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Wikipedia:Jeb Willenbring#0
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Jeb F. Willenbring is a Full Professor and Associate Chair for Graduate Mathematics at the University of Wisconsin-Milwaukee. Most of his research falls within the categories of Representation Theory, Discrete Mathematics, and Mathematical Physics. His current research consists of several collaborations concerning algebraic combinatorics and various aspects of Representation Theory. Willenbring has published multiple papers concerning the nature of Littlewood-Richardson coefficients, a particular focus of his, in collaboration with Professors Pamela Harris, Mark Colorusso, and William Erickson. == Education/Academic Career == Willenbring attended North Dakota State University during his undergraduate years and graduated with a bachelor's degree in 1995. After graduating, he went on to pursue a PhD degree in mathematics from the University of California, San Diego. He worked with his doctoral advisor, Nolan Russell Wallach, to produce his thesis Stability Properties for q-Mulitplicities and Branching Formulas for Representations of the Classical Groups. After receiving his PhD in mathematics in 2000, he accepted an Assistant Professor position at the University of Wisconsin-Milwaukee in 2003. He was promoted to an Associate Professor position in 2008 and to a Full Professor position in 2013. During his time at the University of Wisconsin-Milwaukee, Willenbring has taught numerous classes including Advanced Calculus, Number Theory, and Graduate Algebra and supervised 9 Graduate Student theses to the point of completion. As a legally blind professor, he is a particularly zealous advocate for students with disabilities. As a member of his university's ADA advisory committee, Willenbring successfully advocated to bring an app for visually impaired student to campus, making UWM the first university in its state to offer a free app for the blind and visually impaired. == Publications == Notable publications by Jeb Willenbring include: == Awards == Willenbring has received numerous awards and grants. After graduating from his PhD Program, he taught at Yale University from 2000–2003 and was granted NSF VIGRE funding. In 2013, Willenbring was named as a fellow of the American Mathematics Society as part of its inaugural class of fellows. Recognizing his work towards increasing campus accessibility, Willenbring also received a UWM Accessibility Resource Center Excellence Award in 2016. == References ==
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Wikipedia:Jeffrey B. Remmel#0
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Jeffrey Brian Remmel (October 12, 1948 – September 29, 2017) was an American mathematician employed by the University of California, San Diego. At the time of his death he held a distinguished professorship—his title was Distinguished Professor of Mathematics; he also held a position as a professor of computer science. == Personal life == Remmel was born on October 12, 1948, in Clintonville, Wisconsin. He died aged 68 at Scripps Memorial Hospital in La Jolla, California on September 29, 2017, with a reported cause of death being a heart attack. == Education == Remmel received a Bachelor of Arts degree in mathematics from Swarthmore College in 1970. Later, he received two degrees from Cornell University—a Master of Science in mathematics and a Doctor of Philosophy, also in math (1972 and 1974, respectively). At Cornell, he was advised by Anil Nerode, and his dissertation was entitled Co-recursively Enumerable Structures. == Career == After obtaining his Ph.D., though before he had published a single paper, Remmel joined the faculty of the University of California, San Diego as an assistant professor, where he worked for his entire career. Remmel was noted for his successful publication record in two separate fields—logic, in which he published in mathematical logic; and combinatorics, where he published papers on algebraic combinatorics. He published over 20 papers in logic with Victor W. Marek, and Remmel's more prominent career in combinatorics included over 20 co-authored papers with Sergey Kitaev. A double issue of the Journal of Combinatorics was published in his memory. Remmel's work is highly cited in the fields of vector spaces, including computably enumerable sets and vector spaces. == References == == Notes ==
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Wikipedia:Jehan Adam#0
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Jehan Adam was a French 15th century mathematician. He was secretary to Nicholle Tilhart, who was notary, secretary and auditor of accounts to King Louis XI of France. He published a manuscript in 1475 containing the first use of the terms bymillion and trimillion, which gave rise to the modern terms billion and trillion. His usage referred to the long scale values of 1012 and 1018, respectively. These terms have subsequently been revalued in English to the short scale values 109 and 1012, respectively, although the original values remain in long scale countries. ... item noctes que le premier greton dembas vault ung, le second vault [...here some words seem to be omitted...] cent, le quart vult mille, le Ve vault dix M, le VIe vault cent M, le VIIe vault Milion, Le VIIIe vault dix Million, Le IXe vault cent Millions, Le Xe vault Mil Millions, Le XIe vault dix mil Millions, Le XIIe vault Cent mil Millions, Le XIIIe vault bymillion, Le XIIIIe vault dix bymillions, Le XVe vault cent mil [sic] bymillions, Le XVIe vault mil bymillions, Le XVIIe vault dix Mil bymillions, Le XVIIIe vault cent mil bymillions, Le XIXe vault trimillion, Le XXe vault dix trimillions ... Translation ... Also note that the first counter from the bottom stands for one, the 2nd stands for [...] one hundred, the 4th stands for one thousand, the 5th stands for ten thousand, the 6th stands for one hundred thousand, the 7th stands for a million, the 8th stands for ten millions, the 9th stands for one hundred millions, the 10th stands for one thousand millions, the 11th stands for ten thousand millions, the 12th stands for one hundred thousand million, the 13th stands for a billion, the 14th stands for ten billions, the 15th stands for one hundred [the "mil" in the original is an obvious error] billions, the 16th stands for one thousand billions, the 17th for ten thousand billions, the 18th stands for hundred thousand billions, the 19th stands for a trimillion, the 20th stands for ten trimillions ... == References ==
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Wikipedia:Jennifer Key#0
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Jennifer Denise Key (née Hicks) is a retired South African mathematician whose research has concerned the interconnections between group theory, finite geometry, combinatorial designs, and coding theory. She is a professor emeritus at Clemson University in the US, and an honorary professor at Aberystwyth University in the UK, and the University of KwaZulu-Natal and University of the Western Cape in South Africa. == Education and career == Key graduated with honours from the University of the Witwatersrand in 1963, and went to the University of London for graduate study in mathematics, earning a master's degree in 1967 and completing her Ph.D. in 1969. Her dissertation, Some Topics in Finite Permutation Groups, was supervised by Ascher Wagner. She worked as an academic in England, at the University of Surrey, University of Reading, University of Manchester, and University of Birmingham, before moving to the US in 1990 to take a faculty position at Clemson University. She retired as professor emeritus in 2007. == Book == Key is the author, with Edward F. Assmus Jr., of the book Designs and Their Codes (Cambridge University Press, 1992). == References == == External links == Home page
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Wikipedia:Jennifer Seberry#0
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Jennifer Roma Seberry (also published as Jennifer Seberry Wallis; born 13 February 1944 in Sydney) is an Australian cryptographer, mathematician, and computer scientist, currently a professor at the University of Wollongong, Australia. She was formerly the head of the Department of Computer Science and director of the Centre for Computer Security Research at the university. == Education and career == Seberry attended Parramatta High School and got her BSc at University of New South Wales, 1966; MSc at La Trobe University, 1969; PhD at La Trobe University, 1971 (Computational Mathematics); B.Ec. with two years completed at University of Sydney. Her doctoral advisor was Bertram Mond. Seberry was the first person to teach cryptology at an Australian University (University of Sydney). She was also the first woman Professor of Computer Science in Australia. She was the first woman Reader in Combinatorial Mathematics in Australia. As of 2020 she had supervised 30 doctorates and had 71 academic descendants. Her notable students have included Peter Eades, Mirka Miller, and Deborah Street. == Service == Seberry was a founding member of the University of Sydney's Research Foundation for Information Technology Information Security Group in 1987. The group grew into the Australian Information Security Association, an Australian representative industry body with over 1000 paid members and branches in most capitals. Seberry was one of the founders of the Asiacrypt international conference in 1990 (then called Auscrypt). == Research == Seberry has contributed to the knowledge and use of Hadamard matrices and bent functions for network security. She has published numerous papers on mathematics, cryptography, and computer and network security. She led the team that produced the LOKI and LOKI97 block ciphers and the HAVAL cryptographic hash functions. Seberry is also a co-author of the Py (spelled RU) stream cipher, which was a candidate for the eSTREAM stream cipher project. == References == == External links == Jennifer Seberry's page at University of Wollongong
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Wikipedia:Jennifer Switkes#0
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Jennifer Switkes is a Canadian-American applied mathematician interested in mathematical modeling and operations research, and also known for her volunteer work teaching mathematics in prisons. She is an associate professor of mathematics at California State Polytechnic University, Pomona (Cal Poly Pomona), where she is associate chair of the mathematics department. == Early life and education == Switkes was born in Canada but moved as a child to Northern California. She is a 1994 graduate of Harvey Mudd College, where she completed a double major in mathematics and physics as well as earning credits towards a teaching credential. However, her experience as a student teacher at a middle school convinced her that she was not fully prepared to continue as a teacher, and she returned to graduate school instead. Her doctoral research at Claremont Graduate University concerned mathematical biology, and more specifically mosaic coevolution; her 2000 dissertation, The Geographic Mosaic Theory in Relation to Coevolutionary Interactions, was jointly supervised by Michael E. Moody and John Angus. == Career and volunteer work == Switkes was an instructor at Citrus College and the University of Redlands before becoming a mathematics professor at Cal Poly Pomona in 2001. There, she is known for her project-based education of students, centered around real-world applications of mathematical modeling. Switkes volunteers as an associate pastor at the Orange Coast Free Methodist Church in Costa Mesa, California, and as a mathematics teacher with the Prison Education Project. She has taught mathematics to prison inmates both at the California Rehabilitation Center in Norco, California and in Uganda, where she has traveled repeatedly on church missions, on a 2013 sabbatical visit to Makerere University and on a shorter 2015 visit to teach at the Luzira Maximum Security Prison. As inspiration for her volunteer work she cites a book by Bob Moses, Radical Equations—Civil Rights from Mississippi to the Algebra Project, on the importance of mathematical literacy in escaping underprivileged circumstances. == Recognition == Switkes was one of the winners of the 2015 Inspiring Women in STEM Award of Insight Into Diversity Magazine. In 2019, Switkes won one of the Deborah and Franklin Haimo Awards for Distinguished College or University Teaching of Mathematics, the highest teaching award of the Mathematical Association of America, "for bringing her educational core values of excellence, honor, integrity, love, and purpose to all students, and specifically to traditionally underserved students". The award recognized both her prison volunteer work and her mentorship of undergraduate and master's students at Cal Poly Pomona. She was also honored as an outstanding alumna of Harvey Mudd College in 2019. == References ==
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Wikipedia:Jennifer Taback#0
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Jennifer Taback is an American mathematician whose research focuses on geometric group theory and combinatorial group theory. She is the Isaac Henry Wing Professor of Mathematics and Chair of the Mathematics Department at Bowdoin College in Maine. == Education and career == After earning a bachelor's degree in mathematics at Yale University in 1993, Taback went to the University of Chicago for graduate study in mathematics, earning a master's degree in 1994 and completing her Ph.D. in 1998. Her 1998 doctoral dissertation, Quasi-Isometric Rigity for P S L 2 ( Z [ 1 p ] ) {\displaystyle PSL_{2}{\bigl (}\mathbb {Z} [{\tfrac {1}{p}}]{\bigr )}} , was supervised by Benson Farb. After a postdoctoral stay at the University of California, Berkeley as Charles B. Morrey assistant professor, she became an assistant professor of mathematics at the University at Albany in 1999, moving to her present position at Bowdoin in 2004. She was tenured as an associate professor in 2007, and promoted to full professor in 2012. She was given the Isaac Henry Wing Professorship in 2021; the professorship was endowed in 1906 by a former Bowdoin student. == References == == External links == Home page
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Wikipedia:Jenny McNulty#0
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Jennifer McNulty is an American mathematician and academic administrator, the dean of the College of Arts and Sciences at the University of Alaska Anchorage. Her research is in combinatorics, specializing in matroid theory and graph theory. == Education and career == McNulty majored in both chemistry and mathematics at Providence College, a Catholic university in Rhode Island, graduating in 1985. After earning a master's degree in mathematics at Stony Brook University, she completed her Ph.D. in 1993 at the University of North Carolina at Chapel Hill. Her dissertation, Affine Hyperplane Arrangements and Oriented Matroids, was supervised by Thomas H. Brylawski. She joined the University of Montana as an assistant professor in 1993, earned tenure there in 1997, and was promoted to full professor in 2004. She was chair of the Pacific Northwest Section of the Mathematical Association of America from 1998 to 2002. She became associate dean of the School of Humanities and Sciences at the University of Montana in 2010, and acting dean in 2018. Also in 2018, she traveled to the University of Gondar in Ethiopia as a Fulbright Scholar, working with mathematicians there on curriculum design, mentoring a new group of female faculty with little prior teaching experience that the university had recently hired, and building connections between the University of Gondar and University of Montana. In 2020 she moved from the University of Montana to an affiliated university, the University of Montana Western, as interim provost and vice chancellor of academic and student affairs. She moved to the University of Alaska Anchorage as dean of the College of Arts and Sciences in 2021. == Book == With Gary Gordon of Lafayette College, McNulty is the author of the book Matroids: a Geometric Introduction (Cambridge University Press, 2012). == References ==
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Wikipedia:Jens Erik Fenstad#0
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Jens Erik Fenstad (15 April 1935 – 13 April 2020) was a Norwegian mathematician. Fenstad graduated as mag.scient. from the University of Oslo in 1959, and worked as a research fellow there and at UC Berkeley. He was a professor at the University of Oslo from 1968 to 2003, except for the years 1989 to 1993, when he was vice rector (prorektor). On 29 May 1998 Fensted received an honorary doctorate from the Faculty of Science and Technology at Uppsala University, Sweden. He served in the Executive Committee of the Division for Logic, Methodology and Philosophy of Science of the International Union of History and Philosophy of Science as Treasurer from 1975 to 1983 and as President from 1991 to 1995. He chaired the Norwegian Mathematical Society and the World Commission on Ethics (COMEST) of the UNESCO, and co-founded the Abel Prize. He has also edited the journal Nordisk Matematisk Tidsskrift. He was a member of the Norwegian Academy of Science and Letters. Fenstad resided in Østerås. He died from COVID-19 on 13 April 2020, two days before his 85th birthday. == See also == Influence of non-standard analysis == References ==
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Wikipedia:Jens Kraft#0
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Jens Kraft (1720–1765) was a Dano-Norwegian mathematician and philosopher. He was born in Frederikshald in Norway, but at age 5 he became an orphan and was subsequently raised by his uncle in Thy in Jutland. As a philosopher he introduced the study of ontology to Scandinavian academic circles. He was influenced by Christian Wolff whilst still a student at Copenhagen, attending a lecture by Wolff while visiting Halle. He was appointed professor of philosophy at the Sorø Academy, where he responded to Baumgarten's Metaphsica with his own text metaphysik. Both divided their work into cosmologie, ontologie, psykologie and naturlig theologie. Kraft made a distinction between time and eternity writing that "the finite can never obtain eternity, but it can obtain an infinite time, (Aevum) or a time with beginning but without end." The infinite by contrast has permanence (sempiternité)". == References ==
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Wikipedia:Jensen's inequality#0
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In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier proof of the same inequality for doubly-differentiable functions by Otto Hölder in 1889. Given its generality, the inequality appears in many forms depending on the context, some of which are presented below. In its simplest form the inequality states that the convex transformation of a mean is less than or equal to the mean applied after convex transformation (or equivalently, the opposite inequality for concave transformations). Jensen's inequality generalizes the statement that the secant line of a convex function lies above the graph of the function, which is Jensen's inequality for two points: the secant line consists of weighted means of the convex function (for t ∈ [0,1]), t f ( x 1 ) + ( 1 − t ) f ( x 2 ) , {\displaystyle tf(x_{1})+(1-t)f(x_{2}),} while the graph of the function is the convex function of the weighted means, f ( t x 1 + ( 1 − t ) x 2 ) . {\displaystyle f(tx_{1}+(1-t)x_{2}).} Thus, Jensen's inequality in this case is f ( t x 1 + ( 1 − t ) x 2 ) ≤ t f ( x 1 ) + ( 1 − t ) f ( x 2 ) . {\displaystyle f(tx_{1}+(1-t)x_{2})\leq tf(x_{1})+(1-t)f(x_{2}).} In the context of probability theory, it is generally stated in the following form: if X is a random variable and φ is a convex function, then φ ( E [ X ] ) ≤ E [ φ ( X ) ] . {\displaystyle \varphi (\operatorname {E} [X])\leq \operatorname {E} \left[\varphi (X)\right].} The difference between the two sides of the inequality, E [ φ ( X ) ] − φ ( E [ X ] ) {\displaystyle \operatorname {E} \left[\varphi (X)\right]-\varphi \left(\operatorname {E} [X]\right)} , is called the Jensen gap. == Statements == The classical form of Jensen's inequality involves several numbers and weights. The inequality can be stated quite generally using either the language of measure theory or (equivalently) probability. In the probabilistic setting, the inequality can be further generalized to its full strength. === Finite form === For a real convex function φ {\displaystyle \varphi } , numbers x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} in its domain, and positive weights a i {\displaystyle a_{i}} , Jensen's inequality can be stated as: and the inequality is reversed if φ {\displaystyle \varphi } is concave, which is Equality holds if and only if x 1 = x 2 = ⋯ = x n {\displaystyle x_{1}=x_{2}=\cdots =x_{n}} or φ {\displaystyle \varphi } is linear on a domain containing x 1 , x 2 , ⋯ , x n {\displaystyle x_{1},x_{2},\cdots ,x_{n}} . As a particular case, if the weights a i {\displaystyle a_{i}} are all equal, then (1) and (2) become For instance, the function log(x) is concave, so substituting φ ( x ) = log ( x ) {\displaystyle \varphi (x)=\log(x)} in the previous formula (4) establishes the (logarithm of the) familiar arithmetic-mean/geometric-mean inequality: log ( ∑ i = 1 n x i n ) ≥ ∑ i = 1 n log ( x i ) n {\displaystyle \log \!\left({\frac {\sum _{i=1}^{n}x_{i}}{n}}\right)\geq {\frac {\sum _{i=1}^{n}\log \!\left(x_{i}\right)}{n}}} exp ( log ( ∑ i = 1 n x i n ) ) ≥ exp ( ∑ i = 1 n log ( x i ) n ) {\displaystyle \exp \!\left(\log \!\left({\frac {\sum _{i=1}^{n}x_{i}}{n}}\right)\right)\geq \exp \!\left({\frac {\sum _{i=1}^{n}\log \!\left(x_{i}\right)}{n}}\right)} x 1 + x 2 + ⋯ + x n n ≥ x 1 ⋅ x 2 ⋯ x n n {\displaystyle {\frac {x_{1}+x_{2}+\cdots +x_{n}}{n}}\geq {\sqrt[{n}]{x_{1}\cdot x_{2}\cdots x_{n}}}} A common application has x as a function of another variable (or set of variables) t, that is, x i = g ( t i ) {\displaystyle x_{i}=g(t_{i})} . All of this carries directly over to the general continuous case: the weights ai are replaced by a non-negative integrable function f (x), such as a probability distribution, and the summations are replaced by integrals. === Measure-theoretic form === Let ( Ω , A , μ ) {\displaystyle (\Omega ,A,\mu )} be a probability space. Let f : Ω → R {\displaystyle f:\Omega \to \mathbb {R} } be a μ {\displaystyle \mu } -measurable function and φ : R → R {\displaystyle \varphi :\mathbb {R} \to \mathbb {R} } be convex. Then: φ ( ∫ Ω f d μ ) ≤ ∫ Ω φ ∘ f d μ {\displaystyle \varphi \left(\int _{\Omega }f\,\mathrm {d} \mu \right)\leq \int _{\Omega }\varphi \circ f\,\mathrm {d} \mu } In real analysis, we may require an estimate on φ ( ∫ a b f ( x ) d x ) {\displaystyle \varphi \left(\int _{a}^{b}f(x)\,dx\right)} where a , b ∈ R {\displaystyle a,b\in \mathbb {R} } , and f : [ a , b ] → R {\displaystyle f\colon [a,b]\to \mathbb {R} } is a non-negative Lebesgue-integrable function. In this case, the Lebesgue measure of [ a , b ] {\displaystyle [a,b]} need not be unity. However, by integration by substitution, the interval can be rescaled so that it has measure unity. Then Jensen's inequality can be applied to get φ ( 1 b − a ∫ a b f ( x ) d x ) ≤ 1 b − a ∫ a b φ ( f ( x ) ) d x . {\displaystyle \varphi \left({\frac {1}{b-a}}\int _{a}^{b}f(x)\,dx\right)\leq {\frac {1}{b-a}}\int _{a}^{b}\varphi (f(x))\,dx.} === Probabilistic form === The same result can be equivalently stated in a probability theory setting, by a simple change of notation. Let ( Ω , F , P ) {\displaystyle (\Omega ,{\mathfrak {F}},\operatorname {P} )} be a probability space, X an integrable real-valued random variable and φ {\displaystyle \varphi } a convex function. Then φ ( E [ X ] ) ≤ E [ φ ( X ) ] . {\displaystyle \varphi {\big (}\operatorname {E} [X]{\big )}\leq \operatorname {E} [\varphi (X)].} In this probability setting, the measure μ is intended as a probability P {\displaystyle \operatorname {P} } , the integral with respect to μ as an expected value E {\displaystyle \operatorname {E} } , and the function f {\displaystyle f} as a random variable X. Note that the equality holds if and only if φ {\displaystyle \varphi } is a linear function on some convex set A {\displaystyle A} such that P ( X ∈ A ) = 1 {\displaystyle P(X\in A)=1} (which follows by inspecting the measure-theoretical proof below). === General inequality in a probabilistic setting === More generally, let T be a real topological vector space, and X a T-valued integrable random variable. In this general setting, integrable means that there exists an element E [ X ] {\displaystyle \operatorname {E} [X]} in T, such that for any element z in the dual space of T: E | ⟨ z , X ⟩ | < ∞ {\displaystyle \operatorname {E} |\langle z,X\rangle |<\infty } , and ⟨ z , E [ X ] ⟩ = E [ ⟨ z , X ⟩ ] {\displaystyle \langle z,\operatorname {E} [X]\rangle =\operatorname {E} [\langle z,X\rangle ]} . Then, for any measurable convex function φ and any sub-σ-algebra G {\displaystyle {\mathfrak {G}}} of F {\displaystyle {\mathfrak {F}}} : φ ( E [ X ∣ G ] ) ≤ E [ φ ( X ) ∣ G ] . {\displaystyle \varphi \left(\operatorname {E} \left[X\mid {\mathfrak {G}}\right]\right)\leq \operatorname {E} \left[\varphi (X)\mid {\mathfrak {G}}\right].} Here E [ ⋅ ∣ G ] {\displaystyle \operatorname {E} [\cdot \mid {\mathfrak {G}}]} stands for the expectation conditioned to the σ-algebra G {\displaystyle {\mathfrak {G}}} . This general statement reduces to the previous ones when the topological vector space T is the real axis, and G {\displaystyle {\mathfrak {G}}} is the trivial σ-algebra {∅, Ω} (where ∅ is the empty set, and Ω is the sample space). === A sharpened and generalized form === Let X be a one-dimensional random variable with mean μ {\displaystyle \mu } and variance σ 2 ≥ 0 {\displaystyle \sigma ^{2}\geq 0} . Let φ ( x ) {\displaystyle \varphi (x)} be a twice differentiable function, and define the function h ( x ) ≜ φ ( x ) − φ ( μ ) ( x − μ ) 2 − φ ′ ( μ ) x − μ . {\displaystyle h(x)\triangleq {\frac {\varphi \left(x\right)-\varphi \left(\mu \right)}{\left(x-\mu \right)^{2}}}-{\frac {\varphi '\left(\mu \right)}{x-\mu }}.} Then σ 2 inf φ ″ ( x ) 2 ≤ σ 2 inf h ( x ) ≤ E [ φ ( X ) ] − φ ( E [ X ] ) ≤ σ 2 sup h ( x ) ≤ σ 2 sup φ ″ ( x ) 2 . {\displaystyle \sigma ^{2}\inf {\frac {\varphi ''(x)}{2}}\leq \sigma ^{2}\inf h(x)\leq E\left[\varphi \left(X\right)\right]-\varphi \left(E[X]\right)\leq \sigma ^{2}\sup h(x)\leq \sigma ^{2}\sup {\frac {\varphi ''(x)}{2}}.} In particular, when φ ( x ) {\displaystyle \varphi (x)} is convex, then φ ″ ( x ) ≥ 0 {\displaystyle \varphi ''(x)\geq 0} , and the standard form of Jensen's inequality immediately follows for the case where φ ( x ) {\displaystyle \varphi (x)} is additionally assumed to be twice differentiable. == Proofs == === Intuitive graphical proof === Jensen's inequality can be proved in several ways, and three different proofs corresponding to the different statements above will be offered. Before embarking on these mathematical derivations, however, it is worth analyzing an intuitive graphical argument based on the probabilistic case where X is a real number (see figure). Assuming a hypothetical distribution of X values, one can immediately identify the position of E [ X ] {\displaystyle \operatorname {E} [X]} and its image φ ( E [ X ] ) {\displaystyle \varphi (\operatorname {E} [X])} in the graph. Noticing that for convex mappings Y = φ(x) of some x values the corresponding distribution of Y values is increasingly "stretched up" for increasing values of X, it is easy to see that the distribution of Y is broader in the interval corresponding to X > X0 and narrower in X < X0 for any X0; in particular, this is also true for X 0 = E [ X ] {\displaystyle X_{0}=\operatorname {E} [X]} . Consequently, in this picture the expectation of Y will always shift upwards with respect to the position of φ ( E [ X ] ) {\displaystyle \varphi (\operatorname {E} [X])} . A similar reasoning holds if the distribution of X covers a decreasing portion of the convex function, or both a decreasing and an increasing portion of it. This "proves" the inequality, i.e. φ ( E [ X ] ) ≤ E [ φ ( X ) ] = E [ Y ] , {\displaystyle \varphi (\operatorname {E} [X])\leq \operatorname {E} [\varphi (X)]=\operatorname {E} [Y],} with equality when φ(X) is not strictly convex, e.g. when it is a straight line, or when X follows a degenerate distribution (i.e. is a constant). The proofs below formalize this intuitive notion. === Proof 1 (finite form) === If λ1 and λ2 are two arbitrary nonnegative real numbers such that λ1 + λ2 = 1 then convexity of φ implies ∀ x 1 , x 2 : φ ( λ 1 x 1 + λ 2 x 2 ) ≤ λ 1 φ ( x 1 ) + λ 2 φ ( x 2 ) . {\displaystyle \forall x_{1},x_{2}:\qquad \varphi \left(\lambda _{1}x_{1}+\lambda _{2}x_{2}\right)\leq \lambda _{1}\,\varphi (x_{1})+\lambda _{2}\,\varphi (x_{2}).} This can be generalized: if λ1, ..., λn are nonnegative real numbers such that λ1 + ... + λn = 1, then φ ( λ 1 x 1 + λ 2 x 2 + ⋯ + λ n x n ) ≤ λ 1 φ ( x 1 ) + λ 2 φ ( x 2 ) + ⋯ + λ n φ ( x n ) , {\displaystyle \varphi (\lambda _{1}x_{1}+\lambda _{2}x_{2}+\cdots +\lambda _{n}x_{n})\leq \lambda _{1}\,\varphi (x_{1})+\lambda _{2}\,\varphi (x_{2})+\cdots +\lambda _{n}\,\varphi (x_{n}),} for any x1, ..., xn. The finite form of the Jensen's inequality can be proved by induction: by convexity hypotheses, the statement is true for n = 2. Suppose the statement is true for some n, so φ ( ∑ i = 1 n λ i x i ) ≤ ∑ i = 1 n λ i φ ( x i ) {\displaystyle \varphi \left(\sum _{i=1}^{n}\lambda _{i}x_{i}\right)\leq \sum _{i=1}^{n}\lambda _{i}\varphi \left(x_{i}\right)} for any λ1, ..., λn such that λ1 + ... + λn = 1. One needs to prove it for n + 1. At least one of the λi is strictly smaller than 1 {\displaystyle 1} , say λn+1; therefore by convexity inequality: φ ( ∑ i = 1 n + 1 λ i x i ) = φ ( ( 1 − λ n + 1 ) ∑ i = 1 n λ i 1 − λ n + 1 x i + λ n + 1 x n + 1 ) ≤ ( 1 − λ n + 1 ) φ ( ∑ i = 1 n λ i 1 − λ n + 1 x i ) + λ n + 1 φ ( x n + 1 ) . {\displaystyle {\begin{aligned}\varphi \left(\sum _{i=1}^{n+1}\lambda _{i}x_{i}\right)&=\varphi \left((1-\lambda _{n+1})\sum _{i=1}^{n}{\frac {\lambda _{i}}{1-\lambda _{n+1}}}x_{i}+\lambda _{n+1}x_{n+1}\right)\\&\leq (1-\lambda _{n+1})\varphi \left(\sum _{i=1}^{n}{\frac {\lambda _{i}}{1-\lambda _{n+1}}}x_{i}\right)+\lambda _{n+1}\,\varphi (x_{n+1}).\end{aligned}}} Since λ1 + ... +λn + λn+1 = 1, ∑ i = 1 n λ i 1 − λ n + 1 = 1 {\displaystyle \sum _{i=1}^{n}{\frac {\lambda _{i}}{1-\lambda _{n+1}}}=1} , applying the inductive hypothesis gives φ ( ∑ i = 1 n λ i 1 − λ n + 1 x i ) ≤ ∑ i = 1 n λ i 1 − λ n + 1 φ ( x i ) {\displaystyle \varphi \left(\sum _{i=1}^{n}{\frac {\lambda _{i}}{1-\lambda _{n+1}}}x_{i}\right)\leq \sum _{i=1}^{n}{\frac {\lambda _{i}}{1-\lambda _{n+1}}}\varphi (x_{i})} therefore φ ( ∑ i = 1 n + 1 λ i x i ) ≤ ( 1 − λ n + 1 ) ∑ i = 1 n λ i 1 − λ n + 1 φ ( x i ) + λ n + 1 φ ( x n + 1 ) = ∑ i = 1 n + 1 λ i φ ( x i ) {\displaystyle {\begin{aligned}\varphi \left(\sum _{i=1}^{n+1}\lambda _{i}x_{i}\right)&\leq (1-\lambda _{n+1})\sum _{i=1}^{n}{\frac {\lambda _{i}}{1-\lambda _{n+1}}}\varphi (x_{i})+\lambda _{n+1}\,\varphi (x_{n+1})=\sum _{i=1}^{n+1}\lambda _{i}\varphi (x_{i})\end{aligned}}} We deduce the inequality is true for n + 1, by induction it follows that the result is also true for all integer n greater than 2. In order to obtain the general inequality from this finite form, one needs to use a density argument. The finite form can be rewritten as: φ ( ∫ x d μ n ( x ) ) ≤ ∫ φ ( x ) d μ n ( x ) , {\displaystyle \varphi \left(\int x\,d\mu _{n}(x)\right)\leq \int \varphi (x)\,d\mu _{n}(x),} where μn is a measure given by an arbitrary convex combination of Dirac deltas: μ n = ∑ i = 1 n λ i δ x i . {\displaystyle \mu _{n}=\sum _{i=1}^{n}\lambda _{i}\delta _{x_{i}}.} Since convex functions are continuous, and since convex combinations of Dirac deltas are weakly dense in the set of probability measures (as could be easily verified), the general statement is obtained simply by a limiting procedure. === Proof 2 (measure-theoretic form) === Let g {\displaystyle g} be a real-valued μ {\displaystyle \mu } -integrable function on a probability space Ω {\displaystyle \Omega } , and let φ {\displaystyle \varphi } be a convex function on the real numbers. Since φ {\displaystyle \varphi } is convex, at each real number x {\displaystyle x} we have a nonempty set of subderivatives, which may be thought of as lines touching the graph of φ {\displaystyle \varphi } at x {\displaystyle x} , but which are below the graph of φ {\displaystyle \varphi } at all points (support lines of the graph). Now, if we define x 0 := ∫ Ω g d μ , {\displaystyle x_{0}:=\int _{\Omega }g\,d\mu ,} because of the existence of subderivatives for convex functions, we may choose a {\displaystyle a} and b {\displaystyle b} such that a x + b ≤ φ ( x ) , {\displaystyle ax+b\leq \varphi (x),} for all real x {\displaystyle x} and a x 0 + b = φ ( x 0 ) . {\displaystyle ax_{0}+b=\varphi (x_{0}).} But then we have that φ ∘ g ( ω ) ≥ a g ( ω ) + b {\displaystyle \varphi \circ g(\omega )\geq ag(\omega )+b} for almost all ω ∈ Ω {\displaystyle \omega \in \Omega } . Since we have a probability measure, the integral is monotone with μ ( Ω ) = 1 {\displaystyle \mu (\Omega )=1} so that ∫ Ω φ ∘ g d μ ≥ ∫ Ω ( a g + b ) d μ = a ∫ Ω g d μ + b ∫ Ω d μ = a x 0 + b = φ ( x 0 ) = φ ( ∫ Ω g d μ ) , {\displaystyle \int _{\Omega }\varphi \circ g\,d\mu \geq \int _{\Omega }(ag+b)\,d\mu =a\int _{\Omega }g\,d\mu +b\int _{\Omega }d\mu =ax_{0}+b=\varphi (x_{0})=\varphi \left(\int _{\Omega }g\,d\mu \right),} as desired. === Proof 3 (general inequality in a probabilistic setting) === Let X be an integrable random variable that takes values in a real topological vector space T. Since φ : T → R {\displaystyle \varphi :T\to \mathbb {R} } is convex, for any x , y ∈ T {\displaystyle x,y\in T} , the quantity φ ( x + θ y ) − φ ( x ) θ , {\displaystyle {\frac {\varphi (x+\theta \,y)-\varphi (x)}{\theta }},} is decreasing as θ approaches 0+. In particular, the subdifferential of φ {\displaystyle \varphi } evaluated at x in the direction y is well-defined by ( D φ ) ( x ) ⋅ y := lim θ ↓ 0 φ ( x + θ y ) − φ ( x ) θ = inf θ ≠ 0 φ ( x + θ y ) − φ ( x ) θ . {\displaystyle (D\varphi )(x)\cdot y:=\lim _{\theta \downarrow 0}{\frac {\varphi (x+\theta \,y)-\varphi (x)}{\theta }}=\inf _{\theta \neq 0}{\frac {\varphi (x+\theta \,y)-\varphi (x)}{\theta }}.} It is easily seen that the subdifferential is linear in y (that is false and the assertion requires Hahn-Banach theorem to be proved) and, since the infimum taken in the right-hand side of the previous formula is smaller than the value of the same term for θ = 1, one gets φ ( x ) ≤ φ ( x + y ) − ( D φ ) ( x ) ⋅ y . {\displaystyle \varphi (x)\leq \varphi (x+y)-(D\varphi )(x)\cdot y.} In particular, for an arbitrary sub-σ-algebra G {\displaystyle {\mathfrak {G}}} we can evaluate the last inequality when x = E [ X ∣ G ] , y = X − E [ X ∣ G ] {\displaystyle x=\operatorname {E} [X\mid {\mathfrak {G}}],\,y=X-\operatorname {E} [X\mid {\mathfrak {G}}]} to obtain φ ( E [ X ∣ G ] ) ≤ φ ( X ) − ( D φ ) ( E [ X ∣ G ] ) ⋅ ( X − E [ X ∣ G ] ) . {\displaystyle \varphi (\operatorname {E} [X\mid {\mathfrak {G}}])\leq \varphi (X)-(D\varphi )(\operatorname {E} [X\mid {\mathfrak {G}}])\cdot (X-\operatorname {E} [X\mid {\mathfrak {G}}]).} Now, if we take the expectation conditioned to G {\displaystyle {\mathfrak {G}}} on both sides of the previous expression, we get the result since: E [ [ ( D φ ) ( E [ X ∣ G ] ) ⋅ ( X − E [ X ∣ G ] ) ] ∣ G ] = ( D φ ) ( E [ X ∣ G ] ) ⋅ E [ ( X − E [ X ∣ G ] ) ∣ G ] = 0 , {\displaystyle \operatorname {E} \left[\left[(D\varphi )(\operatorname {E} [X\mid {\mathfrak {G}}])\cdot (X-\operatorname {E} [X\mid {\mathfrak {G}}])\right]\mid {\mathfrak {G}}\right]=(D\varphi )(\operatorname {E} [X\mid {\mathfrak {G}}])\cdot \operatorname {E} [\left(X-\operatorname {E} [X\mid {\mathfrak {G}}]\right)\mid {\mathfrak {G}}]=0,} by the linearity of the subdifferential in the y variable, and the following well-known property of the conditional expectation: E [ ( E [ X ∣ G ] ) ∣ G ] = E [ X ∣ G ] . {\displaystyle \operatorname {E} \left[\left(\operatorname {E} [X\mid {\mathfrak {G}}]\right)\mid {\mathfrak {G}}\right]=\operatorname {E} [X\mid {\mathfrak {G}}].} == Applications and special cases == === Form involving a probability density function === Suppose Ω is a measurable subset of the real line and f(x) is a non-negative function such that ∫ − ∞ ∞ f ( x ) d x = 1. {\displaystyle \int _{-\infty }^{\infty }f(x)\,dx=1.} In probabilistic language, f is a probability density function. Then Jensen's inequality becomes the following statement about convex integrals: If g is any real-valued measurable function and φ {\textstyle \varphi } is convex over the range of g, then φ ( ∫ − ∞ ∞ g ( x ) f ( x ) d x ) ≤ ∫ − ∞ ∞ φ ( g ( x ) ) f ( x ) d x . {\displaystyle \varphi \left(\int _{-\infty }^{\infty }g(x)f(x)\,dx\right)\leq \int _{-\infty }^{\infty }\varphi (g(x))f(x)\,dx.} If g(x) = x, then this form of the inequality reduces to a commonly used special case: φ ( ∫ − ∞ ∞ x f ( x ) d x ) ≤ ∫ − ∞ ∞ φ ( x ) f ( x ) d x . {\displaystyle \varphi \left(\int _{-\infty }^{\infty }x\,f(x)\,dx\right)\leq \int _{-\infty }^{\infty }\varphi (x)\,f(x)\,dx.} This is applied in Variational Bayesian methods. === Example: even moments of a random variable === If g(x) = x2n, and X is a random variable, then g is convex as d 2 g d x 2 ( x ) = 2 n ( 2 n − 1 ) x 2 n − 2 ≥ 0 ∀ x ∈ R {\displaystyle {\frac {d^{2}g}{dx^{2}}}(x)=2n(2n-1)x^{2n-2}\geq 0\quad \forall \ x\in \mathbb {R} } and so g ( E [ X ] ) = ( E [ X ] ) 2 n ≤ E [ X 2 n ] . {\displaystyle g(\operatorname {E} [X])=(\operatorname {E} [X])^{2n}\leq \operatorname {E} [X^{2n}].} In particular, if some even moment 2n of X is finite, X has a finite mean. An extension of this argument shows X has finite moments of every order l ∈ N {\displaystyle l\in \mathbb {N} } dividing n. === Alternative finite form === Let Ω = {x1, ... xn}, and take μ to be the counting measure on Ω, then the general form reduces to a statement about sums: φ ( ∑ i = 1 n g ( x i ) λ i ) ≤ ∑ i = 1 n φ ( g ( x i ) ) λ i , {\displaystyle \varphi \left(\sum _{i=1}^{n}g(x_{i})\lambda _{i}\right)\leq \sum _{i=1}^{n}\varphi (g(x_{i}))\lambda _{i},} provided that λi ≥ 0 and λ 1 + ⋯ + λ n = 1. {\displaystyle \lambda _{1}+\cdots +\lambda _{n}=1.} There is also an infinite discrete form. === Statistical physics === Jensen's inequality is of particular importance in statistical physics when the convex function is an exponential, giving: e E [ X ] ≤ E [ e X ] , {\displaystyle e^{\operatorname {E} [X]}\leq \operatorname {E} \left[e^{X}\right],} where the expected values are with respect to some probability distribution in the random variable X. Proof: Let φ ( x ) = e x {\displaystyle \varphi (x)=e^{x}} in φ ( E [ X ] ) ≤ E [ φ ( X ) ] . {\displaystyle \varphi \left(\operatorname {E} [X]\right)\leq \operatorname {E} \left[\varphi (X)\right].} === Information theory === If p(x) is the true probability density for X, and q(x) is another density, then applying Jensen's inequality for the random variable Y(X) = q(X)/p(X) and the convex function φ(y) = −log(y) gives E [ φ ( Y ) ] ≥ φ ( E [ Y ] ) {\displaystyle \operatorname {E} [\varphi (Y)]\geq \varphi (\operatorname {E} [Y])} Therefore: − D ( p ( x ) ‖ q ( x ) ) = ∫ p ( x ) log ( q ( x ) p ( x ) ) d x ≤ log ( ∫ p ( x ) q ( x ) p ( x ) d x ) = log ( ∫ q ( x ) d x ) = 0 {\displaystyle -D(p(x)\|q(x))=\int p(x)\log \left({\frac {q(x)}{p(x)}}\right)\,dx\leq \log \left(\int p(x){\frac {q(x)}{p(x)}}\,dx\right)=\log \left(\int q(x)\,dx\right)=0} a result called Gibbs' inequality. It shows that the average message length is minimised when codes are assigned on the basis of the true probabilities p rather than any other distribution q. The quantity that is non-negative is called the Kullback–Leibler divergence of q from p, where D ( p ( x ) ‖ q ( x ) ) = ∫ p ( x ) log ( p ( x ) q ( x ) ) d x {\displaystyle D(p(x)\|q(x))=\int p(x)\log \left({\frac {p(x)}{q(x)}}\right)dx} . Since −log(x) is a strictly convex function for x > 0, it follows that equality holds when p(x) equals q(x) almost everywhere. === Rao–Blackwell theorem === If L is a convex function and G {\displaystyle {\mathfrak {G}}} a sub-sigma-algebra, then, from the conditional version of Jensen's inequality, we get L ( E [ δ ( X ) ∣ G ] ) ≤ E [ L ( δ ( X ) ) ∣ G ] ⟹ E [ L ( E [ δ ( X ) ∣ G ] ) ] ≤ E [ L ( δ ( X ) ) ] . {\displaystyle L(\operatorname {E} [\delta (X)\mid {\mathfrak {G}}])\leq \operatorname {E} [L(\delta (X))\mid {\mathfrak {G}}]\quad \Longrightarrow \quad \operatorname {E} [L(\operatorname {E} [\delta (X)\mid {\mathfrak {G}}])]\leq \operatorname {E} [L(\delta (X))].} So if δ(X) is some estimator of an unobserved parameter θ given a vector of observables X; and if T(X) is a sufficient statistic for θ; then an improved estimator, in the sense of having a smaller expected loss L, can be obtained by calculating δ 1 ( X ) = E θ [ δ ( X ′ ) ∣ T ( X ′ ) = T ( X ) ] , {\displaystyle \delta _{1}(X)=\operatorname {E} _{\theta }[\delta (X')\mid T(X')=T(X)],} the expected value of δ with respect to θ, taken over all possible vectors of observations X compatible with the same value of T(X) as that observed. Further, because T is a sufficient statistic, δ 1 ( X ) {\displaystyle \delta _{1}(X)} does not depend on θ, hence, becomes a statistic. This result is known as the Rao–Blackwell theorem. === Risk aversion === The relation between risk aversion and declining marginal utility for scalar outcomes can be stated formally with Jensen's inequality: risk aversion can be stated as preferring a certain outcome u ( E [ x ] ) {\displaystyle u(E[x])} to a fair gamble with potentially larger but uncertain outcome of u ( x ) {\displaystyle u(x)} : u ( E [ x ] ) > E [ u ( x ) ] {\displaystyle u(E[x])>E[u(x)]} . But this is simply Jensen's inequality for a concave u ( x ) {\displaystyle u(x)} : a utility function that exhibits declining marginal utility. == Generalizations == Beyond its classical formulation for real numbers and convex functions, Jensen’s inequality has been extended to the realm of operator theory. In this non‐commutative setting the inequality is expressed in terms of operator convex functions—that is, functions defined on an interval I that satisfy f ( λ x + ( 1 − λ ) y ) ≤ λ f ( x ) + ( 1 − λ ) f ( y ) {\displaystyle f{\bigl (}\lambda x+(1-\lambda )y{\bigr )}\leq \lambda f(x)+(1-\lambda )f(y)} for every pair of self‐adjoint operators x and y (with spectra in I) and every scalar λ ∈ [ 0 , 1 ] {\displaystyle \lambda \in [0,1]} . Hansen and Pedersen established a definitive version of this inequality by considering genuine non‐commutative convex combinations. In particular, if one has an n‑tuple of bounded self‐adjoint operators x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} with spectra in I and an n‑tuple of operators a 1 , … , a n {\displaystyle a_{1},\dots ,a_{n}} satisfying ∑ i = 1 n a i ∗ a i = I , {\displaystyle \sum _{i=1}^{n}a_{i}^{*}a_{i}=I,} then the following operator Jensen inequality holds: f ( ∑ i = 1 n a i ∗ x i a i ) ≤ ∑ i = 1 n a i ∗ f ( x i ) a i . {\displaystyle f{\Bigl (}\sum _{i=1}^{n}a_{i}^{*}x_{i}a_{i}{\Bigr )}\leq \sum _{i=1}^{n}a_{i}^{*}f(x_{i})a_{i}.} This result shows that the convex transformation “respects” non-commutative convex combinations, thereby extending the classical inequality to operators without the need for additional restrictions on the interval of definition. A closely related extension is given by the Jensen trace inequality. For a continuous convex function f defined on I, if one considers self‐adjoint matrices x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} (with spectra in I) and matrices a 1 , … , a n {\displaystyle a_{1},\dots ,a_{n}} satisfying ∑ i = 1 n a i ∗ a i = I {\displaystyle \sum _{i=1}^{n}a_{i}^{*}a_{i}=I} , then one has Tr ( f ( ∑ i = 1 n a i ∗ x i a i ) ) ≤ Tr ( ∑ i = 1 n a i ∗ f ( x i ) a i ) . {\displaystyle \operatorname {Tr} {\Bigl (}f{\Bigl (}\sum _{i=1}^{n}a_{i}^{*}x_{i}a_{i}{\Bigr )}{\Bigr )}\leq \operatorname {Tr} {\Bigl (}\sum _{i=1}^{n}a_{i}^{*}f(x_{i})a_{i}{\Bigr )}.} This inequality naturally extends to C*-algebras equipped with a finite trace and is particularly useful in applications ranging from quantum statistical mechanics to information theory. Furthermore, contractive versions of these operator inequalities are available when one only assumes ∑ i = 1 n a i t a i ≤ I {\displaystyle \sum _{i=1}^{n}a_{i}^{t}a_{i}\leq I} , provided that additional conditions such as f ( 0 ) ≤ 0 {\displaystyle f(0)\leq 0} (when 0 ∈ I) are imposed. Extensions to continuous fields of operators and to settings involving conditional expectations on C-algebras further illustrate the broad applicability of these generalizations. == See also == Karamata's inequality for a more general inequality Popoviciu's inequality Law of averages A proof without words of Jensen's inequality == Notes == == References == David Chandler (1987). Introduction to Modern Statistical Mechanics. Oxford. ISBN 0-19-504277-8. Tristan Needham (1993) "A Visual Explanation of Jensen's Inequality", American Mathematical Monthly 100(8):768–71. Nicola Fusco; Paolo Marcellini; Carlo Sbordone (1996). Analisi Matematica Due. Liguori. ISBN 978-88-207-2675-1. Walter Rudin (1987). Real and Complex Analysis. McGraw-Hill. ISBN 0-07-054234-1. Rick Durrett (2019). Probability: Theory and Examples (5th ed.). Cambridge University Press. p. 430. ISBN 978-1108473682. Retrieved 21 Dec 2020. Sam Savage (2012) The Flaw of Averages: Why We Underestimate Risk in the Face of Uncertainty (1st ed.) Wiley. ISBN 978-0471381976 == External links == Jensen's Operator Inequality of Hansen and Pedersen. "Jensen inequality", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Weisstein, Eric W. "Jensen's inequality". MathWorld. Arthur Lohwater (1982). "Introduction to Inequalities". Online e-book in PDF format.
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Wikipedia:Jerome K. Percus#0
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Jerome Kenneth Percus (born 21 June 1926 in New York City; died 7 March 2021) was a physicist and mathematician known for important contributions to statistical physics, chemical physics, and applied mathematics. In 1958, he published with George J. Yevick a groundbreaking study on the statistical mechanics of classical liquids. They formulated an integral equation (Percus–Yevick equation) that is the foundation for several approximation methods for computing the pair correlation function, and thereby allow the derivation of thermodynamic properties from first principles. == Works == Percus published several books: Combinatorial Methods, Applied Mathematical Sciences 4, Springer 1971 Mathematics of genome analysis, Cambridge UP 2002 Mathematical models in developmental biology, Courant Lectures in Mathematics 26, Courant Institute of Mathematical Sciences 2015 with Stephen Childress == References == == External links == Homepage an der New York University Jerome K. Percus at the Mathematics Genealogy Project Lebensdaten, Publikationen und Akademischer Stammbaum von Jerome K. Percus bei academictree.org, retrieved 22 April 2018. Works by and about Jerome K. Percus in the Deutsche Digitale Bibliothek (German Digital Library)
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Wikipedia:Jerome Ravetz#0
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Jerome (Jerry) Ravetz is a philosopher of science. He is best known for his books analysing scientific knowledge from a social and ethical perspective, focusing on issues of quality. He is the co-author (with Silvio Funtowicz) of the NUSAP notational system and of Post-normal science. He is currently an Associate Fellow at the Institute for Science, Innovation and Society, University of Oxford. == Life and work == Ravetz was born in Philadelphia; his grandfather was a Russian-Jewish immigrant and his father a truck driver and trade union organiser. He attended Central High School and Swarthmore College. He came to England in 1950 on a Fulbright Scholarship to Trinity College, Cambridge, where he studied for a PhD in Pure Mathematics under the supervision of A.S. Besicovitch. In 1955 his passport was taken away, as part of the wave of McCarthyism; it was returned in 1958 and in 1961 he became a British citizen. He taught mathematics at the University of Pennsylvania and then at Durham University. In 1957 he moved to the University of Leeds to join Stephen Toulmin in the establishment of a centre in the History and Philosophy of Science. He stayed at Leeds, eventually becoming a Reader, until taking early retirement in 1983. Since then he has been an independent scholar. He has visited at Utrecht University, Harvard University, the Institute for Advanced Study, the University of California, Santa Cruz, Fudan University (Shanghai), the University of Texas at Dallas, the Carnegie Mellon University and the University of Luxembourg. Over the years he has worked closely with colleagues at the European Commission Joint Research Centre, Ispra, Italy. Ravetz's earliest research, after mathematics, was in the history of the mathematical sciences, with works on Copernicus and Fourier. In the sixties Ravetz, focused on contradictions in the functioning of science and its image, between a narrative of progress and one of depredation science. He had proposed an unusual approach to study these contradictions. He highlighted the craft character of scientific knowledge, a set of procedures and evaluations that are part of the social activity of science. The quality of research, according to Ravetz, is influenced by this social component of scientific knowledge, which is strongly influenced by the professional models communicated through teaching. Due to this, the standard of adequacy is set by norms culturally stablished and shared. His influential book Scientific Knowledge and Its Social Problems went through several English language editions, plus German and Japanese translations, and was republished in 1996. This book raises issues of uncertainty and ethics in the social practice of science. It was an early attempt to recast the philosophy of science for the conditions of 'industrialised science' and to shift the philosophy of science from epistemology to the social and ethical aspects of science. In it he proposed a 'critical science' for a new version of the idealism that had characterised science in the pre-industrial age. In the years around 1970 he was an active member of the British Society for Social Responsibility in Science. From 1973 to 1976 he was Executive Secretary of the Council for Science and Society in London, whose founder was the law reformer Paul Sieghart. He drafted its report on 'The Acceptability of Risks'. From 1977 to 1978, he was a member of the Genetic Manipulation Advisory Group, regulating research in recombinant DNA. Working with Silvio Funtowicz in Leeds he created the NUSAP notational system, described in their book Uncertainty and Quality in Science for Policy (Reidel 1990). This was the stimulus for the development of the 'Guidance' for managing uncertainty, at the Netherlands Environment Agency. They also created the theory of Post-normal science, which applies when 'Facts are uncertain, values in dispute, stakes high and decisions urgent.' A collection of his essays, The merger of knowledge with power: essays in critical science was also published in 1990. With Zia Sardar he co-authored Cyberfutures: Culture and Politics on the Information Superhighway in 1996.[2] His most recent book is The No nonsense guide to science (New Internationalist 2006). His research continues in two main directions: new trends in the social practice of science; and new approaches to the management of uncertainty. On the former, he has co-authored (with Silvio Funtowicz) chapters on 'Science, New Forms of' and 'Peer Review and Quality Control' for the International Encyclopedia of Social and Behavioral Sciences (2015). On the latter he is concerned with the analysis of ignorance and the representation and manipulation of quantitative information where there is 'not even one significant digit'. He has also recently written on the quality control crisis of science. He is currently an Associate Fellow at the Institute for Science, Innovation and Society at the University of Oxford. His research focus is on the impacts of technological progress on science, and new forms of science governance necessary for the protection of society, the environment and science itself. Ravetz studies new institutional models and collective behaviors that rehabilitate the function of science as a solver of the problem of the sustainability of contemporary material society that derives from the ongoing technological development. He was interviewed by the Great Transition Initiative's blog in June 2016. On May 31, 2019, the Institute for Science, Innovation and Society in Oxford has devoted a seminar to honour his 90th birthday. He published a World View piece in the journal Nature on November 19, 2019, entitled Stop the science training that demands ‘don’t ask’. Science governance, unknown unknowns, science as a solver of the problem of the sustainability of society are tackled in the context of present debates. Together with other scholars, Ravetz has contributed to the debate on the COVID-19 pandemic and use of mathematical models. == Quotes == "The activity of modern natural science has transformed our knowledge and control of the world about us; but in the process it has also transformed itself; and it has created problems that natural science alone cannot solve". Scientific Knowledge and its Social Problems, Oxford 1971, p. 9. "Wherever there's a system, there's a racket to beat it." Ibidem, p. 295. "Any artifact will have: intended use, creative new use, incompetent misuse and malevolent abuse". == Bibliography == Ravetz, Jerome R. (1979). Scientific knowledge and its social problems. Oxford: Oxford Univ. Press. ISBN 0-19-519721-6. Ravetz, Jerome R. (1990). The merger of knowledge with power: essays in critical science. London [England]: Mansell. ISBN 0-7201-2021-7. Funtowicz, Silvio O.; Ravetz, Jerome R. (1990). Uncertainty and Quality in Science for Policy. Boston: Kluwer Academic Publishers. ISBN 0-7923-0799-2. Ravetz, Jerome R.; Sardar, Ziauddin (1996). Cyberfutures: culture and politics on the information superhighway. New York: New York University Press. ISBN 0-8147-8058-X. Ravetz, Jerome R. (2005). The No nonsense guide to science. Oxford: New Internationalist. Ravetz, Jerome R. (1965). Astronomy and cosmology in the achievement of Nicolaus Copernicus. Wrocław, Poland: Zakład Narodowy im. Ossolińskich. Bibcode:1965acan.book.....R. Ravetz, Jerome R. (Winter 1997). "In Numbers We Trust". Issues in Science and Technology. 13 (2). == See also == Post-normal science American philosophy List of American philosophers Science wars Gaming the system Goodhart's Law Silvio Funtowicz == References == == External links == 'Crisis, what crisis in Science?' Seminar at the Institute for Science, Innovation and Society, Oxford University, July 30, 2018, hosted by Jerome R. Ravetz with Philip Mirowski and Andrea Saltelli. Reflections on ‘informed critical resistance, reform and the making of futures’, Post-Normal Science Symposium in Barcelona: Address by Jerome Ravetz, 15 November 2018. Science - post normal perspectivs: Jerry Ravetz at 90, a workshop of the Institute for Science, Innovation and Society to honour Jerome R. Ravetz 90th birthday. May 31st, 2019, Oxford. Videos from Science - post normal perspectivs: Jerry Ravetz at 90, a workshop of the Institute for Science, Innovation and Society to honour Jerome R. Ravetz 90th birthday. May 31st, 2019, Oxford., also available as podcast. Matt Finch interviews Jerry Ravetz for Mechanical Dolphin: 'Post-normal science in the time of COVID-19: Discussion with Jerome Ravetz', July 13, 2020. Welcome to the PNS Movement at PNS5 (post-normal science symposium), Florence, September 21, 2020. See the Symposium program here. == External links == Guidance for Uncertainty Assessment and Communication, Netherlands Environmental Agency Jerome Ravetz homepage Recent articles by Ravetz Ongoing discussions of Post-Normal Science can be found on the blog Archived 2008-08-20 at the Wayback Machine More articles and excerpts from J. Ravetz's works
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Wikipedia:Jerrold E. Marsden#0
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Jerrold Eldon Marsden (August 17, 1942 – September 21, 2010) was a Canadian mathematician. He was the Carl F. Braun Professor of Engineering and Control & Dynamical Systems at the California Institute of Technology. Marsden is listed as an ISI highly cited researcher. == Career == Marsden earned his B.Sc. in mathematics at the University of Toronto and his Ph.D. in mathematical physics at Princeton University in 1968 under Arthur S. Wightman. Thereafter, he worked at various universities and research institutes in the US, Canada, the United Kingdom, France and Germany. He was one of the founders of the Fields Institute in Toronto, Canada, and directed it until 1994. At the California Institute of Technology he was the Carl F. Braun Professor of Engineering and Control & Dynamical Systems. Marsden, together with Alan Weinstein, was one of the world leading authorities in mathematical and theoretical classical mechanics. He has laid much of the foundation for symplectic topology. The Marsden-Weinstein quotient is named after him. In 1973, Marsden (along with Arthur E. Fischer) won the Gravitational Research Foundation Prize. He was also the recipient of a Carnegie Fellowship in 1977, and a Miller Professorship in 1981-82. Also in 1981, Marsden won the Jeffery–Williams Prize. And in 1990, he received the Norbert Wiener Prize in Applied Mathematics, jointly awarded by the Society for Industrial and Applied Mathematics (SIAM) and the American Mathematical Society. Marsden was honoured "for his outstanding contributions to the study of differential equations in mechanics: he proved the existence of chaos in specific classical differential equations; his work on the momentum map, from abstract foundations to detailed applications, has had great impact." He was also awarded the Max Planck Research Award for Mathematics and Computer science in 2000. In 2005, he won the prestigious John von Neumann Lecture, which is awarded by SIAM to recognize outstanding contributions to the field of applied mathematical sciences and for their effective communication to the community. In 2006 he was elected Fellow of the Royal Society. In the same year, he also received an honorary doctorate from the University of Surrey. Marsden died of cancer on September 21, 2010. In 2006 a festschrift was published in honor of Marsden's 60th birthday. In 2015 a memorial volume was published in his honor. == Books == Jerrold E. Marsden and Alan Weinstein, Calculus Unlimited, Benjamin-Cummings (1981). J. E. Marsden and A. Weinstein, Calculus, I, II, III', 2nd ed., Springer-Verlag (1985). J. E. Marsden, A. Tromba, and A. Weinstein, Basic Multivariable Calculus, Springer-Verlag (1992). J. E. Marsden and A. Tromba, Vector Calculus, 5th ed., W. H. Freeman (2003). J. E. Marsden and M. Hoffman, Elementary Classical Analysis, 2nd ed., W. H. Freeman (1993) J. E. Marsden and M. Hoffman, Basic Complex Analysis, 3rd ed., W. H. Freeman (1998). Alexandre Chorin and Jerrold E. Marsden, A Mathematical Introduction to Fluid Mechanics, 3rd ed., Springer-Verlag (1993). J. E. Marsden, Applications of Global Analysis in Mathematical Physics, Mathematics Lecture Series, No. 2, Publish or Perish, Inc (1974). Jerrold E. Marsden and Thomas J.R. Hughes, A Short Course in Fluid Mechanics, Mathematics Lecture Series, No. 6, Publish or Perish, Inc (1976). J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Applied Mathematical Sciences, 19 Springer-Verlag (1976). Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, The Mathematical Physics Monograph Series, 1st ed., W.A. Benjamin, Inc (1966). Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, 2nd ed., Addison–Wesley (1987). Ralph Abraham, Jerrold E. Marsden, and Tudor S. Ratiu, Manifolds, Tensor Analysis, and Applications, Springer-Verlag (1988). J. E. Marsden, Lectures on Mechanics, Cambridge University Press (1992). 1st edition, 1989. Jerrold E. Marsden and Thomas J.R. Hughes, Mathematical Foundations of Elasticity, Prentice Hall (1983); Reprinted by Dover Publications (1994). J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Texts in Applied Mathematics, vol. 17, Springer-Verlag (1994). J. E. Marsden, G. Misiolek, J.-P. Ortega, M. Perlmutter, and T. S. Ratiu, Hamiltonian Reduction by Stages, Springer-Verlag (2007). == References == physicists == External links == Jerrold E. Marsden at the Mathematics Genealogy Project J E. Marsden's webpage Laudation Max Planck Research Award
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Wikipedia:Jerry Kervorkian#0
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Jirair "Jerry" Kevorkian (May 14, 1933 – November 9, 2021) was an American applied mathematician and a founding member of the University of Washington's Department of Applied Mathematics. He was recognized for his contributions to asymptotic analysis, perturbation theory, and their applications in aerodynamics and fluid dynamics. Kevorkian co-authored textbooks on multiple scale perturbation methods and partial differential equations. == Early life and education == Jerry Kevorkian was born in Jerusalem on May 14, 1933. He earned his Bachelor’s (1955) and Master’s (1956) degrees in Aeronautical Engineering from the Georgia Institute of Technology. After working as an aerodynamist at General Dynamics and Convair, he pursued a Ph.D. at the California Institute of Technology under the supervision of Julian Cole. He completed his dissertation, *The Uniformly Valid Asymptotic Approximations to the Solutions of Certain Nonlinear Ordinary Differential Equations*, in 1961. == Academic career == Kevorkian joined the faculty at the University of Washington in 1964 as an assistant professor in Aeronautics and Astronautics. In 1971, he became a full professor with joint appointments in Applied Mathematics and Aeronautics. He played a pivotal role in establishing the Department of Applied Mathematics at UW, serving as one of its first chairs. == Research contributions == Kevorkian was an expert on asymptotic methods and perturbation theory. His research contributions include: Multiple Scale Perturbation Methods: Co-authored with Julian Cole a seminal textbook on multiple scale analysis that remains foundational in applied mathematics. Green's Function Approaches to PDEs: Authored a widely used textbook on Green's function methods for solving partial differential equations. Critical Inclination Problem: Contributed to solutions for satellite trajectories under critical inclination conditions using singular perturbation techniques. == Selected publications == Multiple Scale and Singular Perturbation Methods (with Julian Cole) Partial Differential Equations: Analytical Solution Technique == References == == External links == Remembering Professor Jerry Kevorkian - UW Applied Math History of UW Applied Mathematics
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Wikipedia:Jerzy Andrzej Filar#0
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Jerzy Andrzej Filar (born August 30, 1949, in Warsaw, Żoliborz) is an Australian mathematician of Polish origin, known for his significant contributions to operations research, stochastic modelling, game theory, Markov decision processes, perturbation theory, and environmental modelling. == Biography == He received his early education at Primary School No. 1 in Warsaw. In 1964, he moved to Melbourne, Australia, where he attended Kew High School. In 1972, he obtained a B.Sc (Hons) degree in Mathematical Statistics from the University of Melbourne, followed by an M.Sc. in Statistics from Monash University in 1975, and a M.A. in Mathematics from University of Illinois Chicago in 1977. In 1980, he completed his PhD at the University of Illinois Chicago, defending his thesis titled Algorithms for Solving Undiscounted Stochastic Games. His doctoral advisor was T.E.S. Raghavan. == Academic career == Since 1975, Jerzy A. Filar held various academic positions at institutions across Australia and the United States. He began his career as a Tutor at Monash University (1975–1976) before serving as a Visiting assistant professor at the University of Minnesota (1979–1980). He then held faculty positions at Johns Hopkins University (1980–1986) and the University of Maryland, Baltimore County (1986–1991), where he progressed from Associate to Full Professor. In 1992, he joined University of South Australia as Professor of Mathematics and Statistics, later serving as Director of the Centre for Industrial and Applied Mathematics (1994–1998) and Foundation Chair of Mathematics and Statistics (2001–2011). He then became Strategic Professor at Flinders University (2011–2016) and later Professor of Applied Mathematics and Director of the Centre for Applied Resource Mathematics at the University of Queensland (2016–2020). Filar also served as a supervisor or co-supervisor in over thirty Ph.D. dissertations. Filar retired at the end of 2020 and assumed the role of emeritus Professor at the University of Queensland. == Editorial works == He held several editorial roles in academic publishing, including Editor-in-Chief of Environmental Modelling and Assessment (Springer), Honorary Theme Editor for "Mathematical Models" in UNESCO's Encyclopedia of Life Support Systems (EOLSS), and Associate Editor for multiple journals, such as Journal of Mathematical Analysis and Applications (Elsevier), Operations Research (INFORMS, 2011–2015), Dynamic Games and Applications (Springer), and Applicationes Mathematicae, a prestigious Polish journal founded by Hugo Steinhaus. He also served as Co-editor of International Game Theory Reviews (World Scientific) and contributed to the Annals of Dynamic Games (International Society of Dynamic Games). == Research impacts == Filar is an applied mathematician with research interests in operations research, stochastic modelling, optimization, game theory, and environmental modelling. He supervised or co-supervised 31 PhD theses. He authored 148 scholarly works, including articles, books, book chapters, and conference papers. Notable contributions are featured in the following books: Competitive Markov Decision Processes (Springer, 1997; co-authored with K. Vrieze) serves as a textbook on stochastic games and Markov decision processes, under the unifying concept of "Competitive Markov decision processes" for discrete-time models with finite number of states and actions. Analytic Perturbation Theory and Its Applications (SIAM, 2013; co-authored with K.E. Avrachenkov and P.G. Howlett) presents a unified analysis of algebraic systems dependent on small perturbation parameters, applied to fields such as Markov processes and optimization. Hamiltonian Cycle Problem and Markov Chains (Springer, 2012; co-authored with V.S. Borkar, V. Ejov, and G.T. Nguyen). In game theory he introduced the concepts of switching controller stochastic game and games with incompetent players and explored a formulation of dynamic cooperative game. In operations research he pioneered embedding of the famous Hamiltonian cycle problem in Markov decision processes. In environmental modeling he investigated uncertainty propagation and risk sensitivity to small changes in model parameters. This linked to his more theoretical study of perturbation theory. In his final role as Director of the Centre for Applications in Resource Management (CARM), he collaborated with Queensland's Department of Agriculture and Fisheries (DAF) to integrate advanced statistical and mathematical modelling techniques as well as computer simulations, into stock assessments, supporting the Sustainable Fisheries Strategy. This partnership provided effective tools and simulations for evaluating the impacts of harvest policy changes. == Honours and awards == Jerzy Filar received several scholarships, fellowships, and professional recognition throughout his career. In 1985 he was awarded Environmental Science and Engineering Fellowship from the American Association for the Advancement of Science. In 2006, he was granted an Australian Research Council Professorial Fellowship (2006–2010). His professional recognition include the 1981 Dr. Gurdas Chatterjee OPSEARCH Best Paper Award, 1985 Environmental Fellow Worksthe 2003 Fellow of the Australian Mathematical Society, and the 2005 Ren Potts Medal from the Australian Society for Operations Research. In 2007, he received the South Australian Science Excellence Award for Science Leadership and Management. In 2013, he was appointed to the Australian Research Council College of Experts. Filar also secured competitive research grants from organizations such as the National Science Foundation (NSF), Air Force Office of Scientific Research (AFOSR), Australian Research Council (ARC), U.S. Environmental Protection Agency (US EPA), the Defence Science and Technology Group (DSTG), and Sir Keith and Sir Ross Smith Foundation == Science fiction novelist == Since 2021, Filar has held the title of emeritus Professor of Applied Mathematics. His academic career in the United States and Australia has informed his analysis of the interplay between technological advancements and the sustainability of Earth's life support systems. This led to the publication of his science fiction novel, YASMIN: The First Non-Artificial Intelligence Tool, which incorporates mathematical concepts into its narrative. == References ==
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Wikipedia:Jerzy Baksalary#0
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Jerzy Kazimierz Baksalary (25 June 1944 – 8 March 2005) was a Polish mathematician who specialized in mathematical statistics and linear algebra. In 1990 he was appointed professor of mathematical sciences. He authored over 170 academic papers published and won one of the Ministry of National Education awards. == Biography == === Early life and education (1944 – 1988) === Baksalary was born in Poznań, Poland on 25 June 1944. From 1969 to 1988, he worked at the Agricultural University of Poznań. In 1975, Baksalary received a PhD degree from Adam Mickiewicz University in Poznań; his thesis on linear statistical models was supervised by Tadeusz Caliński. He received a Habilitation in 1984, also from Adam Mickiewicz University, where his Habilitationsschrift was also on linear statistical models. === Career (1988 – 2005) === In 1988, Baksalary joined the Tadeusz Kotarbiński Pedagogical University in Zielona Góra, Poland, being the university's rector from 1990 to 1996. In 1990, he became a "Professor of Mathematical Sciences", a title received from the President of Poland. For the 1989–1990 academic year, he moved to the University of Tampere in Finland. Later on, he joined the University of Zielona Góra. === 2005 death and legacy === Baksalary died in Poznań on 8 March 2005. His funeral was held there on 15 March 2005. There, Caliński praised Baksalary for his "contributions to the Poznań school of mathematical statistics and biometry". Memorial events in honor of Baksalary were also held at two conferences after his death: The 14th International Workshop on Matrices and Statistics, held at Massey University in New Zealand from 29 March to 1 April 2005. The Southern Ontario Matrices and Statistics Days, held at the University of Windsor in Canada from 9 to 10 June 2005. == Research == In 1979, Baksalary and Radosław Kala proved that the matrix equation A X − Y B = C {\displaystyle AX-YB=C} has a solution for some matrices X and Y if and only if ( I − A − A ) C ( I − B − B ) = 0 {\displaystyle (I-A^{-}A)C(I-B^{-}B)=0} . (Here, A − {\displaystyle A^{-}} denotes some g-inverse of the matrix A.) This is equivalent to a 1952 result by W. E. Roth on the same equation, which states that the equation has a solution iff the ranks of the block matrices [ A C 0 B ] {\displaystyle {\begin{bmatrix}A&C\\0&B\\\end{bmatrix}}} and [ A 0 0 B ] {\displaystyle {\begin{bmatrix}A&0\\0&B\\\end{bmatrix}}} are equal. In 1980, he and Kala extended this result to the matrix equation A X B + C Y D = E {\displaystyle AXB+CYD=E} , proving that it can be solved if and only if K G K A E = 0 , K A E R D = 0 , K C E R B = 0 , E R B R H = 0 {\displaystyle K_{G}K_{A}E=0,K_{A}ER_{D}=0,K_{C}ER_{B}=0,ER_{B}R_{H}=0} , where G := K A C {\displaystyle G:=K_{A}C} and H := D R B {\displaystyle H:=DR_{B}} .: 146 (Here, the notation K M := I − M M − {\displaystyle K_{M}:=I-MM^{-}} , R M := I − M − M {\displaystyle R_{M}:=I-M^{-}M} is adopted for any matrix M.: 146 ) In 1981, Baksalary and Kala proved that for a Gauss-Markov model { y , X β , V } {\displaystyle \{y,X\beta ,V\}} , where the vector-valued variable has expectation X β {\displaystyle X\beta } and variance V (a dispersion matrix), then for any function F, a best linear unbiased estimator of X β {\displaystyle X\beta } which is a function of F y {\displaystyle Fy} exists iff C ( X ) ⊂ C ( T F ′ ) {\displaystyle C(X)\subset C(TF')} . The condition is equivalent to stating that r ( X ⋮ T F ′ ) = r ( X ) {\displaystyle r(X\vdots TF')=r(X)} , where r ( ⋅ ) {\displaystyle r(\cdot )} denotes the rank of the respective matrix. In 1995, Baksalary and Sujit Kumar Mitra introduced the "left-star" and "right-star" partial orderings on the set of complex matrices, which are defined as follows. The matrix A is below the matrix B in the left-star ordering, written A ∗ < B {\displaystyle A~*<B} , iff A ∗ A = A ∗ B {\displaystyle A^{*}A=A^{*}B} and M ( A ) ⊆ M ( B ) {\displaystyle {\mathcal {M}}(A)\subseteq {\mathcal {M}}(B)} , where M ( ⋅ ) {\displaystyle {\mathcal {M}}(\cdot )} denotes the column span and A ∗ {\displaystyle A^{*}} denotes the conjugate transpose.: 76 Similarly, A is below B in the right-star ordering, written A < ∗ B {\displaystyle A<*~B} , iff A A ∗ = B A ∗ {\displaystyle AA^{*}=BA^{*}} and M ( A ∗ ) ⊆ M ( B ∗ ) {\displaystyle {\mathcal {M}}(A^{*})\subseteq {\mathcal {M}}(B^{*})} .: 76 In 2000, Jerzy Baksalary and Oskar Maria Baksalary characterized all situations when a linear combination P = c 1 P 1 + c 2 P 2 {\displaystyle P=c_{1}P_{1}+c_{2}P_{2}} of two idempotent matrices can itself be idempotent. These include three previously known cases P = P 1 + P 2 {\displaystyle P=P_{1}+P_{2}} , P = P 1 − P 2 {\displaystyle P=P_{1}-P_{2}} , or P = P 2 − P 1 {\displaystyle P=P_{2}-P_{1}} , previously found by Rao and Mitra (1971); and one additional case where c 2 = 1 − c 1 {\displaystyle c_{2}=1-c_{1}} and ( P 1 − P 2 ) 2 = 0 {\displaystyle (P_{1}-P_{2})^{2}=0} . == References ==
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Wikipedia:Jerzy Neyman#0
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Jerzy Spława-Neyman (April 16, 1894 – August 5, 1981; Polish: [ˈjɛʐɨ ˈspwava ˈnɛjman]) was a Polish mathematician and statistician who first introduced the modern concept of a confidence interval into statistical hypothesis testing and, with Egon Pearson, revised Ronald Fisher's null hypothesis testing. Spława-Neyman spent the first part of his professional career at various institutions in Warsaw, Poland, and then at University College London; and the second part, at the University of California, Berkeley. == Life and career == He was born into a Polish family in Bendery, in the Bessarabia Governorate of the Russian Empire, the fourth of four children of Czesław Spława-Neyman and Kazimiera Lutosławska. His family was Roman Catholic, and Neyman served as an altar boy during his early childhood. Later, Neyman would become an agnostic. Neyman's family descended from a long line of Polish nobles and military heroes. He graduated from the Kamieniec Podolski gubernial gymnasium for boys in 1909 under the name Yuri Cheslavovich Neyman. He began studies at Kharkiv University in 1912, where he was taught by Ukrainian probabilist Sergei Natanovich Bernstein. After he read 'Lessons on the integration and the research of the primitive functions' by Henri Lebesgue, he was fascinated with measure and integration. In 1921 he returned to Poland in a program of repatriation of POWs after the Polish-Soviet War. He earned his Doctor of Philosophy degree at University of Warsaw in 1924 for a dissertation titled "On the Applications of the Theory of Probability to Agricultural Experiments". He was examined by Wacław Sierpiński and Stefan Mazurkiewicz, among others. He spent a couple of years in London and Paris on a fellowship to study statistics with Karl Pearson and Émile Borel. After his return to Poland, he established the Biometric Laboratory at the Nencki Institute of Experimental Biology in Warsaw. He published many books dealing with experiments and statistics, and devised the way which the FDA tests medicines today. Neyman proposed and studied randomized experiments in 1923. Furthermore, his paper "On the Two Different Aspects of the Representative Method: The Method of Stratified Sampling and the Method of Purposive Selection", given at the Royal Statistical Society on 19 June 1934, was the groundbreaking event leading to modern scientific sampling. He introduced the confidence interval in his paper in 1937. Another noted contribution is the Neyman–Pearson lemma, the basis of hypothesis testing. He was an Invited Speaker of the ICM in 1928 in Bologna and a Plenary Speaker of the ICM in 1954 in Amsterdam. In 1938 he moved to Berkeley, where he worked for the rest of his life. Thirty-nine students received their Ph.Ds under his advisorship. In 1966 he was awarded the Guy Medal of the Royal Statistical Society and, three years later, the U.S. National Medal of Science. He died in Oakland, California, in 1981. George Dantzig solved two "unsolvable" problems in one's of Neyman's classes. == See also == List of Poles == References == === Citations === === Sources === == External links == O'Connor, John J.; Robertson, Edmund F., "Jerzy Neyman", MacTutor History of Mathematics Archive, University of St Andrews ASA biographical article by Chin Long Chiang Jerzy Neyman — Biographical Memoirs of the National Academy of Sciences Biography of Jerzy Neyman from the Institute for Operations Research and the Management Sciences
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Wikipedia:Jerzy Słupecki#0
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Jerzy Słupecki (1904–1987) was a Polish mathematician and logician. == Life == He attended the seminar of, and wrote a 1938 doctorate under, Jan Łukasiewicz. During WWII he was active in Żegota. In 1963, when at Wroclaw University, where he had been since 1945, he became editor of Studia Logica. == Works == Słupecki showed how the many-valued logics of Łukasiewicz could be included in the theory of Post systems, and gave a functionally complete version of the three-valued logic. In the logic of categorical sentences, he found a rule that made the theory decidable; his work on Aristotle's logic, from 1948, was later reprinted in French. He also continued the work of Stanisław Leśniewski, and wrote on his system ("protothetics") in 1953, in Studia Logica. A survey, "The Logical Works of Jerzy Slupecki", appeared in Studia Logica XLVIII (1989), by Jan Woleński and Jan Zygmunt. He published: Z zagadnień logiki i filozofii: pisma wybrane (1961, editor), selected works of Jan Łukasiewicz Elements of Mathematical Logic and Set Theory (1967) == Notes == == External links == Selection from Jan Woleński and Jan Zygmunt, "Jerzy Słupecki (1904–1987): Life and Work", Studia Logica 48 (1989), 401–411
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Wikipedia:Jesper Møller (mathematician)#0
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Jesper Møller (also written Moller or Moeller; born December 6, 1957) is a Danish mathematician. He is a professor at the Department of Mathematical Sciences, Aalborg University, where he conducts research and teaching in mathematical statistics and probability theory. In 2012, he created the Bachelor and Master's Programmes in Mathematics and Technology at Aalborg University. He was born in Nakskov, Denmark. == Scientific career == Jesper Møller is M.Sc. (cand.scient.) in mathematics and mathematical statistics (Aarhus University), 1984; Ph.D. in mathematical statistics (Aarhus University), 1988; and D.Sc. (dr.scient.; title of dissertation: `Aspects of Spatial Statistics, Stochastic Geometry and Markov Chain Monte Carlo´; Aalborg University), 2000. He was Assistant Professor and Associate Professor at Aarhus University, 1988–1995, Associate Professor at Aalborg University, 1996–2000, and Professor at Aalborg University since 2001. == Research == Jesper Møller's research interests include spatial statistics (in particular statistical models and methods for spatial point processes), stochastic geometry (in particular random tessellations), stochastic simulation (Markov-chain Monte Carlo methods and perfect simulation Archived 2017-06-13 at the Wayback Machine). His research has been supported by project grants from Danish National Research Foundation, Independent Research Fund Denmark, EU, Velux Foundation, and Spar Nord Foundation. He has been Associated Editor of Annals of Applied Probability, Advances in Applied Probability, Bernoulli, and Associate Editor and National Editor of the Scandinavian Journal of Statistics. == Distinctions == Elected Fellow of the Institute of Mathematical Statistics, 1998. Spar Nord Foundation Research Price (DKK 250,000), 2000. Presentation of a read-paper at a Royal Statistical Society Meeting in London, 2005. Plenary Talk of 3 hours at the 21st Nordic Conference on Mathematical Statistics, presenting an invited paper for the Scandinavian Journal of Statistics. Rising Star in the field of Mathematics (Thomson Reuters), 2008. Knight of the Order of Dannebrog, 2015. Professorem Hospitem, Charles University, Prague, since 2020. == Selected publications == Some of his publications are given as follows: J. Møller (1989). Random tessellations in R^d. Advances in Applied Probability, 21:37–73. A. Baddeley and J. Møller (1989). Nearest-neighbour Markov point processes and random sets. International Statistical Review, 2:89–121. J.L. Jensen and J. Møller (1991). Pseudolikelihood for exponential family models of spatial point processes. Annals of Applied Probability, 3:445-461. J. Møller (1994). Lectures on Random Voronoi Tessellations. Lecture Notes in Statistics 87, Springer-Verlag, New York. C.J. Geyer and J. Møller (1994). Simulation procedures and likelihood inference for spatial point processes. Scandinavian Journal of Statistics, 21:359–373. J. Møller, A.R. Syversveen and R.P. Waagepetersen (1998). Log Gaussian Cox processes. Scandinavian Journal of Statistics, 25:451–482. W.S. Kendall and J. Møller (2000). Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes. Advances in Applied Probability, 32:844–865. J. Møller (2003). Shot noise Cox processes. Advances in Applied Probability, 35, 614–640. J. Møller and R.P. Waagepetersen (2004). Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC, Boca Raton. A. Baddeley, R. Turner, J. Møller and M. Hazelton (2005). Residual analysis for spatial point processes (with discussion). Journal of Royal Statistical Society: Series B (Statistical Methodology), 67, 617–666. J. Møller, A.N. Pettitt, K.K. Berthelsen and R.W. Reeves (2006). An efficient Markov chain Monte Carlo method for distributions with intractable normalising constants. Biometrika, 93, 451–458. J. Møller and R.P. Waagepetersen (2007). Modern statistics for spatial point processes (with discussion). Scandinavian Journal of Statistics, 34, 643–711. A. Baddeley, E. Rubak and J. Møller (2011). Score, pseudo-score and residual diagnostics for goodness-of-fit of spatial point process models. Statistical Science, 26, 613–646. F. Lavancier, J. Møller and E. Rubak (2015). Determinantal point process models and statistical inference. Journal of Royal Statistical Society: Series B (Statistical Methodology), 77, 853–877. E. Anderes, J. Møller and J.G. Rasmussen (2020). Isotropic covariance functions on graphs and their edges. To appear in Annals of Statistics. Available at arXiv:1710.01295. == References ==
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Wikipedia:Jessica Purcell#0
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Jessica A. Shepherd Purcell is an American mathematician specializing in low-dimensional topology whose research topics have included hyperbolic Dehn surgery and the Jones polynomial. She is a professor of mathematics at Monash University in Melbourne, Australia. == Education == Purcell credits a high school mathematics teacher, Mr. Pehrson, for her interest in mathematics. She majored in mathematics as an undergraduate at the University of Utah, with a minor in computer science. She was a co-winner of the 1998 Alice T. Schafer Prize for Excellence in Mathematics by an Undergraduate Woman of the Association for Women in Mathematics, and graduated summa cum laude in 1998. After earning a master's degree in mathematics at the University of Michigan in 1999, Purcell completed her Ph.D. at Stanford University in 2004. Her dissertation, Cusp Shapes of Hyperbolic Link Complements and Dehn Filling, was supervised by Steven Kerckhoff. == Career == After postdoctoral research at the University of Texas at Austin and University of Oxford, Purcell became an assistant professor at Brigham Young University in 2007. She was named a Sloan Research Fellow in 2011, and earned tenure at Brigham Young in 2013. After visits to Monash University and at the Institute for Advanced Study as a Von Neumann Fellow, she moved to Monash as an associate professor in 2015, became an ARC Future Fellow in 2017, and was named full professor in 2019. She chaired the Women in Mathematics Special Interest Group of the Australian Mathematical Society for 2018–2019 and 2019–2020. == Books == Purcell is the author of the book Hyperbolic Knot Theory (Graduate Studies in Mathematics 209, American Mathematical Society, 2020). With David Futer and Efstratia Kalfagianni she is a coauthor of Guts of Surfaces and the Colored Jones Polynomial (Lecture Notes in Mathematics 2069, Springer, 2012). == References == == External links == Home page Jessica Purcell publications indexed by Google Scholar Meet Professor Jessica Purcell, Monash University
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Wikipedia:Jessie Forbes Cameron#0
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Jessie Forbes Cameron (1883 – 1968) was a British mathematician who in 1912 became the first woman to complete her doctorate in mathematics at the University of Marburg in Germany. == Life and work == Jessie Cameron was born on 8 January 1883 in Stanley, Scotland, one of eight children whose parents were James Cameron, a school principal at a village school in Perthshire, and his wife Jessie Forbes. After attending the Perth Academy in Scotland, Jessie Cameron studied for four semesters at University of Edinburgh. From 1905 to 1908, she studied mathematics at Newnham College, which is part of University of Cambridge, in England, and earned a Magister degree (MA). There, she was ranked the tenth best in her class (earning her the distinction "10th Wrangler"), passed the "Mathematical Tripos" and graduated with a Bachelor of Arts (BA). === Postgraduate studies === Cameron moved to the University of Göttingen, in Germany, to take two more semesters of math, and finally, she enrolled at the University of Marburg, for three semesters. Under the supervision of distinguished mathematician Kurt Hensel, Cameron wrote her dissertation On the decomposition of a prime number in a composed body. Before her degree was officially completed, however, there was one additional barrier for her to surmount. It seems she had completed the work and won the approval of her advisor, Dr. Hensel, without realizing a lesser-known caveat for graduation from a German university. According to Lorch-Göllner, she received the following letter from the university's Dean of Faculty of Philosophy on 10 November 1911. "Although your doctoral thesis was judged favorably by the representatives of mathematics, especially by Privy Councilor Hensel, your admission to the rigorous examination is unfortunately subject to legal difficulties, which have to be fixed before I make an appointment. According to the regulations of our faculty's doctoral regulations, admission to a doctorate is dependent on proof that at least six semesters were studied at a university in the German Reich or a foreign university set up in the German way. The universities of Great Britain are not among the latter. "You only studied five semesters at German universities and your ten semesters, which you have spent in Scotland and England, cannot be credited easily ... [you will need] a special dispensation from the Minister of Spiritual and Educational Affairs. ... "I wrote to the Minister for this purpose eight days ago. It is hoped that he will give his approval, and then I would presumably be able to schedule the day of rigorosum before Christmas as you requested it."She soon received the Minister's dispensation and passed her exams in "mathematics, physics and philosophy" on 20 December 1911, with the accolade magna cum laude. Thus, Cameron became the first female to earn a PhD in mathematics at that university and her dissertation was published in 1912. On 28 September 1912 Cameron married the lawyer Edward Vincent Thompson, and she returned to Newnham College for a year beginning in 1912 as an "Assistant Lecturer." With this appointment, according to Lorch-Göllner, she was "the only [female] of the first math students after her doctorate - if even temporarily - to work as a mathematics lecturer in a scientific institution." === Later years === In 1913, the couple moved to London so Edward Thompson could pursue his career with a position at the British Treasury. Jessie Thompson gave birth to a daughter and two sons and continued her association with Newnham College until 1927. During the First World War, the Thompson family moved to Berkhamsted, England, where Jessie began working with the British National Council of Women. Jessie Forbes (Cameron) Thompson died on 27 March 1968 in Southwold, England, at the age of 85. == Published work == Cameron, Jessie Forbes. About the decomposition of a prime number in a composed body. University of Marburg, 1912. == Literature == Cameron, Jessie Forbes. In: Newnham College Register, Vol. 1, 1905, pages 184-185. Francesca M. Wilson: Jessie Forbes Thompson (born Cameron), 1883-1968 (Newnham 1905-1909 and 1912-1913). In: Newnham College Roll Letter, Cambridge 1969, pages 63–64. == References == == External links == Renate Tobies (ed.): Despite all male culture Women in mathematics and natural sciences. Frankfurt am Main / New York 1997, page 137
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Wikipedia:Jesús A. De Loera#0
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Jesús Antonio De Loera (born January 18, 1966) is a Mexican-American mathematician at the University of California, Davis, specializing in discrete mathematics and discrete geometry. == Education == De Loera did his undergraduate studies at the National Autonomous University of Mexico, earning a B.S. in mathematics in 1989. After earning a master's degree from Western Michigan University a year later, and another master's degree in 1993 from Cornell University, he finished his doctorate from Cornell in 1995 under the supervision of Bernd Sturmfels. == Career == De Loera joined the UC Davis faculty as an assistant professor in 1998, and was promoted to full professor in 2005. He served on the scientific advisory board for the Institute for Computational and Experimental Research in Mathematics (ICERM) through 2021. == Awards and honors == In 2014, he became a fellow of the American Mathematical Society "for contributions to discrete geometry and combinatorial optimization as well as for service to the profession, including mentoring and diversity". In 2019 he was named a SIAM Fellow "for contributions to discrete geometry and optimization, polynomial algebra, and mathematical software". In 2020 he received the INFORMS optimization society's Farkas Prize "in recognition of his pioneering work at the intersection of discrete mathematics, optimization and algebraic geometry". He delivered an invited plenary address at the 2021 National Math Festival. In 2021, he was elected vice president of the AMS. == References == == External links == Home page
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Wikipedia:Jesús Ildefonso Díaz#0
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Jesús Ildefonso Díaz is a Spanish mathematician who works in partial differential equations. He is a professor at Complutense University of Madrid (UCM) and a member of the Spanish Royal Academy of Sciences. == Biography == Díaz was born in Toledo, Spain on December 11, 1950. He graduated in mathematics from UCM in 1973, and obtained his PhD from the same university in 1976. His Ph.D. thesis advisors were Alberto Dou and Haïm Brezis. == Career == Díaz joined the faculty at UCM as an Associate Professor in Mathematical Analysis in 1978. He moved briefly to the University of Santander in 1980, before returning to UCM as a full professor in 1983. In 1998, he co-founded the journal Revista Matemática de la UCM and served on its editorial board from 1988 to 1995. He founded the Department of Applied Mathematics at the Facultad de Matemáticas of UCM in the early 1980s and led it for several years. In 2006, he founded the Instituto de Matemática Interdisciplinar (IMI), serving as Director from 2006 to 2008 and again from 2012 to 2016. He is an energetic teacher, organizing six summer courses at UCM, two of them with Jacques Louis Lions. Díaz has worked in many areas of applied mathematics, such as theoretical and applied aspects of nonlinear partial differential equations, fluid mechanics models, geophysical models, reaction-diffusion models, elasticity and homogenization models and control theory models, among others. He has also worked in areas closer to pure mathematics, such as nonlinear analysis, focusing on accretive operators, rearrangement and gradient estimates. Other activities include contributions in science history, science communication and scientific management. His mentors and influential colleagues include Philippe Benilan and Jacques Louis Lions. As of July 2019, his research publications included more than 250 papers in research journals, 141 contributions published in proceedings of meetings, seven books, eight book chapters and 20 edited volumes. His popular science works include 43 articles, two memoirs, two contributions to published proceedings and 17 chapters in books. He has collaborated with over 175 researchers. Since 1979, he has been advisor for 20 Ph.D. dissertations (all at UCM). He has 106 scientific "descendants". == Awards and honors == 2015: Grand Prix Jacques-Louis Lions de Mathematique Apliquée of the Acadèmie des Sciences (France). 1997: Member of the Spanish Royal Academy of Sciences. 1996: Doctorate "Honoris Causa" from the University of Pau and Pays de l'Adour. 1980: Mathematics Prize of the Spanish Royal Academy of Sciences, 1980. == Congresses and scientific academies == Díaz was instrumental in the organization of the Spanish Congress of Differential Equations and Applications, an annual national meeting held from 1978 to 1987. This meeting catalyzed discussions in 1988 on the need for a national society. With Antonio Valle (University of Málaga), Díaz co-founded the Sociedad Española de Matemática Aplicada (SEMA), serving as the society's Secretary from its founding in 1991 to 1994, and as its second President, after Valle, from 1994 to 1995. He was influential in the reformation of the Spanish Royal Society of Mathematics in 1997, after it had been inactive for several years. As a member of the Spanish Royal Academy of Sciences, he worked to modernize the academy's activities. He was the founder of the mathematics section of the Journal of the Spanish Royal Academy of Sciences, Series A Mathematics (RACSAM), serving as editor-in-chief from 2001 to 2005. He encouraged links between the academy and the French Academy of Sciences, co-organizing a joint congress "Les Mathématiques et l'environnement", held in Paris. He is responsible for the fact that the word modelizar was accepted by the Royal Academy of the Spanish Language and included in the Spanish Official Dictionary as the correct Spanish translation for the English word modelling. == International relationships == As a result of his activities to connect Spanish and French mathematicians, he was appointed to visiting positions in many French universities, including the Universities of Toulouse, Nancy, Pau, Metz, Tours, Poitiers, Paris VI and Montpellier. He organized several French-Spanish congresses on various topics. He has also been appointed as an honorary member of the Mathematics Research Center, University of Wisconsin–Madison, USA, and Visiting Professor at the University of Rome II and the University of L'Aquila). He organized the NATO Advanced Study Institute on the Mathematics of Models for Climatology and Environment in 1995. From 2009 to 2013 he was European Coordinator of the European Programme FIRST (Fronts and Interfaces in Science and Technologies), with a €4 million budget and activities in 11 countries. == Editorial boards == From September 2003 to June 2014 he was the only Spanish person on the editorial board of the Journal of the European Mathematical Society. He serves on the editorial boards of several other journals including: Advances in Mathematical Science and Applications (since 1991) Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales de Madrid (since 1996) Electronic Differential Equations Journal (since 2001). Differential and Integral Equations (since 2001). Electronic Journal of Mathematical and Physical Sciences (since 2002). Journal of the European Mathematical Society (since 2003). The Annals of the University of Craiova - Mathematics and Computer Science (since 2006). Nonlinear Analysis: Real World Applications (since 2010). Differential Equations and Applications (DEA) (since 2011). Abstract and Applied Analysis (since July 2012). == 60th birthday celebration == An International Congress was organized to celebrate Díaz's 60th birthday, entitled "Non-linear models in partial differential equations: A conference on the occasion of Jesús Ildefonso Díaz's 60th birthday". The meeting took place on June, 14-17, 2011, in the Palacio de Lorenzana, (UCLM) in Toledo, Spain (Ildefonso's birthplace). The members of the Scientific Committee were Professors Haïm Brezis (President), H. Ammann, S.N. Antontsev, A. Friedman, R. Glowinski, A. Liñán, E. Sánchez-Palencia, and R. Temam. Two special issues of the journal Differential Equations and Applications were dedicated to the meeting proceedings (Differential Equations and Applications Vol. 3, Issue 4 (2011) and Vol. 4 Issue 1 (2012)). ISSN 1847-120X. == References == == External links == [1] Jesús Ildefonso Díaz's website.
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Wikipedia:Jet group#0
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In mathematics, a jet group is a generalization of the general linear group which applies to Taylor polynomials instead of vectors at a point. A jet group is a group of jets that describes how a Taylor polynomial transforms under changes of coordinate systems (or, equivalently, diffeomorphisms). == Overview == The k-th order jet group Gnk consists of jets of smooth diffeomorphisms φ: Rn → Rn such that φ(0)=0. The following is a more precise definition of the jet group. Let k ≥ 2. The differential of a function f: Rk → R can be interpreted as a section of the cotangent bundle of RK given by df: Rk → T*Rk. Similarly, derivatives of order up to m are sections of the jet bundle Jm(Rk) = Rk × W, where W = R × ( R ∗ ) k × S 2 ( ( R ∗ ) k ) × ⋯ × S m ( ( R ∗ ) k ) . {\displaystyle W=\mathbf {R} \times (\mathbf {R} ^{*})^{k}\times S^{2}((\mathbf {R} ^{*})^{k})\times \cdots \times S^{m}((\mathbf {R} ^{*})^{k}).} Here R* is the dual vector space to R, and Si denotes the i-th symmetric power. A smooth function f: Rk → R has a prolongation jmf: Rk → Jm(Rk) defined at each point p ∈ Rk by placing the i-th partials of f at p in the Si((R*)k) component of W. Consider a point p = ( x , x ′ ) ∈ J m ( R n ) {\displaystyle p=(x,x')\in J^{m}(\mathbf {R} ^{n})} . There is a unique polynomial fp in k variables and of order m such that p is in the image of jmfp. That is, j k ( f p ) ( x ) = x ′ {\displaystyle j^{k}(f_{p})(x)=x'} . The differential data x′ may be transferred to lie over another point y ∈ Rn as jmfp(y) , the partials of fp over y. Provide Jm(Rn) with a group structure by taking ( x , x ′ ) ∗ ( y , y ′ ) = ( x + y , j m f p ( y ) + y ′ ) {\displaystyle (x,x')*(y,y')=(x+y,j^{m}f_{p}(y)+y')} With this group structure, Jm(Rn) is a Carnot group of class m + 1. Because of the properties of jets under function composition, Gnk is a Lie group. The jet group is a semidirect product of the general linear group and a connected, simply connected nilpotent Lie group. It is also in fact an algebraic group, since the composition involves only polynomial operations. == Notes == == References == Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operations in differential geometry (PDF), Springer-Verlag, archived from the original (PDF) on 2017-03-30, retrieved 2014-05-02 Krupka, Demeter; Janyška, Josef (1990), Lectures on differential invariants, Univerzita J. E. Purkyně V Brně, ISBN 80-210-0165-8 Saunders, D.J. (1989), The geometry of jet bundles, Cambridge University Press, ISBN 0-521-36948-7
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Wikipedia:Jianhong Wu#0
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Jianhong Wu (吴建宏; born in 1964) is a Canadian applied mathematician and the founding director of the Laboratory for Industrial and Applied Mathematics at York University. He is also the inaugural Director of Y-EMERGE [1]. == Education == Wu received his PhD degree in 1987 at age 23. He was the first G. Kaplan Award Postdoctoral Fellow at University of Alberta during 1988–90, and joined York University in 1990. == Career == He holds the life-time title of University Distinguished Research Professor, and was awarded a senior Canada Research Chair in Industrial and Applied Mathematics at York University between 2001-2022. He was awarded the Senior York Research Chair in Industrial and Applied Mathematics in 2022. He was also awarded the NSERC/Sanofi Industrial Research Chair in Vaccine Mathematics, Modelling and Manufacturing in 2017-2022. In 2008, Dr. Wu founded the Centre for Disease Modelling, and in 2016, he became the founding scientific director of The Disaster, Emergency and Rapid Response Simulation initiative (ADERSIM). He is an editor-in-chief of the journal Infectious Disease Modelling. == Awards and recognition == He was awarded by the Alexander von Humboldt Foundation an Alexander van Humboldt Fellow at Justus-Liebig-Universitat Giessen, a Paul Erdos Visiting Professor at the Bolyai Institute, a FAPESP Visiting Research Fellow at Universidade de Sao Paulo. He is a Fellow of the Fields Institute, a Fellow of the Canadian Academy of Health Sciences, and a Fellow of the Royal Society of Canada. Professor Wu is the first recipient of the Canadian Applied and Industrial Mathematical Society's Research Prize, for his "very significant contributions in the area of infinite dimensional differential equations with applications to neural networks and population dynamics". He also received the 2019 CAIMS-Fields Industrial Mathematics Prize "in recognition of their many contributions to dynamical systems in mathematical epidemiology and in particular, their collaborative research with public health professionals in government and industry: applying their expert knowledge to infectious disease mitigation strategies and preparedness.". He received the Queen Elizabeth II Diamond Jubilee Medal from the Government of Canada in 2012. He was awarded an honorary doctorate by the University of Szeged in 2016. He is recognized for his expertise and contribution in the following fields: nonlinear dynamics and delay differential equations; neural networks and pattern recognition; mathematical ecology and epidemiology. He was awarded the 2024-NSERC Synergy Award for Innovation for his contribution to "creating a research and development ecosystem at the forefront of efforts to revolutionize drug and vaccine development and distribution". He was elected as a Fellow of the American Mathematical Society, in the 2025 class of fellows. == Notable projects == During the COVID-19 pandemic, he was asked by the Fields Institute to establish and lead The National COVID-19 Modelling Rapid Response Task Force; and he is appointed to the following provincial and federal committees: The Province of Ontario COVID-19 Modelling Consensus Table; The PHAC's External Modelling Expert Group of the Public Health Agency of Canada; The Canada's Expert Panel on COVID-19 Subcommittee on Modelling chaired by the Chief Science Advisor of Canada; and the COVID-19 Vaccine Modelling Task Group, National Advisory Committee on Immunization. In these capacities, he has been a major player to provide the mathematical modelling and analyses informing public health decision making. He received the Award of Excellence from the Ontario Ministry of Colleagues and Universities in September 2020. Wu co-leads The Mathematics for Public Health (MfPH), along with the Fields Director, V. Kumar Murty. Dr. Wu led several large-scale interdisciplinary projects including: Transmission Dynamics and Spatial Spread of Infectious Diseases: Modelling, Prediction and Control (funded by the Canadian Network of Centers of Excellence/NCE Mitacs/Mprime); Geo-simulation tools for Simulating Spatial-temporal Spread Patterns and Evaluating Health Outcomes of Communicable Diseases (funded by the Canadian Network of Centers of Excellence/NCE Geomatics). Both projects developed a strong academic-public-industry partnership to address key public health issues relevant to emerging infectious diseases including SARS, pandemic influenza, Ebola, Lyme disease and West Nile virus. He also established and co-led the project: Development of an Antimicrobial Resistance Diversity Index (ARDI) to Guide Initiatives and Investment in Public Health, Antimicrobial Stewardship and Infection Control (funded by the Collaborative Health Research Program, a joint effort between the Canadian Institute of Health Research and Natural Sciences and Engineering Research Council of Canada. == References ==
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Wikipedia:Jigu Suanjing#0
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Jigu suanjing (Chinese: 緝古算經, Continuation of Ancient Mathematics) was the work of early Tang dynasty calendarist and mathematician Wang Xiaotong, written some time before the year 626, when he presented his work to the Emperor. Jigu Suanjing was included as one of the requisite texts for Imperial examination; the amount of time required for the study of Jigu Suanjing was three years, the same as for The Nine Chapters on the Mathematical Art and Haidao Suanjing. The book began with presentations to the Emperor, followed by a pursuit problem similar to the one in Jiu Zhang Suan shu, followed by thirteen three-dimensional geometry problems based mostly on engineering construction of astronomic observation tower, dike, barn, excavation of a canal bed etc., and six problems in right angled triangle plane geometry. Apart from the first problem which was solved by arithmetic, the problems deal with the solution of cubic equations, the first known Chinese work to deal with complete cubic equations, as such, it played important roles in the development for solution of high order polynomial equations in the history of Chinese mathematics. Before his time, The Nine Chapters on the Mathematical Art developed algorithm of solving simple cubic equation x 3 = N {\displaystyle x^{3}=N} numerically, often referred to as the "finding the root method". Wang Xiaotong used an algebraic method to solve three-dimensional geometry problems, and his work is a major advance in Algebra in the history of Chinese mathematics. Each problem in Jigu Suanjing follows the same format, the question part begins with "suppose we have such and such,... question:...how many are there?"; followed by "answer:", with concrete numbers; then followed by "The algorithm says:...", in which Wang Xiaotong detailed the reasoning and procedure for the construction of equations, with a terse description of the method of solution. The emphasis of the book is on how to solve engineering problems by construction of mathematical equations from geometric properties of the relevant problem. In Jigu Suanjin, Wang established and solved 25 cubic equations, 23 of them from problem 2 to problem 18 have the form x 3 + p x 2 + q x = N , {\displaystyle x^{3}+px^{2}+qx=N,\,} The remaining two problems 19, and 20 each has a double quadratic equation: x 4 + p x 2 + q = 0 {\displaystyle x^{4}+px^{2}+q=0} Problem 3, two cubic equations: x 3 + 3 c d b − c x 2 + 3 ( a + c ) h d 2 ( H − h ) ( b − c ) x = 6 V d 2 ( H − h ) ( b − c ) {\displaystyle x^{3}+{\frac {3cd}{b-c}}x^{2}+{\frac {3(a+c)hd^{2}}{(H-h)(b-c)}}x={\frac {6Vd^{2}}{(H-h)(b-c)}}} x 3 + 5004 x 2 + 1169953 1 3 x = 41107188 1 3 {\displaystyle x^{3}+5004x^{2}+1169953{\frac {1}{3}}x=41107188{\frac {1}{3}}} ； Problem 4 two cubic equations: x 3 + 62 x 2 + 696 x = 38448 , x = 18 ; {\displaystyle x^{3}+62x^{2}+696x=38448,\quad x=18;} x 3 + 594 x 2 = 682803 , x = 33 ; {\displaystyle x^{3}+594x^{2}=682803,\quad x=33;} Problem 5 x 3 + 15 x 2 + 66 x − 360 , x = 3 {\displaystyle x^{3}+15x^{2}+66x-360,\quad x=3} Problem 7: x 3 + ( D + G ) x 2 + ( D G + D 2 3 ) x = P − D 2 G 3 {\displaystyle x^{3}+(D+G)x^{2}+\left(DG+{\frac {D^{2}}{3}}\right)x=P-{\frac {D^{2}G}{3}}} X + 3 h s D x 2 + 3 ( h s D ) 2 x = P ′ 3 h 2 D 2 {\displaystyle X^{+}3{\frac {hs}{D}}x^{2}+3\left({\frac {hs}{D}}\right)^{2}x={\frac {P'}{3}}{\frac {h^{2}}{D^{2}}}} Problem 8: x 3 + 90 x 2 − 839808 , x = 72 {\displaystyle x^{3}+90x^{2}-839808,\quad x=72} Problem 15: x 3 + S 2 x 2 − P 2 2 S = 0 {\displaystyle x^{3}+{\frac {S}{2}}x^{2}-{\frac {P^{2}}{2S}}=0} 。 Problem 17: x 3 + 5 2 D x 2 + 2 D 2 x = P 2 2 D − D 2 2 {\displaystyle x^{3}+{\frac {5}{2}}Dx^{2}+2D^{2}x={\frac {P^{2}}{2D}}-{\frac {D^{2}}{2}}} Problem 20:"Suppose the long side of a right angle triangle equals to sixteen and a half, the square of the product of the short side and the hypothenuse equals to one hundred sixty four and 14 parts of 25, question, what is the length of the short side ?" Answer: "the length of the short side is eight and four fifth." Algorithm:"Let the square of the square of product as 'shi' (the constant term), and let the square of the long side of right angle triangle be the 'fa' (the coefficient of the y term). Solve by 'finding the root method', then find the square root again." The algorithm is about setting up a double quadratic equation: x 4 + ( 16 1 2 ) 2 x 2 = ( 164 14 15 ) 2 {\displaystyle x^{4}+\left(16{\frac {1}{2}}\right)^{2}x^{2}=\left(164{\frac {14}{15}}\right)^{2}} 。 where x is the short side. Wang's work influence later Chinese mathematicians, like Jia Xian and Qin Jiushao of Song dynasty. == Editions == During the Tang dynasty there were hand-copied Jigu Suanjing in circulation. During the Song dynasty there were 1084 government-printed edition copies. However, by the Ming dynasty the Tang dynasty hand-copied editions and Song dynasty printed editions were almost all lost; only a single copy of a Southern Song print survived. This copy was later obtained by early Qing dynasty publisher Mao Jin, who made an image hand copy (hand-copied character by character, following the printed form closely) of it. Mao Jin's image copy of Jigu Suanjing later became the source for a printed edition during the Qianlong era and was also incorporated into the Siku Quanshu. The Qianlong era printed edition disappeared, and only Mao Jin's image copy edition of Jigu Suanjing survived at the Forbidden City Museum. The copy in the Siku Quanshu still exists. During the Qing dynasty, study of Jigu Suangjing was in vogue; half a dozen books devoted to the study of Jigu Suanjing by mathematicians were published, some of which concentrated on filling the gaps left by many missing characters due to age, and some devoted to the detail elaboration of algorithm either from geometry point of view (Li Huang) or from Tian yuan shu (Zhang Dunren). In 1963, Chinese mathematics historian Qian Baocong published his annotated The Ten Computational Canons, which included Jigu Suanjing. Jigu Suanjing was introduced to the English speaking world by Alexander Wylie in his book Notes on Chinese Literature. == References ==
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Wikipedia:Jill Adler#0
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Jillian Beryl Adler née Smidt (born 31 January 1951 in Johannesburg) is a South African Professor of Mathematics education at the University of the Witwatersrand and the President of the International Commission on Mathematical Instruction (2017–2020). Adler's work has focused on the teaching and learning of mathematics particularly in multilingual classrooms. == Career and impact == Jill Adler was born in Johannesburg, South Africa. She obtained her undergraduate and postgraduate degrees from the University of the Witwatersrand. Currently, she serves as the Chair of Mathematics Education at Wits University and as the President of the International Commission on Mathematical Instruction (2017–2020). Adler's work has focused on the teaching and learning of mathematics particularly in multilingual classrooms as well as the professional development of secondary school maths teachers. She has an A-rating, the highest possible, from the National Research Foundation of South Africa. From 2009 to 2014 Professor Adler was a visiting professor at King's College London. She is a member of the Academy of Science of South Africa. == Awards and honors == ICMI Hans Freudenthal Medal (2015) == References == == External links == O'Connor, John J.; Robertson, Edmund F., "Jill Adler", MacTutor History of Mathematics Archive, University of St Andrews Publications by Jill Adler at ResearchGate
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Wikipedia:Jill Britton#0
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Jill E. Britton (6 November 1944 – 29 February 2016) was a Canadian mathematics educator known for her educational books about mathematics. == Career == Britton was born on 6 November 1944. She taught for many years, at Dawson College in Westmount, Quebec, moving in the late 1980s to Camosun College in Victoria, British Columbia. At Camosun, she taught mathematics mainly to young women aiming to become elementary school teachers. Her own teaching had the goal of changing these students' attitude about mathematics from fear to enthusiasm. She retired in 2015, and died on 29 February 2016. == Publications == Britton was the author of mathematics books including: Introduction to Tessellations (with Dale Seymour, Dale Seymour Publications, 1989) Tessellation Teaching Masters (with Dale Seymour, Dale Seymour Publications, 1989) Teaching Tessellating Art: Activities and Transparency Masters (with Walter Britton, Dale Seymour Publications, 1992) Explorations with Tesselmania! Activities for Math and Art Classrooms (Dale Seymour Publications, 1997) Investigating Patterns: Symmetry and Tessellations (Dale Seymour Publications, 2000) Polyhedra Pastimes (Dale Seymour Publications, 2001) She was also known for her "spectacular web site on symmetry and tessellations". == Recognition == Britton was the 2008–2009 winner of the Teaching Excellence Award of the Association of Canadian Community Colleges. == References == == External links == Archived home page Britton's YouTube channel
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Wikipedia:Jin Akiyama#0
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Jin Akiyama (Japanese: 秋山仁; born 1946) is a Japanese mathematician, known for his appearances on Japanese prime-time television (NHK) presenting magic tricks with mathematical explanations. He is director of the Mathematical Education Research Center at the Tokyo University of Science, and professor emeritus at Tokai University. Akiyama studied mathematics at the Tokyo University of Science, where one of his mentors was Takashi Hamada. He completed a graduate degree at Sophia University under the supervision of Mitio Nagumo, in differential equations, but soon shifted his interests to graph theory. He planned to take a position in Ghana, but after conflict there caused it to be cancelled he joined the faculty at Nippon Ika University, and then moved to the U.S. for 1978 and 1979 to work with Frank Harary at the University of Michigan. In the 1990s, his interests shifted again, from graph theory to discrete geometry. Akiyama is a founder of the Japan Conference on Discrete and Computational Geometry, Graphs, and Games (JCDCG3), the founding managing editor of Graphs and Combinatorics, and the author of the books A Day's Adventure in Math Wonderland (with Mari-Jo Ruiz, World Scientific, 2008), Factors and Factorizations of Graphs (with Mikio Kano, Lecture Notes in Mathematics 2031, Springer, 2011), and Treks Into Intuitive Geometry: The World of Polygons and Polyhedra (with Kiyoko Matsunaga, Springer, 2015). He is also the namesake of a Nintendo DS game, Master Jin Jin's IQ Challenge. Akiyama's lectures sometimes also include musical performances by him, on accordion or xylophone. == References == == External links == Home page
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Wikipedia:Jinde Cao#0
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Jinde Cao is an Endowed Chair Professor at Southeast University, Nanjing, China. He is a Distinguished Professor, the Dean of School of Mathematics and the Director of the Research Center for Complex Systems and Network Sciences at Southeast University. == Education and career == Cao obtained his B.S. in mathematics from Anhui Normal University in 1986. He then studied applied mathematics at Yunnan University, graduating from it with M.S. in 1989 and in 1998 got his PhD in the same field from Sichuan University. From 1996 to 2000, he was a professor at Yunnan University, and from July 2001 to June 2002 served as postdoc at the Department of Automation and Computer-Aided Engineering Faculty of Engineering of the Chinese University of Hong Kong. Between July 2006 and August 2008, Cao was a Royal Society Research Fellow at Brunel University in the United Kingdom and in 2014 became visiting professor at the RMIT Universityin Australia. == Awards and recognitions == Cao was named a Foreign Fellow of the Pakistan Academy of Sciences in 2016, and the same year was awarded IEEE Fellowship for contributions to the analysis of neural networks. The same year, he also became a foreign member of the Academia Europaea and in 2018 became a member of the European Academy of Sciences and Arts. In 2019 was awarded with the Obada Prize. == References == == External links == Jinde Cao publications indexed by Google Scholar
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Wikipedia:Jinkōki#0
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Jinkōki (塵劫記, じんこうき, Permanent Mathematics) is a three-volume work on Japanese mathematics, first edited and published by Yoshida Mitsuyoshi in 1627. Over his lifetime, Mitsuyoshi revised Jinkōki several times. The edition released in the eleventh year of the Kan'ei era (1641) became particularly widespread. The last version personally published by Mitsuyoshi was the Idai (井大), which came out in the eighteenth year of the Kan'ei (1634). Subsequent to that, various editions of Jinkōki were released, one of which includes Shinpen Jinkōki (新編塵劫記). Jinkōki is one of the most popular and influential Japanese mathematics books in history, having influenced Seki Takakazu, Kaibara Ekken, and many other later Japanese mathematicians. It is partly based on the works of Yuan dynasty mathematicians in China. == Etymology == The name Jinkōki is derived from jintenkō (塵点劫), an immeasurably long span of time mentioned in the Lotus Sutra, hence the nuance of permanence in the Sino-Japanese word jinkō and its reflection in English title. == Content == The book contained instructions for dividing and multiplying with a soroban and mathematical problems relevant to merchants and craftsmen. The book also contained several interesting mathematical problems, and was the first Japanese book to use printing in colour. As a result, the Jinkōki became the most popular Japanese mathematics book ever and one of the most widely read books of the Edo period. Mitsuyoshi made reference to everyday problems, such as buying and selling rice. The book was originally published in three volumes, the first of which mainly describes multiplication and division using the soroban. The second and third volumes include an assortment of practical and recreational problems. The included problems are not arranged according to any specific order. The book includes ideas that aimed to keep readers from boredom by adopting a wide variety of problems such as calculations of areas of rice fields, problems related to the construction of rivers and riverbanks, geometric progression, and the Josephus problem. The Shinpen Jinkōki was the most widespread version among the copies of Jinkōki, and widely used as a textbook for use of the soroban throughout the Edo period. In addition to fundamental knowledge such as numerical notation, units, and multiplication tables, it also included slightly more specialised topics, such as methods to find square roots and cube roots, practical calculations of area, currency conversion, and interest calculation. The content covers almost all arithmetic needed in daily life at that time, and it is a characteristic of the book that explanations are given using familiar topics, such as using the increase of mice as an example for geometric progression. Many different versions were published, with slightly modified content, and by the time of the Meiji era, over 400 editions of Jinkōki had been published. == Gallery == == See also == Japanese mathematics == References ==
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Wikipedia:Jiří Matoušek (mathematician)#0
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Jiří (Jirka) Matoušek (10 March 1963 – 9 March 2015) was a Czech mathematician working in computational geometry and algebraic topology. He was a professor at Charles University in Prague and the author of several textbooks and research monographs. == Biography == Matoušek was born in Prague. In 1986, he received his Master's degree at Charles University under Miroslav Katětov. From 1986 until his death he was employed at the Department of Applied Mathematics of Charles University, holding a professor position since 2000. He was also a visiting and later full professor at ETH Zurich. In 1996, he won the European Mathematical Society prize and in 2000 he won the Scientist award of the Learned Society of the Czech Republic. In 1998 he was an Invited Speaker of the International Congress of Mathematicians in Berlin. He became a fellow of the Learned Society of the Czech Republic in 2005. Matoušek's paper on computational aspects of algebraic topology won the Best Paper award at the 2012 ACM Symposium on Discrete Algorithms. Aside from his own academic writing, he has translated the popularization book Mathematics: A Very Short Introduction by Timothy Gowers into Czech. He was a supporter and signatory of the Cost of Knowledge protest. Matoušek died in 2015, aged 51. In 2021, a lecture hall at the Faculty of Mathematics and Physics, Charles University, was named after him. == Books == Invitation to Discrete Mathematics (with Jaroslav Nešetřil). Oxford University Press, 1998. ISBN 978-0-19-850207-4. Translated into French by Delphine Hachez as Introduction aux Mathématiques Discrètes, Springer-Verlag, 2004, ISBN 978-2-287-20010-6. Geometric Discrepancy: An Illustrated Guide. Springer-Verlag, Algorithms and Combinatorics 18, 1999, ISBN 978-3-540-65528-2. Lectures on Discrete Geometry. Springer-Verlag, Graduate Texts in Mathematics, 2002, ISBN 978-0-387-95373-1. Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry. Springer-Verlag, 2003. ISBN 978-3-540-00362-5. Topics in Discrete Mathematics: Dedicated to Jarik Nešetřil on the Occasion of His 60th Birthday (with Martin Klazar, Jan Kratochvíl, Martin Loebl, Robin Thomas, and Pavel Valtr). Springer-Verlag, Algorithms and Combinatorics 26, 2006. ISBN 978-3-540-33698-3. Understanding and Using Linear Programming (with B. Gärtner). Springer-Verlag, Universitext, 2007, ISBN 978-3-540-30697-9. Thirty-three miniatures — Mathematical and algorithmic applications of linear algebra. American Mathematical Society, 2010, ISBN 978-0-8218-4977-4. Approximation Algorithms and Semidefinite Programming (with B. Gärtner). Springer Berlin Heidelberg, 2012, ISBN 978-3-642-22014-2. Mathematics++: Selected Topics Beyond the Basic Courses (with Ida Kantor and Robert Šámal). American Mathematical Society, 2015, ISBN 978-1-4704-2261-5. == See also == Ham sandwich theorem Discrepancy theory Kneser graph == References == == External links == Jiri Matousek home page Jiří Matoušek publications indexed by Microsoft Academic
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Wikipedia:Jiří Rosický (mathematician)#0
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Jiří Rosický (born 1946) is a Czech mathematician. He works on the field of category theory. He is full professor at Masaryk University of Mathematics in the group of Algebra, Topology and Number Theory, which he led for several years. == Life == Jiří Rosický was born in 1946. In 1963–1968, he studied mathematics at the Faculty of Science of the Masaryk University. In 1969, he started to work in the department of algebra and geometry at the Faculty of Science. In 1979, he became head of the department. == Work == His work is in category theory, model theory, abstract homotopy theory, and general algebra. In 1980s he started working on interactions between model theory and category theory. This brought his interest to accessible and locally presentable categories. Along with Jiří Adámek he has written a book on the theory of locally presentable and accessible categories, which quickly became the most cited book on the subject. Closer to the 2000s he became interested in connection between weak factorisation systems and homotopy theory, and he wrote several papers on the subject of combinatorial model categories. He was active in several other topics, ranging from the theory of quantales to many aspects of general category theory: he introduced tangent categories and tangent bundle functors. He serves as an editor for Theory and Applications of Categories. He is the editor-in-chief of the Archivum Mathematicum journal. == References ==
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Wikipedia:Jo Johannis Dronkers#0
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Jo Johannis Dronkers (24 May 1910 – 20 February 1973) was a Dutch mathematician who is notable for the development of mathematical methods for the calculation of tides and tidal currents in estuaries. His work formed much of the mathematical basis for the design of the Delta Works. He attended the 1954 International Congress of Mathematicians in Amsterdam, where his work was discussed by David van Dantzig. == Life and career == === Education and early work === Jo Johannis Dronkers was born in Poortvliet, the son of Willem Dronkers, a mayor, and Cornelia Pieternella van der Slikke. He had a twin brother, Johannis Jo Dronkers (1910-1993). He attended the State High School in Bergen op Zoom and then studied mathematics and physics at Leiden University. He completed his dissertation entitled "On stratifiable congruences" in 1939 under the supervision of Professor Willem van der Woude. Although his work dealt with a problem in pure mathematics, it was notable for containing eleven propositions, five of which concerned tides. Three of these five propositions contained fundamental criticism of the work of Jannis Pieter Mazure, who at that time was one of the most influential people in the field of tidal calculations in the Netherlands. Amongst other propositions, Dronkers proposed a solution to the problem caused by the influence of upstream discharge on the calculation of tidal motion in downstream rivers. Another of Dronkers' propositions concerned an inaccuracy in the work of Hendrik Lorentz contained in Lorentz's report on tidal movements as a consequence of the Zuiderzee Works. === Rijkswaterstaat and the Delta Works === In 1934, Dronkers commenced his career at Rijkswaterstaat, a Dutch governmental agency responsible for water management and civil engineering works. He was assigned to the Sea Arms, Lower Rivers and Coasts service, a division directed by Dr. Johan van Veen, where Dronkers focused on the mathematical study of tides and related hydrodynamic phenomena. His expertise contributed significantly to the post-war restoration efforts associated with the aftermath of the Inundation of Walcheren in 1945. Dronkers' role during this period is vividly described in A. den Doolaard's non-fiction novel Het verjaagde water, in which he is described as "The Calculator." Dronkers' collaborative work with notable engineers like Frank Spaargaren led to several key publications in hydraulic engineering, particularly on topics such as tidal basin closures. His work gained further prominence following the devastating North Sea Flood of 1953, as he became deeply involved in the Delta Works project, a series of major civil engineering projects built to protect the Netherlands from sea flooding. Between 1949 and 1963, Dronkers served at the Central Study Service of RIjkswaterstaat, later renamed the Directorate for Water Management and Water Movement, under the leadership of Ir. J.B. Schijf. Here, as the chief mathematician, he played an integral role in the early studies that shaped the Delta Plan. In 1963, Dronkers took over as head of the hydrological department of the Delta Service, succeeding H.A. Ferguson. His mathematical acumen was crucial in devising tidal calculations for the Delta Works. The earlier Delta Commission Report of 1961, to which Dronkers contributed, focused extensively on storm surges and tidal motion, and included work by other notable Dutch engineers like Pieter Jacobus Wemelsfelder. Dronkers' work was pivotal in transitioning from empirical approaches to more scientifically robust computational methodologies in hydraulic engineering. His analytical techniques allowed for a proactive assessment of hydraulic interventions, thereby laying the groundwork for evidence-based strategic planning in the field. This approach continues to influence much of hydraulic engineering. Dronkers published his magnum opus, Tidal computations in rivers and coastal waters, in 1964. It remains a benchmark in the field of tidal calculation theory, and led to the award of the Conrad Medal by the Royal Netherlands Institute of Engineers to Dronkers in 1965. However, the book’s reception was not without controversy in the hydraulic engineering community. Dronkers' computational approach, though rigorous, was criticized by Johan Van Veen for its complexity and computational demands, which he believed could hinder timely practical applications. Instead, Van Veen proposed the use of an electrical analogue, which led to the use of the Deltar analogue computer. Conversely, the notable Dutch civil engineer Johannes Theodoor Thijsse advocated for a physical model-based approach. == Bibliography == Dronkers, J.J. (1975). "Tidal Theory and Computations". Advances in Hydroscience. 10: 145–230. Archived from the original on 17 June 2023. Retrieved 22 November 2024. Dronkers, Jo Johannis (1939). Over stratifieerbare congruenties [On Stratifiable Congruences] (in Dutch). Amsterdam: H.J. Paris. p. 76. Retrieved 22 November 2024. Dronkers, J.J. (1964). Tidal Computations in Rivers and Coastal Waters. North Holland Publishing Company. Retrieved 22 November 2024. Dronkers, J.J.; Schönfeld, J.C.; Waalewijn, A. (1959). Tidal Computations in Shallow Water - Report on Hydrostatic Levelling across the Westerschelde (Rijkswaterstaat Communications 01 ed.). Rijkswaterstaat. Retrieved 22 November 2024. Dronkers, J.J.; Breusers, H.N.C.; Vinjé, J.J.; Venis, W.A.; Spaargaren, F. (1968). "Closure of Estuarine Channels in Tidal Regions". De Ingenieur. Retrieved 22 November 2024. A list of many of Dronkers' reports is available at the website of the Trésor der Hollandse Waterbouw. == References == == External links == Repository of a number of publications by Dronkers (mostly in Dutch)
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Wikipedia:Joachim Lambek#0
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Joachim "Jim" Lambek (5 December 1922 – 23 June 2014) was a Canadian mathematician. He was Peter Redpath Emeritus Professor of Pure Mathematics at McGill University, where he earned his PhD degree in 1950 with Hans Zassenhaus as advisor. == Biography == Lambek was born in Leipzig, Germany, where he attended a Gymnasium. He came to England in 1938 as a refugee on the Kindertransport. From there he was interned as an enemy alien and deported to a prison work camp in New Brunswick, Canada. There, he began in his spare time a mathematical apprenticeship with Fritz Rothberger, also interned, and wrote the McGill Junior Matriculation in fall of 1941. In the spring of 1942, he was released and settled in Montreal, where he entered studies at McGill University, graduating with an honours mathematics degree in 1945 and an MSc a year later. In 1950, he completed his doctorate under Hans Zassenhaus becoming McGill's first PhD in mathematics. Lambek became assistant professor at McGill; he was made a full professor in 1963. He spent his sabbatical year 1965–66 in at the Institute for Mathematical Research at ETH Zurich, where Beno Eckmann had gathered together a group of researchers interested in algebraic topology and category theory, including Bill Lawvere. There Lambek reoriented his research into category theory. Lambek retired in 1992 but continued his involvement at McGill's mathematics department. In 2000 a festschrift celebrating Lambek's contributions to mathematical structures in computer science was published. On the occasion of Lambek's 90th birthday, a collection Categories and Types in Logic, Language, and Physics was produced in tribute to him. == Scholarly work == Lambek's PhD thesis investigated vector fields using the biquaternion algebra over Minkowski space, as well as semigroup immersion in a group. The second component was published by the Canadian Journal of Mathematics. He later returned to biquaternions when in 1995 he contributed "If Hamilton had prevailed: Quaternions in Physics", which exhibited the Riemann–Silberstein bivector to express the free-space electromagnetic equations. Lambek supervised 17 doctoral students, and has 75 doctoral descendants as of 2020. He has over 100 publications listed in the Mathematical Reviews, including 6 books. His earlier work was mostly in module theory, especially torsion theories, non-commutative localization, and injective modules. One of his earliest papers, Lambek & Moser (1954), proved the Lambek–Moser theorem about integer sequences. In 1963 he published an important result, now known as Lambek's theorem, on character modules characterizing flatness of a module. His more recent work is in pregroups and formal languages; his earliest works in this field were probably Lambek (1958) and Lambek (1979). He is noted, among other things, for the Lambek calculus, an effort to capture mathematical aspects of natural language syntax in logical form, and a work that has been very influential in computational linguistics, as well as for developing the connections between typed lambda calculus and cartesian closed categories (see Curry–Howard–Lambek correspondence). His last works were on pregroup grammar. == Selected works == === Books === Lambek, Joachim (2009) [1966]. Lectures on rings and modules (3rd ed.). Blaisdell Publishing. ISBN 9780821849002. MR 0206032. Fine, N. J.; Gillman, Leonard; — (1966). Rings of quotients of rings of functions. McGill University Press. MR 0200747. — (1966). Completions of categories. Seminar lectures given in 1966 in Zürich. Lecture Notes in Mathematics, No. 24. Berlin, New York: Springer-Verlag. MR 0209330. — (1971). Torsion theories, additive semantics, and rings of quotients. Lecture Notes in Mathematics. Vol. 177. Berlin, New York: Springer-Verlag. MR 0284459. —; Scott, P. J. (1986). Introduction to Higher Order Categorical Logic. Cambridge University Press. ISBN 978-0-521-35653-4. MR 0856915. Anglin, W. S.; — (1995). The heritage of Thales. Undergraduate Texts in Mathematics. Berlin, New York: Springer-Verlag. ISBN 978-0-387-94544-6. MR 1369087. Casadio, Claudia; — (2008). Computational Algebraic Approaches to Natural Language. Polimetrica. ISBN 978-88-7699-125-7. — (2008). From word to sentence: a computational algebraic approach to grammar. Polimetrica. ISBN 978-88-7699-117-2. === Articles === Lambek, Joachim (1951), "The immersibility of a semigroup into a group", Canadian Journal of Mathematics, 3: 34–43, doi:10.4153/CJM-1951-005-8, S2CID 124979541 —; Moser, L. (1954), "Inverse and Complementary Sequences of Natural Numbers", The American Mathematical Monthly, 61 (7): 454–458, doi:10.2307/2308078, ISSN 0002-9890, JSTOR 2308078, MR 0062777 — (1958), "The Mathematics of Sentence Structure" (PDF), The American Mathematical Monthly, 65 (3): 154–170, doi:10.2307/2310058, ISSN 0002-9890, JSTOR 1480361 — (1961), "How to program an infinite abacus", Canadian Mathematical Bulletin, 4 (3): 295–302, doi:10.4153/CMB-1961-032-6, S2CID 121506469 — (1969). "Deductive systems and categories II. Standard constructions and closed categories". Lecture Notes in Mathematics. Vol. 86. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 76–122. doi:10.1007/bfb0079385. ISBN 978-3-540-04605-9. ISSN 0075-8434. — (1972), "Bicommutators of nice injectives", Journal of Algebra, 21: 60–73, doi:10.1016/0021-8693(72)90034-8, ISSN 0021-8693, MR 0301052 — (1972), "Localization and completion", Journal of Pure and Applied Algebra, 2 (4): 343–370, doi:10.1016/0022-4049(72)90011-4, ISSN 0022-4049, MR 0320047 — (1979), "A mathematician looks at Latin conjugation", Theoretical Linguistics, 6 (2): 221–234, doi:10.1515/thli.1979.6.1-3.221, ISSN 0301-4428, MR 0589163, S2CID 62697835 — (1995). "If Hamilton had prevailed: Quaternions and Physics". Mathematical Intelligencer. 17 (4): 7–15. doi:10.1007/BF03024783. S2CID 120099630. Reprinted in Wilson, Robin; Gray, Jeremy, eds. (2001). Mathematical Conversations. Springer-Verlag. ISBN 978-1-4613-0195-0. —; Barr, Michael (2013), In Praise of Quaternions (PDF), McGill University == See also == Cartesian monoid Michael K. Brame == References == == External links == Faculty profile of Joachim Lambek at McGill University Lambek festival (80th anniversary)
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Wikipedia:Joanna Isabel Mayer#0
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Joanna Isabel Mayer (March 6, 1904 – February 28, 1991) was an American mathematician and educator. She was Marquette University's first doctoral student in mathematics and one of the few American women to earn a PhD in mathematics before World War II. == Biography == Mayer was born in Pettis County, Missouri, the third of five children of a parochial school teacher, Anna Poinsignon, and a farmer, Edward John Mayer. The first school she attended was Sacred Heart School in Sedalia, Missouri but then the family moved many times including to Los Angeles, California, Phoenix, Arizona, Portland, Oregon, Kansas City, Missouri, Nashville, Tennessee, Salt Lake City, Utah, and back to San Jose, California where she graduated from Notre Dame High School. Throughout her life, she attended and taught only in Catholic schools. Mayer received her B.A. degree from Dominican College in San Rafael in 1927 with a major in mathematics and a minor in philosophy. While there she was influenced by mathematician Anna Marie Whelan. Mayer immediately enrolled in graduate school at Marquette University in Milwaukee, Wisconsin, and in 1928 she received her MA degree with the thesis titled: A geometric interpretation and classification of the invariants of the binary and ternary conics and cubics, directed by Harvey Pierson Pettit. In 1931, she was the person to earn a doctorate in mathematics at Marquette University, studying again under Pettit's direction with a thesis entitled: Projective Description of Plane Quartic Curves. === Educator === From 1929 to 1930, while she was finishing her doctoral studies, she served as department head at Marymount College in Salina, Kansas. In 1930 the Mathematical Association of America announced her membership. In 1932 she taught in San Jose, California, and from 1937 to 1938 she was an instructor at Seton Hill College (now Seton Hill University), which was a Catholic women's college at the time, located in Greensburg, Pennsylvania. In 1939 she taught at Xavier University in New Orleans, described as "the only Catholic institution among the historically black colleges and universities" at that time. According to Green, "Mayer wrote in a letter to the Marquette graduate school dean in 1956 that she left teaching in 1942 and had supported herself since by buying and selling stocks." During World War II, she worked in Washington, D.C., and in 1950 for the Guided Missiles Committee of the Department of Defense, Pentagon, Washington, D.C. She worked for more than 13 years at the Military Personnel Records Center in St. Louis, Missouri. === Personal life === Mayer died February 28, 1991, in San Jose, California and was buried at the Santa Clara Mission Cemetery. == Memberships == According to Judy Green, Mayer belonged to two professional societies. Mathematical Association of America American Mathematical Society == References ==
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Wikipedia:Joannes Sturmius Mechlinianus#0
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Johannes Storms of Mechelen, also known as Joannes Sturmius Mechlinianus, or simply Sturmius or Mechlinianus (1559–1650) was a Belgian mathematician, physician and poet. == Life == Sturmius was born in Mechelen on 29 August 1559. He studied at Leuven University, graduating Master of Arts before the age of 20, and on 12 June 1579 he was appointed to a benefice of the chaplaincy of St Margaret in Mechelen. In 1585 he was appointed to teach dialectics and metaphysics in Lily College, Leuven, while pursuing further studies in the Faculty of Medicine. In 1591 he graduated licentiate in medicine and was appointed to the university's academic council. He audited the lectures of Adriaan van Roomen. In 1593 Sturmius was appointed to the chair in mathematics vacated by Roomen, graduating doctor of medicine the same year. In 1603 he was appointed regent of Lily College, resigning in 1606 to marry Catherine van Thienen. After her death in 1619, Sturmius took holy orders. He died in Leuven on 9 March 1650, and was buried in the church of St Kwinten. Thomas Fienus (1567–1631) was one of his students. == Works == De Rosa Hierichuntina. Leuven, Gerard Rivius, 1607 De Insulis in civitate Lovaniensi existentibus Physica, seu generalia philosophiae naturalis theoremata e libris Aristotelis desumpta. Leuven, Gerard Rivius, 1610 De cometa anni MDCXVIII. Cologne, 1619. De accurata circuli dimensione et quadratura. Leuven, Fr. Simonis, 1633 Psalterium Β. Mariae Virginis et meditationes piet versibus trochaïchis dimetris illigatae. Leuven, Fr. Simonis, 1633 Fasciculus versuum, tam numeralium. Leuven, Fr. Simonis, 1635 Prognostica pro annis 1631, 1635, 1636, per versus chronicus de singulis mensibus cum quibusdam epigrammatibus. Leuven, Fr. Simonis, 1636 Ad sanctissimae Trinitatis gloriam, versus elegiaci. Leuven, Justus Coppenius, 1639 == References ==
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Wikipedia:Joaquim Gomes de Souza#0
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Joaquim Gomes de Souza "Souzinha" (15 February 1829, in Itapecuru Mirim – 1 June 1864, in London) was a Brazilian mathematician who worked on numerical analysis and differential equations. He was a pioneer on the study of mathematics in Brazil, and was described by José Leite Lopes as "the first great mathematician from Brazil". In 1844, Gomes de Souza enrolled at the Faculdade de Medicina do Rio de Janeiro (now a part of the Federal University of Rio de Janeiro) to study medicine. He had a deep love for the natural sciences, which led him to also be interested in mathematics, and so he started to learn mathematics as a self-taught in parallel with his studies of medicine. In 1848, he obtained his doctorate in mathematics from the Escola Real Militar, with the thesis Dissertação Sobre o Modo de Indagar novos Astros sem o Auxílio das Observações Directas (Dissertation about the means of investigating new celestial objects without the aid of direct observations). He later went to the Sorbonne, in France, where he continued his mathematical studies. He was a personal friend of Cauchy, of whose classes he attended (in one of them, Souza spotted a mathematical mistake by Cauchy, he then asked his license and corrected it on the blackboard). In 1856, he obtained a doctorate in medicine from Paris Faculty of Medicine. In the same year, he presented his mathematical works at the Académie des sciences. Souza held a paid public post in Brazil, and after much time in Europe, he was noticed he should return immediately to Brazil because he had been elected a member of the parliament. Souza had already married Rosa Edith in England and then had to return to Brazil without her. In his book Mélanges de calcul intégral (1882), Souza aimed to obtain a general method to solve PDEs, according to Manfredo do Carmo: "[in his book] He [Souza] employed methods not entirely rigorous and it is not clear exactly how much of his work would remain if submitted to a careful scrutiny; as far as I know, it was never put to such a test." He died at the age of 35, in London. The cause of death was a disease of the lung. C. S. Fernandez and C. M. Souza described his endeavorer in Europe: "He was audacious and fought with insistence for his scientific recognition in Europe. His effort was fruitless, though." == Writings == Resoluções das Equações Numéricas (1850, in Portuguese) Recuel de Memoires d’Analise Mathematiques (1857, in French) Anthologie universelle (1859, in French) Mélanges de calcul intégral (1882, posthumous, in French) == Further reading == Irine Coelho de Araujo, Joaquim Gomes de Souza (1829-1864): A construção de uma imagem de Souzinha, São Paulo, 2012 Carlos Ociran Silva Nascimento, Alguns aspectos da obra matematica de Joaquim Gomes de Souza, Campinas, 2008 == References ==
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Wikipedia:Joceline Lega#0
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Joceline Claude Lega is a French physicist and applied mathematician, interested in nonlinear dynamics. She is a professor in the departments of mathematics, applied mathematics, and epidemiology and biostatistics at the University of Arizona, and editor-in-chief of Physica D. == Education and career == After studying physics at the École Normale Supérieure in Paris from 1984 to 1988, and earning licentiate and maîtrise degrees in physics through Pierre and Marie Curie University in 1985, Lega earned a diplôme d'études approfondies in 1986 and a doctorate in theoretical physics in 1989, both at the University of Nice Sophia Antipolis. Her dissertation was Topological defects associated with the breaking of time translation invariance. She joined the Centre national de la recherche scientifique (CNRS) in 1989, and took a leave from CNRS to join the University of Arizona in 1997. At Arizona, she was the director of the Program in Integrated Science (from 2008 to 2011), and the Institute for Mathematics & Education (from 2009 to 2013). Since 2016 she is the associate head for the Postdoctoral Program in Mathematics. == Recognition == Lega became a Fellow of the Institute of Physics in 2004. In 2017 she was elected as a Fellow of the American Association for the Advancement of Science. == Selected publications == Coullet, P.; Gil, L.; Lega, J. (April 1989), "Defect-mediated turbulence", Physical Review Letters, 62 (14): 1619–1622, Bibcode:1989PhRvL..62.1619C, doi:10.1103/physrevlett.62.1619, PMID 10039721 Coullet, P.; Lega, J.; Houchmanzadeh, B.; Lajzerowicz, J. (September 1990), "Breaking chirality in nonequilibrium systems", Physical Review Letters, 65 (11): 1352–1355, Bibcode:1990PhRvL..65.1352C, doi:10.1103/physrevlett.65.1352, PMID 10042242 Ciliberto, S.; Coullet, P.; Lega, J.; Pampaloni, E.; Perez-Garcia, C. (November 1990), "Defects in roll-hexagon competition" (PDF), Physical Review Letters, 65 (19): 2370–2373, Bibcode:1990PhRvL..65.2370C, doi:10.1103/physrevlett.65.2370, hdl:10171/2241, PMID 10042531 Lega, J.; Moloney, J. V.; Newell, A. C. (November 1994), "Swift-Hohenberg Equation for Lasers", Physical Review Letters, 73 (22): 2978–2981, Bibcode:1994PhRvL..73.2978L, doi:10.1103/physrevlett.73.2978, hdl:10468/4664, PMID 10057251 Lega, J.; Passot, T. (March 2003), "Hydrodynamics of bacterial colonies: A model", Physical Review E, 67 (3): 031906, Bibcode:2003PhRvE..67c1906L, doi:10.1103/physreve.67.031906, PMID 12689100 == References == == External links == Home page
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Wikipedia:Joel Spruck#0
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Joel Spruck (born 1946) is a mathematician, J. J. Sylvester Professor of Mathematics at Johns Hopkins University, whose research concerns geometric analysis and elliptic partial differential equations. He obtained his PhD from Stanford University with the supervision of Robert S. Finn in 1971. == Mathematical contributions == Spruck is well known in the field of elliptic partial differential equations for his series of papers "The Dirichlet problem for nonlinear second-order elliptic equations," written in collaboration with Luis Caffarelli, Joseph J. Kohn, and Louis Nirenberg. These papers were among the first to develop a general theory of second-order elliptic differential equations which are fully nonlinear, with a regularity theory that extends to the boundary. Caffarelli, Nirenberg & Spruck (1985) has been particularly influential in the field of geometric analysis since many geometric partial differential equations are amenable to its methods. With Basilis Gidas, Spruck studied positive solutions of subcritical second-order elliptic partial differential equations of Yamabe type. With Caffarelli, they studied the Yamabe equation on Euclidean space, proving a positive mass-style theorem on the asymptotic behavior of isolated singularities. In 1974, Spruck and David Hoffman extended a mean curvature-based Sobolev inequality of James H. Michael and Leon Simon to the setting of submanifolds of Riemannian manifolds. This has been useful for the study of many analytic problems in geometric settings, such as for Gerhard Huisken's study of mean curvature flow in Riemannian manifolds and for Richard Schoen and Shing-Tung Yau's study of the Jang equation in their resolution of the positive energy theorem in general relativity. In the late 80s, Stanley Osher and James Sethian developed the level-set method as a computational tool in numerical analysis. In collaboration with Lawrence Evans, Spruck pioneered the rigorous study of the level-set flow, as adapted to the mean curvature flow. The level-set approach to mean curvature flow is important in the technical ease with topological change that can occur along the flow. The same approach was independently developed by Yun Gang Chen, Yoshikazu Giga, and Shun'ichi Goto. The works of Evans–Spruck and Chen–Giga–Goto found significant application in Gerhard Huisken and Tom Ilmanen's solution of the Riemannian Penrose inequality of general relativity and differential geometry, where they adopted the level-set approach to the inverse mean curvature flow. In 1994 Spruck was an invited speaker at the International Congress of Mathematicians in Zurich. == Major publications == Hoffman, David; Spruck, Joel. Sobolev and isoperimetric inequalities for Riemannian submanifolds. Comm. Pure Appl. Math. 27 (1974), 715–727. Gidas, B.; Spruck, J. A priori bounds for positive solutions of nonlinear elliptic equations. Comm. Partial Differential Equations 6 (1981), no. 8, 883–901. Gidas, B.; Spruck, J. Global and local behavior of positive solutions of nonlinear elliptic equations. Comm. Pure Appl. Math. 34 (1981), no. 4, 525–598. Caffarelli, L.; Nirenberg, L.; Spruck, J. The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation. Comm. Pure Appl. Math. 37 (1984), no. 3, 369–402. Caffarelli, L.; Kohn, J.J.; Nirenberg, L.; Spruck, J. The Dirichlet problem for nonlinear second-order elliptic equations. II. Complex Monge-Ampère, and uniformly elliptic, equations. Comm. Pure Appl. Math. 38 (1985), no. 2, 209–252. Caffarelli, L.; Nirenberg, L.; Spruck, J. The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian. Acta Math. 155 (1985), no. 3–4, 261–301. Caffarelli, Luis A.; Gidas, Basilis; Spruck, Joel. Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math. 42 (1989), no. 3, 271–297. Evans, L.C.; Spruck, J. Motion of level sets by mean curvature. I. J. Differential Geom. 33 (1991), no. 3, 635–681. Spruck, Joel; Yang, Yi Song. Topological solutions in the self-dual Chern-Simons theory: existence and approximation. Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995), no. 1, 75–97. == Prizes == Simons Fellowship (2012–2013) Fellow of the American Mathematical Society (2013 inauguration) Guggenheim Fellowship (1999–2000) == References == == External links == Joel Spruck publications indexed by Google Scholar
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Wikipedia:Johan Frederik Steffensen#0
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Johan Frederik Steffensen (28 February 1873, in Copenhagen – 20 December 1961) was a Danish mathematician, statistician, and actuary who did research in the fields of calculus of finite differences and interpolation. He was professor of actuarial science at the University of Copenhagen from 1923 to 1943. Steffensen's inequality and Steffensen's method (an iterative numerical method) are named after him. He was an Invited Speaker at the 1912 International Congress of Mathematicians (ICM) in Cambridge, England and at the 1924 ICM in Toronto. ... His more important works included the theory of statistics (1923), interpolation (1925), insurance mathematics (1934) and the calculation of interest (1936). ... He was President of the Danish Actuarial Society in 1922-24 and 1930-33, and of the Danish Mathematical Society in 1930-36 ... == Publications == Factorial moments and discontinuous frequency-functions. Uppsala: Almqvist & Wiksells Boktryckeri-A.B. 1923; 89 pages{{cite book}}: CS1 maint: postscript (link) Interpolation. London: Williams & Wilkins. 1927. == Obituary == Nordlund N. E. (1962). "Johan Frederik Steffensen in memoriam". Nordisk Matematisk Tidsskrift. 10: 105–107. == References == == External links == Johan Frederik Steffensen papirer (Danish)
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Wikipedia:Johan Galtung#0
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Johan Vincent Galtung (24 October 1930 – 17 February 2024) was a Norwegian sociologist and the principal founder of the discipline of peace and conflict studies. He was the main founder of the Peace Research Institute Oslo (PRIO) in 1959 and was its first director until 1970. He also established the Journal of Peace Research in 1964. In 1969, he was appointed to the world's first chair in peace and conflict studies, at the University of Oslo. He resigned his Oslo professorship in 1977 and thereafter held professorships at several other universities; from 1993 to 2000 he taught as Distinguished Professor of Peace Studies at the University of Hawaii. He was the Tun Mahathir Professor of Global Peace at the International Islamic University Malaysia until 2015. == Background == Galtung was born in Oslo. He earned the cand. real. degree in mathematics at the University of Oslo in 1956, and a year later completed the mag. art. (PhD) degree in sociology at the same university. Galtung received the first of thirteen honorary doctorates in 1975. Galtung's father and paternal grandfather were both physicians. The Galtung name has its origins in Hordaland, where his paternal grandfather was born. Nevertheless, his mother, Helga Holmboe, was born in central Norway, in Trøndelag, while his father was born in Østfold, in the south. Galtung was married twice, and had two children by his first wife Ingrid Eide, Harald Galtung and Andreas Galtung, and two by his second wife Fumiko Nishimura, Irene Galtung and Fredrik Galtung. Galtung experienced World War II in German-occupied Norway, and as a 12-year-old saw his father arrested by the Nazis. By 1951, he was already a committed peace mediator, and elected to do 18 months of social service in place of his obligatory military service. After 12 months, Galtung insisted that the remainder of his social service be spent in activities relevant to peace. Galtung died in Stabekk Helsehus og Hospice, Baerum, Norway, on 17 February 2024, at the age of 93. == Career == Upon receiving his mag. art. degree, Galtung moved to Columbia University, in New York City, where he taught for five semesters as an assistant professor in the department of sociology. In 1959, Galtung returned to Oslo, where he founded the Peace Research Institute Oslo (PRIO). He was the institute's director until 1969. In 1964, Galtung led PRIO to establish the first academic journal devoted to Peace Studies: the Journal of Peace Research. In the same year, he assisted in the founding of the International Peace Research Association. In 1969, he left PRIO for a position as professor of peace and conflict research at the University of Oslo, a position he held until 1978. Galtung was the director general of the International University Centre in Dubrovnik and helped to found and lead the World Future Studies Federation. He has held visiting positions at other universities, including Santiago, Chile, the United Nations University in Geneva, Witten/Herdecke University in Germany, and at Columbia, Princeton and the University of Hawaii in the USA. In 2014, he was appointed the first Tun Mahathir Professor of Global Peace at the International Islamic University Malaysia. Economist and fellow peace researcher Kenneth Boulding has said of Galtung that his "output is so large and so varied that it is hard to believe that it comes from a human". He was a member of the Norwegian Academy of Science and Letters. In 1993, he co-founded TRANSCEND: A Peace Development Environment Network. In 1987, he was given the Right Livelihood Award. == Work and views == === Conflict Triangle === In Galtung's 1969 paper, "Violence, Peace and Peace Research", he presents his theory of the Conflict Triangle, a framework used in the study of peace and conflict, with the purpose of defining the three key elements of violence that form this "triangle." The theory is based on the principle that peace must be defined by widely accepted social goals, and that any state of peace is characterized by the absence of violence. When a conflict has features of all three areas of violence, the result is a more consolidated, static state of violence in a social system, which may include a conflict or a nation-state, whereas the absence of these three typologies of violence results in peace. ==== Structural Violence ==== Galtung's concept of structural violence refers to the indirect forms of violence originating from social, economic, and political structures and manifesting primarily as oppression and exploitation.: 16 These indirect forms of violence result in injustices in the distribution of political power and economic benefits.: 16 Rather than conveying a physical image, structural violence is an avoidable impairment of fundamental human needs. Structural violence is increased in situations where low income individuals also suffer in the rank dimensions of education, health, and power. This is due to an overall consolidation of factors in the social structure, resulting in a high correlation between social class and disempowerment. Structural violence can be recognized through its relative stability, having been built into the social structure. This can make structural violence difficult to ascertain, despite its often vast consequences. This concept has been applied in a large number of cases, some of the most notable are listed below. Akhil Gupta argued in 2012 that structural violence has been the key influence in the nature and distribution of extreme suffering in India, driven by the Indian state in its alleged corruption, overly bureaucratic standards of governance used to exclude the middle and working classes from the political system through a system of politicized poverty. Jacklyn Cock's 1989 paper in the Review of African Political Economy applied Galtung's theory of structural violence, analysing the role of militarized society under the apartheid regime of South Africa in the development of patriarchal values that is a form of structural violence against women. Cock found that tacit misdirection of women in society by its leadership focused their energies toward the direct and indirect incorporation of the patriarchal regime in order to maintain the status quo. Mats Utas claimed that even those youth in Liberia indirectly unaffected by direct violence in the civil war of 1989-1996 suffered from structural violence in the form of association with different blocs, leading to poverty, joblessness and marginalisation effects. ==== Cultural Violence ==== Galtung defines cultural violence as ideas, consciousness, language, art, or science that can be used to legitimize or enable direct violence or structural violence.: 16–17 The existence of prevailing or prominent social norms make direct and structural violence seem natural or at least acceptable, and serves to explain how prominent beliefs can become so embedded in a given culture that they function as absolute and inevitable and are reproduced uncritically across generations. Galtung expanded on the concept of cultural violence in a 1990 paper also published in the Journal of Peace Research. This concept has been applied in a limited number of cases, with most occurring after Galtung's follow up paper in 1990, some of the most notable of which are listed below. Johan Galtung has written about Zionism and violence. He has discussed various forms of violence, including structural and cultural violence, in his extensive body of work. Galtung has been critical of Zionism, particularly in the context of the Israeli-Palestinian conflict, and has linked it to broader themes of structural violence. He has framed Zionism within his broader theories of structural and cultural violence, suggesting that the establishment and actions of the state of Israel have contributed to ongoing conflict and suffering in the region. Gregory Phillips argues in his 2003 book, Addictions and Healing in Aboriginal Country, that resistance to the Western medical sphere driven by previous atrocities committed against the Aboriginal community has led to a fierce resistance effort against modern medicine, addiction treatment and perhaps fuels a desire to seek out drugs and illicit substances as a starting point of addiction. Wide scale suspicion against medical practitioners and government representatives has become engendered in the Aboriginal community. In Enduring Violence: Ladina Women's Lives in Guatemala, the 2011 book by Cecilia Menjívar, it is argued that the preexisting cultural conditions of mediania, or half and half, agriculture led to women facing large scale cultural violence due to high rents, low returns and high required investment with additionally harsh conditions due to the conflict in Guatemala. Given the patriarchal culture of Guatemala, any earnings would go to the partner of the working woman, leaving a large poverty gap enshrined in the demographic diversity of the country. The Austrian peace researcher Franz Jedlicka has tried to measure the level of cultural violence in a "Culture of Violence Scale" in 2023. ==== Direct Violence ==== Direct Violence is characterised as having an actor that commits the violence, and is thus able to be traced back to persons as actors. Direct violence shows less stability, given it is subject to the preference sets of individuals, and thus is more easily recognised. Direct violence is the most visible, occurring physically or verbally, and the victim and the offender can be clearly identified. Direct violence is highly interdependent with structural and cultural violence: cultural and structural violence causes direct violence which on the other hand reinforces the former ones. This concept has been applied in a large number of cases, some of which are listed below. A 2011 paper by the International Center for Research on Women (ICRW) demonstrated the widespread nature of child marriage in South Asia. The ICRW highlighted marriage before the age of 18 as a fundamental human rights violation, one that leads to early childbearing, with significantly higher maternal mortality and morbidity rates as well as higher infant mortality rates amongst women. The paper most directly presented evidence to show that child brides are at heightened risk of violence in the home. In Matthew Chandler's 2009 paper on so-called "non-violent" techniques utilised by Hezbollah still include forms of Direct Violence, most notably the threat of violence toward Fouad Siniora's allies after his 2008 order to dismantle the Hezbollah telecommunications network in 2008, which led to the freezing of the order. Further, Hezbollah are argued to have used their operation of social services, in lieu of government operations, as a ransom for support as well as rewarding their fighters with guaranteed healthcare and support for their families. Chandler argues this is due to opposition within the group to harming Lebanese civilians, who they view as "their own", or exacerbating conflict through civil war. In 2005, Steven Wright made the case for Peacekeeping efforts to be regarded as violence due to increasing use of techniques such as pre-interrogation treatment, and the use of non-lethal weapons such as tear gas for crowd dispersal and plastic bullets, which he terms "torture-lite", being increasingly common in peacekeeping manuals across a number of nation-states and supranational organisations. ==== Reinforcing Factors ==== Galtung focuses a section of the paper on the means of direct and structural violence, in particular, developing groups of factors that may be included as types of such forms and methods of maintaining and reinforcing the mechanisms of such violence. In terms of reinforcing factors, Galtung identifies six key areas: Linear Ranking Order Systems in which there is an open and complete ranking of actors leaves no doubt as to the actor who is ranked more highly, and is thus a mechanism of structural violence due to the reinforcement of an existing power dynamic. Acyclical Interaction Pattern Systems in which all actors are connected via a one-way ‘correct’ path of interaction, where outcomes are structurally dependent on using this system in the intended way of its design. This makes structural systems stable, as change can only be achieved through this consolidated power-seeking and power-retaining system. Rank-Centrality Correlation Within the social system, actors that are higher ranked are more central within the system itself, reinforcing their importance to the status quo as well as their incentives to maintain it. (4) System Congruence Social systems are made up of similar components, allowing those who are ranked highly and are successful at mobilising one system shifting from a comparative advantage within one system to an absolute advantage over all systems of desired operation. Rank Concordance Actors that are ranked highly within one metric, such as income, are also ranked highly on other metrics such as education and health. This congruence is also present in actors ranked low within these metrics, and serves to limit mobility within the social system. Interlevel High Rank Coupling Collaboration amongst the highest ranks results in the system being defined in such a way that benefits the most powerful actors, usually through a sub optimally ranked representative (not the highest ranked actor), which limits allegations of system consolidation by the most powerful. Beyond Galtung's initial paper and thesis, scholars have applied the Conflict Triangle to a broad array of conflicts, struggles and occupations since 1969, and retroactively. ==== Criticism of the model ==== Galtung's Conflict Triangle and Peace Research paper are widely cited as the foundational pieces of theory within peace and conflict studies. However, they are not without criticism. Galtung uses very broad definitions of violence, conflict and peace, and applies the terms of mean both direct and indirect, negative and positive, and violence in which one cannot distinguish actors or victims, which serves to limit the direct application of the model itself. Galtung uses a positivist approach, in that he assumes that every rational tenet of the theory can be verified, serving to reject social processes beyond relationships and actions. This approach enforces a paradigm of clear-cut, currently testable propositions as the ‘whole’ of the system, and thus is often deemed reductionist. Galtung also wields an explicit normative orientation in the paper, in which there is a weighting toward evaluative statements that may show bias or simply opinion, or indeed a trend toward the institutions and concepts of peace in the West, which may serve to limit the applicability of the model more widely. === Peacebuilding === Galtung proposes resolving conflicts through peacekeeping, peacemaking, and peacebuilding.: 17 Peacekeeping and peacemaking primarily involve eliminating violence and bringing about immediate peace.: 17 In Galtung's view, addressing the root causes of violence requires peacebuilding which goes beyond ending direct violence to end structural violence and cultural violence.: 17 The peacebuilding structures needed to address the root causes of conflict and support local capacity for peace management and conflict resolution. Galtung has held several significant positions in international research councils and has been an advisor to several international organisations. Since 2004, he has been a member of the Advisory Council of the Committee for a Democratic UN. Galtung is strongly associated with the following concepts: Structural violence – widely defined as the systematic ways in which a regime prevents individuals from achieving their full potential. Institutionalized racism and sexism are examples of this. Negative vs. positive peace – popularized the concept that peace may be more than just the absence of overt violent conflict (negative peace), and will likely include a range of relationships up to a state where nations (or any groupings in conflict) might have collaborative and supportive relationships (positive peace). Though he did not cite them, these terms were, in fact, previously defined and discussed in a series of lectures starting in 1899 by Jane Addams (in her 1907 book she switched to calling it 'newer ideals of peace' but continued to contrast them to the term negative peace), and in 1963 in the letter from a Birmingham jail by Martin Luther King Jr. === Criticism of the United States === In 1973, Galtung criticised the "structural fascism" of the US and other Western countries that make war to secure materials and markets, stating: "Such an economic system is called capitalism, and when it's spread in this way to other countries it's called imperialism", and praised Fidel Castro's Cuba in 1972 for "break[ing] free of imperialism's iron grip". Galtung stated that the US is a "killer country" guilty of "neo-fascist state terrorism" and compared the US to Nazi Germany for bombing Kosovo during the 1999 NATO bombing of Yugoslavia. In an article published in 2004, Galtung predicted that the US empire will "decline and fall" by 2020. He expanded on this hypothesis in his 2009 book titled The Fall of the US Empire - and Then What? Successors, Regionalization or Globalization? US Fascism or US Blossoming?. Following the election of Donald Trump in 2016, Galtung revised forward his theory of American global power decline, citing Trump's deportation policy and critical views of NATO would accelerate it. === Views on Communist regimes === During his career, Galtung statements and views have drawn criticism including his criticism of Western countries during and after the Cold War and what his critics perceived as a positive attitude to the Soviet Union, Cuba and Communist China. A 2007 article by Bruce Bawer published by the City Journal magazine and a subsequent article in February 2009 by Barbara Kay in the National Post criticised Galtung's opinion of China during the rule of Mao Zedong. China, according to Galtung, was "repressive in a certain liberal sense", but he insisted "the whole theory about what an 'open society' is must be rewritten, probably also the theory of 'democracy'—and it will take a long time before the West will be willing to view China as a master teacher in such subjects." Calling Galtung a "lifelong enemy of freedom", Bawer said Galtung discouraged Hungarian resistance against the Soviet invasion in 1956, and criticized his description in 1974 of Aleksandr Solzhenitsyn and Andrei Sakharov as "persecuted elite personages". === Views on Jews and Israel === Galtung recommended that people should read The Protocols of the Elders of Zion, a fabricated antisemitic text purporting to describe a Jewish plan for global domination. In defending his claims that Jews control American media companies, Galtung cited an article published by National Vanguard, a neo-Nazi organization. Galtung's rhetoric has been criticized by Terje Emberland, a historian at the Center for Studies of the Holocaust and Religious Minorities in Oslo, and Øystein Sørensen, a University of Oslo historian known for his scholarship on conspiracy theories. Asked by NRK about his controversial remarks, Galtung reiterated his recommendation that people should read The Protocols of the Elders of Zion. Galtung rejected that he was anti-Semitic. The Israeli newspaper Haaretz accused Galtung in May 2012 of antisemitism for (1) suggesting the possibility of a link between the 2011 Norway attacks and Israel's intelligence agency Mossad; (2) maintaining that "six Jewish companies" control 96% of world media; (3) identifying what he contends are ironic similarities between the banking firm Goldman Sachs and the conspiratorial antisemitic forgery The Protocols of the Elders of Zion; and (4) theorizing, although not justified, antisemitism in post–World War I Germany was a predictable consequence of German Jews holding influential positions. As a result of such statements, TRANSCEND International, an organisation co-founded by Galtung, released a statement in May 2012 attempting to clarify his opinions. On 8 August 2012, the World Peace Academy in Basel, Switzerland announced it was suspending Galtung from its organization, citing what it posited were his "reckless and offensive statements to questions that are specifically sensitive for Jews." Galtung said the claims were "smearing and libel". == Selected awards and recognitions == Dr honoris causa, University of Tampere, 1975, peace studies Dr honoris causa, University of Cluj, 1976, future studies Dr honoris causa, Uppsala University, 1987, Faculty of Social Sciences Dr honoris causa, Soka University, Tokyo, 1990, peace/Buddhism Dr honoris causa, University of Osnabrück, 1995, peace studies Dr honoris causa, University of Torino, 1998, sociology of law Dr honoris causa, FernUniversität Hagen, 2000, philosophy Dr honoris causa, University of Alicante, 2002, sociology Dr honoris causa, Benemérita Universidad Autónoma de Puebla, 2006, law Dr honoris causa, Complutense University, Madrid, 2017, politics and sociology Honorary professor, University of Alicante, Alicante, 1981 Honorary professor, Free University of Berlin, 1984–1993 Honorary professor, Sichuan University, Chengdu, 1986 Honorary professor, Witten/Herdecke University, Witten, 1993 Distinguished professor of peace studies, University of Hawaii, 1993- John Perkins University Distinguished Visiting Professor, 2005- Right Livelihood Award, 1987 First recipient of the Humanist Prize of the Norwegian Humanist Association, 1988 Jamnalal Bajaj International Award for Promoting Gandhian Values, 1993 Brage Prize, 2000 First Morton Deutsch Conflict Resolution Award, 2001 Honorary Prize of the Norwegian Sociological Association, 2001 Premio Hidalgo, Madrid, 2005 Augsburg Golden Book of Peace, 2006 Member of the Norwegian Academy of Science and Letters Honorary member of the Green Party, 2009 Erik Bye Memorial Prize, 2011 == Selected works == Galtung has published more than a thousand articles and over a hundred books. Statistisk hypotesepröving (Statistical hypothesis testing, 1953) Gandhis politiske etikk (Gandhi's political ethics, 1955, with philosopher Arne Næss) Theory and Methods of Social Research (1967) Violence, Peace and Peace Research (1969) Members of Two Worlds (1971) Fred, vold og imperialisme (Peace, violence and imperialism, 1974) Peace: Research – Education – Action (1975) Learning from China? (1977, with Fumiko Nishimura) Europe in the Making (1989) Global Glasnost: Toward a New World Information and Communication Order? (1992, with Richard C. Vincent) Global Projections of Deep-Rooted U.S Pathologies Archived 18 August 2017 at the Wayback Machine (1996) Peace By Peaceful Means: Peace and Conflict, Development and Civilization (1996) Johan uten land. På fredsveien gjennom verden (Johan without land. On the Peace Path Through the World, 2000, autobiography for which he won the Brage Prize) 50 Years: 100 Peace and Conflict Perspectives (2008) Democracy – Peace – Development (2008, with Paul D. Scott) 50 Years: 25 Intellectual Landscapes Explored (2008) Globalizing God: Religion, Spirituality and Peace (2008, with Graeme MacQueen) == See also == Cost of conflict, a tool which attempts to calculate the price of conflict to the human race Democratic peace theory, a theory which posits that democracies are hesitant to engage in armed conflict with other identified democracies Critical race theory, a critical examination of society and culture, to the intersection of race, law, and power == References == == Sources == Boulding, Elise. 1982. "Review: Social Science—For What?: Festschrift for Johan Galtung." Contemporary Sociology. 11(3):323-324. JSTOR Stable URL Boulding, Kenneth E. 1977. "Twelve Friendly Quarrels with Johan Galtung." Journal of Peace Research. 14(1):75-86. JSTOR Stable URL == External links == TRANSCEND: A Peace Development Environment Network Galtung-Institute for Peace Theory and Peace Practice Peace Research Institute Oslo (PRIO) Biography on Right Livelihood Award Lecture transcript and video of Galtung's speech at the Joan B. Kroc Institute for Peace & Justice at the University of San Diego, December 2010 Audio recordings with Johan Galtung in the Online Archive of the Österreichische Mediathek (Interviews and lectures in German). Retrieved 18 September 2019
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Wikipedia:Johan Gielis#0
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Johan Gielis (born July 8, 1962) is a Belgian engineer, scientist, mathematician, and entrepreneur. Gielis is known for his contributions to the field of mathematics, specifically in the area of modeling and geometrical methods. He is best known for developing the concept of the superformula, which is a generalization of the traditional Pythagorean theorem and the equation of the circle, that can generate a wide variety of complex shapes found in nature. == Career == Gielis obtained a degree in horticultural engineering. Later, he changed direction from botany and plant biotechnology to geometry and mathematics. In 2013, Gielis co founded the Antenna Company, in Eindhoven. The company applies the superfomula to develop efficient antennas to transmit data via various frequencies. The company made antenna system for ultra-fast WiFi 6 devices. Antenna systems focus on 2-7 gigaHertz, in line with the IEEE 802.11ax standard and beyond. Other products focus on Internet of Things and mmWave antenna systems. == Superformula == Gielis proposed the superformula in 2003. The superfomula is a generalization of the superellipse. He suggested that it allows for the creation of shapes that can mimic natural forms such as flowers, shells, and other intricate structures. The mathematical equation combines elements of trigonometry and algebra to generate complex and visually appealing patterns. It also allowed for a generalization of minimal surfaces based on a more general notion of the energy functional and allowed for a generalized definition of the Laplacian, and the use of Fourier projection methods to solve boundary value problems. 1 r = | 1 a cos ( m 4 ϕ ) | n 2 + | 1 b sin ( m 4 ϕ ) | n 3 n 1 {\displaystyle {\frac {1}{r}}={\sqrt[{n_{1}}]{\,\left|{\frac {1}{a}}\cos \left({\frac {m}{4}}\phi \right)\right|^{n_{2}}+\left|{\frac {1}{b}}\sin \left({\frac {m}{4}}\phi \right)\right|^{n_{3}}}}} r - distance from the center, Φ - Angle to the x-axis, m - symmetry, n1, n2, n3: - Form, a, b: - expansion (semi-axes) Gielis patented the synthesis of patterns generated by the superformula. The superformula was used in No Man's Sky, an action-adventure survival game developed and published by Hello Games. The formula was also used in the Jewels of the Sea. == Publications == === Books === Modeling in Mathematics Proceedings of the Second Tbilisi-Salerno Workshop on Modeling in Mathematics 2017 Inventing the Circle The geometrical beauty of plants Universal Natural Shapes === Journals === A generic geometric transformation that unifies a wide range of natural and abstract shapes Diatom frustule morphogenesis and function: a multidisciplinary survey Somatic embryogenesis from mature Bambusa balcooa Roxburgh as basis for mass production of elite forestry bamboos Tissue culture strategies for genetic improvement of bamboo Computer implemented tool box systems and methods Superquadrics with rational and irrational symmetry Comparison of dwarf bamboos (Indocalamus sp.) leaf parameters to determine relationship between spatial density of plants and total leaf area per plant A general leaf area geometric formula exists for plants—Evidence from the simplified Gielis equation == References == == External links == Genicap, Johan Gieli's homepage P. Bourke: Supershapes (Superformula) Eric W. Weisstein: Superellipse. (MathWorld—A Wolfram Web Resource) Graphic "super formula": A circle is a square is a fish is a starfish (hot) Supershapes (Java applet) Johan Gielis publications indexed by Google Scholar
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Wikipedia:Johann Friedrich Hennert#0
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Johann Friedrich Hennert (19 October 1733 – 30 March 1813) was a German-born and lectured in mathematics and physics at the University of Utrecht. He was a significant student of Leonhard Euler. He was known for his inclination towards the British school of philosophy. == Work == Hennert held the chair of mathematics at the University of Utrecht until 1805. Hennert was an important figure in the history of Dutch mathematics. He wrote a number of textbooks on differential calculus. Jean Henri van Swinden was one of his most important students. == References == van Berkel, Klaas; van Helden, Albert & Palm, L. C. (1998). A History of Science in the Netherlands: Survey, Themes and Reference. Leiden: Brill. ISBN 90-04-10006-7. Helmers Dini M., Timmerman, Petronella Johanna de, in: Digitaal Vrouwenlexicon van Nederland. == External links == Johann Friedrich Hennert at the Mathematics Genealogy Project
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Wikipedia:Johann Jakob Rebstein#0
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Johann Jakob Rebstein (1840–1907) was a Swiss mathematician and surveyor. == Early life == Rebstein was born on 4 May 1840 in Töss, Switzerland, to his father, a baker and his mother, a doctor.: 131 == Education and career == Rebstein attended post-secondary school in Winterthur, and after graduating in 1860, went on to study for a year at Collège de France.: 131 He was professor of mathematics and physics in Zürich from 1877 to 1898. He was awarded his doctorate in 1895 from the Humboldt University of Berlin for his work Bestimmung aller reellen Minimalflächen, die eine Schaar ebener Curven enthalten, denen auf der Gauss'schen Kugel die Meridiane entsprechen. He is best known for his work in surveying, and for introducing the traverse method in Switzerland. Throughout his career, Rebstein was appointed as surveying expert for a number of cantons, including Thurgau (1863–1881), St. Gallen (1881–1894), Zürich (1886–1892), and Luzern (1894–1907).: 131 In 1868 he was elected to the Swiss Concordat of Geometers, and served as its president from 1887 until his death in 1907.: 132 In 1905 he was awarded an honorary doctorate from the University of Zürich, for "outstanding contributions to actuarial sciences". Rebstein was a member of the organizing committee for the first meeting of the International Congress of Mathematicians.: 79 == Death == Rebstein suffered from kidney disease for the last several years of his life, and died in 1907 in Zürich.: 133 == Publications == Rebstein's publications included:: 132 Lehrbuch über praktische Geometrie mit besonderer Berücksichtigung der Theodolitmessung (1868) Die Kartographie der Schweiz, dargestellt in ihrer historischen Entwicklung (1883) Mitteilungen über die Stadtvermessung von Zürich1 (1892) == See also == List of German-language philosophers == Notes == == References ==
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Wikipedia:Johann Makowsky#0
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Johann (János) A. Makowsky (born March 12, 1948) is a Hungarian-born naturalised Swiss mathematician who works in mathematical logic and the logical foundations of computer science and combinatorics. He studied at ETH Zurich from 1967–73. He was a student in Zürich of Ernst Specker and Hans Läuchli in mathematical logic, (Diploma in Mathematics and Physics 1971, Dr. math.sc. in 1974), of Beno Eckmann (Topology and Geometry) and Volker Strassen (Algorithmics), and in Warsaw of Andrzej Mostowski and Witek Marek, where he spent 1972 as an exchange student. Makowsky held visiting positions at the Banach Center in Warsaw (Poland), Stanford University (USA), Simon Fraser University (Canada), University of Florence (Italy), MIT (USA), Lausanne University and ETH Zurich (Switzerland). He held regular positions at the Free University of Berlin and the Technion – Israel Institute of Technology (Haifa, Israel) where he was a full professor. Among his various contributions are: In model theory, the solution of two open problems in categoricity theory and his study of logics with various interpolation and compactness properties (partially with Saharon Shelah and Jonathan Stavi). In database theory, the first undecidability result of the consequence problem for database dependencies (with Ashok Chandra and Harry Lewis), his work unifying the entity–relationship model and the relational model of databases (with Victor Markowitz), and his work on Boyce–Codd normal form (with E.V. Ravve). In logic programming, his fundamental studies of Horn formulas and their complexity (partially with B. Mahr and A. Itai) In graph algorithms, his unifying approach to tree-width and clique-width via model theory, leading to a general theory of graph polynomials and their definability in various logical formalisms (partially with I. Averbouch, Bruno Courcelle, B. Godlin, T. Kotek, U. Rotics and Boris Zilber). Makowsky was a founding member of the European Association of Computer Science Logic in 1992, its vice-president (2002–2004) and president (2004–2009), and was a member of EACSL's executive council till 2014. During his presidency he established the EACSL Ackermann Award for outstanding PhD theses in computer science logic. In 2008, an event dedicated to Makowsky on his 60th birthday was co-located with the annual meeting of the EACSL. Since 2016, he is a Professor Emeritus at the Faculty of Computer Science at the Technion, and continues his research and teaching and supervising graduate students == References == == External links == Official website Johann A. Makowsky at DBLP Bibliography Server
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Wikipedia:Johannes Boersma#0
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Johannes Boersma (5 December 1937 Marrum – 29 November 2004 Eindhoven) was a Dutch mathematician who specialized in mathematical analysis. His PhD advisor at the University of Groningen was Adriaan Isak van de Vooren. == Selected publications == Boersma, J.; Dempsey, J.P. (1992). "On the evaluation of Legendre's chi-function" (PDF). Mathematics of Computation. 59 (199): 157–163. doi:10.2307/2152987. JSTOR 2152987. == References == Johannes Boersma at the Mathematics Genealogy Project O'Connor, John J.; Robertson, Edmund F., "Johannes Boersma", MacTutor History of Mathematics Archive, University of St Andrews Obituary: Johannes Boersma (1937–2004), a remarkable mathematician by H. K. Kuiken
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Wikipedia:Johannes Buteo#0
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Johannes Buteo (born Jean Borrel, Latinized as Buteonis or given as Boteo, Buteon, Bateon) (c. 1485 – c. 1560) was a French mathematician and logician. Among his contributions was an attempt to calculate the supposed dimensions of Noah's Ark to fit all the world's animals. Buteo was born in Dauphine or possibly Charpey and belonged to the order of St. Anthony. He studied under Oronce Fine and wrote on geometry and exposed Fine publishing a few books Opera Geometrica (1554), Logistica (1559), De quadratura circuli libri duo (1559). He died in a cloister about 1560-64 but some sources suggest he died in Canar in 1572. His contributions included a systematic way of eliminating unknowns in systems of linear equations which he demonstrated in Logistica with three equations and three unknowns. == References ==
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Wikipedia:Johannes Frischauf#0
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Johannes Frischauf (17 September 1837 in Vienna – 7 January 1924 in Graz) was an Austrian mathematician, physicist, astronomer, geodesist and alpinist. == Life and work == Frischauf passed the matura at the Academic Gymnasium in Vienna and in 1857 studied mathematics, physics, astronomy at the University of Vienna, as well as geodesy, chemistry, mechanics at the Technischen Hochschule Vienna. He obtained the doctorate in 1864, and became Privatdozent for mathematics at the University of Vienna and assistant at the observatory of the university. In 1863 he was habilitated in mathematics. Starting in 1863, he was professor at the University of Graz for pure and applied mathematics. He worked together with Ludwig Boltzmann. Frischauf developed a new method of map design and wrote textbooks on arithmetics and geometry – for instance in 1872 and 1876 he wrote summaries of the then current knowledge about non-Euclidean geometry (which he called "absolute geometry"). In 1885 he was elected as a member of the Leopoldina. Starting with 1868, Frischauf pioneered the touristic development of the Sannthaler and Steiner Alps by opening ways and huts. The streets over the Paulitschsattel and the connection Sulzbach-Leutsch were built because of his initiatives. Together with Franz von Juraschek and Mathias Spreiz he was the first to climb the Admonter Reichenstein. Frischauf defended the view at a time of violent national conflicts, that alpinism should not be subordinated under nationalism or religion or political views. He participated in the foundation of the Croatian Mountaineering Association. Frischauf's Funerary urn was placed on the Scheichenspitze in the Dachstein Mountains. His estate is located at the University of Graz. == Works (selection) == Über die Bahn der Asia. In: Sitz. Berichte Kais. Akad. Wiss. Wien, Mat.-nat. Cl. Band 45, 1862, pp. 435–442. Bahnbestimmung des Planeten 67 Asia. In: Sitz. Berichte Kais. Akad. Wiss. Wien, Mat.-nat. Cl. Band 53, 1866, pp. 96–141. Einleitung in die analytische Geometrie. Leuschner & Lubensky, Graz 1871. Zum Rechnen mit unvollständigen Zahlen. In: Zeitschrift math. naturw. Unterr. Band 26, 1895, pp. 161–172. Beiträge zur Landesaufnahme und Kartographie des Erdsphäroids. B. G. Teubner, Leipzig 1919. Hochthor bei Johnsbach. In: Jahrbuch Steir. Gebirgsverein. 1873, p. 41. Reichenstein bei Admont. In: Jahrbuch Steir. Gebirgsverein. 1873, p. 54. Frischauf, J. (1876). Elemente der absoluten Geometrie. Leipzig.{{cite book}}: CS1 maint: location missing publisher (link) Die Sannthaler Alpen. Brockhausen und Bräuser, Wien 1877. Ein Ausflug auf den Monte Baldo.Wien 1883, Wiener Touristen-Führer.11. Das Uskoken-Gebirge. In: Zeitschrift DÖAV (1890), pp. 474–484. [1] Krakau bei Murau. Steirische Sommerfrischen, Band 1, Leuschner & Lubensky, Graz 1896, Hrg. vom Steirischen Gebirgsvereine. == References == == External links == Literature by and about Johannes Frischauf in the German National Library catalogue Martin Fürnkranz: Bibliographie Publications of J. Frischauf in Astrophysics Data System
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Wikipedia:Johannes Kemperman#0
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Johannes Henricus Bernardus Kemperman (July 16, 1924 – June 13, 2011) was a Dutch mathematician. He taught at the University of Rochester for 25 years, and also worked at Purdue University and Rutgers University for ten years, each. Born in Amsterdam, he received his education from the University of Amsterdam. == Selected publications == Kemperman, J. H. B. (1969). "On the optimum rate of transmitting information". Probability and Information Theory. Lecture Notes in Mathematics. Vol. 89. pp. 126–169. doi:10.1007/BFb0079123. ISBN 978-3-540-04608-0. Kemperman, J. H. B. (1963). "A Wiener–Hopf Type Method for a General Random Walk with a Two-Sided Boundary". Annals of Mathematical Statistics. 34 (4): 1168–1193. doi:10.1214/aoms/1177703855. JSTOR 2238330. Kemperman, J. H. B. (1960). "On small sumsets in an abelian group". Acta Mathematica. 103 (1–2): 63–88. doi:10.1007/BF02546525. == References == == External links == Johannes Kemperman at the Mathematics Genealogy Project
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Wikipedia:Johannes Mollerup#0
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Johannes Mollerup (3 December 1872 – 27 June 1937) was a Danish mathematician. Mollerup studied at the University of Copenhagen, and received his doctorate in 1903. Together with Harald Bohr, he developed the Bohr–Mollerup theorem which provides an easy characterization of the gamma function. == References ==
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Wikipedia:Johannes Runnenburg#0
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Johannes Theodorus Runnenburg (19 February 1932 – 16 April 2008) was a Dutch mathematician and professor of probability theory and analysis at the University of Amsterdam from 1962 to 1997. == Biography == Born in Amsterdam he received his MA in Mathematics in 1956 at the University of Amsterdam, and his PhD cum laude in Mathematics and Physics in 1960 with a thesis entitled "On the Use of Markov Processes in One-server Waiting-time Problems and Renewal Theory" advised by Nicolaas Govert de Bruijn. Runnenburg was appointed Lector in Probability theory and analysis at the University of Amsterdam in 1961. In 1962, he was promoted to Professor of Probability theory and analysis, and from 1966 to his retirement in 1997 was Professor of Pure and Applied Mathematics. Among his doctorate students were Gijsbert de Leve (1964), Laurens de Haan (1970), Fred Steutel (1971), Wim Vervaat (1972), August Balkema (1973), Frits Göbel (1974), Arie Hordijk (1974), Aegle Hoekstra (1983), Peter de Jong (1988) and Leo Klein Haneveld (1996). == Publications == Machines Served by a Patrolling Operator. 1957 On the use of Markov processes in one-server waiting-time problems and renewal theory. 1960 Einige voorbeelden van stochastische processen: openbare les Universiteit van Amsterdam. 1961. An Example Illustrating the Possibilities of Renewal Theory and Waiting-time Theory for Markov-dependent Arrival-intervals. 1961 On K.L. Chung's problem of imbedding a time-discrete Markov chain in a time-continuous one for finitely many states. With Carel Louis Scheffer. Amsterdam : Mathematisch Centrum, 1962. == References == == External links == Prof. dr. J.T. Runnenburg, 1932–2008
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Wikipedia:Johannes Sjöstrand#0
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Johannes Sjöstrand (born 1947) is a Swedish mathematician, specializing in partial differential equations and functional analysis. Sjöstrand received his doctorate in 1972 from Lund University under Lars Hörmander. Sjöstrand taught at the University of Paris XI and he is a professor at the University of Burgundy in Dijon. He is a member of the Royal Swedish Academy of Sciences and, since 2017, a member of the American Academy of Arts and Sciences. His research deals with microlocal analysis. He has investigated, inter alia, the Schrödinger equation of an electron in a magnetic field (with a spectrum of the Hofstadter butterfly), resonances in the semiclassical limit, and quantum tunneling in the semiclassical limit. == Selected publications == Operators of principal type with interior boundary conditions. Acta mathematica 130, no. 1 (1973): 1–51. doi:10.1007/BF02392261 with Anders Melin: "Fourier integral operators with complex-valued phase functions." In Fourier integral operators and partial differential equations, pp. 120–223. Springer, Berlin, Heidelberg, 1975. doi:10.1007/BFb0074195 with Richard Melrose: Singularities of boundary value problems. I, Comm. Pure Appl. Math., vol. 31, 1978, pp. 593–619 doi:10.1002/cpa.3160310504; Singularities of boundary value problems. II, Comm. Pure Appl. Math., vol. 35, 1982, pp. 129–168 doi:10.1002/cpa.3160350202 with Melrose: A calculus for Fourier Integral Operators in domains with boundary and applications to the oblique dérivative problem, Comm. in PDE, 2, 1977, pp. 857–935, see Helffer Propagation des singularités pour des problèmes aux limites, Séminaire Bourbaki, Nr. 525, 1978/79 with B. Lascar: Singularités analytiques microlocales, Astérisque 95, 1982 with Bernard Helffer: Multiple wells in the semi-classical limit, Part 1, Communications in PDE, 9, 1984, 337–408 (6 parts altogether, see Robert Didier Analyse semi-classique de l'effet tunnel, Séminaire Bourbaki 665, 1985/86) with Helffer: Résonances en limite semi-classique, Mémoire SMF, Nr. 24–25, 1986 with Helffer: Analyse semi-classique pour l'équation de Harper : (avec application à l'équation de Schrödinger avec champ magnétique), Mémoire SMF, Nr. 34, 1988, Nr. 39, 1989, Nr. 40, 1990 (Parts 1–3) Asymptotique des résonances pour des obstacles, Séminaire Bourbaki, Nr. 724, 1989/90 with Helffer and P. Kerdelhué: Le papillon de Hofstadter revisité, Mémoire SMF, Nr. 43, 1990 with Maciej Zworski: Complex scaling and the distribution of scattering poles, J. Amer. Math. Soc. 4, 1991, 729–769 doi:10.1090/S0894-0347-1991-1115789-9 with Alain Grigis: Microlocal analysis for differential operators: an introduction, Cambridge University Press 1994. with Mouez Dimassi: Spectral asymptotics in the semi-classical limit, Cambridge University Press 1999 with Maciej Zworski: Asymptotic distribution of resonances for convex obstacles. Acta Mathematica 183, no. 2 (1999): 191–253. doi:10.1007/BF02392828 Microlocal Analysis, in: Jean-Paul Pier (ed.): Development of mathematics 1950–2000. Birkhäuser, 2000 Complete asymptotics for correlations of Laplace integrals in the semi-classical limit, Paris, SMF 2000 with Carlos E. Kenig and Gunther Uhlmann: "The Calderón problem with partial data." Annals of mathematics (2007): 567–591. JSTOR 20160036 == References == == External links == Johannes Sjöstrand, IMB, Université de Bourgogne
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Wikipedia:Johannes Widmann#0
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Johannes Widmann (c. 1460 – after 1498) was a German mathematician. The + and - symbols first appeared in print in his book Mercantile Arithmetic or Behende und hüpsche Rechenung auff allen Kauffmanschafft published in Leipzig in 1489 in reference to surpluses and deficits in business problems. Born in Eger, Bohemia, Widmann attended the University of Leipzig in the 1480s. In 1482 he earned his "Baccalaureus" (Bachelor of Art degree) and in 1485 his "Magister" (doctorate). Widman published Behende und hübsche Rechenung auff allen Kauffmanschafft (German; i.e. Nimble and neat calculation in all trades), his work making use of the signs, in Leipzig in 1489. Further editions were published in Pforzheim, Hagenau, and Augsburg. Handwritten entries in a surviving collection show that after earning his "Magister" Widman announced holding lectures on e.g. calculating on the lines of a calculating board and on algebra. There is evidence that the lecture on algebra actually took place, making it the first known university lecture on this topic. Around 1495 Widmann published the Latin writings Algorithmus integrorum cum probis annexis, Algorithmus linealis, Algorithmus minutiarum phisicarum, Algorithmus minutiarum vulgarium, Regula falsi apud philosophantes augmenti et decrementi appellata und Tractatus proportionum plusquam aureus. He died in Leipzig. When Adam Ries was in Erfurt between 1518 and 1522 he got to know Widmann's algebra lecture script (today in the Saxon State Library) wherefrom he took examples for his own writings. == References == == Sources == Barbara Gärtner, Johannes Widmanns „Behende und hübsche Rechenung“. Die Textsorte „Rechenbuch“ in der Frühen Neuzeit, Tübingen 2000. (Germanistische Linguistik. 222.) M. Cantor, Vorlesungen über Geschichte der Mathematik II (Leipzig, 1913), pp. 228s. K. Fogel, Merchants' aids in practical arithmetic from the Middle Ages (Russian), Istor.-Mat. Issled. No. 23 (1978), pp. 235–249; 359. W. Kaunzner and H. Wussing (eds.), Adam Ries, Coss (B.G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1992). Karl Röttel, Johannes Widmann – Am Wendepunkt der Mathematikgeschichte. In: Schatzkammer der Rechenkunst. Annaberg-Buchholz 2008. K. Vogel, Biography in Dictionary of Scientific Biography (New York 1970-1990). Franz Xaver Wilhelm, Zur Biographie des Mathematikers Johann Widmann von Eger. In: Mitteilungen des Vereins für Geschichte der Deutschen in Böhmen, Volume 45 (1907), pp. 429–430.
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Wikipedia:John A. Adam (mathematician)#0
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John Anthony Adam is a British-American applied mathematician known for his work on patterns in nature and on mathematical modeling of the growth patterns of cancer and blood vessels. He is University Professor of Mathematics at Old Dominion University in Virginia. == Education and career == Adam is a 1971 graduate, with first-class honours, from Queen Elizabeth College. He completed his Ph.D. in 1974 at University College London. His dissertation, A Theoretical Study of Magnetohydrodynamic Processes in Solar Active Regions, was jointly supervised by astrochemist Gillian Peach and astrophysicist Carole Jordan. After working as a researcher in theoretical astronomy and applied mathematics, respectively at the University of Sussex and University of St Andrews, he became a lecturer in mathematics at the New University of Ulster in 1978, while also taking a research affiliation at the Dublin Institute for Advanced Studies. In 1983 he came to the US as a Fulbright scholar at the University of Rochester, and in 1984 he moved to Old Dominion University. He was named University Professor there in 1999. == Book == Adam is the author of books including: A Survey of Models for Tumor-Immune System Dynamics (edited with Nicola Bellomo, Springer, 1997) Mathematics in Nature: Modeling Patterns in the Natural World (Princeton University Press, 2003) Guesstimation: Solving the World's Problems on the Back of a Cocktail Napkin (with Lawrence Weinstein, Princeton University Press, 2008) A Mathematical Nature Walk (Princeton University Press, 2009) X and the City: Modeling Aspects of Urban Life (Princeton University Press, 2012) Rays, Waves, and Scattering: Topics in Classical Mathematical Physics (Princeton University Press, 2017) == Recognition == In 2007, the State Council of Higher Education in Virginia (SCHEV) gave Adam their Outstanding Faculty Award. In 2012, Adam won the Carl B. Allendoerfer Award of the Mathematical Association of America for an exposition of blood vessel modeling. == References == == External links == Home page
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Wikipedia:John B. Little (mathematician)#0
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John Brittain Little (born 1956) is a retired American mathematician, the author of several books in algebraic geometry and the history of mathematics. He is distinguished professor emeritus in the departments of mathematics and computer science at the College of the Holy Cross. == Education and career == Little was born January 15, 1956, in Elmira, New York. He majored in mathematics at Haverford College, graduating in 1976, and completed a Ph.D. at Yale University in 1980. His dissertation, Translation Manifolds and the Converse of Abel's Theorem, was supervised by Bernard Saint-Donat. He joined the College of the Holy Cross as an assistant professor in 1980, became an associate professor in 1986, and was promoted to full professor in 2003. He was Anthony and Renee Marlon Professor in the Sciences from 2012 to 2015. == Books == Little's textbooks include: Using Algebraic Geometry (with David Cox and Donal O'Shea, Springer, Graduate Texts in Mathematics, 1998; 2nd ed., 2005) Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra (with David Cox and Donal O'Shea, Springer, 1992; 4th ed., 2015) A Course in Linear Algebra (with David B. Damiano, Dover, 2011) Toric Varieties (with David Cox and Henry K. Schenk, AMS, Graduate Studies in Mathematics, 2011) Modeling and Data Analysis: An Introduction with Environmental Applications (American Mathematical Society, Holy Cross Bookshelf 41, 2019) In his retirement he has shifted his interests to focus more on the history of mathematics and the translation of works in ancient Greek mathematics. His books in this area include: Practical Geometry (Clavius) (Holy Cross Bookshelf 57, 2022) Pappus of Alexandria, Book VIII of the Mathematical Collection (Holy Cross Bookshelf 62, 2023) Pappus of Alexandria, Book III of the Mathematical Collection (Holy Cross Bookshelf 63, 2023) == Recognition == Little's book Ideals, Varieties, and Algorithms was the 2016 winner of the Leroy P. Steele Prize for mathematical exposition. Little received the 2020 Paul R. Halmos – Lester R. Ford Award for his paper "The many lives of the twisted cubic", published in The American Mathematical Monthly. == References == == External links == Home page John B. Little publications indexed by Google Scholar
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Wikipedia:John Brian Helliwell#0
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John Brian Helliwell FRSE FIMA (1924–1992) was a British mathematician and astrophysicist. He was Professor of Engineering Mathematics at Bradford University 1968 to 1985. He is remembered for his work on the behaviour of gases at transonic speeds and upon the action of conductive gases within magnetic fields. == Life == He was born in York on 2 January 1924. He was educated at Goole Grammar School He then studied at the University of Leeds, graduating with a BSc in 1945. His studies were interrupted by the Second World War during which he served first at the Royal Aircraft Establishment in Farnborough then in the gas department of Metropolitan-Vickers in Manchester. He gained his PhD from Leeds in 1949 on the subject of control theory. After his PhD he remained in academia, first at Birmingham and Manchester, then as a Lecturer at the Royal College of Science and Technology in Glasgow, now known as Strathclyde University. In 1963 he was elected a Fellow of the Royal Society of Edinburgh. His proposers were Donald Pack, Benjamin Noble, Reginald Lord, and Patrick Dunbar Ritchie. In 1967 he moved to Bradford as Professor of Engineering Mathematics at Bradford University, until his retirement due to ill-health in 1985. He returned to North Yorkshire where he died on 14 July 1992. == Family == In 1951 he married Joyce Hutchinson. They had three daughters. == References ==
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Wikipedia:John Bryce McLeod#0
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John Bryce McLeod, (23 December 1929 – 20 August 2014) was a British mathematician, who worked on linear and nonlinear partial and ordinary differential equations. == Life and education == McLeod was born in Aberdeen, Scotland, on 23 December 1929. He was educated at Aberdeen Grammar School; the University of Aberdeen, where he took a first in Mathematics and Natural Philosophy in 1950; and Christ Church, Oxford, where he took a first in Mathematics in 1952. He was a Harmsworth Senior Scholar at Merton College, Oxford, from 1955 to 1956. He obtained his PhD in 1959 under the supervision of Edward Charles Titchmarsh at the University of Oxford. He was a junior lecturer in Mathematics at the University of Oxford from 1956 to 1958, and a lecturer in mathematics at the University of Edinburgh from 1958 to 1960. He then returned to Oxford to take up a Fellowship in Pure Mathematics at Wadham College. He remained in Oxford until 1988, becoming a university lecturer in 1970, and a senior research fellow of the Science and Engineering Research Council from 1986 to 1991. In 1988 McLeod took up a professorship at the University of Pittsburgh, where he remained until his retirement in 2007. McLeod married Eunice Third in 1956; they had three sons and a daughter. He died in England on 20 August 2014, aged 84. == Awards and honours == In 1965, he was awarded the Sir Edmund Whittaker Memorial Prize. he was elected a Fellow of the Royal Society of Edinburgh in 1974, and received the Society's Keith Medal in 1987. He was elected a Fellow of the Royal Society (FRS) in 1992. In 2011 he was awarded the Naylor Prize and Lectureship. == References ==
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Wikipedia:John Charles Fields#0
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John Charles Fields, FRS, FRSC (May 14, 1863 – August 9, 1932) was a Canadian mathematician and the founder of the Fields Medal for outstanding achievement in mathematics. == Career == Born in Hamilton, Canada West, to a leather shop owner, Fields graduated from Hamilton Collegiate Institute in 1880 and the University of Toronto in 1884 before leaving for the United States to study at Johns Hopkins University in Baltimore, Maryland. Fields received his Ph.D. in 1887. His thesis, entitled Symbolic Finite Solutions and Solutions by Definite Integrals of the Equation dny/dxn = xmy, was published in the American Journal of Mathematics in 1886. Fields taught for two years at Johns Hopkins before joining the faculty of Allegheny College in Meadville, Pennsylvania. Disillusioned with the state of mathematical research in North America at the time, he left for Europe in 1891, locating primarily in Berlin, Göttingen and Paris, where he associated with some of the greatest mathematical minds of the time, including Karl Weierstrass, Felix Klein, Ferdinand Georg Frobenius and Max Planck. Fields also began a lifelong friendship with Gösta Mittag-Leffler. He began publishing papers on a new topic, algebraic functions, which would prove to be the most fruitful research field of his career. Fields returned to Canada in 1902 to lecture at the University of Toronto. Back in the country of his birth, he worked tirelessly to raise the status of mathematics within academic and public circles. He successfully lobbied the Ontario Legislature for an annual research grant of $75,000 for the university and helped establish the National Research Council of Canada, and the Ontario Research Foundation. Fields served as president of the Royal Canadian Institute from 1919 until 1925, during which time he aspired to mold the institute into a leading centre of scientific research, although with mixed success. His efforts, however, were pivotal in making Toronto the location of the 1924 International Congress of Mathematicians (ICM). He was an Invited Speaker of the ICM in 1912 at Cambridge, in 1924 at Toronto, and in 1928 at Bologna. Fields is best known for his development of the Fields Medal, which is considered by some to be the Nobel Prize of mathematics, although there are differences between the awards. First awarded in 1936, the medal was reintroduced in 1950 and has been awarded every four years since. It is awarded to two, three or four mathematicians under the age of 40 who have made important contributions to the field. Fields began planning the award in the late 1920s but, due to deteriorating health, never saw the implementation of the medal in his lifetime. He died on August 9, 1932, after a three-month illness; in his will, he left $47,000 for the Fields Medal fund. His plan for the Fields medal was completed by J. L. Synge. == Honours == Fields was elected a fellow of the Royal Society of Canada in 1907 and of the Royal Society of London in 1913. The Fields Institute at the University of Toronto was named in his honour. == References == == Further reading == Riehm, Elaine; Hoffman, Frances (2011). Turbulent Times in Mathematics: The Life of J. C. Fields and the History of the Fields Medal. Providence, RI: American Mathematical Society. ISBN 978-0-8218-6914-7. == External links == Fields Institute Biography O'Connor, John J.; Robertson, Edmund F., "John Charles Fields", MacTutor History of Mathematics Archive, University of St Andrews John Charles Fields at the Mathematics Genealogy Project John Charles Fields archival papers held at the University of Toronto Archives and Records Management Services
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Wikipedia:John Clayton Taylor#0
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John Clayton Taylor (born 4 August 1930) is a British mathematical physicist. He is an Emeritus Professor of Mathematical Physics at the Department of Applied Mathematics and Theoretical Physics of the University of Cambridge and an Emeritus Fellow of Robinson College. He is the father of mathematician Richard Taylor. == Education == Taylor earned his PhD from the University of Cambridge in 1956, under the supervision of Richard J. Eden and Abdus Salam. His thesis was entitled Renormalisation and Related Topics in Quantum Field Theory. == Research == Taylor has made contributions to quantum field theory and the physics of elementary particles. His contributions include: the discovery (also made independently by Lev Landau) of singularities in the analytical structure of the Feynman integrals for processes in quantum field theory, the PCAC nature of radioactive decay of the pion and the discovery in 1971 of the so-called Slavnov–Taylor identities, which control symmetry and renormalisation of gauge theories. With various collaborators, in 1980 he discovered that real and virtual infrared divergences do not cancel in QCD as they do in QED. They also showed how these infrared divergences exponentiate. In addition, they contributed to the resummation programme in thermal QCD, simplifying the "hard" part of the effective action. Later, they studied complications arising from the non-polynomial nature of the QCD Hamiltonian in the (unitary) Coulomb gauge. == Books == Gauge Theories of Weak Interactions (1976) Hidden Unity in Nature's Laws (2001) Gauge Theories in the Twentieth Century (2001) == Awards and honours == Taylor was elected a Fellow of the Royal Society (FRS) in 1981. His certificate of election reads: Distinguished for his contributions to the Quantum Theory of Fields and the Physics of Elementary Particles. His important works concern (a) the discovery (also made independently by L.D. Landau) of singularities in the analytical structure of the Feynman integrals for processes in Quantum Field Theory, and (b) the discovery of the so-called Slavnov–Taylor identities in Gauge Theories. He has made significant contributions to Quantum Chromodynamics where his use of the axial gauge has made possible recent advances in "perturbative Q.C.D.". He has also contributed to weak interaction theory, over a long period, and most recently to the elucidation of the gauge structure of the unified weak and electromagnetic interaction. == References ==
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Wikipedia:John Conrad Jaeger#0
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John Conrad Jaeger, FRS (30 July 1907 – 15 May 1979), was an Australian mathematical physicist. == Biography == Jaeger was born in Sydney, Australia, to Carl Jaeger, a cigar manufacturer of German origin. In 1924, Jaeger entered Sydney University at the age of 16, studying engineering, mathematics, and physics, and earned a B.Sc. in 1928. He then spent two additional years at Cambridge University, completing Part II of the Mathematical Tripos and continuing with research in theoretical physics. In 1936, Jaeger began teaching mathematics at the University of Tasmania. He later moved to the University of Sydney, where he was appointed to a chair in mathematics and collaborated with Professor Horatio Scott Carslaw on applying mathematics to heat conduction. Their co-authored textbook, Conduction of Heat in Solids, became a seminal work in the field. After the World War II, Jaeger returned to Tasmania as a senior lecturer and authored several books, including new editions of Operational Methods in Applied Mathematics (with Carslaw in 1948) and Conduction of Heat in Solids (with Carslaw in 1959). He also published An Introduction to Applied Mathematics in 1951 and An Introduction to the Laplace Transformation in 1959. In 1951, Jaeger was invited to chair the new Department of Geophysics at the Australian National University in Canberra. He moved there in 1952, delivering his inaugural lecture in 1953 and initiating several research projects in the following years. The department was later named the Department of Geophysics and Geochemistry. Jaeger retired from ANU in 1972. Jaeger was married twice: first to Sylvia Rees, from whom he divorced in 1950, and later to Martha Elizabeth (Patty) Clarke. He had no children from either marriage. Jaeger died on 15 May 1979 in Canberra and was cremated. In honor of his contributions to Australian Earth science, the Australian Academy of Science annually awards the Jaeger Medal. == Honours and awards == 1930 Mayhew Prize for Mathematics from the Faculty of Mathematics, University of Cambridge 1947 Thomas Ranken Lyle Medal for Physics and Mathematics from the Australian National Research Council 1947 Walter Burfitt Prize from the Royal Society of New South Wales 1954 Elected Fellow of the Australian Academy of Science 1970 Elected Fellow of the Royal Society 1971 Rankine Lecture to the British Geotechnical Association == References ==
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Wikipedia:John D. P. Meldrum#0
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John David Philip Meldrum (18 July 1940 in Rabat, Morocco; died 9 August 2018 in Edinburgh, Scotland) was a British mathematician. Meldrum was an algebraist and his research was mostly related to group theory. == Biography == Meldrum was born in Rabat, Morocco. In 1964 he was appointed as a supernumerary fellow and college lecturer in mathematics at Emmanuel College. Meldrum received his PhD from the University of Cambridge in 1967 on the topic of "Central Series in Wreath Products". His supervisor was Derek Roy Taunt. In 1969 he became a lecturer for mathematics at the University of Edinburgh and in 1982 he was appointed there as a senior lecturer. He died on 9 August 2018 in Edinburgh after a battle with the Parkinson's disease. == Books == === As an author === John D. P. Meldrum (1985). Near-rings and Their Links with Groups. Longman Higher Education. ISBN 978-0273087014. John D. P. Meldrum (1995). Wreath Products of Groups and Semigroups. Chapman and Hall/CRC. ISBN 9780582026933. === As a translator === Meldrum translated the following book by Nicolas Bourbaki: Nicolas Bourbaki (1994). Elements of the History of Mathematics. Translated by John D. P. Meldrum. Springer Verlag. ISBN 978-0387193762. == References ==
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Wikipedia:John Dalgleish Donaldson#0
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John Dalgleish Donaldson (born 5 September 1941), is a Scots-Australian professor and father of Queen Mary of Denmark, the wife of King Frederik X of Denmark. == Family and marriages == John Donaldson was born at Cockenzie and Port Seton in East Lothian, Scotland, the son of Captain Peter Donaldson and his wife, Mary Dalgleish. Captain Donaldson had sailed regularly from Port Seton Harbour, then in 1962 it is recorded the vessel Shearwater navigating the Bass Strait Islands with a cargo of livestock under his command was lost off Ninth Island. He and his crew were saved and there are still remains of the ship on the island today. On 31 August 1963, John Donaldson married his first wife, Henrietta Clark Horne (1942–1997), at Port Seton. They emigrated to Tasmania, Australia, in November of that year. Donaldson's parents, his elder brother Peter and younger sister Roy also emigrated to Tasmania. His father then joined a large maritime trading company as a captain. They had four children, Jane Alison Donaldson (born 26 December 1965), Patricia Anne Donaldson (born 16 March 1968), John Stuart Donaldson (born 9 July 1970) and Mary Elizabeth Donaldson (born 5 February 1972), married in 2004 to Frederik, Crown Prince of Denmark (now King Frederik X). In addition to British citizenship, Donaldson obtained Australian citizenship in 1975. Henrietta died on 20 November 1997, and Donaldson later married Susan Elizabeth Horwood (born 1940) on 5 September 2001. She is a novelist who writes under the names Susan Moody, Susannah James and Susan Madison. == Education == In 1963, Donaldson graduated from the University of Edinburgh with a BSc degree with honours in Mathematics and Physics. After Edinburgh, Donaldson moved to Australia to work under the direction of mathematician Professor David Elliott at the University of Tasmania, taking a PhD in Mathematics in 1967. == Career == After receiving his doctorate in 1967, Donaldson remained at the University of Tasmania as a lecturer in applied mathematics, becoming Dean of UTAS Faculty of Science until retiring in 2003. With the Earl of Dunmore, he served on the Scottish Australian Heritage Council. Professor of Applied Mathematics at the Korea Advanced Institute of Science and Technology (KAIST), Donaldson was previously a Visiting Professor at several universities in Houston, Montreal, then Oxford, from 2004 at Aarhus University and from 2006 also at the University of Copenhagen. A keen rugby player and a member of the University Associates Rugby Union Club, John Donaldson is renown as having captained the last Tasmanian State Rugby Team to play against a touring New Zealand All Black Team, at Queensborough Oval, Hobart, on 25 May 1968. == Honours, title and coat of arms == PhD (UTAS), doctorate received by Dr J. D. Donaldson, 1968 Grand Cross of the Order of the Dannebrog, 2004 Upon the marriage of his daughter to the then Crown Prince Frederik in 2004, Donaldson was appointed to the Order of the Dannebrog. In accordance with the statutes of the Danish Royal Orders, both he and his daughter were granted arms to display in the Chapel of the Royal Orders at Frederiksborg Castle. == Publications == Chick, H.L.; Collis, K.F.; Donaldson, J.D.; Watson, J.M. (1995). Professional development in mathematics for teachers: Who, what, why and how. 15th Biennial Conference of the Australian Association of Mathematics Teachers. Darwin. pp. 65–71. S2CID 180618518. Donaldson, John D.; Jezewski, Donald J. (1977). "An element formulation for perturbed motion about the center of mass". Celestial Mechanics and Dynamical Astronomy. 16 (3): 367–387. Bibcode:1977CeMec..16..367D. doi:10.1007/BF01232661. S2CID 123349501. Roberts, Michael S.; Donaldson, John D.; Rowland, Malcolm (1988). "Models of hepatic elimination: Comparison of stochastic models to describe residence time distributions and to predict the influence of drug distribution, enzyme heterogeneity, and systemic recycling on hepatic elimination". Journal of Pharmacokinetics and Pharmacodynamics. 16 (1): 41–83. doi:10.1007/BF01061862. PMID 3373419. S2CID 360960. Wotherspoon, S.J.; Donaldson, J.D. (1996). "Finite differences and internal tides-representing the boundary". Deep-Sea Research Part I. 43 (6): 949–958. Bibcode:1996DSRI...43..949W. doi:10.1016/0967-0637(96)00029-5. ISSN 0967-0637. Donaldson, John D.; Rahman, Qazi Ibadur (1972). "Inequalities for polynomials with a prescribed zero". Pacific Journal of Mathematics. 41 (2): 375–378. doi:10.2140/pjm.1972.41.375. == Notes and references == Genealogisches Handbuch des Adels, Fürstliche Häuser, Reference: 2004 11 The Ancestry of Mary Elizabeth Donaldson 2006, Reitwiesner, William Addams, Reference: nr.2 == External links == John D. Donaldson @ Mathematics Genealogy Project
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Wikipedia:John David Maitland Wright#0
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John David Maitland Wright (20 May 1942 – 7 August 2023) was a British mathematician, specialising mainly in functional analysis and operator theory. == Early life and education == Wright was born on 20 May 1942, the son of Phyllis (née Harris) and mathematician Sir Edward Maitland Wright. He went to Aberdeen Grammar School and subsequently studied mathematics at the University of Aberdeen. Wright graduated with first class honours from University of Aberdeen in 1964. For his postgraduate studies he went to Christ Church, Oxford where he obtained his doctorate in 1967. For his thesis on measure theory, Wright was awarded the Oxford Senior Mathematical Prize and Johnson Prize. == Career == In 1965 Wright was appointed to a junior college lectureship at Christ Church and in 1968 became the Leathersellers’ Tutorial Fellow in Mathematics at St Catherine’s College, Oxford. In 1971, at the age of 28, he was appointed professor of pure mathematics at the University of Reading and later became head of department. Wright was treasurer of the London Mathematical Society from 1986 to 1994 and deputy director of the Isaac Newton Institute in Cambridge from 1994 to 1996. In 2004 he was appointed to a Sixth Century Chair at the University of Aberdeen which he held until his retirement in 2015. Subsequently he became senior associate research fellow of Christ Church, Oxford until 2021. Wright worked mainly in functional analysis, measure theory, operator theory and related areas, including applications to quantum theory. He published over 130 papers in mathematical journals. A significant part of Wrights research work was on AW*-Algebras and monotone complete C*-algebras culminating in a book coauthored with Kazuyuki Saito. Other notable work include results on Jordan C*-Algebras in 1977 and a solution of the Mackey-Gleason Problem in 1994 (with L.J. Bunce). == Honours == Wright was elected a Fellow of the Royal Society of Edinburgh in 1978 and Foreign (Honorary) Academician of the Società Nazionale di Scienze, Lettere e Arti in Napoli in 1994. == Selected bibliography == Wright, J. D. M. (1977), "Jordan C∗-algebras", Michigan Math. J., 24: 291–302, doi:10.1307/mmj/1029001946, Zbl 0384.46040 Bunce, L.J.; Wright, J. D. M. (1992), "The Mackey-Gleason problem", Bull. Am. Math. Soc., New Series, 26 (2): 288–293, doi:10.1090/S0273-0979-1992-00274-4, Zbl 0759.46054 Saito, K.; Wright, J.D.M. (2015). Monotone complete C*-algebras and generic dynamics. London: Springer. Zbl 1382.46003. == References ==
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Wikipedia:John Edensor Littlewood#0
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John Edensor Littlewood (9 June 1885 – 6 September 1977) was a British mathematician. He worked on topics relating to analysis, number theory, and differential equations and had lengthy collaborations with G. H. Hardy, Srinivasa Ramanujan and Mary Cartwright. == Biography == Littlewood was born on the 9th of June 1885 in Rochester, Kent, the eldest son of Edward Thornton Littlewood and Sylvia Maud (née Ackland). In 1892, his father accepted the headmastership of a school in Wynberg, Cape Town, in South Africa, taking his family there. Littlewood returned to Britain in 1900 to attend St Paul's School in London, studying under Francis Sowerby Macaulay, an influential algebraic geometer. In 1903, Littlewood entered the University of Cambridge, studying in Trinity College. He spent his first two years preparing for the Tripos examinations which qualify undergraduates for a bachelor's degree where he emerged in 1905 as Senior Wrangler bracketed with James Mercer (Mercer had already graduated from the University of Manchester before attending Cambridge). In 1906, after completing the second part of the Tripos, he started his research under Ernest Barnes. One of the problems that Barnes suggested to Littlewood was to prove the Riemann hypothesis, an assignment at which he did not succeed. He was elected a Fellow of Trinity College in 1908. From October 1907 to June 1910, he worked as a Richardson Lecturer in the School of Mathematics at the University of Manchester before returning to Cambridge in October 1910, where he remained for the rest of his career. He was appointed Rouse Ball Professor of Mathematics in 1928, retiring in 1950. He was elected a Fellow of the Royal Society in 1916, awarded the Royal Medal in 1929, the Sylvester Medal in 1943, and the Copley Medal in 1958. He was president of the London Mathematical Society from 1941 to 1943 and was awarded the De Morgan Medal in 1938 and the Senior Berwick Prize in 1960. Littlewood died on 6 September 1977. == Work == Most of Littlewood's work was in the field of mathematical analysis. He began research under the supervision of Ernest William Barnes, who suggested that he attempt to prove the Riemann hypothesis: Littlewood showed that if the Riemann hypothesis is true, then the prime number theorem follows and obtained the error term. This work won him his Trinity fellowship. However, the link between the Riemann hypothesis and the prime number theorem had been known before in Continental Europe, and Littlewood wrote later in his book, A Mathematician's Miscellany that his rediscovery of the result did not shed a positive light on the isolated nature of British mathematics at the time. === Theory of the distribution of prime numbers === In 1914, Littlewood published his first result in the field of analytic number theory concerning the error term of the prime-counting function. If π(x) denotes the number of primes up x, then the prime number theorem implies that π(x) ~ Li(x), where Li ( x ) = ∫ 2 x d t log t {\displaystyle \operatorname {Li} (x)=\int _{2}^{x}{\frac {\mathrm {d} t}{\log t}}} is known as the Eulerian logarithmic integral. Numerical evidence seemed to suggest that π(x) < Li(x) for all x. Littlewood, however proved that the difference π(x) − Li(x) changes sign infinitely often. === Collaboration with G. H. Hardy === Littlewood collaborated for many years with G. H. Hardy. Together they devised the first Hardy–Littlewood conjecture, a strong form of the twin prime conjecture, and the second Hardy–Littlewood conjecture. ==== Ramanujan ==== He also, with Hardy, identified the work of the Indian mathematician Srinivasa Ramanujan as that of a genius and supported him in travelling from India to work at Cambridge. A self-taught mathematician, Ramanujan later became a Fellow of the Royal Society, Fellow of Trinity College, Cambridge, and widely recognised as on a par with other geniuses such as Euler and Jacobi. === Collaboration with Mary Cartwright === In the late 1930s, as the prospect of war loomed, the Department of Scientific and Industrial Research sought the interest of pure mathematicians in the properties of non linear differential equations that were needed by radio engineers and scientists. The problems appealed to Littlewood and Mary Cartwright, and they worked on them independently during the next 20 years. The problems that Littlewood and Cartwright worked on concerned differential equations arising out of early research on radar: their work foreshadowed the modern theory of dynamical systems. Littlewood's 4/3 inequality on bilinear forms was a forerunner of the later Grothendieck tensor norm theory. == Military service WWI – ballistics work == During the Great War, Littlewood served in the Royal Garrison Artillery as a second lieutenant. He made highly significant contributions in the field of ballistics. == Later life == He continued to write papers into his eighties, particularly in analytical areas of what would become the theory of dynamical systems. Littlewood is also remembered for his book of reminiscences, A Mathematician's Miscellany (new edition published in 1986). Among his PhD students were Sarvadaman Chowla, Harold Davenport, and Donald C. Spencer. Spencer reported that in 1941 when he (Spencer) was about to get on the boat that would take him home to the United States, Littlewood reminded him: "n, n alpha, n beta!" (referring to Littlewood's conjecture). Littlewood's collaborative work, carried out by correspondence, covered fields in Diophantine approximation and Waring's problem, in particular. In his other work, he collaborated with Raymond Paley on Littlewood–Paley theory in Fourier theory, and with Cyril Offord in combinatorial work on random sums, in developments that opened up fields that are still intensively studied. In a 1947 lecture, the Danish mathematician Harald Bohr said, "To illustrate to what extent Hardy and Littlewood in the course of the years came to be considered as the leaders of recent English mathematical research, I may report what an excellent colleague once jokingly said: 'Nowadays, there are only three really great English mathematicians: Hardy, Littlewood, and Hardy–Littlewood.' " : xxvii The German mathematician Edmund Landau supposed that Littlewood was a pseudonym that Hardy used for his lesser work and "so doubted the existence of Littlewood that he made a special trip to Great Britain to see the man with his own eyes". He visited Cambridge where he saw much of Hardy but nothing of Littlewood and so considered his conjecture to be proven. A similar story was told about Norbert Wiener, who vehemently denied it in his autobiography. He coined Littlewood's law, which states that individuals can expect "miracles" to happen to them at the rate of about one per month. == Cultural references == John Littlewood is depicted in two films covering the life of Ramanujan – Ramanujan in 2014 portrayed by Michael Lieber and The Man Who Knew Infinity in 2015 portrayed by Toby Jones. == See also == Critical line theorem Littlewood conjecture Littlewood polynomial Littlewood's three principles of real analysis Littlewood's Tauberian theorem Littlewood's 4/3 inequality Littlewood subordination theorem Littlewood–Offord problem Littlewood–Paley theory Hardy–Littlewood circle method Hardy–Littlewood definition Hardy–Littlewood inequality Hardy–Littlewood maximal function Hardy–Littlewood zeta function conjectures Hardy–Littlewood tauberian theorem First Hardy–Littlewood conjecture Second Hardy–Littlewood conjecture Ross–Littlewood paradox Hadamard three-circle theorem Skewes's number == References == === Bibliography === Burkill, J. C. (1978). "John Edensor Littlewood. 9 June 1885–6 September 1977". Biographical Memoirs of Fellows of the Royal Society. 24: 322–326. doi:10.1098/rsbm.1978.0010. JSTOR 769763. S2CID 119754496. Bateman, P; Diamond, P (1978). "John E. Littlewood (1885–1977) An Informal Obituary". The Mathematical Intelligencer. 1 (1): 28–33. doi:10.1007/BF03023041. S2CID 122107252. == Further reading == Littlewood's Miscellany, edited by B. Bollobás, Cambridge University Press; 1986. ISBN 0-521-33702-X (alternative title for A Mathematician's Miscellany) == External links == O'Connor, John J.; Robertson, Edmund F., "John Edensor Littlewood", MacTutor History of Mathematics Archive, University of St Andrews John Edensor Littlewood at the Mathematics Genealogy Project Papers of Littlewood on Number Theory A Mathematicians Miscellany
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Wikipedia:John Friedlander#0
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John Friedlander is a Canadian mathematician specializing in analytic number theory. He received his B.Sc. from the University of Toronto in 1965, an M.A. from the University of Waterloo in 1966, and a Ph.D. from Pennsylvania State University in 1972. He was a lecturer at M.I.T. in 1974–76, and has been on the faculty of the University of Toronto since 1977, where he served as Chair during 1987–91. He has also spent several years at the Institute for Advanced Study. In addition to his individual work, he has been notable for his collaborations with other well-known number theorists, including Enrico Bombieri, William Duke, Andrew Granville, and especially Henryk Iwaniec. In 1997, in joint work with Henryk Iwaniec, Friedlander proved that infinitely many prime numbers can be obtained as the sum of a square and fourth power: a2 + b4. Friedlander and Iwaniec improved Enrico Bombieri's "asymptotic sieve" technique to construct their proof. == Awards and honors == In 1999, Friedlander received the Jeffery–Williams Prize. In 1988, Friedlander became a fellow of the Royal Society of Canada. In 2002, CRM-Fields-PIMS prize In 2012, he became a fellow of the American Mathematical Society. In 2017, he received the Joseph L. Doob prize, jointly with Henryk Iwaniec, for their book Opera de Cribro. == Selected publications == Friedlander, John; Iwaniec, Henryk (2010). Opera de Cribro. Providence: American Mathematical Society. ISBN 978-0-8218-4970-5. == See also == List of University of Waterloo people == References == == External links == John Friedlander at the Mathematics Genealogy Project John Friedlander's profile in Toronto's Focus on Research. On Bombieri's asymptotic sieve
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Wikipedia:John Gray (mathematician)#0
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John Gray, FRS (died 17 July 1769) was a British mathematician. As a young man he taught mathematics at Marischal College, later Aberdeen University. He wrote "A Treatise on Gunnery", dedicated to the Duke of Argyll and published by William Innys (London) in 1731. In collaboration with Andrew Reid and others, he worked to produce a book of abridged Philosophical Transactions of the Royal Society for 1720–1732; which was published by W. Innys and R. Manby in 1732. In March 1732 he was elected a Fellow of the Royal Society, his application citation having described him as being of the Navy Office (or Navy Board), who were responsible for the construction and maintenance of ships in the Royal Dockyards for the Royal Navy. From 1764 to his death he was Rector of Marischal College. He died at his London home in 1769, having asked to be buried at Petersham next to his wife Elizabeth. He left estates in the West Indies to his nephews and 1,000 acres (400 ha) to Gray's Inn of London. == References ==
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Wikipedia:John Grue#0
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John Grue (born 1957) is a Norwegian applied mathematician noted for his contributions to marine hydrodynamics and internal waves. He took the cand.real. degree in 1982 and the dr.philos. degree in 1987, both at the University of Oslo. He stayed at the Massachusetts Institute of Technology from 1987 to 1988, and was appointed associate professor at the University of Oslo in 1990. In 1994 he was promoted to professor. He has later served as guest professor at Harbin Engineering University in China. He was Chair of the Abel Board 2018-2022 of the Abel Prize in mathematics. He is a member of the Norwegian Academy of Science and Letters and the Norwegian Academy of Technological Sciences. == References ==
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Wikipedia:John Harnad#0
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John Harnad (born Hernád János) is a Hungarian-born Canadian mathematical physicist. He did his undergraduate studies at McGill University and his doctorate at the University of Oxford (D.Phil. 1972) under the supervision of John C. Taylor. His research is on integrable systems, gauge theory and random matrices. He is currently Director of the Mathematical Physics group at the Centre de recherches mathématiques (CRM), a national research centre in mathematics at the Université de Montréal and Professor in the Department of Mathematics and Statistics at Concordia University. He is an affiliate member of the Perimeter Institute for Theoretical Physics and was a long-time visiting member of the Princeton Institute for Advanced Study . His work has had a strong impact in several domains of mathematical physics, and his publications are very widely cited. He has made fundamental contributions on: geometrical and topological methods in gauge theory, classical and quantum integrable systems, the spectral theory of random matrices, isomonodromic deformations, the bispectral problem, integrable random processes, transformation groups and symmetries. In 2006, he was recipient of the CAP-CRM Prize in Theoretical and Mathematical Physics "For his deep and lasting contributions to the theory of integrable systems with connections to gauge theory, inverse scattering and random matrices". == References == == External links == Centre de recherches mathématiques Archived 2017-11-12 at the Wayback Machine John Harnad's home page John Harnad at the Mathematics Genealogy Project
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Wikipedia:John Horton Conway#0
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John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician. He was active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches of recreational mathematics, most notably the invention of the cellular automaton called the Game of Life. Born and raised in Liverpool, Conway spent the first half of his career at the University of Cambridge before moving to the United States, where he held the John von Neumann Professorship at Princeton University for the rest of his career. On 11 April 2020, at age 82, he died of complications from COVID-19. == Early life and education == Conway was born on 26 December 1937 in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at a very early age. By the time he was 11, his ambition was to become a mathematician. After leaving sixth form, he studied mathematics at Gonville and Caius College, Cambridge. A "terribly introverted adolescent" in school, he took his admission to Cambridge as an opportunity to transform himself into an extrovert, a change which would later earn him the nickname of "the world's most charismatic mathematician". Conway was awarded a BA in 1959 and, supervised by Harold Davenport, began to undertake research in number theory. Having solved the open problem posed by Davenport on writing numbers as the sums of fifth powers, Conway became interested in infinite ordinals. It appears that his interest in games began during his years studying the Cambridge Mathematical Tripos, where he became an avid backgammon player, spending hours playing the game in the common room. In 1964, Conway was awarded his doctorate and was appointed as College Fellow and Lecturer in Mathematics at Sidney Sussex College, Cambridge. After leaving Cambridge in 1986, he took up the appointment to the John von Neumann Chair of Mathematics at Princeton University. There, he won the Princeton University Pi Day pie-eating contest. == Conway and Martin Gardner == Conway's career was intertwined with that of Martin Gardner. When Gardner featured Conway's Game of Life in his Mathematical Games column in October 1970, it became the most widely read of all his columns and made Conway an instant celebrity. Gardner and Conway had first corresponded in the late 1950s, and over the years Gardner had frequently written about recreational aspects of Conway's work. For instance, he discussed Conway's game of Sprouts (July 1967), Hackenbush (January 1972), and his angel and devil problem (February 1974). In the September 1976 column, he reviewed Conway's book On Numbers and Games and even managed to explain Conway's surreal numbers. Conway was a prominent member of Martin Gardner's Mathematical Grapevine. He regularly visited Gardner and often wrote him long letters summarizing his recreational research. In a 1976 visit, Gardner kept him for a week, pumping him for information on the Penrose tilings which had just been announced. Conway had discovered many (if not most) of the major properties of the tilings. Gardner used these results when he introduced the world to Penrose tiles in his January 1977 column. The cover of that issue of Scientific American features the Penrose tiles and is based on a sketch by Conway. == Major areas of research == === Recreational mathematics === Conway invented the Game of Life, one of the early examples of a cellular automaton. His initial experiments in that field were done with pen and paper, long before personal computers existed. Since Conway's game was popularized by Martin Gardner in Scientific American in 1970, it has spawned hundreds of computer programs, web sites, and articles. It is a staple of recreational mathematics. The LifeWiki is devoted to curating and cataloging the various aspects of the game. From the earliest days, it has been a favorite in computer labs, both for its theoretical interest and as a practical exercise in programming and data display. Conway came to dislike how discussions of him heavily focused on his Game of Life, feeling that it overshadowed deeper and more important things he had done, although he remained proud of his work on it. The game helped to launch a new branch of mathematics, the field of cellular automata. The Game of Life is known to be Turing complete. === Combinatorial game theory === Conway contributed to combinatorial game theory (CGT), a theory of partisan games. He developed the theory with Elwyn Berlekamp and Richard Guy, and also co-authored the book Winning Ways for your Mathematical Plays with them. He also wrote On Numbers and Games (ONAG) which lays out the mathematical foundations of CGT. He was also one of the inventors of the game sprouts, as well as philosopher's football. He developed detailed analyses of many other games and puzzles, such as the Soma cube, peg solitaire, and Conway's soldiers. He came up with the angel problem, which was solved in 2006. He invented a new system of numbers, the surreal numbers, which are closely related to certain games and have been the subject of a mathematical novelette by Donald Knuth. He also invented a nomenclature for exceedingly large numbers, the Conway chained arrow notation. Much of this is discussed in the 0th part of ONAG. === Geometry === In the mid-1960s with Michael Guy, Conway established that there are sixty-four convex uniform polychora excluding two infinite sets of prismatic forms. They discovered the grand antiprism in the process, the only non-Wythoffian uniform polychoron. Conway also suggested a system of notation dedicated to describing polyhedra called Conway polyhedron notation. In the theory of tessellations, he devised the Conway criterion which is a fast way to identify many prototiles that tile the plane. He investigated lattices in higher dimensions and was the first to determine the symmetry group of the Leech lattice. === Geometric topology === In knot theory, Conway formulated a new variation of the Alexander polynomial and produced a new invariant now called the Conway polynomial. After lying dormant for more than a decade, this concept became central to work in the 1980s on the novel knot polynomials. Conway further developed tangle theory and invented a system of notation for tabulating knots, now known as Conway notation, while correcting a number of errors in the 19th-century knot tables and extending them to include all but four of the non-alternating primes with 11 crossings. The Conway knot is named after him. Conway's conjecture that, in any thrackle, the number of edges is at most equal to the number of vertices, is still open. === Group theory === He was the primary author of the ATLAS of Finite Groups giving properties of many finite simple groups. Working with his colleagues Robert Curtis and Simon P. Norton he constructed the first concrete representations of some of the sporadic groups. More specifically, he discovered three sporadic groups based on the symmetry of the Leech lattice, which have been designated the Conway groups. This work made him a key player in the successful classification of the finite simple groups. Based on a 1978 observation by mathematician John McKay, Conway and Norton formulated the complex of conjectures known as monstrous moonshine. This subject, named by Conway, relates the monster group with elliptic modular functions, thus bridging two previously distinct areas of mathematics—finite groups and complex function theory. Monstrous moonshine theory has now been revealed to also have deep connections to string theory. Conway introduced the Mathieu groupoid, an extension of the Mathieu group M12 to 13 points. === Number theory === As a graduate student, he proved one case of a conjecture by Edward Waring, that every integer could be written as the sum of 37 numbers each raised to the fifth power, though Chen Jingrun solved the problem independently before Conway's work could be published. In 1972, Conway proved that a natural generalization of the Collatz problem is algorithmically undecidable. Related to that, he developed the esoteric programming language FRACTRAN. While lecturing on the Collatz conjecture, Terence Tao (who was taught by him in graduate school) mentioned Conway's result and said that he was "always very good at making extremely weird connections in mathematics". === Algebra === Conway wrote a textbook on Stephen Kleene's theory of state machines, and published original work on algebraic structures, focusing particularly on quaternions and octonions. Together with Neil Sloane, he invented the icosians. === Analysis === He invented a base 13 function as a counterexample to the converse of the intermediate value theorem: the function takes on every real value in each interval on the real line, so it has a Darboux property but is not continuous. === Algorithmics === For calculating the day of the week, he invented the Doomsday algorithm. The algorithm is simple enough for anyone with basic arithmetic ability to do the calculations mentally. Conway could usually give the correct answer in under two seconds. To improve his speed, he practised his calendrical calculations on his computer, which was programmed to quiz him with random dates every time he logged on. One of his early books was on finite-state machines. === Theoretical physics === In 2004, Conway and Simon B. Kochen, another Princeton mathematician, proved the free will theorem, a version of the "no hidden variables" principle of quantum mechanics. It states that given certain conditions, if an experimenter can freely decide what quantities to measure in a particular experiment, then elementary particles must be free to choose their spins to make the measurements consistent with physical law. Conway said that "if experimenters have free will, then so do elementary particles." == Personal life and death == Conway was married three times. With his first two wives he had two sons and four daughters. He married Diana in 2001 and had another son with her. He had three grandchildren and two great-grandchildren. On 8 April 2020, Conway developed symptoms of COVID-19. On 11 April, he died in New Brunswick, New Jersey, at the age of 82. == Awards and honours == Conway received the Berwick Prize (1971), was elected a Fellow of the Royal Society (1981), became a fellow of the American Academy of Arts and Sciences in 1992, was the first recipient of the Pólya Prize (LMS) (1987), won the Nemmers Prize in Mathematics (1998) and received the Leroy P. Steele Prize for Mathematical Exposition (2000) of the American Mathematical Society. In 2001 he was awarded an honorary degree from the University of Liverpool, and in 2014 one from Alexandru Ioan Cuza University. His Fellow of the Royal Society nomination in 1981 reads: A versatile mathematician who combines a deep combinatorial insight with algebraic virtuosity, particularly in the construction and manipulation of "off-beat" algebraic structures which illuminate a wide variety of problems in completely unexpected ways. He has made distinguished contributions to the theory of finite groups, to the theory of knots, to mathematical logic (both set theory and automata theory) and to the theory of games (as also to its practice). In 2017 Conway was given honorary membership of the British Mathematical Association. Conferences called Gathering 4 Gardner are held every two years to celebrate the legacy of Martin Gardner, and Conway himself was often a featured speaker at these events, discussing various aspects of recreational mathematics. == Select publications == 1971 – Regular algebra and finite machines. Chapman and Hall, London, 1971, Series: Chapman and Hall mathematics series, ISBN 0412106205. 1976 – On numbers and games. Academic Press, New York, 1976, Series: L.M.S. monographs, 6, ISBN 0121863506. 1979 – On the Distribution of Values of Angles Determined by Coplanar Points (with Paul Erdős, Michael Guy, and H. T. Croft). Journal of the London Mathematical Society, vol. II, series 19, pp. 137–143. 1979 – Monstrous Moonshine (with Simon P. Norton). Bulletin of the London Mathematical Society, vol. 11, issue 2, pp. 308–339. 1982 – Winning Ways for your Mathematical Plays (with Richard K. Guy and Elwyn Berlekamp). Academic Press, ISBN 0120911507. 1985 – Atlas of finite groups (with Robert Turner Curtis, Simon Phillips Norton, Richard A. Parker, and Robert Arnott Wilson). Clarendon Press, New York, Oxford University Press, 1985, ISBN 0198531990. 1988 – Sphere Packings, Lattices, and Groups (with Neil Sloane). Springer-Verlag, New York, Series: Grundlehren der mathematischen Wissenschaften, 290, ISBN 9780387966175. 1995 – Minimal-Energy Clusters of Hard Spheres (with Neil Sloane, R. H. Hardin, and Tom Duff). Discrete & Computational Geometry, vol. 14, no. 3, pp. 237–259. 1996 – The Book of Numbers (with Richard K. Guy). Copernicus, New York, 1996, ISBN 0614971667. 1997 – The Sensual (quadratic) Form (with Francis Yein Chei Fung). Mathematical Association of America, Washington, DC, 1997, Series: Carus mathematical monographs, no. 26, ISBN 1614440255. 2002 – On Quaternions and Octonions (with Derek A. Smith). A. K. Peters, Natick, MA, 2002, ISBN 1568811349. 2008 – The Symmetries of Things (with Heidi Burgiel and Chaim Goodman-Strauss). A. K. Peters, Wellesley, MA, 2008, ISBN 1568812205. == See also == List of things named after John Horton Conway == References == == Sources == Alpert, Mark (1999). Not Just Fun and Games Scientific American, April 1999 Boden, Margaret (2006). Mind As Machine, Oxford University Press, 2006, p. 1271 du Sautoy, Marcus (2008). Symmetry, HarperCollins, p. 308 Guy, Richard K (1983). Conway's Prime Producing Machine Mathematics Magazine, Vol. 56, No. 1 (Jan. 1983), pp. 26–33 Roberts, Siobhan (2015). Genius at play: The curious mind of John Horton Conway. Bloomsbury. ISBN 978-1620405932. O'Connor, John J.; Robertson, Edmund F., "John Horton Conway", MacTutor History of Mathematics Archive, University of St Andrews John Horton Conway at the Mathematics Genealogy Project Princeton University (2009). Bibliography of John H. Conway Archived 27 September 2011 at the Wayback Machine Mathematics Department Seife, Charles (1994). Impressions of Conway The Sciences Schleicher, Dierk (2011), Interview with John Conway, Notices of the AMS == External links == John Horton Conway's publications indexed by the Scopus bibliographic database. (subscription required) Conway, John (20 April 2009). "Proof of the Free Will Theorem" (Video). Archived Lectures. John Conway. Videos. Numberphile. playlist on YouTube Look-and-Say Numbers. Feat John Conway (2014) on YouTube Inventing the Game of Life (2014) on YouTube The Princeton Brick (2014) on YouTube Conway leading a tour of brickwork patterns in Princeton, lecturing on the ordinals and on sums of powers and the Bernoulli numbers necrology by Keith Hartnett in Quanta Magazine, April 20, 2020
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Wikipedia:John Horvath (mathematician)#0
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John Michael Horvath (born János Horváth; 30 July 1924 in Budapest – 12 March 2015) was a Hungarian-American mathematician noted for his contributions to analysis especially in functional analysis and distribution theory. == Education and career == Horvath received his doctorate in 1947 from the University of Budapest as a student of Lipót Fejér and Frigyes Riesz. Four other talented mathematicians also graduated in the class of 1947: János Aczél, Ákos Császár, László Fuchs and István Gál. Together with Horvath, they were referred to as the Big Five. After obtaining his doctorate, he went to the French National Centre for Scientific Research (CNRS) to do research. At the recommendation of John von Neumann and Salomon Lefschetz, in 1951 he came to the newly founded University of Los Andes in Bogota, where he became the first head of the mathematics department and established modern mathematics in Bolivia. At his invitation, Laurent Schwartz and Jean Dieudonné visited the university. His later work on analytic continuations and a general definition of the Convolution of distributions spread the ideas of Schwartz's theory of distributions. In 1957 Horvath finally went to the United States, where he taught at the University of Maryland until 1994 and was then awarded the title of Professor Emeritus. It was Schwartz, who recommended him to write elementary textbook on distribution theory. MathSciNet called his book Topological Vector Spaces and Distributions (1966), "The most readable introduction to the theory of vector spaces available in English and possibly any other language." In 2006, Horvath edited and wrote one of the chapters (Holomorphic Functions) for A Panorama of Hungarian Mathematics in the Twentieth Century. He was an American Mathematical Society since 1958. In 1997, he received a honorary doctorate of the Universidad de los Andes. And in 1998, he became a member of the Hungarian Academy of Sciences, in 2001 he was elected to the Colombian Academy of Sciences. == Works == (1966) Topological Vector Spaces and Distributions Addison-Wesley, Reading, Massachusetts. ISBN 0201029855 Horvath, John (January 2012). 2013 Dover reprint. Courier Corporation. ISBN 978-0-486-48850-9. (2005) A Panorama of Hungarian Mathematics in the Twentieth Century Springer-Verlag Berlin Heidelberg. ISBN 978-3-540-28945-6 == References == == External links == John Horvath at the Mathematics Genealogy Project
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Wikipedia:John Kemp (mathematician)#0
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Prof John Kemp FRSE LLD (1763–1812) was a Scottish mathematician, who settled in the U.S. state of New York for most of his life. == Life == He was born on 10 April 1763 at Achlossan near Aboyne in Aberdeenshire, Scotland, the son of a farmer. He studied Mathematics at Aberdeen University and graduated in 1783. In the same year he emigrated to the United States of America, settling first in Virginia then moving to New York] in 1785. In 1786 he successfully applied to become Professor of Mathematics at Columbia College (later Columbia University), aged only 23. In 1795 he began teaching Geography in addition to Mathematics and in 1799 he also began teaching Natural Philosophy (Physics). In 1792 he was elected a Foreign Fellow of the Royal Society of Edinburgh. His proposers were James Gregory, Daniel Rutherford and John Playfair. He was not (as some records claim) one of the founders of the Society in 1783. In 1810 he visited Lake Erie in relation to an assessment of the then-proposed canal project, and considered it wholly viable. He died in New York on 15 November 1812 following a long illness. He married twice but both wives died. One daughter (by his first marriage) survived him. == References == == External links ==
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Wikipedia:John Madsen (physicist)#0
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Sir John Percival Vissing Madsen FAA (24 March 1879 – 4 October 1969) was an Australian academic, physicist, engineer, mathematician and Army officer. This history of Madsen's activities in Australian science and engineering covers the period 1900–1956 during which applications of electricity, X-ray analysis, standardised measurements for production, radio propagation, radar and radio astronomy, nuclear physics and digital computers made tremendous progress in part due to major advances in fundamental research & also the heightened activity due to two world wars. Central to these activities largely carried out at Sydney University, was the Electrical Engineering Department & the CSIRO National Standards Laboratory/ Radiophysics Laboratory (now The Madsen Building in the University grounds. Madsen's roles encompassed the positions of Lecturer, Assistant Professor, Professor and Emeritus Professor from 1909 up to 1956 when he finally retired as Chairman of the Radio Research Board which he founded in 1927. Until 1946 Madsen was the first and only Professor of Electrical Engineering in any Australian University. == Early life == Madsen was born at Lochinvar, in the Hunter Valley, New South Wales. Madsen attended Sydney High School and was Dux of the school. Madsen graduated from Sydney University with first class honours and University Medals in both Science and Engineering in 1900 and then 1901 when he read physics and mathematics establishing the practice of taking the double degree of BSc and BE and he himself embarking on a lifelong career of applying physics to the foundations of engineering. == Career == In 1901, Madsen was appointed lecturer in mathematics and physics at the University of Adelaide, where he came under the influence of Professor William Henry Bragg with whom he co-operated with and became a lifelong friend. In 1909, Madsen's interest in the practical application of science led him to accept a lectureship in engineering at the University of Sydney, where he became assistant professor in 1912. In recognition of the outstanding Australian radio work done in the previous 25 years the Xth General Assembly was held at Sydney University in August 1952. Madsen was Chairman of the Australian Organising Committee and was also elected President of the Xth Assembly. In 1956, there was great concern in the West that the Soviet Union had taken a significant lead in scientific training of its engineers following the detonation of its hydrogen bomb soviet atomic bomb project and the soon to follow Sputnik 1. To address the situation in Australia, Madsen published a paper through Sydney University setting out recommendations for manpower requirements in the scientific era touching greatly on his own experience of the previous 50 years of promoting engineering with science. In the post war period to the end of the twentieth century, there was a significant change in emerging technology policies moving from a mainly government sponsored "supply" side environment to a business orientated "demand" side market." == Personal life == Madsen was the eldest of the family of four sons and two daughters of Hans Frandsen Madsen and his wife Annie (née Bush). His father, Hans Madsen, born in Denmark, migrated to Australia in 1864 and, while working as a miner, became one of the first pupils of the Ballarat School of Mines. == References ==
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Wikipedia:John McCarthy (mathematician)#0
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John Edward McCarthy (born 20 January 1964) is a mathematician. He is currently the Spencer T. Olin Professor of Arts and Sciences, and former chair of the Department of Mathematics and Statistics at Washington University in St. Louis. He works in operator theory and several complex variables, and applications of mathematics to other areas. He received a B.A. from Trinity College Dublin in 1983, and a Ph.D. from the University of California, Berkeley in 1989. His Ph.D. Advisor was Donald Sarason. He is a native of Ireland. He has worked on Toeplitz operators, spaces of holomorphic functions, Nevanlinna–Pick interpolation, extension theorems in several complex variables, and the mathematics of ultrasound and neuroimaging. In 1995, he, Sheldon Axler and Donald Sarason co-chaired a program at the Mathematical Sciences Research Institute in Holomorphic Spaces. Jim Agler and he wrote the text Pick Interpolation and Hilbert Function Spaces. Honors include the Gilbert de Beauregard Robinson award in 2016 from the Canadian Mathematical Society and being elected a Fellow of the American Mathematical Society in 2018. == Books == (with Sheldon Axler, and Donald Sarason) editors. Holomorphic Spaces, Cambridge University Press 1998 (with Jim Agler) Pick Interpolation and Hilbert function spaces, American Mathematical Society 2002 (with Bob A. Dumas) Transition to Higher Mathematics: Structure and Proof, 1st ed. McGraw Hill 2006; 2nd. ed. Washington University Open Scholarship, 2015 (with Jim Agler and Nicholas Young) ``Operator Analysis: Hilbert space methods in complex analysis, Cambridge University Press, 2020. == References == == External links == McCarthy's Home Page John McCarthy at the Mathematics Genealogy Project Google scholar profile
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Wikipedia:John McKay (mathematician)#0
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John K. S. McKay (18 November 1939 – 19 April 2022) was a British-Canadian mathematician and academic who worked at Concordia University, known for his discovery of monstrous moonshine, his joint construction of some sporadic simple groups, for the McKay conjecture in representation theory, and for the McKay correspondence relating certain finite groups to Lie groups. == Biography == McKay was educated at Dulwich College, earned his Bachelor and Diploma in 1961 and 1962 at the University of Manchester, and his PhD in 1971 from the University of Edinburgh. Since 1974 he worked at Concordia University, since 1979 as a professor in Computer Science. He was elected a fellow of the Royal Society of Canada in 2000, and won the 2003 CRM-Fields-PIMS prize. In April 2007 a Joint Conference was organised by the Université de Montréal and Concordia University honouring four decades of McKay's work. == See also == ADE classification Centre de Recherches Mathématiques == References == == External links == John McKay web page at CRM Archived 17 July 2017 at the Wayback Machine CRM-Fields prize
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Wikipedia:John Meadows Jackson#0
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Dr John Meadows Jackson FRSE FIMA FRAS (1907–1998) was a British mathematician. == Life == He was born in Chorlton-cum-Hardy on the edge of Manchester on 8 February 1907. His mother died during his birth and he was raised by his grandparents. He was educated at Manchester Grammar School having won a scholarship on the basis of his intellect. He obtained two BSc degrees at the University of London: Physics in 1927 and Mathematics in 1928. He won a further scholarship, enabling him to undertake postgraduate studies at Trinity College, Cambridge, under Sir Nevill Mott where he obtained a PhD in Physics in 1933. He then took a job as Assistant Lecturer at Manchester University. He began lecturing in Mathematics at Westfield College in London in 1937. At that time the college was exclusively for women. His career was interrupted by the Second World War. Westfield College was relocated to Oxfordshire in 1939 to avoid bombing. In 1941 he was seconded to the Mine Design Department of the Admiralty based in Helensburgh. The department later relocated to Fettes College in Edinburgh where he worked on the design of mine-sweeping equipment. At the end of the war the family decided not to return to London but to stay in Scotland. In January 1946 he began lecturing in Mathematics at University College, Dundee, initially living with his family in Wormit in Fife on the opposite side of the Tay estuary. He was elected a Fellow of the Royal Society of Edinburgh in 1947. His proposers were Edward Copson, George Dawson Preston, Robert Campbell Garry, and Robert Percival Cook. He retired in 1974 and died at his home, the Cedars on Perth Road in Dundee, on 23 March 1998. == Family == In 1940 he was married to Pat Harris, a lecturer in Chemistry whom he had met at a conference in 1938. They had three daughters: Kathleen, Esther and Margaret. == Publications == Energy Exchange between a Gas and a Solid Surface (1937) == References ==
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Wikipedia:John Mighton#0
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John Mighton, O.C. born in Hamilton, Ontario, Canada on October 2, 1957, is a Canadian mathematician, playwright and best-selling author, who is known for his work to support children's successful math education. Mighton founded JUMP Math as a charity in 2002 and developed the JUMP Math program to address student underachievement in math. Mighton has won national and international awards for his contributions to both math education and Canadian theatre as an internationally recognized playwright. == Education == Mighton earned a BA (Philosophy) at the University of Toronto (1978) and a MA (Philosophy) at McMaster University (1984)., later earning a PhD (Mathematics) at the University of Toronto. He was awarded an NSERC fellowship for his postdoctoral research in knot and graph theory. He is a Fellow of the Fields Institute for Research in Mathematical Sciences and has taught mathematics and math education at the University of Toronto. Mighton has also lectured in philosophy at McMaster University. == Career == Mighton's plays have been performed in theatres worldwide and have been honoured with multiple awards, including the Siminovitch Prize in Theatre, two Governor General’s Literary Awards for Drama, three Dora Awards, and the Chalmers Award. His plays include: Possible Worlds, The Little Years, Body & Soul, Scientific Americans, A Short History of Night, and Half Life. In 1997, Mighton was the math consultant and also played the role of “Tom”, an assistant to the math professor, in the critically acclaimed film, Good Will Hunting. In 1998, Mighton started a small-scale tutoring group to assist children experiencing challenges in math, though he had abandoned the subject himself for years after struggling with math in school and nearly failing first-year calculus in university. In 2002 Mighton founded JUMP (Junior Undiscovered Math Prodigies) Math, a Canadian charitable organization with the mission to enable every child to learn and love math. Mighton has written three books on math, based on his expertise and experiences with JUMP Math, including The Myth of Ability: Nurturing Mathematical Talent in Every Child (2003), which describes his approach and successes with JUMP Math; The End of Ignorance: Multiplying Our Human Potential (2007) as a further exploration of his philosophy and methods; and All Things Being Equal: Why Math is the Key to a Better World (2020). == Awards and honors == Margaret Sinclair Memorial Award for Innovation & Excellence, The Fields Institute for Research in Mathematical Sciences (2022) Adrien Pouliot Award for Outstanding Contributions to Math Education, Canadian Mathematical Society (2022) President’s Award, Professional Engineers Ontario (2020) WISE Award for Innovation in Education (2016) 10th Annual Egerton Ryerson Award for Dedication to Public Education (2016) Schwab Foundation Social Entrepreneur of the Year (2015) Ernst & Young Social Entrepreneur of the Year (2014) Officer of the Order of Canada (2010) Ashoka Fellow (2004) == Publications == Possible Worlds (1990) The Myth of Ability (2003) The End of Ignorance (2007) A Short History of Night (2007) Half-Life (2005) The little years (2012) Kids JUMP for Math, Scientific American Mind (2013) All Things Being Equal: Why Math Is the Key to a Better World (2020) == Filmography == Good Will Hunting (1997): Tom - Lambeau's Teaching Assistant == References == == External links == John Mighton at IMDb About John Mighton at JUMP Math
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Wikipedia:John Montroll#0
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John Montroll is an American origami artist, author, teacher, and mathematician. He has written many books on origami, promoting the single-square, no-cut, no glue approach. Montroll taught mathematics at St. Anselm's Abbey School in Washington, D.C. from 1990 to 2021. == Biography == John Montroll was born in Washington, D.C. He is the son of Elliott Waters Montroll, an American scientist and mathematician. He has a Bachelor of Arts degree in Mathematics from the University of Rochester, a Master of Arts in Electrical Engineering from the University of Michigan, and a Master of Arts in applied mathematics from the University of Maryland. Montroll mastered his first origami book, Isao Honda's How to make Origami, at the age of six, the same age he began creating his own origami animals. He became a member of the Origami Center of America at age twelve. He attended his first origami convention at age 14. In 2021, Montroll retired from his job at St. Anselm's Abbey School in Washington, D.C., where he taught mathematics, as well as an origami class. One of John Montroll's hobbies is whistling. He claims to be able to whistle in five octaves and to have shown this talent at two whistling conventions in Louisburg, North Carolina. John Montroll pioneered modern origami with the publication of his first book, Origami for the Enthusiast; Dover Publications, 1979, which was the first origami book where each model is folded from single square sheet and no cuts. In the same book he introduced the origami term "double rabbit ear fold." He is also known for inventing the "dog base". His work in the field of origami was briefly mentioned in the York series by author Laura Ruby. == Publications == Origami for the Enthusiast; Dover Publications, 1979 ISBN 0486237990 Animal Origami for the Enthusiast; Dover Publications, 1985 ISBN 0486247929 Origami American Style; Zenagraf, 1990 ISBN 0962725404 Origami Sculptures (with Andrew Montroll); Antroll Pub. Co., 1990 ISBN 187765602X Origami Sea Life (with Robert J. Lang); Dover Publications, 1990 ISBN 0486267652 Prehistoric Origami; Dover Publications, 1990 ISBN 0486265889 African Animals in Origami; Dover Publications, 1991 ISBN 0486269779 Easy Origami; Dover Publications, 1992 ISBN 0486272982 Origami Inside-Out; Dover Publications, 1993 ISBN 0486276740 Birds in Origami; Dover Publications, 1995 ISBN 0486283410 North American Animals in Origami; Dover Publications, 1995 ISBN 0486286673 Favorite Animals in Origami; Dover Publications, 1996 ISBN 0486291367 Mythological Creatures and the Chinese Zodiac in Origami; Dover Publications, 1996 ISBN 0486289710 Teach Yourself Origami; Dover Publications, 1998 ISBN 0486401413 Bringing Origami to Life; Dover Publications, 1999 ISBN 0486407144 Dollar Bill Animals in Origami; Dover Publications, 2000 ISBN 0486411575 Bugs and Birds in Origami; Dover Publications, 2001 ISBN 0486417735 A Plethora of Origami Polyhedra; Dover Publications, 2002 ISBN 0486422712 Dollar Bill Origami; Dover Publications, 2003 ISBN 0486429822 A Constellation of Origami Polyhedra; Dover Publications, 2004 ISBN 0486439585 Origami: Birds And Insects; Dover Publications, 2004 ISBN 0486439720 Origami: Wild Animals; Dover Publications, 2004 ISBN 0486439704 Easy Christmas Origami; Dover Publications, 2006 ISBN 0486450244 Christmas Origami; Dover Publications, 2006 ISBN 0486450252 Storytime Origami; Dover Publications, 2009 ISBN 0486467864 Origami Polyhedra Design; AK Peters, 2009 ISBN 1568814585 eZ Origami; Kindle Edition by Antroll Publishing Company (April 17, 2010) eZ Origami; Smashwords Edition by eOrigami Publishing (May 8, 2010) Easy Dollar Bill Origami; Dover Publications (May 20, 2010), ISBN 0486470091 Dinosaur Origami; Dover Publications (June 9, 2010), ISBN 0486477800 Origami Jungle Birds; Kindle Edition by Antroll Publishing Company; 1 edition (June 16, 2010) Origami Under The Sea (with Robert J. Lang); Dover Publications (July 15, 2010) ISBN 0486477843 == Notes == == External links == www.johnmontroll.com Montroll books on Amazon.com
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Wikipedia:John Musinguzi Rujoki#0
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John Musinguzi Rujoki (born c. 1973), is a Ugandan mathematician and corporate executive who serves as the Commissioner-General of the Uganda Revenue Authority (URA), effective 2 April 2020. == Early life and education == Musinguzi was born in Rubirizi, Uganda circa 1973. After attending local primary school, he was admitted to Ntare School, a prestigious all-boys boarding high school in Mbarara city, where he obtained a High School Diploma. He was admitted to Makerere University, Uganda's oldest and largest public university, where he graduated with a Bachelor of Science degree in Mathematics. He went on to study at the University of Greenwich, in the United Kingdom, where he graduated with a Master of Science degree in Computing and Information Systems. == Career == In 2000, Rujoki served as the head of the finance and audit department of the Special Revenue Protection Services (SRPS), a military unit attached to the Uganda Revenue Authority. The SRPS is credited with wiping out smuggling at Ugandan borders and especially on Lake Nalubaale. In April he was appointed to lead at URA, as the new Commissioner General, replacing Doris Akol, who had served one year and five months of her second consecutive four-year contract. Within 60 days of his appointment four senior managers at URA resigned and the URA board "accepted" their resignations. == Other considerations == In addition to his other assignments, Rujoki serves as the Chairman of board of National Information Technology Authority (NITA-U), since October 2019. He is also a senior advisor on Information and Communications Technology (ICT) to the President of Uganda, since 2015. In his capacity as Commissioner General, he sits on the seven-person board of directors of the Uganda Revenue Authority, as his predecessors did. == References == == Succession table at URA == == External links == Website of Uganda Revenue Authority (URA) Website of National Information Technology Authority (NITA-U) Shs195 Billion (approx.US$55 million in 2017) Paint Deal Led To URA Shake-Up As at 1 June 2020.
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Wikipedia:John Nelson Stockwell#0
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John Nelson Robin Stockwell (10 April 1832 in Northampton, Massachusetts – 18 May 1920 in Shaker Heights, Ohio) was an American astronomer and mathematician. == Life and work == John Nelson Stockwell grew up with an uncle and aunt on a farm in Brecksville, Ohio. He showed early mathematical talent and became interested in astronomy at the age of 12, when he experienced his first lunar eclipse. Largely self-taught, he acquired outstanding mathematical knowledge and skill. From 1854 he worked as a human computer for the United States Coast and Geodetic Survey under Benjamin Apthorp Gould, with whom he had been friends for decades. In 1861 he was appointed to a similar position at the United States Naval Observatory in Washington, D.C. In 1876 Stockwell earned a Ph.D. at Western Reserve University in Cleveland, Ohio. There Stockwell became friends with the philanthropist Leonard Case, from whose legacy the Case School of Applied Sciences was founded in 1880. Stockwell became Professor of Mathematics and Astronomy. Case School and Western Reserve University later became Case Western Reserve University. As early as 1887 Stockwell gave up his professorship and subsequently devoted himself exclusively to research. Stockwell was credited with calculating the orbits of numerous celestial bodies and the ecliptic. For example, he calculated the orbit of the asteroid Virginia and the dates of more than a hundred historical lunar and solar eclipses. Stockwell developed theories about the mutual orbital influence of celestial bodies and the calculation of the tides. In 1875 he was elected a Fellow of the American Association for the Advancement of Science, and in 1889 he was elected an Associate Fellow of the American Academy of Arts and Sciences. Stockwell married Sarah Healy (1833–1916) in 1855. The couple had six children, four of whom survived to adulthood. John Nelson Stockwell died on 18 May 1920 in Shaker Heights, Ohio. He was 88 years of age. He is buried in Lake View Cemetery in Cleveland. == Notes == == References == Howe, Charles S. (January 14, 1921), "John Nelson Stockwell", Science, New Series, 53 (1359): 35–37, Bibcode:1921Sci....53...35H, doi:10.1126/science.53.1359.35, JSTOR 1645845, PMID 17815325 See, T. J. J. (1920a), "Obituary Notice of John Nelson Stockwell of Cleveland, Ohio", Publications of the Astronomical Society of the Pacific, 32 (190): 324–325, Bibcode:1920PASP...32..324S, doi:10.1086/123003 See, T. J. J. (1920b), "Historical Notice of John Nelson Stockwell of Cleveland, Ohio", Popular Astronomy, 28 (10): 565–584, Bibcode:1920PA.....28..565S
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Wikipedia:John Overdeck#0
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John Albert Overdeck (born 1969) is an American hedge fund manager. Overdeck is the co-founder and co-chairman of Two Sigma Investments, a New York City-based hedge fund that uses a variety of technological methods, including artificial intelligence, machine learning, and distributed computing, for its trading strategies. In 2016, Overdeck reported $375 million in earnings and as of May 2017, Two Sigma Investments has $41 billion in assets under management. In March 2019, he was named one of the highest-earning hedge fund managers and traders by Forbes. == Early life and education == Overdeck is the son of Dr. John M. Overdeck of Columbia, Maryland, a retired senior mathematician at Fort Meade, Maryland, for the National Security Agency, and the late Betsey Lombard Overdeck, who was a director of Computer Sciences Corporation. Overdeck went to Wilde Lake High School in Columbia, Maryland. In 1986, he was an International Mathematics Olympiad silver medalist. He then earned a bachelor's degree in mathematics and a master's degree in statistics, both from Stanford University. Prior to co-founding Two Sigma Investments, Overdeck was a managing director at D.E. Shaw & Co., and a vice president at Amazon.com. == Two Sigma Investments == Overdeck founded Two Sigma Investments in 2001 with David Siegel and Mark Pickard, who is now retired. The company's main office is in New York City with branch offices in Houston, London, and Hong Kong. In January 2015, the firm had $34.21 billion in assets under management and as of May 2017, the firm had $41 billion. Two Sigma Investments also operates a market-making business, owns a Bermuda reinsurance company, and runs a venture capital arm. == Wealth == In March 2017, Overdeck ranked 9th on Forbes' Highest-Earning Hedge Fund Managers. As of March 2019, his net worth was $6.1 billion, making him one of the highest-earning hedge fund managers and traders by Forbes. == Board memberships == Overdeck is acting chair of the National Museum of Mathematics, director of the Robin Hood Foundation, on the board of trustees of the Institute for Advanced Study, and a board member of the Hamilton Insurance Group. == Personal life == On October 12, 2002, Overdeck married Laura Bilodeau, then a principal consultant for Stanford Research Institute; she went on to found Bedtime Math. They have three children and live in Millburn, New Jersey. Laura Overdeck initiated divorce proceedings in early 2022 citing “irreconcilable differences”. According to The Wall Street Journal, they do not have a prenuptial agreement. Laura Overdeck also sued over the family's lawyers moving various family assets into trusts in Wyoming, saying it appeared to be a way to hide assets from her and the children. Overdeck competed in the 2019 National Museum of Mathematics's Masters Tournament, which raised almost $1 million, and placed third. == Philanthropy == In 2011, Overdeck and his wife Laura established the Overdeck Family Foundation, an organization that funds educational research and STEM education. The foundation's overall goal is to enable all American children to unlock their potential by strengthening learning in and out of school. The Overdecks' education philanthropy focuses on four key issue areas: "early impact", "exceptional educators", "innovative schools", and "inspired minds". They have made gifts to Robin Hood Foundation for work by MDRC to study the long-term effects of various early childhood programs. The Overdecks have also supported the Harlem Children's Zone within its early impact strategy and the Khan Academy. Other grantees have included the NJ STEM Teaching Fellowship, and Governor's School for the Sciences at Drew University in New Jersey. Laura Overdeck is an alumna of Governor's School and was instrumental in saving the program through private donations in 2006. In October 2018, The Overdeck Education Innovation Fund gifted $1 million to be distributed over the next three years by the Woodrow Wilson School of Public and International Affairs for research on education issues. This is the second gift from the Overdeck Family Foundation, the first of which was $1 million donated in 2016. That same month, the Overdeck Family Foundation pledged $3 million to Opportunity Insights, a policy and research institute based at Harvard University that will publish data on the probability that children will escape poverty based on where they were raised. == Honors and recognition == 1986, Silver Medal for the U.S. in the International Mathematical Olympiad in Poland at age 16 2012, Distinguished Alumni of The Johns Hopkins Center for Talented Youth 2017, Golden Plate Award of the American Academy of Achievement 2019, Manager Lifetime Achievement Award at Institutional Investor's 17th annual Hedge Fund Industry Awards == References ==
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Wikipedia:John R. Terry#0
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John R. Terry is a British mathematician, currently an Interdisciplinary Professorial Fellow and EPSRC Established Career Fellow at the University of Birmingham, where he is Director of the Centre for Systems Modelling and Quantitative Biomedicine. He was previously Director of the EPSRC Centre for Predictive Modelling in Healthcare, a £2M initiative funded by the EPSRC. He is well known for the development and application of mathematical techniques in biology and medicine, notably epilepsy and neuroendocrinology. He is a member of the ILAE task force on Network Diseases, as well as previously being a member of the Scientific Advisory Committee of Epilepsy Research UK (2018-2023). He currently serves as Theme Lead for Enabling Technologies of the newly established Epilepsy Research Institute. According to Gateway to Research he has received approaching £13M in research funding, a substantial amount for a mathematician. His research has been recognized internationally, most recently by the University of Melbourne through a Miegunyah Fellowship. == Career == Terry was an undergraduate at the University of Reading before taking a PhD at the University of Surrey. He has held academic positions at Loughborough University, the University of Bristol, the University of Sheffield and the University of Exeter, before joining the University of Birmingham in 2019. During his time at the University of Exeter he established the Centre for Biomedical Modelling and Analysis, of which he was co-director, supported by the Wellcome Trust. In 2018 Terry with Dr Wessel Woldman co-founded Neuronostics, a company focussed on epilepsy diagnosis and management. In 2020 Neuronostics was named national start-up of the year by Medilink UK. In 2021 Neuronostics was one of the final four in the Nature SpinOff Prize. In 2022 Neuronostics was named a winner in the Science StartUp category of the Falling Walls Foundation. In 2024 Neuronostics achieved further international success winning $125,000 in the Epilepsy Foundation Sharktank, whilst Terry himself was recognised for his work with Neuronostics, being awarded Founder of the Year by TechSpark. == Scientific contributions == As a graduate student, Terry focused on synchronization problems in solid state laser systems, before moving into neuroscience during his postdoctoral career. He is most well known for his work in describing the mechanisms of seizures and susceptibility to epilepsy, utilising mathematical models for the first time in the context of diagnosis and surgery. He is also highly cited in the field of neuroendocrinology, where along with Stafford Lightman he has described the pituitary-adrenal interplay responsible for hourly rhythms in the stress responsive hormone cortisol. == References ==
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Wikipedia:John Radford Young#0
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John Radford Young (born 8 April 1799, in Southwark – 5 March 1885, in Peckham) was an English mathematician, professor and author, who was almost entirely self-educated. He was born of humble parents in London. At an early age he became acquainted with Olinthus Gilbert Gregory, who perceived his mathematical ability and assisted him in his studies. In 1823, while working in a private establishment for the deaf, he published An Elementary Treatise on Algebra with a dedication to Gregory. This treatise was followed by a series of elementary works, in which, following in the steps of Robert Woodhouse, Young familiarized English students with continental methods of mathematical analysis. In 1833, he was appointed Professor of Mathematics at Belfast College. When Queen's College, Belfast, opened in 1849, the presbyterian party in control there prevented Young's reappointment as Professor in the new establishment. From that time he devoted himself more completely to the study of mathematical analysis, and made several original discoveries. In 1847, he published in the Transactions of the Cambridge Philosophical Society a paper "On the Principle of Continuity in reference to certain Results of Analysis", and, in 1848, in the Transactions of the Royal Irish Academy a paper "On an Extension of a Theorem of Euler". As early as 1844, he had discovered and published a proof of Newton's rule for determining the number of imaginary roots in an equation. In 1866, he completed his proof, publishing in The Philosophical Magazine a demonstration of a principle which in his earlier paper he had assumed as axiomatic. In 1868, he contributed to the Proceedings of the Royal Irish Academy a memoir "On the Imaginary Roots of Numerical Equations". Young died at Peckham on 5 March 1885. He was married and had at least two sons and four daughters. == Works == An Elementary Treatise on Algebra 1823, 1832, 1834 Elements of Geometry 1827 Elements of Analytical Geometry 1830 An Elementary Essay on the Computation of Logarithms 1830 The Elements of the Differential Calculus 1831 The Elements of the Integral Calculus 1831 The Elements of Mechanics, comprehending Statics and Dynamics 1832 Elements of Plane and Spherical Trigonometry 1833 Theory and Solution of Algebraical Equations 1843 (1st edition: 1835) Mathematical Dissertations for the Use of Students in the Modern Analysis 1841 On the General Principles of Analysis, Part I.: The Analysis of Numerical Equations 1850 An Introductory Treatise on Mensuration 1850 An Introduction to Algebra and to the Solution of Numerical Equations 1851 Rudimentary Treatise on Arithmetic 1858, 1882 A Compendious Course of Mathematics 1855 The Theory and Practice of Navigation and Nautical Astronomy 1856, 1882 Navigation and Nautical Astronomy, 1858 The Mosaic Cosmogony not “adverse to Modern Science 1861 Science elucidative of Scripture and not antagonistic to it 1863 Modern Scepticism Viewed in Relation to Modern Science 1865 == References == Joao Caramalho Domingues (2014). "The repercussion of José Anastácio da Cunha in Britain and the USA in the nineteenth century". BSHM Bulletin. 20 (1): 32–50. doi:10.1080/17498430.2013.802111. hdl:1822/26424. S2CID 54220154. This article is based on a public domain article from Dictionary of National Biography 1885-1900, Vol.63. == External links == E. I. Carlyle, rev. Alan Yoshioka, "Young, John Radford (1799–1885)", Oxford Dictionary of National Biography, Oxford University Press, 2004. John Radford Young, Michael Floy Elements of Geometry with Notes 1833 John Radford Young, Key to the Introduction to Algebra 1854, full-length solutions to An Introduction to Algebra John Radford Young, Sir John Francis Twisden, Alexander Jardine (Esq.), The Mathematical Sciences 1860
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Wikipedia:John Senyonyi#0
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John Musisi Senyonyi is a Ugandan mathematician, academic, evangelist and academic administrator. He is the immediate past Vice Chancellor of Uganda Christian University, a private university that was accredited by the Uganda National Council of Higher Education (UNCHE), in 1997. He was appointed to that position in 2010. Following ten continuous years in that position, he retired, and handed over to Aaron Mushengyezi, in August 2020. == Background and education == He was born in Nakasongola District, in 1956, to Eriakimu Kajja, a schoolteacher and Mrs. Kajja, a full-time housewife. He is one of thirteen siblings. One of his brothers the late Peter Nyombi, Uganda's former Attorney General and former Member of Parliament for "Buruuli County" in Nakasongola District. From 1961 until 1968, Senyonyi attended Nakasongola Primary School and later Bbowa Church of Uganda Primary School. From 1969 until 1972, he attended Ndejje Senior Secondary School for his O-Level education. He then transferred to Kings College Budo, for his A-Level education, from 1973 until 1974. In 1975, he was admitted to the University of Nairobi, where he studied Mathematics, graduating with the degree of Bachelor of Science in Mathematics in 1978. Later, he obtained the Certificate in Management, from the International College of Management, Sydney, in Australia. He also holds the degree of Master of Arts in Theology, from Trinity International University, in the United States, obtained in 1992. His degree of Doctorate in Mathematical Statistics, was obtained from the University of Melbourne, in Australia, in 1984. == Work history == In 1978, Senyonyi returned to Uganda and began lecturing at Makerere University, the largest and oldest of Uganda's public institutions of higher education. He left to study in Australia, between 1979 and 1984, returning to continue teaching at Makerere. He left Makerere in 1988 and for 13 years, until 2000, he traveled the world preaching the gospel with an organisation called African Evangelistic Enterprise. He traveled mainly within Africa and in the United States, but also in Europe. In 2001, he joined Uganda Christian University as the University Chaplain. Two and half years later, in 2003, he transitioned to Deputy Vice Chancellor for Finance and Administration. Three years after that, in 2006, he was appointed to the newly created position of Deputy Vice Chancellor for Development and External Relations. He served in that capacity until September 2010 when he was appointed Vice Chancellor of Uganda Christian University; the second Vice Chancellor in the history of the institution. In 2020 he retired from the University service as the second Vice Chancellor since the University's inception. He remains in the service of God and the church as an active Evangelist. He is married to the Canon Dr. Ruth Senyonyi who has also served the church for many years as the provincial Mothers Union President for the Church of Uganda. == See also == == Other considerations == Senyonyi is a married father with one daughter and three sons. He is an ordained Reverend in the Church of Uganda. == References == == External links == Revival Has Leapt Across An Ocean
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Wikipedia:John Toland (mathematician)#0
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John Francis Toland (born 28 April 1949 in Derry) is an Irish mathematician based in the UK. From 2011 to 2016 he served as director of the Isaac Newton Institute for Mathematical Sciences and N M Rothschild & Sons Professor of Mathematical Sciences at the University of Cambridge. == Education == Toland was educated at St Columb's College in Derry and Queen's University Belfast, where he was awarded a Bachelor of Science degree in 1970. He completed postgraduate study at the University of Sussex, where he was awarded a PhD in 1973 for research on topological methods for nonlinear eigenvalue problems supervised by Charles A. Stuart. == Career and research == From 1982 to 2011 he was professor of mathematics at the University of Bath, where he held an Engineering and Physical Sciences Research Council (EPSRC) Senior Fellowship 1997–2002. In addition from 2002 to 2010 he was scientific director of the International Centre for Mathematical Sciences (ICMS) in Edinburgh. In 2011 he succeeded Sir David Wallace as director of the Isaac Newton Institute for Mathematical Sciences and N M Rothschild & Sons Professor of Mathematical Sciences at the University of Cambridge. In 2016 he was succeeded as director by Professor David Abrahams. His research interests include mathematical analysis and nonlinear partial differential equations with particular interest in the rigorous theory of steady water waves. In 1978, he proved George Gabriel Stokes' conjecture on the existence of gravity waves of maximum height on deep water, a previously open problem in mathematical hydrodynamics which dated back to the 19th century. == Awards and honours == He was elected Fellow of the Royal Society (FRS) in 1999, and a Fellow of the Royal Society of Edinburgh (FRSE) in 2003. He was awarded the London Mathematical Society's Senior Berwick Prize in 2000; and the Royal Society's Sylvester Medal in 2012. He is an Honorary Fellow of University College London and was a Fellow of St John's College, Cambridge October 2011 – September 2016. == References ==
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Wikipedia:John W. Dold#0
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John William Dold (also known as Bill Dold) is an emeritus professor in the department of mathematics in the University of Manchester, a specialist in the field of Fluid Mechanics and Combustion. He was the founder of the journal Combustion Theory and Modelling. == Biography and research == John attended school at the Christian Brothers College, Bulawayo in Zimbabwe. He completed his bachelors (1971–1974) at the University of Zimbabwe and his doctoral degree (1976–1979) at the Cranfield University under the supervision of John Frederick Clarke. He obtained a master's degree (1971–1974) in area studies at the University of London. He then joined the faculty of the mathematics department at the University of Bristol and then joined the faculty of mathematics department at the University of Manchester in 1995. In 1997, he along with Mitchell D. Smooke, founded the journal Combustion Theory and Modelling. He also worked at Greater Manchester Fire and Rescue Service from 2010 to 2013. He served as the vice president of SIAM United Kingdom and Republic of Ireland Section from 1998 to 1999. He also served as the editor for Combustion Theory and Modelling and SIAM Journal on Applied Mathematics. John Dold has made significant contributions in theoretical combustion science, particularly in areas related to partially premixed combustion, fire, ignition, detonations and also in the field of fluid mechanics and wate waves. Kerr–Dold vortex, an exact solution of Navier–Stokes equations is named after him, === Books === David F. Griffiths, John W. Dold, David J. Silvester (2015). Essential Partial Differential Equations: Analytical and Computational Aspects. Springer. ISBN 978-3319225685.{{cite book}}: CS1 maint: multiple names: authors list (link) == See also == == References == == External links == John W. Dold at the Mathematics Genealogy Project Homepage John W. Dold publications indexed by Google Scholar Publications by John W. Dold at ResearchGate
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Wikipedia:John von Neumann#0
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John von Neumann ( von NOY-mən; Hungarian: Neumann János Lajos [ˈnɒjmɒn ˈjaːnoʃ ˈlɒjoʃ]; December 28, 1903 – February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist and engineer. Von Neumann had perhaps the widest coverage of any mathematician of his time, integrating pure and applied sciences and making major contributions to many fields, including mathematics, physics, economics, computing, and statistics. He was a pioneer in building the mathematical framework of quantum physics, in the development of functional analysis, and in game theory, introducing or codifying concepts including cellular automata, the universal constructor and the digital computer. His analysis of the structure of self-replication preceded the discovery of the structure of DNA. During World War II, von Neumann worked on the Manhattan Project. He developed the mathematical models behind the explosive lenses used in the implosion-type nuclear weapon. Before and after the war, he consulted for many organizations including the Office of Scientific Research and Development, the Army's Ballistic Research Laboratory, the Armed Forces Special Weapons Project and the Oak Ridge National Laboratory. At the peak of his influence in the 1950s, he chaired a number of Defense Department committees including the Strategic Missile Evaluation Committee and the ICBM Scientific Advisory Committee. He was also a member of the influential Atomic Energy Commission in charge of all atomic energy development in the country. He played a key role alongside Bernard Schriever and Trevor Gardner in the design and development of the United States' first ICBM programs. At that time he was considered the nation's foremost expert on nuclear weaponry and the leading defense scientist at the U.S. Department of Defense. Von Neumann's contributions and intellectual ability drew praise from colleagues in physics, mathematics, and beyond. Accolades he received range from the Medal of Freedom to a crater on the Moon named in his honor. == Life and education == === Family background === Von Neumann was born in Budapest, Kingdom of Hungary (then part of Austria-Hungary), on December 28, 1903, to a wealthy, non-observant Jewish family. His birth name was Neumann János Lajos. In Hungarian, the family name comes first, and his given names are equivalent to John Louis in English. He was the eldest of three brothers; his two younger siblings were Mihály (Michael) and Miklós (Nicholas). His father Neumann Miksa (Max von Neumann) was a banker and held a doctorate in law. He had moved to Budapest from Pécs at the end of the 1880s. Miksa's father and grandfather were born in Ond (now part of Szerencs), Zemplén County, northern Hungary. John's mother was Kann Margit (Margaret Kann); her parents were Kann Jákab and Meisels Katalin of the Meisels family. Three generations of the Kann family lived in spacious apartments above the Kann-Heller offices in Budapest; von Neumann's family occupied an 18-room apartment on the top floor. On February 20, 1913, Emperor Franz Joseph elevated John's father to the Hungarian nobility for his service to the Austro-Hungarian Empire. The Neumann family thus acquired the hereditary appellation Margittai, meaning "of Margitta" (today Marghita, Romania). The family had no connection with the town; the appellation was chosen in reference to Margaret, as was their chosen coat of arms depicting three marguerites. Neumann János became margittai Neumann János (John Neumann de Margitta), which he later changed to the German Johann von Neumann. === Child prodigy === Von Neumann was a child prodigy who at six years old could divide two eight-digit numbers in his head and converse in Ancient Greek. He, his brothers and his cousins were instructed by governesses. Von Neumann's father believed that knowledge of languages other than their native Hungarian was essential, so the children were tutored in English, French, German and Italian. By age eight, von Neumann was familiar with differential and integral calculus, and by twelve he had read Borel's La Théorie des Fonctions. He was also interested in history, reading Wilhelm Oncken's 46-volume world history series Allgemeine Geschichte in Einzeldarstellungen (General History in Monographs). One of the rooms in the apartment was converted into a library and reading room. Von Neumann entered the Lutheran Fasori Evangélikus Gimnázium in 1914. Eugene Wigner was a year ahead of von Neumann at the school and soon became his friend. Although von Neumann's father insisted that he attend school at the grade level appropriate to his age, he agreed to hire private tutors to give von Neumann advanced instruction. At 15, he began to study advanced calculus under the analyst Gábor Szegő. On their first meeting, Szegő was so astounded by von Neumann's mathematical talent and speed that, as recalled by his wife, he came back home with tears in his eyes. By 19, von Neumann had published two major mathematical papers, the second of which gave the modern definition of ordinal numbers, which superseded Georg Cantor's definition. At the conclusion of his education at the gymnasium, he applied for and won the Eötvös Prize, a national award for mathematics. === University studies === According to his friend Theodore von Kármán, von Neumann's father wanted John to follow him into industry, and asked von Kármán to persuade his son not to take mathematics. Von Neumann and his father decided that the best career path was chemical engineering. This was not something that von Neumann had much knowledge of, so it was arranged for him to take a two-year, non-degree course in chemistry at the University of Berlin, after which he sat for the entrance exam to ETH Zurich, which he passed in September 1923. Simultaneously von Neumann entered Pázmány Péter University, then known as the University of Budapest, as a Ph.D. candidate in mathematics. For his thesis, he produced an axiomatization of Cantor's set theory. In 1926, he graduated as a chemical engineer from ETH Zurich and simultaneously passed his final examinations summa cum laude for his Ph.D. in mathematics (with minors in experimental physics and chemistry) at the University of Budapest. He then went to the University of Göttingen on a grant from the Rockefeller Foundation to study mathematics under David Hilbert. Hermann Weyl remembers how in the winter of 1926–1927 von Neumann, Emmy Noether, and he would walk through "the cold, wet, rain-wet streets of Göttingen" after class discussing hypercomplex number systems and their representations. == Career and private life == Von Neumann's habilitation was completed on December 13, 1927, and he began to give lectures as a Privatdozent at the University of Berlin in 1928. He was the youngest person elected Privatdozent in the university's history. He began writing nearly one major mathematics paper per month. In 1929, he briefly became a Privatdozent at the University of Hamburg, where the prospects of becoming a tenured professor were better, then in October of that year moved to Princeton University as a visiting lecturer in mathematical physics. Von Neumann was baptized a Catholic in 1930. Shortly afterward, he married Marietta Kövesi, who had studied economics at Budapest University. Von Neumann and Marietta had a daughter, Marina, born in 1935; she would become a professor. The couple divorced on November 2, 1937. On November 17, 1938, von Neumann married Klára Dán. In 1933 Von Neumann accepted a tenured professorship at the Institute for Advanced Study in New Jersey, when that institution's plan to appoint Hermann Weyl appeared to have failed. His mother, brothers and in-laws followed von Neumann to the United States in 1939. Von Neumann anglicized his name to John, keeping the German-aristocratic surname von Neumann. Von Neumann became a naturalized U.S. citizen in 1937, and immediately tried to become a lieutenant in the U.S. Army's Officers Reserve Corps. He passed the exams but was rejected because of his age. Klára and John von Neumann were socially active within the local academic community. His white clapboard house on Westcott Road was one of Princeton's largest private residences. He always wore formal suits. He enjoyed Yiddish and "off-color" humor. In Princeton, he received complaints for playing extremely loud German march music; Von Neumann did some of his best work in noisy, chaotic environments. According to Churchill Eisenhart, von Neumann could attend parties until the early hours of the morning and then deliver a lecture at 8:30. He was known for always being happy to provide others of all ability levels with scientific and mathematical advice. Wigner wrote that he perhaps supervised more work (in a casual sense) than any other modern mathematician. His daughter wrote that he was very concerned with his legacy in two aspects: his life and the durability of his intellectual contributions to the world. Many considered him an excellent chairman of committees, deferring rather easily on personal or organizational matters but pressing on technical ones. Herbert York described the many "Von Neumann Committees" that he participated in as "remarkable in style as well as output". The way the committees von Neumann chaired worked directly and intimately with the necessary military or corporate entities became a blueprint for all Air Force long-range missile programs. Many people who had known von Neumann were puzzled by his relationship to the military and to power structures in general. Stanisław Ulam suspected that he had a hidden admiration for people or organizations that could influence the thoughts and decision making of others. He also maintained his knowledge of languages learnt in his youth. He knew Hungarian, French, German and English fluently, and maintained a conversational level of Italian, Yiddish, Latin and Ancient Greek. His Spanish was less perfect. He had a passion for and encyclopedic knowledge of ancient history, and he enjoyed reading Ancient Greek historians in the original Greek. Ulam suspected they may have shaped his views on how future events could play out and how human nature and society worked in general. Von Neumann's closest friend in the United States was the mathematician Stanisław Ulam. Von Neumann believed that much of his mathematical thought occurred intuitively; he would often go to sleep with a problem unsolved and know the answer upon waking up. Ulam noted that von Neumann's way of thinking might not be visual, but more aural. Ulam recalled, "Quite independently of his liking for abstract wit, he had a strong appreciation (one might say almost a hunger) for the more earthy type of comedy and humor". === Illness and death === In 1955, a mass was found near von Neumann's collarbone, which turned out to be cancer originating in the skeleton, pancreas or prostate. (While there is general agreement that the tumor had metastasised, sources differ on the location of the primary cancer.) The malignancy may have been caused by exposure to radiation at Los Alamos National Laboratory. As death neared he asked for a priest, though the priest later recalled that von Neumann found little comfort in receiving the last rites – he remained terrified of death and unable to accept it. Of his religious views, Von Neumann reportedly said, "So long as there is the possibility of eternal damnation for nonbelievers it is more logical to be a believer at the end," referring to Pascal's wager. He confided to his mother, "There probably has to be a God. Many things are easier to explain if there is than if there isn't." He died Roman Catholic on February 8, 1957, at Walter Reed Army Medical Hospital and was buried at Princeton Cemetery. == Mathematics == === Set theory === At the beginning of the 20th century, efforts to base mathematics on naive set theory suffered a setback due to Russell's paradox (on the set of all sets that do not belong to themselves). The problem of an adequate axiomatization of set theory was resolved implicitly about twenty years later by Ernst Zermelo and Abraham Fraenkel. Zermelo–Fraenkel set theory provided a series of principles that allowed for the construction of the sets used in the everyday practice of mathematics, but did not explicitly exclude the possibility of the existence of a set that belongs to itself. In his 1925 doctoral thesis, von Neumann demonstrated two techniques to exclude such sets—the axiom of foundation and the notion of class. The axiom of foundation proposed that every set can be constructed from the bottom up in an ordered succession of steps by way of the Zermelo–Fraenkel principles. If one set belongs to another, then the first must necessarily come before the second in the succession. This excludes the possibility of a set belonging to itself. To demonstrate that the addition of this new axiom to the others did not produce contradictions, von Neumann introduced the method of inner models, which became an essential demonstration instrument in set theory. The second approach to the problem of sets belonging to themselves took as its base the notion of class, and defines a set as a class that belongs to other classes, while a proper class is defined as a class that does not belong to other classes. On the Zermelo–Fraenkel approach, the axioms impede the construction of a set of all sets that do not belong to themselves. In contrast, on von Neumann's approach, the class of all sets that do not belong to themselves can be constructed, but it is a proper class, not a set. Overall, von Neumann's major achievement in set theory was an "axiomatization of set theory and (connected with that) elegant theory of the ordinal and cardinal numbers as well as the first strict formulation of principles of definitions by the transfinite induction". ==== Von Neumann paradox ==== Building on the Hausdorff paradox of Felix Hausdorff (1914), Stefan Banach and Alfred Tarski in 1924 showed how to subdivide a three-dimensional ball into disjoint sets, then translate and rotate these sets to form two identical copies of the same ball; this is the Banach–Tarski paradox. They also proved that a two-dimensional disk has no such paradoxical decomposition. But in 1929, von Neumann subdivided the disk into finitely many pieces and rearranged them into two disks, using area-preserving affine transformations instead of translations and rotations. The result depended on finding free groups of affine transformations, an important technique extended later by von Neumann in his work on measure theory. === Proof theory === With the contributions of von Neumann to sets, the axiomatic system of the theory of sets avoided the contradictions of earlier systems and became usable as a foundation for mathematics, despite the lack of a proof of its consistency. The next question was whether it provided definitive answers to all mathematical questions that could be posed in it, or whether it might be improved by adding stronger axioms that could be used to prove a broader class of theorems. By 1927, von Neumann was involving himself in discussions in Göttingen on whether elementary arithmetic followed from Peano axioms. Building on the work of Ackermann, he began attempting to prove (using the finistic methods of Hilbert's school) the consistency of first-order arithmetic. He succeeded in proving the consistency of a fragment of arithmetic of natural numbers (through the use of restrictions on induction). He continued looking for a more general proof of the consistency of classical mathematics using methods from proof theory. A strongly negative answer to whether it was definitive arrived in September 1930 at the Second Conference on the Epistemology of the Exact Sciences, in which Kurt Gödel announced his first theorem of incompleteness: the usual axiomatic systems are incomplete, in the sense that they cannot prove every truth expressible in their language. Moreover, every consistent extension of these systems necessarily remains incomplete. At the conference, von Neumann suggested to Gödel that he should try to transform his results for undecidable propositions about integers. Less than a month later, von Neumann communicated to Gödel an interesting consequence of his theorem: the usual axiomatic systems are unable to demonstrate their own consistency. Gödel replied that he had already discovered this consequence, now known as his second incompleteness theorem, and that he would send a preprint of his article containing both results, which never appeared. Von Neumann acknowledged Gödel's priority in his next letter. However, von Neumann's method of proof differed from Gödel's, and he was also of the opinion that the second incompleteness theorem had dealt a much stronger blow to Hilbert's program than Gödel thought it did. With this discovery, which drastically changed his views on mathematical rigor, von Neumann ceased research in the foundations of mathematics and metamathematics and instead spent time on problems connected with applications. === Ergodic theory === In a series of papers published in 1932, von Neumann made foundational contributions to ergodic theory, a branch of mathematics that involves the states of dynamical systems with an invariant measure. Of the 1932 papers on ergodic theory, Paul Halmos wrote that even "if von Neumann had never done anything else, they would have been sufficient to guarantee him mathematical immortality". By then von Neumann had already written his articles on operator theory, and the application of this work was instrumental in his mean ergodic theorem. The theorem is about arbitrary one-parameter unitary groups t → V t {\displaystyle {\mathit {t}}\to {\mathit {V_{t}}}} and states that for every vector ϕ {\displaystyle \phi } in the Hilbert space, lim T → ∞ 1 T ∫ 0 T V t ( ϕ ) d t {\textstyle \lim _{T\to \infty }{\frac {1}{T}}\int _{0}^{T}V_{t}(\phi )\,dt} exists in the sense of the metric defined by the Hilbert norm and is a vector ψ {\displaystyle \psi } which is such that V t ( ψ ) = ψ {\displaystyle V_{t}(\psi )=\psi } for all t {\displaystyle t} . This was proven in the first paper. In the second paper, von Neumann argued that his results here were sufficient for physical applications relating to Boltzmann's ergodic hypothesis. He also pointed out that ergodicity had not yet been achieved and isolated this for future work. Later in the year he published another influential paper that began the systematic study of ergodicity. He gave and proved a decomposition theorem showing that the ergodic measure preserving actions of the real line are the fundamental building blocks from which all measure preserving actions can be built. Several other key theorems are given and proven. The results in this paper and another in conjunction with Paul Halmos have significant applications in other areas of mathematics. === Measure theory === In measure theory, the "problem of measure" for an n-dimensional Euclidean space Rn may be stated as: "does there exist a positive, normalized, invariant, and additive set function on the class of all subsets of Rn?" The work of Felix Hausdorff and Stefan Banach had implied that the problem of measure has a positive solution if n = 1 or n = 2 and a negative solution (because of the Banach–Tarski paradox) in all other cases. Von Neumann's work argued that the "problem is essentially group-theoretic in character": the existence of a measure could be determined by looking at the properties of the transformation group of the given space. The positive solution for spaces of dimension at most two, and the negative solution for higher dimensions, comes from the fact that the Euclidean group is a solvable group for dimension at most two, and is not solvable for higher dimensions. "Thus, according to von Neumann, it is the change of group that makes a difference, not the change of space." Around 1942 he told Dorothy Maharam how to prove that every complete σ-finite measure space has a multiplicative lifting; he did not publish this proof and she later came up with a new one. In a number of von Neumann's papers, the methods of argument he employed are considered even more significant than the results. In anticipation of his later study of dimension theory in algebras of operators, von Neumann used results on equivalence by finite decomposition, and reformulated the problem of measure in terms of functions. A major contribution von Neumann made to measure theory was the result of a paper written to answer a question of Haar regarding whether there existed an algebra of all bounded functions on the real number line such that they form "a complete system of representatives of the classes of almost everywhere-equal measurable bounded functions". He proved this in the positive, and in later papers with Stone discussed various generalizations and algebraic aspects of this problem. He also proved by new methods the existence of disintegrations for various general types of measures. Von Neumann also gave a new proof on the uniqueness of Haar measures by using the mean values of functions, although this method only worked for compact groups. He had to create entirely new techniques to apply this to locally compact groups. He also gave a new, ingenious proof for the Radon–Nikodym theorem. His lecture notes on measure theory at the Institute for Advanced Study were an important source for knowledge on the topic in America at the time, and were later published. === Topological groups === Using his previous work on measure theory, von Neumann made several contributions to the theory of topological groups, beginning with a paper on almost periodic functions on groups, where von Neumann extended Bohr's theory of almost periodic functions to arbitrary groups. He continued this work with another paper in conjunction with Bochner that improved the theory of almost periodicity to include functions that took on elements of linear spaces as values rather than numbers. In 1938, he was awarded the Bôcher Memorial Prize for his work in analysis in relation to these papers. In a 1933 paper, he used the newly discovered Haar measure in the solution of Hilbert's fifth problem for the case of compact groups. The basic idea behind this was discovered several years earlier when von Neumann published a paper on the analytic properties of groups of linear transformations and found that closed subgroups of a general linear group are Lie groups. This was later extended by Cartan to arbitrary Lie groups in the form of the closed-subgroup theorem. === Functional analysis === Von Neumann was the first to axiomatically define an abstract Hilbert space. He defined it as a complex vector space with a Hermitian scalar product, with the corresponding norm being both separable and complete. In the same papers he also proved the general form of the Cauchy–Schwarz inequality that had previously been known only in specific examples. He continued with the development of the spectral theory of operators in Hilbert space in three seminal papers between 1929 and 1932. This work cumulated in his Mathematical Foundations of Quantum Mechanics which alongside two other books by Stone and Banach in the same year were the first monographs on Hilbert space theory. Previous work by others showed that a theory of weak topologies could not be obtained by using sequences. Von Neumann was the first to outline a program of how to overcome the difficulties, which resulted in him defining locally convex spaces and topological vector spaces for the first time. In addition several other topological properties he defined at the time (he was among the first mathematicians to apply new topological ideas from Hausdorff from Euclidean to Hilbert spaces) such as boundness and total boundness are still used today. For twenty years von Neumann was considered the 'undisputed master' of this area. These developments were primarily prompted by needs in quantum mechanics where von Neumann realized the need to extend the spectral theory of Hermitian operators from the bounded to the unbounded case. Other major achievements in these papers include a complete elucidation of spectral theory for normal operators, the first abstract presentation of the trace of a positive operator, a generalisation of Riesz's presentation of Hilbert's spectral theorems at the time, and the discovery of Hermitian operators in a Hilbert space, as distinct from self-adjoint operators, which enabled him to give a description of all Hermitian operators which extend a given Hermitian operator. He wrote a paper detailing how the usage of infinite matrices, common at the time in spectral theory, was inadequate as a representation for Hermitian operators. His work on operator theory lead to his most profound invention in pure mathematics, the study of von Neumann algebras and in general of operator algebras. His later work on rings of operators lead to him revisiting his work on spectral theory and providing a new way of working through the geometric content by the use of direct integrals of Hilbert spaces. Like in his work on measure theory he proved several theorems that he did not find time to publish. He told Nachman Aronszajn and K. T. Smith that in the early 1930s he proved the existence of proper invariant subspaces for completely continuous operators in a Hilbert space while working on the invariant subspace problem. With I. J. Schoenberg he wrote several items investigating translation invariant Hilbertian metrics on the real number line which resulted in their complete classification. Their motivation lie in various questions related to embedding metric spaces into Hilbert spaces. With Pascual Jordan he wrote a short paper giving the first derivation of a given norm from an inner product by means of the parallelogram identity. His trace inequality is a key result of matrix theory used in matrix approximation problems. He also first presented the idea that the dual of a pre-norm is a norm in the first major paper discussing the theory of unitarily invariant norms and symmetric gauge functions (now known as symmetric absolute norms). This paper leads naturally to the study of symmetric operator ideals and is the beginning point for modern studies of symmetric operator spaces. Later with Robert Schatten he initiated the study of nuclear operators on Hilbert spaces, tensor products of Banach spaces, introduced and studied trace class operators, their ideals, and their duality with compact operators, and preduality with bounded operators. The generalization of this topic to the study of nuclear operators on Banach spaces was among the first achievements of Alexander Grothendieck. Previously in 1937 von Neumann published several results in this area, for example giving 1-parameter scale of different cross norms on l 2 n ⊗ l 2 n {\displaystyle {\textit {l}}\,_{2}^{n}\otimes {\textit {l}}\,_{2}^{n}} and proving several other results on what are now known as Schatten–von Neumann ideals. === Operator algebras === Von Neumann founded the study of rings of operators, through the von Neumann algebras (originally called W*-algebras). While his original ideas for rings of operators existed already in 1930, he did not begin studying them in depth until he met F. J. Murray several years later. A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. The von Neumann bicommutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as being equal to the bicommutant. After elucidating the study of the commutative algebra case, von Neumann embarked in 1936, with the partial collaboration of Murray, on the noncommutative case, the general study of factors classification of von Neumann algebras. The six major papers in which he developed that theory between 1936 and 1940 "rank among the masterpieces of analysis in the twentieth century"; they collect many foundational results and started several programs in operator algebra theory that mathematicians worked on for decades afterwards. An example is the classification of factors. In addition in 1938 he proved that every von Neumann algebra on a separable Hilbert space is a direct integral of factors; he did not find time to publish this result until 1949. Von Neumann algebras relate closely to a theory of noncommutative integration, something that von Neumann hinted to in his work but did not explicitly write out. Another important result on polar decomposition was published in 1932. === Lattice theory === Between 1935 and 1937, von Neumann worked on lattice theory, the theory of partially ordered sets in which every two elements have a greatest lower bound and a least upper bound. As Garrett Birkhoff wrote, "John von Neumann's brilliant mind blazed over lattice theory like a meteor". Von Neumann combined traditional projective geometry with modern algebra (linear algebra, ring theory, lattice theory). Many previously geometric results could then be interpreted in the case of general modules over rings. His work laid the foundations for some of the modern work in projective geometry. His biggest contribution was founding the field of continuous geometry. It followed his path-breaking work on rings of operators. In mathematics, continuous geometry is a substitute of complex projective geometry, where instead of the dimension of a subspace being in a discrete set 0 , 1 , . . . , n {\displaystyle 0,1,...,{\mathit {n}}} it can be an element of the unit interval [ 0 , 1 ] {\displaystyle [0,1]} . Earlier, Menger and Birkhoff had axiomatized complex projective geometry in terms of the properties of its lattice of linear subspaces. Von Neumann, following his work on rings of operators, weakened those axioms to describe a broader class of lattices, the continuous geometries. While the dimensions of the subspaces of projective geometries are a discrete set (the non-negative integers), the dimensions of the elements of a continuous geometry can range continuously across the unit interval [ 0 , 1 ] {\displaystyle [0,1]} . Von Neumann was motivated by his discovery of von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor. In more pure lattice theoretical work, he solved the difficult problem of characterizing the class of C G ( F ) {\displaystyle {\mathit {CG(F)}}} (continuous-dimensional projective geometry over an arbitrary division ring F {\displaystyle {\mathit {F}}\,} ) in abstract language of lattice theory. Von Neumann provided an abstract exploration of dimension in completed complemented modular topological lattices (properties that arise in the lattices of subspaces of inner product spaces): Dimension is determined, up to a positive linear transformation, by the following two properties. It is conserved by perspective mappings ("perspectivities") and ordered by inclusion. The deepest part of the proof concerns the equivalence of perspectivity with "projectivity by decomposition"—of which a corollary is the transitivity of perspectivity. For any integer n > 3 {\displaystyle n>3} every n {\displaystyle {\mathit {n}}} -dimensional abstract projective geometry is isomorphic to the subspace-lattice of an n {\displaystyle {\mathit {n}}} -dimensional vector space V n ( F ) {\displaystyle V_{n}(F)} over a (unique) corresponding division ring F {\displaystyle F} . This is known as the Veblen–Young theorem. Von Neumann extended this fundamental result in projective geometry to the continuous dimensional case. This coordinatization theorem stimulated considerable work in abstract projective geometry and lattice theory, much of which continued using von Neumann's techniques. Birkhoff described this theorem as follows: Any complemented modular lattice L having a "basis" of n ≥ 4 pairwise perspective elements, is isomorphic with the lattice ℛ(R) of all principal right-ideals of a suitable regular ring R. This conclusion is the culmination of 140 pages of brilliant and incisive algebra involving entirely novel axioms. Anyone wishing to get an unforgettable impression of the razor edge of von Neumann's mind, need merely try to pursue this chain of exact reasoning for himself—realizing that often five pages of it were written down before breakfast, seated at a living room writing-table in a bathrobe. This work required the creation of regular rings. A von Neumann regular ring is a ring where for every a {\displaystyle a} , an element x {\displaystyle x} exists such that a x a = a {\displaystyle axa=a} . These rings came from and have connections to his work on von Neumann algebras, as well as AW*-algebras and various kinds of C*-algebras. Many smaller technical results were proven during the creation and proof of the above theorems, particularly regarding distributivity (such as infinite distributivity), von Neumann developing them as needed. He also developed a theory of valuations in lattices, and shared in developing the general theory of metric lattices. Birkhoff noted in his posthumous article on von Neumann that most of these results were developed in an intense two-year period of work, and that while his interests continued in lattice theory after 1937, they became peripheral and mainly occurred in letters to other mathematicians. A final contribution in 1940 was for a joint seminar he conducted with Birkhoff at the Institute for Advanced Study on the subject where he developed a theory of σ-complete lattice ordered rings. He never wrote up the work for publication. === Mathematical statistics === Von Neumann made fundamental contributions to mathematical statistics. In 1941, he derived the exact distribution of the ratio of the mean square of successive differences to the sample variance for independent and identically normally distributed variables. This ratio was applied to the residuals from regression models and is commonly known as the Durbin–Watson statistic for testing the null hypothesis that the errors are serially independent against the alternative that they follow a stationary first order autoregression. Subsequently, Denis Sargan and Alok Bhargava extended the results for testing whether the errors on a regression model follow a Gaussian random walk (i.e., possess a unit root) against the alternative that they are a stationary first order autoregression. === Other work === In his early years, von Neumann published several papers related to set-theoretical real analysis and number theory. In a paper from 1925, he proved that for any dense sequence of points in [ 0 , 1 ] {\displaystyle [0,1]} , there existed a rearrangement of those points that is uniformly distributed. In 1926 his sole publication was on Prüfer's theory of ideal algebraic numbers where he found a new way of constructing them, thus extending Prüfer's theory to the field of all algebraic numbers, and clarified their relation to p-adic numbers. In 1928 he published two additional papers continuing with these themes. The first dealt with partitioning an interval into countably many congruent subsets. It solved a problem of Hugo Steinhaus asking whether an interval is ℵ 0 {\displaystyle \aleph _{0}} -divisible. Von Neumann proved that indeed that all intervals, half-open, open, or closed are ℵ 0 {\displaystyle \aleph _{0}} -divisible by translations (i.e. that these intervals can be decomposed into ℵ 0 {\displaystyle \aleph _{0}} subsets that are congruent by translation). His next paper dealt with giving a constructive proof without the axiom of choice that 2 ℵ 0 {\displaystyle 2^{\aleph _{0}}} algebraically independent reals exist. He proved that A r = ∑ n = 0 ∞ 2 2 [ n r ] / 2 2 n 2 {\displaystyle A_{r}=\textstyle \sum _{n=0}^{\infty }2^{2^{[nr]}}\!{\big /}\,2^{2^{n^{2}}}} are algebraically independent for r > 0 {\displaystyle r>0} . Consequently, there exists a perfect algebraically independent set of reals the size of the continuum. Other minor results from his early career include a proof of a maximum principle for the gradient of a minimizing function in the field of calculus of variations, and a small simplification of Hermann Minkowski's theorem for linear forms in geometric number theory. Later in his career together with Pascual Jordan and Eugene Wigner he wrote a foundational paper classifying all finite-dimensional formally real Jordan algebras and discovering the Albert algebras while attempting to look for a better mathematical formalism for quantum theory. In 1936 he attempted to further the program of replacing the axioms of his previous Hilbert space program with those of Jordan algebras in a paper investigating the infinite-dimensional case; he planned to write at least one further paper on the topic but never did. Nevertheless, these axioms formed the basis for further investigations of algebraic quantum mechanics started by Irving Segal. == Physics == === Quantum mechanics === Von Neumann was the first to establish a rigorous mathematical framework for quantum mechanics, known as the Dirac–von Neumann axioms, in his influential 1932 work Mathematical Foundations of Quantum Mechanics. After having completed the axiomatization of set theory, he began to confront the axiomatization of quantum mechanics. He realized in 1926 that a state of a quantum system could be represented by a point in a (complex) Hilbert space that, in general, could be infinite-dimensional even for a single particle. In this formalism of quantum mechanics, observable quantities such as position or momentum are represented as linear operators acting on the Hilbert space associated with the quantum system. The physics of quantum mechanics was thereby reduced to the mathematics of Hilbert spaces and linear operators acting on them. For example, the uncertainty principle, according to which the determination of the position of a particle prevents the determination of its momentum and vice versa, is translated into the non-commutativity of the two corresponding operators. This new mathematical formulation included as special cases the formulations of both Heisenberg and Schrödinger. Von Neumann's abstract treatment permitted him to confront the foundational issue of determinism versus non-determinism, and in the book he presented a proof that the statistical results of quantum mechanics could not possibly be averages of an underlying set of determined "hidden variables", as in classical statistical mechanics. In 1935, Grete Hermann published a paper arguing that the proof contained a conceptual error and was therefore invalid. Hermann's work was largely ignored until after John S. Bell made essentially the same argument in 1966. In 2010, Jeffrey Bub argued that Bell had misconstrued von Neumann's proof, and pointed out that the proof, though not valid for all hidden variable theories, does rule out a well-defined and important subset. Bub also suggests that von Neumann was aware of this limitation and did not claim that his proof completely ruled out hidden variable theories. The validity of Bub's argument is, in turn, disputed. Gleason's theorem of 1957 provided an argument against hidden variables along the lines of von Neumann's, but founded on assumptions seen as better motivated and more physically meaningful. Von Neumann's proof inaugurated a line of research that ultimately led, through Bell's theorem and the experiments of Alain Aspect in 1982, to the demonstration that quantum physics either requires a notion of reality substantially different from that of classical physics, or must include nonlocality in apparent violation of special relativity. In a chapter of The Mathematical Foundations of Quantum Mechanics, von Neumann deeply analyzed the so-called measurement problem. He concluded that the entire physical universe could be made subject to the universal wave function. Since something "outside the calculation" was needed to collapse the wave function, von Neumann concluded that the collapse was caused by the consciousness of the experimenter. He argued that the mathematics of quantum mechanics allows the collapse of the wave function to be placed at any position in the causal chain from the measurement device to the "subjective consciousness" of the human observer. In other words, while the line between observer and observed could be drawn in different places, the theory only makes sense if an observer exists somewhere. Although the idea of consciousness causing collapse was accepted by Eugene Wigner, this interpretation never gained acceptance among the majority of physicists. Though theories of quantum mechanics continue to evolve, a basic framework for the mathematical formalism of problems in quantum mechanics underlying most approaches can be traced back to the mathematical formalisms and techniques first used by von Neumann. Discussions about interpretation of the theory, and extensions to it, are now mostly conducted on the basis of shared assumptions about the mathematical foundations. Viewing von Neumann's work on quantum mechanics as a part of the fulfilment of Hilbert's sixth problem, mathematical physicist Arthur Wightman said in 1974 his axiomization of quantum theory was perhaps the most important axiomization of a physical theory to date. With his 1932 book, quantum mechanics became a mature theory in the sense it had a precise mathematical form, which allowed for clear answers to conceptual problems. Nevertheless, von Neumann in his later years felt he had failed in this aspect of his scientific work as despite all the mathematics he developed, he did not find a satisfactory mathematical framework for quantum theory as a whole. ==== Von Neumann entropy ==== Von Neumann entropy is extensively used in different forms (conditional entropy, relative entropy, etc.) in the framework of quantum information theory. Entanglement measures are based upon some quantity directly related to the von Neumann entropy. Given a statistical ensemble of quantum mechanical systems with the density matrix ρ {\displaystyle \rho } , it is given by S ( ρ ) = − Tr ( ρ ln ρ ) . {\displaystyle S(\rho )=-\operatorname {Tr} (\rho \ln \rho ).\,} Many of the same entropy measures in classical information theory can also be generalized to the quantum case, such as Holevo entropy and conditional quantum entropy. Quantum information theory is largely concerned with the interpretation and uses of von Neumann entropy, a cornerstone in the former's development; the Shannon entropy applies to classical information theory. ==== Density matrix ==== The formalism of density operators and matrices was introduced by von Neumann in 1927 and independently, but less systematically by Lev Landau and Felix Bloch in 1927 and 1946 respectively. The density matrix allows the representation of probabilistic mixtures of quantum states (mixed states) in contrast to wavefunctions, which can only represent pure states. ==== Von Neumann measurement scheme ==== The von Neumann measurement scheme, the ancestor of quantum decoherence theory, represents measurements projectively by taking into account the measuring apparatus which is also treated as a quantum object. The 'projective measurement' scheme introduced by von Neumann led to the development of quantum decoherence theories. ==== Quantum logic ==== Von Neumann first proposed a quantum logic in his 1932 treatise Mathematical Foundations of Quantum Mechanics, where he noted that projections on a Hilbert space can be viewed as propositions about physical observables. The field of quantum logic was subsequently inaugurated in a 1936 paper by von Neumann and Garrett Birkhoff, the first to introduce quantum logics, wherein von Neumann and Birkhoff first proved that quantum mechanics requires a propositional calculus substantially different from all classical logics and rigorously isolated a new algebraic structure for quantum logics. The concept of creating a propositional calculus for quantum logic was first outlined in a short section in von Neumann's 1932 work, but in 1936, the need for the new propositional calculus was demonstrated through several proofs. For example, photons cannot pass through two successive filters that are polarized perpendicularly (e.g., horizontally and vertically), and therefore, a fortiori, it cannot pass if a third filter polarized diagonally is added to the other two, either before or after them in the succession, but if the third filter is added between the other two, the photons will indeed pass through. This experimental fact is translatable into logic as the non-commutativity of conjunction ( A ∧ B ) ≠ ( B ∧ A ) {\displaystyle (A\land B)\neq (B\land A)} . It was also demonstrated that the laws of distribution of classical logic, P ∨ ( Q ∧ R ) = {\displaystyle P\lor (Q\land R)={}} ( P ∨ Q ) ∧ ( P ∨ R ) {\displaystyle (P\lor Q)\land (P\lor R)} and P ∧ ( Q ∨ R ) = {\displaystyle P\land (Q\lor R)={}} ( P ∧ Q ) ∨ ( P ∧ R ) {\displaystyle (P\land Q)\lor (P\land R)} , are not valid for quantum theory. The reason for this is that a quantum disjunction, unlike the case for classical disjunction, can be true even when both of the disjuncts are false and this is in turn attributable to the fact that it is frequently the case in quantum mechanics that a pair of alternatives are semantically determinate, while each of its members is necessarily indeterminate. Consequently, the distributive law of classical logic must be replaced with a weaker condition. Instead of a distributive lattice, propositions about a quantum system form an orthomodular lattice isomorphic to the lattice of subspaces of the Hilbert space associated with that system. Nevertheless, he was never satisfied with his work on quantum logic. He intended it to be a joint synthesis of formal logic and probability theory and when he attempted to write up a paper for the Henry Joseph Lecture he gave at the Washington Philosophical Society in 1945 he found that he could not, especially given that he was busy with war work at the time. During his address at the 1954 International Congress of Mathematicians he gave this issue as one of the unsolved problems that future mathematicians could work on. === Fluid dynamics === Von Neumann made fundamental contributions in the field of fluid dynamics, including the classic flow solution to blast waves, and the co-discovery (independently by Yakov Borisovich Zel'dovich and Werner Döring) of the ZND detonation model of explosives. During the 1930s, von Neumann became an authority on the mathematics of shaped charges. Later with Robert D. Richtmyer, von Neumann developed an algorithm defining artificial viscosity that improved the understanding of shock waves. When computers solved hydrodynamic or aerodynamic problems, they put too many computational grid points at regions of sharp discontinuity (shock waves). The mathematics of artificial viscosity smoothed the shock transition without sacrificing basic physics. Von Neumann soon applied computer modelling to the field, developing software for his ballistics research. During World War II, he approached R. H. Kent, the director of the US Army's Ballistic Research Laboratory, with a computer program for calculating a one-dimensional model of 100 molecules to simulate a shock wave. Von Neumann gave a seminar on his program to an audience which included his friend Theodore von Kármán. After von Neumann had finished, von Kármán said "Of course you realize Lagrange also used digital models to simulate continuum mechanics." Von Neumann had been unaware of Lagrange's Mécanique analytique. === Other work === While not as prolific in physics as he was in mathematics, he nevertheless made several other notable contributions. His pioneering papers with Subrahmanyan Chandrasekhar on the statistics of a fluctuating gravitational field generated by randomly distributed stars were considered a tour de force. In this paper they developed a theory of two-body relaxation and used the Holtsmark distribution to model the dynamics of stellar systems. He wrote several other unpublished manuscripts on topics in stellar structure, some of which were included in Chandrasekhar's other works. In earlier work led by Oswald Veblen von Neumann helped develop basic ideas involving spinors that would lead to Roger Penrose's twistor theory. Much of this was done in seminars conducted at the IAS during the 1930s. From this work he wrote a paper with A. H. Taub and Veblen extending the Dirac equation to projective relativity, with a key focus on maintaining invariance with regards to coordinate, spin, and gauge transformations, as a part of early research into potential theories of quantum gravity in the 1930s. In the same time period he made several proposals to colleagues for dealing with the problems in the newly created quantum field theory and for quantizing spacetime; however, both his colleagues and he did not consider the ideas fruitful and did not pursue them. Nevertheless, he maintained at least some interest, in 1940 writing a manuscript on the Dirac equation in de Sitter space. == Economics == === Game theory === Von Neumann founded the field of game theory as a mathematical discipline. He proved his minimax theorem in 1928. It establishes that in zero-sum games with perfect information (i.e., in which players know at each time all moves that have taken place so far), there exists a pair of strategies for both players that allows each to minimize their maximum losses. Such strategies are called optimal. Von Neumann showed that their minimaxes are equal (in absolute value) and contrary (in sign). He improved and extended the minimax theorem to include games involving imperfect information and games with more than two players, publishing this result in his 1944 Theory of Games and Economic Behavior, written with Oskar Morgenstern. The public interest in this work was such that The New York Times ran a front-page story. In this book, von Neumann declared that economic theory needed to use functional analysis, especially convex sets and the topological fixed-point theorem, rather than the traditional differential calculus, because the maximum-operator did not preserve differentiable functions. Von Neumann's functional-analytic techniques—the use of duality pairings of real vector spaces to represent prices and quantities, the use of supporting and separating hyperplanes and convex sets, and fixed-point theory—have been primary tools of mathematical economics ever since. === Mathematical economics === Von Neumann raised the mathematical level of economics in several influential publications. For his model of an expanding economy, he proved the existence and uniqueness of an equilibrium using his generalization of the Brouwer fixed-point theorem. Von Neumann's model of an expanding economy considered the matrix pencil A − λB with nonnegative matrices A and B; von Neumann sought probability vectors p and q and a positive number λ that would solve the complementarity equation p T ( A − λ B ) q = 0 {\displaystyle p^{T}(A-\lambda B)q=0} along with two inequality systems expressing economic efficiency. In this model, the (transposed) probability vector p represents the prices of the goods while the probability vector q represents the "intensity" at which the production process would run. The unique solution λ represents the growth factor which is 1 plus the rate of growth of the economy; the rate of growth equals the interest rate. Von Neumann's results have been viewed as a special case of linear programming, where his model uses only nonnegative matrices. The study of his model of an expanding economy continues to interest mathematical economists. This paper has been called the greatest paper in mathematical economics by several authors, who recognized its introduction of fixed-point theorems, linear inequalities, complementary slackness, and saddlepoint duality. In the proceedings of a conference on von Neumann's growth model, Paul Samuelson said that many mathematicians had developed methods useful to economists, but that von Neumann was unique in having made significant contributions to economic theory itself. The lasting importance of the work on general equilibria and the methodology of fixed point theorems is underscored by the awarding of Nobel prizes in 1972 to Kenneth Arrow, in 1983 to Gérard Debreu, and in 1994 to John Nash who used fixed point theorems to establish equilibria for non-cooperative games and for bargaining problems in his Ph.D. thesis. Arrow and Debreu also used linear programming, as did Nobel laureates Tjalling Koopmans, Leonid Kantorovich, Wassily Leontief, Paul Samuelson, Robert Dorfman, Robert Solow, and Leonid Hurwicz. Von Neumann's interest in the topic began while he was lecturing at Berlin in 1928 and 1929. He spent his summers in Budapest, as did the economist Nicholas Kaldor; Kaldor recommended that von Neumann read a book by the mathematical economist Léon Walras. Von Neumann noticed that Walras's General Equilibrium Theory and Walras's law, which led to systems of simultaneous linear equations, could produce the absurd result that profit could be maximized by producing and selling a negative quantity of a product. He replaced the equations by inequalities, introduced dynamic equilibria, among other things, and eventually produced his paper. === Linear programming === Building on his results on matrix games and on his model of an expanding economy, von Neumann invented the theory of duality in linear programming when George Dantzig described his work in a few minutes, and an impatient von Neumann asked him to get to the point. Dantzig then listened dumbfounded while von Neumann provided an hourlong lecture on convex sets, fixed-point theory, and duality, conjecturing the equivalence between matrix games and linear programming. Later, von Neumann suggested a new method of linear programming, using the homogeneous linear system of Paul Gordan (1873), which was later popularized by Karmarkar's algorithm. Von Neumann's method used a pivoting algorithm between simplices, with the pivoting decision determined by a nonnegative least squares subproblem with a convexity constraint (projecting the zero-vector onto the convex hull of the active simplex). Von Neumann's algorithm was the first interior point method of linear programming. == Computer science == Von Neumann was a founding figure in computing, with significant contributions to computing hardware design, to theoretical computer science, to scientific computing, and to the philosophy of computer science. === Hardware === Von Neumann consulted for the Army's Ballistic Research Laboratory, most notably on the ENIAC project, as a member of its Scientific Advisory Committee. Although the single-memory, stored-program architecture is commonly called von Neumann architecture, the architecture was based on the work of J. Presper Eckert and John Mauchly, inventors of ENIAC and its successor, EDVAC. While consulting for the EDVAC project at the University of Pennsylvania, von Neumann wrote an incomplete First Draft of a Report on the EDVAC. The paper, whose premature distribution nullified the patent claims of Eckert and Mauchly, described a computer that stored both its data and its program in the same address space, unlike the earliest computers which stored their programs separately on paper tape or plugboards. This architecture became the basis of most modern computer designs. Next, von Neumann designed the IAS machine at the Institute for Advanced Study in Princeton, New Jersey. He arranged its financing, and the components were designed and built at the RCA Research Laboratory nearby. Von Neumann recommended that the IBM 701, nicknamed the defense computer, include a magnetic drum. It was a faster version of the IAS machine and formed the basis for the commercially successful IBM 704. === Algorithms === Von Neumann was the inventor, in 1945, of the merge sort algorithm, in which the first and second halves of an array are each sorted recursively and then merged. As part of Von Neumann's hydrogen bomb work, he and Stanisław Ulam developed simulations for hydrodynamic computations. He also contributed to the development of the Monte Carlo method, which used random numbers to approximate the solutions to complicated problems. Von Neumann's algorithm for simulating a fair coin with a biased coin is used in the "software whitening" stage of some hardware random number generators. Because obtaining "truly" random numbers was impractical, von Neumann developed a form of pseudorandomness, using the middle-square method. He justified this crude method as faster than any other method at his disposal, writing that "Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin." He also noted that when this method went awry it did so obviously, unlike other methods which could be subtly incorrect. Stochastic computing was introduced by von Neumann in 1953, but could not be implemented until advances in computing of the 1960s. Around 1950 he was also among the first to talk about the time complexity of computations, which eventually evolved into the field of computational complexity theory. === Cellular automata, DNA and the universal constructor === Von Neumann's mathematical analysis of the structure of self-replication preceded the discovery of the structure of DNA. Ulam and von Neumann are also generally credited with creating the field of cellular automata, beginning in the 1940s, as a simplified mathematical model of biological systems. In lectures in 1948 and 1949, von Neumann proposed a kinematic self-reproducing automaton. By 1952, he was treating the problem more abstractly. He designed an elaborate 2D cellular automaton that would automatically make a copy of its initial configuration of cells. The Von Neumann universal constructor based on the von Neumann cellular automaton was fleshed out in his posthumous Theory of Self Reproducing Automata. The von Neumann neighborhood, in which each cell in a two-dimensional grid has the four orthogonally adjacent grid cells as neighbors, continues to be used for other cellular automata. === Scientific computing and numerical analysis === Considered to be possibly "the most influential researcher in scientific computing of all time", von Neumann made several contributions to the field, both technically and administratively. He developed the Von Neumann stability analysis procedure, still commonly used to avoid errors from building up in numerical methods for linear partial differential equations. His paper with Herman Goldstine in 1947 was the first to describe backward error analysis, although implicitly. He was also one of the first to write about the Jacobi method. At Los Alamos, he wrote several classified reports on solving problems of gas dynamics numerically. However, he was frustrated by the lack of progress with analytic methods for these nonlinear problems. As a result, he turned towards computational methods. Under his influence Los Alamos became the leader in computational science during the 1950s and early 1960s. From this work von Neumann realized that computation was not just a tool to brute force the solution to a problem numerically, but could also provide insight for solving problems analytically, and that there was an enormous variety of scientific and engineering problems towards which computers would be useful, most significant of which were nonlinear problems. In June 1945 at the First Canadian Mathematical Congress he gave his first talk on general ideas of how to solve problems, particularly of fluid dynamics numerically. He also described how wind tunnels were actually analog computers, and how digital computers would replace them and bring a new era of fluid dynamics. Garrett Birkhoff described it as "an unforgettable sales pitch". He expanded this talk with Goldstine into the manuscript "On the Principles of Large Scale Computing Machines" and used it to promote the support of scientific computing. His papers also developed the concepts of inverting matrices, random matrices and automated relaxation methods for solving elliptic boundary value problems. === Weather systems and global warming === As part of his research into possible applications of computers, von Neumann became interested in weather prediction, noting similarities between the problems in the field and those he had worked on during the Manhattan Project. In 1946 von Neumann founded the "Meteorological Project" at the Institute for Advanced Study, securing funding for his project from the Weather Bureau, the US Air Force and US Navy weather services. With Carl-Gustaf Rossby, considered the leading theoretical meteorologist at the time, he gathered a group of twenty meteorologists to work on various problems in the field. However, given his other postwar work he was not able to devote enough time to proper leadership of the project and little was accomplished. This changed when a young Jule Gregory Charney took up co-leadership of the project from Rossby. By 1950 von Neumann and Charney wrote the world's first climate modelling software, and used it to perform the world's first numerical weather forecasts on the ENIAC computer that von Neumann had arranged to be used; von Neumann and his team published the results as Numerical Integration of the Barotropic Vorticity Equation. Together they played a leading role in efforts to integrate sea-air exchanges of energy and moisture into the study of climate. Though primitive, news of the ENIAC forecasts quickly spread around the world and a number of parallel projects in other locations were initiated. In 1955 von Neumann, Charney and their collaborators convinced their funders to open the Joint Numerical Weather Prediction Unit (JNWPU) in Suitland, Maryland, which began routine real-time weather forecasting. Next up, von Neumann proposed a research program for climate modeling: The approach is to first try short-range forecasts, then long-range forecasts of those properties of the circulation that can perpetuate themselves over arbitrarily long periods of time, and only finally to attempt forecast for medium-long time periods which are too long to treat by simple hydrodynamic theory and too short to treat by the general principle of equilibrium theory. Positive results of Norman A. Phillips in 1955 prompted immediate reaction and von Neumann organized a conference at Princeton on "Application of Numerical Integration Techniques to the Problem of the General Circulation". Once again he strategically organized the program as a predictive one to ensure continued support from the Weather Bureau and the military, leading to the creation of the General Circulation Research Section (now the Geophysical Fluid Dynamics Laboratory) next to the JNWPU. He continued work both on technical issues of modelling and in ensuring continuing funding for these projects. During the late 19th century, Svante Arrhenius suggested that human activity could cause global warming by adding carbon dioxide to the atmosphere. In 1955, von Neumann observed that this may already have begun: "Carbon dioxide released into the atmosphere by industry's burning of coal and oil – more than half of it during the last generation – may have changed the atmosphere's composition sufficiently to account for a general warming of the world by about one degree Fahrenheit." His research into weather systems and meteorological prediction led him to propose manipulating the environment by spreading colorants on the polar ice caps to enhance absorption of solar radiation (by reducing the albedo). However, he urged caution in any program of atmosphere modification: What could be done, of course, is no index to what should be done... In fact, to evaluate the ultimate consequences of either a general cooling or a general heating would be a complex matter. Changes would affect the level of the seas, and hence the habitability of the continental coastal shelves; the evaporation of the seas, and hence general precipitation and glaciation levels; and so on... But there is little doubt that one could carry out the necessary analyses needed to predict the results, intervene on any desired scale, and ultimately achieve rather fantastic results. He also warned that weather and climate control could have military uses, telling Congress in 1956 that they could pose an even bigger risk than ICBMs. === Technological singularity hypothesis === The first use of the concept of a singularity in the technological context is attributed to von Neumann, who according to Ulam discussed the "ever accelerating progress of technology and changes in the mode of human life, which gives the appearance of approaching some essential singularity in the history of the race beyond which human affairs, as we know them, could not continue." This concept was later fleshed out in the 1970 book Future Shock by Alvin Toffler. == Defense work == === Manhattan Project === Beginning in the late 1930s, von Neumann developed an expertise in explosions—phenomena that are difficult to model mathematically. During this period, he was the leading authority of the mathematics of shaped charges, leading him to a large number of military consultancies and consequently his involvement in the Manhattan Project. The involvement included frequent trips to the project's secret research facilities at the Los Alamos Laboratory in New Mexico. Von Neumann made his principal contribution to the atomic bomb in the concept and design of the explosive lenses that were needed to compress the plutonium core of the Fat Man weapon that was later dropped on Nagasaki. While von Neumann did not originate the "implosion" concept, he was one of its most persistent proponents, encouraging its continued development against the instincts of many of his colleagues, who felt such a design to be unworkable. He also eventually came up with the idea of using more powerful shaped charges and less fissionable material to greatly increase the speed of "assembly". When it turned out that there would not be enough uranium-235 to make more than one bomb, the implosive lens project was greatly expanded and von Neumann's idea was implemented. Implosion was the only method that could be used with the plutonium-239 that was available from the Hanford Site. He established the design of the explosive lenses required, but there remained concerns about "edge effects" and imperfections in the explosives. His calculations showed that implosion would work if it did not depart by more than 5% from spherical symmetry. After a series of failed attempts with models, this was achieved by George Kistiakowsky, and the construction of the Trinity bomb was completed in July 1945. In a visit to Los Alamos in September 1944, von Neumann showed that the pressure increase from explosion shock wave reflection from solid objects was greater than previously believed if the angle of incidence of the shock wave was between 90° and some limiting angle. As a result, it was determined that the effectiveness of an atomic bomb would be enhanced with detonation some kilometers above the target, rather than at ground level. Von Neumann was included in the target selection committee that was responsible for choosing the Japanese cities of Hiroshima and Nagasaki as the first targets of the atomic bomb. Von Neumann oversaw computations related to the expected size of the bomb blasts, estimated death tolls, and the distance above the ground at which the bombs should be detonated for optimum shock wave propagation. The cultural capital Kyoto was von Neumann's first choice, a selection seconded by Manhattan Project leader General Leslie Groves. However, this target was dismissed by Secretary of War Henry L. Stimson. On July 16, 1945, von Neumann and numerous other Manhattan Project personnel were eyewitnesses to the first test of an atomic bomb detonation, which was code-named Trinity. The event was conducted as a test of the implosion method device, at the Alamogordo Bombing Range in New Mexico. Based on his observation alone, von Neumann estimated the test had resulted in a blast equivalent to 5 kilotons of TNT (21 TJ) but Enrico Fermi produced a more accurate estimate of 10 kilotons by dropping scraps of torn-up paper as the shock wave passed his location and watching how far they scattered. The actual power of the explosion had been between 20 and 22 kilotons. It was in von Neumann's 1944 papers that the expression "kilotons" appeared for the first time. Von Neumann continued unperturbed in his work and became, along with Edward Teller, one of those who sustained the hydrogen bomb project. He collaborated with Klaus Fuchs on further development of the bomb, and in 1946 the two filed a secret patent outlining a scheme for using a fission bomb to compress fusion fuel to initiate nuclear fusion. The Fuchs–von Neumann patent used radiation implosion, but not in the same way as is used in what became the final hydrogen bomb design, the Teller–Ulam design. Their work was, however, incorporated into the "George" shot of Operation Greenhouse, which was instructive in testing out concepts that went into the final design. The Fuchs–von Neumann work was passed on to the Soviet Union by Fuchs as part of his nuclear espionage, but it was not used in the Soviets' own, independent development of the Teller–Ulam design. The historian Jeremy Bernstein has pointed out that ironically, "John von Neumann and Klaus Fuchs, produced a brilliant invention in 1946 that could have changed the whole course of the development of the hydrogen bomb, but was not fully understood until after the bomb had been successfully made." For his wartime services, von Neumann was awarded the Navy Distinguished Civilian Service Award in July 1946, and the Medal for Merit in October 1946. === Post-war work === In 1950, von Neumann became a consultant to the Weapons Systems Evaluation Group, whose function was to advise the Joint Chiefs of Staff and the United States Secretary of Defense on the development and use of new technologies. He also became an adviser to the Armed Forces Special Weapons Project, which was responsible for the military aspects on nuclear weapons. Over the following two years, he became a consultant across the US government. This included the Central Intelligence Agency (CIA), a member of the influential General Advisory Committee of the Atomic Energy Commission, a consultant to the newly established Lawrence Livermore National Laboratory, and a member of the Scientific Advisory Group of the United States Air Force During this time he became a "superstar" defense scientist at the Pentagon. His authority was considered infallible at the highest levels of the US government and military. During several meetings of the advisory board of the US Air Force, von Neumann and Edward Teller predicted that by 1960 the US would be able to build a hydrogen bomb light enough to fit on top of a rocket. In 1953 Bernard Schriever, who was present at the meeting, paid a personal visit to von Neumann at Princeton to confirm this possibility. Schriever enlisted Trevor Gardner, who in turn visited von Neumann several weeks later to fully understand the future possibilities before beginning his campaign for such a weapon in Washington. Now either chairing or serving on several boards dealing with strategic missiles and nuclear weaponry, von Neumann was able to inject several crucial arguments regarding potential Soviet advancements in both these areas and in strategic defenses against American bombers into government reports to argue for the creation of ICBMs. Gardner on several occasions brought von Neumann to meetings with the US Department of Defense to discuss with various senior officials his reports. Several design decisions in these reports such as inertial guidance mechanisms would form the basis for all ICBMs thereafter. By 1954, von Neumann was also regularly testifying to various Congressional military subcommittees to ensure continued support for the ICBM program. However, this was not enough. To have the ICBM program run at full throttle they needed direct action by the President of the United States. They convinced President Eisenhower in a direct meeting in July 1955, which resulted in a presidential directive on September 13, 1955. It stated that "there would be the gravest repercussions on the national security and on the cohesion of the free world" if the Soviet Union developed the ICBM before the US and therefore designated the ICBM project "a research and development program of the highest priority above all others." The Secretary of Defense was ordered to commence the project with "maximum urgency". Evidence would later show that the Soviets indeed were already testing their own intermediate-range ballistic missiles at the time. Von Neumann would continue to meet the President, including at his home in Gettysburg, Pennsylvania, and other high-level government officials as a key advisor on ICBMs until his death. === Atomic Energy Commission === In 1955, von Neumann became a commissioner of the Atomic Energy Commission (AEC), which at the time was the highest official position available to scientists in the government. (While his appointment formally required that he sever all his other consulting contracts, an exemption was made for von Neumann to continue working with several critical military committees after the Air Force and several key senators raised concerns.) He used this position to further the production of compact hydrogen bombs suitable for intercontinental ballistic missile (ICBM) delivery. He involved himself in correcting the severe shortage of tritium and lithium 6 needed for these weapons, and he argued against settling for the intermediate-range missiles that the Army wanted. He was adamant that H-bombs delivered deep into enemy territory by an ICBM would be the most effective weapon possible, and that the relative inaccuracy of the missile would not be a problem with an H-bomb. He said the Russians would probably be building a similar weapon system, which turned out to be the case. While Lewis Strauss was away in the second half of 1955 von Neumann took over as acting chairman of the commission. In his final years before his death from cancer, von Neumann headed the United States government's top-secret ICBM committee, which would sometimes meet in his home. Its purpose was to decide on the feasibility of building an ICBM large enough to carry a thermonuclear weapon. Von Neumann had long argued that while the technical obstacles were sizable, they could be overcome. The SM-65 Atlas passed its first fully functional test in 1959, two years after his death. The more advanced Titan rockets were deployed in 1962. Both had been proposed in the ICBM committees von Neumann chaired. The feasibility of the ICBMs owed as much to improved, smaller warheads that did not have guidance or heat resistance issues as it did to developments in rocketry, and his understanding of the former made his advice invaluable. Von Neumann entered government service primarily because he felt that, if freedom and civilization were to survive, it would have to be because the United States would triumph over totalitarianism from Nazism, Fascism and Soviet Communism. During a Senate committee hearing he described his political ideology as "violently anti-communist, and much more militaristic than the norm". == Personality == === Work habits === Herman Goldstine commented on von Neumann's ability to intuit hidden errors and remember old material perfectly. When he had difficulties he would not labor on them; instead, he would go home and sleep on it and come back later with a solution. This style, 'taking the path of least resistance', sometimes meant that he could go off on tangents. It also meant that if the difficulty was great from the very beginning, he would simply switch to another problem, not trying to find weak spots from which he could break through. At times he could be ignorant of the standard mathematical literature, finding it easier to rederive basic information he needed rather than chase references. After World War II began, he became extremely busy with both academic and military commitments. His habit of not writing up talks or publishing results worsened. He did not find it easy to discuss a topic formally in writing unless it was already mature in his mind; if it was not, he would, in his own words, "develop the worst traits of pedantism and inefficiency". === Mathematical range === The mathematician Jean Dieudonné said that von Neumann "may have been the last representative of a once-flourishing and numerous group, the great mathematicians who were equally at home in pure and applied mathematics and who throughout their careers maintained a steady production in both directions". According to Dieudonné, his specific genius was in analysis and "combinatorics", with combinatorics being understood in a very wide sense that described his ability to organize and axiomize complex works that previously seemed to have little connection with mathematics. His style in analysis followed the German school, based on foundations in linear algebra and general topology. While von Neumann had an encyclopedic background, his range in pure mathematics was not as wide as Poincaré, Hilbert or even Weyl: von Neumann never did significant work in number theory, algebraic topology, algebraic geometry or differential geometry. However, in applied mathematics his work equalled that of Gauss, Cauchy or Poincaré. According to Wigner, "Nobody knows all science, not even von Neumann did. But as for mathematics, he contributed to every part of it except number theory and topology. That is, I think, something unique." Halmos noted that while von Neumann knew lots of mathematics, the most notable gaps were in algebraic topology and number theory; he recalled an incident where von Neumann failed to recognize the topological definition of a torus. Von Neumann admitted to Herman Goldstine that he had no facility at all in topology and he was never comfortable with it, with Goldstine later bringing this up when comparing him to Hermann Weyl, who he thought was deeper and broader. In his biography of von Neumann, Salomon Bochner wrote that much of von Neumann's works in pure mathematics involved finite and infinite dimensional vector spaces, which at the time, covered much of the total area of mathematics. However he pointed out this still did not cover an important part of the mathematical landscape, in particular, anything that involved geometry "in the global sense", topics such as topology, differential geometry and harmonic integrals, algebraic geometry and other such fields. Von Neumann rarely worked in these fields and, as Bochner saw it, had little affinity for them. In one of von Neumann's last articles, he lamented that pure mathematicians could no longer attain deep knowledge of even a fraction of the field. In the early 1940s, Ulam had concocted for him a doctoral-style examination to find weaknesses in his knowledge; von Neumann was unable to answer satisfactorily a question each in differential geometry, number theory, and algebra. They concluded that doctoral exams might have "little permanent meaning". However, when Weyl turned down an offer to write a history of mathematics of the 20th century, arguing that no one person could do it, Ulam thought von Neumann could have aspired to do so. === Preferred problem-solving techniques === Ulam remarked that most mathematicians could master one technique that they then used repeatedly, whereas von Neumann had mastered three: A facility with the symbolic manipulation of linear operators; An intuitive feeling for the logical structure of any new mathematical theory; An intuitive feeling for the combinatorial superstructure of new theories. Although he was commonly described as an analyst, he once classified himself an algebraist, and his style often displayed a mix of algebraic technique and set-theoretical intuition. He loved obsessive detail and had no issues with excess repetition or overly explicit notation. An example of this was a paper of his on rings of operators, where he extended the normal functional notation, ϕ ( x ) {\displaystyle \phi (x)} to ϕ ( ( x ) ) {\displaystyle \phi ((x))} . However, this process ended up being repeated several times, where the final result were equations such as ( ψ ( ( ( ( a ) ) ) ) ) 2 = ϕ ( ( ( ( a ) ) ) ) {\displaystyle (\psi ((((a)))))^{2}=\phi ((((a))))} . The 1936 paper became known to students as "von Neumann's onion" because the equations "needed to be peeled before they could be digested". Overall, although his writings were clear and powerful, they were not clean or elegant. Although powerful technically, his primary concern was more with the clear and viable formation of fundamental issues and questions of science rather than just the solution of mathematical puzzles. According to Ulam, von Neumann surprised physicists by doing dimensional estimates and algebraic computations in his head with fluency Ulam likened to blindfold chess. His impression was that von Neumann analyzed physical situations by abstract logical deduction rather than concrete visualization. === Lecture style === Goldstine compared his lectures to being on glass, smooth and lucid. By comparison, Goldstine thought his scientific articles were written in a much harsher manner, and with much less insight. Halmos described his lectures as "dazzling", with his speech clear, rapid, precise and all encompassing. Like Goldstine, he also described how everything seemed "so easy and natural" in lectures but puzzling on later reflection. He was a quick speaker: Banesh Hoffmann found it very difficult to take notes, even in shorthand, and Albert Tucker said that people often had to ask von Neumann questions to slow him down so they could think through the ideas he was presenting. Von Neumann knew about this and was grateful for his audience telling him when he was going too quickly. Although he did spend time preparing for lectures, he rarely used notes, instead jotting down points of what he would discuss and for how long. === Eidetic memory === Von Neumann was also noted for his eidetic memory, particularly of the symbolic kind. Herman Goldstine writes: One of his remarkable abilities was his power of absolute recall. As far as I could tell, von Neumann was able on once reading a book or article to quote it back verbatim; moreover, he could do it years later without hesitation. He could also translate it at no diminution in speed from its original language into English. On one occasion I tested his ability by asking him to tell me how A Tale of Two Cities started. Whereupon, without any pause, he immediately began to recite the first chapter and continued until asked to stop after about ten or fifteen minutes. Von Neumann was reportedly able to memorize the pages of telephone directories. He entertained friends by asking them to randomly call out page numbers; he then recited the names, addresses and numbers therein. Stanisław Ulam believed that von Neumann's memory was auditory rather than visual. === Mathematical quickness === Von Neumann's mathematical fluency, calculation speed, and general problem-solving ability were widely noted by his peers. Paul Halmos called his speed "awe-inspiring." Lothar Wolfgang Nordheim described him as the "fastest mind I ever met". Enrico Fermi told physicist Herbert L. Anderson: "You know, Herb, Johnny can do calculations in his head ten times as fast as I can! And I can do them ten times as fast as you can, Herb, so you can see how impressive Johnny is!" Edward Teller admitted that he "never could keep up with him", and Israel Halperin described trying to keep up as like riding a "tricycle chasing a racing car." He had an unusual ability to solve novel problems quickly. George Pólya, whose lectures at ETH Zürich von Neumann attended as a student, said, "Johnny was the only student I was ever afraid of. If in the course of a lecture I stated an unsolved problem, the chances were he'd come to me at the end of the lecture with the complete solution scribbled on a slip of paper." When George Dantzig brought von Neumann an unsolved problem in linear programming "as I would to an ordinary mortal", on which there had been no published literature, he was astonished when von Neumann said "Oh, that!", before offhandedly giving a lecture of over an hour, explaining how to solve the problem using the hitherto unconceived theory of duality. A story about von Neumann's encounter with the famous fly puzzle has entered mathematical folklore. In this puzzle, two bicycles begin 20 miles apart, and each travels toward the other at 10 miles per hour until they collide; meanwhile, a fly travels continuously back and forth between the bicycles at 15 miles per hour until it is squashed in the collision. The questioner asks how far the fly traveled in total; the "trick" for a quick answer is to realize that the fly's individual transits do not matter, only that it has been traveling at 15 miles per hour for one hour. As Eugene Wigner tells it, Max Born posed the riddle to von Neumann. The other scientists to whom he had posed it had laboriously computed the distance, so when von Neumann was immediately ready with the correct answer of 15 miles, Born observed that he must have guessed the trick. "What trick?" von Neumann replied. "All I did was sum the geometric series." === Self-doubts === Rota wrote that von Neumann had "deep-seated and recurring self-doubts". John L. Kelley reminisced in 1989 that "Johnny von Neumann has said that he will be forgotten while Kurt Gödel is remembered with Pythagoras, but the rest of us viewed Johnny with awe." Ulam suggests that some of his self-doubts with regard for his own creativity may have come from the fact he had not discovered several important ideas that others had, even though he was more than capable of doing so, giving the incompleteness theorems and Birkhoff's pointwise ergodic theorem as examples. Von Neumann had a virtuosity in following complicated reasoning and had supreme insights, yet he perhaps felt he did not have the gift for seemingly irrational proofs and theorems or intuitive insights. Ulam describes how during one of his stays at Princeton while von Neumann was working on rings of operators, continuous geometries and quantum logic he felt that von Neumann was not convinced of the importance of his work, and only when finding some ingenious technical trick or new approach did he take some pleasure in it. However, according to Rota, von Neumann still had an "incomparably stronger technique" compared to his friend, despite describing Ulam as the more creative mathematician. == Legacy == === Accolades === Nobel Laureate Hans Bethe said "I have sometimes wondered whether a brain like von Neumann's does not indicate a species superior to that of man". Edward Teller observed "von Neumann would carry on a conversation with my 3-year-old son, and the two of them would talk as equals, and I sometimes wondered if he used the same principle when he talked to the rest of us." Peter Lax wrote "Von Neumann was addicted to thinking, and in particular to thinking about mathematics". Eugene Wigner said, "He understood mathematical problems not only in their initial aspect, but in their full complexity." Claude Shannon called him "the smartest person I've ever met", a common opinion. Jacob Bronowski wrote "He was the cleverest man I ever knew, without exception. He was a genius." Due to his wide reaching influence and contributions to many fields, von Neumann is widely considered a polymath. Wigner noted the extraordinary mind that von Neumann had, and he described von Neumann as having a mind faster than anyone he knew, stating that: I have known a great many intelligent people in my life. I knew Max Planck, Max von Laue, and Werner Heisenberg. Paul Dirac was my brother-in-law; Leo Szilard and Edward Teller have been among my closest friends; and Albert Einstein was a good friend, too. And I have known many of the brightest younger scientists. But none of them had a mind as quick and acute as Jancsi von Neumann. I have often remarked this in the presence of those men, and no one ever disputed me."It seems fair to say that if the influence of a scientist is interpreted broadly enough to include impact on fields beyond science proper, then John von Neumann was probably the most influential mathematician who ever lived," wrote Miklós Rédei. Peter Lax commented that von Neumann would have won a Nobel Prize in Economics had he lived longer, and that "if there were Nobel Prizes in computer science and mathematics, he would have been honored by these, too." Rota writes that "he was the first to have a vision of the boundless possibilities of computing, and he had the resolve to gather the considerable intellectual and engineering resources that led to the construction of the first large computer" and consequently that "No other mathematician in this century has had as deep and lasting an influence on the course of civilization." He is widely regarded as one of the greatest and most influential mathematicians and scientists of the 20th century. Neurophysiologist Leon Harmon described him in a similar manner, calling him the only "true genius" he had ever met: "von Neumann's mind was all-encompassing. He could solve problems in any domain. ... And his mind was always working, always restless." While consulting for non-academic projects von Neumann's combination of outstanding scientific ability and practicality gave him a high credibility with military officers, engineers, and industrialists that no other scientist could match. In nuclear missilery he was considered "the clearly dominant advisory figure" according to Herbert York. Economist Nicholas Kaldor said he was "unquestionably the nearest thing to a genius I have ever encountered." Likewise, Paul Samuelson wrote, "We economists are grateful for von Neumann's genius. It is not for us to calculate whether he was a Gauss, or a Poincaré, or a Hilbert. He was the incomparable Johnny von Neumann. He darted briefly into our domain and it has never been the same since." === Honors and awards === Events and awards named in recognition of von Neumann include the annual John von Neumann Theory Prize of the Institute for Operations Research and the Management Sciences, IEEE John von Neumann Medal, and the John von Neumann Prize of the Society for Industrial and Applied Mathematics. Both the crater von Neumann on the Moon and the asteroid 22824 von Neumann are named in his honor. Von Neumann received awards including the Medal for Merit in 1947, the Medal of Freedom in 1956, and the Enrico Fermi Award also in 1956. He was elected a member of multiple honorary societies, including the American Academy of Arts and Sciences and the National Academy of Sciences, and he held eight honorary doctorates. On May 4, 2005, the United States Postal Service issued the American Scientists commemorative postage stamp series, designed by artist Victor Stabin. The scientists depicted were von Neumann, Barbara McClintock, Josiah Willard Gibbs, and Richard Feynman. John von Neumann University was established in Kecskemét, Hungary in 2016, as a successor to Kecskemét College. == Selected works == Von Neumann's first published paper was On the position of zeroes of certain minimum polynomials, co-authored with Michael Fekete and published when von Neumann was 18. At 19, his solo paper On the introduction of transfinite numbers was published. He expanded his second solo paper, An axiomatization of set theory, to create his PhD thesis. His first book, Mathematical Foundations of Quantum Mechanics, was published in 1932. Following this, von Neumann switched from publishing in German to publishing in English, and his publications became more selective and expanded beyond pure mathematics. His 1942 Theory of Detonation Waves contributed to military research, his work on computing began with the unpublished 1946 On the principles of large scale computing machines, and his publications on weather prediction began with the 1950 Numerical integration of the barotropic vorticity equation. Alongside his later papers were informal essays targeted at colleagues and the general public, such as his 1947 The Mathematician, described as a "farewell to pure mathematics", and his 1955 Can we survive technology?, which considered a bleak future including nuclear warfare and deliberate climate change. His complete works have been compiled into a six-volume set. == See also == List of pioneers in computer science Teapot Committee The MANIAC, 2023 book about von Neumann German: Abenteuer eines Mathematikers (English title: Adventures of a Mathematician), biopic about Stanislaw Ulam also features John von Neumann. == Notes == == References == == Further reading == == External links == A more or less complete bibliography of publications of John von Neumann by Nelson H. F. Beebe O'Connor, John J.; Robertson, Edmund F. "John von Neumann". MacTutor History of Mathematics Archive. University of St Andrews. von Neumann's profile at Google Scholar Oral History Project - The Princeton Mathematics Community in the 1930s, contains many interviews that describe contact and anecdotes of von Neumann and others at the Princeton University and Institute for Advanced Study community. Oral history interviews (from the Charles Babbage Institute, University of Minnesota) with: Alice R. Burks and Arthur W. Burks; Eugene P. Wigner; and Nicholas C. Metropolis. zbMATH profile Query for "von neumann" on the digital repository of the Institute for Advanced Study. Von Neumann vs. Dirac on Quantum Theory and Mathematical Rigor – from Stanford Encyclopedia of Philosophy Quantum Logic and Probability Theory - from Stanford Encyclopedia of Philosophy FBI files on John von Neumann released via FOI Biographical video by David Brailsford (John Dunford Professor Emeritus of computer science at the University of Nottingham) John von Neumann: Prophet of the 21st Century 2013 Arte documentary on John von Neumann and his influence in the modern world (in German and French with English subtitles). John von Neumann - A Documentary 1966 detailed documentary by the Mathematical Association of America containing remarks by several of his colleagues including Ulam, Wigner, Halmos, Morgenstern, Bethe, Goldstine, Strauss and Teller.
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Wikipedia:Joint Mathematical Council#0
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The Joint Mathematical Council (JMC) of the United Kingdom was formed in 1963 to "provide co-ordination between the Constituent Societies and generally to promote the advancement of mathematics and the improvement of the teaching of mathematics". The JMC serves as a forum for discussion between societies and for making representations to government and other bodies and responses to their enquiries. It is concerned with all aspects of mathematics at all levels from primary to higher education. == Members == The participating bodies are Adults Learning Mathematics Association of Teachers of Mathematics Association of Mathematics Education Teachers British Society for the History of Mathematics British Society for Research into Learning Mathematics HoDoMS Edinburgh Mathematical Society Institute of Mathematics and its Applications London Mathematical Society Mathematical Association Mathematics in Education and Industry National Association for Numeracy and Mathematics in Colleges National Association of Mathematics Advisers National Numeracy STEM Learning NRICH Operational Research Society Royal Academy of Engineering Royal Statistical Society Scottish Mathematical Council United Kingdom Mathematics Trust The observing bodies are Advisory Committee on Mathematics Education Department for Education (England) Department of Education (Northern Ireland) Education Scotland National Centre for Excellence in Teaching Mathematics Office for Standards in Education The Office of Qualifications and Examinations Regulation The Royal Society Scottish Qualifications Authority Welsh Government Education Directorate == Leadership == The Chair of the JMC is Andy Noyes, Professor of Education at the University of Nottingham and is a member of the Royal Society Advisory Committee on Mathematics Education. == References == Ashley Kent (2000). School Subject Teaching: The History and Future of the Curriculum. Routledge. p. 44. ISBN 0-7494-3377-9. == External links == Web site
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Wikipedia:Joint Mathematics Meetings#0
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The Joint Mathematics Meetings (JMM) is a mathematics conference hosted annually in early January by the American Mathematical Society (AMS). Frequently, several other national mathematics organizations also participate. From 1998 to 2020, the JMM was jointly organized and managed by the AMS and the Mathematical Association of America (MAA). The meeting is the largest gathering of mathematicians in the United States, and the largest annual meeting of mathematicians in the world. For example, more than 6000 people attended the 2017 JMM. Several thousand talks, panels, minicourses, and poster sessions are held each year. The JMM also hosts an Employment Center, which is a focal point for the hiring process of academic mathematicians, especially for liberal arts colleges. Many employers conduct their preliminary interview process at the meeting. Often these interviews take place outside the confines of the conference, so the employers may not appear on the official Employment Center listing. == Future meetings == The next few meetings are scheduled to be held in: Washington, D.C., January 4–7, 2026 Chicago, IL, January 12–15, 2027 == Past meetings == Past years' meetings have been held in: Seattle, WA, January 8–11, 2025 link San Francisco, CA, January 3–6, 2024 link Boston, MA, January 4–7, 2023 link Virtual, April 6–9, 2022 (the originally scheduled meeting in Seattle, WA, January 5–8, had been postponed due to concerns about the Omicron variant of COVID-19) link Virtual, January 6–9, 2021 (the originally scheduled in-person meeting in Washington, D.C. was canceled due to COVID-19) link Denver, CO, January 15–18, 2020 link Baltimore, MD, January 16–19, 2019 link San Diego, CA, January 10-13, 2018 link Atlanta, GA, January 4–7, 2017 link Seattle, WA, January 6–9, 2016, link San Antonio, TX, January 10–13, 2015, link Baltimore, MD, January 15–18, 2014 link San Diego, CA, January 9–12, 2013 link Boston, MA, January 4–7, 2012 link New Orleans, LA, January 6–9, 2011 link San Francisco, CA, January 13–16, 2010 link Washington, D.C., January 5–8, 2009 link San Diego, CA, January 6–9, 2008 link New Orleans, LA, January 5–8, 2007 link San Antonio, TX, January 12–15, 2006 link Atlanta, GA, January 5–8, 2005 link Phoenix, AZ, January 7–10, 2004 link Baltimore, MD, January 15–18, 2003 link San Diego, CA, January 6–9, 2002 link New Orleans, LA, January 10–13, 2001 link Washington, D.C., January 19–22, 2000 link == External links == Listing of national AMS conferences Information about the Employment Center == Gallery == People At the joint Mathematic Meeting == References ==
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Wikipedia:Joint spectral radius#0
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In mathematics, the joint spectral radius is a generalization of the classical notion of spectral radius of a matrix, to sets of matrices. In recent years this notion has found applications in a large number of engineering fields and is still a topic of active research. == General description == The joint spectral radius of a set of matrices is the maximal asymptotic growth rate of products of matrices taken in that set. For a finite (or more generally compact) set of matrices M = { A 1 , … , A m } ⊂ R n × n , {\displaystyle {\mathcal {M}}=\{A_{1},\dots ,A_{m}\}\subset \mathbb {R} ^{n\times n},} the joint spectral radius is defined as follows: ρ ( M ) = lim k → ∞ max { ‖ A i 1 ⋯ A i k ‖ 1 / k : A i ∈ M } . {\displaystyle \rho ({\mathcal {M}})=\lim _{k\to \infty }\max {\{\|A_{i_{1}}\cdots A_{i_{k}}\|^{1/k}:A_{i}\in {\mathcal {M}}\}}.\,} It can be proved that the limit exists and that the quantity actually does not depend on the chosen matrix norm (this is true for any norm but particularly easy to see if the norm is sub-multiplicative). The joint spectral radius was introduced in 1960 by Gian-Carlo Rota and Gilbert Strang, two mathematicians from MIT, but started attracting attention with the work of Ingrid Daubechies and Jeffrey Lagarias. They showed that the joint spectral radius can be used to describe smoothness properties of certain wavelet functions. A wide number of applications have been proposed since then. It is known that the joint spectral radius quantity is NP-hard to compute or to approximate, even when the set M {\displaystyle {\mathcal {M}}} consists of only two matrices with all nonzero entries of the two matrices which are constrained to be equal. Moreover, the question " ρ ≤ 1 ? {\displaystyle \rho \leq 1?} " is an undecidable problem. Nevertheless, in recent years much progress has been done on its understanding, and it appears that in practice the joint spectral radius can often be computed to satisfactory precision, and that it moreover can bring interesting insight in engineering and mathematical problems. == Computation == === Approximation algorithms === In spite of the negative theoretical results on the joint spectral radius computability, methods have been proposed that perform well in practice. Algorithms are even known, which can reach an arbitrary accuracy in an a priori computable amount of time. These algorithms can be seen as trying to approximate the unit ball of a particular vector norm, called the extremal norm. One generally distinguishes between two families of such algorithms: the first family, called polytope norm methods, construct the extremal norm by computing long trajectories of points. An advantage of these methods is that in the favorable cases it can find the exact value of the joint spectral radius and provide a certificate that this is the exact value. The second family of methods approximate the extremal norm with modern optimization techniques, such as ellipsoid norm approximation, semidefinite programming, Sum Of Squares, and conic programming. The advantage of these methods is that they are easy to implement, and in practice, they provide in general the best bounds on the joint spectral radius. === The finiteness conjecture === Related to the computability of the joint spectral radius is the following conjecture: "For any finite set of matrices M ⊂ R n × n , {\displaystyle {\mathcal {M}}\subset \mathbb {R} ^{n\times n},} there is a product A 1 … A t {\displaystyle A_{1}\dots A_{t}} of matrices in this set such that ρ ( M ) = ρ ( A 1 … A t ) 1 / t . {\displaystyle \rho ({\mathcal {M}})=\rho (A_{1}\dots A_{t})^{1/t}.} " In the above equation " ρ ( A 1 … A t ) {\displaystyle \rho (A_{1}\dots A_{t})} " refers to the classical spectral radius of the matrix A 1 … A t . {\displaystyle A_{1}\dots A_{t}.} This conjecture, proposed in 1995, was proven to be false in 2003. The counterexample provided in that reference uses advanced measure-theoretical ideas. Subsequently, many other counterexamples have been provided, including an elementary counterexample that uses simple combinatorial properties matrices and a counterexample based on dynamical systems properties. Recently an explicit counterexample has been proposed in. Many questions related to this conjecture are still open, as for instance the question of knowing whether it holds for pairs of binary matrices. == Applications == The joint spectral radius was introduced for its interpretation as a stability condition for discrete-time switching dynamical systems. Indeed, the system defined by the equations x t + 1 = A t x t , A t ∈ M ∀ t {\displaystyle x_{t+1}=A_{t}x_{t},\quad A_{t}\in {\mathcal {M}}\,\forall t} is stable if and only if ρ ( M ) < 1. {\displaystyle \rho ({\mathcal {M}})<1.} The joint spectral radius became popular when Ingrid Daubechies and Jeffrey Lagarias showed that it rules the continuity of certain wavelet functions. Since then, it has found many applications, ranging from number theory to information theory, autonomous agents consensus, combinatorics on words,... == Related notions == The joint spectral radius is the generalization of the spectral radius of a matrix for a set of several matrices. However, many more quantities can be defined when considering a set of matrices: The joint spectral subradius characterizes the minimal rate of growth of products in the semigroup generated by M {\displaystyle {\mathcal {M}}} . The p-radius characterizes the rate of growth of the L p {\displaystyle L_{p}} average of the norms of the products in the semigroup. The Lyapunov exponent of the set of matrices characterizes the rate of growth of the geometric average. == References == == Further reading == Raphael M. Jungers (2009). The joint spectral radius, Theory and applications. Springer. ISBN 978-3-540-95979-3. Vincent D. Blondel; Michael Karow; Vladimir Protassov; Fabian R. Wirth, eds. (2008). "Linear Algebra and its Applications: special issue on the joint spectral radius". Linear Algebra and Its Applications. 428 (10). Elsevier. Antonio Cicone (2011). "PhD thesis. Spectral Properties of Families of Matrices. Part III" (PDF). Jacques Theys (2005). "PhD thesis. Joint Spectral Radius: Theory and approximations" (PDF). Archived from the original (PDF) on 2007-06-13.
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Wikipedia:Jonathan Bennett (mathematician)#0
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Jonathan Bennett is a British mathematician and Professor of Mathematical Analysis at the University of Birmingham. He was a recipient of the Whitehead Prize of the London Mathematical Society in 2011 for "his foundational work on multilinear inequalities in harmonic and geometric analysis, and for a number of major results in the theory of oscillatory integrals." == Education == In 1995 he graduated with a BA in mathematics from Hertford College at the University of Oxford. He went on to study for a PhD in harmonic analysis under Anthony Carbery at the University of Edinburgh, graduating in 1999. == Career == Bennett has done postdoctoral work at the University of Edinburgh, the Universidad Autonoma de Madrid and Trinity College Dublin. He joined the University of Birmingham in 2005. Bennett is an editor for the journals Mathematika and Quarterly Journal of Mathematics. Bennett is known for his work in harmonic analysis, particularly in applying the methods of heat flow monotonicity and induction-on-scale arguments to prove inequalities arising in harmonic and geometric analysis, in particular for his work (jointly with Anthony Carbery and Terence Tao) on the multilinear Kakeya conjecture. Bennett has an Erdős number of 3, via his collaboration with Tao. == References == == External links == Bennett's page at the Mathematics Genealogy Project. Research published by Bennett. A blog post by Tao explaining induction-on-scales arguments, with reference to his joint work with Bennett and Carbery.
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Wikipedia:Jonathan Borwein#0
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Jonathan Michael Borwein (20 May 1951 – 2 August 2016) was a Scottish mathematician who held an appointment as Laureate Professor of mathematics at the University of Newcastle, Australia. He was a close associate of David H. Bailey, and they have been prominent public advocates of experimental mathematics. Borwein's interests spanned pure mathematics (analysis), applied mathematics (optimization), computational mathematics (numerical and computational analysis), and high performance computing. He authored ten books, including several on experimental mathematics, a monograph on convex functions, and over 400 refereed articles. He was a co-founder in 1995 of software company MathResources, consulting and producing interactive software primarily for school and university mathematics. He was not associated with MathResources at the time of his death. Borwein was also an expert on the number pi and especially its computation. == Early life and education == Borwein was born in St. Andrews, Scotland, in 1951 into a Jewish family. His father was mathematician David Borwein, with whom he collaborated. His brother Peter Borwein was also a mathematician. Borwein was married to Judith (née Roots), and had three daughters. He received his B.A. (Honours Math) from University of Western Ontario in 1971, and his D.Phil. from Oxford University in 1974 as a Rhodes Scholar at Jesus College. == Career == Prior to joining Simon Fraser University in 1993, he worked at Dalhousie University (1974–91), Carnegie-Mellon (1980–82) and the University of Waterloo (1991–93). He was Shrum Professor of Science (1993–2003) and a Canada Research Chair in Information Technology (2001–08) at Simon Fraser University, where he was founding Director of the Centre for Experimental and Constructive Mathematics and developed the Inverse Symbolic Calculator together with his brother and Simon Plouffe. In 2004, he (re-)joined the Faculty of Computer Science at Dalhousie University as a Canada Research Chair in Distributed and Collaborative Research, cross-appointed in Mathematics, while preserving an adjunct appointment at Simon Fraser. Borwein was Governor at large of the Mathematical Association of America (2004–07), was president of the Canadian Mathematical Society (2000–02) and chair of (the Canadian National Science Library) NRC-CISTI Advisory Board (2000–2003). He served as chair of various NATO scientific programs. He was also Chair of the Scientific Advisory Committee of the Australian Mathematical Sciences Institute (AMSI). He chaired the Canadian HPC consortium, later Compute Canada, and the International Mathematical Union's Committee on Electronic Information and Communications (2002–2008). == Awards == Borwein received various awards including the Chauvenet Prize (1993), Fellowship in the Royal Society of Canada (1994), Fellowship in the American Association for the Advancement of Science (2002), an honorary degree from Limoges (1999), and foreign membership in the Bulgarian Academy of Sciences (2003). He was elected as a Fellow to the Australian Academy of Science (2010). In 2014, he became a Fellow of the American Mathematical Society. He was an ISI highly cited mathematician for the period 1981–1999. == See also == Borwein integral Borwein's algorithm List of University of Waterloo people == References == == External links == Jonathan Borwein's homepage Jonathan Borwein memorial website Jonathan Borwein at the Mathematics Genealogy Project The experimental mathematics homepage and blog Researcher ID
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Wikipedia:Jonathan Partington#0
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Jonathan Richard Partington (born 4 February 1955) is an English mathematician who is Emeritus Professor of pure mathematics at the University of Leeds. == Education == Professor Partington was educated at Gresham's School, Holt, and Trinity College, Cambridge, where he completed his PhD thesis entitled "Numerical ranges and the Geometry of Banach Spaces" under the supervision of Béla Bollobás. == Career == Partington works in the area of operator theory and complex analysis, sometimes applied to control theory, and is the author of several books in this area. He was formerly editor-in-chief of the Journal of the London Mathematical Society, a position he held jointly with his Leeds colleague John Truss. Partington's extra-mathematical activities include the invention of the March March march, an annual walk starting at March, Cambridgeshire. He is also known as a writer or co-writer of some of the earliest British text-based computer games, including Acheton, Hamil, Murdac, Avon, Fyleet, Crobe, Sangraal, and SpySnatcher, which started life on the Phoenix computer system at the University of Cambridge Computer Laboratory. These are still available on the IF Archive. == Books == Partington, Jonathan R. (24 February 1989). An Introduction to Hankel Operators. Cambridge University Press. doi:10.1017/cbo9780511623769. ISBN 978-0-521-36611-3. Partington, Jonathan R. (1997). Interpolation, identification, and sampling. Oxford: Clarendon Press. ISBN 0-19-850024-6. OCLC 36681729. Partington, Jonathan R. (15 March 2004). Linear Operators and Linear Systems. Cambridge University Press. doi:10.1017/cbo9780511616693. ISBN 978-0-521-83734-7. Chalendar, Isabelle; Partington, Jonathan R. (18 August 2011). Modern Approaches to the Invariant-Subspace Problem. Cambridge University Press. doi:10.1017/cbo9780511862434. ISBN 978-0-511-86243-4. == References == == External links == Professor Jonathan R. Partington at the University of Leeds Jonathan Partington at the Mathematics Genealogy Project
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Wikipedia:Jonathan Pila#0
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Jonathan Solomon Pila (born 1962) FRS is an Australian mathematician at the University of Oxford University of Melbourne in 1984. He was awarded a PhD from Stanford University in 1988, for research supervised by Peter Sarnak. His dissertation was entitled "Frobenius Maps of Abelian Varieties and Finding Roots of Unity in Finite Fields". In 2010, he received an MA from Oxford. == Career and research == Research interests lie in number theory and model theory. A focus has been applying the theory of o-minimality to Diophantine problems. This work began with an early paper with Enrico Bombieri, and developed through collaborations with Alex Wilkie and Umberto Zannier. The techniques obtained have led to advances in Diophantine problems, including Pila's unconditional proof of the André–Oort conjecture for powers of the modular curve. Work by Pila and Jacob Tsimerman, demonstrated the André–Oort conjecture in the case of the Siegel modular variety. Pila has held posts at Columbia University, McGill University, the University of Bristol and (as a visiting member) the Institute for Advanced Study. Pila also took a substantial break from professional mathematics to work in his family's manufacturing business. Pila has been the Editor of Proceedings of the Edinburgh Mathematical Society, and of Algebra and Number Theory. == Awards and honours == Pila was awarded the Clay Research Award in 2011 for his resolution of the André–Oort conjecture in the case of products of modular curves. In June 2011, he was awarded the Senior Whitehead Prize by the London Mathematical Society. This prize is "awarded in recognition of work in and influence on and service to mathematics; or lecturing gifts." Specifically, the citation recognized "his startling recent work on the Andre-Oort and Manin-Mumford conjectures. The approach he and his collaborators have developed, which combines analytic ideas with model theory, is entirely new and shows great promise for further applications." In addition to the Clay and London Mathematical Society awards, Pila delivered the Arf Lecture in 2011, was awarded the Leverhulme Trust Research Fellowship 2008–2010. and received the Karp Prize in 2013. Pila was elected a Fellow of the Royal Society (FRS) in 2015. In 2022, he received the Rolf Schock Prize in the category of "Mathematics". == References ==
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Wikipedia:Jonathan Rosenhead#0
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Jonathan Vivian Rosenhead (born 21 September 1938) is a British mathematician, operational researcher and Labour Party activist. == Early life and career == Jonathan Rosenhead is the son of mathematician Louis Rosenhead. He studied at St John's College, Cambridge, where he received a B.A. degree in mathematics in 1959. He continued his studies at University College London where he received an M.Sc. degree in statistics in 1961, and an M.A. from Cambridge in 1963. He worked as an operational researcher at United Steel Companies in Sheffield in 1961-63 and at Science in General Management Ltd. (SIGMA) in Croydon in 1963-66 before returning to the academic world. == Academic career == Rosenhead spent 1966-67 at the Wharton School of the University of Pennsylvania and its Management Science Centre, where Russell L. Ackoff was professor. In 1967 he joined the London School of Economics as a lecturer in operational research. He became a senior lecturer in 1981 and professor of operational research in 1987. He retired in 2003. Within operational research (OR), Rosenhead is primarily associated with the development of "soft OR" from the late 1970s, which resulted in the development of a number of new OR methods. He was the editor of the first book to gather a number of problem structuring methods within one volume, Rational analysis for a problematic world, published in 1989. He was president of the Operational Research Society in 1986-87. == Political activism == Rosenhead was a Labour Party candidate for Kensington South at the 1966 general election. His candidacy was endorsed by notable local residents, including playwright Keith Waterhouse, designer Misha Black, writer Baroness Stocks, architect Sir Hugh Casson, writer Brigid Brophy and novelist Lord Snow. However, Rosenhead came a distant second to the Conservatives, who held the seat with a large majority. He was active in the British Society for Social Responsibility in Science over a 20-year period, including a stint as chair of the society. His political activities has included being chairman of the British Committee for the Universities of Palestine (BRICUP) and being information officer for Jewish Voice for Labour. Rosenhead was involved in the campaign against South African apartheid. He was arrested at a May 1972 demonstration against the English rugby team which was departing for a tour in South Africa. He was convicted of a public order offence and fined. In 2020, Rosenhead learned that a fellow protester, who was also arrested at the demonstration, was an undercover police officer working in the Special Demonstration Squad. The fact that the police had told neither the defence nor the court about the officer made the convictions a miscarriage of justice. Rosenhead and several others had their convictions quashed in 2023. == References ==
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Wikipedia:Jordan normal form#0
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In linear algebra, a Jordan normal form, also known as a Jordan canonical form, is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to some basis. Such a matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal (on the superdiagonal), and with identical diagonal entries to the left and below them. Let V be a vector space over a field K. Then a basis with respect to which the matrix has the required form exists if and only if all eigenvalues of the matrix lie in K, or equivalently if the characteristic polynomial of the operator splits into linear factors over K. This condition is always satisfied if K is algebraically closed (for instance, if it is the field of complex numbers). The diagonal entries of the normal form are the eigenvalues (of the operator), and the number of times each eigenvalue occurs is called the algebraic multiplicity of the eigenvalue. If the operator is originally given by a square matrix M, then its Jordan normal form is also called the Jordan normal form of M. Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix. In spite of its name, the normal form for a given M is not entirely unique, as it is a block diagonal matrix formed of Jordan blocks, the order of which is not fixed; it is conventional to group blocks for the same eigenvalue together, but no ordering is imposed among the eigenvalues, nor among the blocks for a given eigenvalue, although the latter could for instance be ordered by weakly decreasing size. The Jordan–Chevalley decomposition is particularly simple with respect to a basis for which the operator takes its Jordan normal form. The diagonal form for diagonalizable matrices, for instance normal matrices, is a special case of the Jordan normal form. The Jordan normal form is named after Camille Jordan, who first stated the Jordan decomposition theorem in 1870. == Overview == === Notation === Some textbooks have the ones on the subdiagonal; that is, immediately below the main diagonal instead of on the superdiagonal. The eigenvalues are still on the main diagonal. === Motivation === An n × n matrix A is diagonalizable if and only if the sum of the dimensions of the eigenspaces is n. Or, equivalently, if and only if A has n linearly independent eigenvectors. Not all matrices are diagonalizable; matrices that are not diagonalizable are called defective matrices. Consider the following matrix: A = [ 5 4 2 1 0 1 − 1 − 1 − 1 − 1 3 0 1 1 − 1 2 ] . {\displaystyle A=\left[{\begin{array}{*{20}{r}}5&4&2&1\\[2pt]0&1&-1&-1\\[2pt]-1&-1&3&0\\[2pt]1&1&-1&2\end{array}}\right].} Including multiplicity, the eigenvalues of A are λ = 1, 2, 4, 4. The dimension of the eigenspace corresponding to the eigenvalue 4 is 1 (and not 2), so A is not diagonalizable. However, there is an invertible matrix P such that J = P−1AP, where J = [ 1 0 0 0 0 2 0 0 0 0 4 1 0 0 0 4 ] . {\displaystyle J={\begin{bmatrix}1&0&0&0\\[2pt]0&2&0&0\\[2pt]0&0&4&1\\[2pt]0&0&0&4\end{bmatrix}}.} The matrix J {\displaystyle J} is almost diagonal. This is the Jordan normal form of A. The section Example below fills in the details of the computation. == Complex matrices == In general, a square complex matrix A is similar to a block diagonal matrix J = [ J 1 ⋱ J p ] {\displaystyle J={\begin{bmatrix}J_{1}&\;&\;\\\;&\ddots &\;\\\;&\;&J_{p}\end{bmatrix}}} where each block Ji is a square matrix of the form J i = [ λ i 1 λ i ⋱ ⋱ 1 λ i ] . {\displaystyle J_{i}={\begin{bmatrix}\lambda _{i}&1&\;&\;\\\;&\lambda _{i}&\ddots &\;\\\;&\;&\ddots &1\\\;&\;&\;&\lambda _{i}\end{bmatrix}}.} So there exists an invertible matrix P such that P−1AP = J is such that the only non-zero entries of J are on the diagonal and the superdiagonal. J is called the Jordan normal form of A. Each Ji is called a Jordan block of A. In a given Jordan block, every entry on the superdiagonal is 1. Assuming this result, we can deduce the following properties: Counting multiplicities, the eigenvalues of J, and therefore of A, are the diagonal entries. Given an eigenvalue λi, its geometric multiplicity is the dimension of ker(A − λi I), where I is the identity matrix, and it is the number of Jordan blocks corresponding to λi. The sum of the sizes of all Jordan blocks corresponding to an eigenvalue λi is its algebraic multiplicity. A is diagonalizable if and only if, for every eigenvalue λ of A, its geometric and algebraic multiplicities coincide. In particular, the Jordan blocks in this case are 1 × 1 matrices; that is, scalars. The Jordan block corresponding to λ is of the form λI + N, where N is a nilpotent matrix defined as Nij = δi,j−1 (where δ is the Kronecker delta). The nilpotency of N can be exploited when calculating f(A) where f is a complex analytic function. For example, in principle the Jordan form could give a closed-form expression for the exponential exp(A). The number of Jordan blocks corresponding to λi of size at least j is dim ker(A − λiI)j − dim ker(A − λiI)j−1. Thus, the number of Jordan blocks of size j is 2 dim ker ( A − λ i I ) j − dim ker ( A − λ i I ) j + 1 − dim ker ( A − λ i I ) j − 1 {\displaystyle 2\dim \ker(A-\lambda _{i}I)^{j}-\dim \ker(A-\lambda _{i}I)^{j+1}-\dim \ker(A-\lambda _{i}I)^{j-1}} Given an eigenvalue λi, its multiplicity in the minimal polynomial is the size of its largest Jordan block. === Example === Consider the matrix A {\displaystyle A} from the example in the previous section. The Jordan normal form is obtained by some similarity transformation: P − 1 A P = J ; {\displaystyle P^{-1}AP=J;} that is, A P = P J . {\displaystyle AP=PJ.} Let P {\displaystyle P} have column vectors p i {\displaystyle p_{i}} , i = 1 , … , 4 {\displaystyle i=1,\ldots ,4} , then A [ p 1 p 2 p 3 p 4 ] = [ p 1 p 2 p 3 p 4 ] [ 1 0 0 0 0 2 0 0 0 0 4 1 0 0 0 4 ] = [ p 1 2 p 2 4 p 3 p 3 + 4 p 4 ] . {\displaystyle A{\begin{bmatrix}p_{1}&p_{2}&p_{3}&p_{4}\end{bmatrix}}={\begin{bmatrix}p_{1}&p_{2}&p_{3}&p_{4}\end{bmatrix}}{\begin{bmatrix}1&0&0&0\\0&2&0&0\\0&0&4&1\\0&0&0&4\end{bmatrix}}={\begin{bmatrix}p_{1}&2p_{2}&4p_{3}&p_{3}+4p_{4}\end{bmatrix}}.} We see that ( A − 1 I ) p 1 = 0 {\displaystyle (A-1I)p_{1}=0} ( A − 2 I ) p 2 = 0 {\displaystyle (A-2I)p_{2}=0} ( A − 4 I ) p 3 = 0 {\displaystyle (A-4I)p_{3}=0} ( A − 4 I ) p 4 = p 3 . {\displaystyle (A-4I)p_{4}=p_{3}.} For i = 1 , 2 , 3 {\displaystyle i=1,2,3} we have p i ∈ ker ( A − λ i I ) {\displaystyle p_{i}\in \ker(A-\lambda _{i}I)} , that is, p i {\displaystyle p_{i}} is an eigenvector of A {\displaystyle A} corresponding to the eigenvalue λ i {\displaystyle \lambda _{i}} . For i = 4 {\displaystyle i=4} , multiplying both sides by ( A − 4 I ) {\displaystyle (A-4I)} gives ( A − 4 I ) 2 p 4 = ( A − 4 I ) p 3 . {\displaystyle (A-4I)^{2}p_{4}=(A-4I)p_{3}.} But ( A − 4 I ) p 3 = 0 {\displaystyle (A-4I)p_{3}=0} , so ( A − 4 I ) 2 p 4 = 0. {\displaystyle (A-4I)^{2}p_{4}=0.} Thus, p 4 ∈ ker ( A − 4 I ) 2 . {\displaystyle p_{4}\in \ker(A-4I)^{2}.} Vectors such as p 4 {\displaystyle p_{4}} are called generalized eigenvectors of A. === Example: Obtaining the normal form === This example shows how to calculate the Jordan normal form of a given matrix. Consider the matrix A = [ 5 4 2 1 0 1 − 1 − 1 − 1 − 1 3 0 1 1 − 1 2 ] {\displaystyle A=\left[{\begin{array}{rrrr}5&4&2&1\\0&1&-1&-1\\-1&-1&3&0\\1&1&-1&2\end{array}}\right]} which is mentioned in the beginning of the article. The characteristic polynomial of A is χ ( λ ) = det ( λ I − A ) = λ 4 − 11 λ 3 + 42 λ 2 − 64 λ + 32 = ( λ − 1 ) ( λ − 2 ) ( λ − 4 ) 2 . {\displaystyle {\begin{aligned}\chi (\lambda )&=\det(\lambda I-A)\\&=\lambda ^{4}-11\lambda ^{3}+42\lambda ^{2}-64\lambda +32\\&=(\lambda -1)(\lambda -2)(\lambda -4)^{2}.\,\end{aligned}}} This shows that the eigenvalues are 1, 2, 4 and 4, according to algebraic multiplicity. The eigenspace corresponding to the eigenvalue 1 can be found by solving the equation Av = 1v. It is spanned by the column vector v = (−1, 1, 0, 0)T. Similarly, the eigenspace corresponding to the eigenvalue 2 is spanned by w = (1, −1, 0, 1)T. Finally, the eigenspace corresponding to the eigenvalue 4 is also one-dimensional (even though this is a double eigenvalue) and is spanned by x = (1, 0, −1, 1)T. So, the geometric multiplicity (that is, the dimension of the eigenspace of the given eigenvalue) of each of the three eigenvalues is one. Therefore, the two eigenvalues equal to 4 correspond to a single Jordan block, and the Jordan normal form of the matrix A is the direct sum J = J 1 ( 1 ) ⊕ J 1 ( 2 ) ⊕ J 2 ( 4 ) = [ 1 0 0 0 0 2 0 0 0 0 4 1 0 0 0 4 ] . {\displaystyle J=J_{1}(1)\oplus J_{1}(2)\oplus J_{2}(4)={\begin{bmatrix}1&0&0&0\\0&2&0&0\\0&0&4&1\\0&0&0&4\end{bmatrix}}.} There are three Jordan chains. Two have length one: {v} and {w}, corresponding to the eigenvalues 1 and 2, respectively. There is one chain of length two corresponding to the eigenvalue 4. To find this chain, calculate ker ( A − 4 I ) 2 = span { [ 1 0 0 0 ] , [ 1 0 − 1 1 ] } {\displaystyle \ker(A-4I)^{2}=\operatorname {span} \,\left\{{\begin{bmatrix}1\\0\\0\\0\end{bmatrix}},\left[{\begin{array}{r}1\\0\\-1\\1\end{array}}\right]\right\}} where I is the 4 × 4 identity matrix. Pick a vector in the above span that is not in the kernel of A − 4I; for example, y = (1,0,0,0)T. Now, (A − 4I)y = x and (A − 4I)x = 0, so {y, x} is a chain of length two corresponding to the eigenvalue 4. The transition matrix P such that P−1AP = J is formed by putting these vectors next to each other as follows P = [ v w x y ] = [ − 1 1 1 1 1 − 1 0 0 0 0 − 1 0 0 1 1 0 ] . {\displaystyle P=\left[{\begin{array}{c|c|c|c}v&w&x&y\end{array}}\right]=\left[{\begin{array}{rrrr}-1&1&1&1\\1&-1&0&0\\0&0&-1&0\\0&1&1&0\end{array}}\right].} A computation shows that the equation P−1AP = J indeed holds. P − 1 A P = J = [ 1 0 0 0 0 2 0 0 0 0 4 1 0 0 0 4 ] . {\displaystyle P^{-1}AP=J={\begin{bmatrix}1&0&0&0\\0&2&0&0\\0&0&4&1\\0&0&0&4\end{bmatrix}}.} If we had interchanged the order in which the chain vectors appeared, that is, changing the order of v, w and {x, y} together, the Jordan blocks would be interchanged. However, the Jordan forms are equivalent Jordan forms. == Generalized eigenvectors == Given an eigenvalue λ, every corresponding Jordan block gives rise to a Jordan chain of linearly independent vectors pi, i = 1, ..., b, where b is the size of the Jordan block. The generator, or lead vector, pb of the chain is a generalized eigenvector such that ( A − λ I ) b p b = 0 {\displaystyle (A-\lambda I)^{b}p_{b}=0} . The vector p 1 = ( A − λ I ) b − 1 p b {\displaystyle p_{1}=(A-\lambda I)^{b-1}p_{b}} is an ordinary eigenvector corresponding to λ. In general, pi is a preimage of pi−1 under A − λ I {\displaystyle A-\lambda I} . So the lead vector generates the chain via multiplication by A − λ I {\displaystyle A-\lambda I} . Therefore, the statement that every square matrix A can be put in Jordan normal form is equivalent to the claim that the underlying vector space has a basis composed of Jordan chains. === A proof === We give a proof by induction that any complex-valued square matrix A may be put in Jordan normal form. Since the underlying vector space can be shown to be the direct sum of invariant subspaces associated with the eigenvalues, A can be assumed to have just one eigenvalue λ. The 1 × 1 case is trivial. Let A be an n × n matrix. The range of A − λ I {\displaystyle A-\lambda I} , denoted by Ran ( A − λ I ) {\displaystyle \operatorname {Ran} (A-\lambda I)} , is an invariant subspace of A. Also, since λ is an eigenvalue of A, the dimension of Ran ( A − λ I ) {\displaystyle \operatorname {Ran} (A-\lambda I)} , r, is strictly less than n, so, by the inductive hypothesis, Ran ( A − λ I ) {\displaystyle \operatorname {Ran} (A-\lambda I)} has a basis {p1, ..., pr} composed of Jordan chains. Next consider the kernel, that is, the subspace ker ( A − λ I ) {\displaystyle \ker(A-\lambda I)} . If Ran ( A − λ I ) ∩ ker ( A − λ I ) = { 0 } , {\displaystyle \operatorname {Ran} (A-\lambda I)\cap \ker(A-\lambda I)=\{0\},} the desired result follows immediately from the rank–nullity theorem. (This would be the case, for example, if A were Hermitian.) Otherwise, if Q = Ran ( A − λ I ) ∩ ker ( A − λ I ) ≠ { 0 } , {\displaystyle Q=\operatorname {Ran} (A-\lambda I)\cap \ker(A-\lambda I)\neq \{0\},} let the dimension of Q be s ≤ r. Each vector in Q is an eigenvector, so Ran ( A − λ I ) {\displaystyle \operatorname {Ran} (A-\lambda I)} must contain s Jordan chains corresponding to s linearly independent eigenvectors. Therefore the basis {p1, ..., pr} must contain s vectors, say {p1, ..., ps}, that are lead vectors of these Jordan chains. We can "extend the chains" by taking the preimages of these lead vectors. (This is the key step.) Let qi be such that ( A − λ I ) q i = p i for i = 1 , … , s . {\displaystyle \;(A-\lambda I)q_{i}=p_{i}{\mbox{ for }}i=1,\ldots ,s.} Finally, we can pick any basis for ker ( A − λ I ) / Q {\displaystyle \ker(A-\lambda I)/Q} and then lift to vectors {z1, ..., zt} in ker ( A − λ I ) {\displaystyle \ker(A-\lambda I)} . Each zi forms a Jordan chain of length 1. We just need to show that the union of {p1, ..., pr}, {z1, ..., zt}, and {q1, ..., qs} forms a basis for the vector space. By the rank-nullity theorem, dim ( ker ( A − λ I ) ) ) = n − r {\displaystyle \dim(\ker(A-\lambda I)))=n-r} , so t = n − r − s {\displaystyle t=n-r-s} , and so the number of vectors in the potential basis is equal to n. To show linear independence, suppose some linear combination of the vectors is 0. Applying A − λ I , {\displaystyle A-\lambda I,} we get some linear combination of pi, with the qi becoming lead vectors among the pi. From linear indepence of pi, it follows that the coefficients of the vectors qi must be zero. Furthermore, no non-trivial linear combination of the zi can equal a linear combination of pi, because then it would belong to Ran ( A − λ I ) {\displaystyle \operatorname {Ran} (A-\lambda I)} and thus Q, which is impossible by the construction of zi. Therefore the coefficients of the zi will also be 0. This leaves in the original linear combination just the pi terms, which are assumed to be linearly independent, and so their coefficients must be zero too. We have found a basis composed of Jordan chains, and this shows A can be put in Jordan normal form. === Uniqueness === It can be shown that the Jordan normal form of a given matrix A is unique up to the order of the Jordan blocks. Knowing the algebraic and geometric multiplicities of the eigenvalues is not sufficient to determine the Jordan normal form of A. Assuming the algebraic multiplicity m(λ) of an eigenvalue λ is known, the structure of the Jordan form can be ascertained by analyzing the ranks of the powers (A − λI)m(λ). To see this, suppose an n × n matrix A has only one eigenvalue λ. So m(λ) = n. The smallest integer k1 such that ( A − λ I ) k 1 = 0 {\displaystyle (A-\lambda I)^{k_{1}}=0} is the size of the largest Jordan block in the Jordan form of A. (This number k1 is also called the index of λ. See discussion in a following section.) The rank of ( A − λ I ) k 1 − 1 {\displaystyle (A-\lambda I)^{k_{1}-1}} is the number of Jordan blocks of size k1. Similarly, the rank of ( A − λ I ) k 1 − 2 {\displaystyle (A-\lambda I)^{k_{1}-2}} is twice the number of Jordan blocks of size k1 plus the number of Jordan blocks of size k1 − 1. The general case is similar. This can be used to show the uniqueness of the Jordan form. Let J1 and J2 be two Jordan normal forms of A. Then J1 and J2 are similar and have the same spectrum, including algebraic multiplicities of the eigenvalues. The procedure outlined in the previous paragraph can be used to determine the structure of these matrices. Since the rank of a matrix is preserved by similarity transformation, there is a bijection between the Jordan blocks of J1 and J2. This proves the uniqueness part of the statement. == Real matrices == If A is a real matrix, its Jordan form can still be non-real. Instead of representing it with complex eigenvalues and ones on the superdiagonal, as discussed above, there exists a real invertible matrix P such that P−1AP = J is a real block diagonal matrix with each block being a real Jordan block. A real Jordan block is either identical to a complex Jordan block (if the corresponding eigenvalue λ i {\displaystyle \lambda _{i}} is real), or is a block matrix itself, consisting of 2×2 blocks (for non-real eigenvalue λ i = a i + i b i {\displaystyle \lambda _{i}=a_{i}+ib_{i}} with given algebraic multiplicity) of the form C i = [ a i − b i b i a i ] {\displaystyle C_{i}=\left[{\begin{array}{rr}a_{i}&-b_{i}\\b_{i}&a_{i}\\\end{array}}\right]} and describe multiplication by λ i {\displaystyle \lambda _{i}} in the complex plane. The superdiagonal blocks are 2×2 identity matrices and hence in this representation the matrix dimensions are larger than the complex Jordan form. The full real Jordan block is given by J i = [ C i I C i ⋱ ⋱ I C i ] . {\displaystyle J_{i}={\begin{bmatrix}C_{i}&I&&\\&C_{i}&\ddots &\\&&\ddots &I\\&&&C_{i}\end{bmatrix}}.} This real Jordan form is a consequence of the complex Jordan form. For a real matrix the nonreal eigenvectors and generalized eigenvectors can always be chosen to form complex conjugate pairs. Taking the real and imaginary part (linear combination of the vector and its conjugate), the matrix has this form with respect to the new basis. == Matrices with entries in a field == Jordan reduction can be extended to any square matrix M whose entries lie in a field K. The result states that any M can be written as a sum D + N where D is semisimple, N is nilpotent, and DN = ND. This is called the Jordan–Chevalley decomposition. Whenever K contains the eigenvalues of M, in particular when K is algebraically closed, the normal form can be expressed explicitly as the direct sum of Jordan blocks. Similar to the case when K is the complex numbers, knowing the dimensions of the kernels of (M − λI)k for 1 ≤ k ≤ m, where m is the algebraic multiplicity of the eigenvalue λ, allows one to determine the Jordan form of M. We may view the underlying vector space V as a K[x]-module by regarding the action of x on V as application of M and extending by K-linearity. Then the polynomials (x − λ)k are the elementary divisors of M, and the Jordan normal form is concerned with representing M in terms of blocks associated to the elementary divisors. The proof of the Jordan normal form is usually carried out as an application to the ring K[x] of the structure theorem for finitely generated modules over a principal ideal domain, of which it is a corollary. == Consequences == One can see that the Jordan normal form is essentially a classification result for square matrices, and as such several important results from linear algebra can be viewed as its consequences. === Spectral mapping theorem === Using the Jordan normal form, direct calculation gives a spectral mapping theorem for the polynomial functional calculus: Let A be an n × n matrix with eigenvalues λ1, ..., λn, then for any polynomial p, p(A) has eigenvalues p(λ1), ..., p(λn). === Characteristic polynomial === The characteristic polynomial of A is p A ( λ ) = det ( λ I − A ) {\displaystyle p_{A}(\lambda )=\det(\lambda I-A)} . Similar matrices have the same characteristic polynomial. Therefore, p A ( λ ) = p J ( λ ) = ∏ i ( λ − λ i ) m i {\textstyle p_{A}(\lambda )=p_{J}(\lambda )=\prod _{i}(\lambda -\lambda _{i})^{m_{i}}} , where λ i {\displaystyle \lambda _{i}} is the ith root of p J {\textstyle p_{J}} and m i {\displaystyle m_{i}} is its multiplicity, because this is clearly the characteristic polynomial of the Jordan form of A. === Cayley–Hamilton theorem === The Cayley–Hamilton theorem asserts that every matrix A satisfies its characteristic equation: if p is the characteristic polynomial of A, then p A ( A ) = 0 {\displaystyle p_{A}(A)=0} . This can be shown via direct calculation in the Jordan form, since if λ i {\displaystyle \lambda _{i}} is an eigenvalue of multiplicity m {\displaystyle m} , then its Jordan block J i {\displaystyle J_{i}} clearly satisfies ( J i − λ i I ) m i = 0 {\displaystyle (J_{i}-\lambda _{i}I)^{m_{i}}=0} . As the diagonal blocks do not affect each other, the ith diagonal block of ( A − λ i I ) m i {\displaystyle (A-\lambda _{i}I)^{m_{i}}} is ( J i − λ i I ) m i {\displaystyle (J_{i}-\lambda _{i}I)^{m_{i}}} ; hence p A ( A ) = ∏ i ( A − λ i I ) m i = 0 {\textstyle p_{A}(A)=\prod _{i}(A-\lambda _{i}I)^{m_{i}}=0} . The Jordan form can be assumed to exist over a field extending the base field of the matrix, for instance over the splitting field of p; this field extension does not change the matrix p(A) in any way. === Minimal polynomial === The minimal polynomial P of a square matrix A is the unique monic polynomial of least degree, m, such that P(A) = 0. Alternatively, the set of polynomials that annihilate a given A form an ideal I in C[x], the principal ideal domain of polynomials with complex coefficients. The monic element that generates I is precisely P. Let λ1, ..., λq be the distinct eigenvalues of A, and si be the size of the largest Jordan block corresponding to λi. It is clear from the Jordan normal form that the minimal polynomial of A has degree Σsi. While the Jordan normal form determines the minimal polynomial, the converse is not true. This leads to the notion of elementary divisors. The elementary divisors of a square matrix A are the characteristic polynomials of its Jordan blocks. The factors of the minimal polynomial m are the elementary divisors of the largest degree corresponding to distinct eigenvalues. The degree of an elementary divisor is the size of the corresponding Jordan block, therefore the dimension of the corresponding invariant subspace. If all elementary divisors are linear, A is diagonalizable. === Invariant subspace decompositions === The Jordan form of a n × n matrix A is block diagonal, and therefore gives a decomposition of the n dimensional Euclidean space into invariant subspaces of A. Every Jordan block Ji corresponds to an invariant subspace Xi. Symbolically, we put C n = ⨁ i = 1 k X i {\displaystyle \mathbb {C} ^{n}=\bigoplus _{i=1}^{k}X_{i}} where each Xi is the span of the corresponding Jordan chain, and k is the number of Jordan chains. One can also obtain a slightly different decomposition via the Jordan form. Given an eigenvalue λi, the size of its largest corresponding Jordan block si is called the index of λi and denoted by v(λi). (Therefore, the degree of the minimal polynomial is the sum of all indices.) Define a subspace Yi by Y i = ker ( λ i I − A ) v ( λ i ) . {\displaystyle Y_{i}=\ker(\lambda _{i}I-A)^{v(\lambda _{i})}.} This gives the decomposition C n = ⨁ i = 1 l Y i {\displaystyle \mathbb {C} ^{n}=\bigoplus _{i=1}^{l}Y_{i}} where l is the number of distinct eigenvalues of A. Intuitively, we glob together the Jordan block invariant subspaces corresponding to the same eigenvalue. In the extreme case where A is a multiple of the identity matrix we have k = n and l = 1. The projection onto Yi and along all the other Yj ( j ≠ i ) is called the spectral projection of A at vi and is usually denoted by P(λi ; A). Spectral projections are mutually orthogonal in the sense that P(λi ; A) P(vj ; A) = 0 if i ≠ j. Also they commute with A and their sum is the identity matrix. Replacing every vi in the Jordan matrix J by one and zeroing all other entries gives P(vi ; J), moreover if U J U−1 is the similarity transformation such that A = U J U−1 then P(λi ; A) = U P(λi ; J) U−1. They are not confined to finite dimensions. See below for their application to compact operators, and in holomorphic functional calculus for a more general discussion. Comparing the two decompositions, notice that, in general, l ≤ k. When A is normal, the subspaces Xi's in the first decomposition are one-dimensional and mutually orthogonal. This is the spectral theorem for normal operators. The second decomposition generalizes more easily for general compact operators on Banach spaces. It might be of interest here to note some properties of the index, ν(λ). More generally, for a complex number λ, its index can be defined as the least non-negative integer ν(λ) such that ker ( A − λ I ) ν ( λ ) = ker ( A − λ I ) m , ∀ m ≥ ν ( λ ) . {\displaystyle \ker(A-\lambda I)^{\nu (\lambda )}=\ker(A-\lambda I)^{m},\;\forall m\geq \nu (\lambda ).} So ν(v) > 0 if and only if λ is an eigenvalue of A. In the finite-dimensional case, ν(v) ≤ the algebraic multiplicity of v. === Plane (flat) normal form === The Jordan form is used to find a normal form of matrices up to conjugacy such that normal matrices make up an algebraic variety of a low fixed degree in the ambient matrix space. Sets of representatives of matrix conjugacy classes for Jordan normal form or rational canonical forms in general do not constitute linear or affine subspaces in the ambient matrix spaces. Vladimir Arnold posed a problem: Find a canonical form of matrices over a field for which the set of representatives of matrix conjugacy classes is a union of affine linear subspaces (flats). In other words, map the set of matrix conjugacy classes injectively back into the initial set of matrices so that the image of this embedding—the set of all normal matrices, has the lowest possible degree—it is a union of shifted linear subspaces. It was solved for algebraically closed fields by Peteris Daugulis. The construction of a uniquely defined plane normal form of a matrix starts by considering its Jordan normal form. == Matrix functions == Iteration of the Jordan chain motivates various extensions to more abstract settings. For finite matrices, one gets matrix functions; this can be extended to compact operators and the holomorphic functional calculus, as described further below. The Jordan normal form is the most convenient for computation of the matrix functions (though it may be not the best choice for computer computations). Let f(z) be an analytical function of a complex argument. Applying the function on a n×n Jordan block J with eigenvalue λ results in an upper triangular matrix: f ( J ) = [ f ( λ ) f ′ ( λ ) f ″ ( λ ) 2 ⋯ f ( n − 1 ) ( λ ) ( n − 1 ) ! 0 f ( λ ) f ′ ( λ ) ⋯ f ( n − 2 ) ( λ ) ( n − 2 ) ! ⋮ ⋮ ⋱ ⋱ ⋮ 0 0 0 f ( λ ) f ′ ( λ ) 0 0 0 0 f ( λ ) ] , {\displaystyle f(J)={\begin{bmatrix}f(\lambda )&f'(\lambda )&{\tfrac {f''(\lambda )}{2}}&\cdots &{\tfrac {f^{(n-1)}(\lambda )}{(n-1)!}}\\0&f(\lambda )&f'(\lambda )&\cdots &{\tfrac {f^{(n-2)}(\lambda )}{(n-2)!}}\\\vdots &\vdots &\ddots &\ddots &\vdots \\0&0&0&f(\lambda )&f'(\lambda )\\0&0&0&0&f(\lambda )\end{bmatrix}},} so that the elements of the k-th superdiagonal of the resulting matrix are f ( k ) ( λ ) k ! {\displaystyle {\tfrac {f^{(k)}(\lambda )}{k!}}} . For a matrix of general Jordan normal form the above expression shall be applied to each Jordan block. The following example shows the application to the power function f(z) = zn: [ λ 1 1 0 0 0 0 λ 1 1 0 0 0 0 λ 1 0 0 0 0 0 λ 2 1 0 0 0 0 λ 2 ] n = [ λ 1 n ( n 1 ) λ 1 n − 1 ( n 2 ) λ 1 n − 2 0 0 0 λ 1 n ( n 1 ) λ 1 n − 1 0 0 0 0 λ 1 n 0 0 0 0 0 λ 2 n ( n 1 ) λ 2 n − 1 0 0 0 0 λ 2 n ] , {\displaystyle {\begin{bmatrix}\lambda _{1}&1&0&0&0\\0&\lambda _{1}&1&0&0\\0&0&\lambda _{1}&0&0\\0&0&0&\lambda _{2}&1\\0&0&0&0&\lambda _{2}\end{bmatrix}}^{n}={\begin{bmatrix}\lambda _{1}^{n}&{\tbinom {n}{1}}\lambda _{1}^{n-1}&{\tbinom {n}{2}}\lambda _{1}^{n-2}&0&0\\0&\lambda _{1}^{n}&{\tbinom {n}{1}}\lambda _{1}^{n-1}&0&0\\0&0&\lambda _{1}^{n}&0&0\\0&0&0&\lambda _{2}^{n}&{\tbinom {n}{1}}\lambda _{2}^{n-1}\\0&0&0&0&\lambda _{2}^{n}\end{bmatrix}},} where the binomial coefficients are defined as ( n k ) = ∏ i = 1 k n + 1 − i i {\textstyle {\binom {n}{k}}=\prod _{i=1}^{k}{\frac {n+1-i}{i}}} . For integer positive n it reduces to standard definition of the coefficients. For negative n the identity ( − n k ) = ( − 1 ) k ( n + k − 1 k ) {\textstyle {\binom {-n}{k}}=(-1)^{k}{\binom {n+k-1}{k}}} may be of use. == Compact operators == A result analogous to the Jordan normal form holds for compact operators on a Banach space. One restricts to compact operators because every point x in the spectrum of a compact operator T is an eigenvalue; The only exception is when x is the limit point of the spectrum. This is not true for bounded operators in general. To give some idea of this generalization, we first reformulate the Jordan decomposition in the language of functional analysis. === Holomorphic functional calculus === Let X be a Banach space, L(X) be the bounded operators on X, and σ(T) denote the spectrum of T ∈ L(X). The holomorphic functional calculus is defined as follows: Fix a bounded operator T. Consider the family Hol(T) of complex functions that is holomorphic on some open set G containing σ(T). Let Γ = {γi} be a finite collection of Jordan curves such that σ(T) lies in the inside of Γ, we define f(T) by f ( T ) = 1 2 π i ∫ Γ f ( z ) ( z − T ) − 1 d z . {\displaystyle f(T)={\frac {1}{2\pi i}}\int _{\Gamma }f(z)(z-T)^{-1}\,dz.} The open set G could vary with f and need not be connected. The integral is defined as the limit of the Riemann sums, as in the scalar case. Although the integral makes sense for continuous f, we restrict to holomorphic functions to apply the machinery from classical function theory (for example, the Cauchy integral formula). The assumption that σ(T) lie in the inside of Γ ensures f(T) is well defined; it does not depend on the choice of Γ. The functional calculus is the mapping Φ from Hol(T) to L(X) given by Φ ( f ) = f ( T ) . {\displaystyle \;\Phi (f)=f(T).} We will require the following properties of this functional calculus: Φ extends the polynomial functional calculus. The spectral mapping theorem holds: σ(f(T)) = f(σ(T)). Φ is an algebra homomorphism. === The finite-dimensional case === In the finite-dimensional case, σ(T) = {λi} is a finite discrete set in the complex plane. Let ei be the function that is 1 in some open neighborhood of λi and 0 elsewhere. By property 3 of the functional calculus, the operator e i ( T ) {\displaystyle e_{i}(T)} is a projection. Moreover, let νi be the index of λi and f ( z ) = ( z − λ i ) ν i . {\displaystyle f(z)=(z-\lambda _{i})^{\nu _{i}}.} The spectral mapping theorem tells us f ( T ) e i ( T ) = ( T − λ i ) ν i e i ( T ) {\displaystyle f(T)e_{i}(T)=(T-\lambda _{i})^{\nu _{i}}e_{i}(T)} has spectrum {0}. By property 1, f(T) can be directly computed in the Jordan form, and by inspection, we see that the operator f(T)ei(T) is the zero matrix. By property 3, f(T) ei(T) = ei(T) f(T). So ei(T) is precisely the projection onto the subspace Ran e i ( T ) = ker ( T − λ i ) ν i . {\displaystyle \operatorname {Ran} e_{i}(T)=\ker(T-\lambda _{i})^{\nu _{i}}.} The relation ∑ i e i = 1 {\displaystyle \sum _{i}e_{i}=1} implies C n = ⨁ i Ran e i ( T ) = ⨁ i ker ( T − λ i ) ν i {\displaystyle \mathbb {C} ^{n}=\bigoplus _{i}\;\operatorname {Ran} e_{i}(T)=\bigoplus _{i}\ker(T-\lambda _{i})^{\nu _{i}}} where the index i runs through the distinct eigenvalues of T. This is the invariant subspace decomposition C n = ⨁ i Y i {\displaystyle \mathbb {C} ^{n}=\bigoplus _{i}Y_{i}} given in a previous section. Each ei(T) is the projection onto the subspace spanned by the Jordan chains corresponding to λi and along the subspaces spanned by the Jordan chains corresponding to vj for j ≠ i. In other words, ei(T) = P(λi;T). This explicit identification of the operators ei(T) in turn gives an explicit form of holomorphic functional calculus for matrices: For all f ∈ Hol(T), f ( T ) = ∑ λ i ∈ σ ( T ) ∑ k = 0 ν i − 1 f ( k ) k ! ( T − λ i ) k e i ( T ) . {\displaystyle f(T)=\sum _{\lambda _{i}\in \sigma (T)}\sum _{k=0}^{\nu _{i}-1}{\frac {f^{(k)}}{k!}}(T-\lambda _{i})^{k}e_{i}(T).} Notice that the expression of f(T) is a finite sum because, on each neighborhood of vi, we have chosen the Taylor series expansion of f centered at vi. === Poles of an operator === Let T be a bounded operator λ be an isolated point of σ(T). (As stated above, when T is compact, every point in its spectrum is an isolated point, except possibly the limit point 0.) The point λ is called a pole of operator T with order ν if the resolvent function RT defined by R T ( λ ) = ( λ − T ) − 1 {\displaystyle R_{T}(\lambda )=(\lambda -T)^{-1}} has a pole of order ν at λ. We will show that, in the finite-dimensional case, the order of an eigenvalue coincides with its index. The result also holds for compact operators. Consider the annular region A centered at the eigenvalue λ with sufficiently small radius ε such that the intersection of the open disc Bε(λ) and σ(T) is {λ}. The resolvent function RT is holomorphic on A. Extending a result from classical function theory, RT has a Laurent series representation on A: R T ( z ) = ∑ − ∞ ∞ a m ( λ − z ) m {\displaystyle R_{T}(z)=\sum _{-\infty }^{\infty }a_{m}(\lambda -z)^{m}} where a − m = − 1 2 π i ∫ C ( λ − z ) m − 1 ( z − T ) − 1 d z {\displaystyle a_{-m}=-{\frac {1}{2\pi i}}\int _{C}(\lambda -z)^{m-1}(z-T)^{-1}dz} and C is a small circle centered at λ. By the previous discussion on the functional calculus, a − m = − ( λ − T ) m − 1 e λ ( T ) {\displaystyle a_{-m}=-(\lambda -T)^{m-1}e_{\lambda }(T)} where e λ {\displaystyle e_{\lambda }} is 1 on B ε ( λ ) {\displaystyle B_{\varepsilon }(\lambda )} and 0 elsewhere. But we have shown that the smallest positive integer m such that a − m ≠ 0 {\displaystyle a_{-m}\neq 0} and a − l = 0 ∀ l ≥ m {\displaystyle a_{-l}=0\;\;\forall \;l\geq m} is precisely the index of λ, ν(λ). In other words, the function RT has a pole of order ν(λ) at λ. == Numerical analysis == If the matrix A has multiple eigenvalues, or is close to a matrix with multiple eigenvalues, then its Jordan normal form is very sensitive to perturbations. Consider for instance the matrix A = [ 1 1 ε 1 ] . {\displaystyle A={\begin{bmatrix}1&1\\\varepsilon &1\end{bmatrix}}.} If ε = 0, then the Jordan normal form is simply [ 1 1 0 1 ] . {\displaystyle {\begin{bmatrix}1&1\\0&1\end{bmatrix}}.} However, for ε ≠ 0, the Jordan normal form is [ 1 + ε 0 0 1 − ε ] . {\displaystyle {\begin{bmatrix}1+{\sqrt {\varepsilon }}&0\\0&1-{\sqrt {\varepsilon }}\end{bmatrix}}.} This ill conditioning makes it very hard to develop a robust numerical algorithm for the Jordan normal form, as the result depends critically on whether two eigenvalues are deemed to be equal. For this reason, the Jordan normal form is usually avoided in numerical analysis; the stable Schur decomposition or pseudospectra are better alternatives. == See also == Canonical basis Canonical form Frobenius normal form Jordan matrix Jordan–Chevalley decomposition Matrix decomposition Modal matrix Weyr canonical form == Notes == == References ==
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Wikipedia:Jordan–Chevalley decomposition#0
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In mathematics, specifically linear algebra, the Jordan–Chevalley decomposition, named after Camille Jordan and Claude Chevalley, expresses a linear operator in a unique way as the sum of two other linear operators which are simpler to understand. Specifically, one part is potentially diagonalisable and the other is nilpotent. The two parts are polynomials in the operator, which makes them behave nicely in algebraic manipulations. The decomposition has a short description when the Jordan normal form of the operator is given, but it exists under weaker hypotheses than are needed for the existence of a Jordan normal form. Hence the Jordan–Chevalley decomposition can be seen as a generalisation of the Jordan normal form, which is also reflected in several proofs of it. It is closely related to the Wedderburn principal theorem about associative algebras, which also leads to several analogues in Lie algebras. Analogues of the Jordan–Chevalley decomposition also exist for elements of Linear algebraic groups and Lie groups via a multiplicative reformulation. The decomposition is an important tool in the study of all of these objects, and was developed for this purpose. In many texts, the potentially diagonalisable part is also characterised as the semisimple part. == Introduction == A basic question in linear algebra is whether an operator on a finite-dimensional vector space can be diagonalised. For example, this is closely related to the eigenvalues of the operator. In several contexts, one may be dealing with many operators which are not diagonalisable. Even over an algebraically closed field, a diagonalisation may not exist. In this context, the Jordan normal form achieves the best possible result akin to a diagonalisation. For linear operators over a field which is not algebraically closed, there may be no eigenvector at all. This latter point is not the main concern dealt with by the Jordan–Chevalley decomposition. To avoid this problem, instead potentially diagonalisable operators are considered, which are those that admit a diagonalisation over some field (or equivalently over the algebraic closure of the field under consideration). The operators which are "the furthest away" from being diagonalisable are nilpotent operators. An operator (or more generally an element of a ring) x {\displaystyle x} is said to be nilpotent when there is some positive integer m ≥ 1 {\displaystyle m\geq 1} such that x m = 0 {\displaystyle x^{m}=0} . In several contexts in abstract algebra, it is the case that the presence of nilpotent elements of a ring make them much more complicated to work with. To some extent, this is also the case for linear operators. The Jordan–Chevalley decomposition "separates out" the nilpotent part of an operator which causes it to be not potentially diagonalisable. So when it exists, the complications introduced by nilpotent operators and their interaction with other operators can be understood using the Jordan–Chevalley decomposition. Historically, the Jordan–Chevalley decomposition was motivated by the applications to the theory of Lie algebras and linear algebraic groups, as described in sections below. == Decomposition of a linear operator == Let K {\displaystyle K} be a field, V {\displaystyle V} a finite-dimensional vector space over K {\displaystyle K} , and T {\displaystyle T} a linear operator over V {\displaystyle V} (equivalently, a matrix with entries from K {\displaystyle K} ). If the minimal polynomial of T {\displaystyle T} splits over K {\displaystyle K} (for example if K {\displaystyle K} is algebraically closed), then T {\displaystyle T} has a Jordan normal form T = S J S − 1 {\displaystyle T=SJS^{-1}} . If D {\displaystyle D} is the diagonal of J {\displaystyle J} , let R = J − D {\displaystyle R=J-D} be the remaining part. Then T = S D S − 1 + S R S − 1 {\displaystyle T=SDS^{-1}+SRS^{-1}} is a decomposition where S D S − 1 {\displaystyle SDS^{-1}} is diagonalisable and S R S − 1 {\displaystyle SRS^{-1}} is nilpotent. This restatement of the normal form as an additive decomposition not only makes the numerical computation more stable, but can be generalised to cases where the minimal polynomial of T {\displaystyle T} does not split. If the minimal polynomial of T {\displaystyle T} splits into distinct linear factors, then T {\displaystyle T} is diagonalisable. Therefore, if the minimal polynomial of T {\displaystyle T} is at least separable, then T {\displaystyle T} is potentially diagonalisable. The Jordan–Chevalley decomposition is concerned with the more general case where the minimal polynomial of T {\displaystyle T} is a product of separable polynomials. Let x : V → V {\displaystyle x:V\to V} be any linear operator on the finite-dimensional vector space V {\displaystyle V} over the field K {\displaystyle K} . A Jordan–Chevalley decomposition of x {\displaystyle x} is an expression of it as a sum x = x s + x n {\displaystyle x=x_{s}+x_{n}} , where x s {\displaystyle x_{s}} is potentially diagonalisable, x n {\displaystyle x_{n}} is nilpotent, and x s x n = x n x s {\displaystyle x_{s}x_{n}=x_{n}x_{s}} . Several proofs are discussed in (Couty, Esterle & Zarouf 2011). Two arguments are also described below. If K {\displaystyle K} is a perfect field, then every polynomial is a product of separable polynomials (since every polynomial is a product of its irreducible factors, and these are separable over a perfect field). So in this case, the Jordan–Chevalley decomposition always exists. Moreover, over a perfect field, a polynomial is separable if and only if it is square-free. Therefore an operator is potentially diagonalisable if and only if its minimal polynomial is square-free. In general (over any field), the minimal polynomial of a linear operator is square-free if and only if the operator is semisimple. (In particular, the sum of two commuting semisimple operators is always semisimple over a perfect field. The same statement is not true over general fields.) The property of being semisimple is more relevant than being potentially diagonalisable in most contexts where the Jordan–Chevalley decomposition is applied, such as for Lie algebras. For these reasons, many texts restrict to the case of perfect fields. === Proof of uniqueness and necessity === That x s {\displaystyle x_{s}} and x n {\displaystyle x_{n}} are polynomials in x {\displaystyle x} implies in particular that they commute with any operator that commutes with x {\displaystyle x} . This observation underlies the uniqueness proof. Let x = x s + x n {\displaystyle x=x_{s}+x_{n}} be a Jordan–Chevalley decomposition in which x s {\displaystyle x_{s}} and (hence also) x n {\displaystyle x_{n}} are polynomials in x {\displaystyle x} . Let x = x s ′ + x n ′ {\displaystyle x=x_{s}'+x_{n}'} be any Jordan–Chevalley decomposition. Then x s − x s ′ = x n ′ − x n {\displaystyle x_{s}-x_{s}'=x_{n}'-x_{n}} , and x s ′ , x n ′ {\displaystyle x_{s}',x_{n}'} both commute with x {\displaystyle x} , hence with x s , x n {\displaystyle x_{s},x_{n}} since these are polynomials in x {\displaystyle x} . The sum of commuting nilpotent operators is again nilpotent, and the sum of commuting potentially diagonalisable operators again potentially diagonalisable (because they are simultaneously diagonalizable over the algebraic closure of K {\displaystyle K} ). Since the only operator which is both potentially diagonalisable and nilpotent is the zero operator it follows that x s − x s ′ = 0 = x n − x n ′ {\displaystyle x_{s}-x_{s}'=0=x_{n}-x_{n}'} . To show that the condition that x {\displaystyle x} have a minimal polynomial which is a product of separable polynomials is necessary, suppose that x = x s + x n {\displaystyle x=x_{s}+x_{n}} is some Jordan–Chevalley decomposition. Letting p {\displaystyle p} be the separable minimal polynomial of x s {\displaystyle x_{s}} , one can check using the binomial theorem that p ( x s + x n ) {\displaystyle p(x_{s}+x_{n})} can be written as x n y {\displaystyle x_{n}y} where y {\displaystyle y} is some polynomial in x s , x n {\displaystyle x_{s},x_{n}} . Moreover, for some ℓ ≥ 1 {\displaystyle \ell \geq 1} , x n ℓ = 0 {\displaystyle x_{n}^{\ell }=0} . Thus p ( x ) ℓ = x n ℓ y ℓ = 0 {\displaystyle p(x)^{\ell }=x_{n}^{\ell }y^{\ell }=0} and so the minimal polynomial of x {\displaystyle x} must divide p ℓ {\displaystyle p^{\ell }} . As p ℓ {\displaystyle p^{\ell }} is a product of separable polynomials (namely of copies of p {\displaystyle p} ), so is the minimal polynomial. === Concrete example for non-existence === If the ground field is not perfect, then a Jordan–Chevalley decomposition may not exist, as it is possible that the minimal polynomial is not a product of separable polynomials. The simplest such example is the following. Let p {\displaystyle p} be a prime number, let k {\displaystyle k} be an imperfect field of characteristic p , {\displaystyle p,} (e. g. k = F p ( t ) {\displaystyle k=\mathbb {F} _{p}(t)} ) and choose a ∈ k {\displaystyle a\in k} that is not a p {\displaystyle p} th power. Let V = k [ X ] / ( X p − a ) 2 , {\displaystyle V=k[X]/\left(X^{p}-a\right)^{2},} let x = X ¯ {\displaystyle x={\overline {X}}} be the image in the quotient and let T {\displaystyle T} be the k {\displaystyle k} -linear operator given by multiplication by x {\displaystyle x} in V {\displaystyle V} . Note that the minimal polynomial is precisely ( X p − a ) 2 {\displaystyle \left(X^{p}-a\right)^{2}} , which is inseparable and a square. By the necessity of the condition for the Jordan–Chevalley decomposition (as shown in the last section), this operator does not have a Jordan–Chevalley decomposition. It can be instructive to see concretely why there is at least no decomposition into a square-free and a nilpotent part. If instead of with the polynomial ( X p − a ) 2 {\displaystyle \left(X^{p}-a\right)^{2}} , the same construction is performed with X p − a {\displaystyle {X^{p}}-a} , the resulting operator T {\displaystyle T} still does not admit a Jordan–Chevalley decomposition by the main theorem. However, T {\displaystyle T} is semi-simple. The trivial decomposition T = T + 0 {\displaystyle T=T+0} hence expresses T {\displaystyle T} as a sum of a semisimple and a nilpotent operator, both of which are polynomials in T {\displaystyle T} . === Elementary proof of existence === This construction is similar to Hensel's lemma in that it uses an algebraic analogue of Taylor's theorem to find an element with a certain algebraic property via a variant of Newton's method. In this form, it is taken from (Geck 2022). Let x {\displaystyle x} have minimal polynomial p {\displaystyle p} and assume this is a product of separable polynomials. This condition is equivalent to demanding that there is some separable q {\displaystyle q} such that q ∣ p {\displaystyle q\mid p} and p ∣ q m {\displaystyle p\mid q^{m}} for some m ≥ 1 {\displaystyle m\geq 1} . By the Bézout lemma, there are polynomials u {\displaystyle u} and v {\displaystyle v} such that u q + v q ′ = 1 {\displaystyle {uq+{vq'}}=1} . This can be used to define a recursion x n + 1 = x n − v ( x n ) q ( x n ) {\displaystyle x_{n+1}=x_{n}-v(x_{n})q(x_{n})} , starting with x 0 = x {\displaystyle x_{0}=x} . Letting X {\displaystyle {\mathfrak {X}}} be the algebra of operators which are polynomials in x {\displaystyle x} , it can be checked by induction that for all n {\displaystyle n} : x n ∈ X {\displaystyle x_{n}\in {\mathfrak {X}}} because in each step, a polynomial is applied, q ( x n ) ∈ q ( x ) 2 n ⋅ X {\displaystyle q(x_{n})\in q(x)^{2^{n}}\cdot {\mathfrak {X}}} because q ( x n + 1 ) = q ( x n ) + q ′ ( x n ) ( x n + 1 − x n ) + ( x n + 1 − x n ) 2 h {\displaystyle q(x_{n+1})=q(x_{n})+q'(x_{n})(x_{n+1}-x_{n})+(x_{n+1}-x_{n})^{2}h} for some h ∈ X {\displaystyle h\in {\mathfrak {X}}} (by the algebraic version of Taylor's theorem). By definition of x n + 1 {\displaystyle x_{n+1}} as well as of u {\displaystyle u} and v {\displaystyle v} , this simplifies to q ( x n + 1 ) = q ( x n ) 2 ( u ( x n ) + v ( x n ) 2 h ) {\displaystyle q(x_{n+1})=q(x_{n})^{2}(u(x_{n})+v(x_{n})^{2}h)} , which indeed lies in q ( x ) 2 n + 1 ⋅ X {\displaystyle q(x)^{2^{n+1}}\cdot {\mathfrak {X}}} by induction hypothesis, x n − x ∈ q ( x ) ⋅ X {\displaystyle x_{n}-x\in q(x)\cdot {\mathfrak {X}}} because x n + 1 − x = ( x n + 1 − x n ) + ( x n − x ) {\displaystyle x_{n+1}-x=(x_{n+1}-x_{n})+(x_{n}-x)} and both terms are in q ( x ) ⋅ X {\displaystyle q(x)\cdot {\mathfrak {X}}} , the first by the preceding point and the second by induction hypothesis. Thus, as soon as 2 n ≥ m {\displaystyle 2^{n}\geq m} , q ( x n ) = 0 {\displaystyle q(x_{n})=0} by the second point since p ∣ q m {\displaystyle p\mid q^{m}} and p ( x ) = 0 {\displaystyle p(x)=0} , so the minimal polynomial of x n {\displaystyle x_{n}} will divide q {\displaystyle q} and hence be separable. Moreover, x n {\displaystyle x_{n}} will be a polynomial in x {\displaystyle x} by the first point and x n − x {\displaystyle x_{n}-x} will be nilpotent by the third point (in fact, ( x n − x ) m = 0 {\displaystyle (x_{n}-x)^{m}=0} ). Therefore, x = x n + ( x − x n ) {\displaystyle x=x_{n}+(x-x_{n})} is then the Jordan–Chevalley decomposition of x {\displaystyle x} . Q.E.D. This proof, besides being completely elementary, has the advantage that it is algorithmic: By the Cayley–Hamilton theorem, p {\displaystyle p} can be taken to be the characteristic polynomial of x {\displaystyle x} , and in many contexts, q {\displaystyle q} can be determined from p {\displaystyle p} . Then v {\displaystyle v} can be determined using the Euclidean algorithm. The iteration of applying the polynomial v q {\displaystyle vq} to the matrix then can be performed until either v ( x n ) q ( x n ) = 0 {\displaystyle v(x_{n})q(x_{n})=0} (because then all later values will be equal) or 2 n {\displaystyle 2^{n}} exceeds the dimension of the vector space on which x {\displaystyle x} is defined (where n {\displaystyle n} is the number of iteration steps performed, as above). === Proof of existence via Galois theory === This proof, or variants of it, is commonly used to establish the Jordan–Chevalley decomposition. It has the advantage that it is very direct and describes quite precisely how close one can get to a Jordan–Chevalley decomposition: If L {\displaystyle L} is the splitting field of the minimal polynomial of x {\displaystyle x} and G {\displaystyle G} is the group of automorphisms of L {\displaystyle L} that fix the base field K {\displaystyle K} , then the set F {\displaystyle F} of elements of L {\displaystyle L} that are fixed by all elements of G {\displaystyle G} is a field with inclusions K ⊆ F ⊆ L {\displaystyle K\subseteq F\subseteq L} (see Galois correspondence). Below it is argued that x {\displaystyle x} admits a Jordan–Chevalley decomposition over F {\displaystyle F} , but not any smaller field. This argument does not use Galois theory. However, Galois theory is required deduce from this the condition for the existence of the Jordan-Chevalley given above. Above it was observed that if x {\displaystyle x} has a Jordan normal form (i. e. if the minimal polynomial of x {\displaystyle x} splits), then it has a Jordan Chevalley decomposition. In this case, one can also see directly that x n {\displaystyle x_{n}} (and hence also x s {\displaystyle x_{s}} ) is a polynomial in x {\displaystyle x} . Indeed, it suffices to check this for the decomposition of the Jordan matrix J = D + R {\displaystyle J=D+R} . This is a technical argument, but does not require any tricks beyond the Chinese remainder theorem. This fact can be used to deduce the Jordan–Chevalley decomposition in the general case. Let L {\displaystyle L} be the splitting field of the minimal polynomial of x {\displaystyle x} , so that x {\displaystyle x} does admit a Jordan normal form over L {\displaystyle L} . Then, by the argument just given, x {\displaystyle x} has a Jordan–Chevalley decomposition x = c ( x ) + ( x − c ( x ) ) {\displaystyle x={c(x)}+{(x-{c(x)})}} where c {\displaystyle c} is a polynomial with coefficients from L {\displaystyle L} , c ( x ) {\displaystyle c(x)} is diagonalisable (over L {\displaystyle L} ) and x − c ( x ) {\displaystyle x-c(x)} is nilpotent. Let σ {\displaystyle \sigma } be a field automorphism of L {\displaystyle L} which fixes K {\displaystyle K} . Then c ( x ) + ( x − c ( x ) ) = x = σ ( x ) = σ ( c ( x ) ) + σ ( x − c ( x ) ) {\displaystyle c(x)+(x-{c(x)})=x={\sigma (x)}={\sigma ({c(x)})}+{\sigma (x-{c(x)})}} Here σ ( c ( x ) ) = σ ( c ) ( x ) {\displaystyle \sigma (c(x))=\sigma (c)(x)} is a polynomial in x {\displaystyle x} , so is x − c ( x ) {\displaystyle x-c(x)} . Thus, σ ( c ( x ) ) {\displaystyle \sigma (c(x))} and σ ( x − c ( x ) ) {\displaystyle \sigma (x-c(x))} commute. Also, σ ( c ( x ) ) {\displaystyle \sigma (c(x))} is potentially diagonalisable and σ ( x − c ( x ) ) {\displaystyle \sigma ({x-c(x)})} is nilpotent. Thus, by the uniqueness of the Jordan–Chevalley decomposition (over L {\displaystyle L} ), σ ( c ( x ) ) = c ( x ) {\displaystyle \sigma (c(x))=c(x)} and σ ( c ( x ) ) = c ( x ) {\displaystyle \sigma (c(x))=c(x)} . Therefore, by definition, x s , x n {\displaystyle x_{s},x_{n}} are endomorphisms (represented by matrices) over F {\displaystyle F} . Finally, since { 1 , x , x 2 , … } {\displaystyle \left\{1,x,x^{2},\dots \right\}} contains an L {\displaystyle L} -basis that spans the space containing x s , x n {\displaystyle x_{s},x_{n}} , by the same argument, we also see that c {\displaystyle c} has coefficients in F {\displaystyle F} . Q.E.D. If the minimal polynomial of x {\displaystyle x} is a product of separable polynomials, then the field extension L / K {\displaystyle L/K} is Galois, meaning that F = K {\displaystyle F=K} . == Relations to the theory of algebras == === Separable algebras === The Jordan–Chevalley decomposition is very closely related to the Wedderburn principal theorem in the following formulation: Usually, the term „separable“ in this theorem refers to the general concept of a separable algebra and the theorem might then be established as a corollary of a more general high-powered result. However, if it is instead interpreted in the more basic sense that every element have a separable minimal polynomial, then this statement is essentially equivalent to the Jordan–Chevalley decomposition as described above. This gives a different way to view the decomposition, and for instance (Jacobson 1979) takes this route for establishing it. Over perfect fields, this result simplifies. Indeed, A / J {\displaystyle A/J} is then always separable in the sense of minimal polynomials: If a ∈ A {\displaystyle a\in A} , then the minimal polynomial p {\displaystyle p} is a product of separable polynomials, so there is a separable polynomial q {\displaystyle q} such that q ∣ p {\displaystyle q\mid p} and p ∣ q m {\displaystyle p\mid q^{m}} for some m ≥ 1 {\displaystyle m\geq 1} . Thus q ( a ) ∈ J {\displaystyle q(a)\in J} . So in A / J {\displaystyle A/J} , the minimal polynomial of a + J {\displaystyle a+J} divides q {\displaystyle q} and is hence separable. The crucial point in the theorem is then not that A / J {\displaystyle A/J} is separable (because that condition is vacuous), but that it is semisimple, meaning its radical is trivial. The same statement is true for Lie algebras, but only in characteristic zero. This is the content of Levi’s theorem. (Note that the notions of semisimple in both results do indeed correspond, because in both cases this is equivalent to being the sum of simple subalgebras or having trivial radical, at least in the finite-dimensional case.) === Preservation under representations === The crucial point in the proof for the Wedderburn principal theorem above is that an element x ∈ A {\displaystyle x\in A} corresponds to a linear operator T x : A → A {\displaystyle T_{x}:A\to A} with the same properties. In the theory of Lie algebras, this corresponds to the adjoint representation of a Lie algebra g {\displaystyle {\mathfrak {g}}} . This decomposed operator has a Jordan–Chevalley decomposition ad ( x ) = ad ( x ) s + ad ( x ) n {\displaystyle \operatorname {ad} (x)=\operatorname {ad} (x)_{s}+\operatorname {ad} (x)_{n}} . Just as in the associative case, this corresponds to a decomposition of x {\displaystyle x} , but polynomials are not available as a tool. One context in which this does makes sense is the restricted case where g {\displaystyle {\mathfrak {g}}} is contained in the Lie algebra g l ( V ) {\displaystyle {\mathfrak {gl}}(V)} of the endomorphisms of a finite-dimensional vector space V {\displaystyle V} over the perfect field K {\displaystyle K} . Indeed, any semisimple Lie algebra can be realised in this way. If x = x s + x n {\displaystyle x=x_{s}+x_{n}} is the Jordan decomposition, then ad ( x ) = ad ( x s ) + ad ( x n ) {\displaystyle \operatorname {ad} (x)=\operatorname {ad} (x_{s})+\operatorname {ad} (x_{n})} is the Jordan decomposition of the adjoint endomorphism ad ( x ) {\displaystyle \operatorname {ad} (x)} on the vector space g {\displaystyle {\mathfrak {g}}} . Indeed, first, ad ( x s ) {\displaystyle \operatorname {ad} (x_{s})} and ad ( x n ) {\displaystyle \operatorname {ad} (x_{n})} commute since [ ad ( x s ) , ad ( x n ) ] = ad ( [ x s , x n ] ) = 0 {\displaystyle [\operatorname {ad} (x_{s}),\operatorname {ad} (x_{n})]=\operatorname {ad} ([x_{s},x_{n}])=0} . Second, in general, for each endomorphism y ∈ g {\displaystyle y\in {\mathfrak {g}}} , we have: If y m = 0 {\displaystyle y^{m}=0} , then ad ( y ) 2 m − 1 = 0 {\displaystyle \operatorname {ad} (y)^{2m-1}=0} , since ad ( y ) {\displaystyle \operatorname {ad} (y)} is the difference of the left and right multiplications by y. If y {\displaystyle y} is semisimple, then ad ( y ) {\displaystyle \operatorname {ad} (y)} is semisimple, since semisimple is equivalent to potentially diagonalisable over a perfect field (if y {\displaystyle y} is diagonal over the basis { b 1 , … , b n } {\displaystyle \{b_{1},\dots ,b_{n}\}} , then ad ( y ) {\displaystyle \operatorname {ad} (y)} is diagonal over the basis consisting of the maps M i j {\displaystyle M_{ij}} with b i ↦ b j {\displaystyle b_{i}\mapsto b_{j}} and b k ↦ 0 {\displaystyle b_{k}\mapsto 0} for k ≠ 0 {\displaystyle k\neq 0} ). Hence, by uniqueness, ad ( x ) s = ad ( x s ) {\displaystyle \operatorname {ad} (x)_{s}=\operatorname {ad} (x_{s})} and ad ( x ) n = ad ( x n ) {\displaystyle \operatorname {ad} (x)_{n}=\operatorname {ad} (x_{n})} . The adjoint representation is a very natural and general representation of any Lie algebra. The argument above illustrates (and indeed proves) a general principle which generalises this: If π : g → g l ( V ) {\displaystyle \pi :{\mathfrak {g}}\to {\mathfrak {gl}}(V)} is any finite-dimensional representation of a semisimple finite-dimensional Lie algebra over a perfect field, then π {\displaystyle \pi } preserves the Jordan decomposition in the following sense: if x = x s + x n {\displaystyle x=x_{s}+x_{n}} , then π ( x s ) = π ( x ) s {\displaystyle \pi (x_{s})=\pi (x)_{s}} and π ( x n ) = π ( x ) n {\displaystyle \pi (x_{n})=\pi (x)_{n}} . === Nilpotency criterion === The Jordan decomposition can be used to characterize nilpotency of an endomorphism. Let k be an algebraically closed field of characteristic zero, E = End Q ( k ) {\displaystyle E=\operatorname {End} _{\mathbb {Q} }(k)} the endomorphism ring of k over rational numbers and V a finite-dimensional vector space over k. Given an endomorphism x : V → V {\displaystyle x:V\to V} , let x = s + n {\displaystyle x=s+n} be the Jordan decomposition. Then s {\displaystyle s} is diagonalizable; i.e., V = ⨁ V i {\textstyle V=\bigoplus V_{i}} where each V i {\displaystyle V_{i}} is the eigenspace for eigenvalue λ i {\displaystyle \lambda _{i}} with multiplicity m i {\displaystyle m_{i}} . Then for any φ ∈ E {\displaystyle \varphi \in E} let φ ( s ) : V → V {\displaystyle \varphi (s):V\to V} be the endomorphism such that φ ( s ) : V i → V i {\displaystyle \varphi (s):V_{i}\to V_{i}} is the multiplication by φ ( λ i ) {\displaystyle \varphi (\lambda _{i})} . Chevalley calls φ ( s ) {\displaystyle \varphi (s)} the replica of s {\displaystyle s} given by φ {\displaystyle \varphi } . (For example, if k = C {\displaystyle k=\mathbb {C} } , then the complex conjugate of an endomorphism is an example of a replica.) Now, Proof: First, since n φ ( s ) {\displaystyle n\varphi (s)} is nilpotent, 0 = tr ( x φ ( s ) ) = ∑ i tr ( s φ ( s ) | V i ) = ∑ i m i λ i φ ( λ i ) {\displaystyle 0=\operatorname {tr} (x\varphi (s))=\sum _{i}\operatorname {tr} \left(s\varphi (s)|V_{i}\right)=\sum _{i}m_{i}\lambda _{i}\varphi (\lambda _{i})} . If φ {\displaystyle \varphi } is the complex conjugation, this implies λ i = 0 {\displaystyle \lambda _{i}=0} for every i. Otherwise, take φ {\displaystyle \varphi } to be a Q {\displaystyle \mathbb {Q} } -linear functional φ : k → Q {\displaystyle \varphi :k\to \mathbb {Q} } followed by Q ↪ k {\displaystyle \mathbb {Q} \hookrightarrow k} . Applying that to the above equation, one gets: ∑ i m i φ ( λ i ) 2 = 0 {\displaystyle \sum _{i}m_{i}\varphi (\lambda _{i})^{2}=0} and, since φ ( λ i ) {\displaystyle \varphi (\lambda _{i})} are all real numbers, φ ( λ i ) = 0 {\displaystyle \varphi (\lambda _{i})=0} for every i. Varying the linear functionals then implies λ i = 0 {\displaystyle \lambda _{i}=0} for every i. ◻ {\displaystyle \square } A typical application of the above criterion is the proof of Cartan's criterion for solvability of a Lie algebra. It says: if g ⊂ g l ( V ) {\displaystyle {\mathfrak {g}}\subset {\mathfrak {gl}}(V)} is a Lie subalgebra over a field k of characteristic zero such that tr ( x y ) = 0 {\displaystyle \operatorname {tr} (xy)=0} for each x ∈ g , y ∈ D g = [ g , g ] {\displaystyle x\in {\mathfrak {g}},y\in D{\mathfrak {g}}=[{\mathfrak {g}},{\mathfrak {g}}]} , then g {\displaystyle {\mathfrak {g}}} is solvable. Proof: Without loss of generality, assume k is algebraically closed. By Lie's theorem and Engel's theorem, it suffices to show for each x ∈ D g {\displaystyle x\in D{\mathfrak {g}}} , x {\displaystyle x} is a nilpotent endomorphism of V. Write x = ∑ i [ x i , y i ] {\textstyle x=\sum _{i}[x_{i},y_{i}]} . Then we need to show: tr ( x φ ( s ) ) = ∑ i tr ( [ x i , y i ] φ ( s ) ) = ∑ i tr ( x i [ y i , φ ( s ) ] ) {\displaystyle \operatorname {tr} (x\varphi (s))=\sum _{i}\operatorname {tr} ([x_{i},y_{i}]\varphi (s))=\sum _{i}\operatorname {tr} (x_{i}[y_{i},\varphi (s)])} is zero. Let g ′ = g l ( V ) {\displaystyle {\mathfrak {g}}'={\mathfrak {gl}}(V)} . Note we have: ad g ′ ( x ) : g → D g {\displaystyle \operatorname {ad} _{{\mathfrak {g}}'}(x):{\mathfrak {g}}\to D{\mathfrak {g}}} and, since ad g ′ ( s ) {\displaystyle \operatorname {ad} _{{\mathfrak {g}}'}(s)} is the semisimple part of the Jordan decomposition of ad g ′ ( x ) {\displaystyle \operatorname {ad} _{{\mathfrak {g}}'}(x)} , it follows that ad g ′ ( s ) {\displaystyle \operatorname {ad} _{{\mathfrak {g}}'}(s)} is a polynomial without constant term in ad g ′ ( x ) {\displaystyle \operatorname {ad} _{{\mathfrak {g}}'}(x)} ; hence, ad g ′ ( s ) : g → D g {\displaystyle \operatorname {ad} _{{\mathfrak {g}}'}(s):{\mathfrak {g}}\to D{\mathfrak {g}}} and the same is true with φ ( s ) {\displaystyle \varphi (s)} in place of s {\displaystyle s} . That is, [ φ ( s ) , g ] ⊂ D g {\displaystyle [\varphi (s),{\mathfrak {g}}]\subset D{\mathfrak {g}}} , which implies the claim given the assumption. ◻ {\displaystyle \square } ==== Real semisimple Lie algebras ==== In the formulation of Chevalley and Mostow, the additive decomposition states that an element X in a real semisimple Lie algebra g with Iwasawa decomposition g = k ⊕ a ⊕ n can be written as the sum of three commuting elements of the Lie algebra X = S + D + N, with S, D and N conjugate to elements in k, a and n respectively. In general the terms in the Iwasawa decomposition do not commute. == Multiplicative decomposition == If x {\displaystyle x} is an invertible linear operator, it may be more convenient to use a multiplicative Jordan–Chevalley decomposition. This expresses x {\displaystyle x} as a product x = x s ⋅ x u {\displaystyle x=x_{s}\cdot x_{u}} , where x s {\displaystyle x_{s}} is potentially diagonalisable, and x u − 1 {\displaystyle x_{u}-1} is nilpotent (one also says that x u {\displaystyle x_{u}} is unipotent). The multiplicative version of the decomposition follows from the additive one since, as x s {\displaystyle x_{s}} is invertible (because the sum of an invertible operator and a nilpotent operator is invertible) x = x s + x n = x s ( 1 + x s − 1 x n ) {\displaystyle x=x_{s}+x_{n}=x_{s}\left(1+x_{s}^{-1}x_{n}\right)} and 1 + x s − 1 x n {\displaystyle 1+x_{s}^{-1}x_{n}} is unipotent. (Conversely, by the same type of argument, one can deduce the additive version from the multiplicative one.) The multiplicative version is closely related to decompositions encountered in a linear algebraic group. For this it is again useful to assume that the underlying field K {\displaystyle K} is perfect because then the Jordan–Chevalley decomposition exists for all matrices. === Linear algebraic groups === Let G {\displaystyle G} be a linear algebraic group over a perfect field. Then, essentially by definition, there is a closed embedding G ↪ G L n {\displaystyle G\hookrightarrow \mathbf {GL} _{n}} . Now, to each element g ∈ G {\displaystyle g\in G} , by the multiplicative Jordan decomposition, there are a pair of a semisimple element g s {\displaystyle g_{s}} and a unipotent element g u {\displaystyle g_{u}} a priori in G L n {\displaystyle \mathbf {GL} _{n}} such that g = g s g u = g u g s {\displaystyle g=g_{s}g_{u}=g_{u}g_{s}} . But, as it turns out, the elements g s , g u {\displaystyle g_{s},g_{u}} can be shown to be in G {\displaystyle G} (i.e., they satisfy the defining equations of G) and that they are independent of the embedding into G L n {\displaystyle \mathbf {GL} _{n}} ; i.e., the decomposition is intrinsic. When G is abelian, G {\displaystyle G} is then the direct product of the closed subgroup of the semisimple elements in G and that of unipotent elements. ==== Real semisimple Lie groups ==== The multiplicative decomposition states that if g is an element of the corresponding connected semisimple Lie group G with corresponding Iwasawa decomposition G = KAN, then g can be written as the product of three commuting elements g = sdu with s, d and u conjugate to elements of K, A and N respectively. In general the terms in the Iwasawa decomposition g = kan do not commute. == References == Chevalley, Claude (1951), Théorie des groupes de Lie. Tome II. Groupes algébriques, Hermann, OCLC 277477632 Couty, Danielle; Esterle, Jean; Zarouf, Rachid (2010), Décomposition effective de Jordan-Chevalley et ses retombées en enseignement. (PDF) (preprint) Couty, Danielle; Esterle, Jean; Zarouf, Rachid (16 June 2011), "Décomposition effective de Jordan-Chevalley et ses retombées en enseignement." (PDF), Gazette des Mathématiciens, no. 129, pp. 29–49 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. Geck, Meinolf (18 Jun 2022), On the Jordan-Chevalley decomposition of a matrix, arXiv:2205.05432 Helgason, Sigurdur (1978), Differential geometry, Lie groups, and symmetric spaces, Academic Press, ISBN 0-8218-2848-7 Humphreys, James E. (1981), Linear Algebraic Groups, Graduate texts in mathematics, vol. 21, Springer, ISBN 0-387-90108-6 Humphreys, James E. (1972), Introduction to Lie Algebras and Representation Theory, Springer, ISBN 978-0-387-90053-7 Jacobson, Nathan (1979) [1962], Lie algebras, Dover, ISBN 0-486-63832-4 Lazard, M. (1954), "Théorie des répliques. Critère de Cartan (Exposé No. 6)", Séminaire "Sophus Lie", 1, archived from the original on 2013-07-04 Mostow, G. D. (1954), "Factor spaces of solvable groups", Ann. of Math., 60 (1): 1–27, doi:10.2307/1969700, JSTOR 1969700 Mostow, G. D. (1973), Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies, vol. 78, Princeton University Press, ISBN 0-691-08136-0 Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001 Serre, Jean-Pierre (1992), Lie algebras and Lie groups: 1964 lectures given at Harvard University, Lecture Notes in Mathematics, vol. 1500 (2nd ed.), Springer-Verlag, ISBN 978-3-540-55008-2 Varadarajan, V. S. (1984), Lie groups, Lie algebras, and their representations, Graduate Texts in Mathematics, vol. 102, Springer-Verlag, ISBN 0-387-90969-9 Waterhouse, William (1979), Introduction to affine group schemes, Graduate Texts in Mathematics, vol. 66, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-6217-6, ISBN 978-0-387-90421-4, MR 0547117
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Wikipedia:Jordan–Pólya number#0
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In mathematics, the Jordan–Pólya numbers are the numbers that can be obtained by multiplying together one or more factorials, not required to be distinct from each other. For instance, 480 {\displaystyle 480} is a Jordan–Pólya number because 480 = 2 ! ⋅ 2 ! ⋅ 5 ! {\displaystyle 480=2!\cdot 2!\cdot 5!} . Every tree has a number of symmetries that is a Jordan–Pólya number, and every Jordan–Pólya number arises in this way as the order of an automorphism group of a tree. These numbers are named after Camille Jordan and George Pólya, who both wrote about them in the context of symmetries of trees. These numbers grow more quickly than polynomials but more slowly than exponentials. As well as in the symmetries of trees, they arise as the numbers of transitive orientations of comparability graphs and in the problem of finding factorials that can be represented as products of smaller factorials. == Sequence and growth rate == The sequence of Jordan–Pólya numbers begins: They form the smallest multiplicatively closed set containing all of the factorials. The n {\displaystyle n} th Jordan–Pólya number grows more quickly than any polynomial of n {\displaystyle n} , but more slowly than any exponential function of n {\displaystyle n} . More precisely, for every ε > 0 {\displaystyle \varepsilon >0} , and every sufficiently large x {\displaystyle x} (depending on ε {\displaystyle \varepsilon } ), the number J ( x ) {\displaystyle J(x)} of Jordan–Pólya numbers up to x {\displaystyle x} obeys the inequalities exp ( 2 − ε ) log x log log x < J ( x ) < exp ( 4 + ε ) log x log log log x log log x . {\displaystyle \exp {\frac {(2-\varepsilon ){\sqrt {\log x}}}{\log \log x}}<J(x)<\exp {\frac {(4+\varepsilon ){\sqrt {\log x}}\log \log \log x}{\log \log x}}.} == Factorials that are products of smaller factorials == Every Jordan–Pólya number n {\displaystyle n} , except 2, has the property that its factorial n ! {\displaystyle n!} can be written as a product of smaller factorials. This can be done simply by expanding n ! = n ⋅ ( n − 1 ) ! {\displaystyle n!=n\cdot (n-1)!} and then replacing n {\displaystyle n} in this product by its representation as a product of factorials. It is conjectured, but unproven, that the only numbers n {\displaystyle n} whose factorial n ! {\displaystyle n!} equals a product of smaller factorials are the Jordan–Pólya numbers (except 2) and the two exceptional numbers 9 and 10, for which 9 ! = 2 ! ⋅ 3 ! ⋅ 3 ! ⋅ 7 ! {\displaystyle 9!=2!\cdot 3!\cdot 3!\cdot 7!} and 10 ! = 6 ! ⋅ 7 ! = 3 ! ⋅ 5 ! ⋅ 7 ! {\displaystyle 10!=6!\cdot 7!=3!\cdot 5!\cdot 7!} . The only other known representation of a factorial as a product of smaller factorials, not obtained by replacing n {\displaystyle n} in the product expansion of n ! {\displaystyle n!} , is 16 ! = 2 ! ⋅ 5 ! ⋅ 14 ! {\displaystyle 16!=2!\cdot 5!\cdot 14!} , but as 16 {\displaystyle 16} is itself a Jordan–Pólya number, it also has the representation 16 ! = 2 ! 4 ⋅ 15 ! {\displaystyle 16!=2!^{4}\cdot 15!} . == See also == Superfactorial, the product of the first n {\displaystyle n} factorials == References ==
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Wikipedia:Jorma Rissanen#0
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Jorma Johannes Rissanen (October 20, 1932 – May 9, 2020) was an information theorist, known for originating the minimum description length (MDL) principle and practical approaches to arithmetic coding for lossless data compression. His work inspired the development of the theory of stochastic chains with memory of variable length. == Education and career == Rissanen was born in Pielisjärvi (now Lieksa) in Finland and grew up in Kemi, a border town between Finland and Sweden. He moved to Helsinki and studied at the Helsinki University of Technology, where he obtained his Master’s degree in electrical engineering in 1956 and licentiate in control theory in 1960. He studied there under Olli Lokki and Hans Blomberg. Rissanen became an IBM researcher since 1960, first in Stockholm, Sweden, while still a Ph.D. student under Hans Blomberg. Most of his PhD work was done remotely as a result and he received his Ph.D. from the Helsinki University of Technology in 1965 with a topic on adaptive control theory. He then moved to IBM Almaden in San Jose, California and stayed with IBM until his retirement in 2002, with a brief interruption in 1974 as a professor of control theory at Linköping University in Sweden. During that time, he became familiar with the work on algorithmic randomness by Andrey Kolmogorov and Per Martin-Löf, which inspired his work on arithmetic coding and MDL, leading to a stream of ground-breaking publications from the late 1970s to the early 1990s. The work on MDL developed into the more general notions of stochastic complexity (about which he wrote an influential book) and universal coding/modeling. After retirement from IBM, he remained professor emeritus of Tampere University of Technology and a fellow of Helsinki Institute for Information Technology. == Awards and recognitions == He was awarded the IEEE Richard W. Hamming Medal in 1993, an IEEE Golden Jubilee Award for Technological Innovation from the IEEE Information Theory Society in 1998, the Kolmogorov Medal of the University of London in 2006, and the IEEE Claude E. Shannon Award in 2009. A Festschrift collection, which includes an interview and substantial biographical information, was published by the Tampere University of Technology in honor of his 75th birthday. == Personal life == Rissanen married Riitta Aberg in 1956, and they have a son Juhani and a daughter Natasha. == Bibliography == Rissanen, Jorma (1989). Stochastic complexity in statistical inquiry. Singapore: World Scientific. ISBN 981-238-549-5. OCLC 52854741. Cybenko, George; O'Leary, Dianne P.; Rissanen, Jorma, eds. (1999). The mathematics of information coding, extraction, and distribution. New York: Springer. ISBN 0-387-98665-0. OCLC 39922074. Rissanen, Jorma (2007). Information and complexity in statistical modeling. New York: Springer. ISBN 978-0-387-68812-1. OCLC 232363255. Rissanen, Jorma (2012). Optimal estimation of parameters. Cambridge: Cambridge University Press. ISBN 978-1-139-51850-5. OCLC 796796345. == References == == External links == Professor Jorma Rissanen page at RHUL Grünwald, Peter; Myllymäki, Petri; Roos, Teemu; Tabus, Ioan (September 2020). "In Memoriam: Jorma J. Rissanen (1932-2020)" (PDF). IEEE Information Theory Society Newsletter. pp. 13–14. Retrieved October 3, 2022.
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