically to the steady state x*—if and only if all eigenvalues of the transition matrix A (whether real or complex) have an absolute value which is less than 1. == Solution of the first-order case == Assume that the equation has been put in the homogeneous form yt = Ayt−1. Then we can iterate and substitute repeatedly from the initial condition y0, which is the initial value of the vector y and which must be known in order to find the solution: y 1 = A y 0 y 2 = A y 1 = A 2 y 0 y 3 = A y 2 = A 3 y 0 {\displaystyle {\begin{aligned}\mathbf {y} _{1}&=\mathbf {Ay} _{0}\\\mathbf {y} _{2}&=\mathbf {Ay} _{1}=\mathbf {A} ^{2}\mathbf {y} _{0}\\\mathbf {y} _{3}&=\mathbf {Ay} _{2}=\mathbf {A} ^{3}\mathbf {y} _{0}\end{aligned}}} and so forth, so that by mathematical induction the solution in terms of t is y t = A t y 0 {\displaystyle \mathbf {y} _{t}=\mathbf {A} ^{t}\mathbf {y} _{0}} Further, if A is diagonalizable, we can rewrite A in terms of its eigenvalues and eigenvectors, giving the solution as y t = P D t P − 1 y 0 , {\displaystyle \mathbf {y} _{t}=\mathbf {PD} ^{t}\mathbf {P} ^{-1}\mathbf {y} _{0},} where P is an n × n matrix whose columns are the eigenvectors of A (assuming the eigenvalues are all distinct) and D is an n × n diagonal matrix whose diagonal elements are the eigenvalues of A. This solution motivates the above stability result: At shrinks to the zero matrix over time if and only if the eigenvalues of A are all less than unity in absolute value. == Extracting the dynamics of a single scalar variable from a first-order matrix system == Starting from the n-dimensional system yt = Ayt−1, we can extract the dynamics of one of the state variables, say y1. The above solution equation for yt shows that the solution for y1,t is in terms of the n eigenvalues of A. Therefore the equation describing the evolution of y1 by itself must have a solution involving those same eigenvalues. This description intuitively motivates the equation of evolution of y1, which is y 1 , t = a 1 y 1 , t − 1 + a 2 y 1 , t − 2 + ⋯ + a n y 1 , t − n {\displaystyle y_{1,t}=a_{1}y_{1,t-1}+a_{2}y_{1,t-2}+\dots +a_{n}y_{1,t-n}} where the parameters ai are from the characteristic equation of the matrix A: λ n − a 1 λ n − 1 − a 2 λ n − 2 − ⋯ − a n λ 0 = 0. {\displaystyle \lambda ^{n}-a_{1}\lambda ^{n-1}-a_{2}\lambda ^{n-2}-\dots -a_{n}\lambda ^{0}=0.} Thus each individual scalar variable of an n-dimensional first-order linear system evolves according to a univariate nth-degree difference equation, which has the same stability property (stable or unstable) as does the matrix difference equation. == Solution and stability of higher-order cases == Matrix difference equations of higher order—that is, with a time lag longer than one period—can be solved, and their stability analyzed, by converting them into first-order form using a block matrix (matrix of matrices). For example, suppose we have the second-order equation x t = A x t − 1 + B x t − 2 {\displaystyle \mathbf {x} _{t}=\mathbf {Ax} _{t-1}+\mathbf {Bx} _{t-2}} with the variable vector x being n × 1 and A and B being n × n. This can be stacked in the form [ x t x t − 1 ] = [ A B I 0 ] [ x t − 1 x t − 2 ] {\displaystyle {\begin{bmatrix}\mathbf {x} _{t}\\\mathbf {x} _{t-1}\\\end{bmatrix}}={\begin{bmatrix}\mathbf {A} &\mathbf {B} \\\mathbf {I} &\mathbf {0} \\\end{bmatrix}}{\begin{bmatrix}\mathbf {x} _{t-1}\\\mathbf {x} _{t-2}\end{bmatrix}}} where I is the n × n identity matrix and 0 is the n × n zero matrix. Then denoting the 2n × 1 stacked vector of current and once-lagged variables as zt and the 2n × 2n block matrix as L, we have as before the solution z t = L t z 0 {\displaystyle \mathbf {z} _{t}=\mathbf {L} ^{t}\mathbf {z} _{0}} Also as before, this stacked equation, and thus the original second-order equation, are stable if and only if all eigenvalues of the matrix L are smaller than unity in absolute value. == Nonlinear matrix difference equations: Riccati equations == In linear-quadratic-Gaussian control, there arises a nonlinear matrix equation for the reverse evolution of a current-and-future-cost matrix, denoted below as H. This equation is called a discrete dynamic Riccati equation, and it arises when a variable vector evolving according to a linear matrix difference equation is controlled by manipulating an exogenous vector in order to optimize a quadratic cost function. This Riccati equation assumes the following, or a similar, form: H t − 1 = K + A ′ H t A − A ′ H t C [ C ′ H t C + R ] − 1 C ′ H t A {\displaystyle \mathbf {H} _{t-1}=\mathbf {K} +\mathbf {A} '\mathbf {H} _{t}\mathbf {A} -\mathbf {A} '\mathbf {H} _{t}\mathbf {C} \left[\mathbf {C} '\mathbf {H} _{t}\mathbf {C} +\mathbf {R} \right]^{-1}\mathbf {C} '\mathbf {H} _{t}\mathbf {A} } where H, K, and A are n × n, C is n × k, R is k × k, n is the number of elements in the vector to be controlled, and k is the number of elements in the control vector. The parameter matrices A and C are from the linear equation, and the parameter matrices K and R are from the quadratic cost function. See here for details. In general this equation cannot be solved analytically for Ht in terms of t; rather, the sequence of values for Ht is found by iterating the Riccati equation. However, it has been shown that this Riccati equation can be solved analytically if R = 0 and n = k + 1, by reducing it to a scalar rational difference equation; moreover, for any k and n if the transition matrix A is nonsingular then the Riccati equation can be solved analytically in terms of the eigenvalues of a matrix, although these may need to be found numerically. In most contexts the evolution of H backwards through time is stable, meaning that H converges to a particular fixed matrix H* which may be irrational even if all the other matrices are rational. See also Stochastic control § Discrete time. A related Riccati equation is X t + 1 = − [ E + B X t ] [ C + A X t ] − 1 {\displaystyle \mathbf {X} _{t+1}=-\left[\mathbf {E} +\mathbf {B} \mathbf {X} _{t}\right]\left[\mathbf {C} +\mathbf {A} \mathbf {X} _{t}\right]^{-1}} in which the matrices X, A, B, C, E are all n × n. This equation can be solved explicitly. Suppose X t = N t D t − 1 , {\displaystyle \mathbf {X} _{t}=\mathbf {N} _{t}\mathbf {D} _{t}^{-1},} which certainly holds for t = 0 with N0 = X0 and with D0 = I. Then using this in the difference equation yields X t + 1 = − [ E + B N t D t − 1 ] D t D t − 1 [ C + A N t D t − 1 ] − 1 = − [ E D t + B N t ] [ [ C + A N t D t − 1 ] D t ] − 1 = − [ E D t + B N t ] [ C D t + A N t ] − 1 = N t + 1 D t + 1 − 1 {\displaystyle {\begin{aligned}\mathbf {X} _{t+1}&=-\left[\mathbf {E} +\mathbf {BN} _{t}\mathbf {D} _{t}^{-1}\right]\mathbf {D} _{t}\mathbf {D} _{t}^{-1}\left[\mathbf {C} +\mathbf {AN} _{t}\mathbf {D} _{t}^{-1}\right]^{-1}\\&=-\left[\mathbf {ED} _{t}+\mathbf {BN} _{t}\right]\left[\left[\mathbf {C} +\mathbf {AN} _{t}\mathbf {D} _{t}^{-1}\right]\mathbf {D} _{t}\right]^{-1}\\&=-\left[\mathbf {ED} _{t}+\mathbf {BN} _{t}\right]\left[\mathbf {CD} _{t}+\mathbf {AN} _{t}\right]^{-1}\\&=\mathbf {N} _{t+1}\mathbf {D} _{t+1}^{-1}\end{aligned}}} so by induction the form X t = N t D t − 1 {\displaystyle \mathbf {X} _{t}=\mathbf {N} _{t}\mathbf {D} _{t}^{-1}} holds for all t. Then the evolution of N and D can be written as [ N t + 1 D t + 1 ] = [ − B − E A C ] [ N t D t ] ≡ J [ N t D t ] {\displaystyle {\begin{bmatrix}\mathbf {N} _{t+1}\\\mathbf {D} _{t+1}\end{bmatrix}}={\begin{bmatrix}-\mathbf {B} &-\mathbf {E} \\\mathbf {A} &\mathbf {C} \end{bmatrix}}{\begin{bmatrix}\mathbf {N} _{t}\\\mathbf {D} _{t}\end{bmatrix}}\equiv \mathbf {J} {\begin{bmatrix}\mathbf {N} _{t}\\\mathbf {D} _{t}\end{bmatrix}}} Thus by induction [ N t D t ] = J t [ N 0 D 0 ] {\displaystyle {\begin{bmatrix}\mathbf {N} _{t}\\\mathbf {D} _{t}\end{bmatrix}}=\mathbf {J} ^{t}{\begin{bmatrix}\mathbf {N} _{0}\\\mathbf {D} _{0}\end{bmatrix}}} == See also == Matrix differential equation Difference equation Linear difference equation Dynamical system Algebraic Riccati equation == References == |
Wikipedia:Matrix factorization of a polynomial#0 | In mathematics, a matrix factorization of a polynomial is a technique for factoring irreducible polynomials with matrices. David Eisenbud proved that every multivariate real-valued polynomial p without linear terms can be written as AB = pI, where A and B are square matrices and I is the identity matrix. Given the polynomial p, the matrices A and B can be found by elementary methods. == Example == The polynomial x2 + y2 is irreducible over R[x,y], but can be written as [ x − y y x ] [ x y − y x ] = ( x 2 + y 2 ) [ 1 0 0 1 ] {\displaystyle \left[{\begin{array}{cc}x&-y\\y&x\end{array}}\right]\left[{\begin{array}{cc}x&y\\-y&x\end{array}}\right]=(x^{2}+y^{2})\left[{\begin{array}{cc}1&0\\0&1\end{array}}\right]} == References == == External links == A Mathematica implementation of an algorithm to matrix-factorize polynomials |
Wikipedia:Matrix norm#0 | In the field of mathematics, norms are defined for elements within a vector space. Specifically, when the vector space comprises matrices, such norms are referred to as matrix norms. Matrix norms differ from vector norms in that they must also interact with matrix multiplication. == Preliminaries == Given a field K {\displaystyle \ K\ } of either real or complex numbers (or any complete subset thereof), let K m × n {\displaystyle \ K^{m\times n}\ } be the K-vector space of matrices with m {\displaystyle m} rows and n {\displaystyle n} columns and entries in the field K . {\displaystyle \ K~.} A matrix norm is a norm on K m × n . {\displaystyle \ K^{m\times n}~.} Norms are often expressed with double vertical bars (like so: ‖ A ‖ {\displaystyle \ \|A\|\ } ). Thus, the matrix norm is a function ‖ ⋅ ‖ : K m × n → R 0 + {\displaystyle \ \|\cdot \|:K^{m\times n}\to \mathbb {R} ^{0+}\ } that must satisfy the following properties: For all scalars α ∈ K {\displaystyle \ \alpha \in K\ } and matrices A , B ∈ K m × n , {\displaystyle \ A,B\in K^{m\times n}\ ,} ‖ A ‖ ≥ 0 {\displaystyle \|A\|\geq 0\ } (positive-valued) ‖ A ‖ = 0 ⟺ A = 0 m , n {\displaystyle \|A\|=0\iff A=0_{m,n}} (definite) ‖ α A ‖ = | α | ‖ A ‖ {\displaystyle \left\|\alpha \ A\right\|=\left|\alpha \right|\ \left\|A\right\|\ } (absolutely homogeneous) ‖ A + B ‖ ≤ ‖ A ‖ + ‖ B ‖ {\displaystyle \|A+B\|\leq \|A\|+\|B\|\ } (sub-additive or satisfying the triangle inequality) The only feature distinguishing matrices from rearranged vectors is multiplication. Matrix norms are particularly useful if they are also sub-multiplicative: ‖ A B ‖ ≤ ‖ A ‖ ‖ B ‖ {\displaystyle \ \left\|AB\right\|\leq \left\|A\right\|\left\|B\right\|\ } Every norm on K n × n {\displaystyle \ K^{n\times n}\ } can be rescaled to be sub-multiplicative; in some books, the terminology matrix norm is reserved for sub-multiplicative norms. == Matrix norms induced by vector norms == Suppose a vector norm ‖ ⋅ ‖ α {\displaystyle \|\cdot \|_{\alpha }} on K n {\displaystyle K^{n}} and a vector norm ‖ ⋅ ‖ β {\displaystyle \|\cdot \|_{\beta }} on K m {\displaystyle K^{m}} are given. Any m × n {\displaystyle m\times n} matrix A induces a linear operator from K n {\displaystyle K^{n}} to K m {\displaystyle K^{m}} with respect to the standard basis, and one defines the corresponding induced norm or operator norm or subordinate norm on the space K m × n {\displaystyle K^{m\times n}} of all m × n {\displaystyle m\times n} matrices as follows: ‖ A ‖ α , β = sup { ‖ A x ‖ β : x ∈ K n such that ‖ x ‖ α ≤ 1 } {\displaystyle \|A\|_{\alpha ,\beta }=\sup\{\|Ax\|_{\beta }:x\in K^{n}{\text{ such that }}\|x\|_{\alpha }\leq 1\}} where sup {\displaystyle \sup } denotes the supremum. This norm measures how much the mapping induced by A {\displaystyle A} can stretch vectors. Depending on the vector norms ‖ ⋅ ‖ α {\displaystyle \|\cdot \|_{\alpha }} , ‖ ⋅ ‖ β {\displaystyle \|\cdot \|_{\beta }} used, notation other than ‖ ⋅ ‖ α , β {\displaystyle \|\cdot \|_{\alpha ,\beta }} can be used for the operator norm. === Matrix norms induced by vector p-norms === If the p-norm for vectors ( 1 ≤ p ≤ ∞ {\displaystyle 1\leq p\leq \infty } ) is used for both spaces K n {\displaystyle K^{n}} and K m , {\displaystyle K^{m},} then the corresponding operator norm is: ‖ A ‖ p = sup { ‖ A x ‖ p : x ∈ K n such that ‖ x ‖ p ≤ 1 } . {\displaystyle \|A\|_{p}=\sup\{\|Ax\|_{p}:x\in K^{n}{\text{ such that }}\|x\|_{p}\leq 1\}.} These induced norms are different from the "entry-wise" p-norms and the Schatten p-norms for matrices treated below, which are also usually denoted by ‖ A ‖ p . {\displaystyle \|A\|_{p}.} Geometrically speaking, one can imagine a p-norm unit ball V p , n = { x ∈ K n : ‖ x ‖ p ≤ 1 } {\displaystyle V_{p,n}=\{x\in K^{n}:\|x\|_{p}\leq 1\}} in K n {\displaystyle K^{n}} , then apply the linear map A {\displaystyle A} to the ball. It would end up becoming a distorted convex shape A V p , n ⊂ K m {\displaystyle AV_{p,n}\subset K^{m}} , and ‖ A ‖ p {\displaystyle \|A\|_{p}} measures the longest "radius" of the distorted convex shape. In other words, we must take a p-norm unit ball V p , m {\displaystyle V_{p,m}} in K m {\displaystyle K^{m}} , then multiply it by at least ‖ A ‖ p {\displaystyle \|A\|_{p}} , in order for it to be large enough to contain A V p , n {\displaystyle AV_{p,n}} . ==== p = 1 or ∞ ==== When p = 1 , {\displaystyle \ p=1\ ,} or p = ∞ , {\displaystyle \ p=\infty \ ,} we have simple formulas. ‖ A ‖ 1 = max 1 ≤ j ≤ n ∑ i = 1 m | a i j | , {\displaystyle \|A\|_{1}=\max _{1\leq j\leq n}\sum _{i=1}^{m}\left|a_{ij}\right|\ ,} which is simply the maximum absolute column sum of the matrix. ‖ A ‖ ∞ = max 1 ≤ i ≤ m ∑ j = 1 n | a i j | , {\displaystyle \|A\|_{\infty }=\max _{1\leq i\leq m}\sum _{j=1}^{n}\left|a_{ij}\right|\ ,} which is simply the maximum absolute row sum of the matrix. For example, for A = [ − 3 5 7 2 6 4 0 2 8 ] , {\displaystyle A={\begin{bmatrix}-3&5&7\\~~2&6&4\\~~0&2&8\\\end{bmatrix}}\ ,} we have that ‖ A ‖ 1 = max { | − 3 | + 2 + 0 , 5 + 6 + 2 , 7 + 4 + 8 } = max { 5 , 13 , 19 } = 19 , {\displaystyle \|A\|_{1}=\max {\bigl \{}\ |{-3}|+2+0\ ,~5+6+2\ ,~7+4+8\ {\bigr \}}=\max {\bigl \{}\ 5\ ,~13\ ,~19\ {\bigr \}}=19\ ,} ‖ A ‖ ∞ = max { | − 3 | + 5 + 7 , 2 + 6 + 4 , 0 + 2 + 8 } = max { 15 , 12 , 10 } = 15 . {\displaystyle \|A\|_{\infty }=\max {\bigl \{}\ |{-3}|+5+7\ ,~2+6+4\ ,~0+2+8\ {\bigr \}}=\max {\bigl \{}\ 15\ ,~12\ ,~10\ {\bigr \}}=15~.} ==== Spectral norm (p = 2) ==== When p = 2 {\displaystyle p=2} (the Euclidean norm or ℓ 2 {\displaystyle \ell _{2}} -norm for vectors), the induced matrix norm is the spectral norm. The two values do not coincide in infinite dimensions — see Spectral radius for further discussion. The spectral radius should not be confused with the spectral norm. The spectral norm of a matrix A {\displaystyle A} is the largest singular value of A {\displaystyle A} , i.e., the square root of the largest eigenvalue of the matrix A ∗ A , {\displaystyle A^{*}A,} where A ∗ {\displaystyle A^{*}} denotes the conjugate transpose of A {\displaystyle A} : ‖ A ‖ 2 = λ max ( A ∗ A ) = σ max ( A ) . {\displaystyle \|A\|_{2}={\sqrt {\lambda _{\max }\left(A^{*}A\right)}}=\sigma _{\max }(A).} where σ max ( A ) {\displaystyle \sigma _{\max }(A)} represents the largest singular value of matrix A . {\displaystyle A.} There are further properties: ‖ A ‖ 2 = sup { x ∗ A y : x ∈ K m , y ∈ K n with ‖ x ‖ 2 = ‖ y ‖ 2 = 1 } . {\textstyle \|A\|_{2}=\sup\{x^{*}Ay:x\in K^{m},y\in K^{n}{\text{ with }}\|x\|_{2}=\|y\|_{2}=1\}.} Proved by the Cauchy–Schwarz inequality. ‖ A ∗ A ‖ 2 = ‖ A A ∗ ‖ 2 = ‖ A ‖ 2 2 {\textstyle \|A^{*}A\|_{2}=\|AA^{*}\|_{2}=\|A\|_{2}^{2}} . Proven by singular value decomposition (SVD) on A {\displaystyle A} . ‖ A ‖ 2 = σ m a x ( A ) ≤ ‖ A ‖ F = ∑ i σ i ( A ) 2 {\textstyle \|A\|_{2}=\sigma _{\mathrm {max} }(A)\leq \|A\|_{\rm {F}}={\sqrt {\sum _{i}\sigma _{i}(A)^{2}}}} , where ‖ A ‖ F {\displaystyle \|A\|_{\textrm {F}}} is the Frobenius norm. Equality holds if and only if the matrix A {\displaystyle A} is a rank-one matrix or a zero matrix. Conversely, ‖ A ‖ F ≤ min ( m , n ) 1 / 2 ‖ A ‖ 2 {\displaystyle \|A\|_{\textrm {F}}\leq \min(m,n)^{1/2}\|A\|_{2}} . ‖ A ‖ 2 = ρ ( A ∗ A ) ≤ ‖ A ∗ A ‖ ∞ ≤ ‖ A ‖ 1 ‖ A ‖ ∞ {\displaystyle \|A\|_{2}={\sqrt {\rho (A^{*}A)}}\leq {\sqrt {\|A^{*}A\|_{\infty }}}\leq {\sqrt {\|A\|_{1}\|A\|_{\infty }}}} . === Matrix norms induced by vector α- and β-norms === We can generalize the above definition. Suppose we have vector norms ‖ ⋅ ‖ α {\displaystyle \|\cdot \|_{\alpha }} and ‖ ⋅ ‖ β {\displaystyle \|\cdot \|_{\beta }} for spaces K n {\displaystyle K^{n}} and K m {\displaystyle K^{m}} respectively; the corresponding operator norm is ‖ A ‖ α , β = sup { ‖ A x ‖ β : x ∈ K n such that ‖ x ‖ α ≤ 1 } {\displaystyle \|A\|_{\alpha ,\beta }=\sup\{\|Ax\|_{\beta }:x\in K^{n}{\text{ such that }}\|x\|_{\alpha }\leq 1\}} In particular, the ‖ A ‖ p {\displaystyle \|A\|_{p}} defined previously is the special case of ‖ A ‖ p , p {\displaystyle \|A\|_{p,p}} . In the special cases of α = 2 {\displaystyle \alpha =2} and β = ∞ {\displaystyle \beta =\infty } , the induced matrix norms can be computed by ‖ A ‖ 2 , ∞ = max 1 ≤ i ≤ m ‖ A i : ‖ 2 , {\displaystyle \|A\|_{2,\infty }=\max _{1\leq i\leq m}\|A_{i:}\|_{2},} where A i : {\displaystyle A_{i:}} is the i-th row of matrix A {\displaystyle A} . In the special cases of α = 1 {\displaystyle \alpha =1} and β = 2 {\displaystyle \beta =2} , the induced matrix norms can be computed by ‖ A ‖ 1 , 2 = max 1 ≤ j ≤ n ‖ A : j ‖ 2 , {\displaystyle \|A\|_{1,2}=\max _{1\leq j\leq n}\|A_{:j}\|_{2},} where A : j {\displaystyle A_{:j}} is the j-th column of matrix A {\displaystyle A} . Hence, ‖ A ‖ 2 , ∞ {\displaystyle \|A\|_{2,\infty }} and ‖ A ‖ 1 , 2 {\displaystyle \|A\|_{1,2}} are the maximum row and column 2-norm of the matrix, respectively. === Properties === Any operator norm is consistent with the vector norms that induce it, giving ‖ A x ‖ β ≤ ‖ A ‖ α , β ‖ x ‖ α . {\displaystyle \|Ax\|_{\beta }\leq \|A\|_{\alpha ,\beta }\|x\|_{\alpha }.} Suppose ‖ ⋅ ‖ α , β {\displaystyle \|\cdot \|_{\alpha ,\beta }} ; ‖ ⋅ ‖ β , γ {\displaystyle \|\cdot \|_{\beta ,\gamma }} ; and ‖ ⋅ ‖ α , γ {\displaystyle \|\cdot \|_{\alpha ,\gamma }} are operator norms induced by the respective pairs of vector norms ( ‖ ⋅ ‖ α , ‖ ⋅ ‖ β ) {\displaystyle (\|\cdot \|_{\alpha },\|\cdot \|_{\beta })} ; ( ‖ ⋅ ‖ β , ‖ ⋅ ‖ γ ) {\displaystyle (\|\cdot \|_{\beta },\|\cdot \|_{\gamma })} ; and ( ‖ ⋅ ‖ α , ‖ ⋅ ‖ γ ) {\displaystyle (\|\cdot \|_{\alpha },\|\cdot \|_{\gamma })} . Then, ‖ A B ‖ α , γ ≤ ‖ A ‖ β , γ ‖ B ‖ α , β ; {\displaystyle \|AB\|_{\alpha ,\gamma }\leq \|A\|_{\beta ,\gamma }\|B\|_{\alpha ,\beta };} this follows from ‖ A B x ‖ γ ≤ ‖ A ‖ β , γ ‖ B x ‖ β ≤ ‖ A ‖ β , γ ‖ B ‖ α , β ‖ x ‖ α {\displaystyle \|ABx\|_{\gamma }\leq \|A\|_{\beta ,\gamma }\|Bx\|_{\beta }\leq \|A\|_{\beta ,\gamma }\|B\|_{\alpha ,\beta }\|x\|_{\alpha }} and sup ‖ x ‖ α = 1 ‖ A B x ‖ γ = ‖ A B ‖ α , γ . {\displaystyle \sup _{\|x\|_{\alpha }=1}\|ABx\|_{\gamma }=\|AB\|_{\alpha ,\gamma }.} === Square matrices === Suppose ‖ ⋅ ‖ α , α {\displaystyle \|\cdot \|_{\alpha ,\alpha }} is an operator norm on the space of square matrices K n × n {\displaystyle K^{n\times n}} induced by vector norms ‖ ⋅ ‖ α {\displaystyle \|\cdot \|_{\alpha }} and ‖ ⋅ ‖ α {\displaystyle \|\cdot \|_{\alpha }} . Then, the operator norm is a sub-multiplicative matrix norm: ‖ A B ‖ α , α ≤ ‖ A ‖ α , α ‖ B ‖ α , α . {\displaystyle \|AB\|_{\alpha ,\alpha }\leq \|A\|_{\alpha ,\alpha }\|B\|_{\alpha ,\alpha }.} Moreover, any such norm satisfies the inequality for all positive integers r, where ρ(A) is the spectral radius of A. For symmetric or hermitian A, we have equality in (1) for the 2-norm, since in this case the 2-norm is precisely the spectral radius of A. For an arbitrary matrix, we may not have equality for any norm; a counterexample would be A = [ 0 1 0 0 ] , {\displaystyle A={\begin{bmatrix}0&1\\0&0\end{bmatrix}},} which has vanishing spectral radius. In any case, for any matrix norm, we have the spectral radius formula: lim r → ∞ ‖ A r ‖ 1 / r = ρ ( A ) . {\displaystyle \lim _{r\to \infty }\|A^{r}\|^{1/r}=\rho (A).} === Energy norms === If the vector norms ‖ ⋅ ‖ α {\displaystyle \|\cdot \|_{\alpha }} and ‖ ⋅ ‖ β {\displaystyle \|\cdot \|_{\beta }} are given in terms of energy norms based on symmetric positive definite matrices P {\displaystyle P} and Q {\displaystyle Q} respectively, the resulting operator norm is given as ‖ A ‖ P , Q = sup { ‖ A x ‖ Q : ‖ x ‖ P ≤ 1 } . {\displaystyle \|A\|_{P,Q}=\sup\{\|Ax\|_{Q}:\|x\|_{P}\leq 1\}.} Using the symmetric matrix square roots of P {\displaystyle P} and Q {\displaystyle Q} respectively, the operator norm can be expressed as the spectral norm of a modified matrix: ‖ A ‖ P , Q = ‖ Q 1 / 2 A P − 1 / 2 ‖ 2 . {\displaystyle \|A\|_{P,Q}=\|Q^{1/2}AP^{-1/2}\|_{2}.} == Consistent and compatible norms == A matrix norm ‖ ⋅ ‖ {\displaystyle \|\cdot \|} on K m × n {\displaystyle K^{m\times n}} is called consistent with a vector norm ‖ ⋅ ‖ α {\displaystyle \|\cdot \|_{\alpha }} on K n {\displaystyle K^{n}} and a vector norm ‖ ⋅ ‖ β {\displaystyle \|\cdot \|_{\beta }} on K m {\displaystyle K^{m}} , if: ‖ A x ‖ β ≤ ‖ A ‖ ‖ x ‖ α {\displaystyle \left\|Ax\right\|_{\beta }\leq \left\|A\right\|\left\|x\right\|_{\alpha }} for all A ∈ K m × n {\displaystyle A\in K^{m\times n}} and all x ∈ K n {\displaystyle x\in K^{n}} . In the special case of m = n and α = β {\displaystyle \alpha =\beta } , ‖ ⋅ ‖ {\displaystyle \|\cdot \|} is also called compatible with ‖ ⋅ ‖ α {\displaystyle \|\cdot \|_{\alpha }} . All induced norms are consistent by definition. Also, any sub-multiplicative matrix norm on K n × n {\displaystyle K^{n\times n}} induces a compatible vector norm on K n {\displaystyle K^{n}} by defining ‖ v ‖ := ‖ ( v , v , … , v ) ‖ {\displaystyle \left\|v\right\|:=\left\|\left(v,v,\dots ,v\right)\right\|} . == "Entry-wise" matrix norms == These norms treat an m × n {\displaystyle m\times n} matrix as a vector of size m ⋅ n {\displaystyle m\cdot n} , and use one of the familiar vector norms. For example, using the p-norm for vectors, p ≥ 1, we get: ‖ A ‖ p , p = ‖ v e c ( A ) ‖ p = ( ∑ i = 1 m ∑ j = 1 n | a i j | p ) 1 / p {\displaystyle \|A\|_{p,p}=\|\mathrm {vec} (A)\|_{p}=\left(\sum _{i=1}^{m}\sum _{j=1}^{n}|a_{ij}|^{p}\right)^{1/p}} This is a different norm from the induced p-norm (see above) and the Schatten p-norm (see below), but the notation is the same. The special case p = 2 is the Frobenius norm, and p = ∞ yields the maximum norm. === L2,1 and Lp,q norms === Let ( a 1 , … , a n ) {\displaystyle (a_{1},\ldots ,a_{n})} be the dimension m columns of matrix A {\displaystyle A} . From the original definition, the matrix A {\displaystyle A} presents n data points in an m-dimensional space. The L 2 , 1 {\displaystyle L_{2,1}} norm is the sum of the Euclidean norms of the columns of the matrix: ‖ A ‖ 2 , 1 = ∑ j = 1 n ‖ a j ‖ 2 = ∑ j = 1 n ( ∑ i = 1 m | a i j | 2 ) 1 / 2 {\displaystyle \|A\|_{2,1}=\sum _{j=1}^{n}\|a_{j}\|_{2}=\sum _{j=1}^{n}\left(\sum _{i=1}^{m}|a_{ij}|^{2}\right)^{1/2}} The L 2 , 1 {\displaystyle L_{2,1}} norm as an error function is more robust, since the error for each data point (a column) is not squared. It is used in robust data analysis and sparse coding. For p, q ≥ 1, the L 2 , 1 {\displaystyle L_{2,1}} norm can be generalized to the L p , q {\displaystyle L_{p,q}} norm as follows: ‖ A ‖ p , q = ( ∑ j = 1 n ( ∑ i = 1 m | a i j | p ) q p ) 1 q . {\displaystyle \|A\|_{p,q}=\left(\sum _{j=1}^{n}\left(\sum _{i=1}^{m}|a_{ij}|^{p}\right)^{\frac {q}{p}}\right)^{\frac {1}{q}}.} === Frobenius norm === When p = q = 2 for the L p , q {\displaystyle L_{p,q}} norm, it is called the Frobenius norm or the Hilbert–Schmidt norm, though the latter term is used more frequently in the context of operators on (possibly infinite-dimensional) Hilbert space. This norm can be defined in various ways: ‖ A ‖ F = ∑ i m ∑ j n | a i j | 2 = trace ( A ∗ A ) = ∑ i = 1 min { m , n } σ i 2 ( A ) , {\displaystyle \|A\|_{\text{F}}={\sqrt {\sum _{i}^{m}\sum _{j}^{n}|a_{ij}|^{2}}}={\sqrt {\operatorname {trace} \left(A^{*}A\right)}}={\sqrt {\sum _{i=1}^{\min\{m,n\}}\sigma _{i}^{2}(A)}},} where the trace is the sum of diagonal entries, and σ i ( A ) {\displaystyle \sigma _{i}(A)} are the singular values of A {\displaystyle A} . The second equality is proven by explicit computation of t r a c e ( A ∗ A ) {\displaystyle \mathrm {trace} (A^{*}A)} . The third equality is proven by singular value decomposition of A {\displaystyle A} , and the fact that the trace is invariant under circular shifts. The Frobenius norm is an extension of the Euclidean norm to K n × n {\displaystyle K^{n\times n}} and comes from the Frobenius inner product on the space of all matrices. The Frobenius norm is sub-multiplicative and is very useful for numerical linear algebra. The sub-multiplicativity of Frobenius norm can be proved using Cauchy–Schwarz inequality. In fact, it is more than sub-multiplicative, as ‖ A B ‖ F ≤ ‖ A ‖ o p ‖ B ‖ F {\displaystyle \|AB\|_{F}\leq \|A\|_{op}\|B\|_{F}} where the operator norm ‖ ⋅ ‖ o p ≤ ‖ ⋅ ‖ F {\displaystyle \|\cdot \|_{op}\leq \|\cdot \|_{F}} . Frobenius norm is often easier to compute than induced norms, and has the useful property of being invariant under rotations (and unitary operations in general). That is, ‖ A ‖ F = ‖ A U ‖ F = ‖ U A ‖ F {\displaystyle \|A\|_{\text{F}}=\|AU\|_{\text{F}}=\|UA\|_{\text{F}}} for any unitary matrix U {\displaystyle U} . This property follows from the cyclic nature of the trace ( trace ( X Y Z ) = trace ( Y Z X ) = trace ( Z X Y ) {\displaystyle \operatorname {trace} (XYZ)=\operatorname {trace} (YZX)=\operatorname {trace} (ZXY)} ): ‖ A U ‖ F 2 = trace ( ( A U ) ∗ A U ) = trace ( U ∗ A ∗ A U ) = trace ( U U ∗ A ∗ A ) = trace ( A ∗ A ) = ‖ A ‖ F 2 , {\displaystyle \|AU\|_{\text{F}}^{2}=\operatorname {trace} \left((AU)^{*}AU\right)=\operatorname {trace} \left(U^{*}A^{*}AU\right)=\operatorname {trace} \left(UU^{*}A^{*}A\right)=\operatorname {trace} \left(A^{*}A\right)=\|A\|_{\text{F}}^{2},} and analogously: ‖ U A ‖ F 2 = trace ( ( U A ) ∗ U A ) = trace ( A ∗ U ∗ U A ) = trace ( A ∗ A ) = ‖ A ‖ F 2 , {\displaystyle \|UA\|_{\text{F}}^{2}=\operatorname {trace} \left((UA)^{*}UA\right)=\operatorname {trace} \left(A^{*}U^{*}UA\right)=\operatorname {trace} \left(A^{*}A\right)=\|A\|_{\text{F}}^{2},} where we have used the unitary nature of U {\displaystyle U} (that is, U ∗ U = U U ∗ = I {\displaystyle U^{*}U=UU^{*}=\mathbf {I} } ). It also satisfies ‖ A ∗ A ‖ F = ‖ A A ∗ ‖ F ≤ ‖ A ‖ F 2 {\displaystyle \|A^{*}A\|_{\text{F}}=\|AA^{*}\|_{\text{F}}\leq \|A\|_{\text{F}}^{2}} and ‖ A + B ‖ F 2 = ‖ A ‖ F 2 + ‖ B ‖ F 2 + 2 Re ( ⟨ A , B ⟩ F ) , {\displaystyle \|A+B\|_{\text{F}}^{2}=\|A\|_{\text{F}}^{2}+\|B\|_{\text{F}}^{2}+2\operatorname {Re} \left(\langle A,B\rangle _{\text{F}}\right),} where ⟨ A , B ⟩ F {\displaystyle \langle A,B\rangle _{\text{F}}} is the Frobenius inner product, and Re is the real part of a complex number (irrelevant for real matrices) === Max norm === The max norm is the elementwise norm in the limit as p = q goes to infinity: ‖ A ‖ max = max i , j | a i j | . {\displaystyle \|A\|_{\max }=\max _{i,j}|a_{ij}|.} This norm is not sub-multiplicative; but modifying the right-hand side to m n max i , j | a i j | {\displaystyle {\sqrt {mn}}\max _{i,j}\vert a_{ij}\vert } makes it so. Note that in some literature (such as Communication complexity), an alternative definition of max-norm, also called the γ 2 {\displaystyle \gamma _{2}} -norm, refers to the factorization norm: γ 2 ( A ) = min U , V : A = U V T ‖ U ‖ 2 , ∞ ‖ V ‖ 2 , ∞ = min U , V : A = U V T max i , j ‖ U i , : ‖ 2 ‖ V j , : ‖ 2 {\displaystyle \gamma _{2}(A)=\min _{U,V:A=UV^{T}}\|U\|_{2,\infty }\|V\|_{2,\infty }=\min _{U,V:A=UV^{T}}\max _{i,j}\|U_{i,:}\|_{2}\|V_{j,:}\|_{2}} == Schatten norms == The Schatten p-norms arise when applying the p-norm to the vector of singular values of a matrix. If the singular values of the m × n {\displaystyle m\times n} matrix A {\displaystyle A} are denoted by σi, then the Schatten p-norm is defined by ‖ A ‖ p = ( ∑ i = 1 min { m , n } σ i p ( A ) ) 1 / p . {\displaystyle \|A\|_{p}=\left(\sum _{i=1}^{\min\{m,n\}}\sigma _{i}^{p}(A)\right)^{1/p}.} These norms again share the notation with the induced and entry-wise p-norms, but they are different. All Schatten norms are sub-multiplicative. They are also unitarily invariant, which means that ‖ A ‖ = ‖ U A V ‖ {\displaystyle \|A\|=\|UAV\|} for all matrices A {\displaystyle A} and all unitary matrices U {\displaystyle U} and V {\displaystyle V} . The most familiar cases are p = 1, 2, ∞. The case p = 2 yields the Frobenius norm, introduced before. The case p = ∞ yields the spectral norm, which is the operator norm induced by the vector 2-norm (see above). Finally, p = 1 yields the nuclear norm (also known as the trace norm, or the Ky Fan 'n'-norm), defined as: ‖ A ‖ ∗ = trace ( A ∗ A ) = ∑ i = 1 min { m , n } σ i ( A ) , {\displaystyle \|A\|_{*}=\operatorname {trace} \left({\sqrt {A^{*}A}}\right)=\sum _{i=1}^{\min\{m,n\}}\sigma _{i}(A),} where A ∗ A {\displaystyle {\sqrt {A^{*}A}}} denotes a positive semidefinite matrix B {\displaystyle B} such that B B = A ∗ A {\displaystyle BB=A^{*}A} . More precisely, since A ∗ A {\displaystyle A^{*}A} is a positive semidefinite matrix, its square root is well defined. The nuclear norm ‖ A ‖ ∗ {\displaystyle \|A\|_{*}} is a convex envelope of the rank function rank ( A ) {\displaystyle {\text{rank}}(A)} , so it is often used in mathematical optimization to search for low-rank matrices. Combining von Neumann's trace inequality with Hölder's inequality for Euclidean space yields a version of Hölder's inequality for Schatten norms for 1 / p + 1 / q = 1 {\displaystyle 1/p+1/q=1} : | trace ( A ∗ B ) | ≤ ‖ A ‖ p ‖ B ‖ q , {\displaystyle \left|\operatorname {trace} (A^{*}B)\right|\leq \|A\|_{p}\|B\|_{q},} In particular, this implies the Schatten norm inequality ‖ A ‖ F 2 ≤ ‖ A ‖ p ‖ A ‖ q . {\displaystyle \|A\|_{F}^{2}\leq \|A\|_{p}\|A\|_{q}.} == Monotone norms == A matrix norm ‖ ⋅ ‖ {\displaystyle \|\cdot \|} is called monotone if it is monotonic with respect to the Loewner order. Thus, a matrix norm is increasing if A ≼ B ⇒ ‖ A ‖ ≤ ‖ B ‖ . {\displaystyle A\preccurlyeq B\Rightarrow \|A\|\leq \|B\|.} The Frobenius norm and spectral norm are examples of monotone norms. == Cut norms == Another source of inspiration for matrix norms arises from considering a matrix as the adjacency matrix of a weighted, directed graph. The so-called "cut norm" measures how close the associated graph is to being bipartite: ‖ A ‖ ◻ = max S ⊆ [ n ] , T ⊆ [ m ] | ∑ s ∈ S , t ∈ T A t , s | {\displaystyle \|A\|_{\Box }=\max _{S\subseteq [n],T\subseteq [m]}{\left|\sum _{s\in S,t\in T}{A_{t,s}}\right|}} where A ∈ Km×n. Equivalent definitions (up to a constant factor) impose the conditions 2|S| > n & 2|T| > m; S = T; or S ∩ T = ∅. The cut-norm is equivalent to the induced operator norm ‖·‖∞→1, which is itself equivalent to another norm, called the Grothendieck norm. To define the Grothendieck norm, first note that a linear operator K1 → K1 is just a scalar, and thus extends to a linear operator on any Kk → Kk. Moreover, given any choice of basis for Kn and Km, any linear operator Kn → Km extends to a linear operator (Kk)n → (Kk)m, by letting each matrix element on elements of Kk via scalar multiplication. The Grothendieck norm is the norm of that extended operator; in symbols: ‖ A ‖ G , k = sup each u j , v j ∈ K k ; ‖ u j ‖ = ‖ v j ‖ = 1 ∑ j ∈ [ n ] , ℓ ∈ [ m ] ( u j ⋅ v j ) A ℓ , j {\displaystyle \|A\|_{G,k}=\sup _{{\text{each }}u_{j},v_{j}\in K^{k};\|u_{j}\|=\|v_{j}\|=1}{\sum _{j\in [n],\ell \in [m]}{(u_{j}\cdot v_{j})A_{\ell ,j}}}} The Grothendieck norm depends on choice of basis (usually taken to be the standard basis) and k. == Equivalence of norms == For any two matrix norms ‖ ⋅ ‖ α {\displaystyle \|\cdot \|_{\alpha }} and ‖ ⋅ ‖ β {\displaystyle \|\cdot \|_{\beta }} , we have that: r ‖ A ‖ α ≤ ‖ A ‖ β ≤ s ‖ A ‖ α {\displaystyle r\|A\|_{\alpha }\leq \|A\|_{\beta }\leq s\|A\|_{\alpha }} for some positive numbers r and s, for all matrices A ∈ K m × n {\displaystyle A\in K^{m\times n}} . In other words, all norms on K m × n {\displaystyle K^{m\times n}} are equivalent; they induce the same topology on K m × n {\displaystyle K^{m\times n}} . This is true because the vector space K m × n {\displaystyle K^{m\times n}} has the finite dimension m × n {\displaystyle m\times n} . Moreover, for every matrix norm ‖ ⋅ ‖ {\displaystyle \|\cdot \|} on R n × n {\displaystyle \mathbb {R} ^{n\times n}} there exists a unique positive real number k {\displaystyle k} such that ℓ ‖ ⋅ ‖ {\displaystyle \ell \|\cdot \|} is a sub-multiplicative matrix norm for every ℓ ≥ k {\displaystyle \ell \geq k} ; to wit, k = sup { ‖ A B ‖ : ‖ A ‖ ≤ 1 , ‖ B ‖ ≤ 1 } . {\displaystyle k=\sup\{\Vert AB\Vert \,:\,\Vert A\Vert \leq 1,\Vert B\Vert \leq 1\}.} A sub-multiplicative matrix norm ‖ ⋅ ‖ α {\displaystyle \|\cdot \|_{\alpha }} is said to be minimal, if there exists no other sub-multiplicative matrix norm ‖ ⋅ ‖ β {\displaystyle \|\cdot \|_{\beta }} satisfying ‖ ⋅ ‖ β < ‖ ⋅ ‖ α {\displaystyle \|\cdot \|_{\beta }<\|\cdot \|_{\alpha }} . === Examples of norm equivalence === Let ‖ A ‖ p {\displaystyle \|A\|_{p}} once again refer to the norm induced by the vector p-norm (as above in the Induced norm section). For matrix A ∈ R m × n {\displaystyle A\in \mathbb {R} ^{m\times n}} of rank r {\displaystyle r} , the following inequalities hold: ‖ A ‖ 2 ≤ ‖ A ‖ F ≤ r ‖ A ‖ 2 {\displaystyle \|A\|_{2}\leq \|A\|_{F}\leq {\sqrt {r}}\|A\|_{2}} ‖ A ‖ F ≤ ‖ A ‖ ∗ ≤ r ‖ A ‖ F {\displaystyle \|A\|_{F}\leq \|A\|_{*}\leq {\sqrt {r}}\|A\|_{F}} ‖ A ‖ max ≤ ‖ A ‖ 2 ≤ m n ‖ A ‖ max {\displaystyle \|A\|_{\max }\leq \|A\|_{2}\leq {\sqrt {mn}}\|A\|_{\max }} 1 n ‖ A ‖ ∞ ≤ ‖ A ‖ 2 ≤ m ‖ A ‖ ∞ {\displaystyle {\frac {1}{\sqrt {n}}}\|A\|_{\infty }\leq \|A\|_{2}\leq {\sqrt {m}}\|A\|_{\infty }} 1 m ‖ A ‖ 1 ≤ ‖ A ‖ 2 ≤ n ‖ A ‖ 1 . {\displaystyle {\frac {1}{\sqrt {m}}}\|A\|_{1}\leq \|A\|_{2}\leq {\sqrt {n}}\|A\|_{1}.} == See also == Dual norm Logarithmic norm == Notes == == References == == Bibliography == James W. Demmel, Applied Numerical Linear Algebra, section 1.7, published by SIAM, 1997. Carl D. Meyer, Matrix Analysis and Applied Linear Algebra, published by SIAM, 2000. [1] John Watrous, Theory of Quantum Information, 2.3 Norms of operators, lecture notes, University of Waterloo, 2011. Kendall Atkinson, An Introduction to Numerical Analysis, published by John Wiley & Sons, Inc 1989 |
Wikipedia:Matrix pencil#0 | In linear algebra, a matrix pencil is a matrix-valued polynomial function defined on a field K {\displaystyle K} , usually the real or complex numbers. == Definition == Let K {\displaystyle K} be a field (typically, K ∈ { R , C } {\displaystyle K\in \{\mathbb {R} ,\mathbb {C} \}} ; the definition can be generalized to rngs), let ℓ ≥ 0 {\displaystyle \ell \geq 0} be a non-negative integer, let n > 0 {\displaystyle n>0} be a positive integer, and let A 0 , A 1 , … , A ℓ {\displaystyle A_{0},A_{1},\dots ,A_{\ell }} be n × n {\displaystyle n\times n} matrices (i. e. A i ∈ M a t ( K , n × n ) {\displaystyle A_{i}\in \mathrm {Mat} (K,n\times n)} for all i = 0 , … , ℓ {\displaystyle i=0,\dots ,\ell } ). Then the matrix pencil defined by A 0 , … , A ℓ {\displaystyle A_{0},\dots ,A_{\ell }} is the matrix-valued function L : K → M a t ( K , n × n ) {\displaystyle L\colon K\to \mathrm {Mat} (K,n\times n)} defined by L ( λ ) = ∑ i = 0 ℓ λ i A i . {\displaystyle L(\lambda )=\sum _{i=0}^{\ell }\lambda ^{i}A_{i}.} The degree of the matrix pencil is defined as the largest integer 0 ≤ k ≤ ℓ {\displaystyle 0\leq k\leq \ell } such that A k ≠ 0 {\displaystyle A_{k}\neq 0} , the n × n {\displaystyle n\times n} zero matrix over K {\displaystyle K} . === Linear matrix pencils === A particular case is a linear matrix pencil L ( λ ) = A − λ B {\displaystyle L(\lambda )=A-\lambda B} (where B ≠ 0 {\displaystyle B\neq 0} ). We denote it briefly with the notation ( A , B ) {\displaystyle (A,B)} , and note that using the more general notation, A 0 = A {\displaystyle A_{0}=A} and A 1 = − B {\displaystyle A_{1}=-B} (not B {\displaystyle B} ). == Properties == A pencil is called regular if there is at least one value of λ {\displaystyle \lambda } such that det ( L ( λ ) ) ≠ 0 {\displaystyle \det(L(\lambda ))\neq 0} ; otherwise it is called singular. We call eigenvalues of a matrix pencil all (complex) numbers λ {\displaystyle \lambda } for which det ( L ( λ ) ) = 0 {\displaystyle \det(L(\lambda ))=0} ; in particular, the eigenvalues of the matrix pencil ( A , I ) {\displaystyle (A,I)} are the matrix eigenvalues of A {\displaystyle A} . For linear pencils in particular, the eigenvalues of the pencil are also called generalized eigenvalues. The set of the eigenvalues of a pencil is called the spectrum of the pencil, and is written σ ( A 0 , … , A ℓ ) {\displaystyle \sigma (A_{0},\dots ,A_{\ell })} . For the linear pencil ( A , B ) {\displaystyle (A,B)} , it is written as σ ( A , B ) {\displaystyle \sigma (A,B)} (not σ ( A , − B ) {\displaystyle \sigma (A,-B)} ). The linear pencil ( A , B ) {\displaystyle (A,B)} is said to have one or more eigenvalues at infinity if B {\displaystyle B} has one or more 0 eigenvalues. == Applications == Matrix pencils play an important role in numerical linear algebra. The problem of finding the eigenvalues of a pencil is called the generalized eigenvalue problem. The most popular algorithm for this task is the QZ algorithm, which is an implicit version of the QR algorithm to solve the eigenvalue problem A x = λ B x {\displaystyle Ax=\lambda Bx} without inverting the matrix B {\displaystyle B} (which is impossible when B {\displaystyle B} is singular, or numerically unstable when it is ill-conditioned). == Pencils generated by commuting matrices == If A B = B A {\displaystyle AB=BA} , then the pencil generated by A {\displaystyle A} and B {\displaystyle B} : consists only of matrices similar to a diagonal matrix, or has no matrices in it similar to a diagonal matrix, or has exactly one matrix in it similar to a diagonal matrix. == See also == Generalized eigenvalue problem Generalized pencil-of-function method Nonlinear eigenproblem Quadratic eigenvalue problem Generalized Rayleigh quotient == Notes == == References == Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Baltimore: Johns Hopkins University Press, ISBN 0-8018-5414-8 Marcus & Minc (1969), A survey of matrix theory and matrix inequalities, Courier Dover Publications Peter Lancaster & Qian Ye (1991) "Variational and numerical methods for symmetric matrix pencils", Bulletin of the Australian Mathematical Society 43: 1 to 17 |
Wikipedia:Matrix sign function#0 | In mathematics, the matrix sign function is a matrix function on square matrices analogous to the complex sign function. It was introduced by J.D. Roberts in 1971 as a tool for model reduction and for solving Lyapunov and Algebraic Riccati equation in a technical report of Cambridge University, which was later published in a journal in 1980. == Definition == The matrix sign function is a generalization of the complex signum function csgn ( z ) = { 1 if R e ( z ) > 0 , − 1 if R e ( z ) < 0 , {\displaystyle \operatorname {csgn} (z)={\begin{cases}1&{\text{if }}\mathrm {Re} (z)>0,\\-1&{\text{if }}\mathrm {Re} (z)<0,\end{cases}}} to the matrix valued analogue csgn ( A ) {\displaystyle \operatorname {csgn} (A)} . Although the sign function is not analytic, the matrix function is well defined for all matrices that have no eigenvalue on the imaginary axis, see for example the Jordan-form-based definition (where the derivatives are all zero). == Properties == Theorem: Let A ∈ C n × n {\displaystyle A\in \mathbb {C} ^{n\times n}} , then csgn ( A ) 2 = I {\displaystyle \operatorname {csgn} (A)^{2}=I} . Theorem: Let A ∈ C n × n {\displaystyle A\in \mathbb {C} ^{n\times n}} , then csgn ( A ) {\displaystyle \operatorname {csgn} (A)} is diagonalizable and has eigenvalues that are ± 1 {\displaystyle \pm 1} . Theorem: Let A ∈ C n × n {\displaystyle A\in \mathbb {C} ^{n\times n}} , then ( I + csgn ( A ) ) / 2 {\displaystyle (I+\operatorname {csgn} (A))/2} is a projector onto the invariant subspace associated with the eigenvalues in the right-half plane, and analogously for ( I − csgn ( A ) ) / 2 {\displaystyle (I-\operatorname {csgn} (A))/2} and the left-half plane. Theorem: Let A ∈ C n × n {\displaystyle A\in \mathbb {C} ^{n\times n}} , and A = P [ J + 0 0 J − ] P − 1 {\displaystyle A=P{\begin{bmatrix}J_{+}&0\\0&J_{-}\end{bmatrix}}P^{-1}} be a Jordan decomposition such that J + {\displaystyle J_{+}} corresponds to eigenvalues with positive real part and J − {\displaystyle J_{-}} to eigenvalue with negative real part. Then csgn ( A ) = P [ I + 0 0 − I − ] P − 1 {\displaystyle \operatorname {csgn} (A)=P{\begin{bmatrix}I_{+}&0\\0&-I_{-}\end{bmatrix}}P^{-1}} , where I + {\displaystyle I_{+}} and I − {\displaystyle I_{-}} are identity matrices of sizes corresponding to J + {\displaystyle J_{+}} and J − {\displaystyle J_{-}} , respectively. == Computational methods == The function can be computed with generic methods for matrix functions, but there are also specialized methods. === Newton iteration === The Newton iteration can be derived by observing that csgn ( x ) = x 2 / x {\displaystyle \operatorname {csgn} (x)={\sqrt {x^{2}}}/x} , which in terms of matrices can be written as csgn ( A ) = A − 1 A 2 {\displaystyle \operatorname {csgn} (A)=A^{-1}{\sqrt {A^{2}}}} , where we use the matrix square root. If we apply the Babylonian method to compute the square root of the matrix A 2 {\displaystyle A^{2}} , that is, the iteration X k + 1 = 1 2 ( X k + A X k − 1 ) {\textstyle X_{k+1}={\frac {1}{2}}\left(X_{k}+AX_{k}^{-1}\right)} , and define the new iterate Z k = A − 1 X k {\displaystyle Z_{k}=A^{-1}X_{k}} , we arrive at the iteration Z k + 1 = 1 2 ( Z k + Z k − 1 ) {\displaystyle Z_{k+1}={\frac {1}{2}}\left(Z_{k}+Z_{k}^{-1}\right)} , where typically Z 0 = A {\displaystyle Z_{0}=A} . Convergence is global, and locally it is quadratic. The Newton iteration uses the explicit inverse of the iterates Z k {\displaystyle Z_{k}} . === Newton–Schulz iteration === To avoid the need of an explicit inverse used in the Newton iteration, the inverse can be approximated with one step of the Newton iteration for the inverse, Z k − 1 ≈ Z k ( 2 I − Z k 2 ) {\displaystyle Z_{k}^{-1}\approx Z_{k}\left(2I-Z_{k}^{2}\right)} , derived by Schulz(de) in 1933. Substituting this approximation into the previous method, the new method becomes Z k + 1 = 1 2 Z k ( 3 I − Z k 2 ) {\displaystyle Z_{k+1}={\frac {1}{2}}Z_{k}\left(3I-Z_{k}^{2}\right)} . Convergence is (still) quadratic, but only local (guaranteed for ‖ I − A 2 ‖ < 1 {\displaystyle \|I-A^{2}\|<1} ). == Applications == === Solutions of Sylvester equations === Theorem: Let A , B , C ∈ R n × n {\displaystyle A,B,C\in \mathbb {R} ^{n\times n}} and assume that A {\displaystyle A} and B {\displaystyle B} are stable, then the unique solution to the Sylvester equation, A X + X B = C {\displaystyle AX+XB=C} , is given by X {\displaystyle X} such that [ − I 2 X 0 I ] = csgn ( [ A − C 0 − B ] ) . {\displaystyle {\begin{bmatrix}-I&2X\\0&I\end{bmatrix}}=\operatorname {csgn} \left({\begin{bmatrix}A&-C\\0&-B\end{bmatrix}}\right).} Proof sketch: The result follows from the similarity transform [ A − C 0 − B ] = [ I X 0 I ] [ A 0 0 − B ] [ I X 0 I ] − 1 , {\displaystyle {\begin{bmatrix}A&-C\\0&-B\end{bmatrix}}={\begin{bmatrix}I&X\\0&I\end{bmatrix}}{\begin{bmatrix}A&0\\0&-B\end{bmatrix}}{\begin{bmatrix}I&X\\0&I\end{bmatrix}}^{-1},} since csgn ( [ A − C 0 − B ] ) = [ I X 0 I ] [ I 0 0 − I ] [ I − X 0 I ] , {\displaystyle \operatorname {csgn} \left({\begin{bmatrix}A&-C\\0&-B\end{bmatrix}}\right)={\begin{bmatrix}I&X\\0&I\end{bmatrix}}{\begin{bmatrix}I&0\\0&-I\end{bmatrix}}{\begin{bmatrix}I&-X\\0&I\end{bmatrix}},} due to the stability of A {\displaystyle A} and B {\displaystyle B} . The theorem is, naturally, also applicable to the Lyapunov equation. However, due to the structure the Newton iteration simplifies to only involving inverses of A {\displaystyle A} and A T {\displaystyle A^{T}} . === Solutions of algebraic Riccati equations === There is a similar result applicable to the algebraic Riccati equation, A H P + P A − P F P + Q = 0 {\displaystyle A^{H}P+PA-PFP+Q=0} . Define V , W ∈ C 2 n × n {\displaystyle V,W\in \mathbb {C} ^{2n\times n}} as [ V W ] = csgn ( [ A H Q F − A ] ) − [ I 0 0 I ] . {\displaystyle {\begin{bmatrix}V&W\end{bmatrix}}=\operatorname {csgn} \left({\begin{bmatrix}A^{H}&Q\\F&-A\end{bmatrix}}\right)-{\begin{bmatrix}I&0\\0&I\end{bmatrix}}.} Under the assumption that F , Q ∈ C n × n {\displaystyle F,Q\in \mathbb {C} ^{n\times n}} are Hermitian and there exists a unique stabilizing solution, in the sense that A − F P {\displaystyle A-FP} is stable, that solution is given by the over-determined, but consistent, linear system V P = − W . {\displaystyle VP=-W.} Proof sketch: The similarity transform [ A H Q F − A ] = [ P − I I 0 ] [ − ( A − F P ) − F 0 ( A − F P ) ] [ P − I I 0 ] − 1 , {\displaystyle {\begin{bmatrix}A^{H}&Q\\F&-A\end{bmatrix}}={\begin{bmatrix}P&-I\\I&0\end{bmatrix}}{\begin{bmatrix}-(A-FP)&-F\\0&(A-FP)\end{bmatrix}}{\begin{bmatrix}P&-I\\I&0\end{bmatrix}}^{-1},} and the stability of A − F P {\displaystyle A-FP} implies that ( csgn ( [ A H Q F − A ] ) − [ I 0 0 I ] ) [ X − I I 0 ] = [ X − I I 0 ] [ 0 Y 0 − 2 I ] , {\displaystyle \left(\operatorname {csgn} \left({\begin{bmatrix}A^{H}&Q\\F&-A\end{bmatrix}}\right)-{\begin{bmatrix}I&0\\0&I\end{bmatrix}}\right){\begin{bmatrix}X&-I\\I&0\end{bmatrix}}={\begin{bmatrix}X&-I\\I&0\end{bmatrix}}{\begin{bmatrix}0&Y\\0&-2I\end{bmatrix}},} for some matrix Y ∈ C n × n {\displaystyle Y\in \mathbb {C} ^{n\times n}} . === Computations of matrix square-root === The Denman–Beavers iteration for the square root of a matrix can be derived from the Newton iteration for the matrix sign function by noticing that A − P I P = 0 {\displaystyle A-PIP=0} is a degenerate algebraic Riccati equation and by definition a solution P {\displaystyle P} is the square root of A {\displaystyle A} . == References == |
Wikipedia:Matt Parker#0 | Matthew Thomas Parker (born 22 December 1980): 20:45 is an Australian recreational mathematician, author, comedian, YouTube personality and science communicator based in the United Kingdom. His book Humble Pi was the first mathematics book in the UK to be a Sunday Times No. 1 bestseller. Parker was the Public Engagement in Mathematics Fellow at Queen Mary University of London. He is a former teacher and has helped popularise mathematics via his tours and videos. == Early life and education == Matt Parker was born in Perth, Western Australia,: 77 and grew up in the northern suburb of Duncraig. He began showing an interest in maths and science from a young age, and at one point was part of his school's titration team. Parker went to the University of Western Australia and started off studying mechanical engineering before he "realized the very real risk of being employable at the end of it." He switched into physics and later mathematics. His love of maths led him to want a job in the subject. While at university, Parker wrote comedy for Pelican, the students' magazine, and produced comedy sketches. Having become interested in comedy, he enrolled on course for stand-up. == Career == After college, Parker taught maths in Australia for a while before moving to London and continuing teaching. He became involved in support education, working with universities and other organizations to arrange maths talks. He later returned to teaching, before stopping after one year. He now helps students communicate mathematics to other people, speaks at schools, does media work, and occasionally writes about maths. His goal is "to get more people more excited about maths." Parker has appeared in numerous YouTube videos, talking about various subjects related to mathematics. He has his own YouTube channel, "Stand-up Maths", with over one million subscribers, and also frequently appears as a guest on other popular channels such as Brady Haran's Numberphile and James May's Head Squeeze (now BritLab). Parker has made videos about unboxing calculators, including the Little Professor; he presents these videos as a member of a fictional "Calculator Appreciation Society". He also appeared in a Tom Scott YouTube video, where they gave tips for users of the London Underground. In 2012, Parker and fellow comedian Timandra Harkness co-wrote a comedy show called Your Days are Numbered: The Maths of Death. They performed the show in Australia, at the Adelaide Fringe and Melbourne International Comedy Festival, on tour around England and in Scotland, at the Edinburgh Festival Fringe. Parker has also toured the UK solo and as part of comedy group Festival of the Spoken Nerd, along with Helen Arney and Steve Mould. His first solo tour, Matt Parker: Number Ninja, finished in July 2013, while his second solo tour, "Matt Parker: Now in 4D", started in late 2014. He has written the book Things to Make and Do in the Fourth Dimension. His second book, Humble Pi, was released in March 2019 and was a Sunday Times #1 bestseller. In 2024, his third book, Love Triangle, was published. In 2014, Parker set up Think Maths, a team of experienced mathematics speakers who visit schools to run workshops and give talks for a wide range of ages and abilities, to show students the wider world of maths beyond school while giving them a chance to develop mathematical thinking skills. In 2016, Parker appeared briefly as a guest on the British comedy panel game quiz show, QI. Parker has appeared on BBC Radio Four's The Infinite Monkey Cage with Robin Ince and Brian Cox. He has also talked about maths-related topics on BBC News, Sky News, Channel4, CBBC, and occasionally writes for The Guardian. On TV, Parker is a regular commentator on Discovery's Outrageous Acts of Science. For the 2019 edition of the Royal Institution Christmas Lectures, televised on BBC Four, Parker assisted presenter Hannah Fry in several segments. In October 2017, Parker started a petition to "Update the UK Traffic Signs Regulations to a geometrically correct football." In a YouTube video, he explained why the current football shape on traffic signs is incorrect and geometrically impossible. Parker described the current signs as a "national embarrassment" and said he hopes the petition will "help raise public awareness and appreciation of geometry." Parker discussed the issue on You Can't Polish A Nerd. According to him, the government initially dismissed the petition because he is a comedian. By November 2017, the petition had gained over 22,000 signatures. The UK government has responded by saying "the current football symbol has a clear meaning and is understood by the public. Changing the design to show accurate geometry is not appropriate in this context." Parker said he felt "like the Department for Transport had not read the petition properly". The official response stated it would be too costly to replace the current signs; however, Parker said he only asked for a "precedent for the new signs". In regards to the exact geometry of a football, Parker said he is "not asking for angles and measurements on the sign, just for it to look more like a football". In 2024, he created a new petition to change the real footballs to look like the signs when viewed from one angle, because “if we can't get the signs changed to match a real football, maybe we can get the football changed to match the signs”. Together with another YouTube mathematics populariser, Vi Hart, Parker won the 2018 Communications Award of the Joint Policy Board for Mathematics for "communicating the excitement of mathematics to a worldwide audience through YouTube videos, TV and radio appearances, book and newspaper writings, and stand-up comedy". Parker hosts two podcasts. The newer of the two, A Podcast of Unnecessary Detail., is co-hosted with Helen Arney and Steve Mould; the three also perform as the comedy troupe Festival of the Spoken Nerd, and have a stage show titled An Evening of Unnecessary Detail, from which the podcast derives its name. In the podcast, each of the hosts brings a topic, usually maths or science related, which they explain to the audience. The first series of six episodes aired during September, 2020 as a way for the troupe to continue to perform their show during the COVID-19 pandemic. A second series of twelve episodes aired during 2022. The older podcast is A Problem Squared, which is formatted as an advice-based podcast, and is co-hosted with author, comedian, and TV presenter Bec Hill. In A Problem Squared, each cohost presents a problem submitted by listeners, which they attempt to exhaustively solve via their own research, sometimes bringing in special guests. New episodes aired monthly from November 2019, to January 2022, and then semimonthly starting in March 2022. === Awards === Parker was awarded the 2020 IMA-LMS Christopher Zeeman Medal in recognition of his "excellence in the communication of mathematics". The award citation highlights work on YouTube, his books, Think Maths, Maths Inspiration, MathsJam, Maths Gear, and his work in broadcast media. On 15 August 2024, the main-belt asteroid 314159 Mattparker was named in his honour. The citation highlights Parker's biennial "Pi Day challenges", stating that they have helped to popularise mathematics. == Recreational mathematics contributions == Parker introduced the recreational mathematics concept of a grafting number, an integer with the property that the square root of the integer, when expressed in base b, will contain the original integer itself before or directly after the decimal point (sequence A232087 in the OEIS). Each odd-numbered year on 14 March, Parker organizes what has been described as "Pi day challenges", where he attempts to calculate the number pi by hand. In 2024, Parker and a team of hundreds of volunteers at City of London School spent six days calculating 139 correct digits of pi by hand, in what he claimed was "the biggest hand calculation in a century". Parker is the namesake of the Parker square, an internet meme consisting of a trivial semimagic square. Parker was attempting to create a magic square made up of all square numbers, however fell quite short of the goal. The semimagic square Parker created uses some numbers more than once, and the diagonal 232 + 372 + 472 sums to 4107, not 3051 as for all the other rows, columns, or diagonal. The Parker Square became a "mascot for people who give it a go, but ultimately fall short". It is also a metaphor for something that is almost right, but is a little off. == Personal life == Parker married the English solar physicist Lucie Green in July 2014. The couple used wedding rings made of meteoric iron. He now lives in Godalming, England. He has a labrador retriever called Skylab who has her own YouTube channel. == Bibliography == Parker, Matt (2015). Things to Make and Do in the Fourth Dimension: A Mathematician's Journey Through Narcissistic Numbers, Optimal Dating Algorithms, at Least Two Kinds of Infinity, and More. Farrar, Straus and Giroux. ISBN 978-0-374-53563-6. Parker, Matt (2019). Humble Pi: A comedy of maths errors. Allen Lane. ISBN 9780241360231. Parker, Matt (2024). Love Triangle: The Life-changing Magic of Trigonometry. Penguin Books. ISBN 9780241505694. == Notes == == References == == External links == Official website Parker's Stand-up Math's channel on YouTube Parker's secondary's channel on YouTube Festival of the Spoken Nerd |
Wikipedia:Matt Visser#0 | Matt Visser () is a mathematics Professor at Victoria University of Wellington, in New Zealand. == Career == Visser completed a PhD at the University of California, Berkeley, supervised by Mary K. Gaillard. Visser's research interests include general relativity, quantum field theory and cosmology. Visser has produced a large number of research papers on the subject of wormholes, gravitational horizons and notably the emerging subject of acoustic metrics. He is the author of the reference book on the current state of wormhole theory, Lorentzian Wormholes — from Einstein to Hawking (1996) and co-editor of Artificial Black Holes (2002). == Awards == In 2013 Visser was awarded the Dan Walls Medal by the New Zealand Institute of Physics. == Books == David L Wiltshire, Matt Visser & Susan Scott, The Kerr Spacetime: Rotating black holes in general relativity (2009) ISBN 978-0-521-88512-6 M Novello, Matt Visser & G E Volovik, Artificial Black Holes (2002) ISBN 978-981-02-4807-9 Matt Visser, Lorentzian Wormholes: From Einstein To Hawking (1995) ISBN 978-1-56396-394-0 == See also == Roman ring == References == == External links == Matt Visser's personalized homepage research papers by Matt Visser on arXiv Matt Visser publications indexed by Google Scholar |
Wikipedia:Matti Jutila#0 | Matti Ilmari Jutila (born 1943) is a mathematician and a professor emeritus at the University of Turku. He researches in the field of analytic number theory. == Education and career == Jutila completed a doctorate at the University of Turku in 1970, with a dissertation related to Linnik's constant supervised by Kustaa Inkeri. Jutila's work has repeatedly succeeded in lowering the upper bound for Linnik's constant. He is the author of a monograph, Lectures on a method in the theory of exponential sums (1987). He has been a member of the Finnish Academy of Science and Letters since 1982. == References == == External links == "Matti Ilmari Jutila author profile". MathSciNet. American Mathematical Society. |
Wikipedia:Maura Mast#0 | Maura B. Mast is an Irish-American mathematician, mathematics educator, and academic administrator, specializing in differential geometry and quantitative reasoning. With Ethan D. Bolker, she is the author of the textbook Common Sense Mathematics. Mast is dean of Fordham College at Rose Hill, part of Fordham University. == Early life and education == Mast is the daughter of Cecil B. Mast (1927–2008), a mathematics professor at the University of Notre Dame in South Bend, Indiana. Her mother was Irish, and Mast has dual Irish and American citizenship. She grew up in South Bend and did her undergraduate studies at Notre Dame, with a double major in mathematics and anthropology. She completed her doctorate in mathematics in 1992 at the University of North Carolina at Chapel Hill. Her dissertation, Closed Geodesics in 2-step Nilmanifolds, concerned the differential geometry of geodesics on curved surfaces, and was supervised by Pat Eberlein. == Career == Mast became a faculty member at the University of Northern Iowa in 1992. After visiting professorships at Northeastern University and Wellesley College, she moved to the University of Massachusetts Boston in 1998. There, in 2009, she became associate vice provost for undergraduate studies. In 2015 she came to Fordham University as the first female dean of Fordham College at Rose Hill. In 2022 she earned and was promoted to rank of full Professor at Fordham University. == Activism == Mast has been an active member of the Clavius Group, a group of Jesuit and lay mathematicians, and is a strong supporter of the Jesuit vision of Catholic spirituality. She has also been a passionate advocate for the advancement of women in mathematics and science, which she writes is "crucial for the future of the country and for women". She has participated in the governance of the Association for Women in Mathematics as Clerk and Executive Committee member of the association. Mast was chair of the Special Interest Group on Quantitative Literacy of the Mathematical Association of America for 2006–2007. == Books == Common Sense Mathematics (with Ethan D. Bolker, Mathematical Association of America, 2016) Women in Mathematics: Celebrating the Centennial of the Mathematical Association of America (edited with Janet Beery, Sarah J. Greenwald, and Jacqueline Jensen-Vallin, Springer, 2017) == Recognition == In 2017 Mast was given the Association for Women in Mathematics Service Award. The Association for Women in Mathematics has included Mast in the 2020 class of AWM Fellows for "her sustained and deep contributions to promoting and encouraging the participation of women in the mathematical sciences through AWM, the Joint Committee on Women, the MAA, and through leadership in academia". == References == |
Wikipedia:Maurice Kraitchik#0 | Maurice Borisovich Kraitchik (21 April 1882 – 19 August 1957) was a Belgian mathematician and populariser. His main interests were the theory of numbers and recreational mathematics. He was born to a Jewish family in Minsk. He wrote several books on number theory during 1922–1930 and after the war, and from 1931 to 1939 edited Sphinx, a periodical devoted to recreational mathematics. During World War II, he emigrated to the United States, where he taught a course at the New School for Social Research in New York City on the general topic of "mathematical recreations." Kraïtchik was agrégé of the Free University of Brussels, engineer at the Société Financière de Transports et d'Entreprises Industrielles (Sofina), and director of the Institut des Hautes Etudes de Belgique. He died in Brussels. Kraïtchik is famous for having inspired the two envelopes problem in 1953, with the following puzzle in La mathématique des jeux: Two people, equally rich, meet to compare the contents of their wallets. Each is ignorant of the contents of the two wallets. The game is as follows: whoever has the least money receives the contents of the wallet of the other (in the case where the amounts are equal, nothing happens). One of the two men can reason: "Suppose that I have the amount A in my wallet. That's the maximum that I could lose. If I win (probability 0.5), the amount that I'll have in my possession at the end of the game will be more than 2A. Therefore the game is favourable to me." The other man can reason in exactly the same way. In fact, by symmetry, the game is fair. Where is the mistake in the reasoning of each man? Among his publications were the following: Théorie des Nombres, Paris: Gauthier-Villars, 1922 Recherches sur la théorie des nombres, Paris: Gauthier-Villars, 1924 La mathématique des jeux ou Récréations mathématiques, Paris: Vuibert, 1930, 566 pages Mathematical Recreations, New York: W. W. Norton, 1942 and London: George Allen & Unwin Ltd, 1943, 328 pages (revised edition New York: Dover, 1953) Alignment Charts, New York: Van Nostrand, 1944 == References == |
Wikipedia:Maurice L'Abbé#0 | Maurice L'Abbé (1920 – July 21, 2006) was a Canadian academic and mathematician. Born in Ottawa, Ontario, L'Abbé obtained his license in mathematics in 1945 from the Université de Montréal, and a doctorate in mathematics from the Princeton University in 1951. He joined the faculty of science in the Université de Montréal becoming an associate professor in 1950 and full professor in 1956. He was director of the Université de Montréal's Department of Mathematics from 1957 to 1968. He was dean of the Faculty of Science from 1964 to 1968 and Vice-Rector for Research from 1968 to 1978. In 1968, he helped to establish the Centre de Recherches Mathématiques, the first mathematical research institute in Canada. == Honours == In 1993, he was made an Officer of the National Order of Quebec. In 1994, he was awarded the Prix Armand-Frappier. == References == |
Wikipedia:Maurice Princet#0 | Maurice Princet (1875 – October 23, 1973) was a French mathematician and actuary who played a role in the birth of cubism. He was an associate of Pablo Picasso, Guillaume Apollinaire, Max Jacob, Jean Metzinger, and Marcel Duchamp. He is known as "le mathématicien du cubisme" ("the mathematician of cubism"). Princet is credited with introducing the work of Henri Poincaré and the concept of the "fourth dimension" to the cubists at the Bateau-Lavoir. Princet brought to Picasso's attention a 1903 book by Esprit Jouffret, Traité élémentaire de géométrie à quatre dimensions (Elementary Treatise on the Geometry of Four Dimensions), a popularization of Poincaré's Science and Hypothesis in which Jouffret described hypercubes and other complex polyhedra in four dimensions and projected them onto the two-dimensional page. Picasso's sketchbooks for Les Demoiselles d'Avignon illustrate Jouffret's influence on the artist's work. In 1907, Princet's wife left him for André Derain, and he drifted away from the circle of artists at the Bateau-Lavoir. But Princet remained close to Metzinger and participated in meetings of the Section d'Or in Puteaux. He gave informal lectures to the group, many of whom were passionate about mathematical order. Princet's influence on the cubists was attested to by his contemporaries. Maurice de Vlaminck wrote, "I witnessed the birth of cubism, its growth, its decline. Picasso was the obstetrician, Guillaume Apollinaire the midwife, Princet the godfather." In 1910, Metzinger said of him, "[Picasso] lays out a free, mobile perspective, from which that ingenious mathematician Maurice Princet has deduced a whole geometry". Later, Metzinger wrote in his memoirs that: Maurice Princet joined us often. Although quite young, thanks to his knowledge of mathematics he had an important job in an insurance company. But, beyond his profession, it was as an artist that he conceptualized mathematics, as an aesthetician that he invoked n-dimensional continuums. He loved to get the artists interested in the new views on space that had been opened up by Schlegel and some others. He succeeded at that. Louis Vauxcelles sarcastically dubbed Princet "the father of cubism": M. Princet has studied at length non-Euclidean geometry and the theorems of Riemann, of which Gleizes and Metzinger speak rather carelessly. Now then, M. Princet one day met M. Max Jacob and confided him one or two of his discoveries relating to the fourth dimension. M. Jacob informed the ingenious M. Picasso of it, and M. Picasso saw there a possibility of new ornamental schemes. M. Picasso explained his intentions to M. Apollinaire, who hastened to write them up in formularies and codify them. The thing spread and propagated. Cubism, the child of M. Princet, was born. Duchamp told Pierre Cabanne, "We weren't mathematicians at all, but we really did believe in Princet". == See also == Mathematics and art Actuary == Notes == == Further reading == |
Wikipedia:Maurice Sion#0 | Maurice Sion (17 October 1927, Skopje – 17 April 2018, Vancouver) was an American and Canadian mathematician, specializing in measure theory and game theory. He is known for Sion's minimax theorem. == Biography == He was born in Skopje (now North Macedonia), to Ladino-speaking Sephardic Jewish parents, Max and Sarah, and spent his early years in Salonika, Greece, Izmir, Turkey and Beirut, Lebanon, before immigrating at the age of 16 with his family to New York. Sion received from New York University his B.A. in 1947 and his M.A. in 1948. He received from the University of California, Berkeley in 1951 his Ph.D. under the supervision of Anthony Morse with thesis On the existence of functions having given partial derivatives on Whitney's curve. Sion was a member of the mathematics faculty at U.C. Berkeley until 1960, when he immigrated to Canada with his wife Emilie and his two children born in the U.S.A. (His two younger children were born in Canada.) From 1960 until he retired in 1989, Maurice Sion was a professor of mathematics at the University of British Columbia. For two academic years from 1957 to 1959 and in the autumn of 1962 he was at the Institute for Advanced Study. He wrote several books on mathematics and served for many years as the head of the University of British Columbia's mathematics department. In 1957 he was the coauthor with Philip Wolfe of a paper with an example of a zero-sum game without a minimax value. Sion was an Invited Speaker at the International Congress of Mathematicians (ICM) in 1970 in Nice and was appointed the Main Organizer for the ICM held in Vancouver in 1974. In 2012 he was elected a Fellow of the American Mathematical Society. Sion was fluent in Spanish, Italian, French, and English. He greatly enjoyed travelling, taking several sabbatical years abroad. His retirement was spent partly in Paris and partly in Vancouver to where he moved back permanently in 2011. He was predeceased by his youngest child. Upon his death he was survived by his widow, three children, and six grandchildren. == Selected publications == === Articles === Sion, Maurice (1960). "On uniformization of sets in topological spaces" (PDF). Trans. Amer. Math. Soc. 96 (2): 237–245. doi:10.1090/s0002-9947-1960-0131506-x. Sion, Maurice (1961). "Continuous images of Borel sets". Proc. Amer. Math. Soc. 12 (3): 385–391. doi:10.1090/S0002-9939-1961-0131508-X. with R. C. Willmott: Sion, M.; Willmott, R. C. (1966). "Hausdorff measures on abstract spaces". Trans. Amer. Math. Soc. 123 (2): 275–309. doi:10.1090/S0002-9947-1966-0200402-7. Sion, Maurice (1969). "Outer measures with values in a topological group". Proceedings of the London Mathematical Society. 3 (1): 89–106. doi:10.1112/plms/s3-19.1.89. === Books === Introduction to the methods of real analysis. New York: Holt, Rinehart and Winston. 1968. Theory of semigroup valued measures. Berlin; New York: Springer-Verlag. 1973. ISBN 9783540378396. == References == |
Wikipedia:Maurice Solovine#0 | Maurice Solovine (21 May 1875 – 13 February 1958) was a Romanian philosopher and mathematician. He is best known for his association with Albert Einstein. == Biography == Solovine was born in Iași, a university city in eastern Romania, near the border with Moldova. As a young student of philosophy in Bern, Solovine applied to study physics with Albert Einstein in response to an advertisement. The two men struck up a close relationship and Einstein was said to say to Solovine a few days after meeting him: "It is not necessary to give you lessons in physics. The discussion about the problems which we face in physics today is much more interesting; simply come to me when you wish. I am pleased to be able to talk to you." One day Solovine suggested reading and debating the works of great authors. Einstein agreed enthusiastically and soon mathematician Conrad Habicht (1876–1958) became involved in what was to be known as the "Akademie Olympia" (Olympia Academy). Often their meetings, held in Einstein's flat, would last until the early morning hours. On one occasion Solovine missed a scheduled meeting in his flat, preferring to listen to a concert in the city. He had prepared a meal for his friends with a note: "Amicis carissimis ova dura et salutem." (To my beloved friends, hard-boiled eggs and greetings). Einstein and Habicht turned his flat upside down after they had eaten the meal. Every piece of furniture was moved and plates, cups, forks, knives and books were scattered all over the flat. The rooms were also covered in smoke from Einstein's pipe and Habicht's cigar. They left a "worthy warning" on the wall: "Amico carissimo fumum spissum et salutem." (To our dearest friend, thick smoke and greetings). Einstein is reputed to have greeted Solovine the next evening with the following words: "You lousy guy, you were cheeky enough to stay away from a meeting of the Akademie to listen to the violin? Barbarian, idiot, stupid one, if you ever let us down this way again you’ll be expelled from the Akademie with shame." This meeting lasted until the morning to make up for lost time. From November 1905, Solovine continued his studies in Lyon. He then moved to Paris, where he helped publish the Revue philosophique between 1908 and 1919 and worked as a freelance translator and editor of classic texts on the history of European philosophy and of contemporary mathematicians. He was the principal translator of works by Einstein from German to French and represented Einstein's financial interest and agreements with publishers in France, for which Einstein paid him a percentage fee. Solovine remained in contact with Einstein with letters dated from 3 May 1906 to 27 February 1955. Solovine published French translations of Greek philosophers. He died in Paris in 1958. == References == Maurice Solovine (ed.) Albert Einstein - Lettres à Maurice Solovine, Paris: Gauthier-Villars, 1956. Akademie Olympia |
Wikipedia:Maurice Vandeweyer#0 | Maurice Vandeweyer (21 February 1945 – 17 August 2021) was a Belgian comic book author and mathematician. He also expressed interest in theatre in the region Entre-Sambre-et-Meuse through chronicles and books. He also wrote about gastronomy. == Biography == Vandeweyer was a professor of mathematics. He closely followed regional cuisine, beer, and wine. He participated in the radio show Gastronomiquement vôtre on Sky Rock and Nostalgie Wallonie. He also worked as a freelance journalist and a columnist for the newspaper L'Avenir. Vandeweyer became a member of SABAM and began writing Contes et légendes de l'Entre-Sambre-et-Meuse. He attributed many of his works to the legends of the Entre-Sambre-et-Meuse region. In 2002, he published Quand l'Entre-Sambre-et-Meuse se met à table. In 2008, he published Histoires de l'Entre-Sambre-et-Meuse with Éditions Racine. In 2013, he wrote a book on Ivan Ivanof, a local Belgian personality and political refugee from Bulgaria. In 2012, Vandeweyer created a puppet theatre with Christophe Delire and Martine Jamin, called the Compagnie du Tradery. For the first staging, he wrote the puppet show La Geste de Guillaume. Maurice Vandeweyer died in Couvin on 17 August 2021 at the age of 76. == Publications == === Books === Contes et légendes de l'Entre-Sambre-et-Meuse (2000) Quand l'Entre-Sambre-et-Meuse se met à table (2002) Autres contes de l'Entre-Sambre-et-Meuse (2004) Histoire de l'Entre-Sambre-et-Meuse (2008) Ivan Ivanov, sa 25e heure (2013) Fernand Platbrood, l'Alter-Native, ça presse pour une énergie durable et équitable (2014) Balades entre châteaux et légendes (2020) === Comic books === Le Comte à la houssette (1980) Landru (1987) Le Fraudeur Calteau (1990) === Theatre === Le Bac (1989) La geste de Guillaume (2012) Djosèf et Françwès (2013) == References == |
Wikipedia:Mautner's lemma#0 | Mautner's lemma in representation theory, named after Austrian-American mathematician Friederich Mautner, states that if G is a topological group and π a unitary representation of G on a Hilbert space H, then for any x in G, which has conjugates yxy−1 converging to the identity element e, for a net of elements y, then any vector v of H invariant under all the π(y) is also invariant under π(x). == References == F. Mautner, Geodesic flows on symmetric Riemannian spaces (1957), Ann. Math. 65, 416-430 |
Wikipedia:Max A. Woodbury#0 | In mathematics, specifically linear algebra, the Woodbury matrix identity – named after Max A. Woodbury – says that the inverse of a rank-k correction of some matrix can be computed by doing a rank-k correction to the inverse of the original matrix. Alternative names for this formula are the matrix inversion lemma, Sherman–Morrison–Woodbury formula or just Woodbury formula. However, the identity appeared in several papers before the Woodbury report. The Woodbury matrix identity is ( A + U C V ) − 1 = A − 1 − A − 1 U ( C − 1 + V A − 1 U ) − 1 V A − 1 , {\displaystyle \left(A+UCV\right)^{-1}=A^{-1}-A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1},} where A, U, C and V are conformable matrices: A is n×n, C is k×k, U is n×k, and V is k×n. This can be derived using blockwise matrix inversion. While the identity is primarily used on matrices, it holds in a general ring or in an Ab-category. The Woodbury matrix identity allows cheap computation of inverses and solutions to linear equations. However, little is known about the numerical stability of the formula. There are no published results concerning its error bounds. Anecdotal evidence suggests that it may diverge even for seemingly benign examples (when both the original and modified matrices are well-conditioned). == Discussion == To prove this result, we will start by proving a simpler one. Replacing A and C with the identity matrix I, we obtain another identity which is a bit simpler: ( I + U V ) − 1 = I − U ( I + V U ) − 1 V . {\displaystyle \left(I+UV\right)^{-1}=I-U\left(I+VU\right)^{-1}V.} To recover the original equation from this reduced identity, replace U {\displaystyle U} by A − 1 U {\displaystyle A^{-1}U} and V {\displaystyle V} by C V {\displaystyle CV} . This identity itself can be viewed as the combination of two simpler identities. We obtain the first identity from I = ( I + P ) − 1 ( I + P ) = ( I + P ) − 1 + ( I + P ) − 1 P , {\displaystyle I=(I+P)^{-1}(I+P)=(I+P)^{-1}+(I+P)^{-1}P,} thus, ( I + P ) − 1 = I − ( I + P ) − 1 P , {\displaystyle (I+P)^{-1}=I-(I+P)^{-1}P,} and similarly ( I + P ) − 1 = I − P ( I + P ) − 1 . {\displaystyle (I+P)^{-1}=I-P(I+P)^{-1}.} The second identity is the so-called push-through identity ( I + U V ) − 1 U = U ( I + V U ) − 1 {\displaystyle (I+UV)^{-1}U=U(I+VU)^{-1}} that we obtain from U ( I + V U ) = ( I + U V ) U {\displaystyle U(I+VU)=(I+UV)U} after multiplying by ( I + V U ) − 1 {\displaystyle (I+VU)^{-1}} on the right and by ( I + U V ) − 1 {\displaystyle (I+UV)^{-1}} on the left. Putting all together, ( I + U V ) − 1 = I − U V ( I + U V ) − 1 = I − U ( I + V U ) − 1 V . {\displaystyle \left(I+UV\right)^{-1}=I-UV\left(I+UV\right)^{-1}=I-U\left(I+VU\right)^{-1}V.} where the first and second equality come from the first and second identity, respectively. === Special cases === When V , U {\displaystyle V,U} are vectors, the identity reduces to the Sherman–Morrison formula. In the scalar case, the reduced version is simply 1 1 + u v = 1 − u v 1 + v u . {\displaystyle {\frac {1}{1+uv}}=1-{\frac {uv}{1+vu}}.} ==== Inverse of a sum ==== If n = k and U = V = In is the identity matrix, then ( A + B ) − 1 = A − 1 − A − 1 ( B − 1 + A − 1 ) − 1 A − 1 = A − 1 − A − 1 ( A B − 1 + I ) − 1 . {\displaystyle {\begin{aligned}\left(A+B\right)^{-1}&=A^{-1}-A^{-1}\left(B^{-1}+A^{-1}\right)^{-1}A^{-1}\\[1ex]&=A^{-1}-A^{-1}\left(AB^{-1}+{I}\right)^{-1}.\end{aligned}}} Continuing with the merging of the terms of the far right-hand side of the above equation results in Hua's identity ( A + B ) − 1 = A − 1 − ( A + A B − 1 A ) − 1 . {\displaystyle \left({A}+{B}\right)^{-1}={A}^{-1}-\left({A}+{A}{B}^{-1}{A}\right)^{-1}.} Another useful form of the same identity is ( A − B ) − 1 = A − 1 + A − 1 B ( A − B ) − 1 , {\displaystyle \left({A}-{B}\right)^{-1}={A}^{-1}+{A}^{-1}{B}\left({A}-{B}\right)^{-1},} which, unlike those above, is valid even if B {\displaystyle B} is singular, and has a recursive structure that yields ( A − B ) − 1 = ∑ k = 0 ∞ ( A − 1 B ) k A − 1 {\displaystyle \left({A}-{B}\right)^{-1}=\sum _{k=0}^{\infty }\left({A}^{-1}{B}\right)^{k}{A}^{-1}} if the spectral radius of A − 1 B {\displaystyle A^{-1}B} is less than one. That is, if the above sum converges then it is equal to ( A − B ) − 1 {\displaystyle (A-B)^{-1}} . This form can be used in perturbative expansions where B is a perturbation of A. === Variations === ==== Binomial inverse theorem ==== If A, B, U, V are matrices of sizes n×n, k×k, n×k, k×n, respectively, then ( A + U B V ) − 1 = A − 1 − A − 1 U B ( B + B V A − 1 U B ) − 1 B V A − 1 {\displaystyle \left(A+UBV\right)^{-1}=A^{-1}-A^{-1}UB\left(B+BVA^{-1}UB\right)^{-1}BVA^{-1}} provided A and B + BVA−1UB are nonsingular. Nonsingularity of the latter requires that B−1 exist since it equals B(I + VA−1UB) and the rank of the latter cannot exceed the rank of B. Since B is invertible, the two B terms flanking the parenthetical quantity inverse in the right-hand side can be replaced with (B−1)−1, which results in the original Woodbury identity. A variation for when B is singular and possibly even non-square: ( A + U B V ) − 1 = A − 1 − A − 1 U ( I + B V A − 1 U ) − 1 B V A − 1 . {\displaystyle (A+UBV)^{-1}=A^{-1}-A^{-1}U(I+BVA^{-1}U)^{-1}BVA^{-1}.} Formulas also exist for certain cases in which A is singular. ==== Pseudoinverse with positive semidefinite matrices ==== In general Woodbury's identity is not valid if one or more inverses are replaced by (Moore–Penrose) pseudoinverses. However, if A {\displaystyle A} and C {\displaystyle C} are positive semidefinite, and V = U H {\displaystyle V=U^{\mathrm {H} }} (implying that A + U C V {\displaystyle A+UCV} is itself positive semidefinite), then the following formula provides a generalization: ( X X H + Y Y H ) + = ( Z Z H ) + + ( I − Y Z + ) H X + H E X + ( I − Y Z + ) , Z = ( I − X X + ) Y , E = I − X + Y ( I − Z + Z ) F − 1 ( X + Y ) H , F = I + ( I − Z + Z ) Y H ( X X H ) + Y ( I − Z + Z ) , {\displaystyle {\begin{aligned}\left(XX^{\mathrm {H} }+YY^{\mathrm {H} }\right)^{+}&=\left(ZZ^{\mathrm {H} }\right)^{+}+\left(I-YZ^{+}\right)^{\mathrm {H} }X^{+\mathrm {H} }EX^{+}\left(I-YZ^{+}\right),\\Z&=\left(I-XX^{+}\right)Y,\\E&=I-X^{+}Y\left(I-Z^{+}Z\right)F^{-1}\left(X^{+}Y\right)^{\mathrm {H} },\\F&=I+\left(I-Z^{+}Z\right)Y^{\mathrm {H} }\left(XX^{\mathrm {H} }\right)^{+}Y\left(I-Z^{+}Z\right),\end{aligned}}} where A + U C U H {\displaystyle A+UCU^{\mathrm {H} }} can be written as X X H + Y Y H {\displaystyle XX^{\mathrm {H} }+YY^{\mathrm {H} }} because any positive semidefinite matrix is equal to M M H {\displaystyle MM^{\mathrm {H} }} for some M {\displaystyle M} . == Derivations == === Direct proof === The formula can be proven by checking that ( A + U C V ) {\displaystyle (A+UCV)} times its alleged inverse on the right side of the Woodbury identity gives the identity matrix: ( A + U C V ) [ A − 1 − A − 1 U ( C − 1 + V A − 1 U ) − 1 V A − 1 ] = { I − U ( C − 1 + V A − 1 U ) − 1 V A − 1 } + { U C V A − 1 − U C V A − 1 U ( C − 1 + V A − 1 U ) − 1 V A − 1 } = { I + U C V A − 1 } − { U ( C − 1 + V A − 1 U ) − 1 V A − 1 + U C V A − 1 U ( C − 1 + V A − 1 U ) − 1 V A − 1 } = I + U C V A − 1 − ( U + U C V A − 1 U ) ( C − 1 + V A − 1 U ) − 1 V A − 1 = I + U C V A − 1 − U C ( C − 1 + V A − 1 U ) ( C − 1 + V A − 1 U ) − 1 V A − 1 = I + U C V A − 1 − U C V A − 1 = I . {\displaystyle {\begin{aligned}&\left(A+UCV\right)\left[A^{-1}-A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}\right]\\={}&\left\{I-U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}\right\}+\left\{UCVA^{-1}-UCVA^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}\right\}\\={}&\left\{I+UCVA^{-1}\right\}-\left\{U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}+UCVA^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}\right\}\\={}&I+UCVA^{-1}-\left(U+UCVA^{-1}U\right)\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}\\={}&I+UCVA^{-1}-UC\left(C^{-1}+VA^{-1}U\right)\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1}\\={}&I+UCVA^{-1}-UCVA^{-1}\\={}&I.\end{aligned}}} === Alternative proofs === == Applications == This identity is useful in certain numerical computations where A−1 has already been computed and it is desired to compute (A + UCV)−1. With the inverse of A available, it is only necessary to find the inverse of C−1 + VA−1U in order to obtain the result using the right-hand side of the identity. If C has a much smaller dimension than A, this is more efficient than inverting A + UCV directly. A common case is finding the inverse of a low-rank update A + UCV of A (where U only has a few columns and V only a few rows), or finding an approximation of the inverse of the matrix A + B where the matrix B can be approximated by a low-rank matrix UCV, for example using the singular value decomposition. This is applied, e.g., in the Kalman filter and recursive least squares methods, to replace the parametric solution, requiring inversion of a state vector sized matrix, with a condition equations based solution. In case of the Kalman filter this matrix has the dimensions of the vector of observations, i.e., as small as 1 in case only one new observation is processed at a time. This significantly speeds up the often real time calculations of the filter. In the case when C is the identity matrix I, the matrix I + V A − 1 U {\displaystyle I+VA^{-1}U} is known in numerical linear algebra and numerical partial differential equations as the capacitance matrix. == See also == Sherman–Morrison formula Schur complement Matrix determinant lemma, formula for a rank-k update to a determinant Invertible matrix Moore–Penrose pseudoinverse § Updating the pseudoinverse == Notes == Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 2.7.3. Woodbury Formula", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8 == External links == Some matrix identities Weisstein, Eric W. "Woodbury formula". MathWorld. |
Wikipedia:Max August Zorn#0 | Max August Zorn (German: [tsɔʁn]; June 6, 1906 – March 9, 1993) was a German mathematician. He was an algebraist, group theorist, and numerical analyst. He is best known for Zorn's lemma, a method used in set theory that is applicable to a wide range of mathematical constructs such as vector spaces, and ordered sets amongst others. Zorn's lemma was first postulated by Kazimierz Kuratowski in 1922, and then independently by Zorn in 1935. == Life and career == Zorn was born in Krefeld, Germany. He attended the University of Hamburg. He received his PhD in April 1930 for a thesis on alternative algebras. He published his findings in Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. Zorn showed that split-octonions could be represented by a mixed-style of matrices called Zorn's vector-matrix algebra. Max Zorn was appointed to an assistant position at the University of Halle. However, he did not have the opportunity to work there for long as he was forced to leave Germany in 1933 because of policies enacted by the Nazis. According to grandson Eric, "[Max] spoke with a raspy, airy voice most of his life. Few people knew why, because he only told the story after significant prodding, but he talked that way because pro-Hitler thugs who objected to his politics, had battered his throat in a 1933 street fight." Zorn immigrated to the United States and was appointed a Sterling Fellow at Yale University. While at Yale, Zorn wrote his paper "A Remark on Method in Transfinite Algebra" that stated his Maximum Principle, later called Zorn's lemma. It requires a set that contains the union of any chain of subsets to have one chain not contained in any other, called the maximal element. He illustrated the principle with applications in ring theory and field extensions. Zorn's lemma is an alternative expression of the axiom of choice, and thus a subject of interest in axiomatic set theory. In 1936 he moved to UCLA and remained until 1946. While at UCLA Zorn revisited his study of alternative rings and proved the existence of the nilradical of certain alternative rings. According to Angus E. Taylor, Max was his most stimulating colleague at UCLA. In 1946 Zorn became a professor at Indiana University, where he taught until retiring in 1971. He was thesis advisor for Israel Nathan Herstein. Zorn died in Bloomington, Indiana, in March 1993, of congestive heart failure. == Family == Max Zorn married Alice Schlottau and they had one son, Jens, and one daughter, Liz. Jens (born June 19, 1931) is an emeritus professor of physics at the University of Michigan and an accomplished sculptor. Max Zorn's grandson Eric Zorn was a columnist for the Chicago Tribune from 1986 until 2021; after retirement Eric Zorn started a newsletter titled The Picayune Sentinel, named after the mathematics newsletter that Max Zorn had distributed during his years at Indiana University. Max's great grandson, Alexander Wolken Zorn, received a PhD in mathematics from the University of California Berkeley in 2018. == References == Steve Carlson (2009) Max Zorn: World Renowned Mathematician and Member Indiana MAA Section Archived 2017-05-10 at the Wayback Machine, from Mathematics Association of America . Darrell Haile (1993) On Max Zorn's Contributions to Mathematics (includes John Ewing, "Zorn's Lemma"), from Memorial Conference at Indiana University, June 1993. == External links == O'Connor, John J.; Robertson, Edmund F., "Max August Zorn", MacTutor History of Mathematics Archive, University of St Andrews Max August Zorn at the Mathematics Genealogy Project |
Wikipedia:Max Euwe#0 | Machgielis "Max" Euwe (Dutch: [ˈøːʋə]; May 20, 1901 – November 26, 1981) was a Dutch chess player, mathematician, author, and chess administrator. He was the fifth player to become World Chess Champion, a title he held from 1935 until 1937. He served as President of FIDE, the World Chess Federation, from 1970 to 1978. == Early years, education, and professional career == Euwe was born in the Watergraafsmeer, in Amsterdam. He studied mathematics at the University of Amsterdam under the founder of intuitionistic logic, L.E.J. Brouwer (who later became his friend and for whom he held a funeral oration), and earned his doctorate in 1926 under Roland Weitzenböck. He taught mathematics, first in Rotterdam, and later at a girls' Lyceum in Amsterdam. After World War II, Euwe became interested in computer programming and was appointed professor in this subject at the universities of Rotterdam and Tilburg, retiring from Tilburg University in 1971. He published a mathematical analysis of the game of chess from an intuitionistic point of view, in which he showed, using the Thue–Morse sequence, that the then-official rules (in 1929) did not exclude the possibility of infinite games. == Early chess career == Euwe played his first tournament at age 10, winning every game. He won every Dutch chess championship that he entered from 1921 until 1952, and won the title again in 1955; his 12 titles are still a record. The only other winners during this period were Salo Landau in 1936, when Euwe, then world champion, did not compete; and Jan Hein Donner in 1954. He became the world amateur chess champion in 1928, at The Hague, with a score of 12/15. Euwe married in 1926, started a family soon afterwards, and could play competitive chess only during school vacations, so his opportunities for top-level international chess competition were limited. But he performed well in the few tournaments and matches for which he could find time, from the early 1920s to mid-1930s. He lost a training match to Alexander Alekhine in the Netherlands in December 1926 / January 1927, with 4½/10 (+2−3=5). The match was played to help Euwe prepare for a future encounter with José Raúl Capablanca, then world champion. Euwe lost both the first and second FIDE Championship matches to Efim Bogoljubow, held in the Netherlands in 1928 and 1928‒29 respectively, scoring 4½/10 in each match (+2−3=5 in the first match; +1−2=7 in the second match). He lost a match to Capablanca in Amsterdam in 1931 with 4/10 (+0−2=8). He won a match against Spielmann in Amsterdam in 1932, 3–1, played to help Euwe prepare for his upcoming match with Salo Flohr. In 1932, Euwe drew a match with Flohr 8–8, and was equal second with Flohr, behind Alekhine, at a major tournament in Bern. According to Reuben Fine, these results established Euwe and Flohr as Alekhine's most credible challengers. At Zürich 1934, Euwe again finished equal second with Flohr, behind Alekhine, and he defeated Alekhine in their game. == World Champion == In 1933, Max Euwe challenged Alekhine to a championship match. Alekhine accepted the challenge for October 1935. Earlier that year, Dutch radio sports journalist Han Hollander asked Capablanca for his views on the forthcoming match. In the rare archival film footage where Capablanca and Euwe both speak, Capablanca replies: "Dr. Alekhine's game is 20% bluff. Dr. Euwe's game is clear and straightforward. Dr. Euwe's game—not so strong as Alekhine's in some respects—is more evenly balanced." Then Euwe gives his assessment in Dutch, explaining that his feelings alternated from optimism to pessimism, but in the previous ten years, their score had been evenly matched at 7–7. On December 15, 1935, after 30 games played in 13 different cities around the Netherlands over a period of 80 days, Euwe defeated Alekhine by 15½–14½, becoming the fifth World Chess Champion. Alekhine quickly went three games ahead, but Euwe managed to even out and eventually win the match. His title gave a huge boost to chess in the Netherlands. It was also the first world championship where the players had seconds to help them with analysis during adjournments. Euwe's win was regarded as a major upset – he reportedly had believed that beating Alekhine was unlikely – and is sometimes attributed to Alekhine's alcoholism. But Salo Flohr, who helped Euwe during the match, thought Alekhine's over-confidence was more of a problem than alcohol; Alekhine himself said he would win easily. Former world champions Vasily Smyslov, Boris Spassky, Anatoly Karpov, and Garry Kasparov later analysed the match and concluded that Euwe deserved to win and that the standard of play was worthy of a world championship. Former World Champion Vladimir Kramnik has said that Euwe won the 1935 match on merit and that the result was not affected by Alekhine's drinking before or during the match. Euwe's performance in the great tournament of Nottingham 1936 (equal third, half a point behind Botvinnik and Capablanca, half a point ahead of Alekhine) indicated he was a worthy champion, even if he was not as dominant as the earlier champions. Reuben Fine wrote, "In the two years before the return match, Euwe's strength increased. Although he never enjoyed the supremacy over his rivals that his predecessors had, he had no superiors in this period." Euwe lost the title to Alekhine in a rematch in 1937, also played in the Netherlands, by the lopsided margin of 15½–9½. Alekhine had given up alcohol and tobacco to prepare for the rematch, although he resumed drinking later. He returned to the sort of form he had shown from 1927 to 1934, when he dominated chess. The match was a real contest initially, but Euwe's play collapsed near the end, and he lost four of the last five games. Fine, who was Euwe's second, attributed the collapse to nervous tension, possibly aggravated by Euwe's attempts to maintain a calm appearance. The two world title matches against Alekhine represent the heart of Euwe's career. Altogether, they played 86 competitive games, and Alekhine had a +28−20=38 lead. Many of Alekhine's wins came early in their series; he was nine years older, and had more experience during that time. The rematch was also one-sided in Alekhine's favour. Until American-born Bobby Fischer won the title in 1972, Euwe was the last World Chess Champion not born in the Russian Empire or Soviet Union. == Later chess career == Euwe finished equal fourth with Alekhine and Reshevsky in the AVRO tournament of 1938 in the Netherlands, which featured the world's top eight players and was an attempt to decide who should challenge Alekhine for the world championship. Euwe also had a major organisational role in the event. He played a match with Paul Keres in the Netherlands in 1939–40, losing 6½–7½. After Alekhine's death in 1946, Euwe was considered by some to have a moral right to the position of world champion, based at least partially on his clear second-place finish in the great tournament at Groningen in 1946, behind Mikhail Botvinnik. But Euwe consented to participate in a five-player tournament to select the new champion, the World Chess Championship 1948. At 47, Euwe was significantly older than the other players, and well past his best. He finished last. In 1950, FIDE granted Euwe the title of international grandmaster on its inaugural list. He took part in the Gijón international tournament in 1951, winning ahead of Pilnik and Rossolimo with a score of +7=2. Euwe's final major tournament was the double round robin Candidates' Tournament in Zürich, 1953, where he finished next to last. He was in the top half of the field after the first half of the tournament, but tired in the second half. Euwe played for the Netherlands in seven Chess Olympiads from 1927 to 1962, a 35-year span, always on first board. He scored 10½/15 at London 1927, 9½/13 at Stockholm 1937 for a bronze medal, 8/12 at Dubrovnik 1950, 7½/13 at Amsterdam 1954, 8½/11 at Munich 1958 for a silver medal at age 57, 6½/16 at Leipzig 1960, and finally 4/7 at Varna 1962. His aggregate was 54½/87 for 62.6 percent. In 1957, Euwe played a short match against 14-year-old future world champion Bobby Fischer, winning one game and drawing the other. His lifetime score against Fischer was one win, one loss, and one draw. Euwe won a total of 102 first prizes in tournaments during his career, many of them local. He became a computer science professor at Tilburg University in 1964. == FIDE President == From 1970 (at age 69) until 1978, Euwe was president of FIDE. As president, he usually did what he considered morally right rather than what was politically expedient. On several occasions this brought him into conflict with the USSR Chess Federation, which thought it had the right to dominate matters because it contributed a very large share of FIDE's budget and Soviet players dominated the world rankings – in effect, they treated chess as an extension of the Cold War. These conflicts included: The events leading up to Bobby Fischer's participation in the World Chess Championship 1972 match against Boris Spassky, which led to Fischer's becoming the first non-Soviet champion since World War II. Euwe thought it important for the game's health and reputation that Fischer have the opportunity to challenge for the title as soon as possible, and interpreted the rules very flexibly to enable Fischer to play in the 1970 Interzonal Tournament, which he won by a commanding score. The defection of Gennadi Sosonko in 1972. The Soviets demanded that Sosonko should be treated as an "unperson", excluded from competitive chess, television or any other event that might be evidence of his defection. When Euwe refused, Soviet players boycotted the 1974 Wijk aan Zee tournament in the Netherlands because Sosonko competed. In 1976, world championship contender Viktor Korchnoi sought political asylum in the Netherlands. In a discussion a few days earlier, Euwe told Korchnoi: "... of course you will retain all your rights ..." and opposed Soviet efforts to prevent Korchnoi from challenging Anatoly Karpov's title in 1978. Later in 1976, Euwe supported FIDE's decision to hold the 1976 Chess Olympiad in Israel, which the Soviet Union did not recognize as a country, although the Soviets had won the 1964 Olympiad which had also been held in Israel. The Central Committee of Communist Party of the Soviet Union then started plotting to depose Euwe as president of FIDE. Euwe lost some of his battles with the Soviets. According to Sosonko, in 1973, he accepted the Soviets' demand that Bent Larsen and Robert Hübner, the two strongest non-Soviet contenders (Fischer was now champion), should play in the Leningrad Interzonal tournament rather than the weaker one in Petrópolis. Larsen and Hübner were eliminated from the competition for the World Championship because Korchnoi and Karpov took the first two places at Leningrad. Some commentators have also questioned whether Euwe did as much as he could have to prevent Fischer from forfeiting his world title in 1975. It is also notable that in 1977, when Rohini Khadilkar became the first female player to compete in the Indian Chess Championship, some players objected to her being in the tournament because she was female; her father wrote to Euwe, and Euwe ruled that female players could not be barred from open chess events. Despite the turbulence of the period, most assessments of Euwe's performance as president of FIDE are sympathetic: Spassky, who had nominated Euwe for the job: "He should certainly not have disqualified Fischer, and he should have been a little tougher with the Soviets ... you get a pile of complicated problems. But Euwe, of course, was the man for the job." Karpov said Euwe was a very good FIDE President, although he did commit one very serious error, rapidly extending the membership of FIDE to many small third-world countries. "But neither he nor I could have foreseen what this would lead to. ... This led not only to the inflation of the grandmaster title, but also to the leadership vacuum at the head of the world of chess." Garry Kasparov was blunter: "... unfortunately, he could not foresee the dangers flowing from a FIDE practically under Soviet dominance." Korchnoi regarded Euwe as the last honorable president of FIDE. Yuri Averbakh, who was a Soviet chess official as well as a grandmaster: "... he always sought to understand the opposing point of view ... Such behavior was in sharp contrast to the behavior of the Soviet delegation leaders ... Max Euwe was, without a doubt, the best President FIDE ever had." Euwe died in 1981, age 80, of a heart attack. Revered around the chess world for his many contributions, he had travelled extensively while FIDE President, bringing many new members into the organisation. == Assessment of Euwe's chess == Euwe was noted for his logical approach and for his knowledge of openings, in which he made major contributions to chess theory. Paradoxically, his two title matches with Alekhine were displays of tactical ferocity from both sides. But the comments by Kmoch and Alekhine (below) may explain this: Euwe "strode confidently into some extraordinarily complex variations" if he thought logic was on his side; and he was extremely good at calculating these variations. On the other hand, he "often lacked the stamina to pull himself out of bad positions". Alekhine was allegedly more frank in his Russian-language articles than in those he wrote in English, French, or German. In his Russian articles he often described Euwe as lacking in originality and in the mental toughness required of a world champion. Sosonko thought Euwe's modesty was a handicap in top-class chess (although Euwe was well aware of how much stronger he was than "ordinary" grandmasters). Vladimir Kramnik also says Euwe anticipated Botvinnik's emphasis on technical preparation, and Euwe was usually in good shape physically because he was a keen sportsman. == Chess books by Euwe == Euwe wrote over 70 chess books, far more than any other world champion; some of the best-known are The Road to Chess Mastery, Judgement and Planning in Chess, The Logical Approach to Chess, and Strategy and Tactics in Chess. Former Soviet grandmaster Sosonko used Euwe and den Hertog's 1927 Practische Schaaklessen as a textbook when teaching in the Leningrad House of Pioneers, and considers it "one of the best chess books ever". Fischer World Champion, an account of the 1972 World Chess Championship match, co-authored by Euwe with Jan Timman, was written in 1972 but not published in English until 2002. Euwe's book From My Games, 1920–1937 was originally published in 1939 by Harcourt, Brace and Company, and was republished by Dover in 1975 (ISBN 0-486-23111-9). He also did not forget children in his published writings. The year he won the World Chess Championship he wrote a book named in Dutch: Oom Jan leert zijn neefje schaken, lit. 'Uncle Jan teaches his nephew to play chess' (EAN 9789043900669). === Bibliography === Strategy and Tactics in Chess. 1937. McKay. My Best Games 1920–1937 My Rise to become World Champion. 2003 [1939]. Hardinge Simpole. Meet The Masters: Pen Portraits to the Greats by a World Champion. 2004 [1940]. Hardinge Simpole. The Hague/Moscow 1948 Match/Tournament for the World Chess Championship. 2013 [1948]. Russell Enterprises. Judgement and Planning in Chess. 1998 [1954]. Batsford. The Logical Approach to Chess. 1982 [1958]. Dover. Chess Master vs. Chess Amateur. with Walter Meiden. 1994 [1963]. Dover. The Middlegame Book One Static Features. with H. Kramer. 1994 [1964]. Hays Pub. The Middlegame Book Two Dynamic & Subjective Features. with H. Kramer. 1994 [1964]. Hays Pub. The Road to Chess Mastery. with Walter Meiden. 1966. David McKay. The Development of Chess Style. with John Nunn. 1997 [1968]. International Chess Enterprises. Fischer World Champion. with Jan Timman. 2009 [1972]. New In Chess. Euwe vs. Alekhine Match 1935. 1973. Chess Digest. A Guide to Chess Endings. with David Hooper. 1976. Dover. Bobby Fischer The Greatest? 1979 [1976]. Sterling. Chess Master vs. Chess Master with Walter Meiden. 1977. McKay. == Legacy == In Amsterdam, there is a Max Euwe Plein (square) (near the Leidseplein) with a large chess set and statue, where the 'Max Euwe Stichting' is located in a former jailhouse. It has a Max Euwe museum and a large collection of chess books. == Honours == In 1936, Euwe was appointed Officer of the Order of Orange-Nassau. In 1979, Euwe was promoted to Commander of the Order of Orange-Nassau. == Notes == == References == Kasparov, Garry (2003). My Great Predecessors, part II. Everyman Chess. ISBN 1-85744-342-X. Winter, Edward, ed. (1981). World Chess Champions. Pergamon Press Ltd. ISBN 0-08-024094-1. == External links == Max Euwe player profile and games at Chessgames.com Machgielis Euwe's biography Max Euwe Centrum, Amsterdam Machgielis (Max) Euwe a short history of Euwe's playing career Albert Silver, "Alekhine-Euwe 1935: powerful images", ChessBase, 13 December 2013. "Max Euwe (1901-81)" by Edward Winter |
Wikipedia:Max Gut#0 | Max Gut (1898–1988) was a Swiss mathematician, specializing in algebraic number theory and group theory. After completing his secondary education at the canton school in Zürich, Gut spent one semester studying law and business at the University of Geneva, but then followed his inclinations to study mathematics. He studied mathematics at the University of Zürich and ETH Zürich and then spent a year studying theoretical physics in Berlin. He received his promotion (Ph.D.) in 1924 from the University of Zürich under Rudolf Fueter. Gut received his habilitation qualification in the summer of 1929 from the University of Zürich, and was appointed there Titularprofessor in 1938. Gut served a two-year term from 1946 to 1947 as president of the Swiss Mathematical Society. He was an Invited Speaker of the ICM in 1932 at Zürich and in 1936 at Oslo. == Selected publications == with Rudolf Fueter: Vorlesungen über die singulären Moduln und die komplexe Multiplikation der elliptischen Funktionen. Vol. 41. BG Teubner, 1927. "Die Zetafunktion, die Klassenzahl und die Kronecker'sche Grenzformel eines beliebigen Kreiskörpers." Commentarii Mathematici Helvetici 1, no. 1 (1929): 160-226. doi:10.1007/BF01208364 "Über die Gradteilerzerlegung in gewissen relativ-ikosaedrischen Zahlkörpern." Commentarii Mathematici Helvetici 7, no. 1 (1934): 103-130. doi:10.1007/BF01292712 "Weitere Untersuchungen über die Primidealzerlegung in gewissen relativ-ikosaedrischen Zahlkörpern." Commentarii Mathematici Helvetici 6, no. 1 (1934): 47-75. doi:10.1007/BF01297322 "Über Erweiterungen von unendlichen algebraischen Zahlkörpern." Commentarii Mathematici Helvetici 9, no. 1 (1936): 136–155. doi:10.1007/BF01258182 "Folgen von Dedekindschen Zetafunktionen." Monatshefte für Mathematik 48, no. 1 (1939): 153–160. doi:10.1007/BF01696173 "Zur Theorie der Klassenkörper der Kreiskörper, insbesondere der Strahlklassenkörper der quadratisch imaginären Zahlkörper." Commentarii Mathematici Helvetici 15, no. 1 (1942): 81-119. doi:10.1007/BF02565635 "Zur Theorie der Strahlklassenkörper der quadratisch reellen Zahlkörper." Commentarii Mathematici Helvetici 16, no. 1 (1943): 37–59. doi:10.1007/BF02568563 "Zur Theorie der Normenreste einer relativ-zyklischen Erweiterung von ungeradem Primzahlgrade." Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich 91 (1946): 17–36. "Eulersche Zahlen und grosser Fermat’scher Satz." Commentarii Mathematici Helvetici 24, no. 1 (1950): 73-99. doi:10.1007/BF02567027 == References == == External links == Literature by and about Max Gut in the German National Library catalogue |
Wikipedia:Max Kelly#0 | Gregory Maxwell "Max" Kelly (5 June 1930 – 26 January 2007) was an Australian mathematician who worked on category theory. == Biography == Kelly was born in Bondi, New South Wales, Australia, on 5 June 1930. He obtained his PhD at Cambridge University in homological algebra in 1957, publishing his first paper in that area in 1959, Single-space axioms for homology theory. He taught in the Pure Mathematics department at the University of Sydney from 1957 to 1966, rising from lecturer to reader. During 1963–1965 he was a visiting fellow at Tulane University and the University of Illinois, where with Samuel Eilenberg he formalized and developed the notion of an enriched category based on intuitions then in the air about making the homsets of a category just as abstract as the objects themselves. He subsequently developed the notion in considerably more detail in his 1982 monograph Basic Concepts of Enriched Category Theory. Let V {\displaystyle {\cal {V}}} be a monoidal category, and denote by V {\displaystyle {\cal {V}}} -Cat the category of V {\displaystyle {\cal {V}}} -enriched categories. Among other things, Kelly showed that V {\displaystyle {\cal {V}}} -Cat has all weighted limits and colimits even when V {\displaystyle {\cal {V}}} does not have all ordinary limits and colimits. He also developed the enriched counterparts of Kan extensions, density of the Yoneda embedding, and essentially algebraic theories. In 1967 Kelly was appointed Professor of Pure Mathematics at the University of New South Wales. In 1972 he was elected a Fellow of the Australian Academy of Science. He returned to the University of Sydney in 1973, serving as Professor of Mathematics until his retirement in 1992. In 2001 he was awarded the Australian government's Centenary Medal. He continued to participate in the department as professorial fellow and professor emeritus until his death at age 76 on 26 January 2007. Kelly worked on many other aspects of category theory besides enriched categories, both individually and in a number of collaborations. His PhD students include Ross Street. == References == == External links == Max Kelly's Perpetual Web Page: a memorial page set up by Kelly's son Simon Kelly. "In Memory of Max Kelly": a post at The n-Category Café, containing praise from his fellow mathematicians G. M. Kelly at DBLP Bibliography Server |
Wikipedia:Max Planck Institute for Mathematics#0 | The Max Planck Institute for Mathematics (German: Max-Planck-Institut für Mathematik, MPIM) is a research institute located in Bonn, Germany. It is named in honor of the German physicist Max Planck and forms part of the Max Planck Society (Max-Planck-Gesellschaft), an association of 84 institutes engaging in fundamental research in the arts and the sciences. The MPIM is the only Max Planck institute specializing in pure mathematics. The Institute was founded by Friedrich Hirzebruch in 1980, having emerged from the collaborative research center "Theoretical Mathematics" (Sonderforschungsbereich "Theoretische Mathematik"). Hirzebruch shaped the institute as its director until his retirement in 1995. Currently, the institute is managed by a board of three directors consisting of Peter Teichner (managing director), Peter Scholze and Dennis Gaitsgory. Friedrich Hirzebruch and Yuri Manin were, and Günter Harder, Werner Ballmann, Gerd Faltings and Don Zagier are, acting as emeriti. == Research == The Max Planck Institute for Mathematics offers mathematicians from around the world the opportunity to visit Bonn and engage in sabbatical work lasting from weeks to several months. This guest program distinguishes the MPIM from other Max Planck institutes, and results in only a limit number of permanent positions and the absence of separate departments within the institute. The research of the members and guests of the institute can be classified into the following areas: Algebraic Geometry and Complex Geometry Algebraic Groups Algebraic Topology Arithmetic Geometry Differential Geometry and Topology Dynamical Systems Global Analysis Mathematical Physics Noncommutative Geometry Number Theory Representation Theory == References == == External links == Homepage Homepage (in German) |
Wikipedia:Max Wyman#0 | Max Wyman (April 14, 1916 – February 9, 1991) was a Canadian mathematician and academic administrator. He served as president of the University of Alberta from 1969 to 1974. He was educated at the University of Alberta (BSc 1937) and California Institute of Technology (PhD magna cum laude). He rejoined his alma mater in 1943, the University of Alberta as a lecturer, and became a professor in 1956. In 1961, he was chair of the Department of Mathematics, and from 1963 to 1965 was the Dean of Science. He was named Vice President (Academic) of the university in 1964, and became president in 1969. He was the first former student of the University to become president. Wyman retired in 1974 and died in 1991. == References == |
Wikipedia:Maximal common divisor#0 | In abstract algebra, particularly ring theory, maximal common divisors are an abstraction of the number theory concept of greatest common divisor (GCD). This definition is slightly more general than GCDs, and may exist in rings in which GCDs do not. Halter-Koch (1998) provides the following definition. d ∈ H {\displaystyle d\in H} is a maximal common divisor of a subset, B ⊂ H {\displaystyle B\subset H} , if the following criteria are met: d | b {\displaystyle d|b} for all b ∈ B {\displaystyle b\in B} Suppose c ∈ H {\displaystyle c\in H} , d | c {\displaystyle d|c} and c | b {\displaystyle c|b} for all b ∈ B {\displaystyle b\in B} . Then c ≃ d {\displaystyle c\simeq d} . == References == |
Wikipedia:Maximal semilattice quotient#0 | In abstract algebra, a branch of mathematics, a maximal semilattice quotient is a commutative monoid derived from another commutative monoid by making certain elements equivalent to each other. Every commutative monoid can be endowed with its algebraic preordering ≤ . By definition, x≤ y holds, if there exists z such that x+z=y. Further, for x, y in M, let x ∝ y {\displaystyle x\propto y} hold, if there exists a positive integer n such that x≤ ny, and let x ≍ y {\displaystyle x\asymp y} hold, if x ∝ y {\displaystyle x\propto y} and y ∝ x {\displaystyle y\propto x} . The binary relation ≍ {\displaystyle \asymp } is a monoid congruence of M, and the quotient monoid M / ≍ {\displaystyle M/{\asymp }} is the maximal semilattice quotient of M. This terminology can be explained by the fact that the canonical projection p from M onto M / ≍ {\displaystyle M/{\asymp }} is universal among all monoid homomorphisms from M to a (∨,0)-semilattice, that is, for any (∨,0)-semilattice S and any monoid homomorphism f: M→ S, there exists a unique (∨,0)-homomorphism g : M / ≍ → S {\displaystyle g\colon M/{\asymp }\to S} such that f=gp. If M is a refinement monoid, then M / ≍ {\displaystyle M/{\asymp }} is a distributive semilattice. == References == A.H. Clifford and G.B. Preston, The Algebraic Theory of Semigroups. Vol. I. Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I. 1961. xv+224 p. |
Wikipedia:Maximilian Janisch#0 | Maximilian Sebastian Janisch (born August 8, 2003) is a Swiss doctoral student in mathematics. Janisch showed aptitude for mathematics at a very young age and was described as a child prodigy or Wunderkind. After passing his high school mathematics exams at the age of nine, at the age of twelve, Janisch became the youngest university student in France, and at the age of eighteen, the youngest doctoral student in Switzerland. == Biography == Janisch is the son of the German professor emeritus of mathematics Thomas Drisch and Monika Janisch, who was awarded a doctorate degree in Business Administration from the University of St. Gallen. Janisch skipped three years of elementary school after achieving an IQ score of 149+ in the WISC test. Janisch was admitted to the high school Gymnasium Immensee at the age of eight, and at the age of nine he passed the Matura in mathematics with top marks. After completing his Matura in mathematics, Janisch asked to attend the courses and exams of the mathematics study program at the ETH Zürich and was denied, receiving significant coverage in Swiss media. Janisch was accepted by the French University of Perpignan in September 2015 as the youngest university student in France in the second year of studies, as it is required to have passed the complete Matura to start university in Switzerland, receiving international media coverage. When Janisch passed the full Swiss Matura in August 2018, he enrolled as a mathematics student at the University of Zurich. Janisch has been a doctoral student at the University of Zurich since December 2021. His doctoral advisor is Ashkan Nikeghbali. There are several documentaries about Janisch, including the cinema documentary Maximilian (director: Nicolas Greinacher), which won a Grand Prix at the Rhode Island International Film Festival in 2018. == References == == External links == Official website Janisch, M at arXiv |
Wikipedia:Maximum and minimum#0 | In mathematical analysis, the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extremum, they may be defined either within a given range (the local or relative extrema) or on the entire domain (the global or absolute extrema) of a function. Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions. As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum. In statistics, the corresponding concept is the sample maximum and minimum. == Definition == A real-valued function f defined on a domain X has a global (or absolute) maximum point at x∗, if f(x∗) ≥ f(x) for all x in X. Similarly, the function has a global (or absolute) minimum point at x∗, if f(x∗) ≤ f(x) for all x in X. The value of the function at a maximum point is called the maximum value of the function, denoted max ( f ( x ) ) {\displaystyle \max(f(x))} , and the value of the function at a minimum point is called the minimum value of the function, (denoted min ( f ( x ) ) {\displaystyle \min(f(x))} for clarity). Symbolically, this can be written as follows: x 0 ∈ X {\displaystyle x_{0}\in X} is a global maximum point of function f : X → R , {\displaystyle f:X\to \mathbb {R} ,} if ( ∀ x ∈ X ) f ( x 0 ) ≥ f ( x ) . {\displaystyle (\forall x\in X)\,f(x_{0})\geq f(x).} The definition of global minimum point also proceeds similarly. If the domain X is a metric space, then f is said to have a local (or relative) maximum point at the point x∗, if there exists some ε > 0 such that f(x∗) ≥ f(x) for all x in X within distance ε of x∗. Similarly, the function has a local minimum point at x∗, if f(x∗) ≤ f(x) for all x in X within distance ε of x∗. A similar definition can be used when X is a topological space, since the definition just given can be rephrased in terms of neighbourhoods. Mathematically, the given definition is written as follows: Let ( X , d X ) {\displaystyle (X,d_{X})} be a metric space and function f : X → R {\displaystyle f:X\to \mathbb {R} } . Then x 0 ∈ X {\displaystyle x_{0}\in X} is a local maximum point of function f {\displaystyle f} if ( ∃ ε > 0 ) {\displaystyle (\exists \varepsilon >0)} such that ( ∀ x ∈ X ) d X ( x , x 0 ) < ε ⟹ f ( x 0 ) ≥ f ( x ) . {\displaystyle (\forall x\in X)\,d_{X}(x,x_{0})<\varepsilon \implies f(x_{0})\geq f(x).} The definition of local minimum point can also proceed similarly. In both the global and local cases, the concept of a strict extremum can be defined. For example, x∗ is a strict global maximum point if for all x in X with x ≠ x∗, we have f(x∗) > f(x), and x∗ is a strict local maximum point if there exists some ε > 0 such that, for all x in X within distance ε of x∗ with x ≠ x∗, we have f(x∗) > f(x). Note that a point is a strict global maximum point if and only if it is the unique global maximum point, and similarly for minimum points. A continuous real-valued function with a compact domain always has a maximum point and a minimum point. An important example is a function whose domain is a closed and bounded interval of real numbers (see the graph above). == Search == Finding global maxima and minima is the goal of mathematical optimization. If a function is continuous on a closed interval, then by the extreme value theorem, global maxima and minima exist. Furthermore, a global maximum (or minimum) either must be a local maximum (or minimum) in the interior of the domain, or must lie on the boundary of the domain. So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the boundary, and take the greatest (or least) one. For differentiable functions, Fermat's theorem states that local extrema in the interior of a domain must occur at critical points (or points where the derivative equals zero). However, not all critical points are extrema. One can often distinguish whether a critical point is a local maximum, a local minimum, or neither by using the first derivative test, second derivative test, or higher-order derivative test, given sufficient differentiability. For any function that is defined piecewise, one finds a maximum (or minimum) by finding the maximum (or minimum) of each piece separately, and then seeing which one is greatest (or least). == Examples == For a practical example, assume a situation where someone has 200 {\displaystyle 200} feet of fencing and is trying to maximize the square footage of a rectangular enclosure, where x {\displaystyle x} is the length, y {\displaystyle y} is the width, and x y {\displaystyle xy} is the area: 2 x + 2 y = 200 {\displaystyle 2x+2y=200} 2 y = 200 − 2 x {\displaystyle 2y=200-2x} 2 y 2 = 200 − 2 x 2 {\displaystyle {\frac {2y}{2}}={\frac {200-2x}{2}}} y = 100 − x {\displaystyle y=100-x} x y = x ( 100 − x ) {\displaystyle xy=x(100-x)} The derivative with respect to x {\displaystyle x} is: d d x x y = d d x x ( 100 − x ) = d d x ( 100 x − x 2 ) = 100 − 2 x {\displaystyle {\begin{aligned}{\frac {d}{dx}}xy&={\frac {d}{dx}}x(100-x)\\&={\frac {d}{dx}}\left(100x-x^{2}\right)\\&=100-2x\end{aligned}}} Setting this equal to 0 {\displaystyle 0} 0 = 100 − 2 x {\displaystyle 0=100-2x} 2 x = 100 {\displaystyle 2x=100} x = 50 {\displaystyle x=50} reveals that x = 50 {\displaystyle x=50} is our only critical point. Now retrieve the endpoints by determining the interval to which x {\displaystyle x} is restricted. Since width is positive, then x > 0 {\displaystyle x>0} , and since x = 100 − y {\displaystyle x=100-y} , that implies that x < 100 {\displaystyle x<100} . Plug in critical point 50 {\displaystyle 50} , as well as endpoints 0 {\displaystyle 0} and 100 {\displaystyle 100} , into x y = x ( 100 − x ) {\displaystyle xy=x(100-x)} , and the results are 2500 , 0 , {\displaystyle 2500,0,} and 0 {\displaystyle 0} respectively. Therefore, the greatest area attainable with a rectangle of 200 {\displaystyle 200} feet of fencing is 50 × 50 = 2500 {\displaystyle 50\times 50=2500} . == Functions of more than one variable == For functions of more than one variable, similar conditions apply. For example, in the (enlargeable) figure on the right, the necessary conditions for a local maximum are similar to those of a function with only one variable. The first partial derivatives as to z (the variable to be maximized) are zero at the maximum (the glowing dot on top in the figure). The second partial derivatives are negative. These are only necessary, not sufficient, conditions for a local maximum, because of the possibility of a saddle point. For use of these conditions to solve for a maximum, the function z must also be differentiable throughout. The second partial derivative test can help classify the point as a relative maximum or relative minimum. In contrast, there are substantial differences between functions of one variable and functions of more than one variable in the identification of global extrema. For example, if a bounded differentiable function f defined on a closed interval in the real line has a single critical point, which is a local minimum, then it is also a global minimum (use the intermediate value theorem and Rolle's theorem to prove this by contradiction). In two and more dimensions, this argument fails. This is illustrated by the function f ( x , y ) = x 2 + y 2 ( 1 − x ) 3 , x , y ∈ R , {\displaystyle f(x,y)=x^{2}+y^{2}(1-x)^{3},\qquad x,y\in \mathbb {R} ,} whose only critical point is at (0,0), which is a local minimum with f(0,0) = 0. However, it cannot be a global one, because f(2,3) = −5. == Maxima or minima of a functional == If the domain of a function for which an extremum is to be found consists itself of functions (i.e. if an extremum is to be found of a functional), then the extremum is found using the calculus of variations. == In relation to sets == Maxima and minima can also be defined for sets. In general, if an ordered set S has a greatest element m, then m is a maximal element of the set, also denoted as max ( S ) {\displaystyle \max(S)} . Furthermore, if S is a subset of an ordered set T and m is the greatest element of S with (respect to order induced by T), then m is a least upper bound of S in T. Similar results hold for least element, minimal element and greatest lower bound. The maximum and minimum function for sets are used in databases, and can be computed rapidly, since the maximum (or minimum) of a set can be computed from the maxima of a partition; formally, they are self-decomposable aggregation functions. In the case of a general partial order, a least element (i.e., one that is less than all others) should not be confused with the minimal element (nothing is lesser). Likewise, a greatest element of a partially ordered set (poset) is an upper bound of the set which is contained within the set, whereas the maximal element m of a poset A is an element of A such that if m ≤ b (for any b in A), then m = b. Any least element or greatest element of a poset is unique, but a poset can have several minimal or maximal elements. If a poset has more than one maximal element, then these elements will not be mutually comparable. In a totally ordered set, or chain, all elements are mutually comparable, so such a set can have at most one minimal element and at most one maximal element. Then, due to mutual comparability, the minimal element will also be the least element, and the maximal element will also be the greatest element. Thus in a totally ordered set, we can simply use the terms minimum and maximum. If a chain is finite, then it will always have a maximum and a minimum. If a chain is infinite, then it need not have a maximum or a minimum. For example, the set of natural numbers has no maximum, though it has a minimum. If an infinite chain S is bounded, then the closure Cl(S) of the set occasionally has a minimum and a maximum, in which case they are called the greatest lower bound and the least upper bound of the set S, respectively. == Argument of the maximum == == See also == Derivative test Infimum and supremum Limit superior and limit inferior Maximum-minimums identity Mechanical equilibrium Mex (mathematics) Saddle point Sample maximum and minimum == Notes == == References == == External links == Thomas Simpson's work on Maxima and Minima at Convergence Application of Maxima and Minima with sub pages of solved problems Jolliffe, Arthur Ernest (1911). "Maxima and Minima" . Encyclopædia Britannica. Vol. 17 (11th ed.). pp. 918–920. |
Wikipedia:Maximum theorem#0 | The maximum theorem provides conditions for the continuity of an optimized function and the set of its maximizers with respect to its parameters. The statement was first proven by Claude Berge in 1959. The theorem is primarily used in mathematical economics and optimal control. == Statement of theorem == Maximum Theorem. Let X {\displaystyle X} and Θ {\displaystyle \Theta } be topological spaces, f : X × Θ → R {\displaystyle f:X\times \Theta \to \mathbb {R} } be a continuous function on the product X × Θ {\displaystyle X\times \Theta } , and C : Θ ⇉ X {\displaystyle C:\Theta \rightrightarrows X} be a compact-valued correspondence such that C ( θ ) ≠ ∅ {\displaystyle C(\theta )\neq \emptyset } for all θ ∈ Θ {\displaystyle \theta \in \Theta } . Define the marginal function (or value function) f ∗ : Θ → R {\displaystyle f^{*}:\Theta \to \mathbb {R} } by f ∗ ( θ ) = sup { f ( x , θ ) : x ∈ C ( θ ) } {\displaystyle f^{*}(\theta )=\sup\{f(x,\theta ):x\in C(\theta )\}} and the set of maximizers C ∗ : Θ ⇉ X {\displaystyle C^{*}:\Theta \rightrightarrows X} by C ∗ ( θ ) = a r g max { f ( x , θ ) : x ∈ C ( θ ) } = { x ∈ C ( θ ) : f ( x , θ ) = f ∗ ( θ ) } {\displaystyle C^{*}(\theta )=\mathrm {arg} \max\{f(x,\theta ):x\in C(\theta )\}=\{x\in C(\theta ):f(x,\theta )=f^{*}(\theta )\}} . If C {\displaystyle C} is continuous (i.e. both upper and lower hemicontinuous) at θ {\displaystyle \theta } , then the value function f ∗ {\displaystyle f^{*}} is continuous, and the set of maximizers C ∗ {\displaystyle C^{*}} is upper-hemicontinuous with nonempty and compact values. As a consequence, the sup {\displaystyle \sup } may be replaced by max {\displaystyle \max } . == Variants == The maximum theorem can be used for minimization by considering the function − f {\displaystyle -f} instead. == Interpretation == The theorem is typically interpreted as providing conditions for a parametric optimization problem to have continuous solutions with regard to the parameter. In this case, Θ {\displaystyle \Theta } is the parameter space, f ( x , θ ) {\displaystyle f(x,\theta )} is the function to be maximized, and C ( θ ) {\displaystyle C(\theta )} gives the constraint set that f {\displaystyle f} is maximized over. Then, f ∗ ( θ ) {\displaystyle f^{*}(\theta )} is the maximized value of the function and C ∗ {\displaystyle C^{*}} is the set of points that maximize f {\displaystyle f} . The result is that if the elements of an optimization problem are sufficiently continuous, then some, but not all, of that continuity is preserved in the solutions. == Proof == Throughout this proof we will use the term neighborhood to refer to an open set containing a particular point. We preface with a preliminary lemma, which is a general fact in the calculus of correspondences. Recall that a correspondence is closed if its graph is closed. Lemma. If A , B : Θ ⇉ X {\displaystyle A,B:\Theta \rightrightarrows X} are correspondences, A {\displaystyle A} is upper hemicontinuous and compact-valued, and B {\displaystyle B} is closed, then A ∩ B : Θ ⇉ X {\displaystyle A\cap B:\Theta \rightrightarrows X} defined by ( A ∩ B ) ( θ ) = A ( θ ) ∩ B ( θ ) {\displaystyle (A\cap B)(\theta )=A(\theta )\cap B(\theta )} is upper hemicontinuous. The continuity of f ∗ {\displaystyle f^{*}} in the maximum theorem is the result of combining two independent theorems together. Theorem 1. If f {\displaystyle f} is upper semicontinuous and C {\displaystyle C} is upper hemicontinuous, nonempty and compact-valued, then f ∗ {\displaystyle f^{*}} is upper semicontinuous. Theorem 2. If f {\displaystyle f} is lower semicontinuous and C {\displaystyle C} is lower hemicontinuous, then f ∗ {\displaystyle f^{*}} is lower semicontinuous. Under the hypotheses of the Maximum theorem, f ∗ {\displaystyle f^{*}} is continuous. It remains to verify that C ∗ {\displaystyle C^{*}} is an upper hemicontinuous correspondence with compact values. Let θ ∈ Θ {\displaystyle \theta \in \Theta } . To see that C ∗ ( θ ) {\displaystyle C^{*}(\theta )} is nonempty, observe that the function f θ : C ( θ ) → R {\displaystyle f_{\theta }:C(\theta )\to \mathbb {R} } by f θ ( x ) = f ( x , θ ) {\displaystyle f_{\theta }(x)=f(x,\theta )} is continuous on the compact set C ( θ ) {\displaystyle C(\theta )} . The Extreme Value theorem implies that C ∗ ( θ ) {\displaystyle C^{*}(\theta )} is nonempty. In addition, since f θ {\displaystyle f_{\theta }} is continuous, it follows that C ∗ ( θ ) {\displaystyle C^{*}(\theta )} a closed subset of the compact set C ( θ ) {\displaystyle C(\theta )} , which implies C ∗ ( θ ) {\displaystyle C^{*}(\theta )} is compact. Finally, let D : Θ ⇉ X {\displaystyle D:\Theta \rightrightarrows X} be defined by D ( θ ) = { x ∈ X : f ( x , θ ) = f ∗ ( θ ) } {\textstyle D(\theta )=\{x\in X:f(x,\theta )=f^{*}(\theta )\}} . Since f {\displaystyle f} is a continuous function, D {\displaystyle D} is a closed correspondence. Moreover, since C ∗ ( θ ) = C ( θ ) ∩ D ( θ ) {\displaystyle C^{*}(\theta )=C(\theta )\cap D(\theta )} , the preliminary Lemma implies that C ∗ {\displaystyle C^{*}} is upper hemicontinuous. ◻ {\displaystyle \square } == Variants and generalizations == A natural generalization from the above results gives sufficient local conditions for f ∗ {\displaystyle f^{*}} to be continuous and C ∗ {\displaystyle C^{*}} to be nonempty, compact-valued, and upper semi-continuous. If in addition to the conditions above, f {\displaystyle f} is quasiconcave in x {\displaystyle x} for each θ {\displaystyle \theta } and C {\displaystyle C} is convex-valued, then C ∗ {\displaystyle C^{*}} is also convex-valued. If f {\displaystyle f} is strictly quasiconcave in x {\displaystyle x} for each θ {\displaystyle \theta } and C {\displaystyle C} is convex-valued, then C ∗ {\displaystyle C^{*}} is single-valued, and thus is a continuous function rather than a correspondence. If f {\displaystyle f} is concave in X × Θ {\displaystyle X\times \Theta } and C {\displaystyle C} has a convex graph, then f ∗ {\displaystyle f^{*}} is concave and C ∗ {\displaystyle C^{*}} is convex-valued. Similarly to above, if f {\displaystyle f} is strictly concave, then C ∗ {\displaystyle C^{*}} is a continuous function. It is also possible to generalize Berge's theorem to non-compact correspondences if the objective function is K-inf-compact. == Examples == Consider a utility maximization problem where a consumer makes a choice from their budget set. Translating from the notation above to the standard consumer theory notation, X = R + l {\displaystyle X=\mathbb {R} _{+}^{l}} is the space of all bundles of l {\displaystyle l} commodities, Θ = R + + l × R + + {\displaystyle \Theta =\mathbb {R} _{++}^{l}\times \mathbb {R} _{++}} represents the price vector of the commodities p {\displaystyle p} and the consumer's wealth w {\displaystyle w} , f ( x , θ ) = u ( x ) {\displaystyle f(x,\theta )=u(x)} is the consumer's utility function, and C ( θ ) = B ( p , w ) = { x | p x ≤ w } {\displaystyle C(\theta )=B(p,w)=\{x\,|\,px\leq w\}} is the consumer's budget set. Then, f ∗ ( θ ) = v ( p , w ) {\displaystyle f^{*}(\theta )=v(p,w)} is the indirect utility function and C ∗ ( θ ) = x ( p , w ) {\displaystyle C^{*}(\theta )=x(p,w)} is the Marshallian demand. Proofs in general equilibrium theory often apply the Brouwer or Kakutani fixed-point theorems to the consumer's demand, which require compactness and continuity, and the maximum theorem provides the sufficient conditions to do so. == See also == Envelope theorem Brouwer fixed point theorem Kakutani fixed point theorem for correspondences Michael selection theorem == Notes == == References == Claude Berge (1963). Topological Spaces. Oliver and Boyd. pp. 115–117. Charalambos D. Aliprantis; Kim C. Border (2006). Infinite Dimensional Analysis: A Hitchhiker's Guide. Springer. pp. 569-571. ISBN 9783540295860. Shouchuan Hu; Nikolas S. Papageorgiou (1997). Handbook of Multivalued Analysis. Vol. 1: Theory. Springer-Science + Business Media, B. V. pp. 82–89. |
Wikipedia:Maximum-minimums identity#0 | In mathematics, the maximum-minimums identity is a relation between the maximum element of a set S of n numbers and the minima of the 2n − 1 non-empty subsets of S. Let S = {x1, x2, ..., xn}. The identity states that max { x 1 , x 2 , … , x n } = ∑ i = 1 n x i − ∑ i < j min { x i , x j } + ∑ i < j < k min { x i , x j , x k } − ⋯ ⋯ + ( − 1 ) n + 1 min { x 1 , x 2 , … , x n } , {\displaystyle {\begin{aligned}\max\{x_{1},x_{2},\ldots ,x_{n}\}&=\sum _{i=1}^{n}x_{i}-\sum _{i<j}\min\{x_{i},x_{j}\}+\sum _{i<j<k}\min\{x_{i},x_{j},x_{k}\}-\cdots \\&\qquad \cdots +\left(-1\right)^{n+1}\min\{x_{1},x_{2},\ldots ,x_{n}\},\end{aligned}}} or conversely min { x 1 , x 2 , … , x n } = ∑ i = 1 n x i − ∑ i < j max { x i , x j } + ∑ i < j < k max { x i , x j , x k } − ⋯ ⋯ + ( − 1 ) n + 1 max { x 1 , x 2 , … , x n } . {\displaystyle {\begin{aligned}\min\{x_{1},x_{2},\ldots ,x_{n}\}&=\sum _{i=1}^{n}x_{i}-\sum _{i<j}\max\{x_{i},x_{j}\}+\sum _{i<j<k}\max\{x_{i},x_{j},x_{k}\}-\cdots \\&\qquad \cdots +\left(-1\right)^{n+1}\max\{x_{1},x_{2},\ldots ,x_{n}\}.\end{aligned}}} For a probabilistic proof, see the reference. == See also == Inclusion–exclusion principle Maxima and minima § In relation to sets == References == Ross, Sheldon M. (2020). A First Course in Probability (Tenth, global ed.). Harlow, United Kingdom: Pearson. pp. 331–333. ISBN 978-1-292-26920-7. |
Wikipedia:Maya Stein#0 | Maya Jakobine Stein is a German mathematician working as a professor at the Department of Mathematical Engineering of the University of Chile. She is also the vice director and the academic director of the Center for Mathematical Modeling of the University of Chile. After earning a degree in mathematics from the University of Hamburg in 2002, she continued for a doctorate in 2005, supervised by Reinhard Diestel. She then spent three years as a postdoctoral researcher at the University of São Paulo with Yoshiharu Kohayakawa, before moving to the University of Chile as an associated researcher in 2008. She obtained an associate professor position in 2016 and was promoted to full professor in 2020. Stein is known for her research in combinatorics, in particular in graph theory, and her interests include extremal and probabilistic combinatorics, Ramsey theory, as well as structural and algorithmic graph theory and infinite graphs. She has more than 60 publications in these areas. She is the vice-chair of the SIAM activity group for discrete mathematics. == Editorial positions == Managing editor for Innovations in Graph Theory Editor-in-Chief for Electronic Journal of Combinatorics Associate editor for SIAM Journal on Discrete Mathematics Associate editor for Orbita Mathematicae == References == |
Wikipedia:Mayuko Yamashita#0 | Mayuko Yamashita (Japanese: 山下 真由子, born 1995) is a Japanese mathematician and mathematical physicist whose research combines the areas of algebraic topology, differential cohomology, and quantum field theory. She is an associate professor at Kyoto University. == Education and career == Yamashita represented Japan in the 2013 International Mathematical Olympiad, earning a silver medal. She studied engineering at the University of Tokyo, earning a bachelor's degree in 2017. She earned a master's degree in mathematical sciences at the University of Tokyo in March 2019, and completed a Ph.D. there in 2022. Her dissertation, Differential models for the Anderson dual to bordism theories and invertible QFT's, was supervised by Yasuyuki Kawahigashi. Meanwhile, in 2019, she became an assistant professor in the Research Institute for Mathematical Sciences of Kyoto University, while only 23 years old. She was promoted to associate professor in 2023 and will be joining the Perimeter Institute for Theoretical Physics in April 2025. == Recognition == Yamashita was one of the recipients of the 2021 Takebe Katahiro Prize for Encouragement of Young Researchers of the Mathematical Society of Japan. She received the 2022 Grand Prize in the Marie Sklodowska Curie Awards of the Japan Science and Technology Agency "for her work on mathematical applications to particle physics". She was named in the Asian Scientist 100 list in 2023 and 2024. She was a 2024 recipient of the Maryam Mirzakhani New Frontiers Prize, associated with the Breakthrough Prize in Mathematics, "for contributions to mathematical physics and index theory".. 2024 she was awarded the Dannie Heineman Prize (Göttingen) of the Göttingen Academy of Sciences and Humanities (today called Niedersächsische Akademie der Wissenschaften zu Göttingen) for her contribution to the precise description of anomalies in quantum field theories. == References == == External links == Home page |
Wikipedia:Mazhamaṅgalaṃ Nārāyaṇan Naṃpūtiri#0 | Mazhamaṅgalaṃ Nārāyaṇan Naṃpūtiri (Nārāyaṇa of Mahiṣamaṅgalṃ) (c. 1540–1610) was an Indian scholar, poet, astrologer and mathematician belonging to the Kerala school of astronomy and mathematics. Nārāyaṇan Naṃpūtiri hailed from Peruvanam village in present-day Thrissur district in Kerala. His father was Mazhamaṅgalaṃ Śaṅkaran Naṃpūtiri himself a respected scholar and writer who had authored a large number of books on astronomy and astrology in the vernacular Malayalam language in an effort to popularize astronomy among the lay public. There is a legend to the effect that in the early days of his life Nārāyaṇan Naṃpūtiri was a spendthrift and lived a wayward life. One day, at the place of the performance of a yāga, he was hugely insulted for his ignorance of veda-s and other scriptures, and being deeply humiliated he left to Chola country, spent several years there and returned as a great scholar in veda-s, śrauta-śāstra and smārtta-sāstra to the amazement of those who had insulted him earlier. Thereafter, he used to be given a respectful and prominent place at the performance in all such rituals, and even after his death, his spiritual presence used to be invoked on the occasion of the performance of the rituals. His highly respected work Smārta-prāyaścitta-vimarśinī stands testimony to his deep knowledge of veda-s and other scriptures. His deep knowledge in astronomy and mathematics is reflected in his works. When Śaṅkara Vāriyar, who was composing an erudite and elaborate commentary on Līlāvatī, had to abandon his work due to some inconveniences while he was half-way through the work the task of completing the commentary was entrusted to Mazhamaṅgalaṃ Nārāyaṇan Naṃpūtiri. The commentary known as Kriyākramakarī is of the joint authorship of Śaṅkara Vāriyar and Mazhamaṅgalaṃ Nārāyaṇan Naṃpūtiri. This commentary has become a very valuable source of information regarding the mathematical achievements of the Kerala school of astronomy and mathematics. Initially Nārāyaṇan Naṃpūtiri was staying at his ancestral home at Peruvanam. Later he moved to Thrissur and took up the position of the chief priest of the local Paramekkavu Bagavathi Temple. While at Thrissur, he married a Brāhmaṇi, a woman of the Nambeesan caste. There was a custom of these Brāhmaṇi-s performing ritualistic singing of devotional songs on the occasions marriage ceremonies. The songs sung during these rituals are called Brāhmaṇi songs. Nārāyaṇan Naṃpūtiri composed as many as five such songs for the benefit of his wife. == Nārāyaṇan Naṃpūtiri's works == === Work on expiatory rituals === Nārāyaṇan Naṃpūtiri's reputation rests mainly on his literary works and on his magnum opus titled Smārta-prāyaścitta-vimarśinī, a work dealing with expiatory rituals. This treatise deals with the expiatory rituals to be carried out in connection with the performance of mega-rituals like Somayajna and is very popular in Kerala. === Astronomy and mathematics === In one of the verses in Smārta-prāyścitta-vimarśinī, the author has described himself as a gaṇitavid (meaning, "scholar in mathematics"). There are not many works authored by him which can be cited in support of this appellation. An important work that can be cited in this context is a part commentary on Bhāskara II's Līlāvatī. The first half of this commentary called Kriyākramakarī (up to verse 199 in Līlāvatī) has been composed by Śaṅkara Vāriyar, a prominent member of the Kerala school of astronomy and mathematics, and the commentary on the remaining verses was composed by Nārāyaṇan Naṃpūtiri. Two other works on astronomy have been identified as authored by Nārāyaṇan Naṃpūtiri: One of them is a brief commentary called Karmadīpikā or Karmapradīpikā on the textual verses in Līlāvatī and the other is a treatise in five chapters, titled Uparāgakriyākrama, on some new methods for the computation of lunar and solar eclipses. === Literary works in Sanskrit === Vyavahāramālā: A work on Indian jurisprudence (Full text is available in the Internet Archive at the link HERE.) Mahiṣamaṅgalaṃ Bhāṇaṃ (Full text is available in the Internet Archive at the link HERE. Full text with translation and detailed explanations in Malayalam is available at the link HERE.) Uttara Rāmaayaṇa Caṃpu Rāsakrīdā Kāvyaṃ === Literary works in Malayalam === == Mazhamaṅgalaṃ Nārāyaṇan Naṃpūtiri: the inventor of pañcāri mēḷaṃ == Pañcāri mēḷaṃ is a percussion ensemble, performed during temple festivals in Kerala, India. It is one of the major forms of ceṇṭa mēḷaṃ (ethnic drum ensemble), and is the best-known and most popular drum ensemble in the kṣtra-vādyaṃ (temple percussion) genre. Pañcāri mēḷaṃ, comprises the instruments ceṇḍa, ilattāḷaṃ, koṃbu and kuzhal. It was Mazhamaṅgalaṃ Nārāyaṇan Naṃpūtiri, with the support of Pandarathil Raman Marar, who conceptualized, designed and helped to perform first the percussion ensemble known as pañcāri mēḷaṃ. According to traditions, the first performance took place in the precincts of Peruvanam Mahadeva Temple. == See also == Mazhamaṅgalaṃ Śaṅkaran Naṃpūtiri List of astronomers and mathematicians of the Kerala school == References == == Additional reading == More detailed information about the works of Mazhamaṅgalam Nārāyaṇan Nampūtiri is available in the classic multi-volume Malayalam work "Kerala Sahithya Charithram" ("History of Literature in Kerala") by Ulloor S. Paramesvara Aiyer: Ulloor S. Paramesvara Aiyer (September 1954). Kerala Sahithya Charithram Part II. Thiruvananthapuram, Kerala, India: Department of Publications, University of Travancore. Retrieved 15 July 2024. (pages 293–301) For a detailed description of the structure of pañcāri mēḷaṃ see: Vishnu Achutha Menon and Boobalakrishnan. N (2020). "The Rhythmic Structure of Melam". The Chitralekha Journal on Art and Design. 4 (2): 1–14. Retrieved 16 July 2024.. |
Wikipedia:Mazhamaṅgalaṃ Śaṅkaran Naṃpūtiri#0 | Mazhamaṅgalaṃ Nārāyaṇan Naṃpūtiri (Nārāyaṇa of Mahiṣamaṅgalṃ) (c. 1540–1610) was an Indian scholar, poet, astrologer and mathematician belonging to the Kerala school of astronomy and mathematics. Nārāyaṇan Naṃpūtiri hailed from Peruvanam village in present-day Thrissur district in Kerala. His father was Mazhamaṅgalaṃ Śaṅkaran Naṃpūtiri himself a respected scholar and writer who had authored a large number of books on astronomy and astrology in the vernacular Malayalam language in an effort to popularize astronomy among the lay public. There is a legend to the effect that in the early days of his life Nārāyaṇan Naṃpūtiri was a spendthrift and lived a wayward life. One day, at the place of the performance of a yāga, he was hugely insulted for his ignorance of veda-s and other scriptures, and being deeply humiliated he left to Chola country, spent several years there and returned as a great scholar in veda-s, śrauta-śāstra and smārtta-sāstra to the amazement of those who had insulted him earlier. Thereafter, he used to be given a respectful and prominent place at the performance in all such rituals, and even after his death, his spiritual presence used to be invoked on the occasion of the performance of the rituals. His highly respected work Smārta-prāyaścitta-vimarśinī stands testimony to his deep knowledge of veda-s and other scriptures. His deep knowledge in astronomy and mathematics is reflected in his works. When Śaṅkara Vāriyar, who was composing an erudite and elaborate commentary on Līlāvatī, had to abandon his work due to some inconveniences while he was half-way through the work the task of completing the commentary was entrusted to Mazhamaṅgalaṃ Nārāyaṇan Naṃpūtiri. The commentary known as Kriyākramakarī is of the joint authorship of Śaṅkara Vāriyar and Mazhamaṅgalaṃ Nārāyaṇan Naṃpūtiri. This commentary has become a very valuable source of information regarding the mathematical achievements of the Kerala school of astronomy and mathematics. Initially Nārāyaṇan Naṃpūtiri was staying at his ancestral home at Peruvanam. Later he moved to Thrissur and took up the position of the chief priest of the local Paramekkavu Bagavathi Temple. While at Thrissur, he married a Brāhmaṇi, a woman of the Nambeesan caste. There was a custom of these Brāhmaṇi-s performing ritualistic singing of devotional songs on the occasions marriage ceremonies. The songs sung during these rituals are called Brāhmaṇi songs. Nārāyaṇan Naṃpūtiri composed as many as five such songs for the benefit of his wife. == Nārāyaṇan Naṃpūtiri's works == === Work on expiatory rituals === Nārāyaṇan Naṃpūtiri's reputation rests mainly on his literary works and on his magnum opus titled Smārta-prāyaścitta-vimarśinī, a work dealing with expiatory rituals. This treatise deals with the expiatory rituals to be carried out in connection with the performance of mega-rituals like Somayajna and is very popular in Kerala. === Astronomy and mathematics === In one of the verses in Smārta-prāyścitta-vimarśinī, the author has described himself as a gaṇitavid (meaning, "scholar in mathematics"). There are not many works authored by him which can be cited in support of this appellation. An important work that can be cited in this context is a part commentary on Bhāskara II's Līlāvatī. The first half of this commentary called Kriyākramakarī (up to verse 199 in Līlāvatī) has been composed by Śaṅkara Vāriyar, a prominent member of the Kerala school of astronomy and mathematics, and the commentary on the remaining verses was composed by Nārāyaṇan Naṃpūtiri. Two other works on astronomy have been identified as authored by Nārāyaṇan Naṃpūtiri: One of them is a brief commentary called Karmadīpikā or Karmapradīpikā on the textual verses in Līlāvatī and the other is a treatise in five chapters, titled Uparāgakriyākrama, on some new methods for the computation of lunar and solar eclipses. === Literary works in Sanskrit === Vyavahāramālā: A work on Indian jurisprudence (Full text is available in the Internet Archive at the link HERE.) Mahiṣamaṅgalaṃ Bhāṇaṃ (Full text is available in the Internet Archive at the link HERE. Full text with translation and detailed explanations in Malayalam is available at the link HERE.) Uttara Rāmaayaṇa Caṃpu Rāsakrīdā Kāvyaṃ === Literary works in Malayalam === == Mazhamaṅgalaṃ Nārāyaṇan Naṃpūtiri: the inventor of pañcāri mēḷaṃ == Pañcāri mēḷaṃ is a percussion ensemble, performed during temple festivals in Kerala, India. It is one of the major forms of ceṇṭa mēḷaṃ (ethnic drum ensemble), and is the best-known and most popular drum ensemble in the kṣtra-vādyaṃ (temple percussion) genre. Pañcāri mēḷaṃ, comprises the instruments ceṇḍa, ilattāḷaṃ, koṃbu and kuzhal. It was Mazhamaṅgalaṃ Nārāyaṇan Naṃpūtiri, with the support of Pandarathil Raman Marar, who conceptualized, designed and helped to perform first the percussion ensemble known as pañcāri mēḷaṃ. According to traditions, the first performance took place in the precincts of Peruvanam Mahadeva Temple. == See also == Mazhamaṅgalaṃ Śaṅkaran Naṃpūtiri List of astronomers and mathematicians of the Kerala school == References == == Additional reading == More detailed information about the works of Mazhamaṅgalam Nārāyaṇan Nampūtiri is available in the classic multi-volume Malayalam work "Kerala Sahithya Charithram" ("History of Literature in Kerala") by Ulloor S. Paramesvara Aiyer: Ulloor S. Paramesvara Aiyer (September 1954). Kerala Sahithya Charithram Part II. Thiruvananthapuram, Kerala, India: Department of Publications, University of Travancore. Retrieved 15 July 2024. (pages 293–301) For a detailed description of the structure of pañcāri mēḷaṃ see: Vishnu Achutha Menon and Boobalakrishnan. N (2020). "The Rhythmic Structure of Melam". The Chitralekha Journal on Art and Design. 4 (2): 1–14. Retrieved 16 July 2024.. |
Wikipedia:Mean value theorem (divided differences)#0 | In mathematical analysis, the mean value theorem for divided differences generalizes the mean value theorem to higher derivatives. == Statement of the theorem == For any n + 1 pairwise distinct points x0, ..., xn in the domain of an n-times differentiable function f there exists an interior point ξ ∈ ( min { x 0 , … , x n } , max { x 0 , … , x n } ) {\displaystyle \xi \in (\min\{x_{0},\dots ,x_{n}\},\max\{x_{0},\dots ,x_{n}\})\,} where the nth derivative of f equals n ! times the nth divided difference at these points: f [ x 0 , … , x n ] = f ( n ) ( ξ ) n ! . {\displaystyle f[x_{0},\dots ,x_{n}]={\frac {f^{(n)}(\xi )}{n!}}.} For n = 1, that is two function points, one obtains the simple mean value theorem. == Proof == Let P {\displaystyle P} be the Lagrange interpolation polynomial for f at x0, ..., xn. Then it follows from the Newton form of P {\displaystyle P} that the highest order term of P {\displaystyle P} is f [ x 0 , … , x n ] x n {\displaystyle f[x_{0},\dots ,x_{n}]x^{n}} . Let g {\displaystyle g} be the remainder of the interpolation, defined by g = f − P {\displaystyle g=f-P} . Then g {\displaystyle g} has n + 1 {\displaystyle n+1} zeros: x0, ..., xn. By applying Rolle's theorem first to g {\displaystyle g} , then to g ′ {\displaystyle g'} , and so on until g ( n − 1 ) {\displaystyle g^{(n-1)}} , we find that g ( n ) {\displaystyle g^{(n)}} has a zero ξ {\displaystyle \xi } . This means that 0 = g ( n ) ( ξ ) = f ( n ) ( ξ ) − f [ x 0 , … , x n ] n ! {\displaystyle 0=g^{(n)}(\xi )=f^{(n)}(\xi )-f[x_{0},\dots ,x_{n}]n!} , f [ x 0 , … , x n ] = f ( n ) ( ξ ) n ! . {\displaystyle f[x_{0},\dots ,x_{n}]={\frac {f^{(n)}(\xi )}{n!}}.} == Applications == The theorem can be used to generalise the Stolarsky mean to more than two variables. == References == |
Wikipedia:Mean-periodic function#0 | In mathematical analysis, the concept of a mean-periodic function is a generalization introduced in 1935 by Jean Delsarte of the concept of a periodic function. Further results were made by Laurent Schwartz and J-P Kahane. == Definition == Consider a continuous complex-valued function f of a real variable. The function f is periodic with period a precisely if for all real x, we have f(x) − f(x − a) = 0. This can be written as ∫ f ( x − t ) d μ ( t ) = 0 ( 1 ) {\displaystyle \int f(x-t)\,d\mu (t)=0\qquad \qquad (1)} where μ {\displaystyle \mu } is the difference between the Dirac measures at 0 and a. The function f is mean-periodic if it satisfies the same equation (1), but where μ {\displaystyle \mu } is some arbitrary nonzero measure with compact (hence bounded) support. Equation (1) can be interpreted as a convolution, so that a mean-periodic function is a function f for which there exists a compactly supported (signed) Borel measure μ {\displaystyle \mu } for which f ∗ μ = 0 {\displaystyle f*\mu =0} . There are several well-known equivalent definitions. == Relation to almost periodic functions == Mean-periodic functions are a separate generalization of periodic functions from the almost periodic functions. For instance, exponential functions are mean-periodic since exp(x+1) − e.exp(x) = 0, but they are not almost periodic as they are unbounded. Still, there is a theorem which states that any uniformly continuous bounded mean-periodic function is almost periodic (in the sense of Bohr). In the other direction, there exist almost periodic functions which are not mean-periodic. == Some basic properties == If f is a mean periodic function, then it is the limit of a certain sequence of exponential polynomials which are finite linear combinations of term t^^n exp(at) where n is any non-negative integer and a is any complex number; also Df is a mean periodic function (ie mean periodic) and if h is an exponential polynomial, then the pointwise product of f and h is mean periodic). If f and g are mean periodic then f + g and the truncated convolution product of f and g is mean periodic. However, the pointwise product of f and g need not be mean periodic. If L(D) is a linear differential operator with constant co-efficients, and L(D)f = g, then f is mean periodic if and only if g is mean periodic. For linear differential difference equations such as Df(t) - af(t - b) = g where a is any complex number and b is a positive real number, then f is mean periodic if and only if g is mean periodic. == Applications == In work related to the Langlands correspondence, the mean-periodicity of certain (functions related to) zeta functions associated to an arithmetic scheme have been suggested to correspond to automorphicity of the related L-function. There is a certain class of mean-periodic functions arising from number theory. == See also == almost periodic functions == References == |
Wikipedia:Mehmet Burak Erdoğan#0 | Mehmet Burak Erdoğan (born 1972) is a Turkish mathematician, scientist, and professor of mathematics. He is a member of the University of Illinois Urbana-Champaign (UIUC) Mathematics Department. == Education == Burak Erdoğan was born in 1972. He attended the high school Kayseri Fen Lisesi in Kayseri, Turkey, for two years before moving to İzmir, Turkey where he finished the high school Atatürk Lisesi in 1989. He ranked third in the country-wide university entrance exams. He graduated from Bilkent University Electrical and Electronics Engineering in Ankara in 1994. He finished his MSc degree in Mathematics department of the same university under the supervision of Iossif Vladimirovich Ostrovskii in 1996. He received his PhD in Mathematics under the supervision of Thomas Wolff at Caltech in 2001. He was the last PhD student of Wolff, who died in the last year of Erdoğan's PhD work. == Academic career == He was a postdoc at the Institute for Advanced Study between 2001-2002 and a visiting assistant professor at University of California, Berkeley between 2002-2004. Since then he is a faculty member in UIUC Mathematics Department. He has been an active researcher, having published more than 50 scientific manuscripts with more than 800 citations. He raised 6 PhD students. == Research areas == His research interests include harmonic analysis and dispersive partial differential equations. === Representative scientific publications === Erdoğan, M.Burak; Tzirakis, Nikolaos (2016). Dispersive partial differential equations, wellposedness and applications. London Mathematical Society Student Texts. Vol. 86. Cambridge University Press. ISBN 9781316602935. Erdog̃an, M. Burak (2005). "A bilinear Fourier extension theorem and applications to the distance set problem". International Mathematics Research Notices. 2005 (23): 1411–1425. doi:10.1155/IMRN.2005.1411.{{cite journal}}: CS1 maint: unflagged free DOI (link) Chapman, Jeremy; Erdoğan, M. Burak; Hart, Derrick; Iosevich, Alex; Koh, Doowon (2012). "Pinned distance sets, k-simplices, Wolff's exponent in finite fields and sum-product estimates". Mathematische Zeitschrift. 271 (1–2): 63–93. arXiv:0903.4218. doi:10.1007/s00209-011-0852-4. S2CID 253711074. Erdoğan, M.Burak; Tzirakis, Nikolaos (2013). "Global smoothing for the periodic KdV evolution". International Mathematics Research Notices. 2013 (20): 4589–4614. arXiv:1103.4190. doi:10.1093/imrn/rns189. Erdoğan, M.Burak; Goldberg, Michael; Schlag, Wilhelm (2008). "Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in R 3 {\displaystyle \mathbb {R} ^{3}} ". Journal of the European Mathematical Society. 10 (2): 507–531. CiteSeerX 10.1.1.235.7269. doi:10.4171/JEMS/120. Erdoğan, M.Burak; Green, William (2022). "The L p {\displaystyle L^{p}} continuity of wave operators for higher order Schrodinger operators". Advances in Mathematics. 404, Part B: 108450. arXiv:2107.09620. doi:10.1016/j.aim.2022.108450. S2CID 237057129. == References == == External links == Mehmet Burak Erdoğan publications indexed by Google Scholar |
Wikipedia:Mehmet Nadir#0 | Mehmet Nadir (1856 – 13 December 1927) was a Turkish mathematician and educator. == Early life == He was born in Sakız island (modern Chios in Greece) then a part of the Ottoman Empire, to a poor family. He was adopted by a sea captain, who would be his father-in-law in the future. He studied in the military high school in Bursa. Then, he went to Istanbul to study in the military college and then the naval academy. After completing his education with honours, he was appointed teacher of mathematics at the naval academy, and later at Darüşşafaka High School, one of the most prominent schools in Istanbul. In 1879, together with one of his friends, he travelled to England for advanced studies. But the details of this period are not known. Probably, he also spent some time in Cyprus. == Educator == After returning to Istanbul, he began teaching in private high schools, and in 1884 he established a high school of his own. It was a junior high school (Turkish: Rüştiye). Later, he added a senior high school section to his school (Turkish: İdadi). Although there were private junior high schools before 1884, he was the founder of the very first private senior high school in Istanbul. This school is now considered as the core of the modern Istanbul High School for Boys (Turkish: İstanbul Erkek Lisesi). He also added a section for the girls and a branch school in Edirne. Another novelty he introduced was the first school newspaper in Istanbul. == Politics and later life == He secretly supported the Young Turks movement. But when Sultan Abdülhamit II arrested 350 Young Turk adherents on the charge of planning a coup in 1896, Mehmet Nadir was forced to resign. After working in a public school in Istanbul in 1903, he was appointed to Aleppo (now in Syria) as the director of education. In 1908, Young Turks came to power and he was exiled to Tripoli (now in Libya) by the Young Turks, who suspected of his betrayal back in 1896. After the Italo-Turkish War, in which Italians invaded Libya, he returned to Istanbul. Although he was also sent to Edirne, he soon returned to Istanbul during the Balkan Wars. After he convinced Young Turks of his innocence, he was appointed mathematics professor in the newly established Girls' University in Istanbul. In 1919 soon after the World War I, he began serving in the Darülfünun (present day, Istanbul University) as a müderris (professor) of mathematics in the newly established branch of Number theory. He died on 27 December 1927. == As a mathematician == Mehmet Nadir is considered as one of the first mathematicians of Turkey. He actively participated in a group of mathematicians of a French mathematics periodical named I’Intermediaire des Mathematiciens. According to Professor Erdal İnönü, a total of 62 papers of him were published in the periodical. One example cited by İnönü is as follows: The whole number solution to the equation x 2 + y 2 − z 2 = u 5 {\displaystyle x^{2}+y^{2}-z^{2}=u^{5}} proposed by Mehmet Nadir is x = b ⋅ ( a 2 + b 2 ) ⋅ ( a 2 − b 2 ) {\displaystyle x=b\cdot (a^{2}+b^{2})\cdot (a^{2}-b^{2})} y = 1 2 ⋅ ( ( a 2 − 1 ) ⋅ ( a 2 + b 2 ) 2 − 4 ⋅ b 2 ) {\displaystyle y={\frac {1}{2}}\cdot ((a^{2}-1)\cdot (a^{2}+b^{2})^{2}-4\cdot b^{2})} z = 1 2 ⋅ ( ( a 2 + 1 ) ⋅ ( a 2 + b 2 ) 2 + 4 ⋅ b 2 ) {\displaystyle z={\frac {1}{2}}\cdot ((a^{2}+1)\cdot (a^{2}+b^{2})^{2}+4\cdot b^{2})} u = a 2 + b 2 {\displaystyle u=a^{2}+b^{2}} where b is a positive integer and a is a positive odd integer. In 1917, Mehmet Nadir published a book on the Number theory. In this book, he proposed an alternative algorithm on divisibility. Well-known German mathematics professor Felix Klein congratulated him for the algorithm. == References == |
Wikipedia:Meike Akveld#0 | Meike Maria Elisabeth Akveld is a Swiss mathematician and textbook author, whose professional interests include knot theory, symplectic geometry, and mathematics education. She is a tenured senior scientist and lecturer in the mathematics and teacher education group in the Department of Mathematics at ETH Zurich. She is also the organizer of the Mathematical Kangaroo competitions in Switzerland, and president of the Association Kangourou sans Frontières, a French-based international society devoted to the popularization of mathematics. == Education == Akveld earned a bachelor's degree from the University of Warwick and took Part III of the Mathematical Tripos at the University of Cambridge. She completed her Ph.D. at ETH Zurich in 2000, with the dissertation Hofer geometry for Lagrangian loops, a Legendrian knot and a travelling wave jointly supervised by Dietmar Salamon and Leonid Polterovich. == Books == Akveld's mathematics books include: Canonical metrics in Kähler geometry (by Tian Gang, based on notes taken by Akveld, Birkhäuser, 2000) Knoten in der Mathematik: Ein Spiel mit Schnüren, Bildern und Formeln (Knots in mathematics: A game with strings, pictures and formulas, in German, Orell Füssli, 2007) Hofer geometry for Lagrangian loops: And a Legendrian Knot and a travelling wave (VDM Verlag, 2008) Integrieren - do it yourself (in German, with Ursula Eisler and Daniel Zogg, Orell Füssli, 2010) Knots Unravelled: From String to Mathematics (with Andrew Jobbings, Arbelos, 2011) Analysis I and Analysis II (in German, with René Sperb, VDF Hochschulverlag, 2012 and 2015) Knopen in de wiskunde (Knots in mathematics, in Dutch, with Ab van der Roest, Epsilon Uitgaven, 2015) Mathe mit dem Känguru 5: Die schönsten Aufgaben von 2015 bis 2019 (Math with the kangaroo 5: The most beautiful problems from 2015 to 2019, in German, with Alexander Unger, Monika Noack, and Robert Geretschläger, Hanser Verlag, 2019) == References == == External links == Home page Meike Akveld publications indexed by Google Scholar |
Wikipedia:Meinhard E. Mayer#0 | Meinhard Edwin Mayer (March 18, 1929 – December 11, 2011) was a Romanian–born American Professor Emeritus of Physics and Mathematics at the University of California, Irvine, which he joined in 1966. == Biography == He was born on March 18, 1929, in Cernăuți. He experienced both the Soviet occupation of Northern Bukovina and, as a Jew, deportation to the Transnistria Governorate. He received his Ph.D. from the University of Bucharest in 1957, where he taught until 1961. He then taught at Brandeis University and Indiana University before moving to the University of California, Irvine (UCI) in 1966, where he taught until his retirement. He also took sabbaticals to various institutes, including the Institut des Hautes Etudes Scientifiques and MIT. He had a deep interest in music, and in Yiddish language and literature. He died in Newport Beach, California, on December 11, 2011. He was survived by his wife Ruth, his children Elma Mayer and Niels Mayer, and his grandchildren Jonathan Mayer, Juniper Woodbury, and Moss Woodbury. == Research == His research interests ranged from geometric methods in gauge theory, to the application of wavelets in turbulence. He was an early contributor (1958) to the theory of vector-bosons (W and Z bosons) and electro-weak unification, which later became the Standard Model, and an early advocate of the use of fiber bundles in gauge theory. He was a co-author (with Gerald Jay Sussman and Jack Wisdom) of Structure and Interpretation of Classical Mechanics, MIT Press, Cambridge, MA, 2001 ISBN 0-262-19455-4 == Notes == == References == Lie Groupoids versus Principal Bundles in Gauge Theories, in Proceedings of the International Conference on Differential-Geometric Methods in Physics, L.-L. Chau and W. Nahm, Eds., Plenum Press, 1990. From Poisson Groupoids to Quantum Groupoids, and Back, in Proceedings of the XIX International Conference on Differential-Geometric Methods in Physics, R. Cianci and U. Bruzzo, Eds. Rapallo, 1990; 12 pages, Springer Verlag, Heidelberg, 1991. Wavelet Transforms and Atmospheric Turbulence, with Carl A. Friehe and Lonnie H. Hudgins, Physical Review Letters, 71, 3279-3282 (November 15, 1993) == External links == Meinhard E. Mayer at the Mathematics Genealogy Project[1] Obituary in Physics Today Web Page (somewhat obsolete) Archived 2007-03-26 at the Wayback Machine Faculty Profile Paul Celan Article An article about the 1908 Yiddish Language Conference (Yiddish and English) QuickTime version (with sound) of a talk On Yiddish and German Poets from Czernowitz at the 2008 La Jolla Yiddish Conference Slides (without sound) of the talk On Yiddish and German Poets from Czernowitz at the 2008 La Jolla Yiddish Conference |
Wikipedia:Melania Alvarez#0 | Melania Alvarez de Adem is a Mexican mathematics educator who works as the Education Coordinator at the Pacific Institute for the Mathematical Sciences (PIMS), and Outreach Coordinator for the Department of Mathematics at the University of British Columbia in Vancouver. == Education == Alvarez grew up in Mexico City, where she completed her undergraduate education at the National Autonomous University of Mexico. She later earned master's degrees in economics and anthropology from the University of Wisconsin–Madison and in operations research from Stanford University. In 2016 she completed a Ph.D. in mathematics education at Simon Fraser University under the supervision of Peter Liljedahl. == Outreach == Alvarez gained interest in helping disadvantaged minorities with mathematics from an incident of racial discrimination that occurred in Madison, where her sixth-grade son (who had inherited his mother's talent for mathematics) was placed in a lower-level mathematics track because of his Hispanic ethnicity. She moved to Vancouver in 2004, and began the PIMS Emerging Indigenous Scholars Summer School Program at UBC in 2007. At UBC, Alvarez is known for the summer mathematics camps she developed to improve the mathematical education of indigenous secondary-school students. She is the 2012 winner of the Adrien Pouliot Award, given by the Canadian Mathematical Society for significant contributions to mathematics education in Canada. As well as her work with indigenous people, Alvarez has been active in organizing mathematics competitions, workshops, and fundraising for mathematics education among the general population. == References == |
Wikipedia:Melanie Johnston-Hollitt#0 | Melanie Johnston-Hollitt (née Johnston; 8 September 1974) is an Australian astrophysicist and professor. She has worked on the design, construction, and international governance of several radio telescopes including the Low Frequency Array (LOFAR), the Murchison Widefield Array (MWA) and the upcoming Square Kilometre Array (SKA). She was the director of the Murchison Widefield Array until December 2020 and is a professor at the Curtin Institute of Radio Astronomy at Curtin University and the International Centre for Radio Astronomy Research. Since August 2020, Melanie Johnston-Hollitt is the director of the Curtin Institute for Data Science. == Early life == Johnston-Hollitt was born on 8 September 1974 in Darwin, Australia. She was born shortly before Cyclone Tracy destroyed the city after which she and her mother were evacuated to Brisbane on December 29, 1974 en route to Townsville. During her early life she lived in various parts of Queensland before her family returned to Darwin in 1979. In 1986 her family moved to Adelaide where she completed high school. Johnston-Hollitt attended the University of Adelaide, where she earned two degrees: one in theoretical and experimental physics and the other in mathematics and computer science. She also completed a BSc with honors in astrophysics under the supervision of Roger Clay, and a doctorate in radio astronomy jointly with the University of Adelaide and the CSIRO Australia Telescope National Facility under the supervision of Ron Ekers and Richard Hunstead. == Career == Johnston-Hollitt moved to Leiden Observatory at the University of Leiden in the Netherlands as the inaugural LOFAR Fellow, joining the group of George Miley to work on the design of the LOFAR telescope. In 2004, she took up a continuing faculty position at the University of Tasmania. In January 2009 she moved to Victoria University of Wellington in New Zealand to establish and lead a new radio astronomy group. Johnston-Hollitt worked on NZ's engagement with the Square Kilometre Array, and she served in an advisory role for the New Zealand Government on the project from April 2009 to January 2018. Shortly after commencing her term advising the NZ Government, it was announced that NZ would join Australia in bidding to host the SKA telescope. Johnston-Hollitt was subsequently an author of the Australia-New Zealand bid to host the SKA, responsible for identifying possible SKA station sites in New Zealand. From 2011, she was a government-appointed member of the preparatory group set-up to establish the SKA, and then director for the SKA from December 2011 to January 2018. For her scientific leadership she was appointed by the SKAO as founding co-chair of the SKA Cosmic Magnetism Science Working group in March 2013, and along with her co-chair, Federica Govoni, defined the early SKA requirements associated with cosmic magnetism, editing the magnetism sections of the SKA Science Case and writing the cosmic magnetism scientific summary paper. Johnston-Hollitt led the 2011 bid for NZ to join the Murchison Widefield Array, a low frequency precursor to the SKA. She joined the MWA Board in 2012, became vice-chair in 2013, and held the position of chair from January 2014 to January 2018. In early 2017 she became founder and chief executive officer (CEO) of Peripety Scientific, an independent research organisation specialising in radio astronomy research and consultancy based in Wellington. She resigned from Victoria University of Wellington in September 2017 and became a professor of Radio Astronomy at Curtin University and director of the Murchison Widefield Array. During her time as chair of the MWA board, Professor Johnston-Hollitt was responsible for the expansion of the MWA project to move from 11 research organisations over 3 countries with 112 individual researchers to 21 research organisations over 6 countries with 270 individual researchers. As MWA Director, she oversaw the third phase of the MWA project realised via the design and funding for the new 'MWAX correlator' - a GPU-based bespoke compute system. After finishing her 3 year term as MWA director in December 2020, Johnston-Hollitt continues to be the lead of the galactic and extragalactic science team in the Curtin Institute for Radio Astronomy, who are exploiting the MWA and other telescopes to uncover the mysteries of the Universe. Further, Professor Johnston-Hollitt has been appointed as the new Director of the Curtin Institute for Data Science (CIDS) in August 2020, leading a multi-disciplinary team of data scientists. The CIDS is a cross-disciplinary research accelerator based on the use of high-performance computing. == Research career == Johnston-Hollitt's primary research interests are cosmic magnetism and observations of galaxy clusters, primarily through the use of radio telescopes. She has authored over 230 publications. She has served on the Editorial Board of Publications of the Astronomical Society of Australia since January 2015, commencing a 3-year term as Editor-in-Chief from January 2018. Johnston-Hollitt has gained funding for design, construction and exploitation of radio telescopes across Europe, Australia and New Zealand. She has held several visiting positions as a guest professor at the Excellence Cluster Universe in Munich, Germany, the Sophia Antipolis University in Nice, France, the University of Bologna, Italy, and the University of Malaya. == Awards == 2002: Union of Radio Scientists International (URSI) Young Scientist Award 2006: Faculty Early Career Research Award – University of Tasmania 2010: Hudson Lecturer – Royal Society of New Zealand == See also == List of female scientists in the 21st century List of University of Adelaide people == References == |
Wikipedia:Melpathur Narayana Bhattathiri#0 | Melpathur Narayana Bhattathiri (Mēlpattūr Nārāyaṇa Bhaṭṭatiri; 1560–1646/1666), third student of Achyuta Pisharati, was a member of Madhava of Sangamagrama's Kerala school of astronomy and mathematics. He was a mathematical linguist (vyakarana). His most important scholarly work, Prakriya-sarvasvam, sets forth an axiomatic system elaborating on the classical system of Panini. However, he is most famous for his masterpiece, Narayaneeyam, a devotional composition in praise of Guruvayurappan (Krishna) that is still sung at Guruvayur Temple. == Birth and education == Bhattathri was from a village named Melpathur at Kurumbathur in Athavanad Panchayat near the Tirur River in 1560 in a Brahmin family, the son of Mathrudattan Bhattathiri, a pandit himself. Bhattathiri studied from his father as a child. He learned the Rig Veda from Madhava, Tarka śāstra (science of debate in Sanskrit) from Damodara and Vyākaraṇa (Sanskrit grammar) from Achyuta Pisharati. He became a pandit by the age of 16. He married his guru Achuta Pisharati's niece and settled at Thrikkandiyur in Tirur.One of the manuscripts of Narayaneeyam says that it was copied by the author's younger brother Matrdatta. The Melpathur family is extinct and it is said that it was merged into the Maravancheri Thekkedathu family. == Works == He wrote on diverse subjects including scientific ones. His works are: Narayaniyam Kriyakrama or Asvalayanakriyakrama Prakriyasarvasva Sripadasaptati (Supposed to be his last work) Dhatukavya Svahasudhakara Matsyavatara Rajasuya Ashtamicampu (fine description of Astami festival celebrated in the month of Krithigai (Nov-Dec) in the Shiva temple at Vaikom in north Travancore. Dutavakya Subhadradharana Pancalisvayamvara Under the patronage of Cochin Vira Keralavarman (1601-1615 A.D) Melputtur wrote the Gosrinagaravarnana and Virakeralaprasasti.: 164 == References == |
Wikipedia:Menachem Oren#0 | Menachem Oren (born Mieczysław Chwojnik; Hebrew: מנחם אורן; 1903 – December 1962) was a Polish-born Israeli chess player and mathematician. Chwojnik was the strongest Cracovian chess player in 1920s, a thrice winner of the Kraków championship (1919, 1925, 1926). He won the Nowy Dziennik tournament in Kraków in 1926. He played in two Polish championships. In 1926, he tied for 8-9th places in Warsaw (first Polish championship). The event was won by Dawid Przepiórka. In 1927, he tied for 5-7th in Łódź. The event was won by Akiba Rubinstein. In 1928, he represented Poland on fourth board at the 2nd Chess Olympiad in The Hague (+4 –3 =4), and won the team bronze medal. He left Kraków for Rivne, Volhynia (then Poland), in the 1930s. He won in the Rivne City championship in 1938. During World War II, he lived in the Soviet Union. After the war, he returned to Poland in 1945/46, and settled in Lower Silesia. In 1949, Chwojnik emigrated, via Czechoslovakia and Austria, to Israel, where he had changed name to Menachem Oren. In 1951, he won the Israeli championship and the Tel Aviv City championship. In 1952, he won again the Tel Aviv City championship. He played thrice for Israel in Chess Olympiads: on board two (+6 –4 =3) at Helsinki 1952, board three(+7 –2 =3) at Amsterdam 1954, and board four (+2 –2 =4) at Moscow 1956. == References == == External links == Menachem Oren player profile and games at Chessgames.com Menachem Oren games at 365Chess.com |
Wikipedia:Menahem Max Schiffer#0 | Menahem Max Schiffer (24 September 1911, Berlin – 11 November 1997)) was a German-born American mathematician who worked in complex analysis, partial differential equations, and mathematical physics. == Biography == Menachem Max Schiffer studied physics from 1930 at the University of Bonn and then at the Humboldt University of Berlin with Max von Laue, Erwin Schrödinger, Walter Nernst, Erhard Schmidt, Issai Schur and Ludwig Bieberbach. In Berlin he worked closely with Issai Schur. In 1934, after being forced by the Nazis to leave the academic world, he immigrated to Mandatory Palestine. On the basis of his prior mathematical publications, Schiffer received a master's degree from the Hebrew University of Jerusalem. In 1938, he received his doctorate under the supervision of Michael Fekete. In his dissertation on Conformal representation and univalent functions he introduced the "Schiffer variation", a method for handling geometric problems in complex analysis. Schiffer married Fanya Rabinivics Schiffer in 1937. His daughter Dinah S. Singer, is an experimental immunologist. == Academic career == In September 1952, he began to teach at Stanford University, along with George Pólya, Charles Loewner, Stefan Bergman, and Gábor Szegő. With Paul Garabedian, Schiffer worked on the Bieberbach conjecture with a proof in 1955 of the special case n=4. He was a speaker at the International Congress of Mathematicians (ICM) in 1950 at Cambridge, Massachusetts, and was a plenary speaker at the ICM in 1958 at Edinburgh with plenary address Extremum Problems and Variational Methods in Conformal Mapping. In 1970 he was elected to the United States National Academy of Sciences. He retired from Stanford University as professor emeritus in 1977. In 1981, Schiffer became a founding member of the World Cultural Council. Never losing his interest in mathematical physics, Schiffer also made important contributions to eigenvalue problems, to partial differential equations, and to the variational theory of “domain functionals” that arise in many classical boundary value problems. And he coauthored a book on general relativity. Schiffer was a prolific author over his entire career, with 135 publications from the 1930s to the 1990s, including four books and around forty different coauthors. He was also an outstanding mathematical stylist, always writing, by his own testimony, with the reader in mind. ... His lectures at Stanford and around the world ranged greatly in subject matter and were widely appreciated. ... At Stanford he often taught graduate courses in applied mathematics and mathematical physics. Students from all departments flocked to them, as did many faculty. Each lecture was a perfect set piece—no pauses, no slips, and no notes. In 1976 he was chosen as one of the first recipients of the Dean's Award for Teaching in the School of Humanities and Sciences. == Selected publications == with Leon Bowden: The role of mathematics in science, Mathematical Association of America 1984 with Stefan Bergman: Kernel functions and elliptic differential equations in mathematical physics, Academic Press 1953 with Donald Spencer: Functionals of finite Riemann Surfaces, Princeton 1954 with Ronald Adler, Maurice Bazin: Introduction to General Relativity, McGraw Hill 1965 xvi+ 451 pp. Illus. 2nd edition. 1975; xiv+ 549 pp.{{cite book}}: CS1 maint: postscript (link) == References == == External links == Literature by and about Menahem Max Schiffer in the German National Library catalogue |
Wikipedia:Menelaus's theorem#0 | In Euclidean geometry, Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle △ABC, and a transversal line that crosses BC, AC, AB at points D, E, F respectively, with D, E, F distinct from A, B, C. A weak version of the theorem states that | A F ¯ F B ¯ | × | B D ¯ D C ¯ | × | C E ¯ E A ¯ | = 1 , {\displaystyle \left|{\frac {\overline {AF}}{\overline {FB}}}\right|\times \left|{\frac {\overline {BD}}{\overline {DC}}}\right|\times \left|{\frac {\overline {CE}}{\overline {EA}}}\right|=1,} where "| |" denotes absolute value (i.e., all segment lengths are positive). The theorem can be strengthened to a statement about signed lengths of segments, which provides some additional information about the relative order of collinear points. Here, the length AB is taken to be positive or negative according to whether A is to the left or right of B in some fixed orientation of the line; for example, A F ¯ F B ¯ {\displaystyle {\tfrac {\overline {AF}}{\overline {FB}}}} is defined as having positive value when F is between A and B and negative otherwise. The signed version of Menelaus's theorem states A F ¯ F B ¯ × B D ¯ D C ¯ × C E ¯ E A ¯ = − 1. {\displaystyle {\frac {\overline {AF}}{\overline {FB}}}\times {\frac {\overline {BD}}{\overline {DC}}}\times {\frac {\overline {CE}}{\overline {EA}}}=-1.} Equivalently, A F ¯ × B D ¯ × C E ¯ = − F B ¯ × D C ¯ × E A ¯ . {\displaystyle {\overline {AF}}\times {\overline {BD}}\times {\overline {CE}}=-{\overline {FB}}\times {\overline {DC}}\times {\overline {EA}}.} Some authors organize the factors differently and obtain the seemingly different relation F A ¯ F B ¯ × D B ¯ D C ¯ × E C ¯ E A ¯ = 1 , {\displaystyle {\frac {\overline {FA}}{\overline {FB}}}\times {\frac {\overline {DB}}{\overline {DC}}}\times {\frac {\overline {EC}}{\overline {EA}}}=1,} but as each of these factors is the negative of the corresponding factor above, the relation is seen to be the same. The converse is also true: If points D, E, F are chosen on BC, AC, AB respectively so that A F ¯ F B ¯ × B D ¯ D C ¯ × C E ¯ E A ¯ = − 1 , {\displaystyle {\frac {\overline {AF}}{\overline {FB}}}\times {\frac {\overline {BD}}{\overline {DC}}}\times {\frac {\overline {CE}}{\overline {EA}}}=-1,} then D, E, F are collinear. The converse is often included as part of the theorem. (Note that the converse of the weaker, unsigned statement is not necessarily true.) The theorem is very similar to Ceva's theorem in that their equations differ only in sign. By re-writing each in terms of cross-ratios, the two theorems may be seen as projective duals. == Proofs == === A standard proof === A proof given by John Wellesley Russell uses Pasch's axiom to consider cases where a line does or does not meet a triangle. First, the sign of the left-hand side will be negative since either all three of the ratios are negative, the case where the line DEF misses the triangle (see diagram), or one is negative and the other two are positive, the case where DEF crosses two sides of the triangle. To check the magnitude, construct perpendiculars from A, B, C to the line DEF and let their lengths be a, b, c respectively. Then by similar triangles it follows that | A F ¯ F B ¯ | = | a b | , | B D ¯ D C ¯ | = | b c | , | C E ¯ E A ¯ | = | c a | . {\displaystyle \left|{\frac {\overline {AF}}{\overline {FB}}}\right|=\left|{\frac {a}{b}}\right|,\quad \left|{\frac {\overline {BD}}{\overline {DC}}}\right|=\left|{\frac {b}{c}}\right|,\quad \left|{\frac {\overline {CE}}{\overline {EA}}}\right|=\left|{\frac {c}{a}}\right|.} Therefore, | A F ¯ F B ¯ | × | B D ¯ D C ¯ | × | C E ¯ E A ¯ | = | a b × b c × c a | = 1. {\displaystyle \left|{\frac {\overline {AF}}{\overline {FB}}}\right|\times \left|{\frac {\overline {BD}}{\overline {DC}}}\right|\times \left|{\frac {\overline {CE}}{\overline {EA}}}\right|=\left|{\frac {a}{b}}\times {\frac {b}{c}}\times {\frac {c}{a}}\right|=1.} For a simpler, if less symmetrical way to check the magnitude, draw CK parallel to AB where DEF meets CK at K. Then by similar triangles | B D ¯ D C ¯ | = | B F ¯ C K ¯ | , | A E ¯ E C ¯ | = | A F ¯ C K ¯ | , {\displaystyle \left|{\frac {\overline {BD}}{\overline {DC}}}\right|=\left|{\frac {\overline {BF}}{\overline {CK}}}\right|,\quad \left|{\frac {\overline {AE}}{\overline {EC}}}\right|=\left|{\frac {\overline {AF}}{\overline {CK}}}\right|,} and the result follows by eliminating CK from these equations. The converse follows as a corollary. Let D, E, F be given on the lines BC, AC, AB so that the equation holds. Let F' be the point where DE crosses AB. Then by the theorem, the equation also holds for D, E, F'. Comparing the two, A F ¯ F B ¯ = A F ′ ¯ F ′ B ¯ . {\displaystyle {\frac {\overline {AF}}{\overline {FB}}}={\frac {\overline {AF'}}{\overline {F'B}}}\ .} But at most one point can cut a segment in a given ratio so F = F'. === A proof using homotheties === The following proof uses only notions of affine geometry, notably homotheties. Whether or not D, E, F are collinear, there are three homotheties with centers D, E, F that respectively send B to C, C to A, and A to B. The composition of the three then is an element of the group of homothety-translations that fixes B, so it is a homothety with center B, possibly with ratio 1 (in which case it is the identity). This composition fixes the line DE if and only if F is collinear with D, E (since the first two homotheties certainly fix DE, and the third does so only if F lies on DE). Therefore D, E, F are collinear if and only if this composition is the identity, which means that the magnitude of the product of the three ratios is 1: D C → D B → × E A → E C → × F B → F A → = 1 , {\displaystyle {\frac {\overrightarrow {DC}}{\overrightarrow {DB}}}\times {\frac {\overrightarrow {EA}}{\overrightarrow {EC}}}\times {\frac {\overrightarrow {FB}}{\overrightarrow {FA}}}=1,} which is equivalent to the given equation. == History == It is uncertain who actually discovered the theorem; however, the oldest extant exposition appears in Spherics by Menelaus. In this book, the plane version of the theorem is used as a lemma to prove a spherical version of the theorem. In Almagest, Ptolemy applies the theorem on a number of problems in spherical astronomy. During the Islamic Golden Age, Muslim scholars devoted a number of works that engaged in the study of Menelaus's theorem, which they referred to as "the proposition on the secants" (shakl al-qatta'). The complete quadrilateral was called the "figure of secants" in their terminology. Al-Biruni's work, The Keys of Astronomy, lists a number of those works, which can be classified into studies as part of commentaries on Ptolemy's Almagest as in the works of al-Nayrizi and al-Khazin where each demonstrated particular cases of Menelaus's theorem that led to the sine rule, or works composed as independent treatises such as: The "Treatise on the Figure of Secants" (Risala fi shakl al-qatta') by Thabit ibn Qurra. Husam al-Din al-Salar's Removing the Veil from the Mysteries of the Figure of Secants (Kashf al-qina' 'an asrar al-shakl al-qatta'), also known as "The Book on the Figure of Secants" (Kitab al-shakl al-qatta') or in Europe as The Treatise on the Complete Quadrilateral. The lost treatise was referred to by Sharaf al-Din al-Tusi and Nasir al-Din al-Tusi. Work by al-Sijzi. Tahdhib by Abu Nasr ibn Iraq. Roshdi Rashed and Athanase Papadopoulos, Menelaus' Spherics: Early Translation and al-Mahani'/al-Harawi's version (Critical edition of Menelaus' Spherics from the Arabic manuscripts, with historical and mathematical commentaries), De Gruyter, Series: Scientia Graeco-Arabica, 21, 2017, 890 pages. ISBN 978-3-11-057142-4 == References == == External links == Alternate proof of Menelaus's theorem, from PlanetMath Menelaus From Ceva Ceva and Menelaus Meet on the Roads Menelaus and Ceva at MathPages Demo of Menelaus's theorem by Jay Warendorff. The Wolfram Demonstrations Project. Weisstein, Eric W. "Menelaus' Theorem". MathWorld. |
Wikipedia:Mers Kutt#0 | Merslau "Mers" Kutt CM (born 1933) is a Canadian inventor, businessman and educator. He is a former professor of mathematics at Queen's University. Through his company, Micro Computer Machines, he is the developer of the world's first keyboard-based portable microcomputer. == Early life == Kutt was born in Winnipeg, Manitoba. He graduated in 1956 from the University of Toronto with a degree in mathematics and physics. == Career == After employment in industry at Phillips, IBM and Honeywell, Kutt worked as a professor of mathematics at Queen's University in Kingston, Ontario during the 1960s. He served as president of the Canadian Information Processing Society, and was director of the University's computing centre. In 1968, after observing the punched card-based input systems in use there at the time to program its mainframe, he partnered with Donald Pamenter to start a company, Consolidated Computer Inc., and produced "Key-Edit", a terminal with a one-line of display, which both streamlined and reduced the cost of the process. This product was the first of its kind, and was sold and used in many countries during the next few years distributed by International Computers Limited and Fujitsu. By 1971, Kutt been forced out of Consolidated's management, and he formed two more companies, Micro Computer Machines and Kutt Systems Inc. As president of these companies he directed the design and manufacture in 1973 of the MCM/70, the world's first complete microprocessor-based, portable personal computer. The device, based on the Intel 8008 processor, was used to edit and execute programs using the APL programming language. The MCM/70 technical specifications were overtaken by products from companies with larger development and marketing budgets, and by 1982 the product was no longer in production. In 1976 Kutt started up another Toronto company, All Computers Inc., which developed improvements to several Intel processors. By 2004, Kutt was the company's only employee; that year he sued Intel, alleging that his patented circuitry had been included in Pentium processors. The suit was dismissed in 2005. In 2006, Kutt was inducted into the Order of Canada. == References == |
Wikipedia:Messenger of Mathematics#0 | The Messenger of Mathematics is a defunct British mathematics journal. The founding editor-in-chief was William Allen Whitworth with Charles Taylor and volumes 1–58 were published between 1872 and 1929. James Whitbread Lee Glaisher was the editor-in-chief after Whitworth. In the nineteenth century, foreign contributions represented 4.7% of all pages of mathematics in the journal. == History == The journal was originally titled Oxford, Cambridge and Dublin Messenger of Mathematics. It was supported by mathematics students and governed by a board of editors composed of members of the universities of Oxford, Cambridge and Dublin (the last being its sole constituent college, Trinity College Dublin). Volumes 1–5 were published between 1862 and 1871. It merged with The Quarterly Journal of Pure and Applied Mathematics to form the Quarterly Journal of Mathematics. == References == == Further reading == Sloan Evans Despeaux (2007). "Launching mathematical research without a formal mandate: The role of university-affiliated journals in Britain, 1837–1870". Historia Mathematica. 34 (1): 89. doi:10.1016/j.hm.2006.02.005. == External links == Messenger of Mathematics, vols. 1–30 (1871–1901) digitized by the Center for Retrospective Digitization. |
Wikipedia:Metavariable#0 | In logic, a metavariable (also metalinguistic variable or syntactical variable) is a symbol or symbol string which belongs to a metalanguage and stands for elements of some object language. For instance, in the sentence Let A and B be two sentences of a language ℒ the symbols A and B are part of the metalanguage in which the statement about the object language ℒ is formulated. John Corcoran considers this terminology unfortunate because it obscures the use of schemata and because such "variables" do not actually range over a domain.: 220 The convention is that a metavariable is to be uniformly substituted with the same instance in all its appearances in a given schema. This is in contrast with nonterminal symbols in formal grammars where the nonterminals on the right of a production can be substituted by different instances. Attempts to formalize the notion of metavariable result in some kind of type theory. == See also == Explicit substitution == Notes == == References == Corcoran, J. (2006). "Schemata: the Concept of Schema in the History of Logic" (PDF). Bulletin of Symbolic Logic. 12 (2): 219–240. doi:10.2178/bsl/1146620060. S2CID 6909703. Hunter, Geoffrey (1996) [1971]. Metalogic: An Introduction to the Metatheory of Standard First-Order Logic. University of California Press (published 1973). ISBN 9780520023567. OCLC 36312727. (accessible to patrons with print disabilities) Shoenfield, Joseph R. (2001) [1967]. Mathematical Logic (2nd ed.). A K Peters. ISBN 978-1-56881-135-2. Tennent, R. D. (2002). Specifying Software: A Hands-On Introduction. Cambridge University Press. ISBN 978-0-521-00401-5. |
Wikipedia:Method of Chester–Friedman–Ursell#0 | In asymptotic analysis, the method of Chester–Friedman–Ursell is a technique to find asymptotic expansions for contour integrals. It was developed as an extension of the steepest descent method for getting uniform asymptotic expansions in the case of coalescing saddle points. The method was published in 1957 by Clive R. Chester, Bernard Friedman and Fritz Ursell. == Method == === Setting === We study integrals of the form I ( α , N ) := ∫ C e − N f ( α , t ) g ( α , t ) d t , {\displaystyle I(\alpha ,N):=\int _{C}e^{-Nf(\alpha ,t)}g(\alpha ,t)dt,} where C {\displaystyle C} is a contour and f , g {\displaystyle f,g} are two analytic functions in the complex variable t {\displaystyle t} and continuous in α {\displaystyle \alpha } . N {\displaystyle N} is a large number. Suppose we have two saddle points t + , t − {\displaystyle t_{+},t_{-}} of f ( α , t ) {\displaystyle f(\alpha ,t)} with multiplicity 1 {\displaystyle 1} that depend on a parameter α {\displaystyle \alpha } . If now an α 0 {\displaystyle \alpha _{0}} exists, such that both saddle points coalescent to a new saddle point t 0 {\displaystyle t_{0}} with multiplicity 2 {\displaystyle 2} , then the steepest descent method no longer gives uniform asymptotic expansions. === Procedure === Suppose there are two simple saddle points t − := t − ( α ) {\displaystyle t_{-}:=t_{-}(\alpha )} and t + := t + ( α ) {\displaystyle t_{+}:=t_{+}(\alpha )} of f {\displaystyle f} and suppose that they coalescent in the point t 0 := t 0 ( α 0 ) {\displaystyle t_{0}:=t_{0}(\alpha _{0})} . We start with the cubic transformation t ↦ w {\displaystyle t\mapsto w} of f ( α , t ) {\displaystyle f(\alpha ,t)} , this means we introduce a new complex variable w {\displaystyle w} and write f ( α , t ) = 1 3 w 3 − η ( α ) w + A ( α ) , {\displaystyle f(\alpha ,t)={\tfrac {1}{3}}w^{3}-\eta (\alpha )w+A(\alpha ),} where the coefficients η := η ( α ) {\displaystyle \eta :=\eta (\alpha )} and A := A ( α ) {\displaystyle A:=A(\alpha )} will be determined later. We have d t d w = w 2 − η f t ( α , t ) , {\displaystyle {\frac {dt}{dw}}={\frac {w^{2}-\eta }{f_{t}(\alpha ,t)}},} so the cubic transformation will be analytic and injective only if d t / d w {\displaystyle dt/dw} and d w / d t {\displaystyle dw/dt} are neither 0 {\displaystyle 0} nor ∞ {\displaystyle \infty } . Therefore t = t − {\displaystyle t=t_{-}} and t = t + {\displaystyle t=t_{+}} must correspond to the zeros of w 2 − η {\displaystyle w^{2}-\eta } , i.e. with w + := η 1 / 2 {\displaystyle w_{+}:=\eta ^{1/2}} and w − := − η 1 / 2 {\displaystyle w_{-}:=-\eta ^{1/2}} . This gives the following system of equations { f ( α , t − ) = − 2 3 η 3 / 2 + A , f ( α , t + ) = 2 3 η 3 / 2 + A , {\displaystyle {\begin{cases}f(\alpha ,t_{-})=-{\frac {2}{3}}\eta ^{3/2}+A,\\f(\alpha ,t_{+})={\frac {2}{3}}\eta ^{3/2}+A,\end{cases}}} we have to solve to determine η {\displaystyle \eta } and A {\displaystyle A} . A theorem by Chester–Friedman–Ursell (see below) says now, that the cubic transform is analytic and injective in a local neighbourhood around the critical point ( α 0 , t 0 ) {\displaystyle (\alpha _{0},t_{0})} . After the transformation the integral becomes I ( α , N ) = e − N A ∫ L exp ( − N ( 1 3 w 3 − η w ) ) h ( α , w ) d w , {\displaystyle I(\alpha ,N)=e^{-NA}\int _{L}\exp \left(-N\left({\tfrac {1}{3}}w^{3}-\eta w\right)\right)h(\alpha ,w)dw,} where L {\displaystyle L} is the new contour for w {\displaystyle w} and h ( α , w ) := g ( α , t ) d t d w = g ( α , t ) w 2 − η f t ( α , t ) . {\displaystyle h(\alpha ,w):=g(\alpha ,t){\frac {dt}{dw}}=g(\alpha ,t){\frac {w^{2}-\eta }{f_{t}(\alpha ,t)}}.} The function h ( α , w ) {\displaystyle h(\alpha ,w)} is analytic at w + ( α ) , w − ( α ) {\displaystyle w_{+}(\alpha ),w_{-}(\alpha )} for α ≠ α 0 {\displaystyle \alpha \neq \alpha _{0}} and also at the coalescing point w 0 {\displaystyle w_{0}} for α 0 {\displaystyle \alpha _{0}} . Here ends the method and one can see the integral representation of the complex Airy function. Chester–Friedman–Ursell note to write h ( α , w ) {\displaystyle h(\alpha ,w)} not as a single power series but instead as h ( α , w ) = ∑ m q m ( α ) ( w 2 − η ) m + ∑ m p m ( α ) w ( w 2 − η ) m {\displaystyle h(\alpha ,w)=\sum \limits _{m}q_{m}(\alpha )(w^{2}-\eta )^{m}+\sum \limits _{m}p_{m}(\alpha )w(w^{2}-\eta )^{m}} to really get asymptotic expansions. === Theorem by Chester–Friedman–Ursell === Let t + := t + ( α ) , t − := t − ( α ) {\displaystyle t_{+}:=t_{+}(\alpha ),t_{-}:=t_{-}(\alpha )} and t 0 := t 0 ( α 0 ) {\displaystyle t_{0}:=t_{0}(\alpha _{0})} be as above. The cubic transformation f ( t , α ) = 1 3 w 3 − η ( α ) w + A ( α ) {\displaystyle f(t,\alpha )={\tfrac {1}{3}}w^{3}-\eta (\alpha )w+A(\alpha )} with the above derived values for η ( α ) {\displaystyle \eta (\alpha )} and A ( α ) {\displaystyle A(\alpha )} , such that t = t ± {\displaystyle t=t_{\pm }} corresponds to u = ± η 1 / 2 {\displaystyle u=\pm \eta ^{1/2}} , has only one branch point w = w ( α , t ) {\displaystyle w=w(\alpha ,t)} , so that for all α {\displaystyle \alpha } in a local neighborhood of α 0 {\displaystyle \alpha _{0}} the transformation is analytic and injective. == Literature == Chester, Clive R.; Friedman, Bernard; Ursell, Fritz (1957). "An extension of the method of steepest descents". Mathematical Proceedings of the Cambridge Philosophical Society. 53 (3). Cambridge University Press. page 604. doi:10.1017/S0305004100032655. Olver, Frank W. J. (1997). Asymptotics and Special Functions. A K Peters/CRC Press. p. 351. doi:10.1201/9781439864548. ISBN 978-0-429-06461-6. Wong, Roderick (2001). Asymptotic Approximations of Integrals. Society for Industrial and Applied Mathematics. doi:10.1137/1.9780898719260. ISBN 978-0-89871-497-5. Temme, Nico M. (2014). Asymptotic Methods For Integrals. Series in Analysis. Vol. 6. World Scientific. doi:10.1142/9195. ISBN 978-981-4612-15-9. == References == |
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