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Wikipedia:Benoit Perthame#0
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Benoit Perthame (born 23 June 1959 in France) is a French mathematician, who deals with non-linear partial differential equations and their applications in biology. He is a professor at Pierre-et-Marie Curie University and at the Laboratoire Jacques-Louis Lions, which he directs. == Career == Perthame studied at the École normale supérieure (ENS) and in 1983, he became an assistant. In 1987, under the supervision of Pierre-Louis Lions, he obtained his habilitation thesis. His thesis focuses on non-linear partial differential equations in optimal control theory, hydrodynamics and kinetic theory. In 1988, he was professor at the University of Orléans, and since 1993, he has been professor at the University of Paris VI and the Institut universitaire de France. From 1997 to 2007, he was in the Department of Mathematics and Applications of the ENS (DMA/ENS) and simultaneously heads the Multi-model and Numerical Methods Project (M3N) of the Institut national de recherche en informatique et en automatique (INRIA). Since 1989, he has been the scientific advisor to this organization and since 1998 has headed the BANG Group (Numerical Analysis of Non-linear Models for Biology and Geophysics). His research interests include mathematical modeling of chemotaxis and movement and self-organization of cells and bacteria, neural networks, tumor growth and chemotherapy, population growth and evolution. == Publications == Perthame, B. (2007). Transport equations in biology. Basel: Birkhäuser. ISBN 978-3-7643-7842-4. OCLC 184984556. Charpentier, Éric; Nikolski, Nikolaï Kapitonovitch; École doctorale de Mathématiques et Informatique (2007). Leçons de mathématiques d'aujourd'hui. Volume 3 (in French). Paris: Cassini. ISBN 978-2-84225-082-9. OCLC 494036904. == Awards and honours == He was a guest speaker at the International Congress of Mathematicians in 1994 in Zurich with a conference lecture entitled Kinetic Equations and Hyperbolic Systems of Conservation Laws and was selected to give a plenary lecture at the ICM 2014 in Seoul (Some aspects of mathematical tumor growth and therapy). In 1989/90, he received the Peccot Prize at the Collège de France, in 1992 the CISI Prize, in 1994 the CNRS Silver Medal and in 1992 the Blaise-Pascal Prize from the French Academy of Sciences. From 1994 to 1999, he was a junior member and then in 2006, a senior member at the Institut universitaire de France. In 2013, he received the Blaise-Pascal medal (de) and in 2015 the Inria Prize from the Academy of Sciences. In 2016, he was elected a member of Academia Europaea. At the end of 2017, he was elected a member of the French Academy of Sciences. Among his doctoral students is Vincent Calvez. == References ==
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Wikipedia:Bent Fuglede#0
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Bent Fuglede (8 October 1925 – 7 December 2023) was a Danish mathematician. == Early life and career == Fuglede was known for his contributions to mathematical analysis, in particular functional analysis, where he proved Fuglede's theorem and stated Fuglede's conjecture. Fuglede graduated from Skt. Jørgens Gymnasium 1943 and received his mag. scient. og cand. mag. in 1948 at the University of Copenhagen after which he studied in USA until 1951. In 1952 he was employed as scientific assistant at Den Polytekniske Læreanstalt and in 1954 as amanuensis at Matematisk Institut University of Copenhagen, in 1958 associate professor, and in 1959 head of department. Fuglede also spent one year in Lund (Sweden) as Nordic docent. Fuglede received his dr.phil. (Ph.D.) in 1960 from the University of Copenhagen; his doctoral advisor was Børge Jessen, after which he became professor of math at Danmarks tekniske Højskole. In 1965 he became professor of math at the University of Copenhagen, where he stayed until his retirement in 1992. Fuglede was a member of the Royal Danish Academy of Sciences and Letters, the Finnish Academy of Science and Letters. and the Bayerische Akademie der Wissenschaften. In 2012 he became a fellow of the American Mathematical Society. == Personal life and death == Fuglende was married to Ólafía Einarsdóttir, an Icelandic archeologist and historian. They had one child together, Einar. Bent Fuglede died in Copenhagen on 7 December 2023, at the age of 98. == Books == Fuglede, Bent (1972). Finely Harmonic Functions. Lecture Notes in Mathematics. Vol. 289. Springer-Verlag. ISBN 978-3-540-06005-5. Fuglede, Bent; Eells, James (2001). Harmonic maps between Riemannian polyhedra. Cambridge tracts in mathematics. Vol. 142. Cambridge University Press. ISBN 0-521-77311-3. == References == == External links == Fuglede's homepage at the University of Copenhagen
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Wikipedia:Bentsion Fleishman#0
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Bentsion Fleishman (Флейшман, Бенцион Шимонович, born 21 November 1923) is a Russian scientist in the field of mathematical statistics, combinatorial analysis and their applications, doctor of physical and mathematical sciences, professor, author of constructive information theory and the theory of potential efficiency. == Life and career == Fleishman was born in Moscow on 21 November 1923. In 1947, he graduated from the Moscow State University, the Department of Probability Theory, headed by A. N. Kolmogorov and was sent to work in the cryptographic service of the Ministry of State Security of the USSR. After his discharge in 1954, he worked at the institutes of the USSR Academy of Sciences: the Institute of Radio Engineering and Electronics (1955–1968) and the Institute of Oceanology (1968–1996). Potential efficiency theory is a synthesis of reliability theory, information theory and game theory, the successor of cybernetics, aiming at biological and engineering systems on general conceptual and mathematical basis. Its main concept – efficiency – is defined as the probability to achieve the goal with limited resources. Other fundamental concepts are purposeful choice and probabilistic feasibility of engineering systems. The theory was first formulated in his book Elements of the Theory of Potential Efficiency of Complex Systems (1971). Fleishman wrote more than a hundred scientific articles and five books. From 1966 to 1996, he was the head of the cybernetics and environmental problems section of the Scientific and Technical Society of Radio Engineering, Electronics and Communications. After moving to the US in 1996, he worked on applications of potential efficiency theory and became a member of the International Society for Risk Analysis. == Bibliography == A. E. Basharinov, B. S. Fleishman, Methods of statistical sequential analysis and their applications, Moscow: Sov. Radio, 1962; B. S. Fleishman, Constructive methods of optimal coding for noisy channels, Moscow: Publishing House of the Academy of Sciences of the USSR, 1963; B. S. Fleishman, Elements of the theory of potential efficiency of complex systems, Moscow: Sov. Radio, 1971 (2nd edition, Smolensk: Oikumene, 2008); B. S. Fleishman, Fundamentals of Systemology, Radio and Communications, Moscow, 1982 (2nd ed. New York: Lulu.com, 2007); B. Fleishman. Stochastic theory of ecological interactions. Ecological Modeling, vol. 17, 1982, p. 65-73; B. Fleishman. Hyperbolic law of reliability and its logarithmic effects in ecology. Ecological Modeling, vol. 55, 1991, p. 75-88; B. Fleishman, Stochastic Theory of Complex Ecological Systems (cap.6). In: B. Patten, S. Jorgenson (eds) Complex Ecology. Prentice Hall PTP, Prentice Hall Inc, A. Simon & Schuster, Englewood Cliffs, New Jersey, 07632, 1995, p. 166-224; B. Fleishman, The Choice is Yours, Moscow: Oikumena, 2000 (2nd ed., New York: Lulu.com, 2008). == References == == External links == Bentsion Fleishman | Official site Геннадий Горелик. Науки о природе и науки об искусности. Семь искусств No.8(135), август 2021
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Wikipedia:Beppo Levi#0
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Beppo Levi (14 May 1875 – 28 August 1961) was an Italian mathematician. He published high-level academic articles and books on mathematics as well as on physics, history, philosophy, and pedagogy. Levi was a member of the Bologna Academy of Sciences and of the Accademia dei Lincei. == Early years == Beppo Levi was born on May 14, 1875, in Turin, Italy to a Jewish family. He was an older brother of Eugenio Elia Levi. Levi obtained his laurea in mathematics in 1896 at age 21 from the University of Turin under Corrado Segre. He was appointed an assistant professor at the University of Turin three months later and shortly thereafter became a full-time Scholar. Levi was appointed Professor at the University of Piacenza in 1901, at the University of Cagliari in 1906, at the University of Parma in 1910, and finally at the University of Bologna in 1928. The years that followed his last appointment saw the rise of Benito Mussolini's power and of antisemitism in Italy, and Levi, being Jewish, was soon expelled from his position at the University of Bologna. He emigrated to Argentina, as did many other European Jews at that time. == Life in Argentina == Levi chose Argentina as a destination because of an invitation by the engineer Cortés Plá, dean of the Facultad de Ciencias Matemáticas, Físico-Químicas y Naturales Aplicadas a la Industria at the Universidad Nacional del Litoral (currently Facultad de Ciencias Exactas, Ingeniería y Agrimensura at the Universidad Nacional de Rosario) in the city of Rosario. Cortés Plá invited Levi to come to Rosario to head the recently created Instituto de Matemática. It was there that Levi did most of his work from 1939 until his death in 1961. While living in Rosario, Levi joined a group of mathematicians that included Luis Santaló, Simón Rubinstein, Juan Olguín, Enrique Ferrari, Fernando and Enrique Gaspar, Mario Castagnino and Edmundo Rofman. In 1940 Levi founded Mathematicae Notae, the first mathematical journal in Argentina. In 1956 he was awarded the Feltrinelli Prize. He died on August 28, 1961, in Rosario, Argentina, and was buried in the Jewish cemetery there. == Mathematical contributions == His early work studied singularities on algebraic curves and surfaces. In particular, he supplied a proof (questioned by some) that a procedure for resolution of singularities on algebraic surfaces terminates in finitely many steps. Later he proved some foundational results concerning Lebesgue integration, including what is commonly known as Beppo Levi's lemma. He also studied the arithmetic of elliptic curves. He classified them up to isomorphism, not only over C, but also over Q. Next he studied what in modern terminology would be the subgroup of rational torsion points on an elliptic curve over Q: he proved that certain groups were realizable and that others were not. He essentially formulated the torsion conjecture for elliptic curves over the rational numbers, providing a complete list of possibilities should be, which was formulated independently by Andrew Ogg about 60 years later and finally proved by Barry Mazur in 1973. == References == === Biographic and general references === Coen, Salvatore (1999), "Beppo Levi: una biografia", in Levi, Beppo (ed.), Opere (Collected Works). Volume I: 1897-1906 (in Italian), Bologna: Edizioni Cremonese (distributed by Unione Matematica Italiana), Zbl 1054.01520. An ample biographical paper of nearly 40 pages, an earlier version of which was published as Coen, Salvatore (1994), "Beppo Levi: la vita", in Coen, Salvatore (ed.), Seminari di geometria, Università di Bologna, Italia, 1991–1993 (in Italian), Bologna: Università degli Studi di Bologna, Dipartimento di Matematica, pp. 193–232, MR 1265762, Zbl 0795.01022. Viola, Tullio (1961), "Necrologio di Beppo Levi", Bollettino della Unione Matematica Italiana, Serie 3, 16 (4): 513–516 === References describing his scientific contributions === Levi, Beppo (1999), Opere di Beppo Levi 1897–1926 [Works of Beppo Levi 1897–1926], vol. 1: 1897–1906, Bologna: Edizioni Cremonese (distributed by Unione Matematica Italiana), Zbl 1054.01520 —— (1999), Opere di Beppo Levi 1897–1926 [Works of Beppo Levi 1897–1926], vol. 2: 1907–1926, Bologna: Edizioni Cremonese (distributed by Unione Matematica Italiana), Zbl 1054.01521. Schappacher, Norbert; Schoof, René (1996), "Beppo Levi and the arithmetic of elliptic curves" (PDF), The Mathematical Intelligencer, 18 (1): 57–69, doi:10.1007/bf03024818, MR 1381581, S2CID 125072148, Zbl 0849.01036 For a freely downloadable offprint from the web site of one of the two authors, see here. == External links == Media related to Beppo Levi at Wikimedia Commons Coen, Salvatore; Lanconelli, Ermanno, eds. (June 12–13, 2008), Convegno Italo – Argentino in onore di Beppo Levi (Italian – Argentinian meeting in honour of Beppo Levi), Bologna{{citation}}: CS1 maint: location missing publisher (link). The web site of a scientific meeting in Bologna, honouring the memory of Beppo Levi. Guerraggio, Angelo; Nastasi, Pietro (2008–2010), "Beppo Levi (1875–1961)", Edizione Nazionale Mathematica Italiana (in Italian), Scuola Normale Superiore, retrieved January 18, 2011 (in Italian). Available from the Edizione Nazionale Mathematica Italiana.
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Wikipedia:Berit Stensønes#0
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Berit Stensønes (19 February 1956 – 5 June 2022) was a Norwegian mathematician specializing in complex analysis and complex dynamics and known for her work on several complex variables. She was a professor of mathematical sciences at the Norwegian University of Science and Technology (NTNU), and a professor emerita at the University of Michigan. == Education == Stensønes completed her Ph.D. in 1985 at Princeton University. Her dissertation, Envelopes of Holomorphy, was supervised by John Erik Fornæss. == Book == With John Erik Fornæss, Stensønes was an author of the book Lectures on Counterexamples in Several Complex Variables (AMS Chelsea Publishing, American Mathematical Society, 1987; reprinted 2007). == Recognition == Stensønes was a member of the Royal Norwegian Society of Sciences and Letters. == References ==
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Wikipedia:Berlekamp's algorithm#0
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In mathematics, particularly computational algebra, Berlekamp's algorithm is a well-known method for factoring polynomials over finite fields (also known as Galois fields). The algorithm consists mainly of matrix reduction and polynomial GCD computations. It was invented by Elwyn Berlekamp in 1967. It was the dominant algorithm for solving the problem until the Cantor–Zassenhaus algorithm of 1981. It is currently implemented in many well-known computer algebra systems. == Overview == Berlekamp's algorithm takes as input a square-free polynomial f ( x ) {\displaystyle f(x)} (i.e. one with no repeated factors) of degree n {\displaystyle n} with coefficients in a finite field F q {\displaystyle \mathbb {F} _{q}} and gives as output a polynomial g ( x ) {\displaystyle g(x)} with coefficients in the same field such that g ( x ) {\displaystyle g(x)} divides f ( x ) {\displaystyle f(x)} . The algorithm may then be applied recursively to these and subsequent divisors, until we find the decomposition of f ( x ) {\displaystyle f(x)} into powers of irreducible polynomials (recalling that the ring of polynomials over a finite field is a unique factorization domain). All possible factors of f ( x ) {\displaystyle f(x)} are contained within the factor ring R = F q [ x ] ⟨ f ( x ) ⟩ . {\displaystyle R={\frac {\mathbb {F} _{q}[x]}{\langle f(x)\rangle }}.} The algorithm focuses on polynomials g ( x ) ∈ R {\displaystyle g(x)\in R} which satisfy the congruence: g ( x ) q ≡ g ( x ) ( mod f ( x ) ) . {\displaystyle g(x)^{q}\equiv g(x){\pmod {f(x)}}.\,} These polynomials form a subalgebra of R (which can be considered as an n {\displaystyle n} -dimensional vector space over F q {\displaystyle \mathbb {F} _{q}} ), called the Berlekamp subalgebra. The Berlekamp subalgebra is of interest because the polynomials g ( x ) {\displaystyle g(x)} it contains satisfy f ( x ) = ∏ s ∈ F q gcd ( f ( x ) , g ( x ) − s ) . {\displaystyle f(x)=\prod _{s\in \mathbb {F} _{q}}\gcd(f(x),g(x)-s).} In general, not every GCD in the above product will be a non-trivial factor of f ( x ) {\displaystyle f(x)} , but some are, providing the factors we seek. Berlekamp's algorithm finds polynomials g ( x ) {\displaystyle g(x)} suitable for use with the above result by computing a basis for the Berlekamp subalgebra. This is achieved via the observation that Berlekamp subalgebra is in fact the kernel of a certain n × n {\displaystyle n\times n} matrix over F q {\displaystyle \mathbb {F} _{q}} , which is derived from the so-called Berlekamp matrix of the polynomial, denoted Q {\displaystyle {\mathcal {Q}}} . If Q = [ q i , j ] {\displaystyle {\mathcal {Q}}=[q_{i,j}]} then q i , j {\displaystyle q_{i,j}} is the coefficient of the j {\displaystyle j} -th power term in the reduction of x i q {\displaystyle x^{iq}} modulo f ( x ) {\displaystyle f(x)} , i.e.: x i q ≡ q i , n − 1 x n − 1 + q i , n − 2 x n − 2 + … + q i , 0 ( mod f ( x ) ) . {\displaystyle x^{iq}\equiv q_{i,n-1}x^{n-1}+q_{i,n-2}x^{n-2}+\ldots +q_{i,0}{\pmod {f(x)}}.\,} With a certain polynomial g ( x ) ∈ R {\displaystyle g(x)\in R} , say: g ( x ) = g n − 1 x n − 1 + g n − 2 x n − 2 + … + g 0 , {\displaystyle g(x)=g_{n-1}x^{n-1}+g_{n-2}x^{n-2}+\ldots +g_{0},\,} we may associate the row vector: g = ( g 0 , g 1 , … , g n − 1 ) . {\displaystyle g=(g_{0},g_{1},\ldots ,g_{n-1}).\,} It is relatively straightforward to see that the row vector g Q {\displaystyle g{\mathcal {Q}}} corresponds, in the same way, to the reduction of g ( x ) q {\displaystyle g(x)^{q}} modulo f ( x ) {\displaystyle f(x)} . Consequently, a polynomial g ( x ) ∈ R {\displaystyle g(x)\in R} is in the Berlekamp subalgebra if and only if g ( Q − I ) = 0 {\displaystyle g({\mathcal {Q}}-I)=0} (where I {\displaystyle I} is the n × n {\displaystyle n\times n} identity matrix), i.e. if and only if it is in the null space of Q − I {\displaystyle {\mathcal {Q}}-I} . By computing the matrix Q − I {\displaystyle {\mathcal {Q}}-I} and reducing it to reduced row echelon form and then easily reading off a basis for the null space, we may find a basis for the Berlekamp subalgebra and hence construct polynomials g ( x ) {\displaystyle g(x)} in it. We then need to successively compute GCDs of the form above until we find a non-trivial factor. Since the ring of polynomials over a field is a Euclidean domain, we may compute these GCDs using the Euclidean algorithm. == Conceptual algebraic explanation == With some abstract algebra, the idea behind Berlekamp's algorithm becomes conceptually clear. We represent a finite field F q {\textstyle \mathbb {F} _{q}} , where q = p m {\textstyle q=p^{m}} for some prime p {\textstyle p} , as F p [ y ] / ( g ( y ) ) {\textstyle \mathbb {F} _{p}[y]/(g(y))} . We can assume that f ( x ) ∈ F q [ x ] {\textstyle f(x)\in \mathbb {F} _{q}[x]} is square free, by taking all possible pth roots and then computing the gcd with its derivative. Now, suppose that f ( x ) = f 1 ( x ) … f n ( x ) {\textstyle f(x)=f_{1}(x)\ldots f_{n}(x)} is the factorization into irreducibles. Then we have a ring isomorphism, σ : F q [ x ] / ( f ( x ) ) → ∏ i F q [ x ] / ( f i ( x ) ) {\textstyle \sigma :\mathbb {F} _{q}[x]/(f(x))\to \prod _{i}\mathbb {F} _{q}[x]/(f_{i}(x))} , given by the Chinese remainder theorem. The crucial observation is that the Frobenius automorphism x → x p {\textstyle x\to x^{p}} commutes with σ {\textstyle \sigma } , so that if we denote Fix p ( R ) = { f ∈ R : f p = f } {\textstyle {\text{Fix}}_{p}(R)=\{f\in R:f^{p}=f\}} , then σ {\textstyle \sigma } restricts to an isomorphism Fix p ( F q [ x ] / ( f ( x ) ) ) → ∏ i = 1 n Fix p ( F q [ x ] / ( f i ( x ) ) ) {\textstyle {\text{Fix}}_{p}(\mathbb {F} _{q}[x]/(f(x)))\to \prod _{i=1}^{n}{\text{Fix}}_{p}(\mathbb {F} _{q}[x]/(f_{i}(x)))} . By finite field theory, Fix p ( F q [ x ] / ( f i ( x ) ) ) {\textstyle {\text{Fix}}_{p}(\mathbb {F} _{q}[x]/(f_{i}(x)))} is always the prime subfield of that field extension. Thus, Fix p ( F q [ x ] / ( f ( x ) ) ) {\textstyle {\text{Fix}}_{p}(\mathbb {F} _{q}[x]/(f(x)))} has p {\textstyle p} elements if and only if f ( x ) {\textstyle f(x)} is irreducible. Moreover, we can use the fact that the Frobenius automorphism is F p {\textstyle \mathbb {F} _{p}} -linear to calculate the fixed set. That is, we note that Fix p ( F q [ x ] / ( f ( x ) ) ) {\textstyle {\text{Fix}}_{p}(\mathbb {F} _{q}[x]/(f(x)))} is a F p {\textstyle \mathbb {F} _{p}} -subspace, and an explicit basis for it can be calculated in the polynomial ring F p [ x , y ] / ( f , g ) {\textstyle \mathbb {F} _{p}[x,y]/(f,g)} by computing ( x i y j ) p {\textstyle (x^{i}y^{j})^{p}} and establishing the linear equations on the coefficients of x , y {\textstyle x,y} polynomials that are satisfied iff it is fixed by Frobenius. We note that at this point we have an efficiently computable irreducibility criterion, and the remaining analysis shows how to use this to find factors. The algorithm now breaks down into two cases: In the case of small p {\textstyle p} we can construct any g ∈ Fix p ( F q [ x ] / ( f ( x ) ) ) ∖ F p {\textstyle g\in {\text{Fix}}_{p}(\mathbb {F} _{q}[x]/(f(x)))\setminus \mathbb {F} _{p}} , and then observe that for some a ∈ F p {\textstyle a\in \mathbb {F} _{p}} there are i , j {\textstyle i,j} so that g − a = 0 mod f i {\textstyle g-a=0\mod f_{i}} and g − a ≠ 0 mod f j {\textstyle g-a\not =0\mod f_{j}} . Such a g − a {\textstyle g-a} has a nontrivial factor in common with f ( x ) {\textstyle f(x)} , which can be computed via the gcd. As p {\textstyle p} is small, we can cycle through all possible a {\textstyle a} . For the case of large primes, which are necessarily odd, one can exploit the fact that a random nonzero element of F p ∗ {\textstyle \mathbb {F} _{p}^{*}} is a square with probability 1 / 2 {\textstyle 1/2} , and that the map x → x p − 1 2 {\textstyle x\to x^{\frac {p-1}{2}}} maps the set of non-zero squares to 1 {\textstyle 1} , and the set of non-squares to − 1 {\textstyle -1} . Thus, if we take a random element g ∈ Fix p ( F q [ x ] / f ( x ) ) {\textstyle g\in {\text{Fix}}_{p}(\mathbb {F} _{q}[x]/f(x))} , then with good probability g p − 1 2 − 1 {\textstyle g^{\frac {p-1}{2}}-1} will have a non-trivial factor in common with f ( x ) {\textstyle f(x)} . For further details one can consult. == Applications == One important application of Berlekamp's algorithm is in computing discrete logarithms over finite fields F p n {\displaystyle \mathbb {F} _{p^{n}}} , where p {\displaystyle p} is prime and n ≥ 2 {\displaystyle n\geq 2} . Computing discrete logarithms is an important problem in public key cryptography and error-control coding. For a finite field, the fastest known method is the index calculus method, which involves the factorisation of field elements. If we represent the field F p n {\displaystyle \mathbb {F} _{p^{n}}} in the usual way - that is, as polynomials over the base field F p {\displaystyle \mathbb {F} _{p}} , reduced modulo an irreducible polynomial of degree n {\displaystyle n} - then this is simply polynomial factorisation, as provided by Berlekamp's algorithm. == Implementation in computer algebra systems == Berlekamp's algorithm may be accessed in the PARI/GP package using the factormod command, and the WolframAlpha [1] website. == See also == Polynomial factorisation Factorization of polynomials over a finite field and irreducibility tests Cantor–Zassenhaus algorithm == References == Berlekamp, Elwyn R. (1967). "Factoring Polynomials Over Finite Fields". Bell System Technical Journal. 46 (8): 1853–1859. doi:10.1002/j.1538-7305.1967.tb03174.x. MR 0219231. BSTJ Later republished in: Berlekamp, Elwyn R. (1968). Algebraic Coding Theory. McGraw Hill. ISBN 0-89412-063-8. Knuth, Donald E (1997). "4.6.2 Factorization of Polynomials". Seminumerical Algorithms. The Art of Computer Programming. Vol. 2 (Third ed.). Reading, Massachusetts: Addison-Wesley. pp. 439–461, 678–691. ISBN 0-201-89684-2.
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Wikipedia:Berlekamp–Rabin algorithm#0
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Elwyn Ralph Berlekamp (September 6, 1940 – April 9, 2019) was a professor of mathematics and computer science at the University of California, Berkeley. Berlekamp was widely known for his work in computer science, coding theory and combinatorial game theory. Berlekamp invented an algorithm to factor polynomials and the Berlekamp switching game, and was one of the inventors of the Berlekamp–Welch algorithm and the Berlekamp–Massey algorithms, which are used to implement Reed–Solomon error correction. He also co-invented the Berlekamp–Rabin algorithm, Berlekamp–Zassenhaus algorithm, and the Berlekamp–Van Lint–Seidel graph. Berlekamp had also been active in investing, and ran Axcom, which became the Renaissance Technologies' Medallion Fund. == Life and education == Berlekamp was born in Dover, Ohio. His family moved to Northern Kentucky, where from 1954 Berlekamp attended Fort Thomas Highlands High School in Fort Thomas, Kentucky. He was elected class president and joined the swim team which practiced naked at the local YMCA pool; Berlekamp was the slowest swimmer but chose swimming because of the low level of competition compared to other sports. He decided to attend the Massachusetts Institute of Technology (MIT) after learning it did not have an American football team. At MIT, his freshman professors included John Forbes Nash Jr. and he was a Putnam Fellow during his senior year in 1961. He completed his bachelor's and master's degrees in electrical engineering in 1962. Berlekamp did internships at Bell Labs in 1960 and 1962, where his boss was John Larry Kelly Jr. Continuing his studies at MIT, he finished his Ph.D. in electrical engineering in 1964; his advisors were Robert G. Gallager, Peter Elias, Claude Shannon, and John Wozencraft. Berlekamp met his wife, Jennifer Wilson, in 1964 after juggling in his apartment and having to apologize for causing a noise disturbance. They had two daughters and a son. He lived in Piedmont, California and died in April 2019 at the age of 78 from complications of pulmonary fibrosis. == Career == Berlekamp was a professor of electrical engineering at the University of California, Berkeley from 1964 until 1966, when he became a mathematics researcher at Bell Labs. In 1971, Berlekamp returned to Berkeley as professor of mathematics and computer science, where he served as the advisor for over twenty doctoral students. He was a member of the National Academy of Engineering (1977) and the National Academy of Sciences (1999). He was elected a Fellow of the American Academy of Arts and Sciences in 1996, and became a fellow of the American Mathematical Society in 2012. In 1991, he received the IEEE Richard W. Hamming Medal, and in 1993, the Claude E. Shannon Award. In 1998, he received a Golden Jubilee Award for Technological Innovation from the IEEE Information Theory Society. Along with Tom M. Rodgers he was one of the founders of Gathering 4 Gardner and was on its board for many years. In the mid-1980s, he was president of Cyclotomics, Inc., a corporation that developed error-correcting code technology. He studied various games, including dots and boxes, fox and geese, and, especially, Go. Berlekamp and co-author David Wolfe described methods for analyzing certain classes of Go endgames in the book Mathematical Go. == Berlekamp and Martin Gardner == Berlekamp was a member of the group of people around the Scientific American columnist Martin Gardner, a close friend. Berlekamp teamed up with John Horton Conway and Richard K. Guy, two other close associates of Gardner, to co-author the book Winning Ways for your Mathematical Plays, leading to his recognition as one of the founders of combinatorial game theory. The dedication of their book says, "To Martin Gardner, who has brought more mathematics to more millions than anyone else." Berlekamp and Gardner both supported recreational mathematics. Conferences called Gathering 4 Gardner (G4G) are held every two years to celebrate the Gardner legacy. Berlekamp was one of the founders of G4G and was on its board of directors for many years. == Selected publications == Block coding with noiseless feedback. Thesis, Massachusetts Institute of Technology, Dept. of Electrical Engineering, 1964. Algebraic Coding Theory, New York: McGraw-Hill, 1968. Revised ed., Aegean Park Press, 1984, ISBN 0-89412-063-8. (with John Horton Conway and Richard K. Guy) Winning Ways for your Mathematical Plays. 1st edition, New York: Academic Press, 2 vols., 1982; vol. 1, hardback: ISBN 0-12-091150-7, paperback: ISBN 0-12-091101-9; vol. 2, hardback: ISBN 0-12-091152-3, paperback: ISBN 0-12-091102-7. 2nd edition, Wellesley, Massachusetts: A. K. Peters Ltd., 4 vols., 2001–2004; vol. 1: ISBN 1-56881-130-6; vol. 2: ISBN 1-56881-142-X; vol. 3: ISBN 1-56881-143-8; vol. 4: ISBN 1-56881-144-6. (with David Wolfe) Mathematical Go. Wellesley, Massachusetts: A. K. Peters Ltd., 1994. ISBN 1-56881-032-6. The Dots-and-Boxes Game. Natick, Massachusetts: A. K. Peters Ltd., 2000. ISBN 1-56881-129-2. == See also == Berlekamp switching game Berlekamp–Zassenhaus algorithm == References == == External links == Elwyn Berlekamp home page at the University of California, Berkeley. Elwyn Berlekamp at the Mathematics Genealogy Project.
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Wikipedia:Berlekamp–Zassenhaus algorithm#0
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Elwyn Ralph Berlekamp (September 6, 1940 – April 9, 2019) was a professor of mathematics and computer science at the University of California, Berkeley. Berlekamp was widely known for his work in computer science, coding theory and combinatorial game theory. Berlekamp invented an algorithm to factor polynomials and the Berlekamp switching game, and was one of the inventors of the Berlekamp–Welch algorithm and the Berlekamp–Massey algorithms, which are used to implement Reed–Solomon error correction. He also co-invented the Berlekamp–Rabin algorithm, Berlekamp–Zassenhaus algorithm, and the Berlekamp–Van Lint–Seidel graph. Berlekamp had also been active in investing, and ran Axcom, which became the Renaissance Technologies' Medallion Fund. == Life and education == Berlekamp was born in Dover, Ohio. His family moved to Northern Kentucky, where from 1954 Berlekamp attended Fort Thomas Highlands High School in Fort Thomas, Kentucky. He was elected class president and joined the swim team which practiced naked at the local YMCA pool; Berlekamp was the slowest swimmer but chose swimming because of the low level of competition compared to other sports. He decided to attend the Massachusetts Institute of Technology (MIT) after learning it did not have an American football team. At MIT, his freshman professors included John Forbes Nash Jr. and he was a Putnam Fellow during his senior year in 1961. He completed his bachelor's and master's degrees in electrical engineering in 1962. Berlekamp did internships at Bell Labs in 1960 and 1962, where his boss was John Larry Kelly Jr. Continuing his studies at MIT, he finished his Ph.D. in electrical engineering in 1964; his advisors were Robert G. Gallager, Peter Elias, Claude Shannon, and John Wozencraft. Berlekamp met his wife, Jennifer Wilson, in 1964 after juggling in his apartment and having to apologize for causing a noise disturbance. They had two daughters and a son. He lived in Piedmont, California and died in April 2019 at the age of 78 from complications of pulmonary fibrosis. == Career == Berlekamp was a professor of electrical engineering at the University of California, Berkeley from 1964 until 1966, when he became a mathematics researcher at Bell Labs. In 1971, Berlekamp returned to Berkeley as professor of mathematics and computer science, where he served as the advisor for over twenty doctoral students. He was a member of the National Academy of Engineering (1977) and the National Academy of Sciences (1999). He was elected a Fellow of the American Academy of Arts and Sciences in 1996, and became a fellow of the American Mathematical Society in 2012. In 1991, he received the IEEE Richard W. Hamming Medal, and in 1993, the Claude E. Shannon Award. In 1998, he received a Golden Jubilee Award for Technological Innovation from the IEEE Information Theory Society. Along with Tom M. Rodgers he was one of the founders of Gathering 4 Gardner and was on its board for many years. In the mid-1980s, he was president of Cyclotomics, Inc., a corporation that developed error-correcting code technology. He studied various games, including dots and boxes, fox and geese, and, especially, Go. Berlekamp and co-author David Wolfe described methods for analyzing certain classes of Go endgames in the book Mathematical Go. == Berlekamp and Martin Gardner == Berlekamp was a member of the group of people around the Scientific American columnist Martin Gardner, a close friend. Berlekamp teamed up with John Horton Conway and Richard K. Guy, two other close associates of Gardner, to co-author the book Winning Ways for your Mathematical Plays, leading to his recognition as one of the founders of combinatorial game theory. The dedication of their book says, "To Martin Gardner, who has brought more mathematics to more millions than anyone else." Berlekamp and Gardner both supported recreational mathematics. Conferences called Gathering 4 Gardner (G4G) are held every two years to celebrate the Gardner legacy. Berlekamp was one of the founders of G4G and was on its board of directors for many years. == Selected publications == Block coding with noiseless feedback. Thesis, Massachusetts Institute of Technology, Dept. of Electrical Engineering, 1964. Algebraic Coding Theory, New York: McGraw-Hill, 1968. Revised ed., Aegean Park Press, 1984, ISBN 0-89412-063-8. (with John Horton Conway and Richard K. Guy) Winning Ways for your Mathematical Plays. 1st edition, New York: Academic Press, 2 vols., 1982; vol. 1, hardback: ISBN 0-12-091150-7, paperback: ISBN 0-12-091101-9; vol. 2, hardback: ISBN 0-12-091152-3, paperback: ISBN 0-12-091102-7. 2nd edition, Wellesley, Massachusetts: A. K. Peters Ltd., 4 vols., 2001–2004; vol. 1: ISBN 1-56881-130-6; vol. 2: ISBN 1-56881-142-X; vol. 3: ISBN 1-56881-143-8; vol. 4: ISBN 1-56881-144-6. (with David Wolfe) Mathematical Go. Wellesley, Massachusetts: A. K. Peters Ltd., 1994. ISBN 1-56881-032-6. The Dots-and-Boxes Game. Natick, Massachusetts: A. K. Peters Ltd., 2000. ISBN 1-56881-129-2. == See also == Berlekamp switching game Berlekamp–Zassenhaus algorithm == References == == External links == Elwyn Berlekamp home page at the University of California, Berkeley. Elwyn Berlekamp at the Mathematics Genealogy Project.
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Wikipedia:Berlin Papyrus 6619#0
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The Berlin Papyrus 6619, simply called the Berlin Papyrus when the context makes it clear, is one of the primary sources of ancient Egyptian mathematics. One of the two mathematics problems on the Papyrus may suggest that the ancient Egyptians knew the Pythagorean theorem. == Description, dating, and provenance == The Berlin Papyrus 6619 is an ancient Egyptian papyrus document from the Middle Kingdom, second half of the 12th (c. 1990–1800 BC) or 13th Dynasty (c. 1800 BC – 1649 BC). The two readable fragments were published by Hans Schack-Schackenburg in 1900 and 1902. == Connection to the Pythagorean theorem == The Berlin Papyrus contains two problems, the first stated as "the area of a square of 100 is equal to that of two smaller squares. The side of one is ½ + ¼ the side of the other." The interest in the question may suggest some knowledge of the Pythagorean theorem, though the papyrus only shows a straightforward solution to a single second degree equation in one unknown. In modern terms, the simultaneous equations x2 + y2 = 100 and x = (3/4)y reduce to the single equation in y: ((3/4)y)2 + y2 = 100, giving the solution y = 8 and x = 6. == See also == List of ancient Egyptian papyri Papyrology Timeline of mathematics Egyptian fraction == References == == External links == Simultaneous equation examples from the Berlin papyrus Two algebra problems compared to RMP algebra Two suggested solutions
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Wikipedia:Berlin workshops on Babylonian mathematics#0
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The Berlin workshops were a series of six workshops that took place between 1983 and 1994 and focused on mathematical conceptualization and notation in a number of early writing systems. Although the names of the workshops varied slightly over time, most included the phase "conceptual development of Babylonian mathematics" and were supported by the Archaische Texte aus Uruk Project at Freie Universität Berlin and the Max Planck Institute for the History of Science. The first meeting was held at the Altorientalisches Seminar und Seminar für Vorderasiatische Altertumskunde on August 5, 1983. Subsequent meetings were held in 1984, 1985, 1988 and 1994. == List of workshops == The workshops played a significant role in advancing the decipherment of Proto-cuneiform and Proto-Elamite numerals as well as the comparative study of early mathematical notation. Workshop on Mathematical Concepts in Babylonian Mathematics Date: August 1-5, 1983. Place: Seminar für Vorderasiatische Altertumskunde und altorientalische Philologie der Freien Universität. Second Workshop on Concept Development in Babylonian Mathematics Date: June 18-22, 1984. Place: Seminar für Vorderasiatische Altertumskunde der Freien Universität Berlin. Third Workshop on Concept Development in Babylonian Mathematics Date: December 9-13, 1985. Place: Seminar für Vorderasiatische Altertumskunde der Freien Universität Berlin. Fourth Workshop on Concept Development in Babylonian Mathematics Date: May 5-9, 1988. Place: Seminar für Vorderasiatische Altertumskunde der Freien Universität Berlin. Fifth Workshop on Mathematical Concepts in Babylonian Mathematics Date: January 21-23, 1994. Place: Seminar für Vorderasiatische Altertumskunde der Freien Universität Berlin. Standardisierung der elektronischen Transliteration von Keilschrifttexten Date: September 7-9, 1994. Place: Max Planck Institute for the History of Science == References ==
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Wikipedia:Bernard Bolzano#0
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Bernard Bolzano (UK: , US: ; German: [bɔlˈtsaːno]; Italian: [bolˈtsaːno]; born Bernardus Placidus Johann Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his liberal views. Bolzano wrote in German, his native language. For the most part, his work came to prominence posthumously. == Family == Bolzano was the son of two pious Catholics. His father, Bernard Pompeius Bolzano, was an Italian who had moved to Prague, where he married Maria Cecilia Maurer who came from Prague's German-speaking family Maurer. Only two of their twelve children lived to adulthood. == Career == When he was ten years old, Bolzano entered the Gymnasium of the Piarists in Prague, which he attended from 1791 to 1796. Bolzano entered the University of Prague in 1796 and studied mathematics, philosophy and physics. Starting in 1800, he also began studying theology, becoming a Catholic priest in 1804. He was appointed to the new chair of philosophy of religion at Prague University in 1805. He proved to be a popular lecturer not only in religion but also in philosophy, and he was elected Dean of the Philosophical Faculty in 1818. Bolzano alienated many faculty and church leaders with his teachings of the social waste of militarism and the needlessness of war. He urged a total reform of the educational, social and economic systems that would direct the nation's interests toward peace rather than toward armed conflict between nations. His political convictions, which he was inclined to share with others with some frequency, eventually proved to be too liberal for the Austrian authorities. On December 24, 1819, he was removed from his professorship (upon his refusal to recant his beliefs) and was exiled to the countryside and then devoted his energies to his writings on social, religious, philosophical, and mathematical matters. Although forbidden to publish in mainstream journals as a condition of his exile, Bolzano continued to develop his ideas and publish them either on his own or in obscure Eastern European journals. In 1842 he moved back to Prague, where he died in 1848. == Mathematical work == Bolzano made several original contributions to mathematics. His overall philosophical stance was that, contrary to much of the prevailing mathematics of the era, it was better not to introduce intuitive ideas such as time and motion into mathematics. To this end, he was one of the earliest mathematicians to begin instilling rigor into mathematical analysis with his three chief mathematical works Beyträge zu einer begründeteren Darstellung der Mathematik (1810), Der binomische Lehrsatz (1816) and Rein analytischer Beweis (1817). These works presented "...a sample of a new way of developing analysis", whose ultimate goal would not be realized until some fifty years later when they came to the attention of Karl Weierstrass. To the foundations of mathematical analysis he contributed the introduction of a fully rigorous ε–δ definition of a mathematical limit. Bolzano was the first to recognize the greatest lower bound property of the real numbers. Like several others of his day, he was skeptical of the possibility of Gottfried Leibniz's infinitesimals, that had been the earliest putative foundation for differential calculus. Bolzano's notion of a limit was similar to the modern one: that a limit, rather than being a relation among infinitesimals, must instead be cast in terms of how the dependent variable approaches a definite quantity as the independent variable approaches some other definite quantity. Bolzano also gave the first purely analytic proof of the fundamental theorem of algebra, which had originally been proven by Gauss from geometrical considerations. He also gave the first purely analytic proof of the intermediate value theorem (also known as Bolzano's theorem). Today he is mostly remembered for the Bolzano–Weierstrass theorem, which Karl Weierstrass developed independently and published years after Bolzano's first proof and which was initially called the Weierstrass theorem until Bolzano's earlier work was rediscovered. == Philosophical work == Bolzano's posthumously published work Paradoxien des Unendlichen (The Paradoxes of the Infinite) (1851) was greatly admired by many of the eminent logicians who came after him, including Charles Sanders Peirce, Georg Cantor, and Richard Dedekind. Bolzano's main claim to fame, however, is his 1837 Wissenschaftslehre (Theory of Science), a work in four volumes that covered not only philosophy of science in the modern sense but also logic, epistemology and scientific pedagogy. The logical theory that Bolzano developed in this work has come to be acknowledged as ground-breaking. Other works are a four-volume Lehrbuch der Religionswissenschaft (Textbook of the Science of Religion) and the metaphysical work Athanasia, a defense of the immortality of the soul. Bolzano also did valuable work in mathematics, which remained virtually unknown until Otto Stolz rediscovered many of his lost journal articles and republished them in 1881. === Wissenschaftslehre (Theory of Science) === In his 1837 Wissenschaftslehre Bolzano attempted to provide logical foundations for all sciences, building on abstractions like part-relation, abstract objects, attributes, sentence-shapes, ideas and propositions in themselves, sums and sets, collections, substances, adherences, subjective ideas, judgments, and sentence-occurrences. These attempts were an extension of his earlier thoughts in the philosophy of mathematics, for example his 1810 Beiträge where he emphasized the distinction between the objective relationship between logical consequences and our subjective recognition of these connections. For Bolzano, it was not enough that we merely have confirmation of natural or mathematical truths, but rather it was the proper role of the sciences (both pure and applied) to seek out justification in terms of the fundamental truths that may or may not appear to be obvious to our intuitions. ==== Introduction to Wissenschaftslehre ==== Bolzano begins his work by explaining what he means by theory of science, and the relation between our knowledge, truths and sciences. Human knowledge, he states, is made of all truths (or true propositions) that men know or have known. However, this is a very small fraction of all the truths that exist, although still too much for one human being to comprehend. Therefore, our knowledge is divided into more accessible parts. Such a collection of truths is what Bolzano calls a science (Wissenschaft). It is important to note that not all true propositions of a science have to be known to men; hence, this is how we can make discoveries in a science. To better understand and comprehend the truths of a science, men have created textbooks (Lehrbuch), which of course contain only the true propositions of the science known to men. But how to know where to divide our knowledge, that is, which truths belong together? Bolzano explains that we will ultimately know this through some reflection, but that the resulting rules of how to divide our knowledge into sciences will be a science in itself. This science, that tells us which truths belong together and should be explained in a textbook, is the Theory of Science (Wissenschaftslehre). ==== Metaphysics ==== In the Wissenschaftslehre, Bolzano is mainly concerned with three realms: (1) The realm of language, consisting in words and sentences. (2) The realm of thought, consisting in subjective ideas and judgements. (3) The realm of logic, consisting in objective ideas (or ideas in themselves) and propositions in themselves. Bolzano devotes a great part of the Wissenschaftslehre to an explanation of these realms and their relations. Two distinctions play a prominent role in his system. First, the distinction between parts and wholes. For instance, words are parts of sentences, subjective ideas are parts of judgments, objective ideas are parts of propositions in themselves. Second, all objects divide into those that exist, which means that they are causally connected and located in time and/or space, and those that do not exist. Bolzano's original claim is that the logical realm is populated by objects of the latter kind. ==== Satz an Sich (proposition in itself) ==== Satz an Sich is a basic notion in Bolzano's Wissenschaftslehre. It is introduced at the very beginning, in section 19. Bolzano first introduces the notions of proposition (spoken or written or thought or in itself) and idea (spoken or written or thought or in itself). "The grass is green" is a proposition (Satz): in this connection of words, something is said or asserted. "Grass", however, is only an idea (Vorstellung). Something is represented by it, but it does not assert anything. Bolzano's notion of proposition is fairly broad: "A rectangle is round" is a proposition — even though it is false by virtue of self-contradiction — because it is composed in an intelligible manner out of intelligible parts. Bolzano does not give a complete definition of a Satz an Sich (i.e. proposition in itself) but he gives us just enough information to understand what he means by it. A proposition in itself (i) has no existence (that is: it has no position in time or place), (ii) is either true or false, independent of anyone knowing or thinking that it is true or false, and (iii) is what is 'grasped' by thinking beings. So a written sentence ('Socrates has wisdom') grasps a proposition in itself, namely the proposition [Socrates has wisdom]. The written sentence does have existence (it has a certain location at a certain time, say it is on your computer screen at this very moment) and expresses the proposition in itself which is in the realm of in itself (i.e. an sich). (Bolzano's use of the term an sich differs greatly from that of Kant; for Kant's use of the term see an sich.) Every proposition in itself is composed out of ideas in themselves (for simplicity, we will use proposition to mean "proposition in itself" and idea to refer to an objective idea or idea in itself). Ideas are negatively defined as those parts of a proposition that are themselves not propositions. A proposition consists of at least three ideas, namely: a subject idea, a predicate idea and the copula (i.e. 'has', or another form of to have). (Though there are propositions which contain propositions, we won't take them into consideration right now.) Bolzano identifies certain types of ideas. There are simple ideas that have no parts (as an example Bolzano uses [something]), but there are also complex ideas that consist of other ideas (Bolzano uses the example of [nothing], which consists of the ideas [not] and [something]). Complex ideas can have the same content (i.e. the same parts) without being the same — because their components are differently connected. The idea [A black pen with blue ink] is different from the idea [A blue pen with black ink] though the parts of both ideas are the same. ==== Ideas and objects ==== It is important to understand that an idea does not need to have an object. Bolzano uses object to denote something that is represented by an idea. An idea that has an object, represents that object. But an idea that does not have an object represents nothing. (Don't get confused here by terminology: an objectless idea is an idea without a representation.) Consider, for further explanation, an example used by Bolzano. The idea [a round square], does not have an object, because the object that ought to be represented is self-contrary. A different example is the idea [nothing] which certainly does not have an object. However, the proposition [the idea of a round square has complexity] has as its subject-idea [the idea of a round square]. This subject-idea does have an object, namely the idea [a round square]. But, that idea does not have an object. Besides objectless ideas, there are ideas that have only one object, e.g. the idea [the first man on the moon] represents only one object. Bolzano calls these ideas 'singular ideas'. Obviously there are also ideas that have many objects (e.g. [the citizens of Amsterdam]) and even infinitely many objects (e.g. [a prime number]). ==== Sensation and simple ideas ==== Bolzano has a complex theory of how we are able to sense things. He explains sensation by means of the term intuition, in German called Anschauung. An intuition is a simple idea, it has only one object (Einzelvorstellung), but besides that, it is also unique (Bolzano needs this to explain sensation). Intuitions (Anschauungen) are objective ideas, they belong to the an sich realm, which means that they don't have existence. As said, Bolzano's argumentation for intuitions is by an explanation of sensation. What happens when you sense a real existing object, for instance a rose, is this: the different aspects of the rose, like its scent and its color, cause in you a change. That change means that before and after sensing the rose, your mind is in a different state. So sensation is in fact a change in your mental state. How is this related to objects and ideas? Bolzano explains that this change, in your mind, is essentially a simple idea (Vorstellung), like, 'this smell' (of this particular rose). This idea represents; it has as its object the change. Besides being simple, this change must also be unique. This is because literally you can't have the same experience twice, nor can two people, who smell the same rose at the same time, have exactly the same experience of that smell (although they will be quite alike). So each single sensation causes a single (new) unique and simple idea with a particular change as its object. Now, this idea in your mind is a subjective idea, meaning that it is in you at a particular time. It has existence. But this subjective idea must correspond to, or has as a content, an objective idea. This is where Bolzano brings in intuitions (Anschauungen); they are the simple, unique and objective ideas that correspond to our subjective ideas of changes caused by sensation. So for each single possible sensation, there is a corresponding objective idea. Schematically the whole process is like this: whenever you smell a rose, its scent causes a change in you. This change is the object of your subjective idea of that particular smell. That subjective idea corresponds to the intuition or Anschauung. ==== Logic ==== According to Bolzano, all propositions are composed out of three (simple or complex) elements: a subject, a predicate and a copula. Instead of the more traditional copulative term 'is', Bolzano prefers 'has'. The reason for this is that 'has', unlike 'is', can connect a concrete term, such as 'Socrates', to an abstract term such as 'baldness'. "Socrates has baldness" is, according to Bolzano, preferable to "Socrates is bald" because the latter form is less basic: 'bald' is itself composed of the elements 'something', 'that', 'has' and 'baldness'. Bolzano also reduces existential propositions to this form: "Socrates exists" would simply become "Socrates has existence (Dasein)". A major role in Bolzano's logical theory is played by the notion of variations: various logical relations are defined in terms of the changes in truth value that propositions incur when their non-logical parts are replaced by others. Logically analytical propositions, for instance, are those in which all the non-logical parts can be replaced without change of truth value. Two propositions are 'compatible' (verträglich) with respect to one of their component parts x if there is at least one term that can be inserted that would make both true. A proposition Q is 'deducible' (ableitbar) from a proposition P, with respect to certain of their non-logical parts, if any replacement of those parts that makes P true also makes Q true. If a proposition is deducible from another with respect to all its non-logical parts, it is said to be 'logically deducible'. Besides the relation of deducibility, Bolzano also has a stricter relation of 'grounding' (Abfolge). This is an asymmetric relation that obtains between true propositions, when one of the propositions is not only deducible from, but also explained by the other. ==== Truth ==== Bolzano distinguishes five meanings the words true and truth have in common usage, all of which Bolzano takes to be unproblematic. The meanings are listed in order of properness: I. Abstract objective meaning: Truth signifies an attribute that may apply to a proposition, primarily to a proposition in itself, namely the attribute on the basis of which the proposition expresses something that in reality is as is expressed. Antonyms: falsity, falseness, falsehood. II. Concrete objective meaning: (a) Truth signifies a proposition that has the attribute truth in the abstract objective meaning. Antonym: (a) falsehood. III. Subjective meaning: (a) Truth signifies a correct judgment. Antonym: (a) mistake. IV. Collective meaning: Truth signifies a body or multiplicity true propositions or judgments (e.g. the biblical truth). V. Improper meaning: True signifies that some object is in reality what some denomination states it to be. (e.g. the true God). Antonyms: false, unreal, illusory. Bolzano's primary concern is with the concrete objective meaning: with concrete objective truths or truths in themselves. All truths in themselves are a kind of propositions in themselves. They do not exist, i.e. they are not spatiotemporally located as thought and spoken propositions are. However, certain propositions have the attribute of being a truth in itself. Being a thought proposition is not a part of the concept of a truth in itself, notwithstanding the fact that, given God's omniscience, all truths in themselves are also thought truths. The concepts 'truth in itself' and 'thought truth' are interchangeable, as they apply to the same objects, but they are not identical. Bolzano offers as the correct definition of (abstract objective) truth: a proposition is true if it expresses something that applies to its object. The correct definition of a (concrete objective) truth must thus be: a truth is a proposition that expresses something that applies to its object. This definition applies to truths in themselves, rather than to thought or known truths, as none of the concepts figuring in this definition are subordinate to a concept of something mental or known. Bolzano proves in §§31–32 of his Wissenschaftslehre three things: There is at least one truth in itself (concrete objective meaning): 1. There are no true propositions (assumption) 2. 1. is a proposition (obvious) 3. 1. is true (assumed) and false (because of 1.) 4. 1. is self-contradictory (because of 3.) 5. 1. is false (because of 4.) 6. There is at least one true proposition (because of 1. and 5.) B. There is more than one truth in itself: 7. There is only one truth in itself, namely A is B (assumption) 8. A is B is a truth in itself (because of 7.) 9. There are no other truths in themselves apart from A is B (because of 7.) 10. 9. is a true proposition/ a truth in itself (because of 7.) 11. There are two truths in themselves (because of 8. and 10.) 12. There is more than one truth in itself (because of 11.) C. There are infinitely many truths in themselves: 13. There are only n truths in themselves, namely A is B .... Y is Z (assumption) 14. A is B .... Y is Z are n truths in themselves (because of 13.) 15. There are no other truths apart from A is B .... Y is Z (because of 13.) 16. 15. is a true proposition/ a truth in itself (because of 13.) 17. There are n+1 truths in themselves (because of 14. and 16.) 18. Steps 1 to 5 can be repeated for n+1, which results in n+2 truths and so on endlessly (because n is a variable) 19. There are infinitely many truths in themselves (because of 18.) ==== Judgments and cognitions ==== A known truth has as its parts (Bestandteile) a truth in itself and a judgment (Bolzano, Wissenschaftslehre §26). A judgment is a thought which states a true proposition. In judging (at least when the matter of the judgment is a true proposition), the idea of an object is being connected in a certain way with the idea of a characteristic (§ 23). In true judgments, the relation between the idea of the object and the idea of the characteristic is an actual/existent relation (§28). Every judgment has as its matter a proposition, which is either true or false. Every judgment exists, but not "für sich". Judgments, namely, in contrast with propositions in themselves, are dependent on subjective mental activity. Not every mental activity, though, has to be a judgment; recall that all judgments have as matter propositions, and hence all judgments need to be either true or false. Mere presentations or thoughts are examples of mental activities which do not necessarily need to be stated (behaupten), and so are not judgments (§ 34). Judgments that have as its matter true propositions can be called cognitions (§36). Cognitions are also dependent on the subject, and so, opposed to truths in themselves, cognitions do permit degrees; a proposition can be more or less known, but it cannot be more or less true. Every cognition implies necessarily a judgment, but not every judgment is necessarily cognition, because there are also judgments that are not true. Bolzano maintains that there are no such things as false cognitions, only false judgments (§34). == Philosophical legacy == Bolzano came to be surrounded by a circle of friends and pupils who spread his thoughts about (the so-called Bolzano Circle), but the effect of his thought on philosophy initially seemed destined to be slight. Alois Höfler (1853–1922), a former student of Franz Brentano and Alexius Meinong, who subsequently become professor of pedagogy at the University of Vienna, created the "missing link between the Vienna Circle and the Bolzano tradition in Austria." Bolzano's work was rediscovered, however, by Edmund Husserl and Kazimierz Twardowski, both students of Brentano. Through them, Bolzano became a formative influence on both phenomenology and analytic philosophy. == Writings == Bolzano: Gesamtausgabe (Bolzano: Collected Works), critical edition edited by Eduard Winter, Jan Berg, Friedrich Kambartel, Bob van Rootselaar, Stuttgart: Fromman-Holzboog, 1969ff. (103 Volumes available, 28 Volumes in preparation). Wissenschaftslehre, 4 vols., 2nd rev. ed. by W. Schultz, Leipzig I–II 1929, III 1980, IV 1931; Critical Edition edited by Jan Berg: Bolzano's Gesamtausgabe, vols. 11–14 (1985–2000). Bernard Bolzano's Grundlegung der Logik. Ausgewählte Paragraphen aus der Wissenschaftslehre, Vols. 1 and 2, with supplementary text summaries, an introduction and indices, edited by F. Kambartel, Hamburg, 1963, 1978². Bolzano, Bernard (1810), Beyträge zu einer begründeteren Darstellung der Mathematik. Erste Lieferung (Contributions to a better grounded presentation of mathematics; Ewald 1996, pp. 174–224 and The Mathematical Works of Bernard Bolzano, 2004, pp. 83–137). Bolzano, Bernard (1817), Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetzes Resultat gewähren, wenigstens eine reele Wurzel der Gleichung liege, Wilhelm Engelmann (Purely analytic proof of the theorem that between any two values which give results of opposite sign, there lies at least one real root of the equation; Ewald 1996, pp. 225–48. Franz Prihonsky (1850), Der Neue Anti-Kant, Bautzen (an assessment of the Critique of Pure Reason by Bolzano, published posthumously by his friend F. Prihonsky).*Bolzano, Bernard (1851), Paradoxien des Unendlichen, C.H. Reclam (Paradoxes of the Infinite; Ewald 1996, pp. 249–92 (excerpt)). Most of Bolzano's work remained in manuscript form, so it had a very small circulation and little influence on the development of the subject. === Translations and compilations === Theory of Science (selection edited and translated by Rolf George, Berkeley and Los Angeles: University of California Press, 1972). Theory of Science (selection edited, with an introduction, by Jan Berg. Translated from the German by Burnham Terrell, Dordrecht and Boston: D. Reidel Publishing Company, 1973). Theory of Science, first complete English translation in four volumes by Rolf George and Paul Rusnock, New York: Oxford University Press, 2014. The Mathematical Works of Bernard Bolzano, translated and edited by Steve Russ, New York: Oxford University Press, 2004 (re-printed 2006). On the Mathematical Method and Correspondence with Exner, translated by Rolf George and Paul Rusnock, Amsterdam: Rodopi, 2004. Selected Writings on Ethics and Politics, translated by Rolf George and Paul Rusnock, Amsterdam: Rodopi, 2007. Franz Prihonsky, The New Anti-Kant, edited by Sandra Lapointe and Clinton Tolley, New York, Palgrave Macmillan, 2014. Russ, S. B. (1980). "A translation of Bolzano's paper on the intermediate value theorem". Historia Mathematica. 7 (2): 156–185. doi:10.1016/0315-0860(80)90036-1. (Translation of Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetzes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege (Prague 1817)) == See also == List of Roman Catholic scientist-clerics == Notes == == References == Boyer, Carl B. (1959), The History of the Calculus and Its Conceptual Development, New York: Dover Publications, MR 0124178. Boyer, Carl B.; Merzbach, Uta C. (1991), A History of Mathematics, New York: John Wiley & Sons, ISBN 978-0-471-54397-8. Ewald, William B., ed. (1996), From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 volumes, Oxford University Press. Künne, Wolfgang (1998), "Bolzano, Bernard", Routledge Encyclopedia of Philosophy, vol. 1, London: Routledge, pp. 823–827. O'Connor, John J.; Robertson, Edmund F. (2005), "Bolzano", MacTutor History of Mathematics archive. == Further reading == Edgar Morscher (1972), "Von Bolzano zu Meinong: Zur Geschichte des logischen Realismus." In: Rudolf Haller (ed.), Jenseits von Sein und Nichtsein: Beiträge zur Meinong-Forschung, Akadem. Druck- u. Verlagsanst., pp. 69–102. Kamila Veverková, Bernard Bolzano: A New Evaluation of His Thought and His Circle, trans. Angelo Shaun Franklin, Lexington Books, 2022. == External links == Morscher, Edgar. "Bernard Bolzano". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy. Šebestík, Jan [in Czech]. "Bolzano's Logic". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy. Bolzano's Philosophy of Mathematical Knowledge entry by Sandra Lapointe in the Internet Encyclopedia of Philosophy The Philosophy of Bernard Bolzano: Logic and Ontology Bernard Bolzano: Bibliography of theEnglish Translations Annotated Bibliography on the Philosophical Work of Bolzano Annotated Bibliography on the Practical Philosophy of Bolzano (religion, aesthetics, politics) Bernard Bolzano at the Mathematics Genealogy Project Works by or about Bernard Bolzano at the Internet Archive Bolzano Collection: Digitized Bolzano's works Volume 1 of Wissenschaftslehre in Google Books Volume 2 of Wissenschaftslehre in Google Books Volumes 3–4 of Wissenschaftslehre in Google Books Volume 1 of Wissenschaftslehre in Archive.org (pages 162 to 243 are missing) Volume 2 of Wissenschaftslehre in Archive.org Volume 4 of Wissenschaftslehre in Archive.org Volume 3 of Wissenschaftslehre in Gallica Volume 4 of Wissenschaftslehre in Gallica
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Wikipedia:Bernard Epstein#0
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Bernard Epstein (10 August 1920, Harrison, New Jersey – 30 March 2005, Montgomery County, Maryland) was an American mathematician and physicist who wrote several widely used textbooks on mathematics. Epstein was the son of Jewish immigrants from Lithuania and Romania, Yitzkhak Aharon Epstein and Sophie-Sarah née Goldenberg, and was the first person in his family to go to college. He received bachelor's and master's degrees in mathematics and physics from New York University and then in 1947 a Ph.D. in applied mathematics, with thesis advisor Maurice Heins, from Brown University with thesis Method for the Solution of the Dirichlet Problem for Certain Types of Domains. In the early 1940s, he worked as a physicist at what is now the National Institute of Standards and Technology. During World War II, he was selected to join the Manhattan Project, which produced the first atomic bombs. After the war, he worked for two years at Harvard University as a research associate, taught mathematics as an associate professor at the University of Pennsylvania, Stanford University and NYU and as a professor at Yeshiva University and then spent 21 years on the faculty of the University of New Mexico as a professor of mathematics until his retirement in 1984. Sabbaticals included Office of Naval Research, London; The Technion in Haifa, Israel; University of Maryland; and Air Force Office of Scientific Research. After retirement, he taught at George Mason University. Epstein was dissertation advisor for the following Ph.D. students: Anne Scheerer, University of Pennsylvania, 1953 William Trench, University of Pennsylvania, 1958 Jack Minker, University of Pennsylvania, 1959 Edwin Sherry, Yeshiva University, 1964 Darrell L. Hicks, University of New Mexico, 1969 Harvey Z. Senter, Yeshiva University Upon his death at age 84, he was survived by his wife, five children, and 16 grandchildren. His sixth child, a daughter, predeceased him. == Selected publications == === Articles === Epstein, Bernard (1947). "Some inequalities relating to conformal mapping upon canonical slit-domains". Bulletin of the American Mathematical Society. 53 (8): 813–819. doi:10.1090/S0002-9904-1947-08895-9. MR 0022259. Epstein, Bernard (1948). "A method for the solution of the Dirichlet problem for certain types of domains". Quarterly of Applied Mathematics. 6 (3): 301–317. doi:10.1090/qam/28692. JSTOR 43633676. with S. Bergman: Bergman, Stefan; Epstein, Bernard (1948). "Determination of a compressible fluid flow past an oval-shaped obstacle". Journal of Mathematics and Physics. 26 (1–4): 195–222. doi:10.1002/sapm1947261195. with J. Lehner: Epstein, Bernard; Lehner, Joseph (1952). "On Ritt's representation of analytic functions as infinite products". Journal of the London Mathematical Society. 27: 30–37. doi:10.1112/jlms/s1-27.1.30. with A. Scheerer: Epstein, Bernard; Scheerer, Anne (1956). "The existence of a generalized Green's function in the plane". Journal d'Analyse Mathématique. 4: 222–235. doi:10.1007/BF02787722. S2CID 124528688. with David S. Greenstein and Jack Minker: "An extremal problem with infinitely many interpolation conditions". Annals of Finnish Academy of Science (Soumalainen Tiedaekatamia Tomituksia), Series A:1 Mathematics 250/10, 1958. with F. Haber: Haber, Fred; Epstein, Bernard (1958). "The parameters of nonlinear devices from harmonic measurements". IRE Transactions on Electron Devices. 5 (1): 26–38. Bibcode:1958ITED....5...26H. doi:10.1109/T-ED.1958.14324. S2CID 51642216. Epstein, Bernard (1958). "The kernel function and conformal invariants". Journal of Mathematics and Mechanics. 7 (6): 925–936. JSTOR 24900650. with I. J. Schoenberg: Epstein, Bernard; Schoenberg, Isaac Jacob (1959). "On a conjecture concerning Schlicht functions". Bulletin of the American Mathematical Society. 65 (4): 273–275. doi:10.1090/S0002-9904-1959-10342-6. MR 0108588. with J. Minker: Epstein, Bernard; Minker, Jack (1960). "Extremal interpolatory problems in the unit disc". Proceedings of the American Mathematical Society. 11 (5): 777–784. doi:10.1090/S0002-9939-1960-0118853-8. MR 0118853. Epstein, Bernard (1962). "On the mean-value property of harmonic functions". Proceedings of the American Mathematical Society. 13 (5): 830. doi:10.1090/S0002-9939-1962-0140700-0. MR 0140700. Epstein, Bernard (1962). "A remark concerning the solution of the Dirichlet problem by finite differences". Mathematics of Computation. 16 (77): 110–112. doi:10.1090/S0025-5718-1962-0132199-3. MR 0132199. with M. M. Schiffer: Epstein, Bernard; Schiffer, Menahem Max (1965). "On the mean-value property of harmonic functions". Journal d'Analyse Mathématique. 14 (1): 109–111. doi:10.1007/BF02806381. S2CID 121605803. Epstein, Bernard (1966). "On a difference equation arising in a learning-theory model". Israel Journal of Mathematics. 4 (3): 145–152. doi:10.1007/BF02760073. S2CID 119940454. with H. Senter: Epstein, Bernard; Senter, Harvey (1967). "The three-dimensional Dirichlet problem associated with a plane lamina". Journal of Mathematics and Physics. 46 (1–4): 425–439. doi:10.1002/sapm1967461425. with J. R. Blum: Blum, Julius R.; Epstein, Bernard (1971). "On the Fourier transforms of an interesting class of measures". Israel Journal of Mathematics. 10 (3): 301–305. doi:10.1007/BF02771647. S2CID 120487512. Epstein, Bernard (1977). "Infinite divisibility of Student's t-distribution". Sankhyā: The Indian Journal of Statistics. Series B. 39 (2): 103–120. JSTOR 25052058. === Books === Partial differential equations: an introduction. 1962. 2nd edition. 1975. Orthogonal families of analytic functions. 1965. Linear functional analysis: introduction to Lebesgue integration and infinite-dimensional problems. 1970. with Liang-shin Hahn: Classical complex analysis. 1996. == References ==
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Wikipedia:Bernard Morin#0
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Bernard Morin (French: [mɔʁɛ̃]; 3 March 1931 in Shanghai, China – 12 March 2018) was a French mathematician, specifically a topologist. == Early life and education == Morin lost his sight at the age of six due to glaucoma, but his blindness did not prevent him from having a successful career in mathematics. He received his Ph.D. in 1972 from the Centre National de la Recherche Scientifique. == Career == Morin was a member of the group that first exhibited an eversion of the sphere, i.e., a homotopy which starts with a sphere and ends with the same sphere but turned inside-out. He also discovered the Morin surface, which is a half-way model for the sphere eversion, and used it to prove a lower bound on the number of steps needed to turn a sphere inside out. Morin discovered the first parametrization of Boy's surface (earlier used as a half-way model), in 1978. His graduate student François Apéry, in 1986, discovered another parametrization of Boy's surface, which conforms to the general method for parametrizing non-orientable surfaces. Morin worked at the Institute for Advanced Study in Princeton, New Jersey. Most of his career, though, he spent at the University of Strasbourg. Morin's surface. == See also == Blind mathematicians: Leonhard Euler, Nicholas Saunderson, Lev Pontryagin, Louis Antoine == References == George K. Francis & Bernard Morin (1980) "Arnold Shapiro's Eversion of the Sphere", Mathematical Intelligencer 2(4):200–3. == External links == Photos of Morin Archived 2019-04-20 at the Wayback Machine with stereolithography models of sphere eversion. The World of Blind Mathematicians, PDF file at the American Mathematical Society's website.
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Wikipedia:Bernard Roy#0
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Bernard Roy (French pronunciation: [bɛʁnaʁ ʁwa]; 15 March 1934 – 28 October 2017) was an emeritus professor at the Université Paris-Dauphine. In 1974 he founded the "Laboratoire d'Analyse et de Modélisation des Systèmes pour l'Aide à la Décision" (Lamsade). He was President of Association of European Operational Research Societies from 1985 to 1986. In 1992 he was awarded the EURO Gold Medal, the highest distinction within Operations Research in Europe. In 2015 he received the EURO Distinguished Service Award. He worked on graph theory and on multi-criteria decision analysis (MCDA), having created the ELECTRE family of methods. The name ELECTRE stands for "ELimination Et Choix Traduisant la REalité". == References == == External links == Biography of Bernard Roy from the Institute for Operations Research and the Management Sciences
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Wikipedia:Bernard Russell Gelbaum#0
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Bernard Russell Gelbaum (died March 22, 2005, Laguna Beach, California) was a mathematician and academic administrator having served as a professor at the University of Minnesota, University of California, Irvine (where he was the first chair of the math department as well as acting dean and associate dean of physical sciences) and as well as emeritus professor in the Department of Mathematics, College of Arts and Sciences, University at Buffalo. When he arrived at Buffalo 1971, he served as vice president for academic affairs as well as being a math professor. == Biography == While still an undergraduate at Columbia University, Gelbaum served as a second lieutenant in the U.S. Signal Corps and was one of the first to liberate the Buchenwald concentration camp. He went on to get his doctorate at Princeton University in 1948. His dissertation, Expansions in Banach Spaces, was supervised by Salomon Bochner. == References ==
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Wikipedia:Bernard Vauquois#0
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Bernard Vauquois ((1929-06-14)June 14, 1929 — (1985-09-30)September 30, 1985) was a French mathematician and computer scientist. He was a pioneer of computer science and machine translation (MT) in France. An astronomer-turned-computer scientist, he is known for his work on the programming language ALGOL 60, and later for extensive work on the theoretical and practical problems of MT, of which the eponymous Vauquois triangle is one of the most widely-known contributions. He was a professor at what would become the Grenoble Alpes University. == Biography == Bernard Vauquois was initially a researcher at French National Centre for Scientific Research (CNRS) from 1952 to 1958 at the Astrophysics Institute of the Meudon Observatory, after completing studies in mathematics, physics, and astronomy. Since 1957, his research program has also focused on methods applied to physics from the perspective of electronic computers, and he has taught programming to physicists. This double interest in astrophysics and electronic computers is reflected in the subject of his thesis and that of the complementary thesis in physical sciences that he defended in 1958. In 1960, at 31 years old, he was appointed professor of computer science at the Grenoble University where, with professors Jean Kuntzmann and Noël Gastinel, he began activities in computer science. At that time, he was also working on the definition of the language ALGOL 60. Also in 1960, he founded the Centre d'Étude pour la Traduction Automatique (CETA), later renamed as Groupe d'Étude pour la Traduction Automatique (GETA) and currently known as GETALP, a team at the Laboratoire d'informatique de Grenoble, and soon showed his gift for rapid understanding, synthesis, and innovation, and his taste for personal communication across linguistic borders and barriers. After visiting a number of centers, mainly in the United States, where machine translation research was conducted, he analyzed the shortcomings of the "first-generation" approach and evaluated the potential of a new generation based on grammar and formal language theory, and proposed a new approach based on a representational "pivot" and the use of (declarative) rule systems that transform a sequential sentence from one level of representation to another. He led the GETA in constructing the first large second-generation system, applied to Russian–French, from 1962 to 1971. At the end of this period, the accumulated experience led him to correct some defects of the "pure" declarative and interlingual approach, and to use heuristic programming methods, implemented with procedural grammars written in LSPLs ("specialized languages for linguistic programming", langages spécialisés pour la programmation linguistique) that were developed under his direction, and integrated into the ARIANE-78 machine translation system. In 1974, when he cofounded the Leibniz laboratory, he proposed "multilevel structure descriptors" (descripteurs de structures multiniveaux) for units larger than sentence translation. This idea, premonitory of later theoretical work (Ray Jackendoff, Gerald Gazdar) is still the cornerstone of all machine translation software built by GETA and the French national TA project. Bernard Vauquois' last contribution was "static grammar" (grammaire statique) in 1982–83, during the ESOPE project, the preparatory phase of the French national MT project. He was a key figure in the field of computational linguistics in France. At CNRS, he was a member of section 22 of the National Committee in 1963: "General Linguistics, Modern Languages and Comparative Literature", and then, in 1969, of section 28: "General Linguistics, Foreign Languages and Literature". Since 1965, he has been vice-president of the Association for Natural Language Processing (ATALA). He was its president from 1966 to 1971. He was also one of the founders, in 1965, of the ICCL (International Committee on Computational Linguistics), which organizes COLING conferences. He was its president from 1969 to 1984. From France, he often collaborated with other countries (notably Canada, the United States, the USSR, Czechoslovakia, Japan, China, Brazil, Malaysia, and Thailand), working on the specification and implementation of grammars and dictionaries. He began cooperating with Malaysia, for example, in 1979, which led to the creation of the Automatic Terjemaan Project, with a first prototype of an English-Malay MT system demonstrated in 1980. == Vauquois triangle == The Vauquois triangle is a conceptual model and diagram illustrating possible approaches to the design of machine translation systems, first proposed in 1968. Different illustrations of the Vauquois triangle, from the most basic to the most detailed. == Legacy == Bernard Vauquois is regarded as a pioneer of machine translation in France. He played a key role in developing the first large-scale second-generation machine translation system, and his work influenced the field of machine translation for many years. He supervised some twenty doctoral theses, most of them concerning formal aspects of natural and artificial languages, with an emphasis on machine translation. The Center for Studies on Automatic Translation, which Vauquois founded in 1960, later became the Group for the Study of Machine Translation and Automated Processing of Languages and Speech (GETALP). It is still a research institution in natural language processing. Vauquois was a prolific writer and speaker, disseminating knowledge about machine translation and related topics. His papers and presentations were instrumental in establishing the field of machine translation in France and beyond. == Publications == Vauquois, Bernard (1973). Traduction automatique (in French). Paris: Gauthier-Villars. Vauquois, Bernard (1967). Introduction à la traduction automatique (in French). Paris: Gauthier-Villars. == References ==
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Wikipedia:Bernd Sturmfels#0
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Bernd Sturmfels (born March 28, 1962, in Kassel, West Germany) is a Professor of Mathematics and Computer Science at the University of California, Berkeley and is a director of the Max Planck Institute for Mathematics in the Sciences in Leipzig since 2017. == Education and career == He received his PhD in 1987 from the University of Washington and the Technische Universität Darmstadt. After two postdoctoral years at the Institute for Mathematics and its Applications in Minneapolis, Minnesota, and the Research Institute for Symbolic Computation in Linz, Austria, he taught at Cornell University, before joining University of California, Berkeley in 1995. His Ph.D. students include Melody Chan, Jesús A. De Loera, Mike Develin, Diane Maclagan, Rekha R. Thomas, Caroline Uhler, and Cynthia Vinzant. == Contributions == Bernd Sturmfels has made contributions to a variety of areas of mathematics, including algebraic geometry, commutative algebra, discrete geometry, Gröbner bases, toric varieties, tropical geometry, algebraic statistics, and computational biology. He has written several highly cited papers in algebra with Dave Bayer. He has authored or co-authored multiple books including Introduction to Tropical Geometry with Diane Maclagan. == Awards and honors == Sturmfels' honors include a National Young Investigator Fellowship, an Alfred P. Sloan Fellowship, and a David and Lucile Packard Fellowship. In 1999 he received a Lester R. Ford Award for his expository article Polynomial equations and convex polytopes. He was awarded a Miller Research Professorship at the University of California Berkeley for 2000–2001. In 2018, he was awarded the George David Birkhoff Prize in Applied Mathematics. In 2012, he became a fellow of the American Mathematical Society. == References == == Further reading == Gallian, Joe; Ivars Peterson (January 2008). ""Mathematicians Have a Different Perspective": An Interview with Bernd Sturmfels" (PDF). MAA FOCUS. 28 (1). Washington, DC: Mathematical Association of America: 4–7. ISSN 0731-2040. Retrieved 2013-09-15. == External links == Homepage at Berkeley Bernd Sturmfels at the Mathematics Genealogy Project Bernd Sturmfels publications indexed by Google Scholar
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Wikipedia:Bernhard Keller#0
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Bernhard Keller (born 1962) is a Swiss mathematician, specializing in algebra. He is a professor at the University of Paris. Keller received in 1990 his PhD from the University of Zurich under Pierre Gabriel with the thesis On Derived Categories. His research is in homological algebra and the representation theory of quivers and finite-dimensional algebras. He has applied triangulated Calabi–Yau categories to the (additive) categorification of cluster algebras. In 2013, he received an honorary degree from the University of Antwerp. In 2014 he received the Sophie Germain Prize. He was an Invited Speaker at the International Congress of Mathematicians in Madrid in 2006, with a talk On differential graded categories. Keller is a fellow of the American Mathematical Society. == Selected works == with Idun Reiten: Keller, Bernhard; Reiten, Idun (2007). "Cluster-tilted algebras are Gorenstein and stably Calabi-Yau". Advances in Mathematics. 211 (1): 123–151. arXiv:math/0512471. doi:10.1016/j.aim.2006.07.013. Keller, Bernhard (2010). "Cluster algebras, quiver representations and triangulated categories". Triangulated Categories. London Mathematical Society Lecture Note Series. Vol. 375. pp. 76–160. arXiv:0807.1960. doi:10.1017/CBO9781139107075.004. ISBN 978-0-521-74431-7. Keller, Bernhard (1996). "Derived categories and their uses". Handbook of Algebra. Vol. 1. pp. 671–701. doi:10.1016/S1570-7954(96)80023-4. ISBN 978-0-444-82212-3. "Exposé Bourbaki 1014 : Algèbres amassées et applications, d'après Fomin-Zelevinsky" (PDF). Astérisque. 339: 63–90. 2011. == References == == External links == Bernhard Keller's homepage
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Wikipedia:Bernhard Neumann#0
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Bernhard Hermann Neumann (15 October 1909 – 21 October 2002) was a German-born British-Australian mathematician, who was a leader in the study of group theory. == Early life and education == After gaining a D.Phil. from Friedrich-Wilhelms Universität in Berlin in 1932 he earned a Ph.D. at the University of Cambridge in 1935 and a Doctor of Science at the University of Manchester in 1954. His doctoral students included Gilbert Baumslag, László Kovács, Michael Newman, and James Wiegold. After war service with the British Army, he became a lecturer at University College, Hull, before moving in 1948 to the University of Manchester, where he spent the next 14 years. In 1954 he received a DSc from the University of Cambridge. In 1962 he migrated to Australia to take up the Foundation Chair of the Department of Mathematics within the Institute of Advanced Studies of the Australian National University (ANU), where he served as head of the department until retiring in 1975. In addition he was a senior research fellow at the CSIRO Division of Mathematics and Statistics from 1975 to 1977 and then honorary research fellow from 1978 until his death in 2002. His wife, Hanna Neumann, and sons, Peter M. Neumann and Walter Neumann, are also notable for their contributions to group theory. He was an invited speaker of the International Congress of Mathematicians in 1936 at Oslo and in 1970 at Nice. He was elected a Fellow of the Royal Society in 1959. In 1994, he was appointed a Companion of the Order of Australia (AC). The Australian Mathematical Society awards a student prize named in his honour. The group-theoretic notion of HNN (Higman-Neumann-Neumann) extension bears the names of Bernard and his wife Hanna, from their joint paper with Graham Higman (who later supervised the PhD of their son Peter). == Career == Assistant lecturer, University College, Cardiff, 1937–40. Army Service, 1940–45. Lecturer, University College, Hull, (now University of Hull), 1946–48 Lecturer, senior lecturer, reader, Manchester, 1948–61 Professor and head of Department of Mathematics, Institute of Advanced Studies, ANU, Canberra, 1962–74; Emeritus Professor, 1975–2002. Senior research fellow, CSIRO Division of Mathematics and Statistics, 1975–77; honorary research fellow, 1978–99. Founding member of the World Cultural Council, 1981. == Awards == 1984 Matthew Flinders Medal and Lecture 1952 Adams Prize, University of Cambridge == References == == External links == Bernhard Neumann at the Mathematics Genealogy Project
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Wikipedia:Bernoulli umbra#0
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In Umbral calculus, the Bernoulli umbra B − {\displaystyle B_{-}} is an umbra, a formal symbol, defined by the relation eval B − n = B n − {\displaystyle \operatorname {eval} B_{-}^{n}=B_{n}^{-}} , where eval {\displaystyle \operatorname {eval} } is the index-lowering operator, also known as evaluation operator and B n − {\displaystyle B_{n}^{-}} are Bernoulli numbers, called moments of the umbra. A similar umbra, defined as eval B + n = B n + {\displaystyle \operatorname {eval} B_{+}^{n}=B_{n}^{+}} , where B 1 + = 1 / 2 {\displaystyle B_{1}^{+}=1/2} is also often used and sometimes called Bernoulli umbra as well. They are related by equality B + = B − + 1 {\displaystyle B_{+}=B_{-}+1} . Along with the Euler umbra, Bernoulli umbra is one of the most important umbras. In Levi-Civita field, Bernoulli umbras can be represented by elements with power series B − = ε − 1 − 1 2 − ε 24 + 3 ε 3 640 − 1525 ε 5 580608 + ⋯ {\displaystyle B_{-}=\varepsilon ^{-1}-{\frac {1}{2}}-{\frac {\varepsilon }{24}}+{\frac {3\varepsilon ^{3}}{640}}-{\frac {1525\varepsilon ^{5}}{580608}}+\dotsb } and B + = ε − 1 + 1 2 − ε 24 + 3 ε 3 640 − 1525 ε 5 580608 + ⋯ {\displaystyle B_{+}=\varepsilon ^{-1}+{\frac {1}{2}}-{\frac {\varepsilon }{24}}+{\frac {3\varepsilon ^{3}}{640}}-{\frac {1525\varepsilon ^{5}}{580608}}+\dotsb } , with lowering index operator corresponding to taking the coefficient of 1 = ε 0 {\displaystyle 1=\varepsilon ^{0}} of the power series. The numerators of the terms are given in OEIS A118050 and the denominators are in OEIS A118051. Since the coefficients of ε − 1 {\displaystyle \varepsilon ^{-1}} are non-zero, the both are infinitely large numbers, B − {\displaystyle B_{-}} being infinitely close (but not equal, a bit smaller) to ε − 1 − 1 / 2 {\displaystyle \varepsilon ^{-1}-1/2} and B + {\displaystyle B_{+}} being infinitely close (a bit smaller) to ε − 1 + 1 / 2 {\displaystyle \varepsilon ^{-1}+1/2} . In Hardy fields (which are generalizations of Levi-Civita field) umbra B + {\displaystyle B_{+}} corresponds to the germ at infinity of the function ψ − 1 ( ln x ) {\displaystyle \psi ^{-1}(\ln x)} while B − {\displaystyle B_{-}} corresponds to the germ at infinity of ψ − 1 ( ln x ) − 1 {\displaystyle \psi ^{-1}(\ln x)-1} , where ψ − 1 ( x ) {\displaystyle \psi ^{-1}(x)} is inverse digamma function. == Exponentiation == Since Bernoulli polynomials is a generalization of Bernoulli numbers, exponentiation of Bernoulli umbra can be expressed via Bernoulli polynomials: eval ( B − + a ) n = B n ( a ) , {\displaystyle \operatorname {eval} (B_{-}+a)^{n}=B_{n}(a),} where a {\displaystyle a} is a real or complex number. This can be further generalized using Hurwitz Zeta function: eval ( B − + a ) p = − p ζ ( 1 − p , a ) . {\displaystyle \operatorname {eval} (B_{-}+a)^{p}=-p\zeta (1-p,a).} From the Riemann functional equation for Zeta function it follows that eval B + − p = eval B + p + 1 2 p π p + 1 sin ( π p / 2 ) Γ ( p ) ( p + 1 ) {\displaystyle \operatorname {eval} \,B_{+}^{-p}=\operatorname {eval} {\frac {B_{+}^{p+1}2^{p}\pi ^{p+1}}{\sin(\pi p/2)\Gamma (p)(p+1)}}} == Derivative rule == Since B 1 + = 1 / 2 {\displaystyle B_{1}^{+}=1/2} and B 1 − = − 1 / 2 {\displaystyle B_{1}^{-}=-1/2} are the only two members of the sequences B n + {\displaystyle B_{n}^{+}} and B n − {\displaystyle B_{n}^{-}} that differ, the following rule follows for any analytic function f ( x ) {\displaystyle f(x)} : f ′ ( x ) = eval ( f ( B + + x ) − f ( B − + x ) ) = eval Δ f ( B − + x ) {\displaystyle f'(x)=\operatorname {eval} (f(B_{+}+x)-f(B_{-}+x))=\operatorname {eval} \Delta f(B_{-}+x)} == Elementary functions of Bernoulli umbra == As a general rule, the following formula holds for any analytic function f ( x ) {\displaystyle f(x)} : eval f ( B − + x ) = D e D − 1 f ( x ) . {\displaystyle \operatorname {eval} f(B_{-}+x)={\frac {D}{e^{D}-1}}f(x).} This allows to derive expressions for elementary functions of Bernoulli umbra. eval cos ( z B − ) = eval cos ( z B + ) = z 2 cot ( z 2 ) {\displaystyle \operatorname {eval} \cos(zB_{-})=\operatorname {eval} \cos(zB_{+})={\frac {z}{2}}\cot \left({\frac {z}{2}}\right)} eval cosh ( z B − ) = eval cosh ( z B + ) = z 2 coth ( z 2 ) {\displaystyle \operatorname {eval} \cosh(zB_{-})=\operatorname {eval} \cosh(zB_{+})={\frac {z}{2}}\coth \left({\frac {z}{2}}\right)} eval e z B − = z e z − 1 {\displaystyle \operatorname {eval} e^{zB_{-}}={\frac {z}{e^{z}-1}}} eval ln ( B − + z ) = ψ ( z ) {\displaystyle \operatorname {eval} \ln(B_{-}+z)=\psi (z)} Particularly, eval ln B + = − γ {\displaystyle \operatorname {eval} \ln B_{+}=-\gamma } eval 1 π ln ( B + − z π B − + z π ) = cot z {\displaystyle \operatorname {eval} {\frac {1}{\pi }}\ln \left({\frac {B_{+}-{\frac {z}{\pi }}}{B_{-}+{\frac {z}{\pi }}}}\right)=\cot z} eval 1 π ln ( B − + 1 / 2 + z π B − + 1 / 2 − z π ) = tan z {\displaystyle \operatorname {eval} {\frac {1}{\pi }}\ln \left({\frac {B_{-}+1/2+{\frac {z}{\pi }}}{B_{-}+1/2-{\frac {z}{\pi }}}}\right)=\tan z} eval cos ( a B − + x ) = a 2 csc ( a 2 ) cos ( a 2 − x ) {\displaystyle \operatorname {eval} \cos(aB_{-}+x)={\frac {a}{2}}\csc \left({\frac {a}{2}}\right)\cos \left({\frac {a}{2}}-x\right)} eval sin ( a B − + x ) = a 2 cot ( a 2 ) sin x − a 2 cos x {\displaystyle \operatorname {eval} \sin(aB_{-}+x)={\frac {a}{2}}\cot \left({\frac {a}{2}}\right)\sin x-{\frac {a}{2}}\cos x} Particularly, eval sin B − = − 1 / 2 {\displaystyle \operatorname {eval} \sin B_{-}=-1/2} , eval sin B + = 1 / 2 {\displaystyle \operatorname {eval} \sin B_{+}=1/2} , == Relations between exponential and logarithmic functions == Bernoulli umbra allows to establish relations between exponential, trigonometric and hyperbolic functions on one side and logarithms, inverse trigonometric and inverse hyperbolic functions on the other side in closed form: eval ( cosh ( 2 x B ± ) − 1 ) = eval x π artanh ( x π B ± ) = eval x π arcoth ( π B ± x ) = x coth ( x ) − 1 {\displaystyle \operatorname {eval} \left(\cosh \left(2xB_{\pm }\right)-1\right)=\operatorname {eval} {\frac {x}{\pi }}\operatorname {artanh} \left({\frac {x}{\pi B_{\pm }}}\right)=\operatorname {eval} {\frac {x}{\pi }}\operatorname {arcoth} \left({\frac {\pi B_{\pm }}{x}}\right)=x\coth(x)-1} eval z 2 π ln ( B + − z 2 π B − + z 2 π ) = eval cos ( z B − ) = eval cos ( z B + ) = z 2 cot ( z 2 ) {\displaystyle \operatorname {eval} {\frac {z}{2\pi }}\ln \left({\frac {B_{+}-{\frac {z}{2\pi }}}{B_{-}+{\frac {z}{2\pi }}}}\right)=\operatorname {eval} \cos(zB_{-})=\operatorname {eval} \cos(zB_{+})={\frac {z}{2}}\cot \left({\frac {z}{2}}\right)} == References ==
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Wikipedia:Bernstein's problem#0
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In differential geometry, Bernstein's problem is as follows: if the graph of a function on Rn−1 is a minimal surface in Rn, does this imply that the function is linear? This is true for n at most 8, but false for n at least 9. The problem is named for Sergei Natanovich Bernstein who solved the case n = 3 in 1914. == Statement == Suppose that f is a function of n − 1 real variables. The graph of f is a surface in Rn, and the condition that this is a minimal surface is that f satisfies the minimal surface equation ∑ i = 1 n − 1 ∂ ∂ x i ∂ f ∂ x i 1 + ∑ j = 1 n − 1 ( ∂ f ∂ x j ) 2 = 0 {\displaystyle \sum _{i=1}^{n-1}{\frac {\partial }{\partial x_{i}}}{\frac {\frac {\partial f}{\partial x_{i}}}{\sqrt {1+\sum _{j=1}^{n-1}\left({\frac {\partial f}{\partial x_{j}}}\right)^{2}}}}=0} Bernstein's problem asks whether an entire function (a function defined throughout Rn−1 ) that solves this equation is necessarily a degree-1 polynomial. == History == Bernstein (1915–1917) proved Bernstein's theorem that a graph of a real function on R2 that is also a minimal surface in R3 must be a plane. Fleming (1962) gave a new proof of Bernstein's theorem by deducing it from the fact that there is no non-planar area-minimizing cone in R3. De Giorgi (1965) showed that if there is no non-planar area-minimizing cone in Rn−1 then the analogue of Bernstein's theorem is true for graphs in Rn, which in particular implies that it is true in R4. Almgren (1966) showed there are no non-planar minimizing cones in R4, thus extending Bernstein's theorem to R5. Simons (1968) showed there are no non-planar minimizing cones in R7, thus extending Bernstein's theorem to R8. He also showed that the surface defined by { x ∈ R 8 : x 1 2 + x 2 2 + x 3 2 + x 4 2 = x 5 2 + x 6 2 + x 7 2 + x 8 2 } {\displaystyle \{x\in \mathbb {R} ^{8}:x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}\}} is a locally stable cone in R8, and asked if it is globally area-minimizing. Bombieri, De Giorgi & Giusti (1969) showed that Simons' cone is indeed globally minimizing, and that in Rn for n≥9 there are graphs that are minimal, but not hyperplanes. Combined with the result of Simons, this shows that the analogue of Bernstein's theorem is true in Rn for n≤8, and false in higher dimensions. == See also == Simons cone == References == Almgren, F. J. (1966), "Some interior regularity theorems for minimal surfaces and an extension of Bernstein's theorem", Annals of Mathematics, Second Series, 84 (2): 277–292, doi:10.2307/1970520, ISSN 0003-486X, JSTOR 1970520, MR 0200816 Bernstein, S. N. (1915–1917), "Sur une théorème de géometrie et ses applications aux équations dérivées partielles du type elliptique", Comm. Soc. Math. Kharkov, 15: 38–45 German translation in Bernstein, Serge (1927), "Über ein geometrisches Theorem und seine Anwendung auf die partiellen Differentialgleichungen vom elliptischen Typus", Mathematische Zeitschrift (in German), 26, Springer Berlin / Heidelberg: 551–558, doi:10.1007/BF01475472, ISSN 0025-5874 Bombieri, Enrico; De Giorgi, Ennio; Giusti, E. (1969), "Minimal cones and the Bernstein problem", Inventiones Mathematicae, 7 (3): 243–268, Bibcode:1969InMat...7..243B, doi:10.1007/BF01404309, ISSN 0020-9910, MR 0250205, S2CID 59816096 De Giorgi, Ennio (1965), "Una estensione del teorema di Bernstein", Ann. Scuola Norm. Sup. Pisa (3), 19: 79–85, MR 0178385 Fleming, Wendell H. (1962), "On the oriented Plateau problem", Rendiconti del Circolo Matematico di Palermo. Serie II, 11: 69–90, doi:10.1007/BF02849427, ISSN 0009-725X, MR 0157263 Sabitov, I. Kh. (2001) [1994], "Bernstein theorem", Encyclopedia of Mathematics, EMS Press Simons, James (1968), "Minimal varieties in riemannian manifolds", Annals of Mathematics, Second Series, 88 (1): 62–105, doi:10.2307/1970556, ISSN 0003-486X, JSTOR 1970556, MR 0233295 Straume, E. (2001) [1994], "Bernstein problem in differential geometry", Encyclopedia of Mathematics, EMS Press == External links == Encyclopaedia of Mathematics article on the Bernstein theorem
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Wikipedia:Bernstein–Kushnirenko theorem#0
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The Bernstein–Kushnirenko theorem (or Bernstein–Khovanskii–Kushnirenko (BKK) theorem), proven by David Bernstein and Anatoliy Kushnirenko in 1975, is a theorem in algebra. It states that the number of non-zero complex solutions of a system of Laurent polynomial equations f 1 = ⋯ = f n = 0 {\displaystyle f_{1}=\cdots =f_{n}=0} is equal to the mixed volume of the Newton polytopes of the polynomials f 1 , … , f n {\displaystyle f_{1},\ldots ,f_{n}} , assuming that all non-zero coefficients of f n {\displaystyle f_{n}} are generic. == Statement == Let A {\displaystyle A} be a finite subset of Z n . {\displaystyle \mathbb {Z} ^{n}.} Consider the subspace L A {\displaystyle L_{A}} of the Laurent polynomial algebra C [ x 1 ± 1 , … , x n ± 1 ] {\displaystyle \mathbb {C} \left[x_{1}^{\pm 1},\ldots ,x_{n}^{\pm 1}\right]} consisting of Laurent polynomials whose exponents are in A {\displaystyle A} . That is: L A = { f | f ( x ) = ∑ α ∈ A c α x α , c α ∈ C } , {\displaystyle L_{A}=\left\{f\,\left|\,f(x)=\sum _{\alpha \in A}c_{\alpha }x^{\alpha },c_{\alpha }\in \mathbb {C} \right\},\right.} where for each α = ( a 1 , … , a n ) ∈ Z n {\displaystyle \alpha =(a_{1},\ldots ,a_{n})\in \mathbb {Z} ^{n}} we have used the shorthand notation x α {\displaystyle x^{\alpha }} to denote the monomial x 1 a 1 ⋯ x n a n . {\displaystyle x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}.} Now take n {\displaystyle n} finite subsets A 1 , … , A n {\displaystyle A_{1},\ldots ,A_{n}} of Z n {\displaystyle \mathbb {Z} ^{n}} , with the corresponding subspaces of Laurent polynomials, L A 1 , … , L A n . {\displaystyle L_{A_{1}},\ldots ,L_{A_{n}}.} Consider a generic system of equations from these subspaces, that is: f 1 ( x ) = ⋯ = f n ( x ) = 0 , {\displaystyle f_{1}(x)=\cdots =f_{n}(x)=0,} where each f i {\displaystyle f_{i}} is a generic element in the (finite dimensional vector space) L A i . {\displaystyle L_{A_{i}}.} The Bernstein–Kushnirenko theorem states that the number of solutions x ∈ ( C ∖ 0 ) n {\displaystyle x\in (\mathbb {C} \setminus 0)^{n}} of such a system is equal to n ! V ( Δ 1 , … , Δ n ) , {\displaystyle n!V(\Delta _{1},\ldots ,\Delta _{n}),} where V {\displaystyle V} denotes the Minkowski mixed volume and for each i , Δ i {\displaystyle i,\Delta _{i}} is the convex hull of the finite set of points A i {\displaystyle A_{i}} . Clearly, Δ i {\displaystyle \Delta _{i}} is a convex lattice polytope; it can be interpreted as the Newton polytope of a generic element of the subspace L A i {\displaystyle L_{A_{i}}} . In particular, if all the sets A i {\displaystyle A_{i}} are the same, A = A 1 = ⋯ = A n , {\displaystyle A=A_{1}=\cdots =A_{n},} then the number of solutions of a generic system of Laurent polynomials from L A {\displaystyle L_{A}} is equal to n ! vol ( Δ ) , {\displaystyle n!\operatorname {vol} (\Delta ),} where Δ {\displaystyle \Delta } is the convex hull of A {\displaystyle A} and vol is the usual n {\displaystyle n} -dimensional Euclidean volume. Note that even though the volume of a lattice polytope is not necessarily an integer, it becomes an integer after multiplying by n ! {\displaystyle n!} . == Trivia == Kushnirenko's name is also spelt Kouchnirenko. David Bernstein is a brother of Joseph Bernstein. Askold Khovanskii has found about 15 different proofs of this theorem. == References == == See also == Bézout's theorem for another upper bound on the number of common zeros of n polynomials in n indeterminates.
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Wikipedia:Bernstein–Sato polynomial#0
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In mathematics, the Bernstein–Sato polynomial is a polynomial related to differential operators, introduced independently by Joseph Bernstein (1971) and Mikio Sato and Takuro Shintani (1972, 1974), Sato (1990). It is also known as the b-function, the b-polynomial, and the Bernstein polynomial, though it is not related to the Bernstein polynomials used in approximation theory. It has applications to singularity theory, monodromy theory, and quantum field theory. Severino Coutinho (1995) gives an elementary introduction, while Armand Borel (1987) and Masaki Kashiwara (2003) give more advanced accounts. == Definition and properties == If f ( x ) {\displaystyle f(x)} is a polynomial in several variables, then there is a non-zero polynomial b ( s ) {\displaystyle b(s)} and a differential operator P ( s ) {\displaystyle P(s)} with polynomial coefficients such that P ( s ) f ( x ) s + 1 = b ( s ) f ( x ) s . {\displaystyle P(s)f(x)^{s+1}=b(s)f(x)^{s}.} The Bernstein–Sato polynomial is the monic polynomial of smallest degree amongst such polynomials b ( s ) {\displaystyle b(s)} . Its existence can be shown using the notion of holonomic D-modules. Kashiwara (1976) proved that all roots of the Bernstein–Sato polynomial are negative rational numbers. The Bernstein–Sato polynomial can also be defined for products of powers of several polynomials (Sabbah 1987). In this case it is a product of linear factors with rational coefficients. Nero Budur, Mircea Mustață, and Morihiko Saito (2006) generalized the Bernstein–Sato polynomial to arbitrary varieties. Note, that the Bernstein–Sato polynomial can be computed algorithmically. However, such computations are hard in general. There are implementations of related algorithms in computer algebra systems RISA/Asir, Macaulay2, and SINGULAR. Daniel Andres, Viktor Levandovskyy, and Jorge Martín-Morales (2009) presented algorithms to compute the Bernstein–Sato polynomial of an affine variety together with an implementation in the computer algebra system SINGULAR. Christine Berkesch and Anton Leykin (2010) described some of the algorithms for computing Bernstein–Sato polynomials by computer. == Examples == If f ( x ) = x 1 2 + ⋯ + x n 2 {\displaystyle f(x)=x_{1}^{2}+\cdots +x_{n}^{2}\,} then ∑ i = 1 n ∂ i 2 f ( x ) s + 1 = 4 ( s + 1 ) ( s + n 2 ) f ( x ) s {\displaystyle \sum _{i=1}^{n}\partial _{i}^{2}f(x)^{s+1}=4(s+1)\left(s+{\frac {n}{2}}\right)f(x)^{s}} so the Bernstein–Sato polynomial is b ( s ) = ( s + 1 ) ( s + n 2 ) . {\displaystyle b(s)=(s+1)\left(s+{\frac {n}{2}}\right).} If f ( x ) = x 1 n 1 x 2 n 2 ⋯ x r n r {\displaystyle f(x)=x_{1}^{n_{1}}x_{2}^{n_{2}}\cdots x_{r}^{n_{r}}} then ∏ j = 1 r ∂ x j n j f ( x ) s + 1 = ∏ j = 1 r ∏ i = 1 n j ( n j s + i ) f ( x ) s {\displaystyle \prod _{j=1}^{r}\partial _{x_{j}}^{n_{j}}\quad f(x)^{s+1}=\prod _{j=1}^{r}\prod _{i=1}^{n_{j}}(n_{j}s+i)\quad f(x)^{s}} so b ( s ) = ∏ j = 1 r ∏ i = 1 n j ( s + i n j ) . {\displaystyle b(s)=\prod _{j=1}^{r}\prod _{i=1}^{n_{j}}\left(s+{\frac {i}{n_{j}}}\right).} The Bernstein–Sato polynomial of x2 + y3 is ( s + 1 ) ( s + 5 6 ) ( s + 7 6 ) . {\displaystyle (s+1)\left(s+{\frac {5}{6}}\right)\left(s+{\frac {7}{6}}\right).} If tij are n2 variables, then the Bernstein–Sato polynomial of det(tij) is given by ( s + 1 ) ( s + 2 ) ⋯ ( s + n ) {\displaystyle (s+1)(s+2)\cdots (s+n)} which follows from Ω ( det ( t i j ) s ) = s ( s + 1 ) ⋯ ( s + n − 1 ) det ( t i j ) s − 1 {\displaystyle \Omega (\det(t_{ij})^{s})=s(s+1)\cdots (s+n-1)\det(t_{ij})^{s-1}} where Ω is Cayley's omega process, which in turn follows from the Capelli identity. == Applications == If f ( x ) {\displaystyle f(x)} is a non-negative polynomial then f ( x ) s {\displaystyle f(x)^{s}} , initially defined for s with non-negative real part, can be analytically continued to a meromorphic distribution-valued function of s by repeatedly using the functional equation f ( x ) s = 1 b ( s ) P ( s ) f ( x ) s + 1 . {\displaystyle f(x)^{s}={1 \over b(s)}P(s)f(x)^{s+1}.} It may have poles whenever b(s + n) is zero for a non-negative integer n. If f(x) is a polynomial, not identically zero, then it has an inverse g that is a distribution; in other words, f g = 1 as distributions. If f(x) is non-negative the inverse can be constructed using the Bernstein–Sato polynomial by taking the constant term of the Laurent expansion of f(x)s at s = −1. For arbitrary f(x) just take f ¯ ( x ) {\displaystyle {\bar {f}}(x)} times the inverse of f ¯ ( x ) f ( x ) . {\displaystyle {\bar {f}}(x)f(x).} The Malgrange–Ehrenpreis theorem states that every differential operator with constant coefficients has a Green's function. By taking Fourier transforms this follows from the fact that every polynomial has a distributional inverse, which is proved in the paragraph above. Pavel Etingof (1999) showed how to use the Bernstein polynomial to define dimensional regularization rigorously, in the massive Euclidean case. The Bernstein-Sato functional equation is used in computations of some of the more complex kinds of singular integrals occurring in quantum field theory Fyodor Tkachov (1997). Such computations are needed for precision measurements in elementary particle physics as practiced for instance at CERN (see the papers citing (Tkachov 1997)). However, the most interesting cases require a simple generalization of the Bernstein-Sato functional equation to the product of two polynomials ( f 1 ( x ) ) s 1 ( f 2 ( x ) ) s 2 {\displaystyle (f_{1}(x))^{s_{1}}(f_{2}(x))^{s_{2}}} , with x having 2-6 scalar components, and the pair of polynomials having orders 2 and 3. Unfortunately, a brute force determination of the corresponding differential operators P ( s 1 , s 2 ) {\displaystyle P(s_{1},s_{2})} and b ( s 1 , s 2 ) {\displaystyle b(s_{1},s_{2})} for such cases has so far proved prohibitively cumbersome. Devising ways to bypass the combinatorial explosion of the brute force algorithm would be of great value in such applications. == Notes == == References == Andres, Daniel; Levandovskyy, Viktor; Martín-Morales, Jorge (2009). "Principal intersection and bernstein-sato polynomial of an affine variety". Proceedings of the 2009 international symposium on Symbolic and algebraic computation. Association for Computing Machinery. pp. 231–238. arXiv:1002.3644. doi:10.1145/1576702.1576735. ISBN 9781605586090. S2CID 2747775. Berkesch, Christine; Leykin, Anton (2010). "Algorithms for Bernstein--Sato polynomials and multiplier ideals". Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation. pp. 99–106. arXiv:1002.1475. doi:10.1145/1837934.1837958. ISBN 9781450301503. S2CID 33730581. Bernstein, Joseph (1971). "Modules over a ring of differential operators. Study of the fundamental solutions of equations with constant coefficients". Functional Analysis and Its Applications. 5 (2): 89–101. doi:10.1007/BF01076413. MR 0290097. S2CID 124605141. Budur, Nero; Mustață, Mircea; Saito, Morihiko (2006). "Bernstein-Sato polynomials of arbitrary varieties". Compositio Mathematica. 142 (3): 779–797. arXiv:math/0408408. Bibcode:2004math......8408B. doi:10.1112/S0010437X06002193 (inactive 20 February 2025). MR 2231202. S2CID 6955564.{{cite journal}}: CS1 maint: DOI inactive as of February 2025 (link) Borel, Armand (1987). Algebraic D-Modules. Perspectives in Mathematics. Vol. 2. Boston, MA: Academic Press. ISBN 0-12-117740-8. Coutinho, Severino C. (1995). A primer of algebraic D-modules. London Mathematical Society Student Texts. Vol. 33. Cambridge, UK: Cambridge University Press. ISBN 0-521-55908-1. Etingof, Pavel (1999). "Note on dimensional regularization". Quantum fields and strings: A course for mathematicians. Vol. 1. Providence, R.I.: American Mathematical Society. pp. 597–607. ISBN 978-0-8218-2012-4. MR 1701608. (Princeton, NJ, 1996/1997) Kashiwara, Masaki (1976). "B-functions and holonomic systems. Rationality of roots of B-functions". Inventiones Mathematicae. 38 (1): 33–53. Bibcode:1976InMat..38...33K. doi:10.1007/BF01390168. MR 0430304. S2CID 17103403. Kashiwara, Masaki (2003). D-modules and microlocal calculus. Translations of Mathematical Monographs. Vol. 217. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-2766-6. MR 1943036. Sabbah, Claude (1987). "Proximité évanescente. I. La structure polaire d'un D-module". Compositio Mathematica. 62 (3): 283–328. MR 0901394. Sato, Mikio; Shintani, Takuro (1972). "On zeta functions associated with prehomogeneous vector spaces". Proceedings of the National Academy of Sciences of the United States of America. 69 (5): 1081–1082. Bibcode:1972PNAS...69.1081S. doi:10.1073/pnas.69.5.1081. JSTOR 61638. MR 0296079. PMC 426633. PMID 16591979. Sato, Mikio; Shintani, Takuro (1974). "On zeta functions associated with prehomogeneous vector spaces". Annals of Mathematics. Second Series. 100 (1): 131–170. doi:10.2307/1970844. JSTOR 1970844. MR 0344230. Sato, Mikio (1990) [1970]. "Theory of prehomogeneous vector spaces (algebraic part)". Nagoya Mathematical Journal. 120: 1–34. doi:10.1017/s0027763000003214. MR 1086566. the English translation of Sato's lecture from Shintani's note Tkachov, Fyodor V. (1997). "Algebraic algorithms for multiloop calculations. The first 15 years. What's next?". Nucl. Instrum. Methods A. 389 (1–2): 309–313. arXiv:hep-ph/9609429. Bibcode:1997NIMPA.389..309T. doi:10.1016/S0168-9002(97)00110-1. S2CID 37109930.
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Wikipedia:Bernt Michael Holmboe#0
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Bernt Michael Holmboe (23 March 1795 – 28 March 1850) was a Norwegian mathematician. He was home-tutored from an early age, and was not enrolled in school until 1810. Following a short period at the Royal Frederick University, which included a stint as assistant to Christopher Hansteen, Holmboe was hired as a mathematics teacher at the Christiania Cathedral School in 1818, where he met the future renowned mathematician Niels Henrik Abel. Holmboe's lasting impact on mathematics worldwide has been said to be his tutoring of Abel, both in school and privately. The two became friends and remained so until Abel's early death. Holmboe moved to the Royal Frederick University in 1826, where he worked until his own death in 1850. Holmboe's significant impact on mathematics in the fledgling Norway was his textbook in two volumes for secondary schools. It was widely used, but faced competition from Christopher Hansteen's alternative offering, sparking what may have been Norway's first debate about school textbooks. == Early life and career == Bernt Michael Holmboe was born in Vang in 1795, the son of vicar Jens Holmboe (1746–1823) and his wife Cathrine Holst (1763–1823). He grew up in Eidsberg with his nine siblings, and was the elder brother of noted philologist Christopher Andreas Holmboe. Holmboe was homeschooled from an early age, but was sent to the Christiania Cathedral School in 1810 to complete his secondary education. There he undertook extracurricular studies in mathematics. He enrolled as a student at the Royal Frederick University in 1814, a turbulent year in Norwegian history. Norway had been a province of Denmark since 1397, but had come under Swedish control in the January 1814 Treaty of Kiel. Following Norway's declaration of independence in the Constitution of 17 May, Sweden responded by waging a military campaign against Norway during the summer of 1814. Holmboe was a spokesperson for the student group opposed to the presence of Swedish troops in the country. Any outspokenness from the student community was highly visible at the time, as the university had only seventeen students as of 1813, its year of establishment. As well as his private studies, Holmboe attended lectures given by Søren Rasmusen. In 1815 he was appointed to the position of scientific assistant under Christopher Hansteen, a lecturer at the university, and delivered some lectures himself. In early 1818, Holmboe became a mathematics teacher at the Christiania Cathedral School, a position that had become vacant in 1817. The school principal, Jacob Rosted, had invited Holmboe's brother, Christopher Andreas, who had also studied mathematics, to take up the position, but he had decided instead to concentrate on philology; Christopher went on to research the Sanskrit language among others. In his teaching, Holmboe drew inspiration from Joseph-Louis Lagrange. === Niels Henrik Abel === At the Christiania Cathedral School, Holmboe met Niels Henrik Abel, then a pupil there. Holmboe quickly discovered Abel's talent, and proclaimed him as a "splendid genius" in his report card. The school's primary focus was on classical education and Latin, and so Holmboe tutored Abel privately. His personal support for Abel has been called "[Holmboe's] most important contribution to mathematics". Holmboe and Abel became close friends. Two of Holmboe's younger brothers studied with Abel, and the three were also on friendly terms. Abel was invited to the Holmboe family residence in Eidsberg on several occasions, including to celebrate Christmas. Abel died from tuberculosis in 1829, at the age of twenty-six. Ten years after Abel's death Holmboe edited and published his complete works in two volumes—Oeuvres complètes de N.H. Abel ('Complete Works of N.H. Abel'). He was the first to do so. == Later life and career == Holmboe published his first textbook in mathematics in 1825. The 274-page book was named Lærebog i Mathematiken. Første Deel (Textbook in Mathematics. Part One). In 1827 he followed with the second volume, Lærebog i Mathematiken. Anden Deel (Textbook in Mathematics. Part Two), consisting of a further 155 pages. He used his own teaching experience as the background for his writing; mainly abstract, the purpose of the books was to instill logical thinking. For instance, in the field geometry, it enticed readers to envisage a figure instead of putting it to paper. The books became widely used, and were reprinted in four and five volumes respectively. In 1826, Holmboe was appointed a lecturer at the Royal Frederick University. Some claimed that he owed his appointment to Abel's absence, as the latter was travelling around in Europe at that time. Holmboe also taught mathematics at a military college, from 1826 until his death, and was promoted to professor at the Royal Frederick University in 1834. His later publications include Stereometrie (Stereometry) (1833), Plan- og sfærisk Trigonometrie (Plan and Spherical Trigonometry) (1834), and Lærebog i den høiere Mathematik (Textbook of Advanced Mathematics) (1849). Holmboe was an influence on other mathematicians as well as Abel, including Ole Jacob Broch (born 1818). At the university, Holmboe again met Christopher Hansteen, who had become a professor there in 1816. In 1835, Hansteen published his own mathematics textbook for secondary schools. A reaction to Holmboe's books and method of teaching, it was more practically oriented. Holmboe wrote a review of the book for the newspaper Morgenbladet, in which he advised schools not to use it. A public debate followed, with contributions from other mathematicians. It has been claimed that this was the first debate on the subject of school textbooks in Norway. Hansteen's textbook was not reprinted. Holmboe also became involved in the field of insurance. From 1832 to 1848 he was a member of Tilsynskomiteen for private forsørgelses- og understøttelsesselskaper, the country's first public committee for the supervision of insurance companies. On the other side of the table, from 1847 Holmboe was a member of the board of directors of the insurance company Gjensidige, founded by his former student Ole Jacob Broch. == Legacy == A Bernt Michael Holmboe Memorial Prize for teachers of mathematics was established in 2005, and is awarded annually. The prize money, NOK 50,000, are provided by the Abel Foundation, which awards the Abel Prize. The prize is administered by the board of the Norwegian Mathematical Society. The current board chairman is Tom Lindstrøm, professor at the University of Oslo. A street at Majorstuen in Oslo, Holmboes gate, has been named after Bernt Michael Holmboe. Before 1879 it was named Hansteens gate, after Christopher Hansteen. == Personal life == Holmboe married twice. He died in 1850. == References and notes == === Notes === === References === == Further reading == Livio, Mario (2005). The Equation That Couldn't be Solved. New York: Simon & Schuster. ISBN 0-7432-5821-5.. See particularly Chapter 4.
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Wikipedia:Bert Peletier#0
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Lambertus Adrianus "Bert" Peletier (29 March 1937 – 16 December 2023) was a Dutch mathematician. He was a professor of analysis and applied mathematics at Leiden University from 1977 until his retirement in 2002. == Life == Lambertus Adrianus Peletier was born on 9 March 1937 in Rijswijk. His grandfather Benjamin Broers was a socialist Dutch minister in the Dutch Reformed Church, who lived with his family during World War II. Peletier's father was an engineer working for Shell plc and took him to his laboratory. He was interested in technique from a young age and desired to pursue a technical study. His maths teacher inspired him to study physics. Peletier thus studied theoretical physics at Delft University of Technology. After graduating, Peletier had the opportunity to study one year at the Massachusetts Institute of Technology and became inspired by academic life. In 1967 he obtained his PhD at Eindhoven University of Technology, with a thesis titled: "On a class of wave equations". He subsequently spent time abroad at the University of Sussex and University of Minnesota and became more inspired by applied mathematics. In 1977 he became professor of analysis and applied mathematics at Leiden University. He retired in 2002. His expertise lay in the field of nonlinear analysis. During the latter part of his career, in the late 1990s, he applied mathematics for the development of medicine. He especially focused on the latter after his retirement and worked intensively with pharmacologists. In 1995 he was one of the founders of the Lorentz Center, an institute for the organization of interactive workshops in the sciences. Peletier was elected a member of the Academia Europaea in 1989. He was elected a member of Royal Netherlands Academy of Arts and Sciences in 1999. Peletier was elected a Fellow of the Society for Industrial and Applied Mathematics in 2009. In April 2013 Peletier was made a Knight in the Order of the Netherlands Lion. Peletier died in Leiden on 16 December 2023, at the age of 86. == References == == External links == Profile at the Mathematics Genealogy Project
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Wikipedia:Bertram Martin Wilson#0
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Prof Bertram Martin Wilson FRSE (14 November 1896, London – 18 March 1935, Dundee, Scotland) was an English mathematician, remembered primarily as a co-editor, along with G. H. Hardy and P. V. Seshu Aiyar, of Srinivasa Ramanujan's Collected Papers. (It seems probable that Wilson did not know about Ramanujan's lost notebook, which was probably passed by G. H. Hardy to G. N. Watson some years after Wilson's death.) == Life == He was born in London on 14 November 1896 the son of Rev Alfred Henry Wilson and his wife, Ellen Elizabeth Vincent. Wilson was educated at King Edward's School, Birmingham and then studied Mathematics at Trinity College, Cambridge, graduating MA. In 1920 he was appointed as a Lecturer in Mathematics at the University of Liverpool, and was promoted to Senior Lecturer in 1926. He remained there for slightly more than thirteen years, working under three professors, Frank Stanton Carey (1860–1928), J. C. Burkhill, and E. C. Titchmarsh. In 1933 Wilson was appointed Professor of Pure and Applied Mathematics at University College, Dundee as successor to John Edward Aloysius Steggall, who retired. His early interests in mathematical analysis are shown in some dozen original papers, published between 1919 and 1924, mainly on the theory of numbers and on integral equations and orthogonal functions, but these were only a small part of his total contribution to mathematics. Subsequently, most of his attention was directed to the study of the remarkable Indian mathematician Srinivasa Ramanujan: he was one of the editors of the Collected Papers, published in 1927, and at the time of his death was still occupied, again as joint editor, in the much greater task of annotating Ramanujan's note-books. He was also on the editorial board of the new Compositio Mathematica and contributed many reviews to the Fortschritte. Sometime in the late 1920s, G. N. Watson and B. M. Wilson began the task of editing Ramanujan's notebooks. The second notebook, being a revised, enlarged edition of the first, was their primary focus. Wilson was assigned Chapters 2–14, and Watson was to examine Chapters 15–21. Wilson devoted his efforts to this task until 1935, when he died from an infection at the early age of 38. Watson wrote over 30 papers inspired by the notebooks before his interest evidently waned in the late 1930s. Thus, the project was never completed. Wilson was elected on 5 March 1934 a Fellow of the Royal Society of Edinburgh. His proposers were Sir Edmund Taylor Whittaker, James Hartley Ashworth, Nicholas Lightfoot and Edward Thomas Copson. In 1934 he gave a talk Ramanujan's Note-Books and their Place in Modern Mathematics at the third Colloquium of the Edinburgh Mathematical Society at the University of St Andrews. Wilson died on 18 March 1935 following a brief illness. == Family == In 1930 he married Margaret Fancourt Mitchell. == Subsequent history for Ramanujan's Notebooks == G. N. Watson and B. M. Wilson never completed their project of editing Ramanujan's notebooks (not including the "lost" notebook), but Bruce C. Berndt completed their project in a 5-volume publication Ramanujan's Notebooks, Parts I—V. The following quote refers to the three notebooks involved in Watson and Wilson's project: .... Ramanujan left three notebooks. The first notebook, totaling 351 pages, contains 16 chapters of loosely organized material with the remainder unorganized. In the organized part, which ends on page 263 ..., Ramanujan wrote on only one side of the paper. Shortly thereafter, Ramanujan began to write on both sides of the page and then returned to the unused reverse sides to record additional material, so that only about 20 of 351 pages are actually blank. The second notebook is a revised enlargement of the first and was probably composed during the nine months that Ramanujan held a scholarship at the University of Madras prior to his departure from England. This notebook contains 21 chapters, comprising 256 pages, followed by 100 pages of miscellaneous material. The third short notebook contains 33 pages of miscellaneous material. Berndt benefited substantially from Wilson's considerable efforts in editing Ramanujan's second notebook. Because some journals require the permission of each author when an article is to be published, for some of Berndt's work he was not permitted to put Wilson or Watson as a coauthor. However, Berndt published several articles with Wilson as a coauthor. == Selected publications == Wilson, B. M. (1923). "Proofs of some formulae enunciated by Ramanujan". Proceedings of the London Mathematical Society. 2. 21 (1): 235–255. doi:10.1112/plms/s2-21.1.235. Wilson, B. M. (1923). "On the manner of divergence of the Legendre series of an integrable function". Proceedings of the London Mathematical Society. 2. 21 (1): 389–400. doi:10.1112/plms/s2-21.1.389. Wilson, B. M. (1924). "An Application of Pfeiffer's Method to a Problem of Hardy and Littlewood". Proceedings of the London Mathematical Society. 2. 22 (1): 248–253. doi:10.1112/plms/s2-22.1.248. Wilson, B. M. (1926). "On Pseudo-Orthogonal Systems of Functions Arising as Solutions of an Integral Equation". Proceedings of the London Mathematical Society. 2. 25 (1): 59–102. doi:10.1112/plms/s2-25.1.59. Wilson, B. M. (1930). "S. Ramanujan". The Mathematical Gazette. 15 (207): 89–94. doi:10.2307/3607411. JSTOR 3607411. S2CID 250432530. == References ==
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Wikipedia:Bertrand Toën#0
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Bertrand Toën (born September 17, 1973 in Millau, France) is a mathematician who works as a director of research at the Centre national de la recherche scientifique (CNRS) at the Paul Sabatier University, Toulouse, France. He received his PhD in 1999 from the Paul Sabatier University, where he was supervised by Carlos Simpson and Joseph Tapia. Toën is a specialist of algebraic geometry. He his best known for his systematic use of homotopical methods in algebraic geometry. Together with Gabriele Vezzosi and Jacob Lurie he has laid the foundations of the subject of derived algebraic geometry and higher category theory. His works establish several contributions to noncommutative algebraic geometry in the sense of Kontsevich and (shifted) symplectic geometry. He was an invited speaker at the International Congress of Mathematicians in 2014, speaking in the section on "Algebraic and Complex Geometry" with a talk "Derived Algebraic Geometry and Deformation Quantization". He was awarded an ERC Advanced Grant in 2016. In 2019 he received the Sophie Germain prize. == References == == External links == Home page Bertrand Toën at the Mathematics Genealogy Project
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Wikipedia:Bertrand's postulate#0
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In number theory, Bertrand's postulate is the theorem that for any integer n > 3 {\displaystyle n>3} , there exists at least one prime number p {\displaystyle p} with n < p < 2 n − 2. {\displaystyle n<p<2n-2.} A less restrictive formulation is: for every n > 1 {\displaystyle n>1} , there is always at least one prime p {\displaystyle p} such that n < p < 2 n . {\displaystyle n<p<2n.} Another formulation, where p n {\displaystyle p_{n}} is the n {\displaystyle n} -th prime, is: for n ≥ 1 {\displaystyle n\geq 1} p n + 1 < 2 p n . {\displaystyle p_{n+1}<2p_{n}.} This statement was first conjectured in 1845 by Joseph Bertrand (1822–1900). Bertrand himself verified his statement for all integers 2 ≤ n ≤ 3 000 000 {\displaystyle 2\leq n\leq 3\,000\,000} . His conjecture was completely proved by Chebyshev (1821–1894) in 1852 and so the postulate is also called the Bertrand–Chebyshev theorem or Chebyshev's theorem. Chebyshev's theorem can also be stated as a relationship with π ( x ) {\displaystyle \pi (x)} , the prime-counting function (number of primes less than or equal to x {\displaystyle x} ): π ( x ) − π ( x 2 ) ≥ 1 , for all x ≥ 2. {\displaystyle \pi (x)-\pi {\bigl (}{\tfrac {x}{2}}{\bigr )}\geq 1,{\text{ for all }}x\geq 2.} == Prime number theorem == The prime number theorem (PNT) implies that the number of primes up to x, π(x), is roughly x/log(x), so if we replace x with 2x then we see the number of primes up to 2x is asymptotically twice the number of primes up to x (the terms log(2x) and log(x) are asymptotically equivalent). Therefore, the number of primes between n and 2n is roughly n/log(n) when n is large, and so in particular there are many more primes in this interval than are guaranteed by Bertrand's postulate. So Bertrand's postulate is comparatively weaker than the PNT. But PNT is a deep theorem, while Bertrand's Postulate can be stated more memorably and proved more easily, and also makes precise claims about what happens for small values of n. (In addition, Chebyshev's theorem was proved before the PNT and so has historical interest.) The similar and still unsolved Legendre's conjecture asks whether for every n ≥ 1, there is a prime p such that n2 < p < (n + 1)2. Again we expect that there will be not just one but many primes between n2 and (n + 1)2, but in this case the PNT does not help: the number of primes up to x2 is asymptotic to x2/log(x2) while the number of primes up to (x + 1)2 is asymptotic to (x + 1)2/log((x + 1)2), which is asymptotic to the estimate on primes up to x2. So, unlike the previous case of x and 2x, we do not get a proof of Legendre's conjecture for large n. Error estimates on the PNT are not (indeed, cannot be) sufficient to prove the existence of even one prime in this interval. In greater detail, the PNT allows to estimate the boundaries for all ε > 0, there exists an S such that for x > S: ( 1 − ε ) x 2 2 log x < π ( x 2 ) < ( 1 + ε ) x 2 2 log x , {\displaystyle (1-\varepsilon ){\frac {x^{2}}{2\log x}}\;<\;\pi (x^{2})\;<\;(1+\varepsilon ){\frac {x^{2}}{2\log x}}\;,} ( 1 − ε ) ( x + 1 ) 2 2 log ( x + 1 ) < π ( ( x + 1 ) 2 ) < ( 1 + ε ) ( x + 1 ) 2 2 log ( x + 1 ) . {\displaystyle (1-\varepsilon ){\frac {(x+1)^{2}}{2\log(x+1)}}\;<\;\pi ((x+1)^{2})\;<\;(1+\varepsilon ){\frac {(x+1)^{2}}{2\log(x+1)}}\;.} The ratio between the lower bound π((x+1)2) and the upper bound of π(x2) is ( x + 1 ) 2 x 2 ⋅ log x log ( x + 1 ) ⋅ 1 − ε 1 + ε . {\displaystyle {\frac {(x+1)^{2}}{x^{2}}}\cdot {\frac {\log x}{\log(x+1)}}\cdot {\frac {1-\varepsilon }{1+\varepsilon }}\;.} Note that since ( x + 1 ) 2 x 2 → 1 {\displaystyle {\frac {(x+1)^{2}}{x^{2}}}\rightarrow 1} when x → ∞ {\displaystyle x\rightarrow \infty } , log x log ( x + 1 ) < 1 {\displaystyle {\frac {\log x}{\log(x+1)}}<1} for all x > 0, and 1 − ε 1 + ε < 1 {\displaystyle {\frac {1-\varepsilon }{1+\varepsilon }}<1} for a fixed ε, there exists an R such that the ratio above is less than 1 for all x > R. Thus, it does not ensure that there exists a prime between π(x2) and π((x+1)2). More generally, these simple bounds are not enough to prove that there exists a prime between π(xn) and π((x+1)n) for any positive integer n > 1. == Generalizations == In 1919, Ramanujan (1887–1920) used properties of the Gamma function to give a simpler proof than Chebyshev's. His short paper included a generalization of the postulate, from which would later arise the concept of Ramanujan primes. Further generalizations of Ramanujan primes have also been discovered; for instance, there is a proof that 2 p i − n > p i for i > k where k = π ( p k ) = π ( R n ) , {\displaystyle 2p_{i-n}>p_{i}{\text{ for }}i>k{\text{ where }}k=\pi (p_{k})=\pi (R_{n})\,,} with pk the kth prime and Rn the nth Ramanujan prime. Other generalizations of Bertrand's postulate have been obtained using elementary methods. (In the following, n runs through the set of positive integers.) In 1973, Denis Hanson proved that there exists a prime between 3n and 4n. In 2006, apparently unaware of Hanson's result, M. El Bachraoui proposed a proof that there exists a prime between 2n and 3n. El Bachraoui's proof is an extension of Erdős's arguments for the primes between n and 2n. Shevelev, Greathouse, and Moses (2013) discuss related results for similar intervals. Bertrand’s postulate over the Gaussian integers is an extension of the idea of the distribution of primes, but in this case on the complex plane. Thus, as Gaussian primes extend over the plane and not only along a line, and doubling a complex number is not simply multiplying by 2 but doubling its norm (multiplying by 1+i), different definitions lead to different results, some are still conjectures, some proven. == Sylvester's theorem == Bertrand's postulate was proposed for applications to permutation groups. Sylvester (1814–1897) generalized the weaker statement with the statement: the product of k consecutive integers greater than k is divisible by a prime greater than k. Bertrand's (weaker) postulate follows from this by taking k = n, and considering the k numbers n + 1, n + 2, up to and including n + k = 2n, where n > 1. According to Sylvester's generalization, one of these numbers has a prime factor greater than k. Since all these numbers are less than 2(k + 1), the number with a prime factor greater than k has only one prime factor, and thus is a prime. Note that 2n is not prime, and thus indeed we now know there exists a prime p with n < p < 2n. == Erdős's theorems == In 1932, Erdős (1913–1996) also published a simpler proof using binomial coefficients and the Chebyshev function ϑ {\displaystyle \vartheta } , defined as: ϑ ( x ) = ∑ p = 2 x log ( p ) , {\displaystyle \vartheta (x)=\sum _{p=2}^{x}\log(p),} where p ≤ x runs over primes. See proof of Bertrand's postulate for the details. Erdős proved in 1934 that for any positive integer k, there is a natural number N such that for all n > N, there are at least k primes between n and 2n. An equivalent statement had been proved in 1919 by Ramanujan (see Ramanujan prime). == Better results == It follows from the prime number theorem that for any real ε > 0 {\displaystyle \varepsilon >0} there is a n 0 > 0 {\displaystyle n_{0}>0} such that for all n > n 0 {\displaystyle n>n_{0}} there is a prime p {\displaystyle p} such that n < p < ( 1 + ε ) n {\displaystyle n<p<(1+\varepsilon )n} . It can be shown, for instance, that lim n → ∞ π ( ( 1 + ε ) n ) − π ( n ) n / log n = ε , {\displaystyle \lim _{n\to \infty }{\frac {\pi ((1+\varepsilon )n)-\pi (n)}{n/\log n}}=\varepsilon ,} which implies that π ( ( 1 + ε ) n ) − π ( n ) {\displaystyle \pi ((1+\varepsilon )n)-\pi (n)} goes to infinity (and, in particular, is greater than 1 for sufficiently large n {\displaystyle n} ). Non-asymptotic bounds have also been proved. In 1952, Jitsuro Nagura proved that for n ≥ 25 {\displaystyle n\geq 25} there is always a prime between n {\displaystyle n} and ( 1 + 1 5 ) n {\displaystyle {\bigl (}1+{\tfrac {1}{5}}{\bigr )}n} . In 1976, Lowell Schoenfeld showed that for n ≥ 2 010 760 {\displaystyle n\geq 2\,010\,760} , there is always a prime p {\displaystyle p} in the open interval n < p < ( 1 + 1 16 597 ) n {\displaystyle n<p<{\bigl (}1+{\tfrac {1}{16\,597}}{\bigr )}n} . In his 1998 doctoral thesis, Pierre Dusart improved the above result, showing that for k ≥ 463 {\displaystyle k\geq 463} , p k + 1 ≤ ( 1 + 1 2 log 2 p k ) p k {\displaystyle p_{k+1}\leq \left(1+{\frac {1}{2\log ^{2}{p_{k}}}}\right)p_{k}} , and in particular for x ≥ 3 275 {\displaystyle x\geq 3\,275} , there exists a prime p {\displaystyle p} in the interval x < p ≤ ( 1 + 1 2 log 2 x ) x {\displaystyle x<p\leq \left(1+{\frac {1}{2\log ^{2}{x}}}\right)x} . In 2010 Pierre Dusart proved that for x ≥ 396 738 {\displaystyle x\geq 396\,738} there is at least one prime p {\displaystyle p} in the interval x < p ≤ ( 1 + 1 25 log 2 x ) x {\displaystyle x<p\leq \left(1+{\frac {1}{25\log ^{2}{x}}}\right)x} . In 2016, Pierre Dusart improved his result from 2010, showing (Proposition 5.4) that if x ≥ 89 693 {\displaystyle x\geq 89\,693} , there is at least one prime p {\displaystyle p} in the interval x < p ≤ ( 1 + 1 log 3 x ) x {\displaystyle x<p\leq \left(1+{\frac {1}{\log ^{3}{x}}}\right)x} . He also shows (Corollary 5.5) that for x ≥ 468 991 632 {\displaystyle x\geq 468\,991\,632} , there is at least one prime p {\displaystyle p} in the interval x < p ≤ ( 1 + 1 5 000 log 2 x ) x {\displaystyle x<p\leq \left(1+{\frac {1}{5\,000\log ^{2}{x}}}\right)x} . Baker, Harman and Pintz proved that there is a prime in the interval [ x − x 0.525 , x ] {\displaystyle [x-x^{0.525},\,x]} for all sufficiently large x {\displaystyle x} . Dudek proved that for all n ≥ e e 33.3 {\displaystyle n\geq e^{e^{33.3}}} , there is at least one prime between n 3 {\displaystyle n^{3}} and ( n + 1 ) 3 {\displaystyle (n+1)^{3}} . Dudek also proved that the Riemann hypothesis implies that for all x ≥ 2 {\displaystyle x\geq 2} there is a prime p {\displaystyle p} satisfying x − 4 π x log x < p ≤ x . {\displaystyle x-{\frac {4}{\pi }}{\sqrt {x}}\log x<p\leq x.} == Consequences == The sequence of primes, along with 1, is a complete sequence; any positive integer can be written as a sum of primes (and 1) using each at most once. The only harmonic number that is an integer is the number 1. == See also == Oppermann's conjecture Prime gap Proof of Bertrand's postulate Ramanujan prime == Notes == == Bibliography == P. Erdős (1934), "A Theorem of Sylvester and Schur", Journal of the London Mathematical Society, 9 (4): 282–288, doi:10.1112/jlms/s1-9.4.282 Jitsuro Nagura (1952), "On the interval containing at least one prime number", Proc. Japan Acad., 28 (4): 177–181, doi:10.3792/pja/1195570997 Chris Caldwell, Bertrand's postulate at Prime Pages glossary. H. Ricardo (2005), "Goldbach's Conjecture Implies Bertrand's Postulate", Amer. Math. Monthly, 112: 492 Hugh L. Montgomery; Robert C. Vaughan (2007). Multiplicative number theory I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97. Cambridge: Cambridge Univ. Press. p. 49. ISBN 978-0-521-84903-6. J. Sondow (2009), "Ramanujan primes and Bertrand's postulate", Amer. Math. Monthly, 116 (7): 630–635, arXiv:0907.5232, doi:10.4169/193009709x458609 == External links == Sondow, Jonathan & Weisstein, Eric W. "Bertrand's Postulate". MathWorld. A proof of the weak version in the Mizar system: http://mizar.org/version/current/html/nat_4.html#T56 Bertrand's postulate − A proof of the weak version at www.dimostriamogoldbach.it/en/
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Wikipedia:Beryl May Dent#0
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Beryl May Dent (10 May 1900 – 9 August 1977) was an English mathematical physicist, technical librarian, and a programmer of early analogue and digital computers to solve electrical engineering problems. She was born in Chippenham, Wiltshire, the eldest daughter of schoolteachers. The family left Chippenham in 1901, after her father became head teacher of the then recently established Warminster County School. In 1923, she graduated from the University of Bristol with First Class Honours in applied mathematics. She was awarded the Ashworth Hallett scholarship by the university and was accepted as a postgraduate student at Newnham College, Cambridge. She returned to Bristol in 1925, after being appointed a researcher in the Physics Department at the University of Bristol, with her salary being paid by the Department of Scientific and Industrial Research. In 1927, John Lennard-Jones was appointed Professor of Theoretical physics, a chair being created for him, with Dent becoming his research assistant in theoretical physics. Lennard‑Jones pioneered the theory of interatomic and intermolecular forces at Bristol and she became one of his first collaborators. They published six papers together from 1926 to 1928, dealing with the forces between atoms and ions, that were to become the foundation of her master's thesis. Later work has shown that the results they obtained had direct application to atomic force microscopy by predicting that non-contact imaging is possible only at small tip-sample separations. In 1930, she joined Metropolitan-Vickers Electrical Company Ltd, Manchester, as a technical librarian for the scientific and technical staff of the research department. She became active in the Association of Special Libraries and Information Bureaux (ASLIB) and was honorary secretary to the founding committee for the Lancashire and Cheshire branch of the association. She served on various ASLIB committees and made conference presentations detailing different aspects of the company's library and information service. She continued to publish scientific papers, contributing numerical methods for solving differential equations by the use of the differential analyser that was built for the University of Manchester and Douglas Hartree. She was the first to develop a detailed reduced major axis method for the best fit of a series of data points. Later in her career she became leader of the computation section at Metropolitan-Vickers, and then a supervisor in the research department for the section that was investigating semiconducting materials. She joined the Women's Engineering Society and published papers on the application of digital computers to electrical design. She retired in 1960, with Isabel Hardwich, later a fellow and president of the Women's Engineering Society, replacing her as section leader for the women in the research department. In 1962, she moved with her mother and sister to Sompting, West Sussex, and died there in 1977. == Early life == Beryl May was born on (1900-05-10)10 May 1900, at Penley Villa, Park Lane, Chippenham, Wiltshire, the eldest daughter of Agnes Dent (1869–1967), née Thornley, and Eustace Edward (1868–1954). She was baptised at St Paul's, Chippenham, on 8 June 1900.: 1 They had married at St Mary's Church, Goosnargh, near Preston, Lancashire, on 27 July 1898. Her mother was educated at the Harris Institute, Preston, passing examinations in science and art. She was a teacher at Attercliffe School, in northeast Sheffield, before moving to Goosnargh School, near her hometown of Preston, where her elder brother and sister, John William and Mary Ann Thornley, were the head teachers. In March 1894, she had applied for the headship at Fairfield School, Cockermouth, making the shortlist, but the board decided to appoint a local candidate. On 18 March 1889, Dent's father was appointed to a teaching assistant position at Portland Road School, in Halifax, West Yorkshire, after completing a teaching apprenticeship with the school board. In the same year, Florence Emily Dent, his elder sister, was appointed head teacher at West Vale girls' school, Stainland Road, Greetland, moving from the Higher Board School at Halifax. In August 1889, he obtained a first class pass in mathematics from the Halifax Mechanics' Institute. He enrolled on a degree course at University College, Aberystwyth, in the Education Day Training College. In January 1894, he was awarded a first by Aberystwyth, and a first in the external University of London examinations. His first teaching post was at Coopers' Company Grammar School, Bow, London, before moving to Chippenham, where he was a senior assistant teacher at the Chippenham County School. In October 1901, Dent's father left Chippenham to become head teacher of the then recently established Warminster County School, that adjoined the Athenaeum Theatre in Warminster. The family moved to Boreham Road, Warminster, where houses were built in the early 19th century. In April 1907, they moved to 22 Portway, Warminster, situated a short distance from the County School and the Athenaeum.: 264 He was elected chair of the Warminster Urban District Council from 1920 to 1922, and remained as head teacher of the County School until his retirement in August 1929. Dent's father was also a regular cast member of the Warminster Operatic Society at the Athenaeum and other venues. Dent and her younger sister, Florence Mary, would often appear with him on stage in such operettas as Snow White and the seven dwarfs and the Princess Ju‑Ju (The Golden Amulet), a Japanese operetta in three acts by Clementine Ward. In Princess Ju‑Ju, she played La La, one of the three maidens attendant on the princess, and sang the first act solo, She must be demure. In act two of the same musical, she performed in the fan dance, Spirits of the Night. She also acted in a scene from Tennyson's Princess at the County School's prize giving day on 16 December 1913. == Education == === Warminster County School (1909–1917) === From 1909, Dent was educated at Warminster County School, where her father was head teacher. At school, she was close friends with her neighbour at Portway, Evelyn Mary Day, the eldest daughter of Henry George Day, a former butler to Colonel Charles Gathorne Gathorne‑Hardy, son of Gathorne Gathorne-Hardy, 1st Earl of Cranbrook. In August 1914, she passed the University of Oxford Junior Local Examination with First Class Honours, and on the strength of her examination result, she was awarded a scholarship by the school. In 1915, she passed the senior examination with second class honours and a distinction in French, and subsequently, her scholarship was renewed. She then joined the sixth form and won the school prize for French in December 1916. In March 1918, she applied for a scholarship in mathematics from Somerville College, one of the first two women's colleges in the University of Oxford. She was highly commended but was not awarded a scholarship nor an exhibition. === University of Bristol (1919–1923) === In 1918, Dent joined the Royal Aircraft Establishment (RAE) at Farnborough, Hampshire. The First World War opened new employment opportunities for women, and RAE was one of the first military establishments to recruit women into engineering, and mathematical and computational research.: 116 : 10 In the same period that Dent was at RAE, Lorna Swain, then mathematics tutor at Newnham College in the University of Cambridge, worked at the establishment on the problem of aircraft propeller vibration.: 84 The Treasury reduced RAE's funding after the end of the war, and consequently, the number of resources and staff available to support research fell significantly.: 58 In 1919, she left RAE after being accepted on to the general Bachelor of Arts (BA) degree course at the University of Bristol. In June 1920, she passed her intermediate examination in French with supplementary courses in Latin, history, and mathematics.: 1 In the following academic year, Dent joined the honours course in mathematics and took an intermediate examination in physics.: 1 After spending the summer of 1921 at her parents' home in Warminster, she returned for the start of the 1921 to 1922 academic year to find that Paul Dirac had joined the mathematics course. The course of mathematics at Bristol University normally lasted three years, but because of Dirac's previous training, the Department of Mathematics had allowed him to join in the second year. They were taught applied mathematics by Henry Ronald Hassé, the then head of the Mathematics Department, and pure mathematics by Peter Fraser. Both of them had come from Cambridge; Fraser had been appointed in 1906 to the staff of the Bristol University College, soon to become the University of Bristol, and Hassé joined him in 1919 as professor of mathematics.: 111 Fraser introduced them to mathematical rigour, projective geometry, and rigorous proofs in differential and integral calculus. Dirac would later say that Peter Fraser was "the best teacher he had ever had." Dent studied four courses in pure mathematics: There was a choice of specialisation in the final year; applied or pure mathematics. As the only official, registered fee-paying student, Dent had the right to choose, and she settled on applied mathematics for the final year. The department could offer only one set of lectures so Dirac also had to follow the same course. Dent studied four courses in applied mathematics: === Newnham College, University of Cambridge (1923–1924) === In June 1923, Dent graduated with Dirac, gaining a Bachelor of Science (BSc) degree in applied mathematics with First Class Honours. On 7 July 1923, she was awarded the Ashworth Hallett scholarship by the University of Bristol and was accepted as a postgraduate student at Newnham College in the University of Cambridge. On her death in 1922, Lilias Sophia Ashworth Hallett left one thousand pounds each to the University of Bristol and Girton College, University of Cambridge, to found scholarships for women.: 259, 261 The University of Bristol scholarship was open to women graduates of a recognised college or university, and worth £45 at the time (equivalent to £3,200 in 2023). She spent a year at Cambridge, leaving in 1924 without further academic qualification. Before 1948, the University of Cambridge denied women graduates a degree, although in the same year as she left Cambridge, Katharine Margaret Wilson was the first woman to be awarded a PhD by the university. == Career == === High School for Girls, Barnsley (1924–1925) === Dent spent the summer of 1924 at her parents' home in Warminster, playing mixed doubles tennis in a tournament organised by the local Women's Unionist Association. In September of the same year, she was appointed an assistant teacher in mathematics at the High School for Girls, in Barnsley, Huddersfield Road, on an annual salary of £250 (equivalent to £18,000 in 2023). Annie Rose Nuttall, the school's head teacher, was a former student at Newnham College. In the early 1920s, women who had studied university level mathematics faced limited employment prospects, as mathematics and engineering professions, other than perhaps school teaching, were dominated by men. Dent resigned her position on 31 August 1925 after being appointed a demonstrator (research assistant) in the Department of Physics at the University of Bristol, with her salary being paid by the Department of Scientific and Industrial Research, the forerunner of the Science and Engineering Research Council (SERC).: 107 === Department of Physics, University of Bristol (1925–1929) === In 1924, the University of Bristol Council had set aside a portion of a bequest from Henry Herbert Wills for the Department of Physics where Arthur Mannering Tyndall was building up a staff for teaching and research in the H. H. Wills Physics Laboratory, Royal Fort House Gardens. From August 1925, John Lennard-Jones, of Trinity College, University of Cambridge, was elected reader in mathematical physics. In March 1927, Lennard‑Jones was appointed Professor of Theoretical physics, a chair being created for him, with Dent becoming his research assistant in theoretical physics.: 24 Lennard‑Jones pioneered the theory of interatomic and intermolecular forces at Bristol and Dent became one of his first collaborators. Lennard‑Jones and Dent published six papers together from 1926 to 1928, dealing with the forces between atoms and ions, with the objective of calculating theoretically the properties of carbonate and nitrate crystals. Dent's thesis for her master's degree, Some theoretical determinations of crystal structure (1927), was the basis of the three papers that followed in 1927; with Lennard‑Jones, "Some theoretical determinations of crystal parameters. CXVI", and with Lennard‑Jones and Sydney Chapman, "Structure of carbonate crystals" and "Part II. Structure of carbonate crystals". On 28 June 1927, she was awarded a MSc degree for her thesis and research work. In 1927, the physics laboratory at Bristol had a surplus of funds, and so it was decided that the funds would be used to provide more technical help. Consequently, Dent was asked to combine her research duties with the post of part-time departmental librarian, the first appointment of librarian in the Department of Physics.: 26 In 1928, Lennard‑Jones and Dent published two papers, "Cohesion at a crystal surface", and with Sydney Chapman, "The change in lattice spacing at a crystal boundary", that studied the force fields on a thin crystal cleavage. Around this time, quantum mechanics was developed to become the standard formulation for atomic physics. Lennard‑Jones left Bristol in 1929 to study the subject for a year as a Rockefeller Fellow at the University of Göttingen. She wrote one last paper before leaving the physics department at Bristol: "The effect of boundary distortion on the surface energy of a crystal" (1929) examined the effect of the polarisation of surface ions in decreasing the surface energy of alkali halides. In November 1929, she was appointed to the position of technical librarian for the scientific and technical staff in the research department at Metropolitan-Vickers, Trafford Park, Manchester.: 14–15 In December 1929, Dent resigned her position at Bristol and it was accepted with regret by the university council. Marjorie Josephine Littleton, the daughter of a local Bristol councillor and a graduate of Girton College, University of Cambridge, was appointed as her replacement on the 1 February 1930. Littleton was later Sir Nevill Mott's co-author and research assistant in the physics department.: 517 In 1930, Lennard‑Jones returned to Bristol, as Dean of the Faculty of Science, and introduced the new quantum theories to the Bristol group. === Metropolitan-Vickers, Trafford Park (1930–1960) === Metropolitan-Vickers was a British heavy industrial firm, well-known for industrial electrical equipment and generators, street lighting, electronics, steam turbines, and diesel locomotives. They built the Metrovick 950, the first commercial transistorised computer. In 1917, a Research and Education Department was established at the Trafford Park site, when the care of the library came within the remit of James George Pearce. He made the library the centre of a new "technical intelligence" section.: 193 In the 1920s, the post of librarian was held by Lucy Stubbs, a former assistant librarian at the University of Birmingham, and past member of the first standing committee of ASLIB.: 228 Stubbs did not possess scientific qualifications, maintaining that a librarian, if assisted by other technical staff, did not need to understand science or engineering.: 193 In 1929, James Steele Park Paton reorganised and expanded the section with Dent succeeding Stubbs as technical librarian on 6 January 1930.: 15 She joined the scientific and technical staff as was one of only two senior women in the research department,: 314 and in contrast to Stubbs, was employed principally for her technical skills.: 193 Dent was honorary secretary to the founding committee for the ASLIB Lancashire and Cheshire branch from 1931 to 1936.: 204–205 In 1932, the branch had twenty-six members and had organised four meetings, including one addressed by Sir Henry Tizard, the then President of ASLIB. After the war, it formed the basis for the Northern Branch of the association.: 412 Technical librarianship emerged as a new scientific career in interwar Britain and rapidly became one of the few types of professional industrial employment that was routinely open to both women and men.: 301 By 1933, Dent reported that the Metropolitan-Vickers library had three thousand engineering volumes and around the same number in pamphlets and patent specifications. Besides covering electrical subjects, the library covered accountancy, employment questions, and subjects of interest to the sales department. It also issued a weekly bulletin, scrutinised patents, handled patents taken out by research staff, and exchanged information with associated companies. Dent continued to publish papers in applied mathematics and contribute to papers on emerging computational technologies. In "On observations of points connected by a linear relation" (1935), she developed a detailed reduced major axis method for line fitting that built on the work of Robert Adcock and Charles Kummell. In 1937, David Myers, then at the Engineering Laboratory at the University of Oxford, asked Douglas Hartree and Arthur Porter to calculate the space charge limitation of secondary current in a triode.: 91 : 96 The calculations relied on some initial numerical integrations that were carried out by Dent on a differential analyser. The results corresponded closely to those obtained experimentally by Myers at Oxford.: 91 : 97 Her knowledge of higher mathematics meant that she was asked to check the mathematics in papers for publication by engineers at Metropolitan-Vickers. For example, Cyril Frederick Gradwell, a graduate of Trinity College, Cambridge, asked her to scrutinise the algebraic part of his work in "The Solution of a problem in disk bending occurring in connexion with gas turbines" (1950). She would later analyse the problem of stress distribution in a thick disk based on a method devised by Philip Pollock, for Richard William Bailey, the former director of the mechanical, metallurgical, and chemical sections of the research department at Metropolitan-Vickers.: 16 Dent was a delegate at the fourteenth International Conference on Documentation and was invited to the Government's conference dinner on 22 September 1938 at the Great Dining Hall of Christ Church, Oxford. In 1939, she was elected to the editing committee of the ASLIB book list. In 1944, she was put in charge of the women working in the research department laboratory at Metropolitan-Vickers, and in 1946, she was promoted to section leader of the new computation section. Her role would bring her into contact with Audrey Stuckes, a materials science researcher in the department, and a graduate of Newnham College, who would later head the physics department at the University of Salford. In 1953, they collaborated on an investigation into the heating effects that occur when a current is passed through a semiconductor that has no barrier layer. Dent suggested methods to solve the equations and computed the numerical integrations. In the following year, she developed the Fourier analysis in "Regenerative Deflection as a Parametrically Excited Resonance Phenomenon" (1954), that calculated the optimal radial oscillations to maintain cyclotron resonance in a synchrocyclotron. The causes of axial spreading of the charged particle beam during extraction were also analysed. Dent joined the Women's Engineering Society and published papers on the application of digital computers to electrical design. With Brian Birtwistle, she wrote programs for the Ferranti Mark 1 (Mark 1) computer at the University of Manchester, that demonstrated that high-speed digital computers could provide considerable assistance to the electrical design engineer. Birtwistle would later have an extensive career in the computer industry, working at, amongst others, Honeywell Information Systems and ADP Network Services. In 1958, she carried out computer calculations for the mechanical engineering team at the Nuclear Power Group, Radbroke Hall. Their paper outlined a procedure for calculating the theoretical deflection (bending) of a circular grid of support girders for a graphite neutron moderator in a gas-cooled reactor. A general expression was derived from the central deflection of the grid and the maximum bending moment on the central cross-beam for a range of grid diameters. In 1959, and a year from retirement, Dent modelled a proposed Zeta circuit on the Mark 1 computer, for Eric Hartill's paper on constructing a high-power pulse transformer and circuit. The cost of the computation was about two thousand pounds (equivalent to £59,000 in 2023), corresponding to around eighty hours of machine time. She retired from Metropolitan-Vickers in May 1960, with Isabel Hardwich, later a fellow and president of the Women's Engineering Society, replacing her as section leader for the women in the research department.: 232 : 243 == Personal life == In the 1920s, Dent was living at Clifton Hill House, the university hall of residence for women in Clifton. May Christophera Staveley was her warden and tutor at Clifton Hill House, and Dent returned to Bristol on 22 December 1934 for Staveley's funeral. Dent was a member of the Clifton Hill House Old Students Association, and secretary and treasurer of the group of former Clifton Hill House students.: 1, 9 She would later write "I was very sorry indeed to leave Bristol and have many happy memories of my time there. I shall miss living at the [Clifton Hill House] Hall very much.": 15 In 1926, Dent was elected treasurer of the University of Bristol's Convocation, the university's alumni association. In 1927, she was one of eleven people elected to the standing committee of the Convocation: 62 She later represented the Manchester branch of the association. Around 1926, Dent was appointed honorary secretary of the Bristol Cheeloo Association. The association's aim was to raise sufficient funds to support a chair of chemistry at Cheeloo University. In an effort to publicise the cause and raise money, she presented to the local branch of the Women's International League in October 1928. In July 1929, in Dent's last year at Bristol, she went on holiday to North Devon with friends that included Gertrude Roxbee, known as "Rox", who had graduated with Dent in 1923 with a BSc in botany.: 12–13 After moving to Manchester in January 1930, Dent found shared lodgings at 10 Montrose Avenue, West Didsbury, in the same house as Roxbee who, at that time, was a teacher at Whalley Range High School.: 15, 50 At weekends, she would ramble to Hebden Bridge, and with Roxbee, learnt to figure skate at the Ice Palace, a former ice rink on Derby Street in Cheetham Hill.: 54 In September 1930, she returned to Bristol for the ninety-eighth conference of the British Association for the Advancement of Science (British Association), meeting her friends at an alumnae association lunch.: 93 In the afternoon of the 4 September 1930, she toured Avonmouth Docks as a conference member,: 94 and in the evening, was invited to a reception held by Walter Bryant, the then lord mayor of Bristol, at the Bristol Museum & Art Gallery.: 94 . On the following day, she visited an aircraft manufacturer at Whitchurch Airport and attended a garden party at Wills Hall.: 94 On the Monday of the conference, Dent was in the audience to see Paul Dirac present his paper on the proton and the structure of matter.: 94 She would later comment:: 94 I heard a striking paper by Dirac, who was a student with me, who is now a very famous person, as I always knew he would be ... I now go about boasting that I was in the same class! Dent's father died on (1954-06-24)24 June 1954, at their shared home, 529 King's Road, Stretford, with the funeral service taking place at St Matthew's Church, Stretford. She had close links to St Matthew's; from 1956 to 1962, she served as a school manager for St Matthew's Church of England Primary School at Poplar Road, Stretford. == Later life and death == In 1962, Dent and her mother moved from Stretford to 1 Cokeham Road, Sompting, a village in the coastal Adur District of West Sussex, between Lancing and Worthing. Her mother died on (1967-04-05)5 April 1967 and was cremated at the Downs Crematorium on 10 April 1967. Dent's sister, Florence Mary, also lived in the house until her death on 13 September 1986(1986-09-13) (aged 84). After a brief period as a teacher at a prep school in Malmesbury, Wiltshire, Florence worked as a secretary for a marine insurance firm attached to Lloyd's of London at 12 Leadenhall Street, commuting into London from Harrow each day.: 57–59 Dent considered herself to be an Anglican layperson who was neither high nor low church. In April 1970, she was elected treasurer of Lancing and Sompting Churches Fraternal (the parish fraternity organisation), and in March 1972, she was elected electoral officer for the parochial church council of St Mary's Church, Sompting. Her Christian faith is perhaps not unexpected, given her father's work for the church in Warminster, and the era she grew up in, where religion pervaded social and political life. However, it is notable that she remained a Christian while pursuing a scientific career. In June 1974, Dent was hospitalised for seven weeks at Southlands Hospital, Shoreham-by-Sea, and after a long period of disablement, she died at Worthing Hospital on 9 August 1977(1977-08-09) (aged 77). The funeral service was held on 12 August 1977 at St Mary's, followed by cremation. Her ashes were interred at Worthing Crematorium, in the Gardens of Rest, towards the Spring Glades, and her entry in the book of remembrance at the crematorium states: Beryl May Dent 1900 – A real Christian loved by all – 1977 The bishop's chair at St Mary's, situated close to the altar, bears a brass plaque with the following inscription: In loving memory of BERYL DENT 1900 – 1977 == Legacy == An archive of Dent's papers, that relate to her life and work in the 1920s in the physics department at the University of Bristol, is held in the Special Collections of the University of Bristol Arts and Social Sciences Library, in Tyndall Avenue, Bristol. Included in that archive is a series of letterbooks, written in the 1930s by members of the Clifton Hill House Old Students' Association, that include news and photographs of Dent, her family, and friends. === Atomic force microscopy === In 1928, Lennard‑Jones and Dent published two papers, "Cohesion at a crystal surface" and "The change in lattice spacing at a crystal boundary", that for the first time, outlined a calculation of the potential of the electric field in a vacuum, produced by a thin sodium chloride crystal surface. They gave an expression for the electric potential produced by a system of point charges in vacuum (although not a real cubic sodium chloride ionic lattice).: 796–797 The expression for the potential in vacuum, φ 0 ( r ) {\displaystyle \varphi _{0}\left(r\right)} , at the point r = {x, y, z}, near the cubic lattice of point ions with different signs, the charge e k {\displaystyle e_{k}} , and the period a (a crystalline solid is distinguished by the fact that the atoms making up the crystal are arranged in a periodic fashion), can be represented in the form:: 797 r ∥ = { x , y } {\displaystyle r_{\parallel }=\left\{x,y\right\}} is the lateral vector that fixes the observation point coordinates in the sample plane. k l , m {\displaystyle k_{l,m}} is the reciprocal lattice vector. s is the number of planes to be calculated inside the crystal; s set to zero would calculate the surface plane. The expression sums the set of potential static charges for the surface and lower planes of the crystal lattice. Lennard‑Jones and Dent showed that this expression forms a rapidly convergent Fourier series.: 797 Harold Eugene Buckley, a crystallographic researcher at the University of Manchester until his death in 1959,: 481 had suggested that their results should be treated with caution. For example, the contraction a crystal plane would suffer under the conditions prescribed would not be the same as that of a similar plane with a solid mass of crystal behind it. Another difficulty arises because calculation of crystal surface field force fields are so great that simplifying assumptions have to be made to render them capable of a solution. Michael Jaycock and Geoffrey Parfitt, then respectively senior lecturer in surface and colloid chemistry at Loughborough University of Technology and professor of chemical engineering at Carnegie Mellon University, concurred with Buckley, noting that "an ideal crystal, in which the ionic positions at the surface were identical to those achieved in the bulk crystal ... is obviously extremely improbable." However, they acknowledged that the Lennard‑Jones and Dent model was singularly elegant, and like most researchers working before the advent of modern computers, they were limited in what could be attempted computationally. Nonetheless, Lennard‑Jones and Dent demonstrated that the force exerted on a single ion, by a surface with evenly distributed positive and negative ions, decreases very rapidly with increasing distance.: 14 Later work by Jason Cleveland, Manfred Radmacher, and Paul Hansma, has shown that this result has direct application to atomic force microscopy by predicting that non-contact imaging is possible only at small tip-sample separations.: 543 === Reduced major axis regression === The theoretical underpinnings of standard least squares regression analysis are based on the assumption that the independent variable (often labelled as x) is measured without error as a design variable. The dependent variable (labeled y) is modeled as having uncertainty or error. Both independent and dependent measurements may have multiple sources of error. Therefore, the underlying least squares regression assumptions can be violated. Reduced major axis (RMA) regression is specifically formulated to handle errors in both the x and y variables.: 1 If the estimate of the ratio of the error variance of y to the error variance of x is denoted by 𝜆, then the reduced major axis method assumes that 𝜆 can be approximated by the ratio of the total variances of x and y. RMA minimizes both vertical and horizontal distances of the data points from the predicted line (by summing areas) rather than the least squares sum of squared vertical (y-axis) distances.: 2 In Dent's 1935 paper on linear regression, entitled "On observations of points connected by a linear relation", she admitted that when the variances in the x and y variables are unknown, "we cannot hope to find the true positions of the observed points, but only their most probable positions." However, by treating the probability of the errors in terms of Gaussian error functions, she contended that this expression may be regarded as "a function of the unknown quantities", or the likelihood function of the data distribution.: 106 Furthermore, she argued that maximising this function to obtain the maximum likelihood estimation,: 5 subject to the condition that the points are collinear, will give the parameters for the line of best fit. She then deduced formulae for the errors in estimating the centroid and the line inclination when the data consists of a single (unrepeated) observation.: 106 Maurice Kendall and Alan Stuart showed that the maximum likelihood estimator of a likelihood function, depending on a parameter θ {\displaystyle \theta } , satisfies the following quadratic equation: where x {\displaystyle x} and y {\displaystyle y} are the X {\displaystyle \mathbf {X} } and Y {\displaystyle \mathbf {Y} } vectors in a covariance matrix giving the covariance between each pair of x and y variables. The superscript T {\displaystyle T} indicates the transpose of the matrix. Using the quadratic formula to solve for the positive root (or zero) of (2):: 183 Inspection of (3) shows that as 𝜆 tends to plus infinity, the positive root tends to:: 183 Correspondingly, as 𝜆 tends to zero, the root tends to:: 183 Dent had solved the maximum likelihood estimator in the case where the covariance matrix is not known.: 1049 Dent's maximum likelihood estimator is the geometric mean of θ x {\displaystyle \theta _{x}} and θ y {\displaystyle \theta _{y}} , equivalent to:: 184 Dennis Lindley repeated Dent's analysis and stated that Dent's geometric mean estimator is not a consistent estimator for the likelihood function,: 235–236, 241 and that the gradient of the estimate will have a bias, and this remains true even if the number of observations tends to infinity.: 15 Subsequently, Theodore Anderson pointed out that the likelihood function has no maximum in this case, and therefore, there is no maximum likelihood estimator.: 3 Kenneth Alva Norton, a former consulting engineer with the then National Bureau of Standards, responded to Lindley, stating Lindley's own methods and assumptions lead to a biased prediction. Furthermore, Albert Madansky, late H. G. B. Alexander professor of business administration at University of Chicago Booth School of Business, noted that Lindley took the wrong root for the quadratic in (2) for the case where x T y {\displaystyle x^{T}y} is negative.: 201–203 Richard J. Smith has stated that Dent was the first to develop a RMA regression method for line fitting that built on the work of Robert Adcock in "A Problem in Least Squares" (1878) and Charles Kummell in "Reduction of observation equations which contain more than one observed quantity" (1879). It is now believed that she was the first to propose what is often called the geometric mean functional relationship estimator of slope, and that her essential arguments can be generalised to any number of variables.: 106 Moreover, although her solution has its theoretical limitations, it is of practical importance, as it likely represents the best a priori estimate if nothing is known about the true error distribution in the model. It is generally much less reasonable to assume that all the error, or residual scatter, is attributable to one of the variables.: 3 === Electrical design using digital computers === In the 1950s, British electrical engineers would rarely use a digital computer, and if they did, it would be to solve some complicated equation outside the scope of analogue computers. To a certain extent, engineers were deterred by the difficulty and the time taken to program a particular problem. Furthermore, the varied and often unique problems that arise in electrical design practice, together with the degree of uncertainty of the numerical data of many problems, accentuated this tendency. On 10 April 1956, Dent and Brian Birtwistle presented their paper, "The digital computer as an aid to the electrical design engineer", to the Convention on Digital Computer Techniques at the Institution of Electrical Engineers. The paper was intended to show, by describing three relatively simple applications, that the digital computer could be a useful aid to the electrical design engineer. The three example problems were: The Ferranti Mark 1 computer at the University of Manchester was used for the calculations in the three problems. Dent was allowed to use the university's library of subroutines, from which the following were taken and incorporated into the programs: The first problem of calculating the impulse voltage distribution on transformer windings took about five hours of machine time. Conversely, a hand calculation, using a method described by Thomas John Lewis in "The Transient Behaviour of Ladder Networks of the Type Representing Transformer and Machine Windings" (1954), took around three months.: 486 The use of a computer in the second problem allowed for a more accurate solution as it was possible to include nonlinear magnetic characteristics in the calculation. In the last problem, the torque and speed curves for the synchronous motors were calculated in around fifteen minutes.: 486 Their paper was one of the first to recognise that high-speed digital computers could provide considerable assistance to the electrical design engineer by carrying out automatically the optimum design of products. Significant research had been devoted to determining a transformer's internal transient voltage distribution. Early attempts were hampered by computational limitations encountered when solving large numbers of coupled differential equations with analogue computers. It was not until Dent, with Hartill and Miles, in "A method of analysis of transformer impulse voltage distribution using a digital computer" (1958), recognised the limitations of the analogue models and developed a digital computer model, and associated program, where non-uniformity in the transformer windings could be introduced and any input voltage applied. == Publications == === Selected papers and academic articles === === Publications detail === == See also == == Footnotes == == References == == Bibliography == Adcock, Robert James (March 1878). Hendricks, Joel Evans (ed.). "A Problem in Least Squares". The Analyst. 5 (2). Princeton: Princeton University: 53–54. doi:10.2307/2635758. hdl:2027/njp.32101040986372. ISSN 0741-7918. JSTOR 2635758. OCLC 277641453. Cleveland, Jason Paul; Radmacher, Manfred; Hansma, Paul Kenneth (September 1994). "Atomic Scale Force Mapping with the Atomic Force Microscope". Applied Sciences. NATO Advanced Study Institute: Series E. 286. Office of Naval Research. Santa Barbara: UC Santa Barbara Physics Department: 1–7. ISSN 0304-9930. OCLC 200640681. Technical Report No. 13. Defense Technical Information Center ADA284940. Retrieved 23 August 2021. Excell, Phyl; Excell, Stanley (September 1979). "Monumental Inscriptions at Sompting Parish Church". Sussex Family Historian. September 1979 to December 1981. Vol. 4, no. 3. Brighton: Sussex Family History Group. pp. 84–85. ISSN 0260-4175. OCLC 3114152. Kummell, Charles Hugo (July 1879). Hendricks, Joel Evans (ed.). "Reduction of observation equations which contain more than one observed quantity". The Analyst. 6 (4). Princeton: Princeton University: 97–105. doi:10.2307/2635646. hdl:2027/hvd.32044102938057. ISSN 0741-7918. JSTOR 2635646. OCLC 277641453. Lewis, Thomas John (October 1954). "The Transient Behaviour of Ladder Networks of the Type Representing Transformer and Machine Windings". Proceedings of the Institution of Electrical Engineers. Part II: Power Engineering. 101 (83). London: Institution of Electrical Engineers: 541–553. doi:10.1049/pi-2.1954.0114. ISSN 0369-8939. Paper No. 1691 Supply Section M. The paper was first received 24 February 1954, and in revised form 21 April 1954. Murray, Janet Horowitz; Stark, Myra, eds. (1985). The Englishwoman's Review of Social and Industrial Questions. The complete run reproduced in facsimile in 41 volumes. January to October 1905. Vol. 36. New York: Garland Publishing. pp. 251–252. hdl:2027/umn.31951002443042e. ISBN 978-0-8240-3761-1. OCLC 13062569. Pollock, Philip John (August 1955). "Discussion on 'The Design of High‑Speed Salient‑Pole A.C. Generators for Water Power Plants'". Proceedings of the Institution of Electrical Engineers. Part A: Power Engineering. 102 (4). London: Institution of Electrical Engineers: 476–482. doi:10.1049/pi-2.1954.0114. ISSN 0369-8882. Retrieved 29 August 2021. North‑Western Centre, at Manchester, 6 May 1952. == Further reading == Byers, Nina; Williams, Gary (2006). Out of the shadows: contributions of twentieth-century women to physics. Cambridge: Cambridge University Press. ISBN 978-0-521-82197-1. OCLC 1050066680. Retrieved 10 December 2022. Dummelow, John (1949). A history of the Metropolitan-Vickers Electrical Company Limited 1899–1949. Manchester: Metropolitan-Vickers Electrical Company Limited. OCLC 5381038. With illustrations. Produced to commemorate the golden jubilee of the Metropolitan-Vickers Electrical Company Limited. A digital edition, digitised by Jim Lawton in 2008, is available at the Sydney Electric Train Society (SETS) website. Fleming, Arthur Percy Morris; Churcher, Brian Andrew Graham; Davies, Leonard John (28 February 1952). "The research laboratories of Associated Electrical Industries Ltd". Proceedings of the Royal Society. B — Biological Sciences. 139 (895). London: Royal Society: 208–235. Bibcode:1952RSPSB.139..208F. doi:10.1098/rspb.1952.0008. PMID 14911826. Napper, Brian (2003). "The Ferranti Mark 1". Computer 50: The University of Manchester Celebrates the Birth of the Modern Computer. Manchester: University of Manchester. Archived from the original on 10 February 2020. Retrieved 4 September 2020. Swinton, Jonathan (4 March 2019). "Women At The Console". Alan Turing's Manchester. Manchester: Deodands. Archived from the original on 27 August 2020. Retrieved 27 August 2020. == External links == Beryl Dent's archive held in the Special Collections of the University of Bristol Arts and Social Sciences Library, in Tyndall Avenue, Bristol. Photograph of the staff working in the research department of Metropolitan-Vickers in 1954 in the collections of the Science and Industry Museum. Dent (front, fifth left) is one of only two women in the department. Arthur Fleming is also pictured (front, centre). The Manchester transistor computer at the Computer History Museum. Beryl May Dent at zbMATH.
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Wikipedia:Besicovitch covering theorem#0
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In mathematical analysis, a Besicovitch cover, named after Abram Samoilovitch Besicovitch, is an open cover of a subset E of the Euclidean space RN by balls such that each point of E is the center of some ball in the cover. The Besicovitch covering theorem asserts that there exists a constant cN depending only on the dimension N with the following property: Given any Besicovitch cover F of a bounded set E, there are cN subcollections of balls A1 = {Bn1}, …, AcN = {BncN} contained in F such that each collection Ai consists of disjoint balls, and E ⊆ ⋃ i = 1 c N ⋃ B ∈ A i B . {\displaystyle E\subseteq \bigcup _{i=1}^{c_{N}}\bigcup _{B\in A_{i}}B.} Let G denote the subcollection of F consisting of all balls from the cN disjoint families A1,...,AcN. The less precise following statement is clearly true: every point x ∈ RN belongs to at most cN different balls from the subcollection G, and G remains a cover for E (every point y ∈ E belongs to at least one ball from the subcollection G). This property gives actually an equivalent form for the theorem (except for the value of the constant). There exists a constant bN depending only on the dimension N with the following property: Given any Besicovitch cover F of a bounded set E, there is a subcollection G of F such that G is a cover of the set E and every point x ∈ E belongs to at most bN different balls from the subcover G. In other words, the function SG equal to the sum of the indicator functions of the balls in G is larger than 1E and bounded on RN by the constant bN, 1 E ≤ S G := ∑ B ∈ G 1 B ≤ b N . {\displaystyle \mathbf {1} _{E}\leq S_{\mathbf {G} }:=\sum _{B\in \mathbf {G} }\mathbf {1} _{B}\leq b_{N}.} == Application to maximal functions and maximal inequalities == Let μ be a Borel non-negative measure on RN, finite on compact subsets and let f {\displaystyle f} be a μ {\displaystyle \mu } -integrable function. Define the maximal function f ∗ {\displaystyle f^{*}} by setting for every x {\displaystyle x} (using the convention ∞ × 0 = 0 {\displaystyle \infty \times 0=0} ) f ∗ ( x ) = sup r > 0 ( μ ( B ( x , r ) ) − 1 ∫ B ( x , r ) | f ( y ) | d μ ( y ) ) . {\displaystyle f^{*}(x)=\sup _{r>0}{\Bigl (}\mu (B(x,r))^{-1}\int _{B(x,r)}|f(y)|\,d\mu (y){\Bigr )}.} This maximal function is lower semicontinuous, hence measurable. The following maximal inequality is satisfied for every λ > 0 : λ μ ( { x : f ∗ ( x ) > λ } ) ≤ b N ∫ | f | d μ . {\displaystyle \lambda \,\mu {\bigl (}\{x:f^{*}(x)>\lambda \}{\bigr )}\leq b_{N}\,\int |f|\,d\mu .} Proof. The set Eλ of the points x such that f ∗ ( x ) > λ {\displaystyle f^{*}(x)>\lambda } clearly admits a Besicovitch cover Fλ by balls B such that ∫ 1 B | f | d μ = ∫ B | f ( y ) | d μ ( y ) > λ μ ( B ) . {\displaystyle \int \mathbf {1} _{B}\,|f|\ d\mu =\int _{B}|f(y)|\,d\mu (y)>\lambda \,\mu (B).} For every bounded Borel subset E´ of Eλ, one can find a subcollection G extracted from Fλ that covers E´ and such that SG ≤ bN, hence λ μ ( E ′ ) ≤ λ ∑ B ∈ G μ ( B ) ≤ ∑ B ∈ G ∫ 1 B | f | d μ = ∫ S G | f | d μ ≤ b N ∫ | f | d μ , {\displaystyle {\begin{aligned}\lambda \,\mu (E')&\leq \lambda \,\sum _{B\in \mathbf {G} }\mu (B)\\&\leq \sum _{B\in \mathbf {G} }\int \mathbf {1} _{B}\,|f|\,d\mu =\int S_{\mathbf {G} }\,|f|\,d\mu \leq b_{N}\,\int |f|\,d\mu ,\end{aligned}}} which implies the inequality above. When dealing with the Lebesgue measure on RN, it is more customary to use the easier (and older) Vitali covering lemma in order to derive the previous maximal inequality (with a different constant). == See also == Vitali covering lemma == References == Besicovitch, A. S. (1945), "A general form of the covering principle and relative differentiation of additive functions, I", Mathematical Proceedings of the Cambridge Philosophical Society, 41 (02): 103–110, doi:10.1017/S0305004100022453. Besicovitch, A. S. (1946), "A general form of the covering principle and relative differentiation of additive functions, II", Mathematical Proceedings of the Cambridge Philosophical Society, 42: 205–235, doi:10.1017/s0305004100022660. DiBenedetto, E (2002), Real analysis, Birkhäuser, ISBN 0-8176-4231-5. Füredi, Zoltán; Loeb, Peter A. (1994), "On the best constant for the Besicovitch covering theorem", Proceedings of the American Mathematical Society, 121 (4): 1063–1073, doi:10.1090/S0002-9939-1994-1249875-4, JSTOR 2161215.
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Wikipedia:Bess Marie Eversull#0
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Bess Marie Eversull Allen (1899 – 1978) was an American mathematician and one of the few women to earn a PhD in mathematics in the United States before World War II. She was the first woman to earn a PhD in mathematics from the University of Cincinnati. == Biography == Bess Eversull was born May 18, 1899, in Elmwood Place, Ohio, a village that was next to Cincinnati but has since become nearly surrounded by that city. Her parents were Warner Solomon Eversull and Olive Magrew. === Education === Eversull attended Woodward High School in Cincinnati and then attended the University of Cincinnati where she graduated with her bachelor's degree in 1921, majoring in both mathematics and English and minoring in French. After graduation, Eversull remained at Cincinnati for her graduate studies. Her 1922 master's thesis and her 1924 doctoral dissertation both concerned triple Fourier series and each document was published soon after it was completed. When she finished her PhD, she was the first doctoral student of mathematician Charles Napoleon Moore and the first woman, and only the third person, to earn a doctorate in mathematics at Cincinnati. For three years (1924–1927) Bess Eversull was an instructor at Smith College until she married a civil engineer, Charles Easton Allen in October 1927 and she took the name Bess Allen. Shortly thereafter, the couple moved to Detroit and for most of the next 20 years, she volunteered in the community except for 15 months during World War II when she worked as a mathematician for a group making films for the U.S. military, from May 1942 until August 1943. Even though she was a married woman, which often disqualified her from finding employment outside the home, she did manage to find some work as a tutor and substitute teacher during that time. In February 1947, Allen took a job as a regular substitute instructor at Wayne University (now Wayne State University). Over the following years, she was promoted to instructor 1948–1950, assistant professor 1950–1959, associate professor 1959–1964, and associate professor emeritus after she retired in 1964. After her retirement, she taught for a few more years at the Detroit Institute of Technology. Bess Allen, who remained childless, died March 18, 1978, at the family home in Detroit, Michigan, and her remains were interred at the Evergreen Cemetery in Detroit. == Memberships == According to Judy Green, Allen belonged to several professional societies. American Mathematical Society Mathematical Association of America American Association for the Advancement of Science American Association of University Women Phi Beta Kappa == References ==
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Wikipedia:Bethany Rose Marsh#0
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Bethany Rose Marsh is a mathematician working in the areas of cluster algebras, representation theory of finite-dimensional algebras, homological algebra, tilting theory, quantum groups, algebraic groups, Lie algebras and Coxeter groups. Marsh currently works at the University of Leeds as a Professor of pure mathematics. She was a EPSRC Leadership Fellow from 2008 to 2014. In addition to her duties at the University of Leeds, Marsh was an editor of the Glasgow Mathematical Journal from 2008 to 2013 and served on the London Mathematical Society editorial board from 2014 to 2018. == Awards == In July 2009, Marsh was awarded the Whitehead Prize by the London Mathematical Society for her work on representation theory, and especially for her research on cluster categories and cluster algebras. == Publications == "MathSciNet". "ArXiv". == References == == External links == Bethany Rose Marsh at the Mathematics Genealogy Project
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Wikipedia:Bettina Richmond#0
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Martha Bettina Richmond (née Zoeller, January 30, 1958 – November 22, 2009) was a German-American mathematician, mathematics textbook author, professor at Western Kentucky University, and murder victim. == Life == Richmond was born in Dresden on January 30, 1958, earned a vordiplom (the German equivalent of a bachelor's degree) from the University of Würzburg,[E] and completed her Ph.D. at Florida State University in 1985. Her doctoral dissertation, Freeness of Hopf algebras over grouplike subalgebras, was supervised by Warren Nichols, a student of Irving Kaplansky. She became a professor at Western Kentucky University, teaching there for 23 years. Topics in her mathematical research included abstract algebra, transformation semigroups, ring theory, and Hopf algebra,[A][B] including the proof of the Nichols–Zoeller freeness theorem in Hopf algebra.[A] With her husband, Thomas Richmond, she was the author of a mathematics textbook, A Discrete Transition to Advanced Mathematics.[C] She also published works in recreational mathematics.[D][E] == Murder == Richmond was stabbed to death on November 22, 2009, in the parking lot of a racquetball facility in downtown Bowling Green, Kentucky. According to the FBI, her murder was likely an opportunistic crime motivated by armed robbery. At the time of her death, she had been on leave from her faculty position to assist her father in Germany. The murder is still unsolved. == Selected publications == == References ==
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Wikipedia:Betty Paërl#0
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Betty Paërl (1935 – 2022) was a Dutch mathematician, writer, dominatrix, and transgender rights activist. == Biography == === Early life and education (1935 – 1968) === Betty Paërl was born in Amsterdam in 1935, and at the age of 22, she married Hetty Paërl (1931 – 2020). She pursued an education in mathematics and subsequently earned her PhD in 1968, ascending to the position of a university lecturer at the University of Amsterdam. === Career before transition (1968 – 1978) === Paërl became increasingly involved in Provo, a Dutch anarchist protest movement. This political activism led her to identify as a communist, a conviction she shared with her wife. The couple served together on the Suriname Committee, an entity established in 1970 to campaign for Suriname's independence from the Netherlands and to show solidarity with the Surinamese people affected by neocolonialism. The committee aimed to enlighten people about the challenges faced by the former colony through activities such as publishing the Suriname Bulletin and informational pamphlets, producing films, and conducting lectures and training sessions. Within this context, Paërl authored four books under her then legal name: "Nederlandse Macht in de Derde Wereld" (Dutch Power in the Third World), Klassenstrijd in Suriname (Class Struggle in Suriname), and Nationalisme of Klassenstrijd (Nationalism or Class Struggle) (1973). She also served as an editor for "De Uitbuiting van Suriname" (The Exploitation of Suriname). In the aftermath of Suriname's independence in 1975, Paërl critiqued the status quo. She asserted that independence should mark the beginning of an extensive decolonial process. She argued that significant foreign companies were exploiting Suriname's resources, and until this practice ceased, the country's independence would remain hollow for its people. The book Class Struggle in Suriname is about a strike in September 1971 that was meant to improve the social and economic situation of workers in Suriname. The book included a poster by Hetty, Paërl's wife at the time. It showed a grotesque ship that stood for Dutch companies that took advantage of Suriname's resources. These works were connected to real political actions and protests in the Netherlands, which helped Suriname's own movements grow stronger. Paërl's work focused on class struggle in the Global South after World War II. For their own benefit, Western countries kept the prices of raw materials low, which kept the Global South from making money. In Suriname, where immigrants from her country faced racism and other problems, she also saw how class and race interacted. Paërl's contributions to anti-colonial activism are important for understanding the ongoing fight for fairness and justice in the Global South. === Career after transition (1978 – 2022) === Between the years 1978 and 1980, Paërl pursued gender transition in the social, legal, and medical areas of life. Paërl has stated that she has been cross-dressing her entire life before she transitioned; however, the desire to fully embody a "feminine persona" did not arise until after she divorced her wife. Paërl has transitioned into a female identity. She also found a newfound enjoyment in sexually charged performances and began going to cabarets in the Rembrandtplein area, where she bought a wig for performances. Her newfound enjoyment led her to experiment with dressing up. She said that working as a sex worker in a brothel gave her a euphoric feeling because it was the first time in her life that she was accepted and approached as a woman on both a sexual and an emotional level. She felt a variety of emotions, some of which were not directly related to the sexual aspect of the experience. She wrote about sadomasochism, advocated for pornography, and worked to eliminate prejudices regarding transgender people, sex work, and sexual pleasure. She advocated for talks about kinks and dildos, and received national attention for her work on the radio, in newspapers, TV, and journals. For instance, she recognized a connection between BDSM and transgender people and held the belief that investigating these roles through SM could assist in making a decision regarding whether or not to transition. After her contract was not renewed at the University of Amsterdam, she went on to pursue a career as a journalist and dominatrix, with mathematics continuing to serve as an important source of motivation for her throughout her life. In 1986, she helped raise money for the gay monument in Amsterdam by offering her escort services. == Selection of works == === Books === Kross M., Megens E., Paërl B., Willems L., eds., 1970. De Uitbuiting van Suriname. Nijmegen: Sun Paërl B., 1971. Nederlandse Macht in de Derde Wereld: Een Inventarisatie van Economische Belangen. Amsterdam: Nederlandse Praktijk van Gennep Paërl B., 1972. Klassenstrijd in Suriname. Nijmegen: Sun Paërl B., 1973. Nationalisme of Klassenstrijd . Leiden: Nesbic-bulletin Paërl B., 1980a. Geluidsfilm Handboek. Amsterdam: Focus Unbehaun K., 1978. Perfecte Filmtrucs. Tr. Paërl B. Amsterdam: Focus. === Journal and newspaper articles === Paërl B., 1984. Betty Paërl over transsexualiteit: de hermafroditische oplossing. Prothese (1984) 20, p. 8-9. Paërl B., 1987. Billen. Sek, 17 (1987) 5 (mei), p. 32. Paërl B., 1985. Fetisjisme: fetisjistiese aspekten van kleding. Slechte Meiden (1985) 4 (sep), p. 8-11. Paërl B., 1980b. Transsexuelen vaak miskend. de Volkskrant, 5 December, p. 13 Paërl B., 1981. Transsexuelen. De Waarheid, 4 May, p. 6 Paërl B., 1984a. Een pleidooi voor de werkers in de seksindustrie. De Waarheid, 7 July, p. 4 Paërl B., 1984b. SM is een vorm van seksuele rebellie: het pervers-feminisme van Pat Califia. Homologie, 6(5), pp. 5–7 Paërl B., Califia P., Arnone J., 1984. Vrouw-zijn is een maatschappelijke constructie. Prothese, 23 October, pp. 6–7 === Films === Brokopondo, 1973. Film. Directed by B. Paërl and H. Paërl. NL. Mariënburg maakt zich vrij, 1973. Film. Directed by B. Paërl and H. Paërl. NL. Rietkappers, 1973. Film. Directed by B. Paërl and H. Paërl. NL. These films premiered at Cinestud in 1973 in Amsterdam. === Translations === Califia P., 1983. Zij imiteren precies de hetero's: spelen met rollen en omkeringen. Tr. Paërl B. Slechte Meiden. (nov), p. 2-7. Unbehaun K., 1978. Perfecte Filmtrucs. Tr. Paërl B. Amsterdam: Focus. == References ==
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Wikipedia:Betty Shannon#0
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Betty Shannon (née Mary Elizabeth Moore) (April 14, 1922 – May 1, 2017) was a mathematician and the main research collaborator of Claude Shannon. Betty inspired and assisted Claude in building some of his most famous inventions. == Life == Shannon was born on April 14, 1922 in New York City to Vilma Ujlaky Moore and James E. Moore. She was awarded a full scholarship to the New Jersey College for Women, where she graduated Phi Beta Kappa after studying mathematics. She worked as a numerical analyst at Bell Labs, where as a computer she supported work on microwaves, and then on radar. She published her own research on "Composing Music by a Stochastic Process"; an "exceptional" accomplishment in an era when it was a "significant and unusual achievement for a woman to get her name on a research report". While at Bell Labs she met the shy and insular Claude Shannon. Claude "didn’t have much patience with people who weren’t as smart as he was" and the two of them got on well. In 1948 he asked her on a date and they ended up dining each night together; they were married in 1949. In addition to her research, Shannon was a member of the Weavers' Guild of Boston, served as Dean of the Guild from 1976 to 1978 and received the Guild's Distinguished Achievement Award. Shannon had three children, Robert James Shannon, Andrew Moore Shannon, and Margarita Shannon, and raised their family in Winchester, Massachusetts. Her oldest son, Robert Shannon, died in 1998 at the age of 45. Betty died on May 1, 2017, at her home at Brookhaven in Lexington, Massachusetts. == References ==
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Wikipedia:Betül Tanbay#0
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Betül Tanbay (born 1960) is a Turkish mathematician, scientist and professor of mathematics at the Boğaziçi University in Istanbul, Turkey and the first woman president of the Turkish Mathematical Society between 2010 and 2016. == Education == Betül Tanbay was born in Istanbul and raised in Ankara until 1977. She graduated from the Lycée Janson de Sailly, Paris, France in 1978. She received her Licence en Mathématiques from Université Louis Pasteur, Strasbourg in 1982, and her PhD in Mathematics under the supervision of Robert Solovay at the University of California, Berkeley in the United States in 1989. == Academic career == She has been a full time member and chairwoman in the Department of Mathematics, at Boğaziçi University, Istanbul. She has been the Vice-Provost for Foreign Affairs and represented her university at the European University Association between 2004 and 2007. She is the founding codirector of the Istanbul Center for Mathematical Sciences (IMBM). She has been in the scientific boards of research institutes such as IMBM, Feza Gürsey Institute, Institut d'Etudes Avancées - Aix Marseille (IMéRA). She was the director of a leading doctoral network project of TÜBİTAK between 2008 and 2012. She has worked as an executive committee member, including presidency, at the Turkish Mathematical Society; as delegate, Ethics Committee member, Raising Awareness Committee member, Executive Committee member at the European Mathematical Society where she was elected vice-president for the 2019-2022 period; and as delegate, member of the Committee for Women in Mathematics at the International Mathematical Union (IMU). In 2025 she was appointed as the Chair of the Governing Board of the IMU-IDM (International Day of Mathematics). === Visiting positions === She has held visiting positions at UC Berkeley and UC Santa Barbara, Université de Bordeaux, Institut de Mathématiques de Jussieu, University of Kansas, Pennsylvania State University and joined the Kadison-Singer Conjecture Workshop at the American Institute of Mathematics. == Research areas == Her research interests include operator algebras and set theory. === Representative scientific journal publications === C. Akemann, J. Anderson, B. Tanbay, Weak Paveability And The Kadison-Singer Problem, Journal of Operator Theory, vol. 71, no. 1, pp. 295–300 (2014). arXiv:1203.2854 B. Tanbay, A Letter on the Kadison-Singer problem, Revue Roumaine des Mathématiques Pures et Appliquées, vol. 59, no. 2, pp. 293–302 (2014). http://imar.ro/journals/Revue_Mathematique/pdfs/2014/2/10.pdf C. Akemann, J. Anderson, B. Tanbay, The Kadison-Singer problem for the direct sum of matrix algebras, Positivity, vol. 16, no. 1, pp 53–66 (2012). https://link.springer.com/content/pdf/10.1007%2Fs11117-010-0109-1.pdf C. Akemann, B. Tanbay, A. Ülger A Note On The Kadison-Singer Problem, Journal of Operator Theory, vol. 63, no. 2, 363–274 (2010). arXiv:0708.2366 == References == == External links == Betül Tanbay's professional home page
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Wikipedia:Beulah Armstrong#0
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Beulah May Armstrong (1895–1965) was an American mathematician and university professor. She was one of the few American women awarded a PhD in math before World War II. == Biography == Beulah was born November 18, 1895 in Sterling, Kansas, the third of five children of Lillie J. Detter and John Allen Armstrong, both Pennsylvania natives. In 1900 the family lived in Enterprise Township, Ford County, Kansas, where her father was a farmer, and by 1910 they had moved to Hutchinson, Kansas. Beulah graduated from Hutchinson High School and then attended Baker University in Baldwin City, a co-educational Methodist Church school in eastern Kansas. After receiving her bachelor 's degree from Baker University in 1917, she received a scholarship to attend the University of Kansas, and in 1918 she earned her master's degree there. Her master's thesis was titled, Simple And Complete K-Points in Continuous and in Modular Projective Spaces. She was offered an additional scholarship to continue her studies at Kansas, but did not accept it apparently because she was keen on doing graduate work at the University of Illinois where she was offered a scholarship for the academic year 1918–1919 and a fellowship for the following year. She earned her PhD there in 1921 under George Abram Miller with the dissertation: Mathematical Induction in Group Theory. She taught at the University of Illinois from 1921 to 1931, was an associate from 1931 to 1945, then became an assistant professor from 1945 to 1959, and finally an associate professor at Illinois from 1959 until her retirement as an associate professor emeritus in 1963. She was heavily involved with advising aspiring math teachers while they studied with her at Illinois and she taught many of the math courses they were scheduled to take. In addition, she was one of the instructors for the University's correspondence courses. She was active in a number of organizations on and off campus. She was secretary-treasurer of the Sigma Xi honor society and was a member of Kappa Delta Pi. She was listed in the Who's Who of American Women in the late 1950s. She died suddenly at age 69 at her home on February 22, 1965, in Urbana, Illinois. She was buried in Fairlawn Cemetery, Hutchinson, Kansas and, in her will, she bequeathed $1,000 to her first alma mater Baker University. == Memberships == Organizational affiliations, according to Green. American Mathematical Society (AMS) Mathematical Association of America (MAA) Sigma Delta Epsilon Sigma Xi Pi Mu Epsilon Phi Beta Kappa == References ==
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Wikipedia:Beulah Russell#0
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Beulah Russell, christened Beatrice Beulah Russell and also known as Bulah Russell, (October 22, 1878 – February 22, 1940) was an American mathematician. == Education == Beulah graduated from Randolph–Macon Woman's College with a Bachelor of Arts in 1903, and graduated from the University of Chicago with a Master of Arts in 1919. == Career == From 1903 to 1905 Beulah taught at Lafayette College as an instructor in mathematics. From 1905 to 1909 she taught at Grenada College as a professor of mathematics. From 1909 to 1925 she taught at Randolph–Macon Woman's College as an instructor of mathematics and an adjunct professor of mathematics. In 1925 she became an associate professor of mathematics at the College of William & Mary. In a 1923 edition of The American Mathematical Monthly it is recorded that Beulah was elected to membership in the Mathematical Association of America. In 1930 Beulah became the first female professor to attend the Edinburgh Mathematical Society Colloquium held in St Andrews, Scotland. == External links == Beulah Russell, "Relation between the definite integral and summation of series", 1919 (University of Chicago Master of Arts Dissertation) Carolyn Lamb Sparks Whittenburg, "President J.A.C. Chandler and the first women faculty at the College of William and Mary", D.Ed. Thesis (College of William & Mary, May 2004). == Further reading == "Beulah Russell Death Mourned By College", The Flat Hat. College of William & Mary. (27 February 1940). == References ==
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Wikipedia:Beurling–Lax theorem#0
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In mathematics, the Beurling–Lax theorem is a theorem due to Beurling (1948) and Lax (1959) which characterizes the shift-invariant subspaces of the Hardy space H 2 ( D , C ) {\displaystyle H^{2}(\mathbb {D} ,\mathbb {C} )} . It states that each such space is of the form θ H 2 ( D , C ) , {\displaystyle \theta H^{2}(\mathbb {D} ,\mathbb {C} ),} for some inner function θ {\displaystyle \theta } . == See also == H2 == References == Ball, J. A. (2001) [1994], "Beurling-Lax theorem", Encyclopedia of Mathematics, EMS Press Beurling, A. (1948), "On two problems concerning linear transformations in Hilbert space", Acta Math., 81: 239–255, doi:10.1007/BF02395019, MR 0027954 Lax, P.D. (1959), "Translation invariant spaces", Acta Math., 101 (3–4): 163–178, doi:10.1007/BF02559553, MR 0105620 Jonathan R. Partington, Linear Operators and Linear Systems, An Analytical Approach to Control Theory, (2004) London Mathematical Society Student Texts 60, Cambridge University Press. Marvin Rosenblum and James Rovnyak, Hardy Classes and Operator Theory, (1985) Oxford University Press.
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Wikipedia:Beyer Professor of Applied Mathematics#0
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The Beyer Chair of Applied Mathematics is an endowed professorial position in the Department of Mathematics, University of Manchester, England. The endowment came from the will of the celebrated locomotive designer and founder of locomotive builder Beyer, Peacock & Company, Charles Frederick Beyer. He was the university's largest single donor. The first appointment in 1881 was of Arthur Schuster who held the position until 1888. After Schuster's departure, the chair of Mathematics to which Horace Lamb had been appointed in 1885 became the Beyer Professorship of Mathematics and remained so until Lamb's retirement in 1920. At this point an existing chair, of Mathematics and Natural Philosophy to which Sydney Chapman had been appointed in 1919, was renamed the Beyer Professorship of Mathematics and Natural Philosophy. After Chapman's resignation, the Beyer title was applied to the chair of Applied Mathematics. There was no incumbent between 1937 and 1945. Most of the holders of the post were elected as Fellows of the Royal Society, an honour bestowed on a small minority of UK mathematics professors. Lamb, Champman, Milne and Goldstein all received the Smith's Prize and indication of early career promise. The other endowed chairs in mathematics at the University of Manchester are the Richardson Chair of Applied Mathematics, and the Fielden Chair of Pure Mathematics as well as the named Sir Horace Lamb Chair. == Beyer Professors == 1881–1888 Arthur Schuster 1888–1920 Horace Lamb 1920–1924 Sydney Chapman, Beyer Professor of Mathematics and Natural Philosophy 1924–1928 Edward Arthur Milne 1929–1937 Douglas Hartree 1945–1950 Sydney Goldstein 1950–1959 James Lighthill 1961–1990 Fritz Ursell 1991–1996 Philip Hall 1996–2017 David Abrahams 2017–2024 Vacant 2024- Paul Glendinning == References ==
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Wikipedia:Bharath Sriraman#0
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Bharath Sriraman (born 1971) is an Indian-born Professor of Mathematical Sciences at the University of Montana – Missoula and an academic editor, known for his contributions to the interdisciplinarity of mathematics-science-arts, theory development in mathematics education, creativity research, the history and philosophy of mathematics ;and gifted education. == Education and honors == Bharath Sriraman attended St. Michael’s School (founded by the Jesuits of St. Michael's Church, Mumbai) from 1976-87. He graduated with a B.S. in mathematics from the University of Alaska Fairbanks in 1995. He obtained his M.S in Mathematics in 1999 followed by a PhD in Mathematical Sciences in 2002 both from Northern Illinois University under the functional analyst Robert Wheeler. In 2009, Northern Illinois University named him as one of 50 "Golden alumni" in the last 50 years for his significant contributions to research in mathematics education, gifted education and interdisciplinary research at the intersection of mathematics-science-arts. He previously received the School Science and Mathematics Association Early Scholar Award in 2007. In 2016 he was the recipient of the University of Montana Distinguished Scholar Award. At the 15th International Congress on Mathematical Education (2024) in Sydney, he chaired the plenary panel on What counts as evidence in mathematics education == Academic and editorial work == Sriraman is the founder and editor-in-chief of The Mathematics Enthusiast, an independent open access journal hosted by University of Montana. He is the co-founder/co-editor-in-chief of two series with Springer Science+Business Media namely Advances in Mathematics Education and Creativity Theory and Action in Education. In addition to editing he is a prolific scholar with over 300 publications to date in numerous areas of research, and held numerous visiting professorships including those as International Fulbright Specialist at institutions in Nordic countries, Eurasia and South America. He also holds an adjunct appointment in the department of Central and Southwest-Asian Studies at the University of Montana == Bibliography of Edited Works (selection) == === Mathematics Education === Sriraman, B., English, L. (2010): Theories of Mathematics Education: Seeking New Frontiers, Springer, Berlin, Germany, ISBN 978-3-642-00741-5 === Interdisciplinary works === Sriraman, Bharath, ed. (2021). Handbook of the Mathematics of the Arts and Sciences. Springerlink. doi:10.1007/978-3-319-70658-0. ISBN 978-3-319-70658-0. S2CID 243647817. Sriraman, Bharath, ed. (2025). Handbook of Visual, Experimental and Computational Mathematics: Bridges through Data. Springerlink. doi:10.1007/978-3-030-93954-0. ISBN 978-3-030-93954-0. === History and Philosophy of Mathematics === Sriraman, Bharath, ed. (2024). Handbook of the History and Philosophy of Mathematical Practice. SpringerNature. doi:10.1007/978-3-031-40846-5. ISBN 978-3-031-40845-8. S2CID 241668074. Sriraman, Bharath, ed. (2017). Humanizing Mathematics and its Philosophy: Essays celebrating the 90th birthday of Reuben Hersh. Springerlink. doi:10.1007/978-3-319-61231-7. ISBN 978-3-319-61230-0. === International mathematics education === Sriraman, B., Bergsten, C., Goodchild, S., Palsdottir, G., et al. (2010): The First Sourcebook on Nordic Research in Mathematics Education: Norway, Sweden, Iceland, Denmark and Contributions from Finland Information Age Publishing, Charlotte, NC.ISBN 978-1-61735-098-6 Sriraman, B., Cai, J., Lee, K., et al. (2015): The First Sourcebook on Asian Research in Mathematics Education: China, Korea, Japan, Singapore, Malaysia and India. Information Age Publishing, Charlotte, NC. ISBN 978-1-681-23277-5 === Creativity === Sriraman, B., &Lee, K. (2011): The Elements of Creativity and Giftedness in Mathematics. Sense Publishers, The Netherlands.ISBN 978-94-6091-439-3 Beghetto, R. &Sriraman, B. (2017): Creative Contradictions in Education: Cross disciplinary paradoxes and perspectives, Springer International, Switzerland, ISBN 978-3-319-21924-0 === Gifted education === Ambrose, D., Sriraman, B., Cross, T. (2013): The Roeper School: A Model for Holistic Development of High Ability, Sense Publishers, Rotterdam, Netherlands, ISBN 978-9-46209417-8 Ambrose, D., Sternberg, R., Sriraman, B.(2011): Confronting Dogmatism in Gifted Education, Routledge, New York, USA, ISBN 978-0-41589446-3 == References == == External links == Official website Bharath Sriraman at the Mathematics Genealogy Project
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Wikipedia:Bhaskara's lemma#0
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Bhaskara's Lemma is an identity used as a lemma during the chakravala method. It states that: N x 2 + k = y 2 ⟹ N ( m x + y k ) 2 + m 2 − N k = ( m y + N x k ) 2 {\displaystyle \,Nx^{2}+k=y^{2}\implies \,N\left({\frac {mx+y}{k}}\right)^{2}+{\frac {m^{2}-N}{k}}=\left({\frac {my+Nx}{k}}\right)^{2}} for integers m , x , y , N , {\displaystyle m,\,x,\,y,\,N,} and non-zero integer k {\displaystyle k} . == Proof == The proof follows from simple algebraic manipulations as follows: multiply both sides of the equation by m 2 − N {\displaystyle m^{2}-N} , add N 2 x 2 + 2 N m x y + N y 2 {\displaystyle N^{2}x^{2}+2Nmxy+Ny^{2}} , factor, and divide by k 2 {\displaystyle k^{2}} . N x 2 + k = y 2 ⟹ N m 2 x 2 − N 2 x 2 + k ( m 2 − N ) = m 2 y 2 − N y 2 {\displaystyle \,Nx^{2}+k=y^{2}\implies Nm^{2}x^{2}-N^{2}x^{2}+k(m^{2}-N)=m^{2}y^{2}-Ny^{2}} ⟹ N m 2 x 2 + 2 N m x y + N y 2 + k ( m 2 − N ) = m 2 y 2 + 2 N m x y + N 2 x 2 {\displaystyle \implies Nm^{2}x^{2}+2Nmxy+Ny^{2}+k(m^{2}-N)=m^{2}y^{2}+2Nmxy+N^{2}x^{2}} ⟹ N ( m x + y ) 2 + k ( m 2 − N ) = ( m y + N x ) 2 {\displaystyle \implies N(mx+y)^{2}+k(m^{2}-N)=(my+Nx)^{2}} ⟹ N ( m x + y k ) 2 + m 2 − N k = ( m y + N x k ) 2 . {\displaystyle \implies \,N\left({\frac {mx+y}{k}}\right)^{2}+{\frac {m^{2}-N}{k}}=\left({\frac {my+Nx}{k}}\right)^{2}.} So long as neither k {\displaystyle k} nor m 2 − N {\displaystyle m^{2}-N} are zero, the implication goes in both directions. (The lemma holds for real or complex numbers as well as integers.) == References == C. O. Selenius, "Rationale of the chakravala process of Jayadeva and Bhaskara II", Historia Mathematica, 2 (1975), 167-184. C. O. Selenius, Kettenbruch theoretische Erklarung der zyklischen Methode zur Losung der Bhaskara-Pell-Gleichung, Acta Acad. Abo. Math. Phys. 23 (10) (1963). George Gheverghese Joseph, The Crest of the Peacock: Non-European Roots of Mathematics (1975). == External links == Introduction to chakravala
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Wikipedia:Bhaskaracharya Pratishthana#0
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Bhaskaracharya Pratishthana is a research and education institute for mathematics in Pune, India, founded by noted Indian-American mathematician professor Shreeram Abhyankar. The institute is named after the great ancient Indian Mathematician Bhaskaracharya (Born in 1114 A.D). It was founded in 1976. It has researchers working in many areas of mathematics, particularly in algebra and number theory. Since 1992, the Pratishthan has also been a recognized center for conducting Regional Mathematics Olympiad (RMO) under the National Board for Higher Mathematics (NBHM) for Maharashtra and Goa Region. Pratishthana publishes the mathematics periodical Bona Mathematica and has published texts in higher and Olympiad mathematics. Besides this the Pratishthan holds annual / biennial conferences/ Workshops in some research areas in higher mathematics attended by Indian/Foreign scholars and Professors. The Pratishthan has organized a number of workshops for research students and college teachers under the aegis of NBHM/NCM. The National Board for Higher Mathematics has greatly helped Pratishthan to enrich its library. The Department of Atomic Energy and the Mathematics Department of S. P. Pune University have rendered active co-operation in holding conferences/workshops. It also conducts the BMTSC exam which is a school-level mathematics competition for students studying in the 5th and the 6th grade. The objectives of the competition are listed below: 1. Identify good students of mathematics at an early age. 2. A pre Olympiad type competition. 3. To enhance Mathematical ability and logical thinking. 4. Nurture programs for successful students to improve their ability. == References == == External links == Official website Home Page of FOSSME project
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Wikipedia:Bhutasamkhya system#0
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The Bhūtasaṃkhyā system is a method of recording numbers in Sanskrit using common nouns having connotations of numerical values. The method was introduced already in astronomical texts in antiquity, but it was expanded and developed during the medieval period. A kind of rebus system, bhūtasaṃkhyā has also been called the "concrete number notation". For example, the number "two" was associated with the word "eye" as every human being has two eyes. Thus every Sanskrit word having the meaning "eye" was used to denote "two". All words synonymous with the meaning "earth" could be used to signify the number "one" as there is only one earth, etc. In the more expansive examples of application, concepts, ideas and objects from all parts of the Sanskrit lexicon were harvested to generate number-connoting words, resulting in a kind of kenning system for numbers. Thus, every Sanskrit word indicating an "arrow" has been used to denote "five" as Kamadeva, the Hindu deity of love, is traditionally depicted as a warrior carrying five arrows of flowers. The term anuṣṭubh has been used to signify "eight" as it is the name of a meter with eight syllables in a foot. Any Sanskrit word for "tooth" could be used to denote 32 as a grown-up man has a full set of 32 teeth. Terms implying "the gods" were used to indicate 33, as there is a tradition of "thirty-three gods" (trāyastriṃśadeva) in certain Hindu and Buddhist texts. A potential user of the system had a multitude of words to choose from for denoting the same number. The mapping from "words" to "numbers" is many-to-one. This has facilitated the embedding of numbers in verses in Indian treatises on mathematics and astronomy. This helped in memorising large tables of numbers required by astronomers and astrologers. Single words indicating smaller numbers were strung together to form phrases and sentences for representing arbitrary large numbers. This formation of large numbers was accomplished by incorporating the decimal place value system into the scheme, where digits are named in ascending order. As an example, in an 18th-century inscription from Kalna, a year is given as bāṇa-vyoma-dharādhar-indu-gaṇite śāke which means "In the Śāka year enumerated by arrow [5], sky [0], mountain [7] and moon [1]", that is, "Śāka 5-0-7-1" = Śāka 1705 = AD 1783. The earliest evidence of this system is found in Yavanajataka, a versification of a Greek astronomical text dated to the early centuries CE. Limited use of Bhutasamkhya is seen in some Puranas, for example Bhagavata Mahatmya of Padma Purana (6.66) uses the word 'nagaaha to refer to "seven days", i.e. naga "mountain" is used as a synonym of "seven" (because of the "seven principal mountains" or kula-giri), a usage already found in medieval recensions of the Surya Siddhanta. It is found throughout the Indian Buddhist Kalacakra Tantra literature. == See also == Aksharapalli Āryabhaṭa numeration Katapayadi system == References == == External links == Bhūtasaṅkhyā, a Bhūtasaṅkhyā encoding-decoding system == Further reading == For a list of words commonly used for the representation of numbers in bhūtasaṃkhyā system see: Terdalkar, Hrishikesh. "Bhūtasaṅkhyā". Sanskrit Activities at IIT Kanpur. Retrieved January 2, 2023. D.C. Sircar (1965). Indian Epigraphy (1 ed.). Delhi: Motilal Banarsidass Publishers Private Limited. pp. 228–234. ISBN 81-208-1166-6. P. V. Kane (1968). History of Dharmaśāstra Volume 5 part 1. pp. 701–703. C. P. Brown (1869). Sanskrit Prosody and Numerical Symbols Explained. pp. 49–54. Related Video [1]
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Wikipedia:Bhāskara I's sine approximation formula#0
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In mathematics, Bhāskara I's sine approximation formula is a rational expression in one variable for the computation of the approximate values of the trigonometric sines discovered by Bhāskara I (c. 600 – c. 680), a seventh-century Indian mathematician. This formula is given in his treatise titled Mahabhaskariya. It is not known how Bhāskara I arrived at his approximation formula. However, several historians of mathematics have put forward different hypotheses as to the method Bhāskara might have used to arrive at his formula. The formula is elegant and simple, and it enables the computation of reasonably accurate values of trigonometric sines without the use of geometry. == Approximation formula == The formula is given in verses 17–19, chapter VII, Mahabhaskariya of Bhāskara I. A translation of the verses is given below: (Now) I briefly state the rule (for finding the bhujaphala and the kotiphala, etc.) without making use of the Rsine-differences 225, etc. Subtract the degrees of a bhuja (or koti) from the degrees of a half circle (that is, 180 degrees). Then multiply the remainder by the degrees of the bhuja or koti and put down the result at two places. At one place subtract the result from 40500. By one-fourth of the remainder (thus obtained), divide the result at the other place as multiplied by the anthyaphala (that is, the epicyclic radius). Thus is obtained the entire bahuphala (or, kotiphala) for the sun, moon or the star-planets. So also are obtained the direct and inverse Rsines. (The reference "Rsine-differences 225" is an allusion to Aryabhata's sine table.) In modern mathematical notations, for an angle x in degrees, this formula gives sin x ∘ ≈ 4 x ( 180 − x ) 40500 − x ( 180 − x ) . {\displaystyle \sin x^{\circ }\approx {\frac {4x(180-x)}{40500-x(180-x)}}.} === Equivalent forms of the formula === Bhāskara I's sine approximation formula can be expressed using the radian measure of angles as follows: sin x ≈ 16 x ( π − x ) 5 π 2 − 4 x ( π − x ) . {\displaystyle \sin x\approx {\frac {16x(\pi -x)}{5\pi ^{2}-4x(\pi -x)}}.} For a positive integer n this takes the following form: sin π n ≈ 16 ( n − 1 ) 5 n 2 − 4 n + 4 . {\displaystyle \sin {\frac {\pi }{n}}\approx {\frac {16(n-1)}{5n^{2}-4n+4}}.} The formula acquires an even simpler form when expressed in terms of the cosine rather than the sine. Using radian measure for angles from − π 2 {\displaystyle -{\frac {\pi }{2}}} to π 2 {\displaystyle {\frac {\pi }{2}}} and putting x = 1 2 π + y {\displaystyle x={\tfrac {1}{2}}\pi +y} , one gets cos y ≈ π 2 − 4 y 2 π 2 + y 2 . {\displaystyle \cos y\approx {\frac {\pi ^{2}-4y^{2}}{\pi ^{2}+y^{2}}}.} To express the previous formula with the constant τ = 2 π , {\displaystyle \tau =2\pi ,} one can use cos y ≈ 1 − 20 y 2 4 y 2 + τ 2 . {\displaystyle \cos y\approx 1-{\frac {20y^{2}}{4y^{2}+\tau ^{2}}}.} Equivalent forms of Bhāskara I's formula have been given by almost all subsequent astronomers and mathematicians of India. For example, Brahmagupta's (598–668 CE) Brhma-Sphuta-Siddhanta (verses 23–24, chapter XIV) gives the formula in the following form: R sin x ∘ ≈ R x ( 180 − x ) 10125 − 1 4 x ( 180 − x ) . {\displaystyle R\sin x^{\circ }\approx {\frac {Rx(180-x)}{10125-{\frac {1}{4}}x(180-x)}}.} Also, Bhāskara II (1114–1185 CE) has given this formula in his Lilavati (Kshetra-vyavahara, Soka No. 48) in the following form: 2 R sin x ∘ ≈ 4 × 2 R × 2 R x × ( 360 R − 2 R x ) 1 4 × 5 × ( 360 R ) 2 − 2 R x × ( 360 R − 2 R x ) = 5760 R x − 32 R x 2 162000 − 720 x + 4 x 2 {\displaystyle 2R\sin x^{\circ }\approx {\frac {4\times 2R\times 2Rx\times (360R-2Rx)}{{\frac {1}{4}}\times 5\times (360R)^{2}-2Rx\times (360R-2Rx)}}={\frac {5760Rx-32Rx^{2}}{162000-720x+4x^{2}}}} The approximation can also be used to derive formulas for inverse cosine and inverse sine: arccos x ≈ π 1 − x 4 + x {\displaystyle \arccos x\approx \pi {\sqrt {\frac {1-x}{4+x}}}} { 0 ≤ x ≤ 1 } {\displaystyle \left\{0\leq x\leq 1\right\}} , and arccos x ≈ π ( 1 − 1 + x 4 − x ) {\displaystyle \arccos x\approx \pi \left(1-{\sqrt {\frac {1+x}{4-x}}}\right)} { − 1 ≤ x ≤ 0 } {\displaystyle \left\{-1\leq x\leq 0\right\}} . arcsin x ≈ π ( 1 2 − 1 − x 4 + x ) {\displaystyle \arcsin x\approx \pi \left({\frac {1}{2}}-{\sqrt {\frac {1-x}{4+x}}}\right)} { 0 ≤ x ≤ 1 } {\displaystyle \left\{0\leq x\leq 1\right\}} , and arcsin x ≈ π ( 1 + x 4 − x − 1 2 ) {\displaystyle \arcsin x\approx \pi \left({\sqrt {\frac {1+x}{4-x}}}-{\frac {1}{2}}\right)} { − 1 ≤ x ≤ 0 } {\displaystyle \left\{-1\leq x\leq 0\right\}} . Alternatively, using absolute values and the Sign function, each pair of functions can be rewritten as such: arccos x ≈ π ( 1 − sgn ( x ) 2 + sgn ( x ) 1 − | x | 4 + | x | ) {\displaystyle \arccos x\approx \pi \left({\frac {1-\operatorname {sgn} \left(x\right)}{2}}+\operatorname {sgn} \left(x\right){\sqrt {\frac {1-\left|x\right|}{4+\left|x\right|}}}\right)} arcsin x ≈ π sgn ( x ) ( 1 2 − 1 − | x | 4 + | x | ) {\displaystyle \arcsin x\approx \pi \operatorname {sgn} \left(x\right)\left({\frac {1}{2}}-{\sqrt {\frac {1-\left|x\right|}{4+\left|x\right|}}}\right)} == Accuracy of the formula == The formula is applicable for values of x° in the range from 0° to 180°. The formula is remarkably accurate in this range. The graphs of sin x and the approximation formula are visually indistinguishable and are nearly identical. One of the accompanying figures gives the graph of the error function, namely, the function sin x ∘ ≈ 4 x ( 180 − x ) 40500 − x ( 180 − x ) {\displaystyle \sin x^{\circ }\approx {\frac {4x(180-x)}{40500-x(180-x)}}} in using the formula. It shows that the maximum absolute error in using the formula is around 0.0016. From a plot of the percentage value of the absolute error, it is clear that the maximum relative error is less than 1.8%. The approximation formula thus gives sufficiently accurate values of sines for most practical purposes. However, it was not sufficient for the more accurate computational requirements of astronomy. The search for more accurate formulas by Indian astronomers eventually led to the discovery of the power series expansions of sin x and cos x by Madhava of Sangamagrama (c. 1350 – c. 1425), the founder of the Kerala school of astronomy and mathematics. == Derivation of the formula == Bhāskara had not indicated any method by which he arrived at his formula. Historians have speculated on various possibilities. No definitive answers have as yet been obtained. Beyond its historical importance of being a prime example of the mathematical achievements of ancient Indian astronomers, the formula is of significance from a modern perspective also. Mathematicians have attempted to derive the rule using modern concepts and tools. Around half a dozen methods have been suggested, each based on a separate set of premises. Most of these derivations use only elementary concepts. === Derivation based on elementary geometry === Let the circumference of a circle be measured in degrees and let the radius R of the circle be also measured in degrees. Choosing a fixed diameter AB and an arbitrary point P on the circle and dropping the perpendicular PM to AB, we can compute the area of the triangle APB in two ways. Equating the two expressions for the area one gets (1/2) AB × PM = (1/2) AP × BP. This gives 1 P M = A B A P × B P . {\displaystyle {\frac {1}{PM}}={\frac {AB}{AP\times BP}}.} Letting x be the length of the arc AP, the length of the arc BP is 180 − x. These arcs are much bigger than the respective chords. Hence one gets 1 P M > 2 R x ( 180 − x ) . {\displaystyle {\frac {1}{PM}}>{\frac {2R}{x(180-x)}}.} One now seeks two constants α and β such that 1 P M = α 2 R x ( 180 − x ) + β . {\displaystyle {\frac {1}{PM}}=\alpha {\frac {2R}{x(180-x)}}+\beta .} It is indeed not possible to obtain such constants. However, one may choose values for α and β so that the above expression is valid for two chosen values of the arc length x. Choosing 30° and 90° as these values and solving the resulting equations, one immediately gets Bhāskara I's sine approximation formula. === Derivation starting with a general rational expression === Assuming that x is in radians, one may seek an approximation to sin x in the following form: sin x ≈ a + b x + c x 2 p + q x + r x 2 . {\displaystyle \sin x\approx {\frac {a+bx+cx^{2}}{p+qx+rx^{2}}}.} The constants a, b, c, p, q and r (only five of them are independent) can be determined by assuming that the formula must be exactly valid when x = 0, π/6, π/2, π, and further assuming that it has to satisfy the property that sin(x) = sin(π − x). This procedure produces the formula expressed using radian measure of angles. === An elementary argument === The part of the graph of sin x in the range from 0° to 180° "looks like" part of a parabola through the points (0, 0) and (180, 0). The general form of such a parabola is k x ( 180 − x ) . {\displaystyle kx(180-x).} The parabola that also passes through (90, 1) (which is the point corresponding to the value sin(90°) = 1) is x ( 180 − x ) 90 × 90 = x ( 180 − x ) 8100 . {\displaystyle {\frac {x(180-x)}{90\times 90}}={\frac {x(180-x)}{8100}}.} The parabola which also passes through (30, 1/2) (which is the point corresponding to the value sin(30°) = 1/2) is x ( 180 − x ) 2 × 30 × 150 = x ( 180 − x ) 9000 . {\displaystyle {\frac {x(180-x)}{2\times 30\times 150}}={\frac {x(180-x)}{9000}}.} These expressions suggest a varying denominator which takes the value 90 × 90 when x = 90 and the value 2 × 30 × 150 when x = 30. That this expression should also be symmetrical about the line x = 90 rules out the possibility of choosing a linear expression in x. Computations involving x(180 − x) might immediately suggest that the expression could be of the form 8100 a + b x ( 180 − x ) . {\displaystyle 8100a+bx(180-x).} A little experimentation (or by setting up and solving two linear equations in a and b) will yield the values a = 5/4, b = −1/4. These give Bhāskara I's sine approximation formula. Karel Stroethoff (2014) offers a similar, but simpler argument for Bhāskara I's choice. He also provides an analogous approximation for the cosine and extends the technique to second and third-order polynomials. == See also == Aryabhata's sine table Madhava's sine table Brahmagupta's interpolation formula == References == == Further references == R.C..Gupta, On derivation of Bhāskara I's formula for the sine, Ganita Bharati 8 (1-4) (1986), 39–41. T. Hayashi, A note on Bhāskara I's rational approximation to sine, Historia Sci. No. 42 (1991), 45–48. K. Stroethoff, Bhāskara's approximation for the sine, The Mathematics Enthusiast, Vol. 11, No. 3 (2014), 485–492.
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Wikipedia:Bhāskara II#0
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Bhāskara II ([bʰɑːskərə]; c.1114–1185), also known as Bhāskarāchārya (lit. 'Bhāskara the teacher'), was an Indian polymath, mathematician, astronomer and engineer. From verses in his main work, Siddhānta Śiromaṇi, it can be inferred that he was born in 1114 in Vijjadavida (Vijjalavida) and living in the Satpura mountain ranges of Western Ghats, believed to be the town of Patana in Chalisgaon, located in present-day Khandesh region of Maharashtra by scholars. In a temple in Maharashtra, an inscription supposedly created by his grandson Changadeva, lists Bhaskaracharya's ancestral lineage for several generations before him as well as two generations after him. Henry Colebrooke who was the first European to translate (1817) Bhaskaracharya II's mathematical classics refers to the family as Maharashtrian Brahmins residing on the banks of the Godavari. Born in a Hindu Deshastha Brahmin family of scholars, mathematicians and astronomers, Bhaskara II was the leader of a cosmic observatory at Ujjain, the main mathematical centre of ancient India. Bhāskara and his works represent a significant contribution to mathematical and astronomical knowledge in the 12th century. He has been called the greatest mathematician of medieval India. His main work, Siddhānta-Śiromaṇi (Sanskrit for "Crown of Treatises"), is divided into four parts called Līlāvatī, Bījagaṇita, Grahagaṇita and Golādhyāya, which are also sometimes considered four independent works. These four sections deal with arithmetic, algebra, mathematics of the planets, and spheres respectively. He also wrote another treatise named Karaṇā Kautūhala. == Date, place and family == Bhāskara gives his date of birth, and date of composition of his major work, in a verse in the Āryā metre: This reveals that he was born in 1036 of the Shaka era (1114 CE), and that he composed the Siddhānta Shiromani when he was 36 years old. Siddhānta Shiromani was completed during 1150 CE. He also wrote another work called the Karaṇa-kutūhala when he was 69 (in 1183). His works show the influence of Brahmagupta, Śrīdhara, Mahāvīra, Padmanābha and other predecessors. Bhaskara lived in Patnadevi located near Patan (Chalisgaon) in the vicinity of Sahyadri. He was born in a Deśastha Rigvedi Brahmin family near Vijjadavida (Vijjalavida). Munishvara (17th century), a commentator on Siddhānta Shiromani of Bhaskara has given the information about the location of Vijjadavida in his work Marīci Tīkā as follows: सह्यकुलपर्वतान्तर्गत भूप्रदेशे महाराष्ट्रदेशान्तर्गतविदर्भपरपर्यायविराटदेशादपि निकटे गोदावर्यां नातिदूरे पंचक्रोशान्तरे विज्जलविडम्। This description locates Vijjalavida in Maharashtra, near the Vidarbha region and close to the banks of Godavari river. However scholars differ about the exact location. Many scholars have placed the place near Patan in Chalisgaon Taluka of Jalgaon district whereas a section of scholars identified it with the modern day Beed city. Some sources identified Vijjalavida as Bijapur or Bidar in Karnataka. Identification of Vijjalavida with Basar in Telangana has also been suggested. Bhāskara is said to have been the head of an astronomical observatory at Ujjain, the leading mathematical centre of medieval India. History records his great-great-great-grandfather holding a hereditary post as a court scholar, as did his son and other descendants. His father Maheśvara (Maheśvaropādhyāya) was a mathematician, astronomer and astrologer, who taught him mathematics, which he later passed on to his son Lokasamudra. Lokasamudra's son helped to set up a school in 1207 for the study of Bhāskara's writings. He died in 1185 CE. == The Siddhānta-Śiromaṇi == === Līlāvatī === The first section Līlāvatī (also known as pāṭīgaṇita or aṅkagaṇita), named after his daughter, consists of 277 verses. It covers calculations, progressions, measurement, permutations, and other topics. === Bijaganita === The second section Bījagaṇita(Algebra) has 213 verses. It discusses zero, infinity, positive and negative numbers, and indeterminate equations including (the now called) Pell's equation, solving it using a kuṭṭaka method. In particular, he also solved the 61 x 2 + 1 = y 2 {\displaystyle 61x^{2}+1=y^{2}} case that was to elude Fermat and his European contemporaries centuries later === Grahaganita === In the third section Grahagaṇita, while treating the motion of planets, he considered their instantaneous speeds. He arrived at the approximation: It consists of 451 verses sin y ′ − sin y ≈ ( y ′ − y ) cos y {\displaystyle \sin y'-\sin y\approx (y'-y)\cos y} for. y ′ {\displaystyle y'} close to y {\displaystyle y} , or in modern notation: d d y sin y = cos y {\displaystyle {\frac {d}{dy}}\sin y=\cos y} . In his words: bimbārdhasya koṭijyā guṇastrijyāhāraḥ phalaṃ dorjyāyorantaram This result had also been observed earlier by Muñjalācārya (or Mañjulācārya) mānasam, in the context of a table of sines. Bhāskara also stated that at its highest point a planet's instantaneous speed is zero. == Mathematics == Some of Bhaskara's contributions to mathematics include the following: A proof of the Pythagorean theorem by calculating the same area in two different ways and then cancelling out terms to get a2 + b2 = c2. In Lilavati, solutions of quadratic, cubic and quartic indeterminate equations are explained. Solutions of indeterminate quadratic equations (of the type ax2 + b = y2). Integer solutions of linear and quadratic indeterminate equations (Kuṭṭaka). The rules he gives are (in effect) the same as those given by the Renaissance European mathematicians of the 17th century. A cyclic Chakravala method for solving indeterminate equations of the form ax2 + bx + c = y. The solution to this equation was traditionally attributed to William Brouncker in 1657, though his method was more difficult than the chakravala method. The first general method for finding the solutions of the problem x2 − ny2 = 1 (so-called "Pell's equation") was given by Bhaskara II. Solutions of Diophantine equations of the second order, such as 61x2 + 1 = y2. This very equation was posed as a problem in 1657 by the French mathematician Pierre de Fermat, but its solution was unknown in Europe until the time of Euler in the 18th century. Solved quadratic equations with more than one unknown, and found negative and irrational solutions. Preliminary concept of mathematical analysis. Preliminary concept of differential calculus, along with preliminary ideas towards integration. preliminary ideas of differential calculus and differential coefficient. Stated Rolle's theorem, a special case of one of the most important theorems in analysis, the mean value theorem. Traces of the general mean value theorem are also found in his works. Calculated the derivative of sine function, although he did not develop the notion of a derivative. (See Calculus section below.) In Siddhanta-Śiromaṇi, Bhaskara developed spherical trigonometry along with a number of other trigonometric results. (See Trigonometry section below.) === Arithmetic === Bhaskara's arithmetic text Līlāvatī covers the topics of definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, plane geometry, solid geometry, the shadow of the gnomon, methods to solve indeterminate equations, and combinations. Līlāvatī is divided into 13 chapters and covers many branches of mathematics, arithmetic, algebra, geometry, and a little trigonometry and measurement. More specifically the contents include: Definitions. Properties of zero (including division, and rules of operations with zero). Further extensive numerical work, including use of negative numbers and surds. Estimation of π. Arithmetical terms, methods of multiplication, and squaring. Inverse rule of three, and rules of 3, 5, 7, 9, and 11. Problems involving interest and interest computation. Indeterminate equations (Kuṭṭaka), integer solutions (first and second order). His contributions to this topic are particularly important, since the rules he gives are (in effect) the same as those given by the renaissance European mathematicians of the 17th century, yet his work was of the 12th century. Bhaskara's method of solving was an improvement of the methods found in the work of Aryabhata and subsequent mathematicians. His work is outstanding for its systematisation, improved methods and the new topics that he introduced. Furthermore, the Lilavati contained excellent problems and it is thought that Bhaskara's intention may have been that a student of 'Lilavati' should concern himself with the mechanical application of the method. === Algebra === His Bījaganita ("Algebra") was a work in twelve chapters. It was the first text to recognize that a positive number has two square roots (a positive and negative square root). His work Bījaganita is effectively a treatise on algebra and contains the following topics: Positive and negative numbers. The 'unknown' (includes determining unknown quantities). Determining unknown quantities. Surds (includes evaluating surds and their square roots). Kuṭṭaka (for solving indeterminate equations and Diophantine equations). Simple equations (indeterminate of second, third and fourth degree). Simple equations with more than one unknown. Indeterminate quadratic equations (of the type ax2 + b = y2). Solutions of indeterminate equations of the second, third and fourth degree. Quadratic equations. Quadratic equations with more than one unknown. Operations with products of several unknowns. Bhaskara derived a cyclic, chakravala method for solving indeterminate quadratic equations of the form ax2 + bx + c = y. Bhaskara's method for finding the solutions of the problem Nx2 + 1 = y2 (the so-called "Pell's equation") is of considerable importance. === Trigonometry === The Siddhānta Shiromani (written in 1150) demonstrates Bhaskara's knowledge of trigonometry, including the sine table and relationships between different trigonometric functions. He also developed spherical trigonometry, along with other interesting trigonometrical results. In particular Bhaskara seemed more interested in trigonometry for its own sake than his predecessors who saw it only as a tool for calculation. Among the many interesting results given by Bhaskara, results found in his works include computation of sines of angles of 18 and 36 degrees, and the now well known formulae for sin ( a + b ) {\displaystyle \sin \left(a+b\right)} and sin ( a − b ) {\displaystyle \sin \left(a-b\right)} . === Calculus === His work, the Siddhānta Shiromani, is an astronomical treatise and contains many theories not found in earlier works. Preliminary concepts of Differential calculus and mathematical analysis, along with a number of results in trigonometry that are found in the work are of particular interest. Evidence suggests Bhaskara was acquainted with some ideas of differential calculus. Bhaskara also goes deeper into the 'differential calculus' and suggests the differential coefficient vanishes at an extremum value of the function, indicating knowledge of the concept of 'infinitesimals'. There is evidence of an early form of Rolle's theorem in his work,though it was stated without a modern formal proof . In this astronomical work he gave one procedure that looks like a precursor to infinitesimal methods. In terms that is if x ≈ y {\displaystyle x\approx y} then sin ( y ) − sin ( x ) ≈ ( y − x ) cos ( y ) {\displaystyle \sin(y)-\sin(x)\approx (y-x)\cos(y)} that is a derivative of sine although he did not develop the notion on derivative. Bhaskara uses this result to work out the position angle of the ecliptic, a quantity required for accurately predicting the time of an eclipse. In computing the instantaneous motion of a planet, the time interval between successive positions of the planets was no greater than a truti, or a 1⁄33750 of a second, and his measure of velocity was expressed in this infinitesimal unit of time. He was aware that when a variable attains the maximum value, its differential vanishes. He also showed that when a planet is at its farthest from the earth, or at its closest, the equation of the centre (measure of how far a planet is from the position in which it is predicted to be, by assuming it is to move uniformly) vanishes. He therefore concluded that for some intermediate position the differential of the equation of the centre is equal to zero. In this result, there are traces of the general mean value theorem, one of the most important theorems in analysis, which today is usually derived from Rolle's theorem. The mean value formula for inverse interpolation of the sine was later founded by Parameshvara in the 15th century in the Lilavati Bhasya, a commentary on Bhaskara's Lilavati. Madhava (1340–1425) and the Kerala School mathematicians (including Parameshvara) from the 14th century to the 16th century expanded on Bhaskara's work. == Astronomy == Using an astronomical model developed by Brahmagupta in the 7th century, Bhāskara accurately defined many astronomical quantities, including, for example, the length of the sidereal year, the time that is required for the Sun to orbit the Earth, as approximately 365.2588 days which is the same as in Surya siddhanta. The modern accepted measurement is 365.25636 days, a difference of 3.5 minutes. His mathematical astronomy text Siddhanta Shiromani is written in two parts: the first part on mathematical astronomy and the second part on the sphere. The twelve chapters of the first part cover topics such as: Mean longitudes of the planets. True longitudes of the planets. The three problems of diurnal rotation. Diurnal motion refers to the apparent daily motion of stars around the Earth, or more precisely around the two celestial poles. It is caused by the Earth's rotation on its axis, so every star apparently moves on a circle that is called the diurnal circle. Syzygies. Lunar eclipses. Solar eclipses. Latitudes of the planets. Sunrise equation. The Moon's crescent. Conjunctions of the planets with each other. Conjunctions of the planets with the fixed stars. The paths of the Sun and Moon. The second part contains thirteen chapters on the sphere. It covers topics such as: Praise of study of the sphere. Nature of the sphere. Cosmography and geography. Planetary mean motion. Eccentric epicyclic model of the planets. The armillary sphere. Spherical trigonometry. Ellipse calculations. First visibilities of the planets. Calculating the lunar crescent. Astronomical instruments. The seasons. Problems of astronomical calculations. == Engineering == The earliest reference to a perpetual motion machine date back to 1150, when Bhāskara II described a wheel that he claimed would run forever. Bhāskara II invented a variety of instruments one of which is Yaṣṭi-yantra. This device could vary from a simple stick to V-shaped staffs designed specifically for determining angles with the help of a calibrated scale. == Legends == In his book Lilavati, he reasons: "In this quantity also which has zero as its divisor there is no change even when many quantities have entered into it or come out [of it], just as at the time of destruction and creation when throngs of creatures enter into and come out of [him, there is no change in] the infinite and unchanging [Vishnu]". === "Behold!" === It has been stated, by several authors, that Bhaskara II proved the Pythagorean theorem by drawing a diagram and providing the single word "Behold!". Sometimes Bhaskara's name is omitted and this is referred to as the Hindu proof, well known by schoolchildren. However, as mathematics historian Kim Plofker points out, after presenting a worked-out example, Bhaskara II states the Pythagorean theorem: Hence, for the sake of brevity, the square root of the sum of the squares of the arm and upright is the hypotenuse: thus it is demonstrated. This is followed by: And otherwise, when one has set down those parts of the figure there [merely] seeing [it is sufficient]. Plofker suggests that this additional statement may be the ultimate source of the widespread "Behold!" legend. == Legacy == A number of institutes and colleges in India are named after him, including Bhaskaracharya Pratishthana in Pune, Bhaskaracharya College of Applied Sciences in Delhi, Bhaskaracharya Institute For Space Applications and Geo-Informatics in Gandhinagar. On 20 November 1981 the Indian Space Research Organisation (ISRO) launched the Bhaskara II satellite honouring the mathematician and astronomer. Invis Multimedia released Bhaskaracharya, an Indian documentary short on the mathematician in 2015. == See also == List of Indian mathematicians Bride's Chair Bījapallava == Notes == == References == === Bibliography === Burton, David M. (2011), The History of Mathematics: An Introduction (7th ed.), McGraw Hill, ISBN 978-0-07-338315-6 Eves, Howard (1990), An Introduction to the History of Mathematics (6th ed.), Saunders College Publishing, ISBN 978-0-03-029558-4 Mazur, Joseph (2005), Euclid in the Rainforest, Plume, ISBN 978-0-452-28783-9 Sarkār, Benoy Kumar (1918), Hindu achievements in exact science: a study in the history of scientific development, Longmans, Green and co. Seal, Sir Brajendranath (1915), The positive sciences of the ancient Hindus, Longmans, Green and co. Colebrooke, Henry T. (1817), Arithmetic and mensuration of Brahmegupta and Bhaskara White, Lynn Townsend (1978), "Tibet, India, and Malaya as Sources of Western Medieval Technology", Medieval religion and technology: collected essays, University of California Press, ISBN 978-0-520-03566-9 Selin, Helaine, ed. (2008), "Astronomical Instruments in India", Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (2nd edition), Springer Verlag Ny, ISBN 978-1-4020-4559-2 Shukla, Kripa Shankar (1984), "Use of Calculus in Hindu Mathematics", Indian Journal of History of Science, 19: 95–104 Pingree, David Edwin (1970), Census of the Exact Sciences in Sanskrit, vol. 146, American Philosophical Society, ISBN 9780871691460 Plofker, Kim (2007), "Mathematics in India", in Katz, Victor J. (ed.), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton University Press, ISBN 9780691114859 Plofker, Kim (2009), Mathematics in India, Princeton University Press, ISBN 9780691120676 Cooke, Roger (1997), "The Mathematics of the Hindus", The History of Mathematics: A Brief Course, Wiley-Interscience, pp. 213–215, ISBN 0-471-18082-3 Poulose, K. G. (1991), K. G. Poulose (ed.), Scientific heritage of India, mathematics, Ravivarma Samskr̥ta granthāvali, vol. 22, Govt. Sanskrit College (Tripunithura, India) Chopra, Pran Nath (1982), Religions and communities of India, Vision Books, ISBN 978-0-85692-081-3 Goonatilake, Susantha (1999), Toward a global science: mining civilizational knowledge, Indiana University Press, ISBN 978-0-253-21182-8 Selin, Helaine; D'Ambrosio, Ubiratan, eds. (2001), "Mathematics across cultures: the history of non-western mathematics", Science Across Cultures, 2, Springer, ISBN 978-1-4020-0260-1 Stillwell, John (2002), Mathematics and its history, Undergraduate Texts in Mathematics, Springer, ISBN 978-0-387-95336-6 Sahni, Madhu (2019), Pedagogy Of Mathematics, Vikas Publishing House, ISBN 978-9353383275 == Further reading == W. W. Rouse Ball. A Short Account of the History of Mathematics, 4th Edition. Dover Publications, 1960. George Gheverghese Joseph. The Crest of the Peacock: Non-European Roots of Mathematics, 2nd Edition. Penguin Books, 2000. O'Connor, John J.; Robertson, Edmund F., "Bhāskara II", MacTutor History of Mathematics Archive, University of St Andrews University of St Andrews, 2000. Ian Pearce. Bhaskaracharya II at the MacTutor archive. St Andrews University, 2002. Pingree, David (1970–1980). "Bhāskara II". Dictionary of Scientific Biography. Vol. 2. New York: Charles Scribner's Sons. pp. 115–120. ISBN 978-0-684-10114-9. == External links == 4to40 Biography
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Wikipedia:Bianca Falcidieno#0
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Bianca Falcidieno is an Italian applied mathematician whose research interests include computer graphics, geometric modeling, shape analysis, and mesh generation; she has been called a pioneer of semantics-driven shape representation. She is retired as a research director for the Italian National Research Council (CNR), where she led the Shape Modeling Group of the Institute for Applied Mathematics and Information Technologies (IMATI). == Education and career == As a master's student, after earning a degree in mathematics in the 1970s, Falcidieno became interested in computer graphics and geometric modeling through a project to visualize mathematical functions for use in teaching mathematical analysis to engineering students. By 1981, she was a permanent researcher for CNR, and founded its first research center for computer graphics. In the 1990s, with Japanese researcher Tosiyasu Kunii, she helped found both the Shape Modeling International conference and the associated journal, the International Journal of Shape Modeling, for which she was editor-in-chief. == Recognition == In 2021, Falcidieno became the inaugural winner of the Shape Modeling International Tosiyasu Kunii Achievement Award, given "for her outstanding career achievements in shape modelling research". She is also a Eurographics Fellow and 2019 Eurographics Gold Medalist, and a Pioneer of the Solid Modeling Association. == References == == External links == Home page Bianca Falcidieno publications indexed by Google Scholar
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Wikipedia:Bibhutibhushan Datta#0
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Bibhutibhushan Datta (Bengali: বিভূতিভূষণ দত্ত, romanized: Bibhūtibhūṣaṇ Datta; also Bibhuti Bhusan Datta; 28 June 1888 – 6 October 1958) was a historian of Indian mathematics. Datta came from a poor Bengali family. He was a student of Ganesh Prasad, studied at the University of Calcutta and secured a master's degree in mathematics in 1914 and a doctorate degree in 1920 in applied mathematics. He taught at Calcutta University where he was a lecturer at the University Science College, and from 1924 to 1929 he was the Rhashbehari Ghosh Professor of Applied Mathematics. During the 1920s and 1930s he created a reputation as an authority on the history of Indian mathematics. He was also deeply interested in Indian philosophy and religion. In 1929 he retired from his professorship and left the university in 1933, and became a sannyasin (an ascetic, a person who has renounced worldly pleasures) in 1938 under the name Swami Vidyaranya. History of Hindu Mathematics: A Source Book, written by him jointly with Avadhesh Narayan Singh (1901–1954) became a standard reference work in the history of Indian mathematics. He also wrote a monograph on the Shulba Sutras. He published more than 70 research papers mostly related to the history of Indian mathematics. In the last years of his life, as Swami Vidyaranya, he lived mainly at Pushkar, a Hindu holy site in Rajasthan. == See also == History of Hindu Mathematics: A Source Book, Datta's magnum opus on Hindu (Indic) mathematics == External links == Biography == References ==
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Wikipedia:Bidiagonal matrix#0
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In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal matrix is [ 3 0 0 2 ] {\displaystyle \left[{\begin{smallmatrix}3&0\\0&2\end{smallmatrix}}\right]} , while an example of a 3×3 diagonal matrix is [ 6 0 0 0 5 0 0 0 4 ] {\displaystyle \left[{\begin{smallmatrix}6&0&0\\0&5&0\\0&0&4\end{smallmatrix}}\right]} . An identity matrix of any size, or any multiple of it is a diagonal matrix called a scalar matrix, for example, [ 0.5 0 0 0.5 ] {\displaystyle \left[{\begin{smallmatrix}0.5&0\\0&0.5\end{smallmatrix}}\right]} . In geometry, a diagonal matrix may be used as a scaling matrix, since matrix multiplication with it results in changing scale (size) and possibly also shape; only a scalar matrix results in uniform change in scale. == Definition == As stated above, a diagonal matrix is a matrix in which all off-diagonal entries are zero. That is, the matrix D = (di,j) with n columns and n rows is diagonal if ∀ i , j ∈ { 1 , 2 , … , n } , i ≠ j ⟹ d i , j = 0. {\displaystyle \forall i,j\in \{1,2,\ldots ,n\},i\neq j\implies d_{i,j}=0.} However, the main diagonal entries are unrestricted. The term diagonal matrix may sometimes refer to a rectangular diagonal matrix, which is an m-by-n matrix with all the entries not of the form di,i being zero. For example: [ 1 0 0 0 4 0 0 0 − 3 0 0 0 ] or [ 1 0 0 0 0 0 4 0 0 0 0 0 − 3 0 0 ] {\displaystyle {\begin{bmatrix}1&0&0\\0&4&0\\0&0&-3\\0&0&0\\\end{bmatrix}}\quad {\text{or}}\quad {\begin{bmatrix}1&0&0&0&0\\0&4&0&0&0\\0&0&-3&0&0\end{bmatrix}}} More often, however, diagonal matrix refers to square matrices, which can be specified explicitly as a square diagonal matrix. A square diagonal matrix is a symmetric matrix, so this can also be called a symmetric diagonal matrix. The following matrix is square diagonal matrix: [ 1 0 0 0 4 0 0 0 − 2 ] {\displaystyle {\begin{bmatrix}1&0&0\\0&4&0\\0&0&-2\end{bmatrix}}} If the entries are real numbers or complex numbers, then it is a normal matrix as well. In the remainder of this article we will consider only square diagonal matrices, and refer to them simply as "diagonal matrices". == Vector-to-matrix diag operator == A diagonal matrix D can be constructed from a vector a = [ a 1 … a n ] T {\displaystyle \mathbf {a} ={\begin{bmatrix}a_{1}&\dots &a_{n}\end{bmatrix}}^{\textsf {T}}} using the diag {\displaystyle \operatorname {diag} } operator: D = diag ( a 1 , … , a n ) . {\displaystyle \mathbf {D} =\operatorname {diag} (a_{1},\dots ,a_{n}).} This may be written more compactly as D = diag ( a ) {\displaystyle \mathbf {D} =\operatorname {diag} (\mathbf {a} )} . The same operator is also used to represent block diagonal matrices as A = diag ( A 1 , … , A n ) {\displaystyle \mathbf {A} =\operatorname {diag} (\mathbf {A} _{1},\dots ,\mathbf {A} _{n})} where each argument Ai is a matrix. The diag operator may be written as diag ( a ) = ( a 1 T ) ∘ I , {\displaystyle \operatorname {diag} (\mathbf {a} )=\left(\mathbf {a} \mathbf {1} ^{\textsf {T}}\right)\circ \mathbf {I} ,} where ∘ {\displaystyle \circ } represents the Hadamard product, and 1 is a constant vector with elements 1. == Matrix-to-vector diag operator == The inverse matrix-to-vector diag operator is sometimes denoted by the identically named diag ( D ) = [ a 1 … a n ] T , {\displaystyle \operatorname {diag} (\mathbf {D} )={\begin{bmatrix}a_{1}&\dots &a_{n}\end{bmatrix}}^{\textsf {T}},} where the argument is now a matrix, and the result is a vector of its diagonal entries. The following property holds: diag ( A B ) = ∑ j ( A ∘ B T ) i j = ( A ∘ B T ) 1 . {\displaystyle \operatorname {diag} (\mathbf {A} \mathbf {B} )=\sum _{j}\left(\mathbf {A} \circ \mathbf {B} ^{\textsf {T}}\right)_{ij}=\left(\mathbf {A} \circ \mathbf {B} ^{\textsf {T}}\right)\mathbf {1} .} == Scalar matrix == A diagonal matrix with equal diagonal entries is a scalar matrix; that is, a scalar multiple λ of the identity matrix I. Its effect on a vector is scalar multiplication by λ. For example, a 3×3 scalar matrix has the form: [ λ 0 0 0 λ 0 0 0 λ ] ≡ λ I 3 {\displaystyle {\begin{bmatrix}\lambda &0&0\\0&\lambda &0\\0&0&\lambda \end{bmatrix}}\equiv \lambda {\boldsymbol {I}}_{3}} The scalar matrices are the center of the algebra of matrices: that is, they are precisely the matrices that commute with all other square matrices of the same size. By contrast, over a field (like the real numbers), a diagonal matrix with all diagonal elements distinct only commutes with diagonal matrices (its centralizer is the set of diagonal matrices). That is because if a diagonal matrix D = diag ( a 1 , … , a n ) {\displaystyle \mathbf {D} =\operatorname {diag} (a_{1},\dots ,a_{n})} has a i ≠ a j , {\displaystyle a_{i}\neq a_{j},} then given a matrix M with m i j ≠ 0 , {\displaystyle m_{ij}\neq 0,} the (i, j) term of the products are: ( D M ) i j = a i m i j {\displaystyle (\mathbf {DM} )_{ij}=a_{i}m_{ij}} and ( M D ) i j = m i j a j , {\displaystyle (\mathbf {MD} )_{ij}=m_{ij}a_{j},} and a j m i j ≠ m i j a i {\displaystyle a_{j}m_{ij}\neq m_{ij}a_{i}} (since one can divide by mij), so they do not commute unless the off-diagonal terms are zero. Diagonal matrices where the diagonal entries are not all equal or all distinct have centralizers intermediate between the whole space and only diagonal matrices. For an abstract vector space V (rather than the concrete vector space Kn), the analog of scalar matrices are scalar transformations. This is true more generally for a module M over a ring R, with the endomorphism algebra End(M) (algebra of linear operators on M) replacing the algebra of matrices. Formally, scalar multiplication is a linear map, inducing a map R → End ( M ) , {\displaystyle R\to \operatorname {End} (M),} (from a scalar λ to its corresponding scalar transformation, multiplication by λ) exhibiting End(M) as a R-algebra. For vector spaces, the scalar transforms are exactly the center of the endomorphism algebra, and, similarly, scalar invertible transforms are the center of the general linear group GL(V). The former is more generally true free modules M ≅ R n , {\displaystyle M\cong R^{n},} for which the endomorphism algebra is isomorphic to a matrix algebra. == Vector operations == Multiplying a vector by a diagonal matrix multiplies each of the terms by the corresponding diagonal entry. Given a diagonal matrix D = diag ( a 1 , … , a n ) {\displaystyle \mathbf {D} =\operatorname {diag} (a_{1},\dots ,a_{n})} and a vector v = [ x 1 ⋯ x n ] T {\displaystyle \mathbf {v} ={\begin{bmatrix}x_{1}&\dotsm &x_{n}\end{bmatrix}}^{\textsf {T}}} , the product is: D v = diag ( a 1 , … , a n ) [ x 1 ⋮ x n ] = [ a 1 ⋱ a n ] [ x 1 ⋮ x n ] = [ a 1 x 1 ⋮ a n x n ] . {\displaystyle \mathbf {D} \mathbf {v} =\operatorname {diag} (a_{1},\dots ,a_{n}){\begin{bmatrix}x_{1}\\\vdots \\x_{n}\end{bmatrix}}={\begin{bmatrix}a_{1}\\&\ddots \\&&a_{n}\end{bmatrix}}{\begin{bmatrix}x_{1}\\\vdots \\x_{n}\end{bmatrix}}={\begin{bmatrix}a_{1}x_{1}\\\vdots \\a_{n}x_{n}\end{bmatrix}}.} This can be expressed more compactly by using a vector instead of a diagonal matrix, d = [ a 1 ⋯ a n ] T {\displaystyle \mathbf {d} ={\begin{bmatrix}a_{1}&\dotsm &a_{n}\end{bmatrix}}^{\textsf {T}}} , and taking the Hadamard product of the vectors (entrywise product), denoted d ∘ v {\displaystyle \mathbf {d} \circ \mathbf {v} } : D v = d ∘ v = [ a 1 ⋮ a n ] ∘ [ x 1 ⋮ x n ] = [ a 1 x 1 ⋮ a n x n ] . {\displaystyle \mathbf {D} \mathbf {v} =\mathbf {d} \circ \mathbf {v} ={\begin{bmatrix}a_{1}\\\vdots \\a_{n}\end{bmatrix}}\circ {\begin{bmatrix}x_{1}\\\vdots \\x_{n}\end{bmatrix}}={\begin{bmatrix}a_{1}x_{1}\\\vdots \\a_{n}x_{n}\end{bmatrix}}.} This is mathematically equivalent, but avoids storing all the zero terms of this sparse matrix. This product is thus used in machine learning, such as computing products of derivatives in backpropagation or multiplying IDF weights in TF-IDF, since some BLAS frameworks, which multiply matrices efficiently, do not include Hadamard product capability directly. == Matrix operations == The operations of matrix addition and matrix multiplication are especially simple for diagonal matrices. Write diag(a1, ..., an) for a diagonal matrix whose diagonal entries starting in the upper left corner are a1, ..., an. Then, for addition, we have diag ( a 1 , … , a n ) + diag ( b 1 , … , b n ) = diag ( a 1 + b 1 , … , a n + b n ) {\displaystyle \operatorname {diag} (a_{1},\,\ldots ,\,a_{n})+\operatorname {diag} (b_{1},\,\ldots ,\,b_{n})=\operatorname {diag} (a_{1}+b_{1},\,\ldots ,\,a_{n}+b_{n})} and for matrix multiplication, diag ( a 1 , … , a n ) diag ( b 1 , … , b n ) = diag ( a 1 b 1 , … , a n b n ) . {\displaystyle \operatorname {diag} (a_{1},\,\ldots ,\,a_{n})\operatorname {diag} (b_{1},\,\ldots ,\,b_{n})=\operatorname {diag} (a_{1}b_{1},\,\ldots ,\,a_{n}b_{n}).} The diagonal matrix diag(a1, ..., an) is invertible if and only if the entries a1, ..., an are all nonzero. In this case, we have diag ( a 1 , … , a n ) − 1 = diag ( a 1 − 1 , … , a n − 1 ) . {\displaystyle \operatorname {diag} (a_{1},\,\ldots ,\,a_{n})^{-1}=\operatorname {diag} (a_{1}^{-1},\,\ldots ,\,a_{n}^{-1}).} In particular, the diagonal matrices form a subring of the ring of all n-by-n matrices. Multiplying an n-by-n matrix A from the left with diag(a1, ..., an) amounts to multiplying the i-th row of A by ai for all i; multiplying the matrix A from the right with diag(a1, ..., an) amounts to multiplying the i-th column of A by ai for all i. == Operator matrix in eigenbasis == As explained in determining coefficients of operator matrix, there is a special basis, e1, ..., en, for which the matrix A takes the diagonal form. Hence, in the defining equation A e j = ∑ i a i , j e i {\textstyle \mathbf {Ae} _{j}=\sum _{i}a_{i,j}\mathbf {e} _{i}} , all coefficients ai, j with i ≠ j are zero, leaving only one term per sum. The surviving diagonal elements, ai, j, are known as eigenvalues and designated with λi in the equation, which reduces to A e i = λ i e i . {\displaystyle \mathbf {Ae} _{i}=\lambda _{i}\mathbf {e} _{i}.} The resulting equation is known as eigenvalue equation and used to derive the characteristic polynomial and, further, eigenvalues and eigenvectors. In other words, the eigenvalues of diag(λ1, ..., λn) are λ1, ..., λn with associated eigenvectors of e1, ..., en. == Properties == The determinant of diag(a1, ..., an) is the product a1⋯an. The adjugate of a diagonal matrix is again diagonal. Where all matrices are square, A matrix is diagonal if and only if it is triangular and normal. A matrix is diagonal if and only if it is both upper- and lower-triangular. A diagonal matrix is symmetric. The identity matrix In and zero matrix are diagonal. A 1×1 matrix is always diagonal. The square of a 2×2 matrix with zero trace is always diagonal. == Applications == Diagonal matrices occur in many areas of linear algebra. Because of the simple description of the matrix operation and eigenvalues/eigenvectors given above, it is typically desirable to represent a given matrix or linear map by a diagonal matrix. In fact, a given n-by-n matrix A is similar to a diagonal matrix (meaning that there is a matrix X such that X−1AX is diagonal) if and only if it has n linearly independent eigenvectors. Such matrices are said to be diagonalizable. Over the field of real or complex numbers, more is true. The spectral theorem says that every normal matrix is unitarily similar to a diagonal matrix (if AA∗ = A∗A then there exists a unitary matrix U such that UAU∗ is diagonal). Furthermore, the singular value decomposition implies that for any matrix A, there exist unitary matrices U and V such that U∗AV is diagonal with positive entries. == Operator theory == In operator theory, particularly the study of PDEs, operators are particularly easy to understand and PDEs easy to solve if the operator is diagonal with respect to the basis with which one is working; this corresponds to a separable partial differential equation. Therefore, a key technique to understanding operators is a change of coordinates—in the language of operators, an integral transform—which changes the basis to an eigenbasis of eigenfunctions: which makes the equation separable. An important example of this is the Fourier transform, which diagonalizes constant coefficient differentiation operators (or more generally translation invariant operators), such as the Laplacian operator, say, in the heat equation. Especially easy are multiplication operators, which are defined as multiplication by (the values of) a fixed function–the values of the function at each point correspond to the diagonal entries of a matrix. == See also == == Notes == == References == == Sources == Horn, Roger Alan; Johnson, Charles Royal (1985), Matrix Analysis, Cambridge University Press, ISBN 978-0-521-38632-6
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Wikipedia:Bidirectional transformation#0
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In computer programming, bidirectional transformations (bx) are programs in which a single piece of code can be run in several ways, such that the same data are sometimes considered as input, and sometimes as output. For example, a bx run in the forward direction might transform input I into output O, while the same bx run backward would take as input versions of I and O and produce a new version of I as its output. Bidirectional model transformations are an important special case in which a model is input to such a program. Some bidirectional languages are bijective. The bijectivity of a language is a severe restriction of its power, because a bijective language is merely relating two different ways to present the very same information. More general is a lens language, in which there is a distinguished forward direction ("get") that takes a concrete input to an abstract output, discarding some information in the process: the concrete state includes all the information that is in the abstract state, and usually some more. The backward direction ("put") takes a concrete state and an abstract state and computes a new concrete state. Lenses are required to obey certain conditions to ensure sensible behaviour. The most general case is that of symmetric bidirectional transformations. Here the two states that are related typically share some information, but each also includes some information that is not included in the other. == Usage == Bidirectional transformations can be used to: Maintain the consistency of several sources of information Provide an 'abstract view' to easily manipulate data and write them back to their source == Definition == Bidirectional transformations fall into various well-studied categories. A lens is a pair of functions g e t : S → V {\displaystyle get:S\rightarrow V} , p u t : S , V → S {\displaystyle put:S,V\rightarrow S} relating a source S {\displaystyle S} and a view V {\displaystyle V} . If these functions obey the three lens laws: PutGet: ∀ s , v . g e t ( p u t ( s , v ) ) = v {\displaystyle \forall s,v.get(put(s,v))=v} GetPut: ∀ s . p u t ( s , g e t ( s ) ) = s {\displaystyle \forall s.put(s,get(s))=s} PutPut: ∀ s , v , v ′ . p u t ( p u t ( s , v ) , v ′ ) = p u t ( s , v ′ ) {\displaystyle \forall s,v,v'.put(put(s,v),v')=put(s,v')} It is called a well-behaved lens. A related notion is that of a prism, in which the signatures of the functions are instead g e t : S → V + 1 {\displaystyle get:S\rightarrow V+1} , p u t : V → S {\displaystyle put:V\rightarrow S} . Unlike a lens, a prism may not always give a view; also unlike a lens, given a prism, a view is sufficient to construct a source. If lenses allow "focusing" (viewing, updating) on a part of a product type, prisms allow focusing (possible viewing, building) on a part of a sum type. Both lenses and prisms, as well as other constructions such as traversals, are more general notion of bidirectional transformations known as optics. == Examples of implementations == Boomerang is a programming language that allows writing lenses to process text data formats bidirectionally Augeas is a configuration management library whose lens language is inspired by the Boomerang project biXid is a programming language for processing XML data bidirectionally XSugar allows translation from XML to non-XML formats == See also == Bidirectionalization Reverse computation Transformation language == References == == External links == "GRACE International Meeting on Bidirectional Transformations". Archived from the original on 12 October 2014. "Bidirectional Transformations". The Bx Wiki. Retrieved 19 February 2024. Pacheco, Hugo; Cunha, Alcino (2012). Multifocal: A strategic bidirectional transformation language for XML schemas (PDF). International Conference on Theory and Practice of Model Transformations. Springer, Berlin, Heidelberg. doi:10.1007/978-3-642-30476-7_6.
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Wikipedia:Bifundamental representation#0
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In mathematics and theoretical physics, a bifundamental representation is a representation obtained as a tensor product of two fundamental or antifundamental representations. For example, the MN-dimensional representation (M,N) of the group S U ( M ) × S U ( N ) {\displaystyle SU(M)\times SU(N)} is a bifundamental representation. These representations occur in quiver diagrams. == References ==
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Wikipedia:Big M method#0
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In operations research, the Big M method is a method of solving linear programming problems using the simplex algorithm. The Big M method extends the simplex algorithm to problems that contain "greater-than" constraints. It does so by associating the constraints with large negative constants which would not be part of any optimal solution, if it exists. == Algorithm == The simplex algorithm is the original and still one of the most widely used methods for solving linear maximization problems. It is obvious that the points with the optimal objective must be reached on a vertex of the simplex which is the shape of feasible region of an LP (linear program). Points on the vertex of the simplex are represented as a basis. So, to apply the simplex algorithm which aims improve the basis until a global optima is reached, one needs to find a feasible basis first. The trivial basis (all problem variables equal to 0) is not always part of the simplex. It is feasible if and only if all the constraints (except non-negativity) are less-than constraints and with positive constant on the right-hand side. The Big M method introduces surplus and artificial variables to convert all inequalities into that form and there by extends the simplex in higher dimensions to be valid in the trivial basis. It is always a vertex due to the positivity constraint on the problem variables inherent in the standard formulation of LP. The "Big M" refers to a large number associated with the artificial variables, represented by the letter M. The steps in the algorithm are as follows: Multiply the inequality constraints to ensure that the right hand side is positive. If the problem is of minimization, transform to maximization by multiplying the objective by −1. For any greater-than constraints, introduce surplus si and artificial variables ai (as shown below). Choose a large positive Value M and introduce a term in the objective of the form −M multiplying the artificial variables. For less-than or equal constraints, introduce slack variables si so that all constraints are equalities. Solve the problem using the usual simplex method. For example, x + y ≤ 100 becomes x + y + s1 = 100, whilst x + y ≥ 100 becomes x + y − s1 + a1 = 100. The artificial variables must be shown to be 0. The function to be maximised is rewritten to include the sum of all the artificial variables. Then row reductions are applied to gain a final solution. The value of M must be chosen sufficiently large so that the artificial variable would not be part of any feasible solution. For a sufficiently large M, the optimal solution contains any artificial variables in the basis (i.e. positive values) if and only if the problem is not feasible. However, the a-priori selection of an appropriate value for M is not trivial. A way to overcome the need to specify the value of M is described in . Other ways to find an initial basis for the simplex algorithm involves solving another linear program in an initial phase. == Other usage == When used in the objective function, the Big M method sometimes refers to formulations of linear optimization problems in which violations of a constraint or set of constraints are associated with a large positive penalty constant, M. In Mixed integer linear optimization the term Big M can also refer to use of a large term in the constraints themselves. For example the logical constraint z = 0 ⟺ x = y {\displaystyle z=0\iff x=y} where z is binary variable (0 or 1) variable refers to ensuring equality of variables only when a certain binary variable takes on one value, but to leave the variables "open" if the binary variable takes on its opposite value. For a sufficiently large M and z binary variable (0 or 1), the constraints x − y ≤ M z {\displaystyle x-y\leq Mz} x − y ≥ − M z {\displaystyle x-y\geq -Mz} ensure that when z = 0 {\displaystyle z=0} then x = y {\displaystyle x=y} . Otherwise, when z = 1 {\displaystyle z=1} , then − M ≤ x − y ≤ M {\displaystyle -M\leq x-y\leq M} , indicating that the variables x and y can have any values so long as the absolute value of their difference is bounded by M {\displaystyle M} (hence the need for M to be "large enough.") Thus it is possible to "encode" the logical constraint into a MILP problem. == See also == Two phase method (linear programming) another approach for solving problems with >= constraints Karush–Kuhn–Tucker conditions, which apply to nonlinear optimization problems with inequality constraints. == External links == Bibliography Griva, Igor; Nash, Stephan G.; Sofer, Ariela (26 March 2009). Linear and Nonlinear Optimization (2nd ed.). Society for Industrial Mathematics. ISBN 978-0-89871-661-0. Discussion Simplex – Big M Method, Lynn Killen, Dublin City University. The Big M Method, businessmanagementcourses.org The Big M Method, Mark Hutchinson The Big-M Method with the Numerical Infinite M, a recently introduced parameterless variant A THREE-PHASE SIMPLEX METHOD FOR INFEASIBLE AND UNBOUNDED LINEAR PROGRAMMING PROBLEMS, Big M method for M=1 == References ==
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Wikipedia:Big O notation#0
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Big O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by German mathematicians Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for Ordnung, meaning the order of approximation. In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; a famous example of such a difference is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates. Big O notation characterizes functions according to their growth rates: different functions with the same asymptotic growth rate may be represented using the same O notation. The letter O is used because the growth rate of a function is also referred to as the order of the function. A description of a function in terms of big O notation usually only provides an upper bound on the growth rate of the function. Associated with big O notation are several related notations, using the symbols o, Ω, ω, and Θ, to describe other kinds of bounds on asymptotic growth rates. == Formal definition == Let f , {\displaystyle f,} the function to be estimated, be a real or complex valued function, and let g , {\displaystyle g,} the comparison function, be a real valued function. Let both functions be defined on some unbounded subset of the positive real numbers, and g ( x ) {\displaystyle g(x)} be non-zero (often, but not necessarily, strictly positive) for all large enough values of x . {\displaystyle x.} One writes f ( x ) = O ( g ( x ) ) as x → ∞ {\displaystyle f(x)=O{\bigl (}g(x){\bigr )}\quad {\text{ as }}x\to \infty } and it is read " f ( x ) {\displaystyle f(x)} is big O of g ( x ) {\displaystyle g(x)} " or more often " f ( x ) {\displaystyle f(x)} is of the order of g ( x ) {\displaystyle g(x)} " if the absolute value of f ( x ) {\displaystyle f(x)} is at most a positive constant multiple of the absolute value of g ( x ) {\displaystyle g(x)} for all sufficiently large values of x . {\displaystyle x.} That is, f ( x ) = O ( g ( x ) ) {\displaystyle f(x)=O{\bigl (}g(x){\bigr )}} if there exists a positive real number M {\displaystyle M} and a real number x 0 {\displaystyle x_{0}} such that | f ( x ) | ≤ M | g ( x ) | for all x ≥ x 0 . {\displaystyle |f(x)|\leq M\ |g(x)|\quad {\text{ for all }}x\geq x_{0}~.} In many contexts, the assumption that we are interested in the growth rate as the variable x {\displaystyle \ x\ } goes to infinity or to zero is left unstated, and one writes more simply that f ( x ) = O ( g ( x ) ) . {\displaystyle f(x)=O{\bigl (}g(x){\bigr )}.} The notation can also be used to describe the behavior of f {\displaystyle f} near some real number a {\displaystyle a} (often, a = 0 {\displaystyle a=0} ): we say f ( x ) = O ( g ( x ) ) as x → a {\displaystyle f(x)=O{\bigl (}g(x){\bigr )}\quad {\text{ as }}\ x\to a} if there exist positive numbers δ {\displaystyle \delta } and M {\displaystyle M} such that for all defined x {\displaystyle x} with 0 < | x − a | < δ , {\displaystyle 0<|x-a|<\delta ,} | f ( x ) | ≤ M | g ( x ) | . {\displaystyle |f(x)|\leq M|g(x)|.} As g ( x ) {\displaystyle g(x)} is non-zero for adequately large (or small) values of x , {\displaystyle x,} both of these definitions can be unified using the limit superior: f ( x ) = O ( g ( x ) ) as x → a {\displaystyle f(x)=O{\bigl (}g(x){\bigr )}\quad {\text{ as }}\ x\to a} if lim sup x → a | f ( x ) | | g ( x ) | < ∞ . {\displaystyle \limsup _{x\to a}{\frac {\left|f(x)\right|}{\left|g(x)\right|}}<\infty .} And in both of these definitions the limit point a {\displaystyle a} (whether ∞ {\displaystyle \infty } or not) is a cluster point of the domains of f {\displaystyle f} and g , {\displaystyle g,} i. e., in every neighbourhood of a {\displaystyle a} there have to be infinitely many points in common. Moreover, as pointed out in the article about the limit inferior and limit superior, the lim sup x → a {\displaystyle \textstyle \limsup _{x\to a}} (at least on the extended real number line) always exists. In computer science, a slightly more restrictive definition is common: f {\displaystyle f} and g {\displaystyle g} are both required to be functions from some unbounded subset of the positive integers to the nonnegative real numbers; then f ( x ) = O ( g ( x ) ) {\displaystyle f(x)=O{\bigl (}g(x){\bigr )}} if there exist positive integer numbers M {\displaystyle M} and n 0 {\displaystyle n_{0}} such that | f ( n ) | ≤ M | g ( n ) | {\displaystyle |f(n)|\leq M|g(n)|} for all n ≥ n 0 . {\displaystyle n\geq n_{0}.} == Example == In typical usage the O notation is asymptotical, that is, it refers to very large x. In this setting, the contribution of the terms that grow "most quickly" will eventually make the other ones irrelevant. As a result, the following simplification rules can be applied: If f(x) is a sum of several terms, if there is one with largest growth rate, it can be kept, and all others omitted. If f(x) is a product of several factors, any constants (factors in the product that do not depend on x) can be omitted. For example, let f(x) = 6x4 − 2x3 + 5, and suppose we wish to simplify this function, using O notation, to describe its growth rate as x approaches infinity. This function is the sum of three terms: 6x4, −2x3, and 5. Of these three terms, the one with the highest growth rate is the one with the largest exponent as a function of x, namely 6x4. Now one may apply the second rule: 6x4 is a product of 6 and x4 in which the first factor does not depend on x. Omitting this factor results in the simplified form x4. Thus, we say that f(x) is a "big O" of x4. Mathematically, we can write f(x) = O(x4). One may confirm this calculation using the formal definition: let f(x) = 6x4 − 2x3 + 5 and g(x) = x4. Applying the formal definition from above, the statement that f(x) = O(x4) is equivalent to its expansion, | f ( x ) | ≤ M x 4 {\displaystyle |f(x)|\leq Mx^{4}} for some suitable choice of a real number x0 and a positive real number M and for all x > x0. To prove this, let x0 = 1 and M = 13. Then, for all x > x0: | 6 x 4 − 2 x 3 + 5 | ≤ 6 x 4 + | 2 x 3 | + 5 ≤ 6 x 4 + 2 x 4 + 5 x 4 = 13 x 4 {\displaystyle {\begin{aligned}|6x^{4}-2x^{3}+5|&\leq 6x^{4}+|2x^{3}|+5\\&\leq 6x^{4}+2x^{4}+5x^{4}\\&=13x^{4}\end{aligned}}} so | 6 x 4 − 2 x 3 + 5 | ≤ 13 x 4 . {\displaystyle |6x^{4}-2x^{3}+5|\leq 13x^{4}.} == Use == Big O notation has two main areas of application: In mathematics, it is commonly used to describe how closely a finite series approximates a given function, especially in the case of a truncated Taylor series or asymptotic expansion. In computer science, it is useful in the analysis of algorithms. In both applications, the function g(x) appearing within the O(·) is typically chosen to be as simple as possible, omitting constant factors and lower order terms. There are two formally close, but noticeably different, usages of this notation: infinite asymptotics infinitesimal asymptotics. This distinction is only in application and not in principle, however—the formal definition for the "big O" is the same for both cases, only with different limits for the function argument. === Infinite asymptotics === Big O notation is useful when analyzing algorithms for efficiency. For example, the time (or the number of steps) it takes to complete a problem of size n might be found to be T(n) = 4n2 − 2n + 2. As n grows large, the n2 term will come to dominate, so that all other terms can be neglected —for instance when n = 500, the term 4n2 is 1000 times as large as the 2n term. Ignoring the latter would have negligible effect on the expression's value for most purposes. Further, the coefficients become irrelevant if we compare to any other order of expression, such as an expression containing a term n3 or n4. Even if T(n) = 1,000,000n2, if U(n) = n3, the latter will always exceed the former once n grows larger than 1,000,000, viz. T(1,000,000) = 1,000,0003 = U(1,000,000). Additionally, the number of steps depends on the details of the machine model on which the algorithm runs, but different types of machines typically vary by only a constant factor in the number of steps needed to execute an algorithm. So the big O notation captures what remains: we write either T ( n ) = O ( n 2 ) {\displaystyle T(n)=O(n^{2})} or T ( n ) ∈ O ( n 2 ) {\displaystyle T(n)\in O(n^{2})} and say that the algorithm has order of n2 time complexity. The sign "=" is not meant to express "is equal to" in its normal mathematical sense, but rather a more colloquial "is", so the second expression is sometimes considered more accurate (see the "Equals sign" discussion below) while the first is considered by some as an abuse of notation. === Infinitesimal asymptotics === Big O can also be used to describe the error term in an approximation to a mathematical function. The most significant terms are written explicitly, and then the least-significant terms are summarized in a single big O term. Consider, for example, the exponential series and two expressions of it that are valid when x is small: e x = 1 + x + x 2 2 ! + x 3 3 ! + x 4 4 ! + ⋯ for all finite x = 1 + x + x 2 2 + O ( x 3 ) as x → 0 = 1 + x + O ( x 2 ) as x → 0 {\displaystyle {\begin{aligned}e^{x}&=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\dotsb &&{\text{for all finite }}x\\[4pt]&=1+x+{\frac {x^{2}}{2}}+O(x^{3})&&{\text{as }}x\to 0\\[4pt]&=1+x+O(x^{2})&&{\text{as }}x\to 0\end{aligned}}} The middle expression (the one with O(x3)) means the absolute-value of the error ex − (1 + x + x2/2) is at most some constant times |x3!| when x is close enough to 0. == Properties == If the function f can be written as a finite sum of other functions, then the fastest growing one determines the order of f(n). For example, f ( n ) = 9 log n + 5 ( log n ) 4 + 3 n 2 + 2 n 3 = O ( n 3 ) as n → ∞ . {\displaystyle f(n)=9\log n+5(\log n)^{4}+3n^{2}+2n^{3}=O(n^{3})\qquad {\text{as }}n\to \infty .} In particular, if a function may be bounded by a polynomial in n, then as n tends to infinity, one may disregard lower-order terms of the polynomial. The sets O(nc) and O(cn) are very different. If c is greater than one, then the latter grows much faster. A function that grows faster than nc for any c is called superpolynomial. One that grows more slowly than any exponential function of the form cn is called subexponential. An algorithm can require time that is both superpolynomial and subexponential; examples of this include the fastest known algorithms for integer factorization and the function nlog n. We may ignore any powers of n inside of the logarithms. The set O(log n) is exactly the same as O(log(nc)). The logarithms differ only by a constant factor (since log(nc) = c log n) and thus the big O notation ignores that. Similarly, logs with different constant bases are equivalent. On the other hand, exponentials with different bases are not of the same order. For example, 2n and 3n are not of the same order. Changing units may or may not affect the order of the resulting algorithm. Changing units is equivalent to multiplying the appropriate variable by a constant wherever it appears. For example, if an algorithm runs in the order of n2, replacing n by cn means the algorithm runs in the order of c2n2, and the big O notation ignores the constant c2. This can be written as c2n2 = O(n2). If, however, an algorithm runs in the order of 2n, replacing n with cn gives 2cn = (2c)n. This is not equivalent to 2n in general. Changing variables may also affect the order of the resulting algorithm. For example, if an algorithm's run time is O(n) when measured in terms of the number n of digits of an input number x, then its run time is O(log x) when measured as a function of the input number x itself, because n = O(log x). === Product === f 1 = O ( g 1 ) and f 2 = O ( g 2 ) ⇒ f 1 f 2 = O ( g 1 g 2 ) {\displaystyle f_{1}=O(g_{1}){\text{ and }}f_{2}=O(g_{2})\Rightarrow f_{1}f_{2}=O(g_{1}g_{2})} f ⋅ O ( g ) = O ( f g ) {\displaystyle f\cdot O(g)=O(fg)} === Sum === If f 1 = O ( g 1 ) {\displaystyle f_{1}=O(g_{1})} and f 2 = O ( g 2 ) {\displaystyle f_{2}=O(g_{2})} then f 1 + f 2 = O ( max ( | g 1 | , | g 2 | ) ) {\displaystyle f_{1}+f_{2}=O(\max(|g_{1}|,|g_{2}|))} . It follows that if f 1 = O ( g ) {\displaystyle f_{1}=O(g)} and f 2 = O ( g ) {\displaystyle f_{2}=O(g)} then f 1 + f 2 ∈ O ( g ) {\displaystyle f_{1}+f_{2}\in O(g)} . === Multiplication by a constant === Let k be a nonzero constant. Then O ( | k | ⋅ g ) = O ( g ) {\displaystyle O(|k|\cdot g)=O(g)} . In other words, if f = O ( g ) {\displaystyle f=O(g)} , then k ⋅ f = O ( g ) . {\displaystyle k\cdot f=O(g).} == Multiple variables == Big O (and little o, Ω, etc.) can also be used with multiple variables. To define big O formally for multiple variables, suppose f {\displaystyle f} and g {\displaystyle g} are two functions defined on some subset of R n {\displaystyle \mathbb {R} ^{n}} . We say f ( x ) is O ( g ( x ) ) as x → ∞ {\displaystyle f(\mathbf {x} ){\text{ is }}O(g(\mathbf {x} ))\quad {\text{ as }}\mathbf {x} \to \infty } if and only if there exist constants M {\displaystyle M} and C > 0 {\displaystyle C>0} such that | f ( x ) | ≤ C | g ( x ) | {\displaystyle |f(\mathbf {x} )|\leq C|g(\mathbf {x} )|} for all x {\displaystyle \mathbf {x} } with x i ≥ M {\displaystyle x_{i}\geq M} for some i . {\displaystyle i.} Equivalently, the condition that x i ≥ M {\displaystyle x_{i}\geq M} for some i {\displaystyle i} can be written ‖ x ‖ ∞ ≥ M {\displaystyle \|\mathbf {x} \|_{\infty }\geq M} , where ‖ x ‖ ∞ {\displaystyle \|\mathbf {x} \|_{\infty }} denotes the Chebyshev norm. For example, the statement f ( n , m ) = n 2 + m 3 + O ( n + m ) as n , m → ∞ {\displaystyle f(n,m)=n^{2}+m^{3}+O(n+m)\quad {\text{ as }}n,m\to \infty } asserts that there exist constants C and M such that | f ( n , m ) − ( n 2 + m 3 ) | ≤ C | n + m | {\displaystyle |f(n,m)-(n^{2}+m^{3})|\leq C|n+m|} whenever either m ≥ M {\displaystyle m\geq M} or n ≥ M {\displaystyle n\geq M} holds. This definition allows all of the coordinates of x {\displaystyle \mathbf {x} } to increase to infinity. In particular, the statement f ( n , m ) = O ( n m ) as n , m → ∞ {\displaystyle f(n,m)=O(n^{m})\quad {\text{ as }}n,m\to \infty } (i.e., ∃ C ∃ M ∀ n ∀ m ⋯ {\displaystyle \exists C\,\exists M\,\forall n\,\forall m\,\cdots } ) is quite different from ∀ m : f ( n , m ) = O ( n m ) as n → ∞ {\displaystyle \forall m\colon ~f(n,m)=O(n^{m})\quad {\text{ as }}n\to \infty } (i.e., ∀ m ∃ C ∃ M ∀ n ⋯ {\displaystyle \forall m\,\exists C\,\exists M\,\forall n\,\cdots } ). Under this definition, the subset on which a function is defined is significant when generalizing statements from the univariate setting to the multivariate setting. For example, if f ( n , m ) = 1 {\displaystyle f(n,m)=1} and g ( n , m ) = n {\displaystyle g(n,m)=n} , then f ( n , m ) = O ( g ( n , m ) ) {\displaystyle f(n,m)=O(g(n,m))} if we restrict f {\displaystyle f} and g {\displaystyle g} to [ 1 , ∞ ) 2 {\displaystyle [1,\infty )^{2}} , but not if they are defined on [ 0 , ∞ ) 2 {\displaystyle [0,\infty )^{2}} . This is not the only generalization of big O to multivariate functions, and in practice, there is some inconsistency in the choice of definition. == Matters of notation == === Equals sign === The statement "f(x) is O[g(x)]" as defined above is usually written as f(x) = O[g(x)]. Some consider this to be an abuse of notation, since the use of the equals sign could be misleading as it suggests a symmetry that this statement does not have. As de Bruijn says, O[x] = O[x2] is true but O[x2] = O[x] is not. Knuth describes such statements as "one-way equalities", since if the sides could be reversed, "we could deduce ridiculous things like n = n2 from the identities n = O[n2] and n2 = O[n2]". In another letter, Knuth also pointed out that the equality sign is not symmetric with respect to such notations [as, in this notation,] mathematicians customarily use the '=' sign as they use the word 'is' in English: Aristotle is a man, but a man isn't necessarily Aristotle. For these reasons, it would be more precise to use set notation and write f(x) ∈ O[g(x)] – read as: "f(x) is an element of O[g(x)]", or "f(x) is in the set O[g(x)]" – thinking of O[g(x)] as the class of all functions h(x) such that |h(x)| ≤ C |g(x)| for some positive real number C. However, the use of the equals sign is customary. === Other arithmetic operators === Big O notation can also be used in conjunction with other arithmetic operators in more complicated equations. For example, h(x) + O(f(x)) denotes the collection of functions having the growth of h(x) plus a part whose growth is limited to that of f(x). Thus, g ( x ) = h ( x ) + O ( f ( x ) ) {\displaystyle g(x)=h(x)+O(f(x))} expresses the same as g ( x ) − h ( x ) = O ( f ( x ) ) . {\displaystyle g(x)-h(x)=O(f(x)).} ==== Example ==== Suppose an algorithm is being developed to operate on a set of n elements. Its developers are interested in finding a function T(n) that will express how long the algorithm will take to run (in some arbitrary measurement of time) in terms of the number of elements in the input set. The algorithm works by first calling a subroutine to sort the elements in the set and then perform its own operations. The sort has a known time complexity of O(n2), and after the subroutine runs the algorithm must take an additional 55n3 + 2n + 10 steps before it terminates. Thus the overall time complexity of the algorithm can be expressed as T(n) = 55n3 + O(n2). Here the terms 2n + 10 are subsumed within the faster-growing O(n2). Again, this usage disregards some of the formal meaning of the "=" symbol, but it does allow one to use the big O notation as a kind of convenient placeholder. === Multiple uses === In more complicated usage, O(·) can appear in different places in an equation, even several times on each side. For example, the following are true for n → ∞ {\displaystyle n\to \infty } : ( n + 1 ) 2 = n 2 + O ( n ) , ( n + O ( n 1 / 2 ) ) ⋅ ( n + O ( log n ) ) 2 = n 3 + O ( n 5 / 2 ) , n O ( 1 ) = O ( e n ) . {\displaystyle {\begin{aligned}(n+1)^{2}&=n^{2}+O(n),\\(n+O(n^{1/2}))\cdot (n+O(\log n))^{2}&=n^{3}+O(n^{5/2}),\\n^{O(1)}&=O(e^{n}).\end{aligned}}} The meaning of such statements is as follows: for any functions which satisfy each O(·) on the left side, there are some functions satisfying each O(·) on the right side, such that substituting all these functions into the equation makes the two sides equal. For example, the third equation above means: "For any function f(n) = O(1), there is some function g(n) = O(en) such that nf(n) = g(n)". In terms of the "set notation" above, the meaning is that the class of functions represented by the left side is a subset of the class of functions represented by the right side. In this use the "=" is a formal symbol that unlike the usual use of "=" is not a symmetric relation. Thus for example nO(1) = O(en) does not imply the false statement O(en) = nO(1). === Typesetting === Big O is typeset as an italicized uppercase "O", as in the following example: O ( n 2 ) {\displaystyle O(n^{2})} . In TeX, it is produced by simply typing 'O' inside math mode. Unlike Greek-named Bachmann–Landau notations, it needs no special symbol. However, some authors use the calligraphic variant O {\displaystyle {\mathcal {O}}} instead. == Orders of common functions == Here is a list of classes of functions that are commonly encountered when analyzing the running time of an algorithm. In each case, c is a positive constant and n increases without bound. The slower-growing functions are generally listed first. The statement f ( n ) = O ( n ! ) {\displaystyle f(n)=O(n!)} is sometimes weakened to f ( n ) = O ( n n ) {\displaystyle f(n)=O\left(n^{n}\right)} to derive simpler formulas for asymptotic complexity. For any k > 0 {\displaystyle k>0} and c > 0 {\displaystyle c>0} , O ( n c ( log n ) k ) {\displaystyle O(n^{c}(\log n)^{k})} is a subset of O ( n c + ε ) {\displaystyle O(n^{c+\varepsilon })} for any ε > 0 {\displaystyle \varepsilon >0} , so may be considered as a polynomial with some bigger order. == Related asymptotic notations == Big O is widely used in computer science. Together with some other related notations, it forms the family of Bachmann–Landau notations. === Little-o notation === Intuitively, the assertion "f(x) is o(g(x))" (read "f(x) is little-o of g(x)" or "f(x) is of inferior order to g(x)") means that g(x) grows much faster than f(x), or equivalently f(x) grows much slower than g(x). As before, let f be a real or complex valued function and g a real valued function, both defined on some unbounded subset of the positive real numbers, such that g(x) is strictly positive for all large enough values of x. One writes f ( x ) = o ( g ( x ) ) as x → ∞ {\displaystyle f(x)=o(g(x))\quad {\text{ as }}x\to \infty } if for every positive constant ε there exists a constant x 0 {\displaystyle x_{0}} such that | f ( x ) | ≤ ε g ( x ) for all x ≥ x 0 . {\displaystyle |f(x)|\leq \varepsilon g(x)\quad {\text{ for all }}x\geq x_{0}.} For example, one has 2 x = o ( x 2 ) {\displaystyle 2x=o(x^{2})} and 1 / x = o ( 1 ) , {\displaystyle 1/x=o(1),} both as x → ∞ . {\displaystyle x\to \infty .} The difference between the definition of the big-O notation and the definition of little-o is that while the former has to be true for at least one constant M, the latter must hold for every positive constant ε, however small. In this way, little-o notation makes a stronger statement than the corresponding big-O notation: every function that is little-o of g is also big-O of g, but not every function that is big-O of g is little-o of g. For example, 2 x 2 = O ( x 2 ) {\displaystyle 2x^{2}=O(x^{2})} but 2 x 2 ≠ o ( x 2 ) {\displaystyle 2x^{2}\neq o(x^{2})} . If g(x) is nonzero, or at least becomes nonzero beyond a certain point, the relation f ( x ) = o ( g ( x ) ) {\displaystyle f(x)=o(g(x))} is equivalent to lim x → ∞ f ( x ) g ( x ) = 0 {\displaystyle \lim _{x\to \infty }{\frac {f(x)}{g(x)}}=0} (and this is in fact how Landau originally defined the little-o notation). Little-o respects a number of arithmetic operations. For example, if c is a nonzero constant and f = o ( g ) {\displaystyle f=o(g)} then c ⋅ f = o ( g ) {\displaystyle c\cdot f=o(g)} , and if f = o ( F ) {\displaystyle f=o(F)} and g = o ( G ) {\displaystyle g=o(G)} then f ⋅ g = o ( F ⋅ G ) . {\displaystyle f\cdot g=o(F\cdot G).} if f = o ( F ) {\displaystyle f=o(F)} and g = o ( G ) {\displaystyle g=o(G)} then f + g = o ( F + G ) {\displaystyle f+g=o(F+G)} It also satisfies a transitivity relation: if f = o ( g ) {\displaystyle f=o(g)} and g = o ( h ) {\displaystyle g=o(h)} then f = o ( h ) . {\displaystyle f=o(h).} Little-o can also be generalized to the finite case: f ( x ) = o ( g ( x ) ) as x → x 0 {\displaystyle f(x)=o(g(x))\quad {\text{ as }}x\to x_{0}} if f ( x ) = α ( x ) g ( x ) {\displaystyle f(x)=\alpha (x)g(x)} for some α ( x ) {\displaystyle \alpha (x)} with lim x → x 0 α ( x ) = 0 {\displaystyle \lim _{x\to x_{0}}\alpha (x)=0} . Or, if g(x) is nonzero in a neighbourhood around x 0 {\displaystyle x_{0}} : f ( x ) = o ( g ( x ) ) as x → x 0 {\displaystyle f(x)=o(g(x))\quad {\text{ as }}x\to x_{0}} if lim x → x 0 f ( x ) g ( x ) = 0 {\displaystyle \lim _{x\to x_{0}}{\frac {f(x)}{g(x)}}=0} . This definition especially useful in the computation of limits using Taylor series. For example: sin x = x − x 3 3 ! + … = x + o ( x 2 ) as x → 0 {\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+\ldots =x+o(x^{2}){\text{ as }}x\to 0} , so lim x → 0 sin x x = lim x → 0 x + o ( x 2 ) x = lim x → 0 1 + o ( x ) = 1 {\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}=\lim _{x\to 0}{\frac {x+o(x^{2})}{x}}=\lim _{x\to 0}1+o(x)=1} === Big Omega notation === Another asymptotic notation is Ω {\displaystyle \Omega } , read "big omega". There are two widespread and incompatible definitions of the statement f ( x ) = Ω ( g ( x ) ) {\displaystyle f(x)=\Omega (g(x))} as x → a {\displaystyle x\to a} , where a is some real number, ∞ {\displaystyle \infty } , or − ∞ {\displaystyle -\infty } , where f and g are real functions defined in a neighbourhood of a, and where g is positive in this neighbourhood. The Hardy–Littlewood definition is used mainly in analytic number theory, and the Knuth definition mainly in computational complexity theory; the definitions are not equivalent. ==== The Hardy–Littlewood definition ==== In 1914 G.H. Hardy and J.E. Littlewood introduced the new symbol Ω , {\displaystyle \ \Omega \ ,} which is defined as follows: f ( x ) = Ω ( g ( x ) ) {\displaystyle f(x)=\Omega {\bigl (}\ g(x)\ {\bigr )}\quad } as x → ∞ {\displaystyle \quad x\to \infty \quad } if lim sup x → ∞ | f ( x ) g ( x ) | > 0 . {\displaystyle \quad \limsup _{x\to \infty }\ \left|{\frac {\ f(x)\ }{g(x)}}\right|>0~.} Thus f ( x ) = Ω ( g ( x ) ) {\displaystyle ~f(x)=\Omega {\bigl (}\ g(x)\ {\bigr )}~} is the negation of f ( x ) = o ( g ( x ) ) . {\displaystyle ~f(x)=o{\bigl (}\ g(x)\ {\bigr )}~.} In 1916 the same authors introduced the two new symbols Ω R {\displaystyle \ \Omega _{R}\ } and Ω L , {\displaystyle \ \Omega _{L}\ ,} defined as: f ( x ) = Ω R ( g ( x ) ) {\displaystyle f(x)=\Omega _{R}{\bigl (}\ g(x)\ {\bigr )}\quad } as x → ∞ {\displaystyle \quad x\to \infty \quad } if lim sup x → ∞ f ( x ) g ( x ) > 0 ; {\displaystyle \quad \limsup _{x\to \infty }\ {\frac {\ f(x)\ }{g(x)}}>0\ ;} f ( x ) = Ω L ( g ( x ) ) {\displaystyle f(x)=\Omega _{L}{\bigl (}\ g(x)\ {\bigr )}\quad } as x → ∞ {\displaystyle \quad x\to \infty \quad } if lim inf x → ∞ f ( x ) g ( x ) < 0 . {\displaystyle \quad ~\liminf _{x\to \infty }\ {\frac {\ f(x)\ }{g(x)}}<0~.} These symbols were used by E. Landau, with the same meanings, in 1924. Authors that followed Landau, however, use a different notation for the same definitions: The symbol Ω R {\displaystyle \ \Omega _{R}\ } has been replaced by the current notation Ω + {\displaystyle \ \Omega _{+}\ } with the same definition, and Ω L {\displaystyle \ \Omega _{L}\ } became Ω − . {\displaystyle \ \Omega _{-}~.} These three symbols Ω , Ω + , Ω − , {\displaystyle \ \Omega \ ,\Omega _{+}\ ,\Omega _{-}\ ,} as well as f ( x ) = Ω ± ( g ( x ) ) {\displaystyle \ f(x)=\Omega _{\pm }{\bigl (}\ g(x)\ {\bigr )}\ } (meaning that f ( x ) = Ω + ( g ( x ) ) {\displaystyle \ f(x)=\Omega _{+}{\bigl (}\ g(x)\ {\bigr )}\ } and f ( x ) = Ω − ( g ( x ) ) {\displaystyle \ f(x)=\Omega _{-}{\bigl (}\ g(x)\ {\bigr )}\ } are both satisfied), are now currently used in analytic number theory. ===== Simple examples ===== We have sin x = Ω ( 1 ) {\displaystyle \sin x=\Omega (1)\quad } as x → ∞ , {\displaystyle \quad x\to \infty \ ,} and more precisely sin x = Ω ± ( 1 ) {\displaystyle \sin x=\Omega _{\pm }(1)\quad } as x → ∞ . {\displaystyle \quad x\to \infty ~.} We have 1 + sin x = Ω ( 1 ) {\displaystyle 1+\sin x=\Omega (1)\quad } as x → ∞ , {\displaystyle \quad x\to \infty \ ,} and more precisely 1 + sin x = Ω + ( 1 ) {\displaystyle 1+\sin x=\Omega _{+}(1)\quad } as x → ∞ ; {\displaystyle \quad x\to \infty \ ;} however 1 + sin x ≠ Ω − ( 1 ) {\displaystyle 1+\sin x\neq \Omega _{-}(1)\quad } as x → ∞ . {\displaystyle \quad x\to \infty ~.} ==== The Knuth definition ==== In 1976 Donald Knuth published a paper to justify his use of the Ω {\displaystyle \Omega } -symbol to describe a stronger property. Knuth wrote: "For all the applications I have seen so far in computer science, a stronger requirement ... is much more appropriate". He defined f ( x ) = Ω ( g ( x ) ) ⇔ g ( x ) = O ( f ( x ) ) {\displaystyle f(x)=\Omega (g(x))\Leftrightarrow g(x)=O(f(x))} with the comment: "Although I have changed Hardy and Littlewood's definition of Ω {\displaystyle \Omega } , I feel justified in doing so because their definition is by no means in wide use, and because there are other ways to say what they want to say in the comparatively rare cases when their definition applies." === Family of Bachmann–Landau notations === The limit definitions assume g ( n ) > 0 {\displaystyle g(n)>0} for sufficiently large n {\displaystyle n} . The table is (partly) sorted from smallest to largest, in the sense that o , O , Θ , ∼ , {\displaystyle o,O,\Theta ,\sim ,} (Knuth's version of) Ω , ω {\displaystyle \Omega ,\omega } on functions correspond to < , ≤ , ≈ , = , {\displaystyle <,\leq ,\approx ,=,} ≥ , > {\displaystyle \geq ,>} on the real line (the Hardy–Littlewood version of Ω {\displaystyle \Omega } , however, doesn't correspond to any such description). Computer science uses the big O {\displaystyle O} , big Theta Θ {\displaystyle \Theta } , little o {\displaystyle o} , little omega ω {\displaystyle \omega } and Knuth's big Omega Ω {\displaystyle \Omega } notations. Analytic number theory often uses the big O {\displaystyle O} , small o {\displaystyle o} , Hardy's ≍ {\displaystyle \asymp } , Hardy–Littlewood's big Omega Ω {\displaystyle \Omega } (with or without the +, − or ± subscripts) and ∼ {\displaystyle \sim } notations. The small omega ω {\displaystyle \omega } notation is not used as often in analysis. === Use in computer science === Informally, especially in computer science, the big O notation often can be used somewhat differently to describe an asymptotic tight bound where using big Theta Θ notation might be more factually appropriate in a given context. For example, when considering a function T(n) = 73n3 + 22n2 + 58, all of the following are generally acceptable, but tighter bounds (such as numbers 2 and 3 below) are usually strongly preferred over looser bounds (such as number 1 below). T(n) = O(n100) T(n) = O(n3) T(n) = Θ(n3) The equivalent English statements are respectively: T(n) grows asymptotically no faster than n100 T(n) grows asymptotically no faster than n3 T(n) grows asymptotically as fast as n3. So while all three statements are true, progressively more information is contained in each. In some fields, however, the big O notation (number 2 in the lists above) would be used more commonly than the big Theta notation (items numbered 3 in the lists above). For example, if T(n) represents the running time of a newly developed algorithm for input size n, the inventors and users of the algorithm might be more inclined to put an upper asymptotic bound on how long it will take to run without making an explicit statement about the lower asymptotic bound. === Other notation === In their book Introduction to Algorithms, Cormen, Leiserson, Rivest and Stein consider the set of functions f which satisfy f ( n ) = O ( g ( n ) ) ( n → ∞ ) . {\displaystyle f(n)=O(g(n))\quad (n\to \infty )~.} In a correct notation this set can, for instance, be called O(g), where O ( g ) = { f : there exist positive constants c and n 0 such that 0 ≤ f ( n ) ≤ c g ( n ) for all n ≥ n 0 } . {\displaystyle O(g)=\{f:{\text{there exist positive constants}}~c~{\text{and}}~n_{0}~{\text{such that}}~0\leq f(n)\leq cg(n){\text{ for all }}n\geq n_{0}\}.} The authors state that the use of equality operator (=) to denote set membership rather than the set membership operator (∈) is an abuse of notation, but that doing so has advantages. Inside an equation or inequality, the use of asymptotic notation stands for an anonymous function in the set O(g), which eliminates lower-order terms, and helps to reduce inessential clutter in equations, for example: 2 n 2 + 3 n + 1 = 2 n 2 + O ( n ) . {\displaystyle 2n^{2}+3n+1=2n^{2}+O(n).} === Extensions to the Bachmann–Landau notations === Another notation sometimes used in computer science is Õ (read soft-O), which hides polylogarithmic factors. There are two definitions in use: some authors use f(n) = Õ(g(n)) as shorthand for f(n) = O(g(n) logk n) for some k, while others use it as shorthand for f(n) = O(g(n) logk g(n)). When g(n) is polynomial in n, there is no difference; however, the latter definition allows one to say, e.g. that n 2 n = O ~ ( 2 n ) {\displaystyle n2^{n}={\tilde {O}}(2^{n})} while the former definition allows for log k n = O ~ ( 1 ) {\displaystyle \log ^{k}n={\tilde {O}}(1)} for any constant k. Some authors write O* for the same purpose as the latter definition. Essentially, it is big O notation, ignoring logarithmic factors because the growth-rate effects of some other super-logarithmic function indicate a growth-rate explosion for large-sized input parameters that is more important to predicting bad run-time performance than the finer-point effects contributed by the logarithmic-growth factor(s). This notation is often used to obviate the "nitpicking" within growth-rates that are stated as too tightly bounded for the matters at hand (since logk n is always o(nε) for any constant k and any ε > 0). Also, the L notation, defined as L n [ α , c ] = e ( c + o ( 1 ) ) ( ln n ) α ( ln ln n ) 1 − α , {\displaystyle L_{n}[\alpha ,c]=e^{(c+o(1))(\ln n)^{\alpha }(\ln \ln n)^{1-\alpha }},} is convenient for functions that are between polynomial and exponential in terms of ln n {\displaystyle \ln n} . == Generalizations and related usages == The generalization to functions taking values in any normed vector space is straightforward (replacing absolute values by norms), where f and g need not take their values in the same space. A generalization to functions g taking values in any topological group is also possible. The "limiting process" x → xo can also be generalized by introducing an arbitrary filter base, i.e. to directed nets f and g. The o notation can be used to define derivatives and differentiability in quite general spaces, and also (asymptotical) equivalence of functions, f ∼ g ⟺ ( f − g ) ∈ o ( g ) {\displaystyle f\sim g\iff (f-g)\in o(g)} which is an equivalence relation and a more restrictive notion than the relationship "f is Θ(g)" from above. (It reduces to lim f / g = 1 if f and g are positive real valued functions.) For example, 2x is Θ(x), but 2x − x is not o(x). == History (Bachmann–Landau, Hardy, and Vinogradov notations) == The symbol O was first introduced by number theorist Paul Bachmann in 1894, in the second volume of his book Analytische Zahlentheorie ("analytic number theory"). The number theorist Edmund Landau adopted it, and was thus inspired to introduce in 1909 the notation o; hence both are now called Landau symbols. These notations were used in applied mathematics during the 1950s for asymptotic analysis. The symbol Ω {\displaystyle \Omega } (in the sense "is not an o of") was introduced in 1914 by Hardy and Littlewood. Hardy and Littlewood also introduced in 1916 the symbols Ω R {\displaystyle \Omega _{R}} ("right") and Ω L {\displaystyle \Omega _{L}} ("left"), precursors of the modern symbols Ω + {\displaystyle \Omega _{+}} ("is not smaller than a small o of") and Ω − {\displaystyle \Omega _{-}} ("is not larger than a small o of"). Thus the Omega symbols (with their original meanings) are sometimes also referred to as "Landau symbols". This notation Ω {\displaystyle \Omega } became commonly used in number theory at least since the 1950s. The symbol ∼ {\displaystyle \sim } , although it had been used before with different meanings, was given its modern definition by Landau in 1909 and by Hardy in 1910. Just above on the same page of his tract Hardy defined the symbol ≍ {\displaystyle \asymp } , where f ( x ) ≍ g ( x ) {\displaystyle f(x)\asymp g(x)} means that both f ( x ) = O ( g ( x ) ) {\displaystyle f(x)=O(g(x))} and g ( x ) = O ( f ( x ) ) {\displaystyle g(x)=O(f(x))} are satisfied. The notation is still currently used in analytic number theory. In his tract Hardy also proposed the symbol ≍ − {\displaystyle \mathbin {\,\asymp \;\;\;\;\!\!\!\!\!\!\!\!\!\!\!\!\!-} } , where f ≍ − g {\displaystyle f\mathbin {\,\asymp \;\;\;\;\!\!\!\!\!\!\!\!\!\!\!\!\!-} g} means that f ∼ K g {\displaystyle f\sim Kg} for some constant K ≠ 0 {\displaystyle K\not =0} . In the 1970s the big O was popularized in computer science by Donald Knuth, who proposed the different notation f ( x ) = Θ ( g ( x ) ) {\displaystyle f(x)=\Theta (g(x))} for Hardy's f ( x ) ≍ g ( x ) {\displaystyle f(x)\asymp g(x)} , and proposed a different definition for the Hardy and Littlewood Omega notation. Two other symbols coined by Hardy were (in terms of the modern O notation) f ≼ g ⟺ f = O ( g ) {\displaystyle f\preccurlyeq g\iff f=O(g)} and f ≺ g ⟺ f = o ( g ) ; {\displaystyle f\prec g\iff f=o(g);} (Hardy however never defined or used the notation ≺ ≺ {\displaystyle \prec \!\!\prec } , nor ≪ {\displaystyle \ll } , as it has been sometimes reported). Hardy introduced the symbols ≼ {\displaystyle \preccurlyeq } and ≺ {\displaystyle \prec } (as well as the already mentioned other symbols) in his 1910 tract "Orders of Infinity", and made use of them only in three papers (1910–1913). In his nearly 400 remaining papers and books he consistently used the Landau symbols O and o. Hardy's symbols ≼ {\displaystyle \preccurlyeq } and ≺ {\displaystyle \prec } (as well as ≍ − {\displaystyle \mathbin {\,\asymp \;\;\;\;\!\!\!\!\!\!\!\!\!\!\!\!\!-} } ) are not used anymore. On the other hand, in the 1930s, the Russian number theorist Ivan Matveyevich Vinogradov introduced his notation ≪ {\displaystyle \ll } , which has been increasingly used in number theory instead of the O {\displaystyle O} notation. We have f ≪ g ⟺ f = O ( g ) , {\displaystyle f\ll g\iff f=O(g),} and frequently both notations are used in the same paper. The big-O originally stands for "order of" ("Ordnung", Bachmann 1894), and is thus a Latin letter. Neither Bachmann nor Landau ever call it "Omicron". The symbol was much later on (1976) viewed by Knuth as a capital omicron, probably in reference to his definition of the symbol Omega. The digit zero should not be used. == See also == Asymptotic computational complexity Asymptotic expansion: Approximation of functions generalizing Taylor's formula Asymptotically optimal algorithm: A phrase frequently used to describe an algorithm that has an upper bound asymptotically within a constant of a lower bound for the problem Big O in probability notation: Op, op Limit inferior and limit superior: An explanation of some of the limit notation used in this article Master theorem (analysis of algorithms): For analyzing divide-and-conquer recursive algorithms using big O notation Nachbin's theorem: A precise method of bounding complex analytic functions so that the domain of convergence of integral transforms can be stated Order of approximation Order of accuracy Computational complexity of mathematical operations == References and notes == === Notes === == Further reading == Hardy, G. H. (1910). Orders of Infinity: The 'Infinitärcalcül' of Paul du Bois-Reymond. Cambridge University Press. Knuth, Donald (1997). "1.2.11: Asymptotic Representations". Fundamental Algorithms. The Art of Computer Programming. Vol. 1 (3rd ed.). Addison-Wesley. ISBN 978-0-201-89683-1. Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001). "3.1: Asymptotic notation". Introduction to Algorithms (2nd ed.). MIT Press and McGraw-Hill. ISBN 978-0-262-03293-3. Sipser, Michael (1997). Introduction to the Theory of Computation. PWS Publishing. pp. 226–228. ISBN 978-0-534-94728-6. Avigad, Jeremy; Donnelly, Kevin (2004). Formalizing O notation in Isabelle/HOL (PDF). International Joint Conference on Automated Reasoning. doi:10.1007/978-3-540-25984-8_27. Black, Paul E. (11 March 2005). Black, Paul E. (ed.). "big-O notation". Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved December 16, 2006. Black, Paul E. (17 December 2004). Black, Paul E. (ed.). "little-o notation". Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved December 16, 2006. Black, Paul E. (17 December 2004). Black, Paul E. (ed.). "Ω". Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved December 16, 2006. Black, Paul E. (17 December 2004). Black, Paul E. (ed.). "ω". Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved December 16, 2006. Black, Paul E. (17 December 2004). Black, Paul E. (ed.). "Θ". Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved December 16, 2006. == External links == Growth of sequences — OEIS (Online Encyclopedia of Integer Sequences) Wiki Introduction to Asymptotic Notations Big-O Notation – What is it good for An example of Big O in accuracy of central divided difference scheme for first derivative A Gentle Introduction to Algorithm Complexity Analysis
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Wikipedia:Biholomorphism#0
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In the mathematical theory of functions of one or more complex variables, and also in complex algebraic geometry, a biholomorphism or biholomorphic function is a bijective holomorphic function whose inverse is also holomorphic. == Formal definition == Formally, a biholomorphic function is a function ϕ {\displaystyle \phi } defined on an open subset U of the n {\displaystyle n} -dimensional complex space Cn with values in Cn which is holomorphic and one-to-one, such that its image is an open set V {\displaystyle V} in Cn and the inverse ϕ − 1 : V → U {\displaystyle \phi ^{-1}:V\to U} is also holomorphic. More generally, U and V can be complex manifolds. As in the case of functions of a single complex variable, a sufficient condition for a holomorphic map to be biholomorphic onto its image is that the map is injective, in which case the inverse is also holomorphic (e.g., see Gunning 1990, Theorem I.11 or Corollary E.10 pg. 57). If there exists a biholomorphism ϕ : U → V {\displaystyle \phi \colon U\to V} , we say that U and V are biholomorphically equivalent or that they are biholomorphic. == Riemann mapping theorem and generalizations == If n = 1 , {\displaystyle n=1,} every simply connected open set other than the whole complex plane is biholomorphic to the unit disc (this is the Riemann mapping theorem). The situation is very different in higher dimensions. For example, open unit balls and open unit polydiscs are not biholomorphically equivalent for n > 1. {\displaystyle n>1.} In fact, there does not exist even a proper holomorphic function from one to the other. == Alternative definitions == In the case of maps f : U → C defined on an open subset U of the complex plane C, some authors (e.g., Freitag 2009, Definition IV.4.1) define a conformal map to be an injective map with nonzero derivative i.e., f’(z)≠ 0 for every z in U. According to this definition, a map f : U → C is conformal if and only if f: U → f(U) is biholomorphic. Notice that per definition of biholomorphisms, nothing is assumed about their derivatives, so, this equivalence contains the claim that a homeomorphism that is complex differentiable must actually have nonzero derivative everywhere. Other authors (e.g., Conway 1978) define a conformal map as one with nonzero derivative, but without requiring that the map be injective. According to this weaker definition, a conformal map need not be biholomorphic, even though it is locally biholomorphic, for example, by the inverse function theorem. For example, if f: U → U is defined by f(z) = z2 with U = C–{0}, then f is conformal on U, since its derivative f’(z) = 2z ≠ 0, but it is not biholomorphic, since it is 2-1. == References == Conway, John B. (1978). Functions of One Complex Variable. Springer-Verlag. ISBN 3-540-90328-3. D'Angelo, John P. (1993). Several Complex Variables and the Geometry of Real Hypersurfaces. CRC Press. ISBN 0-8493-8272-6. Freitag, Eberhard; Busam, Rolf (2009). Complex Analysis. Springer-Verlag. ISBN 978-3-540-93982-5. Gunning, Robert C. (1990). Introduction to Holomorphic Functions of Several Variables, Vol. II. Wadsworth. ISBN 0-534-13309-6. Krantz, Steven G. (2002). Function Theory of Several Complex Variables. American Mathematical Society. ISBN 0-8218-2724-3. This article incorporates material from biholomorphically equivalent on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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Wikipedia:Bijaganita#0
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Bhāskara II ([bʰɑːskərə]; c.1114–1185), also known as Bhāskarāchārya (lit. 'Bhāskara the teacher'), was an Indian polymath, mathematician, astronomer and engineer. From verses in his main work, Siddhānta Śiromaṇi, it can be inferred that he was born in 1114 in Vijjadavida (Vijjalavida) and living in the Satpura mountain ranges of Western Ghats, believed to be the town of Patana in Chalisgaon, located in present-day Khandesh region of Maharashtra by scholars. In a temple in Maharashtra, an inscription supposedly created by his grandson Changadeva, lists Bhaskaracharya's ancestral lineage for several generations before him as well as two generations after him. Henry Colebrooke who was the first European to translate (1817) Bhaskaracharya II's mathematical classics refers to the family as Maharashtrian Brahmins residing on the banks of the Godavari. Born in a Hindu Deshastha Brahmin family of scholars, mathematicians and astronomers, Bhaskara II was the leader of a cosmic observatory at Ujjain, the main mathematical centre of ancient India. Bhāskara and his works represent a significant contribution to mathematical and astronomical knowledge in the 12th century. He has been called the greatest mathematician of medieval India. His main work, Siddhānta-Śiromaṇi (Sanskrit for "Crown of Treatises"), is divided into four parts called Līlāvatī, Bījagaṇita, Grahagaṇita and Golādhyāya, which are also sometimes considered four independent works. These four sections deal with arithmetic, algebra, mathematics of the planets, and spheres respectively. He also wrote another treatise named Karaṇā Kautūhala. == Date, place and family == Bhāskara gives his date of birth, and date of composition of his major work, in a verse in the Āryā metre: This reveals that he was born in 1036 of the Shaka era (1114 CE), and that he composed the Siddhānta Shiromani when he was 36 years old. Siddhānta Shiromani was completed during 1150 CE. He also wrote another work called the Karaṇa-kutūhala when he was 69 (in 1183). His works show the influence of Brahmagupta, Śrīdhara, Mahāvīra, Padmanābha and other predecessors. Bhaskara lived in Patnadevi located near Patan (Chalisgaon) in the vicinity of Sahyadri. He was born in a Deśastha Rigvedi Brahmin family near Vijjadavida (Vijjalavida). Munishvara (17th century), a commentator on Siddhānta Shiromani of Bhaskara has given the information about the location of Vijjadavida in his work Marīci Tīkā as follows: सह्यकुलपर्वतान्तर्गत भूप्रदेशे महाराष्ट्रदेशान्तर्गतविदर्भपरपर्यायविराटदेशादपि निकटे गोदावर्यां नातिदूरे पंचक्रोशान्तरे विज्जलविडम्। This description locates Vijjalavida in Maharashtra, near the Vidarbha region and close to the banks of Godavari river. However scholars differ about the exact location. Many scholars have placed the place near Patan in Chalisgaon Taluka of Jalgaon district whereas a section of scholars identified it with the modern day Beed city. Some sources identified Vijjalavida as Bijapur or Bidar in Karnataka. Identification of Vijjalavida with Basar in Telangana has also been suggested. Bhāskara is said to have been the head of an astronomical observatory at Ujjain, the leading mathematical centre of medieval India. History records his great-great-great-grandfather holding a hereditary post as a court scholar, as did his son and other descendants. His father Maheśvara (Maheśvaropādhyāya) was a mathematician, astronomer and astrologer, who taught him mathematics, which he later passed on to his son Lokasamudra. Lokasamudra's son helped to set up a school in 1207 for the study of Bhāskara's writings. He died in 1185 CE. == The Siddhānta-Śiromaṇi == === Līlāvatī === The first section Līlāvatī (also known as pāṭīgaṇita or aṅkagaṇita), named after his daughter, consists of 277 verses. It covers calculations, progressions, measurement, permutations, and other topics. === Bijaganita === The second section Bījagaṇita(Algebra) has 213 verses. It discusses zero, infinity, positive and negative numbers, and indeterminate equations including (the now called) Pell's equation, solving it using a kuṭṭaka method. In particular, he also solved the 61 x 2 + 1 = y 2 {\displaystyle 61x^{2}+1=y^{2}} case that was to elude Fermat and his European contemporaries centuries later === Grahaganita === In the third section Grahagaṇita, while treating the motion of planets, he considered their instantaneous speeds. He arrived at the approximation: It consists of 451 verses sin y ′ − sin y ≈ ( y ′ − y ) cos y {\displaystyle \sin y'-\sin y\approx (y'-y)\cos y} for. y ′ {\displaystyle y'} close to y {\displaystyle y} , or in modern notation: d d y sin y = cos y {\displaystyle {\frac {d}{dy}}\sin y=\cos y} . In his words: bimbārdhasya koṭijyā guṇastrijyāhāraḥ phalaṃ dorjyāyorantaram This result had also been observed earlier by Muñjalācārya (or Mañjulācārya) mānasam, in the context of a table of sines. Bhāskara also stated that at its highest point a planet's instantaneous speed is zero. == Mathematics == Some of Bhaskara's contributions to mathematics include the following: A proof of the Pythagorean theorem by calculating the same area in two different ways and then cancelling out terms to get a2 + b2 = c2. In Lilavati, solutions of quadratic, cubic and quartic indeterminate equations are explained. Solutions of indeterminate quadratic equations (of the type ax2 + b = y2). Integer solutions of linear and quadratic indeterminate equations (Kuṭṭaka). The rules he gives are (in effect) the same as those given by the Renaissance European mathematicians of the 17th century. A cyclic Chakravala method for solving indeterminate equations of the form ax2 + bx + c = y. The solution to this equation was traditionally attributed to William Brouncker in 1657, though his method was more difficult than the chakravala method. The first general method for finding the solutions of the problem x2 − ny2 = 1 (so-called "Pell's equation") was given by Bhaskara II. Solutions of Diophantine equations of the second order, such as 61x2 + 1 = y2. This very equation was posed as a problem in 1657 by the French mathematician Pierre de Fermat, but its solution was unknown in Europe until the time of Euler in the 18th century. Solved quadratic equations with more than one unknown, and found negative and irrational solutions. Preliminary concept of mathematical analysis. Preliminary concept of differential calculus, along with preliminary ideas towards integration. preliminary ideas of differential calculus and differential coefficient. Stated Rolle's theorem, a special case of one of the most important theorems in analysis, the mean value theorem. Traces of the general mean value theorem are also found in his works. Calculated the derivative of sine function, although he did not develop the notion of a derivative. (See Calculus section below.) In Siddhanta-Śiromaṇi, Bhaskara developed spherical trigonometry along with a number of other trigonometric results. (See Trigonometry section below.) === Arithmetic === Bhaskara's arithmetic text Līlāvatī covers the topics of definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, plane geometry, solid geometry, the shadow of the gnomon, methods to solve indeterminate equations, and combinations. Līlāvatī is divided into 13 chapters and covers many branches of mathematics, arithmetic, algebra, geometry, and a little trigonometry and measurement. More specifically the contents include: Definitions. Properties of zero (including division, and rules of operations with zero). Further extensive numerical work, including use of negative numbers and surds. Estimation of π. Arithmetical terms, methods of multiplication, and squaring. Inverse rule of three, and rules of 3, 5, 7, 9, and 11. Problems involving interest and interest computation. Indeterminate equations (Kuṭṭaka), integer solutions (first and second order). His contributions to this topic are particularly important, since the rules he gives are (in effect) the same as those given by the renaissance European mathematicians of the 17th century, yet his work was of the 12th century. Bhaskara's method of solving was an improvement of the methods found in the work of Aryabhata and subsequent mathematicians. His work is outstanding for its systematisation, improved methods and the new topics that he introduced. Furthermore, the Lilavati contained excellent problems and it is thought that Bhaskara's intention may have been that a student of 'Lilavati' should concern himself with the mechanical application of the method. === Algebra === His Bījaganita ("Algebra") was a work in twelve chapters. It was the first text to recognize that a positive number has two square roots (a positive and negative square root). His work Bījaganita is effectively a treatise on algebra and contains the following topics: Positive and negative numbers. The 'unknown' (includes determining unknown quantities). Determining unknown quantities. Surds (includes evaluating surds and their square roots). Kuṭṭaka (for solving indeterminate equations and Diophantine equations). Simple equations (indeterminate of second, third and fourth degree). Simple equations with more than one unknown. Indeterminate quadratic equations (of the type ax2 + b = y2). Solutions of indeterminate equations of the second, third and fourth degree. Quadratic equations. Quadratic equations with more than one unknown. Operations with products of several unknowns. Bhaskara derived a cyclic, chakravala method for solving indeterminate quadratic equations of the form ax2 + bx + c = y. Bhaskara's method for finding the solutions of the problem Nx2 + 1 = y2 (the so-called "Pell's equation") is of considerable importance. === Trigonometry === The Siddhānta Shiromani (written in 1150) demonstrates Bhaskara's knowledge of trigonometry, including the sine table and relationships between different trigonometric functions. He also developed spherical trigonometry, along with other interesting trigonometrical results. In particular Bhaskara seemed more interested in trigonometry for its own sake than his predecessors who saw it only as a tool for calculation. Among the many interesting results given by Bhaskara, results found in his works include computation of sines of angles of 18 and 36 degrees, and the now well known formulae for sin ( a + b ) {\displaystyle \sin \left(a+b\right)} and sin ( a − b ) {\displaystyle \sin \left(a-b\right)} . === Calculus === His work, the Siddhānta Shiromani, is an astronomical treatise and contains many theories not found in earlier works. Preliminary concepts of Differential calculus and mathematical analysis, along with a number of results in trigonometry that are found in the work are of particular interest. Evidence suggests Bhaskara was acquainted with some ideas of differential calculus. Bhaskara also goes deeper into the 'differential calculus' and suggests the differential coefficient vanishes at an extremum value of the function, indicating knowledge of the concept of 'infinitesimals'. There is evidence of an early form of Rolle's theorem in his work,though it was stated without a modern formal proof . In this astronomical work he gave one procedure that looks like a precursor to infinitesimal methods. In terms that is if x ≈ y {\displaystyle x\approx y} then sin ( y ) − sin ( x ) ≈ ( y − x ) cos ( y ) {\displaystyle \sin(y)-\sin(x)\approx (y-x)\cos(y)} that is a derivative of sine although he did not develop the notion on derivative. Bhaskara uses this result to work out the position angle of the ecliptic, a quantity required for accurately predicting the time of an eclipse. In computing the instantaneous motion of a planet, the time interval between successive positions of the planets was no greater than a truti, or a 1⁄33750 of a second, and his measure of velocity was expressed in this infinitesimal unit of time. He was aware that when a variable attains the maximum value, its differential vanishes. He also showed that when a planet is at its farthest from the earth, or at its closest, the equation of the centre (measure of how far a planet is from the position in which it is predicted to be, by assuming it is to move uniformly) vanishes. He therefore concluded that for some intermediate position the differential of the equation of the centre is equal to zero. In this result, there are traces of the general mean value theorem, one of the most important theorems in analysis, which today is usually derived from Rolle's theorem. The mean value formula for inverse interpolation of the sine was later founded by Parameshvara in the 15th century in the Lilavati Bhasya, a commentary on Bhaskara's Lilavati. Madhava (1340–1425) and the Kerala School mathematicians (including Parameshvara) from the 14th century to the 16th century expanded on Bhaskara's work. == Astronomy == Using an astronomical model developed by Brahmagupta in the 7th century, Bhāskara accurately defined many astronomical quantities, including, for example, the length of the sidereal year, the time that is required for the Sun to orbit the Earth, as approximately 365.2588 days which is the same as in Surya siddhanta. The modern accepted measurement is 365.25636 days, a difference of 3.5 minutes. His mathematical astronomy text Siddhanta Shiromani is written in two parts: the first part on mathematical astronomy and the second part on the sphere. The twelve chapters of the first part cover topics such as: Mean longitudes of the planets. True longitudes of the planets. The three problems of diurnal rotation. Diurnal motion refers to the apparent daily motion of stars around the Earth, or more precisely around the two celestial poles. It is caused by the Earth's rotation on its axis, so every star apparently moves on a circle that is called the diurnal circle. Syzygies. Lunar eclipses. Solar eclipses. Latitudes of the planets. Sunrise equation. The Moon's crescent. Conjunctions of the planets with each other. Conjunctions of the planets with the fixed stars. The paths of the Sun and Moon. The second part contains thirteen chapters on the sphere. It covers topics such as: Praise of study of the sphere. Nature of the sphere. Cosmography and geography. Planetary mean motion. Eccentric epicyclic model of the planets. The armillary sphere. Spherical trigonometry. Ellipse calculations. First visibilities of the planets. Calculating the lunar crescent. Astronomical instruments. The seasons. Problems of astronomical calculations. == Engineering == The earliest reference to a perpetual motion machine date back to 1150, when Bhāskara II described a wheel that he claimed would run forever. Bhāskara II invented a variety of instruments one of which is Yaṣṭi-yantra. This device could vary from a simple stick to V-shaped staffs designed specifically for determining angles with the help of a calibrated scale. == Legends == In his book Lilavati, he reasons: "In this quantity also which has zero as its divisor there is no change even when many quantities have entered into it or come out [of it], just as at the time of destruction and creation when throngs of creatures enter into and come out of [him, there is no change in] the infinite and unchanging [Vishnu]". === "Behold!" === It has been stated, by several authors, that Bhaskara II proved the Pythagorean theorem by drawing a diagram and providing the single word "Behold!". Sometimes Bhaskara's name is omitted and this is referred to as the Hindu proof, well known by schoolchildren. However, as mathematics historian Kim Plofker points out, after presenting a worked-out example, Bhaskara II states the Pythagorean theorem: Hence, for the sake of brevity, the square root of the sum of the squares of the arm and upright is the hypotenuse: thus it is demonstrated. This is followed by: And otherwise, when one has set down those parts of the figure there [merely] seeing [it is sufficient]. Plofker suggests that this additional statement may be the ultimate source of the widespread "Behold!" legend. == Legacy == A number of institutes and colleges in India are named after him, including Bhaskaracharya Pratishthana in Pune, Bhaskaracharya College of Applied Sciences in Delhi, Bhaskaracharya Institute For Space Applications and Geo-Informatics in Gandhinagar. On 20 November 1981 the Indian Space Research Organisation (ISRO) launched the Bhaskara II satellite honouring the mathematician and astronomer. Invis Multimedia released Bhaskaracharya, an Indian documentary short on the mathematician in 2015. == See also == List of Indian mathematicians Bride's Chair Bījapallava == Notes == == References == === Bibliography === Burton, David M. (2011), The History of Mathematics: An Introduction (7th ed.), McGraw Hill, ISBN 978-0-07-338315-6 Eves, Howard (1990), An Introduction to the History of Mathematics (6th ed.), Saunders College Publishing, ISBN 978-0-03-029558-4 Mazur, Joseph (2005), Euclid in the Rainforest, Plume, ISBN 978-0-452-28783-9 Sarkār, Benoy Kumar (1918), Hindu achievements in exact science: a study in the history of scientific development, Longmans, Green and co. Seal, Sir Brajendranath (1915), The positive sciences of the ancient Hindus, Longmans, Green and co. Colebrooke, Henry T. (1817), Arithmetic and mensuration of Brahmegupta and Bhaskara White, Lynn Townsend (1978), "Tibet, India, and Malaya as Sources of Western Medieval Technology", Medieval religion and technology: collected essays, University of California Press, ISBN 978-0-520-03566-9 Selin, Helaine, ed. (2008), "Astronomical Instruments in India", Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (2nd edition), Springer Verlag Ny, ISBN 978-1-4020-4559-2 Shukla, Kripa Shankar (1984), "Use of Calculus in Hindu Mathematics", Indian Journal of History of Science, 19: 95–104 Pingree, David Edwin (1970), Census of the Exact Sciences in Sanskrit, vol. 146, American Philosophical Society, ISBN 9780871691460 Plofker, Kim (2007), "Mathematics in India", in Katz, Victor J. (ed.), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton University Press, ISBN 9780691114859 Plofker, Kim (2009), Mathematics in India, Princeton University Press, ISBN 9780691120676 Cooke, Roger (1997), "The Mathematics of the Hindus", The History of Mathematics: A Brief Course, Wiley-Interscience, pp. 213–215, ISBN 0-471-18082-3 Poulose, K. G. (1991), K. G. Poulose (ed.), Scientific heritage of India, mathematics, Ravivarma Samskr̥ta granthāvali, vol. 22, Govt. Sanskrit College (Tripunithura, India) Chopra, Pran Nath (1982), Religions and communities of India, Vision Books, ISBN 978-0-85692-081-3 Goonatilake, Susantha (1999), Toward a global science: mining civilizational knowledge, Indiana University Press, ISBN 978-0-253-21182-8 Selin, Helaine; D'Ambrosio, Ubiratan, eds. (2001), "Mathematics across cultures: the history of non-western mathematics", Science Across Cultures, 2, Springer, ISBN 978-1-4020-0260-1 Stillwell, John (2002), Mathematics and its history, Undergraduate Texts in Mathematics, Springer, ISBN 978-0-387-95336-6 Sahni, Madhu (2019), Pedagogy Of Mathematics, Vikas Publishing House, ISBN 978-9353383275 == Further reading == W. W. Rouse Ball. A Short Account of the History of Mathematics, 4th Edition. Dover Publications, 1960. George Gheverghese Joseph. The Crest of the Peacock: Non-European Roots of Mathematics, 2nd Edition. Penguin Books, 2000. O'Connor, John J.; Robertson, Edmund F., "Bhāskara II", MacTutor History of Mathematics Archive, University of St Andrews University of St Andrews, 2000. Ian Pearce. Bhaskaracharya II at the MacTutor archive. St Andrews University, 2002. Pingree, David (1970–1980). "Bhāskara II". Dictionary of Scientific Biography. Vol. 2. New York: Charles Scribner's Sons. pp. 115–120. ISBN 978-0-684-10114-9. == External links == 4to40 Biography
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Wikipedia:Bijection, injection and surjection#0
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In mathematics, injections, surjections, and bijections are classes of functions distinguished by the manner in which arguments (input expressions from the domain) and images (output expressions from the codomain) are related or mapped to each other. A function maps elements from its domain to elements in its codomain. Given a function f : X → Y {\displaystyle f\colon X\to Y} : The function is injective, or one-to-one, if each element of the codomain is mapped to by at most one element of the domain, or equivalently, if distinct elements of the domain map to distinct elements in the codomain. An injective function is also called an injection. Notationally: ∀ x , x ′ ∈ X , f ( x ) = f ( x ′ ) ⟹ x = x ′ , {\displaystyle \forall x,x'\in X,f(x)=f(x')\implies x=x',} or, equivalently (using logical transposition), ∀ x , x ′ ∈ X , x ≠ x ′ ⟹ f ( x ) ≠ f ( x ′ ) . {\displaystyle \forall x,x'\in X,x\neq x'\implies f(x)\neq f(x').} The function is surjective, or onto, if each element of the codomain is mapped to by at least one element of the domain; that is, if the image and the codomain of the function are equal. A surjective function is a surjection. Notationally: ∀ y ∈ Y , ∃ x ∈ X , y = f ( x ) . {\displaystyle \forall y\in Y,\exists x\in X,y=f(x).} The function is bijective (one-to-one and onto, one-to-one correspondence, or invertible) if each element of the codomain is mapped to by exactly one element of the domain; that is, if the function is both injective and surjective. A bijective function is also called a bijection. That is, combining the definitions of injective and surjective, ∀ y ∈ Y , ∃ ! x ∈ X , y = f ( x ) , {\displaystyle \forall y\in Y,\exists !x\in X,y=f(x),} where ∃ ! x {\displaystyle \exists !x} means "there exists exactly one x". In any case (for any function), the following holds: ∀ x ∈ X , ∃ ! y ∈ Y , y = f ( x ) . {\displaystyle \forall x\in X,\exists !y\in Y,y=f(x).} An injective function need not be surjective (not all elements of the codomain may be associated with arguments), and a surjective function need not be injective (some images may be associated with more than one argument). The four possible combinations of injective and surjective features are illustrated in the adjacent diagrams. == Injection == A function is injective (one-to-one) if each possible element of the codomain is mapped to by at most one argument. Equivalently, a function is injective if it maps distinct arguments to distinct images. An injective function is an injection. The formal definition is the following. The function f : X → Y {\displaystyle f\colon X\to Y} is injective, if for all x , x ′ ∈ X {\displaystyle x,x'\in X} , f ( x ) = f ( x ′ ) ⇒ x = x ′ . {\displaystyle f(x)=f(x')\Rightarrow x=x'.} The following are some facts related to injections: A function f : X → Y {\displaystyle f\colon X\to Y} is injective if and only if X {\displaystyle X} is empty or f {\displaystyle f} is left-invertible; that is, there is a function g : f ( X ) → X {\displaystyle g\colon f(X)\to X} such that g ∘ f = {\displaystyle g\circ f=} identity function on X. Here, f ( X ) {\displaystyle f(X)} is the image of f {\displaystyle f} . Since every function is surjective when its codomain is restricted to its image, every injection induces a bijection onto its image. More precisely, every injection f : X → Y {\displaystyle f\colon X\to Y} can be factored as a bijection followed by an inclusion as follows. Let f R : X → f ( X ) {\displaystyle f_{R}\colon X\rightarrow f(X)} be f {\displaystyle f} with codomain restricted to its image, and let i : f ( X ) → Y {\displaystyle i\colon f(X)\to Y} be the inclusion map from f ( X ) {\displaystyle f(X)} into Y {\displaystyle Y} . Then f = i ∘ f R {\displaystyle f=i\circ f_{R}} . A dual factorization is given for surjections below. The composition of two injections is again an injection, but if g ∘ f {\displaystyle g\circ f} is injective, then it can only be concluded that f {\displaystyle f} is injective (see figure). Every embedding is injective. == Surjection == A function is surjective or onto if each element of the codomain is mapped to by at least one element of the domain. In other words, each element of the codomain has a non-empty preimage. Equivalently, a function is surjective if its image is equal to its codomain. A surjective function is a surjection. The formal definition is the following. The function f : X → Y {\displaystyle f\colon X\to Y} is surjective, if for all y ∈ Y {\displaystyle y\in Y} , there is x ∈ X {\displaystyle x\in X} such that f ( x ) = y . {\displaystyle f(x)=y.} The following are some facts related to surjections: A function f : X → Y {\displaystyle f\colon X\to Y} is surjective if and only if it is right-invertible; that is, if and only if there is a function g : Y → X {\displaystyle g\colon Y\to X} such that f ∘ g = {\displaystyle f\circ g=} identity function on Y {\displaystyle Y} . (This statement is equivalent to the axiom of choice.) By collapsing all arguments mapping to a given fixed image, every surjection induces a bijection from a quotient set of its domain to its codomain. More precisely, the preimages under f of the elements of the image of f {\displaystyle f} are the equivalence classes of an equivalence relation on the domain of f {\displaystyle f} , such that x and y are equivalent if and only they have the same image under f {\displaystyle f} . As all elements of any one of these equivalence classes are mapped by f {\displaystyle f} on the same element of the codomain, this induces a bijection between the quotient set by this equivalence relation (the set of the equivalence classes) and the image of f {\displaystyle f} (which is its codomain when f {\displaystyle f} is surjective). Moreover, f is the composition of the canonical projection from f to the quotient set, and the bijection between the quotient set and the codomain of f {\displaystyle f} . The composition of two surjections is again a surjection, but if g ∘ f {\displaystyle g\circ f} is surjective, then it can only be concluded that g {\displaystyle g} is surjective (see figure). == Bijection == A function is bijective if it is both injective and surjective. A bijective function is also called a bijection or a one-to-one correspondence (not to be confused with one-to-one function, which refers to injection). A function is bijective if and only if every possible image is mapped to by exactly one argument. This equivalent condition is formally expressed as follows: The function f : X → Y {\displaystyle f\colon X\to Y} is bijective, if for all y ∈ Y {\displaystyle y\in Y} , there is a unique x ∈ X {\displaystyle x\in X} such that f ( x ) = y . {\displaystyle f(x)=y.} The following are some facts related to bijections: A function f : X → Y {\displaystyle f\colon X\to Y} is bijective if and only if it is invertible; that is, there is a function g : Y → X {\displaystyle g\colon Y\to X} such that g ∘ f = {\displaystyle g\circ f=} identity function on X {\displaystyle X} and f ∘ g = {\displaystyle f\circ g=} identity function on Y {\displaystyle Y} . This function maps each image to its unique preimage. The composition of two bijections is again a bijection, but if g ∘ f {\displaystyle g\circ f} is a bijection, then it can only be concluded that f {\displaystyle f} is injective and g {\displaystyle g} is surjective (see the figure at right and the remarks above regarding injections and surjections). The bijections from a set to itself form a group under composition, called the symmetric group. === Cardinality === Suppose that one wants to define what it means for two sets to "have the same number of elements". One way to do this is to say that two sets "have the same number of elements", if and only if all the elements of one set can be paired with the elements of the other, in such a way that each element is paired with exactly one element. Accordingly, one can define two sets to "have the same number of elements"—if there is a bijection between them. In which case, the two sets are said to have the same cardinality. Likewise, one can say that set X {\displaystyle X} "has fewer than or the same number of elements" as set Y {\displaystyle Y} , if there is an injection from X {\displaystyle X} to Y {\displaystyle Y} ; one can also say that set X {\displaystyle X} "has fewer than the number of elements" in set Y {\displaystyle Y} , if there is an injection from X {\displaystyle X} to Y {\displaystyle Y} , but not a bijection between X {\displaystyle X} and Y {\displaystyle Y} . == Examples == It is important to specify the domain and codomain of each function, since by changing these, functions which appear to be the same may have different properties. Injective and surjective (bijective) The identity function idX for every non-empty set X, and thus specifically R → R : x ↦ x . {\displaystyle \mathbf {R} \to \mathbf {R} :x\mapsto x.} R + → R + : x ↦ x 2 {\displaystyle \mathbf {R} ^{+}\to \mathbf {R} ^{+}:x\mapsto x^{2}} , and thus also its inverse R + → R + : x ↦ x . {\displaystyle \mathbf {R} ^{+}\to \mathbf {R} ^{+}:x\mapsto {\sqrt {x}}.} The exponential function exp : R → R + : x ↦ e x {\displaystyle \exp \colon \mathbf {R} \to \mathbf {R} ^{+}:x\mapsto \mathrm {e} ^{x}} (that is, the exponential function with its codomain restricted to its image), and thus also its inverse the natural logarithm ln : R + → R : x ↦ ln x . {\displaystyle \ln \colon \mathbf {R} ^{+}\to \mathbf {R} :x\mapsto \ln {x}.} Here, R + {\displaystyle \mathbf {R} ^{+}} denotes the positive real numbers. R → R : x ↦ x 3 {\displaystyle \mathbf {R} \to \mathbf {R} :x\mapsto x^{3}} Injective and non-surjective The exponential function exp : R → R : x ↦ e x . {\displaystyle \exp \colon \mathbf {R} \to \mathbf {R} :x\mapsto \mathrm {e} ^{x}.} Non-injective and surjective R → R : x ↦ ( x − 1 ) x ( x + 1 ) = x 3 − x . {\displaystyle \mathbf {R} \to \mathbf {R} :x\mapsto (x-1)x(x+1)=x^{3}-x.} R → [ − 1 , 1 ] : x ↦ sin ( x ) . {\displaystyle \mathbf {R} \to [-1,1]:x\mapsto \sin(x).} Non-injective and non-surjective R → R : x ↦ sin ( x ) . {\displaystyle \mathbf {R} \to \mathbf {R} :x\mapsto \sin(x).} R → R : x ↦ x 2 {\displaystyle \mathbf {R} \to \mathbf {R} :x\mapsto x^{2}} == Properties == For every function f, let X be a subset of the domain and Y a subset of the codomain. One has always X ⊆ f−1(f(X)) and f(f−1(Y)) ⊆ Y, where f(X) is the image of X and f−1(Y) is the preimage of Y under f. If f is injective, then X = f−1(f(X)), and if f is surjective, then f(f−1(Y)) = Y. For every function h : X → Y, one can define a surjection H : X → h(X) : x → h(x) and an injection I : h(X) → Y : y → y. It follows that h = I ∘ H {\displaystyle h=I\circ H} . This decomposition as the composition of a surjection and an injection is unique up to an isomorphism, in the sense that, given such a decomposition, there is a unique bijection φ : h ( X ) → H ( X ) {\displaystyle \varphi :h(X)\to H(X)} such that H ( x ) = φ ( h ( x ) ) {\displaystyle H(x)=\varphi (h(x))} and I ( φ ( h ( x ) ) ) = h ( x ) {\displaystyle I(\varphi (h(x)))=h(x)} for every x ∈ X . {\displaystyle x\in X.} == Category theory == In the category of sets, injections, surjections, and bijections correspond precisely to monomorphisms, epimorphisms, and isomorphisms, respectively. == History == The Oxford English Dictionary records the use of the word injection as a noun by S. Mac Lane in Bulletin of the American Mathematical Society (1950), and injective as an adjective by Eilenberg and Steenrod in Foundations of Algebraic Topology (1952). However, it was not until the French Bourbaki group coined the injective-surjective-bijective terminology (both as nouns and adjectives) that they achieved widespread adoption. == See also == Horizontal line test Injective module Permutation == References == == External links == Earliest Uses of Some of the Words of Mathematics: entry on Injection, Surjection and Bijection has the history of Injection and related terms.
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Wikipedia:Bilinear form#0
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In mathematics, a bilinear form is a bilinear map V × V → K on a vector space V (the elements of which are called vectors) over a field K (the elements of which are called scalars). In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately: B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v) B(u, v + w) = B(u, v) + B(u, w) and B(u, λv) = λB(u, v) The dot product on R n {\displaystyle \mathbb {R} ^{n}} is an example of a bilinear form which is also an inner product. An example of a bilinear form that is not an inner product would be the four-vector product. The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms. When K is the field of complex numbers C, one is often more interested in sesquilinear forms, which are similar to bilinear forms but are conjugate linear in one argument. == Coordinate representation == Let V be an n-dimensional vector space with basis {e1, …, en}. The n × n matrix A, defined by Aij = B(ei, ej) is called the matrix of the bilinear form on the basis {e1, …, en}. If the n × 1 matrix x represents a vector x with respect to this basis, and similarly, the n × 1 matrix y represents another vector y, then: B ( x , y ) = x T A y = ∑ i , j = 1 n x i A i j y j . {\displaystyle B(\mathbf {x} ,\mathbf {y} )=\mathbf {x} ^{\textsf {T}}A\mathbf {y} =\sum _{i,j=1}^{n}x_{i}A_{ij}y_{j}.} A bilinear form has different matrices on different bases. However, the matrices of a bilinear form on different bases are all congruent. More precisely, if {f1, …, fn} is another basis of V, then f j = ∑ i = 1 n S i , j e i , {\displaystyle \mathbf {f} _{j}=\sum _{i=1}^{n}S_{i,j}\mathbf {e} _{i},} where the S i , j {\displaystyle S_{i,j}} form an invertible matrix S. Then, the matrix of the bilinear form on the new basis is STAS. == Properties == === Non-degenerate bilinear forms === Every bilinear form B on V defines a pair of linear maps from V to its dual space V∗. Define B1, B2: V → V∗ by This is often denoted as where the dot ( ⋅ ) indicates the slot into which the argument for the resulting linear functional is to be placed (see Currying). For a finite-dimensional vector space V, if either of B1 or B2 is an isomorphism, then both are, and the bilinear form B is said to be nondegenerate. More concretely, for a finite-dimensional vector space, non-degenerate means that every non-zero element pairs non-trivially with some other element: B ( x , y ) = 0 {\displaystyle B(x,y)=0} for all y ∈ V {\displaystyle y\in V} implies that x = 0 and B ( x , y ) = 0 {\displaystyle B(x,y)=0} for all x ∈ V {\displaystyle x\in V} implies that y = 0. The corresponding notion for a module over a commutative ring is that a bilinear form is unimodular if V → V∗ is an isomorphism. Given a finitely generated module over a commutative ring, the pairing may be injective (hence "nondegenerate" in the above sense) but not unimodular. For example, over the integers, the pairing B(x, y) = 2xy is nondegenerate but not unimodular, as the induced map from V = Z to V∗ = Z is multiplication by 2. If V is finite-dimensional then one can identify V with its double dual V∗∗. One can then show that B2 is the transpose of the linear map B1 (if V is infinite-dimensional then B2 is the transpose of B1 restricted to the image of V in V∗∗). Given B one can define the transpose of B to be the bilinear form given by The left radical and right radical of the form B are the kernels of B1 and B2 respectively; they are the vectors orthogonal to the whole space on the left and on the right. If V is finite-dimensional then the rank of B1 is equal to the rank of B2. If this number is equal to dim(V) then B1 and B2 are linear isomorphisms from V to V∗. In this case B is nondegenerate. By the rank–nullity theorem, this is equivalent to the condition that the left and equivalently right radicals be trivial. For finite-dimensional spaces, this is often taken as the definition of nondegeneracy: Given any linear map A : V → V∗ one can obtain a bilinear form B on V via This form will be nondegenerate if and only if A is an isomorphism. If V is finite-dimensional then, relative to some basis for V, a bilinear form is degenerate if and only if the determinant of the associated matrix is zero. Likewise, a nondegenerate form is one for which the determinant of the associated matrix is non-zero (the matrix is non-singular). These statements are independent of the chosen basis. For a module over a commutative ring, a unimodular form is one for which the determinant of the associate matrix is a unit (for example 1), hence the term; note that a form whose matrix determinant is non-zero but not a unit will be nondegenerate but not unimodular, for example B(x, y) = 2xy over the integers. === Symmetric, skew-symmetric, and alternating forms === We define a bilinear form to be symmetric if B(v, w) = B(w, v) for all v, w in V; alternating if B(v, v) = 0 for all v in V; skew-symmetric or antisymmetric if B(v, w) = −B(w, v) for all v, w in V; Proposition Every alternating form is skew-symmetric. Proof This can be seen by expanding B(v + w, v + w). If the characteristic of K is not 2 then the converse is also true: every skew-symmetric form is alternating. However, if char(K) = 2 then a skew-symmetric form is the same as a symmetric form and there exist symmetric/skew-symmetric forms that are not alternating. A bilinear form is symmetric (respectively skew-symmetric) if and only if its coordinate matrix (relative to any basis) is symmetric (respectively skew-symmetric). A bilinear form is alternating if and only if its coordinate matrix is skew-symmetric and the diagonal entries are all zero (which follows from skew-symmetry when char(K) ≠ 2). A bilinear form is symmetric if and only if the maps B1, B2: V → V∗ are equal, and skew-symmetric if and only if they are negatives of one another. If char(K) ≠ 2 then one can decompose a bilinear form into a symmetric and a skew-symmetric part as follows B + = 1 2 ( B + t B ) B − = 1 2 ( B − t B ) , {\displaystyle B^{+}={\tfrac {1}{2}}(B+{}^{\text{t}}B)\qquad B^{-}={\tfrac {1}{2}}(B-{}^{\text{t}}B),} where tB is the transpose of B (defined above). === Reflexive bilinear forms and orthogonal vectors === A bilinear form B is reflexive if and only if it is either symmetric or alternating. In the absence of reflexivity we have to distinguish left and right orthogonality. In a reflexive space the left and right radicals agree and are termed the kernel or the radical of the bilinear form: the subspace of all vectors orthogonal with every other vector. A vector v, with matrix representation x, is in the radical of a bilinear form with matrix representation A, if and only if Ax = 0 ⇔ xTA = 0. The radical is always a subspace of V. It is trivial if and only if the matrix A is nonsingular, and thus if and only if the bilinear form is nondegenerate. Suppose W is a subspace. Define the orthogonal complement W ⊥ = { v ∣ B ( v , w ) = 0 for all w ∈ W } . {\displaystyle W^{\perp }=\left\{\mathbf {v} \mid B(\mathbf {v} ,\mathbf {w} )=0{\text{ for all }}\mathbf {w} \in W\right\}.} For a non-degenerate form on a finite-dimensional space, the map V/W → W⊥ is bijective, and the dimension of W⊥ is dim(V) − dim(W). === Bounded and elliptic bilinear forms === Definition: A bilinear form on a normed vector space (V, ‖⋅‖) is bounded, if there is a constant C such that for all u, v ∈ V, B ( u , v ) ≤ C ‖ u ‖ ‖ v ‖ . {\displaystyle B(\mathbf {u} ,\mathbf {v} )\leq C\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.} Definition: A bilinear form on a normed vector space (V, ‖⋅‖) is elliptic, or coercive, if there is a constant c > 0 such that for all u ∈ V, B ( u , u ) ≥ c ‖ u ‖ 2 . {\displaystyle B(\mathbf {u} ,\mathbf {u} )\geq c\left\|\mathbf {u} \right\|^{2}.} == Associated quadratic form == For any bilinear form B : V × V → K, there exists an associated quadratic form Q : V → K defined by Q : V → K : v ↦ B(v, v). When char(K) ≠ 2, the quadratic form Q is determined by the symmetric part of the bilinear form B and is independent of the antisymmetric part. In this case there is a one-to-one correspondence between the symmetric part of the bilinear form and the quadratic form, and it makes sense to speak of the symmetric bilinear form associated with a quadratic form. When char(K) = 2 and dim V > 1, this correspondence between quadratic forms and symmetric bilinear forms breaks down. == Relation to tensor products == By the universal property of the tensor product, there is a canonical correspondence between bilinear forms on V and linear maps V ⊗ V → K. If B is a bilinear form on V the corresponding linear map is given by In the other direction, if F : V ⊗ V → K is a linear map the corresponding bilinear form is given by composing F with the bilinear map V × V → V ⊗ V that sends (v, w) to v⊗w. The set of all linear maps V ⊗ V → K is the dual space of V ⊗ V, so bilinear forms may be thought of as elements of (V ⊗ V)∗ which (when V is finite-dimensional) is canonically isomorphic to V∗ ⊗ V∗. Likewise, symmetric bilinear forms may be thought of as elements of (Sym2V)* (dual of the second symmetric power of V) and alternating bilinear forms as elements of (Λ2V)∗ ≃ Λ2V∗ (the second exterior power of V∗). If char(K) ≠ 2, (Sym2V)* ≃ Sym2(V∗). == Generalizations == === Pairs of distinct vector spaces === Much of the theory is available for a bilinear mapping from two vector spaces over the same base field to that field Here we still have induced linear mappings from V to W∗, and from W to V∗. It may happen that these mappings are isomorphisms; assuming finite dimensions, if one is an isomorphism, the other must be. When this occurs, B is said to be a perfect pairing. In finite dimensions, this is equivalent to the pairing being nondegenerate (the spaces necessarily having the same dimensions). For modules (instead of vector spaces), just as how a nondegenerate form is weaker than a unimodular form, a nondegenerate pairing is a weaker notion than a perfect pairing. A pairing can be nondegenerate without being a perfect pairing, for instance Z × Z → Z via (x, y) ↦ 2xy is nondegenerate, but induces multiplication by 2 on the map Z → Z∗. Terminology varies in coverage of bilinear forms. For example, F. Reese Harvey discusses "eight types of inner product". To define them he uses diagonal matrices Aij having only +1 or −1 for non-zero elements. Some of the "inner products" are symplectic forms and some are sesquilinear forms or Hermitian forms. Rather than a general field K, the instances with real numbers R, complex numbers C, and quaternions H are spelled out. The bilinear form ∑ k = 1 p x k y k − ∑ k = p + 1 n x k y k {\displaystyle \sum _{k=1}^{p}x_{k}y_{k}-\sum _{k=p+1}^{n}x_{k}y_{k}} is called the real symmetric case and labeled R(p, q), where p + q = n. Then he articulates the connection to traditional terminology: Some of the real symmetric cases are very important. The positive definite case R(n, 0) is called Euclidean space, while the case of a single minus, R(n−1, 1) is called Lorentzian space. If n = 4, then Lorentzian space is also called Minkowski space or Minkowski spacetime. The special case R(p, p) will be referred to as the split-case. === General modules === Given a ring R and a right R-module M and its dual module M∗, a mapping B : M∗ × M → R is called a bilinear form if for all u, v ∈ M∗, all x, y ∈ M and all α, β ∈ R. The mapping ⟨⋅,⋅⟩ : M∗ × M → R : (u, x) ↦ u(x) is known as the natural pairing, also called the canonical bilinear form on M∗ × M. A linear map S : M∗ → M∗ : u ↦ S(u) induces the bilinear form B : M∗ × M → R : (u, x) ↦ ⟨S(u), x⟩, and a linear map T : M → M : x ↦ T(x) induces the bilinear form B : M∗ × M → R : (u, x) ↦ ⟨u, T(x)⟩. Conversely, a bilinear form B : M∗ × M → R induces the R-linear maps S : M∗ → M∗ : u ↦ (x ↦ B(u, x)) and T′ : M → M∗∗ : x ↦ (u ↦ B(u, x)). Here, M∗∗ denotes the double dual of M. == See also == == Citations == == References == == External links == "Bilinear form", Encyclopedia of Mathematics, EMS Press, 2001 [1994] "Bilinear form". PlanetMath. This article incorporates material from Unimodular on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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Wikipedia:Bill Casselman#0
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William Allen Casselman (born November 27, 1941) is an American Canadian mathematician who works in representation theory and automorphic forms. He is a professor emeritus at the University of British Columbia. He is closely connected to the Langlands program and has been involved in posting all of the work of Robert Langlands on the internet. == Career == Casselman did his undergraduate work at Harvard College where his advisor was Raoul Bott and received his Ph.D from Princeton University in 1966 where his advisor was Goro Shimura. He was a visiting scholar at the Institute for Advanced Study in 1974, 1983, and 2001. He emigrated to Canada in 1971 and is a professor emeritus in mathematics at the University of British Columbia. == Research == Casselman specializes in representation theory, automorphic forms, geometric combinatorics, and the structure of algebraic groups. He has an interest in mathematical graphics and has been the graphics editor of the Notices of the American Mathematical Society since January, 2001. == Awards == In 2012, he became one of the inaugural fellows of the American Mathematical Society. == Selected publications == Casselman, Bill (1973). "On some results of Atkin and Lehner". Mathematische Annalen. 201 (4): 301–314. doi:10.1007/BF01428197. S2CID 121867474. Casselman, Bill (1977). "Characters and Jacquet modules". Mathematische Annalen. 230 (2): 101–105. doi:10.1007/BF01370657. ISSN 0025-5831. S2CID 121574262. Casselman, Bill (1980). "The unramified principal series of p-adic groups. I. The Spherical function". Compositio Mathematica. 40 (3): 387–406. Casselman, Bill; Shalika, Joseph (1980). "The unramified principal series of p-adic groups. II. The Whittaker function". Compositio Mathematica. 41 (2): 207–231. Casselman, Bill; Milicic, Dragan (1982). "representations". Duke Mathematical Journal. 49 (4): 869–930. doi:10.1215/S0012-7094-82-04943-2. ISSN 0012-7094. Borel, Armand; Casselman, Bill (1983). "L2-cohomology of locally symmetric manifolds of finite volume". Duke Mathematical Journal. 50 (3): 625–647. doi:10.1215/S0012-7094-83-05029-9. S2CID 122723214. Casselman, Bill; Shahidi, Freydoon (1998). "On irreducibility of standard modules for generic representations". Annales Scientifiques de l'École Normale Supérieure. 31 (4): 561–589. doi:10.1016/S0012-9593(98)80107-9. Casselman, Bill (2005). Mathematical Illustrations: A Manual of Geometry and PostScript. Cambridge University Press. ISBN 0521839211. == References == == External links == Publications of Bill Casselman Bill Casselman's Home Page William Allen Casselman at the Mathematics Genealogy Project
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Wikipedia:Bill Warner (writer)#0
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Bill Warner is the pen name of Bill French (born 1941), a former physics professor and anti-Islam writer. He founded the Center for the Study of Political Islam International, which is based in the Czech Republic. The Southern Poverty Law Center in 2011 described him as one of a core group of ten anti-Islam hardliners in the United States. He has also been described as a part of the counter-jihad movement. == Biography == Warner graduated from North Carolina State University where he got his PhD in physics and mathematics in 1968. He is a former Tennessee State University physics professor. Warner does not have an academic background in religious studies. He participated in the Murfreesboro protests where he spoke to a group of opponents of the mosque and sold his books. The protests included a legal case arguing that Islam is not a religion. == Reception == Middlebury Institute professor and terrorism expert Jeffrey M. Bale refers to Warner as an example of writers who identify Islam with Islamism. According to Bale, these writers relate all the characteristics associated with Islamism with Islam as a whole, alleging that "such characteristics are intrinsic to Islam itself, and therefore that Islamism and jihadism are simply logical extensions - or simple applications in practice - of the authentic tenets and core values of Islam." He argues that, what they "fail to acknowledge is that these particular interpretations are by no means the only possible interpretations of core Islamic doctrines, traditions, and values, nor are they necessarily the most authentic, valid, or widely shared interpretations." This he says, is like claiming that Christian Reconstructionism is identical to Christianity. American Muslim religious liberty lawyer Asma Uddin considers groups like Warner and his organization as anti-Muslim entities that mainstream the idea that Islam is not just a religion but also a political ideology which aids in legitimizing restricting the religious freedom of American Muslims. Warner's organization has said that “Statistics show that Islamic politics is what brought Islam success, not religion” and journalist Uddin described the organization's statement that Islam is mainly a political ideology as "pseudoscience and these quote, unquote ‘think tanks’... are responding to the work of actual legitimate think tanks using the language of statistics." Zafar Iqbal, professor at Pakistan's International Islamic University, has compared Warner to Geert Wilders in that both consider Islam to be a totalitarian political ideology demanding complete submission. Czech politician Jiří Kobza lists 4 publications by Warner in his "list of the most important books that every citizen should read". == Center for the Study of Political Islam International == The Center for the Study of Political Islam International (CSPII), based in the Czech Republic, was formed by Warner and Milan Podlipný from the Czech Republic in 2014. According to its website it has branches in 11 countries. The Southern Poverty Law Center states that the CSPII ran two lectures in the Czech Republic in 2017 where lecturers described "Muslims as being encouraged to promote principles in contradiction with a European understanding of human rights." The Council on American–Islamic Relations has described the CSPI as Islamophobic, listing it as part of the" U.S.-based Islamophobia network’s inner core". The Czech branch distributed one of Warner's books to the Slovak National Council in 2016. Luboš Kropáček, a Czech Islamologist at Charles University stated that "As far as I know, no Orientalist, Arabist or Islamologist pays attention to this man, he has no business in our field of science." Bronislav Ostřanský, a researcher at the Oriental Institute, ASCR, said that his apparently scientific approach impresses many people "including otherwise educated and politically influential personalities" but that he should be quoted "in a professional work in only one context, namely as relevant source material for the study of contemporary Islamophobia." In 2010 the Royal Aal al-Bayt Institute for Islamic Thought commented on Warner's article "Is a nice Muslim a Good Muslim?" which is now on the CSPII site. == See also == Political Islam == Bibliography == Sharia Law for Non-Muslims. Centre for the Study of Political Islam. 2015. ISBN 978-0-9795794-8-6. The Hadith. Center for the Study of Political Islam. 2016. ISBN 978-1-936659-01-2. A Two-Hour Koran. Center for the Study of Political Islam. 2016. ISBN 978-1-936659-02-9. The Life of Mohammed. Center for the Study of Political Islam. 2016. ISBN 978-1-936659-06-7. Measuring Mohammed. Center for the Study of Political Islam. 2019. ISBN 978-1-936659-38-8. == References == == External links == Official website
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Wikipedia:Bin Yu#0
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Bin Yu (Chinese: 郁彬) is a Chinese-American statistician. She is currently Chancellor's Professor in the Departments of Statistics and of Electrical Engineering & Computer Sciences at the University of California, Berkeley. == Biography == Yu earned a bachelor's degree in mathematics in 1984 from Peking University, and went on to pursue graduate studies in statistics at Berkeley, earning a master's degree in 1987 and a Ph.D. in 1990. Her dissertation, Some Results on Empirical Processes and Stochastic Complexity, was jointly supervised by Lucien Le Cam and Terry Speed. After postdoctoral studies at the Mathematical Sciences Research Institute and an assistant professorship at the University of Wisconsin–Madison, she returned to Berkeley as a faculty member in 1993, was tenured in 1997, and became Chancellor's Professor in 2006. She also worked at Bell Labs from 1998 to 2000, while on leave from Berkeley, and has held visiting positions at several other universities. She chaired the Department of Statistics at Berkeley from 2009 to 2012, and was president of the Institute of Mathematical Statistics in 2014. In 2023, she was awarded the COPSS Distinguished Achievement Award and Lectureship. == Research == Yu's work spans many fields including statistics, machine learning, neuroscience, genomics, and remote sensing. Her recent work has focused on data science, including frameworks for veridical data science and interpretable machine learning. Yu has received recent news coverage regarding investigations into the theoretical foundations of deep learning, and work forecasting COVID-19 severity in the US. Other research topics include dictionary learning, non-negative matrix factorization (NMF), EM and deep learning (CNNs and LSTMs), and heterogeneous effect estimation in randomized experiments (X-learner). == Honors and awards == Yu is a fellow of the Institute of Mathematical Statistics, the IEEE, the American Statistical Association, the American Association for the Advancement of Science, the American Academy of Arts and Sciences, and the National Academy of Sciences. In 2012, she was the Tukey Lecturer of the Bernoulli Society for Mathematical Statistics and Probability. In 2018, she was awarded the Elizabeth L. Scott Award. She was invited to give the Breiman lecture at NeurIPS 2019 (formally known as NIPS), on the topic of veridical data science. In 2021, she was awarded an honorary doctorate by the University of Lausanne. And in 2023, she received the COPSS distinguished achievement lecture. == References == == External links == A conversation with Professor Bin Yu By Tao Shi, July 9, 2013
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Wikipedia:Binary expression tree#0
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A binary expression tree is a specific kind of a binary tree used to represent expressions. Two common types of expressions that a binary expression tree can represent are algebraic and boolean. These trees can represent expressions that contain both unary and binary operators. Like any binary tree, each node of a binary expression tree has zero, one, or two children. This restricted structure simplifies the processing of expression trees. == Construction of an expression tree == === Example === The input in postfix notation is: a b + c d e + * * Since the first two symbols are operands, one-node trees are created and pointers to them are pushed onto a stack. For convenience the stack will grow from left to right. The next symbol is a '+'. It pops the two pointers to the trees, a new tree is formed, and a pointer to it is pushed onto the stack. Next, c, d, and e are read. A one-node tree is created for each and a pointer to the corresponding tree is pushed onto the stack. Continuing, a '+' is read, and it merges the last two trees. Now, a '*' is read. The last two tree pointers are popped and a new tree is formed with a '*' as the root. Finally, the last symbol is read. The two trees are merged and a pointer to the final tree remains on the stack. == Algebraic expressions == Algebraic expression trees represent expressions that contain numbers, variables, and unary and binary operators. Some of the common operators are × (multiplication), ÷ (division), + (addition), − (subtraction), ^ (exponentiation), and - (negation). The operators are contained in the internal nodes of the tree, with the numbers and variables in the leaf nodes. The nodes of binary operators have two child nodes, and the unary operators have one child node. == Boolean expressions == Boolean expressions are represented very similarly to algebraic expressions, the only difference being the specific values and operators used. Boolean expressions use true and false as constant values, and the operators include ∧ {\displaystyle \land } (AND), ∨ {\displaystyle \lor } (OR), ¬ {\displaystyle \neg } (NOT). == See also == Expression (mathematics) Term (logic) Context-free grammar Parse tree Abstract syntax tree == References ==
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Wikipedia:Binyamin Amirà#0
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Binyamin A. Amirà (Hebrew: בנימין אמירה; 3 June 1896 – 20 January 1968) was an Israeli mathematician. == Early life and education == Born in 1896 in Mohilev, Russian Empire, Binyamin Amirà immigrated with his family to Tel Aviv in Ottoman Palestine in 1910, where he attended the Herzliya Gymnasium. Amirà went on to study mathematics at the University of Geneva, after which he moved to the University of Göttingen in 1921 to undertake research for his doctorate under the supervision of Edmund Landau. == Academic career == After completing his D.Sc. in 1924, Amirà spent a brief period at the University of Geneva as Privatdozent, after which he followed Landau in 1925 to help him in establishing the Mathematics Institute of the newly-founded Hebrew University in Jerusalem. There he became the institute's first tenured staff member. Amirà founded the Journal d'Analyse Mathématique in 1951, which he edited alongside Ze'ev Nehari and Menahem Schiffer. He retired in 1960. == References ==
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Wikipedia:Bipolar theorem#0
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In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (that is, the polar of the polar) of a set. In convex analysis, the bipolar theorem refers to a necessary and sufficient conditions for a cone to be equal to its bipolar. The bipolar theorem can be seen as a special case of the Fenchel–Moreau theorem.: 76–77 == Preliminaries == Suppose that X {\displaystyle X} is a topological vector space (TVS) with a continuous dual space X ′ {\displaystyle X^{\prime }} and let ⟨ x , x ′ ⟩ := x ′ ( x ) {\displaystyle \left\langle x,x^{\prime }\right\rangle :=x^{\prime }(x)} for all x ∈ X {\displaystyle x\in X} and x ′ ∈ X ′ . {\displaystyle x^{\prime }\in X^{\prime }.} The convex hull of a set A , {\displaystyle A,} denoted by co A , {\displaystyle \operatorname {co} A,} is the smallest convex set containing A . {\displaystyle A.} The convex balanced hull of a set A {\displaystyle A} is the smallest convex balanced set containing A . {\displaystyle A.} The polar of a subset A ⊆ X {\displaystyle A\subseteq X} is defined to be: A ∘ := { x ′ ∈ X ′ : sup a ∈ A | ⟨ a , x ′ ⟩ | ≤ 1 } . {\displaystyle A^{\circ }:=\left\{x^{\prime }\in X^{\prime }:\sup _{a\in A}\left|\left\langle a,x^{\prime }\right\rangle \right|\leq 1\right\}.} while the prepolar of a subset B ⊆ X ′ {\displaystyle B\subseteq X^{\prime }} is: ∘ B := { x ∈ X : sup x ′ ∈ B | ⟨ x , x ′ ⟩ | ≤ 1 } . {\displaystyle {}^{\circ }B:=\left\{x\in X:\sup _{x^{\prime }\in B}\left|\left\langle x,x^{\prime }\right\rangle \right|\leq 1\right\}.} The bipolar of a subset A ⊆ X , {\displaystyle A\subseteq X,} often denoted by A ∘ ∘ {\displaystyle A^{\circ \circ }} is the set A ∘ ∘ := ∘ ( A ∘ ) = { x ∈ X : sup x ′ ∈ A ∘ | ⟨ x , x ′ ⟩ | ≤ 1 } . {\displaystyle A^{\circ \circ }:={}^{\circ }\left(A^{\circ }\right)=\left\{x\in X:\sup _{x^{\prime }\in A^{\circ }}\left|\left\langle x,x^{\prime }\right\rangle \right|\leq 1\right\}.} == Statement in functional analysis == Let σ ( X , X ′ ) {\displaystyle \sigma \left(X,X^{\prime }\right)} denote the weak topology on X {\displaystyle X} (that is, the weakest TVS topology on X {\displaystyle X} making all linear functionals in X ′ {\displaystyle X^{\prime }} continuous). The bipolar theorem: The bipolar of a subset A ⊆ X {\displaystyle A\subseteq X} is equal to the σ ( X , X ′ ) {\displaystyle \sigma \left(X,X^{\prime }\right)} -closure of the convex balanced hull of A . {\displaystyle A.} == Statement in convex analysis == The bipolar theorem:: 54 For any nonempty cone A {\displaystyle A} in some linear space X , {\displaystyle X,} the bipolar set A ∘ ∘ {\displaystyle A^{\circ \circ }} is given by: A ∘ ∘ = cl ( co { r a : r ≥ 0 , a ∈ A } ) . {\displaystyle A^{\circ \circ }=\operatorname {cl} (\operatorname {co} \{ra:r\geq 0,a\in A\}).} === Special case === A subset C ⊆ X {\displaystyle C\subseteq X} is a nonempty closed convex cone if and only if C + + = C ∘ ∘ = C {\displaystyle C^{++}=C^{\circ \circ }=C} when C + + = ( C + ) + , {\displaystyle C^{++}=\left(C^{+}\right)^{+},} where A + {\displaystyle A^{+}} denotes the positive dual cone of a set A . {\displaystyle A.} Or more generally, if C {\displaystyle C} is a nonempty convex cone then the bipolar cone is given by C ∘ ∘ = cl C . {\displaystyle C^{\circ \circ }=\operatorname {cl} C.} == Relation to the Fenchel–Moreau theorem == Let f ( x ) := δ ( x | C ) = { 0 x ∈ C ∞ otherwise {\displaystyle f(x):=\delta (x|C)={\begin{cases}0&x\in C\\\infty &{\text{otherwise}}\end{cases}}} be the indicator function for a cone C . {\displaystyle C.} Then the convex conjugate, f ∗ ( x ∗ ) = δ ( x ∗ | C ∘ ) = δ ∗ ( x ∗ | C ) = sup x ∈ C ⟨ x ∗ , x ⟩ {\displaystyle f^{*}(x^{*})=\delta \left(x^{*}|C^{\circ }\right)=\delta ^{*}\left(x^{*}|C\right)=\sup _{x\in C}\langle x^{*},x\rangle } is the support function for C , {\displaystyle C,} and f ∗ ∗ ( x ) = δ ( x | C ∘ ∘ ) . {\displaystyle f^{**}(x)=\delta (x|C^{\circ \circ }).} Therefore, C = C ∘ ∘ {\displaystyle C=C^{\circ \circ }} if and only if f = f ∗ ∗ . {\displaystyle f=f^{**}.} : 54 == See also == Dual system Fenchel–Moreau theorem – Mathematical theorem in convex analysis − A generalization of the bipolar theorem. Polar set – Subset of all points that is bounded by some given point of a dual (in a dual pairing) == References == == Bibliography == Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
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Wikipedia:Birgit Grodal#0
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Birgit Grodal (24 June 1943 - 4 May 2004), was an economics professor at the University of Copenhagen from 1968 until her death in 2004. == Early life == Birgit Grodal was born on 24 June 1943 in Copenhagen, Denmark. She grew up in Frederiksberg. She was the middle child having both a younger and an older brother. Grodal was interested in mathematics from a young age and used to fill the pages of her psalm book with equations. She was married to Torben Grodal. == Education == Grodal gained her degree (1962) and her masters (1968) in mathematics and physics from the University of Copenhagen. Also at the University of Copenhagen she earned her Ph.D. in mathematics under Werner Fenchel, her dissertation, A critical overview of the present theory on atomless economies, won a gold medal, something which was used to support young scholars in the old Danish academic system. == Research == Birgit Grodal worked on micro-economic theory, mathematical economics, and general equilibrium theory. == Career == Birgit Grodal was elected president of the European Economic Association, but died before she was scheduled to attain her presidency. In 2010 the European Economic Association Council agreed unanimously to institute a prize in her honor to a European-based female economist who has made a significant contribution to the Economics profession. She was also a member of the Econometric Society starting in 1981 and served on the executive committee between 1997 and 2000. == Publications == Her major works include A Second Remark on the Core of an Atomless Economy" published in Econometrica in 1972, and Existence of Approximate Cores with Incomplete Preferences published in Econometrica in 1976. == The Birgit Grodal Award == The Birgit Grodal Award is bestowed bi-annually (even years) in her memory, the winner receives €10,000. Danmarks Nationalbank agreed to sponsor the first three awards. The recipients of the Birgit Grodal Award are: 2012: Hélène Rey 2014: Rachel Griffith 2016: Lucrezia Reichlin 2018: Ekaterina Zhuravskaya 2020: Eliana La Ferrara 2022: Silvana Tenreyro == References ==
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Wikipedia:Bjarni Jónsson#0
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Bjarni Jónsson (February 15, 1920 – September 30, 2016) was an Icelandic mathematician and logician working in universal algebra, lattice theory, model theory and set theory. He was emeritus distinguished professor of mathematics at Vanderbilt University and the honorary editor in chief of Algebra Universalis. He received his PhD in 1946 at UC Berkeley under supervision of Alfred Tarski. In 2012, he became a fellow of the American Mathematical Society. == Work == Jónsson's lemma as well as several mathematical objects are named after him, among them Jónsson algebras, ω-Jónsson functions, Jónsson cardinals, Jónsson terms, Jónsson–Tarski algebras and Jónsson–Tarski duality. == Publications == Jónsson, Bjarni (1972), Topics in Universal Algebra, Lecture Notes in Mathematics, vol. 250, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0058648, ISBN 978-3-540-05722-2, MR 0345895 == References == == Further reading == Kirby A. Baker, Bjarni Jónsson's contributions in algebra, Algebra Universalis, September 1994, Volume 31, Issue 3, pp. 306–336. J. B. Nation, Jónsson's contributions to lattice theory, Algebra Universalis, September 1994, Volume 31, Issue 3, pp. 430–445. == External links == Bjarni Jónsson's homepage
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Wikipedia:Björn Gunnlaugsson#0
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Björn Gunnlaugsson[a] (25 September 1788 – 17 March 1876)[b] was an Icelandic mathematician and cartographer. For the Icelandic Literary Society, he surveyed the country from 1831 to 1843. The results of his work were published in a topographic map of Iceland at a scale of 1:480,000 on four sheets. It was the first complete map of Iceland and, although generally dated to 1844, was not completed until 1848. It was published under the direction of Olaf Nikolas Olsen in Copenhagen. In 1849, a smaller edition on one sheet at a scale of 1:960,000 appeared. For his survey work, Björn received the Order of the Dannebrog in 1846 and the French Légion d'honneur in 1859. == Life == Björn was born at Tannstaðir, a remote farm on the Hrútafjörður in Húnavatnssýsla in north-western Iceland. Although the family was poor, they sent him to school to the local priests, who recognized his intellectual abilities. In 1808, Björn passed an exam in Reykjavík, obtaining a recommendation from the bishop for studying at the University of Copenhagen. But these plans were delayed by the Gunboat War between Denmark-Norway and the United Kingdom. Only after the end of the Napoleonic Wars could Björn travel to Denmark and enrolled in 1817 at the University of Copenhagen, where he studied theology and mathematics. During his studies, he won the university's gold medal for mathematics twice. In 1822, a new post for a school teacher in Danish, mathematics, and history opened at the school of Bessastaðir[c] in Iceland and was offered to Björn. He abandoned his theologic studies and accepted the post, returning to Iceland and taking up his duties as a school teacher on 14 May 1822. When the school was moved to Reykjavík in 1846, Björn followed. Five years later he was appointed chief assistant (Yfirkennari) to the rector. He retired in 1862. Björn Gunnlaugsson was married twice; first to Ragnheiður Bjarnadóttir (died 1834), after her death he married in 1836 Guðlaug Aradóttir (died 1873). == Work == Björn was an exceptional figure in early 19th-century Iceland. The abstract thoughts of this gentle learned man were beyond the grasp of most of his compatriots, who regarded him as an eccentric with few of the practical skills they so highly valued. Yet the simple folk also felt a certain kind of respectful awe towards this scholar. Björn knew the inclination towards the practical of his fellow countrymen well. When he returned to Iceland as a school teacher for mathematics, the curriculum at the grammar school at Bessastaðir covered barely more than the four basic operations addition, subtraction, multiplication and division. In his inauguration speech at the school, he emphasized the practical applications of mathematics. He tried to take mathematics education to a higher level, but failed ultimately as his treatment of the subject was often too abstract for his pupils and he was, according to Benedikz, "not capable of handling a class of mathematical ignoramuses". The only studied mathematician in Iceland in the 19th century, Björn was isolated from the academic community in Europe, and the intellectual environment made him turn to didactics and the applications of mathematics, and also to philosophy. === Uppdráttr Íslands – Map of Iceland === In August 1829, Björn, who had done in the early 1820s cartographic work under the direction of Heinrich Christian Schumacher at Altona, proposed to the Danish government to undertake a land survey of Iceland, and asked for the instruments used by the Danish Navy in their earlier coastal surveys to be sent to Iceland. His request was ignored at first. In 1831, the Literary Society of Iceland decided to sponsor him and helped him obtain the necessary instruments. From 1831 to 1843, Björn spent the summers surveying the country together with one assistant, and in the winters he would draw the maps. He did not lay a new baseline but started from the earlier coastal surveys the Danish Navy had undertaken in the period of 1774 to 1818, extending the triangulation inland. The Literary Society supported him with a yearly grant, and so did the Danish government from 1836 to 1846. Björn's hand-drawn maps were sent to Copenhagen for preparing the publication. Olaf Nikolas Olsen had been appointed as the director of publication; he proposed to publish the map on four sheets, and he probably also defined the scale of 1:480,000 and the conical projection used. The map was published under Olsen's name, with the Literary Society of Iceland as the publisher, and paid by the Danish treasury. The title page was in Danish and French. Although dated 1844, it was probably not completed until 1848. In 1849, a smaller version of the map on one sheet at a scale of 1:960,000 was published. Björn's survey formed the basis of many subsequent maps of Iceland for the next hundred years; new maps based on new surveys appeared only after World War II. It was an immense work, and Björn realized soon enough that one man alone would not be able to triangulate the whole island in his lifetime, and focused his attention on the inhabited areas. Yet he managed to survey a good part of the wilderness, too, even if he had to rely on the accounts of the local population in some remote areas. Björn was well aware of the inaccuracies in some regions; already in 1834, he wrote that one "should neither have too high or too low expectations of the map, nor trust too greatly nor too little in its usefulness or accuracy". The central highlands were sketchy on his map; they were mapped properly for the first time by Þorvaldur Thoroddsen, whose map was printed in 1901. Still, Björn's survey is considered a great advancement, especially given the limited resources he had at his disposition. In recognition of his outstanding survey work Björn Gunnlaugsson was awarded the Knight's Cross of the Order of the Dannebrog in 1846 and also received the Knight's cross of the French Légion d'honneur. The map was also exhibited at the World's fair at Paris in 1878, where it was awarded a prize. === Njóla – "Night" === Njóla is a long didactic theological-philosophical poem Björn wrote mostly during his survey travels when the weather did not permit him to work or in the evenings. It was published originally in the annual report of 1842 of the Bessastaðir school, and then in 1853 with some minor revisions and again in 1884 in Reykjavík. The poem begins by describing a night view of the skies, and then introduces the reader to astronomic distances, explaining how long a cannonball shot from the sun would take to reach each of the planets—and then the next star. In later stanzas, he describes the birth of the universe, covers Newton's laws and explains gravity. He interweaves such physics and mathematics framed as poetry with theological and philosophical musings about the purpose of the universe, the nature of good and evil, and God's intent. === Tölvísi: A mathematical textbook === Towards the end of his teaching career, Björn wrote down the mathematics he would have liked to teach in the Tölvísi, a mathematical textbook unprecedented in Iceland, both in its breadth and depth, but also in the rigour of its proofs. Moreover, it was written in Icelandic; in an attempt to make mathematics more accessible to his fellow countrymen, Björn even tried to find Icelandic names for mathematical concepts that hitherto had only been named using Danish or Latin words. But the work was largely ignored, and Björn's Icelandic terms never caught on. Begun in 1856, volume 1 was published in Reykjavík in 1865 by E. Þórðarson, but its second part was still unpublished in 2003 and existed only as an unpublished manuscript at the National and University Library of Iceland. Gunnlaugsson's first biographers, Melsteð and Jensson, wrote that it was "a book praised by all but read by extremely few". == Notes == a In English, Björn's name is sometimes also given as Bjorn Gunnlaugsson, omitting the diacritic on the Ö. b Björn's birthday is variously given as May 25, September 5 and September 28, see Otto J. Björnsson 1990, p. 3. c Before Björn's arrival, the school at Bessastaðir had had a staff of three teachers only, none of them knowledgeable in mathematics. == References == == Literature == Benedikt S. Benedikz: "The Wise Man with the Child's Heart: Björn Gunnlaugsson, 1788–1876", in Scandinavian Studies 75 (4), pp. 567–590; 2003. ISSN 0036-5637. Kristín Bjarnadóttir: "Fundamental Reasons for Mathematical Education in Iceland", in Bharath Sriraman (Ed): International Perspectives on Social Justice in Mathematics Education (The Montana Mathematics Enthusiast (TMME) Monograph 1), p. 137–150; University of Montana, 2007. ISSN 1551-3440. URL last accessed 2007-09-12. Ágúst H. Bjarnason: "Um Björn Gunnlaugsson", in Timarit Þjóðræknisfélags Íslendinga, vol. 20 (1938), pp. 17–28. Emil Elberling: Gunnlaugsson, Björn, entry in Th. Westrin (ed.): Nordisk familjebok: Konversationslexikon och Realencyklopedi, 2nd ed. (1909), vol. 10, p. 643. URL last accessed 2007-09-19. Björn Gunnlaugsson: De mensura et delineatione Islandiae, Viðey, 1834. Kr. Kaalund: Gunnlaugsson, Bjørn, entry in C. F. Bricka (ed.): Dansk biografisk lexikon: tillige omfattende Norge for Tidsrummet 1537–1814; Gyldendal, Copenhagen 1887–1905; vol. VI (1892), p. 321f. URL last accessed 2007-09-12. Páll Melsteð & Björn Jensson: "Björn Gunnlaugsson", in Andvari, vol. 9 (1883), pp. 3–16. P.M.: Obituary for Björn Gunnlaugsson, Ísafold, 24 March 1876, p. 21. In Icelandic. URL last accessed 2012-11-08. Jökull Sævarsson: Gunnlaugsson's map of Iceland, (with text from Sigurðsson (1982)), Antique maps of Iceland, National and University Library of Iceland. URL last accessed 2007-09-12. Haraldur Sigurðsson: "Iceland on maps.", pp. 7–15 in Kortasafn Háskóla Íslands, Reykjavík 1982. URL last accessed 2007-09-12. Otto J. Björnsson: Brot úr ævi og starfi Björns Gunnlaugssonar riddara og yfirkennara; Reykjavík, Raunvísindastofnun Háskólans, 1990. In Icelandic. Þorvaldur Thoroddsen: Landfræðissaga Íslands; 4 volumes, Copenhagen; S. L. Möller, 1892–1904. In Icelandic. On Björn Gunnlaugsson, see in particular vol. 3, pp. 300ff. == Further reading == Otto J. Björnsson: Varð Gauss á vegi Björns Gunnlaugssonar?; Reykjavík, Raunvísindastofnun Háskólans, 1997. In Icelandic. Halldór Hermannsson: The Cartography of Iceland; Islandica XXI, Fiske Icelandic Collection, Cornell University Library, 1931. Steindór Sigurðsson: "Björn Gunnlaugsson og Uppdráttur Íslands"; Skírnir 111 (1938), pp. 166–173. In Icelandic. == External links == The 1849 edition of Gunnlaugsson's map. Björn Gunnlaugsson: Tøblur yfir Sólarinnar sýnilega gáng á Íslandi Viðeyar Klaustri, 1836. Sun declination tables. In Icelandic. URL last accessed 2007-09-15. N.N.: 200 ára afmælis Björns Gunnlaugsonnar minnst, Morgunblaðið, 24 September 1988, p. 19. Newspaper article on the occasion of the bicentennial of Gunnlaugsson's birthday. In Icelandic. URL last accessed 2007-09-12. Ágústa P. Snæland: "Hvað æðst sýnist í heimi", Lesbók Morgunblaðsins, May 15, 1993. Newspaper article on Njóla, in Icelandic; has an image of the portrait of Gunnlaugsson by Sigurður málari. URL last accessed 2007-09-12.
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Wikipedia:Blaise Pascal#0
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Blaise Pascal (19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic writer. Pascal was a child prodigy who was educated by his father, a tax collector in Rouen. His earliest mathematical work was on projective geometry; he wrote a significant treatise on the subject of conic sections at the age of 16. He later corresponded with Pierre de Fermat on probability theory, strongly influencing the development of modern economics and social science. In 1642, he started some pioneering work on calculating machines (called Pascal's calculators and later Pascalines), establishing him as one of the first two inventors of the mechanical calculator. Like his contemporary René Descartes, Pascal was also a pioneer in the natural and applied sciences. Pascal wrote in defense of the scientific method and produced several controversial results. He made important contributions to the study of fluids, and clarified the concepts of pressure and vacuum by generalising the work of Evangelista Torricelli. Following Torricelli and Galileo Galilei, in 1647 he rebutted the likes of Aristotle and Descartes who insisted that nature abhors a vacuum. He is also credited as the inventor of modern public transportation, having established the carrosses à cinq sols, the first modern public transport service, shortly before his death in 1662. In 1646, he and his sister Jacqueline identified with the religious movement within Catholicism known by its detractors as Jansenism. Following a religious experience in late 1654, he began writing influential works on philosophy and theology. His two most famous works date from this period: the Lettres provinciales and the Pensées, the former set in the conflict between Jansenists and Jesuits. The latter contains Pascal's wager, known in the original as the Discourse on the Machine, a fideistic probabilistic argument for why one should believe in God. In that year, he also wrote an important treatise on the arithmetical triangle. Between 1658 and 1659, he wrote on the cycloid and its use in calculating the volume of solids. Following several years of illness, Pascal died in Paris at the age of 39. == Early life and education == Pascal was born in Clermont-Ferrand, which is in France's Auvergne region, by the Massif Central. He lost his mother, Antoinette Begon, at the age of three. His father, Étienne Pascal, also an amateur mathematician, was a local judge and member of the "Noblesse de Robe". Pascal had two sisters, the younger Jacqueline and the elder Gilberte. === Move to Paris === In 1631, five years after the death of his wife, Étienne Pascal moved with his children to Paris. The newly arrived family soon hired Louise Delfault, a maid who eventually became a key member of the family. Étienne, who never remarried, decided that he alone would educate his children. The young Pascal showed an extraordinary intellectual ability, with an amazing aptitude for mathematics and science. Etienne had tried to keep his son from learning mathematics; but by the age of 12, Pascal had rediscovered, on his own, using charcoal on a tile floor, Euclid’s first thirty-two geometric propositions, and was thus given a copy of Euclid's Elements. ==== Essay on Conics ==== Particularly of interest to Pascal was a work of Desargues on conic sections. Following Desargues' thinking, the 16-year-old Pascal produced, as a means of proof, a short treatise on what was called the Mystic Hexagram, Essai pour les coniques (Essay on Conics) and sent it — his first serious work of mathematics — to Père Mersenne in Paris; it is known still today as Pascal's theorem. It states that if a hexagon is inscribed in a circle (or conic) then the three intersection points of opposite sides lie on a line (called the Pascal line). Pascal's work was so precocious that René Descartes was convinced that Pascal's father had written it. When assured by Mersenne that it was, indeed, the product of the son and not the father, Descartes dismissed it with a sniff: "I do not find it strange that he has offered demonstrations about conics more appropriate than those of the ancients," adding, "but other matters related to this subject can be proposed that would scarcely occur to a 16-year-old child." === Leaving Paris === In France at that time offices and positions could be—and were—bought and sold. In 1631, Étienne sold his position as second president of the Cour des Aides for 65,665 livres. The money was invested in a government bond which provided, if not a lavish, then certainly a comfortable income which allowed the Pascal family to move to, and enjoy, Paris, but in 1638 Cardinal Richelieu, desperate for money to carry on the Thirty Years' War, defaulted on the government's bonds. Suddenly Étienne Pascal's worth had dropped from nearly 66,000 livres to less than 7,300. Like so many others, Étienne was eventually forced to flee Paris because of his opposition to the fiscal policies of Richelieu, leaving his three children in the care of his neighbour Madame Sainctot, a great beauty with an infamous past who kept one of the most glittering and intellectual salons in all France. It was only when Jacqueline performed well in a children's play with Richelieu in attendance that Étienne was pardoned. In time, Étienne was back in good graces with the Cardinal and in 1639 had been appointed the king's commissioner of taxes in the city of Rouen—a city whose tax records, thanks to uprisings, were in utter chaos. === Pascaline === In 1642, in an effort to ease his father's endless, exhausting calculations, and recalculations, of taxes owed and paid (into which work the young Pascal had been recruited), Pascal, not yet 19, constructed a mechanical calculator capable of addition and subtraction, called Pascal's calculator or the Pascaline. Of the eight Pascalines known to have survived, four are held by the Musée des Arts et Métiers in Paris and one more by the Zwinger museum in Dresden, Germany, exhibit two of his original mechanical calculators. Although these machines are pioneering forerunners to a further 400 years of development of mechanical methods of calculation, and in a sense to the later field of computer engineering, the calculator failed to be a great commercial success. Partly because it was still quite cumbersome to use in practice, but probably primarily because it was extraordinarily expensive, the Pascaline became little more than a toy, and a status symbol, for the very rich both in France and elsewhere in Europe. Pascal continued to make improvements to his design through the next decade, and he refers to some 50 machines that were built to his design. He built 20 finished machines over the following 10 years. == Mathematics == === Probability === In 1654, prompted by his friend the Chevalier de Méré, Pascal corresponded with Pierre de Fermat on the subject of gambling problems, and from that collaboration was born the mathematical theory of probability. The specific problem was that of two players who want to finish a game early and, given the current circumstances of the game, want to divide the stakes fairly, based on the chance each has of winning the game from that point. From this discussion, the notion of expected value was introduced. John Ross writes, "Probability theory and the discoveries following it changed the way we regard uncertainty, risk, decision-making, and an individual's and society's ability to influence the course of future events." Pascal, in the Pensées, used a probabilistic argument, Pascal's wager, to justify belief in God and a virtuous life. However, Pascal and Fermat, though doing important early work in probability theory, did not develop the field very far. Christiaan Huygens, learning of the subject from the correspondence of Pascal and Fermat, wrote the first book on the subject. Later figures who continued the development of the theory include Abraham de Moivre and Pierre-Simon Laplace. The work done by Fermat and Pascal into the calculus of probabilities laid important groundwork for Leibniz's formulation of the calculus. === Treatise on the Arithmetical Triangle === Pascal's Traité du triangle arithmétique, written in 1654 but published posthumously in 1665, described a convenient tabular presentation for binomial coefficients which he called the arithmetical triangle, but is now called Pascal's triangle. The triangle can also be represented: He defined the numbers in the triangle by recursion: Call the number in the (m + 1)th row and (n + 1)th column tmn. Then tmn = tm–1,n + tm,n–1, for m = 0, 1, 2, ... and n = 0, 1, 2, ... The boundary conditions are tm,−1 = 0, t−1,n = 0 for m = 1, 2, 3, ... and n = 1, 2, 3, ... The generator t00 = 1. Pascal concluded with the proof, t m n = ( m + n ) ( m + n − 1 ) ⋯ ( m + 1 ) n ( n − 1 ) ⋯ 1 . {\displaystyle t_{mn}={\frac {(m+n)(m+n-1)\cdots (m+1)}{n(n-1)\cdots 1}}.} In the same treatise, Pascal gave an explicit statement of the principle of mathematical induction. In 1654, he proved Pascal's identity relating the sums of the p-th powers of the first n positive integers for p = 0, 1, 2, ..., k. That same year, Pascal had a religious experience, and mostly gave up work in mathematics. === Cycloid === In 1658, Pascal, while suffering from a toothache, began considering several problems concerning the cycloid. His toothache disappeared, and he took this as a heavenly sign to proceed with his research. Eight days later he had completed his essay and, to publicize the results, proposed a contest. Pascal proposed three questions relating to the center of gravity, area and volume of the cycloid, with the winner or winners to receive prizes of 20 and 40 Spanish doubloons. Pascal, Gilles de Roberval and Pierre de Carcavi were the judges, and neither of the two submissions (by John Wallis and Antoine de Lalouvère) were judged to be adequate. While the contest was ongoing, Christopher Wren sent Pascal a proposal for a proof of the rectification of the cycloid; Roberval claimed promptly that he had known of the proof for years. Wallis published Wren's proof (crediting Wren) in Wallis's Tractus Duo, giving Wren priority for the first published proof. == Physics == Pascal contributed to several fields in physics, most notably the fields of fluid mechanics and pressure. In honour of his scientific contributions, the name Pascal has been given to the SI unit of pressure and Pascal's law (an important principle of hydrostatics). He introduced a primitive form of roulette and the roulette wheel in his search for a perpetual motion machine. === Fluid dynamics === His work in the fields of hydrodynamics and hydrostatics centered on the principles of hydraulic fluids. His inventions include the hydraulic press (using hydraulic pressure to multiply force) and the syringe. He proved that hydrostatic pressure depends not on the weight of the fluid but on the elevation difference. He demonstrated this principle by attaching a thin tube to a barrel full of water and filling the tube with water up to the level of the third floor of a building. This caused the barrel to leak, in what became known as Pascal's barrel experiment. === Vacuum === By 1647, Pascal had learned of Evangelista Torricelli's experimentation with barometers. Having replicated an experiment that involved placing a tube filled with mercury upside down in a bowl of mercury, Pascal questioned what force kept some mercury in the tube and what filled the space above the mercury in the tube. At the time, most scientists including Descartes believed in a plenum, i. e. some invisible matter filled all of space, rather than a vacuum ("Nature abhors a vacuum)." This was based on the Aristotelian notion that everything in motion was a substance, moved by another substance. Furthermore, light passed through the glass tube, suggesting a substance such as aether rather than vacuum filled the space. Following more experimentation in this vein, in 1647 Pascal produced Experiences nouvelles touchant le vide ("New experiments with the vacuum"), which detailed basic rules describing to what degree various liquids could be supported by air pressure. It also provided reasons why it was indeed a vacuum above the column of liquid in a barometer tube. This work was followed by Récit de la grande expérience de l'équilibre des liqueurs ("Account of the great experiment on equilibrium in liquids") published in 1648. === First atmospheric pressure vs. altitude experiment === The Torricellian vacuum found that air pressure is equal to the weight of 30 inches of mercury. If air has a finite weight, Earth's atmosphere must have a maximum height. Pascal reasoned that if true, air pressure on a high mountain must be less than at a lower altitude. He lived near the Puy de Dôme mountain, 4,790 feet (1,460 m) tall, but his health was poor so could not climb it. On 19 September 1648, after many months of Pascal's friendly but insistent prodding, Florin Périer, husband of Pascal's elder sister Gilberte, was finally able to carry out the fact-finding mission vital to Pascal's theory. The account, written by Périer, reads: The weather was chancy last Saturday...[but] around five o'clock that morning...the Puy-de-Dôme was visible...so I decided to give it a try. Several important people of the city of Clermont had asked me to let them know when I would make the ascent...I was delighted to have them with me in this great work... ...at eight o'clock we met in the gardens of the Minim Fathers, which has the lowest elevation in town....First I poured 16 pounds of quicksilver...into a vessel...then took several glass tubes...each four feet long and hermetically sealed at one end and opened at the other...then placed them in the vessel [of quicksilver]...I found the quick silver stood at 26" and 3+1⁄2 lines above the quicksilver in the vessel...I repeated the experiment two more times while standing in the same spot...[they] produced the same result each time... I attached one of the tubes to the vessel and marked the height of the quicksilver and...asked Father Chastin, one of the Minim Brothers...to watch if any changes should occur through the day...Taking the other tube and a portion of the quick silver...I walked to the top of Puy-de-Dôme, about 500 fathoms higher than the monastery, where upon experiment...found that the quicksilver reached a height of only 23" and 2 lines...I repeated the experiment five times with care...each at different points on the summit...found the same height of quicksilver...in each case... Pascal replicated the experiment in Paris by carrying a barometer up to the top of the bell tower at the church of Saint-Jacques-de-la-Boucherie, a height of about 50 metres. The mercury dropped two lines. He found with both experiments that an ascent of 7 fathoms lowers the mercury by half a line. Note: Pascal used pouce and ligne for "inch" and "line", and toise for "fathom". In a reply to Étienne Noël, who believed in the plenum, Pascal wrote, echoing contemporary notions of science and falsifiability: "In order to show that a hypothesis is evident, it does not suffice that all the phenomena follow from it; instead, if it leads to something contrary to a single one of the phenomena, that suffices to establish its falsity." Blaise Pascal Chairs are given to outstanding international scientists to conduct their research in the Ile de France region. == Adult life: religion, literature, and philosophy == === Religious conversion === In the winter of 1646, Pascal's 58-year-old father broke his hip when he slipped and fell on an icy street of Rouen; given the man's age and the state of medicine in the 17th century, a broken hip could be a very serious condition, perhaps even fatal. Rouen was home to two of the finest doctors in France, Deslandes and de la Bouteillerie. The elder Pascal "would not let anyone other than these men attend him...It was a good choice, for the old man survived and was able to walk again..." However treatment and rehabilitation took three months, during which time La Bouteillerie and Deslandes had become regular visitors. Both men were followers of Jean Guillebert, proponent of a splinter group from Catholic teaching known as Jansenism. This still fairly small sect was making surprising inroads into the French Catholic community at that time. It espoused rigorous Augustinism. Blaise spoke with the doctors frequently, and after their successful treatment of his father, borrowed from them works by Jansenist authors. In this period, Pascal experienced a sort of "first conversion" and began to write on theological subjects in the course of the following year. Pascal fell away from this initial religious engagement and experienced a few years of what some biographers have called his "worldly period" (1648–54). His father died in 1651 and left his inheritance to Pascal and his sister Jacqueline, for whom Pascal acted as conservator. Jacqueline announced that she would soon become a postulant in the Jansenist convent of Port-Royal. Pascal was deeply affected and very sad, not because of her choice, but because of his chronic poor health; he needed her just as she had needed him. Suddenly there was war in the Pascal household. Blaise pleaded with Jacqueline not to leave, but she was adamant. He commanded her to stay, but that didn't work, either. At the heart of this was...Blaise's fear of abandonment...if Jacqueline entered Port-Royal, she would have to leave her inheritance behind...[but] nothing would change her mind. By the end of October in 1651, a truce had been reached between brother and sister. In return for a healthy annual stipend, Jacqueline signed over her part of the inheritance to her brother. Gilberte had already been given her inheritance in the form of a dowry. In early January, Jacqueline left for Port-Royal. On that day, according to Gilberte concerning her brother, "He retired very sadly to his rooms without seeing Jacqueline, who was waiting in the little parlor..." In early June 1653, after what must have seemed like endless badgering from Jacqueline, Pascal formally signed over the whole of his sister's inheritance to Port-Royal, which, to him, "had begun to smell like a cult." With two-thirds of his father's estate now gone, the 29-year-old Pascal was now consigned to genteel poverty. For a while, Pascal pursued the life of a bachelor. During visits to his sister at Port-Royal in 1654, he displayed contempt for affairs of the world but was not drawn to God. ==== Memorial ==== On the 23 of November, 1654, between 10:30 and 12:30 at night, Pascal had an intense religious experience and immediately wrote a brief note to himself which began: "Fire. God of Abraham, God of Isaac, God of Jacob, not of the philosophers and the scholars..." and concluded by quoting Psalm 119:16: "I will not forget thy word. Amen." He seems to have carefully sewn this document into his coat and always transferred it when he changed clothes; a servant discovered it only by chance after his death. This piece is now known as the Memorial. The story of a carriage accident as having led to the experience described in the Memorial is disputed by some scholars. His belief and religious commitment revitalized, Pascal visited the older of two convents at Port-Royal for a two-week retreat in January 1655. For the next four years, he regularly travelled between Port-Royal and Paris. It was at this point immediately after his conversion when he began writing his first major literary work on religion, the Provincial Letters. == Literature == In literature, Pascal is regarded as one of the most important authors of the French Classical Period and is read today as one of the greatest masters of French prose. His use of satire and wit influenced later polemicists. === The Provincial Letters === Beginning in 1656–57, Pascal published his memorable attack on casuistry, a popular ethical method used by Catholic thinkers in the early modern period (especially the Jesuits, and in particular Antonio Escobar). Pascal denounced casuistry as the mere use of complex reasoning to justify moral laxity and all sorts of sins. The 18-letter series was published between 1656 and 1657 under the pseudonym Louis de Montalte and incensed Louis XIV. The king ordered that the book be shredded and burnt in 1660. In 1661, in the midst of the formulary controversy, the Jansenist school at Port-Royal was condemned and closed down; those involved with the school had to sign a 1656 papal bull condemning the teachings of Jansen as heretical. The final letter from Pascal, in 1657, had defied Alexander VII himself. Even Pope Alexander, while publicly opposing them, nonetheless was persuaded by Pascal's arguments. Aside from their religious influence, the Provincial Letters were popular as a literary work. Pascal's use of humor, mockery, and vicious satire in his arguments made the letters ripe for public consumption, and influenced the prose of later French writers like Voltaire and Jean-Jacques Rousseau. It is in the Provincial Letters that Pascal made his oft-quoted apology for writing a long letter, as he had not had time to write a shorter one. From Letter XVI, as translated by Thomas M'Crie: 'Reverend fathers, my letters were not wont either to be so prolix, or to follow so closely on one another. Want of time must plead my excuse for both of these faults. The present letter is a very long one, simply because I had no leisure to make it shorter.' Charles Perrault wrote of the Letters: "Everything is there—purity of language, nobility of thought, solidity in reasoning, finesse in raillery, and throughout an agrément not to be found anywhere else." === Philosophy === Pascal is arguably best known as a philosopher, considered by some the second greatest French mind behind René Descartes. He was a dualist following Descartes. However, he is also remembered for his opposition to both the rationalism of the likes of Descartes and simultaneous opposition to the main countervailing epistemology, empiricism, preferring fideism. In terms of God, Descartes and Pascal disagreed. Pascal wrote that "I cannot forgive Descartes. In all his philosophy he would have been quite willing to dispense with God, but he couldn't avoid letting him put the world in motion; afterwards he didn't need God anymore". He opposed the rationalism of people like Descartes as applied to the existence of a God, preferring faith as "reason can decide nothing here". For Pascal the nature of God was such that such proofs cannot reveal God. Humans "are in darkness and estranged from God" because "he has hidden Himself from their knowledge". He cared above all about the philosophy of religion. Pascalian theology has grown out of his perspective that humans are, according to Wood, "born into a duplicitous world that shapes us into duplicitous subjects and so we find it easy to reject God continually and deceive ourselves about our own sinfulness". === Philosophy of mathematics === Pascal's major contribution to the philosophy of mathematics came with his De l'Esprit géométrique ("Of the Geometrical Spirit"), originally written as a preface to a geometry textbook for one of the famous Petites écoles de Port-Royal ("Little Schools of Port-Royal"). The work was unpublished until over a century after his death. Here, Pascal looked into the issue of discovering truths, arguing that the ideal of such a method would be to found all propositions on already established truths. At the same time, however, he claimed this was impossible because such established truths would require other truths to back them up—first principles, therefore, cannot be reached. Based on this, Pascal argued that the procedure used in geometry was as perfect as possible, with certain principles assumed and other propositions developed from them. Nevertheless, there was no way to know the assumed principles to be true. Pascal also used De l'Esprit géométrique to develop a theory of definition. He distinguished between definitions which are conventional labels defined by the writer and definitions which are within the language and understood by everyone because they naturally designate their referent. The second type would be characteristic of the philosophy of essentialism. Pascal claimed that only definitions of the first type were important to science and mathematics, arguing that those fields should adopt the philosophy of formalism as formulated by Descartes. In De l'Art de persuader ("On the Art of Persuasion"), Pascal looked deeper into geometry's axiomatic method, specifically the question of how people come to be convinced of the axioms upon which later conclusions are based. Pascal agreed with Montaigne that achieving certainty in these axioms and conclusions through human methods is impossible. He asserted that these principles can be grasped only through intuition, and that this fact underscored the necessity for submission to God in searching out truths. == Pensées == Man is only a reed, the weakest in nature, but he is a thinking reed. Pascal's most influential theological work, referred to posthumously as the Pensées ("Thoughts") is widely considered to be a masterpiece, and a landmark in French prose. When commenting on one particular section (Thought #72), Sainte-Beuve praised it as the finest pages in the French language. Will Durant hailed the Pensées as "the most eloquent book in French prose". The Pensées was not completed before his death. It was to have been a sustained and coherent examination and defense of the Christian faith, with the original title Apologie de la religion Chrétienne ("Defense of the Christian Religion"). The first version of the numerous scraps of paper found after his death appeared in print as a book in 1669 titled Pensées de M. Pascal sur la religion, et sur quelques autres sujets ("Thoughts of M. Pascal on religion, and on some other subjects") and soon thereafter became a classic. One of the Apologie's main strategies was to use the contradictory philosophies of Pyrrhonism and Stoicism, personalized by Montaigne on one hand, and Epictetus on the other, in order to bring the unbeliever to such despair and confusion that he would embrace God. == Last works and death == T. S. Eliot described him during this phase of his life as "a man of the world among ascetics, and an ascetic among men of the world." Pascal's ascetic lifestyle derived from a belief that it was natural and necessary for a person to suffer. In 1659, Pascal fell seriously ill. During his last years, he frequently tried to reject the ministrations of his doctors, saying, "Don't pity me, sickness is the natural state of Christians, because in it we are, as we should always be, in the suffering of evils, in the deprivation of all the goods and pleasures of the senses, free from all the passions that work throughout the course of life, without ambition, without avarice, in the continual expectation of death." Desiring to imitate Jesus’ poverty of spirit, in his spirit of zeal and charity, Pascal said if God allowed him to recover from his illness, he would be resolved to "have no other employment or occupation for the rest of my life than the service of the poor." Louis XIV suppressed the Jansenist movement at Port-Royal in 1661. In response, Pascal wrote one of his final works, Écrit sur la signature du formulaire ("Writ on the Signing of the Form"), exhorting the Jansenists not to give in. Later that year, his sister Jacqueline died, which convinced Pascal to cease his polemics on Jansenism. === Inventor of public transportation === Pascal's last major achievement, returning to his mechanical genius, was inaugurating one of the first land-based public transport services, the carrosses à cinq sols, a network of horse-drawn multi-seat carriages that carried passengers on five fixed routes. Pascal also designated the operation principles which were later used to plan public transportation - the carriages had a fixed route, fixed price (five sols, hence the name), and left even if there were no passengers. The lines were not commercially successful, and the last one closed by 1675. Nonetheless, he has been described as the inventor of public transportation. === Illness and death === In 1662, Pascal's illness became more violent, and his emotional condition had severely worsened since his sister's death. Aware that his health was fading quickly, he sought a move to the hospital for incurable diseases, but his doctors declared that he was too unstable to be carried. In Paris on 18 August 1662, Pascal went into convulsions and received extreme unction. He died the next morning, his last words being "May God never abandon me," and was buried in the cemetery of Saint-Étienne-du-Mont. An autopsy performed after his death revealed grave problems with his stomach and other organs of his abdomen, along with damage to his brain. Despite the autopsy, the cause of his poor health was never precisely determined, though speculation focuses on tuberculosis, stomach cancer, or a combination of the two. The headaches which affected Pascal are generally attributed to his brain lesion. == Legacy == One of the Universities of Clermont-Ferrand, France—Université Blaise Pascal—is named after him. Établissement scolaire français Blaise-Pascal in Lubumbashi, Democratic Republic of the Congo is named after Pascal. The 1969 Eric Rohmer film My Night at Maud's is based on the work of Pascal. Roberto Rossellini directed a filmed biopic, Blaise Pascal, which originally aired on Italian television in 1971. Pascal was a subject of the first edition of the 1984 BBC Two documentary, Sea of Faith, presented by Don Cupitt. The chameleon in the animated film Tangled is named for Pascal. A programming language is named for Pascal. In 2014, Nvidia announced its new Pascal microarchitecture, which is named for Pascal. The first graphics cards featuring Pascal were released in 2016. The 2017 game Nier: Automata has multiple characters named after famous philosophers; one of these is a sentient pacifistic machine named Pascal, who serves as a major supporting character. Pascal creates a village for machines to live peacefully with the androids they are at war with and acts as a parental figure for other machines trying to adapt to their newly-found individuality. The otter in the Animal Crossing series is named for Pascal. The minor planet 4500 Pascal is named in his honor. Pope Paul VI, in encyclical Populorum progressio, issued in 1967, quotes Pascal's Pensées: True humanism points the way toward God and acknowledges the task to which we are called, the task which offers us the real meaning of human life. Man is not the ultimate measure of man. Man becomes truly man only by passing beyond himself. In the words of Pascal: "Man infinitely surpasses man. In 2023, Pope Francis released an apostolic letter, Sublimitas et miseria hominis, dedicated to Blaise Pascal, in commemoration of the fourth centenary of his birth. Pascal influenced both French sociologist Pierre Bourdieu, who named his Pascalian Meditations (1997) after him, and French philosopher Louis Althusser. == Works == Essai pour les coniques [Essay on conics] (1639) Experiences nouvelles touchant le vide [New experiments with the vacuum] (1647) Récit de la grande expérience de l'équilibre des liqueurs [Account of the great experiment on equilibrium in liquids] (1648) Traité du triangle arithmétique [Treatise on the arithmetical triangle] (written c. 1654; publ. 1665) Lettres provinciales [The Provincial Letters] (1656–57) De l'Esprit géométrique [On the Geometrical Spirit] (1657 or 1658) Écrit sur la signature du formulaire (1661) Pensées [Thoughts] (incomplete at death; publ. 1670) Discours sur les passions de l'amour [Discourse on the Passion of Love] (forgery) On the Conversion of the Sinner Ecrits sur la grace [Writings on Grace] == See also == Expected value Gambler's ruin List of pioneers in computer science List of works by Eugène Guillaume Pascal's barrel Pascal distribution Pascal's mugging Pascal's pyramid Pascal's simplex Problem of points Scientific revolution == Notes == == References == == Further reading == == External links == Oeuvres complètes, volume 2 (1858) Paris: Libraire de L Hachette et Cie, link from HathiTrust. Works by Blaise Pascal at Project Gutenberg Works by or about Blaise Pascal at the Internet Archive Works by Blaise Pascal at LibriVox (public domain audiobooks) The Correspondence of Blaise Pascal in EMLO Simpson, David. ""Blaise Pascal"". Internet Encyclopedia of Philosophy. Clarke, Desmond. "Blaise Pascal". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy. Blaise Pascal at the Mathematics Genealogy Project Pensées de Blaise Pascal. Renouard, Paris 1812 (2 vols.) (Digitized) Discussion of the Pascaline, its history, mechanism, surviving examples, and modern replicas at http://things-that-count.net Pascal's Memorial in orig. French/Latin and modern English, trans. Elizabeth T. Knuth. Biography, Bibliography. (in French) Works by Blaise Pascal at Open Library BBC Radio 4. In Our Time: Pascal. Blaise Pascal featured on the 500 French Franc banknote in 1977. Archived 16 April 2009 at the Wayback Machine Blaise Pascal's works: text, concordances and frequency lists "Blaise Pascal" . Catholic Encyclopedia. 1913. Etext of Pascal's Pensées (English, in various formats) Etext of Pascal's Lettres Provinciales (English) Etext of a number of Pascal's minor works (English translation) including, De l'Esprit géométrique and De l'Art de persuader. O'Connor, John J.; Robertson, Edmund F., "Blaise Pascal", MacTutor History of Mathematics Archive, University of St Andrews
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Wikipedia:Blasius of Parma#0
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Biagio Pelacani da Parma; c. 1350 – 1416), known in English as Blasius of Parma, was an Italian philosopher, mathematician and astrologer. He popularised English and French philosophical work in Italy, where he associated both with scholastics and with early Renaissance humanists. He was professor of mathematics at the University of Padua, where he taught from 1382 to 1388; he taught also at the University of Pavia (1374? to 1378, and again 1389 to 1407), and the University of Bologna (1389 to 1382). His students included Vittorino da Feltre. == Works == Blasius around 1390 wrote a work on perspective; it drew on Alhacen, John Pecham, and Witelo. Filippo Brunelleschi may have known of the work of Blasius through Giovanni dell'Abbaco. His Tractatus de Ponderibus was based on Oxford theories on laws of motion taken up from the statics of Jordanus Nemorarius, and introduced them into Italy. He disagreed with the views of Thomas Bradwardine on proportion, and gave a proof of the mean speed theorem. He also wrote on the natural philosophy of Aristotle. == Modern editions == Questiones super tractatus logice magistri Petri Hispani, Paris: Vrin, 2001. Quaestiones circa tractatum proportionum magistri Thome Braduardini, Paris: Vrin, 2006. Questiones super perspectiva communi, Paris: Vrin, 2009. == References == C. B. Schmitt; Quentin Skinner; Eckhard Kessler; Jill Kraye (1988). The Cambridge History of Renaissance Philosophy. Cambridge University Press. ISBN 978-0-521-39748-3. == Notes ==
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Wikipedia:Blaženka Divjak#0
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Blaženka Divjak (born 1 January, 1967) is a Croatian scientist and university professor at the University of Zagreb, Faculty of Organization and Informatics in Varaždin. She served as Minister of Science and Education from 9 June, 2017 until 23 July, 2020. == Political life == Blaženka Divjak has led curricular reform of general education, reform of vocational education and training, enhancement of relevance of higher education and excellence of research in recent years. She was chairing EU Council of ministers for education and Council of ministers for research and space during Croatian presidency (January – June 2020). == Education == She holds a PhD in Mathematics from the University of Zagreb, Faculty of Science and Mathematics. She served as Vice-Rector (academic official) for students and study programs at the University of Zagreb (2010–2014). == Sources ==
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Wikipedia:Board puzzles with algebra of binary variables#0
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Board puzzles with algebra of binary variables ask players to locate the hidden objects based on a set of clue cells and their neighbors marked as variables (unknowns). A variable with value of 1 corresponds to a cell with an object. Conversely, a variable with value of 0 corresponds to an empty cell—no hidden object. == Overview == These puzzles are based on algebra with binary variables taking a pair of values, for example, (no, yes), (false, true), (not exists, exists), (0, 1). It invites the player quickly establish some equations, and inequalities for the solution. The partitioning can be used to reduce the complexity of the problem. Moreover, if the puzzle is prepared in a way that there exists a unique solution only, this fact can be used to eliminate some variables without calculation. The problem can be modeled as binary integer linear programming which is a special case of integer linear programming. == History == Minesweeper, along with its variants, is the most notable example of this type of puzzle. == Algebra with binary variables == Below the letters in the mathematical statements are used as variables where each can take the value either 0 or 1 only. A simple example of an equation with binary variables is given below: a + b = 0 Here there are two variables a and b but one equation. The solution is constrained by the fact that a and b can take only values 0 or 1. There is only one solution here, both a = 0, and b = 0. Another simple example is given below: a + b = 2 The solution is straightforward: a and b must be 1 to make a + b equal to 2. Another interesting case is shown below: a + b + c = 2 a + b ≤ 1 Here, the first statement is an equation and the second statement is an inequality indicating the three possible cases: a = 1 and b = 0, a = 0 and b = 1, and a = 0 and b = 0, The last case causes a contradiction on c by forcing c = 2, which is not possible. Therefore, either first or second case is correct. This leads to the fact that c must be 1. The modification of a large equation into smaller form is not difficult. However, an equation set with binary variables cannot be always solved by applying linear algebra. The following is an example for applying the subtraction of two equations: a + b + c + d = 3 c + d = 1 The first statement has four variables whereas the second statement has only two variables. The latter one means that the sum of c and d is 1. Using this fact on the first statement, the equations above can be reduced to a + b = 2 c + d = 1 == The algebra on a board == A game based on the algebra with binary variables can be visualized in many different ways. One generic way is to represent the right side of an equation as a clue in a cell (clue cell), and the neighbors of a clue cell as variables. A simple case is shown in Figure 1. The neighbors can be assumed to be the up/down, left/right, and corner cells that are sharing an edge or a corner. The white cells may contain a hidden object or nothing. In other words, they are the binary variables. They take place on the left side of the equations. Each clue cell, a cell with blue background in Figure 1, contains a positive number corresponding to the number of its neighbors that have hidden objects. The total number of the objects on the board can be given as an additional clue. The same board with variables marked is shown in Figure 2. === The reduction into a set of equations with binary variables === The main equation is written by using the total number of the hidden objects given. From the first figure this corresponds to the following equation a + b + c + d + e + f + g + h + i + j + k + m = 3 The other equations are composed one by one for each clue cells: a + b + c + e + f + h + i + j = 1 f + g + j + m = 1 h + i + j + k = 2 i + j + m = 2 Although there are several ways to solve the equations above, the following explicit way can be applied: It is known from the equation set that i + j + m = 2. However, since j and m are neighbors of a cell with number 1, the following is true: j + m ≤ 1. This means that the variable i must be 1. Since i = 1 and the variable i is the neighbor to the clue cell with number 1, the variables a, b, c, e, f, h, and j must be zero. The same result can be obtained by replacing i = 1 into the second equation as follows: a + b + c + e + f + h + j = 0. This is equivalent to a = 0, b = 0, c = 0, e = 0, f = 0, h = 0, j = 0. Figure 3 is obtained after Step 1 and Step 2. The grayed cells with '–' are the variables with value 0. The cell with the symbol Δ corresponds to the variable with value 1. The variable k is the only neighbor of the left most clue cell with value 2. This clue cell has one neighbor with an object and only one remaining cell with variable k. Therefore, k must be 1. Similarly, the variable m must be 1 too because it is the only remaining variable neighbor to the right most clue cell with value 2. Since k = 1, m = 1 and i = 1, we complete the marking of three hidden objects therefore d = 0, and g = 0. The final solution is given in Figure 4. === Use of uniqueness === In the example above (Figure 2), the variables a, b, c, and e are the neighbors of the clue cell 1 and they are not neighbors of any other cell. It is obvious that the followings are possible solutions: a = 1, b = 0, c = 0, e = 0 a = 0, b = 1, c = 0, e = 0 a = 0, b = 0, c = 1, e = 0 a = 0, b = 0, c = 0, e = 1 However, if the puzzle is prepared so that we should have one only one (unique) solution, we can set that all these variables a, b, c, and e must be 0. Otherwise there become more than one solutions. === Use of partitioning === Some puzzle configurations may allow the player to use partitioning for complexity reduction. An example is given in Figure 5. Each partition corresponds to a number of the objects hidden. The sum of the hidden objects in the partitions must be equal to the total number of objects hidden on the board. One possible way to determine a partitioning is to choose the lead clue cells which have no common neighbors. The cells outside of the red transparent zones in Figure 5 must be empty. In other words, there are no hidden objects in the all-white cells. Since there must be a hidden object within the upper partition zone, the third row from top shouldn't contain a hidden object. This leads to the fact that the two variable cells on the bottom row around the clue cell must have hidden objects. The rest of the solution is straightforward. === Use of try-and-check method === At some cases, the player can set a variable cell as 1 and check if any inconsistency occurs. The example in Figure 6 shows an inconsistency check. The cell marked with an hidden object Δ is under the test. Its marking leads to the set all the variables (grayed cells) to be 0. This follows the inconsistency. The clue cell marked red with value 1 does not have any remaining neighbor that can include a hidden object. Therefore, the cell under the test must not include a hidden object. In algebraic form we have two equations: a + b + c + d = 1 a + b + c + d + e + f + g = 1 Here a, b, c, and d correspond to the top four grayed cells in Figure 6. The cell with Δ is represented by the variable f, and the other two grayed cells are marked as e and g. If we set f = 1, then a = 0, b = 0, c = 0, d = 0, e = 0, g = 0. The first equation above will have the left hand side equal to 0 while the right hand side has 1. A contradiction. Try-and-check may need to be applied consequently in more than one step on some puzzles in order to reach a conclusion. This is equivalent to binary search algorithm to eliminate possible paths which lead to inconsistency. == Complexity == Because of binary variables, the equation set for the solution does not possess linearity property. In other words, the rank of the equation matrix may not always address the right complexity. The complexity of this class of puzzles can be adjusted in several ways. One of the simplest method is to set a ratio of the number of the clue cells to the total number of the cells on the board. However, this may result a largely varying complexity range for a fixed ratio. Another method is to reduce clue cells based on some problem solving strategies step by step. The complex strategies may be enabled for high complexity levels such as subtracting an equation with another one, or the higher depth of try-and-check steps. When the board size increases, the range of the problem cases increases. The ratio of the number of hidden objects to the total number of cells affects the complexity of the puzzle too. == Notes == == References == Paul Halmos, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). Alexander Schrijver, Theory of Linear and Integer Programming. John Wiley & Sons, 1986. Reprinted in 1999. ISBN 0-471-98232-6. Adam Drozdek, Data Structures and Algorithms in C++, Brooks/Cole, second edition, 2000. ISBN 0-534-37597-9.
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Wikipedia:Boaz Tsaban#0
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Boaz Tsaban (Hebrew: בועז צבאן; born February 1973) is an Israeli mathematician on the faculty of Bar-Ilan University. His research interests include selection principles within set theory and nonabelian cryptology, within mathematical cryptology. == Biography == Boaz Tsaban grew up in Or Yehuda, a city near Tel Aviv. At the age of 16 he was selected with other high school students to attend the first cycle of a special preparation program in mathematics, at Bar-Ilan University, being admitted to regular mathematics courses at the University a year later. He completed his B.Sc., M.Sc. and Ph.D. degrees with highest distinctions. Two years as a post-doctoral fellow at Hebrew University were followed by a three-year Koshland Fellowship at the Weizmann Institute of Science before he joined the Department of Mathematics, Bar-Ilan University in 2007. == Academic career == In the field of selection principles, Tsaban devised the method of omission of intervals for establishing covering properties of sets of real numbers that have certain combinatorial structures. In nonabelian cryptology he devised the algebraic span method that solved a number of computational problems that underlie a number of proposals for nonabelian public-key cryptographic schemes (such as the commutator key exchange). == Awards and recognition == Tsaban's doctoral dissertation, supervised by Hillel Furstenberg, won, with Irit Dinur, the Nessyahu prize for the best Ph.D. in mathematics in Israel in 2003. In 2009 he won the Wolf Foundation Krill Prize for Excellence in Scientific Research. == References == == External links == Boaz Tsaban's homepage at Bar-Ilan University
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Wikipedia:Bob Hearn#0
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Robert Aubrey Hearn is an American ultramarathon runner, computer scientist, and recreational mathematician. == Computer science and recreational mathematics == Hearn is originally from Oklahoma; as a student at Memorial High School (Tulsa, Oklahoma) in the early 1980s, he was passionate about solving the Rubik's Cube. He is a 1987 alumnus of Rice University; at Rice, he was a member of the Marching Owl Band and of Rice's third-place-winning team in the 1986 International Collegiate Programming Contest. Hearn was hired from Rice by StyleWare, a developer of Apple II software. With another Rice student and ICPC contestant, Jeff Erickson, he wrote TopDraw, a black-and-white bitmap drawing program that was purchased by Beagle Bros and became BeagleDraw. StyleWare was purchased by Claris, and with Scott Holdaway, Hearn became one of the two original developers of ClarisWorks, a popular integrated office suite for Apple Macintosh computers. He, Holdaway, and several other ClarisWorks developers founded Gobe Software in 1997. He later became a doctoral student of Erik Demaine at the Massachusetts Institute of Technology, co-advised by Gerald Jay Sussman. His 2006 dissertation invented nondeterministic constraint logic and used it to characterize the computational complexity of many games, puzzles, and reconfiguration problems. He and Demaine turned his dissertation into the 2009 book Games, Puzzles, and Computation. He is a member of the board of directors for Gathering 4 Gardner. == Running == Hearn first began competing in ultra-marathons when he was 42, and has set many US records for his age classes. He was honored in 2022 by a Tennessee House of Representatives Bill that named him "King of the Road" for winning the 2021 Last Annual Vol State Road Race, a 500 km (310 mi) race, of which 468 km (291 mi) passed through Tennessee. In 2023, USA Track & Field named him as their male masters ultrarunner of the year. == Personal life == Hearn is married to Elizabeth H. Hearn, an independent geophysicist, former professor at the University of British Columbia, and program director at the National Science Foundation. == References == == External links == The puzzle of running, Hearn's running blog Bob Hearn Race Results, Ultra Running Magazine Bob Hearn publications indexed by Google Scholar
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Wikipedia:Bochner identity#0
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In mathematics — specifically, differential geometry — the Bochner identity is an identity concerning harmonic maps between Riemannian manifolds. The identity is named after the American mathematician Salomon Bochner. == Statement of the result == Let M and N be Riemannian manifolds and let u : M → N be a harmonic map. Let du denote the derivative (pushforward) of u, ∇ the gradient, Δ the Laplace–Beltrami operator, RiemN the Riemann curvature tensor on N and RicM the Ricci curvature tensor on M. Then 1 2 Δ ( | ∇ u | 2 ) = | ∇ ( d u ) | 2 + ⟨ R i c M ∇ u , ∇ u ⟩ − ⟨ R i e m N ( u ) ( ∇ u , ∇ u ) ∇ u , ∇ u ⟩ . {\displaystyle {\frac {1}{2}}\Delta {\big (}|\nabla u|^{2}{\big )}={\big |}\nabla (\mathrm {d} u){\big |}^{2}+{\big \langle }\mathrm {Ric} _{M}\nabla u,\nabla u{\big \rangle }-{\big \langle }\mathrm {Riem} _{N}(u)(\nabla u,\nabla u)\nabla u,\nabla u{\big \rangle }.} == See also == Bochner's formula == References == Eells, J; Lemaire, L. (1978). "A report on harmonic maps". Bull. London Math. Soc. 10 (1): 1–68. doi:10.1112/blms/10.1.1. MR 0495450. == External links == Weisstein, Eric W. "Bochner identity". MathWorld.
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Wikipedia:Bochner–Kodaira–Nakano identity#0
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In mathematics, the Bochner–Kodaira–Nakano identity is an analogue of the Weitzenböck identity for hermitian manifolds, giving an expression for the antiholomorphic Laplacian of a vector bundle over a hermitian manifold in terms of its complex conjugate and the curvature of the bundle and the torsion of the metric of the manifold. It is named after Salomon Bochner, Kunihiko Kodaira, and Shigeo Nakano. == References == Demailly, Jean-Pierre (1986), "Sur l'identité de Bochner-Kodaira-Nakano en géométrie hermitienne", Séminaire d'analyse P. Lelong-P. Dolbeault-H. Skoda, années 1983/1984, Lecture Notes in Math., vol. 1198, Berlin, New York: Springer-Verlag, pp. 88–97, doi:10.1007/BFb0077045, ISBN 978-3-540-16762-4, MR 0874763 Demailly, Jean-Pierre (2012), Complex Analytic and Differential Geometry (PDF) Kodaira, Kunihiko (1953), "On a differential-geometric method in the theory of analytic stacks", Proceedings of the National Academy of Sciences of the United States of America, 39 (12): 1268–1273, Bibcode:1953PNAS...39.1268K, doi:10.1073/pnas.39.12.1268, ISSN 0027-8424, JSTOR 89226, MR 0066693, PMC 1063947, PMID 16589409
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Wikipedia:Bogdan Gavrilović#0
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Bogdan or Bohdan (Cyrillic: Богдан) is a Slavic masculine name that appears in all Slavic countries as well as Romania and Moldova. It is derived from the Slavic words Bog/Boh (Cyrillic: Бог), meaning "god", and dan (Cyrillic: дан), meaning "given". The name appears to be an early calque from Greek Theodore (Theodotus, Theodosius) or Hebrew Matthew with the same meaning. The name is also used as a surname in Hungary, Ukraine. Bogdana is the feminine version of the name. == Variations == The sound change of 'g' into 'h' occurred in the Ukrainian, Belarusian, Czech and Slovak languages (hence Bohdan). Although the sound change did not occur in Polish, either Bogdan or Bohdan may be used in Poland. Slavic variants include Bulgarian and Serbo-Croatian Božidar (Божидар) and Polish Bożydar, and diminutive forms and nicknames include Boguś, Bodya, Boca, Boci, Boća, Boša, Bogi, Bo, Boga Boga, Boggie. The feminine form is Bogdana or Bohdana, with variants such as Bogdanka. Names with similar meanings include Persian Khodadad, Greek Theodore, Arabic Ataullah, Hebrew Nathaniel, Jonathan, and Matthew, Latin Deodatus, French Dieudonné, and Sanskrit Devadatta. == Name days == Bulgarian: 6 January Croatian: 12 May Czech: 9 November Hungarian: 2 September Moldovian: 19 October Polish: 19 March, 17 July, 10 August or 9 October Slovak: 21 December Orthodox Christianity: 4 March == Given name == === Medieval === Bogdan of Hum (died 1252), Serbian Prince of Hum (fl. until 1249) Bogdan II of Hum, Serbian Prince of Hum (fl. 1312) Bogdan (magnate), a Macedonian magnate confused with Vratko Nemanjić as a hero of Serbian epic poetry. Bogdan I of Moldavia, Voivode of Moldavia (r. 1359–1365), and the House of Bogdan-Muşat (Bogdania was an early name for the principality of Moldavia, named after Bogdan I) Bogdan Kirizmić (fl. 1361–1371), Serbian financial manager in the service of Vukašin Mrnjavčević (fl. 1371) Bogdan (fl. 1363), kaznac in the service of Emperor Uroš V Bogdan (fl. 1407–1413), Serbian state financial manager under Despot Stefan Lazarević, merchant from Prizren and donator to Kalenić monastery Bogdan (fl. 1407), Serbian logothete in the service of Despot Stefan Bogdan, Serbian chancellor in the service of Despot Đurađ Branković (r. 1427–1456) Bogdan II of Moldavia, Voivode of Moldavia (r. 1449–1451) Bogdan (fl. 1469), Bulgarian nobleman from Nikopol Bogdan III the One-Eyed, Voivode of Moldavia (r. 1504–1517) Bohdan Khmelnytsky, Hetman of Ukraine (r. 1648–1657) === Modern === Bogdan Aldea, Romanian footballer Bogdan Andone, Romanian footballer Bogdan Apostu, Romanian footballer Bogdan Baltazar, Romanian banker Bogdan Bălan, Romanian rugby union player Bogdan Baranowski, Polish chemist Bogdan Bogdanović (architect), Serbian architect Bogdan Bogdanović (basketball), Serbian basketball player Bogdan Borusewicz, Polish politician Bogdan Bucurică, Romanian footballer Bogdan Buhuș, Romanian footballer Bohdan Bułakowski, Polish race walker Bogdan Burtea, Romanian scholar Bogdan Cistean, Romanian footballer Bogdan Ciufulescu, Romanian wrestler Bogdan Ciupek, 2022 missile explosion in Poland victim Bogdan Cotolan, Romanian footballer Bogdan Curta, Romanian folk singer Bogdan Daras, Polish sport wrestler Bogdan Diklić, Serbian actor Mihai Bogdan Dobrescu, Romanian boxer Bogdan Filov, Bulgarian archaeologist and politician Bogdan Gavrilović, Serbian mathematician Bogdan Juratoni, Romanian footballer Bogdan Lalić, Croatian chess Grandmaster Bohdan Lepky, Ukrainian writer Bogdan Lobonț, Romanian footballer Bogdan Mandić, Croat Roman Catholic priest Bogdan Maglich, American physicist Bogdan Musiał, Polish-German historian Bogdan Niculescu-Duvăz, Romanian politician Bogdan Olteanu, Romanian politician Bohdan Paczyński, Polish astronomer Bogdan Pătrașcu, Romanian footballer Bogdan Petriceicu Hasdeu, Romanian historian, philologist and politician Bogdan Planić, Serbian footballer Bohdan Pomahač, Czech plastic surgeon Bogdan Raczynski, Polish electronic musician Bogdan Stelea, Romanian footballer Bogdan Stoica, Romanian kickboxer Bogdan Tanjević, Montenegrin basketball coach Bogdan Tirnanić, Serbian journalist and essayists Bogdan Ilić, Serbian YouTuber, rapper, gamer and entertainer Bohdan Tůma, Czech actor and voice actor Bohdan Sláma, Czech director Bohdan Stupka, Ukrainian actor Bohdan Ulihrach, Czech tennis player Bohdan Warchal, Slovak violinist and dirigent Bogdan Zimonjić, Serbian priest and military commander == Surname == The surname Bogdan is one of the most common surnames in the Sisak-Moslavina County of Croatia. Notable people with the surname include: Ádám Bogdán, Hungarian goalkeeper Ana Bogdan, Romanian tennis player Christopher Bogdan, United States Air Force general Denis Bogdan, Russian volleyball player George Bogdan, Romanian physician Goran Bogdan, Croatian actor Henry Bogdan, American bassist and musician Jakub Bogdan, Slovak painter Małgorzata Bogdan, Polish statistician Radu Bogdan, American philosopher Rareș Bogdan, Romanian politician Srećko Bogdan, Croatian footballer Zvonko Bogdan, Serbian composer and singer == See also == All pages with titles containing Bogdan Bogdanski Bogdani, surname meaning son of Bogdan Bogdanov, surname meaning son of Bogdan Bogdanovich (Bogdanović), surname meaning son of Bogdan Bogusław (given name) Bogdan Corporation, a Ukrainian vehicle manufacturer Slavic names == References ==
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Wikipedia:Bohumil Bydžovský#0
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Bohumil Bydžovský (14 March 1880, in Duchcov – 6 May 1969, in Jindřichův Hradec) was a Czech mathematician, specializing in algebraic geometry and algebra. == Education and career == Bydzovsky in 1898 completed his Abitur at the Academic Gymnasium in Prague and then studied mathematics (in particular, geometry taught by Eduard Weyr) and physics at the Charles University in Prague. There Bydzovsky received his Ph.D. (promotion) in 1903 with thesis supervised by Karel Petr. Bydzovksy became a teacher at secondary schools, including the reálce in Prague-Karlín from 1907 to 1910 (with the title of Professor). In 1909 he received his habilitation in mathematics, then lectured at the Polytechnic in Prague, and then in 1911 received his habilitation in engineering. He became in 1917 professor extraordinarius and in 1920 professor ordinarius at the Charles University in Prague. He was in 1930–1931 dean of the Faculty of Sciences and in 1946 rector of the Charles University in Prague. In 1949 he became the chair of the Czechoslovak National Research Council. == Contributions == Bydzovsky wrote undergraduate textbooks in analytic geometry, linear algebra, and algebraic geometry. He did research on infinite groups, the theory of matrices and determinants, and geometric configurations. He also published papers on the history of geometry and mathematics education. == Recognition == He became in 1919 a corresponding member and in 1929 a full member of the Czech Academy of Sciences and Arts and in 1952 a full member of the Czechoslovakian Academy of Sciences. He was an Invited Speaker of the ICM in 1920 in Strasbourg, in 1924 in Toronto, in 1928 in Bologna, and in 1936 in Oslo. == Personal == He married and was the father of two sons. == References == == External links == Photos
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Wikipedia:Bohuslav Hostinský#0
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Bohuslav Hostinský (5 December 1884 – 12 April 1951) was a Czech mathematician and theoretical physicist. == Early life and family == Hostinský was born on 5 December 1884 in the New Town quarter of Prague in Bohemia, Austria-Hungary. His father Otakar Hostinský was a musicologist and professor of aesthetics at Charles University. Bohuslav Hostinský was the eldest of four siblings. He married Emilie Veselíková (1883–?) in Královské Vinohrady on 19 July 1910. According to the police archive, they lived in Královské Vinohrady. They had two children, the daughter Věra and the son Zdeněk, later well-known chess player and a professor of Brno University of Technology == Studies == After graduating from secondary school, Bohuslav Hostinský studied mathematics and physics at the Faculty of Arts of Prague's Charles University. There in 1907 he received his doctorate with a dissertation on Lie spherical geometry and in the same year he became an adjunct professor at the gymnasium in Nový Bydžov, from where in April 1908 he transferred to the gymnasium in Roudnice nad Labem and eventually to Prague's Gymnasium in Kodaňská street. For the academic year 1908–09 he studied at the Sorbonne in Paris; his time there profoundly influenced his research. == Career == In November 1911 he passed the formal examination for the acceptance of his habilitation thesis at Charles University; the examination committee included Karel Petr, Jan Sobotka, and Vincenc Strouhal. On 9 August 1920, Hostinský was appointed a full professor of theoretical physics at the Faculty of Science in Brno's Masaryk University, as well as director of the department. He worked there until his death. Bohuslav Hostinský focused, at the beginning of his career, on differential geometry and then focused on mathematical physics. His research deals with the kinetic theory of gases, probability theory, statistical mechanics, and oscillation theory. He studied the works of the Russian mathematician A. A. Markov and drew attention to them. Hostinský's work on transition probabilities and Markov chains was then further developed by many foreign experts. Laurent Mazliak gave an analysis of the letters exchanged between Hostinský and Wolfgang Doeblin. In a note submitted to the Paris Academy of Sciences in 1928, Hostinský introduced an elementary version of the ergodic theorem for a Markov chain with continuous state [Hostinský 1928]. Hostinský's work on this topic came before the spectacular development of the 1930s at the hands of Andrei N. Kolmogorov and others. Upon reading Hostinský's article, Jacques Hadamard plunged into probability for the first and only time of his life, a period referred to as his "ergodic spring" which ended at the Bologna ICM in September 1928 where Hadamard gave a talk on the ergodic principle [Bru 2003, pp. 158–159]. Between February and June 1928, Hostinský and Hadamard exchanged many letters, published several notes responding to one another, and also met during Hadamardřs journey to Czechoslovakia in May. From this moment, Hostinský acquired real international prestige, and in the 1930s, his little school in Brno became an active research center on Markovian phenomena. Bohuslav Hostinský was four times an invited speaker at the International Congress of Mathematicians — in Cambridge, England in 1912, in Strasbourg in 1920, in Bologna in 1928, and in Zurich in 1932. He published about 140 papers and several monographs. From the establishment of the Faculty of Science in Masaryk University until 1948 (with an interruption from 1934 to 1939), he was the editor of the Spisů (research journal) published by this faculty. He was several times dean of the Faculty of Science and from 1929 to 1930 rector. He was a member of many scientific societies and in 1933 he was elected a member extraordinarius of the Czechoslovak Academy of Sciences. He actively participated in the activities of the Brno chapter of the Union of Czech mathematicians and physicists; in the difficult years from 1942 to 1945 he was the chair of the Brno chapter. == Selected publications == === Articles === Hostinský, B. (1909). "Sur un théorème analogue au théorème de Meusnier" (PDF). Nouvelles annales de mathématiques: Journal des candidats aux écoles polytechnique et normale. 4e série. 9: 399–403. Hostinský, B. (1909). "Sur quelques figures déterminées par les éléments infiniment voisins d'une courbe gauche" (PDF). Journal de mathématiques pures et appliquées. 6e série. 5: 263–326. Hostinský, Bohuslav (1920). "Sur une nouvelle solution du problème de l'aiguille". Bulletin des sciences mathématiques. 44: 126–136. Hostinský, B. (1926). "Sur la méthode des fonctions arbitraires dans le Calcul des probabilités". Acta Mathematica. 49 (1–2): 95–113. doi:10.1007/BF02543850. Hostinský, Bohuslav (1928). "Sur les probabilités relatives aux transformations répétées". Comptes rendus hebdomadaires des séances de l'Académie des Sciences. 186: 59–61. Hostinský, Bohuslav (1932). "Application du Calcul des Probabilités à la Théorie du mouvement Brownien" (PDF). Annales de l'Institut Henri Poincaré. 3 (1): 1–74. Hostinský, B. (1934). "Résolution d'une équation fonctionnelle considérée par M. Hadamard" (PDF). Bulletin de la Société Mathématique de France. 62: 151–166. doi:10.24033/bsmf.1218. Hostinský, Bohuslav (1935). "Sur les progrès récents de la théorie des probabilités". Časopis pro pěstování matematiky a fysiky. 64 (5): 94–106. doi:10.21136/CPMF.1935.121253. Hostinský, Bohuslav (1935). "Principe d'Huyghens" (PDF). Časopis pro pěstování matematiky a fysiky. 64 (6): 230–231. doi:10.21136/CPMF.1935.123580. Hostinský, Bohuslav (1935). "Sur les quatre sommets d'un ovale" (PDF). Časopis pro pěstování matematiky a fysiky. 64 (6): 193. doi:10.21136/CPMF.1935.123587. Hostinský, Bohuslav (1937). "Sur les probabilités relatives aux variables aléatoires liées entre elles Applications diverses" (PDF). Annales de l'Institut Henri Poincaré. 7 (2): 69–119. === Monographs and books === Hostinský, Bohuslav (1915). Diferenciální geometrie křivek a ploch. Knihovna spisů matematických a fysikálních. V Praze : Nákladem Jednoty českých mathematiků a fysiků.{{cite book}}: CS1 maint: publisher location (link) (Differential geometry of curves and surfaces) Hostinský, B. (1924). Mechanika tuhých těles. Knihovna spisů matematických a fyzikálních, JČMF. (Mechanics of rigid bodies) Hostinský, Bohuslav (1926). Sur les probabilités géométriques. Publications de la Faculté des Sciences de l’Université Masaryk; 26 pages{{cite book}}: CS1 maint: postscript (link) with Vito Volterra: Opérations infinitésimales linéaires. Paris: Gauthier-Villars. 1938. == References ==
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Wikipedia:Bonnie Gold#0
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Bonnie Gold (born 1948) is an American mathematician, mathematical logician, philosopher of mathematics, and mathematics educator. She is a professor emerita of mathematics at Monmouth University. == Education and career == Gold completed her Ph.D. in 1976 at Cornell University, under the supervision of Michael D. Morley. She was the chair of the mathematics department at Wabash College before moving to Monmouth, where she also became department chair. == Contributions == The research from Gold's dissertation, Compact and ω {\displaystyle \omega } -compact formulas in L ω 1 , ω {\displaystyle L_{\omega _{1},\omega }} , was later published in the journal Archiv für Mathematische Logik und Grundlagenforschung, and concerned infinitary logic. With Sandra Z. Keith and William A. Marion she co-edited Assessment Practices in Undergraduate Mathematics, published by the Mathematical Association of America (MAA) in 1999. With Roger A. Simons, Gold is also the editor of another book, Proof and Other Dilemmas: Mathematics and Philosophy (MAA, 2008). Her essay "How your philosophy of mathematics impacts your teaching" was selected for inclusion in The Best Writing on Mathematics 2012. In it, she argues that the philosophy of mathematics affects the teaching of mathematics even when the teacher's philosophical principles are implicit and unexamined. == Recognition == In 2012, Gold became the winner of the 22nd Louise Hay Award of the Association for Women in Mathematics for her contributions to mathematics education. The award citation noted her work in educational assessment for undergraduate study in mathematics. == References == == External links == Home page
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Wikipedia:Book of Lemmas#0
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The Book of Lemmas or Book of Assumptions (Arabic Maʾkhūdhāt Mansūba ilā Arshimīdis) is a book attributed to Archimedes by Thābit ibn Qurra, though the authorship of the book is questionable. It consists of fifteen propositions (lemmas) on circles. == History == === Translations === The Book of Lemmas was first introduced in Arabic by Thābit ibn Qurra; he attributed the work to Archimedes. A translation from Arabic into Latin by John Greaves and revised by Samuel Foster (c. 1650) was published in 1659 as Lemmata Archimedis. Another Latin translation by Abraham Ecchellensis and edited by Giovanni A. Borelli was published in 1661 under the name Liber Assumptorum. T. L. Heath translated Heiburg's Latin work into English in his The Works of Archimedes. A more recently discovered manuscript copy of Thābit ibn Qurra's Arabic translation was translated into English by Emre Coşkun in 2018. === Authorship === The original authorship of the Book of Lemmas has been in question because in proposition four, the book refers to Archimedes in third person; however, it has been suggested that it may have been added by the translator. Another possibility is that the Book of Lemmas may be a collection of propositions by Archimedes later collected by a Greek writer. == New geometrical figures == The Book of Lemmas introduces several new geometrical figures. === Arbelos === Archimedes first introduced the arbelos (shoemaker's knife) in proposition four of his book: If AB be the diameter of a semicircle and N any point on AB, and if semicircles be described within the first semicircle and having AN, BN as diameters respectively, the figure included between the circumferences of the three semicircles is "what Archimedes called αρβηλος"; and its area is equal to the circle on PN as diameter, where PN is perpendicular to AB and meets the original semicircle in P. The figure is used in propositions four through eight. In propositions five, Archimedes introduces the Archimedes's twin circles, and in proposition eight, he makes use what would be the Pappus chain, formally introduced by Pappus of Alexandria. === Salinon === Archimedes first introduced the salinon (salt cellar) in proposition fourteen of his book: Let ACB be a semicircle on AB as diameter, and let AD, BE be equal lengths measured along AB from A, B respectively. On AD, BE as diameters describe semicircles on the side towards C, and on DE as diameter a semicircle on the opposite side. Let the perpendicular to AB through O, the centre of the first semicircle, meet the opposite semicircles in C, F respectively. Then shall the area of the figure bounded by the circumferences of all the semicircles be equal to the area of the circle on CF as diameter. Archimedes proved that the salinon and the circle are equal in area. == Propositions == If two circles touch at A, and if CD, EF be parallel diameters in them, ADF is a straight line. Let AB be the diameter of a semicircle, and let the tangents to it at B and at any other point D on it meet in T. If now DE be drawn perpendicular to AB, and if AT, DE meet in F, then DF = FE. Let P be any point on a segment of a circle whose base is AB, and let PN be perpendicular to AB. Take D on AB so that AN = ND. If now PQ be an arc equal to the arc PA, and BQ be joined, then BQ, BD shall be equal. If AB be the diameter of a semicircle and N any point on AB, and if semicircles be described within the first semicircle and having AN, BN as diameters respectively, the figure included between the circumferences of the three semicircles is "what Archimedes called αρβηλος"; and its area is equal to the circle on PN as diameter, where PN is perpendicular to AB and meets the original semicircle in P. Let AB be the diameter of a semicircle, C any point on AB, and CD perpendicular to it, and let semicircles be described within the first semicircle and having AC, CB as diameters. Then if two circles be drawn touching CD on different sides and each touching two of the semicircles, the circles so drawn will be equal. Let AB, the diameter of a semicircle, be divided at C so that AC = 3/2 × CB [or in any ratio]. Describe semicircles within the first semicircle and on AC, CB as diameters, and suppose a circle drawn touching the all three semicircles. If GH be the diameter of this circle, to find relation between GH and AB. If circles are circumscribed about and inscribed in a square, the circumscribed circle is double of the inscribed square. If AB be any chord of a circle whose centre is O, and if AB be produced to C so that BC is equal to the radius; if further CO meets the circle in D and be produced to meet the circle the second time in E, the arc AE will be equal to three times the arc BD. If in a circle two chords AB, CD which do not pass through the centre intersect at right angles, then (arc AD) + (arc CB) = (arc AC) + (arc DB). Suppose that TA, TB are two tangents to a circle, while TC cuts it. Let BD be the chord through B parallel to TC, and let AD meet TC in E. Then, if EH be drawn perpendicular to BD, it will bisect it in H. If two chords AB, CD in a circle intersect at right angles in a point O, not being the centre, then AO2 + BO2 + CO2 + DO2 = (diameter)2. If AB be the diameter of a semicircle, and TP, TQ the tangents to it from any point T, and if AQ, BP be joined meeting in R, then TR is perpendicular to AB. If a diameter AB of a circle meet any chord CD, not a diameter, in E, and if AM, BN be drawn perpendicular to CD, then CN = DM. Let ACB be a semicircle on AB as diameter, and let AD, BE be equal lengths measured along AB from A, B respectively. On AD, BE as diameters describe semicircles on the side towards C, and on DE as diameter a semicircle on the opposite side. Let the perpendicular to AB through O, the centre of the first semicircle, meet the opposite semicircles in C, F respectively. Then shall the area of the figure bounded by the circumferences of all the semicircles be equal to the area of the circle on CF as diameter. Let AB be the diameter of a circle., AC a side of an inscribed regular pentagon, D the middle point of the arc AC. Join CD and produce it to meet BA produced in E; join AC, DB meeting in F, and Draw FM perpendicular to AB. Then EM = (radius of circle). == References ==
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Wikipedia:Book of Optics#0
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The Book of Optics (Arabic: كتاب المناظر, romanized: Kitāb al-Manāẓir; Latin: De Aspectibus or Perspectiva; Italian: Deli Aspecti) is a seven-volume treatise on optics and other fields of study composed by the medieval Arab scholar Ibn al-Haytham, known in the West as Alhazen or Alhacen (965–c. 1040 AD). The Book of Optics presented experimentally founded arguments against the widely held extramission theory of vision (as held by Euclid in his Optica), and proposed the modern intromission theory, the now accepted model that vision takes place by light entering the eye.: 60–67 The book is also noted for its early use of the scientific method, its description of the camera obscura, and its formulation of Alhazen's problem. The book extensively affected the development of optics, physics and mathematics in Europe between the 13th and 17th centuries. == Vision theory == Before the Book of Optics was written, two theories of vision existed. The extramission or emission theory was forwarded by the mathematicians Euclid and Ptolemy, who asserted that certain forms of radiation are emitted from the eyes onto the object which is being seen. When these rays reached the object they allowed the viewer to perceive its color, shape and size. An early version of the intromission theory, held by the followers of Aristotle and Galen, argued that sight was caused by agents, which were transmitted to the eyes from either the object or from its surroundings. Al-Haytham offered many reasons against the extramission theory, pointing to the fact that eyes can be damaged by looking directly at bright lights, such as the sun.: 313–314 He wrote of the low probability that the eye can fill the entirety of space as soon as the eyelids are opened as an observer looks up into the night sky. Using the intromission theory as a foundation, he formed his own theory that an object emits rays of light from every point on its surface which then travel in all directions, thereby allowing some light into a viewer's eyes. According to this theory, the object being viewed is considered to be a compilation of an infinite number of points, from which rays of light are projected. == Light and color theory == In the Book of Optics, al-Haytham hypothesized the existence of primary and secondary light, with primary light being the stronger or more intense of the two. The book describes how the essential form of light comes from self-luminous bodies and that accidental light comes from objects that obtain and emit light from those self-luminous bodies. According to Ibn al-Haytham, primary light comes from self-luminous bodies and secondary light is the light that comes from accidental objects.: 317 Accidental light can only exist if there is a source of primary light. Both primary and secondary light travel in straight lines. Transparency is a characteristic of a body that can transmit light through them, such as air and water, although no body can completely transmit light or be entirely transparent. Opaque objects are those through which light cannot pass through directly, although there are degrees of opaqueness which determine how much light can actually pass through. Opaque objects are struck with light and can become luminous bodies themselves which radiate secondary light. Light can be refracted by going through partially transparent objects and can also be reflected by striking smooth objects such as mirrors, traveling in straight lines in both cases. Al-Haytham presented many experiments in Optics that upheld his theories about light and its transmission. He also wrote that color acts much like light, being a distinct quality of a form and travelling from every point on an object in straight lines. Through experimentation he concluded that color cannot exist without air. == Anatomy of the eye and visual process == As objects radiate light in straight lines in all directions, the eye must also be hit with this light over its outer surface. This idea presented a problem for al-Haytham and his predecessors, as if this was the case, the rays received by the eye from every point on the object would cause a blurred image. Al-Haytham solved this problem using his theory of refraction. He argued that although the object sends an infinite number of rays of light to the eye, only one of these lines falls on the eye perpendicularly: the other rays meet the eye at angles that are not perpendicular. According to al-Haytham, this causes them to be refracted and weakened. He believed that all the rays other than the one that hits the eye perpendicularly are not involved in vision.: 315–316 In al-Haytham's structure of the eye, the crystalline humor is the part that receives light rays from the object and forms a visual cone, with the object being perceived as the base of the cone and the center of the crystalline humor in the eye as the vertex. Other parts of the eye are the aqueous humor in front of the crystalline humor and the vitreous humor at the back. These, however, do not play as critical of a role in vision as the crystalline humor. The crystalline humor transmits the image it perceives to the brain through an optic nerve. == Volumes == Book I deals with al-Haytham's theories on light, colors, and vision. Book II is where al-Haytham presents his theory of visual perception. Book III and Book IV present al-Haytham's ideas on the errors in visual perception with Book VI focusing on errors related to reflection. Book V and Book VI provide experimental evidence for al-Haytham's theories on reflection. Book VII deals with the concept of refraction. == Influence == The Book of Optics was most strongly influenced by Ptolemy's Optics, while the description of the anatomy and physiology of the eye was based upon an account by Galen. The Book of Optics was translated into Latin by an unknown scholar at the end of the 12th (or the beginning of the 13th) century.: 209–210 The work was influential during the Middle Ages.: 86. It was printed by Friedrich Risner in 1572, as part of his collection Opticae thesaurus. This included a book on twilight falsely attributed to Alhazen, as well as a work on optics by Vitello. == See also == History of optics Ibn Sahl Scientific method == English translations == Sabra, A. I., ed. (1983), The Optics of Ibn al-Haytham, Books I–II–III: On Direct Vision. The Arabic text, edited and with Introduction, Arabic-Latin Glossaries and Concordance Tables, Kuwait: National Council for Culture, Arts and Letters Sabra, A. I., ed. (2002), The Optics of Ibn al-Haytham. Edition of the Arabic Text of Books IV–V: On Reflection and Images Seen by Reflection. 2 vols, Kuwait: The National Council for Culture, Arts and Letters The Optics of Ibn al-Haytham. Books I–II–III: On Direct Vision. English Translation and Commentary. 2 vols, Studies of the Warburg Institute, vol. 40, translated by Sabra, A. I., London: The Warburg Institute, University of London, 1989, ISBN 0-85481-072-2 Smith, A. Mark, ed. (2001), "Alhacen's Theory of Visual Perception: A Critical Edition, with English Translation and Commentary, of the First Three Books of Alhacen's De Aspectibus, the Medieval Latin Version of Ibn al-Haytham's Kitāb al-Manāẓir, 2 vols.", Transactions of the American Philosophical Society, 91 (4–5), translated by Smith, Philadelphia: American Philosophical Society, ISBN 0-87169-914-1, OCLC 47168716 Books I–III (2001 – 91(4)) Vol. 1 Commentary and Latin text; – 91(5) Vol 2 English translation, Book I: TOC pp. 339–341, Book II: TOC pp. 415–416, Book III: TOC pp. 559–60, Notes 681ff, Bibl. Smith, A. Mark, ed. (2006), "Alhacen on the principles of reflection: A Critical Edition, with English Translation and Commentary, of books 4 and 5 of Alhacen's De Aspectibus, the Medieval Latin Version of Ibn al-Haytham's Kitāb al-Manāẓir, 2 vols.", Transactions of the American Philosophical Society, 95 (2–3), translated by Smith, Philadelphia: American Philosophical Society 2 vols: . (Philadelphia: American Philosophical Society), 2006 – 95(#2) Books 4–5 Vol. 1 Commentary and Latin text; 95(#3) Vol. 2 English translation, Notes, Bibl. Smith, A. Mark, ed. and trans. (2008) Alhacen on Image-formation and distortion in mirrors : a critical edition, with English translation and commentary, of Book 6 of Alhacen's De aspectibus, [the Medieval Latin version of Ibn al-Haytham's Kitāb al-Manāzir], Transactions of the American Philosophical Society, 2 vols: Vol. 1 98(#1, section 1 – Vol. 1 Commentary and Latin text); 98(#1, section 2 – Vol. 2 English translation). (Philadelphia: American Philosophical Society), 2008. Book 6 (2008) Vol. 1 Commentary and Latin text; Vol. 2 English translation, Notes, Bibl. Smith, A. Mark, ed. and trans. (2010) Alhacen on Refraction : a critical edition, with English translation and commentary, of Book 7 of Alhacen's De aspectibus, [the Medieval Latin version of Ibn al-Haytham's Kitāb al-Manāzir], Transactions of the American Philosophical Society, 2 vols: 100(#3, section 1 – Vol. 1, Introduction and Latin text); 100(#3, section 2 – Vol. 2 English translation). (Philadelphia: American Philosophical Society), 2010. Book 7 (2010) Vol. 1 Commentary and Latin text; Vol. 2 English translation, Notes, Bibl. == References == === Notes === === Citations ===
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Wikipedia:Book on Numbers and Computation#0
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The Book on Numbers and Computation (Chinese: 筭數書; pinyin: Suàn shù shū), or the Writings on Reckoning, is one of the earliest known Chinese mathematical treatises. It was written during the early Western Han dynasty, sometime between 202 BC and 186 BC. It was preserved among the Zhangjiashan Han bamboo texts and contains similar mathematical problems and principles found in the later Eastern Han period text of The Nine Chapters on the Mathematical Art. == Discovery == The text was found in tomb M247 of the burial grounds near Zhangjiashan, Jiangling County, in Hubei province, excavated in December–January 1983–1984. This tomb belonged to an anonymous civil servant in early West Han dynasty. In the tomb were 1200 bamboo strips written in ink. Originally the strips were bound together with string, but the string had rotted away and it took Chinese scholars 17 years to piece together the strips. As well as the mathematical work the strips covered government statutes, law reports and therapeutic gymnastics. On the back of the sixth strip, the top has a black square mark, followed by the three characters 筭數書, which serve as the title of the rolled up book. == Content == The Suàn shù shū consists of 200 strips of bamboo written in ink, 180 strips are intact, the others have rotted. They consist of 69 mathematical problems from a variety of sources, two names Mr Wáng and Mr Yáng were found, probably two of the writers. Each problem has a question, an answer, followed by a method. The problems cover elementary arithmetic; fractions; inverse proportion; factorization of numbers; geometric progressions, in particular interest rate calculations and handling of errors; conversion between different units; the false position method for finding roots and the extraction of approximate square roots; calculation of the volume of various 3-dimensional shapes; relative dimensions of a square and its inscribed circle; calculation of unknown side of rectangle, given area and one side. All the calculations about circumference and area of circle are approximate, equivalent to taking π = 3. Calculations of pi were made more accurate with the work of Liu Xin (c. 46 BC – 23 AD), Zhang Heng (78–139 AD), Liu Hui (fl. 3rd century AD), and Zu Chongzhi (429–500). Prior to discovery the oldest Chinese mathematical text were the Zhoubi Suanjing and The Nine Chapters on the Mathematical Art which dates from around 100 CE. Many topics are covered in both texts, however, error correction problems only appear in the Suàn shù shū, and the last two chapter of the nine chapters have no corresponding material in the Suàn shù shū. The text has been translated to English by Christopher Cullen, director of the Needham Research Institute. == Notes == == References == Christopher Cullen: The Suan shu shu Writings on reckoning, Needham Research Institute, pdf free download [1] Cullen, Christopher (2007). "The Suàn shù shū, "Writings on reckoning": Rewriting the history of early Chinese mathematics in the light of an excavated manuscript". Historia Mathematica. 34: 10–44. doi:10.1016/j.hm.2005.11.006. Dauben, Joseph W. (2004). "The Suan Shu Shu (A Book on Numbers and Computation), A Preliminary Investigation" in Form, Zahl, Ordnung, 151–168. München: Franz Steiner Verlag. ISBN 3-515-08525-4. Dauben, Joseph W. (2007). "Chinese Mathematics" in The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, 187–384. Edited by Victor J. Katz. Princeton: Princeton University Press. ISBN 0-691-11485-4. Guilin Liu, Lisheng Feng, Airong Jiang, and Xiaohui Zheng. (2003). The Development of E-mathematics Resources at Tsinghua University Library (THUL)," in Electronic Information and Communication in Mathematics, 1–13. Edited by Fengshen Bai and Bernd Wegner. Berlin: Springer. ISBN 3-540-40689-1. Stephanie Pain, Histories: China's oldest mathematical puzzles, New Scientist, 30 July 2006. [2] Péng Hào, Zhāngjiāshān Hànjiǎn "Suàn shù shū" zhùshì (The Hàn dynasty book on wooden strips "Suàn shù shū" found at Zhāngjiāshān with a commentary and explanation) Beijing, Science Press, (2001). Wu Wenjun ed, Zhong Guo Shu Xue Shi Da Xi(The Grand Series of History of Chinese Mathematics) vol 1, chapter 2, "Suan Shu Shu". ISBN 7-303-04555-4 Rémi Anicotte (2019). Le Livre sur les calculs effectués avec des bâtonnets – Un manuscrit du -IIe siècle excavé à Zhangjiashan, Paris: Presses de l'Inalco. http://www.inalco.fr/publication/livre-calculs-effectues-batonnets-manuscrit-iie-siecle-excave-zhangjiashan == External links == Full text of book Christopher Cullen: Suan shu shu Writings on reckoning Needham Research Institute
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Wikipedia:Book on the Measurement of Plane and Spherical Figures#0
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The Book on the Measurement of Plane and Spherical Figures (Arabic: كتاب معرفة مساحة الأشكال البسيطة والكريّة, Kitāb maʿrifah masāḥat al-ashkāl al-basīṭah wa-al-kuriyyah) was the most important of the works produced by the Banū Mūsā (three 9th century Persian brothers who worked in Baghdad). A Latin translation by the 12th century Italian astrologer Gerard of Cremona was made, entitled Liber trium fratrum de geometria and Verba filiorum Moysi filii Sekir. The original work in Arabic was edited by the Persian polymath Naṣīr al-Dīn al-Ṭūsī in the 13th century. The original work in Arabic is not extant, but its contents are known from later translations. The treatise, which is about geometry, was similar to two books by Archimedes, On the measurement of the circle and On the sphere and the cylinder. It was used extensively in the Middle Ages, and was quoted by authors such as Thābit ibn Qurra, Ibn al-Haytham, Leonardo Fibonacci (in his Practica geometriae), Jordanus de Nemore, and Roger Bacon. It deals with the geometrical concepts of area and volume, angle trisection, construction, and conic sections. It includes theorems not known to the Greeks. The book was re-published in Latin with an English translation by the American historian Marshall Clagett, who has also summarized how the work influenced mathematicians during the Middle Ages. == See also == Mathematics in the medieval Islamic world == Notes == == References == == Sources == Casulleras, Josep (2007). "Banū Mūsā". In Hockey, Thomas; et al. (eds.). Biographical Encyclopedia of Astronomers. Springer Publishers. pp. 92–94. doi:10.1007/978-0-387-30400-7_1433. ISBN 978-0-387-31022-0. (PDF version) al-Dabbagh, J. (1970). "Banū Mūsā". In Gillispie, Charles Coulston; Holmes, Frederic Lawrence (eds.). Dictionary of Scientific Biography. Vol. 1. New York: Scribner. ISBN 978-0-684-10114-9. OCLC 755137603. Papadopoulos, Athanase (2016). "Roshdi Rashed, Historian of Greek and Arabic Mathematics". Ganita Bharatı (Indian Mathematics). 38 (2). The Indian Society for History of Mathematics: 1–26. Pascual, Lluís (2015). "An Archimedean Proposition Presented by the Brothers Banū Mūsā and Recovered in the Kitāb al-Istikmāl (eleventh century)" (PDF). Suhayl. International Journal for the History of the Exact and Natural Sciences in Islamic Civilisation. 14: 115–143. ISSN 2013-620X. Pingree, David (1988). "Banū Mūsā". Encyclopædia Iranica. == Further reading == Rashed, Roshdi (2012). El-Bizri, Nader (ed.). Founding Figures and Commentators in Arabic Mathematics: A History of Arabic Sciences and Mathematics. Vol. 1. Abingdon, UK; New York: Routledge. ISBN 978-11366-2-000-3. A manuscript facsimile of Kitāb al-mutawassiṭāt kept at Columbia University, New York (via the Internet Archive). The treatise Kitāb maʻrifat masāḥat al-ashkāl al-basīṭah wa-al-kurīyah is located from pp. 253–265 (f. 116 to 122).
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Wikipedia:Borel subalgebra#0
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In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra g {\displaystyle {\mathfrak {g}}} is a maximal solvable subalgebra. The notion is named after Armand Borel. If the Lie algebra g {\displaystyle {\mathfrak {g}}} is the Lie algebra of a complex Lie group, then a Borel subalgebra is the Lie algebra of a Borel subgroup. == Borel subalgebra associated to a flag == Let g = g l ( V ) {\displaystyle {\mathfrak {g}}={\mathfrak {gl}}(V)} be the Lie algebra of the endomorphisms of a finite-dimensional vector space V over the complex numbers. Then to specify a Borel subalgebra of g {\displaystyle {\mathfrak {g}}} amounts to specify a flag of V; given a flag V = V 0 ⊃ V 1 ⊃ ⋯ ⊃ V n = 0 {\displaystyle V=V_{0}\supset V_{1}\supset \cdots \supset V_{n}=0} , the subspace b = { x ∈ g ∣ x ( V i ) ⊂ V i , 1 ≤ i ≤ n } {\displaystyle {\mathfrak {b}}=\{x\in {\mathfrak {g}}\mid x(V_{i})\subset V_{i},1\leq i\leq n\}} is a Borel subalgebra, and conversely, each Borel subalgebra is of that form by Lie's theorem. Hence, the Borel subalgebras are classified by the flag variety of V. == Borel subalgebra relative to a base of a root system == Let g {\displaystyle {\mathfrak {g}}} be a complex semisimple Lie algebra, h {\displaystyle {\mathfrak {h}}} a Cartan subalgebra and R the root system associated to them. Choosing a base of R gives the notion of positive roots. Then g {\displaystyle {\mathfrak {g}}} has the decomposition g = n − ⊕ h ⊕ n + {\displaystyle {\mathfrak {g}}={\mathfrak {n}}^{-}\oplus {\mathfrak {h}}\oplus {\mathfrak {n}}^{+}} where n ± = ∑ α > 0 g ± α {\displaystyle {\mathfrak {n}}^{\pm }=\sum _{\alpha >0}{\mathfrak {g}}_{\pm \alpha }} . Then b = h ⊕ n + {\displaystyle {\mathfrak {b}}={\mathfrak {h}}\oplus {\mathfrak {n}}^{+}} is the Borel subalgebra relative to the above setup. (It is solvable since the derived algebra [ b , b ] {\displaystyle [{\mathfrak {b}},{\mathfrak {b}}]} is nilpotent. It is maximal solvable by a theorem of Borel–Morozov on the conjugacy of solvable subalgebras.) Given a g {\displaystyle {\mathfrak {g}}} -module V, a primitive element of V is a (nonzero) vector that (1) is a weight vector for h {\displaystyle {\mathfrak {h}}} and that (2) is annihilated by n + {\displaystyle {\mathfrak {n}}^{+}} . It is the same thing as a b {\displaystyle {\mathfrak {b}}} -weight vector (Proof: if h ∈ h {\displaystyle h\in {\mathfrak {h}}} and e ∈ n + {\displaystyle e\in {\mathfrak {n}}^{+}} with [ h , e ] = 2 e {\displaystyle [h,e]=2e} and if b ⋅ v {\displaystyle {\mathfrak {b}}\cdot v} is a line, then 0 = [ h , e ] ⋅ v = 2 e ⋅ v {\displaystyle 0=[h,e]\cdot v=2e\cdot v} .) == See also == Borel subgroup Parabolic Lie algebra == References ==
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Wikipedia:Borel's lemma#0
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In mathematics, Borel's lemma, named after Émile Borel, is an important result used in the theory of asymptotic expansions and partial differential equations. == Statement == Suppose U is an open set in the Euclidean space Rn, and suppose that f0, f1, ... is a sequence of smooth functions on U. If I is any open interval in R containing 0 (possibly I = R), then there exists a smooth function F(t, x) defined on I×U, such that ∂ k F ∂ t k | ( 0 , x ) = f k ( x ) , {\displaystyle \left.{\frac {\partial ^{k}F}{\partial t^{k}}}\right|_{(0,x)}=f_{k}(x),} for k ≥ 0 and x in U. == Proof == Proofs of Borel's lemma can be found in many text books on analysis, including Golubitsky & Guillemin (1974) and Hörmander (1990), from which the proof below is taken. Note that it suffices to prove the result for a small interval I = (−ε,ε), since if ψ(t) is a smooth bump function with compact support in (−ε,ε) equal identically to 1 near 0, then ψ(t) ⋅ F(t, x) gives a solution on R × U. Similarly using a smooth partition of unity on Rn subordinate to a covering by open balls with centres at δ⋅Zn, it can be assumed that all the fm have compact support in some fixed closed ball C. For each m, let F m ( t , x ) = t m m ! ⋅ ψ ( t ε m ) ⋅ f m ( x ) , {\displaystyle F_{m}(t,x)={t^{m} \over m!}\cdot \psi \left({t \over \varepsilon _{m}}\right)\cdot f_{m}(x),} where εm is chosen sufficiently small that ‖ ∂ α F m ‖ ∞ ≤ 2 − m {\displaystyle \|\partial ^{\alpha }F_{m}\|_{\infty }\leq 2^{-m}} for |α| < m. These estimates imply that each sum ∑ m ≥ 0 ∂ α F m {\displaystyle \sum _{m\geq 0}\partial ^{\alpha }F_{m}} is uniformly convergent and hence that F = ∑ m ≥ 0 F m {\displaystyle F=\sum _{m\geq 0}F_{m}} is a smooth function with ∂ α F = ∑ m ≥ 0 ∂ α F m . {\displaystyle \partial ^{\alpha }F=\sum _{m\geq 0}\partial ^{\alpha }F_{m}.} By construction ∂ t m F ( t , x ) | t = 0 = f m ( x ) . {\displaystyle \partial _{t}^{m}F(t,x)|_{t=0}=f_{m}(x).} Note: Exactly the same construction can be applied, without the auxiliary space U, to produce a smooth function on the interval I for which the derivatives at 0 form an arbitrary sequence. == See also == Non-analytic smooth function § Application to Taylor series == References == Erdélyi, A. (1956), Asymptotic expansions, Dover Publications, pp. 22–25, ISBN 0486603180 {{citation}}: ISBN / Date incompatibility (help) Golubitsky, M.; Guillemin, V. (1974), Stable mappings and their singularities, Graduate Texts in Mathematics, vol. 14, Springer-Verlag, ISBN 0-387-90072-1 Hörmander, Lars (1990), The analysis of linear partial differential operators, I. Distribution theory and Fourier analysis (2nd ed.), Springer-Verlag, p. 16, ISBN 3-540-52343-X This article incorporates material from Borel lemma on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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Wikipedia:Boris Berezovsky (businessman)#0
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Boris Abramovich Berezovsky (Russian: Борис Абрамович Березовский, Hebrew: בוריס ברזובסקי; 23 January 1946 – 23 March 2013), also known as Platon Elenin, was a Russian business oligarch, government official, engineer and mathematician and a member of the Russian Academy of Sciences. He had the federal state civilian service rank of 1st class Active State Councillor of the Russian Federation. Berezovsky made his fortune in early post-soviet Russia in the 1990s, when the country implemented privatization of state property. He profited from gaining control over assets, including the country's main television channel, Channel One. In 1997, Forbes estimated Berezovsky's wealth at US$3 billion. Berezovsky helped fund Unity, the political party that would form Vladimir Putin's first parliamentary base, and was elected to the Duma in the 1999 Russian legislative election. However, following the Russian presidential election in March 2000, Berezovsky went into opposition and resigned from the Duma. Berezovsky would remain a vocal critic of Putin for the rest of his life. In late 2000, after the Russian Deputy Prosecutor General demanded that Berezovsky appear for questioning, he did not return from abroad and moved to the United Kingdom, which granted him political asylum in 2003. After he moved to Britain, the Russian government took over his television assets, and he divested from other Russian holdings. In Russia, Berezovsky was later convicted in absentia of fraud and embezzlement. The first charges had been brought during Primakov's government in 1999. Despite an Interpol Red Notice for Berezovsky's arrest, Russia repeatedly failed to obtain the extradition of Berezovsky from Britain; the situation became a major point of diplomatic tension between the two countries. In late 2011, an Israeli private investigator ordered the mercenary Indian hack-for-hire firm Appin to hack Berezovsky and his lawyers. In 2012, Berezovsky lost a London High Court case he brought over the ownership of the major oil producer Sibneft, against Roman Abramovich, in which he sought over £3 billion in damages. The court concluded that Berezovsky had never been a co-owner of Sibneft. Berezovsky was found dead in his home, Titness Park, at Sunninghill, near Ascot in Berkshire, on 23 March 2013. A post-mortem examination found that his death was consistent with hanging and that there were no signs of a violent struggle. However, the coroner at the inquest into Berezovsky's death later recorded an open verdict. == Early life, scientific research and engineering experience == Boris Abramovich Berezovsky was born in 1946, in Moscow, to Abram Markovich Berezovsky (1911–1979), an Ashkenazi Jewish civil engineer in construction works, and his wife, Anna Aleksandrovna Gelman (22 November 1923 – 3 September 2013). He studied applied mathematics, receiving his doctorate in 1983. After graduating from the Moscow Forestry Engineering Institute in 1968, Berezovsky worked as an engineer from 1969 until 1987, serving as assistant research officer, research officer and finally the head of a department in the Institute of Control Sciences of the USSR Academy of Sciences. Berezovsky researched optimization and control theory, publishing 16 books and articles between 1975 and 1989. After graduation, he got a job by distribution. Berezovsky later stated that he disliked his initial job, although appreciated his academic entourage that "shaped his personality". As a member of the Communist Party, he did not act as a political dissident. Yet, he disagreed with the official ideology, and sought to exploit his membership for career growth, that helped him to "intuitively compensate" lack of scientific talent.[6] == Political and business career in Russia == === Accumulation of wealth === Alexander Khinshtein (State Duma deputy, member of the United Russia faction) claimed that in 1979 Boris Berezovsky was detained by the OBKhSS authorities in Makhachkala (Dagestan Autonomous Soviet Socialist Republic) for profiteering. In Khinshtein's opinion, Berezovsky has been a KGB officer since 1979. In 1989, Berezovsky took advantage of the opportunities presented by perestroika to found LogoVAZ with Badri Patarkatsishvili and senior managers from Russian automobile manufacturer AvtoVAZ. LogoVAZ developed software for AvtoVAZ, sold Soviet-made cars and serviced foreign cars. The dealership profited from hyperinflation by taking cars on consignment and paying the producer at a later date when the money lost much of its value. One of Berezovsky's early endeavors was All-Russia Automobile Alliance (AVVA), a venture fund he formed in 1993 with Alexander Voloshin (Boris Yeltsin's future Chief of Staff) and AvtoVAZ Chairman Vladimir Kadannikov. Berezovsky controlled about 30% of the company, which raised nearly US$50 million from small investors through a bonded loan to build a plant producing a "people's car". The project did not collect sufficient funds for the plant and the funds were instead invested into AvtoVAZ production, while the debt to investors was swapped for equity. By 2000, AVVA held about one-third of AvtoVAZ. In 1994, Berezovsky was the target of a car bombing incident, but survived the assassination attempt, in which his driver was killed and he himself was injured. Alexander Litvinenko led the FSB investigation into the incident and linked the crime to the resistance of the Soviet-era AvtoVaz management to Berezovsky's growing influence in the Russian automobile market. Berezovsky's involvement in the Russian media began in December 1994, when he gained control over ORT Television (see Channel One (Russia)) to replace the failing Soviet TV Channel 1. He appointed the popular anchorman and producer Vladislav Listyev as CEO of ORT. Three months later Listyev was assassinated amid a fierce struggle for control of advertising sales. Berezovsky was questioned in the police investigation, among many others, but the killers were never found. Under Berezovsky's stewardship, ORT became a major asset of the reformist camp as they prepared to face Communists and nationalists in the upcoming presidential elections. From 1995 to 1997, through the controversial loans-for-shares privatisation auctions, Berezovsky and Patarkatsishvili assisted Roman Abramovich in acquiring control of Sibneft, the sixth-largest Russian oil company, which constituted the bulk of his wealth. In an article in The Washington Post in 2000, Berezovsky revealed that financier George Soros declined an invitation to participate in the acquisition. In 1995, he played a key role in a management reshuffle at Aeroflot and participated in its corporatization, with his close associate Nikolai Glushkov becoming Aeroflot's CFO. In January 1998, it was announced that Sibneft would merge with Mikhail Khodorkovsky's Yukos to create the third-largest oil company in the world. The merger was abandoned five months later amid falling oil prices. === Role in Yeltsin's reelection in 1996 === Berezovsky entered the Kremlin's inner circle in 1993 through arranging for the publication of Yeltsin's memoirs and befriended Valentin Yumashev, the President's ghost-writer. In January 1996, at the World Economic Forum at Davos, Berezovsky liaised with fellow oligarchs to form an alliance – which later became known as the "Davos Pact" – to bankroll Boris Yeltsin's campaign in the upcoming presidential elections. On his return to Moscow, Berezovsky met and befriended Tatyana Dyachenko, Yeltsin's daughter, According to a later profile by The Guardian, "Berezovsky masterminded the 1996 re-election of Boris Yeltsin... He and his billionaire friends coughed up £140 million for Yeltsin's campaign". In the summer of 1996, Berezovsky had emerged as a key advisor to Yeltsin, allied with Anatoly Chubais, opposing a group of hardliners led by General Alexander Korzhakov. One night in June, in the drawing room of Club Logovaz, Berezovsky, Chubais and others plotted the ouster of Korzhakov and other hardliners. On 20 June 1996, Yeltsin fired Korzhakov and two other hawks, leaving the reformers' team in full control of the Kremlin. Firing them was controversial though, as Korzhakov a few days before caught two of Yeltsin's campaign organizers carrying US$500,000 cash without invoices out of the presidential administration building. On 16 June 1996, Yeltsin came first in the first round of elections after forging a tactical alliance with Gen. Alexander Lebed, who finished third. On 3 July, in the runoff vote, he beat the Communist Gennady Zyuganov. His victory was due largely to the support of the TV networks controlled by Gusinsky and Berezovsky (NTV and ORT) and the money from the business elite. The New York Times called Berezovsky the "public spokesman and chief lobbyist for this new elite, which moved from the shadows to respectability in a few short years". === Role in Chechen conflict === On 17 October 1996, Yeltsin dismissed General Alexander Lebed from the position of National Security Advisor amid allegations that he was plotting a coup and secretly mustering a private army. Lebed promptly accused Berezovsky and Gusinsky of engineering his ouster, and formed a coalition with the disgraced General Alexander Korzhakov. The dismissal of Lebed, the architect of the Khasavyurt peace accord, left Yeltsin's Chechen policy in limbo. On 30 October 1996, in a political bombshell, Yeltsin named Ivan Rybkin as his new National Security Advisor and appointed Berezovsky Deputy Secretary in charge of Chechnya with a mandate to oversee the implementation of the Khasavyurt Accord: that is, the withdrawal of Russian forces, the negotiation of a peace treaty, and the preparation of a general election. On 19 December 1996, Berezovsky made headlines by negotiating the release of 21 Russian policeman held hostage by the warlord Salman Raduev amid efforts by radicals from both sides to torpedo peace negotiations. On 12 May 1997, Yeltsin and Maskhadov signed the Russian–Chechen Peace Treaty in the Kremlin. Speaking at a press conference in Moscow, Berezovsky outlined his priorities for the economic reconstruction of Chechnya, particularly the construction of a pipeline for transporting Azerbaijani oil. He called upon the Russian business community to contribute to the rebuilding of the republic, revealing his own donation of US$1 million (some sources mention US$2 million) for a cement factory in Grozny. This payment would come to haunt him years later, when he was accused of funding Chechen terrorists. After his dismissal from the Security Council, Berezovsky vowed to continue his activities in Chechnya as a private individual and maintained contact with Chechen warlords. He was instrumental in the release of 69 hostages, including two Britons, Jon James and Camilla Carr, whom he flew in his private jet to RAF Brize Norton in September 1998. In an interview with Thomas de Waal in 2005, he revealed the involvement of the British Ambassador to Russia, Sir Andrew Wood, and explained that his former negotiations counterpart, the Islamic militant leader Movladi Udugov, helped arrange the Britons' release. Berezovsky had a phone conversation with Movladi Udugov in the spring of 1999, six months before the beginning of fighting in Dagestan. A transcript of that conversation was leaked to a Moscow tabloid on 10 September 1999 and appeared to mention the would-be militants' invasion. It has been the subject of much speculation ever since. As Berezovsky explained later in interviews to de Waal and Goldfarb, Udugov proposed to coordinate the Islamists' incursion into Dagestan, so that a limited Russian response would topple the Chechen president Aslan Maskhadov and establish a new Islamic republic, which would be anti-American but friendly to Russia. Berezovsky said that he disliked the idea but reported Udugov's overture to prime-minister Stepashin. "Udugov and Basayev," he asserted, "conspired with Stepashin and Putin to provoke a war to topple Maskhadov ... but the agreement was for the Russian army to stop at the Terek River. However, Putin double-crossed the Chechens and started an all-out war." === Battle with "Young Reformers" === In March 1997, Berezovsky and Tatyana Dyachenko flew to Nizhniy Novgorod to persuade the city's governor, Boris Nemtsov, to join Chubais' economic team, which became known as the government of Young Reformers. This was the last concerted political action of the "Davos Pact" (see above). Four months later the group split into two cliques fiercely competing for Yeltsin's favour. The clash was precipitated by the privatization auction of the communication utility Svyazinvest, in which Onexim bank of Chubais' loyalist Vladimir Potanin, backed by George Soros, competed with Gusinsky, allied with Spanish Telefónica. An initially commercial dispute swiftly developed into a contest of political wills between Chubais and Berezovsky. Potanin's victory unleashed a bitter media war, in which ORT and NTV accused the Chubais group of fixing the auction in favor of Potanin, whereas Chubais charged Berezovsky with abusing his government position to advance his business interests. Both sides appealed to Yeltsin, who had proclaimed a new era of "fair" privatization "based on strict legislative rules and allowing no deviations". In the end, both sides lost. Berezovsky's media revealed a corrupt scheme whereby a publishing house owned by Onexim Bank paid Chubais and his group hefty advances for a book that was never written. The scandal led to a purge of Chubais' loyalists from the government. Chubais retaliated by persuading Yeltsin to dismiss Boris Berezovsky from the national security council. Berezovsky's service on the Security Council ended on 5 November 1997. Soros called the Berezovsky-Chubais clash a "historical event, in the reality of which I would have never believed, if I had not watched it myself. I saw a fight of the people in the boat floating towards the edge of a waterfall". He argued that the reformist camp never recovered from the wounds sustained in this struggle, setting the political stage for conservative nationalists, and eventually Vladimir Putin. === Philanthropy === In 1991, Berezovsky founded the "Triumph" award, bestowed upon outstanding Russian poets, musicians, artists, directors and ballet dancers. It is reported in the documentary series Captive that Boris Berezovsky, in 1998, was effective in the release of two English aid workers who had been held hostage for ransom in Chechnya for 14 months === The Kremlin Family and Putin's rise to power === In the spring of 1998, Berezovsky made an unexpected political comeback, starting with his appointment, in April 1998, to the position of executive secretary of the Commonwealth of Independent States. He emerged in the centre of a new informal power group – the "Family", a close-knit circle of advisers around Yeltsin, which included Yeltsin's daughter Tatyana and his chief of staff, Yumashev. It was rumoured that no important government appointment could happen without the Family's support. By 1999, the Family also included two of Berezovsky's associates, his former AVVA partner Alexander Voloshin, who replaced Yumashev as Yeltsin's chief of staff, and Roman Abramovich. The principal concern of the Family was finding an "electable" successor to Yeltsin to counter the presidential aspirations of the then–prime minister, Yevgeny Primakov, who was leaning to more statist positions. Political battles between the Family and Primakov's camp dominated the two last years of Yeltsin's presidency. In November 1998, in a televised press conference, five officers of the FSB, led by Lieutenant Colonel Alexander Litvinenko, revealed an alleged plot by their superiors to assassinate Berezovsky. In April 1999, Russia's Prosecutor General, Yury Skuratov, opened an investigation into embezzlement at Aeroflot and issued an arrest warrant for Berezovsky, who called the investigation politically motivated and orchestrated by Primakov. Nikolai Glushkov, Aeroflot's former General Director, later revealed that conflict with Primakov arose from the irritation that Berezovsky's management team caused in the Russian Foreign Intelligence Service, which Primakov headed before becoming prime minister, over firing of thousands of spies, who used Aeroflot as a front organization in Soviet times. The arrest warrant was dropped a week later, after Berezovsky submitted to questioning by the prosecutors. No charges were brought. Yeltsin sacked Primakov's government shortly thereafter and replaced him with Sergey Stepashin as new prime-minister. Vyacheslav Aminov (Russian: Вячеслав Аминов) supported Berezovsky and headed Berezovsky's security service. Vladimir Putin's meteoric rise from relative obscurity to the Russian presidency in the course of a few short months of 1999 has been attributed to his intimacy with the "Family" as a protege of Berezovsky and Yumashev. By the end of 1999, the Family had persuaded Yeltsin to name Putin his political successor and candidate for the presidency. Berezovsky's acquaintance with Putin dated back to the early 1990s, when the latter, as Deputy Mayor of St. Petersburg, helped Logovaz establish a car dealership. They enjoyed friendly relations; on occasion, Berezovsky took Putin skiing with him in Switzerland. In February 1999, when Berezovsky's political standing looked uncertain because of his clash with Primakov over Aeroflot, Putin, then Director of the FSB, made a bold gesture of friendship by showing up at a birthday party for Berezovsky's wife. "I absolutely do not care what Primakov thinks of me", Putin told Berezovsky on that night. That was the beginning of their political alliance. According to the Times, Spanish police discovered that on up to five occasions in 1999, Putin had secretly visited a villa in Spain belonging to Berezovsky. In mid-July 1999, the Family dispatched Berezovsky to Biarritz, where Putin was vacationing, to persuade him to accept the position of prime minister and the role of heir apparent. On 9 August, Yeltsin sacked the government of Sergei Stepashin and appointed Putin prime minister, amid reports that Berezovsky had masterminded the reshuffle. Putin's principal opponents were the former Prime Minister Evgeny Primakov and the Mayor of Moscow Yuri Luzhkov, backed by the Fatherland-All Russia alliance. To counter this group in the Duma elections of 1999, Berezovsky was instrumental in the creation, within the space of a few months, of the Unity party, with no ideology other than its support for Putin. Later, he disclosed that the source of Unity's funding, with Putin's knowledge and consent, was Aeroflot. In the 1999 election, Berezovsky campaigned as a Putin loyalist and won a seat in the Duma, representing the North Caucasian republic of Karachaevo-Cherkessia. During the Duma election campaign Berezovsky's ORT TV served as an extremely effective propaganda machine for the Putin camp, using aggressive attack reporting and programming to denigrate and ridicule Putin's rivals, Primakov and Luzhkov, tactics strongly criticized as undue interference with the media. But Unity got a surprisingly high score in the elections, paving the way for Putin's election victory in spring 2000. === Conflict with Putin and emigration === Berezovsky's disagreements with Putin became public three weeks into Putin's presidency. On 8 May 2000, Berezovsky and Abramovich were spotted together at Putin's invitation-only inauguration ball in Moscow. However, on 31 May, Berezovsky sharply attacked the constitutional reform proposed by the president, which would give the Kremlin the right to dismiss elected governors. On 17 July 2000, Berezovsky resigned from the Duma, saying he "did not want to be involved in the country's ruin and the restoration of an authoritarian regime". In August, Berezovsky's media attacked Putin for the way he handled the sinking of the Kursk submarine, blaming the death of 118 sailors on the Kremlin's reluctance to accept foreign help. In September, Berezovsky alleged that the Kremlin had attempted to expropriate his shares in ORT and announced that he would put his stake into a trust to be controlled by prominent intellectuals. In an article in The Washington Post in 2000, Berezovsky argued that in the absence of a strong civil society and middle class it may sometimes be necessary for capitalists "to interfere directly in the political process" of Russia as a counterweight to ex-Communists "who hate democracy and dream of regaining lost positions." Berezovsky took legal action against the journalist Paul Klebnikov, who accused him of various crimes. In October, in an interview in Le Figaro, Putin announced that he would no longer tolerate criticism of the government by media controlled by the oligarchs. "If necessary we will destroy those instruments that allow this blackmail", he declared. Responding to a question about Berezovsky, he warned that he had a "cudgel" in store for him. "The state has a cudgel in its hands that you use to hit just once, but on the head. We haven't used this cudgel yet. We've just brandished it... [But] the day we get really angry, we won't hesitate to use it." In the same month, Russian prosecutors revived the Aeroflot fraud investigation and Berezovsky was questioned as a witness. On 7 November 2000, Berezovsky, who was travelling abroad, failed to appear for further questioning and announced that he would not return to Russia because of what he described as "constantly intensifying pressure on me by the authorities and President Putin personally. Essentially," he said, "I'm being forced to choose whether to become a political prisoner or a political emigrant." Berezovsky claimed that Putin had made him a suspect in the Aeroflot case simply because ORT had "spoken the truth" about the sinking of the submarine Kursk. In early December, his associate Nikolai Glushkov was arrested in Moscow and Berezovsky dropped the proposal to put ORT stake in trust. === Divestment from Russian holdings === In 2001, the Russian government made a systematic takeover of privately owned television networks, in the course of which Berezovsky, Gusinsky and Patarkatsishvili lost most of their media holdings, prompting one of them to warn of Russia "turning into a banana republic" in a letter to The New York Times. In February, Berezovsky and Patarkatsishvili sold their stake in ORT to Roman Abramovich, who promptly ceded editorial control to the Kremlin. Berezovsky later claimed that there was a secret understanding that Nikolai Glushkov would be released from prison as part of that deal, a promise that was never fulfilled. In April, the government took control of Vladimir Gusinsky's NTV. Berezovsky then moved to acquire a controlling stake in a smaller network, TV-6, made Patarkatsishvili its chairman, and offered employment to hundreds of locked out NTV journalists. Almost immediately, Patarkatshishvili became a target of police investigation and fled the country. In January 2002 a Russian arbitration court forced TV-6 (Russia) into liquidation. The liquidation of TV-6 was precipitated by LUKoil, a partly state-owned minority shareholder, using a piece of legislation that was almost immediately repealed. In 2001, Berezovsky and Patarkatsishvili ended their involvement in Sibneft for a US$1.3 billion fee from Roman Abramovich. This transaction was the subject of a later dispute in the UK commercial courts, with Berezovsky alleging that he had been put under pressure to sell his stake to Abramovich at a fraction of the true value, an allegation that the court rejected. In 2006, Berezovsky sold the Kommersant ("The Businessman") newspaper and his remaining Russian assets. == Exile in Britain == From his new home in the UK, Stanley House, where he and associates including Akhmed Zakayev, Alexander Litvinenko and Alex Goldfarb became known as "the London Circle" of Russian exiles, Berezovsky publicly stated that he was on a mission to bring down Putin "by force" or by bloodless revolution. He established the International Foundation for Civil Liberties (IFCL), to "support the abused and the vulnerable in society – prisoners, national minorities and business people" in Russia and criticized Putin's record in the West. Berezovsky launched a concerted campaign to expose alleged misdeeds of Vladimir Putin, from suppressing freedom of speech to committing war crimes in Chechnya. He also accused Russia's FSB security service of staging the Moscow apartment bombings of 1999 in order to help Putin win the presidency. Many of these activities were funded through the New York-based IFCL, directed by Berezovsky's friend Alex Goldfarb. Berezovsky bought a Belgravia flat, the 125-acre Wentworth Park estate near Virginia Water in Surrey, and for a while owned the 172-acre Hascombe Court estate in Godalming. In 2012, he sold his Wentworth Park house. === Political asylum and extradition proceedings === On 9 September 2003, Berezovsky was granted refugee status and political asylum by the British Home Office which he, according to Alex Goldfarb, welcomed. On 12 September 2003, judge Timothy Workman of Bow Street Magistrates' Court in central London dropped extradition proceedings against Berezovsky, ruling that it would be pointless to pursue the case as the granting of asylum status to Berezovsky made the proceedings redundant. However, when Berezovsky told Reuters in early February 2006 that he was working on plans to overthrow Russian President Vladimir Putin, British Foreign Minister Jack Straw warned the London-based Russian tycoon not to plot against the Russian President while living in Britain. His refugee status could be reviewed if he continued to make such remarks. === Convictions in absentia and investigations abroad === After Berezovsky gained political asylum in Britain, the Russian authorities vigorously pursued various criminal charges against him. This culminated in two trials in absentia. From London, Berezovsky called the trial, which sentenced him to six years in prison, "a farce". In June 2009, the Krasnogorsk City Court near Moscow sentenced Berezovsky to thirteen years imprisonment for defrauding AvtoVAZ of 58 million rubles (US$1.9 million) in the 1990s. Berezovsky was represented by a court-appointed lawyer. In spite of Berezovsky's successes in Britain in fighting off extradition requests and exposing Russian court convictions as politically motivated (see below), some other jurisdictions cooperated with Russian authorities in seizing his property and targeting his financial transactions as money laundering. Berezovsky succeeded in overturning some of these actions. In July 2007, Brazilian prosecutors issued an arrest warrant for Berezovsky in connection with his investment in the Brazilian football club Corinthians. However, a year later the Brazilian Supreme Court cancelled the order and stopped the investigation. On Russian requests, French authorities raided his villa in Nice in search of documents, and seized his two yachts berthed on the French Riviera. However, some months later, the boats were released by a French court. Swiss prosecutors have been assisting their Russian colleagues for over a decade in investigating Berezovsky's finances. === Accusations and libel suits in the UK === Berezovsky's meteoric enrichment and involvement in power struggles have been accompanied by allegations of various crimes from his opponents. After his falling out with Putin and exile to London, these allegations became the recurrent theme of official state-controlled media, earning him comparisons with Leon Trotsky and the Nineteen Eighty-Four character Emmanuel Goldstein. In 1996, Forbes, an American business magazine, published an article by Paul Klebnikov entitled "Godfather of the Kremlin?" with the sub-heading "Power. Politics. Murder. Boris Berezovsky could teach the guys in Sicily a thing or two." The article linked Berezovsky to corruption in the car industry, to the Chechen mafia and to the murder of Vladislav Listyev. In 2000, the House of Lords gave Berezovsky and Nikolai Glushkov permission to sue for libel in the UK courts. Given that only 2,000 of the 785,000 copies sold worldwide were sold in the United Kingdom, this led numerous scholars to cite the case as an example of libel tourism. The case slowly proceeded until the claimants opted to settle when Forbes offered a partial retraction. The following statement appended to the article on the Forbes website summarises: "On 6 March 2003, the resolution of the case was announced in the High Court in London. Forbes stated in open court that (1) it was not the magazine's intention to state that Berezovsky was responsible for the murder of Listiev, only that he had been included in an inconclusive police investigation of the crime; (2) there is no evidence that Berezovsky was responsible for this or any other murder; (3) in light of the English court's ruling, it was wrong to characterize Berezovsky as a mafia boss; and (4) the magazine erred in stating that Glouchkov had been convicted for theft of state property in 1982." Klebnikov elaborated his allegations in his 2000 book Godfather of the Kremlin: Boris Berezovsky and the looting of Russia (the 2001 edition was titled Godfather of the Kremlin: The Decline of Russia in the Age of Gangster Capitalism). In 2006, a UK court awarded Berezovsky £50,000 in libel damages against the Russian private bank Alfa-Bank and its chairman, Mikhail Fridman. Fridman had claimed on a Russian television programme that could be watched in the UK that Berezovsky had threatened him when the two men were competitors for control of the Kommersant publishing house, and that making threats was Berezovsky's usual way of conducting business. The jury rejected the defendants' claim that Fridman's allegations were true. Berezovsky accepted the apology and withdrew his libel suit. In March 2010, Berezovsky, represented by Desmond Browne QC, won a libel case and was awarded £150,000 damages by the High Court in London over allegations that he had been behind the murder of Alexander Litvinenko. The allegations had been broadcast by the Russian state channel RTR Planeta in April 2007 on its programme Vesti Nedeli, which could be viewed from the UK. In his judgement, Mr Justice Eady stated: "I can say unequivocally that there is no evidence before me that Mr Berezovsky had any part in the murder of Mr Litvinenko. Nor, for that matter, do I see any basis for reasonable grounds to suspect him of it." Berezovsky had sued both the channel and a man called Vladimir Terluk, whom Mr Justice Eady agreed was the man who had been interviewed in silhouette by the programme under the pseudonym 'Pyotr'. Terluk had claimed that to further his UK asylum application Berezovsky had approached him to fabricate a murder plot against himself, and that Litvinenko knew of this. Mr Justice Eady accepted that Terluk had not himself alleged Berezovsky's involvement in the murder of Litvinenko, but considered that his own allegations were themselves serious and that there was no truth in any of them. As RTR did not participate in the proceedings, Terluk was left to defend the case himself, receiving significant assistance (as the judge noted) from the Russian prosecutor's office. The Guardian described the 2010 libel case as "almost anarchic at times as officials from the Russian prosecutors' office repeatedly intervened despite not being party to proceedings. So obvious was their intention that when one of their mobile phones went off in court one day, Desmond Browne quipped: 'That must be Mr Putin on the line.'" The defendants appealed to the Court of Appeal but the appeal was dismissed, Lord Justice Laws giving a judgment with which the Chancellor of the High Court and Lady Justice Rafferty agreed. The Lord Justice described a witness statement of Andrei Lugovoi, newly adduced by the defendants, as 'not sensibly capable of belief'. === High Court case against Abramovich === In 2011, Berezovsky brought a civil case against Roman Abramovich in the High Court of Justice in London, accusing Abramovich of blackmail, breach of trust and breach of contract, and seeking over £3 billion in damages. This became the largest civil court case in British legal history. Berezovsky's claimed past ownership of Sibneft – which constituted the bulk of his fortune – was put into question by Abramovich, who in a statement to the High Court in London asserted that Berezovsky had never owned shares in Sibneft, and that US$1.3 billion paid in 2001 ostensibly for his stake in the company was actually in recognition of Berezovsky's "political assistance and protection" during the creation of Sibneft in 1995. The hearings, which started on 3 October 2011, examined Berezovsky's US$5.5 billion claim against Abramovich for damages arising from the sale of his assets under alleged "threats and intimidation". In late 2011, Israeli private detective Tamir Mori ordered the mercenary Indian hack-for-hire firm Appin to hack more than 40 targets, including Berezovsky and his lawyers. On 31 August 2012, the High Court found for Abramovich. The High Court judge stated that because of the nature of the evidence, the case hinged on whether to believe Berezovsky or Abramovich's evidence. In her ruling, the judge observed: "On my analysis of the entirety of the evidence, I found Mr. Berezovsky an unimpressive, and inherently unreliable, witness, who regarded truth as a transitory, flexible concept, which could be moulded to suit his current purposes. ... I regret to say that the bottom line of my analysis of Mr. Berezovsky's credibility is that he would have said almost anything to support his case.": 16–18 She ruled that the monies paid represented a final payment in discharge of all obligations. === Business and personal activities in exile === Berezovsky conducted business with Neil Bush, the younger brother of the US President George W. Bush. Berezovsky was an investor in Bush's Ignite! Learning, an educational software corporation, since at least 2003. In 2005, Neil Bush met with Berezovsky in Latvia, causing tension with Russia due to Berezovsky's fugitive status. Neil Bush was also seen with Berezovsky in his box at an Arsenal F.C. match at the Emirates Stadium in London. There had been speculation that the relationship had become a cause of tension in Russo-American bilateral relations. It had been speculated that Berezovsky's wealth was depleted with the onset of the late 2000s recession. According to the Sunday Times Rich List, in 2011 his net worth was about US$900 million. === Appeals for regime change === In September 2005, Berezovsky said in an interview with the BBC: "I'm sure that Putin doesn't have the chance to survive, even to the next election in 2008. I am doing everything in my power to limit his time frame, and I am really thinking of returning to Russia after Putin collapses, which he will." In January 2006, Berezovsky stated in an interview to a Moscow-based radio station that he was working on overthrowing the administration of Vladimir Putin by force. Berezovsky also accused Putin of being "a gangster" and the "terrorist number one". Berezovsky declared that he was plotting the overthrow of President Putin on 13 April 2007 during an interview The Guardian conducted: "We need to use force to change this regime. It isn't possible to change this regime through democratic means. There can be no change without force, pressure." He also admitted that during the last six years he had struggled hard to "destroy the positive image of Putin" and said that "Putin has created an authoritarian regime against the Russian constitution. ... I don't know how it will happen, but authoritarian regimes only collapse by force." Soon after Berezovsky's 2007 statement, Garry Kasparov, a figure in the opposition movement The Other Russia and leader of the United Civil Front, wrote the following on his website: "Berezovsky has lived in emigration for many years and no longer has significant influence upon the political processes which take place in Russian society. His extravagant proclamations are simply a method of attracting attention. Furthermore, for the overwhelming majority of Russian people he was a political symbol of the 90s, one of the 'bad blokes' enriching themselves behind the back of president Yeltsin. The informational noise around Berezovsky was specifically beneficial for the Kremlin, which was trying to compromise Russia's real opposition. Berezovsky has not had and does not have any relation to Other Russia or the United Civil Front." Berezovsky responded in June 2007 by saying that "there is not one significant politician in Russia whom he has not financed" and that this included members of Other Russia. The managing director of the United Civil Front, in turn, said that the organization would consider suing Berezovsky over these allegations, but the lawsuit has never been brought before the court. The Russian Prosecutor General's Office had launched a criminal investigation against Berezovsky to determine whether his comments could be considered a "seizure of power by force", as outlined in the Russian Criminal Code. If convicted, an offender faces up to twenty years imprisonment. The British Foreign Office denounced Berezovsky's statements, warning him that his status of a political refugee might be reconsidered, should he continue to make similar remarks. Furthermore, Scotland Yard had announced that it would investigate whether Berezovsky's statements violated the law. However, in the following July, the Crown Prosecution Service announced that Berezovsky would not face charges in the UK for his comments. Kremlin officials called it a "disturbing moment" in Anglo-Russian relations. === Involvement in the 2004 Ukrainian presidential election === In September 2005, the former president of Ukraine, Leonid Kravchuk, accused Berezovsky of having financed Viktor Yushchenko's 2004 Ukrainian presidential election campaign, and provided copies of documents showing money transfers from companies he claimed were controlled by Berezovsky to companies controlled by Yuschenko's official backers. Berezovsky claimed that he met Yushchenko's representatives in London before the election, and that the money was transferred from his companies, but he declined to confirm or deny that the companies that received the money were used in Yushchenko's campaign. Financing of election campaigns by foreign citizens is illegal in Ukraine. In November 2005, Berezovsky also claimed he had heavily financed Ukraine's Orange Revolution (that had followed the presidential election). In September 2007, Berezovsky launched lawsuits against two Ukrainian politicians, Oleksandr Tretyakov, a former presidential aid, and David Zhvaniya, a former emergencies minister. Berezovsky was suing the men for nearly US$23 million, accusing them of misusing the money he had allocated in 2004 to fund the Orange Revolution. Yushchenko has denied Berezovsky financed his election campaign. Berezovsky called on Ukrainian business to support Yushchenko in the 2010 presidential election of January 2010 as a guarantor of debarment of property redistribution after the election. On 10 December 2009, the Ukrainian minister of interior affairs Yuriy Lutsenko stated that if the Russian interior ministry requested it, Berezovsky would be detained upon arriving in Ukraine. In February 2012, in an interview for the independent Russian TV Rain channel Berezovsky reiterated that he had personally provided approximately $50 million to the Ukrainian Orange revolution. David Zhvania (Russian: Давид Важаевич Жвания) and Oleksandr Tretiakov were among the ones who allegedly received money. === Persona non-grata in Latvia since October 2005 === In October 2005, Latvian Prime Minister Aigars Kalvītis signed a decree placing Berezovsky on the list of personae non gratae. The exact reasons for blacklisting Berezovsky were not disclosed. Kalvitis called Berezovsky a "threat" to national security. Previously, the National Security Council of Latvia took the decision to recommend that Berezovsky be barred from travelling to Latvia. The decision to bar the one-time Russian oligarch came swiftly after Berezovsky's trip to Riga in September 2005. Berezovsky was in Riga along with Neil Bush to discuss a project with Latvian businessmen. The Baltic News Service quoted Berezovsky saying that he believes Latvia's decision to declare him persona non-grata was the result of intense pressure by Russia and structures linked with George Soros, the US business magnate who had had acrimonious relations with Berezovsky. Kalvitis however denied the theory that the banning came on pressure from the Kremlin or the White House. == Alleged assassination attempts in London == === Alleged 2003 plot === According to Alexander Litvinenko, a Russian Federal Security Service (FSB) officer in London was preparing to assassinate Berezovsky with a binary weapon in September 2003. This alleged plot was reported to British police. Hazel Blears, then a Home Office Minister, said that inquiries made [into these claims] were "unable to either substantiate this information or find evidence of any criminal offences having been committed". === Alleged 2007 plot === In June 2007, Berezovsky said he fled Britain on the advice of Scotland Yard, amid reports that he was the target of an assassination attempt by a suspected Russian hitman. On 18 July 2007, the British tabloid The Sun reported that the alleged would-be assassin was captured by the police at the Hilton Hotel in Park Lane. They reported that the suspect, arrested by the anti-terrorist police after being tracked for a week by MI5, was deported back to Russia when no weapons were found and there was not enough evidence to charge him with any offence. In addition, they said British police placed a squad of uniformed officers around Akhmed Zakayev's house in north London, and also phoned Litvinenko's widow, Marina, to urge her to take greater security precautions. Berezovsky again accused Vladimir Putin of being behind a plot to assassinate him. The Kremlin had denied similar claims in the past. According to the interview given by a high-ranking British security official on BBC Two in July 2008, the alleged Russian agent, known as "A", was of Chechen nationality. He was identified by Kommersant as the Chechen mobster Movladi Atlangeriyev; after returning to Russia, Atlangeriyev was forcibly disappeared in January 2008 by unknown men in Moscow. == Death of friends and associates in London == === Death of Alexander Litvinenko in November 2006 === Alexander Litvinenko, one of Berezovsky's closest associates, was murdered in London in November 2006 with a rare radioactive poison, Polonium 210. The British authorities charged a former FSB officer and head of security at ORT, Andrey Lugovoy, with the murder and requested his extradition, which Russia refused. Several Russian diplomats were expelled from the UK over the case. The UK government has not publicly expressed a view on the matter, but allegations that the murder was sponsored by the Russian state have been expressed by "sources in the UK government", according to the BBC, and by officials of the US Department of State, as revealed by leaked diplomatic cables; they were reflected in a 2008 resolution by the US Congress. The intricate details of the murder, the relationship between Litvinenko and Berezosvsky, and the implications of the case have been described in the 2007 book, Death of a Dissident: The Poisoning of Alexander Litvinenko and the Return of the KGB by Alex Goldfarb with Marina Litvinenko. An alternative, more dubious narrative – that the murder was orchestrated by Berezovsky and his associate Alex Goldfarb with the aim of "framing" the Russian government and discrediting it on the global stage – has aired in Russian state-funded media by Lugovoy, by Litvinenko's Italy-based father, by Nikita Chekulin and by Russian officials. Berezovsky won a UK libel suit against Russian State Television over these allegations in 2010 (see above), following which he commented, "I trust the conclusions of the British investigators that the trail leads to Russia and I hope that one day justice will prevail." === Death of Badri Patarkatsishvili in February 2008 === In the evening of Tuesday, 12 February 2008, Georgia's richest man, billionaire Arkady "Badri" Patarkatsishvili, a close friend and long-time business partner of Berezovsky, collapsed and died in his bedroom after a family dinner at Downside Manor, his mansion in Leatherhead, Surrey, England, at the age of 52. Patarkatsishvili, who as a presidential candidate had also been campaigning to oust Georgia's President Mikhail Saakashvili, spent his last day in the City of London office of international law firm Debevoise and Plimpton. He was preparing along with his lawyer Lord Goldsmith QC and fellow exiles, the Russians Nikolai Glushkov and Yuli Dubov. Shortly after dining at Downside Manor, Patarkatsishvili told his family he felt unwell and went upstairs to his bedroom where he was found unconscious after a heart attack. Resuscitation attempts were unsuccessful. As in any other case of unexpected death, Surrey police treated the case as "suspicious" and launched an official investigation. Preliminary reports indicated a heart attack as the cause of death. Berezovsky described the death of his closest friend as "a terrible tragedy". == Death == On 23 March 2013, Berezovsky was found dead at his home, Titness Park, at Sunninghill, near Ascot in Berkshire. His body was found by a bodyguard in a locked bathroom, with a ligature around his neck. His death was announced in a post on Facebook, by his son-in-law. Alexander Dobrovinsky, a lawyer who had represented Berezovsky, wrote that he may have committed suicide, adding that Berezovsky had fallen into debt after losing the lawsuit against Abramovich, and had spent the final few months of his life selling his possessions to cover his court costs. Berezovsky was also said to have recently been depressed and to have isolated himself from friends. He reportedly suffered from depression and was taking antidepressant drugs; a day prior to his death he told a reporter in London that he had nothing left to live for. The Thames Valley Police classified his death as "unexplained" and launched a formal investigation into the circumstances behind it. Specialists in chemical, biological, radiological and nuclear materials were deployed to Berezovsky's home as a "precaution". These specialists later "found nothing of concern". Berezovsky was buried on 8 May 2013 in a private ceremony at Brookwood Cemetery in Surrey. The burial timing had been changed on several occasions to try to avoid interest from the Russian media. A post-mortem examination carried out by the Home Office pathologist found the cause of death was consistent with hanging and there was nothing pointing to a violent struggle. At the March 2014 inquest into the death, however, Berezovsky's daughter Elizaveta introduced a report by German pathologist Bernd Brinkmann, with whom she had shared the autopsy photos, noting that the ligature mark on her father's neck was circular rather than V-shaped as is commonly the case with hanging victims, and called the coroner's attention to a statement by one of the responding paramedics who found it strange that Berezovsky's face was purple, rather than pale as hanging victims usually are. The body also had a fresh wound on the back of the head and a fractured rib (injuries police believed Berezovsky could have suffered in the process of falling as he hanged himself). An unidentified fingerprint was found near the body, and one paramedic's radiation alarm sounded as they entered the house. Pathologist Bernd Brinkmann said that he did not believe that the marks on Berezovsky's neck were a result of hanging. Following the inquest the coroner, Peter Bedford, recorded an open verdict commenting, "I am not saying Mr Berezovsky took his own life, I am not saying Mr Berezovsky was unlawfully killed. What I am saying is that the burden of proof sets such a high standard it is impossible for me to say." === Apology to Putin === After Berezovsky's death, a spokesman for President Putin reported that Berezovsky had sent a letter to the Russian president, asking for permission to return to Russia and asking "forgiveness for his mistakes". Some of Berezovsky's associates doubted the letter's existence, claiming that it was out of character. However, his girlfriend at the time, Katerina Sabirova, later confirmed in an interview that he did in fact send the letter: I said that they will publish it and you will look bad. And that it won't help. He answered that it was all the same to him, that in any case all sins were blamed on him and that this was his only chance. It was claimed by anonymous sources that rival Roman Abramovich delivered the letter to Putin personally, having received an apology from Berezovsky himself. Both Putin's chief of staff, Sergei Ivanov, and Abramovich's spokesman alluded to the letter being passed by a "certain person", but did not go into details due to the personal nature of the issue. == Publications by Berezovsky == Berezovsky was a doctor of technical sciences and author of many academic papers and studies such as "Binary relations in multi-criteria optimizations" and "Multi-criteria optimization: mathematical aspects". In the mathematical review index MathSciNet, B. A. Berezovsky is credited with 16 publications from 1975 to 1989 on operations research and mathematical programming, earning 9 citations in other publications. Most cited is the book The Problem of Optimal Choice with A. V. Gnedin (Nauka, Moscow 1984), devoted to secretary problems. Aside from his academic publications, he frequently authored articles and gave interviews; these are collected in The Art of the Impossible (3 volumes). He continued to contribute articles while in exile, taking a highly critical view of Russia's political leaders. == Works about Berezovsky == In 1996, the Russian-American journalist Paul Klebnikov wrote a highly critical article entitled "Godfather of the Kremlin?" on Berezovsky and the state of Russia more generally, in response to which Berezovsky sued Forbes in the UK;: 7 in 2001, he expanded his article into a book entitled Godfather of the Kremlin, alternatively subtitled The Decline of Russia in the Age of Gangster Capitalism and Boris Berezovsky and the looting of Russia. On 9 July 2004, while leaving the Forbes office in Moscow, unknown assailants fired at Klebnikov from a slowly moving car. He was shot four times and died later in hospital. The same day Berezovsky, in the words of investigative journalist Richard Behar, "whipped out his tongue from its holster and publicly called the 41-year-old editor of Forbes Russia 'a dishonest reporter'". The books Secret Diary of a Russian Oligarch and How to get rid of an Oligarch or Who Beat Berezovsky by Sasha Nerozina (friend of the Berezovsky family and a spokeswoman of Berezovsky's wife Galina) were published in Russia and other former Soviet states in 2013 and 2014 by Olma Media Publishing House. Yuli Dubov, a close business associate of Berezovsky, wrote a novel based on Berezovsky's life which provided the basis for the 2002 film Tycoon. Like Berezovsky, he fled to London and successfully fought extradition to Russia. Judge Timothy Workman of Bow Street Magistrates' Court in central London dropped extradition proceedings against Yuly Dubov in October 2003. Alex Goldfarb, a microbiologist and activist who became acquainted with Berezovsky in the 1990s and later worked for him, provides snapshots of Berezovsky at crucial moments as background to his 2007 account of the Litvinenko murder case, co-written with Marina Litvinenko, Death of a Dissident: The Poisoning of Alexander Litvinenko and the Return of the KGB. David E. Hoffman of The Washington Post wrote The Oligarchs: Wealth and Power in the New Russia, which provides a comparative treatment of Berezovsky and several of his fellow so-called business oligarchs. Ben Mezrich wrote Once Upon a Time in Russia: The Rise of the Oligarchs—A True Story of Ambition, Wealth, Betrayal, and Murder, which provides a comparative narrative of Berezovsky and Roman Abramovich through their careers, friendship, and ultimate rivalry. In 2017, the Russian book The age of Berezovsky (also known as The time of Berezovsky) was published by Corpus (an imprint of AST), in which Petr Aven – a friend of Berezovsky – interviewed various people who were close to Berezovsky at different times, including Leonid Boguslavsky, Yuli Dubov, Galina Besharova, Yelena Gorbunova, Yuri Shefler, Anatoly Chubais, Mikhail Fridman, Valentin Yumashev, Sergey Dorenko, Eugene Shvidler, Vladimir Posner, Alexander Goldfarb, Alexander Voloshin, Stanislav Belkovsky and Yuri Felshtinsky. In the book Aven opined that Berezovsky "played a particularly important role in two episodes of new Russian history: the 1996 election and the struggle against Primakov and Luzhkov in 1999". A documentary about Berezovsky's efforts to undermine Putin from his exile in UK was shown on BBC Two in December 2005. Berezovsky features in a painting by the Russian artist Ilya Glazunov, displayed in Moscow's Ilya Glazunov Gallery. According to the Rough Guide, "The Market of Our Democracy shows Yeltsin waving a conductor's baton as two lesbians kiss and the oligarch Berezovsky flaunts a sign reading 'I will buy Russia', while charlatans rob a crowd of refugees and starving children." Berezovsky also features as a character in the opera The Life and Death of Alexander Litvinenko by Alexander Woolf to a libretto by David Pountney, which was premiered in July 2021 at Grange Park Opera. Patriots premiered at the Almeida Theatre in Islington, London, in May 2022, following the life of Berezovsky from the president's inner circle to public enemy number one. Tom Hollander played Berezovsky. The play was written by Peter Morgan and directed by Rupert Goold. It played a limited run from 2 July 2022 until 20 August. It transferred to the West End in 2023. In the 2022 ITVX miniseries Litvinenko, Berezovsky was portrayed by Nikolai Tsankov. == See also == List of unsolved deaths Russian oligarchs Semibankirschina Tycoon (2002 film) == References ==
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Wikipedia:Boris Kashin#0
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Boris Sergeevich Kashin (Russian: Борис Сергеевич Кашин; born July 3, 1951, in Moscow) is a Russian mathematician, Academician of the Russian Academy of Sciences (since 2011), Doctor of Sciences, Professor at the MSU Faculty of Mechanics and Mathematics. Member of the Central Committee of the Communist Party of the Russian Federation since 2000. He graduated from the MSU Faculty of Mechanics and Mathematics in 1973. And then entered to the Steklov Mathematical Institute, where he currently works. In 1976 he defended his Candidate's Dissertation. In 1977 he defended his doctoral dissertation. In 1990 he received the title of Professor. Member of the Communist Party of the Soviet Union since 1980. He was elected a corresponding member of the Russian Academy of Sciences in 1997. He is currently Editor-in-Chief of the journal Matematicheskii Sbornik. In 2012, as a deputy of the State Duma, Kashin was among the initiators of the bill of the Dima Yakovlev law. == References ==
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Wikipedia:Boris Khesin#0
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Boris Aronovich Khesin (in Russian: Борис Аронович Хесин, born in 1964) is a Russian and Canadian mathematician working on infinite-dimensional Lie groups, Poisson geometry and hydrodynamics. He has held positions at the University of California, Berkeley, Yale University, and currently is a professor at the University of Toronto. Khesin obtained his Ph.D. from Moscow State University in 1990 under the supervision of Vladimir Arnold (Thesis: Normal forms and versal deformations of evolution differential equations). From 1990 to 1992 he was Morrey Assistant Professor at the University of California at Berkeley and from 1992 to 1996 Assistant Professor at Yale University. In 1997/98 and in 2012 he worked at the Institute for Advanced Study. In 1996 he became Associate Professor and in 2002 Professor at the University of Toronto. He is an editor of the Complete Works of Vladimir Arnold. In 1997 he was awarded the Aisenstadt Prize. "Professor Khesin is recognized for his work in Poisson geometry and infinite-dimensional group theory, and for his remarkable geometric intuition applied to problems of topological hydrodynamics and double-loop groups. He has also done fundamental work in bifurcation theory. He has proven R. Thom's rule, called the rule of "seven elementary catastrophes", in dynamical systems. In addition, Professor Khesin discovered the "logarithm of the derivative", a notion of elegant simplicity providing a link between the theory of determinants and the theory of infinite-dimensional integrable systems." From 1997 to 2001 he was a recipient of the Sloan Research Fellowship. == References ==
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Wikipedia:Boris Korenblum#0
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Boris Isaac Korenblum (Russian: Борис Исаакович Коренблюм, 12 August 1923, Odessa, now Ukraine – 15 December 2011, Slingerlands, New York) was a Soviet-Israeli-American mathematician, specializing in mathematical analysis. Boris Korenblum was a child prodigy in music, languages, and mathematics. He started as a violinist at the famous School of Stolyarsky in Odessa. After he won a young mathematicians competition, the family was given an apartment in Kiev, an extraordinary event. Boris was given a mentor, a local mathematics professor, who would peremptorily supervise his course of self study. To the great chagrin of his mother, Boris decided against pursuing a music career. In June 1941, when the war began, he volunteered, not yet having reached the draft age, for the Soviet Army. Because of his fluency in German, he served in a reconnaissance unit. Some of his tasks was going to the enemy lines to capture a prisoner for interrogation. He was also the one to interpret to his commanding officers when a prisoner was taken. Once, refusing to beat a prisoner who was talking already, he quarrelled with the superior officer, and was punished by being sent to a penal battalion. There, he served with, and made fast wartime friends with, some rough characters, many of whom were discharged from penal colonies "to wash with their blood the offences against the Motherland." He later told his family that this experience, together with the inevitable maturing during a bloody war, made a man out of a soft city boy with a doting Jewish mother. After some time, the need for competent interpreters saw him return to his unit, where he served with distinction to the end of the war. His awards, including an Order of the Red Banner, were taken from him when he emigrated to Israel in November 1973. Coming home from the war, he passed all the exams for the undergraduate degree in mathematics in a few of months, and was admitted for graduate study at the Institute of Mathematics of the National Academy of Sciences of Ukraine where he received in 1947 his Candidate of Sciences degree (PhD) under the direction of Evgeny Yakovlevich Remez. Korenblum received his Russian Doctorate of Sciences (habilitation) from Moscow State University in 1956. Around 1952, during the height of the anti-Semitic campaign, he was dismissed from the Institute of Mathematics, along with all other Jewish and half-Jewish scientists. Subsequently, he became a Professor of Mathematics at the Institute of Civil Engineering, thanks to the heroic efforts of Professor Yurii D. Sokolov (1896–1971), who risked his own position in the politically charged atmosphere of the time. Boris Korenblum remained there until his emigration to Israel. In 1958, he published a little-noticed paper outlining the basic concept and theory for computed tomography (CT) with S.I. Tetelbaum. This was five years before A.M. Cormack's seminal paper in the West, which laid the theoretical foundation for CT and earned him the Nobel Prize in Physiology or Medicine in 1979. From 1974 to 1977 Korenblum was a professor of mathematics at Tel Aviv University. In 1977 he was at the Institute for Advanced Study in Princeton, New Jersey. He was a professor at the University at Albany, SUNY from 1977 until his retirement in 2009 as professor emeritus. Korenblum's research dealt with classical harmonic analysis, functional analysis, Banach algebras, and complex analysis. He was an Invited Speaker at the ICM in 1978 in Helsinki. In November 2003 a conference was held in Barcelona to celebrate the occasion of his 80th birthday. He is survived by his wife, his children, and a granddaughter. == Selected publications == "About one scheme of tomography", with S.I. Tetelbaum and A. A.Tyutin, Proc. of Higher Edu. Institutions - Radiophysics 1958, Volume 1, No. 3 “A generalization of Wiener's Tauberian theorem and harmonic analysis of rapidly increasing functions”, Proc. (Trudy) Moscow Math. Soc., 1958, v. 7, 121–148. “Closed ideals of ring An, Func. Anal. and Applic. (Moscow), 1972, v. 6, 38–52. Korenblum, Boris (1975). "An extension of the Nevanlinna theory". Acta Mathematica. 135: 187–219. doi:10.1007/BF02392019. Korenblum, Boris (1977). "A Beurling-type theorem". Acta Mathematica. 138: 265–293. doi:10.1007/BF02392318. Korenblum, Boris (1983). "Some problems in potential theory and the notion of harmonic entropy" (PDF). Bull. Amer. Math. Soc. (N.S.). 8 (3): 459–462. doi:10.1090/S0273-0979-1983-15120-0. Korenblum, B. (1983). "A generalization of two classical convergence tests for Fourier series, and some new Banach spaces of functions" (PDF). Bull. Amer. Math. Soc. (N.S.). 9 (2): 215–218. doi:10.1090/S0273-0979-1983-15160-1. with Edward Thomas: Korenblum, Boris; Thomas, Edward (1983). "An inequality with applications in potential theory". Trans. Amer. Math. Soc. 279 (2): 525–536. doi:10.1090/S0002-9947-1983-0709566-X. Korenblum, B. (1985). "BMO estimates and radial growth of Bloch functions". Bull. Amer. Math. Soc. (N.S.). 12: 99–102. doi:10.1090/S0273-0979-1985-15302-9. Korenblum, Boris (1985). "On a class of Banach spaces of functions associated with the notion of entropy". Trans. Amer. Math. Soc. 290 (2): 527–553. doi:10.1090/S0002-9947-1985-0792810-2. with Leon Brown: “Cyclic vectors in A–∞, Proc. Amer. Math. Soc., 1987, v. 101, 137–138. doi:10.1090/S0002-9939-1988-0915731-9 with Joaquim Bruna: Bruna, Joaquim; Korenblum, Boris (1987). "A note on Calderón-Zygmund singular integral convolution operators" (PDF). Bull. Amer. Math. Soc. (N.S.). 16 (2): 271–273. doi:10.1090/S0273-0979-1987-15515-7. “Transformation of zero sets by contractive operators in the Bergman space”, Bull. Sci. Math. (2), 1990, v. 114, 385–394. “A maximum principle for the Bergman space”, Publicacions Math., 1991, v. 35, 479–486. JSTOR 43736335 with Kendall Richards: Korenblum, Boris; Richards, Kendall (1993). "Majorization and domination in the Bergman space". Proc. Amer. Math. Soc. 117: 153–158. doi:10.1090/S0002-9939-1993-1113643-3. with R. O’Neil, K. Richards, and K. Zhu: Korenblum, B.; o'Neil, R.; Richards, K.; Zhu, K. (1993). "Totally monotone functions with applications to the Bergman space". Trans. Amer. Math. Soc. 337 (2): 795–806. doi:10.1090/S0002-9947-1993-1118827-0. with Kehe Zhu: "An application of Tauberian theorems to Toeplitz operators." Journal of Operator Theory, 1995, 353–361. JSTOR 24714916 with A. Mascuilli and J. Panariello: Korenblum, B.; Mascuilli, A.; Panariello, J. (1998). "A generalization of Carleman's uniqueness theorem and a discrete Phragmén-Lindelöf theorem". Proc. Amer. Math. Soc. 126 (7): 2025–2032. doi:10.1090/S0002-9939-98-04239-7. with Håkan Hedenmalm and Kehe Zhu: Theory of Bergman Spaces, Springer, 2000. with John C. Racquet: “Concurrence of Uniqueness and Boundedness Conditions for Regular Sequences”, Complex Variables, 2000, v. 41, 231–239. doi:10.1080/17476930008815251 with Catherine Beneteau: “Jensen type inequalities and radial null sets”, Analysis, 2001, v. 21, 99–105. with Emmanuel Rashba: “Classical properties of low-dimensional conductors”, Phys. Rev. Lett., 2002, v. 89, no. 9. doi:10.1103/PhysRevLett.89.096803 with C. Beneteau: “Some coefficient estimates for Hp functions”, Proc. of the International Conference in Karmiel (Israel), 2004. == References == == External links == Korenblum, Boris I., mathnet.ru A Slideshow in Memory of Boris Korenblum (page by Daniel Korenblum, use left and right arrow keys to change slides)
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Wikipedia:Boris Levitan#0
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Boris Levitan (7 June 1914 – 4 April 2004) was a mathematician who worked on almost periodic functions, Sturm–Liouville operators and inverse scattering. Levitan was born in Berdyansk (southeastern Ukraine), and grew up in Kharkiv. He graduated from Kharkov University in 1936. In 1938, he submitted his PhD thesis "Some Generalization of Almost Periodic Function" under the supervision of Naum Akhiezer. He then defended the habilitation thesis "Theory of Generalized Translation Operators". Levitan was drafted into the army at the beginning of World War II in 1941, and served until 1944. From 1944 to 1961, he worked at the Dzerzhinsky Military Academy, and from 1961 until about 1992 at Moscow University. In 1992, he emigrated to the United States. During the last years of his life, he worked for the University of Minnesota. == References ==
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Wikipedia:Boris Tsirelson#0
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Boris Semyonovich Tsirelson (May 4, 1950 – January 21, 2020) (Hebrew: בוריס סמיונוביץ' צירלסון, Russian: Борис Семёнович Цирельсон) was a Russian–Israeli mathematician and Professor of Mathematics at Tel Aviv University in Israel, as well as a Wikipedia editor. == Biography == Tsirelson was born in Leningrad to a Russian Jewish family. From his father Simeon's side, he was the great-nephew of rabbi Yehuda Leib Tsirelson, chief rabbi of Bessarabia from 1918 to 1941, and a prominent posek and Jewish leader. He obtained his Master of Science from the University of Leningrad and remained there to pursue graduate studies. He obtained his Ph.D. in 1975, with thesis "General properties of bounded Gaussian processes and related questions" written under the direction of Ildar Abdulovich Ibragimov. Later, he participated in the refusenik movement, but only received permission to immigrate to Israel in 1991. From then until 2017, he was a professor at Tel-Aviv University. He has also worked on fault-tolerant cellular automaton. In 1998 he was an Invited Speaker at the International Congress of Mathematicians in Berlin. == Contributions to mathematics == Tsirelson made notable contributions to probability theory and functional analysis. These include: Tsirelson's bound, in quantum mechanics, is an inequality, related to the issue of quantum nonlocality. Tsirelson space is an example of a reflexive Banach space in which neither a l p space nor a c0 space can be embedded. The Tsirelson's drift, a counterexample in the theory of stochastic differential equations, is an SDE which has a weak solution but no strong solution. The Gaussian isoperimetric inequality (proved by Vladimir Sudakov and Tsirelson, and independently by Christer Borell), stating that affine halfspaces are the isoperimetric sets for the Gaussian measure. == Death == Tsirelson died on January 21, 2020, at the age of 69. == References == == External links == Tsirelson's homepage, at Tel Aviv University Mourning page, at Tel Aviv University
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Wikipedia:Boris Zilber#0
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Boris Zilber (Russian: Борис Иосифович Зильбер, born 1949) is a Soviet-British mathematician who works in mathematical logic, specifically model theory. He is a emeritus professor of mathematical logic at the University of Oxford. He obtained his doctorate (Candidate of Sciences) from the Novosibirsk State University in 1975 under the supervision of Mikhail Taitslin and his habilitation (Doctor of Sciences) from the Saint Petersburg State University in 1986. Zilber received the Senior Berwick Prize (2004) and the Pólya Prize (2015) from the London Mathematical Society. He also gave the Tarski Lectures in 2002. == Research == Zilber is well known for his seminal work around several fundamental problems in mathematics, mostly in the broad area of geometric model theory. In particular, his trichotomy conjecture on the nature of strongly minimal sets has been extremely influential in geometric stability theory. Although it is false in full generality (refuted by Ehud Hrushovski), it holds in many important settings, e.g. Zariski geometries, and has been successfully applied to several problems including the Mordell-Lang conjecture for function fields. Zilber's work on model theory of complex exponentiation led him to propose several influential conjectures including the Quasiminimality conjecture, the Existential Closedness conjecture, and the Conjecture on Intersections with Tori. == See also == Zilber-Pink conjecture Existential Closedness conjecture Schanuel's conjecture == References == == External links == Prof. Zilber's homepage
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Wikipedia:Bose–Mesner algebra#0
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In mathematics, a Bose–Mesner algebra is a special set of matrices which arise from a combinatorial structure known as an association scheme, together with the usual set of rules for combining (forming the products of) those matrices, such that they form an associative algebra, or, more precisely, a unitary commutative algebra. Among these rules are: the result of a product is also within the set of matrices, there is an identity matrix in the set, and taking products is commutative. Bose–Mesner algebras have applications in physics to spin models, and in statistics to the design of experiments. They are named for R. C. Bose and Dale Marsh Mesner. == Definition == Let X be a set of v elements. Consider a partition of the 2-element subsets of X into n non-empty subsets, R1, ..., Rn such that: given an x ∈ X {\displaystyle x\in X} , the number of y ∈ X {\displaystyle y\in X} such that { x , y } ∈ R i {\displaystyle \{x,y\}\in R_{i}} depends only on i (and not on x). This number will be denoted by vi, and given x , y ∈ X {\displaystyle x,y\in X} with { x , y } ∈ R k {\displaystyle \{x,y\}\in R_{k}} , the number of z ∈ X {\displaystyle z\in X} such that { x , z } ∈ R i {\displaystyle \{x,z\}\in R_{i}} and { z , y } ∈ R j {\displaystyle \{z,y\}\in R_{j}} depends only on i,j and k (and not on x and y). This number will be denoted by p i j k {\displaystyle p_{ij}^{k}} . This structure is enhanced by adding all pairs of repeated elements of X and collecting them in a subset R0. This enhancement permits the parameters i, j, and k to take on the value of zero, and lets some of x,y or z be equal. A set with such an enhanced partition is called an association scheme. One may view an association scheme as a partition of the edges of a complete graph (with vertex set X) into n classes, often thought of as color classes. In this representation, there is a loop at each vertex and all the loops receive the same 0th color. The association scheme can also be represented algebraically. Consider the matrices Di defined by: ( D i ) x , y = { 1 , if ( x , y ) ∈ R i , 0 , otherwise. ( 1 ) {\displaystyle (D_{i})_{x,y}={\begin{cases}1,&{\text{if }}\left(x,y\right)\in R_{i},\\0,&{\text{otherwise.}}\end{cases}}\qquad (1)} Let A {\displaystyle {\mathcal {A}}} be the vector space consisting of all matrices ∑ ∑ i = 0 n a i D i {\displaystyle \sideset {}{_{i=0}^{n}}\sum a_{i}D_{i}} , with a i {\displaystyle a_{i}} complex. The definition of an association scheme is equivalent to saying that the D i {\displaystyle D_{i}} are v × v (0,1)-matrices which satisfy D i {\displaystyle D_{i}} is symmetric, ∑ i = 0 n D i = J {\displaystyle \sum _{i=0}^{n}D_{i}=J} (the all-ones matrix), D 0 = I , {\displaystyle D_{0}=I,} D i D j = ∑ k = 0 n p i j k D k = D j D i , i , j = 0 , … , n . {\displaystyle D_{i}D_{j}=\sum _{k=0}^{n}p_{ij}^{k}D_{k}=D_{j}D_{i},\qquad i,j=0,\ldots ,n.} The (x,y)-th entry of the left side of 4. is the number of two colored paths of length two joining x and y (using "colors" i and j) in the graph. Note that the rows and columns of D i {\displaystyle D_{i}} contain v i {\displaystyle v_{i}} 1s: D i J = J D i = v i J . ( 2 ) {\displaystyle D_{i}J=JD_{i}=v_{i}J.\qquad (2)} From 1., these matrices are symmetric. From 2., D 0 , … , D n {\displaystyle D_{0},\ldots ,D_{n}} are linearly independent, and the dimension of A {\displaystyle {\mathcal {A}}} is n + 1 {\displaystyle n+1} . From 4., A {\displaystyle {\mathcal {A}}} is closed under multiplication, and multiplication is always associative. This associative commutative algebra A {\displaystyle {\mathcal {A}}} is called the Bose–Mesner algebra of the association scheme. Since the matrices in A {\displaystyle {\mathcal {A}}} are symmetric and commute with each other, they can be simultaneously diagonalized. This means that there is a matrix S {\displaystyle S} such that to each A ∈ A {\displaystyle A\in {\mathcal {A}}} there is a diagonal matrix Λ A {\displaystyle \Lambda _{A}} with S − 1 A S = Λ A {\displaystyle S^{-1}AS=\Lambda _{A}} . This means that A {\displaystyle {\mathcal {A}}} is semi-simple and has a unique basis of primitive idempotents J 0 , … , J n {\displaystyle J_{0},\ldots ,J_{n}} . These are complex n × n matrices satisfying J i 2 = J i , i = 0 , … , n , ( 3 ) {\displaystyle J_{i}^{2}=J_{i},i=0,\ldots ,n,\qquad (3)} J i J k = 0 , i ≠ k , ( 4 ) {\displaystyle J_{i}J_{k}=0,i\neq k,\qquad (4)} ∑ i = 0 n J i = I . ( 5 ) {\displaystyle \sum _{i=0}^{n}J_{i}=I.\qquad (5)} The Bose–Mesner algebra has two distinguished bases: the basis consisting of the adjacency matrices D i {\displaystyle D_{i}} , and the basis consisting of the irreducible idempotent matrices J k {\displaystyle J_{k}} . By definition, there exist well-defined complex numbers such that D i = ∑ k = 0 n p i ( k ) J k , ( 6 ) {\displaystyle D_{i}=\sum _{k=0}^{n}p_{i}(k)J_{k},\qquad (6)} and | X | J k = ∑ i = 0 n q k ( i ) D i . ( 7 ) {\displaystyle |X|J_{k}=\sum _{i=0}^{n}q_{k}\left(i\right)D_{i}.\qquad (7)} The p-numbers p i ( k ) {\displaystyle p_{i}(k)} , and the q-numbers q k ( i ) {\displaystyle q_{k}(i)} , play a prominent role in the theory. They satisfy well-defined orthogonality relations. The p-numbers are the eigenvalues of the adjacency matrix D i {\displaystyle D_{i}} . == Theorem == The eigenvalues of p i ( k ) {\displaystyle p_{i}(k)} and q k ( i ) {\displaystyle q_{k}(i)} , satisfy the orthogonality conditions: ∑ k = 0 n μ i p i ( k ) p ℓ ( k ) = v v i δ i ℓ , ( 8 ) {\displaystyle \sum _{k=0}^{n}\mu _{i}p_{i}(k)p_{\ell }(k)=vv_{i}\delta _{i\ell },\quad (8)} ∑ k = 0 n μ i q k ( i ) q ℓ ( i ) = v μ k δ k ℓ . ( 9 ) {\displaystyle \sum _{k=0}^{n}\mu _{i}q_{k}(i)q_{\ell }(i)=v\mu _{k}\delta _{k\ell }.\quad (9)} Also μ j p i ( j ) = v i q j ( i ) , i , j = 0 , … , n . ( 10 ) {\displaystyle \mu _{j}p_{i}(j)=v_{i}q_{j}(i),\quad i,j=0,\ldots ,n.\quad (10)} In matrix notation, these are P T Δ μ P = v Δ v , ( 11 ) {\displaystyle P^{T}\Delta _{\mu }P=v\Delta _{v},\quad (11)} Q T Δ v Q = v Δ μ , ( 12 ) {\displaystyle Q^{T}\Delta _{v}Q=v\Delta _{\mu },\quad (12)} where Δ v = diag { v 0 , v 1 , … , v n } , Δ μ = diag { μ 0 , μ 1 , … , μ n } . {\displaystyle \Delta _{v}=\operatorname {diag} \{v_{0},v_{1},\ldots ,v_{n}\},\qquad \Delta _{\mu }=\operatorname {diag} \{\mu _{0},\mu _{1},\ldots ,\mu _{n}\}.} == Proof of theorem == The eigenvalues of D i D ℓ {\displaystyle D_{i}D_{\ell }} are p i ( k ) p ℓ ( k ) {\displaystyle p_{i}(k)p_{\ell }(k)} with multiplicities μ k {\displaystyle \mu _{k}} . This implies that v v i δ i ℓ = trace D i D ℓ = ∑ k = 0 n μ i p i ( k ) p ℓ ( k ) , ( 13 ) {\displaystyle vv_{i}\delta _{i\ell }=\operatorname {trace} D_{i}D_{\ell }=\sum _{k=0}^{n}\mu _{i}p_{i}(k)p_{\ell }(k),\quad (13)} which proves Equation ( 8 ) {\displaystyle \left(8\right)} and Equation ( 11 ) {\displaystyle \left(11\right)} , Q = v P − 1 = Δ v − 1 P T Δ μ , ( 14 ) {\displaystyle Q=vP^{-1}=\Delta _{v}^{-1}P^{T}\Delta _{\mu },\quad (14)} which gives Equations ( 9 ) {\displaystyle (9)} , ( 10 ) {\displaystyle (10)} and ( 12 ) {\displaystyle (12)} . ◻ {\displaystyle \Box } There is an analogy between extensions of association schemes and extensions of finite fields. The cases we are most interested in are those where the extended schemes are defined on the n {\displaystyle n} -th Cartesian power X = F n {\displaystyle X={\mathcal {F}}^{n}} of a set F {\displaystyle {\mathcal {F}}} on which a basic association scheme ( F , K ) {\displaystyle \left({\mathcal {F}},K\right)} is defined. A first association scheme defined on X = F n {\displaystyle X={\mathcal {F}}^{n}} is called the n {\displaystyle n} -th Kronecker power ( F , K ) ⊗ n {\displaystyle \left({\mathcal {F}},K\right)_{\otimes }^{n}} of ( F , K ) {\displaystyle \left({\mathcal {F}},K\right)} . Next the extension is defined on the same set X = F n {\displaystyle X={\mathcal {F}}^{n}} by gathering classes of ( F , K ) ⊗ n {\displaystyle \left({\mathcal {F}},K\right)_{\otimes }^{n}} . The Kronecker power corresponds to the polynomial ring F [ X ] {\displaystyle F\left[X\right]} first defined on a field F {\displaystyle \mathbb {F} } , while the extension scheme corresponds to the extension field obtained as a quotient. An example of such an extended scheme is the Hamming scheme. Association schemes may be merged, but merging them leads to non-symmetric association schemes, whereas all usual codes are subgroups in symmetric Abelian schemes. == See also == Association scheme == Notes == == References == Bailey, Rosemary A. (2004), Association schemes: Designed experiments, algebra and combinatorics, Cambridge Studies in Advanced Mathematics, vol. 84, Cambridge University Press, p. 387, ISBN 978-0-521-82446-0, MR 2047311 Bannai, Eiichi; Ito, Tatsuro (1984), Algebraic combinatorics I: Association schemes, Menlo Park, CA: The Benjamin/Cummings Publishing Co., Inc., pp. xxiv+425, ISBN 0-8053-0490-8, MR 0882540 Bannai, Etsuko (2001), "Bose–Mesner algebras associated with four-weight spin models", Graphs and Combinatorics, 17 (4): 589–598, doi:10.1007/PL00007251, S2CID 41255028 Bose, R. C.; Mesner, D. M. (1959), "On linear associative algebras corresponding to association schemes of partially balanced designs", Annals of Mathematical Statistics, 30 (1): 21–38, doi:10.1214/aoms/1177706356, JSTOR 2237117, MR 0102157 Cameron, P. J.; van Lint, J. H. (1991), Designs, Graphs, Codes and their Links, Cambridge: Cambridge University Press, ISBN 0-521-42385-6 Camion, P. (1998), "Codes and association schemes: Basic properties of association schemes relevant to coding", in Pless, V. S.; Huffman, W. C. (eds.), Handbook of coding theory, The Netherlands: Elsevier Delsarte, P.; Levenshtein, V. I. (1998), "Association schemes and coding theory", IEEE Transactions on Information Theory, 44 (6): 2477–2504, doi:10.1109/18.720545 MacWilliams, F. J.; Sloane, N. J. A. (1978), The theory of error-correcting codes, New York: Elsevier Nomura, K. (1997), "An algebra associated with a spin model", Journal of Algebraic Combinatorics, 6 (1): 53–58, doi:10.1023/A:1008644201287
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Wikipedia:Boualem Khouider#0
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Boualem Khouider is an Algerian-Canadian applied mathematician, climate scientist, academic, and author. He is a professor, and former Chair of Mathematics and Statistics at the University of Victoria. Khouider has published more than 100 papers with his most recognizable contributions being in the applied mathematics, atmospheric science, as well as climate modelling. He is the author of a book Models for Tropical Climate Dynamics: Waves, Clouds, and Precipitation, and is the editor of several edited volumes. He is also a Senior Fellow of Institute for Pure and Applied Mathematics (IPAM) at the University of California Los Angeles, a senior advisor of Center for Prototype Climate Models at NYU Abu-Dhabi Institute, and holds editorial appointments as an Editor for Mathematics of Climate and Weather Forecasting, and as Associate Editor for AIMS Mathematics, == Education == Khouider received a "High studies diploma" (DES) in Mathematical Analysis of Partial Differential Equations from the University of Sciences and Technology Houari Boumedienne in 1990. He then enrolled at University of Montreal, and earned his Master's and doctoral degrees in Applied Mathematics in 1997 and 2002, respectively. == Career == Khouider began his academic career as an Assistant Lecturer at Ecole Nationale Polytechnique in 1990, and was subsequently appointed as an Assistant Lecturer at the University of Sciences and Technology Houari Boumedienne from 1992 until 1994. During this time period, he also held concurrent appointments as lecturer at Ecole Nationale Naval, and Institut National de Formation en Batiment. In 1995, he held a brief appointment as lecturer at Ecole Militaire Polytechnique, before being appointed by the University of Montreal as a Teaching Assistant, and by Center of Research on Applied Computations (Cerca) as a Research Assistant from 1996 to 2000. Following this, he was appointed as a Research Associate in Courant Institute of Mathematical Sciences at New York University until 2003. He held his next appointment at the University of Victoria as Assistant Professor Mathematics and Statistics in 2003, and was promoted to associate professor in 2008, and to Full Professor in 2013. Khouider's contribution to the community includes his role as an organizer of the conferences and workshops, especially the ones in Banff and Oberwolfach. He has also conducted invited lectures and presentations at various professional institutions, including New York University, DongHua University, and the University of Sciences and Technology Houari Boumedienne. == Research == Khouider's work mainly focuses on applied mathematics, in particular in the fields of computational fluid dynamics, earth System modelling, sea-ice dynamics modelling, tropical meteorology, tropical extra-tropical interactions, organized convection and convectively coupled waves. His research work has been supported by numerous professional organizations, including Natural Sciences and Engineering Research Council (NSERC), Canadian Foundation for Climate and Atmospheric Research, Indian Institute of Tropical Meteorology, and Pacific Institute for Mathematical Sciences. === Atmospheric science === Khouider conducted a detailed dynamical analysis for the linear waves in 2006, while developing a systematic model convective parameterization focused on highlighting the dynamic role of the three cloud types, congestus, stratiform, and deep convective cumulus clouds, in terms of exploring the dynamics of large-scale convectively coupled Kelvin waves, westward-propagating two-day waves, and the Madden–Julian oscillation. In a companion paper, he presented a report on idealized nonlinear numerical simulations in the context of the developed model. Furthermore, he highlighted how the adequate representation of the dominant intraseasonal and synoptic-scale variability in the tropics, characterized by the Madden–Julian oscillation (MJO) and convectively coupled waves, continues to be a problem in current operational general circulation models (GCMs). Khouider discussed the role of environmental moisture in terms of the deepening of cumulus convection. It was found out that the mixing of water vapor by subgrid-scale turbulence has a significant impact on cloud depth, while the mixing of sensible heat has comparatively a negligible impact. In another study, he presented a paradigm model, and provided the applications of the stochastic multicloud framework in terms of improving deterministic parameterizations with clear deficiencies. Moreover, he along with co-worker, demonstrated the linear stability results for the multicloud model on an equatorial beta plane. === Climate modelling === In 2010, Khouider developed a stochastic multicloud model to represent the missing variability that occurred in global climate models due to unresolved features of organized tropical convection. Furthermore, he coupled stochastic multicloud model to a simple tropical climate model consisting of a system of ODEs, and highlighted the dynamical features of the coupled model. Using a statistical inference method based on the Bayesian paradigm, he estimated the stochasticity of convection in terms of Giga-LES data. In his study conducted in 2019, he demonstrated the stochastic parametrization of organized convection, and also explored the performance of the stochastic multicloud model in a stand-alone mode where the cloud model is forced directly by the observed predictors without feedback into the environmental variables. Later on, he successfully implemented the stochastic multicloud model in the CFSv2 GCM, used by the Indian Institute for Tropical Meteorology which led to huge improvements of the climate model in terms of simulating tropical modes of variability, including the MJO, monsoon intra-seasonal oscillations, and convectively coupled equatorial waves. Khouider also proposed a non-oscillatory balanced numerical scheme with application to preserve geostrophic steady states with minimal ad hoc dissipation by using state of the art numerical methods for each piece. Furthermore, he explored and highlighted the role of stratiform heating in the context of scale-selection of organized tropical convection over the monsoon trough, while using aquaplanet version of a coarse-resolution atmospheric general circulation model coupled to a stochastic multicloud cumulus parameterization scheme. In his paper titled "Climate science in the tropics: waves, vortices and PDEs," he presented a review on the interdisciplinary contributions over the last decade through the modus operandi of applied mathematics to most frequently occurring scientific problems. He discussed novel multiscale equations, PDEs, and numerical algorithms with the purpose to persuade mathematicians and physicists to do research in this particular area of study. In a recent study, he conducted a comparative analysis between four theories of the Madden-Julian Oscillation (MJO), which lead to a realization that theoretical thinking of the MJO is diverse and understanding of MJO dynamics needs to be further advanced. == Awards and honors == 2009-10 - Distinguished Visitor, Courant Institute for Mathematical Sciences, New York University 2010 - Senior Fellow, Institute for Pure and Applied Mathematics (IPAM), University of California Los Angeles 2015 - Research Excellence Award, Faculty of Science == Bibliography == === Books === Models for Tropical Climate Dynamics: Waves, Clouds, and Precipitation (2019) ISBN 9783030177751 === Selected articles === Majda, A. J., & Khouider, B. (2002). Stochastic and mesoscopic models for tropical convection. Proceedings of the National Academy of Sciences, 99(3), 1123–1128. Khouider, B., & Majda, A. J. (2006). A simple multicloud parameterization for convectively coupled tropical waves. Part I: Linear analysis. Journal of the atmospheric sciences, 63(4), 1308–1323. Khouider, B., & Majda, A. J. (2007). A simple multicloud parameterization for convectively coupled tropical waves. Part II: Nonlinear simulations. Journal of the atmospheric sciences, 64(2), 381–400. Majda, A. J., Franzke, C., & Khouider, B. (2008). An applied mathematics perspective on stochastic modelling for climate. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 366(1875), 2427–2453. Khouider, B., Biello, J., & Majda, A. J. (2010). A stochastic multicloud model for tropical convection. Communications in Mathematical Sciences, 8(1), 187–216. == References ==
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Wikipedia:Bounded set#0
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In mathematical analysis and related areas of mathematics, a set is called bounded if all of its points are within a certain distance of each other. Conversely, a set which is not bounded is called unbounded. The word "bounded" makes no sense in a general topological space without a corresponding metric. Boundary is a distinct concept; for example, a circle (not to be confused with a disk) in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary. A bounded set is not necessarily a closed set and vice versa. For example, a subset S of a 2-dimensional real space R2 constrained by two parabolic curves x2 + 1 and x2 − 1 defined in a Cartesian coordinate system is closed by the curves but not bounded (so unbounded). == Definition in the real numbers == A set S of real numbers is called bounded from above if there exists some real number k (not necessarily in S) such that k ≥ s for all s in S. The number k is called an upper bound of S. The terms bounded from below and lower bound are similarly defined. A set S is bounded if it has both upper and lower bounds. Therefore, a set of real numbers is bounded if it is contained in a finite interval. == Definition in a metric space == A subset S of a metric space (M, d) is bounded if there exists r > 0 such that for all s and t in S, we have d(s, t) < r. The metric space (M, d) is a bounded metric space (or d is a bounded metric) if M is bounded as a subset of itself. Total boundedness implies boundedness. For subsets of Rn the two are equivalent. A metric space is compact if and only if it is complete and totally bounded. A subset of Euclidean space Rn is compact if and only if it is closed and bounded. This is also called the Heine-Borel theorem. == Boundedness in topological vector spaces == In topological vector spaces, a different definition for bounded sets exists which is sometimes called von Neumann boundedness. If the topology of the topological vector space is induced by a metric which is homogeneous, as in the case of a metric induced by the norm of normed vector spaces, then the two definitions coincide. == Boundedness in order theory == A set of real numbers is bounded if and only if it has an upper and lower bound. This definition is extendable to subsets of any partially ordered set. Note that this more general concept of boundedness does not correspond to a notion of "size". A subset S of a partially ordered set P is called bounded above if there is an element k in P such that k ≥ s for all s in S. The element k is called an upper bound of S. The concepts of bounded below and lower bound are defined similarly. (See also upper and lower bounds.) A subset S of a partially ordered set P is called bounded if it has both an upper and a lower bound, or equivalently, if it is contained in an interval. Note that this is not just a property of the set S but also one of the set S as subset of P. A bounded poset P (that is, by itself, not as subset) is one that has a least element and a greatest element. Note that this concept of boundedness has nothing to do with finite size, and that a subset S of a bounded poset P with as order the restriction of the order on P is not necessarily a bounded poset. A subset S of Rn is bounded with respect to the Euclidean distance if and only if it bounded as subset of Rn with the product order. However, S may be bounded as subset of Rn with the lexicographical order, but not with respect to the Euclidean distance. A class of ordinal numbers is said to be unbounded, or cofinal, when given any ordinal, there is always some element of the class greater than it. Thus in this case "unbounded" does not mean unbounded by itself but unbounded as a subclass of the class of all ordinal numbers. == See also == Bounded domain Bounded function Local boundedness Order theory Totally bounded == References == Bartle, Robert G.; Sherbert, Donald R. (1982). Introduction to Real Analysis. New York: John Wiley & Sons. ISBN 0-471-05944-7. Richtmyer, Robert D. (1978). Principles of Advanced Mathematical Physics. New York: Springer. ISBN 0-387-08873-3.
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Wikipedia:Bounding point#0
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In functional analysis, a branch of mathematics, a bounding point of a subset of a vector space is a conceptual extension of the boundary of a set. == Definition == Let A {\displaystyle A} be a subset of a vector space X {\displaystyle X} . Then x ∈ X {\displaystyle x\in X} is a bounding point for A {\displaystyle A} if it is neither an internal point for A {\displaystyle A} nor its complement. == References ==
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Wikipedia:Box counting#0
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Box counting is a method of gathering data for analyzing complex patterns by breaking a dataset, object, image, etc. into smaller and smaller pieces, typically "box"-shaped, and analyzing the pieces at each smaller scale. The essence of the process has been compared to zooming in or out using optical or computer based methods to examine how observations of detail change with scale. In box counting, however, rather than changing the magnification or resolution of a lens, the investigator changes the size of the element used to inspect the object or pattern (see Figure 1). Computer based box counting algorithms have been applied to patterns in 1-, 2-, and 3-dimensional spaces. The technique is usually implemented in software for use on patterns extracted from digital media, although the fundamental method can be used to investigate some patterns physically. The technique arose out of and is used in fractal analysis. It also has application in related fields such as lacunarity and multifractal analysis. == The method == Theoretically, the intent of box counting is to quantify fractal scaling, but from a practical perspective this would require that the scaling be known ahead of time. This can be seen in Figure 1 where choosing boxes of the right relative sizes readily shows how the pattern repeats itself at smaller scales. In fractal analysis, however, the scaling factor is not always known ahead of time, so box counting algorithms attempt to find an optimized way of cutting a pattern up that will reveal the scaling factor. The fundamental method for doing this starts with a set of measuring elements—boxes—consisting of an arbitrary number, called E {\displaystyle \mathrm {E} } here for convenience, of sizes or calibres, which we will call the set of ϵ {\displaystyle \epsilon } s. Then these ϵ {\displaystyle \epsilon } -sized boxes are applied to the pattern and counted. To do this, for each ϵ {\displaystyle \epsilon } in E {\displaystyle \mathrm {E} } , a measuring element that is typically a 2-dimensional square or 3-dimensional box with side length corresponding to ϵ {\displaystyle \epsilon } is used to scan a pattern or data set (e.g., an image or object) according to a predetermined scanning plan to cover the relevant part of the data set, recording, i.e.,counting, for each step in the scan relevant features captured within the measuring element. === The data === The relevant features gathered during box counting depend on the subject being investigated and the type of analysis being done. Two well-studied subjects of box counting, for instance, are binary (meaning having only two colours, usually black and white) and gray-scale digital images (i.e., jpegs, tiffs, etc.). Box counting is generally done on patterns extracted from such still images in which case the raw information recorded is typically based on features of pixels such as a predetermined colour value or range of colours or intensities. When box counting is done to determine a fractal dimension known as the box counting dimension, the information recorded is usually either yes or no as to whether or not the box contained any pixels of the predetermined colour or range (i.e., the number of boxes containing relevant pixels at each ϵ {\displaystyle \epsilon } is counted). For other types of analysis, the data sought may be the number of pixels that fall within the measuring box, the range or average values of colours or intensities, the spatial arrangement amongst pixels within each box, or properties such as average speed (e.g., from particle flow). === Scan types === Every box counting algorithm has a scanning plan that describes how the data will be gathered, in essence, how the box will be moved over the space containing the pattern. A variety of scanning strategies has been used in box counting algorithms, where a few basic approaches have been modified in order to address issues such as sampling, analysis methods, etc. ==== Fixed grid scans ==== The traditional approach is to scan in a non-overlapping regular grid or lattice pattern. To illustrate, Figure 2a shows the typical pattern used in software that calculates box counting dimensions from patterns extracted into binary digital images of contours such as the fractal contour illustrated in Figure 1 or the classic example of the coastline of Britain often used to explain the method of finding a box counting dimension. The strategy simulates repeatedly laying a square box as though it were part of a grid overlaid on the image, such that the box for each ϵ {\displaystyle \epsilon } never overlaps where it has previously been (see Figure 4). This is done until the entire area of interest has been scanned using each ϵ {\displaystyle \epsilon } and the relevant information has been recorded. When used to find a box counting dimension, the method is modified to find an optimal covering. ==== Sliding box scans ==== Another approach that has been used is a sliding box algorithm, in which each box is slid over the image overlapping the previous placement. Figure 2b illustrates the basic pattern of scanning using a sliding box. The fixed grid approach can be seen as a sliding box algorithm with the increments horizontally and vertically equal to ϵ {\displaystyle \epsilon } . Sliding box algorithms are often used for analyzing textures in lacunarity analysis and have also been applied to multifractal analysis. ==== Subsampling and local dimensions ==== Box counting may also be used to determine local variation as opposed to global measures describing an entire pattern. Local variation can be assessed after the data have been gathered and analyzed (e.g., some software colour codes areas according to the fractal dimension for each subsample), but a third approach to box counting is to move the box according to some feature related to the pixels of interest. In local connected dimension box counting algorithms, for instance, the box for each ϵ {\displaystyle \epsilon } is centred on each pixel of interest, as illustrated in Figure 2c. == Methodological considerations == The implementation of any box counting algorithm has to specify certain details such as how to determine the actual values in E {\displaystyle \mathrm {E} } , including the minimum and maximum sizes to use and the method of incrementing between sizes. Many such details reflect practical matters such as the size of a digital image but also technical issues related to the specific analysis that will be performed on the data. Another issue that has received considerable attention is how to approximate the so-called "optimal covering" for determining box counting dimensions and assessing multifractal scaling. === Edge effects === One known issue in this respect is deciding what constitutes the edge of the useful information in a digital image, as the limits employed in the box counting strategy can affect the data gathered. === Scaling box size === The algorithm has to specify the type of increment to use between box sizes (e.g., linear vs exponential), which can have a profound effect on the results of a scan. === Grid orientation === As Figure 4 illustrates, the overall positioning of the boxes also influences the results of a box count. One approach in this respect is to scan from multiple orientations and use averaged or optimized data. To address various methodological considerations, some software is written so users can specify many such details, and some includes methods such as smoothing the data after the fact to be more amenable to the type of analysis being done. == See also == Fractal analysis Fractal dimension Minkowski–Bouligand dimension Multifractal analysis Lacunarity == References ==
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Wikipedia:Box-counting content#0
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In mathematics, the box-counting content is an analog of Minkowski content. == Definition == Let A {\displaystyle A} be a bounded subset of m {\displaystyle m} -dimensional Euclidean space R m {\displaystyle \mathbb {R} ^{m}} such that the box-counting dimension D B {\displaystyle D_{B}} exists. The upper and lower box-counting contents of A {\displaystyle A} are defined by B ∗ ( A ) := lim sup x → ∞ N B ( A , x ) x D B and B ∗ ( A ) := lim inf x → ∞ N B ( A , x ) x D B {\displaystyle {\mathcal {B}}^{*}(A):=\limsup _{x\rightarrow \infty }{\frac {N_{B}(A,x)}{x^{D_{B}}}}\quad \quad {\text{and}}\quad \quad {\mathcal {B}}_{*}(A):=\liminf _{x\rightarrow \infty }{\frac {N_{B}(A,x)}{x^{D_{B}}}}} where N B ( A , x ) {\displaystyle N_{B}(A,x)} is the maximum number of disjoint closed balls with centers a ∈ A {\displaystyle a\in A} and radii x − 1 > 0 {\displaystyle x^{-1}>0} . If B ∗ ( A ) = B ∗ ( A ) {\displaystyle {\mathcal {B}}^{*}(A)={\mathcal {B}}_{*}(A)} , then the common value, denoted B ( A ) {\displaystyle {\mathcal {B}}(A)} , is called the box-counting content of A {\displaystyle A} . If 0 < B ∗ ( A ) < B ∗ ( A ) < ∞ {\displaystyle 0<{\mathcal {B}}_{*}(A)<{\mathcal {B}}^{*}(A)<\infty } , then A {\displaystyle A} is said to be box-counting measurable. == Examples == Let I = [ 0 , 1 ] {\displaystyle I=[0,1]} denote the unit interval. Note that the box-counting dimension dim B I {\displaystyle \dim _{B}I} and the Minkowski dimension dim M I {\displaystyle \dim _{M}I} coincide with a common value of 1; i.e. dim B I = dim M I = 1. {\displaystyle \dim _{B}I=\dim _{M}I=1.} Now observe that N B ( I , x ) = ⌊ x / 2 ⌋ + 1 {\displaystyle N_{B}(I,x)=\lfloor x/2\rfloor +1} , where ⌊ y ⌋ {\displaystyle \lfloor y\rfloor } denotes the integer part of y {\displaystyle y} . Hence I {\displaystyle I} is box-counting measurable with B ( I ) = 1 / 2 {\displaystyle {\mathcal {B}}(I)=1/2} . By contrast, I {\displaystyle I} is Minkowski measurable with M ( I ) = 1 {\displaystyle {\mathcal {M}}(I)=1} . == See also == Box counting == References == Dettmers, Kristin; Giza, Robert; Morales, Rafael; Rock, John A.; Knox, Christina (January 2017). "A survey of complex dimensions, measurability, and the lattice/nonlattice dichotomy". Discrete and Continuous Dynamical Systems - Series S. 10 (2): 213–240. arXiv:1510.06467. doi:10.3934/dcdss.2017011.
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Wikipedia:Brahmagupta's interpolation formula#0
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Brahmagupta's interpolation formula is a second-order polynomial interpolation formula developed by the Indian mathematician and astronomer Brahmagupta (598–668 CE) in the early 7th century CE. The Sanskrit couplet describing the formula can be found in the supplementary part of Khandakadyaka a work of Brahmagupta completed in 665 CE. The same couplet appears in Brahmagupta's earlier Dhyana-graha-adhikara, which was probably written "near the beginning of the second quarter of the 7th century CE, if not earlier." Brahmagupta was one of the first to describe and use an interpolation formula using second-order differences. Brahmagupta's interpolation formula is equivalent to modern-day second-order Newton–Stirling interpolation formula. Mathematicians prior to Brahmagupta used a simple linear interpolation formula. The linear interpolation formula to compute f(a) is f ( a ) = f r + t D r {\displaystyle f(a)=f_{r}+tD_{r}} where t = a − x r h {\displaystyle t={\frac {a-x_{r}}{h}}} . For the computation of f(a), Brahmagupta replaces Dr with another expression which gives more accurate values and which amounts to using a second-order interpolation formula. == Brahmagupta's description of the scheme == In Brahmagupta's terminology the difference Dr is the gatakhanda, meaning past difference or the difference that was crossed over, the difference Dr+1 is the bhogyakhanda which is the difference yet to come. Vikala is the amount in minutes by which the interval has been covered at the point where we want to interpolate. In the present notations it is a − xr. The new expression which replaces fr+1 − fr is called sphuta-bhogyakhanda. The description of sphuta-bhogyakhanda is contained in the following Sanskrit couplet (Dhyana-Graha-Upadesa-Adhyaya, 17; Khandaka Khadyaka, IX, 8): This has been translated using Bhattolpala's (10th century CE) commentary as follows: Multiply the vikala by the half the difference of the gatakhanda and the bhogyakhanda and divide the product by 900. Add the result to half the sum of the gatakhanda and the bhogyakhanda if their half-sum is less than the bhogyakhanda, subtract if greater. (The result in each case is sphuta-bhogyakhanda the correct tabular difference.) This formula was originally stated for the computation of the values of the sine function for which the common interval in the underlying base table was 900 minutes or 15 degrees. So the reference to 900 is in fact a reference to the common interval h. == In modern notation == Brahmagupta's method computation of shutabhogyakhanda can be formulated in modern notation as follows: sphuta-bhogyakhanda = D r + D r − 1 2 ± t | D r − D r − 1 | 2 . {\displaystyle \displaystyle ={\frac {D_{r}+D_{r-1}}{2}}\pm t{\frac {|D_{r}-D_{r-1}|}{2}}.} The ± sign is to be taken according to whether 1/2(Dr + Dr+1) is less than or greater than Dr+1, or equivalently, according to whether Dr < Dr+1 or Dr > Dr+1. Brahmagupta's expression can be put in the following form: sphuta-bhogyakhanda = D r + D r − 1 2 + t D r − D r − 1 2 . {\displaystyle \displaystyle ={\frac {D_{r}+D_{r-1}}{2}}+t{\frac {D_{r}-D_{r-1}}{2}}.} This correction factor yields the following approximate value for f(a): f ( a ) = f r + t × sphuta-bhogyakhanda = f r + t D r + D r − 1 2 + t 2 D r − D r − 1 2 . {\displaystyle {\begin{aligned}f(a)&=f_{r}+t\times {\text{sphuta-bhogyakhanda}}\\&=f_{r}+t{\frac {D_{r}+D_{r-1}}{2}}+t^{2}{\frac {D_{r}-D_{r-1}}{2}}.\end{aligned}}} This is Stirling's interpolation formula truncated at the second-order differences. It is not known how Brahmagupta arrived at his interpolation formula. Brahmagupta has given a separate formula for the case where the values of the independent variable are not equally spaced. == See also == Brahmagupta's identity Brahmagupta matrix Brahmagupta–Fibonacci identity == References ==
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Wikipedia:Braided vector space#0
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In mathematics, a braided vector space V {\displaystyle \;V} is a vector space together with an additional structure map τ {\displaystyle \tau } symbolizing interchanging of two vector tensor copies: τ : V ⊗ V ⟶ V ⊗ V {\displaystyle \tau :\;V\otimes V\longrightarrow V\otimes V} such that the Yang–Baxter equation is fulfilled. Hence drawing tensor diagrams with τ {\displaystyle \tau } an overcrossing the corresponding composed morphism is unchanged when a Reidemeister move is applied to the tensor diagram and thus they present a representation of the braid group. As first example, every vector space is braided via the trivial braiding (simply flipping). A superspace has a braiding with negative sign in braiding two odd vectors. More generally, a diagonal braiding means that for a V {\displaystyle V} -base x i {\displaystyle x_{i}} we have τ ( x i ⊗ x j ) = q i j ( x j ⊗ x i ) {\displaystyle \tau (x_{i}\otimes x_{j})=q_{ij}(x_{j}\otimes x_{i})} A good source for braided vector spaces entire braided monoidal categories with braidings between any objects τ V , W {\displaystyle \tau _{V,W}} , most importantly the modules over quasitriangular Hopf algebras and Yetter–Drinfeld modules over finite groups (such as Z 2 {\displaystyle \mathbb {Z} _{2}} above) If V {\displaystyle V} additionally possesses an algebra structure inside the braided category ("braided algebra") one has a braided commutator (e.g. for a superspace the anticommutator): [ x , y ] τ := μ ( ( x ⊗ y ) − τ ( x ⊗ y ) ) μ ( x ⊗ y ) := x y {\displaystyle \;[x,y]_{\tau }:=\mu ((x\otimes y)-\tau (x\otimes y))\qquad \mu (x\otimes y):=xy} Examples of such braided algebras (and even Hopf algebras) are the Nichols algebras, that are by definition generated by a given braided vectorspace. They appear as quantum Borel part of quantum groups and often (e.g. when finite or over an abelian group) possess an arithmetic root system, multiple Dynkin diagrams and a PBW-basis made up of braided commutators just like the ones in semisimple Lie algebras. == References ==
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Wikipedia:Brailey Sims#0
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Brailey Sims (born 26 October 1947) is an Australian mathematician born and educated in Newcastle, New South Wales. He received his BSc from the University of Newcastle (Australia) in 1969 and, under the supervision of J. R. Giles, a PhD from the same university in 1972. He was on the faculty of the University of New England (Australia) from 1972 to 1989. In 1990 he took up an appointment at the University of Newcastle (Australia). where he was Head of Mathematics from 1997 to 2000. He is best known for his work in nonlinear analysis and especially metric fixed point theory and its connections with Banach and metric space geometry, and for his efforts to promote and enhance mathematics at the secondary and tertiary level. == Publications == His most cited publications are: Mustafa Z, Sims B. A new approach to generalized metric spaces. Journal of Nonlinear and convex Analysis. 2006 Jan 1;7(2):289. According to Google Scholar, it has been cited 1154 times. Kirk WA, Sims B. Handbook of metric fixed point theory. Australian Mathematical Society GAZETTE. 2002;29(2). According to Google Scholar, this article has been cited 604 times Mustafa Z, Sims B. Some remarks concerning D-metric spaces. In Proceedings of the International Conference on Fixed Point Theory and Applications 2003 Jul 13 (pp. 189–198). According to Google Scholar, this article has been cited 432 times Mustafa Z, Sims B. Fixed point theorems for contractive mappings in complete-metric spaces. Fixed point theory and Applications. 2009 Dec;2009:1-0. According to Google Scholar, this article has been cited 418 times Dhompongsa S, Kirk WA, Sims B. Fixed points of uniformly Lipschitzian mappings. Nonlinear analysis: theory, methods & applications. 2006 Aug 15;65(4):762-72. According to Google Scholar, this article has been cited 297 times == References ==
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