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Wikipedia:Walther Mayer#0	Walther Mayer (11 March 1887 – 10 September 1948) was an Austrian mathematician, born in Graz, Austria-Hungary. With Leopold Vietoris he is the namesake of the Mayer–Vietoris sequence in topology. He served as an assistant to Albert Einstein and subsequently worked with him several years as a close collaborator, leading to the nickname "Einstein's calculator". == Biography == Mayer studied at the Federal Institute of Technology in Zürich and the University of Paris before receiving his doctorate in 1912 from the University of Vienna; his thesis concerned the Fredholm integral equation. He served in the military between 1914 and 1919, during which he found time to complete a habilitation on differential geometry. Because he was Jewish, he had little opportunity for an academic career in Austria, and left the country; however, in 1926, with help from Einstein, he returned to a position at the University of Vienna as Privatdozent (lecturer). He made a name for himself in topology with the Mayer–Vietoris sequence, and with an axiomatic treatment of homology predating the Eilenberg–Steenrod axioms. He also published a book on Riemannian geometry in 1930, the second volume of a textbook on differential geometry that had been started by Adalbert Duschek with a volume on curves and surfaces. In 1929, on the recommendation of Richard von Mises, he became Albert Einstein's assistant with the explicit understanding that he work with him on distant parallelism, and from 1931 to 1936, he collaborated with Albert Einstein on the theory of relativity. In 1933, after Hitler's assumption of power, he followed Einstein to the United States and became an associate in mathematics at the Institute for Advanced Study in Princeton, New Jersey. He continued working on mathematics at the Institute, and died in Princeton in 1948. == Selected publications == with Adalbert Duschek: Lehrbuch der Differentialgeometrie. 2 vols., Teubner 1930. vol. 1 vol. 2 Über abstrakte Topologie. In: Monatshefte für Mathematik. vol. 36, 1929, pp. 1–42 (Mayer-Vietoris-Sequenzen) with T. Y. Thomas: Foundations of the theory of Lie groups. In: Annals of Mathematics. 36, 1935, 770–822. Die Differentialgeometrie der Untermannigfaltigkeiten des Rn konstanter Krümmung. Transactions of the American Mathematical Society 38 no. 2, 1935: 267–309. with T. Y. Thomas: Fields of parallel vectors in non-analytic manifolds in the large. Compositio Mathematica, vol. 5, 1938: pp. 198-207. with Herbert Busemann: "On the foundations of calculus of variations." Transactions of the American Mathematical Society 49, no. 2, 1941: 173-198 A new homology theory. In: Annals of Mathematics. vol. 43, 1942, pp. 370–380, 594–605. The Duality Theory and the Basic Isomorphisms of Group Systems and Nets and Co-Nets of Group Systems. In: Annals of Mathematics. vol. 46, 1945, pp. 1–28 On Products in Topology. In: Annals of Mathematics. vol. 46, 1945, pp. 29–57. Duality Theorems. In: Fundamenta Mathematicae 35, 1948, 188–202. == References == == External links == Portrait of Walther Mayer (1940), United States Holocaust Memorial Museum
Wikipedia:Wanda Szmielew#0	Wanda Szmielew née Montlak (5 April 1918 – 27 August 1976) was a Polish mathematical logician who first proved the decidability of the first-order theory of abelian groups. == Life == Wanda Montlak was born on 5 April 1918 in Warsaw. She completed high school in 1935 and married, taking the name Szmielew. In the same year she entered the University of Warsaw, where she studied logic under Adolf Lindenbaum, Jan Łukasiewicz, Kazimierz Kuratowski, and Alfred Tarski. Her research at this time included work on the axiom of choice, but it was interrupted by the 1939 Invasion of Poland. Szmielew became a surveyor during World War II, during which time she continued her research on her own, developing a decision procedure based on quantifier elimination for the theory of abelian groups. She also taught for the Polish underground. After the liberation of Poland, Szmielew took a position at the University of Łódź, which was founded in May 1945. In 1947, she published her paper on the axiom of choice, earned a master's degree from the University of Warsaw, and moved to Warsaw as a senior assistant. In 1949 and 1950, Szmielew visited the University of California, Berkeley, where Tarski had found a permanent position after being exiled from Poland for the war. She lived in the home of Tarski and his wife as Tarski's mistress, leaving her husband behind in Poland, and completed a Ph.D. at Berkeley in 1950 under Tarski's supervision, with her dissertation consisting of her work on abelian groups. For the 1955 journal publication of these results, Tarski convinced Szmielew to rephrase her work in terms of his theory of arithmetical functions, a decision that caused this work to be described by Solomon Feferman as "unreadable". Later work by Eklof & Fischer (1972) re-proved Szmielew's result using more standard model-theoretic techniques. Returning to Warsaw as an assistant professor, her interests shifted to the foundations of geometry. With Karol Borsuk, she published a text on the subject in 1955 (translated into English in 1960), and another monograph, published posthumously in 1981 and (in English translation) 1983. She died of cancer on 27 August 1976 in Warsaw. == Selected publications == Szmielew, Wanda (1947), "On choices from finite sets", Fundamenta Mathematicae, 34 (1): 75–80, doi:10.4064/fm-34-1-75-80, ISSN 0016-2736, MR 0022539. Szmielew, Wanda (1955), "Elementary properties of Abelian groups", Fundamenta Mathematicae, 41 (2): 203–271, doi:10.4064/fm-41-2-203-271, ISSN 0016-2736, MR 0072131. Borsuk, Karol; Szmielew, Wanda (1955), Podstawy geometrii, Warsawa: Państwowe Wydawnictwo Naukowe, MR 0071791. Translated as Borsuk, Karol; Szmielew, Wanda (1960), Foundations of geometry: Euclidean and Bolyai-Lobachevskian geometry; projective geometry, Revised English translation, New York: Interscience Publishers, Inc., MR 0143072. Szmielew, Wanda (1981), Od geometrii afinicznej do euklidesowej, Biblioteka Matematyczna [Mathematics Library], vol. 55, Warsaw: Państwowe Wydawnictwo Naukowe (PWN), p. 172, ISBN 83-01-01374-5, MR 0664205. Translated as Szmielew, Wanda (1983), From affine to Euclidean geometry, Warsaw: PWN—Polish Scientific Publishers, ISBN 90-277-1243-3, MR 0720548. Schwabhäuser, W.; Szmielew, W.; Tarski, A. (1983), Metamathematische Methoden in der Geometrie, Hochschultext [University Textbooks], Berlin: Springer-Verlag, doi:10.1007/978-3-642-69418-9, ISBN 3-540-12958-8, MR 0731370. == References ==
Wikipedia:Wang algebra#0	In algebra and network theory, a Wang algebra is a commutative algebra A {\displaystyle A} , over a field or (more generally) a commutative unital ring, in which A {\displaystyle A} has two additional properties:(Rule i) For all elements x of A {\displaystyle A} , x + x = 0 (universal additive nilpotency of degree 1).(Rule ii) For all elements x of A {\displaystyle A} , x⋅x = 0 (universal multiplicative nilpotency of degree 1). == History and applications == Rules (i) and (ii) were originally published by K. T. Wang (Wang Ki-Tung, 王季同) in 1934 as part of a method for analyzing electrical networks. From 1935 to 1940, several Chinese electrical engineering researchers published papers on the method. The original Wang algebra is the Grassman algebra over the finite field mod 2. At the 57th annual meeting of the American Mathematical Society, held on December 27–29, 1950, Raoul Bott and Richard Duffin introduced the concept of a Wang algebra in their abstract (number 144t) The Wang algebra of networks. They gave an interpretation of the Wang algebra as a particular type of Grassman algebra mod 2. In 1969 Wai-Kai Chen used the Wang algebra formulation to give a unification of several different techniques for generating the trees of a graph. The Wang algebra formulation has been used to systematically generate King-Altman directed graph patterns. Such patterns are useful in deriving rate equations in the theory of enzyme kinetics. According to Guo Jinhai, professor in the Institute for the History of Natural Sciences of the Chinese Academy of Sciences, Wang Ki Tung's pioneering method of analyzing electrical networks significantly promoted electrical engineering not only in China but in the rest of the world; the Wang algebra formulation is useful in electrical networks for solving problems involving topological methods, graph theory, and Hamiltonian cycles. == Wang Algebra and the Spanning Trees of a Graph == The Wang Rules for Finding all Spanning Trees of a Graph G For each node write the sum of all the edge-labels that meet that node. Leave out one node and take the product of the sums of labels for all the remaining nodes. Expand the product in 2. using the Wang algebra. The terms in the sum of the expansion obtained in 3. are in 1-1 correspondence with the spanning trees in the graph. == References ==
Wikipedia:Warren Ewens#0	Warren John Ewens (born 23 January 1937 in Canberra) is an Australian-born mathematician who has been Professor of Biology at the University of Pennsylvania since 1997. (He also held that position 1972–1977.) He concentrates his research on the mathematical, statistical and theoretical aspects of population genetics. Ewens has worked in mathematical population genetics, computational biology, and evolutionary population genetics. He introduced Ewens's sampling formula. Ewens received a B.A. (1958) and M.A. (1960) in Mathematical Statistics from the University of Melbourne, where he was a resident student at Trinity College, and a Ph.D. from the Australian National University (1963) under P. A. P. Moran. He first joined the department of biology at the University of Pennsylvania in 1972, and in 2006 was named the Christopher H. Browne Distinguished Professor of Biology. Positions held include: 1967–1972 Foundation Chair and Professor of Mathematics at La Trobe University 1972–1977 Professor of Biology at the University of Pennsylvania 1978–1996 Chair and Professor of Mathematics at Monash University 1997– Professor of Biology at the University of Pennsylvania Ewens is a Fellow of the Royal Society and the Australian Academy of Science. He is also the recipient of the Australian Statistical Society's E.J. Pitman Medal (1996), and Oxford University's Weldon Memorial Prize. His teaching and mentoring at the University of Pennsylvania have also been recognized by awards. Ewens also participates in the Genomics and Computational Biology (GCB) Ph.D. program of the University of Pennsylvania School of Medicine. Since 2006, he has taught statistics at the University of Pennsylvania's Wharton School. In 2022, Ewens was appointed Officer of the Order of Australia in the 2022 Queen's Birthday Honours for "distinguished service to biology and data science, to research, and to tertiary education". == Publications == Ewens has produced many publications; the following is a small selection: Ewens W.J. (1972). "The sampling theory of selectively neutral alleles". Theoretical Population Biology. 3 (1): 87–112. Bibcode:1972TPBio...3...87E. doi:10.1016/0040-5809(72)90035-4. PMID 4667078. Ewens W.J. (2004). Mathematical Population Genetics (2nd ed.). Springer-Verlag, New York. ISBN 0-387-20191-2. 'Statistical Methods in Bioinformatics: An Introduction (Statistics for Biology and Health)' 'Kingman and mathematical population genetics' in 'Probability and mathematical genetics' edited by N.H. Bingham and C.M. Goldie 'Genetics and Analysis of Quantitative Traits', American Journal of Human Biology 1999 'On estimating P values by Monte Carlo methods' American Journal of Human Genetics 2003 'Sam Karlin and the stochastic theory of evolutionary population genetics' Theoretical Population Biology 2009 == See also == Ewens's sampling formula (the multivariate Ewens distribution) == References == == External links == Warren Ewens at the Mathematics Genealogy Project Ewens, Warren John (1937 – ), Encyclopedia of Australian Science 2010 Ewens, W. J. (Warren John) at trove.nla.gov.au 'Pitman Medal Awarded to W. J. Ewens', Australian Journal of Statistics, vol. 39, no. 1, 1997, pp. 1–4. Ewens' page at the University of Pennsylvania Warren Ewens named Christopher H. Browne Distinguished Professor of Biology
Wikipedia:Warsaw School (mathematics)#0	Warsaw School of Mathematics is the name given to a group of mathematicians who worked at Warsaw, Poland, in the two decades between the World Wars, especially in the fields of logic, set theory, point-set topology and real analysis. They published in the journal Fundamenta Mathematicae, founded in 1920—one of the world's first specialist pure-mathematics journals. It was in this journal, in 1933, that Alfred Tarski—whose illustrious career would a few years later take him to the University of California, Berkeley—published his celebrated theorem on the undefinability of the notion of truth. Notable members of the Warsaw School of Mathematics have included: Wacław Sierpiński Kazimierz Kuratowski Edward Marczewski Bronisław Knaster Zygmunt Janiszewski Stefan Mazurkiewicz Stanisław Saks Karol Borsuk Roman Sikorski Nachman Aronszajn Samuel Eilenberg Additionally, notable logicians of the Lwów–Warsaw School of Logic, working at Warsaw, have included: Stanisław Leśniewski Adolf Lindenbaum Alfred Tarski Jan Łukasiewicz Andrzej Mostowski Helena Rasiowa Fourier analysis has been advanced at Warsaw by: Aleksander Rajchman Antoni Zygmund Józef Marcinkiewicz Otton M. Nikodym Jerzy Spława-Neyman == See also == Polish School of Mathematics Kraków School of Mathematics Lwów School of Mathematics == References ==
Wikipedia:Warwick Tucker#0	Warwick Tucker is an Australian mathematician at Monash University (previously deputy Chair and Chair at the Department of Mathematics at Uppsala University 2009–2020) who works on dynamical systems, chaos theory and computational mathematics. He is a recipient of the 2002 R. E. Moore Prize, and the 2004 EMS Prize. Tucker obtained his Ph.D. in 1998 at Uppsala University (thesis: The Lorenz attractor exists) with Lennart Carleson as advisor. In 2002, Tucker succeeded in solving an important open problem that had been posed by Stephen Smale (the fourteenth problem on Smale's list of problems). He was an invited speaker at the conference Dynamics, Equations and Applications in Kraków in 2019. == References == == External links == Warwick Tucker publications indexed by Google Scholar Publications by Warwick Tucker at ResearchGate
Wikipedia:Watson's lemma#0	In mathematics, Watson's lemma, proved by G. N. Watson (1918, p. 133), has significant application within the theory on the asymptotic behavior of integrals. == Statement of the lemma == Let 0 < T ≤ ∞ {\displaystyle 0<T\leq \infty } be fixed. Assume φ ( t ) = t λ g ( t ) {\displaystyle \varphi (t)=t^{\lambda }\,g(t)} , where g ( t ) {\displaystyle g(t)} has an infinite number of derivatives in the neighborhood of t = 0 {\displaystyle t=0} , with g ( 0 ) ≠ 0 {\displaystyle g(0)\neq 0} , and λ > − 1 {\displaystyle \lambda >-1} . Suppose, in addition, either that \| φ ( t ) \| < K e b t ∀ t > 0 , {\displaystyle \|\varphi (t)\|<Ke^{bt}\ \forall t>0,} where K , b {\displaystyle K,b} are independent of t {\displaystyle t} , or that ∫ 0 T \| φ ( t ) \| d t < ∞ . {\displaystyle \int _{0}^{T}\|\varphi (t)\|\,\mathrm {d} t<\infty .} Then, it is true that for all positive x {\displaystyle x} that \| ∫ 0 T e − x t φ ( t ) d t \| < ∞ {\displaystyle \left\|\int _{0}^{T}e^{-xt}\varphi (t)\,\mathrm {d} t\right\|<\infty } and that the following asymptotic equivalence holds: ∫ 0 T e − x t φ ( t ) d t ∼ ∑ n = 0 ∞ g ( n ) ( 0 ) Γ ( λ + n + 1 ) n ! x λ + n + 1 , ( x > 0 , x → ∞ ) . {\displaystyle \int _{0}^{T}e^{-xt}\varphi (t)\,\mathrm {d} t\sim \ \sum _{n=0}^{\infty }{\frac {g^{(n)}(0)\ \Gamma (\lambda +n+1)}{n!\ x^{\lambda +n+1}}},\ \ (x>0,\ x\rightarrow \infty ).} See, for instance, Watson (1918) for the original proof or Miller (2006) for a more recent development. == Proof == We will prove the version of Watson's lemma which assumes that \| φ ( t ) \| {\displaystyle \|\varphi (t)\|} has at most exponential growth as t → ∞ {\displaystyle t\to \infty } . The basic idea behind the proof is that we will approximate g ( t ) {\displaystyle g(t)} by finitely many terms of its Taylor series. Since the derivatives of g {\displaystyle g} are only assumed to exist in a neighborhood of the origin, we will essentially proceed by removing the tail of the integral, applying Taylor's theorem with remainder in the remaining small interval, then adding the tail back on in the end. At each step we will carefully estimate how much we are throwing away or adding on. This proof is a modification of the one found in Miller (2006). Let 0 < T ≤ ∞ {\displaystyle 0<T\leq \infty } and suppose that φ {\displaystyle \varphi } is a measurable function of the form φ ( t ) = t λ g ( t ) {\displaystyle \varphi (t)=t^{\lambda }g(t)} , where λ > − 1 {\displaystyle \lambda >-1} and g {\displaystyle g} has an infinite number of continuous derivatives in the interval [ 0 , δ ] {\displaystyle [0,\delta ]} for some 0 < δ < T {\displaystyle 0<\delta <T} , and that \| φ ( t ) \| ≤ K e b t {\displaystyle \|\varphi (t)\|\leq Ke^{bt}} for all δ ≤ t ≤ T {\displaystyle \delta \leq t\leq T} , where the constants K {\displaystyle K} and b {\displaystyle b} are independent of t {\displaystyle t} . We can show that the integral is finite for x {\displaystyle x} large enough by writing ( 1 ) ∫ 0 T e − x t φ ( t ) d t = ∫ 0 δ e − x t φ ( t ) d t + ∫ δ T e − x t φ ( t ) d t {\displaystyle (1)\quad \int _{0}^{T}e^{-xt}\varphi (t)\,\mathrm {d} t=\int _{0}^{\delta }e^{-xt}\varphi (t)\,\mathrm {d} t+\int _{\delta }^{T}e^{-xt}\varphi (t)\,\mathrm {d} t} and estimating each term. For the first term we have \| ∫ 0 δ e − x t φ ( t ) d t \| ≤ ∫ 0 δ e − x t \| φ ( t ) \| d t ≤ ∫ 0 δ \| φ ( t ) \| d t {\displaystyle \left\|\int _{0}^{\delta }e^{-xt}\varphi (t)\,\mathrm {d} t\right\|\leq \int _{0}^{\delta }e^{-xt}\|\varphi (t)\|\,\mathrm {d} t\leq \int _{0}^{\delta }\|\varphi (t)\|\,\mathrm {d} t} for x ≥ 0 {\displaystyle x\geq 0} , where the last integral is finite by the assumptions that g {\displaystyle g} is continuous on the interval [ 0 , δ ] {\displaystyle [0,\delta ]} and that λ > − 1 {\displaystyle \lambda >-1} . For the second term we use the assumption that φ {\displaystyle \varphi } is exponentially bounded to see that, for x > b {\displaystyle x>b} , \| ∫ δ T e − x t φ ( t ) d t \| ≤ ∫ δ T e − x t \| φ ( t ) \| d t ≤ K ∫ δ T e ( b − x ) t d t ≤ K ∫ δ ∞ e ( b − x ) t d t = K e ( b − x ) δ x − b . {\displaystyle {\begin{aligned}\left\|\int _{\delta }^{T}e^{-xt}\varphi (t)\,\mathrm {d} t\right\|&\leq \int _{\delta }^{T}e^{-xt}\|\varphi (t)\|\,\mathrm {d} t\\&\leq K\int _{\delta }^{T}e^{(b-x)t}\,\mathrm {d} t\\&\leq K\int _{\delta }^{\infty }e^{(b-x)t}\,\mathrm {d} t\\&=K\,{\frac {e^{(b-x)\delta }}{x-b}}.\end{aligned}}} The finiteness of the original integral then follows from applying the triangle inequality to ( 1 ) {\displaystyle (1)} . We can deduce from the above calculation that ( 2 ) ∫ 0 T e − x t φ ( t ) d t = ∫ 0 δ e − x t φ ( t ) d t + O ( x − 1 e − δ x ) {\displaystyle (2)\quad \int _{0}^{T}e^{-xt}\varphi (t)\,\mathrm {d} t=\int _{0}^{\delta }e^{-xt}\varphi (t)\,\mathrm {d} t+O\left(x^{-1}e^{-\delta x}\right)} as x → ∞ {\displaystyle x\to \infty } . By appealing to Taylor's theorem with remainder we know that, for each integer N ≥ 0 {\displaystyle N\geq 0} , g ( t ) = ∑ n = 0 N g ( n ) ( 0 ) n ! t n + g ( N + 1 ) ( t ∗ ) ( N + 1 ) ! t N + 1 {\displaystyle g(t)=\sum _{n=0}^{N}{\frac {g^{(n)}(0)}{n!}}\,t^{n}+{\frac {g^{(N+1)}(t^{})}{(N+1)!}}\,t^{N+1}} for 0 ≤ t ≤ δ {\displaystyle 0\leq t\leq \delta } , where 0 ≤ t ∗ ≤ t {\displaystyle 0\leq t^{}\leq t} . Plugging this in to the first term in ( 2 ) {\displaystyle (2)} we get ( 3 ) ∫ 0 δ e − x t φ ( t ) d t = ∫ 0 δ e − x t t λ g ( t ) d t = ∑ n = 0 N g ( n ) ( 0 ) n ! ∫ 0 δ t λ + n e − x t d t + 1 ( N + 1 ) ! ∫ 0 δ g ( N + 1 ) ( t ∗ ) t λ + N + 1 e − x t d t . {\displaystyle {\begin{aligned}(3)\quad \int _{0}^{\delta }e^{-xt}\varphi (t)\,\mathrm {d} t&=\int _{0}^{\delta }e^{-xt}t^{\lambda }g(t)\,\mathrm {d} t\\&=\sum _{n=0}^{N}{\frac {g^{(n)}(0)}{n!}}\int _{0}^{\delta }t^{\lambda +n}e^{-xt}\,\mathrm {d} t+{\frac {1}{(N+1)!}}\int _{0}^{\delta }g^{(N+1)}(t^{})\,t^{\lambda +N+1}e^{-xt}\,\mathrm {d} t.\end{aligned}}} To bound the term involving the remainder we use the assumption that g ( N + 1 ) {\displaystyle g^{(N+1)}} is continuous on the interval [ 0 , δ ] {\displaystyle [0,\delta ]} , and in particular it is bounded there. As such we see that \| ∫ 0 δ g ( N + 1 ) ( t ∗ ) t λ + N + 1 e − x t d t \| ≤ sup t ∈ [ 0 , δ ] \| g ( N + 1 ) ( t ) \| ∫ 0 δ t λ + N + 1 e − x t d t < sup t ∈ [ 0 , δ ] \| g ( N + 1 ) ( t ) \| ∫ 0 ∞ t λ + N + 1 e − x t d t = sup t ∈ [ 0 , δ ] \| g ( N + 1 ) ( t ) \| Γ ( λ + N + 2 ) x λ + N + 2 . {\displaystyle {\begin{aligned}\left\|\int _{0}^{\delta }g^{(N+1)}(t^{})\,t^{\lambda +N+1}e^{-xt}\,\mathrm {d} t\right\|&\leq \sup _{t\in [0,\delta ]}\left\|g^{(N+1)}(t)\right\|\int _{0}^{\delta }t^{\lambda +N+1}e^{-xt}\,\mathrm {d} t\\&<\sup _{t\in [0,\delta ]}\left\|g^{(N+1)}(t)\right\|\int _{0}^{\infty }t^{\lambda +N+1}e^{-xt}\,\mathrm {d} t\\&=\sup _{t\in [0,\delta ]}\left\|g^{(N+1)}(t)\right\|\,{\frac {\Gamma (\lambda +N+2)}{x^{\lambda +N+2}}}.\end{aligned}}} Here we have used the fact that ∫ 0 ∞ t a e − x t d t = Γ ( a + 1 ) x a + 1 {\displaystyle \int _{0}^{\infty }t^{a}e^{-xt}\,\mathrm {d} t={\frac {\Gamma (a+1)}{x^{a+1}}}} if x > 0 {\displaystyle x>0} and a > − 1 {\displaystyle a>-1} , where Γ {\displaystyle \Gamma } is the gamma function. From the above calculation we see from ( 3 ) {\displaystyle (3)} that ( 4 ) ∫ 0 δ e − x t φ ( t ) d t = ∑ n = 0 N g ( n ) ( 0 ) n ! ∫ 0 δ t λ + n e − x t d t + O ( x − λ − N − 2 ) {\displaystyle (4)\quad \int _{0}^{\delta }e^{-xt}\varphi (t)\,\mathrm {d} t=\sum _{n=0}^{N}{\frac {g^{(n)}(0)}{n!}}\int _{0}^{\delta }t^{\lambda +n}e^{-xt}\,\mathrm {d} t+O\left(x^{-\lambda -N-2}\right)} as x → ∞ {\displaystyle x\to \infty } . We will now add the tails on to each integral in ( 4 ) {\displaystyle (4)} . For each n {\displaystyle n} we have ∫ 0 δ t λ + n e − x t d t = ∫ 0 ∞ t λ + n e − x t d t − ∫ δ ∞ t λ + n e − x t d t = Γ ( λ + n + 1 ) x λ + n + 1 − ∫ δ ∞ t λ + n e − x t d t , {\displaystyle {\begin{aligned}\int _{0}^{\delta }t^{\lambda +n}e^{-xt}\,\mathrm {d} t&=\int _{0}^{\infty }t^{\lambda +n}e^{-xt}\,\mathrm {d} t-\int _{\delta }^{\infty }t^{\lambda +n}e^{-xt}\,\mathrm {d} t\\[5pt]&={\frac {\Gamma (\lambda +n+1)}{x^{\lambda +n+1}}}-\int _{\delta }^{\infty }t^{\lambda +n}e^{-xt}\,\mathrm {d} t,\end{aligned}}} and we will show that the remaining integrals are exponentially small. Indeed, if we make the change of variables t = s + δ {\displaystyle t=s+\delta } we get ∫ δ ∞ t λ + n e − x t d t = ∫ 0 ∞ ( s + δ ) λ + n e − x ( s + δ ) d s = e − δ x ∫ 0 ∞ ( s + δ ) λ + n e − x s d s ≤ e − δ x ∫ 0 ∞ ( s + δ ) λ + n e − s d s {\displaystyle {\begin{aligned}\int _{\delta }^{\infty }t^{\lambda +n}e^{-xt}\,\mathrm {d} t&=\int _{0}^{\infty }(s+\delta )^{\lambda +n}e^{-x(s+\delta )}\,\mathrm {d} s\\[5pt]&=e^{-\delta x}\int _{0}^{\infty }(s+\delta )^{\lambda +n}e^{-xs}\,\mathrm {d} s\\[5pt]&\leq e^{-\delta x}\int _{0}^{\infty }(s+\delta )^{\lambda +n}e^{-s}\,\mathrm {d} s\end{aligned}}} for x ≥ 1 {\displaystyle x\geq 1} , so that ∫ 0 δ t λ + n e − x t d t = Γ ( λ + n + 1 ) x λ + n + 1 + O ( e − δ x ) as x → ∞ . {\displaystyle \int _{0}^{\delta }t^{\lambda +n}e^{-xt}\,\mathrm {d} t={\frac {\Gamma (\lambda +n+1)}{x^{\lambda +n+1}}}+O\left(e^{-\delta x}\right){\text{ as }}x\to \infty .} If we substitute this last result into ( 4 ) {\displaystyle (4)} we find that ∫ 0 δ e − x t φ ( t ) d t = ∑ n = 0 N g ( n ) ( 0 ) Γ ( λ + n + 1 ) n ! x λ + n + 1 + O ( e − δ x ) + O ( x − λ − N − 2 ) = ∑ n = 0 N g ( n ) ( 0 ) Γ ( λ + n + 1 ) n ! x λ + n + 1 + O ( x − λ − N − 2 ) {\displaystyle {\begin{aligned}\int _{0}^{\delta }e^{-xt}\varphi (t)\,\mathrm {d} t&=\sum _{n=0}^{N}{\frac {g^{(n)}(0)\ \Gamma (\lambda +n+1)}{n!\ x^{\lambda +n+1}}}+O\left(e^{-\delta x}\right)+O\left(x^{-\lambda -N-2}\right)\\&=\sum _{n=0}^{N}{\frac {g^{(n)}(0)\ \Gamma (\lambda +n+1)}{n!\ x^{\lambda +n+1}}}+O\left(x^{-\lambda -N-2}\right)\end{aligned}}} as x → ∞ {\displaystyle x\to \infty } . Finally, substituting this into ( 2 ) {\displaystyle (2)} we conclude that ∫ 0 T e − x t φ ( t ) d t = ∑ n = 0 N g ( n ) ( 0 ) Γ ( λ + n + 1 ) n ! x λ + n + 1 + O ( x − λ − N − 2 ) + O ( x − 1 e − δ x ) = ∑ n = 0 N g ( n ) ( 0 ) Γ ( λ + n + 1 ) n ! x λ + n + 1 + O ( x − λ − N − 2 ) {\displaystyle {\begin{aligned}\int _{0}^{T}e^{-xt}\varphi (t)\,\mathrm {d} t&=\sum _{n=0}^{N}{\frac {g^{(n)}(0)\ \Gamma (\lambda +n+1)}{n!\ x^{\lambda +n+1}}}+O\left(x^{-\lambda -N-2}\right)+O\left(x^{-1}e^{-\delta x}\right)\\&=\sum _{n=0}^{N}{\frac {g^{(n)}(0)\ \Gamma (\lambda +n+1)}{n!\ x^{\lambda +n+1}}}+O\left(x^{-\lambda -N-2}\right)\end{aligned}}} as x → ∞ {\displaystyle x\to \infty } . Since this last expression is true for each integer N ≥ 0 {\displaystyle N\geq 0} we have thus shown that ∫ 0 T e − x t φ ( t ) d t ∼ ∑ n = 0 ∞ g ( n ) ( 0 ) Γ ( λ + n + 1 ) n ! x λ + n + 1 {\displaystyle \int _{0}^{T}e^{-xt}\varphi (t)\,\mathrm {d} t\sim \sum _{n=0}^{\infty }{\frac {g^{(n)}(0)\ \Gamma (\lambda +n+1)}{n!\ x^{\lambda +n+1}}}} as x → ∞ {\displaystyle x\to \infty } , where the infinite series is interpreted as an asymptotic expansion of the integral in question. == Example == When 0 < a < b {\displaystyle 0<a<b} , the confluent hypergeometric function of the first kind has the integral representation 1 F 1 ( a , b , x ) = Γ ( b ) Γ ( a ) Γ ( b − a ) ∫ 0 1 e x t t a − 1 ( 1 − t ) b − a − 1 d t , {\displaystyle {}_{1}F_{1}(a,b,x)={\frac {\Gamma (b)}{\Gamma (a)\Gamma (b-a)}}\int _{0}^{1}e^{xt}t^{a-1}(1-t)^{b-a-1}\,\mathrm {d} t,} where Γ {\displaystyle \Gamma } is the gamma function. The change of variables t = 1 − s {\displaystyle t=1-s} puts this into the form 1 F 1 ( a , b , x ) = Γ ( b ) Γ ( a ) Γ ( b − a ) e x ∫ 0 1 e − x s ( 1 − s ) a − 1 s b − a − 1 d s , {\displaystyle {}_{1}F_{1}(a,b,x)={\frac {\Gamma (b)}{\Gamma (a)\Gamma (b-a)}}\,e^{x}\int _{0}^{1}e^{-xs}(1-s)^{a-1}s^{b-a-1}\,ds,} which is now amenable to the use of Watson's lemma. Taking λ = b − a − 1 {\displaystyle \lambda =b-a-1} and g ( s ) = ( 1 − s ) a − 1 {\displaystyle g(s)=(1-s)^{a-1}} , Watson's lemma tells us that ∫ 0 1 e − x s ( 1 − s ) a − 1 s b − a − 1 d s ∼ Γ ( b − a ) x a − b as x → ∞ with x > 0 , {\displaystyle \int _{0}^{1}e^{-xs}(1-s)^{a-1}s^{b-a-1}\,ds\sim \Gamma (b-a)x^{a-b}\quad {\text{as }}x\to \infty {\text{ with }}x>0,} which allows us to conclude that 1 F 1 ( a , b , x ) ∼ Γ ( b ) Γ ( a ) x a − b e x as x → ∞ with x > 0. {\displaystyle {}_{1}F_{1}(a,b,x)\sim {\frac {\Gamma (b)}{\Gamma (a)}}\,x^{a-b}e^{x}\quad {\text{as }}x\to \infty {\text{ with }}x>0.} == References == Miller, P.D. (2006), Applied Asymptotic Analysis, Providence, RI: American Mathematical Society, p. 467, ISBN 978-0-8218-4078-8. Watson, G. N. (1918), "The harmonic functions associated with the parabolic cylinder", Proceedings of the London Mathematical Society, vol. 2, no. 17, pp. 116–148, doi:10.1112/plms/s2-17.1.116. Ablowitz, M. J., Fokas, A. S. (2003). Complex variables: introduction and applications. Cambridge University Press.
Wikipedia:Wave front set#0	In mathematical analysis, more precisely in microlocal analysis, the wave front (set) WF(f) characterizes the singularities of a generalized function f, not only in space, but also with respect to its Fourier transform at each point. The term "wave front" was coined by Lars Hörmander around 1970. == Introduction == In more familiar terms, WF(f) tells not only where the function f is singular (which is already described by its singular support), but also how or why it is singular, by being more exact about the direction in which the singularity occurs. This concept is mostly useful in dimensions at least two, since in one dimension there are only two possible directions. The complementary notion of a function being non-singular in a direction is microlocal smoothness. Intuitively, as an example, consider a function ƒ whose singular support is concentrated on a smooth curve in the plane at which the function has a jump discontinuity. In the direction tangent to the curve, the function remains smooth. By contrast, in the direction normal to the curve, the function has a singularity. To decide on whether the function is smooth in another direction v, one can try to smooth the function out by averaging in directions perpendicular to v. If the resulting function is smooth, then we regard ƒ to be smooth in the direction of v. Otherwise, v is in the wavefront set. Formally, in Euclidean space, the wave front set of ƒ is defined as the complement of the set of all pairs (x0,v) such that there exists a test function ϕ ∈ C c ∞ {\displaystyle \phi \in C_{c}^{\infty }} with ϕ {\displaystyle \phi } (x0) ≠ 0 and an open cone Γ containing v such that the estimate \| ( ϕ f ) ∧ ( ξ ) \| ≤ C N ( 1 + \| ξ \| ) − N for all ξ ∈ Γ {\displaystyle \|(\phi f)^{\wedge }(\xi )\|\leq C_{N}(1+\|\xi \|)^{-N}\quad {\mbox{for all }}\ \xi \in \Gamma } holds for all positive integers N. Here ( ϕ f ) ∧ {\displaystyle (\phi f)^{\wedge }} denotes the Fourier transform. Observe that the wavefront set is conical in the sense that if (x,v) ∈ Wf(ƒ), then (x,λv) ∈ Wf(ƒ) for all λ > 0. In the example discussed in the previous paragraph, the wavefront set is the set-theoretic complement of the image of the tangent bundle of the curve inside the tangent bundle of the plane. Because the definition involves cutoff by a compactly supported function, the notion of a wave front set can be transported to any differentiable manifold X. In this more general situation, the wave front set is a closed conical subset of the cotangent bundle T(X), since the ξ variable naturally localizes to a covector rather than a vector. The wave front set is defined such that its projection on X is equal to the singular support of the function. == Definition == In Euclidean space, the wave front set of a distribution ƒ is defined as W F ( f ) = { ( x , ξ ) ∈ R n × R n ∣ ξ ∈ Σ x ( f ) } {\displaystyle {\rm {WF}}(f)=\{(x,\xi )\in \mathbb {R} ^{n}\times \mathbb {R} ^{n}\mid \xi \in \Sigma _{x}(f)\}} where Σ x ( f ) {\displaystyle \Sigma _{x}(f)} is the singular fibre of ƒ at x. The singular fibre is defined to be the complement of all directions ξ {\displaystyle \xi } such that the Fourier transform of f, localized at x, is sufficiently regular when restricted to an open cone containing ξ {\displaystyle \xi } . More precisely, a direction v is in the complement of Σ x ( f ) {\displaystyle \Sigma _{x}(f)} if there is a compactly supported smooth function φ with φ(x) ≠ 0 and an open cone Γ containing v such that the following estimate holds for each positive integer N: \| ( ϕ f ) ∧ ( ξ ) \| < c N ( 1 + \| ξ \| ) − N f o r a l l ξ ∈ Γ . {\displaystyle \|(\phi f)^{\wedge }(\xi )\|<c_{N}(1+\|\xi \|)^{-N}\quad {\rm {for~all}}\ \xi \in \Gamma .} Once such an estimate holds for a particular cutoff function φ at x, it also holds for all cutoff functions with smaller support, possibly for a different open cone containing v. On a differentiable manifold M, using local coordinates x , ξ {\displaystyle x,\xi } on the cotangent bundle, the wave front set WF(f) of a distribution ƒ can be defined in the following general way: W F ( f ) = { ( x , ξ ) ∈ T ∗ ( X ) ∣ ξ ∈ Σ x ( f ) } {\displaystyle {\rm {WF}}(f)=\{(x,\xi )\in T^{}(X)\mid \xi \in \Sigma _{x}(f)\}} where the singular fibre Σ x ( f ) {\displaystyle \Sigma _{x}(f)} is again the complement of all directions ξ {\displaystyle \xi } such that the Fourier transform of f, localized at x, is sufficiently regular when restricted to a conical neighbourhood of ξ {\displaystyle \xi } . The problem of regularity is local, so it can be checked in the local coordinate system, using the Fourier transform on the x variables. The required regularity estimate transforms well under diffeomorphism, and so the notion of regularity is independent of the choice of local coordinates. === Generalizations === The notion of a wave front set can be adapted to accommodate other notions of regularity of a function. Localized can here be expressed by saying that f is truncated by some smooth cutoff function not vanishing at x. (The localization process could be done in a more elegant fashion, using germs.) More concretely, this can be expressed as ξ ∉ Σ x ( f ) ⟺ ξ = 0 or ∃ ϕ ∈ D x , ∃ V ∈ V ξ : ϕ f ^ \| V ∈ O ( V ) {\displaystyle \xi \notin \Sigma _{x}(f)\iff \xi =0{\text{ or }}\exists \phi \in {\mathcal {D}}_{x},\ \exists V\in {\mathcal {V}}_{\xi }:{\widehat {\phi f}}\|_{V}\in O(V)} where D x {\displaystyle {\mathcal {D}}_{x}} are compactly supported smooth functions not vanishing at x, V ξ {\displaystyle {\mathcal {V}}_{\xi }} are conical neighbourhoods of ξ {\displaystyle \xi } , i.e. neighbourhoods V such that c ⋅ V ⊂ V {\displaystyle c\cdot V\subset V} for all c > 0 {\displaystyle c>0} , u ^ \| V {\displaystyle {\widehat {u}}\|_{V}} denotes the Fourier transform of the (compactly supported generalized) function u, restricted to V, O : Ω → O ( Ω ) {\displaystyle O:\Omega \to O(\Omega )} is a fixed presheaf of functions (or distributions) whose choice enforces the desired regularity of the Fourier transform. Typically, sections of O are required to satisfy some growth (or decrease) condition at infinity, e.g. such that ( 1 + \| ξ \| ) s v ( ξ ) {\displaystyle (1+\|\xi \|)^{s}v(\xi )} belong to some Lp space. This definition makes sense, because the Fourier transform becomes more regular (in terms of growth at infinity) when f is truncated with the smooth cutoff ϕ {\displaystyle \phi } . The most difficult "problem", from a theoretical point of view, is finding the adequate sheaf O characterizing functions belonging to a given subsheaf E of the space G of generalized functions. === Example === If we take G = D′ the space of Schwartz distributions and want to characterize distributions which are locally C ∞ {\displaystyle C^{\infty }} functions, we must take for O(Ω) the classical function spaces called O′M(Ω) in the literature. Then the projection on the first component of a distribution's wave front set is nothing else than its classical singular support, i.e. the complement of the set on which its restriction would be a smooth function. === Applications === The wave front set is useful, among others, when studying propagation of singularities by pseudodifferential operators. The propagation of singularities theorem characterizes the wave front set. == See also == FBI transform Singular spectrum Essential support == References == Lars Hörmander, Fourier integral operators I, Acta Math. 127 (1971), pp. 79–183. Hörmander, Lars (1990), The Analysis of Linear Partial Differential Equations I: Distribution Theory and Fourier Analysis, Grundlehren der mathematischen Wissenschaften, vol. 256 (2nd ed.), Springer, pp. 251–279, ISBN 0-387-52345-6 Chapter VIII, Spectral Analysis of Singularities
Wikipedia:Waynflete Professor of Pure Mathematics#0	The Waynflete Professorships are four professorial fellowships at the University of Oxford endowed by Magdalen College and named in honour of the college founder William of Waynflete, who had a great interest in science. These professorships are statutory professorships of the University, that is, they are professorships established in the university's regulations, and which are by those regulations attached to Magdalen College in particular. The oldest professorship is the Waynflete Professor of Metaphysical Philosophy. The three science professorships were created following the recommendation of the University Commission in 1857, in recognition of William of Waynflete's lifetime support of science. The professorships are the Waynflete Professor of Chemistry, the Waynflete Professor of Physiology, and the Waynflete Professor of Pure Mathematics. == Waynflete Professors of Metaphysical Philosophy == This Waynflete Professorship is one of five statutory professorships in philosophy at the University of Oxford, the other four being the Wykeham Professorship in Logic, the White’s Professorship of Moral Philosophy, the Wilde Professor of Mental Philosophy, as well as the untitled professorship in Ancient Philosophy. 1859–1867 Henry Longueville Mansel 1867–1889 Henry William Chandler 1889–1910 Thomas Case 1910–1935 John Alexander Smith 1935–1941 Robin G. Collingwood 1945–1967 Gilbert Ryle 1968–1987 P. F. Strawson 1989–2000 Christopher Peacocke 2003–2006 Dorothy Edgington 2006–2015 John Hawthorne 2016– Ofra Magidor == Waynflete Professors of Chemistry == The four heads of the Dyson Perrins Laboratory were four consecutive Waynflete Professors of Chemistry, from its foundation in 1916 as the University's research centre for organic chemistry to its relocation in 2003. 1865–1872 Sir Benjamin Collins Brodie, 2nd Baronet 1872–1912 William Odling 1912–1930 William Henry Perkin, Jr., first head of Dyson Perrins Laboratory; 1930–1954 Sir Robert Robinson 1954–1978 Ewart Jones 1978–2005 Sir Jack Baldwin, last head of Dyson Perrins Laboratory; 2006–2021 Stephen G. Davies, ex-Chairman of Chemistry. 2022– Véronique Gouverneur, from 1 November == Waynflete Professors of Physiology == 1882–1905 John Scott Burdon-Sanderson 1905–1913 Francis Gotch 1913–1935 Charles Scott Sherrington 1936–1939 John Mellanby 1940–1960 Edward George Tandy Liddell 1960–1967 George Lindor Brown 1968–1979 David Whitteridge 1979–2007 Colin Blakemore 2007– Gero Miesenböck == Waynflete Professors of Pure Mathematics == 1892–1921 Edwin Bailey Elliott 1922–1945 Arthur Lee Dixon 1947–1960 J. H. C. Whitehead 1960–1984 Graham Higman 1984–2006 Daniel Quillen 2007–2013 Raphaël Rouquier 2013– Ben Green == Notes ==
Wikipedia:Weierstrass Nullstellensatz#0	In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros", or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields. This relationship was discovered by David Hilbert, who proved the Nullstellensatz in his second major paper on invariant theory in 1893 (following his seminal 1890 paper in which he proved Hilbert's basis theorem). == Formulation == Let k {\displaystyle k} be a field (such as the rational numbers) and K {\displaystyle K} be an algebraically closed field extension of k {\displaystyle k} (such as the complex numbers). Consider the polynomial ring k [ X 1 , … , X n ] {\displaystyle k[X_{1},\ldots ,X_{n}]} and let I {\displaystyle I} be an ideal in this ring. The algebraic set V ( I ) {\displaystyle \mathrm {V} (I)} defined by this ideal consists of all n {\displaystyle n} -tuples x = ( x 1 , … , x n ) {\displaystyle \mathbf {x} =(x_{1},\dots ,x_{n})} in K n {\displaystyle K^{n}} such that f ( x ) = 0 {\displaystyle f(\mathbf {x} )=0} for all f {\displaystyle f} in I {\displaystyle I} . Hilbert's Nullstellensatz states that if p is some polynomial in k [ X 1 , … , X n ] {\displaystyle k[X_{1},\ldots ,X_{n}]} that vanishes on the algebraic set V ( I ) {\displaystyle \mathrm {V} (I)} , i.e. p ( x ) = 0 {\displaystyle p(\mathbf {x} )=0} for all x {\displaystyle \mathbf {x} } in V ( I ) {\displaystyle \mathrm {V} (I)} , then there exists a natural number r {\displaystyle r} such that p r {\displaystyle p^{r}} is in I {\displaystyle I} . An immediate corollary is the weak Nullstellensatz: The ideal I ⊆ k [ X 1 , … , X n ] {\displaystyle I\subseteq k[X_{1},\ldots ,X_{n}]} contains 1 if and only if the polynomials in I {\displaystyle I} do not have any common zeros in Kn. Specializing to the case k = K = C , n = 1 {\displaystyle k=K=\mathbb {C} ,n=1} , one immediately recovers a restatement of the fundamental theorem of algebra: a polynomial P in C [ X ] {\displaystyle \mathbb {C} [X]} has a root in C {\displaystyle \mathbb {C} } if and only if deg P ≠ 0. For this reason, the (weak) Nullstellensatz has been referred to as a generalization of the fundamental theorem of algebra for multivariable polynomials. The weak Nullstellensatz may also be formulated as follows: if I is a proper ideal in k [ X 1 , … , X n ] , {\displaystyle k[X_{1},\ldots ,X_{n}],} then V(I) cannot be empty, i.e. there exists a common zero for all the polynomials in the ideal in every algebraically closed extension of k. This is the reason for the name of the theorem, the full version of which can be proved easily from the 'weak' form using the Rabinowitsch trick. The assumption of considering common zeros in an algebraically closed field is essential here; for example, the elements of the proper ideal (X2 + 1) in R [ X ] {\displaystyle \mathbb {R} [X]} do not have a common zero in R . {\displaystyle \mathbb {R} .} With the notation common in algebraic geometry, the Nullstellensatz can also be formulated as I ( V ( J ) ) = J {\displaystyle {\hbox{I}}({\hbox{V}}(J))={\sqrt {J}}} for every ideal J. Here, J {\displaystyle {\sqrt {J}}} denotes the radical of J and I(U) is the ideal of all polynomials that vanish on the set U. In this way, taking k = K {\displaystyle k=K} we obtain an order-reversing bijective correspondence between the algebraic sets in Kn and the radical ideals of K [ X 1 , … , X n ] . {\displaystyle K[X_{1},\ldots ,X_{n}].} In fact, more generally, one has a Galois connection between subsets of the space and subsets of the algebra, where "Zariski closure" and "radical of the ideal generated" are the closure operators. As a particular example, consider a point P = ( a 1 , … , a n ) ∈ K n {\displaystyle P=(a_{1},\dots ,a_{n})\in K^{n}} . Then I ( P ) = ( X 1 − a 1 , … , X n − a n ) {\displaystyle I(P)=(X_{1}-a_{1},\ldots ,X_{n}-a_{n})} . More generally, I = ⋂ ( a 1 , … , a n ) ∈ V ( I ) ( X 1 − a 1 , … , X n − a n ) . {\displaystyle {\sqrt {I}}=\bigcap _{(a_{1},\dots ,a_{n})\in V(I)}(X_{1}-a_{1},\dots ,X_{n}-a_{n}).} Conversely, every maximal ideal of the polynomial ring K [ X 1 , … , X n ] {\displaystyle K[X_{1},\ldots ,X_{n}]} (note that K {\displaystyle K} is algebraically closed) is of the form ( X 1 − a 1 , … , X n − a n ) {\displaystyle (X_{1}-a_{1},\ldots ,X_{n}-a_{n})} for some a 1 , … , a n ∈ K {\displaystyle a_{1},\ldots ,a_{n}\in K} . As another example, an algebraic subset W in Kn is irreducible (in the Zariski topology) if and only if I ( W ) {\displaystyle I(W)} is a prime ideal. == Proofs == There are many known proofs of the theorem. Some are non-constructive, such as the first one. Others are constructive, as based on algorithms for expressing 1 or pr as a linear combination of the generators of the ideal. === Using Zariski's lemma === Zariski's lemma asserts that if a field is finitely generated as an associative algebra over a field K, then it is a finite field extension of K (that is, it is also finitely generated as a vector space). If m {\displaystyle {\mathfrak {m}}} is a maximal ideal of K [ X 1 , … , X n ] {\displaystyle K[X_{1},\ldots ,X_{n}]} for algebraically closed K, then Zariski's lemma implies that K [ X 1 , … , X n ] / m {\displaystyle K[X_{1},\ldots ,X_{n}]/{\mathfrak {m}}} is a finite field extension of K, and thus, by algebraic closure, must be K. From this, it follows that there is an a = ( a 1 , … , a n ) ∈ K n {\displaystyle a=(a_{1},\dots ,a_{n})\in K^{n}} such that X i − a i ∈ m {\displaystyle X_{i}-a_{i}\in {\mathfrak {m}}} for i = 1 , … , n {\displaystyle i=1,\dots ,n} . In other words, m ⊇ m a = ( X 1 − a 1 , … , X n − a n ) {\displaystyle {\mathfrak {m}}\supseteq {\mathfrak {m}}_{a}=(X_{1}-a_{1},\ldots ,X_{n}-a_{n})} for some a = ( a 1 , … , a n ) ∈ K n {\displaystyle a=(a_{1},\dots ,a_{n})\in K^{n}} . But m a {\displaystyle {\mathfrak {m}}_{a}} is clearly maximal, so m = m a {\displaystyle {\mathfrak {m}}={\mathfrak {m}}_{a}} . This is the weak Nullstellensatz: every maximal ideal of K [ X 1 , … , X n ] {\displaystyle K[X_{1},\ldots ,X_{n}]} for algebraically closed K is of the form m a = ( X 1 − a 1 , … , X n − a n ) {\displaystyle {\mathfrak {m}}_{a}=(X_{1}-a_{1},\ldots ,X_{n}-a_{n})} for some a = ( a 1 , … , a n ) ∈ K n {\displaystyle a=(a_{1},\dots ,a_{n})\in K^{n}} . Because of this close relationship, some texts refer to Zariski's lemma as the weak Nullstellensatz or as the 'algebraic version' of the weak Nullstellensatz. The full Nullstellensatz can also be proved directly from Zariski's lemma without employing the Rabinowitsch trick. Here is a sketch of a proof using this lemma. Let A = K [ X 1 , … , X n ] {\displaystyle A=K[X_{1},\ldots ,X_{n}]} (K an algebraically closed field), J an ideal of A, and V = V ( J ) {\displaystyle V=\mathrm {V} (J)} the common zeros of J in K n {\displaystyle K^{n}} . Clearly, J ⊆ I ( V ) {\displaystyle {\sqrt {J}}\subseteq \mathrm {I} (V)} , where I ( V ) {\displaystyle \mathrm {I} (V)} is the ideal of polynomials in A vanishing on V. To show the opposite inclusion, let f ∉ J {\displaystyle f\not \in {\sqrt {J}}} . Then f ∉ p {\displaystyle f\not \in {\mathfrak {p}}} for some prime ideal p ⊇ J {\displaystyle {\mathfrak {p}}\supseteq J} in A. Let R = ( A / p ) [ 1 / f ¯ ] {\displaystyle R=(A/{\mathfrak {p}})[1/{\bar {f}}]} , where f ¯ {\displaystyle {\bar {f}}} is the image of f under the natural map A → A / p {\displaystyle A\to A/{\mathfrak {p}}} , and m {\displaystyle {\mathfrak {m}}} be a maximal ideal in R. By Zariski's lemma, R / m {\displaystyle R/{\mathfrak {m}}} is a finite extension of K, and thus, is K since K is algebraically closed. Let x i {\displaystyle x_{i}} be the images of X i {\displaystyle X_{i}} under the natural map A → A / p → R → R / m ≅ K {\displaystyle A\to A/{\mathfrak {p}}\to R\to R/{\mathfrak {m}}\cong K} . It follows that, by construction, x = ( x 1 , … , x n ) ∈ V {\displaystyle x=(x_{1},\ldots ,x_{n})\in V} but f ( x ) ≠ 0 {\displaystyle f(x)\neq 0} , so f ∉ I ( V ) {\displaystyle f\notin \mathrm {I} (V)} . === Using resultants === The following constructive proof of the weak form is one of the oldest proofs (the strong form results from the Rabinowitsch trick, which is also constructive). The resultant of two polynomials depending on a variable x and other variables is a polynomial in the other variables that is in the ideal generated by the two polynomials, and has the following properties: if one of the polynomials is monic in x, every zero (in the other variables) of the resultant may be extended into a common zero of the two polynomials. The proof is as follows. If the ideal is principal, generated by a non-constant polynomial p that depends on x, one chooses arbitrary values for the other variables. The fundamental theorem of algebra asserts that this choice can be extended to a zero of p. In the case of several polynomials p 1 , … , p n , {\displaystyle p_{1},\ldots ,p_{n},} a linear change of variables allows to suppose that p 1 {\displaystyle p_{1}} is monic in the first variable x. Then, one introduces n − 1 {\displaystyle n-1} new variables u 2 , … , u n , {\displaystyle u_{2},\ldots ,u_{n},} and one considers the resultant R = Res x ⁡ ( p 1 , u 2 p 2 + ⋯ + u n p n ) . {\displaystyle R=\operatorname {Res} _{x}(p_{1},u_{2}p_{2}+\cdots +u_{n}p_{n}).} As R is in the ideal generated by p 1 , … , p n , {\displaystyle p_{1},\ldots ,p_{n},} the same is true for the coefficients in R of the monomials in u 2 , … , u n . {\displaystyle u_{2},\ldots ,u_{n}.} So, if 1 is in the ideal generated by these coefficients, it is also in the ideal generated by p 1 , … , p n . {\displaystyle p_{1},\ldots ,p_{n}.} On the other hand, if these coefficients have a common zero, this zero can be extended to a common zero of p 1 , … , p n , {\displaystyle p_{1},\ldots ,p_{n},} by the above property of the resultant. This proves the weak Nullstellensatz by induction on the number of variables. === Using Gröbner bases === A Gröbner basis is an algorithmic concept that was introduced in 1973 by Bruno Buchberger. It is presently fundamental in computational geometry. A Gröbner basis is a special generating set of an ideal from which most properties of the ideal can easily be extracted. Those that are related to the Nullstellensatz are the following: An ideal contains 1 if and only if its reduced Gröbner basis (for any monomial ordering) is 1. The number of the common zeros of the polynomials in a Gröbner basis is strongly related to the number of monomials that are irreducibles by the basis. Namely, the number of common zeros is infinite if and only if the same is true for the irreducible monomials; if the two numbers are finite, the number of irreducible monomials equals the numbers of zeros (in an algebraically closed field), counted with multiplicities. With a lexicographic monomial order, the common zeros can be computed by solving iteratively univariate polynomials (this is not used in practice since one knows better algorithms). Strong Nullstellensatz: a power of p belongs to an ideal I if and only the saturation of I by p produces the Gröbner basis 1. Thus, the strong Nullstellensatz results almost immediately from the definition of the saturation. == Generalizations == The Nullstellensatz is subsumed by a systematic development of the theory of Jacobson rings, which are those rings in which every radical ideal is an intersection of maximal ideals. Given Zariski's lemma, proving the Nullstellensatz amounts to showing that if k is a field, then every finitely generated k-algebra R (necessarily of the form R = k [ t 1 , ⋯ , t n ] / I {\textstyle R=k[t_{1},\cdots ,t_{n}]/I} ) is Jacobson. More generally, one has the following theorem: Let R {\displaystyle R} be a Jacobson ring. If S {\displaystyle S} is a finitely generated R-algebra, then S {\displaystyle S} is a Jacobson ring. Furthermore, if n ⊆ S {\displaystyle {\mathfrak {n}}\subseteq S} is a maximal ideal, then m := n ∩ R {\displaystyle {\mathfrak {m}}:={\mathfrak {n}}\cap R} is a maximal ideal of R {\textstyle R} , and S / n {\displaystyle S/{\mathfrak {n}}} is a finite extension of R / m {\displaystyle R/{\mathfrak {m}}} . Other generalizations proceed from viewing the Nullstellensatz in scheme-theoretic terms as saying that for any field k and nonzero finitely generated k-algebra R, the morphism S p e c R → S p e c k {\textstyle \mathrm {Spec} \,R\to \mathrm {Spec} \,k} admits a section étale-locally (equivalently, after base change along S p e c L → S p e c k {\textstyle \mathrm {Spec} \,L\to \mathrm {Spec} \,k} for some finite field extension L / k {\textstyle L/k} ). In this vein, one has the following theorem: Any faithfully flat morphism of schemes f : Y → X {\textstyle f:Y\to X} locally of finite presentation admits a quasi-section, in the sense that there exists a faithfully flat and locally quasi-finite morphism g : X ′ → X {\textstyle g:X'\to X} locally of finite presentation such that the base change f ′ : Y × X X ′ → X ′ {\textstyle f':Y\times _{X}X'\to X'} of f {\textstyle f} along g {\textstyle g} admits a section. Moreover, if X {\textstyle X} is quasi-compact (resp. quasi-compact and quasi-separated), then one may take X ′ {\textstyle X'} to be affine (resp. X ′ {\textstyle X'} affine and g {\textstyle g} quasi-finite), and if f {\textstyle f} is smooth surjective, then one may take g {\textstyle g} to be étale. Serge Lang gave an extension of the Nullstellensatz to the case of infinitely many generators: Let κ {\textstyle \kappa } be an infinite cardinal and let K {\textstyle K} be an algebraically closed field whose transcendence degree over its prime subfield is strictly greater than κ {\displaystyle \kappa } . Then for any set S {\textstyle S} of cardinality κ {\textstyle \kappa } , the polynomial ring A = K [ x i ] i ∈ S {\textstyle A=K[x_{i}]_{i\in S}} satisfies the Nullstellensatz, i.e., for any ideal J ⊂ A {\textstyle J\subset A} we have that J = I ( V ( J ) ) {\displaystyle {\sqrt {J}}={\hbox{I}}({\hbox{V}}(J))} . == Effective Nullstellensatz == In all of its variants, Hilbert's Nullstellensatz asserts that some polynomial g belongs or not to an ideal generated, say, by f1, ..., fk; we have g = f r in the strong version, g = 1 in the weak form. This means the existence or the non-existence of polynomials g1, ..., gk such that g = f1g1 + ... + fkgk. The usual proofs of the Nullstellensatz are not constructive, non-effective, in the sense that they do not give any way to compute the gi. It is thus a rather natural question to ask if there is an effective way to compute the gi (and the exponent r in the strong form) or to prove that they do not exist. To solve this problem, it suffices to provide an upper bound on the total degree of the gi: such a bound reduces the problem to a finite system of linear equations that may be solved by usual linear algebra techniques. Any such upper bound is called an effective Nullstellensatz. A related problem is the ideal membership problem, which consists in testing if a polynomial belongs to an ideal. For this problem also, a solution is provided by an upper bound on the degree of the gi. A general solution of the ideal membership problem provides an effective Nullstellensatz, at least for the weak form. In 1925, Grete Hermann gave an upper bound for ideal membership problem that is doubly exponential in the number of variables. In 1982 Mayr and Meyer gave an example where the gi have a degree that is at least double exponential, showing that every general upper bound for the ideal membership problem is doubly exponential in the number of variables. Since most mathematicians at the time assumed the effective Nullstellensatz was at least as hard as ideal membership, few mathematicians sought a bound better than double-exponential. In 1987, however, W. Dale Brownawell gave an upper bound for the effective Nullstellensatz that is simply exponential in the number of variables. Brownawell's proof relied on analytic techniques valid only in characteristic 0, but, one year later, János Kollár gave a purely algebraic proof, valid in any characteristic, of a slightly better bound. In the case of the weak Nullstellensatz, Kollár's bound is the following: Let f1, ..., fs be polynomials in n ≥ 2 variables, of total degree d1 ≥ ... ≥ ds. If there exist polynomials gi such that f1g1 + ... + fsgs = 1, then they can be chosen such that deg ⁡ ( f i g i ) ≤ max ( d s , 3 ) ∏ j = 1 min ( n , s ) − 1 max ( d j , 3 ) . {\displaystyle \deg(f_{i}g_{i})\leq \max(d_{s},3)\prod _{j=1}^{\min(n,s)-1}\max(d_{j},3).} This bound is optimal if all the degrees are greater than 2. If d is the maximum of the degrees of the fi, this bound may be simplified to max ( 3 , d ) min ( n , s ) . {\displaystyle \max(3,d)^{\min(n,s)}.} An improvement due to M. Sombra is deg ⁡ ( f i g i ) ≤ 2 d s ∏ j = 1 min ( n , s ) − 1 d j . {\displaystyle \deg(f_{i}g_{i})\leq 2d_{s}\prod _{j=1}^{\min(n,s)-1}d_{j}.} His bound improves Kollár's as soon as at least two of the degrees that are involved are lower than 3. == Projective Nullstellensatz == We can formulate a certain correspondence between homogeneous ideals of polynomials and algebraic subsets of a projective space, called the projective Nullstellensatz, that is analogous to the affine one. To do that, we introduce some notations. Let R = k [ t 0 , … , t n ] . {\displaystyle R=k[t_{0},\ldots ,t_{n}].} The homogeneous ideal, R + = ⨁ d ⩾ 1 R d {\displaystyle R_{+}=\bigoplus _{d\geqslant 1}R_{d}} is called the maximal homogeneous ideal (see also irrelevant ideal). As in the affine case, we let: for a subset S ⊆ P n {\displaystyle S\subseteq \mathbb {P} ^{n}} and a homogeneous ideal I of R, I P n ⁡ ( S ) = { f ∈ R + ∣ f = 0 on S } , V P n ⁡ ( I ) = { x ∈ P n ∣ f ( x ) = 0 for all f ∈ I } . {\displaystyle {\begin{aligned}\operatorname {I} _{\mathbb {P} ^{n}}(S)&=\{f\in R_{+}\mid f=0{\text{ on }}S\},\\\operatorname {V} _{\mathbb {P} ^{n}}(I)&=\{x\in \mathbb {P} ^{n}\mid f(x)=0{\text{ for all }}f\in I\}.\end{aligned}}} By f = 0 on S {\displaystyle f=0{\text{ on }}S} we mean: for every homogeneous coordinates ( a 0 : ⋯ : a n ) {\displaystyle (a_{0}:\cdots :a_{n})} of a point of S we have f ( a 0 , … , a n ) = 0 {\displaystyle f(a_{0},\ldots ,a_{n})=0} . This implies that the homogeneous components of f are also zero on S and thus that I P n ⁡ ( S ) {\displaystyle \operatorname {I} _{\mathbb {P} ^{n}}(S)} is a homogeneous ideal. Equivalently, I P n ⁡ ( S ) {\displaystyle \operatorname {I} _{\mathbb {P} ^{n}}(S)} is the homogeneous ideal generated by homogeneous polynomials f that vanish on S. Now, for any homogeneous ideal I ⊆ R + {\displaystyle I\subseteq R_{+}} , by the usual Nullstellensatz, we have: I = I P n ⁡ ( V P n ⁡ ( I ) ) , {\displaystyle {\sqrt {I}}=\operatorname {I} _{\mathbb {P} ^{n}}(\operatorname {V} _{\mathbb {P} ^{n}}(I)),} and so, like in the affine case, we have: There exists an order-reversing one-to-one correspondence between proper homogeneous radical ideals of R and subsets of P n {\displaystyle \mathbb {P} ^{n}} of the form V P n ⁡ ( I ) . {\displaystyle \operatorname {V} _{\mathbb {P} ^{n}}(I).} The correspondence is given by I P n {\displaystyle \operatorname {I} _{\mathbb {P} ^{n}}} and V P n . {\displaystyle \operatorname {V} _{\mathbb {P} ^{n}}.} == Analytic Nullstellensatz (Rückert’s Nullstellensatz) == The Nullstellensatz also holds for the germs of holomorphic functions at a point of complex n-space C n . {\displaystyle \mathbb {C} ^{n}.} Precisely, for each open subset U ⊆ C n , {\displaystyle U\subseteq \mathbb {C} ^{n},} let O C n ( U ) {\displaystyle {\mathcal {O}}_{\mathbb {C} ^{n}}(U)} denote the ring of holomorphic functions on U; then O C n {\displaystyle {\mathcal {O}}_{\mathbb {C} ^{n}}} is a sheaf on C n . {\displaystyle \mathbb {C} ^{n}.} The stalk O C n , 0 {\displaystyle {\mathcal {O}}_{\mathbb {C} ^{n},0}} at, say, the origin can be shown to be a Noetherian local ring that is a unique factorization domain. If f ∈ O C n , 0 {\displaystyle f\in {\mathcal {O}}_{\mathbb {C} ^{n},0}} is a germ represented by a holomorphic function f ~ : U → C {\displaystyle {\widetilde {f}}:U\to \mathbb {C} } , then let V 0 ( f ) {\displaystyle V_{0}(f)} be the equivalence class of the set { z ∈ U ∣ f ~ ( z ) = 0 } , {\displaystyle \left\{z\in U\mid {\widetilde {f}}(z)=0\right\},} where two subsets X , Y ⊆ C n {\displaystyle X,Y\subseteq \mathbb {C} ^{n}} are considered equivalent if X ∩ U = Y ∩ U {\displaystyle X\cap U=Y\cap U} for some neighborhood U of 0. Note V 0 ( f ) {\displaystyle V_{0}(f)} is independent of a choice of the representative f ~ . {\displaystyle {\widetilde {f}}.} For each ideal I ⊆ O C n , 0 , {\displaystyle I\subseteq {\mathcal {O}}_{\mathbb {C} ^{n},0},} let V 0 ( I ) {\displaystyle V_{0}(I)} denote V 0 ( f 1 ) ∩ ⋯ ∩ V 0 ( f r ) {\displaystyle V_{0}(f_{1})\cap \dots \cap V_{0}(f_{r})} for some generators f 1 , … , f r {\displaystyle f_{1},\ldots ,f_{r}} of I. It is well-defined; i.e., is independent of a choice of the generators. For each subset X ⊆ C n {\displaystyle X\subseteq \mathbb {C} ^{n}} , let I 0 ( X ) = { f ∈ O C n , 0 ∣ V 0 ( f ) ⊃ X } . {\displaystyle I_{0}(X)=\left\{f\in {\mathcal {O}}_{\mathbb {C} ^{n},0}\mid V_{0}(f)\supset X\right\}.} It is easy to see that I 0 ( X ) {\displaystyle I_{0}(X)} is an ideal of O C n , 0 {\displaystyle {\mathcal {O}}_{\mathbb {C} ^{n},0}} and that I 0 ( X ) = I 0 ( Y ) {\displaystyle I_{0}(X)=I_{0}(Y)} if X ∼ Y {\displaystyle X\sim Y} in the sense discussed above. The analytic Nullstellensatz then states: for each ideal I ⊆ O C n , 0 {\displaystyle I\subseteq {\mathcal {O}}_{\mathbb {C} ^{n},0}} , I = I 0 ( V 0 ( I ) ) {\displaystyle {\sqrt {I}}=I_{0}(V_{0}(I))} where the left-hand side is the radical of I. == See also == Artin–Tate lemma Combinatorial Nullstellensatz Differential Nullstellensatz Real radical Restricted power series#Tate algebra, an analog of Hilbert's Nullstellensatz holds for Tate algebras. Stengle's Positivstellensatz Weierstrass Nullstellensatz == Notes == == References == Almira, Jose María (2007). "Nullstellensatz revisited" (PDF). Rend. Semin. Mat. Univ. Politec. Torino. 65 (3): 365–369. Atiyah, M.F.; Macdonald, I.G. (1994). Introduction to Commutative Algebra. Addison-Wesley. ISBN 0-201-40751-5. Eisenbud, David (1999). Commutative Algebra With a View Toward Algebraic Geometry. Graduate Texts in Mathematics. Vol. 150. New York: Springer-Verlag. ISBN 978-0-387-94268-1. Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157 Hilbert, David (1893). "Ueber die vollen Invariantensysteme". Mathematische Annalen. 42 (3): 313–373. doi:10.1007/BF01444162. Huybrechts, Daniel (2005). Complex Geometry: An Introduction. Springer. ISBN 3-540-21290-6. Mukai, Shigeru (2003). An Introduction to Invariants and Moduli. Cambridge studies in advanced mathematics. Vol. 81. William Oxbury (trans.). p. 82. ISBN 0-521-80906-1. Zariski, Oscar; Samuel, Pierre (1960). Commutative algebra. Volume II. Berlin. ISBN 978-3-662-27753-9. {{cite book}}: ISBN / Date incompatibility (help)CS1 maint: location missing publisher (link)
Wikipedia:Weight function#0	A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is a weighted sum or weighted average. Weight functions occur frequently in statistics and analysis, and are closely related to the concept of a measure. Weight functions can be employed in both discrete and continuous settings. They can be used to construct systems of calculus called "weighted calculus" and "meta-calculus". == Discrete weights == === General definition === In the discrete setting, a weight function w : A → R + {\displaystyle w\colon A\to \mathbb {R} ^{+}} is a positive function defined on a discrete set A {\displaystyle A} , which is typically finite or countable. The weight function w ( a ) := 1 {\displaystyle w(a):=1} corresponds to the unweighted situation in which all elements have equal weight. One can then apply this weight to various concepts. If the function f : A → R {\displaystyle f\colon A\to \mathbb {R} } is a real-valued function, then the unweighted sum of f {\displaystyle f} on A {\displaystyle A} is defined as ∑ a ∈ A f ( a ) ; {\displaystyle \sum _{a\in A}f(a);} but given a weight function w : A → R + {\displaystyle w\colon A\to \mathbb {R} ^{+}} , the weighted sum or conical combination is defined as ∑ a ∈ A f ( a ) w ( a ) . {\displaystyle \sum _{a\in A}f(a)w(a).} One common application of weighted sums arises in numerical integration. If B is a finite subset of A, one can replace the unweighted cardinality \|B\| of B by the weighted cardinality ∑ a ∈ B w ( a ) . {\displaystyle \sum _{a\in B}w(a).} If A is a finite non-empty set, one can replace the unweighted mean or average 1 \| A \| ∑ a ∈ A f ( a ) {\displaystyle {\frac {1}{\|A\|}}\sum _{a\in A}f(a)} by the weighted mean or weighted average ∑ a ∈ A f ( a ) w ( a ) ∑ a ∈ A w ( a ) . {\displaystyle {\frac {\sum _{a\in A}f(a)w(a)}{\sum _{a\in A}w(a)}}.} In this case only the relative weights are relevant. === Statistics === Weighted means are commonly used in statistics to compensate for the presence of bias. For a quantity f {\displaystyle f} measured multiple independent times f i {\displaystyle f_{i}} with variance σ i 2 {\displaystyle \sigma _{i}^{2}} , the best estimate of the signal is obtained by averaging all the measurements with weight w i = 1 / σ i 2 {\textstyle w_{i}=1/{\sigma _{i}^{2}}} , and the resulting variance is smaller than each of the independent measurements σ 2 = 1 / ∑ i w i {\textstyle \sigma ^{2}=1/\sum _{i}w_{i}} . The maximum likelihood method weights the difference between fit and data using the same weights w i {\displaystyle w_{i}} . The expected value of a random variable is the weighted average of the possible values it might take on, with the weights being the respective probabilities. More generally, the expected value of a function of a random variable is the probability-weighted average of the values the function takes on for each possible value of the random variable. In regressions in which the dependent variable is assumed to be affected by both current and lagged (past) values of the independent variable, a distributed lag function is estimated, this function being a weighted average of the current and various lagged independent variable values. Similarly, a moving average model specifies an evolving variable as a weighted average of current and various lagged values of a random variable. === Mechanics === The terminology weight function arises from mechanics: if one has a collection of n {\displaystyle n} objects on a lever, with weights w 1 , … , w n {\displaystyle w_{1},\ldots ,w_{n}} (where weight is now interpreted in the physical sense) and locations x 1 , … , x n {\displaystyle {\boldsymbol {x}}_{1},\dotsc ,{\boldsymbol {x}}_{n}} , then the lever will be in balance if the fulcrum of the lever is at the center of mass ∑ i = 1 n w i x i ∑ i = 1 n w i , {\displaystyle {\frac {\sum _{i=1}^{n}w_{i}{\boldsymbol {x}}_{i}}{\sum _{i=1}^{n}w_{i}}},} which is also the weighted average of the positions x i {\displaystyle {\boldsymbol {x}}_{i}} . == Continuous weights == In the continuous setting, a weight is a positive measure such as w ( x ) d x {\displaystyle w(x)\,dx} on some domain Ω {\displaystyle \Omega } , which is typically a subset of a Euclidean space R n {\displaystyle \mathbb {R} ^{n}} , for instance Ω {\displaystyle \Omega } could be an interval [ a , b ] {\displaystyle [a,b]} . Here d x {\displaystyle dx} is Lebesgue measure and w : Ω → R + {\displaystyle w\colon \Omega \to \mathbb {R} ^{+}} is a non-negative measurable function. In this context, the weight function w ( x ) {\displaystyle w(x)} is sometimes referred to as a density. === General definition === If f : Ω → R {\displaystyle f\colon \Omega \to \mathbb {R} } is a real-valued function, then the unweighted integral ∫ Ω f ( x ) d x {\displaystyle \int _{\Omega }f(x)\ dx} can be generalized to the weighted integral ∫ Ω f ( x ) w ( x ) d x {\displaystyle \int _{\Omega }f(x)w(x)\,dx} Note that one may need to require f {\displaystyle f} to be absolutely integrable with respect to the weight w ( x ) d x {\displaystyle w(x)\,dx} in order for this integral to be finite. === Weighted volume === If E is a subset of Ω {\displaystyle \Omega } , then the volume vol(E) of E can be generalized to the weighted volume ∫ E w ( x ) d x , {\displaystyle \int _{E}w(x)\ dx,} === Weighted average === If Ω {\displaystyle \Omega } has finite non-zero weighted volume, then we can replace the unweighted average 1 v o l ( Ω ) ∫ Ω f ( x ) d x {\displaystyle {\frac {1}{\mathrm {vol} (\Omega )}}\int _{\Omega }f(x)\ dx} by the weighted average ∫ Ω f ( x ) w ( x ) d x ∫ Ω w ( x ) d x {\displaystyle {\frac {\displaystyle \int _{\Omega }f(x)\,w(x)\,dx}{\displaystyle \int _{\Omega }w(x)\,dx}}} === Bilinear form === If f : Ω → R {\displaystyle f\colon \Omega \to {\mathbb {R} }} and g : Ω → R {\displaystyle g\colon \Omega \to {\mathbb {R} }} are two functions, one can generalize the unweighted bilinear form ⟨ f , g ⟩ := ∫ Ω f ( x ) g ( x ) d x {\displaystyle \langle f,g\rangle :=\int _{\Omega }f(x)g(x)\ dx} to a weighted bilinear form ⟨ f , g ⟩ w := ∫ Ω f ( x ) g ( x ) w ( x ) d x . {\displaystyle {\langle f,g\rangle }_{w}:=\int _{\Omega }f(x)g(x)\ w(x)\ dx.} See the entry on orthogonal polynomials for examples of weighted orthogonal functions. == See also == Center of mass Numerical integration Orthogonality Weighted mean Linear combination Kernel (statistics) Measure (mathematics) Riemann–Stieltjes integral Weighting Window function == References ==
Wikipedia:Weighted arithmetic mean#0	The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics. If all the weights are equal, then the weighted mean is the same as the arithmetic mean. While weighted means generally behave in a similar fashion to arithmetic means, they do have a few counterintuitive properties, as captured for instance in Simpson's paradox. == Examples == === Basic example === Given two school classes — one with 20 students, one with 30 students — and test grades in each class as follows: Morning class = {62, 67, 71, 74, 76, 77, 78, 79, 79, 80, 80, 81, 81, 82, 83, 84, 86, 89, 93, 98} Afternoon class = {81, 82, 83, 84, 85, 86, 87, 87, 88, 88, 89, 89, 89, 90, 90, 90, 90, 91, 91, 91, 92, 92, 93, 93, 94, 95, 96, 97, 98, 99} The mean for the morning class is 80 and the mean of the afternoon class is 90. The unweighted mean of the two means is 85. However, this does not account for the difference in number of students in each class (20 versus 30); hence the value of 85 does not reflect the average student grade (independent of class). The average student grade can be obtained by averaging all the grades, without regard to classes (add all the grades up and divide by the total number of students): x ¯ = 4300 50 = 86. {\displaystyle {\bar {x}}={\frac {4300}{50}}=86.} Or, this can be accomplished by weighting the class means by the number of students in each class. The larger class is given more "weight": x ¯ = ( 20 × 80 ) + ( 30 × 90 ) 20 + 30 = 86. {\displaystyle {\bar {x}}={\frac {(20\times 80)+(30\times 90)}{20+30}}=86.} Thus, the weighted mean makes it possible to find the mean average student grade without knowing each student's score. Only the class means and the number of students in each class are needed. === Convex combination example === Since only the relative weights are relevant, any weighted mean can be expressed using coefficients that sum to one. Such a linear combination is called a convex combination. Using the previous example, we would get the following weights: 20 20 + 30 = 0.4 {\displaystyle {\frac {20}{20+30}}=0.4} 30 20 + 30 = 0.6 {\displaystyle {\frac {30}{20+30}}=0.6} Then, apply the weights like this: x ¯ = ( 0.4 × 80 ) + ( 0.6 × 90 ) = 86. {\displaystyle {\bar {x}}=(0.4\times 80)+(0.6\times 90)=86.} == Mathematical definition == Formally, the weighted mean of a non-empty finite tuple of data ( x 1 , x 2 , … , x n ) {\displaystyle \left(x_{1},x_{2},\dots ,x_{n}\right)} , with corresponding non-negative weights ( w 1 , w 2 , … , w n ) {\displaystyle \left(w_{1},w_{2},\dots ,w_{n}\right)} is x ¯ = ∑ i = 1 n w i x i ∑ i = 1 n w i , {\displaystyle {\bar {x}}={\frac {\sum \limits _{i=1}^{n}w_{i}x_{i}}{\sum \limits _{i=1}^{n}w_{i}}},} which expands to: x ¯ = w 1 x 1 + w 2 x 2 + ⋯ + w n x n w 1 + w 2 + ⋯ + w n . {\displaystyle {\bar {x}}={\frac {w_{1}x_{1}+w_{2}x_{2}+\cdots +w_{n}x_{n}}{w_{1}+w_{2}+\cdots +w_{n}}}.} Therefore, data elements with a high weight contribute more to the weighted mean than do elements with a low weight. The weights may not be negative in order for the equation to work. Some may be zero, but not all of them (since division by zero is not allowed). The formulas are simplified when the weights are normalized such that they sum up to 1, i.e., ∑ i = 1 n w i ′ = 1 {\textstyle \sum \limits _{i=1}^{n}{w_{i}'}=1} . For such normalized weights, the weighted mean is equivalently: x ¯ = ∑ i = 1 n w i ′ x i {\displaystyle {\bar {x}}=\sum \limits _{i=1}^{n}{w_{i}'x_{i}}} . One can always normalize the weights by making the following transformation on the original weights: w i ′ = w i ∑ j = 1 n w j {\displaystyle w_{i}'={\frac {w_{i}}{\sum \limits _{j=1}^{n}{w_{j}}}}} . The ordinary mean 1 n ∑ i = 1 n x i {\textstyle {\frac {1}{n}}\sum \limits _{i=1}^{n}{x_{i}}} is a special case of the weighted mean where all data have equal weights. If the data elements are independent and identically distributed random variables with variance σ 2 {\displaystyle \sigma ^{2}} , the standard error of the weighted mean, σ x ¯ {\displaystyle \sigma _{\bar {x}}} , can be shown via uncertainty propagation to be: σ x ¯ = σ ∑ i = 1 n w i ′ 2 {\textstyle \sigma _{\bar {x}}=\sigma {\sqrt {\sum \limits _{i=1}^{n}w_{i}'^{2}}}} === Variance-defined weights === For the weighted mean of a list of data for which each element x i {\displaystyle x_{i}} potentially comes from a different probability distribution with known variance σ i 2 {\displaystyle \sigma _{i}^{2}} , all having the same mean, one possible choice for the weights is given by the reciprocal of variance: w i = 1 σ i 2 . {\displaystyle w_{i}={\frac {1}{\sigma _{i}^{2}}}.} The weighted mean in this case is: x ¯ = ∑ i = 1 n ( x i σ i 2 ) ∑ i = 1 n 1 σ i 2 = ∑ i = 1 n ( x i ⋅ w i ) ∑ i = 1 n w i , {\displaystyle {\bar {x}}={\frac {\sum _{i=1}^{n}\left({\dfrac {x_{i}}{\sigma _{i}^{2}}}\right)}{\sum _{i=1}^{n}{\dfrac {1}{\sigma _{i}^{2}}}}}={\frac {\sum _{i=1}^{n}\left(x_{i}\cdot w_{i}\right)}{\sum _{i=1}^{n}w_{i}}},} and the standard error of the weighted mean (with inverse-variance weights) is: σ x ¯ = 1 ∑ i = 1 n σ i − 2 = 1 ∑ i = 1 n w i , {\displaystyle \sigma _{\bar {x}}={\sqrt {\frac {1}{\sum _{i=1}^{n}\sigma _{i}^{-2}}}}={\sqrt {\frac {1}{\sum _{i=1}^{n}w_{i}}}},} Note this reduces to σ x ¯ 2 = σ 0 2 / n {\displaystyle \sigma _{\bar {x}}^{2}=\sigma _{0}^{2}/n} when all σ i = σ 0 {\displaystyle \sigma _{i}=\sigma _{0}} . It is a special case of the general formula in previous section, σ x ¯ 2 = ∑ i = 1 n w i ′ 2 σ i 2 = ∑ i = 1 n σ i − 4 σ i 2 ( ∑ i = 1 n σ i − 2 ) 2 . {\displaystyle \sigma _{\bar {x}}^{2}=\sum _{i=1}^{n}{w_{i}'^{2}\sigma _{i}^{2}}={\frac {\sum _{i=1}^{n}{\sigma _{i}^{-4}\sigma _{i}^{2}}}{\left(\sum _{i=1}^{n}\sigma _{i}^{-2}\right)^{2}}}.} The equations above can be combined to obtain: x ¯ = σ x ¯ 2 ∑ i = 1 n x i σ i 2 . {\displaystyle {\bar {x}}=\sigma _{\bar {x}}^{2}\sum _{i=1}^{n}{\frac {x_{i}}{\sigma _{i}^{2}}}.} The significance of this choice is that this weighted mean is the maximum likelihood estimator of the mean of the probability distributions under the assumption that they are independent and normally distributed with the same mean. == Statistical properties == === Expectancy === The weighted sample mean, x ¯ {\displaystyle {\bar {x}}} , is itself a random variable. Its expected value and standard deviation are related to the expected values and standard deviations of the observations, as follows. For simplicity, we assume normalized weights (weights summing to one). If the observations have expected values E ( x i ) = μ i , {\displaystyle E(x_{i})={\mu _{i}},} then the weighted sample mean has expectation E ( x ¯ ) = ∑ i = 1 n w i ′ μ i . {\displaystyle E({\bar {x}})=\sum _{i=1}^{n}{w_{i}'\mu _{i}}.} In particular, if the means are equal, μ i = μ {\displaystyle \mu _{i}=\mu } , then the expectation of the weighted sample mean will be that value, E ( x ¯ ) = μ . {\displaystyle E({\bar {x}})=\mu .} === Variance === ==== Simple i.i.d. case ==== When treating the weights as constants, and having a sample of n observations from uncorrelated random variables, all with the same variance and expectation (as is the case for i.i.d random variables), then the variance of the weighted mean can be estimated as the multiplication of the unweighted variance by Kish's design effect (see proof): Var ⁡ ( y ¯ w ) = σ ^ y 2 w 2 ¯ w ¯ 2 {\displaystyle \operatorname {Var} ({\bar {y}}_{w})={\hat {\sigma }}_{y}^{2}{\frac {\overline {w^{2}}}{{\bar {w}}^{2}}}} With σ ^ y 2 = ∑ i = 1 n ( y i − y ¯ ) 2 n − 1 {\displaystyle {\hat {\sigma }}_{y}^{2}={\frac {\sum _{i=1}^{n}(y_{i}-{\bar {y}})^{2}}{n-1}}} , w ¯ = ∑ i = 1 n w i n {\displaystyle {\bar {w}}={\frac {\sum _{i=1}^{n}w_{i}}{n}}} , and w 2 ¯ = ∑ i = 1 n w i 2 n {\displaystyle {\overline {w^{2}}}={\frac {\sum _{i=1}^{n}w_{i}^{2}}{n}}} However, this estimation is rather limited due to the strong assumption about the y observations. This has led to the development of alternative, more general, estimators. ==== Survey sampling perspective ==== From a model based perspective, we are interested in estimating the variance of the weighted mean when the different y i {\displaystyle y_{i}} are not i.i.d random variables. An alternative perspective for this problem is that of some arbitrary sampling design of the data in which units are selected with unequal probabilities (with replacement).: 306 In Survey methodology, the population mean, of some quantity of interest y, is calculated by taking an estimation of the total of y over all elements in the population (Y or sometimes T) and dividing it by the population size – either known ( N {\displaystyle N} ) or estimated ( N ^ {\displaystyle {\hat {N}}} ). In this context, each value of y is considered constant, and the variability comes from the selection procedure. This in contrast to "model based" approaches in which the randomness is often described in the y values. The survey sampling procedure yields a series of Bernoulli indicator values ( I i {\displaystyle I_{i}} ) that get 1 if some observation i is in the sample and 0 if it was not selected. This can occur with fixed sample size, or varied sample size sampling (e.g.: Poisson sampling). The probability of some element to be chosen, given a sample, is denoted as P ( I i = 1 ∣ Some sample of size n ) = π i {\displaystyle P(I_{i}=1\mid {\text{Some sample of size }}n)=\pi _{i}} , and the one-draw probability of selection is P ( I i = 1 \| one sample draw ) = p i ≈ π i n {\displaystyle P(I_{i}=1\|{\text{one sample draw}})=p_{i}\approx {\frac {\pi _{i}}{n}}} (If N is very large and each p i {\displaystyle p_{i}} is very small). For the following derivation we'll assume that the probability of selecting each element is fully represented by these probabilities.: 42, 43, 51 I.e.: selecting some element will not influence the probability of drawing another element (this doesn't apply for things such as cluster sampling design). Since each element ( y i {\displaystyle y_{i}} ) is fixed, and the randomness comes from it being included in the sample or not ( I i {\displaystyle I_{i}} ), we often talk about the multiplication of the two, which is a random variable. To avoid confusion in the following section, let's call this term: y i ′ = y i I i {\displaystyle y'_{i}=y_{i}I_{i}} . With the following expectancy: E [ y i ′ ] = y i E [ I i ] = y i π i {\displaystyle E[y'_{i}]=y_{i}E[I_{i}]=y_{i}\pi _{i}} ; and variance: V [ y i ′ ] = y i 2 V [ I i ] = y i 2 π i ( 1 − π i ) {\displaystyle V[y'_{i}]=y_{i}^{2}V[I_{i}]=y_{i}^{2}\pi _{i}(1-\pi _{i})} . When each element of the sample is inflated by the inverse of its selection probability, it is termed the π {\displaystyle \pi } -expanded y values, i.e.: y ˇ i = y i π i {\displaystyle {\check {y}}_{i}={\frac {y_{i}}{\pi _{i}}}} . A related quantity is p {\displaystyle p} -expanded y values: y i p i = n y ˇ i {\displaystyle {\frac {y_{i}}{p_{i}}}=n{\check {y}}_{i}} .: 42, 43, 51, 52 As above, we can add a tick mark if multiplying by the indicator function. I.e.: y ˇ i ′ = I i y ˇ i = I i y i π i {\displaystyle {\check {y}}'_{i}=I_{i}{\check {y}}_{i}={\frac {I_{i}y_{i}}{\pi _{i}}}} In this design based perspective, the weights, used in the numerator of the weighted mean, are obtained from taking the inverse of the selection probability (i.e.: the inflation factor). I.e.: w i = 1 π i ≈ 1 n × p i {\displaystyle w_{i}={\frac {1}{\pi _{i}}}\approx {\frac {1}{n\times p_{i}}}} . ==== Variance of the weighted sum (pwr-estimator for totals) ==== If the population size N is known we can estimate the population mean using Y ¯ ^ known N = Y ^ p w r N ≈ ∑ i = 1 n w i y i ′ N {\displaystyle {\hat {\bar {Y}}}_{{\text{known }}N}={\frac {{\hat {Y}}_{pwr}}{N}}\approx {\frac {\sum _{i=1}^{n}w_{i}y'_{i}}{N}}} . If the sampling design is one that results in a fixed sample size n (such as in pps sampling), then the variance of this estimator is: Var ⁡ ( Y ¯ ^ known N ) = 1 N 2 n n − 1 ∑ i = 1 n ( w i y i − w y ¯ ) 2 {\displaystyle \operatorname {Var} \left({\hat {\bar {Y}}}_{{\text{known }}N}\right)={\frac {1}{N^{2}}}{\frac {n}{n-1}}\sum _{i=1}^{n}\left(w_{i}y_{i}-{\overline {wy}}\right)^{2}} An alternative term, for when the sampling has a random sample size (as in Poisson sampling), is presented in Sarndal et al. (1992) as:: 182 Var ⁡ ( Y ¯ ^ pwr (known N ) ) = 1 N 2 ∑ i = 1 n ∑ j = 1 n ( Δ ˇ i j y ˇ i y ˇ j ) {\displaystyle \operatorname {Var} ({\hat {\bar {Y}}}_{{\text{pwr (known }}N{\text{)}}})={\frac {1}{N^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}\left({\check {\Delta }}_{ij}{\check {y}}_{i}{\check {y}}_{j}\right)} With y ˇ i = y i π i {\displaystyle {\check {y}}_{i}={\frac {y_{i}}{\pi _{i}}}} . Also, C ( I i , I j ) = π i j − π i π j = Δ i j {\displaystyle C(I_{i},I_{j})=\pi _{ij}-\pi _{i}\pi _{j}=\Delta _{ij}} where π i j {\displaystyle \pi _{ij}} is the probability of selecting both i and j.: 36 And Δ ˇ i j = 1 − π i π j π i j {\displaystyle {\check {\Delta }}_{ij}=1-{\frac {\pi _{i}\pi _{j}}{\pi _{ij}}}} , and for i=j: Δ ˇ i i = 1 − π i π i π i = 1 − π i {\displaystyle {\check {\Delta }}_{ii}=1-{\frac {\pi _{i}\pi _{i}}{\pi _{i}}}=1-\pi _{i}} .: 43 If the selection probability are uncorrelated (i.e.: ∀ i ≠ j : C ( I i , I j ) = 0 {\displaystyle \forall i\neq j:C(I_{i},I_{j})=0} ), and when assuming the probability of each element is very small, then: Var ⁡ ( Y ¯ ^ pwr (known N ) ) = 1 N 2 ∑ i = 1 n ( w i y i ) 2 {\displaystyle \operatorname {Var} ({\hat {\bar {Y}}}_{{\text{pwr (known }}N{\text{)}}})={\frac {1}{N^{2}}}\sum _{i=1}^{n}\left(w_{i}y_{i}\right)^{2}} ==== Variance of the weighted mean (π-estimator for ratio-mean) ==== The previous section dealt with estimating the population mean as a ratio of an estimated population total ( Y ^ {\displaystyle {\hat {Y}}} ) with a known population size ( N {\displaystyle N} ), and the variance was estimated in that context. Another common case is that the population size itself ( N {\displaystyle N} ) is unknown and is estimated using the sample (i.e.: N ^ {\displaystyle {\hat {N}}} ). The estimation of N {\displaystyle N} can be described as the sum of weights. So when w i = 1 π i {\displaystyle w_{i}={\frac {1}{\pi _{i}}}} we get N ^ = ∑ i = 1 n w i I i = ∑ i = 1 n I i π i = ∑ i = 1 n 1 ˇ i ′ {\displaystyle {\hat {N}}=\sum _{i=1}^{n}w_{i}I_{i}=\sum _{i=1}^{n}{\frac {I_{i}}{\pi _{i}}}=\sum _{i=1}^{n}{\check {1}}'_{i}} . With the above notation, the parameter we care about is the ratio of the sums of y i {\displaystyle y_{i}} s, and 1s. I.e.: R = Y ¯ = ∑ i = 1 N y i π i ∑ i = 1 N 1 π i = ∑ i = 1 N y ˇ i ∑ i = 1 N 1 ˇ i = ∑ i = 1 N w i y i ∑ i = 1 N w i {\displaystyle R={\bar {Y}}={\frac {\sum _{i=1}^{N}{\frac {y_{i}}{\pi _{i}}}}{\sum _{i=1}^{N}{\frac {1}{\pi _{i}}}}}={\frac {\sum _{i=1}^{N}{\check {y}}_{i}}{\sum _{i=1}^{N}{\check {1}}_{i}}}={\frac {\sum _{i=1}^{N}w_{i}y_{i}}{\sum _{i=1}^{N}w_{i}}}} . We can estimate it using our sample with: R ^ = Y ¯ ^ = ∑ i = 1 N I i y i π i ∑ i = 1 N I i 1 π i = ∑ i = 1 N y ˇ i ′ ∑ i = 1 N 1 ˇ i ′ = ∑ i = 1 N w i y i ′ ∑ i = 1 N w i 1 i ′ = ∑ i = 1 n w i y i ′ ∑ i = 1 n w i 1 i ′ = y ¯ w {\displaystyle {\hat {R}}={\hat {\bar {Y}}}={\frac {\sum _{i=1}^{N}I_{i}{\frac {y_{i}}{\pi _{i}}}}{\sum _{i=1}^{N}I_{i}{\frac {1}{\pi _{i}}}}}={\frac {\sum _{i=1}^{N}{\check {y}}'_{i}}{\sum _{i=1}^{N}{\check {1}}'_{i}}}={\frac {\sum _{i=1}^{N}w_{i}y'_{i}}{\sum _{i=1}^{N}w_{i}1'_{i}}}={\frac {\sum _{i=1}^{n}w_{i}y'_{i}}{\sum _{i=1}^{n}w_{i}1'_{i}}}={\bar {y}}_{w}} . As we moved from using N to using n, we actually know that all the indicator variables get 1, so we could simply write: y ¯ w = ∑ i = 1 n w i y i ∑ i = 1 n w i {\displaystyle {\bar {y}}_{w}={\frac {\sum _{i=1}^{n}w_{i}y_{i}}{\sum _{i=1}^{n}w_{i}}}} . This will be the estimand for specific values of y and w, but the statistical properties comes when including the indicator variable y ¯ w = ∑ i = 1 n w i y i ′ ∑ i = 1 n w i 1 i ′ {\displaystyle {\bar {y}}_{w}={\frac {\sum _{i=1}^{n}w_{i}y'_{i}}{\sum _{i=1}^{n}w_{i}1'_{i}}}} .: 162, 163, 176 This is called a Ratio estimator and it is approximately unbiased for R.: 182 In this case, the variability of the ratio depends on the variability of the random variables both in the numerator and the denominator - as well as their correlation. Since there is no closed analytical form to compute this variance, various methods are used for approximate estimation. Primarily Taylor series first-order linearization, asymptotics, and bootstrap/jackknife.: 172 The Taylor linearization method could lead to under-estimation of the variance for small sample sizes in general, but that depends on the complexity of the statistic. For the weighted mean, the approximate variance is supposed to be relatively accurate even for medium sample sizes.: 176 For when the sampling has a random sample size (as in Poisson sampling), it is as follows:: 182 V ( y ¯ w ) ^ = 1 ( ∑ i = 1 n w i ) 2 ∑ i = 1 n w i 2 ( y i − y ¯ w ) 2 {\displaystyle {\widehat {V({\bar {y}}_{w})}}={\frac {1}{(\sum _{i=1}^{n}w_{i})^{2}}}\sum _{i=1}^{n}w_{i}^{2}(y_{i}-{\bar {y}}_{w})^{2}} . If π i ≈ p i n {\displaystyle \pi _{i}\approx p_{i}n} , then either using w i = 1 π i {\displaystyle w_{i}={\frac {1}{\pi _{i}}}} or w i = 1 p i {\displaystyle w_{i}={\frac {1}{p_{i}}}} would give the same estimator, since multiplying w i {\displaystyle w_{i}} by some factor would lead to the same estimator. It also means that if we scale the sum of weights to be equal to a known-from-before population size N, the variance calculation would look the same. When all weights are equal to one another, this formula is reduced to the standard unbiased variance estimator. We have (at least) two versions of variance for the weighted mean: one with known and one with unknown population size estimation. There is no uniformly better approach, but the literature presents several arguments to prefer using the population estimation version (even when the population size is known).: 188 For example: if all y values are constant, the estimator with unknown population size will give the correct result, while the one with known population size will have some variability. Also, when the sample size itself is random (e.g.: in Poisson sampling), the version with unknown population mean is considered more stable. Lastly, if the proportion of sampling is negatively correlated with the values (i.e.: smaller chance to sample an observation that is large), then the un-known population size version slightly compensates for that. For the trivial case in which all the weights are equal to 1, the above formula is just like the regular formula for the variance of the mean (but notice that it uses the maximum likelihood estimator for the variance instead of the unbiased variance. I.e.: dividing it by n instead of (n-1)). ==== Bootstrapping validation ==== It has been shown, by Gatz et al. (1995), that in comparison to bootstrapping methods, the following (variance estimation of ratio-mean using Taylor series linearization) is a reasonable estimation for the square of the standard error of the mean (when used in the context of measuring chemical constituents):: 1186 σ x ¯ w 2 ^ = n ( n − 1 ) ( n w ¯ ) 2 [ ∑ ( w i x i − w ¯ x ¯ w ) 2 − 2 x ¯ w ∑ ( w i − w ¯ ) ( w i x i − w ¯ x ¯ w ) + x ¯ w 2 ∑ ( w i − w ¯ ) 2 ] {\displaystyle {\widehat {\sigma _{{\bar {x}}_{w}}^{2}}}={\frac {n}{(n-1)(n{\bar {w}})^{2}}}\left[\sum (w_{i}x_{i}-{\bar {w}}{\bar {x}}_{w})^{2}-2{\bar {x}}_{w}\sum (w_{i}-{\bar {w}})(w_{i}x_{i}-{\bar {w}}{\bar {x}}_{w})+{\bar {x}}_{w}^{2}\sum (w_{i}-{\bar {w}})^{2}\right]} where w ¯ = ∑ w i n {\displaystyle {\bar {w}}={\frac {\sum w_{i}}{n}}} . Further simplification leads to σ x ¯ 2 ^ = n ( n − 1 ) ( n w ¯ ) 2 ∑ w i 2 ( x i − x ¯ w ) 2 {\displaystyle {\widehat {\sigma _{\bar {x}}^{2}}}={\frac {n}{(n-1)(n{\bar {w}})^{2}}}\sum w_{i}^{2}(x_{i}-{\bar {x}}_{w})^{2}} Gatz et al. mention that the above formulation was published by Endlich et al. (1988) when treating the weighted mean as a combination of a weighted total estimator divided by an estimator of the population size, based on the formulation published by Cochran (1977), as an approximation to the ratio mean. However, Endlich et al. didn't seem to publish this derivation in their paper (even though they mention they used it), and Cochran's book includes a slightly different formulation.: 155 Still, it's almost identical to the formulations described in previous sections. ==== Replication-based estimators ==== Because there is no closed analytical form for the variance of the weighted mean, it was proposed in the literature to rely on replication methods such as the Jackknife and Bootstrapping.: 321 ==== Other notes ==== For uncorrelated observations with variances σ i 2 {\displaystyle \sigma _{i}^{2}} , the variance of the weighted sample mean is σ x ¯ 2 = ∑ i = 1 n w i ′ 2 σ i 2 {\displaystyle \sigma _{\bar {x}}^{2}=\sum _{i=1}^{n}{w_{i}'^{2}\sigma _{i}^{2}}} whose square root σ x ¯ {\displaystyle \sigma _{\bar {x}}} can be called the standard error of the weighted mean (general case). Consequently, if all the observations have equal variance, σ i 2 = σ 0 2 {\displaystyle \sigma _{i}^{2}=\sigma _{0}^{2}} , the weighted sample mean will have variance σ x ¯ 2 = σ 0 2 ∑ i = 1 n w i ′ 2 , {\displaystyle \sigma _{\bar {x}}^{2}=\sigma _{0}^{2}\sum _{i=1}^{n}{w_{i}'^{2}},} where 1 / n ≤ ∑ i = 1 n w i ′ 2 ≤ 1 {\textstyle 1/n\leq \sum _{i=1}^{n}{w_{i}'^{2}}\leq 1} . The variance attains its maximum value, σ 0 2 {\displaystyle \sigma _{0}^{2}} , when all weights except one are zero. Its minimum value is found when all weights are equal (i.e., unweighted mean), in which case we have σ x ¯ = σ 0 / n {\textstyle \sigma _{\bar {x}}=\sigma _{0}/{\sqrt {n}}} , i.e., it degenerates into the standard error of the mean, squared. Because one can always transform non-normalized weights to normalized weights, all formulas in this section can be adapted to non-normalized weights by replacing all w i ′ = w i ∑ i = 1 n w i {\displaystyle w_{i}'={\frac {w_{i}}{\sum _{i=1}^{n}{w_{i}}}}} . == Related concepts == === Weighted sample variance === Typically when a mean is calculated it is important to know the variance and standard deviation about that mean. When a weighted mean μ ∗ {\displaystyle \mu ^{}} is used, the variance of the weighted sample is different from the variance of the unweighted sample. The biased weighted sample variance σ ^ w 2 {\displaystyle {\hat {\sigma }}_{\mathrm {w} }^{2}} is defined similarly to the normal biased sample variance σ ^ 2 {\displaystyle {\hat {\sigma }}^{2}} : σ ^ 2 = ∑ i = 1 N ( x i − μ ) 2 N σ ^ w 2 = ∑ i = 1 N w i ( x i − μ ∗ ) 2 ∑ i = 1 N w i {\displaystyle {\begin{aligned}{\hat {\sigma }}^{2}\ &={\frac {\sum \limits _{i=1}^{N}\left(x_{i}-\mu \right)^{2}}{N}}\\{\hat {\sigma }}_{\mathrm {w} }^{2}&={\frac {\sum \limits _{i=1}^{N}w_{i}\left(x_{i}-\mu ^{}\right)^{2}}{\sum _{i=1}^{N}w_{i}}}\end{aligned}}} where ∑ i = 1 N w i = 1 {\displaystyle \sum _{i=1}^{N}w_{i}=1} for normalized weights. If the weights are frequency weights (and thus are random variables), it can be shown that σ ^ w 2 {\displaystyle {\hat {\sigma }}_{\mathrm {w} }^{2}} is the maximum likelihood estimator of σ 2 {\displaystyle \sigma ^{2}} for iid Gaussian observations. For small samples, it is customary to use an unbiased estimator for the population variance. In normal unweighted samples, the N in the denominator (corresponding to the sample size) is changed to N − 1 (see Bessel's correction). In the weighted setting, there are actually two different unbiased estimators, one for the case of frequency weights and another for the case of reliability weights. ==== Frequency weights ==== If the weights are frequency weights (where a weight equals the number of occurrences), then the unbiased estimator is: s 2 = ∑ i = 1 N w i ( x i − μ ∗ ) 2 ∑ i = 1 N w i − 1 {\displaystyle s^{2}\ ={\frac {\sum \limits _{i=1}^{N}w_{i}\left(x_{i}-\mu ^{}\right)^{2}}{\sum _{i=1}^{N}w_{i}-1}}} This effectively applies Bessel's correction for frequency weights. For example, if values { 2 , 2 , 4 , 5 , 5 , 5 } {\displaystyle \{2,2,4,5,5,5\}} are drawn from the same distribution, then we can treat this set as an unweighted sample, or we can treat it as the weighted sample { 2 , 4 , 5 } {\displaystyle \{2,4,5\}} with corresponding weights { 2 , 1 , 3 } {\displaystyle \{2,1,3\}} , and we get the same result either way. If the frequency weights { w i } {\displaystyle \{w_{i}\}} are normalized to 1, then the correct expression after Bessel's correction becomes s 2 = ∑ i = 1 N w i ∑ i = 1 N w i − 1 ∑ i = 1 N w i ( x i − μ ∗ ) 2 {\displaystyle s^{2}\ ={\frac {\sum _{i=1}^{N}w_{i}}{\sum _{i=1}^{N}w_{i}-1}}\sum _{i=1}^{N}w_{i}\left(x_{i}-\mu ^{}\right)^{2}} where the total number of samples is ∑ i = 1 N w i {\displaystyle \sum _{i=1}^{N}w_{i}} (not N {\displaystyle N} ). In any case, the information on total number of samples is necessary in order to obtain an unbiased correction, even if w i {\displaystyle w_{i}} has a different meaning other than frequency weight. The estimator can be unbiased only if the weights are not standardized nor normalized, these processes changing the data's mean and variance and thus leading to a loss of the base rate (the population count, which is a requirement for Bessel's correction). ==== Reliability weights ==== If the weights are instead reliability weights (non-random values reflecting the sample's relative trustworthiness, often derived from sample variance), we can determine a correction factor to yield an unbiased estimator. Assuming each random variable is sampled from the same distribution with mean μ {\displaystyle \mu } and actual variance σ actual 2 {\displaystyle \sigma _{\text{actual}}^{2}} , taking expectations we have, E ⁡ [ σ ^ 2 ] = ∑ i = 1 N E ⁡ [ ( x i − μ ) 2 ] N = E ⁡ [ ( X − E ⁡ [ X ] ) 2 ] − 1 N E ⁡ [ ( X − E ⁡ [ X ] ) 2 ] = ( N − 1 N ) σ actual 2 E ⁡ [ σ ^ w 2 ] = ∑ i = 1 N w i E ⁡ [ ( x i − μ ∗ ) 2 ] V 1 = E ⁡ [ ( X − E ⁡ [ X ] ) 2 ] − V 2 V 1 2 E ⁡ [ ( X − E ⁡ [ X ] ) 2 ] = ( 1 − V 2 V 1 2 ) σ actual 2 {\displaystyle {\begin{aligned}\operatorname {E} [{\hat {\sigma }}^{2}]&={\frac {\sum \limits _{i=1}^{N}\operatorname {E} [(x_{i}-\mu )^{2}]}{N}}\\&=\operatorname {E} [(X-\operatorname {E} [X])^{2}]-{\frac {1}{N}}\operatorname {E} [(X-\operatorname {E} [X])^{2}]\\&=\left({\frac {N-1}{N}}\right)\sigma _{\text{actual}}^{2}\\\operatorname {E} [{\hat {\sigma }}_{\mathrm {w} }^{2}]&={\frac {\sum \limits _{i=1}^{N}w_{i}\operatorname {E} [(x_{i}-\mu ^{})^{2}]}{V_{1}}}\\&=\operatorname {E} [(X-\operatorname {E} [X])^{2}]-{\frac {V_{2}}{V_{1}^{2}}}\operatorname {E} [(X-\operatorname {E} [X])^{2}]\\&=\left(1-{\frac {V_{2}}{V_{1}^{2}}}\right)\sigma _{\text{actual}}^{2}\end{aligned}}} where V 1 = ∑ i = 1 N w i {\displaystyle V_{1}=\sum _{i=1}^{N}w_{i}} and V 2 = ∑ i = 1 N w i 2 {\displaystyle V_{2}=\sum _{i=1}^{N}w_{i}^{2}} . Therefore, the bias in our estimator is ( 1 − V 2 V 1 2 ) {\displaystyle \left(1-{\frac {V_{2}}{V_{1}^{2}}}\right)} , analogous to the ( N − 1 N ) {\displaystyle \left({\frac {N-1}{N}}\right)} bias in the unweighted estimator (also notice that V 1 2 / V 2 = N e f f {\displaystyle \ V_{1}^{2}/V_{2}=N_{eff}} is the effective sample size). This means that to unbias our estimator we need to pre-divide by 1 − ( V 2 / V 1 2 ) {\displaystyle 1-\left(V_{2}/V_{1}^{2}\right)} , ensuring that the expected value of the estimated variance equals the actual variance of the sampling distribution. The final unbiased estimate of sample variance is: s w 2 = σ ^ w 2 1 − ( V 2 / V 1 2 ) = ∑ i = 1 N w i ( x i − μ ∗ ) 2 V 1 − ( V 2 / V 1 ) , {\displaystyle {\begin{aligned}s_{\mathrm {w} }^{2}\ &={\frac {{\hat {\sigma }}_{\mathrm {w} }^{2}}{1-(V_{2}/V_{1}^{2})}}\\[4pt]&={\frac {\sum \limits _{i=1}^{N}w_{i}(x_{i}-\mu ^{})^{2}}{V_{1}-(V_{2}/V_{1})}},\end{aligned}}} where E ⁡ [ s w 2 ] = σ actual 2 {\displaystyle \operatorname {E} [s_{\mathrm {w} }^{2}]=\sigma _{\text{actual}}^{2}} . The degrees of freedom of this weighted, unbiased sample variance vary accordingly from N − 1 down to 0. The standard deviation is simply the square root of the variance above. As a side note, other approaches have been described to compute the weighted sample variance. === Weighted sample covariance === In a weighted sample, each row vector x i {\displaystyle \mathbf {x} _{i}} (each set of single observations on each of the K random variables) is assigned a weight w i ≥ 0 {\displaystyle w_{i}\geq 0} . Then the weighted mean vector μ ∗ {\displaystyle \mathbf {\mu ^{}} } is given by μ ∗ = ∑ i = 1 N w i x i ∑ i = 1 N w i . {\displaystyle \mathbf {\mu ^{}} ={\frac {\sum _{i=1}^{N}w_{i}\mathbf {x} _{i}}{\sum _{i=1}^{N}w_{i}}}.} And the weighted covariance matrix is given by: C = ∑ i = 1 N w i ( x i − μ ∗ ) T ( x i − μ ∗ ) V 1 . {\displaystyle \mathbf {C} ={\frac {\sum _{i=1}^{N}w_{i}\left(\mathbf {x} _{i}-\mu ^{}\right)^{T}\left(\mathbf {x} _{i}-\mu ^{}\right)}{V_{1}}}.} Similarly to weighted sample variance, there are two different unbiased estimators depending on the type of the weights. ==== Frequency weights ==== If the weights are frequency weights, the unbiased weighted estimate of the covariance matrix C {\displaystyle \textstyle \mathbf {C} } , with Bessel's correction, is given by: C = ∑ i = 1 N w i ( x i − μ ∗ ) T ( x i − μ ∗ ) V 1 − 1 . {\displaystyle \mathbf {C} ={\frac {\sum _{i=1}^{N}w_{i}\left(\mathbf {x} _{i}-\mu ^{}\right)^{T}\left(\mathbf {x} _{i}-\mu ^{}\right)}{V_{1}-1}}.} This estimator can be unbiased only if the weights are not standardized nor normalized, these processes changing the data's mean and variance and thus leading to a loss of the base rate (the population count, which is a requirement for Bessel's correction). ==== Reliability weights ==== In the case of reliability weights, the weights are normalized: V 1 = ∑ i = 1 N w i = 1. {\displaystyle V_{1}=\sum _{i=1}^{N}w_{i}=1.} (If they are not, divide the weights by their sum to normalize prior to calculating V 1 {\displaystyle V_{1}} : w i ′ = w i ∑ i = 1 N w i {\displaystyle w_{i}'={\frac {w_{i}}{\sum _{i=1}^{N}w_{i}}}} Then the weighted mean vector μ ∗ {\displaystyle \mathbf {\mu ^{}} } can be simplified to μ ∗ = ∑ i = 1 N w i x i . {\displaystyle \mathbf {\mu ^{}} =\sum _{i=1}^{N}w_{i}\mathbf {x} _{i}.} and the unbiased weighted estimate of the covariance matrix C {\displaystyle \mathbf {C} } is: C = ∑ i = 1 N w i ( ∑ i = 1 N w i ) 2 − ∑ i = 1 N w i 2 ∑ i = 1 N w i ( x i − μ ∗ ) T ( x i − μ ∗ ) = ∑ i = 1 N w i ( x i − μ ∗ ) T ( x i − μ ∗ ) V 1 − ( V 2 / V 1 ) . {\displaystyle {\begin{aligned}\mathbf {C} &={\frac {\sum _{i=1}^{N}w_{i}}{\left(\sum _{i=1}^{N}w_{i}\right)^{2}-\sum _{i=1}^{N}w_{i}^{2}}}\sum _{i=1}^{N}w_{i}\left(\mathbf {x} _{i}-\mu ^{}\right)^{T}\left(\mathbf {x} _{i}-\mu ^{}\right)\\&={\frac {\sum _{i=1}^{N}w_{i}\left(\mathbf {x} _{i}-\mu ^{}\right)^{T}\left(\mathbf {x} _{i}-\mu ^{}\right)}{V_{1}-(V_{2}/V_{1})}}.\end{aligned}}} The reasoning here is the same as in the previous section. Since we are assuming the weights are normalized, then V 1 = 1 {\displaystyle V_{1}=1} and this reduces to: C = ∑ i = 1 N w i ( x i − μ ∗ ) T ( x i − μ ∗ ) 1 − V 2 . {\displaystyle \mathbf {C} ={\frac {\sum _{i=1}^{N}w_{i}\left(\mathbf {x} _{i}-\mu ^{}\right)^{T}\left(\mathbf {x} _{i}-\mu ^{}\right)}{1-V_{2}}}.} If all weights are the same, i.e. w i / V 1 = 1 / N {\displaystyle w_{i}/V_{1}=1/N} , then the weighted mean and covariance reduce to the unweighted sample mean and covariance above. === Vector-valued estimates === The above generalizes easily to the case of taking the mean of vector-valued estimates. For example, estimates of position on a plane may have less certainty in one direction than another. As in the scalar case, the weighted mean of multiple estimates can provide a maximum likelihood estimate. We simply replace the variance σ 2 {\displaystyle \sigma ^{2}} by the covariance matrix C {\displaystyle \mathbf {C} } and the arithmetic inverse by the matrix inverse (both denoted in the same way, via superscripts); the weight matrix then reads: W i = C i − 1 . {\displaystyle \mathbf {W} _{i}=\mathbf {C} _{i}^{-1}.} The weighted mean in this case is: x ¯ = C x ¯ ( ∑ i = 1 n W i x i ) , {\displaystyle {\bar {\mathbf {x} }}=\mathbf {C} _{\bar {\mathbf {x} }}\left(\sum _{i=1}^{n}\mathbf {W} _{i}\mathbf {x} _{i}\right),} (where the order of the matrix–vector product is not commutative), in terms of the covariance of the weighted mean: C x ¯ = ( ∑ i = 1 n W i ) − 1 , {\displaystyle \mathbf {C} _{\bar {\mathbf {x} }}=\left(\sum _{i=1}^{n}\mathbf {W} _{i}\right)^{-1},} For example, consider the weighted mean of the point [1 0] with high variance in the second component and [0 1] with high variance in the first component. Then x 1 := [ 1 0 ] ⊤ , C 1 := [ 1 0 0 100 ] {\displaystyle \mathbf {x} _{1}:={\begin{bmatrix}1&0\end{bmatrix}}^{\top },\qquad \mathbf {C} _{1}:={\begin{bmatrix}1&0\\0&100\end{bmatrix}}} x 2 := [ 0 1 ] ⊤ , C 2 := [ 100 0 0 1 ] {\displaystyle \mathbf {x} _{2}:={\begin{bmatrix}0&1\end{bmatrix}}^{\top },\qquad \mathbf {C} _{2}:={\begin{bmatrix}100&0\\0&1\end{bmatrix}}} then the weighted mean is: x ¯ = ( C 1 − 1 + C 2 − 1 ) − 1 ( C 1 − 1 x 1 + C 2 − 1 x 2 ) = [ 0.9901 0 0 0.9901 ] [ 1 1 ] = [ 0.9901 0.9901 ] {\displaystyle {\begin{aligned}{\bar {\mathbf {x} }}&=\left(\mathbf {C} _{1}^{-1}+\mathbf {C} _{2}^{-1}\right)^{-1}\left(\mathbf {C} _{1}^{-1}\mathbf {x} _{1}+\mathbf {C} _{2}^{-1}\mathbf {x} _{2}\right)\\[5pt]&={\begin{bmatrix}0.9901&0\\0&0.9901\end{bmatrix}}{\begin{bmatrix}1\\1\end{bmatrix}}={\begin{bmatrix}0.9901\\0.9901\end{bmatrix}}\end{aligned}}} which makes sense: the [1 0] estimate is "compliant" in the second component and the [0 1] estimate is compliant in the first component, so the weighted mean is nearly [1 1]. === Accounting for correlations === In the general case, suppose that X = [ x 1 , … , x n ] T {\displaystyle \mathbf {X} =[x_{1},\dots ,x_{n}]^{T}} , C {\displaystyle \mathbf {C} } is the covariance matrix relating the quantities x i {\displaystyle x_{i}} , x ¯ {\displaystyle {\bar {x}}} is the common mean to be estimated, and J {\displaystyle \mathbf {J} } is a design matrix equal to a vector of ones [ 1 , … , 1 ] T {\displaystyle [1,\dots ,1]^{T}} (of length n {\displaystyle n} ). The Gauss–Markov theorem states that the estimate of the mean having minimum variance is given by: σ x ¯ 2 = ( J T W J ) − 1 , {\displaystyle \sigma _{\bar {x}}^{2}=(\mathbf {J} ^{T}\mathbf {W} \mathbf {J} )^{-1},} and x ¯ = σ x ¯ 2 ( J T W X ) , {\displaystyle {\bar {x}}=\sigma _{\bar {x}}^{2}(\mathbf {J} ^{T}\mathbf {W} \mathbf {X} ),} where: W = C − 1 . {\displaystyle \mathbf {W} =\mathbf {C} ^{-1}.} === Decreasing strength of interactions === Consider the time series of an independent variable x {\displaystyle x} and a dependent variable y {\displaystyle y} , with n {\displaystyle n} observations sampled at discrete times t i {\displaystyle t_{i}} . In many common situations, the value of y {\displaystyle y} at time t i {\displaystyle t_{i}} depends not only on x i {\displaystyle x_{i}} but also on its past values. Commonly, the strength of this dependence decreases as the separation of observations in time increases. To model this situation, one may replace the independent variable by its sliding mean z {\displaystyle z} for a window size m {\displaystyle m} . z k = ∑ i = 1 m w i x k + 1 − i . {\displaystyle z_{k}=\sum _{i=1}^{m}w_{i}x_{k+1-i}.} === Exponentially decreasing weights === In the scenario described in the previous section, most frequently the decrease in interaction strength obeys a negative exponential law. If the observations are sampled at equidistant times, then exponential decrease is equivalent to decrease by a constant fraction 0 < Δ < 1 {\displaystyle 0<\Delta <1} at each time step. Setting w = 1 − Δ {\displaystyle w=1-\Delta } we can define m {\displaystyle m} normalized weights by w i = w i − 1 V 1 , {\displaystyle w_{i}={\frac {w^{i-1}}{V_{1}}},} where V 1 {\displaystyle V_{1}} is the sum of the unnormalized weights. In this case V 1 {\displaystyle V_{1}} is simply V 1 = ∑ i = 1 m w i − 1 = 1 − w m 1 − w , {\displaystyle V_{1}=\sum _{i=1}^{m}{w^{i-1}}={\frac {1-w^{m}}{1-w}},} approaching V 1 = 1 / ( 1 − w ) {\displaystyle V_{1}=1/(1-w)} for large values of m {\displaystyle m} . The damping constant w {\displaystyle w} must correspond to the actual decrease of interaction strength. If this cannot be determined from theoretical considerations, then the following properties of exponentially decreasing weights are useful in making a suitable choice: at step ( 1 − w ) − 1 {\displaystyle (1-w)^{-1}} , the weight approximately equals e − 1 ( 1 − w ) = 0.39 ( 1 − w ) {\displaystyle {e^{-1}}(1-w)=0.39(1-w)} , the tail area the value e − 1 {\displaystyle e^{-1}} , the head area 1 − e − 1 = 0.61 {\displaystyle {1-e^{-1}}=0.61} . The tail area at step n {\displaystyle n} is ≤ e − n ( 1 − w ) {\displaystyle \leq {e^{-n(1-w)}}} . Where primarily the closest n {\displaystyle n} observations matter and the effect of the remaining observations can be ignored safely, then choose w {\displaystyle w} such that the tail area is sufficiently small. === Weighted averages of functions === The concept of weighted average can be extended to functions. Weighted averages of functions play an important role in the systems of weighted differential and integral calculus. === Correcting for over- or under-dispersion === Weighted means are typically used to find the weighted mean of historical data, rather than theoretically generated data. In this case, there will be some error in the variance of each data point. Typically experimental errors may be underestimated due to the experimenter not taking into account all sources of error in calculating the variance of each data point. In this event, the variance in the weighted mean must be corrected to account for the fact that χ 2 {\displaystyle \chi ^{2}} is too large. The correction that must be made is σ ^ x ¯ 2 = σ x ¯ 2 χ ν 2 {\displaystyle {\hat {\sigma }}_{\bar {x}}^{2}=\sigma _{\bar {x}}^{2}\chi _{\nu }^{2}} where χ ν 2 {\displaystyle \chi _{\nu }^{2}} is the reduced chi-squared: χ ν 2 = 1 ( n − 1 ) ∑ i = 1 n ( x i − x ¯ ) 2 σ i 2 ; {\displaystyle \chi _{\nu }^{2}={\frac {1}{(n-1)}}\sum _{i=1}^{n}{\frac {(x_{i}-{\bar {x}})^{2}}{\sigma _{i}^{2}}};} The square root σ ^ x ¯ {\displaystyle {\hat {\sigma }}_{\bar {x}}} can be called the standard error of the weighted mean (variance weights, scale corrected). When all data variances are equal, σ i = σ 0 {\displaystyle \sigma _{i}=\sigma _{0}} , they cancel out in the weighted mean variance, σ x ¯ 2 {\displaystyle \sigma _{\bar {x}}^{2}} , which again reduces to the standard error of the mean (squared), σ x ¯ 2 = σ 2 / n {\displaystyle \sigma _{\bar {x}}^{2}=\sigma ^{2}/n} , formulated in terms of the sample standard deviation (squared), σ 2 = ∑ i = 1 n ( x i − x ¯ ) 2 n − 1 . {\displaystyle \sigma ^{2}={\frac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}{n-1}}.} == See also == == Notes == == References == == Further reading == Bevington, Philip R (1969). Data Reduction and Error Analysis for the Physical Sciences. New York, N.Y.: McGraw-Hill. OCLC 300283069. Strutz, T. (2010). Data Fitting and Uncertainty (A practical introduction to weighted least squares and beyond). Vieweg+Teubner. ISBN 978-3-8348-1022-9. == External links == David Terr. "Weighted Mean". MathWorld. Tool to calculate Weighted Average
Wikipedia:Weighted geometric mean#0	In statistics, the weighted geometric mean is a generalization of the geometric mean using the weighted arithmetic mean. Given a sample x = ( x 1 , x 2 … , x n ) {\displaystyle x=(x_{1},x_{2}\dots ,x_{n})} and weights w = ( w 1 , w 2 , … , w n ) {\displaystyle w=(w_{1},w_{2},\dots ,w_{n})} , it is calculated as: x ¯ = ( ∏ i = 1 n x i w i ) 1 / ∑ i = 1 n w i = exp ⁡ ( ∑ i = 1 n w i ln ⁡ x i ∑ i = 1 n w i ) {\displaystyle {\bar {x}}=\left(\prod _{i=1}^{n}x_{i}^{w_{i}}\right)^{1/\sum _{i=1}^{n}w_{i}}=\quad \exp \left({\frac {\sum _{i=1}^{n}w_{i}\ln x_{i}}{\sum _{i=1}^{n}w_{i}\quad }}\right)} The second form above illustrates that the logarithm of the geometric mean is the weighted arithmetic mean of the logarithms of the individual values. If all the weights are equal, the weighted geometric mean simplifies to the ordinary unweighted geometric mean. == References == == See also == Average Central tendency Summary statistics Weighted arithmetic mean Weighted harmonic mean == External links == Non-Newtonian calculus website
Wikipedia:Weil algebra#0	The term "Weil algebra" is also sometimes used to mean a finite-dimensional real local Artinian ring. In mathematics, the Weil algebra of a Lie algebra g, introduced by Cartan (1951) based on unpublished work of André Weil, is a differential graded algebra given by the Koszul algebra Λ(g)⊗S(g) of its dual g*. == References == Cartan, Henri (1951), "Notions d'algèbre différentielle; application aux groupes de Lie et aux variétés où opère un groupe de Lie", Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège, pp. 15–27, MR 0042426 Reprinted in (Guillemin & Sternberg 1999) Guillemin, Victor W.; Sternberg, Shlomo (1999), Supersymmetry and equivariant de Rham theory, Mathematics Past and Present, Berlin, New York: Springer-Verlag, ISBN 978-3-540-64797-3, MR 1689252
Wikipedia:Weil conjectures#0	In mathematics, the Weil conjectures were highly influential proposals by André Weil (1949). They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory. The conjectures concern the generating functions (known as local zeta functions) derived from counting points on algebraic varieties over finite fields. A variety V over a finite field with q elements has a finite number of rational points (with coordinates in the original field), as well as points with coordinates in any finite extension of the original field. The generating function has coefficients derived from the numbers Nk of points over the extension field with qk elements. Weil conjectured that such zeta functions for smooth varieties are rational functions, satisfy a certain functional equation, and have their zeros in restricted places. The last two parts were consciously modelled on the Riemann zeta function, a kind of generating function for prime integers, which obeys a functional equation and (conjecturally) has its zeros restricted by the Riemann hypothesis. The rationality was proved by Bernard Dwork (1960), the functional equation by Alexander Grothendieck (1965), and the analogue of the Riemann hypothesis by Pierre Deligne (1974). == Background and history == The earliest antecedent of the Weil conjectures is by Carl Friedrich Gauss and appears in section VII of his Disquisitiones Arithmeticae (Mazur 1974), concerned with roots of unity and Gaussian periods. In article 358, he moves on from the periods that build up towers of quadratic extensions, for the construction of regular polygons; and assumes that p is a prime number congruent to 1 modulo 3. Then there is a cyclic cubic field inside the cyclotomic field of pth roots of unity, and a normal integral basis of periods for the integers of this field (an instance of the Hilbert–Speiser theorem). Gauss constructs the order-3 periods, corresponding to the cyclic group (Z/pZ)× of non-zero residues modulo p under multiplication and its unique subgroup of index three. Gauss lets R {\displaystyle {\mathfrak {R}}} , R ′ {\displaystyle {\mathfrak {R}}'} , and R ″ {\displaystyle {\mathfrak {R}}''} be its cosets. Taking the periods (sums of roots of unity) corresponding to these cosets applied to exp(2πi/p), he notes that these periods have a multiplication table that is accessible to calculation. Products are linear combinations of the periods, and he determines the coefficients. He sets, for example, ( R R ) {\displaystyle ({\mathfrak {R}}{\mathfrak {R}})} equal to the number of elements of Z/pZ which are in R {\displaystyle {\mathfrak {R}}} and which, after being increased by one, are also in R {\displaystyle {\mathfrak {R}}} . He proves that this number and related ones are the coefficients of the products of the periods. To see the relation of these sets to the Weil conjectures, notice that if α and α + 1 are both in R {\displaystyle {\mathfrak {R}}} , then there exist x and y in Z/pZ such that x3 = α and y3 = α + 1; consequently, x3 + 1 = y3. Therefore ( R R ) {\displaystyle ({\mathfrak {R}}{\mathfrak {R}})} is related to the number of solutions to x3 + 1 = y3 in the finite field Z/pZ. The other coefficients have similar interpretations. Gauss's determination of the coefficients of the products of the periods therefore counts the number of points on these elliptic curves, and as a byproduct he proves the analog of the Riemann hypothesis. The Weil conjectures in the special case of algebraic curves were conjectured by Emil Artin (1924). The case of curves over finite fields was proved by Weil, finishing the project started by Hasse's theorem on elliptic curves over finite fields. Their interest was obvious enough from within number theory: they implied upper bounds for exponential sums, a basic concern in analytic number theory (Moreno 2001). What was really eye-catching, from the point of view of other mathematical areas, was the proposed connection with algebraic topology. Given that finite fields are discrete in nature, and topology speaks only about the continuous, the detailed formulation of Weil (based on working out some examples) was striking and novel. It suggested that geometry over finite fields should fit into well-known patterns relating to Betti numbers, the Lefschetz fixed-point theorem and so on. The analogy with topology suggested that a new homological theory be set up applying within algebraic geometry. This took two decades (it was a central aim of the work and school of Alexander Grothendieck) building up on initial suggestions from Serre. The rationality part of the conjectures was proved first by Bernard Dwork (1960), using p-adic methods. Grothendieck (1965) and his collaborators established the rationality conjecture, the functional equation and the link to Betti numbers by using the properties of étale cohomology, a new cohomology theory developed by Grothendieck and Michael Artin for attacking the Weil conjectures, as outlined in Grothendieck (1960). Of the four conjectures, the analogue of the Riemann hypothesis was the hardest to prove. Motivated by the proof of Serre (1960) of an analogue of the Weil conjectures for Kähler manifolds, Grothendieck envisioned a proof based on his standard conjectures on algebraic cycles (Kleiman 1968). However, Grothendieck's standard conjectures remain open (except for the hard Lefschetz theorem, which was proved by Deligne by extending his work on the Weil conjectures), and the analogue of the Riemann hypothesis was proved by Deligne (1974), using the étale cohomology theory but circumventing the use of standard conjectures by an ingenious argument. Deligne (1980) found and proved a generalization of the Weil conjectures, bounding the weights of the pushforward of a sheaf. == Statement of the Weil conjectures == Suppose that X is a non-singular n-dimensional projective algebraic variety over the field Fq with q elements. The zeta function ζ(X, s) of X is by definition ζ ( X , s ) = exp ⁡ ( ∑ m = 1 ∞ N m m q − m s ) {\displaystyle \zeta (X,s)=\exp \left(\sum _{m=1}^{\infty }{\frac {N_{m}}{m}}q^{-ms}\right)} where Nm is the number of points of X defined over the degree m extension Fqm of Fq. The Weil conjectures state: 1. (Rationality) ζ(X, s) is a rational function of T = q−s. More precisely, ζ(X, s) can be written as a finite alternating product ∏ i = 0 2 n P i ( q − s ) ( − 1 ) i + 1 = P 1 ( T ) ⋯ P 2 n − 1 ( T ) P 0 ( T ) ⋯ P 2 n ( T ) , {\displaystyle \prod _{i=0}^{2n}P_{i}(q^{-s})^{(-1)^{i+1}}={\frac {P_{1}(T)\dotsb P_{2n-1}(T)}{P_{0}(T)\dotsb P_{2n}(T)}},} where each Pi(T) is an integral polynomial. Furthermore, P0(T) = 1 − T, P2n(T) = 1 − qnT, and for 1 ≤ i ≤ 2n − 1, Pi(T) factors over C as ∏ j ( 1 − α i j T ) {\displaystyle \textstyle \prod _{j}(1-\alpha _{ij}T)} for some numbers αij. 2. (Functional equation and Poincaré duality) The zeta function satisfies ζ ( X , n − s ) = ± q n E / 2 − E s ζ ( X , s ) {\displaystyle \zeta (X,n-s)=\pm q^{nE/2-Es}\zeta (X,s)} or equivalently ζ ( X , q − n T − 1 ) = ± q n E / 2 T E ζ ( X , T ) {\displaystyle \zeta (X,q^{-n}T^{-1})=\pm q^{nE/2}T^{E}\zeta (X,T)} where E is the Euler characteristic of X. In particular, for each i, the numbers α2n−i,1, α2n−i,2, ... equal the numbers qn/αi,1, qn/αi,2, ... in some order. 3. (Riemann hypothesis) \|αi,j\| = qi/2 for all 1 ≤ i ≤ 2n − 1 and all j. This implies that all zeros of Pk(T) lie on the "critical line" of complex numbers s with real part k/2. 4. (Betti numbers) If X is a (good) "reduction mod p" of a non-singular projective variety Y defined over a number field embedded in the field of complex numbers, then the degree of Pi is the ith Betti number of the space of complex points of Y. == Examples == === The projective line === The simplest example (other than a point) is to take X to be the projective line. The number of points of X over a field with qm elements is just Nm = qm + 1 (where the "+ 1" comes from the "point at infinity"). The zeta function is just 1 ( 1 − q − s ) ( 1 − q 1 − s ) . {\displaystyle {\frac {1}{(1-q^{-s})(1-q^{1-s})}}.} It is easy to check all parts of the Weil conjectures directly. For example, the corresponding complex variety is the Riemann sphere and its initial Betti numbers are 1, 0, 1. === Projective space === It is not much harder to do n-dimensional projective space. The number of points of X over a field with qm elements is just Nm = 1 + qm + q2m + ⋯ + qnm. The zeta function is just 1 ( 1 − q − s ) ( 1 − q 1 − s ) … ( 1 − q n − s ) . {\displaystyle {\frac {1}{(1-q^{-s})(1-q^{1-s})\dots (1-q^{n-s})}}.} It is again easy to check all parts of the Weil conjectures directly. (Complex projective space gives the relevant Betti numbers, which nearly determine the answer.) The number of points on the projective line and projective space are so easy to calculate because they can be written as disjoint unions of a finite number of copies of affine spaces. It is also easy to prove the Weil conjectures for other spaces, such as Grassmannians and flag varieties, which have the same "paving" property. === Elliptic curves === These give the first non-trivial cases of the Weil conjectures (proved by Hasse). If E is an elliptic curve over a finite field with q elements, then the number of points of E defined over the field with qm elements is 1 − αm − βm + qm, where α and β are complex conjugates with absolute value √q. The zeta function is ( 1 − α q − s ) ( 1 − β q − s ) ( 1 − q − s ) ( 1 − q 1 − s ) . {\displaystyle {\frac {(1-\alpha q^{-s})(1-\beta q^{-s})}{(1-q^{-s})(1-q^{1-s})}}.} The Betti numbers are given by the torus, 1,2,1, and the numerator is a quadratic. === Hyperelliptic curves === As an example, consider the hyperelliptic curve C : y 2 + y = x 5 , {\displaystyle C:y^{2}+y=x^{5},} which is of genus g = 2 {\displaystyle g=2} and dimension n = 1 {\displaystyle n=1} . At first viewed as a curve C / Q {\displaystyle C/\mathbb {Q} } defined over the rational numbers Q {\displaystyle \mathbb {Q} } , this curve has good reduction at all primes 5 ≠ q ∈ P {\displaystyle 5\neq q\in \mathbb {P} } . So, after reduction modulo q ≠ 5 {\displaystyle q\neq 5} , one obtains a hyperelliptic curve C / F q : y 2 + h ( x ) y = f ( x ) {\displaystyle C/{\bf {F}}_{q}:y^{2}+h(x)y=f(x)} of genus 2, with h ( x ) = 1 , f ( x ) = x 5 ∈ F q [ x ] {\displaystyle h(x)=1,f(x)=x^{5}\in {\bf {F}}_{q}[x]} . Taking q = 41 {\displaystyle q=41} as an example, the Weil polynomials P i ( T ) {\displaystyle P_{i}(T)} , i = 0 , 1 , 2 , {\displaystyle i=0,1,2,} and the zeta function of C / F 41 {\displaystyle C/{\bf {F}}_{41}} assume the form ζ ( C / F 41 , s ) = P 1 ( T ) P 0 ( T ) ⋅ P 2 ( T ) = 1 − 9 ⋅ T + 71 ⋅ T 2 − 9 ⋅ 41 ⋅ T 3 + 41 2 ⋅ T 4 ( 1 − T ) ( 1 − 41 ⋅ T ) . {\displaystyle \zeta (C/{\bf {F}}_{41},s)={\frac {P_{1}(T)}{P_{0}(T)\cdot P_{2}(T)}}={\frac {1-9\cdot T+71\cdot T^{2}-9\cdot 41\cdot T^{3}+41^{2}\cdot T^{4}}{(1-T)(1-41\cdot T)}}.} The values c 1 = − 9 {\displaystyle c_{1}=-9} and c 2 = 71 {\displaystyle c_{2}=71} can be determined by counting the numbers of solutions ( x , y ) {\displaystyle (x,y)} of y 2 + y = x 5 {\displaystyle y^{2}+y=x^{5}} over F 41 {\displaystyle {\bf {F}}_{41}} and F 41 2 {\displaystyle {\bf {F}}_{41^{2}}} , respectively, and adding 1 to each of these two numbers to allow for the point at infinity ∞ {\displaystyle \infty } . This counting yields N 1 = 33 {\displaystyle N_{1}=33} and N 2 = 1743 {\displaystyle N_{2}=1743} . It follows: c 1 = N 1 − 1 − q = 33 − 1 − 41 = − 9 {\displaystyle c_{1}=N_{1}-1-q=33-1-41=-9} and c 2 = ( N 2 − 1 − q 2 + c 1 2 ) / 2 = ( 1743 − 1 − 41 2 + ( − 9 ) 2 ) / 2 = 71. {\displaystyle c_{2}=(N_{2}-1-q^{2}+c_{1}^{2})/2=(1743-1-41^{2}+(-9)^{2})/2=71.} The zeros of P 1 ( T ) {\displaystyle P_{1}(T)} are z 1 := 0.12305 + − 1 ⋅ 0.09617 {\displaystyle z_{1}:=0.12305+{\sqrt {-1}}\cdot 0.09617} and z 2 := − 0.01329 + − 1 ⋅ 0.15560 {\displaystyle z_{2}:=-0.01329+{\sqrt {-1}}\cdot 0.15560} (the decimal expansions of these real and imaginary parts are cut off after the fifth decimal place) together with their complex conjugates z 3 := z ¯ 1 {\displaystyle z_{3}:={\bar {z}}_{1}} and z 4 := z ¯ 2 {\displaystyle z_{4}:={\bar {z}}_{2}} . So, in the factorisation P 1 ( T ) = ∏ j = 1 4 ( 1 − α 1 , j T ) {\displaystyle P_{1}(T)=\prod _{j=1}^{4}(1-\alpha _{1,j}T)} , we have α 1 , j = 1 / z j {\displaystyle \alpha _{1,j}=1/z_{j}} . As stated in the third part (Riemann hypothesis) of the Weil conjectures, \| α 1 , j \| = 41 {\displaystyle \|\alpha _{1,j}\|={\sqrt {41}}} for j = 1 , 2 , 3 , 4 {\displaystyle j=1,2,3,4} . The non-singular, projective, complex manifold that belongs to C / Q {\displaystyle C/\mathbb {Q} } has the Betti numbers B 0 = 1 , B 1 = 2 g = 4 , B 2 = 1 {\displaystyle B_{0}=1,B_{1}=2g=4,B_{2}=1} . As described in part four of the Weil conjectures, the (topologically defined!) Betti numbers coincide with the degrees of the Weil polynomials P i ( T ) {\displaystyle P_{i}(T)} , for all primes q ≠ 5 {\displaystyle q\neq 5} : d e g ( P i ) = B i , i = 0 , 1 , 2 {\displaystyle {\rm {deg}}(P_{i})=B_{i},\,i=0,1,2} . === Abelian surfaces === An Abelian surface is a two-dimensional Abelian variety. This is, they are projective varieties that also have the structure of a group, in a way that is compatible with the group composition and taking inverses. Elliptic curves represent one-dimensional Abelian varieties. As an example of an Abelian surface defined over a finite field, consider the Jacobian variety X := Jac ( C / F 41 ) {\displaystyle X:={\text{Jac}}(C/{\bf {F}}_{41})} of the genus 2 curve C / F 41 : y 2 + y = x 5 , {\displaystyle C/{\bf {F}}_{41}:y^{2}+y=x^{5},} which was introduced in the section on hyperelliptic curves. The dimension of X {\displaystyle X} equals the genus of C {\displaystyle C} , so n = 2 {\displaystyle n=2} . There are algebraic integers α 1 , … , α 4 {\displaystyle \alpha _{1},\ldots ,\alpha _{4}} such that the polynomial P ( x ) = ∏ j = 1 4 ( x − α j ) {\displaystyle P(x)=\prod _{j=1}^{4}(x-\alpha _{j})} has coefficients in Z {\displaystyle \mathbb {Z} } ; M m := \| Jac ( C / F 41 m ) \| = ∏ j = 1 4 ( 1 − α j m ) {\displaystyle M_{m}:=\|{\text{Jac}}(C/{\bf {F}}_{41^{m}})\|=\prod _{j=1}^{4}(1-\alpha _{j}^{m})} for all m ∈ N {\displaystyle m\in \mathbb {N} } ; and \| α j \| = 41 {\displaystyle \|\alpha _{j}\|={\sqrt {41}}} for j = 1 , … , 4 {\displaystyle j=1,\ldots ,4} . The zeta-function of X {\displaystyle X} is given by ζ ( X , s ) = ∏ i = 0 4 P i ( q − s ) ( − 1 ) i + 1 = P 1 ( T ) ⋅ P 3 ( T ) P 0 ( T ) ⋅ P 2 ( T ) ⋅ P 4 ( T ) , {\displaystyle \zeta (X,s)=\prod _{i=0}^{4}P_{i}(q^{-s})^{(-1)^{i+1}}={\frac {P_{1}(T)\cdot P_{3}(T)}{P_{0}(T)\cdot P_{2}(T)\cdot P_{4}(T)}},} where q = 41 {\displaystyle q=41} , T = q − s = d e f exp ( − s ⋅ log ( 41 ) ) {\displaystyle T=q^{-s}\,{\stackrel {\rm {def}}{=}}\,{\text{exp}}(-s\cdot {\text{log}}(41))} , and s {\displaystyle s} represents the complex variable of the zeta-function. The Weil polynomials P i ( T ) {\displaystyle P_{i}(T)} have the following specific form (Kahn 2020): P i ( T ) = ∏ 1 ≤ j 1 < j 2 < … < j i − 1 < j i ≤ 4 ( 1 − α j 1 ⋅ … ⋅ α j i T ) {\displaystyle P_{i}(T)=\prod _{1\leq j_{1}<j_{2}<\ldots <j_{i-1}<j_{i}\leq 4}(1-\alpha _{j_{1}}\cdot \ldots \cdot \alpha _{j_{i}}T)} for i = 0 , 1 , … , 4 {\displaystyle i=0,1,\ldots ,4} , and P 1 ( T ) = ∏ j = 1 4 ( 1 − α j T ) = 1 − 9 ⋅ T + 71 ⋅ T 2 − 9 ⋅ 41 ⋅ T 3 + 41 2 ⋅ T 4 {\displaystyle P_{1}(T)=\prod _{j=1}^{4}(1-\alpha _{j}T)=1-9\cdot T+71\cdot T^{2}-9\cdot 41\cdot T^{3}+41^{2}\cdot T^{4}} is the same for the curve C {\displaystyle C} (see section above) and its Jacobian variety X {\displaystyle X} . This is, the inverse roots of P i ( T ) {\displaystyle P_{i}(T)} are the products α j 1 ⋅ … ⋅ α j i {\displaystyle \alpha _{j_{1}}\cdot \ldots \cdot \alpha _{j_{i}}} that consist of i {\displaystyle i} many, different inverse roots of P 1 ( T ) {\displaystyle P_{1}(T)} . Hence, all coefficients of the polynomials P i ( T ) {\displaystyle P_{i}(T)} can be expressed as polynomial functions of the parameters c 1 = − 9 {\displaystyle c_{1}=-9} , c 2 = 71 {\displaystyle c_{2}=71} and q = 41 {\displaystyle q=41} appearing in P 1 ( T ) = 1 + c 1 T + c 2 T 2 + q c 1 T 3 + q 2 T 4 . {\displaystyle P_{1}(T)=1+c_{1}T+c_{2}T^{2}+qc_{1}T^{3}+q^{2}T^{4}.} Calculating these polynomial functions for the coefficients of the P i ( T ) {\displaystyle P_{i}(T)} shows that P 0 ( T ) = 1 − T P 1 ( T ) = 1 − 3 2 ⋅ T + 71 ⋅ T 2 − 3 2 ⋅ 41 ⋅ T 3 + 41 2 ⋅ T 4 P 2 ( T ) = ( 1 − 41 ⋅ T ) 2 ⋅ ( 1 + 11 ⋅ T + 3 ⋅ 7 ⋅ 41 ⋅ T 2 + 11 ⋅ 41 2 ⋅ T 3 + 41 4 ⋅ T 4 ) P 3 ( T ) = 1 − 3 2 ⋅ 41 ⋅ T + 71 ⋅ 41 2 ⋅ T 2 − 3 2 ⋅ 41 4 ⋅ T 3 + 41 6 ⋅ T 4 P 4 ( T ) = 1 − 41 2 ⋅ T {\displaystyle {\begin{alignedat}{2}P_{0}(T)&=1-T\\P_{1}(T)&=1-3^{2}\cdot T+71\cdot T^{2}-3^{2}\cdot 41\cdot T^{3}+41^{2}\cdot T^{4}\\P_{2}(T)&=(1-41\cdot T)^{2}\cdot (1+11\cdot T+3\cdot 7\cdot 41\cdot T^{2}+11\cdot 41^{2}\cdot T^{3}+41^{4}\cdot T^{4})\\P_{3}(T)&=1-3^{2}\cdot 41\cdot T+71\cdot 41^{2}\cdot T^{2}-3^{2}\cdot 41^{4}\cdot T^{3}+41^{6}\cdot T^{4}\\P_{4}(T)&=1-41^{2}\cdot T\end{alignedat}}} Polynomial P 1 {\displaystyle P_{1}} allows for calculating the numbers of elements of the Jacobian variety Jac ( C ) {\displaystyle {\text{Jac}}(C)} over the finite field F 41 {\displaystyle {\bf {F}}_{41}} and its field extension F 41 2 {\displaystyle {\bf {F}}_{41^{2}}} : M 1 = d e f \| Jac ( C / F 41 ) \| = P 1 ( 1 ) = ∏ j = 1 4 [ 1 − α j T ] T = 1 = [ 1 − 9 ⋅ T + 71 ⋅ T 2 − 9 ⋅ 41 ⋅ T 3 + 41 2 ⋅ T 4 ] T = 1 = 1 − 9 + 71 − 9 ⋅ 41 + 41 2 = 1375 = 5 3 ⋅ 11 , and M 2 = d e f \| Jac ( C / F 41 2 ) \| = ∏ j = 1 4 [ 1 − α j 2 T ] T = 1 = [ 1 + 61 ⋅ T + 3 ⋅ 587 ⋅ T 2 + 61 ⋅ 41 2 ⋅ T 3 + 41 4 ⋅ T 4 ] T = 1 = 2930125 = 5 3 ⋅ 11 ⋅ 2131. {\displaystyle {\begin{alignedat}{2}M_{1}&\;{\overset {\underset {\mathrm {def} }{}}{=}}\;\|{\text{Jac}}(C/{\bf {F}}_{41})\|=P_{1}(1)=\prod _{j=1}^{4}[1-\alpha _{j}T]_{T=1}\\&=[1-9\cdot T+71\cdot T^{2}-9\cdot 41\cdot T^{3}+41^{2}\cdot T^{4}]_{T=1}=1-9+71-9\cdot 41+41^{2}=1375=5^{3}\cdot 11{\text{, and}}\\M_{2}&\;{\overset {\underset {\mathrm {def} }{}}{=}}\;\|{\text{Jac}}(C/{\bf {F}}_{41^{2}})\|=\prod _{j=1}^{4}[1-\alpha _{j}^{2}T]_{T=1}\\&=[1+61\cdot T+3\cdot 587\cdot T^{2}+61\cdot 41^{2}\cdot T^{3}+41^{4}\cdot T^{4}]_{T=1}=2930125=5^{3}\cdot 11\cdot 2131.\end{alignedat}}} The inverses α i , j {\displaystyle \alpha _{i,j}} of the zeros of P i ( T ) {\displaystyle P_{i}(T)} do have the expected absolute value of 41 i / 2 {\displaystyle 41^{i/2}} (Riemann hypothesis). Moreover, the maps α i , j ⟼ 41 2 / α i , j , {\displaystyle \alpha _{i,j}\longmapsto 41^{2}/\alpha _{i,j},} j = 1 , … , deg ⁡ P i , {\displaystyle j=1,\ldots ,\deg P_{i},} correlate the inverses of the zeros of P i ( T ) {\displaystyle P_{i}(T)} and the inverses of the zeros of P 4 − i ( T ) {\displaystyle P_{4-i}(T)} . A non-singular, complex, projective, algebraic variety Y {\displaystyle Y} with good reduction at the prime 41 to X = Jac ( C / F 41 ) {\displaystyle X={\text{Jac}}(C/{\bf {F}}_{41})} must have Betti numbers B 0 = B 4 = 1 , B 1 = B 3 = 4 , B 2 = 6 {\displaystyle B_{0}=B_{4}=1,B_{1}=B_{3}=4,B_{2}=6} , since these are the degrees of the polynomials P i ( T ) . {\displaystyle P_{i}(T).} The Euler characteristic E {\displaystyle E} of X {\displaystyle X} is given by the alternating sum of these degrees/Betti numbers: E = 1 − 4 + 6 − 4 + 1 = 0 {\displaystyle E=1-4+6-4+1=0} . By taking the logarithm of ζ ( Jac ( C / F 41 ) , s ) = exp ⁡ ( ∑ m = 1 ∞ M m m ( 41 − s ) m ) = ∏ i = 0 4 P i ( 41 − s ) ( − 1 ) i + 1 = P 1 ( T ) ⋅ P 3 ( T ) P 0 ( T ) ⋅ P 2 ( T ) ⋅ P 4 ( T ) , {\displaystyle \zeta ({\text{Jac}}(C/{\bf {F}}_{41}),s)\,=\,\exp \left(\sum _{m=1}^{\infty }{\frac {M_{m}}{m}}(41^{-s})^{m}\right)\,=\,\prod _{i=0}^{4}\,P_{i}(41^{-s})^{(-1)^{i+1}}={\frac {P_{1}(T)\cdot P_{3}(T)}{P_{0}(T)\cdot P_{2}(T)\cdot P_{4}(T)}},} it follows that ∑ m = 1 ∞ M m m ( 41 − s ) m = log ⁡ ( P 1 ( T ) ⋅ P 3 ( T ) P 0 ( T ) ⋅ P 2 ( T ) ⋅ P 4 ( T ) ) = 1375 ⋅ T + 2930125 / 2 ⋅ T 2 + 4755796375 / 3 ⋅ T 3 + 7984359145125 / 4 ⋅ T 4 + 13426146538750000 / 5 ⋅ T 5 + O ( T 6 ) . {\displaystyle {\begin{alignedat}{2}\sum _{m=1}^{\infty }&{\frac {M_{m}}{m}}(41^{-s})^{m}\,=\,\log \left({\frac {P_{1}(T)\cdot P_{3}(T)}{P_{0}(T)\cdot P_{2}(T)\cdot P_{4}(T)}}\right)\\&=1375\cdot T+2930125/2\cdot T^{2}+4755796375/3\cdot T^{3}+7984359145125/4\cdot T^{4}+13426146538750000/5\cdot T^{5}+O(T^{6}).\end{alignedat}}} Aside from the values M 1 {\displaystyle M_{1}} and M 2 {\displaystyle M_{2}} already known, you can read off from this Taylor series all other numbers M m {\displaystyle M_{m}} , m ∈ N {\displaystyle m\in \mathbb {N} } , of F 41 m {\displaystyle {\bf {F}}_{41^{m}}} -rational elements of the Jacobian variety, defined over F 41 {\displaystyle {\bf {F}}_{41}} , of the curve C / F 41 {\displaystyle C/{\bf {F}}_{41}} : for instance, M 3 = 4755796375 = 5 3 ⋅ 11 ⋅ 61 ⋅ 56701 {\displaystyle M_{3}=4755796375=5^{3}\cdot 11\cdot 61\cdot 56701} and M 4 = 7984359145125 = 3 4 ⋅ 5 3 ⋅ 11 ⋅ 2131 ⋅ 33641 {\displaystyle M_{4}=7984359145125=3^{4}\cdot 5^{3}\cdot 11\cdot 2131\cdot 33641} . In doing so, m 1 \| m 2 {\displaystyle m_{1}\|m_{2}} always implies M m 1 \| M m 2 {\displaystyle M_{m_{1}}\|M_{m_{2}}} since then, Jac ( C / F 41 m 1 ) {\displaystyle {\text{Jac}}(C/{\bf {F}}_{41^{m_{1}}})} is a subgroup of Jac ( C / F 41 m 2 ) {\displaystyle {\text{Jac}}(C/{\bf {F}}_{41^{m_{2}}})} . == Weil cohomology == Weil suggested that the conjectures would follow from the existence of a suitable "Weil cohomology theory" for varieties over finite fields, similar to the usual cohomology with rational coefficients for complex varieties. His idea was that if F is the Frobenius automorphism over the finite field, then the number of points of the variety X over the field of order qm is the number of fixed points of Fm (acting on all points of the variety X defined over the algebraic closure). In algebraic topology the number of fixed points of an automorphism can be worked out using the Lefschetz fixed-point theorem, given as an alternating sum of traces on the cohomology groups. So if there were similar cohomology groups for varieties over finite fields, then the zeta function could be expressed in terms of them. The first problem with this is that the coefficient field for a Weil cohomology theory cannot be the rational numbers. To see this consider the case of a supersingular elliptic curve over a finite field of characteristic p. The endomorphism ring of this is an order in a quaternion algebra over the rationals, and should act on the first cohomology group, which should be a 2-dimensional vector space over the coefficient field by analogy with the case of a complex elliptic curve. However a quaternion algebra over the rationals cannot act on a 2-dimensional vector space over the rationals. The same argument eliminates the possibility of the coefficient field being the reals or the p-adic numbers, because the quaternion algebra is still a division algebra over these fields. However it does not eliminate the possibility that the coefficient field is the field of ℓ-adic numbers for some prime ℓ ≠ p, because over these fields the division algebra splits and becomes a matrix algebra, which can act on a 2-dimensional vector space. Grothendieck and Michael Artin managed to construct suitable cohomology theories over the field of ℓ-adic numbers for each prime ℓ ≠ p, called ℓ-adic cohomology. == Grothendieck's proofs of three of the four conjectures == By the end of 1964 Grothendieck together with Artin and Jean-Louis Verdier (and the earlier 1960 work by Dwork) proved the Weil conjectures apart from the most difficult third conjecture above (the "Riemann hypothesis" conjecture) (Grothendieck 1965). The general theorems about étale cohomology allowed Grothendieck to prove an analogue of the Lefschetz fixed-point formula for the ℓ-adic cohomology theory, and by applying it to the Frobenius automorphism F he was able to prove the conjectured formula for the zeta function: ζ ( s ) = P 1 ( T ) ⋯ P 2 n − 1 ( T ) P 0 ( T ) P 2 ( T ) ⋯ P 2 n ( T ) {\displaystyle \zeta (s)={\frac {P_{1}(T)\cdots P_{2n-1}(T)}{P_{0}(T)P_{2}(T)\cdots P_{2n}(T)}}} where each polynomial Pi is the determinant of I − TF on the ℓ-adic cohomology group Hi. The rationality of the zeta function follows immediately. The functional equation for the zeta function follows from Poincaré duality for ℓ-adic cohomology, and the relation with complex Betti numbers of a lift follows from a comparison theorem between ℓ-adic and ordinary cohomology for complex varieties. More generally, Grothendieck proved a similar formula for the zeta function (or "generalized L-function") of a sheaf F0: Z ( X 0 , F 0 , t ) = ∏ x ∈ \| X 0 \| det ( 1 − F x ∗ t deg ⁡ ( x ) ∣ F 0 ) − 1 {\displaystyle Z(X_{0},F_{0},t)=\prod _{x\in \|X_{0}\|}\det(1-F_{x}^{}t^{\deg(x)}\mid F_{0})^{-1}} as a product over cohomology groups: Z ( X 0 , F 0 , t ) = ∏ i det ( 1 − F ∗ t ∣ H c i ( F ) ) ( − 1 ) i + 1 {\displaystyle Z(X_{0},F_{0},t)=\prod _{i}\det(1-F^{}t\mid H_{c}^{i}(F))^{(-1)^{i+1}}} The special case of the constant sheaf gives the usual zeta function. == Deligne's first proof of the Riemann hypothesis conjecture == Verdier (1974), Serre (1975), Katz (1976) and Freitag & Kiehl (1988) gave expository accounts of the first proof of Deligne (1974). Much of the background in ℓ-adic cohomology is described in (Deligne 1977). Deligne's first proof of the remaining third Weil conjecture (the "Riemann hypothesis conjecture") used the following steps: === Use of Lefschetz pencils === Grothendieck expressed the zeta function in terms of the trace of Frobenius on ℓ-adic cohomology groups, so the Weil conjectures for a d-dimensional variety V over a finite field with q elements depend on showing that the eigenvalues α of Frobenius acting on the ith ℓ-adic cohomology group Hi(V) of V have absolute values \|α\| = qi/2 (for an embedding of the algebraic elements of Qℓ into the complex numbers). After blowing up V and extending the base field, one may assume that the variety V has a morphism onto the projective line P1, with a finite number of singular fibers with very mild (quadratic) singularities. The theory of monodromy of Lefschetz pencils, introduced for complex varieties (and ordinary cohomology) by Lefschetz (1924), and extended by Grothendieck (1972) and Deligne & Katz (1973) to ℓ-adic cohomology, relates the cohomology of V to that of its fibers. The relation depends on the space Ex of vanishing cycles, the subspace of the cohomology Hd−1(Vx) of a non-singular fiber Vx, spanned by classes that vanish on singular fibers. The Leray spectral sequence relates the middle cohomology group of V to the cohomology of the fiber and base. The hard part to deal with is more or less a group H1(P1, jE) = H1c(U,E), where U is the points the projective line with non-singular fibers, and j is the inclusion of U into the projective line, and E is the sheaf with fibers the spaces Ex of vanishing cycles. === The key estimate === The heart of Deligne's proof is to show that the sheaf E over U is pure, in other words to find the absolute values of the eigenvalues of Frobenius on its stalks. This is done by studying the zeta functions of the even powers Ek of E and applying Grothendieck's formula for the zeta functions as alternating products over cohomology groups. The crucial idea of considering even k powers of E was inspired by the paper Rankin (1939), who used a similar idea with k = 2 for bounding the Ramanujan tau function. Langlands (1970, section 8) pointed out that a generalization of Rankin's result for higher even values of k would imply the Ramanujan conjecture, and Deligne realized that in the case of zeta functions of varieties, Grothendieck's theory of zeta functions of sheaves provided an analogue of this generalization. The poles of the zeta function of Ek are found using Grothendieck's formula Z ( U , E k , T ) = det ( 1 − F ∗ T ∣ H c 1 ( E k ) ) det ( 1 − F ∗ T ∣ H c 0 ( E k ) ) det ( 1 − F ∗ T ∣ H c 2 ( E k ) ) {\displaystyle Z(U,E^{k},T)={\frac {\det(1-F^{}T\mid H_{c}^{1}(E^{k}))}{\det(1-F^{}T\mid H_{c}^{0}(E^{k}))\det(1-F^{}T\mid H_{c}^{2}(E^{k}))}}} and calculating the cohomology groups in the denominator explicitly. The H0c term is usually just 1 as U is usually not compact, and the H2c can be calculated explicitly as follows. Poincaré duality relates H2c(Ek) to H0(Ek), which is in turn the space of covariants of the monodromy group, which is the geometric fundamental group of U acting on the fiber of Ek at a point. The fiber of E has a bilinear form induced by cup product, which is antisymmetric if d is even, and makes E into a symplectic space. (This is a little inaccurate: Deligne did later show that E∩E⊥ = 0 by using the hard Lefschetz theorem, this requires the Weil conjectures, and the proof of the Weil conjectures really has to use a slightly more complicated argument with E/E∩E⊥ rather than E.) An argument of Kazhdan and Margulis shows that the image of the monodromy group acting on E, given by the Picard–Lefschetz formula, is Zariski dense in a symplectic group and therefore has the same invariants, which are well known from classical invariant theory. Keeping track of the action of Frobenius in this calculation shows that its eigenvalues are all qk(d−1)/2+1, so the zeta function of Z(Ek,T) has poles only at T = 1/qk(d−1)/2+1. The Euler product for the zeta function of Ek is Z ( E k , T ) = ∏ x 1 Z ( E x k , T ) {\displaystyle Z(E^{k},T)=\prod _{x}{\frac {1}{Z(E_{x}^{k},T)}}} If k is even then all the coefficients of the factors on the right (considered as power series in T) are non-negative; this follows by writing 1 det ( 1 − T deg ⁡ ( x ) F x ∣ E k ) = exp ⁡ ( ∑ n > 0 T n n Trace ⁡ ( F x n ∣ E ) k ) {\displaystyle {\frac {1}{\det(1-T^{\deg(x)}F_{x}\mid E^{k})}}=\exp \left(\sum _{n>0}{\frac {T^{n}}{n}}\operatorname {Trace} (F_{x}^{n}\mid E)^{k}\right)} and using the fact that the traces of powers of F are rational, so their k powers are non-negative as k is even. Deligne proves the rationality of the traces by relating them to numbers of points of varieties, which are always (rational) integers. The powers series for Z(Ek, T) converges for T less than the absolute value 1/qk(d−1)/2+1 of its only possible pole. When k is even the coefficients of all its Euler factors are non-negative, so that each of the Euler factors has coefficients bounded by a constant times the coefficients of Z(Ek, T) and therefore converges on the same region and has no poles in this region. So for k even the polynomials Z(Ekx, T) have no zeros in this region, or in other words the eigenvalues of Frobenius on the stalks of Ek have absolute value at most qk(d−1)/2+1. This estimate can be used to find the absolute value of any eigenvalue α of Frobenius on a fiber of E as follows. For any integer k, αk is an eigenvalue of Frobenius on a stalk of Ek, which for k even is bounded by q1+k(d−1)/2. So \| α k \| ≤ q k ( d − 1 ) / 2 + 1 {\displaystyle \|\alpha ^{k}\|\leq q^{k(d-1)/2+1}} As this is true for arbitrarily large even k, this implies that \| α \| ≤ q ( d − 1 ) / 2 . {\displaystyle \|\alpha \|\leq q^{(d-1)/2}.} Poincaré duality then implies that \| α \| = q ( d − 1 ) / 2 . {\displaystyle \|\alpha \|=q^{(d-1)/2}.} === Completion of the proof === The deduction of the Riemann hypothesis from this estimate is mostly a fairly straightforward use of standard techniques and is done as follows. The eigenvalues of Frobenius on H1c(U,E) can now be estimated as they are the zeros of the zeta function of the sheaf E. This zeta function can be written as an Euler product of zeta functions of the stalks of E, and using the estimate for the eigenvalues on these stalks shows that this product converges for \|T\| < q−d/2−1/2, so that there are no zeros of the zeta function in this region. This implies that the eigenvalues of Frobenius on E are at most qd/2+1/2 in absolute value (in fact it will soon be seen that they have absolute value exactly qd/2). This step of the argument is very similar to the usual proof that the Riemann zeta function has no zeros with real part greater than 1, by writing it as an Euler product. The conclusion of this is that the eigenvalues α of the Frobenius of a variety of even dimension d on the middle cohomology group satisfy \| α \| ≤ q d / 2 + 1 / 2 {\displaystyle \|\alpha \|\leq q^{d/2+1/2}} To obtain the Riemann hypothesis one needs to eliminate the 1/2 from the exponent. This can be done as follows. Applying this estimate to any even power Vk of V and using the Künneth formula shows that the eigenvalues of Frobenius on the middle cohomology of a variety V of any dimension d satisfy \| α k \| ≤ q k d / 2 + 1 / 2 {\displaystyle \|\alpha ^{k}\|\leq q^{kd/2+1/2}} As this is true for arbitrarily large even k, this implies that \| α \| ≤ q d / 2 {\displaystyle \|\alpha \|\leq q^{d/2}} Poincaré duality then implies that \| α \| = q d / 2 . {\displaystyle \|\alpha \|=q^{d/2}.} This proves the Weil conjectures for the middle cohomology of a variety. The Weil conjectures for the cohomology below the middle dimension follow from this by applying the weak Lefschetz theorem, and the conjectures for cohomology above the middle dimension then follow from Poincaré duality. == Deligne's second proof == Deligne (1980) found and proved a generalization of the Weil conjectures, bounding the weights of the pushforward of a sheaf. In practice it is this generalization rather than the original Weil conjectures that is mostly used in applications, such as the hard Lefschetz theorem. Much of the second proof is a rearrangement of the ideas of his first proof. The main extra idea needed is an argument closely related to the theorem of Jacques Hadamard and Charles Jean de la Vallée Poussin, used by Deligne to show that various L-series do not have zeros with real part 1. A constructible sheaf on a variety over a finite field is called pure of weight β if for all points x the eigenvalues of the Frobenius at x all have absolute value N(x)β/2, and is called mixed of weight ≤ β if it can be written as repeated extensions by pure sheaves with weights ≤ β. Deligne's theorem states that if f is a morphism of schemes of finite type over a finite field, then Rif! takes mixed sheaves of weight ≤ β to mixed sheaves of weight ≤ β + i. The original Weil conjectures follow by taking f to be a morphism from a smooth projective variety to a point and considering the constant sheaf Qℓ on the variety. This gives an upper bound on the absolute values of the eigenvalues of Frobenius, and Poincaré duality then shows that this is also a lower bound. In general Rif! does not take pure sheaves to pure sheaves. However it does when a suitable form of Poincaré duality holds, for example if f is smooth and proper, or if one works with perverse sheaves rather than sheaves as in Beilinson, Bernstein & Deligne (1982). Inspired by the work of Witten (1982) on Morse theory, Laumon (1987) found another proof, using Deligne's ℓ-adic Fourier transform, which allowed him to simplify Deligne's proof by avoiding the use of the method of Hadamard and de la Vallée Poussin. His proof generalizes the classical calculation of the absolute value of Gauss sums using the fact that the norm of a Fourier transform has a simple relation to the norm of the original function. Kiehl & Weissauer (2001) used Laumon's proof as the basis for their exposition of Deligne's theorem. Katz (2001) gave a further simplification of Laumon's proof, using monodromy in the spirit of Deligne's first proof. Kedlaya (2006) gave another proof using the Fourier transform, replacing etale cohomology with rigid cohomology. == Applications == Deligne (1980) was able to prove the hard Lefschetz theorem over finite fields using his second proof of the Weil conjectures. Deligne (1971) had previously shown that the Ramanujan–Petersson conjecture follows from the Weil conjectures. Deligne (1974, section 8) used the Weil conjectures to prove estimates for exponential sums. 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The order of the Fourier coefficients of integral modular forms", Proceedings of the Cambridge Philosophical Society, 35 (3): 357–372, Bibcode:1939PCPS...35..357R, doi:10.1017/S0305004100021101, MR 0000411, S2CID 251097961 Serre, Jean-Pierre (1960), "Analogues kählériens de certaines conjectures de Weil", Annals of Mathematics, Second Series, 71 (2), The Annals of Mathematics, Vol. 71, No. 2: 392–394, doi:10.2307/1970088, ISSN 0003-486X, JSTOR 1970088, MR 0112163 Serre, Jean-Pierre (1975), "Valeurs propers des endomorphismes de Frobenius [d'après P. Deligne]", Séminaire Bourbaki vol. 1973/74 Exposés 436–452, Lecture Notes in Mathematics, vol. 431, pp. 190–204, doi:10.1007/BFb0066371, ISBN 978-3-540-07023-8 Verdier, Jean-Louis (1974), "Indépendance par rapport a ℓ des polynômes caractéristiques des endomorphismes de frobenius de la cohomologie ℓ-adique", Séminaire Bourbaki vol. 1972/73 Exposés 418–435, Lecture Notes in Mathematics, vol. 383, Springer Berlin / Heidelberg, pp. 98–115, doi:10.1007/BFb0057304, ISBN 978-3-540-06796-2 Weil, André (1949), "Numbers of solutions of equations in finite fields", Bulletin of the American Mathematical Society, 55 (5): 497–508, doi:10.1090/S0002-9904-1949-09219-4, ISSN 0002-9904, MR 0029393 Reprinted in Oeuvres Scientifiques/Collected Papers by André Weil ISBN 0-387-90330-5 Witten, Edward (1982), "Supersymmetry and Morse theory", Journal of Differential Geometry, 17 (4): 661–692, doi:10.4310/jdg/1214437492, ISSN 0022-040X, MR 0683171 == External links == == References ==
Wikipedia:Weitzenböck identity#0	In mathematics, in particular in differential geometry, mathematical physics, and representation theory, a Weitzenböck identity, named after Roland Weitzenböck, expresses a relationship between two second-order elliptic operators on a manifold with the same principal symbol. Usually Weitzenböck formulae are implemented for G-invariant self-adjoint operators between vector bundles associated to some principal G-bundle, although the precise conditions under which such a formula exists are difficult to formulate. This article focuses on three examples of Weitzenböck identities: from Riemannian geometry, spin geometry, and complex analysis. == Riemannian geometry == In Riemannian geometry there are two notions of the Laplacian on differential forms over an oriented compact Riemannian manifold M. The first definition uses the divergence operator δ defined as the formal adjoint of the de Rham operator d: ∫ M ⟨ α , δ β ⟩ := ∫ M ⟨ d α , β ⟩ {\displaystyle \int _{M}\langle \alpha ,\delta \beta \rangle :=\int _{M}\langle d\alpha ,\beta \rangle } where α is any p-form and β is any (p + 1)-form, and ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is the metric induced on the bundle of (p + 1)-forms. The usual form Laplacian is then given by Δ = d δ + δ d . {\displaystyle \Delta =d\delta +\delta d.} On the other hand, the Levi-Civita connection supplies a differential operator ∇ : Ω p M → Ω 1 M ⊗ Ω p M , {\displaystyle \nabla :\Omega ^{p}M\rightarrow \Omega ^{1}M\otimes \Omega ^{p}M,} where ΩpM is the bundle of p-forms. The Bochner Laplacian is given by Δ ′ = ∇ ∗ ∇ {\displaystyle \Delta '=\nabla ^{}\nabla } where ∇ ∗ {\displaystyle \nabla ^{}} is the adjoint of ∇ {\displaystyle \nabla } . This is also known as the connection or rough Laplacian. The Weitzenböck formula then asserts that Δ ′ − Δ = A {\displaystyle \Delta '-\Delta =A} where A is a linear operator of order zero involving only the curvature. The precise form of A is given, up to an overall sign depending on curvature conventions, by A = 1 2 ⟨ R ( θ , θ ) # , # ⟩ + Ric ⁡ ( θ , # ) , {\displaystyle A={\frac {1}{2}}\langle R(\theta ,\theta )\#,\#\rangle +\operatorname {Ric} (\theta ,\#),} where R is the Riemann curvature tensor, Ric is the Ricci tensor, θ : T ∗ M ⊗ Ω p M → Ω p + 1 M {\displaystyle \theta :T^{}M\otimes \Omega ^{p}M\rightarrow \Omega ^{p+1}M} is the map that takes the wedge product of a 1-form and p-form and gives a (p+1)-form, # : Ω p + 1 M → T ∗ M ⊗ Ω p M {\displaystyle \#:\Omega ^{p+1}M\rightarrow T^{}M\otimes \Omega ^{p}M} is the universal derivation inverse to θ on 1-forms. == Spin geometry == If M is an oriented spin manifold with Dirac operator ð, then one may form the spin Laplacian Δ = ð2 on the spin bundle. On the other hand, the Levi-Civita connection extends to the spin bundle to yield a differential operator ∇ : S M → T ∗ M ⊗ S M . {\displaystyle \nabla :SM\rightarrow T^{}M\otimes SM.} As in the case of Riemannian manifolds, let Δ ′ = ∇ ∗ ∇ {\displaystyle \Delta '=\nabla ^{}\nabla } . This is another self-adjoint operator and, moreover, has the same leading symbol as the spin Laplacian. The Weitzenböck formula yields: Δ ′ − Δ = − 1 4 S c {\displaystyle \Delta '-\Delta =-{\frac {1}{4}}Sc} where Sc is the scalar curvature. This result is also known as the Lichnerowicz formula. == Complex differential geometry == If M is a compact Kähler manifold, there is a Weitzenböck formula relating the ∂ ¯ {\displaystyle {\bar {\partial }}} -Laplacian (see Dolbeault complex) and the Euclidean Laplacian on (p,q)-forms. Specifically, let Δ = ∂ ¯ ∗ ∂ ¯ + ∂ ¯ ∂ ¯ ∗ , {\displaystyle \Delta ={\bar {\partial }}^{}{\bar {\partial }}+{\bar {\partial }}{\bar {\partial }}^{},} and Δ ′ = − ∑ k ∇ k ∇ k ¯ {\displaystyle \Delta '=-\sum _{k}\nabla _{k}\nabla _{\bar {k}}} in a unitary frame at each point. According to the Weitzenböck formula, if α ∈ Ω ( p , q ) M {\displaystyle \alpha \in \Omega ^{(p,q)}M} , then Δ ′ α − Δ α = A ( α ) {\displaystyle \Delta ^{\prime }\alpha -\Delta \alpha =A(\alpha )} where A {\displaystyle A} is an operator of order zero involving the curvature. Specifically, if α = α i 1 i 2 … i p j ¯ 1 j ¯ 2 … j ¯ q {\displaystyle \alpha =\alpha _{i_{1}i_{2}\dots i_{p}{\bar {j}}_{1}{\bar {j}}_{2}\dots {\bar {j}}_{q}}} in a unitary frame, then A ( α ) = − ∑ k , j s Ric j ¯ α k ¯ ⁡ α i 1 i 2 … i p j ¯ 1 j ¯ 2 … k ¯ … j ¯ q {\displaystyle A(\alpha )=-\sum _{k,j_{s}}\operatorname {Ric} _{{\bar {j}}_{\alpha }}^{\bar {k}}\alpha _{i_{1}i_{2}\dots i_{p}{\bar {j}}_{1}{\bar {j}}_{2}\dots {\bar {k}}\dots {\bar {j}}_{q}}} with k in the s-th place. == Other Weitzenböck identities == In conformal geometry there is a Weitzenböck formula relating a particular pair of differential operators defined on the tractor bundle. See Branson, T. and Gover, A.R., "Conformally Invariant Operators, Differential Forms, Cohomology and a Generalisation of Q-Curvature", Communications in Partial Differential Equations, 30 (2005) 1611–1669. == See also == Bochner identity Bochner–Kodaira–Nakano identity Laplacian operators in differential geometry == References == Griffiths, Philip; Harris, Joe (1978), Principles of algebraic geometry, Wiley-Interscience (published 1994), ISBN 978-0-471-05059-9
Wikipedia:Wenxian Shen#0	Wenxian Shen is a Chinese-American mathematician known for her work in topological dynamics, almost-periodicity, waves and other spatial patterns in dynamical systems. She is Don Logan Chair of Mathematics at Auburn University. == Education == Shen graduated from Zhejiang Normal University in 1982, and earned a master's degree at Peking University in 1987. She completed a Ph.D. in mathematics at the Georgia Institute of Technology in 1992, with the dissertation Stability and Bifurcation of Traveling Wave Solutions supervised by Shui-Nee Chow. == Books == Shen is the coauthor of two monographs, Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflows (with Yingfei Yi, American Mathematical Society, 1998), and Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications (with Janusz Mierczyński, CRC Press, 2008). == References == == External links == Home page Wenxian Shen publications indexed by Google Scholar
Wikipedia:Werner Fenchel#0	Moritz Werner Fenchel (German: [ˈfɛnçəl]; 3 May 1905 – 24 January 1988) was a German-Danish mathematician known for his contributions to geometry and to optimization theory. Fenchel established the basic results of convex analysis and nonlinear optimization theory which would, in time, serve as the foundation for nonlinear programming. A German-born Jew and early refugee from Nazi suppression of intellectuals, Fenchel lived most of his life in Denmark. Fenchel's monographs and lecture notes are considered influential. == Biography == === Early life and education === Fenchel was born on 3 May 1905 in Berlin, Germany, his younger brother was the Israeli film director and architect Heinz Fenchel. Fenchel studied mathematics and physics at the University of Berlin between 1923 and 1928. He wrote his doctorate thesis in geometry (Über Krümmung und Windung geschlossener Raumkurven) under Ludwig Bieberbach. === Professorship in Germany === From 1928 to 1933, Fenchel was Professor E. Landau's Assistant at the University of Göttingen. During a one-year leave (on Rockefeller Fellowship) between 1930 and 1931, Fenchel spent time in Rome with Tullio Levi-Civita, as well as in Copenhagen with Harald Bohr and Tommy Bonnesen. He visited Denmark again in 1932. === Professorship in exile === Fenchel taught at Göttingen until 1933, when the Nazi discrimination laws led to mass-firings of Jews. Fenchel emigrated to Denmark somewhere between April and September 1933, ultimately obtaining a position at the University of Copenhagen. In December 1933, Fenchel married fellow German refugee mathematician Käte Sperling. When Germany occupied Denmark, Fenchel and roughly eight-thousand other Danish Jews received refuge in Sweden, where he taught (between 1943 and 1945) at the Danish School in Lund. After the Allied powers' liberation of Denmark, Fenchel returned to Copenhagen. === Professorship postwar === In 1946, Fenchel was elected a member of the Royal Danish Academy of Sciences and Letters. On leave between 1949 and 1951, Fenchel taught in the U.S. at the University of Southern California, Stanford University, and Princeton University. From 1952 to 1956 Fenchel was the professor in mechanics at the Polytechnic in Copenhagen. From 1956 to 1974 he was the professor in mathematics at the University of Copenhagen. === Last years, death, legacy === Professor Fenchel died on 24 January 1988. == Geometric contributions == === Convex geometry === === Optimization theory === Fenchel lectured on "Convex Sets, Cones, and Functions" at Princeton University in the early 1950s. His lecture notes shaped the field of convex analysis, according to the monograph Convex Analysis of R. T. Rockafellar. === Hyperbolic geometry === == Books == == See also == == References == == External links == Werner Fenchel at the Mathematics Genealogy Project Werner Fenchel website – contains CV, biography, links to archive, etc.
Wikipedia:Weyl algebra#0	In abstract algebra, the Weyl algebras are abstracted from the ring of differential operators with polynomial coefficients. They are named after Hermann Weyl, who introduced them to study the Heisenberg uncertainty principle in quantum mechanics. In the simplest case, these are differential operators. Let F {\displaystyle F} be a field, and let F [ x ] {\displaystyle F[x]} be the ring of polynomials in one variable with coefficients in F {\displaystyle F} . Then the corresponding Weyl algebra consists of differential operators of form f m ( x ) ∂ x m + f m − 1 ( x ) ∂ x m − 1 + ⋯ + f 1 ( x ) ∂ x + f 0 ( x ) {\displaystyle f_{m}(x)\partial _{x}^{m}+f_{m-1}(x)\partial _{x}^{m-1}+\cdots +f_{1}(x)\partial _{x}+f_{0}(x)} This is the first Weyl algebra A 1 {\displaystyle A_{1}} . The n-th Weyl algebra A n {\displaystyle A_{n}} are constructed similarly. Alternatively, A 1 {\displaystyle A_{1}} can be constructed as the quotient of the free algebra on two generators, q and p, by the ideal generated by ( [ p , q ] − 1 ) {\displaystyle ([p,q]-1)} . Similarly, A n {\displaystyle A_{n}} is obtained by quotienting the free algebra on 2n generators by the ideal generated by ( [ p i , q j ] − δ i , j ) , ∀ i , j = 1 , … , n {\displaystyle ([p_{i},q_{j}]-\delta _{i,j}),\quad \forall i,j=1,\dots ,n} where δ i , j {\displaystyle \delta _{i,j}} is the Kronecker delta. More generally, let ( R , Δ ) {\displaystyle (R,\Delta )} be a partial differential ring with commuting derivatives Δ = { ∂ 1 , … , ∂ m } {\displaystyle \Delta =\lbrace \partial _{1},\ldots ,\partial _{m}\rbrace } . The Weyl algebra associated to ( R , Δ ) {\displaystyle (R,\Delta )} is the noncommutative ring R [ ∂ 1 , … , ∂ m ] {\displaystyle R[\partial _{1},\ldots ,\partial _{m}]} satisfying the relations ∂ i r = r ∂ i + ∂ i ( r ) {\displaystyle \partial _{i}r=r\partial _{i}+\partial _{i}(r)} for all r ∈ R {\displaystyle r\in R} . The previous case is the special case where R = F [ x 1 , … , x n ] {\displaystyle R=F[x_{1},\ldots ,x_{n}]} and Δ = { ∂ x 1 , … , ∂ x n } {\displaystyle \Delta =\lbrace \partial _{x_{1}},\ldots ,\partial _{x_{n}}\rbrace } where F {\displaystyle F} is a field. This article discusses only the case of A n {\displaystyle A_{n}} with underlying field F {\displaystyle F} characteristic zero, unless otherwise stated. The Weyl algebra is an example of a simple ring that is not a matrix ring over a division ring. It is also a noncommutative example of a domain, and an example of an Ore extension. == Motivation == The Weyl algebra arises naturally in the context of quantum mechanics and the process of canonical quantization. Consider a classical phase space with canonical coordinates ( q 1 , p 1 , … , q n , p n ) {\displaystyle (q_{1},p_{1},\dots ,q_{n},p_{n})} . These coordinates satisfy the Poisson bracket relations: { q i , q j } = 0 , { p i , p j } = 0 , { q i , p j } = δ i j . {\displaystyle \{q_{i},q_{j}\}=0,\quad \{p_{i},p_{j}\}=0,\quad \{q_{i},p_{j}\}=\delta _{ij}.} In canonical quantization, one seeks to construct a Hilbert space of states and represent the classical observables (functions on phase space) as self-adjoint operators on this space. The canonical commutation relations are imposed: [ q ^ i , q ^ j ] = 0 , [ p ^ i , p ^ j ] = 0 , [ q ^ i , p ^ j ] = i ℏ δ i j , {\displaystyle [{\hat {q}}_{i},{\hat {q}}_{j}]=0,\quad [{\hat {p}}_{i},{\hat {p}}_{j}]=0,\quad [{\hat {q}}_{i},{\hat {p}}_{j}]=i\hbar \delta _{ij},} where [ ⋅ , ⋅ ] {\displaystyle [\cdot ,\cdot ]} denotes the commutator. Here, q ^ i {\displaystyle {\hat {q}}_{i}} and p ^ i {\displaystyle {\hat {p}}_{i}} are the operators corresponding to q i {\displaystyle q_{i}} and p i {\displaystyle p_{i}} respectively. Erwin Schrödinger proposed in 1926 the following: q j ^ {\displaystyle {\hat {q_{j}}}} with multiplication by x j {\displaystyle x_{j}} . p ^ j {\displaystyle {\hat {p}}_{j}} with − i ℏ ∂ x j {\displaystyle -i\hbar \partial _{x_{j}}} . With this identification, the canonical commutation relation holds. == Constructions == The Weyl algebras have different constructions, with different levels of abstraction. === Representation === The Weyl algebra A n {\displaystyle A_{n}} can be concretely constructed as a representation. In the differential operator representation, similar to Schrödinger's canonical quantization, let q j {\displaystyle q_{j}} be represented by multiplication on the left by x j {\displaystyle x_{j}} , and let p j {\displaystyle p_{j}} be represented by differentiation on the left by ∂ x j {\displaystyle \partial _{x_{j}}} . In the matrix representation, similar to the matrix mechanics, A 1 {\displaystyle A_{1}} is represented by P = [ 0 1 0 0 ⋯ 0 0 2 0 ⋯ 0 0 0 3 ⋯ ⋮ ⋮ ⋮ ⋮ ⋱ ] , Q = [ 0 0 0 0 … 1 0 0 0 ⋯ 0 1 0 0 ⋯ ⋮ ⋮ ⋮ ⋮ ⋱ ] {\displaystyle P={\begin{bmatrix}0&1&0&0&\cdots \\0&0&2&0&\cdots \\0&0&0&3&\cdots \\\vdots &\vdots &\vdots &\vdots &\ddots \end{bmatrix}},\quad Q={\begin{bmatrix}0&0&0&0&\ldots \\1&0&0&0&\cdots \\0&1&0&0&\cdots \\\vdots &\vdots &\vdots &\vdots &\ddots \end{bmatrix}}} === Generator === A n {\displaystyle A_{n}} can be constructed as a quotient of a free algebra in terms of generators and relations. One construction starts with an abstract vector space V (of dimension 2n) equipped with a symplectic form ω. Define the Weyl algebra W(V) to be W ( V ) := T ( V ) / ( ( v ⊗ u − u ⊗ v − ω ( v , u ) , for v , u ∈ V ) ) , {\displaystyle W(V):=T(V)/(\!(v\otimes u-u\otimes v-\omega (v,u),{\text{ for }}v,u\in V)\!),} where T(V) is the tensor algebra on V, and the notation ( ( ) ) {\displaystyle (\!()\!)} means "the ideal generated by". In other words, W(V) is the algebra generated by V subject only to the relation vu − uv = ω(v, u). Then, W(V) is isomorphic to An via the choice of a Darboux basis for ω. A n {\displaystyle A_{n}} is also a quotient of the universal enveloping algebra of the Heisenberg algebra, the Lie algebra of the Heisenberg group, by setting the central element of the Heisenberg algebra (namely [q, p]) equal to the unit of the universal enveloping algebra (called 1 above). === Quantization === The algebra W(V) is a quantization of the symmetric algebra Sym(V). If V is over a field of characteristic zero, then W(V) is naturally isomorphic to the underlying vector space of the symmetric algebra Sym(V) equipped with a deformed product – called the Groenewold–Moyal product (considering the symmetric algebra to be polynomial functions on V∗, where the variables span the vector space V, and replacing iħ in the Moyal product formula with 1). The isomorphism is given by the symmetrization map from Sym(V) to W(V) a 1 ⋯ a n ↦ 1 n ! ∑ σ ∈ S n a σ ( 1 ) ⊗ ⋯ ⊗ a σ ( n ) . {\displaystyle a_{1}\cdots a_{n}\mapsto {\frac {1}{n!}}\sum _{\sigma \in S_{n}}a_{\sigma (1)}\otimes \cdots \otimes a_{\sigma (n)}~.} If one prefers to have the iħ and work over the complex numbers, one could have instead defined the Weyl algebra above as generated by qi and iħ∂qi (as per quantum mechanics usage). Thus, the Weyl algebra is a quantization of the symmetric algebra, which is essentially the same as the Moyal quantization (if for the latter one restricts to polynomial functions), but the former is in terms of generators and relations (considered to be differential operators) and the latter is in terms of a deformed multiplication. Stated in another way, let the Moyal star product be denoted f ⋆ g {\displaystyle f\star g} , then the Weyl algebra is isomorphic to ( C [ x 1 , … , x n ] , ⋆ ) {\displaystyle (\mathbb {C} [x_{1},\dots ,x_{n}],\star )} . In the case of exterior algebras, the analogous quantization to the Weyl one is the Clifford algebra, which is also referred to as the orthogonal Clifford algebra. The Weyl algebra is also referred to as the symplectic Clifford algebra. Weyl algebras represent for symplectic bilinear forms the same structure that Clifford algebras represent for non-degenerate symmetric bilinear forms. === D-module === The Weyl algebra can be constructed as a D-module. Specifically, the Weyl algebra corresponding to the polynomial ring R [ x 1 , . . . , x n ] {\displaystyle R[x_{1},...,x_{n}]} with its usual partial differential structure is precisely equal to Grothendieck's ring of differential operations D A R n / R {\displaystyle D_{\mathbb {A} _{R}^{n}/R}} . More generally, let X {\displaystyle X} be a smooth scheme over a ring R {\displaystyle R} . Locally, X → R {\displaystyle X\to R} factors as an étale cover over some A R n {\displaystyle \mathbb {A} _{R}^{n}} equipped with the standard projection. Because "étale" means "(flat and) possessing null cotangent sheaf", this means that every D-module over such a scheme can be thought of locally as a module over the n th {\displaystyle n^{\text{th}}} Weyl algebra. Let R {\displaystyle R} be a commutative algebra over a subring S {\displaystyle S} . The ring of differential operators D R / S {\displaystyle D_{R/S}} (notated D R {\displaystyle D_{R}} when S {\displaystyle S} is clear from context) is inductively defined as a graded subalgebra of End S ⁡ ( R ) {\displaystyle \operatorname {End} _{S}(R)} : D R 0 = R {\displaystyle D_{R}^{0}=R} D R k = { d ∈ End S ⁡ ( R ) : [ d , a ] ∈ D R k − 1 for all a ∈ R } . {\displaystyle D_{R}^{k}=\left\{d\in \operatorname {End} _{S}(R):[d,a]\in D_{R}^{k-1}{\text{ for all }}a\in R\right\}.} Let D R {\displaystyle D_{R}} be the union of all D R k {\displaystyle D_{R}^{k}} for k ≥ 0 {\displaystyle k\geq 0} . This is a subalgebra of End S ⁡ ( R ) {\displaystyle \operatorname {End} _{S}(R)} . In the case R = S [ x 1 , . . . , x n ] {\displaystyle R=S[x_{1},...,x_{n}]} , the ring of differential operators of order ≤ n {\displaystyle \leq n} presents similarly as in the special case S = C {\displaystyle S=\mathbb {C} } but for the added consideration of "divided power operators"; these are operators corresponding to those in the complex case which stabilize Z [ x 1 , . . . , x n ] {\displaystyle \mathbb {Z} [x_{1},...,x_{n}]} , but which cannot be written as integral combinations of higher-order operators, i.e. do not inhabit D A Z n / Z {\displaystyle D_{\mathbb {A} _{\mathbb {Z} }^{n}/\mathbb {Z} }} . One such example is the operator ∂ x 1 [ p ] : x 1 N ↦ ( N p ) x 1 N − p {\displaystyle \partial _{x_{1}}^{[p]}:x_{1}^{N}\mapsto {N \choose p}x_{1}^{N-p}} . Explicitly, a presentation is given by D S [ x 1 , … , x ℓ ] / S n = S ⟨ x 1 , … , x ℓ , { ∂ x i , ∂ x i [ 2 ] , … , ∂ x i [ n ] } 1 ≤ i ≤ ℓ ⟩ {\displaystyle D_{S[x_{1},\dots ,x_{\ell }]/S}^{n}=S\langle x_{1},\dots ,x_{\ell },\{\partial _{x_{i}},\partial _{x_{i}}^{[2]},\dots ,\partial _{x_{i}}^{[n]}\}_{1\leq i\leq \ell }\rangle } with the relations [ x i , x j ] = [ ∂ x i [ k ] , ∂ x j [ m ] ] = 0 {\displaystyle [x_{i},x_{j}]=[\partial _{x_{i}}^{[k]},\partial _{x_{j}}^{[m]}]=0} [ ∂ x i [ k ] , x j ] = { ∂ x i [ k − 1 ] if i = j 0 if i ≠ j {\displaystyle [\partial _{x_{i}}^{[k]},x_{j}]=\left\{{\begin{matrix}\partial _{x_{i}}^{[k-1]}&{\text{if }}i=j\\0&{\text{if }}i\neq j\end{matrix}}\right.} ∂ x i [ k ] ∂ x i [ m ] = ( k + m k ) ∂ x i [ k + m ] when k + m ≤ n {\displaystyle \partial _{x_{i}}^{[k]}\partial _{x_{i}}^{[m]}={k+m \choose k}\partial _{x_{i}}^{[k+m]}~~~~~{\text{when }}k+m\leq n} where ∂ x i [ 0 ] = 1 {\displaystyle \partial _{x_{i}}^{[0]}=1} by convention. The Weyl algebra then consists of the limit of these algebras as n → ∞ {\displaystyle n\to \infty } .: Ch. IV.16.II When S {\displaystyle S} is a field of characteristic 0, then D R 1 {\displaystyle D_{R}^{1}} is generated, as an R {\displaystyle R} -module, by 1 and the S {\displaystyle S} -derivations of R {\displaystyle R} . Moreover, D R {\displaystyle D_{R}} is generated as a ring by the R {\displaystyle R} -subalgebra D R 1 {\displaystyle D_{R}^{1}} . In particular, if S = C {\displaystyle S=\mathbb {C} } and R = C [ x 1 , . . . , x n ] {\displaystyle R=\mathbb {C} [x_{1},...,x_{n}]} , then D R 1 = R + ∑ i R ∂ x i {\displaystyle D_{R}^{1}=R+\sum _{i}R\partial _{x_{i}}} . As mentioned, A n = D R {\displaystyle A_{n}=D_{R}} . == Properties of An == Many properties of A 1 {\displaystyle A_{1}} apply to A n {\displaystyle A_{n}} with essentially similar proofs, since the different dimensions commute. === General Leibniz rule === In particular, [ q , q m p n ] = − n q m p n − 1 {\textstyle [q,q^{m}p^{n}]=-nq^{m}p^{n-1}} and [ p , q m p n ] = m q m − 1 p n {\textstyle [p,q^{m}p^{n}]=mq^{m-1}p^{n}} . === Degree === This allows A 1 {\displaystyle A_{1}} to be a graded algebra, where the degree of ∑ m , n c m , n q m p n {\displaystyle \sum _{m,n}c_{m,n}q^{m}p^{n}} is max ( m + n ) {\displaystyle \max(m+n)} among its nonzero monomials. The degree is similarly defined for A n {\displaystyle A_{n}} . That is, it has no two-sided nontrivial ideals and has no zero divisors. === Derivation === That is, any derivation D {\textstyle D} is equal to [ ⋅ , f ] {\textstyle [\cdot ,f]} for some f ∈ A n {\textstyle f\in A_{n}} ; any f ∈ A n {\textstyle f\in A_{n}} yields a derivation [ ⋅ , f ] {\textstyle [\cdot ,f]} ; if f , f ′ ∈ A n {\textstyle f,f'\in A_{n}} satisfies [ ⋅ , f ] = [ ⋅ , f ′ ] {\textstyle [\cdot ,f]=[\cdot ,f']} , then f − f ′ ∈ F {\textstyle f-f'\in F} . The proof is similar to computing the potential function for a conservative polynomial vector field on the plane. == Representation theory == === Zero characteristic === In the case that the ground field F has characteristic zero, the nth Weyl algebra is a simple Noetherian domain. It has global dimension n, in contrast to the ring it deforms, Sym(V), which has global dimension 2n. It has no finite-dimensional representations. Although this follows from simplicity, it can be more directly shown by taking the trace of σ(q) and σ(Y) for some finite-dimensional representation σ (where [q,p] = 1). t r ( [ σ ( q ) , σ ( Y ) ] ) = t r ( 1 ) . {\displaystyle \mathrm {tr} ([\sigma (q),\sigma (Y)])=\mathrm {tr} (1)~.} Since the trace of a commutator is zero, and the trace of the identity is the dimension of the representation, the representation must be zero dimensional. In fact, there are stronger statements than the absence of finite-dimensional representations. To any finitely generated An-module M, there is a corresponding subvariety Char(M) of V × V∗ called the 'characteristic variety' whose size roughly corresponds to the size of M (a finite-dimensional module would have zero-dimensional characteristic variety). Then Bernstein's inequality states that for M non-zero, dim ⁡ ( char ⁡ ( M ) ) ≥ n {\displaystyle \dim(\operatorname {char} (M))\geq n} An even stronger statement is Gabber's theorem, which states that Char(M) is a co-isotropic subvariety of V × V∗ for the natural symplectic form. === Positive characteristic === The situation is considerably different in the case of a Weyl algebra over a field of characteristic p > 0. In this case, for any element D of the Weyl algebra, the element Dp is central, and so the Weyl algebra has a very large center. In fact, it is a finitely generated module over its center; even more so, it is an Azumaya algebra over its center. As a consequence, there are many finite-dimensional representations which are all built out of simple representations of dimension p. == Generalizations == The ideals and automorphisms of A 1 {\displaystyle A_{1}} have been well-studied. The moduli space for its right ideal is known. However, the case for A n {\displaystyle A_{n}} is considerably harder and is related to the Jacobian conjecture. For more details about this quantization in the case n = 1 (and an extension using the Fourier transform to a class of integrable functions larger than the polynomial functions), see Wigner–Weyl transform. Weyl algebras and Clifford algebras admit a further structure of a *-algebra, and can be unified as even and odd terms of a superalgebra, as discussed in CCR and CAR algebras. === Affine varieties === Weyl algebras also generalize in the case of algebraic varieties. Consider a polynomial ring R = C [ x 1 , … , x n ] I . {\displaystyle R={\frac {\mathbb {C} [x_{1},\ldots ,x_{n}]}{I}}.} Then a differential operator is defined as a composition of C {\displaystyle \mathbb {C} } -linear derivations of R {\displaystyle R} . This can be described explicitly as the quotient ring Diff ( R ) = { D ∈ A n : D ( I ) ⊆ I } I ⋅ A n . {\displaystyle {\text{Diff}}(R)={\frac {\{D\in A_{n}\colon D(I)\subseteq I\}}{I\cdot A_{n}}}.} == See also == Jacobian conjecture Dixmier conjecture == Notes == == References == Coutinho, S. C. (1995). A Primer of Algebraic D-Modules. Cambridge [England] ; New York, NY, USA: Cambridge University Press. doi:10.1017/cbo9780511623653. ISBN 978-0-521-55119-9. Coutinho, S. C. (1997). "The Many Avatars of a Simple Algebra". The American Mathematical Monthly. 104 (7): 593–604. doi:10.1080/00029890.1997.11990687. ISSN 0002-9890. Dirac, P. A. M. (1926). "On Quantum Algebra". Mathematical Proceedings of the Cambridge Philosophical Society. 23 (4): 412–418. doi:10.1017/S0305004100015231. ISSN 0305-0041. Helmstetter, J.; Micali, A. (2008). Quadratic Mappings and Clifford Algebras. Basel ; Boston: Birkhäuser. ISBN 978-3-7643-8605-4. OCLC 175285188. Landsman, N.P. (2007). "BETWEEN CLASSICAL AND QUANTUM". Philosophy of Physics. Elsevier. doi:10.1016/b978-044451560-5/50008-7. ISBN 978-0-444-51560-5. Lounesto, P.; Ablamowicz, R. (2004). Clifford Algebras. Boston: Springer Science & Business Media. ISBN 0-8176-3525-4. Micali, A.; Boudet, R.; Helmstetter, J. (1992). Clifford Algebras and their Applications in Mathematical Physics. Dordrecht: Springer Science & Business Media. ISBN 0-7923-1623-1. de Traubenberg, M. Rausch; Slupinski, M. J.; Tanasa, A. (2006). "Finite-dimensional Lie subalgebras of the Weyl algebra". J. Lie Theory. 16: 427–454. arXiv:math/0504224. Traves, Will (2010). "Differential Operations on Grassmann Varieties". In Campbell, H.; Helminck, A.; Kraft, H.; Wehlau, D. (eds.). Symmetry and Spaces. Progress in Mathematics. Vol. 278. Birkhäuse. pp. 197–207. doi:10.1007/978-0-8176-4875-6_10. ISBN 978-0-8176-4875-6. Tsit Yuen Lam (2001). A first course in noncommutative rings. Graduate Texts in Mathematics. Vol. 131 (2nd ed.). Springer. p. 6. ISBN 978-0-387-95325-0. Berest, Yuri; Wilson, George (September 1, 2000). "Automorphisms and ideals of the Weyl algebra". Mathematische Annalen. 318 (1): 127–147. arXiv:math/0102190. doi:10.1007/s002080000115. ISSN 0025-5831. Cannings, R.C.; Holland, M.P. (1994). "Right Ideals of Rings of Differential Operators". Journal of Algebra. 167 (1). Elsevier BV: 116–141. doi:10.1006/jabr.1994.1179. ISSN 0021-8693. Lebruyn, L. (1995). "Moduli Spaces for Right Ideals of the Weyl Algebra". Journal of Algebra. 172 (1). Elsevier BV: 32–48. doi:10.1006/jabr.1995.1046. hdl:10067/123950151162165141. ISSN 0021-8693.
Wikipedia:Weyl expansion#0	In physics, the Weyl expansion, also known as the Weyl identity or angular spectrum expansion, expresses an outgoing spherical wave as a linear combination of plane waves. In a Cartesian coordinate system, it can be denoted as e − j k 0 r r = 1 j 2 π ∫ − ∞ ∞ ∫ − ∞ ∞ d k x d k y e − j ( k x x + k y y ) e − j k z \| z \| k z {\displaystyle {\frac {e^{-jk_{0}r}}{r}}={\frac {1}{j2\pi }}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }dk_{x}dk_{y}e^{-j(k_{x}x+k_{y}y)}{\frac {e^{-jk_{z}\|z\|}}{k_{z}}}} , where k x {\displaystyle k_{x}} , k y {\displaystyle k_{y}} and k z {\displaystyle k_{z}} are the wavenumbers in their respective coordinate axes: k 0 = k x 2 + k y 2 + k z 2 {\displaystyle k_{0}={\sqrt {k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}}} . The expansion is named after Hermann Weyl, who published it in 1919. The Weyl identity is largely used to characterize the reflection and transmission of spherical waves at planar interfaces; it is often used to derive the Green's functions for Helmholtz equation in layered media. The expansion also covers evanescent wave components. It is often preferred to the Sommerfeld identity when the field representation is needed to be in Cartesian coordinates. The resulting Weyl integral is commonly encountered in microwave integrated circuit analysis and electromagnetic radiation over a stratified medium; as in the case for Sommerfeld integral, it is numerically evaluated. As a result, it is used in calculation of Green's functions for method of moments for such geometries. Other uses include the descriptions of dipolar emissions near surfaces in nanophotonics, holographic inverse scattering problems, Green's functions in quantum electrodynamics and acoustic or seismic waves. == See also == Angular spectrum method Fourier optics Green's function Plane wave expansion Sommerfeld identity == References == == Sources == Aki, Keiiti; Richards, Paul G. (2002). Quantitative Seismology (2 ed.). Sausalito: University Science Books. ISBN 9781891389634. Chew, Weng Cho (1990). Waves and Fields in Inhomogeneous Media. New York: Van Nostrand Reinhold. ISBN 9780780347496. Kinayman, Noyan; Aksun, M. I. (2005). Modern Microwave Circuits. Norwood: Artech House. ISBN 9781844073832. Novotny, Lukas; Hecht, Bert (2012). Principles of Nano-Optics. Norwood: Cambridge University Press. ISBN 9780511794193.
Wikipedia:Weyl's inequality#0	In linear algebra, Weyl's inequality is a theorem about the changes to eigenvalues of an Hermitian matrix that is perturbed. It can be used to estimate the eigenvalues of a perturbed Hermitian matrix. == Weyl's inequality about perturbation == Let A , B {\textstyle A,B} be Hermitian on inner product space V {\textstyle V} with dimension n {\textstyle n} , with spectrum ordered in descending order λ 1 ≥ . . . ≥ λ n {\textstyle \lambda _{1}\geq ...\geq \lambda _{n}} . Note that these eigenvalues can be ordered, because they are real (as eigenvalues of Hermitian matrices). Weyl's inequality states that the spectrum of Hermitian matrices is stable under perturbation. Specifically, we have: In jargon, it says that λ k {\displaystyle \lambda _{k}} is Lipschitz-continuous on the space of Hermitian matrices with operator norm. == Weyl's inequality between eigenvalues and singular values == Let A ∈ C n × n {\displaystyle A\in \mathbb {C} ^{n\times n}} have singular values σ 1 ( A ) ≥ ⋯ ≥ σ n ( A ) ≥ 0 {\displaystyle \sigma _{1}(A)\geq \cdots \geq \sigma _{n}(A)\geq 0} and eigenvalues ordered so that \| λ 1 ( A ) \| ≥ ⋯ ≥ \| λ n ( A ) \| {\displaystyle \|\lambda _{1}(A)\|\geq \cdots \geq \|\lambda _{n}(A)\|} . Then \| λ 1 ( A ) ⋯ λ k ( A ) \| ≤ σ 1 ( A ) ⋯ σ k ( A ) {\displaystyle \|\lambda _{1}(A)\cdots \lambda _{k}(A)\|\leq \sigma _{1}(A)\cdots \sigma _{k}(A)} For k = 1 , … , n {\displaystyle k=1,\ldots ,n} , with equality for k = n {\displaystyle k=n} . == Applications == === Estimating perturbations of the spectrum === Assume that R {\displaystyle R} is small in the sense that its spectral norm satisfies ‖ R ‖ 2 ≤ ϵ {\displaystyle \\|R\\|_{2}\leq \epsilon } for some small ϵ > 0 {\displaystyle \epsilon >0} . Then it follows that all the eigenvalues of R {\displaystyle R} are bounded in absolute value by ϵ {\displaystyle \epsilon } . Applying Weyl's inequality, it follows that the spectra of the Hermitian matrices M and N are close in the sense that \| μ i − ν i \| ≤ ϵ ∀ i = 1 , … , n . {\displaystyle \|\mu _{i}-\nu _{i}\|\leq \epsilon \qquad \forall i=1,\ldots ,n.} Note, however, that this eigenvalue perturbation bound is generally false for non-Hermitian matrices (or more accurately, for non-normal matrices). For a counterexample, let t > 0 {\displaystyle t>0} be arbitrarily small, and consider M = [ 0 0 1 / t 2 0 ] , N = M + R = [ 0 1 1 / t 2 0 ] , R = [ 0 1 0 0 ] . {\displaystyle M={\begin{bmatrix}0&0\\1/t^{2}&0\end{bmatrix}},\qquad N=M+R={\begin{bmatrix}0&1\\1/t^{2}&0\end{bmatrix}},\qquad R={\begin{bmatrix}0&1\\0&0\end{bmatrix}}.} whose eigenvalues μ 1 = μ 2 = 0 {\displaystyle \mu _{1}=\mu _{2}=0} and ν 1 = + 1 / t , ν 2 = − 1 / t {\displaystyle \nu _{1}=+1/t,\nu _{2}=-1/t} do not satisfy \| μ i − ν i \| ≤ ‖ R ‖ 2 = 1 {\displaystyle \|\mu _{i}-\nu _{i}\|\leq \\|R\\|_{2}=1} . === Weyl's inequality for singular values === Let M {\displaystyle M} be a p × n {\displaystyle p\times n} matrix with 1 ≤ p ≤ n {\displaystyle 1\leq p\leq n} . Its singular values σ k ( M ) {\displaystyle \sigma _{k}(M)} are the p {\displaystyle p} positive eigenvalues of the ( p + n ) × ( p + n ) {\displaystyle (p+n)\times (p+n)} Hermitian augmented matrix [ 0 M M ∗ 0 ] . {\displaystyle {\begin{bmatrix}0&M\\M^{*}&0\end{bmatrix}}.} Therefore, Weyl's eigenvalue perturbation inequality for Hermitian matrices extends naturally to perturbation of singular values. This result gives the bound for the perturbation in the singular values of a matrix M {\displaystyle M} due to an additive perturbation Δ {\displaystyle \Delta } : \| σ k ( M + Δ ) − σ k ( M ) \| ≤ σ 1 ( Δ ) {\displaystyle \|\sigma _{k}(M+\Delta )-\sigma _{k}(M)\|\leq \sigma _{1}(\Delta )} where we note that the largest singular value σ 1 ( Δ ) {\displaystyle \sigma _{1}(\Delta )} coincides with the spectral norm ‖ Δ ‖ 2 {\displaystyle \\|\Delta \\|_{2}} . == Notes == == References == Matrix Theory, Joel N. Franklin, (Dover Publications, 1993) ISBN 0-486-41179-6 "Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen", H. Weyl, Math. Ann., 71 (1912), 441–479
Wikipedia:Weyr canonical form#0	In mathematics, in linear algebra, a Weyr canonical form (or, Weyr form or Weyr matrix) is a square matrix which (in some sense) induces "nice" properties with matrices it commutes with. It also has a particularly simple structure and the conditions for possessing a Weyr form are fairly weak, making it a suitable tool for studying classes of commuting matrices. A square matrix is said to be in the Weyr canonical form if the matrix has the structure defining the Weyr canonical form. The Weyr form was discovered by the Czech mathematician Eduard Weyr in 1885. The Weyr form did not become popular among mathematicians and it was overshadowed by the closely related, but distinct, canonical form known by the name Jordan canonical form. The Weyr form has been rediscovered several times since Weyr’s original discovery in 1885. This form has been variously called as modified Jordan form, reordered Jordan form, second Jordan form, and H-form. The current terminology is credited to Shapiro who introduced it in a paper published in the American Mathematical Monthly in 1999. Recently several applications have been found for the Weyr matrix. Of particular interest is an application of the Weyr matrix in the study of phylogenetic invariants in biomathematics. == Definitions == === Basic Weyr matrix === === Definition === A basic Weyr matrix with eigenvalue λ {\displaystyle \lambda } is an n × n {\displaystyle n\times n} matrix W {\displaystyle W} of the following form: There is an integer partition n 1 + n 2 + ⋯ + n r = n {\displaystyle n_{1}+n_{2}+\cdots +n_{r}=n} of n {\displaystyle n} with n 1 ≥ n 2 ≥ ⋯ ≥ n r ≥ 1 {\displaystyle n_{1}\geq n_{2}\geq \cdots \geq n_{r}\geq 1} such that, when W {\displaystyle W} is viewed as an r × r {\displaystyle r\times r} block matrix ( W i j ) {\displaystyle (W_{ij})} , where the ( i , j ) {\displaystyle (i,j)} block W i j {\displaystyle W_{ij}} is an n i × n j {\displaystyle n_{i}\times n_{j}} matrix, the following three features are present: The main diagonal blocks W i i {\displaystyle W_{ii}} are the n i × n i {\displaystyle n_{i}\times n_{i}} scalar matrices λ I {\displaystyle \lambda I} for i = 1 , … , r {\displaystyle i=1,\ldots ,r} . The first superdiagonal blocks W i , i + 1 {\displaystyle W_{i,i+1}} are full column rank n i × n i + 1 {\displaystyle n_{i}\times n_{i+1}} matrices in reduced row-echelon form (that is, an identity matrix followed by zero rows) for i = 1 , … , r − 1 {\displaystyle i=1,\ldots ,r-1} . All other blocks of W are zero (that is, W i j = 0 {\displaystyle W_{ij}=0} when j ≠ i , i + 1 {\displaystyle j\neq i,i+1} ). In this case, we say that W {\displaystyle W} has Weyr structure ( n 1 , n 2 , … , n r ) {\displaystyle (n_{1},n_{2},\ldots ,n_{r})} . === Example === The following is an example of a basic Weyr matrix. In this matrix, n = 9 {\displaystyle n=9} and n 1 = 4 , n 2 = 2 , n 3 = 2 , n 4 = 1 {\displaystyle n_{1}=4,n_{2}=2,n_{3}=2,n_{4}=1} . So W {\displaystyle W} has the Weyr structure ( 4 , 2 , 2 , 1 ) {\displaystyle (4,2,2,1)} . Also, and === General Weyr matrix === === Definition === Let W {\displaystyle W} be a square matrix and let λ 1 , … , λ k {\displaystyle \lambda _{1},\ldots ,\lambda _{k}} be the distinct eigenvalues of W {\displaystyle W} . We say that W {\displaystyle W} is in Weyr form (or is a Weyr matrix) if W {\displaystyle W} has the following form: where W i {\displaystyle W_{i}} is a basic Weyr matrix with eigenvalue λ i {\displaystyle \lambda _{i}} for i = 1 , … , k {\displaystyle i=1,\ldots ,k} . === Example === The following image shows an example of a general Weyr matrix consisting of three basic Weyr matrix blocks. The basic Weyr matrix in the top-left corner has the structure (4,2,1) with eigenvalue 4, the middle block has structure (2,2,1,1) with eigenvalue -3 and the one in the lower-right corner has the structure (3, 2) with eigenvalue 0. == Relation between Weyr and Jordan forms == The Weyr canonical form W = P − 1 J P {\displaystyle W=P^{-1}JP} is related to the Jordan form J {\displaystyle J} by a simple permutation P {\displaystyle P} for each Weyr basic block as follows: The first index of each Weyr subblock forms the largest Jordan chain. After crossing out these rows and columns, the first index of each new subblock forms the second largest Jordan chain, and so forth. == The Weyr form is canonical == That the Weyr form is a canonical form of a matrix is a consequence of the following result: Each square matrix A {\displaystyle A} over an algebraically closed field is similar to a Weyr matrix W {\displaystyle W} which is unique up to permutation of its basic blocks. The matrix W {\displaystyle W} is called the Weyr (canonical) form of A {\displaystyle A} . == Computation of the Weyr canonical form == === Reduction to the nilpotent case === Let A {\displaystyle A} be a square matrix of order n {\displaystyle n} over an algebraically closed field and let the distinct eigenvalues of A {\displaystyle A} be λ 1 , λ 2 , … , λ k {\displaystyle \lambda _{1},\lambda _{2},\ldots ,\lambda _{k}} . The Jordan–Chevalley decomposition theorem states that A {\displaystyle A} is similar to a block diagonal matrix of the form A = [ λ 1 I + N 1 λ 2 I + N 2 ⋱ λ k I + N k ] = [ λ 1 I λ 2 I ⋱ λ k I ] + [ N 1 N 2 ⋱ N k ] = D + N {\displaystyle A={\begin{bmatrix}\lambda _{1}I+N_{1}&&&\\&\lambda _{2}I+N_{2}&&\\&&\ddots &\\&&&\lambda _{k}I+N_{k}\\\end{bmatrix}}={\begin{bmatrix}\lambda _{1}I&&&\\&\lambda _{2}I&&\\&&\ddots &\\&&&\lambda _{k}I\\\end{bmatrix}}+{\begin{bmatrix}N_{1}&&&\\&N_{2}&&\\&&\ddots &\\&&&N_{k}\\\end{bmatrix}}=D+N} where D {\displaystyle D} is a diagonal matrix, N {\displaystyle N} is a nilpotent matrix, and [ D , N ] = 0 {\displaystyle [D,N]=0} , justifying the reduction of N {\displaystyle N} into subblocks N i {\displaystyle N_{i}} . So the problem of reducing A {\displaystyle A} to the Weyr form reduces to the problem of reducing the nilpotent matrices N i {\displaystyle N_{i}} to the Weyr form. This leads to the generalized eigenspace decomposition theorem. === Reduction of a nilpotent matrix to the Weyr form === Given a nilpotent square matrix A {\displaystyle A} of order n {\displaystyle n} over an algebraically closed field F {\displaystyle F} , the following algorithm produces an invertible matrix C {\displaystyle C} and a Weyr matrix W {\displaystyle W} such that W = C − 1 A C {\displaystyle W=C^{-1}AC} . Step 1 Let A 1 = A {\displaystyle A_{1}=A} Step 2 Compute a basis for the null space of A 1 {\displaystyle A_{1}} . Extend the basis for the null space of A 1 {\displaystyle A_{1}} to a basis for the n {\displaystyle n} -dimensional vector space F n {\displaystyle F^{n}} . Form the matrix P 1 {\displaystyle P_{1}} consisting of these basis vectors. Compute P 1 − 1 A 1 P 1 = [ 0 B 2 0 A 2 ] {\displaystyle P_{1}^{-1}A_{1}P_{1}={\begin{bmatrix}0&B_{2}\\0&A_{2}\end{bmatrix}}} . A 2 {\displaystyle A_{2}} is a square matrix of size n {\displaystyle n} − nullity ( A 1 ) {\displaystyle (A_{1})} . Step 3 If A 2 {\displaystyle A_{2}} is nonzero, repeat Step 2 on A 2 {\displaystyle A_{2}} . Compute a basis for the null space of A 2 {\displaystyle A_{2}} . Extend the basis for the null space of A 2 {\displaystyle A_{2}} to a basis for the vector space having dimension n {\displaystyle n} − nullity ( A 1 ) {\displaystyle (A_{1})} . Form the matrix P 2 {\displaystyle P_{2}} consisting of these basis vectors. Compute P 2 − 1 A 2 P 2 = [ 0 B 3 0 A 3 ] {\displaystyle P_{2}^{-1}A_{2}P_{2}={\begin{bmatrix}0&B_{3}\\0&A_{3}\end{bmatrix}}} . A 2 {\displaystyle A_{2}} is a square matrix of size n {\displaystyle n} − nullity ( A 1 ) {\displaystyle (A_{1})} − nullity ( A 2 ) {\displaystyle (A_{2})} . Step 4 Continue the processes of Steps 1 and 2 to obtain increasingly smaller square matrices A 1 , A 2 , A 3 , … {\displaystyle A_{1},A_{2},A_{3},\ldots } and associated invertible matrices P 1 , P 2 , P 3 , … {\displaystyle P_{1},P_{2},P_{3},\ldots } until the first zero matrix A r {\displaystyle A_{r}} is obtained. Step 5 The Weyr structure of A {\displaystyle A} is ( n 1 , n 2 , … , n r ) {\displaystyle (n_{1},n_{2},\ldots ,n_{r})} where n i {\displaystyle n_{i}} = nullity ( A i ) {\displaystyle (A_{i})} . Step 6 Compute the matrix P = P 1 [ I 0 0 P 2 ] [ I 0 0 P 3 ] ⋯ [ I 0 0 P r ] {\displaystyle P=P_{1}{\begin{bmatrix}I&0\\0&P_{2}\end{bmatrix}}{\begin{bmatrix}I&0\\0&P_{3}\end{bmatrix}}\cdots {\begin{bmatrix}I&0\\0&P_{r}\end{bmatrix}}} (here the I {\displaystyle I} 's are appropriately sized identity matrices). Compute X = P − 1 A P {\displaystyle X=P^{-1}AP} . X {\displaystyle X} is a matrix of the following form: X = [ 0 X 12 X 13 ⋯ X 1 , r − 1 X 1 r 0 X 23 ⋯ X 2 , r − 1 X 2 r ⋱ ⋯ 0 X r − 1 , r 0 ] {\displaystyle X={\begin{bmatrix}0&X_{12}&X_{13}&\cdots &X_{1,r-1}&X_{1r}\\&0&X_{23}&\cdots &X_{2,r-1}&X_{2r}\\&&&\ddots &\\&&&\cdots &0&X_{r-1,r}\\&&&&&0\end{bmatrix}}} . Step 7 Use elementary row operations to find an invertible matrix Y r − 1 {\displaystyle Y_{r-1}} of appropriate size such that the product Y r − 1 X r , r − 1 {\displaystyle Y_{r-1}X_{r,r-1}} is a matrix of the form I r , r − 1 = [ I O ] {\displaystyle I_{r,r-1}={\begin{bmatrix}I\\O\end{bmatrix}}} . Step 8 Set Q 1 = {\displaystyle Q_{1}=} diag ( I , I , … , Y r − 1 − 1 , I ) {\displaystyle (I,I,\ldots ,Y_{r-1}^{-1},I)} and compute Q 1 − 1 X Q 1 {\displaystyle Q_{1}^{-1}XQ_{1}} . In this matrix, the ( r , r − 1 ) {\displaystyle (r,r-1)} -block is I r , r − 1 {\displaystyle I_{r,r-1}} . Step 9 Find a matrix R 1 {\displaystyle R_{1}} formed as a product of elementary matrices such that R 1 − 1 Q 1 − 1 X Q 1 R 1 {\displaystyle R_{1}^{-1}Q_{1}^{-1}XQ_{1}R_{1}} is a matrix in which all the blocks above the block I r , r − 1 {\displaystyle I_{r,r-1}} contain only 0 {\displaystyle 0} 's. Step 10 Repeat Steps 8 and 9 on column r − 1 {\displaystyle r-1} converting ( r − 1 , r − 2 ) {\displaystyle (r-1,r-2)} -block to I r − 1 , r − 2 {\displaystyle I_{r-1,r-2}} via conjugation by some invertible matrix Q 2 {\displaystyle Q_{2}} . Use this block to clear out the blocks above, via conjugation by a product R 2 {\displaystyle R_{2}} of elementary matrices. Step 11 Repeat these processes on r − 2 , r − 3 , … , 3 , 2 {\displaystyle r-2,r-3,\ldots ,3,2} columns, using conjugations by Q 3 , R 3 , … , Q r − 2 , R r − 2 , Q r − 1 {\displaystyle Q_{3},R_{3},\ldots ,Q_{r-2},R_{r-2},Q_{r-1}} . The resulting matrix W {\displaystyle W} is now in Weyr form. Step 12 Let C = P 1 diag ( I , P 2 ) ⋯ diag ( I , P r − 1 ) Q 1 R 1 Q 2 ⋯ R r − 2 Q r − 1 {\displaystyle C=P_{1}{\text{diag}}(I,P_{2})\cdots {\text{diag}}(I,P_{r-1})Q_{1}R_{1}Q_{2}\cdots R_{r-2}Q_{r-1}} . Then W = C − 1 A C {\displaystyle W=C^{-1}AC} . == Applications of the Weyr form == Some well-known applications of the Weyr form are listed below: The Weyr form can be used to simplify the proof of Gerstenhaber’s Theorem which asserts that the subalgebra generated by two commuting n × n {\displaystyle n\times n} matrices has dimension at most n {\displaystyle n} . A set of finite matrices is said to be approximately simultaneously diagonalizable if they can be perturbed to simultaneously diagonalizable matrices. The Weyr form is used to prove approximate simultaneous diagonalizability of various classes of matrices. The approximate simultaneous diagonalizability property has applications in the study of phylogenetic invariants in biomathematics. The Weyr form can be used to simplify the proofs of the irreducibility of the variety of all k-tuples of commuting complex matrices. == References ==
Wikipedia:Whitney extension theorem#0	In mathematics, in particular in mathematical analysis, the Whitney extension theorem is a partial converse to Taylor's theorem. Roughly speaking, the theorem asserts that if A is a closed subset of a Euclidean space, then it is possible to extend a given function of A in such a way as to have prescribed derivatives at the points of A. It is a result of Hassler Whitney. == Statement == A precise statement of the theorem requires careful consideration of what it means to prescribe the derivative of a function on a closed set. One difficulty, for instance, is that closed subsets of Euclidean space in general lack a differentiable structure. The starting point, then, is an examination of the statement of Taylor's theorem. Given a real-valued Cm function f(x) on Rn, Taylor's theorem asserts that for each a, x, y ∈ Rn, there is a function Rα(x,y) approaching 0 uniformly as x,y → a such that where the sum is over multi-indices α. Let fα = Dαf for each multi-index α. Differentiating (1) with respect to x, and possibly replacing R as needed, yields where Rα is o(\|x − y\|m−\|α\|) uniformly as x,y → a. Note that (2) may be regarded as purely a compatibility condition between the functions fα which must be satisfied in order for these functions to be the coefficients of the Taylor series of the function f. It is this insight which facilitates the following statement: Theorem. Suppose that fα are a collection of functions on a closed subset A of Rn for all multi-indices α with \| α \| ≤ m {\displaystyle \|\alpha \|\leq m} satisfying the compatibility condition (2) at all points x, y, and a of A. Then there exists a function F(x) of class Cm such that: F = f0 on A. DαF = fα on A. F is real-analytic at every point of Rn − A. Proofs are given in the original paper of Whitney (1934), and in Malgrange (1967), Bierstone (1980) and Hörmander (1990). == Extension in a half space == Seeley (1964) proved a sharpening of the Whitney extension theorem in the special case of a half space. A smooth function on a half space Rn,+ of points where xn ≥ 0 is a smooth function f on the interior xn for which the derivatives ∂α f extend to continuous functions on the half space. On the boundary xn = 0, f restricts to smooth function. By Borel's lemma, f can be extended to a smooth function on the whole of Rn. Since Borel's lemma is local in nature, the same argument shows that if Ω {\displaystyle \Omega } is a (bounded or unbounded) domain in Rn with smooth boundary, then any smooth function on the closure of Ω {\displaystyle \Omega } can be extended to a smooth function on Rn. Seeley's result for a half line gives a uniform extension map E : C ∞ ( R + ) → C ∞ ( R ) , {\displaystyle \displaystyle {E:C^{\infty }(\mathbf {R} ^{+})\rightarrow C^{\infty }(\mathbf {R} ),}} which is linear, continuous (for the topology of uniform convergence of functions and their derivatives on compacta) and takes functions supported in [0,R] into functions supported in [−R,R] To define E , {\displaystyle E,} set E ( f ) ( x ) = ∑ m = 1 ∞ a m f ( − b m x ) φ ( − b m x ) ( x < 0 ) , {\displaystyle \displaystyle {E(f)(x)=\sum _{m=1}^{\infty }a_{m}f(-b_{m}x)\varphi (-b_{m}x)\,\,\,(x<0),}} where φ is a smooth function of compact support on R equal to 1 near 0 and the sequences (am), (bm) satisfy: b m > 0 {\displaystyle b_{m}>0} tends to ∞ {\displaystyle \infty } ; ∑ a m b m j = ( − 1 ) j {\displaystyle \sum a_{m}b_{m}^{j}=(-1)^{j}} for j ≥ 0 {\displaystyle j\geq 0} with the sum absolutely convergent. A solution to this system of equations can be obtained by taking b n = 2 n {\displaystyle b_{n}=2^{n}} and seeking an entire function g ( z ) = ∑ m = 1 ∞ a m z m {\displaystyle g(z)=\sum _{m=1}^{\infty }a_{m}z^{m}} such that g ( 2 j ) = ( − 1 ) j . {\displaystyle g\left(2^{j}\right)=(-1)^{j}.} That such a function can be constructed follows from the Weierstrass theorem and Mittag-Leffler theorem. It can be seen directly by setting W ( z ) = ∏ j ≥ 1 ( 1 − z / 2 j ) , {\displaystyle W(z)=\prod _{j\geq 1}(1-z/2^{j}),} an entire function with simple zeros at 2 j . {\displaystyle 2^{j}.} The derivatives W '(2j) are bounded above and below. Similarly the function M ( z ) = ∑ j ≥ 1 ( − 1 ) j W ′ ( 2 j ) ( z − 2 j ) {\displaystyle M(z)=\sum _{j\geq 1}{(-1)^{j} \over W^{\prime }(2^{j})(z-2^{j})}} meromorphic with simple poles and prescribed residues at 2 j . {\displaystyle 2^{j}.} By construction g ( z ) = W ( z ) M ( z ) {\displaystyle \displaystyle {g(z)=W(z)M(z)}} is an entire function with the required properties. The definition for a half space in Rn by applying the operator E to the last variable xn. Similarly, using a smooth partition of unity and a local change of variables, the result for a half space implies the existence of an analogous extending map C ∞ ( Ω ¯ ) → C ∞ ( R n ) {\displaystyle \displaystyle {C^{\infty }({\overline {\Omega }})\rightarrow C^{\infty }(\mathbf {R} ^{n})}} for any domain Ω {\displaystyle \Omega } in Rn with smooth boundary. == See also == The Kirszbraun theorem gives extensions of Lipschitz functions. Tietze extension theorem – Continuous maps on a closed subset of a normal space can be extended Hahn–Banach theorem – Theorem on extension of bounded linear functionals == Notes == == References == McShane, Edward James (1934), "Extension of range of functions", Bull. Amer. Math. Soc., 40 (12): 837–842, doi:10.1090/s0002-9904-1934-05978-0, MR 1562984, Zbl 0010.34606 Whitney, Hassler (1934), "Analytic extensions of differentiable functions defined in closed sets", Transactions of the American Mathematical Society, 36 (1), American Mathematical Society: 63–89, doi:10.2307/1989708, JSTOR 1989708 Bierstone, Edward (1980), "Differentiable functions", Bulletin of the Brazilian Mathematical Society, 11 (2): 139–189, doi:10.1007/bf02584636 Malgrange, Bernard (1967), Ideals of differentiable functions, Tata Institute of Fundamental Research Studies in Mathematics, vol. 3, Oxford University Press Seeley, R. T. (1964), "Extension of C∞ functions defined in a half space", Proc. Amer. Math. Soc., 15: 625–626, doi:10.1090/s0002-9939-1964-0165392-8 Hörmander, Lars (1990), The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis, Springer-Verlag, ISBN 3-540-00662-1 Chazarain, Jacques; Piriou, Alain (1982), Introduction to the Theory of Linear Partial Differential Equations, Studies in Mathematics and Its Applications, vol. 14, Elsevier, ISBN 0444864520 Ponnusamy, S.; Silverman, Herb (2006), Complex variables with applications, Birkhäuser, ISBN 0-8176-4457-1 Fefferman, Charles (2005), "A sharp form of Whitney's extension theorem", Annals of Mathematics, 161 (1): 509–577, doi:10.4007/annals.2005.161.509, MR 2150391
Wikipedia:Wild problem#0	In the mathematical areas of linear algebra and representation theory, a problem is wild if it contains the problem of classifying pairs of square matrices up to simultaneous similarity. Examples of wild problems are classifying indecomposable representations of any quiver that is neither a Dynkin quiver (i.e. the underlying undirected graph of the quiver is a (finite) Dynkin diagram) nor a Euclidean quiver (i.e., the underlying undirected graph of the quiver is an affine Dynkin diagram). Necessary and sufficient conditions have been proposed to check the simultaneously block triangularization and diagonalization of a finite set of matrices under the assumption that each matrix is diagonalizable over the field of the complex numbers. == See also == Semi-invariant of a quiver == References ==
Wikipedia:Wilf Malcolm#0	Wilfred Gordon Malcolm (29 November 1933 – 6 October 2018) was a New Zealand mathematician and university administrator. He was professor of pure mathematics at Victoria University of Wellington from the mid 1970s, until serving as vice-chancellor of the University of Waikato between 1985 and 1994. == Biography == Born in Feilding on 29 November 1933, Malcolm was educated at Feilding Agricultural High School. He went on to study at Wellington Teachers' College and Victoria University College, graduating Master of Arts with first-class honours in 1957. He won a Shirtcliffe Fellowship, which enabled him to take parts II and III of the Mathematical Tripos, specialising in algebra and topology, at the University of Cambridge. While in England, Malcolm married Edmée Ruth Prebensen. Malcolm returned to Victoria, where he took up a lecturership in pure mathematics. Between 1964 and 1966, he spent time away from the university, working as the general secretary of the Inter-Varsity Fellowship of Evangelical Unions. However, he returned to lecturing at Victoria in 1967, and was promoted to senior lecturer the following year. In 1972, he completed his PhD thesis, titled Ultraproducts and higher order models, supervised by George Hughes and Max Cresswell from the Department of Philosophy, and C.J. Seelye from the Mathematics Department. In 1985, Malcolm moved to the University of Waikato to take up the vice-chancellorship, serving in that role until 1994. Malcolm was awarded the New Zealand 1990 Commemoration Medal in 1990. In the 1994 Queen's Birthday Honours, he was appointed a Commander of the Order of the British Empire, for services to tertiary education. The following year, he was conferred with an honorary doctorate by the University of Waikato. The Wilf Malcolm Institute of Educational Research at Waikato was named in his honour in 2002, in recognition of his contribution to education. Malcolm died in Auckland on 6 October 2018. == References ==
Wikipedia:Wilfred Cockcroft#0	Sir Wilfred Halliday Cockcroft (7 June 1923 – 27 September 1999) was an eminent mathematics educator from the University of Hull. == Early life == He attended Keighley Boys' Grammar School, now called Beckfoot Oakbank, and studied Mathematics at Balliol College, Oxford. During WWII he worked in radar. == Career == === Mathematics report === In 1978 he was commissioned by the then Labour government to chair a comprehensive inquiry into the teaching of mathematics in primary and secondary schools in England and Wales. The committee of inquiry produced its report in 1982, published as Mathematics Counts but widely known as "the Cockcroft report". === Examinations === From 1983 to 1988 he was Chairman and Chief Executive of the Secondary Exams Council. == Personal life == He married in 1949, with two sons, and later married in 1982. Cockcroft was knighted in 1983, and in May 1984 was awarded an honorary degree from the Open University as Doctor of the University. == References == == External links == "Mathematics Counts". London: Her Majesty's Stationery Office. 1982. Archived from the original on 20 January 2013. Tony Crilly (2001), Memories of Sir Wilfred Cockcroft 1923–1999, The Mathematical Gazette 85 (502), 72. JSTOR 3620472 Obituary, Bulletin of the London Mathematical Society (2005), 37, 149-155 Short biography, and account of archives at Hull University, Access to Archives
Wikipedia:Wilfried Imrich#0	Wilfried Imrich (born 25 May 1941) is an Austrian mathematician working mainly in graph theory. He is known for his work on graph products, and authored the books Product Graphs: Structure and Recognition (Wiley, 2000, with Sandi Klavžar), Topics in graph theory: Graphs and their Cartesian Products (AK Peters, 2008, with Klavžar and Douglas F. Rall), and Handbook of Product Graphs (2nd ed., CRC, 2011, with Klavžar and Richard Hammack). Imrich earned his doctorate from the University of Vienna in 1965, under the joint supervision of Nikolaus Hofreiter and Edmund Hlawka. He has worked as a researcher for IBM in Vienna, as an assistant professor at TU Wien and the University at Albany, SUNY, as a postdoctoral researcher at Lomonosov University, and, since 1973, as a full professor at the University of Leoben in Austria. He retired in 2009, becoming a professor emeritus at Leoben. He is on board of advisors of the journal Ars Mathematica Contemporanea. Since 2012 he has been a member of the Academia Europaea. == References == == External links == Personal web site "Wilfried Imrich's 75th Birthday Colloquium", Sandi Klavžar, Ars Math Contemp 11(2016) XXI
Wikipedia:William Feller#0	William "Vilim" Feller (July 7, 1906 – January 14, 1970), born Vilibald Srećko Feller, was a Croatian–American mathematician specializing in probability theory. == Early life and education == Feller was born in Zagreb to Ida Oemichen-Perc, a Croatian–Austrian Catholic, and Eugen Viktor Feller, son of a Polish–Jewish father (David Feller) and an Austrian mother (Elsa Holzer). Eugen Feller was a famous chemist and created Elsa fluid named after his mother. According to Gian-Carlo Rota, Eugen Feller's surname was a "Slavic tongue twister", which William changed at the age of twenty. This claim appears to be false. His forename, Vilibald, was chosen by his Catholic mother for the saint day of his birthday. == Career and later life == Feller held a docent position at the University of Kiel beginning in 1928. Because he refused to sign a Nazi oath, he fled the Nazis and went to Copenhagen, Denmark in 1933. He also lectured in Sweden (Stockholm and Lund). As a refugee in Sweden, Feller reported being troubled by increasing fascism at the universities. He reported that the mathematician Torsten Carleman would offer his opinion that Jews and foreigners should be executed. After marrying a former student from Kiel, Clara Mary Nielsen, in 1938, he moved with her to the US in 1939. In that year he joined Brown University as an associate professor; he became a US citizen in 1944. He moved to Cornell University in 1945 and to Princeton University in 1950. At Princeton, he became Eugene Higgins Professor of Mathematics. He remained there until his death on January 14, 1970. == Work == The works of Feller are contained in 104 papers and two books on a variety of topics such as mathematical analysis, theory of measurement, functional analysis, geometry, and differential equations in addition to his work in mathematical statistics and probability. Feller was one of the greatest probabilists of the twentieth century. He is remembered for his championing of probability theory as a branch of mathematical analysis in Sweden and the United States. In the middle of the 20th century, probability theory was popular in France and Russia, while mathematical statistics was more popular in the United Kingdom and the United States, according to the Swedish statistician, Harald Cramér. His two-volume textbook on probability theory and its applications was called "the most successful treatise on probability ever written" by Gian-Carlo Rota. By stimulating his colleagues and students in Sweden and then in the United States, Feller helped establish research groups studying the analytic theory of probability. In his research, Feller contributed to the study of the relationship between Markov chains and differential equations, where his theory of generators of one-parameter semigroups of stochastic processes gave rise to the theory of "Feller operators". Numerous topics relating to probability are named after him, including Feller processes, Feller's explosion test, Feller–Brown movement, and the Lindeberg–Feller theorem. Feller made fundamental contributions to renewal theory, Tauberian theorems, random walks, diffusion processes, and the law of the iterated logarithm. Feller was among those early editors who launched the journal Mathematical Reviews. == Books == An Introduction to Probability Theory and its Applications, Volume I, 3rd edition (1968); 1st edn. (1950); 2nd edn. (1957) An Introduction to Probability Theory and its Applications, Volume II, 2nd edition (1971) == Recognition == In 1949, Feller was named a Fellow of the American Statistical Association. He was elected to the American Academy of Arts and Sciences in 1958, the United States National Academy of Sciences in 1960, and the American Philosophical Society in 1966. Feller won the National Medal of Science in 1969. He was president of the Institute of Mathematical Statistics. == See also == Feller condition Beta distribution Compound Poisson distribution Gillespie algorithm Kolmogorov equations Poisson point process Stability (probability) St. Petersburg paradox Stochastic process == References == == External links == William Feller at the Mathematics Genealogy Project A biographical memoir by Murray Rosenblatt Croatian Giants of Science - in Croatian O'Connor, John J.; Robertson, Edmund F., "William Feller", MacTutor History of Mathematics Archive, University of St Andrews "Fine Hall in its golden age: Remembrances of Princeton in the early fifties" by Gian-Carlo Rota. Contains a section on Feller at Princeton. Feller Matriculation Form giving personal details
Wikipedia:William G. McCallum#0	William G. McCallum (born 1956 in Sydney, Australia) is a retired University Distinguished Professor of Mathematics at the University of Arizona. His professional interests include arithmetical algebraic geometry and mathematics education. == Education and professional work == McCallum received a bachelor's degree from the University of New South Wales in 1977, and his Ph.D. in Mathematics from Harvard University in 1984, under the supervision of Barry Mazur. He was a postdoctoral researcher at the University of California, Berkeley and the Mathematical Sciences Research Institute in Berkeley before joining the University of Arizona in 1987. He became University Distinguished Professor from 2006 until his retirement in 2018, and headed the Department of Mathematics from 2009 to 2013. McCallum helped found the Harvard Calculus Consortium with other mathematicians including Andrew M. Gleason and Deborah Hughes Hallett. He led the development of the Common Core standards for mathematics from 2009 until the standards were first released in 2010. In 2014 he founded a company, eventually known as Illustrative Mathematics, to develop teaching resources for the mathematics standards that he helped develop. == Selected honors and awards == In 2012, McCallum became a Fellow of the American Mathematical Society. In the same year he received the inaugural Mary P. Dolciani Award for distinguished contributions to mathematical education, administered by the Mathematical Association of America, and the AMS Distinguished Public Service Award of the American Mathematical Society, "for his energetic and effective efforts in promoting improvements to mathematics education" == References ==
Wikipedia:William H. Bossert#0	William H. Bossert (born 1937) is an American mathematician. He is the David B. Arnold, Jr. Professor of Science, Emeritus at Harvard University. He was the housemaster of Lowell House for 23 years. He received his PhD from Harvard in 1963. == Publications == With Edward O. Wilson A primer of population biology (1971) == References ==
Wikipedia:William Hamilton Martin#0	In September 1960, two U.S. National Security Agency (NSA) cryptologists, William Hamilton Martin and Bernon F. Mitchell, defected to the Soviet Union. A secret 1963 NSA study said that: "Beyond any doubt, no other event has had, or is likely to have in the future, a greater impact on the Agency's security program." Martin and Mitchell met while serving in the U.S. Navy in Japan in the early 1950s and both joined the NSA on the same day in 1957. They defected together to the Soviet Union in 1960 and, at a Moscow press conference, revealed and denounced various U.S. policies, especially provocative incursions into the air space of other nations and spying on America's own allies. Underscoring their apprehension of nuclear war, they said: "we would attempt to crawl to the moon if we thought it would lessen the threat of an atomic war." Within days of the press conference, citing a trusted source, Congressman Francis E. Walter, chairman of the House Un-American Activities Committee (HUAC), said Martin and Mitchell were "sex deviates", prompting sensational press coverage. U.S. officials at the National Security Council privately shared their assumption that the two were part of a traitorous homosexual network. Classified NSA investigations, on the other hand, determined the pair had "greatly inflated opinions concerning their intellectual attainments and talents" and had defected to satisfy social aspirations. The House Un-American Activities Committee publicly intimated its interpretation of the relationship between Martin and Mitchell as homosexual and that reading guided the Pentagon's discussion of the defection for decades. == Early lives and careers == William Hamilton Martin (May 27, 1931 – January 17, 1987) was born in Columbus, Georgia. His family soon moved to Washington state where his father was president of the Ellensburg Chamber of Commerce. He graduated from Ellensburg High School after two years. After studies at Central Washington College of Education (now Central Washington University), he earned a degree in mathematics from the University of Washington in Seattle in 1947. He enlisted in the United States Navy and served from 1951 to 1954, working as a cryptologist with the Naval Security Group in Japan. As hobbies, Martin played chess and collected Japanese sword handles (tsuka). Bernon F. Mitchell (March 11, 1929 – November 12, 2001) was born and raised in Eureka, California, and enlisted in the US Navy after one year of college. He gained experience as a cryptologist during a tour of duty in the Navy from 1951 to 1954, serving in Japan with the Naval Security Group at Kami Seya. He stayed on in Japan for another year, working for the Army Security Agency. Following his Navy service, in 1957 he earned a bachelor's degree in statistics at Stanford University. Martin and Mitchell became friends during their Navy service at the Naval communications intercept facility at Kami Seya, Japan. They kept in touch as each returned to school after their Navy service and encountered one another again when each was recruited into the National Security Agency (NSA) in 1957. Their years at the NSA were uneventful. Martin gained enough recognition that he was twice awarded scholarships for study toward a master's degree. == Background to defection == Mitchell and Martin became disturbed by what they learned of American incursions into foreign airspace and realized that Congress was unaware of those NSA-sponsored flights. In February 1959, in violation of NSA rules, they tried to report what they knew to a Congressman, Ohio Democrat Wayne Hays, who had expressed frustration with the information he was receiving from the NSA. In December 1959, the pair visited Cuba, without notifying their superiors as required by NSA procedures. According to Galina Mitchell, Bernon Mitchell's widow, Bernon Mitchell and William Martin were not communists; they chose the Soviet Union for two reasons: there women could work equally with men, and because the USSR had a developed chess culture. == Defection == On June 25, 1960, Mitchell and Martin left the U.S. for Mexico. They traveled from there to Havana and then sailed on a Russian freighter to the Soviet Union. On August 5, the Pentagon announced that they had not returned from vacation and said: "there is a likelihood that they have gone behind the Iron Curtain." On September 6, 1960, they appeared at a joint news conference at the House of Journalists in Moscow and announced they had requested asylum and Soviet citizenship. During the conference, the defectors made public for the first time the mission and activities of the NSA in a prepared statement written, they said: "without consulting the Government of the Soviet Union". It said that: "the United States Government is as unscrupulous as it has accused the Soviet Government of being". They also said: Our main dissatisfaction concerned some of the practices the United States uses in gathering intelligence information ... deliberately violating the airspace of other nations ... intercepting and deciphering the secret communications of its own allies ... Perhaps United States hostility towards Communism arises out of a feeling of insecurity engendered by Communist achievements in science, culture and industry. As we know from our previous experience working at N.S.A., the United States successfully reads the secure communications of more than forty nations, including its own allies. They particularly attacked the views of General Thomas S. Power who had recently told a Congressional committee that the U.S. needed to maintain a nuclear first-strike capability and Senator Barry Goldwater's opposition to banning nuclear tests and negotiating a disarmament treaty. By contrast, they said: "we would attempt to crawl to the moon if we thought it would lessen the threat of an atomic war." The U.S. had recently admitted sending reconnaissance flights over foreign countries in recent years, but Martin and Mitchell said they knew from their Navy service that such flights had occurred as early as 1952–1954. They detailed a U.S. C-130 flight over Soviet Armenia that the Soviets brought down. They contended that it was designed to gain an understanding of Soviet defenses, and that it therefore represented an American interest in attacking the Soviets rather than defending against them. They also complained of restrictions on freedom in the U.S., such as government confiscation of mail, particularly the freedom of those who are "not theists" or "whose political convictions are unpopular". In an interview with the Soviet news agency Tass in December 1960, they expressed their belief that American espionage against Russia, U.S. allies, and neutral nations would continue unchanged despite the inauguration of a new American president in January 1961. In response, the American government called Mitchell and Martin's charges "completely false". The Department of Defense called them "turncoats" and "tools of Soviet propaganda", "one mentally sick and both obviously confused". It also characterized their positions at the NSA as "junior mathematicians". === Initial allegations of homosexuality === The New York Times described them as "long-time bachelor friends" and reported they smiled at each other only when they described the social advantages they anticipated in the Soviet Union, where, their prepared statement said: "The talents of women are encouraged and utilized to a much greater extent in the Soviet Union than in the United States. We feel that this enriches Soviet society and makes Soviet women more desirable as mates." The issue of the pair's sexuality was raised and dismissed by the government: "Representative Francis E. Walter, Democrat of Pennsylvania [and chairman of the House Un-American Activities Committee], denied that he had made an allegation, reported by a news agency, that one of the men had been described as a homosexual in a report by the Federal Bureau of Investigation." A Pentagon spokesman told reporters that there was nothing in Mitchell and Martin's personnel records to suggest homosexuality or sexual perversion. The next day, Congressman Walter explicitly stated that a source he trusted had told him that the two defectors were "known to their acquaintances as 'sex deviates'". That charge was promptly picked up by the press and resulted immediately in stories about homosexuals recruiting "other sexual deviates" for jobs in the federal government. Hearst newspapers referred to Martin and Mitchell as "two defecting blackmailed homosexual specialists" and a "love team". Time reported that a review of security checks turned up a Mitchell visit to a psychiatrist "presumably out of concern for homosexual tendencies". == Later years == According to a later government report, Martin—who was fluent in Russian—studied at Leningrad University (now Saint Petersburg State University), and used the name Vladimir Sokolodsky. He married a Soviet citizen whom he divorced in 1963. He later told a Russian newspaper that his defection had been "foolhardy". He also expressed disappointment that the Russians did not trust him with important work. He occasionally sought the help of American visitors in arranging for repatriation, including Donald Duffy, vice president of the Kaiser Foundation, and bandleader Benny Goodman. On another occasion he told an American that before defecting he had believed the vision of Russia presented by propaganda publications like USSR and Soviet Life. By 1975, a source told the U.S. government Martin was "totally on the skids." In 1979, he inquired at the American consulate about repatriation. As a result, his case was examined and he was stripped of his American citizenship. He was next denied permission to immigrate to the U.S. and then denied a tourist visa. Martin eventually left the Soviet Union and died of cancer in Mexico on January 17, 1987, at Tijuana's Hospital Del Mar. He was buried in the U.S. Less is known of Mitchell. Having renounced his U.S. citizenship, he remained in the Soviet Union. He married Galina Vladimirovna Yakovleva, a member of the piano department faculty at the Leningrad Conservatory. He worked at the Bonch-Bruevich Institute in Leningrad as a programmer. In 1979, Bernon Mitchell tried to emigrate to the United States with his wife Galina. After this became known, Galina was expelled from the ranks of the Communist Party of the Soviet Union. After reviewing the documents, Bernon Mitchell was denied entry into the United States. The visa was issued only to Galina, who refused to emigrate without her husband. Mitchell died of a heart attack in November 2001, and was buried in St. Petersburg. == Government response == The defections had another life inside the U.S. intelligence community. At a meeting of the National Security Council in October 1960, officials considered a response to the Martin-Mitchell affair. Attorney General William P. Rogers believed that the Soviets had a list of homosexuals to use in their recruiting and blackmail efforts, that Martin and Mitchell were part of "an organized group". Several at the meeting thought polygraph tests would help prevent the hiring of homosexuals. President Eisenhower himself wanted a central authority to coordinate all government lists of homosexuals. In order to prevent another occurrence, the NSA needed to understand what motivated the defectors. Their initial investigation turned up little of interest. Notes of psychological counseling sessions from the 1940s described Martin as "brilliant but emotionally immature" and offered a diagnosis of "beginning character neurosis with schizoid tendencies" and mentioned he was likely "sadistic". Mitchell had told the NSA when questioned not long after starting work at the Agency that he had experimented sexually as a teenager with dogs and chickens. The immediate NSA response focused on sexual issues. In July 1961 the Agency announced that it had purged 26 employees it identified as "sexual deviates" though it added the qualification that "not all were homosexuals". Yet a series of NSA investigations gave little credit to the role of sexuality in Mitchell and Martin's defection. In 1961, an NSA report called them "close friends and somewhat anti-social", "egotistical, arrogant and insecure young men whose place in society was much lower than they believed they deserved", with "greatly inflated opinions concerning their intellectual attainments and talents". In 1963, another NSA report found "no clear motive", that they had not been recruited by foreigners, and termed the defection "impulsive". NSA files obtained by journalists at the Seattle Weekly in 2007 cited definitive testimony on the part of women acquaintances who attested to their heterosexuality. The only perversions recorded were Martin's "all-controlling sadomasochism". He had occasionally watched women having sex or had sex himself with multiple female partners. In 1962, Congressman Walters' House Un-American Activities Committee (HUAC) concluded its 13-month investigation and issued a report on the defections. Where Mitchell had told his psychiatrist that he had affairs with both men and women and was not troubled about his sexual identity, the report referred to his "homosexual problems". The report never identified a rationale for the Mitchell and Martin defections, but focused on the inadequacy of the investigations that granted them security clearances despite evidence of "homosexuality or other sexual abnormality", atheism, and Communist sympathies on the part of one or both of the men. The report made a series of recommendations with respect to NSA hiring practices and security investigations that were promptly adopted by the Agency. Later government analyses went beyond the characterizations in the HUAC account, unaware of the NSA's unpublished analysis. Despite the contrary evidence, a 1991 study by the Pentagon's Defense Security Service—still in use in 2007—called Martin and Mitchell "publicly known homosexuals". == See also == Project HOMERUN Lavender scare Perry Fellwock == Notes == == References == == Additional sources == Barrett, David M. (2009). "Secrecy, Security, and Sex: The NSA, Congress, and the Martin–Mitchell Defections". International Journal of Intelligence and CounterIntelligence. 22 (4): 699–729. doi:10.1080/08850600903143320. S2CID 154112700. Obituary, Eureka Times-Standard, November 2001
Wikipedia:William Hanson Dodge#0	William Hanson Dodge (March 5, 1866 – February 1, 1932) was an American photographer in the late nineteenth and early twentieth centuries, living in Lowell, Massachusetts. His son, Harold F. Dodge, a noted mathematician, was a pioneer in the field of statistical quality control. William Dodge was employed by the Lowell Manufacturing Company; this firm was later purchased by the Bigelow Carpet Company. His work as a designer and color mixer was augmented by his hobby: photography. Dodge was a skilled photographer: in 1894 he won a bronze medal at a competition in New York City for a composition entitled “Winter.” This photo appears in W. I. Lincoln Adams's book Sunlight and Shadow, where it was included as an example of “successful landscape work”; other photographers whose work is featured in the book include Alfred Stieglitz and H. P. Robinson. Later that year, another of his photographs at an exhibition in New York, “December Morning,” was highly praised by a reviewer in American Amateur Photographer. Another activity for Dodge was bicycle racing. Winning many ribbons as a racer, he belonged to the Spindle City Club, of Lowell, Massachusetts, taking many long trips with this group. He also served as a referee and official at races throughout the region during the 1890s. During these years, Dodge was also active in the Knights of Pythias, a men's fraternal organization, serving as chancellor and master of works. In the early twentieth century, William Dodge turned to politics. He was elected to the Lowell Common Council in 1903. In 1905 he was elected president of the council, a year-long position, after which he served on the Board of Aldermen for a year. Around 1910 William Dodge moved to Detroit, where he opened a business selling X-ray equipment to doctors and hospitals. His sales trips frequently took him to Chicago. He died in Detroit on February 1, 1932. == References ==
Wikipedia:William Kingdon Clifford#0	William Kingdon Clifford (4 May 1845 – 3 March 1879) was a British mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his honour. The operations of geometric algebra have the effect of mirroring, rotating, translating, and mapping the geometric objects that are being modelled to new positions. Clifford algebras in general and geometric algebra in particular have been of ever increasing importance to mathematical physics, geometry, and computing. Clifford was the first to suggest that gravitation might be a manifestation of an underlying geometry. In his philosophical writings he coined the expression mind-stuff. == Biography == Born in Exeter, William Clifford was educated at Doctor Templeton's Academy on Bedford Circus and showed great promise at school. He went on to King's College London (at age 15) and Trinity College, Cambridge, where he was elected fellow in 1868, after being Second Wrangler in 1867 and second Smith's prizeman. In 1870, he was part of an expedition to Italy to observe the solar eclipse of 22 December 1870. During that voyage he survived a shipwreck along the Sicilian coast. In 1871, he was appointed professor of mathematics and mechanics at University College London, and in 1874 became a fellow of the Royal Society. He was also a member of the London Mathematical Society and the Metaphysical Society. Clifford married Lucy Lane on 7 April 1875, with whom he had two children. Clifford enjoyed entertaining children and wrote a collection of fairy stories, The Little People. === Death and legacy === In 1876, Clifford suffered a breakdown, probably brought on by overwork. He taught and administered by day, and wrote by night. A half-year holiday in Algeria and Spain allowed him to resume his duties for 18 months, after which he collapsed again. He went to the island of Madeira to recover, but died there of tuberculosis after a few months, leaving a widow with two children. Clifford and his wife are buried in London's Highgate Cemetery, near the graves of George Eliot and Herbert Spencer, just north of the grave of Karl Marx. The academic journal Advances in Applied Clifford Algebras publishes on Clifford's legacy in kinematics and abstract algebra. == Mathematics == "Clifford was above all and before all a geometer." The discovery of non-Euclidean geometry opened new possibilities in geometry in Clifford's era. The field of intrinsic differential geometry was born, with the concept of curvature broadly applied to space itself as well as to curved lines and surfaces. Clifford was very much impressed by Bernhard Riemann’s 1854 essay "On the hypotheses which lie at the bases of geometry". In 1870, he reported to the Cambridge Philosophical Society on the curved space concepts of Riemann, and included speculation on the bending of space by gravity. Clifford's translation of Riemann's paper was published in Nature in 1873. His report at Cambridge, "On the Space-Theory of Matter", was published in 1876, anticipating Albert Einstein's general relativity by 40 years. Clifford elaborated elliptic space geometry as a non-Euclidean metric space. Equidistant curves in elliptic space are now said to be Clifford parallels. Clifford's contemporaries considered him acute and original, witty and warm. He often worked late into the night, which may have hastened his death. He published papers on a range of topics including algebraic forms and projective geometry and the textbook Elements of Dynamic. His application of graph theory to invariant theory was followed up by William Spottiswoode and Alfred Kempe. === Algebras === In 1878, Clifford published a seminal work, building on Grassmann's extensive algebra. He had succeeded in unifying the quaternions, developed by William Rowan Hamilton, with Grassmann's outer product (aka the exterior product). He understood the geometric nature of Grassmann's creation, and that the quaternions fit cleanly into the algebra Grassmann had developed. The versors in quaternions facilitate representation of rotation. Clifford laid the foundation for a geometric product, composed of the sum of the inner product and Grassmann's outer product. The geometric product was eventually formalized by the Hungarian mathematician Marcel Riesz. The inner product equips geometric algebra with a metric, fully incorporating distance and angle relationships for lines, planes, and volumes, while the outer product gives those planes and volumes vector-like properties, including a directional bias. Combining the two brought the operation of division into play. This greatly expanded our qualitative understanding of how objects interact in space. Crucially, it also provided the means for quantitatively calculating the spatial consequences of those interactions. The resulting geometric algebra, as he called it, eventually realized the long sought goal of creating an algebra that mirrors the movements and projections of objects in 3-dimensional space. Moreover, Clifford's algebraic schema extends to higher dimensions. The algebraic operations have the same symbolic form as they do in 2 or 3-dimensions. The importance of general Clifford algebras has grown over time, while their isomorphism classes - as real algebras - have been identified in other mathematical systems beyond simply the quaternions. The realms of real analysis and complex analysis have been expanded through the algebra H of quaternions, thanks to its notion of a three-dimensional sphere embedded in a four-dimensional space. Quaternion versors, which inhabit this 3-sphere, provide a representation of the rotation group SO(3). Clifford noted that Hamilton's biquaternions were a tensor product H ⊗ C {\displaystyle H\otimes C} of known algebras, and proposed instead two other tensor products of H: Clifford argued that the "scalars" taken from the complex numbers C might instead be taken from split-complex numbers D or from the dual numbers N. In terms of tensor products, H ⊗ D {\displaystyle H\otimes D} produces split-biquaternions, while H ⊗ N {\displaystyle H\otimes N} forms dual quaternions. The algebra of dual quaternions is used to express screw displacement, a common mapping in kinematics. == Philosophy == As a philosopher, Clifford's name is chiefly associated with two phrases of his coining, mind-stuff and the tribal self. The former symbolizes his metaphysical conception, suggested to him by his reading of Baruch Spinoza, which Clifford (1878) defined as follows: That element of which, as we have seen, even the simplest feeling is a complex, I shall call Mind-stuff. A moving molecule of inorganic matter does not possess mind or consciousness; but it possesses a small piece of mind-stuff. When molecules are so combined together as to form the film on the under side of a jelly-fish, the elements of mind-stuff which go along with them are so combined as to form the faint beginnings of Sentience. When the molecules are so combined as to form the brain and nervous system of a vertebrate, the corresponding elements of mind-stuff are so combined as to form some kind of consciousness; that is to say, changes in the complex which take place at the same time get so linked together that the repetition of one implies the repetition of the other. When matter takes the complex form of a living human brain, the corresponding mind-stuff takes the form of a human consciousness, having intelligence and volition. Regarding Clifford's concept, Sir Frederick Pollock wrote: Briefly put, the conception is that mind is the one ultimate reality; not mind as we know it in the complex forms of conscious feeling and thought, but the simpler elements out of which thought and feeling are built up. The hypothetical ultimate element of mind, or atom of mind-stuff, precisely corresponds to the hypothetical atom of matter, being the ultimate fact of which the material atom is the phenomenon. Matter and the sensible universe are the relations between particular organisms, that is, mind organized into consciousness, and the rest of the world. This leads to results which would in a loose and popular sense be called materialist. But the theory must, as a metaphysical theory, be reckoned on the idealist side. To speak technically, it is an idealist monism. Tribal self, on the other hand, gives the key to Clifford's ethical view, which explains conscience and the moral law by the development in each individual of a 'self,' which prescribes the conduct conducive to the welfare of the 'tribe.' Much of Clifford's contemporary prominence was due to his attitude toward religion. Animated by an intense love of his conception of truth and devotion to public duty, he waged war on such ecclesiastical systems as seemed to him to favour obscurantism, and to put the claims of sect above those of human society. The alarm was greater, as theology was still unreconciled with Darwinism; and Clifford was regarded as a dangerous champion of the anti-spiritual tendencies then imputed to modern science. There has also been debate on the extent to which Clifford's doctrine of 'concomitance' or 'psychophysical parallelism' influenced John Hughlings Jackson's model of the nervous system and, through him, the work of Janet, Freud, Ribot, and Ey. === Ethics === In his 1877 essay, The Ethics of Belief, Clifford argues that it is immoral to believe things for which one lacks evidence. He describes a ship-owner who planned to send to sea an old and not well-built ship full of passengers. The ship-owner had doubts suggested to him that the ship might not be seaworthy: "These doubts preyed upon his mind, and made him unhappy." He considered having the ship refitted even though it would be expensive. At last, "he succeeded in overcoming these melancholy reflections." He watched the ship depart, "with a light heart…and he got his insurance money when she went down in mid-ocean and told no tales." Clifford argues that the ship-owner was guilty of the deaths of the passengers even though he sincerely believed the ship was sound: "[H]e had no right to believe on such evidence as was before him." Moreover, he contends that even in the case where the ship successfully reaches the destination, the decision remains immoral, because the morality of the choice is defined forever once the choice is made, and actual outcome, defined by blind chance, doesn't matter. The ship-owner would be no less guilty: his wrongdoing would never be discovered, but he still had no right to make that decision given the information available to him at the time. Clifford famously concludes with what has come to be known as Clifford's principle: "it is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence." As such, he is arguing in direct opposition to religious thinkers for whom 'blind faith' (i.e. belief in things in spite of the lack of evidence for them) was a virtue. This paper was famously attacked by pragmatist philosopher William James in his "Will to Believe" lecture. Often these two works are read and published together as touchstones for the debate over evidentialism, faith, and overbelief. == Premonition of relativity == Though Clifford never constructed a full theory of spacetime and relativity, there are some remarkable observations he made in print that foreshadowed these modern concepts: In his book Elements of Dynamic (1878), he introduced "quasi-harmonic motion in a hyperbola". He wrote an expression for a parametrized unit hyperbola, which other authors later used as a model for relativistic velocity. Elsewhere he states: The geometry of rotors and motors…forms the basis of the whole modern theory of the relative rest (Static) and the relative motion (Kinematic and Kinetic) of invariable systems. This passage makes reference to biquaternions, though Clifford made these into split-biquaternions as his independent development. The book continues with a chapter "On the bending of space", the substance of general relativity. Clifford also discussed his views in On the Space-Theory of Matter in 1876. In 1910, William Barrett Frankland quoted the Space-Theory of Matter in his book on parallelism: "The boldness of this speculation is surely unexcelled in the history of thought. Up to the present, however, it presents the appearance of an Icarian flight." Years later, after general relativity had been advanced by Albert Einstein, various authors noted that Clifford had anticipated Einstein. Hermann Weyl (1923), for instance, mentioned Clifford as one of those who, like Bernhard Riemann, anticipated the geometric ideas of relativity. In 1940, Eric Temple Bell published The Development of Mathematics, in which he discusses the prescience of Clifford on relativity: Bolder even than Riemann, Clifford confessed his belief (1870) that matter is only a manifestation of curvature in a space-time manifold. This embryonic divination has been acclaimed as an anticipation of Einstein's (1915–16) relativistic theory of the gravitational field. The actual theory, however, bears but slight resemblance to Clifford's rather detailed creed. As a rule, those mathematical prophets who never descend to particulars make the top scores. Almost anyone can hit the side of a barn at forty yards with a charge of buckshot. John Archibald Wheeler, during the 1960 International Congress for Logic, Methodology, and Philosophy of Science (CLMPS) at Stanford, introduced his geometrodynamics formulation of general relativity by crediting Clifford as the initiator. In The Natural Philosophy of Time (1961), Gerald James Whitrow recalls Clifford's prescience, quoting him in order to describe the Friedmann–Lemaître–Robertson–Walker metric in cosmology. Cornelius Lanczos (1970) summarizes Clifford's premonitions: [He] with great ingenuity foresaw in a qualitative fashion that physical matter might be conceived as a curved ripple on a generally flat plane. Many of his ingenious hunches were later realized in Einstein's gravitational theory. Such speculations were automatically premature and could not lead to anything constructive without an intermediate link which demanded the extension of 3-dimensional geometry to the inclusion of time. The theory of curved spaces had to be preceded by the realization that space and time form a single four-dimensional entity. Likewise, Banesh Hoffmann (1973) writes: Riemann, and more specifically Clifford, conjectured that forces and matter might be local irregularities in the curvature of space, and in this they were strikingly prophetic, though for their pains they were dismissed at the time as visionaries. In 1990, Ruth Farwell and Christopher Knee examined the record on acknowledgement of Clifford's foresight. They conclude that "it was Clifford, not Riemann, who anticipated some of the conceptual ideas of General Relativity." To explain the lack of recognition of Clifford's prescience, they point out that he was an expert in metric geometry, and "metric geometry was too challenging to orthodox epistemology to be pursued." In 1992, Farwell and Knee continued their study of Clifford and Riemann:[They] hold that once tensors had been used in the theory of general relativity, the framework existed in which a geometrical perspective in physics could be developed and allowed the challenging geometrical conceptions of Riemann and Clifford to be rediscovered. == Selected writings == 1872. On the aims and instruments of scientific thought, 524–41. 1876 [1870]. On the Space-Theory of Matter. 1877. "The Ethics of Belief." Contemporary Review 29:289. 1878. Elements of Dynamic: An Introduction to the Study of Motion And Rest In Solid And Fluid Bodies. Book I: "Translations" Book II: "Rotations" Book III: "Strains" 1878. "Applications of Grassmann's Extensive Algebra." American Journal of Mathematics 1(4):353. 1879: Seeing and Thinking—includes four popular science lectures: "The Eye and the Brain" "The Eye and Seeing" "The Brain and Thinking" "Of Boundaries in General" 1879. Lectures and Essays I & II, with an introduction by Sir Frederick Pollock. 1881. "Mathematical fragments" (facsimiles). 1882. Mathematical Papers, edited by Robert Tucker, with an introduction by Henry J. S. Smith. 1885. The Common Sense of the Exact Sciences, completed by Karl Pearson. 1887. Elements of Dynamic 2. == Quotations == "I…hold that in the physical world nothing else takes place but this variation [of the curvature of space]." "There is no scientific discoverer, no poet, no painter, no musician, who will not tell you that he found ready made his discovery or poem or picture—that it came to him from outside, and that he did not consciously create it from within." "It is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence." "If a man, holding a belief which he was taught in childhood or persuaded of afterwards, keeps down and pushes away any doubts which arise about it in his mind, purposely avoids the reading of books and the company of men that call in question or discuss it, and regards as impious those questions which cannot easily be asked without disturbing it—the life of that man is one long sin against mankind." "I was not, and was conceived. I loved and did a little work. I am not and grieve not." == See also == Bessel–Clifford function Clifford's principle Clifford analysis Clifford gates Clifford bundle Clifford module Clifford number Motor Rotor Simplex Split-biquaternion Will to Believe Doctrine == References == === Notes === === Citations === This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "Clifford, William Kingdon". Encyclopædia Britannica. Vol. 6 (11th ed.). Cambridge University Press. p. 506. == Further reading == Chisholm, M. (1997). "William Kingdon Clifford (1845-1879) and his wife Lucy (1846-1929)". Advances in Applied Clifford Algebras. 7S: 27–41. Archived from the original on 3 August 2020. Retrieved 22 April 2007. (The on-line version lacks the article's photographs.) Chisholm, M. (2002). Such Silver Currents - The Story of William and Lucy Clifford, 1845-1929. Cambridge, UK: The Lutterworth Press. ISBN 978-0-7188-3017-5. Farwell, Ruth; Knee, Christopher (1990). "The End of the Absolute: a nineteenth century contribution to General Relativity". Studies in History and Philosophy of Science. 21 (1): 91–121. Bibcode:1990SHPSA..21...91F. doi:10.1016/0039-3681(90)90016-2. Macfarlane, Alexander (1916). Lectures on Ten British Mathematicians of the Nineteenth Century. New York: John Wiley and Sons. Lectures on Ten British Mathematicians of the Nineteenth Century. (See especially pages 78–91) Madigan, Timothy J. (2010). W.K. Clifford and "The Ethics of Belief Cambridge Scholars Press, Cambridge, UK 978-1847-18503-7. Penrose, Roger (2004). The Road to Reality: A Complete Guide to the Laws of the Universe. Alfred A. Knopf. ISBN 9780679454434. (See especially Chapter 11) Stephen, Leslie; Pollock, Frederick (1879). Lectures and Essays by the Late William Kingdon Clifford, F.R.S. Vol. 1. New York: Macmillan and Company. Stephen, Leslie; Pollock, Frederick (1879). Lectures and Essays by the Late William Kingdon Clifford, F.R.S. Vol. 2. New York: Macmillan and Company. == External links == Works by William Kingdon Clifford at Project Gutenberg William and Lucy Clifford (with pictures) O'Connor, John J.; Robertson, Edmund F., "William Kingdon Clifford", MacTutor History of Mathematics Archive, University of St Andrews Works by or about William Kingdon Clifford at the Internet Archive Works by William Kingdon Clifford at LibriVox (public domain audiobooks) Clifford, William Kingdon, William James, and A.J. Burger (Ed.), The Ethics of Belief. Joe Rooney William Kingdon Clifford, Department of Design and Innovation, the Open University, London.
Wikipedia:William Metzler#0	William Henry Metzler (1863–1943) was a Canadian mathematician. == Career == He was born in Odessa, Canada West on 18 September 1863. He studied mathematics at the University of Toronto under Henry Taber from 1886, graduating in 1888 and then continuing as a postgraduate. He gained his doctorate in 1892. In 1895 he was appointed professor of mathematics at Syracuse University, then became dean of the graduate school. From 1923–1933, he was dean and professor of mathematics at the New York State College of Teachers in Albany, New York. In 1902 he was elected a Fellow of the Royal Society of Edinburgh. His proposers were David Henry Marshall, Robert Wenley, John George Adami, and James Douglas Hamilton Dickson. He died in Syracuse, New York, on 14 April 1943. == Publications == On the Roots of Matrices (1892) Homogenous Strains (1893) On the Rank of a Matrix (1913) == References ==
Wikipedia:William S. Burnside#0	This English mathematician is sometimes confused with the Irish mathematician William S. Burnside (1839–1920). William Burnside (2 July 1852 – 21 August 1927) was an English mathematician. He is known mostly as an early researcher in the theory of finite groups. Burnside was born in London in 1852. He went to school at Christ's Hospital until 1871 and attended St. John's and Pembroke Colleges at the University of Cambridge, where he was the Second Wrangler (bracketed with George Chrystal) in 1875. He lectured at Cambridge for the following ten years, before being appointed professor of mathematics at the Royal Naval College in Greenwich. While this was a little outside the main centres of British mathematical research, Burnside remained a very active researcher, publishing more than 150 papers in his career. Burnside's early research was in applied mathematics. This work was of sufficient distinction to merit his election as a fellow of the Royal Society in 1893, though it is little remembered today. Around the same time as his election his interests turned to the study of finite groups. This was not a widely studied subject in Britain in the late 19th century, and it took some years for his research in this area to gain widespread recognition. The central part of Burnside's group theory work was in the area of group representations, where he helped to develop some of the foundational theory, complementing, and sometimes competing with, the work of Ferdinand Georg Frobenius, who began his research in the subject during the 1890s. One of Burnside's best known contributions to group theory is his paqb theorem, which shows that every finite group whose order is divisible by fewer than three distinct primes is solvable. In 1897 Burnside's classic work Theory of Groups of Finite Order was published. The second edition (pub. 1911) was for many decades the standard work in the field. A major difference between the editions was the inclusion of character theory in the second. Burnside is also remembered for the formulation of Burnside's problem that concerns the question of bounding the size of a group if there are fixed bounds both on the order of all of its elements and the number of elements needed to generate it, and also for Burnside's lemma (a formula relating the number of orbits of a permutation group acting on a set with the number of fixed points of each of its elements) though the latter had been discovered earlier and independently by Frobenius and Augustin Cauchy. He received an honorary doctorate (D.Sc.) from the University of Dublin in June 1901. In addition to his mathematical work, Burnside was a noted rower. While he was a lecturer at Cambridge, he also coached the rowing crew team. In fact, his obituary in The Times took more interest in his athletic career, calling him "one of the best known Cambridge athletes of his day". He is buried at the West Wickham Parish Church in South London. == Books == Theory of groups of finite order (2nd ed.). Cambridge University Press. 1911; xxiv+512 p.{{cite book}}: CS1 maint: postscript (link) Forsyth, A. R., ed. (1936). Theory of probability. Cambridge University Press; reprint of 1928 first edition, based on a manuscript almost completed by Burnside shortly before his death{{cite book}}: CS1 maint: postscript (link) Neumann, Peter M.; Mann, A. J. S.; Tompson, Julia C., eds. (2004). The collected papers of William Burnside. Vol. 1: Commentary on Burnside's life and work, papers 1883–1899, Volume 2: Papers 1900–1926. Clarendon Press/Oxford University Press. ISBN 978-0-19-850585-3; 1584 pages in 2-volume set{{cite book}}: CS1 maint: postscript (link); Burnside, William (2004). Vol. 1 (as separate book). ISBN 978-0-19-850586-0; 788 pages in vol. 1.{{cite book}}: CS1 maint: postscript (link) Burnside, William (2004). Vol. 2 (as separate book). ISBN 978-0-19-850587-7. == Notes == == References == Curtis, Charles W. (2003), Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer, History of Mathematics, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2677-5, MR 1715145 Review Wagner, Ascher; Mosenthal, Verity (1978), "A bibliography of William Burnside (1852–1927)" (PDF), Historia Mathematica, 5 (3): 307–312, doi:10.1016/0315-0860(78)90116-7 == External links == Works by or about William Burnside at Wikisource Works by William Burnside at Project Gutenberg Works by or about William Burnside at the Internet Archive O'Connor, John J.; Robertson, Edmund F., "William Burnside", MacTutor History of Mathematics Archive, University of St Andrews William Burnside at the Mathematics Genealogy Project
Wikipedia:William Yslas Vélez#0	William "Bill" Yslas Vélez is an American mathematician, a current Emeritus Professor at the University of Arizona, and an Elected Fellow of the American Association for the Advancement of Science. From 1992–96, Vélez served as the president of Society for the Advancement of Chicanos/Hispanics and Native Americans in Science (SACNAS). At the University of Arizona he graduated with B.Sc. in 1968, M.Sc. in 1972, and Ph.D. in 1975. His doctoral thesis was written under the supervision of H. B. Mann. Vélez became a Fellow of the American Mathematical Society in January 2013. In 2017, he was selected as a fellow of the Association for Women in Mathematics in the inaugural class. In 2014, Vélez won the M. Gweneth Humphreys Award of the Association for Women in Mathematics for his mentorship of mathematics students and particularly of women in mathematics. == Notable publications == === Patents === Method and apparatus for suppressing interference from bandspread communication signals, (1996). Method and apparatus for suppressing linear amplitude interference from bandspread communication signals, (1996). Simplified interference suppressor, (2000). Adaptive processor integrator for interference suppression, (2001). == References ==
Wikipedia:Winfried Scharlau#0	Winfried Scharlau (12 August 1940, in Berlin – 26 November 2020) was a German mathematician. == Biography == Scharlau received his doctorate in 1967 from the University of Bonn. His doctoral thesis Quadratische Formen und Galois-Cohomologie (Quadratic Forms and Galois Cohomology) was supervised by Friedrich Hirzebruch. Scharlau was at the Institute for Advanced Study for the academic year 1969–1970 and in spring 1972. From 1970 he was a professor (most recent Institutsdirektor) at the University of Münster, from where he retired. Scharlau's research deals with number theory and, in particular, the theory of quadratic forms, about which he wrote a 1985 monograph Quadratic and Hermitian Forms in Springer's series Grundlehren der mathematischen Wissenschaften. Scharlau was also an amateur ornithologist and author of two novels, I megali istoria - die große Geschichte (2nd edition 2001), set on the Greek island of Naxos, and Scharife (2001), set on the island of Zanzibar in the 19th century. He also deals with the history of mathematics and wrote, with Hans Opolka, a historically-oriented introduction to number theory. Their book presents, among other topics, the analytical class number formula of Dirichlet and the geometry of the numbers in the 19th century. Scharlau wrote a multi-part biography of Alexander Grothendieck. Scharlau was a corresponding member of the Göttingen Academy of Sciences and Humanities. From 1991 to 1992 he was president of the German Mathematical Society. In 1974 he was invited as speaker with talks On subspaces of inner product spaces at the International Congress of Mathematicians in Vancouver. He was the father of the cognitive psychologist Ingrid Scharlau. == Selected publications == with Friedrich Hirzebruch: Einführung in die Funktionalanalysis, BI, Mannheim 1971, ISBN 978-3-411-00296-2, Hirzebruch Collection (PDF) Scharlau, Winfried (1972). "Quadratic reciprocity laws". Journal of Number Theory. 4 (1): 78–97. Bibcode:1972JNT.....4...78S. doi:10.1016/0022-314X(72)90012-1. ISSN 0022-314X. with H.-G. Quebbemann and Manfred Schultz: Quadratic and Hermitian forms in additive and abelian categories, Journal of Algebra vol. 59, no. 2, 264-289 1979 with Manfred Knebusch: Algebraic theory of quadratic forms: generic methods and Pfister forms, vol. 1. Birkhäuser, 1980 (DMV Seminary, notes taken by Heisook Lee) with Hans Opolka: Von Fermat bis Minkowski. Eine Vorlesung über die Zahlentheorie und ihre Entwicklung. Springer, Berlin 1980 Richard Dedekind 1831/1981. Vieweg, Braunschweig [et alia loca] 1981 Quadratic and Hermitian Forms. Grundlehren der Mathematischen Wissenschaften 270. Springer, Berlin [et alia loca] 1985 Mathematische Institute in Deutschland 1800–1945. Dokumente zur Geschichte der Mathematik. Vieweg, Braunschweig [et alia loca] 1990 Schulwissen Mathematik. Ein Überblick. Vieweg, Braunschweig [et alia loca] 1994 Beiträge zur Vogelwelt der südlichen Ägäis. Verlag C. Lienau, 1999 Scharlau, Winfried (2000). "On the history of the algebraic theory of quadratic forms". Quadratic Forms and Their Applications. Contemporary Mathematics. Vol. 272. pp. 229–259. doi:10.1090/conm/272/04406. ISBN 9780821827796. ISSN 1098-3627. Mathematik für Naturwissenschaftler. LIT Verlag, Münster 2005 Winfried Scharlau (2011), Wer ist Alexander Grothendieck? Anarchie, Mathematik, Spiritualität, Einsamkeit (in German), vol. Teil 1. Anarchie (3. ed.), Norderstedt: Books on Demand, ISBN 978-3-8423-7147-7 Winfried Scharlau (2010), Wer ist Alexander Grothendieck? Anarchie, Mathematik, Spiritualität, Einsamkeit (in German), vol. Teil 3. Spiritualität, Norderstedt: Books on Demand, ISBN 978-3-8391-4939-3 Winfried Scharlau (2016), Das Glück, Mathematiker zu sein. Friedrich Hirzebruch und seine Zeit (in German), Wiesbaden: Springer Spektrum, ISBN 978-3-658-14756-3 == References == == External links == Literature by and about Winfried Scharlau in the German National Library catalogue website at the University of Münster https://www.trauer.ms/traueranzeige/winfried-scharlau-2020-11-26-havixbeck-9312841
Wikipedia:Wirtinger derivatives#0	In complex analysis of one and several complex variables, Wirtinger derivatives (sometimes also called Wirtinger operators), named after Wilhelm Wirtinger who introduced them in 1927 in the course of his studies on the theory of functions of several complex variables, are partial differential operators of the first order which behave in a very similar manner to the ordinary derivatives with respect to one real variable, when applied to holomorphic functions, antiholomorphic functions or simply differentiable functions on complex domains. These operators permit the construction of a differential calculus for such functions that is entirely analogous to the ordinary differential calculus for functions of real variables. == Historical notes == === Early days (1899–1911): the work of Henri Poincaré === Wirtinger derivatives were used in complex analysis at least as early as in the paper (Poincaré 1899), as briefly noted by Cherry & Ye (2001, p. 31) and by Remmert (1991, pp. 66–67). In the third paragraph of his 1899 paper, Henri Poincaré first defines the complex variable in C n {\displaystyle \mathbb {C} ^{n}} and its complex conjugate as follows { x k + i y k = z k x k − i y k = u k 1 ⩽ k ⩽ n . {\displaystyle {\begin{cases}x_{k}+iy_{k}=z_{k}\\x_{k}-iy_{k}=u_{k}\end{cases}}\qquad 1\leqslant k\leqslant n.} Then he writes the equation defining the functions V {\displaystyle V} he calls biharmonique, previously written using partial derivatives with respect to the real variables x k , y q {\displaystyle x_{k},y_{q}} with k , q {\displaystyle k,q} ranging from 1 to n {\displaystyle n} , exactly in the following way d 2 V d z k d u q = 0 {\displaystyle {\frac {d^{2}V}{dz_{k}\,du_{q}}}=0} This implies that he implicitly used definition 2 below: to see this it is sufficient to compare equations 2 and 2' of (Poincaré 1899, p. 112). Apparently, this paper was not noticed by early researchers in the theory of functions of several complex variables: in the papers of Levi-Civita (1905), Levi (1910) (and Levi 1911) and of Amoroso (1912) all fundamental partial differential operators of the theory are expressed directly by using partial derivatives respect to the real and imaginary parts of the complex variables involved. In the long survey paper by Osgood (1966) (first published in 1913), partial derivatives with respect to each complex variable of a holomorphic function of several complex variables seem to be meant as formal derivatives: as a matter of fact when Osgood expresses the pluriharmonic operator and the Levi operator, he follows the established practice of Amoroso, Levi and Levi-Civita. === The work of Dimitrie Pompeiu in 1912 and 1913: a new formulation === According to Henrici (1993, p. 294), a new step in the definition of the concept was taken by Dimitrie Pompeiu: in the paper (Pompeiu 1912), given a complex valued differentiable function (in the sense of real analysis) of one complex variable g ( z ) {\displaystyle g(z)} defined in the neighbourhood of a given point z 0 ∈ C , {\displaystyle z_{0}\in \mathbb {C} ,} he defines the areolar derivative as the following limit ∂ g ∂ z ¯ ( z 0 ) = d e f lim r → 0 1 2 π i r 2 ∮ Γ ( z 0 , r ) g ( z ) d z , {\displaystyle {{\frac {\partial g}{\partial {\bar {z}}}}(z_{0})}\mathrel {\overset {\mathrm {def} }{=}} \lim _{r\to 0}{\frac {1}{2\pi ir^{2}}}\oint _{\Gamma (z_{0},r)}g(z)\mathrm {d} z,} where Γ ( z 0 , r ) = ∂ D ( z 0 , r ) {\displaystyle \Gamma (z_{0},r)=\partial D(z_{0},r)} is the boundary of a disk of radius r {\displaystyle r} entirely contained in the domain of definition of g ( z ) , {\displaystyle g(z),} i.e. his bounding circle. This is evidently an alternative definition of Wirtinger derivative respect to the complex conjugate variable: it is a more general one, since, as noted a by Henrici (1993, p. 294), the limit may exist for functions that are not even differentiable at z = z 0 . {\displaystyle z=z_{0}.} According to Fichera (1969, p. 28), the first to identify the areolar derivative as a weak derivative in the sense of Sobolev was Ilia Vekua. In his following paper, Pompeiu (1913) uses this newly defined concept in order to introduce his generalization of Cauchy's integral formula, the now called Cauchy–Pompeiu formula. === The work of Wilhelm Wirtinger === The first systematic introduction of Wirtinger derivatives seems due to Wilhelm Wirtinger in the paper Wirtinger 1927 in order to simplify the calculations of quantities occurring in the theory of functions of several complex variables: as a result of the introduction of these differential operators, the form of all the differential operators commonly used in the theory, like the Levi operator and the Cauchy–Riemann operator, is considerably simplified and consequently easier to handle. The paper is deliberately written from a formal point of view, i.e. without giving a rigorous derivation of the properties deduced. == Formal definition == Despite their ubiquitous use, it seems that there is no text listing all the properties of Wirtinger derivatives: however, fairly complete references are the short course on multidimensional complex analysis by Andreotti (1976, pp. 3–5), the monograph of Gunning & Rossi (1965, pp. 3–6), and the monograph of Kaup & Kaup (1983, p. 2,4) which are used as general references in this and the following sections. === Functions of one complex variable === Definition 1. Consider the complex plane C ≡ R 2 = { ( x , y ) ∣ x , y ∈ R } {\displaystyle \mathbb {C} \equiv \mathbb {R} ^{2}=\{(x,y)\mid x,y\in \mathbb {R} \}} (in a sense of expressing a complex number z = x + i y {\displaystyle z=x+iy} for real numbers x {\displaystyle x} and y {\displaystyle y} ). The Wirtinger derivatives are defined as the following linear partial differential operators of first order: ∂ ∂ z = 1 2 ( ∂ ∂ x − i ∂ ∂ y ) ∂ ∂ z ¯ = 1 2 ( ∂ ∂ x + i ∂ ∂ y ) {\displaystyle {\begin{aligned}{\frac {\partial }{\partial z}}&={\frac {1}{2}}\left({\frac {\partial }{\partial x}}-i{\frac {\partial }{\partial y}}\right)\\{\frac {\partial }{\partial {\bar {z}}}}&={\frac {1}{2}}\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right)\end{aligned}}} Clearly, the natural domain of definition of these partial differential operators is the space of C 1 {\displaystyle C^{1}} functions on a domain Ω ⊆ R 2 , {\displaystyle \Omega \subseteq \mathbb {R} ^{2},} but, since these operators are linear and have constant coefficients, they can be readily extended to every space of generalized functions. === Functions of n > 1 complex variables === Definition 2. Consider the Euclidean space on the complex field C n = R 2 n = { ( x , y ) = ( x 1 , … , x n , y 1 , … , y n ) ∣ x , y ∈ R n } . {\displaystyle \mathbb {C} ^{n}=\mathbb {R} ^{2n}=\left\{\left(\mathbf {x} ,\mathbf {y} \right)=\left(x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n}\right)\mid \mathbf {x} ,\mathbf {y} \in \mathbb {R} ^{n}\right\}.} The Wirtinger derivatives are defined as the following linear partial differential operators of first order: { ∂ ∂ z 1 = 1 2 ( ∂ ∂ x 1 − i ∂ ∂ y 1 ) ⋮ ∂ ∂ z n = 1 2 ( ∂ ∂ x n − i ∂ ∂ y n ) , { ∂ ∂ z ¯ 1 = 1 2 ( ∂ ∂ x 1 + i ∂ ∂ y 1 ) ⋮ ∂ ∂ z ¯ n = 1 2 ( ∂ ∂ x n + i ∂ ∂ y n ) . {\displaystyle {\begin{cases}{\frac {\partial }{\partial z_{1}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x_{1}}}-i{\frac {\partial }{\partial y_{1}}}\right)\\\qquad \vdots \\{\frac {\partial }{\partial z_{n}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x_{n}}}-i{\frac {\partial }{\partial y_{n}}}\right)\\\end{cases}},\qquad {\begin{cases}{\frac {\partial }{\partial {\bar {z}}_{1}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x_{1}}}+i{\frac {\partial }{\partial y_{1}}}\right)\\\qquad \vdots \\{\frac {\partial }{\partial {\bar {z}}_{n}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x_{n}}}+i{\frac {\partial }{\partial y_{n}}}\right)\\\end{cases}}.} As for Wirtinger derivatives for functions of one complex variable, the natural domain of definition of these partial differential operators is again the space of C 1 {\displaystyle C^{1}} functions on a domain Ω ⊂ R 2 n , {\displaystyle \Omega \subset \mathbb {R} ^{2n},} and again, since these operators are linear and have constant coefficients, they can be readily extended to every space of generalized functions. == Relation with complex differentiation == When a function f {\displaystyle f} is complex differentiable at a point, the Wirtinger derivative ∂ f / ∂ z {\displaystyle \partial f/\partial z} agrees with the complex derivative d f / d z {\displaystyle df/dz} . This follows from the Cauchy-Riemann equations. For the complex function f ( z ) = u ( z ) + i v ( z ) {\displaystyle f(z)=u(z)+iv(z)} which is complex differentiable ∂ f ∂ z = 1 2 ( ∂ f ∂ x − i ∂ f ∂ y ) = 1 2 ( ∂ u ∂ x + i ∂ v ∂ x − i ∂ u ∂ y + ∂ v ∂ y ) = ∂ u ∂ z + i ∂ v ∂ z = d f d z {\displaystyle {\begin{aligned}{\frac {\partial f}{\partial z}}&={\frac {1}{2}}\left({\frac {\partial f}{\partial x}}-i{\frac {\partial f}{\partial y}}\right)\\&={\frac {1}{2}}\left({\frac {\partial u}{\partial x}}+i{\frac {\partial v}{\partial x}}-i{\frac {\partial u}{\partial y}}+{\frac {\partial v}{\partial y}}\right)\\&={\frac {\partial u}{\partial z}}+i{\frac {\partial v}{\partial z}}={\frac {df}{dz}}\end{aligned}}} where the third equality uses the first definition of Wirtinger's derivatives for u {\displaystyle u} and v {\displaystyle v} . It can also be done through actual application of the Cauchy-Riemann equations. ∂ f ∂ z = 1 2 ( ∂ f ∂ x − i ∂ f ∂ y ) = 1 2 ( ∂ u ∂ x + i ∂ v ∂ x − i ∂ u ∂ y + ∂ v ∂ y ) = 1 2 ( ∂ u ∂ x + i ∂ v ∂ x + i ∂ v ∂ x + ∂ u ∂ x ) = ∂ u ∂ x + i ∂ v ∂ x = d f d z {\displaystyle {\begin{aligned}{\frac {\partial f}{\partial z}}&={\frac {1}{2}}\left({\frac {\partial f}{\partial x}}-i{\frac {\partial f}{\partial y}}\right)\\&={\frac {1}{2}}\left({\frac {\partial u}{\partial x}}+i{\frac {\partial v}{\partial x}}-i{\frac {\partial u}{\partial y}}+{\frac {\partial v}{\partial y}}\right)\\&={\frac {1}{2}}\left({\frac {\partial u}{\partial x}}+i{\frac {\partial v}{\partial x}}+i{\frac {\partial v}{\partial x}}+{\frac {\partial u}{\partial x}}\right)\\&={\frac {\partial u}{\partial x}}+i{\frac {\partial v}{\partial x}}={\frac {df}{dz}}\end{aligned}}} The final equality comes from it being one of four equivalent formulations of the complex derivative through partial derivatives of the components. The second Wirtinger derivative is also related with complex differentiation; ∂ f ∂ z ¯ = 0 {\displaystyle {\frac {\partial f}{\partial {\bar {z}}}}=0} is equivalent to the Cauchy-Riemann equations in a complex form. == Basic properties == In the present section and in the following ones it is assumed that z ∈ C n {\displaystyle z\in \mathbb {C} ^{n}} is a complex vector and that z ≡ ( x , y ) = ( x 1 , … , x n , y 1 , … , y n ) {\displaystyle z\equiv (x,y)=(x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n})} where x , y {\displaystyle x,y} are real vectors, with n ≥ 1: also it is assumed that the subset Ω {\displaystyle \Omega } can be thought of as a domain in the real euclidean space R 2 n {\displaystyle \mathbb {R} ^{2n}} or in its isomorphic complex counterpart C n . {\displaystyle \mathbb {C} ^{n}.} All the proofs are easy consequences of definition 1 and definition 2 and of the corresponding properties of the derivatives (ordinary or partial). === Linearity === Lemma 1. If f , g ∈ C 1 ( Ω ) {\displaystyle f,g\in C^{1}(\Omega )} and α , β {\displaystyle \alpha ,\beta } are complex numbers, then for i = 1 , … , n {\displaystyle i=1,\dots ,n} the following equalities hold ∂ ∂ z i ( α f + β g ) = α ∂ f ∂ z i + β ∂ g ∂ z i ∂ ∂ z ¯ i ( α f + β g ) = α ∂ f ∂ z ¯ i + β ∂ g ∂ z ¯ i {\displaystyle {\begin{aligned}{\frac {\partial }{\partial z_{i}}}\left(\alpha f+\beta g\right)&=\alpha {\frac {\partial f}{\partial z_{i}}}+\beta {\frac {\partial g}{\partial z_{i}}}\\{\frac {\partial }{\partial {\bar {z}}_{i}}}\left(\alpha f+\beta g\right)&=\alpha {\frac {\partial f}{\partial {\bar {z}}_{i}}}+\beta {\frac {\partial g}{\partial {\bar {z}}_{i}}}\end{aligned}}} === Product rule === Lemma 2. If f , g ∈ C 1 ( Ω ) , {\displaystyle f,g\in C^{1}(\Omega ),} then for i = 1 , … , n {\displaystyle i=1,\dots ,n} the product rule holds ∂ ∂ z i ( f ⋅ g ) = ∂ f ∂ z i ⋅ g + f ⋅ ∂ g ∂ z i ∂ ∂ z ¯ i ( f ⋅ g ) = ∂ f ∂ z ¯ i ⋅ g + f ⋅ ∂ g ∂ z ¯ i {\displaystyle {\begin{aligned}{\frac {\partial }{\partial z_{i}}}(f\cdot g)&={\frac {\partial f}{\partial z_{i}}}\cdot g+f\cdot {\frac {\partial g}{\partial z_{i}}}\\{\frac {\partial }{\partial {\bar {z}}_{i}}}(f\cdot g)&={\frac {\partial f}{\partial {\bar {z}}_{i}}}\cdot g+f\cdot {\frac {\partial g}{\partial {\bar {z}}_{i}}}\end{aligned}}} This property implies that Wirtinger derivatives are derivations from the abstract algebra point of view, exactly like ordinary derivatives are. === Chain rule === This property takes two different forms respectively for functions of one and several complex variables: for the n > 1 case, to express the chain rule in its full generality it is necessary to consider two domains Ω ′ ⊆ C m {\displaystyle \Omega '\subseteq \mathbb {C} ^{m}} and Ω ″ ⊆ C p {\displaystyle \Omega ''\subseteq \mathbb {C} ^{p}} and two maps g : Ω ′ → Ω {\displaystyle g:\Omega '\to \Omega } and f : Ω → Ω ″ {\displaystyle f:\Omega \to \Omega ''} having natural smoothness requirements. ==== Functions of one complex variable ==== Lemma 3.1 If f , g ∈ C 1 ( Ω ) , {\displaystyle f,g\in C^{1}(\Omega ),} and g ( Ω ) ⊆ Ω , {\displaystyle g(\Omega )\subseteq \Omega ,} then the chain rule holds ∂ ∂ z ( f ∘ g ) = ( ∂ f ∂ z ∘ g ) ∂ g ∂ z + ( ∂ f ∂ z ¯ ∘ g ) ∂ g ¯ ∂ z ∂ ∂ z ¯ ( f ∘ g ) = ( ∂ f ∂ z ∘ g ) ∂ g ∂ z ¯ + ( ∂ f ∂ z ¯ ∘ g ) ∂ g ¯ ∂ z ¯ {\displaystyle {\begin{aligned}{\frac {\partial }{\partial z}}(f\circ g)&=\left({\frac {\partial f}{\partial z}}\circ g\right){\frac {\partial g}{\partial z}}+\left({\frac {\partial f}{\partial {\bar {z}}}}\circ g\right){\frac {\partial {\bar {g}}}{\partial z}}\\{\frac {\partial }{\partial {\bar {z}}}}(f\circ g)&=\left({\frac {\partial f}{\partial z}}\circ g\right){\frac {\partial g}{\partial {\bar {z}}}}+\left({\frac {\partial f}{\partial {\bar {z}}}}\circ g\right){\frac {\partial {\bar {g}}}{\partial {\bar {z}}}}\end{aligned}}} ==== Functions of n > 1 complex variables ==== Lemma 3.2 If g ∈ C 1 ( Ω ′ , Ω ) {\displaystyle g\in C^{1}(\Omega ',\Omega )} and f ∈ C 1 ( Ω , Ω ″ ) , {\displaystyle f\in C^{1}(\Omega ,\Omega ''),} then for i = 1 , … , n {\displaystyle i=1,\dots ,n} the following form of the chain rule holds ∂ ∂ z i ( f ∘ g ) = ∑ j = 1 n ( ∂ f ∂ z j ∘ g ) ∂ g j ∂ z i + ∑ j = 1 n ( ∂ f ∂ z ¯ j ∘ g ) ∂ g ¯ j ∂ z i ∂ ∂ z ¯ i ( f ∘ g ) = ∑ j = 1 n ( ∂ f ∂ z j ∘ g ) ∂ g j ∂ z ¯ i + ∑ j = 1 n ( ∂ f ∂ z ¯ j ∘ g ) ∂ g ¯ j ∂ z ¯ i {\displaystyle {\begin{aligned}{\frac {\partial }{\partial z_{i}}}\left(f\circ g\right)&=\sum _{j=1}^{n}\left({\frac {\partial f}{\partial z_{j}}}\circ g\right){\frac {\partial g_{j}}{\partial z_{i}}}+\sum _{j=1}^{n}\left({\frac {\partial f}{\partial {\bar {z}}_{j}}}\circ g\right){\frac {\partial {\bar {g}}_{j}}{\partial z_{i}}}\\{\frac {\partial }{\partial {\bar {z}}_{i}}}\left(f\circ g\right)&=\sum _{j=1}^{n}\left({\frac {\partial f}{\partial z_{j}}}\circ g\right){\frac {\partial g_{j}}{\partial {\bar {z}}_{i}}}+\sum _{j=1}^{n}\left({\frac {\partial f}{\partial {\bar {z}}_{j}}}\circ g\right){\frac {\partial {\bar {g}}_{j}}{\partial {\bar {z}}_{i}}}\end{aligned}}} === Conjugation === Lemma 4. If f ∈ C 1 ( Ω ) , {\displaystyle f\in C^{1}(\Omega ),} then for i = 1 , … , n {\displaystyle i=1,\dots ,n} the following equalities hold ( ∂ f ∂ z i ) ¯ = ∂ f ¯ ∂ z ¯ i ( ∂ f ∂ z ¯ i ) ¯ = ∂ f ¯ ∂ z i {\displaystyle {\begin{aligned}{\overline {\left({\frac {\partial f}{\partial z_{i}}}\right)}}&={\frac {\partial {\bar {f}}}{\partial {\bar {z}}_{i}}}\\{\overline {\left({\frac {\partial f}{\partial {\bar {z}}_{i}}}\right)}}&={\frac {\partial {\bar {f}}}{\partial z_{i}}}\end{aligned}}} == See also == CR–function Dolbeault complex Dolbeault operator Pluriharmonic function == Notes == == References ==
Wikipedia:Wirtinger's inequality for functions#0	For other inequalities named after Wirtinger, see Wirtinger's inequality. In the mathematical field of analysis, the Wirtinger inequality is an important inequality for functions of a single variable, named after Wilhelm Wirtinger. It was used by Adolf Hurwitz in 1901 to give a new proof of the isoperimetric inequality for curves in the plane. A variety of closely related results are today known as Wirtinger's inequality, all of which can be viewed as certain forms of the Poincaré inequality. == Theorem == There are several inequivalent versions of the Wirtinger inequality: Let y be a continuous and differentiable function on the interval [0, L] with average value zero and with y(0) = y(L). Then ∫ 0 L y ( x ) 2 d x ≤ L 2 4 π 2 ∫ 0 L y ′ ( x ) 2 d x , {\displaystyle \int _{0}^{L}y(x)^{2}\,\mathrm {d} x\leq {\frac {L^{2}}{4\pi ^{2}}}\int _{0}^{L}y'(x)^{2}\,\mathrm {d} x,} and equality holds if and only if y(x) = c sin ⁠2π(x − α)/L⁠ for some numbers c and α. Let y be a continuous and differentiable function on the interval [0, L] with y(0) = y(L) = 0. Then ∫ 0 L y ( x ) 2 d x ≤ L 2 π 2 ∫ 0 L y ′ ( x ) 2 d x , {\displaystyle \int _{0}^{L}y(x)^{2}\,\mathrm {d} x\leq {\frac {L^{2}}{\pi ^{2}}}\int _{0}^{L}y'(x)^{2}\,\mathrm {d} x,} and equality holds if and only if y(x) = c sin ⁠πx/L⁠ for some number c. Let y be a continuous and differentiable function on the interval [0, L] with average value zero. Then ∫ 0 L y ( x ) 2 d x ≤ L 2 π 2 ∫ 0 L y ′ ( x ) 2 d x . {\displaystyle \int _{0}^{L}y(x)^{2}\,\mathrm {d} x\leq {\frac {L^{2}}{\pi ^{2}}}\int _{0}^{L}y'(x)^{2}\,\mathrm {d} x.} and equality holds if and only if y(x) = c cos ⁠πx/L⁠ for some number c. Despite their differences, these are closely related to one another, as can be seen from the account given below in terms of spectral geometry. They can also all be regarded as special cases of various forms of the Poincaré inequality, with the optimal Poincaré constant identified explicitly. The middle version is also a special case of the Friedrichs inequality, again with the optimal constant identified. === Proofs === The three versions of the Wirtinger inequality can all be proved by various means. This is illustrated in the following by a different kind of proof for each of the three Wirtinger inequalities given above. In each case, by a linear change of variables in the integrals involved, there is no loss of generality in only proving the theorem for one particular choice of L. ==== Fourier series ==== Consider the first Wirtinger inequality given above. Take L to be 2π. Since Dirichlet's conditions are met, we can write y ( x ) = 1 2 a 0 + ∑ n ≥ 1 ( a n sin ⁡ n x π + b n cos ⁡ n x π ) , {\displaystyle y(x)={\frac {1}{2}}a_{0}+\sum _{n\geq 1}\left(a_{n}{\frac {\sin nx}{\sqrt {\pi }}}+b_{n}{\frac {\cos nx}{\sqrt {\pi }}}\right),} and the fact that the average value of y is zero means that a0 = 0. By Parseval's identity, ∫ 0 2 π y ( x ) 2 d x = ∑ n = 1 ∞ ( a n 2 + b n 2 ) {\displaystyle \int _{0}^{2\pi }y(x)^{2}\,\mathrm {d} x=\sum _{n=1}^{\infty }(a_{n}^{2}+b_{n}^{2})} and ∫ 0 2 π y ′ ( x ) 2 d x = ∑ n = 1 ∞ n 2 ( a n 2 + b n 2 ) {\displaystyle \int _{0}^{2\pi }y'(x)^{2}\,\mathrm {d} x=\sum _{n=1}^{\infty }n^{2}(a_{n}^{2}+b_{n}^{2})} and since the summands are all nonnegative, the Wirtinger inequality is proved. Furthermore, it is seen that equality holds if and only if an = bn = 0 for all n ≥ 2, which is to say that y(x) = a1 sin x + b1 cos x. This is equivalent to the stated condition by use of the trigonometric addition formulas. ==== Integration by parts ==== Consider the second Wirtinger inequality given above. Take L to be π. Any differentiable function y(x) satisfies the identity y ( x ) 2 + ( y ′ ( x ) − y ( x ) cot ⁡ x ) 2 = y ′ ( x ) 2 − d d x ( y ( x ) 2 cot ⁡ x ) . {\displaystyle y(x)^{2}+{\big (}y'(x)-y(x)\cot x{\big )}^{2}=y'(x)^{2}-{\frac {d}{dx}}{\big (}y(x)^{2}\cot x{\big )}.} Integration using the fundamental theorem of calculus and the boundary conditions y(0) = y(π) = 0 then shows ∫ 0 π y ( x ) 2 d x + ∫ 0 π ( y ′ ( x ) − y ( x ) cot ⁡ x ) 2 d x = ∫ 0 π y ′ ( x ) 2 d x . {\displaystyle \int _{0}^{\pi }y(x)^{2}\,\mathrm {d} x+\int _{0}^{\pi }{\big (}y'(x)-y(x)\cot x{\big )}^{2}\,\mathrm {d} x=\int _{0}^{\pi }y'(x)^{2}\,\mathrm {d} x.} This proves the Wirtinger inequality, since the second integral is clearly nonnegative. Furthermore, equality in the Wirtinger inequality is seen to be equivalent to y′(x) = y(x) cot x, the general solution of which (as computed by separation of variables) is y(x) = c sin x for an arbitrary number c. There is a subtlety in the above application of the fundamental theorem of calculus, since it is not the case that y(x)2 cot x extends continuously to x = 0 and x = π for every function y(x). This is resolved as follows. It follows from the Hölder inequality and y(0) = 0 that \| y ( x ) \| = \| ∫ 0 x y ′ ( x ) d x \| ≤ ∫ 0 x \| y ′ ( x ) \| d x ≤ x ( ∫ 0 x y ′ ( x ) 2 d x ) 1 / 2 , {\displaystyle \|y(x)\|=\left\|\int _{0}^{x}y'(x)\,\mathrm {d} x\right\|\leq \int _{0}^{x}\|y'(x)\|\,\mathrm {d} x\leq {\sqrt {x}}\left(\int _{0}^{x}y'(x)^{2}\,\mathrm {d} x\right)^{1/2},} which shows that as long as ∫ 0 π y ′ ( x ) 2 d x {\displaystyle \int _{0}^{\pi }y'(x)^{2}\,\mathrm {d} x} is finite, the limit of ⁠1/x⁠ y(x)2 as x converges to zero is zero. Since cot x < ⁠1/x⁠ for small positive values of x, it follows from the squeeze theorem that y(x)2 cot x converges to zero as x converges to zero. In exactly the same way, it can be proved that y(x)2 cot x converges to zero as x converges to π. ==== Functional analysis ==== Consider the third Wirtinger inequality given above. Take L to be 1. Given a continuous function f on [0, 1] of average value zero, let Tf denote the function u on [0, 1] which is of average value zero, and with u′′ + f = 0 and u′(0) = u′(1) = 0. From basic analysis of ordinary differential equations with constant coefficients, the eigenvalues of T are (kπ)−2 for nonzero integers k, the largest of which is then π−2. Because T is a bounded and self-adjoint operator, it follows that ∫ 0 1 T f ( x ) 2 d x ≤ π − 2 ∫ 0 1 f ( x ) T f ( x ) d x = 1 π 2 ∫ 0 1 ( T f ) ′ ( x ) 2 d x {\displaystyle \int _{0}^{1}Tf(x)^{2}\,\mathrm {d} x\leq \pi ^{-2}\int _{0}^{1}f(x)Tf(x)\,\mathrm {d} x={\frac {1}{\pi ^{2}}}\int _{0}^{1}(Tf)'(x)^{2}\,\mathrm {d} x} for all f of average value zero, where the equality is due to integration by parts. Finally, for any continuously differentiable function y on [0, 1] of average value zero, let gn be a sequence of compactly supported continuously differentiable functions on (0, 1) which converge in L2 to y′. Then define y n ( x ) = ∫ 0 x g n ( z ) d z − ∫ 0 1 ∫ 0 w g n ( z ) d z d w . {\displaystyle y_{n}(x)=\int _{0}^{x}g_{n}(z)\,\mathrm {d} z-\int _{0}^{1}\int _{0}^{w}g_{n}(z)\,\mathrm {d} z\,\mathrm {d} w.} Then each yn has average value zero with yn′(0) = yn′(1) = 0, which in turn implies that −yn′′ has average value zero. So application of the above inequality to f = −yn′′ is legitimate and shows that ∫ 0 1 y n ( x ) 2 d x ≤ 1 π 2 ∫ 0 1 y n ′ ( x ) 2 d x . {\displaystyle \int _{0}^{1}y_{n}(x)^{2}\,\mathrm {d} x\leq {\frac {1}{\pi ^{2}}}\int _{0}^{1}y_{n}'(x)^{2}\,\mathrm {d} x.} It is possible to replace yn by y, and thereby prove the Wirtinger inequality, as soon as it is verified that yn converges in L2 to y. This is verified in a standard way, by writing y ( x ) − y n ( x ) = ∫ 0 x ( y n ′ ( z ) − g n ( z ) ) d z − ∫ 0 1 ∫ 0 w ( y n ′ ( z ) − g n ( z ) ) d z d w {\displaystyle y(x)-y_{n}(x)=\int _{0}^{x}{\big (}y_{n}'(z)-g_{n}(z){\big )}\,\mathrm {d} z-\int _{0}^{1}\int _{0}^{w}(y_{n}'(z)-g_{n}(z){\big )}\,\mathrm {d} z\,\mathrm {d} w} and applying the Hölder or Jensen inequalities. This proves the Wirtinger inequality. In the case that y(x) is a function for which equality in the Wirtinger inequality holds, then a standard argument in the calculus of variations says that y must be a weak solution of the Euler–Lagrange equation y′′(x) + y(x) = 0 with y′(0) = y′(1) = 0, and the regularity theory of such equations, followed by the usual analysis of ordinary differential equations with constant coefficients, shows that y(x) = c cos πx for some number c. To make this argument fully formal and precise, it is necessary to be more careful about the function spaces in question. == Spectral geometry == In the language of spectral geometry, the three versions of the Wirtinger inequality above can be rephrased as theorems about the first eigenvalue and corresponding eigenfunctions of the Laplace–Beltrami operator on various one-dimensional Riemannian manifolds: the first eigenvalue of the Laplace–Beltrami operator on the Riemannian circle of length L is ⁠4π2/L2⁠, and the corresponding eigenfunctions are the linear combinations of the two coordinate functions. the first Dirichlet eigenvalue of the Laplace–Beltrami operator on the interval [0, L] is ⁠π2/L2⁠ and the corresponding eigenfunctions are given by c sin ⁠πx/L⁠ for arbitrary nonzero numbers c. the first Neumann eigenvalue of the Laplace–Beltrami operator on the interval [0, L] is ⁠π2/L2⁠ and the corresponding eigenfunctions are given by c cos ⁠πx/L⁠ for arbitrary nonzero numbers c. These can also be extended to statements about higher-dimensional spaces. For example, the Riemannian circle may be viewed as the one-dimensional version of either a sphere, real projective space, or torus (of arbitrary dimension). The Wirtinger inequality, in the first version given here, can then be seen as the n = 1 case of any of the following: the first eigenvalue of the Laplace–Beltrami operator on the unit-radius n-dimensional sphere is n, and the corresponding eigenfunctions are the linear combinations of the n + 1 coordinate functions. the first eigenvalue of the Laplace–Beltrami operator on the n-dimensional real projective space (with normalization given by the covering map from the unit-radius sphere) is 2n + 2, and the corresponding eigenfunctions are the restrictions of the homogeneous quadratic polynomials on Rn + 1 to the unit sphere (and then to the real projective space). the first eigenvalue of the Laplace–Beltrami operator on the n-dimensional torus (given as the n-fold product of the circle of length 2π with itself) is 1, and the corresponding eigenfunctions are arbitrary linear combinations of n-fold products of the eigenfunctions on the circles. The second and third versions of the Wirtinger inequality can be extended to statements about first Dirichlet and Neumann eigenvalues of the Laplace−Beltrami operator on metric balls in Euclidean space: the first Dirichlet eigenvalue of the Laplace−Beltrami operator on the unit ball in Rn is the square of the smallest positive zero of the Bessel function of the first kind J(n − 2)/2. the first Neumann eigenvalue of the Laplace−Beltrami operator on the unit ball in Rn is the square of the smallest positive zero of the first derivative of the Bessel function of the first kind Jn/2. == Application to the isoperimetric inequality == In the first form given above, the Wirtinger inequality can be used to prove the isoperimetric inequality for curves in the plane, as found by Adolf Hurwitz in 1901. Let (x, y) be a differentiable embedding of the circle in the plane. Parametrizing the circle by [0, 2π] so that (x, y) has constant speed, the length L of the curve is given by ∫ 0 2 π x ′ ( t ) 2 + y ′ ( t ) 2 d t {\displaystyle \int _{0}^{2\pi }{\sqrt {x'(t)^{2}+y'(t)^{2}}}\,\mathrm {d} t} and the area A enclosed by the curve is given (due to Stokes theorem) by − ∫ 0 2 π y ( t ) x ′ ( t ) d t . {\displaystyle -\int _{0}^{2\pi }y(t)x'(t)\,\mathrm {d} t.} Since the integrand of the integral defining L is assumed constant, there is L 2 2 π − 2 A = ∫ 0 2 π ( x ′ ( t ) 2 + y ′ ( t ) 2 + 2 y ( t ) x ′ ( t ) ) d t {\displaystyle {\frac {L^{2}}{2\pi }}-2A=\int _{0}^{2\pi }{\big (}x'(t)^{2}+y'(t)^{2}+2y(t)x'(t){\big )}\,\mathrm {d} t} which can be rewritten as ∫ 0 2 π ( x ′ ( t ) + y ( t ) ) 2 d t + ∫ 0 2 π ( y ′ ( t ) 2 − y ( t ) 2 ) d t . {\displaystyle \int _{0}^{2\pi }{\big (}x'(t)+y(t){\big )}^{2}\,\mathrm {d} t+\int _{0}^{2\pi }{\big (}y'(t)^{2}-y(t)^{2}{\big )}\,\mathrm {d} t.} The first integral is clearly nonnegative. Without changing the area or length of the curve, (x, y) can be replaced by (x, y + z) for some number z, so as to make y have average value zero. Then the Wirtinger inequality can be applied to see that the second integral is also nonnegative, and therefore L 2 4 π ≥ A , {\displaystyle {\frac {L^{2}}{4\pi }}\geq A,} which is the isoperimetric inequality. Furthermore, equality in the isoperimetric inequality implies both equality in the Wirtinger inequality and also the equality x′(t) + y(t) = 0, which amounts to y(t) = c1 sin(t – α) and then x(t) = c1 cos(t – α) + c2 for arbitrary numbers c1 and c2. These equations mean that the image of (x, y) is a round circle in the plane. == References == Brezis, Haim (2011). Functional analysis, Sobolev spaces and partial differential equations. Universitext. New York: Springer. doi:10.1007/978-0-387-70914-7. ISBN 978-0-387-70913-0. MR 2759829. Zbl 1220.46002. Chavel, Isaac (1984). Eigenvalues in Riemannian geometry. Pure and Applied Mathematics. Vol. 115. Orlando, FL: Academic Press. doi:10.1016/s0079-8169(08)x6051-9. ISBN 0-12-170640-0. MR 0768584. Zbl 0551.53001. Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1952). Inequalities (Second edition of 1934 original ed.). Cambridge University Press. MR 0046395. Zbl 0047.05302. Hurwitz, A. (1901). "Sur le problème des isopérimètres". Comptes Rendus des Séances de l'Académie des Sciences. 132: 401–403. JFM 32.0386.01. Stein, Elias M.; Weiss, Guido (1971). Introduction to Fourier analysis on Euclidean spaces. Princeton Mathematical Series. Vol. 32. Princeton, NJ: Princeton University Press. MR 0304972. Zbl 0232.42007.
Wikipedia:Witold Kosiński#0	Witold Kosiński (August 13, 1946 in Kraków – March 14, 2014 in Warsaw) was a Polish mathematician and computer scientist. He was the lead inventor and main propagator of Ordered Fuzzy Numbers (now named after him: Kosiński's Fuzzy Numbers). For many years Professor Witold Kosiński was associated with the Institute of Fundamental Technological Research of the Polish Academy of Sciences. He has also worked as the Vice-Chancellor of the Polish-Japanese Institute of Information Technology - PJIIT (now called Polish-Japanese Academy of Information Technology) in Warsaw and the Head of the Artificial Systems Division at the PJIIT. Finally, he was a lecturer at the Faculty of Mathematics, Physics and Technical Sciences, the Kazimierz Wielki University in Bydgoszcz. Professor Kosiński was a researcher specialising in continuum mechanics, thermodynamics, and wave propagation as well as in mathematical foundations of information technology and particularly in artificial intelligence, fuzzy logic and evolutionary algorithms. His fields of research have also comprised applied mathematics and partial differential equations of hyperbolic type as well as neural networks and computational intelligence. He was a scientist, mentor to scientific staff and several generations of students, as well as an active athlete. == Education and career == Professor Kosiński defended his Master's Thesis, "On the existence of functions of two variables satisfying some differential inequality", at the Faculty of Mathematics and Mechanics at the University of Warsaw in 1969. Three years later, in 1972, he obtained a Doctor of Science degree and then in 1984 a further dr hab. ("doktor habilitowany") degree (see: Habilitation) at the Institute of the Fundamental Technological Research in the Polish Academy of Sciences (IPPT PAN). He was elevated to the degree of Professor in 1993 through a formal nomination by the President of the Republic of Poland. Over 25 years (1973–1999) he has worked at the Institute of Fundamental Technological Research in the Polish Academy of Science in Warsaw; first as an Assistant, later as an Associate Professor and finally (in 1993) as a Full Professor. Between 1986 and 1999 he headed the Division of Optical and Computer Methods in Mechanics IPPT PAN (SPOKoMM). In 1999 he obtained the position of Vice-Chancellor (scientific affairs) (“Vice-Rektor”) at the Polish-Japanese Institute of Information Technology (PJIIT) in Warsaw, a position that he held till 2005. At the PJIIT he was also the Head of Artificial Systems Division and of the Research Center. In addition he was a member of the PJIIT Senate and of the Council of the Faculty of Information Technology PJIIT. In 1996 he joined the Department of Environmental Mechanics at The Higher Pedagogical School in Bydgoszcz. In 2005, with the establishment of the Kazimierz Wielki University in Bydgoszcz, Kosiński became a Head of the Department of Database Systems and Computational Intelligence at the Faculty of Mathematics, Physics and Technical Sciences at the Institute of Mechanics and Applied Computer Science at that University. In 2009, he became a Chairman of the Council of this Institute. He managed several scientific projects financed by the Polish State Committee for Scientific Research (KBN). He participated in numerous international conferences and worked as a contract lecturer in Poland (e.g. at Bialystok University of Technology and Warsaw University of Life Sciences) and abroad. == International collaboration == Between 1975–1976 he was in the US as a National Science Foundation post-doctoral research fellow at the Division of Materials Engineering, the University of Iowa. Later as a research fellow of Alexander von Humboldt Foundation he undertook several research visits to numerous German scientific institutes, incl. Institute for Applied Mathematics of the University of Bonn (1983–1985), Institute for Applied Mathematics of the Heidelberg University and Institute of Mechanics of the Technische Universität Darmstadt (1988). Subsequently, he became a visiting professor at the following institutions: LMM, Universite Pierre et Marie Curie (Paris VI), at Universite d'Aix – Marseille III, France, Department of Mathematical and Computer Sciences, Loyola University New Orleans, USA, Dublin Institute for Advanced Studies, Ireland, Nagoya University, Japan, and departments of mathematics of the following universities: Genova, Ferrara, Catania, Napoli, Potenza, Univ. Roma "La Sapienza" and Terza, Italy and Rostock, Germany. In addition, he was a research fellow of the Japan International Cooperation Agency (JICA) and participated in several research and training programs in Japan. == Books and work as a supervisor == Kosiński was an editor of many volumes of collective works and conference materials and an author of two monographs: W. Kosiński: Field Singularities and Wave Analysis in Continuum Mechanics. Ellis Horwood Series: Mathematics and Its Applications, Ellis Horwood Ltd., Chichester, Halsted Press: a Division of John Wiley & Sons, New York Chichester Brisbane Toronto, PWN – Polish Scientific Publishers, Warsaw (1986) W. Kosiński: Wstęp do teorii osobliwości pola i analizy fal. PWN, Warsaw – Poznań (1981) as well as over 230 of other scientific publications. He was a supervisor of 11 Ph.D. theses (10 of which dealt with informatics) and a number of Engineering Diploma works and Master Theses. He was a member of editorial boards of several journals as well as a member of numerous Polish and international scientific associations. Between 2000–2011 he was an Editor-in-Chief of the Annales Societatis Mathematicae Polonae Series III. Mathematica Applicanda (a journal of the Polish Mathematical Society). == References ==
Wikipedia:Witold Nowacki#0	Prof Witold Nowacki HFRSE PPAS (1911–1986) was a Polish mathematician and expert on the mechanics of elasticity and thermoelasticity. He served as President of the Polish Academy of Sciences from 1978 to 1980 and was the first President of the Society of the Interaction of Mathematics and Mechanics. == Life == He was born in Zakrzewo in Poland on 20 July 1911, the son of Ludwik Nowacki and his wife, Bronislawa Czyzewska. He studied at Gdańsk Polytechnic graduating in 1934. In the Second World War he served as an officer in the Polish Army but was captured in 1939 and spent the entire war as a German prisoner of war. In 1945 he was accepted as a Professor at Gdańsk University of Technology lecturing in the Strength of Materials. In 1952 he transferred to Warsaw Polytechnic lecturing in Building Mechanics and from 1955 taught Elasticity and Plasticity at Warsaw University. In 1979 he was elected an Honorary Fellow of the Royal Society of Edinburgh. Nowacki retired in 1980. == Personal life == In 1936, Nowacki married Janina Sztabianka. On 23 August 1986, Nowacki died in Warsaw, Poland. == Publications == Thermoelasticity (1962) Dynamics of Elastic Systems (1963) Theory of Micropolar Elasticity (1970) Trends in Elasticity and Thermoelasticity (1971) Autobiographical Notes (1985) Theory of Asymmetric Elasticity (1986) == References ==
Wikipedia:Witten zeta function#0	In mathematics, the Witten zeta function, is a function associated to a root system that encodes the degrees of the irreducible representations of the corresponding Lie group. These zeta functions were introduced by Don Zagier who named them after Edward Witten's study of their special values (among other things). Note that in, Witten zeta functions do not appear as explicit objects in their own right. == Definition == If G {\displaystyle G} is a compact semisimple Lie group, the associated Witten zeta function is (the meromorphic continuation of) the series ζ G ( s ) = ∑ ρ 1 ( dim ⁡ ρ ) s , {\displaystyle \zeta _{G}(s)=\sum _{\rho }{\frac {1}{(\dim \rho )^{s}}},} where the sum is over equivalence classes of irreducible representations of G {\displaystyle G} . In the case where G {\displaystyle G} is connected and simply connected, the correspondence between representations of G {\displaystyle G} and of its Lie algebra, together with the Weyl dimension formula, implies that ζ G ( s ) {\displaystyle \zeta _{G}(s)} can be written as ∑ m 1 , … , m r > 0 ∏ α ∈ Φ + 1 ⟨ α ∨ , m 1 λ 1 + ⋯ + m r λ r ⟩ s , {\displaystyle \sum _{m_{1},\dots ,m_{r}>0}\prod _{\alpha \in \Phi ^{+}}{\frac {1}{\langle \alpha ^{\lor },m_{1}\lambda _{1}+\cdots +m_{r}\lambda _{r}\rangle ^{s}}},} where Φ + {\displaystyle \Phi ^{+}} denotes the set of positive roots, { λ i } {\displaystyle \{\lambda _{i}\}} is a set of simple roots and r {\displaystyle r} is the rank. == Examples == ζ S U ( 2 ) ( s ) = ζ ( s ) {\displaystyle \zeta _{SU(2)}(s)=\zeta (s)} , the Riemann zeta function. ζ S U ( 3 ) ( s ) = ∑ x = 1 ∞ ∑ y = 1 ∞ 1 ( x y ( x + y ) / 2 ) s . {\displaystyle \zeta _{SU(3)}(s)=\sum _{x=1}^{\infty }\sum _{y=1}^{\infty }{\frac {1}{(xy(x+y)/2)^{s}}}.} == Abscissa of convergence == If G {\displaystyle G} is simple and simply connected, the abscissa of convergence of ζ G ( s ) {\displaystyle \zeta _{G}(s)} is r / κ {\displaystyle r/\kappa } , where r {\displaystyle r} is the rank and κ = \| Φ + \| {\displaystyle \kappa =\|\Phi ^{+}\|} . This is a theorem due to Alex Lubotzky and Michael Larsen. A new proof is given by Jokke Häsä and Alexander Stasinski which yields a more general result, namely it gives an explicit value (in terms of simple combinatorics) of the abscissa of convergence of any "Mellin zeta function" of the form ∑ x 1 , … , x r = 1 ∞ 1 P ( x 1 , … , x r ) s , {\displaystyle \sum _{x_{1},\dots ,x_{r}=1}^{\infty }{\frac {1}{P(x_{1},\dots ,x_{r})^{s}}},} where P ( x 1 , … , x r ) {\displaystyle P(x_{1},\dots ,x_{r})} is a product of linear polynomials with non-negative real coefficients. == Singularities and values of the Witten zeta function associated to SU(3) == ζ S U ( 3 ) {\displaystyle \zeta _{SU(3)}} is absolutely convergent in { s ∈ C , ℜ ( s ) > 2 / 3 } {\displaystyle \{s\in \mathbb {C} ,\Re (s)>2/3\}} , and it can be extended meromorphicaly in C {\displaystyle \mathbb {C} } . Its singularities are in { 2 3 } ∪ { 1 2 − k , k ∈ N } , {\displaystyle {\Bigl \{}{\frac {2}{3}}{\Bigr \}}\cup {\Bigl \{}{\frac {1}{2}}-k,k\in \mathbb {N} {\Bigr \}},} and all of those singularities are simple poles. In particular, the values of ζ S U ( 3 ) ( s ) {\displaystyle \zeta _{SU(3)}(s)} are well defined at all integers, and have been computed by Kazuhiro Onodera. At s = 0 {\displaystyle s=0} , we have ζ S U ( 3 ) ( 0 ) = 1 3 , {\displaystyle \zeta _{SU(3)}(0)={\frac {1}{3}},} and ζ S U ( 3 ) ′ ( 0 ) = log ⁡ ( 2 4 / 3 π ) . {\displaystyle \zeta _{SU(3)}'(0)=\log(2^{4/3}\pi ).} Let a ∈ N ∗ {\displaystyle a\in \mathbb {N} ^{*}} be a positive integer. We have ζ S U ( 3 ) ( a ) = 2 a + 2 1 + ( − 1 ) a 2 ∑ k = 0 [ a / 2 ] ( 2 a − 2 k − 1 a − 1 ) ζ ( 2 k ) ζ ( 3 a − k ) . {\displaystyle \zeta _{SU(3)}(a)={\frac {2^{a+2}}{1+(-1)^{a}2}}\sum _{k=0}^{[a/2]}{2a-2k-1 \choose a-1}\zeta (2k)\zeta (3a-k).} If a is odd, then ζ S U ( 3 ) {\displaystyle \zeta _{SU(3)}} has a simple zero at s = − a , {\displaystyle s=-a,} and ζ S U ( 3 ) ′ ( − a ) = 2 − a + 1 ( a ! ) 2 ( 2 a + 1 ) ! ζ ′ ( − 3 a − 1 ) + 2 − a + 2 ∑ k = 0 ( a − 1 ) / 2 ( a 2 k ) ζ ( − a − 2 k ) ζ ′ ( − 2 a + 2 k ) . {\displaystyle \zeta _{SU(3)}'(-a)={\frac {2^{-a+1}(a!)^{2}}{(2a+1)!}}\zeta '(-3a-1)+2^{-a+2}\sum _{k=0}^{(a-1)/2}{a \choose 2k}\zeta (-a-2k)\zeta '(-2a+2k).} If a is even, then ζ S U ( 3 ) {\displaystyle \zeta _{SU(3)}} has a zero of order 2 {\displaystyle 2} at s = − a , {\displaystyle s=-a,} and ζ S U ( 3 ) ″ ( − a ) = 2 − a + 2 ∑ k = 0 a / 2 ( a 2 k ) ζ ′ ( − a − 2 k ) ζ ′ ( − 2 a + 2 k ) . {\displaystyle \zeta _{SU(3)}''(-a)=2^{-a+2}\sum _{k=0}^{a/2}{a \choose 2k}\zeta '(-a-2k)\zeta '(-2a+2k).} == References ==
Wikipedia:Wojciech Samotij#0	Wojciech Samotij (Polish: [ˈvɔjt͡ɕɛx saˈmɔtij]) is a Polish mathematician and a full professor at the School of Mathematical Sciences at the Tel Aviv University. He is known for his work in combinatorics, additive number theory, Ramsey theory and graph theory. == Education and career == He studied at the University of Wrocław where in 2007 he obtained his Master of Science degrees in mathematics and computer science. He received his PhD in 2011 at University of Illinois at Urbana-Champaign on the basis of his dissertation titled Extremal Problems In Pseudo-random Graphs And Asymptotic Enumeration and written under the supervision of József Balogh. Between 2010 and 2014, he was a fellow of the Trinity College, Cambridge at the University of Cambridge. Currently, he is an associate professor at Tel Aviv University. He published his scientific work in such journals as Random Structures & Algorithms, Journal of the American Mathematical Society, or Israel Journal of Mathematics. == Awards == He received the 2013 Kuratowski Prize conferred jointly by the Polish Academy of Sciences and the Polish Mathematical Society to young mathematicians under the age of 30. The same year, he won the 2013 European Prize in Combinatorics. Samotij is also the recipient of the 2016 George Pólya Prize and the 2022 Erdős Prize. In 2024 he was awarded the Leroy P. Steele Prize for Seminal Contribution to Research jointly with József Balogh and Robert Morris. == Selected publications == with József Balogh, Robert Morris, and Lutz Warnke: Balogh, József; Morris, Robert; Samotij, Wojciech; Warnke, Lutz (2016), "The typical structure of sparse K r + 1 {\displaystyle K_{r+1}} -free graphs", Transactions of the American Mathematical Society, 368: 6439–6485, arXiv:1307.5967, doi:10.1090/tran/6552, S2CID 17878868 with József Balogh and Robert Morris: Balogh, József; Morris, Robert; Samotij, Wojciech (2015), "Independent sets in hypergraphs", Journal of the American Mathematical Society, 28 (3): 669–709, arXiv:1204.6530, doi:10.1090/S0894-0347-2014-00816-X with Noga Alon, József Balogh, and Robert Morris: Alon, Noga; Balogh, József; Morris, Robert; Samotij, Wojciech (January 2014), "A refinement of the Cameron-Erdős conjecture", Proceedings of the London Mathematical Society, 108 (1): 44–72, arXiv:1202.5200, doi:10.1112/plms/pdt033, S2CID 1536592 with Noga Alon, József Balogh, and Robert Morris: Alon, Noga; Balogh, József; Morris, Robert; Samotij, Wojciech (January 2014), "Counting sum-free sets in Abelian groups", Israel Journal of Mathematics, 199: 309–344, arXiv:1201.6654, doi:10.1007/s11856-013-0067-y with Ron Peled: Peled, Ron; Samotij, Wojciech (August 2014), "Odd cutsets and the hard-core model on Z d {\displaystyle \mathbb {Z} ^{d}} ", Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 50 (3): 975–998, arXiv:1106.3594, doi:10.1214/12-AIHP535 with József Balogh: Balogh, József; Samotij, Wojciech (April 2011), "The number of K s , t {\displaystyle K_{s,t}} -free graphs", Journal of the London Mathematical Society, 83 (2): 368–388, doi:10.1112/jlms/jdq086, S2CID 7455668 == See also == List of Polish mathematicians Combinatorics Ramsey theory == References ==
Wikipedia:Wojciech Szpankowski#0	Wojciech Szpankowski (born February 18, 1952, in Wapno) is the Saul Rosen Professor of Computer Science at Purdue University. He is known for his work in analytic combinatorics, analysis of algorithms and analytic information theory. He is the director of the NSF Science and Technology Center for Science of Information. == Biography == Szpankowski received his MS and PhD in Electrical Engineering and Computer Science from the Technical University of Gdańsk in 1970 and 1980 respectively. == Awards and honors == Fellow of IEEE The Erskine Fellow Flajolet Lecture Prize Humboldt Research Award == References ==
Wikipedia:Wojciech Zaremba#0	Wojciech Zaremba (born 30 November 1988) is a Polish computer scientist and founding team member of OpenAI (2016–present). He initially led OpenAI's work on robotics, notably creating a robotic arm capable of solving Rubik's Cube. When the team was dissolved in 2020, he began leading teams working on OpenAI's GPT models, GitHub Copilot, and Codex. == Early life and education == Zaremba was born in Kluczbork, Poland. At a young age, he won local competitions and awards in mathematics, computer science, chemistry and physics. In 2007, Zaremba represented Poland in the International Mathematical Olympiad in Vietnam, and won a silver medal. Zaremba studied at the University of Warsaw and École Polytechnique mathematics and computer science, and graduated in 2013 with two master's degrees in mathematics. He then began his PhD at New York University (NYU) in deep learning under the supervision of Yann LeCun and Rob Fergus. Zaremba graduated and received his PhD in 2016. == Career == During his bachelor studies, he spent time at NVIDIA before the deep learning era. His PhD was divided between Google Brain where he spent a year, and Facebook Artificial Intelligence Research where he spent another year. During his stay at Google, he co-authored work on adversarial examples for neural networks. This result created the field of adversarial attacks on neural networks. His PhD is focused on matching capabilities of neural networks with the algorithmic power of programmable computers. In 2015, Zaremba became one of the co-founders of OpenAI, an artificial intelligence research company whose stated mission is to ensure that artificial general intelligence "benefits all of humanity". In OpenAl, Zaremba has worked as robotics research manager until 2020, then managing the development of GitHub Copilot, and Codex, and GPT models underlying ChatGPT. Zaremba sits on the advisory board of Growbots, a Silicon Valley startup company aiming to automate sales processes with the use of machine learning and artificial intelligence. He is also on the advisory board of the Qualia Research Institute. == Honors and awards == Listed among the most influential Polish under 30s, Polish edition of Forbes magazine 2017 Google Fellowship 2015 Silver Medal in 48th International Mathematical Olympiad, Vietnam == References ==
Wikipedia:Wolfgang Sternberg#0	Wolfgang Sternberg (1887–1953) was a German-American mathematician. He completed his doctoral dissertation in 1912 at Breslau and his habilitation in 1920 at Heidelberg University. In 1935 Sternberg was dismissed by the non-Aryan laws from his post at Breslau, and briefly went to Palestine. However, in Palestine he was unhappy, due to various factors, such as "Jewish nationalism", lack of knowledge of Hebrew, and bad climate. He worked at the Hebrew University without renumeration, and actively sought possibilities to emigrate to other places, including the Soviet Union. Eventually Sternberg left Palestine for Prague, and in 1939 he managed to reach the United States where he had great difficulty finding employment despite help from Richard Courant. Eventually, he went through a number of temporary jobs at the Cornell University and other places, until his early retirement in 1948. Sternberg made contributions to potential theory, integral equations, and probability theory. == References == As of this edit, this article uses content from "Mathematicians Going East", authored by Pasha Zusmanovich, which is licensed in a way that permits reuse under the Creative Commons Attribution-ShareAlike 4.0 International License, but not under the GFDL. All relevant terms must be followed.
Wikipedia:Workshop on Numerical Ranges and Numerical Radii#0	Workshop on Numerical Ranges and Numerical Radii (WONRA) is a biennial workshop series on numerical ranges and numerical radii which began in 1992. == About == Numerical ranges and numerical radii are useful in the study of matrix and operator theory. These topics have applications in many subjects in pure and applied mathematics, such as quadratic forms, Banach spaces, dilation theory, control theory, numerical analysis, quantum information science. == History == In the early 1970s, numerical range workshops were organized by Frank Bonsall and John Duncan. More activities were started in early 1990s, including the biennial workshop series, which began in 1992, and special issues devoted to this workshop were published. === Workshops === === Symposium in conferences === == References == == External links == WONRA 2008 – Williamsburg, VA, USA WONRA 2010 – Krakow, Poland WONRA 2012 – Kaohsiung, Taiwan WONRA 2014 – Sanya, China WONRA 2016 – Taipei, Taiwan WONRA 2018 – Munich, Germany WONRA 2019 - Kawagoe, Japan WONRA 2023 - Coimbra, Portugal
Wikipedia:Worley noise#0	Worley noise, also called Voronoi noise and cellular noise, is a noise function introduced by Steven Worley in 1996. Worley noise is an extension of the Voronoi diagram that outputs a real value at a given coordinate that corresponds to the distance of the nth nearest seed (usually n=1) and the seeds are distributed evenly through the region. Worley noise is used to create procedural textures. Worley noise of Euclidean distance is differentiable and continuous everywhere except on the edges of the Voronoi diagram of the set of seeds and on the location of the seeds. == Basic algorithm == The algorithm chooses random points in space (2- or 3-dimensional) and then for every location in space takes the distances Fn to the nth-closest point (e.g. the second-closest point) and uses combinations of those to control color information (note that Fn+1 > Fn). More precisely: Randomly distribute feature points in space organized as grid cells. In practice this is done on the fly without storage (as a procedural noise). The original method considered a variable number of seed points per cell so as to mimic a Poisson disc, but many implementations just put one. This is an optimization that limits the number of terms that will be compared At run time, extract the distances Fn from the given location to the closest seed point. For F1 It is only necessary to find the value of the seeds location in the grid cell being sampled and the grid cells adjacent to that grid cell. Noise W(x) is formally the vector of distances, plus possibly the corresponding seed ids, user-combined so as to produce a color. == See also == Fractal Voronoi diagram Perlin noise Simplex noise == References == == Further reading == Worley, Steven (1996). A cellular texture basis function (PDF). Proceedings of the 23rd annual conference on computer graphics and interactive techniques. acm.org. pp. 291–294. ISBN 0-89791-746-4. David S. Ebert; F. Kenton Musgrave; Darwyn Peachey; Ken Perlin; Steve Worley (2002). Texturing and Modeling: A Procedural Approach. Morgan Kaufmann. pp. 135–155. ISBN 978-1-55860-848-1. == External links == Detailed description on how to implement cell noise A version with the color plates appended at the end
Wikipedia:Wrangler (University of Cambridge)#0	At the University of Cambridge in England, a "Wrangler" is a student who gains first-class honours in the Mathematical Tripos competition. The highest-scoring student is the Senior Wrangler, the second highest is the Second Wrangler, and so on. By contrast, the person who achieves the lowest exam marks while still earning a third-class honours degree (that is, while still earning an honours degree at all) is known as the wooden spoon. == History == Until 1909, the university made the rankings public. Since 1910, it has publicly revealed only the class of degree gained by each student. An examiner reveals the identity of the Senior Wrangler "unofficially" by tipping his hat when reading out the person's name, but other rankings are communicated to each student privately. Therefore, the names of only some 20th-century Senior Wranglers (such as Crispin Nash-Williams, Christopher Budd, Frank P. Ramsey, Donald Coxeter, Kevin Buzzard, Jayant Narlikar, George Reid and Ben J. Green) have become publicly known. Another notable was Philippa Fawcett. She was educated at Newnham College, Cambridge, which had been co-founded by her mother. In 1890, Fawcett became the first woman to obtain the top score in the Cambridge Mathematical Tripos exams. Her score was 13 per cent higher than the second-highest score. When the women's list was announced, Fawcett was described as "above the senior wrangler", but she did not receive the title of senior wrangler, as at that time only men could receive degrees and therefore only men were eligible for the Senior Wrangler title. The results were always highly publicised, with the top scorers receiving great acclaim. Women had been allowed to take the Tripos since 1881, after Charlotte Angas Scott was unofficially ranked as eighth wrangler. It was recorded that "virtually every high wrangler (for whom records exist) participated in some form of regular physical exercise to preserve his strength and stamina." Obtaining the position of a highly ranked Wrangler created many opportunities for the individual's subsequent profession. Such individuals would often become Fellows initially, before moving on to other professions. Throughout the United Kingdom and the British Empire, university mathematics professors were often among the top three Wranglers. The order of Wranglers was widely publicised and shaped the public perception of mathematics as being the most intellectually challenging of all subjects. According to Andrew Warwick, author of Masters of Theory, the term "Senior Wrangler" became "synonymous with academic supremacy". == Past wranglers == Top marks in the Cambridge mathematics exam did not always guarantee the Senior Wrangler success in life; the exams were largely a test of speed in applying familiar rules, and some of the most inventive and original students of Mathematics at Cambridge did not come top of their class. Lord Kelvin was second, William Henry Bragg was third, Augustus De Morgan and G. H. Hardy were fourth, Adam Sedgwick fifth, Bertrand Russell seventh, Thomas Robert Malthus ninth, John Maynard Keynes twelfth, and some fared even worse: Klaus Roth was not even a wrangler. Joan Clarke, who helped to break the Nazi Enigma code at Bletchley Park, was a wrangler at Cambridge and earned a double first in mathematics, although she was prevented from receiving a full degree based on the university's policy of awarding degrees only to men. That policy was abandoned in 1948. The present Astronomer Royal, Martin Rees, a wrangler, went on to become one of the world's leading scientists, also holding the illustrious posts of Master of Trinity College, Cambridge, and President of the Royal Society, and being a member of the Order of Merit. == Optimes == Students who achieve second-class and third-class mathematics degrees are known as Senior Optimes (second-class) and Junior Optimes (third-class). Cambridge did not divide its examination classification in mathematics into 2:1s and 2:2s until 1995 but now there are Senior Optimes Division 1 and Senior Optimes Division 2. == In fiction == "The Senior Wrangler" is a member of the faculty of Unseen University in Terry Pratchett's Discworld series of novels. Roger Hamley, a character in Elizabeth Gaskell's Wives and Daughters, achieved the rank of Senior Wrangler. Vivie Warren, the headstrong heroine of George Bernard Shaw's Mrs. Warren's Profession (1893) and daughter of the play's infamous madam, tied with the Third Wrangler, settling for that place because she recognized that "it was not worth [her] while to face the grind" because she did not intend an academic career for herself. "Wrangler" is a jargon term applied to codebreakers in some of John Le Carré's spy novels, such as Tinker Tailor Soldier Spy. Thomas Jericho, the main character of Robert Harris's book Enigma, was Senior Wrangler in 1938. In Ford Madox Ford's Parade's End, reference is made to the fact that Christopher Tietjens left Cambridge as "a mere Second Wrangler". In Rumer Godden's In This House of Brede, Dame Agnes is noted to have been Eighth Wrangler before entering the abbey. In C S Forester's book, The General, a member of the main character's staff (the deputy assistant quartermaster-general, Spiller) is described as a Second Wrangler. In Bram Stoker's The Judge's House, the main character Malcom Malcomson is looking for a quiet place to stay whilst preparing his Mathematical Tripos examinations. Mrs Witham, the inn's landlady, warns Malcom about the judge's house, but the charwoman, Mrs Dempster, dispels these fears – explaining she is not afraid of 'bogies' because they are only rats. Malcom replies: "Mrs. Dempster, [...] you know more than a Senior Wrangler! And let me say, that, as a mark of esteem for your indubitable soundness of head and heart, I shall, when I go, give you possession of this house, and let you stay here by yourself for the last two months of my tenancy, for four weeks will serve my purpose." == See also == List of mathematics awards == Notes == == References == Galton, Francis (1869). "Classification of Men According to their Natural Gifts". pp. 14–36. Peter Groenewegen (2003). A Soaring Eagle: Alfred Marshall 1842-1924. Cheltenham: Edward Elgar. ISBN 1-85898-151-4. gives the story about Rayleigh; Alfred Marshall was the commoner who came second to Rayleigh. C. M. Neale (1907) The Senior Wranglers of the University of Cambridge. Available online == External links == Information on the wranglers in the period 1860–1940 can be extracted from the BritMath database: BritMath Many of the wranglers who made careers in mathematics can be identified by searching on "wrangler" in: The MacTutor History of Mathematics archive Cambridge Mathematical Tripos: Wooden Spoons
Wikipedia:Wu's method of characteristic set#0	Wenjun Wu's method is an algorithm for solving multivariate polynomial equations introduced in the late 1970s by the Chinese mathematician Wen-Tsun Wu. This method is based on the mathematical concept of characteristic set introduced in the late 1940s by J.F. Ritt. It is fully independent of the Gröbner basis method, introduced by Bruno Buchberger (1965), even if Gröbner bases may be used to compute characteristic sets. Wu's method is powerful for mechanical theorem proving in elementary geometry, and provides a complete decision process for certain classes of problem. It has been used in research in his laboratory (KLMM, Key Laboratory of Mathematics Mechanization in Chinese Academy of Science) and around the world. The main trends of research on Wu's method concern systems of polynomial equations of positive dimension and differential algebra where Ritt's results have been made effective. Wu's method has been applied in various scientific fields, like biology, computer vision, robot kinematics and especially automatic proofs in geometry. == Informal description == Wu's method uses polynomial division to solve problems of the form: ∀ x , y , z , … I ( x , y , z , … ) ⟹ f ( x , y , z , … ) {\displaystyle \forall x,y,z,\dots I(x,y,z,\dots )\implies f(x,y,z,\dots )\,} where f is a polynomial equation and I is a conjunction of polynomial equations. The algorithm is complete for such problems over the complex domain. The core idea of the algorithm is that you can divide one polynomial by another to give a remainder. Repeated division results in either the remainder vanishing (in which case the I implies f statement is true), or an irreducible remainder is left behind (in which case the statement is false). More specifically, for an ideal I in the ring k[x1, ..., xn] over a field k, a (Ritt) characteristic set C of I is composed of a set of polynomials in I, which is in triangular shape: polynomials in C have distinct main variables (see the formal definition below). Given a characteristic set C of I, one can decide if a polynomial f is zero modulo I. That is, the membership test is checkable for I, provided a characteristic set of I. == Ritt characteristic set == A Ritt characteristic set is a finite set of polynomials in triangular form of an ideal. This triangular set satisfies certain minimal condition with respect to the Ritt ordering, and it preserves many interesting geometrical properties of the ideal. However it may not be its system of generators. === Notation === Let R be the multivariate polynomial ring k[x1, ..., xn] over a field k. The variables are ordered linearly according to their subscript: x1 < ... < xn. For a non-constant polynomial p in R, the greatest variable effectively presenting in p, called main variable or class, plays a particular role: p can be naturally regarded as a univariate polynomial in its main variable xk with coefficients in k[x1, ..., xk−1]. The degree of p as a univariate polynomial in its main variable is also called its main degree. === Triangular set === A set T of non-constant polynomials is called a triangular set if all polynomials in T have distinct main variables. This generalizes triangular systems of linear equations in a natural way. === Ritt ordering === For two non-constant polynomials p and q, we say p is smaller than q with respect to Ritt ordering and written as p <r q, if one of the following assertions holds: (1) the main variable of p is smaller than the main variable of q, that is, mvar(p) < mvar(q), (2) p and q have the same main variable, and the main degree of p is less than the main degree of q, that is, mvar(p) = mvar(q) and mdeg(p) < mdeg(q). In this way, (k[x1, ..., xn],<r) forms a well partial order. However, the Ritt ordering is not a total order: there exist polynomials p and q such that neither p <r q nor p >r q. In this case, we say that p and q are not comparable. The Ritt ordering is comparing the rank of p and q. The rank, denoted by rank(p), of a non-constant polynomial p is defined to be a power of its main variable: mvar(p)mdeg(p) and ranks are compared by comparing first the variables and then, in case of equality of the variables, the degrees. === Ritt ordering on triangular sets === A crucial generalization on Ritt ordering is to compare triangular sets. Let T = { t1, ..., tu} and S = { s1, ..., sv} be two triangular sets such that polynomials in T and S are sorted increasingly according to their main variables. We say T is smaller than S w.r.t. Ritt ordering if one of the following assertions holds there exists k ≤ min(u, v) such that rank(ti) = rank(si) for 1 ≤ i < k and tk <r sk, u > v and rank(ti) = rank(si) for 1 ≤ i ≤ v. Also, there exists incomparable triangular sets w.r.t Ritt ordering. === Ritt characteristic set === Let I be a non-zero ideal of k[x1, ..., xn]. A subset T of I is a Ritt characteristic set of I if one of the following conditions holds: T consists of a single nonzero constant of k, T is a triangular set and T is minimal w.r.t Ritt ordering in the set of all triangular sets contained in I. A polynomial ideal may possess (infinitely) many characteristic sets, since Ritt ordering is a partial order. == Wu characteristic set == The Ritt–Wu process, first devised by Ritt, subsequently modified by Wu, computes not a Ritt characteristic but an extended one, called Wu characteristic set or ascending chain. A non-empty subset T of the ideal ⟨F⟩ generated by F is a Wu characteristic set of F if one of the following condition holds T = {a} with a being a nonzero constant, T is a triangular set and there exists a subset G of ⟨F⟩ such that ⟨F⟩ = ⟨G⟩ and every polynomial in G is pseudo-reduced to zero with respect to T. Wu characteristic set is defined to the set F of polynomials, rather to the ideal ⟨F⟩ generated by F. Also it can be shown that a Ritt characteristic set T of ⟨F⟩ is a Wu characteristic set of F. Wu characteristic sets can be computed by Wu's algorithm CHRST-REM, which only requires pseudo-remainder computations and no factorizations are needed. Wu's characteristic set method has exponential complexity; improvements in computing efficiency by weak chains, regular chains, saturated chain were introduced == Decomposing algebraic varieties == An application is an algorithm for solving systems of algebraic equations by means of characteristic sets. More precisely, given a finite subset F of polynomials, there is an algorithm to compute characteristic sets T1, ..., Te such that: V ( F ) = W ( T 1 ) ∪ ⋯ ∪ W ( T e ) , {\displaystyle V(F)=W(T_{1})\cup \cdots \cup W(T_{e}),} where W(Ti) is the difference of V(Ti) and V(hi), here hi is the product of initials of the polynomials in Ti. == See also == Regular chain Mathematics-Mechanization Platform == References == P. Aubry, M. Moreno Maza (1999) Triangular Sets for Solving Polynomial Systems: a Comparative Implementation of Four Methods. J. Symb. Comput. 28(1–2): 125–154 David A. Cox, John B. Little, Donal O'Shea. Ideals, Varieties, and Algorithms. 2007. Hua-Shan, Liu (24 August 2005). "WuRittSolva: Implementation of Wu-Ritt Characteristic Set Method". Wolfram Library Archive. Wolfram. Retrieved 17 November 2012. Heck, André (2003). Introduction to Maple (3. ed.). New York: Springer. pp. 105, 508. ISBN 9780387002309. Ritt, J. (1966). Differential Algebra. New York, Dover Publications. Dongming Wang (1998). Elimination Methods. Springer-Verlag, Wien, Springer-Verlag Dongming Wang (2004). Elimination Practice, Imperial College Press, London ISBN 1-86094-438-8 Wu, W. T. (1984). Basic principles of mechanical theorem proving in elementary geometries. J. Syst. Sci. Math. Sci., 4, 207–35 Wu, W. T. (1987). A zero structure theorem for polynomial equations solving. MM Research Preprints, 1, 2–12 Xiaoshan, Gao; Chunming, Yuan; Guilin, Zhang (2009). "Ritt-Wu's characteristic set method for ordinary difference polynomial systems with arbitrary ordering". Acta Mathematica Scientia. 29 (4): 1063–1080. CiteSeerX 10.1.1.556.9549. doi:10.1016/S0252-9602(09)60086-2. == External links == wsolve Maple package The Characteristic Set Method
Wikipedia:Wucao Suanjing#0	Wucao Suanjing (五曹算經; Mathematical Manual of the Five Administrative Departments) is one of the books in the collection of mathematical texts assembled by Li Chunfeng and collectively referred to as The Ten Computational Canons by later writers. The text was designed for the teaching of those entering the five government departments of agriculture, war, accounts, granary and treasury. There is a chapter relating to each one of these departments. The text contains some formulas to find the areas of different shapes of fields. Though the formulas give approximately correct answers, they are actually incorrect. This incorrectness motivated further mathematical work. The mathematics involved does not go much beyond the processes of multiplication and division. == An approximation formula given in Wucao suanjing == Wucao suanjing contains an interesting approximate formula to find the area of a quadrilateral. This formula, known as "Surveyor's Rule" appears in the ancient mathematical literature of Mesopotamia, Egypt, Europe, Arabia and India. The formula can be stated thus: Area of a quadrilateral = (a + c)(b + d)/4 where a, b, c, d are the lengths of the sides of the quadrilateral. == References ==
Wikipedia:Wujing Suanshu#0	Wujing Suanshu (五經算術; translated as Mathematical Procedures of the Five Canons or Arithmetic methods in the Five Classics) is a 6th-century Chinese mathematical text written by Zhen Luan (535 – 566). During the early Tang dynasty, the text was selected to be part of the collection Ten Computational Canons. == References ==
Wikipedia:Władysław Matwin#0	Władysław Matwin (17 July 1916 – 21 October 2012) was a Polish politician, journalist and mathematician who was one of the pioneers of computer science in Poland. == Biography == After his parents divorced, he and his mother found themselves in Poznań, where he studied economics. At that time he belonged to the Communist Party of Poland and the Young Communist League of Poland (KZMP). He was secretary of the KZMP District Committee. In January 1935, he was arrested for Communist activities and sentenced to three years in prison. After being released, he went to Czechoslovakia, where he studied chemistry in Brno. He returned to Poland in the spring of 1939. He volunteered to join the army, but was considered a dangerous criminal and was banned from serving in the Polish army. During World War II he stayed on the territory of the Soviet Union. First, he worked as a miner and later studied at the Metallurgy Institute at night. For a short time he was in the Red Army, from which - due to his origin - he was removed. Later he worked in railway construction. He then stayed in Tbilisi. Later he joined First Polish Army army in Ryazan, where he taught politics at the officer's school. In 1944 he belonged to the corps of political and educational officers of the 1st Tadeusz Kościuszko Infantry Division. In 1944 he was sent to Tehran, where the Union of Polish Patriots (in which he was active) created an outpost whose main task was to reach the local Polish community through radio broadcasts and newspapers, and above all the soldiers of Władysław Anders. In 1945, Władysław Matwin was summoned to Moscow, where he became chargé d'affaires at the Polish embassy. In 1946 he returned to Poland and served as an instructor of the Central Committee. In 1947 and 1948 he was the first secretary of the Provincial Committee of the Polish Workers' Party in Wrocław. He also stayed for a year in Davos, Switzerland, where he had his eyes treated (he was in danger of losing his sight as a result of disease). Together with the PPR, he joined the Polish United Workers' Party, sitting until June 1964 in its Central Committee (he also held the position of the first secretary of the Provincial Committee in Wrocław, which he held until 1949). In the 1950s, he was associated with the Puławy faction. From 1949 to 1952 he was the chairman of the Main Board of the Union of Polish Youth. From December 1952 to February 1954 he was the first secretary of the Warsaw Committee of the PZPR. From 1954 to March 1956 and again from November 1956 to March 1957 he was editor-in-chief of Trybuna Ludu. From November 1954 to January 1955 he headed the Organizational Department of the Central Committee of the PZPR, and then until November 1963 he was secretary of the Central Committee (until March 1956 responsible for education ). In 1957, he was again sent to Wrocław, where he took the position of the First Secretary of the Provincial Committee of the PZPR, holding the position until his retirement from politics 1963. In 1963 he began studying mathematics and in 1966 he obtained a diploma in automata theory. The following year, he became the director of the Central Center for Management Staff Improvement, but in 1968 he lost this position for not agreeing to the demand to remove employees of Jewish origin from the institute. He took the job of a senior technologist in Włochy, and in 1970 - at the Institute of Mathematical Machines; in 1973 he became the director of the Department of Electronic Computing Technology. From 1976 to 1991 he worked part-time at the Systems Research Institute of the Polish Academy of Sciences. Władysław Matwin died in October 2012, being one of the last remaining politicians of the pre-war Polish Communist Party. == References ==
Wikipedia:Władysław Zajączkowski#0	Władysław Zajączkowski (April 12, 1837, in Strzyżów near the Rzeszów – October 7, 1898, in Lwów) was a Polish mathematician. Professor of Warsaw Main School, Imperial University of Warsaw (now University of Warsaw), Technical Academy in Lviv (now Lviv Polytechnic; twice a rector). Member of Polish Academy of Learning and French Academy of Sciences. He was specialising mainly in mathematical analysis and differential equations. == References ==
Wikipedia:Włodzimierz Marek Tulczyjew#0	Włodzimierz Marek Tulczyjew (18 June 1931 – 4 December 2022) was a Polish-Italian physicist and mathematician, known for his contributions to the geometric formulation of classical mechanics and field theory. He was a professor emeritus of mathematical methods of physics at the University of Camerino and a member of the Academy of Sciences of Turin. == Biography == Tulczyjew was born in Włodawa, a small town in eastern Poland, in 1931. From 1933 to 1943, he lived in Ostrów Lubelski, where his father worked as an accountant. In 1943, he was deported with his family to Germany, where he worked in an armaments factory. In June 1945, he returned to Lublin. Later on he moved to Warsaw where, in 1952, he graduated with a diploma from the State Telecommunication Technical School. From 1952 to 1956, he studied in the Faculty of Mathematics and Physics at the University of Warsaw. During his studies, Tulczyjew joined a group centered around Leopold Infeld (Jerzy Plebański, Andrzej Trautman, Iwo Białynicki-Birula, Stanisław Bażański, and others). He obtained his Ph.D. in 1959 and his D.Sc. in 1965 at the University of Warsaw under the supervision of Andrzej Trautman. Then he became an assistant professor there. Tulczyjew's research was highly valued by Infeld, who, in his posthumously published memoirs, referred to Włodek as his most outstanding student. After Infeld's death (January 15, 1968), Tulczyjew decided to emigrate. He left Poland on September 28, 1968, and through Rome reached Canada, at the University of Calgary. In the late 1980s, Tulczyjew took early retirement in Canada and relocated to Camerino, Italy. He was appointed as a professor per chiara fama (a distinguished position) of Mathematical Methods of Physics at the University of Camerino. He soon began collaborating with Giuseppe Marmo, a professor at Federico II University in Naples, and with the National Institute of Nuclear Physics. In the 2000s, he initiated new collaborations with scholars at the University of Bari, including Fiorella Barone and Margherita Barile, and mentored a PhD student, Antonio De Nicola. He retired in 2006, but remained active in research and teaching until his death in 2022. == Scientific work == Tulczyjew's main research interest was the geometry of classical mechanics and field theory, especially the symplectic and multisymplectic structures that underlie the Hamiltonian and Lagrangian formulations, and the Legendre transformation that connects them. Among his notable contributions are the Tulczyjew triple, the Tulczyjew symplectic structure, the Tulczyjew-Dedecker differential, and the Tulczyjew isomorphism. He also worked on general relativity, gauge theories, quantum mechanics, and differential geometry. In his habilitation thesis, Tulczyjew presented a scheme of relativistic quantum mechanics as a scattering theory, where antiparticles are described as particles moving backward in time. His work prefigured geometric quantization, later developed by Jean-Marie Souriau and Bertram Kostant. Tulczyjew was also one of the first to recognize the relashionship between Utiyama's theory and Yang-Mills field theory. Seeking the proper formulation of relativistic quantum mechanics led him back to the foundations of classical theories, especially to classical mechanics and the fundamentals of variational calculus (or rather the variational description of physical systems). Tulczyjew's approach to relativistic quantum mechanics led him to revisit the foundations of classical mechanics and the variational calculus, where he sought a geometric formulation. His work inspired several colleagues, particularly at the University of Warsaw, to explore multisymplectic geometry and its applications in variational calculus. Notable collaborators include Jerzy Kijowski, Wiktor Szczyrba, Jacek Komorowski, and Krzysztof Gawędzki. During his time in Canada, Tulczyjew continued to work on the dynamics of charged particles and the inverse problem of variational calculus, which seeks the conditions for a system of partial differential equations to be derived from a Lagrangian. He solved this problem by constructing a double variational complex and proving Poincaré’s Lemma for this structure. His work was highly regarded by contemporaries such as André Lichnerowicz, Alexandre Vinogradov, Paul Dedecker. Ian Anderson attributed the discovery of the variational bicomplex to both Tulczyjew and Vinogradov independently. In 1974, Tulczyjew published a seminal paper on Hamiltonian and Lagrangian systems, providing a full geometric interpretation of the Legendre transformation. Later on, Tulczyjew formulated his innovative vision of variational principles in physics by providing a conceptual framework. The basic geometric structure associated with the Legendre transformation is now known as the Tulczyjew triple. Tulczyjew's contributions were recognized internationally, and in 1981, he was elected as a foreign member of the Accademia delle Scienze di Torino. Tulczyjew published around 100 scientific papers and several books, including A symplectic framework for field theories (1979), and Geometric Formulation of Physical Theories (1989). He collaborated with many distinguished mathematicians and physicists. == Honors and awards == Tulczyjew was elected as a member of several academies and societies, such as: The Academy of Sciences of Turin (1981). The Polish Academy of Sciences (1990). The Gold Cross of Merit awarded by the President of Poland (2014). == References ==
Wikipedia:Xavier Tolsa#0	Xavier Tolsa (born 1966) is a Catalan mathematician, specializing in analysis. Tolsa is a professor at the Autonomous University of Barcelona and at the Institució Catalana de Recerca i Estudis Avançats (ICREA), the Catalan Institute for Advanced Scientific Studies. Tolsa does research on harmonic analysis (Calderón-Zygmund theory), complex analysis, geometric measure theory, and potential theory. Specifically, he is known for his research on analytic capacity and removable sets. He solved the problem of A. G. Vitushkin about the semi-additivity of analytic capacity. This enabled him to solve an even older problem of Paul Painlevé on the geometric characterization of removable sets. Tolsa succeeded in solving the Painlevé problem by using the concept of so-called curvatures of measures introduced by Mark Melnikov in 1995. Tolsa's proof involves estimates of Cauchy transforms. He has also done research on the so-called David-Semmes problem involving Riesz transforms and rectifiability. In 2002 he was awarded the Salem Prize. In 2006 in Madrid he was an Invited Speaker at the ICM with talk Analytic capacity, rectifiability, and the Cauchy integral. He received in 2004 the EMS Prize and was an Invited Lecturer at the 2004 ECM with talk Painlevé's problem, analytic capacity and curvature of measures. In 2013 he received the Ferran Sunyer i Balaguer Prize for his monograph Analytic capacity, the Cauchy transform, and non-homogeneous Calderón-Zygmund theory (Birkhäuser Verlag, 2013}. In 2019 he received the Rei Jaume I prize for his contributions to Mathematics. == Selected publications == Tolsa, Xavier (2000). "Principal Values for the Cauchy Integral and Rectifiability". Proceedings of the American Mathematical Society. 128 (7): 2111–2119. doi:10.1090/S0002-9939-00-05264-3. JSTOR 119706. Tolsa, Xavier (2003). "Painlevé's problem and the semiadditivity of analytic capacity". Acta Mathematica. 190: 105–149. arXiv:math/0204027. doi:10.1007/BF02393237. Nazarov, Fedor; Volberg, Alexander; Tolsa, Xavier (2014). "On the uniform rectifiability of AD-regular measures with bounded Riesz transform operator: the case of codimension 1". Acta Mathematica. 213 (2): 237–321. arXiv:1212.5229. doi:10.1007/s11511-014-0120-7. ISSN 0001-5962. == References ==
Wikipedia:Xhezair Teliti#0	Xhezair Teliti (born 17 February 1948, in Kavajë) is a professor of mathematics and has served as chief of the Department of Mathematics at Tirana University since 2008. He was Albania's Minister of Education from 1993–1996. == Career == His field of study is Functional Analysis and Theory of Mass and Integration. Teliti is author of many text-books: "Teoria e Funksioneve te Variablit Real, I, II(Theory of Functions of Real Variable)", 1980, Tirana; "Përgjithësimi i Konceptit të Integrali(Generalization of the Concept of Integral)", 1981; "Teoria Konstruktive e Funksioneve(The Constructive Theory of Functions)", P. Pilika, Xh. Teliti – 1984; "Përmbledhje Problemash në Analizën Funksional(Summary of Problems for Functional Analysis)", 1989, Tirana; "Probleme dhe Ushtrime të Analizës Matematike (Problems and Exercises of Mathematical Analysis", 1997, Tirana; "Teoria e Masës dhe e Integrimit(Theory of Mass and Integration)", 1997, Tirana; "Problema në Teorinë e Masës e të Integrimit(Problems for the Theory of Mass and Integration)", 1998, Tirana; "Topologjia e Përgjithshme dhe Analiza Funksionale(General Topology and Funbctional Analysis)", 2002, Tirana; "Elemente Strukturorë dhe Topologjikë në Hapësirat R dhe R (n) (Topological and Structural Elements in R and R (n) spaces", 2008, Tirana. Prof. Teliti has also written many articles in the Bulletin of Natural Sciences at the University. == References ==
Wikipedia:Xiahou Yang Suanjing#0	Xiahou Yang Suanjing (Xiahou Yang's Mathematical Manual) is a mathematical treatise attributed to the fifth century CE Chinese mathematician Xiahou Yang. However, some historians are of the opinion that Xiahou Yang Suanjing was not written by Xiahou Yang. It is one of the books in The Ten Computational Canons, a collection of mathematical texts assembled by Li Chunfeng and used as the official mathematical for the imperial examinations. Though little is known about the period of the author, there is some evidence which more or less conclusively establishes the date of the work. These suggest 468 CE as the latest possible date for the work to be written and 425 CE as the earliest date. == Contents == The treatise is divided into three parts and these are spoken of as the higher, the middle and the lower sections. The first chapter contains 19 problems, the second chapter contains 29 problems and the last chapter contains 44 problems. As in all the older Chinese books, no technical rules are given, and the problems are simply followed by the answers, occasionally with brief explanations. === Section 1 === In the first section the five operations of addition, subtraction, multiplication, division, and square and cube roots are given. The work on division is subdivided into (1) "ordinary division"; (2) "division by ten, hundred, and so on," especially intended for work in mensuration; (3) "division by simplification" (yo ch'ut). The last problem in the section is as follows: "There are 1843 k'o, 8 t'ow, 3 ho of coarse rice. A contract requires that this be exchanged for refined rice at the rate of 1 k'o, 4 t'ow for 3 k'o. How much refined rice must be given?" The answer is 860 k'o, 534 ho. The solution is given as follows: "Multiply the given number by 1 k'o, 4 t'ow and divide by 3 k'o and you will obtain the result." (1 k'o = 10 t'ow = 100 ho) Fractions are also mentioned, special names being given to the four most common ones, as follows: 1/2 is called chung p'an (even part) 1/3 is called shaw p'an (small part) 2/3 is called thai p'an (large part) 1/4 is called joh p'an (weak part) === Section 2 === In the second section there are twenty-eight applied problems relating to taxes, commissions, and such questions as concern the division by army officers of loot and food (silk, rice, wine, soy sauce, vinegar, and the like) among their soldiers. === Section 3 === The third section contains forty-two problems. The translations of some of these problems are given below. "Now for 1 pound of gold one gets 1200 pieces of silk. How many can you get for 1 ounce?" Answer: For 1 ounce you get exactly 75 pieces. Solution: Take the given number of pieces, have it divided by 16 ounces, and you will obtain the answer. (Chinese pound was divided into 16 ounces.) "Now you have 192 ounces of silk. How many choo have you?" Answer: Four thousand six hundred eight. (It appears that in obtaining the given solution to the problem, pound was divided into 24 choos.) "Now 2000 packages of cash must be carried to the town at the rate of 10 cash per bundle. How much will be given to the mandarin and how much to the carrier?" Answer: 1980 packages and 198 2/101 cash to the mandarin; 19 packages and 801 98/101 to the carrier. Solution: Take the total number as the dividend, and 1 package plus 10 cash as the divisor. "Out of 3485 ounces of silk how many pieces of satin can be made, 5 ounces being required for each piece?" Answer: 697. Solution: Multiply the number of ounces by 2 and go back by one row. Dividing by 5 will also give the answer. "Now they build a wall, high 3 rods, broad 5 feet at the upper part and 15 feet at the lower part; the length 100 rods. For a 2-foot square a man works 1 day. How many days are required?" Answer: 75,000. Solution: Take-half the sum of the upper and lower breadths, have it multiplied by the height and length; the product will be the dividend. As the divisor you will use the square of the given 2 feet. == References ==
Wikipedia:Xiaojun Chen#0	Xiaojun Chen is a Chinese applied mathematician, Chair Professor of Applied Mathematics at Hong Kong Polytechnic University. Her research interests include nonsmooth and nonconvex optimization, complementarity theory, and stochastic equilibrium problems. == Education and career == Chen completed her Ph.D. in 1987 at Xi'an Jiaotong University. At Hong Kong Polytechnic University, she was head of the applied mathematics department from 2013 to 2019. Since 2020 she has directed the University Research Facility in Big Data Analytics, and co-directed the CAS AMSS-PolyU Joint Laboratory of Applied Mathematics == Recognition == Chen was named a SIAM Fellow in the 2021 class of fellows, "for contributions to optimization, stochastic variational inequalities, and nonsmooth analysis". She was named to the 2023 class of Fellows of the American Mathematical Society, "for contributions to mathematical optimization, stochastic variational inequalities, and the analysis of nondifferentiable functions". == References == == External links == Xiaojun Chen publications indexed by Google Scholar
Wikipedia:Xiaoying Han#0	Xiaoying (Maggie) Han is a Chinese mathematician whose research concerns random dynamical systems, stochastic differential equations, and actuarial science. She is Marguerite Scharnagle Endowed Professor in Mathematics at Auburn University. == Education and career == Han graduated from the University of Science and Technology of China in 2001. She completed her Ph.D. in 2007 at the University at Buffalo. Her dissertation, Interlayer Mixing in Thin Film Growth, was supervised by Brian J. Spencer. She joined the Auburn faculty in 2007. She was promoted to full professor in 2017, and given the Marguerite Scharnagle Endowed Professorship in 2018. In 2020 she was named a Fulbright Scholar, funding her for a research visit to Brazil in 2021. == Books == Han is the co-author of three books: Applied Nonautonomous and Random Dynamical Systems (with T. Caraballo, Springer, 2016) Attractors Under Discretisation (with P. E. Kloeden, Springer, 2017) Random Ordinary Differential Equations and Their Numerical Solution (with P. E. Kloeden, Springer, 2017) == References == == External links == Home page Xiaoying Han publications indexed by Google Scholar
Wikipedia:Xiaoyu Luo#0	Xiaoyu Luo (Chinese: 罗小玉, born 1960) is a Chinese and British applied mathematician who studies biomechanics, fluid dynamics, and the interactions of fluid flows with soft biological tissues. She is a professor of applied mathematics at the University of Glasgow. == Education and career == Luo was born in the UK but grew up in Xi'an in a family of artists. After earning bachelor's and master's degrees in theoretical mechanics at Xi'an Jiaotong University, in 1982 and 1985 respectively, she became a lecturer at Xi'an Jiaotong University. There, she studied for a Ph.D. from 1987 until 1990, with a visit to the UK through a joint doctoral program with the University of Sheffield. When she earned her Ph.D. at Xi'an Jiaotong University in 1990, she became the first woman to do so. She moved to the UK in 1992 to become a postdoctoral researcher at the University of Leeds. She worked as a lecturer in engineering at Queen Mary and Westfield College from 1997 to 2000, and in mechanical engineering at the University of Sheffield from 2000 to 2004, before becoming a senior lecturer in mathematics at the University of Glasgow in 2005. She was promoted to professor in 2008, the first female professor of applied mathematics at Glasgow. In 2014 she was named a chair professor at Northwestern Polytechnical University in Xi'an. She has also been a visitor to the International Center for Applied Mechanics at Xi'an Jiaotong University. == Recognition == Luo became a Fellow of the Institution of Mechanical Engineers in 2004 and a Fellow of the Royal Society of Edinburgh in 2014. == References == == External links == Home page Xiaoyu Luo publications indexed by Google Scholar
Wikipedia:Xu-Jia Wang#0	Xu-Jia Wang (Chinese: 汪徐家; pinyin: Wāng Xújiā; born September 1963) is a Chinese-Australian mathematician. He is a professor of mathematics at the Australian National University and a fellow of the Australian Academy of Science. He joined Westlake University in Hangzhou, China in September 2024 as full-time Chair Professor of Mathematics. == Biography == Wang was born in Chun'an County, Zhejiang province, China. Wang obtained his B.S. in 1983 and his Ph.D. in 1990 from the Department of Mathematics of Zhejiang University (ZJU) in Hangzhou. After completing his PhD, Wang served as lecturer and associate professor, at ZJU before departing for ANU In 1995. Wang is a Professor in the Centre for Mathematics and its Applications and Mathematical Sciences Institute of Australian National University. Wang is well known for his work on differential equations, especially non-linear partial differential equations and their geometrical and transportational applications. == Honors and awards == Australian Mathematical Society Medal (2002) invited speaker, 2002 International Congress of Mathematicians Morningside Gold Medal of Mathematics, 2007 Fellow of the Australian Academy of Science (2009). Australian Laureate Fellowship (2013) == Publications (selected) == Wang, Xu-Jia (2011). "Convex solutions to the mean curvature flow". Annals of Mathematics. 173 (3): 1185–1239. arXiv:math/0404326. doi:10.4007/annals.2011.173.3.1. (with Aram Karakhanyan) Karakhanyan, Aram; Wang, Xu-Jia (2010). "On the reflector shape design". Journal of Differential Geometry. 84 (3): 561–610. doi:10.4310/jdg/1279114301. (with Guji Tian) Tian, Gu-Ji; Wang, Xu-Jia (2010). "Moser-Trudinger type inequalities for the Hessian equation". Journal of Functional Analysis. 259 (8): 1974–2002. doi:10.1016/j.jfa.2010.06.009. See also: Hessian equation (with Kai-Seng Chou) Chou, Kai-Seng; Wang, Xu-Jia (2006). "The Lp-Minkowski problem and the Minkowski problem in centroaffine geometry". Advances in Mathematics. 205 (1): 33–83. doi:10.1016/j.aim.2005.07.004. hdl:1885/28586. Wang, Xu-Jia (2006). "Schauder estimates for elliptic and parabolic equations". Chinese Annals of Mathematics, Series B. 27 (6): 637–642. doi:10.1007/s11401-006-0142-3. hdl:1885/28861. (with Neil Trudinger) Trudinger, Neil; Wang, Xu-Jia (2005). "The affine Plateau problem". Journal of the American Mathematical Society. 18 (2): 253–289. doi:10.1090/S0894-0347-05-00475-3. (with Neil Trudinger and Xi-Nan Ma) Ma, Xi-Nan; Trudinger, Neil S.; Wang, Xu-Jia (2005). "Regularity of Potential Functions of the Optimal Transportation Problem". Archive for Rational Mechanics and Analysis. 177 (2): 151–183. Bibcode:2005ArRMA.177..151M. doi:10.1007/s00205-005-0362-9. (with Xiaohua Zhu) Wang, Xu-Jia; Zhu, Xiaohua (2004). "Kähler-Ricci solitons on toric manifolds with positive first Chern class". Advances in Mathematics. 188 (1): 87–103. doi:10.1016/j.aim.2003.09.009. == References == == External links == Xu-Jia Wang's homepage at ANU The Australian Mathematical Society Medal 浙大数学博士汪徐家当选为澳大利亚科学院院士 (in Chinese) The Australian Academy of Science
Wikipedia:Xuan tu#0	Xuan tu or Hsuan thu (simplified Chinese: 弦图; traditional Chinese: 絃圖; pinyin: xuántú; Wade–Giles: hsüan2 tʻu2) is a diagram given in the ancient Chinese astronomical and mathematical text Zhoubi Suanjing indicating a proof of the Pythagorean theorem. Zhoubi Suanjing is one of the oldest Chinese texts on mathematics. The exact date of composition of the book has not been determined. Some estimates of the date range as far back as 1100 BCE, while others estimate the date as late as 200 CE. However, from astronomical evidence available in the book it would appear that much of the material in the book is from the time of Confucius, that is, the 6th century BCE. Hsuan thu represents one of the earliest known proofs of the Pythagorean theorem and also one of the simplest. The text in Zhoubi Suanjing accompanying the diagram has been translated as follows: The art of numbering proceeds from the circle and the square. The circle is derived from the square and the square from the rectangle (literally, the T-square or the carpenter's square). The rectangle originates from the fact that 9x9 = 81 (that is, the multiplication table or properties of numbers as such). Thus, let us cut a rectangle (diagonally) and make the width 3 (units) wide and the height 4 (units) long. The diagonal between the two corners will then be 5 (units) long. Now after drawing a square on the diagonal, circumscribe it by half-rectangles like that which has been left outside, so as to form a (square) plate. Thus the (four) outer half-rectangles of width 3, length 4 and diagonal 5, together make two rectangles (of area 24); then (when this is subtracted from the square plate of area 24) the remainder is of area 25. This (process) is called "piling up the rectangles" (chi chu). The hsuan thu diagram makes use of the 3,4,5 right triangle to demonstrate the Pythagorean theorem. However the Chinese people seems to have generalized its conclusion to all right triangles. The hsuan thu diagram, in its generalized form can be found in the writings of the Indian mathematician Bhaskara II (c. 1114–1185). The description of this diagram appears in verse 129 of Bijaganita of Bhaskara II. There is a legend that Bhaskara's proof of the Pythagorean theorem consisted of only just one word, namely, "Behold!". However, using the notations of the diagram, the theorem follows from the following equation: c 2 = ( a − b ) 2 + 4 ( 1 2 a b ) = a 2 + b 2 . {\displaystyle c^{2}=(a-b)^{2}+4({\tfrac {1}{2}}ab)=a^{2}+b^{2}.} == References ==
Wikipedia:YBC 7289#0	YBC 7289 is a Babylonian clay tablet notable for containing an accurate sexagesimal approximation to the square root of 2, the length of the diagonal of a unit square. This number is given to the equivalent of six decimal digits, "the greatest known computational accuracy ... in the ancient world". The tablet is believed to be the work of a student in southern Mesopotamia from some time between 1800 and 1600 BC. == Content == The tablet depicts a square with its two diagonals. One side of the square is labeled with the sexagesimal number 30. The diagonal of the square is labeled with two sexagesimal numbers. The first of these two, 1;24,51,10 represents the fraction ⁠305,470/216,000⁠ ≈ 1.414213, a numerical approximation of the square root of two that is off by less than one part in two million. The second of the two numbers is 42;25,35 = ⁠30547/720⁠ ≈ 42.426. This number is the result of multiplying 30 by the given approximation to the square root of two, and approximates the length of the diagonal of a square of side length 30. Because the Babylonian sexagesimal notation did not indicate which digit had which place value, one alternative interpretation is that the number on the side of the square is 30⁄60 = 1⁄2. Under this alternative interpretation, the number on the diagonal is ⁠30,547/43,200⁠ ≈ 0.70711, a close numerical approximation of 1 2 {\displaystyle {\frac {1}{\sqrt {2}}}} , the length of the diagonal of a square of side length 1⁄2, that is also off by less than one part in two million. David Fowler and Eleanor Robson write, "Thus we have a reciprocal pair of numbers with a geometric interpretation…". They point out that, while the importance of reciprocal pairs in Babylonian mathematics makes this interpretation attractive, there are reasons for skepticism. The reverse side is partly erased, but Robson believes it contains a similar problem concerning the diagonal of a rectangle whose two sides and diagonal are in the ratio 3:4:5. == Interpretation == Although YBC 7289 is frequently depicted (as in the photo) with the square oriented diagonally, the standard Babylonian conventions for drawing squares would have made the sides of the square vertical and horizontal, with the numbered side at the top. The small round shape of the tablet, and the large writing on it, suggests that it was a "hand tablet" of a type typically used for rough work by a student who would hold it in the palm of his hand. The student would likely have copied the sexagesimal value of the square root of 2 from a table of constants, but an iterative procedure for computing this value can be found in another Babylonian tablet, BM 96957 + VAT 6598. A table of constants that includes the same approximation of the square root of 2 as YBC 7289 is the tablet YBC 7243. The constant appears on line 10 of the table along with the inscription, "the diagonal of a square". The mathematical significance of this tablet was first recognized by Otto E. Neugebauer and Abraham Sachs in 1945. The tablet "demonstrates the greatest known computational accuracy obtained anywhere in the ancient world", the equivalent of six decimal digits of accuracy. Other Babylonian tablets include the computations of areas of hexagons and heptagons, which involve the approximation of more complicated algebraic numbers such as 3 {\displaystyle {\sqrt {3}}} . The same number 3 {\displaystyle {\sqrt {3}}} can also be used in the interpretation of certain ancient Egyptian calculations of the dimensions of pyramids. However, the much greater numerical precision of the numbers on YBC 7289 makes it more clear that they are the result of a general procedure for calculating them, rather than merely being an estimate. The same sexagesimal approximation to 2 {\displaystyle {\sqrt {2}}} , 1;24,51,10, was used much later by Greek mathematician Claudius Ptolemy in his Almagest. Ptolemy did not explain where this approximation came from and it may be assumed to have been well known by his time. == Provenance and curation == It is unknown where in Mesopotamia YBC 7289 comes from, but its shape and writing style make it likely that it was created in southern Mesopotamia, sometime between 1800BC and 1600BC. At Yale, the Institute for the Preservation of Cultural Heritage has produced a digital model of the tablet, suitable for 3D printing. The original tablet is currently kept in the Yale Babylonian Collection at Yale University. == See also == Babylonian mathematics Plimpton 322 IM 67118 == External links == Cuneiform Digital Library Initiative, The CDLI Collection YBC 7289 YBC 7243 Yale Peabody Museum, Babylonian Collection YBC 7289 YBC 7243 == References ==
Wikipedia:Yael Dowker#0	Yael Naim Dowker (Hebrew: יעל נעים דוקר; born Yael Naim; 30 October 1919 – 28 January 2016) was an Israeli-born English mathematician, prominent especially due to her work in the fields of measure theory, ergodic theory and topological dynamics. == Biography == Yael Naim (later Dowker) was born in Tel Aviv. She left for the United States to study at Johns Hopkins University in Baltimore, Maryland. In 1941, as a graduate student, she met Clifford Hugh Dowker, a Canadian topologist working as an instructor there. The couple married in 1944. From 1943 to 1946 they worked together at the Radiation Laboratory at Massachusetts Institute of Technology. Clifford also worked as a civilian adviser for the United States Air Force during World War II. Dowker did her doctorate at Radcliffe College (in Cambridge, Massachusetts) under Witold Hurewicz (a Polish mathematician known for the Hurewicz theorem). She published her thesis Invariant measure and the ergodic theorems in 1947 and received her Ph.D. in 1948. In the period between 1948 and 1949, she did post-doctoral work at the Institute for Advanced Study, located in Princeton, New Jersey. A few years after the war, McCarthyism became a common phenomenon in the academic world, with several of the Dowker couple's friends in the mathematical community harassed and one arrested. In 1950, they emigrated to the United Kingdom. In 1951 Dowker was appointed as assistant lecturer at the University of Manchester, and later went to the Imperial College London, where she was the first female reader within the department. While there, among the students she advised was Bill Parry, who published his thesis in 1960. She also cooperated on some of her work with the Hungarian mathematician Paul Erdős (Erdős' number of one). She worked with her husband with gifted children who were having difficulties at school for the National association for gifted children. == Legacy == The best PhD award at Imperial College London is given in her name each year. == Works == Invariant measure and the ergodic theorems, Duke Math. J. 14 (1947), 1051–1061 Finite and σ {\displaystyle \sigma } -finite measures, Annals of Mathematics, 54 (1951), 595–608 The mean and transitive points of homeomorphisms, Annals of Mathematics, 58 (1953), 123–133 On limit sets in dynamical systems, Proc. London Math. Soc. 4 (1954), 168–176 (with Friedlander, F. G.) On minimal sets in dynamical systems, Quart. J. Math. Oxford Ser. (2) 7 (1956), 5–16 Some examples in ergodic theory, Proc. London Math. Soc. 9 (1959), 227–241 (with Erdős, Paul) == References == == External links == "Yael N. Dowker". Institute for Advanced Study. 1948-09-20. Retrieved 2018-02-13. "Yael Dowker". The Mathematics Genealogy Project. 2017-04-04. Retrieved 2018-02-13.
Wikipedia:Yael Karshon#0	Yael Karshon (Hebrew: יעל קרשון; born 1964) is an Israeli and Canadian mathematician who has been described as "one of Canada's leading experts in symplectic geometry". She works as a professor at the University of Toronto Mississauga and Tel Aviv University . == Education and career == Karshon took part in the 1982 International Mathematical Olympiad, on the Israeli team. She earned her Ph.D. in 1993 from Harvard University under the supervision of Shlomo Sternberg. After working as a C. L. E. Moore instructor at the Massachusetts Institute of Technology, and then earning tenure at the Hebrew University of Jerusalem, she moved to the University of Toronto Mississauga in 2002. == Selected publications == Karshon is the author of the monographs Periodic Hamiltonian flows on four dimensional manifolds (Memoirs of the American Mathematical Society 672, 1999), which completely classified the Hamiltonian actions of the circle group on four-dimensional compact manifolds. With Viktor Ginzburg and Victor Guillemin, she also wrote Moment maps, cobordisms, and Hamiltonian group actions (Mathematical Surveys and Monographs 98, American Mathematical Society, 2002), which surveys "symplectic geometry in the context of equivariant cobordism". == Awards and honours == Karshon won the Krieger–Nelson Prize in 2008. == Personal == Karshon is from Israel, and lived in the US for ten years, eventually becoming a permanent resident. She took Canadian citizenship in 2011. From her marriage to mathematician Dror Bar-Natan she has two sons. == References == == External links == Home page Yael Karshon in the Oberwolfach Photo Collection
Wikipedia:Yair Censor#0	Yair Censor (Hebrew: יאיר צנזור; born November 29, 1943) is an Israeli mathematician and a professor at the University of Haifa, specializing in computational mathematics and optimization, as well as applications of these fields, in particular to medical imaging and radiation therapy treatment planning. == Biography == Yair Censor was born in Rishon LeZion. After serving in the IDF, he studied at the Technion in Haifa, where he earned his D.Sc. in 1975 under the supervision of Professor Adi Ben-Israel. == Academic career == Censor joined the department of mathematics at the University of Haifa in 1979, and became full professor in 1989. His research focuses on mathematical aspects of Intensity-Modulated Radiation Therapy (IMRT). In 2002, he founded the Center for Computational Mathematics and Scientific Computation at the University of Haifa. In recent years he is involved with research about the Superiorization Methodology. Together with S.A. Zenios, he co-authored the book Parallel Optimization: Theory, Algorithms, and Applications (Oxford University Press, New York, NY, USA, 1997), for which he received the 1999 ICS (INFORMS Computing Society) Prize for Research Excellence in the Interface Between Operations Research and Computer Science. Active in the struggle to preserve the academic freedom of the research universities in Israel, Censor was one of the founders of the Inter-Senate Committee (ISC) of the Universities for the Protection of Academic Independence. == See also == Education in Israel == References == == External links == Yair Censor website Convergence and Perturbation Resilience of Dynamic String-Averaging Projection Methods An iterative approach to plan combination in radiotherapy
Wikipedia:Yakov Geronimus#0	Yakov Lazarevich Geronimus, sometimes spelled J. Geronimus (Russian: Я́ков Лазаре́вич Геро́нимус; February 6, 1898, Rostov – July 17, 1984, Kharkov) was a Russian mathematician known for contributions to theoretical mechanics and the study of orthogonal polynomials. The Geronimus polynomials are named after him. == References == Yakov Geronimus at the Mathematics Genealogy Project Geronimus, Yakov Lazarevich (1898–1984) Golinskii, L. (1999), "On the Scientific Legacy of Ya. L. Geronimus" (PDF), Self-Similar Systems (Dubna, 1998), Joint Inst. Nuclear Res., Dubna, pp. 273–281
Wikipedia:Yakov Pesin#0	Yakov Borisovich Pesin (Russian: Яков Борисович Песин) was born in Moscow, Russia (former USSR) on December 12, 1946. Pesin is currently a Distinguished Professor in the Department of Mathematics and the Director of the Anatole Katok Center for Dynamical Systems and Geometry at the Pennsylvania State University (PSU). His primary areas of research are the theory of dynamical systems with an emphasis on smooth ergodic theory, dimension theory in dynamical systems, and Riemannian geometry, as well as mathematical and statistical physics. == Professional life and education == Pesin became interested in mathematics in school but his real involvement began when he entered the boarding school with emphasis on teaching physics and mathematics that was organized by Andrei Kolmogorov. Following his graduation of the school (with honors) in 1965, he successfully passed entry exams to the Department of Mechanics and Mathematics (or "Mech-Mat") of Moscow State University. Pesin graduated from Moscow State University (also with honors) in 1970, receiving a master's degree in mathematics. His master thesis advisor was Yakov Sinai. Pesin naturally wanted to pursue a PhD in mathematics but faced significant challenges due to the oppressive nature and anti-Semitic policies of the Soviet regime. Thus, he was not permitted to continue his study at the university graduate school and was subsequently assigned to work at a research institute in Moscow (for a more complete historical account of the anti-Semitic sentiment in the Soviet mathematics establishment during this period see the article). Since Pesin always dreamed to be a "pure" mathematician, under the circumstances, he chose to combine his work at the institute with his after-hours research in mathematics and within a few years after graduation, he made a number of outstanding breakthroughs in the theory of smooth dynamical systems. His research at this time was conducted under the supervision of his PhD advisor, Dmitry Anosov, and also Anatole Katok. In 1989 Pesin immigrated to the United States with his family. He first worked as a visiting Professor in the Department of Mathematics at the University of Chicago before getting the position of Full Professor at Penn State University. In 2003 Pesin received the title of Distinguished Professor of mathematics. Yakov Pesin is married to Natasha Pesin who while in Russia worked for several years as a senior editor in the division of mathematics at the "Prosvechenie" ("Education") Publishing House in Moscow. After moving to the US she started a new career as a ceramicist (see her artworks at www.natashapesin.com). Yakov Pesin also has two daughters, Elena and Irina who reside in the US. == Research accomplishments == Yakov Pesin is famous for several fundamental discoveries in the theory of dynamical systems (relevant references can be found on Pesin's website). 1) In a joint work with Michael Brin "Flows of frames on manifolds of negative curvature" (Russian Math. Surveys, 1973), Pesin laid down the foundations of partial hyperbolicity theory. As an application, they studied ergodic properties of the frame flows on manifolds of negative curvature. In a later work with Yakov Sinai "Gibbs measures for partially hyperbolic attractors" (Ergodic Theory and Dynamical Systems, 1983) Pesin constructed a special class of u-measures for partially hyperbolic systems which are a direct analog in this setting of the famous Sinai-Ruelle-Bowen (SRB) measures. 2) Pesin's greatest contribution to dynamics is creation of non-uniform hyperbolicity theory, which is commonly known as Pesin Theory. This theory serves as the mathematical foundation for the principal phenomenon known as "deterministic chaos" – the appearance of highly irregular chaotic motions in completely deterministic dynamical systems. Among the highlights of this theory is the formula for the Kolmogorov-Sinai entropy of the system (also known as Pesin entropy formula). His main article on this topic "Characteristic Lyapunov exponents and smooth ergodic theory" (Russian Mathematical Surveys, 1977) has a very high number of citations in mathematical literature and beyond (in physics, biology, etc.). 3) Pesin's later work on non-uniform hyperbolicity includes establishing presence of systems with non-zero Lyapunov exponents on any manifold; a proof of the Eckmann—Ruelle conjecture; the study of the essential coexistence phenomenon of regular and chaotic dynamics; constructions of SRB measures for hyperbolic attractors with singularities, partially hyperbolic and non-uniformly hyperbolic attractors; and effecting thermodynamic formalism for some classes of non-uniformly hyperbolic dynamical systems. 4) Pesin designed a construction (known also as the Caratheodory-Pesin construction) that allows one to introduce and study various dimension-type characteristics of dynamical systems. Among other things his work reveals "dimension nature" of many of the well-known thermodynamics invariants such as metric and topological entropies and topological pressure. It also provides a unified approach to describe various dimension spectra and related multi-fractal formalism (see ). 5) Pesin's work in Mathematical Physics includes the study of Coupled Map Lattices associated with infinite chains of hyperbolic systems as well as the ones generated by some diffusion-type PDEs such as FitzHu-Nagumo and Belousov-Zhabotinsky equations. == Teaching == Yakov Pesin holds a tenured faculty position at the Pennsylvania State University, where he has advised numerous PhD students on their thesis. In addition to his regular teaching responsibilities, he designed and taught courses at the special MASS (Mathematics Advanced Study Semester) program on Dynamical Systems and Analytic and Projective Geometry. He has also delivered mini-courses at numerous International Mathematical Schools. == Honors and recognition == In 1986 Yakov Pesin was invited to speak at the International Congress of Mathematicians (ICM) in Berkeley, CA, but Soviet authorities did not allow him to travel to the US. Nevertheless, his talk on "Ergodic properties and dimension-like characteristics of strange attractors that are close to hyperbolic" was published in the proceedings of the ICM in 1987. In 2012 Yakov Pesin became a Fellow of the American Mathematical Society (in its inaugural class) and in 2019 he was elected a (foreign) member of the European Academy—Academia Europaea. He was elected to the American Academy of Arts and Sciences in 2023. Yakov Pesin was invited to give many distinguished lectures including Invited Address at SIAM Annual Meeting (Kansas City, 1996), Invited Address, at the AMS Annual Meeting (Ohio State University, 2001), and Bernoulli Lecture at the Centre Interfacultaire Bernoulli, Ecole Polytechnique Federale de Lausanne, Switzerland (2013). == References == == External links == Anatole Katok Center for Dynamical Systems and Geometry at Penn State: http://www.math.psu.edu/dynsys/ Yakov Pesin website: http://www.math.psu.edu/pesin
Wikipedia:Yan Rachinsky#0	Yan Zbignevich Rachinsky (Russian: Ян Збигневич Рачинский, also spelt Jan Raczynski, born 6 December 1958 in Moscow, USSR) is a Russian human rights activist, programmer and mathematician. He has been a human rights activist since the late 1980s when he first became involved in the work and activities of Memorial, a human rights organization examining the crimes of Stalin's regime. When the long-serving chairman of International Memorial, Arseny Roginsky, died in 2018, the board elected Rachinsky as his successor. Memorial was awarded the Nobel Peace Prize in 2022, and Rachinsky received the prize on its behalf. == Family == Rachinsky's grandfather Sigismund Raczyński was Polish; his grandmother Rebecca (Rivka) Fyalka (1888-1975) was a prominent member of the Socialist Revolutionary Party. She was sentenced to 13 years "hard labour" (katorga) by a field tribunal after the 1905 Revolution. She began her sentence in 1907 in Buryatia (east Siberia) and was sent into permanent exile in 1910. She escaped and after the February 1917 Revolution was elected to the Soviet of Workers and Soldiers Deputies in Svobodny (Amur Region). == Work with Memorial HRC == In 1990-1995, Raczynski worked extensively with the Memorial Human Rights Centre (HRC), travelling to many hotspots in and around Russia: Karabakh in Azerbaijan; Transnistria in Moldova; and the Prigorodny district of North Ossetia. He was a member of the organisation's team of observers during the first Chechen conflict (1994-1996). == International recognition == In April 2011, Raczynski was awarded the Order of Merit of the Republic of Poland for his research on the 1940 massacres in the Smolensk, Tver and Kharkov Regions of the USSR of POWs and others from the occupied territories of eastern Poland. == "Victims of Political Terror" == Over the past 15 years, Raczynski has served as director of the project to assemble a single resource from the information scattered between the numerous Books of Remembrance compiled and published in Russia since the early 1990s. By 2016, its fifth edition, an online database entitled "Victims of Political Terror in the USSR", contained the names of about three million victims of the Soviet regime: those who were deported, imprisoned or executed from 1918 onwards. This impressive figure was estimated to represent only a quarter of those who would qualify for rehabilitation under the terms of the October 1991 Law. A controversy arose in August 2021 when Israeli historian Aron Schneer publicly announced that Nazi collaborators guilty of war crimes had been included in the database as "victims of political terror". In December 2021, Raczynski responded on behalf of Memorial to Vladimir Putin: on 10 December the Russian president publicly named three Latvian polizei, who had already been excluded from the database in September 2021. Raczynski and Memorial suggested that the Russian authorities should express some appreciation for Memorial's work in compiling such an extensive database. In 2015, formulating the State program for the Commemoration of the Victims of Political Repression, President Putin had talked of creating a unified database of victims. In January 2021 he instructed the FSB, Ministry of Internal Affairs and other relevant bodies to report back on this proposal in early October 2021. By the end of that year, however, nothing more was known of their activities. Memorial, meanwhile, was hampered as before by a lack of access to the archival materials at the disposal of the police and security services. == Closure of Memorial == In mid-November 2021 lawsuits were brought against International Memorial and Memorial HRC in the RF Supreme Court and the Moscow City Court, respectively. After a number of hearings, the Moscow courts ruled on two consecutive days, 28 and 29 December 2021, that both organisations should dissolve. "We never counted on love from the State," commented Raczynski. There were international protests, and a petition in many languages attracted tens of thousands of signatures worldwide. As board chairman of International Memorial, Raczynski said that the organisation would appeal against the decision, but it was forced to close. Memorial was awarded the Nobel Peace Prize in 2022. Raczynski told the BBC that he was ordered to turn down the prize by the Russian authorities. He received the prize on behalf of Memorial and gave the Nobel Lecture. == See also == Chechnya Katyn massacre (1940) in the USSR. Memorial (society) Arseny Roginsky == Bibliography == Alexander Cherkasov and Oleg Orlov, Russia and Chechnya: a trail of crimes and errors, 1998 Zvenya publishers: Moscow, 9785787000214, 398 pp (in Russian). == References and notes == == External links ==
Wikipedia:Yan Soibelman#0	Iakov (Yan) Soibelman (Russian: Яков Семенович Сойбельман) born 15 April 1956 (Kiev, USSR) is a Russian American mathematician, professor at Kansas State University (Manhattan, USA), member of the Kyiv Mathematical Society (Ukraine), founder of Manhattan Mathematical Olympiad. == Scientific work == Yan Soibelman is a specialist in theory of quantum groups, representation theory and symplectic geometry. He introduced the notion of quantum Weyl group, studied representation theory of the algebras of functions on compact quantum groups, and meromorphic braided monoidal categories. His long term collaboration with Maxim Kontsevich is devoted to various aspects of homological mirror symmetry, a proof of Deligne conjecture about operations on the cohomological Hochschild complex, a direct construction of Calabi-Yau varieties based on SYZ conjecture and non-archimedean geometry, and more recently to Donaldson-Thomas theory. Together with Kontsevich he laid the foundation and developed the theory of motivic Donaldson-Thomas invariants. Kontsevich-Soibelman wall-crossing formula for Donaldson-Thomas invariants (a.k.a BPS invariants) found important applications in physics. They also introduced the notion of Cohomological Hall algebra which has numerous applications in geometric representation theory and quantum physics. == See also == bio Yan Soibelman's papers posted to math archives Manhattan Mathematical Olympiad, past years problems == References ==
Wikipedia:Yaroslav Lopatynskyi#0	Yaroslav Borysovych Lopatynskyi (1906–1981) was a Soviet mathematician. Born in Tbilisi, Lopatinskii acquired wide acclaim for his contributions to the theory of differential equations. He is especially known for his condition of stability for boundary-value problems in elliptic equations and for initial boundary-value problems in evolution PDEs. == See also == Lev Lopatinsky == References == http://www-history.mcs.st-and.ac.uk/Biographies/Lopatynsky.html
Wikipedia:Yash Mittal#0	Yashaswini Deval Mittal (born 1941) is a retired mathematician specializing in probability theory and mathematical statistics. She is a professor emerita of mathematics at the University of Arizona. Mittal has a Ph.D. from the University of California, Los Angeles, completed in 1972. Her dissertation, Limiting Behaviour of Maxima in Stationary Gaussian Process, was supervised by Don Ylvisaker. In 1986, she became the first female program director for probability theory at the National Science Foundation, in the same year that Nancy Flournoy became its first female program director for statistics. She was named a Fellow of the Institute of Mathematical Statistics in 1988, "for outstanding and noteworthy contributions to probability theory and its applications, for work in extreme value theory, and for dedicated and conscientious service to the profession and to the IMS". In her retirement from Arizona, she has become an avid origami folder. == Selected publications == Mittal, Y.; Ylvisaker, D. (1975), "Limit distributions for the maxima of stationary Gaussian processes", Stochastic Processes and Their Applications, 3: 1–18, doi:10.1016/0304-4149(75)90002-2, MR 0413243 Stuetzle, Werner; Mittal, Yashaswini (1979), "Some comments on the asymptotic behavior of robust smoothers", in Gasser, Th.; Rosenblatt, M. (eds.), Smoothing Techniques for Curve Estimation, Proceedings of a Workshop held in Heidelberg, April 2–4, 1979, Lecture Notes in Mathematics, vol. 757, Berlin: Springer, pp. 191–195, doi:10.1007/BFb0098497, ISBN 3-540-09706-6, MR 0564259 Good, I. J.; Mittal, Y. (1987), "The amalgamation and geometry of two-by-two contingency tables", Annals of Statistics, 15 (2): 694–711, doi:10.1214/aos/1176350369, MR 0888434 Mittal, Yashaswini (1991), "Homogeneity of subpopulations and Simpson's paradox", Journal of the American Statistical Association, 86 (413): 167–172, doi:10.2307/2289727, JSTOR 2289727 == References ==
Wikipedia:Yasuo Akizuki#0	Yasuo Akizuki (23 August 1902 – 11 July 1984) was a Japanese mathematician. He was a professor at Kyoto University. Alongside Wolfgang Krull, Oscar Zariski, and Masayoshi Nagata, he is famous for his early work in commutative algebra. In particular, he is most well known in helping to demonstrate Akizuki–Hopkins–Levitzki theorem. == Life == Yasuo Akizuki was born on 23 August 1902 in Wakayama. In 1926, Akizuki graduated Faculty of Mathematics, Department of Science, Kyoto Imperial University. He was inaugurated as a professor of Kyoto University in 1948. == See also == Jacob Levitzki == References ==
Wikipedia:Yasuyuki Kawahigashi#0	Yasuyuki Kawahigashi (河東泰之, born 1962), formerly known as Yasuyuki Asano, is a Japanese mathematician and a professor at the Graduate School of Mathematical Sciences, the University of Tokyo. His primary area of expertise is operator algebra theory. == Career == Born in Ōta, Tokyo, Kawahigashi was raised in a family where his father worked for an oil company and his mother was a teacher at a Kumon learning centre. Kawahigashi graduated from Azabu High School in 1981, where he was classmates with Hiraku Nakajima. He matriculated at the University of Tokyo in 1981. During his years as an undergraduate, he worked for ASCII and authored several bestselling software books, supporting himself through royalties. Kawahigashi completed his undergraduate degree at UTokyo's Department of Mathematics in 1985. He then went on to study at the University of California, Los Angeles in 1985 under the supervision of Masamichi Takesaki, a proponent of the Tomita–Takesaki theory. he earned his master's degree from UCLA the following year. He earned his first Ph.D. from UCLA in 1989 and the second from UTokyo in 1990. He has held various academic positions at various institutions before attaining his current role. === Major Contributions === According to Kawahigashi, his most notable achievement so far is the completion of the classification theory for local conformal nets with central charge less than one, co-authored with Italian mathematician Roberto Longo in 2004. == Supervision == Kawahigashi has supervised Narutaka Ozawa as a doctoral candidate. Although educated at the Department of Physics of UTokyo, Yoshiko Ogata joined Kawahigashi's seminar as a student, and she worked with him at the Department of Mathematics at the university for more than a decade. Kawahigashi believes that one should never stop questioning, researching, or consulting others until complete understanding is achieved, asserting that 'It is absurd for someone who cannot do this to be in a doctoral programme. == References ==
Wikipedia:Yboon García Ramos#0	Yboon Victoria García Ramos is a Peruvian mathematician with a Ph.D. from the Université des Antilles et de la Guyane and the National University of Engineering.[2] Currently, García is an ordinary professor and researcher at Universidad del Pacífico. == Education == Her interest in science led her to pursue a career in mathematics. She began her studies at the National University of Engineering where she continued until earning a Master's degree in Mathematics. She earned her Ph.D. at the Université des Antilles et de la Guyane and the National University of Engineering in operator theory. She completed a postdoctoral fellowship at the Center for Mathematical Modeling at the University of Chile. == Career == García has academic expertise in applied mathematics, mathematical economics, and nonlinear analysis. She has served as a thesis advisor for undergraduate students at UNI, and is currently a researcher at the Instituto de Matemática y Ciencias Afines (IMCA). She is a member of both the Peruvian Mathematical Society and the Société mathématique de France. In recognition of her contributions to science and technology, she has been accredited as a qualified researcher by SINACYT (REGINA-CONCYTEC). As coordinator of the Mathematics II course in the Department of Economics at Universidad del Pacífico, she has continued to foster mathematical education and research. === Research and publications === Her research has significantly contributed to the study of monotone operators, quasiconvex optimization, and variational analysis, with applications in variational inequality problems and game theory. In 2003, García co-authored an article with the co-founder of IMCA titled "Applications of the Ky Fan Conjecture." The paper presents Ky Fan's lemma as a tool for establishing existence results in various fields, including variational inequality problems, equilibrium problems, and game theory. In 2014, she played a key role as the principal organizer of the Latin American Workshop on Optimization and Control. Previously, between 2011 and 2012, she served as the scientific lead of the Peruvian team in the Stic-Amsud OVIMINE project, further solidifying her impact in international research collaborations. In 2016, she published her first book, Cálculo Diferencial e Integral in collaboration with Oswaldo Velásquez. It covers differential and integral calculus while also introducing fundamental mathematical concepts and their applications in economics and business science. == Works == === Books === García, Yboon (2016). Cálculo diferencial e integral (1a ed.). Lima: Universidad del Pacífico. ISBN 978-9972-57-369-9. === Research contributions === García-Ramos has published various articles in scientific journals of high impact in the area of convex analysis and mathematical programming and optimization. Her most relevant publications are: Bueno, Orestes; Cotrina, John; García, Yboon (2023). "Existence and uniqueness of maximal elements for preference relations: Variational approach". Journal of Optimization Theory and Applications. 198 (3): 1246–1263. arXiv:2301.12329. doi:10.1007/s10957-023-02251-y. ISSN 0022-3239. Flores-Bazán, F.; García, Y.; Hadjisavvas, N. (1 September 2021). "Characterizing quasiconvexity of the pointwise infimum of a family of arbitrary translations of quasiconvex functions, with applications to sums and quasiconvex optimization". Mathematical Programming. 189 (1): 315–337. doi:10.1007/s10107-021-01647-w. ISSN 1436-4646. Aussel, D.; García, Y. (1 January 2014). "On extensions of kenderov's single-valuedness result for monotone maps and quasimonotone maps". SIAM Journal on Optimization. 24 (2): 702–713. doi:10.1137/120880215. ISSN 1052-6234. Aussel, D.; Garcia, Y.; Hadjisavvas, N. (2010). "Single-Directional Property of Multivalued Maps and Variational Systems". SIAM Journal on Optimization. 20 (3): 1274–1285. doi:10.1137/080735618. ISSN 1052-6234. García, Yboon; Sosa, Wilfredo (2003). "Aplicaciones de Lema de Ky Fan". TECNIA. 13 (2): 75–86. ISSN 2309-0413. == References ==
Wikipedia:Yevgeny Dyakonov#0	Evgenii Georgievich Dyakonov (Russian: Евгений Георгиевич Дьяконов) (July 2, 1935 – August 11, 2006) was a Russian mathematician. Dyakonov was a Ph.D. student of Sergei Sobolev. He worked at the Moscow State University. He authored over hundred papers and several books. Dyakonov was recognized for his pioneering work in the 60s–80s on efficient spectrally equivalent preconditioning for linear systems and eigenvalue problems. In the last decade, strengthened Sobolev spaces became Dyakonov's main topic of research, e.g., (Dyakonov, 2004). == References == Dyakonov, E.G. (1996). Optimization in solving elliptic problems. CRC-Press. pp. 592. ISBN 978-0-8493-2872-5. Dyakonov, E.G. (2004), "On spectral problems in energy spaces on composite manifolds with singular geometry of the blocks: I", Differential Equations, 4 (7): 934–946, doi:10.1023/B:DIEQ.0000047024.18841.44, S2CID 195368332 == External links == Evgenii D'yakonov — scientific works on the website Math-Net.Ru Yevgeny Dyakonov at zbMATH Yevgeny Dyakonov at the Mathematics Genealogy Project NA Digest, V. 06, # 33 obituary on NA Digest by Andrew Knyazev.
Wikipedia:Yigu yanduan#0	Yigu yanduan (益古演段 Old Mathematics in Expanded Sections) is a 13th-century mathematical work by Yuan dynasty mathematician Li Zhi. == Overview == Yigu yanduan was based on Northern Song mathematician Jiang Zhou's (蒋周) Yigu Ji (益古集 Collection of Old Mathematics) which is not extant. However, from fragments quoted in Yang Hui's work The Complete Algorithms of Acreage (田亩比类算法大全), this lost mathematical treatise Yigu Ji was about solving area problems with geometry. Li Zhi used the examples of Yigu Ji to introduce the art of Tian yuan shu to newcomers to this field. Although Li Zhi's previous monograph Ceyuan haijing also used Tian yuan shu, it is harder to understand than Yigu yanduan. Yigu yanduan was later collected into Siku Quanshu. Yigu yanduan consists of three volumes with 64 problems solved using Tian yuan sh] in parallel with the geometrical method. Li Zhi intended to introduce students to the art of Tian yuan shu through ancient geometry. Yigu yanduan together with Ceyuan haijing are considered major contributions to Tian yuan shu by Li Zhi. These two works are also considered as the earliest extant documents on Tian yuans shu. All the 64 problems followed more or less the same format, starting with a question (问), followed by an answer (答曰), a diagram, then an algorithm (术), in which Li Zhi explained step by step how to set up algebra equation with Tian yuan shu, then followed by geometrical interpretation (Tiao duan shu). The order of arrangement of Tian yuan shu equation in Yigu yanduan is the reverse of that in Ceyuan haijing, i.e., here with the constant term at top, followed by first order tian yuan, second order tian yuan, third order tian yuan etc. This later arrangement conformed with contemporary convention of algebra equation( for instance, Qin Jiushao's Mathematical Treatise in Nine Sections), and later became a norm. Yigu yanduan was first introduced to the English readers by the British Protestant Christian missionary to China, Alexander Wylie who wrote:Yi koo yen t'wan...written in 1282 consists of 64 geometrical problem, illustrated the principle of Plane Measurement, Evolution and other rules, the whole being developed by means of T'een yuen. In 1913 Van Hée translated all 64 problems in Yigu yanduan into French. == Volume I == Problem 1 to 22, all about the mathematics of a circle embedded in a square. Example: problem 8 There is a square field, with a circular pool in the middle, given that the land is 13.75 mu, and the sum of the circumferences of the square field and the circular pool equals to 300 steps, what is the circumferences of the square and circle respective ? Anwwer: The circumference of the square is 240 steps, the circumference of the circle is 60 steps. Method: set up tian yuan one (celestial element 1) as the diameter of the circle, x TAI multiply it by 3 to get the circumference of the circle 3x (pi ~~3) TAI subtract this from the sum of circumferences to obtain the circumference of the square 300 − 3 x {\displaystyle 300-3x} TAI The square of it equals to 16 times the area of the square ( 300 − 3 x ) ∗ ( 300 − 3 x ) = 90000 − 1800 x + 9 x 2 {\displaystyle (300-3x)(300-3x)=90000-1800x+9x^{2}} TAI Again set up tian yuan 1 as the diameter of circle, square it up and multiplied by 12 to get 16 times the area of circle as TAI subtract from 16 time square area we have 16 times area of land TAI put it at right hand side and put 16 times 13.75 mu = 16 13.75 240 =52800 steps at left, after cancellation, we get − 3 x 2 − 1800 x + 37200 = 0 : {\displaystyle -3x^{2}-1800x+37200=0:} TAI Solve this equation to get diameter of circle = 20 steps, circumference of circle = 60 steps == Volume II == Problem 23 to 42, 20 problems in all solving geometry of rectangle embedded in circle with tian yuan shu Example, problem 35 Suppose we have a circular field with a rectangular water pool in the center, and the distance of a corner to the circumference is 17.5 steps, and the sum of length and width of the pool is 85 steps, what is the diameter of the circle, the length and width of the pool ? Answer: The diameter of the circle is one hundred steps, the length of pool is 60 steps, and the width 25 steps. Method: Let tian yuan one as the diagonal of rectangle, then the diameter of circle is tian yuan one plus 17.52 x + 35 {\displaystyle x+35} multiply the square of diameter with π ≈ 3 {\displaystyle \pi \approx 3} equals to four times the area of the circle: 3 ( x + 35 ) 2 = 3 x 2 + 210 x + 3675 {\displaystyle 3(x+35)^{2}=3x^{2}+210x+3675} subtracting four times the area of land to obtain: four times the area of pool = 3 x 2 + 210 x + 3675 − 4 x 6000 {\displaystyle 3x^{2}+210x+3675-4x6000} = 3 x 2 + 210 x − 20325 {\displaystyle 3x^{2}+210x-20325} now The square of the sum of length and width of the pool =8585 =7225 which is four times the pool area plus the square of the difference of its length and width ( ( L − W ) 2 {\displaystyle (L-W)^{2}} ) Further double the pool area plus ( L − W ) 2 {\displaystyle (L-W)^{2}} equals to L 2 + W 2 {\displaystyle L^{2}+W^{2}} = the square of the diagonal of the pool thus ( four time pool area + the square of its dimension difference) - (twice the pool area + square if its dimension difference) equals 7225 − x 2 {\displaystyle 7225-x^{2}} = twice the pool area so four times the area of pool = 2 ( 7225 − x 2 ) {\displaystyle 2(7225-x^{2})} equate this with the four times pool area obtained above 2 ( 7225 − x 2 ) {\displaystyle 2(7225-x^{2})} = 3 x 2 + 210 x − 20325 {\displaystyle 3x^{2}+210x-20325} we get a quadratic equation 5 x 2 + 210 x − 34775 {\displaystyle 5x^{2}+210x-34775} =0 Solve this equation to get diagonal of pool =65 steps diameter of circle =65 +217.5 =100 steps Length - width =35 steps Length + width =85 steps Length =60 steps Width =25 steps == Volume III == Problem 42 to 64, altogether 22 questions about the mathematics of more complex diagrams Q: fifty-fourth. There is a square field, with a rectangular water pool lying on its diagonal. The area outside the pool is one thousand one hundred fifty paces. Given that from the corners of the field to the straight sides of the pool are fourteen paces and nineteen paces. What is the area of the square field, what is the length and width of the pool? Answer: The area of the square field is 40 square paces, the length of the pool is thirty five paces, and the width is twenty five paces. Let the width of the pool be Tianyuan 1. TAI Add the width of the pool to twice the distance from field corner to short long side of pool equals to the length of diagonal of the field x+38 TAI Square it to obtain the area of square with the length of the pool diagonal as its sides x 2 + 76 x + 1444 {\displaystyle x^{2}+76x+1444} TAI The length of pool minus the width of pool multiplied by 2 = 2 (19-14) = 10 Pool length = pool width +10: x+10 TAI Pool area = pool with times pool length : x(x+10) = x 2 + 10 x {\displaystyle x^{2}+10x} TAI Area of pool times 乘 1.96 ( the square root of 2) =1.4 one has 1.96 x 2 + 19.6 x {\displaystyle 1.96x^{2}+19.6x} tai Area of diagonal square subtract area of pool multiplied 1.96 equals to area of land times 1.96: x 2 + 76 x + 1444 {\displaystyle x^{2}+76x+1444} - 1.96 x 2 + 19.6 x = {\displaystyle 1.96x^{2}+19.6x=} ： − 0.96 x 2 + 56.4 x + 1444 {\displaystyle -0.96x^{2}+56.4x+1444} TAI Occupied plot times 1.96 =1150 * 1.96 =2254= − 0.96 x 2 + 56.4 x + 1444 {\displaystyle -0.96x^{2}+56.4x+1444} hence = − 0.96 x 2 + 56.4 x − 810 {\displaystyle -0.96x^{2}+56.4x-810} : TAI Solve this equation and we obtain width of pool 25 paces therefore pool length =pool width +10 =35 paces length of pool =45 paces == References == == Further reading == Yoshio Mikami The Development of Mathematics in China and Japan, p. 81 Annotated Yigu yanduan by Qing dynasty mathematician Li Rui.
Wikipedia:Yilin Wang#0	Wang Yilin (Chinese: 王宜林; pinyin: Wáng Yílín; born September 1956) is a Chinese business and oil magnate who was the chairman of the Board of China National Petroleum Corporation (CNPC) and the chairman of the Board of PetroChina. Wang, as the country's most influential business leader, has accompanied Chinese Communist Party general secretary Xi Jinping during many state visits, including to the UK, France, Kazakhstan, Russia, UAE etc. == Biography == In April 2011, Wang assumed the role of Chairman of China National Offshore Oil Corporation. In November 2012, he was elected member of the CCP's 18th Central Commission for Discipline Inspection. Wang Yilin began to serve Chairman of CNPC in April 2015, and he started to hold a concurrent post as Chairman of PetroChina in June 2015. In July 2017, Wang Yiling, the Chairman of CNPC, serves as the head of the Chinese National Delegation to the World Petroleum Congress in Istanbul. On March 13, Wang was elected co-chairman of the Economic Committee of the Chinese People's Political Consultative Conference (PCC 2018). === Downfall === In July 2024, Wang was expelled from the CCP for discipline violations. == References ==
Wikipedia:Yingda Cheng#0	Yingda Cheng (Chinese: 程颖达, born 1983) is a Chinese-American applied mathematician specializing in scientific computation and numerical analysis, including Galerkin methods for the computational solution of differential equations and the simulation of nonlinear optics and plasma physics. She is a professor of mathematics at Virginia Tech. == Education and career == Cheng is originally from Hefei, where she was born in 1983. She graduated from the University of Science and Technology of China in 2003, and earned a master's degree in applied mathematics at Brown University in 2004. She completed her Ph.D. at Brown in 2007. Her dissertation, Discontinuous Galerkin Methods for Hamilton–Jacobi Equations and Equations with Higher Order Derivatives, was supervised by Chi-Wang Shu. After postdoctoral research with Irene M. Gamba at the University of Texas at Austin, she joined Michigan State University as an assistant professor of mathematics in 2011. She was promoted to associate professor in 2016 and full professor in 2021. In 2023 she was the Knut and Alice Wallenberg Foundation Visiting Professor at Uppsala University in Sweden, and moved to her present position as a professor of mathematics at Virginia Tech, affiliated with the Computational Modeling & Data Analytic Program. == Recognition == Cheng was the 2023 recipient of the Germund Dahlquist Prize of the Society for Industrial and Applied Mathematics, in recognition of her research "on discontinuous Galerkin methods, including structure preservation and sparse grid methods for kinetic and transport equations". == References == == External links == Home page Yingda Cheng publications indexed by Google Scholar
Wikipedia:Yiqun Lisa Yin#0	Yiqun Lisa Yin (Chinese: 殷益群; pinyin: Yīn Yìqún) is a Chinese-American cryptographer and independent security consultant. Yin is known for breaking the SHA-1 cryptographic hash function, for developing the RC6 block cipher, and for her service as editor of the IEEE P1363 project for the standardization of public-key cryptography. == Education and career == Yin was a student at Peking University from 1985 to 1989, and earned a bachelor's degree in applied mathematics there. She went to the Massachusetts Institute of Technology for graduate study, and completed her Ph.D. there in applied mathematics in 1994. Her dissertation, Teaching, Learning, and Exploration, concerned computational learning theory and online algorithms; it was supervised by Michael Sipser. She worked as a researcher at RSA Laboratories from 1994 to 1999, and as director of security technologies at NTT's Palo Alto Laboratory for Multimedia Communications from 1999 to 2002, before becoming an independent consultant. She also worked as a visiting researcher at Princeton University and Tsinghua University. From 2016 to 2019, Yin was the chief security officer and chief cryptographer of Symbiont. == Contributions == Yin was the editor of the IEEE P1363 project for the standardization of public-key cryptography. With Ron Rivest, Matt Robshaw, and Ray Sidney, she was one of the designers of RC6, a block cipher with symmetric keys that was one of five finalists for the 1997–2000 Advanced Encryption Standard competition. In 2005, with Wang Xiaoyun and Hongbo Yu, Yin demonstrated an unexpected high probability of collisions (two different data values with the same hash) in the SHA-1 cryptographic hash function, originally designed by the National Security Agency. Their work caused SHA-1 to be considered as broken, and it has since fallen out of use. == References == == External links == Home page at MIT Yiqun Lisa Yin publications indexed by Google Scholar
Wikipedia:Yitzhak Katznelson#0	Yitzhak Katznelson (Hebrew: יצחק כצנלסון; born 1934) is an Israeli mathematician. Katznelson was born in Jerusalem. He received his doctoral degree from the University of Paris in 1956. He is a professor of mathematics at Stanford University. He is the author of An Introduction to Harmonic Analysis, which won the Steele Prize for Mathematical Exposition in 2002. In 2012 he became a fellow of the American Mathematical Society. == References == == External links == An Introduction to Harmonic Analysis
Wikipedia:Yoneda product#0	In algebra, the Yoneda product (named after Nobuo Yoneda) is the pairing between Ext groups of modules: Ext n ⁡ ( M , N ) ⊗ Ext m ⁡ ( L , M ) → Ext n + m ⁡ ( L , N ) {\displaystyle \operatorname {Ext} ^{n}(M,N)\otimes \operatorname {Ext} ^{m}(L,M)\to \operatorname {Ext} ^{n+m}(L,N)} induced by Hom ⁡ ( N , M ) ⊗ Hom ⁡ ( M , L ) → Hom ⁡ ( N , L ) , f ⊗ g ↦ g ∘ f . {\displaystyle \operatorname {Hom} (N,M)\otimes \operatorname {Hom} (M,L)\to \operatorname {Hom} (N,L),\,f\otimes g\mapsto g\circ f.} Specifically, for an element ξ ∈ Ext n ⁡ ( M , N ) {\displaystyle \xi \in \operatorname {Ext} ^{n}(M,N)} , thought of as an extension ξ : 0 → N → E 0 → ⋯ → E n − 1 → M → 0 , {\displaystyle \xi :0\rightarrow N\rightarrow E_{0}\rightarrow \cdots \rightarrow E_{n-1}\rightarrow M\rightarrow 0,} and similarly ρ : 0 → M → F 0 → ⋯ → F m − 1 → L → 0 ∈ Ext m ⁡ ( L , M ) , {\displaystyle \rho :0\rightarrow M\rightarrow F_{0}\rightarrow \cdots \rightarrow F_{m-1}\rightarrow L\rightarrow 0\in \operatorname {Ext} ^{m}(L,M),} we form the Yoneda (cup) product ξ ⌣ ρ : 0 → N → E 0 → ⋯ → E n − 1 → F 0 → ⋯ → F m − 1 → L → 0 ∈ Ext m + n ⁡ ( L , N ) . {\displaystyle \xi \smile \rho :0\rightarrow N\rightarrow E_{0}\rightarrow \cdots \rightarrow E_{n-1}\rightarrow F_{0}\rightarrow \cdots \rightarrow F_{m-1}\rightarrow L\rightarrow 0\in \operatorname {Ext} ^{m+n}(L,N).} Note that the middle map E n − 1 → F 0 {\displaystyle E_{n-1}\rightarrow F_{0}} factors through the given maps to M {\displaystyle M} . We extend this definition to include m , n = 0 {\displaystyle m,n=0} using the usual functoriality of the Ext ∗ ⁡ ( ⋅ , ⋅ ) {\displaystyle \operatorname {Ext} ^{}(\cdot ,\cdot )} groups. == Applications == === Ext Algebras === Given a commutative ring R {\displaystyle R} and a module M {\displaystyle M} , the Yoneda product defines a product structure on the groups Ext ∙ ( M , M ) {\displaystyle {\text{Ext}}^{\bullet }(M,M)} , where Ext 0 ( M , M ) = Hom R ( M , M ) {\displaystyle {\text{Ext}}^{0}(M,M)={\text{Hom}}_{R}(M,M)} is generally a non-commutative ring. This can be generalized to the case of sheaves of modules over a ringed space, or ringed topos. === Grothendieck duality === In Grothendieck's duality theory of coherent sheaves on a projective scheme i : X ↪ P k n {\displaystyle i:X\hookrightarrow \mathbb {P} _{k}^{n}} of pure dimension r {\displaystyle r} over an algebraically closed field k {\displaystyle k} , there is a pairing Ext O X p ( O X , F ) × Ext O X r − p ( F , ω X ∙ ) → k {\displaystyle {\text{Ext}}_{{\mathcal {O}}_{X}}^{p}({\mathcal {O}}_{X},{\mathcal {F}})\times {\text{Ext}}_{{\mathcal {O}}_{X}}^{r-p}({\mathcal {F}},\omega _{X}^{\bullet })\to k} where ω X {\displaystyle \omega _{X}} is the dualizing complex ω X = E x t O P n − r ( i ∗ F , ω P ) {\displaystyle \omega _{X}={\mathcal {Ext}}_{{\mathcal {O}}_{\mathbb {P} }}^{n-r}(i_{}{\mathcal {F}},\omega _{\mathbb {P} })} and ω P = O P ( − ( n + 1 ) ) {\displaystyle \omega _{\mathbb {P} }={\mathcal {O}}_{\mathbb {P} }(-(n+1))} given by the Yoneda pairing. === Deformation theory === The Yoneda product is useful for understanding the obstructions to a deformation of maps of ringed topoi. For example, given a composition of ringed topoi X → f Y → S {\displaystyle X\xrightarrow {f} Y\to S} and an S {\displaystyle S} -extension j : Y → Y ′ {\displaystyle j:Y\to Y'} of Y {\displaystyle Y} by an O Y {\displaystyle {\mathcal {O}}_{Y}} -module J {\displaystyle J} , there is an obstruction class ω ( f , j ) ∈ Ext 2 ( L X / Y , f ∗ J ) {\displaystyle \omega (f,j)\in {\text{Ext}}^{2}(\mathbf {L} _{X/Y},f^{}J)} which can be described as the yoneda product ω ( f , j ) = f ∗ ( e ( j ) ) ⋅ K ( X / Y / S ) {\displaystyle \omega (f,j)=f^{}(e(j))\cdot K(X/Y/S)} where K ( X / Y / S ) ∈ Ext 1 ( L X / Y , L Y / S ) f ∗ ( e ( j ) ) ∈ Ext 1 ( f ∗ L Y / S , f ∗ J ) {\displaystyle {\begin{aligned}K(X/Y/S)&\in {\text{Ext}}^{1}(\mathbf {L} _{X/Y},\mathbf {L} _{Y/S})\\f^{}(e(j))&\in {\text{Ext}}^{1}(f^{}\mathbf {L} _{Y/S},f^{*}J)\end{aligned}}} and L X / Y {\displaystyle \mathbf {L} _{X/Y}} corresponds to the cotangent complex. == See also == Ext functor Derived category Deformation theory Kodaira–Spencer map == References == May, J. Peter. "Notes on Tor and Ext" (PDF). == External links == Universality of Ext functor using Yoneda extensions
Wikipedia:Yoshie Katsurada#0	Yoshie Katsurada (Japanese: 桂田芳枝, 3 September 1911 – 10 May 1980) was a Japanese mathematician specializing in differential geometry. She became the first Japanese woman to earn a doctorate in mathematics, in 1950, and the first to obtain an imperial university professorship in mathematics, in 1967. == Life == Katsurada was born in Akaigawa, Hokkaido on 3 September 1911, a daughter of an elementary school principal. In high school in Otaru, she took special instruction in mathematics from a boys' mathematics instructor. Graduating from high school in 1929, she began auditing classes at the Tokyo Physics School, a predecessor to the Tokyo University of Science, in 1931. She began working as an administrative assistant in the Hokkaido University Department of Mathematics in 1936. In 1938 she began study in mathematics at Tokyo Woman's Christian University, withdrawing in 1940 to transfer to Hokkaido University. She graduated from Hokkaido University in 1942, and in the same year became an assistant professor there. In 1950, she completed a doctorate in mathematics at Hokkaido University, under the supervision of Shoji Kawaguchi, becoming the first Japanese woman to earn a doctorate in mathematics, and earning a promotion to associate professor. She remained at Hokkaido University for the remainder of her career, with research visits to Sapienza University of Rome, ETH Zurich, and the University of California, Berkeley. She was promoted to full professor in 1967, the first female professor in mathematics at a former imperial university. She retired in 1975, and died on 10 May 1980. == Research == Katsurada's early research, from the beginning of her studies into the mid-1950s, primarily concerned line elements; this was the primary interest of her advisor Shoji Kawaguchi, with whom she continued to collaborate on this subject. After visiting Heinz Hopf at ETH Zurich in 1957–1958, she shifted interests to submanifolds and hypersurfaces in Riemannian manifolds, publishing well-regarded work in this area. == Recognition == Several papers in the 1972 volume of the Hokkaido Mathematical Journal are dedicated to Katsurada in honor of her 60th birthday. Katsurada was given the Hokkaido Culture Award in 1973. == References ==
Wikipedia:Yoshiko Ogata#0	Yoshiko Ogata (Japanese: 緒方芳子) is a Japanese mathematical physicist whose research concerns quantum statistical mechanics, quantum information theory, and the quantum many-body problem. She is a professor of mathematics at the Research Institute for Mathematical Sciences, Kyoto University. == Education and career == Ogata studied physics at the University of Tokyo at both the undergraduate and graduate level. After completing her Ph.D., and postdoctoral research at Aix-Marseille University and the University of California, Davis, she became a faculty member at Kyushu University. She returned to the University of Tokyo as a professor in 2009. == Recognition == Ogata won the 2007 Takebe Katahiro Prize for Encouragement of Young Researchers, and the 2010 2nd Inoue Science Research Award. In 2014, she won the Young Scientists' Prize of the Commendation for Science and Technology of the Japanese Ministry of Education, Culture, Sports, Science and Technology, for "her researches on operator algebras and their applications to quantum statistical mechanics". She was the 2022 winner of the Autumn Prize of the Mathematical Society of Japan. She was one of the 2021 winners of the Henri Poincaré Prize, honored for her "groundbreaking work on the mathematical theory of quantum spin systems, ranging from the formulation of Onsager reciprocity relations to innovative contributions to the theory of matrix product states and of symmetry-protected topological phases of infinite quantum spin chains". She was an invited speaker at the 2022 International Congress of Mathematicians. She won the 44th Saruhashi Prize in 2024 for "Mathematical Studies of Quantum Many-Body Systems". She won the Asahi Prize in 2025 (fiscal 2024) for her research on "mathematical problems arising from quantum statistical mechanics". == References ==
Wikipedia:Yoshiko Wakabayashi#0	Yoshiko Wakabayashi (born 21 May 1950) is a Brazilian computer scientist and applied mathematician whose research interests include combinatorial optimization, polyhedral combinatorics, packing problems, and graph algorithms. She is a professor in the department of computer science and institute of mathematics and statistics at the University of São Paulo. == Education and career == After earning bachelor's and master's degrees in applied mathematics at the University of São Paulo in 1972 and 1977 respectively, Wakabayashi went to the University of Augsburg in Germany for doctoral study in applied mathematics, completing her doctorate (Dr. rer. nat.) in 1986. Her dissertation, Aggregation of Binary Relations: Algorithmic and Polyhedral Investigations, was supervised by Martin Grötschel. She became an assistant professor at the University of São Paulo in 1977, associate professor in 1995, and full professor in 2006. == Recognition == Wakabayashi was named as a commander of the National Order of Scientific Merit in 2010. She was elected to the Academia de Ciências do Estado de São Paulo in 2012, and to the Brazilian Academy of Sciences in 2019. In 2020 the Brazilian Computer Society gave her their prize for scientific merit. == References == == External links == Home page Yoshiko Wakabayashi publications indexed by Google Scholar
Wikipedia:Yoshio Shimamoto#0	Yoshio Shimamoto was a nuclear physicist who also did work in mathematics and computer science. While at Brookhaven National Laboratory (1954-1987), he designed the logic for the MERLIN digital computer in 1958, and served as chairman of the Applied Mathematics Department from 1964 to 1975. Shimamoto researched in combinatorial mathematics, the economics of outer continental shelf oil and gas lease sales (on behalf of the U.S. Geological Survey), the architecture of supercomputers, and the linking of computers for parallel processing. During the 1970s, he worked with Heinrich Heesch and Karl Durre on methods for a computer-aided proof of the four color theorem, using computer programs to apply Heesch's notion of "discharging" to eliminate 4-colorable cases. A proof of the Four Color Theorem, which he presented in 1971, was later shown to be flawed, but it served as the basis for further work. Born in Hawaii in 1924, Shimamoto served with the U.S. Army Signal Corps and Strategic Bombing Survey in Japan, during World War II. He died in New Jersey on August 27, 2009. == References ==
Wikipedia:Young symmetrizer#0	In mathematics, a Young symmetrizer is an element of the group algebra of the symmetric group S n {\displaystyle S_{n}} whose natural action on tensor products V ⊗ n {\displaystyle V^{\otimes n}} of a complex vector space V {\displaystyle V} has as image an irreducible representation of the group of invertible linear transformations G L ( V ) {\displaystyle GL(V)} . All irreducible representations of G L ( V ) {\displaystyle GL(V)} are thus obtained. It is constructed from the action of S n {\displaystyle S_{n}} on the vector space V ⊗ n {\displaystyle V^{\otimes n}} by permutation of the different factors (or equivalently, from the permutation of the indices of the tensor components). A similar construction works over any field but in characteristic p (in particular over finite fields) the image need not be an irreducible representation. The Young symmetrizers also act on the vector space of functions on Young tableau and the resulting representations are called Specht modules which again construct all complex irreducible representations of the symmetric group while the analogous construction in prime characteristic need not be irreducible. The Young symmetrizer is named after British mathematician Alfred Young. == Definition == Given a finite symmetric group Sn and specific Young tableau λ corresponding to a numbered partition of n, and consider the action of S n {\displaystyle S_{n}} given by permuting the boxes of λ {\displaystyle \lambda } . Define two permutation subgroups P λ {\displaystyle P_{\lambda }} and Q λ {\displaystyle Q_{\lambda }} of Sn as follows: P λ = { g ∈ S n : g preserves each row of λ } {\displaystyle P_{\lambda }=\{g\in S_{n}:g{\text{ preserves each row of }}\lambda \}} and Q λ = { g ∈ S n : g preserves each column of λ } . {\displaystyle Q_{\lambda }=\{g\in S_{n}:g{\text{ preserves each column of }}\lambda \}.} Corresponding to these two subgroups, define two vectors in the group algebra C S n {\displaystyle \mathbb {C} S_{n}} as a λ = ∑ g ∈ P λ e g {\displaystyle a_{\lambda }=\sum _{g\in P_{\lambda }}e_{g}} and b λ = ∑ g ∈ Q λ sgn ⁡ ( g ) e g {\displaystyle b_{\lambda }=\sum _{g\in Q_{\lambda }}\operatorname {sgn}(g)e_{g}} where e g {\displaystyle e_{g}} is the unit vector corresponding to g, and sgn ⁡ ( g ) {\displaystyle \operatorname {sgn}(g)} is the sign of the permutation. The product c λ := a λ b λ = ∑ g ∈ P λ , h ∈ Q λ sgn ⁡ ( h ) e g h {\displaystyle c_{\lambda }:=a_{\lambda }b_{\lambda }=\sum _{g\in P_{\lambda },h\in Q_{\lambda }}\operatorname {sgn}(h)e_{gh}} is the Young symmetrizer corresponding to the Young tableau λ. Each Young symmetrizer corresponds to an irreducible representation of the symmetric group, and every irreducible representation can be obtained from a corresponding Young symmetrizer. (If we replace the complex numbers by more general fields the corresponding representations will not be irreducible in general.) == Construction == Let V be any vector space over the complex numbers. Consider then the tensor product vector space V ⊗ n = V ⊗ V ⊗ ⋯ ⊗ V {\displaystyle V^{\otimes n}=V\otimes V\otimes \cdots \otimes V} (n times). Let Sn act on this tensor product space by permuting the indices. One then has a natural group algebra representation C S n → End ⁡ ( V ⊗ n ) {\displaystyle \mathbb {C} S_{n}\to \operatorname {End} (V^{\otimes n})} on V ⊗ n {\displaystyle V^{\otimes n}} (i.e. V ⊗ n {\displaystyle V^{\otimes n}} is a right C S n {\displaystyle \mathbb {C} S_{n}} module). Given a partition λ of n, so that n = λ 1 + λ 2 + ⋯ + λ j {\displaystyle n=\lambda _{1}+\lambda _{2}+\cdots +\lambda _{j}} , then the image of a λ {\displaystyle a_{\lambda }} is Im ⁡ ( a λ ) := V ⊗ n a λ ≅ Sym λ 1 ⁡ V ⊗ Sym λ 2 ⁡ V ⊗ ⋯ ⊗ Sym λ j ⁡ V . {\displaystyle \operatorname {Im} (a_{\lambda }):=V^{\otimes n}a_{\lambda }\cong \operatorname {Sym} ^{\lambda _{1}}V\otimes \operatorname {Sym} ^{\lambda _{2}}V\otimes \cdots \otimes \operatorname {Sym} ^{\lambda _{j}}V.} For instance, if n = 4 {\displaystyle n=4} , and λ = ( 2 , 2 ) {\displaystyle \lambda =(2,2)} , with the canonical Young tableau { { 1 , 2 } , { 3 , 4 } } {\displaystyle \{\{1,2\},\{3,4\}\}} . Then the corresponding a λ {\displaystyle a_{\lambda }} is given by a λ = e id + e ( 1 , 2 ) + e ( 3 , 4 ) + e ( 1 , 2 ) ( 3 , 4 ) . {\displaystyle a_{\lambda }=e_{\text{id}}+e_{(1,2)}+e_{(3,4)}+e_{(1,2)(3,4)}.} For any product vector v 1 , 2 , 3 , 4 := v 1 ⊗ v 2 ⊗ v 3 ⊗ v 4 {\displaystyle v_{1,2,3,4}:=v_{1}\otimes v_{2}\otimes v_{3}\otimes v_{4}} of V ⊗ 4 {\displaystyle V^{\otimes 4}} we then have v 1 , 2 , 3 , 4 a λ = v 1 , 2 , 3 , 4 + v 2 , 1 , 3 , 4 + v 1 , 2 , 4 , 3 + v 2 , 1 , 4 , 3 = ( v 1 ⊗ v 2 + v 2 ⊗ v 1 ) ⊗ ( v 3 ⊗ v 4 + v 4 ⊗ v 3 ) . {\displaystyle v_{1,2,3,4}a_{\lambda }=v_{1,2,3,4}+v_{2,1,3,4}+v_{1,2,4,3}+v_{2,1,4,3}=(v_{1}\otimes v_{2}+v_{2}\otimes v_{1})\otimes (v_{3}\otimes v_{4}+v_{4}\otimes v_{3}).} Thus the set of all a λ v 1 , 2 , 3 , 4 {\displaystyle a_{\lambda }v_{1,2,3,4}} clearly spans Sym 2 ⁡ V ⊗ Sym 2 ⁡ V {\displaystyle \operatorname {Sym} ^{2}V\otimes \operatorname {Sym} ^{2}V} and since the v 1 , 2 , 3 , 4 {\displaystyle v_{1,2,3,4}} span V ⊗ 4 {\displaystyle V^{\otimes 4}} we obtain V ⊗ 4 a λ = Sym 2 ⁡ V ⊗ Sym 2 ⁡ V {\displaystyle V^{\otimes 4}a_{\lambda }=\operatorname {Sym} ^{2}V\otimes \operatorname {Sym} ^{2}V} , where we wrote informally V ⊗ 4 a λ ≡ Im ⁡ ( a λ ) {\displaystyle V^{\otimes 4}a_{\lambda }\equiv \operatorname {Im} (a_{\lambda })} . Notice also how this construction can be reduced to the construction for n = 2 {\displaystyle n=2} . Let 1 ∈ End ⁡ ( V ⊗ 2 ) {\displaystyle \mathbb {1} \in \operatorname {End} (V^{\otimes 2})} be the identity operator and S ∈ End ⁡ ( V ⊗ 2 ) {\displaystyle S\in \operatorname {End} (V^{\otimes 2})} the swap operator defined by S ( v ⊗ w ) = w ⊗ v {\displaystyle S(v\otimes w)=w\otimes v} , thus 1 = e id {\displaystyle \mathbb {1} =e_{\text{id}}} and S = e ( 1 , 2 ) {\displaystyle S=e_{(1,2)}} . We have that e id + e ( 1 , 2 ) = 1 + S {\displaystyle e_{\text{id}}+e_{(1,2)}=\mathbb {1} +S} maps into Sym 2 ⁡ V {\displaystyle \operatorname {Sym} ^{2}V} , more precisely 1 2 ( 1 + S ) {\displaystyle {\frac {1}{2}}(\mathbb {1} +S)} is the projector onto Sym 2 ⁡ V {\displaystyle \operatorname {Sym} ^{2}V} . Then 1 4 a λ = 1 4 ( e id + e ( 1 , 2 ) + e ( 3 , 4 ) + e ( 1 , 2 ) ( 3 , 4 ) ) = 1 4 ( 1 ⊗ 1 + S ⊗ 1 + 1 ⊗ S + S ⊗ S ) = 1 2 ( 1 + S ) ⊗ 1 2 ( 1 + S ) {\displaystyle {\frac {1}{4}}a_{\lambda }={\frac {1}{4}}(e_{\text{id}}+e_{(1,2)}+e_{(3,4)}+e_{(1,2)(3,4)})={\frac {1}{4}}(\mathbb {1} \otimes \mathbb {1} +S\otimes \mathbb {1} +\mathbb {1} \otimes S+S\otimes S)={\frac {1}{2}}(\mathbb {1} +S)\otimes {\frac {1}{2}}(\mathbb {1} +S)} which is the projector onto Sym 2 ⁡ V ⊗ Sym 2 ⁡ V {\displaystyle \operatorname {Sym} ^{2}V\otimes \operatorname {Sym} ^{2}V} . The image of b λ {\displaystyle b_{\lambda }} is Im ⁡ ( b λ ) ≅ ⋀ μ 1 V ⊗ ⋀ μ 2 V ⊗ ⋯ ⊗ ⋀ μ k V {\displaystyle \operatorname {Im} (b_{\lambda })\cong \bigwedge ^{\mu _{1}}V\otimes \bigwedge ^{\mu _{2}}V\otimes \cdots \otimes \bigwedge ^{\mu _{k}}V} where μ is the conjugate partition to λ. Here, Sym i ⁡ V {\displaystyle \operatorname {Sym} ^{i}V} and ⋀ j V {\displaystyle \bigwedge ^{j}V} are the symmetric and alternating tensor product spaces. The image C S n c λ {\displaystyle \mathbb {C} S_{n}c_{\lambda }} of c λ = a λ ⋅ b λ {\displaystyle c_{\lambda }=a_{\lambda }\cdot b_{\lambda }} in C S n {\displaystyle \mathbb {C} S_{n}} is an irreducible representation of Sn, called a Specht module. We write Im ⁡ ( c λ ) = V λ {\displaystyle \operatorname {Im} (c_{\lambda })=V_{\lambda }} for the irreducible representation. Some scalar multiple of c λ {\displaystyle c_{\lambda }} is idempotent, that is c λ 2 = α λ c λ {\displaystyle c_{\lambda }^{2}=\alpha _{\lambda }c_{\lambda }} for some rational number α λ ∈ Q . {\displaystyle \alpha _{\lambda }\in \mathbb {Q} .} Specifically, one finds α λ = n ! / dim ⁡ V λ {\displaystyle \alpha _{\lambda }=n!/\dim V_{\lambda }} . In particular, this implies that representations of the symmetric group can be defined over the rational numbers; that is, over the rational group algebra Q S n {\displaystyle \mathbb {Q} S_{n}} . Consider, for example, S3 and the partition (2,1). Then one has c ( 2 , 1 ) = e 123 + e 213 − e 321 − e 312 . {\displaystyle c_{(2,1)}=e_{123}+e_{213}-e_{321}-e_{312}.} If V is a complex vector space, then the images of c λ {\displaystyle c_{\lambda }} on spaces V ⊗ d {\displaystyle V^{\otimes d}} provides essentially all the finite-dimensional irreducible representations of GL(V). == See also == Representation theory of the symmetric group == Notes == == References == William Fulton. Young Tableaux, with Applications to Representation Theory and Geometry. Cambridge University Press, 1997. Lecture 4 of Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103. Bruce E. Sagan. The Symmetric Group. Springer, 2001.
Wikipedia:Young tableau#0	In mathematics, a Young tableau (; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups and to study their properties. Young tableaux were introduced by Alfred Young, a mathematician at Cambridge University, in 1900. They were then applied to the study of the symmetric group by Georg Frobenius in 1903. Their theory was further developed by many mathematicians, including Percy MacMahon, W. V. D. Hodge, G. de B. Robinson, Gian-Carlo Rota, Alain Lascoux, Marcel-Paul Schützenberger and Richard P. Stanley. == Definitions == Note: this article uses the English convention for displaying Young diagrams and tableaux. === Diagrams === A Young diagram (also called a Ferrers diagram, particularly when represented using dots) is a finite collection of boxes, or cells, arranged in left-justified rows, with the row lengths in non-increasing order. Listing the number of boxes in each row gives a partition λ of a non-negative integer n, the total number of boxes of the diagram. The Young diagram is said to be of shape λ, and it carries the same information as that partition. Containment of one Young diagram in another defines a partial ordering on the set of all partitions, which is in fact a lattice structure, known as Young's lattice. Listing the number of boxes of a Young diagram in each column gives another partition, the conjugate or transpose partition of λ; one obtains a Young diagram of that shape by reflecting the original diagram along its main diagonal. There is almost universal agreement that in labeling boxes of Young diagrams by pairs of integers, the first index selects the row of the diagram, and the second index selects the box within the row. Nevertheless, two distinct conventions exist to display these diagrams, and consequently tableaux: the first places each row below the previous one, the second stacks each row on top of the previous one. Since the former convention is mainly used by Anglophones while the latter is often preferred by Francophones, it is customary to refer to these conventions respectively as the English notation and the French notation; for instance, in his book on symmetric functions, Macdonald advises readers preferring the French convention to "read this book upside down in a mirror" (Macdonald 1979, p. 2). This nomenclature probably started out as jocular. The English notation corresponds to the one universally used for matrices, while the French notation is closer to the convention of Cartesian coordinates; however, French notation differs from that convention by placing the vertical coordinate first. The figure on the right shows, using the English notation, the Young diagram corresponding to the partition (5, 4, 1) of the number 10. The conjugate partition, measuring the column lengths, is (3, 2, 2, 2, 1). ==== Arm and leg length ==== In many applications, for example when defining Jack functions, it is convenient to define the arm length aλ(s) of a box s as the number of boxes to the right of s in the diagram λ in English notation. Similarly, the leg length lλ(s) is the number of boxes below s. The hook length of a box s is the number of boxes to the right of s or below s in English notation, including the box s itself; in other words, the hook length is aλ(s) + lλ(s) + 1. === Tableaux === A Young tableau is obtained by filling in the boxes of the Young diagram with symbols taken from some alphabet, which is usually required to be a totally ordered set. Originally that alphabet was a set of indexed variables x1, x2, x3..., but now one usually uses a set of numbers for brevity. In their original application to representations of the symmetric group, Young tableaux have n distinct entries, arbitrarily assigned to boxes of the diagram. A tableau is called standard if the entries in each row and each column are increasing. The number of distinct standard Young tableaux on n entries is given by the involution numbers 1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, ... (sequence A000085 in the OEIS). In other applications, it is natural to allow the same number to appear more than once (or not at all) in a tableau. A tableau is called semistandard, or column strict, if the entries weakly increase along each row and strictly increase down each column. Recording the number of times each number appears in a tableau gives a sequence known as the weight of the tableau. Thus the standard Young tableaux are precisely the semistandard tableaux of weight (1,1,...,1), which requires every integer up to n to occur exactly once. In a standard Young tableau, the integer k {\displaystyle k} is a descent if k + 1 {\displaystyle k+1} appears in a row strictly below k {\displaystyle k} . The sum of the descents is called the major index of the tableau. === Variations === There are several variations of this definition: for example, in a row-strict tableau the entries strictly increase along the rows and weakly increase down the columns. Also, tableaux with decreasing entries have been considered, notably, in the theory of plane partitions. There are also generalizations such as domino tableaux or ribbon tableaux, in which several boxes may be grouped together before assigning entries to them. === Skew tableaux === A skew shape is a pair of partitions (λ, μ) such that the Young diagram of λ contains the Young diagram of μ; it is denoted by λ/μ. If λ = (λ1, λ2, ...) and μ = (μ1, μ2, ...), then the containment of diagrams means that μi ≤ λi for all i. The skew diagram of a skew shape λ/μ is the set-theoretic difference of the Young diagrams of λ and μ: the set of squares that belong to the diagram of λ but not to that of μ. A skew tableau of shape λ/μ is obtained by filling the squares of the corresponding skew diagram; such a tableau is semistandard if entries increase weakly along each row, and increase strictly down each column, and it is standard if moreover all numbers from 1 to the number of squares of the skew diagram occur exactly once. While the map from partitions to their Young diagrams is injective, this is not the case for the map from skew shapes to skew diagrams; therefore the shape of a skew diagram cannot always be determined from the set of filled squares only. Although many properties of skew tableaux only depend on the filled squares, some operations defined on them do require explicit knowledge of λ and μ, so it is important that skew tableaux do record this information: two distinct skew tableaux may differ only in their shape, while they occupy the same set of squares, each filled with the same entries. Young tableaux can be identified with skew tableaux in which μ is the empty partition (0) (the unique partition of 0). Any skew semistandard tableau T of shape λ/μ with positive integer entries gives rise to a sequence of partitions (or Young diagrams), by starting with μ, and taking for the partition i places further in the sequence the one whose diagram is obtained from that of μ by adding all the boxes that contain a value ≤ i in T; this partition eventually becomes equal to λ. Any pair of successive shapes in such a sequence is a skew shape whose diagram contains at most one box in each column; such shapes are called horizontal strips. This sequence of partitions completely determines T, and it is in fact possible to define (skew) semistandard tableaux as such sequences, as is done by Macdonald (Macdonald 1979, p. 4). This definition incorporates the partitions λ and μ in the data comprising the skew tableau. == Overview of applications == Young tableaux have numerous applications in combinatorics, representation theory, and algebraic geometry. Various ways of counting Young tableaux have been explored and lead to the definition of and identities for Schur functions. Many combinatorial algorithms on tableaux are known, including Schützenberger's jeu de taquin and the Robinson–Schensted–Knuth correspondence. Lascoux and Schützenberger studied an associative product on the set of all semistandard Young tableaux, giving it the structure called the plactic monoid (French: le monoïde plaxique). In representation theory, standard Young tableaux of size k describe bases in irreducible representations of the symmetric group on k letters. The standard monomial basis in a finite-dimensional irreducible representation of the general linear group GLn are parametrized by the set of semistandard Young tableaux of a fixed shape over the alphabet {1, 2, ..., n}. This has important consequences for invariant theory, starting from the work of Hodge on the homogeneous coordinate ring of the Grassmannian and further explored by Gian-Carlo Rota with collaborators, de Concini and Procesi, and Eisenbud. The Littlewood–Richardson rule describing (among other things) the decomposition of tensor products of irreducible representations of GLn into irreducible components is formulated in terms of certain skew semistandard tableaux. Applications to algebraic geometry center around Schubert calculus on Grassmannians and flag varieties. Certain important cohomology classes can be represented by Schubert polynomials and described in terms of Young tableaux. == Applications in representation theory == Young diagrams are in one-to-one correspondence with irreducible representations of the symmetric group over the complex numbers. They provide a convenient way of specifying the Young symmetrizers from which the irreducible representations are built. Many facts about a representation can be deduced from the corresponding diagram. Below, we describe two examples: determining the dimension of a representation and restricted representations. In both cases, we will see that some properties of a representation can be determined by using just its diagram. Young tableaux are involved in the use of the symmetric group in quantum chemistry studies of atoms, molecules and solids. Young diagrams also parametrize the irreducible polynomial representations of the general linear group GLn (when they have at most n nonempty rows), or the irreducible representations of the special linear group SLn (when they have at most n − 1 nonempty rows), or the irreducible complex representations of the special unitary group SUn (again when they have at most n − 1 nonempty rows). In these cases semistandard tableaux with entries up to n play a central role, rather than standard tableaux; in particular it is the number of those tableaux that determines the dimension of the representation. === Dimension of a representation === The dimension of the irreducible representation πλ of the symmetric group Sn corresponding to a partition λ of n is equal to the number of different standard Young tableaux that can be obtained from the diagram of the representation. This number can be calculated by the hook length formula. A hook length hook(x) of a box x in Young diagram Y(λ) of shape λ is the number of boxes that are in the same row to the right of it plus those boxes in the same column below it, plus one (for the box itself). By the hook-length formula, the dimension of an irreducible representation is n! divided by the product of the hook lengths of all boxes in the diagram of the representation: dim ⁡ π λ = n ! ∏ x ∈ Y ( λ ) hook ⁡ ( x ) . {\displaystyle \dim \pi _{\lambda }={\frac {n!}{\prod _{x\in Y(\lambda )}\operatorname {hook} (x)}}.} The figure on the right shows hook-lengths for all boxes in the diagram of the partition 10 = 5 + 4 + 1. Thus dim ⁡ π λ = 10 ! 7 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 1 ⋅ 5 ⋅ 3 ⋅ 2 ⋅ 1 ⋅ 1 = 288. {\displaystyle \dim \pi _{\lambda }={\frac {10!}{7\cdot 5\cdot 4\cdot 3\cdot 1\cdot 5\cdot 3\cdot 2\cdot 1\cdot 1}}=288.} Similarly, the dimension of the irreducible representation W(λ) of GLr corresponding to the partition λ of n (with at most r parts) is the number of semistandard Young tableaux of shape λ (containing only the entries from 1 to r), which is given by the hook-length formula: dim ⁡ W ( λ ) = ∏ ( i , j ) ∈ Y ( λ ) r + j − i hook ⁡ ( i , j ) , {\displaystyle \dim W(\lambda )=\prod _{(i,j)\in Y(\lambda )}{\frac {r+j-i}{\operatorname {hook} (i,j)}},} where the index i gives the row and j the column of a box. For instance, for the partition (5,4,1) we get as dimension of the corresponding irreducible representation of GL7 (traversing the boxes by rows): dim ⁡ W ( λ ) = 7 ⋅ 8 ⋅ 9 ⋅ 10 ⋅ 11 ⋅ 6 ⋅ 7 ⋅ 8 ⋅ 9 ⋅ 5 7 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 1 ⋅ 5 ⋅ 3 ⋅ 2 ⋅ 1 ⋅ 1 = 66528. {\displaystyle \dim W(\lambda )={\frac {7\cdot 8\cdot 9\cdot 10\cdot 11\cdot 6\cdot 7\cdot 8\cdot 9\cdot 5}{7\cdot 5\cdot 4\cdot 3\cdot 1\cdot 5\cdot 3\cdot 2\cdot 1\cdot 1}}=66528.} === Restricted representations === A representation of the symmetric group on n elements, Sn is also a representation of the symmetric group on n − 1 elements, Sn−1. However, an irreducible representation of Sn may not be irreducible for Sn−1. Instead, it may be a direct sum of several representations that are irreducible for Sn−1. These representations are then called the factors of the restricted representation (see also induced representation). The question of determining this decomposition of the restricted representation of a given irreducible representation of Sn, corresponding to a partition λ of n, is answered as follows. One forms the set of all Young diagrams that can be obtained from the diagram of shape λ by removing just one box (which must be at the end both of its row and of its column); the restricted representation then decomposes as a direct sum of the irreducible representations of Sn−1 corresponding to those diagrams, each occurring exactly once in the sum. == See also == Robinson–Schensted correspondence Schur–Weyl duality == Notes == == References == William Fulton. Young Tableaux, with Applications to Representation Theory and Geometry. Cambridge University Press, 1997, ISBN 0-521-56724-6. Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103. Lecture 4 Howard Georgi, Lie Algebras in Particle Physics, 2nd Edition - Westview Macdonald, I. G. Symmetric functions and Hall polynomials. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 1979. viii+180 pp. ISBN 0-19-853530-9 MR553598 Laurent Manivel. Symmetric Functions, Schubert Polynomials, and Degeneracy Loci. American Mathematical Society. Greene, Curtis; Nijenhuis, Albert; Wilf, Herbert S. (1979). "A probabilistic proof of a formula for the number of Young tableaux of a given shape". Advances in Mathematics. 31 (1). Amsterdam: Elsevier: 104–109. doi:10.1016/0001-8708(79)90023-9. MR 0521470. Zbl 0398.05008. Jean-Christophe Novelli, Igor Pak, Alexander V. Stoyanovskii, "A direct bijective proof of the Hook-length formula", Discrete Mathematics and Theoretical Computer Science 1 (1997), pp. 53–67. Bruce E. Sagan. The Symmetric Group. Springer, 2001, ISBN 0-387-95067-2 Vinberg, E.B. (2001) [1994], "Young tableau", Encyclopedia of Mathematics, EMS Press Yong, Alexander (February 2007). "What is...a Young Tableau?" (PDF). Notices of the American Mathematical Society. 54 (2): 240–241. Retrieved 2008-01-16. Predrag Cvitanović, Group Theory: Birdtracks, Lie's, and Exceptional Groups. Princeton University Press, 2008. == External links == Eric W. Weisstein. "Ferrers Diagram". From MathWorld—A Wolfram Web Resource. Eric W. Weisstein. "Young Tableau." From MathWorld—A Wolfram Web Resource. Semistandard tableaux entry in the FindStat database Standard tableaux entry in the FindStat database
Wikipedia:Young's lattice#0	In mathematics, Young's lattice is a lattice that is formed by all integer partitions. It is named after Alfred Young, who, in a series of papers On quantitative substitutional analysis, developed the representation theory of the symmetric group. In Young's theory, the objects now called Young diagrams and the partial order on them played a key, even decisive, role. Young's lattice prominently figures in algebraic combinatorics, forming the simplest example of a differential poset in the sense of Stanley (1988). It is also closely connected with the crystal bases for affine Lie algebras. == Definition == Young's lattice is a lattice (and hence also a partially ordered set) Y formed by all integer partitions ordered by inclusion of their Young diagrams (or Ferrers diagrams). == Significance == The traditional application of Young's lattice is to the description of the irreducible representations of symmetric groups Sn for all n, together with their branching properties, in characteristic zero. The equivalence classes of irreducible representations may be parametrized by partitions or Young diagrams, the restriction from Sn+1 to Sn is multiplicity-free, and the representation of Sn with partition p is contained in the representation of Sn+1 with partition q if and only if q covers p in Young's lattice. Iterating this procedure, one arrives at Young's semicanonical basis in the irreducible representation of Sn with partition p, which is indexed by the standard Young tableaux of shape p. == Properties == The poset Y is graded: the minimal element is ∅, the unique partition of zero, and the partitions of n have rank n. This means that given two partitions that are comparable in the lattice, their ranks are ordered in the same sense as the partitions, and there is at least one intermediate partition of each intermediate rank. The poset Y is a lattice. The meet and join of two partitions are given by the intersection and the union of the corresponding Young diagrams. Because it is a lattice in which the meet and join operations are represented by intersections and unions, it is a distributive lattice. If a partition p covers k elements of Young's lattice for some k then it is covered by k + 1 elements. All partitions covered by p can be found by removing one of the "corners" of its Young diagram (boxes at the end both of their row and of their column). All partitions covering p can be found by adding one of the "dual corners" to its Young diagram (boxes outside the diagram that are the first such box both in their row and in their column). There is always a dual corner in the first row, and for each other dual corner there is a corner in the previous row, whence the stated property. If distinct partitions p and q both cover k elements of Y then k is 0 or 1, and p and q are covered by k elements. In plain language: two partitions can have at most one (third) partition covered by both (their respective diagrams then each have one box not belonging to the other), in which case there is also one (fourth) partition covering them both (whose diagram is the union of their diagrams). Saturated chains between ∅ and p are in a natural bijection with the standard Young tableaux of shape p: the diagrams in the chain add the boxes of the diagram of the standard Young tableau in the order of their numbering. More generally, saturated chains between q and p are in a natural bijection with the skew standard tableaux of skew shape p/q. The Möbius function of Young's lattice takes values 0, ±1. It is given by the formula μ ( q , p ) = { ( − 1 ) \| p \| − \| q \| if the skew diagram p / q is a disconnected union of squares (no common edges); 0 otherwise . {\displaystyle \mu (q,p)={\begin{cases}(-1)^{\|p\|-\|q\|}&{\text{if the skew diagram }}p/q{\text{ is a disconnected union of squares}}\\&{\text{(no common edges);}}\\[10pt]0&{\text{otherwise}}.\end{cases}}} == Dihedral symmetry == Conventionally, Young's lattice is depicted in a Hasse diagram with all elements of the same rank shown at the same height above the bottom. Suter (2002) has shown that a different way of depicting some subsets of Young's lattice shows some unexpected symmetries. The partition n + ⋯ + 3 + 2 + 1 {\displaystyle n+\cdots +3+2+1} of the nth triangular number has a Ferrers diagram that looks like a staircase. The largest elements whose Ferrers diagrams are rectangular that lie under the staircase are these: 1 + ⋯ ⋯ ⋯ + 1 ⏟ n terms 2 + ⋯ ⋯ + 2 ⏟ n − 1 terms 3 + ⋯ + 3 ⏟ n − 2 terms ⋮ n ⏟ 1 term {\displaystyle {\begin{array}{c}\underbrace {1+\cdots \cdots \cdots +1} _{n{\text{ terms}}}\\\underbrace {2+\cdots \cdots +2} _{n-1{\text{ terms}}}\\\underbrace {3+\cdots +3} _{n-2{\text{ terms}}}\\\vdots \\\underbrace {{}\quad n\quad {}} _{1{\text{ term}}}\end{array}}} Partitions of this form are the only ones that have only one element immediately below them in Young's lattice. Suter showed that the set of all elements less than or equal to these particular partitions has not only the bilateral symmetry that one expects of Young's lattice, but also rotational symmetry: the rotation group of order n + 1 acts on this poset. Since this set has both bilateral symmetry and rotational symmetry, it must have dihedral symmetry: the (n + 1)st dihedral group acts faithfully on this set. The size of this set is 2n. For example, when n = 4, then the maximal element under the "staircase" that have rectangular Ferrers diagrams are 1 + 1 + 1 + 1 2 + 2 + 2 3 + 3 4 The subset of Young's lattice lying below these partitions has both bilateral symmetry and 5-fold rotational symmetry. Hence the dihedral group D5 acts faithfully on this subset of Young's lattice. == See also == Young–Fibonacci lattice Bratteli diagram == References == Misra, Kailash C.; Miwa, Tetsuji (1990). "Crystal base for the basic representation of U q ( s l ^ ( n ) ) {\displaystyle U_{q}({\widehat {\mathfrak {sl}}}(n))} ". Communications in Mathematical Physics. 134 (1): 79–88. Bibcode:1990CMaPh.134...79M. doi:10.1007/BF02102090. S2CID 120298905. Sagan, Bruce (2000). The Symmetric Group. Berlin: Springer. ISBN 0-387-95067-2. Stanley, Richard P. (1988). "Differential posets". Journal of the American Mathematical Society. 1 (4): 919–961. doi:10.2307/1990995. JSTOR 1990995. Suter, Ruedi (2002). "Young's lattice and dihedral symmetries". European Journal of Combinatorics. 23 (2): 233–238. doi:10.1006/eujc.2001.0541.
Wikipedia:Yousef Saad#0	Yousef Saad (born 1950) in Algiers, Algeria from Boghni, Tizi Ouzou, Kabylia is an I.T. Distinguished Professor of Computer Science in the Department of Computer Science and Engineering at the University of Minnesota. He holds the William Norris Chair for Large-Scale Computing since January 2006. He is known for his contributions to the matrix computations, including the iterative methods for solving large sparse linear algebraic systems, eigenvalue problems, and parallel computing. He is listed as an ISI highly cited researcher in mathematics, is the most cited author in the journal Numerical Linear Algebra with Applications, and is the author of the highly cited book Iterative Methods for Sparse Linear Systems. He is a SIAM fellow (class of 2010) and a fellow of the AAAS (2011). In 2023, he won the John von Neumann Prize. == Education and career == Saad received his B.S. degree in mathematics from the University of Algiers, Algeria in 1970. He then joined University of Grenoble for the doctoral program and obtained a junior doctorate, 'Doctorat de troisieme cycle' in 1974 and a higher doctorate, 'Doctorat d’Etat' in 1983. During the course of his academic career, he has held various positions, including Research Scientist in the Computer Science Department at Yale University (1981–1983), Associate Professor in the University of Tizi-Ouzou in Algeria (1983–1984), Research Scientist at the Computer Science Department at Yale University (1984–1986), and Associate Professor in the Mathematics Department at University of Illinois at Urbana-Champaign (1986–1988). He also worked as a Senior Scientist in the Research Institute for Advanced Computer Science (RIACS) during 1980–1990. Saad joined University of Minnesota as a Professor in the Department of Computer Science in 1990. At Minnesota, he held the position of Head of the Department of Computer Science and Engineering between January 1997 and June 2000. Currently, he is the I. T. Distinguished Professor of Computer Science at University of Minnesota. == Books == Saad is the author of a couple of influential books in linear algebra and matrix computation which include Numerical Methods for Large Eigenvalue Problems, Halstead Press, 1992. Iterative Methods for Sparse Linear Systems, 2nd ed., Society for Industrial and Applied Mathematics, Philadelphia, 2003. He has also co-edited the following article collections: D. L. Boley, D. G. Truhlar, Y. Saad, R. E. Wyatt, and L. E. Collins, Practical Iterative Methods for Large Scale Computations. North Holland, Amsterdam, 1989. D. E. Keyes, Y. Saad, and D. G. Truhlar, Domain-Based Parallelism and Problem Decomposition Methods in Computational Science and Engineering. SIAM, Philadelphia, 1995. A. Ferreira, J. Rolim, Y. Saad, and T. Yang, Parallel Algorithms for Irregularly Structured Problems, Proceedings of Third International Workshop, IRREGULAR’96 Santa Barbara, CA USA, 1996. Lecture Notes in Computer Science, No 1117. Springer Verlag, 1996. M. W. Berry, K. A. Gallivan, E. Gallopoulos, A. Grama, B. Philippe, Y. Saad, and F. Saied, High-Performance Scientific Computing: Algorithms and Applications. Springer, 2012. == References == == External links == Yousef Saad at the Mathematics Genealogy Project
Wikipedia:Yuan Wang (control theorist)#0	Yuan Wang (Chinese: 王沅) is a Chinese-American mathematician specializing in control theory and known for her research on input-to-state stability. She is a professor of mathematics at Florida Atlantic University, chair of the university's Department of Mathematical Sciences, and a moderator for the arXiv mathematical preprint repository in the areas of optimization and control (math.OC) and systems and control (cs.SY). == Education and career == Wang studied mathematics at Shandong University in China, graduating with a bachelor's degree in 1982. She completed a Ph.D. in 1990 at Rutgers University, with the dissertation Algebraic Differential Equations and Nonlinear Control Systems supervised by Eduardo D. Sontag. She joined Florida Atlantic University as an assistant professor of mathematics in 1990. She was promoted to associate professor in 1995 and full professor in 2000. == Recognition == Wang was named as an IEEE Fellow in 2013, "for contributions to stability and control of nonlinear systems". == References == == External links == Yuan Wang publications indexed by Google Scholar
Wikipedia:Yuktibhāṣā#0	Yuktibhāṣā (Malayalam: യുക്തിഭാഷ, lit. 'Rationale'), also known as Gaṇita-yukti-bhāṣā: xxi and Gaṇitanyāyasaṅgraha (English: Compendium of Astronomical Rationale), is a major treatise on mathematics and astronomy, written by the Indian astronomer Jyesthadeva of the Kerala school of mathematics around 1530. The treatise, written in Malayalam, is a consolidation of the discoveries by Madhava of Sangamagrama, Nilakantha Somayaji, Parameshvara, Jyeshtadeva, Achyuta Pisharati, and other astronomer-mathematicians of the Kerala school. It also exists in a Sanskrit version, with unclear author and date, composed as a rough translation of the Malayalam original. The work contains proofs and derivations of the theorems that it presents. Modern historians used to assert, based on the works of Indian mathematics that first became available, that early Indian scholars in astronomy and computation lacked in proofs, but Yuktibhāṣā demonstrates otherwise. Some of its important topics include the infinite series expansions of functions; power series, including of π and π/4; trigonometric series of sine, cosine, and arctangent; Taylor series, including second and third order approximations of sine and cosine; radii, diameters and circumferences. Yuktibhāṣā mainly gives rationale for the results in Nilakantha's Tantra Samgraha. It is considered an early text to give some ideas of calculus like Taylor and infinite series of some trigonometric functions, predating Newton and Leibniz by two centuries. however they did not combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the powerful problem-solving tool we have today. The treatise was largely unnoticed outside India, as it was written in the local language of Malayalam. In modern times, due to wider international cooperation in mathematics, the wider world has taken notice of the work. For example, both Oxford University and the Royal Society of Great Britain have given attribution to pioneering mathematical theorems of Indian origin that predate their Western counterparts. == Contents == Yuktibhāṣā contains most of the developments of the earlier Kerala school, particularly Madhava and Nilakantha. The text is divided into two parts – the former deals with mathematical analysis and the latter with astronomy. Beyond this, the continuous text does not have any further division into subjects or topics, so published editions divide the work into chapters based on editorial judgment.: xxxvii === Mathematics === This subjects treated in the mathematics part of the Yuktibhāṣā can be divided into seven chapters:: xxxvii parikarma: logistics (the eight mathematical operations) daśapraśna: ten problems involving logistics bhinnagaṇita: arithmetic of fractions trairāśika: rule of three kuṭṭakāra: pulverisation (linear indeterminate equations) paridhi-vyāsa: relation between circumference and diameter: infinite series and approximations for the ratio of the circumference and diameter of a circle jyānayana: derivation of Rsines: infinite series and approximations for sines. The first four chapters of the contain elementary mathematics, such as division, the Pythagorean theorem, square roots, etc. Novel ideas are not discussed until the sixth chapter on circumference of a circle. Yuktibhāṣā contains a derivation and proof for the power series of inverse tangent, discovered by Madhava. In the text, Jyesthadeva describes Madhava's series in the following manner: The first term is the product of the given sine and radius of the desired arc divided by the cosine of the arc. The succeeding terms are obtained by a process of iteration when the first term is repeatedly multiplied by the square of the sine and divided by the square of the cosine. All the terms are then divided by the odd numbers 1, 3, 5, .... The arc is obtained by adding and subtracting respectively the terms of odd rank and those of even rank. It is laid down that the sine of the arc or that of its complement whichever is the smaller should be taken here as the given sine. Otherwise the terms obtained by this above iteration will not tend to the vanishing magnitude. In modern mathematical notation, r θ = r sin ⁡ θ cos ⁡ θ − r 3 sin 3 ⁡ θ cos 3 ⁡ θ + r 5 sin 5 ⁡ θ cos 5 ⁡ θ − r 7 sin 7 ⁡ θ cos 7 ⁡ θ + ⋯ {\displaystyle r\theta ={r{\frac {\sin \theta }{\cos \theta }}}-{\frac {r}{3}}{\frac {\sin ^{3}\theta }{\cos ^{3}\theta }}+{\frac {r}{5}}{\frac {\sin ^{5}\theta }{\cos ^{5}\theta }}-{\frac {r}{7}}{\frac {\sin ^{7}\theta }{\cos ^{7}\theta }}+\cdots } or, expressed in terms of tangents, θ = tan ⁡ θ − 1 3 tan 3 ⁡ θ + 1 5 tan 5 ⁡ θ − ⋯ , {\displaystyle \theta =\tan \theta -{\frac {1}{3}}\tan ^{3}\theta +{\frac {1}{5}}\tan ^{5}\theta -\cdots \ ,} which in Europe was conventionally called Gregory's series after James Gregory, who rediscovered it in 1671. The text also contains Madhava's infinite series expansion of π which he obtained from the expansion of the arc-tangent function. π 4 = 1 − 1 3 + 1 5 − 1 7 + ⋯ + ( − 1 ) n 2 n + 1 + ⋯ , {\displaystyle {\frac {\pi }{4}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots +{\frac {(-1)^{n}}{2n+1}}+\cdots \ ,} which in Europe was conventionally called Leibniz's series, after Gottfried Leibniz who rediscovered it in 1673. Using a rational approximation of this series, he gave values of the number π as 3.14159265359, correct to 11 decimals, and as 3.1415926535898, correct to 13 decimals. The text describes two methods for computing the value of π. First, obtain a rapidly converging series by transforming the original infinite series of π. By doing so, the first 21 terms of the infinite series π = 12 ( 1 − 1 3 ⋅ 3 + 1 5 ⋅ 3 2 − 1 7 ⋅ 3 3 + ⋯ ) {\displaystyle \pi ={\sqrt {12}}\left(1-{1 \over 3\cdot 3}+{1 \over 5\cdot 3^{2}}-{1 \over 7\cdot 3^{3}}+\cdots \right)} was used to compute the approximation to 11 decimal places. The other method was to add a remainder term to the original series of π. The remainder term n 2 + 1 4 n 3 + 5 n {\textstyle {\frac {n^{2}+1}{4n^{3}+5n}}} was used in the infinite series expansion of π 4 {\displaystyle {\frac {\pi }{4}}} to improve the approximation of π to 13 decimal places of accuracy when n=76. Apart from these, the Yuktibhāṣā contains many elementary and complex mathematical topics, including, Proofs for the expansion of the sine and cosine functions The sum and difference formulae for sine and cosine Integer solutions of systems of linear equations (solved using a system known as kuttakaram) Geometric derivations of series Early statements of Taylor series for some functions === Astronomy === Chapters eight to seventeen deal with subjects of astronomy: planetary orbits, celestial spheres, ascension, declination, directions and shadows, spherical triangles, ellipses, and parallax correction. The planetary theory described in the book is similar to that later adopted by Danish astronomer Tycho Brahe. The topics covered in the eight chapters are computation of mean and true longitudes of planets, Earth and celestial spheres, fifteen problems relating to ascension, declination, longitude, etc., determination of time, place, direction, etc., from gnomonic shadow, eclipses, Vyatipata (when the sun and moon have the same declination), visibility correction for planets and phases of the moon. Specifically,: xxxviii grahagati: planetary motion, bhagola: sphere of the zodiac, madhyagraha: mean planets, sūryasphuṭa: true sun, grahasphuṭa: true planets bhū-vāyu-bhagola: spheres of the earth, atmosphere, and asterisms, ayanacalana: precession of the equinoxes pañcadaśa-praśna: fifteen problems relating to spherical triangles dig-jñāna: orientation, chāyā-gaṇita: shadow computations, lagna: rising point of the ecliptic, nati-lambana: parallaxes of latitude and longitude grahaṇa: eclipse vyatīpāta visibility correction of planets moon's cusps and phases of the moon == Modern editions == The importance of Yuktibhāṣā was brought to the attention of modern scholarship by C. M. Whish in 1832 through a paper published in the Transactions of the Royal Asiatic Society of Great Britain and Ireland. The mathematics part of the text, along with notes in Malayalam, was first published in 1948 by Rama Varma Thampuran and Akhileswara Aiyar. The first critical edition of the entire Malayalam text, alongside an English translation and detailed explanatory notes, was published in two volumes by Springer in 2008. A third volume, containing a critical edition of the Sanskrit Ganitayuktibhasa, was published by the Indian Institute of Advanced Study, Shimla in 2009. This edition of Yuktibhasa has been divided into two volumes: Volume I deals with mathematics and Volume II treats astronomy. Each volume is divided into three parts: First part is an English translation of the relevant Malayalam part of Yuktibhasa, second part contains detailed explanatory notes on the translation, and in the third part the text in the Malayalam original is reproduced. The English translation is by K.V. Sarma and the explanatory notes are provided by K. Ramasubramanian, M. D. Srinivas, and M. S. Sriram. An open access edition of Yuktibhasa is published by Sayahna Foundation in 2020. == See also == Ganita-yukti-bhasa Madhava's correction term Indian mathematics Kerala School == References == == External links == Biography of Jyesthadeva – School of Mathematics and Statistics University of St Andrews, Scotland
Wikipedia:Yulij Ilyashenko#0	Yulij Sergeevich Ilyashenko (Юлий Сергеевич Ильяшенко, 4 November 1943, Moscow) is a Russian mathematician, specializing in dynamical systems, differential equations, and complex foliations. Ilyashenko received in 1969 from Moscow State University his Russian candidate degree (Ph.D.) under Evgenii Landis and Vladimir Arnold. Ilyashenko was a professor at Moscow State University, an academic at Steklov Institute, and also taught at the Independent University of Moscow. He became a professor at Cornell University. His research deals with, among other things, what he calls the "infinitesimal Hilbert's sixteenth problem", which asks what one can say about the number and location of the boundary cycles of planar polynomial vector fields. The problem is not yet completely solved. Ilyashenko attacked the problem using new techniques of complex analysis (such as functional cochains). He proved that planar polynomial vector fields have only finitely many limit cycles. Jean Écalle independently proved the same result, and an earlier attempted proof by Henri Dulac (in 1923) was shown to be defective by Ilyashenko in the 1970s. He was an Invited Speaker of the ICM in 1978 at Helsinki and in 1990 with talk Finiteness theorems for limit cycles at Kyoto. In 2017 he was elected a Fellow of the American Mathematical Society. == Selected publications == Finiteness theorems for limit cycles, American Mathematical Society Translations, 1991 (also published in Russian Mathematical Surveys, 45, 1990, 143–200) with Weigu Li: Nonlocal Bifurcations, Mathematical Surveys and Monographs, AMS 1998 with S. Yakovenko: Lectures on analytic differential equations, AMS 2007 as editor with Yakovenko: Concerning the Hilbert 16th Problem, AMS 1995 as editor: Nonlinear Stokes Phenomena, Advances in Soviet Mathematics 14, AMS 1993 as editor with Christiane Rousseau: Normal Forms, Bifurcations and Finiteness Problems in Differential Equations, Proceedings of a NATO seminar, Montreal, 2002, Kluwer, 2004 article by Ilyashenko: Selected topics in differential equations with real and complex time, 317–354 with Anton Gorodetski: Certain new robust properties of invariant sets and attractors of dynamical systems, Functional Analysis and Applications, vol. 33, no. 2, 1999, pp. 16–32. doi:10.1007/BF02465190 Ilyashenko, Yu (2000). "Hilbert-type numbers for Abel equations, growth and zeros of holomorphic functions". Nonlinearity. 13 (4): 1337. Bibcode:2000Nonli..13.1337I. doi:10.1088/0951-7715/13/4/319. S2CID 250887845. with G. Buzzard and S. Hruska: Kupka-Smale theorem for polynomial automorphisms of C 2 {\displaystyle C^{2}} and persistence of heteroclinic intersections, Inventiones Mathematicae, vol. 161, 2005, pp. 45–89 doi:10.1007/s00222-004-0418-8 == References == == External links == Media related to Yulij Ilyashenko at Wikimedia Commons mathnet.ru
Wikipedia:Yupana#0	A Yūpa (यूप), or Yūpastambha, was a Vedic sacrificial pillar used in Ancient India. It is one of the most important elements of the Vedic rituals for animal sacrifice. The execution of a victim (generally an animal), who was tied at the yūpa, was meant to bring prosperity to everyone. Most yūpa, and all from the Vedic period, were in wood, and have not survived. The few stone survivals seem to be a later type of memorial using the form of the wooden originals. The Isapur Yupa, the most complete, replicates in stone the rope used to tether the animal. The topmost section is missing; texts describe a "wheel-like headpiece made of perishable material", representing the sun, but the appearance of that is rather unclear from the Gupta period coins that are the best other visual evidence. == Isapur Yūpa == The Isapur Yūpa, now in the Mathura Museum, was found at Isapur (27.5115°N 77.6893°E / 27.5115; 77.6893) in the vicinity of Mathura, and has an inscription in the name of the third century CE Kushan ruler Vāsishka, and mentions the erection of the Yūpa pillar for a sacrificial session. == Yūpa in coinage == During the Gupta Empire period, the Ashvamedha scene of a horse tied to a yūpa sacrificial post appears on the coinage of Samudragupta. On the reverse, the queen is holding a chowrie for the fanning of the horse and a needle-like pointed instrument, with legend "One powerful enough to perform the Ashvamedha sacrifice". == Yūpa inscription in Indonesia == The oldest known Sanskrit inscriptions in the Nusantara are those on seven stone pillars, or Yūpa ("sacrificial posts"), found in the eastern part of Borneo, in the historical area of Kutai, East Kalimantan province. They were written by Brahmins using the early Pallava script, in the Sanskrit language, to commemorate sacrifices held by a generous mighty king called Mulavarman who ruled the Kutai Martadipura Kingdom, the first Hindu kingdom in present Indonesia. Based on palaeographical grounds, they have been dated to the second half of the 4th century CE. They attest to the emergence of an Indianized state in the Indonesian archipelago prior to 400 CE. In addition to Mulavarman, the reigning king, the inscriptions mention the names of his father Aswawarman and his grandfather Kudungga (the founder of the Kutai Martadipura Kingdom). Aswawarman is the first of the line to bear a Sanskrit name in the Yupa which indicates that he was probably the first to adhere to Hinduism. === Text === The four Yupa inscriptions founded are classified as "Muarakaman"s and has been translated by language experts as follows: === Translation === Translation according to the Indonesia University of Education: The Yupas are now kept in the National Museum of Indonesia in Jakarta. == References ==
Wikipedia:Yuri A. Kuznetsov#0	Yuri A. Kuznetsov is a Russian-American mathematician currently the M. D. Anderson Chair Professor of Mathematics at University of Houston and Editor-in-Chief of Journal of Numerical Mathematics. == References ==
Wikipedia:Yuri Babayev#0	Yuri Nikolayevich Babayev (Russian: Юрий Николаевич Бабаев; 21 May 1928 – 6 October 1986), k.N, was a Soviet physicist who spent a long career in the former Soviet program of nuclear weapons, and known as one of the principles who designed the Tsar Bomba, the largest-ever nuclear weapon. == Early life == He was born in Moscow. His family was evacuated during the battles of the Eastern Front (World War II), first to Chelyabinsk then to Leninabad (now Khujand). He did well at school despite the hardships. He graduated with honours from the faculty of Physics of Moscow State University in 1950. He entered the Soviet weapons programme as one of its youngest scientists, a senior laboratory assistant in Andrei Sakharov's group at Arzamas-16 (also known as KB-11), now known as All-Russian Scientific Research Institute of Experimental Physics (VNIIEF), in Sarov, Nizhny Novgorod region. In 1953, he received the Stalin Prize for his part in the work to develop the Soviet union's first thermonuclear weapon, the RDS-6 which was detonated in 1953; this was the first of several state awards for his work advancing nuclear weapons. With fellow physicist Yuri Trutnev, he proposed a new design in 1955 for a two-stage thermonuclear device with much-improved features, followed by theoretical development and finally completion in 1958. He frequently took part in testing weapons he had helped to develop. He received his Ph.D. in nuclear engineering in 1960. In 1962, he became a doctor of technical sciences and senior research worker. In 1964 he was promoted to head of his department and deputy head of VNIIEF. His work also encompassed development of low-radiation-yield nuclear charges for civilian purposes – for example to make reservoirs – and nuclear-pumped lasers. He was also interested in the effects of radiation on humans and the environment. Many scientists were trained under his direction as Chair of the Academic Council at KB-11. He was elected to the Soviet Academy of Sciences in November 1968. He was buried at Kuntsevo Cemetery in Moscow. == Awards == 1953: Stalin Prize. 1956 & 1962: Order of the Red Banner of Labour. 1959: Lenin Prize. 1962: Hero of Socialist Labour & Order of Lenin. 1975: Medal "For Labour Valour". 2000: (posthumously) State Prize of the Russian Federation. == References ==
Wikipedia:Yuri Manin#0	Yuri Ivanovich Manin (Russian: Ю́рий Ива́нович Ма́нин; 16 February 1937 – 7 January 2023) was a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics. == Life and career == Manin was born on 16 February 1937 in Simferopol, Crimean ASSR, Soviet Union. He received a doctorate in 1960 at the Steklov Mathematics Institute as a student of Igor Shafarevich. He became a professor at the Max-Planck-Institut für Mathematik in Bonn, where he was director from 1992 to 2005 and then director emeritus. He was also a Trustee Chair Professor at Northwestern University from 2002 to 2011. He had over the years more than 40 doctoral students, including Vladimir Berkovich, Mariusz Wodzicki, Alexander Beilinson, Ivan Cherednik, Alexei Skorobogatov, Vladimir Drinfeld, Mikhail Kapranov, Vyacheslav Shokurov, Ralph Kaufmann, Victor Kolyvagin, Alexander A. Voronov, and Hà Huy Khoái. Manin died on 7 January 2023. == Research == Manin's early work included papers on the arithmetic and formal groups of abelian varieties, the Mordell conjecture in the function field case, and algebraic differential equations. The Gauss–Manin connection is a basic ingredient of the study of cohomology in families of algebraic varieties. He developed the Manin obstruction, indicating the role of the Brauer group in accounting for obstructions to the Hasse principle via Grothendieck's theory of global Azumaya algebras, setting off a generation of further work. Manin pioneered the field of arithmetic topology (along with John Tate, David Mumford, Michael Artin, and Barry Mazur). He also formulated the Manin conjecture, which predicts the asymptotic behaviour of the number of rational points of bounded height on algebraic varieties. In mathematical physics, Manin wrote on Yang–Mills theory, quantum information, and mirror symmetry. He was one of the first to propose the idea of a quantum computer in 1980 with his book Computable and Uncomputable. He wrote a book on cubic surfaces and cubic forms, showing how to apply both classical and contemporary methods of algebraic geometry, as well as nonassociative algebra. == Awards == He was awarded the Brouwer Medal in 1987, the first Nemmers Prize in Mathematics in 1994, the Schock Prize of the Royal Swedish Academy of Sciences in 1999, the Cantor Medal of the German Mathematical Society in 2002, the King Faisal International Prize in 2002, and the Bolyai Prize of the Hungarian Academy of Sciences in 2010. In 1990, he became a foreign member of the Royal Netherlands Academy of Arts and Sciences. He was a member of eight other academies of science and was also an honorary member of the London Mathematical Society. == Selected works == Mathematics as metaphor – selected essays. American Mathematical Society. 2009. "Rational points of algebraic curves over function fields". AMS translations 1966 (Mordell conjecture for function fields). Manin, Yu I. (1965). "Algebraic topology of algebraic varieties". Russian Mathematical Surveys. 20 (6): 183–192. Bibcode:1965RuMaS..20..183M. doi:10.1070/RM1965v020n06ABEH001192. S2CID 250895773. Frobenius manifolds, quantum cohomology, and moduli spaces. American Mathematical Society. 1999. Quantum groups and non commutative geometry. Montreal: Centre de Recherches Mathématiques. 1988. Topics in non-commutative geometry. Princeton University Press. 1991. ISBN 9780691635781. Gauge field theory and complex geometry. Grundlehren der mathematischen Wissenschaften. Springer. 1988. Cubic forms - algebra, geometry, arithmetics. North Holland. 1986. A course in mathematical logic. Springer. 1977., second expanded edition with new chapters by the author and Boris Zilber, Springer 2010. Computable and Uncomputable. Moscow. 1980.{{cite book}}: CS1 maint: location missing publisher (link) Mathematics and physics. Birkhäuser. 1981. Manin, Yu. I. (1984). "New dimensions in geometry". Arbeitstagung. Lectures Notes in Mathematics. Vol. 1111. Bonn: Springer. pp. 59–101. doi:10.1007/BFb0084585. ISBN 978-3-540-15195-1. Manin, Yuri; Kostrikin, Alexei I. (1989). Linear algebra and geometry. London, England: Gordon and Breach. doi:10.1201/9781466593480. ISBN 9780429073816. S2CID 124713118. Manin, Yuri; Gelfand, Sergei (1994). Homological algebra. Encyclopedia of Mathematical Sciences. Springer. Manin, Yuri; Gelfand, Sergei Gelfand (1996). Methods of Homological algebra. Springer Monographs in Mathematics. Springer. doi:10.1007/978-3-662-12492-5. ISBN 978-3-642-07813-2. Manin, Yuri; Kobzarev, Igor (1989). Elementary Particles: mathematics, physics and philosophy. Dordrecht: Kluwer. Manin, Yuri; Panchishkin, Alexei A. (1995). Introduction to Number theory. Springer. Manin, Yuri I. (2001). "Moduli, Motives, Mirrors". European Congress of Mathematics. Progress in Mathematics. Barcelona. pp. 53–73. doi:10.1007/978-3-0348-8268-2_4. hdl:21.11116/0000-0004-357E-4. ISBN 978-3-0348-9497-5.{{cite book}}: CS1 maint: location missing publisher (link) Classical computing, quantum computing and Shor´s factoring algorithm (PDF). Bourbaki Seminar. 1999.{{cite book}}: CS1 maint: location missing publisher (link) Rademacher, Hans; Toeplitz, Otto (2002). Von Zahlen und Figuren [From Numbers and Figures] (in German). doi:10.1007/978-3-662-36239-6. ISBN 978-3-662-35411-7. Manin, Yuri; Marcolli, Matilde (2002). "Holography principle and arithmetic of algebraic curves". Advances in Theoretical and Mathematical Physics. 5 (3). Max-Planck-Institut für Mathematik, Bonn: International Press: 617–650. arXiv:hep-th/0201036. doi:10.4310/ATMP.2001.v5.n3.a6. S2CID 25731842. Manin, Yu. I. (December 1991). "Three-dimensional hyperbolic geometry as ∞-adic Arakelov geometry". Inventiones Mathematicae. 104 (1): 223–243. Bibcode:1991InMat.104..223M. doi:10.1007/BF01245074. S2CID 121350567. Mathematik, Kunst und Zivilisation [Mathematics, Art and Civilisation]. Die weltweit besten mathematischen Artikel im 21. Jahrhundert. Vol. 3. e-enterprise. 2014. ISBN 978-3-945059-15-9. == See also == Arithmetic topology Noncommutative residue == References == == Further reading == Némethi, A. (April 2011). "Yuri Ivanovich Manin" (PDF). Acta Mathematica Hungarica. 133 (1–2): 1–13. doi:10.1007/s10474-011-0151-x. Jean-Paul Pier (1 January 2000). Development of Mathematics 1950–2000. Springer Science & Business Media. p. 1116. ISBN 978-3-7643-6280-5. == External links == Manin's page at Max-Planck-Institut für Mathematik website Good Proofs are Proofs that Make us Wiser, interview by Martin Aigner and Vasco A. Schmidt Biography Interviewed by David Eisenbud for Simons Foundation "Science Lives" Fedor Bogomolov; Yuri Tschinkel, eds. (December 2023). "Memorial Article for Yuri Manin" (PDF). Notices of the American Mathematical Society. 70 (11): 1831–1849. doi:10.1090/noti2814.
Wikipedia:Yuri Nesterenko (mathematician)#0	Yuri Valentinovich Nesterenko (Russian: Ю́рий Валенти́нович Нестере́нко; born 5 December 1946 in Kharkov, USSR, now Ukraine) is a Soviet and Russian mathematician who has written papers in algebraic independence theory and transcendental number theory. In 1997, he was awarded the Ostrowski Prize for his proof that the numbers π and eπ are algebraically independent. In fact, he proved the stronger result: the numbers π, eπ, and Γ(1/4) are algebraically independent over Q. the numbers π, e π 3 {\displaystyle e^{\pi {\sqrt {3}}}} , and Γ(1/3) are algebraically independent over Q. for all positive integers n, the numbers π, e π n {\displaystyle e^{\pi {\sqrt {n}}}} are algebraically independent over Q. He is a professor at Moscow State University, where he completed the mechanical-mathematical program in 1969, then the doctorate program (Soviet habilitation) in 1973, and became a professor of the Number Theory Department in 1992. He studied under Andrei Borisovich Shidlovskii. Nesterenko's students have included Wadim Zudilin. == Publications == Nesterenko, Y. (1996). "Modular Functions and Transcendence Problems". Comptes Rendus de l'Académie des Sciences, Série I. 322 (10): 909–914. == References == == External links == Ostrowski Foundation (August 1998). "Nesterenko and Pisier Share Ostrowski Prize" (PDF). Notices of the AMS. A picture Web page at Moscow State University (in Russian); switch to Windows-1251 encoding if your browser does not render correctly. Yuri Nesterenko at the Mathematics Genealogy Project
Wikipedia:Yuri Ofman#0	Yuri Petrovich Ofman (Russian: Ю́рий Петро́вич Офман, born 1939) is a Russian mathematician who works in computational complexity theory. He obtained his Doctorate from Moscow State University, where he was advised by Andrey Kolmogorov. He did important early work on parallel algorithms for prefix sums and their application in the design of Boolean circuits for addition. == Publications == "О приближенной реализации непрерывных функций на автоматах" [On the approximate realization of continuous functions on automata]. Doklady Akademii Nauk SSSR. 152 (4): 823–826. 1963. "Об алгоритмической сложности дискретных функций" [On the algorithmic complexity of discrete functions]. Doklady Akademii Nauk SSSR. 145 (1): 48–51. 1962. Translated in Soviet Physics Doklady. 7: 589.{{cite journal}}: CS1 maint: untitled periodical (link) Anatolii A. Karatsuba and Yu. P. Ofman (1962), "Умножение многозначных чисел на автоматах" ("Multiplication of Many-Digital Numbers by Automatic Computers"), Doklady Akademii Nauk SSSR, vol. 146, pages 293–294. (Published by A. N. Kolmogorov, with two separate results by the two authors.) Yu. P. Ofman (1965), "A universal automaton". Transactions of the Moscow Mathemathematical Society, volume 14, pages 200–215. == References ==

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