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Wikipedia:Richard P. Brent#0	Richard Peirce Brent is an Australian mathematician and computer scientist. He is an emeritus professor at the Australian National University. From March 2005 to March 2010 he was a Federation Fellow at the Australian National University. His research interests include number theory (in particular factorisation), random number generators, computer architecture, and analysis of algorithms. In 1973, he published a root-finding algorithm (an algorithm for solving equations numerically) which is now known as Brent's method. In 1975 he and Eugene Salamin independently conceived the Salamin–Brent algorithm, used in high-precision calculation of π {\displaystyle \pi } . At the same time, he showed that all the elementary functions (such as log(x), sin(x) etc.) can be evaluated to high precision in the same time as π {\displaystyle \pi } (apart from a small constant factor) using the arithmetic-geometric mean of Carl Friedrich Gauss. In 1979 he showed that the first 75 million complex zeros of the Riemann zeta function lie on the critical line, providing some experimental evidence for the Riemann hypothesis. In 1980 he and Nobel laureate Edwin McMillan found a new algorithm for high-precision computation of the Euler–Mascheroni constant γ {\displaystyle \gamma } using Bessel functions, and showed that γ {\displaystyle \gamma } can not have a simple rational form p/q (where p and q are integers) unless q is extremely large (greater than 1015000). In 1980 he and John Pollard factored the eighth Fermat number using a variant of the Pollard rho algorithm. He later factored the tenth and eleventh Fermat numbers using Lenstra's elliptic curve factorisation algorithm. In 2002, Brent, Samuli Larvala and Paul Zimmermann discovered a very large primitive trinomial over GF(2): x 6972593 + x 3037958 + 1. {\displaystyle x^{6972593}+x^{3037958}+1.} The degree 6972593 is the exponent of a Mersenne prime. In 2009 and 2016, Brent and Paul Zimmermann discovered some even larger primitive trinomials, for example: x 43112609 + x 3569337 + 1. {\displaystyle x^{43112609}+x^{3569337}+1.} The degree 43112609 is again the exponent of a Mersenne prime. The highest degree trinomials found were three trinomials of degree 74,207,281, also a Mersenne prime exponent. In 2011, Brent and Paul Zimmermann published Modern Computer Arithmetic (Cambridge University Press), a book about algorithms for performing arithmetic, and their implementation on modern computers. Brent is a Fellow of the Association for Computing Machinery, the IEEE, SIAM and the Australian Academy of Science. In 2005, he was awarded the Hannan Medal by the Australian Academy of Science. In 2014, he was awarded the Moyal Medal by Macquarie University. == See also == Brent–Kung adder == References == == External links == Richard Brent's home page Richard P. Brent at the Mathematics Genealogy Project
Wikipedia:Richard von Mises#0	Richard Martin Edler von Mises (German: [fɔn ˈmiːzəs]; 19 April 1883 – 14 July 1953) was an Austrian scientist and mathematician who worked on solid mechanics, fluid mechanics, aerodynamics, aeronautics, statistics and probability theory. He held the position of Gordon McKay Professor of Aerodynamics and Applied Mathematics at Harvard University. He described his work in his own words shortly before his death as: practical analysis, integral and differential equations, mechanics, hydrodynamics and aerodynamics, constructive geometry, probability calculus, statistics and philosophy. Although best known for his mathematical work, von Mises also contributed to the philosophy of science as a neo-positivist and empiricist, following the line of Ernst Mach. Historians of the Vienna Circle of logical empiricism recognize a "first phase" from 1907 through 1914 with Philipp Frank, Hans Hahn, and Otto Neurath. His older brother, Ludwig von Mises, held an opposite point of view with respect to positivism and epistemology. His brother developed praxeology, an a priori view. During his time in Istanbul, Mises maintained close contact with Philipp Frank, a logical positivist and Professor of Physics in Prague until 1938. His literary interests included the Austrian novelist Robert Musil and the poet Rainer Maria Rilke, on whom he became a recognized expert. == Life == Von Mises was born in Lemberg, Austria-Hungary into a Jewish family, eighteen months after his brother Ludwig von Mises, who became a prominent economist of the Austrian School, a heterodox school of economics. His parents were Arthur Edler von Mises, a doctor of technical sciences who worked for the Austrian State Railways, and Adele Landau. Richard and Ludwig had a younger brother, Karl von Mises, who died as an infant from Scarlet Fever. Richard attended the Akademisches Gymnasium in Vienna, from which he graduated with honors in Latin and mathematics in autumn 1901. After graduating in mathematics, physics and engineering from the Vienna University of Technology, he was appointed as Georg Hamel's assistant in Brünn (Brno). In 1905, still a student, he published an article on the geometry of curves called "Zur konstruktiven Infinitesimalgeometrie der ebenen Kurven," in the prestigious Zeitschrift für Mathematik und Physik. In 1908, von Mises was awarded a doctorate from Vienna (his dissertation was on "the determination of flywheel masses in crank drives") and he received his habilitation from Brünn (Brno) (on "Theory of the Waterwheels") to lecture on engineering. In 1909, at 26, he was appointed professor of applied mathematics in Straßburg, then part of the German Empire (later Strasbourg, Alsace, France) and received Prussian citizenship. His application for a teaching position at the Brno University of Technology was interrupted by the First World War. Before the war he had already become a pilot and lectured on aircraft design, and in 1913 at Strasbourg he gave the first university course on powered flight. At the outbreak of war it was natural for him to join the Austro-Hungarian army as a test pilot and a flying instructor. In 1915, he supervised the construction of a 600-horsepower (450 kW) aircraft – the "Mises-Flugzeug" (Mises aircraft) for the Austrian army. It was completed in 1916 but never saw active service. After the war, von Mises held the new chair of hydrodynamics and aerodynamics at the Dresden Technische Hochschule. In 1919 he was appointed director and full professor at the new Institute of Applied Mathematics created at the behest of Erhard Schmidt at the University of Berlin. In 1921 he founded the journal Zeitschrift für Angewandte Mathematik und Mechanik and became its editor. With the rise of the National Socialist Party to power in 1933, Mises felt his position threatened, despite his First World War military service. He moved to Turkey, where he assumed the new chair of pure and applied mathematics at the University of Istanbul. In 1939 he accepted a position in the United States, where in 1944 he was appointed as Gordon McKay Professor of Aerodynamics and Applied Mathematics at Harvard University. In 1943 he married Hilda Geiringer, a mathematician who had been his assistant at the Institute and moved with him to Turkey and then to the U.S. In 1950, von Mises declined honorary membership from the Communist-dominated East German Academy of Science. == Contributions == In aerodynamics, von Mises made advances in boundary-layer flow theory and airfoil design. He developed the distortion energy theory of stress, an important factor in material strength calculations. His ideas were not universally well received, although Alexander Ostrowski had said of him: "Only with the appointment of Richard von Mises to the University of Berlin did the first serious German school of applied mathematics with a broad sphere of influence come into existence. Von Mises was an incredibly dynamic person and at the same time amazingly versatile like Runge. He was especially well versed in the realm of technology." and also wrote "Because of his dynamic personality his occasional major blunders were somehow tolerated. One has even forgiven him his theory of probability." Yet Andrey Kolmogorov, whose rival axiomatisation was better received, was less severe: "The basis for the applicability of the results of the mathematical theory of probability to real 'random phenomena' must depend on some form of the frequency concept of probability, the unavoidable nature of which has been established by von Mises in a spirited manner." In probability theory, he proposed the famous "birthday problem". He also defined the impossibility of a gambling system. In solid mechanics, von Mises contributed to the theory of plasticity by formulating the von Mises yield criterion, independently of Tytus Maksymilian Huber. He is often credited for the Principle of Maximum Plastic Dissipation. The Gesellschaft für Angewandte Mathematik und Mechanik (Society of Applied Mathematics and Mechanics) awards a Richard von Mises Prize since 1989. == Bibliography == === Books === === Articles === R. v. Mises, "Zur konstruktiven Infinitesimalgeometrie der ebenen Kurven," Zeitschrift für Mathematik und Physik, 52, 1905, pp. 44–85. R. v. Mises, "Zur Theorie der Regulatoren", Elektrotechnik und Maschinenbau 37, 1908, pp. 783–789. == See also == Birthday problem Impossibility of a gambling system Bernstein–von Mises theorem Cramér–von Mises criterion von Mises distribution == Notes == == References == Biography in Dictionary of Scientific Biography, New York, 1970–1990. Biography in Encyclopædia Britannica. === Further reading === A. Basch, "Richard von Mises zum 70. Geburtstag", Osterreich. Ing.-Arch. 7, 1953, pp. 73–76. B. Bernhardt, "Skizzen zu Leben und Werk von Richard Mises", in Österreichische Mathematik und Physik, Wien, Zentralbibliothek für Physik, 1993, pp. 51–62. H. Bernhardt, "Zum Leben und Wirken des Mathematikers Richard von Mises", NTM Schr. Geschichte Natur. Tech. Medizin 16 (2), 1979, pp. 40–49. G. Birkhoff, "Richard von Mises' years at Harvard", Zeitschrift für Angewandte Mathematik und Mechanik 63 (7), 1983, pp. 283–284. L. Collatz, "Richard von Mises als numerischer Mathematiker", Zeitschrift für Angewandte Mathematik und Mechanik (7), 1983, pp. 278–280. H. Cramér, "Richard von Mises' work in probability and statistics", Ann. Math. Statistics 24, 1953, pp. 657–662. D. v. Dalen, "The War of the Frogs and the Mice or the Crisis of the 'Mathematische Annalen'", The Mathematical Intelligencer 12 (1990), No.4, pp. 17–31. Gaye Erginoz, "An Emigrant Scientist in Istanbul University: Richard Martin Edler von Mises (1883-1953)",2011, Almagest H. Föllmer and K. Küchler, "Richard von Mises", in Mathematics in Berlin, Berlin, 1998, pp. 55–60. J. Förste, "Zum 100. Geburtstag von Richard von Mises", Zeitschrift für Angewandte Mathematik und Mechanik 63 (7), 1983, p. 277. P. Frank, "The work of Richard von Mises: 1883–1953", Science 119, 1954, pp. 823–824. A. Haussner, "Geschichte der Deutschen Technischen Hochschule in Brünn 1849–1924." In Festschrift der Deutschen Technischen Hochschule in Brünn zur Feier ihres fünfundsiebzigjährigen Bestandes im Mai 1924, Verlag der Deutschen Technischen Hochschule, Brünn, 1924, pp. 5–92. G. S. S. Ludford, "Mechanics in the applied- mathematical world of von Mises", Zeitschrift für Angewandte Mathematik und Mechanik 63 (7), 1983, pp. 281–282. R. Sauer, "Nachruf: Richard von Mises", Bayer. Akad. Wiss. Jbuch. 1953, pp. 194–197. R. Sauer, "Richard von Mises 19. 4. 1883 – 14. 7. 1953" (in German), Bayer. Akad. Wiss. Jbuch. 1953, pp. 194–197 M. Schield and T. Burnham. "Von Mises’ Frequentist Approach to Probability." 2008 American Statistical Association Proceedings of the Section on Statistical Education. pp. 2187-2194. See www.statlit.org/pdf/2008SchieldBurnhamASA.pdf R. Siegmund-Schultze, "Hilda Geiringer von Mises, Charlier Series, Ideology, and the human side of the emancipation of applied mathematics at the University of Berlin during the 1920s", Historia Mathematica 20, 1993, 364–381. P. Sisma, "Georg Hamel and Richard von Mises in Brno", Historia Mathematica, 29, 2002, pp. 176–192. A. Szafarz, "Richard von Mises: l'échec d'une axiomatique", Dialectica 38 (4), 1984, pp. 311–317. M. van Lambalgen, "Randomness and foundations of probability: von Mises' axiomatisation of random sequences", in Statistics, probability and game theory, Hayward, CA, 1996, pp. 347–367. J. Weinhold, "Zur Geschichte der Deutschen Technischen Hochschule in Brünn, Rückblicke und Vergleiche", Südetendeutsche Akademie der Wissenschaften und Künste, Naturwissenschaftliche Klasse, 1991. == External links == O'Connor, John J.; Robertson, Edmund F., "Richard von Mises", MacTutor History of Mathematics Archive, University of St Andrews Richard von Mises at the Mathematics Genealogy Project Biography, in Czech
Wikipedia:Richardson Professor of Applied Mathematics#0	The Richardson Chair of Applied Mathematics is an endowed professorial position in the School of Mathematics, University of Manchester, England. The chair was founded by an endowment of £3,600 from one John Richardson, in 1890. The endowment was originally used to support the Richardson Lectureship in Mathematics. One holder of the Richardson Lectureship was John Edensor Littlewood (1907–1910). The position lapsed in 1918, but was resurrected as a lectureship in Pure Mathematics between 1935 and 1944. There was then a further hiatus until the establishment of the Richardson Chair of Applied Mathematics in 1998. The current holder (since 1998) is Nicholas Higham. A complete list of Richardson Lecturers and Professors is as follows: F. T. Swanwick (1891–1907) Lecturer in Mathematics J. E. Littlewood (1907–1910) Lecturer in Mathematics H. R. Hasse (1910–1912) Lecturer in Mathematics W. D. Evans (1912–1918) Lecturer in Mathematics W. N. Bailey (1935–1944) Lecturer in Pure Mathematics N. J. Higham (1998– ) Professor of Applied Mathematics The School of Mathematics has three other endowed chairs, the others being the Beyer Chair, the Fielden Chair of Pure Mathematics and the Sir Horace Lamb Chair. == References ==
Wikipedia:Richardson's theorem#0	In mathematics, Richardson's theorem establishes the undecidability of the equality of real numbers defined by expressions involving integers, π, and exponential and sine functions. It was proved in 1968 by the mathematician and computer scientist Daniel Richardson of the University of Bath. Specifically, the class of expressions for which the theorem holds is that generated by rational numbers, the number π, the number ln 2, the variable x, the operations of addition, subtraction, multiplication, composition, and the sin, exp, and abs functions. For some classes of expressions generated by other primitives than in Richardson's theorem, there exist algorithms that can determine whether an expression is zero. == Statement of the theorem == Richardson's theorem can be stated as follows: Let E be a set of expressions that represent R → R {\displaystyle \mathbb {R} \to \mathbb {R} } functions. Suppose that E includes these expressions: x (representing the identity function) ex (representing the exponential functions) sin x (representing the sin function) all rational numbers, ln 2, and π (representing constant functions that ignore their input and produce the given number as output) Suppose E is also closed under a few standard operations. Specifically, suppose that if A and B are in E, then all of the following are also in E: A + B (representing the pointwise addition of the functions that A and B represent) A − B (representing pointwise subtraction) AB (representing pointwise multiplication) A∘B (representing the composition of the functions represented by A and B) Then the following decision problems are unsolvable: Deciding whether an expression A in E represents a function that is nonnegative everywhere If E includes also the expression \|x\| (representing the absolute value function), deciding whether an expression A in E represents a function that is zero everywhere If E includes an expression B representing a function whose antiderivative has no representative in E, deciding whether an expression A in E represents a function whose antiderivative can be represented in E. (Example: e a x 2 {\displaystyle e^{ax^{2}}} has an antiderivative in the elementary functions if and only if a = 0.) == Extensions == After Hilbert's tenth problem was solved in 1970, B. F. Caviness observed that the use of ex and ln 2 could be removed. Wang later noted that under the same assumptions under which the question of whether there was x with A(x) < 0 was insolvable, the question of whether there was x with A(x) = 0 was also insolvable. Miklós Laczkovich removed also the need for π and reduced the use of composition. In particular, given an expression A(x) in the ring generated by the integers, x, sin xn, and sin(x sin xn) (for n ranging over positive integers), both the question of whether A(x) > 0 for some x and whether A(x) = 0 for some x are unsolvable. By contrast, the Tarski–Seidenberg theorem says that the first-order theory of the real field is decidable, so it is not possible to remove the sine function entirely. == See also == Constant problem – Problem of deciding whether an expression equals zero Elementary function – A kind of mathematical function Tarski's high school algebra problem – Mathematical problem == References == == Further reading == Petkovšek, Marko; Wilf, Herbert S.; Zeilberger, Doron (1996). A = B. A. K. Peters. p. 212. ISBN 1-56881-063-6. Archived from the original on 2006-01-29. == External links == Weisstein, Eric W. "Richardson's theorem". MathWorld.
Wikipedia:Rick Durrett#0	Richard Timothy Durrett is an American mathematician known for his research and books on mathematical probability theory, stochastic processes and their application to mathematical ecology and population genetics. == Education and career == He received his BS and MS at Emory University in 1972 and 1973 and his Ph.D. at Stanford University in 1976 under advisor Donald Iglehart. From 1976 to 1985 he taught at UCLA. From 1985 until 2010 was on the faculty at Cornell University, where his students included Claudia Neuhauser. Since 2010, Durrett has been a professor at Duke University. He was elected to the United States National Academy of Sciences in 2007. In 2012 he became a fellow of the American Mathematical Society. Durrett is the founder of the Cornell Probability Summer Schools. == Selected publications == === Books === Durrett, R. Probability. Theory and examples. Wadsworth & Brooks/Cole, Pacific Grove, CA (1991). 453 pp. ISBN 0-534-13206-5 ; 4th edition, 2010 Durrett, R. Probability models for DNA sequence evolution. Springer-Verlag, New York (2002). 240 pp. ISBN 0-387-95435-X ; 2nd edition, 2008 Durrett, R. Stochastic Calculus: A Practical Introduction. CRC Press (1996). 341 pp. ISBN 0-8493-8071-5 Durrett, R. Random Graph Dynamics. Cambridge University Press (2006). 222 pp. ISBN 0-521-86656-1 === Papers === Durrett, R. (1988). "Crabgrass, measles and gypsy moths: An introduction to modern probability" (PDF). Bulletin of the American Mathematical Society. 18 (2): 117–144. doi:10.1090/S0273-0979-1988-15625-X. ISSN 0273-0979. Durrett, R.; Levin, S. (1994). "The Importance of Being Discrete (and Spatial)". Theoretical Population Biology. 46 (3): 363–394. Bibcode:1994TPBio..46..363D. doi:10.1006/tpbi.1994.1032. ISSN 0040-5809. (This article has over 1100 citations.) == References == == External links == Personal Home Page at Duke University. Cornell Probability Summer Schools.
Wikipedia:Rida Laraki#0	Rida Laraki is a Moroccan researcher, professor, and engineer in the fields of game theory, social choice, theoretical economics, optimization, learning, and operations research at the French National Centre for Scientific Research. == Life == Born in 1974, Rida Laraki studied in Morocco and passed his baccalaureate in 1992. After attending preparatory classes at the Mohammed V high school, he joined the École Polytechnique in Paris (X93). He also represented Morocco at the International Mathematics Olympiads in Moscow in 1992 and in Istanbul in 1993. He obtained his engineering degree from Polytechnique in 1996. Four years later, in 2000, he obtained a doctorate in mathematics from the Pierre and Marie Curie University. He joined the CNRS in 2001 and was a lecturer at Polytechnique for around ten years. He took up the position of lecturer at the École Polytechnique in 2006. Since 2013, he has been director of computer science research at the Laboratory for Analysis and Modeling of Systems for Decision Support (LAMSADE) of the CNRS, and honorary professor at the University of Liverpool in 2017. He is best known for having designed a collective decision method, called majority judgment, in 2007, with another CNRS researcher, Michel Balinski. In 2011, he and Balinski published a book with MIT Press presenting this new voting method. He also wrote a book on game theory for Springer Editions in 2019. == Majority judgment == The majority judgment developed by Rida Laraki and Michel Balinski is a voting method based on voting by values, or mention (very good, fair, to be rejected...) ultimately obtaining a "majority grade". It is distinguished by determining the winner by the median rather than the average. It can be applied to political votes but also, for example, to wine rankings. It allows voters to express themselves on all choices. == Bibliography == Majority Judgment: Measuring, Ranking, and Electing == References ==
Wikipedia:Ridge function#0	In mathematics, a ridge function is any function f : R d → R {\displaystyle f:\mathbb {R} ^{d}\rightarrow \mathbb {R} } that can be written as the composition of an univariate function g : R → R {\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} } , that is called a profile function, with an affine transformation, given by a direction vector a ∈ R d {\displaystyle a\in \mathbb {R} ^{d}} with shift b ∈ R {\displaystyle b\in \mathbb {R} } . Then, the ridge function reads f ( x ) = g ( x ⊤ a + b ) {\displaystyle f(x)=g(x^{\top }a+b)} for x ∈ R d {\displaystyle x\in \mathbb {R} ^{d}} . Coinage of the term 'ridge function' is often attributed to B.F. Logan and L.A. Shepp. == Relevance == A ridge function is not susceptible to the curse of dimensionality, making it an instrumental tool in various estimation problems. This is a direct result of the fact that ridge functions are constant in d − 1 {\displaystyle d-1} directions: Let a 1 , … , a d − 1 {\displaystyle a_{1},\dots ,a_{d-1}} be d − 1 {\displaystyle d-1} independent vectors that are orthogonal to a {\displaystyle a} , such that these vectors span d − 1 {\displaystyle d-1} dimensions. Then f ( x + ∑ k = 1 d − 1 c k a k ) = g ( x ⋅ a + ∑ k = 1 d − 1 c k a k ⋅ a ) = g ( x ⋅ a + ∑ k = 1 d − 1 c k 0 ) = g ( x ⋅ a ) = f ( x ) {\displaystyle f\left({\boldsymbol {x}}+\sum _{k=1}^{d-1}c_{k}{\boldsymbol {a}}_{k}\right)=g\left({\boldsymbol {x}}\cdot {\boldsymbol {a}}+\sum _{k=1}^{d-1}c_{k}{\boldsymbol {a}}_{k}\cdot {\boldsymbol {a}}\right)=g\left({\boldsymbol {x}}\cdot {\boldsymbol {a}}+\sum _{k=1}^{d-1}c_{k}0\right)=g({\boldsymbol {x}}\cdot {\boldsymbol {a}})=f({\boldsymbol {x}})} for all c i ∈ R , 1 ≤ i < d {\displaystyle c_{i}\in \mathbb {R} ,1\leq i<d} . In other words, any shift of x {\displaystyle {\boldsymbol {x}}} in a direction perpendicular to a {\displaystyle {\boldsymbol {a}}} does not change the value of f {\displaystyle f} . Ridge functions play an essential role in amongst others projection pursuit, generalized linear models, and as activation functions in neural networks. For a survey on ridge functions, see. For books on ridge functions, see. == References ==
Wikipedia:Ridge regression#0	Ridge regression (also known as Tikhonov regularization, named for Andrey Tikhonov) is a method of estimating the coefficients of multiple-regression models in scenarios where the independent variables are highly correlated. It has been used in many fields including econometrics, chemistry, and engineering. It is a method of regularization of ill-posed problems. It is particularly useful to mitigate the problem of multicollinearity in linear regression, which commonly occurs in models with large numbers of parameters. In general, the method provides improved efficiency in parameter estimation problems in exchange for a tolerable amount of bias (see bias–variance tradeoff). The theory was first introduced by Hoerl and Kennard in 1970 in their Technometrics papers "Ridge regressions: biased estimation of nonorthogonal problems" and "Ridge regressions: applications in nonorthogonal problems". Ridge regression was developed as a possible solution to the imprecision of least square estimators when linear regression models have some multicollinear (highly correlated) independent variables—by creating a ridge regression estimator (RR). This provides a more precise ridge parameters estimate, as its variance and mean square estimator are often smaller than the least square estimators previously derived. == Overview == In the simplest case, the problem of a near-singular moment matrix X T X {\displaystyle \mathbf {X} ^{\mathsf {T}}\mathbf {X} } is alleviated by adding positive elements to the diagonals, thereby decreasing its condition number. Analogous to the ordinary least squares estimator, the simple ridge estimator is then given by β ^ R = ( X T X + λ I ) − 1 X T y {\displaystyle {\hat {\boldsymbol {\beta }}}_{R}=\left(\mathbf {X} ^{\mathsf {T}}\mathbf {X} +\lambda \mathbf {I} \right)^{-1}\mathbf {X} ^{\mathsf {T}}\mathbf {y} } where y {\displaystyle \mathbf {y} } is the regressand, X {\displaystyle \mathbf {X} } is the design matrix, I {\displaystyle \mathbf {I} } is the identity matrix, and the ridge parameter λ ≥ 0 {\displaystyle \lambda \geq 0} serves as the constant shifting the diagonals of the moment matrix. It can be shown that this estimator is the solution to the least squares problem subject to the constraint β T β = c {\displaystyle {\boldsymbol {\beta }}^{\mathsf {T}}{\boldsymbol {\beta }}=c} , which can be expressed as a Lagrangian minimization: β ^ R = argmin β ( y − X β ) T ( y − X β ) + λ ( β T β − c ) {\displaystyle {\hat {\boldsymbol {\beta }}}_{R}={\text{argmin}}_{\boldsymbol {\beta }}\,\left(\mathbf {y} -\mathbf {X} {\boldsymbol {\beta }}\right)^{\mathsf {T}}\left(\mathbf {y} -\mathbf {X} {\boldsymbol {\beta }}\right)+\lambda \left({\boldsymbol {\beta }}^{\mathsf {T}}{\boldsymbol {\beta }}-c\right)} which shows that λ {\displaystyle \lambda } is nothing but the Lagrange multiplier of the constraint. In fact, there is a one-to-one relationship between c {\displaystyle c} and β {\displaystyle \beta } and since, in practice, we do not know c {\displaystyle c} , we define λ {\displaystyle \lambda } heuristically or find it via additional data-fitting strategies, see Determination of the Tikhonov factor. Note that, when λ = 0 {\displaystyle \lambda =0} , in which case the constraint is non-binding, the ridge estimator reduces to ordinary least squares. A more general approach to Tikhonov regularization is discussed below. == History == Tikhonov regularization was invented independently in many different contexts. It became widely known through its application to integral equations in the works of Andrey Tikhonov and David L. Phillips. Some authors use the term Tikhonov–Phillips regularization. The finite-dimensional case was expounded by Arthur E. Hoerl, who took a statistical approach, and by Manus Foster, who interpreted this method as a Wiener–Kolmogorov (Kriging) filter. Following Hoerl, it is known in the statistical literature as ridge regression, named after ridge analysis ("ridge" refers to the path from the constrained maximum). == Tikhonov regularization == Suppose that for a known real matrix A {\displaystyle A} and vector b {\displaystyle \mathbf {b} } , we wish to find a vector x {\displaystyle \mathbf {x} } such that A x = b , {\displaystyle A\mathbf {x} =\mathbf {b} ,} where x {\displaystyle \mathbf {x} } and b {\displaystyle \mathbf {b} } may be of different sizes and A {\displaystyle A} may be non-square. The standard approach is ordinary least squares linear regression. However, if no x {\displaystyle \mathbf {x} } satisfies the equation or more than one x {\displaystyle \mathbf {x} } does—that is, the solution is not unique—the problem is said to be ill posed. In such cases, ordinary least squares estimation leads to an overdetermined, or more often an underdetermined system of equations. Most real-world phenomena have the effect of low-pass filters in the forward direction where A {\displaystyle A} maps x {\displaystyle \mathbf {x} } to b {\displaystyle \mathbf {b} } . Therefore, in solving the inverse-problem, the inverse mapping operates as a high-pass filter that has the undesirable tendency of amplifying noise (eigenvalues / singular values are largest in the reverse mapping where they were smallest in the forward mapping). In addition, ordinary least squares implicitly nullifies every element of the reconstructed version of x {\displaystyle \mathbf {x} } that is in the null-space of A {\displaystyle A} , rather than allowing for a model to be used as a prior for x {\displaystyle \mathbf {x} } . Ordinary least squares seeks to minimize the sum of squared residuals, which can be compactly written as ‖ A x − b ‖ 2 2 , {\displaystyle \left\\|A\mathbf {x} -\mathbf {b} \right\\|_{2}^{2},} where ‖ ⋅ ‖ 2 {\displaystyle \\|\cdot \\|_{2}} is the Euclidean norm. In order to give preference to a particular solution with desirable properties, a regularization term can be included in this minimization: ‖ A x − b ‖ 2 2 + ‖ Γ x ‖ 2 2 = ‖ ( A Γ ) x − ( b 0 ) ‖ 2 2 {\displaystyle \left\\|A\mathbf {x} -\mathbf {b} \right\\|_{2}^{2}+\left\\|\Gamma \mathbf {x} \right\\|_{2}^{2}=\left\\|{\begin{pmatrix}A\\\Gamma \end{pmatrix}}\mathbf {x} -{\begin{pmatrix}\mathbf {b} \\{\boldsymbol {0}}\end{pmatrix}}\right\\|_{2}^{2}} for some suitably chosen Tikhonov matrix Γ {\displaystyle \Gamma } . In many cases, this matrix is chosen as a scalar multiple of the identity matrix ( Γ = α I {\displaystyle \Gamma =\alpha I} ), giving preference to solutions with smaller norms; this is known as L2 regularization. In other cases, high-pass operators (e.g., a difference operator or a weighted Fourier operator) may be used to enforce smoothness if the underlying vector is believed to be mostly continuous. This regularization improves the conditioning of the problem, thus enabling a direct numerical solution. An explicit solution, denoted by x ^ {\displaystyle {\hat {\mathbf {x} }}} , is given by x ^ = ( A T A + Γ T Γ ) − 1 A T b = ( ( A Γ ) T ( A Γ ) ) − 1 ( A Γ ) T ( b 0 ) . {\displaystyle {\hat {\mathbf {x} }}=\left(A^{\mathsf {T}}A+\Gamma ^{\mathsf {T}}\Gamma \right)^{-1}A^{\mathsf {T}}\mathbf {b} =\left({\begin{pmatrix}A\\\Gamma \end{pmatrix}}^{\mathsf {T}}{\begin{pmatrix}A\\\Gamma \end{pmatrix}}\right)^{-1}{\begin{pmatrix}A\\\Gamma \end{pmatrix}}^{\mathsf {T}}{\begin{pmatrix}\mathbf {b} \\{\boldsymbol {0}}\end{pmatrix}}.} The effect of regularization may be varied by the scale of matrix Γ {\displaystyle \Gamma } . For Γ = 0 {\displaystyle \Gamma =0} this reduces to the unregularized least-squares solution, provided that (ATA)−1 exists. Note that in case of a complex matrix A {\displaystyle A} , as usual the transpose A T {\displaystyle A^{\mathsf {T}}} has to be replaced by the Hermitian transpose A H {\displaystyle A^{\mathsf {H}}} . L2 regularization is used in many contexts aside from linear regression, such as classification with logistic regression or support vector machines, and matrix factorization. === Application to existing fit results === Since Tikhonov Regularization simply adds a quadratic term to the objective function in optimization problems, it is possible to do so after the unregularised optimisation has taken place. E.g., if the above problem with Γ = 0 {\displaystyle \Gamma =0} yields the solution x ^ 0 {\displaystyle {\hat {\mathbf {x} }}_{0}} , the solution in the presence of Γ ≠ 0 {\displaystyle \Gamma \neq 0} can be expressed as: x ^ = B x ^ 0 , {\displaystyle {\hat {\mathbf {x} }}=B{\hat {\mathbf {x} }}_{0},} with the "regularisation matrix" B = ( A T A + Γ T Γ ) − 1 A T A {\displaystyle B=\left(A^{\mathsf {T}}A+\Gamma ^{\mathsf {T}}\Gamma \right)^{-1}A^{\mathsf {T}}A} . If the parameter fit comes with a covariance matrix of the estimated parameter uncertainties V 0 {\displaystyle V_{0}} , then the regularisation matrix will be B = ( V 0 − 1 + Γ T Γ ) − 1 V 0 − 1 , {\displaystyle B=(V_{0}^{-1}+\Gamma ^{\mathsf {T}}\Gamma )^{-1}V_{0}^{-1},} and the regularised result will have a new covariance V = B V 0 B T . {\displaystyle V=BV_{0}B^{\mathsf {T}}.} In the context of arbitrary likelihood fits, this is valid, as long as the quadratic approximation of the likelihood function is valid. This means that, as long as the perturbation from the unregularised result is small, one can regularise any result that is presented as a best fit point with a covariance matrix. No detailed knowledge of the underlying likelihood function is needed. === Generalized Tikhonov regularization === For general multivariate normal distributions for x {\displaystyle \mathbf {x} } and the data error, one can apply a transformation of the variables to reduce to the case above. Equivalently, one can seek an x {\displaystyle \mathbf {x} } to minimize ‖ A x − b ‖ P 2 + ‖ x − x 0 ‖ Q 2 , {\displaystyle \left\\|A\mathbf {x} -\mathbf {b} \right\\|_{P}^{2}+\left\\|\mathbf {x} -\mathbf {x} _{0}\right\\|_{Q}^{2},} where we have used ‖ x ‖ Q 2 {\displaystyle \left\\|\mathbf {x} \right\\|_{Q}^{2}} to stand for the weighted norm squared x T Q x {\displaystyle \mathbf {x} ^{\mathsf {T}}Q\mathbf {x} } (compare with the Mahalanobis distance). In the Bayesian interpretation P {\displaystyle P} is the inverse covariance matrix of b {\displaystyle \mathbf {b} } , x 0 {\displaystyle \mathbf {x} _{0}} is the expected value of x {\displaystyle \mathbf {x} } , and Q {\displaystyle Q} is the inverse covariance matrix of x {\displaystyle \mathbf {x} } . The Tikhonov matrix is then given as a factorization of the matrix Q = Γ T Γ {\displaystyle Q=\Gamma ^{\mathsf {T}}\Gamma } (e.g. the Cholesky factorization) and is considered a whitening filter. This generalized problem has an optimal solution x ∗ {\displaystyle \mathbf {x} ^{}} which can be written explicitly using the formula x ∗ = ( A T P A + Q ) − 1 ( A T P b + Q x 0 ) , {\displaystyle \mathbf {x} ^{}=\left(A^{\mathsf {T}}PA+Q\right)^{-1}\left(A^{\mathsf {T}}P\mathbf {b} +Q\mathbf {x} _{0}\right),} or equivalently, when Q is not a null matrix: x ∗ = x 0 + ( A T P A + Q ) − 1 ( A T P ( b − A x 0 ) ) . {\displaystyle \mathbf {x} ^{}=\mathbf {x} _{0}+\left(A^{\mathsf {T}}PA+Q\right)^{-1}\left(A^{\mathsf {T}}P\left(\mathbf {b} -A\mathbf {x} _{0}\right)\right).} == Lavrentyev regularization == In some situations, one can avoid using the transpose A T {\displaystyle A^{\mathsf {T}}} , as proposed by Mikhail Lavrentyev. For example, if A {\displaystyle A} is symmetric positive definite, i.e. A = A T > 0 {\displaystyle A=A^{\mathsf {T}}>0} , so is its inverse A − 1 {\displaystyle A^{-1}} , which can thus be used to set up the weighted norm squared ‖ x ‖ P 2 = x T A − 1 x {\displaystyle \left\\|\mathbf {x} \right\\|_{P}^{2}=\mathbf {x} ^{\mathsf {T}}A^{-1}\mathbf {x} } in the generalized Tikhonov regularization, leading to minimizing ‖ A x − b ‖ A − 1 2 + ‖ x − x 0 ‖ Q 2 {\displaystyle \left\\|A\mathbf {x} -\mathbf {b} \right\\|_{A^{-1}}^{2}+\left\\|\mathbf {x} -\mathbf {x} _{0}\right\\|_{Q}^{2}} or, equivalently up to a constant term, x T ( A + Q ) x − 2 x T ( b + Q x 0 ) . {\displaystyle \mathbf {x} ^{\mathsf {T}}\left(A+Q\right)\mathbf {x} -2\mathbf {x} ^{\mathsf {T}}\left(\mathbf {b} +Q\mathbf {x} _{0}\right).} This minimization problem has an optimal solution x ∗ {\displaystyle \mathbf {x} ^{}} which can be written explicitly using the formula x ∗ = ( A + Q ) − 1 ( b + Q x 0 ) , {\displaystyle \mathbf {x} ^{}=\left(A+Q\right)^{-1}\left(\mathbf {b} +Q\mathbf {x} _{0}\right),} which is nothing but the solution of the generalized Tikhonov problem where A = A T = P − 1 . {\displaystyle A=A^{\mathsf {T}}=P^{-1}.} The Lavrentyev regularization, if applicable, is advantageous to the original Tikhonov regularization, since the Lavrentyev matrix A + Q {\displaystyle A+Q} can be better conditioned, i.e., have a smaller condition number, compared to the Tikhonov matrix A T A + Γ T Γ . {\displaystyle A^{\mathsf {T}}A+\Gamma ^{\mathsf {T}}\Gamma .} == Regularization in Hilbert space == Typically discrete linear ill-conditioned problems result from discretization of integral equations, and one can formulate a Tikhonov regularization in the original infinite-dimensional context. In the above we can interpret A {\displaystyle A} as a compact operator on Hilbert spaces, and x {\displaystyle x} and b {\displaystyle b} as elements in the domain and range of A {\displaystyle A} . The operator A ∗ A + Γ T Γ {\displaystyle A^{}A+\Gamma ^{\mathsf {T}}\Gamma } is then a self-adjoint bounded invertible operator. == Relation to singular-value decomposition and Wiener filter == With Γ = α I {\displaystyle \Gamma =\alpha I} , this least-squares solution can be analyzed in a special way using the singular-value decomposition. Given the singular value decomposition A = U Σ V T {\displaystyle A=U\Sigma V^{\mathsf {T}}} with singular values σ i {\displaystyle \sigma _{i}} , the Tikhonov regularized solution can be expressed as x ^ = V D U T b , {\displaystyle {\hat {x}}=VDU^{\mathsf {T}}b,} where D {\displaystyle D} has diagonal values D i i = σ i σ i 2 + α 2 {\displaystyle D_{ii}={\frac {\sigma _{i}}{\sigma _{i}^{2}+\alpha ^{2}}}} and is zero elsewhere. This demonstrates the effect of the Tikhonov parameter on the condition number of the regularized problem. For the generalized case, a similar representation can be derived using a generalized singular-value decomposition. Finally, it is related to the Wiener filter: x ^ = ∑ i = 1 q f i u i T b σ i v i , {\displaystyle {\hat {x}}=\sum _{i=1}^{q}f_{i}{\frac {u_{i}^{\mathsf {T}}b}{\sigma _{i}}}v_{i},} where the Wiener weights are f i = σ i 2 σ i 2 + α 2 {\displaystyle f_{i}={\frac {\sigma _{i}^{2}}{\sigma _{i}^{2}+\alpha ^{2}}}} and q {\displaystyle q} is the rank of A {\displaystyle A} . == Determination of the Tikhonov factor == The optimal regularization parameter α {\displaystyle \alpha } is usually unknown and often in practical problems is determined by an ad hoc method. A possible approach relies on the Bayesian interpretation described below. Other approaches include the discrepancy principle, cross-validation, L-curve method, restricted maximum likelihood and unbiased predictive risk estimator. Grace Wahba proved that the optimal parameter, in the sense of leave-one-out cross-validation minimizes G = RSS τ 2 = ‖ X β ^ − y ‖ 2 [ Tr ⁡ ( I − X ( X T X + α 2 I ) − 1 X T ) ] 2 , {\displaystyle G={\frac {\operatorname {RSS} }{\tau ^{2}}}={\frac {\left\\|X{\hat {\beta }}-y\right\\|^{2}}{\left[\operatorname {Tr} \left(I-X\left(X^{\mathsf {T}}X+\alpha ^{2}I\right)^{-1}X^{\mathsf {T}}\right)\right]^{2}}},} where RSS {\displaystyle \operatorname {RSS} } is the residual sum of squares, and τ {\displaystyle \tau } is the effective number of degrees of freedom. Using the previous SVD decomposition, we can simplify the above expression: RSS = ‖ y − ∑ i = 1 q ( u i ′ b ) u i ‖ 2 + ‖ ∑ i = 1 q α 2 σ i 2 + α 2 ( u i ′ b ) u i ‖ 2 , {\displaystyle \operatorname {RSS} =\left\\|y-\sum _{i=1}^{q}(u_{i}'b)u_{i}\right\\|^{2}+\left\\|\sum _{i=1}^{q}{\frac {\alpha ^{2}}{\sigma _{i}^{2}+\alpha ^{2}}}(u_{i}'b)u_{i}\right\\|^{2},} RSS = RSS 0 + ‖ ∑ i = 1 q α 2 σ i 2 + α 2 ( u i ′ b ) u i ‖ 2 , {\displaystyle \operatorname {RSS} =\operatorname {RSS} _{0}+\left\\|\sum _{i=1}^{q}{\frac {\alpha ^{2}}{\sigma _{i}^{2}+\alpha ^{2}}}(u_{i}'b)u_{i}\right\\|^{2},} and τ = m − ∑ i = 1 q σ i 2 σ i 2 + α 2 = m − q + ∑ i = 1 q α 2 σ i 2 + α 2 . {\displaystyle \tau =m-\sum _{i=1}^{q}{\frac {\sigma _{i}^{2}}{\sigma _{i}^{2}+\alpha ^{2}}}=m-q+\sum _{i=1}^{q}{\frac {\alpha ^{2}}{\sigma _{i}^{2}+\alpha ^{2}}}.} == Relation to probabilistic formulation == The probabilistic formulation of an inverse problem introduces (when all uncertainties are Gaussian) a covariance matrix C M {\displaystyle C_{M}} representing the a priori uncertainties on the model parameters, and a covariance matrix C D {\displaystyle C_{D}} representing the uncertainties on the observed parameters. In the special case when these two matrices are diagonal and isotropic, C M = σ M 2 I {\displaystyle C_{M}=\sigma _{M}^{2}I} and C D = σ D 2 I {\displaystyle C_{D}=\sigma _{D}^{2}I} , and, in this case, the equations of inverse theory reduce to the equations above, with α = σ D / σ M {\displaystyle \alpha ={\sigma _{D}}/{\sigma _{M}}} . == Bayesian interpretation == Although at first the choice of the solution to this regularized problem may look artificial, and indeed the matrix Γ {\displaystyle \Gamma } seems rather arbitrary, the process can be justified from a Bayesian point of view. Note that for an ill-posed problem one must necessarily introduce some additional assumptions in order to get a unique solution. Statistically, the prior probability distribution of x {\displaystyle x} is sometimes taken to be a multivariate normal distribution. For simplicity here, the following assumptions are made: the means are zero; their components are independent; the components have the same standard deviation σ x {\displaystyle \sigma _{x}} . The data are also subject to errors, and the errors in b {\displaystyle b} are also assumed to be independent with zero mean and standard deviation σ b {\displaystyle \sigma _{b}} . Under these assumptions the Tikhonov-regularized solution is the most probable solution given the data and the a priori distribution of x {\displaystyle x} , according to Bayes' theorem. If the assumption of normality is replaced by assumptions of homoscedasticity and uncorrelatedness of errors, and if one still assumes zero mean, then the Gauss–Markov theorem entails that the solution is the minimal unbiased linear estimator. == See also == LASSO estimator is another regularization method in statistics. Elastic net regularization Matrix regularization == Notes == == References == == Further reading == Gruber, Marvin (1998). Improving Efficiency by Shrinkage: The James–Stein and Ridge Regression Estimators. Boca Raton: CRC Press. ISBN 0-8247-0156-9. Kress, Rainer (1998). "Tikhonov Regularization". Numerical Analysis. New York: Springer. pp. 86–90. ISBN 0-387-98408-9. Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. (2007). "Section 19.5. Linear Regularization Methods". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8. Saleh, A. K. Md. Ehsanes; Arashi, Mohammad; Kibria, B. M. Golam (2019). Theory of Ridge Regression Estimation with Applications. New York: John Wiley & Sons. ISBN 978-1-118-64461-4. Taddy, Matt (2019). "Regularization". Business Data Science: Combining Machine Learning and Economics to Optimize, Automate, and Accelerate Business Decisions. New York: McGraw-Hill. pp. 69–104. ISBN 978-1-260-45277-8.
Wikipedia:Riemann–Lebesgue lemma#0	In mathematics, the Riemann–Lebesgue lemma, named after Bernhard Riemann and Henri Lebesgue, states that the Fourier transform or Laplace transform of an L1 function vanishes at infinity. It is of importance in harmonic analysis and asymptotic analysis. == Statement == Let f ∈ L 1 ( R n ) {\displaystyle f\in L^{1}(\mathbb {R} ^{n})} be an integrable function, i.e. f : R n → C {\displaystyle f\colon \mathbb {R} ^{n}\rightarrow \mathbb {C} } is a measurable function such that ‖ f ‖ L 1 = ∫ R n \| f ( x ) \| d x < ∞ , {\displaystyle \\|f\\|_{L^{1}}=\int _{\mathbb {R} ^{n}}\|f(x)\|\mathrm {d} x<\infty ,} and let f ^ {\displaystyle {\hat {f}}} be the Fourier transform of f {\displaystyle f} , i.e. f ^ : R n → C , ξ ↦ ∫ R n f ( x ) e − i x ⋅ ξ d x . {\displaystyle {\hat {f}}\colon \mathbb {R} ^{n}\rightarrow \mathbb {C} ,\ \xi \mapsto \int _{\mathbb {R} ^{n}}f(x)\mathrm {e} ^{-\mathrm {i} x\cdot \xi }\mathrm {d} x.} Then f ^ {\displaystyle {\hat {f}}} vanishes at infinity: \| f ^ ( ξ ) \| → 0 {\displaystyle \|{\hat {f}}(\xi )\|\to 0} as \| ξ \| → ∞ {\displaystyle \|\xi \|\to \infty } . Because the Fourier transform of an integrable function is continuous, the Fourier transform f ^ {\displaystyle {\hat {f}}} is a continuous function vanishing at infinity. If C 0 ( R n ) {\displaystyle C_{0}(\mathbb {R} ^{n})} denotes the vector space of continuous functions vanishing at infinity, the Riemann–Lebesgue lemma may be formulated as follows: The Fourier transformation maps L 1 ( R n ) {\displaystyle L^{1}(\mathbb {R} ^{n})} to C 0 ( R n ) {\displaystyle C_{0}(\mathbb {R} ^{n})} . === Proof === We will focus on the one-dimensional case n = 1 {\displaystyle n=1} , the proof in higher dimensions is similar. First, suppose that f {\displaystyle f} is continuous and compactly supported. For ξ ≠ 0 {\displaystyle \xi \neq 0} , the substitution x → x + π ξ {\displaystyle \textstyle x\to x+{\frac {\pi }{\xi }}} leads to f ^ ( ξ ) = ∫ R f ( x ) e − i x ξ d x = ∫ R f ( x + π ξ ) e − i x ξ e − i π d x = − ∫ R f ( x + π ξ ) e − i x ξ d x {\displaystyle {\hat {f}}(\xi )=\int _{\mathbb {R} }f(x)\mathrm {e} ^{-\mathrm {i} x\xi }\mathrm {d} x=\int _{\mathbb {R} }f\left(x+{\frac {\pi }{\xi }}\right)\mathrm {e} ^{-\mathrm {i} x\xi }\mathrm {e} ^{-\mathrm {i} \pi }\mathrm {d} x=-\int _{\mathbb {R} }f\left(x+{\frac {\pi }{\xi }}\right)\mathrm {e} ^{-\mathrm {i} x\xi }\mathrm {d} x} . This gives a second formula for f ^ ( ξ ) {\displaystyle {\hat {f}}(\xi )} . Taking the mean of both formulas, we arrive at the following estimate: \| f ^ ( ξ ) \| ≤ 1 2 ∫ R \| f ( x ) − f ( x + π ξ ) \| d x {\displaystyle \|{\hat {f}}(\xi )\|\leq {\frac {1}{2}}\int _{\mathbb {R} }\left\|f(x)-f\left(x+{\frac {\pi }{\xi }}\right)\right\|\mathrm {d} x} . Because f {\displaystyle f} is continuous, \| f ( x ) − f ( x + π ξ ) \| {\displaystyle \left\|f(x)-f\left(x+{\tfrac {\pi }{\xi }}\right)\right\|} converges to 0 {\displaystyle 0} as \| ξ \| → ∞ {\displaystyle \|\xi \|\to \infty } for all x ∈ R {\displaystyle x\in \mathbb {R} } . Thus, \| f ^ ( ξ ) \| {\displaystyle \|{\hat {f}}(\xi )\|} converges to 0 as \| ξ \| → ∞ {\displaystyle \|\xi \|\to \infty } due to the dominated convergence theorem. If f {\displaystyle f} is an arbitrary integrable function, it may be approximated in the L 1 {\displaystyle L^{1}} norm by a compactly supported continuous function. For ε > 0 {\displaystyle \varepsilon >0} , pick a compactly supported continuous function g {\displaystyle g} such that ‖ f − g ‖ L 1 ≤ ε {\displaystyle \\|f-g\\|_{L^{1}}\leq \varepsilon } . Then lim sup ξ → ± ∞ \| f ^ ( ξ ) \| ≤ lim sup ξ → ± ∞ \| ∫ ( f ( x ) − g ( x ) ) e − i x ξ d x \| + lim sup ξ → ± ∞ \| ∫ g ( x ) e − i x ξ d x \| ≤ ε + 0 = ε . {\displaystyle \limsup _{\xi \rightarrow \pm \infty }\|{\hat {f}}(\xi )\|\leq \limsup _{\xi \to \pm \infty }\left\|\int (f(x)-g(x))\mathrm {e} ^{-\mathrm {i} x\xi }\,\mathrm {d} x\right\|+\limsup _{\xi \rightarrow \pm \infty }\left\|\int g(x)\mathrm {e} ^{-\mathrm {i} x\xi }\,\mathrm {d} x\right\|\leq \varepsilon +0=\varepsilon .} Because this holds for any ε > 0 {\displaystyle \varepsilon >0} , it follows that \| f ^ ( ξ ) \| → 0 {\displaystyle \|{\hat {f}}(\xi )\|\to 0} as \| ξ \| → ∞ {\displaystyle \|\xi \|\to \infty } . == Other versions == The Riemann–Lebesgue lemma holds in a variety of other situations. If f ∈ L 1 [ 0 , ∞ ) {\displaystyle f\in L^{1}[0,\infty )} , then the Riemann–Lebesgue lemma also holds for the Laplace transform of f {\displaystyle f} , that is, ∫ 0 ∞ f ( t ) e − t z d t → 0 {\displaystyle \int _{0}^{\infty }f(t)\mathrm {e} ^{-tz}\mathrm {d} t\to 0} as \| z \| → ∞ {\displaystyle \|z\|\to \infty } within the half-plane R e ( z ) ≥ 0 {\displaystyle \mathrm {Re} (z)\geq 0} . A version holds for Fourier series as well: if f {\displaystyle f} is an integrable function on a bounded interval, then the Fourier coefficients f ^ k {\displaystyle {\hat {f}}_{k}} of f {\displaystyle f} tend to 0 as k → ± ∞ {\displaystyle k\to \pm \infty } . This follows by extending f {\displaystyle f} by zero outside the interval, and then applying the version of the Riemann–Lebesgue lemma on the entire real line. However, the Riemann–Lebesgue lemma does not hold for arbitrary distributions. For example, the Dirac delta function distribution formally has a finite integral over the real line, but its Fourier transform is a constant and does not vanish at infinity. == Applications == The Riemann–Lebesgue lemma can be used to prove the validity of asymptotic approximations for integrals. Rigorous treatments of the method of steepest descent and the method of stationary phase, amongst others, are based on the Riemann–Lebesgue lemma. == References == Bochner S., Chandrasekharan K. (1949). Fourier Transforms. Princeton University Press. Weisstein, Eric W. "Riemann–Lebesgue Lemma". MathWorld.
Wikipedia:Rien Kaashoek#0	Marinus Adriaan "Rien" Kaashoek (November 10, 1937 – November 21, 2024) was a Dutch mathematician, and Emeritus Professor Analysis and Operator Theory at the Vrije Universiteit in Amsterdam. == Biography == Born in Ridderkerk, Kaashoek has studied mathematics at the Leiden University, where he received his PhD in 1964 under supervision of Adriaan Zaanen. Kaashoek had started his academic career as assistant at the Leiden University from 1959 to 1962, and Junior Staff Member from 1962 to 1965. In 1966 he started as senior lecturer at the Vrije Universiteit, where in 1969 he was appointed professor. Among his doctoral students is Harm Bart (1973). Kaashoek has been at the University of Maryland, College Park in 1975, at the University of Calgary in 1987, at the Ben-Gurion University of the Negev in 1987 and at the Tel Aviv University on various occasions. He was a member of the honorary editorial board of the Journal Integral Equations and Operator Theory, and has been appointed Knight in the Order of the Dutch Lion (Ridder in de Orde van de Nederlandse Leeuw) in 2002. In 2014 he received an honorary doctorate of North-West University in South-Africa (Potchefstroom campus). He was elected honorary member (erelid) of the Royal Dutch Mathematical Society (Koninklijk Wiskundig Genootschap) on 22 March 2016. M. A. Kaashoek was one of the early supporters of the International Workshop on Operator Theory and its Applications (IWOTA), which was started in 1981. From the beginning, M. A. Kaashoek and J. W. Helton served as vice presidents of the IWOTA Steering Committee. In addition, M. A. Kaashoek organized the third IWOTA in 1985, the first time this series of conferences took place in Europe. Moreover, he maintained a website documenting the complete IWOTA series. Kaashoek died on November 21, 2024, at the age of 86. == Work == Kaashoeks research interests are in the field of 'the "analysis and Operator Theory, and various connections between Operator Theory, Matrix Theory and Mathematical Systems Theory. In particular, Wiener–Hopf integral equations and Toeplitz operators, their nonstationary variants, and other structured operators, such as continuous operator analogs of Bézout and resultant matrices. State space methods for problems in analysis. Also metric constrained interpolation problems and completion problems for partially given operators, including relaxed commutant lifting problems." == Publications == Kaashoek has authored and co-authored ten books A selection: 1974. Locally compact semi-algebras : with applications to spectral theory of positive operators. With Trevor West. Amsterdam; London : North-Holland Publishing Co. 1993. Classes of linear operators, Volume 1 and 2. With Israel Gohberg and Seymour Goldberg. Birkhäuser. 2003. Basic classes of linear operators. With Israel Gohberg and Seymour Goldberg. Springer. Articles, a selection: Harm Bart, et al. "Factorizations of transfer functions." SIAM Journal on Control and Optimization 18.6 (1980): 675–696. Israel Gohberg and Marinus Adriaan Kaashoek. "Time varying linear systems with boundary conditions and integral operators. I. The transfer operator and its properties." Integral Equations and Operator Theory 7.3 (1984): 325–391. == References == == External links == Rien Kaashoek at the Vrije Universiteit
Wikipedia:Rigid transformation#0	In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points. The rigid transformations include rotations, translations, reflections, or any sequence of these. Reflections are sometimes excluded from the definition of a rigid transformation by requiring that the transformation also preserve the handedness of objects in the Euclidean space. (A reflection would not preserve handedness; for instance, it would transform a left hand into a right hand.) To avoid ambiguity, a transformation that preserves handedness is known as a rigid motion, a Euclidean motion, or a proper rigid transformation. In dimension two, a rigid motion is either a translation or a rotation. In dimension three, every rigid motion can be decomposed as the composition of a rotation and a translation, and is thus sometimes called a rototranslation. In dimension three, all rigid motions are also screw motions (this is Chasles' theorem). In dimension at most three, any improper rigid transformation can be decomposed into an improper rotation followed by a translation, or into a sequence of reflections. Any object will keep the same shape and size after a proper rigid transformation. All rigid transformations are examples of affine transformations. The set of all (proper and improper) rigid transformations is a mathematical group called the Euclidean group, denoted E(n) for n-dimensional Euclidean spaces. The set of rigid motions is called the special Euclidean group, and denoted SE(n). In kinematics, rigid motions in a 3-dimensional Euclidean space are used to represent displacements of rigid bodies. According to Chasles' theorem, every rigid transformation can be expressed as a screw motion. == Formal definition == A rigid transformation is formally defined as a transformation that, when acting on any vector v, produces a transformed vector T(v) of the form where RT = R−1 (i.e., R is an orthogonal transformation), and t is a vector giving the translation of the origin. A proper rigid transformation has, in addition, which means that R does not produce a reflection, and hence it represents a rotation (an orientation-preserving orthogonal transformation). Indeed, when an orthogonal transformation matrix produces a reflection, its determinant is −1. == Distance formula == A measure of distance between points, or metric, is needed in order to confirm that a transformation is rigid. The Euclidean distance formula for Rn is the generalization of the Pythagorean theorem. The formula gives the distance squared between two points X and Y as the sum of the squares of the distances along the coordinate axes, that is d ( X , Y ) 2 = ( X 1 − Y 1 ) 2 + ( X 2 − Y 2 ) 2 + ⋯ + ( X n − Y n ) 2 = ( X − Y ) ⋅ ( X − Y ) . {\displaystyle d\left(\mathbf {X} ,\mathbf {Y} \right)^{2}=\left(X_{1}-Y_{1}\right)^{2}+\left(X_{2}-Y_{2}\right)^{2}+\dots +\left(X_{n}-Y_{n}\right)^{2}=\left(\mathbf {X} -\mathbf {Y} \right)\cdot \left(\mathbf {X} -\mathbf {Y} \right).} where X = (X1, X2, ..., Xn) and Y = (Y1, Y2, ..., Yn), and the dot denotes the scalar product. Using this distance formula, a rigid transformation g : Rn → Rn has the property, d ( g ( X ) , g ( Y ) ) 2 = d ( X , Y ) 2 . {\displaystyle d(g(\mathbf {X} ),g(\mathbf {Y} ))^{2}=d(\mathbf {X} ,\mathbf {Y} )^{2}.} == Translations and linear transformations == A translation of a vector space adds a vector d to every vector in the space, which means it is the transformation It is easy to show that this is a rigid transformation by showing that the distance between translated vectors equal the distance between the original vectors: d ( v + d , w + d ) 2 = ( v + d − w − d ) ⋅ ( v + d − w − d ) = ( v − w ) ⋅ ( v − w ) = d ( v , w ) 2 . {\displaystyle d(\mathbf {v} +\mathbf {d} ,\mathbf {w} +\mathbf {d} )^{2}=(\mathbf {v} +\mathbf {d} -\mathbf {w} -\mathbf {d} )\cdot (\mathbf {v} +\mathbf {d} -\mathbf {w} -\mathbf {d} )=(\mathbf {v} -\mathbf {w} )\cdot (\mathbf {v} -\mathbf {w} )=d(\mathbf {v} ,\mathbf {w} )^{2}.} A linear transformation of a vector space, L : Rn → Rn, preserves linear combinations, L ( V ) = L ( a v + b w ) = a L ( v ) + b L ( w ) . {\displaystyle L(\mathbf {V} )=L(a\mathbf {v} +b\mathbf {w} )=aL(\mathbf {v} )+bL(\mathbf {w} ).} A linear transformation L can be represented by a matrix, which means where [L] is an n×n matrix. A linear transformation is a rigid transformation if it satisfies the condition, d ( [ L ] v , [ L ] w ) 2 = d ( v , w ) 2 , {\displaystyle d([L]\mathbf {v} ,[L]\mathbf {w} )^{2}=d(\mathbf {v} ,\mathbf {w} )^{2},} that is d ( [ L ] v , [ L ] w ) 2 = ( [ L ] v − [ L ] w ) ⋅ ( [ L ] v − [ L ] w ) = ( [ L ] ( v − w ) ) ⋅ ( [ L ] ( v − w ) ) . {\displaystyle d([L]\mathbf {v} ,[L]\mathbf {w} )^{2}=([L]\mathbf {v} -[L]\mathbf {w} )\cdot ([L]\mathbf {v} -[L]\mathbf {w} )=([L](\mathbf {v} -\mathbf {w} ))\cdot ([L](\mathbf {v} -\mathbf {w} )).} Now use the fact that the scalar product of two vectors v.w can be written as the matrix operation vTw, where the T denotes the matrix transpose, we have d ( [ L ] v , [ L ] w ) 2 = ( v − w ) T [ L ] T [ L ] ( v − w ) . {\displaystyle d([L]\mathbf {v} ,[L]\mathbf {w} )^{2}=(\mathbf {v} -\mathbf {w} )^{\mathsf {T}}[L]^{\mathsf {T}}[L](\mathbf {v} -\mathbf {w} ).} Thus, the linear transformation L is rigid if its matrix satisfies the condition [ L ] T [ L ] = [ I ] , {\displaystyle [L]^{\mathsf {T}}[L]=[I],} where [I] is the identity matrix. Matrices that satisfy this condition are called orthogonal matrices. This condition actually requires the columns of these matrices to be orthogonal unit vectors. Matrices that satisfy this condition form a mathematical group under the operation of matrix multiplication called the orthogonal group of n×n matrices and denoted O(n). Compute the determinant of the condition for an orthogonal matrix to obtain det ( [ L ] T [ L ] ) = det [ L ] 2 = det [ I ] = 1 , {\displaystyle \det \left([L]^{\mathsf {T}}[L]\right)=\det[L]^{2}=\det[I]=1,} which shows that the matrix [L] can have a determinant of either +1 or −1. Orthogonal matrices with determinant −1 are reflections, and those with determinant +1 are rotations. Notice that the set of orthogonal matrices can be viewed as consisting of two manifolds in Rn×n separated by the set of singular matrices. The set of rotation matrices is called the special orthogonal group, and denoted SO(n). It is an example of a Lie group because it has the structure of a manifold. == See also == Deformation (mechanics) Motion (geometry) Rigid body dynamics == References ==
Wikipedia:Riho Terras (mathematician)#0	Riho Terras (June 13, 1939 – November 28, 2005) was an Estonian-American mathematician. He was born in Tartu, Estonia, and moved to Ulm, Germany, before starting school. In 1951, he emigrated to the United States along with his mother. In 1965, he was given the Milton Abramowitz award for his studies at the University of Maryland. He finished his PhD in 1970 at the University of Illinois Urbana-Champaign. He is known for the Terras theorem about the Collatz conjecture, published in 1976, which proved that the conjecture holds for "almost all" numbers and established bounds for the conjecture. He married fellow mathematician Audrey Terras. == References ==
Wikipedia:Rimhak Ree#0	Rimhak Ree (Korean: 이임학; December 18, 1922 – January 9, 2005), alternatively Im-hak Ree, was a Korean Canadian mathematician. He contributed in the field of group theory, most notably with the concept of the Ree group in (Ree 1960, 1961). == Early life == Ree received his early education in Hamhung, Kankyōnan-dō, Korea, Empire of Japan (now in North Korea). He attended the Hamhung #1 Public Ordinary School (함흥 제 1공립보통학교), and in 1934 entered the Hamhung Public High School (함흥공립고등보통학교). He went onto Keijō Imperial University, where he studied physics, which was an unusual choice for Koreans at the time. Ree graduated in 1944 with a physics degree; he then went to Fengtian, Manchukuo (today Shenyang, Liaoning in the People's Republic of China) to work for an aircraft company. == Career == After the surrender of Japan in 1945 and the end of Japanese rule in Korea, Ree returned to his home country and in 1947 took up a teaching position in the mathematics department at Seoul National University as an assistant professor. Later that year, in Namdaemun Market, Ree found an issue of the Bulletin of the American Mathematical Society, which proposedly was left by an American soldier. On the Bulletin was the paper 'Note on power series', in which Max Zorn solved a problem about the convergence of certain power series with complex coefficients. In the paper, Zorn posed a question of whether the same result held for power series with real coefficients. Ree solved the problem and sent the solution to Max Zorn. When Zorn received Ree's solution, it was sent to the Bulletin of the American Mathematical Society to be published in 1949 with the title 'On a problem of Max Zorn' and become the first mathematical paper published by a Korean in an international journal. During the Korean War, he fled south to Busan, and in 1953 he was awarded a Canadian scholarship to allow him to study for a Ph.D. degree at the University of British Columbia in Vancouver, Canada. He completed his dissertation on Witt algebras in 1955. His thesis advisor was Stephen Arthur Jennings. Following the award of his doctorate, Ree was appointed as a lecturer at Montana State University, despite facing several problems regarding his labour permission and nationality. In mid-1955, Ree received a grant from the National Research Council of Canada and he worked with Jennings on Lie algebras. In 1958, he published a solution to a problem of Paul Erdős regarding a certain class of irrational numbers. Ree's two most renowned papers were written from 1960 to 1961, in which he suggested a Lie type group over a finite field now named after him. In 1962 after being promoted to an assistant professor in mathematics at University of British Columbia, he was granted an academic year which he spent in Yale. He was elected a member of Royal Society of Canada in 1964. == Personal life == === Family === Ree had two daughters Erran and Hiran from his first marriage. He later married Rhoda Mah, a doctor and the daughter of John Ming Mah, who owned Northwest Food Products, a manufacturers of noodles. She would go on to work as staff physician for Canadian Pacific Airlines. Rimhak and Rhoda's first son Ronald was followed by another son Robert in December 1971. They also had a third son Richard. Ree died on January 9, 2005, in Vancouver, Canada. === Statelessness === Around the time Ree received his doctorate, his passport was approaching its expiration date, so he approached the South Korean consulate in San Francisco to extend it, but instead the consular officer confiscated his passport and ordered him to return to South Korea. Ree refused the order, which caused him considerable difficulty, but in the end the Canadian government treated him as a de facto stateless person and granted him permanent residency in Canada. Afterwards, he continued to work at the University of British Columbia. Though Ree secured his immigration status in Canada, he continued to encounter difficulties with the South Korean government. Ree's family was divided by the Korean War, with his father, older sister, and other relatives having stayed in their hometown of Hamhung. Hamhung was the site of a munitions factory built during Japanese rule, making the city a frequent target for bombing by the United States Air Force during the Korean War, and Lee did not know if any of his relatives there had survived the war. He visited North Korea using his Canadian passport various times for academic exchanges, but he was not able to travel freely in North Korea and thus had no success in making contact with his relatives; furthermore, his visits to North Korea led South Korea's Park Chung Hee military government to place an entry ban on him. Ree requested help from Erdős, who as an internationally-famous Hungarian citizen faced fewer restrictions on travel or communication in either capitalist or communist countries. Finally, in the 1980s, Erdős was able to make contact with several relatives of Ree's with the help of Hungary's Ministry of Foreign Affairs and the country's embassy in Pyongyang, and sent Ree an envelope containing their letters, photographs, and addresses. Ree was so excited by the news that he forwarded the envelope to his mother and younger sister in South Korea, which reportedly resulted in them being investigated by South Korea's intelligence services. Ree remained banned from South Korea until 1996, when the ban was cancelled as he was invited to the 50th anniversary ceremony of the Korean Mathematical Society. According to his colleagues, Rimhak Ree identified his nationality as "Joseon", which is a former name of Korea as well as a current autonym of North Korea. == Publications == Ree, Rimhak (1960), "A family of simple groups associated with the simple Lie algebra of type (G2)", Bulletin of the American Mathematical Society, 66: 508–510, doi:10.1090/S0002-9904-1960-10523-X, ISSN 0002-9904, MR 0125155 Ree, Rimhak (1961), "A family of simple groups associated with the simple Lie algebra of type (F4)", Bulletin of the American Mathematical Society, 67: 115–116, doi:10.1090/S0002-9904-1961-10527-2, ISSN 0002-9904, MR 0125155 == Notes == == References == Carol Tretkoff; Marvin Tretkoff (January 1979), "On a theorem of Rimhak Ree about permutations", Journal of Combinatorial Theory, Series A, 26 (1): 84–86, doi:10.1016/0097-3165(79)90056-6 주진순 [Ju Jin-sun] (March 2007), "세계적인 수학자 이임학 형을 그리워하며" (PDF), Newsletter of the Korean Mathematical Society, vol. 112, no. 1, pp. 2–4, retrieved 2010-10-08 "News from the Departments", Canadian Mathematical Bulletin, 6 (3): 465, 1963, retrieved 2018-05-30 == External links == Rimhak Ree at the Mathematics Genealogy Project
Wikipedia:Ring of symmetric functions#0	In algebra and in particular in algebraic combinatorics, the ring of symmetric functions is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity. This ring serves as universal structure in which relations between symmetric polynomials can be expressed in a way independent of the number n of indeterminates (but its elements are neither polynomials nor functions). Among other things, this ring plays an important role in the representation theory of the symmetric group. The ring of symmetric functions can be given a coproduct and a bilinear form making it into a positive selfadjoint graded Hopf algebra that is both commutative and cocommutative. == Symmetric polynomials == The study of symmetric functions is based on that of symmetric polynomials. In a polynomial ring in some finite set of indeterminates, a polynomial is called symmetric if it stays the same whenever the indeterminates are permuted in any way. More formally, there is an action by ring automorphisms of the symmetric group Sn on the polynomial ring in n indeterminates, where a permutation acts on a polynomial by simultaneously substituting each of the indeterminates for another according to the permutation used. The invariants for this action form the subring of symmetric polynomials. If the indeterminates are X1, ..., Xn, then examples of such symmetric polynomials are X 1 + X 2 + ⋯ + X n , {\displaystyle X_{1}+X_{2}+\cdots +X_{n},\,} X 1 3 + X 2 3 + ⋯ + X n 3 , {\displaystyle X_{1}^{3}+X_{2}^{3}+\cdots +X_{n}^{3},\,} and X 1 X 2 ⋯ X n . {\displaystyle X_{1}X_{2}\cdots X_{n}.\,} A somewhat more complicated example is X13X2X3 + X1X23X3 + X1X2X33 + X13X2X4 + X1X23X4 + X1X2X43 + ... where the summation goes on to include all products of the third power of some variable and two other variables. There are many specific kinds of symmetric polynomials, such as elementary symmetric polynomials, power sum symmetric polynomials, monomial symmetric polynomials, complete homogeneous symmetric polynomials, and Schur polynomials. == The ring of symmetric functions == Most relations between symmetric polynomials do not depend on the number n of indeterminates, other than that some polynomials in the relation might require n to be large enough in order to be defined. For instance the Newton's identity for the third power sum polynomial p3 leads to p 3 ( X 1 , … , X n ) = e 1 ( X 1 , … , X n ) 3 − 3 e 2 ( X 1 , … , X n ) e 1 ( X 1 , … , X n ) + 3 e 3 ( X 1 , … , X n ) , {\displaystyle p_{3}(X_{1},\ldots ,X_{n})=e_{1}(X_{1},\ldots ,X_{n})^{3}-3e_{2}(X_{1},\ldots ,X_{n})e_{1}(X_{1},\ldots ,X_{n})+3e_{3}(X_{1},\ldots ,X_{n}),} where the e i {\displaystyle e_{i}} denote elementary symmetric polynomials; this formula is valid for all natural numbers n, and the only notable dependency on it is that ek(X1,...,Xn) = 0 whenever n < k. One would like to write this as an identity p 3 = e 1 3 − 3 e 2 e 1 + 3 e 3 {\displaystyle p_{3}=e_{1}^{3}-3e_{2}e_{1}+3e_{3}} that does not depend on n at all, and this can be done in the ring of symmetric functions. In that ring there are nonzero elements ek for all integers k ≥ 1, and any element of the ring can be given by a polynomial expression in the elements ek. === Definitions === A ring of symmetric functions can be defined over any commutative ring R, and will be denoted ΛR; the basic case is for R = Z. The ring ΛR is in fact a graded R-algebra. There are two main constructions for it; the first one given below can be found in (Stanley, 1999), and the second is essentially the one given in (Macdonald, 1979). ==== As a ring of formal power series ==== The easiest (though somewhat heavy) construction starts with the ring of formal power series R [ [ X 1 , X 2 , . . . ] ] {\displaystyle R[[X_{1},X_{2},...]]} over R in infinitely (countably) many indeterminates; the elements of this power series ring are formal infinite sums of terms, each of which consists of a coefficient from R multiplied by a monomial, where each monomial is a product of finitely many finite powers of indeterminates. One defines ΛR as its subring consisting of those power series S that satisfy S is invariant under any permutation of the indeterminates, and the degrees of the monomials occurring in S are bounded. Note that because of the second condition, power series are used here only to allow infinitely many terms of a fixed degree, rather than to sum terms of all possible degrees. Allowing this is necessary because an element that contains for instance a term X1 should also contain a term Xi for every i > 1 in order to be symmetric. Unlike the whole power series ring, the subring ΛR is graded by the total degree of monomials: due to condition 2, every element of ΛR is a finite sum of homogeneous elements of ΛR (which are themselves infinite sums of terms of equal degree). For every k ≥ 0, the element ek ∈ ΛR is defined as the formal sum of all products of k distinct indeterminates, which is clearly homogeneous of degree k. ==== As an algebraic limit ==== Another construction of ΛR takes somewhat longer to describe, but better indicates the relationship with the rings R[X1,...,Xn]Sn of symmetric polynomials in n indeterminates. For every n there is a surjective ring homomorphism ρn from the analogous ring R[X1,...,Xn+1]Sn+1 with one more indeterminate onto R[X1,...,Xn]Sn, defined by setting the last indeterminate Xn+1 to 0. Although ρn has a non-trivial kernel, the nonzero elements of that kernel have degree at least n + 1 {\displaystyle n+1} (they are multiples of X1X2...Xn+1). This means that the restriction of ρn to elements of degree at most n is a bijective linear map, and ρn(ek(X1,...,Xn+1)) = ek(X1,...,Xn) for all k ≤ n. The inverse of this restriction can be extended uniquely to a ring homomorphism φn from R[X1,...,Xn]Sn to R[X1,...,Xn+1]Sn+1, as follows for instance from the fundamental theorem of symmetric polynomials. Since the images φn(ek(X1,...,Xn)) = ek(X1,...,Xn+1) for k = 1,...,n are still algebraically independent over R, the homomorphism φn is injective and can be viewed as a (somewhat unusual) inclusion of rings; applying φn to a polynomial amounts to adding all monomials containing the new indeterminate obtained by symmetry from monomials already present. The ring ΛR is then the "union" (direct limit) of all these rings subject to these inclusions. Since all φn are compatible with the grading by total degree of the rings involved, ΛR obtains the structure of a graded ring. This construction differs slightly from the one in (Macdonald, 1979). That construction only uses the surjective morphisms ρn without mentioning the injective morphisms φn: it constructs the homogeneous components of ΛR separately, and equips their direct sum with a ring structure using the ρn. It is also observed that the result can be described as an inverse limit in the category of graded rings. That description however somewhat obscures an important property typical for a direct limit of injective morphisms, namely that every individual element (symmetric function) is already faithfully represented in some object used in the limit construction, here a ring R[X1,...,Xd]Sd. It suffices to take for d the degree of the symmetric function, since the part in degree d of that ring is mapped isomorphically to rings with more indeterminates by φn for all n ≥ d. This implies that for studying relations between individual elements, there is no fundamental difference between symmetric polynomials and symmetric functions. === Defining individual symmetric functions === The name "symmetric function" for elements of ΛR is a misnomer: in neither construction are the elements functions, and in fact, unlike symmetric polynomials, no function of independent variables can be associated to such elements (for instance e1 would be the sum of all infinitely many variables, which is not defined unless restrictions are imposed on the variables). However the name is traditional and well established; it can be found both in (Macdonald, 1979), which says (footnote on p. 12) The elements of Λ (unlike those of Λn) are no longer polynomials: they are formal infinite sums of monomials. We have therefore reverted to the older terminology of symmetric functions. (here Λn denotes the ring of symmetric polynomials in n indeterminates), and also in (Stanley, 1999). To define a symmetric function one must either indicate directly a power series as in the first construction, or give a symmetric polynomial in n indeterminates for every natural number n in a way compatible with the second construction. An expression in an unspecified number of indeterminates may do both, for instance e 2 = ∑ i < j X i X j {\displaystyle e_{2}=\sum _{i<j}X_{i}X_{j}\,} can be taken as the definition of an elementary symmetric function if the number of indeterminates is infinite, or as the definition of an elementary symmetric polynomial in any finite number of indeterminates. Symmetric polynomials for the same symmetric function should be compatible with the homomorphisms ρn (decreasing the number of indeterminates is obtained by setting some of them to zero, so that the coefficients of any monomial in the remaining indeterminates is unchanged), and their degree should remain bounded. (An example of a family of symmetric polynomials that fails both conditions is ∏ i = 1 n X i {\displaystyle \textstyle \prod _{i=1}^{n}X_{i}} ; the family ∏ i = 1 n ( X i + 1 ) {\displaystyle \textstyle \prod _{i=1}^{n}(X_{i}+1)} fails only the second condition.) Any symmetric polynomial in n indeterminates can be used to construct a compatible family of symmetric polynomials, using the homomorphisms ρi for i < n to decrease the number of indeterminates, and φi for i ≥ n to increase the number of indeterminates (which amounts to adding all monomials in new indeterminates obtained by symmetry from monomials already present). The following are fundamental examples of symmetric functions. The monomial symmetric functions mα. Suppose α = (α1,α2,...) is a sequence of non-negative integers, only finitely many of which are non-zero. Then we can consider the monomial defined by α: Xα = X1α1X2α2X3α3.... Then mα is the symmetric function determined by Xα, i.e. the sum of all monomials obtained from Xα by symmetry. For a formal definition, define β ~ α to mean that the sequence β is a permutation of the sequence α and set m α = ∑ β ∼ α X β . {\displaystyle m_{\alpha }=\sum \nolimits _{\beta \sim \alpha }X^{\beta }.} This symmetric function corresponds to the monomial symmetric polynomial mα(X1,...,Xn) for any n large enough to have the monomial Xα. The distinct monomial symmetric functions are parametrized by the integer partitions (each mα has a unique representative monomial Xλ with the parts λi in weakly decreasing order). Since any symmetric function containing any of the monomials of some mα must contain all of them with the same coefficient, each symmetric function can be written as an R-linear combination of monomial symmetric functions, and the distinct monomial symmetric functions therefore form a basis of ΛR as an R-module. The elementary symmetric functions ek, for any natural number k; one has ek = mα where X α = ∏ i = 1 k X i {\displaystyle \textstyle X^{\alpha }=\prod _{i=1}^{k}X_{i}} . As a power series, this is the sum of all distinct products of k distinct indeterminates. This symmetric function corresponds to the elementary symmetric polynomial ek(X1,...,Xn) for any n ≥ k. The power sum symmetric functions pk, for any positive integer k; one has pk = m(k), the monomial symmetric function for the monomial X1k. This symmetric function corresponds to the power sum symmetric polynomial pk(X1,...,Xn) = X1k + ... + Xnk for any n ≥ 1. The complete homogeneous symmetric functions hk, for any natural number k; hk is the sum of all monomial symmetric functions mα where α is a partition of k. As a power series, this is the sum of all monomials of degree k, which is what motivates its name. This symmetric function corresponds to the complete homogeneous symmetric polynomial hk(X1,...,Xn) for any n ≥ k. The Schur functions sλ for any partition λ, which corresponds to the Schur polynomial sλ(X1,...,Xn) for any n large enough to have the monomial Xλ. There is no power sum symmetric function p0: although it is possible (and in some contexts natural) to define p 0 ( X 1 , … , X n ) = ∑ i = 1 n X i 0 = n {\displaystyle \textstyle p_{0}(X_{1},\ldots ,X_{n})=\sum _{i=1}^{n}X_{i}^{0}=n} as a symmetric polynomial in n variables, these values are not compatible with the morphisms ρn. The "discriminant" ( ∏ i < j ( X i − X j ) ) 2 {\displaystyle \textstyle (\prod _{i<j}(X_{i}-X_{j}))^{2}} is another example of an expression giving a symmetric polynomial for all n, but not defining any symmetric function. The expressions defining Schur polynomials as a quotient of alternating polynomials are somewhat similar to that for the discriminant, but the polynomials sλ(X1,...,Xn) turn out to be compatible for varying n, and therefore do define a symmetric function. === A principle relating symmetric polynomials and symmetric functions === For any symmetric function P, the corresponding symmetric polynomials in n indeterminates for any natural number n may be designated by P(X1,...,Xn). The second definition of the ring of symmetric functions implies the following fundamental principle: If P and Q are symmetric functions of degree d, then one has the identity P = Q {\displaystyle P=Q} of symmetric functions if and only if one has the identity P(X1,...,Xd) = Q(X1,...,Xd) of symmetric polynomials in d indeterminates. In this case one has in fact P(X1,...,Xn) = Q(X1,...,Xn) for any number n of indeterminates. This is because one can always reduce the number of variables by substituting zero for some variables, and one can increase the number of variables by applying the homomorphisms φn; the definition of those homomorphisms assures that φn(P(X1,...,Xn)) = P(X1,...,Xn+1) (and similarly for Q) whenever n ≥ d. See a proof of Newton's identities for an effective application of this principle. == Properties of the ring of symmetric functions == === Identities === The ring of symmetric functions is a convenient tool for writing identities between symmetric polynomials that are independent of the number of indeterminates: in ΛR there is no such number, yet by the above principle any identity in ΛR automatically gives identities the rings of symmetric polynomials over R in any number of indeterminates. Some fundamental identities are ∑ i = 0 k ( − 1 ) i e i h k − i = 0 = ∑ i = 0 k ( − 1 ) i h i e k − i for all k > 0 , {\displaystyle \sum _{i=0}^{k}(-1)^{i}e_{i}h_{k-i}=0=\sum _{i=0}^{k}(-1)^{i}h_{i}e_{k-i}\quad {\mbox{for all }}k>0,} which shows a symmetry between elementary and complete homogeneous symmetric functions; these relations are explained under complete homogeneous symmetric polynomial. k e k = ∑ i = 1 k ( − 1 ) i − 1 p i e k − i for all k ≥ 0 , {\displaystyle ke_{k}=\sum _{i=1}^{k}(-1)^{i-1}p_{i}e_{k-i}\quad {\mbox{for all }}k\geq 0,} the Newton identities, which also have a variant for complete homogeneous symmetric functions: k h k = ∑ i = 1 k p i h k − i for all k ≥ 0. {\displaystyle kh_{k}=\sum _{i=1}^{k}p_{i}h_{k-i}\quad {\mbox{for all }}k\geq 0.} === Structural properties of ΛR === Important properties of ΛR include the following. The set of monomial symmetric functions parametrized by partitions form a basis of ΛR as a graded R-module, those parametrized by partitions of d being homogeneous of degree d; the same is true for the set of Schur functions (also parametrized by partitions). ΛR is isomorphic as a graded R-algebra to a polynomial ring R[Y1,Y2, ...] in infinitely many variables, where Yi is given degree i for all i > 0, one isomorphism being the one that sends Yi to ei ∈ ΛR for every i. There is an involutory automorphism ω of ΛR that interchanges the elementary symmetric functions ei and the complete homogeneous symmetric function hi for all i. It also sends each power sum symmetric function pi to (−1)i−1pi, and it permutes the Schur functions among each other, interchanging sλ and sλt where λt is the transpose partition of λ. Property 2 is the essence of the fundamental theorem of symmetric polynomials. It immediately implies some other properties: The subring of ΛR generated by its elements of degree at most n is isomorphic to the ring of symmetric polynomials over R in n variables; The Hilbert–Poincaré series of ΛR is ∏ i = 1 ∞ 1 1 − t i {\displaystyle \textstyle \prod _{i=1}^{\infty }{\frac {1}{1-t^{i}}}} , the generating function of the integer partitions (this also follows from property 1); For every n > 0, the R-module formed by the homogeneous part of ΛR of degree n, modulo its intersection with the subring generated by its elements of degree strictly less than n, is free of rank 1, and (the image of) en is a generator of this R-module; For every family of symmetric functions (fi)i>0 in which fi is homogeneous of degree i and gives a generator of the free R-module of the previous point (for all i), there is an alternative isomorphism of graded R-algebras from R[Y1,Y2, ...] as above to ΛR that sends Yi to fi; in other words, the family (fi)i>0 forms a set of free polynomial generators of ΛR. This final point applies in particular to the family (hi)i>0 of complete homogeneous symmetric functions. If R contains the field Q {\displaystyle \mathbb {Q} } of rational numbers, it applies also to the family (pi)i>0 of power sum symmetric functions. This explains why the first n elements of each of these families define sets of symmetric polynomials in n variables that are free polynomial generators of that ring of symmetric polynomials. The fact that the complete homogeneous symmetric functions form a set of free polynomial generators of ΛR already shows the existence of an automorphism ω sending the elementary symmetric functions to the complete homogeneous ones, as mentioned in property 3. The fact that ω is an involution of ΛR follows from the symmetry between elementary and complete homogeneous symmetric functions expressed by the first set of relations given above. The ring of symmetric functions ΛZ is the Exp ring of the integers Z. It is also a lambda-ring in a natural fashion; in fact it is the universal lambda-ring in one generator. === Generating functions === The first definition of ΛR as a subring of R [ [ X 1 , X 2 , . . . ] ] {\displaystyle R[[X_{1},X_{2},...]]} allows the generating functions of several sequences of symmetric functions to be elegantly expressed. Contrary to the relations mentioned earlier, which are internal to ΛR, these expressions involve operations taking place in R[[X1,X2,...;t]] but outside its subring ΛR[[t]], so they are meaningful only if symmetric functions are viewed as formal power series in indeterminates Xi. We shall write "(X)" after the symmetric functions to stress this interpretation. The generating function for the elementary symmetric functions is E ( t ) = ∑ k ≥ 0 e k ( X ) t k = ∏ i = 1 ∞ ( 1 + X i t ) . {\displaystyle E(t)=\sum _{k\geq 0}e_{k}(X)t^{k}=\prod _{i=1}^{\infty }(1+X_{i}t).} Similarly one has for complete homogeneous symmetric functions H ( t ) = ∑ k ≥ 0 h k ( X ) t k = ∏ i = 1 ∞ ( ∑ k ≥ 0 ( X i t ) k ) = ∏ i = 1 ∞ 1 1 − X i t . {\displaystyle H(t)=\sum _{k\geq 0}h_{k}(X)t^{k}=\prod _{i=1}^{\infty }\left(\sum _{k\geq 0}(X_{i}t)^{k}\right)=\prod _{i=1}^{\infty }{\frac {1}{1-X_{i}t}}.} The obvious fact that E ( − t ) H ( t ) = 1 = E ( t ) H ( − t ) {\displaystyle E(-t)H(t)=1=E(t)H(-t)} explains the symmetry between elementary and complete homogeneous symmetric functions. The generating function for the power sum symmetric functions can be expressed as P ( t ) = ∑ k > 0 p k ( X ) t k = ∑ k > 0 ∑ i = 1 ∞ ( X i t ) k = ∑ i = 1 ∞ X i t 1 − X i t = t E ′ ( − t ) E ( − t ) = t H ′ ( t ) H ( t ) {\displaystyle P(t)=\sum _{k>0}p_{k}(X)t^{k}=\sum _{k>0}\sum _{i=1}^{\infty }(X_{i}t)^{k}=\sum _{i=1}^{\infty }{\frac {X_{i}t}{1-X_{i}t}}={\frac {tE'(-t)}{E(-t)}}={\frac {tH'(t)}{H(t)}}} ((Macdonald, 1979) defines P(t) as Σk>0 pk(X)tk−1, and its expressions therefore lack a factor t with respect to those given here). The two final expressions, involving the formal derivatives of the generating functions E(t) and H(t), imply Newton's identities and their variants for the complete homogeneous symmetric functions. These expressions are sometimes written as P ( t ) = − t d d t log ⁡ ( E ( − t ) ) = t d d t log ⁡ ( H ( t ) ) , {\displaystyle P(t)=-t{\frac {d}{dt}}\log(E(-t))=t{\frac {d}{dt}}\log(H(t)),} which amounts to the same, but requires that R contain the rational numbers, so that the logarithm of power series with constant term 1 is defined (by log ⁡ ( 1 − t S ) = − ∑ i > 0 1 i ( t S ) i {\displaystyle \textstyle \log(1-tS)=-\sum _{i>0}{\frac {1}{i}}(tS)^{i}} ). == Specializations == Let Λ {\displaystyle \Lambda } be the ring of symmetric functions and R {\displaystyle R} a commutative algebra with unit element. An algebra homomorphism φ : Λ → R , f ↦ f ( φ ) {\displaystyle \varphi :\Lambda \to R,\quad f\mapsto f(\varphi )} is called a specialization. Example: Given some real numbers a 1 , … , a k {\displaystyle a_{1},\dots ,a_{k}} and f ( x 1 , x 2 , … , ) ∈ Λ {\displaystyle f(x_{1},x_{2},\dots ,)\in \Lambda } , then the substitution x 1 = a 1 , … , x k = a k {\displaystyle x_{1}=a_{1},\dots ,x_{k}=a_{k}} and x j = 0 , ∀ j > k {\displaystyle x_{j}=0,\forall j>k} is a specialization. Let f ∈ Λ {\displaystyle f\in \Lambda } , then ps ⁡ ( f ) := f ( 1 , q , q 2 , q 3 , … ) {\displaystyle \operatorname {ps} (f):=f(1,q,q^{2},q^{3},\dots )} is called principal specialization. == See also == Newton's identities Quasisymmetric function == References == Macdonald, I. G. Symmetric functions and Hall polynomials. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 1979. viii+180 pp. ISBN 0-19-853530-9 MR553598 Macdonald, I. G. Symmetric functions and Hall polynomials. Second edition. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. x+475 pp. ISBN 0-19-853489-2 MR1354144 Stanley, Richard P. Enumerative Combinatorics, Vol. 2, Cambridge University Press, 1999. ISBN 0-521-56069-1 (hardback) ISBN 0-521-78987-7 (paperback).
Wikipedia:Ringel–Hall algebra#0	In mathematics, a Ringel–Hall algebra is a generalization of the Hall algebra, studied by Claus Michael Ringel (1990). It has a basis of equivalence classes of objects of an abelian category, and the structure constants for this basis are related to the numbers of extensions of objects in the category. == References == Lusztig, George (1991), "Quivers, perverse sheaves, and quantized enveloping algebras", Journal of the American Mathematical Society, 4 (2): 365–421, CiteSeerX 10.1.1.454.3334, doi:10.1090/S0894-0347-1991-1088333-2, JSTOR 2939279, MR 1088333 Ringel, Claus Michael (1990), "Hall algebras and quantum groups", Inventiones Mathematicae, 101 (3): 583–591, Bibcode:1990InMat.101..583R, doi:10.1007/BF01231516, MR 1062796, S2CID 120480847 Schiffmann, Olivier (2006). "Lectures on Hall algebras". arXiv:math/0611617. == External links == Hubery, Andrew W., Introduction to Ringel–Hall algebras (PDF), Bielefeld University
Wikipedia:Risch algorithm#0	In symbolic computation, the Risch algorithm is a method of indefinite integration used in some computer algebra systems to find antiderivatives. It is named after the American mathematician Robert Henry Risch, a specialist in computer algebra who developed it in 1968. The algorithm transforms the problem of integration into a problem in algebra. It is based on the form of the function being integrated and on methods for integrating rational functions, radicals, logarithms, and exponential functions. Risch called it a decision procedure, because it is a method for deciding whether a function has an elementary function as an indefinite integral, and if it does, for determining that indefinite integral. However, the algorithm does not always succeed in identifying whether or not the antiderivative of a given function in fact can be expressed in terms of elementary functions. The complete description of the Risch algorithm takes over 100 pages. The Risch–Norman algorithm is a simpler, faster, but less powerful variant that was developed in 1976 by Arthur Norman. Some significant progress has been made in computing the logarithmic part of a mixed transcendental-algebraic integral by Brian L. Miller. == Description == The Risch algorithm is used to integrate elementary functions. These are functions obtained by composing exponentials, logarithms, radicals, trigonometric functions, and the four arithmetic operations (+ − × ÷). Laplace solved this problem for the case of rational functions, as he showed that the indefinite integral of a rational function is a rational function and a finite number of constant multiples of logarithms of rational functions . The algorithm suggested by Laplace is usually described in calculus textbooks; as a computer program, it was finally implemented in the 1960s. Liouville formulated the problem that is solved by the Risch algorithm. Liouville proved by analytical means that if there is an elementary solution g to the equation g′ = f then there exist constants αi and functions ui and v in the field generated by f such that the solution is of the form g = v + ∑ i < n α i ln ⁡ ( u i ) {\displaystyle g=v+\sum _{i<n}\alpha _{i}\ln(u_{i})} Risch developed a method that allows one to consider only a finite set of functions of Liouville's form. The intuition for the Risch algorithm comes from the behavior of the exponential and logarithm functions under differentiation. For the function f eg, where f and g are differentiable functions, we have ( f ⋅ e g ) ′ = ( f ′ + f ⋅ g ′ ) ⋅ e g , {\displaystyle \left(f\cdot e^{g}\right)^{\prime }=\left(f^{\prime }+f\cdot g^{\prime }\right)\cdot e^{g},\,} so if eg were in the result of an indefinite integration, it should be expected to be inside the integral. Also, as ( f ⋅ ( ln ⁡ g ) n ) ′ = f ′ ( ln ⁡ g ) n + n f g ′ g ( ln ⁡ g ) n − 1 {\displaystyle \left(f\cdot (\ln g)^{n}\right)^{\prime }=f^{\prime }\left(\ln g\right)^{n}+nf{\frac {g^{\prime }}{g}}\left(\ln g\right)^{n-1}} then if (ln g)n were in the result of an integration, then only a few powers of the logarithm should be expected. == Problem examples == Finding an elementary antiderivative is very sensitive to details. For instance, the following algebraic function (posted to sci.math.symbolic by Henri Cohen in 1993) has an elementary antiderivative, as Wolfram Mathematica since version 13 shows (however, Mathematica does not use the Risch algorithm to compute this integral): f ( x ) = x x 4 + 10 x 2 − 96 x − 71 , {\displaystyle f(x)={\frac {x}{\sqrt {x^{4}+10x^{2}-96x-71}}},} namely: F ( x ) = − 1 8 ln ( ( x 6 + 15 x 4 − 80 x 3 + 27 x 2 − 528 x + 781 ) x 4 + 10 x 2 − 96 x − 71 − ( x 8 + 20 x 6 − 128 x 5 + 54 x 4 − 1408 x 3 + 3124 x 2 + 10001 ) ) + C . {\displaystyle {\begin{aligned}F(x)=-{\frac {1}{8}}\ln &\,{\Big (}(x^{6}+15x^{4}-80x^{3}+27x^{2}-528x+781){\sqrt {x^{4}+10x^{2}-96x-71}}{\Big .}\\&{}-{\Big .}(x^{8}+20x^{6}-128x^{5}+54x^{4}-1408x^{3}+3124x^{2}+10001){\Big )}+C.\end{aligned}}} But if the constant term 71 is changed to 72, it is not possible to represent the antiderivative in terms of elementary functions, as FriCAS also shows. Some computer algebra systems may here return an antiderivative in terms of non-elementary functions (i.e. elliptic integrals), which are outside the scope of the Risch algorithm. For example, Mathematica returns a result with the functions EllipticPi and EllipticF. Integrals in the form ∫ x + A x 4 + a x 3 + b x 2 + c x + d d x {\displaystyle \int {\frac {x+A}{\sqrt {x^{4}+ax^{3}+bx^{2}+cx+d}}}\,dx} were solved by Chebyshev (and in what cases it is elementary), but the strict proof for it was ultimately done by Zolotarev. The following is a more complex example that involves both algebraic and transcendental functions: f ( x ) = x 2 + 2 x + 1 + ( 3 x + 1 ) x + ln ⁡ x x x + ln ⁡ x ( x + x + ln ⁡ x ) . {\displaystyle f(x)={\frac {x^{2}+2x+1+(3x+1){\sqrt {x+\ln x}}}{x\,{\sqrt {x+\ln x}}\left(x+{\sqrt {x+\ln x}}\right)}}.} In fact, the antiderivative of this function has a fairly short form that can be found using substitution u = x + x + ln ⁡ x {\displaystyle u=x+{\sqrt {x+\ln x}}} (SymPy can solve it while FriCAS fails with "implementation incomplete (constant residues)" error in Risch algorithm): F ( x ) = 2 ( x + ln ⁡ x + ln ⁡ ( x + x + ln ⁡ x ) ) + C . {\displaystyle F(x)=2\left({\sqrt {x+\ln x}}+\ln \left(x+{\sqrt {x+\ln x}}\right)\right)+C.} Some Davenport "theorems" are still being clarified. For example in 2020 a counterexample to such a "theorem" was found, where it turns out that an elementary antiderivative exists after all. == Implementation == Transforming Risch's theoretical algorithm into an algorithm that can be effectively executed by a computer was a complex task which took a long time. The case of the purely transcendental functions (which do not involve roots of polynomials) is relatively easy and was implemented early in most computer algebra systems. The first implementation was done by Joel Moses in Macsyma soon after the publication of Risch's paper. The case of purely algebraic functions was partially solved and implemented in Reduce by James H. Davenport – for simplicity it could only deal with square roots and repeated square roots and not general radicals or other non-quadratic algebraic relations between variables. The general case was solved and almost fully implemented in Scratchpad, a precursor of Axiom, by Manuel Bronstein, there is Axiom's fork FriCAS, with active Risch and other algorithm development on github. However, the implementation did not include some of the branches for special cases completely. Currently in 2025, there is no known full implementation of the Risch algorithm. == Decidability == The Risch algorithm applied to general elementary functions is not an algorithm but a semi-algorithm because it needs to check, as a part of its operation, if certain expressions are equivalent to zero (constant problem), in particular in the constant field. For expressions that involve only functions commonly taken to be elementary it is not known whether an algorithm performing such a check exists (current computer algebra systems use heuristics); moreover, if one adds the absolute value function to the list of elementary functions, then it is known that no such algorithm exists; see Richardson's theorem. This issue also arises in the polynomial division algorithm; this algorithm will fail if it cannot correctly determine whether coefficients vanish identically. Virtually every non-trivial algorithm relating to polynomials uses the polynomial division algorithm, the Risch algorithm included. If the constant field is computable, i.e., for elements not dependent on x, then the problem of zero-equivalence is decidable, so the Risch algorithm is a complete algorithm. Examples of computable constant fields are ℚ and ℚ(y), i.e., rational numbers and rational functions in y with rational-number coefficients, respectively, where y is an indeterminate that does not depend on x. This is also an issue in the Gaussian elimination matrix algorithm (or any algorithm that can compute the nullspace of a matrix), which is also necessary for many parts of the Risch algorithm. Gaussian elimination will produce incorrect results if it cannot correctly determine whether a pivot is identically zero. == See also == Axiom (computer algebra system) Closed-form expression Incomplete gamma function Lists of integrals Liouville's theorem (differential algebra) Nonelementary integral Symbolic integration == Notes == == References == Bronstein, Manuel (1990). "Integration of elementary functions". Journal of Symbolic Computation. 9 (2): 117–173. doi:10.1016/s0747-7171(08)80027-2. Bronstein, Manuel (1998). "Symbolic Integration Tutorial" (PDF). ISSAC'98, Rostock (August 1998) and Differential Algebra Workshop, Rutgers. Bronstein, Manuel (2005). Symbolic Integration I. Springer. ISBN 3-540-21493-3. Davenport, James H. (1981). On the integration of algebraic functions. Lecture Notes in Computer Science. Vol. 102. Springer. ISBN 978-3-540-10290-8. Geddes, Keith O.; Czapor, Stephen R.; Labahn, George (1992). Algorithms for computer algebra. Boston, MA: Kluwer Academic Publishers. pp. xxii+585. Bibcode:1992afca.book.....G. doi:10.1007/b102438. ISBN 0-7923-9259-0. Moses, Joel (2012). "Macsyma: A personal history". Journal of Symbolic Computation. 47 (2): 123–130. doi:10.1016/j.jsc.2010.08.018. Risch, R. H. (1969). "The problem of integration in finite terms". Transactions of the American Mathematical Society. 139. American Mathematical Society: 167–189. doi:10.2307/1995313. JSTOR 1995313. Risch, R. H. (1970). "The solution of the problem of integration in finite terms". Bulletin of the American Mathematical Society. 76 (3): 605–608. doi:10.1090/S0002-9904-1970-12454-5. Rosenlicht, Maxwell (1972). "Integration in finite terms". American Mathematical Monthly. 79 (9). Mathematical Association of America: 963–972. doi:10.2307/2318066. JSTOR 2318066. == External links == Bhatt, Bhuvanesh. "Risch Algorithm". MathWorld.
Wikipedia:Rob Eastaway#0	Rob Eastaway is an English author. He is active in the popularisation of mathematics and was awarded the Zeeman medal in 2017 for excellence in the promotion of maths. He is best known for his books, including the bestselling Why Do Buses Come in Threes? and Maths for Mums and Dads. His first book was What is a Googly?, an explanation of cricket for Americans and other newcomers to the game. Eastaway is a keen cricketer and was one of the originators of the International Rankings of Cricketers. He is also a puzzle setter and adviser for New Scientist magazine and he has appeared frequently on BBC Radio 4 and 5 Live. He is the director of Maths Inspiration, a national programme of maths lectures for teenagers which involves some of the UK’s leading maths speakers. He was president of the UK Mathematical Association for 2007/2008. He is a former pupil of The King's School, Chester, and has a degree in engineering and management science from the University of Cambridge. == Books == 1992: What is a Googly? 1995: The Guinness Book of Mindbenders, co-author David Wells 1998: Why do Buses Come in Threes?, co-author Jeremy Wyndham, foreword by Tim Rice 1999: The Memory Kit 2002: How Long is a Piece of String?, co-author Jeremy Wyndham 2004: How to Remember 2005: How to Take a Penalty, co-author John Haigh 2007: How to Remember (Almost) Everything Ever 2007: Out of the Box 2008: How Many Socks Make a Pair? 2009: Improve Your Memory Today, with Dr Hilary Jones 2010: Maths for Mums and Dads, co-author Mike Askew 2011: The Hidden Mathematics of Sport (new edition of Beating the Odds) 2013: More Maths for Mums and Dads, co-author Mike Askew 2016: Maths on the Go, co-author Mike Askew 2017: Any ideas? Tips and Techniques to Help You Think Creatively 2018: 100 Maddening Mindbending Puzzles 2019: Maths On The Back of an Envelope 2023: Headscratchers - The New Scientist Puzzle Book 2024: Much Ado About Numbers == References == == External links == Rob Eastaway's Official Website Maths Inspiration Website Eastaway, Rob. "Zequals". Numberphile. Brady Haran. [1]
Wikipedia:Rob Reitzen#0	Rob Reitzen is an American mathematician and professional gambler. == Biography == Reitzen attended University of California, Los Angeles, where he studied mathematics and probabilities. He was initially interested in poker. However, his focus shifted to blackjack after discovering and studying Lawrence Revere's book, Playing Blackjack as a Business. In the early 1980s, coinciding with the rise of home computing, Reitzen collaborated with a colleague who later became influential in horse betting technology in Hong Kong. Together, they utilized emerging computer technology to develop new blackjack techniques. These techniques included sophisticated card counting systems, shuffle tracking, and improved methods for memorizing card sequences. In the early 1990s, Reitzen co-founded CORE, a company that specialized in providing financial and operational support to Native American casinos, including supplying both the bankroll and dealers for blackjack games. Retizen also specialized in hold'em poker, developing a mathematical strategy that resulted in forming a successful team under his guidance, which achieved notable success on the Full Tilt Poker platform. Later, Reitzen developed another mathematical technique with John Wayne and Darrell Miers that was provided as a service to casinos as well as stock and futures markets. In 1997, Reitzen's methods and achievements were notably featured in a Esquire magazine article titled "Fleecing Las Vegas." The article described his use of a technique known as "The Hammer" at the blackjack tables of Caesars Palace, where he reportedly won $500,000. "The Hammer" involved a combination of card counting, shuffle tracking, ace location, and card sequence memorization. In 2019, Reitzen was inducted into the Blackjack Hall of Fame. Reitzen is the founder of Random Order Inc. He is also the co-founder of StyleScan, a B2B software specializing in AI. == Recognition == 2019: Blackjack Hall of Fame == References ==
Wikipedia:Robert Aumann#0	Robert John Aumann (Yisrael Aumann, Hebrew: ישראל אומן; born June 8, 1930) is an Israeli-American mathematician, and a member of the United States National Academy of Sciences. He is a professor at the Center for the Study of Rationality in the Hebrew University of Jerusalem. He also holds a visiting position at Stony Brook University, and is one of the founding members of the Stony Brook Center for Game Theory. Aumann received the Nobel Memorial Prize in Economic Sciences in 2005 for his work on conflict and cooperation through game theory analysis. He shared the prize with Thomas Schelling. == Early life and education == Aumann was born in Frankfurt am Main, Germany, and fled to the United States with his family in 1938, two weeks before the Kristallnacht pogrom. He attended the Rabbi Jacob Joseph School, a yeshiva high school in New York City. Aumann graduated from the City College of New York in 1950 with a B.S. in mathematics. He received his M.S. in 1952, and his Ph.D. in Mathematics in 1955, both from the Massachusetts Institute of Technology. His doctoral dissertation, Asphericity of Alternating Linkages, concerned knot theory. His advisor was George Whitehead, Jr. == Academic career == In 1956 he joined the Mathematics faculty of the Hebrew University of Jerusalem and has been a visiting professor at Stony Brook University since 1989. He has held visiting professorship at the University of California, Berkeley (1971, 1985–1986), Stanford University (1975–1976, 1980–1981), and Universite Catholique de Louvain (1972, 1978, 1984). === Mathematical and scientific contribution === Aumann's greatest contribution was in the realm of repeated games, which are situations in which players encounter the same situation over and over again. Aumann was the first to define the concept of correlated equilibrium in game theory, which is a type of equilibrium in non-cooperative games that is more flexible than the classical Nash equilibrium. Furthermore, Aumann has introduced the first purely formal account of the notion of common knowledge in game theory. He collaborated with Lloyd Shapley on the Aumann–Shapley value. He is also known for Aumann's agreement theorem, in which he argues that under his given conditions, two Bayesian rationalists with common prior beliefs cannot agree to disagree. Aumann and Maschler used game theory to analyze Talmudic dilemmas. They were able to solve the mystery about the "division problem", a long-standing dilemma of explaining the Talmudic rationale in dividing the heritage of a late husband to his three wives depending on the worth of the heritage compared to its original worth. The article in that matter was dedicated to a son of Aumann, Shlomo, who was killed during the 1982 Lebanon War, while serving as a tank gunner in the Israel Defense Forces's armored corps. Aumann's Ph.D. students include David Schmeidler, Sergiu Hart, Abraham Neyman, and Yair Tauman. === Torah codes controversy === Aumann has entered the controversy of Bible codes research. In his position as both a religious Jew and a man of science, the codes research holds special interest to him. He has partially vouched for the validity of the "Great Rabbis Experiment" by Doron Witztum, Eliyahu Rips, and Yoav Rosenberg, which was published in Statistical Science. Aumann not only arranged for Rips to give a lecture on Torah codes in the Israel Academy of Sciences and Humanities, but sponsored the Witztum-Rips-Rosenberg paper for publication in the Proceedings of the National Academy of Sciences. The academy requires a member to sponsor any publication in its Proceedings; the paper was turned down however. In 1996, a committee consisting of Robert J. Aumann, Dror Bar-Natan, Hillel Furstenberg, Isaak Lapides, and Rips, was formed to examine the results that had been reported by H.J. Gans regarding the existence of "encoded" text in the bible foretelling events that took place many years after the Bible was written. The committee performed two additional tests in the spirit of the Gans experiments. Both tests failed to confirm the existence of the putative code. After a long analysis of the experiment and the dynamics of the controversy, stating for example that "almost everybody included [in the controversy] made up their mind early in the game" Aumann concluded: "A priori, the thesis of the Codes research seems wildly improbable... Research conducted under my own supervision failed to confirm the existence of the codes – though it also did not establish their non-existence. So I must return to my a priori estimate, that the Codes phenomenon is improbable". == Political views == These are some of the themes of Aumann's Nobel lecture, named "War and Peace": War is not irrational, but must be scientifically studied in order to be understood, and eventually conquered; Repeated game study de-emphasizes the "now" for the sake of the "later"; Simplistic peacemaking can cause war, while an arms race, credible war threats and mutually assured destruction can reliably prevent war. Aumann is a member of Professors for a Strong Israel (PSI), a right-wing political group. Aumann opposed the disengagement from Gaza in 2005 claiming that it was a crime against Gush Katif settlers and a serious threat to the security of Israel. Aumann drew on a case in game theory called the Blackmailer Paradox to argue that giving land to the Arabs is strategically foolish based on the mathematical theory. By presenting an unyielding demand, he claims that the Arab states will force Israel to "yield to blackmail due to the perception that it will leave the negotiating room with nothing if it is inflexible". As a result of his political views, and his use of his research to justify them, the decision to give him the Nobel prize was criticized in the European press. A petition to cancel his prize garnered signatures from 1,000 academics worldwide. In a speech to the religious Zionist youth movement, Bnei Akiva, Aumann argued that Israel is in "deep trouble" due to his belief that anti-Zionist Satmar Jews might have been right in their condemnation of the original Zionist movement. "I fear the Satmars were right", he said, and quoted a verse from Psalm 127: "Unless the Lord builds a house, its builders toil on it in vain." Aumann feels that the historical Zionist establishment failed to transmit its message to its successors, because it was secular. The only way that Zionism can survive, according to Aumann, is if it has a religious basis. In 2008, Aumann joined the right-wing religious Zionist Ahi political party, which was led at the time by Effi Eitam and Yitzhak Levy. == Personal life == Aumann married Esther Schlesinger in April 1955 in Brooklyn. They had met in 1953, when Esther, who was from Israel, was visiting the United States. The couple had five children; the oldest, Shlomo, a student in Yeshivat Shaalvim, was killed in action while serving as a tank gunner in the Israel Defense Forces's armored corps in the 1982 Lebanon War. Machon Shlomo Aumann, an institute affiliated with Shaalvim that republishes old manuscripts of Jewish legal texts, was named after him. Esther died of ovarian cancer in October 1998. In late November 2005, Aumann married Esther's widowed sister, Batya Cohn. Aumann is a cousin of the late Oliver Sacks. == Honours and awards == 1974: Foreign Honorary Member of the American Academy of Arts and Sciences 1983: Harvey Prize in Science and Technology. 1994: Israel Prize for economics. 1998: Erwin Plein Nemmers Prize in Economics from Northwestern University. 2002: The EMET Prize in the Social Sciences category, for Economics 2005: Nobel Memorial Prize in Economic Sciences (share US$1.3 million prize with Thomas Schelling). 2006: Yakir Yerushalayim (Worthy Citizen of Jerusalem) award from the city of Jerusalem. == Publications == 1956: Asphericity of alternating knots, Annals of Mathematics 64: 374–92 doi:10.2307/1969980 1958: (with Joseph Kruskal) The Coefficients in an Allocation Problem, Naval Research Logistics 1960: Acceptable Points in Games of Perfect Information, Pacific Journal of Mathematics 10 (1960), pp. 381–417 1974: (with L.S. Shapley) Values of Non-Atomic Games, Princeton University Press 1981: (with Y. Tauman and S. Zamir) Game Theory, volumes 1 & 2 (in Hebrew), Everyman's University, Tel Aviv 1989: Lectures on Game Theory, Underground Classics in Economics, Westview Press 1992, 1994, 2002: (coedited with Sergiu Hart) Handbook of Game Theory with Economic Applications, volumes 1,2 & 3 Elsevier 1995: (with M. Maschler) Repeated Games with Incomplete Information, MIT Press 2000: Collected Papers, volumes 1 & 2, MIT Press. 2015: (with I. Arieli) The Logic of Backward Induction, Journal of Economic Theory 159 (2015), pp. 443–464 == See also == List of Israel Prize recipients List of Israeli Nobel laureates List of Jewish Nobel laureates List of economists == References == == External links == Official homepage Robert J. Aumann on Nobelprize.org
Wikipedia:Robert B. Lisek#0	Robert B. Lisek is a Polish artist and mathematician who focuses on systems and processes, conducts a research in the area theory of ordered sets in relation with logic, algebra and combinatorics; his artistic practice draws upon conceptual art, radical art strategies, hacktivism, bioart, software art. == Works == Lisek is an artist whose work is focused on systems and processes (computer, biological and social), disrupting the language of these systems, including rules, commands, errors and by using worms and computer viruses. Lisek is currently researching problems of security, privacy and identity in networked societies. He built NEST – Citizens Intelligent Agency, a piece of software for searching hidden patterns and links between people, groups, events, objects and places. Lisek is also a scientist focused on the computational complexity theory, graph theory and order theory. He studied at the Department of Logic of Wroclaw University, at the Fine Art Academy in Wroclaw and the PWSTiTV film school in Łódź. His research interest is also artificial general intelligence (AGI). He examines, among others: problem of self-reference, mathematical induction, probabilistic techniques and recurrent AI self-improvement. He is also working on human enhancement: extensions through the use of radical transgressive methods that arise at the intersection of disciplines such as AGI, bioengineering, and political and social sciences. He has prepared an anthology entitled Transhuman. Lisek is a founder of Institute for Research in Science and Art, Fundamental Research Lab and an ACCESS art symposium. == Exhibitions == Lisek exhibits, lectures, and conducts workshops worldwide. His projects include among others: NEST – FILE Electronic Language International Festival, São Paulo NEST – ARCO International Contemporary Art Fair, Madrid FLOAT – Lower Manhattan Cultural Council, NY FLOAT – Harvestworks Digital Media Art Center, NY SPECTRUM – Leto Gallery, Warsaw WWAI – SIGGRAPH 2005, Los Angeles Falsecodes – Red Gate Gallery & Planetary Collegium, Beijing GENGINE – Zacheta National Gallery, Warsaw FLEXTEXT – CiberArt Bilbao FLEXTEXT – Medi@terra – Byzantine Museum, Athens FXT – ACA Japan Media Festival, Tokyo STACK – ISEA 02, Nagoya SSSPEAR – 17th Meridian, WRO Art Center, Wroclaw The New Art Fest (2020), digital art festival, Lisbon == References == == External links == Networked_Performance — GESPENST / WIDMO / SPECTRE [Warsaw]. turbulence.org, July 16, 2011 Archived 2011-07-16 at the Wayback Machine http://www.recyklingidei.pl/vitcheva_wypadek_laboratorium http://portal.acm.org/citation.cfm?id=1186796.1186804 http://music.columbia.edu/organism/?s=robert+lisek http://knowledgetoday.org/wiki/index.php/ICCS07/72 http://fundamental.art.pl/ http://lisek.art.pl/gespenst.html
Wikipedia:Robert D. Hough#0	Robert D. Hough is an American born mathematician specializing in number theory, probability, and discrete mathematics. He is currently an associate professor of mathematics at Stony Brook University. == Early life and education == Hough holds BS in Math, MS in CS, and PhD in Math degrees from Stanford University. He completed his PhD under Kannan Soundararajan in 2012. Hough was a post-doctoral researcher at Cambridge University and Oxford University in the United Kingdom working with Ben Green from 2013 to 2015, and was a post-doctoral member of the Institute for Advanced Study, Princeton, New Jersey from 2015 to 2016. == Career == Hough joined Stony Brook University as an assistant professor in 2016 and has been an associate professor of mathematics since 2022. == Achievements == Hough won the Mathematical Association of America's David P. Robbins Prize at the Joint Math Meetings in 2017. The prize was given for finding the solution of a problem imposed by Paul Erdős. In February 2020, Hough won the Sloan Research Fellowship. He also won a Trustees Faculty Award from Stony Brook University. == References ==
Wikipedia:Robert Dautray#0	Robert Dautray (French pronunciation: [ʁɔbɛʁ dotʁɛ]; 1 February 1928 – 20 August 2023) was a French engineer, scientific director of the French Commissariat à l'Energie Atomique (CEA) and High Commissioner for Atomic Energy. He was a member of the French Academy of Sciences, section mechanical and computer sciences, and of the French Academy of Technology. == Biography == Ignace Robert Kouchelevitz was born on 1 February 1928, in the 10th arrondissement of Paris, France, to a Belarusian father who came to France in 1905 and a Ukrainian mother who came to France in 1902, he escaped the Holocaust during the Second World War. After the war, he prepared as a free candidate for the entrance exam to the École nationale des arts et métiers. He was awarded the top promotion in the promotion that entered Paris in 1945 (Pa45 promotion). On the advice of his professors, he passed the École polytechnique exam in 1949, where he graduated as a major, then joined the CEA Saclay in the mathematical physics department headed by Jacques Yvon, Jules Horowitz, Albert Messiah, Anatole Abragam, Claude Bloch, and others. Scientific Director of the CEA, he contributes to the development of atomic applications after scientific work on isotopic regulation and the construction of experimental reactors (Grenoble high flux reactor). He is working on the process of separating uranium isotopes. He is the director of the Phébus large laser program. Robert Dautray was High Commissioner for Atomic Energy from 1993 to 1998. Robert Dautray recounted his memories, especially his difficult youth, in his book of Memoirs, published in 2007. Chairman of the Scientific Programs Committee of the National Space Center (CNES). Dautray also addressed the problems of climate change (radiative transfer: greenhouse effect. Dautray died in Paris on 20 August 2023, at the age of 95. == Scientific work == Almost all of Robert Dautray's professional activity has been devoted to the physical sciences contributing to nuclear energy, both in reactor physics (reactor control and command, breeder reactor physics, Pegasus research reactors, high-flow reactors from the Von Laue Langevin Institute, etc.) and in the physics of reactors.) and upstream of the fuel cycle (control command of the uranium isotope separation plant) as well as downstream of this cycle (formation and physics of plutonium and other actinides isotopes, descendants of fission products, activated structure nuclei, etc.). In addition, Robert Dautray participated in the establishment of the basic physical sciences for the sciences of high densities and powers of materials and electromagnetic radiation (state equations, opacity, radiative transfer, discontinuities of high velocity flows, interface instabilities, laser implosions, thermonuclear reactions, non-linear neutronics of high velocity media of nuclei making the neutron transport and plasma physics equations non-linear, etc.). Robert Dautray contributed to the development of the mathematical methods necessary to model these phenomena. Robert Dautray co-chaired with the EDF Studies and Research Department the CEA/EDF digital analysis summer schools. There was a controversy over the attribution to Dautray of the paternity of the French H-bomb. Experts dispute it, highlighting Michel Carayol's work. == Selected works == Scientific works, co-authored by Dautray, non-exhaustive list: Statics and dynamics of nuclear power plants and engines; service of mathematical physics, published by the course qhe RD taught at CEA/Saclay/ INSTN. 1957. The Pegasus fuel material test reactor, in collaboration with Pierre Arditti and R. Raievski, CEA Cadarache, United Nations Conference in Geneva, 1960. Project for a high neutron flux reactor as part of work with Paul Ageron et al, CEA: 1964 Project studies for the German French High flux reactor ILL, in collaboration with H. Beckurts. Proceedings Los Alamos and Santa Fe, New Mexico conference, pages 281-310, USA.1966. Conference on plasma physics and controlled nuclear fusion, CEA, Limeil, 1984. Monte Carlo methods and applications in neutronics, photonics and statistical physics in collaboration with Alcouffe et al. Springer-verlag, Lectures notes in physics, volume 240, 1985. The greenhouse effect and its climatic consequences: Chairmanship of the working group of the Academy of Sciences that drafted the report (number 25), 1990. Laser/matter interaction work at CEA/Limeil, in collaboration with Berthier et al. IAEA CN/50 Kyoto, 1990 Civil nuclear energy in the context of climate change. RD report to the Academy of Sciences TEC/DOC Lavoisier; 330 pages, 2001. Plutonium isotopes and their descendants by neutron absorption and/or decay. RD Report to the Academy of Sciences, 238 pages, TEC/DOC Lavoisier, 2005. Security and hostile use of nuclear energy: from physics to biology. RD report to the Academy of Sciences, 176 pages, TEC/DOC Lavoisier, 2007. Energy: towards nuclear breeders installations before the end of the century? In collaboration with J. Friedel. Comptes rendus de l'académie des sciences, Mécanique volume 335, pages 51–74, Elsevier., 2007. Supergenerators; the state of materials at high irradiation, high local power and temperature, their gradients and mechanical properties, adapted to the resulting stresses, in collaboration with J. Friedel, CR of the Academy of Sciences, Mechanics, volume 338, pages 649-655, Elsevier, 2010. The long term future for civilian nuclear power generation in France. The case for breeder reactors; novelties and issues; CR de l'Académie des sciences, Mécanique; volume 338, pages 369-387, Elsevier, 2011. Reflections on the future of nuclear energy, from today's France to tomorrow's world, from the 2nd to the 3rd generation, with J. Friedel and Y. Bréchet, CR of the Academy of Sciences, Physics, 2012, Elsevier. Science of nuclear safety post Fukushima; in collaboration with E. Brézin et al. CR Physique, volume13, pages 337-382, Elsevier, 2012. Control and limit the dispersion of radioactive products from nuclear power plants in the event of an accident. RD with J. Friedel and Y. Bréchet, CR Physique, volume 15, pages 481-508, Elsevier, 2014. == Participation in scientific works == "Mathematical analysis and numerical computation for science and technology", in collaboration with JL Lions and his colleagues, CEA/Masson". 4000-page book, first published in three volumes in the CEA collection by Masson, then reissued in eight volumes, CEA INSTN collection published by Eyrolles, and reissued in six volumes in English by Springer; then in paperback by Springer. "Probabilistic methods for the equations of physics" with P. L. Lions, R. Sentis, M. Cessenat, G. Ledanois and E. Pardoux, 1989. "La fusion thermonucléaire inertielle par laser", with JP Watteau et al, in five volumes CEA Eyrolles collection, 1993. == Distinctions == Légion d'Honneur: Grand Croix of the Légion d'Honneur (2007); Member of the French Academy of Sciences (section of mechanical and computer sciences) since February 7, 1977; Laplace Prize of the French Academy of Sciences: 1951 Lamb Prize of the French Academy of Sciences: 1975 Nessim Habif Prize for Arts et Métiers (1977) Fellow of the Los Alamos Laboratory Edward Teller Medal for laser fusion at the Lawrence Livermore National Laboratory (1993) == References ==
Wikipedia:Robert Ditchburn#0	Robert William Ditchburn (14 January 1903 – 8 April 1987) was an English physicist whose career started as Erasmus Smith's Professor of Natural and Experimental Philosophy at Trinity College Dublin (1929-1946), and ended at the University of Reading, where he worked hard to build up the physics department. == Life and career == Ditchburn was born in Waterloo, near Liverpool, England, and was educated first at Liverpool University, taking a physics degree there in 1922. He then went to Trinity College, Cambridge, earning BA (1924) and a PhD (1928) for research done under J. J. Thomson at the Cavendish Laboratory. He successfully competed for a Fellowship at TCD in 1928, and the following year moved to Ireland to become Erasmus Smith's Professor of Natural and Experimental Philosophy. In 1930 he was elected a member of the Royal Irish Academy and delivered one of the Donnellan Lectures in 1945. Apart from a few years back in England at the Admiralty Research Laboratory in Teddington during WWII, he remained in Dublin until 1946. He then became professor and head of the department of physics at Reading University, where he remained until 1968. While there, he focussed on building up the department, and set up the J.J. Thomson Physical Laboratory. He authored the book Light (Interscience Publishers, Inc, 1953). His own research included work on photoionization, the optical properties of solids and the effects of eye movements on visual perception, in particular methods for stabilizing retinal images. In 1962, he became a Fellow of the Royal Society. He was very active in retirement, both as a consultant for the diamond industry, and working for nuclear disarmament in Pugwash movement. He published the book Eye Movements and Visual Perception (Clarendon Press, 1973) and in 1983 he was awarded the C. E. K. Mees Medal by The Optical Society "for his lengthy career in many disciplines of optics and for his enrichment of optical knowledge". In 1960 he got the Thomas Young Orator Prize. == References ==
Wikipedia:Robert Edmund Edwards#0	Robert Edmund Edwards (1926–2000), usually cited simply as R. E. Edwards, was a British-born Australian mathematician who specialized in functional analysis. He is the author of several volumes in Springer's Graduate Texts in Mathematics. He received his PhD at Birkbeck College, University of London in 1951 under Lionel Cooper. His dissertation topic was Theory of Normed Rings, and Translations in Function Spaces. He continued to teach there as a lecturer until 1959, and then spent a few years at Manchester, before migrating to Australia in 1961, where he worked at the Institute of Advanced Studies at ANU as a professorial fellow (1961-1970) and professor of mathematics (1970-1978). == Selected publications == Functional Analysis: Theory and Applications. Holt, Rinehart, and Winston, 1965; revised edition Dover Publications, 1995; ISBN 0-486-68143-2. with Garth Ian Gaudry: Littlewood-Paley and multiplier theory. Springer-Verlag, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 90, 1977. 2012 pbk reprint A Formal Background to Mathematics: Volume 1, Logic, Sets and Numbers. Springer-Verlag, 1979; ISBN 978-0-387-90431-3. A Formal Background to Mathematics: Volume 2, A Critical Approach to Elementary Analysis. Springer-Verlag, 1980; ISBN 978-0-387-90513-6. Graduate Texts in Mathematics: Fourier Series, A Modern Introduction. Volumes 1 and 2. Holt, Rinehart, and Winston, 1967; 2nd edition, Springer-Verlag, 1982; ISBN 978-1-4613-8158-7. Integration and Harmonic Analysis on Compact Groups. Cambridge University Press, 1972; ISBN 978-0-521-09717-8. == References ==
Wikipedia:Robert Goldblatt#0	Robert Ian Goldblatt (born 1949) is a mathematical logician who is emeritus Professor in the School of Mathematics and Statistics at Victoria University, Wellington, New Zealand. His doctoral advisor was Max Cresswell. His most popular books are Logics of Time and Computation and Topoi: the Categorial Analysis of Logic. He has also written a graduate level textbook on hyperreal numbers which is an introduction to nonstandard analysis. In 1987 he took "a trip on Einstein's train" to develop hyperbolic orthogonality, the geometry of relativity of simultaneity. He has been Coordinating Editor of The Journal of Symbolic Logic and a Managing Editor of Studia Logica. He was elected Fellow and Councillor of the Royal Society of New Zealand, President of the New Zealand Mathematical Society, and represented New Zealand to the International Mathematical Union. In 2012 the Royal Society of New Zealand awarded him the Jones Medal for lifetime achievement in mathematics. == Books and handbook chapters == 1979: Topoi: The Categorial Analysis of Logic, North-Holland. Revised edition 1984. Dover Publications edition 2006. Internet edition, Project Euclid. Benjamin C. Pierce recommends it as an "excellent beginner book", praising it for the use of simple set-theoretic examples and motivating intuitions, but noted that it "is sometimes criticized by category theorists for being misleading on some aspects of the subject, and for presenting long and difficult proofs where simple ones are available." But the preface of the Dover edition observes (p. xv) that "This is a book about logic, rather than category theory per se. It aims to explain, in an introductory way, how certain logical ideas are illuminated by a category-theoretic perspective." 1982: Axiomatising the Logic of Computer Programming, Lecture Notes in Computer Science 130, Springer-Verlag. 1987: Orthogonality and Spacetime Geometry, Universitext Springer-Verlag ISBN 0-387-96519-X MR0888161 1987: Logics of Time and Computation. CSLI Lecture Notes, 7. Stanford University, Center for the Study of Language and Information MR1191162. Second edition 1992. 1993: Mathematics of Modality, CSLI Publications, ISBN 978-1-881526-24-7 MR1317099 1998: Lectures on the Hyperreals: An Introduction to Nonstandard Analysis. Graduate Texts in Mathematics, 188. Springer-Verlag. Reviewer Perry Smith for MathSciNet wrote: "The author's ideas on how to achieve both intelligibility and rigor, explained in the preface, will be useful reading for anyone intending to teach nonstandard analysis." 2006: "Mathematical Modal Logic: a View of its Evolution" in Modalities in the Twentieth Century, Volume 7 of the Handbook of the History of Logic, edited by Dov M. Gabbay and John Woods, Elsevier, pp. 1–98. 2011: Quantifiers, Propositions and Identity: Admissible Semantics for Quantified Modal and Substructural Logics, Cambridge University Press and the Association for Symbolic Logic. == See also == Influence of non-standard analysis == References == == External links == Home page Robert Goldblatt at the Mathematics Genealogy Project
Wikipedia:Robert J. Berman#0	Robert J. Berman is a Swedish mathematical scientist currently at Chalmers University and was awarded the Göran Gustafsson Prize in 2017. Berman is known for his constributions to the K-stability of Fano varieties. == References ==
Wikipedia:Robert Liptser#0	Robert Sh. Liptser (Russian: Роберт Шевелевич Липцер; Hebrew: רוברט ליפצר, 20 March 1936 – 2 January 2019) was a Russian-Israeli mathematician who made contributions to the theory and applications of stochastic processes, in particular to martingales, stochastic control and nonlinear filtering. == Biography == Liptser was born in Kirovograd, Ukraine and spent his youth in Odesa, Ukraine. In 1959 he graduated with the M.Sc. degree in electrical engineering from Moscow Aviation Institute and in 1965 he graduated with the second M.Sc in mathematics from Faculty of Mechanics and Mathematics of Moscow State University. In 1968 he obtained his Ph.D. degree from Moscow Institute of Physics and Technology (MIPT). He held a Professor position at MIPT and worked in the Institute of Control Sciences until 1990, when he joined the Institute for Information Transmission Problems as the head of the Stochastic Dynamic Systems Laboratory. In 1993 emigrated to Israel and lived in Kfar Saba. In Israel he held a Professor position at the School of Electrical Engineering in Tel Aviv University, until his retirement in 2005. == Research == Liptser made several important contributions to the theory of martingales and to their applications in engineering and statistics. This includes his study of the conditionally Gaussian processes, which play an important role in the separation principle in stochastic control. He coauthored a number of influential books. His monograph "Statistics of Random processes: General Theory and Applications", written together with Albert Shiryaev in 1974, has become internationally renowned reference textbook among scholars, working in stochastic analysis and related fields. == References == == External links == Robert Sh. Liptser at the Mathematics Genealogy Project Monash Probability Conference in Honor of Robert Liptser's 80th Birthday, 26–29 April 2016 Statistics And Control Of Stochastic Processes: The Liptser Festschrift at Google Books Robert Shevilevich Liptser, biographical sketch in Russian Institute of Control Sciences of Russian Academy of Sciences Robert Shevilevich Liptser, obituary in Theory of Probability and its Applications Abramov, V. M., Miller, B. M., Rubinovich, E. Ya. and Chigansky, P. Yu. Development of the theory of stochastic control and filtering in the works of R. Sh. Liptser, Automatika i Telemekhanika, issue 3 (2020), 1--13. (In Russian.)
Wikipedia:Robert M. Anderson (mathematician)#0	Robert Murdoch Anderson (born 1951) is Professor of Economics and of Mathematics at the University of California, Berkeley. He is director of the Center for Risk Management Research, University of California, Berkeley and he was chair of the University of California Academic Senate 2011–12. He is also the co-director for the Consortium for Data Analytics in Risk at UC Berkeley. == Research == Anderson's nonstandard construction of Brownian motion is a single object which, when viewed from a nonstandard perspective, has all the formal properties of a discrete random walk; however, when viewed from a measure-theoretic perspective, it is a standard Brownian motion. This permits a pathwise definition of the Itô Integral and pathwise solutions of stochastic differential equations. Anderson's contributions to mathematical economics are primarily within General Equilibrium Theory. Some of this work uses nonstandard analysis, but much of it provides simple elementary treatments that generalize work that had originally been done using sophisticated mathematical machinery. The best known of these papers is the 1978 Econometrica article cited, which establishes by elementary means a very general theorem on the cores of exchange economies. In the 2008 Econometrica article cited, Anderson and Raimondo provide the first satisfactory proof of existence of equilibrium in a continuous-time securities market with more than one agent. The paper also provides a convergence theorem relating the equilibria of discrete-time securities markets to those of continuous-time securities markets. It uses Anderson's nonstandard construction of Brownian and properties of real analytic functions. Recently, Anderson has focused on the analysis of investment strategies, and his work relies on both theoretical considerations and empirical analysis. In an article published in the Financial Analysts Journal in 2012 and cited below, Anderson, Bianchi and Goldberg found that long-term returns to risk parity strategies, which have acquired tens of billions of dollars in assets under management in the wake of the global financial crisis, are not materially different from the returns to more transparent strategies once realistic financing and trading costs are taken into account; they do well in some periods and poorly in others. A subsequent investigation by the same research team found that returns to dynamically levered strategies such as risk parity are highly unpredictable due to high sensitivity of strategy performance to a key risk factor: the co-movement of leverage with return to the underlying portfolio that is levered. == Selected publications == Anderson, Robert M.: A nonstandard representation for Brownian motion and Ito integration. Israel Journal of Mathematics 25(1976), 15–46. Anderson, Robert M.: An elementary core equivalence theorem. Econometrica 46(1978), 1483–1487. Anderson, Robert M. and Salim Rashid: A Nonstandard Characterization of Weak Convergence, Proceedings of the American Mathematical Society 69(1978), 327-332 Anderson, Robert M.: Star-finite representations of measure spaces. Trans. Amer. Math. Soc. 271 (1982), no. 2, 667–687. MathSciNet review: "In nonstandard analysis, -finite sets are infinite sets which nonetheless possess the formal properties of finite sets. They permit a synthesis of continuous and discrete theories in many areas of mathematics, including probability theory, functional analysis, and mathematical economics. -finite models are particularly useful in building new models of economic or probabilistic processes." here Anderson, Robert M.: Nonstandard analysis with applications to economics. Handbook of mathematical economics, Vol. IV, 2145–2208, Handbooks in Econom. 1, North-Holland, Amsterdam, 1991. Anderson, Robert M. and William R. Zame: Genericity with Infinitely Many Parameters, Advances in Theoretical Economics 1(2001), Article 1. Anderson, Robert M. and Roberto C. Raimondo: Equilibrium in continuous-time financial markets: Endogenously dynamically complete markets, Econometrica 76(2008), 841–907. Anderson, Robert M., Stephen W. Bianchi and Lisa R. Goldberg: Will My Risk Parity Strategy Outperform? Financial Analysts Journal 68(2012), no. 6, 75–93. == Personal life == Anderson is gay and has worked to attain greater equality for same-sex couples in academia. In 1991, he spoke at the Stanford University Faculty Senate, countering the claims of committee chair Professor Alain Enthoven that granting the same benefits to domestic partners of gay faculty members as to the spouses of heterosexual faculty would cost the university millions of dollars and thus be untenable. As the Chair of the University of California Academic Council during the Occupy Wall Street protests of 2011, Anderson also spoke out against police violence on the campus of UC Davis, pledging the Council's "opposition to the state’s disinvestment in higher education, which is at the root of the student protests." == See also == Influence of non-standard analysis == References == == External links == Robert M. Anderson's Home Page Robert M. Anderson at the Mathematics Genealogy Project
Wikipedia:Robert McLachlan (mathematician)#0	Robert Iain McLachlan (born 1964) is a New Zealand mathematician and Distinguished Professor in the School of Fundamental Sciences, Massey University, New Zealand. His research in geometric integration encompasses both pure and applied mathematics, modelling the structure of systems such as liquids, climate cycles, and quantum mechanics. He also writes for the public on the subject of climate change policy. == Academic career == McLachlan was born in Christchurch, New Zealand in 1964, and studied mathematics at the University of Canterbury, graduating with a BSc (Hons) First Class in 1984. One formative experience was in his last year of high school, where he had free rein to experiment with assembly language programming on the school PDP-11/10. McLachlan went on to graduate work in numerical analysis in 1986. He received a PhD from Caltech (the California Institute of Technology) in 1990, supervised by Herbert Keller in computational fluid dynamics, with a thesis titled "Separated Viscous Flows via Multigrid". He then worked as a postdoctoral fellow at the University of Colorado Boulder in what was then the new field of symplectic geometry. After meeting Jürgen Moser, who was visiting Boulder at the time, McLachlan spent six months on a postdoctoral fellowship in Switzerland, at the Swiss Federal Institute of Technology in Zurich. McLachlan joined Massey University in Palmerston North in 1994, and began a collaboration with Reinout Quispel at La Trobe University that resulted in over 26 publications on geometric integration. In 2002 he became Professor of Applied Mathematics at Massey, and spent a year's sabbatical at the University of Geneva, working with Gerhard Wanner and Ernst Hairer, and the Norwegian Academy of Sciences in Oslo. In 2007 he won the prestigious Germund Dahlquist Prize, the first mathematician from the southern hemisphere to do so. From 2008 to 2012, along with Stephen Marsland and Matt Perlmutter, he worked on a Marsden grant project "Geodesics in diffeomorphism groups: geometry and applications", designing efficient numerical integrators that preserved the geometric properties of systems. In 2013, McLachlan was the LMS-NZMS Aitken Lecturer, delivering talks on geometric numerical integration to six UK universities. McLachlan is a Fellow of the New Zealand Mathematical Society (NZMS), and in 1998 organised the first of the annual Manawatu-Wellington Applied Mathematics Conferences. He was president of the NZMS in 2008–2009 and vice president in 2010, and edited the New Zealand Journal of Mathematics for six years. Since 2016 he has been a Distinguished Professor in Massey's School of Fundamental Sciences. == Research == McLachlan is a world leader in the field of geometric integration, a technique for the reliable simulation of large-scale complex systems, and in particular the use of symplectic techniques in the numerical analysis of differential equations. This field, which McLachlan helped found in the 1990s, builds into its approach the underlying geometric structure of data sets. Because it allows the simulation of large systems, it has the potential for solving practical problems in fields as disparate as the structure of liquids, climate cycles, the motion of the solar system, particles in circular accelerators, chaos in dynamical systems, and weather forecasting. For example, during Hurricane Sandy in 2012, the European Centre for Medium-Range Weather Forecasts, using geometric integration models, correctly predicted the hurricane would suddenly turn 90 degrees towards New York six days in advance. McLachlan's methods have been used in computational science to examine a possible celestial origin of the ice ages, biological models, and the dynamics of flexible structures. His research contributed to a solar system simulation that revised the dates of geophysical epochs by millions of years. Although his work in geometric numerical integration has a wide range of real-world applications, he considers himself a pure mathematician. == Awards and fellowships == Fellow of the New Zealand Mathematical Society (2001) Fellow of the Royal Society of New Zealand (2002) NZ Association of Scientists Research Medal (2003) NZMS Research Award (2005) NZIMA Maclaurin Fellowship (2005) SIAM Germund Dahlquist Prize (2007) James Cook Research Fellow (2012) Research fellow at the Isaac Newton Institute, Cambridge, UK Research fellow at the Mathematical Sciences Research Institute, Berkeley, USA Research fellow at the Centre for Advanced Study at the Norwegian Academy of Science and Letters, Oslo Visiting fellow, Mathematical Research Institute of Oberwolfach, Germany == Science communication == As an advocate for action on climate change McLachlan writes frequently for the public, in media like Scientific American's blog, the blog Planetary Ecology and the Royal Society Te Apārangi's Sciblogs. His writing focuses on consequences of climate change, the benefits of wind turbines, electric cars, and climate policy. == Selected research == McLachlan, Robert I.; Offen, Christian (2019). "Symplectic integration of boundary value problems". Numerical Algorithms. 81 (4): 1219–1233. arXiv:1804.09042. doi:10.1007/s11075-018-0599-7. S2CID 52297714. McLachlan, Robert I.; Modin, Klas; Verdier, Olivier (2014). "Symplectic integrators for spin systems". Physical Review E. 89 (6): 061301. arXiv:1402.4114. Bibcode:2014PhRvE..89f1301M. doi:10.1103/PhysRevE.89.061301. PMID 25019718. S2CID 16838116. McLachlan, Robert I.; Quispel, G. Reinout W. (2006). "Geometric integrators for ODEs". Journal of Physics A: Mathematical and General. 39 (19): 5251–5285. Bibcode:2006JPhA...39.5251M. doi:10.1088/0305-4470/39/19/S01. McLachlan, Robert I.; Quispel, G. Reinout W. (2002). "Splitting methods". Acta Numerica. 11: 341–434. doi:10.1017/S0962492902000053. S2CID 229168188. McLachlan, Robert I. (1995). "On the Numerical Integration of Ordinary Differential Equations by Symmetric Composition Methods". SIAM Journal on Scientific Computing. 16 (1): 151–168. Bibcode:1995SJSC...16..151M. doi:10.1137/0916010. Braunstein, Samuel L.; McLachlan, Robert I. (1987). "Generalized squeezing". Physical Review A. 35 (4): 1659–1667. Bibcode:1987PhRvA..35.1659B. doi:10.1103/PhysRevA.35.1659. PMID 9898327. S2CID 42118827. == References == == External links == Planetary Ecology blog Massey University staff web page Google Scholar profile Robert McLachlan on SciBlogs McLachlan's web page of geometric integration images
Wikipedia:Robert Moody#0	Robert Vaughan Moody, (; born November 28, 1941) is a Canadian mathematician. He is the co-discoverer of Kac–Moody algebra, a Lie algebra, usually infinite-dimensional, that can be defined through a generalized root system. "Almost simultaneously in 1967, Victor Kac in the USSR and Robert Moody in Canada developed what was to become Kac–Moody algebra. Kac and Moody noticed that if Wilhelm Killing's conditions were relaxed, it was still possible to associate to the Cartan matrix a Lie algebra which, necessarily, would be infinite dimensional." - A. J. Coleman Born in Great Britain, he received a Bachelor of Arts in Mathematics in 1962 from the University of Saskatchewan, a Master of Arts in Mathematics in 1964 from the University of Toronto, and a Ph.D. in Mathematics in 1966 from the University of Toronto. In 1966, he joined the Department of Mathematics as an assistant professor in the University of Saskatchewan. In 1970, he was appointed an associate professor and a professor in 1976. In 1989, he joined the University of Alberta as a professor in the Department of Mathematics. In 1999, he was made an Officer of the Order of Canada. In 1980, he was made a fellow of the Royal Society of Canada. In 1996 Moody and Kac were co-winners of the Wigner Medal. == Selected works == Moody, R. V. (1967). "Lie algebras associated with generalized Cartan matrices" (PDF). Bull. Amer. Math. Soc. 73 (2): 217–222. doi:10.1090/s0002-9904-1967-11688-4. MR 0207783. Moody, R. V. (1975). "Macdonald identities and Euclidean Lie algebras". Proc. Amer. Math. Soc. 48 (1): 43–52. doi:10.1090/s0002-9939-1975-0442048-2. MR 0442048. with S. Berman: Berman, S.; Moody, R. V. (1979). "Lie algebra multiplicities". Proc. Amer. Math. Soc. 76 (2): 223–228. doi:10.1090/s0002-9939-1979-0537078-x. MR 0537078. with J. Patera: Moody, R. V.; Patera, J. (1982). "Fast recursion formula for weight multiplicities" (PDF). Bull. Amer. Math. Soc. (N.S.). 7 (1): 237–242. doi:10.1090/s0273-0979-1982-15021-2. MR 0656202. with Bremner & Patera: Tables of weight space multiplicities, Marcel Dekker 1983 with A. Pianzola: Moody, R. V.; Pianzola, A. (1989). "On infinite root systems". Trans. Amer. Math. Soc. 315 (2): 661–696. doi:10.1090/s0002-9947-1989-0964901-8. MR 0964901. with S. Kass, J. Patera, & R. Slansky: Affine Lie Algebras, weight multiplicities and branching rules, 2 vols., University of California Press 1991 vol. 1 books.google with Pianzola: Lie algebras with triangular decompositions, Canadian Mathematical Society Series, John Wiley 1995 with Baake & Grimm: Die verborgene Ordnung der Quasikristalle, Spektrum, Feb. 2002; What is Aperiodic Order?, Eng. trans. on arxiv.org == Notes == == References == Robert Moody at the Mathematics Genealogy Project "Robert Vaughan Moody's Home Page". "Robert Vaughan Moody Curriculum Vitae". Retrieved March 7, 2006.
Wikipedia:Robert Schatten#0	Robert Schatten (January 28, 1911 – August 26, 1977) was an American mathematician. Robert Schatten was born to a Jewish family in Lviv. His intellectual origins were at Lwów School of Mathematics, particularly well known for fundamental contributions to functional analysis. His entire family was murdered during World War II, he himself emigrated to the United States. In 1933 he got magister degree at Jan Kazimierz University of Lwów, and in 1939 he got master's degree at Columbia University. Supervised by Francis Joseph Murray, he got doctorate degree in 1942 for the thesis "On the Direct Product of Banach Spaces". Shortly after being appointed to a junior professorship, he joined the United States army where during training he suffered a back injury which affected him for the remainder of his life. In 1943 he was appointed to an assistant professorship at University of Vermont. At National Research Council, by two years he worked with John von Neumann and Nelson Dunford. In 1946, he went to the University of Kansas, first as extraordinary professor until 1952 and then as ordinary professor until 1961. He stayed at Institute for Advanced Study in 1950 and 1952–1953, at University of Southern California in 1960–1961, and at State University of New York in 1961–1962. In 1962 he became professor at Hunter College, where he stayed until his death. Schatten widely studied tensor products of Banach spaces. In functional analysis, he is the namesake of the Schatten norm and the Schatten class operators. His doctoral students included Elliott Ward Cheney, Jr. at University of Kansas, and Peter Falley and Charles Masiello at City University of New York. Schatten died in New York City in 1977. == Further reading == A Theory of Cross-Spaces. Annals of Mathematics Studies, ISBN 0-691-08396-7 Norm Ideals of Completely Continuous Operators. Ergebnisse der Mathematik und ihrer Grenzgebiete, 2. Folge, ISBN 3-540-04806-5 == References == == External links == Robert Schatten at the Mathematics Genealogy Project
Wikipedia:Robert Steinberg#0	Robert Steinberg (May 25, 1922, Soroca, Bessarabia, Romania (present-day Moldova) – May 25, 2014) was a mathematician at the University of California, Los Angeles. He introduced the Steinberg representation, the Lang–Steinberg theorem, the Steinberg group in algebraic K-theory, Steinberg's formula in representation theory, and the Steinberg groups in Lie theory that yield finite simple groups over finite fields. == Biography == Born in Soroca (then in the Kingdom of Romania, today in Moldova), Steinberg's parents settled in Canada very soon after his birth. Steinberg studied under Richard Brauer and he received his Ph.D. in mathematics from the University of Toronto in 1948. Steinberg joined the Mathematics Department at UCLA the same year. He retired from UCLA in 1992. == Awards == Steinberg was an invited speaker at the International Congress of Mathematicians in 1966, won the Steele Prize in 1985, was elected to the United States National Academy of Sciences in 1985, and won the Jeffery–Williams Prize in 1990. In 2003, the Journal of Algebra published a special issue to celebrate Robert Steinberg's 80th birthday. == Selected publications == Steinberg, R. (1951). "A geometric approach to the representations of the full linear group over a Galois field". Transactions of the American Mathematical Society. 71 (2): 274–282. doi:10.1090/s0002-9947-1951-0043784-0. MR 0043784. Steinberg, Robert (1959). "Finite reflection groups". Transactions of the American Mathematical Society. 91 (3): 493–504. doi:10.1090/s0002-9947-1959-0106428-2. MR 0106428. Steinberg, Robert (1961). "A general Clebsch–Gordan theorem". Bulletin of the American Mathematical Society. 67 (4): 406–407. doi:10.1090/s0002-9904-1961-10644-7. MR 0126508. Steinberg, Robert (1962). "Complete sets of representations of algebras". Proceedings of the American Mathematical Society. 13 (5): 746–747. doi:10.1090/s0002-9939-1962-0141710-x. MR 0141710. Steinberg, Robert (1962). "A closure property of sets of vectors". Transactions of the American Mathematical Society. 105: 118–125. doi:10.1090/s0002-9947-1962-0165040-x. MR 0165040. Steinberg, Robert (1962). "Générateurs, relations et revêtements de groupes algébriques". Colloq. Théorie des Groupes Algébriques (in French). Bruxelles: Gauthier-Villars: 113–127. MR 0153677. Zbl 0272.20036. "Differential equations invariant under finite reflections". Transactions of the American Mathematical Society. 112: 392–400. 1964. doi:10.1090/s0002-9947-1964-0167535-3. MR 0167535. Steinberg, Robert (2017) [1968], Lectures on Chevalley groups, University Lecture Series, vol. 66, American Mathematical Society, ISBN 978-1-4704-3105-1, MR 0466335 Steinberg, Robert (1974), Conjugacy classes in algebraic groups, Lecture Notes in Mathematics, vol. 366, Berlin-New York: Springer-Verlag, doi:10.1007/BFb0067854, ISBN 978-3-540-06657-6, MR 0352279 Steinberg, Robert (1988). "An occurrence of the Robinson–Schensted correspondence". Journal of Algebra. 113 (2): 523–528. doi:10.1016/0021-8693(88)90177-9. R. Steinberg, Collected Papers, Amer. Math. Soc. (1997), ISBN 0-8218-0576-2. == References == == External links == Robert Steinberg at the Mathematics Genealogy Project
Wikipedia:Robert Vallée#0	Robert Vallée (5 October 1922 in Poitiers, France – 1 January 2017, Paris, France) was a French cyberneticist and mathematician. He was Professor at the Paris 13 University (University of Paris-Nord) and president of the World Organization of Systems and Cybernetics (WOSC). At the beginning of the 1950s, Vallée wrote his first publications on what he named "opérateur d'observation" (which means in English "operator of observation"). The latter, in the simplest case, allows a cybernetic system to observe the state of its environment and itself. Thereafter, on the basis of these results, a decisional operator will be able to indicate the action to be taken. The two stages of perception and decision are distinguished by "intellectual convenience", but it is interesting to gather them into a unique operator, known as "pragmatic". A decision is influenced by the observation of events, but also by past perceptions. That means that, in the observation made at a given moment, traces of past observations are also present. Eventually, these processes follow one another in a loop. Vallée defined the study of this situation with the term "epistemo-praxeology", underlining the existing link between knowledge (episteme), resulting from observation, and action (praxis). Regarding the observation problem, Vallée was also interested in information theory. Vallée also nourished private interests in sociological problems as well as in history. The first led him to describe a cybernetic creature covering the whole surface of the globe with its communication net (1952), an idea which has also been proposed (under the name of "cybionte", 1975) by Joël de Rosnay. He also wrote articles devoted to historical aspects of cybernetics and systems, referring to René Descartes, Louis de Broglie, and Norbert Wiener. == Biography == Vallée was born on 5 October 1922, in Poitiers, (France), as the son of professors in history. In 1969 he married the editor and translator Nicole Georges-Lévy. With his wife, Robert Vallée contributed to a translation, in French, of Norbert Wiener's book, Cybernetics: Or Control and Communication in the Animal and the Machine. Towards the end of the 1920s and during the 1930s, Vallée attended the college of Angoulême where, in 1940, he obtained a bachelor's degree in Latin-Greek, mathematics, and philosophy. Between 1944 and 1946, he was a student at the École Polytechnique in Paris. During the summer of 1954, he took part in the "Foreign Students Summer Project" of the Massachusetts Institute of Technology (with Norbert Wiener and Armand Siegel). In 1961 he became a Doctor of Science in mathematics with a thesis on an extension of the general relativity of Kaluza-Klein, under the direction of André Lichnerowicz (University of Paris). During his career, Vallée occupied several positions. Between 1956 and 1958, he was Associate-Director of the Institute Blaise Pascal in Paris. From 1961 to 1971, he was university lecturer in mathematics at the École Polytechnique and at the University of Franche-Comté (1962–1971) where he subsequently became Professor. Between 1971 and 1987 he was a Professor at the University of Paris-Nord where he was also dean of the Faculty of Economics from 1973 to 1975 and president of the Department of Economical mathematics from 1975 to 1987. In 1987, the University of Paris-Nord conferred on him the title of professor emeritus. Vallée also gave a doctoral course on dynamic systems at the University of Paris 1 Pantheon-Sorbonne between 1975 and 1987. Vallée was active within several associations and organizations, in particular: Founder of the Cercle d’Etudes Cybernétiques (President Louis de Broglie), 1950. Member of the Council of the Société Mathématique de France, 1964–1967. General Director of the Institut de Sciences Mathématiques et Economiques Appliquées (President François Perroux), 1980–1982. President of the Collège de Systémique de l'Association Française pour la Cybernétique Economique et Technique (AFCET), 1981–1984. Member of the council of the French Association of Theoretical Biology, 1984–1988. Representative of the AFCET, (later CET), at the International Federation for Systems Research (IFSR), 1986. Participant in several Fuschl Conversations (International Systems Institute and IFSR), 1986–1996. General Director (1987) then President (2003) of the World Organization of Systems and Cybernetics (WOSC, founder J. Rose), 1987. Member of the council of the International Association of Cybernetics, 1987–2000. Member of the council of the Association Française de Science des Systèmes Cybernétiques, Cognitifs, et Techniques (AFSCET), 1999. He also was a member of the International Society for the Systems Sciences, the American Society for Cybernetics, the Tutmonda Asocio pri Kibernetiko, Informatiko kaj Sistemiko (TAKIS), and of the international league of scientists for the use of the French language. In his career several titles were conferred on him: From 1987 to 1999, Vallée was chief editor of the Revue Internationale de Systémique (AFCET) as well as member of the editorial boards of Kybernetes (official review of the WOSC), Economies et Sociétés (ISMEA), International Journal for Biological Systems, Cybernetics and Human Knowing, Grundlagenstudien aus Kybernetik und Geisteswissenschaft (TAKIS), Robotica, and Res-Systemica (electronic journal of the French Association of Science of Systems (AFSCET) and the European Union of Systemics). == References == == Notes == (in French) Robert Vallée, Cognition et Système, l'Interdisciplinaire, Lyon-Limonest, 1995. Robert Vallée, 30th Anniversary Cyberprofile, Kybernetes, 32, 3, 2003, pp. 449–453. Robert Vallée. "De la connaissance à l'action, Automates Intelligents, 82" (in French). Retrieved 22 July 2009.
Wikipedia:Robert W. Brooks#0	Robert Wolfe Brooks (Washington, D.C., September 16, 1952 – Montreal, September 5, 2002) was a mathematician known for his work in spectral geometry, Riemann surfaces, circle packings, and differential geometry. == Biography == Brooks was born in 1952 in Washington, D.C. and grew up in Bethesda, Maryland, where he graduated in 1970 from Walt Whitman High School. In 1974 he completed his Masters thesis from Harvard University; his thesis "Russell, Poincaré, and the foundations of geometry" won him the Bowdoin Prize for Essays in the Natural Sciences in 1975. He received his Ph.D. from Harvard University in 1977; his thesis, The smooth cohomology of groups of diffeomorphisms, was written under the supervision of Raoul Bott. He then undertook postdoctoral studies with J. Peter Matelski at the State University of New York at Stony Brook, where they created pictures of fractals, leading to Benoit Mandelbrot's creation of the Mandelbrot set in 1980. He worked at the University of Maryland (1979–1984), then at the University of Southern California, and then, from 1995, at the Technion in Haifa. Brooks died from heart attack during a visit to Montreal, Canada and was buried in Sde Yehoshua cemetery in Haifa. He was survived by his parents David and Harriet Brooks, his wife Sharon and four children. His eldest son Shimon Brooks is a mathematics professor at Bar-Ilan University. == Work == In an influential paper (Brooks 1981), Brooks proved that the bounded cohomology of a topological space is isomorphic to the bounded cohomology of its fundamental group. == Honors == Bowdoin Prize for Essays in the Natural Sciences, 1975 Alfred P. Sloan fellowship Guastella fellowship == Selected publications == Brooks, Robert (1981). "Some remarks on bounded cohomology". Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978). Ann. of Math. Stud. Vol. 97. Princeton, N.J.: Princeton Univ. Press. pp. 53–63. MR 0624804. Brooks, Robert (1981). "A relation between growth and the spectrum of the Laplacian". Mathematische Zeitschrift. 178 (4): 501–508. doi:10.1007/BF01174771. MR 0638814. S2CID 122114581. Brooks, Robert (1981). "The fundamental group and the spectrum of the Laplacian". Commentarii Mathematici Helvetici. 56 (4): 581–598. doi:10.1007/BF02566228. MR 0656213. S2CID 121175762. Brooks, Robert (1988). "Constructing isospectral manifolds". American Mathematical Monthly. 95 (9): 823–839. doi:10.1080/00029890.1988.11972094. MR 0967343. Reviewer Maung Min-Oo for MathSciNet wrote: "This is a well written survey article on the construction of isospectral manifolds which are not isometric with emphasis on hyperbolic Riemann surfaces of constant negative curvature." Brooks, Robert, "Form in Topology", The Magicians of Form, ed. by Robert M. Weiss. Laurelhurst Publications, 2003. == References == == External links == Memorial page (Technion) Robert W. Brooks at the Mathematics Genealogy Project
Wikipedia:Robert Waddington (mathematician)#0	Robert Waddington (died 1779) was a mathematician, astronomer and teacher of navigation. He is best known as one of the observers appointed by the Royal Society to observe the 1761 transit of Venus with Nevil Maskelyne on the island of Saint Helena. On that voyage they made successful use of the lunar-distance method of establishing longitude at sea. Waddington subsequently taught the method at his academy in London and published a navigation manual, A Practical Method for Finding the Longitude and Latitude of a Ship at Sea, by Observations of the Moon (1763). == Life before 1761 == Little is known about Waddington's early life, although he was probably the "Mr. Rob. Waddington of Hull" who appeared in The Gentleman's Diary or The Mathematical Repository in 1758 and Benjamin Martin's General Magazine of Arts and Sciences in 1759. The papers of the gentleman astronomer Nathaniel Pigott show that Waddington was living in his household, in Whitton, Middlesex, immediately before his appointment as an observer by the Royal Society. It is from these manuscripts that we have a detailed knowledge of Waddington's experience of the Saint Helena expedition and his subsequent attempts to forge a career as a teacher, mathematical practitioner and longitude projector. == Voyage to Saint Helena == In 1760 Waddington was appointed by the Royal Society to accompany Nevil Maskelyne on a voyage to Saint Helena as one of two expeditions being organised by the Society, and paid for by George II, to observe the 1761 Transit of Venus. The expedition's equipment was ordered by Maskelyne and the Society arranged transport from the East India Company. The Directors of the East India Company in London wrote to The Governor of St Helena on 31 December 1760 to inform them that "Revd. Mr. Nevil Maskelyne and Mr. Robert Waddington take passage on the Prince Henry to St. Helena. As this is done to make some improvements in Astronomy which will be of general utility the two last named gentlemen are upon their arrival and during their stay to be accommodated by you in a suitable manner with diet and apartments at the Company's expense and you are to give them all the assistance as to materials, workmen, and whatsoever else the service they are employed upon may require." The reply confirmed their readiness to help and stated "We have already erected an observatory for them in the country". In the event, Waddington and Maskelyne's view of the transit of Venus on 6 July 1761 was thwarted by clouds. However, the voyages to and from the island proved to be very significant to the subsequent careers of both men as they used it to make longitude determinations by the lunar-distance method testing the accuracy of lunar tables calculated by Tobias Mayer. Maskelyne's log of the voyage records their efforts to do this, with the assistance of officers from the ship, and refers to the use of "Mr Waddington's Quadrant", a Hadley's quadrant adapted by him to better facilitate lunar-distance observations Waddington kept his own account of the voyages, and seems to have produced more accurate observations than Maskelyne. He told Piggott that their observations "finds a Practical & Certain Method" of determining longitude "& may be depended upon to one Degree of Longitude". This wording and degree of accuracy echoes that in the 1714 Longitude Act, which offered a reward of £10,000 for methods that kept or found longitude to within a degree. == Subsequent career == Waddington returned from Saint Helena before Maskelyne, who stayed on to attempt observations of the parallax of Sirius. He arrived back in London on 21 September 1761 and began to forge a career around providing teaching and texts on navigation, particularly the lunar-distance method of finding longitude. Waddington, having announced publicly by his method longitude "may be generally obtained to less than half a Degree, and always to less than One Degree", hoped that he might be in a position to receive a reward from the Board of Longitude. His contribution was a method of observing and computing the data, but was based on Mayer's lunar tables and the earlier publication of precomputed tables and rules by Nicolas Louis de Lacaille. His 1763 Practical Method for Finding the Longitude included such instructions and tables, which he claimed would reduce the necessary computation from several hours to three-quarters of an hour. Maskelyne, on his return to London, produced his own version of such a text in the same year, The British Mariner's Guide, although never seems to have expected a reward from the Board of Longitude for his effort. Waddington styled himself "Teacher of Mathematics at the Mathematical Academy in Three Tun-court, Mile's-lane, London", where he taught officers of the East India Company and also sold Hadley's quadrants. In 1763 he attempted, unsuccessfully, to gain employment from the Royal Society. By 1764 he had moved to Rolls Building, Chancery Lane. He reported to Piggott his hopes that he and Maskelyne might gain the interest of the Board of Longitude but his only formal approach to them seems to have been in 1771, when a petition was read at a meeting on 11 May stating that he had much improved, and successfully trialled, the steering compass, or binnacle. In 1777 he published his longest and most comprehensive work, An Epitome of Theoretical and Practical Navigation. As The Edinburgh Magazine reported in 1778, he claimed that the book could "teach the young navigator every particular essential to his art, without his being under the necessity of having recourse to any other author." Waddington's will, dated 12 May 1775 and proved posthumously in 1779, states his address as Downing Street, Westminster. His wife, Margaret, was the beneficiary. == Further reading == Taylor, E.G.R. (1966). The Mathematical Practitioners of Hanoverian England 1714-1840. Cambridge University Press. Howse, Derek (1989). Nevil Maskelyne: The Seaman's Astronomer. Cambridge University Press. == References ==
Wikipedia:Robert William Chapman (engineer)#0	Sir Robert William Chapman MIEAust (27 December 1866 – 27 February 1942) was an Australian mathematician and engineer. == History == Chapman was born in Stony Stratford in Buckinghamshire, England, eldest son of Charles Chapman (c. 1838 – 14 September 1921), a currier from Melbourne, Australia, and his wife Matilda, née Harrison (c. 1840 – 23 October 1933). His parents returned to Melbourne in 1876, where he was educated at Wesley College and the University of Melbourne, graduating MA and BCE with first class honours in Physics and Mathematics. In 1888, at the recommendation of Professor William Bragg, he was appointed a lecturer in Mathematics and Physics at Adelaide University. He was appointed Professor of Engineering in 1907 and served as (Sir Thomas) Elder Professor of Mathematics and Mechanics from 1910 when Professor Bragg was appointed to the Cavendish chair of physics in the University of Leeds. Chapman then returned to his previous post in 1919. He served as Vice-Chancellor during the absence of Sir William Mitchell. He retired in 1937. He was appointed president of the School of Mines council in 1939 on the death of Sir Langdon Bonython. His research work included: (With Thomas Roberts (1845–1920)) Breakage of locomotive and other railway axles Established a laboratory where local stone and timber could be tested Distribution of stress in steel reinforcing rods in concrete Effects of building a dam on the Mundoo Channel, Lake Alexandrina Use of brown coal from Leigh Creek, including briquettes == Recognition and memberships == He was elected to the Royal Society of South Australia in 1888 and to the Australian Association for the Advancement of Science. He was a founding member of the South Australian Institute of Engineers and a foundation member of the Institution of Engineers Australia in 1921, in 1918 the first chairman of its South Australian division and the third Federal President. He was awarded their Peter Nicol Russell Memorial Medal in 1928 and made honorary life member in 1932. He was awarded Melbourne University's Kernot Medal in 1927. He was president of the Astronomical Society of South Australia for a record 32 years and elected a Fellow of the Royal Astronomical Society in 1902? 1909?. He was a member of the council of the Australasian Institute of Mining Engineers and in 1920 elected president of the Australasian Institute of Mining and Metallurgy. He was a member of the South Australian Institute of Surveyors from 1912 and president 1917–1929 He was a member of the councils of both the University of Adelaide and the School of Mines. On his retirement from the University of Adelaide in 1937 he was made Emeritus Professor; his portrait, by Ivor Hele hangs in the School of Engineering's Chapman Lecture Theatre, which was named for him. He was appointed CMG in 1927 and knighted in 1938. == Family == On 14 February 1889 he married Eva Maud Hall, who survived him; they had six sons and two daughters: Robert Hall Chapman (6 January 1890 – 10 May 1953) engaged to Florence Muriel Day, but she married footballer Charlie Perry. Robert Hall Chapman married May Warren Knox. He was chief engineer, later commissioner, with the South Australian Railways. Charles George Chapman (19 November 1891 – April 1916), was a lieutenant with the Royal Engineers, killed in Mesopotamia during World War I. Eva Florence Chapman (5 December 1893 – ) married Essington Day on 19 March 1919, lived at Burnside Lilian Eleanor Chapman (1895– ) married Edgar Bills of Orroroo on 4 October 1922, lived at Peterborough Walter Harrison Chapman (1896– ) of Bulolo, New Guinea James Douglas Chapman (1901– ) married Gwendolen Ruth Johnston on 7 January 1929. He was an engineer with the Adelaide City Council Ernest Stirling Chapman (1907– ), dentist in Clare Leslie Drake Chapman (8 December 1911– ) engaged to Ellen Rose McBeath. He was engineer associated with the Goolwa barrage == References == == External links == Works by or about Robert William Chapman at Wikisource Media related to Robert William Chapman at Wikimedia Commons
Wikipedia:Robert William Genese#0	Robert William Genese (1848, Dublin – 1928) was an Irish mathematician whose career was spent in Wales. == Early life and education == Genese was born on Westland Row a street on the south side of Dublin on 8 May 1848. From St John's College of the University of Cambridge, Genese received in 1871 his bachelor's degree (with rank eighth Wrangler in the Tripos) and in 1874 his master's degree. == Professional life == Following an unsuccessful application for the Chair of Mathematics at Aberystwyth in 1872, he taught at the Training College in Carmarthen. He finally secured the professorship at Aberystwyth in 1879, and held it until 1919. Along the way his title became Professor of Mathematics and Astronomy. Genese introduced into the United Kingdom the ideas of Hermann Grassmann (advancing the use of vector analysis). In his 1941 book The calculus of extensions, Henry Forder published numerous examples in vector analysis taken from Genese's posthumous notes. (Genese's notes were left to the Mathematical Association and then given in 1929 to Forder by E. H. Neville.) Genese was an Invited Speaker of the ICM in 1904 in Heidelberg with talk On some useful theorems in the continued multiplication of a regressive product in real four-point space and in 1908 in Rome with talk The method of reciprocal polars applied to forces in space. == Selected publications == "Suggestions for the Practical Treatment of the Standard Cubic Equation, and a Contribution to the Theory of Substitution." The Mathematical Gazette 9, no. 129 (1917): 65–69. doi:10.2307/3603498 "On the Theory of the Plane Complex with Simple Geometrical and Kinematical Illustrations." The Mathematical Gazette 11, no. 164 (1923): 293–301. doi:10.2307/3603761 "A Simple Exposition of Grassmann's Methods." The Mathematical Gazette 13, no. 189 (1927): 373–391. doi:10.2307/3604329 == References == == External links == biographical details, Ceredigion County Council
Wikipedia:Robert Wolak#0	Robert Antoni Wolak (born September 19, 1955) is a Polish mathematician, habilitated doctor of mathematical sciences. He specializes in differential geometry, foliation theory and differential topology. Associate professor of the Department of Geometry of the Institute of Mathematics, Faculty of Mathematics and Computer Science of the Jagiellonian University. == References ==
Wikipedia:Robert Woodhouse#0	Robert Woodhouse (28 April 1773 – 23 December 1827) was a British mathematician and astronomer. == Biography == === Early life and education === Robert Woodhouse was born on 28 April 1773 in Norwich, Norfolk, the son of Robert Woodhouse, linen draper, and Judith Alderson, the daughter of a Unitarian minister from Lowestoft. Robert junior was baptised at St George's Church, Colegate, Norwich, on 19 May, 1773. A younger son, John Thomas Woodhouse, was born in 1780. The brothers were educated at the Paston School in North Walsham, 24 kilometres (15 mi) north of Norwich. In May 1790 Woodhouse was admitted to Gonville and Caius College, Cambridge, the college where Paston pupils were traditionally sent. In 1795 he graduated as the Senior Wrangler (ranked first among the mathematics undergraduates at the university), and took the First Smith's Prize. He obtained his Master's degree at Cambridge in 1798. === Marriage and career at Cambridge === Woodhouse was a fellow of the college from 1798 to 1823, after which he resigned so as to be able to marry Harriet, the daughter of William Wilkin, a Norwich architect. They were married on 20 February 1823; the marriage produced a son, also named Robert. Harriet Woodhouse died at Cambridge on 31 March 1826. Woodhouse was elected a Fellow of the Royal Society on 16 December 1802. His earliest work, entitled the Principles of Analytical Calculation, was published at Cambridge in 1803. In this he explained the differential notation and strongly pressed the employment of it; but he severely criticised the methods used by continental writers, and their constant assumption of non-evident principles. In 1809 Woodhouse published a textbook covering planar trigonometry and spherical trigonometry and the next year a historical treatise on the calculus of variations and isoperimetrical problems. He next produced an astronomy; of which the first book (usually bound in two volumes), on practical and descriptive astronomy, was issued in 1812, and the second book, containing an account of the treatment of physical astronomy by Pierre-Simon Laplace and other continental writers, was issued in 1818. Woodhouse became the Lucasian Professor of Mathematics in 1820, but the small income caused him to resign the professorship in 1822 and instead accept the better paid post as the Plumian professor in the university. As Plumian Professor he was responsible for installing and adjusting the transit instruments and clocks at the Cambridge Observatory. Woodhouse did not exercise much influence on the majority of his contemporaries, and the movement might have died away for the time being if it had not been for the advocacy of George Peacock, Charles Babbage, and John Herschel, who formed the Analytical Society, with the object of advocating the general use in the university of analytical methods and of the differential notation. Woodhouse was the first director of the newly built observatory at Cambridge, a post he held until his death in 1827. On his death in Cambridge he was buried in Caius College chapel. == Notes == == References == == Sources == Becher, Harvey W. (1980). "Woodhouse, Babbage, Peacock and Modern Algebra". Historia Mathematica. 7 (4): 389–400. doi:10.1016/0315-0860(80)90003-8. Becher, Harvey W. (2004). "Woodhouse, Robert". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/29926. OCLC 56568095. Retrieved 24 September 2021. (Subscription or UK public library membership required.) (subscription may be required or content may be available in libraries that are in the UK) Guicciardini, Niccolò (1989). "Robert Woodhouse". The Development of Newtonian Calculus in Britain 1700–1800. New York: Cambridge University Press. ISBN 0-521-36466-3. Rouse Ball, W.W. (1912). A Short Account Of The History Of Mathematics. London: MacMillan and Co. Ltd. OCLC 844389098. Woodhouse, Robert (1825). "Some account of the transit instrument made by Mr. Dollond, and lately put up at the Cambridge Observatory". Philosophical Transactions of the Royal Society of London: 418–428. === Further reading === Harman, P. M. (1988). "Newton to Maxwell: The 'Principia' and British Physics". Notes and Records of the Royal Society of London. 42 (1: Newton's 'Principia' and Its Legacy): 75–96. Bibcode:1988npl..conf...75H. doi:10.1098/rsnr.1988.0008. JSTOR 531370. S2CID 122622492. Johnson, W. (1995). "Contributors to Improving the Teaching of Calculus in Early 19th-Century England". Notes and Records of the Royal Society of London. 49 (1): 93–103. doi:10.1098/rsnr.1995.0006. JSTOR 531886. S2CID 145534544. == External links == Facsimile of Woodhouse's certificate of election to the Royal Society === Works === 1803: Principles of Analytical Calculation 1809: A Treatise on Plane and Spherical Trigonometry (5th edition 1827) 1810: A Treatise on Isoperimetric Problems and the Calculus of Variations 1818: An Elementary Treatise on Physical Astronomy, volume 1 1818: An Elementary Treatise on Astronomy, volume 2 1821: A Treatise on Astronomy, Theoretical and Practical
Wikipedia:Roberta Frances Johnson#0	Roberta Frances Johnson (January 22, 1902–October 12, 1988) was an American mathematician and one of the few women to earn a PhD in that subject in the United States before World War II. She joined the faculty of Wilson College, Colorado State University and University of Colorado. == Biography == Johnson was born in Philadelphia, Pennsylvania to Mary Wallace (Abdill) and Jesse B. Johnson and attended Philadelphia's Frankford High School. In 1925, she graduated in mathematics (with departmental honors) with a minor in history from Wilson College, in Chambersburg, Pennsylvania. Following her studies, she taught mathematics at Chambersburg High School for three years and then taught both math and history at the high school in Newfoundland, Pennsylvania. Financed by fellowships, including one from Wilson College, she studied at Cornell University and earned her master's degree with her major and minor in mathematics. Her thesis was titled: Certain properties and a classification of nets of conics and was supervised by mathematician Virgil Snyder. In 1933, she received her doctorate with her major subject in geometry, her first minor in analysis and her second minor in philosophy. Her dissertation in algebraic geometry, also directed by Virgil Snyder, was titled Involutions of Order 2 Associated with Surfaces of Genera P(A)=P(G)=0, P(2)=1, P(3)=0, and was published in the American Journal of Mathematics. == Educator == Although she had hoped to teach at Cornell, Johnson was asked to return to Wilson College when the head of the mathematics department became ill. Initially, Johnson was to substitute for about one month but that assignment was extended when the department head took a leave of absence. She stayed at Wilson on a temporary assignment for 1933–1934, but then she remained at the College for 25 years, first as an instructor until 1935, then assistant professor until 1944, when she was named associate professor. She was named department head after the former chair retired in 1946. In 1957, she grew unhappy with her role at Wilson and began looking for work elsewhere mentioning in a letter that part of her reasoning was a "refusal to accept the injustice of being passed over when promotions are made." When Wilson administrators learned of her search, they quickly elevated her position to full professor with a tenure contract but the promotion came too late to change her mind. In 1958, Johnson moved to Fort Collins, Colorado, to teach as an associate professor at Colorado State University. There she directed the master's theses of about eight students. She stayed at Colorado State until 1967 when she retired as associate professor emeritus. The following spring the University of Colorado at Denver hired her as an associate professor, and she taught there for the next three years. Johnson died October 12, 1988, of bone cancer at her Fort Collins home. == Memberships == According to Green, Johnson was active in several organizations. American Mathematical Society Mathematical Association of America Sigma Delta Epsilon Phi Beta Kappa Phi Kappa Phi Sigma Xi == References ==
Wikipedia:Roberto Markarian#0	Roberto Markarian Abrahamian (12 December 1946) is a Uruguayan mathematician of Armenian descent, expert in dynamical systems and chaos theory. == Biography == He started studying at the University of the Republic in the 1960s. During the civic-military dictatorship he was arrested due to political reasons. Later he went to Brazil, where he graduated from the Federal University of Rio Grande do Sul. Later on, his degree was validated in Uruguay. Markarian served as rector of the University of the Republic (2014-2018). He is brother of the football coach Sergio Markarián. == References ==
Wikipedia:Robin Williams (mathematician)#0	Robert Martin Williams (30 March 1919 – 18 March 2013), generally known as Robin Williams, was a New Zealand mathematician, academic administrator and public servant. He served as vice chancellor of the University of Otago from 1967 to 1972, and of the Australian National University from 1973 to 1975. Between 1975 and 1981, he was chair of the State Services Commission. == Early life and family == Born in Christchurch in 1919, Williams was educated at Christ's College and went on to study at Canterbury University College, graduating MA with first-class honours in mathematics and mathematical physics in 1941. On 15 July 1944, Williams married Mary Thorpe in Wellington, and the couple went on to have three children. == Career == Williams worked in the applied mathematics laboratory of the Department of Scientific and Industrial Research. During World War II, he worked at the University of California, Berkeley on the Manhattan Project in 1944–45 on the separation of uranium. After the war, he graduated from St. John's College, Cambridge with a Bachelor of Arts (1946) and PhD (1949). He was a Harkness Fellow at Princeton in 1957. In 1963, Williams moved to work as an administrator at the State Services Commission. From 1967 to 1972, Williams was vice chancellor of the University of Otago, before accepting the same position at the Australian National University in Canberra, where he remained until 1975. That year, he was appointed chair of the State Services Commission, based in Wellington, serving in that role until 1981. In 1971 he succeeded Dr K. J. Sheen as Director-General of Education in New Zealand. == Honours and awards == Williams was elected to the American Philosophical Society in 1967. In 1972, Williams was conferred with an honorary LLD degree by the University of Otago. In the 1973 New Year Honours, he was appointed a Commander of the Order of the British Empire, for services to science, administration and education, and in the 1981 Queen's Birthday Honours, he was made a Companion of the Order of the Bath. == Death == Williams died in Wellington in 2013, aged 93. His funeral was held at Old St Paul's, and he was buried at Mākara Cemetery. == References == == Sources == Henderson, Alan (1990). The Quest for Efficiency:The Origins of the State Services Commission. State Services Commission. ISBN 0-477-05538-9.
Wikipedia:Robinson–Schensted–Knuth correspondence#0	In mathematics, the Robinson–Schensted–Knuth correspondence, also referred to as the RSK correspondence or RSK algorithm, is a combinatorial bijection between matrices A with non-negative integer entries and pairs (P,Q) of semistandard Young tableaux of equal shape, whose size equals the sum of the entries of A. More precisely the weight of P is given by the column sums of A, and the weight of Q by its row sums. It is a generalization of the Robinson–Schensted correspondence, in the sense that taking A to be a permutation matrix, the pair (P,Q) will be the pair of standard tableaux associated to the permutation under the Robinson–Schensted correspondence. The Robinson–Schensted–Knuth correspondence extends many of the remarkable properties of the Robinson–Schensted correspondence, notably its symmetry: transposition of the matrix A results in interchange of the tableaux P,Q. == The Robinson–Schensted–Knuth correspondence == === Introduction === The Robinson–Schensted correspondence is a bijective mapping between permutations and pairs of standard Young tableaux, both having the same shape. This bijection can be constructed using an algorithm called Schensted insertion, starting with an empty tableau and successively inserting the values σ1, ..., σn of the permutation σ at the numbers 1, 2, ..., n; these form the second line when σ is given in two-line notation: σ = ( 1 2 … n σ 1 σ 2 … σ n ) {\displaystyle \sigma ={\begin{pmatrix}1&2&\ldots &n\\\sigma _{1}&\sigma _{2}&\ldots &\sigma _{n}\end{pmatrix}}} . The first standard tableau P is the result of successive insertions; the other standard tableau Q records the successive shapes of the intermediate tableaux during the construction of P. The Schensted insertion easily generalizes to the case where σ has repeated entries; in that case the correspondence will produce a semistandard tableau P rather than a standard tableau, but Q will still be a standard tableau. The definition of the RSK correspondence reestablishes symmetry between the P and Q tableaux by producing a semistandard tableau for Q as well. === Two-line arrays === The two-line array (or generalized permutation) wA corresponding to a matrix A is defined as w A = ( i 1 i 2 … i m j 1 j 2 … j m ) {\displaystyle w_{A}={\begin{pmatrix}i_{1}&i_{2}&\ldots &i_{m}\\j_{1}&j_{2}&\ldots &j_{m}\end{pmatrix}}} in which for any pair (i,j) that indexes an entry Ai,j of A, there are Ai,j columns equal to ( i j ) {\displaystyle {\tbinom {i}{j}}} , and all columns are in lexicographic order, which means that i 1 ≤ i 2 ≤ i 3 ⋯ ≤ i m {\displaystyle i_{1}\leq i_{2}\leq i_{3}\cdots \leq i_{m}} , and if i r = i s {\displaystyle i_{r}=i_{s}\,} and r ≤ s {\displaystyle r\leq s} then j r ≤ j s {\displaystyle j_{r}\leq j_{s}} . ==== Example ==== The two-line array corresponding to A = ( 1 0 2 0 2 0 1 1 0 ) {\displaystyle A={\begin{pmatrix}1&0&2\\0&2&0\\1&1&0\end{pmatrix}}} is w A = ( 1 1 1 2 2 3 3 1 3 3 2 2 1 2 ) {\displaystyle w_{A}={\begin{pmatrix}1&1&1&2&2&3&3\\1&3&3&2&2&1&2\end{pmatrix}}} === Definition of the correspondence === By applying the Schensted insertion algorithm to the bottom line of this two-line array, one obtains a pair consisting of a semistandard tableau P and a standard tableau Q0, where the latter can be turned into a semistandard tableau Q by replacing each entry b of Q0 by the b-th entry of the top line of wA. One thus obtains a bijection from matrices A to ordered pairs, (P,Q) of semistandard Young tableaux of the same shape, in which the set of entries of P is that of the second line of wA, and the set of entries of Q is that of the first line of wA. The number of entries j in P is therefore equal to the sum of the entries in column j of A, and the number of entries i in Q is equal to the sum of the entries in row i of A. ==== Example ==== In the above example, the result of applying the Schensted insertion to successively insert 1,3,3,2,2,1,2 into an initially empty tableau results in a tableau P, and an additional standard tableau Q0 recoding the successive shapes, given by P = 1 1 2 2 2 3 3 , Q 0 = 1 2 3 7 4 5 6 , {\displaystyle P\quad =\quad {\begin{matrix}1&1&2&2\\2&3\\3\end{matrix}},\qquad Q_{0}\quad =\quad {\begin{matrix}1&2&3&7\\4&5\\6\end{matrix}},} and after replacing the entries 1,2,3,4,5,6,7 in Q0 successively by 1,1,1,2,2,3,3 one obtains the pair of semistandard tableaux P = 1 1 2 2 2 3 3 , Q = 1 1 1 3 2 2 3 . {\displaystyle P\quad =\quad {\begin{matrix}1&1&2&2\\2&3\\3\end{matrix}},\qquad Q\quad =\quad {\begin{matrix}1&1&1&3\\2&2\\3\end{matrix}}.} ==== Direct definition of the RSK correspondence ==== The above definition uses the Schensted algorithm, which produces a standard recording tableau Q0, and modifies it to take into account the first line of the two-line array and produce a semistandard recording tableau; this makes the relation to the Robinson–Schensted correspondence evident. It is natural however to simplify the construction by modifying the shape recording part of the algorithm to directly take into account the first line of the two-line array; it is in this form that the algorithm for the RSK correspondence is usually described. This simply means that after every Schensted insertion step, the tableau Q is extended by adding, as entry of the new square, the b-th entry ib of the first line of wA, where b is the current size of the tableaux. That this always produces a semistandard tableau follows from the property (first observed by Knuth) that for successive insertions with an identical value in the first line of wA, each successive square added to the shape is in a column strictly to the right of the previous one. Here is a detailed example of this construction of both semistandard tableaux. Corresponding to a matrix A = ( 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ) {\displaystyle A={\begin{pmatrix}0&0&0&0&0&0&0\\0&0&0&1&0&1&0\\0&0&0&1&0&0&0\\0&0&0&0&0&0&1\\0&0&0&0&1&0&0\\0&0&1&1&0&0&0\\0&0&0&0&0&0&0\\1&0&0&0&0&0&0\\\end{pmatrix}}} one has the two-line array w A = ( 2 2 3 4 5 6 6 8 4 6 4 7 5 3 4 1 ) . {\displaystyle w_{A}={\begin{pmatrix}2&2&3&4&5&6&6&8\\4&6&4&7&5&3&4&1\end{pmatrix}}.} The following table shows the construction of both tableaux for this example == Combinatorial properties of the RSK correspondence == === The case of permutation matrices === If A {\displaystyle A} is a permutation matrix then RSK outputs standard Young Tableaux (SYT), P , Q {\displaystyle P,Q} of the same shape λ {\displaystyle \lambda } . Conversely, if P , Q {\displaystyle P,Q} are SYT having the same shape λ {\displaystyle \lambda } , then the corresponding matrix A {\displaystyle A} is a permutation matrix. As a result of this property by simply comparing the cardinalities of the two sets on the two sides of the bijective mapping we get the following corollary: Corollary 1: For each n ≥ 1 {\displaystyle n\geq 1} we have ∑ λ ⊢ n ( t λ ) 2 = n ! {\displaystyle \sum _{\lambda \vdash n}(t_{\lambda })^{2}=n!} where λ ⊢ n {\displaystyle \lambda \vdash n} means λ {\displaystyle \lambda } varies over all partitions of n {\displaystyle n} and t λ {\displaystyle t_{\lambda }} is the number of standard Young tableaux of shape λ {\displaystyle \lambda } . === Symmetry === Let A {\displaystyle A} be a matrix with non-negative entries. Suppose the RSK algorithm maps A {\displaystyle A} to ( P , Q ) {\displaystyle (P,Q)} then the RSK algorithm maps A T {\displaystyle A^{T}} to ( Q , P ) {\displaystyle (Q,P)} , where A T {\displaystyle A^{T}} is the transpose of A {\displaystyle A} . In particular for the case of permutation matrices, one recovers the symmetry of the Robinson–Schensted correspondence: Theorem 2: If the permutation σ {\displaystyle \sigma } corresponds to a triple ( λ , P , Q ) {\displaystyle (\lambda ,P,Q)} , then the inverse permutation, σ − 1 {\displaystyle \sigma ^{-1}} , corresponds to ( λ , Q , P ) {\displaystyle (\lambda ,Q,P)} . This leads to the following relation between the number of involutions on S n {\displaystyle S_{n}} with the number of tableaux that can be formed from S n {\displaystyle S_{n}} (An involution is a permutation that is its own inverse): Corollary 2: The number of tableaux that can be formed from { 1 , 2 , 3 , … , n } {\displaystyle \{1,2,3,\ldots ,n\}} is equal to the number of involutions on { 1 , 2 , 3 , … , n } {\displaystyle \{1,2,3,\ldots ,n\}} . Proof: If π {\displaystyle \pi } is an involution corresponding to ( P , Q ) {\displaystyle (P,Q)} , then π = π − {\displaystyle \pi =\pi ^{-}} corresponds to ( Q , P ) {\displaystyle (Q,P)} ; hence P = Q {\displaystyle P=Q} . Conversely, if π {\displaystyle \pi } is any permutation corresponding to ( P , P ) {\displaystyle (P,P)} , then π − {\displaystyle \pi ^{-}} also corresponds to ( P , P ) {\displaystyle (P,P)} ; hence π = π − {\displaystyle \pi =\pi ^{-}} . So there is a one-one correspondence between involutions π {\displaystyle \pi } and tableaux P {\displaystyle P} The number of involutions on { 1 , 2 , 3 , … , n } {\displaystyle \{1,2,3,\ldots ,n\}} is given by the recurrence: a ( n ) = a ( n − 1 ) + ( n − 1 ) a ( n − 2 ) {\displaystyle a(n)=a(n-1)+(n-1)a(n-2)\,} Where a ( 1 ) = 1 , a ( 2 ) = 2 {\displaystyle a(1)=1,a(2)=2} . By solving this recurrence we can get the number of involutions on { 1 , 2 , 3 , … , n } {\displaystyle \{1,2,3,\ldots ,n\}} , I ( n ) = n ! ∑ k = 0 ⌊ n / 2 ⌋ 1 2 k k ! ( n − 2 k ) ! {\displaystyle I(n)=n!\sum _{k=0}^{\lfloor n/2\rfloor }{\frac {1}{2^{k}k!(n-2k)!}}} === Symmetric matrices === Let A = A T {\displaystyle A=A^{T}} and let the RSK algorithm map the matrix A {\displaystyle A} to the pair ( P , P ) {\displaystyle (P,P)} , where P {\displaystyle P} is an SSYT of shape α {\displaystyle \alpha } . Let α = ( α 1 , α 2 , … ) {\displaystyle \alpha =(\alpha _{1},\alpha _{2},\ldots )} where the α i {\displaystyle \alpha _{i}} are non-negative integers and ∑ i α i < ∞ {\textstyle \sum _{i}\alpha _{i}<\infty } . Then the map A ⟼ P {\displaystyle A\longmapsto P} establishes a bijection between symmetric matrices with r o w ( A ) = α {\displaystyle \mathrm {row} (A)=\alpha } and SSYT's of weight α {\displaystyle \alpha } . == Applications of the RSK correspondence == === Cauchy's identity === The Robinson–Schensted–Knuth correspondence provides a direct bijective proof of the following celebrated identity for symmetric functions: ∏ i , j ( 1 − x i y j ) − 1 = ∑ λ s λ ( x ) s λ ( y ) {\displaystyle \prod _{i,j}(1-x_{i}y_{j})^{-1}=\sum _{\lambda }s_{\lambda }(x)s_{\lambda }(y)} where s λ {\displaystyle s_{\lambda }} are Schur functions. === Kostka numbers === Fix partitions μ , ν ⊢ n {\displaystyle \mu ,\nu \vdash n} , then ∑ λ ⊢ n K λ μ K λ ν = N μ ν {\displaystyle \sum _{\lambda \vdash n}K_{\lambda \mu }K_{\lambda \nu }=N_{\mu \nu }} where K λ μ {\displaystyle K_{\lambda \mu }} and K λ ν {\displaystyle K_{\lambda \nu }} denote the Kostka numbers and N μ ν {\displaystyle N_{\mu \nu }} is the number of matrices A {\displaystyle A} , with non-negative elements, with r o w ( A ) = μ {\displaystyle \mathrm {row} (A)=\mu } and c o l u m n ( A ) = ν {\displaystyle \mathrm {column} (A)=\nu } . == References == Brualdi, Richard A. (2006). Combinatorial matrix classes. Encyclopedia of Mathematics and Its Applications. Vol. 108. Cambridge: Cambridge University Press. pp. 135–162. ISBN 0-521-86565-4. Zbl 1106.05001.
Wikipedia:Rod Downey#0	Rodney Graham Downey (born 20 September 1957) is a New Zealand and Australian mathematician and computer scientist, an emeritus professor in the School of Mathematics and Statistics at Victoria University of Wellington in New Zealand. He is known for his work in mathematical logic and computational complexity theory, and in particular for founding the field of parameterised complexity together with Michael Fellows. == Biography == Downey earned a bachelor's degree at the University of Queensland in 1978, and then went on to graduate school at Monash University, earning a doctorate in 1982 under the supervision of John Crossley. After holding teaching and visiting positions at the Chisholm Institute of Technology, Western Illinois University, the National University of Singapore, and the University of Illinois at Urbana-Champaign, he came to New Zealand in 1986 as a lecturer at Victoria University. He was promoted to reader in 1991, was given a personal chair at Victoria in 1995, and retired in 2023. Downey was president of the New Zealand Mathematical Society from 2001 to 2003. == Publications == Downey is the co-author of six books: Parameterized Complexity (with Michael Fellows, Springer, 1999) Algorithmic Randomness and Complexity (with D. Hirschfeldt, Springer, 2010) Fundamentals of Parameterized Complexity (with Michael Fellows, Springer, 2013) Minimal Weak Truth Table Degrees and Computably Enumerable Turing Degrees (with Keng Meng Ng and David Reed Solomon, Memoirs American Mathematical Society, Vol. 2184, 2020) A Hierarchy of Turing Degrees (with Noam Greenberg, Annals of Mathematics Studies No. 206, Princeton University Press, 2020) Computability and Complexity: Foundations and Tools for Pursuing Scientific Applications, (Springer-Verlag Texts in Computer Science, 2024) He is also the author or co-author of around 300 research papers, including a highly cited sequence of four papers with Michael Fellows and Karl Abrahamson setting the foundation for the study of parameterised complexity. == Awards and honours == In 1990, Downey won the Hamilton Research Award from the Royal Society of New Zealand. In 1992, Downey won the Research Award of the New Zealand Mathematical Society "for penetrating and prolific investigations that have made him a leading expert in many aspects of recursion theory, effective algebra and complexity". In 1994, he won the New Zealand Association of Scientists Research Award, and became a fellow of the Royal Society of New Zealand in 1996. In 2006, he became the first New Zealand-based mathematician to give an Invited Lecture at the International Congress of Mathematicians. He has also given invited lectures at the International Congress of Logic, Methodology and Philosophy of Science and the ACM Conference on Computational Complexity. He was elected as an ACM Fellow in 2007 "for contributions to computability and complexity theory", becoming the second ACM Fellow in New Zealand, and in the same year was elected as a fellow of the New Zealand Mathematical Society. Also in 2007 he was awarded a James Cook Research Fellowship for research on the nature of computation. In 2010 he won the Shoenfield Prize (for articles) of the Association for Symbolic Logic for his work with Denis Hirschfeldt, Andre Nies, and Sebastiaan Terwijn on randomness. In 2011, the Royal Society of New Zealand gave him their Hector Medal "for his outstanding, internationally acclaimed work in recursion theory, computational complexity, and other aspects of mathematical logic and combinatorics." In 2012, he became a fellow of the American Mathematical Society. In 2013, he became a Fellow of the Australian Mathematical Society. In 2014, he was awarded the Nerode Prize from the European Association for Theoretical Computer Science, jointly with Hans Bodlaender, Michael Fellows, Danny Hermelin, Lance Fortnow and Rahul Santhanam for their work on kernelization lower bounds. In October 2016, Downey received a distinguished Humboldt Research Award for his academic contributions. With Denis Hirschfeldt, Downey won another Shoenfield Prize from the Association for Symbolic Logic, this time the 2016 book prize for Algorithmic Randomness and Complexity. In 2018, Downey delivered the Gödel Lecture of the Association for Symbolic Logic, titled Algorithmic randomness, at the European Summer Meeting at Udine, Italy. The same year, Downey was awarded the Rutherford Medal, the highest honour awarded by the Royal Society of New Zealand, "for his pre-eminent revolutionary research into computability, including development of the theory of parameterised complexity and the algorithmic study of randomness." In 2022, Downey was awarded the New Zealand Association of von Humboldt Fellows Research Award for research over the preceding five years. In 2023, Downey was awarded the S. Barry Cooper Prize from the Association for Computability in Europe. This award is awarded every two to three years "to a researcher who has contributed to a broad understanding and foundational study of computability by outstanding results, by seminal and lasting theory building, by exceptional service to the research communities involved, or by a combination of these." In 2024, Downey was awarded the New Zealand Mathematics Society Kalman Prize "for a single publication of original research, which may be an article, monograph or book, having appeared within the last 5 calendar years: 2019-2024". This publication was the monograph "A Hierarchy of Turing Degrees" published in the Annals of Mathematics Studies, jointly written with Noam Greenberg. == References == == External links == Home page at Victoria University of Wellington
Wikipedia:Rod calculus#0	Rod calculus or rod calculation was the mechanical method of algorithmic computation with counting rods in China from the Warring States to Ming dynasty before the counting rods were increasingly replaced by the more convenient and faster abacus. Rod calculus played a key role in the development of Chinese mathematics to its height in the Song dynasty and Yuan dynasty, culminating in the invention of polynomial equations of up to four unknowns in the work of Zhu Shijie. == Hardware == The basic equipment for carrying out rod calculus is a bundle of counting rods and a counting board. The counting rods are usually made of bamboo sticks, about 12 cm- 15 cm in length, 2mm to 4 mm diameter, sometimes from animal bones, or ivory and jade (for well-heeled merchants). A counting board could be a table top, a wooden board with or without grid, on the floor or on sand. In 1971 Chinese archaeologists unearthed a bundle of well-preserved animal bone counting rods stored in a silk pouch from a tomb in Qian Yang county in Shanxi province, dated back to the first half of Han dynasty (206 BC – 8AD). In 1975 a bundle of bamboo counting rods was unearthed. The use of counting rods for rod calculus flourished in the Warring States, although no archaeological artefacts were found earlier than the Western Han dynasty (the first half of Han dynasty; however, archaeologists did unearth software artefacts of rod calculus dated back to the Warring States); since the rod calculus software must have gone along with rod calculus hardware, there is no doubt that rod calculus was already flourishing during the Warring States more than 2,200 years ago. == Software == The key software required for rod calculus was a simple 45 phrase positional decimal multiplication table used in China since antiquity, called the nine-nine table, which were learned by heart by pupils, merchants, government officials and mathematicians alike. == Rod numerals == === Displaying numbers === Rod numerals is the only numeric system that uses different placement combination of a single symbol to convey any number or fraction in the Decimal System. For numbers in the units place, every vertical rod represent 1. Two vertical rods represent 2, and so on, until 5 vertical rods, which represents 5. For number between 6 and 9, a biquinary system is used, in which a horizontal bar on top of the vertical bars represent 5. The first row are the number 1 to 9 in rod numerals, and the second row is the same numbers in horizontal form. For numbers larger than 9, a decimal system is used. Rods placed one place to the left of the units place represent 10 times that number. For the hundreds place, another set of rods is placed to the left which represents 100 times of that number, and so on. As shown in the adjacent image, the number 231 is represented in rod numerals in the top row, with one rod in the units place representing 1, three rods in the tens place representing 30, and two rods in the hundreds place representing 200, with a sum of 231. When doing calculation, usually there was no grid on the surface. If rod numerals two, three, and one is placed consecutively in the vertical form, there's a possibility of it being mistaken for 51 or 24, as shown in the second and third row of the adjacent image. To avoid confusion, number in consecutive places are placed in alternating vertical and horizontal form, with the units place in vertical form, as shown in the bottom row on the right. === Displaying zeroes === In Rod numerals, zeroes are represented by a space, which serves both as a number and a place holder value. Unlike in Hindu-Arabic numerals, there is no specific symbol to represent zero. Before the introduction of a written zero, in addition to a space to indicate no units, the character in the subsequent unit column would be rotated by 90°, to reduce the ambiguity of a single zero. For example 107 (𝍠 𝍧) and 17 (𝍩𝍧) would be distinguished by rotation, in addition to the space, though multiple zero units could lead to ambiguity, e.g. 1007 (𝍩 𝍧), and 10007 (𝍠 𝍧). In the adjacent image, the number zero is merely represented with a space. === Negative and positive numbers === Song mathematicians used red to represent positive numbers and black for negative numbers. However, another way is to add a slash to the last place to show that the number is negative. === Decimal fraction === The Mathematical Treatise of Sunzi used decimal fraction metrology. The unit of length was 1 chi, 1 chi = 10 cun, 1 cun = 10 fen, 1 fen = 10 li, 1 li = 10 hao, 10 hao = 1 shi, 1 shi = 10 hu. 1 chi 2 cun 3 fen 4 li 5 hao 6 shi 7 hu is laid out on counting board as where is the unit measurement chi. Southern Song dynasty mathematician Qin Jiushao extended the use of decimal fraction beyond metrology. In his book Mathematical Treatise in Nine Sections, he formally expressed 1.1446154 day as 日 He marked the unit with a word “日” (day) underneath it. == Addition == Rod calculus works on the principle of addition. Unlike Arabic numerals, digits represented by counting rods have additive properties. The process of addition involves mechanically moving the rods without the need of memorising an addition table. This is the biggest difference with Arabic numerals, as one cannot mechanically put 1 and 2 together to form 3, or 2 and 3 together to form 5. The adjacent image presents the steps in adding 3748 to 289: Place the augend 3748 in the first row, and the addend 289 in the second. Calculate from LEFT to RIGHT, from the 2 of 289 first. Take away two rods from the bottom add to 7 on top to make 9. Move 2 rods from top to bottom 8, carry one to forward to 9, which becomes zero and carries to 3 to make 4, remove 8 from bottom row. Move one rod from 8 on top row to 9 on bottom to form a carry one to next rank and add one rod to 2 rods on top row to make 3 rods, top row left 7. Result 3748+289=4037 The rods in the augend change throughout the addition, while the rods in the addend at the bottom "disappear". == Subtraction == === Without borrowing === In situation in which no borrowing is needed, one only needs to take the number of rods in the subtrahend from the minuend. The result of the calculation is the difference. The adjacent image shows the steps in subtracting 23 from 54. === Borrowing === In situations in which borrowing is needed such as 4231–789, one need use a more complicated procedure. The steps for this example are shown on the left. Place the minuend 4231 on top, the subtrahend 789 on the bottom. Calculate from the left to the right. Borrow 1 from the thousands place for a ten in the hundreds place, minus 7 from the row below, the difference 3 is added to the 2 on top to form 5. The 7 on the bottom is subtracted, shown by the space. Borrow 1 from the hundreds place, which leaves 4. The 10 in the tens place minus the 8 below results in 2, which is added to the 3 above to form 5. The top row now is 3451, the bottom 9. Borrow 1 from the 5 in the tens place on top, which leaves 4. The 1 borrowed from the tens is 10 in the units place, subtracting 9 which results in 1, which are added to the top to form 2. With all rods in the bottom row subtracted, the 3442 in the top row is then, the result of the calculation == Multiplication == Sunzi Suanjing described in detail the algorithm of multiplication. On the left are the steps to calculate 38×76: Place the multiplicand on top, the multiplier on bottom. Line up the units place of the multiplier with the highest place of the multiplicand. Leave room in the middle for recording. Start calculating from the highest place of the multiplicand (in the example, calculate 30×76, and then 8×76). Using the multiplication table 3 times 7 is 21. Place 21 in rods in the middle, with 1 aligned with the tens place of the multiplier (on top of 7). Then, 3 times 6 equals 18, place 18 as it is shown in the image. With the 3 in the multiplicand multiplied totally, take the rods off. Move the multiplier one place to the right. Change 7 to horizontal form, 6 to vertical. 8×7 = 56, place 56 in the second row in the middle, with the units place aligned with the digits multiplied in the multiplier. Take 7 out of the multiplier since it has been multiplied. 8×6 = 48, 4 added to the 6 of the last step makes 10, carry 1 over. Take off 8 of the units place in the multiplicand, and take off 6 in the units place of the multiplier. Sum the 2380 and 508 in the middle, which results in 2888: the product. == Division == The animation on the left shows the steps for calculating ⁠309/7⁠ = 44⁠1/7⁠. Place the dividend, 309, in the middle row and the divisor, 7, in the bottom row. Leave space for the top row. Move the divisor, 7, one place to the left, changing it to horizontal form. Using the Chinese multiplication table and division, 30÷7 equals 4 remainder 2. Place the quotient, 4, in the top row and the remainder, 2, in the middle row. Move the divisor one place to the right, changing it to vertical form. 29÷7 equals 4 remainder 1. Place the quotient, 4, on top, leaving the divisor in place. Place the remainder in the middle row in place of the dividend in this step. The result is the quotient is 44 with a remainder of 1 The Sunzi algorithm for division was transmitted in toto by al Khwarizmi to Islamic country from Indian sources in 825AD. Al Khwarizmi's book was translated into Latin in the 13th century, The Sunzi division algorithm later evolved into Galley division in Europe. The division algorithm in Abu'l-Hasan al-Uqlidisi's 925AD book Kitab al-Fusul fi al-Hisab al-Hindi and in 11th century Kushyar ibn Labban's Principles of Hindu Reckoning were identical to Sunzu's division algorithm. == Fractions == If there is a remainder in a place value decimal rod calculus division, both the remainder and the divisor must be left in place with one on top of another. In Liu Hui's notes to Jiuzhang suanshu (2nd century BCE), the number on top is called "shi" (实), while the one at bottom is called "fa" (法). In Sunzi Suanjing, the number on top is called "zi" (子) or "fenzi" (lit., son of fraction), and the one on the bottom is called "mu" (母) or "fenmu" (lit., mother of fraction). Fenzi and Fenmu are also the modern Chinese name for numerator and denominator, respectively. As shown on the right, 1 is the numerator remainder, 7 is the denominator divisor, formed a fraction ⁠1/7⁠. The quotient of the division ⁠309/7⁠ is 44 + ⁠1/7⁠. Liu Hui used a lot of calculations with fractions in Haidao Suanjing. This form of fraction with numerator on top and denominator at bottom without a horizontal bar in between, was transmitted to Arabic country in an 825AD book by al Khwarizmi via India, and in use by 10th century Abu'l-Hasan al-Uqlidisi and 15th century Jamshīd al-Kāshī's work "Arithematic Key". === Addition === ⁠1/3⁠ + ⁠2/5⁠ Put the two numerators 1 and 2 on the left side of counting board, put the two denominators 3 and 5 at the right hand side Cross multiply 1 with 5, 2 with 3 to get 5 and 6, replace the numerators with the corresponding cross products. Multiply the two denominators 3 × 5 = 15, put at bottom right Add the two numerators 5 and 6 = 11 put on top right of counting board. Result: ⁠1/3⁠ + ⁠2/5⁠ = ⁠11/15⁠ === Subtraction === ⁠8/9⁠ − ⁠1/5⁠ Put down the rod numeral for numerators 1 and 8 at left hand side of a counting board Put down the rods for denominators 5 and 9 at the right hand side of a counting board Cross multiply 1 × 9 = 9, 5 × 8 = 40, replace the corresponding numerators Multiply the denominators 5 × 9 = 45, put 45 at the bottom right of counting board, replace the denominator 5 Subtract 40 − 9 = 31, put on top right. Result: ⁠8/9⁠ − ⁠1/5⁠ = ⁠31/45⁠ === Multiplication === 3⁠1/3⁠ × 5⁠2/5⁠ Arrange the counting rods for 3⁠1/3⁠ and 5⁠2/5⁠ on the counting board as shang, shi, fa tabulation format. shang times fa add to shi: 3 × 3 + 1 = 10; 5 × 5 + 2 = 27 shi multiplied by shi:10 × 27 = 270 fa multiplied by fa:3 × 5 = 15 shi divided by fa: 3⁠1/3⁠ × 5⁠2/5⁠ = 18 === Highest common factor and fraction reduction === The algorithm for finding the highest common factor of two numbers and reduction of fraction was laid out in Jiuzhang suanshu. The highest common factor is found by successive division with remainders until the last two remainders are identical. The animation on the right illustrates the algorithm for finding the highest common factor of ⁠32,450,625/59,056,400⁠ and reduction of a fraction. In this case the hcf is 25. Divide the numerator and denominator by 25. The reduced fraction is ⁠1,298,025/2,362,256⁠. === Interpolation === Calendarist and mathematician He Chengtian (何承天) used fraction interpolation method, called "harmonisation of the divisor of the day" (调日法) to obtain a better approximate value than the old one by iteratively adding the numerators and denominators a "weaker" fraction with a "stronger fraction". Zu Chongzhi's legendary π = ⁠355/113⁠ could be obtained with He Chengtian's method == System of linear equations == Chapter Eight Rectangular Arrays of Jiuzhang suanshu provided an algorithm for solving System of linear equations by method of elimination: Problem 8-1: Suppose we have 3 bundles of top quality cereals, 2 bundles of medium quality cereals, and a bundle of low quality cereal with accumulative weight of 39 dou. We also have 2, 3 and 1 bundles of respective cereals amounting to 34 dou; we also have 1,2 and 3 bundles of respective cereals, totaling 26 dou. Find the quantity of top, medium, and poor quality cereals. In algebra, this problem can be expressed in three system equations with three unknowns. { 3 x + 2 y + z = 39 2 x + 3 y + z = 34 x + 2 y + 3 z = 26 {\displaystyle {\begin{cases}3x+2y+z=39\\2x+3y+z=34\\x+2y+3z=26\end{cases}}} This problem was solved in Jiuzhang suanshu with counting rods laid out on a counting board in a tabular format similar to a 3x4 matrix: Algorithm: Multiply the center column with right column top quality number. Repeatedly subtract right column from center column, until the top number of center column=0. multiply the left column with the value of top row of right column. Repeatedly subtract right column from left column, until the top number of left column=0. After applying above elimination algorithm to the reduced center column and left column, the matrix was reduced to triangular shape. The amount of one bundle of low quality cereal = 99 36 = 2 3 4 {\displaystyle ={\frac {99}{36}}=2{\frac {3}{4}}} From which the amount of one bundle of top and medium quality cereals can be found easily: One bundle of top quality cereals=9 dou 1 4 {\displaystyle {\frac {1}{4}}} One bundle of medium cereal=4 dou 1 4 {\displaystyle {\frac {1}{4}}} == Extraction of Square root == Algorithm for extraction of square root was described in Jiuzhang suanshu and with minor difference in terminology in Sunzi Suanjing. The animation shows the algorithm for rod calculus extraction of an approximation of the square root 234567 ≈ 484 311 968 {\displaystyle {\sqrt {234567}}\approx 484{\tfrac {311}{968}}} from the algorithm in chap 2 problem 19 of Sunzi Suanjing: Now there is a square area 234567, find one side of the square. The algorithm is as follows: Set up 234567 on the counting board, on the second row from top, named shi Set up a marker 1 at 10000 position at the 4th row named xia fa Estimate the first digit of square root to be counting rod numeral 4, put on the top row (shang) hundreds position, Multiply the shang 4 with xiafa 1, put the product 4 on 3rd row named fang fa Multiply shang with fang fa deduct the product 4x4=16 from shi: 23-16=7, remain numeral 7. double up the fang fa 4 to become 8, shift one position right, and change the vertical 8 into horizontal 8 after moved right. Move xia fa two position right. Estimate second digit of shang as 8: put numeral 8 at tenth position on top row. Multiply xia fa with the new digit of shang, add to fang fa . 8 calls 8 =64, subtract 64 from top row numeral "74", leaving one rod at the most significant digit. double the last digit of fang fa 8, add to 80 =96 Move fang fa96 one position right, change convention;move xia fa "1" two position right. Estimate 3rd digit of shang to be 4. Multiply new digit of shang 4 with xia fa 1, combined with fang fa to make 964. subtract successively 49=36,46=24,44=16 from the shi, leaving 311 double the last digit 4 of fang fa into 8 and merge with fang fa result 234567 ≈ 484 311 968 {\displaystyle {\sqrt {234567}}\approx 484{\tfrac {311}{968}}} North Song dynasty mathematician Jia Xian developed an additive multiplicative algorithm for square root extraction, in which he replaced the traditional "doubling" of "fang fa" by adding shang digit to fang fa digit, with same effect. == Extraction of cubic root == Jiuzhang suanshu vol iv "shaoguang" provided algorithm for extraction of cubic root. 〔一九〕今有積一百八十六萬八百六十七尺。問為立方幾何？答曰：一百二十三尺。 problem 19: We have a 1860867 cubic chi, what is the length of a side ? Answer:123 chi. 1860867 3 = 123 {\displaystyle {\sqrt[{3}]{1860867}}=123} North Song dynasty mathematician Jia Xian invented a method similar to simplified form of Horner scheme for extraction of cubic root. The animation at right shows Jia Xian's algorithm for solving problem 19 in Jiuzhang suanshu vol 4. == Polynomial equation == North Song dynasty mathematician Jia Xian invented Horner scheme for solving simple 4th order equation of the form x 4 = a {\displaystyle x^{4}=a} South Song dynasty mathematician Qin Jiushao improved Jia Xian's Horner method to solve polynomial equation up to 10th order. The following is algorithm for solving − x 4 + 15245 x 2 − 6262506.25 = 0 {\displaystyle -x^{4}+15245x^{2}-6262506.25=0} in his Mathematical Treatise in Nine Sections vol 6 problem 2. This equation was arranged bottom up with counting rods on counting board in tabular form Algorithm: Arrange the coefficients in tabular form, constant at shi, coeffienct of x at shang lian, the coeffiecnt of ⁠ x 4 {\displaystyle x^{4}} ⁠ at yi yu;align the numbers at unit rank. Advance shang lian two ranks Advance yi yu three ranks Estimate shang=20 let xia lian =shang yi yu let fu lian=shang yi yu merge fu lian with shang lian let fang=shang shang lian subtract shangfang from shi add shang yi yu to xia lian retract xia lian 3 ranks, retract yi yu 4 ranks The second digit of shang is 0 merge shang lian into fang merge yi yu into xia lian Add yi yu to fu lian, subtract the result from fang, let the result be denominator find the highest common factor =25 and simplify the fraction 32450625 59056400 {\displaystyle {\frac {32450625}{59056400}}} solution x = 20 1298205 2362256 {\displaystyle x=20{\frac {1298205}{2362256}}} == Tian Yuan shu == Yuan dynasty mathematician Li Zhi developed rod calculus into Tian yuan shu Example Li Zhi Ceyuan haijing vol II, problem 14 equation of one unknown: − x 2 − 680 x + 96000 = 0 {\displaystyle -x^{2}-680x+96000=0} 元 == Polynomial equations of four unknowns == Mathematician Zhu Shijie further developed rod calculus to include polynomial equations of 2 to four unknowns. For example, polynomials of three unknowns: Equation 1: − y − z − y 2 ∗ x − x + x y z = 0 {\displaystyle -y-z-y^{2}*x-x+xyz=0} 太 Equation 2: − y − z + x − x 2 + x z = 0 {\displaystyle -y-z+x-x^{2}+xz=0} Equation 3: y 2 − z 2 + x 2 = 0 ; {\displaystyle y^{2}-z^{2}+x^{2}=0;} 太 After successive elimination of two unknowns, the polynomial equations of three unknowns was reduced to a polynomial equation of one unknown: x 4 − 6 x 3 + 4 x 2 + 6 x − 5 = 0 {\displaystyle x^{4}-6x^{3}+4x^{2}+6x-5=0} Solved x=5; Which ignores 3 other answers, 2 are repeated. == See also == Chinese mathematics Counting rods == References == Lam Lay Yong (蓝丽蓉) Ang Tian Se (洪天赐), Fleeting Footsteps, World Scientific ISBN 981-02-3696-4 Jean Claude Martzloff, A History of Chinese Mathematics ISBN 978-3-540-33782-9
Wikipedia:Roderick S. C. Wong#0	Roderick S. C. Wong (born October 2, 1944) is a mathematician who works in classical analysis. His research mainly focuses on asymptotic analysis, singular perturbation theory, special functions and orthogonal polynomials, integral transforms, integral equations, and ordinary differential equations. He is currently a chair professor at City University of Hong Kong and director of the Liu Bie Ju Centre for Mathematical Sciences. == Education and career == Wong obtained his BA degree in mathematics from San Diego State College in 1965 and his PhD from the University of Alberta in 1969. He started his career as an assistant professor in the Department of Mathematics at the University of Manitoba and was promoted to full professor in 1979. He received a Killam Research Fellowship (from the Canada Council) from 1982 to 1984 and became a Fellow of the Royal Society of Canada in 1993. He was appointed the head of the Department of Applied Mathematics in 1986, a post which he held until he left the University of Manitoba in 1994. After spending almost 30 years in Canada, Wong returned to Hong Kong and joined City University of Hong Kong in early 1994 to take up the post of professor of mathematics. He was concurrently appointed head of the then newly formed Department of Mathematics. In 1995, he led the efforts for the establishment of the Liu Bie Ju Centre for Mathematical Sciences and was appointed director of the centre. He was appointed to dean of College of Science and Engineering in 1998, and became vice-president (Research & Technology) and dean of graduate studies in 2006. He later took up the position of vice-president for development & external relations and had been very successful in soliciting pledges of funds for the university. He established the William Benter Prize in Applied Mathematics in 2010. Over the years, the award has been getting more and more recognized internationally and has become one of the highly acclaimed accolades in the field of applied mathematics. Wong has received many awards and honours for his achievements and contribution over the years. He is a Foreign Member of Accademia delle Scienze di Torino in Italy, a member of the European Academy of Sciences, and a Founding Fellow of Hong Kong Institution of Science. He was awarded the Chevalier dans l’Ordre National de la Légion d’Honneur. == Professional activities == Chairman, Selection Committee of the William Benter Prize in Applied Mathematics, 2008–present == Books (authored or edited) == R. Wong (2001). Asymptotic Approximations of Integrals. Philadelphia, PA: eprinted by SIAM. doi:10.1137/1.9780898719260. ISBN 978-0-89871-497-5. R. Wong (2010). Lecture Notes on Applied Analysis. Series in Analysis. Vol. 5. World Scientific. doi:10.1142/7475. ISBN 978-981-4287-74-6. R. Wong with Richard Beals (mathematician) (2010). Special Functions: A Graduate Text. Cambridge University Press. doi:10.1017/CBO9780511762543. ISBN 978-051-1762-54-3. R Wong with Richard Beals (mathematician) (2016). Special Functions and Orthogonal Polynomials. Cambridge University Press. ISBN 978-110-7106-98-7. D. Dai, H. – H. Dai, T. Yang and D. X. Zhou (Editors) (2016). The Selected Works of Roderick S.C. Wong. World Scientific. doi:10.1142/9489. ISBN 978-981-4656-04-7. {{cite book}}: \|author= has generic name (help)CS1 maint: multiple names: authors list (link) R. Wong, ed. (1990). Asymptotic and Computational Analysis. Marcel Dekker. ISBN 978-082-4783-47-1. R. Wong, ed. (2000). Selected Papers of F. W. J. Olver. World Scientific Series in 20th Century Mathematics. Vol. 7. World Scientific. doi:10.1142/4251. ISBN 978-981-02-4106-3. R. Wong (Editor, with F. Cucker) (2000). The Collected Papers of Stephen Smale. World Scientific, Singapore. doi:10.1142/4424. ISBN 978-981-02-4307-4. {{cite book}}: \|author= has generic name (help) R. Wong (Editor, with C. Dunkl and Mourad Ismail) (2000). Special Functions. World Scientific, Singapore. doi:10.1142/4502. ISBN 978-981-02-4393-7. {{cite book}}: \|author= has generic name (help) R. Wong (Editor, with H. Chen) (2004). Differential Equations and Asymptotic Theory in Mathematical Physics. World Scientific, Singapore. doi:10.1142/5667. ISBN 978-981-256-055-1. {{cite book}}: \|author= has generic name (help) == References == == External links == Liu Bie Centre for Mathematical Sciences William Benter Prize in Applied Mathematics Video of Liu Bie Ju Mathematical Centre broadcast at International Congress of Mathematicians 2018, Brazil on YouTube
Wikipedia:Rodion Kuzmin#0	Rodion Osievich Kuzmin (Russian: Родион Осиевич Кузьмин, 9 November 1891, Riabye village in the Haradok district – 24 March 1949, Leningrad) was a Soviet mathematician, known for his works in number theory and analysis. His name is sometimes transliterated as Kusmin. He was an Invited Speaker of the ICM in 1928 in Bologna. == Selected results == In 1928, Kuzmin solved the following problem due to Gauss (see Gauss–Kuzmin distribution): if x is a random number chosen uniformly in (0, 1), and x = 1 k 1 + 1 k 2 + ⋯ {\displaystyle x={\frac {1}{k_{1}+{\frac {1}{k_{2}+\cdots }}}}} is its continued fraction expansion, find a bound for Δ n ( s ) = P { x n ≤ s } − log 2 ⁡ ( 1 + s ) , {\displaystyle \Delta _{n}(s)=\mathbb {P} \left\{x_{n}\leq s\right\}-\log _{2}(1+s),} where x n = 1 k n + 1 + 1 k n + 2 + ⋯ . {\displaystyle x_{n}={\frac {1}{k_{n+1}+{\frac {1}{k_{n+2}+\cdots }}}}.} Gauss showed that Δn tends to zero as n goes to infinity, however, he was unable to give an explicit bound. Kuzmin showed that \| Δ n ( s ) \| ≤ C e − α n , {\displaystyle \|\Delta _{n}(s)\|\leq Ce^{-\alpha {\sqrt {n}}}~,} where C,α > 0 are numerical constants. In 1929, the bound was improved to C 0.7n by Paul Lévy. In 1930, Kuzmin proved that numbers of the form ab, where a is algebraic and b is a real quadratic irrational, are transcendental. In particular, this result implies that Gelfond–Schneider constant 2 2 = 2.6651441426902251886502972498731 … {\displaystyle 2^{\sqrt {2}}=2.6651441426902251886502972498731\ldots } is transcendental. See Gelfond–Schneider theorem for later developments. He is also known for the Kusmin-Landau inequality: If f {\displaystyle f} is continuously differentiable with monotonic derivative f ′ {\displaystyle f'} satisfying ‖ f ′ ( x ) ‖ ≥ λ > 0 {\displaystyle \Vert f'(x)\Vert \geq \lambda >0} (where ‖ ⋅ ‖ {\displaystyle \Vert \cdot \Vert } denotes the Nearest integer function) on a finite interval I {\displaystyle I} , then ∑ n ∈ I e 2 π i f ( n ) ≪ λ − 1 . {\displaystyle \sum _{n\in I}e^{2\pi if(n)}\ll \lambda ^{-1}.} == Notes == == External links == Rodion Kuzmin at the Mathematics Genealogy Project (The chronology there is apparently wrong, since J. V. Uspensky lived in USA from 1929.)
Wikipedia:Rodolfo H. Torres#0	Rodolfo Humberto Torres is an Argentinian American mathematician specializing in harmonic analysis who works as the Vice Chancellor for Research and Economic Development and a Distinguished Professor of Mathematics at the University of California, Riverside. Torres did his undergraduate studies at the National University of Rosario in Argentina, completing a licenciatura there in 1984. He earned his doctorate in 1989 from Washington University in St. Louis, with a dissertation entitled On the Boundedness of Certain Operators with Singular Kernels on Distribution Spaces and supervised by Björn D. Jawerth. In 2012 he became one of the inaugural fellows of the American Mathematical Society. He was named a Distinguished Professor in 2016. As well as his work in pure mathematics, Torres has also published works on light scattering mechanisms for the colorings of birds and insects. == References == == External links == Home page Rodolfo H. Torres publications indexed by Google Scholar
Wikipedia:Rogemar Mamon#0	Rogemar Sombong Mamon, is a Canadian mathematician, quant, and academic. He is a co-editor of the IMA Journal of Management Mathematics published by Oxford University Press since 2009. Mamon is known for his contributions to the developments and applications of regime-switching framework useful in economic, financial and actuarial modeling. Majority of his works promote regime-switching paradigms modulated by either discrete- or continuous-time hidden Markov models (HMM). A recurrent theme of his research is dynamic parameter estimation via HMM filtering recursions. He also made contributions in the areas of derivative pricing, asset allocation, risk measurement, filtering to remove noise from data as well as inverse problems in quantitative finance. He was the lead editor of the handbook Hidden Markov Models in Finance, published by Springer. In 2010, he and two co-authors won the Society of Actuaries Award for the Best Paper published in the North American Actuarial Journal. Since 2006, he has taught, conducted research and held administrative roles at the University of Western Ontario, and garnered recognitions for excellence in teaching and research. Previously, he held academic positions at Brunel University, London, UK; University of British Columbia; University of Waterloo; and University of Alberta. He spent short-term research visits at several institutions including the Isaac Newton Institute for Mathematical Sciences, University of Cambridge, England; Maxwell Institute for Mathematical Sciences, Scotland; Centre for Mathematical Physics and Stochastics, University of Aarhus, Denmark; Institute for Mathematics and its Applications, University of Minnesota, USA; University of Adelaide, Australia; University of Wollongong in New South Wales, Australia; and Centro de Investigacion en Matematicas, Mexico. Mamon holds professional designations conferred by various British learned societies. He is a Fellow and Chartered Mathematician of the Institute of Mathematics and its Applications; Chartered Scientist of the Science Council; and Fellow of the Higher Education Academy. He is also a Fellow of the Royal Society of Arts and the Royal Statistical Society, and was an elected member of the London Mathematical Society. He began PhD studies in Mathematical Finance at the University of Alberta, and completed his dissertation during a research visit at the University of Adelaide, Australia. He was supervised by Robert J. Elliott making him a mathematical descendant of Godfrey Harold Hardy, Sir Isaac Newton and Galileo Galilei. == References ==
Wikipedia:Roger Carter (mathematician)#0	Roger William Carter (25 August 1934 – 21 February 2022) was a British mathematician who was emeritus professor at the University of Warwick. He defined Carter subgroups and wrote the standard reference Simple Groups of Lie Type. He obtained his PhD at the University of Cambridge in 1960 and his dissertation was entitled Some Contributions to the Theory of Finite Soluble Groups, with Derek Taunt as thesis advisor. Carter died in Wirral on 21 February 2022, at the age of 87. == Publications == Carter, Roger W. (1961). "Nilpotent self normalizing subgroups of soluble groups". Mathematische Zeitschrift. 75 (2): 136–139. doi:10.1007/BF01211016. S2CID 120448397. Simple Groups of Lie Type by Roger W. Carter, ISBN 0-471-50683-4 Lie Algebras of Finite and Affine Type (Cambridge Studies in Advanced Mathematics) by Roger Carter, ISBN 0-521-85138-6 Carter, Roger W. (1993), Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, Wiley Classics Library, Chichester: Wiley, ISBN 978-0-471-94109-5 == References == Roger W. Carter at the Mathematics Genealogy Project Roger Carter Obituary
Wikipedia:Roger Collingwood#0	Roger Collingwood (fl. 1513) was a British mathematician. == Biography == Collingwood was elected a fellow of Queens' College, Cambridge, in 1497, being then B.A., and proceeded M.A. two years later. He had the college title for orders on 7 Aug. 1497, was dean of his college in 1504, and obtained a license on 16 Sept. 1507 to travel on the continent during four years for the purpose of studying canon law. On the expiration of that term it was stipulated that he was to resign his fellowship, and his name, accordingly, disappears from the college books after 1509–10. He acted, however, as proctor of the university in 1513. Under the name of 'Carbo-in-ligno' Collingwood wrote an unfinished treatise entitled 'Arithmetica Experimentalis,' which he dedicated, in the character of a former pupil, to Richard Fox, bishop of Winchester. The manuscript is preserved in the library of Corpus Christi College, Oxford. == References == This article incorporates text from a publication now in the public domain: Clerke, Agnes Mary (1887). "Collingwood, Roger". In Stephen, Leslie (ed.). Dictionary of National Biography. Vol. 11. London: Smith, Elder & Co.
Wikipedia:Roger Cotes#0	Roger Cotes (10 July 1682 – 5 June 1716) was an English mathematician, known for working closely with Isaac Newton by proofreading the second edition of his famous book, the Principia, before publication. He also devised the quadrature formulas known as Newton–Cotes formulas, which originated from Newton's research, and made a geometric argument that can be interpreted as a logarithmic version of Euler's formula. He was the first Plumian Professor at Cambridge University from 1707 until his death. == Early life == Cotes was born in Burbage, Leicestershire. His parents were Robert, the rector of Burbage, and his wife, Grace, née Farmer. Roger had an elder brother, Anthony (born 1681), and a younger sister, Susanna (born 1683), both of whom died young. At first Roger attended Leicester School, where his mathematical talent was recognised. His aunt Hannah had married Rev. John Smith, and Smith took on the role of tutor to encourage Roger's talent. The Smiths' son, Robert Smith, became a close associate of Roger Cotes throughout his life. Cotes later studied at St Paul's School in London and entered Trinity College, Cambridge, in 1699. He graduated BA in 1702 and MA in 1706. == Astronomy == Roger Cotes's contributions to modern computational methods lie heavily in the fields of astronomy and mathematics. Cotes began his educational career with a focus on astronomy. He became a fellow of Trinity College in 1707, and at age 26 he became the first Plumian Professor of Astronomy and Experimental Philosophy. On his appointment to professor, he opened a subscription list in an effort to provide an observatory for Trinity. Unfortunately, the observatory was still unfinished when Cotes died, and was demolished in 1797. In correspondence with Isaac Newton, Cotes designed a heliostat telescope with a mirror revolving by clockwork. He recomputed the solar and planetary tables of Giovanni Domenico Cassini and John Flamsteed, and he intended to create tables of the moon's motion, based on Newtonian principles. Finally, in 1707 he formed a school of physical sciences at Trinity in partnership with William Whiston. == The Principia == From 1709 to 1713, Cotes became heavily involved with the second edition of Newton's Principia, a book that explained Newton's theory of universal gravitation. The first edition of Principia had only a few copies printed and was in need of revision to include Newton's works and principles of lunar and planetary theory. Newton at first had a casual approach to the revision, since he had all but given up scientific work. However, through the vigorous passion displayed by Cotes, Newton's scientific hunger was once again reignited. The two spent nearly three and half years collaborating on the work, in which they fully deduce, from Newton's laws of motion, the theory of the moon, the equinoxes, and the orbits of comets. Only 750 copies of the second edition were printed although pirated copies from Amsterdam were also distributed to meet the demand for the work. As a reward to Cotes, he was given a share of the profits and 12 copies of his own. Cotes's original contribution to the work was a preface which supported the scientific superiority of Newton's principles over the then popular vortex theory of gravity advocated by René Descartes. Cotes concluded that the Newton's law of gravitation was confirmed by observation of celestial phenomena that were inconsistent with the vortex theory. == Mathematics == Cotes's major original work was in mathematics, especially in the fields of integral calculus, logarithms, and numerical analysis. He published only one scientific paper in his lifetime, titled Logometria, in which he successfully constructs the logarithmic spiral. After his death, many of Cotes's mathematical papers were edited by his cousin Robert Smith and published in a book, Harmonia mensurarum. Cotes's additional works were later published in Thomas Simpson's The Doctrine and Application of Fluxions. Although Cotes's style was somewhat obscure, his systematic approach to integration and mathematical theory was highly regarded by his peers. Cotes discovered an important theorem on the n-th roots of unity, foresaw the method of least squares, and discovered a method for integrating rational fractions with binomial denominators. He was also praised for his efforts in numerical methods, especially in interpolation methods and his table construction techniques. He was regarded as one of the few British mathematicians capable of following the powerful work of Sir Isaac Newton. == Death and assessment == Cotes died from a violent fever in Cambridge in 1716 at the early age of 33. Isaac Newton remarked, "If he had lived we would have known something." == See also == Cotes's spiral Extended Euclidean algorithm Newton–Cotes formulas Lituus (mathematics) == References == == Sources == [Anon.] "Cotes, Roger" . Encyclopædia Britannica. Vol. 7 (11th ed.). 1911. Cohen, I. B. (1971). Introduction to Newton's "Principia". Harvard: Harvard University Press. ISBN 0-674-46193-2. Edleston, J., ed. (1850). Correspondence of Sir Isaac Newton and Professor Cotes. via Internet Archive Gowing, R. (2002). Roger Cotes: Natural Philosopher. London: Cambridge University Press. ISBN 0-521-52649-3. Koyré, A. (1965). Newtonian Studies. London: Chapman & Hall. pp. 273–82. ISBN 0-412-42300-6. Price, D. J. (1952). "The early observatory instruments of Trinity College, Cambridge". Annals of Science. 8: 1–12. doi:10.1080/00033795200200012. Turnbull, H. W.; et al. (1975–1976). The Correspondence of Isaac Newton (7 vols ed.). London: Cambridge University Press. vols.5–6. Whitman, A., ed. (1972). Isaac Newton's Philosophiae Naturalis Principia Mathematica: The Third Edition (1726) with Variant Readings. London: Cambridge University Press. pp. 817–26. ISBN 0-521-07960-8. == External links == "Harmonia Mensurarum". MathPages. Retrieved 7 September 2007.- A more complete account of Cotes's involvement with Principia, followed by an even more thorough discussion of his mathematical work. Roger Cotes at the Mathematics Genealogy Project O'Connor, John J.; Robertson, Edmund F., "Roger Cotes", MacTutor History of Mathematics Archive, University of St Andrews Meli, D. B. (2004) "Cotes, Roger (1682–1716)", Oxford Dictionary of National Biography, Oxford University Press, retrieved 7 September 2007 (subscription or UK public library membership required)
Wikipedia:Roger Fletcher (mathematician)#0	Roger Fletcher FRS FRSE (29 January 1939 – 15 July 2016) was a British mathematician and professor at University of Dundee. He was a Fellow of the Society for Industrial and Applied Mathematics (SIAM) and was elected as a Fellow of the Royal Society in 2003. In 2006, he won the Lagrange Prize from SIAM. In 2008, he was awarded a Royal Medal of the Royal Society of Edinburgh. == See also == BFGS method Davidon–Fletcher–Powell formula Nonlinear conjugate gradient method == Bibliography == Practical methods of optimization, Wiley, 1987, ISBN 978-0-471-91547-8; Wiley, 2000, ISBN 978-0-471-49463-8 == References ==
Wikipedia:Roger Godement#0	Roger Godement (French: [ɡɔdmɑ̃]; 1 October 1921 – 21 July 2016) was a French mathematician, known for his work in functional analysis as well as his expository books. == Biography == Godement started as a student at the École normale supérieure in 1940, where he became a student of Henri Cartan. He started research into harmonic analysis on locally compact abelian groups, finding a number of major results; this work was in parallel but independent of similar investigations in the USSR and Japan. Work on the abstract theory of spherical functions published in 1952 proved very influential in subsequent work, particularly that of Harish-Chandra. The isolation of the concept of square-integrable representation is attributed to him. The Godement compactness criterion in the theory of arithmetic groups was a conjecture of his. He later worked with Jacquet on the zeta function of a simple algebra. He was an active member of the Bourbaki group in the early 1950s, and subsequently gave a number of significant Bourbaki seminars. He also took part in the Cartan seminar. His book Topologie Algébrique et Théorie des Faisceaux from 1958 was, as he said, a very unoriginal idea for the time (that is, to write an exposition of sheaf theory); as a non-specialist, he managed to write an enduring classic. It introduced the technical method of flasque resolutions, nowadays called Godement resolutions. It has also been credited as the place in which a comonad can first be discerned. He also wrote texts on Lie groups, abstract algebra and mathematical analysis. == Selected publications == Godement, Roger (1952). "A theory of spherical functions. I". Transactions of the American Mathematical Society. 73 (3): 496–556. doi:10.2307/1990805. JSTOR 1990805. MR 0052444. Topologie algébrique et théorie des faisceaux. Hermann 1958, 1960. Cours d´algèbre. Hermann 1963, 1966. Algebra. Hermann 1968. Godement, R. (1970), Notes on Jacquet–Langlands' theory, Institute for Advanced Study Godement, Roger; Jacquet, Hervé (1972). Zeta functions of simple algebras. Lecture Notes in Mathematics. Vol. 260. Berlin-New York: Springer-Verlag. MR 0342495. Analyse mathématique. 4 vols., Springer-Verlag 1998–2001. == See also == Commutation theorem for traces Plancherel theorem for spherical functions Standard L-function == References == == External links == Roger Godement at the Mathematics Genealogy Project O'Connor, John J.; Robertson, Edmund F., "Roger Godement", MacTutor History of Mathematics Archive, University of St Andrews
Wikipedia:Roger J-B Wets#0	Roger Jean-Baptiste Robert Wets (February 1937 – April 1, 2025) was a Belgian stochastic programming and a leader in variational analysis who publishes as Roger J-B Wets. His research, expositions, graduate students, and his collaboration with R. Tyrrell Rockafellar have had a profound influence on optimization theory, computations, and applications. Since 2009, Wets has been a distinguished research professor at the mathematics department of the University of California, Davis. == Schooling and positions == Roger Wets attended high school in Belgium, after which he worked for his family while earning his Licence in applied economics from Université de Bruxelles (Brussels, Belgium) in 1961. He was encouraged by Jacques H. Drèze to study optimization with George Dantzig at the program in operations research at the University of California, Berkeley. Dantzig and mathematician–statistician David Blackwell jointly supervised Wets's dissertation. In 1965 Wets befriended R. Tyrrell Rockafellar, whom Wets introduced to stochastic optimization, starting a collaboration of many decades. He worked at Boeing Scientific Research Labs, 1964–1970 and was Ford Professor at the University of Chicago, 1970–1972 before being appointed Professor at the Mathematics Department of the University of Kentucky and then University Research Professor (1977–78). While at the International Institute for Applied Systems Analysis (IIASA) in Austria, during 1980–1984, he led research in decision-making in uncertainty, returning as an acting leader in 1985–1987; during that time, Wets and Rockafellar developed the progressive-hedging algorithm for stochastic programming. The University of California, Davis named him Professor (1984–1997), Distinguished Professor, and Distinguished Research Professor of Mathematics (2009–). === Awards and contributions === Wets was awarded a George B. Dantzig Prize for "original research that has had a major impact on the field of mathematical programming" by the Society for Industrial and Applied Mathematics (SIAM) and the Mathematical Programming Society (MPS, now the Mathematical Optimization Society). In 1994, the Dantzig Prize was awarded to Wets and also to the French pioneer in nonsmooth computational-optimization, Claude Lemaréchal. Wets's contributions included developing set-valued analysis, including metric spaces of sets, which he used to study the convergence of epigraphs; Wets's ideas of epigraphical convergence was used to study the convergence iterative methods of stochastic optimization and has had applications in the approximation theory of statistics. A metric theory of finite-dimensional epigraphical convergence ("cosmic convergence") appears in Variational analysis. Wets and his coauthor R. Tyrrell Rockafellar were awarded the 1997 Frederick W. Lanchester Prize by the Institute for Operations Research and the Management Sciences (INFORMS) for their monograph Variational Analysis, which was published in November 1997 and copyrighted in 1998. With Rockafellar, Wets proposed, studied, and implemented the progressive-hedging algorithm for stochastic programming. Besides his theoretical and computational contributions, Wets has worked with applications on lake ecology (IIASA), finance (Frank Russel investment system), and developmental economics (World Bank). He also consulted with the development of professional stochastic-optimization software (IBM). Wets died on 1 April, 2025, at the age of 87–88. == See also == Pompeiu–Hausdorff distance == References == === Sources === Aardal, Karen (July 1995). "Optima interview Roger J.-B. (sic.) Wets" (PDF). Optima: Mathematical Programming Society Newsletter. 46. Mathematical Programming Society: 3–5. Anonymous, COSP (1 November 2004). Roger J-B Wets (PDF). Pioneers in Stochastic Programming. Committee on Stochastic Programming (COSP). Archived from the original (PDF) on 3 March 2012. Retrieved 12 March 2012. Anonymous, INFORMS (1998). "Roger J-B Wets, Past awards: 1977 Frederick W. Lanchester Prize: Winner". Institute for Operations Research and the Management Sciences (INFORMS). Archived from the original on 24 May 2012. Retrieved 12 March 2012. Rockafellar, R. Tyrrell; Wets, Roger J-B (2005) [1996]. Variational Analysis (PDF). Grundlehren der mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences). Vol. 317 (third corrected printing ed.). Berlin: Springer-Verlag. pp. xiv+733. doi:10.1007/978-3-642-02431-3. ISBN 978-3-540-62772-2. MR 1491362. Retrieved 12 March 2012. Vicente, L .N. (2004). "An Interview with R. Tyrrell Rockafellar" (PDF). SIAG/Opt News and Views. 15 (1). Society for Industrial and Applied Mathematics (SIAM), Special Interest Group in Optimization: 9–14. Retrieved 12 March 2012. Dantzig Prize Committee (1994). "Citation of Roger Wets (for the George Dantzig Prize, 1994)" (PDF). Optima: Mathematical Programming Society Newsletter. 44. Mathematical Programming Society: 5. Retrieved 12 March 2012. Wets, Roger J-B (31 December 2011). "ROGER J-B WETS (Curriculum Vitae)" (PDF). Department of Mathematics, University of California, Davis. Retrieved 12 March 2012. Wets, Roger J-B (31 December 2011b). "ROGER J-B WETS : Biography-Summary" (PDF). Department of Mathematics, University of California, Davis. Retrieved 12 March 2012. == External links == Homepage of Roger J-B Wets at the Mathematics Department of the University of California, Davis. Contains biography, research overviews, lectures and presentations.
Wikipedia:Roger Wolcott Richardson#0	Roger Wolcott Richardson (30 May 1930 – 15 June 1993) was a mathematician noted for his work in representation theory and geometry. He was born in Baton Rouge, Louisiana, and educated at Louisiana State University, Harvard University and University of Michigan, Ann Arbor where he obtained a Ph.D. in 1958 under the supervision of Hans Samelson. After a postdoc appointment at Princeton University, he accepted a faculty position at the University of Washington in Seattle. He emigrated to the United Kingdom in 1970, taking up a chair at Durham University. In 1978 he moved to the Australian National University in Canberra, where he stayed as faculty until his death. Richardson's best known result states that if P is a parabolic subgroup of a reductive group, then P has a dense orbit on its nilradical, i.e., one whose closure is the whole space. This orbit is now universally known as the Richardson orbit. In 1997 the Cambridge University Press published Algebraic Groups and Lie Groups: A Volume of Papers in Honour of the Late R. W. Richardson, which was organized by a committee of 5 mathematicians selected by the Australian Mathematical Society. The volume's preface has several paragraphs about Richardson's research. == Publications == Nijenhuis, Albert; Richardson Jr., Roger W. (1966). "Cohomology and deformations in graded Lie algebras". Bulletin of the American Mathematical Society. 72 (1): 1–29. doi:10.1090/s0002-9904-1966-11401-5. MR 0195995. == See also == Prehomogeneous vector space == External links == Roger Wolcott Richardson at the Mathematics Genealogy Project Mathematical Reviews analysis == References ==
Wikipedia:Rogers–Ramanujan continued fraction#0	The Rogers–Ramanujan continued fraction is a continued fraction discovered by Rogers (1894) and independently by Srinivasa Ramanujan, and closely related to the Rogers–Ramanujan identities. It can be evaluated explicitly for a broad class of values of its argument. == Definition == Given the functions G ( q ) {\displaystyle G(q)} and H ( q ) {\displaystyle H(q)} appearing in the Rogers–Ramanujan identities, and assume q = e 2 π i τ {\displaystyle q=e^{2\pi i\tau }} , G ( q ) = ∑ n = 0 ∞ q n 2 ( 1 − q ) ( 1 − q 2 ) ⋯ ( 1 − q n ) = ∑ n = 0 ∞ q n 2 ( q ; q ) n = 1 ( q ; q 5 ) ∞ ( q 4 ; q 5 ) ∞ = ∏ n = 1 ∞ 1 ( 1 − q 5 n − 1 ) ( 1 − q 5 n − 4 ) = q j 60 2 F 1 ( − 1 60 , 19 60 ; 4 5 ; 1728 j ) = q ( j − 1728 ) 60 2 F 1 ( − 1 60 , 29 60 ; 4 5 ; − 1728 j − 1728 ) = 1 + q + q 2 + q 3 + 2 q 4 + 2 q 5 + 3 q 6 + ⋯ {\displaystyle {\begin{aligned}G(q)&=\sum _{n=0}^{\infty }{\frac {q^{n^{2}}}{(1-q)(1-q^{2})\cdots (1-q^{n})}}=\sum _{n=0}^{\infty }{\frac {q^{n^{2}}}{(q;q)_{n}}}={\frac {1}{(q;q^{5})_{\infty }(q^{4};q^{5})_{\infty }}}\\[6pt]&=\prod _{n=1}^{\infty }{\frac {1}{(1-q^{5n-1})(1-q^{5n-4})}}\\[6pt]&={\sqrt[{60}]{q\,j}}\,\,_{2}F_{1}\left(-{\tfrac {1}{60}},{\tfrac {19}{60}};{\tfrac {4}{5}};{\tfrac {1728}{j}}\right)\\[6pt]&={\sqrt[{60}]{q\left(j-1728\right)}}\,_{2}F_{1}\left(-{\tfrac {1}{60}},{\tfrac {29}{60}};{\tfrac {4}{5}};-{\tfrac {1728}{j-1728}}\right)\\[6pt]&=1+q+q^{2}+q^{3}+2q^{4}+2q^{5}+3q^{6}+\cdots \end{aligned}}} and, H ( q ) = ∑ n = 0 ∞ q n 2 + n ( 1 − q ) ( 1 − q 2 ) ⋯ ( 1 − q n ) = ∑ n = 0 ∞ q n 2 + n ( q ; q ) n = 1 ( q 2 ; q 5 ) ∞ ( q 3 ; q 5 ) ∞ = ∏ n = 1 ∞ 1 ( 1 − q 5 n − 2 ) ( 1 − q 5 n − 3 ) = 1 q 11 j 11 60 2 F 1 ( 11 60 , 31 60 ; 6 5 ; 1728 j ) = 1 q 11 ( j − 1728 ) 11 60 2 F 1 ( 11 60 , 41 60 ; 6 5 ; − 1728 j − 1728 ) = 1 + q 2 + q 3 + q 4 + q 5 + 2 q 6 + 2 q 7 + ⋯ {\displaystyle {\begin{aligned}H(q)&=\sum _{n=0}^{\infty }{\frac {q^{n^{2}+n}}{(1-q)(1-q^{2})\cdots (1-q^{n})}}=\sum _{n=0}^{\infty }{\frac {q^{n^{2}+n}}{(q;q)_{n}}}={\frac {1}{(q^{2};q^{5})_{\infty }(q^{3};q^{5})_{\infty }}}\\[6pt]&=\prod _{n=1}^{\infty }{\frac {1}{(1-q^{5n-2})(1-q^{5n-3})}}\\[6pt]&={\frac {1}{\sqrt[{60}]{q^{11}j^{11}}}}\,_{2}F_{1}\left({\tfrac {11}{60}},{\tfrac {31}{60}};{\tfrac {6}{5}};{\tfrac {1728}{j}}\right)\\[6pt]&={\frac {1}{\sqrt[{60}]{q^{11}\left(j-1728\right)^{11}}}}\,_{2}F_{1}\left({\tfrac {11}{60}},{\tfrac {41}{60}};{\tfrac {6}{5}};-{\tfrac {1728}{j-1728}}\right)\\[6pt]&=1+q^{2}+q^{3}+q^{4}+q^{5}+2q^{6}+2q^{7}+\cdots \end{aligned}}} with the coefficients of the q-expansion being OEIS: A003114 and OEIS: A003106, respectively, where ( a ; q ) ∞ {\displaystyle (a;q)_{\infty }} denotes the infinite q-Pochhammer symbol, j is the j-function, and 2F1 is the hypergeometric function. The Rogers–Ramanujan continued fraction is then R ( q ) = q 11 60 H ( q ) q − 1 60 G ( q ) = q 1 5 ∏ n = 1 ∞ ( 1 − q 5 n − 1 ) ( 1 − q 5 n − 4 ) ( 1 − q 5 n − 2 ) ( 1 − q 5 n − 3 ) = q 1 / 5 ∏ n = 1 ∞ ( 1 − q n ) ( n \| 5 ) = q 1 / 5 1 + q 1 + q 2 1 + q 3 1 + ⋱ {\displaystyle {\begin{aligned}R(q)&={\frac {q^{\frac {11}{60}}H(q)}{q^{-{\frac {1}{60}}}G(q)}}=q^{\frac {1}{5}}\prod _{n=1}^{\infty }{\frac {(1-q^{5n-1})(1-q^{5n-4})}{(1-q^{5n-2})(1-q^{5n-3})}}=q^{1/5}\prod _{n=1}^{\infty }(1-q^{n})^{(n\|5)}\\[8pt]&={\cfrac {q^{1/5}}{1+{\cfrac {q}{1+{\cfrac {q^{2}}{1+{\cfrac {q^{3}}{1+\ddots }}}}}}}}\end{aligned}}} ( n ∣ m ) {\displaystyle (n\mid m)} is the Jacobi symbol. One should be careful with notation since the formulas employing the j-function j {\displaystyle j} will be consistent with the other formulas only if q = e 2 π i τ {\displaystyle q=e^{2\pi i\tau }} (the square of the nome) is used throughout this section since the q-expansion of the j-function (as well as the well-known Dedekind eta function) uses q = e 2 π i τ {\displaystyle q=e^{2\pi i\tau }} . However, Ramanujan, in his examples to Hardy and given below, used the nome q = e π i τ {\displaystyle q=e^{\pi i\tau }} instead. == Special values == If q is the nome or its square, then q − 1 60 G ( q ) {\displaystyle q^{-{\frac {1}{60}}}G(q)} and q 11 60 H ( q ) {\displaystyle q^{\frac {11}{60}}H(q)} , as well as their quotient R ( q ) {\displaystyle R(q)} , are related to modular functions of τ {\displaystyle \tau } . Since they have integral coefficients, the theory of complex multiplication implies that their values for τ {\displaystyle \tau } involving an imaginary quadratic field are algebraic numbers that can be evaluated explicitly. === Examples of R(q) === Given the general form where Ramanujan used the nome q = e π i τ {\displaystyle q=e^{\pi i\tau }} , R ( q ) = q 1 / 5 1 + q 1 + q 2 1 + q 3 1 + ⋱ {\displaystyle R(q)={\cfrac {q^{1/5}}{1+{\cfrac {q}{1+{\cfrac {q^{2}}{1+{\cfrac {q^{3}}{1+\ddots }}}}}}}}} f when τ = i {\displaystyle \tau =i} , R ( e − π ) = e − π 5 1 + e − π 1 + e − 2 π 1 + ⋱ = 1 2 φ ( 5 − φ 3 / 2 ) ( 5 4 + φ 3 / 2 ) = 0.511428 … {\displaystyle R{\big (}e^{-\pi }{\big )}={\cfrac {e^{-{\frac {\pi }{5}}}}{1+{\cfrac {e^{-\pi }}{1+{\cfrac {e^{-2\pi }}{1+\ddots }}}}}}={\tfrac {1}{2}}\varphi \,({\sqrt {5}}-\varphi ^{3/2})({\sqrt[{4}]{5}}+\varphi ^{3/2})=0.511428\dots } when τ = 2 i {\displaystyle \tau =2i} , R ( e − 2 π ) = e − 2 π 5 1 + e − 2 π 1 + e − 4 π 1 + ⋱ = 5 4 φ 1 / 2 − φ = 0.284079 … {\displaystyle R{\big (}e^{-2\pi }{\big )}={\cfrac {e^{-{\frac {2\pi }{5}}}}{1+{\cfrac {e^{-2\pi }}{1+{\cfrac {e^{-4\pi }}{1+\ddots }}}}}}={{\sqrt[{4}]{5}}\,\varphi ^{1/2}-\varphi }=0.284079\dots } when τ = 4 i {\displaystyle \tau =4i} , R ( e − 4 π ) = e − 4 π 5 1 + e − 4 π 1 + e − 8 π 1 + ⋱ = 1 2 φ ( 5 − φ 3 / 2 ) ( − 5 4 + φ 3 / 2 ) = 0.081002 … {\displaystyle R{\big (}e^{-4\pi }{\big )}={\cfrac {e^{-{\frac {4\pi }{5}}}}{1+{\cfrac {e^{-4\pi }}{1+{\cfrac {e^{-8\pi }}{1+\ddots }}}}}}={\tfrac {1}{2}}\varphi \,({\sqrt {5}}-\varphi ^{3/2})(-{\sqrt[{4}]{5}}+\varphi ^{3/2})=0.081002\dots } when τ = 2 5 i {\displaystyle \tau =2{\sqrt {5}}i} , R ( e − 2 5 π ) = e − 2 π 5 1 + e − 2 π 5 1 + e − 4 π 5 1 + ⋱ = 5 1 + ( 5 3 / 4 ( φ − 1 ) 5 / 2 − 1 ) 1 / 5 − φ = 0.0602094 … {\displaystyle R{\big (}e^{-2{\sqrt {5}}\pi }{\big )}={\cfrac {e^{-{\frac {2\pi }{\sqrt {5}}}}}{1+{\cfrac {e^{-2\pi {\sqrt {5}}}}{1+{\cfrac {e^{-4\pi {\sqrt {5}}}}{1+\ddots }}}}}}={\frac {\sqrt {5}}{1+{\big (}5^{3/4}(\varphi -1)^{5/2}-1{\big )}^{1/5}}}-\varphi =0.0602094\dots } when τ = 5 i {\displaystyle \tau =5i} , R ( e − 5 π ) = e − π 1 + e − 5 π 1 + e − 10 π 1 + ⋱ = 1 + φ 2 φ + ( 1 2 ( 4 − φ − 3 φ − 1 ) ( 3 φ 3 / 2 − 5 4 ) ) 1 / 5 − φ = 0.0432139 … {\displaystyle R{\big (}e^{-5\pi }{\big )}={\cfrac {e^{-\pi }}{1+{\cfrac {e^{-5\pi }}{1+{\cfrac {e^{-10\pi }}{1+\ddots }}}}}}={\frac {1+\varphi ^{2}}{\varphi +{\big (}{\frac {1}{2}}(4-\varphi -3{\sqrt {\varphi -1}})(3\varphi ^{3/2}-{\sqrt[{4}]{5}}){\big )}^{1/5}}}-\varphi =0.0432139\dots } when τ = 10 i {\displaystyle \tau =10i} , R ( e − 10 π ) = e − 2 π 1 + e − 10 π 1 + e − 20 π 1 + ⋱ = 1 + φ 2 φ + ( 3 1 + φ 2 − 4 − φ ) 1 / 5 − φ = 0.00186744 … {\displaystyle R{\big (}e^{-10\pi }{\big )}={\cfrac {e^{-2\pi }}{1+{\cfrac {e^{-10\pi }}{1+{\cfrac {e^{-20\pi }}{1+\ddots }}}}}}={\frac {1+\varphi ^{2}}{\varphi +{\big (}3{\sqrt {1+\varphi ^{2}}}-4-\varphi {\big )}^{1/5}}}-\varphi =0.00186744\dots } when τ = 20 i {\displaystyle \tau =20i} , R ( e − 20 π ) = e − 4 π 1 + e − 20 π 1 + e − 40 π 1 + ⋱ = 1 + φ 2 φ + ( 1 2 ( 4 − φ − 3 φ − 1 ) ( 3 φ 3 / 2 + 5 4 ) ) 1 / 5 − φ = 0.00000348734 … {\displaystyle R{\big (}e^{-20\pi }{\big )}={\cfrac {e^{-4\pi }}{1+{\cfrac {e^{-20\pi }}{1+{\cfrac {e^{-40\pi }}{1+\ddots }}}}}}={\frac {1+\varphi ^{2}}{\varphi +{\big (}{\frac {1}{2}}(4-\varphi -3{\sqrt {\varphi -1}})(3\varphi ^{3/2}+{\sqrt[{4}]{5}}){\big )}^{1/5}}}-\varphi =0.00000348734\dots } and φ = 1 + 5 2 {\displaystyle \varphi ={\tfrac {1+{\sqrt {5}}}{2}}} is the golden ratio. Note that R ( e − 2 π ) {\displaystyle R{\big (}e^{-2\pi }{\big )}} is a positive root of the quartic equation, x 4 + 2 x 3 − 6 x 2 − 2 x + 1 = 0 {\displaystyle x^{4}+2x^{3}-6x^{2}-2x+1=0} while R ( e − π ) {\displaystyle R{\big (}e^{-\pi }{\big )}} and R ( e − 4 π ) {\displaystyle R{\big (}e^{-4\pi }{\big )}} are two positive roots of a single octic, y 4 + 2 φ 4 y 3 + 6 φ 2 y 2 − 2 φ 4 y + 1 = 0 {\displaystyle y^{4}+2\varphi ^{4}y^{3}+6\varphi ^{2}y^{2}-2\varphi ^{4}y+1=0} (since φ {\displaystyle \varphi } has a square root) which explains the similarity of the two closed-forms. More generally, for positive integer m, then R ( e − 2 π / m ) {\displaystyle R(e^{-2\pi /m})} and R ( e − 2 π m ) {\displaystyle R(e^{-2\pi \,m})} are two roots of the same equation as well as, [ R ( e − 2 π / m ) + φ ] [ R ( e − 2 π m ) + φ ] = 5 φ {\displaystyle {\bigl [}R(e^{-2\pi /m})+\varphi {\bigr ]}{\bigl [}R(e^{-2\pi \,m})+\varphi {\bigr ]}={\sqrt {5}}\,\varphi } The algebraic degree k of R ( e − π n ) {\displaystyle R(e^{-\pi \,n})} for n = 1 , 2 , 3 , 4 , … {\displaystyle n=1,2,3,4,\dots } is k = 8 , 4 , 32 , 8 , … {\displaystyle k=8,4,32,8,\dots } (OEIS: A082682). Incidentally, these continued fractions can be used to solve some quintic equations as shown in a later section. === Examples of G(q) and H(q) === Interestingly, there are explicit formulas for G ( q ) {\displaystyle G(q)} and H ( q ) {\displaystyle H(q)} in terms of the j-function j ( τ ) {\displaystyle j(\tau )} and the Rogers-Ramanujan continued fraction R ( q ) {\displaystyle R(q)} . However, since j ( τ ) {\displaystyle j(\tau )} uses the nome's square q = e 2 π i τ {\displaystyle q=e^{2\pi \,i\tau }} , then one should be careful with notation such that j ( τ ) , G ( q ) , H ( q ) {\displaystyle j(\tau ),\,G(q),\,H(q)} and r = R ( q ) {\displaystyle r=R(q)} use the same q {\displaystyle q} . G ( q ) = ∏ n = 1 ∞ 1 ( 1 − q 5 n − 1 ) ( 1 − q 5 n − 4 ) = q 1 / 60 j ( τ ) 1 / 60 ( r 20 − 228 r 15 + 494 r 10 + 228 r 5 + 1 ) 1 / 20 {\displaystyle {\begin{aligned}G(q)&=\prod _{n=1}^{\infty }{\frac {1}{(1-q^{5n-1})(1-q^{5n-4})}}\\[6pt]&=q^{1/60}{\frac {j(\tau )^{1/60}}{(r^{20}-228r^{15}+494r^{10}+228r^{5}+1)^{1/20}}}\end{aligned}}} H ( q ) = ∏ n = 1 ∞ 1 ( 1 − q 5 n − 2 ) ( 1 − q 5 n − 3 ) = − 1 q 11 / 60 ( r 20 − 228 r 15 + 494 r 10 + 228 r 5 + 1 ) 11 / 20 j ( τ ) 11 / 60 ( r 10 + 11 r 5 − 1 ) {\displaystyle {\begin{aligned}H(q)&=\prod _{n=1}^{\infty }{\frac {1}{(1-q^{5n-2})(1-q^{5n-3})}}\\[6pt]&={\frac {-1}{q^{11/60}}}{\frac {(r^{20}-228r^{15}+494r^{10}+228r^{5}+1)^{11/20}}{j(\tau )^{11/60}\,(r^{10}+11r^{5}-1)}}\end{aligned}}} Of course, the secondary formulas imply that q − 1 / 60 G ( q ) {\displaystyle q^{-1/60}G(q)} and q 11 / 60 H ( q ) {\displaystyle q^{11/60}H(q)} are algebraic numbers (though normally of high degree) for τ {\displaystyle \tau } involving an imaginary quadratic field. For example, the formulas above simplify to, G ( e − 2 π ) = ( e − 2 π ) 1 / 60 1 ( 5 φ ) 1 / 4 1 R ( e − 2 π ) = 1.00187093 … H ( e − 2 π ) = 1 ( e − 2 π ) 11 / 60 1 ( 5 φ ) 1 / 4 R ( e − 2 π ) = 1.00000349 … {\displaystyle {\begin{aligned}G(e^{-2\pi })&=(e^{-2\pi })^{1/60}{\frac {1}{(5\,\varphi )^{1/4}}}{\frac {1}{\sqrt {R(e^{-2\pi })}}}\\[6pt]&=1.00187093\dots \\[6pt]H(e^{-2\pi })&={\frac {1}{(e^{-2\pi })^{11/60}}}{\frac {1}{(5\,\varphi )^{1/4}}}{\sqrt {R(e^{-2\pi })}}\\[6pt]&=1.00000349\ldots \end{aligned}}} and, G ( e − 4 π ) = ( e − 4 π ) 1 / 60 1 ( 5 φ 3 ) 1 / 4 ( φ + 5 4 ) 1 / 4 1 R ( e − 4 π ) = 1.000003487354 … H ( e − 4 π ) = 1 ( e − 4 π ) 11 / 60 1 ( 5 φ 3 ) 1 / 4 ( φ + 5 4 ) 1 / 4 R ( e − 4 π ) = 1.000000000012 … {\displaystyle {\begin{aligned}G(e^{-4\pi })&=(e^{-4\pi })^{1/60}{\frac {1}{(5\,\varphi ^{3})^{1/4}\,(\varphi +{\sqrt[{4}]{5}})^{1/4}}}{\frac {1}{\sqrt {R(e^{-4\pi })}}}\\[6pt]&=1.000003487354\dots \\[6pt]H(e^{-4\pi })&={\frac {1}{(e^{-4\pi })^{11/60}}}{\frac {1}{(5\,\varphi ^{3})^{1/4}\,(\varphi +{\sqrt[{4}]{5}})^{1/4}}}{\sqrt {R(e^{-4\pi })}}\\[6pt]&=1.000000000012\dots \end{aligned}}} and so on, with φ {\displaystyle \varphi } as the golden ratio. == Derivation of special values == === Tangential sums === In the following we express the essential theorems of the Rogers-Ramanujan continued fractions R and S by using the tangential sums and tangential differences: a ⊕ b = tan ⁡ [ arctan ⁡ ( a ) + arctan ⁡ ( b ) ] = a + b 1 − a b {\displaystyle a\oplus b=\tan {\bigl [}\arctan(a)+\arctan(b){\bigr ]}={\frac {a+b}{1-ab}}} c ⊖ d = tan ⁡ [ arctan ⁡ ( c ) − arctan ⁡ ( d ) ] = c − d 1 + c d {\displaystyle c\ominus d=\tan {\bigl [}\arctan(c)-\arctan(d){\bigr ]}={\frac {c-d}{1+cd}}} The elliptic nome and the complementary nome have this relationship to each other: ln ⁡ ( q ) ln ⁡ ( q 1 ) = π 2 {\displaystyle \ln(q)\ln(q_{1})=\pi ^{2}} The complementary nome of a modulus k is equal to the nome of the Pythagorean complementary modulus: q 1 ( k ) = q ( k ′ ) = q ( 1 − k 2 ) {\displaystyle q_{1}(k)=q(k')=q({\sqrt {1-k^{2}}})} These are the reflection theorems for the continued fractions R and S: The letter Φ {\displaystyle \Phi } represents the Golden number exactly: Φ = 1 2 ( 5 + 1 ) = cot ⁡ [ 1 2 arctan ⁡ ( 2 ) ] = 2 cos ⁡ ( 1 5 π ) {\displaystyle \Phi ={\tfrac {1}{2}}({\sqrt {5}}+1)=\cot[{\tfrac {1}{2}}\arctan(2)]=2\cos({\tfrac {1}{5}}{\pi })} Φ − 1 = 1 2 ( 5 − 1 ) = tan ⁡ [ 1 2 arctan ⁡ ( 2 ) ] = 2 sin ⁡ ( 1 10 π ) {\displaystyle \Phi ^{-1}={\tfrac {1}{2}}({\sqrt {5}}-1)=\tan[{\tfrac {1}{2}}\arctan(2)]=2\sin({\tfrac {1}{10}}{\pi })} The theorems for the squared nome are constructed as follows: Following relations between the continued fractions and the Jacobi theta functions are given: === Derivation of Lemniscatic values === Into the now shown theorems certain values are inserted: S [ exp ⁡ ( − π ) ] ⊕ S [ exp ⁡ ( − π ) ] = Φ {\displaystyle S{\bigl [}\exp(-\pi ){\bigr ]}\oplus S{\bigl [}\exp(-\pi ){\bigr ]}=\Phi } Therefore following identity is valid: In an analogue pattern we get this result: R [ exp ⁡ ( − 2 π ) ] ⊕ R [ exp ⁡ ( − 2 π ) ] = Φ − 1 {\displaystyle R{\bigl [}\exp(-2\pi ){\bigr ]}\oplus R{\bigl [}\exp(-2\pi ){\bigr ]}=\Phi ^{-1}} Therefore following identity is valid: Furthermore we get the same relation by using the above mentioned theorem about the Jacobi theta functions: S [ exp ⁡ ( − π ) ] ⊕ R [ exp ⁡ ( − 2 π ) ] = S ( q ) ⊕ R ( q 2 ) [ q = exp ⁡ ( − π ) ] = {\displaystyle S{\bigl [}\exp(-\pi ){\bigr ]}\oplus R{\bigl [}\exp(-2\pi ){\bigr ]}=S(q)\oplus R(q^{2}){\bigl [}q=\exp(-\pi ){\bigr ]}=} = ϑ 00 ( q 1 / 5 ) 2 − ϑ 00 ( q ) 2 5 ϑ 00 ( q 5 ) 2 − ϑ 00 ( q ) 2 [ q = exp ⁡ ( − π ) ] = 1 {\displaystyle ={\frac {\vartheta _{00}(q^{1/5})^{2}-\vartheta _{00}(q)^{2}}{5\,\vartheta _{00}(q^{5})^{2}-\vartheta _{00}(q)^{2}}}{\bigl [}q=\exp(-\pi ){\bigr ]}=1} This result appears because of the Poisson summation formula and this equation can be solved in this way: R [ exp ⁡ ( − 2 π ) ] = 1 ⊖ S [ exp ⁡ ( − π ) ] = 1 ⊖ tan ⁡ [ 1 4 π − 1 4 arctan ⁡ ( 2 ) ] = tan ⁡ [ 1 4 arctan ⁡ ( 2 ) ] {\displaystyle R{\bigl [}\exp(-2\pi ){\bigr ]}=1\ominus S{\bigl [}\exp(-\pi ){\bigr ]}=1\ominus \tan {\bigl [}{\tfrac {1}{4}}\pi -{\tfrac {1}{4}}\arctan(2){\bigr ]}=\tan {\bigl [}{\tfrac {1}{4}}\arctan(2){\bigr ]}} By taking the other mentioned theorem about the Jacobi theta functions a next value can be determined: R [ exp ⁡ ( − π ) ] ⊖ R [ exp ⁡ ( − 2 π ) ] = R ( q ) ⊖ R ( q 2 ) [ q = exp ⁡ ( − π ) ] = {\displaystyle R{\bigl [}\exp(-\pi ){\bigr ]}\ominus R{\bigl [}\exp(-2\pi ){\bigr ]}=R(q)\ominus R(q^{2}){\bigl [}q=\exp(-\pi ){\bigr ]}=} = ϑ 01 ( q ) 2 − ϑ 01 ( q 1 / 5 ) 2 5 ϑ 01 ( q 5 ) 2 − ϑ 01 ( q ) 2 [ q = exp ⁡ ( − π ) ] = 5 4 − 1 5 4 + 1 = 5 4 ⊖ 1 = tan ⁡ [ arctan ⁡ ( 5 4 ) − 1 4 π ] {\displaystyle ={\frac {\vartheta _{01}(q)^{2}-\vartheta _{01}(q^{1/5})^{2}}{5\,\vartheta _{01}(q^{5})^{2}-\vartheta _{01}(q)^{2}}}{\bigl [}q=\exp(-\pi ){\bigr ]}={\frac {{\sqrt[{4}]{5}}-1}{{\sqrt[{4}]{5}}+1}}={\sqrt[{4}]{5}}\ominus 1=\tan {\bigl [}\arctan({\sqrt[{4}]{5}}\,)-{\tfrac {1}{4}}\pi {\bigr ]}} That equation chain leads to this tangential sum: R [ exp ⁡ ( − π ) ] = R [ exp ⁡ ( − 2 π ) ] ⊕ tan ⁡ [ arctan ⁡ ( 5 4 ) − 1 4 π ] {\displaystyle R{\bigl [}\exp(-\pi ){\bigr ]}=R{\bigl [}\exp(-2\pi ){\bigr ]}\oplus \tan {\bigl [}\arctan({\sqrt[{4}]{5}}\,)-{\tfrac {1}{4}}\pi {\bigr ]}} And therefore following result appears: In the next step we use the reflection theorem for the continued fraction R again: R [ exp ⁡ ( − π ) ] ⊕ R [ exp ⁡ ( − 4 π ) ] = Φ − 1 {\displaystyle R{\bigl [}\exp(-\pi ){\bigr ]}\oplus R{\bigl [}\exp(-4\pi ){\bigr ]}=\Phi ^{-1}} R [ exp ⁡ ( − 4 π ) ] = tan ⁡ [ 1 2 arctan ⁡ ( 2 ) ] ⊖ R [ exp ⁡ ( − π ) ] {\displaystyle R{\bigl [}\exp(-4\pi ){\bigr ]}=\tan {\bigl [}{\tfrac {1}{2}}\arctan(2){\bigr ]}\ominus R{\bigl [}\exp(-\pi ){\bigr ]}} And a further result appears: === Derivation of Non-Lemniscatic values === The reflection theorem is now used for following values: R [ exp ⁡ ( − 2 π ) ] ⊕ R [ exp ⁡ ( − 2 2 π ) ] = Φ − 1 {\displaystyle R{\bigl [}\exp(-{\sqrt {2}}\,\pi ){\bigr ]}\oplus R{\bigl [}\exp(-2{\sqrt {2}}\,\pi ){\bigr ]}=\Phi ^{-1}} The Jacobi theta theorem leads to a further relation: R [ exp ⁡ ( − 2 π ) ] ⊖ R [ exp ⁡ ( − 2 2 π ) ] = R ( q ) ⊖ R ( q 2 ) [ q = exp ⁡ ( − 2 π ) ] = {\displaystyle R{\bigl [}\exp(-{\sqrt {2}}\,\pi ){\bigr ]}\ominus R{\bigl [}\exp(-2{\sqrt {2}}\,\pi ){\bigr ]}=R(q)\ominus R(q^{2}){\bigl [}q=\exp(-{\sqrt {2}}\,\pi ){\bigr ]}=} = ϑ 01 ( q ) 2 − ϑ 01 ( q 1 / 5 ) 2 5 ϑ 01 ( q 5 ) 2 − ϑ 01 ( q ) 2 [ q = exp ⁡ ( − 2 π ) ] = tan ⁡ [ 2 arctan ⁡ ( 1 3 5 − 1 3 6 30 + 4 5 3 + 1 3 6 30 − 4 5 3 ) − 1 4 π ] {\displaystyle ={\frac {\vartheta _{01}(q)^{2}-\vartheta _{01}(q^{1/5})^{2}}{5\,\vartheta _{01}(q^{5})^{2}-\vartheta _{01}(q)^{2}}}{\bigl [}q=\exp(-{\sqrt {2}}\,\pi ){\bigr ]}=\tan {\bigl [}2\arctan({\tfrac {1}{3}}{\sqrt {5}}-{\tfrac {1}{3}}{\sqrt[{3}]{6{\sqrt {30}}+4{\sqrt {5}}}}+{\tfrac {1}{3}}{\sqrt[{3}]{6{\sqrt {30}}-4{\sqrt {5}}}}\,)-{\tfrac {1}{4}}\pi {\bigr ]}} By tangential adding the now mentioned two theorems we get this result: R [ exp ⁡ ( − 2 π ) ] ⊕ R [ exp ⁡ ( − 2 π ) ] = Φ − 1 ⊕ tan ⁡ [ 2 arctan ⁡ ( 1 3 5 − 1 3 6 30 + 4 5 3 + 1 3 6 30 − 4 5 3 ) − 1 4 π ] {\displaystyle R{\bigl [}\exp(-{\sqrt {2}}\,\pi ){\bigr ]}\oplus R{\bigl [}\exp(-{\sqrt {2}}\,\pi ){\bigr ]}=\Phi ^{-1}\oplus \tan {\bigl [}2\arctan({\tfrac {1}{3}}{\sqrt {5}}-{\tfrac {1}{3}}{\sqrt[{3}]{6{\sqrt {30}}+4{\sqrt {5}}}}+{\tfrac {1}{3}}{\sqrt[{3}]{6{\sqrt {30}}-4{\sqrt {5}}}}\,)-{\tfrac {1}{4}}\pi {\bigr ]}} By tangential substraction that result appears: R [ exp ⁡ ( − 2 2 π ) ] ⊕ R [ exp ⁡ ( − 2 2 π ) ] = Φ − 1 ⊖ tan ⁡ [ 2 arctan ⁡ ( 1 3 5 − 1 3 6 30 + 4 5 3 + 1 3 6 30 − 4 5 3 ) − 1 4 π ] {\displaystyle R{\bigl [}\exp(-2{\sqrt {2}}\,\pi ){\bigr ]}\oplus R{\bigl [}\exp(-2{\sqrt {2}}\,\pi ){\bigr ]}=\Phi ^{-1}\ominus \tan {\bigl [}2\arctan({\tfrac {1}{3}}{\sqrt {5}}-{\tfrac {1}{3}}{\sqrt[{3}]{6{\sqrt {30}}+4{\sqrt {5}}}}+{\tfrac {1}{3}}{\sqrt[{3}]{6{\sqrt {30}}-4{\sqrt {5}}}}\,)-{\tfrac {1}{4}}\pi {\bigr ]}} In an alternative solution way we use the theorem for the squared nome: R [ exp ⁡ ( − 2 π ) ] 2 R [ exp ⁡ ( − 2 2 π ) ] − 1 ⊕ R [ exp ⁡ ( − 2 π ) ] R [ exp ⁡ ( − 2 2 π ) ] 2 = 1 {\displaystyle R{\bigl [}\exp(-{\sqrt {2}}\,\pi ){\bigr ]}^{2}R{\bigl [}\exp(-2{\sqrt {2}}\,\pi ){\bigr ]}^{-1}\oplus R{\bigl [}\exp(-{\sqrt {2}}\,\pi ){\bigr ]}R{\bigl [}\exp(-2{\sqrt {2}}\,\pi ){\bigr ]}^{2}=1} { R [ exp ⁡ ( − 2 π ) ] 2 R [ exp ⁡ ( − 2 2 π ) ] − 1 + 1 } { R [ exp ⁡ ( − 2 π ) ] R [ exp ⁡ ( − 2 2 π ) ] 2 + 1 } = 2 {\displaystyle {\bigl \{}R{\bigl [}\exp(-{\sqrt {2}}\,\pi ){\bigr ]}^{2}R{\bigl [}\exp(-2{\sqrt {2}}\,\pi ){\bigr ]}^{-1}+1{\bigr \}}{\bigl \{}R{\bigl [}\exp(-{\sqrt {2}}\,\pi ){\bigr ]}R{\bigl [}\exp(-2{\sqrt {2}}\,\pi ){\bigr ]}^{2}+1{\bigr \}}=2} Now the reflection theorem is taken again: R [ exp ⁡ ( − 2 2 π ) ] = Φ − 1 ⊖ R [ exp ⁡ ( − 2 π ) ] {\displaystyle R{\bigl [}\exp(-2{\sqrt {2}}\,\pi ){\bigr ]}=\Phi ^{-1}\ominus R{\bigl [}\exp(-{\sqrt {2}}\,\pi ){\bigr ]}} R [ exp ⁡ ( − 2 2 π ) ] = 1 − Φ R [ exp ⁡ ( − 2 π ) ] Φ + R [ exp ⁡ ( − 2 π ) ] {\displaystyle R{\bigl [}\exp(-2{\sqrt {2}}\,\pi ){\bigr ]}={\frac {1-\Phi R{\bigl [}\exp(-{\sqrt {2}}\,\pi ){\bigr ]}}{\Phi +R{\bigl [}\exp(-{\sqrt {2}}\,\pi ){\bigr ]}}}} The insertion of the last mentioned expression into the squared nome theorem gives that equation: { R [ exp ⁡ ( − 2 π ) ] 2 Φ + R [ exp ⁡ ( − 2 π ) ] 1 − Φ R [ exp ⁡ ( − 2 π ) ] + 1 } ⟨ R [ exp ⁡ ( − 2 π ) ] { 1 − Φ R [ exp ⁡ ( − 2 π ) ] } 2 { Φ + R [ exp ⁡ ( − 2 π ) ] } 2 + 1 ⟩ = 2 {\displaystyle {\biggl \{}R{\bigl [}\exp(-{\sqrt {2}}\,\pi ){\bigr ]}^{2}{\frac {\Phi +R{\bigl [}\exp(-{\sqrt {2}}\,\pi ){\bigr ]}}{1-\Phi R{\bigl [}\exp(-{\sqrt {2}}\,\pi ){\bigr ]}}}+1{\biggr \}}{\biggl \langle }R{\bigl [}\exp(-{\sqrt {2}}\,\pi ){\bigr ]}{\frac {{\bigl \{}1-\Phi R{\bigl [}\exp(-{\sqrt {2}}\,\pi ){\bigr ]}{\bigr \}}^{2}}{{\bigl \{}\Phi +R{\bigl [}\exp(-{\sqrt {2}}\,\pi ){\bigr ]}{\bigr \}}^{2}}}+1{\biggr \rangle }=2} Erasing the denominators gives an equation of sixth degree: R [ exp ⁡ ( − 2 π ) ] 6 + 2 Φ − 2 R [ exp ⁡ ( − 2 π ) ] 5 − 5 Φ − 1 R [ exp ⁡ ( − 2 π ) ] 4 + {\displaystyle R{\bigl [}\exp(-{\sqrt {2}}\,\pi ){\bigr ]}^{6}+2\,\Phi ^{-2}R{\bigl [}\exp(-{\sqrt {2}}\,\pi ){\bigr ]}^{5}-{\sqrt {5}}\,\Phi ^{-1}R{\bigl [}\exp(-{\sqrt {2}}\,\pi ){\bigr ]}^{4}+} + 2 5 Φ R [ exp ⁡ ( − 2 π ) ] 3 + 5 Φ − 1 R [ exp ⁡ ( − 2 π ) ] 2 + 2 Φ − 2 R [ exp ⁡ ( − 2 π ) ] − 1 = 0 {\displaystyle +2\,{\sqrt {5}}\,\Phi R{\bigl [}\exp(-{\sqrt {2}}\,\pi ){\bigr ]}^{3}+{\sqrt {5}}\,\Phi ^{-1}R{\bigl [}\exp(-{\sqrt {2}}\,\pi ){\bigr ]}^{2}+2\,\Phi ^{-2}R{\bigl [}\exp(-{\sqrt {2}}\,\pi ){\bigr ]}-1=0} The solution of this equation is the already mentioned solution: R [ exp ⁡ ( − 2 π ) ] = tan ⁡ [ arctan ⁡ ( 1 3 5 − 1 3 6 30 + 4 5 3 + 1 3 6 30 − 4 5 3 ) − 1 4 arccot ⁡ ( 2 ) ] {\displaystyle R{\bigl [}\exp(-{\sqrt {2}}\,\pi ){\bigr ]}=\tan {\bigl [}\arctan({\tfrac {1}{3}}{\sqrt {5}}-{\tfrac {1}{3}}{\sqrt[{3}]{6{\sqrt {30}}+4{\sqrt {5}}}}+{\tfrac {1}{3}}{\sqrt[{3}]{6{\sqrt {30}}-4{\sqrt {5}}}}\,)-{\tfrac {1}{4}}\operatorname {arccot}(2){\bigr ]}} == Relation to modular forms == R ( q ) {\displaystyle R(q)} can be related to the Dedekind eta function, a modular form of weight 1/2, as, 1 R ( q ) − R ( q ) = η ( τ 5 ) η ( 5 τ ) + 1 {\displaystyle {\frac {1}{R(q)}}-R(q)={\frac {\eta ({\frac {\tau }{5}})}{\eta (5\tau )}}+1} 1 R 5 ( q ) − R 5 ( q ) = [ η ( τ ) η ( 5 τ ) ] 6 + 11 {\displaystyle {\frac {1}{R^{5}(q)}}-R^{5}(q)=\left[{\frac {\eta (\tau )}{\eta (5\tau )}}\right]^{6}+11} The Rogers-Ramanujan continued fraction can also be expressed in terms of the Jacobi theta functions. Recall the notation, ϑ 10 ( 0 ; τ ) = θ 2 ( q ) = ∑ n = − ∞ ∞ q ( n + 1 / 2 ) 2 ϑ 00 ( 0 ; τ ) = θ 3 ( q ) = ∑ n = − ∞ ∞ q n 2 ϑ 01 ( 0 ; τ ) = θ 4 ( q ) = ∑ n = − ∞ ∞ ( − 1 ) n q n 2 {\displaystyle {\begin{aligned}\vartheta _{10}(0;\tau )&=\theta _{2}(q)=\sum _{n=-\infty }^{\infty }q^{(n+1/2)^{2}}\\\vartheta _{00}(0;\tau )&=\theta _{3}(q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}\\\vartheta _{01}(0;\tau )&=\theta _{4}(q)=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{n^{2}}\end{aligned}}} The notation θ n {\displaystyle \theta _{n}} is slightly easier to remember since θ 2 4 + θ 4 4 = θ 3 4 {\displaystyle \theta _{2}^{4}+\theta _{4}^{4}=\theta _{3}^{4}} , with even subscripts on the LHS. Thus, R ( x ) = tan ⁡ { 1 2 arccot ⁡ [ 1 2 + θ 4 ( x 1 / 5 ) [ 5 θ 4 ( x 5 ) 2 − θ 4 ( x ) 2 ] 2 θ 4 ( x 5 ) [ θ 4 ( x ) 2 − θ 4 ( x 1 / 5 ) 2 ] ] } {\displaystyle R(x)=\tan {\biggl \{}{\frac {1}{2}}\operatorname {arccot} {\biggl [}{\frac {1}{2}}+{\frac {\theta _{4}(x^{1/5})[5\,\theta _{4}(x^{5})^{2}-\theta _{4}(x)^{2}]}{2\,\theta _{4}(x^{5})[\theta _{4}(x)^{2}-\theta _{4}(x^{1/5})^{2}]}}{\biggr ]}{\biggr \}}} R ( x ) = tan ⁡ { 1 2 arccot ⁡ [ 1 2 + ( θ 2 ( x 1 / 10 ) θ 3 ( x 1 / 10 ) θ 4 ( x 1 / 10 ) 2 3 θ 2 ( x 5 / 2 ) θ 3 ( x 5 / 2 ) θ 4 ( x 5 / 2 ) ) 1 / 3 ] } {\displaystyle R(x)=\tan {\biggl \{}{\frac {1}{2}}\operatorname {arccot} {\biggl [}{\frac {1}{2}}+{\bigg (}{\frac {\theta _{2}(x^{1/10})\,\theta _{3}(x^{1/10})\,\theta _{4}(x^{1/10})}{2^{3}\,\theta _{2}(x^{5/2})\,\theta _{3}(x^{5/2})\,\theta _{4}(x^{5/2})}}{\bigg )}^{1/3}{\biggr ]}{\biggr \}}} R ( x ) = tan ⁡ { 1 2 arctan ⁡ [ 1 2 − θ 4 ( x ) 2 2 θ 4 ( x 5 ) 2 ] } 1 / 5 × tan ⁡ { 1 2 arccot ⁡ [ 1 2 − θ 4 ( x ) 2 2 θ 4 ( x 5 ) 2 ] } 2 / 5 {\displaystyle R(x)=\tan {\biggl \{}{\frac {1}{2}}\arctan {\biggl [}{\frac {1}{2}}-{\frac {\theta _{4}(x)^{2}}{2\,\theta _{4}(x^{5})^{2}}}{\biggr ]}{\biggr \}}^{1/5}\times \tan {\biggl \{}{\frac {1}{2}}\operatorname {arccot} {\biggl [}{\frac {1}{2}}-{\frac {\theta _{4}(x)^{2}}{2\,\theta _{4}(x^{5})^{2}}}{\biggr ]}{\biggr \}}^{2/5}} R ( x ) = tan ⁡ { 1 2 arctan ⁡ [ 1 2 − θ 4 ( x 1 / 2 ) 2 2 θ 4 ( x 5 / 2 ) 2 ] } 2 / 5 × cot ⁡ { 1 2 arccot ⁡ [ 1 2 − θ 4 ( x 1 / 2 ) 2 2 θ 4 ( x 5 / 2 ) 2 ] } 1 / 5 {\displaystyle R(x)=\tan {\biggl \{}{\frac {1}{2}}\arctan {\biggl [}{\frac {1}{2}}-{\frac {\theta _{4}(x^{1/2})^{2}}{2\,\theta _{4}(x^{5/2})^{2}}}{\biggr ]}{\biggr \}}^{2/5}\times \cot {\biggl \{}{\frac {1}{2}}\operatorname {arccot} {\biggl [}{\frac {1}{2}}-{\frac {\theta _{4}(x^{1/2})^{2}}{2\,\theta _{4}(x^{5/2})^{2}}}{\biggr ]}{\biggr \}}^{1/5}} Note, however, that theta functions normally use the nome q = eiπτ, while the Dedekind eta function uses the square of the nome q = e2iπτ, thus the variable x has been employed instead to maintain consistency between all functions. For example, let τ = − 1 {\displaystyle \tau ={\sqrt {-1}}} so x = e − π {\displaystyle x=e^{-\pi }} . Plugging this into the theta functions, one gets the same value for all three R(x) formulas which is the correct evaluation of the continued fraction given previously, R ( e − π ) = 1 2 φ ( 5 − φ 3 / 2 ) ( 5 4 + φ 3 / 2 ) = 0.511428 … {\displaystyle R{\big (}e^{-\pi }{\big )}={\frac {1}{2}}\varphi \,({\sqrt {5}}-\varphi ^{3/2})({\sqrt[{4}]{5}}+\varphi ^{3/2})=0.511428\dots } One can also define the elliptic nome, q ( k ) = exp ⁡ [ − π K ( 1 − k 2 ) / K ( k ) ] {\displaystyle q(k)=\exp {\big [}-\pi K({\sqrt {1-k^{2}}})/K(k){\big ]}} The small letter k describes the elliptic modulus and the big letter K describes the complete elliptic integral of the first kind. The continued fraction can then be also expressed by the Jacobi elliptic functions as follows: R ( q ( k ) ) = tan ⁡ { 1 2 arctan ⁡ y } 1 / 5 tan ⁡ { 1 2 arccot ⁡ y } 2 / 5 = { y 2 + 1 − 1 y } 1 / 5 { y [ 1 y 2 + 1 − 1 ] } 2 / 5 {\displaystyle R{\big (}q(k){\big )}=\tan {\biggl \{}{\frac {1}{2}}\arctan y{\biggr \}}^{1/5}\tan {\biggl \{}{\frac {1}{2}}\operatorname {arccot} y{\biggr \}}^{2/5}=\left\{{\frac {{\sqrt {y^{2}+1}}-1}{y}}\right\}^{1/5}\left\{y\left[{\sqrt {{\frac {1}{y^{2}}}+1}}-1\right]\right\}^{2/5}} with y = 2 k 2 sn [ 2 5 K ( k ) ; k ] 2 sn [ 4 5 K ( k ) ; k ] 2 5 − k 2 sn [ 2 5 K ( k ) ; k ] 2 sn [ 4 5 K ( k ) ; k ] 2 . {\displaystyle y={\frac {2k^{2}\,{\text{sn}}[{\tfrac {2}{5}}K(k);k]^{2}\,{\text{sn}}[{\tfrac {4}{5}}K(k);k]^{2}}{5-k^{2}\,{\text{sn}}[{\tfrac {2}{5}}K(k);k]^{2}\,{\text{sn}}[{\tfrac {4}{5}}K(k);k]^{2}}}.} == Relation to j-function == One formula involving the j-function and the Dedekind eta function is this: j ( τ ) = ( x 2 + 10 x + 5 ) 3 x {\displaystyle j(\tau )={\frac {(x^{2}+10x+5)^{3}}{x}}} where x = [ 5 η ( 5 τ ) η ( τ ) ] 6 . {\displaystyle x=\left[{\frac {{\sqrt {5}}\,\eta (5\tau )}{\eta (\tau )}}\right]^{6}.\,} Since also, 1 R 5 ( q ) − R 5 ( q ) = [ η ( τ ) η ( 5 τ ) ] 6 + 11 {\displaystyle {\frac {1}{R^{5}(q)}}-R^{5}(q)=\left[{\frac {\eta (\tau )}{\eta (5\tau )}}\right]^{6}+11} Eliminating the eta quotient x {\displaystyle x} between the two equations, one can then express j(τ) in terms of r = R ( q ) {\displaystyle r=R(q)} as, j ( τ ) = − ( r 20 − 228 r 15 + 494 r 10 + 228 r 5 + 1 ) 3 r 5 ( r 10 + 11 r 5 − 1 ) 5 j ( τ ) − 1728 = − ( r 30 + 522 r 25 − 10005 r 20 − 10005 r 10 − 522 r 5 + 1 ) 2 r 5 ( r 10 + 11 r 5 − 1 ) 5 {\displaystyle {\begin{aligned}&j(\tau )=-{\frac {(r^{20}-228r^{15}+494r^{10}+228r^{5}+1)^{3}}{r^{5}(r^{10}+11r^{5}-1)^{5}}}\\[6pt]&j(\tau )-1728=-{\frac {(r^{30}+522r^{25}-10005r^{20}-10005r^{10}-522r^{5}+1)^{2}}{r^{5}(r^{10}+11r^{5}-1)^{5}}}\end{aligned}}} where the numerator and denominator are polynomial invariants of the icosahedron. Using the modular equation between R ( q ) {\displaystyle R(q)} and R ( q 5 ) {\displaystyle R(q^{5})} , one finds that, j ( 5 τ ) = − ( r 20 + 12 r 15 + 14 r 10 − 12 r 5 + 1 ) 3 r 25 ( r 10 + 11 r 5 − 1 ) j ( 5 τ ) − 1728 = − ( r 30 + 18 r 25 + 75 r 20 + 75 r 10 − 18 r 5 + 1 ) 2 r 25 ( r 10 + 11 r 5 − 1 ) {\displaystyle {\begin{aligned}&j(5\tau )=-{\frac {(r^{20}+12r^{15}+14r^{10}-12r^{5}+1)^{3}}{r^{25}(r^{10}+11r^{5}-1)}}\\[6pt]&j(5\tau )-1728=-{\frac {(r^{30}+18r^{25}+75r^{20}+75r^{10}-18r^{5}+1)^{2}}{r^{25}(r^{10}+11r^{5}-1)}}\end{aligned}}} Let z = r 5 − 1 r 5 {\displaystyle z=r^{5}-{\frac {1}{r^{5}}}} , then j ( 5 τ ) = − ( z 2 + 12 z + 16 ) 3 z + 11 {\displaystyle j(5\tau )=-{\frac {\left(z^{2}+12z+16\right)^{3}}{z+11}}} where z ∞ = − [ 5 η ( 25 τ ) η ( 5 τ ) ] 6 − 11 , z 0 = − [ η ( τ ) η ( 5 τ ) ] 6 − 11 , z 1 = [ η ( 5 τ + 2 5 ) η ( 5 τ ) ] 6 − 11 , z 2 = − [ η ( 5 τ + 4 5 ) η ( 5 τ ) ] 6 − 11 , z 3 = [ η ( 5 τ + 6 5 ) η ( 5 τ ) ] 6 − 11 , z 4 = − [ η ( 5 τ + 8 5 ) η ( 5 τ ) ] 6 − 11 {\displaystyle {\begin{aligned}&z_{\infty }=-\left[{\frac {{\sqrt {5}}\,\eta (25\tau )}{\eta (5\tau )}}\right]^{6}-11,\ z_{0}=-\left[{\frac {\eta (\tau )}{\eta (5\tau )}}\right]^{6}-11,\ z_{1}=\left[{\frac {\eta ({\frac {5\tau +2}{5}})}{\eta (5\tau )}}\right]^{6}-11,\\[6pt]&z_{2}=-\left[{\frac {\eta ({\frac {5\tau +4}{5}})}{\eta (5\tau )}}\right]^{6}-11,\ z_{3}=\left[{\frac {\eta ({\frac {5\tau +6}{5}})}{\eta (5\tau )}}\right]^{6}-11,\ z_{4}=-\left[{\frac {\eta ({\frac {5\tau +8}{5}})}{\eta (5\tau )}}\right]^{6}-11\end{aligned}}} which in fact is the j-invariant of the elliptic curve, y 2 + ( 1 + r 5 ) x y + r 5 y = x 3 + r 5 x 2 {\displaystyle y^{2}+(1+r^{5})xy+r^{5}y=x^{3}+r^{5}x^{2}} parameterized by the non-cusp points of the modular curve X 1 ( 5 ) {\displaystyle X_{1}(5)} . == Functional equation == For convenience, one can also use the notation r ( τ ) = R ( q ) {\displaystyle r(\tau )=R(q)} when q = e2πiτ. While other modular functions like the j-invariant satisfies, j ( − 1 τ ) = j ( τ ) {\displaystyle j(-{\tfrac {1}{\tau }})=j(\tau )} and the Dedekind eta function has, η ( − 1 τ ) = − i τ η ( τ ) {\displaystyle \eta (-{\tfrac {1}{\tau }})={\sqrt {-i\tau }}\,\eta (\tau )} the functional equation of the Rogers–Ramanujan continued fraction involves the golden ratio φ {\displaystyle \varphi } , r ( − 1 τ ) = 1 − φ r ( τ ) φ + r ( τ ) {\displaystyle r(-{\tfrac {1}{\tau }})={\frac {1-\varphi \,r(\tau )}{\varphi +r(\tau )}}} Incidentally, r ( 7 + i 10 ) = i {\displaystyle r({\tfrac {7+i}{10}})=i} == Modular equations == There are modular equations between R ( q ) {\displaystyle R(q)} and R ( q n ) {\displaystyle R(q^{n})} . Elegant ones for small prime n are as follows. For n = 2 {\displaystyle n=2} , let u = R ( q ) {\displaystyle u=R(q)} and v = R ( q 2 ) {\displaystyle v=R(q^{2})} , then v − u 2 = ( v + u 2 ) u v 2 . {\displaystyle v-u^{2}=(v+u^{2})uv^{2}.} For n = 3 {\displaystyle n=3} , let u = R ( q ) {\displaystyle u=R(q)} and v = R ( q 3 ) {\displaystyle v=R(q^{3})} , then ( v − u 3 ) ( 1 + u v 3 ) = 3 u 2 v 2 . {\displaystyle (v-u^{3})(1+uv^{3})=3u^{2}v^{2}.} For n = 5 {\displaystyle n=5} , let u = R ( q ) {\displaystyle u=R(q)} and v = R ( q 5 ) {\displaystyle v=R(q^{5})} , then v ( v 4 − 3 v 3 + 4 v 2 − 2 v + 1 ) = ( v 4 + 2 v 3 + 4 v 2 + 3 v + 1 ) u 5 . {\displaystyle v(v^{4}-3v^{3}+4v^{2}-2v+1)=(v^{4}+2v^{3}+4v^{2}+3v+1)u^{5}.} Or equivalently for n = 5 {\displaystyle n=5} , let u = R ( q ) {\displaystyle u=R(q)} and v = R ( q 5 ) {\displaystyle v=R(q^{5})} and φ = 1 + 5 2 {\displaystyle \varphi ={\tfrac {1+{\sqrt {5}}}{2}}} , then u 5 = v ( v 2 − φ 2 v + φ 2 ) ( v 2 − φ − 2 v + φ − 2 ) ( v 2 + v + φ 2 ) ( v 2 + v + φ − 2 ) . {\displaystyle u^{5}={\frac {v\,(v^{2}-\varphi ^{2}v+\varphi ^{2})(v^{2}-\varphi ^{-2}v+\varphi ^{-2})}{(v^{2}+v+\varphi ^{2})(v^{2}+v+\varphi ^{-2})}}.} For n = 11 {\displaystyle n=11} , let u = R ( q ) {\displaystyle u=R(q)} and v = R ( q 11 ) {\displaystyle v=R(q^{11})} , then u v ( u 10 + 11 u 5 − 1 ) ( v 10 + 11 v 5 − 1 ) = ( u − v ) 12 . {\displaystyle uv(u^{10}+11u^{5}-1)(v^{10}+11v^{5}-1)=(u-v)^{12}.} Regarding n = 5 {\displaystyle n=5} , note that v 10 + 11 v 5 − 1 = ( v 2 + v − 1 ) ( v 4 − 3 v 3 + 4 v 2 − 2 v + 1 ) ( v 4 + 2 v 3 + 4 v 2 + 3 v + 1 ) . {\displaystyle v^{10}+11v^{5}-1=(v^{2}+v-1)(v^{4}-3v^{3}+4v^{2}-2v+1)(v^{4}+2v^{3}+4v^{2}+3v+1).} == Other results == Ramanujan found many other interesting results regarding R ( q ) {\displaystyle R(q)} . Let a , b ∈ R + {\displaystyle a,b\in \mathbb {R} ^{+}} , and φ {\displaystyle \varphi } as the golden ratio. If a b = π 2 {\displaystyle ab=\pi ^{2}} then, [ R ( e − 2 a ) + φ ] [ R ( e − 2 b ) + φ ] = 5 φ . {\displaystyle {\bigl [}R(e^{-2a})+\varphi {\bigl ]}{\bigl [}R(e^{-2b})+\varphi {\bigr ]}={\sqrt {5}}\,\varphi .} If 5 a b = π 2 {\displaystyle 5ab=\pi ^{2}} then, [ R 5 ( e − 2 a ) + φ 5 ] [ R 5 ( e − 2 b ) + φ 5 ] = 5 5 φ 5 . {\displaystyle {\bigl [}R^{5}(e^{-2a})+\varphi ^{5}{\bigl ]}{\bigl [}R^{5}(e^{-2b})+\varphi ^{5}{\bigr ]}=5{\sqrt {5}}\,\varphi ^{5}.} The powers of R ( q ) {\displaystyle R(q)} also can be expressed in unusual ways. For its cube, R 3 ( q ) = α β {\displaystyle R^{3}(q)={\frac {\alpha }{\beta }}} where α = ∑ n = 0 ∞ q 2 n 1 − q 5 n + 2 − ∑ n = 0 ∞ q 3 n + 1 1 − q 5 n + 3 , {\displaystyle \alpha =\sum _{n=0}^{\infty }{\frac {q^{2n}}{1-q^{5n+2}}}-\sum _{n=0}^{\infty }{\frac {q^{3n+1}}{1-q^{5n+3}}},} β = ∑ n = 0 ∞ q n 1 − q 5 n + 1 − ∑ n = 0 ∞ q 4 n + 3 1 − q 5 n + 4 . {\displaystyle \beta =\sum _{n=0}^{\infty }{\frac {q^{n}}{1-q^{5n+1}}}-\sum _{n=0}^{\infty }{\frac {q^{4n+3}}{1-q^{5n+4}}}.} For its fifth power, let w = R ( q ) R 2 ( q 2 ) {\displaystyle w=R(q)R^{2}(q^{2})} , then, R 5 ( q ) = w ( 1 − w 1 + w ) 2 , R 5 ( q 2 ) = w 2 ( 1 + w 1 − w ) {\displaystyle R^{5}(q)=w\left({\frac {1-w}{1+w}}\right)^{2},\;\;R^{5}(q^{2})=w^{2}\left({\frac {1+w}{1-w}}\right)} == Quintic equations == The general quintic equation in Bring-Jerrard form: x 5 − 5 x − 4 a = 0 {\displaystyle x^{5}-5x-4a=0} for every real value a > 1 {\displaystyle a>1} can be solved in terms of Rogers-Ramanujan continued fraction R ( q ) {\displaystyle R(q)} and the elliptic nome q ( k ) = exp ⁡ [ − π K ( 1 − k 2 ) / K ( k ) ] . {\displaystyle q(k)=\exp {\big [}-\pi K({\sqrt {1-k^{2}}})/K(k){\big ]}.} To solve this quintic, the elliptic modulus must first be determined as k = tan ⁡ [ 1 4 π − 1 4 arccsc ⁡ ( a 2 ) ] . {\displaystyle k=\tan[{\tfrac {1}{4}}\pi -{\tfrac {1}{4}}\operatorname {arccsc}(a^{2})].} Then the real solution is x = 2 − { 1 − R [ q ( k ) ] } { 1 + R [ q ( k ) 2 ] } R [ q ( k ) ] R [ q ( k ) 2 ] 4 cot ⁡ ⟨ 4 arctan ⁡ { S } ⟩ − 3 4 = 2 − { 1 − R [ q ( k ) ] } { 1 + R [ q ( k ) 2 ] } R [ q ( k ) ] R [ q ( k ) 2 ] 2 S − 1 + 2 S + 1 + 1 S − S − 3 4 . {\displaystyle {\begin{aligned}x&={\frac {2-{\bigl \{}1-R[q(k)]{\bigr \}}{\bigl \{}1+R[q(k)^{2}]{\bigr \}}}{{\sqrt {R[q(k)]\,R[q(k)^{2}]}}\,{\sqrt[{4}]{4\cot \langle 4\arctan\{S\}\rangle -3}}}}\\&={\frac {2-{\bigl \{}1-R[q(k)]{\bigr \}}{\bigl \{}1+R[q(k)^{2}]{\bigr \}}}{{\sqrt {R[q(k)]R[q(k)^{2}]}}\,{\sqrt[{4}]{{\frac {2}{S-1}}+{\frac {2}{S+1}}+{\frac {1}{S}}-S-3}}}}.\end{aligned}}} where S = R [ q ( k ) ] R 2 [ q ( k ) 2 ] . {\displaystyle S=R[q(k)]\,R^{2}[q(k)^{2}].} . Recall in the previous section the 5th power of R ( q ) {\displaystyle R(q)} can be expressed by S {\displaystyle S} : R 5 [ q ( k ) ] = S ( 1 − S 1 + S ) 2 {\displaystyle R^{5}[q(k)]=S\left({\frac {1-S}{1+S}}\right)^{2}} === Example 1 === x 5 − x − 1 = 0 {\displaystyle x^{5}-x-1=0} Transform to, ( 5 4 x ) 5 − 5 ( 5 4 x ) − 4 ( 5 4 5 4 ) = 0 {\displaystyle ({\sqrt[{4}]{5}}x)^{5}-5({\sqrt[{4}]{5}}x)-4({\tfrac {5}{4}}{\sqrt[{4}]{5}})=0} thus, a = 5 4 5 4 {\displaystyle a={\tfrac {5}{4}}{\sqrt[{4}]{5}}} k = tan ⁡ [ 1 4 π − 1 4 arccsc ⁡ ( a 2 ) ] = 5 5 / 4 + 25 5 − 16 5 5 / 4 + 25 5 + 16 {\displaystyle k=\tan[{\tfrac {1}{4}}\pi -{\tfrac {1}{4}}\operatorname {arccsc}(a^{2})]={\tfrac {5^{5/4}+{\sqrt {25{\sqrt {5}}-16}}}{5^{5/4}+{\sqrt {25{\sqrt {5}}+16}}}}} q ( k ) = 0.0851414716 … {\displaystyle q(k)=0.0851414716\dots } R [ q ( k ) ] = 0.5633613184 … {\displaystyle R[q(k)]=0.5633613184\dots } R [ q ( k ) 2 ] = 0.3706122329 … {\displaystyle R[q(k)^{2}]=0.3706122329\dots } and the solution is: x = 2 − { 1 − R [ q ( k ) ] } { 1 + R [ q ( k ) 2 ] } R [ q ( k ) ] R [ q ( k ) 2 ] 20 cot ⁡ ⟨ 4 arctan ⁡ { R [ q ( k ) ] R [ q ( k ) 2 ] 2 } ⟩ − 15 4 = 1.167303978 … {\displaystyle x={\frac {2-{\bigl \{}1-R[q(k)]{\bigr \}}{\bigl \{}1+R[q(k)^{2}]{\bigr \}}}{{\sqrt {R[q(k)]\,R[q(k)^{2}]}}\,{\sqrt[{4}]{20\cot \langle 4\arctan\{R[q(k)]\,R[q(k)^{2}]^{2}\}\rangle -15}}}}=1.167303978\dots } and can not be represented by elementary root expressions. === Example 2 === x 5 − 5 x − 4 ( 81 32 4 ) = 0 {\displaystyle x^{5}-5x-4{\Bigl (}{\sqrt[{4}]{\tfrac {81}{32}}}{\Bigr )}=0} thus, a = 81 32 4 {\displaystyle a={\sqrt[{4}]{\tfrac {81}{32}}}} Given the more familiar continued fractions with closed-forms, r 1 = R ( e − π ) = 1 2 φ ( 5 − φ 3 / 2 ) ( 5 4 + φ 3 / 2 ) = 0.511428 … {\displaystyle r_{1}=R{\big (}e^{-\pi }{\big )}={\tfrac {1}{2}}\varphi \,({\sqrt {5}}-\varphi ^{3/2})({\sqrt[{4}]{5}}+\varphi ^{3/2})=0.511428\dots } r 2 = R ( e − 2 π ) = 5 4 φ 1 / 2 − φ = 0.284079 … {\displaystyle r_{2}=R{\big (}e^{-2\pi }{\big )}={\sqrt[{4}]{5}}\,\varphi ^{1/2}-\varphi =0.284079\dots } r 4 = R ( e − 4 π ) = 1 2 φ ( 5 − φ 3 / 2 ) ( − 5 4 + φ 3 / 2 ) = 0.081002 … {\displaystyle r_{4}=R{\big (}e^{-4\pi }{\big )}={\tfrac {1}{2}}\varphi \,({\sqrt {5}}-\varphi ^{3/2})(-{\sqrt[{4}]{5}}+\varphi ^{3/2})=0.081002\dots } with golden ratio φ = 1 + 5 2 {\displaystyle \varphi ={\tfrac {1+{\sqrt {5}}}{2}}} and the solution simplifies to x = 5 4 2 − { 1 − r 1 } { 1 + r 2 } r 1 r 2 20 cot ⁡ ⟨ 4 arctan ⁡ { r 1 r 2 2 } ⟩ − 15 4 = 5 4 2 − { 1 − r 2 } { 1 + r 4 } r 2 r 4 20 cot ⁡ ⟨ 4 arctan ⁡ { r 2 r 4 2 } ⟩ − 15 4 = 8 4 = 1.681792 … {\displaystyle {\begin{aligned}x&={\sqrt[{4}]{5}}\,{\frac {2-{\bigl \{}1-r_{1}{\bigr \}}{\bigl \{}1+r_{2}{\bigr \}}}{{\sqrt {r_{1}\,r_{2}}}\,{\sqrt[{4}]{20\cot \langle 4\arctan\{r_{1}\,r_{2}^{2}\}\rangle -15}}}}\\[6pt]&={\sqrt[{4}]{5}}\,{\frac {2-{\bigl \{}1-r_{2}{\bigr \}}{\bigl \{}1+r_{4}{\bigr \}}}{{\sqrt {r_{2}\,r_{4}}}\,{\sqrt[{4}]{20\cot \langle 4\arctan\{r_{2}\,r_{4}^{2}\}\rangle -15}}}}\\[6pt]&={\sqrt[{4}]{8}}=1.681792\dots \end{aligned}}} == References == Rogers, L. J. (1894), "Second Memoir on the Expansion of certain Infinite Products", Proc. London Math. Soc., s1-25 (1): 318–343, doi:10.1112/plms/s1-25.1.318 Berndt, B. C.; Chan, H. H.; Huang, S. S.; Kang, S. Y.; Sohn, J.; Son, S. H. (1999), "The Rogers–Ramanujan continued fraction" (PDF), Journal of Computational and Applied Mathematics, 105 (1–2): 9–24, doi:10.1016/S0377-0427(99)00033-3 == External links == Weisstein, Eric W. "Rogers-Ramanujan Identities". MathWorld. Weisstein, Eric W. "Rogers-Ramanujan Continued Fraction". MathWorld.
Wikipedia:Rogers–Ramanujan identities#0	In mathematics, the Rogers–Ramanujan identities are two identities related to basic hypergeometric series and integer partitions. The identities were first discovered and proved by Leonard James Rogers (1894), and were subsequently rediscovered (without a proof) by Srinivasa Ramanujan some time before 1913. Ramanujan had no proof, but rediscovered Rogers's paper in 1917, and they then published a joint new proof (Rogers & Ramanujan 1919). Issai Schur (1917) independently rediscovered and proved the identities. == Definition == The Rogers–Ramanujan identities are G ( q ) = ∑ n = 0 ∞ q n 2 ( q ; q ) n = 1 ( q ; q 5 ) ∞ ( q 4 ; q 5 ) ∞ = 1 + q + q 2 + q 3 + 2 q 4 + 2 q 5 + 3 q 6 + ⋯ {\displaystyle G(q)=\sum _{n=0}^{\infty }{\frac {q^{n^{2}}}{(q;q)_{n}}}={\frac {1}{(q;q^{5})_{\infty }(q^{4};q^{5})_{\infty }}}=1+q+q^{2}+q^{3}+2q^{4}+2q^{5}+3q^{6}+\cdots } (sequence A003114 in the OEIS) and H ( q ) = ∑ n = 0 ∞ q n 2 + n ( q ; q ) n = 1 ( q 2 ; q 5 ) ∞ ( q 3 ; q 5 ) ∞ = 1 + q 2 + q 3 + q 4 + q 5 + 2 q 6 + ⋯ {\displaystyle H(q)=\sum _{n=0}^{\infty }{\frac {q^{n^{2}+n}}{(q;q)_{n}}}={\frac {1}{(q^{2};q^{5})_{\infty }(q^{3};q^{5})_{\infty }}}=1+q^{2}+q^{3}+q^{4}+q^{5}+2q^{6}+\cdots } (sequence A003106 in the OEIS). Here, ( a ; q ) n {\displaystyle (a;q)_{n}} denotes the q-Pochhammer symbol. == Combinatorial interpretation == Consider the following: q n 2 ( q ; q ) n {\displaystyle {\frac {q^{n^{2}}}{(q;q)_{n}}}} is the generating function for partitions with exactly n {\displaystyle n} parts such that adjacent parts have difference at least 2. 1 ( q ; q 5 ) ∞ ( q 4 ; q 5 ) ∞ {\displaystyle {\frac {1}{(q;q^{5})_{\infty }(q^{4};q^{5})_{\infty }}}} is the generating function for partitions such that each part is congruent to either 1 or 4 modulo 5. q n 2 + n ( q ; q ) n {\displaystyle {\frac {q^{n^{2}+n}}{(q;q)_{n}}}} is the generating function for partitions with exactly n {\displaystyle n} parts such that adjacent parts have difference at least 2 and such that the smallest part is at least 2. 1 ( q 2 ; q 5 ) ∞ ( q 3 ; q 5 ) ∞ {\displaystyle {\frac {1}{(q^{2};q^{5})_{\infty }(q^{3};q^{5})_{\infty }}}} is the generating function for partitions such that each part is congruent to either 2 or 3 modulo 5. The Rogers–Ramanujan identities could be now interpreted in the following way. Let n {\displaystyle n} be a non-negative integer. The number of partitions of n {\displaystyle n} such that the adjacent parts differ by at least 2 is the same as the number of partitions of n {\displaystyle n} such that each part is congruent to either 1 or 4 modulo 5. The number of partitions of n {\displaystyle n} such that the adjacent parts differ by at least 2 and such that the smallest part is at least 2 is the same as the number of partitions of n {\displaystyle n} such that each part is congruent to either 2 or 3 modulo 5. Alternatively, The number of partitions of n {\displaystyle n} such that with k {\displaystyle k} parts the smallest part is at least k {\displaystyle k} is the same as the number of partitions of n {\displaystyle n} such that each part is congruent to either 1 or 4 modulo 5. The number of partitions of n {\displaystyle n} such that with k {\displaystyle k} parts the smallest part is at least k + 1 {\displaystyle k+1} is the same as the number of partitions of n {\displaystyle n} such that each part is congruent to either 2 or 3 modulo 5. == Application to partitions == Since the terms occurring in the identity are generating functions of certain partitions, the identities make statements about partitions (decompositions) of natural numbers. The number sequences resulting from the coefficients of the Maclaurin series of the Rogers–Ramanujan functions G and H are special partition number sequences of level 5: G ( x ) = 1 ( x ; x 5 ) ∞ ( x 4 ; x 5 ) ∞ = 1 + ∑ n = 1 ∞ P G ( n ) x n {\displaystyle G(x)={\frac {1}{(x;x^{5})_{\infty }(x^{4};x^{5})_{\infty }}}=1+\sum _{n=1}^{\infty }P_{G}(n)x^{n}} H ( x ) = 1 ( x 2 ; x 5 ) ∞ ( x 3 ; x 5 ) ∞ = 1 + ∑ n = 1 ∞ P H ( n ) x n {\displaystyle H(x)={\frac {1}{(x^{2};x^{5})_{\infty }(x^{3};x^{5})_{\infty }}}=1+\sum _{n=1}^{\infty }P_{H}(n)x^{n}} The number sequence P G ( n ) {\displaystyle P_{G}(n)} (sequence A003114 in the OEIS)) represents the number of possibilities for the affected natural number n to decompose this number into summands of the patterns 5a + 1 or 5a + 4 with a ∈ N 0 {\displaystyle \mathbb {N} _{0}} . Thus P G ( n ) {\displaystyle P_{G}(n)} gives the number of decays of an integer n in which adjacent parts of the partition differ by at least 2, equal to the number of decays in which each part is equal to 1 or 4 mod 5 is. And the number sequence P H ( n ) {\displaystyle P_{H}(n)} (sequence A003106 in the OEIS)) analogously represents the number of possibilities for the affected natural number n to decompose this number into summands of the patterns 5a + 2 or 5a + 3 with a ∈ N 0 {\displaystyle \mathbb {N} _{0}} . Thus P H ( n ) {\displaystyle P_{H}(n)} gives the number of decays of an integer n in which adjacent parts of the partition differ by at least 2 and in which the smallest part is greater than or equal to 2 is equal the number of decays whose parts are equal to 2 or 3 mod 5. This will be illustrated as examples in the following two tables: == Rogers–Ramanujan continued fractions R and S == === Definition of the continued fractions === The following continued fraction R ( q ) {\displaystyle R(q)} is called Rogers–Ramanujan continued fraction, Continuing fraction S ( q ) {\displaystyle S(q)} is called alternating Rogers–Ramanujan continued fraction! The factor q 1 5 {\displaystyle q^{\frac {1}{5}}} creates a quotient of module functions and it also makes these shown continued fractions modular: This definition applies for the continued fraction mentioned: R ( q ) = q 1 / 5 ( q ; q 5 ) ∞ ( q 4 ; q 5 ) ∞ ( q 2 ; q 5 ) ∞ ( q 3 ; q 5 ) ∞ {\displaystyle R(q)=q^{1/5}{\frac {(q;q^{5})_{\infty }(q^{4};q^{5})_{\infty }}{(q^{2};q^{5})_{\infty }(q^{3};q^{5})_{\infty }}}} R ( q ) = q 1 / 5 ∏ k = 0 ∞ ( 1 − q 5 k + 1 ) ( 1 − q 5 k + 4 ) ( 1 − q 5 k + 2 ) ( 1 − q 5 k + 3 ) = q 1 / 5 H ( q ) G ( q ) {\displaystyle R(q)=q^{1/5}\prod _{k=0}^{\infty }{\frac {(1-q^{5k+1})(1-q^{5k+4})}{(1-q^{5k+2})(1-q^{5k+3})}}=q^{1/5}{\frac {H(q)}{G(q)}}} This is the definition of the Ramanujan theta function: f ( a , b ) = ∑ k = − ∞ ∞ a k ( k + 1 ) 2 b k ( k − 1 ) 2 {\displaystyle f(a,b)=\sum _{k=-\infty }^{\infty }a^{\frac {k(k+1)}{2}}b^{\frac {k(k-1)}{2}}} With this function, the continued fraction R can be created this way: R ( q ) = q 1 / 5 f ( − q , − q 4 ) f ( − q 2 , − q 3 ) {\displaystyle R(q)=q^{1/5}{\frac {f(-q,-q^{4})}{f(-q^{2},-q^{3})}}} . The connection between the continued fraction and the Rogers–Ramanujan functions was already found by Rogers in 1894 (and later independently by Ramanujan). The continued fraction can also be expressed by the Dedekind eta function: R ( q ) = tan ⁡ { 1 2 arccot ⁡ [ η W ( q 1 / 5 ) 2 η W ( q 5 ) + 1 2 ] } {\displaystyle R(q)=\tan {\biggl \{}{\frac {1}{2}}\operatorname {arccot} {\biggl [}{\frac {\eta _{W}(q^{1/5})}{2\eta _{W}(q^{5})}}+{\frac {1}{2}}{\biggr ]}{\biggr \}}} The alternating continued fraction S ( q ) {\displaystyle S(q)} has the following identities to the remaining Rogers–Ramanujan functions and to the Ramanujan theta function described above: S ( q ) = q 1 / 5 H ( − q ) G ( − q ) {\displaystyle S(q)=q^{1/5}{\frac {H(-q)}{G(-q)}}} S ( q ) = q 1 / 5 f ( q , − q 4 ) f ( − q 2 , q 3 ) {\displaystyle S(q)=q^{1/5}{\frac {f(q,-q^{4})}{f(-q^{2},q^{3})}}} S ( q ) = R ( q 4 ) R ( q ) R ( q 2 ) {\displaystyle S(q)={\frac {R(q^{4})}{R(q)R(q^{2})}}} S ( q ) = q 1 / 5 G ( q ) G ( q 2 ) H ( q 4 ) H ( q ) H ( q 2 ) G ( q 4 ) {\displaystyle S(q)=q^{1/5}{\frac {G(q)G(q^{2})H(q^{4})}{H(q)H(q^{2})G(q^{4})}}} === Identities with Jacobi theta functions === The following definitions are valid for the Jacobi "Theta-Nullwert" functions: ϑ 00 ( x ) = 1 + 2 ∑ n = 1 ∞ x ◻ ( n ) {\displaystyle \vartheta _{00}(x)=1+2\sum _{n=1}^{\infty }x^{\Box (n)}} ϑ 01 ( x ) = 1 − 2 ∑ n = 1 ∞ ( − 1 ) n + 1 x ◻ ( n ) {\displaystyle \vartheta _{01}(x)=1-2\sum _{n=1}^{\infty }(-1)^{n+1}x^{\Box (n)}} ϑ 10 ( x ) = 2 x 1 / 4 + 2 x 1 / 4 ∑ n = 1 ∞ x 2 △ ( n ) {\displaystyle \vartheta _{10}(x)=2x^{1/4}+2x^{1/4}\sum _{n=1}^{\infty }x^{2\bigtriangleup (n)}} And the following product definitions are identical to the total definitions mentioned: ϑ 00 ( x ) = ∏ n = 1 ∞ ( 1 − x 2 n ) ( 1 + x 2 n − 1 ) 2 {\displaystyle \vartheta _{00}(x)=\prod _{n=1}^{\infty }(1-x^{2n})(1+x^{2n-1})^{2}} ϑ 01 ( x ) = ∏ n = 1 ∞ ( 1 − x 2 n ) ( 1 − x 2 n − 1 ) 2 {\displaystyle \vartheta _{01}(x)=\prod _{n=1}^{\infty }(1-x^{2n})(1-x^{2n-1})^{2}} ϑ 10 ( x ) = 2 x 1 / 4 ∏ n = 1 ∞ ( 1 − x 2 n ) ( 1 + x 2 n ) 2 {\displaystyle \vartheta _{10}(x)=2x^{1/4}\prod _{n=1}^{\infty }(1-x^{2n})(1+x^{2n})^{2}} These three so-called theta zero value functions are linked to each other using the Jacobian identity: ϑ 10 ( x ) = ϑ 00 ( x ) 4 − ϑ 01 ( x ) 4 4 {\displaystyle \vartheta _{10}(x)={\sqrt[{4}]{\vartheta _{00}(x)^{4}-\vartheta _{01}(x)^{4}}}} The mathematicians Edmund Taylor Whittaker and George Neville Watson discovered these definitional identities. The Rogers–Ramanujan continued fraction functions R ( x ) {\displaystyle R(x)} and S ( x ) {\displaystyle S(x)} have these relationships to the theta Nullwert functions: R ( x ) = tan ⁡ ⟨ 1 2 arccot ⁡ { ϑ 01 ( x 1 / 5 ) [ 5 ϑ 01 ( x 5 ) 2 − ϑ 01 ( x ) 2 ] 2 ϑ 01 ( x 5 ) [ ϑ 01 ( x ) 2 − ϑ 01 ( x 1 / 5 ) 2 ] + 1 2 } ⟩ {\displaystyle R(x)=\tan {\biggl \langle }{\frac {1}{2}}\operatorname {arccot} {\biggl \{}{\frac {\vartheta _{01}(x^{1/5})[5\,\vartheta _{01}(x^{5})^{2}-\vartheta _{01}(x)^{2}]}{2\,\vartheta _{01}(x^{5})[\vartheta _{01}(x)^{2}-\vartheta _{01}(x^{1/5})^{2}]}}+{\frac {1}{2}}{\biggr \}}{\biggr \rangle }} S ( x ) = tan ⁡ ⟨ 1 2 arccot ⁡ { ϑ 00 ( x 1 / 5 ) [ 5 ϑ 00 ( x 5 ) 2 − ϑ 00 ( x ) 2 ] 2 ϑ 00 ( x 5 ) [ ϑ 00 ( x 1 / 5 ) 2 − ϑ 00 ( x ) 2 ] − 1 2 } ⟩ {\displaystyle S(x)=\tan {\biggl \langle }{\frac {1}{2}}\operatorname {arccot} {\biggl \{}{\frac {\vartheta _{00}(x^{1/5})[5\,\vartheta _{00}(x^{5})^{2}-\vartheta _{00}(x)^{2}]}{2\,\vartheta _{00}(x^{5})[\vartheta _{00}(x^{1/5})^{2}-\vartheta _{00}(x)^{2}]}}-{\frac {1}{2}}{\biggr \}}{\biggr \rangle }} The element of the fifth root can also be removed from the elliptic nome of the theta functions and transferred to the external tangent function. In this way, a formula can be created that only requires one of the three main theta functions: R ( x ) = tan ⁡ { 1 2 arctan ⁡ [ 1 2 − ϑ 01 ( x ) 2 2 ϑ 01 ( x 5 ) 2 ] } 1 / 5 tan ⁡ { 1 2 arccot ⁡ [ 1 2 − ϑ 01 ( x ) 2 2 ϑ 01 ( x 5 ) 2 ] } 2 / 5 {\displaystyle R(x)=\tan {\biggl \{}{\frac {1}{2}}\arctan {\biggl [}{\frac {1}{2}}-{\frac {\vartheta _{01}(x)^{2}}{2\vartheta _{01}(x^{5})^{2}}}{\biggr ]}{\biggr \}}^{1/5}\tan {\biggl \{}{\frac {1}{2}}\operatorname {arccot} {\biggl [}{\frac {1}{2}}-{\frac {\vartheta _{01}(x)^{2}}{2\vartheta _{01}(x^{5})^{2}}}{\biggr ]}{\biggr \}}^{2/5}} S ( x ) = tan ⁡ { 1 2 arctan ⁡ [ ϑ 00 ( x ) 2 2 ϑ 00 ( x 5 ) 2 − 1 2 ] } 1 / 5 cot ⁡ { 1 2 arccot ⁡ [ ϑ 00 ( x ) 2 2 ϑ 00 ( x 5 ) 2 − 1 2 ] } 2 / 5 {\displaystyle S(x)=\tan {\biggl \{}{\frac {1}{2}}\arctan {\biggl [}{\frac {\vartheta _{00}(x)^{2}}{2\vartheta _{00}(x^{5})^{2}}}-{\frac {1}{2}}{\biggr ]}{\biggr \}}^{1/5}\cot {\biggl \{}{\frac {1}{2}}\operatorname {arccot} {\biggl [}{\frac {\vartheta _{00}(x)^{2}}{2\vartheta _{00}(x^{5})^{2}}}-{\frac {1}{2}}{\biggr ]}{\biggr \}}^{2/5}} == Modular modified functions of G and H == === Definition of the modular form of G and H === An elliptic function is a modular function if this function in dependence on the elliptic nome as an internal variable function results in a function, which also results as an algebraic combination of Legendre's elliptic modulus and its complete elliptic integrals of the first kind in the K and K' form. The Legendre's elliptic modulus is the numerical eccentricity of the corresponding ellipse. If you set q = e 2 π i τ {\displaystyle q=e^{2\pi i\tau }} (where the imaginary part of τ ∈ C {\displaystyle \tau \in \mathbb {C} } is positive), following two functions are modular functions! G M ( q ) = q − 1 60 G ( q ) {\displaystyle G_{M}(q)=q^{\frac {-1}{60}}G(q)} H M ( q ) = q 11 60 H ( q ) {\displaystyle H_{M}(q)=q^{\frac {11}{60}}H(q)} If q = e2πiτ, then q−1/60G(q) and q11/60H(q) are modular functions of τ. For the Rogers–Ramanujan continued fraction R(q) this formula is valid based on the described modular modifications of G and H: R ( q ) = H M ( q ) G M ( q ) {\displaystyle R(q)={\frac {H_{M}(q)}{G_{M}(q)}}} === Special values === These functions have the following values for the reciprocal of Gelfond's constant and for the square of this reciprocal: G M [ exp ⁡ ( − π ) ] = 2 − 1 / 2 5 − 1 / 4 ( 5 − 1 ) 1 / 4 ( 5 4 + 1 ) 1 / 2 R [ exp ⁡ ( − π ) ] − 1 / 2 = = 2 1 / 4 5 − 1 / 8 Φ 1 / 2 cos ⁡ [ 1 4 arctan ⁡ ( 2 ) + 1 2 arcsin ⁡ ( Φ − 2 ) ] {\displaystyle {\begin{aligned}G_{M}{\bigl [}\exp(-\pi ){\bigr ]}&=2^{-1/2}5^{-1/4}({\sqrt {5}}-1)^{1/4}({\sqrt[{4}]{5}}+1)^{1/2}R{\bigl [}\exp(-\pi ){\bigr ]}^{-1/2}=\\[4pt]&=2^{1/4}\,5^{-1/8}\,\Phi ^{1/2}\,{\color {blue}\cos {\bigl [}{\tfrac {1}{4}}\arctan(2)+{\tfrac {1}{2}}\arcsin(\Phi ^{-2}){\bigr ]}}\end{aligned}}} H M [ exp ⁡ ( − π ) ] = 2 − 1 / 2 5 − 1 / 4 ( 5 − 1 ) 1 / 4 ( 5 4 + 1 ) 1 / 2 R [ exp ⁡ ( − π ) ] 1 / 2 = = 2 1 / 4 5 − 1 / 8 Φ 1 / 2 sin ⁡ [ 1 4 arctan ⁡ ( 2 ) + 1 2 arcsin ⁡ ( Φ − 2 ) ] {\displaystyle {\begin{aligned}H_{M}{\bigl [}\exp(-\pi ){\bigr ]}&=2^{-1/2}5^{-1/4}({\sqrt {5}}-1)^{1/4}({\sqrt[{4}]{5}}+1)^{1/2}R{\bigl [}\exp(-\pi ){\bigr ]}^{1/2}=\\[4pt]&=2^{1/4}\,5^{-1/8}\,\Phi ^{1/2}\,{\color {blue}\sin {\bigl [}{\tfrac {1}{4}}\arctan(2)+{\tfrac {1}{2}}\arcsin(\Phi ^{-2}){\bigr ]}}\end{aligned}}} G M [ exp ⁡ ( − 2 π ) ] = 10 − 1 / 4 ( 5 − 1 ) 1 / 4 R [ exp ⁡ ( − 2 π ) ] − 1 / 2 = = 2 1 / 2 5 − 1 / 8 cos ⁡ [ 1 4 arctan ⁡ ( 2 ) ] {\displaystyle {\begin{aligned}G_{M}{\bigl [}\exp(-2\pi ){\bigr ]}&=10^{-1/4}({\sqrt {5}}-1)^{1/4}R{\bigl [}\exp(-2\pi ){\bigr ]}^{-1/2}=\\[4pt]&=2^{1/2}\,5^{-1/8}\,{\color {blue}\cos {\bigl [}{\tfrac {1}{4}}\arctan(2){\bigr ]}}\end{aligned}}} H M [ exp ⁡ ( − 2 π ) ] = 10 − 1 / 4 ( 5 − 1 ) 1 / 4 R [ exp ⁡ ( − 2 π ) ] 1 / 2 = = 2 1 / 2 5 − 1 / 8 sin ⁡ [ 1 4 arctan ⁡ ( 2 ) ] {\displaystyle {\begin{aligned}H_{M}{\bigl [}\exp(-2\pi ){\bigr ]}&=10^{-1/4}({\sqrt {5}}-1)^{1/4}R{\bigl [}\exp(-2\pi ){\bigr ]}^{1/2}=\\[4pt]&=2^{1/2}\,5^{-1/8}\,{\color {blue}\sin {\bigl [}{\tfrac {1}{4}}\arctan(2){\bigr ]}}\end{aligned}}} The Rogers–Ramanujan continued fraction takes the following ordinate values for these abscissa values: == Dedekind eta function identities == === Derivation by the geometric mean === Given are the mentioned definitions of G M {\displaystyle G_{M}} and H M {\displaystyle H_{M}} in this already mentioned way: G M ( q ) = q − 1 60 1 ( q ; q 5 ) ∞ ( q 4 ; q 5 ) ∞ {\displaystyle G_{M}(q)=q^{\frac {-1}{60}}{\frac {1}{(q;q^{5})_{\infty }(q^{4};q^{5})_{\infty }}}} H M ( q ) = q 11 60 1 ( q 2 ; q 5 ) ∞ ( q 3 ; q 5 ) ∞ {\displaystyle H_{M}(q)=q^{\frac {11}{60}}{\frac {1}{(q^{2};q^{5})_{\infty }(q^{3};q^{5})_{\infty }}}} The Dedekind eta function identities for the functions G and H result by combining only the following two equation chains: The quotient is the Rogers Ramanujan continued fraction accurately: H M ( q ) ÷ G M ( q ) = R ( q ) {\displaystyle H_{M}(q)\div G_{M}(q)=R(q)} But the product leads to a simplified combination of Pochhammer operators: H M ( q ) G M ( q ) = q 1 / 6 1 ( q ; q 5 ) ∞ ( q 4 ; q 5 ) ∞ ( q 2 ; q 5 ) ∞ ( q 3 ; q 5 ) ∞ = {\displaystyle H_{M}(q)\,G_{M}(q)=q^{1/6}{\frac {1}{(q;q^{5})_{\infty }(q^{4};q^{5})_{\infty }(q^{2};q^{5})_{\infty }(q^{3};q^{5})_{\infty }}}=} = q 1 / 6 ( q 5 ; q 5 ) ∞ ( q ; q ) ∞ = η W ( q 5 ) η W ( q ) {\displaystyle =q^{1/6}{\frac {(q^{5};q^{5})_{\infty }}{(q;q)_{\infty }}}={\frac {\eta _{W}(q^{5})}{\eta _{W}(q)}}} The geometric mean of these two equation chains directly lead to following expressions in dependence of the Dedekind eta function in their Weber form: G M ( q ) = η W ( q 5 ) 1 / 2 η W ( q ) − 1 / 2 R ( q ) − 1 / 2 {\displaystyle G_{M}(q)=\eta _{W}(q^{5})^{1/2}\eta _{W}(q)^{-1/2}R(q)^{-1/2}} H M ( q ) = η W ( q 5 ) 1 / 2 η W ( q ) − 1 / 2 R ( q ) 1 / 2 {\displaystyle H_{M}(q)=\eta _{W}(q^{5})^{1/2}\eta _{W}(q)^{-1/2}R(q)^{1/2}} In this way the modulated functions G M {\displaystyle G_{M}} and H M {\displaystyle H_{M}} are represented directly using only the continued fraction R and the Dedekind eta function quotient! With the Pochhammer products alone, the following identity then applies to the non-modulated functions G and H: G ( q ) = ( q ; q 5 ) ∞ − 1 ( q 4 ; q 5 ) ∞ − 1 = ( q 5 ; q 5 ) ∞ 1 / 2 ( q ; q ) ∞ − 1 / 2 [ H ( q ) G ( q ) ] − 1 / 2 {\displaystyle G(q)=(q;q^{5})_{\infty }^{-1}(q^{4};q^{5})_{\infty }^{-1}=(q^{5};q^{5})_{\infty }^{1/2}(q;q)_{\infty }^{-1/2}{\biggl [}{\frac {H(q)}{G(q)}}{\biggr ]}^{-1/2}} H ( q ) = ( q 2 ; q 5 ) ∞ − 1 ( q 3 ; q 5 ) ∞ − 1 = ( q 5 ; q 5 ) ∞ 1 / 2 ( q ; q ) ∞ − 1 / 2 [ H ( q ) G ( q ) ] 1 / 2 {\displaystyle H(q)=(q^{2};q^{5})_{\infty }^{-1}(q^{3};q^{5})_{\infty }^{-1}=(q^{5};q^{5})_{\infty }^{1/2}(q;q)_{\infty }^{-1/2}{\biggl [}{\frac {H(q)}{G(q)}}{\biggr ]}^{1/2}} === Pentagonal number theorem === For the Dedekind eta function according to Weber's definition these formulas apply: η W ( x ) = 2 − 1 / 6 ϑ 10 ( x ) 1 / 6 ϑ 00 ( x ) 1 / 6 ϑ 01 ( x ) 2 / 3 {\displaystyle \eta _{W}(x)=2^{-1/6}\vartheta _{10}(x)^{1/6}\vartheta _{00}(x)^{1/6}\vartheta _{01}(x)^{2/3}} η W ( x ) = 2 − 1 / 3 ϑ 10 ( x 1 / 2 ) 1 / 3 ϑ 00 ( x 1 / 2 ) 1 / 3 ϑ 01 ( x 1 / 2 ) 1 / 3 {\displaystyle \eta _{W}(x)=2^{-1/3}\vartheta _{10}(x^{1/2})^{1/3}\vartheta _{00}(x^{1/2})^{1/3}\vartheta _{01}(x^{1/2})^{1/3}} η W ( x ) = x 1 / 24 ∏ n = 1 ∞ ( 1 − x n ) = x 1 / 24 ( x ; x ) ∞ {\displaystyle \eta _{W}(x)=x^{1/24}\prod _{n=1}^{\infty }(1-x^{n})=x^{1/24}(x;x)_{\infty }} η W ( x ) = x 1 / 24 { 1 + ∑ n = 1 ∞ [ − x Fn ( 2 n − 1 ) − x Kr ( 2 n − 1 ) + x Fn ( 2 n ) + x Kr ( 2 n ) ] } {\displaystyle \eta _{W}(x)=x^{1/24}{\biggl \{}1+\sum _{n=1}^{\infty }{\bigl [}-x^{{\text{Fn}}(2n-1)}-x^{{\text{Kr}}(2n-1)}+x^{{\text{Fn}}(2n)}+x^{{\text{Kr}}(2n)}{\bigr ]}{\biggr \}}} η W ( x ) = x 1 / 24 { 1 + ∑ n = 1 ∞ P ( n ) x n } − 1 {\displaystyle \eta _{W}(x)=x^{1/24}{\biggl \{}1+\sum _{n=1}^{\infty }\mathrm {P} (n)\,x^{n}{\biggr \}}^{-1}} The fourth formula describes the pentagonal number theorem because of the exponents! These basic definitions apply to the pentagonal numbers and the card house numbers: Fn ( z ) = 1 2 z ( 3 z − 1 ) {\displaystyle {\text{Fn}}(z)={\tfrac {1}{2}}z(3z-1)} Kr ( z ) = 1 2 z ( 3 z + 1 ) {\displaystyle {\text{Kr}}(z)={\tfrac {1}{2}}z(3z+1)} The fifth formula contains the Regular Partition Numbers as coefficients. The Regular Partition Number Sequence P ( n ) {\displaystyle \mathrm {P} (n)} itself indicates the number of ways in which a positive integer number n {\displaystyle n} can be split into positive integer summands. For the numbers n = 1 {\displaystyle n=1} to n = 5 {\displaystyle n=5} , the associated partition numbers P {\displaystyle P} with all associated number partitions are listed in the following table: === Further Dedekind eta identities === The following further simplification for the modulated functions G M {\displaystyle G_{M}} and H M {\displaystyle H_{M}} can be undertaken. This connection applies especially to the Dedekind eta function from the fifth power of the elliptic nome: η W ( q 5 ) η W ( q ) = η W ( q 2 ) 4 η W ( q ) 4 ϑ 01 ( q 5 ) ϑ 01 ( q ) [ 5 ϑ 01 ( q 5 ) 2 4 ϑ 01 ( q ) 2 − 1 4 ] − 1 {\displaystyle {\frac {\eta _{W}(q^{5})}{\eta _{W}(q)}}={\frac {\eta _{W}(q^{2})^{4}}{\eta _{W}(q)^{4}}}\,{\frac {\vartheta _{01}(q^{5})}{\vartheta _{01}(q)}}{\biggl [}{\frac {5\,\vartheta _{01}(q^{5})^{2}}{4\,\vartheta _{01}(q)^{2}}}-{\frac {1}{4}}{\biggr ]}^{-1}} These two identities with respect to the Rogers–Ramanujan continued fraction were given for the modulated functions G M {\displaystyle G_{M}} and H M {\displaystyle H_{M}} : G M ( q ) = η W ( q 5 ) 1 / 2 η W ( q ) − 1 / 2 R ( q ) − 1 / 2 {\displaystyle G_{M}(q)=\eta _{W}(q^{5})^{1/2}\eta _{W}(q)^{-1/2}R(q)^{-1/2}} H M ( q ) = η W ( q 5 ) 1 / 2 η W ( q ) − 1 / 2 R ( q ) 1 / 2 {\displaystyle H_{M}(q)=\eta _{W}(q^{5})^{1/2}\eta _{W}(q)^{-1/2}R(q)^{1/2}} The combination of the last three formulas mentioned results in the following pair of formulas: === Reduced Weber modular function === The Weber modular functions in their reduced form are an efficient way of computing the values of the Rogers–Ramanujan functions: First of all we introduce the reduced Weber modular functions in that pattern: w R n ( ε ) = 2 ( n − 1 ) / 4 [ q ( ε ) n ; q ( ε ) 2 n ] ∞ [ q ( ε ) ; q ( ε ) 2 ] ∞ n {\displaystyle w_{Rn}(\varepsilon )={\frac {2^{(n-1)/4}[q(\varepsilon )^{n};q(\varepsilon )^{2n}]_{\infty }}{[q(\varepsilon );q(\varepsilon )^{2}]_{\infty }^{n}}}} w R 5 ( ε ) = 2 [ q ( ε ) 5 ; q ( ε ) 10 ] ∞ [ q ( ε ) ; q ( ε ) 2 ] ∞ 5 {\displaystyle w_{R5}(\varepsilon )={\frac {2[q(\varepsilon )^{5};q(\varepsilon )^{10}]_{\infty }}{[q(\varepsilon );q(\varepsilon )^{2}]_{\infty }^{5}}}} This function fulfills following equation of sixth degree: Therefore this w R 5 {\displaystyle w_{R5}} function is an algebraic function indeed. But along with the Abel–Ruffini theorem this function in relation to the eccentricity can not be represented by elementary expressions. However there are many values that in fact can be expressed elementarily. Four examples shall be given for this: First example: Second example: Third example: Fourth example: For that function, a further expression is valid: w R 5 ( ε ) = 5 ϑ 01 [ q ( k ) 5 ] 2 2 ϑ 01 [ q ( k ) ] 2 − 1 2 {\displaystyle w_{R5}(\varepsilon )={\frac {5\,\vartheta _{01}[q(k)^{5}]^{2}}{2\,\vartheta _{01}[q(k)]^{2}}}-{\frac {1}{2}}} === Exact eccentricity identity for the functions G and H === In this way the accurate eccentricity dependent formulas for the functions G and H can be generated: Following Dedekind eta function quotient has this eccentricity dependency: η W [ q ( ε ) 2 ] η W [ q ( ε ) ] = 2 − 1 / 4 tan ⁡ [ 2 arctan ⁡ ( ε ) ] 1 / 12 {\displaystyle {\frac {\eta _{W}[q(\varepsilon )^{2}]}{\eta _{W}[q(\varepsilon )]}}=2^{-1/4}\tan {\bigl [}2\arctan(\varepsilon ){\bigr ]}^{1/12}} This is the eccentricity dependent formula for the continued fraction R: R [ q ( ε ) ] = tan ⁡ { 1 2 arctan ⁡ [ w R 5 ( ε ) − 2 2 w R 5 ( ε ) + 1 ] } 1 / 5 tan ⁡ { 1 2 arccot ⁡ [ w R 5 ( ε ) − 2 2 w R 5 ( ε ) + 1 ] } 2 / 5 {\displaystyle R[q(\varepsilon )]=\tan {\biggl \{}{\frac {1}{2}}\arctan {\biggl [}{\frac {w_{R5}(\varepsilon )-2}{2\,w_{R5}(\varepsilon )+1}}{\biggr ]}{\biggr \}}^{1/5}\tan {\biggl \{}{\frac {1}{2}}\operatorname {arccot} {\biggl [}{\frac {w_{R5}(\varepsilon )-2}{2\,w_{R5}(\varepsilon )+1}}{\biggr ]}{\biggr \}}^{2/5}} The last three now mentioned formulas will be inserted into the final formulas mentioned in the section above: On the left side of the balances the functions G M ( q ) {\displaystyle G_{M}(q)} and H M ( q ) {\displaystyle H_{M}(q)} in relation to the elliptic nome function q ( ε ) {\displaystyle q(\varepsilon )} are written down directly. And on the right side an algebraic combination of the eccentricity ε {\displaystyle \varepsilon } is formulated. Therefore these functions G M ( q ) = q − 1 / 60 G ( q ) {\displaystyle G_{M}(q)=q^{-1/60}G(q)} and H M ( q ) = q 11 / 60 H ( q ) {\displaystyle H_{M}(q)=q^{11/60}H(q)} are modular functions indeed! == Application to quintic equations == === Discovery of the corresponding modulus by Charles Hermite === The general case of quintic equations in the Bring–Jerrard form has a non-elementary solution based on the Abel–Ruffini theorem and will now be explained using the elliptic nome of the corresponding modulus, described by the lemniscate elliptic functions in a simplified way. x 5 + 5 x = 4 c {\displaystyle x^{5}+5\,x=4\,c} The real solution for all real values c ∈ R {\displaystyle c\in \mathbb {R} } can be determined as follows: x = S ⟨ q { ctlh ⁡ [ 1 2 aclh ⁡ ( c ) ] 2 } ⟩ 2 − R ⟨ q { ctlh ⁡ [ 1 2 aclh ⁡ ( c ) ] 2 } 2 ⟩ S ⟨ q { ctlh ⁡ [ 1 2 aclh ⁡ ( c ) ] 2 } ⟩ 2 × × 1 − R ⟨ q { ctlh ⁡ [ 1 2 aclh ⁡ ( c ) ] 2 } 2 ⟩ S ⟨ q { ctlh ⁡ [ 1 2 aclh ⁡ ( c ) ] 2 } ⟩ R ⟨ q { ctlh ⁡ [ 1 2 aclh ⁡ ( c ) ] 2 } 2 ⟩ 2 × × ϑ 00 ⟨ q { ctlh ⁡ [ 1 2 aclh ⁡ ( c ) ] 2 } 5 ⟩ ϑ 00 ⟨ q { ctlh ⁡ [ 1 2 aclh ⁡ ( c ) ] 2 } 1 / 5 ⟩ 2 − 5 ϑ 00 ⟨ q { ctlh ⁡ [ 1 2 aclh ⁡ ( c ) ] 2 } 5 ⟩ 3 4 ϑ 10 ⟨ q { ctlh ⁡ [ 1 2 aclh ⁡ ( c ) ] 2 } ⟩ ϑ 01 ⟨ q { ctlh ⁡ [ 1 2 aclh ⁡ ( c ) ] 2 } ⟩ ϑ 00 ⟨ q { ctlh ⁡ [ 1 2 aclh ⁡ ( c ) ] 2 } ⟩ {\displaystyle {\begin{aligned}x={}&{\frac {S{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}{\bigr \rangle }^{2}-R{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}^{2}{\bigr \rangle }}{S{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}{\bigr \rangle }^{2}}}\times \\[4pt]&{}\times {\frac {1-R{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}^{2}{\bigr \rangle }\,S{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}{\bigr \rangle }}{R{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}^{2}{\bigr \rangle }^{2}}}\times \\[4pt]&{}\times {\frac {\vartheta _{00}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}^{5}{\bigr \rangle }\,\vartheta _{00}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}^{1/5}{\bigr \rangle }^{2}-5\,\vartheta _{00}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}^{5}{\bigr \rangle }^{3}}{4\,\vartheta _{10}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}{\bigr \rangle }\,\vartheta _{01}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}{\bigr \rangle }\,\vartheta _{00}{\bigl \langle }q\{\operatorname {ctlh} [{\tfrac {1}{2}}\operatorname {aclh} (c)]^{2}\}{\bigr \rangle }}}\end{aligned}}} Alternatively, the same solution can be presented in this way: x = 5 ϑ 00 ( Q 5 ) 3 − ϑ 00 ( Q 5 ) ϑ 00 ( Q ) 2 4 ϑ 10 ( Q ) ϑ 01 ( Q ) ϑ 00 ( Q ) × S ( Q ) 2 + R ( Q 2 ) S ( Q ) × [ R ( Q 2 ) S ( Q ) + R ( Q 2 ) + S ( Q ) − 1 ] w i t h Q = q { c t l h [ 1 2 aclh ⁡ ( c ) ] 2 } {\displaystyle {\begin{aligned}x={}&{\frac {5\,\vartheta _{00}(Q^{5})^{3}-\vartheta _{00}(Q^{5})\,\vartheta _{00}(Q)^{2}}{4\,\vartheta _{10}(Q)\,\vartheta _{01}(Q)\,\vartheta _{00}(Q)}}\times {\frac {S(Q)^{2}+R(Q^{2})}{S(Q)}}\times {\bigl [}R(Q^{2})S(Q)+R(Q^{2})+S(Q)-1{\bigr ]}\\[4pt]&\mathrm {with} \,\,Q=q{\bigl \{}\mathrm {ctlh} {\bigl [}{\tfrac {1}{2}}\operatorname {aclh} (c){\bigr ]}^{2}{\bigr \}}\end{aligned}}} The mathematician Charles Hermite determined the value of the elliptic modulus k in relation to the coefficient of the absolute term of the Bring–Jerrard form. In his essay "Sur la résolution de l'Équation du cinquiéme degré Comptes rendus" he described the calculation method for the elliptic modulus in terms of the absolute term. The Italian version of his essay "Sulla risoluzione delle equazioni del quinto grado" contains exactly on page 258 the upper Bring–Jerrard equation formula, which can be solved directly with the functions based on the corresponding elliptic modulus. This corresponding elliptic modulus can be worked out by using the square of the Hyperbolic lemniscate cotangent. For the derivation of this, please see the Wikipedia article lemniscate elliptic functions! The elliptic nome of this corresponding modulus is represented here with the letter Q: Q = q { c t l h [ 1 2 aclh ⁡ ( c ) ] 2 } = {\displaystyle Q=q{\bigl \{}\mathrm {ctlh} {\bigl [}{\tfrac {1}{2}}\operatorname {aclh} (c){\bigr ]}^{2}{\bigr \}}=} = q [ ( c 4 + 1 + 1 + c ) ( 2 c 2 + 2 + 2 c 4 + 1 ) − 1 / 2 ] {\displaystyle =q{\bigl [}{\bigl (}{\sqrt {{\sqrt {c^{4}+1}}+1}}+c{\bigr )}{\bigl (}2c^{2}+2+2{\sqrt {c^{4}+1}}\,{\bigr )}^{-1/2}{\bigr ]}} The abbreviation ctlh expresses the Hyperbolic Lemniscate Cotangent and the abbreviation aclh represents the Hyperbolic Lemniscate Areacosine! === Calculation examples === Two examples of this solution algorithm are now mentioned: First calculation example: Second calculation example: == Applications in Physics == The Rogers–Ramanujan identities appeared in Baxter's solution of the hard hexagon model in statistical mechanics. The demodularized standard form of the Ramanujan's continued fraction unanchored from the modular form is as follows:: H ( q ) G ( q ) = [ 1 + q 1 + q 2 1 + q 3 1 + ⋯ ] {\displaystyle {\frac {H(q)}{G(q)}}=\left[1+{\frac {q}{1+{\frac {q^{2}}{1+{\frac {q^{3}}{1+\cdots }}}}}}\right]} == Relations to affine Lie algebras and vertex operator algebras == James Lepowsky and Robert Lee Wilson were the first to prove Rogers–Ramanujan identities using completely representation-theoretic techniques. They proved these identities using level 3 modules for the affine Lie algebra s l 2 ^ {\displaystyle {\widehat {{\mathfrak {sl}}_{2}}}} . In the course of this proof they invented and used what they called Z {\displaystyle Z} -algebras. Lepowsky and Wilson's approach is universal, in that it is able to treat all affine Lie algebras at all levels. It can be used to find (and prove) new partition identities. First such example is that of Capparelli's identities discovered by Stefano Capparelli using level 3 modules for the affine Lie algebra A 2 ( 2 ) {\displaystyle A_{2}^{(2)}} . == See also == Rogers polynomials Continuous q-Hermite polynomials == References == Rogers, L. J.; Ramanujan, Srinivasa (1919), "Proof of certain identities in combinatory analysis.", Cambr. Phil. Soc. Proc., 19: 211–216, Reprinted as Paper 26 in Ramanujan's collected papers Rogers, L. J. (1892), "On the expansion of some infinite products", Proc. London Math. Soc., 24 (1): 337–352, doi:10.1112/plms/s1-24.1.337, JFM 25.0432.01 Rogers, L. J. (1893), "Second Memoir on the Expansion of certain Infinite Products", Proc. London Math. Soc., 25 (1): 318–343, doi:10.1112/plms/s1-25.1.318 Rogers, L. J. (1894), "Third Memoir on the Expansion of certain Infinite Products", Proc. London Math. Soc., 26 (1): 15–32, doi:10.1112/plms/s1-26.1.15 Schur, Issai (1917), "Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbrüche", Sitzungsberichte der Berliner Akademie: 302–321 W.N. Bailey, Generalized Hypergeometric Series, (1935) Cambridge Tracts in Mathematics and Mathematical Physics, No. 32, Cambridge University Press, Cambridge. George Gasper and Mizan Rahman, Basic Hypergeometric Series, 2nd Edition, (2004), Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge. ISBN 0-521-83357-4. Bruce C. Berndt, Heng Huat Chan, Sen-Shan Huang, Soon-Yi Kang, Jaebum Sohn, Seung Hwan Son, The Rogers–Ramanujan Continued Fraction, J. Comput. Appl. Math. 105 (1999), pp. 9–24. Cilanne Boulet, Igor Pak, A Combinatorial Proof of the Rogers–Ramanujan and Schur Identities, Journal of Combinatorial Theory, Ser. A, vol. 113 (2006), 1019–1030. Slater, L. J. (1952), "Further identities of the Rogers–Ramanujan type", Proceedings of the London Mathematical Society, Series 2, 54 (2): 147–167, doi:10.1112/plms/s2-54.2.147, ISSN 0024-6115, MR 0049225 James Lepowsky and Robert L. Wilson, Construction of the affine Lie algebra A 1 ( 1 ) {\displaystyle A_{1}^{(1)}} , Comm. Math. Phys. 62 (1978) 43-53. James Lepowsky and Robert L. Wilson, A new family of algebras underlying the Rogers–Ramanujan identities, Proc. Natl. Acad. Sci. USA 78 (1981), 7254-7258. James Lepowsky and Robert L. Wilson, The structure of standard modules, I: Universal algebras and the Rogers–Ramanujan identities, Invent. Math. 77 (1984), 199-290. James Lepowsky and Robert L. Wilson, The structure of standard modules, II: The case A 1 ( 1 ) {\displaystyle A_{1}^{(1)}} , principal gradation, Invent. Math. 79 (1985), 417-442. Stefano Capparelli, Vertex operator relations for affine algebras and combinatorial identities, Thesis (Ph.D.)–Rutgers The State University of New Jersey - New Brunswick. 1988. 107 pp. == External links == Weisstein, Eric W. "Rogers–Ramanujan Identities". MathWorld. Weisstein, Eric W. "Rogers–Ramanujan Continued Fraction". MathWorld.
Wikipedia:Roland Glowinski#0	Roland Glowinski (9 March 1937 – 26 January 2022) was a French-American mathematician. He obtained his PhD in 1970 from Jacques-Louis Lions and was known for his work in applied mathematics, in particular numerical solution and applications of partial differential equations and variational inequalities. He was a member of the French Academy of Sciences and held an endowed chair at the University of Houston from 1985. Glowinski wrote many books on the subject of mathematics. In 2012, he became a fellow of the American Mathematical Society. == Selected publications == with Jacques-Louis Lions and Raymond Trémolières: Numerical Analysis of variational inequalities, North Holland 1981 2011 pbk edition Numerical methods for nonlinear variational problems, Springer Verlag 1984, 2008; 2013 pbk edition with Michel Fortin: Augmented Lagrangian methods : applications to the numerical solution of boundary-value problems, North Holland 1983 with Patrick Le Tallec: Augmented Lagrangian and operator-splitting methods in nonlinear mechanics, Society for Industrial and Applied Mathematics 1989 Glowinski, R. (2003). Ciarlet, P. G.; Lions, J. L. (eds.). Numerical analysis for fluids (Part 3). Finite element methods for incompressible viscous flows. Handbook of Numerical Analysis, Vol. IX. North-Holland. ISBN 9780444512246. with Jacques-Louis Lions and Jiwen He: Exact and approximate controllability for distributed parameter systems: a numerical approach, Cambridge University Press 2008 == References == == External links == homepage short biography
Wikipedia:Roland Richardson#0	Roland George Dwight Richardson (born May 14, 1878, Dartmouth, Nova Scotia; died July 17, 1949, Antigonish, Nova Scotia) was a prominent Canadian-American mathematician chiefly known for his work building the math department at Brown University and as Secretary of the American Mathematical Society. == Early life == Richardson was the son of George J. Richardson (1828–1898), a teacher, and Rebecca Newcomb Richardson (1837–1923). The family lived in several different towns in Nova Scotia during Richardson's youth. After completing high school, Richardson taught school in the small village of Margaretsville, Nova Scotia. In 1896 Richardson entered Acadia University; after graduating in 1898, he returned to his teaching job in Margaretsville. From 1899 to 1902 he was the principal of the high school in tiny Westport, Nova Scotia. There he met his future wife Louise MacHattie, whom he married in 1908. == Career in mathematics == In 1902 Richardson entered Yale University, earning an AB in 1903 and a Masters in 1904. He became an instructor at Yale in the Math department and began research under Professor James Pierpont. In 1906 Richardson was awarded a PhD by Yale for his thesis on "Improper Multiple Integrals". In 1907 he was appointed assistant professor of mathematics at Brown University, with the stipulation that he first spend a study year in Gottingen, Germany. By 1915 Richardson had become a full professor and the head of the mathematics department at Brown. In 1926 he was also given the position of Dean of the Graduate School at Brown. Under Roland's leadership Brown's graduate program was recognized when Brown was elected to the elite Association of American Universities in 1933. Richardson was the Secretary of the American Mathematical Society in 1921 and held the job until 1940. During his time, Raymond Clare Archibald wrote in his article on Richardson, "No American mathematician was more widely known among his colleagues and the careers of scores of them were notably promoted by his time-consuming activities in their behalf." He was credited with helping many European mathematicians concerned about conditions in Europe move to America during the 1930s. At the start of World War II Richardson organized accelerated applied mathematics courses at Brown for servicemen as the "Program of Advanced Instruction and Research in Applied Mechanics", recruiting German mathematician William Prager to lead it. This led to the founding of a new "Quarterly of Applied Mathematics" edited at Brown in 1943. After the war the program was converted into a new graduate division of applied mathematics. From 1943 to 1946 he was a member of the applied mathematics panel of the National Defense Research Committee. == Family and death == Richardson died while on a fishing trip to his native Nova Scotia and was buried in Camp Hill Cemetery in Halifax. Richardson and his wife had one child, George Webdell Richardson (b. July 7, 1920). == Recognition == Richardson received a number of honorary degrees. Acadia University awarded him a Doctor of Civil Law in 1931, Lehigh University gave him an LLD in 1941, and Brown University an LLD on his retirement in 1948. Richardson was elected a member of the American Academy of Arts and Sciences in 1914 and served as vice president 1945–9. == References ==
Wikipedia:Rolando Chuaqui#0	Rolando Basim Chuaqui Kettlun (December 30, 1935–April 23, 1994) was a Chilean mathematician who worked on the foundations of probabilities and foundations of mathematics. Throughout his lifetime, he published two books and over 50 journal articles in mathematics and logic. He also spearheaded the creation and expansion of mathematics departments across multiple Chilean universities. == Biography == Chuaqui was born into a Syrian immigrant family from Homs in Syria. He entered the University of Chile in 1953 to study medicine. He obtained a Ph.D. in Logic and the Methodology of Science, an interdisciplinary program between the Department of Mathematics and Department of Philosophy, from the University of California, Berkeley in 1965. His doctoral advisor was David Blackwell. Chuaqui returned to Chile after graduating, serving as a professor at the University of Chile and then the Pontifical Catholic University of Chile. During his time at the Pontifical Catholic, he advised three doctoral students. Chuaqui held several visiting positions, including at UCLA (1967), Princeton University (1970), University of São Paulo (1971 and 1982), University of California, Berkeley (1973–74), University of Campinas (1976, 1977 and 1978), Stanford University (1984), and San José State University (1986-89). He was a long-term collaborator of Patrick Suppes, with whom he worked on non-standard analysis and measurement in sciences. In 1986, he proposed a mathematical formulation for pragmatic truth. == Honors and awards == He was awarded a Guggenheim Fellowship in Mathematics in 1983. Since 1999, a series of annual research conferences in Chile, known as the Jornadas Rolando Chuaqui Kettlun, is held in his memory. The Pontifical Catholic University of Chile also has a building named after him, which houses its Department of Mathematics. == References ==
Wikipedia:Rolf Nevanlinna#0	Rolf Herman Nevanlinna (né Neovius; 22 October 1895 – 28 May 1980) was a Finnish mathematician who made significant contributions to complex analysis. == Background == Nevanlinna was born Rolf Herman Neovius, becoming Nevanlinna in 1906 when his father changed the family name. The Neovius-Nevanlinna family contained many mathematicians: Edvard Engelbert Neovius (Rolf's grandfather) taught mathematics and topography at a military academy; Edvard Rudolf Neovius (Rolf's uncle) was a professor of mathematics at the University of Helsinki from 1883 to 1900; Lars Theodor Neovius-Nevanlinna (Rolf's uncle) was an author of mathematical textbooks; and Otto Wilhelm Neovius-Nevanlinna (Rolf's father) was a physicist, astronomer and mathematician. After Otto obtained his Ph.D. in physics from the University of Helsinki, he studied at the Pulkovo Observatory with the German astronomer Herman Romberg, whose daughter, Margarete Henriette Louise Romberg, he married in 1892. Otto and Margarete then settled in Joensuu, where Otto taught physics, and there their four children were born: Frithiof (born 1894; also a mathematician), Rolf (born 1895), Anna (born 1896) and Erik (born 1901). == Education == Nevanlinna began his formal education at the age of 7. Having already been taught to read and write by his parents, he went straight into the second grade but still found the work boring and soon refused to attend the school. He was then homeschooled before being sent to a grammar school in 1903 when the family moved to Helsinki, where his father took up a new post as a teacher at Helsinki High School. At the new school, Nevanlinna studied French and German in addition to the languages he already spoke: Finnish and Swedish. He also attended an orchestra school and had a love of music, which was encouraged by his mother: Margarete was an excellent pianist and Frithiof and Rolf would lie under the piano and listen to her playing. At 13 they went to orchestra school and became accomplished musicians – Frithiof on the cello and Rolf on the violin. Through free tickets from the orchestra school they got to know and love the music of the great composers, Bach, Beethoven, Brahms, Schubert, Schumann, Chopin and Liszt, as well as the early symphonies of Sibelius. Rolf first met Sibelius's music in 1907, when he heard his Third Symphony. Although later he met Hilbert, Einstein, Thomas Mann and other famous people, Rolf said that none had had such a strong effect on him as Sibelius. The boys played trios with their mother and their love of music – in particular of chamber music – lasted all their lives. Nevanlinna then progressed onto the Helsinki High School, where his main interests were classics and mathematics. He was taught by a number of teachers during this time but the best of them all was his own father, who taught him physics and mathematics. He graduated in 1913 having performed very well, although he was not the top student of his year. He then went beyond the school syllabus in the summer of 1913 when he read Ernst Leonard Lindelöf's Introduction to Higher Analysis; from that time on, Nevanlinna had an enthusiastic interest in mathematical analysis. (Lindelöf was also a cousin of Nevanlinna's father, and so a part of the Neovius-Nevanlinna mathematical family.) Nevanlinna began his studies at the University of Helsinki in 1913, and received his Master of Philosophy in mathematics in 1917. Lindelöf taught at the university and Nevanlinna was further influenced by him. During his time at the University of Helsinki, World War I was underway and Nevanlinna wanted to join the 27th Jäger Battalion, but his parents convinced him to continue with his studies. He did however join the White Guard in the Finnish Civil War, but did not see active military action. In 1919, Nevanlinna presented his thesis, entitled Über beschränkte Funktionen die in gegebenen Punkten vorgeschriebene Werte annehmen ("On limited functions prescribed values at given points"), to Lindelöf, his doctoral advisor. The thesis, which was on complex analysis, was of high quality and Nevanlinna was awarded his Doctor of Philosophy on 2 June 1919. == Career == When Nevanlinna earned his doctorate in 1919, there were no university posts available so he became a school teacher. His brother, Frithiof, had received his doctorate in 1918 but likewise was unable to take up a post at a university, and instead began working as a mathematician for an insurance company. Frithiof recruited Rolf to the company, and Nevanlinna worked for the company and as a school teacher until he was appointed a Docent of Mathematics at the University of Helsinki in 1922. During this time, he had been contacted by Edmund Landau and requested to move to Germany to work at the University of Göttingen, but did not accept. After his appointment as Docent of Mathematics, he gave up his insurance job but did not resign his position as school teacher until he received a newly created full professorship at the university in 1926. Despite this heavy workload, it was between the years of 1922–25 that he developed what would become to be known as Nevanlinna theory. From 1947 Nevanlinna had a chair in the University of Zurich, which he held on a half-time basis after receiving in 1948 a permanent position as one of the 12 salaried Academicians in the newly created Academy of Finland. Rolf Nevanlinna's most important mathematical achievement is the value distribution theory of meromorphic functions. The roots of the theory go back to the result of Émile Picard in 1879, showing that a non-constant complex-valued function which is analytic in the entire complex plane assumes all complex values save at most one. In the early 1920s Rolf Nevanlinna, partly in collaboration with his brother Frithiof, extended the theory to cover meromorphic functions, i.e. functions analytic in the plane except for isolated points in which the Laurent series of the function has a finite number of terms with a negative power of the variable. Nevanlinna's value distribution theory or Nevanlinna theory is crystallised in its two Main Theorems. Qualitatively, the first one states that if a value is assumed less frequently than average, then the function comes close to that value more often than average. The Second Main Theorem, more difficult than the first one, states roughly that there are relatively few values which the function assumes less often than average. Rolf Nevanlinna's article Zur Theorie der meromorphen Funktionen which contains the Main Theorems was published in 1925 in the journal Acta Mathematica. Hermann Weyl has called it "one of the few great mathematical events of the [twentieth] century." Nevanlinna gave a fuller account of the theory in the monographs Le théoreme de Picard – Borel et la théorie des fonctions méromorphes (1929) and Eindeutige analytische Funktionen (1936). Nevanlinna theory touches also on a class of functions called the Nevanlinna class, or functions of "bounded type". When the Winter War broke out (1939), Nevanlinna was invited to join the Finnish Army's Ballistics Office to assist in improving artillery firing tables. These tables had been based on a calculation technique developed by General Vilho Petter Nenonen, but Nevanlinna now came up with a new method which made them considerably faster to compile. In recognition of his work he was awarded the Order of the Cross of Liberty, Second Class, and throughout his life he held this honour in especial esteem. Among Rolf Nevanlinna's later interests in mathematics were the theory of Riemann surfaces (the monograph Uniformisierung in 1953) and functional analysis (Absolute analysis in 1959, written in collaboration with his brother Frithiof). Nevanlinna also published in Finnish a book on the foundations of geometry and a semipopular account of the Theory of Relativity. His Finnish textbook on the elements of complex analysis, Funktioteoria (1963), written together with Veikko Paatero, has appeared in German, English and Russian translations. Rolf Nevanlinna supervised at least 28 doctoral theses. His first and most famous doctoral student was Lars Ahlfors, one of the first two Fields Medal recipients. The research for which Ahlfors was awarded the prize (proving the Denjoy Conjecture, now known as the Denjoy–Carleman–Ahlfors theorem) was strongly based on Nevanlinna's work. Nevanlinna's work was recognised in the form of honorary degrees which he held from the universities of Heidelberg, the University of Bucharest, the University of Giessen, the Free University of Berlin, the University of Glasgow, the University of Uppsala, the University of Istanbul and the University of Jyväskylä. He was an honorary member of several learned societies, among them the London Mathematical Society and the Hungarian Academy of Sciences. — The 1679 Nevanlinna main belt asteroid is named after him. == Administrative activities == From 1954, Rolf Nevanlinna chaired the committee which set about the first computer project in Finland. Rolf Nevanlinna served as President of the International Mathematical Union (IMU) from 1959 to 1963 and as President of the International Congress of Mathematicians (ICM) in 1962. In 1964, Nevanlinna's connections with President Urho Kekkonen were instrumental in bringing about a total reorganization of the Academy of Finland. From 1965 to 1970 Nevanlinna was Chancellor of the University of Turku. == Political activities == Although Nevanlinna did not participate actively in politics, he was known to sympathise with the right-wing Patriotic People's Movement and, partly because of his half-German parentage, was also sympathetic towards Nazi Germany; with many mathematics professors fired in the 1930s due to the Nuremberg Laws, mathematicians sympathetic to the Nazi policies were sought as replacements, and Nevanlinna accepted a position as professor at the University of Göttingen in 1936 and 1937. His sympathy towards the Nazis led to his removal from his position as Rector of the University of Helsinki after Finland made peace with the Soviet Union in 1944. In the spring of 1941, Finland contributed a Volunteer Battalion to the Waffen-SS. In 1942, a committee was established for the Volunteer Battalion to take care of the battalion's somewhat strained relations with its German commanders, and Nevanlinna was chosen to be the chairman of the committee, as he was a person respected in Germany but loyal to Finland. He stated in his autobiography that he accepted this role due to a "sense of duty". Nevanlinna's collaboration with Nazi Germany did not prevent mathematical contacts with Allied countries; after World War II, the Soviet mathematical community was isolated from the Western mathematical community and the International Colloquium on Function Theory in Helsinki in 1957, directed by Nevanlinna, was one of the first post-war occasions when Soviet mathematicians could contact their Western colleagues in person. In 1965, Nevanlinna was an honorary guest at a function theory congress in Soviet Armenia. == IMU Abacus Medal (formerly Nevanlinna Prize) == When the IMU in 1981 decided to create a prize, similar to the Fields Medal, in theoretical computer science and the funding for the prize was secured from Finland, the Union decided to give Nevanlinna's name to the prize; the Rolf Nevanlinna Prize is awarded every four years at the ICM. In 2018, the General Assembly of the IMU approved a resolution to remove Nevanlinna's name from the prize. Starting in 2022 the prize has been called the IMU Abacus Medal. == See also == Harmonic measure Nevanlinna theory Nevanlinna class (functions of bounded type) Nevanlinna function Nevanlinna invariant Nevanlinna–Pick interpolation Nevanlinna's criterion Nevanlinna Prize == References == == Sources == Lehto, Olli (2008). Erhabene Welten: Das Leben Rolf Nevanlinnas [High Worlds: The life of Rolf Nevanlinna] (in German). Translated by Manfred Stern. Birkhäuser. ISBN 978-3-7643-7701-4. == External links == Media related to Rolf Nevanlinna at Wikimedia Commons Rolf Nevanlinna at the Mathematics Genealogy Project Nevanlinna, Rolf. National Biography of Finland.
Wikipedia:Roman abacus#0	The Ancient Romans developed the Roman hand abacus, a portable, but less capable, base-10 version of earlier abacuses like those that were used by the Greeks and Babylonians. == Origin == The Roman abacus was the first portable calculating device for engineers, merchants, and presumably tax collectors. It greatly reduced the time needed to perform the basic operations of arithmetic using Roman numerals. Karl Menninger said: For more extensive and complicated calculations, such as those involved in Roman land surveys, there was, in addition to the hand abacus, a true reckoning board with unattached counters or pebbles. The Etruscan cameo and the Greek predecessors, such as the Salamis Tablet and the Darius Vase, give us a good idea of what it must have been like, although no actual specimens of the true Roman counting board are known to be extant. But language, the most reliable and conservative guardian of a past culture, has come to our rescue once more. Above all, it has preserved the fact of the unattached counters so faithfully that we can discern this more clearly than if we possessed an actual counting board. What the Greeks called psephoi, the Romans called calculi. The Latin word calx means 'pebble' or 'gravel stone'; calculi are thus little stones (used as counters). Both the Roman abacus and the Chinese suanpan have been used since ancient times. With one bead above and four below the bar, the systematic configuration of the Roman abacus is comparable to the modern Japanese soroban, although the soroban was historically derived from the suanpan. == Layout == The Late Roman hand abacus shown here as a reconstruction contains seven longer and seven shorter grooves used for whole number counting, the former having up to four beads in each, and the latter having just one. The rightmost two grooves were for fractional counting. The abacus was made of a metal plate where the beads ran in slots. The size was such that it could fit in a modern shirt pocket. \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|X\| CCC\|ƆƆƆ CC\|ƆƆ C\|Ɔ C X I Ө \| \| --- --- --- --- --- --- --- --- S \|O\| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| Ɔ \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \| \| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| 2 \|O\| \|O\| \|O\| The lower groove marked I indicates units, X tens, and so on up to millions. The beads in the upper shorter grooves denote fives (five units, five tens, etc.), resembling a bi-quinary coded decimal place value system. Computations are made by means of beads which it is believed would have been slid up and down the grooves to indicate the value of each column. The upper slots contained a single bead, while the lower slots contained four beads, the only exceptions being the two rightmost columns, column 2 marked Ө and column 1 with three symbols down the side of a single slot or beside three separate slots with Ɛ, 3 or S or a symbol like the £ sign but without the horizontal bar beside the top slot, a backwards C beside the middle slot and a 2 symbol beside the bottom slot, depending on the example abacus and the source which could be Friedlein, Menninger or Ifrah. These latter two slots are for mixed-base math, a development unique to the Roman hand abacus described in following sections. The longer slot with five beads below the Ө position allowed for the counting of 1⁄12 of a whole unit called an uncia (from which the English words inch and ounce are derived), making the abacus useful for Roman measures and Roman currency. The first column was either a single slot with 4 beads or 3 slots with one, one and two beads respectively top to bottom. In either case, three symbols were included beside the single slot version or one symbol per slot for the three slot version. Many measures were aggregated by twelfths. Thus the Roman pound ('libra'), consisted of 12 ounces (unciae) (1 uncia = 28 grams). A measure of volume, congius, consisted of 12 heminae (1 hemina = 0.273 litres). The Roman foot (pes), was 12 inches (300 mm) (unciae) (1 uncia = 2.43 cm). The actus, the standard furrow length when plowing, was 120 pedes. There were however other measures in common use - for example the sextarius was two heminae. The as, the principal copper coin in Roman currency, was also divided into 12 unciae. Again, the abacus was ideally suited for counting currency. == Symbols and usage == The first column was arranged either as a single slot with three different symbols or as three separate slots with one, one and two beads or counters respectively and a distinct symbol for each slot. It is most likely that the rightmost slot or slots were used to enumerate fractions of an uncia and these were, from top to bottom, 1⁄2 s, 1⁄4 s and 1⁄12 s of an uncia. The upper character in this slot (or the top slot where the rightmost column is three separate slots) is the character most closely resembling that used to denote a semuncia or 1/24. The name semuncia denotes 1⁄2 of an uncia or 1⁄24 of the base unit, the As. Likewise, the next character is that used to indicate a sicilicus or 1⁄48 of an As, which is 1⁄4 of an uncia. These two characters are to be found in the table of Roman fractions on page 75 of Graham Flegg's book. Finally, the last or lower character is most similar but not identical to the character in Flegg's table to denote 1⁄144 of an As, the dimidio sextula, which is the same as 1⁄12 of an uncia. This is however even more strongly supported by Gottfried Friedlein in the table at the end of the book which summarizes the use of a very extensive set of alternative formats for different values including that of fractions. In the entry in this table numbered 14 referring back to (Zu) 48, he lists different symbols for the semuncia (1/24), the sicilicus (1/48), the sextula (1/72), the dimidia sextula (1/144), and the scriptulum (1/288). Of prime importance, he specifically notes the formats of the semuncia, sicilicus and sextula as used on the Roman bronze abacus, "auf dem chernan abacus". The semuncia is the symbol resembling a capital "S", but he also includes the symbol that resembles a numeral three with horizontal line at the top, the whole rotated 180 degrees. It is these two symbols that appear on samples of abacus in different museums. The symbol for the sicilicus is that found on the abacus and resembles a large right single quotation mark spanning the entire line height. The most important symbol is that for the sextula, which resembles very closely a cursive digit 2. Now, as stated by Friedlein, this symbol indicates the value of 1/72 of an As. However, he stated specifically in the penultimate sentence of section 32 on page 23, the two beads in the bottom slot each have a value of 1/72. This would allow this slot to represent only 1/72 (i.e. 1/6 × 1/12 with one bead) or 1/36 (i.e. 2/6 × 1/12 = 1/3 × 1/12 with two beads) of an uncia respectively. This contradicts all existing documents that state this lower slot was used to count thirds of an uncia (i.e. 1/3 and 2/3 × 1/12 of an As. This results in two opposing interpretations of this slot, that of Friedlein and that of many other experts such as Ifrah, and Menninger who propose the one and two thirds usage. There is however a third possibility. If this symbol refers to the total value of the slot (i.e. 1⁄72 of an as), then each of the two counters can only have a value of half this or 1⁄144 of an as or 1⁄12 of an uncia. This then suggests that these two counters did in fact count twelfths of an uncia and not thirds of an uncia. Likewise, for the top and upper middle, the symbols for the semuncia and sicilicus could also indicate the value of the slot itself and since there is only one bead in each, would be the value of the bead also. This would allow the symbols for all three of these slots to represent the slot value without involving any contradictions. A further argument which suggests the lower slot represents twelfths rather than thirds of an uncia is best described by the figure above. The diagram above assumes for ease that one is using fractions of an uncia as a unit value equal to one. If the beads in the lower slot of column I represent thirds, then the beads in the three slots for fractions of 1⁄12 of an uncia cannot show all values from 1⁄12 of an uncia to 11⁄12 of an uncia. In particular, it would not be possible to represent 1⁄12, 2⁄12 and 5⁄12. Furthermore, this arrangement would allow for seemingly unnecessary values of 13⁄12, 14⁄12 and 17⁄12. Even more significant, it is logically impossible for there to be a rational progression of arrangements of the beads in step with unit increasing values of twelfths. Likewise, if each of the beads in the lower slot is assumed to have a value of 1⁄6 of an uncia, there is again an irregular series of values available to the user, no possible value of 1⁄12 and an extraneous value of 13⁄12. It is only by employing a value of 1⁄12 for each of the beads in the lower slot that all values of twelfths from 1⁄12 to 11⁄12 can be represented and in a logical ternary, binary, binary progression for the slots from bottom to top. This can be best appreciated by reference to the figure above. Alternative usages of the beads in the lower slot It can be argued that the beads in this first column could have been used as originally believed and widely stated, i.e. as 1⁄2, 1⁄4 and 1⁄3 and 2⁄3, completely independently of each other. However this is more difficult to support in the case where this first column is a single slot with the three inscribed symbols. To complete the known possibilities, in one example found by this author, the first and second columns were transposed. It would not be unremarkable if the makers of these instruments produced output with minor differences, since the vast number of variations in modern calculators provide a compelling example. What can be deduced from these Roman abacuses, is the undeniable proof that Romans were using a device that exhibited a decimal, place-value system, and the inferred knowledge of a zero value as represented by a column with no beads in a counted position. Furthermore, the biquinary-like nature of the integer portion allowed for direct transcription from and to the written Roman numerals. No matter what the true usage was, what cannot be denied by the very format of the abacus is that if not yet proven, these instruments provide very strong arguments in favour of far greater facility with practical mathematics known and practised by the Romans in this authors view. The reconstruction of a Roman hand abacus in the Cabinet, supports this. The replica Roman hand abacus at, shown alone here, plus the description of a Roman abacus on page 23 of Die Zahlzeichen und das elementare Rechnen der Griechen und Römer und des christlichen provides further evidence of such devices. == References == == Further reading == Stephenson, Stephen K. (July 7, 2010), Ancient Computers, IEEE Global History Network, retrieved 2011-07-02
Wikipedia:Romanesco broccoli#0	Romanesco broccoli (also known as broccolo romanesco, romanesque cauliflower, or simply romanesco) is in fact a cultivar of the cauliflower (Brassica oleracea var. botrytis), not broccoli (Brassica oleracea var. italica). It is one of two types of broccoflower. It is an edible flower bud of the species Brassica oleracea, which also includes regular broccoli and cauliflower. It is chartreuse in color and has a striking form that naturally approximates a fractal. Romanesco has a nutty flavor and a firmer texture than white cauliflower or broccoli when cooked. == Description == Romanesco superficially resembles a cauliflower, but it is chartreuse in color, with the form of a natural fractal. Nutritionally, romanesco is rich in vitamin C, vitamin K, dietary fiber, and carotenoids. === Fractal structure === The inflorescence (the bud) is self-similar in character, with the branched meristems making up a logarithmic spiral, giving a form approximating a natural fractal; each bud is composed of a series of smaller buds, all arranged in yet another logarithmic spiral. This self-similar pattern continues at smaller levels. The pattern is only an approximate fractal since the pattern eventually terminates when the feature size becomes sufficiently small. The number of spirals on the head of Romanesco broccoli is a Fibonacci number. The causes of its differences in appearance from the normal cauliflower and broccoli have been modeled as an extension of the preinfloresence stage of bud growth. A 2021 paper has ascribed this phenomenon to perturbations of floral gene networks that causes the development of meristems into flowers to fail, but instead to repeat itself in a self-similar way. == See also == Phyllotaxis == References == == External links == Media related to Romanesco broccoli at Wikimedia Commons The dictionary definition of Romanesco at Wiktionary Malatesta, M.; Davey, J.C. (1996). "Cultivar Identification Within Broccoli, Brassica Oleracea L. Var. Italica Plenck And Cauliflower, Brassica Oleracea Var. Botrytis L.". Acta Hortic. 407 (407): 109–114. doi:10.17660/ActaHortic.1996.407.12. Fractal Food: Self-Similarity on the Supermarket Shelf (John Walker, March 2005) Procedural fractal 3-D generation of Romanesco broccoli with RenderMan (Aleksandar Rodić, 2009)
Wikipedia:Romano Scozzafava#0	Romano Scozzafava (born November 12, 1935) is an Italian mathematician known for his contributions to subjective probability along the lines of Bruno de Finetti, based on the concept of coherence. He taught Probability Calculus at the Engineering Faculty of the Sapienza University of Rome from 1979 to his retirement (at the end of 2009). Scozzafava has conducted significant research on Bayesian inference, statistical physics, artificial intelligence, and fuzzy set theory in terms of coherent conditional probability. He has written six books and over 200 papers on these subjects. Throughout his career, he actively participated in politics as a supporter of the Italian Radical Party and of “Associazione Luca Coscioni” for Freedom of Scientific Research. == Education and early career == Scozzafava graduated in Mathematics in 1961 at the Sapienza University of Rome. He was given a fellowship at Istituto Superiore Poste Telecomunicazioni and then one at CNEN (Comitato Nazionale Energia Nucleare) in 1962. For the next five years, he conducted research at CNEN and during this time, he wrote several articles on the application of Mathematics in Physics. In 1967, he received his academic teaching habilitation ("libera docenza") in Mathematical Methods in Physics, which was confirmed in 1973. == Later career == In 1967, Scozzafava began teaching at the University of Perugia. He taught there until 1969, when he left to join University of Florence as assistant professor of Mathematical Analysis. At this time, the focus of his research began shifting towards Algebra, and mainly towards Statistics and Probability. Scozzafava joined University of Lecce as full professor in 1976. After teaching at University of Lecce for three years, he left to join Sapienza University of Rome in 1979. Over two decades of career, he received several research grants from Ministry of Education and Research and National Council of Research to conduct research and write papers in the field of Bayesian Statistics, Probability and Artificial Intelligence. While teaching at Sapienza, he taught also at the Universities of Ancona, L'Aquila and Perugia. In 1994 and 1997 he served as the director of the International School of Mathematics G. Stampacchia of the Ettore Majorana Foundation and Centre for Scientific Culture in Erice, Sicily. From 2001 to 2009 he organized the international school ReasonPark (Reasoning under Partial Knowledge). He has been a visiting professor in University of Edinburgh, Eindhoven University of Technology, Karl Marx University of Budapest, Somali National University, University of Warwick (UK), University of North Carolina, Chapel Hill (USA), Virginia Polytechnic Institute and State University (USA), University of Canterbury (New Zealand) and University of Economics, Prague. He has been the editor of Rendiconti di matematica, Pure Mathematics and Applications, Induzioni and Cognitive Processing. He has been Elected Member of the International Statistical Institute and Coordinator of the Dottorato di Ricerca (Ph.D.) in Modelli e Metodi Matematici per la Tecnologia e la Società, University La Sapienza, Roma. He has been Guest Editor of special issues of Soft Computing in 1999 and of Annals of Mathematics and Artificial Intelligence in 2002. A special issue of the International Journal of Approximate Reasoning was dedicated to Scozzafava to celebrate his 70th birthday. An international workshop was also organized on the same occasion in his honour. Due to his involvement in politics, he has also written some papers on the connections among mathematics, politics, elections and scientific reasoning. == Selected bibliography == === Papers === Bayesian Inference and Inductive Logic. Scientia. (1980) Subjective Probability versus Belief Functions in Artificial Intelligence. International Journal of General Systems. (1994) Probabilistic Background for the Management of Uncertainty in Artificial Intelligence. European Journal of Engineering Education. (1995) Characterization of Coherent Conditional Probabilities as a Tool for Their Assessment and Extension. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems. (1996) The Role of Probability in Statistical Physics. Transport Theory and Statistical Physics. (2000) The Role of Coherence in Eliciting and Handling Imprecise Probabilities and Its Application to Medical Diagnosis. Information Sciences. (2000) Partial Algebraic Conditional Spaces. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems. (2004) Conditional Probability and Fuzzy Information. Computational Statistics & Data Analysis. (2006) Nonconglomerative Coherent Conditional Probabilities in Statistical Inference. Statistical Methods & Applications. (2007) Fuzzy Inclusion and Similarity through Coherent Conditional Probability. Fuzzy Sets and Systems. (2009) Inferential Processes Leading to Possibility and Necessity. Information Sciences. (2013) == Books and books chapters == La Probabilità Soggettiva e Le Sue Applicazioni, MASSON, Milano (1989) Matematica di Base, MASSON, Milano (1992). Calcolo delle Probabilità in Matematica per Docenti Scuole Secondarie Superiori, CUD-MPI (1992) Mathematical Models for Handling Partial Knowledge in Artificial Intelligence, Plenum Press, New York (1995) Primi Passi in Probabilità e Statistica. Teoria ed Esercizi, Zanichelli, Bologna (1996) Incertezza e Probabilità, Zanichelli, Bologna (2001) Probabilistic Logic in a Coherent Setting, Springer (2002) Vaghezza e Verosimiglianza in Statistica e Demografia, Un Ricordo di Enzo Lombardo Tra Scienza e Cultura, TIPAR (2007) Possibility Measures in Probabilistic Inference in Soft Methodology for Handling Variability and Imprecision, Springer (2008) The Membership of a Fuzzy Set as Coherent Conditional Probability in On Fuzziness: A Homage to Lofti A. Zadeh, Springer (2013) == References == == External links == Romano Scozzafava's personal home page
Wikipedia:Ron Aharoni#0	Ron Aharoni (Hebrew: רון אהרוני; born 1952) is an Israeli mathematician, working in finite and infinite combinatorics. Aharoni is a professor at the Technion – Israel Institute of Technology, where he received his Ph.D. in mathematics in 1979. With Nash-Williams and Shelah he generalized Hall's marriage theorem by obtaining the right transfinite conditions for infinite bipartite graphs. He subsequently proved the appropriate versions of the Kőnig theorem and the Menger theorem for infinite graphs (the latter with Eli Berger). Aharoni is the author of several nonspecialist books; the most successful is Arithmetic for Parents, a book helping parents and elementary school teachers in teaching basic mathematics. He also wrote a book on the connections between Mathematics, poetry and beauty and on philosophy, The Cat That is not There. His book, "Man detaches meaning", is on a mechanism common to jokes and poetry. His last to date book is Circularity: A Common Secret to Paradoxes, Scientific Revolutions and Humor, which binds together mathematics, philosophy and the secrets of humor. == Books == 1. Arithmetic for Parents, A book for grownups on children's mathematics, Schocken Press 2004 2. Mathematics, poetry and beauty (in Hebrew), Hakibutz Hameuchad 2008. 3. The cat that is not there - a non-philosophical book on philosophy, Magness Press (The Hebrew University Publishing House), 2009. 4. Man detaches meaning - poems, jokes and in between, Hakibutz Hameuchad 2011. 5. Mathematics, Poetry and Beauty (in English), World Scientific Publishing 2014. 6. Arithmetic for Parents (Revised Edition), World Scientific Publishing 2015 7. Circularity: A Common Secret to Paradoxes, Scientific Revolutions and Humor, World Scientific Publishing 2016. == References == == External links == Ron Aharoni's home page on Elementary school mathematics Vicious circles -- confusing, instructive, amusing? Ron Aharoni: What I learnt in elementary school, Address at the British Mathematical Colloquium, Birmingham, 2003 Ron Aharoni: The Cat That is Not There, Magnes Press, December 2009. Ron Aharoni: The cat that is not there, a summary [1]
Wikipedia:Ron Goldman (mathematician)#0	Ronald Neil Goldman is a Professor of Computer Science at Rice University in Houston, Texas. Professor Goldman received his B.S. in Mathematics from the Massachusetts Institute of Technology in 1968 and his M.A. and Ph.D. in Mathematics from Johns Hopkins University in 1973. Goldman's current research interests lie in the mathematical representation, manipulation, and analysis of shape using computers. His work includes research in computer-aided geometric design, solid modeling, computer graphics, and splines. He is particularly interested in algorithms for polynomial and piecewise polynomial curves and surfaces, and he is currently investigating applications of algebraic and differential geometry to geometric modeling. He has published over a hundred articles in journals, books, and conference proceedings on these and related topics. Before returning to academia, Goldman worked for 10 years in industry solving problems in computer graphics, geometric modeling, and computer aided design. He served as a mathematician at Manufacturing Data Systems Inc., where he helped to implement one of the first industrial solid modeling systems. Later he worked as a senior design engineer at Ford Motor Company, enhancing the capabilities of their corporate graphics and computer-aided design software. From Ford he moved on to Control Data Corporation, where he was a principal consultant for the development group devoted to computer-aided design and manufacture. His responsibilities included database design, algorithms, education, acquisitions, and research. Goldman left Control Data Corporation in 1987 to become an associate professor of computer science at the University of Waterloo in Ontario, Canada. He joined the faculty at Rice University in Houston, Texas as a professor of computer science in July 1990. == Selected publications == Goldman, Ron (2002). Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling. Morgan Kaufmann. ISBN 9781558603547. Goldman, Ron (2009). An Integrated Introduction to Computer Graphics and Geometric Modeling. CRC Press. ISBN 9781439803349. == References ==
Wikipedia:Ronald Coifman#0	Ronald Raphael Coifman (Hebrew: רונלד קויפמן; born June 29, 1941) is a Sterling professor of Mathematics at Yale University. Coifman earned a doctorate from the University of Geneva in 1965, supervised by Jovan Karamata. Coifman is a member of the American Academy of Arts and Sciences, the Connecticut Academy of Science and Engineering, and the National Academy of Sciences. He is a recipient of the 1996 DARPA Sustained Excellence Award, the 1996 Connecticut Science Medal, the 1999 Pioneer Award of the International Society for Industrial and Applied Science, and the 1999 National Medal of Science. Prior to teaching at Yale, Coifman taught at Washington University in St. Louis and the University of Chicago. In 2013, he co-founded ThetaRay, a cyber security and big data analytics company. In 2018, he received the Rolf Schock Prize for Mathematics. In 2024 he was awarded the George David Birkhoff Prize. == References == == External links == Scientific Data Has Become So Complex, We Have to Invent New Math to Deal With It, Wired
Wikipedia:Ronald Does#0	Ronaldus Joannes Michael Maria "Ronald" Does (born 13 January 1955 in Haarlem) is a Dutch mathematician, known for several contributions to statistics and Lean Six Sigma. His research interests include control charts, Lean Six Sigma, and the integration of industrial statistics in services and healthcare. Since 1991 he has been holding a full-professorship in Industrial Statistics at the University of Amsterdam, first at its Korteweg-de Vries Institute for Mathematics, and since April 2009 at the department of Operations Management. In 2007 he has been appointed as fellow of the American Society for Quality and in 2014 as fellow of the American Statistical Association. He is the managing director and founder of the Institute for Business and Industrial Statistics of the University of Amsterdam. Since 2011 he has also been director of the Executive Programmes of the University of Amsterdam. == Books == "Lean Six Sigma Stap voor Stap" (2008, in Dutch) "Lean Six Sigma for Services and Healthcare" (2012) == References == == External links == Homepage (Dutch)
Wikipedia:Ronald Venetiaan#0	Ronald Runaldo Venetiaan (born 18 June 1936) is a former politician who served as the sixth president of Suriname. == Biography == Venetiaan was born in Paramaribo. In 1955, Venetiaan left Suriname to study mathematics and physics at the University of Leiden. In 1964, he obtained his doctorandus, and returned to Suriname to become a mathematics and physics teacher. In 1973 Venetiaan was Minister of Education for the National Party of Suriname (NPS) in the government of Henck Arron. He was disposed by the 1980 Surinamese coup d'état. Venetiaan decided to teach at the Anton de Kom University. In 1987, Venetiaan returned to politics as the Chairman of the National Party of Suriname, and as the Minister of Education. His first term as president ran from 1991 to 1996, after which he lost in the elections to Jules Wijdenbosch. In 2000 however, he regained his former position on the New Front banner, receiving an absolute majority of 37 from 51 votes in the Parliament. In 2005 he was re-elected to serve a third term as president and sworn in on August 12, 2005. Venetiaan relinquished the Chair of the NPS to Gregory Rusland in 2012, and retired from politics in 2013. Venetiaan thought that it was time that the younger generation take over. == Personal life == Venetiaan is a mathematician beside all political activity. His surname means Venetian (a person from Venice) in Dutch. Venetiaan published his first poetry under the pseudonym Vene in Mamio (1962). Most of his work was never published but was performed in theatre plays. Venetiaan had also used the pseudonym Krumanty. Venetiaan is a collaborator on Chan Santokhi's We gaan Suriname redden (We are going to save Suriname) of 2020. Ronald Venetiaan is married to Liesbeth Vanenburg, and has three daughters and one son. == Honours == Suriname: Grand Cordon of the Honorary Order of the Yellow Star - 2020 Grand Cordon of the Honorary Order of the Palm - 1991 Commander of the Honorary Order of the Yellow Star - 1978 Brazil: Collar of the Order of the Southern Cross - 1995 Netherlands: Knight Grand Cross of the Order of Orange-Nassau - 1978 Venezuela: Collar of the Order of the Liberator - 1993 == References == == External links == Poetry of Venetiaan at Digital Library for Dutch Literature (in Dutch and Sranan Tongo)
Wikipedia:Ronen Eldan#0	Ronen Eldan (Hebrew: רונן אלדן) is an Israeli mathematician, working at OpenAI. Previously, Eldan was a professor at the Weizmann Institute of Science working on probability theory, mathematical analysis, theoretical computer science and the theory of machine learning. He received the 2018 Erdős Prize, the 2022 Blavatnik Award for Young Scientists and the 2023 New Horizons Breakthrough Prize in Mathematics. He was a speaker at the 2022 International Congress of Mathematicians. == Selected works == Eldan, Ronen (19 February 2011). "A Polynomial Number of Random Points Does Not Determine the Volume of a Convex Body". Discrete & Computational Geometry. 46 (1). Springer Science and Business Media LLC: 29–47. arXiv:0903.2634. doi:10.1007/s00454-011-9328-x. ISSN 0179-5376. S2CID 16096886. Eldan, Ronen (22 March 2013). "Thin Shell Implies Spectral Gap Up to Polylog via a Stochastic Localization Scheme". Geometric and Functional Analysis. 23 (2). Springer Science and Business Media LLC: 532–569. arXiv:1203.0893. doi:10.1007/s00039-013-0214-y. ISSN 1016-443X. S2CID 253637768. Eldan, Ronen (30 October 2014). "A two-sided estimate for the Gaussian noise stability deficit". Inventiones Mathematicae. 201 (2). Springer Science and Business Media LLC: 561–624. arXiv:1307.2781. doi:10.1007/s00222-014-0556-6. ISSN 0020-9910. S2CID 253737938. Sébastien Bubeck, Ronen Eldan: “Multi-scale exploration of convex functions and bandit convex optimization”, 2015; arXiv:1507.06580. Sébastien Bubeck, Ronen Eldan, Yin Tat Lee: “Kernel-based methods for bandit convex optimization”, 2016; arXiv:1607.03084. Eldan, Ronen; Lee, James R. (1 April 2018). "Regularization under diffusion and anticoncentration of the information content". Duke Mathematical Journal. 167 (5). Duke University Press. arXiv:1410.3887. doi:10.1215/00127094-2017-0048. ISSN 0012-7094. S2CID 119657905. == Awards == Haim Nessyahu Prize for Mathematics (2013) Erdős Prize in Mathematics (2018) Blavatnik Award for Young Scientists (2022) New Horizons Breakthrough Prize in Mathematics (2023) == References ==
Wikipedia:Ronny Hadani#0	Ronny Hadani (Hebrew: רוני הדני) is an Israeli-American mathematician, specializing in representation theory and harmonic analysis, with applications to signal processing. He is known for developing Orthogonal Time Frequency and Space (OTFS) modulating techniques, a method used for making wireless 5G communications faster, that is also being considered for use in 6G technology. The technology is being used by several wireless 5G related companies and Cohere Technologies, a company he has co-founded. == Early life and education == Hadani received his MS degree in Computer Science from the Weizmann Institute of Science in 1999 under the supervision of David Harel. He received his Ph.D. degree in Pure Mathematics from Tel-Aviv University in 2006 under the supervision of Joseph Bernstein. == Career == === Academia === From the years 2006 to 2009, Hadani held an L.E. Dickson Postdoctoral Fellowship of Mathematics at the University of Chicago. Since 2009, he has been an associate professor of mathematics at the University of Texas, Austin. === Cohere Technologies === In 2010, Hadani co-founded Cohere Technologies with Shlomo Rakib. The company is a Silicon Valley wireless startup that focuses on wireless improvements using OTFS and the Delay-Doppler model to improve FDD/TDD spectrum performance with channel detection, estimation, prediction, and precoding software for 4G and 5G networks, compliant with 3GPP standards and O-RAN. In December 2022, during the inaugural 6G Evolution Summit event opening keynote, Fierce Wireless moderator referred to Hadani as “The Father of OTFS.” == Patents == Hadani has been granted over 70 OTFS related patents, which include a communications method employing Orthogonal Time Frequency Space (OTFS) shifting and spectral shaping, which allows users to transmit and receive at least one frame of data ([D]) over a wireless communications link. He has also patented methods of operating and implementing wireless OTFS communications systems. His OTFS technology has been tested by companies such as C Spire, 5TONIC, Telefónica, and Deutsche Telekom, == Selected publications == === Papers in journals === Hadani, Ronny; Gurevich, Shamgar (July 2014). "The categorical weil representation". Journal of Symplectic Geometry. arXiv:1108.0351. Fish, Alexander; Gurevich, Shamgar; Hadani, Ronny; Sayeed, Akbar M.; Schwartz, Oded (November 2013). "Delay-doppler channel estimation in almost linear complexity". IEEE Transactions on Information Theory. 59 (11): 7632–7644. arXiv:1208.4405. doi:10.1109/TIT.2013.2273931. S2CID 230975. Hadani, Ronny; Gurevich, Shamgar (2012). "The Weil representation in characteristic two" (PDF). Advances in Mathematics. 230 (3): 894–926. doi:10.1016/j.aim.2012.03.008. S2CID 17843124. Hadani, Ronny; Gurevich, Shamgar (2008). "The geometric Weil representation". Advances in Mathematics. 13 (3). arXiv:math/0610818. Bibcode:2006math.....10818G. Gurevich, Shamgar; Hadani, Ronny; Sochen, Nir (August 2008). "The Finite Harmonic Oscillator and Its Applications to Sequences, Communication, and Radar". IEEE Transactions on Information Theory. 54 (9): 4239–4253. arXiv:0808.1495. doi:10.1109/TIT.2008.926440. S2CID 6037080. == Academic works == According to Google Scholar, Hadani has published over 75 research papers and patents. His works have been cited over 3300 times. === Patents === OTFS methods of data channel characterization and uses thereof, R Hadani, SS Rakib, US Patent 9,668,148, Cited 188 Times. Signal modulation method resistant to echo reflections and frequency offsets, SS Rakib, R Hadani. US Patent 9,083,595, Cited 183 Times. Modulation and equalization in an orthonormal time-frequency shifting communications system, R Hadani, SS Rakib, US Patent 9,590,779, Cited 182 times. Communications method employing orthonormal time-frequency shifting and spectral shaping, R Hadani, SS Rakib, US Patent 8,547,988, Cited 96 Times. === Academic papers === Orthogonal time frequency space modulation, R Hadani, S Rakib, M Tsatsanis, A Monk, AJ Goldsmith, AF Molisch, 2017 IEEE Wireless Communications and Networking Conference (WCNC), 1-6, Cited 228 Times. Viewing angle classification of cryo-electron microscopy images using eigenvectors, A Singer, Z Zhao, Y Shkolnisky, R Hadani, SIAM Journal on Imaging Sciences 4 (2), 723-759, Cited 92 Times. == References == == External links == Cohere Technologies Ronny Hadani on LinkedIn Communication Theory & Systems: Ronny Hadani Representation Theoretic Patterns in 3D Cryo-Electron Microscopy - Prof. Ronny Hadani
Wikipedia:Rosa Donat#0	Rosa María Donat Beneito (born 1960) is a Spanish applied mathematician whose research involves numerical methods for partial differential equations, particularly multiresolution methods for problems modeling fluid dynamics with shock waves or with high Mach number. She is a professor of applied mathematics and vice rector for innovation and transfer at the University of Valencia, and former president of the Spanish Society of Applied Mathematics. == Education and career == Donat was born in 1960 in La Font de la Figuera. After earning a degree in mathematal sciences from the University of Valencia in 1984, she traveled to the University of California, Los Angeles in 1985 as a Fulbright Scholar, earning a master's degree there in 1987 and a Ph.D. in mathematics in 1990. Her doctoral dissertation, Studies on Error Propagation Into Regions of Smoothness for Certain Nonlinear Approximations to Hyperbolic Equations, was supervised by Stanley Osher. She has held a tenured position at the University of Valencia since 1993, and was given a professorial chair there in 2008. She was elected as president of the Spanish Society of Applied Mathematics in 2016, becoming the society's first woman president and holding office from 2016 to 2020. == References ==
Wikipedia:Rosa María Farfán#0	Rosa María Farfán Márquez is a Mexican researcher in social epistemology and mathematics education, affiliated with CINVESTAV in the Instituto Politécnico Nacional. == Education and career == Farfán has been a researcher for CINVESTAV since 1985. She completed a doctorate through CINVESTAV in 1993, with the dissertation Construcción de la noción de convergencia en ámbitos fenomenológicos vinculados a la ingeniería: Estudio de caso, jointly supervised by Carlos Ímaz Janhke and Fernando Hitt. She was a postdoctoral researcher at Paris Diderot University before returning to CINVESTAV. She became the founding editor in chief of the journal Revista Latinoamericana de Investigación en Matemática Educativa (Relime) in 1997, remaining editor until 2007. == Recognition == Farfán was elected to the Mexican Academy of Sciences in 2001. == References == == External links == Rosa María Farfán publications indexed by Google Scholar
Wikipedia:Rosamund Sutherland#0	Rosamund Sutherland (née Hatfield, 1947–2019) was a British mathematics educator. She was a professor emeritus at the University of Bristol, and the former head of the school of education at Bristol. == Education and career == Sutherland was born in Birmingham; her mother taught geography and her father was a physicist. The family moved to south Wales when she was young, and after attending Haberdashers' Monmouth School for Girls she became a student at the University of Bristol, where she met and married her husband, mechanical and biomedical engineer Ian Sutherland. She worked briefly as a computer programmer, and then as a researcher at the University of Bristol while her husband completed his doctorate. After she and her family moved to Hertfordshire, she taught for The Open University and the Borehamwood College of Further Education. Through her position at The Open University she came to work with Celia Hoyles, who encouraged her to become an academic researcher in a project combining mathematics education with computer programming in the Logo programming language. She worked at the University of London from 1983 until 1995, when she was given a Chair in Education at the University of Bristol. At Bristol, in 1997, she chaired a national committee that helped bring algebra to a more prominent position in secondary-school mathematics education. She was the head of the school from 2003 to 2006, and again in 2014. She also played a key role in improving educational opportunities for underprivileged youth in south Bristol. She died on 26 January 2019. == Books == Sutherland was the author or coauthor of several books or booklets on mathematics education, including: Logo Mathematics in the Classroom (Routledge / Chapman & Hall, 1989) Exploring Mathematics with Spreadsheets (with Lulu Healy, Blackwell, 1992) Key Aspects of Teaching Algebra in Schools (with John Mason, QCA, 2002) A Comparative Study of Algebra Curricula (QCA, 2002) Screenplay: Children and Computing in the Home (with Keri Facer, John Furlong, and Ruth Furlong, RoutledgeFalmer, 2003) Teaching for Learning Mathematics (Open University Press / McGraw Hill, 2007) Improving Classroom Learning With ICT (with Susan Robertson and Peter John, Routledge, 2009) Education and Social Justice in a Digital Age (Bristol University Press, 2014) She also edited books including: Theory of Didactical Situations in Mathematics (Didactique des Mathématiques, 1970–1990) (by Guy Brousseau, edited and translated by Balacheff, Cooper, Sutherland, and Warfield, Kluwer, 1997) Learning and Teaching Where Worldviews Meet (edited with Guy Claxton and Andrew Pollard, Trentham, 2004) == References == == External links == Rosamund Sutherland publications indexed by Google Scholar
Wikipedia:Rosella Kanarik#0	Rosella Kanarik (1909–2014) was an American mathematics educator and was one of the first women to receive a Ph.D. in math from the University of Pittsburgh. She was one of the few women to earn a doctorate in math before World War II. == Biography == Rosella Kanarik was born February 7, 1909 in Bartfa, then part of Hungary, as the oldest of two children. Her parents were Sarah Schondorf and Albert Kanarik, both of Bartfa. Her father immigrated to the United States ahead of his family and then he returned to Hungary to marry Sarah in November 1907. In July 1912, her father returned to the U.S., followed by Rosella and her mother in September 1913. Her younger brother Edgar was born in Pittsburgh in 1926. She attended public high schools, first at Wadleigh High School in New York City followed by Fifth Avenue High School in Pittsburgh, where she graduated with highest honors in 1926. She went on to earn three degrees from the University of Pittsburgh (BA 1930, MA 1931, PhD 1934). Her doctoral dissertation on Fundamental regions in S4 for the Hessian group was supervised by Montgomery Culver. Jobs were scarce when she graduated during The Great Depression so Rosella took a teaching position at a Pittsburgh high school (1932–1936). In 1936, she got married and by 1937, the couple were living in California where their first child was born. Rosella Kanarik taught in high schools in Los Angeles from 1939 to 1946 when she was hired as a lecturer at the University of Southern California where there was a great need for math instructors because of World War II. She held that position until 1952 and taught many levels of math courses while she worked there. In 1953, she became the first female to be named a member of the math department at Los Angeles City College and spent the remainder of her career there. She retired only when she reached the mandated age of 65. While she lived, the mathematics department of the Los Angeles City College awarded an annual merit-based grant called the Rosella Kanarik scholarship, which was funded through a "generous donation" by Kanarik. In her retirement, Kanarik volunteered as a tutor for numerous high school and college students without cost to them. Her organizational affiliations included AMS, MAA, NCTM and the Sigma Xi honor society. She was also a member of the UCLA Affiliates, the Los Angeles Women's Architectural League, the Brandeis University National Women's Committee and Pioneer Women. == Personal life == On July 25, 1936, Rosella married her cousin Emery Kanarik, who was born in 1909 in Bardejov Spa, Czechoslovakia and attended the College of the City of New York, (now part of City University of New York) 1926–1928. They had two children, both born in Los Angeles, Richard (born 1937), and Susan Carol (born 1940). Emery Kanarik died in 1992. Rosella died at home in Los Angeles on April 19, 2014 at the age of 105. == References ==
Wikipedia:Rosemary Renaut#0	Rosemary Anne Renaut is a British and American computational mathematician whose research interests include inverse problems and regularization with applications to medical imaging and seismic analysis. She is a professor in the School of Mathematical and Statistical Sciences at Arizona State University. == Education and career == Renaut earned a bachelor's degree in 1980 at Durham University and then studied for Part III of the Mathematical Tripos in applied mathematics at the University of Cambridge. She completed her Ph.D. at Cambridge in 1985. Her dissertation, Numerical Solution of Hyperbolic Partial Differential Equations, was supervised by Arieh Iserles. After postdoctoral research at RWTH Aachen University in Germany and the Chr. Michelsen Institute in Norway, she joined the Arizona State University faculty as an assistant professor in 1987. She was promoted to associate professor in 1991 and full professor in 1996, and chaired the Department of Mathematics from 1997 to 2001. She has also visited multiple other institutions, including a term as John von Neumann Professor at the Technical University of Munich in 2001–2002, and terms as program director for computational mathematics and mathematical biology at the National Science Foundation from 2008 to 2011 and 2014 to 2017. == Recognition == Renaut has been a Fellow of the Institute of Mathematics and its Applications since 1996. She was elected as a Fellow of the Society for Industrial and Applied Mathematics, in the 2022 Class of SIAM Fellows, "for contributions to ill-posed inverse problems and regularization, geophysical and medical imaging, and high order numerical methods". == References == == External links == Home page Rosemary Renaut publications indexed by Google Scholar
Wikipedia:Rosenbrock function#0	In mathematical optimization, the Rosenbrock function is a non-convex function, introduced by Howard H. Rosenbrock in 1960, which is used as a performance test problem for optimization algorithms. It is also known as Rosenbrock's valley or Rosenbrock's banana function. The global minimum is inside a long, narrow, parabolic-shaped flat valley. To find the valley is trivial. To converge to the global minimum, however, is difficult. The function is defined by f ( x , y ) = ( a − x ) 2 + b ( y − x 2 ) 2 {\displaystyle f(x,y)=(a-x)^{2}+b(y-x^{2})^{2}} It has a global minimum at ( x , y ) = ( a , a 2 ) {\displaystyle (x,y)=(a,a^{2})} , where f ( x , y ) = 0 {\displaystyle f(x,y)=0} . Usually, these parameters are set such that a = 1 {\displaystyle a=1} and b = 100 {\displaystyle b=100} . Only in the trivial case where a = 0 {\displaystyle a=0} the function is symmetric and the minimum is at the origin. == Multidimensional generalizations == Two variants are commonly encountered. One is the sum of N / 2 {\displaystyle N/2} uncoupled 2D Rosenbrock problems, and is defined only for even N {\displaystyle N} s: f ( x ) = f ( x 1 , x 2 , … , x N ) = ∑ i = 1 N / 2 [ 100 ( x 2 i − 1 2 − x 2 i ) 2 + ( x 2 i − 1 − 1 ) 2 ] . {\displaystyle f(\mathbf {x} )=f(x_{1},x_{2},\dots ,x_{N})=\sum _{i=1}^{N/2}\left[100(x_{2i-1}^{2}-x_{2i})^{2}+(x_{2i-1}-1)^{2}\right].} This variant has predictably simple solutions. A second, more involved variant is f ( x ) = ∑ i = 1 N − 1 [ 100 ( x i + 1 − x i 2 ) 2 + ( 1 − x i ) 2 ] where x = ( x 1 , … , x N ) ∈ R N . {\displaystyle f(\mathbf {x} )=\sum _{i=1}^{N-1}[100(x_{i+1}-x_{i}^{2})^{2}+(1-x_{i})^{2}]\quad {\mbox{where}}\quad \mathbf {x} =(x_{1},\ldots ,x_{N})\in \mathbb {R} ^{N}.} has exactly one minimum for N = 3 {\displaystyle N=3} (at ( 1 , 1 , 1 ) {\displaystyle (1,1,1)} ) and exactly two minima for 4 ≤ N ≤ 7 {\displaystyle 4\leq N\leq 7} —the global minimum at ( 1 , 1 , . . . , 1 ) {\displaystyle (1,1,...,1)} and a local minimum near x ^ = ( − 1 , 1 , … , 1 ) {\displaystyle {\hat {\mathbf {x} }}=(-1,1,\dots ,1)} . This result is obtained by setting the gradient of the function equal to zero, noticing that the resulting equation is a rational function of x {\displaystyle x} . For small N {\displaystyle N} the polynomials can be determined exactly and Sturm's theorem can be used to determine the number of real roots, while the roots can be bounded in the region of \| x i \| < 2.4 {\displaystyle \|x_{i}\|<2.4} . For larger N {\displaystyle N} this method breaks down due to the size of the coefficients involved. == Stationary points == Many of the stationary points of the function exhibit a regular pattern when plotted. This structure can be exploited to locate them. == Optimization examples == The Rosenbrock function can be efficiently optimized by adapting appropriate coordinate system without using any gradient information and without building local approximation models (in contrast to many derivate-free optimizers). The following figure illustrates an example of 2-dimensional Rosenbrock function optimization by adaptive coordinate descent from starting point x 0 = ( − 3 , − 4 ) {\displaystyle x_{0}=(-3,-4)} . The solution with the function value 10 − 10 {\displaystyle 10^{-10}} can be found after 325 function evaluations. Using the Nelder–Mead method from starting point x 0 = ( − 1 , 1 ) {\displaystyle x_{0}=(-1,1)} with a regular initial simplex a minimum is found with function value 1.36 ⋅ 10 − 10 {\displaystyle 1.36\cdot 10^{-10}} after 185 function evaluations. The figure below visualizes the evolution of the algorithm. == See also == Test functions for optimization == References == == External links == Rosenbrock function plot in 3D Weisstein, Eric W. "Rosenbrock Function". MathWorld.
Wikipedia:Ross Honsberger#0	Ross Honsberger (1929–2016) was a Canadian mathematician and author on recreational mathematics. == Life == Honsberger studied mathematics at the University of Toronto, with a bachelor's degree, and then worked for ten years as a teacher in Toronto, before continuing his studies at the University of Waterloo (master's degree). Since 1964 he had been on the faculty of mathematics, where he later became a professor emeritus. He dealt with combinatorics and optimization, especially with mathematics education. He developed education courses, for example, on combinatorial geometry, frequently held lectures for students and math teachers, and was editor of the Ontario Secondary School Mathematics Bulletin. He wrote numerous books on elementary mathematics (geometry, number theory, combinatorics, probability theory), and recreational mathematics (often at the Mathematical Association of America, MAA), with him in his own words using the book by Hans Rademacher and Otto Toeplitz of numbers and figures as a model. Frequent were his expositions of problems at the International Mathematical Olympiads and other competitions. Edsger W. Dijkstra called his Mathematical Gems "delightful". == Books == Ingenuity in Mathematics, New Mathematical Library, Random House / Singer 1970 Mathematical Gems, MAA 1973, 2003 (Mathematical Expositions Dolciani Vol.1), German Mathematical gems of elementary combinatorics, number theory and geometry, Wiley, 1990, ISBN 3-528-08474-X, Chapter "The Story of Louis Posa". Mathematical Gems 2, MAA 1975 (Vol.2 Dolciani Mathematical Expositions) Mathematical Gems 3, MAA 1985, 1991 (vol.9 Dolciani Mathematical Expositions) Mathematical Morsels, MAA 1978 (Vol.3 Dolciani Mathematical Expositions) More Mathematical Morsels, MAA 1991 (Dolciani Bd.10 Mathematical Expositions) Mathematical Plums, MAA 1979 (vol.4 Dolciani Mathematical Expositions) Mathematical Chestnuts from around the world, MAA 2001 (Dolciani Bd.24 Mathematical Expositions) Mathematical Diamonds, MAA 2003 In Pólya's Footsteps, MAA 1997 (Dolciani Bd.19 Mathematical Expositions) Episodes in nineteenth and twentieth century euclidean geometry, MAA 1995 From Erdos to Kiev – Problems of Olympiad Caliber, MAA 1997 Mathematical Delights, MAA 2004 (Dolciani Mathematics Expositions Bd.28) == References == == External links == Ross Honsberger at a website of the University of Waterloo
Wikipedia:Ross Street#0	Ross Howard Street (born 29 September 1945, Sydney) is an Australian mathematician specialising in category theory. == Biography == Street completed his undergraduate and postgraduate study at the University of Sydney, where his dissertation advisor was Max Kelly. He is an emeritus professor of mathematics at Macquarie University, a fellow of the Australian Mathematical Society (1995), and was elected Fellow of the Australian Academy of Science in 1989. He was awarded the Edgeworth David Medal of the Royal Society of New South Wales in 1977, and the Australian Mathematical Society's George Szekeres Medal in 2012. == References == == External links == Personal webpage, maths.mq.edu.au Ross Street at the Mathematics Genealogy Project Ross Street publications indexed by Google Scholar
Wikipedia:Rostislav Grigorchuk#0	Rostislav Ivanovich Grigorchuk (Russian: Ростислав Иванович Григорчук; Ukrainian: Ростисла́в Iва́нович Григорчу́к; b. February 23, 1953) is a mathematician working in different areas of mathematics including group theory, dynamical systems, geometry and computer science. He holds the rank of Distinguished Professor in the Mathematics Department of Texas A&M University. Grigorchuk is particularly well known for having constructed, in a 1984 paper, the first example of a finitely generated group of intermediate growth, thus answering an important problem posed by John Milnor in 1968. This group is now known as the Grigorchuk group and it is one of the important objects studied in geometric group theory, particularly in the study of branch groups, automaton groups and iterated monodromy groups. Grigorchuk is one of the pioneers of asymptotic group theory as well as of the theory of dynamically defined groups. He introduced the notion of branch groups and developed the foundations of the related theory. Grigorchuk, together with his collaborators and students, initiated the theory of groups generated by finite Mealy type automata, interpreted them as groups of fractal type, developed the theory of groups acting on rooted trees, and found numerous applications of these groups in various fields of mathematics including functional analysis, topology, spectral graph theory, dynamical systems and ergodic theory. == Biographical data == Grigorchuk was born on February 23, 1953, in Ternopil Oblast, now Ukraine (in 1953 part of the USSR). He received his undergraduate degree in 1975 from Moscow State University. He obtained a PhD (Candidate of Science) in Mathematics in 1978, also from Moscow State University, where his thesis advisor was Anatoly M. Stepin. Grigorchuk received a habilitation (Doctor of Science) degree in Mathematics in 1985 at the Steklov Institute of Mathematics in Moscow. During the 1980s and 1990s, Rostislav Grigorchuk held positions at the Moscow State University of Transportation, and subsequently at the Steklov Institute of Mathematics and Moscow State University. In 2002 Grigorchuk joined the faculty of Texas A&M University as a Professor of Mathematics, and he was promoted to the rank of Distinguished Professor in 2008. Rostislav Grigorchuk gave an invited address at the 1990 International Congress of Mathematicians in Kyoto an AMS Invited Address at the March 2004 meeting of the American Mathematical Society in Athens, Ohio and a plenary talk at the 2004 Winter Meeting of the Canadian Mathematical Society. Grigorchuk is the Editor-in-Chief of the journal "Groups, Geometry and Dynamics", published by the European Mathematical Society, and is or was a member of the editorial boards of the journals "Mathematical Notes", "International Journal of Algebra and Computation", "Journal of Modern Dynamics", "Geometriae Dedicata", "Ukrainian Mathematical Journal", "Algebra and Discrete Mathematics", "Carpathian Mathematical Publications", "Bukovinian Mathematical Journal", and "Matematychni Studii". == Mathematical contributions == Grigorchuk is most well known for having constructed the first example of a finitely generated group of intermediate growth which now bears his name and is called the Grigorchuk group (sometimes it is also called the first Grigorchuk group since Grigorchuk constructed several other groups that are also commonly studied). This group has growth that is faster than polynomial but slower than exponential. Grigorchuk constructed this group in a 1980 paper and proved that it has intermediate growth in a 1984 article. This result answered a long-standing open problem posed by John Milnor in 1968 about the existence of finitely generated groups of intermediate growth. Grigorchuk's group has a number of other remarkable mathematical properties. It is a finitely generated infinite residually finite 2-group (that is, every element of the group has a finite order which is a power of 2). It is also the first example of a finitely generated group that is amenable but not elementary amenable, thus providing an answer to another long-standing problem, posed by Mahlon Day in 1957. Also Grigorchuk's group is "just infinite": that is, it is infinite but every proper quotient of this group is finite. Grigorchuk's group is a central object in the study of the so-called branch groups and automata groups. These are finitely generated groups of automorphisms of rooted trees that are given by particularly nice recursive descriptions and that have remarkable self-similar properties. The study of branch, automata and self-similar groups has been particularly active in the 1990s and 2000s and a number of unexpected connections with other areas of mathematics have been discovered there, including dynamical systems, differential geometry, Galois theory, ergodic theory, random walks, fractals, Hecke algebras, bounded cohomology, functional analysis, and others. In particular, many of these self-similar groups arise as iterated monodromy groups of complex polynomials. Important connections have been discovered between the algebraic structure of self-similar groups and the dynamical properties of the polynomials in question, including encoding their Julia sets. Much of Grigorchuk's work in the 1990s and 2000s has been on developing the theory of branch, automata and self-similar groups and on exploring these connections. For example, Grigorchuk, with co-authors, obtained a counter-example to the conjecture of Michael Atiyah about L2-betti numbers of closed manifolds. Grigorchuk is also known for his contributions to the general theory of random walks on groups and the theory of amenable groups, particularly for obtaining in 1980 what is commonly known (see for example) as Grigorchuk's co-growth criterion of amenability for finitely generated groups. == Awards and honors == In 1979 Rostislav Grigorchuk was awarded the Moscow Mathematical Society. In 1991 he obtained Fulbright Senior Scholarship, Columbia University, New York. In 2003 an international group theory conference in honor of Grigorchuk's 50th birthday was held in Gaeta, Italy. Special anniversary issues of the "International Journal of Algebra and Computation", the journal "Algebra and Discrete Mathematics" and the book "Infinite Groups: Geometric, Combinatorial and Dynamical Aspects" were dedicated to Grigorchuk's 50th birthday. In 2009 Grigorchuk R.I. was awarded the Association of Former Students Distinguished Achievement in Research, Texas A&M University. In 2012 he became a fellow of the American Mathematical Society. In 2015 Rostislav Grigorchuk was awarded the AMS Leroy P. Steele Prize for Seminal Contribution to Research. In 2020 Grigorchuk R.I. received the Humboldt Research Award by Germany’s Alexander von Humboldt Foundation. == See also == Geometric group theory Growth of groups Iterated monodromy group Amenable groups Grigorchuk group == References == == External links == Web-page of Rostislav Grigorchuk at Texas A&M University Groups and Dynamics at Texas A&M University
Wikipedia:Rot operator#0	In vector calculus, the curl, also known as rotor, is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation. The curl of a field is formally defined as the circulation density at each point of the field. A vector field whose curl is zero is called irrotational. The curl is a form of differentiation for vector fields. The corresponding form of the fundamental theorem of calculus is Stokes' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve. The notation curl F is more common in North America. In the rest of the world, particularly in 20th century scientific literature, the alternative notation rot F is traditionally used, which comes from the "rate of rotation" that it represents. To avoid confusion, modern authors tend to use the cross product notation with the del (nabla) operator, as in ∇ × F {\displaystyle \nabla \times \mathbf {F} } , which also reveals the relation between curl (rotor), divergence, and gradient operators. Unlike the gradient and divergence, curl as formulated in vector calculus does not generalize simply to other dimensions; some generalizations are possible, but only in three dimensions is the geometrically defined curl of a vector field again a vector field. This deficiency is a direct consequence of the limitations of vector calculus; on the other hand, when expressed as an antisymmetric tensor field via the wedge operator of geometric calculus, the curl generalizes to all dimensions. The circumstance is similar to that attending the 3-dimensional cross product, and indeed the connection is reflected in the notation ∇ × {\displaystyle \nabla \times } for the curl. The name "curl" was first suggested by James Clerk Maxwell in 1871 but the concept was apparently first used in the construction of an optical field theory by James MacCullagh in 1839. == Definition == The curl of a vector field F, denoted by curl F, or ∇ × F {\displaystyle \nabla \times \mathbf {F} } , or rot F, is an operator that maps Ck functions in R3 to Ck−1 functions in R3, and in particular, it maps continuously differentiable functions R3 → R3 to continuous functions R3 → R3. It can be defined in several ways, to be mentioned below: One way to define the curl of a vector field at a point is implicitly through its components along various axes passing through the point: if u ^ {\displaystyle \mathbf {\hat {u}} } is any unit vector, the component of the curl of F along the direction u ^ {\displaystyle \mathbf {\hat {u}} } may be defined to be the limiting value of a closed line integral in a plane perpendicular to u ^ {\displaystyle \mathbf {\hat {u}} } divided by the area enclosed, as the path of integration is contracted indefinitely around the point. More specifically, the curl is defined at a point p as ( ∇ × F ) ( p ) ⋅ u ^ = d e f lim A → 0 1 \| A \| ∮ C ( p ) F ⋅ d r {\displaystyle (\nabla \times \mathbf {F} )(p)\cdot \mathbf {\hat {u}} \ {\overset {\underset {\mathrm {def} }{}}{{}={}}}\lim _{A\to 0}{\frac {1}{\|A\|}}\oint _{C(p)}\mathbf {F} \cdot \mathrm {d} \mathbf {r} } where the line integral is calculated along the boundary C of the area A containing point p, \|A\| being the magnitude of the area. This equation defines the component of the curl of F along the direction u ^ {\displaystyle \mathbf {\hat {u}} } . The infinitesimal surfaces bounded by C have u ^ {\displaystyle \mathbf {\hat {u}} } as their normal. C is oriented via the right-hand rule. The above formula means that the component of the curl of a vector field along a certain axis is the infinitesimal area density of the circulation of the field in a plane perpendicular to that axis. This formula does not a priori define a legitimate vector field, for the individual circulation densities with respect to various axes a priori need not relate to each other in the same way as the components of a vector do; that they do indeed relate to each other in this precise manner must be proven separately. To this definition fits naturally the Kelvin–Stokes theorem, as a global formula corresponding to the definition. It equates the surface integral of the curl of a vector field to the above line integral taken around the boundary of the surface. Another way one can define the curl vector of a function F at a point is explicitly as the limiting value of a vector-valued surface integral around a shell enclosing p divided by the volume enclosed, as the shell is contracted indefinitely around p. More specifically, the curl may be defined by the vector formula ( ∇ × F ) ( p ) = d e f lim V → 0 1 \| V \| ∮ S n ^ × F d S {\displaystyle (\nabla \times \mathbf {F} )(p){\overset {\underset {\mathrm {def} }{}}{{}={}}}\lim _{V\to 0}{\frac {1}{\|V\|}}\oint _{S}\mathbf {\hat {n}} \times \mathbf {F} \ \mathrm {d} S} where the surface integral is calculated along the boundary S of the volume V, \|V\| being the magnitude of the volume, and n ^ {\displaystyle \mathbf {\hat {n}} } pointing outward from the surface S perpendicularly at every point in S. In this formula, the cross product in the integrand measures the tangential component of F at each point on the surface S, and points along the surface at right angles to the tangential projection of F. Integrating this cross product over the whole surface results in a vector whose magnitude measures the overall circulation of F around S, and whose direction is at right angles to this circulation. The above formula says that the curl of a vector field at a point is the infinitesimal volume density of this "circulation vector" around the point. To this definition fits naturally another global formula (similar to the Kelvin-Stokes theorem) which equates the volume integral of the curl of a vector field to the above surface integral taken over the boundary of the volume. Whereas the above two definitions of the curl are coordinate free, there is another "easy to memorize" definition of the curl in curvilinear orthogonal coordinates, e.g. in Cartesian coordinates, spherical, cylindrical, or even elliptical or parabolic coordinates: ( curl ⁡ F ) 1 = 1 h 2 h 3 ( ∂ ( h 3 F 3 ) ∂ u 2 − ∂ ( h 2 F 2 ) ∂ u 3 ) , ( curl ⁡ F ) 2 = 1 h 3 h 1 ( ∂ ( h 1 F 1 ) ∂ u 3 − ∂ ( h 3 F 3 ) ∂ u 1 ) , ( curl ⁡ F ) 3 = 1 h 1 h 2 ( ∂ ( h 2 F 2 ) ∂ u 1 − ∂ ( h 1 F 1 ) ∂ u 2 ) . {\displaystyle {\begin{aligned}&(\operatorname {curl} \mathbf {F} )_{1}={\frac {1}{h_{2}h_{3}}}\left({\frac {\partial (h_{3}F_{3})}{\partial u_{2}}}-{\frac {\partial (h_{2}F_{2})}{\partial u_{3}}}\right),\\[5pt]&(\operatorname {curl} \mathbf {F} )_{2}={\frac {1}{h_{3}h_{1}}}\left({\frac {\partial (h_{1}F_{1})}{\partial u_{3}}}-{\frac {\partial (h_{3}F_{3})}{\partial u_{1}}}\right),\\[5pt]&(\operatorname {curl} \mathbf {F} )_{3}={\frac {1}{h_{1}h_{2}}}\left({\frac {\partial (h_{2}F_{2})}{\partial u_{1}}}-{\frac {\partial (h_{1}F_{1})}{\partial u_{2}}}\right).\end{aligned}}} The equation for each component (curl F)k can be obtained by exchanging each occurrence of a subscript 1, 2, 3 in cyclic permutation: 1 → 2, 2 → 3, and 3 → 1 (where the subscripts represent the relevant indices). If (x1, x2, x3) are the Cartesian coordinates and (u1, u2, u3) are the orthogonal coordinates, then h i = ( ∂ x 1 ∂ u i ) 2 + ( ∂ x 2 ∂ u i ) 2 + ( ∂ x 3 ∂ u i ) 2 {\displaystyle h_{i}={\sqrt {\left({\frac {\partial x_{1}}{\partial u_{i}}}\right)^{2}+\left({\frac {\partial x_{2}}{\partial u_{i}}}\right)^{2}+\left({\frac {\partial x_{3}}{\partial u_{i}}}\right)^{2}}}} is the length of the coordinate vector corresponding to ui. The remaining two components of curl result from cyclic permutation of indices: 3,1,2 → 1,2,3 → 2,3,1. == Usage == In practice, the two coordinate-free definitions described above are rarely used because in virtually all cases, the curl operator can be applied using some set of curvilinear coordinates, for which simpler representations have been derived. The notation ∇ × F {\displaystyle \nabla \times \mathbf {F} } has its origins in the similarities to the 3-dimensional cross product, and it is useful as a mnemonic in Cartesian coordinates if ∇ {\displaystyle \nabla } is taken as a vector differential operator del. Such notation involving operators is common in physics and algebra. Expanded in 3-dimensional Cartesian coordinates (see Del in cylindrical and spherical coordinates for spherical and cylindrical coordinate representations), ∇ × F {\displaystyle \nabla \times \mathbf {F} } is, for F {\displaystyle \mathbf {F} } composed of [ F x , F y , F z ] {\displaystyle [F_{x},F_{y},F_{z}]} (where the subscripts indicate the components of the vector, not partial derivatives): ∇ × F = \| ı ^ ȷ ^ k ^ ∂ ∂ x ∂ ∂ y ∂ ∂ z F x F y F z \| {\displaystyle \nabla \times \mathbf {F} ={\begin{vmatrix}{\boldsymbol {\hat {\imath }}}&{\boldsymbol {\hat {\jmath }}}&{\boldsymbol {\hat {k}}}\\[5mu]{\dfrac {\partial }{\partial x}}&{\dfrac {\partial }{\partial y}}&{\dfrac {\partial }{\partial z}}\\[5mu]F_{x}&F_{y}&F_{z}\end{vmatrix}}} where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively. This expands as follows: ∇ × F = ( ∂ F z ∂ y − ∂ F y ∂ z ) ı ^ + ( ∂ F x ∂ z − ∂ F z ∂ x ) ȷ ^ + ( ∂ F y ∂ x − ∂ F x ∂ y ) k ^ {\displaystyle \nabla \times \mathbf {F} =\left({\frac {\partial F_{z}}{\partial y}}-{\frac {\partial F_{y}}{\partial z}}\right){\boldsymbol {\hat {\imath }}}+\left({\frac {\partial F_{x}}{\partial z}}-{\frac {\partial F_{z}}{\partial x}}\right){\boldsymbol {\hat {\jmath }}}+\left({\frac {\partial F_{y}}{\partial x}}-{\frac {\partial F_{x}}{\partial y}}\right){\boldsymbol {\hat {k}}}} Although expressed in terms of coordinates, the result is invariant under proper rotations of the coordinate axes but the result inverts under reflection. In a general coordinate system, the curl is given by ( ∇ × F ) k = 1 g ε k ℓ m ∇ ℓ F m {\displaystyle (\nabla \times \mathbf {F} )^{k}={\frac {1}{\sqrt {g}}}\varepsilon ^{k\ell m}\nabla _{\ell }F_{m}} where ε denotes the Levi-Civita tensor, ∇ the covariant derivative, g {\displaystyle g} is the determinant of the metric tensor and the Einstein summation convention implies that repeated indices are summed over. Due to the symmetry of the Christoffel symbols participating in the covariant derivative, this expression reduces to the partial derivative: ( ∇ × F ) = 1 g R k ε k ℓ m ∂ ℓ F m {\displaystyle (\nabla \times \mathbf {F} )={\frac {1}{\sqrt {g}}}\mathbf {R} _{k}\varepsilon ^{k\ell m}\partial _{\ell }F_{m}} where Rk are the local basis vectors. Equivalently, using the exterior derivative, the curl can be expressed as: ∇ × F = ( ⋆ ( d F ♭ ) ) ♯ {\displaystyle \nabla \times \mathbf {F} =\left(\star {\big (}{\mathrm {d} }\mathbf {F} ^{\flat }{\big )}\right)^{\sharp }} Here ♭ and ♯ are the musical isomorphisms, and ★ is the Hodge star operator. This formula shows how to calculate the curl of F in any coordinate system, and how to extend the curl to any oriented three-dimensional Riemannian manifold. Since this depends on a choice of orientation, curl is a chiral operation. In other words, if the orientation is reversed, then the direction of the curl is also reversed. == Examples == === Example 1 === Suppose the vector field describes the velocity field of a fluid flow (such as a large tank of liquid or gas) and a small ball is located within the fluid or gas (the center of the ball being fixed at a certain point). If the ball has a rough surface, the fluid flowing past it will make it rotate. The rotation axis (oriented according to the right hand rule) points in the direction of the curl of the field at the center of the ball, and the angular speed of the rotation is half the magnitude of the curl at this point. The curl of the vector field at any point is given by the rotation of an infinitesimal area in the xy-plane (for z-axis component of the curl), zx-plane (for y-axis component of the curl) and yz-plane (for x-axis component of the curl vector). This can be seen in the examples below. === Example 2 === The vector field F ( x , y , z ) = y ı ^ − x ȷ ^ {\displaystyle \mathbf {F} (x,y,z)=y{\boldsymbol {\hat {\imath }}}-x{\boldsymbol {\hat {\jmath }}}} can be decomposed as F x = y , F y = − x , F z = 0. {\displaystyle F_{x}=y,F_{y}=-x,F_{z}=0.} Upon visual inspection, the field can be described as "rotating". If the vectors of the field were to represent a linear force acting on objects present at that point, and an object were to be placed inside the field, the object would start to rotate clockwise around itself. This is true regardless of where the object is placed. Calculating the curl: ∇ × F = 0 ı ^ + 0 ȷ ^ + ( ∂ ∂ x ( − x ) − ∂ ∂ y y ) k ^ = − 2 k ^ {\displaystyle \nabla \times \mathbf {F} =0{\boldsymbol {\hat {\imath }}}+0{\boldsymbol {\hat {\jmath }}}+\left({\frac {\partial }{\partial x}}(-x)-{\frac {\partial }{\partial y}}y\right){\boldsymbol {\hat {k}}}=-2{\boldsymbol {\hat {k}}}} The resulting vector field describing the curl would at all points be pointing in the negative z direction. The results of this equation align with what could have been predicted using the right-hand rule using a right-handed coordinate system. Being a uniform vector field, the object described before would have the same rotational intensity regardless of where it was placed. === Example 3 === For the vector field F ( x , y , z ) = − x 2 ȷ ^ {\displaystyle \mathbf {F} (x,y,z)=-x^{2}{\boldsymbol {\hat {\jmath }}}} the curl is not as obvious from the graph. However, taking the object in the previous example, and placing it anywhere on the line x = 3, the force exerted on the right side would be slightly greater than the force exerted on the left, causing it to rotate clockwise. Using the right-hand rule, it can be predicted that the resulting curl would be straight in the negative z direction. Inversely, if placed on x = −3, the object would rotate counterclockwise and the right-hand rule would result in a positive z direction. Calculating the curl: ∇ × F = 0 ı ^ + 0 ȷ ^ + ∂ ∂ x ( − x 2 ) k ^ = − 2 x k ^ . {\displaystyle {\nabla }\times \mathbf {F} =0{\boldsymbol {\hat {\imath }}}+0{\boldsymbol {\hat {\jmath }}}+{\frac {\partial }{\partial x}}\left(-x^{2}\right){\boldsymbol {\hat {k}}}=-2x{\boldsymbol {\hat {k}}}.} The curl points in the negative z direction when x is positive and vice versa. In this field, the intensity of rotation would be greater as the object moves away from the plane x = 0. === Further examples === In a vector field describing the linear velocities of each part of a rotating disk in uniform circular motion, the curl has the same value at all points, and this value turns out to be exactly two times the vectorial angular velocity of the disk (oriented as usual by the right-hand rule). More generally, for any flowing mass, the linear velocity vector field at each point of the mass flow has a curl (the vorticity of the flow at that point) equal to exactly two times the local vectorial angular velocity of the mass about the point. For any solid object subject to an external physical force (such as gravity or the electromagnetic force), one may consider the vector field representing the infinitesimal force-per-unit-volume contributions acting at each of the points of the object. This force field may create a net torque on the object about its center of mass, and this torque turns out to be directly proportional and vectorially parallel to the (vector-valued) integral of the curl of the force field over the whole volume. Of the four Maxwell's equations, two—Faraday's law and Ampère's law—can be compactly expressed using curl. Faraday's law states that the curl of an electric field is equal to the opposite of the time rate of change of the magnetic field, while Ampère's law relates the curl of the magnetic field to the current and the time rate of change of the electric field. == Identities == In general curvilinear coordinates (not only in Cartesian coordinates), the curl of a cross product of vector fields v and F can be shown to be ∇ × ( v × F ) = ( ( ∇ ⋅ F ) + F ⋅ ∇ ) v − ( ( ∇ ⋅ v ) + v ⋅ ∇ ) F . {\displaystyle \nabla \times \left(\mathbf {v\times F} \right)={\Big (}\left(\mathbf {\nabla \cdot F} \right)+\mathbf {F\cdot \nabla } {\Big )}\mathbf {v} -{\Big (}\left(\mathbf {\nabla \cdot v} \right)+\mathbf {v\cdot \nabla } {\Big )}\mathbf {F} \ .} Interchanging the vector field v and ∇ operator, we arrive at the cross product of a vector field with curl of a vector field: v × ( ∇ × F ) = ∇ F ( v ⋅ F ) − ( v ⋅ ∇ ) F , {\displaystyle \mathbf {v\ \times } \left(\mathbf {\nabla \times F} \right)=\nabla _{\mathbf {F} }\left(\mathbf {v\cdot F} \right)-\left(\mathbf {v\cdot \nabla } \right)\mathbf {F} \ ,} where ∇F is the Feynman subscript notation, which considers only the variation due to the vector field F (i.e., in this case, v is treated as being constant in space). Another example is the curl of a curl of a vector field. It can be shown that in general coordinates ∇ × ( ∇ × F ) = ∇ ( ∇ ⋅ F ) − ∇ 2 F , {\displaystyle \nabla \times \left(\mathbf {\nabla \times F} \right)=\mathbf {\nabla } (\mathbf {\nabla \cdot F} )-\nabla ^{2}\mathbf {F} \ ,} and this identity defines the vector Laplacian of F, symbolized as ∇2F. The curl of the gradient of any scalar field φ is always the zero vector field ∇ × ( ∇ φ ) = 0 {\displaystyle \nabla \times (\nabla \varphi )={\boldsymbol {0}}} which follows from the antisymmetry in the definition of the curl, and the symmetry of second derivatives. The divergence of the curl of any vector field is equal to zero: ∇ ⋅ ( ∇ × F ) = 0. {\displaystyle \nabla \cdot (\nabla \times \mathbf {F} )=0.} If φ is a scalar valued function and F is a vector field, then ∇ × ( φ F ) = ∇ φ × F + φ ∇ × F {\displaystyle \nabla \times (\varphi \mathbf {F} )=\nabla \varphi \times \mathbf {F} +\varphi \nabla \times \mathbf {F} } == Generalizations == The vector calculus operations of grad, curl, and div are most easily generalized in the context of differential forms, which involves a number of steps. In short, they correspond to the derivatives of 0-forms, 1-forms, and 2-forms, respectively. The geometric interpretation of curl as rotation corresponds to identifying bivectors (2-vectors) in 3 dimensions with the special orthogonal Lie algebra s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} of infinitesimal rotations (in coordinates, skew-symmetric 3 × 3 matrices), while representing rotations by vectors corresponds to identifying 1-vectors (equivalently, 2-vectors) and s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} , these all being 3-dimensional spaces. === Differential forms === In 3 dimensions, a differential 0-form is a real-valued function f ( x , y , z ) {\displaystyle f(x,y,z)} ; a differential 1-form is the following expression, where the coefficients are functions: a 1 d x + a 2 d y + a 3 d z ; {\displaystyle a_{1}\,dx+a_{2}\,dy+a_{3}\,dz;} a differential 2-form is the formal sum, again with function coefficients: a 12 d x ∧ d y + a 13 d x ∧ d z + a 23 d y ∧ d z ; {\displaystyle a_{12}\,dx\wedge dy+a_{13}\,dx\wedge dz+a_{23}\,dy\wedge dz;} and a differential 3-form is defined by a single term with one function as coefficient: a 123 d x ∧ d y ∧ d z . {\displaystyle a_{123}\,dx\wedge dy\wedge dz.} (Here the a-coefficients are real functions of three variables; the wedge products, e.g. d x ∧ d y {\displaystyle {\text{d}}x\wedge {\text{d}}y} , can be interpreted as oriented plane segments, d x ∧ d y = − d y ∧ d x {\displaystyle {\text{d}}x\wedge {\text{d}}y=-{\text{d}}y\wedge {\text{d}}x} , etc.) The exterior derivative of a k-form in R3 is defined as the (k + 1)-form from above—and in Rn if, e.g., ω ( k ) = ∑ 1 ≤ i 1 < i 2 < ⋯ < i k ≤ n a i 1 , … , i k d x i 1 ∧ ⋯ ∧ d x i k , {\displaystyle \omega ^{(k)}=\sum _{1\leq i_{1}<i_{2}<\cdots <i_{k}\leq n}a_{i_{1},\ldots ,i_{k}}\,dx_{i_{1}}\wedge \cdots \wedge dx_{i_{k}},} then the exterior derivative d leads to d ω ( k ) = ∑ j = 1 i 1 < ⋯ < i k n ∂ a i 1 , … , i k ∂ x j d x j ∧ d x i 1 ∧ ⋯ ∧ d x i k . {\displaystyle d\omega ^{(k)}=\sum _{\scriptstyle {j=1} \atop \scriptstyle {i_{1}<\cdots <i_{k}}}^{n}{\frac {\partial a_{i_{1},\ldots ,i_{k}}}{\partial x_{j}}}\,dx_{j}\wedge dx_{i_{1}}\wedge \cdots \wedge dx_{i_{k}}.} The exterior derivative of a 1-form is therefore a 2-form, and that of a 2-form is a 3-form. On the other hand, because of the interchangeability of mixed derivatives, ∂ 2 ∂ x i ∂ x j = ∂ 2 ∂ x j ∂ x i , {\displaystyle {\frac {\partial ^{2}}{\partial x_{i}\,\partial x_{j}}}={\frac {\partial ^{2}}{\partial x_{j}\,\partial x_{i}}},} and antisymmetry, d x i ∧ d x j = − d x j ∧ d x i {\displaystyle dx_{i}\wedge dx_{j}=-dx_{j}\wedge dx_{i}} the twofold application of the exterior derivative yields 0 {\displaystyle 0} (the zero k + 2 {\displaystyle k+2} -form). Thus, denoting the space of k-forms by Ω k ( R 3 ) {\displaystyle \Omega ^{k}(\mathbb {R} ^{3})} and the exterior derivative by d one gets a sequence: 0 ⟶ d Ω 0 ( R 3 ) ⟶ d Ω 1 ( R 3 ) ⟶ d Ω 2 ( R 3 ) ⟶ d Ω 3 ( R 3 ) ⟶ d 0. {\displaystyle 0\,{\overset {d}{\longrightarrow }}\;\Omega ^{0}\left(\mathbb {R} ^{3}\right)\,{\overset {d}{\longrightarrow }}\;\Omega ^{1}\left(\mathbb {R} ^{3}\right)\,{\overset {d}{\longrightarrow }}\;\Omega ^{2}\left(\mathbb {R} ^{3}\right)\,{\overset {d}{\longrightarrow }}\;\Omega ^{3}\left(\mathbb {R} ^{3}\right)\,{\overset {d}{\longrightarrow }}\,0.} Here Ω k ( R n ) {\displaystyle \Omega ^{k}(\mathbb {R} ^{n})} is the space of sections of the exterior algebra Λ k ( R n ) {\displaystyle \Lambda ^{k}(\mathbb {R} ^{n})} vector bundle over Rn, whose dimension is the binomial coefficient ( n k ) {\displaystyle {\binom {n}{k}}} ; note that Ω k ( R 3 ) = 0 {\displaystyle \Omega ^{k}(\mathbb {R} ^{3})=0} for k > 3 {\displaystyle k>3} or k < 0 {\displaystyle k<0} . Writing only dimensions, one obtains a row of Pascal's triangle: 0 → 1 → 3 → 3 → 1 → 0 ; {\displaystyle 0\rightarrow 1\rightarrow 3\rightarrow 3\rightarrow 1\rightarrow 0;} the 1-dimensional fibers correspond to scalar fields, and the 3-dimensional fibers to vector fields, as described below. Modulo suitable identifications, the three nontrivial occurrences of the exterior derivative correspond to grad, curl, and div. Differential forms and the differential can be defined on any Euclidean space, or indeed any manifold, without any notion of a Riemannian metric. On a Riemannian manifold, or more generally pseudo-Riemannian manifold, k-forms can be identified with k-vector fields (k-forms are k-covector fields, and a pseudo-Riemannian metric gives an isomorphism between vectors and covectors), and on an oriented vector space with a nondegenerate form (an isomorphism between vectors and covectors), there is an isomorphism between k-vectors and (n − k)-vectors; in particular on (the tangent space of) an oriented pseudo-Riemannian manifold. Thus on an oriented pseudo-Riemannian manifold, one can interchange k-forms, k-vector fields, (n − k)-forms, and (n − k)-vector fields; this is known as Hodge duality. Concretely, on R3 this is given by: 1-forms and 1-vector fields: the 1-form ax dx + ay dy + az dz corresponds to the vector field (ax, ay, az). 1-forms and 2-forms: one replaces dx by the dual quantity dy ∧ dz (i.e., omit dx), and likewise, taking care of orientation: dy corresponds to dz ∧ dx = −dx ∧ dz, and dz corresponds to dx ∧ dy. Thus the form ax dx + ay dy + az dz corresponds to the "dual form" az dx ∧ dy + ay dz ∧ dx + ax dy ∧ dz. Thus, identifying 0-forms and 3-forms with scalar fields, and 1-forms and 2-forms with vector fields: grad takes a scalar field (0-form) to a vector field (1-form); curl takes a vector field (1-form) to a pseudovector field (2-form); div takes a pseudovector field (2-form) to a pseudoscalar field (3-form) On the other hand, the fact that d2 = 0 corresponds to the identities ∇ × ( ∇ f ) = 0 {\displaystyle \nabla \times (\nabla f)=\mathbf {0} } for any scalar field f, and ∇ ⋅ ( ∇ × v ) = 0 {\displaystyle \nabla \cdot (\nabla \times \mathbf {v} )=0} for any vector field v. Grad and div generalize to all oriented pseudo-Riemannian manifolds, with the same geometric interpretation, because the spaces of 0-forms and n-forms at each point are always 1-dimensional and can be identified with scalar fields, while the spaces of 1-forms and (n − 1)-forms are always fiberwise n-dimensional and can be identified with vector fields. Curl does not generalize in this way to 4 or more dimensions (or down to 2 or fewer dimensions); in 4 dimensions the dimensions are so the curl of a 1-vector field (fiberwise 4-dimensional) is a 2-vector field, which at each point belongs to 6-dimensional vector space, and so one has ω ( 2 ) = ∑ i < k = 1 , 2 , 3 , 4 a i , k d x i ∧ d x k , {\displaystyle \omega ^{(2)}=\sum _{i<k=1,2,3,4}a_{i,k}\,dx_{i}\wedge dx_{k},} which yields a sum of six independent terms, and cannot be identified with a 1-vector field. Nor can one meaningfully go from a 1-vector field to a 2-vector field to a 3-vector field (4 → 6 → 4), as taking the differential twice yields zero (d2 = 0). Thus there is no curl function from vector fields to vector fields in other dimensions arising in this way. However, one can define a curl of a vector field as a 2-vector field in general, as described below. === Curl geometrically === 2-vectors correspond to the exterior power Λ2V; in the presence of an inner product, in coordinates these are the skew-symmetric matrices, which are geometrically considered as the special orthogonal Lie algebra s o {\displaystyle {\mathfrak {so}}} (V) of infinitesimal rotations. This has (n2) = ⁠1/2⁠n(n − 1) dimensions, and allows one to interpret the differential of a 1-vector field as its infinitesimal rotations. Only in 3 dimensions (or trivially in 0 dimensions) we have n = ⁠1/2⁠n(n − 1), which is the most elegant and common case. In 2 dimensions the curl of a vector field is not a vector field but a function, as 2-dimensional rotations are given by an angle (a scalar – an orientation is required to choose whether one counts clockwise or counterclockwise rotations as positive); this is not the div, but is rather perpendicular to it. In 3 dimensions the curl of a vector field is a vector field as is familiar (in 1 and 0 dimensions the curl of a vector field is 0, because there are no non-trivial 2-vectors), while in 4 dimensions the curl of a vector field is, geometrically, at each point an element of the 6-dimensional Lie algebra s o ( 4 ) {\displaystyle {\mathfrak {so}}(4)} . The curl of a 3-dimensional vector field which only depends on 2 coordinates (say x and y) is simply a vertical vector field (in the z direction) whose magnitude is the curl of the 2-dimensional vector field, as in the examples on this page. Considering curl as a 2-vector field (an antisymmetric 2-tensor) has been used to generalize vector calculus and associated physics to higher dimensions. == Inverse == In the case where the divergence of a vector field V is zero, a vector field W exists such that V = curl(W). This is why the magnetic field, characterized by zero divergence, can be expressed as the curl of a magnetic vector potential. If W is a vector field with curl(W) = V, then adding any gradient vector field grad(f) to W will result in another vector field W + grad(f) such that curl(W + grad(f)) = V as well. This can be summarized by saying that the inverse curl of a three-dimensional vector field can be obtained up to an unknown irrotational field with the Biot–Savart law. == See also == Helmholtz decomposition Hiptmair–Xu preconditioner Del in cylindrical and spherical coordinates Vorticity == References == == Further reading == Korn, Granino Arthur and Theresa M. Korn (January 2000). Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. New York: Dover Publications. pp. 157–160. ISBN 0-486-41147-8. Schey, H. M. (1997). Div, Grad, Curl, and All That: An Informal Text on Vector Calculus. New York: Norton. ISBN 0-393-96997-5. == External links == "Curl", Encyclopedia of Mathematics, EMS Press, 2001 [1994] "Multivariable calculus". mathinsight.org. Retrieved February 12, 2022. "Divergence and Curl: The Language of Maxwell's Equations, Fluid Flow, and More". June 21, 2018. Archived from the original on 2021-11-24 – via YouTube.
Wikipedia:Rota's basis conjecture#0	In linear algebra and matroid theory, Rota's basis conjecture is an unproven conjecture concerning rearrangements of bases, named after Gian-Carlo Rota. It states that, if X is either a vector space of dimension n or more generally a matroid of rank n, with n disjoint bases Bi, then it is possible to arrange the elements of these bases into an n × n matrix in such a way that the rows of the matrix are exactly the given bases and the columns of the matrix are also bases. That is, it should be possible to find a second set of n disjoint bases Ci, each of which consists of one element from each of the bases Bi. == Examples == Rota's basis conjecture has a simple formulation for points in the Euclidean plane: it states that, given three triangles with distinct vertices, with each triangle colored with one of three colors, it must be possible to regroup the nine triangle vertices into three "rainbow" triangles having one vertex of each color. The triangles are all required to be non-degenerate, meaning that they do not have all three vertices on a line. To see this as an instance of the basis conjecture, one may use either linear independence of the vectors ( x i , y i , 1 {\displaystyle x_{i},y_{i},1} ) in a three-dimensional real vector space (where ( x i , y i {\displaystyle x_{i},y_{i}} ) are the Cartesian coordinates of the triangle vertices) or equivalently one may use a matroid of rank three in which a set S of points is independent if either \|S\| ≤ 2 or S forms the three vertices of a non-degenerate triangle. For this linear algebra and this matroid, the bases are exactly the non-degenerate triangles. Given the three input triangles and the three rainbow triangles, it is possible to arrange the nine vertices into a 3 × 3 matrix in which each row contains the vertices of one of the single-color triangles and each column contains the vertices of one of the rainbow triangles. Analogously, for points in three-dimensional Euclidean space, the conjecture states that the sixteen vertices of four non-degenerate tetrahedra of four different colors may be regrouped into four rainbow tetrahedra. == Partial results == The statement of Rota's basis conjecture was first published by Huang & Rota (1994), crediting it (without citation) to Rota in 1989. The basis conjecture has been proven for paving matroids (for all n) and for the case n ≤ 3 (for all types of matroid). For arbitrary matroids, it is possible to arrange the basis elements into a matrix the first Ω(√n) columns of which are bases. The basis conjecture for linear algebras over fields of characteristic zero and for even values of n would follow from another conjecture on Latin squares by Alon and Tarsi. Based on this implication, the conjecture is known to be true for linear algebras over the real numbers for infinitely many values of n. == Related problems == In connection with Tverberg's theorem, Bárány & Larman (1992) conjectured that, for every set of r (d + 1) points in d-dimensional Euclidean space, colored with d + 1 colors in such a way that there are r points of each color, there is a way to partition the points into rainbow simplices (sets of d + 1 points with one point of each color) in such a way that the convex hulls of these sets have a nonempty intersection. For instance, the two-dimensional case (proven by Bárány and Larman) with r = 3 states that, for every set of nine points in the plane, colored with three colors and three points of each color, it is possible to partition the points into three intersecting rainbow triangles, a statement similar to Rota's basis conjecture which states that it is possible to partition the points into three non-degenerate rainbow triangles. The conjecture of Bárány and Larman allows a collinear triple of points to be considered as a rainbow triangle, whereas Rota's basis conjecture disallows this; on the other hand, Rota's basis conjecture does not require the triangles to have a common intersection. Substantial progress on the conjecture of Bárány and Larman was made by Blagojević, Matschke & Ziegler (2009). == See also == Rota's conjecture, a different conjecture by Rota about linear algebra and matroids == References == == External links == Rota's basis conjecture, Open Problem Garden.
Wikipedia:Rotation number#0	In mathematics, the rotation number is an invariant of homeomorphisms of the circle. == History == It was first defined by Henri Poincaré in 1885, in relation to the precession of the perihelion of a planetary orbit. Poincaré later proved a theorem characterizing the existence of periodic orbits in terms of rationality of the rotation number. == Definition == Suppose that f : S 1 → S 1 {\displaystyle f:S^{1}\to S^{1}} is an orientation-preserving homeomorphism of the circle S 1 = R / Z . {\displaystyle S^{1}=\mathbb {R} /\mathbb {Z} .} Then f may be lifted to a homeomorphism F : R → R {\displaystyle F:\mathbb {R} \to \mathbb {R} } of the real line, satisfying F ( x + m ) = F ( x ) + m {\displaystyle F(x+m)=F(x)+m} for every real number x and every integer m. The rotation number of f is defined in terms of the iterates of F: ω ( f ) = lim n → ∞ F n ( x ) − x n . {\displaystyle \omega (f)=\lim _{n\to \infty }{\frac {F^{n}(x)-x}{n}}.} Henri Poincaré proved that the limit exists and is independent of the choice of the starting point x. The lift F is unique modulo integers, therefore the rotation number is a well-defined element of ⁠ R / Z . {\displaystyle \mathbb {R} /\mathbb {Z} .} ⁠ Intuitively, it measures the average rotation angle along the orbits of f. == Example == If f {\displaystyle f} is a rotation by 2 π N {\displaystyle 2\pi N} (where 0 < N < 1 {\displaystyle 0<N<1} ), then F ( x ) = x + N , {\displaystyle F(x)=x+N,} and its rotation number is N {\displaystyle N} (cf. irrational rotation). == Properties == The rotation number is invariant under topological conjugacy, and even monotone topological semiconjugacy: if f and g are two homeomorphisms of the circle and h ∘ f = g ∘ h {\displaystyle h\circ f=g\circ h} for a monotone continuous map h of the circle into itself (not necessarily homeomorphic) then f and g have the same rotation numbers. It was used by Poincaré and Arnaud Denjoy for topological classification of homeomorphisms of the circle. There are two distinct possibilities. The rotation number of f is a rational number p/q (in the lowest terms). Then f has a periodic orbit, every periodic orbit has period q, and the order of the points on each such orbit coincides with the order of the points for a rotation by p/q. Moreover, every forward orbit of f converges to a periodic orbit. The same is true for backward orbits, corresponding to iterations of f –1, but the limiting periodic orbits in forward and backward directions may be different. The rotation number of f is an irrational number θ. Then f has no periodic orbits (this follows immediately by considering a periodic point x of f). There are two subcases. There exists a dense orbit. In this case f is topologically conjugate to the irrational rotation by the angle θ and all orbits are dense. Denjoy proved that this possibility is always realized when f is twice continuously differentiable. There exists a Cantor set C invariant under f. Then C is a unique minimal set and the orbits of all points both in forward and backward direction converge to C. In this case, f is semiconjugate to the irrational rotation by θ, and the semiconjugating map h of degree 1 is constant on components of the complement of C. The rotation number is continuous when viewed as a map from the group of homeomorphisms (with C0 topology) of the circle into the circle. == See also == Circle map Denjoy diffeomorphism Poincaré section Poincaré recurrence Poincaré–Bendixson theorem == References == Herman, Michael Robert (December 1979). "Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations" [On the Differentiable Conjugation of Diffeomorphisms from the Circle to Rotations]. Publications Mathématiques de l'IHÉS (in French). 49: 5–233. doi:10.1007/BF02684798. S2CID 118356096., also SciSpace for smaller file size in pdf ver 1.3 Poincaré, Henri (1885). "Sur les courbes définies par les équations différentielles (III)". Journal de Mathématiques Pures et Appliquées (in French). 1: 167–244. == External links == Michał Misiurewicz (ed.). "Rotation theory". Scholarpedia. Weisstein, Eric W. "Map Winding Number". From MathWorld--A Wolfram Web Resource.
Wikipedia:Rotation of axes in two dimensions#0	In mathematics, a rotation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x′y′-Cartesian coordinate system in which the origin is kept fixed and the x′ and y′ axes are obtained by rotating the x and y axes counterclockwise through an angle θ {\displaystyle \theta } . A point P has coordinates (x, y) with respect to the original system and coordinates (x′, y′) with respect to the new system. In the new coordinate system, the point P will appear to have been rotated in the opposite direction, that is, clockwise through the angle θ {\displaystyle \theta } . A rotation of axes in more than two dimensions is defined similarly. A rotation of axes is a linear map and a rigid transformation. == Motivation == Coordinate systems are essential for studying the equations of curves using the methods of analytic geometry. To use the method of coordinate geometry, the axes are placed at a convenient position with respect to the curve under consideration. For example, to study the equations of ellipses and hyperbolas, the foci are usually located on one of the axes and are situated symmetrically with respect to the origin. If the curve (hyperbola, parabola, ellipse, etc.) is not situated conveniently with respect to the axes, the coordinate system should be changed to place the curve at a convenient and familiar location and orientation. The process of making this change is called a transformation of coordinates. The solutions to many problems can be simplified by rotating the coordinate axes to obtain new axes through the same origin. == Derivation == The equations defining the transformation in two dimensions, which rotates the xy axes counterclockwise through an angle θ {\displaystyle \theta } into the x′y′ axes, are derived as follows. In the xy system, let the point P have polar coordinates ( r , α ) {\displaystyle (r,\alpha )} . Then, in the x′y′ system, P will have polar coordinates ( r , α − θ ) {\displaystyle (r,\alpha -\theta )} . Using trigonometric functions, we have and using the standard trigonometric formulae for differences, we have Substituting equations (1) and (2) into equations (3) and (4), we obtain Equations (5) and (6) can be represented in matrix form as [ x ′ y ′ ] = [ cos ⁡ θ sin ⁡ θ − sin ⁡ θ cos ⁡ θ ] [ x y ] , {\displaystyle {\begin{bmatrix}x'\\y'\end{bmatrix}}={\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}},} which is the standard matrix equation of a rotation of axes in two dimensions. The inverse transformation is or [ x y ] = [ cos ⁡ θ − sin ⁡ θ sin ⁡ θ cos ⁡ θ ] [ x ′ y ′ ] . {\displaystyle {\begin{bmatrix}x\\y\end{bmatrix}}={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{bmatrix}}{\begin{bmatrix}x'\\y'\end{bmatrix}}.} === Examples in two dimensions === ==== Example 1 ==== Find the coordinates of the point P 1 = ( x , y ) = ( 3 , 1 ) {\displaystyle P_{1}=(x,y)=({\sqrt {3}},1)} after the axes have been rotated through the angle θ 1 = π / 6 {\displaystyle \theta _{1}=\pi /6} , or 30°. Solution: x ′ = 3 cos ⁡ ( π / 6 ) + 1 sin ⁡ ( π / 6 ) = ( 3 ) ( 3 / 2 ) + ( 1 ) ( 1 / 2 ) = 2 {\displaystyle x'={\sqrt {3}}\cos(\pi /6)+1\sin(\pi /6)=({\sqrt {3}})({\sqrt {3}}/2)+(1)(1/2)=2} y ′ = 1 cos ⁡ ( π / 6 ) − 3 sin ⁡ ( π / 6 ) = ( 1 ) ( 3 / 2 ) − ( 3 ) ( 1 / 2 ) = 0. {\displaystyle y'=1\cos(\pi /6)-{\sqrt {3}}\sin(\pi /6)=(1)({\sqrt {3}}/2)-({\sqrt {3}})(1/2)=0.} The axes have been rotated counterclockwise through an angle of θ 1 = π / 6 {\displaystyle \theta _{1}=\pi /6} and the new coordinates are P 1 = ( x ′ , y ′ ) = ( 2 , 0 ) {\displaystyle P_{1}=(x',y')=(2,0)} . Note that the point appears to have been rotated clockwise through π / 6 {\displaystyle \pi /6} with respect to fixed axes so it now coincides with the (new) x′ axis. ==== Example 2 ==== Find the coordinates of the point P 2 = ( x , y ) = ( 7 , 7 ) {\displaystyle P_{2}=(x,y)=(7,7)} after the axes have been rotated clockwise 90°, that is, through the angle θ 2 = − π / 2 {\displaystyle \theta _{2}=-\pi /2} , or −90°. Solution: [ x ′ y ′ ] = [ cos ⁡ ( − π / 2 ) sin ⁡ ( − π / 2 ) − sin ⁡ ( − π / 2 ) cos ⁡ ( − π / 2 ) ] [ 7 7 ] = [ 0 − 1 1 0 ] [ 7 7 ] = [ − 7 7 ] . {\displaystyle {\begin{bmatrix}x'\\y'\end{bmatrix}}={\begin{bmatrix}\cos(-\pi /2)&\sin(-\pi /2)\\-\sin(-\pi /2)&\cos(-\pi /2)\end{bmatrix}}{\begin{bmatrix}7\\7\end{bmatrix}}={\begin{bmatrix}0&-1\\1&0\end{bmatrix}}{\begin{bmatrix}7\\7\end{bmatrix}}={\begin{bmatrix}-7\\7\end{bmatrix}}.} The axes have been rotated through an angle of θ 2 = − π / 2 {\displaystyle \theta _{2}=-\pi /2} , which is in the clockwise direction and the new coordinates are P 2 = ( x ′ , y ′ ) = ( − 7 , 7 ) {\displaystyle P_{2}=(x',y')=(-7,7)} . Again, note that the point appears to have been rotated counterclockwise through π / 2 {\displaystyle \pi /2} with respect to fixed axes. == Rotation of conic sections == The most general equation of the second degree has the form Through a change of coordinates (a rotation of axes and a translation of axes), equation (9) can be put into a standard form, which is usually easier to work with. It is always possible to rotate the coordinates at a specific angle so as to eliminate the x′y′ term. Substituting equations (7) and (8) into equation (9), we obtain where If θ {\displaystyle \theta } is selected so that cot ⁡ 2 θ = ( A − C ) / B {\displaystyle \cot 2\theta =(A-C)/B} we will have B ′ = 0 {\displaystyle B'=0} and the x′y′ term in equation (10) will vanish. When a problem arises with B, D and E all different from zero, they can be eliminated by performing in succession a rotation (eliminating B) and a translation (eliminating the D and E terms). === Identifying rotated conic sections === A non-degenerate conic section given by equation (9) can be identified by evaluating B 2 − 4 A C {\displaystyle B^{2}-4AC} . The conic section is: an ellipse or a circle, if B 2 − 4 A C < 0 {\displaystyle B^{2}-4AC<0} ; a parabola, if B 2 − 4 A C = 0 {\displaystyle B^{2}-4AC=0} ; a hyperbola, if B 2 − 4 A C > 0 {\displaystyle B^{2}-4AC>0} . == Generalization to several dimensions == Suppose a rectangular xyz-coordinate system is rotated around its z axis counterclockwise (looking down the positive z axis) through an angle θ {\displaystyle \theta } , that is, the positive x axis is rotated immediately into the positive y axis. The z coordinate of each point is unchanged and the x and y coordinates transform as above. The old coordinates (x, y, z) of a point Q are related to its new coordinates (x′, y′, z′) by [ x ′ y ′ z ′ ] = [ cos ⁡ θ sin ⁡ θ 0 − sin ⁡ θ cos ⁡ θ 0 0 0 1 ] [ x y z ] . {\displaystyle {\begin{bmatrix}x'\\y'\\z'\end{bmatrix}}={\begin{bmatrix}\cos \theta &\sin \theta &0\\-\sin \theta &\cos \theta &0\\0&0&1\end{bmatrix}}{\begin{bmatrix}x\\y\\z\end{bmatrix}}.} Generalizing to any finite number of dimensions, a rotation matrix A {\displaystyle A} is an orthogonal matrix that differs from the identity matrix in at most four elements. These four elements are of the form a i i = a j j = cos ⁡ θ {\displaystyle a_{ii}=a_{jj}=\cos \theta } and a i j = − a j i = sin ⁡ θ , {\displaystyle a_{ij}=-a_{ji}=\sin \theta ,} for some θ {\displaystyle \theta } and some i ≠ j. == Example in several dimensions == === Example 3 === Find the coordinates of the point P 3 = ( w , x , y , z ) = ( 1 , 1 , 1 , 1 ) {\displaystyle P_{3}=(w,x,y,z)=(1,1,1,1)} after the positive w axis has been rotated through the angle θ 3 = π / 12 {\displaystyle \theta _{3}=\pi /12} , or 15°, into the positive z axis. Solution: [ w ′ x ′ y ′ z ′ ] = [ cos ⁡ ( π / 12 ) 0 0 sin ⁡ ( π / 12 ) 0 1 0 0 0 0 1 0 − sin ⁡ ( π / 12 ) 0 0 cos ⁡ ( π / 12 ) ] [ w x y z ] ≈ [ 0.96593 0.0 0.0 0.25882 0.0 1.0 0.0 0.0 0.0 0.0 1.0 0.0 − 0.25882 0.0 0.0 0.96593 ] [ 1.0 1.0 1.0 1.0 ] = [ 1.22475 1.00000 1.00000 0.70711 ] . {\displaystyle {\begin{aligned}{\begin{bmatrix}w'\\x'\\y'\\z'\end{bmatrix}}&={\begin{bmatrix}\cos(\pi /12)&0&0&\sin(\pi /12)\\0&1&0&0\\0&0&1&0\\-\sin(\pi /12)&0&0&\cos(\pi /12)\end{bmatrix}}{\begin{bmatrix}w\\x\\y\\z\end{bmatrix}}\\[4pt]&\approx {\begin{bmatrix}0.96593&0.0&0.0&0.25882\\0.0&1.0&0.0&0.0\\0.0&0.0&1.0&0.0\\-0.25882&0.0&0.0&0.96593\end{bmatrix}}{\begin{bmatrix}1.0\\1.0\\1.0\\1.0\end{bmatrix}}={\begin{bmatrix}1.22475\\1.00000\\1.00000\\0.70711\end{bmatrix}}.\end{aligned}}} == See also == Rotation Rotation (mathematics) == Notes == == References == Anton, Howard (1987), Elementary Linear Algebra (5th ed.), New York: Wiley, ISBN 0-471-84819-0 Beauregard, Raymond A.; Fraleigh, John B. (1973), A First Course In Linear Algebra: With Optional Introduction to Groups, Rings, and Fields, Boston: Houghton Mifflin Co., ISBN 0-395-14017-X Burden, Richard L.; Faires, J. Douglas (1993), Numerical Analysis (5th ed.), Boston: Prindle, Weber and Schmidt, ISBN 0-534-93219-3 Protter, Murray H.; Morrey, Charles B. Jr. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading: Addison-Wesley, LCCN 76087042
Wikipedia:Rouse Ball Professor of Mathematics#0	The Rouse Ball Professorship of Mathematics is one of the senior chairs in the Mathematics Departments at the University of Cambridge and the University of Oxford. The two positions were founded in 1927 by a bequest from the mathematician W. W. Rouse Ball. At Cambridge, this bequest was made with the "hope (but not making it in any way a condition) that it might be found practicable for such Professor or Reader to include in his or her lectures and treatment historical and philosophical aspects of the subject." == List of Rouse Ball Professors at Cambridge == 1928–1950 John Edensor Littlewood 1950–1958 Abram Samoilovitch Besicovitch 1958–1969 Harold Davenport 1971–1993 John G. Thompson 1994–1997 Nigel Hitchin 1998–2020 William Timothy Gowers 2023– Wendelin Werner == List of Rouse Ball Professors at Oxford == The chair at Oxford was established with a £25,000 bequest and was initially advertised by the University as a Chair in Mathematical Physics. The Rouse Ball Professor is now hosted at the university's Mathematical Institute, and holds a Fellowship at Wadham College. 1928–1950 E. A. Milne 1952–1972 Charles Coulson 1973–1999 Roger Penrose, Emeritus Rouse Ball Professor of Mathematics 1999–2020 Philip Candelas, Emeritus Rouse Ball Professor of Mathematics 2020– Luis Fernando Alday, currently Rouse Ball Professor of Mathematics == See also == Rouse Ball Professor of English Law == References ==
Wikipedia:Row and column spaces#0	In linear algebra, the column space (also called the range or image) of a matrix A is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image or range of the corresponding matrix transformation. Let F {\displaystyle F} be a field. The column space of an m × n matrix with components from F {\displaystyle F} is a linear subspace of the m-space F m {\displaystyle F^{m}} . The dimension of the column space is called the rank of the matrix and is at most min(m, n). A definition for matrices over a ring R {\displaystyle R} is also possible. The row space is defined similarly. The row space and the column space of a matrix A are sometimes denoted as C(AT) and C(A) respectively. This article considers matrices of real numbers. The row and column spaces are subspaces of the real spaces R n {\displaystyle \mathbb {R} ^{n}} and R m {\displaystyle \mathbb {R} ^{m}} respectively. == Overview == Let A be an m-by-n matrix. Then rank(A) = dim(rowsp(A)) = dim(colsp(A)), rank(A) = number of pivots in any echelon form of A, rank(A) = the maximum number of linearly independent rows or columns of A. If the matrix represents a linear transformation, the column space of the matrix equals the image of this linear transformation. The column space of a matrix A is the set of all linear combinations of the columns in A. If A = [a1 ⋯ an], then colsp(A) = span({a1, ..., an}). Given a matrix A, the action of the matrix A on a vector x returns a linear combination of the columns of A with the coordinates of x as coefficients; that is, the columns of the matrix generate the column space. === Example === Given a matrix J: J = [ 2 4 1 3 2 − 1 − 2 1 0 5 1 6 2 2 2 3 6 2 5 1 ] {\displaystyle J={\begin{bmatrix}2&4&1&3&2\\-1&-2&1&0&5\\1&6&2&2&2\\3&6&2&5&1\end{bmatrix}}} the rows are r 1 = [ 2 4 1 3 2 ] {\displaystyle \mathbf {r} _{1}={\begin{bmatrix}2&4&1&3&2\end{bmatrix}}} , r 2 = [ − 1 − 2 1 0 5 ] {\displaystyle \mathbf {r} _{2}={\begin{bmatrix}-1&-2&1&0&5\end{bmatrix}}} , r 3 = [ 1 6 2 2 2 ] {\displaystyle \mathbf {r} _{3}={\begin{bmatrix}1&6&2&2&2\end{bmatrix}}} , r 4 = [ 3 6 2 5 1 ] {\displaystyle \mathbf {r} _{4}={\begin{bmatrix}3&6&2&5&1\end{bmatrix}}} . Consequently, the row space of J is the subspace of R 5 {\displaystyle \mathbb {R} ^{5}} spanned by { r1, r2, r3, r4 }. Since these four row vectors are linearly independent, the row space is 4-dimensional. Moreover, in this case it can be seen that they are all orthogonal to the vector n = [6, −1, 4, −4, 0] (n is an element of the kernel of J ), so it can be deduced that the row space consists of all vectors in R 5 {\displaystyle \mathbb {R} ^{5}} that are orthogonal to n. == Column space == === Definition === Let K be a field of scalars. Let A be an m × n matrix, with column vectors v1, v2, ..., vn. A linear combination of these vectors is any vector of the form c 1 v 1 + c 2 v 2 + ⋯ + c n v n , {\displaystyle c_{1}\mathbf {v} _{1}+c_{2}\mathbf {v} _{2}+\cdots +c_{n}\mathbf {v} _{n},} where c1, c2, ..., cn are scalars. The set of all possible linear combinations of v1, ..., vn is called the column space of A. That is, the column space of A is the span of the vectors v1, ..., vn. Any linear combination of the column vectors of a matrix A can be written as the product of A with a column vector: A [ c 1 ⋮ c n ] = [ a 11 ⋯ a 1 n ⋮ ⋱ ⋮ a m 1 ⋯ a m n ] [ c 1 ⋮ c n ] = [ c 1 a 11 + ⋯ + c n a 1 n ⋮ c 1 a m 1 + ⋯ + c n a m n ] = c 1 [ a 11 ⋮ a m 1 ] + ⋯ + c n [ a 1 n ⋮ a m n ] = c 1 v 1 + ⋯ + c n v n {\displaystyle {\begin{array}{rcl}A{\begin{bmatrix}c_{1}\\\vdots \\c_{n}\end{bmatrix}}&=&{\begin{bmatrix}a_{11}&\cdots &a_{1n}\\\vdots &\ddots &\vdots \\a_{m1}&\cdots &a_{mn}\end{bmatrix}}{\begin{bmatrix}c_{1}\\\vdots \\c_{n}\end{bmatrix}}={\begin{bmatrix}c_{1}a_{11}+\cdots +c_{n}a_{1n}\\\vdots \\c_{1}a_{m1}+\cdots +c_{n}a_{mn}\end{bmatrix}}=c_{1}{\begin{bmatrix}a_{11}\\\vdots \\a_{m1}\end{bmatrix}}+\cdots +c_{n}{\begin{bmatrix}a_{1n}\\\vdots \\a_{mn}\end{bmatrix}}\\&=&c_{1}\mathbf {v} _{1}+\cdots +c_{n}\mathbf {v} _{n}\end{array}}} Therefore, the column space of A consists of all possible products Ax, for x ∈ Kn. This is the same as the image (or range) of the corresponding matrix transformation. ==== Example ==== If A = [ 1 0 0 1 2 0 ] {\displaystyle A={\begin{bmatrix}1&0\\0&1\\2&0\end{bmatrix}}} , then the column vectors are v1 = [1, 0, 2]T and v2 = [0, 1, 0]T. A linear combination of v1 and v2 is any vector of the form c 1 [ 1 0 2 ] + c 2 [ 0 1 0 ] = [ c 1 c 2 2 c 1 ] {\displaystyle c_{1}{\begin{bmatrix}1\\0\\2\end{bmatrix}}+c_{2}{\begin{bmatrix}0\\1\\0\end{bmatrix}}={\begin{bmatrix}c_{1}\\c_{2}\\2c_{1}\end{bmatrix}}} The set of all such vectors is the column space of A. In this case, the column space is precisely the set of vectors (x, y, z) ∈ R3 satisfying the equation z = 2x (using Cartesian coordinates, this set is a plane through the origin in three-dimensional space). === Basis === The columns of A span the column space, but they may not form a basis if the column vectors are not linearly independent. Fortunately, elementary row operations do not affect the dependence relations between the column vectors. This makes it possible to use row reduction to find a basis for the column space. For example, consider the matrix A = [ 1 3 1 4 2 7 3 9 1 5 3 1 1 2 0 8 ] . {\displaystyle A={\begin{bmatrix}1&3&1&4\\2&7&3&9\\1&5&3&1\\1&2&0&8\end{bmatrix}}.} The columns of this matrix span the column space, but they may not be linearly independent, in which case some subset of them will form a basis. To find this basis, we reduce A to reduced row echelon form: [ 1 3 1 4 2 7 3 9 1 5 3 1 1 2 0 8 ] ∼ [ 1 3 1 4 0 1 1 1 0 2 2 − 3 0 − 1 − 1 4 ] ∼ [ 1 0 − 2 1 0 1 1 1 0 0 0 − 5 0 0 0 5 ] ∼ [ 1 0 − 2 0 0 1 1 0 0 0 0 1 0 0 0 0 ] . {\displaystyle {\begin{bmatrix}1&3&1&4\\2&7&3&9\\1&5&3&1\\1&2&0&8\end{bmatrix}}\sim {\begin{bmatrix}1&3&1&4\\0&1&1&1\\0&2&2&-3\\0&-1&-1&4\end{bmatrix}}\sim {\begin{bmatrix}1&0&-2&1\\0&1&1&1\\0&0&0&-5\\0&0&0&5\end{bmatrix}}\sim {\begin{bmatrix}1&0&-2&0\\0&1&1&0\\0&0&0&1\\0&0&0&0\end{bmatrix}}.} At this point, it is clear that the first, second, and fourth columns are linearly independent, while the third column is a linear combination of the first two. (Specifically, v3 = −2v1 + v2.) Therefore, the first, second, and fourth columns of the original matrix are a basis for the column space: [ 1 2 1 1 ] , [ 3 7 5 2 ] , [ 4 9 1 8 ] . {\displaystyle {\begin{bmatrix}1\\2\\1\\1\end{bmatrix}},\;\;{\begin{bmatrix}3\\7\\5\\2\end{bmatrix}},\;\;{\begin{bmatrix}4\\9\\1\\8\end{bmatrix}}.} Note that the independent columns of the reduced row echelon form are precisely the columns with pivots. This makes it possible to determine which columns are linearly independent by reducing only to echelon form. The above algorithm can be used in general to find the dependence relations between any set of vectors, and to pick out a basis from any spanning set. Also finding a basis for the column space of A is equivalent to finding a basis for the row space of the transpose matrix AT. To find the basis in a practical setting (e.g., for large matrices), the singular-value decomposition is typically used. === Dimension === The dimension of the column space is called the rank of the matrix. The rank is equal to the number of pivots in the reduced row echelon form, and is the maximum number of linearly independent columns that can be chosen from the matrix. For example, the 4 × 4 matrix in the example above has rank three. Because the column space is the image of the corresponding matrix transformation, the rank of a matrix is the same as the dimension of the image. For example, the transformation R 4 → R 4 {\displaystyle \mathbb {R} ^{4}\to \mathbb {R} ^{4}} described by the matrix above maps all of R 4 {\displaystyle \mathbb {R} ^{4}} to some three-dimensional subspace. The nullity of a matrix is the dimension of the null space, and is equal to the number of columns in the reduced row echelon form that do not have pivots. The rank and nullity of a matrix A with n columns are related by the equation: rank ⁡ ( A ) + nullity ⁡ ( A ) = n . {\displaystyle \operatorname {rank} (A)+\operatorname {nullity} (A)=n.\,} This is known as the rank–nullity theorem. === Relation to the left null space === The left null space of A is the set of all vectors x such that xTA = 0T. It is the same as the null space of the transpose of A. The product of the matrix AT and the vector x can be written in terms of the dot product of vectors: A T x = [ v 1 ⋅ x v 2 ⋅ x ⋮ v n ⋅ x ] , {\displaystyle A^{\mathsf {T}}\mathbf {x} ={\begin{bmatrix}\mathbf {v} _{1}\cdot \mathbf {x} \\\mathbf {v} _{2}\cdot \mathbf {x} \\\vdots \\\mathbf {v} _{n}\cdot \mathbf {x} \end{bmatrix}},} because row vectors of AT are transposes of column vectors vk of A. Thus ATx = 0 if and only if x is orthogonal (perpendicular) to each of the column vectors of A. It follows that the left null space (the null space of AT) is the orthogonal complement to the column space of A. For a matrix A, the column space, row space, null space, and left null space are sometimes referred to as the four fundamental subspaces. === For matrices over a ring === Similarly the column space (sometimes disambiguated as right column space) can be defined for matrices over a ring K as ∑ k = 1 n v k c k {\displaystyle \sum \limits _{k=1}^{n}\mathbf {v} _{k}c_{k}} for any c1, ..., cn, with replacement of the vector m-space with "right free module", which changes the order of scalar multiplication of the vector vk to the scalar ck such that it is written in an unusual order vector–scalar. == Row space == === Definition === Let K be a field of scalars. Let A be an m × n matrix, with row vectors r1, r2, ..., rm. A linear combination of these vectors is any vector of the form c 1 r 1 + c 2 r 2 + ⋯ + c m r m , {\displaystyle c_{1}\mathbf {r} _{1}+c_{2}\mathbf {r} _{2}+\cdots +c_{m}\mathbf {r} _{m},} where c1, c2, ..., cm are scalars. The set of all possible linear combinations of r1, ..., rm is called the row space of A. That is, the row space of A is the span of the vectors r1, ..., rm. For example, if A = [ 1 0 2 0 1 0 ] , {\displaystyle A={\begin{bmatrix}1&0&2\\0&1&0\end{bmatrix}},} then the row vectors are r1 = [1, 0, 2] and r2 = [0, 1, 0]. A linear combination of r1 and r2 is any vector of the form c 1 [ 1 0 2 ] + c 2 [ 0 1 0 ] = [ c 1 c 2 2 c 1 ] . {\displaystyle c_{1}{\begin{bmatrix}1&0&2\end{bmatrix}}+c_{2}{\begin{bmatrix}0&1&0\end{bmatrix}}={\begin{bmatrix}c_{1}&c_{2}&2c_{1}\end{bmatrix}}.} The set of all such vectors is the row space of A. In this case, the row space is precisely the set of vectors (x, y, z) ∈ K3 satisfying the equation z = 2x (using Cartesian coordinates, this set is a plane through the origin in three-dimensional space). For a matrix that represents a homogeneous system of linear equations, the row space consists of all linear equations that follow from those in the system. The column space of A is equal to the row space of AT. === Basis === The row space is not affected by elementary row operations. This makes it possible to use row reduction to find a basis for the row space. For example, consider the matrix A = [ 1 3 2 2 7 4 1 5 2 ] . {\displaystyle A={\begin{bmatrix}1&3&2\\2&7&4\\1&5&2\end{bmatrix}}.} The rows of this matrix span the row space, but they may not be linearly independent, in which case the rows will not be a basis. To find a basis, we reduce A to row echelon form: r1, r2, r3 represents the rows. [ 1 3 2 2 7 4 1 5 2 ] → r 2 − 2 r 1 → r 2 [ 1 3 2 0 1 0 1 5 2 ] → r 3 − r 1 → r 3 [ 1 3 2 0 1 0 0 2 0 ] → r 3 − 2 r 2 → r 3 [ 1 3 2 0 1 0 0 0 0 ] → r 1 − 3 r 2 → r 1 [ 1 0 2 0 1 0 0 0 0 ] . {\displaystyle {\begin{aligned}{\begin{bmatrix}1&3&2\\2&7&4\\1&5&2\end{bmatrix}}&\xrightarrow {\mathbf {r} _{2}-2\mathbf {r} _{1}\to \mathbf {r} _{2}} {\begin{bmatrix}1&3&2\\0&1&0\\1&5&2\end{bmatrix}}\xrightarrow {\mathbf {r} _{3}-\,\,\mathbf {r} _{1}\to \mathbf {r} _{3}} {\begin{bmatrix}1&3&2\\0&1&0\\0&2&0\end{bmatrix}}\\&\xrightarrow {\mathbf {r} _{3}-2\mathbf {r} _{2}\to \mathbf {r} _{3}} {\begin{bmatrix}1&3&2\\0&1&0\\0&0&0\end{bmatrix}}\xrightarrow {\mathbf {r} _{1}-3\mathbf {r} _{2}\to \mathbf {r} _{1}} {\begin{bmatrix}1&0&2\\0&1&0\\0&0&0\end{bmatrix}}.\end{aligned}}} Once the matrix is in echelon form, the nonzero rows are a basis for the row space. In this case, the basis is { [1, 3, 2], [2, 7, 4] }. Another possible basis { [1, 0, 2], [0, 1, 0] } comes from a further reduction. This algorithm can be used in general to find a basis for the span of a set of vectors. If the matrix is further simplified to reduced row echelon form, then the resulting basis is uniquely determined by the row space. It is sometimes convenient to find a basis for the row space from among the rows of the original matrix instead (for example, this result is useful in giving an elementary proof that the determinantal rank of a matrix is equal to its rank). Since row operations can affect linear dependence relations of the row vectors, such a basis is instead found indirectly using the fact that the column space of AT is equal to the row space of A. Using the example matrix A above, find AT and reduce it to row echelon form: A T = [ 1 2 1 3 7 5 2 4 2 ] ∼ [ 1 2 1 0 1 2 0 0 0 ] . {\displaystyle A^{\mathrm {T} }={\begin{bmatrix}1&2&1\\3&7&5\\2&4&2\end{bmatrix}}\sim {\begin{bmatrix}1&2&1\\0&1&2\\0&0&0\end{bmatrix}}.} The pivots indicate that the first two columns of AT form a basis of the column space of AT. Therefore, the first two rows of A (before any row reductions) also form a basis of the row space of A. === Dimension === The dimension of the row space is called the rank of the matrix. This is the same as the maximum number of linearly independent rows that can be chosen from the matrix, or equivalently the number of pivots. For example, the 3 × 3 matrix in the example above has rank two. The rank of a matrix is also equal to the dimension of the column space. The dimension of the null space is called the nullity of the matrix, and is related to the rank by the following equation: rank ⁡ ( A ) + nullity ⁡ ( A ) = n , {\displaystyle \operatorname {rank} (A)+\operatorname {nullity} (A)=n,} where n is the number of columns of the matrix A. The equation above is known as the rank–nullity theorem. === Relation to the null space === The null space of matrix A is the set of all vectors x for which Ax = 0. The product of the matrix A and the vector x can be written in terms of the dot product of vectors: A x = [ r 1 ⋅ x r 2 ⋅ x ⋮ r m ⋅ x ] , {\displaystyle A\mathbf {x} ={\begin{bmatrix}\mathbf {r} _{1}\cdot \mathbf {x} \\\mathbf {r} _{2}\cdot \mathbf {x} \\\vdots \\\mathbf {r} _{m}\cdot \mathbf {x} \end{bmatrix}},} where r1, ..., rm are the row vectors of A. Thus Ax = 0 if and only if x is orthogonal (perpendicular) to each of the row vectors of A. It follows that the null space of A is the orthogonal complement to the row space. For example, if the row space is a plane through the origin in three dimensions, then the null space will be the perpendicular line through the origin. This provides a proof of the rank–nullity theorem (see dimension above). The row space and null space are two of the four fundamental subspaces associated with a matrix A (the other two being the column space and left null space). === Relation to coimage === If V and W are vector spaces, then the kernel of a linear transformation T: V → W is the set of vectors v ∈ V for which T(v) = 0. The kernel of a linear transformation is analogous to the null space of a matrix. If V is an inner product space, then the orthogonal complement to the kernel can be thought of as a generalization of the row space. This is sometimes called the coimage of T. The transformation T is one-to-one on its coimage, and the coimage maps isomorphically onto the image of T. When V is not an inner product space, the coimage of T can be defined as the quotient space V / ker(T). == See also == Euclidean subspace == References & Notes == == Further reading == Anton, Howard (1987), Elementary Linear Algebra (5th ed.), New York: Wiley, ISBN 0-471-84819-0 Axler, Sheldon Jay (1997), Linear Algebra Done Right (2nd ed.), Springer-Verlag, ISBN 0-387-98259-0 Banerjee, Sudipto; Roy, Anindya (June 6, 2014), Linear Algebra and Matrix Analysis for Statistics (1st ed.), CRC Press, ISBN 978-1-42-009538-8 Beauregard, Raymond A.; Fraleigh, John B. (1973), A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields, Boston: Houghton Mifflin Company, ISBN 0-395-14017-X Lay, David C. (August 22, 2005), Linear Algebra and Its Applications (3rd ed.), Addison Wesley, ISBN 978-0-321-28713-7 Leon, Steven J. (2006), Linear Algebra With Applications (7th ed.), Pearson Prentice Hall Meyer, Carl D. (February 15, 2001), Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics (SIAM), ISBN 978-0-89871-454-8, archived from the original on March 1, 2001 Poole, David (2006), Linear Algebra: A Modern Introduction (2nd ed.), Brooks/Cole, ISBN 0-534-99845-3 Strang, Gilbert (July 19, 2005), Linear Algebra and Its Applications (4th ed.), Brooks Cole, ISBN 978-0-03-010567-8 == External links == Weisstein, Eric W. "Row Space". MathWorld. Weisstein, Eric W. "Column Space". MathWorld. Gilbert Strang, MIT Linear Algebra Lecture on the Four Fundamental Subspaces at Google Video, from MIT OpenCourseWare Khan Academy video tutorial Lecture on column space and nullspace by Gilbert Strang of MIT Row Space and Column Space
Wikipedia:Row and column vectors#0	In linear algebra, a column vector with ⁠ m {\displaystyle m} ⁠ elements is an m × 1 {\displaystyle m\times 1} matrix consisting of a single column of ⁠ m {\displaystyle m} ⁠ entries, for example, x = [ x 1 x 2 ⋮ x m ] . {\displaystyle {\boldsymbol {x}}={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{m}\end{bmatrix}}.} Similarly, a row vector is a 1 × n {\displaystyle 1\times n} matrix for some ⁠ n {\displaystyle n} ⁠, consisting of a single row of ⁠ n {\displaystyle n} ⁠ entries, a = [ a 1 a 2 … a n ] . {\displaystyle {\boldsymbol {a}}={\begin{bmatrix}a_{1}&a_{2}&\dots &a_{n}\end{bmatrix}}.} (Throughout this article, boldface is used for both row and column vectors.) The transpose (indicated by T) of any row vector is a column vector, and the transpose of any column vector is a row vector: [ x 1 x 2 … x m ] T = [ x 1 x 2 ⋮ x m ] {\displaystyle {\begin{bmatrix}x_{1}\;x_{2}\;\dots \;x_{m}\end{bmatrix}}^{\rm {T}}={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{m}\end{bmatrix}}} and [ x 1 x 2 ⋮ x m ] T = [ x 1 x 2 … x m ] . {\displaystyle {\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{m}\end{bmatrix}}^{\rm {T}}={\begin{bmatrix}x_{1}\;x_{2}\;\dots \;x_{m}\end{bmatrix}}.} The set of all row vectors with n entries in a given field (such as the real numbers) forms an n-dimensional vector space; similarly, the set of all column vectors with m entries forms an m-dimensional vector space. The space of row vectors with n entries can be regarded as the dual space of the space of column vectors with n entries, since any linear functional on the space of column vectors can be represented as the left-multiplication of a unique row vector. == Notation == To simplify writing column vectors in-line with other text, sometimes they are written as row vectors with the transpose operation applied to them. x = [ x 1 x 2 … x m ] T {\displaystyle {\boldsymbol {x}}={\begin{bmatrix}x_{1}\;x_{2}\;\dots \;x_{m}\end{bmatrix}}^{\rm {T}}} or x = [ x 1 , x 2 , … , x m ] T {\displaystyle {\boldsymbol {x}}={\begin{bmatrix}x_{1},x_{2},\dots ,x_{m}\end{bmatrix}}^{\rm {T}}} Some authors also use the convention of writing both column vectors and row vectors as rows, but separating row vector elements with commas and column vector elements with semicolons (see alternative notation 2 in the table below). == Operations == Matrix multiplication involves the action of multiplying each row vector of one matrix by each column vector of another matrix. The dot product of two column vectors a, b, considered as elements of a coordinate space, is equal to the matrix product of the transpose of a with b, a ⋅ b = a ⊺ b = [ a 1 ⋯ a n ] [ b 1 ⋮ b n ] = a 1 b 1 + ⋯ + a n b n , {\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {a} ^{\intercal }\mathbf {b} ={\begin{bmatrix}a_{1}&\cdots &a_{n}\end{bmatrix}}{\begin{bmatrix}b_{1}\\\vdots \\b_{n}\end{bmatrix}}=a_{1}b_{1}+\cdots +a_{n}b_{n}\,,} By the symmetry of the dot product, the dot product of two column vectors a, b is also equal to the matrix product of the transpose of b with a, b ⋅ a = b ⊺ a = [ b 1 ⋯ b n ] [ a 1 ⋮ a n ] = a 1 b 1 + ⋯ + a n b n . {\displaystyle \mathbf {b} \cdot \mathbf {a} =\mathbf {b} ^{\intercal }\mathbf {a} ={\begin{bmatrix}b_{1}&\cdots &b_{n}\end{bmatrix}}{\begin{bmatrix}a_{1}\\\vdots \\a_{n}\end{bmatrix}}=a_{1}b_{1}+\cdots +a_{n}b_{n}\,.} The matrix product of a column and a row vector gives the outer product of two vectors a, b, an example of the more general tensor product. The matrix product of the column vector representation of a and the row vector representation of b gives the components of their dyadic product, a ⊗ b = a b ⊺ = [ a 1 a 2 a 3 ] [ b 1 b 2 b 3 ] = [ a 1 b 1 a 1 b 2 a 1 b 3 a 2 b 1 a 2 b 2 a 2 b 3 a 3 b 1 a 3 b 2 a 3 b 3 ] , {\displaystyle \mathbf {a} \otimes \mathbf {b} =\mathbf {a} \mathbf {b} ^{\intercal }={\begin{bmatrix}a_{1}\\a_{2}\\a_{3}\end{bmatrix}}{\begin{bmatrix}b_{1}&b_{2}&b_{3}\end{bmatrix}}={\begin{bmatrix}a_{1}b_{1}&a_{1}b_{2}&a_{1}b_{3}\\a_{2}b_{1}&a_{2}b_{2}&a_{2}b_{3}\\a_{3}b_{1}&a_{3}b_{2}&a_{3}b_{3}\\\end{bmatrix}}\,,} which is the transpose of the matrix product of the column vector representation of b and the row vector representation of a, b ⊗ a = b a ⊺ = [ b 1 b 2 b 3 ] [ a 1 a 2 a 3 ] = [ b 1 a 1 b 1 a 2 b 1 a 3 b 2 a 1 b 2 a 2 b 2 a 3 b 3 a 1 b 3 a 2 b 3 a 3 ] . {\displaystyle \mathbf {b} \otimes \mathbf {a} =\mathbf {b} \mathbf {a} ^{\intercal }={\begin{bmatrix}b_{1}\\b_{2}\\b_{3}\end{bmatrix}}{\begin{bmatrix}a_{1}&a_{2}&a_{3}\end{bmatrix}}={\begin{bmatrix}b_{1}a_{1}&b_{1}a_{2}&b_{1}a_{3}\\b_{2}a_{1}&b_{2}a_{2}&b_{2}a_{3}\\b_{3}a_{1}&b_{3}a_{2}&b_{3}a_{3}\\\end{bmatrix}}\,.} == Matrix transformations == An n × n matrix M can represent a linear map and act on row and column vectors as the linear map's transformation matrix. For a row vector v, the product vM is another row vector p: v M = p . {\displaystyle \mathbf {v} M=\mathbf {p} \,.} Another n × n matrix Q can act on p, p Q = t . {\displaystyle \mathbf {p} Q=\mathbf {t} \,.} Then one can write t = pQ = vMQ, so the matrix product transformation MQ maps v directly to t. Continuing with row vectors, matrix transformations further reconfiguring n-space can be applied to the right of previous outputs. When a column vector is transformed to another column vector under an n × n matrix action, the operation occurs to the left, p T = M v T , t T = Q p T , {\displaystyle \mathbf {p} ^{\mathrm {T} }=M\mathbf {v} ^{\mathrm {T} }\,,\quad \mathbf {t} ^{\mathrm {T} }=Q\mathbf {p} ^{\mathrm {T} },} leading to the algebraic expression QM vT for the composed output from vT input. The matrix transformations mount up to the left in this use of a column vector for input to matrix transformation. == See also == Covariance and contravariance of vectors Index notation Vector of ones Single-entry vector Standard unit vector Unit vector == Notes == == References == Axler, Sheldon Jay (1997), Linear Algebra Done Right (2nd ed.), Springer-Verlag, ISBN 0-387-98259-0 Lay, David C. (August 22, 2005), Linear Algebra and Its Applications (3rd ed.), Addison Wesley, ISBN 978-0-321-28713-7 Meyer, Carl D. (February 15, 2001), Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics (SIAM), ISBN 978-0-89871-454-8, archived from the original on March 1, 2001 Poole, David (2006), Linear Algebra: A Modern Introduction (2nd ed.), Brooks/Cole, ISBN 0-534-99845-3 Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International Leon, Steven J. (2006), Linear Algebra With Applications (7th ed.), Pearson Prentice Hall
Wikipedia:Row equivalence#0	In linear algebra, two matrices are row equivalent if one can be changed to the other by a sequence of elementary row operations. Alternatively, two m × n matrices are row equivalent if and only if they have the same row space. The concept is most commonly applied to matrices that represent systems of linear equations, in which case two matrices of the same size are row equivalent if and only if the corresponding homogeneous systems have the same set of solutions, or equivalently the matrices have the same null space. Because elementary row operations are reversible, row equivalence is an equivalence relation. It is commonly denoted by a tilde (~). There is a similar notion of column equivalence, defined by elementary column operations; two matrices are column equivalent if and only if their transpose matrices are row equivalent. Two rectangular matrices that can be converted into one another allowing both elementary row and column operations are called simply equivalent. == Elementary row operations == An elementary row operation is any one of the following moves: Swap: Swap two rows of a matrix. Scale: Multiply a row of a matrix by a nonzero constant. Pivot: Add a multiple of one row of a matrix to another row. Two matrices A and B are row equivalent if it is possible to transform A into B by a sequence of elementary row operations. == Row space == The row space of a matrix is the set of all possible linear combinations of its row vectors. If the rows of the matrix represent a system of linear equations, then the row space consists of all linear equations that can be deduced algebraically from those in the system. Two m × n matrices are row equivalent if and only if they have the same row space. For example, the matrices ( 1 0 0 0 1 1 ) and ( 1 0 0 1 1 1 ) {\displaystyle {\begin{pmatrix}1&0&0\\0&1&1\end{pmatrix}}\;\;\;\;{\text{and}}\;\;\;\;{\begin{pmatrix}1&0&0\\1&1&1\end{pmatrix}}} are row equivalent, the row space being all vectors of the form ( a b b ) {\displaystyle {\begin{pmatrix}a&b&b\end{pmatrix}}} . The corresponding systems of homogeneous equations convey the same information: x = 0 y + z = 0 and x = 0 x + y + z = 0. {\displaystyle {\begin{matrix}x=0\\y+z=0\end{matrix}}\;\;\;\;{\text{and}}\;\;\;\;{\begin{matrix}x=0\\x+y+z=0.\end{matrix}}} In particular, both of these systems imply every equation of the form a x + b y + b z = 0. {\displaystyle ax+by+bz=0.\,} == Equivalence of the definitions == The fact that two matrices are row equivalent if and only if they have the same row space is an important theorem in linear algebra. The proof is based on the following observations: Elementary row operations do not affect the row space of a matrix. In particular, any two row equivalent matrices have the same row space. Any matrix can be reduced by elementary row operations to a matrix in reduced row echelon form. Two matrices in reduced row echelon form have the same row space if and only if they are equal. This line of reasoning also proves that every matrix is row equivalent to a unique matrix with reduced row echelon form. == Additional properties == Because the null space of a matrix is the orthogonal complement of the row space, two matrices are row equivalent if and only if they have the same null space. The rank of a matrix is equal to the dimension of the row space, so row equivalent matrices must have the same rank. This is equal to the number of pivots in the reduced row echelon form. A matrix is invertible if and only if it is row equivalent to the identity matrix. Matrices A and B are row equivalent if and only if there exists an invertible matrix P such that A=PB. == See also == Elementary row operations Row space Basis (linear algebra) Row reduction (Reduced) row echelon form == References == === Bibliography === Axler, Sheldon Jay (1997), Linear Algebra Done Right (2nd ed.), Springer-Verlag, ISBN 0-387-98259-0 Lay, David C. (August 22, 2005), Linear Algebra and Its Applications (3rd ed.), Addison Wesley, ISBN 978-0-321-28713-7 Meyer, Carl D. (February 15, 2001), Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics (SIAM), ISBN 978-0-89871-454-8, archived from the original on March 1, 2001 Poole, David (2006), Linear Algebra: A Modern Introduction (2nd ed.), Brooks/Cole, ISBN 0-534-99845-3 Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International Leon, Steven J. (2006), Linear Algebra With Applications (7th ed.), Pearson Prentice Hall Roman, Steven (2008). Advanced Linear Algebra. Graduate Texts in Mathematics. Vol. 135 (3rd ed.). Springer Science+Business Media, LLC. ISBN 978-0-387-72828-5. == External links ==
Wikipedia:Rudolf Kruse#0	Rudolf Kruse (born 12 September 1952 in Rotenburg/Wümme) is a German computer scientist and mathematician. == Education and professional career == Rudolf Kruse obtained his diploma (Mathematics) degree in 1979 from the TU Braunschweig, Germany, and a PhD in Mathematics in 1980 as well as the venia legendi in Mathematics in 1984 from the same university. Following a stay at the Fraunhofer Society, in 1986 he joined the University of Braunschweig as a professor of computer science. From 1996–2017 he was a professor at the Department of Computer Science of the Otto-von-Guericke Universität Magdeburg where he has been leading the computational intelligence research group. Since October 2017 he has been an emeritus professor. == Research activities == He has carried out research and projects in data science, artificial intelligence, fuzzy systems, and computational intelligence. His research group was very successful in various industrial applications. Rudolf Kruse has coauthored 40 books as well as more than 450 refereed technical papers in various scientific areas. He is a fellow of the International Fuzzy Systems Association (IFSA), fellow of the European Association for Artificial Intelligence: EurAI) and fellow of the Institute of Electrical and Electronics Engineers: (IEEE). == References == == External links == Information about Rudolf Kruse and his work DBLP computer science bibliography of Rudolf Kruse Website Computational Intelligence group at Otto-von-Guericke University Magdeburg
Wikipedia:Rudranath Capildeo#0	Rudranath Capildeo (pronounced [rud̪rənɑːt̪ʰə kəpiləd̪eːoː]; 2 February 1920 – 12 May 1970) was a Trinidadian and Tobagonian politician, mathematician and barrister. He was a member of the prominent Hindu Indo-Trinidadian Capildeo family. Capildeo was the leader of the Democratic Labour Party (DLP) from 1960 to 1969 and the first Leader of the Opposition in the Parliament of the independent Trinidad and Tobago from 1962 to 1967. He was also a faculty member at the University of London, eventually holding the position of Reader of Mathematics. He was awarded the Trinity Cross, the nation's highest award, in 1969. == Early years and education == Rudranath Capildeo was born on 2 February 1920 into a Brahmin Hindu Indo-Trinidadian family at Anand Bhavan (translation: Mansion of Eternal Bliss; aka Lion House) on the Main Road in the city of Chaguanas in Caroni County in the then British-ruled Trinidad and Tobago. His father was Pundit Capil Deo Dubey, an indentured laborer who emigrated from Mahadeva Dubey Village, Gorakhpur district, North-Western Provinces, British India (present-day Mahadeva Dubey, Maharajganj district, Gorakhpur division, Uttar Pradesh, India) in 1894, and his mother was Soogee Capildeo (née Gobin). Capildeo was the youngest child of the prominent Capildeo family. He was educated at Queen's Royal College in Port of Spain where he won an island scholarship in 1938. He attended the University of London where he obtained his BSc in Mathematics and Physics in 1943, his MSc in Mathematics in 1945, and his PhD in Mathematical Physics in 1948, his thesis being entitled "The flexure problem in elasticity". == Career == Capildeo held lectureships at the University of London, including at both University College London and at Westfield College. He also taught briefly at Queen's Royal College (1945) and was Principal of the Polytechnic Institute in Port of Spain in 1959. Capildeo's entry into politics in the late 1950s was because the political figures who entered the DLP in 1957 did not trust each other, and could only agree on him. Though he left Trinidad in 1939 to study medicine, he changed his course of study, focusing on applied mathematics and physics. He was committed to understanding the nature of space and time, and this sparked his interest in understanding Einstein's Theory of Special Relativity. This work led to several new theories, which had practical implications in aerodynamics and space. They included "The Flexure Problem in Elasticity" (Ph.D. thesis) and his study on the "Theory of Rotation and Gravity" named Capildeo's Theory, which had applications in early outer-space expeditions in the 1960s and 1970s. His political career was unusual, since he was active only during election campaigns (in 1961 and 1967) and during the summer months. His conduct of the last pre-independence electoral campaign, in 1961, was also unusual, beginning with his declaration that because he understood Einstein he could "compress the time" necessary to undo what the sitting government had done. Not only did Capildeo produce many significant mathematical theories and a book on Vector Algebra and Mechanics in 1967, he also studied law in London in 1956. Two years later he was admitted to practice as a barrister-at-law in Trinidad. He founded and led the Democratic Labour Party (DLP) in 1960, and became Leader of the Opposition in the Trinidad & Tobago Parliament (1960–67). With Eric Williams as Prime Minister, both men laid the foundation for an independent Trinidad and Tobago. Capildeo was also responsible for having the freedom of worship included in the Constitution of Trinidad and Tobago and Service Commissions because he felt that service commissions would ensure equality and fairness in the appointment of people to public office. The Rudranath Capildeo Learning Resource Centre (RCLRC) is located in McBean Village, Couva, Trinidad. == Personal life == He was married to Ruth Goodchild in 1944 and they had one son named Rudy Capildeo. He also has a daughter, Anne Saraswati Gasteen Capildeo, born was in 1959. He was the younger brother of Simbhoonath Capildeo, brother-in-law of Seepersad Naipaul, uncle of Nobel Prize-winning author V.S. Naipaul, Shiva Naipaul, and Surendranath Capildeo, uncle-in-law of Nadira Naipaul, and grand uncle of Vahni Capildeo and Neil Bissoondath. == References ==
Wikipedia:Rudy Horne#0	Rudy Lee Horne (1968 – 2017) was an American mathematician and professor of mathematics at Morehouse College. He worked on dynamical systems, including nonlinear waves. He was the mathematics consultant for the film Hidden Figures. == Early life and education == Horne grew up in the south side of Chicago. His father worked at Sherwin-Williams. He graduated from Crete-Monee High School. He completed a double degree in mathematics and physics at the University of Oklahoma in 1991. He joined the University of Colorado Boulder for his postgraduate studies, earning a master's in physics in 1994 and in mathematics in 1996. He completed his doctorate, Collision induced timing jitter and four-wave mixing in wavelength division multiplexing soliton systems, in 2001 which was supervised by Mark J. Ablowitz. He was the first African American to graduate from the University of Colorado Boulder Department of Applied Mathematics. == Career and research == After completing his PhD, Horne had a position at the California State University, East Bay. before working as postdoctoral researcher at the University of North Carolina at Chapel Hill, with Chris Jones. Horne joined Florida State University in 2005. Horne joined Morehouse College in 2010 and was promoted to associate professor of mathematics in 2015. He continued to study four-wave mixing. His work considered nonlinear optical phenomena. He uncovered effects in parity-time symmetric systems. Horne was recommended to serve as a mathematics consultant for Hidden Figures by Morehouse College. He worked closely with Theodore Melfi ensured the actors knew how to pronounce "Euler's". He spent four months working with 20th Century Fox. In particular, Horne worked with Taraji P. Henson on the mathematics she required for her role as Katherine Johnson. He taught the cast how to get excited by mathematics. His handwriting is on screen during a scene at the beginning of the film where Katherine Johnson solves a quadratic equation. He appeared on the interview series In the Know. Horne completed a Mathematical Association of America Maths Fest tour where he discussed the mathematics in Hidden Figures, focusing on the calculations that concerned Glenn's orbit around in 1962. He appeared on NPR's Closer Look. He died on December 11, 2017, after surgery for a torn aorta. The University of Colorado Boulder established a Rudy Lee Horne Memorial Fellowship in his honour. He was described as a "rock star", inspiring generations of black students. He was awarded the National Association of Mathematicians (NAM) lifetime achievement award posthumously in 2018, and was recognized by Mathematically Gifted & Black as a Black History Month 2018 Honoree. == References ==
Wikipedia:Ruggiero Torelli#0	Ruggiero Torelli (7 June 1884 in Naples – 9 September 1915) was an Italian mathematician who introduced Torelli's theorem, a classical result of algebraic geometry over the complex number field. == Publications == Ruggiero Torelli (1913). "Sulle varietà di Jacobi". Rendiconti della Reale accademia nazionale dei Lincei. 22 (5): 98–103. Torelli, Ruggiero (1995), Ciliberto, Ciro; Ribenboim, Paulo; Sernesi, Edoardo (eds.), Collected papers of Ruggiero Torelli, Queen's Papers in Pure and Applied Mathematics, vol. 101, Kingston, ON: Queen's University, ISBN 0-88911-707-1, MR 1374332 == See also == Torelli group == References == Severi, Francesco (1916), "Ruggiero Torelli", Bollettino di bibliografia e storia delle scienze matematiche, Obituary, 18: 11–21 == External links == Biography Biography in Italian
Wikipedia:Rule of Sarrus#0	In matrix theory, the rule of Sarrus is a mnemonic device for computing the determinant of a 3 × 3 {\displaystyle 3\times 3} matrix named after the French mathematician Pierre Frédéric Sarrus. Consider a 3 × 3 {\displaystyle 3\times 3} matrix M = [ a b c d e f g h i ] {\displaystyle M={\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}}} then its determinant can be computed by the following scheme. Write out the first two columns of the matrix to the right of the third column, giving five columns in a row. Then add the products of the diagonals going from top to bottom (solid) and subtract the products of the diagonals going from bottom to top (dashed). This yields det ( M ) = \| a b c d e f g h i \| = a e i + b f g + c d h − c e g − b d i − a f h . {\displaystyle {\begin{aligned}\det(M)={\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}=aei+bfg+cdh-ceg-bdi-afh.\end{aligned}}} A similar scheme based on diagonals works for 2 × 2 {\displaystyle 2\times 2} matrices: \| a b c d \| = a d − b c {\displaystyle {\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc} Both are special cases of the Leibniz formula, which however does not yield similar memorization schemes for larger matrices. Sarrus' rule can also be derived using the Laplace expansion of a 3 × 3 {\displaystyle 3\times 3} matrix. Another way of thinking of Sarrus' rule is to imagine that the matrix is wrapped around a cylinder, such that the right and left edges are joined. == References == == External links == Sarrus' rule at Planetmath Linear Algebra: Rule of Sarrus of Determinants at khanacademy.org
Wikipedia:Ruma Falk#0	Ruma Falk (Hebrew: רומה פלק, née Oren-Aharonovich, 1932–2020) was an Israeli psychologist and philosopher of mathematics known for her work on probability theory and human understanding of probability and statistics. Falk was born in Jerusalem, and educated at the Herzliya Hebrew Gymnasium and Hebrew University of Jerusalem. She completed her PhD on the perception of chance at the Hebrew University in 1975 under the supervision of Amos Tversky, and became a professor there. She was married to Raphael Falk, a geneticist and historian of science. Falk won the George Pólya Award of the Mathematical Association of America with Maya Bar-Hillel in 1984 for their joint work on probability. She died on August 15, 2020. == Selected works == Falk was the author of books including: Understanding Probability and Statistics: A Book of Problems (A K Peters, 1993) אתגרים לתאים האפורים (Challenges to the Gray Cells, Poalim Library Publishing, 2004) יש בעיה! (There is a Problem, Poalim Library Publishing, 2013) Many Faces of the Gambler's Fallacy: Subjective Randomness and Its Diverse Manifestations (self-published, 2016) She also created a board game, ברירה וסיכוי (Choice and Chance). Her other publications include: Bar-Hillel, Maya; Falk, Ruma (March 1982), "Some teasers concerning conditional probabilities", Cognition, 11 (2): 109–122, doi:10.1016/0010-0277(82)90021-x, PMID 7198956, S2CID 44509163 Falk, Ruma; Bar-Hillel, Maya (June 1983), "Probabilistic dependence between events", The Two-Year College Mathematics Journal, 14 (3): 240–247, doi:10.2307/3027094, JSTOR 3027094 Falk, Ruma; Greenbaum, Charles W. (February 1995), "Significance tests die hard", Theory & Psychology, 5 (1): 75–98, doi:10.1177/0959354395051004, S2CID 143583538 Falk, Ruma; Konold, Clifford (1997), "Making sense of randomness: Implicit encoding as a basis for judgment.", Psychological Review, 104 (2): 301–318, doi:10.1037/0033-295x.104.2.301 == References ==
Wikipedia:Ruriko Yoshida#0	Ruriko (Rudy) Yoshida is a Japanese-American mathematician and statistician whose research topics have ranged from abstract mathematical problems in algebraic combinatorics to optimized camera placement in sensor networks and the phylogenomics of fungi. She works at the Naval Postgraduate School in Monterey, California as a professor of operations research. She was promoted as a rank of professor on July 1, 2023. == Early life and education == Yoshida grew up in Japan. Despite a love of mathematics that began in middle school, she was discouraged from studying mathematics by her teachers, and in response dropped out of her Japanese high school and took the high school equivalency examination instead. In order to continue her study of mathematics, she moved to the US, and after studying at a junior college, transferred to the University of California, Berkeley. Her parents, who had been supporting her financially, stopped their support when they learned that she was studying mathematics instead of business, and she put herself through school working both as a grader in the mathematics department and in the university's police department. She graduated with a bachelor's degree in mathematics in 2000. She went to the University of California, Davis for graduate study, under the supervision of Jesús A. De Loera. De Loera had been a student of Berkeley professor Bernd Sturmfels, and Yoshida also considers Sturmfels to be an academic mentor. Part of her work there involved implementing a method of Alexander Barvinok for counting integer points in convex polyhedra by decomposing the input into cones, and her 2004 dissertation was Barvinok's Rational Functions: Algorithms and Applications to Optimization, Statistics, and Algebra. == Career == After completing her doctorate, Yoshida returned to the University of California, Berkeley as a postdoctoral researcher, working with Lior Pachter in the Center for Pure and Applied Mathematics, and attended Duke University for more postdoctoral research as an assistant research professor of mathematics, working with Mark L. Huber. Yoshida became an assistant professor of statistics at the University of Kentucky in 2006, and was promoted to a tenured associate professor in 2012. In 2016 she moved to her present position at the Naval Postgraduate School, moving there to be closer to her husband's family in California. She has also returned to Japan as a visitor to the Institute of Statistical Mathematics. She is also known as an accomplished teacher of mathematics and statistics as is testified by the provost's announcement and the list of her students . == References == == External links == Home page Ruriko Yoshida publications indexed by Google Scholar
Wikipedia:Ruslan Smelyansky#0	Ruslan Smelyansky (Russian: Русла́н Леони́дович Смеля́нский) (born 1950) is a Russian mathematician, Dr. Sc., Professor, a professor at the Faculty of Computer Science at the Moscow State University, Corresponding Member of the Russian Academy of Sciences. He defended the thesis «Analysis of the performance of multiprocessor systems based on the invariant behavior of programs» for the degree of Doctor of Physical and Mathematical Sciences (1990). He is the author of six books. == References == == Bibliography == Grigoriev, Evgeny (2010). Faculty of Computational Mathematics and Cybernetics: History and Modernity: A Biographical Directory. Moscow: Publishing house of Moscow University. pp. 442–444. ISBN 978-5-211-05838-5. == External links == Annals of the Moscow University(in Russian) MSU CMC(in Russian) Scientific works of Ruslan Smelyansky Scientific works of Ruslan Smelyansky(in English)
Wikipedia:Ruth Baker#0	Ruth Madoc (born Margaret Ruth Llewellyn Baker; 16 April 1943 – 9 December 2022) was a British actress who had a career on stage and screen spanning over 60 years. She was best known for her role as Gladys Pugh in the BBC television comedy Hi-de-Hi! (1980–1988), for which she received a BAFTA TV award nomination for Best Light Entertainment Performance. == Early life and education == Madoc was born Margaret Ruth Llewellyn Baker on 16 April 1943 in Norwich, daughter of George Baker and Iris (née Williams), who worked in healthcare, her father as an administrator and her mother as a nurse. They ran a "poor law" institution for people with severe learning difficulties. Her parents travelled around Britain for much of her childhood, and she was brought up by her Welsh grandmother Etta Williams and her English grandfather at Llansamlet in Swansea. She later trained at the Royal Academy of Dramatic Art (RADA) in London. == Career == In 1971, Madoc played Fruma Sarah in the film version of the musical Fiddler on the Roof, and in 1972 she appeared as Mrs Dai Bread Two in the film version of Under Milk Wood. She appeared regularly in the entertainment programme Poems and Pints on BBC Wales. She provided one of the alien voices in the Cadbury's Smash commercials in the 1970s, and made a brief appearance in the 1977 film The Prince and the Pauper (aka Crossed Swords). Madoc appeared in many theatre productions, including the stage version of Under Milk Wood, Steel Magnolias, Agatha Christie thrillers (And Then There Were None), the musical Annie, and many pantomime parts. She appeared twice at the Royal Variety Performance, once in 1982 and again in 1986. Madoc was best known for her portrayal of Gladys Pugh, one of the lead roles in the television sitcom Hi-de-Hi! (1980-1988), for which she received a BAFTA TV award nomination for Best Light Entertainment Performance. The comedy was set in the fictional 1950s-type holiday camp Maplins. Madoc's recurring role centred on her unrequited love for the camp entertainment manager Professor Jeffrey Fairbrother (Simon Cadell), and she was notable for her announcements on the camp tannoy with her signature three notes played on a mini xylophone. In 2004 she appeared in the reality television programme Cariad@Iaith on S4C, in which celebrities went on an intensive course in the Welsh language. In 2005 she appeared as Daffyd Thomas's mother in the second series of BBC sketch show Little Britain. In 2007 Madoc appeared in the fourth series of LivingTV reality show I'm Famous and Frightened! which she went on to win. Also in 2007 she appeared as a fictional version of herself in episode 2 of the BBC Radio 2 comedy Buy Me Up TV. In 2008 she appeared at the Pavilion Theatre in Rhyl, playing the bad fairy in the pantomime Sleeping Beauty, with Sonia and Rebecca Trehearn. Madoc returned to situation comedy in 2009 and appeared in Big Top on BBC1, alongside Amanda Holden, John Thomson and Tony Robinson. In January 2015, Madoc appeared as the fairy godmother in the pantomime Cinderella at the Palace Theatre, Mansfield. In September 2019 she re-joined the cast in the autumn tour of Calendar Girls: The Musical, after recovering from an injury earlier in the year; the show opened at Bournemouth Pavilion on 17 September and ended on 23 November at Chichester Festival Theatre. == Recognition and honours == In 1984, Madoc was the subject of This Is Your Life when she was surprised by Eamonn Andrews. In 1993 she played Mrs Bardell in Pickwick at the Chichester Festival Theatre. Madoc was awarded an honorary degree by Swansea University in July 2006. == Personal life and death == In 2010, Madoc investigated her family history for the BBC Wales programme Coming Home and learned that she was a distant cousin of British Prime Minister David Lloyd George on her father's side. She had starred in The Life and Times of David Lloyd George in 1981. Madoc's first husband was the actor Philip Madoc, with whom she appeared in the 1981 TV serial The Life and Times of David Lloyd George. They had a son and a daughter, and were married for 20 years, but divorced in 1981. In 1982, she married her second husband, John Jackson, with whom she bought a home in Glynneath in 2002. They were married until his death in September 2021. In December 2022, Madoc was set to appear in the pantomime Aladdin at the Princess Theatre, Torquay. However, on 8 December, a statement posted to Madoc's Instagram account confirmed she had suffered a fall earlier in the week and was unable to appear in the production. After undergoing surgery, Madoc died the following day, 9 December, in hospital, at the age of 79. == Filmography == === Television === === Films === == References == == External links == Ruth Madoc at IMDb Official website Ruth Madoc discography at Discogs
Wikipedia:Ruth Kellerhals#0	Ruth Kellerhals (born 17 July 1957) is a Swiss mathematician at the University of Fribourg, whose field of study is hyperbolic geometry, geometric group theory and polylogarithm identities. == Biography == As a child, she went to a gymnasium in Basel and then studied at the University of Basel, graduating in 1982 with a diploma directed by Heinz Huber "On finiteness of the isometry group of a compact negatively curved Riemannian manifold". She received her PhD in 1988, from the same university, with a thesis entitled "On the volumes of hyperbolic polytopes in dimensions three and four". Her advisor was Hans-Christoph Im Hof. During the year 1983–84 she also studied at the University of Grenoble (Fourier Institute). In 1995 she received her habilitation from the University of Bonn, where she worked at the Max Planck Institute for Mathematics since 1989 until 1995. There, she was an assistant with Professor Friedrich Hirzebruch. Since 1995 she has been an assistant professor at the University of Göttingen, and since 1999 a distinguished professor at the University of Bordeaux 1. In 2000 she became a professor at the University of Fribourg, Switzerland, where she was in 1998 to 1999 as a visiting professor. == Research == Her main research fields include hyperbolic geometry, geometric group theory, geometry of discrete groups (especially reflection groups, Coxeter groups), convex and polyhedral geometry, volumes of hyperbolic polytopes, manifolds and polylogarithms. She does historical research into the works and life of Ludwig Schläfli, a Swiss geometer. She has been a guest researcher at MSRI, IHES, Mittag-Leffler Institute, the State University of New York at Stony Brook, RIMS in Kyoto, Osaka City University, ETH Zürich, the University of Bern and the University of Auckland. Also she visited numerous research institutes and universities in Helsinki, Berlin and Budapest. == Selected works == Guglielmetti, Rafael; Jacquemet, Matthieu; Kellerhals, Ruth (2016). "On commensurable hyperbolic Coxeter groups". Geometriae Dedicata. 183: 143–167. doi:10.1007/s10711-016-0151-7. Kellerhals, Ruth; Kolpakov, Alexander (2014). "The minimal growth rate of cocompact Coxeter groups in hyperbolic 3-space". Canadian Journal of Mathematics. 66 (2): 354–372. doi:10.4153/CJM-2012-062-3. Kellerhals, Ruth (2012). "Scissors congruence, the golden ratio and volumes in hyperbolic 5-space". Discrete & Computational Geometry. 47 (3): 629–658. doi:10.1007/s00454-012-9397-5. Kellerhals, Ruth; Perren, Geneviève (2011). "On the growth of cocompact hyperbolic Coxeter groups". European Journal of Combinatorics. 32 (8): 1299–1316. arXiv:0910.4103. doi:10.1016/j.ejc.2011.03.020. Kellerhals, Ruth (2004). "On the structure of hyperbolic manifolds". Israel Journal of Mathematics. 143: 361–379. doi:10.1007/BF02803507. S2CID 119557013. Kellerhals, Ruth (1998). "Ball packings in spaces of constant curvature and the simplicial density function". Journal für die reine und angewandte Mathematik. 494: 189–203. doi:10.1515/crll.1998.006. Kellerhals, Ruth (1998). "Volumes of cusped hyperbolic manifolds". Topology. 37 (4): 719–734. doi:10.1016/S0040-9383(97)00052-9. Kellerhals, Ruth (1995). "Volumes in hyperbolic 5-space". Geometric and Functional Analysis. 5 (4): 640–667. doi:10.1007/BF01902056. S2CID 53356944. Kellerhals, Ruth (1992). "On the volumes of hyperbolic 5-orthoschemes and the trilogarithm". Commentarii Mathematici Helvetici. 67: 648–663. doi:10.1007/BF02566523. S2CID 120505581. Kellerhals, Ruth (1989). "On the volume of hyperbolic polyhedra". Mathematische Annalen. 285 (4): 541–569. doi:10.1007/BF01452047. S2CID 55942104. == References == == External links == Homepage
Wikipedia:Ruth Lawrence#0	Ruth Elke Lawrence-Neimark (Hebrew: רות אלקה לורנס-נאימרק, born 2 August 1971) is a British–Israeli mathematician and a professor of mathematics at the Einstein Institute of Mathematics, Hebrew University of Jerusalem, and a researcher in knot theory and algebraic topology. In the public eye, she is best known for having been a child prodigy in mathematics. == Early life == Ruth Lawrence was born in Brighton, England. Her parents, Harry Lawrence and Sylvia Greybourne, were both computer consultants. When Lawrence was five, her father gave up his job so that he could educate her at home. She is Jewish. == Education == At the age of nine, Lawrence gained an O-level in mathematics, setting a new age record (later surpassed in 2001 when Arran Fernandez successfully sat GCSE mathematics aged five). Also at the age of nine she achieved a Grade A at A-level pure mathematics. In 1981 Lawrence passed the Oxford University entrance examination in mathematics, joining St Hugh's College in 1983 at the age of 12. At Oxford, her father continued to be actively involved in her education, accompanying her to all lectures and some tutorials. Lawrence completed her bachelor's degree in two years, instead of the normal three, and graduated in 1985 at the age of 13 with a congratulatory first and special commendation. Attracting considerable press interest, she became the youngest British person to gain a first-class degree, and the youngest to graduate from the University of Oxford in modern times. Lawrence followed her first degree with a bachelor's degree in physics in 1986 and a Doctor of Philosophy (DPhil) degree in mathematics at Oxford in June 1989, at the age of 17. Her doctoral thesis title was Homology representations of braid groups and her thesis adviser was Sir Michael Atiyah. == Academic career == Lawrence and her father moved to America for Lawrence's first academic post, which was at Harvard University, where she became a junior fellow in 1990 at the age of 19. In 1993, she moved to the University of Michigan, where she became an associate professor with tenure in 1997. In 1998, Lawrence married Ariyeh Neimark, a mathematician at the Hebrew University of Jerusalem, and adopted the name Ruth Lawrence-Neimark. The following year, she moved to Israel with him and took up the post of associate professor of mathematics at the Einstein Institute of Mathematics, a part of the Hebrew University of Jerusalem. == Research == Lawrence's 1990 paper "Homological representations of the Hecke algebra", in Communications in Mathematical Physics, introduced, among other things, certain novel linear representations of the braid group — known as Lawrence–Krammer representations. In papers published in 2000 and 2001, Daan Krammer and Stephen Bigelow established the faithfulness of Lawrence's representation. This result goes by the phrase "braid groups are linear." == Awards and honors == In 2012 she became a fellow of the American Mathematical Society. == Selected publications == Griniasty, Itay; Lawrence, Ruth (2019). "An explicit symmetric DGLA model of a triangle". Higher Structures. 3 (1): 1–16. arXiv:1802.02795. MR 3939044. Lawrence, Ruth; Sullivan, Dennis (2014). "A formula for topology/deformations and its significance". Fundamenta Mathematicae. 225 (1): 229–242. arXiv:math/0610949. doi:10.4064/fm225-1-10. MR 3205571. Lawrence, R. J. (1990). "Homological representations of the Hecke algebra". Communications in Mathematical Physics. 135 (1): 141–191. doi:10.1007/BF02097660. MR 1086755. Lawrence, Ruth; Zagier, Don (1999). "Modular forms and quantum invariants of 3-manifolds". Asian Journal of Mathematics. 3 (1): 93–107. doi:10.4310/AJM.1999.v3.n1.a5. MR 1701924. Lawrence, Ruth; Rozansky, Lev (1999). "Witten-Reshetikhin-Turaev invariants of Seifert manifolds". Communications in Mathematical Physics. 205 (2): 287–314. doi:10.1007/s002200050678. hdl:2027.42/41998. MR 1712599. == References == == External links == Ruth Lawrence's home page at the Hebrew University of Jerusalem

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