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Victor Lomonosov
Victor Lomonosov (7 February 1946 – 29 March 2018) was a Russian-American mathematician known for his work in functional analysis. In operator theory, he is best known for his work in 1973 on the invariant subspace problem, which was described by Walter Rudin in his classical book on Functional Analysis as "Lomonosov's spectacular invariant subspace theorem".[1] The Theorem Lomonosov gives a very short proof, using the Schauder fixed point theorem, that if the bounded linear operator T on a Banach space commutes with a non-zero compact operator then T has a non-trivial invariant subspace.[2] Lomonosov has also published on the Bishop–Phelps theorem[3] and Burnside's Theorem [4]
Lomonosov received his master's degree from the Moscow State University in 1969 and his Ph.D. from National University of Kharkiv in 1974 (adviser Vladimir Matsaev). He was appointed at the rank of Associate Professor at Kent State University in the fall of1991, becoming Professor at the same university in 1999.
References
1. Rudin, Walter (1991) [1973]. Functional Analysis (2nd ed.). New York: McGraw-Hill. ISBN 0-07-100944-2.
2. Lomonosov, V. I. (1973). "Invariant subspaces of the family of operators that commute with a completely continuous operator". Akademija Nauk SSSR. Funkcional' Nyi Analiz I Ego Prilozenija. 7 (3): 55–56. doi:10.1007/BF01080698. MR 0420305. S2CID 121421267.
3. Lomonosov, Victor (2000). "A counterexample to the Bishop-Phelps theorem in complex spaces". Israel Journal of Mathematics. 115: 25–28. doi:10.1007/bf02810578. S2CID 53646715.
4. Lomonosov, Victor (1991). "An extension of Burnside's theorem to infinite-dimensional spaces". Israel Journal of Mathematics. 75 (2–3): 329–339. doi:10.1007/bf02776031. S2CID 120120695.
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Victor Mazurov
Victor Danilovich Mazurov (Russian: Виктор Данилович Мазуров; born January 31, 1943) is a Russian mathematician. He is well known for his works in group theory and is the founder of the Novosibirsk school of finite groups. Mazurov is a Corresponding Member of the Russian Academy of Sciences.
Mazurov's parents Daniil Petrovich and Evstolia Ivanovna were teachers.[1] Victor went to elementary school in a village of Kuvashi and finished high school with highest honors in Zlatoust. He then moved to Sverdlovsk (now Yekaterinburg) to study mathematics in Ural State University. His advisers in Sverdlovsk were Victor Busarkin and Albert Starostin.[2] In 1963 Mazurov married his university classmate Nadezhda Khomenko. After graduating in 1965, they moved to Novosibirsk where Mazurov joined the research staff of the Sobolev Institute of Mathematics (Russian: Институт математики СО РАН).
Mazurov is an editor (with Evgenyj Khukhro) of the "Kourovka Notebook",[3] a periodically updated collection of over 1,000 open problems in Group Theory.
Mazurov obtained several results that contributed to the proof of the classification of finite simple groups, also known as the Enormous Theorem[4] and considered one of the greatest achievements in mathematics of the 20th century.
He is one of the initial group of fellows of the American Mathematical Society.[5]
References
1. Sib Math Journal, Vol. 54(1), 2013 http://math.nsc.ru/LBRT/a4/Mazurov/rus/bio_ru.htm
2. Victor Mazurov at the Mathematics Genealogy Project
3. Khukhro, E. I.; Mazurov, V. D. (2014). "Unsolved Problems in Group Theory. The Kourovka Notebook". arXiv:1401.0300 [math.GR].
4. "Enormous Theorem".
5. Jackson, Allyn (May 2013), "Fellows of the AMS: Inaugural Class" (PDF), Notices of the American Mathematical Society, 60 (5): 631–637
External links
• Victor D. Mazurov Personal webpage at the Sobolev Institute of Mathematics
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Victor Săhleanu
Victor Aurelian Săhleanu (Romanian: [ˈviktor a.ureliˈan səhˈle̯anu]; 19 January 1924 – 26 August 1997) was a Romanian physician and anthropologist. He was a leading figure in anthropology in his country from the late 1960s until his death.
Biography
Education and early career
Săhleanu was born in Gura Humorului, in the Bukovina region of the Kingdom of Romania. After finishing secondary school at the Aron Pumnul High School in Cernăuți, he entered the medical faculty of the University of Bucharest, from which he graduated in 1948. At that point, with the onset of the Communist regime, the institution became the Carol Davila University of Medicine and Pharmacy. Between 1944 and 1946, he took part-time courses at the literature and philosophy faculty as well, but did not earn a degree. In 1949, he became a doctor of medicine and surgery, with a thesis on "Considerations regarding Field Medicine". Ștefan-Marius Milcu presided over the doctoral committee, which also included Constantin Ion Parhon. He began working in hospitals while still a student, and during 1946, was a junior teaching assistant in the anatomical pathology department.[1]
After graduation in 1948, he won a competition to become an intern at the Parhon-led endocrinology institute, where he was also a researcher from 1954 to 1961. He worked in endocrinology for a total of seventeen years, during which he founded the institute's morphopathology laboratory. From 1950 to 1952, he was a peer reviewer at the Milcu-led anthropology collective, a section of the endocrinology institute that was essentially a continuation of the Francisc Rainer-founded anthropology institute. In 1954, he signed up for part-time classes at the physics and mathematics faculty in Bucharest, graduating in 1961.[1]
Involvement in anthropology and legacy
In 1963, he became a primary care endocrinologist and, at the request of Eugen A. Pora, began teaching courses in biophysics and biomathematics at Babeș-Bolyai University in Cluj. In 1965, he earned the title of Doctor of Science. The same year, he was transferred from the endocrinology to the geriatrics institute. In 1969, he was transferred from Babeș-Bolyai and named adjunct scientific director at Bucharest's center for anthropological research;[1] from that point until his death, he was at the forefront of anthropology in Romania.[2] In 1974, when the center became a laboratory within the Victor Babeș institute, Săhleanu became its director, serving for eight years.[1]
In 1982, due to the so-called "Transcendental Meditation Affair", he was excluded from scientific life. His works were withdrawn from libraries, his name could no longer appear in books or publications, and he was transferred to work as a doctor at a hospital in the Titan neighborhood. In 1984, aged 60, he retired upon his request. Between 1982 and 1984, he taught postgraduate courses in anthropology at a United Nations demographic center in Bucharest.[1] As an anthropologist, he developed an interdisciplinary approach to the field that combined biology and culture, exploring the relationship between anatomical features and their behavioral, symbolic and cultural significance. In 1980, he was editor-in-chief of Romania's first atlas of biological anthropology.[2]
In February 1990, after the fall of the regime, he was restored as head of the Romanian Academy's anthropological research center, by government decree. He died in 1997, following complications from a cerebral hemorrhage.[3] He and his wife Zoe, a pediatrician, had two sons: Adrian George, who became a philologist and psychoanalyst; and Valentin, later an architect.[2]
Săhleanu published over 2000 articles and 60 books, in fields that included methodology, medical psychology and psychoanalysis, ethics, aesthetics and the history of medicine and science.[2] He kept a diary, from age 17 until his final days, that reached over 25,000 pages.[4] He was also an essayist and poet, publishing volumes in 1961, 1972, 1977 and 1997; and was among the founders of the Romanian Society of Writer and Journalist Physicians.[5]
Alexandru Ofrim stated that Săhleanu wrote communist propaganda against erotic pleasure.[6][7]
• With his mother at a family reunion in 1935
• As a high school student
• At his wedding in 1948
• In his office at the anthropology institute
Notes
1. Ciuhuța, p.9
2. Kozma, p.34
3. Ciuhuța, p.10
4. Kozma, p.35
5. Ciuhuța, p.12
6. Ofrim, Alexandru (18 September 2008). "Tot ce trebuia să știm despre sex - din cărți". Dilema veche (in Romanian). Retrieved 15 February 2019.
7. "Private Life and Social Practices during the Golden Age". Muzeul Municipiului București – Site Oficial. Retrieved 1 December 2020.
References
• (in Romanian) Andrei Kozma, Cristiana Glavce, Constantin Bălăceanu-Stolnici (eds.), Antropologie și mediu. Editura Niculescu, Bucharest, 2014, ISBN 978-973-748-859-6
• Mircea Ștefan Ciuhuța, "Victor Săhleanu, personalitate de prim rang în antropologia românească", p. 9-13
• Andrei Kozma, "Victor Aurelian Săhleanu, poetul om de știință", p. 34-9
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Victor Thébault
Victor Michael Jean-Marie Thébault (1882–1960) was a French mathematician best known for propounding three problems in geometry. The name Thébault's theorem is used by some authors to refer to the first of these problems and by others to refer to the third.
Thébault was born on March 6, 1882, in Ambrières-les-Grand (today a part of Ambrières-les-Vallées, Mayenne) in the northwest of France. He got his education at a teacher's college in Laval, where he studied from 1898 to 1901. After his graduation he taught for three years at Pré-en-Pail until he got a professorship at technical school in Ernée. In 1909 he placed first in a competitive exams, which yielded him a certificate to work as a science professor at teachers' colleges. Thébault however found a professor's salary insufficient to support his large family and hence he left teaching to become a factory superintendent at Ernée from 1910 to 1923. In 1924 he became a chief insurance inspector in Le Mans, a position he held until his retirement in 1940. During his retirement he lived in Tennie. He died on March 19, 1960, shortly after a severe stroke and was survived by his wife, five sons and a daughter.[1]
Despite leaving teaching Thébault stayed active in mathematics with number theory and geometry being his main areas of interest. He published a large number of articles in math journals all over the world and aside from regular articles he also contributed many original problems and solutions to their problem sections. He published over 1000 original problems in various mathematical magazines[2] and his contributions to the problem section of the American Mathematical Monthly alone comprise over 600 problems and solutions. In recognition of his contributions the French government bestowed two titles on him. In 1932 he became an Officier de L'Instruction Publique and in a 1935 a Chevalier de l'Order de Couronne de Belgium.[1]
Notes
1. C. W. Trigg: Victor Thebault 1882-1960. Mathematics Magazine, Vol. 33, No. 5 (May - Jun., 1960) (JSTOR)
2. Alexander Ostermann, Gerhard Wanner: Geometry by Its History. Springer, 2012, p. 181
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Victor Batyrev
Victor Vadimovich Batyrev (Виктор Вадимович Батырев, born 31 August 1961, Moscow)[1] is a Russian mathematician, specializing in algebraic and arithmetic geometry and its applications to mathematical physics. He is a professor at the University of Tübingen.
Biography
Batyrev studied mathematics from 1978 to 1985 at Moscow State University. From 1991 he was at the University of Essen, where he earned his habilitation in 1993. Since 1996 he has been a professor at the University of Tübingen.[2]
He received in 1994 the Gottschalk-Diederich-Baedeker Prize. In 1995 he received the Heinz Maier-Leibnitz Prize for his habilitation thesis Hodge Theory of Hypersurfaces in Toric Varieties and Recent Developments in Quantum Physics. In 1998 he was an invited speaker at the International Congress of Mathematicians in Berlin and gave a talk Mirror Symmetry and Toric Geometry.[3] In 2003 he was elected a member of the Heidelberger Akademie der Wissenschaften.[4]
Selected publications
• Batyrev, V. V.; Manin, Yu. I. (1990). "Sur le nombre des points rationnels de hauté borné des variétés algébriques". Mathematische Annalen. 286 (1–3): 27–43. doi:10.1007/bf01453564. S2CID 119945673.
• Batyrev, Victor V. (1993). "Quantum cohomology rings of toric manifolds". arXiv:alg-geom/9310004. Bibcode:1993alg.geom.10004B. {{cite journal}}: Cite journal requires |journal= (help)
• Batyrev, Victor V. (1994). "Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties". Journal of Algebraic Geometry: 493–535. arXiv:alg-geom/9310003. Bibcode:1993alg.geom.10003B.
• Batyrev, Victor V.; Borisov, Lev A. (1996). "Mirror duality and string theoretic Hodge numbers". Inventiones Mathematicae. 126 (1): 183–203. arXiv:alg-geom/9509009. Bibcode:1996InMat.126..183B. doi:10.1007/s002220050093. S2CID 55227866.
• Batyrev, Victor V.; Dais, Dimitrios I. (1996). "Strong McKay correspondence, string-theoretic Hodge numbers and mirror symmetry". Topology. 35 (4): 901–929. arXiv:alg-geom/9410001. doi:10.1016/0040-9383(95)00051-8. S2CID 15604511.
• Batyrev, Victor V. (1997). "Stringy Hodge numbers of varieties with Gorenstein canonical singularities". arXiv:alg-geom/9711008. Bibcode:1997alg.geom.11008B. {{cite journal}}: Cite journal requires |journal= (help)
• Batyrev, Victor V.; Tschinkel, Yuri (1998). "Manin's conjecture for toric manifolds". Journal of Algebraic Geometry. 7: 15–53. arXiv:alg-geom/9510014. Bibcode:1995alg.geom.10014B.
References
1. Jahrbuch der Heidelberger Akademie der Wissenschaften 2010
2. homepage of Victor Batyrev at the University of Tübingen
3. Batyrev, Victor V. (1998). "Mirror symmetry and toric geometry". Doc. Math. (Bielefeld) Extra Vol. ICM Berlin, 1998, vol. II. pp. 239–248.
4. page for Batyrev at homepage of Heidelberger Akademie der Wissenschaften
External links
• mathnet.ru
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Victor W. Marek
Victor Witold Marek, formerly Wiktor Witold Marek known as Witek Marek (born 22 March 1943) is a Polish mathematician and computer scientist working in the fields of theoretical computer science and mathematical logic.
Biography
Victor Witold Marek studied mathematics at the Faculty of Mathematics and Physics of the University of Warsaw. Supervised by Andrzej Mostowski, he received both a magister degree in mathematics in 1964 and a doctoral degree in mathematics in 1968. He completed habilitation in mathematics in 1972.
In 1970–1971, Marek was a postdoctoral researcher at Utrecht University, the Netherlands, where he worked under Dirk van Dalen. In 1967–1968 as well as in 1973–1975, he was a researcher at the Institute of Mathematics of the Polish Academy of Sciences, Warsaw, Poland. In 1979–1980 and 1982–1983 he worked at the Venezuelan Institute of Scientific Research. In 1976, he was appointed an Assistant Professor of Mathematics at the University of Warsaw.
In 1983, he was appointed a professor of computer science at the University of Kentucky. In 1989–1990, he was a Visiting Professor of Mathematics at Cornell University, Ithaca, New York. In 2001–2002, he was a visitor at the Department of Mathematics of the University of California, San Diego.
In 2013, Professor Marek was the Chair of the Program Committee of the scientific conference commemorating Andrzej Mostowski's Centennial.
Legacy
Teaching
He has supervised a number of graduate theses and projects. He was an advisor of 16 doctoral candidates both in mathematics and computer science. In particular, he advised dissertations in mathematics by Małgorzata Dubiel-Lachlan, Roman Kossak, Adam Krawczyk, Tadeusz Kreid, Roman Murawski, Andrzej Pelc, Zygmunt Ratajczyk, Marian Srebrny, and Zygmunt Vetulani. In computer science his students were V. K. Cody Bumgardner, Waldemar W. Koczkodaj, Witold Lipski, Joseph Oldham, Inna Pivkina, Michał Sobolewski , Paweł Traczyk, and Zygmunt Vetulani. These individuals have worked in various institutions of higher education in Canada, France, Poland, and the United States.
Mathematics
He investigated a number of areas in the foundations of mathematics, for instance infinitary combinatorics (large cardinals), metamathematics of set theory, the hierarchy of constructible sets,[1] models of second-order arithmetic,[2] the impredicative theory of Kelley–Morse classes.[3] He proved that the so-called Fraïssé conjecture (second-order theories of countable ordinals are all different) is entailed by Gödel's axiom of constructibility. Together with Marian Srebrny, he investigated properties of gaps in a constructible universe.
Computer science
He studied logical foundations of computer science. In the early 1970s, in collaboration with Zdzisław Pawlak,[4][5] he investigated Pawlak's information storage and retrieval systems,[6] which then was a widely studied concept, especially in Eastern Europe. These systems were essentially single-table relational databases, but unlike Codd's relational databases were bags rather than sets of records. These investigations, in turn, led Pawlak to the concept of rough set,[5] studied by Marek and Pawlak in 1981.[7] The concept of rough set, in computer science, statistics, topology, universal algebra, combinatorics, and modal logic, turned out to be an expressive language for describing, and especially manipulating an incomplete information.
Logic
In the area of nonmonotonic logics, a group of logics related to artificial intelligence, he focused on investigations of Reiter's default logic,[8] and autoepistemic logic of R. Moore. These investigations led to a form of logic programming called answer set programming[9] a computational knowledge representation formalism, studied both in Europe and in the United States. Together with Mirosław Truszczynski, he proved that the problem of existence of stable models of logic programs is NP-complete. In a stronger formalism admitting function symbols, along with Nerode and Remmel he showed that the analogous problem is Σ1
1
-complete.
Publications
V. W. Marek is an author of over 180 scientific papers in the area of foundations of mathematics and of computer science. He was also an editor of numerous proceedings of scientific meetings. Additionally, he authored or coauthored several books. These include:
• Logika i Podstawy Matematyki w Zadaniach (jointly with Janusz Onyszkiewicz)
• Logic and Foundations of Mathematics in problems (jointly with Janusz Onyszkiewicz)
• Analiza Kombinatoryczna (jointly with W. Lipski),
• Nonmonotonic Logic – Context-dependent Reasoning (jointly with M. Truszczyński),
• Introduction to Mathematics of Satisfiability.
References
1. W. Marek and M. Srebrny, Gaps in constructible universe, Annals of Mathematical Logic, 6:359–394, 1974.
2. K.R. Apt and W. Marek, Second order arithmetic and related topics, Annals of Mathematical Logic, 6:177–229, 1974
3. W. Marek, On the metamathematics of impredicative set theory. Dissertationes Mathematicae 98, 45 pages, 1973
4. Z. Pawlak, Mathematical foundations of information retrieval. Institute of Computer Sciences, Polish Academy of Sciences, Technical Report 101, 8 pages, 1973
5. Z. Pawlak, Rough sets. Institute of Computer Science, Polish Academy of Sciences, Technical Report 431, 12 pages, 1981
6. W. Marek and Z. Pawlak On the foundations of information retrieval. Bull. Acad. Pol. Sci. 22:447–452, 1974
7. W. Marek and Z. Pawlak. Rough sets and information systems, Institute of Computer Science, Technical Report 441, Polish Academy of Sciences, 15 pages, 1981
8. M.Denecker, V.W. Marek and M. Truszczynski, Uniform semantic treatment of default and autoepistemic logics. Artificial Intelligence. 143:79–122, 2003
9. V.W. Marek and M. Truszczynski, Stable logic programming – an alternative logic programming paradigm. In: 25 years of Logic Programming Paradigm, pages 375–398, Springer-Verlag, 1999
External links
• Personal page of Dr. V.W. Marek at the University of Kentucky
• Papers online
• Slides and other scientific materials
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Victor Pan
Victor Yakovlevich Pan (Russian: Пан Виктор Яковлевич) is a Soviet and American mathematician and computer scientist, known for his research on algorithms for polynomials and matrix multiplication.
Education and career
Pan earned his Ph.D. at Moscow University in 1964, under the supervision of Anatoli Georgievich Vitushkin,[1] and continued his work at the Soviet Academy of Sciences. During that time, he published a number of significant papers and became known informally as "polynomial Pan" for his pioneering work in the area of polynomial computations. In late 1970s, he immigrated to the United States and held positions at several institutions including IBM Research. Since 1988, he has taught at Lehman College of the City University of New York.[2]
Contributions
Victor Pan is an expert in computational complexity and has developed a number of new algorithms. One of his notable early results is a proof that the number of multiplications in Horner's method is optimal.[CVP]
In the theory of matrix multiplication algorithms, Pan in 1978 published an algorithm with running time $O(n^{2.795})$. This was the first improvement over the Strassen algorithm after nearly a decade, and kicked off a long line of improvements in fast matrix multiplication that later included the Coppersmith–Winograd algorithm and subsequent developments.[SNO] He wrote the text How to Multiply Matrices Faster (Springer, 1984) surveying early developments in this area.[3][HMM] His 1982 algorithm[P82] still held the record in 2020 for the fastest "practically useful" matrix multiplication algorithm (i.e., with a small base size and manageable hidden constants).[4] In 1998, with his student Xiaohan Huang, Pan showed that matrix multiplication algorithms can take advantage of rectangular matrices with unbalanced aspect ratios, multiplying them more quickly than the time bounds one would obtain using square matrix multiplication algorithms.[FRM]
Since that work, Pan has returned to symbolic and numeric computation and to an earlier theme of his research, computations with polynomials. He developed fast algorithms for the numerical computation of polynomial roots,[UP] and, with Bernard Mourrain, algorithms for multivariate polynomials based on their relations to structured matrices.[5][MPD] He also authored or co-authored several more books, on matrix and polynomial computation,[6][PMC] structured matrices,[7][SMP] and on numerical root-finding procedures.[8][NMR]
Recognition
Pan was appointed Distinguished Professor at Lehman College in 2000.[2]
In 2013 he became a fellow of the American Mathematical Society, for "contributions to the mathematical theory of computation".[9]
Selected publications
Research papers
CVP.
Pan, V. Ja. (1966), "On means of calculating values of polynomials", Russian Math. Surveys, 21: 105–136, doi:10.1070/rm1966v021n01abeh004147, MR 0207178, S2CID 250869179
SNO.
Pan, V. Ya. (October 1978), "Strassen's algorithm is not optimal: Trilinear technique of aggregating, uniting and canceling for constructing fast algorithms for matrix operations", Proceedings of the 19th Annual Symposium on Foundations of Computer Science (FOCS 1978), IEEE, doi:10.1109/sfcs.1978.34, S2CID 14348408
P82.
Pan, Victor Y. (1982), "Trilinear aggregating with implicit canceling for a new acceleration of matrix multiplication", Computers and Mathematics with Applications, 8: 23–34, doi:10.1016/0898-1221(82)90037-2, MR 0644547
FRM.
Huang, Xiaohan; Pan, Victor Y. (1998), "Fast rectangular matrix multiplication and applications", Journal of Complexity, 14 (2): 257–299, doi:10.1006/jcom.1998.0476, MR 1629113
MPD.
Mourrain, Bernard; Pan, Victor Y. (2000), "Multivariate polynomials, duality, and structured matrices" (PDF), Journal of Complexity, 16 (1): 110–180, doi:10.1006/jcom.1999.0530, MR 1762401 (winner, J. Complexity best paper award)[5]
UP.
Pan, Victor Y. (2002), "Univariate polynomials: nearly optimal algorithms for numerical factorization and root-finding", Journal of Symbolic Computation, 33 (5): 701–733, doi:10.1006/jsco.2002.0531, MR 1919911
Books
HMM.
Pan, Victor (1984), How to Multiply Matrices Faster, Lecture Notes in Computer Science, vol. 179, Berlin: Springer-Verlag, doi:10.1007/3-540-13866-8, ISBN 3-540-13866-8, S2CID 5280107[3]
PMC.
Bini, Dario; Pan, Victor Y. (1994), Polynomial and Matrix Computations, Vol. I: Fundamental Algorithms, Progress in Theoretical Computer Science, Boston, MA: Birkhäuser, doi:10.1007/978-1-4612-0265-3, ISBN 0-8176-3786-9, S2CID 30728536[6]
SMP.
Pan, Victor Y. (2001), Structured Matrices and Polynomials: Unified Superfast Algorithms, New York: Springer-Verlag, doi:10.1007/978-1-4612-0129-8, ISBN 0-8176-4240-4[7]
NMR.
McNamee, J. M.; Pan, V. Y. (2013), Numerical Methods for Roots of Polynomials, Part II, Studies in Computational Mathematics, vol. 16, Amsterdam: Elsevier/Academic Press, ISBN 978-0-444-52730-1[8]
References
1. Victor Pan at the Mathematics Genealogy Project
2. Victor Pan of Lehman mathematics faculty selected as Distinguished Professor, Lehman College, archived from the original on 2018-02-14
3. Reviews of How to Multiply Matrices Faster:
• Gladwell, Ian (1986), Mathematical Reviews, Lecture Notes in Computer Science, 179, doi:10.1007/3-540-13866-8, ISBN 978-3-540-13866-2, MR 0765701, S2CID 5280107{{citation}}: CS1 maint: untitled periodical (link)
• Coppersmith, Don (July 1986), SIAM Review, 28 (2): 250–252, doi:10.1137/1028072, JSTOR 2030488{{citation}}: CS1 maint: untitled periodical (link)
• Probert, Robert L. (November–December 1986), American Scientist, 74 (6): 682, JSTOR 27854420{{citation}}: CS1 maint: untitled periodical (link)
4. Karstadt, Elaye; Schwartz, Oded (2020), "Matrix multiplication, a little faster", Journal of the ACM, 67 (1): 1–31, doi:10.1145/3364504, MR 4061328, S2CID 211041916
5. "Best paper awards", Journal of Complexity, retrieved 2018-10-16
6. Reviews of Polynomial and Matrix Computations:
• Gupta, Murli M. (1995), Mathematical Reviews, doi:10.1007/978-1-4612-0265-3, ISBN 978-1-4612-6686-0, MR 1289412, S2CID 30728536{{citation}}: CS1 maint: untitled periodical (link)
• Tate, Stephen R. (June 1995), ACM SIGACT News, 26 (2): 26–27, doi:10.1145/202840.606473, S2CID 4740448{{citation}}: CS1 maint: untitled periodical (link)
• Eberly, Wayne (March 1996), SIAM Review, 38 (1): 161–165, doi:10.1137/1038020, JSTOR 2132983{{citation}}: CS1 maint: untitled periodical (link)
• Higham, Nicholas J. (April 1996), Mathematics of Computation, 65 (214): 888–889, JSTOR 2153629{{citation}}: CS1 maint: untitled periodical (link)
• Emiris, I. Z.; Galligo, A. (September 1996), ACM SIGSAM Bulletin, 30 (3): 21–23, doi:10.1145/240065.570109, S2CID 14598227{{citation}}: CS1 maint: untitled periodical (link)
7. Review of Structured Matrices and Polynomials:
• Melman, Aaron (2002), Mathematical Reviews, doi:10.1007/978-1-4612-0129-8, ISBN 978-1-4612-6625-9, MR 1843842{{citation}}: CS1 maint: untitled periodical (link)
8. Review of Numerical Methods for Roots of Polynomials, Part II:
• Proinov, Petko D., Mathematical Reviews, MR 3293902{{citation}}: CS1 maint: untitled periodical (link)
9. "List of Fellows of the American Mathematical Society", American Mathematical Society, retrieved 22 May 2015
External links
• Victor Pan publications indexed by Google Scholar
• Profile in American Scientist
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Victoria Howle
Victoria E. Howle is an American applied mathematician specializing in numerical linear algebra and known as one of the developers of the Trilinos open-source software library for scientific computing. She is an associate professor in the Department of Mathematics and Statistics at Texas Tech University.
Education and career
Howle graduated from Rutgers University in 1988 with a bachelor's degree in English literature.[1] She earned her Ph.D. in 2001 from Cornell University. Her dissertation, Efficient Iterative Methods for Ill-Conditioned Linear and Nonlinear Network Problems, was supervised by Stephen Vavasis.[2]
After working as a researcher at Sandia National Laboratories from 2000 to 2007, she took a faculty position at Texas Tech in 2007.[1]
Service and recognition
Howle was one of the inaugural winners of the AWM Service Award of the Association for Women in Mathematics, in 2013.[3][4] The award honored her service to the association, including founding its annual essay contest in which students write biographies of women mathematicians.[1][4]
References
1. Curriculum vitae (PDF), September 13, 2019, retrieved 2020-05-13
2. Victoria Howle at the Mathematics Genealogy Project
3. "AWM Service Award" (PDF), AWM Awards Given in San Diego, Notices of the American Mathematical Society, 60 (5): 616–617, May 2013
4. Association for Women in Mathematics Service Award 2013, Association for Women in Mathematics, retrieved 2020-05-13
External links
• Home page
• Victoria Howle publications indexed by Google Scholar
Authority control: Academics
• MathSciNet
• Mathematics Genealogy Project
| Wikipedia |
Vicumpriya Perera
Vicumpriya Perera (Sinhala: විකුම්ප්රිය පෙරේරා) is a Sri Lankan born mathematician, lyricist, poet and music producer.[1][2] He has published three books of Sinhala poetry, Mekunu Satahan (Sinhala: මැකුනු සටහන්) in 2001,[3] Paa Satahan (Sinhala: පා සටහන්) in 2013, [4][5] and Mawbime Suwandha (Sinhala: මව්බිමේ සුවඳ) in 2023.[6] He has written over 200 songs and has produced eleven Sinhala song albums. He currently works as a mathematics professor in Ohio, US.
Vicumpriya Perera
Born19 February ????
Sri Lanka
NationalitySri Lankan
EducationAnanda College, St. Anthony's College, Wattala
Occupation(s)lyricist, poet, mathematician
Life and career
Vicumpriya Perera is originally from Wattala, Sri Lanka. He is a graduate of St. Anthony's College, Wattala and Ananda College, Maradana, Sri Lanka. He received a Bachelor of Science degree in Mathematics with first class honors from University of Colombo, Sri Lanka and continued his graduate studies at Indiana University - Purdue University at Indianapolis. He obtained a doctorate degree from Purdue University in Pure Mathematics with research concentrating on operator algebras and functional analysis in 1993.[7]
Vicumpriya Perera lives in Ohio, US, where he has worked as a mathematics professor at Kent State University (Trumbull campus) since 1998.[8][9] He works in operator algebra, which is an area of pure mathematics.
List of albums
The following is a list of the songs albums that Vicumpriya Perera has produced. Vicumpriya Perera was the sole composer of the lyrics of all of them.
TitleSinhalaYearNo. of songsSinger(s)Music directorsCitation(s)
Paa Satahanපා සටහන්200820Bhadraji Mahinda Jayatilaka, Praneeth Mash, Sumith Vanniarachchi, and Indeevari Abeywardena.Bhadraji Mahinda Jayatilaka, Shantha Gunaratne, and Lassana Jayasekara.[10]
Weli Aetayakවැලි ඇටයක්200918Nalin Jayawardena, with duet singers Santhuri Waidyasekera, Sangeeth Wickramasinghe, Ananda Waidyasekera, Nijamali Jayawardena, and Sanduni Rashmika.Sangeeth Wickramasinghe, Ananda Waidyasekera, Rohan Jayawardena, Rukshan Karunanayake, Jayanga Dedigama and Sanuka Wickramasinghe.[11]
Ukusu Esඋකුසු ඇස්201018Bhadraji Mahinda Jayatilaka, Nalin Jayawardena, Chimes of the Seventies, Vidarshana Kodagoda, Dammika Tissarachchi, Isuru Roshan, Anura Dias, Channa and Ravindra Kasturisinghe, Mahesh Fernando, Indeevari Abeywardena, and Praneeth Mash.Rohan Jayawardena, Sangeeth Wickramasinghe, Ananda Waidyasekera, Rukshan Karunanayake, and Shantha Gunaratne.[12]
Niwaadu Kaleනිවාඩු කාලේ201116Sanduni Rashmika, with duet singers Nalin Jayawardena, Santhuri Waidyasekera, and Jayanga Dedigama.Title song composed by Bhadraji Mahinda Jayatilaka and directed by Sangeeth Wickramasinghe. Other songs directed by Ananda Waidyasekera.[13]
Viduli Eliyakවිදුලි එලියක්201216Nilupuli Dilhara, with duet singers Keerthi Pasquel, Nimal Gunasekera, Nalin Jayawardena, and Sanduni Rashmika.Ananda Waidyasekera. Bhadraji Mahinda Jayatilaka composed music for one song in the album as well.[14]
Mal Renuwakමල් රේණුවක්201316Nalin Jayawardena, with duet singers Amilaa Nadeeshani (second runner-up of Sirasa Superstar season 2 (2007)), Rupa Indumathi, Bhadraji Mahinda Jayatilaka, Walter Fernando, and Thilini Athukorala.Rohana Weerasinghe, Navaratne Gamage, Sarath De Alwis, H M Jayawardena, Nimal Mendis, Ernest Soysa, Bhadraji Mahinda Jayatilaka, Rohan Jayawardena, Mervin Priyantha, Ananda Waidyasekera, Sangeeth Wickramasinghe and Rukshan Karunanayake.[15][16][17]
Siththaruwananiසිත්තරුවාණනි201416Nalin Jayawardena, with duet singer Nimanthi Chamodini (Sri Lankan reality musical show star)Sangeeth Wickramasinghe[18][19][20]
Indikalaa Pem Medurakඉඳිකලා පෙම් මැදුරක්201516Devananda Waidyasekera, Chandrakumar Kandanarachchi, Thyaga N Edward, Walter Fernando, Ajith Ariyarathna, Nalin Jayawardena, Athula Sri Gamage, Srilal Fonseka, and Praneeth Shiwanka PereraAnanda Waidyasekera[21]
Ithiri Giyaada Aadareඉතිරී ගියාද ආදරේ201616Nalin Jayawardena, Rohan Jayawardena, Dhammika Tissaarachchi, Minali Gamage, Amanda Perera, Sangeeth Wickramasinghe and Renuka WickramasingheRohan Jayawardena, Nimal Mendis, Bhadraji Mahinda Jayatilaka, Ananda Waidyasekera, Sangeeth Wickramasinghe, Rukshan Karunanayake, and Jayanga Dedigama[22]
Mawbime Suwandhaමව්බිමේ සුවඳ201814Sangeeth Nipun Professor Sanath Nandasiri, Visharada Edward Jayakody, Anuradha Nandasiri, Dayan Witharana, Swarnalatha Kaveeshwara, Visharada Charitha Priyadarshani Peiris, Visharada Sarath Peiris, Nirasha Ratnayake, Nadeesha Dayaratne, Visharada Harshana Disanayake, and Upendra PiyasenaVisharada Sarath Peiris[23]
Minpasu Aayeමින්පසු ආයේ202015Nalin Jayawardena, Shashika Srimali and Dhammika EdussooriyaDhammika Edussooriya[24]
Siththaruwanani included songs from the sinhala classical musical genre (sarala gee). Instrumentalists for this album consisted of Sri Lankan musicians Mahendra Pasquel, Sarath Fernando, Dhananjaya Somasiri, Janaka Bogoda, Susil Amarasinghe, Rohana Dharmakeerthi, Shelton Wijesekera, and Dilusha Ravindranath.[20]
Other productions
In 2005, Vicumpriya Perera (along with Nalin Jayawardena, and Jaanaka Wimaladharma) produced a compact disc set, Dhammapadaya (Sinhala: ධම්මපදය), under the Lanka Heritage label.[25] The set contained four discs, and consisted of complete the Dhammapada stanzas in the original Pali language followed by the Sinhala translations chanted by venerable Beruwala Siri Sobhitha Thero of the Sri Lanka Buddhist Vihara in Perth, Australia.[lower-alpha 1] In 2006, this disc set had an English release called Dhammapada.[25] This version had the original Dhammapada stanzas (again in Pali) followed by the English translations written and rendered by Dr. Gil Fronsdal,[27] director and resident teacher Insight Meditation Center, Redwood City, California, US.
In 2012 Vicumpriya Perera (in collaboration with Nalin Jayawardena) produced a Sinhala Audiobook called Kulageyin Kulageyata (Sinhala: කුලගෙයින් කුලගෙයට) under the Lanka Heritage, LLC. The book was written in 2009 by Bhadraji Mahinda Jayatilaka, who provided most of the voice work . The audiobook has a total length of five compact discs, and was published by Sarasavi Publishers,[28] Nugegoda, Sri Lanka.
Notes
1. Dhammapada is a widely read Buddhist scripture containing 423 pali verses spanning into 26 chapters called varga.[26]
References
1. "Digital legacy of Sinhala songs". Sunday Observer. Retrieved 5 December 2017.
2. "Vicumpriya Perera – mathematician, translator and electronic recorder". Sunday Island. Retrieved 2 December 2013.
3. "Mekunu Satahan ebook". Retrieved 9 December 2013.
4. Perera, Vicumpriya (2013). Paa Satahan. Nugegoda, Sri Lanka: Sarasavi Publishers. ISBN 9789556717921.
5. "A homeland in cyberspace". Sunday Observer. Retrieved 5 December 2017.
6. "Mawbime Suwandha". Retrieved 9 April 2023.
7. Perera, Vicumpriya (1993). Real Valued Spectral Flow in a Type II-[infinity] Factor. Purdue University.
8. "Vic Perera". Retrieved 5 December 2017.
9. "Vicumpriya Perera". Retrieved 6 December 2017.
10. Paa Satahan – Bhadraji Mahinda Jayatilaka
11. "Weli Aetayak': Nalin and Vicumpriya's joint venture". Sunday Times. Retrieved 2 December 2013.
12. Ukusu Es
13. Niwadu Kaale – Sanduni Rashmika
14. Viduli Eliyak – Nilupuli Dilhara
15. Mal Renuwak – Nalin Jayawardena
16. Thilakarathne, Indeewara. "Depicting life through songs". Ceylon Today. Retrieved 2 December 2013.
17. Madugalle, Dushyantha. "Promoting Lankan culture in and out of diaspora". Sunday Observer. Retrieved 20 December 2013.
18. Siththaruwanani – Nalin Jayawardena
19. Withanachchi, Thinani. "Nalin in Sri Lanka to release 11th CD". Sarasaviya. Retrieved 12 September 2014.
20. "Voice from Australia, Lyrics from America and Music from Sri Lanka". Sarasaviya. Retrieved 12 September 2014.
21. Vicumpriya Perera Lyrics, Vol. 8: Indikalaa Pem Medurak
22. Vicumpriya Perera Lyrics 9: Ithiri Giyaada Aadare
23. Lyrical Compositions of Dr. Vicumpriya Perera, Vol.10: Fragrances Of The Motherland
24. Minpasu Aaye - Lyrical Compositions of Vicumpriya Perera 11
25. "Dhammapada at Lanka Heritage website". Retrieved 2 December 2013.
26. Müller, F. Max (1881). The Dhammapada (Sacred Books of the East, Vol. X). Oxford University Press.
27. Gil Fronsdal, and Jack Kornfield (foreword) (2005). The Dhammapada: A New Translation of the Buddhist Classic with Annotations, Boston: Shambhala. ISBN 1-59030-211-7.
28. "Sarasavi Prakashakayo". Retrieved 2 December 2013.
External links
• Lyrics of Vicumpriya Perera
• Vicumpriya Perera music on Google Play
• Home Page of Insight Meditation Center of Redwood City, CA
Authority control: Academics
• MathSciNet
• Mathematics Genealogy Project
| Wikipedia |
Vidyadhar P. Godambe
Vidyadhar Prabhakar Godambe FRSC (1 June 1926 – 9 June 2016) was an Indian statistician. He was a Distinguished Professor Emeritus at the University of Waterloo. Godambe was known for formulating and developing a theory of estimating equations.
Vidyadhar P. Godambe
Born(1926-06-01)1 June 1926
Pune, India
Died9 June 2016(2016-06-09) (aged 90)
Known forsurvey sampling
estimating equations
Academic background
EducationB.Sc., Fergusson College
M.Sc., Statistics, 1950, Bombay University
PhD., 1958, University of London
ThesisRobust and non-parametric inference and other general criteria for statistical decisions (1958)
Doctoral advisorGeorge Alfred Barnard
Academic work
InstitutionsDominion Bureau of Statistics
Rashtrasant Tukadoji Maharaj Nagpur University
Bombay University
Johns Hopkins University
University of Michigan
University of Waterloo
Early life and education
Godambe was born on 1 June 1926, in Pune, India as the second oldest of four children.[1] He was frail and sickly growing up so he attended the local school from age five to 10.[2] Godambe later attended Nutan Marathi Vidyalaya in Pune and Fergusson College for his Bachelor of Science in mathematics.[1]
After earning his Master's degree, Godambe accepted a position in the Bureau of Economics and Statistics with the Government of Bombay. While there, he submitted papers for publication in the Journal of the Royal Statistical Society and Bulletin of the Bureau of Economics and Statistics, Bombay.[2] Godambe shortly thereafter left Bombay to pursue a PhD at the University of London, and accepted a fellowship at the University of California, Berkeley. Upon his return, and completion of his thesis, Godambe was appointed a Senior Research Fellow at the Indian Statistical Institute and Professor and Head of the Statistics Department in Nagpur.[1]
Career
Godambe eventually left Nagpur and accepted a position at Bombay University as a professor for one year. He then moved to North America and worked at the Dominion Bureau of Statistics, alongside Ivan Fellegi, then taught at Johns Hopkins University, University of Michigan, and finally the University of Waterloo.[3] Godambe began his career at the University of Waterloo in July 1967 as a visiting professor in Statistics and Actuarial Sciences but was granted tenure as Professor in July 1969.[4] A few years later, in 1971, Godambe and Mary Thompson read a paper to the Royal Statistical Society entitled ‘Bayes, fiducial, and frequency aspects of statistical inference in survey sampling.[3]
From there, "Godambe’s paradox" was invented. Based on the paper published in the Journal of the Royal Statistical Society, he demonstrated that the likelihood principle implies that inference should be independent of the sampling design in general, which led to the development of model theory in survey sampling.[3] His method of estimating equations argued that all statistical inferences should adhere to his "ancillarity principle."[5]
Awards and honours
In 1987, Godambe was honoured with the Statistical Society of Canada (SSC) Gold Medal[6] and was later named an honorary member.[7] In 1991, he was appointed a Distinguished Professor Emeritus at the University of Waterloo.[8] In 2002, Godambe was elected a Fellow of the Royal Society of Canada.[9][10]
References
1. "OBITUARY:V.P. Godambe, 1926–2016". bulletin.imstat.org. Retrieved 2 December 2019.
2. Thompson, Mary E. (November 2002). "A Conversation with V. P. Godambe". Statistical Science. 17 (4): 458–466. doi:10.1214/ss/1049993204. JSTOR 3182767.
3. "A Conversation with V.P. Godambe". ssc.ca. 12 March 2010. Retrieved 2 December 2019.
4. "Remembering Vidyadhar Godambe; other notes". uwaterloo.ca. 5 July 2016. Retrieved 2 December 2019.
5. Christian Genest; Mark J. Schervish (December 1985). "Resolution of Godambe's Paradox". The Canadian Journal of Statistics. 13 (4): 293–297. doi:10.2307/3314949. JSTOR 3314949.
6. "Vidyadhar Prabhakar Godambe, SSC Gold Medalist 1987". ssc.ca. Retrieved 2 December 2019.
7. "Vidyadhar Prabhakar Godambe, Honorary Member 2001". ssc.ca. Retrieved 2 December 2019.
8. "1960 - 1999". uwaterloo.ca. Retrieved 2 December 2019.
9. "Notices of the American Mathematical Society" (PDF). ams.org. December 2002. p. 1399. Retrieved 2 December 2019.
10. "Two named to Royal Society of Canada". bulletin.uwaterloo.ca. 28 June 2002. Retrieved 2 December 2019.
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| Wikipedia |
Viennot's geometric construction
In mathematics, Viennot's geometric construction (named after Xavier Gérard Viennot) gives a diagrammatic interpretation of the Robinson–Schensted correspondence in terms of shadow lines. It has a generalization to the Robinson–Schensted–Knuth correspondence, which is known as the matrix-ball construction.
The construction
Starting with a permutation $\sigma \in S_{n}$, written in two-line notation, say:
$\sigma ={\begin{pmatrix}1&2&\cdots &n\\\sigma _{1}&\sigma _{2}&\cdots &\sigma _{n}\end{pmatrix}},$
one can apply the Robinson–Schensted correspondence to this permutation, yielding two standard Young tableaux of the same shape, P and Q. P is obtained by performing a sequence of insertions, and Q is the recording tableau, indicating in which order the boxes were filled.
Viennot's construction starts by plotting the points $(i,\sigma _{i})$ in the plane, and imagining there is a light that shines from the origin, casting shadows straight up and to the right. This allows consideration of the points which are not shadowed by any other point; the boundary of their shadows then forms the first shadow line. Removing these points and repeating the procedure, one obtains all the shadow lines for this permutation. Viennot's insight is then that these shadow lines read off the first rows of P and Q (in fact, even more than that; these shadow lines form a "timeline", indicating which elements formed the first rows of P and Q after the successive insertions). One can then repeat the construction, using as new points the previous unlabelled corners, which allows to read off the other rows of P and Q.
Animation
For example consider the permutation
$\sigma ={\begin{pmatrix}1&2&3&4&5&6&7&8\\3&8&1&2&4&7&5&6\end{pmatrix}}.$
Then Viennot's construction goes as follows:
Applications
One can use Viennot's geometric construction to prove that if $\sigma $ corresponds to the pair of tableaux P,Q under the Robinson–Schensted correspondence, then $\sigma ^{-1}$ corresponds to the switched pair Q,P. Indeed, taking $\sigma $ to $\sigma ^{-1}$ reflects Viennot's construction in the $y=x$-axis, and this precisely switches the roles of P and Q.
See also
• Plactic monoid
• Jeu de taquin
References
• Bruce E. Sagan. The Symmetric Group. Springer, 2001.
| Wikipedia |
Viète's formula
In mathematics, Viète's formula is the following infinite product of nested radicals representing twice the reciprocal of the mathematical constant π:
${\frac {2}{\pi }}={\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots $
This article is about a formula for π. For formulas for symmetric functions of the roots, see Vieta's formulas.
It can also be represented as:
${\frac {2}{\pi }}=\prod _{n=1}^{\infty }\cos {\frac {\pi }{2^{n+1}}}$
The formula is named after François Viète, who published it in 1593.[1] As the first formula of European mathematics to represent an infinite process,[2] it can be given a rigorous meaning as a limit expression,[3] and marks the beginning of mathematical analysis. It has linear convergence, and can be used for calculations of π,[4] but other methods before and since have led to greater accuracy. It has also been used in calculations of the behavior of systems of springs and masses,[5] and as a motivating example for the concept of statistical independence.
The formula can be derived as a telescoping product of either the areas or perimeters of nested polygons converging to a circle. Alternatively, repeated use of the half-angle formula from trigonometry leads to a generalized formula, discovered by Leonhard Euler, that has Viète's formula as a special case. Many similar formulas involving nested roots or infinite products are now known.
Significance
François Viète (1540–1603) was a French lawyer, privy councillor to two French kings, and amateur mathematician. He published this formula in 1593 in his work Variorum de rebus mathematicis responsorum, liber VIII. At this time, methods for approximating π to (in principle) arbitrary accuracy had long been known. Viète's own method can be interpreted as a variation of an idea of Archimedes of approximating the circumference of a circle by the perimeter of a many-sided polygon,[1] used by Archimedes to find the approximation[6]
${\frac {223}{71}}<\pi <{\frac {22}{7}}.$
By publishing his method as a mathematical formula, Viète formulated the first instance of an infinite product known in mathematics,[7][8] and the first example of an explicit formula for the exact value of π.[9][10] As the first representation in European mathematics of a number as the result of an infinite process rather than of a finite calculation,[11] Eli Maor highlights Viète's formula as marking the beginning of mathematical analysis[2] and Jonathan Borwein calls its appearance "the dawn of modern mathematics".[12]
Using his formula, Viète calculated π to an accuracy of nine decimal digits.[4] However, this was not the most accurate approximation to π known at the time, as the Persian mathematician Jamshīd al-Kāshī had calculated π to an accuracy of nine sexagesimal digits and 16 decimal digits in 1424.[12] Not long after Viète published his formula, Ludolph van Ceulen used a method closely related to Viète's to calculate 35 digits of π, which were published only after van Ceulen's death in 1610.[12]
Beyond its mathematical and historical significance, Viète's formula can be used to explain the different speeds of waves of different frequencies in an infinite chain of springs and masses, and the appearance of π in the limiting behavior of these speeds.[5] Additionally, a derivation of this formula as a product of integrals involving the Rademacher system, equal to the integral of products of the same functions, provides a motivating example for the concept of statistical independence.[13]
Interpretation and convergence
Viète's formula may be rewritten and understood as a limit expression[3]
$\lim _{n\rightarrow \infty }\prod _{i=1}^{n}{\frac {a_{i}}{2}}={\frac {2}{\pi }}$
where
${\begin{aligned}a_{1}&={\sqrt {2}}\\a_{n}&={\sqrt {2+a_{n-1}}}.\end{aligned}}$
For each choice of $n$, the expression in the limit is a finite product, and as $n$ gets arbitrarily large these finite products have values that approach the value of Viète's formula arbitrarily closely. Viète did his work long before the concepts of limits and rigorous proofs of convergence were developed in mathematics; the first proof that this limit exists was not given until the work of Ferdinand Rudio in 1891.[1][14]
The rate of convergence of a limit governs the number of terms of the expression needed to achieve a given number of digits of accuracy. In Viète's formula, the numbers of terms and digits are proportional to each other: the product of the first n terms in the limit gives an expression for π that is accurate to approximately 0.6n digits.[4][15] This convergence rate compares very favorably with the Wallis product, a later infinite product formula for π. Although Viète himself used his formula to calculate π only with nine-digit accuracy, an accelerated version of his formula has been used to calculate π to hundreds of thousands of digits.[4]
Related formulas
Viète's formula may be obtained as a special case of a formula for the sinc function that has often been attributed to Leonhard Euler[16], more than a century later:[1]
${\frac {\sin x}{x}}=\cos {\frac {x}{2}}\cdot \cos {\frac {x}{4}}\cdot \cos {\frac {x}{8}}\cdots $
Substituting x = π/2 in this formula yields:[17]
${\frac {2}{\pi }}=\cos {\frac {\pi }{4}}\cdot \cos {\frac {\pi }{8}}\cdot \cos {\frac {\pi }{16}}\cdots $
Then, expressing each term of the product on the right as a function of earlier terms using the half-angle formula:
$\cos {\frac {x}{2}}={\sqrt {\frac {1+\cos x}{2}}}$
gives Viète's formula.[9]
It is also possible to derive from Viète's formula a related formula for π that still involves nested square roots of two, but uses only one multiplication:[18]
$\pi =\lim _{k\to \infty }2^{k}\underbrace {\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+\cdots +{\sqrt {2}}}}}}}}}}}} _{k{\text{ square roots}}},$
which can be rewritten compactly as
${\begin{aligned}\pi &=\lim _{k\to \infty }2^{k}{\sqrt {2-a_{k}}}\\[5px]a_{1}&=0\\a_{k}&={\sqrt {2+a_{k-1}}}.\end{aligned}}$
Many formulae for π and other constants such as the golden ratio are now known, similar to Viète's in their use of either nested radicals or infinite products of trigonometric functions.[8][18][19][20][21][22][23][24]
Derivation
Viète obtained his formula by comparing the areas of regular polygons with 2n and 2n + 1 sides inscribed in a circle.[1][2] The first term in the product, √2/2, is the ratio of areas of a square and an octagon, the second term is the ratio of areas of an octagon and a hexadecagon, etc. Thus, the product telescopes to give the ratio of areas of a square (the initial polygon in the sequence) to a circle (the limiting case of a 2n-gon). Alternatively, the terms in the product may be instead interpreted as ratios of perimeters of the same sequence of polygons, starting with the ratio of perimeters of a digon (the diameter of the circle, counted twice) and a square, the ratio of perimeters of a square and an octagon, etc.[25]
Another derivation is possible based on trigonometric identities and Euler's formula. Repeatedly applying the double-angle formula
$\sin x=2\sin {\frac {x}{2}}\cos {\frac {x}{2}},$
leads to a proof by mathematical induction that, for all positive integers n,
$\sin x=2^{n}\sin {\frac {x}{2^{n}}}\left(\prod _{i=1}^{n}\cos {\frac {x}{2^{i}}}\right).$
The term 2n sin x/2n goes to x in the limit as n goes to infinity, from which Euler's formula follows. Viète's formula may be obtained from this formula by the substitution x = π/2.[9][13]
References
1. Beckmann, Petr (1971). A History of π (2nd ed.). Boulder, Colorado: The Golem Press. pp. 94–95. ISBN 978-0-88029-418-8. MR 0449960.
2. Maor, Eli (2011). Trigonometric Delights. Princeton, New Jersey: Princeton University Press. pp. 50, 140. ISBN 978-1-4008-4282-7.
3. Eymard, Pierre; Lafon, Jean Pierre (2004). "2.1 Viète's infinite product". The Number pi. Translated by Wilson, Stephen S. Providence, Rhode Island: American Mathematical Society. pp. 44–46. ISBN 978-0-8218-3246-2. MR 2036595.
4. Kreminski, Rick (2008). "π to thousands of digits from Vieta's formula". Mathematics Magazine. 81 (3): 201–207. doi:10.1080/0025570X.2008.11953549. JSTOR 27643107. S2CID 125362227.
5. Cullerne, J. P.; Goekjian, M. C. Dunn (December 2011). "Teaching wave propagation and the emergence of Viète's formula". Physics Education. 47 (1): 87–91. doi:10.1088/0031-9120/47/1/87. S2CID 122368450.
6. Beckmann 1971, p. 67.
7. De Smith, Michael J. (2006). Maths for the Mystified: An Exploration of the History of Mathematics and Its Relationship to Modern-day Science and Computing. Leicester: Matador. p. 165. ISBN 978-1905237-81-4.
8. Moreno, Samuel G.; García-Caballero, Esther M. (2013). "On Viète-like formulas". Journal of Approximation Theory. 174: 90–112. doi:10.1016/j.jat.2013.06.006. MR 3090772.
9. Morrison, Kent E. (1995). "Cosine products, Fourier transforms, and random sums". The American Mathematical Monthly. 102 (8): 716–724. arXiv:math/0411380. doi:10.2307/2974641. JSTOR 2974641. MR 1357488.
10. Oldham, Keith B.; Myland, Jan C.; Spanier, Jerome (2010). An Atlas of Functions: with Equator, the Atlas Function Calculator. New York: Springer. p. 15. doi:10.1007/978-0-387-48807-3. ISBN 978-0-387-48807-3.
11. Very similar infinite trigonometric series for $\pi $ appeared earlier in Indian mathematics, in the work of Madhava of Sangamagrama (c. 1340 – 1425), but were not known in Europe until much later. See: Plofker, Kim (2009). "7.3.1 Mādhava on the circumference and arcs of the circle". Mathematics in India. Princeton, New Jersey: Princeton University Press. pp. 221–234. ISBN 978-0-691-12067-6.
12. Borwein, Jonathan M. (2013). "The life of Pi: From Archimedes to ENIAC and beyond" (PDF). In Sidoli, Nathan; Van Brummelen, Glen (eds.). From Alexandria, Through Baghdad: Surveys and Studies in the Ancient Greek and Medieval Islamic Mathematical Sciences in Honor of J. L. Berggren. Berlin & Heidelberg: Springer. pp. 531–561. doi:10.1007/978-3-642-36736-6_24. ISBN 978-3-642-36735-9.
13. Kac, Mark (1959). "Chapter 1: From Vieta to the notion of statistical independence". Statistical Independence in Probability, Analysis and Number Theory. Carus Mathematical Monographs. Vol. 12. New York: John Wiley & Sons for the Mathematical Association of America. pp. 1–12. MR 0110114.
14. Rudio, F. (1891). "Ueber die Convergenz einer von Vieta herrührenden eigentümlichen Produktentwicklung" [On the convergence of a special product expansion due to Vieta]. Historisch-litterarische Abteilung der Zeitschrift für Mathematik und Physik (in German). 36: 139–140. JFM 23.0263.02.
15. Osler, Thomas J. (2007). "A simple geometric method of estimating the error in using Vieta's product for π". International Journal of Mathematical Education in Science and Technology. 38 (1): 136–142. doi:10.1080/00207390601002799. S2CID 120145020.
16. Euler, Leonhard (1738). "De variis modis circuli quadraturam numeris proxime exprimendi" [On various methods for expressing the quadrature of a circle with verging numbers]. Commentarii Academiae Scientiarum Petropolitanae (in Latin). 9: 222–236. Translated into English by Thomas W. Polaski. See final formula. The same formula is also in Euler, Leonhard (1783). "Variae observationes circa angulos in progressione geometrica progredientes" [Various observations about angles proceeding in geometric progression]. Opuscula Analytica (in Latin). 1: 345–352. Translated into English by Jordan Bell, arXiv:1009.1439. See the formula in numbered paragraph 3.
17. Wilson, Robin J. (2018). Euler's pioneering equation: the most beautiful theorem in mathematics (PDF) (First ed.). Oxford, United Kingdom. pp. 57–58. ISBN 9780198794929.{{cite book}}: CS1 maint: location missing publisher (link)
18. Servi, L. D. (2003). "Nested square roots of 2". The American Mathematical Monthly. 110 (4): 326–330. doi:10.2307/3647881. JSTOR 3647881. MR 1984573.
19. Nyblom, M. A. (2012). "Some closed-form evaluations of infinite products involving nested radicals". Rocky Mountain Journal of Mathematics. 42 (2): 751–758. doi:10.1216/RMJ-2012-42-2-751. MR 2915517.
20. Levin, Aaron (2006). "A geometric interpretation of an infinite product for the lemniscate constant". The American Mathematical Monthly. 113 (6): 510–520. doi:10.2307/27641976. JSTOR 27641976. MR 2231136.
21. Levin, Aaron (2005). "A new class of infinite products generalizing Viète's product formula for π". The Ramanujan Journal. 10 (3): 305–324. doi:10.1007/s11139-005-4852-z. MR 2193382. S2CID 123023282.
22. Osler, Thomas J. (2007). "Vieta-like products of nested radicals with Fibonacci and Lucas numbers". Fibonacci Quarterly. 45 (3): 202–204. MR 2437033.
23. Stolarsky, Kenneth B. (1980). "Mapping properties, growth, and uniqueness of Vieta (infinite cosine) products". Pacific Journal of Mathematics. 89 (1): 209–227. doi:10.2140/pjm.1980.89.209. MR 0596932.
24. Allen, Edward J. (1985). "Continued radicals". The Mathematical Gazette. 69 (450): 261–263. doi:10.2307/3617569. JSTOR 3617569. S2CID 250441699.
25. Rummler, Hansklaus (1993). "Squaring the circle with holes". The American Mathematical Monthly. 100 (9): 858–860. doi:10.2307/2324662. JSTOR 2324662. MR 1247533.
External links
• Viète's Variorum de rebus mathematicis responsorum, liber VIII (1593) on Google Books. The formula is on the second half of p. 30.
| Wikipedia |
Vieta's formulas
In mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. They are named after François Viète (more commonly referred to by the Latinised form of his name, "Franciscus Vieta").
For a method for computing π, see Viète's formula.
Basic formulas
Any general polynomial of degree n
$P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}$
(with the coefficients being real or complex numbers and an ≠ 0) has n (not necessarily distinct) complex roots r1, r2, ..., rn by the fundamental theorem of algebra. Vieta's formulas relate the polynomial's coefficients to signed sums of products of the roots r1, r2, ..., rn as follows:
${\begin{cases}r_{1}+r_{2}+\dots +r_{n-1}+r_{n}=-{\dfrac {a_{n-1}}{a_{n}}}\\[1ex](r_{1}r_{2}+r_{1}r_{3}+\cdots +r_{1}r_{n})+(r_{2}r_{3}+r_{2}r_{4}+\cdots +r_{2}r_{n})+\cdots +r_{n-1}r_{n}={\dfrac {a_{n-2}}{a_{n}}}\\[1ex]{}\quad \vdots \\[1ex]r_{1}r_{2}\cdots r_{n}=(-1)^{n}{\dfrac {a_{0}}{a_{n}}}.\end{cases}}$
(*)
Vieta's formulas can equivalently be written as
$\sum _{1\leq i_{1}<i_{2}<\cdots <i_{k}\leq n}\left(\prod _{j=1}^{k}r_{i_{j}}\right)=(-1)^{k}{\frac {a_{n-k}}{a_{n}}}$
for k = 1, 2, ..., n (the indices ik are sorted in increasing order to ensure each product of k roots is used exactly once).
The left-hand sides of Vieta's formulas are the elementary symmetric polynomials of the roots.
Vieta's system (*) can be solved by Newton's method through an explicit simple iterative formula, the Durand-Kerner method.
Generalization to rings
Vieta's formulas are frequently used with polynomials with coefficients in any integral domain R. Then, the quotients $a_{i}/a_{n}$ belong to the field of fractions of R (and possibly are in R itself if $a_{n}$ happens to be invertible in R) and the roots $r_{i}$ are taken in an algebraically closed extension. Typically, R is the ring of the integers, the field of fractions is the field of the rational numbers and the algebraically closed field is the field of the complex numbers.
Vieta's formulas are then useful because they provide relations between the roots without having to compute them.
For polynomials over a commutative ring that is not an integral domain, Vieta's formulas are only valid when $a_{n}$ is not a zero-divisor and $P(x)$ factors as $a_{n}(x-r_{1})(x-r_{2})\dots (x-r_{n})$. For example, in the ring of the integers modulo 8, the quadratic polynomial $P(x)=x^{2}-1$ has four roots: 1, 3, 5, and 7. Vieta's formulas are not true if, say, $r_{1}=1$ and $r_{2}=3$, because $P(x)\neq (x-1)(x-3)$. However, $P(x)$ does factor as $(x-1)(x-7)$ and also as $(x-3)(x-5)$, and Vieta's formulas hold if we set either $r_{1}=1$ and $r_{2}=7$ or $r_{1}=3$ and $r_{2}=5$.
Example
Vieta's formulas applied to quadratic and cubic polynomials:
The roots $r_{1},r_{2}$ of the quadratic polynomial $P(x)=ax^{2}+bx+c$ satisfy
$r_{1}+r_{2}=-{\frac {b}{a}},\quad r_{1}r_{2}={\frac {c}{a}}.$
The first of these equations can be used to find the minimum (or maximum) of P; see Quadratic equation § Vieta's formulas.
The roots $r_{1},r_{2},r_{3}$ of the cubic polynomial $P(x)=ax^{3}+bx^{2}+cx+d$ satisfy
$r_{1}+r_{2}+r_{3}=-{\frac {b}{a}},\quad r_{1}r_{2}+r_{1}r_{3}+r_{2}r_{3}={\frac {c}{a}},\quad r_{1}r_{2}r_{3}=-{\frac {d}{a}}.$
Proof
Vieta's formulas can be proved by expanding the equality
$a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}=a_{n}(x-r_{1})(x-r_{2})\cdots (x-r_{n})$
(which is true since $r_{1},r_{2},\dots ,r_{n}$ are all the roots of this polynomial), multiplying the factors on the right-hand side, and identifying the coefficients of each power of $x.$
Formally, if one expands $(x-r_{1})(x-r_{2})\cdots (x-r_{n}),$ the terms are precisely $(-1)^{n-k}r_{1}^{b_{1}}\cdots r_{n}^{b_{n}}x^{k},$ where $b_{i}$ is either 0 or 1, accordingly as whether $r_{i}$ is included in the product or not, and k is the number of $r_{i}$ that are included, so the total number of factors in the product is n (counting $x^{k}$ with multiplicity k) – as there are n binary choices (include $r_{i}$ or x), there are $2^{n}$ terms – geometrically, these can be understood as the vertices of a hypercube. Grouping these terms by degree yields the elementary symmetric polynomials in $r_{i}$ – for xk, all distinct k-fold products of $r_{i}.$
As an example, consider the quadratic
$f(x)=a_{2}x^{2}+a_{1}x+a_{0}=a_{2}(x-r_{1})(x-r_{2})=a_{2}(x^{2}-x(r_{1}+r_{2})+r_{1}r_{2}).$
Comparing identical powers of $x$, we find $a_{2}=a_{2}$, $a_{1}=-a_{2}(r_{1}+r_{2})$ and $a_{0}=a_{2}(r_{1}r_{2})$, with which we can for example identify $r_{1}+r_{2}=-a_{1}/a_{2}$ and $r_{1}r_{2}=a_{0}/a_{2}$, which are Vieta's formula's for $n=2$.
History
As reflected in the name, the formulas were discovered by the 16th-century French mathematician François Viète, for the case of positive roots.
In the opinion of the 18th-century British mathematician Charles Hutton, as quoted by Funkhouser,[1] the general principle (not restricted to positive real roots) was first understood by the 17th-century French mathematician Albert Girard:
...[Girard was] the first person who understood the general doctrine of the formation of the coefficients of the powers from the sum of the roots and their products. He was the first who discovered the rules for summing the powers of the roots of any equation.
See also
• Content (algebra)
• Descartes' rule of signs
• Newton's identities
• Gauss–Lucas theorem
• Properties of polynomial roots
• Rational root theorem
• Symmetric polynomial and elementary symmetric polynomial
References
1. (Funkhouser 1930)
• "Viète theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Funkhouser, H. Gray (1930), "A short account of the history of symmetric functions of roots of equations", American Mathematical Monthly, Mathematical Association of America, 37 (7): 357–365, doi:10.2307/2299273, JSTOR 2299273
• Vinberg, E. B. (2003), A course in algebra, American Mathematical Society, Providence, R.I, ISBN 0-8218-3413-4
• Djukić, Dušan; et al. (2006), The IMO compendium: a collection of problems suggested for the International Mathematical Olympiads, 1959–2004, Springer, New York, NY, ISBN 0-387-24299-6
| Wikipedia |
François Viète
François Viète, Seigneur de la Bigotière (Latin: Franciscus Vieta; 1540 – 23 February 1603), commonly known by his mononym, Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to his innovative use of letters as parameters in equations. He was a lawyer by trade, and served as a privy councillor to both Henry III and Henry IV of France.
François Viète
Born1540
Fontenay-le-Comte, Kingdom of France
Died23 February 1603 (aged 62–63)
Paris, Kingdom of France
NationalityFrench
Other namesFranciscus Vieta
EducationUniversity of Poitiers
(LL.B., 1559)
Known forNew algebra (the first symbolic algebra)
Vieta's formulas
Viète's formula
Scientific career
FieldsAstronomy, mathematics (algebra and trigonometry)
Notable studentsAlexander Anderson
InfluencesPeter Ramus
Gerolamo Cardano[1]
InfluencedPierre de Fermat
René Descartes[2]
Signature
Biography
Early life and education
Viète was born at Fontenay-le-Comte in present-day Vendée. His grandfather was a merchant from La Rochelle. His father, Etienne Viète, was an attorney in Fontenay-le-Comte and a notary in Le Busseau. His mother was the aunt of Barnabé Brisson, a magistrate and the first president of parliament during the ascendancy of the Catholic League of France.
Viète went to a Franciscan school and in 1558 studied law at Poitiers, graduating as a Bachelor of Laws in 1559. A year later, he began his career as an attorney in his native town.[3] From the outset, he was entrusted with some major cases, including the settlement of rent in Poitou for the widow of King Francis I of France and looking after the interests of Mary, Queen of Scots.
Serving Parthenay
In 1564, Viète entered the service of Antoinette d'Aubeterre, Lady Soubise, wife of Jean V de Parthenay-Soubise, one of the main Huguenot military leaders and accompanied him to Lyon to collect documents about his heroic defence of that city against the troops of Jacques of Savoy, 2nd Duke of Nemours just the year before.
The same year, at Parc-Soubise, in the commune of Mouchamps in present-day Vendée, Viète became the tutor of Catherine de Parthenay, Soubise's twelve-year-old daughter. He taught her science and mathematics and wrote for her numerous treatises on astronomy and trigonometry, some of which have survived. In these treatises, Viète used decimal numbers (twenty years before Stevin's paper) and he also noted the elliptic orbit of the planets,[4] forty years before Kepler and twenty years before Giordano Bruno's death.
John V de Parthenay presented him to King Charles IX of France. Viète wrote a genealogy of the Parthenay family and following the death of Jean V de Parthenay-Soubise in 1566 his biography.
In 1568, Antoinette, Lady Soubise, married her daughter Catherine to Baron Charles de Quellenec and Viète went with Lady Soubise to La Rochelle, where he mixed with the highest Calvinist aristocracy, leaders like Coligny and Condé and Queen Jeanne d’Albret of Navarre and her son, Henry of Navarre, the future Henry IV of France.
In 1570, he refused to represent the Soubise ladies in their infamous lawsuit against the Baron De Quellenec, where they claimed the Baron was unable (or unwilling) to provide an heir.
First steps in Paris
In 1571, he enrolled as an attorney in Paris, and continued to visit his student Catherine. He regularly lived in Fontenay-le-Comte, where he took on some municipal functions. He began publishing his Universalium inspectionum ad Canonem mathematicum liber singularis and wrote new mathematical research by night or during periods of leisure. He was known to dwell on any one question for up to three days, his elbow on the desk, feeding himself without changing position (according to his friend, Jacques de Thou).[5]
In 1572, Viète was in Paris during the St. Bartholomew's Day massacre. That night, Baron De Quellenec was killed after having tried to save Admiral Coligny the previous night. The same year, Viète met Françoise de Rohan, Lady of Garnache, and became her adviser against Jacques, Duke of Nemours.
In 1573, he became a councillor of the Parliament of Brittany, at Rennes, and two years later, he obtained the agreement of Antoinette d'Aubeterre for the marriage of Catherine of Parthenay to Duke René de Rohan, Françoise's brother.
In 1576, Henri, duc de Rohan took him under his special protection, recommending him in 1580 as "maître des requêtes". In 1579, Viète finished the printing of his Universalium inspectionum (Mettayer publisher), published as an appendix to a book of two trigonometric tables (Canon mathematicus, seu ad triangula, the "canon" referred to by the title of his Universalium inspectionum, and Canonion triangulorum laterum rationalium). A year later, he was appointed maître des requêtes to the parliament of Paris, committed to serving the king. That same year, his success in the trial between the Duke of Nemours and Françoise de Rohan, to the benefit of the latter, earned him the resentment of the tenacious Catholic League.
Exile in Fontenay
Between 1583 and 1585, the League persuaded Henry III to release Viète, Viète having been accused of sympathy with the Protestant cause. Henry of Navarre, at Rohan's instigation, addressed two letters to King Henry III of France on March 3 and April 26, 1585, in an attempt to obtain Viète's restoration to his former office, but he failed.[3]
Viète retired to Fontenay and Beauvoir-sur-Mer, with François de Rohan. He spent four years devoted to mathematics, writing his New Algebra (1591).
Code-breaker to two kings
In 1589, Henry III took refuge in Blois. He commanded the royal officials to be at Tours before 15 April 1589. Viète was one of the first who came back to Tours. He deciphered the secret letters of the Catholic League and other enemies of the king. Later, he had arguments with the classical scholar Joseph Juste Scaliger. Viète triumphed against him in 1590.
After the death of Henry III, Viète became a privy councillor to Henry of Navarre, now Henry IV.[6]: 75–77 He was appreciated by the king, who admired his mathematical talents. Viète was given the position of councillor of the parlement at Tours. In 1590, Viète discovered the key to a Spanish cipher, consisting of more than 500 characters, and this meant that all dispatches in that language which fell into the hands of the French could be easily read.[7]
Henry IV published a letter from Commander Moreo to the King of Spain. The contents of this letter, read by Viète, revealed that the head of the League in France, Charles, Duke of Mayenne, planned to become king in place of Henry IV. This publication led to the settlement of the Wars of Religion. The King of Spain accused Viète of having used magical powers. In 1593, Viète published his arguments against Scaliger. Beginning in 1594, he was appointed exclusively deciphering the enemy's secret codes.
Gregorian calendar
In 1582, Pope Gregory XIII published his bull Inter gravissimas and ordered Catholic kings to comply with the change from the Julian calendar, based on the calculations of the Calabrian doctor Aloysius Lilius, aka Luigi Lilio or Luigi Giglio. His work was resumed, after his death, by the scientific adviser to the Pope, Christopher Clavius.
Viète accused Clavius, in a series of pamphlets (1600), of introducing corrections and intermediate days in an arbitrary manner, and misunderstanding the meaning of the works of his predecessor, particularly in the calculation of the lunar cycle. Viète gave a new timetable, which Clavius cleverly refuted,[8] after Viète's death, in his Explicatio (1603).
It is said that Viète was wrong. Without doubt, he believed himself to be a kind of "King of Times" as the historian of mathematics, Dhombres, claimed.[9] It is true that Viète held Clavius in low esteem, as evidenced by De Thou:
He said that Clavius was very clever to explain the principles of mathematics, that he heard with great clarity what the authors had invented, and wrote various treatises compiling what had been written before him without quoting its references. So, his works were in a better order which was scattered and confused in early writings.
The Adriaan van Roomen problem
In 1596, Scaliger resumed his attacks from the University of Leyden. Viète replied definitively the following year. In March that same year, Adriaan van Roomen sought the resolution, by any of Europe's top mathematicians, to a polynomial equation of degree 45. King Henri IV received a snub from the Dutch ambassador, who claimed that there was no mathematician in France. He said it was simply because some Dutch mathematician, Adriaan van Roomen, had not asked any Frenchman to solve his problem.
Viète came, saw the problem, and, after leaning on a window for a few minutes, solved it. It was the equation between sin(x) and sin(x/45). He resolved this at once, and said he was able to give at the same time (actually the next day) the solution to the other 22 problems to the ambassador. "Ut legit, ut solvit," he later said. Further, he sent a new problem back to Van Roomen, for resolution by Euclidean tools (rule and compass) of the lost answer to the problem first set by Apollonius of Perga. Van Roomen could not overcome that problem without resorting to a trick (see detail below).
Final years
In 1598, Viète was granted special leave. Henry IV, however, charged him to end the revolt of the Notaries, whom the King had ordered to pay back their fees. Sick and exhausted by work, he left the King's service in December 1602 and received 20,000 écu, which were found at his bedside after his death.
A few weeks before his death, he wrote a final thesis on issues of cryptography, whose memory made obsolete all encryption methods of the time. He died on 23 February 1603, as De Thou wrote,[10] leaving two daughters, Jeanne, whose mother was Barbe Cottereau, and Suzanne, whose mother was Julienne Leclerc. Jeanne, the eldest, died in 1628, having married Jean Gabriau, a councillor of the parliament of Brittany. Suzanne died in January 1618 in Paris.
The cause of Viète's death is unknown. Alexander Anderson, student of Viète and publisher of his scientific writings, speaks of a "praeceps et immaturum autoris fatum." (meeting an untimely end).[7][11]
Work and thought
Background
At the end of the 16th century, mathematics was placed under the dual aegis of Greek geometry and the Arabic procedures for resolution. At the time of Viète, algebra therefore oscillated between arithmetic, which gave the appearance of a list of rules; and geometry, which seemed more rigorous. Meanwhile, Italian mathematicians Luca Pacioli, Scipione del Ferro, Niccolò Fontana Tartaglia, Gerolamo Cardano, Lodovico Ferrari, and especially Raphael Bombelli (1560) all developed techniques for solving equations of the third degree, which heralded a new era.
On the other hand, from the German school of Coss, the Welsh mathematician Robert Recorde (1550) and the Dutchman Simon Stevin (1581) brought an early algebraic notation: the use of decimals and exponents. However, complex numbers remained at best a philosophical way of thinking. Descartes, almost a century after their invention, used them as imaginary numbers. Only positive solutions were considered and using geometrical proof was common.
The mathematician's task was in fact twofold. It was necessary to produce algebra in a more geometrical way (i.e. to give it a rigorous foundation), and it was also necessary to make geometry more algebraic, allowing for analytical calculation in the plane. Viète and Descartes solved this dual task in a double revolution.
Viète's symbolic algebra
Firstly, Viète gave algebra a foundation as strong as that of geometry. He then ended the algebra of procedures (al-Jabr and al-Muqabala), creating the first symbolic algebra, and claiming that with it, all problems could be solved (nullum non problema solvere).[12][13]
In his dedication of the Isagoge to Catherine de Parthenay, Viète wrote:
"These things which are new are wont in the beginning to be set forth rudely and formlessly and must then be polished and perfected in succeeding centuries. Behold, the art which I present is new, but in truth so old, so spoiled and defiled by the barbarians, that I considered it necessary, in order to introduce an entirely new form into it, to think out and publish a new vocabulary, having gotten rid of all its pseudo-technical terms..."[14]
Viète did not know "multiplied" notation (given by William Oughtred in 1631) or the symbol of equality, =, an absence which is more striking because Robert Recorde had used the present symbol for this purpose since 1557, and Guilielmus Xylander had used parallel vertical lines since 1575.[7] Note also the use of a 'u' like symbol with a number above it for an unknown to a given power by Rafael Bombelli in 1572.[15]
Viète had neither much time, nor students able to brilliantly illustrate his method. He took years in publishing his work (he was very meticulous), and most importantly, he made a very specific choice to separate the unknown variables, using consonants for parameters and vowels for unknowns. In this notation he perhaps followed some older contemporaries, such as Petrus Ramus, who designated the points in geometrical figures by vowels, making use of consonants, R, S, T, etc., only when these were exhausted.[7] This choice proved unpopular with future mathematicians and Descartes, among others, preferred the first letters of the alphabet to designate the parameters and the latter for the unknowns.
Viète also remained a prisoner of his time in several respects. First, he was heir of Ramus and did not address the lengths as numbers. His writing kept track of homogeneity, which did not simplify their reading. He failed to recognize the complex numbers of Bombelli and needed to double-check his algebraic answers through geometrical construction. Although he was fully aware that his new algebra was sufficient to give a solution, this concession tainted his reputation.
However, Viète created many innovations: the binomial formula, which would be taken by Pascal and Newton, and the coefficients of a polynomial to sums and products of its roots, called Viète's formula.
Geometric algebra
Viète was well skilled in most modern artifices, aiming at the simplification of equations by the substitution of new quantities having a certain connection with the primitive unknown quantities. Another of his works, Recensio canonica effectionum geometricarum, bears a modern stamp, being what was later called an algebraic geometry—a collection of precepts how to construct algebraic expressions with the use of ruler and compass only. While these writings were generally intelligible, and therefore of the greatest didactic importance, the principle of homogeneity, first enunciated by Viète, was so far in advance of his times that most readers seem to have passed it over. That principle had been made use of by the Greek authors of the classic age; but of later mathematicians only Hero, Diophantus, etc., ventured to regard lines and surfaces as mere numbers that could be joined to give a new number, their sum.[7]
The study of such sums, found in the works of Diophantus, may have prompted Viète to lay down the principle that quantities occurring in an equation ought to be homogeneous, all of them lines, or surfaces, or solids, or supersolids — an equation between mere numbers being inadmissible. During the centuries that have elapsed between Viète's day and the present, several changes of opinion have taken place on this subject. Modern mathematicians like to make homogeneous such equations as are not so from the beginning, in order to get values of a symmetrical shape. Viète himself did not see that far; nevertheless, he indirectly suggested the thought. He also conceived methods for the general resolution of equations of the second, third and fourth degrees different from those of Scipione dal Ferro and Lodovico Ferrari, with which he had not been acquainted. He devised an approximate numerical solution of equations of the second and third degrees, wherein Leonardo of Pisa must have preceded him, but by a method which was completely lost.[7]
Above all, Viète was the first mathematician who introduced notations for the problem (and not just for the unknowns).[12] As a result, his algebra was no longer limited to the statement of rules, but relied on an efficient computational algebra, in which the operations act on the letters and the results can be obtained at the end of the calculations by a simple replacement. This approach, which is the heart of contemporary algebraic method, was a fundamental step in the development of mathematics.[16] With this, Viète marked the end of medieval algebra (from Al-Khwarizmi to Stevin) and opened the modern period.
The logic of species
Being wealthy, Viète began to publish at his own expense, for a few friends and scholars in almost every country of Europe, the systematic presentation of his mathematic theory, which he called "species logistic" (from species: symbol) or art of calculation on symbols (1591).[17]
He described in three stages how to proceed for solving a problem:
• As a first step, he summarized the problem in the form of an equation. Viète called this stage the Zetetic. It denotes the known quantities by consonants (B, D, etc.) and the unknown quantities by the vowels (A, E, etc.)
• In a second step, he made an analysis. He called this stage the Poristic. Here mathematicians must discuss the equation and solve it. It gives the characteristic of the problem, porisma (corrollary), from which we can move to the next step.
• In the last step, the exegetical analysis, he returned to the initial problem which presents a solution through a geometrical or numerical construction based on porisma.
Among the problems addressed by Viète with this method is the complete resolution of the quadratic equations of the form $X^{2}+Xb=c$ and third-degree equations of the form $X^{3}+aX=b$ (Viète reduced it to quadratic equations). He knew the connection between the positive roots of an equation (which, in his day, were alone thought of as roots) and the coefficients of the different powers of the unknown quantity (see Viète's formulas and their application on quadratic equations). He discovered the formula for deriving the sine of a multiple angle, knowing that of the simple angle with due regard to the periodicity of sines. This formula must have been known to Viète in 1593.[7]
Viète's formula
Main article: Viète's formula
In 1593, based on geometrical considerations and through trigonometric calculations perfectly mastered, he discovered the first infinite product in the history of mathematics by giving an expression of π, now known as Viète's formula:[18]
$\pi =2\times {\frac {2}{\sqrt {2}}}\times {\frac {2}{\sqrt {2+{\sqrt {2}}}}}\times {\frac {2}{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}\times {\frac {2}{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}}}\times \cdots $
He provides 10 decimal places of π by applying the Archimedes method to a polygon with 6 × 216 = 393,216 sides.
Adriaan van Roomen's problem
This famous controversy is told by Tallemant des Réaux in these terms (46th story from the first volume of Les Historiettes. Mémoires pour servir à l’histoire du XVIIe siècle):
"In the times of Henri the fourth, a Dutchman called Adrianus Romanus, a learned mathematician, but not so good as he believed, published a treatise in which he proposed a question to all the mathematicians of Europe, but did not ask any Frenchman. Shortly after, a state ambassador came to the King at Fontainebleau. The King took pleasure in showing him all the sights, and he said people there were excellent in every profession in his kingdom. 'But, Sire,' said the ambassador, 'you have no mathematician, according to Adrianus Romanus, who didn't mention any in his catalog.' 'Yes, we have,' said the King. 'I have an excellent man. Go and seek Monsieur Viette,' he ordered. Vieta, who was at Fontainebleau, came at once. The ambassador sent for the book from Adrianus Romanus and showed the proposal to Vieta, who had arrived in the gallery, and before the King came out, he had already written two solutions with a pencil. By the evening he had sent many other solutions to the ambassador."
This suggests that the Adrien van Roomen problem is an equation of 45°, which Viète recognized immediately as a chord of an arc of 8° (${\tfrac {1}{45}}$ turn). It was then easy to determine the following 22 positive alternatives, the only valid ones at the time.
When, in 1595, Viète published his response to the problem set by Adriaan van Roomen, he proposed finding the resolution of the old problem of Apollonius, namely to find a circle tangent to three given circles. Van Roomen proposed a solution using a hyperbola, with which Viète did not agree, as he was hoping for a solution using Euclidean tools.
Viète published his own solution in 1600 in his work Apollonius Gallus. In this paper, Viète made use of the center of similitude of two circles.[7] His friend De Thou said that Adriaan van Roomen immediately left the University of Würzburg, saddled his horse and went to Fontenay-le-Comte, where Viète lived. According to De Thou, he stayed a month with him, and learned the methods of the new algebra. The two men became friends and Viète paid all van Roomen's expenses before his return to Würzburg.
This resolution had an almost immediate impact in Europe and Viète earned the admiration of many mathematicians over the centuries. Viète did not deal with cases (circles together, these tangents, etc.), but recognized that the number of solutions depends on the relative position of the three circles and outlined the ten resulting situations. Descartes completed (in 1643) the theorem of the three circles of Apollonius, leading to a quadratic equation in 87 terms, each of which is a product of six factors (which, with this method, makes the actual construction humanly impossible).[19]
Religious and political beliefs
Viète was accused of Protestantism by the Catholic League, but he was not a Huguenot. His father was, according to Dhombres.[20] Indifferent in religious matters, he did not adopt the Calvinist faith of Parthenay, nor that of his other protectors, the Rohan family. His call to the parliament of Rennes proved the opposite. At the reception as a member of the court of Brittany, on 6 April 1574, he read in public a statement of Catholic faith.[20]
Nevertheless, Viète defended and protected Protestants his whole life, and suffered, in turn, the wrath of the League. It seems that for him, the stability of the state was to be preserved and that under this requirement, the King's religion did not matter. At that time, such people were called "Politicals."
Furthermore, at his death, he did not want to confess his sins. A friend had to convince him that his own daughter would not find a husband, were he to refuse the sacraments of the Catholic Church. Whether Viète was an atheist or not is a matter of debate.[20]
Publications
Chronological list
• Between 1564 and 1568, Viète prepared for his student, Catherine de Parthenay, some textbooks of astronomy and trigonometry and a treatise that was never published: Harmonicon coeleste.
• In 1579, the trigonometric tables Canon mathematicus, seu ad triangula, published together with a table of rational-sided triangles Canonion triangulorum laterum rationalium, and a book of trigonometry Universalium inspectionum ad canonem mathematicum – which he published at his own expense and with great printing difficulties. This text contains many formulas on the sine and cosine and is unusual in using decimal numbers. The trigonometric tables here exceeded those of Regiomontanus (Triangulate Omnimodis, 1533) and Rheticus (1543, annexed to De revolutionibus of Copernicus). (Alternative scan of a 1589 reprint)
• In 1589, Deschiffrement d'une lettre escripte par le Commandeur Moreo au Roy d'Espaigne son maître.
• In 1590, Deschiffrement description of a letter by the Commander Moreo at Roy Espaigne of his master, Tours: Mettayer.
• In 1591:
• In artem analyticem isagoge (Introduction to the art of analysis), also known as Algebra Nova (New Algebra) Tours: Mettayer, in 9 folio; the first edition of the Isagoge.
• Zeteticorum libri quinque. Tours: Mettayer, in 24 folio; which are the five books of Zetetics, a collection of problems from Diophantus solved using the analytical art.
• Between 1591 and 1593, Effectionum geometricarum canonica recensio. Tours: Mettayer, in 7 folio.
• In 1593:
• Vietae Supplementum geometriae. Tours: Francisci, in 21 folio.
• Francisci Vietae Variorum de rebus responsorum mathematics liber VIII. Tours: Mettaye, in 49 folio; about the challenges of Scaliger.
• Variorum de rebus mathematicis responsorum liber VIII; the "Eighth Book of Varied Responses" in which he talks about the problems of the trisection of the angle (which he acknowledges that it is bound to an equation of third degree) of squaring the circle, building the regular heptagon, etc.
• In 1594, Munimen adversus nova cyclometrica. Paris: Mettayer, in quarto, 8 folio; again, a response against Scaliger.
• In 1595, Ad problema quod omnibus mathematicis totius orbis construendum proposuit Adrianus Romanus, Francisci Vietae responsum. Paris: Mettayer, in quarto, 16 folio; about the Adriaan van Roomen problem.
• In 1600:
• De numerosa potestatum ad exegesim resolutione. Paris: Le Clerc, in 36 folio; work that provided the means for extracting roots and solutions of equations of degree at most 6.
• Francisci Vietae Apollonius Gallus. Paris: Le Clerc, in quarto, 13 folio; where he referred to himself as the French Apollonius.
• Between 1600 and 1602:
• Fontenaeensis libellorum supplicum in Regia magistri relatio Kalendarii vere Gregoriani ad ecclesiasticos doctores exhibita Pontifici Maximi Clementi VIII. Paris: Mettayer, in quarto, 40 folio.
• Francisci Vietae adversus Christophorum Clavium expostulatio. Paris: Mettayer, in quarto, 8 folio; his theses against Clavius.
Posthumous publications
• 1612:
• Supplementum Apollonii Galli edited by Marin Ghetaldi.
• Supplementum Apollonii Redivivi sive analysis problematis bactenus desiderati ad Apollonii Pergaei doctrinam a Marino Ghetaldo Patritio Regusino hujusque non ita pridem institutam edited by Alexander Anderson.
• 1615:
• Ad Angularum Sectionem Analytica Theoremata F. Vieta primum excogitata at absque ulla demonstratione ad nos transmissa, iam tandem demonstrationibus confirmata edited by Alexander Anderson.
• Pro Zetetico Apolloniani problematis a se jam pridem edito in supplemento Apollonii Redivivi Zetetico Apolloniani problematis a se jam pridem edito; in qua ad ea quae obiter inibi perstrinxit Ghetaldus respondetur edited by Alexander Anderson
• Francisci Vietae Fontenaeensis, De aequationum — recognitione et emendatione tractatus duo per Alexandrum Andersonum edited by Alexander Anderson
• 1617: Animadversionis in Franciscum Vietam, a Clemente Cyriaco nuper editae brevis diakrisis edited by Alexander Anderson
• 1619: Exercitationum Mathematicarum Decas Prima edited by Alexander Anderson
• 1631: In artem analyticem isagoge. Eiusdem ad logisticem speciosam notae priores, nunc primum in lucem editae. Paris: Baudry, in 12 folio; the second edition of the Isagoge, including the posthumously published Ad logisticem speciosam notae priores.
Reception and influence
During the ascendancy of the Catholic League, Viète's secretary was Nathaniel Tarporley, perhaps one of the more interesting and enigmatic mathematicians of 16th-century England. When he returned to London, Tarporley became one of the trusted friends of Thomas Harriot.
Apart from Catherine de Parthenay, Viète's other notable students were: French mathematician Jacques Aleaume, from Orleans, Marino Ghetaldi of Ragusa, Jean de Beaugrand and the Scottish mathematician Alexander Anderson. They illustrated his theories by publishing his works and continuing his methods. At his death, his heirs gave his manuscripts to Peter Aleaume.[21] We give here the most important posthumous editions:
• In 1612: Supplementum Apollonii Galli of Marino Ghetaldi.
• From 1615 to 1619: Animadversionis in Franciscum Vietam, Clemente a Cyriaco nuper by Alexander Anderson
• Francisci Vietae Fontenaeensis ab aequationum recognitione et emendatione Tractatus duo Alexandrum per Andersonum. Paris, Laquehay, 1615, in 4, 135 p. The death of Alexander Anderson unfortunately halted the publication.
• In 1630, an Introduction en l'art analytic ou nouvelle algèbre ('Introduction to the analytic art or modern algebra),[22] translated into French and commentary by mathematician J. L. Sieur de Vaulezard. Paris, Jacquin.
• The Five Books of François Viette's Zetetic (Les cinq livres des zététiques de François Viette), put into French, and commented increased by mathematician J. L. Sieur de Vaulezard. Paris, Jacquin, p. 219.
The same year, there appeared an Isagoge by Antoine Vasset (a pseudonym of Claude Hardy), and the following year, a translation into Latin of Beaugrand, which Descartes would have received.
In 1648, the corpus of mathematical works printed by Frans van Schooten, professor at Leiden University (Elzevirs presses). He was assisted by Jacques Golius and Mersenne.
The English mathematicians Thomas Harriot and Isaac Newton, and the Dutch physicist Willebrord Snellius, the French mathematicians Pierre de Fermat and Blaise Pascal all used Viète's symbolism.
About 1770, the Italian mathematician Targioni Tozzetti, found in Florence Viète's Harmonicon coeleste. Viète had written in it: Describat Planeta Ellipsim ad motum anomaliae ad Terram. (That shows he adopted Copernicus' system and understood before Kepler the elliptic form of the orbits of planets.)[23]
In 1841, the French mathematician Michel Chasles was one of the first to reevaluate his role in the development of modern algebra.
In 1847, a letter from François Arago, perpetual secretary of the Academy of Sciences (Paris), announced his intention to write a biography of François Viète.
Between 1880 and 1890, the polytechnician Fréderic Ritter, based in Fontenay-le-Comte, was the first translator of the works of François Viète and his first contemporary biographer with Benjamin Fillon.
Descartes' views on Viète
Thirty-four years after the death of Viète, the philosopher René Descartes published his method and a book of geometry that changed the landscape of algebra and built on Viète's work, applying it to the geometry by removing its requirements of homogeneity. Descartes, accused by Jean Baptiste Chauveau, a former classmate of La Flèche, explained in a letter to Mersenne (1639 February) that he never read those works.[24] Descartes accepted the Viète's view of mathematics for which the study shall stress the self-evidence of the results that Descartes implemented translating the symbolic algebra in geometric reasoning.[25] The locution mathesis universalis was derived from van Roomen's works.[25]
"I have no knowledge of this surveyor and I wonder what he said, that we studied Viète's work together in Paris, because it is a book which I cannot remember having seen the cover, while I was in France."
Elsewhere, Descartes said that Viète's notations were confusing and used unnecessary geometric justifications. In some letters, he showed he understands the program of the Artem Analyticem Isagoge; in others, he shamelessly caricatured Viète's proposals. One of his biographers, Charles Adam,[26] noted this contradiction:
"These words are surprising, by the way, for he (Descartes) had just said a few lines earlier that he had tried to put in his geometry only what he believed "was known neither by Vieta nor by anyone else". So he was informed of what Viète knew; and he must have read his works previously."
Current research has not shown the extent of the direct influence of the works of Viète on Descartes. This influence could have been formed through the works of Adriaan van Roomen or Jacques Aleaume at the Hague, or through the book by Jean de Beaugrand.[27]
In his letters to Mersenne, Descartes consciously minimized the originality and depth of the work of his predecessors. "I began," he says, "where Vieta finished". His views emerged in the 17th century and mathematicians won a clear algebraic language without the requirements of homogeneity. Many contemporary studies have restored the work of Parthenay's mathematician, showing he had the double merit of introducing the first elements of literal calculation and building a first axiomatic for algebra.[28]
Although Viète was not the first to propose notation of unknown quantities by letters - Jordanus Nemorarius had done this in the past - we can reasonably estimate that it would be simplistic to summarize his innovations for that discovery and place him at the junction of algebraic transformations made during the late sixteenth – early 17th century.
See also
• Michael Stifel
• Rafael Bombelli
Notes
1. Jacqueline A. Stedall, From Cardano's Great Art to Lagrange's Reflections: Filling a Gap in the History of Algebra, European Mathematical Society, 2011, p. 20.
2. H. Ben-Yami, Descartes' Philosophical Revolution: A Reassessment, Palgrave Macmillan, 2015, p. 179: "[Descartes'] work in mathematics was apparently influenced by Vieta's, despite his denial of any acquaintance with the latter’s work."
3. Cantor 1911, p. 57.
4. Goldstein, Bernard R. (1998), "What's new in Kepler's new astronomy?", in Earman, John; Norton, John D. (eds.), The Cosmos of Science: Essays of Exploration, Pittsburgh-Konstanz series in the philosophy and history of science, University of Pittsburgh Press, pp. 3–23, ISBN 9780822972013. See in particular p. 21: "an unpublished manuscript by Viète includes a mathematical discussion of an ellipse in a planetary model".
5. Kinser, Sam. The works of Jacques-Auguste de Thou. Google Books
6. Bashmakova, I. G., & Smirnova, G. S., The Beginnings and Evolution of Algebra (Washington, D.C.: Mathematical Association of America, 2000), pp. 75–77
7. Cantor 1911, p. 58.
8. Clavius, Christophorus. Operum mathematicorum tomus quintus continens Romani Christophorus Clavius, published by Anton Hierat, Johann Volmar, place Royale Paris, in 1612
9. Otte, Michael; Panza, Marco. Analysis and synthesis in mathematics. Google Books
10. De thou (from University of Saint Andrews) Archived 2008-07-08 at the Wayback Machine
11. Ball, Walter William Rouse. A short account of the history of mathematics. Google Books
12. H. J. M. Bos : Redefining geometrical exactness: Descartes' transformation Google Books
13. Jacob Klein: Greek mathematical thought and the origin of algebra, Google Books
14. Hadden, Richard W. (1994), On the Shoulders of Merchants: Exchange and the Mathematical Conception of Nature in Early Modern Europe, New York: State University of New York Press, ISBN 0-585-04483-X.
15. Stedall, Jacqueline Anne (2000). A large discourse concerning algebra: John Wallis's 1685 Treatise of algebra (Thesis). The Open University Press.
16. Helena M. Pycior : Symbols, Impossible Numbers, and Geometric Entanglements: British Algebra... Google books
17. Peter Murphy, Peter Murphy (LL. B.) : Evidence, proof, and facts: a book of sources, Google Books
18. Variorum de rebus Mathèmaticis Reíponíorum Liber VIII
19. Henk J.M. Bos: Descartes, Elisabeth and Apollonius’ Problem. In The Correspondence of René Descartes 1643, Quæstiones Infinitæ, pages 202–212. Zeno Institute of Philosophy, Utrecht, Theo Verbeek edition, Erik-Jan Bos and Jeroen van de Ven, 2003
20. Dhombres, Jean. François Viète et la Réforme. Available at cc-parthenay.fr Archived 2007-09-11 at the Wayback Machine (in French)
21. De Thou, Jacques-Auguste available at L'histoire universelle (fr) and at Universal History (en) Archived 2008-07-08 at the Wayback Machine
22. Viète, François (1983). The Analytic Art, translated by T. Richard Witmer. Kent, Ohio: The Kent State University Press.
23. Article about Harmonicon coeleste: Adsabs.harvard.edu "The Planetary Theory of François Viète, Part 1".
24. Letter from Descartes to Mersenne. (PDF) Pagesperso-orange.fr, February 20, 1639 (in French)
25. Bullynck, Maarten (2018). The 'Everyday' in Mathematics On the usability of mathematical practices for doing history. pp. 11, 10. Archived from the original on July 9, 2020. {{cite book}}: |website= ignored (help)
26. Archive.org, Charles Adam, Vie et Oeuvre de Descartes Paris, L Cerf, 1910, p 215.
27. Chikara Sasaki. Descartes' mathematical thought p.259
28. For example: Hairer, E (2008). Analysis by its history. New York: Springer. p. 6. ISBN 9780387770314.
Bibliography
• Bailey Ogilvie, Marilyn; Harvey, Joy Dorothy. The Biographical Dictionary of Women in Science: L–Z. Google Books. p 985.
• Bachmakova, Izabella G., Slavutin, E.I. “ Genesis Triangulorum de François Viète et ses recherches dans l’analyse indéterminée ”, Archives for History of Exact Science, 16 (4), 1977, 289-306.
• Bashmakova, Izabella Grigorievna; Smirnova Galina S; Shenitzer, Abe. The Beginnings and Evolution of Algebra. Google Books. pp. 75–.
• Biard, Joel; Rāshid, Rushdī. Descartes et le Moyen Age. Paris: Vrin, 1998. Google Books (in French)
• Burton, David M (1985). The History of Mathematics: An Introduction. Newton, Massachusetts: Allyn and Bacon, Inc.
• Cajori, F. (1919). A History of Mathematics. pp. 152 and onward.
• Calinger, Ronald (ed.) (1995). Classics of Mathematics. Englewood Cliffs, New Jersey: Prentice–Hall, Inc.
• Calinger, Ronald. Vita mathematica. Mathematical Association of America. Google Books
• Chabert, Jean-Luc; Barbin, Évelyne; Weeks, Chris. A History of Algorithms. Google Books
• Derbyshire, John (2006). Unknown Quantity a Real and Imaginary History of Algebra. Scribd.com Archived 2009-12-21 at the Wayback Machine
• Eves, Howard (1980). Great Moments in Mathematics (Before 1650). The Mathematical Association of America. Google Books
• Grisard, J. (1968) François Viète, mathématicien de la fin du seizième siècle: essai bio-bibliographique (Thèse de doctorat de 3ème cycle) École Pratique des Hautes Études, Centre de Recherche d'Histoire des Sciences et des Techniques, Paris. (in French)
• Godard, Gaston. François Viète (1540–1603), Father of Modern Algebra. Université de Paris-VII, France, Recherches vendéennes. ISSN 1257-7979 (in French)
• W. Hadd, Richard. On the shoulders of merchants. Google Books
• Hofmann, Joseph E (1957). The History of Mathematics, translated by F. Graynor and H. O. Midonick. New York, New York: The Philosophical Library.
• Joseph, Anthony. Round tables. European Congress of Mathematics. Google Books
• Michael Sean Mahoney (1994). The mathematical career of Pierre de Fermat (1601–1665). Google Books
• Jacob Klein. Die griechische Logistik und die Entstehung der Algebra in: Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Abteilung B: Studien, Band 3, Erstes Heft, Berlin 1934, p. 18–105 and Zweites Heft, Berlin 1936, p. 122–235; translated in English by Eva Brann as: Greek Mathematical Thought and the Origin of Algebra. Cambridge, Mass. 1968, ISBN 0-486-27289-3
• Mazur, Joseph (2014). Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers. Princeton, New Jersey: Princeton University Press.
• Nadine Bednarz, Carolyn Kieran, Lesley Lee. Approaches to algebra. Google Books
• Otte, Michael; Panza, Marco. Analysis and Synthesis in Mathematics. Google Books
• Pycior, Helena M. Symbols, Impossible Numbers, and Geometric Entanglements. Google Books
• Francisci Vietae Opera Mathematica, collected by F. Van Schooten. Leyde, Elzévir, 1646, p. 554 Hildesheim-New-York: Georg Olms Verlag (1970). (in Latin)
• The intégral corpus (excluding Harmonicon) was published by Frans van Schooten, professor at Leyde as Francisci Vietæ. Opera mathematica, in unum volumen congesta ac recognita, opera atque studio Francisci a Schooten, Officine de Bonaventure et Abraham Elzevier, Leyde, 1646. Gallica.bnf.fr (pdf). (in Latin)
• Stillwell, John. Mathematics and its history. Google Books
• Varadarajan, V. S. (1998). Algebra in Ancient and Modern Times The American Mathematical Society. Google Books
Attribution
• This article incorporates text from a publication now in the public domain: Cantor, Moritz (1911). "Vieta, François". In Chisholm, Hugh (ed.). Encyclopædia Britannica. Vol. 28 (11th ed.). Cambridge University Press. pp. 57–58.
External links
• Literature by and about François Viète in the German National Library catalogue
• François Viète at Library of Congress
• O'Connor, John J.; Robertson, Edmund F., "François Viète", MacTutor History of Mathematics Archive, University of St Andrews
• New Algebra (1591) online
• Francois Viète: Father of Modern Algebraic Notation
• The Lawyer and the Gambler
• About Tarporley
• Site de Jean-Paul Guichard (in French)
• L'algèbre nouvelle (in French)
• "About the Harmonicon" (PDF). Archived from the original (PDF) on 2011-08-07. Retrieved 2009-06-18. (200 KB). (in French)
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| Wikipedia |
Vieta jumping
In number theory, Vieta jumping, also known as root flipping, is a proof technique. It is most often used for problems in which a relation between two integers is given, along with a statement to prove about its solutions. In particular, it can be used to produce new solutions of a quadratic Diophantine equation from known ones. There exist multiple variations of Vieta jumping, all of which involve the common theme of infinite descent by finding new solutions to an equation using Vieta's formulas.
History
Vieta jumping is a classical method in the theory of quadratic Diophantine equations and binary quadratic forms. For example, it was used in the analysis of Markov equation back in 1879 and in the 1953 paper of Mills.[1]
In 1988, the method came to the attention to mathematical olympiad problems in the light of the first olympiad problem to use it in a solution that was proposed for the International Mathematics Olympiad and assumed to be the most difficult problem on the contest:[2][3]
Let a and b be positive integers such that ab + 1 divides a2 + b2. Show that ${\frac {a^{2}+b^{2}}{ab+1}}$ is the square of an integer.[4]
Arthur Engel wrote the following about the problem's difficulty:
Nobody of the six members of the Australian problem committee could solve it. Two of the members were husband and wife George and Esther Szekeres, both famous problem solvers and problem creators. Since it was a number theoretic problem it was sent to the four most renowned Australian number theorists. They were asked to work on it for six hours. None of them could solve it in this time. The problem committee submitted it to the jury of the XXIX IMO marked with a double asterisk, which meant a superhard problem, possibly too hard to pose. After a long discussion, the jury finally had the courage to choose it as the last problem of the competition. Eleven students gave perfect solutions.
Among the eleven students receiving the maximum score for solving this problem were Ngô Bảo Châu, Ravi Vakil, Zvezdelina Stankova, and Nicușor Dan.[5] Emanouil Atanassov (from Bulgaria) solved the problem in a paragraph and received a special prize.[6]
Standard Vieta jumping
The concept of standard Vieta jumping is a proof by contradiction, and consists of the following four steps:[7]
1. Assume toward a contradiction that some solution (a1, a2, ...) exists that violates the given requirements.
2. Take the minimal such solution according to some definition of minimality.
3. Replace some ai by a variable x in the formulas, and obtain an equation for which ai is a solution.
4. Using Vieta's formulas, show that this implies the existence of a smaller solution, hence a contradiction.
Example
Problem #6 at IMO 1988: Let a and b be positive integers such that ab + 1 divides a2 + b2. Prove that a2 + b2/ab + 1 is a perfect square.[8][9]
1. Fix some value k that is a non-square positive integer. Assume there exist positive integers (a, b) for which k = a2 + b2/ab + 1.
2. Let (A, B) be positive integers for which k = A2 + B2/AB + 1 and such that A + B is minimized, and without loss of generality assume A ≥ B.
3. Fixing B, replace A with the variable x to yield x2 – (kB)x + (B2 – k) = 0. We know that one root of this equation is x1 = A. By standard properties of quadratic equations, we know that the other root satisfies x2 = kB – A and x2 = B2 – k/A.
4. The first expression for x2 shows that x2 is an integer, while the second expression implies that x2 ≠ 0 since k is not a perfect square. From x22 + B2/x2B + 1 = k > 0 it further follows that x2B > −1, and hence x2 is a positive integer. Finally, A ≥ B implies that x2 = B2 − k/A < B2/A ≤ A, hence x2 < A, and thus x2 + B < A + B, which contradicts the minimality of A + B.
Constant descent Vieta jumping
The method of constant descent Vieta jumping is used when we wish to prove a statement regarding a constant k having something to do with the relation between a and b. Unlike standard Vieta jumping, constant descent is not a proof by contradiction, and it consists of the following four steps:[10]
1. The equality case is proven so that it may be assumed that a > b.
2. b and k are fixed and the expression relating a, b, and k is rearranged to form a quadratic with coefficients in terms of b and k, one of whose roots is a. The other root, x2 is determined using Vieta's formulas.
3. For all (a, b) above a certain base case, show that 0 < x2 < b < a and that x2 is an integer. Thus, while maintaining the same k, we may replace (a, b) with (b, x2) and repeat this process until we arrive at the base case.
4. Prove the statement for the base case, and as k has remained constant through this process, this is sufficient to prove the statement for all ordered pairs.
Example
Let a and b be positive integers such that ab divides a2 + b2 + 1. Prove that 3ab = a2 + b2 + 1.[11]
1. If a = b, a2 dividing 2a2 + 1 implies that a2 divides 1, and hence the positive integers a = b = 1, and 3(1)(1) = 12 + 12 + 1. So, without loss of generality, assume that a > b.
2. For any (a, b) satisfying the given condition, let k = a2 + b2 + 1/ab and rearrange and substitute to get x2 − (kb) x + (b2 + 1) = 0. One root to this quadratic is a, so by Vieta's formulas the other root may be written as follows: x2 = kb − a = b2 + 1/a.
3. The first equation shows that x2 is an integer and the second that it is positive. Because a > b and they are both integers, a ≥ b + 1, and hence ab ≥ b2 + b; As long as b > 1, we always have ab > b2 + 1, and therefore x2 = b2 + 1/a < b. Thus, while maintaining the same k, we may replace (a, b) with (b, x2) and repeat this process until we arrive at the base case.
4. The base case we arrive at is the case where b = 1. For (a, 1) to satisfy the given condition, a must divide a2 + 2, which implies that a divides 2, making a either 1 or 2. The first case is eliminated because a = b. In the second case, k = a2 + b2 + 1/ab = 6/2 = 3. As k has remained constant throughout this process of Vieta jumping, this is sufficient to show that for any (a, b) satisfying the given condition, k will always equal 3.
Geometric interpretation
Vieta jumping can be described in terms of lattice points on hyperbolas in the first quadrant.[2] The same process of finding smaller roots is used instead to find lower lattice points on a hyperbola while remaining in the first quadrant. The procedure is as follows:
1. From the given condition we obtain the equation of a family of hyperbolas that are unchanged by switching x and y so that they are symmetric about the line y = x.
2. Prove the desired statement for the intersections of the hyperbolas and the line y = x.
3. Assume there is some lattice point (x, y) on some hyperbola and without loss of generality x < y. Then by Vieta's formulas, there is a corresponding lattice point with the same x-coordinate on the other branch of the hyperbola, and by reflection through y = x a new point on the original branch of the hyperbola is obtained.
4. It is shown that this process produces lower points on the same branch and can be repeated until some condition (such as x = 0) is achieved. Then by substitution of this condition into the equation of the hyperbola, the desired conclusion will be proven.
Example
This method can be applied to problem #6 at IMO 1988: Let a and b be positive integers such that ab + 1 divides a2 + b2. Prove that a2 + b2/ab + 1 is a perfect square.
1. Let a2 + b2/ab + 1 = q and fix the value of q. If q = 1, q is a perfect square as desired. If q = 2, then (a-b)2 = 2 and there is no integral solution a, b. When q > 2, the equation x2 + y2 − qxy − q = 0 defines a hyperbola H and (a,b) represents an integral lattice point on H.
2. If (x,x) is an integral lattice point on H with x > 0, then (since q is integral) one can see that x = 1). This proposition's statement is then true for the point (x,x).
3. Now let P = (x, y) be a lattice point on a branch H with x, y > 0 and x ≠ y (as the previous remark covers the case x = y). By symmetry, we can assume that x < y and that P is on the higher branch of H. By applying Vieta's Formulas, (x, qx − y) is a lattice point on the lower branch of H. Let y′ = qx − y. From the equation for H, one sees that 1 + x y′ > 0. Since x > 0, it follows that y′ ≥ 0. Hence the point (x, y′) is in the first quadrant. By reflection, the point (y′, x) is also a point in the first quadrant on H. Moreover from Vieta's formulas, yy′ = x2 - q, and y′ = x2 - q/ y. Combining this equation with x < y, one can show that y′ < x. The new constructed point Q = (y′, x) is then in the first quadrant, on the higher branch of H, and with smaller x,y-coordinates than the point P we started with.
4. The process in the previous step can be repeated whenever the point Q has a positive x-coordinate. However, since the x-coordinates of these points will form a decreasing sequence of non-negative integers, the process can only be repeated finitely many times before it produces a point Q = (0, y) on the upper branch of H; by substitution, q = y2 is a square as required.
See also
• Vieta's formulas
• Proof by contradiction
• Infinite descent
• Markov number
• Apollonian gasket
Notes
1. Mills 1953.
2. Arthur Engel (1998). Problem Solving Strategies. Problem Books in Mathematics. Springer. p. 127. doi:10.1007/b97682. ISBN 978-0-387-98219-9.
3. "The Return of the Legend of Question Six". Numberphile. August 16, 2016. Archived from the original on 2021-12-20 – via YouTube.
4. "International Mathematical Olympiad". www.imo-official.org. Retrieved 29 September 2020.
5. "Results of the 1988 International Mathematical Olympiad". Imo-official.org. Retrieved 2013-03-03.
6. "Individual ranking of Emanouil Atanassov". International Mathematical Olympiad.
7. Yimin Ge (2007). "The Method of Vieta Jumping" (PDF). Mathematical Reflections. 5.
8. "AoPS Forum – One of my favourites problems, yeah!". Artofproblemsolving.com. Retrieved 2023-01-08.
9. K. S. Brown. "N = (x^2 + y^2)/(1+xy) is a Square". MathPages.com. Retrieved 2016-09-26.
10. "AoPS Forum — Lemur Numbers". Artofproblemsolving.com. Retrieved 2023-01-08.
11. "AoPS Forum – x*y | x^2+y^2+1". Artofproblemsolving.com. 2005-06-07. Retrieved 2023-01-08.
External links
• Vieta Root Jumping at Brilliant.org
• Mills, W. H. (1953). "A system of quadratic Diophantine equations". Pacific J. Math. 3 (1): 209–220.
| Wikipedia |
Vietnamese numerals
Historically Vietnamese has two sets of numbers: one is etymologically native Vietnamese; the other uses Sino-Vietnamese vocabulary. In the modern language the native Vietnamese vocabulary is used for both everyday counting and mathematical purposes. The Sino-Vietnamese vocabulary is used only in fixed expressions or in Sino-Vietnamese words, in a similar way that Latin and Greek numerals are used in modern English (e.g., the bi- prefix in bicycle).
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List of numeral systems
For numbers up to one million, native Vietnamese terms is often used the most, whilst mixed Sino-Vietnamese origin words and native Vietnamese words are used for units of one million or above.
Concept
For non-official purposes prior to the 20th century, Vietnamese had a writing system known as Hán-Nôm. Sino-Vietnamese numbers were written in Chữ Hán and native vocabulary was written in Chữ Nôm. Hence, there are two concurrent system in Vietnamese nowadays in the romanized script, one for native Vietnamese and one for Sino-Vietnamese.
In the modern Vietnamese writing system, numbers are written as Arabic numerals or in the romanized script Chữ Quốc ngữ (một, hai, ba), which had a Chữ Nôm character. Less common for numbers under one million are the numbers of Sino-Vietnamese origin (nhất [1], nhị [2], tam [3]), using Chữ Hán (classical Chinese characters). Chữ Hán and Chữ Nôm has all but become obsolete in the Vietnamese language, with the Latin-style of reading, writing, and pronouncing native Vietnamese and Sino-Vietnamese being wide spread instead, when France occupied Vietnam. Chữ Hán can still be seen in traditional temples or traditional literature or in cultural artefacts. The Hán-Nôm Institute resides in Hanoi, Vietnam.
Basic figures
The following table is an overview of the basic Vietnamese numeric figures, provided in both native and Sino-Vietnamese counting systems. The form that is highlighted in green is the most widely used in all purposes whilst the ones highlighted in blue are seen as archaic but may still be in use. There are slight differences between the Hanoi and Saigon dialects of Vietnamese, readings between each are differentiated below.
Number Native Vietnamese Sino-Vietnamese Notes
Chữ quốc ngữ Chữ Nôm Chữ quốc ngữ Hán tự
0 không 空 linh 空 • 〇（零） The foreign-language borrowed word "zêrô (zêro, dê-rô)" is often used in physics-related publications, or colloquially.
1 một 𠬠 nhất 一（壹）
2 hai 𠄩 nhị 二（貳）
3 ba 𠀧 tam 三（叄）
4 bốn 𦊚 tứ 四（肆） In the ordinal number system, the Sino-Vietnamese "tư／四" is more systematic; as the digit 4 appears after the number 20 when counting upwards, the Sino-Vietnamese "tư／四" is more commonly used.
5 năm 𠄼 ngũ 五（伍） In numbers above ten that end in five (such as 115, 25, 1055), five is alternatively pronounced as "lăm／𠄻" to avoid possible confusion with "năm／𢆥", a homonym of năm, meaning "year". Exceptions to this rule are numbers ending in 05 (such as 605, 9405).
6 sáu 𦒹 lục 六（陸）
7 bảy 𦉱 thất 七（柒） In some Vietnamese dialects, it is also read as "bẩy".
8 tám 𠔭 bát 八（捌）
9 chín 𠃩 cửu 九（玖）
10 mười • một chục 𨒒 thập 十（拾） Chục is used colloquially. "Ten eggs" may be called một chục quả trứng rather than mười quả trứng. It's also used in compounds like mươi instead of mười (e.g.: hai mươi/chục "twenty").
100 trăm • một trăm 𤾓 • 𠬠𤾓 bách (bá) 百（佰） The Sino-Vietnamese "bách／百" is commonly used as a morpheme (in compound words), and is rarely used in the field of mathematics as a digit. Example: "bách phát bách trúng／百發百中".
1,000 nghìn (ngàn) • một nghìn (ngàn) 𠦳 • 𠬠𠦳 thiên 千（仟） The Sino-Vietnamese "thiên／千" is commonly used as a morpheme, but rarely used in a mathematical sense, however only in counting bricks, it is used. Example: "thiên kim／千金". "nghìn" is the standard word in Northern Vietnam, whilst "ngàn" is the word used in the South.
10,000 mười nghìn (ngàn) 𨒒𠦳 vạn • một vạn 萬 • 𠬠萬 The "một／𠬠" within "một vạn／𠬠萬" is a native Vietnamese (intrinsic term) morpheme. This was officially used in Vietnamese in the past, however, this unit has become less common after 1945, but in counting bricks, it is still widely used. The borrowed native pronunciation muôn for 萬 is still used in slogans such as "muôn năm" (ten thousand years/endless).
100,000 trăm nghìn (ngàn) • một trăm nghìn (ngàn) 𤾓𠦳 • 𠬠𤾓𠦳 ức • một ức • mười vạn[1] 億 • 𠬠億 • 𨒒萬 The "mười／𨒒" and "một／𠬠" within "mười vạn／𨒒萬" and "một ức／𠬠億" are native Vietnamese (intrinsic term) morphemes.
1,000,000 (none) (none) triệu • một triệu • một trăm vạn[2] 兆 • 𠬠兆 • 𠬠𤾓萬 The "một／𠬠" and "trăm／𤾓" within "một triệu／𠬠兆" and "một trăm vạn／𠬠𤾓萬" are native Vietnamese (intrinsic term) morphemes.
10,000,000 (mixed usage of Sino-Vietnamese and native Vietnamese systems) (mixed usage of Sino-Vietnamese and native Vietnamese systems) mười triệu 𨒒兆 The "mười／𨒒" within "mười triệu／𨒒兆" is a native Vietnamese (intrinsic term) morpheme.
100,000,000 (mixed usage of Sino-Vietnamese and native Vietnamese systems) (mixed usage of Sino-Vietnamese and native Vietnamese systems) trăm triệu 𤾓兆 The "trăm／𤾓" within "trăm triệu／𤾓兆" is a native Vietnamese (intrinsic term) morpheme.
1,000,000,000 (none) (none) tỷ 秭[3]
1012 (mixed usage of Sino-Vietnamese and native Vietnamese systems) (mixed usage of Sino-Vietnamese and native Vietnamese systems) nghìn (ngàn) tỷ 𠦳秭
1015 (none) (none) triệu tỷ 兆秭
1018 (none) (none) tỷ tỷ 秭秭
Some other features of Vietnamese numerals include the following:
• Outside of fixed Sino-Vietnamese expressions, Sino-Vietnamese words are usually used in combination with native Vietnamese words. For instance, "mười triệu" combines native "mười" and Sino-Vietnamese "triệu".
• Modern Vietnamese separates place values in thousands instead of myriads. For example, "123123123" is recorded in Vietnamese as "một trăm hai mươi ba triệu một trăm hai mươi ba nghìn (ngàn) một trăm hai mươi ba, or '123 million, 123 thousand and 123'.[4] Meanwhile, in Chinese, Japanese & Korean, the same number is rendered as "1億2312萬3123" (1 hundred-million, 2312 ten-thousand and 3123).
• Sino-Vietnamese numbers are not in frequent use in modern Vietnamese. Sino-Vietnamese numbers such as "vạn／萬" 'ten thousand', "ức／億" 'hundred-thousand' and "triệu／兆" 'million' are used for figures exceeding one thousand, but with the exception of "triệu" are becoming less commonly used. Number values for these words are used for each numeral increasing tenfold in digit value, 億 being the number for 105, 兆 for 106, et cetera. However, Triệu in Vietnamese and 兆 in Modern Chinese now have different values.
Other figures
NumberChữ quốc ngữHán-NômNotes
11 mười một𨒒𠬠
12 mười hai • một tá𨒒𠄩 • 𠬠打"một tá／𠬠打" is often used within mathematics-related occasions, to which "tá" represents the foreign loanword "dozen".
14 mười bốn • mười tư𨒒𦊚 • 𨒒四"mười tư／𨒒四" is often used within literature-related occasions, to which "tư／四" forms part of the Sino-Vietnamese vocabulary.
15 mười lăm𨒒𠄻Here, five is pronounced "lăm／𠄻", or also "nhăm／𠄶" by some speakers in the north.
19 mười chín𨒒𠃩
20 hai mươi • hai chục𠄩𨒒 • 𠄩𨔿
21 hai mươi mốt𠄩𨒒𠬠For numbers which include the digit 1 from 21 to 91, the number 1 is pronounced "mốt".
24 hai mươi tư𠄩𨒒四When the digit 4 appears in numbers after 20 as the last digit of a 3-digit group, it is more common to use "tư／四".
25 hai mươi lăm𠄩𨒒𠄻Here, five is pronounced "lăm".
50 năm mươi • năm chục𠄼𨒒 • 𠄼𨔿When "𨒒" (10) appears after the number 20, the pronunciation changes to "mươi".
101 một trăm linh một • một trăm lẻ một𠬠𤾓零𠬠 • 𠬠𤾓𥘶𠬠"Một trăm linh một／𠬠𤾓零𠬠" is the Northern form, where "linh／零" forms part of the Sino-Vietnamese vocabulary; "một trăm lẻ một／𠬠𤾓𥘶𠬠" is commonly used in the Southern and Central dialect groups of Vietnam.
1001 một nghìn (ngàn) không trăm linh một • một nghìn (ngàn) không trăm lẻ một𠬠𠦳空𤾓零𠬠 • 𠬠𠦳空𤾓𥘶𠬠When the hundreds digit is occupied by a zero, these are expressed using "không trăm／空𤾓".
10055 mười nghìn (ngàn) không trăm năm mươi lăm𨒒𠦳空𤾓𠄼𨒒𠄻
• When the number 1 appears after 20 in the unit digit, the pronunciation changes to "mốt".
• When the number 4 appears after 20 in the unit digit, it is more common to use Sino-Vietnamese "tư／四".
• When the number 5 appears after 10 in the unit digit, the pronunciation changes to "lăm／𠄻".
• When "mười" appears after 20, the pronunciation changes to "mươi".
Ordinal numbers
Vietnamese ordinal numbers are generally preceded by the prefix "thứ-", which is a Sino-Vietnamese word which corresponds to "次-". For the ordinal numbers of one and four, the Sino-Vietnamese readings "nhất／一" and "tư／四" are more commonly used; two is occasionally rendered using the Sino-Vietnamese "nhị／二". In all other cases, the native Vietnamese number is used.
In formal cases, the ordinal number with the structure "đệ (第) + Sino-Vietnamese numbers" is used, especially in calling the generation of monarches, with an example being Nữ vương Elizabeth đệ nhị/女王 Elizabeth 第二 (Queen Elizabeth II).
Ordinal numberChữ quốc ngữHán-Nôm
1stthứ nhất次一
2ndthứ hai • thứ nhì次𠄩 • 次二
3rdthứ ba次𠀧
4ththứ tư次四
5ththứ năm次𠄼
nththứ "n"次「n」
Footnotes
1. Tu dien Han Viet Thieu Chuu：「(1): ức, mười vạn là một ức.」
2. Tu dien Han Viet Thieu Chuu：「(3): triệu, một trăm vạn.」
3. Hán Việt Từ Điển Trích Dẫn 漢越辭典摘引：「Một ngàn lần một triệu là một tỉ 秭 (*). Tức là 1.000.000.000. § Ghi chú: Ngày xưa, mười vạn 萬 là một ức 億, một vạn ức là một tỉ 秭.」
4. Triệu means one million in Vietnamese, but the Chinese number that is the source of the Vietnamese word, "兆" (Mandarin zhào), is officially rendered as 1012 in Taiwan, and commonly designated as 106 in the People's Republic of China (See various scale systems).
See also
• Japanese numerals, Korean numerals, Chinese numerals
| Wikipedia |
Vietoris–Rips complex
In topology, the Vietoris–Rips complex, also called the Vietoris complex or Rips complex, is a way of forming a topological space from distances in a set of points. It is an abstract simplicial complex that can be defined from any metric space M and distance δ by forming a simplex for every finite set of points that has diameter at most δ. That is, it is a family of finite subsets of M, in which we think of a subset of k points as forming a (k − 1)-dimensional simplex (an edge for two points, a triangle for three points, a tetrahedron for four points, etc.); if a finite set S has the property that the distance between every pair of points in S is at most δ, then we include S as a simplex in the complex.
History
The Vietoris–Rips complex was originally called the Vietoris complex, for Leopold Vietoris, who introduced it as a means of extending homology theory from simplicial complexes to metric spaces.[1] After Eliyahu Rips applied the same complex to the study of hyperbolic groups, its use was popularized by Mikhail Gromov (1987), who called it the Rips complex.[2] The name "Vietoris–Rips complex" is due to Jean-Claude Hausmann (1995).[3]
Relation to Čech complex
The Vietoris–Rips complex is closely related to the Čech complex (or nerve) of a set of balls, which has a simplex for every finite subset of balls with nonempty intersection. In a geodesically convex space Y, the Vietoris–Rips complex of any subspace X ⊂ Y for distance δ has the same points and edges as the Čech complex of the set of balls of radius δ/2 in Y that are centered at the points of X. However, unlike the Čech complex, the Vietoris–Rips complex of X depends only on the intrinsic geometry of X, and not on any embedding of X into some larger space.
As an example, consider the uniform metric space M3 consisting of three points, each at unit distance from each other. The Vietoris–Rips complex of M3, for δ = 1, includes a simplex for every subset of points in M3, including a triangle for M3 itself. If we embed M3 as an equilateral triangle in the Euclidean plane, then the Čech complex of the radius-1/2 balls centered at the points of M3 would contain all other simplexes of the Vietoris–Rips complex but would not contain this triangle, as there is no point of the plane contained in all three balls. However, if M3 is instead embedded into a metric space that contains a fourth point at distance 1/2 from each of the three points of M3, the Čech complex of the radius-1/2 balls in this space would contain the triangle. Thus, the Čech complex of fixed-radius balls centered at M3 differs depending on which larger space M3 might be embedded into, while the Vietoris–Rips complex remains unchanged.
If any metric space X is embedded in an injective metric space Y, the Vietoris–Rips complex for distance δ and X coincides with the Čech complex of the balls of radius δ/2 centered at the points of X in Y. Thus, the Vietoris–Rips complex of any metric space M equals the Čech complex of a system of balls in the tight span of M.
Relation to unit disk graphs and clique complexes
The Vietoris–Rips complex for δ = 1 contains an edge for every pair of points that are at unit distance or less in the given metric space. As such, its 1-skeleton is the unit disk graph of its points. It contains a simplex for every clique in the unit disk graph, so it is the clique complex or flag complex of the unit disk graph.[4] More generally, the clique complex of any graph G is a Vietoris–Rips complex for the metric space having as points the vertices of G and having as its distances the lengths of the shortest paths in G.
Other results
If M is a closed Riemannian manifold, then for sufficiently small values of δ the Vietoris–Rips complex of M, or of spaces sufficiently close to M, is homotopy equivalent to M itself.[5]
Chambers, Erickson & Worah (2008) describe efficient algorithms for determining whether a given cycle is contractible in the Rips complex of any finite point set in the Euclidean plane.
Applications
As with unit disk graphs, the Vietoris–Rips complex has been applied in computer science to model the topology of ad hoc wireless communication networks. One advantage of the Vietoris–Rips complex in this application is that it can be determined only from the distances between the communication nodes, without having to infer their exact physical locations. A disadvantage is that, unlike the Čech complex, the Vietoris–Rips complex does not directly provide information about gaps in communication coverage, but this flaw can be ameliorated by sandwiching the Čech complex between two Vietoris–Rips complexes for different values of δ.[6] An implementation of Vietoris-Rips complexes can be found in the TDAstats R package.[7]
Vietoris–Rips complexes have also been applied for feature-extraction in digital image data; in this application, the complex is built from a high-dimensional metric space in which the points represent low-level image features.[8]
The collection of all Vietoris-Rips complexes is a commonly applied construction in persistent homology and topological data analysis, and is known as the Rips filtration.[9]
Notes
1. Vietoris (1927); Lefschetz (1942); Hausmann (1995); Reitberger (2002).
2. Hausmann (1995); Reitberger (2002).
3. Reitberger (2002).
4. Chambers, Erickson & Worah (2008).
5. Hausmann (1995), Latschev (2001).
6. de Silva & Ghrist (2006), Muhammad & Jadbabaie (2007).
7. Wadhwa et al. 2018.
8. Carlsson, Carlsson & de Silva (2006).
9. Dey, Tamal K.; Shi, Dayu; Wang, Yusu (2019-01-30). "SimBa: An Efficient Tool for Approximating Rips-filtration Persistence via Simplicial Batch Collapse". ACM Journal of Experimental Algorithmics. 24: 1.5:1–1.5:16. doi:10.1145/3284360. ISSN 1084-6654. S2CID 216028146.
References
• Carlsson, Erik; Carlsson, Gunnar; de Silva, Vin (2006), "An algebraic topological method for feature identification" (PDF), International Journal of Computational Geometry and Applications, 16 (4): 291–314, doi:10.1142/S021819590600204X, S2CID 5831809, archived from the original (PDF) on 2019-03-04.
• Chambers, Erin W.; Erickson, Jeff; Worah, Pratik (2008), "Testing contractibility in planar Rips complexes", Proceedings of the 24th Annual ACM Symposium on Computational Geometry, pp. 251–259, CiteSeerX 10.1.1.296.6424, doi:10.1145/1377676.1377721, S2CID 8072058.
• Chazal, Frédéric; Oudot, Steve (2008), "Towards persistence-based reconstruction in euclidean spaces", Proceedings of the twenty-fourth annual symposium on Computational geometry, pp. 232–241, arXiv:0712.2638, doi:10.1145/1377676.1377719, ISBN 978-1-60558-071-5, S2CID 1020710{{citation}}: CS1 maint: date and year (link).
• de Silva, Vin; Ghrist, Robert (2006), "Coordinate-free coverage in sensor networks with controlled boundaries via homology", The International Journal of Robotics Research, 25 (12): 1205–1222, doi:10.1177/0278364906072252, S2CID 10210836.
• Gromov, Mikhail (1987), "Hyperbolic groups", Essays in group theory, Mathematical Sciences Research Institute Publications, vol. 8, Springer-Verlag, pp. 75–263.
• Hausmann, Jean-Claude (1995), "On the Vietoris–Rips complexes and a cohomology theory for metric spaces", Prospects in Topology: Proceedings of a conference in honour of William Browder, Annals of Mathematics Studies, vol. 138, Princeton University Press, pp. 175–188, MR 1368659.
• Latschev, Janko (2001), "Vietoris-Rips complexes of metric spaces near a closed Riemannian manifold", Archiv der Mathematik, 77 (6): 522–528, doi:10.1007/PL00000526, MR 1879057, S2CID 119878137.
• Lefschetz, Solomon (1942), Algebraic Topology, New York: Amer. Math. Soc., p. 271, MR 0007093.
• Muhammad, A.; Jadbabaie, A. (2007), "Dynamic coverage verification in mobile sensor networks via switched higher order Laplacians" (PDF), in Broch, Oliver (ed.), Robotics: Science and Systems, MIT Press.
• Reitberger, Heinrich (2002), "Leopold Vietoris (1891–2002)" (PDF), Notices of the American Mathematical Society, 49 (20).
• Vietoris, Leopold (1927), "Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen", Mathematische Annalen, 97 (1): 454–472, doi:10.1007/BF01447877, S2CID 121172198.
• Wadhwa, Raoul; Williamson, Drew; Dhawan, Andrew; Scott, Jacob (2018), "TDAstats: R pipeline for computing persistent homology in topological data analysis", Journal of Open Source Software, 3 (28): 860, Bibcode:2018JOSS....3..860R, doi:10.21105/joss.00860, PMC 7771879, PMID 33381678
| Wikipedia |
Leopold Vietoris
Leopold Vietoris (/viːˈtɔːrɪs/; German: [viːˈtoːʀɪs]; 4 June 1891 – 9 April 2002) was an Austrian mathematician, World War I veteran and supercentenarian. He was born in Radkersburg and died in Innsbruck.
Leopold Vietoris
Leopold Vietoris on his 110th birthday
Born(1891-06-04)4 June 1891
Bad Radkersburg, Styria
Austria-Hungary
Died(2002-04-09)9 April 2002
(aged 110 years, 309 days)
Innsbruck, Tyrol
Austria
NationalityAustrian
Alma materTU Wien
University of Vienna
Known forContributions to topology
Being a supercentenarian
Spouse(s)Klara Riccabona (m. 1928–1935) (her death)
Maria Josefa Vincentia Vietoris, born von Riccabona zu Reichenfels (m. 1936–2002) (her death)
Children6
Scientific career
FieldsMathematics
InstitutionsUniversity of Innsbruck
Doctoral advisorsGustav Ritter von Escherich
Wilhelm Wirtinger
He was known for his contributions to topology—notably the Mayer–Vietoris sequence—and other fields of mathematics, his interest in mathematical history and for being a keen alpinist.
Biography
Vietoris studied mathematics and geometry at the Vienna University of Technology.[1] He was drafted in 1914 in World War I and was wounded in September that same year.[1] On 4 November 1918, one week before the Armistice of Villa Giusti, he became an Italian prisoner of war.[1] After returning to Austria, he attended the University of Vienna, where he earned his PhD in 1920, with a thesis written under the supervision of Gustav von Escherich and Wilhelm Wirtinger.[1][2]
In autumn 1928 he married his first wife Klara Riccabona, who later died while giving birth to their sixth daughter.[1] In 1936 he married Klara's sister, Maria Riccabona.[1]
Vietoris was survived by his six daughters, 17 grandchildren, and 30 great-grandchildren.[3]
He lends his name to a few mathematical concepts:
• Vietoris topology (see topological space)
• Vietoris homology (see homology theory)
• Mayer–Vietoris sequence
• Vietoris–Begle mapping theorem
• Vietoris–Rips complex
Vietoris remained scientifically active in his later years, even writing one paper on trigonometric sums at the age of 103.[4]
Vietoris lived to be 110 years and 309 days old, and became the oldest verified Austrian man ever.[5]
Decorations and awards
• Austrian Decoration for Science and Art (1973)
• Grand Gold Decoration for Services to the Republic of Austria (1981)
• Honorary member of the German Mathematical Society (1992)
Notes
1. Reitberger, Heinrich (November 2002). "Leopold Vietoris (1891–2002)" (PDF). American Mathematical Society. Retrieved 5 September 2003.
2. Leopold Vietoris at the Mathematics Genealogy Project
3. "Professor Dr. Leopold Vietoris" (PDF). Geo Imagining. Retrieved 11 October 2009.
4. Reitberger, Heinrich (November 2002). "Leopold Vietoris (1891–2002)" (PDF). Notices of the American Mathematical Society. 49 (10): 1235.
5. "Verified Supercentenarians (Ranked By Age) Gerontology Research Group". 1 January 2014. Retrieved 28 February 2019.
References
• Weibel, Peter, ed. (2005). Beyond Art: A Third Culture: A Comparative Study in Cultures, Art and Science in 20th Century Austria and Hungary. Springer Science & Business Media. p. 260. ISBN 978-3-211-24562-0.
External links
• O'Connor, John J.; Robertson, Edmund F., "Leopold Vietoris", MacTutor History of Mathematics Archive, University of St Andrews
Authority control
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| Wikipedia |
Vietoris–Begle mapping theorem
The Vietoris–Begle mapping theorem is a result in the mathematical field of algebraic topology. It is named for Leopold Vietoris and Edward G. Begle. The statement of the theorem, below, is as formulated by Stephen Smale.
Theorem
Let $X$ and $Y$ be compact metric spaces, and let $f:X\to Y$ be surjective and continuous. Suppose that the fibers of $f$ are acyclic, so that
${\tilde {H}}_{r}(f^{-1}(y))=0,$ for all $0\leq r\leq n-1$ and all $y\in Y$,
with ${\tilde {H}}_{r}$ denoting the $r$th reduced Vietoris homology group. Then, the induced homomorphism
$f_{*}:{\tilde {H}}_{r}(X)\to {\tilde {H}}_{r}(Y)$
is an isomorphism for $r\leq n-1$ and a surjection for $r=n$.
Note that as stated the theorem doesn't hold for homology theories like singular homology. For example, Vietoris homology groups of the closed topologist's sine curve and of a segment are isomorphic (since the first projects onto the second with acyclic fibers). But the singular homology differs, since the segment is path connected and the topologist's sine curve is not.
References
• "Leopold Vietoris (1891–2002)", Notices of the American Mathematical Society, vol. 49, no. 10 (November 2002) by Heinrich Reitberger
| Wikipedia |
Viewpoints: Mathematical Perspective and Fractal Geometry in Art
Viewpoints: Mathematical Perspective and Fractal Geometry in Art is a textbook on mathematics and art. It was written by mathematicians Marc Frantz and Annalisa Crannell, and published in 2011 by the Princeton University Press (ISBN 9780691125923). The Basic Library List Committee of the Mathematical Association of America has recommended it for inclusion in undergraduate mathematics libraries.[1]
Topics
The first seven chapters of the book concern perspectivity, while its final two concern fractals and their geometry.[1][2] Topics covered within the chapters on perspectivity include coordinate systems for the plane and for Euclidean space, similarity, angles, and orthocenters, one-point and multi-point perspective, and anamorphic art.[1][3] In the fractal chapters, the topics include self-similarity, exponentiation, and logarithms, and fractal dimension. Beyond this mathematical material, the book also describes methods for artists to depict scenes in perspective, and for viewers of art to understand the perspectives in the artworks they see,[1] for instance by finding the optimal point from which to view an artwork.[2] The chapters are ordered by difficulty, and begin with experiments that the students can perform on their own to motivate the material in each chapter.[3]
The book is heavily illustrated by artworks and photography (such as the landscapes of Ansel Adams) and includes a series of essays or interviews by contemporary artists on the mathematical content of their artworks.[1][3] An appendix contains suggestions aimed at teachers of this material.[3]
Audience and reception
Viewpoints is intended as a textbook for mathematics classes aimed at undergraduate liberal arts students,[1][2][4] as a way to show these students how geometry can be used in their everyday life.[2] However, it could even be used for high school art students,[2][3] and reviewer Paul Kelley writes that "it will be of value to anyone interested in an elementary introduction to the mathematics and practice of perspective drawing".[2] It differs from many other liberal arts mathematics textbooks in its relatively narrow focus on geometry and perspective, and its avoidance of more well-covered ground in mathematics and the arts such as symmetry and the geometry of polyhedra.[2]
Although reviewer Blake Mellor complains that the connection between the material on perspective and on fractal geometry "feels forced", he concludes that "this is an excellent text".[4] Reviewer Paul Kelley writes that the book's "step-by-step progression" through its topics makes it "readable [and] easy-to-follow", and that "Students can learn a great deal from this book."[2] Reviewer Alexander Bogomolny calls it "an elegant fusion of mathematical ideas and practical aspects of fine art".[1]
References
1. Bogomolny, Alexander (September 2011), "Review of Viewpoints", MAA Reviews, Mathematical Association of America
2. Kelley, Paul (December 2012 – January 2013), "Review of Viewpoints", The Mathematics Teacher, 106 (5): 399, doi:10.5951/mathteacher.106.5.0398, JSTOR 10.5951/mathteacher.106.5.0398
3. Marchetti, Elena (February 2015), "Review of Viewpoints", Nexus Network Journal, 17 (2): 685–687, doi:10.1007/s00004-015-0237-9
4. Mellor, Blake (December 2011), "Review of Viewpoints", Journal of Mathematics and the Arts, 5 (4): 221–222, doi:10.1080/17513472.2011.624443
Mathematics and art
Concepts
• Algorithm
• Catenary
• Fractal
• Golden ratio
• Hyperboloid structure
• Minimal surface
• Paraboloid
• Perspective
• Camera lucida
• Camera obscura
• Plastic number
• Projective geometry
• Proportion
• Architecture
• Human
• Symmetry
• Tessellation
• Wallpaper group
Forms
• Algorithmic art
• Anamorphic art
• Architecture
• Geodesic dome
• Islamic
• Mughal
• Pyramid
• Vastu shastra
• Computer art
• Fiber arts
• 4D art
• Fractal art
• Islamic geometric patterns
• Girih
• Jali
• Muqarnas
• Zellij
• Knotting
• Celtic knot
• Croatian interlace
• Interlace
• Music
• Origami
• Sculpture
• String art
• Tiling
Artworks
• List of works designed with the golden ratio
• Continuum
• Mathemalchemy
• Mathematica: A World of Numbers... and Beyond
• Octacube
• Pi
• Pi in the Sky
Buildings
• Cathedral of Saint Mary of the Assumption
• Hagia Sophia
• Pantheon
• Parthenon
• Pyramid of Khufu
• Sagrada Família
• Sydney Opera House
• Taj Mahal
Artists
Renaissance
• Paolo Uccello
• Piero della Francesca
• Leonardo da Vinci
• Vitruvian Man
• Albrecht Dürer
• Parmigianino
• Self-portrait in a Convex Mirror
19th–20th
Century
• William Blake
• The Ancient of Days
• Newton
• Jean Metzinger
• Danseuse au café
• L'Oiseau bleu
• Giorgio de Chirico
• Man Ray
• M. C. Escher
• Circle Limit III
• Print Gallery
• Relativity
• Reptiles
• Waterfall
• René Magritte
• La condition humaine
• Salvador Dalí
• Crucifixion
• The Swallow's Tail
• Crockett Johnson
Contemporary
• Max Bill
• Martin and Erik Demaine
• Scott Draves
• Jan Dibbets
• John Ernest
• Helaman Ferguson
• Peter Forakis
• Susan Goldstine
• Bathsheba Grossman
• George W. Hart
• Desmond Paul Henry
• Anthony Hill
• Charles Jencks
• Garden of Cosmic Speculation
• Andy Lomas
• Robert Longhurst
• Jeanette McLeod
• Hamid Naderi Yeganeh
• István Orosz
• Hinke Osinga
• Antoine Pevsner
• Tony Robbin
• Alba Rojo Cama
• Reza Sarhangi
• Oliver Sin
• Hiroshi Sugimoto
• Daina Taimiņa
• Roman Verostko
• Margaret Wertheim
Theorists
Ancient
• Polykleitos
• Canon
• Vitruvius
• De architectura
Renaissance
• Filippo Brunelleschi
• Leon Battista Alberti
• De pictura
• De re aedificatoria
• Piero della Francesca
• De prospectiva pingendi
• Luca Pacioli
• De divina proportione
• Leonardo da Vinci
• A Treatise on Painting
• Albrecht Dürer
• Vier Bücher von Menschlicher Proportion
• Sebastiano Serlio
• Regole generali d'architettura
• Andrea Palladio
• I quattro libri dell'architettura
Romantic
• Samuel Colman
• Nature's Harmonic Unity
• Frederik Macody Lund
• Ad Quadratum
• Jay Hambidge
• The Greek Vase
Modern
• Owen Jones
• The Grammar of Ornament
• Ernest Hanbury Hankin
• The Drawing of Geometric Patterns in Saracenic Art
• G. H. Hardy
• A Mathematician's Apology
• George David Birkhoff
• Aesthetic Measure
• Douglas Hofstadter
• Gödel, Escher, Bach
• Nikos Salingaros
• The 'Life' of a Carpet
Publications
• Journal of Mathematics and the Arts
• Lumen Naturae
• Making Mathematics with Needlework
• Rhythm of Structure
• Viewpoints: Mathematical Perspective and Fractal Geometry in Art
Organizations
• Ars Mathematica
• The Bridges Organization
• European Society for Mathematics and the Arts
• Goudreau Museum of Mathematics in Art and Science
• Institute For Figuring
• Mathemalchemy
• National Museum of Mathematics
Related
• Droste effect
• Mathematical beauty
• Patterns in nature
• Sacred geometry
• Category
| Wikipedia |
Viggo Brun
Viggo Brun (13 October 1885 – 15 August 1978) was a Norwegian professor, mathematician and number theorist. [1]
Viggo Brun
Born13 October 1885
Lier, Norway
Died15 August 1978
Drøbak, Norway
CitizenshipNorway
Known forBrun's Theorem, Brun Sieve
Scientific career
FieldsNumber Theory
Contributions
In 1915, he introduced a new method, based on Legendre's version of the sieve of Eratosthenes, now known as the Brun sieve, which addresses additive problems such as Goldbach's conjecture and the twin prime conjecture. He used it to prove that there exist infinitely many integers n such that n and n+2 have at most nine prime factors, and that all large even integers are the sum of two numbers with at most nine prime factors.[2]
He also showed that the sum of the reciprocals of twin primes converges to a finite value, now called Brun's constant: by contrast, the sum of the reciprocals of all primes is divergent. He developed a multi-dimensional continued fraction algorithm in 1919–1920 and applied this to problems in musical theory. He also served as praeses of the Royal Norwegian Society of Sciences and Letters in 1946.[3]
Biography
Brun was born at Lier in Buskerud, Norway. He studied at the University of Oslo and began research at the University of Göttingen in 1910. In 1923, Brun became a professor at the
Technical University in Trondheim and in 1946 a professor at the University of Oslo.[4]
He retired in 1955 at the age of 70 and died in 1978 (at 92 years-old) at Drøbak in Akershus, Norway.[5]
See also
• Brun's theorem
• Brun-Titchmarsh theorem
• Brun sieve
• Sieve theory
References
1. "Viggo Brun". numbertheory.org. 18 June 2003. Retrieved January 1, 2017.
2. J J O'Connor; E F Robertson. "Viggo Brun". School of Mathematics and Statistics, University of St Andrews, Scotland. Retrieved January 1, 2017.
3. Bratberg, Terje (1996). "Vitenskapsselskapet". In Arntzen, Jon Gunnar (ed.). Trondheim byleksikon. Oslo: Kunnskapsforlaget. pp. 599–600. ISBN 82-573-0642-8.
4. "Viggo Brun". Store norske leksikon. Retrieved January 1, 2017.
5. Bent Birkeland. "Viggo Brun". Norsk biografisk leksikon. Retrieved January 1, 2017.
Other sources
• H. Halberstam and H. E. Richert, Sieve methods, Academic Press (1974) ISBN 0-12-318250-6. Gives an account of Brun's sieve.
• C.J. Scriba, Viggo Brun, Historia Mathematica 7 (1980) 1–6.
• C.J. Scriba, Zur Erinnerung an Viggo Brun, Mitt. Math. Ges. Hamburg 11 (1985) 271-290
External links
• Brun's Constant
• Brun's Pure Sieve
• Viggo Brun personal archive exists at NTN University Library Dorabiblioteket
Authority control
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| Wikipedia |
Vi Hart
Victoria Hart (born 1988),[2] commonly known as Vi Hart (/ˈvaɪ hɑːrt, ˈviː hɑːrt/),[3] is an American mathematician and YouTuber. They describe themself as a "recreational mathemusician" and are well-known for creating mathematical videos on YouTube[4][5][6] and popularizing mathematics.[7][8] Hart founded the virtual reality research group eleVR and has co-authored several research papers on computational geometry and the mathematics of paper folding.[9][10]
Vi Hart
Hart in 2012, sitting on top of a finished project
Born
Victoria Hart
1988 (age 34–35)
NationalityAmerican
Occupation(s)YouTube personality, educator, inventor
Known forMathematical/musical YouTube videos
YouTube information
Channel
• Vihart
Years active2009–present
Genres
• Education
• Music
Subscribers1.44 million[1]
Total views149 million[1]
Creator Awards
100,000 subscribers
1,000,000 subscribers
Last updated: 16 Jul 2023
Together with another YouTube mathematics popularizer, Matt Parker, Hart won the 2018 Communications Award of the Joint Policy Board for Mathematics for "entertaining, thought-provoking mathematics and music videos on YouTube that explain mathematical concepts through doodles".[11]
Early life and education
Hart is the child of mathematical sculptor George W. Hart, and received a degree in music at Stony Brook University.[4] Hart identifies as "gender agnostic";[12] in a video released in 2015, they spoke about their lack of gender identity—including lacking non-binary identities such as agender—and their attitude to gendered terms such as pronouns as a "linguistic game" that they were not interested in playing. They indicated that they have no preference and do not care which pronouns they are called by.[13]
Career
Hart's career as a mathematics popularizer began in 2010 with a video series about "doodling in math class". After these recreational mathematics videos — which introduced topics like fractal dimensions — grew popular, they were featured in The New York Times and on National Public Radio,[4][14] eventually gaining the support of the Khan Academy and making videos for the educational site as their "Resident Mathemusician".[15][16] Many of Hart's videos combined mathematics and music, such as "Twelve tones", which was called "deliriously and delightfully profound" by Salon.[17]
Together with Henry Segerman, Hart wrote "The Quaternion Group as a Symmetry Group", which was included in the anthology The Best Writing on Mathematics 2015.[18]
In 2014, Hart founded a research group called eleVR, with Emily Eifler and Andrea Hawksley, to research virtual reality (VR). The group created VR videos, and had also collaborated on educational computer games.[19][20][21][3][22] They created the game Hypernom, where the player has to eat part of 4 dimensional polytopes which are stereographically projected into 3D and viewed using a virtual reality headset.[23][24] In June, eleVR released an open source web video player that worked with the Oculus Rift.[25] In the same year Hart created the playable blog post Parable of the Polygons with Nicky Case. The game was based on economist Thomas Schelling's Dynamic Models of Segregation.[20][26] In May 2016, eleVR joined Y Combinator Research (YCR) as part of the Human Advancement Research Community (HARC) project[27], in which Hart was listed as a Principal Investigator.[28]
Hart is a Senior Research Project Manager at Microsoft.[29]
References
1. "About Vihart". YouTube.
2. "Khan Academy's mathemusician Vi Hart brings dull lessons to life". Wired. Retrieved January 27, 2016.
3. "FAQ". Vi Hart.com. Archived from the original on December 13, 2014. Retrieved December 12, 2014.
4. Chang, Kenneth (January 17, 2011), "Bending and Stretching Classroom Lessons to Make Math Inspire", The New York Times.
5. Bell, Melissa (December 17, 2010), "Making math magic: Vi Hart doodles her lessons", The Washington Post.
6. Krulwich, Robert (December 16, 2010), I Hate Math! (Not After This, You Won't), NPR
7. "Weird geometry: Art enters the hyperbolic realm". New Scientist. Retrieved January 4, 2023.
8. "Parable of the Polygons". Parable of the Polygons. Retrieved January 4, 2023.
9. Vi Hart at DBLP Bibliography Server . Retrieved March 29, 2014.
10. "Reshaping the Universe: VR Landscapes Explore Mind-Bending Geometry". Live Science. March 29, 2017.
11. "Vi Hart and Matt Parker to Receive 2018 JPBM Communications Awards", News, Events and Announcements, American Mathematical Society, December 8, 2017
12. Hart, Vi [@vihartvihart] (April 30, 2014). "Fun fact: I consider myself gender agnostic. "Person," not "Woman," please. I respect your religion, but don't like having it pushed on me" (Tweet). Archived from the original on March 5, 2016 – via Twitter.
13. Hart, Vi (June 8, 2015). On Gender (Online video). YouTube.
14. "I Hate Math! (Not After This, You Won't)". NPR.org. Retrieved November 12, 2016.
15. Khan Academy (January 3, 2012), Announcement, retrieved January 7, 2018
16. Gans, Joshua (January 24, 2012). "Learning on Speed". Harvard Business Review. Retrieved January 8, 2018.
17. Leonard, Andrew (June 28, 2013). "The mad genius of Vi Hart". Salon. Retrieved January 8, 2018.
18. Hart, Vi; Segerman, Henry (January 12, 2016). "The Quaternion Group as a Symmetry Group". In Pitici, Mircea (ed.). The Best Writing on Mathematics 2015. Princeton University Press. pp. 141–153. arXiv:1404.6596. Bibcode:2014arXiv1404.6596H. ISBN 9781400873371.
19. "About Us". eleVR. Retrieved December 12, 2014.
20. Case, Nicky; Hart, Vi. "Parable of the Polygons". Retrieved December 12, 2014.
21. Bhatia, Aatish (December 8, 2014). "Empirical Zeal How Small Biases Lead to a Divided World: An Interactive Exploration of Racial Segregation". Wired.
22. "Introducing eleVR – Vi Hart". vihart.com. Archived from the original on May 23, 2022. Retrieved November 28, 2017.
23. Lawson-Perfect, Christian (July 31, 2015). "Hypernom". The Aperiodical. Retrieved April 5, 2016.
24. Hart, Vi; Hawksley, Andrea; Segerman, Henry; Bosch, Marc ten (July 21, 2015). "Hypernom: Mapping VR Headset Orientation to S^3". Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture. pp. 387–390. arXiv:1507.05707. Bibcode:2015arXiv150705707H.
25. "eleVR: the first web video player for virtual reality".
26. Farokhmanesh, Megan (December 11, 2014). "A visual guide to bias, as explained by adorable shapes". Polygon.
27. "eleVR leaving YCR – elevr". elevr.com.
28. Altman, Sam (May 11, 2016). "HARC". Y Combinator Blog. Retrieved June 20, 2016.
29. Allen, Danielle (April 21, 2020). "Roadmap to Pandemic Resilience" (PDF). Edmond J. Safra Center for Ethics. Harvard University. Retrieved April 21, 2020.
External links
Wikimedia Commons has media related to Vi Hart.
• Vi Hart's channel on YouTube
• Vi Hart's second's channel on YouTube
• "Vi Hart". Khan Academy. Archived from the original on January 28, 2016.{{cite web}}: CS1 maint: unfit URL (link)
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| Wikipedia |
Vikram Bhagvandas Mehta
Vikram Bhagvandas Mehta (August 15, 1946 – June 4, 2014) was an Indian mathematician who worked on algebraic geometry and vector bundles. Together with Annamalai Ramanathan he introduced the notion of Frobenius split varieties, which led to the solution of several problems about Schubert varieties.[1] He is also known to have worked, from the 2000s onward, on the fundamental group scheme. It was precisely in the year 2002 when he and Subramanian published a proof of a conjecture by Madhav V. Nori[2] that brought back into the limelight the theory of an object that until then had met with little success.[3]
Awards
The Council of Scientific and Industrial Research awarded him the Shanti Swarup Bhatnagar Prize for Science and Technology in 1991 for his work in algebraic geometry.[4]
References
1. Bhagvandas Mehta, Vikram; Ramanathan, Annamalai (July 1985). "Frobenius splitting and cohomology vanishing for Schubert varieties". Annals of Mathematics. 122 (1): 27–40. doi:10.2307/1971368. ISSN 0003-486X. JSTOR 1971368 – via JSTOR.
2. M. V. Nori On the Representations of the Fundamental Group, Compositio Mathematica, Vol. 33, Fasc. 1, (1976), p. 29-42
3. V. B. Mehta, S. Subramanian On the Fundamental Group Scheme, Inventiones mathematicae, 148, 143-150 (2002)
4. "Awardee Details: Shanti Swarup Bhatnagar Prize". ssbprize.gov.in. Retrieved 19 October 2020.
External links
• Vikram Bhagvandas Mehta citation
Recipients of Shanti Swarup Bhatnagar Prize for Science and Technology in Mathematical Science
1950s–70s
• K. S. Chandrasekharan & C. R. Rao (1959)
• K. G. Ramanathan (1965)
• A. S. Gupta & C. S. Seshadri (1972)
• P. C. Jain & M. S. Narasimhan (1975)
• K. R. Parthasarathy & S. K. Trehan (1976)
• M. S. Raghunathan (1977)
• E. M. V. Krishnamurthy (1978)
• S. Raghavan & S. Ramanan (1979)
1980s
• R. Sridharan (1980)
• J. K. Ghosh (1981)
• B. L. S. Prakasa Rao & J. B. Shukla (1982)
• I. B. S. Passi & Phoolan Prasad (1983)
• S. K. Malik & R. Parthasarathy (1985)
• T. Parthasarathy & U. B. Tewari (1986)
• Raman Parimala & T. N. Shorey (1987)
• M. B. Banerjee & K. B. Sinha (1988)
• Gopal Prasad (1989)
1990s
• R. Balasubramanian & S. G. Dani (1990)
• V. B. Mehta & A. Ramanathan (1991)
• Maithili Sharan (1992)
• Karmeshu & Navin M. Singhi (1993)
• N. Mohan Kumar (1994)
• Rajendra Bhatia (1995)
• V. S. Sunder (1996)
• Subhashis Nag & T. R. Ramadas (1998)
• Rajeeva Laxman Karandikar (1999)
2000s
• Rahul Mukerjee (2000)
• Gadadhar Misra & T. N. Venkataramana (2001)
• Dipendra Prasad & S. Thangavelu (2002)
• Manindra Agrawal & V. Srinivas (2003)
• Arup Bose & Sujatha Ramdorai (2004)
• Probal Chaudhuri & K. H. Paranjape (2005)
• Vikraman Balaji & Indranil Biswas (2006)
• B. V. Rajarama Bhat (2007)
• Rama Govindarajan (2007)
• Jaikumar Radhakrishnan (2008)
• Suresh Venapally (2009)
2010s
• Mahan Mitra & Palash Sarkar (2011)
• Siva Athreya & Debashish Goswami (2012)
• Eknath Prabhakar Ghate (2013)
• Kaushal Kumar Verma (2014)
• K Sandeep & Ritabrata Munshi (2015)
• Amalendu Krishna (2016)
• Naveen Garg (2016)
• (Not awarded) (2017)
• Amit Kumar & Nitin Saxena (2018)
• Neena Gupta & Dishant Mayurbhai Pancholi (2019)
2020s
• Rajat Subhra Hazra (2020)
• U. K. Anandavardhanan (2020)
• Anish Ghosh (2021)
• Saket Saurabh (2021)
Authority control: Academics
• MathSciNet
• Mathematics Genealogy Project
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| Wikipedia |
Viktor Maslov (mathematician)
Viktor Pavlovich Maslov (Russian: Виктор Павлович Маслов; 15 June 1930 – 3 August 2023) was a Russian mathematical physicist. He was a member of the Russian Academy of Sciences. He obtained his doctorate in physico-mathematical sciences in 1957.[2] His main fields of interest were quantum theory, idempotent analysis, non-commutative analysis, superfluidity, superconductivity, and phase transitions. He was editor-in-chief of Mathematical Notes and Russian Journal of Mathematical Physics.
Viktor Maslov
Виктор Маслов
Born
Viktor Pavlovich Maslov
(1930-06-15)15 June 1930
Moscow, Russian SFSR, USSR
Died3 August 2023(2023-08-03) (aged 93)
NationalityRussian
CitizenshipRussian
Alma materLomonosov Moscow State University
Known forMaslov index
Spouse
Lê Vũ Anh
(m. 1975–1981)
AwardsState Prize of the USSR (1978); A.M.Lyapunov Gold Medal (USSR Academy of Science, 1982); Lenin Prize (1985); State Prize of the Russian Federation (1997); Demidov prize (2000); Independent Russian Triumph Prize (2002); State Prize of the Russian Federation (2013)[1]
Scientific career
Fieldsphysico-mathematics
InstitutionsLomonosov Moscow State University
Doctoral advisorSergei Fomin[2]
The Maslov index is named after him. He also introduced the concept of Lagrangian submanifold.[2]
Early life and career
Viktor Pavlovich Maslov was born in Moscow on 15 June 1930. He was the son of statistician Pavel Maslov and researcher Izolda Lukomskaya, and the grandson of the economist and agriculturalist Petr Maslov. At the beginning of World War II, he was evacuated to Kazan with his mother, grandmother and other members of his mother's family.[3]
In 1953 he graduated from the Physics Department of the Moscow State University and taught at the university. In 1957 he defended his Ph.D. thesis and in 1966, his doctoral dissertation. In 1984, he was elected an academician within Department of Mathematics of the Academy of Sciences of the USSR.[4]
From 1968 to 1998, he headed the Department of Applied Mathematics at the Moscow Institute of Electronics and Mathematics. From 1992 to 2016, he was in charge of the Department of Quantum Statistics and Field Theory of the Physics Faculty of Moscow State University.[4]
Maslov headed the laboratory of the mechanics of natural disasters at the Institute for Problems in Mechanics of the Russian Academy of Sciences. He was a research professor at the Department of Applied Mathematics at Moscow Institute of Electronics and Mathematics of Higher School of Economics.[4]
Scientific acitivity
Maslov was known as a prominent specialist in the field of mathematical physics, differential equations, functional analysis, mechanics and quantum physics. He developed asymptotic methods that are widely applied to equations arising in quantum mechanics, field theory, statistical physics and abstract mathematics, that bear his name.[5]
Maslov's asymptotic methods are closely related to such problems as the theory of a self-consistent field in quantum and classical statistics, superfluidity and superconductivity, quantization of solitons, quantum field theory in strong external fields and in curved space-time, the method of expansion in the inverse number of particle types. In 1983, he attended the International Congress of Mathematicians in Warsaw, where he presented a plenary report "Non-standard characteristics of asymptotic problems".[6]
Maslov dealt with the problems of liquid and gas, carried out fundamental research on the problems of magnetohydrodynamics related to the dynamo problem. He also made calculations for the emergency unit of the Chernobyl nuclear power plant during the 1986 disaster. In 1991, he made model and forecasts of the economic situation in Russia.[6]
From the early 1990s, he worked on the use of equations of mathematical physics in economics and financial analysis. In particular, he managed to predict the 1998 Russian financial crisis, and even earlier, the collapse of the economic and, as a consequence, the collapse of the political system of the USSR.[6]
In 2008, Maslov in his own words predicted a global recession in the late 2000s. He calculated the critical number of U.S. debt, and found out that a crisis should break out in the near future. In the calculations, he used equations similar to the equations of phase transition in physics. In the mid-1980s, Maslov introduced the term tropical mathematics, in which the operations of the conditional optimization problem were considered.[7]
Personal life
In the early 1970s, he met Lê Vũ Anh, the daughter of Lê Duẩn, then General Secretary of the Communist Party of Vietnam, when she was a student at the Faculty of Physics in Moscow State University. The romance was considered scandalous because Vietnamese students studying abroad were not allowed to have romantic relationships with foreigners and anyone caught would have to be disciplined and may be sent back to Vietnam. In order to avoid trouble, she returned home to marry a Vietnamese student from the same university and wanted to stay in Vietnam to forget her love affair with Maslov. However, she was forced by her father to return to USSR to complete her studies.[8]
When she and her husband returned to Moscow, Anh realized that she did not love her husband and could not forget her former lover. She decided to live separately from her husband and secretly went back and forth with Maslov. After being pregnant for the second time, after having a miscarriage for the first time, Anh had enough energy to ask her husband for a divorce in order to be able to marry Maslov. In 1975, she and Maslov married. She gave birth to a daughter on 31 October 1977 named Lena. Meeting her father by chance when he went to USSR for a state visit, Anh confessed all her love affairs. Lê Duẩn did not accept it and tried to lure her back to the country. However, Anh gradually reconciled with her family.[8]
After giving birth to her second daughter Tania, Anh gave birth to her son, Anton, on 1981. Anh died shortly after giving birth to her son, due to hemorrhage.[9]
Immediately after Anh died, a dispute over custody of his three children with his wife's family occurred. An official from the Vietnamese Communist Party's Central Committee took over the communication between Maslov and Anh's family. Both sides proposed a compromise solution, Maslov kept his daughters and son would be returned to Lê Duẩn. Maslov only allowed his son to go to Vietnam for two years. But after the deadline, his son never returned to him. Maslov had to fight for two more years before Lê Duẩn accepted to bring his grandson to meet his father.[10]
However, the son that Maslov met was no longer Anton Maslov as before, but a Vietnamese citizen with the new name Nguyễn An Hoàn and he was unable to speak Russian. According to Maslov, Lê Duẩn did not intend to return the child, but also hoped to bring back his daughters. Fearing the loss of his children, Maslov contacted the son of the President of the Supreme Soviet of the Soviet Union Andrei Gromyko, a close friend of Soviet leader Mikhail Gorbachev. He was advised to write to Gorbachev and was promised to convince Gorbachev to read it. After a massive legal struggle, Lê Duẩn gave up the idea of taking him and his children back.[9]
His children later resided in England and the Netherlands, where they were highly successful in their respective professions.[9]
Maslov later re-married a woman named Irina, who was at the same age as his ex-wife Anh. Irina is a linguist and she received the title of Associate Doctor of Science in 1991. For the last three decades, he lived in Troitsk.[9]
Viktor Maslow died on 3 August 2023, at the age of 93.[11]
Selected books
• Maslov, V. P. (1972). Théorie des perturbations et méthodes asymptotiques. Dunod; 384 pages{{cite book}}: CS1 maint: postscript (link)[12]
• Karasëv, M. V.; Maslov, V. P.: Nonlinear Poisson brackets. Geometry and quantization. Translated from the Russian by A. Sossinsky [A. B. Sosinskiĭ] and M. Shishkova. Translations of Mathematical Monographs, 119. American Mathematical Society, Providence, RI, 1993.[13]
• Kolokoltsov, Vassili N.; Maslov, Victor P.: Idempotent analysis and its applications. Translation of Idempotent analysis and its application in optimal control (Russian), "Nauka" Moscow, 1994. Translated by V. E. Nazaikinskii. With an appendix by Pierre Del Moral. Mathematics and its Applications, 401. Kluwer Academic Publishers Group, Dordrecht, 1997.
• Maslov, V. P.; Fedoriuk, M. V.: Semi-classical approximation in quantum mechanics. Translated from the Russian by J. Niederle and J. Tolar. Mathematical Physics and Applied Mathematics, 7. Contemporary Mathematics, 5. D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1981.[14]
This book was cited over 700 times at Google Scholar in 2011.
• Maslov, V. P. Operational methods. Translated from the Russian by V. Golo, N. Kulman and G. Voropaeva. Mir Publishers, Moscow, 1976.
References
1. Viktor Maslov, HSE
2. "Fiftieth Anniversary of research and teaching by Viktor Pavlovich Maslov" (PDF).
3. "The Central Database of Shoah Victims' Names". www.yvng.yadvashem.org. Retrieved 2021-07-29.{{cite web}}: CS1 maint: url-status (link)
4. "Маслов, Виктор Павлович". www.tass.ru. Retrieved 2021-07-29.{{cite web}}: CS1 maint: url-status (link)
5. Proceedings of the International Congress of Mathematicians. August 16-24, 1983, Warszawa
6. "Академику Маслову Виктору Павловичу - 90 лет!". www.ras.ru. 2020-06-15. Retrieved 2021-07-29.{{cite web}}: CS1 maint: url-status (link)
7. Medvedev, Yuri (2009-03-12). "Он рассчитал катастрофy". www.rg.ru. Retrieved 2021-07-29.{{cite web}}: CS1 maint: url-status (link)
8. "Về câu chuyện tình của con gái Tổng Bí thư Lê Duẩn với viện sĩ khoa học Nga". www.cand.com.vn. 2016-08-26. Retrieved 2021-07-29.{{cite web}}: CS1 maint: url-status (link)
9. "Hồi ký của VS Maslov về mối tình với Lê Vũ Anh". www.nguoivietodessa.com. 2016-08-28. Retrieved 2021-07-29.{{cite web}}: CS1 maint: url-status (link)
10. "Câu chuyện tình buồn bí mật của Lê Vũ Anh con gái ông Lê Duẩn lấy chồng người Nga". www.ttx.vanganh.org. Retrieved 2021-07-29.{{cite web}}: CS1 maint: url-status (link)
11. Умер Виктор Павлович Маслов (in Russian)
12. Streater, R. F. (1975). "Review of Théorie des perturbations et méthodes asymptotiques by V. P. Maslov". Bulletin of the London Mathematical Society. 7 (3): 334. doi:10.1112/blms/7.3.334. ISSN 0024-6093.
13. Libermann, P. (1996). "Book Review: Nonlinear Poisson brackets, geometry and quantization". Bulletin of the American Mathematical Society. 33: 101–106. doi:10.1090/S0273-0979-96-00619-2.
14. Blattner, Robert J.; Ralston, James (1983). "joint review of Lagrangian analysis and quantum mechanics, a mathematical structure related to asymptotic expansions and the Maslow index by Jean Leray; Semi-classical approximation in quantum mechanics by V. P. Maslow and M. V. Fedoriuk". Bulletin of the American Mathematical Society. 9 (3): 387–397. doi:10.1090/S0273-0979-1983-15224-2.
External links
• Viktor Maslov at the Mathematics Genealogy Project
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| Wikipedia |
Viktor Valentinovich Novozhilov
Viktor Valentinovich Novozhilov (Russian: Виктор Валентинович Новожилов) (27 October [O.S. 15 October] 1892 – 15 August 1970) was a Soviet economist and mathematician, known for his development of techniques for the mathematical analysis of economic phenomena. He was awarded the Lenin Prize (1965) and served as head of the Laboratory for Economic Assessment Systems at the Leningrad office of the Central Economic Mathematical Institute.
Viktor Valentinovich Novozhilov
Born(1892-10-27)27 October 1892
Kharkov, Kharkov Governorate, Russian Empire (now Ukraine)
Died15 August 1970(1970-08-15) (aged 77)
Leningrad, Soviet Union
Alma materSt. Volodymyr Kyiv University
Scientific career
FieldsEconomics
Biography
Novozhilov graduated from high school with a gold medal in 1911 and entered the St. Volodymyr Kyiv University, which he completed in 1915, becoming an assistant professor in political economics and statistics. In 1922, he moved to the Leningrad Polytechnic Institute, where he served as the head of the department of Auto Industry Economics from 1938 to 1951. From 1951 to 1966, Novozhilov was the head of the Statistics Department at the Leningrad Engineering and Economics Institute.
He was part of the government-sponsored team engaged in economic reform analysis in the 1920s in the Soviet Union. He performed extensive research in the field of economic analysis for agriculture and made specific recommendations regarding optimal investment levels in a socialist agricultural setting.
External links
• Paper on Novozhilov's contribution
• Alternative encyclopedic article
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| Wikipedia |
William Feller
William "Vilim" Feller (July 7, 1906 – January 14, 1970), born Vilibald Srećko Feller, was a Croatian–American mathematician specializing in probability theory.
William Feller
Born
Vilibald Srećko Feller
July 7, 1906 (1906-07-07)
Zagreb, Austro-Hungarian Monarchy (now Croatia)
DiedJanuary 14, 1970 (1970-01-15) (aged 63)
New York City, US
NationalityCroatian–American
Alma materUniversity of Zagreb
University of Göttingen
Known forFeller process
Feller's coin-tossing constants
Feller-continuous process
Feller's paradox
Feller's theorem
Feller–Pareto distribution
Feller–Tornier constant
Feller–Miyadera–Phillips theorem
Proof by intimidation
Stars and bars
AwardsNational Medal of Science (USA) in Mathematical, Statistical, and Computational Sciences (1969)
Scientific career
FieldsMathematician
InstitutionsUniversity of Kiel
University of Copenhagen
University of Stockholm
University of Lund
Brown University
Cornell University
Princeton University
Doctoral advisorRichard Courant
Doctoral studentsPatrick Billingsley
George Forsythe
Robert Kurtz
Henry McKean
Lawrence Shepp
Hale Trotter
Benjamin Weiss
David A. Freedman
InfluencesStanko Vlögel
Signature
Early life and education
Feller was born in Zagreb to Ida Oemichen-Perc, a Croatian–Austrian Catholic, and Eugen Viktor Feller, son of a Polish–Jewish father (David Feller) and an Austrian mother (Elsa Holzer).[1]
Eugen Feller was a famous chemist and created Elsa fluid named after his mother. According to Gian-Carlo Rota, Eugen Feller's surname was a "Slavic tongue twister", which William changed at the age of twenty.[2] This claim appears to be false. His forename, Vilibald, was chosen by his Catholic mother for the saint day of his birthday.[3]
Work
Feller held a docent position at the University of Kiel beginning in 1928. Because he refused to sign a Nazi oath,[4] he fled the Nazis and went to Copenhagen, Denmark in 1933. He also lectured in Sweden (Stockholm and Lund).[5] As a refugee in Sweden, Feller reported being troubled by increasing fascism at the universities. He reported that the mathematician Torsten Carleman would offer his opinion that Jews and foreigners should be executed.[6]
Finally, in 1939 he arrived in the U.S., where he became a citizen in 1944 and was on the faculty at Brown and Cornell. In 1950 he became a professor at Princeton University.
The works of Feller are contained in 104 papers and two books on a variety of topics such as mathematical analysis, theory of measurement, functional analysis, geometry, and differential equations in addition to his work in mathematical statistics and probability.
Feller was one of the greatest probabilists of the twentieth century. He is remembered for his championing of probability theory as a branch of mathematical analysis in Sweden and the United States. In the middle of the 20th century, probability theory was popular in France and Russia, while mathematical statistics was more popular in the United Kingdom and the United States, according to the Swedish statistician, Harald Cramér.[7] His two-volume textbook on probability theory and its applications was called "the most successful treatise on probability ever written" by Gian-Carlo Rota.[8] By stimulating his colleagues and students in Sweden and then in the United States, Feller helped establish research groups studying the analytic theory of probability. In his research, Feller contributed to the study of the relationship between Markov chains and differential equations, where his theory of generators of one-parameter semigroups of stochastic processes gave rise to the theory of "Feller operators".
Results
Numerous topics relating to probability are named after him, including Feller processes, Feller's explosion test, Feller–Brown movement, and the Lindeberg–Feller theorem. Feller made fundamental contributions to renewal theory, Tauberian theorems, random walks, diffusion processes, and the law of the iterated logarithm. Feller was among those early editors who launched the journal Mathematical Reviews.
Notable books
• An Introduction to Probability Theory and its Applications, Volume I, 3rd edition (1968); 1st edn. (1950);[9] 2nd edn. (1957)[10]
• An Introduction to Probability Theory and its Applications, Volume II, 2nd edition (1971)
Recognition
In 1949, Feller was named a Fellow of the American Statistical Association.[11] He was elected to the American Academy of Arts and Sciences in 1958, the United States National Academy of Sciences in 1960, and the American Philosophical Society in 1966.[12][13][14] Feller won the National Medal of Science in 1969. He was president of the Institute of Mathematical Statistics.
See also
• Feller condition
• Beta distribution
• Compound Poisson distribution
• Gillespie algorithm
• Kolmogorov equations
• Poisson point process
• Stability (probability)
• St. Petersburg paradox
• Stochastic process
References
1. Zubrinic, Darko (2006). "William Feller (1906-1970)". Croatianhistory.net. Accessed 3 July 2018.
2. Rota, Gian-Carlo (1996). Indiscrete Thoughts. Birkhäuser. ISBN 0-8176-3866-0.
3. O'Connor, John J.; Robertson, Edmund F., "William Feller", MacTutor History of Mathematics Archive, University of St Andrews
4. "Biography of William Feller". History of William Feller. Retrieved 2006-06-27.
5. Siegmund-Schultze, Reinhard (2009). Mathematicians fleeing from Nazi Germany: Individual fates and global impact. Princeton, New Jersey: Princeton University Press. pp. xxviii+471. ISBN 978-0-691-14041-4. MR 2522825.
6. (Siegmund-Schultze 2009, p. 135)
7. Preface to his Mathematical Methods of Statistics.
8. Page 199: Indiscrete Thoughts.
9. Wolfowitz, J. (1951). "Review: An introduction to probability theory and its applications, Vol. I, 1st ed., by W. Feller" (PDF). Bull. Amer. Math. Soc. 57 (2): 156–159. doi:10.1090/s0002-9904-1951-09491-4.
10. "Review: An introduction to probability theory and its applications, Vol. I, 2nd ed., by W. Feller" (PDF). Bull. Amer. Math. Soc. 64 (6): 393. 1958. doi:10.1090/s0002-9904-1958-10252-9.
11. "View/Search Fellows of the ASA". American Statistical Association. Retrieved 2016-07-22.
12. "William Feller". American Academy of Arts & Sciences. Retrieved 2022-09-29.
13. "William Feller". www.nasonline.org. Retrieved 2022-09-29.
14. "APS Member History". search.amphilsoc.org. Retrieved 2022-09-29.
External links
Wikiquote has quotations related to William Feller.
• William Feller at the Mathematics Genealogy Project
• A biographical memoir by Murray Rosenblatt
• Croatian Giants of Science - in Croatian
• O'Connor, John J.; Robertson, Edmund F., "William Feller", MacTutor History of Mathematics Archive, University of St Andrews
• "Fine Hall in its golden age: Remembrances of Princeton in the early fifties" by Gian-Carlo Rota. Contains a section on Feller at Princeton.
• Feller Matriculation Form giving personal details
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| Wikipedia |
Villarceau circles
In geometry, Villarceau circles (/viːlɑːrˈsoʊ/) are a pair of circles produced by cutting a torus obliquely through the center at a special angle.
Given an arbitrary point on a torus, four circles can be drawn through it. One is in a plane parallel to the equatorial plane of the torus and another perpendicular to that plane (these are analogous to lines of latitude and longitude on the Earth). The other two are Villarceau circles. They are obtained as the intersection of the torus with a plane that passes through the center of the torus and touches it tangentially at two antipodal points. If one considers all these planes, one obtains two families of circles on the torus. Each of these families consists of disjoint circles that cover each point of the torus exactly once and thus forms a 1-dimensional foliation of the torus.
The Villarceau circles are named after the French astronomer and mathematician Yvon Villarceau (1813–1883) who wrote about them in 1848.
Mannheim (1903) showed that the Villarceau circles meet all of the parallel circular cross-sections of the torus at the same angle, a result that he said a Colonel Schoelcher had presented at a congress in 1891.
Example
Consider a horizontal torus in xyz space, centered at the origin and with major radius 5 and minor radius 3. That means that the torus is the locus of some vertical circles of radius three whose centers are on a circle of radius five in the horizontal xy plane. Points on this torus satisfy this equation:
$0=(x^{2}+y^{2}+z^{2}+16)^{2}-100(x^{2}+y^{2}).\,\!$
Slicing with the z = 0 plane produces two concentric circles, x2 + y2 = 22 and x2 + y2 = 82, the outer and inner equator. Slicing with the x = 0 plane produces two side-by-side circles, (y − 5)2 + z2 = 32 and (y + 5)2 + z2 = 32.
Two example Villarceau circles can be produced by slicing with the plane 3x = 4z. One is centered at (0, +3, 0) and the other at (0, −3, 0); both have radius five. They can be written in parametric form as
$(x,y,z)=(4\cos \vartheta ,+3+5\sin \vartheta ,3\cos \vartheta )\,\!$
and
$(x,y,z)=(4\cos \vartheta ,-3+5\sin \vartheta ,3\cos \vartheta ).\,\!$
The slicing plane is chosen to be tangent to the torus at two points while passing through its center. It is tangent at (16⁄5, 0, 12⁄5) and at (−16⁄5, 0, −12⁄5). The angle of slicing is uniquely determined by the dimensions of the chosen torus. Rotating any one such plane around the z-axis gives all of the Villarceau circles for that torus.
Existence and equations
A proof of the circles’ existence can be constructed from the fact that the slicing plane is tangent to the torus at two points. One characterization of a torus is that it is a surface of revolution. Without loss of generality, choose a coordinate system so that the axis of revolution is the z axis. Begin with a circle of radius r in the xz plane, centered at (R, 0, 0).
$0=(x-R)^{2}+z^{2}-r^{2}\,\!$
Sweeping replaces x by (x2 + y2)1/2, and clearing the square root produces a quartic equation.
$0=(x^{2}+y^{2}+z^{2}+R^{2}-r^{2})^{2}-4R^{2}(x^{2}+y^{2}).\,\!$
The cross-section of the swept surface in the xz plane now includes a second circle.
$0=(x+R)^{2}+z^{2}-r^{2}\,\!$
This pair of circles has two common internal tangent lines, with slope at the origin found from the right triangle with hypotenuse R and opposite side r (which has its right angle at the point of tangency). Thus z/x equals ±r / (R2 − r2)1/2, and choosing the plus sign produces the equation of a plane bitangent to the torus.
$0=xr-z{\sqrt {R^{2}-r^{2}}}\,\!$
By symmetry, rotations of this plane around the z axis give all the bitangent planes through the center. (There are also horizontal planes tangent to the top and bottom of the torus, each of which gives a “double circle”, but not Villarceau circles.)
$0=xr\cos \varphi +yr\sin \varphi -z{\sqrt {R^{2}-r^{2}}}\,\!$
We can calculate the intersection of the plane(s) with the torus analytically, and thus show that the result is a symmetric pair of circles, one of which is a circle of radius R centered at
$(-r\sin \varphi ,r\cos \varphi ,0).\,\!$
A treatment along these lines can be found in Coxeter (1969).
A more abstract — and more flexible — approach was described by Hirsch (2002), using algebraic geometry in a projective setting. In the homogeneous quartic equation for the torus,
$0=(x^{2}+y^{2}+z^{2}+R^{2}w^{2}-r^{2}w^{2})^{2}-4R^{2}w^{2}(x^{2}+y^{2}),\,\!$
setting w to zero gives the intersection with the “plane at infinity”, and reduces the equation to
$0=(x^{2}+y^{2}+z^{2})^{2}.\,\!$
This intersection is a double point, in fact a double point counted twice. Furthermore, it is included in every bitangent plane. The two points of tangency are also double points. Thus the intersection curve, which theory says must be a quartic, contains four double points. But we also know that a quartic with more than three double points must factor (it cannot be irreducible), and by symmetry the factors must be two congruent conics. Hirsch extends this argument to any surface of revolution generated by a conic, and shows that intersection with a bitangent plane must produce two conics of the same type as the generator when the intersection curve is real.
Filling space
The torus plays a central role in the Hopf fibration of the 3-sphere, S3, over the ordinary sphere, S2, which has circles, S1, as fibers. When the 3-sphere is mapped to Euclidean 3-space by stereographic projection, the inverse image of a circle of latitude on S2 under the fiber map is a torus, and the fibers themselves are Villarceau circles.[1] Banchoff has explored such a torus with computer graphics imagery. One of the unusual facts about the circles is that each links through all the others, not just in its own torus but in the collection filling all of space; Berger has a discussion and drawing.[2]
See also
• Toric section
• Vesica piscis
Citations
1. Dorst 2019, §6. Hopf Fibration and Stereographic Projection from 4D.
2. Berger 1987, pp. 304–305, §18.9: Villarceau circles and parataxy.
References
• Banchoff, Thomas F. (1990). Beyond the Third Dimension. Scientific American Library. ISBN 978-0-7167-5025-3.
• Berger, Marcel (1987). Geometry II. Springer. ISBN 978-3-540-17015-0.
• Coxeter, H. S. M. (1969). Introduction to Geometry (2/e ed.). Wiley. pp. 132–133. ISBN 978-0-471-50458-0.
• Hirsch, Anton (2002). "Extension of the 'Villarceau-Section' to Surfaces of Revolution with a Generating Conic". Journal for Geometry and Graphics. Lemgo, Germany: Heldermann Verlag. 6 (2): 121–132. ISSN 1433-8157.
• Mannheim, M. A. (1903). "Sur le théorème de Schoelcher". Nouvelles Annales de Mathématiques. Paris: Carilian-Gœury et Vor. Dalmont. 4th series, volume 3: 105–107.
• Stachel, Hellmuth (2002). "Remarks on A. Hirsch's Paper concerning Villarceau Sections". Journal for Geometry and Graphics. Lemgo, Germany: Heldermann Verlag. 6 (2): 133–139. ISSN 1433-8157.
• Yvon Villarceau, Antoine Joseph François (1848). "Théorème sur le tore". Nouvelles Annales de Mathématiques. Série 1. Paris: Gauthier-Villars. 7: 345–347. OCLC: 2449182.
• Dorst, Leo (2019). "Conformal Villarceau Rotors". Advances in Applied Clifford Algebras. 29 (44). doi:10.1007/s00006-019-0960-5. S2CID 253592159.
External links
Wikimedia Commons has media related to Villarceau circles.
• Weisstein, Eric W. "Villarceau Circles". MathWorld.
• Flat Torus in the Three-Sphere
• (in French) The circles of the torus (Les cercles du tore)
| Wikipedia |
Ville's inequality
In probability theory, Ville's inequality provides an upper bound on the probability that a supermartingale exceeds a certain value. The inequality is named after Jean Ville, who proved it in 1939.[1][2][3][4] The inequality has applications in statistical testing.
Statement
Let $X_{0},X_{1},X_{2},\dots $ be a non-negative supermartingale. Then, for any real number $a>0,$
$\operatorname {P} \left[\sup _{n\geq 0}X_{n}\geq a\right]\leq {\frac {\operatorname {E} [X_{0}]}{a}}\ .$
The inequality is a generalization of Markov's inequality.
References
1. Ville, Jean (1939). Etude Critique de la Notion de Collectif (PDF) (Thesis).
2. Durrett, Rick (2019). Probability Theory and Examples (Fifth ed.). Exercise 4.8.2: Cambridge University Press.{{cite book}}: CS1 maint: location (link)
3. Howard, Steven R. (2019). Sequential and Adaptive Inference Based on Martingale Concentration (Thesis).
4. Choi, K. P. (1988). "Some sharp inequalities for Martingale transforms". Transactions of the American Mathematical Society. 307 (1): 279–300. doi:10.1090/S0002-9947-1988-0936817-3. S2CID 121892687.
| Wikipedia |
Vilma Mesa
Vilma María Mesa Narváez (born 1963)[1] is a Colombian-American mathematics educator whose research topics have included secondary-school curriculum development, college-level calculus instruction, mathematics in community colleges, international perspectives in mathematics education, and inquiry-based learning.[2] She is a professor of education and mathematics at the University of Michigan, where she is affiliated with the Center for the Study of Higher and Post-secondary Education.[3]
Education and career
Mesa earned her bachelor's degrees in computer science and mathematics at the University of Los Andes (Colombia) in 1986 and 1987, respectively,[4] and became a computer programmer for the Colombian government and in the private sector.[3] From 1988 to 1995 she worked as a researcher at the University of Los Andes,[4] working in mathematics education and authoring textbooks on mathematics and statistics for applications including engineering and social sciences.[3]
In 1996, she began graduate study in mathematics education at the University of Georgia, where she earned her master's degree in 1996 and completed her Ph.D. in 2000.[4] Her dissertation, Conceptions of Function Promoted by Seventh- and Eighth-Grade Textbooks from Eighteen Countries, was jointly advised by Jeremy Kilpatrick and Edward Arthur Azoff.[5]
After postdoctoral research at the University of Michigan, she stayed on at the University of Michigan as a coordinator for the master's program in curriculum development and as an instructional consultant until she was hired in 2005 as an assistant professor of mathematics education in the School of Education. She was tenured in 2014 and added a joint appointment in the university's mathematics department in 2015.[4]
In 2016, she visited the University of Santiago, Chile as a Fulbright Scholar.[4][3]
Recognition
Mesa is the 2022 winner of the Louise Hay Award for Contributions to Mathematics Education, where she was recognised "for her distinguished contributions to mathematics education research at the collegiate level, for her teaching and mentorship, and as an advocate for access to mathematics for women and members of underprivileged populations."[2]
References
1. Birth year from WorldCat identities, retrieved 2022-02-02
2. "Louise Hay Award, 2022 Winner: Vilma Mesa", Awards, Association for Women in Mathematics, 1 February 2022, retrieved 2022-02-02
3. "Vilma Mesa", Calendar 2019, Lathisms, retrieved 2022-02-02
4. Curriculum vitae, retrieved 2022-02-02
5. Vilma Mesa at the Mathematics Genealogy Project
External links
• Home page
• Vilma Mesa publications indexed by Google Scholar
Authority control: Academics
• Mathematics Genealogy Project
| Wikipedia |
Vincent's theorem
In mathematics, Vincent's theorem—named after Alexandre Joseph Hidulphe Vincent—is a theorem that isolates the real roots of polynomials with rational coefficients.
Even though Vincent's theorem is the basis of the fastest method for the isolation of the real roots of polynomials, it was almost totally forgotten, having been overshadowed by Sturm's theorem; consequently, it does not appear in any of the classical books on the theory of equations (of the 20th century), except for Uspensky's book. Two variants of this theorem are presented, along with several (continued fractions and bisection) real root isolation methods derived from them.
Sign variation
Let c0, c1, c2, ... be a finite or infinite sequence of real numbers. Suppose l < r and the following conditions hold:
1. If r = l+1 the numbers cl and cr have opposite signs.
2. If r ≥ l+2 the numbers cl+1, ..., cr−1 are all zero and the numbers cl and cr have opposite signs.
This is called a sign variation or sign change between the numbers cl and cr.
When dealing with the polynomial p(x) in one variable, one defines the number of sign variations of p(x) as the number of sign variations in the sequence of its coefficients.
Two versions of this theorem are presented: the continued fractions version due to Vincent,[1][2][3] and the bisection version due to Alesina and Galuzzi.[4][5]
Vincent's theorem: Continued fractions version (1834 and 1836)
If in a polynomial equation with rational coefficients and without multiple roots, one makes successive transformations of the form
$x=a_{1}+{\frac {1}{x'}},\quad x'=a_{2}+{\frac {1}{x''}},\quad x''=a_{3}+{\frac {1}{x'''}},\ldots $
where $a_{1},a_{2},a_{3},\ldots $ are any positive numbers greater than or equal to one, then after a number of such transformations, the resulting transformed equation either has zero sign variations or it has a single sign variation. In the first case there is no root, whereas in the second case there is a single positive real root. Furthermore, the corresponding root of the proposed equation is approximated by the finite continued fraction:[1][2][3]
$a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+{\cfrac {1}{\ddots }}}}}}$
Moreover, if infinitely many numbers $a_{1},a_{2},a_{3},\ldots $ satisfying this property can be found, then the root is represented by the (infinite) corresponding continued fraction.
The above statement is an exact translation of the theorem found in Vincent's original papers;[1][2][3] however, the following remarks are needed for a clearer understanding:
• If $f_{n}(x)$ denotes the polynomial obtained after n substitutions (and after removing the denominator), then there exists N such that for all $n\geq N$ either $f_{n}(x)$ has no sign variation or it has one sign variation. In the latter case $f_{n}(x)$ has a single positive real root for all $n\geq N$.
• The continued fraction represents a positive root of the original equation, and the original equation may have more than one positive root. Moreover, assuming $a_{1}\geq 1$, we can only obtain a root of the original equation that is > 1. To obtain an arbitrary positive root we need to assume that $a_{1}\geq 0$.
• Negative roots are obtained by replacing x by −x, in which case the negative roots become positive.
Vincent's theorem: Bisection version (Alesina and Galuzzi 2000)
Let p(x) be a real polynomial of degree deg(p) that has only simple roots. It is possible to determine a positive quantity δ so that for every pair of positive real numbers a, b with $|b-a|<\delta $, every transformed polynomial of the form
$f(x)=(1+x)^{\deg(p)}p\left({\frac {a+bx}{1+x}}\right)$
(1)
has exactly 0 or 1 sign variations. The second case is possible if and only if p(x) has a single root within (a, b).
The Alesina–Galuzzi "a_b roots test"
From equation (1) the following criterion is obtained for determining whether a polynomial has any roots in the interval (a, b):
Perform on p(x) the substitution
$x\leftarrow {\frac {a+bx}{1+x}}$
and count the number of sign variations in the sequence of coefficients of the transformed polynomial; this number gives an upper bound on the number of real roots p(x) has inside the open interval (a, b). More precisely, the number ρab(p) of real roots in the open interval (a, b)—multiplicities counted—of the polynomial p(x) in R[x], of degree deg(p), is bounded above by the number of sign variations varab(p), where
$\operatorname {var} _{ab}(p)=\operatorname {var} \left((1+x)^{\deg(p)}p\left({\frac {a+bx}{1+x}}\right)\right),$
$\operatorname {var} _{ab}(p)=\operatorname {var} _{ba}(p)\geq \rho _{ab}(p).$
As in the case of Descartes' rule of signs if varab(p) = 0 it follows that ρab(p) = 0 and if varab(p) = 1 it follows that ρab(p) = 1.
A special case of the Alesina–Galuzzi "a_b roots test" is Budan's "0_1 roots test".
Sketch of a proof
A detailed discussion of Vincent's theorem, its extension, the geometrical interpretation of the transformations involved and three different proofs can be found in the work by Alesina and Galuzzi.[4][5] A fourth proof is due to Ostrowski[6] who rediscovered a special case of a theorem stated by Obreschkoff,[7] p. 81, in 1920–1923.
To prove (both versions of) Vincent's theorem Alesina and Galuzzi show that after a series of transformations mentioned in the theorem, a polynomial with one positive root eventually has one sign variation. To show this, they use the following corollary to the theorem by Obreschkoff of 1920–1923 mentioned earlier; that is, the following corollary gives the necessary conditions under which a polynomial with one positive root has exactly one sign variation in the sequence of its coefficients; see also the corresponding figure.
Corollary. (Obreschkoff's cone or sector theorem, 1920–1923[7] p. 81): If a real polynomial has one simple root x0, and all other (possibly multiple) roots lie in the sector
$S_{\sqrt {3}}=\left\{x=-\alpha +i\beta \ :\ |\beta |\leq {\sqrt {3}}|\alpha |,\alpha >0\right\}$ :\ |\beta |\leq {\sqrt {3}}|\alpha |,\alpha >0\right\}}
then the sequence of its coefficients has exactly one sign variation.
Consider now the Möbius transformation
$M(x)={\frac {ax+b}{cx+d}},\qquad a,b,c,d\in \mathbb {Z} _{>0}$
and the three circles shown in the corresponding figure; assume that a/c < b/d.
• The (yellow) circle
$\left|x-{\tfrac {1}{2}}\left({\tfrac {a}{c}}+{\tfrac {b}{d}}\right)\right|={\tfrac {1}{2}}\left({\tfrac {b}{d}}-{\tfrac {a}{c}}\right)$
whose diameter lies on the real axis, with endpoints a/c and b/d, is mapped by the inverse Möbius transformation
$M^{-1}(x)={\frac {dx-b}{-cx+a}}$
onto the imaginary axis. For example the point
${\tfrac {1}{2}}\left({\tfrac {a}{c}}+{\tfrac {b}{d}}\right)+{\tfrac {i}{2}}\left({\tfrac {b}{d}}-{\tfrac {a}{c}}\right)$
gets mapped onto the point −i d/c. The exterior points get mapped onto the half-plane with Re(x) < 0.
• The two circles (only their blue crescents are visible) with center
${\tfrac {1}{2}}\left({\tfrac {a}{c}}+{\tfrac {b}{d}}\right)\pm {\tfrac {i}{2{\sqrt {3}}}}\left({\tfrac {b}{d}}-{\tfrac {a}{c}}\right)$
and radius
${\tfrac {1}{\sqrt {3}}}\left({\tfrac {b}{d}}-{\tfrac {a}{c}}\right)$
are mapped by the inverse Möbius transformation
$M^{-1}(x)={\frac {dx-b}{-cx+a}}$
onto the lines Im(x) = ±√3 Re(x). For example the point
${\tfrac {1}{2}}\left({\tfrac {a}{c}}+{\tfrac {b}{d}}\right)-{\tfrac {3i}{2{\sqrt {3}}}}\left({\tfrac {b}{d}}-{\tfrac {a}{c}}\right)$
gets mapped to the point
${\tfrac {-d}{2c}}\left(1-i{\sqrt {3}}\right).$
The exterior points (those outside the eight-shaped figure) get mapped onto the $S_{\sqrt {3}}$ sector.
From the above it becomes obvious that if a polynomial has a single positive root inside the eight-shaped figure and all other roots are outside of it, it presents one sign variation in the sequence of its coefficients. This also guarantees the termination of the process.
Historical background
Early applications of Vincent's theorem
In his fundamental papers,[1][2][3] Vincent presented examples that show precisely how to use his theorem to isolate real roots of polynomials with continued fractions. However the resulting method had exponential computing time, a fact that mathematicians must have realized then, as was realized by Uspensky[8] p. 136, a century later.
The exponential nature of Vincent's algorithm is due to the way the partial quotients ai (in Vincent's theorem) are computed. That is, to compute each partial quotient ai (that is, to locate where the roots lie on the x-axis) Vincent uses Budan's theorem as a "no roots test"; in other words, to find the integer part of a root Vincent performs successive substitutions of the form x ← x+1 and stops only when the polynomials p(x) and p(x+1) differ in the number of sign variations in the sequence of their coefficients (i.e. when the number of sign variations of p(x+1) is decreased).
See the corresponding diagram where the root lies in the interval (5, 6). It can be easily inferred that, if the root is far away from the origin, it takes a lot of time to find its integer part this way, hence the exponential nature of Vincent's method. Below there is an explanation of how this drawback is overcome.
Disappearance of Vincent's theorem
Vincent was the last author in the 19th century to use his theorem for the isolation of the real roots of a polynomial.
The reason for that was the appearance of Sturm's theorem in 1827, which solved the real root isolation problem in polynomial time, by defining the precise number of real roots a polynomial has in a real open interval (a, b). The resulting (Sturm's) method for computing the real roots of polynomials has been the only one widely known and used ever since—up to about 1980, when it was replaced (in almost all computer algebra systems) by methods derived from Vincent's theorem, the fastest one being the Vincent–Akritas–Strzeboński (VAS) method.[9]
Serret included in his Algebra,[10] pp 363–368, Vincent's theorem along with its proof and directed all interested readers to Vincent's papers for examples on how it is used. Serret was the last author to mention Vincent's theorem in the 19th century.
Comeback of Vincent's theorem
In the 20th century Vincent's theorem cannot be found in any of the theory of equations books; the only exceptions are the books by Uspensky[8] and Obreschkoff,[7] where in the second there is just the statement of the theorem.
It was in Uspensky's book[8] that Akritas found Vincent's theorem and made it the topic of his Ph.D. Thesis "Vincent's Theorem in Algebraic Manipulation", North Carolina State University, USA, 1978. A major achievement at the time was getting hold of Vincent's original paper of 1836, something that had eluded Uspensky—resulting thus in a great misunderstanding. Vincent's original paper of 1836 was made available to Akritas through the commendable efforts (interlibrary loan) of a librarian in the Library of the University of Wisconsin–Madison, USA.
Real root isolation methods derived from Vincent's theorem
Isolation of the real roots of a polynomial is the process of finding open disjoint intervals such that each contains exactly one real root and every real root is contained in some interval. According to the French school of mathematics of the 19th century, this is the first step in computing the real roots, the second being their approximation to any degree of accuracy; moreover, the focus is on the positive roots, because to isolate the negative roots of the polynomial p(x) replace x by −x (x ← −x) and repeat the process.
The continued fractions version of Vincent's theorem can be used to isolate the positive roots of a given polynomial p(x) of degree deg(p). To see this, represent by the Möbius transformation
$M(x)={\frac {ax+b}{cx+d}},\qquad a,b,c,d\in \mathbb {N} $
the continued fraction that leads to a transformed polynomial
$f(x)=(cx+d)^{\deg(p)}p\left({\frac {ax+b}{cx+d}}\right)$
(2)
with one sign variation in the sequence of its coefficients. Then, the single positive root of f(x) (in the interval (0, ∞)) corresponds to that positive root of p(x) that is in the open interval with endpoints ${\frac {b}{d}}$ and ${\frac {a}{c}}$. These endpoints are not ordered and correspond to M(0) and M(∞) respectively.
Therefore, to isolate the positive roots of a polynomial, all that must be done is to compute—for each root—the variables a, b, c, d of the corresponding Möbius transformation
$M(x)={\frac {ax+b}{cx+d}}$
that leads to a transformed polynomial as in equation (2), with one sign variation in the sequence of its coefficients.
Crucial Observation: The variables a, b, c, d of a Möbius transformation
$M(x)={\frac {ax+b}{cx+d}}$
(in Vincent's theorem) leading to a transformed polynomial—as in equation (2)—with one sign variation in the sequence of its coefficients can be computed:
• either by continued fractions, leading to the Vincent–Akritas–Strzebonski (VAS) continued fractions method,[9]
• or by bisection, leading to (among others) the Vincent–Collins–Akritas (VCA) bisection method.[11]
The "bisection part" of this all important observation appeared as a special theorem in the papers by Alesina and Galuzzi.[4][5]
All methods described below (see the article on Budan's theorem for their historical background) need to compute (once) an upper bound, ub, on the values of the positive roots of the polynomial under consideration. Exception is the VAS method where additionally lower bounds, lb, must be computed at almost every cycle of the main loop. To compute the lower bound lb of the polynomial p(x) compute the upper bound ub of the polynomial $x^{\deg(p)}p\left({\frac {1}{x}}\right)$ and set $lb={\frac {1}{ub}}$.
Excellent (upper and lower) bounds on the values of just the positive roots of polynomials have been developed by Akritas, Strzeboński and Vigklas based on previous work by Doru Stefanescu. They are described in P. S. Vigklas' Ph.D. Thesis[12] and elsewhere.[13] These bounds have already been implemented in the computer algebra systems Mathematica, SageMath, SymPy, Xcas etc.
All three methods described below follow the excellent presentation of François Boulier,[14] p. 24.
Continued fractions method
Only one continued fractions method derives from Vincent's theorem. As stated above, it started in the 1830s when Vincent presented, in the papers[1][2][3] several examples that show how to use his theorem to isolate the real roots of polynomials with continued fractions. However the resulting method had exponential computing time. Below is an explanation of how this method evolved.
Vincent–Akritas–Strzeboński (VAS, 2005)
This is the second method (after VCA) developed to handle the exponential behavior of Vincent's method.
The VAS continued fractions method is a direct implementation of Vincent's theorem. It was originally presented by Vincent from 1834 to 1938 in the papers [1][2][3] in a exponential form; namely, Vincent computed each partial quotient ai by a series of unit increments ai ← ai + 1, which are equivalent to substitutions of the form x ← x + 1.
Vincent's method was converted into its polynomial complexity form by Akritas, who in his 1978 Ph.D. Thesis (Vincent's theorem in algebraic manipulation, North Carolina State University, USA) computed each partial quotient ai as the lower bound, lb, on the values of the positive roots of a polynomial. This is called the ideal positive lower root bound that computes the integer part of the smallest positive root (see the corresponding figure). To wit, now set ai ← lb or, equivalently, perform the substitution x ← x + lb, which takes about the same time as the substitution x ← x + 1.
Finally, since the ideal positive lower root bound does not exist, Strzeboński[15] introduced in 2005 the substitution $x\leftarrow lb_{computed}*x$, whenever $lb_{computed}>16$; in general $lb>lb_{computed}$ and the value 16 was determined experimentally. Moreover, it has been shown[15] that the VAS (continued fractions) method is faster than the fastest implementation of the VCA (bisection) method,[16] a fact that was confirmed[17] independently; more precisely, for the Mignotte polynomials of high degree VAS is about 50,000 times faster than the fastest implementation of VCA.
In 2007, Sharma[18] removed the hypothesis of the ideal positive lower bound and proved that VAS is still polynomial in time.
VAS is the default algorithm for root isolation in Mathematica, SageMath, SymPy, Xcas.
For a comparison between Sturm's method and VAS use the functions realroot(poly) and time(realroot(poly)) of Xcas. By default, to isolate the real roots of poly realroot uses the VAS method; to use Sturm's method write realroot(sturm, poly). See also the External links for an application by A. Berkakis for Android devices that does the same thing.
Here is how VAS(p, M) works, where for simplicity Strzeboński's contribution is not included:
• Let p(x) be a polynomial of degree deg(p) such that p(0) ≠ 0. To isolate its positive roots, associate with p(x) the Möbius transformation M(x) = x and repeat the following steps while there are pairs {p(x), M(x)} to be processed.
• Use Descartes' rule of signs on p(x) to compute, if possible, (using the number var of sign variations in the sequence of its coefficients) the number of its roots inside the interval (0, ∞). If there are no roots return the empty set, ∅ whereas if there is one root return the interval (a, b), where a = min(M(0), M(∞)), and b = max(M(0), M(∞)); if b = ∞ set b = ub, where ub is an upper bound on the values of the positive roots of p(x).[12][13]
• If there are two or more sign variations Descartes' rule of signs implies that there may be zero, one or more real roots inside the interval (0, ∞); in this case consider separately the roots of p(x) that lie inside the interval (0, 1) from those inside the interval (1, ∞). A special test must be made for 1.
• To guarantee that there are roots inside the interval (0, 1) the ideal lower bound, lb is used; that is the integer part of the smallest positive root is computed with the help of the lower bound,[12][13] $lb_{computed}$, on the values of the positive roots of p(x). If $lb_{computed}>1$, the substitution $x\leftarrow x+lb_{computed}$ is performed to p(x) and M(x), whereas if $lb_{computed}\leq 1$ use substitution(s) x ← x+1 to find the integer part of the root(s).
• To compute the roots inside the interval (0, 1) perform the substitution $x\leftarrow {\frac {1}{1+x}}$ to p(x) and M(x) and process the pair
$\left\{(1+x)^{\deg(p)}p\left({\tfrac {1}{1+x}}\right),M({\tfrac {1}{1+x}})\right\},$
whereas to compute the roots in the interval (1, ∞) perform the substitution x ← x + 1 to p(x) and M(x) and process the pair {p(1 + x), M(1 + x)}. It may well turn out that 1 is a root of p(x), in which case, M(1) is a root of the original polynomial and the isolation interval reduces to a point.
Below is a recursive presentation of VAS(p, M).
VAS(p, M):
Input: A univariate, square-free polynomial $p(x)\in \mathbb {Z} [x],p(0)\neq 0$, of degree deg(p), and the Möbius transformation
$M(x)={\frac {ax+b}{cx+d}}=x,\qquad a,b,c,d\in \mathbb {N} .$
Output: A list of isolating intervals of the positive roots of p(x).
1 var ← the number of sign variations of p(x) // Descartes' rule of signs;
2 if var = 0 then RETURN ∅;
3 if var = 1 then RETURN {(a, b)} // a = min(M(0), M(∞)), b = max(M(0), M(∞)), but if b = ∞ set b = ub, where ub is an upper bound on the values of the positive roots of p(x);
4 lb ← the ideal lower bound on the positive roots of p(x);
5 if lb ≥ 1 then p ← p(x + lb), M ← M(x + lb);
6 p01 ← (x + 1)deg(p) p(1/x + 1), M01 ← M(1/x + 1) // Look for real roots in (0, 1);
7 m ← M(1) // Is 1 a root?
8 p1∞ ← p(x + 1), M1∞ ← M(x + 1) // Look for real roots in (1, ∞);
9 if p(1) ≠ 0 then
10 RETURN VAS(p01, M01) ∪ VAS(p1∞, M1∞)
11 else
12 RETURN VAS(p01, M01) ∪ {[m, m]} ∪ VAS(p1∞, M1∞)
13 end
Remarks
• For simplicity Strzeboński's contribution is not included.
• In the above algorithm with each polynomial there is associated a Möbius transformation M(x).
• In line 1 Descartes' rule of signs is applied.
• If lines 4 and 5 are removed from VAS(p, M) the resulting algorithm is Vincent's exponential one.
• Any substitution performed on the polynomial p(x) is also performed on the associated Möbius transformation M(x) (lines 5 6 and 8).
• The isolating intervals are computed from the Möbius transformation in line 3, except for integer roots computed in line 7 (also 12).
Example of VAS(p, M)
We apply the VAS method to p(x) = x3 − 7x + 7 (note that: M(x) = x).
Iteration 1
VAS(x3 − 7x + 7, x)
1 var ← 2 // the number of sign variations in the sequence of coefficients of p(x) = x3 − 7x + 7
4 lb ← 1 // the ideal lower bound—found using lbcomputed and substitution(s) x ← x + 1
5 p ← x3 + 3x2 − 4x + 1, M ← x + 1
6 p01 ← x3 − x2 − 2x + 1, M01 ← x + 2/x + 1
7 m ← 1
8 p1∞ ← x3 + 6x2 + 5x + 1, M1∞ ← x + 2
10 RETURN VAS(x3 − x2 − 2x + 1, x + 2/x + 1) ∪ VAS(x3 + 6x2 + 5x + 1, x + 2)
List of isolation intervals: { }.
List of pairs {p, M} to be processed:
$\left\{\left\{x^{3}-x^{2}-2x+1,{\tfrac {x+2}{x+1}}\right\},\{x^{3}+6x^{2}+5x+1,x+2\}\right\}.$
Remove the first and process it.
Iteration 2
VAS(x3 − x2 − 2x + 1, x + 2/x + 1)
1 var ← 2 // the number of sign variations in the sequence of coefficients of p(x) = x3 − x2 − 2x + 1
4 lb ← 0 // the ideal lower bound—found using lbcomputed and substitution(s) x ← x + 1
6 p01 ← x3 + x2 − 2x − 1, M01 ← 2x + 3/x + 1
7 m ← 3/2
8 p1∞ ← x3 + 2x2 − x − 1, M1∞ ← x + 3/x + 2
10 RETURN VAS(x3 + x2 − 2x − 1, 2x + 3/x + 2) ∪ VAS(x3 + 2x2 − x − 1, x + 3/x + 2)
List of isolation intervals: { }.
List of pairs {p, M} to be processed:
$\left\{\left\{x^{3}+x^{2}-2x-1,{\tfrac {2x+3}{x+2}}\right\},\left\{x^{3}+2x^{2}-x-1,{\tfrac {x+3}{x+2}}\right\},\{x^{3}+6x^{2}+5x+1,x+2\}\right\}.$
Remove the first and process it.
Iteration 3
VAS(x3 + x2 − 2x − 1, 2x + 3/x + 2)
1 var ← 1 // the number of sign variations in the sequence of coefficients of p(x) = x3 + x2 − 2x − 1
3 RETURN {(3/2, 2)}
List of isolation intervals: {(3/2, 2)}.
List of pairs {p, M} to be processed:
$\left\{\left\{x^{3}+2x^{2}-x-1,{\tfrac {x+3}{x+2}}\right\},\{x^{3}+6x^{2}+5x+1,x+2\}\right\}.$
Remove the first and process it.
Iteration 4
VAS(x3 + 2x2 − x − 1, x + 3/x + 2)
1 var ← 1 // the number of sign variations in the sequence of coefficients of p(x) = x3 + 2x2 − x − 1
3 RETURN {(1, 3/2)}
List of isolation intervals: {(1, 3/2), (3/2, 2)}.
List of pairs {p, M} to be processed:
$\left\{\left\{x^{3}+6x^{2}+5x+1,x+2\right\}\right\}.$
Remove the first and process it.
Iteration 5
VAS(x3 + 6x2 + 5x + 1, x + 2)
1 var ← 0 // the number of sign variations in the sequence of coefficients of p(x) = x3 + 6x2 + 5x + 1
2 RETURN ∅
List of isolation intervals: {(1, 3/2), (3/2, 2)}.
List of pairs {p, M} to be processed: ∅.
Finished.
Conclusion
Therefore, the two positive roots of the polynomial p(x) = x3 − 7x + 7 lie inside the isolation intervals (1, 3/2) and (3/2, 2)}. Each root can be approximated by (for example) bisecting the isolation interval it lies in until the difference of the endpoints is smaller than 10−6; following this approach, the roots turn out to be ρ1 = 1.3569 and ρ2 = 1.69202.
Bisection methods
There are various bisection methods derived from Vincent's theorem; they are all presented and compared elsewhere.[19] Here the two most important of them are described, namely, the Vincent–Collins–Akritas (VCA) method and the Vincent–Alesina–Galuzzi (VAG) method.
The Vincent–Alesina–Galuzzi (VAG) method is the simplest of all methods derived from Vincent's theorem but has the most time consuming test (in line 1) to determine if a polynomial has roots in the interval of interest; this makes it the slowest of the methods presented in this article.
By contrast, the Vincent–Collins–Akritas (VCA) method is more complex but uses a simpler test (in line 1) than VAG. This along with certain improvements[16] have made VCA the fastest bisection method.
Vincent–Collins–Akritas (VCA, 1976)
This was the first method developed to overcome the exponential nature of Vincent's original approach, and has had quite an interesting history as far as its name is concerned. This method, which isolates the real roots, using Descartes' rule of signs and Vincent's theorem, had been originally called modified Uspensky's algorithm by its inventors Collins and Akritas.[11] After going through names like "Collins–Akritas method" and "Descartes' method" (too confusing if ones considers Fourier's article[20]), it was finally François Boulier, of Lille University, who gave it the name Vincent–Collins–Akritas (VCA) method,[14] p. 24, based on the fact that "Uspensky's method" does not exist[21] and neither does "Descartes' method".[22] The best implementation of this method is due to Rouillier and Zimmerman,[16] and to this date, it is the fastest bisection method. It has the same worst case complexity as Sturm's algorithm, but is almost always much faster. It has been implemented in Maple's RootFinding package.
Here is how VCA(p, (a, b)) works:
• Given a polynomial porig(x) of degree deg(p), such that porig(0) ≠ 0, whose positive roots must be isolated, first compute an upper bound,[12][13] ub on the values of these positive roots and set p(x) = porig(ub * x) and (a, b) = (0, ub). The positive roots of p(x) all lie in the interval (0, 1) and there is a bijection between them and the roots of porig(x), which all lie in the interval (a, b) = (0, ub) (see the corresponding figure); this bijection is expressed by α(a,b) = a +α(0,1)(b − a). Likewise, there is a bijection between the intervals (0, 1) and (0, ub).
• Repeat the following steps while there are pairs {p(x), (a, b)} to be processed.
• Use Budan's "0_1 roots test" on p(x) to compute (using the number var of sign variations in the sequence of its coefficients) the number of its roots inside the interval (0, 1). If there are no roots return the empty set, ∅ and if there is one root return the interval (a, b).
• If there are two or more sign variations Budan's "0_1 roots test" implies that there may be zero, one, two or more real roots inside the interval (0, 1). In this case cut it in half and consider separately the roots of p(x) inside the interval (0, 1/2)—and that correspond to the roots of porig(x) inside the interval (a, 1/2(a + b)) from those inside the interval (1/2, 1) and correspond to the roots of porig(x) inside the interval (1/2(a + b), b); that is, process, respectively, the pairs
$\left\{2^{\deg(p)}p({\tfrac {x}{2}}),(a,{\tfrac {1}{2}}(a+b))\right\},\quad \left\{2^{\deg(p)}p({\tfrac {1}{2}}(x+1)),({\tfrac {1}{2}}(a+b),b)\right\}$
(see the corresponding figure). It may well turn out that 1/2 is a root of p(x), in which case 1/2(a + b) is a root of porig(x) and the isolation interval reduces to a point.
Below is a recursive presentation of the original algorithm VCA(p, (a, b)).
VCA(p, (a, b))
Input: A univariate, square-free polynomial p(ub * x) ∈ Z[x], p(0) ≠ 0 of degree deg(p), and the open interval (a, b) = (0, ub), where ub is an upper bound on the values of the positive roots of p(x). (The positive roots of p(ub * x) are all in the open interval (0, 1)).
Output: A list of isolating intervals of the positive roots of p(x)
1 var ← the number of sign variations of (x + 1)deg(p)p(1/x + 1) // Budan's "0_1 roots test";
2 if var = 0 then RETURN ∅;
3 if var = 1 then RETURN {(a, b)};
4 p01/2 ← 2deg(p)p(x/2) // Look for real roots in (0, 1/2);
5 m ← 1/2(a + b) // Is 1/2 a root?
6 p1/21 ← 2deg(p)p(x + 1/2) // Look for real roots in (1/2, 1);
7 if p(1/2) ≠ 0 then
8 RETURN VCA (p01/2, (a, m)) ∪ VCA (p1/21, (m, b))
9 else
10 RETURN VCA (p01/2, (a, m)) ∪ {[m, m]} ∪ VCA (p1/21, (m, b))
11 end
Remark
• In the above algorithm with each polynomial there is associated an interval (a, b). As shown elsewhere,[22] p. 11, a Möbius transformation can also be associated with each polynomial in which case VCA looks more like VAS.
• In line 1 Budan's "0_1 roots test" is applied.
Example of VCA(p, (a,b))
Given the polynomial porig(x) = x3 − 7x + 7 and considering as an upper bound[12][13] on the values of the positive roots ub = 4 the arguments of the VCA method are: p(x) = 64x3 − 28x + 7 and (a, b) = (0, 4).
Iteration 1
1 var ← 2 // the number of sign variations in the sequence of coefficients of (x + 1)3p(1/x + 1) = 7x3 − 7x2 − 35x + 43
4 p01/2 ← 64x3 − 112x + 56
5 m ← 2
6 p1/21 ← 64x3 + 192x2 + 80x + 8
7 p(1/2) = 1
8 RETURN VCA(64x3 − 112x + 56, (0, 2)) ∪ VCA(64x3 + 192x2 + 80x + 8, (2, 4))
List of isolation intervals: { }.
List of pairs {p, I} to be processed:
$\left\{\left\{64x^{3}-112x+56,(0,2)\right\},\left\{64x^{3}+192x^{2}+80x+8,(2,4)\right\}\right\}.$
Remove the first and process it.
Iteration 2
VCA(64x3 − 112x + 56, (0, 2))
1 var ← 2 // the number of sign variations in the sequence of coefficients of (x + 1)3p(1/x + 1) = 56x3 + 56x2 − 56x + 8
4 p01/2 ← 64x3 − 448x + 448
5 m ← 1
6 p1/21 ← 64x3 + 192x2 − 256x + 64
7 p(1/2) = 8
8 RETURN VCA(64x3 − 448x + 448, (0, 1)) ∪ VCA(64x3 + 192x2 − 256x + 64, (1, 2))
List of isolation intervals: { }.
List of pairs {p, I} to be processed:
$\left\{\left\{64x^{3}-448x+448,(0,1)\right\},\left\{64x^{3}+192x^{2}-256x+64,(1,2)\right\},\left\{64x^{3}+192x^{2}+80x+8,(2,4)\right\}\right\}.$
Remove the first and process it.
Iteration 3
VCA(64x3 − 448x + 448, (0, 1))
1 var ← 0 // the number of sign variations in the sequence of coefficients of (x + 1)3p(1/x + 1) = 448x3 + 896x2 + 448x + 64
2 RETURN ∅
List of isolation intervals: { }.
List of pairs {p, I} to be processed:
$\left\{\left\{64x^{3}+192x^{2}-256x+64,(1,2)\right\},\left\{64x^{3}+192x^{2}+80x+8,(2,4)\right\}\right\}.$
Remove the first and process it.
Iteration 4
VCA(64x3 + 192x2 − 256x + 64, (1, 2))
1 var ← 2 // the number of sign variations in the sequence of coefficients of (x + 1)3p(1/x + 1) = 64x3 − 64x2 − 128x + 64
4 p01/2 ← 64x3 + 384x2 − 1024x + 512
5 m ← 3/2
6 p1/21 ← 64x3 + 576x2 − 64x + 64
7 p(1/2) = −8
8 RETURN VCA(64x3 + 384x2 − 1024x + 512, (1, 3/2)) ∪ VCA(64x3 + 576x2 − 64x − 64, (3/2, 2))
List of isolation intervals: { }.
List of pairs {p, I} to be processed:
$\left\{\left\{64x^{3}+384x^{2}-1024x+512,\left(1,{\tfrac {3}{2}}\right)\right\},\left\{64x^{3}+576x^{2}-64x-64,\left({\tfrac {3}{2}},2\right)\right\},\left\{64x^{3}+192x^{2}+80x+8,(2,4)\right\}\right\}.$
Remove the first and process it.
Iteration 5
VCA(64x3 + 384x2 − 1024x + 512, (1, 3/2))
1 var ← 1 // the number of sign variations in the sequence of coefficients of (x + 1)3p(1/x + 1) = 512x3 + 512x2 − 128x − 64
3 RETURN {(1, 3/2)}
List of isolation intervals: {(1, 3/2)}.
List of pairs {p, I} to be processed:
$\left\{\left\{64x^{3}+576x^{2}-64x-64,\left({\tfrac {3}{2}},2\right)\right\},\left\{64x^{3}+192x^{2}+80x+8,(2,4)\right\}\right\}.$
Remove the first and process it.
Iteration 6
VCA(64x3 + 576x2 − 64x − 64, (3/2, 2))
1 var ← 1 // the number of sign variations in the sequence of coefficients of (x + 1)3p(1/x + 1) = −64x3 − 256x2 + 256x + 512
3 RETURN {(3/2, 2)}
List of isolation intervals: {(1, 3/2), (3/2, 2)}.
List of pairs {p, I} to be processed:
$\left\{\left\{64x^{3}+192x^{2}+80x+8,(2,4)\right\}\right\}.$
Remove the first and process it.
Iteration 7
VCA(64x3 + 192x2 + 80x + 8, (2, 4))
1 var ← 0 // the number of sign variations in the sequence of coefficients of (x + 1)3p(1/x + 1) = 8x3 + 104x2 + 376x + 344
2 RETURN ∅
List of isolation intervals: {(1, 3/2), (3/2, 2)}.
List of pairs {p, I} to be processed: ∅.
Finished.
Conclusion
Therefore, the two positive roots of the polynomial p(x) = x3 − 7x + 7 lie inside the isolation intervals (1, 3/2) and (3/2, 2)}. Each root can be approximated by (for example) bisecting the isolation interval it lies in until the difference of the endpoints is smaller than 10−6; following this approach, the roots turn out to be ρ1 = 1.3569 and ρ2 = 1.69202.
Vincent–Alesina–Galuzzi (VAG, 2000)
This was developed last and is the simplest real root isolation method derived from Vincent's theorem.
Here is how VAG(p, (a, b)) works:
• Given a polynomial p(x) of degree deg(p), such that p(0) ≠ 0, whose positive roots must be isolated, first compute an upper bound,[12][13] ub on the values of these positive roots and set (a, b) = (0, ub). The positive roots of p(x) all lie in the interval (a, b).
• Repeat the following steps while there are intervals (a, b) to be processed; in this case the polynomial p(x) stays the same.
• Use the Alesina–Galuzzi "a_b roots test" on p(x) to compute (using the number var of sign variations in the sequence of its coefficients) the number of its roots inside the interval (a, b). If there are no roots return the empty set, ∅ and if there is one root return the interval (a, b).
• If there are two or more sign variations the Alesina–Galuzzi "a_b roots test" implies that there may be zero, one, two or more real roots inside the interval (a, b). In this case cut it in half and consider separately the roots of p(x) inside the interval (a, 1/2(a + b)) from those inside the interval (1/2(a + b), b); that is, process, respectively, the intervals (a, 1/2(a + b)) and (1/2(a + b), b). It may well turn out that 1/2(a + b) is a root of p(x), in which case the isolation interval reduces to a point.
Below is a recursive presentation of VAG(p, (a, b)).
VAG(p, (a, b))
Input: A univariate, square-free polynomial p(x) ∈ Z[x], p(0) ≠ 0 of degree deg(p) and the open interval (a, b) = (0, ub), where ub is an upper bound on the values of the positive roots of p(x).
Output: A list of isolating intervals of the positive roots of p(x).
1 var ← the number of sign variations of (x + 1)deg(p) p(a + bx/1 + x) // The Alesina–Galuzzi "a_b roots test";
2 if var = 0 then RETURN ∅;
3 if var = 1 then RETURN {(a, b)};
4 m ← 1/2(a + b) // Subdivide the interval (a, b) in two equal parts;
5 if p(m) ≠ 0 then
6 RETURN VAG(p, (a, m)) ∪ VAG(p, (m, b))
7 else
8 RETURN VAG(p, (a, m)) ∪ {[m, m]} ∪ VAG(p, (m, b))
9 end
Remarks
• Compared to VCA the above algorithm is extremely simple; by contrast, VAG uses the time consuming "a_b roots test" and that makes it much slower than VCA.[19]
• As Alesina and Galuzzi point out,[5] p. 189, there is a variant of this algorithm due to Donato Saeli. Saeli suggested that the mediant of the endpoints be used instead of their midpoint 1/2(a + b). However, it has been shown[19] that using the mediant of the endpoints is in general much slower than the "mid-point" version.
Example of VAG(p, (a,b))
Given the polynomial p(x) = x3 − 7x + 7 and considering as an upper bound[12][13] on the values of the positive roots ub = 4 the arguments of VAG are: p(x) = x3 − 7x + 7 and (a, b) = (0, 4).
Iteration 1
1 var ← 2 // the number of sign variations in the sequence of coefficients of (x + 1)3p(4x/x + 1) = 43x3 − 35x2 − 7x + 7
4 m ← 1/2(0 + 4) = 2
5 p(m) = 1
8 RETURN VAG(x3 − 7x + 7, (0, 2)) ∪ VAG(x3 − 7x + 7, (2, 4)
List of isolation intervals: {}.
List of intervals to be processed: {(0, 2), (2, 4)}.
Remove the first and process it.
Iteration 2
VAG(x3 − 7x + 7, (0, 2))
1 var ← 2 // the number of sign variations in the sequence of coefficients of (x + 1)3p(2x/x + 1) = x3 − 7x2 + 7x + 7
4 m ← 1/2(0 + 2) = 1
5 p(m) = 1
8 RETURN VAG(x3 − 7x + 7, (0, 1)) ∪ VAG(x3 − 7x + 7, (1, 2)
List of isolation intervals: {}.
List of intervals to be processed: {(0, 1), (1, 2), (2, 4)}.
Remove the first and process it.
Iteration 3
VAG(x3 − 7x + 7, (0, 1))
1 var ← 0 // the number of sign variations in the sequence of coefficients of (x + 1)3p(x/x + 1) = x3 + 7x2 + 14x + 7
2 RETURN ∅
List of isolation intervals: {}.
List of intervals to be processed: {(1, 2), (2, 4)}.
Remove the first and process it.
Iteration 4
VAG(x3 − 7x + 7, (1, 2))
1 var ← 2 // the number of sign variations in the sequence of coefficients of (x + 1)3p(2x + 1/x + 1) = x3 − 2x2 − x + 1
4 m ← 1/2(1 + 2) = 3/2
5 p(m) = −1/8
8 RETURN VAG(x3 − 7x + 7, (1, 3/2)) ∪ VAG(x3 − 7x + 7, (3/2, 2))
List of isolation intervals: {}.
List of intervals to be processed: {(1, 3/2), (3/2, 2), (2, 4)}.
Remove the first and process it.
Iteration 5
VAG(x3 − 7x + 7, (1, 3/2))
1 var ← 1 // the number of sign variations in the sequence of coefficients of 23(x + 1)3p(3/2x + 1/x + 1) = x3 + 2x2 − 8x − 8
3 RETURN (1, 3/2)
List of isolation intervals: {(1, 3/2)}.
List of intervals to be processed: {(3/2, 2), (2, 4)}.
Remove the first and process it.
Iteration 6
VAG(x3 − 7x + 7, (3/2, 2))
1 var ← 1 // the number of sign variations in the sequence of coefficients of 23(x + 1)3p(2x + 3/2/x + 1) = 8x3 + 4x2 − 4x − 1
3 RETURN (3/2, 2)
List of isolation intervals: {(1, 3/2), (3/2, 2)}.
List of intervals to be processed: {(2, 4)}.
Remove the first and process it.
Iteration 7
VAG(x3 − 7x + 7, (2, 4))
1 var ← 0 // the number of sign variations in the sequence of coefficients of (x + 1)3p(4x + 2/x + 1) = 344x3 + 376x2 + 104x + 8
2 RETURN ∅
List of isolation intervals: {(1, 3/2), (3/2, 2)}.
List of intervals to be processed: ∅.
Finished.
Conclusion
Therefore, the two positive roots of the polynomial p(x) = x3 − 7x + 7 lie inside the isolation intervals (1, 3/2) and (3/2, 2)}. Each root can be approximated by (for example) bisecting the isolation interval it lies in until the difference of the endpoints is smaller than 10−6; following this approach, the roots turn out to be ρ1 = 1.3569 and ρ2 = 1.69202.
See also
• Properties of polynomial roots
• Root-finding algorithm
• Vieta's formulas
• Newton's method
References
1. Vincent, Alexandre Joseph Hidulphe (1834). "Mémoire sur la résolution des équations numériques". Mémoires de la Société Royale des Sciences, de l' Agriculture et des Arts, de Lille: 1–34.
2. Vincent, Alexandre Joseph Hidulphe (1836). "Note sur la résolution des équations numériques" (PDF). Journal de Mathématiques Pures et Appliquées. 1: 341–372.
3. Vincent, Alexandre Joseph Hidulphe (1838). "Addition à une précédente note relative à la résolution des équations numériques" (PDF). Journal de Mathématiques Pures et Appliquées. 3: 235–243.
4. Alesina, Alberto; Massimo Galuzzi (1998). "A new proof of Vincent's theorem". L'Enseignement Mathématique. 44 (3–4): 219–256.
5. Alesina, Alberto; Massimo Galuzzi (2000). "Vincent's Theorem from a Modern Point of View" (PDF). Categorical Studies in Italy 2000, Rendiconti del Circolo Matematico di Palermo, Serie II, N. 64: 179–191.
6. Ostrowski, A. M. (1950). "Note on Vincent's theorem". Annals of Mathematics. Second Series. 52 (3): 702–707. doi:10.2307/1969443. JSTOR 1969443.
7. Obreschkoff, Nikola (1963). Verteilung und Berechnung der Nullstellen reeller Polynome. Berlin: VEB Deutscher Verlag der Wissenschaften.
8. Uspensky, James Victor (1948). Theory of Equations. New York: McGraw–Hill Book Company.
9. Akritas, Alkiviadis G.; A.W. Strzeboński; P.S. Vigklas (2008). "Improving the performance of the continued fractions method using new bounds of positive roots" (PDF). Nonlinear Analysis: Modelling and Control. 13 (3): 265–279. doi:10.15388/NA.2008.13.3.14557.
10. Serret, Joseph A. (1877). Cours d'algèbre supérieure. Tome I. Gauthier-Villars.
11. Collins, George E.; Alkiviadis G. Akritas (1976). "Polynomial real root isolation using Descarte's rule of signs". Polynomial Real Root Isolation Using Descartes' Rule of Signs. SYMSAC '76, Proceedings of the third ACM symposium on Symbolic and algebraic computation. Yorktown Heights, NY, USA: ACM. pp. 272–275. doi:10.1145/800205.806346. ISBN 9781450377904. S2CID 17003369.
12. Vigklas, Panagiotis, S. (2010). Upper bounds on the values of the positive roots of polynomials (PDF). Ph. D. Thesis, University of Thessaly, Greece.{{cite book}}: CS1 maint: multiple names: authors list (link)
13. Akritas, Alkiviadis, G. (2009). "Linear and Quadratic Complexity Bounds on the Values of the Positive Roots of Polynomials". Journal of Universal Computer Science. 15 (3): 523–537.{{cite journal}}: CS1 maint: multiple names: authors list (link)
14. Boulier, François (2010). Systèmes polynomiaux : que signifie " résoudre " ? (PDF). Université Lille 1.
15. Akritas, Alkiviadis G.; Adam W. Strzeboński (2005). "A Comparative Study of Two Real Root Isolation Methods" (PDF). Nonlinear Analysis: Modelling and Control. 10 (4): 297–304. doi:10.15388/NA.2005.10.4.15110.
16. Rouillier, F.; P. Zimmerman (2004). "Efficient isolation of polynomial's real roots". Journal of Computational and Applied Mathematics. 162: 33–50. doi:10.1016/j.cam.2003.08.015.
17. Tsigaridas, Elias P.; Emiris, Ioannis Z. (2006). "Univariate polynomial real root isolation: Continued fractions revisited". In Azar, Yossi; Erlebach, Thomas (eds.). Algorithms – ESA 2006, 14th Annual European Symposium, Zurich, Switzerland, September 11–13, 2006, Proceedings. Lecture Notes in Computer Science. Vol. 4168. Springer. pp. 817–828. arXiv:cs/0604066. doi:10.1007/11841036_72.
18. Sharma, Vikram (2007). Complexity Analysis of Algorithms in Algebraic Computation (PDF). Ph.D. Thesis, Courant Institute of Mathematical Sciences, New York University,USA.
19. Akritas, Alkiviadis G.; Adam W. Strzeboński; Panagiotis S. Vigklas (2008). "On the Various Bisection Methods Derived from Vincent's Theorem". Serdica Journal of Computing. 2 (1): 89–104. doi:10.55630/sjc.2008.2.89-104. hdl:10525/376. S2CID 126142131.
20. Fourier, Jean Baptiste Joseph (1820). "Sur l'usage du théorème de Descartes dans la recherche des limites des racines". Bulletin des Sciences, par la Société Philomatique de Paris: 156–165.
21. Akritas, Alkiviadis G. (1986). "There is no "Uspensky's method."". There's no "Uspensky's Method". In: Proceedings of the fifth ACM Symposium on Symbolic and Algebraic Computation (SYMSAC '86, Waterloo, Ontario, Canada), pp. 88–90. pp. 88–90. doi:10.1145/32439.32457. ISBN 0897911997. S2CID 15446040.
22. Akritas, Alkiviadis G. (2008). There is no "Descartes' method". In: M.J.Wester and M. Beaudin (Eds), Computer Algebra in Education, AullonaPress, USA, pp. 19–35. ISBN 9780975454190.
External links
• Berkakis, Antonis: RealRoots, a free App for Android devices to compare Sturm's method and VAS
• https://play.google.com/store/apps/details?id=org.kde.necessitas.berkakis.realroots
| Wikipedia |
Vincent Léotaud
Vincent Léotaud (1595 – 1672) was a French Jesuit mathematician.[1][2]
Vincent Léotaud
Born1595
Vallouise
Died1672 (aged 76–77)
Embrun
OccupationMathematician
In his work Examen circuli quadraturae he affirmed the impossibility of squaring the circle, against the opinion of Grégoire de Saint-Vincent.[3][4]
Works
• Examen circuli quadraturae (in Latin). Vol. 1. Lyon: Guillaume Barbier. 1654.
• Examen circuli quadraturae (in Latin). Vol. 2. Lyon: Guillaume Barbier. 1654.
• Institutionum arithmeticarum libri quatuor (in Latin). Lyon: Guillaume Barbier. 1660.
• Cyclomathia seu Multiplex circuli contemplatio, tribus libris comprehensa (in Latin). Lyon: Benoit Coral. 1663.
• Magnetologia; in qua exponitur noua de magneticis philosophia (in Latin). Lyon: Laurent Anisson. 1668.
References
1. Léotaud, Vincent (1595-1672) (in French). Bibliothèque nationale de France.
2. "Léotaud, Vincent". Consortium of European Research Libraries.
3. Robson, Eleanor; Stedall, Jacqueline, eds. (2009). The Oxford Handbook of the History of Mathematics. Oxford University Press. p. 554. ISBN 9780199213122.
4. Zupanov, Ines G., ed. (2019). The Oxford Handbook of the Jesuits. Oxford University Press. p. 650. ISBN 9780190639631.
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Vincentio Reinieri
Vincentio (Vincenzio, Vincenzo) Reinieri (Renieri, Reiner) (30 March 1606 – 5 November 1647) was an Italian mathematician and astronomer. He was a friend and disciple of Galileo Galilei.
Vincentio Reinieri
Born(1606-03-30)30 March 1606
Died5 November 1642(1642-11-05) (aged 36)
NationalityItalian
Occupation(s)Mathematician, astronomer
Biography
Born at Genoa, he was a member of the Olivetan order. His order sent him to Rome in 1623. He met Galileo at Siena in 1633. Galileo had Reinieri update and attempt to improve his astronomical tables of the motions of Jupiter's moons, revising these tables for prediction of the positions of these satellites.
Reinieri's work led him to Arcetri, where he befriended Vincenzo Viviani. Reinieri enjoyed the same spirit of inquiry and love of debate as his mentor. On 5 February 1641 Reinieri wrote to Galileo from Pisa: "Not infrequently I am in some battle with the Peripatetic gentlemen, particularly when I note that those fattest with ignorance least appreciate your worth, and I have just given the head of one of those a good scrubbing." (Drake, p. 413-4)
Reinieri became professor of mathematics at the University of Pisa on the death of Dino Peri. He also taught Greek there. His astronomical work consisted of adding new observations of Jupiter's moons to Galileo's. To some degree, Reinieri improved the Galilean tables on the motions of these satellites. Before his death, Galileo decided to place all of the papers containing his observations and calculations in the hands of Reinieri. Reinieri was to finish and revise them.
Reinieri's observations of Jupiter's moons remained unpublished at the time of his premature death at Pisa in 1647. He was succeeded to the chair of mathematics by Famiano Michelini (c. 1600-1666).
Legacy
On Reinieri's death, papers concerning longitude entrusted to him by Galileo are said to have been stolen by a man named Giuseppe Agostini (Fahie, p. 374). However, scholars such as Antonio Favaro doubt whether this theft actually occurred (see Antonio Favaro, Documenti inediti per la Storia dei MSS. Galileiani, Rome, 1886, pp. 8–14).
The crater Reiner on the Moon is named after him.
Latin works
• Expugnata Hierusalem, poema, Publisher: Maceratae, Apud Petrum Salvionum (1628)
• Tabulae mediceae secundorum mobilium universales quibus per unicum prosthaphaereseon orbis canonem planetarum calculus exhibetur. Non solum tychonicè iuxta Rudolphinas Danicas & Lansbergianas, sed etiam iuxta Prutenicas Alphonsinas & Ptolemaicas, Publisher: Florentiae, typis nouis Amatoris Massae & Laurentij de Landis (1639)
• Tabulae Mediceae secundorum mobilium uniuersales (in Latin). Firenze: Amadore Massi & Lorenzo Landi. 1639.
• Tabulæ motuum cælestium universales : serenissimi magni ducis etruriæ Ferdinandi II. auspicijs primo editæ, & Mediceæ nuncupati, nunc vero auctæ, recognitæ, atque... Bernardini Fernandez de Velasco... iussu, ac sumptibus recusæ...Publisher: Florentiæ : typis Amatoris Massæ Foroliuien., 1647
Sources
• Drake, Stillman, Galileo at Work: His Scientific Biography (Chicago: University of Chicago Press, 1978), 464. ISBN 0-226-16226-5
• Fahie, J.J., Galileo: His Life and Work (London: John Murray, 1903), 374-5. - Google Books
Further reading
• A Selection from Italian Prose Writers: with a double translation: for the use of students of the Italian language on the Hamiltonian system, London, Hunt and Clark, 1828 - Google Books. Letters of Galileo to Renieri: pp. 142–147 (no images for remainder of letter), and pp. 242–253 (no images pp, 246-250).
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Vincenzo Brunacci
Vincenzo Brunacci (3 March 1768 – 16 June 1818) was an Italian mathematician born in Florence.[1] He was professor of Matematica sublime (infinitesimal calculus) in Pavia. He transmitted Lagrange's ideas to his pupils, including Ottaviano Fabrizio Mossotti, Antonio Bordoni and Gabrio Piola.
Vincenzo Brunacci
Born(1768-03-03)3 March 1768
Florence, Italy
Died16 June 1818(1818-06-16) (aged 50)
Pavia, Italy
NationalityItalian
Alma materUniversity of Pisa
Known forContributions to infinitesimal calculus
Scientific career
FieldsMathematics
Doctoral advisorPietro Paoli
Other academic advisorsSebastiano Canovai
Doctoral studentsOttaviano Fabrizio Mossotti
Antonio Bordoni
Gabrio Piola
Biography
He studied medicine, astronomy and mathematics at the University of Pisa. In 1788 he earned his laurea and the same year he started teaching mathematics at the Naval Institute of Livorno. In 1796, when Napoleon entered Italy, he endorsed the new order. Upon the reinstatement of the Austrian rule, he moved to France between 1799 and 1800. On returning he attained a chair at the University of Pisa. In 1801 he moved to the University of Pavia with the office of professor of infinitesimal calculus and become its dean.
Brunacci believed that Lagrange's approach, developed in the "Théorie des fonctions analytiques", was the correct one and that the infinitesimal concept was to be banned from analysis and mechanics. In Brunacci's university teaching infinitesimal calculus differently from Lagrange's principles was even prohibited as a rule. Brunacci passed his idea of analysis on to his students, among which Fabrizio Ottaviano Mossotti, Gabrio Piola and Antonio Bordoni.
He cooperated with the public administration, in 1805 he was in the Committee for the Naviglio Pavese (Pavia Canal) project and the following year as inspector of Waters and Roads.
In 1809 he joined the Committee for the new measurements and weights system and from 1811 he was inspector general of Public Education for the entire Italian Kingdom.
He died in Pavia in 1818.
Writings
• Opuscolo analitico, (1792).
• Calcolo integrale delle equazioni lineari, (1798).
• Corso di matematica sublime, in four volumes, Firenze, (1804–1807).
• Elementi di algebra e di geometria, in two volumes, Firenze, (1809).
• Trattato dell'ariete idraulico, (1810).
• Quale tra le pratiche usate in Italia per la dispensa delle acque è la più convenevole, e quali precauzioni ed artifizi dovrebbero aggiungersi per intieramente perfezionarla riducendo le antiche alle nuove misure metriche (in Italian). Verona: Mainardi. 1814.
• Trattato di navigazione (in Italian). Vol. 2. Milano: Stamperia reale. 1817.
• Trattato di navigazione, 1817
Notes
1. An Italian short biography Vincenzo Brunacci in Edizione Nazionale Mathematica Italiana online.
External links
• Vincenzo Brunacci at the Mathematics Genealogy Project
• An Italian short biography Vincenzo Brunacci in Edizione Nazionale Mathematica Italiana online.
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Vincenzo Flauti
Vincenzo Flauti (1782–1863) was an Italian mathematician.
Vincenzo Flauti
Born(1782-04-04)4 April 1782
Naples, Kingdom of Naples, today Italy
Died20 June 1863(1863-06-20) (aged 81)
Naples, Italy
Alma materUniversity of Naples
Scientific career
FieldsMathematics
InstitutionsUniversity of Naples
InfluencedNicola Fergola
Life and work
Flauti studied at the Liceo del Salvatore, the school led by Nicola Fergola. Although he began medical studies, he changed them to mathematics influenced by his master Fergola. He taught at the University of Naples from 1803 to 1860, succeeding Fergola in his chair in 1812.
In 1860, when the Kingdom of the Two Sicilies was conquered by Giuseppe Garibaldi and was incorporated into the Kingdom of Italy, Flauti was excluded from the Academy of Sciences of Naples and from his docent duties, because he had been a supporter of the Bourbon monarchy.
Flauti was the leader of the synthetic school of mathematics founded by Fergola.[1] In 1807, jointly with Felice Giannattasio, he was entrusted by the Bourbon government to write a mathematics textbook for all schoolchildren in the kingdom.[2]
References
1. Mazzotti 2002, p. 141.
2. Ferraro 2008, p. 108.
Bibliography
• Ferraro, Giovanni (2008). "Manuali di geometria elementare nella Napoli preunitaria (1806–1860)" (PDF). History of Education & Children's Literature (in Italian). 3 (2): 103–139. ISSN 1971-1093.
• Ferraro, Giovanni (2012). "Excellens in arte non debet mori". HAL (in Italian): 1–16.
• Mazzotti, Massimo (1998). "The Geometers of God: Mathematics and Reaction in the Kingdom of Naples" (PDF). Isis. 89 (4): 674–701. doi:10.1086/384160. hdl:10036/31212. ISSN 0021-1753. JSTOR 236738. S2CID 143956681.
• Mazzotti, Massimo (2002). "The Making of the Modern Engineer". In Mordechai Feingold (ed.). History of Universities: Volume XVII. Oxford University Press. pp. 121–161. ISBN 978-0-19-925636-5.
External links
• O'Connor, John J.; Robertson, Edmund F., "Vincenzo Flauti", MacTutor History of Mathematics Archive, University of St Andrews
• Menghini, Marta (1997). "FLAUTI, Vincenzo". Dizionario Biografico degli Italiani. Retrieved December 13, 2018.
• "VINCENZO FLAUTI". Matematica PRISTEM – Università Bocconi. Retrieved December 18, 2018.
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Vincenzo Mollame
Vincenzo Mollame (Naples, 4 July 1848 – Catania, 23 June 1912) was an Italian mathematician.
Mollame was privately tutored by Achille Sanni and then studied Mathematics at the University of Naples Federico II. After obtaining his degree, he became a high-school teacher, first at Benevento and after that at Naples, starting in 1878. He became a professor at the University of Catania in 1880 and remained there for the rest of his career, having retired in 1911, a few months before his death.[1]
His research area was the theory of equations and he proved in 1890 that when a cubic polynomial with rational coefficients has three real roots but it is irreducible in Q[x] (the so-called casus irreducibilis), then the roots cannot be expressed from the coefficients using real radicals alone, that is, complex non-real numbers must be involved if one expresses the roots from the coefficients using radicals,[2] probably unaware of the fact that Pierre Wantzel had already proved it in 1843. Molleme's research activity stopped in 1896, due to health problems.
Mollame was the author of a textbook on determinants.[3]
Notes
1. Marchisotto, Elena Anne; Smith, James (2007), "Life and works", The Legacy of Mario Pieri in Geometry and Arithmetic, Birkhäuser, ISBN 978-0-8176-3210-6
2. Mollame, Vincenzo (1890), "Sul casus irreductibilis dell'equazione cubica", Rendiconto dell'Accademia delle scienze fisiche e matematiche (Sezione della società Reale di Napoli) II (in Italian), 4: 167–171
3. Mollame, Vincenzo (1878), I determinanti e loro applicazioni all' algebra ed alla geometria analitica (in Italian), Tipografia dell'Accademia reale delle scienze
External links
• Short biography (in Italian)
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Vinculum (symbol)
A vinculum (from Latin vinculum 'fetter, chain, tie') is a horizontal line used in mathematical notation for various purposes. It may be placed as an overline (or underline) over (or under) a mathematical expression to indicate that the expression is to be considered grouped together. Historically, vincula were extensively used to group items together, especially in written mathematics, but in modern mathematics this function has almost entirely been replaced by the use of parentheses.[1] It was also used to mark Roman numerals whose values are multiplied by 1,000.[2] Today, however, the common usage of a vinculum to indicate the repetend of a repeating decimal[3][4] is a significant exception and reflects the original usage.
${\overline {\rm {AB}}}$
line segment from A to B
1⁄7 = 0.142857
repeated 0.1428571428571428571...
${\overline {a+bi}}$
complex conjugate
$Y={\overline {\rm {AB}}}$
boolean NOT (A AND B)
${\sqrt[{n}]{ab+2}}$
radical ab + 2
$a-{\overline {b+c}}$ = a − (b + c)
bracketing function
Vinculum usage
History
The vinculum, in its general use, was introduced by Frans van Schooten in 1646 as he edited the works of François Viète (who had himself not used this notation). However, earlier versions, such as using an underline as Chuquet did in 1484, or in limited form as Descartes did in 1637, using it only in relation to the radical sign, were common.[5]
Usage
Modern
A vinculum can indicate a line segment where A and B are the endpoints:
• ${\overline {\rm {AB}}}.$
A vinculum can indicate the repetend of a repeating decimal value:
• 1⁄7 = 0.142857 = 0.1428571428571428571...
A vinculum can indicate the complex conjugate of a complex number:
• ${\overline {2+3i}}=2-3i$
Logarithm of a number less than 1 can conveniently be represented using vinculum:
• $\log 2=0.301\Rightarrow \log 0.2={\overline {1}}.301=-0.699$
In Boolean algebra, a vinculum may be used to represent the operation of inversion (also known as the NOT function):
• $Y={\overline {\rm {AB}}},$
meaning that Y is false only when both A and B are both true - or by extension, Y is true when either A or B is false.
Similarly, it is used to show the repeating terms in a periodic continued fraction. Quadratic irrational numbers are the only numbers that have these.
Historical
Formerly its main use was as a notation to indicate a group (a bracketing device serving the same function as parentheses):
$a-{\overline {b+c}},$
meaning to add b and c first and then subtract the result from a, which would be written more commonly today as a − (b + c). Parentheses, used for grouping, are only rarely found in the mathematical literature before the eighteenth century. The vinculum was used extensively, usually as an overline, but Chuquet in 1484 used the underline version.[6]
In India, the use of this notation is still tested in primary school.[7]
As a part of a radical
The vinculum is used as part of the notation of a radical to indicate the radicand whose root is being indicated. In the following, the quantity $ab+2$ is the whole radicand, and thus has a vinculum over it:
${\sqrt[{n}]{ab+2}}.$
In 1637 Descartes was the first to unite the German radical sign √ with the vinculum to create the radical symbol in common use today.[8]
The symbol used to indicate a vinculum need not be a line segment (overline or underline); sometimes braces can be used (pointing either up or down).[9]
Encodings
Main article: Overline § Implementations
In Unicode
• U+0305 ◌̅ COMBINING OVERLINE
TeX
In LaTeX, a text <text> can be overlined with $\overline{\mbox{<text>}}$. The inner \mbox{} is necessary to override the math-mode (here invoked by the dollar signs) which the \overline{} demands.
See also
• Overline § Math and science similar-looking symbols
• Overline § Implementations in word processing and text editing software
• Underline
References
1. Cajori, Florian (2012) [1928]. A History of Mathematical Notations. Vol. I. Dover. p. 384. ISBN 978-0-486-67766-8.
2. Ifrah, Georges (2000). The Universal History of Numbers: From Prehistory to the Invention of the Computer. Translated by David Bellos, E. F. Harding, Sophie MENGNIU , Ian Monk. John Wiley & Sons.
3. Childs, Lindsay N. (2009). A Concrete Introduction to Higher Algebra (3rd ed.). Springer. pp. 183-188.
4. Conférence Intercantonale de l'Instruction Publique de la Suisse Romande et du Tessin (2011). Aide-mémoire. Mathématiques 9-10-11. LEP. pp. 20–21.
5. Cajori 2012, p. 386
6. Cajori 2012, pp. 390–391
7. https://www.khanacademy.org/math/middle-school-math-india/x888d92141b3e0e09:bridge-7th/x888d92141b3e0e09:untitled-302/e/b7-bodmas-1
8. Cajori 2012, p. 208
9. Abbott, Jacob (1847) [1847], Vulgar and decimal fractions (The Mount Vernon Arithmetic Part II), p. 27
External links
• Weisstein, Eric W. "Periodic Continued Fraction". MathWorld.
• Weisstein, Eric W. "Vinculum". MathWorld.
| Wikipedia |
Diffiety
In mathematics, a diffiety (/dəˈfaɪəˌtiː/) is a geometrical object which plays the same role in the modern theory of partial differential equations that algebraic varieties play for algebraic equations, that is, to encode the space of solutions in a more conceptual way. The term was coined in 1984 by Alexandre Mikhailovich Vinogradov as portmanteau from differential variety.[1]
Not to be confused with Diffeology.
Intuitive definition
In algebraic geometry the main objects of study (varieties) model the space of solutions of a system of algebraic equations (i.e. the zero locus of a set of polynomials), together with all their "algebraic consequences". This means that, applying algebraic operations to this set (e.g. adding those polynomials to each other or multiplying them with any other polynomials) will give rise to the same zero locus. In other words, one can actually consider the zero locus of the algebraic ideal generated by the initial set of polynomials.
When dealing with differential equations, apart from applying algebraic operations as above, one has also the option to differentiate the starting equations, obtaining new differential constraints. Therefore, the differential analogue of a variety should be the space of solutions of a system of differential equations, together with all their "differential consequences". Instead of considering the zero locus of an algebraic ideal, one needs therefore to work with a differential ideal.
An elementary diffiety will consist therefore of the infinite prolongation ${\mathcal {E}}^{\infty }$of a differential equation ${\mathcal {E}}\subset J^{k}(E,m)$, together with an extra structure provided by a special distribution. Elementary diffieties play the same role in the theory of differential equations as affine algebraic varieties do in the theory of algebraic equations. Accordingly, just like varieties or schemes are composed of irreducible affine varieties or affine schemes, one defines a (non-elementary) diffiety as an object that locally looks like an elementary diffiety.
Formal definition
The formal definition of a diffiety, which relies on the geometric approach to differential equations and their solutions, requires the notions of jets of submanifolds, prolongations, and Cartan distribution, which are recalled below.
Jet spaces of submanifolds
Let $E$ be an $(m+e)$-dimensional smooth manifold. Two $m$-dimensional submanifolds $M$, $M'$ of $E$ are tangent up to order $k$ at the point $p\in M\cap M'\subset E$ if one can locally describe both submanifolds as zeroes of functions defined in a neighbourhood of $p$, whose derivatives at $p$ agree up to order $k$.
One can show that being tangent up to order $k$ is a coordinate-invariant notion and an equivalence relation.[2] One says also that $M$ and $M'$ have same $k$-th order jet at $p$, and denotes their equivalence class by $[M]_{p}^{k}$ or $j_{p}^{k}M$.
The $k$-jet space of $k$-submanifolds of $E$, denoted by $J^{k}(E,m)$, is defined as the set of all $k$-jets of $m$-dimensional submanifolds of $E$ at all points of $E$:
$J^{k}(E,m):=\{[M]_{p}^{k}~|~p\in M,~{\text{dim}}(M)=m,M\subset E\ {\text{ submanifold}}\}$
As any given jet $[M]_{p}^{k}$ is locally determined by the derivatives up to order $k$ of the functions describing $M$ around $p$, one can use such functions to build local coordinates $(x^{i},u_{\sigma }^{j})$ and provide $J^{k}(E,m)$ with a natural structure of smooth manifold.[2]
For instance, for $k=1$ one recovers just points in $E$ and for $k=1$ one recovers the Grassmannian of $n$-dimensional subspaces of $TE$. More generally, all the projections $J^{k}(E)\to J^{k-1}E$ are fibre bundles.
As a particular case, when $E$ has a structure of fibred manifold over an $n$-dimensional manifold $X$, one can consider submanifolds of $E$ given by the graphs of local sections of $\pi :E\to X$. Then the notion of jet of submanifolds boils down to the standard notion of jet of sections, and the jet bundle $J^{k}(\pi )$ turns out to be an open and dense subset of $J^{k}(E,m)$.[3]
Prolongations of submanifolds
The $k$-jet prolongation of a submanifold $M\subseteq E$ is
$j^{k}(M):M\rightarrow J^{k}(E,m),\quad p\mapsto [M]_{p}^{k}$
The map $j^{k}(M)$ is a smooth embedding and its image $M^{k}:={\text{im}}(j^{k}(M))$, called the prolongation of the submanifold $M$, is a submanifold of $J^{k}(E,m)$ diffeomorphic to $M$.
Cartan distribution on jet spaces
A space of the form $T_{\theta }(M^{k})$, where $M$ is any submanifold of $E$ whose prolongation contains the point $\theta \in J^{k}(E,m)$, is called an $R$-plane (or jet plane, or Cartan plane) at $\theta $. The Cartan distribution on the jet space $J^{k}(E,m)$ is the distribution ${\mathcal {C}}\subseteq T(J^{k}(E,m))$ defined by
${\mathcal {C}}:J^{k}(E,m)\rightarrow TJ^{k}(E,m),\qquad \theta \mapsto {\mathcal {C}}_{\theta }\subset T_{\theta }(J^{k}(E,m))$
where ${\mathcal {C}}_{\theta }$ is the span of all $R$-planes at $\theta \in J^{k}(E,m)$.[4]
Differential equations
A differential equation of order $k$ on the manifold $E$ is a submanifold ${\mathcal {E}}\subset J^{k}(E,m)$; a solution is defined to be an $m$-dimensional submanifold $S\subset {\mathcal {E}}$ such that $S^{k}\subseteq {\mathcal {E}}$. When $E$ is a fibred manifold over $X$, one recovers the notion of partial differential equations on jet bundles and their solutions, which provide a coordinate-free way to describe the analogous notions of mathematical analysis. While jet bundles are enough to deal with many equations arising in geometry, jet spaces of submanifolds provide a greater generality, used to tackle several PDEs imposed on submanifolds of a given manifold, such as Lagrangian submanifolds and minimal surfaces.
As in the jet bundle case, the Cartan distribution is important in the algebro-geometric approach to differential equations because it allows to encode solutions in purely geometric terms. Indeed, a submanifold $S\subset {\mathcal {E}}$ is a solution if and only if it is an integral manifold for ${\mathcal {C}}$, i.e. $T_{\theta }S\subset {\mathcal {C}}_{\theta }$ for all $\theta \in S$.
One can also look at the Cartan distribution of a PDE ${\mathcal {E}}\subset J^{k}(E,m)$ more intrinsically, defining
${\mathcal {C}}({\mathcal {E}}):=\{{\mathcal {C}}_{\theta }\cap T_{\theta }({\mathcal {E}})~|~\theta \in {\mathcal {E}}\}$
In this sense, the pair $({\mathcal {E}},{\mathcal {C}}({\mathcal {E}}))$ encodes the information about the solutions of the differential equation ${\mathcal {E}}$.
Prolongations of PDEs
Given a differential equation ${\mathcal {E}}\subset J^{l}(E,m)$ of order $l$, its $k$-th prolongation is defined as
${\mathcal {E}}^{k}:=J^{k}({\mathcal {E}},m)\cap J^{k+l}(E,m)\subseteq J^{k+l}(E,m)$
where both $J^{k}({\mathcal {E}},m)$ and $J^{k+l}(E,m)$ are viewed as embedded submanifolds of $J^{k}(J^{l}(E,m),m)$, so that their intersection is well-defined. However, such an intersection is not necessarily a manifold again, hence ${\mathcal {E}}^{k}$ may not be an equation of order $k+l$. One therefore usually requires ${\mathcal {E}}$ to be "nice enough" such that at least its first prolongation is indeed a submanifold of $J^{k+1}(E,m)$.
Below we will assume that the PDE is formally integrable, i.e. all prolongations ${\mathcal {E}}^{k}$ are smooth manifolds and all projections ${\mathcal {E}}^{k}\to {\mathcal {E}}^{k-1}$ are smooth surjective submersions. Note that a suitable version of Cartan–Kuranishi prolongation theorem guarantees that, under minor regularity assumptions, checking the smoothness of a finite number of prolongations is enough. Then the inverse limit of the sequence $\{{\mathcal {E}}^{k}\}_{k\in \mathbb {N} }$ extends the definition of prolongation to the case when $k$ goes to infinity, and the space ${\mathcal {E}}^{\infty }$ has the structure of a profinite-dimensional manifold.[5]
Definition of a diffiety
An elementary diffiety is a pair $({\mathcal {E}}^{\infty },{\mathcal {C}}({\mathcal {E}}^{\infty }))$ where ${\mathcal {E}}\subset J^{k}(E,m)$ is a $k$-th order differential equation on some manifold, ${\mathcal {E}}^{\infty }$ its infinite prolongation and ${\mathcal {C}}({\mathcal {E}}^{\infty })$ its Cartan distribution. Note that, unlike in the finite case, one can show that the Cartan distribution ${\mathcal {C}}({\mathcal {E}}^{\infty })$ is $m$-dimensional and involutive. However, due to the infinite-dimensionality of the ambient manifold, the Frobenius theorem does not hold, therefore ${\mathcal {C}}({\mathcal {E}}^{\infty })$ is not integrable
A diffiety is a triple $({\mathcal {O}},{\mathcal {F}}({\mathcal {O}}),{\mathcal {C}}({\mathcal {O}}))$, consisting of
• a (generally infinite-dimensional) manifold ${\mathcal {O}}$
• the algebra of its smooth functions ${\mathcal {F}}({\mathcal {O}})$
• a finite-dimensional distribution ${\mathcal {C}}({\mathcal {O}})$,
such that $({\mathcal {O}},{\mathcal {F}}({\mathcal {O}}),{\mathcal {C}}({\mathcal {O}}))$ is locally of the form $({\mathcal {E}}^{\infty },{\mathcal {F}}({\mathcal {E}}^{\infty }),{\mathcal {C}}({\mathcal {E}}^{\infty }))$, where $({\mathcal {E}}^{\infty },{\mathcal {C}}({\mathcal {E}}^{\infty }))$ is an elementary diffiety and ${\mathcal {F}}({\mathcal {E}}^{\infty })$ denotes the algebra of smooth functions on ${\mathcal {E}}^{\infty }$. Here locally means a suitable localisation with respect to the Zariski topology corresponding to the algebra ${\mathcal {F}}({\mathcal {O}})$.
The dimension of ${\mathcal {C}}({\mathcal {O}})$ is called dimension of the diffiety and its denoted by $\mathrm {Dim} ({\mathcal {O}})$, with a capital D (to distinguish it from the dimension of ${\mathcal {O}}$ as a manifold).
Morphisms of diffieties
A morphism between two diffieties $({\mathcal {O}},{\mathcal {F}}({\mathcal {O}}),{\mathcal {C}}({\mathcal {O}}))$ and $({\mathcal {O}}',{\mathcal {F}}({\mathcal {O}}'),{\mathcal {C}}({\mathcal {O'}}))$ consists of a smooth map $\Phi :{\mathcal {O}}\rightarrow {\mathcal {O}}'$ :{\mathcal {O}}\rightarrow {\mathcal {O}}'} whose pushforward preserves the Cartan distribution, i.e. such that, for every point $\theta \in {\mathcal {O}}$, one has $d_{\theta }\Phi ({\mathcal {C}}_{\theta })\subseteq {\mathcal {C}}_{\Phi (\theta )}$.
Diffieties together with their morphisms define the category of differential equations.[3]
Applications
Vinogradov sequence
The Vinogradov ${\mathcal {C}}$-spectral sequence (or, for short, Vinogradov sequence) is a spectral sequence associated to a diffiety, which can be used to investigate certain properties of the formal solution space of a differential equation by exploiting its Cartan distribution ${\mathcal {C}}$.[6]
Given a diffiety $({\mathcal {O}},{\mathcal {F}}({\mathcal {O}}),{\mathcal {C}}({\mathcal {O}}))$, consider the algebra of differential forms over ${\mathcal {O}}$
$\Omega ({\mathcal {O}}):=\sum _{i\geq 0}\Omega ^{i}({\mathcal {O}})$
and the corresponding de Rham complex:
$C^{\infty }({\mathcal {O}})\longrightarrow \Omega ^{1}({\mathcal {O}})\longrightarrow \Omega ^{2}({\mathcal {O}})\longrightarrow \cdots $
Its cohomology groups $H^{i}({\mathcal {O}}):={\text{ker}}({\text{d}}_{i})/{\text{im}}({\text{d}}_{i-1})$ contain some structural information about the PDE; however, due to the Poincaré Lemma, they all vanish locally. In order to extract much more and even local information, one thus needs to take the Cartan distribution into account and introduce a more sophisticated sequence. To this end, let
${\mathcal {C}}\Omega ({\mathcal {O}})=\sum _{i\geq 0}{\mathcal {C}}\Omega ^{i}({\mathcal {O}})\subseteq \Omega ({\mathcal {O}})$
be the submodule of differential forms over ${\mathcal {O}}$ whose restriction to the distribution ${\mathcal {C}}$ vanishes, i.e.
${\mathcal {C}}\Omega ^{p}({\mathcal {O}}):=\{w\in \Omega ^{p}({\mathcal {O}})\mid w(X_{1},\cdots ,X_{p})=0\quad \forall ~X_{1},\ldots ,X_{p}\in {\mathcal {C}}({\mathcal {O}})\}.$
Note that ${\mathcal {C}}\Omega ^{i}({\mathcal {O}})\subseteq \Omega ^{i}({\mathcal {O}})$ is actually a differential ideal since it is stable w.r.t. to the de Rham differential, i.e. ${\text{d}}({\mathcal {C}}\Omega ^{i}({\mathcal {O}}))\subset {\mathcal {C}}\Omega ^{i+1}({\mathcal {O}})$.
Now let ${\mathcal {C}}^{k}\Omega ({\mathcal {O}})$ be its $k$-th power, i.e. the linear subspace of ${\mathcal {C}}\Omega $ generated by $w_{1}\wedge \cdots \wedge w_{k},~w_{i}\in {\mathcal {C}}\Omega $. Then one obtains a filtration
$\Omega ({\mathcal {O}})\supset {\mathcal {C}}\Omega ({\mathcal {O}})\supset {\mathcal {C}}^{2}\Omega ({\mathcal {O}})\supset \cdots $
and since all ideals ${\mathcal {C}}^{k}\Omega $ are stable, this filtration completely determines the following spectral sequence:
${\mathcal {C}}E({\mathcal {O}})=\{E_{r}^{p,q},{\text{d}}_{r}^{p,q}\}\qquad {\text{where}}\qquad E_{0}^{p,q}:={\frac {{\mathcal {C}}^{p}\Omega ^{p+q}({\mathcal {O}})}{{\mathcal {C}}^{p+1}\Omega ^{p+q}({\mathcal {O}})}},\qquad {\text{and}}\qquad E_{r+1}^{p,q}:=H(E_{r}^{p,q},d_{r}^{p,q}).$
The filtration above is finite in each degree, i.e. for every $k\geq 0$
$\Omega ^{k}({\mathcal {O}})\supset {\mathcal {C}}^{1}\Omega ^{k}({\mathcal {O}})\supset \cdots \supset {\mathcal {C}}^{k+1}\Omega ^{k}({\mathcal {O}})=0,$
so that the spectral sequence converges to the de Rham cohomology $H({\mathcal {O}})$ of the diffiety. One can therefore analyse the terms of the spectral sequence order by order to recover information on the original PDE. For instance:[7]
• $E_{1}^{0,n}$ corresponds to action functionals constrained by the PDE ${\mathcal {E}}$. In particular, for ${\mathcal {L}}\in E_{1}^{0,n}$, the corresponding Euler-Lagrange equation is ${\text{d}}_{1}^{0,n}{\mathcal {L}}=0$.
• $E_{1}^{0,n-1}$ corresponds to conservation laws for solutions of ${\mathcal {E}}$.
• $E_{2}$ is interpreted as characteristic classes of bordisms of solutions of ${\mathcal {E}}$.
Many higher-order terms do not have an interpretation yet.
Variational bicomplex
As a particular case, starting with a fibred manifold $\pi :E\to X$ and its jet bundle $J^{k}(\pi )$ instead of the jet space $J^{k}(E,m)$, instead of the ${\mathcal {C}}$-spectral sequence one obtains the slightly less general variational bicomplex. More precisely, any bicomplex determines two spectral sequences: one of the two spectral sequences determined by the variational bicomplex is exactly the Vinogradov ${\mathcal {C}}$-spectral sequence. However, the variational bicomplex was developed independently from the Vinogradov sequence.[8][9]
Similarly to the terms of the spectral sequence, many terms of the variational bicomplex can be given a physical interpretation in classical field theory: for example, one obtains cohomology classes corresponding to action functionals, conserved currents, gauge charges, etc.[10]
Secondary calculus
Vinogradov developed a theory, known as secondary calculus, to formalise in cohomological terms the idea of a differential calculus on the space of solutions of a given system of PDEs (i.e. the space of integral manifolds of a given diffiety).[11][12][13][3]
In other words, secondary calculus provides substitutes for functions, vector fields, differential forms, differential operators, etc., on a (generically) very singular space where these objects cannot be defined in the usual (smooth) way on the space of solution. Furthermore, the space of these new objects are naturally endowed with the same algebraic structures of the space of the original objects.[14]
More precisely, consider the horizontal De Rham complex ${\overline {\Omega }}^{\bullet }({\mathcal {O}}):=\Gamma (\wedge ^{\bullet }{\mathcal {C(O)}}^{*})$ of a diffiety, which can be seen as the leafwise de Rham complex of the involutive distribution ${\mathcal {C(O)}}$or, equivalently, the Lie algebroid complex of the Lie algebroid ${\mathcal {C(O)}}$. Then the complex ${\overline {\Omega }}^{\bullet }({\mathcal {O}})$ becomes naturally a commutative DG algebra together with a suitable differential ${\overline {d}}$. Then, possibly tensoring with the normal bundle ${\mathcal {V}}:=T{\mathcal {O}}/{\mathcal {C(O)}}\to {\mathcal {O}}$, its cohomology is used to define the following "secondary objects":
• secondary functions are elements of the cohomology ${\overline {H}}^{\bullet }({\mathcal {O}})=H^{\bullet }({\overline {\Omega }}^{\bullet }({\mathcal {O}}),{\overline {d}})$, which is naturally a commutative DG algebra (it is actually the first page of the ${\mathcal {C}}$-spectral sequence discussed above);
• secondary vector fields are elements of the cohomology ${\overline {H}}^{\bullet }({\mathcal {O}},{\mathcal {V}})=H^{\bullet }({\overline {\Omega }}^{\bullet }({\mathcal {O}}\otimes {\mathcal {V}}),{\overline {d}})$, which is naturally a Lie algebra; moreover, it forms a graded Lie-Rinehart algebra together with ${\overline {H}}^{\bullet }({\mathcal {O}})$;
• secondary differential $p$-forms are elements of the cohomology ${\overline {H}}^{\bullet }({\mathcal {O}},\wedge ^{p}{\mathcal {V}}^{*})=H^{\bullet }({\overline {\Omega }}^{\bullet }({\mathcal {O}}\otimes \wedge ^{p}{\mathcal {V}}^{*}),{\overline {d}})$, which is naturally a commutative DG algebra.
Secondary calculus can also be related to the covariant Phase Space, i.e. the solution space of the Euler-Lagrange equations associated to a Lagrangian field theory.[15]
See also
• Secondary calculus and cohomological physics
• Partial differential equations on Jet bundles
• Differential ideal
• Differential calculus over commutative algebras
Another way of generalizing ideas from algebraic geometry is differential algebraic geometry.
References
1. Vinogradov, A. M. (March 1984). "Local symmetries and conservation laws". Acta Applicandae Mathematicae. 2 (1): 21–78. doi:10.1007/BF01405491. ISSN 0167-8019. S2CID 121860845.
2. Saunders, D. J. (1989). The Geometry of Jet Bundles. London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press. doi:10.1017/cbo9780511526411. ISBN 978-0-521-36948-0.
3. Vinogradov, A. M. (2001). Cohomological analysis of partial differential equations and secondary calculus. Providence, R.I.: American Mathematical Society. ISBN 0-8218-2922-X. OCLC 47296188.
4. Krasil'shchik, I. S.; Lychagin, V. V.; Vinogradov, A. M. (1986). Geometry of jet spaces and nonlinear partial differential equations. Adv. Stud. Contemp. Math., N. Y. Vol. 1. New York etc.: Gordon and Breach Science Publishers. ISBN 978-2-88124-051-5.
5. Güneysu, Batu; Pflaum, Markus J. (2017-01-10). "The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs". SIGMA. Symmetry, Integrability and Geometry: Methods and Applications. 13: 003. arXiv:1308.1005. Bibcode:2017SIGMA..13..003G. doi:10.3842/SIGMA.2017.003. S2CID 15871902.
6. Vinogradov, A. M. (1978). "A spectral sequence associated with a nonlinear differential equation and algebro-geometric foundations of Lagrangian field theory with constraints". Soviet Math. Dokl. (in Russian). 19: 144–148 – via Math-Net.Ru.
7. Symmetries and conservation laws for differential equations of mathematical physics. A. V. Bocharov, I. S. Krasilʹshchik, A. M. Vinogradov. Providence, R.I.: American Mathematical Society. 1999. ISBN 978-1-4704-4596-6. OCLC 1031947580.{{cite book}}: CS1 maint: others (link)
8. Tulczyjew, W. M. (1980). García, P. L.; Pérez-Rendón, A.; Souriau, J. M. (eds.). "The Euler-Lagrange resolution". Differential Geometrical Methods in Mathematical Physics. Lecture Notes in Mathematics. Berlin, Heidelberg: Springer. 836: 22–48. doi:10.1007/BFb0089725. ISBN 978-3-540-38405-2.
9. Tsujishita, Toru (1982). "On variation bicomplexes associated to differential equations". Osaka Journal of Mathematics. 19 (2): 311–363. ISSN 0030-6126.
10. "variational bicomplex in nLab". ncatlab.org. Retrieved 2021-12-11.
11. Vinogradov, A.M. (1984-04-30). "The b-spectral sequence, Lagrangian formalism, and conservation laws. I. The linear theory". Journal of Mathematical Analysis and Applications. 100 (1): 1–40. doi:10.1016/0022-247X(84)90071-4.
12. Vinogradov, A. M. (1984-04-30). "The b-spectral sequence, Lagrangian formalism, and conservation laws. II. The nonlinear theory". Journal of Mathematical Analysis and Applications. 100 (1): 41–129. doi:10.1016/0022-247X(84)90072-6. ISSN 0022-247X.
13. Henneaux, Marc; Krasil′shchik, Joseph; Vinogradov, Alexandre, eds. (1998). Secondary Calculus and Cohomological Physics. Contemporary Mathematics. Vol. 219. Providence, Rhode Island: American Mathematical Society. doi:10.1090/conm/219. ISBN 978-0-8218-0828-3.
14. Vitagliano, Luca (2014). "On the strong homotopy Lie–Rinehart algebra of a foliation". Communications in Contemporary Mathematics. 16 (6): 1450007. arXiv:1204.2467. doi:10.1142/S0219199714500072. ISSN 0219-1997. S2CID 119704524.
15. Vitagliano, Luca (2009-04-01). "Secondary calculus and the covariant phase space". Journal of Geometry and Physics. 59 (4): 426–447. arXiv:0809.4164. Bibcode:2009JGP....59..426V. doi:10.1016/j.geomphys.2008.12.001. ISSN 0393-0440. S2CID 21787052.
External links
• The Diffiety Institute (frozen since 2010)
• The Levi-Civita Institute (successor of above site)
• Geometry of Differential Equations
• Differential Geometry and PDEs
Manifolds (Glossary)
Basic concepts
• Topological manifold
• Atlas
• Differentiable/Smooth manifold
• Differential structure
• Smooth atlas
• Submanifold
• Riemannian manifold
• Smooth map
• Submersion
• Pushforward
• Tangent space
• Differential form
• Vector field
Main results (list)
• Atiyah–Singer index
• Darboux's
• De Rham's
• Frobenius
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• Hopf–Rinow
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• Whitney embedding
Maps
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manifolds
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• Manifold with boundary
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• Poisson
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• (Pseudo−, Sub−) Riemannian
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Bundles
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• Classification of manifolds
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Generalizations
• Banach manifold
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• K-theory
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• over commutative algebras
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| Wikipedia |
Asymmetry
Asymmetry is the absence of, or a violation of, symmetry (the property of an object being invariant to a transformation, such as reflection).[1] Symmetry is an important property of both physical and abstract systems and it may be displayed in precise terms or in more aesthetic terms.[2] The absence of or violation of symmetry that are either expected or desired can have important consequences for a system.
This article is about the absence of symmetry. For a specific use in mathematics, see asymmetric relation. For other uses, see Asymmetry (disambiguation).
In organisms
Due to how cells divide in organisms, asymmetry in organisms is fairly usual in at least one dimension, with biological symmetry also being common in at least one dimension.
Louis Pasteur proposed that biological molecules are asymmetric because the cosmic [i.e. physical] forces that preside over their formation are themselves asymmetric. While at his time, and even now, the symmetry of physical processes are highlighted, it is known that there are fundamental physical asymmetries, starting with time.
Asymmetry in biology
Asymmetry is an important and widespread trait, having evolved numerous times in many organisms and at many levels of organisation (ranging from individual cells, through organs, to entire body-shapes). Benefits of asymmetry sometimes have to do with improved spatial arrangements, such as the left human lung being smaller, and having one fewer lobes than the right lung to make room for the asymmetrical heart. In other examples, division of function between the right and left half may have been beneficial and has driven the asymmetry to become stronger. Such an explanation is usually given for mammal hand or paw preference (handedness), an asymmetry in skill development in mammals. Training the neural pathways in a skill with one hand (or paw) may take less effort than doing the same with both hands.[3]
Nature also provides several examples of handedness in traits that are usually symmetric. The following are examples of animals with obvious left-right asymmetries:
• Most snails, because of torsion during development, show remarkable asymmetry in the shell and in the internal organs.[4]
• Male fiddler crabs have one big claw and one small claw.[5]
• The narwhal's tusk is a left incisor which can grow up to 10 feet in length and forms a left-handed helix.[6]
• Flatfish have evolved to swim with one side upward, and as a result have both eyes on one side of their heads.[7]
• Several species of owls exhibit asymmetries in the size and positioning of their ears, which is thought to help locate prey.[8]
• Many animals (ranging from insects to mammals) have asymmetric male genitalia. The evolutionary cause behind this is, in most cases, still a mystery.[9]
As an indicator of unfitness
• Certain disturbances during the development of the organism, resulting in birth defects.
• Injuries after cell division that cannot be biologically repaired, such as a lost limb from an accident.
Since birth defects and injuries are likely to indicate poor health of the organism, defects resulting in asymmetry often put an animal at a disadvantage when it comes to finding a mate. For example, a greater degree of facial symmetry is seen as more attractive in humans, especially in the context of mate selection. In general, there is a correlation between symmetry and fitness-related traits such as growth rate, fecundity and survivability for many species. This means that, through sexual selection, individuals with greater symmetry (and therefore fitness) tend to be preferred as mates, as they are more likely to produce healthy offspring.[10]
In structures
Pre-modern architectural styles tended to place an emphasis on symmetry, except where extreme site conditions or historical developments lead away from this classical ideal. To the contrary, modernist and postmodern architects became much more free to use asymmetry as a design element.
While most bridges employ a symmetrical form due to intrinsic simplicities of design, analysis and fabrication and economical use of materials, a number of modern bridges have deliberately departed from this, either in response to site-specific considerations or to create a dramatic design statement.
Some asymmetrical structures
• Eastern span replacement of the San Francisco – Oakland Bay Bridge
• Puente de la Mujer
• Auditorio de Tenerife
• Blohm & Voss BV 141 aircraft
• A proa, a form of outrigger canoe
In fire protection
In fire-resistance rated wall assemblies, used in passive fire protection, including, but not limited to, high-voltage transformer fire barriers, asymmetry is a crucial aspect of design. When designing a facility, it is not always certain, that in the event of fire, which side a fire may come from. Therefore, many building codes and fire test standards outline, that a symmetrical assembly, need only be tested from one side, because both sides are the same. However, as soon as an assembly is asymmetrical, both sides must be tested and the test report is required to state the results for each side. In practical use, the lowest result achieved is the one that turns up in certification listings. Neither the test sponsor, nor the laboratory can go by an opinion or deduction as to which side was in more peril as a result of contemplated testing and then test only one side. Both must be tested in order to be compliant with test standards and building codes.
In mathematics
See also: Symmetry in mathematics
In mathematics, asymmetry can arise in various ways. Examples include asymmetric relations, asymmetry of shapes in geometry, asymmetric graphs et cetera.
Asymmetric Relation
An asymmetric relation is a binary relation $R$ defined on a set of elements such that if $aRb$ holds for elements $a$ and $b$, then $bRa$ must be false. Stated differently, an asymmetric relation is characterized by a necessary absence of symmetry of the relation in the opposite direction.
Inequalities exemplify asymmetric relations. Consider elements $a$ and $b$. If $a$ is less than $b$ ($a<b$), then $a$ cannot be greater than $b$ ($a\ngtr b$).[11] This highlights how the relations "less than", and similarly "greater than", are not symmetric.
In contrast, if $a$ is equal to $b$ ($a=b$), then $b$ is also equal to $a$ ($b=a$). Thus the binary relation "equal to" is a symmetric one.
In chemistry
Certain molecules are chiral; that is, they cannot be superposed upon their mirror image. Chemically identical molecules with different chirality are called enantiomers; this difference in orientation can lead to different properties in the way they react with biological systems.
In physics
Asymmetry arises in physics in a number of different realms.
Thermodynamics
The original non-statistical formulation of thermodynamics was asymmetrical in time: it claimed that the entropy in a closed system can only increase with time. This was derived from the Second Law (any of the two, Clausius' or Lord Kelvin's statement can be used since they are equivalent) and using the Clausius' Theorem (see Kerson Huang ISBN 978-0471815181). The later theory of statistical mechanics, however, is symmetric in time. Although it states that a system significantly below maximum entropy is very likely to evolve towards higher entropy, it also states that such a system is very likely to have evolved from higher entropy.
Particle physics
Symmetry is one of the most powerful tools in particle physics, because it has become evident that practically all laws of nature originate in symmetries. Violations of symmetry therefore present theoretical and experimental puzzles that lead to a deeper understanding of nature. Asymmetries in experimental measurements also provide powerful handles that are often relatively free from background or systematic uncertainties.
Parity violation
Until the 1950s, it was believed that fundamental physics was left-right symmetric; i.e., that interactions were invariant under parity. Although parity is conserved in electromagnetism, strong interactions and gravity, it turns out to be violated in weak interactions. The Standard Model incorporates parity violation by expressing the weak interaction as a chiral gauge interaction. Only the left-handed components of particles and right-handed components of antiparticles participate in weak interactions in the Standard Model. A consequence of parity violation in particle physics is that neutrinos have only been observed as left-handed particles (and antineutrinos as right-handed particles).
In 1956–1957 Chien-Shiung Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson found a clear violation of parity conservation in the beta decay of cobalt-60. Simultaneously, R. L. Garwin, Leon Lederman, and R. Weinrich modified an existing cyclotron experiment and immediately verified parity violation.
CP violation
After the discovery of the violation of parity in 1956–57, it was believed that the combined symmetry of parity (P) and simultaneous charge conjugation (C), called CP, was preserved. For example, CP transforms a left-handed neutrino into a right-handed antineutrino. In 1964, however, James Cronin and Val Fitch provided clear evidence that CP symmetry was also violated in an experiment with neutral kaons.
CP violation is one of the necessary conditions for the generation of a baryon asymmetry in the early universe.
Combining the CP symmetry with simultaneous time reversal (T) produces a combined symmetry called CPT symmetry. CPT symmetry must be preserved in any Lorentz invariant local quantum field theory with a Hermitian Hamiltonian. As of 2006, no violations of CPT symmetry have been observed.
Baryon asymmetry of the universe
The baryons (i.e., the protons and neutrons and the atoms that they comprise) observed so far in the universe are overwhelmingly matter as opposed to anti-matter. This asymmetry is called the baryon asymmetry of the universe.
Isospin violation
Isospin is the symmetry transformation of the weak interactions. The concept was first introduced by Werner Heisenberg in nuclear physics based on the observations that the masses of the neutron and the proton are almost identical and that the strength of the strong interaction between any pair of nucleons is the same, independent of whether they are protons or neutrons. This symmetry arises at a more fundamental level as a symmetry between up-type and down-type quarks. Isospin symmetry in the strong interactions can be considered as a subset of a larger flavor symmetry group, in which the strong interactions are invariant under interchange of different types of quarks. Including the strange quark in this scheme gives rise to the Eightfold Way scheme for classifying mesons and baryons.
Isospin is violated by the fact that the masses of the up and down quarks are different, as well as by their different electric charges. Because this violation is only a small effect in most processes that involve the strong interactions, isospin symmetry remains a useful calculational tool, and its violation introduces corrections to the isospin-symmetric results.
In collider experiments
Because the weak interactions violate parity, collider processes that can involve the weak interactions typically exhibit asymmetries in the distributions of the final-state particles. These asymmetries are typically sensitive to the difference in the interaction between particles and antiparticles, or between left-handed and right-handed particles. They can thus be used as a sensitive measurement of differences in interaction strength and/or to distinguish a small asymmetric signal from a large but symmetric background.
• A forward-backward asymmetry is defined as AFB=(NF-NB)/(NF+NB), where NF is the number of events in which some particular final-state particle is moving "forward" with respect to some chosen direction (e.g., a final-state electron moving in the same direction as the initial-state electron beam in electron-positron collisions), while NB is the number of events with the final-state particle moving "backward". Forward-backward asymmetries were used by the LEP experiments to measure the difference in the interaction strength of the Z boson between left-handed and right-handed fermions, which provides a precision measurement of the weak mixing angle.
• A left-right asymmetry is defined as ALR=(NL-NR)/(NL+NR), where NL is the number of events in which some initial- or final-state particle is left-polarized, while NR is the corresponding number of right-polarized events. Left-right asymmetries in Z boson production and decay were measured at the Stanford Linear Collider using the event rates obtained with left-polarized versus right-polarized initial electron beams. Left-right asymmetries can also be defined as asymmetries in the polarization of final-state particles whose polarizations can be measured; e.g., tau leptons.
• A charge asymmetry or particle-antiparticle asymmetry is defined in a similar way. This type of asymmetry has been used to constrain the parton distribution functions of protons at the Tevatron from events in which a produced W boson decays to a charged lepton. The asymmetry between positively and negatively charged leptons as a function of the direction of the W boson relative to the proton beam provides information on the relative distributions of up and down quarks in the proton. Particle-antiparticle asymmetries are also used to extract measurements of CP violation from B meson and anti-B meson production at the BaBar and Belle experiments.
Lexical
Asymmetry is also relevant to grammar and linguistics, especially in the contexts of lexical analysis and transformational grammar.
Enumeration example: In English, there are grammatical rules for specifying coordinate items in an enumeration or series. Similar rules exist for programming languages and mathematical notation. These rules vary, and some require lexical asymmetry to be considered grammatically correct.
For example, in standard written English:
We sell domesticated cats, dogs, and goldfish. ### in-line asymmetric and grammatical
We sell domesticated animals (cats, dogs, goldfish). ### in-line symmetric and grammatical
We sell domesticated animals (cats, dogs, goldfish,). ### in-line symmetric and ungrammatical
We sell domesticated animals: ### outline symmetric and grammatical
- cats
- dogs
- goldfish
In fiction
• In Transformers: Rise of the Beasts, Apelinq's left shoulder seems to be asymmetrical to the right, being slightly more bulked up.
• Bagugan: New Vestroia featuresMira, the Token Girl, wearing a catsuit that goes halfway past the knee on the left side and barely past the waist on the right.
• Several of the title character's outfits in Cardcaptor Sakura have one short and one long stocking.
• Digimon features Angewoman wearing one glove, and also many Digimon including Zudomon owning asymmetric designs.
• In Dragon Ball, Cooler's troops all wear a single shoulder pad on their left side.
See also
• Information asymmetry
• Asymmetric multiprocessing
References
1. "Definition of ASYMMETRY". www.merriam-webster.com. 2023-07-19. Retrieved 2023-07-23.
2. "Definition of SYMMETRY". www.merriam-webster.com. 2023-07-22. Retrieved 2023-07-23.
3. Baofu, Peter (19 Mar 2009). The Future of Post-Human Geometry: A Preface to a New Theory of Infinity, Symmetry, and Dimensionality. p. 149. ISBN 978-1-4438-0524-7.
4. "Surprising Start for Snail Asymmetry". www.science.org. Retrieved 2023-06-04.
5. "Fiddler Crabs". biology-assets.anu.edu.au. Retrieved 2023-06-04.
6. Kingsley, Michael C.S.; Ramsay, Malcolm A. (September 1988). "The Spiral in the Tusk of the Narwhal". Arctic. 41 (3): 1. JSTOR 40510720 – via JSTOR.
7. Friedman, Matt (2008-07-10). "The evolutionary origin of flatfish asymmetry". Nature. 454 (7201): 209–212. doi:10.1038/nature07108. ISSN 1476-4687. PMID 18615083.
8. "Owl hearing | BTO - British Trust for Ornithology". www.bto.org. Retrieved 2023-06-04.
9. Schilthuizen, Menno (2013). "Something gone awry: unsolved mysteries in the evolution of asymmetric animal genitalia". Animal Biology. 63 (1): 1–20. doi:10.1163/15707563-00002398.
10. Little, Anthony C.; Jones, Benedict C.; DeBruine, Lisa M. (2011-06-12). "Facial attractiveness: evolutionary based research". Philosophical Transactions of the Royal Society B: Biological Sciences. 366 (1571): 1638–1659. doi:10.1098/rstb.2010.0404. ISSN 0962-8436. PMC 3130383. PMID 21536551.
11. Introduction to Set Theory, Third Edition, Revised and Expanded: Hrbacek, Jech.
Further reading
• Gardner, Martin (1990), The New Ambidextrous Universe: Symmetry and Asymmetry from Mirror Reflections to Superstrings, 3rd edition, W.H.Freeman & Co Ltd.
• Jan, Yuh-Nung; Yeh Jan, Lily (1999). "Asymmetry across species". Nature Cell Biology. 1 (2): E42–E44. doi:10.1038/10036. PMID 10559895. S2CID 9399564.
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| Wikipedia |
Violet B. Haas
Violet Bushwick Haas (November 23, 1926 – January 21, 1986) was an American applied mathematician specializing in control theory and optimal estimation who became a professor of electrical engineering at Purdue University College of Engineering.
Violet B. Haas
BornNovember 23, 1926
Brooklyn, New York, U.S.
DiedJanuary 21, 1986(1986-01-21) (aged 59)
Lafayette, Indiana, U.S.
Alma materBrooklyn College
Massachusetts Institute of Technology
SpouseFelix Haas
Children3
Scientific career
FieldsControl theory, optimal estimation
InstitutionsUniversity of Detroit Mercy
Purdue University College of Engineering
Doctoral advisorNorman Levinson
Early life and education
Haas was born November 23, 1926 in Brooklyn.[1] She completed a A.B. in mathematics[2] at Brooklyn College in 1947.[1] Haas earned a M.S. (1949) and Ph.D. (1951) in mathematics from the MIT Department of Mathematics (MIT).[1] Her dissertation was titled Singular perturbations of an ordinary differential equation. Norman Levinson was her doctoral advisor.[3] She met her future husband, Felix Haas, a fellow mathematician at MIT.[4] Haas was selected as an American Association of University Women Vassie James Hill Fellow in 1951.[2][4] She was a member of Phi Beta Kappa, Sigma Xi, and Eta Kappa Nu.[2]
Career
Haas was a lecturer at Immaculata College from 1952 to 1955 and an instructor at the University of Connecticut from 1955 to 1956.[2] She was a faculty member at the University of Detroit from 1957 to 1962.[2] Haas also taught at Wayne State University.[4]
She joined the faculty at Purdue University in January 1962[1] as an assistant professor in the college of electrical engineering and computer engineering.[4] By 1978, Haas was a full professor of electrical engineering in the Purdue University College of Engineering.[4][2] Her areas of expertise included optimal control, nonlinear control, and optimal estimation.[2] Due to nepotism rules (her husband was a fellow mathematician), Haas took a position in electrical engineering rather than mathematics.[5]
Haas advocated for women in STEM fields. Some of her earlier academic environments were hostile to women.[5] In a few instances, she was the only department member excluded from grant proposals. This had largely improved by the early 1980s.[5] For 15 years, Haas was the counselor of the Purdue University student chapter of the Society of Women Engineers.[5]
Haas joined the Association for Women in Mathematics in 1975, serving as a coordinator for the speakers' bureau.[5] She was a member of the Institute of Electrical and Electronics Engineers (IEEE) committee on professional opportunities for women and the American Society for Engineering Education (ASEE) constituent committee on women in engineering.[5] For Haas' support and encouragement of women students in engineering, in 1977, she was elected as one of five "Very Important Women" on campus by the Association of Women Students.[6] In 1977, Haas received the D.D. Ewing Award as an outstanding teacher in the Purdue School of Electrical Engineering.[7] She received the 1978 Helen B. Schleman Medallion Award for her service and encouragement of women in academic and professional areas.[7] In the 1970s, Haas was a nominee for the distinguished science award of the Society of Women Engineers.[7]
From 1983 to 1984, Haas was a visiting professor at Massachusetts Institute of Technology through the National Science Foundation visiting professorships for women program. In this position, she was a full time researcher investigating control theory and infinite dimensional control problems.[5]
Haas was active in the control systems community and was on the program committee of the American Control Conference. She was also the Society for Industrial and Applied Mathematics (SIAM) representative to the IEEE conference on decision and control.[5] Mathematician and colleague Pamela G. Coxson stated that Haas' involvement "increased the participation of the mathematical community in these two annual conferences."[5] She is included in biographical listings of Who's Who in the Midwest, Who's Who of American Women, and American Men and Women of Science.[6] She was a former editor of the Women in Engineering Students Newsletter.[1] Haas was a member of the American Association of University Women,[6] League of Women Voters, YWCA, and served on the board of directors of the Lafayette Symphony.[1][2]
Haas and sociologist Carolyn C. Perrucci co-edited the book Women in Scientific and Engineering Professions. University of Michigan Press. 1984. ISBN 978-0-472-10049-1.[4]
Eight months after leaving MIT in 1984, Haas was diagnosed with a brain tumor and was soon unconscious.[5]
Personal life
Haas resided in West Lafayette, Indiana.[1] She was married to Felix Haas.[1] They had a daughter and two sons.[1]
Haas was unconscious from a brain tumor from 1984 until her death on the morning of January 21, 1986 at St. Elizabeth Hospital.[1][6]
In 1990, the Council on the Status of Women at Purdue University established Violet B. Haas award that recognizes people who promote the status of women at the university.[4]
References
Citations
1. Journal & Courier 1986, p. 12.
2. The Indianapolis Star 1986, p. 55.
3. Haas 1951.
4. Purdue University Archives and Special Collections.
5. Coxson 1986, p. 2.
6. Society for Industrial and Applied Mathematics 1986, p. 10.
7. Journal & Courier 1978, p. 6.
Bibliography
• "Obituary". Society for Industrial and Applied Mathematics News. March 1986.
• Coxson, Pamela G. (July 1986). "In Remembrance of Violet Bushwick Haas (1926-1986)". Association for Women in Mathematics Newsletter. Vol. 16, no. 4. pp. 2–3. Retrieved 2022-04-19.
• Haas, Violet B. (1951). Singular perturbations of an ordinary differential equation (Ph.D. thesis). MIT Department of Mathematics. OCLC 30350947.
• "Prof. Haas receives Schleman award for service". Journal & Courier. 1978-05-11. p. 6. Retrieved 2022-04-19.
• "Dr. Violet Haas, 59, dies; was on Purdue faculty". Journal & Courier. 1986-01-22. p. 12. Retrieved 2022-04-19.
• "Haas, Violet B." Purdue University Archives and Special Collections.
• "Memorial rites Friday for Professor Violet Haas". The Indianapolis Star. 1986-01-23. p. 55. Retrieved 2022-04-19.
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| Wikipedia |
Viorel P. Barbu
Viorel P. Barbu (born 14 June 1941) is a Romanian mathematician, specializing in partial differential equations, control theory, and stochastic differential equations.
Biography
He was born in Deleni, Vaslui County, Romania.[1] He attended the Mihail Kogălniceanu High School in Vaslui and then the Costache Negruzzi National College in Iași. Barbu completed his undergraduate degree at the Alexandru Ioan Cuza University of Iași in 1964,[1] and his Ph.D. at the same university in 1969.[2] His doctoral advisor was Adolf Haimovici; his dissertation thesis was titled Regularity Theory of Pseudodifferential Operators.[2] He became a professor at the University of Iași in 1980.[1] His Ph.D. students there included Gheorghe Moroșanu and Daniel Tătaru.[2]
In 1993, he was elected a titular member of the Romanian Academy.[3] In 2011 he was awarded the Order of the Star of Romania, Knight rank by President Traian Băsescu.[4]
Bibliography
Some of his books and papers are:[5][6][7]
• Analysis And Control Of Nonlinear Infinite Dimensional Systems
• Optimization, Optimal Control and Partial Differential Equations
• Nonlinear semigroups and differential equations in Banach spaces
• Hamilton-Jacobi Equations on Hilbert Space
• Stochastic Porous Media Equations
• Nonlinear Differential Equations of Monotone Types in Banach Spaces
• Convexity and Optimization in Banach Spaces
• Optimal Control of Variational Inequalities
References
1. "Pagina personală Acad. Viorel Barbu". uaic.ro.
2. Viorel P. Barbu at the Mathematics Genealogy Project
3. (in Romanian) Membrii Academiei Române din 1866 până în prezent at the Romanian Academy site
4. "Decret nr. 639 din 7 iulie 2011 privind conferirea unor decorații". legislatie.just.ro (in Romanian). July 7, 2011. Retrieved May 1, 2021.
5. "Books by Viorel Barbu (Author of Differential Equations)". goodreads.com.
6. Viorel P. Barbu publications indexed by Google Scholar
7. "Viorel Barbu (Universitatea Alexandru Ioan Cuza, Iași) on ResearchGate - Expertise: Analysis, Civil Engineering, Structural Engineering". researchgate.net.
External links
• Official website
• Official website
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Virasoro conformal block
In two-dimensional conformal field theory, Virasoro conformal blocks (named after Miguel Ángel Virasoro) are special functions that serve as building blocks of correlation functions. On a given punctured Riemann surface, Virasoro conformal blocks form a particular basis of the space of solutions of the conformal Ward identities. Zero-point blocks on the torus are characters of representations of the Virasoro algebra; four-point blocks on the sphere reduce to hypergeometric functions in special cases, but are in general much more complicated. In two dimensions as in other dimensions, conformal blocks play an essential role in the conformal bootstrap approach to conformal field theory.
Definition
Definition from OPEs
Using operator product expansions (OPEs), an $N$-point function on the sphere can be written as a combination of three-point structure constants, and universal quantities called $N$-point conformal blocks.[1][2]
Given an $N$-point function, there are several types of conformal blocks, depending on which OPEs are used. In the case $N=4$, there are three types of conformal blocks, corresponding to three possible decompositions of the same four-point function. Schematically, these decompositions read
$\left\langle V_{1}V_{2}V_{3}V_{4}\right\rangle =\sum _{s}C_{12s}C_{s34}{\mathcal {F}}_{s}^{\text{(s-channel)}}=\sum _{t}C_{14t}C_{t23}{\mathcal {F}}_{t}^{\text{(t-channel)}}=\sum _{u}C_{13u}C_{24u}{\mathcal {F}}_{u}^{\text{(u-channel)}}\ ,$
where $C$ are structure constants and ${\mathcal {F}}$ are conformal blocks. The sums are over representations of the conformal algebra that appear in the CFT's spectrum. OPEs involve sums over the spectrum, i.e. over representations and over states in representations, but the sums over states are absorbed in the conformal blocks.
In two dimensions, the symmetry algebra factorizes into two copies of the Virasoro algebra, called left-moving and right-moving. If the fields are factorized too, then the conformal blocks factorize as well, and the factors are called Virasoro conformal blocks. Left-moving Virasoro conformal blocks are locally holomorphic functions of the fields' positions $z_{i}$; right-moving Virasoro conformal blocks are the same functions of ${\bar {z}}_{i}$. The factorization of a conformal block into Virasoro conformal blocks is of the type
${\mathcal {F}}_{s_{L}\otimes s_{R}}^{\text{(s-channel)}}(\{z_{i}\})={\mathcal {F}}_{s_{L}}^{\text{(s-channel, Virasoro)}}(\{z_{i}\}){\mathcal {F}}_{s_{R}}^{\text{(s-channel, Virasoro)}}(\{{\bar {z}}_{i}\})\ ,$
where $s_{L},s_{R}$ are representations of the left- and right-moving Virasoro algebras respectively.
Definition from Virasoro Ward identities
Conformal Ward identities are the linear equations that correlation functions obey, as a result of conformal symmetry.
In two dimensions, conformal Ward identities decompose into left-moving and right-moving Virasoro Ward identities. Virasoro conformal blocks are solutions of the Virasoro Ward identities.[3][4]
OPEs define specific bases of Virasoro conformal blocks, such as the s-channel basis in the case of four-point blocks. The blocks that are defined from OPEs are special cases of the blocks that are defined from Ward identities.
Properties
Any linear holomorphic equation that is obeyed by a correlation function, must also hold for the corresponding conformal blocks. In addition, specific bases of conformal blocks come with extra properties that are not inherited from the correlation function.
Conformal blocks that involve only primary fields have relatively simple properties. Conformal blocks that involve descendant fields can then be deduced using local Ward identities. An s-channel four-point block of primary fields depends on the four fields' conformal dimensions $\Delta _{i},$ on their positions $z_{i},$ and on the s-channel conformal dimension $\Delta _{s}$. It can be written as ${\mathcal {F}}_{\Delta _{s}}^{(s)}(\Delta _{i}|\{z_{i}\}),$ where the dependence on the Virasoro algebra's central charge is kept implicit.
Linear equations
From the corresponding correlation function, conformal blocks inherit linear equations: global and local Ward identities, and BPZ equations if at least one field is degenerate.[2]
In particular, in an $N$-point block on the sphere, global Ward identities reduce the dependence on the $N$ field positions to a dependence on $N-3$ cross-ratios. In the case $N=4,$
${\mathcal {F}}_{\Delta _{s}}^{(s)}(\{\Delta _{i}\}|\{z_{i}\})=z_{23}^{\Delta _{1}-\Delta _{2}-\Delta _{3}+\Delta _{4}}z_{13}^{-2\Delta _{1}}z_{34}^{\Delta _{1}+\Delta _{2}-\Delta _{3}-\Delta _{4}}z_{24}^{-\Delta _{1}-\Delta _{2}+\Delta _{3}-\Delta _{4}}{\mathcal {F}}_{\Delta _{s}}^{(s)}(\{\Delta _{i}\}|z),$
where $z_{ij}=z_{i}-z_{j},$ and
$z={\frac {z_{12}z_{34}}{z_{13}z_{24}}}$
is the cross-ratio, and the reduced block ${\mathcal {F}}_{\Delta _{s}}^{(s)}(\{\Delta _{i}\}|z)$ coincides with the original block where three positions are sent to $(0,\infty ,1),$
${\mathcal {F}}_{\Delta _{s}}^{(s)}(\{\Delta _{i}\}|z)={\mathcal {F}}_{\Delta _{s}}^{(s)}(\{\Delta _{i}\}|z,0,\infty ,1).$
Singularities
Like correlation functions, conformal blocks are singular when two fields coincide. Unlike correlation functions, conformal blocks have very simple behaviours at some of these singularities. As a consequence of their definition from OPEs, s-channel four-point blocks obey
${\mathcal {F}}_{\Delta _{s}}^{(s)}(\{\Delta _{i}\}|z){\underset {z\to 0}{=}}z^{\Delta _{s}-\Delta _{1}-\Delta _{2}}\left(1+\sum _{n=1}^{\infty }c_{n}z^{n}\right),$
for some coefficients $c_{n}.$ On the other hand, s-channel blocks have complicated singular behaviours at $z=1,\infty $: it is t-channel blocks that are simple at $z=1$, and u-channel blocks that are simple at $z=\infty .$
In a four-point block that obeys a BPZ differential equation, $z=0,1,\infty $ are regular singular points of the differential equation, and $\Delta _{s}-\Delta _{1}-\Delta _{2}$ is a characteristic exponent of the differential equation. For a differential equation of order $n$, the $n$ characteristic exponents correspond to the $n$ values of $\Delta _{s}$ that are allowed by the fusion rules.
Field permutations
Permutations of the fields $V_{i}(z_{i})$ leave the correlation function
$\left\langle \prod _{i=1}^{N}V_{i}(z_{i})\right\rangle $
invariant, and therefore relate different bases of conformal blocks with one another. In the case of four-point blocks, t-channel blocks are related to s-channel blocks by[2]
${\mathcal {F}}_{\Delta }^{(t)}(\Delta _{1},\Delta _{2},\Delta _{3},\Delta _{4}|z_{1},z_{2},z_{3},z_{4})={\mathcal {F}}_{\Delta }^{(s)}(\Delta _{1},\Delta _{4},\Delta _{3},\Delta _{2}|z_{1},z_{4},z_{3},z_{2}),$
or equivalently
${\mathcal {F}}_{\Delta }^{(t)}(\Delta _{1},\Delta _{2},\Delta _{3},\Delta _{4}|z)={\mathcal {F}}_{\Delta }^{(s)}(\Delta _{1},\Delta _{4},\Delta _{3},\Delta _{2}|1-z).$
Fusing matrix
The change of bases from s-channel to t-channel four-point blocks is characterized by the fusing matrix (or fusion kernel) $F$, such that
${\mathcal {F}}_{\Delta _{s}}^{(s)}(\{\Delta _{i}\}|\{z_{i}\})=\int _{i\mathbb {R} }dP_{t}\ F_{\Delta _{s},\Delta _{t}}{\begin{bmatrix}\Delta _{2}&\Delta _{3}\\\Delta _{1}&\Delta _{4}\end{bmatrix}}{\mathcal {F}}_{\Delta _{t}}^{(t)}(\{\Delta _{i}\}|\{z_{i}\}).$
The fusing matrix is a function of the central charge and conformal dimensions, but it does not depend on the positions $z_{i}.$ The momentum $P_{t}$ is defined in terms of the dimension $\Delta _{t}$ by
$\Delta ={\frac {c-1}{24}}-P^{2}.$
The values $P\in i\mathbb {R} $ correspond to the spectrum of Liouville theory.
We also need to introduce two parameters $Q,b$ related to the central charge $c$,
$c=1+6Q^{2},\qquad Q=b+b^{-1}.$
Assuming $c\notin (-\infty ,1)$ and $P_{i}\in i\mathbb {R} $, the explicit expression of the fusing matrix is[5]
${\begin{aligned}F_{\Delta _{s},\Delta _{t}}&{\begin{bmatrix}\Delta _{2}&\Delta _{3}\\\Delta _{1}&\Delta _{4}\end{bmatrix}}=\\&=\left(\prod _{\pm }{\frac {\Gamma _{b}(Q\pm 2P_{s})}{\Gamma _{b}(\pm 2P_{t})}}\right){\frac {\Xi _{+}(P_{1},P_{4},P_{t})\Xi _{+}(P_{2},P_{3},P_{t})}{\Xi _{-}(P_{1},P_{2},P_{s})\Xi _{-}(P_{3},P_{4},P_{s})}}\times \\&\quad \times \int _{{\frac {Q}{4}}+i\mathbb {R} }du\ S_{b}\left(u-P_{12s}\right)S_{b}\left(u-P_{s34}\right)S_{b}\left(u-P_{23t}\right)S_{b}\left(u-P_{t14}\right)\\&\qquad \times S_{b}\left({\tfrac {Q}{2}}-u+P_{1234}\right)S_{b}\left({\tfrac {Q}{2}}-u+P_{st13}\right)S_{b}\left({\tfrac {Q}{2}}-u+P_{st24}\right)S_{b}\left({\tfrac {Q}{2}}-u\right)\end{aligned}}$
where $\Gamma _{b}$ is a double gamma function,
${\begin{aligned}S_{b}(x)&={\frac {\Gamma _{b}(x)}{\Gamma _{b}(Q-x)}}\\[6pt]\Xi _{\epsilon }(P_{1},P_{2},P_{3})&=\prod _{\underset {\epsilon _{1}\epsilon _{2}\epsilon _{3}=\epsilon }{\epsilon _{1},\epsilon _{2},\epsilon _{3}=\pm }}\Gamma _{b}\left({\tfrac {Q}{2}}+\sum _{i}\epsilon _{i}P_{i}\right)\\[6pt]P_{ijk}&=P_{i}+P_{j}+P_{k}\end{aligned}}$
Although its expression is simpler in terms of momentums $P_{i}$ than in terms of conformal dimensions $\Delta _{i}$, the fusing matrix is really a function of $\Delta _{i}$, i.e. a function of $P_{i}$ that is invariant under $P_{i}\to -P_{i}$. In the expression for the fusing matrix, the integral is a hyperbolic Barnes integral. Up to normalization, the fusing matrix coincides with Ruijsenaars' hypergeometric function, with the arguments $P_{s},P_{t}$ and parameters $b,b^{-1},P_{1},P_{2},P_{3},P_{4}$.[6]
In $N$-point blocks on the sphere, the change of bases between two sets of blocks that are defined from different sequences of OPEs can always be written in terms of the fusing matrix, and a simple matrix that describes the permutation of the first two fields in an s-channel block,[3]
${\mathcal {F}}_{\Delta _{s}}^{(s)}(\Delta _{1},\Delta _{2},\Delta _{3},\Delta _{4}|z_{1},z_{2},z_{3},z_{4})=e^{i\pi (\Delta _{s}-\Delta _{1}-\Delta _{2})}{\mathcal {F}}_{\Delta _{s}}^{(s)}(\Delta _{2},\Delta _{1},\Delta _{3},\Delta _{4}|z_{2},z_{1},z_{3},z_{4}).$
Computation of conformal blocks
From the definition
The definition from OPEs leads to an expression for an s-channel four-point conformal block as a sum over states in the s-channel representation, of the type [7]
${\mathcal {F}}_{\Delta _{s}}^{\text{(s)}}(\{\Delta _{i}\}|z)=z^{\Delta _{s}-\Delta _{1}-\Delta _{2}}\sum _{L,L'}z^{|L|}f_{12s}^{L}Q_{L,L'}^{s}f_{43s}^{L'}\ .$
The sums are over creation modes $L,L'$ of the Virasoro algebra, i.e. combinations of the type $L=\prod _{i}L_{-n_{i}}$ of Virasoro generators with $1\leq n_{1}\leq n_{2}\leq \cdots $, whose level is $|L|=\sum n_{i}$. Such generators correspond to basis states in the Verma module with the conformal dimension $\Delta _{s}$. The coefficient $f_{12s}^{L}$ is a function of $\Delta _{1},\Delta _{2},\Delta _{s},L$, which is known explicitly. The matrix element $Q_{L,L'}^{s}$ is a function of $c,\Delta _{s},L,L'$ which vanishes if $|L|\neq |L'|$, and diverges for $|L|=N$ if there is a null vector at level $N$. Up to $|L|=1$, this reads
${\mathcal {F}}_{\Delta _{s}}^{\text{(s)}}(\{\Delta _{i}\}|z)=z^{\Delta _{s}-\Delta _{1}-\Delta _{2}}{\Bigg \{}1+{\frac {(\Delta _{s}+\Delta _{1}-\Delta _{2})(\Delta _{s}+\Delta _{4}-\Delta _{3})}{2\Delta _{s}}}z+O(z^{2}){\Bigg \}}\ .$
(In particular, $Q_{L_{-1},L_{-1}}^{s}={\frac {1}{2\Delta _{s}}}$ does not depend on the central charge $c$.)
Zamolodchikov's recursive representation
In Alexei Zamolodchikov's recursive representation of four-point blocks on the sphere, the cross-ratio $z$ appears via the nome
$q=\exp -\pi {\frac {F({\frac {1}{2}},{\frac {1}{2}},1,1-z)}{F({\frac {1}{2}},{\frac {1}{2}},1,z)}}{\underset {z\to 0}{=}}{\frac {z}{16}}+{\frac {z^{2}}{32}}+O(z^{3})\quad \iff \quad z={\frac {\theta _{2}(q)^{4}}{\theta _{3}(q)^{4}}}{\underset {q\to 0}{=}}16q-128q^{2}+O(q^{3})$
where $F$ is the hypergeometric function, and we used the Jacobi theta functions
$\theta _{2}(q)=2q^{\frac {1}{4}}\sum _{n=0}^{\infty }q^{n(n+1)}\quad ,\quad \theta _{3}(q)=\sum _{n\in {\mathbb {Z} }}q^{n^{2}}$
The representation is of the type
${\mathcal {F}}_{\Delta }^{(s)}(\{\Delta _{i}\}|z)=(16q)^{\Delta -{\frac {1}{4}}Q^{2}}z^{{\frac {1}{4}}Q^{2}-\Delta _{1}-\Delta _{2}}(1-z)^{{\frac {1}{4}}Q^{2}-\Delta _{1}-\Delta _{4}}\theta _{3}(q)^{3Q^{2}-4(\Delta _{1}+\Delta _{2}+\Delta _{3}+\Delta _{4})}H_{\Delta }(\{\Delta _{i}\}|q)\ .$
The function $H_{\Delta }(\{\Delta _{i}\}|q)$ is a power series in $q$, which is recursively defined by
$H_{\Delta }(\{\Delta _{i}\}|q)=1+\sum _{m,n=1}^{\infty }{\frac {(16q)^{mn}}{\Delta -\Delta _{(m,n)}}}R_{m,n}H_{\Delta _{(m,-n)}}(\{\Delta _{i}\}|q)\ .$
In this formula, the positions $\Delta _{(m,n)}$ of the poles are the dimensions of degenerate representations, which correspond to the momentums
$P_{(m,n)}={\frac {1}{2}}\left(mb+nb^{-1}\right)\ .$
The residues $R_{m,n}$ are given by
$R_{m,n}={\frac {2P_{(0,0)}P_{(m,n)}}{\prod _{r=1-m}^{m}\prod _{s=1-n}^{n}2P_{(r,s)}}}\prod _{r{\overset {2}{=}}1-m}^{m-1}\prod _{s{\overset {2}{=}}1-n}^{n-1}\prod _{\pm }(P_{2}\pm P_{1}+P_{(r,s)})(P_{3}\pm P_{4}+P_{(r,s)})\ ,$
where the superscript in ${\overset {2}{=}}$ indicates a product that runs by increments of $2$. The recursion relation for $H_{\Delta }(\{\Delta _{i}\}|q)$ can be solved, giving rise to an explicit (but impractical) formula.[2][8]
While the coefficients of the power series $H_{\Delta }(\{\Delta _{i}\}|q)$ need not be positive in unitary theories, the coefficients of $\prod _{k=1}^{\infty }(1-q^{2k})^{-{\frac {1}{2}}}H_{\Delta }(\{\Delta _{i}\}|q)$ are positive, due to this combination's interpretation in terms of sums of states in the pillow geometry.[9]
The recursive representation can be seen as an expansion around $\Delta =\infty $. It is sometimes called the $\Delta $-recursion, in order to distinguish it from the $c$-recursion: another recursive representation, also due to Alexei Zamolodchikov, which expands around $c=\infty $. Both representations can be generalized to $N$-point Virasoro conformal blocks on arbitrary Riemann surfaces.[10]
From the relation to instanton counting
The Alday–Gaiotto–Tachikawa relation between two-dimensional conformal field theory and supersymmetric gauge theory, more specifically, between the conformal blocks of Liouville theory and Nekrasov partition functions[11] of supersymmetric gauge theories in four dimensions, leads to combinatorial expressions for conformal blocks as sums over Young diagrams. Each diagram can be interpreted as a state in a representation of the Virasoro algebra, times an abelian affine Lie algebra.[12]
Special cases
Zero-point blocks on the torus
A zero-point block does not depend on field positions, but it depends on the moduli of the underlying Riemann surface. In the case of the torus
${\frac {\mathbb {C} }{\mathbb {Z} +\tau \mathbb {Z} }},$
that dependence is better written through $q=e^{2\pi i\tau }$ and the zero-point block associated to a representation ${\mathcal {R}}$ of the Virasoro algebra is
$\chi _{\mathcal {R}}(\tau )=\operatorname {Tr} _{\mathcal {R}}q^{L_{0}-{\frac {c}{24}}},$
where $L_{0}$ is a generator of the Virasoro algebra. This coincides with the character of ${\mathcal {R}}.$ The characters of some highest-weight representations are:[1]
• Verma module with conformal dimension $\Delta ={\tfrac {c-1}{24}}-P^{2}$:
$\chi _{P}(\tau )={\frac {q^{-P^{2}}}{\eta (\tau )}},$
where $\eta (\tau )$ is the Dedekind eta function.
• Degenerate representation with the momentum $P_{(r,s)}$:
$\chi _{(r,s)}(\tau )=\chi _{P_{(r,s)}}(\tau )-\chi _{P_{(r,-s)}}(\tau ).$
• Fully degenerate representation at rational $b^{2}=-{\tfrac {p}{q}}$:
$\chi _{(r,s)}(\tau )=\sum _{k\in \mathbb {Z} }\left(\chi _{P_{(r,s)}+ik{\sqrt {pq}}}(\tau )-\chi _{P_{(r,-s)}+ik{\sqrt {pq}}}(\tau )\right).$
The characters transform linearly under the modular transformations:
$\tau \to {\frac {a\tau +b}{c\tau +d}},\qquad {\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in SL_{2}(\mathbb {Z} ).$
In particular their transformation under $\tau \to -{\tfrac {1}{\tau }}$ is described by the modular S-matrix. Using the S-matrix, constraints on a CFT's spectrum can be derived from the modular invariance of the torus partition function, leading in particular to the ADE classification of minimal models.[13]
One-point blocks on the torus
An arbitrary one-point block on the torus can be written in terms of a four-point block on the sphere at a different central charge. This relation maps the modulus of the torus to the cross-ratio of the four points' positions, and three of the four fields on the sphere have the fixed momentum $P_{(0,{\frac {1}{2}})}={\tfrac {1}{4b}}$:[14][15]
$H_{P'}^{\text{torus}}(P_{1}|q^{2})=H_{P}\left(\left.{\tfrac {1}{4b}},P_{2},{\tfrac {1}{4b}},{\tfrac {1}{4b}}\right|q\right)\quad {\text{with}}\quad \left\{{\begin{array}{l}b={\frac {b'}{\sqrt {2}}}\\P_{2}={\frac {P_{1}}{\sqrt {2}}}\\P={\sqrt {2}}P'\end{array}}\right.$
where
• $H_{P_{s}}\left(\left.P_{1},P_{2},P_{3},P_{4}\right|q\right)$ is the non-trivial factor of the sphere four-point block in Zamolodchikov's recursive representation, written in terms of momentums $P_{i}$ instead of dimensions $\Delta _{i}$.
• $H_{P}^{\text{torus}}(P_{1}|q)$ is the non-trivial factor of the torus one-point block ${\mathcal {F}}_{\Delta }^{\text{torus}}(\Delta _{1}|q)=q^{\Delta -{\frac {c-1}{24}}}\eta (q)^{-1}H_{\Delta }^{\text{torus}}(\Delta _{1}|q)$, where $\eta (q)$ is the Dedekind eta function, the modular parameter $\tau $ of the torus is such that $q=e^{2\pi i\tau }$, and the field on the torus has the dimension $\Delta _{1}$.
The recursive representation of one-point blocks on the torus is[16]
$H_{\Delta }^{\text{torus}}(\Delta _{1}|q)=1+\sum _{m,n=1}^{\infty }{\frac {q^{mn}}{\Delta -\Delta _{(m,n)}}}R_{m,n}^{\text{torus}}H_{\Delta _{(m,-n)}}^{\text{torus}}(\Delta _{1}|q)\ ,$
where the residues are
$R_{m,n}^{\text{torus}}={\frac {2P_{(0,0)}P_{(m,n)}}{\prod _{r=1-m}^{m}\prod _{s=1-n}^{n}2P_{(r,s)}}}\prod _{r{\overset {2}{=}}1-2m}^{2m-1}\prod _{s{\overset {2}{=}}1-2n}^{2n-1}\left(P_{1}+P_{(r,s)}\right)\ .$
Under modular transformations, one-point blocks on the torus behave as
${\mathcal {F}}_{P}^{\text{torus}}\left(P_{1}|-{\tfrac {1}{\tau }}\right)=\int _{i\mathbb {R} }dP'\ S_{P,P'}(P_{1}){\mathcal {F}}_{P'}^{\text{torus}}\left(P_{1}|\tau \right)\ ,$
where the modular kernel is[17][18]
$S_{P,P'}(P_{1})={\frac {2^{-{\frac {5}{2}}}}{S_{b}({\frac {Q}{2}}+P_{1})}}\prod _{\pm }{\frac {\Gamma _{b}(Q\pm 2P)}{\Gamma _{b}(\pm 2P')}}{\frac {\Gamma _{b}({\frac {Q}{2}}-P_{1}\pm 2P')}{\Gamma _{b}({\frac {Q}{2}}-P_{1}\pm 2P)}}\int _{i\mathbb {R} }du\ e^{4\pi iPu}\prod _{\pm ,\pm }S_{b}\left({\tfrac {Q}{4}}+{\tfrac {P_{1}}{2}}\pm u\pm P'\right)\ .$
Hypergeometric blocks
For a four-point function on the sphere
$\left\langle V_{\langle 2,1\rangle }(x)\prod _{i=1}^{3}V_{\Delta _{i}}(z_{i})\right\rangle $
where one field has a null vector at level two, the second-order BPZ equation reduces to the hypergeometric equation. A basis of solutions is made of the two s-channel conformal blocks that are allowed by the fusion rules, and these blocks can be written in terms of the hypergeometric function,
${\begin{aligned}{\mathcal {F}}_{P_{1}+\epsilon {\frac {b}{2}}}^{(s)}(z)&=z^{{\frac {1}{2}}+{\frac {b^{2}}{2}}+b\epsilon P_{1}}(1-z)^{{\frac {1}{2}}+{\frac {b^{2}}{2}}+bP_{3}}\\&\times F\left({\tfrac {1}{2}}+b(\epsilon P_{1}+P_{2}+P_{3}),{\tfrac {1}{2}}+b(\epsilon P_{1}-P_{2}+P_{3}),1+2b\epsilon P_{1},z\right),\end{aligned}}$
with $\epsilon \in \{+,-\}.$ Another basis is made of the two t-channel conformal blocks,
${\begin{aligned}{\mathcal {F}}_{P_{3}+\epsilon {\frac {b}{2}}}^{(t)}(z)&=z^{{\frac {1}{2}}+{\frac {b^{2}}{2}}+bP_{1}}(1-z)^{{\frac {1}{2}}+{\frac {b^{2}}{2}}+b\epsilon P_{3}}\\&\times F\left({\tfrac {1}{2}}+b(P_{1}+P_{2}+\epsilon P_{3}),{\tfrac {1}{2}}+b(P_{1}-P_{2}+\epsilon P_{3}),1+2b\epsilon P_{3},1-z\right).\end{aligned}}$
The fusing matrix is the matrix of size two such that
${\mathcal {F}}_{P_{1}+\epsilon _{1}{\frac {b}{2}}}^{(s)}(x)=\sum _{\epsilon _{3}=\pm }F_{\epsilon _{1},\epsilon _{3}}{\mathcal {F}}_{P_{3}+\epsilon _{3}{\frac {b}{2}}}^{(t)}(x),$
whose explicit expression is
$F_{\epsilon _{1},\epsilon _{3}}={\frac {\Gamma (1-2b\epsilon _{1}P_{1})\Gamma (2b\epsilon _{3}P_{3})}{\prod _{\pm }\Gamma ({\frac {1}{2}}+b(-\epsilon _{1}P_{1}\pm P_{2}+\epsilon _{3}P_{3}))}}.$
Hypergeometric conformal blocks play an important role in the analytic bootstrap approach to two-dimensional CFT.[19][20]
Solutions of the Painlevé VI equation
If $c=1,$ then certain linear combinations of s-channel conformal blocks are solutions of the Painlevé VI nonlinear differential equation.[21] The relevant linear combinations involve sums over sets of momentums of the type $P_{s}+i\mathbb {Z} .$ This allows conformal blocks to be deduced from solutions of the Painlevé VI equation and vice versa. This also leads to a relatively simple formula for the fusing matrix at $c=1.$[22] Curiously, the $c=\infty $ limit of conformal blocks is also related to the Painlevé VI equation.[23] The relation between the $c=\infty $ and the $c=1$ limits, mysterious on the conformal field theory side, is explained naturally in the context of four dimensional gauge theories, using blowup equations,[24][25] and can be generalized to more general pairs $c,c'$of central charges.
Generalizations
Other representations of the Virasoro algebra
The Virasoro conformal blocks that are described in this article are associated to a certain type of representations of the Virasoro algebra: highest-weight representations, in other words Verma modules and their cosets.[2] Correlation functions that involve other types of representations give rise to other types of conformal blocks. For example:
• Logarithmic conformal field theory involves representations where the Virasoro generator $L_{0}$ is not diagonalizable, which give rise to blocks that depend logarithmically on field positions.
• Representations can be built from states on which some annihilation modes of the Virasoro algebra act diagonally, rather than vanishing. The corresponding conformal blocks have been called irregular conformal blocks.[26]
Larger symmetry algebras
In a theory whose symmetry algebra is larger than the Virasoro algebra, for example a WZW model or a theory with W-symmetry, correlation functions can in principle be decomposed into Virasoro conformal blocks, but that decomposition typically involves too many terms to be useful. Instead, it is possible to use conformal blocks based on the larger algebra: for example, in a WZW model, conformal blocks based on the corresponding affine Lie algebra, which obey Knizhnik–Zamolodchikov equations.
References
1. P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, 1997, ISBN 0-387-94785-X
2. Ribault, Sylvain (2014). "Conformal field theory on the plane". arXiv:1406.4290 [hep-th].
3. Moore, Gregory; Seiberg, Nathan (1989). "Classical and quantum conformal field theory". Communications in Mathematical Physics. 123 (2): 177–254. Bibcode:1989CMaPh.123..177M. doi:10.1007/BF01238857. S2CID 122836843.
4. Teschner, Joerg (2017). "A guide to two-dimensional conformal field theory". arXiv:1708.00680 [hep-th].
5. Teschner, J.; Vartanov, G. S. (2012). "6j symbols for the modular double, quantum hyperbolic geometry, and supersymmetric gauge theories". arXiv:1202.4698 [hep-th].
6. Roussillon, Julien (2021). "The Virasoro fusion kernel and Ruijsenaars' hypergeometric function". Letters in Mathematical Physics. 111 (1): 7. arXiv:2006.16101. Bibcode:2021LMaPh.111....7R. doi:10.1007/s11005-020-01351-4. PMC 7796901. PMID 33479555.
7. Marshakov, A.; Mironov, A.; Morozov, A. (2009). "On Combinatorial Expansions of Conformal Blocks". Theoretical and Mathematical Physics. 164: 831–852. arXiv:0907.3946. doi:10.1007/s11232-010-0067-6. S2CID 16017224.
8. Perlmutter, Eric (2015). "Virasoro conformal blocks in closed form". Journal of High Energy Physics. 2015 (8): 88. arXiv:1502.07742. Bibcode:2015JHEP...08..088P. doi:10.1007/JHEP08(2015)088. S2CID 54075672.
9. Maldacena, Juan; Simmons-Duffin, David; Zhiboedov, Alexander (2015-09-11). "Looking for a bulk point". arXiv:1509.03612 [hep-th].
10. Cho, Minjae; Collier, Scott; Yin, Xi (2017). "Recursive Representations of Arbitrary Virasoro Conformal Blocks". arXiv:1703.09805 [hep-th].
11. Nekrasov, Nikita (2004). "Seiberg-Witten Prepotential from Instanton Counting". Advances in Theoretical and Mathematical Physics. 7 (5): 831–864. arXiv:hep-th/0206161. doi:10.4310/ATMP.2003.v7.n5.a4. S2CID 2285041.
12. Alba, Vasyl A.; Fateev, Vladimir A.; Litvinov, Alexey V.; Tarnopolskiy, Grigory M. (2011). "On Combinatorial Expansion of the Conformal Blocks Arising from AGT Conjecture". Letters in Mathematical Physics. 98 (1): 33–64. arXiv:1012.1312. Bibcode:2011LMaPh..98...33A. doi:10.1007/s11005-011-0503-z. S2CID 119143670.
13. A. Cappelli, J-B. Zuber, "A-D-E Classification of Conformal Field Theories", Scholarpedia
14. Fateev, V. A.; Litvinov, A. V.; Neveu, A.; Onofri, E. (2009-02-08). "Differential equation for four-point correlation function in Liouville field theory and elliptic four-point conformal blocks". Journal of Physics A: Mathematical and Theoretical. 42 (30): 304011. arXiv:0902.1331. Bibcode:2009JPhA...42D4011F. doi:10.1088/1751-8113/42/30/304011. S2CID 16106733.
15. Hadasz, Leszek; Jaskolski, Zbigniew; Suchanek, Paulina (2010). "Modular bootstrap in Liouville field theory". Physics Letters B. 685 (1): 79–85. arXiv:0911.4296. Bibcode:2010PhLB..685...79H. doi:10.1016/j.physletb.2010.01.036. S2CID 118625083.
16. Fateev, V. A.; Litvinov, A. V. (2010). "On AGT conjecture". Journal of High Energy Physics. 2010 (2): 014. arXiv:0912.0504. Bibcode:2010JHEP...02..014F. doi:10.1007/JHEP02(2010)014. S2CID 118561574.
17. Teschner, J. (2003-08-05). "From Liouville Theory to the Quantum Geometry of Riemann Surfaces". arXiv:hep-th/0308031.
18. Nemkov, Nikita (2015-04-16). "On modular transformations of non-degenerate toric conformal blocks". Journal of High Energy Physics. 1510: 039. arXiv:1504.04360. doi:10.1007/JHEP10(2015)039. S2CID 73549642.
19. Teschner, Joerg. (1995). "On the Liouville three-point function". Physics Letters B. 363 (1–2): 65–70. arXiv:hep-th/9507109. Bibcode:1995PhLB..363...65T. doi:10.1016/0370-2693(95)01200-A. S2CID 15910029.
20. Migliaccio, Santiago; Ribault, Sylvain (2018). "The analytic bootstrap equations of non-diagonal two-dimensional CFT". Journal of High Energy Physics. 2018 (5): 169. arXiv:1711.08916. Bibcode:2018JHEP...05..169M. doi:10.1007/JHEP05(2018)169. S2CID 119385003.
21. Gamayun, O.; Iorgov, N.; Lisovyy, O. (2012). "Conformal field theory of Painlevé VI". Journal of High Energy Physics. 2012 (10): 038. arXiv:1207.0787. Bibcode:2012JHEP...10..038G. doi:10.1007/JHEP10(2012)038. S2CID 119610935.
22. Iorgov, N.; Lisovyy, O.; Tykhyy, Yu. (2013). "Painlevé VI connection problem and monodromy of c = 1 conformal blocks". Journal of High Energy Physics. 2013 (12): 029. arXiv:1308.4092. Bibcode:2013JHEP...12..029I. doi:10.1007/JHEP12(2013)029. S2CID 56401903.
23. Litvinov, Alexey; Lukyanov, Sergei; Nekrasov, Nikita; Zamolodchikov, Alexander (2014). "Classical conformal blocks and Painlevé VI". Journal of High Energy Physics. 2014 (7): 144. arXiv:1309.4700. Bibcode:2014JHEP...07..144L. doi:10.1007/JHEP07(2014)144. S2CID 119710593.
24. Nekrasov, Nikita (2020). "Blowups in BPS/CFT correspondence, and Painlevé VI". arXiv:2007.03646. {{cite journal}}: Cite journal requires |journal= (help)
25. Jeong, Saebyeok; Nekrasov, Nikita (2020). "Riemann-Hilbert correspondence and blown up surface defects". Journal of High Energy Physics. 2020 (12): 006. arXiv:2007.03660. Bibcode:2020JHEP...12..006J. doi:10.1007/JHEP12(2020)006. S2CID 220381427.
26. Gaiotto, D.; Teschner, J. (2012). "Irregular singularities in Liouville theory and Argyres-Douglas type gauge theories". Journal of High Energy Physics. 2012 (12): 50. arXiv:1203.1052. Bibcode:2012JHEP...12..050G. doi:10.1007/JHEP12(2012)050. S2CID 118380071.
| Wikipedia |
Virasoro conjecture
In algebraic geometry, the Virasoro conjecture states that a certain generating function encoding Gromov–Witten invariants of a smooth projective variety is fixed by an action of half of the Virasoro algebra. The Virasoro conjecture is named after theoretical physicist Miguel Ángel Virasoro. Tohru Eguchi, Kentaro Hori, and Chuan-Sheng Xiong (1997) proposed the Virasoro conjecture as a generalization of Witten's conjecture. Ezra Getzler (1999) gave a survey of the Virasoro conjecture.
References
• Getzler, Ezra (1999), "The Virasoro conjecture for Gromov-Witten invariants", in Wiśniewski, Jarosław; Szurek, Michał; Pragacz, Piotr (eds.), Algebraic geometry: Hirzebruch 70 (Warsaw, 1998), Contemporary Mathematics, vol. 241, Providence, R.I.: American Mathematical Society, pp. 147–176, arXiv:math/9812026, Bibcode:1998math.....12026G, doi:10.1090/conm/241/03634, ISBN 978-0-8218-1149-8, MR 1718143
• Eguchi, Tohru; Hori, Kentaro; Xiong, Chuan-Sheng (1997), "Quantum cohomology and Virasoro algebra", Physics Letters B, 402 (1): 71–80, arXiv:hep-th/9703086, Bibcode:1997PhLB..402...71E, doi:10.1016/S0370-2693(97)00401-2, ISSN 0370-2693, MR 1454328
| Wikipedia |
Virasoro group
In abstract algebra, the Virasoro group or Bott–Virasoro group (often denoted by Vir)[1] is an infinite-dimensional Lie group defined as the universal central extension of the group of diffeomorphisms of the circle. The corresponding Lie algebra is the Virasoro algebra, which has a key role in conformal field theory (CFT) and string theory.
The group is named after Miguel Ángel Virasoro and Raoul Bott.
Background
An orientation-preserving diffeomorphism of the circle $S^{1}$, whose points are labelled by a real coordinate $x$ subject to the identification $x\sim x+2\pi $, is a smooth map $f:\mathbb {R} \to \mathbb {R} :x\mapsto f(x)$ such that $f(x+2\pi )=f(x)+2\pi $ and $f'(x)>0$. The set of all such maps spans a group, with multiplication given by the composition of diffeomorphisms. This group is the universal cover of the group of orientation-preserving diffeomorphisms of the circle, denoted as ${\widetilde {\text{Diff}}}{}^{+}(S^{1})$.
Definition
The Virasoro group is the universal central extension of ${\widetilde {\text{Diff}}}{}^{+}(S^{1})$.[2]: sect. 4.4 The extension is defined by a specific two-cocycle, which is a real-valued function ${\mathsf {C}}(f,g)$ of pairs of diffeomorphisms. Specifically, the extension is defined by the Bott–Thurston cocycle:
${\mathsf {C}}(f,g)\equiv -{\frac {1}{48\pi }}\int _{0}^{2\pi }\log {\big [}f'{\big (}g(x){\big )}{\big ]}{\frac {g''(x)\,{\text{d}}x}{g'(x)}}.$
In these terms, the Virasoro group is the set ${\widetilde {\text{Diff}}}{}^{+}(S^{1})\times \mathbb {R} $ of all pairs $(f,\alpha )$, where $f$ is a diffeomorphism and $\alpha $ is a real number, endowed with the binary operation
$(f,\alpha )\cdot (g,\beta )={\big (}f\circ g,\alpha +\beta +{\mathsf {C}}(f,g){\big )}.$
This operation is an associative group operation. This extension is the only central extension of the universal cover of the group of circle diffeomorphisms, up to trivial extensions.[2] The Virasoro group can also be defined without the use explicit coordinates or an explicit choice of cocycle to represent the central extension, via a description the universal cover of the group.[2]
Virasoro algebra
Main article: Virasoro algebra
The Lie algebra of the Virasoro group is the Virasoro algebra. As a vector space, the Lie algebra of the Virasoro group consists of pairs $(\xi ,\alpha )$, where $\xi =\xi (x)\partial _{x}$ is a vector field on the circle and $\alpha $ is a real number as before. The vector field, in particular, can be seen as an infinitesimal diffeomorphism $f(x)=x+\epsilon \xi (x)$. The Lie bracket of pairs $(\xi ,\alpha )$ then follows from the multiplication defined above, and can be shown to satisfy[3]: sect. 6.4
${\big [}(\xi ,\alpha ),(\zeta ,\beta ){\big ]}={\bigg (}[\xi ,\zeta ],-{\frac {1}{24\pi }}\int _{0}^{2\pi }{\text{d}}x\,\xi (x)\zeta '''(x){\bigg )}$
where the bracket of vector fields on the right-hand side is the usual one: $[\xi ,\zeta ]=(\xi (x)\zeta '(x)-\zeta (x)\xi '(x))\partial _{x}$. Upon defining the complex generators
$L_{m}\equiv {\Big (}-ie^{imx}\partial _{x},-{\frac {i}{24}}\delta _{m,0}{\Big )},\qquad Z\equiv (0,-i),$
the Lie bracket takes the standard textbook form of the Virasoro algebra:[4]
$[L_{m},L_{n}]=(m-n)L_{m+n}+{\frac {Z}{12}}m(m^{2}-1)\delta _{m+n}.$
The generator $Z$ commutes with the whole algebra. Since its presence is due to a central extension, it is subject to a superselection rule which guarantees that, in any physical system having Virasoro symmetry, the operator representing $Z$ is a multiple of the identity. The coefficient in front of the identity is then known as a central charge.
Properties
Since each diffeomorphism $f$ must be specified by infinitely many parameters (for instance the Fourier modes of the periodic function $f(x)-x$), the Virasoro group is infinite-dimensional.
Coadjoint representation
The Lie bracket of the Virasoro algebra can be viewed as a differential of the adjoint representation of the Virasoro group. Its dual, the coadjoint representation of the Virasoro group, provides the transformation law of a CFT stress tensor under conformal transformations. From this perspective, the Schwarzian derivative in this transformation law emerges as a consequence of the Bott–Thurston cocycle; in fact, the Schwarzian is the so-called Souriau cocycle (referring to Jean-Marie Souriau) associated with the Bott–Thurston cocycle.[2]
References
1. Bahns, Dorothea; Bauer, Wolfram; Witt, Ingo (2016-02-11). Quantization, PDEs, and Geometry: The Interplay of Analysis and Mathematical Physics. Birkhäuser. ISBN 978-3-319-22407-7.
2. Guieu, Laurent; Roger, Claude (2007), L'algèbre et le groupe de Virasoro, Montréal: Centre de Recherches Mathématiques, ISBN 978-2921120449
3. Oblak, Blagoje (2016), BMS Particles in Three Dimensions, Springer Theses, Springer Theses, arXiv:1610.08526, doi:10.1007/978-3-319-61878-4, ISBN 978-3319618784, S2CID 119321869
4. Di Francesco, P.; Mathieu, P.; Sénéchal, D. (1997), Conformal Field Theory, New York: Springer Verlag, doi:10.1007/978-1-4612-2256-9, ISBN 9780387947853
| Wikipedia |
Virbhadra–Ellis lens equation
The Virbhadra-Ellis lens equation [1] in astronomy and mathematics relates to the angular positions of an unlensed source $\left(\beta \right)$, the image $\left(\theta \right)$, the Einstein bending angle of light $({\hat {\alpha }})$, and the angular diameter lens-source $\left(D_{ds}\right)$ and observer-source $\left(D_{s}\right)$ distances.
$\tan \beta =\tan \theta -{\frac {D_{ds}}{D_{s}}}\left[\tan \theta +\tan \left({\hat {\alpha }}-\theta \right)\right]$.
This lens equation is useful for studying gravitational lensing in a strong gravitational field.
References
1. Virbhadra, K. S.; Ellis, George F. R. (2000-09-08). "Schwarzschild black hole lensing". Physical Review D. American Physical Society (APS). 62 (8): 084003. arXiv:astro-ph/9904193. Bibcode:2000PhRvD..62h4003V. doi:10.1103/physrevd.62.084003. ISSN 0556-2821. S2CID 15956589.
| Wikipedia |
Virginia Newell
Virginia Kimbrough Newell (born October 7, 1917) is an American mathematics educator, author, politician, and centenarian.[1]
Early life and education
Virginia Kimbrough was born on October 7, 1917 in Advance, North Carolina,[1] one of nine children. Although her family was African American, she grew up playing with the white children in a white neighborhood; her father, a builder, had the right to vote because he had a white ancestor,[2] and both of her parents had studied at Shaw University, without finishing a degree.[3] Kimbrough learned arithmetic helping her father in his measurements, and won a mathematics competition in elementary school.[2]
Her family sent her away to live with a great aunt, so that she could obtain a better education at Atkins High School (North Carolina). There, she learned mathematics from teachers Togo West and Beatrice Armstead, earning straight A's and becoming a teacher's assistant.[3] After graduating in 1936,[4] she obtained scholarships from many colleges,[3] and chose to major in mathematics at Talladega College, a historically black college in Alabama.[1] Many of her teachers there had previously taught at Ivy League universities, and had come to Talladega to teach because of mandatory retirement at their former employers.[3]
She later earned a master's degree from New York University,[1] and took courses from the University of Wisconsin, Atlanta University, University of Chicago, and North Carolina State College.[5] She completed a doctorate in education at the University of Sarasota in 1976, with the dissertation Development of mathematics self-instructional learning packages with activities from the newspaper for prospective elementary school teachers enrolled at Winston-Salem State University.[6]
Mathematics
After college, Kimbrough returned to Atkins High School as a mathematics teacher.[7] There, in 1943,[8] she married George Newell, who had been her biology teacher at the same school, changing her name to Virginia Newell. They both taught at several institutions in Atlanta and Raleigh, North Carolina,[7] including Washington Graded and High School, John W. Ligon High School,[1] and Shaw University, where Virginia Newell was an associate professor of mathematics from 1960 to 1965.[5][2]
In 1965,[1] they both settled at Winston-Salem State University, where Virginia Newell became a mathematics professor.[7] At Winston-Salem State University, she chaired the mathematics department,[1] helped bring computers to the university and found the computer science program,[7] becoming founding chair of the computer science department in 1979.[8] She spearheaded several initiatives for middle school students, including the Math and Science Academy of Excellence, the New Directions for our Youth program aimed at preventing dropouts, and the Best Choice Center for after-school education. She was a co-founder and president of the North Carolina Council of Teachers on Mathematics[9]
In 1980, Newell became one of the coauthors of Black Mathematicians and Their Works (with Joella Gipson, L. Waldo Rich, and Beauregard Stubblefield, Dorrance & Company),[10] the first book to highlight the contributions of African American mathematicians. She was also editor of the newsletter of the National Association of Mathematicians, an organization for African American mathematicians, from 1974 into the 1980s.[8]
She retired after 20 years of service at Winston-Salem State, circa 1985,[1] as professor emerita.[7]
Politics and later life
As part of the 1972 US presidential campaign, Newell was co-chair of the Shirley Chisholm campaign in North Carolina.[1] In 1977, Newell was elected (with Vivian Burke) as one of the first two African American women to become aldermen of Winston-Salem, North Carolina; she represented its East Ward.[4] She served in that position for 16 years.[1]
Recognition
The computer science center at Winston-Salem State University is named for Newell, as is one of the streets in Winston-Salem, Virginia Newell Lane.[1]
In 2017, Newell was given the Order of the Long Leaf Pine, the highest honor of the governor of North Carolina. In 2018, the National Association of Mathematicians gave her their Centenarian Award.[8] In 2019, Newell was given the YWCA Women of Vision Lifetime Achievement Award.[9] She was listed in 2021 as a Black History Month Honoree by the Mathematically Gifted and Black website.[8]
References
1. "Happy birthday, Dr. Virginia Kimbrough", Congressional Record, 163 (159), 4 October 2017
2. Barr, Matthew, Oral history interview with Virginia Newell, University of North Carolina at Greensboro
3. Sua, Lou Sanders (2012), But Your Mother Was An Activist: Black Women's Activism in North Carolina (PDF) (Doctoral dissertation), University of North Carolina at Greensboro
4. Elam, Bridget (7 October 2020), "Virginia Newell turns 103", Winston-Salem Chronicle
5. "Associate professors", Shaw University Bulletin, XXX (1): 16, July 1961
6. WorldCat catalog entry for Development of mathematics self-instructional learning packages with activities from the newspaper for prospective elementary school teachers enrolled at Winston-Salem State University, retrieved 2021-09-28
7. Drabble, Jenny (8 October 2017), "Former Winston-Salem elected official turns 100", Winston-Salem Journal
8. "Dr. Virginia Newell", Black History Month 2021 Honoree, Mathematically Gifted and Black, 2021, retrieved 2021-09-28
9. Vickers, Talitha (24 April 2019), "YWCA Women of Vision: Lifetime Achievement Award recipient Virginia Newell", WXII 12 News, WXII
10. Reviews of Black Mathematicians and their Works:
• Goins, Edray (February 2021), "Mathematical comfort food", The American Mathematical Monthly, 128 (2): 188, doi:10.1080/00029890.2021.1853445
• Kenschaft, Patricia Clark (1997), "What next? A meta-history of black mathematicians", African Americans in mathematics: Proceedings of the second conference for African-American researchers in the mathematical sciences held at DIMACS, Piscataway, NJ, USA, June 26–28, 1996, Providence, RI: American Mathematical Society, pp. 183–186, ISBN 0-8218-0678-5, Zbl 1155.01347; review, p. 185
• Sims, Janet L. (Summer 1981), The Journal of Negro History, 66 (2): 160–161, doi:10.2307/2717293, JSTOR 2717293{{citation}}: CS1 maint: untitled periodical (link)
• Sonnabend, Tom (November 1980), The Mathematics Teacher, 73 (8): 629, JSTOR 27962208{{citation}}: CS1 maint: untitled periodical (link)
• Zaslavsky, Claudia (February 1983), Historia Mathematica, 10 (1): 105–115, doi:10.1016/0315-0860(83)90049-6{{citation}}: CS1 maint: untitled periodical (link)
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Virginia Kiryakova
Virginia S. Kiryakova (née Virdzhinia Stoinova Hristova) is a Bulgarian mathematician known for her work on the fractional calculus, on special functions in fractional calculus including the Mittag-Leffler functions, and on the history of calculus. She is a professor in the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences.
Education and career
As a high school student, Kiryakova competed for Bulgaria in the 1969 International Mathematical Olympiad, earning a bronze medal.[1][2] She graduated from Sofia University in 1975 with a combined bachelor's and master's degree in mathematics, and in the same year became a researcher in the Institute of Mathematics and Informatics. She earned a Ph.D. in 1987, with the thesis Generalized Operators of Integration and Differentiation of Fractional Order and Applications,[1] and completed a Dr.Sc. (habilitation) in 2010, with the thesis Generalized Fractional Calculus and Applications in Analysis, supervised by Ivan Dimovski.[1][3]
She is editor-in-chief of the journals Fractional Calculus and Applied Analysis and International Journal of Applied Mathematics.[1]
Selected publications
Kiryakova is the author of the research monograph Generalized Fractional Calculus and Applications (1993).[4] She has also coauthored highly cited work on the history of calculus.[5]
Recognition
Kiryakova won the 1996 Academic Prize for Mathematical Sciences of Bulgarian Academy of Sciences.[1] In 2012, at the 5th Symposium on Fractional Differentiation and its Applications, she was given the FDA Dissemination Award, for her "dissemination of fractional calculus among the scientific community, industry and society" over the previous five years.[1][6]
References
1. Curriculum vitae (PDF), 2017, retrieved 2022-02-24
2. Bulgaria in the 11th IMO, 1969, International Mathematical Olympiad, retrieved 2022-02-24
3. Virginia Kiryakova at the Mathematics Genealogy Project
4. Generalized Fractional Calculus and Applications (Pitman Research Notes in Mathematics Series 301, Longman Scientific and John Wiley & Sons, 1993). Reviews: Anatoly Kilbas (1995), MR1265940; S.L.Kalla, Zbl 0882.26003; A. C. McBride (1995), Proc. Edinburgh Math. Soc, doi:10.1017/S0013091500006325
5. Machado, J. Tenreiro; Kiryakova, Virginia; Mainardi, Francesco (2011), "Recent history of fractional calculus", Communications in Nonlinear Science and Numerical Simulation, 16 (3): 1140–1153, Bibcode:2011CNSNS..16.1140M, doi:10.1016/j.cnsns.2010.05.027, hdl:10400.22/4149, MR 2736622
6. "Awards", 5th Symposium on Fractional Differentiation and its Applications, Hohai University, archived from the original on 2017-10-03
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• Home page
• Virginia Kiryakova publications indexed by Google Scholar
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Virginia Ragsdale
Virginia Ragsdale (December 13, 1870 – June 4, 1945) was a teacher and mathematician specializing in algebraic curves. She is most known as the creator of the Ragsdale conjecture.
Virginia Ragsdale
Virginia Ragsdale
Born(1870-12-13)December 13, 1870
Jamestown, North Carolina, US
DiedJune 4, 1945(1945-06-04) (aged 74)
Greensboro, North Carolina, US
Alma materGuilford College
Bryn Mawr College
Known forRagsdale conjecture
Scientific career
FieldsMathematics
InstitutionsWoman's College in Greensboro
ThesisOn the Arrangement of the Real Branches of Plane Algebraic Curves (1904)
Doctoral advisorCharlotte Scott
Early life
Ragsdale was born on a farm in Jamestown, North Carolina the third child of John Sinclair Ragsdale and Emily Jane Idol.[1] John was an officer in the Civil War, a teacher in the Flint Hill School, and later a state legislator.[1]
Virginia Ragsdale descended from Godfrey Ragsdale, a settler of the new Jamestown colony. Jamestown was raided by a native-American tribe in 1644 led by the uncle of Pocahontas, during which Godfrey and his wife were killed, but their infant son, Godfrey, Jr., survived. Ragsdale was then descended from the infant.[2]
Virginia documented her early years in a paper titled "Our Early Home and Childhood", writing:
One of my earliest recollections was a little trundle bed where Ida [her sister] and I slept together. … The house had no conveniences. Water had to be carried from a spring at the foot of the hill, milk and butter were kept there, washing was done there.
In the first years or two, there were three or four boarders, boys or young men, who came to attend Father's school.
Grandma (Idol), mother and Aunt Julia had all done their bit before and during the war, weaving blankets (and) jeans for men's suits, which were sold to Greensboro merchants in exchange for silk and other goods.
— Virginia Ragsdale, Our Early Home and Childhood[1]
Study
As a junior, Ragsdale entered Salem Academy, and graduated in 1887 as valedictorian with an extra diploma in piano.[2] Ragsdale attended Guilford College in Greensboro, North Carolina, where she earned her B.S. in 1892.[2] She was active in student life, establishing a Y.M.C.A. on campus, expanding collegiate athletics, and contributing to the formation the Guilford's Alumni Association.[2]
Ragsdale was awarded the first scholarship from Bryn Mawr College for the top scholar Guilford College.[1] She studied physics at Bryn Mawr College, obtaining an A.B. degree in 1896.[3] She was elected European fellow for the class of 1896, but waited a year before traveling, working as an assistant demonstrator in physics and mathematics graduate student at Bryn Mawr.[3]
Together with two of her colleagues (including Emilie Martin),[4] she spent 1897-98 abroad at the University of Göttingen, attending lectures of Felix Klein and David Hilbert.[3] After her return to the United States, she taught in Baltimore for three years until a second scholarship, by the Baltimore Association for the Promotion of University Education of Women,[3] permitted her to return to Bryn Mawr college to complete her Ph.D. under the direction of Charlotte Scott.[2]
Her dissertation, "On the Arrangement of the Real Branches of Plane Algebraic Curves," was published in 1906 by the American Journal of Mathematics.[2] Her dissertation addressed the 16th of Hilbert's problems, for which Ragsdale formulated a conjecture that provided an upper bound on the number of topological circles of a certain type.[2] Her result is called the Ragsdale conjecture; it was an open problem for 90 years until counterexamples were derived by Oleg Viro (1979) and Ilya Itenberg (1994).[2]
Career
After completing her degree, Ragsdale taught in New York City and Dr. Sach's School for Girls until 1905.[3] She was head of the Baldwin School in Bryn Mawr from 1906 to 1911, and a reader for Charlotte Scott from 1908 to 1910.[3] Ragsdale returned to North Carolina in 1911 to accept a mathematics position at Woman's College in Greensboro (now known as the University of North Carolina at Greensboro).[2] She remained there for almost two decades and served as department head from 1926 to 1928.[2] She encouraged the school to buy a telescope, and the math department to add statistics to the curriculum.[2]
In 1928, she retired from teaching in order to care for her mother's health and help manage the family farm.[3] After the death of her mother in 1934, she built a house at Guilford College, where she spent her last years gardening, working with furniture,[2] working on family genealogy, holding book clubs, and visiting with students.[1] Upon her death, she donated her house to Guilford College, where it housed the faculty, alumni, and visitors.[2] In 1965 President of Guilford Grimsley Hobbs moved into Ragsdale's house, and it has been the home of the college's president ever since.[1]
See also
• Ragsdale conjecture
• Algebraic curves
• Emilie Martin
References
1. Brooks, Carol (March 21, 2012). "Virginia Ragsdale: From farm girl to Ph.D." Jamestown News. Retrieved 3 January 2014.
2. De Loera, Jesús; Wicklin, Frederick J. "Biographies of Women in Mathematics: Virginia Ragsdale". Anges Scott College. Retrieved 3 January 2014.
3. Green, Judy; LaDuke, Jeanne (2009). Pioneering Women in American Mathematics: The Pre-1940 PhD's. Providence, Rhode Island: American Mathematical Society. https://books.google.com/books?id=jUrq3bUvQlYC&pg=PA271 pp. 271–272]. ISBN 978-0-8218-4376-5. Biography on p.503-505 of the Supplementary Material at AMS
4. Green & LaDuke (2009), p. 235.
External links
• Works by or about Virginia Ragsdale at Internet Archive
• Virginia Ragsdale at the Mathematics Genealogy Project
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Virginia Torczon
Virginia Joanne Torczon is an American applied mathematician and computer scientist known for her research on nonlinear optimization methods including pattern search. She is dean of graduate studies and research, and chancellor professor of computer science, at the College of William & Mary.[1]
Education and career
Torczon majored in history as an undergraduate at Wesleyan University.[2] She earned her Ph.D. in mathematical sciences in 1989 from Rice University.[2][3] Her dissertation, Multi-Directional Search: a Direct Search Algorithm for Parallel Machines, was supervised by John E. Dennis.[3]
Before becoming dean of graduate studies and research at William & Mary, she was the first female chair of the computer science department there.[4]
Recognition
Torczon's paper "On the Convergence of Pattern Search Algorithms" won the inaugural Society for Industrial and Applied Mathematics (SIAM) Outstanding Paper Prize for the best paper published in a SIAM journal in 1999.[5]
References
1. "Virginia Torczon", Computer Science Faculty, College of William & Mary, retrieved 2021-02-18
2. "Virginia Torczon", Parallel Profile, Parallel Computing Research, 5 (1), Winter 1997
3. Virginia Torczon at the Mathematics Genealogy Project
4. Berard, Adrienne (November 11, 2018), "Women in computer science: Taking the 'brogrammer' out of the algorithm", Williamsburg Yorktown Daily
5. The SIAM Outstanding Paper Prizes, Society for Industrial and Applied Mathematics, retrieved 2021-02-18
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Virginia Vassilevska Williams
Virginia Vassilevska Williams (née Virginia Panayotova Vassilevska)[1] is a theoretical computer scientist and mathematician known for her research in computational complexity theory and algorithms. She is currently the Steven and Renee Finn Career Development Associate Professor of Electrical Engineering and Computer Science at the Massachusetts Institute of Technology.[2] She is notable for her breakthrough results in fast matrix multiplication,[3] for her work on dynamic algorithms,[4] and for helping to develop the field of fine-grained complexity.[5]
Virginia Vassilevska Williams
Vassilevska Williams at Oberwolfach, 2012
NationalityBulgarian American
Alma mater
• Carnegie Mellon University (PhD, 2008)
• Caltech (BS, 2003)
Known for
• Matrix multiplication
• Graph algorithms
• Dynamic algorithms
• Fine-grained complexity theory
Scientific career
Fields
• Complexity theory
• Algorithms
Institutions
• MIT
• Stanford
• UC Berkeley
• Institute for Advanced Study
Doctoral advisorGuy Blelloch
Education and career
Williams is originally from Bulgaria, and attended a German-language high school in Sofia.[6] She graduated from the California Institute of Technology in 2003, and completed her Ph.D. at Carnegie Mellon University in 2008.[1] Her dissertation, Efficient Algorithms for Path Problems in Weighted Graphs, was supervised by Guy Blelloch.[7]
After postdoctoral research at the Institute for Advanced Study and University of California, Berkeley, Williams became an assistant professor of computer science at Stanford University in 2013.[1] She moved to MIT as an associate professor in 2017.[2]
Research
In 2011, Williams found an algorithm for multiplying two $n\times n$ matrices in time $O(n^{2.373})$. This improved a previous time bound for matrix multiplication algorithms, the Coppersmith–Winograd algorithm, that had stood as the best known for 24 years. Her initial improvement was independent of Andrew Stothers, who also improved the same bound a year earlier; after learning of Stothers' work, she combined ideas from both methods to improve his bound as well.[8][3] As of 2020, her work also establishes the current best-known algorithm for matrix multiplication with Josh Alman, in time $O(n^{2.3728596})$.[9]
Recognition
Williams was an NSF Computing Innovation Fellow for 2009–2011,[1] and won a Sloan Research Fellowship in 2017.[2] She was an invited speaker at the 2018 International Congress of Mathematicians, speaking in the section on Mathematical Aspects of Computer Science.[10]
Personal life
Williams is the daughter of applied mathematicians Panayot Vassilevski and Tanya Kostova-Vassilevska.[11] She is married to Ryan Williams, also a computer science professor at MIT; they have worked together in the field of fine-grained complexity.[6]
References
1. Curriculum vitae (PDF), retrieved 2018-02-24
2. Three EECS professors receive 2017 Sloan Research Fellowships, Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science, February 22, 2017
3. Virginia Vassilevska Williams (2012), "Multiplying Matrices Faster than Coppersmith-Winograd", in Howard J. Karloff and Toniann Pitassi (ed.), Proceedings of the 44th Symposium on Theory of Computing (STOC), ACM, pp. 887–898, doi:10.1145/2213977.2214056, S2CID 14350287
4. Abboud, Amir; Williams, Virginia Vassilevska (2014), "Popular Conjectures Imply Strong Lower Bounds for Dynamic Problems", 2014 IEEE 55th Annual Symposium on Foundations of Computer Science, pp. 434–443, arXiv:1402.0054, doi:10.1109/FOCS.2014.53, ISBN 978-1-4799-6517-5, S2CID 2267837
5. Williams, V. V. (2019), "On Some Fine-Grained Questions in Algorithms and Complexity", Proceedings of the International Congress of Mathematicians (ICM 2018): 3447–3487, doi:10.1142/9789813272880_0188, ISBN 978-981-327-287-3, S2CID 19282287
6. Matheson, Rob (January 7, 2020), "Finding the true potential of algorithms: Using mathematical theory, Virginia Williams coaxes algorithms to run faster or proves they've hit their maximum speed", MIT News, retrieved 2021-12-18
7. Virginia Vassilevska Williams at the Mathematics Genealogy Project
8. Aron, Jacob (December 9, 2011), "Key mathematical tool sees first advance in 24 years", New Scientist
9. Hartnett, Kevin (March 23, 2021), "Matrix Multiplication Inches Closer to Mythic Goal", Quanta Magazine, retrieved 2021-04-01
10. "Speakers", ICM 2018, archived from the original on 2017-12-15, retrieved 2018-02-24
11. "Vassilevska, Williams to wed", Engagements, Hartselle Enquirer, August 28, 2008, retrieved 2022-07-10
External links
• Home page
• Virginia Vassilevska Williams publications indexed by Google Scholar
• Virginia Vassilevska Williams at DBLP Bibliography Server
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Virginia Warfield
Virginia "Ginger" Patricia McShane Warfield is an American mathematician and mathematical educator. She received the Louise Hay Award from the Association for Women in Mathematics in 2007.[1]
Virginia Warfield
Born
Virginia Patricia McShane
NationalityAmerican
Alma materBrown University,
AwardsLouise Hay Award
Scientific career
InstitutionsUniversity of Washington
Doctoral advisorWendell Fleming
Education
Warfield's father was mathematician Edward J. McShane.[2] She received her Ph.D. in mathematics from Brown University in 1971. Her doctoral advisor was Wendell Fleming and the title of her dissertation was A Stochastic Maximum Principle.[3]
Career
While making contributions to the field of stochastic analysis after her Ph.D., Warfield became more and more engrossed by the problems of mathematics education. She worked with Project SEED, a highly regarded mathematics program whose goal was to promote sense-making mathematical activities for fourth through sixth graders. She addressed issues of teacher preparation and enhancement. She collaborated with the French mathematician Guy Brousseau, a pioneer in the “didactics of mathematics,” the scientific study of issues in mathematics teaching and learning.[4] She has been an active member of the Association for Women in Mathematics (AWM). She has chaired the Education Committee, has served as Education Column Editor for the AWM Newsletter, and was elected as a Member-at-large to the Executive Committee. She has been a member of the Mathematical Association of America’s committees on Professional Development and Mathematical Education of Teachers.[4]
Books
Warfield is the author of the book Invitation to Didactique (self-published, 2007, and Springer Briefs in Education, 2014)[5] and the co-author of Teaching Fractions through Situations: A Fundamental Experiment (with Guy Brousseau and Nadine Brousseau, Springer 2013).[6]
References
1. Kelley, Peter. "Teacher's teacher: 'Ginger' Warfield wins national math education award". UW News. Retrieved 7 April 2019.
2. New York Times:Edward McShane, 85, Mathematician, Dies; June 06, 1989
3. "Virginia P. Warfield". Mathematics Genealogy Project. Retrieved 7 April 2019.
4. "Virginia McShane Warfield Honored With Hay Award". Association for Women in Mathematics. Retrieved 7 April 2019.
5. Lancaster, Stephen (January 16, 2008). "Review of Invitation to Didactique". MAA Reviews. Mathematical Association of America.
6. Selden, Annie (February 22, 2014). "Review of Teaching Fractions through Situations". MAA Reviews. Mathematical Association of America.
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Virginie Bonnaillie-Noël
Virginie Bonnaillie-Noël (born on 3 October 1976 in Calais) is a French mathematician and research director specializing in numerical analysis. Her research topics concern partial differential equations, asymptotic, spectral and numerical analysis of problems arising from physics or mechanics.
Virginie Bonnaillie-Noël
Born
Virginie Bonnaillie
3 October 1976
Calais, France
NationalityFrench
Alma materParis-Sud University
Ecole Normale Supérieure Paris-Saclay
Occupation(s)Mathematician and research director
Known forIrène-Joliot-Curie Prize
Life and work
After completing her preparatory classes at Faidherbe high school in Lille between 1994 and 1997, Bonnaillie-Noël entered Paris-Sud University where she obtained a bachelor's degree in 1998 and a master's degree in 1999. The same year, she was admitted to the Ecole Normale Supérieure Paris-Saclay (ENS) where she obtained the aggregation in 2000 with the option of numerical analysis. In 2001, she obtained a Diploma of Advanced Studies (DEA) in Numerical Analysis and Partial Differential Equations.[1][2]
Between 2001 and 2003 she completed a thesis (as Virginie Bonnaillie) under the supervision of François Alouges and Bernard Helffer titled Mathematical analysis of superconductivity in a corner domain: semi-classical and numerical methods, which explored interdisciplinarity between physics and mathematics, at the border of numerical analysis, partial differential equations and spectral theory.[1][3]
In 2004, she joined the Mathematical Research Institute of Rennes (IRMAR) as a research fellow. In 2011, she obtained authorization to direct research at the University of Rennes I. In 2014, she left IRMAR to direct research in the Mathematics and Applications Department of ENS.[1]
In addition to her official work, Bonnaillie-Noël has frequently spoken about gender parity in science and research to encourage more young people to participate in the sciences.[1]
Distinctions
• 2008: CNRS bronze medal[1][4]
• 2009: Irène-Joliot-Curie Prize in the young female scientist category[1][2][5]
• 2011: Chevalier of the National Order of Merit by Cédric Villani[2]
• 2021: Officer of the National Order of Merit[6]
Selected publications
• Bonnaillie-Noël, V., & Dauge, M. (2006, August). Asymptotics for the low-lying eigenstates of the Schrödinger operator with magnetic field near corners. In Annales Henri Poincaré (Vol. 7, No. 5, pp. 899-931). Birkhäuser-Verlag.
• Bonnaillie-Noël, V., & Fournais, S. (2007). Superconductivity in domains with corners. Reviews in Mathematical Physics, 19(06), 607-637.
• Bonnaillie-Noël, V., Dambrine, M., Tordeux, S., & Vial, G. (2009). Interactions between moderately close inclusions for the Laplace equation. Mathematical Models and Methods in Applied Sciences, 19(10), 1853-1882.
• Bonnaillie-Noël, V., Dambrine, M., Hérau, F., & Vial, G. (2010). On generalized Ventcel's type boundary conditions for Laplace operator in a bounded domain. SIAM journal on mathematical analysis, 42(2), 931-945.
• Bonnaillie-Noël, V., Helffer, B., & Vial, G. (2010). Numerical simulations for nodal domains and spectral minimal partitions. ESAIM: Control, Optimisation and Calculus of Variations, 16(1), 221-246.
References
1. "PRIX DE LA JEUNE FEMME SCIENTIFIQUE, Virginie BONNAILLIE-NOËL" (PDF). 2009. Retrieved 2022-07-22.
2. Jeune, Cecile Le (September 11, 2012). "ENS Rennes - Remise de l'insigne de Chevalier de l'Ordre National du mérite à Virginie Bonnaillie-Noël par Cédric Villani". ENS Rennes (in French). Retrieved 2022-07-22.
3. Bonnaillie, Virginie. "Thesis".
4. "INSMI - Institut national des sciences mathématiques et de leurs interactions - Remise de la médaille de bronze du CNRS à Virginie Bonnaillie-Noël". archive.wikiwix.com. Retrieved 2022-07-22.
5. "Les lauréates du Prix Irène Joliot-Curie 2009 - ESR : enseignementsup-recherche.gouv.fr". www.enseignementsup-recherche.gouv.fr (in French). Retrieved 2022-07-22.
6. "Promotions". www.legifrance.gouv.fr. Retrieved 2022-07-22.
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Virtual knot
In knot theory, a virtual knot is a generalization of knots in 3-dimensional Euclidean space, R3, to knots in thickened surfaces $\Sigma \times [0,1]$ modulo an equivalence relation called stabilization/destabilization. Here $\Sigma $ is required to be closed and oriented. Virtual knots were first introduced by Kauffman (1999).
Unsolved problem in mathematics:
[Extension of Jones polynomial to general 3-manifolds.] Can the original Jones polynomial, which is defined for 1-links in the 3-sphere (the 3-ball, the 3-space R3), be extended for 1-links in any 3-manifold?
(more unsolved problems in mathematics)
Overview
In the theory of classical knots, knots can be considered equivalence classes of knot diagrams under the Reidemeister moves. Likewise a virtual knot can be considered an equivalence of virtual knot diagrams that are equivalent under generalized Reidemeister moves. Virtual knots allow for the existence of, for example, knots whose Gauss codes which could not exist in 3-dimensional Euclidean space. A virtual knot diagram is a 4-valent planar graph, but each vertex is now allowed to be a classical crossing or a new type called virtual. The generalized moves show how to manipulate such diagrams to obtain an equivalent diagram; one move called the semi-virtual move involves both classical and virtual crossings, but all the other moves involve only one variety of crossing.
Virtual knots are important, and there is a strong relation between Quantum Field Theory and virtual knots.
Virtual knots themselves are fascinating objects, and having many connections to other areas of mathematics. Virtual knots have many exciting connections with other fields of knots theory. The unsolved problem shown is an important motivation to the study of virtual knots.
See section 1.1 of this paper [KOS] [1] for the background and the history of this problem. Kauffman submitted a solution in the case of the product manifold of closed oriented surface and the closed interval, by introducing virtual 1-knots .[2] It is open in the other cases. Witten’s path integral for Jones polynomial is written for links in any compact 3-manifold formally, but the calculus is not done even in physics level in any case other than the 3-sphere (the 3-ball, the 3-space R3). This problem is also open in physics level. In the case of Alexander polynomial, this problem is solved.
A classical knot can also be considered an equivalence class of Gauss diagrams under certain moves coming from the Reidemeister moves. Not all Gauss diagrams are realizable as knot diagrams, but by considering all equivalence classes of Gauss diagrams we obtain virtual knots.
A classical knot can be considered an ambient isotopy class of embeddings of the circle into a thickened 2-sphere. This can be generalized by considering such classes of embeddings into thickened higher-genus surfaces. This is not quite what we want since adding a handle to a (thick) surface will create a higher-genus embedding of the original knot. The adding of a handle is called stabilization and the reverse process destabilization. Thus a virtual knot can be considered an ambient isotopy class of embeddings of the circle into thickened surfaces with the equivalence given by (de)stabilization.
Some basic theorems relating classical and virtual knots:
• If two classical knots are equivalent as virtual knots, they are equivalent as classical knots.
• There is an algorithm to determine if a virtual knot is classical.
• There is an algorithm to determine if two virtual knots are equivalent.
It is important that there is a relation among the following. See the paper [KOS] cited above and below.
• Virtual equivalence of virtual 1-knot diagrams, which is a set of virtual 1-knots.
• Welded equivalence of virtual 1-knot diagrams
• Rotational welded equivalence of virtual 1-knot diagrams
• Fiberwise equivalence of virtual 1-knot diagrams
Virtual 2-knots are also defined. See the paper cited above.
See also
• Knots and graphs
References
1. Kauffman, L.H; Ogasa, E; Schneider, J (2018), A spinning construction for virtual 1-knots and 2-knots, and the fiberwise and welded equivalence of virtual 1-knots, arXiv:1808.03023
2. Kauffman, L.E. (1998), Talks at MSRI Meeting in January 1997, AMS Meeting at University of Maryland, College Park in March 1997, Isaac Newton Institute Lecture in November 1997, Knots in Hellas Meeting in Delphi, Greece in July 1998, APCTP-NANKAI Symposium on Yang-Baxter Systems, Non-Linear Models and Applications at Seoul, Korea in October 1998, and Kauffman's paper1999 cited below., arXiv:math/9811028
• Boden, Hans; Nagel, Matthias (2017). "Concordance group of virtual knots". Proceedings of the American Mathematical Society. 145 (12): 5451–5461. doi:10.1090/proc/13667. S2CID 119139769.
• Carter, J. Scott; Kamada, Seiichi; Saito, Masahico (2002). "Stable equivalence of knots on surfaces and virtual knot cobordisms. Knots 2000 Korea, Vol. 1 (Yongpyong)". J. Knot Theory Ramifications. 11 (3): 311–322.
• Carter, J. Scott; Silver, Daniel; Williams, Susan (2014). "Invariants of links in thickened surfaces". Algebraic & Geometric Topology. 14 (3): 1377–1394. doi:10.2140/agt.2014.14.1377. S2CID 53137201.
• Dye, Heather A (2016). An Invitation to Knot Theory : Virtual and Classical (First ed.). Chapman and Hall/CRC. ISBN 9781315370750.
• Goussarov, Mikhail; Polyak, Michael; Viro, Oleg (2000). "Finite-type invariants of classical and virtual knots". Topology. 39 (5): 1045–1068. arXiv:math/9810073. doi:10.1016/S0040-9383(99)00054-3. S2CID 8871411.
• Kamada, Naoko; Kamda, Seiichi (2000). "Abstract link diagrams and virtual knots". Journal of Knot Theory and Its Ramifications. 9 (1): 93–106. doi:10.1142/S0218216500000049.
• Kauffman, Louis H. (1999). "Virtual knot theory" (PDF). European Journal of Combinatorics. 20 (7): 663–690. doi:10.1006/eujc.1999.0314. ISSN 0195-6698. MR 1721925. S2CID 5993431.
• Kauffman, Louis H.; Manturov, Vassily Olegovich (2005). "Virtual Knots and Links". arXiv:math.GT/0502014.
• Kuperberg, Greg (2003). "What is a virtual link?". Algebraic & Geometric Topology. 3: 587–591. doi:10.2140/agt.2003.3.587. S2CID 16803280.
• Manturov, Vassily (2004). Knot Theory. CRC Press. ISBN 978-0-415-31001-7.
• Manturov, Vassily Olegovich (2004). "Virtual knots and infinite dimensional Lie algebras". Acta Applicandae Mathematicae. 83 (3): 221–233. doi:10.1023/B:ACAP.0000038944.29820.5e. S2CID 124019548.
• Turaev, Vladimir (2008). "Cobordism of knots on surfaces". Journal of Topology. 1 (2): 285–305. arXiv:math/0703055. doi:10.1112/jtopol/jtn002. S2CID 17888102.
External links
• A Table of Virtual Knots
• Elementary explanation with diagrams
| Wikipedia |
Virtually Haken conjecture
In topology, an area of mathematics, the virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is virtually Haken. That is, it has a finite cover (a covering space with a finite-to-one covering map) that is a Haken manifold.
After the proof of the geometrization conjecture by Perelman, the conjecture was only open for hyperbolic 3-manifolds.
The conjecture is usually attributed to Friedhelm Waldhausen in a paper from 1968,[1] although he did not formally state it. This problem is formally stated as Problem 3.2 in Kirby's problem list.
A proof of the conjecture was announced on March 12, 2012 by Ian Agol in a seminar lecture he gave at the Institut Henri Poincaré. The proof appeared shortly thereafter in a preprint which was eventually published in Documenta Mathematica.[2] The proof was obtained via a strategy by previous work of Daniel Wise and collaborators, relying on actions of the fundamental group on certain auxiliary spaces (CAT(0) cube complexes)[3] It used as an essential ingredient the freshly-obtained solution to the surface subgroup conjecture by Jeremy Kahn and Vladimir Markovic.[4][5] Other results which are directly used in Agol's proof include the Malnormal Special Quotient Theorem of Wise[6] and a criterion of Nicolas Bergeron and Wise for the cubulation of groups.[7]
In 2018 related results were obtained by Piotr Przytycki and Daniel Wise proving that mixed 3-manifolds are also virtually special, that is they can be cubulated into a cube complex with a finite cover where all the hyperplanes are embedded which by the previous mentioned work can be made virtually Hanken[8][9]
See also
• Virtually fibered conjecture
• Surface subgroup conjecture
• Ehrenpreis conjecture
Notes
1. Waldhausen, Friedhelm (1968). "On irreducible 3-manifolds which are sufficiently large". Annals of Mathematics. 87 (1): 56–88. doi:10.2307/1970594. JSTOR 1970594. MR 0224099.
2. Agol, Ian (2013). With an appendix by Ian Agol, Daniel Groves, and Jason Manning. "The virtual Haken Conjecture". Doc. Math. 18: 1045–1087. MR 3104553.
3. Haglund, Frédéric; Wise, Daniel (2012). "A combination theorem for special cube complexes". Annals of Mathematics. 176 (3): 1427–1482. doi:10.4007/annals.2012.176.3.2. MR 2979855.
4. Kahn, Jeremy; Markovic, Vladimir (2012). "Immersing almost geodesic surfaces in a closed hyperbolic three manifold". Annals of Mathematics. 175 (3): 1127–1190. arXiv:0910.5501. doi:10.4007/annals.2012.175.3.4. MR 2912704. S2CID 32593851.
5. Kahn, Jeremy; Markovic, Vladimir (2012). "Counting essential surfaces in a closed hyperbolic three-manifold". Geometry & Topology. 16 (1): 601–624. arXiv:1012.2828. doi:10.2140/gt.2012.16.601. MR 2916295.
6. Daniel T. Wise, The structure of groups with a quasiconvex hierarchy, https://docs.google.com/file/d/0B45cNx80t5-2NTU0ZTdhMmItZTIxOS00ZGUyLWE0YzItNTEyYWFiMjczZmIz/edit?pli=1
7. Bergeron, Nicolas; Wise, Daniel T. (2012). "A boundary criterion for cubulation". American Journal of Mathematics. 134 (3): 843–859. arXiv:0908.3609. doi:10.1353/ajm.2012.0020. MR 2931226. S2CID 14128842.
8. Przytycki, Piotr; Wise, Daniel (2017-10-19). "Mixed 3-manifolds are virtually special". Journal of the American Mathematical Society. 31 (2): 319–347. doi:10.1090/jams/886. ISSN 0894-0347. S2CID 39611341.
9. "Piotr Przytycki and Daniel Wise receive 2022 Moore Prize". American Mathematical Society.
References
• Dunfield, Nathan; Thurston, William (2003), "The virtual Haken conjecture: experiments and examples", Geometry and Topology, 7: 399–441, arXiv:math/0209214, doi:10.2140/gt.2003.7.399, MR 1988291, S2CID 6265421.
• Kirby, Robion (1978), "Problems in low dimensional manifold theory.", Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), vol. 7, pp. 273–312, ISBN 9780821867891, MR 0520548.
External links
• Klarreich, Erica (2012-10-02). "Getting Into Shapes: From Hyperbolic Geometry to Cube Complexes and Back". Quanta Magazine.
| Wikipedia |
Virtually
In mathematics, especially in the area of abstract algebra that studies infinite groups, the adverb virtually is used to modify a property so that it need only hold for a subgroup of finite index. Given a property P, the group G is said to be virtually P if there is a finite index subgroup $H\leq G$ such that H has property P.
For a definition of the term "virtually", see the Wiktionary entry virtually.
Common uses for this would be when P is abelian, nilpotent, solvable or free. For example, virtually solvable groups are one of the two alternatives in the Tits alternative, while Gromov's theorem states that the finitely generated groups with polynomial growth are precisely the finitely generated virtually nilpotent groups.
This terminology is also used when P is just another group. That is, if G and H are groups then G is virtually H if G has a subgroup K of finite index in G such that K is isomorphic to H.
In particular, a group is virtually trivial if and only if it is finite. Two groups are virtually equal if and only if they are commensurable.
Examples
Virtually abelian
The following groups are virtually abelian.
• Any abelian group.
• Any semidirect product $N\rtimes H$ where N is abelian and H is finite. (For example, any generalized dihedral group.)
• Any semidirect product $N\rtimes H$ where N is finite and H is abelian.
• Any finite group (since the trivial subgroup is abelian).
Virtually nilpotent
• Any group that is virtually abelian.
• Any nilpotent group.
• Any semidirect product $N\rtimes H$ where N is nilpotent and H is finite.
• Any semidirect product $N\rtimes H$ where N is finite and H is nilpotent.
Gromov's theorem says that a finitely generated group is virtually nilpotent if and only if it has polynomial growth.
Virtually polycyclic
Main article: virtually polycyclic group
Virtually free
• Any free group.
• Any virtually cyclic group.
• Any semidirect product $N\rtimes H$ where N is free and H is finite.
• Any semidirect product $N\rtimes H$ where N is finite and H is free.
• Any free product $H*K$, where H and K are both finite. (For example, the modular group $\operatorname {PSL} (2,\mathbb {Z} )$.)
It follows from Stalling's theorem that any torsion-free virtually free group is free.
Others
The free group $F_{2}$ on 2 generators is virtually $F_{n}$ for any $n\geq 2$ as a consequence of the Nielsen–Schreier theorem and the Schreier index formula.
The group $\operatorname {O} (n)$ is virtually connected as $\operatorname {SO} (n)$ has index 2 in it.
References
Look up virtually in Wiktionary, the free dictionary.
• Schneebeli, Hans Rudolf (1978). "On virtual properties and group extensions". Mathematische Zeitschrift. 159: 159–167. doi:10.1007/bf01214488. Zbl 0358.20048.
| Wikipedia |
Virtually fibered conjecture
In the mathematical subfield of 3-manifolds, the virtually fibered conjecture, formulated by American mathematician William Thurston, states that every closed, irreducible, atoroidal 3-manifold with infinite fundamental group has a finite cover which is a surface bundle over the circle.
A 3-manifold which has such a finite cover is said to virtually fiber. If M is a Seifert fiber space, then M virtually fibers if and only if the rational Euler number of the Seifert fibration or the (orbifold) Euler characteristic of the base space is zero.
The hypotheses of the conjecture are satisfied by hyperbolic 3-manifolds. In fact, given that the geometrization conjecture is now settled, the only case needed to be proven for the virtually fibered conjecture is that of hyperbolic 3-manifolds.
The original interest in the virtually fibered conjecture (as well as its weaker cousins, such as the virtually Haken conjecture) stemmed from the fact that any of these conjectures, combined with Thurston's hyperbolization theorem, would imply the geometrization conjecture. However, in practice all known attacks on the "virtual" conjecture take geometrization as a hypothesis, and rely on the geometric and group-theoretic properties of hyperbolic 3-manifolds.
The virtually fibered conjecture was not actually conjectured by Thurston. Rather, he posed it as a question and has stated that it was intended as a challenge and not meant to indicate he believed it, although he wrote that "[t]his dubious-sounding question seems to have a definite chance for a positive answer".[1]
The conjecture was finally settled in the affirmative in a series of papers from 2009 to 2012. In a posting on the ArXiv on 25 Aug 2009,[2] Daniel Wise implicitly implied (by referring to a then-unpublished longer manuscript) that he had proven the conjecture for the case where the 3-manifold is closed, hyperbolic, and Haken. This was followed by a survey article in Electronic Research Announcements in Mathematical Sciences.[3][4][5][6] have followed, including the aforementioned longer manuscript by Wise.[7] In March 2012, during a conference at Institut Henri Poincaré in Paris, Ian Agol announced he could prove the virtually Haken conjecture for closed hyperbolic 3-manifolds .[8] Taken together with Daniel Wise's results, this implies the virtually fibered conjecture for all closed hyperbolic 3-manifolds.
See also
• Virtually Haken conjecture
• Surface subgroup conjecture
• Ehrenpreis conjecture
• positive virtual Betti number conjecture
Notes
1. Thurston 1982, p. 380.
2. Bergeron, Nicolas; Wise, Daniel T. (2009). "A boundary criterion for cubulation". arXiv:0908.3609. {{cite journal}}: Cite journal requires |journal= (help)
3. Wise, Daniel (2009). "Research announcement: The structure of groups with a quasiconvex hierarchy". Electronic Research Announcements in Mathematical Sciences. 16: 44–55. doi:10.3934/era.2009.16.44.
4. Haglund, Frédéric; Wise, Daniel (2012). "A combination theorem for special cube complexes". Annals of Mathematics. 176 (3): 1427–1482. doi:10.4007/annals.2012.176.3.2.
5. Christopher Hruska, G. C.; Wise, Daniel T. (2014). "Finiteness properties of cubulated groups". Compositio Mathematica. 150 (3): 453–506. arXiv:1209.1074. doi:10.1112/S0010437X13007112. S2CID 119341019.
6. Hsu, Tim; Wise, Daniel T. (2015). "Cubulating malnormal amalgams". Inventiones Mathematicae. 199 (2): 293–331. Bibcode:2015InMat.199..293H. doi:10.1007/s00222-014-0513-4. S2CID 122292998.
7. Wise, Daniel T. The structure of groups with a quasiconvex hierarchy (PDF).
8. Agol, Ian; Groves, Daniel; Manning, Jason (2012). "The virtual Haken conjecture". arXiv:1204.2810. {{cite journal}}: Cite journal requires |journal= (help)
References
• Thurston, William P. (1982). "Three dimensional manifolds, Kleinian groups and hyperbolic geometry". Bulletin of the American Mathematical Society. 6 (3): 357–382. doi:10.1090/S0273-0979-1982-15003-0.
• D. Gabai, On 3-manifold finitely covered by surface bundles, Low Dimensional Topology and Kleinian Groups (ed: D.B.A. Epstein), London Mathematical Society Lecture Note Series vol 112 (1986), p. 145-155.
• Agol, Ian (2008). "Criteria for virtual fibering". Journal of Topology. 1 (2): 269–284. arXiv:0707.4522. doi:10.1112/jtopol/jtn003. S2CID 3028314.
External links
• Klarreich, Erica (2012-10-02). "Getting Into Shapes: From Hyperbolic Geometry to Cube Complexes and Back". Quanta Magazine.
| Wikipedia |
Michael Viscardi
Michael Anthony Viscardi (born February 22, 1989 in Plano, Texas) of San Diego, California is an American mathematician who, as a highschooler, won the 2005 Siemens Competition and Davidson Fellowship with a mathematical project on the Dirichlet problem, whose applications include describing the flow of heat across a metal surface, winning $100,000 and $50,000 in scholarships, respectively.[1][2] Viscardi's theorem is an expansion of the 19th-century work of Peter Gustav Lejeune Dirichlet.[3] He was also named a finalist with the same project in the Intel Science Talent Search. Viscardi placed Best of Category in Mathematics at the International Science and Engineering Fair (ISEF) in May 2006. Viscardi also qualified for the United States of America Mathematical Olympiad and the Junior Science and Humanities Symposium.
Michael Viscardi
Born (1989-02-22) February 22, 1989
Plano, Texas, United States
NationalityAmerican
Alma materHarvard University
Massachusetts Institute of Technology
Known forSiemens Competition winner
Awards2010 Hoopes Prize
Scientific career
FieldsMathematics
Doctoral advisorRoman Bezrukavnikov
Other academic advisorsShing-Tung Yau
Joe Harris
Life
Viscardi was homeschooled for high school, supplemented with mathematics classes at the University of California, San Diego.[4][5] He is also a pianist and violinist, and onetime concertmaster of the San Diego Youth Symphony.[5]
Viscardi is a member of the Harvard College class of 2010.[6] He graduated summa cum laude from Harvard, receiving a 2010 Thomas T. Hoopes, Class of 1919, Prize, and earning the 2011 Morgan Prize honorable mention for his senior thesis "Alternate Compactifications of the Moduli Space of Genus One Maps".[7] He worked as a postdoc at UC Berkeley from 2016 to 2018.[8]
Selected publication
• ———; Ebenfelt, Peter (2007), "An Explicit Solution to the Dirichlet Problem with Rational Holomorphic Data in Terms of a Riemann Mapping", Computational Methods and Function Theory, 7 (1): 127–140, doi:10.1007/BF03321636, S2CID 120812150.
References
1. Briggs, Tracey Wong (December 5, 2005), "Problems no problem for Siemens winners", USA Today.
2. Rodriguez, Juan-Carlos (December 6, 2005), "California teen wins science competition", Seattle Times.
3. "Teen Updates 19th-Century Mathematical Law", ABC News, December 9, 2005.
4. "Homeschooled teen wins top science honor", Associated Press, 2005
5. "Mathematics Student Wins the Siemens-Westinghouse Competition", FOCUS, Mathematical Association of America, vol. 26, no. 1, p. 3, January 2006
6. Herchel Smith Research Fellows to begin this summer
7. Viscardi, Michael (2010). "Alternate compactifications of the moduli space of genus one maps". arXiv:1005.1431 [math.AG].
8. Viscardi's webpage at Berkeley
External links
• Viscardi's website at MIT
• Michael Viscardi: Person of the Week
• Michael's Presentation
• Biography at Davidson Institute site
Authority control: Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
| Wikipedia |
Viscosity solution
In mathematics, the viscosity solution concept was introduced in the early 1980s by Pierre-Louis Lions and Michael G. Crandall as a generalization of the classical concept of what is meant by a 'solution' to a partial differential equation (PDE). It has been found that the viscosity solution is the natural solution concept to use in many applications of PDE's, including for example first order equations arising in dynamic programming (the Hamilton–Jacobi–Bellman equation), differential games (the Hamilton–Jacobi–Isaacs equation) or front evolution problems,[1][2] as well as second-order equations such as the ones arising in stochastic optimal control or stochastic differential games.
The classical concept was that a PDE
$F(x,u,Du,D^{2}u)=0$
over a domain $x\in \Omega $ has a solution if we can find a function u(x) continuous and differentiable over the entire domain such that $x$, $u$, $Du$, $D^{2}u$ satisfy the above equation at every point.
If a scalar equation is degenerate elliptic (defined below), one can define a type of weak solution called viscosity solution. Under the viscosity solution concept, u does not need to be everywhere differentiable. There may be points where either $Du$ or $D^{2}u$ does not exist and yet u satisfies the equation in an appropriate generalized sense. The definition allows only for certain kind of singularities, so that existence, uniqueness, and stability under uniform limits, hold for a large class of equations.
Definition
There are several equivalent ways to phrase the definition of viscosity solutions. See for example the section II.4 of Fleming and Soner's book[3] or the definition using semi-jets in the Users Guide.[4]
Degenerate elliptic
An equation $F(x,u,Du,D^{2}u)=0$ in a domain $\Omega $ is defined to be degenerate elliptic if for any two symmetric matrices $X$ and $Y$ such that $Y-X$ is positive definite, and any values of $x\in \Omega $, $u\in \mathbb {R} $ and $p\in \mathbb {R} ^{n}$, we have the inequality $F(x,u,p,X)\geq F(x,u,p,Y)$. For example, $-\Delta u=0$ (where $\Delta $ denotes the Laplacian) is degenerate elliptic since in this case, $F(x,u,p,X)=-{\text{trace}}(X)$, and the trace of $X$ is the sum of its eigenvalues. Any real first- order equation is degenerate elliptic.
Viscosity subsolution
An upper semicontinuous function $u$ in $\Omega $ is defined to be a subsolution of the above degenerate elliptic equation in the viscosity sense if for any point $x_{0}\in \Omega $ and any $C^{2}$ function $\phi $ such that $\phi (x_{0})=u(x_{0})$ and $\phi \geq u$ in a neighborhood of $x_{0}$, we have $F(x_{0},\phi (x_{0}),D\phi (x_{0}),D^{2}\phi (x_{0}))\leq 0$.
Viscosity supersolution
A lower semicontinuous function $u$ in $\Omega $ is defined to be a supersolution of the above degenerate elliptic equation in the viscosity sense if for any point $x_{0}\in \Omega $ and any $C^{2}$ function $\phi $ such that $\phi (x_{0})=u(x_{0})$ and $\phi \leq u$ in a neighborhood of $x_{0}$, we have $F(x_{0},\phi (x_{0}),D\phi (x_{0}),D^{2}\phi (x_{0}))\geq 0$.
Viscosity solution
A continuous function u is a viscosity solution of the PDE $F(x,u,Du,D^{2}u)=0$ in $\Omega $ if it is both a supersolution and a subsolution. Note that the boundary condition in the viscosity sense has not been discussed here.
Example
Consider the boundary value problem $|u'(x)|=1$, or $F(u')=|u'|-1=0$, on $(-1,1)$ with boundary conditions $u(-1)=u(1)=0$. Then, the function $u(x)=1-|x|$ is a viscosity solution.
Indeed, note that the boundary conditions are satisfied classically, and $|u'(x)|=1$ is well-defined in the interior except at $x=0$. Thus, it remains to show that the conditions for viscosity subsolution and viscosity supersolution hold at $x=0$. Suppose that $\phi (x)$ is any function differentiable at $x=0$ with $\phi (0)=u(0)=1$ and $\phi (x)\geq u(x)$ near $x=0$. From these assumptions, it follows that $\phi (x)-\phi (0)\geq -|x|$. For positive $x$, this inequality implies $\lim _{x\to 0^{+}}{\frac {\phi (x)-\phi (0)}{x}}\geq -1$, using that $|x|/x=sgn(x)=1$ for $x>0$. On the other hand, for $x<0$, we have that $\lim _{x\to 0^{-}}{\frac {\phi (x)-\phi (0)}{x}}\leq 1$. Because $\phi $ is differentiable, the left and right limits agree and are equal to $\phi '(0)$, and we therefore conclude that $|\phi '(0)|\leq 1$, i.e., $F(\phi '(0))\leq 0$. Thus, $u$ is a viscosity subsolution. Moreover, the fact that $u$ is a supersolution holds vacuously, since there is no function $\phi (x)$ differentiable at $x=0$ with $\phi (0)=u(0)=1$ and $\phi (x)\leq u(x)$ near $x=0$. This implies that $u$ is a viscosity solution.
In fact, one may prove that $u$ is the unique viscosity solution for such problem. The uniqueness part involves a more refined argument.
Discussion
The previous boundary value problem is an eikonal equation in a single spatial dimension with $f=1$, where the solution is known to be the signed distance function to the boundary of the domain. Note also in the previous example, the importance of the sign of $F$. In particular, the viscosity solution to the PDE $-F=0$ with the same boundary conditions is $u(x)=|x|-1$. This can be explained by observing that the solution $u(x)=1-|x|$ is the limiting solution of the vanishing viscosity problem $F(u')=[u']^{2}-1=\epsilon u''$ as $\epsilon $ goes to zero, while $u(x)=|x|-1$ is the limit solution of the vanishing viscosity problem $-F(u')=1-[u']^{2}=\epsilon u''$.[5] One can readily confirm that $u_{\epsilon }(x)=\epsilon [\ln(\cosh(1/\epsilon ))-\ln(\cosh(x/\epsilon ))]$ solves the PDE $F(u')=[u']^{2}-1=\epsilon u''$ for each $\epsilon >0$. Further, the family of solutions $u_{\epsilon }$ converges toward the solution $u=1-|x|$ as $\epsilon $ vanishes (see Figure).
Basic properties
The three basic properties of viscosity solutions are existence, uniqueness and stability.
• The uniqueness of solutions requires some extra structural assumptions on the equation. Yet it can be shown for a very large class of degenerate elliptic equations.[4] It is a direct consequence of the comparison principle. Some simple examples where comparison principle holds are
1. $u+H(x,\nabla u)=0$ with H uniformly continuous in both variables.
2. (Uniformly elliptic case) $F(D^{2}u,Du,u)=0$ so that $F$ is Lipschitz with respect to all variables and for every $r\leq s$ and $X\geq Y$, $F(Y,p,s)\geq F(X,p,r)+\lambda ||X-Y||$ for some $\lambda >0$.
• The existence of solutions holds in all cases where the comparison principle holds and the boundary conditions can be enforced in some way (through barrier functions in the case of a Dirichlet boundary condition). For first order equations, it can be obtained using the vanishing viscosity method[6][2] or for most equations using Perron's method.[7][8][2] There is a generalized notion of boundary condition, in the viscosity sense. The solution to a boundary problem with generalized boundary conditions is solvable whenever the comparison principle holds.[4]
• The stability of solutions in $L^{\infty }$ holds as follows: a locally uniform limit of a sequence of solutions (or subsolutions, or supersolutions) is a solution (or subsolution, or supersolution). More generally, the notions of viscosity sub- and supersolution are also conserved by half-relaxed limits.[4]
History
The term viscosity solutions first appear in the work of Michael G. Crandall and Pierre-Louis Lions in 1983 regarding the Hamilton–Jacobi equation.[6] The name is justified by the fact that the existence of solutions was obtained by the vanishing viscosity method. The definition of solution had actually been given earlier by Lawrence C. Evans in 1980.[9] Subsequently the definition and properties of viscosity solutions for the Hamilton–Jacobi equation were refined in a joint work by Crandall, Evans and Lions in 1984.[10]
For a few years the work on viscosity solutions concentrated on first order equations because it was not known whether second order elliptic equations would have a unique viscosity solution except in very particular cases. The breakthrough result came with the method introduced by Robert Jensen in 1988 to prove the comparison principle using a regularized approximation of the solution which has a second derivative almost everywhere (in modern versions of the proof this is achieved with sup-convolutions and Alexandrov theorem).[11]
In subsequent years the concept of viscosity solution has become increasingly prevalent in analysis of degenerate elliptic PDE. Based on their stability properties, Barles and Souganidis obtained a very simple and general proof of convergence of finite difference schemes.[12] Further regularity properties of viscosity solutions were obtained, especially in the uniformly elliptic case with the work of Luis Caffarelli.[13] Viscosity solutions have become a central concept in the study of elliptic PDE. In particular, Viscosity solutions are essential in the study of the infinity Laplacian.[14]
In the modern approach, the existence of solutions is obtained most often through the Perron method.[4] The vanishing viscosity method is not practical for second order equations in general since the addition of artificial viscosity does not guarantee the existence of a classical solution. Moreover, the definition of viscosity solutions does not generally involve physical viscosity. Nevertheless, while the theory of viscosity solutions is sometimes considered unrelated to viscous fluids, irrotational fluids can indeed be described by a Hamilton-Jacobi equation.[15] In this case, viscosity corresponds to the bulk viscosity of an irrotational, incompressible fluid. Other names that were suggested were Crandall–Lions solutions, in honor to their pioneers, $L^{\infty }$-weak solutions, referring to their stability properties, or comparison solutions, referring to their most characteristic property.
References
1. Dolcetta, I.; Lions, P., eds. (1995). Viscosity Solutions and Applications. Berlin: Springer. ISBN 3-540-62910-6.
2. Tran, Hung V. (2021). Hamilton-Jacobi Equations : Theory and Applications. Providence, Rhode Island. ISBN 978-1-4704-6511-7. OCLC 1240263322.{{cite book}}: CS1 maint: location missing publisher (link)
3. Wendell H. Fleming, H. M . Soner, (2006), Controlled Markov Processes and Viscosity Solutions. Springer, ISBN 978-0-387-26045-7.
4. Crandall, Michael G.; Ishii, Hitoshi; Lions, Pierre-Louis (1992), "User's guide to viscosity solutions of second order partial differential equations", Bulletin of the American Mathematical Society, New Series, 27 (1): 1–67, arXiv:math/9207212, Bibcode:1992math......7212C, doi:10.1090/S0273-0979-1992-00266-5, ISSN 0002-9904, S2CID 119623818
5. Barles, Guy (2013). "An Introduction to the Theory of Viscosity Solutions for First-Order Hamilton–Jacobi Equations and Applications". Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications. Lecture Notes in Mathematics. Vol. 2074. Berlin: Springer. pp. 49–109. doi:10.1007/978-3-642-36433-4_2. ISBN 978-3-642-36432-7. S2CID 55804130.
6. Crandall, Michael G.; Lions, Pierre-Louis (1983), "Viscosity solutions of Hamilton-Jacobi equations", Transactions of the American Mathematical Society, 277 (1): 1–42, doi:10.2307/1999343, ISSN 0002-9947, JSTOR 1999343
7. Ishii, Hitoshi (1987), "Perron's method for Hamilton-Jacobi equations", Duke Mathematical Journal, 55 (2): 369–384, doi:10.1215/S0012-7094-87-05521-9, ISSN 0012-7094
8. Ishii, Hitoshi (1989), "On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs", Communications on Pure and Applied Mathematics, 42 (1): 15–45, doi:10.1002/cpa.3160420103, ISSN 0010-3640
9. Evans, Lawrence C. (1980), "On solving certain nonlinear partial differential equations by accretive operator methods", Israel Journal of Mathematics, 36 (3): 225–247, doi:10.1007/BF02762047, ISSN 0021-2172
10. Crandall, Michael G.; Evans, Lawrence C.; Lions, Pierre-Louis (1984), "Some properties of viscosity solutions of Hamilton–Jacobi equations", Transactions of the American Mathematical Society, 282 (2): 487–502, doi:10.2307/1999247, ISSN 0002-9947, JSTOR 1999247
11. Jensen, Robert (1988), "The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations", Archive for Rational Mechanics and Analysis, 101 (1): 1–27, Bibcode:1988ArRMA.101....1J, doi:10.1007/BF00281780, ISSN 0003-9527, S2CID 5776251
12. Barles, G.; Souganidis, P. E. (1991), "Convergence of approximation schemes for fully nonlinear second order equations", Asymptotic Analysis, 4 (3): 271–283, doi:10.3233/ASY-1991-4305, ISSN 0921-7134
13. Caffarelli, Luis A.; Cabré, Xavier (1995), Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, vol. 43, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0437-7
14. Crandall, Michael G.; Evans, Lawrence C.; Gariepy, Ronald F. (2001), "Optimal Lipschitz extensions and the infinity Laplacian", Calculus of Variations and Partial Differential Equations, 13 (2): 123–129, doi:10.1007/s005260000065, S2CID 1529607
15. Westernacher-Schneider, John Ryan; Markakis, Charalampos; Tsao, Bing Jyun (2020). "Hamilton-Jacobi hydrodynamics of pulsating relativistic stars". Classical and Quantum Gravity. 37 (15): 155005. arXiv:1912.03701. Bibcode:2020CQGra..37o5005W. doi:10.1088/1361-6382/ab93e9. S2CID 208909879.
| Wikipedia |
Visibility graph analysis
In architecture, visibility graph analysis (VGA) is a method of analysing the inter-visibility connections within buildings or urban networks. Visibility graph analysis was developed from the architectural theory of space syntax by Turner et al. (2001), and is applied through the construction of a visibility graph within the open space of a plan.
"Visibility analysis" redirects here. Not to be confused with Visibility (geometry).
Visibility graph analysis uses various measures from the theory of small-world networks and centrality in network theory in order to assess perceptual qualities of space and the possible usage of it.
Visibility graph analysis was firstly implemented in Turner's Depthmap software and is now widely used by both academics and practitioners through the open source and multi-platform depthmapX developed by Tasos Varoudis.
Another opensource and multi-platform software that implements visibility graphs is topologicpy developed by Wassim Jabi.
See also
• Fuzzy architectural spatial analysis
• Isovist
• Spatial network analysis software
• Viewshed analysis
References
• A. Turner; Doxa, M.; O'Sullivan, D.; Penn, A. (2001). "From isovists to visibility graphs: a methodology for the analysis of architectural space" (PDF). Environment and Planning B. 28 (1): 103–121. doi:10.1068/b2684.
| Wikipedia |
Visibility graph
In computational geometry and robot motion planning,[1] a visibility graph is a graph of intervisible locations, typically for a set of points and obstacles in the Euclidean plane. Each node in the graph represents a point location, and each edge represents a visible connection between them. That is, if the line segment connecting two locations does not pass through any obstacle, an edge is drawn between them in the graph. When the set of locations lies in a line, this can be understood as an ordered series. Visibility graphs have therefore been extended to the realm of time series analysis.
Applications
Visibility graphs may be used to find Euclidean shortest paths among a set of polygonal obstacles in the plane: the shortest path between two obstacles follows straight line segments except at the vertices of the obstacles, where it may turn, so the Euclidean shortest path is the shortest path in a visibility graph that has as its nodes the start and destination points and the vertices of the obstacles.[2] Therefore, the Euclidean shortest path problem may be decomposed into two simpler subproblems: constructing the visibility graph, and applying a shortest path algorithm such as Dijkstra's algorithm to the graph. For planning the motion of a robot that has non-negligible size compared to the obstacles, a similar approach may be used after expanding the obstacles to compensate for the size of the robot.[2] Lozano-Pérez & Wesley (1979) attribute the visibility graph method for Euclidean shortest paths to research in 1969 by Nils Nilsson on motion planning for Shakey the robot, and also cite a 1973 description of this method by Russian mathematicians M. B. Ignat'yev, F. M. Kulakov, and A. M. Pokrovskiy.
Visibility graphs may also be used to calculate the placement of radio antennas, or as a tool used within architecture and urban planning through visibility graph analysis.
The visibility graph of a set of locations that lie in a line can be interpreted as a graph-theoretical representation of a time series.[3] This particular case builds a bridge between time series, dynamical systems and graph theory.
Characterization
The visibility graph of a simple polygon has the polygon's vertices as its point locations, and the exterior of the polygon as the only obstacle. Visibility graphs of simple polygons must be Hamiltonian graphs: the boundary of the polygon forms a Hamiltonian cycle in the visibility graph. It is known that not all visibility graphs induce a simple polygon. However, an efficient algorithmic characterization of the visibility graphs of simple polygons remains unknown. These graphs do not fall into many known families of well-structured graphs: they might not be perfect graphs, circle graphs, or chordal graphs.[4] An exception to this phenomenon is that the visibility graphs of simple polygons are cop-win graphs.[5]
Related problems
The art gallery problem is the problem of finding a small set of points such that all other non-obstacle points are visible from this set. Certain forms of the art gallery problem may be interpreted as finding a dominating set in a visibility graph.
The bitangents of a system of polygons or curves are lines that touch two of them without penetrating them at their points of contact. The bitangents of a set of polygons form a subset of the visibility graph that has the polygon's vertices as its nodes and the polygons themselves as the obstacles. The visibility graph approach to the Euclidean shortest path problem may be sped up by forming a graph from the bitangents instead of using all visibility edges, since a Euclidean shortest path may only enter or leave the boundary of an obstacle along a bitangent.[6]
See also
• Visibility graph analysis
• Fuzzy architectural spatial analysis
• Space syntax
Notes
1. Niu, Hanlin; Savvaris, Al; Tsourdos, Antonios; Ji, Ze (2019). "Voronoi-Visibility Roadmap-based Path Planning Algorithm for Unmanned Surface Vehicles". Journal of Navigation. 72 (04): 850–874. doi:10.1017/S0373463318001005. ISSN 0373-4633.
2. de Berg et al. (2000), sections 5.1 and 5.3; Lozano-Pérez & Wesley (1979).
3. Lacasa, Lucas; Luque, Bartolo; Ballesteros, Fernando; Luque, Jordi; Nuño, Juan Carlos (2008). "From time series to complex networks: The visibility graph". Proceedings of the National Academy of Sciences. 105 (13): 4972–4975. arXiv:0810.0920. doi:10.1073/pnas.0709247105. PMC 2278201. PMID 18362361.
4. Ghosh, S. K. (1997-03-01). "On recognizing and characterizing visibility graphs of simple polygons". Discrete & Computational Geometry. 17 (2): 143–162. doi:10.1007/BF02770871. ISSN 0179-5376.
5. Lubiw, Anna; Snoeyink, Jack; Vosoughpour, Hamideh (2017). "Visibility graphs, dismantlability, and the cops and robbers game". Computational Geometry. 66: 14–27. arXiv:1601.01298. doi:10.1016/j.comgeo.2017.07.001. MR 3693353.
6. de Berg et al. (2000), p. 316.
References
• de Berg, Mark; van Kreveld, Marc; Overmars, Mark; Schwarzkopf, Otfried (2000), "Chapter 15: Visibility Graphs", Computational Geometry (2nd ed.), Springer-Verlag, pp. 307–317, ISBN 978-3-540-65620-3.
• Lozano-Pérez, Tomás; Wesley, Michael A. (1979), "An algorithm for planning collision-free paths among polyhedral obstacles", Communications of the ACM, 22 (10): 560–570, doi:10.1145/359156.359164, S2CID 17397594.
External links
• VisiLibity: A free open source C++ library of floating-point visibility algorithms and supporting data types. This software can be used for calculating visibility graphs of polygonal environments with polygonal holes. A Matlab interface is also included.
| Wikipedia |
Visibility (geometry)
In geometry, visibility is a mathematical abstraction of the real-life notion of visibility.
Given a set of obstacles in the Euclidean space, two points in the space are said to be visible to each other, if the line segment that joins them does not intersect any obstacles. (In the Earth's atmosphere light follows a slightly curved path that is not perfectly predictable, complicating the calculation of actual visibility.)
Computation of visibility is among the basic problems in computational geometry and has applications in computer graphics, motion planning, and other areas.
Concepts and problems
• Point visibility
• Edge visibility[1][2]
• Visibility polygon
• Weak visibility
• Art gallery problem or museum problem
• Visibility graph
• Visibility graph of vertical line segments
• Watchman route problem
• Computer graphics applications:
• Hidden surface determination
• Hidden line removal
• z-buffering
• portal engine
• Star-shaped polygon
• Kernel of a polygon
• Isovist
• Viewshed
• Zone of Visual Influence
• Painter's algorithm
References
• O'Rourke, Joseph (1987). Art Gallery Theorems and Algorithms. Oxford University Press. ISBN 0-19-503965-3.
• Ghosh, Subir Kumar (2007). Visibility Algorithms in the Plane. Cambridge University Press. ISBN 978-0-521-87574-5.
• Mark de Berg, Marc van Kreveld, Mark Overmars, and Otfried Schwarzkopf (2000). Computational Geometry (2nd revised ed.). Springer-Verlag. ISBN 3-540-65620-0. 1st edition (1987).{{cite book}}: CS1 maint: multiple names: authors list (link) Chapter 15: "Visibility graphs"
1. D. Avis and G. T. Toussaint, "An optimal algorithm for determining the visibility of a polygon from an edge," IEEE Transactions on Computers, vol. C-30, No. 12, December 1981, pp. 910-914.
2. E. Roth, G. Panin and A. Knoll, "Sampling feature points for contour tracking with graphics hardware", "In International Workshop on Vision, Modeling and Visualization (VMV)", Konstanz, Germany, October 2008.
External links
Software
• VisiLibity: A free open source C++ library of floating-point visibility algorithms and supporting data types
| Wikipedia |
Visual calculus
Visual calculus, invented by Mamikon Mnatsakanian (known as Mamikon), is an approach to solving a variety of integral calculus problems.[1] Many problems that would otherwise seem quite difficult yield to the method with hardly a line of calculation, often reminiscent of what Martin Gardner called "aha! solutions" or Roger Nelsen a proof without words.[2][3]
Description
Mamikon devised his method in 1959 while an undergraduate, first applying it to a well-known geometry problem: find the area of a ring (annulus), given the length of a chord tangent to the inner circumference. Perhaps surprisingly, no additional information is needed; the solution does not depend on the ring's inner and outer dimensions.
The traditional approach involves algebra and application of the Pythagorean theorem. Mamikon's method, however, envisions an alternate construction of the ring: first the inner circle alone is drawn, then a constant-length tangent is made to travel along its circumference, "sweeping out" the ring as it goes.
Now if all the (constant-length) tangents used in constructing the ring are translated so that their points of tangency coincide, the result is a circular disk of known radius (and easily computed area). Indeed, since the inner circle's radius is irrelevant, one could just as well have started with a circle of radius zero (a point)—and sweeping out a ring around a circle of zero radius is indistinguishable from simply rotating a line segment about one of its endpoints and sweeping out a disk.
Mamikon's insight was to recognize the equivalence of the two constructions; and because they are equivalent, they yield equal areas. Moreover, so long as it is given that the tangent length is constant, the two starting curves need not be circular—a finding not easily proven by more traditional geometric methods. This yields Mamikon's theorem:
The area of a tangent sweep is equal to the area of its tangent cluster, regardless of the shape of the original curve.
Applications
Area of a cycloid
The area of a cycloid can be calculated by considering the area between it and an enclosing rectangle. These tangents can all be clustered to form a circle. If the circle generating the cycloid has radius r then this circle also has radius r and area πr2. The area of the rectangle is 2r × 2πr = 4πr2. Therefore the area of the cycloid is 3πr2: it is 3 times the area of the generating circle.
The tangent cluster can be seen to be a circle because the cycloid is generated by a circle and the tangent to the cycloid will be at right angle to the line from the generating point to the rolling point. Thus the tangent and the line to the contact point form a right-angled triangle in the generating circle. This means that clustered together the tangents will describe the shape of the generating circle.[5]
See also
• Cavalieri's principle
• Hodograph – This is a related construct that maps the velocity of a point using a polar diagram.
• The Method of Mechanical Theorems
• Pappus's centroid theorem
• Planimeter
References
1. Visual Calculus Mamikon Mnatsakanian
2. Nelsen, Roger B. (1993). Proofs without Words, Cambridge University Press. ISBN 978-0-88385-700-7.
3. Martin Gardner (1978) Aha! Insight, W.H. Freeman & Company; ISBN 0-7167-1017-X
4. Haunsperger, Deanna; Kennedy, Stephen (2006). The Edge of the Universe: Celebrating Ten Years of Math Horizons. ISBN 9780883855553. Retrieved May 9, 2017.
5. Apostol, Mnatsakanian (2012). New Horizons in Geometry. Mathematical Association of America. ISBN 9781614442103.
External links
• ProjMath Mamikon
• Proof without Words from MathWorld
• Wolfram Interactive Demonstration of Mamikon's theorem
| Wikipedia |
Tristan Needham
Tristan Needham is a British mathematician and professor of mathematics at the University of San Francisco.
Education, career and publications
Tristan is the son of social anthropologist Rodney Needham of Oxford, England. He attended the Dragon School. Later Needham attended the University of Oxford and studied physics at Merton College, and then transferred to the Mathematical Institute where he studied under Roger Penrose. He obtained his D.Phil. in 1987 and in 1989 took up his post at University of San Francisco.[1][2]
In 1993 he published A Visual Explanation of Jensen's inequality.[3] The following year he published The Geometry of Harmonic Functions, which won the Carl B. Allendoerfer Award for 1995.[4][5]
Needham wrote the book Visual Complex Analysis, which has received positive reviews.[6] Though it is described as a "radical first course in complex analysis aimed at undergraduates", writing in Mathematical Reviews D.H. Armitage said that "the book will be appreciated most by those who already know some complex analysis."[7] In fact Douglas Hofstadter wrote "Needham's work of art with its hundreds and hundreds of beautiful figures á la Latta, brings complex analysis alive in an unprecedented manner".[8] Hofstadter had studied complex analysis at Stanford with Gordon Latta, and he recalled "Latta's amazingly precise and elegant blackboard diagrams". In 2001 a German language version, translated by Norbert Herrmann and Ina Paschen, was published by R. Oldenbourg Verlag, Munich.
In 2021, Needham published Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts (Princeton University Press)[9]. (The original title was Visual Differential Geometry.) Much of this material was already developed in the writing of Visual Complex Analysis.
See also
• Amplitwist
Bibliography
• Needham, Tristan. Visual Complex Analysis. The Clarendon Press, Oxford University Press, New York, 1997 ISBN 0-19-853447-7.[10][11]
• Needham, Tristan. Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts. Princeton University Press, Princeton, 2021 ISBN 9780691203706.[12]
Notes
1. Faculty profile Archived 2012-06-07 at the Wayback Machine from University of San Francisco
2. University of San Francisco website – History of the Sciences: Changing Course.
3. Needham, Tristan (1993). "A Visual Explanation of Jensen's Inequality". The American Mathematical Monthly. 100 (8): 768–771. doi:10.2307/2324783. JSTOR 2324783.
4. Needham, Tristan (1994). "The Geometry of Harmonic Functions". Mathematics Magazine. 67 (2): 92–108. doi:10.1080/0025570X.1994.11996195. ISSN 0025-570X.
5. Allendoerfer Award from Mathematics Association of America
6. Frank A. Farris (1998) American Mathematical Monthly, 105(6):570: "Visual Complex Analysis will show you the field of complex analysis in a way you almost certainly have not seen it before".
7. Review of Visual Complex Analysis from Mathematical Reviews
8. Preface page xvi of Chris Pritchard (2003) Changing Shape of Geometry, Cambridge University Press ISBN 0521531624
9. Needham, Tristan (2021). Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts. Princeton University Press. ISBN 9780691203690.
10. Farris, Frank A. (1998-01-01). "Review of Visual Complex Analysis". The American Mathematical Monthly. 105 (6): 570–576. doi:10.2307/2589427. JSTOR 2589427.
11. Shiu, P. (1999-01-01). "Review of Visual Complex Analysis". The Mathematical Gazette. 83 (496): 182–183. doi:10.2307/3618747. JSTOR 3618747.
12. Bultheel, Adhemar (2021-01-10). "Book Review: Visual Differential Geometry and Forms (T. Needham)". MAA Publications. Mathematical Association of America (MAA). Retrieved 2022-09-03.
External links
• Tristan Needham at the Mathematics Genealogy Project
• Author website for the book Visual Complex Analysis
• Princeton University Press website for the book Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts
• Author website (including Errata) for the book Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts
Authority control
International
• ISNI
• VIAF
National
• France
• BnF data
• Italy
• Israel
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• Czech Republic
Academics
• MathSciNet
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• zbMATH
Other
• IdRef
| Wikipedia |
Visual Turing Test
The Visual Turing Test,[1] it is “an operator-assisted device that produces a stochastic sequence of binary questions from a given test image”.[1] The query engine produces a sequence of questions that have unpredictable answers given the history of questions. The test is only about vision and does not require any natural language processing. The job of the human operator is to provide the correct answer to the question or reject it as ambiguous. The query generator produces questions such that they follow a “natural story line”, similar to what humans do when they look at a picture.
History
Research in computer vision dates back to the 1960s when Seymour Papert first attempted to solve the problem. This unsuccessful attempt was referred to as the Summer Vision Project. The reason why it was not successful was because computer vision is more complicated than what people think. The complexity is in alignment with the human visual system. Roughly 50% of the human brain is devoted in processing vision, which indicates that it is a difficult problem.
Later there were attempts to solve the problems with models inspired by the human brain. Perceptrons by Frank Rosenblatt, which is a form of the neural networks, was one of the first such approaches. These simple neural networks could not live up to their expectations and had certain limitations due to which they were not considered in future research.
Later with the availability of the hardware and some processing power the research shifted to image processing which involves pixel-level operations, like finding edges, de-noising images or applying filters to name a few. There was some great progress in this field but the problem of vision which was to make the machines understand the images was still not being addressed. During this time the neural networks also resurfaced as it was shown that the limitations of the perceptrons can be overcome by Multi-layer perceptrons. Also in the early 1990s convolutional neural networks were born which showed great results on digit recognition but did not scale up well on harder problems.
The late 1990s and early 2000s saw the birth of modern computer vision. One of the reasons this happened was due to the availability of key, feature extraction and representation algorithms. Features along with the already present machine learning algorithms were used to detect, localise and segment objects in Images.
While all these advancements were being made, the community felt the need to have standardised datasets and evaluation metrics so the performances can be compared. This led to the emergence of challenges like the Pascal VOC challenge and the ImageNet challenge. The availability of standard evaluation metrics and the open challenges gave directions to the research. Better algorithms were introduced for specific tasks like object detection and classification.
Visual Turing Test aims to give a new direction to the computer vision research which would lead to the introduction of systems that will be one step closer to understanding images the way humans do.
Current evaluation practices
A large number of datasets have been annotated and generalised to benchmark performances of difference classes of algorithms to assess different vision tasks (e.g., object detection/recognition) on some image domain (e.g., scene images).
One of the most famous datasets in computer vision is ImageNet which is used to assess the problem of object level Image classification. ImageNet is one of the largest annotated datasets available and has over one million images. The other important vision task is object detection and localisation which refers to detecting the object instance in the image and providing the bounding box coordinates around the object instance or segmenting the object. The most popular dataset for this task is the Pascal dataset. Similarly there are other datasets for specific tasks like the H3D[2] dataset for human pose detection, Core dataset to evaluate the quality of detected object attributes such as colour, orientation, and activity.
Having these standard datasets has helped the vision community to come up with extremely well performing algorithms for all these tasks. The next logical step is to create a larger task encompassing of these smaller subtasks. Having such a task would lead to building systems that would understand images, as understanding images would inherently involve detecting objects, localising them and segmenting them.
Details
The Visual Turing Test (VTT) unlike the Turing test has a query engine system which interrogates a computer vision system in the presence of a human co-ordinator.
It is a system that generates a random sequence of binary questions specific to the test image, such that the answer to any question k is unpredictable given the true answers to the previous k − 1 questions (also known as history of questions).
The test happens in the presence of a human operator who serves two main purposes: removing the ambiguous questions and providing the correct answers to the unambiguous questions. Given an Image infinite possible binary questions can be asked and a lot of them are bound to be ambiguous. These questions if generated by the query engine are removed by the human moderator and instead the query engine generates another question such that the answer to it is unpredictable given the history of the questions.
The aim of the Visual Turing Test is to evaluate the Image understanding of a computer system, and an important part of image understanding is the story line of the image. When humans look at an image, they do not think that there is a car at ‘x’ pixels from the left and ‘y’ pixels from the top, but instead they look at it as a story, for e.g. they might think that there is a car parked on the road, a person is exiting the car and heading towards a building. The most important elements of the story line are the objects and so to extract any story line from an image the first and the most important task is to instantiate the objects in it, and that is what the query engine does.
Query engine
The query engine is the core of the Visual Turing Test and it comprises two main parts : Vocabulary and Questions
Vocabulary
Vocabulary is a set of words that represent the elements of the images. This vocabulary when used with appropriate grammar leads to a set of questions. The grammar is defined in the next section in a way that it leads to a space of binary questions.
The vocabulary ${\mathcal {V}}$ consist of three components:
1. Types of Objects ${\mathcal {T}}$
2. Type-dependent attributes of objects ${\mathcal {A}}(t)$
3. Type-dependent relationships between two objects ${\mathcal {R}}(t,t')$
For Images of urban street scenes the types of objects include people, vehicle and buildings. Attributes refer to the properties of these objects, for e.g. female, child, wearing a hat or carrying something, for people and moving, parked, stopped, one tire visible or two tires visible for vehicles. Relationships between each pair of object classes can be either “ordered” or “unordered”. The unordered relationships may include talking, walking together and the ordered relationships include taller, closer to the camera, occluding, being occluded etc.
Additionally all of this vocabulary is used in context of rectangular image regions w \in W which allow for the localisation of objects in the image. An extremely large number of such regions are possible and this complicates the problem, so for this test, regions at specific scales are only used which include 1/16 the size of image, 1/4 the size of image, 1/2 the size of image or larger.
Questions
The question space is composed of four types of questions:
• Existence questions: The aim of the existence questions is to find new objects in the image that have not been uniquely identified previously.
They are of the form :
Qexist = 'Is there an instance of an object of type t with attributes A partially visible in region w that was not previously instantiated?'
• Uniqueness questions: A uniqueness question tries to uniquely identify an object to instantiate it.
Quniq = 'Is there a unique instance of an object of type t with attributes A partially visible in region w that was not previously instantiated?'
The uniqueness questions along with the existence questions form the instantiation questions. As mentioned earlier instantiating objects leads to other interesting questions and eventually a story line. Uniqueness questions follow the existence questions and a positive answer to it leads to instantiation of an object.
• Attribute questions: An attribute question tries to find more about the object once it has been instantiated. Such questions can query about a single attribute, conjunction of two attributes or disjunction of two attributes.
Qatt(ot) = {'Does object ot have attribute a?' , 'Does object ot have attribute a1 or attribute a2?' , 'Does object ot have attribute a1 and attribute a2?'}
• Relationship questions: Once multiple objects have been instantiated, a relationship question explores the relationship between pairs of objects.
Qrel(ot,ot') = 'Does object ot have relationship r with object ot'?'
Implementation details
As mentioned before the core of the Visual Turing Test is the query generator which generates a sequence of binary questions such that the answer to any question k is unpredictable given the correct answers to the previous k − 1 questions. This is a recursive process, given a history of questions and their correct answers, the query generator either stops because there are no more unpredictable questions, or randomly selects an unpredictable question and adds it to the history.
The question space defined earlier implicitly imposes a constraint on the flow of the questions. To make it more clear this means that the attribute and relationship questions can not precede the instantiation questions. Only when the objects have been instantiated, can they be queried about their attributes and relations to other previously instantiated objects. Thus given a history we can restrict the possible questions that can follow it, and this set of questions are referred to as the candidate questions $Q_{\text{can}}$.
The task is to choose an unpredictable question from these candidate questions such that it conforms with the question flow that we will describe in the next section. For this, find the unpredictability of every question among the candidate questions.
Let $H$ be a binary random variable, where $H(I)=1$, if the history $H$ is valid for the Image $I$ and $0$ otherwise. Let $q\in Q$ can be the proposed question, and $X_{q}$ be the answer to the question $q$.
Then, find the conditional probability of getting the answer Xq to the question q given the history H.
$P_{H}(X_{q}=x)={\frac {P\{I:H(I)=1,X_{q}(I)=x\}}{P\{I:H(I)=1\}}}$
Given this probability the measure of the unpredictability is given by:
$\rho _{H}(q)=|P_{H}(X_{Q}=1)-0.5|$
The closer $\rho _{H}(q)$ is to 0, the more unpredictable the question is. $\rho _{H}(q)$ for every question is calculated. The questions for which $\rho _{H}(q)<\epsilon $, are the set of almost unpredictable questions and the next question is randomly picked from these.
Question flow
As discussed in the previous section there is an implicit ordering in the question space, according to which the attribute questions come after the instantiation questions and the relationship questions come after the attribute questions, once multiple objects have been instantiated.
Therefore, the query engine follows a loop structure where it first instantiates an object with the existence and uniqueness questions, then queries about its attributes, and then the relationship questions are asked for that object with all the previously instantiated objects.
Look-ahead search
It is clear that the interesting questions about the attributes and the relations come after the instantiation questions, and so the query generator aims at instantiating as many objects as possible.
Instantiation questions are composed of both the existence and the uniqueness questions, but it is the uniqueness questions that actually instantiate an object if they get a positive response. So if the query generator has to randomly pick an instantiation question, it prefers to pick an unpredictable uniqueness question if present. If such a question is not present, the query generator picks an existence question such that it will lead to a uniqueness question with a high probability in the future. Thus the query generator performs a look-ahead search in this case.
Story line
An integral part of the ultimate aim of building systems that can understand images the way humans do, is the story line. Humans try to figure out a story line in the Image they see. The query generator achieves this by a continuity in the question sequences.
This means that once the object has been instantiated it tries to explore it in more details. Apart from finding its attributes and relation to the other objects, localisation is also an important step. Thus, as a next step the query generator tries to localise the object in the region it was first identified, so it restricts the set of instantiation questions to the regions within the original region.
Simplicity preference
Simplicity preference states that the query generator should pick simpler questions over the more complicated ones. Simpler questions are the ones that have fewer attributes in them. So this gives an ordering to the questions based on the number of attributes, and the query generator prefers the simpler ones.
Estimating predictability
To select the next question in the sequence, VTT has to estimate the predictability of every proposed question. This is done using the annotated training set of Images. Each Image is annotated with bounding box around the objects and labelled with the attributes, and pairs of objects are labelled with the relations.
Consider each question type separately:
1. Instantiation questions: The conditional probability estimator for instantiation questions can be represented as:
$\quad {\widehat {P}}(X_{q}=1)={\frac {\#\{I\in T,H(I)=1,X_{q}(I)=1\}}{\#\{I\in T,H(I)=1\}}}$
The question is only considered if the denominator is at least 80 images. The condition of $H(I)=1$ is very strict and may not be true for a large number of Images, as every question in the history eliminates approximately half of the candidates (Images in this case). As a result, the history is pruned and the questions which may not alter the conditional probability are eliminated. Having a shorter history lets us consider a larger number of Images for the probability estimation.
The history pruning is done in two stages:
• In the first stage all the attribute and relationship questions are removed, under the assumption that the presence and instantiation of objects only depends on other objects and not their attributes or relations. Also, all the existence questions referring to regions disjoint from the region being referred to in the proposed question, are dropped with the assumption being that the probability of the presence of an object at a location $w$ does not change with the presence or absence of objects at locations other than $w$. And finally all the uniqueness questions with a negative response referring to regions disjointed from the region being referred to in the proposed question, are dropped with the assumption that the uniqueness questions with a positive response if dropped can alter the response of the future instantiation questions. The history of questions obtained after this first stage of pruning can be referred to as $H_{q}'$.
• In the second stage an image-by-image pruning is performed. Let $q_{i}$ be a uniqueness question in $H$ that has not been pruned and is preserved in $H_{q}'$. If this question is in context of a region which is disjoint from the region being referenced in the proposed question, then the expected answer to this question will be $1$, because of the constraints in the first stage. But if the actual answer to this question for the training image is $0$, then that training image is not considered for the probability estimation, and the question $q_{i}$ is also dropped. The final history of questions after this is ${\tilde {H}}(q,I)$, and the probability is given by:
$\quad {\widehat {P}}(X_{q}=1)={\frac {\#\{I\in T,{\tilde {H}}(q,I)=1,X_{q}(I)=1\}}{\#\{I\in T,{\tilde {H}}(q,I)=1\}}}$
2. Attribute questions: The probability estimator for attribute questions is dependent on the number of labeled objects rather than the images unlike the instantiation questions.
Consider an attribute question of the form : ‘Does object ot have attribute a?’, where $o_{t}$ is an object of type $t$ and $a\in A_{t}$. Let $A$ be the set of attributes already known to belong to $o_{t}$ because of the history. Let ${\mathcal {O}}_{\mathbb {T} }$ be the set of all the annotated objects (ground truth) in the training set, and for each $o\in {\mathcal {O}}_{\mathbb {T} }$, let ${\mathcal {T}}_{\mathbb {T} }(o)$ be the type of object, and ${\mathcal {A}}_{\mathbb {T} }(o)$ be the set of attributes belonging to $o$. Then the estimator is given by:
$\quad P(X_{q}=1)={\frac {\#\{o\in {\mathcal {O}}_{\mathbb {T} }:{\mathcal {T}}_{\mathbb {T} }(o)=t,A\cup \{a\}\subseteq {\mathcal {A}}_{\mathbb {T} }(o)\}}{\#\{o\in {\mathcal {O}}_{\mathbb {T} }:{\mathcal {T}}_{\mathbb {T} }(o)=t,A\subseteq {\mathcal {A}}_{\mathbb {T} }(o)\}}}$
This is basically the ratio of the number of times the object $o$ of type $t$ with attributes $A\cup \{a\}$ occurs in the training data, to the number of times the object $o$ of type $t$ with attributes $A$ occurs in the training data. A high number of attributes in $A$ leads to a sparsity problem similar to the instantiation questions. To deal with it we partition the attributes into subsets that are approximately independent conditioned on belonging to the object $o_{t}$. For e.g. for $t={}$person, attributes like crossing a street and standing still are not independent, but both are fairly independent of the sex of the person, whether the person is child or adult, and whether they are carrying something or not. These conditional independencies reduce the size of the set $A$, and thereby overcome the problem of sparsity.
3. Relationship questions: The approach for relationship questions is the same as the attribute questions, where instead of the number of objects, number of pair of objects is considered and for the independence assumption, the relationships that are independent of the attributes of the related objects and the relationships that are independent of each other are included.
Example
Detailed example sequences can be found here.[3]
Dataset
The Images considered for the Geman et al.[1] work are that of ‘Urban street scenes’ dataset,[1] which has scenes of streets from different cities across the world. This why the types of objects are constrained to people and vehicles for this experiment.
Another dataset introduced by the Max Planck Institute for Informatics is known as DAQUAR[4][5] dataset which has real world images of indoor scenes. But they[4] propose a different version of the visual Turing test which takes on a holistic approach and expects the participating system to exhibit human like common sense.
Conclusion
This is a very recent work published on March 9, 2015, in the journal Proceedings of the National Academy of Sciences, by researchers from Brown University and Johns Hopkins University. It evaluates how the computer vision systems understand the Images as compared to humans. Currently the test is written and the interrogator is a machine because having an oral evaluation by a human interrogator gives the humans an undue advantage of being subjective, and also expects real time answers.
The Visual Turing Test is expected to give a new direction to the computer vision research. Companies like Google and Facebook are investing millions of dollars into computer vision research, and are trying to build systems that closely resemble the human visual system. Recently Facebook announced its new platform M, which looks at an image and provides a description of it to help the visually impaired.[6] Such systems might be able to perform well on the VTT.
References
1. Geman, Donald; Geman, Stuart; Hallonquist, Neil; Younes, Laurent (2015-03-24). "Visual Turing test for computer vision systems". Proceedings of the National Academy of Sciences. 112 (12): 3618–3623. Bibcode:2015PNAS..112.3618G. doi:10.1073/pnas.1422953112. ISSN 0027-8424. PMC 4378453. PMID 25755262.
2. "H3D". www.eecs.berkeley.edu. Retrieved 2015-11-19.
3. "Visual Turing Test | Division of Applied Mathematics". www.brown.edu. Retrieved 2015-11-19.
4. "Max-Planck-Institut für Informatik: Visual Turing Challenge". www.mpi-inf.mpg.de. Retrieved 2015-11-19.
5. Malinowski, Mateusz; Fritz, Mario (2014-10-29). "Towards a Visual Turing Challenge". arXiv:1410.8027 [cs.AI].
6. Metz, Cade (27 October 2015). "Facebook's AI Can Caption Photos for the Blind on Its Own". WIRED. Retrieved 2015-11-19.
| Wikipedia |
Implicit curve
In mathematics, an implicit curve is a plane curve defined by an implicit equation relating two coordinate variables, commonly x and y. For example, the unit circle is defined by the implicit equation $x^{2}+y^{2}=1$. In general, every implicit curve is defined by an equation of the form
$F(x,y)=0$
for some function F of two variables. Hence an implicit curve can be considered as the set of zeros of a function of two variables. Implicit means that the equation is not expressed as a solution for either x in terms of y or vice versa.
If $F(x,y)$ is a polynomial in two variables, the corresponding curve is called an algebraic curve, and specific methods are available for studying it.
Plane curves can be represented in Cartesian coordinates (x, y coordinates) by any of three methods, one of which is the implicit equation given above. The graph of a function is usually described by an equation $y=f(x)$ in which the functional form is explicitly stated; this is called an explicit representation. The third essential description of a curve is the parametric one, where the x- and y-coordinates of curve points are represented by two functions x(t), y(t) both of whose functional forms are explicitly stated, and which are dependent on a common parameter $t.$
Examples of implicit curves include:
1. a line: $x+2y-3=0,$
2. a circle: $x^{2}+y^{2}-4=0,$
3. the semicubical parabola: $x^{3}-y^{2}=0,$
4. Cassini ovals $(x^{2}+y^{2})^{2}-2c^{2}(x^{2}-y^{2})-(a^{4}-c^{4})=0$ (see diagram),
5. $\sin(x+y)-\cos(xy)+1=0$ (see diagram).
The first four examples are algebraic curves, but the last one is not algebraic. The first three examples possess simple parametric representations, which is not true for the fourth and fifth examples. The fifth example shows the possibly complicated geometric structure of an implicit curve.
The implicit function theorem describes conditions under which an equation $F(x,y)=0$ can be solved implicitly for x and/or y – that is, under which one can validly write $x=g(y)$ or $y=f(x)$. This theorem is the key for the computation of essential geometric features of the curve: tangents, normals, and curvature. In practice implicit curves have an essential drawback: their visualization is difficult. But there are computer programs enabling one to display an implicit curve. Special properties of implicit curves make them essential tools in geometry and computer graphics.
An implicit curve with an equation $F(x,y)=0$ can be considered as the level curve of level 0 of the surface $z=F(x,y)$ (see third diagram).
Slope and curvature
In general, implicit curves fail the vertical line test (meaning that some values of x are associated with more than one value of y) and so are not necessarily graphs of functions. However, the implicit function theorem gives conditions under which an implicit curve locally is given by the graph of a function (so in particular it has no self-intersections). If the defining relations are sufficiently smooth then, in such regions, implicit curves have well defined slopes, tangent lines, normal vectors, and curvature.
There are several possible ways to compute these quantities for a given implicit curve. One method is to use implicit differentiation to compute the derivatives of y with respect to x. Alternatively, for a curve defined by the implicit equation $F(x,y)=0$, one can express these formulas directly in terms of the partial derivatives of $F$. In what follows, the partial derivatives are denoted $F_{x}$ (for the derivative with respect to x), $F_{y}$, $F_{xx}$ (for the second partial with respect to x), $F_{xy}$ (for the mixed second partial), $F_{yy}.$
Tangent and normal vector
A curve point $(x_{0},y_{0})$ is regular if the first partial derivatives $F_{x}(x_{0},y_{0})$ and $F_{y}(x_{0},y_{0})$ are not both equal to 0.
The equation of the tangent line at a regular point $(x_{0},y_{0})$ is
$F_{x}(x_{0},y_{0})(x-x_{0})+F_{y}(x_{0},y_{0})(y-y_{0})=0,$
so the slope of the tangent line, and hence the slope of the curve at that point, is
${\text{slope}}=-{\frac {F_{x}(x_{0},y_{0})}{F_{y}(x_{0},y_{0})}}.$
If $F_{y}(x,y)=0\neq F_{x}(x,y)$ at $(x_{0},y_{0}),$ the curve is vertical at that point, while if both $F_{y}(x,y)=0$ and $F_{x}(x,y)=0$ at that point then the curve is not differentiable there, but instead is a singular point – either a cusp or a point where the curve intersects itself.
A normal vector to the curve at the point is given by
$\mathbf {n} (x_{0},y_{0})=(F_{x}(x_{0},y_{0}),F_{y}(x_{0},y_{0}))$
(here written as a row vector).
Curvature
For readability of the formulas, the arguments $(x_{0},y_{0})$ are omitted. The curvature $\kappa $ at a regular point is given by the formula
$\kappa ={\frac {-F_{y}^{2}F_{xx}+2F_{x}F_{y}F_{xy}-F_{x}^{2}F_{yy}}{(F_{x}^{2}+F_{y}^{2})^{3/2}}}$.[1]
Derivation of the formulas
The implicit function theorem guarantees within a neighborhood of a point $(x_{0},y_{0})$ the existence of a function $f$ such that $F(x,f(x))=0$. By the chain rule, the derivatives of function $f$ are
$f'(x)=-{\frac {F_{x}(x,f(x))}{F_{y}(x,f(x))}}$ and $f''(x)={\frac {-F_{y}^{2}F_{xx}+2F_{x}F_{y}F_{xy}-F_{x}^{2}F_{yy}}{F_{y}^{3}}}$
(where the arguments $(x,f(x))$ on the right side of the second formula are omitted for ease of reading).
Inserting the derivatives of function $f$ into the formulas for a tangent and curvature of the graph of the explicit equation $y=f(x)$ yields
$y=f(x_{0})+f'(x_{0})(x-x_{0})$ (tangent)
$\kappa (x_{0})={\frac {f''(x_{0})}{(1+f'(x_{0})^{2})^{3/2}}}$ (curvature).
Advantage and disadvantage of implicit curves
Disadvantage
The essential disadvantage of an implicit curve is the lack of an easy possibility to calculate single points which is necessary for visualization of an implicit curve (see next section).
Advantages
1. Implicit representations facilitate the computation of intersection points: If one curve is represented implicitly and the other parametrically the computation of intersection points needs only a simple (1-dimensional) Newton iteration, which is contrary to the cases implicit-implicit and parametric-parametric (see Intersection).
2. An implicit representation $F(x,y)=0$ gives the possibility of separating points not on the curve by the sign of $F(x,y)$. This may be helpful for example applying the false position method instead of a Newton iteration.
3. It is easy to generate curves which are almost geometrically similar to the given implicit curve $F(x,y)=0,$ by just adding a small number: $F(x,y)-c=0$ (see section #Smooth approximations).
Applications of implicit curves
Within mathematics implicit curves play a prominent role as algebraic curves. In addition, implicit curves are used for designing curves of desired geometrical shapes. Here are two examples.
Convex polygons
A smooth approximation of a convex polygon can be achieved in the following way: Let $g_{i}(x,y)=a_{i}x+b_{i}y+c_{i}=0,\ i=1,\dotsc ,n$ be the equations of the lines containing the edges of the polygon such that for an inner point of the polygon $g_{i}$ is positive. Then a subset of the implicit curve
$F(x,y)=g_{1}(x,y)\cdots g_{n}(x,y)-c=0$
with suitable small parameter $c$ is a smooth (differentiable) approximation of the polygon. For example, the curves
$F(x,y)=(x+1)(-x+1)y(-x-y+2)(x-y+2)-c=0$ for $c=0.03,\dotsc ,0.6$
contain smooth approximations of a polygon with 5 edges (see diagram).
Pairs of lines
In case of two lines
$F(x,y)=g_{1}(x,y)g_{2}(x,y)-c=0$
one gets
a pencil of parallel lines, if the given lines are parallel or
the pencil of hyperbolas, which have the given lines as asymptotes.
For example, the product of the coordinate axes variables yields the pencil of hyperbolas $xy-c=0,\ c\neq 0$, which have the coordinate axes as asymptotes.
Others
If one starts with simple implicit curves other than lines (circles, parabolas,...) one gets a wide range of interesting new curves. For example,
$F(x,y)=y(-x^{2}-y^{2}+1)-c=0$
(product of a circle and the x-axis) yields smooth approximations of one half of a circle (see picture), and
$F(x,y)=(-x^{2}-(y+1)^{2}+4)(-x^{2}-(y-1)^{2}+4)-c=0$
(product of two circles) yields smooth approximations of the intersection of two circles (see diagram).
Blending curves
In CAD one uses implicit curves for the generation of blending curves,[2][3] which are special curves establishing a smooth transition between two given curves. For example,
$F(x,y)=(1-\mu )f_{1}f_{2}-\mu (g_{1}g_{2})^{3}=0$
generates blending curves between the two circles
$f_{1}(x,y)=(x-x_{1})^{2}+y^{2}-r_{1}^{2}=0,$
$f_{2}(x,y)=(x-x_{2})^{2}+y^{2}-r_{2}^{2}=0.$
The method guarantees the continuity of the tangents and curvatures at the points of contact (see diagram). The two lines
$g_{1}(x,y)=x-x_{1}=0,\ g_{2}(x,y)=x-x_{2}=0$
determine the points of contact at the circles. Parameter $\mu $ is a design parameter. In the diagram, $\mu =0.05,\dotsc ,0.2$.
Equipotential curves of two point charges
Equipotential curves of two equal point charges at the points $P_{1}=(1,0),\;P_{2}=(-1,0)$ can be represented by the equation
$f(x,y)={\frac {1}{|PP_{1}|}}+{\frac {1}{|PP_{2}|}}-c$
$={\frac {1}{\sqrt {(x-1)^{2}+y^{2}}}}+{\frac {1}{\sqrt {(x+1)^{2}+y^{2}}}}-c=0.$
The curves are similar to Cassini ovals, but they are not such curves.
Visualization of an implicit curve
To visualize an implicit curve one usually determines a polygon on the curve and displays the polygon. For a parametric curve this is an easy task: One just computes the points of a sequence of parametric values. For an implicit curve one has to solve two subproblems:
1. determination of a first curve point to a given starting point in the vicinity of the curve,
2. determination of a curve point starting from a known curve point.
In both cases it is reasonable to assume $\operatorname {grad} F\neq (0,0)$. In practice this assumption is violated at single isolated points only.
Point algorithm
For the solution of both tasks mentioned above it is essential to have a computer program (which we will call ${\mathsf {CPoint}}$), which, when given a point $Q_{0}=(x_{0},y_{0})$ near an implicit curve, finds a point $P$ that is exactly on the curve:
(P1) for the start point is $j=0$
(P2) repeat
$(x_{j+1},y_{j+1})=(x_{j},y_{j})-{\frac {F(x_{j},y_{j})}{F_{x}(x_{j},y_{j})^{2}+F_{y}(x_{j},y_{j})^{2}}}\,\left(F_{x}(x_{j},y_{j}),F_{y}(x_{j},y_{j})\right)$
( Newton step for function $g(t)=F\left(x_{j}+tF_{x}(x_{j},y_{j}),y_{j}+tF_{y}(x_{j},y_{j})\right)\ .$)
(P3) until the distance between the points $(x_{j+1},y_{j+1}),\,(x_{j},y_{j})$ is small enough.
(P4) $P=(x_{j+1},y_{j+1})$ is the curve point near the start point $Q_{0}$.
Tracing algorithm
In order to generate a nearly equally spaced polygon on the implicit curve one chooses a step length $s$ and
(T1) chooses a suitable starting point in the vicinity of the curve
(T2) determines a first curve point $P_{1}$ using program ${\mathsf {CPoint}}$
(T3) determines the tangent (see above), chooses a starting point on the tangent using step length $s$ (see diagram) and determines a second curve point $P_{2}$ using program ${\mathsf {CPoint}}$ .
$\cdots $
Because the algorithm traces the implicit curve it is called a tracing algorithm. The algorithm traces only connected parts of the curve. If the implicit curve consists of several parts it has to be started several times with suitable starting points.
Raster algorithm
If the implicit curve consists of several or even unknown parts, it may be better to use a rasterisation algorithm. Instead of exactly following the curve, a raster algorithm covers the entire curve in so many points that they blend together and look like the curve.
(R1) Generate a net of points (raster) on the area of interest of the x-y-plane.
(R2) For every point $P$ in the raster, run the point algorithm ${\mathsf {CPoint}}$ starting from P, then mark its output.
If the net is dense enough, the result approximates the connected parts of the implicit curve. If for further applications polygons on the curves are needed one can trace parts of interest by the tracing algorithm.
Implicit space curves
Any space curve which is defined by two equations
${\begin{matrix}F(x,y,z)=0,\\G(x,y,z)=0\end{matrix}}$
is called an implicit space curve.
A curve point $(x_{0},y_{0},z_{0})$ is called regular if the cross product of the gradients $F$ and $G$ is not $(0,0,0)$ at this point:
$\mathbf {t} (x_{0},y_{0},z_{0})=\operatorname {grad} F(x_{0},y_{0},z_{0})\times \operatorname {grad} G(x_{0},y_{0},z_{0})\neq (0,0,0);$
otherwise it is called singular. Vector $\mathbf {t} (x_{0},y_{0},z_{0})$ is a tangent vector of the curve at point $(x_{0},y_{0},z_{0}).$
Examples:
$(1)\quad x+y+z-1=0\ ,\ x-y+z-2=0$
is a line.
$(2)\quad x^{2}+y^{2}+z^{2}-4=0\ ,\ x+y+z-1=0$
is a plane section of a sphere, hence a circle.
$(3)\quad x^{2}+y^{2}-1=0\ ,\ x+y+z-1=0$
is an ellipse (plane section of a cylinder).
$(4)\quad x^{2}+y^{2}+z^{2}-16=0\ ,\ (y-y_{0})^{2}+z^{2}-9=0$
is the intersection curve between a sphere and a cylinder.
For the computation of curve points and the visualization of an implicit space curve see Intersection.
See also
• Implicit surface
References
1. Goldman, R. (2005). "Curvature formulas for implicit curves and surfaces". Computer Aided Geometric Design. 22 (7): 632. CiteSeerX 10.1.1.413.3008. doi:10.1016/j.cagd.2005.06.005.
2. C. Hoffmann & J. Hopcroft: The potential method for blending surfaces and corners in G. Farin (Ed) Geometric-Modeling, SIAM, Philadelphia, pp. 347-365
3. E. Hartmann: Blending of implicit surfaces with functional splines, CAD,Butterworth-Heinemann, Volume 22 (8), 1990, p. 500-507
4. G. Taubin: Distance Approximations for Rastering Implicit Curves. ACM Transactions on Graphics, Vol. 13, No. 1, 1994.
• Gomes, A., Voiculescu, I., Jorge, J., Wyvill, B., Galbraith, C.: Implicit Curves and Surfaces: Mathematics, Data Structures and Algorithms, 2009, Springer-Verlag London, ISBN 978-1-84882-405-8
• C:L: Bajaj, C.M. Hoffmann, R.E. Lynch: Tracing surface intersections, Comp. Aided Geom. Design 5 (1988), 285-307.
• Geometry and Algorithms for COMPUTER AIDED DESIGN
External links
Wikimedia Commons has media related to Implicit curves.
• Famous Curves
| Wikipedia |
Implicit surface
In mathematics, an implicit surface is a surface in Euclidean space defined by an equation
$F(x,y,z)=0.$
An implicit surface is the set of zeros of a function of three variables. Implicit means that the equation is not solved for x or y or z.
The graph of a function is usually described by an equation $z=f(x,y)$ and is called an explicit representation. The third essential description of a surface is the parametric one: $(x(s,t),y(s,t),z(s,t))$, where the x-, y- and z-coordinates of surface points are represented by three functions $x(s,t)\,,y(s,t)\,,z(s,t)$ depending on common parameters $s,t$. Generally the change of representations is simple only when the explicit representation $z=f(x,y)$ is given: $z-f(x,y)=0$ (implicit), $(s,t,f(s,t))$ (parametric).
Examples:
1. The plane $x+2y-3z+1=0.$
2. The sphere $x^{2}+y^{2}+z^{2}-4=0.$
3. The torus $(x^{2}+y^{2}+z^{2}+R^{2}-a^{2})^{2}-4R^{2}(x^{2}+y^{2})=0.$
4. A surface of genus 2: $2y(y^{2}-3x^{2})(1-z^{2})+(x^{2}+y^{2})^{2}-(9z^{2}-1)(1-z^{2})=0$ (see diagram).
5. The surface of revolution $x^{2}+y^{2}-(\ln(z+3.2))^{2}-0.02=0$ (see diagram wineglass).
For a plane, a sphere, and a torus there exist simple parametric representations. This is not true for the fourth example.
The implicit function theorem describes conditions under which an equation $F(x,y,z)=0$ can be solved (at least implicitly) for x, y or z. But in general the solution may not be made explicit. This theorem is the key to the computation of essential geometric features of a surface: tangent planes, surface normals, curvatures (see below). But they have an essential drawback: their visualization is difficult.
If $F(x,y,z)$ is polynomial in x, y and z, the surface is called algebraic. Example 5 is non-algebraic.
Despite difficulty of visualization, implicit surfaces provide relatively simple techniques to generate theoretically (e.g. Steiner surface) and practically (see below) interesting surfaces.
Formulas
Throughout the following considerations the implicit surface is represented by an equation $F(x,y,z)=0$ where function $F$ meets the necessary conditions of differentiability. The partial derivatives of $F$ are $F_{x},F_{y},F_{z},F_{xx},\ldots $.
Tangent plane and normal vector
A surface point $(x_{0},y_{0},z_{0})$ is called regular if and only if the gradient of $F$ at $(x_{0},y_{0},z_{0})$ is not the zero vector $(0,0,0)$, meaning
$(F_{x}(x_{0},y_{0},z_{0}),F_{y}(x_{0},y_{0},z_{0}),F_{z}(x_{0},y_{0},z_{0}))\neq (0,0,0)$.
If the surface point $(x_{0},y_{0},z_{0})$ is not regular, it is called singular.
The equation of the tangent plane at a regular point $(x_{0},y_{0},z_{0})$ is
$F_{x}(x_{0},y_{0},z_{0})(x-x_{0})+F_{y}(x_{0},y_{0},z_{0})(y-y_{0})+F_{z}(x_{0},y_{0},z_{0})(z-z_{0})=0,$
and a normal vector is
$\mathbf {n} (x_{0},y_{0},z_{0})=(F_{x}(x_{0},y_{0},z_{0}),F_{y}(x_{0},y_{0},z_{0}),F_{z}(x_{0},y_{0},z_{0}))^{T}.$
Normal curvature
In order to keep the formula simple the arguments $(x_{0},y_{0},z_{0})$ are omitted:
$\kappa _{n}={\frac {\mathbf {v} ^{\top }H_{F}\mathbf {v} }{\|\operatorname {grad} F\|}}$
is the normal curvature of the surface at a regular point for the unit tangent direction $\mathbf {v} $. $H_{F}$ is the Hessian matrix of $F$ (matrix of the second derivatives).
The proof of this formula relies (as in the case of an implicit curve) on the implicit function theorem and the formula for the normal curvature of a parametric surface.
Applications of implicit surfaces
As in the case of implicit curves it is an easy task to generate implicit surfaces with desired shapes by applying algebraic operations (addition, multiplication) on simple primitives.
Equipotential surface of point charges
The electrical potential of a point charge $q_{i}$ at point $\mathbf {p} _{i}=(x_{i},y_{i},z_{i})$ generates at point $\mathbf {p} =(x,y,z)$ the potential (omitting physical constants)
$F_{i}(x,y,z)={\frac {q_{i}}{\|\mathbf {p} -\mathbf {p} _{i}\|}}.$
The equipotential surface for the potential value $c$ is the implicit surface $F_{i}(x,y,z)-c=0$ which is a sphere with center at point $\mathbf {p} _{i}$.
The potential of $4$ point charges is represented by
$F(x,y,z)={\frac {q_{1}}{\|\mathbf {p} -\mathbf {p} _{1}\|}}+{\frac {q_{2}}{\|\mathbf {p} -\mathbf {p} _{2}\|}}+{\frac {q_{3}}{\|\mathbf {p} -\mathbf {p} _{3}\|}}+{\frac {q_{4}}{\|\mathbf {p} -\mathbf {p} _{4}\|}}.$
For the picture the four charges equal 1 and are located at the points $(\pm 1,\pm 1,0)$. The displayed surface is the equipotential surface (implicit surface) $F(x,y,z)-2.8=0$.
Constant distance product surface
A Cassini oval can be defined as the point set for which the product of the distances to two given points is constant (in contrast, for an ellipse the sum is constant). In a similar way implicit surfaces can be defined by a constant distance product to several fixed points.
In the diagram metamorphoses the upper left surface is generated by this rule: With
${\begin{aligned}F(x,y,z)={}&{\Big (}{\sqrt {(x-1)^{2}+y^{2}+z^{2}}}\cdot {\sqrt {(x+1)^{2}+y^{2}+z^{2}}}\\&\qquad \cdot {\sqrt {x^{2}+(y-1)^{2}+z^{2}}}\cdot {\sqrt {x^{2}+(y+1)^{2}+z^{2}}}{\Big )}\end{aligned}}$
the constant distance product surface $F(x,y,z)-1.1=0$ is displayed.
Metamorphoses of implicit surfaces
A further simple method to generate new implicit surfaces is called metamorphosis of implicit surfaces:
For two implicit surfaces $F_{1}(x,y,z)=0,F_{2}(x,y,z)=0$ (in the diagram: a constant distance product surface and a torus) one defines new surfaces using the design parameter $\mu \in [0,1]$:
$F(x,y,z)=\mu F_{1}(x,y,z)+(1-\mu )F_{2}(x,y,z)=0$
In the diagram the design parameter is successively $\mu =0,\,0.33,\,0.66,\,1$ .
Smooth approximations of several implicit surfaces
$\Pi $-surfaces [1] can be used to approximate any given smooth and bounded object in $R^{3}$ whose surface is defined by a single polynomial as a product of subsidiary polynomials. In other words, we can design any smooth object with a single algebraic surface. Let us denote the defining polynomials as $f_{i}\in \mathbb {R} [x_{1},\ldots ,x_{n}](i=1,\ldots ,k)$. Then, the approximating object is defined by the polynomial
$F(x,y,z)=\prod _{i}f_{i}(x,y,z)-r$[1]
where $r\in \mathbb {R} $ stands for the blending parameter that controls the approximating error.
Analogously to the smooth approximation with implicit curves, the equation
$F(x,y,z)=F_{1}(x,y,z)\cdot F_{2}(x,y,z)\cdot F_{3}(x,y,z)-r=0$
represents for suitable parameters $c$ smooth approximations of three intersecting tori with equations
${\begin{aligned}F_{1}=(x^{2}+y^{2}+z^{2}+R^{2}-a^{2})^{2}-4R^{2}(x^{2}+y^{2})=0,\\[3pt]F_{2}=(x^{2}+y^{2}+z^{2}+R^{2}-a^{2})^{2}-4R^{2}(x^{2}+z^{2})=0,\\[3pt]F_{3}=(x^{2}+y^{2}+z^{2}+R^{2}-a^{2})^{2}-4R^{2}(y^{2}+z^{2})=0.\end{aligned}}$
(In the diagram the parameters are $R=1,\,a=0.2,\,r=0.01.$)
Visualization of implicit surfaces
There are various algorithms for rendering implicit surfaces,[2] including the marching cubes algorithm.[3] Essentially there are two ideas for visualizing an implicit surface: One generates a net of polygons which is visualized (see surface triangulation) and the second relies on ray tracing which determines intersection points of rays with the surface.[4] The intersection points can be approximated by sphere tracing, using a signed distance function to find the distance to the surface.[5]
See also
• Implicit curve
References
1. Adriano N. Raposo; Abel J.P. Gomes (2019). "Pi-surfaces: products of implicit surfaces towards constructive composition of 3D objects". WSCG 2019 27. International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision. arXiv:1906.06751.
2. Jules Bloomenthal; Chandrajit Bajaj; Brian Wyvill (15 August 1997). Introduction to Implicit Surfaces. Morgan Kaufmann. ISBN 978-1-55860-233-5.
3. Ian Stephenson (1 December 2004). Production Rendering: Design and Implementation. Springer Science & Business Media. ISBN 978-1-85233-821-3.
4. Eric Haines, Tomas Akenine-Moller: Ray Tracing Gems, Springer, 2019, ISBN 978-1-4842-4427-2
5. Hardy, Alexandre; Steeb, Willi-Hans (2008). Mathematical Tools in Computer Graphics with C# Implementations. World Scientific. ISBN 978-981-279-102-3.
Further reading
• Gomes, A., Voiculescu, I., Jorge, J., Wyvill, B., Galbraith, C.: Implicit Curves and Surfaces: Mathematics, Data Structures and Algorithms, 2009, Springer-Verlag London, ISBN 978-1-84882-405-8
• Thorpe: Elementary Topics in Differential Geometry, Springer-Verlag, New York, 1979, ISBN 0-387-90357-7
External links
• Sultanow: Implizite Flächen
• Hartmann: Geometry and Algorithms for COMPUTER AIDED DESIGN
• GEOMVIEW
• K3Dsurf: 3d surface generator
• SURF: Visualisierung algebraischer Flächen
Dimension
Dimensional spaces
• Vector space
• Euclidean space
• Affine space
• Projective space
• Free module
• Manifold
• Algebraic variety
• Spacetime
Other dimensions
• Krull
• Lebesgue covering
• Inductive
• Hausdorff
• Minkowski
• Fractal
• Degrees of freedom
Polytopes and shapes
• Hyperplane
• Hypersurface
• Hypercube
• Hyperrectangle
• Demihypercube
• Hypersphere
• Cross-polytope
• Simplex
• Hyperpyramid
Dimensions by number
• Zero
• One
• Two
• Three
• Four
• Five
• Six
• Seven
• Eight
• n-dimensions
See also
• Hyperspace
• Codimension
Category
| Wikipedia |
Vitale's random Brunn–Minkowski inequality
In mathematics, Vitale's random Brunn–Minkowski inequality is a theorem due to Richard Vitale that generalizes the classical Brunn–Minkowski inequality for compact subsets of n-dimensional Euclidean space Rn to random compact sets.
Statement of the inequality
Let X be a random compact set in Rn; that is, a Borel–measurable function from some probability space (Ω, Σ, Pr) to the space of non-empty, compact subsets of Rn equipped with the Hausdorff metric. A random vector V : Ω → Rn is called a selection of X if Pr(V ∈ X) = 1. If K is a non-empty, compact subset of Rn, let
$\|K\|=\max \left\{\left.\|v\|_{\mathbb {R} ^{n}}\right|v\in K\right\}$
and define the set-valued expectation E[X] of X to be
$\mathrm {E} [X]=\{\mathrm {E} [V]|V{\mbox{ is a selection of }}X{\mbox{ and }}\mathrm {E} \|V\|<+\infty \}.$
Note that E[X] is a subset of Rn. In this notation, Vitale's random Brunn–Minkowski inequality is that, for any random compact set X with $E[\|X\|]<+\infty $,
$\left(\mathrm {vol} _{n}\left(\mathrm {E} [X]\right)\right)^{1/n}\geq \mathrm {E} \left[\mathrm {vol} _{n}(X)^{1/n}\right],$
where "$vol_{n}$" denotes n-dimensional Lebesgue measure.
Relationship to the Brunn–Minkowski inequality
If X takes the values (non-empty, compact sets) K and L with probabilities 1 − λ and λ respectively, then Vitale's random Brunn–Minkowski inequality is simply the original Brunn–Minkowski inequality for compact sets.
References
• Gardner, Richard J. (2002). "The Brunn-Minkowski inequality" (PDF). Bull. Amer. Math. Soc. (N.S.). 39 (3): 355–405 (electronic). doi:10.1090/S0273-0979-02-00941-2.
• Vitale, Richard A. (1990). "The Brunn-Minkowski inequality for random sets". J. Multivariate Anal. 33 (2): 286–293. doi:10.1016/0047-259X(90)90052-J.
Lp spaces
Basic concepts
• Banach & Hilbert spaces
• Lp spaces
• Measure
• Lebesgue
• Measure space
• Measurable space/function
• Minkowski distance
• Sequence spaces
L1 spaces
• Integrable function
• Lebesgue integration
• Taxicab geometry
L2 spaces
• Bessel's
• Cauchy–Schwarz
• Euclidean distance
• Hilbert space
• Parseval's identity
• Polarization identity
• Pythagorean theorem
• Square-integrable function
$L^{\infty }$ spaces
• Bounded function
• Chebyshev distance
• Infimum and supremum
• Essential
• Uniform norm
Maps
• Almost everywhere
• Convergence almost everywhere
• Convergence in measure
• Function space
• Integral transform
• Locally integrable function
• Measurable function
• Symmetric decreasing rearrangement
Inequalities
• Babenko–Beckner
• Chebyshev's
• Clarkson's
• Hanner's
• Hausdorff–Young
• Hölder's
• Markov's
• Minkowski
• Young's convolution
Results
• Marcinkiewicz interpolation theorem
• Plancherel theorem
• Riemann–Lebesgue
• Riesz–Fischer theorem
• Riesz–Thorin theorem
For Lebesgue measure
• Isoperimetric inequality
• Brunn–Minkowski theorem
• Milman's reverse
• Minkowski–Steiner formula
• Prékopa–Leindler inequality
• Vitale's random Brunn–Minkowski inequality
Applications & related
• Bochner space
• Fourier analysis
• Lorentz space
• Probability theory
• Quasinorm
• Real analysis
• Sobolev space
• *-algebra
• C*-algebra
• Von Neumann
Measure theory
Basic concepts
• Absolute continuity of measures
• Lebesgue integration
• Lp spaces
• Measure
• Measure space
• Probability space
• Measurable space/function
Sets
• Almost everywhere
• Atom
• Baire set
• Borel set
• equivalence relation
• Borel space
• Carathéodory's criterion
• Cylindrical σ-algebra
• Cylinder set
• 𝜆-system
• Essential range
• infimum/supremum
• Locally measurable
• π-system
• σ-algebra
• Non-measurable set
• Vitali set
• Null set
• Support
• Transverse measure
• Universally measurable
Types of Measures
• Atomic
• Baire
• Banach
• Besov
• Borel
• Brown
• Complex
• Complete
• Content
• (Logarithmically) Convex
• Decomposable
• Discrete
• Equivalent
• Finite
• Inner
• (Quasi-) Invariant
• Locally finite
• Maximising
• Metric outer
• Outer
• Perfect
• Pre-measure
• (Sub-) Probability
• Projection-valued
• Radon
• Random
• Regular
• Borel regular
• Inner regular
• Outer regular
• Saturated
• Set function
• σ-finite
• s-finite
• Signed
• Singular
• Spectral
• Strictly positive
• Tight
• Vector
Particular measures
• Counting
• Dirac
• Euler
• Gaussian
• Haar
• Harmonic
• Hausdorff
• Intensity
• Lebesgue
• Infinite-dimensional
• Logarithmic
• Product
• Projections
• Pushforward
• Spherical measure
• Tangent
• Trivial
• Young
Maps
• Measurable function
• Bochner
• Strongly
• Weakly
• Convergence: almost everywhere
• of measures
• in measure
• of random variables
• in distribution
• in probability
• Cylinder set measure
• Random: compact set
• element
• measure
• process
• variable
• vector
• Projection-valued measure
Main results
• Carathéodory's extension theorem
• Convergence theorems
• Dominated
• Monotone
• Vitali
• Decomposition theorems
• Hahn
• Jordan
• Maharam's
• Egorov's
• Fatou's lemma
• Fubini's
• Fubini–Tonelli
• Hölder's inequality
• Minkowski inequality
• Radon–Nikodym
• Riesz–Markov–Kakutani representation theorem
Other results
• Disintegration theorem
• Lifting theory
• Lebesgue's density theorem
• Lebesgue differentiation theorem
• Sard's theorem
For Lebesgue measure
• Isoperimetric inequality
• Brunn–Minkowski theorem
• Milman's reverse
• Minkowski–Steiner formula
• Prékopa–Leindler inequality
• Vitale's random Brunn–Minkowski inequality
Applications & related
• Convex analysis
• Descriptive set theory
• Probability theory
• Real analysis
• Spectral theory
| Wikipedia |
Vitali convergence theorem
In real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a generalization of the better-known dominated convergence theorem of Henri Lebesgue. It is a characterization of the convergence in Lp in terms of convergence in measure and a condition related to uniform integrability.
Preliminary definitions
Let $(X,{\mathcal {A}},\mu )$ be a measure space, i.e. $\mu :{\mathcal {A}}\to [0,\infty ]$ :{\mathcal {A}}\to [0,\infty ]} is a set function such that $\mu (\emptyset )=0$ and $\mu $ is countably-additive. All functions considered in the sequel will be functions $f:X\to \mathbb {K} $, where $\mathbb {K} =\mathbb {R} $ or $\mathbb {C} $. We adopt the following definitions according to Bogachev's terminology.[1]
• A set of functions ${\mathcal {F}}\subset L^{1}(X,{\mathcal {A}},\mu )$ is called uniformly integrable if $\lim _{M\to +\infty }\sup _{f\in {\mathcal {F}}}\int _{\{|f|>M\}}|f|\,d\mu =0$, i.e $\forall \ \varepsilon >0,\ \exists \ M_{\varepsilon }>0:\sup _{f\in {\mathcal {F}}}\int _{\{|f|\geq M_{\varepsilon }\}}|f|\,d\mu <\varepsilon $.
• A set of functions ${\mathcal {F}}\subset L^{1}(X,{\mathcal {A}},\mu )$ is said to have uniformly absolutely continuous integrals if $\lim _{\mu (A)\to 0}\sup _{f\in {\mathcal {F}}}\int _{A}|f|\,d\mu =0$, i.e. $\forall \ \varepsilon >0,\ \exists \ \delta _{\varepsilon }>0,\ \forall \ A\in {\mathcal {A}}:\mu (A)<\delta _{\varepsilon }\Rightarrow \sup _{f\in {\mathcal {F}}}\int _{A}|f|\,d\mu <\varepsilon $. This definition is sometimes used as a definition of uniform integrability. However, it differs from the definition of uniform integrability given above.
When $\mu (X)<\infty $, a set of functions ${\mathcal {F}}\subset L^{1}(X,{\mathcal {A}},\mu )$ is uniformly integrable if and only if it is bounded in $L^{1}(X,{\mathcal {A}},\mu )$ and has uniformly absolutely continuous integrals. If, in addition, $\mu $ is atomless, then the uniform integrability is equivalent to the uniform absolute continuity of integrals.
Finite measure case
Let $(X,{\mathcal {A}},\mu )$ be a measure space with $\mu (X)<\infty $. Let $(f_{n})\subset L^{p}(X,{\mathcal {A}},\mu )$ and $f$ be an ${\mathcal {A}}$-measurable function. Then, the following are equivalent :
1. $f\in L^{p}(X,{\mathcal {A}},\mu )$ and $(f_{n})$ converges to $f$ in $L^{p}(X,{\mathcal {A}},\mu )$ ;
2. The sequence of functions $(f_{n})$ converges in $\mu $-measure to $f$ and $(|f_{n}|^{p})_{n\geq 1}$ is uniformly integrable ;
For a proof, see Bogachev's monograph "Measure Theory, Volume I".[1]
Infinite measure case
Let $(X,{\mathcal {A}},\mu )$ be a measure space and $1\leq p<\infty $. Let $(f_{n})_{n\geq 1}\subseteq L^{p}(X,{\mathcal {A}},\mu )$ and $f\in L^{p}(X,{\mathcal {A}},\mu )$. Then, $(f_{n})$ converges to $f$ in $L^{p}(X,{\mathcal {A}},\mu )$ if and only if the following holds :
1. The sequence of functions $(f_{n})$ converges in $\mu $-measure to $f$ ;
2. $(f_{n})$ has uniformly absolutely continuous integrals;
3. For every $\varepsilon >0$, there exists $X_{\varepsilon }\in {\mathcal {A}}$ such that $\mu (X_{\varepsilon })<\infty $ and $\sup _{n\geq 1}\int _{X\setminus X_{\varepsilon }}|f_{n}|^{p}\,d\mu <\varepsilon .$
When $\mu (X)<\infty $, the third condition becomes superfluous (one can simply take $X_{\varepsilon }=X$) and the first two conditions give the usual form of Lebesgue-Vitali's convergence theorem originally stated for measure spaces with finite measure. In this case, one can show that conditions 1 and 2 imply that the sequence $(|f_{n}|^{p})_{n\geq 1}$ is uniformly integrable.
Converse of the theorem
Let $(X,{\mathcal {A}},\mu )$ be measure space. Let $(f_{n})_{n\geq 1}\subseteq L^{1}(X,{\mathcal {A}},\mu )$ and assume that $\lim _{n\to \infty }\int _{A}f_{n}\,d\mu $ exists for every $A\in {\mathcal {A}}$. Then, the sequence $(f_{n})$ is bounded in $L^{1}(X,{\mathcal {A}},\mu )$ and has uniformly absolutely continuous integrals. In addition, there exists $f\in L^{1}(X,{\mathcal {A}},\mu )$ such that $\lim _{n\to \infty }\int _{A}f_{n}\,d\mu =\int _{A}f\,d\mu $ for every $A\in {\mathcal {A}}$.
When $\mu (X)<\infty $, this implies that $(f_{n})$ is uniformly integrable.
For a proof, see Bogachev's monograph "Measure Theory, Volume I".[1]
Citations
1. Bogachev, Vladimir I. (2007). Measure Theory Volume I. New York: Springer. pp. 267–271. ISBN 978-3-540-34513-8.
Measure theory
Basic concepts
• Absolute continuity of measures
• Lebesgue integration
• Lp spaces
• Measure
• Measure space
• Probability space
• Measurable space/function
Sets
• Almost everywhere
• Atom
• Baire set
• Borel set
• equivalence relation
• Borel space
• Carathéodory's criterion
• Cylindrical σ-algebra
• Cylinder set
• 𝜆-system
• Essential range
• infimum/supremum
• Locally measurable
• π-system
• σ-algebra
• Non-measurable set
• Vitali set
• Null set
• Support
• Transverse measure
• Universally measurable
Types of Measures
• Atomic
• Baire
• Banach
• Besov
• Borel
• Brown
• Complex
• Complete
• Content
• (Logarithmically) Convex
• Decomposable
• Discrete
• Equivalent
• Finite
• Inner
• (Quasi-) Invariant
• Locally finite
• Maximising
• Metric outer
• Outer
• Perfect
• Pre-measure
• (Sub-) Probability
• Projection-valued
• Radon
• Random
• Regular
• Borel regular
• Inner regular
• Outer regular
• Saturated
• Set function
• σ-finite
• s-finite
• Signed
• Singular
• Spectral
• Strictly positive
• Tight
• Vector
Particular measures
• Counting
• Dirac
• Euler
• Gaussian
• Haar
• Harmonic
• Hausdorff
• Intensity
• Lebesgue
• Infinite-dimensional
• Logarithmic
• Product
• Projections
• Pushforward
• Spherical measure
• Tangent
• Trivial
• Young
Maps
• Measurable function
• Bochner
• Strongly
• Weakly
• Convergence: almost everywhere
• of measures
• in measure
• of random variables
• in distribution
• in probability
• Cylinder set measure
• Random: compact set
• element
• measure
• process
• variable
• vector
• Projection-valued measure
Main results
• Carathéodory's extension theorem
• Convergence theorems
• Dominated
• Monotone
• Vitali
• Decomposition theorems
• Hahn
• Jordan
• Maharam's
• Egorov's
• Fatou's lemma
• Fubini's
• Fubini–Tonelli
• Hölder's inequality
• Minkowski inequality
• Radon–Nikodym
• Riesz–Markov–Kakutani representation theorem
Other results
• Disintegration theorem
• Lifting theory
• Lebesgue's density theorem
• Lebesgue differentiation theorem
• Sard's theorem
For Lebesgue measure
• Isoperimetric inequality
• Brunn–Minkowski theorem
• Milman's reverse
• Minkowski–Steiner formula
• Prékopa–Leindler inequality
• Vitale's random Brunn–Minkowski inequality
Applications & related
• Convex analysis
• Descriptive set theory
• Probability theory
• Real analysis
• Spectral theory
| Wikipedia |
Vitali–Carathéodory theorem
In mathematics, the Vitali–Carathéodory theorem is a result in real analysis that shows that, under the conditions stated below, integrable functions can be approximated in L1 from above and below by lower- and upper-semicontinuous functions, respectively. It is named after Giuseppe Vitali and Constantin Carathéodory.
Statement of the theorem
Let X be a locally compact Hausdorff space equipped with a Borel measure, µ, that is finite on every compact set, outer regular, and tight when restricted to any Borel set that is open or of finite mass. If f is an element of L1(µ) then, for every ε > 0, there are functions u and v on X such that u ≤ f ≤ v, u is upper-semicontinuous and bounded above, v is lower-semicontinuous and bounded below, and
$\int _{X}(v-u)\,\mathrm {d} \mu <\varepsilon .$
References
• Rudin, Walter (1986). Real and Complex Analysis (third ed.). McGraw-Hill. pp. 56–57. ISBN 978-0-07-054234-1.
| Wikipedia |
Vitali covering lemma
In mathematics, the Vitali covering lemma is a combinatorial and geometric result commonly used in measure theory of Euclidean spaces. This lemma is an intermediate step, of independent interest, in the proof of the Vitali covering theorem. The covering theorem is credited to the Italian mathematician Giuseppe Vitali.[1] The theorem states that it is possible to cover, up to a Lebesgue-negligible set, a given subset E of Rd by a disjoint family extracted from a Vitali covering of E.
Vitali covering lemma
There are two basic version of the lemma, a finite version and an infinite version. Both lemmas can be proved in the general setting of a metric space, typically these results are applied to the special case of the Euclidean space $\mathbb {R} ^{d}$. In both theorems we will use the following notation: if $ B=B(x,r)$ is a ball and $c\in \mathbb {R} $, we will write $cB$ for the ball $ B(x,cr)$.
Finite version
Theorem (Finite Covering Lemma). Let $B_{1},\dots ,B_{n}$ be any finite collection of balls contained in an arbitrary metric space. Then there exists a subcollection $B_{j_{1}},B_{j_{2}},\dots ,B_{j_{m}}$ of these balls which are disjoint and satisfy
$B_{1}\cup B_{2}\cup \dots \cup B_{n}\subseteq 3B_{j_{1}}\cup 3B_{j_{2}}\cup \dots \cup 3B_{j_{m}}.$
Proof: Without loss of generality, we assume that the collection of balls is not empty; that is, n > 0. Let $B_{j_{1}}$ be the ball of largest radius. Inductively, assume that $B_{j_{1}},\dots ,B_{j_{k}}$ have been chosen. If there is some ball in $B_{1},\dots ,B_{n}$ that is disjoint from $B_{j_{1}}\cup B_{j_{2}}\cup \dots \cup B_{j_{k}}$, let $B_{j_{k+1}}$ be such ball with maximal radius (breaking ties arbitrarily), otherwise, we set m := k and terminate the inductive definition.
Now set $ X:=\bigcup _{k=1}^{m}3\,B_{j_{k}}$. It remains to show that $B_{i}\subset X$ for every $i=1,2,\dots ,n$. This is clear if $i\in \{j_{1},\dots ,j_{m}\}$. Otherwise, there necessarily is some $k\in \{1,\dots ,m\}$ such that $B_{i}$ intersects $B_{j_{k}}$ and the radius of $B_{j_{k}}$ is at least as large as that of $B_{i}$. The triangle inequality then easily implies that $B_{i}\subset 3\,B_{j_{k}}\subset X$, as needed. This completes the proof of the finite version.
Infinite version
Theorem (Infinite Covering Lemma). Let $\mathbf {F} $ be an arbitrary collection of balls in a separable metric space such that
$R:=\sup \,\{\mathrm {rad} (B):B\in \mathbf {F} \}<\infty $
where $\mathrm {rad} (B)$ denotes the radius of the ball B. Then there exists a countable sub-collection $\mathbf {G} \subset \mathbf {F} $ such that the balls of $\mathbf {G} $ are pairwise disjoint, and satisfy
$\bigcup _{B\in \mathbf {F} }B\subseteq \bigcup _{C\in \mathbf {G} }5\,C.$
And moreover, each $B\in \mathbf {F} $ intersects some $C\in \mathbf {G} $ with $B\subset 5C$.
Proof: Consider the partition of F into subcollections Fn, n ≥ 0, defined by
$\mathbf {F} _{n}=\{B\in \mathbf {F} :2^{-n-1}R<{\text{rad}}(B)\leq 2^{-n}R\}.$
That is, $ \mathbf {F} _{n}$ consists of the balls B whose radius is in (2−n−1R, 2−nR]. A sequence Gn, with Gn ⊂ Fn, is defined inductively as follows. First, set H0 = F0 and let G0 be a maximal disjoint subcollection of H0 (such a subcollection exists by Zorn's lemma). Assuming that G0,...,Gn have been selected, let
$\mathbf {H} _{n+1}=\{B\in \mathbf {F} _{n+1}:\ B\cap C=\emptyset ,\ \ \forall C\in \mathbf {G} _{0}\cup \mathbf {G} _{1}\cup \dots \cup \mathbf {G} _{n}\},$
and let Gn+1 be a maximal disjoint subcollection of Hn+1. The subcollection
$\mathbf {G} :=\bigcup _{n=0}^{\infty }\mathbf {G} _{n}$ :=\bigcup _{n=0}^{\infty }\mathbf {G} _{n}}
of F satisfies the requirements of the theorem: G is a disjoint collection, and is thus countable since the given metric space is separable. Moreover, every ball B ∈ F intersects a ball C ∈ G such that B ⊂ 5 C.
Indeed, if we are given some $B\in \mathbf {F} $, there must be some n be such that B belongs to Fn. Either B does not belong to Hn, which implies n > 0 and means that B intersects a ball from the union of G0, ..., Gn−1, or B ∈ Hn and by maximality of Gn, B intersects a ball in Gn. In any case, B intersects a ball C that belongs to the union of G0, ..., Gn. Such a ball C must have a radius larger than 2−n−1R. Since the radius of B is less than or equal to 2−nR, we can conclude by the triangle inequality that B ⊂ 5 C, as claimed. From this $\bigcup _{B\in \mathbf {F} }B\subseteq \bigcup _{C\in \mathbf {G} }5\,C$ immediately follows, completing the proof.[2]
Remarks
• In the infinite version, the initial collection of balls can be countable or uncountable. In a separable metric space, any pairwise disjoint collection of balls must be countable. In a non-separable space, the same argument shows a pairwise disjoint subfamily exists, but that family need not be countable.
• The result may fail if the radii are not bounded: consider the family of all balls centered at 0 in Rd; any disjoint subfamily consists of only one ball B, and 5 B does not contain all the balls in this family.
• The constant 5 is not optimal. If the scale c−n, c > 1, is used instead of 2−n for defining Fn, the final value is 1 + 2c instead of 5. Any constant larger than 3 gives a correct statement of the lemma, but not 3.
• Using a finer analysis, when the original collection F is a Vitali covering of a subset E of Rd, one shows that the subcollection G, defined in the above proof, covers E up to a Lebesgue-negligible set.[3]
Applications and method of use
An application of the Vitali lemma is in proving the Hardy–Littlewood maximal inequality. As in this proof, the Vitali lemma is frequently used when we are, for instance, considering the d-dimensional Lebesgue measure, $\lambda _{d}$, of a set E ⊂ Rd, which we know is contained in the union of a certain collection of balls $\{B_{j}:j\in J\}$, each of which has a measure we can more easily compute, or has a special property one would like to exploit. Hence, if we compute the measure of this union, we will have an upper bound on the measure of E. However, it is difficult to compute the measure of the union of all these balls if they overlap. By the Vitali lemma, we may choose a subcollection $\left\{B_{j}:j\in J'\right\}$ which is disjoint and such that $ \bigcup _{j\in J'}5B_{j}\supset \bigcup _{j\in J}B_{j}\supset E$. Therefore,
$\lambda _{d}(E)\leq \lambda _{d}{\biggl (}\bigcup _{j\in J}B_{j}{\biggr )}\leq \lambda _{d}{\biggl (}\bigcup _{j\in J'}5B_{j}{\biggr )}\leq \sum _{j\in J'}\lambda _{d}(5B_{j}).$
Now, since increasing the radius of a d-dimensional ball by a factor of five increases its volume by a factor of 5d, we know that
$\sum _{j\in J'}\lambda _{d}(5B_{j})=5^{d}\sum _{j\in J'}\lambda _{d}(B_{j})$
and thus
$\lambda _{d}(E)\leq 5^{d}\sum _{j\in J'}\lambda _{d}(B_{j}).$
Vitali covering theorem
In the covering theorem, the aim is to cover, up to a "negligible set", a given set E ⊆ Rd by a disjoint subcollection extracted from a Vitali covering for E : a Vitali class or Vitali covering ${\mathcal {V}}$ for E is a collection of sets such that, for every x ∈ E and δ > 0, there is a set U in the collection ${\mathcal {V}}$ such that x ∈ U and the diameter of U is non-zero and less than δ.
In the classical setting of Vitali,[1] the negligible set is a Lebesgue negligible set, but measures other than the Lebesgue measure, and spaces other than Rd have also been considered, as is shown in the relevant section below.
The following observation is useful: if ${\mathcal {V}}$ is a Vitali covering for E and if E is contained in an open set Ω ⊆ Rd, then the subcollection of sets U in ${\mathcal {V}}$ that are contained in Ω is also a Vitali covering for E.
Vitali's covering theorem for the Lebesgue measure
The next covering theorem for the Lebesgue measure λd is due to Lebesgue (1910). A collection ${\mathcal {V}}$ of measurable subsets of Rd is a regular family (in the sense of Lebesgue) if there exists a constant C such that
$\operatorname {diam} (V)^{d}\leq C\,\lambda _{d}(V)$
for every set V in the collection ${\mathcal {V}}$.
The family of cubes is an example of regular family ${\mathcal {V}}$, as is the family ${\mathcal {V}}(m)$ of rectangles in R2 such that the ratio of sides stays between m−1 and m, for some fixed m ≥ 1. If an arbitrary norm is given on Rd, the family of balls for the metric associated to the norm is another example. To the contrary, the family of all rectangles in R2 is not regular.
Theorem — Let E ⊆ Rd be a measurable set with finite Lebesgue measure, and let ${\mathcal {V}}$ be a regular family of closed subsets of Rd that is a Vitali covering for E. Then there exists a finite or countably infinite disjoint subcollection $\{U_{j}\}\subseteq {\mathcal {V}}$ such that
$\lambda _{d}{\biggl (}E\setminus \bigcup _{j}U_{j}{\biggr )}=0.$
The original result of Vitali (1908) is a special case of this theorem, in which d = 1 and ${\mathcal {V}}$ is a collection of intervals that is a Vitali covering for a measurable subset E of the real line having finite measure.
The theorem above remains true without assuming that E has finite measure. This is obtained by applying the covering result in the finite measure case, for every integer n ≥ 0, to the portion of E contained in the open annulus Ωn of points x such that n < |x| < n+1.[4]
A somewhat related covering theorem is the Besicovitch covering theorem. To each point a of a subset A ⊆ Rd, a Euclidean ball B(a, ra) with center a and positive radius ra is assigned. Then, as in the Vitali theorem, a subcollection of these balls is selected in order to cover A in a specific way. The main differences with the Vitali covering theorem are that on one hand, the disjointness requirement of Vitali is relaxed to the fact that the number Nx of the selected balls containing an arbitrary point x ∈ Rd is bounded by a constant Bd depending only upon the dimension d; on the other hand, the selected balls do cover the set A of all the given centers.[5]
Vitali's covering theorem for the Hausdorff measure
One may have a similar objective when considering Hausdorff measure instead of Lebesgue measure. The following theorem applies in that case.[6]
Theorem — Let Hs denote s-dimensional Hausdorff measure, let E ⊆ Rd be an Hs-measurable set and ${\mathcal {V}}$ a Vitali class of closed sets for E. Then there exists a (finite or countably infinite) disjoint subcollection $\{U_{j}\}\subseteq {\mathcal {V}}$ such that either
$H^{s}\left(E\setminus \bigcup _{j}U_{j}\right)=0$
or
$\sum _{j}\operatorname {diam} (U_{j})^{s}=\infty .$
Furthermore, if E has finite s-dimensional Hausdorff measure, then for any ε > 0, we may choose this subcollection {Uj} such that
$H^{s}(E)\leq \sum _{j}\mathrm {diam} (U_{j})^{s}+\varepsilon .$
This theorem implies the result of Lebesgue given above. Indeed, when s = d, the Hausdorff measure Hs on Rd coincides with a multiple of the d-dimensional Lebesgue measure. If a disjoint collection $\{U_{j}\}$ is regular and contained in a measurable region B with finite Lebesgue measure, then
$\sum _{j}\operatorname {diam} (U_{j})^{d}\leq C\sum _{j}\lambda _{d}(U_{j})\leq C\,\lambda _{d}(B)<+\infty $
which excludes the second possibility in the first assertion of the previous theorem. It follows that E is covered, up to a Lebesgue-negligible set, by the selected disjoint subcollection.
From the covering lemma to the covering theorem
The covering lemma can be used as intermediate step in the proof of the following basic form of the Vitali covering theorem.
Theorem — For every subset E of Rd and every Vitali cover of E by a collection F of closed balls, there exists a disjoint subcollection G which covers E up to a Lebesgue-negligible set.
Proof: Without loss of generality, one can assume that all balls in F are nondegenerate and have radius less than or equal to 1. By the infinite form of the covering lemma, there exists a countable disjoint subcollection $\mathbf {G} $ of F such that every ball B ∈ F intersects a ball C ∈ G for which B ⊂ 5 C. Let r > 0 be given, and let Z denote the set of points z ∈ E that are not contained in any ball from G and belong to the open ball B(r) of radius r, centered at 0. It is enough to show that Z is Lebesgue-negligible, for every given r.
Let $\mathbf {G} _{r}=\{C_{n}\}_{n}$ denote the subcollection of those balls in G that meet B(r). Note that $\mathbf {G} _{r}$ may be finite or countably infinite. Let z ∈ Z be fixed. For each N, z does not belong to the closed set $K=\bigcup _{n\leq N}C_{n}$ by the definition of Z. But by the Vitali cover property, one can find a ball B ∈ F containing z, contained in B(r), and disjoint from K. By the property of G, the ball B intersects some ball $C_{i}\in \mathbf {G} $ and is contained in $5C_{i}$. But because K and B are disjoint, we must have i > N. So $z\in 5C_{i}$ for some i > N, and therefore
$Z\subset \bigcup _{n>N}5C_{n}.$
This gives for every N the inequality
$\lambda _{d}(Z)\leq \sum _{n>N}\lambda _{d}(5C_{n})=5^{d}\sum _{n>N}\lambda _{d}(C_{n}).$
But since the balls of $\mathbf {G} _{r}$ are contained in B(r+2), and these balls are disjoint we see
$\sum _{n}\lambda _{d}(C_{n})<\infty .$
Therefore, the term on the right side of the above inequality converges to 0 as N goes to infinity, which shows that Z is negligible as needed.[7]
Infinite-dimensional spaces
The Vitali covering theorem is not valid in infinite-dimensional settings. The first result in this direction was given by David Preiss in 1979:[8] there exists a Gaussian measure γ on an (infinite-dimensional) separable Hilbert space H so that the Vitali covering theorem fails for (H, Borel(H), γ). This result was strengthened in 2003 by Jaroslav Tišer: the Vitali covering theorem in fact fails for every infinite-dimensional Gaussian measure on any (infinite-dimensional) separable Hilbert space.[9]
See also
• Besicovitch covering theorem
Notes
1. (Vitali 1908).
2. The proof given is based on (Evans & Gariepy 1992, section 1.5.1)
3. See the "From the covering lemma to the covering theorem" section of this entry.
4. See (Evans & Gariepy 1992).
5. Vitali (1908) allowed a negligible error.
6. (Falconer 1986).
7. The proof given is based on (Natanson 1955), with some notation taken from (Evans & Gariepy 1992).
8. (Preiss 1979).
9. (Tišer 2003).
References
• Evans, Lawrence C.; Gariepy, Ronald F. (1992), Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, Boca Raton, FL: CRC Press, pp. viii+268, ISBN 0-8493-7157-0, MR 1158660, Zbl 0804.28001
• Falconer, Kenneth J. (1986), The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge: Cambridge University Press, pp. xiv+162, ISBN 0-521-25694-1, MR 0867284, Zbl 0587.28004
• "Vitali theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Lebesgue, Henri (1910), "Sur l'intégration des fonctions discontinues", Annales Scientifiques de l'École Normale Supérieure, 27: 361–450, doi:10.24033/asens.624, JFM 41.0457.01
• Natanson, I. P (1955), Theory of functions of a real variable, New York: Frederick Ungar Publishing Co., p. 277, MR 0067952, Zbl 0064.29102
• Preiss, David (1979), "Gaussian measures and covering theorems", Commentatione Mathematicae Universitatis Carolinae, 20 (1): 95–99, ISSN 0010-2628, MR 0526149, Zbl 0386.28015
• Stein, Elias M.; Shakarchi, Rami (2005), Real analysis. Measure theory, integration, and Hilbert spaces, Princeton Lectures in Analysis, III, Princeton, NJ: Princeton University Press, pp. xx+402, ISBN 0-691-11386-6, MR 2129625, Zbl 1081.28001
• Tišer, Jaroslav (2003), "Vitali covering theorem in Hilbert space", Transactions of the American Mathematical Society, 355 (8): 3277–3289 (electronic), doi:10.1090/S0002-9947-03-03296-3, MR 1974687, Zbl 1042.28014
• Vitali, Giuseppe (1908) [17 December 1907], "Sui gruppi di punti e sulle funzioni di variabili reali", Atti dell'Accademia delle Scienze di Torino (in Italian), 43: 75–92, JFM 39.0101.05 (Title translation) "On groups of points and functions of real variables" is the paper containing the first proof of Vitali covering theorem.
| Wikipedia |
Vitali set
In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by Giuseppe Vitali in 1905.[1] The Vitali theorem is the existence theorem that there are such sets. There are uncountably many Vitali sets, and their existence depends on the axiom of choice. In 1970, Robert Solovay constructed a model of Zermelo–Fraenkel set theory without the axiom of choice where all sets of real numbers are Lebesgue measurable, assuming the existence of an inaccessible cardinal (see Solovay model).[2]
Measurable sets
Certain sets have a definite 'length' or 'mass'. For instance, the interval [0, 1] is deemed to have length 1; more generally, an interval [a, b], a ≤ b, is deemed to have length b − a. If we think of such intervals as metal rods with uniform density, they likewise have well-defined masses. The set [0, 1] ∪ [2, 3] is composed of two intervals of length one, so we take its total length to be 2. In terms of mass, we have two rods of mass 1, so the total mass is 2.
There is a natural question here: if E is an arbitrary subset of the real line, does it have a 'mass' or 'total length'? As an example, we might ask what is the mass of the set of rational numbers between 0 and 1, given that the mass of the interval [0, 1] is 1. The rationals are dense in the reals, so any value between and including 0 and 1 may appear reasonable.
However the closest generalization to mass is sigma additivity, which gives rise to the Lebesgue measure. It assigns a measure of b − a to the interval [a, b], but will assign a measure of 0 to the set of rational numbers because it is countable. Any set which has a well-defined Lebesgue measure is said to be "measurable", but the construction of the Lebesgue measure (for instance using Carathéodory's extension theorem) does not make it obvious whether non-measurable sets exist. The answer to that question involves the axiom of choice.
Construction and proof
A Vitali set is a subset $V$ of the interval $[0,1]$ of real numbers such that, for each real number $r$, there is exactly one number $v\in V$ such that $v-r$ is a rational number. Vitali sets exist because the rational numbers $\mathbb {Q} $ form a normal subgroup of the real numbers $\mathbb {R} $ under addition, and this allows the construction of the additive quotient group $\mathbb {R} /\mathbb {Q} $ of these two groups which is the group formed by the cosets $r+\mathbb {Q} $ of the rational numbers as a subgroup of the real numbers under addition. This group $\mathbb {R} /\mathbb {Q} $ consists of disjoint "shifted copies" of $\mathbb {Q} $ in the sense that each element of this quotient group is a set of the form $r+\mathbb {Q} $ for some $r$ in $\mathbb {R} $. The uncountably many elements of $\mathbb {R} /\mathbb {Q} $ partition $\mathbb {R} $ into disjoint sets, and each element is dense in $\mathbb {R} $. Each element of $\mathbb {R} /\mathbb {Q} $ intersects $[0,1]$, and the axiom of choice guarantees the existence of a subset of $[0,1]$ containing exactly one representative out of each element of $\mathbb {R} /\mathbb {Q} $. A set formed this way is called a Vitali set.
Every Vitali set $V$ is uncountable, and $v-u$ is irrational for any $u,v\in V,u\neq v$.
Non-measurability
A Vitali set is non-measurable. To show this, we assume that $V$ is measurable and we derive a contradiction. Let $q_{1},q_{2},\dots $ be an enumeration of the rational numbers in $[-1,1]$ (recall that the rational numbers are countable). From the construction of $V$, note that the translated sets $V_{k}=V+q_{k}=\{v+q_{k}:v\in V\}$, $k=1,2,\dots $ are pairwise disjoint, and further note that
$[0,1]\subseteq \bigcup _{k}V_{k}\subseteq [-1,2].$
To see the first inclusion, consider any real number $r$ in $[0,1]$ and let $v$ be the representative in $V$ for the equivalence class $[r]$; then $r-v=q_{i}$ for some rational number $q_{i}$ in $[-1,1]$ which implies that $r$ is in $V_{i}$.
Apply the Lebesgue measure to these inclusions using sigma additivity:
$1\leq \sum _{k=1}^{\infty }\lambda (V_{k})\leq 3.$
Because the Lebesgue measure is translation invariant, $\lambda (V_{k})=\lambda (V)$ and therefore
$1\leq \sum _{k=1}^{\infty }\lambda (V)\leq 3.$
But this is impossible. Summing infinitely many copies of the constant $\lambda (V)$ yields either zero or infinity, according to whether the constant is zero or positive. In neither case is the sum in $[1,3]$. So $V$ cannot have been measurable after all, i.e., the Lebesgue measure $\lambda $ must not define any value for $\lambda (V)$.
See also
• Banach–Tarski paradox – Geometric theorem
• Carathéodory's criterion – necessary and sufficient condition for a measurable setPages displaying wikidata descriptions as a fallback
• Non-measurable set – Set which cannot be assigned a meaningful "volume"
• Outer measure – Mathematical function
References
1. Vitali, Giuseppe (1905). "Sul problema della misura dei gruppi di punti di una retta". Bologna, Tip. Gamberini e Parmeggiani.
2. Solovay, Robert M. (1970), "A model of set-theory in which every set of reals is Lebesgue measurable", Annals of Mathematics, Second Series, 92 (1): 1–56, doi:10.2307/1970696, ISSN 0003-486X, JSTOR 1970696, MR 0265151
Bibliography
• Herrlich, Horst (2006). Axiom of Choice. Springer. p. 120. ISBN 9783540309895.
• Vitali, Giuseppe (1905). "Sul problema della misura dei gruppi di punti di una retta". Bologna, Tip. Gamberini e Parmeggiani.
Measure theory
Basic concepts
• Absolute continuity of measures
• Lebesgue integration
• Lp spaces
• Measure
• Measure space
• Probability space
• Measurable space/function
Sets
• Almost everywhere
• Atom
• Baire set
• Borel set
• equivalence relation
• Borel space
• Carathéodory's criterion
• Cylindrical σ-algebra
• Cylinder set
• 𝜆-system
• Essential range
• infimum/supremum
• Locally measurable
• π-system
• σ-algebra
• Non-measurable set
• Vitali set
• Null set
• Support
• Transverse measure
• Universally measurable
Types of Measures
• Atomic
• Baire
• Banach
• Besov
• Borel
• Brown
• Complex
• Complete
• Content
• (Logarithmically) Convex
• Decomposable
• Discrete
• Equivalent
• Finite
• Inner
• (Quasi-) Invariant
• Locally finite
• Maximising
• Metric outer
• Outer
• Perfect
• Pre-measure
• (Sub-) Probability
• Projection-valued
• Radon
• Random
• Regular
• Borel regular
• Inner regular
• Outer regular
• Saturated
• Set function
• σ-finite
• s-finite
• Signed
• Singular
• Spectral
• Strictly positive
• Tight
• Vector
Particular measures
• Counting
• Dirac
• Euler
• Gaussian
• Haar
• Harmonic
• Hausdorff
• Intensity
• Lebesgue
• Infinite-dimensional
• Logarithmic
• Product
• Projections
• Pushforward
• Spherical measure
• Tangent
• Trivial
• Young
Maps
• Measurable function
• Bochner
• Strongly
• Weakly
• Convergence: almost everywhere
• of measures
• in measure
• of random variables
• in distribution
• in probability
• Cylinder set measure
• Random: compact set
• element
• measure
• process
• variable
• vector
• Projection-valued measure
Main results
• Carathéodory's extension theorem
• Convergence theorems
• Dominated
• Monotone
• Vitali
• Decomposition theorems
• Hahn
• Jordan
• Maharam's
• Egorov's
• Fatou's lemma
• Fubini's
• Fubini–Tonelli
• Hölder's inequality
• Minkowski inequality
• Radon–Nikodym
• Riesz–Markov–Kakutani representation theorem
Other results
• Disintegration theorem
• Lifting theory
• Lebesgue's density theorem
• Lebesgue differentiation theorem
• Sard's theorem
For Lebesgue measure
• Isoperimetric inequality
• Brunn–Minkowski theorem
• Milman's reverse
• Minkowski–Steiner formula
• Prékopa–Leindler inequality
• Vitale's random Brunn–Minkowski inequality
Applications & related
• Convex analysis
• Descriptive set theory
• Probability theory
• Real analysis
• Spectral theory
Real numbers
• 0.999...
• Absolute difference
• Cantor set
• Cantor–Dedekind axiom
• Completeness
• Construction
• Decidability of first-order theories
• Extended real number line
• Gregory number
• Irrational number
• Normal number
• Rational number
• Rational zeta series
• Real coordinate space
• Real line
• Tarski axiomatization
• Vitali set
| Wikipedia |
Vitali–Hahn–Saks theorem
In mathematics, the Vitali–Hahn–Saks theorem, introduced by Vitali (1907), Hahn (1922), and Saks (1933), proves that under some conditions a sequence of measures converging point-wise does so uniformly and the limit is also a measure.
Statement of the theorem
If $(S,{\mathcal {B}},m)$ is a measure space with $m(S)<\infty ,$ and a sequence $\lambda _{n}$ of complex measures. Assuming that each $\lambda _{n}$ is absolutely continuous with respect to $m,$ and that a for all $B\in {\mathcal {B}}$ the finite limits exist $\lim _{n\to \infty }\lambda _{n}(B)=\lambda (B).$ Then the absolute continuity of the $\lambda _{n}$ with respect to $m$ is uniform in $n,$ that is, $\lim _{B}m(B)=0$ implies that $\lim _{B}\lambda _{n}(B)=0$ uniformly in $n.$ Also $\lambda $ is countably additive on ${\mathcal {B}}.$
Preliminaries
Given a measure space $(S,{\mathcal {B}},m),$ a distance can be constructed on ${\mathcal {B}}_{0},$ the set of measurable sets $B\in {\mathcal {B}}$ with $m(B)<\infty .$ This is done by defining
$d(B_{1},B_{2})=m(B_{1}\Delta B_{2}),$ where $B_{1}\Delta B_{2}=(B_{1}\setminus B_{2})\cup (B_{2}\setminus B_{1})$ is the symmetric difference of the sets $B_{1},B_{2}\in {\mathcal {B}}_{0}.$
This gives rise to a metric space ${\tilde {{\mathcal {B}}_{0}}}$ by identifying two sets $B_{1},B_{2}\in {\mathcal {B}}_{0}$ when $m(B_{1}\Delta B_{2})=0.$ Thus a point ${\overline {B}}\in {\tilde {{\mathcal {B}}_{0}}}$ with representative $B\in {\mathcal {B}}_{0}$ is the set of all $B_{1}\in {\mathcal {B}}_{0}$ such that $m(B\Delta B_{1})=0.$
Proposition: ${\tilde {{\mathcal {B}}_{0}}}$ with the metric defined above is a complete metric space.
Proof: Let
$\chi _{B}(x)={\begin{cases}1,&x\in B\\0,&x\notin B\end{cases}}$
Then
$d(B_{1},B_{2})=\int _{S}|\chi _{B_{1}}(s)-\chi _{B_{2}}(x)|dm$
This means that the metric space ${\tilde {{\mathcal {B}}_{0}}}$ can be identified with a subset of the Banach space $L^{1}(S,{\mathcal {B}},m)$.
Let $B_{n}\in {\mathcal {B}}_{0}$, with
$\lim _{n,k\to \infty }d(B_{n},B_{k})=\lim _{n,k\to \infty }\int _{S}|\chi _{B_{n}}(x)-\chi _{B_{k}}(x)|dm=0$
Then we can choose a sub-sequence $\chi _{B_{n'}}$ such that $\lim _{n'\to \infty }\chi _{B_{n'}}(x)=\chi (x)$ exists almost everywhere and $\lim _{n'\to \infty }\int _{S}|\chi (x)-\chi _{B_{n'}(x)}|dm=0$. It follows that $\chi =\chi _{B_{\infty }}$ for some $B_{\infty }\in {\mathcal {B}}_{0}$ (furthermore $\chi (x)=1$ if and only if $\chi _{B_{n'}}(x)=1$ for $n'$ large enough, then we have that $B_{\infty }=\liminf _{n'\to \infty }B_{n'}={\bigcup _{n'=1}^{\infty }}\left({\bigcap _{m=n'}^{\infty }}B_{m}\right)$ the limit inferior of the sequence) and hence $\lim _{n\to \infty }d(B_{\infty },B_{n})=0.$ Therefore, ${\tilde {{\mathcal {B}}_{0}}}$ is complete.
Proof of Vitali-Hahn-Saks theorem
Each $\lambda _{n}$ defines a function ${\overline {\lambda }}_{n}({\overline {B}})$ on ${\tilde {\mathcal {B}}}$ by taking ${\overline {\lambda }}_{n}({\overline {B}})=\lambda _{n}(B)$. This function is well defined, this is it is independent on the representative $B$ of the class ${\overline {B}}$ due to the absolute continuity of $\lambda _{n}$ with respect to $m$. Moreover ${\overline {\lambda }}_{n}$ is continuous.
For every $\epsilon >0$ the set
$F_{k,\epsilon }=\{{\overline {B}}\in {\tilde {\mathcal {B}}}:\ \sup _{n\geq 1}|{\overline {\lambda }}_{k}({\overline {B}})-{\overline {\lambda }}_{k+n}({\overline {B}})|\leq \epsilon \}$
is closed in ${\tilde {\mathcal {B}}}$, and by the hypothesis $\lim _{n\to \infty }\lambda _{n}(B)=\lambda (B)$ we have that
${\tilde {\mathcal {B}}}=\bigcup _{k=1}^{\infty }F_{k,\epsilon }$
By Baire category theorem at least one $F_{k_{0},\epsilon }$ must contain a non-empty open set of ${\tilde {\mathcal {B}}}$. This means that there is ${\overline {B_{0}}}\in {\tilde {\mathcal {B}}}$ and a $\delta >0$ such that
$d(B,B_{0})<\delta $
implies $\sup _{n\geq 1}|{\overline {\lambda }}_{k_{0}}({\overline {B}})-{\overline {\lambda }}_{k_{0}+n}({\overline {B}})|\leq \epsilon $
On the other hand, any $B\in {\mathcal {B}}$ with $m(B)\leq \delta $ can be represented as $B=B_{1}\setminus B_{2}$ with $d(B_{1},B_{0})\leq \delta $ and $d(B_{2},B_{0})\leq \delta $. This can be done, for example by taking $B_{1}=B\cup B_{0}$ and $B_{2}=B_{0}\setminus (B\cap B_{0})$. Thus, if $m(B)\leq \delta $ and $k\geq k_{0}$ then
${\begin{aligned}|\lambda _{k}(B)|&\leq |\lambda _{k_{0}}(B)|+|\lambda _{k_{0}}(B)-\lambda _{k}(B)|\\&\leq |\lambda _{k_{0}}(B)|+|\lambda _{k_{0}}(B_{1})-\lambda _{k}(B_{1})|+|\lambda _{k_{0}}(B_{2})-\lambda _{k}(B_{2})|\\&\leq |\lambda _{k_{0}}(B)|+2\epsilon \end{aligned}}$
Therefore, by the absolute continuity of $\lambda _{k_{0}}$ with respect to $m$, and since $\epsilon $ is arbitrary, we get that $m(B)\to 0$ implies $\lambda _{n}(B)\to 0$ uniformly in $n.$ In particular, $m(B)\to 0$ implies $\lambda (B)\to 0.$
By the additivity of the limit it follows that $\lambda $ is finitely-additive. Then, since $\lim _{m(B)\to 0}\lambda (B)=0$ it follows that $\lambda $ is actually countably additive.
References
• Hahn, H. (1922), "Über Folgen linearer Operationen", Monatsh. Math. (in German), 32: 3–88, doi:10.1007/bf01696876
• Saks, Stanislaw (1933), "Addition to the Note on Some Functionals", Transactions of the American Mathematical Society, 35 (4): 965–970, doi:10.2307/1989603, JSTOR 1989603
• Vitali, G. (1907), "Sull' integrazione per serie", Rendiconti del Circolo Matematico di Palermo (in Italian), 23: 137–155, doi:10.1007/BF03013514
• Yosida, K. (1971), Functional Analysis, Springer, pp. 70–71, ISBN 0-387-05506-1
Measure theory
Basic concepts
• Absolute continuity of measures
• Lebesgue integration
• Lp spaces
• Measure
• Measure space
• Probability space
• Measurable space/function
Sets
• Almost everywhere
• Atom
• Baire set
• Borel set
• equivalence relation
• Borel space
• Carathéodory's criterion
• Cylindrical σ-algebra
• Cylinder set
• 𝜆-system
• Essential range
• infimum/supremum
• Locally measurable
• π-system
• σ-algebra
• Non-measurable set
• Vitali set
• Null set
• Support
• Transverse measure
• Universally measurable
Types of Measures
• Atomic
• Baire
• Banach
• Besov
• Borel
• Brown
• Complex
• Complete
• Content
• (Logarithmically) Convex
• Decomposable
• Discrete
• Equivalent
• Finite
• Inner
• (Quasi-) Invariant
• Locally finite
• Maximising
• Metric outer
• Outer
• Perfect
• Pre-measure
• (Sub-) Probability
• Projection-valued
• Radon
• Random
• Regular
• Borel regular
• Inner regular
• Outer regular
• Saturated
• Set function
• σ-finite
• s-finite
• Signed
• Singular
• Spectral
• Strictly positive
• Tight
• Vector
Particular measures
• Counting
• Dirac
• Euler
• Gaussian
• Haar
• Harmonic
• Hausdorff
• Intensity
• Lebesgue
• Infinite-dimensional
• Logarithmic
• Product
• Projections
• Pushforward
• Spherical measure
• Tangent
• Trivial
• Young
Maps
• Measurable function
• Bochner
• Strongly
• Weakly
• Convergence: almost everywhere
• of measures
• in measure
• of random variables
• in distribution
• in probability
• Cylinder set measure
• Random: compact set
• element
• measure
• process
• variable
• vector
• Projection-valued measure
Main results
• Carathéodory's extension theorem
• Convergence theorems
• Dominated
• Monotone
• Vitali
• Decomposition theorems
• Hahn
• Jordan
• Maharam's
• Egorov's
• Fatou's lemma
• Fubini's
• Fubini–Tonelli
• Hölder's inequality
• Minkowski inequality
• Radon–Nikodym
• Riesz–Markov–Kakutani representation theorem
Other results
• Disintegration theorem
• Lifting theory
• Lebesgue's density theorem
• Lebesgue differentiation theorem
• Sard's theorem
For Lebesgue measure
• Isoperimetric inequality
• Brunn–Minkowski theorem
• Milman's reverse
• Minkowski–Steiner formula
• Prékopa–Leindler inequality
• Vitale's random Brunn–Minkowski inequality
Applications & related
• Convex analysis
• Descriptive set theory
• Probability theory
• Real analysis
• Spectral theory
| Wikipedia |
Vitaly Khonik
Khonik Vitaly Alexandrovich (Russian: Хоник Виталий Александрович; born 17 December 1955) is a Russian physicist, doctor of physics and mathematics, professor, head of a laboratory researching the physics of non-crystalline materials, and head of the Department of General Physics at Voronezh State Pedagogical University (VSPU). He was born in Kemerovo, USSR.[1]
Vitaly Khonik
Born17 December 1955 (1955-12-17) (age 67)
CitizenshipRussian Federation
EducationDoctor of Science (physics and mathematics)
Alma materVoronezh State Technical University (VSTU)
AwardsHonored Worker in Higher Professional Education
Scientific career
InstitutionsVoronezh State Pedagogical University
Websitehosting.vspu.ac.ru/~khonik
His laboratory collaborates with the Institute of Solid State Physics of the Russian Academy of Sciences, the Institute of Physics of the Slovak Academy of Sciences, the Institut für Materialphysik in Germany and the School of Mechanics and Civil Architecture of Northwestern Polytechical University in China.
Education, academic degrees and titles
• 1994 - Professor
• 1992 - Doctor of Science (physics & mathematics), focusing on solid state physics
• 1991 - Senior researcher in solid state physics
• 1983 - Candidate for a doctoral degree in solid state physics
• 1978 - Graduated from Voronezh Polytechnic Institute (VPI), majoring in solid state physics
Employment history
• 2010 to present - Head of the Department of General Physics at VSPU
• 1992 to 2010 - Professor at VSPU
• 1992 - Associate professor at VSPU
• 1991-1992 - Associate professor at VPI
• 1985-1991 - Senior researcher at VPI
• 1984-1985 - Junior researcher at VPI
• 1981-1983 - Doctoral student at VPI
• 1978-1981 - Engineer and physicist at VPI
Academic awards
• Awarded the title "Soros Professor" in 1997, 1998 and 1999.
• Honored Worker in Higher Professional Education (2011).
International experience
• July 2019 - Visiting professor at Northwestern Polytechical University, Xi'an, China
• July 2018 - Visiting professor at Northwestern Polytechical University, Xi'an, China
• October 2016 - Visiting professor at the Institute of Physics, Chinese Academy of Sciences, Beijing, China
• August 2012 - Visiting professor at the department of physics, University of Illinois at Urbana-Champaign, USA
• May 2009 - Guest professor at the school of materials science, Harbin Institute of Technology, China
• April 2007 – Guest professor at Roskilde University, Denmark
• January 2007 to February 2007 – Visiting scholar at the physics department, University of Illinois at Urbana-Champaign, USA
• January 2006 to March 2006 – Scholar of the Japanese Society for the Promotion of Science (JSPS) at the graduate school of natural science and technology of Kanazawa University, Japan
• January 2005 to February 2005 – Visiting scholar at the physics department, University of Illinois at Urbana-Champaign, USA
• April 2003 to August 2003 – Visiting scholar at the physics department, University of Illinois at Urbana-Champaign, USA
• October 2002 to December 2002 – Scholar of the German Service for Academic Exchanges (DAAD), Technical University Carolo-Wilhelmina, Braunschweig, Germany
• May 1999 to April 2000 – Associate professor of the mechanical system engineering department, Kanazawa University, Kanazawa, Japan
• Visiting professor at the Institute of Experimental Physics, Slovak Academy of Sciences, Kosice, Slovakia (two to four week visits in 1996, 1998 and 2001)
International conferences and workshops
• Internal Friction and Ultrasonic Attenuation (ICIFUAS, Italy 1993, France 1996, Spain 2002)
• Mechanical Spectroscopy (Poland 2000)
• Structure of Non-Crystalline Solids (Czech Republic 1996)
• 18th International Congress on Glass (USA 1998)
• Physics of Amorphous Solids: Mechanical Properties and Plasticity (France, Les Houches, March 2010 )
• ACAM Workshop on Multiscale Modelling of Amorphous Materials: from Structure to Mechnical Properties (Dublin, Ireland, July 2011)
• 8th International Discussion Meeting on Relaxations in Complex Systems (Wisla, Poland, July 2017).
Major scientific projects
• Ministry of Education and Science of the Russian Federation, No 3.114.2014/К, "Nature of relaxation phenomena in non-crystalline metallic materials - new theoretical concepts and experiments", 2014–2016.
• Ministry of Education and Science of the Russian Federation, No 3.1310.2017/4.6, "Shear elasticity relaxation as a fundamental basis for the description and prediction of the physical properties of amorphous alloys", 2017–2019.
• Russian Science Foundation, No 20-62-46003, "Amorphous alloys: a new approach to the understanding of the defect structure and its influence on physical properties", 2020 – present.
References
1. "Professor V.A. Khonik". hosting.vspu.ac.ru. Retrieved 2020-11-22.
External links
• "List of papers". hosting.vspu.ac.ru. Retrieved 2020-11-22.
| Wikipedia |
Vittorio Francesco Stancari
Vittorio Francesco Stancari (1678 – 1709) was a professor of mathematics at the University of Bologna who undertook research into the measurement of sounds, and into optics and hydrostatics.
Vittorio Francesco Stancari
Born1678
Bologna, Papal States
Died1709
Bologna, Papal States
NationalityBolognese
OccupationMathematician
Known forMeasurement of the pitch of sounds
Career
Vittorio Francesco Stancari was born in Bologna in 1678. In 1698 he became a professor of mathematics at the University of Bologna.[1] Stancari was one of a group of young men at the University who became interested in the techniques of Cartesian geometry and differential calculus, and who engaged in experiments and astronomical observation. Others were Eustachio Manfredi, his brother Gabriele Manfredi and Giuseppe Sentenziola Verzaglia. Of these, Gabriele Manfredi developed the most advanced understanding of mathematics.[2] Stancari was awarded the chair of infinitesimal calculus in Bologna in 1708.[3] He died in Bologna in 1709, aged about 31.[1]
Work
Stancari's dissertations and manuscripts show that he applied Leibnizian calculus to problems of physics, hydrodynamics, meteorology and mechanics.[3] He was also aware of Sir Isaac Newton's Principia Mathematica, and discussed Newton in lectures before the Accademia degli Inquieti in Bologna.[4]
Stancari developed a method of measuring the pitch of sound in 1706, using foil that was excited into vibration by rotating toothed wheels.[1] Working in the observatory founded by Count Marsigli,[5] Stancari and Eustachio Manfredi discovered the comet C/1707 W1 in the evening of 25 November 1707. They described it as visible to the naked eye, white, irregular and with a short, faint tail. It had the same apparent size as Jupiter.[6]
Stancari experimented with Guillaume Amontons' air thermometer, where air in the bulb pushes up a column of mercury as it expands due to rising temperature. He discovered that the humidity of the air in the bulb had a significant effect on the readings.[7]
Bibliography
• Stancari, Vittorio Francesco; Manfredi, Eustachio (1713). Victorij Francisci Stancarij philosophiae doctoris Bononiensis et patrio archigymnasio analyticae lectoris Schedae mathematicae: post ejus obitum collectae ejusdem observationes astronomicae. Typis Jo: Petri Barbiroli sub signo Rose propè Archigymnasium. Retrieved 21 January 2013.
• Stancari, Vittorio Francesco (1733). Lettera del signor Vittorio Francesco Stancari ... in cui parla della figura del seme del GebelIndi guardata e disaminata col microscopio, dell'ovaja delle anguille, del camaleonte e suoi occhi, come le uova empiatrate poco o nulla traspirino de'fonti o pozzi osservati sulle cime de'monti. Retrieved 21 January 2013.
References
Citations
1. Stancari, Vittorio Francesco – Treccani.
2. Feingold & Brotons 2006, p. 133.
3. Olschki 1996, p. 307.
4. Feingold 2004, p. 81.
5. Dizionario biografico universale, Volume 5, by Felice Scifoni, Publisher Davide Passagli, Florence (1849); page 172.
6. Kronk 1999, p. 389.
7. Camuffo & Jones 2002, p. 299.
Sources
• Camuffo, Dario; Jones, Phil D. (31 May 2002). Improved Understanding of Past Climatic Variability from Early Daily European Instrumental Sources. Springer. ISBN 978-1-4020-0556-5. Retrieved 21 January 2013.
• Feingold, Mordechai (2004). The Newtonian moment: Isaac Newton and the making of modern culture. New York Public Library. ISBN 978-0-19-517735-0. Retrieved 21 January 2013.
• Feingold, Mordechai; Brotons, Víctor Navarro (1 January 2006). Universities And Science in the Early Modern Period. Springer. ISBN 978-1-4020-3975-1. Retrieved 21 January 2013.
• Kronk, Gary W. (28 September 1999). Cometography: Volume 1, Ancient–1799: A Catalog of Comets. Cambridge University Press. ISBN 978-0-521-58504-0. Retrieved 21 January 2013.
• Olschki, L. S. (1996). Physis; rivista internazionale di storia della scienza. Consiglio Nazionale delle Ricerche. Retrieved 21 January 2013.
• Stancari, Vittorio Francesco. Treccani. Retrieved 2013-01-21.
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Vittorio Grünwald
Vittorio Grünwald (Verona, Italy, 13 June 1855 – Florence, Italy, March 1943) was an Italian professor of mathematics and German language. His father Guglielmo (Willhelm) Grünwald (son of Aronne and Regina) was Hungarian, his mother Fortuna Marini (daughter of Mandolino Marini and Ricca Bassani) was Italian. In 1861 he moved to Hungary with his family, then came back in 1877 to Verona, later in November 1885 they moved to Brescia, and then to Venice. He studied at the Technische Universität Wien, where he graduated in mathematics. After coming back to Italy, he taught mathematics and German language in several schools (such as in Livorno and Venice), and then he settled in Florence.
Vittorio Grünwald
Vittorio Grünwald
Born(1855-06-13)13 June 1855
Verona, Italy
DiedMarch 1943 (1943-04) (aged 87)
Firenze, Italy
He married Dora Olschky, born in Berlin, and had three kids: Marta Grünwald, Beniamino (Benno) Grünwald, and Emanuele Grünwald.
He was a librarian and a teacher at the Rabbinical College of Florence. He died at 88 in Florence, a few months before Nazi's persecutions hit Jewish families in Central Italy. He published several contributions in mathematics, including a seminal work on negative numerical bases. He also published an Italian-German vocabulary.
References
• Vittorio Grünwald. Saggio di aritmetica non decimale con applicazioni del calcolo duodecima/e e trigesimale a problemi sui numeri complessi (Verona, 1884)
• Vittorio Grünwald. Intorno all'aritmetica dei sistemi numerici a base negativa con particolare riguardo al sistema numerico a base negativo-decimale per lo studio delle sue analogie coll'aritmetica ordinaria (decimale), Giornale di Matematiche di Battaglini (1885), 203-221, 367
• Vittorio Grünwald and Garibaldi Menotti Gatti, Vocabolario delle lingue Italiana e Tedesca. Ed. Belforte.
• Gianfranco Di Segni, In ricordo del prof. Vittorio Grünwald, Firenze Ebraica, Anno 25 n. 5, Settembre-Ottobre 2012.
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Vittorio Siri
Vittorio Siri or Francesco Siri (1608–1685) was an Italian mathematician, monk and historian.[1]
Vittorio Siri
Title page of the third volume of Vittorio Siri's Il Mercurio, 1652, etched by Stefano della Bella
Born
Francesco Siri
(1608-11-02)2 November 1608
Parma, Duchy of Parma
Died6 October 1685(1685-10-06) (aged 76)
Paris, Kingdom of France
Occupations
• Christian monk
• Historian
• Diplomat
Parent(s)Ottavio Siri
Maria Caterina Siri
Writing career
LanguageItalian, Latin
Notable worksIl Mercurio overo historia de' correnti tempi
Life
Siri was born in Parma, and studied at the Benedictine convent of San Giovanni Evangelista, Parma, where he pronounced his vows on December 25, 1625. At first, he specialized in geometry, and taught mathematics in Venice.[2] There he befriended the French ambassador and took a liking to political matters.[2]
In 1640, Siri published a book about the occupation of Casale Monferrato (Il politico soldato Monferrino) defending the French position. This earned him the patronage of Cardinal Richelieu, who granted him access to the French archives. Based on what he found in the archives, Siri set up to publish Il Mercurio overo historia de' correnti tempi ('Mercury, or the History of Current Times'), a monumental work in 15 volumes, published in Venice between 1644 and 1682 and translated into French by Jean Baptiste Requier.[3] Besides the Mercurio Politico Siri wrote another historical work, entitled Memorie Recondite, which fills eight volumes. Both these works contain a vast number of valuable authentic documents.[4]
In 1648 Genoese historian and polygraph Giovanni Battista Birago Avogadro offended Siri by publishing a survey of Europe in the year 1642 which he called Mercurio veridico, an undisguised slight of the latter's Mercurio, whose second volume appeared that same year. The affront was answered by Siri in 1653 with a whole book that enumerated Birago's mistakes and charged him with dishonesty (Bollo di D. Vittorio Siri).[5]
Cardinal Mazarin honored Siri with a pension and the title of Counsellor of State, chaplain and historian of the king of France. Siri therefore moved to France in 1649 and from 1655 he lived at the court. In the meanwhile he served as the representative in France of the duke of Parma and wrote newsletters for that duke as well as for the rival duke of Modena.[6] He died in Paris on 6 October 1685.[2]
Works
• Problemata et theoremata geometrica et mecanica, Bologna, 1633.
• Siri, Vittorio (1634). Propositiones mathematicae (in Latin). Bologna: Nicolò Tebaldini.
• Il politico Soldato Monferrino, ovvero discorso politico sopra gli affari di Casale published under the pseudonym Capitano Latino Verità, Casale (Venice), 1640.
• Il Mercurio overo historia de' correnti tempi in 15 volumes in-4°, 1644–1682.
• Memorie recondite in 8 volumes in-4°, 1676-79.
Bibliography
• Affò, Ireneo (1797). Memorie degli scrittori e letterati parmigiani. Vol. V. Parma. pp. 205–336.
References
1. "Siri, Vittorio (1608–1685) in Cerl Thesaurus".
2. "Vittorio Siri in Treccani.it".
3. Mercure de Vittorio Siri, conseiller d'État et historiographe de sa majesté très chretienne, contenant l'histoire generale de l'Europe, depuis 1640 jusqu'en 1655, Didot, Paris (voll. 1-2), Durand, Paris (voll. 3-18) 1756-1759.
4. Jean Le Clerc. Bibliothèque Choisie. Vol. IV. p. 158.
5. Ilan Rachum (1995). "Italian Historians and the Emergence of the Term 'Revolution', 1644–1659". History. 80 (259): 197–198. doi:10.1111/j.1468-229X.1995.tb01666.x.
6. Brendan Maurice Dooley (1999). The Social History of Skepticism. Experience and Doubt in Early Modern Culture. Johns Hopkins University Press. p. 98. ISBN 978-0801861420.
External links
• Ceccarelli, Alessia (2018). "SIRI, Vittorio". Dizionario Biografico degli Italiani, Volume 92: Semino–Sisto IV (in Italian). Rome: Istituto dell'Enciclopedia Italiana. ISBN 978-8-81200032-6.
• Villani, Stefano (2001). "La prima rivoluzione inglese nelle pagine del 'Mercurio' di Vittorio Siri". L'Informazione politica in Italia (Secoli XVI-XVIII). Atti del seminario organizzato dalla Scuola Normale Superiore di Pisa e dal Dipartimento di Storia moderna e contemporanea dell'Università di Pisa. Pisa, 23 e 24 giugno 1997. Pisa: Scuola Normale Superiore: 137–172.
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Vivanti–Pringsheim theorem
The Vivanti–Pringsheim theorem is a mathematical statement in complex analysis, that determines a specific singularity for a function described by certain type of power series. The theorem was originally formulated by Giulio Vivanti in 1893 and proved in the following year by Alfred Pringsheim.
More precisely the theorem states the following:
A complex function defined by a power series
$f(z)=\sum _{n=0}^{\infty }a_{n}z^{n}$
with non-negative real coefficients $a_{n}$ and a radius of convergence $R$ has a singularity at $z=R$.
A simple example is the (complex) geometric series
$f(z)=\sum _{n=0}^{\infty }z^{n}={\frac {1}{1-z}}$
with a singularity at $z=1$.
References
• Reinhold Remmert: The Theory of Complex Functions. Springer Science & Business Media, 1991, ISBN 9780387971957, p. 235
• I-hsiung Lin: Classical Complex Analysis: A Geometric Approach (Volume 2). World Scientific Publishing Company, 2010, ISBN 9789813101074, p. 45
| Wikipedia |
Viveka Erlandsson
Viveka Erlandsson is a Swedish[1] mathematician specialising in low-dimensional topology and geometry, and known in particular for extending the work of Maryam Mirzakhani on counting geodesics on hyperbolic manifolds.[2][3] She is a lecturer at the University of Bristol.
Education and career
Erlandsson earned a bachelor's degree in applied mathematics from San Francisco State University in 2004, and continued at the same university for a master's degree in 2006. She became a lecturer at Baruch College and Hunter College in the City University of New York system, while pursuing a doctorate in mathematics through the Graduate Center of the City University of New York, which she completed in 2013.[4] Her dissertation, The Margulis region in hyperbolic 4-space, was supervised by Ara Basmajian.[5]
After postdoctoral research at Aalto University and the University of Helsinki in Finland, she became a lecturer in mathematics at the University of Bristol in 2017.[4]
Book
Erlandsson is the coauthor of the book Geodesic Currents and Mirzakhani’s Curve Counting, with Juan Souto, to be published by Springer in 2022.[3][4]
Recognition
Erlandsson is the 2021 winner of the Anne Bennett Prize of the London Mathematical Society, given to her "for her outstanding achievements in geometry and topology and her inspirational active role in promoting women mathematicians".[2][3]
References
1. Curriculum vitae (PDF), Aalto University, 2016, retrieved 2022-02-04
2. Anne Bennett Prize: citation for Viveka Erlandsson (PDF), London Mathematical Society, 2021, retrieved 2022-02-04
3. Alumna Viveka Erlandsson wins the Anne Bennett Prize from the London Mathematical Society, CUNY Graduate Center, retrieved 2022-02-04
4. Curriculum vitae, retrieved 2022-02-04
5. Viveka Erlandsson at the Mathematics Genealogy Project
External links
• Home page
• Viveka Erlandsson publications indexed by Google Scholar
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Viviane Baladi
Viviane Baladi (born 23 May 1963) is a mathematician who works as a director of research at the Centre national de la recherche scientifique (CNRS) in France. Originally Swiss, she has become a naturalized citizen of France.[1] Her research concerns dynamical systems.
Viviane Baladi
Baladi at Oberwolfach, 2009
Born (1963-05-23) 23 May 1963
Switzerland
NationalitySwiss
Alma materUniversity of Geneva
Scientific career
FieldsMathematics
Doctoral advisorJean-Pierre Eckmann
Education and career
Baladi earned master's degrees in mathematics and computer science in 1986 from the University of Geneva.[1] She stayed in Geneva for her doctoral studies, finishing a Ph.D. in 1989 under the supervision of Jean-Pierre Eckmann, with a dissertation concerning the zeta functions of dynamical systems.[2]
She worked at CNRS beginning in 1990, with a leave of absence from 1993 to 1999 when she taught at ETH Zurich and the University of Geneva. She also spent a year as a professor at the University of Copenhagen in 2012–2013.[1]
Books
She is the author of the book Positive Transfer Operators and Decay of Correlation (Advanced Series in Nonlinear Dynamics 16, World Scientific, 2000)[3] and of Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps: A Functional Approach (Ergebnisse der Mathematik und ihrer Grenzgebiete 68, Springer, 2018).[4]
Recognition
She was an invited speaker at the International Congress of Mathematicians in 2014, speaking in the section on "Dynamical Systems and Ordinary Differential Equations".[5] She became a member of the Academia Europaea in 2018.[6] Baladi was awarded the CNRS Silver Medal in 2019.[7]
References
1. Curriculum vitae: Viviane Baladi, Centre national de la recherche scientifique, retrieved 2015-10-14.
2. Viviane Baladi at the Mathematics Genealogy Project.
3. Review of Positive Transfer Operators and Decay of Correlation by Jérôme Buzzi (2001), MR1793194.
4. Reviews of Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps: Claudio Bonanno, MR3837132; Kazuhiro Sakai, Zbl 1405.37001
5. ICM Plenary and Invited Speakers since 1897, International Mathematical Union, retrieved 2015-10-01.
6. List of members, Academia Europaea, retrieved 2020-10-02
7. Médaille d'argent du CNRS, 26 January 2023
External links
• Home page
• Viviane Baladi publications indexed by Google Scholar
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Viviani's theorem
Viviani's theorem, named after Vincenzo Viviani, states that the sum of the distances from any interior point to the sides of an equilateral triangle equals the length of the triangle's altitude.[1] It is a theorem commonly employed in various math competitions, secondary school mathematics examinations, and has wide applicability to many problems in the real world.
Proof
This proof depends on the readily-proved proposition that the area of a triangle is half its base times its height—that is, half the product of one side with the altitude from that side.[2]
Let ABC be an equilateral triangle whose height is h and whose side is a.
Let P be any point inside the triangle, and u, s, t the distances of P from the sides. Draw a line from P to each of A, B, and C, forming three triangles PAB, PBC, and PCA.
Now, the areas of these triangles are ${\frac {u\cdot a}{2}}$, ${\frac {s\cdot a}{2}}$, and ${\frac {t\cdot a}{2}}$. They exactly fill the enclosing triangle, so the sum of these areas is equal to the area of the enclosing triangle. So we can write:
${\frac {u\cdot a}{2}}+{\frac {s\cdot a}{2}}+{\frac {t\cdot a}{2}}={\frac {h\cdot a}{2}}$
and thus
$u+s+t=h$
Q.E.D.
Converse
The converse also holds: If the sum of the distances from an interior point of a triangle to the sides is independent of the location of the point, the triangle is equilateral.[3]
Applications
Further information: Ternary plot
Viviani's theorem means that lines parallel to the sides of an equilateral triangle give coordinates for making ternary plots, such as flammability diagrams.
More generally, they allow one to give coordinates on a regular simplex in the same way.
Extensions
Parallelogram
The sum of the distances from any interior point of a parallelogram to the sides is independent of the location of the point. The converse also holds: If the sum of the distances from a point in the interior of a quadrilateral to the sides is independent of the location of the point, then the quadrilateral is a parallelogram.[3]
The result generalizes to any 2n-gon with opposite sides parallel. Since the sum of distances between any pair of opposite parallel sides is constant, it follows that the sum of all pairwise sums between the pairs of parallel sides, is also constant. The converse in general is not true, as the result holds for an equilateral hexagon, which does not necessarily have opposite sides parallel.
Regular polygon
If a polygon is regular (both equiangular and equilateral), the sum of the distances to the sides from an interior point is independent of the location of the point. Specifically, it equals n times the apothem, where n is the number of sides and the apothem is the distance from the center to a side.[3][4] However, the converse does not hold; the non-square parallelogram is a counterexample.[3]
Equiangular polygon
The sum of the distances from an interior point to the sides of an equiangular polygon does not depend on the location of the point.[1]
Convex polygon
A necessary and sufficient condition for a convex polygon to have a constant sum of distances from any interior point to the sides is that there exist three non-collinear interior points with equal sums of distances.[1]
Regular polyhedron
The sum of the distances from any point in the interior of a regular polyhedron to the sides is independent of the location of the point. However, the converse does not hold, not even for tetrahedra.[3]
References
1. Abboud, Elias (2010). "On Viviani's Theorem and its Extensions". College Mathematics Journal. 43 (3): 203–211. arXiv:0903.0753. doi:10.4169/074683410X488683. S2CID 118912287.
2. Claudi Alsina, Roger B. Nelsen: Charming Proofs: A Journey Into Elegant Mathematics. MAA 2010, ISBN 9780883853481, p. 96 (excerpt (Google), p. 96, at Google Books)
3. Chen, Zhibo; Liang, Tian (2006). "The converse of Viviani's theorem". The College Mathematics Journal. 37 (5): 390–391. doi:10.2307/27646392. JSTOR 27646392.
4. Pickover, Clifford A. (2009). The Math Book. Stirling. p. 150. ISBN 978-1402788291.
Further reading
• Gueron, Shay; Tessler, Ran (2002). "The Fermat-Steiner problem". Amer. Math. Monthly. 109 (5): 443–451. doi:10.2307/2695644. JSTOR 2695644.
• Samelson, Hans (2003). "Proof without words: Viviani's theorem with vectors". Math. Mag. 76 (3): 225. doi:10.2307/3219327. JSTOR 3219327.
• Chen, Zhibo; Liang, Tian (2006). "The converse of Viviani's theorem". The College Mathematics Journal. 37 (5): 390–391. doi:10.2307/27646392. JSTOR 27646392.
• Kawasaki, Ken-Ichiroh; Yagi, Yoshihiro; Yanagawa, Katsuya (2005). "On Viviani's theorem in three dimensions". Math. Gaz. 89 (515): 283–287. doi:10.1017/S002555720017785X. JSTOR 3621243. S2CID 126113074.
• Zhou, Li (2012). "Viviani polytopes and Fermat Points". Coll. Math. J. 43 (4): 309–312. arXiv:1008.1236. CiteSeerX 10.1.1.740.7670. doi:10.4169/college.math.j.43.4.309. S2CID 117039483.
External links
• Weisstein, Eric W. "Viviani's Theorem". MathWorld.
• Li Zhou, Viviani Polytopes and Fermat Points
• "Viviani's Theorem: What is it?". at Cut the knot.
• Warendorff, Jay. "Viviani's Theorem". the Wolfram Demonstrations Project.
• "A variation of Viviani's theorem & some generalizations". at Dynamic Geometry Sketches, an interactive dynamic geometry sketch.
• Abboud, Elias (2017). "Loci of points inspired by Viviani's theorem". arXiv:1701.07339 [math.HO].
• Armstrong, Addie; McQuillan, Dan (2017). "Specialization, generalization, and a new proof of Viviani's theorem". arXiv:1701.01344 [math.HO].
| Wikipedia |
Vivien Kirk
Vivien Kirk is a New Zealand mathematician who studies dynamical systems. She is a professor of mathematics at the University of Auckland, where she also serves as associate dean,[2] and was president of the New Zealand Mathematical Society for 2017–2019.[3]
Vivien Kirk
Vivien Kirk in 1990
Academic background
Alma materUniversity of Auckland
University of Cambridge
Doctoral advisorNigel Weiss
Academic work
Doctoral studentsAlona Ben-Tal[1]
Education and career
After earning bachelor's and master's degrees at the University of Auckland, Kirk went to the University of Cambridge for doctoral studies.[2] She completed her Ph.D. in 1990; her dissertation, Destruction of tori in dissipative flows, was supervised by Nigel Weiss.[4]
She was a postdoctoral researcher at the University of California, Berkeley and at the California Institute of Technology.[2] Kirk's notable students include Alona Ben-Tal.[5]
Books
Kirk is the co-author of the books Mathematical Analysis of Complex Cellular Activity (Springer, 2015) and Models of Calcium Signalling (Springer, 2016).
Recognition
In 2017, Kirk won the Miriam Dell Excellence in Science Mentoring Award of New Zealand's Association for Women in the Sciences, in part for her efforts in founding and running a series of annual workshops for young women in mathematics and physics since 2007.[6]
References
1. Ben-Tal, Alona (2001). A Study of Symmetric Forced Oscillators (Doctoral thesis). ResearchSpace@Auckland.
2. Professor Vivien Kirk, University of Auckland, retrieved 2022-02-09
3. Presidents of the Society, New Zealand Mathematical Society, retrieved 2018-10-11
4. Vivien Kirk at the Mathematics Genealogy Project
5. Ben-Tal, Alona (2001). A Study of Symmetric Forced Oscillators (Doctoral thesis). ResearchSpace@Auckland.
6. Association for Women in the Sciences (1 December 2007), Mathematician Vivien Kirk Recognised for Mentoring Others – via Scoop
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Vivienne Malone-Mayes
Vivienne Lucille Malone-Mayes (February 10, 1932 – June 9, 1995) was an American mathematician and professor. Malone-Mayes studied properties of functions, as well as methods of teaching mathematics.[1] She was the fifth African-American woman to gain a PhD in mathematics in the United States, and the first African-American member of the faculty of Baylor University.
Vivienne Malone-Mayes
Born
Vivienne Malone
(1932-02-10)February 10, 1932
Waco, Texas, US
DiedJune 9, 1995(1995-06-09) (aged 63)
Waco, Texas, US
Alma mater
• Fisk University
• University of Texas
Known forFirst African-American full-time mathematics professor at Baylor University
Scientific career
Fieldsmathematics
Institutions
• Paul Quinn College
• Baylor University
Early life and education
Vivienne Lucille Malone was born on February 10, 1932, in Waco, Texas, to Pizarro and Vera Estelle Allen Malone.[1] She encountered educational challenges associated with growing up in an African-American community in the South, including racially segregated schools,[2] but the encouragement of her parents, both educators, led her to avidly pursue her own education. She graduated from A. J. Moore High School in 1948. She entered Fisk University at the age of 16 where she earned a bachelor's degree (1952) and a master's degree (1954). Vivienne switched from medicine to mathematics after she began studying under Evelyn Boyd Granville and Lee Lorch. Granville was one of the first of five African-American women to earn her Ph.D. in mathematics.[1] When she was in grade 6 she would get bullied by teachers and students. When she would get a low grade all of the teachers and the students would make stereotypes because of her skin color and the fact that she got a low grade, an example of a stereotype that she got a lot was " see i told you she would fail all, of those people do." she would always feel like she had let down everybody together with her being the only black woman in her class and all of her classmates ignoring her made it very difficult for her.
Career
After earning her master's, she chaired the Mathematics department at Paul Quinn College for seven years and then at Bishop College for one year before deciding to take further graduate mathematics course. She was refused admission at Baylor University due to segregation and instead attend summer courses at the University of Texas. After another year of teaching she decided to attend the University of Texas full-time as a graduate student. She was the only African American and only woman in the class, and at first her classmates ignored her. She was not allowed to teach, was unable to attend professor Robert Lee Moore's lectures, and could not join off-campus meetings because they were held in a coffee shop which could not, under Texas law, serve African Americans. She wrote, "My mathematical isolation was complete", and that "it took a faith in scholarship almost beyond measure to endure the stress of earning a Ph.D. degree as a Black, female graduate student".[3] She participated in civil rights demonstrations, and her friends and colleagues Etta Falconer and Lee Lorch wrote on her death that "With skill, integrity, steadfastness and love she fought racism and sexism her entire life, never yielding to the pressures or problems which beset her path".[4]
As an educator, Malone-Mayes's developed novel methods of teaching mathematics including a program using self-paced audio-tutorials. Her mathematical research was in the field of functional analysis, particularly characterizing the growth properties of ranges of nonlinear operators. Malone-Mayes graduated in 1966, with a dissertation entitled "A structure problem in asymptotic analysis".[2] Her doctoral supervisor was Don E. Edmondson.[2]
Following graduation, Malone-Mayes was hired as a full-time professor in the mathematics department at Baylor University. Her research there continued to focus on functional analysis; of her two papers, one studies summability methods for the moment problem as operators on sequence spaces[5][6] and the other studies the long-term behavior of a certain linear ordinary differential equation.[7] Nonetheless, her research was sufficiently innovative for her to qualify for federal grants to support her work,[2] and the latter paper was published in the prestigious Proceedings of the American Mathematical Society.[7] She was soon a full professor.[2]
Malone-Mayes had a successful, lengthy career and served on several boards and committees of note, retiring in 1994 due to ill health.[1] She was the fifth African-American woman to be allowed in the White House.[1]
Memberships
She was a member of the board of directors of the National Association of Mathematicians. She was elected Director-at-large for the Texas section of Mathematical Association of America and served as director of the High School Lecture Program for the Texas section.
She was also active in her local community as a lifetime member of New Hope Baptist Church. She served on boards of directors for Cerebral Palsy, Goodwill Industries, and Family Counseling and Children. She was on the Texas State Advisory Council for Construction of Community Mental Health Centers and served on the board of the Heart of Texas Region Mental Health and Mental Retardation Center.[1]
Vivienne Malone-Mayes was a member of Delta Sigma Theta sorority and served as President of Waco Alumnae Chapter.
Legacy and awards
After Lillian K. Bradley in 1960, Malone-Mayes became one of the first African-American women to receive a PhD in mathematics from University of Texas (and fifth African-American woman in the United States). She was the first African-American member of the faculty at Baylor University, and the first African-American person elected to Executive Committee of the Association of Women in Mathematics.
The student congress of Baylor voted her the "Outstanding Faculty Member of the Year" in 1971.
Personal
Malone-Mayes married James Mayes in 1952,[1] and had a daughter, Patsyanne Mayes Wheeler.[4] She died of a heart attack, in Waco, on June 9, 1995, at the age of 63.[4] She is buried in Greenwood Cemetery.[8]
References
1. Vivienne Malone-Mayes Papers #2072, The Texas Collection.
2. Warren, Wini (1999). Black women scientists in the United States. Bloomington, Ind. [u.a.]: Indiana University Press. pp. 193–195. ISBN 0253336031. wini warren black women scientists.
3. Case, Bettye Anne; Leggett, Anne M. (31 May 2016). Complexities: Women in mathematics. Princeton University Press. ISBN 978-1400880164.
4. Falconer, Etta; Lorch, Lee (Nov–Dec 1995). "Vivienne Malone-Mayes: In Memoriam". AWM Newsletter. Vol. 25, no. 6. AWM. Retrieved 11 March 2017.
5. Mayes, Vivienne; Rhoades, B. E. (1980). "Some properties of the Leininger generalized Hausdorff matrix". Houston Journal of Mathematics. 6 (2). CiteSeerX 10.1.1.589.1897.
6. Galanopoulos, Petros; Siskaki, Aristomenos G. (Fall 2001). "Hausdorff matrices and composition operators" (pdf). Illinois Journal of Mathematics. 45 (3): 757–773. doi:10.1215/ijm/1258138149 – via Project Euclid.
7. Mayes, Vivienne (September 1969). "Some Steady State Properties of equation 1". Proceedings of the American Mathematical Society. 22 (3): 672–677. doi:10.2307/2037456. JSTOR 2037456. Retrieved 11 March 2017.
8. Minutaglio, Bill (2021). A Single Star and Bloody Knuckles: A History of Politics and Race in Texas. University of Texas Press. pp. 57–62. ISBN 9781477310366.
• Notable Women in Mathematics, a Biographical Dictionary, edited by Charlene Morrow and Teri Perl, Greenwood Press, 1998, pp. 133–137.
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• zbMATH
| Wikipedia |
Vizing's theorem
In graph theory, Vizing's theorem states that every simple undirected graph may be edge colored using a number of colors that is at most one larger than the maximum degree Δ of the graph. At least Δ colors are always necessary, so the undirected graphs may be partitioned into two classes: "class one" graphs for which Δ colors suffice, and "class two" graphs for which Δ + 1 colors are necessary. A more general version of Vizing's theorem states that every undirected multigraph without loops can be colored with at most Δ+µ colors, where µ is the multiplicity of the multigraph.[1] The theorem is named for Vadim G. Vizing who published it in 1964.
Discovery
The theorem discovered by Russian mathematician Vadim G. Vizing was published in 1964 when Vizing was working in Novosibirsk and became known as Vizing's theorem.[2] Indian mathematician R. P. Gupta independently discovered the theorem, while undertaking his doctorate (1965-1967).[3][4]
Examples
When Δ = 1, the graph G must itself be a matching, with no two edges adjacent, and its edge chromatic number is one. That is, all graphs with Δ(G) = 1 are of class one.
When Δ = 2, the graph G must be a disjoint union of paths and cycles. If all cycles are even, they can be 2-edge-colored by alternating the two colors around each cycle. However, if there exists at least one odd cycle, then no 2-edge-coloring is possible. That is, a graph with Δ = 2 is of class one if and only if it is bipartite.
Proof
This proof is inspired by Diestel (2000).
Let G = (V, E) be a simple undirected graph. We proceed by induction on m, the number of edges. If the graph is empty, the theorem trivially holds. Let m > 0 and suppose a proper (Δ+1)-edge-coloring exists for all G − xy where xy ∈ E.
We say that color α ∈ {1,...,Δ+1} is missing in x ∈ V with respect to proper (Δ+1)-edge-coloring c if c(xy) ≠ α for all y ∈ N(x). Also, let α/β-path from x denote the unique maximal path starting in x with α-colored edge and alternating the colors of edges (the second edge has color β, the third edge has color α and so on), its length can be 0. Note that if c is a proper (Δ+1)-edge-coloring of G then every vertex has a missing color with respect to c.
Suppose that no proper (Δ+1)-edge-coloring of G exists. This is equivalent to this statement:
(1) Let xy ∈ E and c be arbitrary proper (Δ+1)-edge-coloring of G − xy and α be missing from x and β be missing from y with respect to c. Then the α/β-path from y ends in x.
This is equivalent, because if (1) doesn't hold, then we can interchange the colors α and β on the α/β-path and set the color of xy to be α, thus creating a proper (Δ+1)-edge-coloring of G from c. The other way around, if a proper (Δ+1)-edge-coloring exists, then we can delete xy, restrict the coloring and (1) won't hold either.
Now, let xy0 ∈ E and c0 be a proper (Δ+1)-edge-coloring of G − xy0 and α be missing in x with respect to c0. We define y0,...,yk to be a maximal sequence of neighbours of x such that c0(xyi) is missing in yi−1 with respect to c0 for all 0 < i ≤ k.
We define colorings c1,...,ck as
ci(xyj)=c0(xyj+1) for all 0 ≤ j < i,
ci(xyi) not defined,
ci(e)=c0(e) otherwise.
Then ci is a proper (Δ+1)-edge-coloring of G − xyi due to definition of y0,...,yk. Also, note that the missing colors in x are the same with respect to ci for all 0 ≤ i ≤ k.
Let β be the color missing in yk with respect to c0, then β is also missing in yk with respect to ci for all 0 ≤ i ≤ k. Note that β cannot be missing in x, otherwise we could easily extend ck, therefore an edge with color β is incident to x for all cj. From the maximality of k, there exists 1 ≤ i < k such that c0(xyi) = β. From the definition of c1,...,ck this holds:
c0(xyi) = ci−1(xyi) = ck(xyi−1) = β
Let P be the α/β-path from yk with respect to ck. From (1), P has to end in x. But α is missing in x, so it has to end with an edge of color β. Therefore, the last edge of P is yi−1x. Now, let P' be the α/β-path from yi−1 with respect to ci−1. Since P' is uniquely determined and the inner edges of P are not changed in c0,...,ck, the path P' uses the same edges as P in reverse order and visits yk. The edge leading to yk clearly has color α. But β is missing in yk, so P' ends in yk. Which is a contradiction with (1) above.
Classification of graphs
Several authors have provided additional conditions that classify some graphs as being of class one or class two, but do not provide a complete classification. For instance, if the vertices of the maximum degree Δ in a graph G form an independent set, or more generally if the induced subgraph for this set of vertices is a forest, then G must be of class one.[5]
Erdős & Wilson (1977) showed that almost all graphs are of class one. That is, in the Erdős–Rényi model of random graphs, in which all n-vertex graphs are equally likely, let p(n) be the probability that an n-vertex graph drawn from this distribution is of class one; then p(n) approaches one in the limit as n goes to infinity. For more precise bounds on the rate at which p(n) converges to one, see Frieze et al. (1988).
Planar graphs
Vizing (1965) showed that a planar graph is of class one if its maximum degree is at least eight. In contrast, he observed that for any maximum degree in the range from two to five, there exist planar graphs of class two. For degree two, any odd cycle is such a graph, and for degree three, four, and five, these graphs can be constructed from platonic solids by replacing a single edge by a path of two adjacent edges.
In Vizing's planar graph conjecture, Vizing (1965) states that all simple, planar graphs with maximum degree six or seven are of class one, closing the remaining possible cases. Independently, Zhang (2000) and Sanders & Zhao (2001) partially proved Vizing's planar graph conjecture by showing that all planar graphs with maximum degree seven are of class one. Thus, the only case of the conjecture that remains unsolved is that of maximum degree six. This conjecture has implications for the total coloring conjecture.
The planar graphs of class two constructed by subdivision of the platonic solids are not regular: they have vertices of degree two as well as vertices of higher degree. The four color theorem (proved by Appel & Haken (1976)) on vertex coloring of planar graphs, is equivalent to the statement that every bridgeless 3-regular planar graph is of class one (Tait 1880).
Graphs on nonplanar surfaces
In 1969, Branko Grünbaum conjectured that every 3-regular graph with a polyhedral embedding on any two-dimensional oriented manifold such as a torus must be of class one. In this context, a polyhedral embedding is a graph embedding such that every face of the embedding is topologically a disk and such that the dual graph of the embedding is simple, with no self-loops or multiple adjacencies. If true, this would be a generalization of the four color theorem, which was shown by Tait to be equivalent to the statement that 3-regular graphs with a polyhedral embedding on a sphere are of class one. However, Kochol (2009) showed the conjecture to be false by finding snarks that have polyhedral embeddings on high-genus orientable surfaces. Based on this construction, he also showed that it is NP-complete to tell whether a polyhedrally embedded graph is of class one.[6]
Algorithms
Misra & Gries (1992) describe a polynomial time algorithm for coloring the edges of any graph with Δ + 1 colors, where Δ is the maximum degree of the graph. That is, the algorithm uses the optimal number of colors for graphs of class two, and uses at most one more color than necessary for all graphs. Their algorithm follows the same strategy as Vizing's original proof of his theorem: it starts with an uncolored graph, and then repeatedly finds a way of recoloring the graph in order to increase the number of colored edges by one.
More specifically, suppose that uv is an uncolored edge in a partially colored graph. The algorithm of Misra and Gries may be interpreted as constructing a directed pseudoforest P (a graph in which each vertex has at most one outgoing edge) on the neighbors of u: for each neighbor p of u, the algorithm finds a color c that is not used by any of the edges incident to p, finds the vertex q (if it exists) for which edge uq has color c, and adds pq as an edge to P. There are two cases:
• If the pseudoforest P constructed in this way contains a path from v to a vertex w that has no outgoing edges in P, then there is a color c that is available both at u and w. Recoloring edge uw with color c allows the remaining edge colors to be shifted one step along this path: for each vertex p in the path, edge up takes the color that was previously used by the successor of p in the path. This leads to a new coloring that includes edge uv.
• If, on the other hand, the path starting from v in the pseudoforest P leads to a cycle, let w be the neighbor of u at which the path joins the cycle, let c be the color of edge uw, and let d be a color that is not used by any of the edges at vertex u. Then swapping colors c and d on a Kempe chain either breaks the cycle or the edge on which the path joins the cycle, leading to the previous case.
With some simple data structures to keep track of the colors that are used and available at each vertex, the construction of P and the recoloring steps of the algorithm can all be implemented in time O(n), where n is the number of vertices in the input graph. Since these steps need to be repeated m times, with each repetition increasing the number of colored edges by one, the total time is O(mn).
In an unpublished technical report, Gabow et al. (1985) claimed a faster $O(m{\sqrt {n}}\log n)$ time bound for the same problem of coloring with Δ + 1 colors.
History
In both Gutin & Toft (2000) and Soifer (2008), Vizing mentions that his work was motivated by a theorem of Shannon (1949) showing that multigraphs could be colored with at most (3/2)Δ colors. Although Vizing's theorem is now standard material in many graph theory textbooks, Vizing had trouble publishing the result initially, and his paper on it appears in an obscure journal, Diskret. Analiz.[7]
See also
• Brooks' theorem relating vertex colorings to maximum degree
Notes
1. Berge, Claude; Fournier, Jean Claude (1991). "A short proof for a generalization of Vizing's theorem". Journal of graph theory. Wiley Online Library. 15: 333--336.
2. Vizing (1965)
3. Stiebitz, Michael; Scheide, Diego; Toft, Bjarne; Favrholdt, Lene M. (2012). Graph Edge Coloring: Vizing's Theorem and Goldberg's Conjecture. Wiley Series in Discrete Mathematics and Optimization. John Wiley & Sons, Inc., Hoboken, NJ. p. xii. ISBN 978-1-118-09137-1. MR 2975974.
4. Toft, B; Wilson, R (11 March 2021). "A brief history of edge-colorings – with personal reminiscences". Discrete Mathematics Letters. 6: 38–46. doi:10.47443/dml.2021.s105.
5. Fournier (1973).
6. Kochol (2010).
7. The full name of this journal was Akademiya Nauk SSSR. Sibirskoe Otdelenie. Institut Matematiki. Diskretny˘ı Analiz. Sbornik Trudov. It was renamed Metody Diskretnogo Analiza in 1980 (the name given for it in Gutin & Toft (2000)) and discontinued in 1991 .
References
• Appel, K.; Haken, W. (1976), "Every planar map is four colorable", Bulletin of the American Mathematical Society, 82 (5): 711–712, doi:10.1090/S0002-9904-1976-14122-5, MR 0424602.
• Diestel, Reinhard (2000), Graph Theory (PDF), Berlin, New York: Springer-Verlag, pp. 103–104.
• Erdős, Paul; Wilson, Robin J. (1977), "Note on the chromatic index of almost all graphs" (PDF), Journal of Combinatorial Theory, Series B, 23 (2–3): 255–257, doi:10.1016/0095-8956(77)90039-9.
• Fournier, Jean-Claude (1973), "Colorations des arêtes d'un graphe", Cahiers du Centre d'Études de Recherche Opérationnelle, 15: 311–314, MR 0349458.
• Frieze, Alan M.; Jackson, B.; McDiarmid, C. J. H.; Reed, B. (1988), "Edge-colouring random graphs", Journal of Combinatorial Theory, Series B, 45 (2): 135–149, doi:10.1016/0095-8956(88)90065-2, MR 0961145.
• Gabow, Harold N.; Nishizeki, Takao; Kariv, Oded; Leven, Daniel; Terada, Osamu (1985), Algorithms for edge-coloring graphs, Tech. Report TRECIS-8501, Tohoku University.
• Gutin, Gregory; Toft, Bjarne (December 2000), "Interview with Vadim G. Vizing" (PDF), European Mathematical Society Newsletter, 38: 22–23.
• Kochol, Martin (2009), "Polyhedral embeddings of snarks in orientable surfaces", Proceedings of the American Mathematical Society, vol. 137, pp. 1613–1619.
• Kochol, Martin (2010), "Complexity of 3-edge-coloring in the class of cubic graphs with a polyhedral embedding in an orientable surface", Discrete Applied Mathematics, 158 (16): 1856–1860, doi:10.1016/j.dam.2010.06.019, MR 2679785.
• Misra, J.; Gries, David (1992), "A constructive proof of Vizing's Theorem", Information Processing Letters, 41 (3): 131–133, doi:10.1016/0020-0190(92)90041-S.
• Sanders, Daniel P.; Zhao, Yue (2001), "Planar graphs of maximum degree seven are class I", Journal of Combinatorial Theory, Series B, 83 (2): 201–212, doi:10.1006/jctb.2001.2047.
• Shannon, Claude E. (1949), "A theorem on coloring the lines of a network", J. Math. Physics, 28 (1–4): 148–151, doi:10.1002/sapm1949281148, MR 0030203.
• Soifer, Alexander (2008), The Mathematical Coloring Book, Springer-Verlag, pp. 136–137, ISBN 978-0-387-74640-1.
• Tait, P. G. (1880), "Remarks on the colourings of maps", Proc. R. Soc. Edinburgh, 10: 729, doi:10.1017/S0370164600044643.
• Vizing, V. G. (1964), "On an estimate of the chromatic class of a p-graph", Diskret. Analiz., 3: 25–30, MR 0180505.
• Vizing, V. G. (1965), "Critical graphs with given chromatic class", Metody Diskret. Analiz., 5: 9–17. (In Russian.)
• Zhang, Limin (2000), "Every planar graph with maximum degree 7 is of class 1", Graphs and Combinatorics, 16 (4): 467–495, doi:10.1007/s003730070009, S2CID 10945647.
External links
• Proof of Vizing's theorem at PlanetMath.
| Wikipedia |
Vlad Voroninski
Vlad Y. Voroninski (born 21 March 1985) is a Russian-American mathematician and entrepreneur.
Vlad Y. Voroninski
Born (1985-03-21) 21 March 1985
Novosibirsk, Russia
NationalityRussian, American
Alma materUC Berkeley
UCLA
Known forPhase Retrieval, Deep Compressive Sensing
AwardsSIAM Outstanding Paper Prize (2014)
Bernhard Friedman Memorial Prize (2013)
Scientific career
FieldsMathematics, Entrepreneurship, Artificial Intelligence
InstitutionsMIT, Helm.ai
Doctoral advisorEmmanuel Candes, John A. Strain
Academic biography
Voroninski received his B.S. and M.A degrees in Applied Mathematics from UCLA in 2008, summa cum laude.[1] He earned his Ph.D. in mathematics from UC Berkeley in 2013, under the supervision of Emmanuel Candes and John Strain.[2] He was on the faculty at the MIT Mathematics Department from 2013 to 2016.[3]
Research
Voroninski's PhD thesis kicked off the study of phase retrieval in the applied mathematics community, by providing the PhaseLift algorithm along with the first mathematical recovery guarantees for phase retrieval.[4] His research has also led to solutions to open problems in computer vision, quantum operator theory, optimization and the theory of deep learning and compressive sensing.[5][6][7][8] More recently, Voroninski's research connected the fields of deep learning and inverse problems, resolving the sample complexity bottleneck for compressive phase retrieval.[9]
Awards and honors
Voroninski was awarded the 2014 SIAM Outstanding Paper Prize,[10][11] given to works that "exhibit originality, for example, papers that bring a fresh look at an existing field or that open up new areas of applied mathematics".[10] His PhD thesis was awarded the university-wide Bernhard Friedman Memorial Prize from UC Berkeley.[12][13] In addition he has received the SIAM Student Paper Prize and SIGEST Review Awards from SIAM.
He received the George E.G. Sherwood Prize from the UCLA Mathematics Department in 2008, which is awarded to the top graduating senior, as well as the Computing Research Association Outstanding Undergraduate Award in 2007.[14]
Entrepreneurship
From 2014 to 2016, Voroninski was the founding Chief Scientist at Sift Security, a cybersecurity machine learning startup which was acquired by Netskope in 2018.[15] As of 2016, Voroninski is the CEO and co-founder of Helm.ai, a stealth mode AI software startup focusing on autonomous driving.[16]
References
1. "UCLA Professor and Alumni Win SIAM Prizes | UCLA Department of Mathematics". www.math.ucla.edu. Retrieved 9 September 2019.
2. "Emmanuel Candes | STUDENTS". statweb.stanford.edu. Retrieved 6 September 2019.
3. "Faculty Awards | MIT Mathematics". 15 March 2015. Archived from the original on 2015-03-15. Retrieved 9 September 2019.
4. Hassibi, Babak; Eldar, Yonina C.; Jaganathan, Kishore (26 October 2015). "Phase Retrieval: An Overview of Recent Developments". arXiv:1510.07713. Bibcode:2015arXiv151007713J. {{cite journal}}: Cite journal requires |journal= (help)
5. Boumal, Nicolas; Voroninski, Vlad; Bandeira, Afonso (2016). "The non-convex Burer-Monteiro approach works on smooth semidefinite programs". Advances in Neural Information Processing Systems. Curran Associates, Inc. 29: 2757–2765. arXiv:1606.04970. Bibcode:2016arXiv160604970B. Retrieved 8 September 2019.
6. Mixon, Dustin G. (31 May 2017). "Global Guarantees for Enforcing Deep Generative Priors by Empirical Risk". Short, Fat Matrices. Retrieved 6 September 2019.
7. Mixon, Dustin G. (6 June 2013). "Determination of all pure quantum states from a minimal number of observables". Short, Fat Matrices. Retrieved 6 September 2019.
8. "ShapeFit and ShapeKick for Robust, Scalable Structure from Motion" (PDF). Retrieved 6 September 2019.
9. Hand, Paul; Leong, Oscar; Voroninski, Vlad (2018). "Phase Retrieval Under a Generative Prior". Advances in Neural Information Processing Systems. Curran Associates, Inc. 31: 9136–9146. Retrieved 6 September 2019.
10. "SIAM Outstanding Paper Prizes". SIAM. Retrieved 6 September 2019.
11. "UCLA Professor and Alumni Win SIAM Prizes | UCLA Department of Mathematics". www.math.ucla.edu. Retrieved 6 September 2019.
12. "Graduate Student Honors" (PDF). math.berkeley.edu. Retrieved 9 September 2019.
13. "Bernard Friedman Memorial Prize in Applied Mathematics | Department of Mathematics at University of California Berkeley". math.berkeley.edu. Retrieved 6 September 2019.
14. "Graduating Students Take Top Honors" (PDF). math.ucla.edu. Retrieved 9 September 2019.
15. Miller, Ron. "Netskope nabs Sift Security to enhance infrastructure cloud security". TechCrunch. Retrieved 6 September 2019.
16. Herger, Mario (17 December 2018). "Helm.ai 62nd Company With California Test License for Autonomous Cars". The Last Driver License Holder... Retrieved 6 September 2019.
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• Mathematics Genealogy Project
| Wikipedia |
Vladas Sidoravicius
Vladas Sidoravicius (1963, Vilnius, Lithuania – 23 May 2019, Shanghai) was a Lithuanian-Brazilian mathematician, specializing in probability theory.[1][2][3]
Education and career
At Vilnius University, Sidoravicius graduated in mathematics with Diplom in 1985 and Magister degree in 1986.[2] At Lomonosov State University he matriculated in 1986 and received his doctoral degree in 1990 with thesis advisor Vadim Aleksandrovich Malyshev.[4] At Heidelberg University and at Paris Dauphine University, Sidoravicius was a postdoc from 1991 to 1993.[2] In the early 1990s he gained an international reputation for his research in probability theory.[3] In 1993 he moved to Brazil.[1] He became a naturalized Brazilian citizen and was a full professor at the Instituto de Matemática Pura e Aplicada (IMPA) in Rio de Janeiro from 1999 to 2015, when he moved to China.[2] At New York University Shanghai (NYU Shanghai), he was a professor of mathematics and also served as the deputy director of NYU Shanghai's NYU-ECNU (East China Normal University) Institute of Mathematical Sciences from 2015 until his death in 2019 at age 55.[3]
Sidoravicius was the author or co-author of over 100 articles in refereed journals. He was a frequent collaborator of Harry Kesten. Their 2008 article A Shape Theory for the Spread of an Infection is particularly noteworthy.[3][5]
In 2014 Sidoravicius was an invited speaker at the International Congress of Mathematicians in Seoul.[6] In 2019 the XXIII Escola Brasileira de Probabilidade (XXIII Brazilian School of Probability) was dedicated to his memory.[7]
Selected publications
Articles
• Kesten, H.; Sidoravicius, V.; Zhang, Yu (1998). "Almost all words are seen in critical site percolation on the triangular lattice" (PDF). Electronic Journal of Probability. 3 (10): 1–75. doi:10.1214/EJP.v3-32.
• Kesten, H.; Sidoravicius, V. (2003). "Branching random walk with catalysts" (PDF). Electronic Journal of Probability. 8 (5): 1–51. doi:10.1214/EJP.v8-127.
• Sidoravicius, Vladas; Sznitman, Alain-Sol (2004). "Quenched invariance principles for walks on clusters of percolation or among random conductances". Probability Theory and Related Fields. 129 (2): 219–244. doi:10.1007/s00440-004-0336-0. S2CID 120061442.
• Kesten, Harry; Sidoravicius, Vladas (2005). "The spread of a rumor or infection in a moving population". The Annals of Probability. 33 (6): 2402–2462. arXiv:math/0312496. doi:10.1214/009117905000000413.
• Alexander, Kenneth S.; Sidoravicius, Vladas (2006). "Pinning of polymers and interfaces by random potentials". The Annals of Applied Probability. 16 (2): 636–669. arXiv:math/0501028. doi:10.1214/105051606000000015.
• Kesten, Harry; Sidoravicius, Vladas (2006). "A phase transition in a model for the spread of an infection". Illinois Journal of Mathematics. 50 (1–4): 547–634. doi:10.1215/ijm/1258059486. 2009
• Sidoravicius, V.; Sznitman, A. S. (2010). "Connectivity bounds for the vacant set of random interlacements". Annales de l'Institut Henri Poincaré B. 46 (4): 976–990. arXiv:0908.2206. Bibcode:2010AIHPB..46..976S. doi:10.1214/09-AIHP335. S2CID 6853534.
• Ellwood, David; Newman, Charles; Sidoravicius, Vladas; Werner, Wendelin (2012). "Probability and Statistical Physics in Two and More Dimensions". Clay Mathematical Proceedings. 15. CiteSeerX 10.1.1.680.4319; XIV Brazilian School of Probability , July 11–August 7, 2010{{cite journal}}: CS1 maint: postscript (link)
Books
• Sidoravicius, V., ed. (31 August 2009). New Trends in Mathematical Physics: Selected contributions of the XVth International Congress on Mathematical Physics. Springer Science & Business Media. ISBN 978-90-481-2810-5.
• Sidoravicius, V., ed. (2002). In and Out of Equilibrium: Probability with a Physics Flavor. Progress in Probability. Vol. 51. Birkhäuser Basel. pp. vii+472. doi:10.1007/978-1-4612-0063-5. ISBN 978-1-4612-6595-5.
• Sidoravicius, V., ed. (2019). Sojourns in Probability Theory and Statistical Physics: A Festschrift for Charles M. Newman. Proceedings in Mathematics & Statistics. Vol. (3 vols.). Springer.
• Vol. I. Spin Glasses and Statistical Mechanics.
• Vol. II. Brownian Web and Percolation.
• Vol. III. Interacting Particle Systems and Random Walks.
References
1. "Vladas Sidoravicius (1963–2019)". ICMC-USP, University of São Paulo. 17 August 2019.
2. "Mathematician Vladas Sidoravicius". Instituto de Matemática Pura e Aplicada (IMPA). 24 May 2019.
3. "In Memoriam: Professor of Mathematics Vladas Sidoravicius, 1963–2019". Shanghai NUY. 29 May 2019.
4. Vladas Sidoravicius at the Mathematics Genealogy Project
5. Kesten, H.; Sidoravicius, V. (2008). "A shape theorem for the spread of an infection" (PDF). Annals of Mathematics. 167 (3): 701–766. doi:10.4007/annals.2008.167.701.
6. Sidoravicius, Vladas. "Criticality and Phase Transitions: five favorite pieces". Proceedings of the ICM, Seoul 2014. Vol. 4. pp. 199–224.
7. "XXIII Brazilian School of Probability". 17 August 2019.
External links
• "Clay Mathematics Institute 2010 Summer School - Vladas Sidoravicius". YouTube. Instituto de Matemática Pura e Aplicada. 7 June 2018.
• "ICM2014 VideoSeries IL 12.11: Vladas Sidoravicius on Aug20Wed". YouTube. Seoul ICM VOD. 20 August 2014.
• "Random walks in growing domains - recurrence vs transience by Vladas Sidoravicius". YouTube. International Centre for Theoretical Sciences. 15 April 2019.
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• IdRef
| Wikipedia |
Vladimir Abramovich Rokhlin
Vladimir Abramovich Rokhlin (Russian: Влади́мир Абра́мович Ро́хлин) (23 August 1919 – 3 December 1984) was a Soviet mathematician, who made numerous contributions in algebraic topology, geometry, measure theory, probability theory, ergodic theory and entropy theory.[1]
Vladimir Abramovich Rokhlin
Vladimir Rokhlin in Leningrad, 1966.
Born(1919-08-23)23 August 1919
Baku, Azerbaijan
Died3 December 1984(1984-12-03) (aged 65)
Leningrad, Soviet Union
CitizenshipCitizen of the Soviet Union
EducationMoscow State University (1935-1941)
Known for
• Rokhlin lemma
• Rokhlin partitions
• Rokhlin's theorem
• Standard probability space
ChildrenVladimir Rokhlin, Jr.
Scientific career
FieldsMathematics
InstitutionsLeningrad State University
Academic advisorsAbraham Plessner
Notable students
• Yakov Eliashberg
• Mikhail Gromov
• Nikolai V. Ivanov
• Viatcheslav M. Kharlamov
• Anatoly Vershik
• Oleg Viro
Life
Vladimir Abramovich Rokhlin was born in Baku, Azerbaijan, to a wealthy Jewish family.[2] His mother, Henrietta Emmanuilovna Levenson, had studied medicine in France (she died in Baku in 1923, believed to have been killed during civil unrest provoked by an epidemic). His maternal grandmother, Clara Levenson, had been one of the first female doctors in Russia. His maternal grandfather Emmanuil Levenson was a wealthy businessman (he was also the illegitimate father of Korney Chukovsky, who was thus Henrietta's half-brother). Vladimir Rokhlin's father Abram Veniaminovich Rokhlin was a well-known social democrat (he was imprisoned during Stalin's Great Purge, and executed in 1941).[3]
Vladimir Rokhlin entered Moscow State University in 1935. His advisor was Abraham Plessner. He volunteered for the army in 1941, leading to four years as a prisoner of a German war camp. During this time he was able to hide his Jewish origins from the Nazis. Rokhlin was liberated by the Soviet military in January 1945. He then served as a German language translator for the 5th Army of the Belorussian front. In May 1945 he was sent to a Soviet 'verification camp' for former prisoners of war. In January 1946 he was transferred to another camp to determine if he was an "enemy of the Soviet." Rokhlin was cleared in June 1946 but was forced to remain in the camp as a guard. Due to intercession by mathematicians Andrey Kolmogorov and Lev Pontryagin, he was released in December 1946 and allowed to return to Moscow, after which he returned to mathematics.
In 1959 Rokhlin joined Leningrad State University as a faculty member. He died in 1984 in Leningrad. His students include Viatcheslav Kharlamov, Yakov Eliashberg, Mikhail Gromov, Nikolai V. Ivanov, Anatoly Vershik and Oleg Viro.[4]
Work
Rokhlin's contributions to topology include Rokhlin's theorem, a result of 1952 on the signature of 4-manifolds. He also worked in the theory of characteristic classes, homotopy theory, cobordism theory, and in the topology of real algebraic varieties.
In measure theory, Rokhlin introduced what are now called Rokhlin partitions. He introduced the notion of standard probability space, and characterised such spaces up to isomorphism mod 0. He also proved the famous Rokhlin lemma.
Family
His son Vladimir Rokhlin, Jr. is a well-known mathematician and computer scientist at Yale University.
Rokhlin's uncle was Korney Chukovsky, a well-known Russian poet, most famous for his popular children's books.
See also
• Gudkov's conjecture
Notes
1. see Vershik (1989), Arnol'd (1986)
2. see Turaev & Vershik (2001), p. 8
3. see Turaev & Vershik (2001), p. 1
4. Vladimir Abramovich Rokhlin at the Mathematics Genealogy Project
References
• Arnol'd, Vladimir; et al. (1986), "Vladimir Abramovich Rokhlin (obituary)", Russ. Math. Surv., 41 (3): 189–195, Bibcode:1986RuMaS..41R.189A, doi:10.1070/RM1986v041n03ABEH003331, MR 0854242, S2CID 250910259
• Turaev, Vladimir G.; Vershik, Anatoly M., eds. (2001), Topology, ergodic theory, real algebraic geometry: Rokhlin's Memorial, American Mathematical Society Translations: Series 2, vol. 202, Providence, RI: American Mathematical Society, doi:10.1090/trans2/202, hdl:11693/48903, ISBN 9780821827406, MR 1819175
• Vershik, Anatoly M. (1989), "Vladimir Abramovich Rokhlin—A biographical tribute (23.8.1919–3.12.1984)", Ergodic Theory and Dynamical Systems, Cambridge University Press, 9 (4): 629–641, doi:10.1017/S0143385700005265, MR 1036901
External links
• Vladimir Abramovich Rokhlin at the Mathematics Genealogy Project
• Novikov, Sergei. "Rokhlin". www.mccme.ru (in Russian). Archived from the original on 15 May 2011. Retrieved 10 May 2006.
• Rokhlin, Vladimir A. (1981). "A lecture about teaching mathematics to non-mathematicians, Part I." mathfoolery.wordpress.com.
• O'Connor, John J.; Robertson, Edmund F., "Vladimir Abramovich Rokhlin", MacTutor History of Mathematics Archive, University of St Andrews
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Vladimir Buslaev
Vladimir Savel'evich Buslaev (Владимир Савельевич Буслаев, 19 April 1937, Leningrad[1] – 14 March 2012) was a Russian mathematical physicist.[2]
Education
Buslajew received his Ph.D. (Russian candidate degree) in 1963 from the University of Leningrad under Olga Ladyzhenskaya with thesis Short-Wave Asymptotics of Diffraction Problems in Convex Domains.[3] He was a professor at Saint Petersburg State University.
Contributions
He did research on mathematical problems of diffraction and the WKB method.[4]
Recognition
In 1963 he received the prize of the Leningrad Mathematical Society.
In 1983 he was an invited speaker at the International Congress of Mathematicians in Warsaw and gave a talk Regularization of many particle scattering. He received an honorary doctorate from the Université Paris Nord. In 2000 he received the State Prize of the Russian Federation and he was an Honoured Scientist of the Russian Federation. In 2000 he gave a plenary lecture (Adiabatic perturbations of linear periodic problems) at the annual meeting of the German Mathematical Society in Dresden.[5]
Selected publications
• with Vladimir Borisovich Matveev: Buslaev, V. S.; Matveev, V. B. (1970). "Wave operators for the Schrödinger equation with a slowly decreasing potential". Teoreticheskaya I Matematicheskaya Fizika. 2 (3): 367–376. Bibcode:1970TMP.....2..266B. doi:10.1007/BF01038047. S2CID 121852629.
• Buslaev, V. S. (1987). "Semiclassical approximation for equations with periodic coefficients". Russian Mathematical Surveys. 42 (6): 97–125. Bibcode:1987RuMaS..42...97B. doi:10.1070/RM1987v042n06ABEH001502. S2CID 250803586.
• with Vincenzo Grecchi: Buslaev, V.; Grecchi, V. (1993). "Equivalence of unstable anharmonic oscillators and double wells". Journal of Physics A: Mathematical and General. 26 (20): 5541–5549. Bibcode:1993JPhA...26.5541B. doi:10.1088/0305-4470/26/20/035.
• with Catherine Sulem: Buslaev, Vladimir S.; Sulem, Catherine (2003). "On asymptotic stability of solitary waves for nonlinear Schrödinger equations". Annales de l'Institut Henri Poincaré C. 20 (3): 419–475. Bibcode:2003AIHPC..20..419B. doi:10.1016/S0294-1449(02)00018-5.
References
1. "professors". www.ioffe.ru.
2. "Staff — Buslaev Vladimir Savelievich (1937 - 2012)". math.nw.ru.
3. Vladimir Savel'evich Buslaev at the Mathematics Genealogy Project
4. "On the mathematical work of Vladimir Savel'evich Buslaev". St. Petersburg Mathematical Journal. 25: 151–174. 2014. doi:10.1090/S1061-0022-2014-01283-6. MR 3114847.
5. "L206". www.math.tu-dresden.de.
External links
• Памяти Владимира Савельевича Буслаева (Journal of the Saint Petersburg State University, Number 3847, 14. April 2012)
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Vladimir Gennadievich Sprindzuk
Vladimir Gennadievich Sprindzuk (Russian Владимир Геннадьевич Спринджук, Belarusian Уладзімір Генадзевіч Спрынджук, 22 July 1936, Minsk – 26 July 1987) was a Soviet-Belarusian number theorist.
Education and career
Sprindzuk studied from 1954 at Belarusian State University and from 1959 at the University of Vilnius. There he received in 1963 his Ph.D. with Jonas Kubilius as primary advisor and Yuri Linnik as secondary advisor and with thesis entitled (in Russian) "Метрические теоремы о дыяфантавых приближение алгебраическими числами ограниченной степени" (Metric Theorems of Diophantine Approximations and Approximations by Algebraic Numbers of Bounded Degree).[1] In 1965 he received his Russian doctorate of sciences (Doctor Nauk) from the State University of Leningrad with thesis entitled (in Russian) "Проблема Малера в метрической теории чисел" (The Mahler Problem in the Metric Theory of Numbers). In 1969 he became a professor and head of the academic division of number theory at the Mathematical Institute of the National Academy of Sciences of Belarus in Minsk and lectured at the Belarusian State University in Minsk. He was a visiting professor at the University of Paris, at the Polish Academy of Sciences and at the Slovak Academy of Sciences.
Sprindzuk's research deals with Diophantine approximation, Diophantine equations and transcendental numbers. While a first year undergraduate student, he published his first paper, in which he solved a problem of Aleksandr Khinchin, and wrote to Khinchin about the solution. Another important influence was the Leningrad number theorist Yuri Linnik, who was Sprindzuk's advisor for his Russian doctorate of sciences. In 1965 Sprindzuk proved a conjecture of Mahler, that almost all real numbers are S-numbers of Type 1 — Mahler had previously proved that almost all real numbers are S-numbers.[2] Sprindzuk generalized an important theorem proved by Wolfgang M. Schmidt.[3]
In the late sixties V. Sprindzuk began studying the theory of transcendental numbers and Diophantine equations. In 1969-71 he investigated the arithmetic properties of the Siegel hypergeometric E- functions with algebraic parameters and defined a wider class of E*-functions. His detailed studies of the Thue equation in algebraic number fields proved to be useful for the effective solution of a wide class of Diophantine equations and allowed him to study the possibility of effective approximations to algebraic numbers both in archimedean and non-archimedean domains. Sprindzuk's results are based on the connections between linear forms of logarithms in different norms. He observed that if a linear form is p-adically "not too small" then it cannot be too small in any other norm, be it archimedean or non-archimedean. A quantitative variant of this criterion led Sprindzuk to several effective results concerning the representation of numbers by binary forms, estimates for the magnitude of maximal prime factor of a binary form and the rational approximations to algebraic integers. He discovered in particular, a relation between the magnitude of the solutions of Diophantine equations and the number of classes of ideals, as well as some constructions of algebraic fields with the large class number.[4]
He was elected in 1969 a corresponding member and in 1986 a full member of the National Academy of Sciences of Belarus. Beginning in 1970 he was on the editorial staff of Acta Arithmetica. In 1970 he was an Invited Speaker at the ICM in Nice with talk New applications of analytic and p-adic methods in diophantine approximations.[5]
The theory of transcendental numbers, initiated by Liouville in 1844, has been enriched greatly in recent years. Among the relevant profound contributions are those of A. Baker, W. M. Schmidt and V. G. Sprindzuk.[6]
Selected publications
Articles
• "Achievements and problems in the theory of Diophantine approximations". Russian Math. Surveys. 35 (4): 1–80. 1980. doi:10.1070/RM1980v035n04ABEH001861.
Books
• Mahler’s Problem in metric number theory. American Mathematical Society 1969 (translation from Russian original, Minsk 1967)
• Metric theory of Diophantine approximations. Winston and Sons, Washington D.C. 1979 (translation from Russian original, published Nauka, Moscow 1977)
• Classical Diophantine Equations. Springer, Lecture Notes in Mathematics vol. 1559, 1993 (translation from Russian original, Moscow 1982)[7]
References
1. Vladimir Genadjevich Sprindzuk at the Mathematics Genealogy Project
2. Bugeaud, Yann (2004). "3.1 Mahler's Classification". Approximation by Algebraic Numbers. Cambridge University Press. p. 43. ISBN 9781139455671.
3. Schmidt, W. M. (1996) [1980]. Diophantine Approximations. Springer. p. 62. ISBN 9783540097624.
4. Obituary from numbertheory.org
5. "New applications of analytic and p-adic methods in diophantine approximations" (PDF). Actes, Congrès intern. Math. Vol. Tome 1. 1970. pp. 505–509.
6. Turán, Paul (1970). "The work of Alan Baker". Actes, Congrès intern. Math. Vol. Tome 1. pp. 3–5. ISBN 9789810231170.
7. Sprindžuk, Vladimir G. Classical diophantine equations. 1993.
External links
• Sprinzduk's publication list from numbertheory.org
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Vladimir Drinfeld
Vladimir Gershonovich Drinfeld (Ukrainian: Володи́мир Ге́ршонович Дрінфельд; Russian: Влади́мир Ге́ршонович Дри́нфельд; born February 14, 1954), surname also romanized as Drinfel'd, is a renowned mathematician from the former USSR, who emigrated to the United States and is currently working at the University of Chicago.
Vladimir Drinfeld
Born (1954-02-14) February 14, 1954
Kharkov, Ukrainian SSR, Soviet Union
Alma materMoscow State University
Known forDrinfeld center
Drinfeld double
Drinfeld level structure
Drinfeld module
Drinfeld reciprocity
Drinfeld upper half plane
Drinfeld twist
Drinfeld–Sokolov reduction
Drinfeld–Sokolov–Wilson equation
ADHM construction
Manin–Drinfeld theorem
Yetter–Drinfeld category
Chiral algebra
Chiral homology
Quantum groups
Geometric Langlands correspondence
Grothendieck–Teichmüller group
Lie-* algebra
Opers
Quantum affine algebra
Quantized enveloping algebra
Quasi-bialgebra
Quasi-triangular quasi-Hopf algebra
Ruziewicz problem
Tate modules
AwardsFields Medal (1990)
Wolf Prize (2018)
Shaw Prize (2023)
Scientific career
FieldsMathematics
InstitutionsUniversity of Chicago
Doctoral advisorYuri Manin
Drinfeld's work connected algebraic geometry over finite fields with number theory, especially the theory of automorphic forms, through the notions of elliptic module and the theory of the geometric Langlands correspondence. Drinfeld introduced the notion of a quantum group (independently discovered by Michio Jimbo at the same time) and made important contributions to mathematical physics, including the ADHM construction of instantons, algebraic formalism of the quantum inverse scattering method, and the Drinfeld–Sokolov reduction in the theory of solitons.
He was awarded the Fields Medal in 1990.[1] In 2016, he was elected to the National Academy of Sciences.[2] In 2018 he received the Wolf Prize in Mathematics.[3] In 2023 he was awarded the Shaw Prize in Mathematical Sciences.[4]
Biography
Drinfeld was born into a Jewish[5] mathematical family, in Kharkiv, Ukrainian SSR, Soviet Union in 1954. In 1969, at the age of 15, Drinfeld represented the Soviet Union at the International Mathematics Olympiad in Bucharest, Romania, and won a gold medal with the full score of 40 points. He was, at the time, the youngest participant to achieve a perfect score, a record that has since been surpassed by only four others including Sergei Konyagin and Noam Elkies. Drinfeld entered Moscow State University in the same year and graduated from it in 1974. Drinfeld was awarded the Candidate of Sciences degree in 1978 and the Doctor of Sciences degree from the Steklov Institute of Mathematics in 1988. He was awarded the Fields Medal in 1990. From 1981 till 1999 he worked at the Verkin Institute for Low Temperature Physics and Engineering (Department of Mathematical Physics). Drinfeld moved to the United States in 1999 and has been working at the University of Chicago since January 1999.
Contributions to mathematics
In 1974, at the age of twenty, Drinfeld announced a proof of the Langlands conjectures for GL2 over a global field of positive characteristic. In the course of proving the conjectures, Drinfeld introduced a new class of objects that he called "elliptic modules" (now known as Drinfeld modules). Later, in 1983, Drinfeld published a short article that expanded the scope of the Langlands conjectures. The Langlands conjectures, when published in 1967, could be seen as a sort of non-abelian class field theory. It postulated the existence of a natural one-to-one correspondence between Galois representations and some automorphic forms. The "naturalness" is guaranteed by the essential coincidence of L-functions. However, this condition is purely arithmetic and cannot be considered for a general one-dimensional function field in a straightforward way. Drinfeld pointed out that instead of automorphic forms one can consider automorphic perverse sheaves or automorphic D-modules. "Automorphicity" of these modules and the Langlands correspondence could be then understood in terms of the action of Hecke operators.
Drinfeld has also worked in mathematical physics. In collaboration with his advisor Yuri Manin, he constructed the moduli space of Yang–Mills instantons, a result that was proved independently by Michael Atiyah and Nigel Hitchin. Drinfeld coined the term "quantum group" in reference to Hopf algebras that are deformations of simple Lie algebras, and connected them to the study of the Yang–Baxter equation, which is a necessary condition for the solvability of statistical mechanical models. He also generalized Hopf algebras to quasi-Hopf algebras and introduced the study of Drinfeld twists, which can be used to factorize the R-matrix corresponding to the solution of the Yang–Baxter equation associated with a quasitriangular Hopf algebra.
Drinfeld has also collaborated with Alexander Beilinson to rebuild the theory of vertex algebras in a coordinate-free form, which have become increasingly important to two-dimensional conformal field theory, string theory, and the geometric Langlands program. Drinfeld and Beilinson published their work in 2004 in a book titled "Chiral Algebras."[6]
See also
• Drinfeld reciprocity
• Drinfeld upper half plane
• Manin–Drinfeld theorem
• Quantum group
• Chiral algebra
• Quasitriangular Hopf algebra
• Ruziewicz problem
Notes
1. O'Connor, J. J.; Robertson, E. F. "Vladimir Gershonovich Drinfeld". Biographies. School of Mathematics and Statistics University of St Andrews, Scotland. Retrieved 21 May 2012.
2. National Academy of Sciences Members and Foreign Associates Elected, News from the National Academy of Sciences, National Academy of Sciences, May 3, 2016, retrieved 2016-05-14.
3. Jerusalem Post - Wolf Prizes 2018
4. Shaw Prize 2023
5. Vladimir Gershonovich Drinfeld
6. Beilinson, Alexander; Drinfeld, Vladimir (2004). Chiral Algebras. Providence, R.I.: American Mathematical Society. ISBN 0-8218-3528-9. OCLC 53896661.
References
• O'Connor, John J.; Robertson, Edmund F., "Vladimir Drinfeld", MacTutor History of Mathematics Archive, University of St Andrews
• Victor Ginzburg, Preface to the special volume of Transformation Groups (vol 10, 3–4, December 2005, Birkhäuser) on occasion of Vladimir Drinfeld's 50th birthday, pp 277–278, doi:10.1007/s00031-005-0400-6
• Report by Manin
External links
• Vladimir Drinfeld at the Mathematics Genealogy Project
• Vladimir Drinfeld's results at International Mathematical Olympiad
• Langlands Seminar homepage
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• Enrico Costa and Gerald Fishman (2011)
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• Daniel Eisenstein, Shaun Cole and John A. Peacock (2014)
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and medicine
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Vladimir Arnold
Vladimir Igorevich Arnold (alternative spelling Arnol'd, Russian: Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010)[3][4][1] was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, he made revolutionary and deep contributions in several areas including geometrical theory of dynamical systems theory, algebra, catastrophe theory, topology, algebraic geometry, symplectic geometry, symplectic topology, differential equations, classical mechanics, differential geometric approach to hydrodynamics, geometric analysis and singularity theory, including posing the ADE classification problem, since his first main result—the solution of Hilbert's thirteenth problem in 1957 at the age of 19. He co-founded two new branches of mathematics—KAM theory, and topological Galois theory (this, with his student Askold Khovanskii). He is widely regarded as one of the greatest mathematicians of all time.
Vladimir Arnold
Vladimir Arnold in 2008
Born(1937-06-12)12 June 1937
Odesa, Ukrainian SSR, Soviet Union
Died3 June 2010(2010-06-03) (aged 72)
Paris, France
NationalitySoviet Union, Russian
Alma materMoscow State University
Known forADE classification
Arnold's cat map
Arnold conjecture
Arnold diffusion
Arnold's rouble problem
Arnold's spectral sequence
Arnold tongue
ABC flow
Arnold–Givental conjecture
Gömböc
Gudkov's conjecture
Hilbert's thirteenth problem
KAM theorem
Kolmogorov–Arnold theorem
Liouville–Arnold theorem
Topological Galois theory
Mathematical Methods of Classical Mechanics
AwardsShaw Prize (2008)
State Prize of the Russian Federation (2007)
Wolf Prize (2001)
Dannie Heineman Prize for Mathematical Physics (2001)
Harvey Prize (1994)
RAS Lobachevsky Prize (1992)
Crafoord Prize (1982)
Lenin Prize (1965)
Scientific career
FieldsMathematics
InstitutionsParis Dauphine University
Steklov Institute of Mathematics
Independent University of Moscow
Moscow State University
Doctoral advisorAndrey Kolmogorov
Doctoral students
• Alexander Givental
• Victor Goryunov
• Sabir Gusein-Zade
• Emil Horozov
• Boris Khesin[1]
• Askold Khovanskii
• Nikolay Nekhoroshev
• Boris Shapiro
• Alexander Varchenko
• Victor Vassiliev
• Vladimir Zakalyukin[2]
Arnold was also known as a popularizer of mathematics. Through his lectures, seminars, and as the author of several textbooks (such as the famous Mathematical Methods of Classical Mechanics) and popular mathematics books, he influenced many mathematicians and physicists.[5][6] Many of his books were translated into English. His views on education were particularly opposed to those of Bourbaki.
Biography
Vladimir Igorevich Arnold was born on 12 June 1937 in Odesa, Soviet Union (now Odesa, Ukraine). His father was Igor Vladimirovich Arnold (1900–1948), a mathematician. His mother was Nina Alexandrovna Arnold (1909–1986, née Isakovich), a Jewish art historian.[4] While a school student, Arnold once asked his father on the reason why the multiplication of two negative numbers yielded a positive number, and his father provided an answer involving the field properties of real numbers and the preservation of the distributive property. Arnold was deeply disappointed with this answer, and developed an aversion to the axiomatic method that lasted through his life.[7] When Arnold was thirteen, his uncle Nikolai B. Zhitkov,[8] who was an engineer, told him about calculus and how it could be used to understand some physical phenomena, this contributed to spark his interest for mathematics, and he started to study by himself the mathematical books his father had left to him, which included some works of Leonhard Euler and Charles Hermite.[9]
While a student of Andrey Kolmogorov at Moscow State University and still a teenager, Arnold showed in 1957 that any continuous function of several variables can be constructed with a finite number of two-variable functions, thereby solving Hilbert's thirteenth problem.[10] This is the Kolmogorov–Arnold representation theorem.
After graduating from Moscow State University in 1959, he worked there until 1986 (a professor since 1965), and then at Steklov Mathematical Institute.
He became an academician of the Academy of Sciences of the Soviet Union (Russian Academy of Science since 1991) in 1990.[11] Arnold can be said to have initiated the theory of symplectic topology as a distinct discipline. The Arnold conjecture on the number of fixed points of Hamiltonian symplectomorphisms and Lagrangian intersections were also a major motivation in the development of Floer homology.
In 1999 he suffered a serious bike accident in Paris, resulting in traumatic brain injury, and though he regained consciousness after a few weeks, he had amnesia and for some time could not even recognize his own wife at the hospital,[12] but he went on to make a good recovery.[13]
Arnold worked at the Steklov Mathematical Institute in Moscow and at Paris Dauphine University up until his death. As of 2006 he was reported to have the highest citation index among Russian scientists,[14] and h-index of 40. His students include Alexander Givental, Victor Goryunov, Sabir Gusein-Zade, Emil Horozov, Boris Khesin, Askold Khovanskii, Nikolay Nekhoroshev, Boris Shapiro, Alexander Varchenko, Victor Vassiliev and Vladimir Zakalyukin.[2]
To his students and colleagues Arnold was known also for his sense of humour. For example, once at his seminar in Moscow, at the beginning of the school year, when he usually was formulating new problems, he said:
There is a general principle that a stupid man can ask such questions to which one hundred wise men would not be able to answer. In accordance with this principle I shall formulate some problems.[15]
Death
Arnold died of acute pancreatitis[16] on 3 June 2010 in Paris, nine days before his 73rd birthday.[17] He was buried on 15 June in Moscow, at the Novodevichy Monastery.[18]
In a telegram to Arnold's family, Russian President Dmitry Medvedev stated:
The death of Vladimir Arnold, one of the greatest mathematicians of our time, is an irretrievable loss for world science. It is difficult to overestimate the contribution made by academician Arnold to modern mathematics and the prestige of Russian science.
Teaching had a special place in Vladimir Arnold's life and he had great influence as an enlightened mentor who taught several generations of talented scientists.
The memory of Vladimir Arnold will forever remain in the hearts of his colleagues, friends and students, as well as everyone who knew and admired this brilliant man.[19]
Popular mathematical writings
Arnold is well known for his lucid writing style, combining mathematical rigour with physical intuition, and an easy conversational style of teaching and education. His writings present a fresh, often geometric approach to traditional mathematical topics like ordinary differential equations, and his many textbooks have proved influential in the development of new areas of mathematics. The standard criticism about Arnold's pedagogy is that his books "are beautiful treatments of their subjects that are appreciated by experts, but too many details are omitted for students to learn the mathematics required to prove the statements that he so effortlessly justifies." His defense was that his books are meant to teach the subject to "those who truly wish to understand it" (Chicone, 2007).[20]
Arnold was an outspoken critic of the trend towards high levels of abstraction in mathematics during the middle of the last century. He had very strong opinions on how this approach—which was most popularly implemented by the Bourbaki school in France—initially had a negative impact on French mathematical education, and then later on that of other countries as well.[21][22] Arnold was very interested in the history of mathematics.[23] In an interview,[22] he said he had learned much of what he knew about mathematics through the study of Felix Klein's book Development of Mathematics in the 19th Century —a book he often recommended to his students.[24] He studied the classics, most notably the works of Huygens, Newton and Poincaré,[25] and many times he reported to have found in their works ideas that had not been explored yet.[26]
Mathematical work
Arnold worked on dynamical systems theory, catastrophe theory, topology, algebraic geometry, symplectic geometry, differential equations, classical mechanics, hydrodynamics and singularity theory.[5] Michèle Audin described him as "a geometer in the widest possible sense of the word" and said that "he was very fast to make connections between different fields".[27]
Hilbert's thirteenth problem
The problem is the following question: can every continuous function of three variables be expressed as a composition of finitely many continuous functions of two variables? The affirmative answer to this general question was given in 1957 by Vladimir Arnold, then only nineteen years old and a student of Andrey Kolmogorov. Kolmogorov had shown in the previous year that any function of several variables can be constructed with a finite number of three-variable functions. Arnold then expanded on this work to show that only two-variable functions were in fact required, thus answering the Hilbert's question when posed for the class of continuous functions.[28]
Dynamical systems
See also: Kolmogorov–Arnold–Moser theorem and Arnold diffusion
Moser and Arnold expanded the ideas of Kolmogorov (who was inspired by questions of Poincaré) and gave rise to what is now known as Kolmogorov–Arnold–Moser theorem (or "KAM theory"), which concerns the persistence of some quasi-periodic motions (nearly integrable Hamiltonian systems) when they are perturbed. KAM theory shows that, despite the perturbations, such systems can be stable over an infinite period of time, and specifies what the conditions for this are.[29]
In 1964, Arnold introduced the Arnold web, the first example of a stochastic web.[30][31]
Singularity theory
In 1965, Arnold attended René Thom's seminar on catastrophe theory. He later said of it: "I am deeply indebted to Thom, whose singularity seminar at the Institut des Hautes Etudes Scientifiques, which I frequented throughout the year 1965, profoundly changed my mathematical universe."[32] After this event, singularity theory became one of the major interests of Arnold and his students.[33] Among his most famous results in this area is his classification of simple singularities, contained in his paper "Normal forms of functions near degenerate critical points, the Weyl groups of Ak,Dk,Ek and Lagrangian singularities".[34][35][36]
Fluid dynamics
In 1966, Arnold published "Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits", in which he presented a common geometric interpretation for both the Euler's equations for rotating rigid bodies and the Euler's equations of fluid dynamics, this effectively linked topics previously thought to be unrelated, and enabled mathematical solutions to many questions related to fluid flows and their turbulence.[37][38][39]
Real algebraic geometry
In the year 1971, Arnold published "On the arrangement of ovals of real plane algebraic curves, involutions of four-dimensional smooth manifolds, and the arithmetic of integral quadratic forms",[40] which gave new life to real algebraic geometry. In it, he made major advances in the direction of a solution to Gudkov's conjecture, by finding a connection between it and four-dimensional topology.[41] The conjecture was to be later fully solved by V. A. Rokhlin building on Arnold's work.[42][43]
Symplectic geometry
The Arnold conjecture, linking the number of fixed points of Hamiltonian symplectomorphisms and the topology of the subjacent manifolds, was the motivating source of many of the pioneer studies in symplectic topology.[44][45]
Topology
According to Victor Vassiliev, Arnold "worked comparatively little on topology for topology's sake." And he was rather motivated by problems on other areas of mathematics where topology could be of use. His contributions include the invention of a topological form of the Abel–Ruffini theorem and the initial development of some of the consequent ideas, a work which resulted in the creation of the field of topological Galois theory in the 1960s.[46][47]
Theory of plane curves
According to Marcel Berger, Arnold revolutionized plane curves theory.[48] Among his contributions are the Arnold invariants of plane curves.[49]
Other
Arnold conjectured the existence of the gömböc.[50]
Honours and awards
• Lenin Prize (1965, with Andrey Kolmogorov),[51] "for work on celestial mechanics."
• Crafoord Prize (1982, with Louis Nirenberg),[52] "for contributions to the theory of non-linear differential equations."
• Elected member of the United States National Academy of Sciences in 1983).[53]
• Foreign Honorary Member of the American Academy of Arts and Sciences (1987)[54]
• Elected a Foreign Member of the Royal Society (ForMemRS) of London in 1988.[1]
• Elected member of the American Philosophical Society in 1990.[55]
• Lobachevsky Prize of the Russian Academy of Sciences (1992)[56]
• Harvey Prize (1994), "for basic contribution to the stability theory of dynamical systems, his pioneering work on singularity theory and seminal contributions to analysis and geometry."
• Dannie Heineman Prize for Mathematical Physics (2001), "for his fundamental contributions to our understanding of dynamics and of singularities of maps with profound consequences for mechanics, astrophysics, statistical mechanics, hydrodynamics and optics."[57]
• Wolf Prize in Mathematics (2001), "for his deep and influential work in a multitude of areas of mathematics, including dynamical systems, differential equations, and singularity theory."[58]
• State Prize of the Russian Federation (2007),[59] "for outstanding success in mathematics."
• Shaw Prize in mathematical sciences (2008, with Ludwig Faddeev), "for their contributions to mathematical physics."
The minor planet 10031 Vladarnolda was named after him in 1981 by Lyudmila Georgievna Karachkina.[60]
The Arnold Mathematical Journal, published for the first time in 2015, is named after him.[61]
The Arnold Fellowships, of the London Institute are named after him.[62][63]
He was a plenary speaker at both the 1974 and 1983 International Congress of Mathematicians in Vancouver and Warsaw, respectively.[64]
Fields Medal omission
Even though Arnold was nominated for the 1974 Fields Medal, which was then viewed as the highest honour a mathematician could receive, interference from the Soviet government led to it being withdrawn. Arnold's public opposition to the persecution of dissidents had led him into direct conflict with influential Soviet officials, and he suffered persecution himself, including not being allowed to leave the Soviet Union during most of the 1970s and 1980s.[65][66]
Selected bibliography
• 1966: Arnold, Vladimir (1966). "Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits" (PDF). Annales de l'Institut Fourier. 16 (1): 319–361. doi:10.5802/aif.233.
• 1978: Ordinary Differential Equations, The MIT Press ISBN 0-262-51018-9.[67][68][69]
• 1985: Arnold, V. I.; Gusein-Zade, S. M.; Varchenko, A. N. (1985). Singularities of Differentiable Maps, Volume I: The Classification of Critical Points Caustics and Wave Fronts. Monographs in Mathematics. Vol. 82. Birkhäuser. doi:10.1007/978-1-4612-5154-5. ISBN 978-1-4612-9589-1.
• 1988: Arnold, V. I.; Gusein-Zade, S. M.; Varchenko, A. N. (1988). Arnold, V. I; Gusein-Zade, S. M; Varchenko, A. N (eds.). Singularities of Differentiable Maps, Volume II: Monodromy and Asymptotics of Integrals. Monographs in Mathematics. Vol. 83. Birkhäuser. doi:10.1007/978-1-4612-3940-6. ISBN 978-1-4612-8408-6. S2CID 131768406.
• 1988: Arnold, V.I. (1988). Geometrical Methods in the Theory of Ordinary Differential Equations. Grundlehren der mathematischen Wissenschaften. Vol. 250 (2nd ed.). Springer. doi:10.1007/978-1-4612-1037-5. ISBN 978-1-4612-6994-6.
• 1989: Arnold, V.I. (1989). Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics. Vol. 60 (2nd ed.). Springer. doi:10.1007/978-1-4757-2063-1. ISBN 978-1-4419-3087-3.[70][71]
• 1989 Арнольд, В. И. (1989). Гюйгенс и Барроу, Ньютон и Гук - Первые шаги математического анализа и теории катастроф. М.: Наука. p. 98. ISBN 5-02-013935-1.
• 1989: (with A. Avez) Ergodic Problems of Classical Mechanics, Addison-Wesley ISBN 0-201-09406-1.
• 1990: Huygens and Barrow, Newton and Hooke: Pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals, Eric J.F. Primrose translator, Birkhäuser Verlag (1990) ISBN 3-7643-2383-3.[72][73][74]
• 1991: Arnolʹd, Vladimir Igorevich (1991). The Theory of Singularities and Its Applications. Cambridge University Press. ISBN 9780521422802.
• 1995:Topological Invariants of Plane Curves and Caustics,[75] American Mathematical Society (1994) ISBN 978-0-8218-0308-0
• 1998: "On the teaching of mathematics" (Russian) Uspekhi Mat. Nauk 53 (1998), no. 1(319), 229–234; translation in Russian Math. Surveys 53(1): 229–236.
• 1999: (with Valentin Afraimovich) Bifurcation Theory And Catastrophe Theory Springer ISBN 3-540-65379-1
• 2001: "Tsepniye Drobi" (Continued Fractions, in Russian), Moscow (2001).
• 2004: Teoriya Katastrof (Catastrophe Theory,[76] in Russian), 4th ed. Moscow, Editorial-URSS (2004), ISBN 5-354-00674-0.
• 2004: Vladimir I. Arnold, ed. (15 November 2004). Arnold's Problems (2nd ed.). Springer-Verlag. ISBN 978-3-540-20748-1.
• 2004: Arnold, Vladimir I. (2004). Lectures on Partial Differential Equations. Universitext. Springer. doi:10.1007/978-3-662-05441-3. ISBN 978-3-540-40448-4.[77][78]
• 2007: Yesterday and Long Ago, Springer (2007), ISBN 978-3-540-28734-6.
• 2013: Arnold, Vladimir I. (2013). Itenberg, Ilia; Kharlamov, Viatcheslav; Shustin, Eugenii I. (eds.). Real Algebraic Geometry. Unitext. Vol. 66. Springer. doi:10.1007/978-3-642-36243-9. ISBN 978-3-642-36242-2.[79]
• 2014: V. I. Arnold (2014). Mathematical Understanding of Nature: Essays on Amazing Physical Phenomena and Their Understanding by Mathematicians. American Mathematical Society. ISBN 978-1-4704-1701-7.
• 2015: Experimental Mathematics. American Mathematical Society (translated from Russian, 2015).
• 2015: Lectures and Problems: A Gift to Young Mathematicians, American Math Society, (translated from Russian, 2015)
Collected works
• 2010: A. B. Givental; B. A. Khesin; J. E. Marsden; A. N. Varchenko; V. A. Vassilev; O. Ya. Viro; V. M. Zakalyukin (editors). Collected Works, Volume I: Representations of Functions, Celestial Mechanics, and KAM Theory (1957–1965). Springer
• 2013: A. B. Givental; B. A. Khesin; A. N. Varchenko; V. A. Vassilev; O. Ya. Viro; (editors). Collected Works, Volume II: Hydrodynamics, Bifurcation Theory, and Algebraic Geometry (1965–1972). Springer.
• 2016: Givental, A.B., Khesin, B., Sevryuk, M.B., Vassiliev, V.A., Viro, O.Y. (Eds.). Collected Works, Volume III: Singularity Theory 1972–1979. Springer.
• 2018: Givental, A.B., Khesin, B., Sevryuk, M.B., Vassiliev, V.A., Viro, O.Y. (Eds.). Collected Works, Volume IV: Singularities in Symplectic and Contact Geometry 1980–1985. Springer.
• 2022 (To be published, September 2022): Alexander B. Givental, Boris A. Khesin, Mikhail B. Sevryuk, Victor A. Vassiliev, Oleg Ya. Viro (Eds.). Collected Works, Volume VI: Dynamics, Combinatorics, and Invariants of Knots, Curves, and Wave Fronts 1992–1995. Springer.
See also
• List of things named after Vladimir Arnold
• Independent University of Moscow
• Geometric mechanics
References
1. Khesin, Boris; Tabachnikov, Sergei (2018). "Vladimir Igorevich Arnold. 12 June 1937 – 3 June 2010". Biographical Memoirs of Fellows of the Royal Society. 64: 7–26. doi:10.1098/rsbm.2017.0016. ISSN 0080-4606.
2. Vladimir Arnold at the Mathematics Genealogy Project
3. Mort d'un grand mathématicien russe, AFP (Le Figaro)
4. Gusein-Zade, Sabir M.; Varchenko, Alexander N (December 2010), "Obituary: Vladimir Arnold (12 June 1937 – 3 June 2010)" (PDF), Newsletter of the European Mathematical Society, 78: 28–29
5. O'Connor, John J.; Robertson, Edmund F., "Vladimir Arnold", MacTutor History of Mathematics Archive, University of St Andrews
6. Bartocci, Claudio; Betti, Renato; Guerraggio, Angelo; Lucchetti, Roberto; Williams, Kim (2010). Mathematical Lives: Protagonists of the Twentieth Century From Hilbert to Wiles. Springer. p. 211. ISBN 9783642136061.
7. Vladimir I. Arnold (2007). Yesterday and Long Ago. Springer. pp. 19–26. ISBN 978-3-540-28734-6.
8. Swimming Against the Tide, p. 3
9. Табачников, С. Л. . "Интервью с В.И.Арнольдом", Квант, 1990, Nº 7, pp. 2–7. (in Russian)
10. Daniel Robertz (13 October 2014). Formal Algorithmic Elimination for PDEs. Springer. p. 192. ISBN 978-3-319-11445-3.
11. Great Russian Encyclopedia (2005), Moscow: Bol'shaya Rossiyskaya Enciklopediya Publisher, vol. 2.
12. Arnold: Yesterday and Long Ago (2010)
13. Polterovich and Scherbak (2011)
14. List of Russian Scientists with High Citation Index
15. "Vladimir Arnold". The Daily Telegraph. London. 12 July 2010.
16. Kenneth Chang (11 June 2010). "Vladimir Arnold Dies at 72; Pioneering Mathematician". The New York Times. Retrieved 12 June 2013.
17. "Number's up as top mathematician Vladimir Arnold dies". Herald Sun. 4 June 2010. Retrieved 6 June 2010.
18. "From V. I. Arnold's web page". Retrieved 12 June 2013.
19. "Condolences to the family of Vladimir Arnold". Presidential Press and Information Office. 15 June 2010. Retrieved 1 September 2011.
20. Carmen Chicone (2007), Book review of "Ordinary Differential Equations", by Vladimir I. Arnold. Springer-Verlag, Berlin, 2006. SIAM Review 49(2):335–336. (Chicone mentions the criticism but does not agree with it.)
21. See and other essays in .
22. An Interview with Vladimir Arnol'd, by S. H. Lui, AMS Notices, 1991.
23. Oleg Karpenkov. "Vladimir Igorevich Arnold"
24. B. Khesin and S. Tabachnikov, Tribute to Vladimir Arnold, Notices of the AMS, 59:3 (2012) 378–399.
25. Goryunov, V.; Zakalyukin, V. (2011), "Vladimir I. Arnold", Moscow Mathematical Journal, 11 (3).
26. See for example: Arnold, V. I.; Vasilev, V. A. (1989), "Newton's Principia read 300 years later" and Arnold, V. I. (2006); "Forgotten and neglected theories of Poincaré".
27. "Vladimir Igorevich Arnold and the Invention of Symplectic Topology", chapter I in the book Contact and Symplectic Topology (editors: Frédéric Bourgeois, Vincent Colin, András Stipsicz)
28. Ornes, Stephen (14 January 2021). "Mathematicians Resurrect Hilbert's 13th Problem". Quanta Magazine.
29. Szpiro, George G. (29 July 2008). Poincare's Prize: The Hundred-Year Quest to Solve One of Math's Greatest Puzzles. Penguin. ISBN 9781440634284.
30. Phase Space Crystals, by Lingzhen Guo https://iopscience.iop.org/book/978-0-7503-3563-8.pdf
31. Zaslavsky web map, by George Zaslavsky http://www.scholarpedia.org/article/Zaslavsky_web_map
32. "Archived copy" (PDF). Archived from the original (PDF) on 14 July 2015. Retrieved 22 February 2015.{{cite web}}: CS1 maint: archived copy as title (link)
33. "Resonance – Journal of Science Education | Indian Academy of Sciences" (PDF).
34. Note: It also appears in another article by him, but in English: Local Normal Forms of Functions, http://www.maths.ed.ac.uk/~aar/papers/arnold15.pdf
35. Dirk Siersma; Charles Wall; V. Zakalyukin (30 June 2001). New Developments in Singularity Theory. Springer Science & Business Media. p. 29. ISBN 978-0-7923-6996-7.
36. Landsberg, J. M.; Manivel, L. (2002). "Representation theory and projective geometry". arXiv:math/0203260.
37. Terence Tao (22 March 2013). Compactness and Contradiction. American Mathematical Soc. pp. 205–206. ISBN 978-0-8218-9492-7.
38. MacKay, Robert Sinclair; Stewart, Ian (19 August 2010). "VI Arnold obituary". The Guardian.
39. IAMP News Bulletin, July 2010, pp. 25–26
40. Note: The paper also appears with other names, as in http://perso.univ-rennes1.fr/marie-francoise.roy/cirm07/arnold.pdf
41. A. G. Khovanskii; Aleksandr Nikolaevich Varchenko; V. A. Vasiliev (1997). Topics in Singularity Theory: V. I. Arnold's 60th Anniversary Collection (preface). American Mathematical Soc. p. 10. ISBN 978-0-8218-0807-8.
42. Khesin, Boris A.; Tabachnikov, Serge L. (10 September 2014). Arnold: Swimming Against the Tide. p. 159. ISBN 9781470416997.
43. Degtyarev, A. I.; Kharlamov, V. M. (2000). "Topological properties of real algebraic varieties: Du coté de chez Rokhlin". Russian Mathematical Surveys. 55 (4): 735–814. arXiv:math/0004134. Bibcode:2000RuMaS..55..735D. doi:10.1070/RM2000v055n04ABEH000315. S2CID 250775854.
44. "Arnold and Symplectic Geometry", by Helmut Hofer
45. "Vladimir Igorevich Arnold and the invention of symplectic topology", by Michèle Audin https://web.archive.org/web/20160303175152/http://www-irma.u-strasbg.fr/~maudin/Arnold.pdf
46. "Topology in Arnold's work", by Victor Vassiliev
47. http://www.ams.org/journals/bull/2008-45-02/S0273-0979-07-01165-2/S0273-0979-07-01165-2.pdf Bulletin (New Series) of The American Mathematical Society Volume 45, Number 2, April 2008, pp. 329–334
48. A Panoramic View of Riemannian Geometry, by Marcel Berger, pp.24-25
49. Extrema of Arnold's invariants of curves on surfaces, by Vladimir Chernov https://math.dartmouth.edu/~chernov-china/
50. Mackenzie, Dana (29 December 2010). What's Happening in the Mathematical Sciences. American Mathematical Soc. p. 104. ISBN 9780821849996.
51. O. Karpenkov, "Vladimir Igorevich Arnold", Internat. Math. Nachrichten, no. 214, pp. 49–57, 2010. (link to arXiv preprint)
52. Harold M. Schmeck Jr. (27 June 1982). "American and Russian Share Prize in Mathematics". The New York Times.
53. "Vladimir I. Arnold". www.nasonline.org. Retrieved 14 April 2022.
54. "Book of Members, 1780–2010: Chapter A" (PDF). American Academy of Arts and Sciences. Retrieved 25 April 2011.
55. "APS Member History". search.amphilsoc.org. Retrieved 14 April 2022.
56. D. B. Anosov, A. A. Bolibrukh, Lyudvig D. Faddeev, A. A. Gonchar, M. L. Gromov, S. M. Gusein-Zade, Yu. S. Il'yashenko, B. A. Khesin, A. G. Khovanskii, M. L. Kontsevich, V. V. Kozlov, Yu. I. Manin, A. I. Neishtadt, S. P. Novikov, Yu. S. Osipov, M. B. Sevryuk, Yakov G. Sinai, A. N. Tyurin, A. N. Varchenko, V. A. Vasil'ev, V. M. Vershik and V. M. Zakalyukin (1997) . "Vladimir Igorevich Arnol'd (on his sixtieth birthday)". Russian Mathematical Surveys, Volume 52, Number 5. (translated from the Russian by R. F. Wheeler)
57. American Physical Society – 2001 Dannie Heineman Prize for Mathematical Physics Recipient
58. The Wolf Foundation – Vladimir I. Arnold Winner of Wolf Prize in Mathematics
59. Названы лауреаты Государственной премии РФ Kommersant 20 May 2008.
60. Lutz D. Schmadel (10 June 2012). Dictionary of Minor Planet Names. Springer Science & Business Media. p. 717. ISBN 978-3-642-29718-2.
61. Editorial (2015), "Journal Description Arnold Mathematical Journal", Arnold Mathematical Journal, 1 (1): 1–3, doi:10.1007/s40598-015-0006-6.
62. "Arnold Fellowships".
63. Fink, Thomas (July 2022). "Britain is rescuing academics from Vladimir Putin's clutches". The Telegraph.
64. "International Mathematical Union (IMU)". Archived from the original on 24 November 2017. Retrieved 22 May 2015.
65. Martin L. White (2015). "Vladimir Igorevich Arnold". Encyclopædia Britannica.
66. Thomas H. Maugh II (23 June 2010). "Vladimir Arnold, noted Russian mathematician, dies at 72". The Washington Post. Retrieved 18 March 2015.
67. Sacker, Robert J. (1 August 1975). "Ordinary Differential Equations". Technometrics. 17 (3): 388–389. doi:10.1080/00401706.1975.10489355. ISSN 0040-1706.
68. Kapadia, Devendra A. (March 1995). "Ordinary differential equations, by V. I. Arnold. Pp 334. DM 78. 1992. ISBN 3-540-54813-0 (Springer)". The Mathematical Gazette. 79 (484): 228–229. doi:10.2307/3620107. ISSN 0025-5572. JSTOR 3620107. S2CID 125723419.
69. Chicone, Carmen (2007). "Review of Ordinary Differential Equations". SIAM Review. 49 (2): 335–336. ISSN 0036-1445. JSTOR 20453964.
70. Review by Ian N. Sneddon (Bulletin of the American Mathematical Society, Vol. 2): http://www.ams.org/journals/bull/1980-02-02/S0273-0979-1980-14755-2/S0273-0979-1980-14755-2.pdf
71. Review by R. Broucke (Celestial Mechanics, Vol. 28): Bibcode:1982CeMec..28..345A.
72. Kazarinoff, N. (1 September 1991). "Huygens and Barrow, Newton and Hooke: Pioneers in Mathematical Analysis and Catastrophe Theory from Evolvents to Quasicrystals (V. I. Arnol'd)". SIAM Review. 33 (3): 493–495. doi:10.1137/1033119. ISSN 0036-1445.
73. Thiele, R. (1 January 1993). "Arnol'd, V. I., Huygens and Barrow, Newton and Hooke. Pioneers in Mathematical Analysis and Catastrophe Theory from Evolvents to Quasicrystals. Basel etc., Birkhäuser Verlag 1990. 118 pp., sfr 24.00. ISBN 3-7643-2383-3". Journal of Applied Mathematics and Mechanics. 73 (1): 34. Bibcode:1993ZaMM...73S..34T. doi:10.1002/zamm.19930730109. ISSN 1521-4001.
74. Heggie, Douglas C. (1 June 1991). "V. I. Arnol'd, Huygens and Barrow, Newton and Hooke, translated by E. J. F. Primrose (Birkhäuser Verlag, Basel 1990), 118 pp., 3 7643 2383 3, sFr 24". Proceedings of the Edinburgh Mathematical Society. Series 2. 34 (2): 335–336. doi:10.1017/S0013091500007240. ISSN 1464-3839.
75. Goryunov, V. V. (1 October 1996). "V. I. Arnold Topological invariants of plane curves and caustics (University Lecture Series, Vol. 5, American Mathematical Society, Providence, RI, 1995), 60pp., paperback, 0 8218 0308 5, £17.50". Proceedings of the Edinburgh Mathematical Society. Series 2. 39 (3): 590–591. doi:10.1017/S0013091500023348. ISSN 1464-3839.
76. Bernfeld, Stephen R. (1 January 1985). "Review of Catastrophe Theory". SIAM Review. 27 (1): 90–91. doi:10.1137/1027019. JSTOR 2031497.
77. Guenther, Ronald B.; Thomann, Enrique A. (2005). Renardy, Michael; Rogers, Robert C.; Arnold, Vladimir I. (eds.). "Featured Review: Two New Books on Partial Differential Equations". SIAM Review. 47 (1): 165–168. ISSN 0036-1445. JSTOR 20453608.
78. Groves, M. (2005). "Book Review: Vladimir I. Arnold, Lectures on Partial Differential Equations. Universitext". Journal of Applied Mathematics and Mechanics. 85 (4): 304. Bibcode:2005ZaMM...85..304G. doi:10.1002/zamm.200590023. ISSN 1521-4001.
79. Review by Fernando Q. Gouvêa of Real Algebraic Geometry by Arnold https://www.maa.org/press/maa-reviews/real-algebraic-geometry
Further reading
• Khesin, Boris; Tabachnikov, Serge (Coordinating Editors). "Tribute to Vladimir Arnold", Notices of the American Mathematical Society, March 2012, Volume 59, Number 3, pp. 378–399.
• Khesin, Boris; Tabachnikov, Serge (Coordinating Editors). "Memories of Vladimir Arnold", Notices of the American Mathematical Society, April 2012, Volume 59, Number 4, pp. 482–502.
• Boris A. Khesin; Serge L. Tabachnikov (2014). Arnold: Swimming Against the Tide. American Mathematical Society. ISBN 978-1-4704-1699-7.
• Leonid Polterovich; Inna Scherbak (7 September 2011). "V.I. Arnold (1937–2010)". Jahresbericht der Deutschen Mathematiker-Vereinigung. 113 (4): 185–219. doi:10.1365/s13291-011-0027-6. S2CID 122052411.
• "Features: "Knotted Vortex Lines and Vortex Tubes in Stationary Fluid Flows"; "On Delusive Nodal Sets of Free Oscillations"" (PDF). EMS Newsletter (96): 26–48. June 2015. ISSN 1027-488X.
External links
Wikimedia Commons has media related to Vladimir Arnold.
Wikiquote has quotations related to Vladimir Arnold.
• V. I. Arnold's web page
• Personal web page
• V. I. Arnold lecturing on Continued Fractions
• A short curriculum vitae
• On Teaching Mathematics, text of a talk espousing Arnold's opinions on mathematical instruction
• Topology of Plane Curves, Wave Fronts, Legendrian Knots, Sturm Theory and Flattenings of Projective Curves
• Problems from 5 to 15, a text by Arnold for school students, available at the IMAGINARY platform
• Vladimir Arnold at the Mathematics Genealogy Project
• S. Kutateladze, Arnold Is Gone
• В.Б.Демидовичем (2009), МЕХМАТЯНЕ ВСПОМИНАЮТ 2: В.И.Арнольд, pp. 25–58
• Author profile in the database zbMATH
Laureates of the Wolf Prize in Mathematics
1970s
• Israel Gelfand / Carl L. Siegel (1978)
• Jean Leray / André Weil (1979)
1980s
• Henri Cartan / Andrey Kolmogorov (1980)
• Lars Ahlfors / Oscar Zariski (1981)
• Hassler Whitney / Mark Krein (1982)
• Shiing-Shen Chern / Paul Erdős (1983/84)
• Kunihiko Kodaira / Hans Lewy (1984/85)
• Samuel Eilenberg / Atle Selberg (1986)
• Kiyosi Itô / Peter Lax (1987)
• Friedrich Hirzebruch / Lars Hörmander (1988)
• Alberto Calderón / John Milnor (1989)
1990s
• Ennio de Giorgi / Ilya Piatetski-Shapiro (1990)
• Lennart Carleson / John G. Thompson (1992)
• Mikhail Gromov / Jacques Tits (1993)
• Jürgen Moser (1994/95)
• Robert Langlands / Andrew Wiles (1995/96)
• Joseph Keller / Yakov G. Sinai (1996/97)
• László Lovász / Elias M. Stein (1999)
2000s
• Raoul Bott / Jean-Pierre Serre (2000)
• Vladimir Arnold / Saharon Shelah (2001)
• Mikio Sato / John Tate (2002/03)
• Grigory Margulis / Sergei Novikov (2005)
• Stephen Smale / Hillel Furstenberg (2006/07)
• Pierre Deligne / Phillip A. Griffiths / David B. Mumford (2008)
2010s
• Dennis Sullivan / Shing-Tung Yau (2010)
• Michael Aschbacher / Luis Caffarelli (2012)
• George Mostow / Michael Artin (2013)
• Peter Sarnak (2014)
• James G. Arthur (2015)
• Richard Schoen / Charles Fefferman (2017)
• Alexander Beilinson / Vladimir Drinfeld (2018)
• Jean-François Le Gall / Gregory Lawler (2019)
2020s
• Simon K. Donaldson / Yakov Eliashberg (2020)
• George Lusztig (2022)
• Ingrid Daubechies (2023)
Mathematics portal
Shaw Prize laureates
Astronomy
• Jim Peebles (2004)
• Geoffrey Marcy and Michel Mayor (2005)
• Saul Perlmutter, Adam Riess and Brian Schmidt (2006)
• Peter Goldreich (2007)
• Reinhard Genzel (2008)
• Frank Shu (2009)
• Charles Bennett, Lyman Page and David Spergel (2010)
• Enrico Costa and Gerald Fishman (2011)
• David C. Jewitt and Jane Luu (2012)
• Steven Balbus and John F. Hawley (2013)
• Daniel Eisenstein, Shaun Cole and John A. Peacock (2014)
• William J. Borucki (2015)
• Ronald Drever, Kip Thorne and Rainer Weiss (2016)
• Simon White (2017)
• Jean-Loup Puget (2018)
• Edward C. Stone (2019)
• Roger Blandford (2020)
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Vladimir Bogachev
Vladimir Igorevich Bogachev (Russian: Владимир Игоревич Богачёв; born in 1961) is an eminent Russian mathematician and Full Professor of the Department of Mechanics and Mathematics of the Lomonosov Moscow State University. He is an expert in measure theory, probability theory, infinite-dimensional analysis and partial differential equations arising in mathematical physics. [1][2] His research was distinguished by several awards including the medal and the prize of the Academy of Sciences of the Soviet Union (1990); Award of the Japan Society for the Promotion of Science (2000); the Doob Lecture of the Bernoulli Society (2017);[3] and the Kolmogorov Prize of the Russian Academy of Sciences (2018).[4]
Vladimir Bogachev is one of the most cited Russian mathematicians. He is the author of more than 200 publications and 12 monographs. His total citation index by MathSciNet is 2960, with h-index=23 (by September 2021)[5]
Biography
Bogachev graduated with honours from Moscow State University (1983). In 1986, he received his PhD (Candidate of Sciences in Russia) under the supervision of Prof. O. G. Smolyanov.[6]
Awards
• The medal and the prize of the Academy of Sciences of the Soviet Union (1990)
• Award of the Japan Society for the Promotion of Science (2000)
• The Doob Lecture of the Bernoulli Society (2017)
• The Kolmogorov Prize of the Russian Academy of Sciences (2018)
Scientific contributions
In 1984, V. Bogachev resolved three Aronszajn's problems on infinite-dimensional probability distributions and answered a famous question of I. M. Gelfand posed about 25 years before that. In 1992, Vladimir Bogachev proved T. Pitcher’s conjecture (stated in 1961) on the differentiability of the distributions of diffusion processes. In 1995, he proved (with Michael Röckner) the famous Shigekawa conjecture on the absolute continuity of invariant measures of diffusion processes. In 1999, in a joint work with Sergio Albeverio and Röckner, Professor Bogachev resolved the well-known problem of S. R. S. Varadhan on the uniqueness of stationary distributions, which had remained open for about 20 years.
A remarkable achievement of Vladimir Bogachev is the recently obtained (2021) answer to the question of Andrey Kolmogorov (posed in 1931) on the uniqueness of the solution to the Cauchy problem: it is shown that the Cauchy problem with a unit diffusion coefficient and locally bounded drift has a unique probabilistic solution on $\mathbb {R} ^{1}$, and in $\mathbb {R} ^{>1}$ this is not true even for smooth drift.[7]
Main Publications
Papers
• Bogachev V.I., Röckner M. Regularity of invariant measures on finite and infinite dimensional spaces and applications. J. Funct. Anal., V. 133, N 1, P. 168–223 (1995)
• Albeverio S., Bogachev V.I., Röckner M. On uniqueness of invariant measures for finite and infinite dimensional diffusions. Comm. Pure Appl. Math., V. 52, P. 325–362 (1999)
• Bogachev V.I., Krasovitskii T.I., Shaposhnikov S.V. On uniqueness of probability solutions of the Fokker–Planck–Kolmogorov equation, Sb. Math., V. 212, N 6, P. 745–781 (2021)
Books
• Bogachev V.I. Gaussian measures. American Mathematical Society, Rhode Island, 1998
• Bogachev V.I. Measure theory. V. 1,2. Springer-Verlag, Berlin, 2007
• Bogachev V.I. Weak convergence of measures, American Mathematical Society, Rhode Island, 2018.
References
1. Bogachev Vladimir: on his 60th birthday
2. Higher School of Economics
3. The Doob Lecture of the Bernoulli Society
4. the Kolmogorov Prize of the Russian Academy of Sciences
5. MathSciNet
6. Vladimir Bogachev at the Mathematics Genealogy Project
7. Bogachev Vladimir: on his 60th birthday
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Vladimir Kondrashov
Vladimir Iosifovich Kondrashov (Владимир Иосифович Кондрашов; 2 February 1909 – 26 February 1971) was a Soviet mathematician most well known for proving the Rellich–Kondrachov theorem that shows that the embedding of certain Sobolev spaces into Lp spaces is compact. His name has also been transliterated as V.I. Kondrachoff, W. Kondrachov, or V.I. Kondrašov.[1]
Life
Kondrashov was born on 2 February [O.S. 20 January] 1909 in Moscow.[2] He graduated with a PhD from Moscow State University under the supervision of Sergei Sobolev in 1941. As a postdoc at Steklov Institute, where he obtained a DSc in 1950, he organised the Moscow Seminar on the Theory of Functions of Several Variables.[3][4] For the final 20 years of his life until his death on 26 February 1971, he worked at the Moscow Engineering and Physics Institute.[2]
References
1. "MR: Kondrashov, Vladimir Iosifovich - 478194". mathscinet.ams.org. Archived from the original on 24 July 2019. Retrieved 26 July 2019.
2. Kudryavtsev, L D; Lizorkin, P I; Nikol'skii, Sergei M; Sobolev, S L (30 April 1972). "VLADIMIR IOSIFOVICH KONDRASHOV (obituary)". Russian Mathematical Surveys. 27 (2): 83–90. Bibcode:1972RuMaS..27...83K. doi:10.1070/rm1972v027n02abeh001372. ISSN 0036-0279. S2CID 250813276.
3. Pietsch, Albrecht, ed. (2007). History of Banach Spaces and Linear Operators. Boston, MA: Birkhäuser Boston. p. 606. doi:10.1007/978-0-8176-4596-0_8. ISBN 9780817645960.
4. "Steklov Mathematical Institute". www.mi-ras.ru. Retrieved 26 July 2019.
External links
• Vladimir Kondrashov at the Mathematics Genealogy Project
• Photo of Kondrashov
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Vladimir Kondratiev
Vladimir Aleksandrovich Kondratiev (Russian: Владимир Александрович Кондратьев; 2 July 1935 – 11 March 2010) was a Russian mathematician and professor.[1] He worked particularly in the field of ordinary differential equations and partial differential equations.[2]
Prizes
• USSR State Prize (1988)
• Petrovsky Prize of the Russian Academy of Sciences (1998)
• Lomonosov Prize of Moscow State University (2009)
Publications
Books
• Borsuk, Michail; Kondratiev, Vladimir (2006). Elliptic Boundary Value Problems of Second Order in Piecewise Smooth Domains. North-Holland Mathematical Library. Vol. 69. North Holland. ISBN 9780080461731.
Weblinks
• All-Russian Mathematical Portal entry
References
1. "Кондратьев Владимир Александрович". Moscow State University. Retrieved 8 November 2022.
2. "Kondratiev, Vladimir Aleksandrovich". Steklov Mathematical Institute RAS. Retrieved 5 August 2022.
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Vladimir Popov (mathematician)
Vladimir Leonidovich Popov (Russian: Влади́мир Леони́дович Попо́в; born 3 September 1946) is a Russian mathematician working in the invariant theory and the theory of transformation groups.[1]
Vladimir Leonidovich Popov
Born (1946-09-03) 3 September 1946
Moscow
NationalityRussian
Academic career
FieldMathematics
Information at IDEAS / RePEc
Education and career
In 1969 he graduated from the Faculty of Mechanics and Mathematics of Moscow State University. In 1972 he received his Candidate of Sciences degree (PhD) with thesis Стабильность действия алгебраических групп и арифметика квазиоднородных многообразий (Stability of the action of algebraic groups and the arithmetic of quasi-homogeneous varieties). In 1984 he received his Russian Doctor of Sciences degree (habilitation) with thesis Группы, образующие, сизигии и орбиты в теории инвариантов (Groups, generators, syzygies and orbits in the theory of invariants).[2]
He is a member of the Steklov Institute of Mathematics and a professor of the National Research University – Higher School of Economics.[1] In 1986, he was an invited speaker at the International Congress of Mathematicians (Berkeley, USA),[3] and in 2008–2010 he was a core member of the panel for Section 2, "Algebra" of the Program Committee for the 2010 International Congress of Mathematicians (Hyderabad, India).[4]
In 1987 he published a proof of a conjecture of Claudio Procesi and Hanspeter Kraft.[5] In 2006, with Nicole Lemire and Zinovy Reichstein, Popov published a solution to a problem posed by Domingo Luna in 1973.[6]
Awards
In 2012, he was elected a member of the inaugural class of Fellows of the American Mathematical Society[7] which recognizes mathematicians who have made significant contributions to the field.
In 2016, he was elected a corresponding member of the Russian Academy of Sciences.
Books
• Popov, Vladimir L. (1982). Discrete complex reflection groups. Utrecht: Communications of the Mathematical Institute Rijksuniversiteit Utrecht, Vol. 15.
• Popov, Vladimir L. (1992). Groups, generators, syzygies, and orbits in invariant theory. Providence RI: Translations of Mathematical Monographs, Vol. 100, Providence RI: Amer. Math. Soc. ISBN 0-8218-4557-8.[8]
• Popov, V. L.; Vinberg, E. B. (1994). "Invariant Theory". Algebraic Geometry IV. Encyclopaedia of Mathematical Sciences. Vol. 55. Berlin; Heidelberg: Springer. pp. 123–278. doi:10.1007/978-3-662-03073-8_2. ISBN 978-3-642-08119-4.
• Popov, Vladimir L. (2004). Algebraic transformation groups and algebraic varieties: proceedings of the conference Interesting algebraic varieties arising in algebraic transformation group theory held at the Erwin Schrödinger Institute, Vienna, October 22-26, 2001. Berlin New York: Springer. ISBN 9783540208389.
References
1. "Попов Владимир Леонидович". math-net.ru.
2. "Vladimir Popov". HSE University.
3. Popov, V. L. "Modern developments in invariant theory" (PDF). In: Proc. Intern. Congr. Math. Berkley, California. 1986. Vol. 1. pp. 394–406.
4. "Popov Vladimir Leonidovich". All-Russian Mathematical Portal. Retrieved 28 August 2016.
5. Popov, V L (1987). "Contraction of the actions of reductive algebraic groups". Mathematics of the USSR-Sbornik. 58 (2): 311–335. doi:10.1070/SM1987v058n02ABEH003106. ISSN 0025-5734.
6. Lemire, Nicole; Popov, Vladimir L.; Reichstein, Zinovy (2006). "Cayley groups". Journal of the American Mathematical Society. 19 (4): 921–967. doi:10.1090/S0894-0347-06-00522-4. S2CID 9987646.
7. List of Fellows of the American Mathematical Society, retrieved 16 November 2013.
8. Schwarz, Gerald W. (1993). "Book Review: Groups, generators, syzygies, and orbits in invariant theory". Bulletin of the American Mathematical Society. 29 (2): 299–305. doi:10.1090/S0273-0979-1993-00433-6.
External links
• Personal profile: Vladimir Leonidovich Popov Steklov Institute of Mathematics
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Vladimir Markov (mathematician)
Vladimir Andreyevich Markov (Russian: Влади́мир Андре́евич Ма́рков; May 8, 1871 – January 18, 1897) was a mathematician, known for proving the Markov brothers' inequality with his older brother Andrey Markov. He was from the Russian Empire. He died of tuberculosis at the age of 25.[1]
Vladimir Andreyevich Markov
Born(1871-05-08)8 May 1871
Died18 January 1897(1897-01-18) (aged 25)
NationalityRussian
Scientific career
FieldsMathematics
Doctoral advisorPafnuty Chebyshev
Notes
1. В Комиссии по истории физико-математических наук. Заседание памяти Владимира Андреевича Маркова. УМН (in Russian). 9 (4(62)): 256–258. 1954.
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Vladimir Entov
Vladimir Markovich Entov (January 8, 1937 – April 10, 2008) was an applied mathematician and physicist.
Vladimir Entov
Born
Vladimir Markovich Entov
(1937-01-08)January 8, 1937
Moscow, USSR
DiedApril 10, 2008(2008-04-10) (aged 71)
Rockville, Maryland, United States
Alma materGubkin Russian State University of Oil and Gas
Scientific career
FieldsApplied Mathematics, Physics
InstitutionsInstitute for Problems in Mechanics of the Russian Academy of Sciences, Gubkin Russian State University of Oil and Gas
Notes
List of publications
Biography
During his high school years, Entov won multiple awards at the all-Union Physics Olympiads. In 1954 he graduated from a high school with a Gold Medal (valedictorian). He applied to the Physics Department of the Moscow State University, but was rejected on ideological grounds. The same year he began his study at the Mechanical Department of the Moscow Institute for Oil and Gas.
In his second year, he approached Professor I.A. Charny and asked him for a research assignment. This was a beginning of their close personal and scientific friendship, which lasted until the death of Professor Charny.
Upon his graduation from the Moscow Institute for Oil and Gas in 1959, Entov began a long-distance course of study at the mathematical department of the Moscow State University, while working in the Institute For Drilling Technology.
In 1961 he began his graduate study with Professor I.A.Charny, and in 1965 he successfully defended his dissertation thesis “Non- stationary problems of the non-linear filtration”, earning the title of “Candidate of Science" (roughly equivalent to PhD).
Beginning in 1971 and until his death, Entov worked at the Institute for Problems in Mechanics of the Soviet Academy of Science. In 1972 he was awarded the title of Doctor of Sciences for his dissertation thesis "Hydrodynamic theory of filtration of anomalous fluids", completed at the Institute for Problems in Mechanics. In his years at the Institute, Vladimir supervised the theses of 24 Candidates of Science and six Doctors of Science. He worked as a scientific supervisor at the Laboratory of Applied Continuum Mechanics until the last years of his life.
In addition, since 1983 he served as a Professor of Applied Mathematics and Computer Modeling of the Moscow Institute of Oil and Gas (now known as the Gubkin Russian State University of Oil and Gas).
Since 1993, Vladimir was actively involved in research and teaching activity in France (Institut de Physique du Globe de Paris), the UK (University of Cambridge and University of Oxford), and the USA (WPI, University of Stanford, MIT, UMN).
Professor Entov was a Corresponding Member of The Russian Academy of the Natural Sciences, a member of the Russian National Committee for the Theoretical and Applied Mechanics, a member of the International Society for the Interaction of Mechanics and Mathematics, and a member of the editorial board of the European Journal of Applied Mathematics (Cambridge University Press).
Works published (partial list)
Works include:[1]
1. Теория нестационарной фильтрации жидкости и газа // М., "Недра", 1972 (совместно с Г.И.Баренблаттом, В.М.Рыжиком)
2. Гидродинамическая теория фильтрации аномальных жидкостей // М., "Наука", 1975 (совместно с М.Г.Бернадинером)
3. Движение жидкостей и газов в природных пластах // М., "Недра", 1984 (совместно с Г.И.Баренблаттом, В.М.Рыжиком)
4. Гидродинамика в бурении // М., "Недра", 1985 (совместно с А.Х. Мирзаджанзаде)
5. Качественные методы в механике сплошных сред // М., "Наука", 1989 (совместно с Р.В.Гольдштейном)
6. Гидродинамика процессов повышения нефтеотдачи // М., "Недра", 1989 (совместно с А.Ф.Зазовским)
7. Математическая теория целиков остаточной вязкопластичной нефти // Томск, Издательство Томского Университета, 1989 (совместно с В.Н.Панковым и С.В.Панько)
8. Fluids Flow through Natural Rocks // Dordrecht, "Kluwer Academic Publishers", 1990 (with G.I. Barenblatt, V.M.Ryzhik)
9. Qualitative Methods in Continuum Mechanics // New York, "Longman Scientific & Technical", 1994 (with R.V. Goldstein)
10. Mechanics of Continua and Its Application to Gas and Oil Productions // Moscow, "Moscow Nedra", 2008 (with E.V. Glivenko)
References
1. List of publications
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Vladimir Alekseev (mathematician)
Vladimir Mikhailovich Alekseev (Владимир Михайлович Алексеев, sometimes transliterated as "Alexeyev" or "Alexeev", 17 June 1932, Bykovo, Ramensky District, Moscow Oblast – 1 December 1980) was a Russian mathematician who specialized in celestial mechanics and dynamical systems.[1]
He attended secondary school in Moscow at one of the special schools of mathematics affiliated with Moscow State University and participated in several mathematical olympiads. From 1950 he studied at the Faculty of Mathematics and Mechanics at the Moscow State University, where he worked as a student of Andrei Kolmogorov on the asymptotic behavior in the three-body problem of celestial mechanics. Already as an undergraduate, Alekseev proved significant new results on quasi-random motion associated with the three-body problem. This was the subject of his dissertation for the Russian candidate degree (Ph.D.) and then his dissertation in 1969 for the Russian doctorate (higher doctoral degree). From 1957 he taught at Moscow State University.[1]
In 1970 Alekseev was an Invited Speaker with talk Sur l´allure finale du mouvement dans le problème de trois corps at the ICM in Nice.[2]
Over a 20-year period, he conducted 3 ongoing seminars: with Yakov Sinai on dynamical systeme, with V. A. Egorov on celestial mechanics, and with M. Zelikin and V. M. Tikhomirov on variational problems and optimal control.[1]
Selected publications
• Symbolic dynamics (Russian), Kiev 1976
• with V. M. Tikhomirov, S. Fomin: Optimal Control, New York: Consultants Bureau 1987 (trans. from the Russian by V. M. Volosov)
• "A theorem on an integral inequality and some of its applications" by V. M. Alekseev in Thirteen papers on dynamical systems by V. M. Alekseev & 14 other authors, American Mathematical Society 1981 doi:10.1090/trans2/089
• with E. M. Galeev, V. M. Tikhomirov: Recueil de problèmes d'optimisation (French), Moscow, MIR 1987
References
1. D. Anosov, V. Arnold, A. N. Kolmogorov, Y. Sinai et al., (Obituary in Russian) Mathematical Surveys, vol. 36, 1981, pp. 201–206, Russian on mathnet.ru
2. Alexeyev, V. M. "Sur l´allure finale du mouvement dans le problème de trois corps". Actes du Congrès international des mathématiciens (Nice, 1970). Vol. 2. pp. 893–907.
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Vladimir Miklyukov
Vladimir Michaelovich Miklyukov (Russian: Миклюков, Владимир Михайлович, also spelled Miklioukov or Mikljukov) (8 January 1944 – October 2013) was a Russian educator in mathematics, and head of the Superslow Process workgroup based at Volgograd State University.[3]
Vladimir Michaelovich Miklyukov
Miklyukov, 2007
Born8 January 1944
DiedOctober 2013 (aged 69)
NationalityRussian
Alma materDonetsk National University
Known forfounding Superslow Processes laboratory
AwardsDistinguished Scientist of Russian Federation[1]
Scientific career
FieldsMathematics
InstitutionsVolgograd State University
Brigham Young University
Doctoral advisorGeorgy Dmitrievich Suvorov[2]
Biography
In 1970, as a student of Georgy D. Suvorov at Donetsk National University, he defended his Ph.D. thesis Theory of Quasiconformal Mappings in Space.[2] In 1981 Miklyukov and his family moved to Volgograd. He was transferred to the newly built Volgograd State University where he became chairman of the Department of Mathematical Analysis and Theory of Functions.[4]
His scientific research focused on geometrical analysis. At the same time, he was studying zero mean curvature surfaces in Euclidean and pseudo-Euclidean spaces, nonlinear elliptic type partial differential equations and quasiregular mappings of Riemannian manifolds. The main results of that work were related to the following groups of questions:
• The external geometrical structure of zero mean curvature surfaces in Euclidean and pseudo-euclidean spaces; spacelike tubes and bands of zero mean curvature, their stability and instability with respect to small deformations, their life-time, branches, connections between branch points and Lorentz invariant characteristics of surfaces;
• Phragmén-Lindelöf type theorems for differential forms; Ahlfors type theorems for differential forms with finite or infinite number of different asymptotic tracts; generalizations of Wiman theorem of forms, applications to quasiregular mappings on manifolds; applications of isoperimetric methods to the Phragmén–Lindelöf principle for quasiregular mappings on manifolds.
From 1998-2000 Miklyukov was a visiting professor at Brigham Young University.[5] In 2004 he concentrated on studying of the mathematical theory of superslow processes and differential forms in micro- and nanoflows, and founded the Laboratory of Superslow Processes. In 2009 Miklyukov was named a Distinguished Scientist of Russian Federation.[1]
Publications
• Miklyukov, Vladimir (2008). Introduction to Nonsmooth Analysis (PDF) (in Russian) (2 ed.). Volgograd: VolSU. ISBN 978-5-9669-0457-9. Archived from the original (PDF) on 2011-07-22. Retrieved 2009-11-16.
• —— (2007). Geometrical Analysis. Differential Forms, Almost-solutions, Almost Quasiconformal Mappings (PDF) (in Russian). Volgograd: VolSU. ISBN 978-5-9669-0268-1.
• —— (2006). Introduction to Nonsmooth Analysis. Volgograd: VolSU. ISBN 5-9669-0209-7.
• —— (2005). Conformal Mapping of Irregular Surfaces and Its Application (PDF) (in Russian). Volgograd: VolSU. ISBN 5-9669-0071-X. Archived from the original (PDF) on 2011-07-22. Retrieved 2009-11-15.
• Miklyukov, Vladimir; Klyachin, Vladimir A. (2004). Tubes and Bands in Space-Time. Anniversary series: The Scientists of the Volga (in Russian). Volgograd: VolSU. ISBN 5-85534-971-3.
References
1. "Presidential Decree No.160" (in Russian). 14 February 2009. Archived from the original on 2 March 2012.
2. Vladimir Miklyukov at the Mathematics Genealogy Project
3. Announcements of the workgroup "Superslow Processes"
4. "Faculty of Mathematics (Past And Present)..." (in Russian). Department of Mathematics and Computer Science, Volgograd State University. Archived from the original on 2011-08-21.
5. Miklyukov, Vladimir; Vuorinen, Matti (September 1999). "Hardy's Inequality for W0 1,p-Functions on Riemannian Manifolds". Proceedings of the American Mathematical Society. American Mathematical Society. 127 (9): 2745–2754. doi:10.1090/S0002-9939-99-04849-2. JSTOR 119576. MR 1600117.
External links
• Official website of Milyukov (in Russian)
• Vladimir Miklyukov's obituary (in Russian)
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Vladimir Platonov
Vladimir Petrovich Platonov (Belarusian: Уладзімір Пятровіч Платонаў, Uladzimir Piatrovic Platonau; Russian: Влади́мир Петро́вич Плато́нов, Vladimir Platonov) (born December 1, 1939, Stayki village, Vitebsk Region, Belarusian SSR) is a Soviet, Belarusian and Russian mathematician. He is an expert in algebraic geometry and topology and member of the Russian Academy of Science.[1][2][3]
From 1992–2004 he worked at research centers in the United States, Canada and Germany.[1]
Education
In 1961 Platonov graduated with highest distinction from Belarus State University. In 1963 he received his Ph.D. from the Academy of Sciences of Belarus. In 1967, Platonov received his Doctor of Science degree from the Academy of Sciences of USSR.
Career
At age 28 Platonov received a title of full professor at Belarus State. This made him the youngest full professor in the nation's history. In 1972 he became an Academician of the National Academy of Sciences of Belarus and its President (1987–1993).[4] He has been an Academician of the Russian/USSR Academy of Sciences since 1987. He was the Director of the Institute of Mathematics of the Academy of Sciences of Belarus from 1977 to 1992.
Research
His interests are algebra, algebraic geometry and number theory. He solved the Strong approximation problem, developed the reduced K-theory and solved the Tannaka–Artin problem.[5] He solved the Kneser–Tits and Grothendieck problems. Together with F. Grunewald he solved the arithmeticity problem for finite extensions of arithmetic groups and the rigidity problem for arithmetic subgroups of algebraic groups with radical. Platonov solved the rationality problem for spinor varieties and the Dieudonne problem on spinor norms.
Platonov was an invited speaker of the International Congresses of Mathematicians in Vancouver (1974), Helsinki (1978) and the European Congress of Mathematicians in Budapest (1996).
He is a member of the Canadian Mathematical Society and from 1993 to 2001 was a Professor of the Faculty of Mathematics of the University of Waterloo in Waterloo, Ontario, Canada.
He is the author, with Andrei Rapinchuk, of Algebraic Groups and Number Theory.[6]
He currently works as a Chief Science Officer of Scientific Research Institute of System Development (NIISI RAN).[7]
Assault conviction
On November 9, 1999, Platonov appeared in court on a bail hearing on a charge of attempted murder for an attack on his wife.[8] He was convicted of assault.[9] The court gave him a conditional sentence of two years. In September 2001, Platonov took early retirement as a professor of the University of Waterloo.
Awards
• 1968: Lenin Komsomol Prize, for a series of works in topological group theory
• 1978: Lenin Prize in Science and Technology, for a fundamental series of works "Arithmetics of Algebraic Groups and Reduced K-Theory" ("Арифметика алгебраических групп и приведенная К-теория")
• 1993: Humboldt Prize
See also
• List of University of Waterloo people
References
1. Платонов Владимир Петрович, profile as the Russian Academy of Science website, President of the Belarus Academy of Science (1987–1992).
2. Владимир Петрович Платонов (К 60-летию со дня рождения). ("Vladimir Petrovich Platonov (on the occasion of the 60th anniversary)"), Известия Национальной академии наук Беларуси СЕРИЯ ФИЗИКО-МАТЕМАТИЧЕСКИХ НАУК, No. 1, 2000, pp. 135–136
3. Academician Vladimir P. Platonov Archived 2004-10-20 at the Wayback Machine, Belarusian Academy of Sciences
4. "Academician Vladimir P. Platonov". Belarusian Academy of Sciences. Archived from the original on 2004-10-20. Retrieved 2004-08-17.
5. V. P. Platonov, "The Tanaka–Artin problem and reduced K-theory," Izv. Akad. Nauk SSSR,. Ser. Mat., 40:2, 1976, pp. 227–261 ((in Russian) Проблема Таннака–Артина и приведенная K-теория)
6. Vladimir Platonov, Andrei Rapinchuk, Algebraic Groups and Number Theory, Academic Press, 1993, ISBN 978-0-12-558180-6, (in Russian) Алгебраические группы и теория чисел, УМН, 1992, No. 47:2 (284), pp. 117–141
7. (in Russian) Vladimir Platonov at NIISI website
8. "Prof charged with attempted murder". Waterloo Daily Bulletin. Nov 9, 1999.
9. "Statement on Prof. Vladimir Platonov". University of Waterloo Communications & Public Affairs.
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Vladimir Retakh
Vladimir Solomonovich Retakh (Russian: Ретах Владимир Соломонович; 20 May 1948) is a Russian-American mathematician who made important contributions to Noncommutative algebra and combinatorics among other areas.
Vladimir Retakh
Born(1948-05-20)May 20, 1948
Chișinău, Soviet Union
NationalitySoviet Union
United States
Alma materMoscow State V. I. Lenin Pedagogical Institute
Known forQuasideterminant and Noncommutative symmetric function
Scientific career
FieldsMathematics
InstitutionsRutgers University
Doctoral advisorsDmitrii Abramovich Raikov
Biography
Retakh graduated in 1970 from the Moscow State Pedagogical University. Beginning as an undergraduate Retakh regularly attended lectures and seminars at the Moscow State University most notably the Gelfand seminars.[1] He obtained his PhD in 1973 under the mentorship of Dmitrii Abramovich Raikov. He joined the Gelfand group in 1986.
His first position was at the central Research Institute for Engineering Buildings and later obtained his first academic position at the Council for Cybernetics of the Soviet Academy of Sciences in 1989. While at the Council for Cybernetics of the Soviet Academy of Sciences in 1990, Retakh had started working with Gelfand on their new program on Noncommutative determinants. Prior to immigrating to the US in 1993 he also held a position at the Scientific Research Institute of System Development
Research
Retakh's other contributions include:
• Contributions to the theory of general hypergeometric functions
• Contributions to the theory of Lie–Massey operators
• Instigated the study of homotopical properties of categories of extensions based on the Retakh isomorphism [2]
• Introduction of noncommutative determinants, also known as quasideterminants
• Introduction of noncommutative symmetric functions
• The introduction of noncommutative Plücker coordinates
• Noncommutative integrable systems
Recognition
He was included in the 2019 class of fellows of the American Mathematical Society "for contributions to noncommutative algebra and noncommutative algebraic geometry".[3]
References
1. On Israel Moiseevich Gelfand
2. Monoidal Categories and the Gerstenhaber Bracket in Hochschild Cohomology by Reiner Hermann, Memoirs of the American Mathematical Society, 2016
3. 2019 Class of the Fellows of the AMS, American Mathematical Society, retrieved 2018-11-07
• Retakh V. (2010). "Israel Moiseevich Gelfand" (PDF). Newsletter of the European Mathematical Society. 63 (76): 25–27. Bibcode:2010PhT....63h..63R. doi:10.1063/1.3480085. ISSN 1027-488X.
• Gelʹfand, I. M.; Graev, M. I.; Retakh, V. S. (1992). "General hypergeometric systems of equations and series of hypergeometric type. (Russian. Russian summary) Uspekhi Mat. Nauk 47" (76): 25–27. {{cite journal}}: Cite journal requires |journal= (help)
• "Publications of Vladimir Retakh". Rutgers, The State University of New Jersey. 2011-01-08.
• Ретах В. (2009-12-25). "Об Израиле Моисеевиче Гельфанде". МЦНМО. Archived from the original on 2010-12-28. Retrieved 2012-04-15.
• "Ретах Владимир Соломонович : Публикации в базе данных Math-Net.Ru". mathnet.ru – Общероссийский математический портал. Retrieved 2012-04-15.
• "Israel Moiseevich Gelfand, Part I" (PDF). American Mathematical Society.
• "Israel Moiseevich Gelfand, Part II" (PDF). American Mathematical Society.
• "Gelfand Centennial Conference: A View of 21st Century Mathematics, MIT, Cambridge, Massachusetts". 2013.
• Etingof, Pavel, Retakh, Vladimir S., Singer, I. M. (2013). The Unity of Mathematics. In Honor of the Ninetieth Birthday of I.M. Gelfand.{{cite book}}: CS1 maint: uses authors parameter (link)
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Vladimir Tretyakov (mathematician)
Vladimir Evgenyevich Tretyakov (Russian: Влади́мир Евге́ньевич Третьяко́в; 12 December 1936 – 8 January 2021) was a Russian mathematician. He was the rector of the Ural State University from 1993 to 2006 and was its President until his death on 8 January 2021.[1][2]
Vladimir Tretyakov
Born12 December 1936
Tula
Died8 January 2021 (aged 84)
Yekaterinburg
Alma mater
• Ural State University
Occupation
• Mathematician
Awards
• Order of Honour
• Medal "For the Development of Virgin Lands"
• Medal "Veteran of Labour"
• Honoured Higher education employee of the Russian Federation
• Jubilee Medal "In Commemoration of the 100th Anniversary of the Birth of Vladimir Ilyich Lenin"
Academic career
Doctoral advisorNikolay Krasovsky
Position heldrector (1993–2006)
References
1. "Президент УрГУ Владимир Третьяков награжден орденом Почета". API Information Agency. 25 July 2008. Retrieved 5 November 2010.
2. "В Екатеринбурге скончался бывший ректор УрГУ Владимир Третьяков". 8 January 2021. Retrieved 8 January 2021.
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