Split (2)

train · 465k rows

text stringlengths 9 7.94M
William Gasarch William Ian Gasarch (/ɡəˈsɑːrʃ/ gə-SARSH;[1] born 1959[2]) is an American computer scientist known for his work in computational complexity theory, computability theory, computational learning theory, and Ramsey theory. He is currently a professor at the University of Maryland Department of Computer Science with an affiliate appointment in Mathematics. William Ian Gasarch Professor Bill Gasarch at UMD Born1959 (age 63–64) NationalityAmerican Alma materStony Brook University Harvard University Known forComputational complexity theory Computability theory Computational learning theory Ramsey theory Scientific career FieldsComputer science InstitutionsUniversity of Maryland, College Park Doctoral advisorHarry R. Lewis Websitewww.cs.umd.edu/~gasarch http://blog.computationalcomplexity.org/ As of 2015 he has supervised over 40 high school students on research projects, including Jacob Lurie. He has co-blogged on computational complexity with Lance Fortnow since 2007. He was book review editor for ACM SIGACT NEWS from 1997 to 2015. Education Gasarch received his doctorate in computer science from Harvard in 1985, advised by Harry R. Lewis. His thesis was titled Recursion-Theoretic Techniques in Complexity Theory and Combinatorics.[3] He was hired into a tenure track professorial job at the University of Maryland in the Fall of 1985. He was promoted to Associate Professor with Tenure in 1991, and to Full Professor in 1998. Work Gasarch co-founded (with Richard Beigel) the field of Bounded Queries in Recursion Theory[4] and has written many papers in the area capped off by a book on the topic co-authored with Georgia Martin, titled Bounded Queries in Recursion Theory.[5] He has published books such as Problems with a Point,[6] a book with a broad view on mathematics and theoretical computer science which he co-authored with Clyde Kruskal and includes works by other professors such as David Eppstein.[7] He also co-founded the subfield of recursion-theoretic inductive inference named Learning via Queries[8] with Carl Smith. More recently he has been more involved with combinatorics, notably Ramsey Theory.[9][10][11] He has written three surveys of what theorists think of the P vs NP problem: in 2002, 2012, and 2019.[12][13][14] In 2020 he wrote Mathematical Muffin Morsels: Nobody Wants a Small Piece with Erik Metz, Jacob Prinz, and Daniel Smolyak. [15] Blog Lance Fortnow began writing a blog on theoretical computer science with an emphasis on complexity theory in 2002.[16] Gasarch was a frequent guest blogger until 2007 when he became an official co-blogger. References 1. "Rectangle Free Colorings – William Gasarch". YouTube. May 8, 2017. Retrieved 12 October 2022. 2. "Still Typecasting from Dagstuhl". Computational Complexity Weblog. Lance Fortnow and William Gasarch. Retrieved 27 September 2018. 3. William Gasarch at the Mathematics Genealogy Project 4. http://www.cs.umd.edu/~gasarch/papers/gems.pdf Gems in the Field of Bounded Queries by William Gasarch, 2003 5. https://www.springer.com/us/book/9780817639662 Bounded Queries in Recursion Theory (with Georgia Martin), Birkhauser, 1999 6. https://www.worldscientific.com/worldscibooks/10.1142/11261 Problems with a Point Exploring Math and Computer Science, 2019 7. https://www.worldscientific.com/doi/abs/10.1142/9789813279735_0014 Chapter 14: Is This Problem Too Hard for a High School Math Competition?, 2019 8. http://www.cs.umd.edu/~gasarch/papers/lvqsur.pdf A Survey of Inductive Inference with an Emphasis on Queries, Gasarch and Smith, 1997 9. Gasarch, William; Haeupler, Bernhard (2011). "Lower Bounds on the van der Waerden Numbers: Randomized- and Deterministic-Constructive". Electronic Journal of Combinatorics. 18 (64). arXiv:1005.3749. doi:10.37236/551. S2CID 534179. 10. Gasarch, William; Haeupler, Bernhard (2010). "Rectangle Free Coloring of Grids". arXiv:1005.3750 [math.CO]. 11. Gasarch, William; Haeupler, Bernhard (2011). "Proving programs terminate using well orderings, Ramsey Theory, and Matrices". arXiv:1108.3347 [math.CO]. 12. Hemaspaandra, Lane A. (2002-06-01). "SIGACT news complexity theory column 36". ACM SIGACT News. 33 (2): 34–47. doi:10.1145/564585.564599. ISSN 0163-5700. S2CID 36828694. 13. Hemaspaandra, Lane A. (2012-06-11). "SIGACT news complexity theory column 74". ACM SIGACT News. 43 (2): 51–52. doi:10.1145/2261417.2261433. ISSN 0163-5700. S2CID 52847337. 14. Gasarch, William I. (2019-03-13). "Guest Column: The Third P=?NP Poll". ACM SIGACT News. 50 (1): 38–59. doi:10.1145/3319627.3319636. ISSN 0163-5700. S2CID 83458626. 15. Gasarch, William; Metz, Erik; Prinz, Jacob; Smolyak, Daniel (28 May 2020). Mathematical Muffin Morsels: Nobody Wants A Small Piece. World Scientific. ISBN 978-981-12-1519-3. 16. http://blog.computationalcomplexity.org/ Computational Complexity Weblog External links • Gasarch's Homepage Authority control International • ISNI • VIAF National • France • BnF data • Israel • United States • Czech Republic • Netherlands Academics • DBLP • Google Scholar • MathSciNet • Mathematics Genealogy Project • Scopus • zbMATH
William Gentle (headmaster) William Gentle FRSE (19 January 1877 – 31 March 1964) was a Scottish mathematician and physicist, and headmaster of George Heriot's School from 1926 to 1942. He was President of the Science Teachers Association in Scotland.[1] Life He was born in Dundee on 19 January 1877 the third child of Marion Drummond Brand from Torryburn in Fife and William Gentle (1838–1890), a printer. He was raised at 12 Garland Place in Dundee.[2] He attended Rosebank School then Morgan Academy. In 1889 his family moved to 2 Blackwood Crescent in Edinburgh, a Victorian flat in the south side of the city. He then attended George Heriot's School going on to study mathematics and other sciences at the University of Edinburgh, graduating in 1903 with a BSc. He studied geology under Professor James Geikie and mathematics under Professor George Chrystal. In autumn 1903 he returned to George Heriot's School as a teacher of mathematics and physics.[3][4] He was elected a Fellow of the Royal Society of Edinburgh in 1908. His proposers were James Gordon MacGregor, David Fowler Lowe, Thomas Burns and John Brown Clark.[5] In the First World War he served as an officer in the Royal Field Artillery, and was wounded by shell-fire at Messines Ridge in 1917. Thereafter he concerned himself with supply of food to the troops, at one point finding himself responsible for feeding 7,000 men.[3] He returned to George Heriot's School after the war. In 1926 he succeeded John Brown Clark as headmaster at a salary of £1,000 per annum. He retired in 1942 and was succeeded by William Carnon. He died in Edinburgh on 31 March 1964. Family In 1924 he married Jessie Currie Ainslie. References 1. "Gentle biography". www-history.mcs.st-andrews.ac.uk. Retrieved 19 January 2018. 2. Dundee Post Office Directory 1877-78 3. "Gentle biography". www-groups.dcs.st-and.ac.uk. Retrieved 19 January 2018. 4. "Slater's Royal National Commercial Directory of Scotland, 1903, Part 2". National Library of Scotland. Retrieved 19 January 2018. 5. Biographical Index of Former Fellows of the Royal Society of Edinburgh 1783–2002 (PDF). The Royal Society of Edinburgh. July 2006. ISBN 0-902-198-84-X.
William Goldman (mathematician) William Mark Goldman (born 1955 in Kansas City, Missouri) is a professor of mathematics at the University of Maryland, College Park (since 1986). He received a B.A. in mathematics from Princeton University in 1977, and a Ph.D. in mathematics from the University of California, Berkeley in 1980. William Goldman William Goldman at Bar-Ilan University in 2008 Born (1955-11-17) November 17, 1955 Kansas City, United States NationalityAmerican Alma materPrinceton University University of California, Berkeley Scientific career FieldsMathematics InstitutionsUniversity of Maryland-College Park Doctoral advisorsMorris Hirsch William Thurston Research contributions Goldman has investigated geometric structures, in various incarnations, on manifolds since his undergraduate thesis, "Affine manifolds and projective geometry on manifolds", supervised by William Thurston and Dennis Sullivan. This work led to work with Morris Hirsch and David Fried on affine structures on manifolds, and work in real projective structures on compact surfaces. In particular he proved that the space of convex real projective structures on a closed orientable surface of genus $g>1$ is homeomorphic to an open cell of dimension $16g-16$. With Suhyoung Choi, he proved that this space is a connected component (the "Hitchin component") of the space of equivalence classes of representations of the fundamental group in ${\rm {SL}}(3,\mathbb {R} )$. Combining this result with Suhyoung Choi's convex decomposition theorem, this led to a complete classification of convex real projective structures on compact surfaces. His doctoral dissertation, "Discontinuous groups and the Euler class" (supervised by Morris W. Hirsch), characterizes discrete embeddings of surface groups in ${\rm {PSL}}(3,\mathbb {R} )$ in terms of maximal Euler class, proving a converse to the Milnor–Wood inequality for flat bundles. Shortly thereafter he showed that the space of representations of the fundamental group of a closed orientable surface of genus $g>1$ in ${\rm {PSL}}(3,\mathbb {R} )$ has $4g-3$ connected components, distinguished by the Euler class. With David Fried, he classified compact quotients of Euclidean 3-space by discrete groups of affine transformations, showing that all such manifolds are finite quotients of torus bundles over the circle. The noncompact case is much more interesting, as Grigory Margulis found complete affine manifolds with nonabelian free fundamental group. In his 1990 doctoral thesis, Todd Drumm found examples which are solid handlebodies using polyhedra which have since been called "crooked planes." Goldman found examples (non-Euclidean nilmanifolds and solvmanifolds) of closed 3-manifolds which fail to admit flat conformal structures. Generalizing Scott Wolpert's work on the Weil–Petersson symplectic structure on the space of hyperbolic structures on surfaces, he found an algebraic-topological description of a symplectic structure on spaces of representations of a surface group in a reductive Lie group. Traces of representations of the corresponding curves on the surfaces generate a Poisson algebra, whose Lie bracket has a topological description in terms of the intersections of curves. Furthermore, the Hamiltonian vector fields of these trace functions define flows generalizing the Fenchel–Nielsen flows on Teichmüller space. This symplectic structure is invariant under the natural action of the mapping class group, and using the relationship between Dehn twists and the generalized Fenchel–Nielsen flows, he proved the ergodicity of the action of the mapping class group on the SU(2)-character variety with respect to symplectic Lebesgue measure. Following suggestions of Pierre Deligne, he and John Millson proved that the variety of representations of the fundamental group of a compact Kähler manifold has singularities defined by systems of homogeneous quadratic equations. This leads to various local rigidity results for actions on Hermitian symmetric spaces. With John Parker, he examined the complex hyperbolic ideal triangle group representations. These are representations of hyperbolic ideal triangle groups to the group of holomorphic isometries of the complex hyperbolic plane such that each standard generator of the triangle group maps to a complex reflection and the products of pairs of generators to parabolics. The space of representations for a given triangle group (modulo conjugacy) is parametrized by a half-open interval. They showed that the representations in a particular range were discrete and conjectured that a representation would be discrete if and only if it was in a specified larger range. This has become known as the Goldman–Parker conjecture and was eventually proven by Richard Schwartz. Professional service Goldman also heads a research group at the University of Maryland called the Experimental Geometry Lab, a team developing software (primarily in Mathematica) to explore geometric structures and dynamics in low dimensions. He served on the Board of Governors for The Geometry Center at the University of Minnesota from 1994 to 1996. He served as Editor-In-Chief of Geometriae Dedicata from 2003 until 2013. Awards and honors In 2012 he became a fellow of the American Mathematical Society.[1] Publications • Goldman, William M. (1999). Complex hyperbolic geometry. Oxford Mathematical Monographs. Oxford Science Publications. New York: The Clarendon Press, Oxford University Press. xx+316 pp. ISBN 0-19-853793-X. MR 1695450. • Goldman, William M.; Xia, Eugene Z. (2008). "Rank one Higgs bundles and representations of fundamental groups of Riemann surfaces". Memoirs of the American Mathematical Society. 193 (904): viii+69 pp. arXiv:math/0402429. doi:10.1090/memo/0904. ISSN 0065-9266. MR 2400111. S2CID 2865489. References 1. List of Fellows of the American Mathematical Society, retrieved 2013-01-19. External links • Faculty page at the University of Maryland, College Park Authority control International • ISNI • VIAF National • Norway • France • BnF data • Catalonia • Germany • Israel • United States • Netherlands Academics • DBLP • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
William B. Gragg William B. Gragg (1936–2016) ended his career as an Emeritus Professor in the Department of Applied Mathematics at the Naval Postgraduate School. He has made fundamental contributions in numerical analysis, particularly the areas of numerical linear algebra and numerical methods for ordinary differential equations. William B. Gragg Born(1936-11-02)2 November 1936 Bakersfield, California Died25 December 2016(2016-12-25) (aged 80) Monterey, California NationalityAmerican Alma materUCLA Known forGragg Extrapolation Scientific career FieldsMathematics InstitutionsNaval Postgraduate School ThesisRepeated extrapolation to the limit in the numerical solution of ordinary differential equations (1964) Doctoral advisorPeter Henrici He received his PhD at UCLA in 1964 under the direction of Peter Henrici. His dissertation work resulted in the Gragg Extrapolation method[1] for the numerical solution of ordinary differential equations (sometimes also called the Bulirsch–Stoer algorithm). Gragg is also well known for his work on the QR algorithm for unitary Hessenberg matrices, on updating the QR factorization,[2] superfast solution of Toeplitz systems,[3] parallel algorithms for solving eigenvalue problems,[4][5] as well as his exposition on the Pade table and its relation to a large number of algorithms in numerical analysis.[6] References 1. http://epubs.siam.org/doi/pdf/10.1137/0702030 On extrapolation algorithms for ordinary initial value problems, WB Gragg SINUM, vol. 2, no. 3, 1965. 2. Daniel, J. W.; Gragg, W. B.; Kaufman, L.; Stewart, G. W. (1976). "Reorthogonalization and stable algorithms for updating the Gram-Schmidt factorization". Math. Comp. 30 (136): 772–795. doi:10.1090/S0025-5718-1976-0431641-8. 3. Ammar, Gregory S.; Gragg, William B. (1988). "Superfast Solution of Real Positive Definite Toeplitz Systems". SIAM Journal on Matrix Analysis and Applications. 9: 61–76. CiteSeerX 10.1.1.64.8032. doi:10.1137/0609005. hdl:10945/30445. 4. Article title A Parallel Divide and Conquer Algorithm for the Generalized Real Symmetric Definite Tridiagonal Eigenproblem, C.F. Borges and W.B.Gragg, 1992 5. Gragg, W. B.; Reichel, L. (1990). "A divide and conquer method for unitary and orthogonal eigenproblems". Numerische Mathematik. 57: 695–718. doi:10.1007/BF01386438. hdl:10945/29823. S2CID 53684596. 6. Gragg, W. B. (1972). "The Padé Table and Its Relation to Certain Algorithms of Numerical Analysis". SIAM Review. 14: 1–62. doi:10.1137/1014001. External links • William B. Gragg at the Mathematics Genealogy Project Authority control International • VIAF National • Germany Academics • MathSciNet • Mathematics Genealogy Project • zbMATH
William H. Bossert William H. Bossert (born 1937) is an American mathematician. He is the David B. Arnold, Jr. Professor of Science, Emeritus at Harvard University. He was the housemaster of Lowell House for 23 years.[1][2] He received his PhD from Harvard in 1963.[3] Publications With Edward O. Wilson A primer of population biology (1971)[4] References 1. Flaherty, Julie (April 15, 1998). "Harvard Breaks a Housemaster Mold" – via NYTimes.com. 2. "William H. Bossert \| Harvard John A. Paulson School of Engineering and Applied Sciences". www.seas.harvard.edu. 3. "William Bossert - The Mathematics Genealogy Project". www.genealogy.math.ndsu.nodak.edu. Retrieved 2021-12-31. 4. "A Primer of Population Biology". Oxford University Press. June 1, 1971 – via Oxford University Press. Authority control International • ISNI • VIAF National • Norway • France • BnF data • Germany • Israel • United States • Japan • Czech Republic • Greece • Netherlands Academics • CiNii • MathSciNet • Mathematics Genealogy Project
William Jack (mathematician) William Jack FRSE (29 May 1834 – 20 March 1924) was a Scottish mathematician and journalist. He was Editor of the Glasgow Herald newspaper from 1870 to 1876, and Professor of Mathematics at the University of Glasgow from 1879 until 1909.[1] Life He was born on 29 May 1834 in Stewarton in Ayrshire the son of Robert Jack of Irvine. He was educated at Irvine Academy, going on to study mathematics at the University of Glasgow, graduating with an MA in 1853. He then continued his studies at the University of Cambridge, graduating with a second MA in 1859. From 1860 to 1866 he was HM Inspector of Schools for Scotland. In 1866 he accepted the post of Professor of Natural Philosophy (Physics) at Owens College in Manchester and held this position until 1870 when he moved to Glasgow as Editor of the Glasgow Herald newspaper. He left in 1876 to run Macmillan & Co, a London publisher,[2] and in 1879 joined the staff of Glasgow University as Professor of Mathematics. In 1875, he was elected a Fellow of the Royal Society of Edinburgh. His proposers were, William Thomson, Lord Kelvin, James Thomson Bottomley, Allen Thomson and Peter Guthrie Tait.[3] In 1875 the University of Glasgow awarded him an honorary LLD and in 1902 the University of Manchester awarded him an honorary doctorate (DSc). In his final working years he lived on the campus of the University of Glasgow, at 10 The College.[4] He died on 20 March 1924.[5] Family He was married to Agnes Jane Nichol (1837–1901), daughter of John Pringle Nichol and sister of Professor John Nichol. Their children included sons William Tullis Jack (b.1862), William Robert Jack (b.1866), Adolphus Alfred Jack (b.1868), and a daughter Agnes Elizabeth Jack (b.1871).[6] References 1. "University of Glasgow :: Story :: Biography of William Jack". Universitystory.gla.ac.uk. 23 March 1924. Retrieved 12 March 2017. 2. "William Jack". Glasgowwestaddress.co.uk. Retrieved 12 March 2017. 3. Biographical Index of Former Fellows of the Royal Society of Edinburgh 1783–2002 (PDF). The Royal Society of Edinburgh. July 2006. ISBN 0-902-198-84-X. 4. Glasgow Post Office directory 1905–6 5. "Professor Jack. Death of distinguished mathematician. Thirty years in Glasgow University". The Glasgow Herald. 21 March 1924. p. 5. Retrieved 6 January 2016. 6. "William Jack (1834 - 1924) - Genealogy". Geni.com. 20 March 1924. Retrieved 12 March 2017. Authority control International • VIAF National • United States Other • IdRef
William James Macdonald William James Macdonald FRSE was born on 14 December 1851 in Scotland.[1] He is known for being a pioneer of the introduction of modern geometry to the mathematical curriculum in schools and for being one of the founding members of the Edinburgh Mathematical Society.[1][2][3] William James Macdonald Born(1851-12-14)December 14, 1851 DiedDecember 29, 1941(1941-12-29) (aged 90) Alma materUniversity of St Andrews Scientific career InstitutionsMadras College, Merchiston Castle School, Daniel Stewart’s College Biography Macdonald was born in Huntly, Aberdeenshire, but moved to the coastal town of St Andrews when he was young.[1] There he got an education in Madras College,[1][4][5] and became dux of the college in 1868.[1] After completing his school education he entered the University of St Andrews, where he studied a variety of subjects including mathematics, English literature, Latin, Greek, chemistry, and philosophy.[1][6] While there Macdonald won many prizes, including the Miller prize given to the student who did the best work in 1870, 1871, and 1872, the Gray prize in 1872 for an essay on spectrum analysis, and the Arnott prize, also in 1872.[1][6] After graduating, Macdonald was appointed assistant to the Mathematics Department in Madras College, but only taught there for a short time before accepting a role as Mathematics Master at Merchiston Castle school in Edinburgh.[1][6] He soon after accepted a role at Daniel Stewart’s College where he spent the rest of his career.[1][6] Between 1898 and 1899, he was the president of the Scottish Secondary Teachers' Association.[1] He died in Edinburgh on 29 December 1941.[1][6] Accomplishments Macdonald was a pioneer in the introduction of modern geometry to the mathematical curriculum. He wrote Higher Geometry: Containing an Introduction to Modern Geometry and Elementary Geometrical Conics,[7] a text which was widely used in schools and colleges to teach geometry.[1] He was a founding member of the Edinburgh Mathematical Society,[1][3] and was honoured by the society when he was elected as president for 1887-88 session.[8] On the 1st of February 1886 he accepted a fellowship to the Royal Society of Edinburgh[9] after being proposed by William Swan, John Sturgeon Mackay, George Chrystal, and Sir Thomas Muir.[1] In June 1914, he was offered the degree of LL.D by the Senatus Academicus of the University of St Andrews, but he respectfully declined the honour.[1][6] References 1. "William James Macdonald - Biography". Maths History. Retrieved 2022-06-23. 2. Macdonald, W. J. (1888). "Office-Bearers". Proceedings of the Edinburgh Mathematical Society. 7: 1. doi:10.1017/S0013091500030200. ISSN 0013-0915. S2CID 251062914. 3. "EMS Founder Members". Maths History. Retrieved 2022-06-24. 4. "Biographies". www.madrascollegearchive.org.uk. Retrieved 2022-06-24. 5. "MCM Christmas 1912". www.madrascollegearchive.org.uk. Retrieved 2022-06-24. 6. "Biographical Register 1747-1897". arts.st-andrews.ac.uk. Retrieved 2022-06-24. 7. Macdonald, William James (1897). Higher Geometry: Containing an Introduction to Modern Geometry and Elementary Geometrical Conics (2nd ed.). Edinburgh: James Thin. 8. Rankin, R. A. (1983). "The first hundred years (1883–1983)". Proceedings of the Edinburgh Mathematical Society. 26 (2): 147. doi:10.1017/S0013091500016849. ISSN 0013-0915. S2CID 178104090. 9. Royal Society of Edinburgh (1886). Proceedings of the Royal Society of Edinburgh (Volume 13 ed.). Edinburgh. p. 596.{{cite book}}: CS1 maint: location missing publisher (link)
William John Greenstreet William John Greenstreet (1861–1930) was an English mathematician who was editor of The Mathematical Gazette for more than thirty years.[1] William John Greenstreet Born(1861-08-18)18 August 1861 Shorncliffe Army Camp, Folkestone, Kent Died28 June 1930(1930-06-28) (aged 68) Burghfield Common, West Berkshire Alma materSt John's College, Cambridge SpouseEthel de Medina Spender Parent(s)Thomas Greenstreet and Catherine S Greenstreet Scientific career FieldsMathematics InstitutionsMarling School Life and work Greenstreet was son of a Royal Artillery's Sergeant. He was educated at Southwark and he entered St John's College, Cambridge in 1879, graduating there in 1883. Then he was mathematics professor in different schools in Framlingham, East Riding and Cardiff before he became Head Master at Marling School in 1891. In 1910 he retired to Burghfield Common with the intention of devoting to literary work.[2] Greenstreet was founding member of the Mathematical Association and he started the Association's Library given a large collection of books.[3] References 1. Macaulay et al. 1930, p. 181. 2. Macaulay et al. 1930, p. 182. 3. Goodstein 1974, p. 100. Bibliography • Goodstein, R.L. (1974). "The Mathematical Association Library at the University of Leicester". The British Journal for the History of Science. 7 (1): 100–103. doi:10.1017/S0007087400013066. ISSN 0007-0874. S2CID 144469175. • Macaulay, I.F.S.; Neville, E.H.; Pendlebury, C.; Spender, J.A.; Anderson, W.C.F. (1930). "Obituary: W. J. Greenstreet". The Mathematical Gazette. 15 (209): 181–186. doi:10.1017/S0025557200137258. ISSN 0025-5572. JSTOR 3605787. S2CID 185174470. External links • O'Connor, John J.; Robertson, Edmund F., "William John Greenstreet", MacTutor History of Mathematics Archive, University of St Andrews Authority control International • ISNI • VIAF National • Germany • Israel • United States • Netherlands Academics • zbMATH
William J. LeVeque William Judson LeVeque (August 9, 1923 – December 1, 2007) was an American mathematician and administrator who worked primarily in number theory. He was executive director of the American Mathematical Society during the 1970s and 1980s when that organization was growing rapidly and greatly increasing its use of computers in academic publishing. William J. LeVeque Born(1923-08-09)August 9, 1923 Boulder, Colorado DiedDecember 1, 2007(2007-12-01) (aged 84) Alma materCornell University Known forNumber theory AwardsSLA PMA Division Award Scientific career FieldsMathematics InstitutionsHarvard University University of Michigan Claremont Graduate School American Mathematical Society Doctoral advisorBurton W. Jones, Mark Kac Doctoral studentsUnderwood Dudley Life and education LeVeque was born August 9, 1923, in Boulder, Colorado. He received his Bachelor of Arts degree from the University of Colorado in 1944, and a master's degree in 1945 and a Ph.D. in 1947 from Cornell University.[1] He was an instructor at Harvard University from 1947 to 1949, then started at University of Michigan as an instructor and rose to professor. In 1970 he moved to the Claremont Graduate School. In 1977 he became executive director of the American Mathematical Society and remained there until his retirement in 1988.[1] After retirement LeVeque and his wife, Ann, took up sailing and lived on their sailboat for three years while they traveled from Narragansett Bay to Grenada. They then moved to Bainbridge Island, Washington, where he kept active in volunteer activities for the rest of his life.[2] He died December 1, 2007.[3] His son Randall J. LeVeque is a well known applied mathematician. Work LeVeque's research interest was number theory, specifically transcendental numbers, uniform distribution, and Diophantine approximation.[3] He wrote a number of number theory textbooks and reference books, which influenced the development of number theory in the United States. A long-term project was to update Leonard Eugene Dickson's History of the Theory of Numbers. This project eventually produced a six-volume collection titled Reviews in Number Theory.[3] The Special Libraries Association's Physics-Astronomy-Mathematics Division awarded LeVeque its Division Award in 1978 for his contributions to the bibliography of mathematics.[4] The American Mathematical Society grew rapidly during LeVeque's time as executive director (1977–1988). Revenues tripled from $5 million in 1977 to $14.9 million in 1988. The Society began computerizing at a rapid rate during this period, with Mathematical Reviews first becoming available electronically through existing academic dial-up services; this system later evolved into MathSciNet. Most of the headquarters staff received computer terminals for use in the new operations.[3] Selected publications • LeVeque, William J. (2002) [1956]. Topics in Number Theory, Volumes I and II. New York: Dover Publications. ISBN 978-0-486-42539-9. • Leveque, William J. (1990) [1962]. Elementary Theory of Numbers. New York: Dover Publications. ISBN 978-0-486-66348-7. • LeVeque, William J., ed. (1969). Studies in number theory. Mathematical Association of America. OCLC 17369. • LeVeque, William J., ed. (1974). Reviews in number theory, as printed in Mathematical reviews, 1940 through 1972, volumes 1-44 inclusive. American Mathematical Society. OCLC 948152. (6 volumes) • Leveque, William J. (1996) [1977]. Fundamentals of Number Theory. New York: Dover Publications. ISBN 978-0-486-68906-7. Further reading • LeVeque, William J. (July–August 1988). "The AMS—Then, Now, and Soon" (PDF). Notices of the American Mathematical Society. 35 (6): 785–789. Retrieved 2009-01-01. A retrospective by LeVeque of his work at the American Mathematical Society. Notes 1. "William J. LeVeque Retires as Executive Director" (PDF). Notices of the American Mathematical Society. 35 (6): 783–784. July–August 1988. Retrieved 2009-01-01. 2. "Current Obituaries: William J. LeVeque". Cook Family Funeral Home. Retrieved 2 January 2009. 3. Maxwell, James W. (November 2008). "William J. LeVeque (1923–2007)" (PDF). Notices of the American Mathematical Society. 55 (10): 1261–1262. Retrieved 2009-01-01. 4. "PAM Division Award Winners List". Special Libraries Association. 2007-12-18. Retrieved 2 January 2009. External links • William J. LeVeque at the Mathematics Genealogy Project Authority control International • ISNI • VIAF National • Norway • France • BnF data • Catalonia • Germany • Israel • United States • Sweden • Netherlands Academics • CiNii • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie Other • SNAC • IdRef
William Lloyd Garrison Williams William Lloyd Garrison Williams (3 October 1888 - 31 January 1976) was an American-Canadian Quaker and mathematician, known for the founding of the Canadian Mathematical Society and overseeing Elbert Frank Cox's doctorate in mathematics. William Lloyd Garrison Williams Born(1888-10-03)October 3, 1888 Friendship, Kansas DiedJanuary 31, 1976(1976-01-31) (aged 87) NationalityAmerican-Canadian Other namesLloyd Williams Education • Haverford College B.A. • University of Oxford (1913, M.A.) • University of Chicago (1925, Ph.D) OccupationMathematician Known for • Canadian Mathematical Congress • Elbert Frank Cox Ph.D PartnerAnne Skyes (1917) Children2 Personal life Williams was born in Friendship, Kansas to Amanda Dunreath Truex and Nathan Williams.[1] After the death of his mother, he was taken in by his father's first wife's family, the Tominsons, in Indiana. Williams married Anne Skyes, a pianist from Cincinnati, in 1917.[2] Together, they had two daughters, Hester and mathematician Christine Ayoub.[3] Academic career After attending the Quaker Academy, Williams taught in North Dakota.[2] He then studied Classics at Haverford.[1] Subsequently, he was awarded a Rhodes Scholarship, where he studied mathematics at Oxford from 1910 to 1913.[4] Once obtaining an M.A. he took on a faculty position at the Miami University.[2] During the summers, he did Ph.D. work at the University of Chicago.[4] He wrote his Ph.D. on Fundamental Systems of Formal Modular Seminvariants of the Binary Cubic, published in 1920. He then taught briefly at Gettysburg College and William and Mary, before relocating to Cornell.[1] In 1924, Williams moved again, this time to teach at McGill, to develop their graduate program. He remained there until his retirement in 1954.[5] He was awarded honorary degrees by the University of Montreal, University of Manitoba, Dalhousie University, and Mount Allison University.[1] Elbert Frank Cox Williams supervised the Ph.D. of Elbert Frank Cox, the first African American to get a mathematics doctorate, during his time at Cornell. When Williams took up his position at McGill, Cox followed him. Cox was awarded his degree in 1925, utilizing the Erastus Brooks Fellowship.[1] Unsatisfied with the limited amount of recognition Cox received, Williams petitioned international universities to recognise his student, eventually convincing the Sendai University, Japan.[6] Canadian Mathematical Congress Williams founded the Canadian Mathematical Society, formerly the Canadian Mathematical Congress, in 1945. Williams dreamed of a forum to bring Canadian mathematicians together, regardless of race or creed, especially after attending the Toronto Congress in 1924 where all mathematicians of the Central Powers had been excluded.[3] He worked as the treasurer from its founding in 1945 to 1963.[1] He was particularly successful in achieving support from insurance companies, although he was noted by many to be quite zealous and friendly and found support from many.[7] Currently the Jeffrey-Williams Prize is named in his honour, and awarded to mathematicians who have made outstanding contributions to mathematical research by the Canadian Mathematical Society.[5] Religious beliefs A devoted Quaker, Williams was instrumental in the Montreal Quaker community during the 1930s and beyond.[8] These beliefs influenced his views of racial equality and hard work. Williams pioneered Quaker causes. He was a member of the Board of the Canadian Friends Service Committee and chairman of the committee from 1959 until 1963.[3] He founded the Montreal “Save the Children” Fund in 1944.[9] He additionally helped with the purchasing of a building for Montreal friends to use as a meeting house, although it is no longer in use.[2] He is noted for being an official of the Quakers of Montreal.[3] References 1. O'Connor, J J; Robertson, E F (November 2006). "MacTutor: William Lloyd Garrison Williams". MacTutor. School of Mathematics and Statistics, University of St Andrews, Scotland.{{cite web}}: CS1 maint: url-status (link) 2. Ayoub, Christine (2014). Memories of the Quaker past : stories of thirty-seven senior Quakers. Xlibris Corporation. [State College, PA]. pp. 5–20. ISBN 978-1-4691-6254-6. OCLC 870842742.{{cite book}}: CS1 maint: location missing publisher (link) 3. "My father, Lloyd Williams". Quakers In Canada. Retrieved 2022-06-23. 4. Donaldson, James A.; Fleming, Richard J. (2000-02-01). "Elbert F. Cox: An Early Pioneer". The American Mathematical Monthly. 107 (2): 105–128. doi:10.1080/00029890.2000.12005169. ISSN 0002-9890. S2CID 833917. 5. "Jeffery-Williams Prize". CMS-SMC. Retrieved 2022-06-23. 6. "Elbert Frank Cox, first Black to earn a Ph.D. in mathematics". www.math.buffalo.edu. Retrieved 2022-06-23. 7. MATHEMATICAL CULTURES : the london meetings 2012 2014. [Place of publication not identified]: BIRKHAUSER. 2018. ISBN 978-3-319-80379-1. OCLC 1040201215. 8. Lawson, David (15 December 1977). "A View of Montreal Meeting" (PDF). Friend's Journal. Philadelphia, PA: Friends Publishing Company. 23 (21): 660. 9. Zavitz-Bond, Jane (1997). "Book Review: My Father, Llyod Williams" (PDF). Canadian Quaker History Journal. Toronto, Ontario: Canadian Friends Historical Association (62): 31. ISSN 0319-3934.
William Lowell Putnam Mathematical Competition The William Lowell Putnam Mathematical Competition, often abbreviated to Putnam Competition, is an annual mathematics competition for undergraduate college students enrolled at institutions of higher learning in the United States and Canada (regardless of the students' nationalities). It awards a scholarship and cash prizes ranging from $250 to $2,500 for the top students and $5,000 to $25,000 for the top schools, plus one of the top five individual scorers (designated as Putnam Fellows) is awarded a scholarship of up to $12,000 plus tuition at Harvard University (Putnam Fellow Prize Fellowship),[1] the top 100 individual scorers have their names mentioned in the American Mathematical Monthly (alphabetically ordered within rank), and the names and addresses of the top 500 contestants are mailed to all participating institutions. It is widely considered to be the most prestigious university-level mathematics competition in the world, and its difficulty is such that the median score is often zero (out of 120) despite being attempted by students specializing in mathematics.[2][3] The competition was founded in 1927 by Elizabeth Lowell Putnam in memory of her husband William Lowell Putnam, who was an advocate of intercollegiate intellectual competition. The competition has been offered annually since 1938 and is administered by the Mathematical Association of America.[4] Competition layout The Putnam competition takes place on the first Saturday in December, and consists of two three-hour sittings separated by a lunch break. The competition is supervised by faculty members at the participating schools. Each one consists of twelve challenging problems. The problems cover a range of advanced material in undergraduate mathematics, including concepts from group theory, set theory, graph theory, lattice theory, and number theory. [5] Each of the twelve questions is worth 10 points, and the most frequent scores above zero are 10 points for a complete solution, 9 points for a nearly complete solution, and 1 point for the beginnings of a solution. In earlier years, the twelve questions were worth one point each, with no partial credit given. The competition is considered to be very difficult: it is typically attempted by students specializing in mathematics, but the median score is usually zero or one point out of 120 possible, and there have been only five perfect scores as of 2021.[6][7] In 2003, of the 3,615 students competing, 1,024 (28%) scored 10 or more points, and 42 points was sufficient to make the top percentile. At a participating college, any student who wishes to take part in the competition may (limited by the number of spots a school receives); but until 2019 the school's official team consisted of three individuals whom it designated in advance. Until 2019, a team's score was the sum of the ranks of its three team members, with the lowest cumulative rank winning. It was entirely possible, even commonplace at some institutions, for the eventual results to show that the "wrong" team was picked—i.e. that some students not on the official team outscored an official team member. For example, in 2010, MIT had two of the top five scorers in the competition and seven of the top 24, while Caltech had just one student in the top five and only four in the top 24; yet Caltech took first place among teams while MIT took second.[8] In 2019 the rules of the competition changed, with a school's team consisting of its top three scorers, and team ranks determined by comparing the sums of the scores of the team members. [5] Awards The top five teams win $25,000, $20,000, $15,000, $10,000, and $5,000, in that order, with team members receiving $1,000, $800, $600, $400, and $200, respectively. The top five individual scorers are named Putnam Fellows and awarded $2,500. The school with the first-place team receives an award of $25,000. Each first-place team member, as well as the winner of the Elizabeth Lowell Putnam Prize, receives $1,000. Sixth through 15th place individuals receive $1,000 and the next ten receive $250. The names of the top 100 students are published in the American Mathematical Monthly, and competition results are published in early April of the year following the competition.[9] Many Putnam Fellows have gone on to become distinguished researchers in mathematics and other fields, including three Fields Medalists—John Milnor (also an Abel Prize laureate), David Mumford, and Daniel Quillen—and two Nobel laureates in physics—Richard Feynman and Kenneth Wilson. Winners Top-scoring teams Year First Second Third Fourth Fifth 1938 Toronto UC Berkeley Columbia 1939 Brooklyn College MIT Mississippi Woman's 1940 Toronto Yale Columbia 1941 Brooklyn College UPenn MIT 1942 Toronto Yale MIT City College of NY 1946 Toronto MIT Brooklyn College Carnegie Tech 1947 Harvard Yale Columbia UPenn 1948 Brooklyn College Toronto Harvard City College of NY and McGill 1949 Harvard Toronto Carnegie Tech City College of NY 1950 Caltech Harvard NYU Toronto 1951 Cornell Harvard Cooper Union City College of NY 1952 Queen's Brooklyn Polytech Harvard MIT 1953 Harvard City College of NY Cornell UC Berkeley 1954 Cornell Harvard MIT Toronto 1955 Harvard Toronto Yale Kenyon 1956 Harvard Columbia Queen's MIT 1957 Harvard Columbia Cornell Caltech 1958 (Spring) Brooklyn Polytech Harvard Toronto Manitoba 1958 (Fall) Harvard Toronto Caltech Cornell 1959 Brooklyn Polytech Caltech Toronto Harvard Case Tech 1960 UC Berkeley Harvard MIT Michigan State Cornell 1961 Michigan State MIT Caltech Harvard Dartmouth 1962 Caltech Dartmouth Harvard Queen's UCLA 1963 Michigan State Brooklyn College UPenn Caltech MIT 1964 Caltech MIT Harvard Case Tech UC Berkeley 1965 Harvard MIT Toronto Princeton Caltech 1966 Harvard MIT Chicago Michigan Princeton 1967 Michigan State Caltech Harvard MIT Michigan 1968 MIT Waterloo UCLA Michigan State Kansas 1969 MIT Rice Chicago Harvard Yale 1970 Chicago MIT Toronto Illinois Tech Caltech 1971 Caltech Chicago Harvard UC Davis MIT 1972 Caltech Oberlin Harvard Swarthmore MIT 1973 Caltech British Columbia Chicago Harvard Princeton 1974 Waterloo Chicago Caltech MIT British Columbia 1975 Caltech Chicago MIT Princeton Harvard 1976 Caltech Washington U in StL Princeton Case Western Reserve and MIT 1977 Washington U in StL UC Davis Caltech Princeton MIT 1978 Case Western Reserve Washington U in StL Waterloo Harvard Caltech 1979 MIT Caltech Princeton Stanford Waterloo 1980 Washington U in StL Harvard Maryland Chicago UC Berkeley 1981 Washington U in StL Princeton Harvard Stanford Maryland 1982 Harvard Waterloo Caltech Yale Princeton 1983 Caltech Washington U in StL Waterloo Princeton Chicago 1984 UC Davis and Washington U in StL Harvard Princeton Yale 1985 Harvard Princeton UC Berkeley Rice Waterloo 1986 Harvard Washington U in StL UC Berkeley Yale MIT 1987 Harvard Princeton Carnegie Mellon UC Berkeley MIT 1988 Harvard Princeton Rice Waterloo Caltech 1989 Harvard Princeton Waterloo Yale Rice 1990 Harvard Duke Waterloo Yale Washington U in StL 1991 Harvard Waterloo Harvey Mudd Stanford Yale 1992 Harvard Toronto Waterloo Princeton Cornell 1993 Duke Harvard Miami University MIT Michigan 1994 Harvard Cornell MIT Princeton Waterloo 1995 Harvard Cornell MIT Toronto Princeton 1996 Duke Princeton Harvard Washington U in StL Caltech 1997 Harvard Duke Princeton MIT Washington U in StL 1998 Harvard MIT Princeton Caltech Waterloo 1999 Waterloo Harvard Duke Michigan Chicago 2000 Duke MIT Harvard Caltech Toronto 2001 Harvard MIT Duke UC Berkeley Stanford 2002 Harvard Princeton Duke UC Berkeley Stanford 2003 MIT Harvard Duke Caltech Harvey Mudd 2004 MIT Princeton Duke Waterloo Caltech 2005 Harvard Princeton Duke MIT Waterloo 2006 Princeton Harvard MIT Toronto Chicago 2007 Harvard Princeton MIT Stanford Duke 2008 Harvard Princeton MIT Stanford Caltech 2009 MIT Harvard Caltech Stanford Princeton 2010 Caltech MIT Harvard UC Berkeley Waterloo 2011 Harvard Carnegie Mellon Caltech Stanford MIT 2012 Harvard MIT UCLA Stony Brook Carnegie Mellon 2013 MIT Carnegie Mellon Stanford Harvard Caltech 2014 MIT Harvard RPI Waterloo Carnegie Mellon 2015 MIT Carnegie Mellon Princeton Stanford Harvard 2016 Carnegie Mellon Princeton Harvard MIT Stanford 2017 MIT Harvard Princeton Toronto UCLA 2018 Harvard MIT UCLA Columbia Stanford 2019 MIT Harvard Stanford UCLA Waterloo 2021 MIT Princeton Harvard Stanford UCLA 2022 MIT Harvard Stanford Maryland Yale Teams ranked by historical performance Below is a table of teams by the number of appearances in the top five and number of titles. The following table lists Teams finishing in Top Five (as of 2021 competition): Top Five Team (s) 67 Harvard 52 MIT 33 Caltech 32 Princeton 20 Waterloo 19 Toronto 16 Stanford 13 Yale 12 Duke 11 Chicago, Washington University in St. Louis 10 UC Berkeley, Cornell 9 Carnegie Mellon (including former Carnegie Tech) 7 UCLA 6 Columbia 5 Brooklyn College, City College of New York, Michigan State 4 Case Western Reserve (including former Case Tech), Michigan, Rice 3 Brooklyn Polytech, UC Davis, Queen's, Penn, Maryland 2 British Columbia, Dartmouth, Harvey Mudd 1 Cooper Union, Illinois Tech, Kansas, Kenyon, Manitoba, McGill, Miami University, RPI NYU, Oberlin, Stony Brook, Swarthmore, William Carey (under former name of Mississippi Woman's) For a recent analysis, the following table lists teams that finished in the top five since 2000 (as of 2021 competition): Top Five Team (s) 21 Harvard, MIT 13 Stanford 11 Princeton 8 Caltech 7 Duke 6 Carnegie Mellon 5 UCLA, Waterloo 3 UC Berkeley, Toronto 1 Chicago, Harvey Mudd, Stony Brook, Yale, RPI, Columbia, Maryland The following table lists Teams with First place finishes (as of 2021 competition): First Place Team (s) 30 Harvard 12 MIT 10 Caltech 4 Toronto, Washington University in St. Louis 3 Brooklyn College, Duke, Michigan State 2 Brooklyn Polytech, Cornell, Waterloo 1 UC Berkeley, UC Davis, Carnegie Mellon, Case Western Reserve, Chicago, Princeton, Queen's Putnam Fellows Since the first competition, the top five (or six, in case of a tie) scorers in the competition have been named Putnam Fellows. Within the top five, Putnam Fellows are not ranked. Students are not allowed to participate in the Putnam Competition more than four times. For example, if a high school senior chooses to officially participate, he/she effectively chooses to forfeit one of his/her years of eligibility in college (see Gabriel Carroll). This makes it even more of a remarkable feat to become a Putnam Fellow four times. In the history of the Competition, only eight students have been Putnam Fellows four times, with twenty-three others winning the award three times. The following table lists these students: Name School Years Don Coppersmith MIT 1968 1969 1970 1971 Arthur Rubin Purdue, Caltech 1970 1971 1972 1973 Bjorn M. Poonen Harvard 1985 1986 1987 1988 Ravi D. Vakil Toronto 1988 1989 1990 1991 Gabriel D. Carroll UC Berkeley, Harvard 2000 2001 2002 2003 Reid W. Barton MIT 2001 2002 2003 2004 Daniel Kane MIT 2003 2004 2005 2006 Brian R. Lawrence Caltech 2007 2008 2010 2011 Edward L. Kaplan Carnegie Tech 1939 1940 1941 Andrew M. Gleason Yale 1940 1941 1942 Donald J. Newman City College of NY 1948 1949 1950 James B. Herreshoff IV UC Berkeley 1951 1952 1953 Samuel Jacob Klein City College of NY 1953 1959 1960 Randall L. Dougherty UC Berkeley 1978 1979 1980 Eric D. Carlson Michigan State 1980 1982 1983 David W. Ash Waterloo 1981 1982 1983 Noam D. Elkies Columbia 1982 1983 1984 David J. Grabiner Princeton 1986 1987 1988 David J. Moews Harvard 1986 1987 1988 J. P. Grossman Toronto 1993 1994 1995 Kiran S. Kedlaya Harvard 1993 1994 1995 Lenhard L. Ng Harvard 1993 1994 1995 Ciprian Manolescu Harvard 1997 1998 2000 Aaron C. Pixton Princeton 2004 2005 2007 Yufei Zhao MIT 2006 2008 2009 Arnav Tripathy Harvard 2007 2008 2009 Seok Hyeong Lee Stanford 2008 2010 2011 Evan M. O'Dorney UC Berkeley, Harvard 2011 2012 2013 Zipei Nie MIT 2012 2013 2014 David H. Yang MIT 2013 2014 2015 Yunkun Zhou MIT 2015 2016 2017 Shengtong Zhang MIT 2018 2019 2021 The following table lists all Putnam fellows from 1938 to present, with the years they placed in the top five.[10] Ioana Dumitriu was the first woman to become a Putnam Fellow, in 1996. Name (School) Year (s) George W. Mackey (Rice) 1938 Irving Kaplansky (Toronto) 1938 Michael J. Norris (College of St. Thomas) 1938 Robert W. Gibson (Fort Hays Kansas State) 1938 Bernard Sherman (Brooklyn College) 1938, 1939 Abraham Hillman (Brooklyn College) 1939 Richard P. Feynman (MIT) 1939 William Nierenberg (City College of NY) 1939 Edward L. Kaplan (Carnegie Tech) 1939, 1940, 1941 John Cotton Maynard (Toronto) 1940 Robert Maughan Snow (George Washington) 1940 W. J. R. Crosby (Toronto) 1940 Andrew M. Gleason (Yale) 1940, 1941, 1942 Paul C. Rosenbloom (UPenn) 1941 Richard F. Arens (UCLA) 1941 Samuel I. Askovitz (UPenn) 1941 Harold Victor Lyons (Toronto) 1942 Harvey Cohn (City College of NY) 1942 Melvin A. Preston (Toronto) 1942 Warren S. Loud (MIT) 1942 Donald A. Fraser (Toronto) 1946 Eugenio Calabi (MIT) 1946 Felix Browder (MIT) 1946 J. Arthur Greenwood (Harvard) 1946 Maxwell A. Rosenlicht (Columbia) 1946, 1947 Clarence Wilson Hewlett, Jr. (Harvard) 1947 William Turanski (UPenn) 1947 Eoin L. Whitney (Alberta) 1947, 1948 W. Forrest Stinespring (Harvard) 1947, 1949 George F. D. Duff (Toronto) 1948 Harry Gonshor (McGill) 1948 Leonard Geller (Brooklyn College) 1948 Robert L. Mills (Columbia) 1948 Donald J. Newman (City College of NY) 1948, 1949, 1950 Ariel Zemach (Harvard) 1949 David L. Yarmush (Harvard) 1949 John W. Milnor (Princeton) 1949, 1950 John P. Mayberry (Toronto) 1950 Richard J. Semple (Toronto) 1950 Z. Alexander Melzak (British Columbia) 1950 Arthur P. Dempster (Toronto) 1951 Harold Widom (City College of NY) 1951 Herbert C. Kranzer (NYU) 1951 Peter John Redmond (Cooper Union) 1951 James B. Herreshoff IV (UC Berkeley) 1951, 1952, 1953 Eugene R. Rodemich (Washington U in StL) 1952 Gerhard Rayna (Harvard) 1952 Richard G. Swan (Princeton) 1952 Walter Lewis Baily, Jr. (MIT) 1952 Marshall L. Freimer (Harvard) 1953 Norman Bauman (Harvard) 1953 Tai Tsun Wu (Minnesota) 1953 Samuel Jacob Klein (City College of NY) 1953, 1959, 1960 Benjamin Muckenhoupt (Harvard) 1954 James Daniel Bjorken (MIT) 1954 Leonard Evens (Cornell) 1954 William P. Hanf (UC Berkeley) 1954 Kenneth G. Wilson (Harvard) 1954, 1956 Howard C. Rumsey, Jr. (Caltech) 1955 Jack Towber (Brooklyn College) 1955 David B. Mumford (Harvard) 1955, 1956 Trevor Barker (Kenyon) 1955, 1956 Everett C. Dade (Harvard) 1955, 1957 Richard Michael Friedberg (Harvard) 1956 David M. Bloom (Columbia) 1956, 1957 J. Ian Richards (Minnesota) 1957 Richard T. Bumby (MIT) 1957 Rohit J. Parikh (Harvard) 1957 David R. Brillinger (Toronto) Spring 1958 Donald J. C. Bures (Queen's) Spring 1958 Lawrence A. Shepp (Brooklyn Polytech) Spring 1958 Richard M. Dudley (Harvard) Spring 1958 Joseph Lipman (Toronto) Spring 1958, Fall 1958 Alan Gaisford Waterman (San Diego State) Fall 1958 John Rex Forrester Hewett (Toronto) Fall 1958 Robert C. Hartshorne (Harvard) Fall 1958 Alfred W. Hales (Caltech) Fall 1958, 1959 Daniel G. Quillen (Harvard) 1959 Donald Passman (Brooklyn Polytech) 1959 Donald S. Gorman (Harvard) 1959 Martin Isaacs (Brooklyn Polytech) 1959 Stephen L. Adler (Harvard) 1959 Stephen Lichtenbaum (Harvard) 1959 Jon H. Folkman (UC Berkeley) 1960 Louis Jaeckel (UCLA) 1960 Melvin Hochster (Harvard) 1960 William R. Emerson (Caltech) 1960 Barry Wolk (Manitoba) 1961 Elwyn R. Berlekamp (MIT) 1961 Edward Anton Bender (Caltech) 1961, 1962 John Hathaway Lindsey (Caltech) 1961, 1962 William C. Waterhouse (Harvard) 1961, 1962 John William Wood (Harvard) 1962 Robert S. Strichartz (Dartmouth) 1962 Joel H. Spencer (MIT) 1963 Lawrence A. Zalcman (Dartmouth) 1963 Lawrence J. Corwin (Harvard) 1963 Robert E. Greene (Michigan State) 1963 Stephen E. Crick, Jr. (Michigan State) 1963 Barry B. MacKichan (Harvard) 1964 Fred William Roush (UNC Chapel Hill) 1964 Roger E. Howe (Harvard) 1964 Rufus (Robert) Bowen (UC Berkeley) 1964, 1965 Vern Sheridan Poythress (Caltech) 1964 Andreas R. Blass (Detroit) 1965 Barry Simon (Harvard) 1965 Daniel Fendel (Harvard) 1965 Lon M. Rosen (Toronto) 1965 Marshall W. Buck (Harvard) 1966 Robert E. Maas (Santa Clara) 1966 Robert S. Winternitz (MIT) 1966 Theodore C. Chang (MIT) 1966 Richard C. Schroeppel (MIT) 1966, 1967 David R. Haynor (Harvard) 1967 Dennis A. Hejhal (Chicago) 1967 Don B. Zagier (MIT) 1967 Peter L. Montgomery (UC Berkeley) 1967 Dean G. Huffman (Yale) 1968 Gerald S. Gras (MIT) 1968 Neal Koblitz (Harvard) 1968 Gerald A. Edgar (UC Santa Barbara) 1968, 1969 Don Coppersmith (MIT) 1968, 1969, 1970, 1971 Alan R. Beale (Rice) 1969 Steven K. Winkler (MIT) 1969, 1970 Robert A. Oliver (Chicago) 1969, 1970 Jeffrey Lagarias (MIT) 1970 Jockum Aniansson (Yale) 1970 Arthur Rubin (Purdue, Caltech) 1970, 1971, 1972, 1973 Dale Peterson (Yale) 1971 David Shucker (Swarthmore) 1971 Robert Israel (Chicago) 1971 Michael Yoder (Caltech) 1971, 1972 Arthur Rothstein (Reed) 1972 David Vogan (Chicago) 1972 Dean Hickerson (UC Davis) 1972 Ira Gessel (Harvard) 1972 Angelos J. Tsirimokos (Princeton) 1973 Matthew L. Ginsberg (Wesleyan) 1973 Peter G. De Buda (Toronto) 1973 David J. Anick (MIT) 1973, 1975 Grant M. Roberts (Waterloo) 1974 James B. Saxe (Union) 1974 Karl C. Rubin (Princeton) 1974 Philip N. Strenski (Armstrong State) 1974 Thomas G. Goodwillie (Harvard) 1974, 1975 Ernest S. Davis (MIT) 1975 Franklin T. Adams (Chicago) 1975 Christopher L. Henley (Caltech) 1975, 1976 David J. Wright (Cornell) 1976 Nathaniel S. Kuhn (Harvard) 1976 Paul M. Herdig (Case Western Reserve) 1976 Philip I. Harrington (Washington U in StL) 1976 Steven T. Tschantz (UC Berkeley) 1976, 1978 Adam L. Stephanides (Chicago) 1977, 1981 Michael Roberts (MIT) 1977 Paul A. Vojta (Minnesota) 1977 Stephen W. Modzelewski (Harvard) 1977 Russell D. Lyons (Case Western Reserve) 1977, 1978 Mark R. Kleiman (Princeton) 1978 Peter W. Shor (Caltech) 1978 Randall L. Dougherty (UC Berkeley) 1978, 1979, 1980 Charles H. Walter (Princeton) 1979 Mark G. Pleszkoch (Virginia) 1979 Miller Puckette (MIT) 1979 Richard Mifflin (Rice) 1979 Daniel J. Goldstein (Chicago) 1980 Laurence E. Penn (Harvard) 1980 Michael Raship (Harvard) 1980 Eric D. Carlson (Michigan State) 1980, 1982, 1983 Robin A. Pemantle (UC Berkeley) 1981 Scott R. Fluhrer (Case Western Reserve) 1981 David W. Ash (Waterloo) 1981, 1982, 1983 Michael J. Larsen (Harvard) 1981, 1983 Brian R. Hunt (Maryland) 1982 Edward A. Shpiz (Washington U in StL) 1982 Noam D. Elkies (Columbia) 1982, 1983, 1984 Gregg N. Patruno (Princeton) 1983 Benji N. Fisher (Harvard) 1984 Daniel W. Johnson (Rose-Hulman Tech) 1984 Richard A. Stong (Washington U in StL) 1984 Michael Reid (Harvard) 1984, 1987 Everett W. Howe (Caltech) 1985 Keith A. Ramsay (Chicago) 1985 Martin V. Hildebrand (Williams) 1985 Douglas S. Jungreis (Harvard) 1985, 1986 Bjorn M. Poonen (Harvard) 1985, 1986, 1987, 1988 David I. Zuckerman (Harvard) 1986 Waldemar P. Horwat (MIT) 1986 David J. Grabiner (Princeton) 1986, 1987, 1988 David J. Moews (Harvard) 1986, 1987, 1988 Constantin S. Teleman (Harvard) 1987 John S. Tillinghast (UC Davis) 1987 Jeremy A. Kahn (Harvard) 1988 Ravi D. Vakil (Toronto) 1988, 1989, 1990, 1991 Andrew H. Kresch (Yale) 1989 Christos A. Athanasiadis (MIT) 1989 Colin M. Springer (Waterloo) 1989 Sihao Wu (Yale) 1989 William P. Cross (Caltech) 1989 Jordan Lampe (UC Berkeley) 1990 Raymond M. Sidney (Harvard) 1990 Eric K. Wepsic (Harvard) 1990, 1991 Jordan S. Ellenberg (Harvard) 1990, 1992 Joshua B. Fischman (Princeton) 1991 Xi Chen (Missouri–Rolla) 1991 Samuel A. Kutin (Harvard) 1991, 1992 Jeffrey M. Vanderkam (Duke) 1992 Serban M. Nacu (Harvard) 1992 Adam M. Logan (Princeton) 1992, 1993 Craig B. Gentry (Duke) 1993 Wei-Hwa Huang (Caltech) 1993 J. P. Grossman (Toronto) 1993, 1994, 1995 Kiran S. Kedlaya (Harvard) 1993, 1994, 1995 Lenhard L. Ng (Harvard) 1993, 1994, 1995 William R. Mann (Princeton) 1994 Jeremy L. Bem (Cornell) 1994, 1996 Sergey V. Levin (Harvard) 1995 Yevgeniy Dodis (NYU) 1995 Dragos N. Oprea (Harvard) 1996 Ioana Dumitriu (NYU) 1996 Robert D. Kleinberg (Cornell) 1996 Stephen S. Wang (Harvard) 1996 Daniel K. Schepler (Washington U in StL) 1996, 1997 Ovidiu Savin (Pittsburgh) 1997 Patrick K. Corn (Harvard) 1997 Samuel Grushevsky (Harvard) 1997 Mike L. Develin (Harvard) 1997, 1998 Ciprian Manolescu (Harvard) 1997, 1998, 2000 Ari M. Turner (Princeton) 1998 Nathan G. Curtis (Duke) 1998 Kevin D. Lacker (Duke) 1998, 2001 Christopher C. Mihelich (Harvard) 1999 Colin A. Percival (Simon Fraser) 1999 Davesh Maulik (Harvard) 1999 Derek I.E. Kisman (Waterloo) 1999 Sabin Cautis (Waterloo) 1999 Abhinav Kumar (MIT) 1999, 2000 Pavlo Pylyavskyy (MIT) 2000 Alexander B. Schwartz (Harvard) 2000, 2002 Gabriel D. Carroll (UC Berkeley, Harvard) 2000, 2001, 2002, 2003 George Lee, Jr. (Harvard) 2001 Jan K. Siwanowicz (City College of NY) 2001 Reid W. Barton (MIT) 2001, 2002, 2003, 2004 Deniss Cebikins (MIT) 2002 Melanie E. Wood (Duke) 2002 Ralph C. Furmaniak (Waterloo) 2003 Ana Caraiani (Princeton) 2003, 2004 Daniel M. Kane (MIT) 2003, 2004, 2005, 2006 Vladimir V. Barzov (MIT) 2004 Aaron Pixton (Princeton) 2004, 2005, 2007 Oleg I. Golberg (MIT) 2005 Matthew M. Ince (MIT) 2005 Ricky I. Liu (Harvard) 2005 Tiankai Liu (Harvard) 2005, 2006 Hansheng Diao (MIT) 2006 Po-Ru Loh (Caltech) 2006 Yufei Zhao (MIT) 2006, 2008, 2009 Jason C. Bland (Caltech) 2007 Brian R. Lawrence (Caltech) 2007, 2008, 2010, 2011 Qingchun Ren (MIT) 2007, 2009 Xuancheng Shao (MIT) 2007 Arnav Tripathy (Harvard) 2007, 2008, 2009 Seok Hyeong Lee (Stanford) 2008, 2010, 2011 Bohua Zhan (MIT) 2008 William A. Johnson (U of Washington) 2009 Xiaosheng Mu (Yale) 2009, 2011 Yu Deng (MIT) 2010 Colin P. Sandon (MIT) 2010 Alex (Lin) Zhai (Harvard) 2010 Samuel S. Elder (Caltech) 2011 Evan M. O'Dorney (Harvard) 2011, 2012, 2013 Benjamin P. Gunby (MIT) 2012 Eric K. Larson (Harvard) 2012 Mitchell M. Lee (MIT) 2012, 2013 Zipei Nie (MIT) 2012, 2013, 2014 Bobby C. Shen (MIT) 2013, 2014 David H. Yang (MIT) 2013, 2014, 2015 Ravi Jagadeesan (Harvard) 2014 Mark A. Sellke (MIT) 2014 Lingfu Zhang (MIT) 2014 Pakawut Jiradilok (Harvard) 2015 Bumsoo Kim (Princeton) 2015 Gyujin Oh (Stanford) 2015 Daniel Spivak (Waterloo) 2015 Yunkun Zhou (MIT) 2015, 2016, 2017 Joshua D. Brakensiek (Carnegie Mellon) 2016 Dong Ryul Kim (Harvard) 2016, 2018 Thomas E. Swayze (Carnegie Mellon) 2016 Samuel Zbarsky (Carnegie Mellon) 2016 David Stoner (Harvard) 2017, 2018 Ömer Cerrahoğlu (MIT) 2017 Jiyang Gao (MIT) 2017 Junyao Peng (MIT) 2017 Ashwin Sah (MIT) 2017, 2019 Yuan Yao (MIT) 2018, 2019 Shengtong Zhang (MIT) 2018, 2019, 2021 Shyam Narayanan (Harvard) 2018 Kevin Sun (MIT) 2019 Daniel Zhu (MIT) 2019, 2021, 2022 Andrew Gu (MIT) 2021 Michael Ren (MIT) 2021 Edward Wan (MIT) 2021 Mingyang Deng (MIT) 2022 Papon Lapate (MIT) 2022 Brian Liu (MIT) 2022 Luke Robitaille (MIT) 2022 Elizabeth Lowell Putnam Award winners Since 1992, the Elizabeth Lowell Putnam Award has been available to be awarded to a female participant with a high score, with three awards being made for the first time in 2019.[11] The year(s) in which they were Fellows are in bold. Ioana Dumitriu was the first woman to become a Putnam Fellow, in 1996. NameSchoolYear (s) Dana PascoviciDartmouth1992 Ruth A. Britto-PacumioMIT1994 Ioana DumitriuNYU1995, 1996, 1997 Wai Ling YeeWaterloo1999 Melanie E. WoodDuke2001, 2002 Ana CaraianiPrinceton2003, 2004 Alison B. MillerHarvard2005, 2006, 2007 Viktoriya KrakovnaToronto2008 Yinghui WangMIT2011 Fei SongVirginia2011 Xiao WuYale2013 Simona DiaconuPrinceton2016 Ni YanUCLA2017 Danielle WangMIT2015, 2018 Laura PiersonHarvard2019 Qi QiMIT2019 Hanzhi ZhengStanford2019 Dain KimMIT2021 Binwei YanMIT2022 See also • List of mathematics awards References 1. "William Lowell Putnam Competition Prizes". math.scu.edu. Santa Clara University. 2. "Putnam Competition \| Mathematical Association of America". www.maa.org. Retrieved 18 April 2018. 3. David Arney; George Rosenstein. "The Harvard-United States Military Academy Mathematics Competition of 1933: Genesis of the William Lowell Putnam Mathematical Competition". www.westpoint.edu. Retrieved 18 April 2018. 4. "History of the Putnam Prize". Santa Clara University – Mathematics & Computer Science. Retrieved 7 November 2021. 5. "Putnam Competition \| Mathematical Association of America". www.maa.org. Retrieved Mar 27, 2020. 6. "82nd Putnam Competition Announces Top Students in Undergraduate Mathematics". newsroom.maa.org. Retrieved 2022-12-01. 7. "MIT students dominate annual Putnam Mathematical Competition". MIT News \| Massachusetts Institute of Technology. Retrieved 2022-12-01. 8. Fujimori, Jessica (3 May 2016). "At Putnam, students rise to the challenge". news.mit.edu. MIT News. Retrieved 18 April 2018. 9. Weisstein, Eric (17 April 2018). "Putnam Mathematical Competition". mathworld.wolfram.com. Wolfram Web Resources. Retrieved 18 April 2018. 10. "List of Previous Putnam Winners" (PDF). Mathematical Association of America. Retrieved December 30, 2020. 11. "Top Students and Teams Announced in Putnam Competition". www.maa.org. Mathematical Association of America. External links Wikimedia Commons has media related to Putnam Fellows. • William Lowell Putnam Mathematical Competition results • William Lowell Putnam Competition problems, solutions, and results archive • Archive of Problems 1938–2003 • Searchable data base for information about careers of Putnam Fellows • A comprehensive history of the Putnam competition An electronic update of Gallian's 2004 paper (PDF) American mathematics Organizations • AMS • MAA • SIAM • AMATYC • AWM Institutions • AIM • CIMS • IAS • ICERM • IMA • IPAM • MBI • SLMath • SAMSI • Geometry Center Competitions • MATHCOUNTS • AMC • AIME • USAMO • MOP • Putnam Competition • Integration Bee
William Kantor William M. Kantor (born September 19, 1944) is an American mathematician who works in finite group theory and finite geometries, particularly in computational aspects of these subjects. William Kantor William Kantor in Oberwolfach 2011 Born William M. Kantor (1944-09-19) September 19, 1944 Other namesBill Kantor Scientific career FieldsMathematics InstitutionsUniversity of Oregon, University of Illinois at Chicago Thesis2-transitive symmetric designs (1968) Doctoral advisorsPeter Dembowski, Richard Hubert Bruck Notable studentsMark Ronan Websitehttps://pages.uoregon.edu/kantor/ Education and career Kantor graduated with a bachelor's degree from Brooklyn College in 1964.[1] He went on to graduate studies at the University of Wisconsin, receiving his PhD in 1968 under the supervision of Peter Dembowski and R. H. Bruck.[2] He then worked at the University of Illinois at Chicago from 1968 to 1971 before moving in 1971 to the University of Oregon, where he remained for the rest of his career. Kantor's research mostly involves finite groups, often in relation to finite geometries and computation. Algorithms developed by him have found use, for example, in the GAP computer algebra system.[3] Kantor has written over 170 papers,[4] and has advised 7 PhD students.[2] Significant publications Books and monographs • Kantor, W. M. (1979). Classical groups from a nonclassical viewpoint. Oxford University, Mathematical Institute, Oxford. MR 0578539. • Kantor, William M.; Seress, Ákos (2001). "Black box classical groups". Memoirs of the American Mathematical Society. 149 (708): 0. CiteSeerX 10.1.1.294.1011. doi:10.1090/memo/0708. ISSN 0065-9266. MR 1804385. Journal articles • Calderbank, R.; Kantor, W. M. (1986). "The Geometry of Two-Weight Codes". Bulletin of the London Mathematical Society. 18 (2): 97–122. doi:10.1112/blms/18.2.97. ISSN 0024-6093. MR 0818812. • Kantor, William M.; Lubotzky, Alexander (1990). "The probability of generating a finite classical group". Geometriae Dedicata. 36 (1). doi:10.1007/BF00181465. ISSN 0046-5755. MR 1065213. S2CID 6771. • Kantor, William M. (1985). "Homogeneous designs and geometric lattices". Journal of Combinatorial Theory, Series A. 38 (1): 66–74. doi:10.1016/0097-3165(85)90022-6. ISSN 0097-3165. MR 0773556. Awards and honors • In 2013, Kantor was named a fellow of the American Mathematical Society as a member of the inaugural class of fellows.[5] • In 2004, a conference "Finite geometries, groups, and computation" was held in honor of Kantor's 60th birthday.[6] • In 1998, Kantor gave an invited talk at the International Congress of Mathematicians in Berlin.[7][8] References 1. William Kantor on LinkedIn 2. William M. Kantor at the Mathematics Genealogy Project 3. "GAP source code". Official GAP system website. Retrieved September 27, 2019. 4. "William Kantor author profile". MathSciNet. American Mathematical Society. 5. "Fellows of the AMS: Inaugural Class" (PDF). Notices of the American Mathematical Society. 60 (5): 631–637. May 2013. Retrieved September 24, 2019. 6. Hulpke, Alexander; Liebler, Robert; Penttila, Tim; Seress, Ákos, eds. (2006). Finite geometries, groups, and computation. Berlin New York: Walter de Gruyter. ISBN 978-3-11-019974-1. MR 2256928. OCLC 181078514. 7. "Invited Lectures in Algebra at the Berlin ICM". Retrieved September 24, 2019. 8. Kantor, William M. (1998). "Simple groups in computational group theory". Doc. Math. (Bielefeld) Extra Vol. ICM Berlin, 1998, vol. II. pp. 77–86. Authority control International • ISNI • VIAF National • Norway • France • BnF data • Catalonia • Germany • Israel • United States • Netherlands Academics • DBLP • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie Other • IdRef
William Makeham William Matthew Makeham (11 September 1826 – 17 November 1891) was an English actuary and mathematician. Makeham was responsible for proposing the age-independent Makeham term in the Gompertz–Makeham law of mortality that, together with the exponentially age-dependent Gompertz term, was one of the most effective theories to describe human mortality.[1] Makeham was responsible for two important studies on human mortality:[2] • Makeham (1860). "On the Law of Mortality and the Construction of Annuity Tables". The Assurance Magazine, and Journal of the Institute of Actuaries. 8 (6): 301–310. doi:10.1017/S204616580000126X. • Makeham (1874). "On an Application of the Theory of the Composition of Decremental Forces". Journal of the Institute of Actuaries. 18: 317–322. He had one wife, Hepzibah Reed, and seven children, William, Amy, Elizabeth, Thomas, Frederick, Emily, and George. References 1. Dale, Andrew I. (1999). A History of Inverse Probability: From Thomas Bayes to Karl Pearson (2nd ed.). Springer. pp. 489 ff. ISBN 9780387988078. 2. Wolfram Mathworld, "Makeham Curve"
William Marrat William Marrat (1772–1852) was an English printer, publisher and educator, known as a mathematician and antiquarian. Life Born at Sibsey, Lincolnshire, on 6 April 1772, Marrat was self-taught through wide reading and study of modern language. While at Boston, Lincolnshire, he for some years worked as a printer and publisher. In 1811–12 he, in conjunction with Pishey Thompson, ran The Enquirer, or Literary, Mathematical, and Philosophical Repository, Boston.[1] At other times Marrat was a teacher of mathematics, in Lincolnshire and elsewhere. He lived in New York City from 1817 to 1820, and edited there The Scientific Journal (imprint "Perth Amboy, N. J. and New York", 1818, nine numbers). He returned to England, and at Liverpool, where he settled in 1821.[1] George Boole taught at his school, in 1833.[2] From 1833 to 1836 Marrat was mathematical tutor in a school at Exeter, but on the death of his wife he returned to Liverpool. He died suddenly there on 26 March 1852, and was buried at the necropolis near that city.[1] Works Marrat was for fifty years a contributor to mathematical serials, including The Ladies' Diary and The Gentlemen's Diary, The Receptacle, The Student, and the Leeds Correspondent. His first book was An Introduction to the Theory and Practice of Mechanics, Boston, 1810, pp. 468. During 1814–16 he wrote The History of Lincolnshire, which came out in parts, and after three volumes had been published, it was stopped: Marrat alleged this was a consequence of Sir Joseph Banks's refusal to allow access to his papers. In 1816 his Historical Description of Stamford was published at Lincoln.[1] An anonymous Geometrical System of Conic Sections, Cambridge, 1822, was ascribed to Marrat in the catalogue of the Liverpool Free Library. He compiled Lunar Tables, Liverpool, 1823, and wrote The Elements of Mechanical Philosophy, 1825. At this period he compiled the Liverpool Tide Table, and was a contributor to Blackwood's Magazine.[1] Family His son Frederick Price Marrat was known as a conchologist.[1] Notes 1. Lee, Sidney, ed. (1893). "Marrat, William" . Dictionary of National Biography. Vol. 36. London: Smith, Elder & Co. 2. Des MacHale (1985). George Boole: his life and work. Boole Press. p. 19. ISBN 978-0-906783-05-4. Attribution • This article incorporates text from a publication now in the public domain: Lee, Sidney, ed. (1893). "Marrat, William". Dictionary of National Biography. Vol. 36. London: Smith, Elder & Co.
W. T. Martin William Ted Martin (June 4, 1911 – May 30, 2004) was an American mathematician, who worked on mathematical analysis, several complex variables, and probability theory. He is known for the Cameron–Martin theorem and for his 1948 book Several complex variables, co-authored with Salomon Bochner. W. T. Martin Born(1911-06-04)June 4, 1911 Arkansas DiedMay 30, 2004(2004-05-30) (aged 92) NationalityAmerican Alma materUniversity of Illinois Scientific career FieldsMathematics Doctoral advisorRobert Carmichael Biography He was born on June 4, 1911, in Arkansas. W. T. Martin received his B.A. in mathematics from the University of Arkansas in 1930. He did graduate work at the University of Illinois at Urbana–Champaign, where he received his M.A. in 1931 and his Ph.D. in 1934 under the direction of Robert Carmichael.[1] He studied under a National Research Council postdoctoral fellowship at the Institute for Advanced Study in Princeton from 1934 to 1936.[2] In 1936 Martin became an instructor at MIT and in 1938 a faculty member there. He collaborated with several fellow MIT faculty members, notably Norbert Wiener, R. H. Cameron, Stefan Bergman, and Salomon Bochner. During the 1940s Martin and R. H. Cameron wrote a series of papers extending Norbert Wiener's early work on mathematical models of Brownian motion.[3] During the 1950s W. T. Martin wrote with Salomon Bochner a series of papers that proved basic results in the theory of several complex variables. Martin was the department head for the MIT mathematics department from 1947 to 1968. During this time he oversaw the hiring of 24 faculty members in the mathematics department. He initiated MIT's C. L. E. Moore Instructorship Program in 1949.[4] He spent his entire career at MIT, except for the years from 1943 to 1946, when he left MIT to become the head of the mathematics department of Syracuse University[5] and, in the academic year 1951–1952, when he was on sabbatical at the Institute for Advanced Study.[2] Martin did important editorial work and co-authored three influential books: Several complex variables (1948), Elementary differential equations (1956), and Differential space, quantum space, and prediction (1966).[5] Beginning in 1961, Martin involved himself in developing math curricula for English-speaking African nations, serving as chair of the Steering Committee of the Education Development Center's African Mathematics Program and visited Africa regularly from 1961 to 1975.[6] He retired to Block Island and died on May 30, 2004.[5] Selected publications • with Norbert Wiener: Wiener, N.; Martin, W. T. (1937). "Taylor's series of entire functions of smooth growth". Duke Math. J. 3 (2): 213–223. doi:10.1215/s0012-7094-37-00314-4. MR 1545980. • with Norbert Wiener: "Taylor's series of smooth growth in the unit circle". Duke Math. J. 4 (2): 384–392. 1938. doi:10.1215/s0012-7094-38-00430-2. MR 1546059. • with Stefan Bergman: Bergman, Stefan; Martin, W. T. (1940). "A modified moment problem in two variables". Duke Math. J. 6 (2): 389–407. doi:10.1215/s0012-7094-40-00630-5. MR 0001993. • Martin, W. T. (1944). "Mappings by means of systems of analytic functions of several complex variables". Bull. Amer. Math. Soc. 50 (1): 5–19. doi:10.1090/s0002-9904-1944-08043-9. MR 0009641. • with R. H. Cameron: Cameron, R. H.; Martin, W. T. (1944). "Transformations of Wiener integrals under translations". The Annals of Mathematics. 45 (2): 386–396. doi:10.2307/1969276. JSTOR 1969276. (2nd most cited of all Cameron and Martin's papers) • with R. H. Cameron: Cameron, R. H.; Martin, W. T. (1947). "The orthogonal development of non-linear functionals in series of Fourier–Hermite functionals". The Annals of Mathematics. 48 (2): 385–392. doi:10.2307/1969178. JSTOR 1969178. (most cited of all Cameron and Martin's papers) • with Salomon Bochner: Several complex variables. Princeton, N. J.: Princeton University Press. 1948. (216 pages) • with Eric Reissner: Elementary differential equations. Cambridge, Massachusetts: Addison-Wesley. 1956, 260 pages{{cite book}}: CS1 maint: postscript (link); 2nd edn. Reading, Mass.: Addison-Wesley. 1961, 331 pages{{cite book}}: CS1 maint: postscript (link); Reprinting of 2nd edn. NY: Dover. 1986. ISBN 0486650243. • as co-editor with editors Norbert Wiener, Armand Siegel, and Bayard Rankin: Differential space, quantum systems, and prediction. Cambridge, Massachusetts: M.I.T. Press. 1966. (176 pages, essays) References 1. W. T. Martin at the Mathematics Genealogy Project 2. Martin, William T., Community of Scholars Profile, IAS 3. Kac, Mark (1985). Enigmas of Chance. New York: Harper & Row. p. 113. ISBN 0520059867. 4. Jackson, Allyn (Sep 2004). "William Ted Martin (1911 – 2004)" (PDF). Notices of the AMS. 51 (8): 919. 5. "Longtime math department head Ted Martin dies at age 92". MITnews. 4 June 2004. 6. Jackson, Allyn (Sep 2004). "William Ted Martin (1911 – 2004)" (PDF). Notices of the AMS. 51 (8): 919. Authority control International • FAST • ISNI • VIAF National • France • BnF data • Germany • Israel • United States • Czech Republic • Netherlands Academics • CiNii • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie Other • SNAC • IdRef
William G. McCallum William G. McCallum (born 1956 in Sydney, Australia) is a University Distinguished Professor of Mathematics and was Head of the Department of Mathematics at the University of Arizona from 2009 to 2013. Education and professional work He was educated at North Sydney Boys High School.[1] He received his Ph.D. in Mathematics from Harvard University in 1984, under the supervision of Barry Mazur. After spending two years at the University of California, Berkeley, and one at the Mathematical Sciences Research Institute in Berkeley, he joined the faculty at the University of Arizona in 1987. In 1989 he joined the Harvard calculus consortium, and is the lead author of the consortium's multivariable calculus and college algebra texts. In 1993–94 he spent a year at the Institut des Hautes Études Scientifiques, and in 1995–96 he spent a year at the Institute for Advanced Study on a Centennial Fellowship from the American Mathematical Society. In 2006 he founded the Institute for Mathematics & Education at the University of Arizona. He was Director of the Institute until 2009 and again starting in 2013. In 2009–2010 he was one of the lead writers for the Common Core State Standards in Mathematics. His professional interests include arithmetical algebraic geometry and mathematics education. Selected honors and awards • 2012: Fellow of the American Mathematical Society.[2] • 2012: The Mary P. Dolciani Award, administered by the Mathematical Association of America • 2012: The American Mathematical Society Distinguished Public Service Award • 2006: University of Arizona College of Science Galileo Circle Fellow. • 2005: National Science Foundation’s Director's Award for Distinguished Teaching Scholars • 1996: The University of Arizona College of Science Innovation in Teaching Award • 1995: The American Mathematical Society Centennial Research Fellowship. Current projects • Institute for Mathematics and Education • Common Core State Standards in Mathematics • Illustrative Mathematics Project • Standards Progressions for the Common Core • Tools for the Common Core Blog • The Klein Project • Mathematical Models at the University of Arizona References 1. NSBHS Higher School Certificate 1973 2. List of Fellows of the American Mathematical Society, retrieved 2015-01-12 External links • The U.S. Common Core State Standards, paper presented at ICME 12, Seoul, Korea (slides for this talk) • Restoring and Balancing, in Usiskin, Anderson, and Zotto (eds), Future Curricular Trends in School Algebra and Geometry, Information Age Publishing (2010) Authority control International • ISNI • VIAF National • France • BnF data • Catalonia • Israel • United States Academics • CiNii • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
William Menasco William W. Menasco is a topologist and a professor at the University at Buffalo. He is best known for his work in knot theory. Biography Menasco received his B.A. from the University of California, Los Angeles in 1975, and his Ph.D. from the University of California, Berkeley in 1981, where his advisor was Robion Kirby. He served as assistant professor at Rutgers University from 1981 to 1984. He then taught as a visiting professor at the University at Buffalo where he became an assistant professor in 1985, an associate professor in 1991. In 1994 he became a professor at the University at Buffalo where he currently serves.[1] Work Menasco proved that a link with an alternating diagram, such as an alternating link, will be non-split if and only if the diagram is connected. Menasco, along with Morwen Thistlethwaite proved the Tait flyping conjecture, which states that, given any two reduced alternating diagrams $D1,D2$ of an oriented, prime alternating link, $D1$ may be transformed to $D2$ by means of a sequence of certain simple moves called flypes.[2] References 1. Homepage of William W. Menasco (continued) 2. Weisstein, Eric W. "Tait's Knot Conjectures". MathWorld. Authority control International • ISNI • VIAF • WorldCat National • Germany • United States • Czech Republic Academics • CiNii • DBLP • MathSciNet • Mathematics Genealogy Project • Scopus • zbMATH Other • IdRef
William Mollison (mathematician) William Loudon Mollison (19 September 1851 – 10 March 1929)[1] was a Scottish mathematician and academic.[2] From 1915 to 1929, he was Master of Clare College, Cambridge.[3] William Mollison 38th Master of Clare College, Cambridge In office March 1915 – March 1929 Preceded byEdward Atkinson Succeeded byGodfrey Wilson Personal details Born(1851-09-19)19 September 1851 Aberdeen, Scotland Died10 March 1929(1929-03-10) (aged 77) London, England Resting placeAscension Parish Burial Ground, Cambridge SpouseEllen Mayhew ChildrenThree EducationAberdeen Grammar School Alma materUniversity of Aberdeen Clare College, Cambridge Early life and education Mollison was born on 19 September 1851 in Aberdeen, Scotland.[4][5] He was educated at Aberdeen Grammar School, then an all-boys grammar school.[6] He studied mathematics and natural philosophy at the University of Aberdeen, graduating in 1872 with a first class degree.[7] That year, he was awarded the Ferguson Scholarship by Aberdeen and matriculated into Clare College, Cambridge to continue his mathematical studies.[4][6] He became a Foundation Scholar in 1873.[4] His private tutor while at Cambridge was Edward Routh.[8] He graduated from the University of Cambridge in 1876 as the Second Wrangler.[4][8][9] Career On 29 April 1876, Mollison was elected a Fellow of Clare College, Cambridge.[10] He was an examiner for the University of St Andrews between 1876 and 1880.[5] He was a mathematics lecturer at Jesus College, Cambridge from 1877 to 1882, and at Clare College from 1882.[6] In addition to his college teaching, he was a private tutor or "coach" in mathematics.[4] Due to ill health, he moved from teaching a large number of students, privately and through his college, into administration.[4] He was appointed junior tutor of Clare College in 1880,[6] and was made its senior tutor in May 1894.[11] He was elected a member of the Council of the Senate of the University Of Cambridge in 1892,[4] and appointed Secretary of the General Board of Studies of the University in 1904: he stepped down from both these posts in 1920.[6] He served as locum tenens for the then Master (Edward Atkinson) from 1913 to 1915.[6] Mollison was unanimously elected as Atkinson successor as the 38th Master of Clare College, Cambridge in March 1915.[12] Personal life Mollison was married to Ellen Mayhew.[6] They had one son and two daughters,[13][1] one of whom pre-deceased him.[1][6] His wife died in 1917, and he provided the endowment for the Mayhew Prize, a mathematics prize awarded by the University of Cambridge, in her honour.[6] His son, William Mayhew Mollison, was a distinguished ear, nose and throat surgeon,[14] and his son Patrick Mollison, a noted haematologist.[15] Mollison died on 10 March 1929 in London, England; he was aged 77.[7][6] His funeral was held at the chapel of Clare College, Cambridge,[4] and he was buried in the Ascension Parish Burial Ground alongside his wife.[4] References 1. { The Times (London, England), Monday, 11 March 1929; pg. 21; Issue 45148.} 2. "Mollison, William Loudon". Who's Who & Who Was Who. Vol. 1920–2016 (April 2014 online ed.). A & C Black. Retrieved 18 November 2018. (Subscription or UK public library membership required.) 3. Alumni Cantabrigienses: A Biographical List of All Known Students, Graduates and Holders of Office at the University of Cambridge, from the Earliest Times to 1900, John Venn/John Archibald Venn Cambridge University Press > (10 volumes 1922 to 1953) Part II. 1752–1900 Vol. iv. Kahlenberg – Oyler, (1947) p434 4. Knewstubb, Peter (2012). "William Loudon Mollison (1851–1929)" (PDF). Clare Association Annual: 65–67. Retrieved 18 November 2018. 5. The New International Year Book. New York City, NY: Dodd, Mead and Company. 1930. p. 581. 6. "Mollison, William Loudon". Who Was Who. Oxford University Press. 1 December 2007. doi:10.1093/ww/9780199540884.013.U214230. Retrieved 18 November 2018. 7. "Births, Deaths, Marriages and Obituaries – Master of Clare". Aberdeen Press and Journal. No. 23148. 12 March 1929. p. 11. 8. "CAMBRIDGE UNIVERSITY – Mathematical Tripos". The Leeds Mercury. No. 11795. 29 January 1876. 9. Tanner, Joseph Robson (1917). The historical register of the University of Cambridge, being a supplement to the Calendar with a record of University offices, honours and distinctions to the year 1910 (PDF). Cambridge University Press. Retrieved 18 November 2018. 10. "Scotchmen at Cambridge University". The Dundee Courier & Argus. No. 7103. 1 May 1876. 11. "The Tutorship of Clare". Cambridge Independent Press. No. 4065. 4 May 1894. p. 5. 12. "New Master of Clare". Cambridge Independent Press. No. 5155. 2 April 1915. p. 5. 13. "1891 Census Swanage, Dorset, England RG12/1641 page 2". Retrieved 16 September 2020. 14. "Mollison, William Mayhew (1878–1967)". Plarr's Lives of the Fellows. The Royal College of Surgeons of England. 18 September 2014. Retrieved 18 November 2018. 15. "Professor Patrick Mollison". The Daily Telegraph. 18 January 2012. Retrieved 18 November 2018. External links • Portraits of William Loudon Mollison at the National Portrait Gallery, London Masters of Clare College, Cambridge • Walter de Thaxted • Ralph Kerdington • Nicholas de Brunne • John de Donewich • John de Charteresse • William Radwinter • William Wymbyll • William Gull • William Wilflete • John Millington • Thomas Stoyll • Richard Stubbs • Gabriel Silvester • William Woodruff • Edmund Natures • John Crayford • Roland Swynbourne • John Madew • Roland Swynbourne • Thomas Bayly • Edward Leeds • Thomas Byng • William Smith • Robert Scott • Thomas Paske • Ralph Cudworth • Theophilus Dillingham • Thomas Paske • Theophilus Dillingham • Samuel Blythe • William Grigg • Charles Morgan • John Wilcox • Peter Godard • John Torkington • William Webb • Edward Atkinson • William Mollison • Godfrey Wilson • Henry Thirkill • Eric Ashby • Robin Matthews • Bob Hepple • Anthony Badger • Anthony Grabiner, Baron Grabiner • Loretta Minghella University of Cambridge People • Chancellor • The Lord Sainsbury of Turville • Acting Vice-Chancellor • Anthony Freeling • List of University of Cambridge people Colleges • Christ's • Churchill • Clare • Clare Hall • Corpus Christi • Darwin • Downing • Emmanuel • Fitzwilliam • Girton • Gonville and Caius • Homerton • Hughes Hall • Jesus • King’s • Lucy Cavendish • Magdalene • Murray Edwards • Newnham • Pembroke • Peterhouse • Queens’ • Robinson • St Catharine’s • St Edmund’s • St John’s • Selwyn • Sidney Sussex • Trinity • Trinity Hall • Wolfson Faculties and departments, by school Arts and Humanities • English • Anglo-Saxon Norse and Celtic • Architecture • CRASSH • Existential Risk • Classics • Divinity • Music • Philosophy Biological sciences • Genetics • Gurdon Institute • Physiology, Development and Neuroscience • Plant Sciences • Botanic Garden • Sainsbury Laboratory • Stem Cell Institute Clinical Medicine • Autism Research Centre • Cancer Unit • Cognition and Brain Sciences Unit • Mitochondrial Biology Unit • Oncology • Brain Imaging Centre Humanities and Social sciences • Economics • Education • History • History and Philosophy of Science • Human, Social, and Political Science • Archaeological Research • Politics and International Studies • Law • Criminology Physical sciences • Astronomy • Chemistry • Earth Sciences • Palaeoclimate Research • Geography • Scott Polar Research Institute • Mathematical Sciences • Mathematics • Theoretical Cosmology • Materials Science and Metallurgy • Physics Technology • Business School • Alternative Finance • India & Global Business • Chemical Engineering and Biotechnology • Computer Science and Technology • Engineering • Manufacturing Others • ADC Theatre • Institute of Continuing Education • Library • Press (journals) Student life • Students' Union • Graduate Union • Air Squadron • Amateur Dramatic Club • Apostles • BlueSci • Cam FM • Christian Union • Conservatives • Footlights • May Week • May Ball • Labour Club • Liberal Democrats • Light Entertainment Society • Moral Sciences Club • Musical Society • Philosophical Society • Railway Club • SCA • Spaceflight • Union Society • Cambridge University Wine Society • Varsity (student newspaper) • The Mays Sport • Association football • Rules • Aussie Rules • Boxing • Cricket • Cross Country • Dancing • Golf • Handball • Ice Hockey • Real Tennis • Rifle shooting • Rowing • Openweight Men (CUBC) • Lightweight Men (CULRC) • Women (CUWBC) • Rugby union • Tennis • Competitions • Cuppers • The Boat Race • Women's Boat Race • Henley Boat Races • The Varsity Polo Match • Rugby League Varsity Match • Rugby Union Varsity Match • University Cricket Match • University Golf Match Affiliates • Alan Turing Institute • Cambridge Theological Federation • Cambridge University Health Partners • Cambridge University Hospitals NHS Foundation Trust • Cambridgeshire and Peterborough NHS Foundation Trust • Royal Papworth Hospital NHS Foundation Trust • Partner institutions • Animal Health Trust • Babraham Institute • British Antarctic Survey • CCDC • EMBL-EBI • Laboratory of Molecular Biology • NIAB • Wellcome Sanger Institute Museums • Fitzwilliam Museum • Hamilton Kerr Institute • Kettle's Yard • Museum of Archaeology and Anthropology • Museum of Classical Archaeology • Museum of Zoology • Sedgwick Museum of Earth Sciences • Polar Museum • Whipple Museum of the History of Science Related • Awards and prizes • Cambridge Zero • Regent House • Senate House • Cambridge University Council • Fictional Cambridge colleges • Cambridge University Reporter • Category Authority control: Academics • zbMATH
William Neile William Neile (7 December 1637 – 24 August 1670) was an English mathematician and founder member of the Royal Society. His major mathematical work, the rectification of the semicubical parabola, was carried out when he was aged nineteen, and was published by John Wallis. By carrying out the determination of arc lengths on a curve given algebraically, in other words by extending to algebraic curves generally with Cartesian geometry a basic concept from differential geometry, it represented a major advance in what would become infinitesimal calculus. His name also appears as Neil. Life Neile was born at Bishopsthorpe, the eldest son of Sir Paul Neile MP for Ripon and Newark. His grandfather was Richard Neile, the Archbishop of York.[1] He entered Wadham College, Oxford as a gentleman-commoner in 1652, matriculating in 1655. He was taught by John Wilkins and Seth Ward.[1] In 1657, he became a student at the Middle Temple. In the same year he gave his exact rectification of the semicubical parabola and communicated his discovery to William Brouncker, Christopher Wren and others connected with Gresham College. His demonstration was published by Wallis in De Cycloide (1659). The general formula for rectification by definite integral was in effect discovered by Hendrik van Heuraet in 1659. In 1673 Wallis asserted that Christiaan Huyghens, who was advancing his own claim to have influenced Heuraet, was also slighting the priority of Neile. [1][2] Neile was elected a fellow of the Royal Society on 7 January 1663 and a member of the council on 11 April 1666. He entered the debate on the theory of motion, as a critic of the empiricist stance of other members. His own theory of motion was held up from publication by unfavourable peer review by Wallis, in 1667; a revision was communicated to the society on 29 April 1669. Neile objected to Wren's 1668 work on collision as lacking discussion of causality: he asked for discussion of the nature of momentum. His own work was much influenced by ideas drawn from the De Corpore of Thomas Hobbes.[1][3][4] He made astronomical observations with instruments erected on the roof of his father's residence, the “Hill House” (later called Waltham Place) at White Waltham in Berkshire, where he died at the age of 32. A white marble monument in the parish church of White Waltham commemorates him and an inscribed slab in the floor marks his burial-place. He belonged to the privy council of King Charles II.[1] Notes 1. "Neile, William" . Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900. 2. "Hendrik van Heuraet - Biography". 3. Andrew Pyle (editor), Dictionary of Seventeenth Century British Philosophers (2000), article on Neile, pp. 602-3. 4. Jon Parkin, Science, Religion and Politics in Restoration England: Richard Cumberland's De Legibus Naturae (1999), pp. 136-7. References • This article incorporates text from a publication now in the public domain: "Neile, William". Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900. External links • O'Connor, John J.; Robertson, Edmund F., "William Neile", MacTutor History of Mathematics Archive, University of St Andrews Authority control: Academics • zbMATH
William P. Byers William Paul Byers (born 1943) is a Canadian mathematician and philosopher; professor emeritus in mathematics and statistics at Concordia University in Montreal, Quebec, Canada. He completed a BSc ('64), and an MSc ('65) from McGill University, and obtained his PhD ('69) from the University of California, Berkeley. His dissertation, Anosov Flows, was supervised by Stephen Smale.[1] His area of interests include dynamical systems and the philosophy of mathematics. Books Byers is the author of three books on mathematics: • How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics (Princeton University Press, 2007)[2] • The Blind Spot: Science and the Crisis of Uncertainty (Princeton University Press, 2011)[3] • Deep Thinking: What Mathematics Can Teach Us About the Mind (World Scientific, 2015)[4] See also • List of people from Montreal References 1. William P. Byers at the Mathematics Genealogy Project 2. Reviews of How Mathematicians Think: • Grattan-Guinness, Ivor, zbMATH, Zbl 1123.00003{{citation}}: CS1 maint: untitled periodical (link) • Rogovchenko, Svitlana P., zbMATH, Zbl 1191.00009{{citation}}: CS1 maint: untitled periodical (link) • Chaitin, Gregory (July 25, 2007), "Review", New Scientist • Hersh, Reuben (December 2007), "Review" (PDF), Notices of the American Mathematical Society, 54 (11): 1496–1499 • Kennard, Lee (April 2008), Math Horizons, 15 (4): 24–25, doi:10.1080/10724117.2008.11974771, JSTOR 25678755, S2CID 125265631{{citation}}: CS1 maint: untitled periodical (link) • Wolff, Ken (May 2008), The Mathematics Teacher, 101 (9): 696, JSTOR 20876252{{citation}}: CS1 maint: untitled periodical (link) • Baylis, John (March 2009), The Mathematical Gazette, 93 (526): 170–173, doi:10.1017/S0025557200184463, JSTOR 40378699, S2CID 185022987{{citation}}: CS1 maint: untitled periodical (link) • Keyfitz, Barbara Lee (Winter 2009), University of Toronto Quarterly, 78 (1): 141–143, doi:10.1353/utq.0.0372, S2CID 120506445{{citation}}: CS1 maint: untitled periodical (link) 3. Reviews of The Blind Spot: • "Review", Quill & Quire, Ontario Arts Council, 26 July 2011 • "Nonfiction book review", Publishers Weekly • Jubin, Brenda (April 2011), "Review", Seeking Alpha • "Review", Kirkus Reviews, May 2011 • Taylor, Paul (April 2012), "Review", Mathematics Today, Institute of Mathematics & its Applications • Sears, Ruthmae (October 2012), The Mathematics Teacher, 106 (3): 238, doi:10.5951/mathteacher.106.3.0238, JSTOR 10.5951/mathteacher.106.3.0238{{citation}}: CS1 maint: untitled periodical (link) • Edwards, Matthew (February 2013), "Review", The Actuary 4. Review of Deep Thinking: • Bultheel, Adhemar (February 2015), "Review", EMS Reviews, European Mathematical Society External links Wikimedia Commons has media related to William P. Byers. • Roberts, Russ (May 16, 2011). "Byers on The Blind Spot, Science, and Uncertainty". EconTalk. Library of Economics and Liberty. Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • United States • Czech Republic • Korea • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
William Payne (mathematician) William Payne (unknown – c. 1779) was an English mathematician[1] and the author of books about mathematics, draughts, and whist. Payne was the brother of prominent London bookseller Thomas Payne, who sold his works and published some of them.[1][2][3] Payne's first book, An Introduction to the Game of Draughts, was published in 1756.[1][2] The dedication and preface were written by Samuel Johnson.[2][3] Payne's second book, An Introduction to Geometry: Containing the Most Useful Propositions in Euclid, & Other Authors, was published in 1767.[4] The book Maxims for Playing the Game of Whist; With All Necessary Calculations, and Laws of the Game was published anonymously in 1773; published by his brother Thomas, it is believed to have been written by William Payne.[5] References 1. Courtney, A bibliography, p. 74. 2. Hanley, p. 181. 3. Boswell, p. 192. 4. Payne, William (1767). An Introduction to Geometry: Containing the Most Useful Propositions in Euclid, & Other Authors. London: H. Hart. 5. Courtney, English whist, p. 360. Sources • Courtney, William Prideaux (1894). English whist and English whist players. Richard Bentley and Son. • Courtney, William Prideaux; Smith, David Nichol (ed.) (1915). A bibliography of Samuel Johnson. Clarendon Press. • Boswell, James (1888). The Life of Samuel Johnson. Swan Schonnenheim, Lowrey & Co. • Hanley, Brian J (2001). Samuel Johnson as Book Reviewer: A Duty to Examine the Labors of the Learned. University of Delaware Press.
William Playfair William Playfair (22 September 1759 – 11 February 1823), a Scottish engineer and political economist, served as a secret agent on behalf of Great Britain during its war with France.[1] The founder of graphical methods of statistics,[2] Playfair invented several types of diagrams: in 1786 the line, area and bar chart of economic data, and in 1801 the pie chart and circle graph, used to show part-whole relations.[3] As a secret agent, Playfair reported on the French Revolution and organized a clandestine counterfeiting operation in 1793 to collapse the French currency. William Playfair Born(1759-09-22)September 22, 1759 Benvie, Forfarshire, Scotland Died11 February 1823(1823-02-11) (aged 63) London, England Known forinventor of statistical graphs, writer on political economy, and secret agent for Great Britain FamilyJohn Playfair (brother) James Playfair (brother) William Henry Playfair (nephew) Biography William Playfair was born in 1759 in Scotland. He was the fourth son (named after his grandfather) of the Reverend James Playfair of the parish of Liff & Benvie near the city of Dundee in Scotland; his notable brothers were architect James Playfair and mathematician John Playfair. His father died in 1772 when he was 13, leaving the eldest brother John to care for the family and his education. After his apprenticeship with Andrew Meikle, the inventor of the threshing machine, Playfair became draftsman and personal assistant to James Watt at the Boulton and Watt steam engine manufactory in Soho, Birmingham.[4] Playfair had a variety of careers. He was in turn a millwright, engineer, draftsman, accountant, inventor, silversmith, merchant, investment broker, economist, statistician, pamphleteer, translator, publicist, land speculator, convict, banker, ardent royalist, editor, blackmailer and journalist. On leaving Watt's company in 1782, he set up a silversmithing business and shop in London, which failed. In 1787 he moved to Paris, taking part in the storming of the Bastille two years later. After the French revolution, Playfair played a role in the Scioto Land sale to French settlers in the Ohio River Valley.[1] He returned to London in 1793, where he opened a "security bank", which also failed. From 1775 he worked as a writer and pamphleteer and did some engineering work.[4] In the 1790s, Playfair informed the British government on events in France and proposed various clandestine operations to bring down the French government. At the end of the 1790s he was imprisoned for debt in the Fleet Prison, being released in 1802.[1] Work Ian Spence and Howard Wainer in 2001 describe Playfair as "engineer, political economist and scoundrel" while "Eminent Scotsmen" calls him an "ingenious mechanic and miscellaneous writer".[5] It compares his career with the glorious one of his older brother John Playfair, the distinguished Edinburgh mathematics professor, and draws a moral about the importance of "steadiness and consistency of plan" as well as of "genius". Bruce Berkowitz in 2018 provides a detailed portrait of Playfair as an "ambitious, audacious, and woefully imperfect British patriot" who undertook the "most complex covert operation anyone had ever conceived".[1] Bar chart Two decades before Playfair's first achievements, in 1765 Joseph Priestley had created the innovation of the first timeline charts, in which individual bars were used to visualise the life span of a person, and the whole can be used to compare the life spans of multiple persons. According to James R. Beniger and Robyn (1978) "Priestley's timelines proved a commercial success and a popular sensation, and went through dozens of editions".[6] These timelines directly inspired Playfair's invention of the bar chart, which first appeared in his Commercial and Political Atlas, published in 1786. According to Beniger and Robyn (1978) "Playfair was driven to this invention by a lack of data. In his Atlas he had collected a series of 34 plates about the import and export from different countries over the years, which he presented as line graphs or surface charts: line graphs shaded or tinted between abscissa and function. Because Playfair lacked the necessary series data for Scotland, he graphed its trade data for a single year as a series of 34 bars, one for each of 17 trading partners".[6] In this bar chart Scotland's imports and exports from and to 17 countries in 1781 are represented. "This bar chart was the first quantitative graphical form that did not locate data either in space, as had coordinates and tables, or time, as had Priestley's timelines. It constitutes a pure solution to the problem of discrete quantitative comparison".[6] The idea of representing data as a series of bars had earlier (14th century) been published by Jacobus de Sancto Martino and attributed to Nicole Oresme. Oresme used the bars to generate a graph of velocity against continuously varying time. Playfair's use of bars was to generate a chart of discrete measurements.[7] Graphics Playfair, who argued that charts communicated better than tables of data, has been credited with inventing the line, bar, area, and pie charts. His time-series plots are still presented as models of clarity. Playfair first published The Commercial and Political Atlas in London in 1786. It contained 43 time-series plots and one bar chart, a form apparently introduced in this work. It has been described as the first major work to contain statistical graphs. Playfair's Statistical Breviary, published in London in 1801, contains what is generally credited as the first pie chart.[8][9][10] From 1809 until 1811, he published the massive "British Family Antiquity, Illustrative of the Origin and Progress of the Rank, honours and personal merit of the nobility of the United Kingdom. Accompanied with an Elegant Set of Chronological Charts." The work was 9 large volumes in 11 parts; Volume six contained a suite of 12 plates of which 10 are in two states, coloured and uncoloured, and 9 large folding tables, partly hand coloured. This was an important work on genealogy published in a very limited edition. In it, Playfair sought to show the true character and heroism of the British nobility and that the Monarchy, particularly the British Monarchy, is the true defender of liberty. The volumes are separated into the peerage and baronetage of England, Scotland and Ireland. Counterfeiting operation In 1793 Playfair as secret agent devised a clandestine plan that he presented to Henry Dundas, who was Home Secretary soon to become Britain's Secretary of State for War. Playfair proposed to "fabricate one hundred millions of assignats (the French currency) and spread them in France by every means in my power." He saw the counterfeiting plan as the lesser of two evils: "That there are two ways of combatting the French nation the forces of which are measured by men and money. Their assignats are their money and it is better to destroy this paper founded upon an iniquitous extortion and a villainous deception than to shed the blood of men." Playfair forged the assignats at Haughton Castle in Northumberland and distributed them according to an elaborate plan. The plan apparently worked: by 1795 the French assignat had become worthless and the ensuing chaos undermined the French government. Playfair never told anyone about the operation.[1] Playfair cycle The following quotation, known as the "Playfair cycle," has achieved notoriety as it pertains to the "Tytler cycle": :...wealth and power have never been long permanent in any place. ...they travel over the face of the earth, something like a caravan of merchants. On their arrival, every thing is found green and fresh; while they remain all is bustle and abundance, and, when gone, all is left trampled down, barren, and bare.[11] Works • 1785. The Increase of Manufactures, Commerce, and Finance, with the Extension of Civil Liberty, Proposed in Regulations for the Interest of Money. London: G.J. & J. Robinson. • 1786. The Commercial and Political Atlas: Representing, by Means of Stained Copper-Plate Charts, the Progress of the Commerce, Revenues, Expenditure and Debts of England during the Whole of the Eighteenth Century. • 1787. Joseph and Benjamin, a Conversation Translated from a French Manuscript. London: J. Murray. • 1793. Thoughts on the Present State of French Politics, and the Necessity and Policy of Diminishing France, for Her Internal Peace, and to Secure the Tranquillity of Europe. London: J. Stockdale. • 1793. A general view of the actual force and resources of France, in January, M. DCC. XCIII: to which is added, a table, shewing the depreciation of assignats, arising from their increase in quantity. J. Stockdale. • 1796. The History of Jacobinism, Its Crimes, Cruelties and Perfidies: Comprising an Inquiry into the Manner of Disseminating, under the Appearance of Philosophy and Virtue, Principles which are Equally Subversive of Order, Virtue, Religion, Liberty and Happiness. Vol. I.. Philadelphia: W. Cobbett. • 1796. For the Use of the Enemies of England, a Real Statement of the Finances and Resources of Great Britain • 1798. Lineal arithmetic, Applied to Shew the Progress of the Commerce and Revenue of England During the Present Century. A. Paris. • 1799. Stricture on the Asiatic Establishments of Great Britain, With a View to an Enquiry into the True Interests of the East India Company. Bunney & Gold. • 1801. Statistical Breviary; Shewing, on a Principle Entirely New, the Resources of Every State and Kingdom in Europe. London: Wallis. • 1805. An Inquiry into the Permanent Causes of the Decline and Fall of Powerful and Wealthy Nations. London: Greenland & Norris. • 1805. European commerce, shewing new and secure channels of trade with the continent of Europe... • 1805. Statistical Account of the United States of America by D. F. Donnant. London: J. Whiting. William Playfair, Trans. • 1807. European Commerce, Shewing New and Secure Channels of Trade with the Continent of Europe. Vol. I.. Philadelphia: J. Humphreys. • 1808. Inevitable Consequences of a Reform in Parliament • 1809. A Fair and Candid Address to the Nobility and Baronets of the United Kingdom; Accompanied with Illustrations and Proofs of the Advantage of Hereditary Rank and Title in a Free Country • 1811. British Family Antiquity: Index to the 9 Volumes of William Playfair's Family Antiquity of the British Nobility • 1813. Outlines of a Plan for a New and Solid Balance of Power in Europe. J. Stockdale. • 1814. Political Portraits in This New Æra, Vol. II . London: C. Chapple. • 1816. Supplementary Volume to Political Portraits in This New Æra. London: C. Chapple. • 1818. The History of England, from the Revolution in 1688 to the Death of George II. Vol. II. R. Scholey. • 1819. France as it Is, Not Lady Morgan's France, Vol. I. London: C. Chapple. • 1820. France as it Is, Not Lady Morgan's France, Vol. II. London: C. Chapple. References 1. Berkowitz, Bruce (2018). Playfair: The True Story of the British Secret Agent Who Changed How We See the World. ISBN 978-1-942695-04-2. 2. Paul J. FitzPatrick (1960). "Leading British Statisticians of the Nineteenth Century". In: Journal of the American Statistical Association, Vol. 55, No. 289 (Mar. 1960), pp. 38–70. 3. Michael Friendly (2008). "Milestones in the history of thematic cartography, statistical graphics, and data visualization" Archived 26 September 2018 at the Wayback Machine. pp 13–14. Retrieved 7 July 2008. 4. Ian Spence and Howard Wainer (1997). "Who Was Playfair?". In: Chance 10, p. 35–37. 5. Ian Spence and Howard Wainer (2001). "William Playfair". In: Statisticians of the Centuries. C.C. Heyde and E. Seneta (eds.) New York: Springer. pp. 105–110. 6. James R. Beniger and Dorothy L. Robyn (1978). "Quantitative graphics in statistics: A brief history". In: The American Statistician. 32: pp. 1–11. 7. Der, Geoff; Everitt, Brian S. (2014). A Handbook of Statistical Graphics Using SAS ODS. Chapman and Hall - CRC. p. 4. ISBN 978-1-584-88784-3. William Playfair, for example, is often credited with inventing the bar chart (see Chapter 3) in the last part of the 18th century, although a Frenchman, Nicole Oresme, used a bar chart in a 14th century publication, The Latitude of Forms to plot the velocity of a constantly accelerating object against time. But it was Playfair who popularized the idea of graphic depiction of quantitative information. 8. Edward R. Tufte (2001). The Visual Display of Quantitative Information. Cheshire, CT: Graphics Press, p. 44. 9. Ian Spence (2005). "No Humble Pie: The Origins and Usage of a statistical Chart" Archived 20 March 2007 at the Wayback Machine. In: Journal of Educational and Behavioral Statistics. Winter 2005, 30 (4), 353–368. 10. Playfair, William; Wainer, Howard; Spence, Ian (2005). Playfair's Commercial and Political Atlas and Statistical Breviary. Cambridge University Press. ISBN 9780521855549. 11. William Playfair (1807). An Inquiry into the Permanent Causes of the Decline and Fall of Powerful and Wealthy Nations, p. 102. External links Wikimedia Commons has media related to William Playfair. • Playfair, William (1759–1823) at oxforddnb.com • William PLAYFAIR b. 22 September 1759 - d. 11 February 1823 at statprob.com • "Biographical Dictionary of Eminent Scotsmen" • Works by William Playfair at Project Gutenberg • Works by or about William Playfair at Internet Archive Visualization of technical information Fields • Biological data visualization • Chemical imaging • Crime mapping • Data visualization • Educational visualization • Flow visualization • Geovisualization • Information visualization • Mathematical visualization • Medical imaging • Molecular graphics • Product visualization • Scientific visualization • Software visualization • Technical drawing • User interface design • Visual culture • Volume visualization Image types • Chart • Diagram • Engineering drawing • Graph of a function • Ideogram • Map • Photograph • Pictogram • Plot • Sankey diagram • Schematic • Skeletal formula • Statistical graphics • Table • Technical drawings • Technical illustration People Pre-19th century • Edmond Halley • Charles-René de Fourcroy • Joseph Priestley • Gaspard Monge 19th century • Charles Dupin • Adolphe Quetelet • André-Michel Guerry • William Playfair • August Kekulé • Charles Joseph Minard • Luigi Perozzo • Francis Amasa Walker • John Venn • Oliver Byrne • Matthew Sankey • Charles Booth • Georg von Mayr • John Snow • Florence Nightingale • Karl Wilhelm Pohlke • Toussaint Loua • Francis Galton Early 20th century • Edward Walter Maunder • Otto Neurath • W. E. B. Du Bois • Henry Gantt • Arthur Lyon Bowley • Howard G. Funkhouser • John B. Peddle • Ejnar Hertzsprung • Henry Norris Russell • Max O. Lorenz • Fritz Kahn • Harry Beck • Erwin Raisz Mid 20th century • Jacques Bertin • Rudolf Modley • Arthur H. Robinson • John Tukey • Mary Eleanor Spear • Edgar Anderson • Howard T. Fisher Late 20th century • Borden Dent • Nigel Holmes • William S. Cleveland • George G. Robertson • Bruce H. McCormick • Catherine Plaisant • Stuart Card • Pat Hanrahan • Edward Tufte • Ben Shneiderman • Michael Friendly • Howard Wainer • Clifford A. Pickover • Lawrence J. Rosenblum • Thomas A. DeFanti • George Furnas • Sheelagh Carpendale • Cynthia Brewer • Miriah Meyer • Jock D. Mackinlay • Alan MacEachren • David Goodsell • Michael Maltz • Leland Wilkinson • Alfred Inselberg Early 21st century • Ben Fry • Hans Rosling • Christopher R. Johnson • David McCandless • Mauro Martino • John Maeda • Tamara Munzner • Jeffrey Heer • Gordon Kindlmann • Hanspeter Pfister • Manuel Lima • Aaron Koblin • Martin Krzywinski • Bang Wong • Jessica Hullman • Hadley Wickham • Polo Chau • Fernanda Viégas • Martin Wattenberg • Claudio Silva • Ade Olufeko • Moritz Stefaner Related topics • Cartography • Chartjunk • Computer graphics • in computer science • CPK coloring • Graph drawing • Graphic design • Graphic organizer • Imaging science • Information graphics • Information science • Misleading graph • Neuroimaging • Patent drawing • Scientific modelling • Spatial analysis • Visual analytics • Visual perception • Volume cartography • Volume rendering • Information art Authority control International • FAST • ISNI • VIAF National • Norway • Spain • France • BnF data • Germany • Israel • Belgium • United States • Czech Republic • Netherlands • Poland Academics • zbMATH People • Trove Other • SNAC • IdRef
William Raoul Reagle Transue William Raoul Reagle Transue (January 31, 1937 – December 17, 2008) was an American mathematician and topologist. He is the son of mathematician William Reagle Transue and Monique Serpette who moved from her native France to the US in 1936. Bill, as he was known, earned his bachelor's degree from Harvard University in 1958, and his Ph.D. in mathematics from The University of Georgia in 1967 under Billy Joe Ball. He was a professor of mathematics at Auburn University from 1967 until his retirement over 30 years later. References • William Raoul Reagle Transue at the Mathematics Genealogy Project Authority control International • VIAF Academics • MathSciNet • Mathematics Genealogy Project
William S. Massey William Schumacher Massey (August 23, 1920[1] – June 17, 2017) was an American mathematician, known for his work in algebraic topology. The Massey product is named for him. He worked also on the formulation of spectral sequences by means of exact couples, and wrote several textbooks, including A Basic Course in Algebraic Topology (ISBN 0-387-97430-X). William Schumacher Massey Born(1920-08-20)August 20, 1920 Granville, Illinois, United States DiedJune 17, 2017(2017-06-17) (aged 96) Hamden, Connecticut, U.S. NationalityAmerican Alma materUniversity of Chicago Princeton University Known forMassey product Blakers–Massey theorem Exact couple SpouseEthel H. Massey Children3 Scientific career FieldsTopology InstitutionsBrown University Yale University ThesisClassification of mappings of an (n + 1)-dimensional space into an n-sphere (1948) Doctoral advisorNorman Steenrod Military career AllegianceUnited States Service/branchUnited States Navy Years of service1942–1945 Life William Massey was born in Granville, Illinois, in 1920, the son of Robert and Alma Massey, and grew up in Peoria. He was an undergraduate student at the University of Chicago. After serving as a meteorologist aboard aircraft carriers in the United States Navy for 4 years during World War II, he received a Ph.D. degree from Princeton University in 1949.[2] His dissertation, entitled Classification of mappings of an $(n+1)$-dimensional space into an n-sphere, was written under the direction of Norman Steenrod. He spent two additional years at Princeton as a post-doctoral research assistant.[3] He then taught for ten years on the faculty of Brown University. In 1958 he was elected to the American Academy of Arts and Sciences.[4] From 1960 till his retirement he was a professor at Yale University. He died on June 17, 2017, in Hamden, Connecticut. He had 23 PhD students, including Donald Kahn, Larry Smith, and Robert Greenblatt. Selected works • Algebraic topology: an introduction. NY: Harcourt, Brace & World. 1967; xix+261 pp.{{cite book}}: CS1 maint: postscript (link) 4th corrected printing. 1977. • Homology and cohomology theory. Monographs and Textbooks in Pure and Applied Mathematics. Vol. 46. New York: Marcel Dekker. 1978; xiv+412 pp.{{cite book}}: CS1 maint: postscript (link)[5] • Singular homology theory. Graduate Texts in Mathematics. Springer-Verlag. 1980; xii+265 pp.{{cite book}}: CS1 maint: postscript (link)[6] • A basic course in algebraic topology. Springer. 1991. ISBN 9780387974309. 3rd corrected printing. 1997. • Massey, William S. (1952). "Exact couples in algebraic topology. I, II". Annals of Mathematics. Second Series. 56: 363–396. doi:10.2307/1969805. JSTOR 1969805. MR 0052770. See also • Blakers–Massey theorem • Exact couple • Massey product External links • Address at Yale References 1. Massey, William S. "Indiana, Marriages, 1811–195". familysearch.org. Retrieved 1 November 2013. 2. "William Massey obituary". New Haven Register. June 20, 2017. Retrieved July 8, 2022. 3. William S. Massey at the Mathematics Genealogy Project 4. "In Memoriam: William S. Massey, 1920–2017". math.yale.edu. Department of Mathematics, Yale University. June 30, 2017. Retrieved July 8, 2022. 5. Ewing, John H. (1979). "Review: Homology and cohomology theory by W. S. Massey" (PDF). Bulletin of the American Mathematical Society. New Series. 1 (6): 985–989. doi:10.1090/s0273-0979-1979-14707-4. 6. Vick, James W. (1981). "Review: Singular homology theory by W. S. Massey" (PDF). Bulletin of the American Mathematical Society. New Series. 4 (2): 229–233. doi:10.1090/s0273-0979-1981-14892-8. Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • United States • Czech Republic • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
William Shaw (mathematician) William Shaw (born 14 May 1958) is a British mathematician, and formerly professor of the mathematics and computation of risk at University College London.[1][2] He is a consultant on financial derivatives, an author of a primary book on using Mathematica to model financial derivatives, formerly co-Editor-in-Chief of the journal Applied Mathematical Finance. Shaw studied at King's College, Cambridge, where he studied mathematics; he was Wrangler and earned a B.A. in 1980. In 1981 he won the Mayhew Prize[3] for his performance on the Cambridge Mathematical Tripos. In 1984 he received a D.Phil. (PhD) in mathematical physics from Wolfson College, Oxford. From 1984 to 1987 he was a research fellow at Clare College, Cambridge and C.L.E. Moore Instructor at the Massachusetts Institute of Technology. From 1987 to 1990, he worked for Smith Associates in Guildford, and ECL in Henley-on Thames. From 1991 to 2002 he was a lecturer in mathematics at Balliol College, Oxford. In 2002 he moved to St Catherine's College, Oxford, where he was University Lecturer in financial mathematics. In 2006 he moved to a Professorship at King's College London and in 2011 to a Professorship at UCL. He returned to the financial industry in 2012 and remained a visiting professor at UCL until 2017. Books • Applied Mathematica: Getting Started, Getting it Done by W.T. Shaw and J. Tigg. Addison-Wesley, 1993. • Modelling Financial Derivatives with Mathematica by W.T. Shaw, Cambridge University Press, 1998. • Complex Analysis with Mathematica by W.T. Shaw, Cambridge University Press, 2006. References 1. "Professor William T. Shaw". University College London. 2. "Profile: William T. Shaw". ResearchGate. 3. Mayhew Prize External links • William Shaw's former UCL web-page • Entry in Mathematics Genealogy Project • LinkedIn profile Authority control: Academics • MathSciNet • Mathematics Genealogy Project • zbMATH
William James Sidis William James Sidis (/ˈsaɪdɪs/; April 1, 1898 – July 17, 1944) was an American child prodigy with exceptional mathematical and linguistic skills. He wrote the book The Animate and the Inanimate, published in 1925 (written around 1920), in which he speculated about the origin of life in the context of thermodynamics. William James Sidis Sidis at his Harvard graduation (1914) Born(1898-04-01)April 1, 1898 Manhattan, New York City, U.S. DiedJuly 17, 1944(1944-07-17) (aged 46) Boston, Massachusetts, U.S. Other names • John W. Shattuck • Frank Folupa • Parker Greene • Jacob Marmor Alma materHarvard University Rice Institute Harvard Law School Notable work • The Animate and the Inanimate (1925) • The Tribes and the States (c. 1935) His father, psychiatrist Boris Sidis raised his son according to certain principles as he wished his son to be gifted. Sidis became famous first for his precocity and later for his eccentricity and withdrawal from public life. Eventually, he avoided mathematics altogether, writing on other subjects under a number of pseudonyms. He entered Harvard at age 11 and, as an adult, was said to have an extremely high IQ, and to be conversant in about 25 languages and dialects. Some of these statements have not been verified, but many of his contemporaries, including Norbert Wiener, Daniel Frost Comstock and William James had agreed that he was extremely intelligent. Biography Parents and upbringing (1898–1908) Sidis was born to Jewish emigrants from Ukraine,[1] on April 1, 1898, in New York City. His father, Boris Sidis, had emigrated in 1887 to escape political and anti-semitic persecution.[2]: 2–4 His mother, Sarah (Mandelbaum) Sidis, and her family had fled the pogroms in the late 1880s.[2]: 7 Sarah attended Boston University and graduated from its School of Medicine in 1897.[3] William was named after his godfather, Boris's friend and colleague, the American philosopher William James. Boris was a psychiatrist and published numerous books and articles, performing pioneering work in abnormal psychology. He was a polyglot, and his son William would also become one at a young age. Sidis's parents believed in nurturing a precocious and fearless love of knowledge, although their methods of parenting were criticized in the media and retrospectively.[4][2]: 281, Epilogue Sidis could read The New York Times at 18 months.[2]: 23 By age eight, he had reportedly taught himself eight languages (Latin, Greek, French, Russian, German, Hebrew, Turkish, and Armenian) and invented another, which he called "Vendergood". Harvard University and college life (1909–1914) Although the university had previously refused to let his father enroll him at age 9 because he was still a child, Sidis set a record in 1909 by becoming the youngest person to enroll at Harvard University. In early 1910, Sidis's mastery of higher mathematics was such that he lectured the Harvard Mathematical Club on four-dimensional bodies, attracting nationwide attention.[5][6] Notable child prodigy cybernetics pioneer Norbert Wiener who also attended Harvard at the time and knew Sidis, later stated in his book Ex-Prodigy: "The talk would have done credit to a first or second-year graduate student of any age...talk represented the triumph of the unaided efforts of a very brilliant child."[7] MIT physics professor Daniel F. Comstock was full of praises: "Karl Friedrich Gauss is the only example in history, of all prodigies, whom Sidis resembles. I predict that young Sidis will be a great astronomical mathematician. He will evolve new theories and invent new ways of calculating astronomical phenomena. I believe he will be a great mathematician, the leader in that science in the future."[2] Sidis began taking a full-time course load in 1910 and earned his Bachelor of Arts degree, cum laude, on June 18, 1914, at age 16.[8] Shortly after graduation, he told reporters "I want to live the perfect life. The only way to live the perfect life is to live it in seclusion". He granted an interview to a reporter from the Boston Herald. The paper reported Sidis's vows to remain celibate and never to marry, as he said women did not appeal to him. Later he developed a strong affection for Martha Foley, one year older than him. He later enrolled at Harvard Graduate School of Arts and Sciences. Teaching and further education (1915–1919) After a group of Harvard students threatened Sidis physically, his parents secured him a job at the William Marsh Rice Institute for the Advancement of Letters, Science, and Art (now Rice University) in Houston, Texas, as a mathematics teaching assistant. He arrived at Rice in December 1915 at the age of 17. He was a graduate fellow working toward his doctorate. Sidis taught three classes: Euclidean geometry, non-Euclidean geometry, and freshman math (he wrote a textbook for the Euclidean geometry course in Greek).[2]: 112 After less than a year, frustrated with the department, his teaching requirements, and his treatment by students older than himself, Sidis left his post and returned to New England. When a friend later asked him why he had left, he replied, "I never knew why they gave me the job in the first place—I'm not much of a teacher. I didn't leave: I was asked to go." Sidis abandoned his pursuit of a graduate degree in mathematics and enrolled at the Harvard Law School in September 1916, but withdrew in good standing in his final year in March 1919.[9] Politics and arrest (1919–1921) In 1919, shortly after his withdrawal from law school, Sidis was arrested for participating in a socialist May Day parade in Boston that turned violent. He was sentenced to 18 months in prison under the Sedition Act of 1918 by Roxbury Municipal Court Judge Albert F. Hayden. Sidis's arrest featured prominently in newspapers, as his early graduation from Harvard had garnered considerable local celebrity status. During the trial, Sidis stated that he had been a conscientious objector to the World War I draft, was a socialist, and did not believe in a god like the "big boss of the Christians," but rather in something that is in a way apart from a human being.[10][11] He later developed his own libertarian philosophy based on individual rights and "the American social continuity".[12][13] His father arranged with the district attorney to keep Sidis out of prison before his appeal came to trial; his parents, instead, held him in their sanatorium in New Hampshire for a year. They took him to California, where he spent another year.[14] At the sanatorium, his parents set about "reforming" him and threatened him with transfer to an insane asylum.[14] Later life (1921–1944) After returning to the East Coast in 1921, Sidis was determined to live an independent and private life. He only took work running adding machines or other fairly menial tasks. He worked in New York City and became estranged from his parents. It took years before he was cleared legally to return to Massachusetts, and he was concerned for years about his risk of arrest. He obsessively collected streetcar transfers, wrote self-published periodicals, and taught small circles of interested friends his version of American history. In 1933, Sidis passed a Civil Service exam in New York, but scored a low ranking of 254.[15] In a private letter, Sidis wrote that this was "not so encouraging".[15] In 1935, he wrote an unpublished manuscript, The Tribes and the States, which traces Native American contributions to American democracy.[16] In 1944, Sidis won a settlement from The New Yorker for an article published in 1937.[17] He had alleged it contained many false statements.[18] Under the title "Where Are They Now?", James Thurber pseudonymously described Sidis's life as lonely, in a "hall bedroom in Boston's shabby South End".[19] Lower courts had dismissed Sidis as a public figure with no right to challenge personal publicity. He lost an appeal of an invasion of privacy lawsuit at the United States Court of Appeals for the Second Circuit in 1940 over the same article. Judge Charles Edward Clark expressed sympathy for Sidis, who claimed that the publication had exposed him to "public scorn, ridicule, and contempt" and caused him "grievous mental anguish [and] humiliation," but found that the court was not disposed to "afford to all the intimate details of private life an absolute immunity from the prying of the press".[20] Sidis died from a cerebral hemorrhage in 1944 in Boston at age 46.[21] Publications and research From writings on cosmology, to Native American history, to Notes on the Collection of Transfers, and several purported lost texts on anthropology, philology, and transportation systems, Sidis covered a broad range of subjects. Some of his ideas concerned cosmological reversibility[22] and "social continuity".[23] In The Animate and the Inanimate (1925), Sidis predicted the existence of regions of space where the second law of thermodynamics operates in reverse to the temporal direction experienced in our local area. Everything outside of what we call a galaxy would be such a region. Sidis said that the matter in this region would not generate light. Sidis's The Tribes and the States (c. 1935) employs the pseudonym "John W. Shattuck", purporting to give a 100,000-year history of the Settlement of the Americas, from prehistoric times to 1828.[24] In this text, he suggests that "there were red men at one time in Europe as well as in America".[25] Sidis was a "peridromophile", a term he coined for people fascinated with transportation research and streetcar systems. He wrote a treatise on streetcar transfers under the pseudonym of "Frank Folupa" that identified means of increasing public transport usage.[26] For this work, in 1926 he was invited to speak at the inaugural "genius meeting" hosted by Winifred Sackville Stoner's League for Fostering Genius in Tuckahoe, New York.[27] In 1930, Sidis received a patent for a rotary perpetual calendar that took into account leap years.[28] The Animate and the Inanimate Sidis wrote The Animate and the Inanimate to elaborate his thoughts on the origin of life, cosmology, and the potential reversibility of the second law of thermodynamics through Maxwell's Demon, among other things. It was published in 1925;[29] however, it has been suggested that Sidis was working on the theory as early as 1916.[30] One motivation for the theory appears to be to explain psychologist and philosopher William James's "reserve energy" theory which proposed that a "reserve energy" could be used by people subjected to extreme conditions. Sidis's own "forced prodigy" upbringing was a result of testing the theory. The work is one of the few that Sidis did not write under a pseudonym. In The Animate and the Inanimate, Sidis states that the universe is infinite, and contains sections of "negative tendencies" where[31] the laws of physics are reversed, juxtaposed with "positive tendencies", which swap over epochs of time. Sidis states that there was no "origin of life", but that life has always existed and that it has only changed through evolution. Sidis adopted Eduard Pflüger's cyanogen based life theory, and Sidis cites "organic" things such as almonds (his example) that have cyanogen that does not kill. Because cyanogen is normally highly toxic, almonds are a strange anomaly. Sidis describes his theory as a fusion of the mechanistic model of life and the vitalist model of life, as well as entertaining the notion of life coming to earth from asteroids (as advanced by Lord Kelvin and Hermann von Helmholtz). Sidis also states that functionally speaking, stars are "alive" and undergo an eternally repeating light-dark cycle, reversing the second law in the dark portion of the cycle.[32] Sidis's theory was ignored upon release,[17] only to be found in an attic in 1979. Upon this discovery, Buckminster Fuller (who was a classmate of Sidis) commented on The Animate and the Inanimate:[33] Imagine my excitement and joy on being handed this xerox of Sidis's 1925 book, in which he clearly predicts the black hole. In fact, I find his whole book, The Animate and the Inanimate to be a fine cosmological piece. I find him focusing on the same subjects that fascinate me, and coming to about the same conclusions as those I have published in SYNERGETICS, and will be publishing in SYNERGETICS Volume II, which has already gone to the press. As a Harvard man of a generation later, I hope you will become as excited as I am at this discovery that Sidis did go on after college to do the most magnificent thinking and writing." — Buckminster Fuller Vendergood language Sidis created a constructed language called Vendergood in his second book, the Book of Vendergood, which he wrote at the age of 8. The language was mostly based on Latin and Greek, but also drew on German and French and other Romance languages.[2] It distinguished between eight moods: indicative, potential, imperative absolute, subjunctive, imperative, infinitive, optative, and Sidis's own strongeable.[2]: 41 One of its chapters is titled "Imperfect and Future Indicative Active". Other parts explain the origin of Roman numerals. It uses base 12 instead of base 10: • eis – 'one' • duet – 'two' • tre – 'three' • guar – 'four' • quin – 'five' • sex – 'six' • sep – 'seven' • oo (oe?) – 'eight' • non – 'nine' • ecem – 'ten' • elevenos – 'eleven' • dec – 'twelve' • eidec (eis, dec) – 'thirteen' Most of the examples are presented in the form of tests: 1. 'Do I love the young man?' = Amevo (-)ne the neania? 2. 'The bowman obscures.' = The toxoteis obscurit. 3. 'I am learning Vendergood.' = (Euni) disceuo Vendergood. 4. 'What do you learn?' (sing.). = Quen diseois-nar? 5. 'I obscure ten farmers.' = Obscureuo ecem agrieolai.[2]: 42–43 The Tribes and the States The Tribes and the States outlines the history of the Native Americans, focusing on the Northeastern tribes and continuing up to the mid-19th century. It was written around 1935 but never completed, and remained unpublished at the time of Sidis's death. Sidis wrote the history under the pseudonym "John W. Shattuck". Much of the history was taken from wampum belts; Sidis explained, "The weaving of wampum belts is a sort of writing by means of belts of colored beads, in which the various designs of beads denoted different ideas according to a definitely accepted system, which could be read by anyone acquainted with wampum language, irrespective of what the spoken language is. Records and treaties are kept in this manner, and individuals could write letters to one another in this way."[34] Much of the book is centered on the influence of Native Americans on migrating Europeans and the effect of Native Americans on the formation of the United States. It describes the origination of the federations that were to be an important event to the Founding Fathers. Legacy After his death, Helena Sidis said that her brother had an IQ reported in Abraham Sperling's 1946 book Psychology for the Millions as "the very highest that had ever been obtained",[35] but some of his biographers, such as Amy Wallace, exaggerated his IQ.[15] Sperling wrote:[35] Helena Sidis told me that a few years before his death, her brother Bill took an intelligence test with a psychologist. His score was the very highest that had ever been obtained. In terms of I. Q., the psychologist related that the figure would be between 250 and 300. Late in life William Sidis took general intelligence tests for Civil Service positions in New York and Boston. His phenomenal ratings are a matter of record. It has been acknowledged that Helena and William's mother Sarah had a reputation for exaggerated statements about the Sidis family.[15] Helena had falsely stated that the Civil Service exam William took in 1933 was an IQ test and that his ranking of 254 was an IQ score of 254.[15] It is speculated that the number "254" was actually William's placement on the list after he passed the Civil Service exam, as he stated in a letter sent to his family.[36] Helena also said that "Billy knew all the languages in the world, while my father only knew 27. I wonder if there were any Billy didn't know."[15] This statement was not backed by any source outside the Sidis family, and Sarah Sidis also made the improbable statement in her 1950 book The Sidis Story that William could learn a language in just one day.[15] Boris Sidis had once dismissed tests of intelligence as "silly, pedantic, absurd, and grossly misleading".[37] Regardless of the exaggerations, Sidis was judged by contemporaries such as MIT Physics professor Daniel Frost Comstock and American mathematician Norbert Wiener (who wrote about Sidis in his autobiography) to have had genuine ability.[38][2]: 54 [39] Sidis's life and work, particularly his ideas about Native Americans, are extensively discussed in Robert M. Pirsig's book Lila: An Inquiry into Morals (1991).[40] Sidis is also discussed in Ex-Prodigy, an autobiography by mathematician Norbert Wiener (1894–1964), who was a prodigy himself.[41] The Danish author Morten Brask wrote a novel that was a fictional account based on Sidis's life; The Perfect Life of William Sidis was published in Denmark in 2011. Another novel based on his biography was published by the German author Klaus Caesar Zehrer in 2017.[42] In education discussions The debate about Sidis's manner of upbringing occurred within a larger discourse about the best way to educate children. Newspapers criticized Boris Sidis's child-rearing methods. Most educators of the day believed that schools should expose children to common experiences to create good citizens. Most psychologists thought intelligence was hereditary, a position that precluded early childhood education at home.[43] The difficulties Sidis encountered in dealing with the social structure of a collegiate setting may have shaped opinion against allowing such children to rapidly advance through higher education in his day. Research indicates that a challenging curriculum can relieve social and emotional difficulties commonly experienced by gifted children.[44] Embracing these findings, several colleges incorporated procedures for early entrance. The Davidson Institute for Talent Development has developed a guidebook on the topic.[45] Sidis was portrayed derisively in the The New York Times in 1909, as "a wonderfully successful result of a scientific forcing experiment".[4] His mother later maintained that newspaper accounts of her son bore little resemblance to him. Bibliography • The Animate and the Inanimate (1925) • The Tribes and the States (c. 1935) (PDF file) References 1. Heinze, Andrew R. (2006). Jews and the American Soul: Human Nature in the Twentieth Century. Princeton University Press. ISBN 978-0-691-12775-0. 2. Wallace, Amy (1986). The prodigy: a biography of William James Sidis, the world's greatest child prodigy. London: Macmillan. ISBN 978-0333432235. 3. "History of Homeopathy and Its Institutions in America By William Harvey King, M.D., LL.D. Presented by Sylvain Cazalet". Homeoint.org. Retrieved May 25, 2011. 4. "Sidis Could Read at Two Years Old; Youngest Harvard Undergraduate Under Father's Scientific Forcing Process Almost from Birth. Good Typewriter at Four; At 5 Composed Text Book on Anatomy, in Grammar School at 6, Then Studied German, French, Latin, and Russian". The New York Times. October 18, 1909. p. 7. 5. Montour, Kathleen (April 1977). "William James Sidis, the broken twig". American Psychologist. 32 (4): 265–279. doi:10.1037/0003-066X.32.4.265. 6. "Wonderful Boys of History Compared With Sidis. All Except Macaulay Showed Special Ability in Mathematics. Instances of Boys Having 'Universal Genius'". The New York Times. January 16, 1910. p. SM11. Retrieved November 26, 2014. 7. Renselle, Doug. "A Review of Kathleen Montour's William James Sidis, The Broken Twig". Quantonics.com. Retrieved February 13, 2020. 8. "Harvard College, 1952". Retrieved November 26, 2014 – via Sidis.net. 9. "Harvard Transcripts". Retrieved May 25, 2011 – via Sidis.net. 10. "Sidis Gets Year and Half in Jail". Boston Herald. May 14, 1919. Retrieved January 12, 2018. 'Do you believe in a god?' 'No.' Atty. Connolly then asked the court what God he meant, whereupon Judge Hayden replied, God Almighty. Here Sidis said that the kind of a God that he did not believe in was the 'big boss of the Christians,' adding that he believed in something that is in a way apart from a human being. 11. Mahony, Dan. "Frequently Asked Questions About W. J. Sidis". Retrieved January 12, 2018. Was he religious? 'He espoused no religion, but said that... the kind of a God he did not believe in was the "big boss of the Christians", adding that he believed in something that is in a way apart from a human being (Boston Herald, May 14, 1919).' 12. Sidis, William James (June 1938). "Libertarian". Continuity News. Cambridge, Massachusetts (2): 4. 13. Sidis, William James. "The Concept of Rights". American Independence Society. Retrieved November 26, 2014. {{cite journal}}: Cite journal requires \|journal= (help) 14. "Railroading in the Past". Retrieved May 25, 2011 – via Sidis.net. 15. "The Logics – Was William James Sidis the Smartest Man on Earth". Thelogics.org. Archived from the original on December 20, 2014. Retrieved November 26, 2014. 16. Johansen, Bruce E. (Fall 1989). "William James Sidis' 'Tribes and States': An Unpublished Exploration of Native American Contributions to Democracy". Northeast Indian Quarterly. 6 (3): 24–29 – via eric.ed.gov. 17. Bates, Stephen (2011). "The Prodigy and the Press: William James Sidis, Anti-Intellectualism, and Standards of Success". J&MC Quarterly. 88 (2): 374–397. doi:10.1177/107769901108800209. ISSN 1077-6990. S2CID 145637498. 18. "Sidis vs New Yorker". Sidis.net. February 29, 2008. Retrieved May 25, 2011. 19. LaMay, Craig L. (2003). Journalism and the Debate Over Privacy. LEA's Communication Series. Mahwah, New Jersey: Lawrence Erlbaum Associates, Inc. p. 63. ISBN 978-0-8058-4626-3. 20. Seitz, Robert N. (2002). "Review of Amy Wallace, The Prodigy (1986)". High IQ News. Archived from the original on June 2, 2008. Retrieved February 5, 2016. 21. Smith, Shirley (July 19, 1944). "Letter to the Editor". Boston Traveler. Retrieved May 25, 2011 – via Sidis.net. 22. Sidis, William James (1925). "The Animate and the Inanimate". Boston: The Gorham Press. {{cite journal}}: Cite journal requires \|journal= (help) 23. Sidis, William James. "Continuity News". Archived from the original on May 7, 2016. Retrieved August 12, 2008 – via Sidis.net. 24. "The Tribes and the States, Table of Contents". Sidis.net. Retrieved May 25, 2011. 25. "The Tribes and the States, Native American history". Sidis.net. Retrieved May 25, 2011. 26. "Notes on the Collection of Transfers". June 20, 1926. Retrieved May 25, 2011 – via Sidis.net. 27. Bates, Stephen (June 20, 1926). "Youthful Prodigies at Genius Meeting" (PDF). The New York Times. p. 8. Retrieved January 16, 2023. 28. U.S. Patent 1784117A, Perpetual Calendar, December 9, 1930 29. "The Animate and the Inanimate". Sidis.net. Retrieved August 23, 2019. 30. "Letter to Huxley". Sidis.net. Retrieved August 23, 2019. 31. "The Animate and the Inanimate : William James Sidis". Archived from the original on December 10, 2000. Retrieved August 29, 2019. 32. "ANIM11". October 24, 2000. Archived from the original on October 24, 2000. Retrieved August 29, 2019. 33. "Bucky Ltr". March 3, 2001. Archived from the original on March 3, 2001. Retrieved August 29, 2019. 34. William James Sidis, 'The Tribes And The States: 100,000-Year History of North America' (via sidis.net) 35. Sperling, Abraham Paul (1947) [July 1946]. Psychology for the Millions. New York: Frederick Fell. pp. 332–339. Retrieved November 26, 2014. 36. Gowdy, Larry Neal (October 20, 2013). "Myths, Facts, Lies, and Humor About William James Sidis – Part One". thelogics.org. Retrieved March 4, 2016. A letter written by William Sidis stated that he had taken a civil service exam, that he passed the state clerical exam, and that he was number 254 on the list; "not so encouraging". It may never be known if Sidis actually did take an IQ test, and it may never be known if the 250–300 number arrived from Sidis's placement in the job pool. 37. "Foundations of Normal and Abnormal psychology". Sidis.net. Retrieved May 25, 2011. 38. Manley, Jared L.; (James Thurber) (August 14, 1937). "Where Are They Now? April Fool!". The New Yorker. pp. 22–26. Retrieved February 13, 2020 – via sidis.net. 39. Pirsig, Robert M. (1991). Lila. p. 55. Retrieved February 13, 2020 – via sidis.net. 40. "Lila: An Inquiry into Morals". barnesandnoble.com. Barnes & Noble. Retrieved April 1, 2019. 41. Ex-Prodigy. 1964. ISBN 978-0262230117. Retrieved April 1, 2019. {{cite book}}: \|website= ignored (help) 42. Zehrer, Klaus Cäsar (2017). Das Genie (in German). Zürich: Diogenes Verlag. ISBN 978-3-257-06998-3. 43. Kett, Joseph F. (1978). "Curing the Disease of Precocity". The American Journal of Sociology. 84 (suppl): S183–S211. doi:10.1086/649240. ISSN 0002-9602. JSTOR 3083227. S2CID 144509596. 44. Neihart, Maureen; Reis, Sally M.; Robinson, Nancy M.; Moon, Sidney M., eds. (2002). The Social and Emotional Development of Gifted Children: What Do We Know. National Association for Gifted Children (Prufrock Press, Inc.). pp. 286–287. 45. Considering the Options: A Guidebook for Investigating Early College Entrance (PDF). Print.ditd.org. Retrieved November 26, 2014. Sources • Wallace, Amy (1986). The Prodigy: a Biography of William James Sidis, America's Greatest Child Prodigy. New York: E.P. Dutton & Co. ISBN 0-525-24404-2. External links • Sidis Archives at Sidis.net • Article about William James Sidis at "The Straight Dope" Authority control International • FAST • ISNI • VIAF National • Germany • Israel • United States • Netherlands Other • SNAC
William S. Burnside William Snow Burnside (20 December 1839 – 11 March 1920) was an Irish mathematician whose entire career was spent at Trinity College Dublin (TCD). He is chiefly remembered for the book The Theory of Equations: With an Introduction to the Theory of Binary Algebraic Forms (1881)[1] and his long tenure as Erasmus Smith's Professor of Mathematics at TCD . He is sometimes confused with his rough contemporary, the English mathematician William Burnside.[2] William S. Burnside Born(1839-12-20)December 20, 1839 DiedMarch 11, 1920(1920-03-11) (aged 80) NationalityIrish Academic background Alma materTrinity College Dublin Academic work DisciplineMathematics InstitutionsTrinity College Dublin William Snow Burnside was born at Corcreevy House, near Fivemiletown, Tyrone, to William Smyth Burnside (1810–1884, Chancellor of Clogher Cathedral) and Anne Henderson (1808–1881).[3] He studied mathematics under George Salmon at TCD (BA 1861, MA 1866, Fellowship 1871), and taught there until his retirement in 1917. He served as Erasmus Smiths's Professor of Mathematics for many decades (1879–1913), and co-authored the influential 1881 book The Theory of Equations: With an Introduction to the Theory of Binary Algebraic Forms with his TCD colleague Arthur William Panton (1843–1906). It ran to at least 7 editions, and was reissued by Dover Books in 1960. TCD awarded him DSc in 1891. He lived one and a half miles away from campus, on Raglan Road, and was allegedly "the last man to regularly arrive in College on horseback"[4] References 1. co-authored with Arthur William Panton 2. William Snow Burnside Obituary, Irish Times, 13 March 1920 3. Burnside Family Genealogy Library Ireland 4. The Collected Papers of William Burnside: Commentary on Burnside's life by William Burnside Authority control International • ISNI • VIAF National • Israel • United States • Sweden
William Spence (mathematician) William Spence (born 31 July 1777 in Greenock, Scotland – died 22 May 1815 in Glasgow, Scotland) was a Scottish mathematician who published works on the fields of logarithmic functions, algebraic equations and their relation to integral and differential calculus respectively. William Spence Born31 July 1777 Greenock,Scotland Died20 May 1815 (aged 37) Glasgow, Scotland Parents • Ninian Spence (father) • Sarah Townsend (mother) Notes A D D Craik, The 'Mathematical Essays' of William Spence (1777–1815), Historia Mathematica 40: 4 (2013), 386–422. Image accredited to the Watt Institute, Inverclyde Council Early life, family, and personal life Spence was the second son to Ninian Spence and his wife Sarah Townsend. Ninian Spence ran a coppersmith business, and the Spence family were a prominent family in Greenock at the time.[1][2] From an early age, Spence was characterised as having a docile and reasonable nature, with him being mature for his age.[3] At school he formed a life-long friendship with John Galt, who documented much of his life and his works posthumously.[3][4] Despite having received a formal education until he was a teenager, Spence never attended university, instead he moved to Glasgow where he lodged with a friend of his fathers, learning the skills of a manufacturer.[1] Two years after his father's death in 1795, Spence returned to Greenock in 1797.[1] With the support of Galt and others, he established a small literary society, wherein once a month they read a range of essays on varying subjects, this society met frequently until 1804.[5] After this, Spence visited many places in England, he lived in London for a few months where, in 1809, he published his first work.[1] In 1814, he published his second work, getting married in the same year – Spence intended to live in London, and began his journey back before becoming ill, having travelled as far as Glasgow, he died in his sleep due to illness.[1][5] Spence held an interest in musical composition, and played the flute.[4] Published works Spence published An Essay on the Theory of the Various Orders of Logarithmic Transcendents: With an Inquiry Into Their Applications to the Integral Calculus and the Summation of Series in 1809.[6] Throughout his work, he displayed a familiarity with the work of Lagrange and Arbogast, which is notable since at the time very few were familiar with their works.[1][7] In his preface he derived the binomial theorem and mainly focused on the properties and analytic applications of the series:[1][6][7] $\pm x/1^{n}-x^{2}/2^{n}\pm x^{3}/3^{n}-...$ which he denoted with $L_{n}(1\pm x)$.[6] He went on further to derive nine general properties of this function in a table.[6] Spence also wrote on presenting analytical mathematics without the need of demonstrating the practical applications of such work.[6] Spence continues to write that the functions $L_{n}(1\pm x)$ can be expressed as iterations of the previous n term: $L_{1}(1\pm x)=\int \pm dx/1\pm x$, $L_{2}(1\pm x)=\int (dx/x)L_{1}(1\pm x)$, . . . , $L_{n}(1\pm x)=\int (dx/x)L_{n-1}(1\pm x)$ for all values of x.[6] Spence goes on to calculate the values of: $L_{2}(x)=-\int _{0}^{x}{\frac {\ln(1-t)}{t}}\operatorname {d} \!t$ to nine decimal places, in a table, for all integer values of $1+x$ from 1 to 100, the first ever of its kind.[1][6] These functions became known as the polylogarithm functions, with this particular case often called Spence's function after Spence. Later on he also created a similar table for $\tan ^{-1}x$.[6][7] Spence published his last work, Outlines of a theory of Algebraical Equations, deduced from the principles of Harriott, and extended to the fluxional or differential calculus was published in 1814.[8] In which he took a systematic approach to solving equations up to the fourth degree using symmetrical functions of the roots.[7][8] After Spence's death, John Herschel edited Mathematical Essays by the late William Spence, which was published in 1819, with John Galt writing a biography on Spence.[3][9] Legacy Spence's work was noted to be remarkable at the time, with John Herschel, his acquaintance and one of Britain's leading mathematicians at the time, had referenced it in one of his later publications Consideration of various points of analysis, which prompted Herschel to edit Spence's manuscripts.[1][10] Spence was held in such high regard by Galt, and later Herschel that they published a collection of his individual essays in 1819.[1][11] Posthumously, his work was met with appreciation from his contemporaries, with a review in the ninety-fourth number of the Quarterly Review (reproduced in Galt's The Literary and Miscellanies of John Galt, Volume 1) that described his first work in 1809 as: " [The] first formal essay in our language on any distinct and considerable branch of the integral calculus, which has appeared since… Hellinsʼs papers on the ‘Rectification of the Conic Sections".[1][12][13] References 1. Craik, Alex D.D. (October 2013). "Polylogarithms, functional equations and more: The elusive essays of William Spence (1777–1815)". Historia Mathematica. 40 (4): 386–422. doi:10.1016/j.hm.2013.06.002. 2. "Greenock - Towns - Scottish Directories - National Library of Scotland". digital.nls.uk. Retrieved 24 June 2022. 3. Galt, J. (May 1819). "THE LATE MR. WILLIAM SPENCE". The Monthly Magazine. 47 (325): 373–375. ProQuest 4520067. 4. Spence, William (1819). Mathematical Essays, by the Late William Spence, Esq. Edited by John F. W. Herschel, Esq. With a Biographical Sketch of the Author. Thomas and George Underwood, 32, Fleet Street. OCLC 1021878949. 5. Galt, John (1833). The autobiography of John Galt. Key & Biddle. 6. Spence, William (1809). An Essay on the Theory of the Various Orders of Logarithmic Transcendents: With an Inquiry Into Their Applications to the Integral Calculus and the Summation of Series. John Murray and Archibald Constable and Company. OCLC 10156665. 7. "William Spence – Biography". Maths History. Retrieved 24 June 2022. 8. Spence, William (1814). Outlines of a theory of Algebraical Equations, deduced from the principles of Harriott, and extended to the fluxional or differential calculus. OCLC 1063204490. 9. Spence, William (1819). Mathematical Essays, by the Late William Spence, Esq. Edited by John F. W. Herschel, Esq. With a Biographical Sketch of the Author. Thomas and George Underwood, 32, Fleet Street. OCLC 1021878949. 10. "XXII. Consideration of various points of analysis". Philosophical Transactions of the Royal Society of London. 104: 440–468. 31 December 1814. doi:10.1098/rstl.1814.0023. S2CID 111328500. 11. "John Herschel Correspondence". historydb.adlerplanetarium.org. Retrieved 28 June 2022. 12. Galt, John (1834). The Literary Life and Miscellanies of John Galt. W. Blackwood. 13. "XVIL. Of the rectification of the conic sections". Philosophical Transactions of the Royal Society of London. 92: 448–476. 31 December 1802. doi:10.1098/rstl.1802.0020. S2CID 110222385.
William Thomson (mathematician) Sir William Thomson FRSE LLD (1856–1947) was a 19th/20th century Scottish mathematician and physicist primarily working as a university administrator in South Africa. Life He was born on New Year's Eve, 31 December 1856, in the village of Kirkton of Mailler in Perthshire. He was educated at Perth Academy then studied mathematics and physics at the University of Edinburgh. He graduated with a BSc and MA in 1878 and began assisting in lectures at the University.[1] In 1882, aged 26, he was elected a Fellow of the Royal Society of Edinburgh. His proposers were George Chrystal, Peter Guthrie Tait, Alexander Crum Brown and Sir William Turner.[2] In 1883 he succeeded Prof George Gordon as Professor of Mathematics at Stellenbosch University in South Africa.[3] In 1895 he succeeded Rev James Cameron as University Registrar at the University of the Cape of Good Hope. In 1918 he transferred to the same role in the newly created University of South Africa and in 1922 moved to the University of the Witwatersrand. He was knighted by King George V in 1922 for services to university education. He retired in 1928 and died in Simonstown near Cape Town on 6 August 1947. Family In 1884 he married Annie Catherine van der Riet. They had two daughters. Publications • Mensuration in 9th edition of Encyclopædia Britannica (1878) • Introduction to Determinants (1881) • Algebra for the Use of Schools and Colleges (1886) • Textbook of Geometrical Deductions (1891) • Elementary Algebra (1901) References 1. "S2A3 Biographical Database of Southern African Science". s2a3.org.za. Retrieved 14 December 2018. 2. Biographical Index of Former Fellows of the Royal Society of Edinburgh 1783–2002 (PDF). The Royal Society of Edinburgh. July 2006. ISBN 0 902 198 84 X. 3. "Thomson_William summary". www-history.mcs.st-andrews.ac.uk. Retrieved 14 December 2018. Authority control International • VIAF National • United States Academics • MathSciNet • zbMATH
William Thurston William Paul Thurston (October 30, 1946 – August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. William Thurston Thurston in 1991 Born William Paul Thurston (1946-10-30)October 30, 1946 Washington, D.C., United States DiedAugust 21, 2012(2012-08-21) (aged 65) Rochester, New York, United States NationalityAmerican Alma materNew College of Florida University of California, Berkeley Known forThurston's geometrization conjecture Thurston's theory of surfaces Milnor–Thurston kneading theory AwardsFields Medal (1982) Oswald Veblen Prize in Geometry (1976) Alan T. Waterman Award (1979) National Academy of Sciences (1983) Doob Prize (2005) Leroy P. Steele Prize (2012). Scientific career FieldsMathematics InstitutionsCornell University University of California, Davis Mathematical Sciences Research Institute University of California, Berkeley Princeton University Massachusetts Institute of Technology Institute for Advanced Study ThesisFoliations of three-manifolds which are circle bundles (1972) Doctoral advisorMorris Hirsch Doctoral studentsRichard Canary Benson Farb David Gabai William Goldman Richard Kenyon Steven Kerckhoff Yair Minsky Igor Rivin Oded Schramm Richard Schwartz Danny Calegari Thurston was a professor of mathematics at Princeton University, University of California, Davis, and Cornell University. He was also a director of the Mathematical Sciences Research Institute. Early life and education William Thurston was born in Washington, D.C., to Margaret Thurston (née Martt), a seamstress, and Paul Thurston, an aeronautical engineer.[1] William Thurston suffered from congenital strabismus as a child, causing issues with depth perception.[1] His mother worked with him as a toddler to reconstruct three-dimensional images from two-dimensional ones.[1] He received his bachelor's degree from New College in 1967 as part of its inaugural class.[1][2] For his undergraduate thesis, he developed an intuitionist foundation for topology.[3] Following this, he received a doctorate in mathematics from the University of California, Berkeley under Morris Hirsch, with his thesis Foliations of Three-Manifolds which are Circle Bundles in 1972.[1][4] Career After completing his Ph.D., Thurston spent a year at the Institute for Advanced Study,[1][5] then another year at the Massachusetts Institute of Technology as an assistant professor.[1] In 1974, Thurston was appointed a full professor at Princeton University.[1][6] He returned to Berkeley in 1991 to be a professor (1991-1996) and was also director of the Mathematical Sciences Research Institute (MSRI) from 1992 to 1997.[1][7] He was on the faculty at UC Davis from 1996 until 2003, when he moved to Cornell University.[1] Thurston was an early adopter of computing in pure mathematics research.[1] He inspired Jeffrey Weeks to develop the SnapPea computing program.[1] During Thurston's directorship at MSRI, the institute introduced several innovative educational programs that have since become standard for research institutes.[1] His Ph.D. students include Danny Calegari, Richard Canary, David Gabai, William Goldman, Benson Farb, Richard Kenyon, Steven Kerckhoff, Yair Minsky, Igor Rivin, Oded Schramm, Richard Schwartz, William Floyd, and Jeffrey Weeks.[8] Research Foliations His early work, in the early 1970s, was mainly in foliation theory. His more significant results include: • The proof that every Haefliger structure on a manifold can be integrated to a foliation (this implies, in particular, that every manifold with zero Euler characteristic admits a foliation of codimension one). • The construction of a continuous family of smooth, codimension-one foliations on the three-sphere whose Godbillon–Vey invariant (after Claude Godbillon and Jacques Vey) takes every real value. • With John N. Mather, he gave a proof that the cohomology of the group of homeomorphisms of a manifold is the same whether the group is considered with its discrete topology or its compact-open topology. In fact, Thurston resolved so many outstanding problems in foliation theory in such a short period of time that it led to an exodus from the field, where advisors counselled students against going into foliation theory,[9] because Thurston was "cleaning out the subject" (see "On Proof and Progress in Mathematics", especially section 6[10]). The geometrization conjecture Main article: Geometrization conjecture His later work, starting around the mid-1970s, revealed that hyperbolic geometry played a far more important role in the general theory of 3-manifolds than was previously realised. Prior to Thurston, there were only a handful of known examples of hyperbolic 3-manifolds of finite volume, such as the Seifert–Weber space. The independent and distinct approaches of Robert Riley and Troels Jørgensen in the mid-to-late 1970s showed that such examples were less atypical than previously believed; in particular their work showed that the figure-eight knot complement was hyperbolic. This was the first example of a hyperbolic knot. Inspired by their work, Thurston took a different, more explicit means of exhibiting the hyperbolic structure of the figure-eight knot complement. He showed that the figure-eight knot complement could be decomposed as the union of two regular ideal hyperbolic tetrahedra whose hyperbolic structures matched up correctly and gave the hyperbolic structure on the figure-eight knot complement. By utilizing Haken's normal surface techniques, he classified the incompressible surfaces in the knot complement. Together with his analysis of deformations of hyperbolic structures, he concluded that all but 10 Dehn surgeries on the figure-eight knot resulted in irreducible, non-Haken non-Seifert-fibered 3-manifolds. These were the first such examples; previously it had been believed that except for certain Seifert fiber spaces, all irreducible 3-manifolds were Haken. These examples were actually hyperbolic and motivated his next theorem. Thurston proved that in fact most Dehn fillings on a cusped hyperbolic 3-manifold resulted in hyperbolic 3-manifolds. This is his celebrated hyperbolic Dehn surgery theorem. To complete the picture, Thurston proved a hyperbolization theorem for Haken manifolds. A particularly important corollary is that many knots and links are in fact hyperbolic. Together with his hyperbolic Dehn surgery theorem, this showed that closed hyperbolic 3-manifolds existed in great abundance. The hyperbolization theorem for Haken manifolds has been called Thurston's Monster Theorem, due to the length and difficulty of the proof. Complete proofs were not written up until almost 20 years later. The proof involves a number of deep and original insights which have linked many apparently disparate fields to 3-manifolds. Thurston was next led to formulate his geometrization conjecture. This gave a conjectural picture of 3-manifolds which indicated that all 3-manifolds admitted a certain kind of geometric decomposition involving eight geometries, now called Thurston model geometries. Hyperbolic geometry is the most prevalent geometry in this picture and also the most complicated. The conjecture was proved by Grigori Perelman in 2002–2003.[11][12] Density conjecture Thurston and Dennis Sullivan generalized Lipman Bers' density conjecture from singly degenerate Kleinian surface groups to all finitely generated Kleinian groups in the late 1970s and early 1980s.[13][14] The conjecture states that every finitely generated Kleinian group is an algebraic limit of geometrically finite Kleinian groups, and was independently proven by Ohshika and Namazi–Souto in 2011 and 2012 respectively.[13][14] Orbifold theorem In his work on hyperbolic Dehn surgery, Thurston realized that orbifold structures naturally arose. Such structures had been studied prior to Thurston, but his work, particularly the next theorem, would bring them to prominence. In 1981, he announced the orbifold theorem, an extension of his geometrization theorem to the setting of 3-orbifolds. Two teams of mathematicians around 2000 finally finished their efforts to write down a complete proof, based mostly on Thurston's lectures given in the early 1980s in Princeton. His original proof relied partly on Richard S. Hamilton's work on the Ricci flow. Awards and honors In 1976, Thurston and James Harris Simons shared the Oswald Veblen Prize in Geometry.[1] Thurston received the Fields Medal in 1982 for "revolutioniz[ing] [the] study of topology in 2 and 3 dimensions, showing interplay between analysis, topology, and geometry" and "contribut[ing] [the] idea that a very large class of closed 3-manifolds carry a hyperbolic structure."[15][16] In 2005, Thurston won the first American Mathematical Society Book Prize, for Three-dimensional Geometry and Topology. The prize "recognizes an outstanding research book that makes a seminal contribution to the research literature".[17] He was awarded the 2012 Leroy P. Steele Prize by the American Mathematical Society for seminal contribution to research. The citation described his work as having "revolutionized 3-manifold theory".[18] Personal life Thurston and his first wife, Rachel Findley, had three children: Dylan, Nathaniel, and Emily.[6] Dylan was a MOSP participant (1988–90)[19] and is a mathematician at Indiana University Bloomington.[20] Thurston had two children with his second wife, Julian Muriel Thurston: Hannah Jade and Liam.[6] Thurston died on August 21, 2012, in Rochester, New York, of a sinus mucosal melanoma that was diagnosed in 2011.[6][21][7] Selected publications • William Thurston, The geometry and topology of three-manifolds, Princeton lecture notes (1978–1981). • William Thurston, Three-dimensional geometry and topology. Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, New Jersey, 1997. x+311 pp. ISBN 0-691-08304-5 • William Thurston, Hyperbolic structures on 3-manifolds. I. Deformation of acylindrical manifolds. Ann. of Math. (2) 124 (1986), no. 2, 203–246. • William Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), 357–381. • William Thurston, On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431 • Epstein, David B. A.; Cannon, James W.; Holt, Derek F.; Levy, Silvio V. F.; Paterson, Michael S.; Thurston, William P. Word Processing in Groups. Jones and Bartlett Publishers, Boston, Massachusetts, 1992. xii+330 pp. ISBN 0-86720-244-0[22] • Eliashberg, Yakov M.; Thurston, William P. Confoliations. University Lecture Series, 13. American Mathematical Society, Providence, Rhode Island and Providence Plantations, 1998. x+66 pp. ISBN 0-8218-0776-5 • William Thurston, On proof and progress in mathematics. Bull. Amer. Math. Soc. (N.S.) 30 (1994) 161–177 • William P. Thurston, "Mathematical education". Notices of the AMS 37:7 (September 1990) pp 844–850 See also • Automatic group • Cannon–Thurston map • Circle packing theorem • Hyperbolic volume • Hyperbolic Dehn surgery • Thurston boundary • Milnor–Thurston kneading theory • Misiurewicz–Thurston points • Nielsen–Thurston classification • Normal surface • Orbifold notation • Thurston norm • Thurston's double limit theorem • Thurston elliptization conjecture • Thurston's geometrization conjecture • Thurston's height condition • Thurston's orbifold theorem • Earthquake theorem References 1. Gabai, David; Kerckhoff, Steven (2015). "William P. Thurston, 1946–2012" (PDF). Notices of the American Mathematical Society. 62 (11): 1318–1332. doi:10.1090/noti1300. Archived (PDF) from the original on 2022-10-09. 2. Kelley, Susan (Aug 24, 2012). "World-renowned mathematician William Thurston dies at 65". Retrieved 2023-01-11. 3. See p. 3 in Laudenbach, François; Papadopoulos, Athanase (2019). "W. P. Thurston and French mathematics". arXiv:1912.03115 [math.GT]. 4. "William Thurston – the Mathematics Genealogy Project". 5. "Institute for Advanced Study: A Community of Scholars". Ias.edu. Retrieved 2013-09-06. 6. Leslie Kaufman (August 23, 2012). "William P. Thurston, Theoretical Mathematician, Dies at 65". New York Times. p. B15. 7. "William P. Thurston, 1946-2012". American Mathematical Society. August 22, 2012. Retrieved March 25, 2022. 8. "William Thurston – the Mathematics Genealogy Project". 9. "The Mathematical Legacy of William Thurston (1946–2012)". 10. Thurston, William P. (April 1994). "On Proof and Progress in Mathematics". Bulletin of the American Mathematical Society. 30 (2): 161–177. arXiv:math/9404236. Bibcode:1994math......4236T. doi:10.1090/S0273-0979-1994-00502-6. 11. Perelman, Grisha (2003-03-10). "Ricci flow with surgery on three-manifolds". arXiv:math/0303109. 12. Kleiner, Bruce; Lott, John (2008-11-06). "Notes on Perelman's papers". Geometry & Topology. 12 (5): 2587–2855. arXiv:math/0605667. doi:10.2140/gt.2008.12.2587. ISSN 1364-0380. 13. Namazi, Hossein; Souto, Juan (2012). "Non-realizability and ending laminations: Proof of the density conjecture". Acta Mathematica. 209 (2): 323–395. doi:10.1007/s11511-012-0088-0. ISSN 0001-5962. S2CID 10138438. 14. Ohshika, Ken'ichi (2011). "Realising end invariants by limits of minimally parabolic, geometrically finite groups". Geometry and Topology. 15 (2): 827–890. arXiv:math/0504546. doi:10.2140/gt.2011.15.827. ISSN 1364-0380. S2CID 14463721. Archived from the original on May 25, 2014. Retrieved March 24, 2022. 15. "William P. Thurston, 1946–2012". 30 August 2012. Retrieved 18 August 2014. 16. "Fields Medals and Nevanlinna Prize 1982". mathunion.org. International Mathematical Union. 17. "William P. Thurston Receives 2005 AMS Book Prize". Retrieved 2008-06-26. 18. "AMS prize booklet 2012" (PDF). Archived (PDF) from the original on 2022-10-09. 19. "YEAR 1990" (PDF). USAMO Archive. Retrieved 30 January 2023. 20. Thurston, Dylan P., ed. (2020). What's Next? The Mathematical Legacy of William P. Thurston. Princeton University Press. ISBN 978-0-691-16776-3. 21. "Department mourns loss of friend and colleague, Bill Thurston", Cornell University 22. Reviews of Word Processing in Groups: B. N. Apanasov, Zbl 0764.20017; Gilbert Baumslag, Bull. AMS, doi:10.1090/S0273-0979-1994-00481-1; D. E. Cohen, Bull LMS, doi:10.1112/blms/25.6.614; Richard M. Thomas, MR1161694 Further reading • Gabai, David; Kerckhoff, Steve (Coordinating Editors). "William P. Thurston, 1946–2012" (part 2), Notices of the American Mathematical Society, January 2015, Volume 63, Number 1, pp. 31–41. External links • Media related to William Thurston at Wikimedia Commons Wikiquote has quotations related to William Thurston. • William Thurston at the Mathematics Genealogy Project • O'Connor, John J.; Robertson, Edmund F., "William Thurston", MacTutor History of Mathematics Archive, University of St Andrews • Thurston's page at Cornell • Tribute and remembrance page at Cornell • Etienne Ghys : La géométrie et la mode • "Landau Lectures \| Prof. Thurston \| Part 1 \| 1995/6". YouTube. Hebrew University of Jerusalem. April 8, 2014. • "Landau Lectures \| Prof. Thurston \| Part 2 \| 1995/6". YouTube. Hebrew University of Jerusalem. April 8, 2014. • "Landau Lectures \| Prof. Thurston \| Part 3 \| 1995/6". YouTube. Hebrew University of Jerusalem. April 8, 2014. • "The Mystery of 3-Manifolds - William Thurston". YouTube. PoincareDuality. November 27, 2011. 2010 Clay Research Conference • Goldman, William (May 9, 2013). "William Thurston: A Mathematical Perspective". YouTube. UMD Mathematics. William Goldman (U. of Maryland), Collloquium, Department of Mathematics, Howard University, 25 January 2013 Fields Medalists • 1936 Ahlfors • Douglas • 1950 Schwartz • Selberg • 1954 Kodaira • Serre • 1958 Roth • Thom • 1962 Hörmander • Milnor • 1966 Atiyah • Cohen • Grothendieck • Smale • 1970 Baker • Hironaka • Novikov • Thompson • 1974 Bombieri • Mumford • 1978 Deligne • Fefferman • Margulis • Quillen • 1982 Connes • Thurston • Yau • 1986 Donaldson • Faltings • Freedman • 1990 Drinfeld • Jones • Mori • Witten • 1994 Bourgain • Lions • Yoccoz • Zelmanov • 1998 Borcherds • Gowers • Kontsevich • McMullen • 2002 Lafforgue • Voevodsky • 2006 Okounkov • Perelman • Tao • Werner • 2010 Lindenstrauss • Ngô • Smirnov • Villani • 2014 Avila • Bhargava • Hairer • Mirzakhani • 2018 Birkar • Figalli • Scholze • Venkatesh • 2022 Duminil-Copin • Huh • Maynard • Viazovska • Category • Mathematics portal Recipients of the Oswald Veblen Prize in Geometry • 1964 Christos Papakyriakopoulos • 1964 Raoul Bott • 1966 Stephen Smale • 1966 Morton Brown and Barry Mazur • 1971 Robion Kirby • 1971 Dennis Sullivan • 1976 William Thurston • 1976 James Harris Simons • 1981 Mikhail Gromov • 1981 Shing-Tung Yau • 1986 Michael Freedman • 1991 Andrew Casson and Clifford Taubes • 1996 Richard S. Hamilton and Gang Tian • 2001 Jeff Cheeger, Yakov Eliashberg and Michael J. Hopkins • 2004 David Gabai • 2007 Peter Kronheimer and Tomasz Mrowka; Peter Ozsváth and Zoltán Szabó • 2010 Tobias Colding and William Minicozzi; Paul Seidel • 2013 Ian Agol and Daniel Wise • 2016 Fernando Codá Marques and André Neves • 2019 Xiuxiong Chen, Simon Donaldson and Song Sun Authority control International • FAST • ISNI • VIAF National • Norway • France • BnF data • Catalonia • Germany • Israel • United States • Japan • Czech Republic • Netherlands Academics • DBLP • MathSciNet • Mathematics Genealogy Project • zbMATH Other • SNAC • IdRef
William A. Veech William A. Veech was the Edgar O. Lovett Professor of Mathematics at Rice University[1] until his death. His research concerned dynamical systems; he is particularly known for his work on interval exchange transformations, and is the namesake of the Veech surface. He died unexpectedly on August 30, 2016 in Houston, Texas.[2] Education Veech graduated from Dartmouth College in 1960,[1] and earned his Ph.D. in 1963 from Princeton University under the supervision of Salomon Bochner.[1][3] Contributions An interval exchange transformation is a dynamical system defined from a partition of the unit interval into finitely many smaller intervals, and a permutation on those intervals. Veech and Howard Masur independently discovered that, for almost every partition and every irreducible permutation, these systems are uniquely ergodic, and also made contributions to the theory of weak mixing for these systems.[4] The Rauzy–Veech–Zorich induction map, a function from and to the space of interval exchange transformations is named in part after Veech: Rauzy defined the map, Veech constructed an infinite invariant measure for it, and Zorich strengthened Veech's result by making the measure finite.[5] The Veech surface and the related Veech group are named after Veech, as is the Veech dichotomy according to which geodesic flow on the Veech surface is either periodic or ergodic.[6] Veech played a role in the Nobel-prize-winning discovery of buckminsterfullerene in 1985 by a team of Rice University chemists including Richard Smalley. At that time, Veech was chair of the Rice mathematics department, and was asked by Smalley to identify the shape that the chemists had determined for this molecule. Veech answered, "I could explain this to you in a number of ways, but what you've got there, boys, is a soccer ball."[7][8] Veech is the author of A Second Course in Complex Analysis (W. A. Benjamin, 1967; Dover, 2008, ISBN 9780486462943).[9][10][11] Awards and honors In 2012, Veech became one of the inaugural fellows of the American Mathematical Society.[12] References 1. Faculty profile, Rice University, retrieved 2015-03-01. 2. Todd, Hannah. "Former math department chair passes away". Rice Thresher. Retrieved 29 September 2016. 3. William A. Veech at the Mathematics Genealogy Project 4. Hunt, B. R.; Kaloshin, V. Yu. (2010), "Prevalence", in Broer, H.; Takens, F.; Hasselblatt, B. (eds.), Handbook of Dynamical Systems, Volume 3, Elsevier, pp. 43–88, ISBN 9780080932262. See in particular p. 51. 5. Bufetov, Alexander I. (2006), "Decay of correlations for the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations and the central limit theorem for the Teichmüller flow on the moduli space of abelian differentials", Journal of the American Mathematical Society, 19 (3): 579–623, arXiv:math/0506222, doi:10.1090/S0894-0347-06-00528-5, MR 2220100, S2CID 15755696. 6. Smillie, John; Weiss, Barak (2008), "Veech's dichotomy and the lattice property", Ergodic Theory and Dynamical Systems, 28 (6): 1959–1972, doi:10.1017/S0143385708000114, MR 2465608, S2CID 42112090. 7. Edelson, Edward (August 1991), "Buckyball: the magic molecule", Popular Science: 52–57, 87. The quote is on p. 55. 8. Ball, Philip (1996), Designing the Molecular World: Chemistry at the Frontier, Princeton Science Library, Princeton University Press, p. 46, ISBN 9780691029009. 9. Review of A Second Course in Complex Analysis by E. Hille, MR0220903. 10. Wenzel, H., "W. A. Veech, A Second Course in Complex Analysis", Book Reviews, Journal of Applied Mathematics and Mechanics, 48 (7): 502–503, Bibcode:1968ZaMM...48..502W, doi:10.1002/zamm.19680480725. 11. Stenger, Allen (April 24, 2008), "A Second Course in Complex Analysis, William A. Veech", MAA Reviews, Mathematical Association of America. 12. List of Fellows of the American Mathematical Society, retrieved 2015-03-01. Authority control International • ISNI • VIAF National • Israel • United States • Netherlands Academics • CiNii • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
William Wallace (mathematician) William Wallace FRSE MInstCE FRAS LLD (23 September 1768 – 28 April 1843) was a Scottish mathematician and astronomer who invented the eidograph (an improved pantograph). Life Wallace was born at Dysart in Fife, the son of Alexander Wallace, a leather manufacturer, and his wife, Janet Simson.[1] He received his school education in Dysart and Kirkcaldy. In 1784 his family moved to Edinburgh, where he himself was set to learn the trade of a bookbinder.[2] In 1790 he appears as "William Wallace, bookbinder" living and trading at Cowgatehead, at the east end of the Grassmarket.[3] His taste for mathematics had already developed itself, and he made such use of his leisure hours that before the completion of his apprenticeship he had made considerable acquirements in geometry, algebra and astronomy. He was further assisted in his studies by John Robison (1739–1805) and John Playfair, to whom his abilities had become known.[2] After various changes of situation, dictated mainly by a desire to gain time for study, he became assistant teacher of mathematics in the Perth Academy in 1794. This post he exchanged in 1803 for a mathematical mastership in the Royal Military College at Great Marlow, in which post he continued after it moved to Sandhurst, with a recommendation by Playfair.[2] In 1804 he was elected a Fellow of the Royal Society of Edinburgh.[2] His proposers were John Playfair, Thomas Charles Hope and William Wright.[4] In 1819 he was chosen to succeed John Playfair in the chair of mathematics at Edinburgh. Playfair's second chair (in Natural Philosophy) was taken by John Leslie. Wallace developed a reputation for being an excellent teacher. Among his students was Mary Somerville. In 1838 he retired from the university due to ill health.[2] In his final years he lived at 6 Lauriston Lane on the south side of Edinburgh.[5] He died in Edinburgh aged 74 and was buried in Greyfriars Kirkyard. The grave lies on the north-facing retaining wall in the centre of the northern section. Mathematical contributions In his earlier years Wallace was an occasional contributor to Leybourne's Mathematical Repository and the Gentleman's Mathematical Companion. Between 1801 and 1810 he contributed articles on "Algebra", "Conic Sections", "Trigonometry", and several others in mathematical and physical science to the fourth edition of the Encyclopædia Britannica, and some of these were retained in subsequent editions from the fifth to the eighth inclusive. He was also the author of the principal mathematical articles in the Edinburgh Encyclopædia, edited by David Brewster. He also contributed many important papers to the Transactions of the Royal Society of Edinburgh.[2] He mainly worked in the field of geometry and in 1799 became the first to publish the concept of the Simson line, which erroneously was attributed to Robert Simson.[6] In 1807 he proved a result about polygons with an equal area, that later became known as the Bolyai–Gerwien theorem.[7] His most important contribution to British mathematics however was, that he was one of the first mathematicians introducing and promoting the advancement of the continental European version of calculus in Britain.[6] Other works Wallace also worked in astronomy and invented the eidograph, a mechanical device for scaling drawings.[6][8] Books • A Geometrical Treatise on the Conic Sections with an Appendix Containing Formulae for their Quadrature. (1838) • Geometrical Theorems and Analytical Formulae with their application to the Solution of Certain Geodetical Problems and an Appendix. (1839) Family Wallace was married to Janet Kerr (1775–1824). His daughter, Margaret Wallace, married the mathematician Thomas Galloway. His sons included the Rev Alexander Wallace (1803–1842) and Archibald C. Wallace (1806–1830). He also appears to have had a son William Wallace (1784–1864) when William was aged only 16. The mother is not clear. References 1. "William Wallace - Biography". Maths History. 2. Chisholm 1911. 3. Williamson's Edinburgh Directory 1790 4. Biographical Index of Former Fellows of the Royal Society of Edinburgh 1783–2002 (PDF). The Royal Society of Edinburgh. July 2006. ISBN 978-0-902198-84-5. 5. Edinburgh and Leith Post Office directory 1835-36 6. O'Connor, John J.; Robertson, Edmund F., "William Wallace (mathematician)", MacTutor History of Mathematics Archive, University of St Andrews 7. Ian Stewart: From Here to Infinity. Oxford University Press 1996 (3. edition), ISBN 978-0-19-283202-3, p. 169 (restricted online copy, p. 169, at Google Books) 8. Gerard L'Estrange Turner: Nineteenth-Century Scientific Instruments. University of California Press 1983, ISBN 0-520-05160-2, p. 280 (online copy, p. 280, at Google Books) Sources • This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "Wallace, William". Encyclopædia Britannica. Vol. 28 (11th ed.). Cambridge University Press. p. 278. External links • O'Connor, John J.; Robertson, Edmund F., "William Wallace (mathematician)", MacTutor History of Mathematics Archive, University of St Andrews • Short biographical note on William Wallace in the Gazetteer for Scotland Authority control International • FAST • ISNI • VIAF National • Germany • Israel • United States • Netherlands • Poland Academics • zbMATH People • Deutsche Biographie Other • SNAC • IdRef
William Whiston William Whiston (9 December 1667 – 22 August 1752) was an English theologian, historian, natural philosopher, and mathematician, a leading figure in the popularisation of the ideas of Isaac Newton. He is now probably best known for helping to instigate the Longitude Act in 1714 (and his attempts to win the rewards that it promised) and his important translations of the Antiquities of the Jews and other works by Josephus (which are still in print). He was a prominent exponent of Arianism and wrote A New Theory of the Earth. William Whiston Born(1667-12-09)9 December 1667 Norton-juxta-Twycross, Leicestershire, England Died22 August 1752(1752-08-22) (aged 84) Lyndon, Rutland, England NationalityEnglish Alma materClare College, Cambridge Known forTranslating the works of Josephus, catastrophism, isoclinic maps, work on longitude Scientific career FieldsMathematics, theology InstitutionsClare College, Cambridge Academic advisorsIsaac Newton Robert Herne Notable studentsJames Jurin InfluencesDavid Gregory Isaac Newton Signature Whiston succeeded his mentor Newton as Lucasian Professor of Mathematics at the University of Cambridge. In 1710 he lost the professorship and was expelled from the university as a result of his unorthodox religious views. Whiston rejected the notion of eternal torment in hellfire, which he viewed as absurd, cruel, and an insult to God. What especially pitted him against church authorities was his denial of the doctrine of the Trinity, which he believed had pagan origins. Early life and career Whiston was born to Josiah Whiston (1622–1685) and Katherine Rosse (1639–1701) at Norton-juxta-Twycross, in Leicestershire, where his father was rector. His mother was daughter of the previous rector at Norton-juxta-Twycross, Gabriel Rosse. Josiah Whiston was a presbyterian, but retained his rectorship after the Stuart Restoration in 1660. William Whiston was educated privately, for his health, and so that he could act as amanuensis to his blind father.[1][2] He studied at Queen Elizabeth Grammar School at Tamworth, Staffordshire. After his father's death, he entered Clare College, Cambridge as a sizar in 1686. He applied himself to mathematical study, was awarded the degree of Bachelor of Arts (BA) (1690), and AM (1693), and was elected Fellow in 1691 and probationary senior Fellow in 1693.[1][3] William Lloyd ordained Whiston at Lichfield in 1693. In 1694, claiming ill health, he resigned his tutorship at Clare to Richard Laughton, chaplain to John Moore, the bishop of Norwich, and swapped positions with him. He now divided his time between Norwich, Cambridge and London. In 1698 Moore gave him the living of Lowestoft where he became rector. In 1699 he resigned his Fellowship of Clare College and left to marry.[1] Whiston first met Isaac Newton in 1694 and attended some of his lectures, though he first found them, by his own admission, incomprehensible. Encouraged after reading a paper by David Gregory on Newtonian philosophy, he set out to master Newton's Principia mathematica thereafter. He and Newton became friends.[1] In 1701 Whiston resigned his living to become Isaac Newton's substitute, giving the Lucasian lectures at Cambridge.[2] He succeeded Newton as Lucasian professor in 1702. There followed a period of joint research with Roger Cotes, appointed with Whiston's patronage to the Plumian professorship in 1706. Students at the Cotes–Whiston experimental philosophy course included Stephen Hales, William Stukeley, and Joseph Wasse.[4] Newtonian theologian In 1707 Whiston was Boyle lecturer; this lecture series was at the period a significant opportunity for Newton's followers, including Richard Bentley and Samuel Clarke, to express their views, especially in opposition to the rise of deism.[5] The "Newtonian" line came to include, with Bentley, Clarke and Whiston in particular, a defence of natural law by returning to the definition of Augustine of Hippo of a miracle (a cause of human wonderment), rather than the prevailing concept of a divine intervention against nature, which went back to Anselm. This move was intended to undermine arguments of deists and sceptics.[6] The Boyle lectures dwelt on the connections between biblical prophecies, dramatic physical events such as floods and eclipses, and their explanations in terms of science.[7] On the other hand, Whiston was alive to possible connections of prophecy with current affairs: the War of the Spanish Succession, and later the Jacobite rebellions.[8] Whiston supported a qualified biblical literalism: the literal meaning should be the default, unless there was a good reason to think otherwise.[9] This view again went back to Augustine. Newton's attitude to the cosmogony of Thomas Burnet reflected on the language of the Genesis creation narrative; as did Whiston's alternative cosmogony. Moses as author of Genesis was not necessarily writing as a natural philosopher, nor as a law-giver, but for a particular audience.[10] The new cosmogonies of Burnet, Whiston and John Woodward were all criticised for their disregard of the biblical account, by John Arbuthnot, John Edwards and William Nicolson in particular.[11] The title for Whiston's Boyle lectures was The Accomplishment of Scripture Prophecies. Rejecting typological interpretation of biblical prophecy, he argued that the meaning of a prophecy must be unique. His views were later challenged by Anthony Collins.[12] There was a more immediate attack by Nicholas Clagett in 1710.[13] One reason prophecy was topical was the Camisard movement that saw French exiles ("French prophets") in England. Whiston had started writing on the millenarianism that was integral to the Newtonian theology, and wanted to distance his views from theirs, and in particular from those of John Lacy.[14] Meeting the French prophets in 1713, Whiston developed the view that the charismatic gift of revelation could be demonic possession.[15] Tensions with Newton It is no longer assumed that Whiston's Memoirs are completely trustworthy on the matter of his personal relations with Newton. One view is that the relationship was never very close, Bentley being more involved in Whiston's appointment to the Lucasian chair; and that it deteriorated as soon as Whiston began to write on prophecy, publishing Essay on the Revelation of St John (1706).[14] This work proclaimed the millennium for the year 1716.[16] Whiston's 1707 edition of Newton's Arithmetica Universalis did nothing to improve matters. Newton himself was heavily if covertly involved in the 1722 edition, nominally due to John Machin, making many changes.[17] In 1708–9 Whiston was engaging Thomas Tenison and John Sharp as archbishops in debates on the Trinity. There is evidence from Hopton Haynes that Newton reacted by pulling back from publication on the issue;[18] his antitrinitarian views, from the 1690s, were finally published in 1754 as An Historical Account of Two Notable Corruptions of Scripture. Whiston was never a Fellow of the Royal Society. In conversation with Edmond Halley he blamed his reputation as a "heretick". Also, though, he claimed Newton had disliked having an independent-minded disciple; and was unnaturally cautious and suspicious by nature.[19] Expelled Arian Whiston's route to rejection of the Nicene Creed, the historical orthodox position against Arianism, began early in his tenure of the Lucasian chair as he followed hints from Samuel Clarke. He read also in Louis Ellies Dupin, and the Explication of Gospel Theism (1706) of Richard Brocklesby.[20] His study of the Apostolic Constitutions then convinced him that Arianism was the creed of the early church.[2] The general election of 1710 brought the Tories solid political power for a number of years, up to the Hanoverian succession of 1714. Their distrust of theological innovation had a direct impact on Whiston, as well as others of similar views. His heterodoxy was notorious.[21] In 1710 he was deprived of his professorship and expelled from the university.[2] The matter was not allowed to rest there: Whiston tried to get a hearing before Convocation. He did have defenders even in the high church ranks, such as George Smalridge.[22] For political reasons, this development would have been divisive at the time. Queen Anne made a point of twice "losing" the papers in the case.[23] After her death in 1714 the intended hearing was allowed to drop.[24] The party passions of these years found an echo in Henry Sacheverell's attempt to exclude Whiston from his church of St Andrew's, Holborn, taking place in 1719.[24][25] "Primitive Christianity" Whiston founded a society for promoting primitive Christianity, lecturing in support of his theories in halls and coffee-houses at London, Bath, and Tunbridge Wells.[2] Those he involved included Thomas Chubb,[26] Thomas Emlyn,[27] John Gale,[28] Benjamin Hoadley,[29] Arthur Onslow,[29] and Thomas Rundle.[30] There were meetings at Whiston's house from 1715 to 1717; Hoadley avoided coming, as did Samuel Clarke, though invited.[31] A meeting with Clarke, Hoadley, John Craig and Gilbert Burnet the younger had left these leading latitudinarians unconvinced about Whiston's reliance on the Apostolical Constitutions.[32] Franz Wokenius wrote a 1728 Latin work on Whiston's view of primitive Christianity.[33] His challenge to the teachings of Athanasius meant that Whiston was commonly considered heretical on many points. On the other hand, he was a firm believer in supernatural aspects of Christianity. He defended prophecy and miracle. He supported anointing the sick and touching for the king's evil. His dislike of rationalism in religion also made him one of the numerous opponents of Hoadley's Plain Account of the Nature and End of the Sacrament. He was fervent in his views of ecclesiastical government and discipline, derived from the Apostolical Constitutions.[2] Around 1747, when his clergyman began to read the Athanasian Creed, which Whiston did not believe in, he physically left the church and the Anglican communion, becoming a Baptist.[2] By the 1720s, some dissenters and early Unitarians viewed Whiston as a role model.[1] Lecturer and popular author Whiston began lecturing on natural philosophy in London. He gave regular courses at coffee houses, particularly Button's, and also at the Censorium, a set of riverside meeting rooms in London run by Richard Steele.[34] At Button's, he gave courses of demonstration lectures on astronomical and physical phenomena, and Francis Hauksbee the younger worked with him on experimental demonstrations. His passing remarks on religious topics were sometimes objected to, for example by Henry Newman writing to Steele.[35][36] His lectures were often accompanied by publications. In 1712, he published, with John Senex, a chart of the Solar System showing numerous paths of comets.[37] In 1715, he lectured on the total solar eclipse of 3 May 1715 (which fell in April Old Style in England); Whiston lectured on it at the time, in Covent Garden, and later, as a natural event and as a portent.[38] By 1715 Whiston had also become adept at newspaper advertising.[39] He frequently lectured to the Royal Society. Longitude In 1714, he was instrumental in the passing of the Longitude Act, which established the Board of Longitude. In collaboration with Humphrey Ditton he published A New Method for Discovering the Longitude, both at Sea and Land,[40] which was widely referenced and discussed. For the next forty years he continued to propose a range of methods to solve the longitude reward, which earned him widespread ridicule, particularly from the group of writers known as the Scriblerians.[41][42] In one proposal for using magnetic dip to find longitude he produced one of the first isoclinic maps of southern England in 1719 and 1721. In 1734, he proposed using the eclipses of Jupiter's satellites.[43] Broader natural philosophy Whiston's A New Theory of the Earth from its Original to the Consummation of All Things (1696) was an articulation of creationism and flood geology. It held that the global flood of Noah had been caused by a comet. The work obtained the praise of John Locke, who classed the author among those who, if not adding much to our knowledge, "At least bring some new things to our thoughts."[2] He was an early advocate, along with Edmond Halley, of the periodicity of comets; he also held that comets were responsible for past catastrophes in Earth's history. In 1736, he caused widespread anxiety among London's citizens when he predicted the world would end on 16 October that year because a comet would hit the earth.[44] William Wake as Archbishop of Canterbury officially denied this prediction to calm the public. There was no consensus within the Newtonians as to how far mechanical causes could be held responsible for key events of sacred history: John Keill was at the opposite extreme to Whiston in minimising such causes.[45] As a natural philosopher, Whiston's speculations respected no boundary with his theological views. He saw the creation of man as an intervention in the natural order. He picked up on Arthur Ashley Sykes's advice to Samuel Clarke to omit an eclipse and earthquake mentioned by Phlegon of Tralles from future editions of Clarke's Boyle lectures, these events being possibly synchronous with Christ's crucifixion. Whiston published The Testimony of Phlegon Vindicated in 1732.[46] Views The series of Moyer Lectures often made Whiston's unorthodox views a particular target.[47] Whiston held that Song of Solomon was apocryphal and that the Book of Baruch was not.[2] He modified the biblical Ussher chronology, setting the Creation at 4010 BCE.[48] He challenged Newton's system of The Chronology of Ancient Kingdoms Amended (1728). Westfall absolves Whiston of the charge that he pushed for the posthumous publication of the Chronology just to attack it, commenting that the heirs were in any case looking to publish manuscripts of Newton, who died in 1727.[49] Whiston's advocacy of clerical monogamy is referenced in Oliver Goldsmith's novel The Vicar of Wakefield. His last "famous discovery, or rather revival of Dr Giles Fletcher, the Elder's," which he mentions in his autobiography, was the identification of the Tatars with the lost tribes of Israel.[2] Personal life Whiston married Ruth, daughter of George Antrobus, his headmaster at Tamworth school. He had a happy family life and died in Lyndon Hall, Rutland, at the home of his son-in-law, Samuel Barker, on 22 August 1752.[1] He was survived by his children Sarah, William, George, and John.[50] Works Whiston's later life was spent in continual controversy: theological, mathematical, chronological, and miscellaneous. He vindicated his estimate of the Apostolical Constitutions and the Arian views he had derived from them in his Primitive Christianity Revived (5 vols., 1711–1712). In 1713 he produced a reformed liturgy. His Life of Samuel Clarke appeared in 1730.[2] In 1727 he published a two volume work called Authentik Record belonging to the Old and New Testament. This was a collection of translations and essays on various deuterocanonical books, pseudepigrapha and other essays with a translation if relevant.[2] Whiston translated the complete works of Josephus into English, and published them along with his own notes and dissertations under the title The Genuine Works of Flavius Josephus the Jewish Historian in 1737. This translation was based on the same Greek edition of Josephus' works used by Siwart Haverkamp in his prior translation.[51] The text on which Whiston's translation of Josephus is based is, reputedly, one which had many errors in transcription.[52] In 1745 he published his Primitive New Testament (on the basis of Codex Bezae and Codex Claromontanus). Whiston left memoirs (3 vols., 1749–1750). These do not contain the account of the proceedings taken against him at Cambridge for his antitrinitarianism, which was published separately at the time.[2] Editions • New theory of the Earth. London: Robert Roberts. 1696. • New theory of the Earth (in German). Frankfurt am Main: Christian Gottlieb Ludwig. 1713. See also • Noah's Flood • Catastrophism • Biblical prophecy • Dorsa Whiston, named after him References 1. "Whiston, William". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/29217. (Subscription or UK public library membership required.) 2. One or more of the preceding sentences incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "Whiston, William". Encyclopædia Britannica. Vol. 28 (11th ed.). Cambridge University Press. p. 597. 3. "Whiston, William (WHSN686W)". A Cambridge Alumni Database. University of Cambridge. 4. Knox, Kevin C. (6 November 2003). From Newton to Hawking: A History of Cambridge University's Lucasian Professors of Mathematics. Cambridge University Press. p. 145. ISBN 978-0-521-66310-6. Retrieved 28 May 2013. 5. Shank, J. B. (2008). The Newton Wars and the Beginning of the French Enlightenment. University of Chicago Press. pp. 128–29. ISBN 978-0-226-74947-1. Retrieved 22 May 2013. 6. Shaw, Jane (2006). Miracles in Enlightenment England. Yale University Press. pp. 31, 171. ISBN 978-0-300-11272-6. Retrieved 22 May 2013. 7. Andrew Pyle (editor), The Dictionary of Seventeenth Century British Philosophers (2000), Thoemmes Press (two volumes), article Whiston, William, p. 875. 8. Sara Schechner; Sara Schechner Genuth (1999). Comets, popular culture, and the birth of modern cosmology. Princeton University Press. p. 292. ISBN 978-0-691-00925-4. Retrieved 23 May 2013. 9. Kidd, Colin (1999). British Identities before Nationalism. Cambridge University Press. p. 45. ISBN 978-1-139-42572-8. Retrieved 22 May 2013. 10. Poole, William (2010). The World Makers: Scientists of the Restoration and the Search for the Origins of the Earth. Peter Lang. p. 68. ISBN 978-1-906165-08-6. Retrieved 22 May 2013. 11. Stephen Gaukroger; John Schuster; John Sutton (2002). Descartes' Natural Philosophy. Taylor & Francis. pp. 176–77. ISBN 978-0-203-46301-7. Retrieved 23 May 2013. 12. Henk J. M. Nellen, ed. (1994). Hugo Grotius, Theologian: Essays in Honour of G. H. M. Posthumus Meyjes. Brill. p. 195. ISBN 978-90-04-10000-8. Retrieved 22 May 2013. 13. Stephen, Leslie, ed. (1887). "Clagett, Nicholas (1654–1727)" . Dictionary of National Biography. Vol. 10. London: Smith, Elder & Co. 14. Jed Zachary Buchwald; Mordechai Feingold (2012). Newton and the origin of civilization. Princeton University Press. p. 336. ISBN 978-0-691-15478-7. Retrieved 22 May 2013. 15. J.E. Force; S. Hutton (2004). Newton and Newtonianism: New Studies. Springer. p. 179 note 102. ISBN 978-1-4020-1969-2. Retrieved 25 May 2013. 16. Jacob, Margaret C. (1976). The Newtonians and the English Revolution 1689–1720. Harvester Press. pp. 132–33. 17. D. T. Whiteside, ed. (2008). The Mathematical Papers of Isaac Newton. Cambridge University Press. p. 14. ISBN 978-0-521-04584-1. Retrieved 22 May 2013. 18. J.E. Force; S. Hutton (2004). Newton and Newtonianism: New Studies. Springer. p. 109. ISBN 978-1-4020-1969-2. Retrieved 25 May 2013. 19. Richard H. Popkin, ed. (1999). The Pimlico History of Western Philosophy. Pimlico. p. 427. ISBN 0-7126-6534-X. 20. Wiles, Maurice (1996). Archetypal Heresy: Arianism Through the Centuries. Oxford University Press. pp. 94–. ISBN 978-0-19-826927-4. Retrieved 22 May 2013. 21. Gibson, William (2004). The Enlightenment Prelate: Benjamin Hoadly, 1767–1761. James Clarke & Co. pp. 121–23. ISBN 978-0-227-67978-4. Retrieved 22 May 2013. 22. William Gibson; Robert G.. Ingram (2005). Religious Identities in Britain: 1660– 1832. Ashgate Publishing, Ltd. pp. 47–48. ISBN 978-0-7546-3209-2. Retrieved 22 May 2013. 23. William Gibson; William Gibson (2002). The Church of England 1688–1832: Unity and Accord. Taylor & Francis. p. 81. ISBN 978-0-203-13462-7. Retrieved 22 May 2013. 24. Lee, Sidney, ed. (1900). "Whiston, William" . Dictionary of National Biography. Vol. 61. London: Smith, Elder & Co. 25. Steele, John M. (2012). Ancient Astronomical Observations and the Study of the Moon's Motion (1691–1757). Springer. p. 24. ISBN 978-1-4614-2149-8. Retrieved 22 May 2013. 26. Probyn, Clive. "Chubb, Thomas". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/5378. (Subscription or UK public library membership required.) 27. McLachlan, H. J. "Emlyn, Thomas". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/8793. (Subscription or UK public library membership required.) 28. Benedict, Jim. "Gale, John". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/10292. (Subscription or UK public library membership required.) 29. Force, James E. (2002). William Whiston: Honest Newtonian. Cambridge University Press. p. 27. ISBN 978-0-521-52488-9. Retrieved 21 May 2013. 30. Acheson, Alan R. "Rundle, Thomas". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/24279. (Subscription or UK public library membership required.) 31. Sheehan, Jonathan (2005). "The" Enlightenment Bible: Translation, Scholarship, Culture. Princeton University Press. p. 35 note 21. ISBN 978-0-691-11887-1. Retrieved 22 May 2013. 32. Gibson, William (2004). The Enlightenment Prelate: Benjamin Hoadly, 1767-1761. James Clarke & Co. p. 122. ISBN 978-0-227-67978-4. Retrieved 22 May 2013. 33. Wokenius, Franz (1728). Christianismus primaevus quem Guil. Whistonus modo non-probando restituendum dictitat sed Apostolus Paulus breviter quasi in tabula depinxit ... Retrieved 25 May 2013. 34. Margaret C. Jacob; Larry Stewart (2009). Practical Matter: Newton's Science in the Service of Industry and Empire 1687–1851. Harvard University Press. p. 64. ISBN 978-0-674-03903-2. Retrieved 21 May 2013. 35. O'Connor, John J.; Robertson, Edmund F., "London Coffee houses and mathematics", MacTutor History of Mathematics Archive, University of St Andrews 36. Stewart, Larry. "Hauksbee, Francis". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/12619. (Subscription or UK public library membership required.) 37. Thomas Hockey; Katherine Bracher; Marvin Bolt; Virginia Trimble; Richard Jarrell; JoAnn Palmeri; Jordan D. Marché; Thomas Williams; F. Jamil Ragep, eds. (2007). Biographical Encyclopedia of Astronomers. Springer. p. 1213. ISBN 978-0-387-30400-7. Retrieved 25 May 2013. 38. Knox, Kevin C. (2003). From Newton to Hawking: A History of Cambridge University's Lucasian Professors of Mathematics. Cambridge University Press. p. 162. ISBN 978-0-521-66310-6. Retrieved 22 May 2013. 39. Wigelsworth, Jeffrey R. (2010). Selling Science in the Age of Newton: Advertising and the Commoditization of Knowledge. Ashgate Publishing, Ltd. p. 137. ISBN 978-1-4094-2310-2. Retrieved 25 May 2013. 40. Ditton, William Whiston; Ditton, Humphrey (1714). A New Method for Discovering the Longitude, both at Sea and Land. John Phillips. Retrieved 15 April 2015. 41. For example, Jonathan Swift's 1714 "Ode, to Musick. On the Longitude", including numerous references to bepissing and beshitting upon both Whiston and Ditton. 42. S.D. Snobelen, "William Whiston: Natural Philosopher, Prophet, Primitive Christian" (Cambridge Univ. PhD Thesis, 2000) 43. Mr Whiston's Project for finding the Longitude (MSS/79/130.2), Board of Longitude project, University of Cambridge Digital Library 44. "This Month in Physics History". Retrieved 16 October 2018. 45. Poole, William (2010). The World Makers: Scientists of the Restoration and the Search for the Origins of the Earth. Peter Lang. p. 72. ISBN 978-1-906165-08-6. Retrieved 25 May 2013. 46. Force, James E. (1985). William Whiston: Honest Newtonian. Cambridge University Press. p. 181 note 128. ISBN 978-0-521-26590-4. Retrieved 25 May 2013. 47. J.E. Force; S. Hutton (2004). Newton and Newtonianism: New Studies. Springer. p. 102. ISBN 978-1-4020-1969-2. Retrieved 25 May 2013. 48. Davis A. Young; Ralph Stearley (2008). The Bible, Rocks and Time: Geological Evidence for the Age of the Earth. InterVarsity Press. p. 67. ISBN 978-0-8308-2876-0. Retrieved 23 May 2013. 49. Westfall, Richard S. (1983). Never at Rest: A Biography of Isaac Newton. Cambridge University Press. pp. 815 note 112. ISBN 978-0-521-27435-7. Retrieved 25 May 2013. 50. Farrell, Maureen (1981). William Whiston. New York: Arno Press. pp. 46–47. 51. "The genuine works of Flavius Josephus the Jewish historian". University of Chicago. Retrieved 16 June 2023. 52. Josephus (1981). Josephus Complete Works. Translated by William Whiston. Grand Rapids, Michigan: Kregel Publications. p. xi (Foreword). ISBN 0-8254-2951-X. Further reading • Farrell, Maureen (1981). William Whiston. New York: Arno Press. • Force, James E. (2002). William Whiston: Honest Newtonian. Cambridge: Cambridge University Press. • Rouse Ball, W. W. (2009) [1889]. A History of the Study of Mathematics at Cambridge University. Cambridge University Press. pp. 83–85. ISBN 978-1-108-00207-3. External links • Media related to William Whiston at Wikimedia Commons • Biography of William Whiston at the LucasianChair.org, the homepage of the Lucasian Chair of Mathematics at Cambridge University • Bibliography for William Whiston Archived 10 May 2012 at the Wayback Machine at the LucasianChair.org the homepage of the Lucasian Chair of Mathematics at Cambridge University • Whiston's MacTutor Biography • Works by William Whiston at Project Gutenberg • Works by or about William Whiston at Internet Archive • Works by William Whiston at LibriVox (public domain audiobooks) • William Whiston at the Mathematics Genealogy Project • "Account of Newton", Collection of Authentick Records (1728), pp. 1070–1082 • "The Works of Flavius Josephus" translated by William Whiston • "William Whiston and the Deluge" by Immanuel Velikovsky • "Whiston's Flood" • Whiston biography at Chambers' Book of Days • Some of Whiston's views on biblical prophecy Archived 25 April 2013 at the Wayback Machine • "William Whiston, The Universal Deluge, and a Terrible Specracle" by Roomet Jakapi • Collection of Authentick Records by Whiston at the Newton Project Archived 4 October 2006 at the Wayback Machine • William Whiston, 1667–1752 Archived 29 September 2007 at the Wayback Machine • Collection of William Whiston portraits at England's National Portrait Gallery • Primitive New Testament • William Whiston \| Portraits From the Past • A New Theory of the Earth (1696) – full digital facsimile at Linda Hall Library Lucasian Professors of Mathematics • Isaac Barrow (1664) • Isaac Newton (1669) • William Whiston (1702) • Nicholas Saunderson (1711) • John Colson (1739) • Edward Waring (1760) • Isaac Milner (1798) • Robert Woodhouse (1820) • Thomas Turton (1822) • George Biddell Airy (1826) • Charles Babbage (1828) • Joshua King (1839) • George Stokes (1849) • Joseph Larmor (1903) • Paul Dirac (1932) • James Lighthill (1969) • Stephen Hawking (1979) • Michael Green (2009) • Michael Cates (2015) Authority control International • FAST • ISNI • VIAF National • Spain • 2 • France • BnF data • Germany • Israel • Belgium • United States • Sweden • Czech Republic • Australia • Greece • Netherlands • Poland • Portugal • Vatican Academics • CiNii • MathSciNet • Mathematics Genealogy Project People • Deutsche Biographie • Trove Other • SNAC • IdRef
William Whyburn William Marvin Whyburn (12 November 1901 – 5 May 1972) was an American mathematician who worked on ordinary differential equations. His work focussed on a multitude of topics including, boundary value problems, properties of Green’s function and properties of Green’s matrix.[1] William Whyburn Born(1901-11-12)12 November 1901 Lewisville, Texas Died5 May 1972(1972-05-05) (aged 70) Greenville, North Carolina Resting placeChinn's Chapel Cemetery, Copper Canyon, Denton County, Texas, USA NationalityAmerican SpouseMarie Barfield ChildrenWilla Whyburn, Clifton Whyburn Biography Early life William Marvin Whyburn was born in Lewisville, Texas, on 12 November 1901.[1] He was the son of farmers Thomas Whyburn and Eugenia Elizabeth McLeod. He had a brother Gordon Whyburn, also a mathematician who primarily studied topology. He attended Bethel School where he would study until he was 14, when he sat an entrance exam and was accepted into North Texas State College in 1916.[2] Undergraduate education Whyburn studied for four years at North Texas State College, where after he would study mathematics at the University of Texas. He attained his Bachelor of Arts degree in 1922 and went on to achieve a Master of Arts degree in mathematics the following year.[2] In 1923 Whyburn married Marie Barfield. Marie was also a student at the University of Texas. Together they had two children, Willa Whyburn and Clifton Whyburn. Clifton also studied mathematics.[2] Undergraduate teaching While studying at North Texas State College (1918–1920) Whyburn taught at different schools in Denton County. One of the students taught by Whyburn was famous mathematician Samuel S. Wilks.[2] In 1923/24 Whyburn taught full-time at South Park Junior College, Beaumont then he held an assistant professor role at Texas A&M College the years after. Again in 1925/26 Whyburn was an associate professor at the Texas Technological College in Lubbock, Texas. Whyburn was given the Louis Lipsitz fellowship for the academic year 1926/27, which allowed him to study full-time.[2] Postgraduate career Whyburn continued to study at the University of Texas for his Ph.D. under the supervision of his advisor Hyman Joseph Ettlinger.[3] After the publication of his thesis Linear Boundary Value Problems for Ordinary Differential Equations and Their Associated Difference Equations he was awarded his doctorate in June 1927.[4] Additionally, in the three years before this publication Whyburn published three other papers, two of which were on Green’s function. Whyburn published two more papers in 1927 before spending the 1927/28 academic year at Harvard university as a National Research Fellow.[2] Whyburn was assigned as an Assistant Professor of Mathematics at the University of California, Los Angeles in 1928. Ten years later Whyburn was made a full professor in 1938 as well as being the chairman for the Mathematics Department for a seven-year tenure beginning in 1937.[2] Whyburn was the chairman of the supervisors committee for Engineering, Science, Management War Training Programs during the second world war. Throughout the war he wrote a paper and a book about mathematics as its applied in war. Whyburn was given the role of president of the Texas Technological College in 1944. In this role he would help improve the educational profile of the school to other major educational bodies such as the American Association of Universities and American Association of University Women. As a result of his work, the college gained recognition from governmental agencies, reflecting his presidential impact.[2] In 1948 Whyburn resigned from his position at the Texas Technological College as he was appointed Kenan Professor of Mathematics at the University of North Carolina at Chapel Hill, where he would further be appointed as chairman of the Mathematics department. After serving three years as Vice President for research from 1957-1960, Whyburn retired in 1967. He was then appointed as the Frensley Professor of Mathematics at the Southern Methodist University in Dallas. Whyburn retired from this position in 1970 before working a part-time teaching position at East Carolina University, North Carolina.[1][2] As a teacher Whyburn was focussed on the students perception and put them first. He would be methodical in how he approached different students and their areas of postgraduate research whilst supervising. He supervised the Ph.D. of the following students: Leonard P. Burton, Albert Deal, Bertram Drucker, Garett Etgen, Paul Herwitz, Sandra Hilt, A. Keith Hinds, Nathaniel Macon, Edward J. Pellicciaro, Tullio Pignani, Clay Campbell Ross, David Showalter and Frank Stellard.[3] Whyburn died of a heart attack on 5 May 1972 in Greenville, North Carolina.[5][6] He is buried in Chinn's Chapel Cemetery, Copper Canyon, Texas[7] Selected publications • "An Extension of the Definition of the Green's Function in One Dimension"(1924)[8] • "Second-Order Differential Systems With Integral and k-Point Boundary Conditions" (1928)[9] • "Functional Properties of the Solutions of Differential Systems" (1930)[10] • "Differential Equations with General Boundary Conditions" (1942)[11] • "A Nonlinear Boundary Value Problem For Second Order Differential Systems" (1955)[12] • "Complexes of Differential Systems" (1972)[13] References 1. "William Whyburn - Biography". Maths History. Retrieved 2022-06-16. 2. Reid, W.T. (1973). "William M. Whyburn" (PDF). Bulletin of the American Mathematical Society. 3. "William Whyburn - The Mathematics Genealogy Project". www.genealogy.math.ndsu.nodak.edu. Retrieved 2022-07-02. 4. "Mathematics Alumni". web.ma.utexas.edu. Retrieved 2022-07-02. 5. North Carolina, State Board of Health (June 12, 1972). "Certificate of Death: William Marvin Whyburn". 6. "Prof. William Whyburn Dead; Mathematician at Chapel Hill". The New York Times. 1972-05-07. ISSN 0362-4331. Retrieved 2022-07-02. 7. "William Marvin Whyburn (1901-1972)". www.findagrave.com. Retrieved 2022-06-30. 8. Whyburn, W. M. (1924). "An Extension of the Definition of the Green's Function in One Dimension". Annals of Mathematics. 26 (1/2): 125–130. doi:10.2307/1967748. ISSN 0003-486X. JSTOR 1967748. MR 1502681 – via jstor.org. 9. Whyburn, William M. (1928). "Second-Order Differential Systems With Integral and k-Point Boundary Conditions" (PDF). Transactions of the American Mathematical Society. 30 (4): 630–640. MR 1501451 – via ams.org. 10. Whyburn, William M. (1930). "Functional Properties of the Solutions of Differential Systems" (PDF). Transactions of the American Mathematical Society. 32 (3): 502–508. doi:10.1090/S0002-9947-1930-1501548-9. MR 1501548 – via ams.org. 11. Whyburn, William M. (1942). "Differential Equations with General Boundary Conditions" (PDF). Bulletin of the American Mathematical Society. 48 (10): 692–704. doi:10.1090/S0002-9904-1942-07760-3. MR 0007192. S2CID 51822059 – via ams.org. 12. Whyburn, William M. (1955). "A nonlinear boundary value problem for second order differential systems". Pacific Journal of Mathematics. 5: 147–160. doi:10.2140/pjm.1955.5.147. MR 0069368 – via projecteuclid.org. 13. Whyburn, William M.; Williams, Richard K. (1972-03-01). "Complexes of differential systems". Journal of Differential Equations. 11 (2): 299–306. Bibcode:1972JDE....11..299W. doi:10.1016/0022-0396(72)90046-0. ISSN 0022-0396. MR 0294752. Authority control International • FAST • ISNI • VIAF National • Israel • United States Academics • MathSciNet • Mathematics Genealogy Project • zbMATH
William Wilson Hunter Sir William Wilson Hunter KCSI CIE (15 July 1840 – 6 February 1900)[1] was a Scottish historian, statistician, a compiler and a member of the Indian Civil Service. Sir William Wilson Hunter Born(1840-07-15)15 July 1840 Glasgow, Scotland, UK Died6 February 1900(1900-02-06) (aged 59) Oaken Holt, England, UK NationalityBritish Alma materUniversity of Glasgow Scientific career FieldsHistory, statistics InstitutionsIndian Civil Service University of Calcutta He is most known for The Imperial Gazetteer of India on which he started working in 1869, and which was eventually published in nine volumes in 1881, then fourteen, and later as a twenty-six volume set after his death. Early life and education William Wilson Hunter was born on 15 July 1840 in Glasgow, Scotland, to Andrew Galloway Hunter, a Glasgow manufacturer. He was the second of his father's three sons. In 1854 he started his education at the 'Quaker Seminary' at Queenswood, Hampshire and a year later he joined The Glasgow Academy. He was educated at the University of Glasgow (BA 1860), Paris and Bonn, acquiring a knowledge of Sanskrit, LL.D., before passing first in the final examination for the Indian Civil Service in 1862.[2] Career He reached Bengal Presidency in November 1862 and was appointed assistant magistrate and collector of Birbhum, in the lower provinces of Bengal, where he began collecting local traditions and records, which formed the materials for his publication, entitled The Annals of Rural Bengal,[2] which influenced the historical romances of Bankim Chandra Chattopadhyay.[3] He also compiled A Comparative Dictionary of the Non-Aryan Languages of India, a glossary of dialects based mainly upon the collections of Brian Houghton Hodgson, which according to the Encyclopædia Britannica Eleventh Edition, "testifies to the industry of the writer but contains much immature philological speculation".[2] In 1869 Lord Mayo, the then governor-general, asked Hunter to submit a scheme for a comprehensive statistical survey of India. The work involved the compilation of a number of local gazetteers, in various stages of progress, and their consolidation in a condensed form upon a single and uniform plan.[2] There was unhappiness with the scope and completeness of the earlier surveys conducted by administrators such as Buchanan, and Hunter determined to model his efforts on the Ain-i-Akbari and Description de l'Égypte. Hunter said that "It was my hope to make a memorial of England's work in India, more lasting, because truer and more complete, than these monuments of Mughal Empire and of French ambition."[4] In response to Mayo's question on 30 May 1871 of whether the Indian Muslims are "bound by their religion to rebel against the Queen" Hunter completed his influential work The Indian Musalmans in mid-June 1871 and later published it as a book in mid-August of the same year.[5][6] In it, Hunter concluded that the majority of the Indian Muslim scholars rejected the idea of rebelling against the Government because of their opinion that the condition for religious war, i.e. the absence of protection and liberty between Muslims and infidel rulers, did not exist in British India; and that "there is no jihad in a country where protection is afforded".[7] In 1872 Hunter published his history of Orissa. The third International Sanitary Conference held at Constantinople in 1866 declared Hindu and Muslim pilgrimages to be 'the most powerful of all the causes which conduce to the development and propagation of Cholera epidemics'. Hunter echoing the view described the 'squalid pilgrim army of Jagannath' as[8] with its rags and hair and skin freighted with vermin and impregnated with infection, may any year slay thousands of the most talented and beautiful of our age in Vienna, London, or Washington. He embarked on a series of tours throughout the country,[4] and he supervised the A Statistical Account of Bengal (20 volumes, 1875–1877)[9] and a similar work for Assam (2 volumes, 1879).[10] Hunter wrote that Under this system, the materials for the whole of British India have now been collected, in several Provinces the work of compilation has rapidly advanced, and everywhere it is well in hand. During the same period the first Census of India has been taken, and furnished a vast accession to our knowledge of the people. The materials now amassed form a Statistical Survey of a continent with a population exceeding that of all Europe, Russia excepted."[11] The statistical accounts, covering the 240 administrative districts, comprised 128 volumes and these were condensed into the nine volumes of The Imperial Gazetteer of India, which was published in 1881.[4] The Gazetteer was revised in later series, the second edition comprising 14 volumes published between 1885 and 1887, while the third comprised 26 volumes, including an atlas, and was published in 1908 under the editorship of Herbert Hope Risley, William Stevenson Meyer, Richard Burn and James Sutherland Cotton.[12] Again according to the Encyclopædia Britannica Eleventh Edition, Hunter "adopted a transliteration of vernacular place-names, by which means the correct pronunciation is ordinarily indicated; but hardly sufficient allowance was made for old spellings consecrated by history and long usage."[13] Hunter's own article on India was published in 1880 as A Brief History of the Indian Peoples, and has been widely translated and utilized in Indian schools. A revised form was issued in 1895, under the title of The Indian Empire: its People, History and Products. Hunter later said that Nothing is more costly than ignorance. I believe that, in spite of its many defects, this work will provide a memorable episode in the long battle against ignorance; a breakwater against the tide of prejudice and false opinions flowing down upon us from the past, and the foundation for a truer and wider knowledge of India in time to come. Its aim has been not literary graces, nor scientific discovery, nor antiquarian research; but an earnest endeavour to render India better governed, because better understood.[4] Hunter contributed the articles "Bombay", "Calcutta", "Dacca", "Delhi" and "Mysore" to the 9th edition (1875–89) of the Encyclopædia Britannica.[14] In 1882 Hunter, as a member of the governor-general's council, presided over the Commission on Indian Education; in 1886 he was elected vice-chancellor of the University of Calcutta. In 1887 he retired from the service, was created KCSI, and settled at Oaken Holt, near Oxford.[15] He was on the governing body of Abingdon School from 1895 until his death in 1900.[16] On 13 March 1889 Philip Lyttelton Gell the then Secretary to the Delegates of the Clarendon Press, wrote to Hunter about a project which has been for some time under the consideration of the Delegates, to publish a series giving the salient features of Indian History in the Biographies of successive Generals and Administrators.[17] Gell arranged the publication of the series by June 1889; with Hunter receiving £75 for each volume, and the author £25. Gell's experience of the earlier unsaleable Sacred Books of the East and financial constraints forced the Rulers of India to end at 28 volumes in spite of Hunter's disappointment about the same.[18] Hunter himself contributed the volumes on Dalhousie (1890)[19] and Mayo (1891)[20] to the series. He had previously written an official Life of Lord Mayo, which was published on 19 November 1875 in two volumes with a second edition appearing in 1876.[21] He also wrote a weekly article on Indian affairs for The Times. But the great task to which he applied himself on his settlement in England was a history upon a large scale of the British Dominion in India, two volumes of which only had appeared when he died, carrying the reader barely down to 1700. He was much hindered by the confused state of his materials, a portion of which he arranged and published in 1894 as Bengal Manuscript Records, in three volumes.[15] Hunter dedicated his 1892 work Bombay 1885 to 1890: A Study in Indian Administration to Florence Nightingale.[22] His later works include the novel titled The Old Missionary (1895, described on the title-page as "revised from The Contemporary Review"),[23] and The Thackerays in India (1897). John F. Riddick describes Hunter's The Old Missionary as one of the "three significant works" produced by Anglo-Indian writers on Indian missionaries along with The Hosts of the Lord (1900) by Flora Annie Steel and Idolatry (1909) by Alice Perrin.[24] In the winter of 1898–1899, in consequence of the fatigue incurred in a journey to the Caspian and back, on a visit to the sick-bed of one of his two sons, Hunter was stricken down by a severe attack of influenza, which affected his heart. He died at Oaken Holt on 6 February 1900.[15] S. C. Mittal believes that Hunter "represented the official mind of the bureaucratic Victorian historians in India", of whom James Talboys Wheeler and Alfred Comyn Lyall were other examples.[25] Bibliography Works • A Comparative Dictionary of the Languages of India and High Asia: With a Dissertation. Based on the Hodgson Lists, Official Records, and Mss. Trübner and Company. 1868. • Annals of Rural Bengal. Smith, Elder & Co. 1868. • The Indian Musalmans: Are They Bound in Conscience to Rebel Against the Queen?. Trübner and Company. 1871.[26] • Orissa, Or, The Vicissitudes of an Indian Province Under Native and British Rule. Smith, Elder and Company. 1872. • A Statistical Account of Bengal. London: Trübner & Co. 1875–1879. (20 volumes) • A Statistical Account of Assam. 1879. (2 volumes) • A Brief history of the Indian peoples. Oxford: Clarendon Press. 1880. • The Imperial Gazetteer of India. 1908–1909. (3rd ed. 26 vols; 1st ed. 9 vols, 1881; 2nd ed. 14 vols, 1885–87) • The Indian Empire: Its People, History, and Products London (Second ed.). Trübner & Co. 1886. ISBN 9788120615816. • Bombay, 1885-1890: A Study in Indian Administration. Frowde. 1892. • The Marquess of Dalhousie. 1894. • State Education for the People in America, Europe, India, and Australia: With Papers on the Education of Women, Technical Instruction, and Payment by Results. C. W. Bardeen. 1895. • The Thackerays in India and Some Calcutta Graves. London: Henry Frowde. 1897.[27] • Williams Jackson, A. V., ed. (1906). History of India: From the first European settlements to the founding of the English East India Company . History of India. Vol. 6. London: Grolier Society. • Williams Jackson, A. V., ed. (1907). History of India: The European struggle for Indian supremacy in the seventeenth century . History of India. Vol. 7. London: Grolier Society. Works about Hunter • Francis Henry Bennett Skrine (1901). Life of Sir William Wilson Hunter, K.C.S.I. Longmans, Green. See also • The Imperial Gazetteer of India • Hunterian transliteration • Census of India prior to independence References 1. "Obituary: Sir William Wilson Hunter, K. C. S. I., C. I. E.". The Geographical Journal. 15 (3): 289–290. March 1900. JSTOR 1774698. 2. Chisholm 1911, p. 945. 3. Chatterjee, Rimi B. (2004). "'Every Line for India': The Oxford University Press and the Rise and Fall of the Rulers of India Series". In Gupta, Abhijit; Chakravorty, Swapan (eds.). Print Areas: Book History in India. Permanent Black. pp. 77, 93. ISBN 978-81-7824-082-4. 4. Marriott, John (2003). The other empire: metropolis, India and progress in the colonial imagination. Manchester University Press. p. 209. ISBN 978-0-7190-6018-2. Retrieved 7 December 2011. 5. v. L. B. (1872). "De Mohammedanen in Hindostan. —Our Indian Musalmans: Are they bound in conscience to rebel against the Queen? by W. W. Hunter". Bijdragen tot de Taal-, Land- en Volkenkunde van Nederlandsch-Indië. 18 (2): 121–122. JSTOR 25736656. 6. Ali, M. Mohar (1980). "Hunter's "Indian Musalmans": A Re-Examination of Its Background". The Journal of the Royal Asiatic Society of Great Britain and Ireland. 112 (1): 30–51. doi:10.1017/S0035869X00135889. JSTOR 25211084. S2CID 154830629. 7. Bonney, R. (2004) Jihad: From Qur'an to Bin Laden, Hampshire: Palgrave Macmillan, pp. 193-194 8. Thomas R. Metcalf (27 February 1997). Ideologies of the Raj. Cambridge University Press. p. 175. ISBN 978-0-521-58937-6. 9. "A Statistical Account of Bengal by W. W. Hunter". The North American Review. 127 (264): 339–342. September–October 1878. JSTOR 25100678. 10. "Hunter, Sir William Wilson". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/14237. (Subscription or UK public library membership required.) 11. Nicholas B. Dirks (2003). Castes of Mind: Colonialism and the Making of Modern India. Permanent Black. p. 199. ISBN 978-81-7824-072-5. 12. Henry Scholberg (1970). The District Gazetteers of British India: A Bibliography. Zug, Switzerland: Inter Documentation Company. ISBN 9780800212650. 13. Chisholm 1911, pp. 945–946. 14. Important Contributors to the Britannica, 9th and 10th Editions. 1902encyclopedia.com. Retrieved 20 April 2018. 15. Chisholm 1911, p. 946. 16. "School Notes" (PDF). The Abingdonian. 17. Chatterjee, Rimi B. (2004). "'Every Line for India': The Oxford University Press and the Rise and Fall of the Rulers of India Series". In Gupta, Abhijit; Chakravorty, Swapan (eds.). Print Areas: Book History in India. Permanent Black. p. 81. ISBN 978-81-7824-082-4. 18. Chatterjee, Rimi B. (2004). "'Every Line for India': The Oxford University Press and the Rise and Fall of the Rulers of India Series". In Gupta, Abhijit; Chakravorty, Swapan (eds.). Print Areas: Book History in India. Permanent Black. pp. 81, 87. ISBN 978-81-7824-082-4. 19. H. P. (1891). "The Marquess of Dalhousie by William Wilson Hunter". Revue Historique. 47 (2): 387–393. JSTOR 40938228. 20. H. P. (1892). "The Earl of Mayo by William Wilson Hunter". Revue Historique. 48 (2): 387–400. JSTOR 40939452. 21. Satish Chandra Mittal (1 January 1996). India Distorted: A Study of British Historians on India. M.D. Publications Pvt. Ltd. p. 199. ISBN 978-81-7533-018-4. 22. Florence Nightingale (6 December 2007). Florence Nightingale on Social Change in India: Collected Works of Florence Nightingale. Wilfrid Laurier University Press. p. 841. ISBN 978-0-88920-495-9. 23. "The Old Missionary". The Spectator. 5 October 1895. p. 19. Retrieved 23 December 2014. 24. John F. Riddick (1 January 2006). The History of British India: A Chronology. Greenwood Publishing Group. p. 179. ISBN 978-0-313-32280-8. 25. Mittal, Satish Chandra (1996). India Distorted: A Study of British Historians on India. Vol. 2. M.D. Publications. p. 170. ISBN 9788175330184. 26. "In the book 'The Indian Musalmans' by William Wilson Hunter, the author has mentioned the Indians Muslims who rebelled against the British empire as 'Wahhabis', so did only Salafis/Ahlul Hadith fight against the British? – Quora". quora.com. Retrieved 27 March 2021. 27. "Review: The Thackerays in India and Some Calcutta Graves by Sir William Wilson Hunter". The Athenæum (3613): 111–112. 23 January 1897. Attribution: • This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "Hunter, Sir William Wilson". Encyclopædia Britannica. Vol. 13 (11th ed.). Cambridge University Press. pp. 945–946. External links • Works by William Wilson Hunter at Project Gutenberg • Works by or about William Wilson Hunter at Internet Archive Vice chancellors of the University of Calcutta • James William Colvile • William Ritchie • Claudius James Erskine • Henry Sumner Maine • W. S. Seton-Karr • Edward Clive Bayley • Arthur Hobhouse • William Markby • Alexander Arbuthnot • Arthur Wilson • H. J. Reynolds • C. P. Ilbert • William Wilson Hunter • William Comer Petheram • Gooroodass Banerjee • Jones Quain Pigot • Alfred Woodley Croft • E. J. Trevelyan • Francis William Maclean • Thomas Raleigh • Alexander Pedler • Asutosh Mookerjee • Deva Prosad Sarbadhicary • Lancelot Sanderson • Nilratan Sircar • Asutosh Mookerjee • Bhupendra Nath Bose • William Ewart Greaves • Jadunath Sarkar • W. S. Urquhart • Hassan Suhrawardy • Syama Prasad Mookerjee • Sir Azizul Huque • Bidhan Chandra Roy • Radhabinod Pal • Pramathanath Banerjee • Charu Chandra Biswas • Sambhunath Banerjee • Jnan Chandra Ghosh • Nirmal Kumar Sidhanta • Subodh Mitra • Surajit Chandra Lahiri • Bidhubhushan Malik • S. N. Sen • Sushil Kumar Mukherjee • Ramendra Kumar Podder • Santosh Bhattacharyya • Bhaskarananda Ray Chaudhuri • Rathindra Narayan Basu • Asis Kumar Banerjee • Suranjan Das • Sugata Marjit • Sonali Chakravarti Banerjee Authority control International • FAST • ISNI • VIAF National • France • BnF data • Germany • Israel • United States • Czech Republic • Australia • Greece • Netherlands • Poland • Portugal • Vatican People • Deutsche Biographie • Trove Other • SNAC • IdRef
William Woolsey Johnson William Woolsey Johnson (1841–1927) was an American mathematician, who was one of the founders of the American Mathematical Society. William Woolsey Johnson Picture by Thomas Eakins Born(1841-06-23)June 23, 1841 Owego (NY) DiedMay 14, 1927(1927-05-14) (aged 85) Baltimore (MD) Resting placeWoodlawn Cemetery (Baltimore) 39.325200°N 76.726093°W / 39.325200; -76.726093 Alma materYale University Scientific career FieldsMathematics InstitutionsUnited States Naval Academy St. John's College Kenyon College Life and work Johnson, son of a farmer of Tioga County, New York, studied at Yale University where he received his BA in 1862. After two years serving in the Nautical Almanac Office in Cambridge, Massachusetts, he began his academic career as assistant professor in the Naval Academy in Newport, Rhode Island, but soon transferred to Annapolis, Maryland, from 1864 to 1869. In 1870 he was appointed professor of mathematics at Kenyon College and since 1872 at St. John's College (Annapolis).[1] In 1881 he returned to the Naval Academy as full professor where he remained until his retirement in 1921. He served as one of the five members of the Council of the American Mathematical Society for the 1892–1893 term[2] and he was one of the impulsors of the birth of the Bulletin of the Society[3] and one of his main first contributors. Johnson is mainly remembered by his books on differential calculus, basing it on related rates.[4] He is also known to be the first on probing the conditions of solvability of the 15 puzzle.[5] Selected publications Articles • Johnson, W. Woolsey (1891). "Octonary numeration". Bull. Amer. Math. Soc. 1: 1–6. doi:10.1090/S0002-9904-1891-00015-2. • Johnson, W. Woolsey (1892). "The mechanical axioms or laws of motion". Bull. Amer. Math. Soc. 1 (6): 129–139. doi:10.1090/S0002-9904-1892-00051-1. • Johnson, W. Woolsey (1893). "On Peters's formula for probable error". Bull. Amer. Math. Soc. 2 (4): 57–61. doi:10.1090/S0002-9904-1893-00107-9. • Johnson, W. Woolsey (1893). "A case of non-euclidian geometry". Bull. Amer. Math. Soc. 2 (7): 158–161. doi:10.1090/S0002-9904-1893-00130-4. • Johnson, W. Woolsey (1894). "Gravitation and absolute units of force". Bull. Amer. Math. Soc. 3 (8): 197–199. doi:10.1090/S0002-9904-1894-00210-9. • Johnson, W. Woolsey (1895). "Kinetic stability of central orbits". Bull. Amer. Math. Soc. 1 (8): 193–196. doi:10.1090/S0002-9904-1895-00272-4. • Johnson, W. Woolsey (1906). "Note on the numerical transcendents Sn and sn =Sn-1". Bull. Amer. Math. Soc. 12 (10): 477–482. doi:10.1090/S0002-9904-1906-01374-X. Books • A treatise on ordinary and partial differential equations. 1881. 3rd edition. 1893. • An elementary treatise on the differential calculus, founded on the method of rates or fluxions. 1889. • The theory of errors and method of least squares. 1893. • Theoretical mechanics. An elementary treatise. 1901. References 1. Dwight 1874, p. 298. 2. Austin, Barry & Berman 2010, p. 145. 3. Fiske 2000, p. 5. 4. Austin, Barry & Berman 2010, p. 144. 5. Hendrixson 2011, p. 6. Bibliography • Austin, Bill; Barry, Don; Berman, David (2010). "The Lengthening Shadow: The Story of Related Rates". In Caren L. Diefenderfer; Roger B. Nelsen (eds.). The Calculus Collection: A Resource for AP and Beyond. Mathematical Association of America. pp. 139–148. ISBN 978-0-88385-761-8. • Dwight, Benjamin Woodbridge (1874). The History of the Descendants of John Dwight of Dedham, Mass. John F. Trow & Son. pp. 298. • Fiske, Thomas S. (2000). "Mathematical progress in America". Bulletin of the AMS. 37 (1): 3–8. doi:10.1090/S0273-0979-99-00799-5. ISSN 0273-0979. PMID 17781208. • Hendrixson, Lisa Rose (2011). Variations of the 15 Puzzle. Ohio Link (Thesis). External links • O'Connor, John J.; Robertson, Edmund F. "William Woolsey Johnson". MacTutor History of Mathematics Archive. University of St Andrews. • William Woolsey Johnson papers. Manuscripts and Archives Repository, Yale University. Authority control International • FAST • ISNI • VIAF National • Germany • Israel • United States • Japan • Czech Republic • Australia • Netherlands Academics • CiNii • MathSciNet • zbMATH People • Trove Other • IdRef
William Jones (mathematician) William Jones, FRS (1675 – 1 July 1749[1]) was a Welsh mathematician, most noted for his use of the symbol π (the Greek letter Pi) to represent the ratio of the circumference of a circle to its diameter. He was a close friend of Sir Isaac Newton and Sir Edmund Halley. In November 1711, he became a Fellow of the Royal Society, and was later its vice-president.[2] William Jones Portrait of William Jones by William Hogarth, 1740 National Portrait Gallery Born1675 Llanfihangel Tre'r Beirdd, Isle of Anglesey Died3 July 1749 (aged 73–74) London, England Part of a series of articles on the mathematical constant π 3.1415926535897932384626433... Uses • Area of a circle • Circumference • Use in other formulae Properties • Irrationality • Transcendence Value • Less than 22/7 • Approximations • Madhava's correction term • Memorization People • Archimedes • Liu Hui • Zu Chongzhi • Aryabhata • Madhava • Jamshīd al-Kāshī • Ludolph van Ceulen • François Viète • Seki Takakazu • Takebe Kenko • William Jones • John Machin • William Shanks • Srinivasa Ramanujan • John Wrench • Chudnovsky brothers • Yasumasa Kanada History • Chronology • A History of Pi In culture • Indiana Pi Bill • Pi Day Related topics • Squaring the circle • Basel problem • Six nines in π • Other topics related to π Biography William Jones was born the son of Siôn Siôr (John George Jones) and Elizabeth Rowland in the parish of Llanfihangel Tre'r Beirdd, about 4 miles (6.4 km) west of Benllech on the Isle of Anglesey in Wales. He attended a charity school at Llanfechell, also on the Isle of Anglesey, where his mathematical talents were spotted by the local landowner Lord Bulkeley, who arranged for him to work in a merchant's counting-house in London.[3] His main patrons were the Bulkeley family of north Wales, and later the Earl of Macclesfield.[4] Jones initially served at sea, teaching mathematics on board Navy ships between 1695 and 1702, where he became very interested in navigation and published A New Compendium of the Whole Art of Navigation in 1702,[3] dedicated to a benefactor John Harris.[5] In this work he applied mathematics to navigation, studying methods of calculating position at sea. After his voyages were over he became a mathematics teacher in London, both in coffee houses and as a private tutor to the son of the future Earl of Macclesfield and also the future Baron Hardwicke. He also held a number of undemanding posts in government offices with the help of his former pupils. Jones published Synopsis Palmariorum Matheseos in 1706, a work which was intended for beginners and which included theorems on differential calculus and infinite series. This used π for the ratio of circumference to diameter, following earlier abbreviations for the Greek word periphery (περιφέρεια) by William Oughtred and others.[6][7][8][9][10] His 1711 work Analysis per quantitatum series, fluxiones ac differentias introduced the dot notation for differentiation in calculus.[11] He was noticed and befriended by two of Britain's foremost mathematicians – Edmund Halley and Sir Isaac Newton – and was elected a fellow of the Royal Society in 1711. He later became the editor and publisher of many of Newton's manuscripts and built up an extraordinary library that was one of the greatest collections of books on science and mathematics ever known, and only recently fully dispersed.[12] He married twice, firstly the widow of his counting-house employer, whose property he inherited on her death, and secondly, in 1731, Mary, the 22-year-old daughter of cabinet-maker George Nix, with whom he had two surviving children. His son, also named William Jones and born in 1746, was a renowned philologist who established links between Latin, Greek and Sanskrit, leading to the concept of the Indo-European language group.[13] References 1. Roberts, Gareth Ffowc (2020). Cyfri'n Cewri. University Press Wales. p. 57. ISBN 978-1786835949. 2. "Library and Archive catalogue". Royal Society. Retrieved 1 November 2010. 3. "Jones biography". University of St. Andrews. Retrieved 12 December 2010. 4. Cyfri'n Cewri by Gareth Ffowc Roberts; University of Wales Press (2020); p. 14. 5. William Jones (1702). A New Compendium of the Whole Art of Navigation. Retrieved 3 February 2011. 6. Jones, William (1706). Synopsis Palmariorum Matheseos : or, a New Introduction to the Mathematics. pp. 243, 263. 7. Rothman, Patricia (7 July 2009). "William Jones and his Circle: The Man who invented Pi". History Today. Retrieved 6 October 2017. 8. Roberts, Gareth Ffowc (14 March 2015). "Pi Day 2015: meet the man who invented π". The Guardian. ISSN 0261-3077. Retrieved 6 October 2017. 9. Bogart, Steven. "What is pi, and how did it originate?". Scientific American. Archived from the original on 6 October 2017. Retrieved 6 October 2017. 10. Archibald, R. C. (1921). "Historical Notes on the Relation $e^{-(\pi /2)}=i^{i}$". The American Mathematical Monthly. 28 (3): 121. doi:10.2307/2972388. JSTOR 2972388. It was probably suggested to Jones by Oughtred who employed the symbol in a different sense. 11. Garland Hampton Cannon (1990). The life and mind Oriental Jones. Retrieved 3 February 2011. 12. "How a farm boy from Wales gave the world pi". The Conversation. Retrieved 14 March 2017. 13. Roberts, Gareth Ffowc (14 March 2015). "Pi Day 2015: meet the man who invented π". The Guardian. Retrieved 14 March 2015. External links • William Jones and other important Welsh mathematicians • William Jones and his Circle: The Man who invented Pi • Pi Day 2015: meet the man who invented π Sir Isaac Newton Publications • Fluxions (1671) • De Motu (1684) • Principia (1687) • Opticks (1704) • Queries (1704) • Arithmetica (1707) • De Analysi (1711) Other writings • Quaestiones (1661–1665) • "standing on the shoulders of giants" (1675) • Notes on the Jewish Temple (c. 1680) • "General Scholium" (1713; "hypotheses non fingo" ) • Ancient Kingdoms Amended (1728) • Corruptions of Scripture (1754) Contributions • Calculus • fluxion • Impact depth • Inertia • Newton disc • Newton polygon • Newton–Okounkov body • Newton's reflector • Newtonian telescope • Newton scale • Newton's metal • Spectrum • Structural coloration Newtonianism • Bucket argument • Newton's inequalities • Newton's law of cooling • Newton's law of universal gravitation • post-Newtonian expansion • parameterized • gravitational constant • Newton–Cartan theory • Schrödinger–Newton equation • Newton's laws of motion • Kepler's laws • Newtonian dynamics • Newton's method in optimization • Apollonius's problem • truncated Newton method • Gauss–Newton algorithm • Newton's rings • Newton's theorem about ovals • Newton–Pepys problem • Newtonian potential • Newtonian fluid • Classical mechanics • Corpuscular theory of light • Leibniz–Newton calculus controversy • Newton's notation • Rotating spheres • Newton's cannonball • Newton–Cotes formulas • Newton's method • generalized Gauss–Newton method • Newton fractal • Newton's identities • Newton polynomial • Newton's theorem of revolving orbits • Newton–Euler equations • Newton number • kissing number problem • Newton's quotient • Parallelogram of force • Newton–Puiseux theorem • Absolute space and time • Luminiferous aether • Newtonian series • table Personal life • Woolsthorpe Manor (birthplace) • Cranbury Park (home) • Early life • Later life • Apple tree • Religious views • Occult studies • Scientific Revolution • Copernican Revolution Relations • Catherine Barton (niece) • John Conduitt (nephew-in-law) • Isaac Barrow (professor) • William Clarke (mentor) • Benjamin Pulleyn (tutor) • John Keill (disciple) • William Stukeley (friend) • William Jones (friend) • Abraham de Moivre (friend) Depictions • Newton by Blake (monotype) • Newton by Paolozzi (sculpture) • Isaac Newton Gargoyle • Astronomers Monument Namesake • Newton (unit) • Newton's cradle • Isaac Newton Institute • Isaac Newton Medal • Isaac Newton Telescope • Isaac Newton Group of Telescopes • XMM-Newton • Sir Isaac Newton Sixth Form • Statal Institute of Higher Education Isaac Newton • Newton International Fellowship Categories Isaac Newton Premier Grand Lodge of England Active 1717–1813, united with the Ancient Grand Lodge of England (1751–1813) to create the United Grand Lodge of England (1813–present) Grand Masters • Anthony Sayer (1717–1718) • George Payne (1718–1719) • John Theophilus Desaguliers (1719–1720) • George Payne (1720–1721) • Duke of Montagu (1721–1723) • Duke of Wharton (1723) • Earl of Dalkeith (1723–1724) • Duke of Richmond (1724) • Lord Paisley (1724–1725) • Earl of Inchiquin (1726–1727) • Baron Colerane (1727–1728) • Baron Kingston (1728–1730) • Duke of Norfolk (1730–1731) • Baron Lovell (1731–1732) • Viscount Montagu (1732–1733) • Earl of Strathmore and Kinghorne (1733–1734) • Earl of Crawford (1734–1735) • Lord Weymouth (1735–1736) • Earl of Loudoun (1736–1737) • Earl of Darnley (1737–1738) • Marquis of Carnarvon (1738–1739) • Baron Raymond (1739–1740) • Earl of Kintore (1740–1741) • Earl of Morton (1741–1742) • Baron Ward (1742–1744) • Lord Cranstoun (1744–1747) • Baron Byron (1747–1752) • Baron Carysfort (1752–1753) • Marquis of Carnarvon (1754–1757) • Lord Aberdour (1757–1762) • Earl Ferrers (1762–1764) • Baron Blayney (1764–1767) • Duke of Beaufort (1767–1772) • Baron Petre (1772–1777) • Duke of Manchester (1777–1782) • Duke of Cumberland (1782–1790) • George, Prince of Wales (1792–1813) • Duke of Sussex (1813) Related articles • History of Freemasonry • Antient Grand Lodge of England • United Grand Lodge of England • James Anderson's The Constitutions of the Free-Masons (1723) • Freemasons' Tavern • Freemasons' Hall, London • Royal Society • Society of Antiquaries of London • Royal College of Physicians • Worshipful Society of Apothecaries • Spalding Gentlemen's Society • Newtonianism • English Enlightenment • Order of the Bath • Walpole ministries • Whiggism (Kit-Cat Club) • Gormogons • Hellfire Club • Foundling Hospital • Unlawful Societies Act 1799 Members • James Anderson • John Byrom • William Stukeley • William Jones • Earl of Chesterfield • Charles Delafaye • Baron Carpenter • William Billers • Sir Thomas Prendergast, 2nd Baronet • Brook Taylor • Martin Folkes • John Arbuthnot • Charles Cox • Earl Cornwallis • Richard Cantillon • John Machin • William Rutty • James Vernon • John Senex • James Thornhill • Earl of Macclesfield • John Browne • James Jurin • James Douglas • Alexander Stuart • Ephraim Chambers • Richard Manningham • Frank Nicholls • Richard Rawlinson • Charles Stanhope • Lord James Cavendish • Earl of Hopetoun • William Richardson • William Becket • John Anstis • Duke of Ancaster • Charles Hayes • Edmund Prideaux • George Shelvocke • John Woodward • John Ward • John Baptist Grano • Baron King • Jacques Leblon • Adolphus Oughton • Sir Robert Rich, 4th Baronet • Viscount Cobham • Francis Columbine • Hugh Warburton • Earl of Pembroke • Viscount Townshend • Martin Bladen • Earl Waldegrave • Duke of Kingston • Earl of Burlington • Earl of Essex • Duke of Queensberry • Earl of Deloraine • Earl of Portmore • Duke of Marlborough • Baron Baltimore • Duke of Atholl • Marquess of Lothian • Earl of Balcarres • Earl of Winchilsea • Sir Arthur Acheson, 5th Baronet • Sir Robert Lawley, 4th Baronet • Alexander Brodie • William Hogarth • Charles Labelye • Walter Calverley-Blackett • Frederick, Prince of Wales • Thomas Wright • Edward Gibbon • Baron Hervey • Thomas Dunckerley • William Preston • Marquess of Hastings • James Moore Smythe • Robert Boyle-Walsingham • Sir Robert de Cornwall • Batty Langley • Thomas Arne • John Soane • Joseph Banks • Johan Zoffany • John Coustos • Hipólito da Costa • Meyer Löw Schomberg • Joseph Salvador • Sampson Eardley • Moses Mendez • Meyer Solomon • Moses Montefiore • Nathan Mayer Rothschild Prime ministers • Robert Walpole • Henry Pelham • Duke of Newcastle Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • Belgium • United States • Netherlands People • Deutsche Biographie Other • IdRef
William of Soissons William of Soissons; French: Guillaume de Soissons; was a French logician who lived in Paris in the 12th century. He belonged to a school of logicians, called the Parvipontians.[1] William of Soissons fundamental logical problem and solution William of Soissons[2] seems to have been the first one to answer the question, "Why is a contradiction not accepted in logic reasoning?" by the principle of explosion. Exposing a contradiction was already in the ancient days of Plato a way of showing that some reasoning was wrong, but there was no explicit argument as to why contradictions were incorrect. William of Soissons gave a proof in which he showed that from a contradiction any assertion can be inferred as true.[1] In example from: It is raining (P) and it is not raining (¬P) you may infer that there are trees on the moon (or whatever else)(E). In symbolic language: P & ¬P → E. If a contradiction makes anything true then it makes it impossible to say anything meaningful: whatever you say, its contradiction is also true. C. I. Lewis's reconstruction of his proof William's contemporaries compared his proof with a siege engine (12th century).[3] Clarence Irving Lewis[4] formalized this proof as follows:[5] Proof V : or & : and → : inference P : proposition ¬ P : denial of P P &¬ P : contradiction. E : any possible assertion (Explosion). (1) P &¬ P → P (If P and ¬ P are both true then P is true) (2) P → P∨E (If P is true then P or E is true) (3) P &¬ P → P∨E (If P and ¬ P are both true then P or E are true (from (2)) (4) P &¬ P → ¬P (If P and ¬ P are both true then ¬P is true) (5) P &¬ P → (P∨E) &¬P (If P and ¬ P are both true then (P∨E) is true (from (3)) and ¬P is true (from (4))) (6) (P∨E) &¬P → E (If (P∨E) is true and ¬P is true then E is true) (7) P &¬ P → E (From (5) and (6) one after the other follows (7)) Acceptance and criticism in later ages In the 15th century this proof was rejected by a school in Cologne. They didn't accept step (6).[6] In 19th-century classical logic, the Principle of Explosion was widely accepted as self-evident, e.g. by logicians like George Boole and Gottlob Frege, though the formalization of the Soissons proof by Lewis provided additional grounding for the Principle of Explosion. References 1. Graham Priest, 'What's so bad about contradictions?' in Priest, Beall and Armour-Garb, The Law of Non-Contradiction, p. 25, Clarendon Press, Oxford, 2011. 2. His writings are lost, see: The Metalogicon of John Salisbury. A Twelfth-Century Defense of the Verbal and Logical Arts of the Trivium, translated with an Introduction and Notes by Daniel D. McGarry, Gloucester (Mass.), Peter Smith, 1971, Book II, Chapter 10, pp. 98-99. 3. William Kneale and Martha Kneale, The Development of Logic, Clarendon Press Oxford, 1962, p. 201. 4. C. I. Lewis and C. H. Langford, Symbolic Logic, New York, The Century Co, 1932. 5. Christopher J. Martin, William’s Machine, Journal of Philosophy, 83, 1986, pp. 564 – 572. In particular p. 565 6. "Paraconsistent Logic (Stanford Encyclopedia of Philosophy)". Plato.stanford.edu. Retrieved 2017-12-18.
William Oughtred William Oughtred (5 March 1574 – 30 June 1660),[1] also Owtred, Uhtred, etc., was an English mathematician and Anglican clergyman.[2][3] After John Napier invented logarithms and Edmund Gunter created the logarithmic scales (lines, or rules) upon which slide rules are based, Oughtred was the first to use two such scales sliding by one another to perform direct multiplication and division. He is credited with inventing the slide rule in about 1622.[4] He also introduced the "×" symbol for multiplication and the abbreviations "sin" and "cos" for the sine and cosine functions.[5] William Oughtred William Oughtred engraving by Wenceslaus Hollar Born5 March 1574 Eton, Buckinghamshire, England Died30 June 1660(1660-06-30) (aged 86) Albury, Surrey, England EducationEton College Alma materKing's College, Cambridge Known for • Slide rule • Multiplication "×" sign Scientific career FieldsMathematician InstitutionsKing's College, Cambridge Notable students • John Wallis • Christopher Wren • Richard Delamain • Seth Ward Clerical life Education The son of Benjamin Oughtred of Eton in Buckinghamshire (now part of Berkshire), William was born there on 5 March 1574/75 and was educated at Eton College, where his father, a writing-master, was one of his teachers.[6] Oughtred had a passion for mathematics, and would often stay awake at nights to learn while others were sleeping.[7] He then attended King's College, Cambridge, where he graduated BA in 1596/97 and MA in 1600, holding a fellowship in the college from 1595 to 1603.[8] He composed a Funeral Ode in Latin for Sir William More of Loseley Park in 1600.[9] Rector at Guildford and at Shalford Admitted to holy orders, he left the University of Cambridge about 1603, when as "Master" William Oughtred he held the rectorate of St Mary's Church, Guildford, Surrey.[10] At the presentation of the lay patron George Austen, gent., he was instituted as vicar at Shalford near Wonersh, in the neighbourhood of Guildford in western Surrey, on 2 July 1605.[11] On 20 February 1606, at Shalford, Oughtred married Christsgift Caryll, a member of the Caryll family seated at Great Tangley Hall at Shalford.[12] The Oughtreds had twelve children, William, Henry, Henry (the first Henry died as a baby), Benjamin, Simon, Margaret, Judith, Edward, Elizabeth, Anne, George, and John. Two of the sons, Benjamin and John, shared their father's interest in instruments and became watchmakers.[13] Oughtred's wife was a niece of Simon Caryll of Tangley and his wife Lady Elizabeth Aungier (married 1607), daughter of Sir Francis Aungier. Oughtred was a witness to Simon Caryll's will, made 1618,[14][15] and through two further marriages Elizabeth remained matriarch and dowager of Great Tangley until her death in about 1650.[16][17] Elizabeth's brother Gerald, 2nd Baron Aungier of Longford, was married to Jane, daughter of Sir Edward Onslow of Knowle, Surrey in 1638. Oughtred praised Gerald (whom he taught) as a man of great piety and learning, skilled in Latin, Greek, Hebrew and other oriental languages.[18][19] In January 1610 Sir George More, patron of Compton church adjacent to Loseley Park, granted the advowson (right of presentation of the minister) to Oughtred, when it should next fall vacant, though Oughtred was not thereby empowered to present himself to the living.[20] This was soon after Sir George More became reconciled to the marriage of his daughter Anne to the poet John Donne, which had occurred secretly in 1601. Rector of Albury Oughtred was presented by Sir (Edward) Randall (lord of the manor) to the rectory of Albury, near Guildford in Surrey and instituted on 16 October 1610,[21] vacating Shalford on 18 January 1611.[22] In January 1615/16 Sir George More re-granted the advowson of Compton church (still occupied) in trust to Roger Heath and Simon Caryll, to present Oughtred himself, or any other person whom Oughtred should nominate, when the vacancy should arise.[23] Soon afterwards Oughtred was approached by John Tichborne seeking his own nomination, and entering an agreement to pay him a sum of money upon certain days. Before this could be completed the incumbent died (November 1618), and Oughtred sought for himself to be presented, preaching several times at Compton, having the first fruits sequestered to his use, and, after four months, asking the patron to present him. However, Tichborne offered to complete the agreed payment at once, and was accordingly presented by the trustees in May 1619 (Simon Caryll dying in that year): but before he could be admitted, the Crown interposed a different candidate because the contract between Oughtred and Tichborne was deemed by Sir Henry Yelverton clearly to be Simoniacal.[20] Oughtred therefore remained at Albury,[24] serving as rector there for fifty years.[25][26] William Lilly, that celebrated astrologer, knew Oughtred and claimed in his autobiography to have intervened on his behalf to prevent his ejection from his living by Parliament in 1646: "About this time, the most famous mathematician of all Europe, Mr. William Oughtred, parson of Aldbury in Surry, was in danger of sequestration by the Committee of or for plundered ministers; (Ambo-dexters they were;)[27] several inconsiderable articles were deposed and sworn against him, material enough to have sequestered him, but that, upon his day of hearing, I applied myself to Sir Bolstrode Whitlock, and all my own old friends, who in such numbers appeared in his behalf, that though the chairman and many other Presbyterian members were stiff against him, yet he was cleared by the major number."[28] Of his portrait (aged 73, 1646) engraved by Wenceslas Hollar, prefixed to the Clavis Mathematica, John Evelyn remarked that it "extreamly resembles him", and that it showed "that calm and placid Composure, which seemed to proceed from, and be the result of some happy ἕυρησις and Invention".[29] William Oughtred died at Albury in 1660, a month after the restoration of Charles II. A staunch supporter of the royalty, he is said to have died of joy at the knowledge of the return of the King. He was buried in Old St Peter and St Paul's Church, Albury.[30] Autobiographical information is contained in his address "To the English gentrie" in his Just Apologie of c. 1634.[31] Mathematician Oughtred developed his interest in mathematics early in life, and devoted whatever spare time his academic studies allowed him to it. Among the short tracts added to the 1647/48 editions of the Clavis Mathematica was one describing a natural and easy way of delineating sun-dials upon any surface, however positioned, which the author states he invented in his 23rd year (1597/98), which is to say, during his fellowship at King's College, Cambridge. His early preoccupation was to find a portable instrument or dial by which to find the hour, he tried various contrivances, but never to his satisfaction. "At last, considering that all manner of questions concerning the first motions were performed most properly by the Globe itself, rectified to the present elevation by the help of a moveable Azimuth; he projected the Globe upon the plane of the Horizon, and applied to it at the center, which was therein the Zenith, an Index with projected degrees, for the moveable Azimuth."[2] This projection answered his search, but then he had to invent theorems, problems and methods to calculate sections and intersections of large circles, which he could not find by instruments, not having access to any of sufficient size. In this way he drew out his findings, presenting one example to Bishop Thomas Bilson (who had ordained him), and another, in about 1606, to a certain noble lady, for whom he wrote notes for its use. In London, in spring 1618, Oughtred visited his friend Henry Briggs at Gresham College, and was introduced to Edmund Gunter, Reader in Astronomy, then occupying Dr Brooks's rooms. He showed Gunter his "Horizontall Instrument", who questioned him closely about it and spoke very approvingly. Soon afterwards Gunter sent him a print taken from a brass instrument made by Elias Allen, after Oughtred's written instructions (which Allen preserved).[2] When, in 1632, Richard Delamain the elder claimed that invention for himself,[32][33] William Robinson wrote to Oughtred: "I cannot but wonder at the indiscretion of Rich. Delamain, who being conscious to himself that he is but the pickpurse of another man's wit, would thus inconsiderately provoke and awake a sleeping lion..."[34] Around 1628 he was appointed by the Earl of Arundel to instruct his son William Howard in mathematics.[24] Some of Oughtred's mathematical correspondence survives, and is printed in Bayle's General Dictionary,[2] and (with some editorial omissions restored) in Dr Rigaud's Correspondence of Scientific Men.[35] William Alabaster wrote to him in 1633 to propose the quadrature of the circle by consideration of the fourth chapter of the Book of Ezekiel.[36] In 1634 he corresponded with the French architect François Derand, and (among others) with Sir Charles Cavendish (1635), Johannes Banfi Hunyades (1637), William Gascoigne (1640)[37] and Dr John Twysden, M.D. (1650).[38] Oughtred offered free mathematical tuition to pupils, among them Richard Delamain and Jonas Moore, and his teaching influenced a generation of mathematicians. Seth Ward resided with Oughtred for six months to learn contemporary mathematics, and the physician Charles Scarborough also stayed at Albury: John Wallis and Christopher Wren corresponded with him.[39] Another Albury pupil was Robert Wood, who helped him to see the Clavis through the press.[40] Isaac Newton's high opinion of Oughtred is expressed in his letter of 1694 to Nathaniel Hawes, where he quotes him extensively, calling him "a Man whose judgement (if any man's) may safely be relyed upon... that very good and judicious man, Mr Oughtred".[41] The first edition of John Wallis's foundational text on infinitesimal calculus, Arithmetica Infinitorum (1656), carries a long letter of dedication to William Oughtred.[42] Publications Clavis Mathematicæ (1631) William Oughtred's most important work was first published in 1631, in Latin, under the title Arithemeticæ in Numeris et Speciebus Institutio, quae tum Logisticæ, tum Analyticæ, atque adeus totius Mathematicæ quasi Clavis est (i.e. "The Foundation of Arithmetic in Numbers and Kinds, which is as it were the Key of the Logistic, then of the Analytic, and so of the whole Mathematic(s)"). It was dedicated to William Howard, son of Oughtred's patron Thomas Howard, 14th Earl of Arundel.[43] This is a textbook on elementary algebra. It begins with a discussion of the Hindu-Arabic notation of decimal fractions and later introduces multiplication and division sign abbreviations of decimal fractions. Oughtred also discussed two ways to perform long division and introduced the "~" symbol, in terms of mathematics, expressing the difference between two variables. Clavis Mathematicae became a classic, reprinted in several editions. It was used as a textbook by John Wallis and Isaac Newton among others. A concise work, it argued for a less verbose style in mathematics, and greater dependence on symbols. Drawing on François Viète (though not explicitly), Oughtred also innovated freely with symbols, introducing not only the multiplication sign as now used universally,[44] but also the proportion sign (double colon ::).[45] The first edition, 1631, contained 20 chapters and 88 pages including algebra and various fundamentals of mathematics.[46] The work was recast for the New Key, which appeared first in an English edition of 1647, The Key of the Mathematicks New Forged and Filed, dedicated to Sir Richard Onslow and to his son Arthur Onslow (son and grandson of Sir Edward), and then in a Latin edition of 1648, entitled Clavis Mathematica Denuo Limata, sive potius Fabricata (i.e. "The Mathematical Key Newly Filed, or rather Made"), in which the preface was removed and the book was reduced by one chapter. In the English Foreword, Oughtred explains that the intention had always been to provide the ingenious reader with an Ariadne's thread through the intricate labyrinth of these studies, but that his earlier, highly compressed style had been found difficult by some, and was now further elucidated.[47] These editions contained additional tracts on the resolution of adfected equations proposed in numbers, and other materials necessary for the use of decimal parts and logarithms, as well as his work on delineating sundials.[48] The last lifetime edition (third) was in 1652, and posthumous editions (as Clavis Mathematicæ: i.e. "The Key of Mathematic(s)") appeared in 1667 and 1693 (Latin), and in 1694 (English). The work gained popularity around 15 years after it first appeared, as mathematics took a greater role in higher education. Wallis wrote the introduction to his 1652 edition, and used it to publicise his skill as cryptographer;[49] in another, Oughtred promoted the talents of Wren. The Circles of Proportion and the Horizontal Instrument (1632) This work[50] was used by Oughtred in manuscript before it was edited for publication by his pupil, William Forster.[51] Here Oughtred introduced the abbreviations for trigonometric functions. It contains his description and instructions for the use of his important invention, the slide rule, a mechanical means of finding logarithmic results.[52] Two of Oughtred's students, William Forster and Richard Delamaine the elder, are concerned with the story of this book.[54] As instructor to the Earl of Arundel's son, Oughtred had the use of a room in Arundel House, the Earl's residence in the Strand, in London. He gave free instruction there to Richard Delamaine, whom he found to be too dependent on mathematical instruments to get a proper grasp of the theory behind them. Another student of his, Forster, who came to him as a beginner during the 1620s, was therefore taught without reference to instruments so that he should have a true grounding.[55] However, during the long vacation of 1630 Forster (who taught mathematics from a house in St Clement Danes churchyard, on the Westminster side of Temple Bar, in the same locality as Elias Allen's shop), while staying with Oughtred at Albury, asked him about Gunter's Ruler, and was shown two instruments used by his master, including Oughtred's circular slide rule.[56] Oughtred then said to Forster: "... the true way of Art is not by Instruments, but by Demonstration: and ... it is a preposterous course of vulgar Teachers, to begin with Instruments, and not with the Sciences, and so in-stead of Artists, to make their Schollers only doers of tricks, and as it were Iuglers: to the despite of Art, losse of precious time, and betraying of willing and industrious wits, unto ignorance, and idlenesse. ... the use of Instruments is indeed excellent, if a man be an Artist: but contemptible, being set and opposed to Art. And lastly, ... he meant to commend to me, the skill of Instruments, but first he would have me well instructed in the Sciences."[56] Forster obtained Oughtred's permission to translate, edit and publish the description, explanations and instructions which Oughtred had in manuscript, finishing his work in 1632.[56] Meanwhile Delamaine, who had also been shown the instruments, and had copied a text sent by Oughtred to his instrument-maker Elias Allen, was writing-up his own description and account. Delamaine came off the press first, in two separate tracts,[57] claiming himself to be the inventor, and dedicating the prior treatise to King Charles I. He went so far as to show his page-proofs to Oughtred as they were being prepared, and dismissed his objections,[55] printing some derogatory comments aimed at Forster and Oughtred in his Foreword. Forster, who dedicated The Circles of Proportion to the famous intellectual Sir Kenelm Digby, observed that another person had hastily anticipated Oughtred's publication.[56] It was left to Oughtred himself to publish his Just Apologie explaining the priority of his inventions and writings, and showing the behaviour of Delamaine.[55][58] It is stated in Cajori's book that John Napier was the first person ever to use to the decimal point and comma, but Bartholomaeus Pitiscus was really the first to do so.[59] Trigonometria, with Canones Sinuum (1657) Trigonometria, Hoc est, Modus Computandi Triangulorum Latera & Angulos was a collection compiled from Oughtred's papers by Richard Stokes and Arthur Haughton.[60] It contains about 36 pages of writing. Here the abbreviations for the trigonometric functions are explained in further detail consisting of mathematical tables.[7] It carries a frontispiece portrait of Oughtred similar to that by Wenceslas Hollar, but re-engraved by William Faithorne, and depicted as aged 83, and with a short epigram by "R.S." beneath. Longer verses addressed to Oughtred are prefixed by Christopher Wase. Opuscula Mathematica (1677) A miscellaneous collection of his hitherto unpublished mathematical papers (in Latin) was edited and published by his friend Sir Charles Scarborough in 1677.[61][62] The treatises contained are on these subjects: • Institutiones Mechanicæ. • De variis corporum generibus gravitate et magnitudine comparatis. • Automata • Quæstiones Diophanti Alexandrini Lib. 3 • De Triangulis planis rectangulis • De Divisione Superficierum • Musicæ Elementa • De Propugnaculorum Munitionibus • Sectiones Angulares Inventions Slide rule Oughtred's invention of the slide rule consisted of taking a single "rule", already known to Gunter, and simplifying the method of employing it. Gunter required the use of a pair of dividers to lay off distances on his rule; Oughtred made the step of sliding two rules past each other to achieve the same ends.[63] His original design of some time in the 1620s was for a circular slide rule; but he was not the first into print with this idea, which was published by Delamain in 1630. The conventional design of a sliding middle section for a linear rule was an invention of the 1650s.[64] Double horizontal sun dial At the age of 23, Oughtred invented the double horizontal sundial, now named the Oughtred type after him.[65] A short description The description and use of the double Horizontall Dyall (16 pp.) was added to a 1653 edition (in English translation) of the pioneer book on recreational mathematics, Récréations Mathématiques (1624) by Hendrik van Etten, a pseudonym of Jean Leurechon.[66] The translation itself is no longer attributed to Oughtred, but (probably) to Francis Malthus.[67] Universal equinoctial ring dial Oughtred also invented the Universal equinoctial ring dial.[68] Occult interests According to his contemporaries, Oughtred had interests in alchemy and astrology.[69] The Hermetic science remained a philosophical touchstone among many reputable scientists of his time, and his student Thomas Henshaw copied a Diary and "Practike" given to him by his teacher.[70] He was well-acquainted with the astrologer William Lilly who, as noted above, helped to prevent his ejection from his living in 1646. John Aubrey: Astrology and Geomancy John Aubrey states that (despite their political differences) Sir Richard Onslow, son of Sir Edward, also defended Oughtred against ejection in 1646. He adds that Oughtred was an astrologer, and successful in the use of natal astrology, and used to say that he did not know why it should be effective, but believed that some "genius" or "spirit" assisted. According to Aubrey, Elias Ashmole possessed the original copy in Oughtred's handwriting of his rational division of the twelve houses of the zodiac. Oughtred penned an approving testimonial, dated 16 October 1659, to the foot of the English abstract of The Cabal of the Twelve Houses Astrological by "Morinus" (Jean-Baptiste Morin) which George Wharton inserted in his Almanac for 1659.[71] Aubrey suggests that Oughtred was happy to allow the country people to believe that he was capable of conjuring. Aubrey himself had seen a copy of Christopher Cattan's work on Geomancy[72] annotated by Oughtred.[73] He reported that Oughtred had told Bishop Ward and Elias Ashmole that he had received sudden intuitions or solutions to problems when standing in particular places, or leaning against a particular oak or ash tree, "as if infused by a divine genius", after having pondered those problems unsuccessfully for months or years.[74] Elias Ashmole: Freemasonry Oughtred was well-known to Elias Ashmole, as Ashmole stated in a note to Lilly's autobiographical sketch: "This gentleman I was very well acquainted with, having lived at the house over-against his, at Aldbury in Surrey, three or four years. E.A."[28] The biography of Ashmole in the Biographia Britannica (1747)[75] called forth the supposition that Oughtred was a participant in Ashmole's admission to freemasonry in 1646. Friedrich Nicolai, in both sections of his Essay (on the Templar and Masonic Orders) of 1783, associated Oughtred, Lilly, Wharton and other Astrologers in the formation of the order of Free and Accepted Masons in Warrington and London.[76] The statement was reinforced through repetition by Thomas De Quincey,[77] and elaborated by Jean-Marie Ragon,[78] but was debunked in A.G. Mackey's History of Freemasonry (1906).[79] Ashmole noted that he paid a visit to "Mr. Oughtred, the famous mathematician", on 15 September 1654, about three weeks after the Astrologers' Feast of that year.[80] John Evelyn: Millenarianism Oughtred expressed millenarian views to John Evelyn in 1655: "Came that renowned mathematician, Mr. Oughtred, to see me, I sending my coach to bring him to Wotton, being now very aged. Among other discourse, he told me he thought water to be the philosopher's first matter, and that he was well persuaded of the possibility of their elixir; he believed the sun to be a material fire, the moon a continent, as appears by the late selenographers; he had strong apprehensions of some significant event to happen the following year, from the calculation of difference with the diluvian period; and added that it might possibly be to convert the Jews by our Saviour's visible appearance, or to judge the world; and therefore, his word was, Parate in occursum;[81] he said Original Sin was not met with in the Greek Fathers, yet he believed the thing; this was from some discourse on Dr. Taylor's late book, which I had lent him."[82] Oughtred Society Oughtred's name is remembered in the Oughtred Society, a group formed in the United States in 1991 for collectors of slide rules. It produces the twice-yearly Journal of the Oughtred Society and holds meetings and auctions for its members.[83][84] References 1. Smith, David Eugene (1923). History of Mathematics. Vol. 1. p. 392. ISBN 9780486204291. 2. 'Oughtred (William)', in P. Bayle, translated and revised by J.P. Bernard, T. Birch and J. Lockman, A General Dictionary, Historical and Critical, (James Bettenham, for G. Strachan and J. Clarke, London 1734/1739), Vol. VIII, pp. 77-86 (Google). 3. F. Willmoth, 'Oughtred, William (bap. 1575, d. 1660)', Oxford Dictionary of National Biography (2004); superseding J.B. Mullinger, 'Oughtred, William (1575-1660)', Dictionary of National Biography (1885-1900), vol. 42. 4. Smith, David E. (1958). History of Mathematics. Courier Corporation. p. 205. ISBN 9780486204307. 5. Florian Cajori (1919). A History of Mathematics. Macmillan. p. 157. cajori william-oughtred multiplication. 6. Wallis, P.J. (1968). "William Oughtred's 'Circles of Proportion' and 'Trigonometries'". Transactions of the Cambridge Bibliographical Society. 4 (5): 372–382. JSTOR 41154471. 7. F. Cajori, William Oughtred, a Great Seventeenth-Century Teacher of Mathematics (Open Court Publishing Company, Chicago 1916), pp. 12-14 (Internet Archive). 8. "Oughtred, William (OTRT592W)". A Cambridge Alumni Database. University of Cambridge. 9. 'Funeral ode by William Outhred', Surrey History Centre, ref. 6729/7/129 (Discovery Catalogue). 10. Church of England Clergy database, Liber Cleri detail, CCEd Record ID: 199392, from British Library Harleian MS 595. 11. Church of England Clergy database, Episcopal Register of Thomas Bilson (Winchester), Appointment Record ("Owtred") CCEd Record ID: 59030. 12. ODNB, and see Aubrey's Brief Lives, Ed. Oliver Lawson Dick (Ann Arbor, Michigan 1962), pp. 222–224. 13. J.J. O'Connor and E.F. Robertson, "Biographies: William Oughtred", MacTutor History of Mathematics archive, Last Update, 2017 (University of St Andrews). 14. London Metropolitan Archive ref. DW/PA/5/1619/22. 15. William "Owtred" also witnessed the first codicil (8 November 1620) to the Will of George Austen of Shalford (P.C.C. 1621, Dale quire). 16. W. Bruce Bannerman (ed.), The Visitations of the county of Surrey taken in the years 1530, 1572 and 1623, Harleian Society Vol. XLIII (1899), pp. 88–89 (Internet Archive). 17. Cf. Will of John Machell, Gentleman of Wonersh (P.C.C. 1647, Fine quire); Will of Elizabeth Machell of Wonersh (P.C.C. 1650, Pembroke quire). 18. E.W. Brayley, J. Britton and E.W Brayley jnr., A Topographical History of Surrey (G. Willis, London 1850), vol. II, p. 49 (Google). 19. Gerald is sometimes referred to as "Gerard", but the name derives from his mother's family of Fitzgerald, seated at Hatchlands Park in East Clandon. 20. "Yelverton, Sir Henry. Opinion that simony was involved in the contract between William Oughtred and John Tichborne for presentation to the church of Compton in Surrey", Papers of the More family of Loseley Park, Surrey, 1488-1682 (bulk 1538-1630), Surrey History Centre, ref. Z/407/Lb.668.4 (Discovery Catalogue). View original document at Folger Shakespeare Library. 21. Church of England Clergy database, Episcopal Register of Thomas Bilson (Winchester), Appointment Record, CCEd Record ID: 59103. The advowson was in the lord of the manor, who was Sir Edward. 22. Church of England Clergy database, Episcopal Register of Thomas Bilson (Winchester), Vacancy Evidence Record ("Outhred"), CCEd Record ID: 59115 23. This was just one year after the ordination of John Donne (23 January 1615) and 6 months before his institution as rector of Sevenoaks (12 July 1616). The Clergy database, CCEd Ordination record ID 119823, and Appointment Evidence Record ID 114560. 24. Chisholm, Hugh, ed. (1911). "Oughtred, William" . Encyclopædia Britannica. Vol. 20 (11th ed.). Cambridge University Press. p. 378. 25. J. and J.A. Venn, Alumni Cantabrigienses Part 1 Vol. III (Cambridge University Press 1924), p. 288 (Internet Archive) (appointment 1610). 26. "Parishes: Albury", in H.E. Malden (ed.), A History of the County of Surrey, Volume 3 (V.C.H./HMSO, London 1911), pp. 72-77 (British History Online): "was rector from 1610 to 1660". 27. "Ambo-dexters", in the figurative meaning, i.e. their allegiances swayed according to their own advantage. 28. William Lilly's History of his Life and Times, from the year 1602 to 1681 (Published London 1715), Reprint (Charles Baldwyn, London 1822), pp. 135-37 (Internet Archive). 29. J. Evelyn, Numismata: A Disccourse of Medals, Ancient and Modern... To which is added, A Discourse concerning Physiognomy (Benjamin Tooke, London 1697), p. 341 (Google). 30. "Parishes: Albury", in H.E. Malden (ed.), A History of the County of Surrey, Volume 3 (V.C.H./HMSO, London 1911), pp. 72-77 (British History Online, accessed 6 December 2018). 31. (W. Oughtred), To the English gentrie, and all others studious of the mathematicks which shall bee readers hereof. The just apologie of Wil: Oughtred, against the slaunderous insimulations of Richard Delamain, in a pamphlet called Grammelogia, or the Mathematicall Ring, or Mirifica Logarithmorum Projectio Circularis (A. Mathewes, London ?1634). Full text at Umich/eebo. Extracts in F. Cajori (1915) (Further reading). 32. M. Selinger, Teaching Mathematics (1994), p. 142. 33. "The Galileo Project". Galileo.rice.edu. Retrieved 31 October 2012. 34. 'VII: W. Robinson to Oughtred', in S.J. Rigaud (ed.), Correspondence of Scientific Men of the Seventeenth Century, 2 vols (University Press, Oxford 1841), I pp. 11-14 (Google). 35. Letters II-XXXVI, in S.J. Rigaud (ed.), Correspondence of Scientific Men of the Seventeenth Century, 2 vols (University Press, Oxford 1841), I, pp. 3-92 (Google). 36. Bayle, General Dictionary, VIII, p. 80, note, col. b (Google). 37. "DSpace at Cambridge: Letter from William Gascoigne to William Oughtred". Dspace.cam.ac.uk. 13 June 2007. Retrieved 31 October 2012. {{cite journal}}: Cite journal requires \|journal= (help) 38. "Janus: Oughtred, William (? 1574-1660) mathematician". Janus.lib.cam.ac.uk. Retrieved 31 October 2012. 39. H.M. Pycior, Symbols, Impossible Numbers, and Geometric Entanglements: British Algebra Through the Commentaries on Newton's Universal Arithmetick (1997), p. 42. 40. T.C. Barnard, Cromwellian Ireland: English Government and Reform in Ireland 1649–1660 (2000), p. 223. 41. 'Appendix. No. XXV. Newton to Hawes', in J. Edleston (ed.), Correspondence of Sir Isaac Newton and Professor Cotes, Including Letters of Other Eminent Men (John W. Parker, London/John Deighton, Cambridge 1850), pp. 279-92, at pp. 291-92 (Google). 42. J. Wallis, Arithmetica Infinitorum, sive Nova Methodus Inquirendi in Curvisineorum Quadraturam (Typis Leon: Lichfield Academiae Typographi, Impensis Tho. Robinson, Oxford 1656), unpaginated front matter (Internet Archive). 43. (Londini, apud Thomam Harperum, 1631): see full page-views at Google. 44. F. Cajori, 'The cross X as a symbol for multiplication', Nature, Vol. XCIV (1914), Abstract, pp. 363-64 (journal's webpage). 45. Helena Mary Pycior, Symbols, Impossible Numbers, and Geometric Entanglements: British Algebra Through the Commentaries on Newton's Universal Arithmetick (1997), p. 48. 46. Cajori, Florian (1915). "The Works of William Oughtred". The Monist. 25 (3): 441–466. doi:10.5840/monist191525315. JSTOR 27900548. 47. Bayle, 'Oughtred (William)', General Dictionary, VIII, p. 78, note col. a (Google). 48. G.[W.] Oughtred, Clavis Mathematica Denuo Limata, sive potius Fabricata (Londini, Excudebat Thomas Harper, sumptibus Thome Whitakeri, apud quem venales sunt in Cœmiterio D. Pauli, 1648); Full page views at Google. 49. "Oxford Figures, Chapter 1: 800 years of mathematical traditions". Mathematical Institute – University of Oxford. 17 September 2007. Archived from the original on 26 October 2012. Retrieved 31 October 2012. 50. W. Oughtred, ed. W. Forster, The Circles of Proportion and the Horizontall Instrument. The former shewing the maner how to work proportions both simple and compound: and the ready and easy resolving of quæstions both in arithmetic, geometrie, & astronomie: and is newly increased with an additament for navigation. All which rules may also be wrought with the penne by arithmetic, and the canon of triangles. The latter teaching how to work most quæstions, which may be performed by the globe: and to delineat dialls upon any kind of plaine. Both invented, and the uses of both written in Latine by Mr. W. O. Translated into English and set forth for the publique benefit by William Forster First issue (Printed for Elias Allen maker of these and all other Mathematical Instruments and are to be sold at his shop over against St Clements church without Temple Bar, London 1632); Second issue (Printed by Augustine Mathewes, and are to bee sold by Nic: Bourne at the Royall Exchange, London 1633), full text at Umich/eebo. 51. Stephen, Leslie, ed. (1889). "Forster, William (fl.1632)" . Dictionary of National Biography. Vol. 20. London: Smith, Elder & Co. 52. Ball, W. W. Rouse (1917). "Review of William Oughtred: a great Seventeenth-century Teacher of Mathematics". Science Progress (1916-1919). 11 (44): 694–695. JSTOR 43426914. 53. The captions "Versus Septentrionem", "versus Meridiem", suggest the opposite orientation, but the river Thames (to the south) is clearly visible in the "Septentrio" scene. 54. A.J. Turner, 'William Oughtred, Richard Delamain and the Horizontal Instrument in seventeenth-century England', Annali dell'Istituto e Museo di storia della scienza di Firenze vol. 6 pt. 2 (1981), pp. 99-125. 55. (W. Oughtred), To the English gentrie, and all others studious of the mathematicks which shall bee readers hereof. The Just Apologie of Wil: Oughtred, against the slaunderous insimulations of Richard Delamain, in a pamphlet called Grammelogia, or The Mathematicall Ring, or Mirifica Logarithmorum Projectio Circularis (A. Mathewes, London ?1634). Full text at Umich/eebo. 56. "To the Honourable and Renowned for vertue, learning, and true valour, Sir Kenelme Digbye, Knight (dated 1632)", in W. Oughtred, ed. W. Forster, The Circles of Proportion and the Horizontall Instrvment (1632, second issue 1633), unpaginated front matter; also here (Umich/eebo). 57. R. Delamaine, ''Grammelogia, or, The mathematicall ring extracted from the logarythmes, and projected circular (printed 1631), full text at Umich/eebo: R. Delamaine, The Making, Description, and Use of a small portable Instrument called a Horizontall Quadrant (London: Printed [by Thomas Cotes] for Richard Hawkins and are to be sold at his shop in Chancery lane neere Sarjants Inne, 1632), full text at Umich/eebo. 58. F.J. Swetz, 'Mathematical Treasure: Oughtred's Defense of His Slide Rule', Convergence (Online Periodical of the Mathematical Association of America), August 2018 (MAA). 59. L.C. Karpinsky, 'Review: William Oughtred, a Great Seventeenth-Century Teacher of Mathematics, by F. Cajori', The American Mathematical Monthly, Vol. 24 part 1, pp. 29-30 (jstor, open pdf). Cajori is also chastised for his having mis-spelled the name of Erasmus O. Schreckenfuchs. 60. W. Oughtred, ed. R. Stokes and A. Haughton, Trigonometria, Hoc est, Modus Computandi Triangulorum Latera & Angulos Ex Canone Mathematico traditus et demonstratus... Una cum Tabulis Sinuum, Tangent & Secant, &c. (Londini, Typis R. & L.W. Leybourn, Impensis Thomæ Johnson, apud quem væneunt sub signo Clavis Aureæ in Cœmiterio S. Pauli, 1657); full page views at Internet Archive. 61. 'William Oughtred', in O. Manning and Bray, The History and Antiquities of the County of Surrey (John White/John Nichols and Son, London 1804-14), II, pp. 132-33 (Google). 62. W. Oughtred, ed. C. Scarborough, Gulielmi Oughtredi Ætonensis, quondam Collegii Regalis Cantabrigia Socii, Opuscula Mathematica hactenus inedita (E Theatro Sheldoniano, Oxford 1677); Full pageviews at Google. 63. "Slide Rules". Hpmuseum.org. Retrieved 31 October 2012. 64. "The slide rule – a forgotten tool". Powerhouse Museum Collection. Retrieved 31 October 2012. 65. "Harvard University – Department of History of Science". Dssmhi1.fas.harvard.edu. Archived from the original on 20 February 2012. Retrieved 31 October 2012. 66. W. Oughtred, 'The Description, and use of the double Horizontall Diall', in H. van Etten, Mathematicall recreations. Or, A collection of many problemes, extracted out of the ancient and modern philosophers (William Leake, London 1653), unnumbered pages, full text at Umich/eebo. 67. Heefer, Albrecht. "Récréations Mathématiques (1624) A Study on its Authorship, Sources and Influence" (PDF). logica.ugent.be. 68. "Royal Museums Greenwich". 69. Keith Thomas, Religion and the Decline of Magic (1973), p. 322 and 452n. 70. D.R. Dickson, 'Thomas Henshaw and Sir Robert Paston's pursuit of the Red Elixir: an early collaboration between Fellows of the Royal Society', Notes and Records of the Royal Society of London, Vol. 51, No. 1 (Jan. 1997), pp. 57-76, at pp. 67-72. 71. 'The Cabal of the Twelve Houses Astrological', collected in J. Gadbury (ed.), The Works of that Late Most Excellent Philosopher and Astronomer, Sir George Wharton, bar. collected into one volume (M.H. for John Leigh, London 1683), pp. 189-208, at p. 208 (Google). 72. La Geomance du Seigneur Christofe de Cattan, Gentilhomme Genevoys. Livre non moins plaisant et recreatif. Avec la roüe de Pythagoras (Gilles Gilles, Paris 1558). Full text (page views) at Internet Archive. 73. Oughtred may have possessed the English translation by Francis Sparry, The Geomancie of Maister Christopher Catton, a Booke no lesse pleasant and recreative, then of a wittie invention (London 1591). 74. 'William Oughtred, 1575-1660', in R. Barber (ed.), John Aubrey - Brief Lives: A selection based upon existing contemporary portraits (Folio Society, London 1975), 232-37. 75. W. Oldys (ed.), Biographia Britannica: or, the Lives of the most eminent persons who have flourished in Great Britain and Ireland, 6 vols (W. Innys (etc.), London 1747-1766), I, pp. 223-36, at p. 224, note E, 'Collections relating to the history of Free-Masons', and pp. 228-29 (Google). 76. C.F. Nicolai, Versuch über die Beschuldigungen welche dem Tempelherrenorden gemacht worden, und über dessen Geheimniß: Nebst einem Anhange über das Entstehen der Freymaurergesellschaft (Nicolai, Berlin und Stettin 1782), Theil I, at p. 188; Theil II, pp. 191-196 (Google). 77. T. De Quincey, 'Historico-Critical Inquiry into the Origins of the Rosicrucians and the Free-Masons', in D. Masson (ed.), The Collected Writings of Thomas De Quincey, New and Enlarged Edition, Vol. XIII: Tales and Prose Phantasies (Adam and Charles Black, Edinburgh 1890), at pp. 425-26 (Google). 78. J.M. Ragon, Orthodoxie maçonnique: suivie de la Maçonnerie occulte et de l'initiation hermétique (E. Dentu, Paris 1853), pp. 28-33, pp. 99-108, and passim (Google). 79. A.G. Mackey, The History of Freemasonry, 2 volumes (The Masonic History Company, New York and London 1906), II, pp. 306, 316-18 (Internet Archive). 80. 'The Life of Elias Ashmole, Esq.', in The Lives of Those Eminent Antiquaries Elias Ashmole, Esquire, and Mr William Lilly, Written by Themselves (T. Davies, London 1774), at p. 321 (Google). 81. I.e. "Praeparare in occursum Dei tui, Israel" (Book of Amos, Chapter IV, v. 12): "Prepare to meet thy God, O Israel". 82. 'Entry for 28 August 1655', in W. Bray (ed.), The Diary of John Evelyn, with a Biographical introduction by the editor, and a special introduction by Richard Garnett, LL.D., 2 vols (M. Walter Dunne, New York and London 1901), I, pp. 305-06 (Internet Archive, Retrieved 5 December 2018). 83. "The Oughtred Society". The Oughtred Society. Retrieved 18 March 2015. 84. "Brochure" (PDF). The Oughtred Society. Retrieved 18 March 2015. Further reading • Cajori, Florian (1916). William Oughtred: A Great Seventeenth-Century Teacher of Mathematics. The Open Court Publishing Company. • Florian Cajori (1915), "The Life of William Oughtred", The Open Court, Vol. XXIX no. 8 (Chicago, August 1915), p. 711, pp. 449-59 (pdf) • Jacqueline Anne Stedall, Ariadne's Thread: The Life and Times of Oughtred's Clavis, Annals of Science, Volume 57, Issue 1 January 2000, pp. 27–60. doi:10.1080/000337900296290 External links • Media related to William Oughtred at Wikimedia Commons • O'Connor, John J.; Robertson, Edmund F., "William Oughtred", MacTutor History of Mathematics Archive, University of St Andrews • Galileo Project page • The Oughtred Society inspired by Oughtred and dedicated to the history and preservation of slide rules. • Answers.com article with additional material on Oughtred. • Account of Oughtred by John Aubrey • William Oughtred's "Key of the Mathematics" (John Salusbury's English translation of Oughtred's "Clavis Mathematicae"). Authority control International • FAST • ISNI • VIAF National • Germany • Italy • Israel • Belgium • United States • Czech Republic • Korea • Netherlands • Portugal • Vatican Academics • MathSciNet • zbMATH People • Deutsche Biographie Other • SNAC • IdRef
Williams's p + 1 algorithm In computational number theory, Williams's p + 1 algorithm is an integer factorization algorithm, one of the family of algebraic-group factorisation algorithms. It was invented by Hugh C. Williams in 1982. It works well if the number N to be factored contains one or more prime factors p such that p + 1 is smooth, i.e. p + 1 contains only small factors. It uses Lucas sequences to perform exponentiation in a quadratic field. It is analogous to Pollard's p − 1 algorithm. Algorithm Choose some integer A greater than 2 which characterizes the Lucas sequence: $V_{0}=2,V_{1}=A,V_{j}=AV_{j-1}-V_{j-2}$ where all operations are performed modulo N. Then any odd prime p divides $\gcd(N,V_{M}-2)$ whenever M is a multiple of $p-(D/p)$, where $D=A^{2}-4$ and $(D/p)$ is the Jacobi symbol. We require that $(D/p)=-1$, that is, D should be a quadratic non-residue modulo p. But as we don't know p beforehand, more than one value of A may be required before finding a solution. If $(D/p)=+1$, this algorithm degenerates into a slow version of Pollard's p − 1 algorithm. So, for different values of M we calculate $\gcd(N,V_{M}-2)$, and when the result is not equal to 1 or to N, we have found a non-trivial factor of N. The values of M used are successive factorials, and $V_{M}$ is the M-th value of the sequence characterized by $V_{M-1}$. To find the M-th element V of the sequence characterized by B, we proceed in a manner similar to left-to-right exponentiation: x := B y := (B ^ 2 − 2) mod N for each bit of M to the right of the most significant bit do if the bit is 1 then x := (x × y − B) mod N y := (y ^ 2 − 2) mod N else y := (x × y − B) mod N x := (x ^ 2 − 2) mod N V := x Example With N=112729 and A=5, successive values of $V_{M}$ are: V1 of seq(5) = V1! of seq(5) = 5 V2 of seq(5) = V2! of seq(5) = 23 V3 of seq(23) = V3! of seq(5) = 12098 V4 of seq(12098) = V4! of seq(5) = 87680 V5 of seq(87680) = V5! of seq(5) = 53242 V6 of seq(53242) = V6! of seq(5) = 27666 V7 of seq(27666) = V7! of seq(5) = 110229. At this point, gcd(110229-2,112729) = 139, so 139 is a non-trivial factor of 112729. Notice that p+1 = 140 = 22 × 5 × 7. The number 7! is the lowest factorial which is multiple of 140, so the proper factor 139 is found in this step. Using another initial value, say A = 9, we get: V1 of seq(9) = V1! of seq(9) = 9 V2 of seq(9) = V2! of seq(9) = 79 V3 of seq(79) = V3! of seq(9) = 41886 V4 of seq(41886) = V4! of seq(9) = 79378 V5 of seq(79378) = V5! of seq(9) = 1934 V6 of seq(1934) = V6! of seq(9) = 10582 V7 of seq(10582) = V7! of seq(9) = 84241 V8 of seq(84241) = V8! of seq(9) = 93973 V9 of seq(93973) = V9! of seq(9) = 91645. At this point gcd(91645-2,112729) = 811, so 811 is a non-trivial factor of 112729. Notice that p−1 = 810 = 2 × 5 × 34. The number 9! is the lowest factorial which is multiple of 810, so the proper factor 811 is found in this step. The factor 139 is not found this time because p−1 = 138 = 2 × 3 × 23 which is not a divisor of 9! As can be seen in these examples we do not know in advance whether the prime that will be found has a smooth p+1 or p−1. Generalization Based on Pollard's p − 1 and Williams's p+1 factoring algorithms, Eric Bach and Jeffrey Shallit developed techniques to factor n efficiently provided that it has a prime factor p such that any kth cyclotomic polynomial Φk(p) is smooth.[1] The first few cyclotomic polynomials are given by the sequence Φ1(p) = p−1, Φ2(p) = p+1, Φ3(p) = p2+p+1, and Φ4(p) = p2+1. References 1. Bach, Eric; Shallit, Jeffrey (1989). "Factoring with Cyclotomic Polynomials" (PDF). Mathematics of Computation. American Mathematical Society. 52 (185): 201–219. doi:10.1090/S0025-5718-1989-0947467-1. JSTOR 2008664. • Williams, H. C. (1982), "A p+1 method of factoring", Mathematics of Computation, 39 (159): 225–234, doi:10.2307/2007633, JSTOR 2007633, MR 0658227 External links • P + 1 factorization method at Prime Wiki Number-theoretic algorithms Primality tests • AKS • APR • Baillie–PSW • Elliptic curve • Pocklington • Fermat • Lucas • Lucas–Lehmer • Lucas–Lehmer–Riesel • Proth's theorem • Pépin's • Quadratic Frobenius • Solovay–Strassen • Miller–Rabin Prime-generating • Sieve of Atkin • Sieve of Eratosthenes • Sieve of Pritchard • Sieve of Sundaram • Wheel factorization Integer factorization • Continued fraction (CFRAC) • Dixon's • Lenstra elliptic curve (ECM) • Euler's • Pollard's rho • p − 1 • p + 1 • Quadratic sieve (QS) • General number field sieve (GNFS) • Special number field sieve (SNFS) • Rational sieve • Fermat's • Shanks's square forms • Trial division • Shor's Multiplication • Ancient Egyptian • Long • Karatsuba • Toom–Cook • Schönhage–Strassen • Fürer's Euclidean division • Binary • Chunking • Fourier • Goldschmidt • Newton-Raphson • Long • Short • SRT Discrete logarithm • Baby-step giant-step • Pollard rho • Pollard kangaroo • Pohlig–Hellman • Index calculus • Function field sieve Greatest common divisor • Binary • Euclidean • Extended Euclidean • Lehmer's Modular square root • Cipolla • Pocklington's • Tonelli–Shanks • Berlekamp • Kunerth Other algorithms • Chakravala • Cornacchia • Exponentiation by squaring • Integer square root • Integer relation (LLL; KZ) • Modular exponentiation • Montgomery reduction • Schoof • Trachtenberg system • Italics indicate that algorithm is for numbers of special forms
William Ruggles William Ruggles (September 5, 1797 – September 10, 1877) was a professor at George Washington University.[1][2] Biography William Ruggles was born in Rochester, Massachusetts, about fifty miles south of present-day Boston, on Tuesday September 5, 1797. He was the son of Elisha Ruggles and Mary Clap who also parented six other children: Nathaniel, Micah, Henry, Charles, James, and Lucy. William was the second youngest child in his family. Not much is known about his childhood growing up in Massachusetts until he enrolled in Brown University; where he later graduated from, at the age of twenty-three, in the class of 1820. Two years after graduating from Brown University, Ruggles became a tutor at Columbian College. On February 9, 1821 Congress chartered Columbian College, a nonsectarian school but with Baptist sponsorship that would not become the George Washington University until January 23, 1904. In 1824, two years after he became a tutor, Ruggles became a Professor of Mathematics and Natural Philosophy, a position he would retain until 1865. In 1827 William Ruggles became the chair of both mathematics and natural philosophy. In 1865 Ruggles was made professor emeritus. He continued to lecture on political economy and civil polity from 1865 to 1874. He died at Schooley's Mountain, New Jersey, on Monday, September 10, 1877, five days after his eightieth birthday.[1][2] Legacy Ruggles was a man of many admirable characteristics including loyalty, conscientiousness, and morality, which are shown not only through the words of those who describe him but through his actions as well. The best example of William Ruggles' loyalty was shown in his dedication to the Columbian College itself. Until 1985, no person had a longer stay at the George Washington University. Ruggles came to the George Washington University in 1822 and stayed until his death, a record at the time, of fifty-five and a half years, which has only been surpassed once by Elmer Louis Kayser who came to the university as a student in 1914 and stayed until his eventual death in 1985. William Ruggles was involved at the Columbian College (George Washington University) from almost the beginning of the Columbian College. He lived through the drab years of 1826 through 1842 where mounting debt and pressure threatened to shut down the Columbian College and rejoiced harder than anyone when that debt was lifted. He served under the first six presidents of the institution and even served as acting president three times. He saw a large student body dwindle down to a handful of students due to the US civil war. Since the college buildings were being used as a hospital in the war efforts, classes were taught in the homes of professors. Lastly, Professor Ruggles had the enjoyment of being a contemporary to University President Welling. Ruggles was the owner of over thirty historical and archival books which are now located in the Gelman Library Archives at George Washington University. The subjects include theology, philosophy, psychology, economics, mathematics (algebra and geometry), and chemistry, all of which Ruggles either taught or was passionate about. Ruggles was described by President Welling, the second president of the George Washington University: When the Board was tardy in paying salaries or when it embarked on some policy he opposed, his resignation was always forthcoming .... A member of no religious denomination, but dealing almost exclusively with Baptists, he observed on the basis of attitudes, that he was perhaps the better Christian .... [he was] a man of great conscientiousness, high intelligence and blameless character. His excellent portrait in the University collection suggests a very wise man who perhaps was not always loved but who was respected. — President Welling The Board of Trustees, in adopting resolutions in appreciation of his services, declared Ruggles in similar words to Welling, saying, "We hereby testify and record our exulted sense of the virtues which adorned his private character, the unselfish zeal he brought to the performance of all his duties and the inestimable value of the manifold and multiform services which he rendered to the College during the long period of his connection with its history." William Ruggles was noted for his generous contributions to charities and missionary bequests. His accomplishments and contributions were honored and recognized when he received an honorary LL.D. from Brown University in 1852, the same school that he graduated from more than thirty years prior Upon reviewing letters that Ruggles had written, it can be seen that he was full of charity towards his students. While he may be described as someone who was "not always loved but who was always respected," it can be seen that Ruggles did have great affection towards the students he taught. On November 24, 1837, Ruggles wrote a letter to Joel R. Poinsett about the character of a young man named John D. Kuntz, who was expecting to make appointment as a cadet at West Point Military Academy. In the letter Ruggles goes on to praise the young boy and even go as far as to call him, "a young man of sound principles." William Ruggles' folders were filled with many such letters, expressing his kind opinions on the young students that he respected and cared for. Although it is noted that Ruggles was a man of spirit, he was a member of no religion despite the fact that he dealt almost exclusively with the Baptist church. Ruggles shared very many correspondences with Reverend Elon Galusha with letters dating from November 1825 all the way until April 1832, while it is unclear what the two wrote about it clear that these letters dealt with Ruggles faith. In a letter preserved to a different Reverend, Ruggles states that, "Oh, whatever else is taken from me, may I have a share in the great inheritance purchased by Jesus Christ for those who love him and are regenerated by the Holy Spirit!" Although William Ruggles died 1877, a reprint of his obituary appeared in the Faculty Newsletter volume 2, number 1, Spring 1965 as part of a series of called GW Footnotes. It was titled the 55-Year Professor and was the first in a series of anecdotes from the university's past written by the university historian. Ruggles Prizes The Ruggles Prizes are awarded annually for excellence in mathematics to a candidate for a bachelor's degree. The prizes were initiated in 1859, and consists of two gold medals. They are awarded "upon examination to the best two scholars in Mathematics."[3] References 1. Elmer Louis Kayser (1970). Bricks without straw. Appleton-Century-Crofts. At the end of 1873, Professor William Ruggles asked to be relieved of the duties of his chair and they were ... 1877, at Schooley's Mountain, New Jersey 2. Lyle Slovick (2006-12-11). "William Ruggles". The GW and Foggy Bottom Historical Encyclopedia. Retrieved 2008-05-13. Born in Rochester, Massachusetts, September 5, 1797, he died at Schooley's Mountain, New Jersey, September 10, 1877. Ruggles was connected with the College and University for fifty-five years. A graduate of Brown in the class of 1820, he became a tutor in Columbian College two years later and Professor of Mathematics and Natural Philosophy from 1824 to 1865 when he was made Professor Emeritus in those fields. 3. George Washington University Bulletin. George Washington University. 1905. The Ruggles Prizes, for excellence in Mathematics, founded by Professor William Ruggles, LL. D., consist of two gold medals, annually awarded upon examination to the best two scholars in Mathematics. ... Authority control International • FAST • VIAF National • United States Other • SNAC
William Wales (astronomer) William Wales (1734? – 29 December 1798) was a British mathematician and astronomer who sailed on Captain Cook's second voyage of discovery, then became Master of the Royal Mathematical School at Christ's Hospital and a Fellow of the Royal Society. Early life Wales was born around 1734 to John and Sarah Wales and was baptised in Warmfield (near the West Yorkshire town of Wakefield) that year. As a youth, according to the historian John Cawte Beaglehole, Wales travelled south in the company of a Mr Holroyd, who became a plumber in the service of George III.[1] By the mid-1760s, Wales was contributing to The Ladies' Diary. In 1765 he married Mary Green, sister of the astronomer Charles Green.[1] In 1765, Wales was employed by the Astronomer Royal Nevil Maskelyne as a computer, calculating ephemerides that could be used to establish the longitude of a ship, for Maskelyne's Nautical Almanac.[2] 1769 transit of Venus and wintering at Hudson Bay As part of the plans of the Royal Society to make observations of the June 1769 transit of Venus, which would lead to an accurate determination of the astronomical unit (the distance between the Earth and the Sun), Wales and an assistant, Joseph Dymond, were sent to Prince of Wales Fort on Hudson Bay to observe the transit,[3] with the pair being offered a reward of £200 for a successful conclusion to their expedition.[1] Other Royal Society expeditions associated with the 1769 transit were Cook's first voyage to the Pacific, with observations of the transit being made at Tahiti, and the expedition of Jeremiah Dixon and William Bayly to Norway. Due to winter pack ice making the journey impossible during the winter months, Wales and Dymond were obliged to begin their journey in the summer of 1768, setting sail on 23 June. Ironically, Wales when volunteering to make a journey to observe the transit, had requested that he be sent to a more hospitable location.[4] The party arrived at Prince of Wales Fort in August 1768.[5] Due to the scarcity of building materials at the chosen site, the party had to bring not only astronomical instruments, but the materials required for the construction of living quarters.[5] On their arrival, the pair constructed two "Portable Observatories", which had been designed by the engineer John Smeaton.[6] Construction work occupied the pair for a month and then they settled in for the long winter season. When the day of the transit, 3 June 1769, finally arrived, the pair were lucky to have a reasonably clear day and they were able to observe the transit at around local midday. However, the two astronomers' results for the time of first contact, when Venus first appeared to cross the disc of the Sun, differed by 11 seconds; the discrepancy was to prove a cause of upset for Wales.[4] They were to stay in Canada for another three months before making the return voyage to England, thus becoming the first scientists to spend the winter at Hudson Bay.[7] On his return, Wales was still upset by the difference in the observations and refused to present his findings to the Royal Society until March 1770; however, his report of the expedition, including the astronomical results as well as other climatic and botanical observations, met with approval and he was invited by James Cook to join his next expedition.[4] Captain Cook's second circumnavigation voyage Wales and William Bayly were appointed by the Board of Longitude to accompany James Cook on his second voyage of 1772–75,[3] with Wales accompanying Cook aboard the Resolution. Wales' brother-in-law Charles Green, had been the astronomer appointed by the Royal Society to observe the 1769 transit of Venus but had died during the return leg of Cook's first voyage.[8] The primary objective of Wales and Bayly was to test Larcum Kendall's K1 chronometer, based on the H4 of John Harrison.[8] Wales compiled a log book of the voyage, recording locations and conditions, the use and testing of the instruments entrusted to him, as well as making many observations of the people and places encountered on the voyage.[9] Later life Following his return, Wales was commissioned in 1778 to write the official astronomical account of Cook's first voyage.[10] Wales became Master of the Royal Mathematical School at Christ's Hospital and was elected a Fellow of the Royal Society in 1776.[2][7] Amongst Wales' pupils at Christ's Hospital were Samuel Taylor Coleridge and Charles Lamb.[5] It has been suggested that Wales' accounts of his journeys might have influenced Coleridge when writing his poem The Rime of the Ancient Mariner.[11] The writer Leigh Hunt, another of Wales' pupils, remembered him as "a good man, of plain simple manners, with a heavy large person and a benign countenance".[12] He was appointed as Secretary of the Board of Longitude in 1795, serving in that position until his death in 1798.[10][13] He was nominated by the First Lord of the Admiralty, Earl Spencer, and his appointment confirmed 5 December 1795.[14] Recognition of his work During his voyage of 1791–95, George Vancouver, who had studied astronomy under Wales as a midshipman on HMS Resolution during Cook's second circumnavigation, named Wales Point, a cape at the entrance to Portland Inlet on the coast of British Columbia, in honour of his tutor; the name was later applied to the nearby Wales Island by an official at the British Hydrographic Office.[15] In his journal, Vancouver recorded his gratitude and indebtedness to Wales's tutelage "for that information which has enabled me to traverse and delineate these lonely regions."[16] Wales featured on a New Hebrides (now Vanuatu) postage stamp of 1974 commemorating the 200th anniversary of Cook's discovery of the islands.[8] The asteroid 15045 Walesdymond, discovered in 1998, was named after Wales and Dymond.[17] Works by William Wales • "Journal of a voyage, made by order of the Royal Society, to Churchill River, on the North-west Coast of Hudson's Bay". Philosophical Transactions of the Royal Society of London. 60: 109–136. 1771. • The Method of Finding the Longitude by timekeepers London: 1794. See also • European and American voyages of scientific exploration • Wales, Wendy (2015). Captain Cook’s Computer: the life of William Wales, F.R.S. (1734-1798). Hame House. ISBN 978-09933758-0-4. Notes 1. Wendy Wales. "William Wales' First Voyage". Cook's Log. Captain Cook Society. Retrieved 10 September 2009. 2. Mary Croarken (September 2002). "Providing longitude for all – The eighteenth-century computers of the Nautical Almanac". Journal for Maritime Research. Retrieved 6 August 2009. 3. "William Wales". State Library of New South Wales. Retrieved 6 August 2009. 4. Hudon, Daniel (February 2004). "A (Not So) Brief History of the Transits of Venus". Journal of the Royal Astronomical Society of Canada. 98 (1): 11–13. Retrieved 18 February 2022. 5. Fernie, J. Donald (September–October 1998). "Transits, Travels and Tribulations, IV: Life on the High Arctic". American Scientist. 86 (5): 422. doi:10.1511/1998.37.3396. 6. Steven van Roode. "Historical observations of the transit of Venus". Retrieved 10 August 2009. 7. Glyndwr Williams. "Wales, William". Dictionary of Canadian Biography Online. Retrieved 6 August 2009. 8. "William Wales". Ian Ridpath. Retrieved 6 August 2009. 9. Wales, William. "Log book of HMS 'Resolution'". Cambridge Digital Library. Retrieved 28 May 2013. 10. Orchison, Wayne (2007). Hockey, Thomas A. (ed.). The Biographical Encyclopedia of Astronomers: A-L. p. 1189. ISBN 978-0-387-31022-0. 11. Christopher Ondaatje (15 March 2002). "From Fu Man Chu to a grizzly end". Times Higher Education. Retrieved 11 August 2009. 12. Hunt, Leigh (1828). Lord Byron and some of his comtemporanies with recollections of the author's life and of his visit to Italy. Colburn. p. 352. 13. The Philosophical Transactions of the Royal Society of London, from Their Commencement, in 1665, to the Year 1800: 1763–1769. Royal Society. 1809. p. 683. 14. "Papers of the Board of Longitude : Confirmed minutes of the Board of Longitude, 1780-1801 (5 December 1795)". Cambridge Digital Library. Retrieved 15 January 2017. 15. "Wales Island Cannery". Porcher Island Cannery. Retrieved 10 August 2009. 16. "Captain George Vancouver". Discover Vancouver. Retrieved 10 August 2009. 17. "15045 Walesdymond (1998 XY21)". JPL Small-Body Database Browser. Retrieved 10 August 2009. Sources • Who's Who in Science (Marquis Who's Who Inc, Chicago Ill. 1968) ISBN 0-8379-1001-3 • Francis Lucian Reid "William Wales (ca. 1734–1798): playing the astronomer", Studies in History and Philosophy of Science, 39 (2008) 170–175 External links • "Wales, William" . Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900. • Journal of a Voyage, Made by Order of the Royal Society, to Churchill River, on the North-West Coast of Hudson's Bay; Of Thirteen Months Residence in That Country; and of the Voyage Back to England; In the Years 1768 and 1769: By William Wales • Extracts of William Wales's Journal kept on his voyage aboard HMS Resolution • Full digitised version of Wales' Logbook from his voyage on HMS Resolution • The Original Astronomical Observations, Made in the Course of a Voyage...in the Resolution and Adventure – Results of Wales' work published in 1777 • Article on Wales compiled for Captain Cook Society • The Transit of William Wales Educational comic book produced by the Hudson's Bay Company for Canadian high school students Captain James Cook Voyages • First voyage (1769 transit of Venus observed from Tahiti) • Second voyage • Third voyage (Death of James Cook) Vessels • HMS Adventure • HMS Discovery • HMS Eagle • HMS Endeavour • HMS Grenville • HMS Pembroke • HMS Resolution Associates • Joseph Banks • William Bayly • William Bligh • Alexander Buchan • James Burney • Charles Clerke • James Colnett • Alexander Dalrymple • Georg Forster • Johann Reinhold Forster • Tobias Furneaux • John Gore • Charles Green • Zachary Hickes • James King • John Ledyard • David Nelson • Omai • Hugh Palliser • Sydney Parkinson • Nathaniel Portlock • Edward Riou • Henry Roberts • David Samwell • Daniel Solander • Herman Spöring • William Taylor • James Trevenen • William Wales • John Watts • John Webber • Thomas Willis Artworks • Zoffany's Death of Cook • Statue in Christchurch • Statue in The Mall, London • Hawaii Sesquicentennial half dollar Books • An Account of the Voyages • A Journal of a Voyage to the South Seas • Characteres generum plantarum • A Voyage Round the World • Observations Made During a Voyage Round the World Related • Birthplace Museum • Cooks' Cottage • James Cook Collection: Australian Museum • Memorial Museum • Puhi Kai Iti / Cook Landing Site Authority control International • FAST • ISNI • VIAF National • Chile • France • BnF data • Germany • Italy • Belgium • United States • Czech Republic • Australia • Netherlands Artists • ULAN People • Trove Other • SNAC • IdRef
Willmore conjecture In differential geometry, the Willmore conjecture is a lower bound on the Willmore energy of a torus. It is named after the English mathematician Tom Willmore, who conjectured it in 1965.[2] A proof by Fernando Codá Marques and André Neves was announced in 2012 and published in 2014.[1][3] Willmore energy Main article: Willmore energy Let v : M → R3 be a smooth immersion of a compact, orientable surface. Giving M the Riemannian metric induced by v, let H : M → R be the mean curvature (the arithmetic mean of the principal curvatures κ1 and κ2 at each point). In this notation, the Willmore energy W(M) of M is given by $W(M)=\int _{M}H^{2}\,dA.$ It is not hard to prove that the Willmore energy satisfies W(M) ≥ 4π, with equality if and only if M is an embedded round sphere. Statement Calculation of W(M) for a few examples suggests that there should be a better bound than W(M) ≥ 4π for surfaces with genus g(M) > 0. In particular, calculation of W(M) for tori with various symmetries led Willmore to propose in 1965 the following conjecture, which now bears his name For every smooth immersed torus M in R3, W(M) ≥ 2π2. In 1982, Peter Wai-Kwong Li and Shing-Tung Yau proved the conjecture in the non-embedded case, showing that if $f:\Sigma \to S^{3}$ is an immersion of a compact surface, which is not an embedding, then W(M) is at least 8π.[4] In 2012, Fernando Codá Marques and André Neves proved the conjecture in the embedded case, using the Almgren–Pitts min-max theory of minimal surfaces.[3][1] Martin Schmidt claimed a proof in 2002,[5] but it was not accepted for publication in any peer-reviewed mathematical journal (although it did not contain a proof of the Willmore conjecture, he proved some other important conjectures in it). Prior to the proof of Marques and Neves, the Willmore conjecture had already been proved for many special cases, such as tube tori (by Willmore himself), and for tori of revolution (by Langer & Singer).[6] References 1. Marques, Fernando C.; Neves, André (2014). "Min-max theory and the Willmore conjecture". Annals of Mathematics. 179: 683–782. arXiv:1202.6036. doi:10.4007/annals.2014.179.2.6. MR 3152944. 2. Willmore, Thomas J. (1965). "Note on embedded surfaces". Analele Ştiinţifice ale Universităţii "Al. I. Cuza" din Iaşi, Secţiunea I a Matematică. 11B: 493–496. MR 0202066. 3. Frank Morgan (2012) "Math Finds the Best Doughnut", The Huffington Post 4. Li, Peter; Yau, Shing Tung (1982). "A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces". Inventiones Mathematicae. 69 (2): 269–291. doi:10.1007/BF01399507. MR 0674407. 5. Schmidt, Martin U. (2002). "A proof of the Willmore conjecture". arXiv:math/0203224. 6. Langer, Joel; Singer, David (1984). "Curves in the hyperbolic plane and mean curvature of tori in 3-space". The Bulletin of the London Mathematical Society. 16 (5): 531–534. doi:10.1112/blms/16.5.531. MR 0751827.
Willmore energy In differential geometry, the Willmore energy is a quantitative measure of how much a given surface deviates from a round sphere. Mathematically, the Willmore energy of a smooth closed surface embedded in three-dimensional Euclidean space is defined to be the integral of the square of the mean curvature minus the Gaussian curvature. It is named after the English geometer Thomas Willmore. Definition Expressed symbolically, the Willmore energy of S is: ${\mathcal {W}}=\int _{S}H^{2}\,dA-\int _{S}K\,dA$ where $H$ is the mean curvature, $K$ is the Gaussian curvature, and dA is the area form of S. For a closed surface, by the Gauss–Bonnet theorem, the integral of the Gaussian curvature may be computed in terms of the Euler characteristic $\chi (S)$ of the surface, so $\int _{S}K\,dA=2\pi \chi (S),$ which is a topological invariant and thus independent of the particular embedding in $\mathbb {R} ^{3}$ that was chosen. Thus the Willmore energy can be expressed as ${\mathcal {W}}=\int _{S}H^{2}\,dA-2\pi \chi (S)$ An alternative, but equivalent, formula is ${\mathcal {W}}={1 \over 4}\int _{S}(k_{1}-k_{2})^{2}\,dA$ where $k_{1}$ and $k_{2}$ are the principal curvatures of the surface. Properties The Willmore energy is always greater than or equal to zero. A round sphere has zero Willmore energy. The Willmore energy can be considered a functional on the space of embeddings of a given surface, in the sense of the calculus of variations, and one can vary the embedding of a surface, while leaving it topologically unaltered. Critical points A basic problem in the calculus of variations is to find the critical points and minima of a functional. For a given topological space, this is equivalent to finding the critical points of the function $\int _{S}H^{2}\,dA$ since the Euler characteristic is constant. One can find (local) minima for the Willmore energy by gradient descent, which in this context is called Willmore flow. For embeddings of the sphere in 3-space, the critical points have been classified:[1] they are all conformal transforms of minimal surfaces, the round sphere is the minimum, and all other critical values are integers greater than 4$\pi $. They are called Willmore surfaces. Willmore flow The Willmore flow is the geometric flow corresponding to the Willmore energy; it is an $L^{2}$-gradient flow. $e[{\mathcal {M}}]={\frac {1}{2}}\int _{\mathcal {M}}H^{2}\,\mathrm {d} A$ where H stands for the mean curvature of the manifold ${\mathcal {M}}$. Flow lines satisfy the differential equation: $\partial _{t}x(t)=-\nabla {\mathcal {W}}[x(t)]\,$ where $x$ is a point belonging to the surface. This flow leads to an evolution problem in differential geometry: the surface ${\mathcal {M}}$ is evolving in time to follow variations of steepest descent of the energy. Like surface diffusion it is a fourth-order flow, since the variation of the energy contains fourth derivatives. Applications • Cell membranes tend to position themselves so as to minimize Willmore energy.[2] • Willmore energy is used in constructing a class of optimal sphere eversions, the minimax eversions. See also • Willmore conjecture Notes 1. Bryant, Robert L. (1984), "A duality theorem for Willmore surfaces", Journal of Differential Geometry, 20 (1): 23–53, doi:10.4310/jdg/1214438991, MR 0772125. 2. Müller, Stefan; Röger, Matthias (May 2014). "Confined structures of least bending energy". Journal of Differential Geometry. 97 (1): 109–139. doi:10.4310/jdg/1404912105. References • Willmore, T. J. (1992), "A survey on Willmore immersions", Geometry and Topology of Submanifolds, IV (Leuven, 1991), River Edge, NJ: World Scientific, pp. 11–16, MR 1185712.
Wilson matrix Wilson matrix is the following $4\times 4$ matrix having integers as elements:[1][2][3][4][5] $W={\begin{bmatrix}5&7&6&5\\7&10&8&7\\6&8&10&9\\5&7&9&10\end{bmatrix}}$ This is the coefficient matrix of the following system of linear equations considered in a paper by J. Morris published in 1946:[6] ${\text{(S1)}}\quad {\begin{aligned}5x+7y+6z+5u&=23\\7x+10y+8z+7u&=32\\6x+8y+10z+9u&=33\\5x+7y+9z+10u&=31\end{aligned}}$ Morris ascribes the source of the set of equations to one T. S. Wilson but no details about Wilson have been provided. The particular system of equations was used by Morris to illustrate the concept of ill-conditioned system of equations. The matrix $W$ has been used as an example and for test purposes in many research papers and books over the years. John Todd has referred to $W$ as “the notorious matrix W of T. S. Wilson”.[1] Properties 1. $W$ is a symmetric matrix. 2. $W$ is positive definite. 3. The determinant of $W$ is $1$. 4. The inverse of $W$ is $W^{-1}={\begin{bmatrix}68&-41&-17&10\\-41&25&10&-6\\-17&10&5&-3\\10&-6&-3&2\end{bmatrix}}$ 5. The characteristic polynomial of $W$ is $\lambda ^{4}-35\lambda ^{3}+146\lambda ^{2}-100\lambda +1$. 6. The eigenvalues of $W$ are $\quad 0.01015004839789187,\quad 0.8431071498550294,\quad 3.858057455944953,\quad 30.28868534580213$. 7. Since $W$ is symmetric, the 2-norm condition number of $W$ is $\kappa _{2}(W)=({\text{max eigen value}})/({\text{min eigen value}})=30.28868534580213/0.01015004839789187=2984.09270167549$. 8. The solution of the system of equations $(S1)$ is $x=y=z=u=1$. 9. The Cholesky factorisation of $W$ is $W=R^{T}R$ where $R={\begin{bmatrix}{\sqrt {5}}&{\frac {7}{\sqrt {5}}}&{\frac {6}{\sqrt {5}}}&{\sqrt {5}}\\0&{\frac {1}{\sqrt {5}}}&-{\frac {2}{\sqrt {5}}}&0\\0&0&{\sqrt {2}}&{\frac {3}{\sqrt {2}}}\\0&0&0&{\frac {1}{\sqrt {2}}}\end{bmatrix}}$. 10. $W$ has the factorisation $W=LDL^{T}$ where $L={\begin{bmatrix}1&0&0&0\\{\frac {7}{5}}&1&0&0\\{\frac {6}{5}}&-2&1&0\\1&0&{\frac {3}{2}}&1\end{bmatrix}},\quad D={\begin{bmatrix}5&0&0&0\\0&{\frac {1}{5}}&0&0\\0&0&2&0\\0&0&0&{\frac {1}{2}}\end{bmatrix}}$. 11. $W$ has the factorisation $W=Z^{T}Z$ with $Z$ as the integer matrix[7] $Z={\begin{bmatrix}2&3&2&2\\1&1&2&1\\0&0&1&2\\0&0&1&1\end{bmatrix}}$. Research problems spawned by Wilson matrix A consideration of the condition number of the Wilson matrix has spawned several interesting research problems relating to condition numbers of matrices in certain special classes of matrices having some or all the special features of the Wilson matrix. In particular, the following special classes of matrices have been studied:[1] 1. $S=$ the set of $4\times 4$ nonsingular, symmetric matrices with integer entries between 1 and 10. 2. $P=$ the set of $4\times 4$ positive definite, symmetric matrices with integer entries between 1 and 10. An exhaustive computation of the condition numbers of the matrices in the above sets has yielded the following results: 1. Among the elements of $S$, the maximum condition number is $7.6119\times 10^{4}$ and this maximum is attained by the matrix ${\begin{bmatrix}2&7&10&10\\7&10&10&9\\10&10&10&1\\10&9&1&10\end{bmatrix}}$. 2. Among the elements of $P$, the maximum condition number is $3.5529\times 10^{4}$ and this maximum is attained by the matrix ${\begin{bmatrix}9&1&1&5\\1&10&1&9\\1&1&10&1\\5&9&1&10\end{bmatrix}}$. References 1. Nick Higham (June 2021). "What Is the Wilson Matrix?". What Is the Wilson Matrix?. Retrieved 24 May 2022. 2. Nicholas J. Higham and Matthew C. Lettington (2022). "Optimizing and Factorizing the Wilson Matrix". The American Mathematical Monthly. 129 (5): 454–465. doi:10.1080/00029890.2022.2038006. S2CID 233322415. Retrieved 24 May 2022. (An eprint of the paper is available here) 3. Cleve Moler. "Reviving Wilson's Matrix". Cleve’s Corner: Cleve Moler on Mathematics and Computing. MathWorks. Retrieved 24 May 2022. 4. Carl Erik Froberg (1969). Introduction to Numerical Analysis (2 ed.). Reading, Mass.: Addison-Wesley. 5. Robert T Gregory and David L Karney (1978). A Collection of Matrices for Testing Computational Algorithms. Huntington, New York: Robert Krieger Publishing Company. p. 57. 6. J Morris (1946). "An escalator process for the solution of linear simultaneous equations". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 37:265 (265): 106–120. doi:10.1080/14786444608561331. Retrieved 19 May 2022. 7. Nicholas J. Higham, Matthew C. Lettington, Karl Michael Schmidt (2021). "nteger matrix factorisations, superalgebras and the quadratic form obstruction". Linear Algebra and Its Applications. 622: 250–267. doi:10.1016/j.laa.2021.03.028. S2CID 232146938.{{cite journal}}: CS1 maint: multiple names: authors list (link)
Wilson polynomials In mathematics, Wilson polynomials are a family of orthogonal polynomials introduced by James A. Wilson (1980) that generalize Jacobi polynomials, Hahn polynomials, and Charlier polynomials. They are defined in terms of the generalized hypergeometric function and the Pochhammer symbols by $p_{n}(t^{2})=(a+b)_{n}(a+c)_{n}(a+d)_{n}{}_{4}F_{3}\left({\begin{matrix}-n&a+b+c+d+n-1&a-t&a+t\\a+b&a+c&a+d\end{matrix}};1\right).$ See also • Askey–Wilson polynomials are a q-analogue of Wilson polynomials. References • Wilson, James A. (1980), "Some hypergeometric orthogonal polynomials", SIAM Journal on Mathematical Analysis, 11 (4): 690–701, doi:10.1137/0511064, ISSN 0036-1410, MR 0579561 • Koornwinder, T.H. (2001) [1994], "Wilson polynomials", Encyclopedia of Mathematics, EMS Press
Wilson prime In number theory, a Wilson prime is a prime number $p$ such that $p^{2}$ divides $(p-1)!+1$, where "$!$ !} " denotes the factorial function; compare this with Wilson's theorem, which states that every prime $p$ divides $(p-1)!+1$. Both are named for 18th-century English mathematician John Wilson; in 1770, Edward Waring credited the theorem to Wilson,[1] although it had been stated centuries earlier by Ibn al-Haytham.[2] Wilson prime Named afterJohn Wilson No. of known terms3 First terms5, 13, 563 OEIS index • A007540 • Wilson primes: primes $p$ such that $(p-1)!\equiv -1\ (\operatorname {mod} {p^{2}})$ The only known Wilson primes are 5, 13, and 563 (sequence A007540 in the OEIS). Costa et al. write that "the case $p=5$ is trivial", and credit the observation that 13 is a Wilson prime to Mathews (1892).[3][4] Early work on these numbers included searches by N. G. W. H. Beeger and Emma Lehmer,[5][3][6] but 563 was not discovered until the early 1950s, when computer searches could be applied to the problem.[3][7][8] If any others exist, they must be greater than 2 × 1013.[3] It has been conjectured that infinitely many Wilson primes exist, and that the number of Wilson primes in an interval $[x,y]$ is about $\log \log _{x}y$.[9] Several computer searches have been done in the hope of finding new Wilson primes.[10][11][12] The Ibercivis distributed computing project includes a search for Wilson primes.[13] Another search was coordinated at the Great Internet Mersenne Prime Search forum.[14] Generalizations Wilson primes of order n Wilson's theorem can be expressed in general as $(n-1)!(p-n)!\equiv (-1)^{n}\ {\bmod {p}}$ for every integer $n\geq 1$ and prime $p\geq n$. Generalized Wilson primes of order n are the primes p such that $p^{2}$ divides $(n-1)!(p-n)!-(-1)^{n}$. It was conjectured that for every natural number n, there are infinitely many Wilson primes of order n. The smallest generalized Wilson primes of order $n$ are: 5, 2, 7, 10429, 5, 11, 17, ... (The next term > 1.4 × 107) (sequence A128666 in the OEIS) Near-Wilson primes pB 1282279+20 1306817−30 1308491−55 1433813−32 1638347−45 1640147−88 1647931+14 1666403+99 1750901+34 1851953−50 2031053−18 2278343+21 2313083+15 2695933−73 3640753+69 3677071−32 3764437−99 3958621+75 5062469+39 5063803+40 6331519+91 6706067+45 7392257+40 8315831+3 8871167−85 9278443−75 9615329+27 9756727+23 10746881−7 11465149−62 11512541−26 11892977−7 12632117−27 12893203−53 14296621+2 16711069+95 16738091+58 17879887+63 19344553−93 19365641+75 20951477+25 20972977+58 21561013−90 23818681+23 27783521−51 27812887+21 29085907+9 29327513+13 30959321+24 33187157+60 33968041+12 39198017−7 45920923−63 51802061+4 53188379−54 56151923−1 57526411−66 64197799+13 72818227−27 87467099−2 91926437−32 92191909+94 93445061−30 93559087−3 94510219−69 101710369−70 111310567+22 117385529−43 176779259+56 212911781−92 216331463−36 253512533+25 282361201+24 327357841−62 411237857−84 479163953−50 757362197−28 824846833+60 866006431−81 1227886151−51 1527857939−19 1636804231+64 1686290297+18 1767839071+8 1913042311−65 1987272877+5 2100839597−34 2312420701−78 2476913683+94 3542985241−74 4036677373−5 4271431471+83 4296847931+41 5087988391+51 5127702389+50 7973760941+76 9965682053−18 10242692519−97 11355061259−45 11774118061−1 12896325149+86 13286279999+52 20042556601+27 21950810731+93 23607097193+97 24664241321+46 28737804211−58 35525054743+26 41659815553+55 42647052491+10 44034466379+39 60373446719−48 64643245189−21 66966581777+91 67133912011+9 80248324571+46 80908082573−20 100660783343+87 112825721339+70 231939720421+41 258818504023+4 260584487287−52 265784418461−78 298114694431+82 A prime $p$ satisfying the congruence $(p-1)!\equiv -1+Bp\ (\operatorname {mod} {p^{2}})$ with small $\|B\|$ can be called a near-Wilson prime. Near-Wilson primes with $B=0$ are bona fide Wilson primes. The table on the right lists all such primes with $\|B\|\leq 100$ from 106 up to 4×1011.[3] Wilson numbers A Wilson number is a natural number $n$ such that $W(n)\equiv 0\ (\operatorname {mod} {n^{2}})$, where $W(n)=\pm 1+\prod _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}{k},$ and where the $\pm 1$ term is positive if and only if $n$ has a primitive root and negative otherwise.[15] For every natural number $n$, $W(n)$ is divisible by $n$, and the quotients (called generalized Wilson quotients) are listed in OEIS: A157249. The Wilson numbers are 1, 5, 13, 563, 5971, 558771, 1964215, 8121909, 12326713, 23025711, 26921605, 341569806, 399292158, ... (sequence A157250 in the OEIS) If a Wilson number $n$ is prime, then $n$ is a Wilson prime. There are 13 Wilson numbers up to 5×108.[16] See also • PrimeGrid • Table of congruences • Wall–Sun–Sun prime • Wieferich prime • Wolstenholme prime References 1. Edward Waring, Meditationes Algebraicae (Cambridge, England: 1770), page 218 (in Latin). In the third (1782) edition of Waring's Meditationes Algebraicae, Wilson's theorem appears as problem 5 on page 380. On that page, Waring states: "Hanc maxime elegantem primorum numerorum proprietatem invenit vir clarissimus, rerumque mathematicarum peritissimus Joannes Wilson Armiger." (A man most illustrious and most skilled in mathematics, Squire John Wilson, found this most elegant property of prime numbers.) 2. O'Connor, John J.; Robertson, Edmund F. "Abu Ali al-Hasan ibn al-Haytham". MacTutor History of Mathematics Archive. University of St Andrews. 3. Costa, Edgar; Gerbicz, Robert; Harvey, David (2014). "A search for Wilson primes". Mathematics of Computation. 83 (290): 3071–3091. arXiv:1209.3436. doi:10.1090/S0025-5718-2014-02800-7. MR 3246824. S2CID 6738476. 4. Mathews, George Ballard (1892). "Example 15". Theory of Numbers, Part 1. Deighton & Bell. p. 318. 5. Lehmer, Emma (April 1938). "On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson" (PDF). Annals of Mathematics. 39 (2): 350–360. doi:10.2307/1968791. JSTOR 1968791. Retrieved 8 March 2011. 6. Beeger, N. G. W. H. (1913–1914). "Quelques remarques sur les congruences $r^{p-1}\equiv 1\ (\operatorname {mod} {p^{2}})$ et $(p-1)!\equiv -1\ (\operatorname {mod} {p^{2}})$". The Messenger of Mathematics. 43: 72–84. 7. Wall, D. D. (October 1952). "Unpublished mathematical tables" (PDF). Mathematical Tables and Other Aids to Computation. 6 (40): 238. doi:10.2307/2002270. JSTOR 2002270. 8. Goldberg, Karl (1953). "A table of Wilson quotients and the third Wilson prime". J. London Math. Soc. 28 (2): 252–256. doi:10.1112/jlms/s1-28.2.252. 9. The Prime Glossary: Wilson prime 10. McIntosh, R. (9 March 2004). "WILSON STATUS (Feb. 1999)". E-Mail to Paul Zimmermann. Retrieved 6 June 2011. 11. Crandall, Richard E.; Dilcher, Karl; Pomerance, Carl (1997). "A search for Wieferich and Wilson primes". Math. Comput. 66 (217): 433–449. Bibcode:1997MaCom..66..433C. doi:10.1090/S0025-5718-97-00791-6. See p. 443. 12. Ribenboim, P.; Keller, W. (2006). Die Welt der Primzahlen: Geheimnisse und Rekorde (in German). Berlin Heidelberg New York: Springer. p. 241. ISBN 978-3-540-34283-0. 13. "Ibercivis site". Archived from the original on 2012-06-20. Retrieved 2011-03-10. 14. Distributed search for Wilson primes (at mersenneforum.org) 15. see Gauss's generalization of Wilson's theorem 16. Agoh, Takashi; Dilcher, Karl; Skula, Ladislav (1998). "Wilson quotients for composite moduli" (PDF). Math. Comput. 67 (222): 843–861. Bibcode:1998MaCom..67..843A. doi:10.1090/S0025-5718-98-00951-X. Further reading • Crandall, Richard E.; Pomerance, Carl (2001). Prime Numbers: A Computational Perspective. Springer-Verlag. p. 29. ISBN 978-0-387-94777-8. • Pearson, Erna H. (1963). "On the Congruences (p − 1)! ≡ −1 and 2p−1 ≡ 1 (mod p2)" (PDF). Math. Comput. 17: 194–195. External links • The Prime Glossary: Wilson prime • Weisstein, Eric W. "Wilson prime". MathWorld. • Status of the search for Wilson primes Prime number classes By formula • Fermat (22n + 1) • Mersenne (2p − 1) • Double Mersenne (22p−1 − 1) • Wagstaff (2p + 1)/3 • Proth (k·2n + 1) • Factorial (n! ± 1) • Primorial (pn# ± 1) • Euclid (pn# + 1) • Pythagorean (4n + 1) • Pierpont (2m·3n + 1) • Quartan (x4 + y4) • Solinas (2m ± 2n ± 1) • Cullen (n·2n + 1) • Woodall (n·2n − 1) • Cuban (x3 − y3)/(x − y) • Leyland (xy + yx) • Thabit (3·2n − 1) • Williams ((b−1)·bn − 1) • Mills (⌊A3n⌋) By integer sequence • Fibonacci • Lucas • Pell • Newman–Shanks–Williams • Perrin • Partitions • Bell • Motzkin By property • Wieferich (pair) • Wall–Sun–Sun • Wolstenholme • Wilson • Lucky • Fortunate • Ramanujan • Pillai • Regular • Strong • Stern • Supersingular (elliptic curve) • Supersingular (moonshine theory) • Good • Super • Higgs • Highly cototient • Unique Base-dependent • Palindromic • Emirp • Repunit (10n − 1)/9 • Permutable • Circular • Truncatable • Minimal • Delicate • Primeval • Full reptend • Unique • Happy • Self • Smarandache–Wellin • Strobogrammatic • Dihedral • Tetradic Patterns • Twin (p, p + 2) • Bi-twin chain (n ± 1, 2n ± 1, 4n ± 1, …) • Triplet (p, p + 2 or p + 4, p + 6) • Quadruplet (p, p + 2, p + 6, p + 8) • k-tuple • Cousin (p, p + 4) • Sexy (p, p + 6) • Chen • Sophie Germain/Safe (p, 2p + 1) • Cunningham (p, 2p ± 1, 4p ± 3, 8p ± 7, ...) • Arithmetic progression (p + a·n, n = 0, 1, 2, 3, ...) • Balanced (consecutive p − n, p, p + n) By size • Mega (1,000,000+ digits) • Largest known • list Complex numbers • Eisenstein prime • Gaussian prime Composite numbers • Pseudoprime • Catalan • Elliptic • Euler • Euler–Jacobi • Fermat • Frobenius • Lucas • Somer–Lucas • Strong • Carmichael number • Almost prime • Semiprime • Sphenic number • Interprime • Pernicious Related topics • Probable prime • Industrial-grade prime • Illegal prime • Formula for primes • Prime gap First 60 primes • 2 • 3 • 5 • 7 • 11 • 13 • 17 • 19 • 23 • 29 • 31 • 37 • 41 • 43 • 47 • 53 • 59 • 61 • 67 • 71 • 73 • 79 • 83 • 89 • 97 • 101 • 103 • 107 • 109 • 113 • 127 • 131 • 137 • 139 • 149 • 151 • 157 • 163 • 167 • 173 • 179 • 181 • 191 • 193 • 197 • 199 • 211 • 223 • 227 • 229 • 233 • 239 • 241 • 251 • 257 • 263 • 269 • 271 • 277 • 281 List of prime numbers
Wilson quotient The Wilson quotient W(p) is defined as: $W(p)={\frac {(p-1)!+1}{p}}$ If p is a prime number, the quotient is an integer by Wilson's theorem; moreover, if p is composite, the quotient is not an integer. If p divides W(p), it is called a Wilson prime. The integer values of W(p) are (sequence A007619 in the OEIS): W(2) = 1 W(3) = 1 W(5) = 5 W(7) = 103 W(11) = 329891 W(13) = 36846277 W(17) = 1230752346353 W(19) = 336967037143579 ... It is known that[1] $W(p)\equiv B_{2(p-1)}-B_{p-1}{\pmod {p}},$ $p-1+ptW(p)\equiv pB_{t(p-1)}{\pmod {p^{2}}},$ where $B_{k}$ is the k-th Bernoulli number. Note that the first relation comes from the second one by subtraction, after substituting $t=1$ and $t=2$. See also • Fermat quotient References 1. Lehmer, Emma (1938). "On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson". Annals of Mathematics. 39 (2): 350–360. doi:10.2307/1968791. JSTOR 1968791. External links • MathWorld: Wilson Quotient
Wilson's theorem In algebra and number theory, Wilson's theorem states that a natural number n > 1 is a prime number if and only if the product of all the positive integers less than n is one less than a multiple of n. That is (using the notations of modular arithmetic), the factorial $(n-1)!=1\times 2\times 3\times \cdots \times (n-1)$ satisfies $(n-1)!\ \equiv \;-1{\pmod {n}}$ exactly when n is a prime number. In other words, any number n is a prime number if, and only if, (n − 1)! + 1 is divisible by n.[1] History This theorem was stated by Ibn al-Haytham (c. 1000 AD),[2] and, in the 18th century, by the English mathematician John Wilson.[3] Edward Waring announced the theorem in 1770, although neither he nor his student Wilson could prove it. Lagrange gave the first proof in 1771.[4] There is evidence that Leibniz was also aware of the result a century earlier, but he never published it.[5] Example For each of the values of n from 2 to 30, the following table shows the number (n − 1)! and the remainder when (n − 1)! is divided by n. (In the notation of modular arithmetic, the remainder when m is divided by n is written m mod n.) The background color is blue for prime values of n, gold for composite values. Table of factorial and its remainder modulo n $n$$(n-1)!$ (sequence A000142 in the OEIS) $(n-1)!\ {\bmod {\ }}n$ (sequence A061006 in the OEIS) 211 322 462 5244 61200 77206 850400 9403200 103628800 11362880010 12399168000 1347900160012 1462270208000 15871782912000 1613076743680000 172092278988800016 183556874280960000 19640237370572800018 201216451004088320000 2124329020081766400000 22510909421717094400000 23112400072777760768000022 24258520167388849766400000 256204484017332394393600000 26155112100433309859840000000 274032914611266056355840000000 28108888694504183521607680000000 2930488834461171386050150400000028 3088417619937397019545436160000000 Proofs The proofs (for prime moduli) below use the fact that the residue classes modulo a prime number are a field—see the article prime field for more details.[6] Lagrange's theorem, which states that in any field a polynomial of degree n has at most n roots, is needed for all the proofs. Composite modulus If n is composite it is divisible by some prime number q, where 2 ≤ q ≤ n − 2. Because $q$ divides $n$, let $n=qk$ for some integer $k$. Suppose for the sake of contradiction that $(n-1)!$ were congruent to −1 (mod n) where n is composite. Then (n-1)! would also be congruent to −1 (mod q) as $(n-1)!\equiv -1\ ({\text{mod}}\ n)$ implies that $(n-1)!=nm-1=(qk)m-1=q(km)-1$ for some integer $m$ which shows (n-1)! being congruent to -1 (mod q). But (n − 1)! ≡ 0 (mod q) by the fact that q is a term in (n-1)! making (n-1)! a multiple of q. A contradiction is now reached. In fact, more is true. With the sole exception of 4, where 3! = 6 ≡ 2 (mod 4), if n is composite then (n − 1)! is congruent to 0 (mod n). The proof is divided into two cases: First, if n can be factored as the product of two unequal numbers, n = ab, where 2 ≤ a < b ≤ n − 2, then both a and b will appear in the product 1 × 2 × ... × (n − 1) = (n − 1)! and (n − 1)! will be divisible by n. If n has no such factorization, then it must be the square of some prime q, q > 2. But then 2q < q2 = n, both q and 2q will be factors of (n − 1)!, and again n divides (n − 1)!. Elementary proof The result is trivial when p = 2, so assume p is an odd prime, p ≥ 3. Since the residue classes (mod p) are a field, every non-zero a has a unique multiplicative inverse, a−1. Lagrange's theorem implies that the only values of a for which a ≡ a−1 (mod p) are a ≡ ±1 (mod p) (because the congruence a2 ≡ 1 can have at most two roots (mod p)). Therefore, with the exception of ±1, the factors of (p − 1)! can be arranged in disjoint pairs such that product of each pair is congruent to 1 modulo p. This proves Wilson's theorem. For example, for p = 11, one has $10!=[(1\cdot 10)]\cdot [(2\cdot 6)(3\cdot 4)(5\cdot 9)(7\cdot 8)]\equiv [-1]\cdot [1\cdot 1\cdot 1\cdot 1]\equiv -1{\pmod {11}}.$ Proof using Fermat's little theorem Again, the result is trivial for p = 2, so suppose p is an odd prime, p ≥ 3. Consider the polynomial $g(x)=(x-1)(x-2)\cdots (x-(p-1)).$ g has degree p − 1, leading term xp − 1, and constant term (p − 1)!. Its p − 1 roots are 1, 2, ..., p − 1. Now consider $h(x)=x^{p-1}-1.$ h also has degree p − 1 and leading term xp − 1. Modulo p, Fermat's little theorem says it also has the same p − 1 roots, 1, 2, ..., p − 1. Finally, consider $f(x)=g(x)-h(x).$ f has degree at most p − 2 (since the leading terms cancel), and modulo p also has the p − 1 roots 1, 2, ..., p − 1. But Lagrange's theorem says it cannot have more than p − 2 roots. Therefore, f must be identically zero (mod p), so its constant term is (p − 1)! + 1 ≡ 0 (mod p). This is Wilson's theorem. Proof using the Sylow theorems It is possible to deduce Wilson's theorem from a particular application of the Sylow theorems. Let p be a prime. It is immediate to deduce that the symmetric group $S_{p}$ has exactly $(p-1)!$ elements of order p, namely the p-cycles $C_{p}$. On the other hand, each Sylow p-subgroup in $S_{p}$ is a copy of $C_{p}$. Hence it follows that the number of Sylow p-subgroups is $n_{p}=(p-2)!$. The third Sylow theorem implies $(p-2)!\equiv 1{\pmod {p}}.$ Multiplying both sides by (p − 1) gives $(p-1)!\equiv p-1\equiv -1{\pmod {p}},$ that is, the result. Applications Primality tests In practice, Wilson's theorem is useless as a primality test because computing (n − 1)! modulo n for large n is computationally complex, and much faster primality tests are known (indeed, even trial division is considerably more efficient). Used in the other direction, to determine the primality of the successors of large factorials, it is indeed a very fast and effective method. This is of limited utility, however. Quadratic residues Using Wilson's Theorem, for any odd prime p = 2m + 1, we can rearrange the left hand side of $1\cdot 2\cdots (p-1)\ \equiv \ -1\ {\pmod {p}}$ to obtain the equality $1\cdot (p-1)\cdot 2\cdot (p-2)\cdots m\cdot (p-m)\ \equiv \ 1\cdot (-1)\cdot 2\cdot (-2)\cdots m\cdot (-m)\ \equiv \ -1{\pmod {p}}.$ This becomes $\prod _{j=1}^{m}\ j^{2}\ \equiv (-1)^{m+1}{\pmod {p}}$ or $(m!)^{2}\equiv (-1)^{m+1}{\pmod {p}}.$ We can use this fact to prove part of a famous result: for any prime p such that p ≡ 1 (mod 4), the number (−1) is a square (quadratic residue) mod p. For this, suppose p = 4k + 1 for some integer k. Then we can take m = 2k above, and we conclude that (m!)2 is congruent to (−1) (mod p). Formulas for primes Wilson's theorem has been used to construct formulas for primes, but they are too slow to have practical value. p-adic gamma function Wilson's theorem allows one to define the p-adic gamma function. Gauss' generalization Gauss’ generalization of Wilson’s Theorem states that if $n$ is four, an odd prime power, or twice an odd prime power, then the product of relatively prime integers less than itself add one is divisible by $n$. It goes further to say that otherwise, the same product subtract one is divisible by $n$. To state Gauss' Generalization of Wilson's Theorem, we use the Euler's totient function, denoted $\varphi (n)$, which is defined as the number of positive integers less than or equal to $n$ which are also relatively prime with $n$. Let's call such numbers $a_{i}$, where $\gcd(a_{i},n)=1$. Gauss proved given an odd prime $p$ and some integer $\alpha >0$, then $\prod _{k=1}^{\varphi (n)}\!\!a_{k}\ ={\begin{cases}-1{\pmod {n}}&{\text{if }}n=4,\;p^{\alpha },\;2p^{\alpha }\\\;\;\,1{\pmod {n}}&{\text{otherwise}}\end{cases}}$. First, let's note this is the proof for cases $n>2$, since the results are trivial for $n=\{1,2\}$. For all $a_{i}$, we know there exist some $a_{j}$, where $i\neq j$ and $\gcd(a_{j},n)=1$, such that $a_{i}a_{j}=1$. This allows us to pair each of the elements together with its inverse. We are left now with $a_{i}$ being its own inverse. So in other words $a_{i}$ is a root of $f(x)=x^{2}-1$ in $\mathbb {Z} /n\mathbb {Z} $, and $f(x)=(x-1)(x+1)$, in the polynomial ring $\mathbb {Z} /n\mathbb {Z} [X]$. If $a_{i}$ is a root, it follows that $n-a_{i}$ is also a root. Our objective is to show that the number of roots is divisible by four, unless $n=4,n=p^{\alpha }$, or $n=2p^{\alpha }$. Let's consider $n=2$. Then we notice we have one root since $1\equiv -1{\pmod {2}}$. Consider $n=4$. Then, it is clear there are two roots, specifically, $x\equiv 1{\pmod {4}}$ and $x\equiv -1{\pmod {4}}$. Say $n=p^{\alpha }$. It is again clear there are two solutions. We now consider $n=2^{\beta },\beta >2$. If one of the factors of $f(x)$ is divisible by 2, so is the other. Take the factor $(x+1)$ to be divisible by $2^{1}$. Then, it follows that there are 4 distinct roots of $f(x)$, namely $x\equiv 1{\pmod {n}}$, $x\equiv 1+2^{\beta -1}{\pmod {n}}$, $x\equiv -1{\pmod {n}}$, and $x\equiv -1-2^{\beta -1}{\pmod {n}}$, when $n=2^{\beta },\beta >2$. Finally, let's look at the general case where $n=2^{\beta }p_{1}{^{\gamma _{1}}}p_{2}{^{\gamma _{2}}}\dots p_{k}{^{\gamma _{k}}}$. We find 2 roots of $f(x)$ over each $\mathbb {Z} /2^{\beta }\mathbb {Z} $ and $\mathbb {Z} /p_{r}^{\gamma _{r}}\mathbb {Z} $, except when $\beta >2$. Using the Chinese remainder theorem, we find that when $n$ is not divisible by 2, we have a total of $2^{k}$ solutions of $f(x)$. Assuming $\beta =1$, in $\mathbb {Z} /2\mathbb {Z} $, we have one root, so we still have a total of $2^{k}$ solutions. When $\beta =2$, we have 2 roots in $\mathbb {Z} /4\mathbb {Z} $, so there are a total of $2^{k+1}$ roots of $f(x)$. For all cases where $\beta >2$, there are 4 roots in $\mathbb {Z} /2^{\beta }\mathbb {Z} $ with a total of $2^{k+2}$ solutions. This shows that the number of roots are divisible by 4, unless $n=4,n=p^{\alpha }$, or $n=2p^{\alpha }$. Say $a_{i}$ is a root of $f(x)$ in $\mathbb {Z} /n\mathbb {Z} $. Then $a_{i}(n-a_{i})=-a_{i}^{2}=-1$. So, if the number of roots of $f(x)$ is divisible by 4, then we can say the product of the roots if 1. Otherwise, we can say the product is -1. So we can conclude that $\prod _{k=1}^{\varphi (n)}\!\!a_{k}\ ={\begin{cases}-1{\pmod {n}}&{\text{if }}n=4,\;p^{\alpha },\;2p^{\alpha }\\\;\;\,1{\pmod {n}}&{\text{otherwise}}\end{cases}}$. See also • Wilson prime • Table of congruences Notes 1. The Universal Book of Mathematics. David Darling, p. 350. 2. O'Connor, John J.; Robertson, Edmund F., "Abu Ali al-Hasan ibn al-Haytham", MacTutor History of Mathematics Archive, University of St Andrews 3. Edward Waring, Meditationes Algebraicae (Cambridge, England: 1770), page 218 (in Latin). In the third (1782) edition of Waring's Meditationes Algebraicae, Wilson's theorem appears as problem 5 on page 380. On that page, Waring states: "Hanc maxime elegantem primorum numerorum proprietatem invenit vir clarissimus, rerumque mathematicarum peritissimus Joannes Wilson Armiger." (A man most illustrious and most skilled in mathematics, Squire John Wilson, found this most elegant property of prime numbers.) 4. Joseph Louis Lagrange, "Demonstration d'un théorème nouveau concernant les nombres premiers" (Proof of a new theorem concerning prime numbers), Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres (Berlin), vol. 2, pages 125–137 (1771). 5. Giovanni Vacca (1899) "Sui manoscritti inediti di Leibniz" (On unpublished manuscripts of Leibniz), Bollettino di bibliografia e storia delle scienze matematiche ... (Bulletin of the bibliography and history of mathematics), vol. 2, pages 113–116; see page 114 (in Italian). Vacca quotes from Leibniz's mathematical manuscripts kept at the Royal Public Library in Hanover (Germany), vol. 3 B, bundle 11, page 10: Original : Inoltre egli intravide anche il teorema di Wilson, come risulta dall'enunciato seguente: "Productus continuorum usque ad numerum qui antepraecedit datum divisus per datum relinquit 1 (vel complementum ad unum?) si datus sit primitivus. Si datus sit derivativus relinquet numerum qui cum dato habeat communem mensuram unitate majorem." Egli non giunse pero a dimostrarlo. Translation : In addition, he [Leibniz] also glimpsed Wilson's theorem, as shown in the following statement: "The product of all integers preceding the given integer, when divided by the given integer, leaves 1 (or the complement of 1?) if the given integer be prime. If the given integer be composite, it leaves a number which has a common factor with the given integer [which is] greater than one." However, he didn't succeed in proving it. See also: Giuseppe Peano, ed., Formulaire de mathématiques, vol. 2, no. 3, page 85 (1897). 6. Landau, two proofs of thm. 78 References The Disquisitiones Arithmeticae has been translated from Gauss's Ciceronian Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes. • Gauss, Carl Friedrich; Clarke, Arthur A. (1986), Disquisitiones Arithemeticae (2nd corrected ed.), New York: Springer, ISBN 0-387-96254-9(translated into English){{citation}}: CS1 maint: postscript (link). • Gauss, Carl Friedrich; Maser, H. (1965), Untersuchungen über hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (2nd ed.), New York: Chelsea, ISBN 0-8284-0191-8(translated into German){{citation}}: CS1 maint: postscript (link). • Landau, Edmund (1966), Elementary Number Theory, New York: Chelsea. • Ore, Øystein (1988). Number Theory and its History. Dover. pp. 259–271. ISBN 0-486-65620-9. External links • "Wilson theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994] • Weisstein, Eric W. "Wilson's Theorem". MathWorld. • Mizar system proof: http://mizar.org/version/current/html/nat_5.html#T22 • [1] 1. Ohana, Andrew. "A Generalization of Wilson's Theorem" (PDF).
Wim Blok Willem Johannes "Wim" Blok (1947–2003) was a Dutch logician who made major contributions to algebraic logic, universal algebra, and modal logic. His important achievements over the course of his career include "a brilliant demonstration of the fact that various techniques and results that originated in universal algebra can be used to prove significant and deep theorems in modal logic."[1] Blok began his career in 1973 as an algebraist investigating the varieties of interior algebras at the University of Illinois at Chicago. Following the 1976 completion of his Ph.D. on that topic, he continued on to study more general varieties of modal algebras. As an algebraist, Blok "was recognised by the modal logic community as one of the most influential modal logicians" by the end of the 1970s.[2] He published many papers in the Reports on Mathematical Logic, served as a member on their editorial board, and was one of their guest editors.[1] Along with Don Pigozzi, Wim Blok co-authored the monograph Algebraizable Logics, which began the field now known as abstract algebraic logic.[3] He died in a car accident on November 30, 2003.[1] See also • Abstract algebraic logic • Blok–Esakia isomorphism • Leibniz operator References 1. Font, Josep Maria (May 2006). "In Memory of Wim Blok". Reports on Mathematical Logic (Special Issue). CiteSeerX 10.1.1.103.2741. 2. Rautenberg, W.; Wolter, F.; Zakharyaschev, M. (June 2006). "Willem Blok and Modal Logic". Studia Logica. 83 (1–3): 15–30. doi:10.1007/s11225-006-8296-2. S2CID 17091670. Retrieved June 24, 2016 – via ResearchGate. 3. Raftery, James G. (March 2004). "Willem Blok's Work in Algebraic Logic". Studia Logica. 76 (2): 155–160. doi:10.1023/B:STUD.0000032083.45504.62. JSTOR 20016583. S2CID 37825139. Authority control International • ISNI • VIAF National • Norway • Germany • Israel • United States • Czech Republic • Netherlands Academics • CiNii • zbMATH • 2 Other • IdRef
Wiman's sextic In mathematics, Wiman's sextic is a degree 6 plane curve with four nodes studied by Anders Wiman (1896). It is given by the equation (in homogeneous coordinates) $x^{6}+y^{6}+z^{6}+(x^{2}+y^{2}+z^{2})(x^{4}+y^{4}+z^{4})=12x^{2}y^{2}z^{2}$ Its normalization is a genus 6 curve with automorphism group isomorphic to the symmetric group S5. References • Inoue, Naoki; Kato, Fumiharu (2005), "On the geometry of Wiman's sextic", Journal of Mathematics of Kyoto University, 45 (4): 743–757, doi:10.1215/kjm/1250281655, ISSN 0023-608X, MR 2226628 • Wiman, A. (1896), "Zur Theorie der endlichen Gruppen von birationalen Transformationen in der Ebene", Mathematische Annalen, 48 (1–2): 195–240, doi:10.1007/BF01446342, S2CID 123516972
Winding number In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of turns. For certain open plane curves, the number of turns may be non-integer. The winding number depends on the orientation of the curve, and it is negative if the curve travels around the point clockwise. Not to be confused with Map winding number. Mathematical analysis → Complex analysis Complex analysis Complex numbers • Real number • Imaginary number • Complex plane • Complex conjugate • Unit complex number Complex functions • Complex-valued function • Analytic function • Holomorphic function • Cauchy–Riemann equations • Formal power series Basic theory • Zeros and poles • Cauchy's integral theorem • Local primitive • Cauchy's integral formula • Winding number • Laurent series • Isolated singularity • Residue theorem • Conformal map • Schwarz lemma • Harmonic function • Laplace's equation Geometric function theory People • Augustin-Louis Cauchy • Leonhard Euler • Carl Friedrich Gauss • Jacques Hadamard • Kiyoshi Oka • Bernhard Riemann • Karl Weierstrass • Mathematics portal Winding numbers are fundamental objects of study in algebraic topology, and they play an important role in vector calculus, complex analysis, geometric topology, differential geometry, and physics (such as in string theory). Intuitive description Suppose we are given a closed, oriented curve in the xy plane. We can imagine the curve as the path of motion of some object, with the orientation indicating the direction in which the object moves. Then the winding number of the curve is equal to the total number of counterclockwise turns that the object makes around the origin. When counting the total number of turns, counterclockwise motion counts as positive, while clockwise motion counts as negative. For example, if the object first circles the origin four times counterclockwise, and then circles the origin once clockwise, then the total winding number of the curve is three. Using this scheme, a curve that does not travel around the origin at all has winding number zero, while a curve that travels clockwise around the origin has negative winding number. Therefore, the winding number of a curve may be any integer. The following pictures show curves with winding numbers between −2 and 3: $\cdots $ −2 −1 0 $\cdots $ 1 2 3 Formal definition Let $\gamma :[0,1]\to \mathbb {C} \setminus \{a\}$ :[0,1]\to \mathbb {C} \setminus \{a\}} be a continuous closed path on the plane minus one point. The winding number of $\gamma $ around $a$ is the integer ${\text{wind}}(\gamma ,a)=s(1)-s(0),$ where $(\rho ,s)$ is the path written in polar coordinates, i.e. the lifted path through the covering map $p:\mathbb {R} _{>0}\times \mathbb {R} \to \mathbb {C} \setminus \{a\}:(\rho _{0},s_{0})\mapsto a+\rho _{0}e^{i2\pi s_{0}}.$ The winding number is well defined because of the existence and uniqueness of the lifted path (given the starting point in the covering space) and because all the fibers of $p$ are of the form $\rho _{0}\times (s_{0}+\mathbb {Z} )$ (so the above expression does not depend on the choice of the starting point). It is an integer because the path is closed. Alternative definitions Winding number is often defined in different ways in various parts of mathematics. All of the definitions below are equivalent to the one given above: Alexander numbering A simple combinatorial rule for defining the winding number was proposed by August Ferdinand Möbius in 1865[1] and again independently by James Waddell Alexander II in 1928.[2] Any curve partitions the plane into several connected regions, one of which is unbounded. The winding numbers of the curve around two points in the same region are equal. The winding number around (any point in) the unbounded region is zero. Finally, the winding numbers for any two adjacent regions differ by exactly 1; the region with the larger winding number appears on the left side of the curve (with respect to motion down the curve). Differential geometry In differential geometry, parametric equations are usually assumed to be differentiable (or at least piecewise differentiable). In this case, the polar coordinate θ is related to the rectangular coordinates x and y by the equation: $d\theta ={\frac {1}{r^{2}}}\left(x\,dy-y\,dx\right)\quad {\text{where }}r^{2}=x^{2}+y^{2}.$ Which is found by differentiating the following definition for θ: $\theta (t)=\arctan {\bigg (}{\frac {y(t)}{x(t)}}{\bigg )}$ By the fundamental theorem of calculus, the total change in θ is equal to the integral of dθ. We can therefore express the winding number of a differentiable curve as a line integral: ${\text{wind}}(\gamma ,0)={\frac {1}{2\pi }}\oint _{\gamma }\,\left({\frac {x}{r^{2}}}\,dy-{\frac {y}{r^{2}}}\,dx\right).$ The one-form dθ (defined on the complement of the origin) is closed but not exact, and it generates the first de Rham cohomology group of the punctured plane. In particular, if ω is any closed differentiable one-form defined on the complement of the origin, then the integral of ω along closed loops gives a multiple of the winding number. Complex analysis Winding numbers play a very important role throughout complex analysis (c.f. the statement of the residue theorem). In the context of complex analysis, the winding number of a closed curve $\gamma $ in the complex plane can be expressed in terms of the complex coordinate z = x + iy. Specifically, if we write z = reiθ, then $dz=e^{i\theta }dr+ire^{i\theta }d\theta $ and therefore ${\frac {dz}{z}}={\frac {dr}{r}}+i\,d\theta =d[\ln r]+i\,d\theta .$ As $\gamma $ is a closed curve, the total change in $\ln(r)$ is zero, and thus the integral of $ {\frac {dz}{z}}$ is equal to $i$ multiplied by the total change in $\theta $. Therefore, the winding number of closed path $\gamma $ about the origin is given by the expression[3] ${\frac {1}{2\pi i}}\oint _{\gamma }{\frac {dz}{z}}\,.$ More generally, if $\gamma $ is a closed curve parameterized by $t\in [\alpha ,\beta ]$, the winding number of $\gamma $ about $z_{0}$, also known as the index of $z_{0}$ with respect to $\gamma $, is defined for complex $z_{0}\notin \gamma ([\alpha ,\beta ])$ as[4] $\mathrm {Ind} _{\gamma }(z_{0})={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {d\zeta }{\zeta -z_{0}}}={\frac {1}{2\pi i}}\int _{\alpha }^{\beta }{\frac {\gamma '(t)}{\gamma (t)-z_{0}}}dt.$ This is a special case of the famous Cauchy integral formula. Some of the basic properties of the winding number in the complex plane are given by the following theorem:[5] Theorem. Let $\gamma :[\alpha ,\beta ]\to \mathbb {C} $ :[\alpha ,\beta ]\to \mathbb {C} } be a closed path and let $\Omega $ be the set complement of the image of $\gamma $, that is, $\Omega :=\mathbb {C} \setminus \gamma ([\alpha ,\beta ])$ :=\mathbb {C} \setminus \gamma ([\alpha ,\beta ])} . Then the index of $z$ with respect to $\gamma $, $\mathrm {Ind} _{\gamma }:\Omega \to \mathbb {C} ,\ \ z\mapsto {\frac {1}{2\pi i}}\oint _{\gamma }{\frac {d\zeta }{\zeta -z}},$ is (i) integer-valued, i.e., $\mathrm {Ind} _{\gamma }(z)\in \mathbb {Z} $ for all $z\in \Omega $; (ii) constant over each component (i.e., maximal connected subset) of $\Omega $; and (iii) zero if $z$ is in the unbounded component of $\Omega $. As an immediate corollary, this theorem gives the winding number of a circular path $\gamma $ about a point $z$. As expected, the winding number counts the number of (counterclockwise) loops $\gamma $ makes around $z$: Corollary. If $\gamma $ is the path defined by $\gamma (t)=a+re^{int},\ \ 0\leq t\leq 2\pi ,\ \ n\in \mathbb {Z} $, then $\mathrm {Ind} _{\gamma }(z)={\begin{cases}n,&\|z-a\|<r;\\0,&\|z-a\|>r.\end{cases}}$ Topology In topology, the winding number is an alternate term for the degree of a continuous mapping. In physics, winding numbers are frequently called topological quantum numbers. In both cases, the same concept applies. The above example of a curve winding around a point has a simple topological interpretation. The complement of a point in the plane is homotopy equivalent to the circle, such that maps from the circle to itself are really all that need to be considered. It can be shown that each such map can be continuously deformed to (is homotopic to) one of the standard maps $S^{1}\to S^{1}:s\mapsto s^{n}$, where multiplication in the circle is defined by identifying it with the complex unit circle. The set of homotopy classes of maps from a circle to a topological space form a group, which is called the first homotopy group or fundamental group of that space. The fundamental group of the circle is the group of the integers, Z; and the winding number of a complex curve is just its homotopy class. Maps from the 3-sphere to itself are also classified by an integer which is also called the winding number or sometimes Pontryagin index. Turning number One can also consider the winding number of the path with respect to the tangent of the path itself. As a path followed through time, this would be the winding number with respect to the origin of the velocity vector. In this case the example illustrated at the beginning of this article has a winding number of 3, because the small loop is counted. This is only defined for immersed paths (i.e., for differentiable paths with nowhere vanishing derivatives), and is the degree of the tangential Gauss map. This is called the turning number, rotation number,[6] rotation index[7] or index of the curve, and can be computed as the total curvature divided by 2π. Polygons Further information: Density (polytope) § Polygons In polygons, the turning number is referred to as the polygon density. For convex polygons, and more generally simple polygons (not self-intersecting), the density is 1, by the Jordan curve theorem. By contrast, for a regular star polygon {p/q}, the density is q. Space curves Turning number cannot be defined for space curves as degree requires matching dimensions. However, for locally convex, closed space curves, one can define tangent turning sign as $(-1)^{d}$, where $d$ is the turning number of the stereographic projection of its tangent indicatrix. Its two values correspond to the two non-degenerate homotopy classes of locally convex curves.[8] [9] Winding number and Heisenberg ferromagnet equations The winding number is closely related with the (2 + 1)-dimensional continuous Heisenberg ferromagnet equations and its integrable extensions: the Ishimori equation etc. Solutions of the last equations are classified by the winding number or topological charge (topological invariant and/or topological quantum number). Applications Point in polygon Further information: Point in polygon § Winding number algorithm A point's winding number with respect to a polygon can be used to solve the point in polygon (PIP) problem – that is, it can be used to determine if the point is inside the polygon or not. Generally, the ray casting algorithm is a better alternative to the PIP problem as it does not require trigonometric functions, contrary to the winding number algorithm. Nevertheless, the winding number algorithm can be sped up so that it too, does not require calculations involving trigonometric functions.[10] The sped-up version of the algorithm, also known as Sunday's algorithm, is recommendable in cases where non-simple polygons should also be accounted for. See also • Argument principle • Coin rotation paradox • Linking coefficient • Nonzero-rule • Polygon density • Residue theorem • Schläfli symbol • Topological degree theory • Topological quantum number • Twist (mathematics) • Wilson loop • Writhe References 1. Möbius, August (1865). "Über die Bestimmung des Inhaltes eines Polyëders". Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften, Mathematisch-Physische Klasse. 17: 31–68. 2. Alexander, J. W. (April 1928). "Topological Invariants of Knots and Links". Transactions of the American Mathematical Society. 30 (2): 275–306. doi:10.2307/1989123. JSTOR 1989123. 3. Weisstein, Eric W. "Contour Winding Number". MathWorld. Retrieved 7 July 2022. 4. Rudin, Walter (1976). Principles of Mathematical Analysis. McGraw-Hill. p. 201. ISBN 0-07-054235-X. 5. Rudin, Walter (1987). Real and Complex Analysis (3rd ed.). McGraw-Hill. p. 203. ISBN 0-07-054234-1. 6. Abelson, Harold (1981). Turtle Graphics: The Computer as a Medium for Exploring Mathematics. MIT Press. p. 24. 7. Do Carmo, Manfredo P. (1976). "5. Global Differential Geometry". Differential Geometry of Curves and Surfaces. Prentice-Hall. p. 393. ISBN 0-13-212589-7. 8. Feldman, E. A. (1968). "Deformations of closed space curves". Journal of Differential Geometry. 2 (1): 67–75. doi:10.4310/jdg/1214501138. S2CID 116999463. 9. Minarčík, Jiří; Beneš, Michal (2022). "Nondegenerate homotopy and geometric flows". Homology, Homotopy and Applications. 24 (2): 255–264. doi:10.4310/HHA.2022.v24.n2.a12. S2CID 252274622. 10. Sunday, Dan (2001). "Inclusion of a Point in a Polygon". Archived from the original on 26 January 2013. External links • Winding number at PlanetMath. Topology Fields • General (point-set) • Algebraic • Combinatorial • Continuum • Differential • Geometric • low-dimensional • Homology • cohomology • Set-theoretic • Digital Key concepts • Open set / Closed set • Interior • Continuity • Space • compact • Connected • Hausdorff • metric • uniform • Homotopy • homotopy group • fundamental group • Simplicial complex • CW complex • Polyhedral complex • Manifold • Bundle (mathematics) • Second-countable space • Cobordism Metrics and properties • Euler characteristic • Betti number • Winding number • Chern number • Orientability Key results • Banach fixed-point theorem • De Rham cohomology • Invariance of domain • Poincaré conjecture • Tychonoff's theorem • Urysohn's lemma • Category • Mathematics portal • Wikibook • Wikiversity • Topics • general • algebraic • geometric • Publications Authority control: National • Germany
Windmill graph In the mathematical field of graph theory, the windmill graph Wd(k,n) is an undirected graph constructed for k ≥ 2 and n ≥ 2 by joining n copies of the complete graph Kk at a shared universal vertex. That is, it is a 1-clique-sum of these complete graphs.[1] Windmill graph The Windmill graph Wd(5,4). Verticesn(k – 1) + 1 Edgesnk(k − 1)/2 Radius1 Diameter2 Girth3 if k > 2 Chromatic numberk Chromatic indexn(k – 1) NotationWd(k,n) Table of graphs and parameters Properties It has n(k – 1) + 1 vertices and nk(k − 1)/2 edges,[2] girth 3 (if k > 2), radius 1 and diameter 2. It has vertex connectivity 1 because its central vertex is an articulation point; however, like the complete graphs from which it is formed, it is (k – 1)-edge-connected. It is trivially perfect and a block graph. Special cases By construction, the windmill graph Wd(3,n) is the friendship graph Fn, the windmill graph Wd(2,n) is the star graph Sn and the windmill graph Wd(3,2) is the butterfly graph. Labeling and colouring The windmill graph has chromatic number k and chromatic index n(k – 1). Its chromatic polynomial can be deduced from the chromatic polynomial of the complete graph and is equal to $x\prod _{i=1}^{k-1}(x-i)^{n}.$ The windmill graph Wd(k,n) is proved not graceful if k > 5.[3] In 1979, Bermond has conjectured that Wd(4,n) is graceful for all n ≥ 4.[4] Through an equivalence with perfect difference families, this has been proved for n ≤ 1000. [5] Bermond, Kotzig, and Turgeon proved that Wd(k,n) is not graceful when k = 4 and n = 2 or n = 3, and when k = 5 and n = 2.[6] The windmill Wd(3,n) is graceful if and only if n ≡ 0 (mod 4) or n ≡ 1 (mod 4).[7] Gallery References 1. Gallian, J. A. (3 January 2007). "A dynamic survey of graph labeling" (PDF). Electronic Journal of Combinatorics. DS6: 1–58. MR 1668059. 2. Weisstein, Eric W. "Windmill Graph". MathWorld. 3. Koh, K. M.; Rogers, D. G.; Teo, H. K.; Yap, K. Y. (1980). "Graceful graphs: some further results and problems". Congressus Numerantium. 29: 559–571. MR 0608456. 4. Bermond, J.-C. (1979). "Graceful graphs, radio antennae and French windmills". In Wilson, Robin J. (ed.). Graph theory and combinatorics (Proc. Conf., Open Univ., Milton Keynes, 1978). Research notes in mathematics. Vol. 34. Pitman. pp. 18–37. ISBN 978-0273084358. MR 0587620. OCLC 757210583. 5. Ge, G.; Miao, Y.; Sun, X. (2010). "Perfect difference families, perfect difference matrices, and related combinatorial structures". Journal of Combinatorial Designs. 18 (6): 415–449. doi:10.1002/jcd.20259. MR 2743134. S2CID 120800012. 6. Bermond, J.-C.; Kotzig, A.; Turgeon, J. (1978). "On a combinatorial problem of antennas in radioastronomy". In Hajnal, A.; Sos, Vera T. (eds.). Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. I. Colloquia mathematica Societatis János Bolyai. Vol. 18. North-Holland. pp. 135–149. ISBN 978-0-444-85095-9. MR 0519261. 7. Bermond, J.-C.; Brouwer, A. E.; Germa, A. (1978). "Systèmes de triplets et différences associées". Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976). Colloques internationaux du Centre National de la Recherche Scientifique. Vol. 260. Éditions du Centre national de la recherche scientifique. pp. 35–38. ISBN 978-2-222-02070-7. MR 0539936.
Wine/water paradox The wine/water paradox is an apparent paradox in probability theory. It is stated by Michael Deakin as follows: A mixture is known to contain a mix of wine and water in proportions such that the amount of wine divided by the amount of water is a ratio $x$ lying in the interval $1/3\leq x\leq 3$ (i.e. 25-75% wine). We seek the probability, $P^{\ast }$ say, that $x\leq 2$. (i.e. less than or equal to 66%.) The core of the paradox is in finding consistent and justifiable simultaneous prior distributions for $x$ and ${\frac {1}{x}}$.[1] Calculation This calculation is the demonstration of the paradoxical conclusion when making use of the principle of indifference. To recapitulate, We do not know $x$, the wine to water ratio. When considering the numbers above, it is only known that it lies in an interval between the minimum of one quarter wine over three quarters water on one end (i.e. 25% wine), to the maximum of three quarters wine over one quarter water on the other (i.e. 75% wine). In term of ratios, $ x_{\mathrm {min} }={\frac {1/4}{3/4}}={\frac {1}{3}}$ resp. $ x_{\mathrm {max} }={\frac {3/4}{1/4}}=3$. Now, making use of the principle of indifference, we may assume that $x$ is uniformly distributed. Then the chance of finding the ratio $x$ below any given fixed threshold $x_{t}$, with $x_{\mathrm {min} }<x_{t}<x_{\mathrm {max} }$, should linearly depend on the value $x_{t}$. So the probability value is the number $\operatorname {Prob} \{x\leq x_{t}\}={\frac {x_{t}-x_{\mathrm {min} }}{x_{\mathrm {max} }-x_{\mathrm {min} }}}={\frac {1}{8}}(3x_{t}-1).$ As a function of the threshold value $x_{t}$, this is the linearly growing function that is $0$ resp. $1$ at the end points $ x_{\mathrm {min} }$ resp. the larger $ x_{\mathrm {max} }$. Consider the threshold $x_{t}=2$, as in the example of the original formulation above. This is two parts wine vs. one part water, i.e. 66% wine. With this we conclude that $\operatorname {Prob} \{x\leq 2\}={\frac {1}{8}}(3\cdot 2-1)={\frac {5}{8}}$. Now consider $y={\frac {1}{x}}$, the inverted ratio of water to wine but the equivalent wine/water mixture threshold. It lies between the inverted bounds. Again using the principle of indifference, we get $\operatorname {Prob} \{y\geq y_{t}\}={\frac {x_{\mathrm {max} }(1-x_{\mathrm {min} }\,y_{t})}{x_{\mathrm {max} }-x_{\mathrm {min} }}}={\frac {3}{8}}(3-y_{t})$. This is the function which is $0$ resp. $1$ at the end points ${\tfrac {1}{x_{\mathrm {min} }}}$ resp. the smaller $ {\tfrac {1}{x_{\mathrm {max} }}}$. Now taking the corresponding threshold $ y_{t}={\frac {1}{x_{t}}}={\frac {1}{2}}$ (also half as much water as wine). We conclude that $\operatorname {Prob} \left\{y\geq {\tfrac {1}{2}}\right\}={\frac {3}{8}}{\frac {3\cdot 2-1}{2}}={\frac {15}{16}}={\frac {3}{2}}{\frac {5}{8}}$. The second probability always exceeds the first by a factor of $ {\frac {x_{\mathrm {max} }}{x_{t}}}\geq 1$. For our example the numbers is $ {\frac {3}{2}}$. Paradoxical conclusion Since $ y={\frac {1}{x}}$, we get ${\frac {5}{8}}=\operatorname {Prob} \{x\leq 2\}=P^{*}=\operatorname {Prob} \left\{y\geq {\frac {1}{2}}\right\}={\frac {15}{16}}>{\frac {5}{8}}$, a contradiction. References 1. Deakin, Michael A. B. (December 2005). "The Wine/Water Paradox: background, provenance and proposed resolutions". Australian Mathematical Society Gazette. 33 (3): 200–205.
Winifred Sargent Winifred Lydia Caunden Sargent (8 May 1905 – October 1979) was an English mathematician. She studied at Newnham College, Cambridge and carried out research into Lebesgue integration, fractional integration and differentiation and the properties of BK-spaces. Winifred Sargent Born(1905-05-08)8 May 1905 Ambergate, England Died(1979-10-00)October 1979 London, England Alma materNewnham College, Cambridge Scientific career FieldsMathematics, Numerical integration, Functional analysis InfluencesLancelot Stephen Bosanquet Early life Sargent was born into a Quaker family, daughter of Henry Sargent and Edith, his second wife, growing up in Fritchley, Derbyshire. She attended Ackworth School, a private school for Quakers, from 1915 to 1919. She then won a scholarship to attend The Mount School, York, another Quaker school, and later the Herbert Strutt School. In 1923, while there, she won a Derby scholarship, a State Scholarship, and a Mary Ewart scholarship to attend Newnham College, Cambridge and study mathematics in 1924. While at Newnham she won further awards: an Arthur Hugh Clough Scholarship in 1927, a Mary Ewart Travelling Scholarship and a Goldsmiths Company Senior Studentship both in 1928. She graduated with a First class degree and remained at Cambridge conducting research but was unsatisfied by her progress and left to teach mathematics at Bolton High School. Academic career Sargent's first publication was in 1929, On Young's criteria for the convergence of Fourier series and their conjugates, published in the Mathematical Proceedings of the Cambridge Philosophical Society. In 1931 she was appointed an Assistant Lecturer at Westfield College and became a member of the London Mathematical Society in January 1932.[1] in 1936 she moved to Royal Holloway, University of London, at the time both women's colleges. In 1939 she became a doctoral student of Lancelot Bosanquet, but World War II broke out, preventing his formal supervision from continuing. In 1941 Sargent was promoted to lecturer at Royal Holloway, moving to Bedford College in 1948. She served on the Mathematical Association teaching committee from 1950 to 1954.[2] In 1954 she was awarded the degree of Sc.D. (Doctor of Science) by Cambridge and was given the title of Reader. While at the University of London she supervised Alan J. White in 1959.[3][4] Bosanquet started a weekly seminar in mathematics in 1947, which Sargent attended without absence for twenty years until her retirement in 1967. She rarely presented at it, and did not attend mathematical conferences, despite being a compelling speaker. Mathematical results Much of Sargent's mathematical research involved studying types of integral, building on work done on Lebesgue integration and the Riemann integral. She produced results relating to the Perron and Denjoy integrals and Cesàro summation. Her final three papers consider BK-spaces or Banach coordinate spaces, proving a number of interesting results.[5] For example, her 1936 paper[6] proves a version of Rolle's theorem for Denjoy–Perron integrable functions using different techniques from the standard proofs:[7] as in much of Dr. Sargent's work, the arguments are pushed as far as they will go and counter examples given to show that the results are the best possible. Her 1953 paper[8] established several important results on summability kernels and is referenced in two textbooks on functional analysis.[9] Her papers in 1950 and 1957 contributed to fractional integration and differentiation theory.[10] In her obituary, her work is described as being:[11] marked by its exceptional lucidity, its exactness of expression and by the decisiveness of her results. She made important contributions to a field in which the complexity of the structure can only be revealed by subtle arguments. Papers • Sargent, Miss W. L. C. (1929). "On Young's criteria for the convergence of Fourier series and their conjugates". Mathematical Proceedings of the Cambridge Philosophical Society. 25 (1): 26–30. Bibcode:1929PCPS...25...26S. doi:10.1017/S030500410001851X. • Sargent, W. L. C. (1935). "The Borel derivates of a function". Proceedings of the London Mathematical Society. Second Series. 38 (1): 180–196. doi:10.1112/plms/s2-38.1.180. • Sargent, W. L. C. (1936). "On the Cesàro derivates of a function". Proceedings of the London Mathematical Society. Second Series. 40 (1): 235–254. doi:10.1112/plms/s2-40.1.235. • Sargent, W. L. C. (1942a). "A descriptive definition of Cesàro–Perron integrals". Proceedings of the London Mathematical Society. Second Series. 47 (1): 212–247. doi:10.1112/plms/s2-47.1.212. • Sargent, W. L. C. (1942b). "On sufficient conditions for a function integrable in the Cesàro–Perron sense to be monotonic". The Quarterly Journal of Mathematics. Oxford Series. 12 (1): 148–153. doi:10.1093/qmath/os-12.1.148. • Sargent, W. L. C. (1946a). "On the order of magnitude of the Fourier coefficients of a function integrable in the CλL sense". Journal of the London Mathematical Society. First Series. 21 (3): 198–203. doi:10.1112/jlms/s1-21.3.198. • Sargent, W. L. C. (1946b). "A mean value theorem involving Cesàro means". Proceedings of the London Mathematical Society. Second Series. 49 (1): 227–240. doi:10.1112/plms/s2-49.3.227. • Sargent, W. L. C. (1948a). "On the integrability of a product". Journal of the London Mathematical Society. First Series. 23 (1): 28–34. doi:10.1112/jlms/s1-23.1.28. hdl:10338.dmlcz/127918. • Sargent, W. L. C. (1948b). "On the summability (C) of allied series and the existence of $(CP)\int \limits _{0}^{\pi }{\frac {f(x+t)-f(x-t)}{t}}\,dt$". Proceedings of the London Mathematical Society. Second Series. 50 (1): 330–348. doi:10.1112/plms/s2-50.5.330. • Sargent, W. L. C. (1949). "On fractional integrals of a function integrable in the Cesàro-Perron sense". Proceedings of the London Mathematical Society. Second Series. 51 (1): 46–80. doi:10.1112/plms/s2-51.1.46. • Sargent, W. L. C. (1950a). "On linear functionals in spaces of conditionally integrable functions". The Quarterly Journal of Mathematics. Oxford Second Series. 1 (1): 288–298. Bibcode:1950QJMat...1..288S. doi:10.1093/qmath/1.1.288. hdl:10338.dmlcz/140532. • Sargent, W. L. C. (1950b). "On the continuity (C) and integrability (CP) of fractional integrals". Proceedings of the London Mathematical Society. Second Series. 52 (1): 253–270. doi:10.1112/plms/s2-52.4.253. • Sargent, W. L. C. (1950c). "On generalized derivatives and Cesàro–Denjoy integrals". Proceedings of the London Mathematical Society. Second Series. 52 (1): 365–376. doi:10.1112/plms/s2-52.5.365. • Sargent, W. L. C. (1951a). "Some properties of Cλ-continuous functions". Journal of the London Mathematical Society. First Series. 26 (2): 116–121. doi:10.1112/jlms/s1-26.2.116. • Sargent, W. L. C. (1951b). "On the integrability of a product (II)". Journal of the London Mathematical Society. First Series. 26 (4): 278–285. doi:10.1112/jlms/s1-26.4.278. • Sargent, W. L. C. (1951c). "2213. On the differentiation of a function of a function". The Mathematical Gazette. 35 (312): 121–122. doi:10.2307/3609346. JSTOR 3609346. • Sargent, W. L. C. (1952a). "On the summability of infinite integrals". Journal of the London Mathematical Society. First Series. 27 (4): 401–413. doi:10.1112/jlms/s1-27.4.401. • Sargent, W. L. C. (1952b). "Book review: Éléments de Mathématiques. XII by N. Bourbaki". The Mathematical Gazette. 36 (317): 216–217. doi:10.2307/3608266. JSTOR 3608266. • Sargent, W. L. C. (1952b). "Book review: Vorlesungen über Fouriersche Integrale by S. Bochner". The Mathematical Gazette. 36 (317): 217–218. doi:10.2307/3608268. JSTOR 3608268. • Sargent, W. L. C. (1953). "On some theorems of Hahn, Banach and Steinhaus". Journal of the London Mathematical Society. First Series. 28 (4): 438–451. doi:10.1112/jlms/s1-28.4.438. • Sargent, W. L. C. (1954). "Book review: Volume and Integral by W. W. Rogosinski". The Mathematical Gazette. 38 (323): 67. doi:10.2307/3609800. JSTOR 3609800. • Sargent, W. L. C. (1955). "On the transform $y_{x}(s)=\int \limits _{0}^{\infty }x(t)k_{s}(t)dt$". Journal of the London Mathematical Society. First Series. 30 (4): 401–416. doi:10.1112/jlms/s1-30.4.401. • Sargent, W. L. C. (1957a). "Some summability factor theorems for infinite integrals". Journal of the London Mathematical Society. First Series. 32 (4): 387–396. doi:10.1112/jlms/s1-32.4.387. • Sargent, W. L. C. (1957b). "On some cases of distinction between integrals and series". Proceedings of the London Mathematical Society. Third Series. 7 (1): 249–264. doi:10.1112/plms/s3-7.1.249. • Sargent, W. L. C. (1960). "Some sequence spaces related to the lp spaces". Journal of the London Mathematical Society. First Series. 35 (2): 161–171. doi:10.1112/jlms/s1-35.2.161. • Sargent, W. L. C. (1961). "Some analogues and extensions of Marcinkiewicz's interpolation problem". Proceedings of the London Mathematical Society. Third Series. 11 (1): 457–468. doi:10.1112/plms/s3-11.1.457. • Sargent, W. L. C. (1964). "On sectionally bounded BK-spaces". Mathematische Zeitschrift. 83 (1): 57–66. doi:10.1007/BF01111108. S2CID 119753065. • Sargent, W. L. C. (1966). "On compact matrix transformations between sectionally bounded BK-spaces". Journal of the London Mathematical Society. First Series. 41 (1): 79–87. doi:10.1112/jlms/s1-41.1.79. Notes 1. Dixon 1932, p. 81. 2. j. t. c (1950). "Report of the Meeting of the Teaching Committee. 5th January 1950". The Mathematical Gazette. 34 (307): 5–7. doi:10.1017/S0025557200023469. JSTOR 3610867., p. 6. 3. White 1961, p. 319. 4. "Alan J. White". Mathematics Genealogy Project. Department of Mathematics, North Dakota State University. Retrieved 15 October 2015. 5. Sargent 1961, Sargent 1964, Sargent 1966. 6. Sargent 1936, pp. 239–240. 7. Eggleston 1981, pp. 173–174. 8. Sargent 1953. 9. Swartz, Charles (1992). An introduction to functional analysis. CRC Press. pp. 102–104. ISBN 978-0824786434. and Orlicz, Władysław (1992). Linear functional analysis. World Scientific Publishing. p. 125. ISBN 978-9810208530. 10. Sargent 1950a, Sargent 1950b and Sargent 1957a. 11. Eggleston 1981, p. 175. References • Dixon, Prof. A. C. (1932). "Records of proceedings at meetings. Session November, 1931–June, 1932". Journal of the London Mathematical Society. First Series. 7 (2): 81–82. doi:10.1112/jlms/s1-7.2.81. • Eggleston, H.G. (1981). "Winifred L. C. Sargent". Bulletin of the London Mathematical Society. 13 (2): 173–176. doi:10.1112/blms/13.2.173. • O'Connor, JJ; Robertson, EF. "Winifred Lydia Caunden Sargent Biography". MacTutor History of Mathematics archive. University of St Andrews. Retrieved 13 October 2015. • Ogilvie, Marilyn; Harvey, Joy (2000). The Biographical Dictionary of Women in Science: Pioneering Lives From Ancient Times to the Mid-20th Century. Routledge. pp. 1152–1153. ISBN 978-0415920384. • White, A. J. (1961). "On the restricted Cesàro summability of double Fourier series". Transactions of the American Mathematical Society. 99 (2): 308–319. doi:10.2307/1993402. JSTOR 1993402. Authority control: Academics • MathSciNet • Mathematics Genealogy Project • zbMATH
Winifred Asprey Winifred "Tim" Alice Asprey (April 8, 1917 – October 19, 2007) was an American mathematician and computer scientist. She was one of only around 200 women to earn PhDs in mathematics from American universities during the 1940s, a period of women's underrepresentation in mathematics at this level.[1] She was involved in developing the close contact between Vassar College and IBM that led to the establishment of the first computer science lab at Vassar.[1] Winifred Asprey Born(1917-04-08)April 8, 1917 Sioux City, Iowa DiedOctober 19, 2007(2007-10-19) (aged 90) Poughkeepsie (town), New York Alma materVassar College University of Iowa Scientific career FieldsMathematics, computer science Doctoral advisorEdward Wilson Chittenden InfluencesGrace Hopper Family Asprey was born in Sioux City, Iowa; her parents were Gladys Brown Asprey, Vassar class of 1905, and Peter Asprey Jr.[2] She had two brothers, actinide and fluorine chemist Larned B. Asprey (1919–2005), a signer of the Szilárd petition, and military historian and writer Robert B. Asprey (1923–2009) who dedicated several of his books to his sister Winifred.[3][4] Education and work Asprey attended Vassar College in Poughkeepsie, New York, where she earned her undergraduate degree in 1938. As a student there, Asprey met Grace Hopper, the "First Lady of Computing," who taught mathematics at the time. After graduating, Asprey taught at several private schools in New York City and Chicago before going on to earn her MS and PhD degrees from the University of Iowa in 1942 and 1945, respectively.[2] Her doctoral advisor was the topologist Edward Wilson Chittenden.[5] Asprey returned to Vassar College as a professor. By then, Grace Hopper had moved to Philadelphia to work on UNIVAC (Universal Automatic Computer) project. Asprey became interested in computing and visited Hopper to learn about the foundations of computer architecture.[2] Asprey believed that computers would be an essential part of a liberal arts education. At Vassar, Asprey taught mathematics and computer science for 38 years and was the chair of the mathematics department from 1957 until her retirement in 1982.[6] She created the first Computer Science courses at Vassar, the first being taught in 1963, and secured funds for the college's first computer, making Vassar the second college in the nation to acquire an IBM System/360 computer in 1967.[7] Asprey connected with researchers at IBM and other research centers and lobbied for computer science at Vassar. In 1989, due to her contributions, the computer center she started was renamed the Asprey Advanced Computation Laboratory.[2] References 1. Margaret Anne Marie Murray (2001). Women Becoming Mathematicians: Creating a Professional Identity in Post-World War II America. MIT Press. ISBN 0-262-63246-2. 2. "Winifred "Tim" Asprey, computer pioneer and longtime professor at Vassar College, dies at 90". Vassar Office of College Relations. 3. "Dr. Larned "Larry" Brown Asprey". Obituaries. Albuquerque Journal(. March 11, 2005. 4. "New College Receives Gift from Estate of Robert B. Asprey". New College of Florida. June 15, 2009. Retrieved December 10, 2009. 5. Winifred Asprey at the Mathematics Genealogy Project 6. "Scientists in the News". Science. American Association for the Advancement of Science. 125 (3257): 1080–1081. 1957. doi:10.1126/science.125.3257.1077. JSTOR 1752434. PMID 17756202. 7. "Winifred Asprey: Into the Future". Vassar Office of College Relations. Archived from the original on May 2, 2016. Retrieved January 31, 2014. External links • Profile at Vassar College Innovators Gallery • Winifred Asprey Papers at Vassar College Archives and Special Collections Library Authority control International • ISNI • VIAF National • United States • Australia Academics • Mathematics Genealogy Project People • Trove
Peter Winkler Peter Mann Winkler is a research mathematician, author of more than 125 research papers in mathematics[1] and patent holder in a broad range of applications, ranging from cryptography to marine navigation.[2] His research areas include discrete mathematics, theory of computation and probability theory. He is currently a professor of mathematics and computer science at Dartmouth College.[3] Peter Winkler studied mathematics at Harvard University and later received his PhD in 1975 from Yale University under the supervision of Angus McIntyre.[4] He has also served as an assistant professor at Stanford, full professor and chair at Emory and as a mathematics research director at Bell Labs and Lucent Technologies.[2] He was visiting professor at the Technische Universität Darmstadt.[5] He has published three books on mathematical puzzles: Mathematical Puzzles: A connoisseur's collection (A K Peters, 2004, ISBN 978-1-56881-201-4), Mathematical Mind-Benders (A K Peters, 2007, ISBN 978-1-56881-336-3), and Mathematical Puzzles (A K Peters, 2021, ISBN 978-0-36720-693-2). And he is widely considered to be a pre eminent scholar in this domain. He was the Visiting Distinguished Chair for Public Dissemination of Mathematics at the National Museum of Mathematics (MoMath), gave topical talks at the Gathering 4 Gardner conferences, and wrote novel papers related to some of these puzzles. Winkler's book Bridge at the Enigma Club[6] was a runner up for the 2011 Master Point Press Book Of The Year award.[7] Also in 2011, Winkler received the David P. Robbins Prize of the Mathematical Association of America as coauthor of one of two papers[8] in the American Mathematical Monthly. Paul Erdős anecdote According to a story included in Chapter One of "The Man Who Loved Only Numbers / The Story of Paul Erdös and the Search for Mathematical Truth",[9] Paul Erdős attended the bar mitzvah celebration for Peter Winkler's twins, and Winkler's mother-in-law tried to throw Erdős out. [Quote:] "Erdös came to my twins' bar mitzvah, notebook in hand," said Peter Winkler, a colleague of Graham's at AT&T. "He also brought gifts for my children--he loved kids--and behaved himself very well. But my mother-in-law tried to throw him out. She thought he was some guy who wandered in off the street, in a rumpled suit, carrying a pad under his arm. It is entirely possible that he proved a theorem or two during the ceremony."[9] References 1. Publication list from Winkler's home page at Dartmouth. 2. Information listed on Peter Winkler's homepage at Dartmouth. 3. Dartmouth mathematics faculty listing. 4. Peter Winkler at the Mathematics Genealogy Project. 5. "Humboldt network profile of Peter Winkler". www.humboldt-foundation.de. Retrieved 2019-09-17.{{cite web}}: CS1 maint: url-status (link) 6. The Bridge World Bookstore Bridge at the Enigma Club by Peter Winkler 7. The 2011 Master Point Press Book Of The Year Award 2014 IPBA Handbook, p. 176 8. "Overhang", American Mathematical Monthly, vol. 116, January 2009 (Online) "Maximum Overhang", American Mathematical Monthly, vol. 116, December 2009 (Online) 9. Hoffman, Paul (15 July 1998). The Man Who Loved Only Numbers / The Story of Paul Erdös and the Search for Mathematical Truth. "The Man Who Loved Only Numbers" was published in hardcover by: Hyperion Books and a later edition was published by The New York Times Book Company. ISBN 0-7868-6362-5. Retrieved November 23, 2017.{{cite book}}: CS1 maint: url-status (link) External links • Peter Winkler at the Mathematics Genealogy Project Authority control International • ISNI • VIAF • 2 National • Germany • Israel • United States • Czech Republic Academics • DBLP • MathSciNet • Mathematics Genealogy Project • ORCID • Scopus • zbMATH Other • IdRef
Winnie Li Wen-Ch'ing (Winnie) Li (Chinese: 李文卿; born December 25, 1948) is a Taiwanese-American mathematician and a Distinguished Professor of Mathematics at Pennsylvania State University.[1] She is a number theorist, with research focusing on the theory of automorphic forms and applications of number theory to coding theory and spectral graph theory. In particular, she has applied her research results in automorphic forms and number theory to construct efficient communication networks called Ramanujan graphs and Ramanujan complexes. Wen-Ch'ing Li 李文卿 Born (1948-12-25) December 25, 1948 Awards • Chern Prize (2010) • Fellow, American Mathematical Society (2012) • Noether Lecture (2015) Academic background Alma materNational Taiwan University, University of California, Berkeley Doctoral advisorAndrew Ogg Academic work DisciplineMathematics InstitutionsHarvard University, University of Illinois at Chicago, Pennsylvania State University Professional career Li did her undergraduate studies at National Taiwan University, graduating in 1970;[1][2] at NTU, she was a classmate of other notable female mathematicians Fan Chung, Sun-Yung Alice Chang and Jang-Mei Wu.[3] She earned a doctorate from the University of California, Berkeley in 1974, under the supervision of Andrew Ogg.[1][2][4] Before joining the PSU faculty in 1979, she was a Benjamin Pierce assistant professor at Harvard University for 3.5 years from 1974 to 1977, and a tenure-track assistant professor at the University of Illinois at Chicago from 1978 to 1979.[1][2] She was also the director of the National Center of Theoretical Sciences in Taiwan from 2009 to 2014.[1][2] Awards and honors In 2010, Li was the winner of the Chern Prize, given every three years to an outstanding Chinese mathematician.[5] In 2012 she became a fellow of the American Mathematical Society.[6] She was chosen to give the 2015 Noether Lecture.[7] References 1. Winnie Li Named Distinguished Professor, Pennsylvania State University, 30 January 2012, retrieved 2013-02-02. 2. Staff biography, National Center of Theoretical Sciences, retrieved 2013-02-02. 3. Fan Chung Graham, Association for Women in Mathematics, retrieved 2013-02-02. 4. Wen-Ching Winnie Li at the Mathematics Genealogy Project 5. Wen Ching Li Awarded the 2010 Chern Prize in Mathematics, Pennsylvania State University, 30 January 2012, retrieved 2013-02-02. 6. List of Fellows of the American Mathematical Society, retrieved 2013-01-27. 7. “Modular Forms for Congruence and Noncongruence Subgroups” External links • Home page • Ramdorai, Sujatha (January 2015). "Interview with Wen-Ching (Winnie) Li" (PDF). Asia Pacific Mathematics Newsletter. pp. 18–21. Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • United States • Netherlands Academics • DBLP • MathSciNet • Mathematics Genealogy Project • 2 • zbMATH People • Deutsche Biographie Other • IdRef
Winning Ways for Your Mathematical Plays Winning Ways for Your Mathematical Plays (Academic Press, 1982) by Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy is a compendium of information on mathematical games. It was first published in 1982 in two volumes. The first volume introduces combinatorial game theory and its foundation in the surreal numbers; partizan and impartial games; Sprague–Grundy theory and misère games. The second volume applies the theorems of the first volume to many games, including nim, sprouts, dots and boxes, Sylver coinage, philosopher's phutball, fox and geese. A final section on puzzles analyzes the Soma cube, Rubik's Cube, peg solitaire, and Conway's Game of Life. A republication of the work by A K Peters split the content into four volumes. Editions • 1st edition, New York: Academic Press, 2 vols., 1982; vol. 1, hardback: ISBN 0-12-091150-7, paperback: ISBN 0-12-091101-9; vol. 2, hardback: ISBN 0-12-091152-3, paperback: ISBN 0-12-091102-7. • 2nd edition, Wellesley, Massachusetts: A. K. Peters Ltd., 4 vols., 2001–2004; vol. 1: ISBN 1-56881-130-6; vol. 2: ISBN 1-56881-142-X; vol. 3: ISBN 1-56881-143-8; vol. 4: ISBN 1-56881-144-6. Games mentioned in the book This is a partial list of the games mentioned in the book. Note: Misère games not included • Hackenbush • Blue-Red Hackenbush • Blue-Red-Green Hackenbush (Introduced as Hackenbush Hotchpotch in the book) • Childish Hackenbush • Ski-Jumps • Toads-and-Frogs • Cutcake • Maundy Cake • (2nd Unnamed Cutcake variant by Dean Hickerson) • Hotcake • Coolcakes • Baked Alaska • Eatcake • Turn-and-Eatcake • Col • Snort • Nim (Green Hackenbush) • Prim • Dim • Lasker's Nim • Seating Couples • Northcott's Game (Poker-Nim) • The White Knight • Wyt Queens (Wythoff's Game) • Kayles • Double Kayles • Quadruple Kayles • Dawson's Chess • Dawson's Kayles • Treblecross • Grundy's Game • Mrs. Grundy • Domineering • No Highway • De Bono's L-Game • Snakes-and-Ladders (Adders-and-Ladders) • Jelly Bean Game • Dividing Rulers Reviews • Games[1] See also • On Numbers and Games by John H. Conway, one of the three coauthors of Winning Ways References 1. https://archive.org/details/games-32-1982-October/page/n57/mode/2up
Winograd schema challenge The Winograd schema challenge (WSC) is a test of machine intelligence proposed in 2012 by Hector Levesque, a computer scientist at the University of Toronto. Designed to be an improvement on the Turing test, it is a multiple-choice test that employs questions of a very specific structure: they are instances of what are called Winograd schemas, named after Terry Winograd, professor of computer science at Stanford University.[1] On the surface, Winograd schema questions simply require the resolution of anaphora: the machine must identify the antecedent of an ambiguous pronoun in a statement. This makes it a task of natural language processing, but Levesque argues that for Winograd schemas, the task requires the use of knowledge and commonsense reasoning.[2] Nuance Communications announced in July 2014 that it would sponsor an annual WSC competition, with a prize of $25,000 for the best system that could match human performance.[3] However, the prize is no longer offered. Background The Winograd Schema Challenge was proposed in the spirit of the Turing test. Proposed by Alan Turing in 1950, the Turing test plays a central role in the philosophy of artificial intelligence. Turing proposed that, instead of debating whether a machine can think, the science of AI should be concerned with demonstrating intelligent behavior, which can be tested. But the exact nature of the test Turing proposed has come under scrutiny, especially since an AI chatbot named Eugene Goostman claimed to pass it in 2014. One of the major concerns with the Turing test is that a machine could easily pass the test with brute force and/or trickery, rather than true intelligence.[4] The Winograd schema challenge was proposed in 2012 in part to ameliorate the problems that came to light with the nature of the programs that performed well on the test.[5] Turing's original proposal was what he called the imitation game, which involves free-flowing, unrestricted conversations in English between human judges and computer programs over a text-only channel (such as teletype). In general, the machine passes the test if interrogators are not able to tell the difference between it and a human in a five-minute conversation.[4] Weaknesses of the Turing test The performance of Eugene Goostman exhibited some of the Turing test's problems. Levesque identifies several major issues,[2] summarized as follows:[6] • Deception: The machine is forced to construct a false identity, which is not part of intelligence. • Conversation: A lot of interaction may qualify as "legitimate conversation"—jokes, clever asides, points of order—without requiring intelligent reasoning. • Evaluation: Humans make mistakes and judges often would disagree on the results. Winograd schemas The key factor in the WSC is the special format of its questions, which are derived from Winograd schemas. Questions of this form may be tailored to require knowledge and commonsense reasoning in a variety of domains. They must also be carefully written not to betray their answers by selectional restrictions or statistical information about the words in the sentence. Origin The first cited example of a Winograd schema (and the reason for their name) is due to Terry Winograd:[7] The city councilmen refused the demonstrators a permit because they [feared/advocated] violence. The choices of "feared" and "advocated" turn the schema into its two instances: The city councilmen refused the demonstrators a permit because they feared violence. The city councilmen refused the demonstrators a permit because they advocated violence. The schema challenge question is, "Does the pronoun 'they' refer to the city councilmen or the demonstrators?" Switching between the two instances of the schema changes the answer. The answer is immediate for a human reader but proves difficult to emulate in machines. Levesque[2] argues that knowledge plays a central role in these problems: the answer to this schema has to do with our understanding of the typical relationships between and behavior of councilmen and demonstrators. Since the original proposal of the Winograd schema challenge, Ernest Davis, a professor at New York University, has compiled a list of over 140 Winograd schemas from various sources as examples of the kinds of questions that should appear on the Winograd schema challenge.[8] Formal description A Winograd schema challenge question consists of three parts: 1. A sentence or brief discourse that contains the following: • Two noun phrases of the same semantic class (male, female, inanimate, or group of objects or people), • An ambiguous pronoun that may refer to either of the above noun phrases, and • A special word and alternate word, such that if the special word is replaced with the alternate word, the natural resolution of the pronoun changes. 2. A question asking the identity of the ambiguous pronoun, and 3. Two answer choices corresponding to the noun phrases in question. A machine will be given the problem in a standardized form which includes the answer choices, thus making it a binary decision problem. Advantages The Winograd schema challenge has the following purported advantages: • Knowledge and commonsense reasoning are required to solve them. • Winograd schemas of varying difficulty may be designed, involving anything from simple cause-and-effect relationships to complex narratives of events. • They may be constructed to test reasoning ability in specific domains (e.g., social/psychological or spatial reasoning). • There is no need for human judges.[5] Pitfalls One difficulty with the Winograd schema challenge is the development of the questions. They need to be carefully tailored to ensure that they require commonsense reasoning to solve. For example, Levesque[5] gives the following example of a so-called Winograd schema that is "too easy": The women stopped taking pills because they were [pregnant/carcinogenic]. Which individuals were [pregnant/carcinogenic]? The answer to this question can be determined on the basis of selectional restrictions: in any situation, pills do not get pregnant, women do; women cannot be carcinogenic, but pills can. Thus this answer could be derived without the use of reasoning, or any understanding of the sentences' meaning—all that is necessary is data on the selectional restrictions of pregnant and carcinogenic. Activity In 2016 and 2018, Nuance Communications sponsored a competition, offering a grand prize of $25,000 for the top scorer above 90% (for comparison, humans correctly answer to 92–96% of WSC questions[9]). However, nobody came close to winning the prize in 2016 and the 2018 competition was cancelled for lack of prospects;[10] the prize is no longer offered.[11] The Twelfth International Symposium on the Logical Formalizations of Commonsense Reasoning was held on March 23–25, 2015 at the AAAI Spring Symposium Series at Stanford University, with a special focus on the Winograd schema challenge. The organizing committee included Leora Morgenstern (Leidos), Theodore Patkos (The Foundation for Research & Technology Hellas), and Robert Sloan (University of Illinois at Chicago).[12] The 2016 Winograd Schema Challenge was run on July 11, 2016 at IJCAI-16. There were four contestants. The first round of the contest was to solve PDPs—pronoun disambiguation problems, adapted from literary sources, not constructed as pairs of sentences.[13] The highest score achieved was 58% correct, by Quan Liu et al, of the University of Science and Technology, China.[14] Hence, by the rules of that challenge, no prizes were awarded, and the challenge did not proceed to the second round. The organizing committee in 2016 was Leora Morgenstern, Ernest Davis, and Charles Ortiz.[15] In 2017, a neural association model designed for commonsense knowledge acquisition achieved 70% accuracy on 70 manually selected problems from the original 273 Winograd schema dataset.[16] In June 2018, a score of 63.7% accuracy was achieved on the full dataset using an ensemble of recurrent neural network language models,[17] marking the first use of deep neural networks that learn from independent corpora to acquire common sense knowledge. In 2019 a score of 90.1%, was achieved on the original Winograd schema dataset by fine-tuning of the BERT language model with appropriate WSC-like training data to avoid having to learn commonsense reasoning.[9] The general language model GPT-3 achieved a score of 88.3% without specific fine-tuning in 2020.[18] A more challenging, adversarial "Winogrande" dataset of 44,000 problems was designed in 2019. This dataset consists of fill-in-the-blank style sentences, as opposed to the pronoun format of previous datasets.[9] A version of the Winograd schema challenge is one part of the GLUE (General Language Understanding Evaluation) benchmark collection of challenges in automated natural-language understanding.[19] References 1. Ackerman, Evan (29 July 2014). "Can Winograd Schemas Replace Turing Test for Defining Human-level AI". IEEE Spectrum. Retrieved 29 October 2014. 2. Levesque, H. J. (2014). "On our best behaviour". Artificial Intelligence. 212: 27–35. doi:10.1016/j.artint.2014.03.007. 3. "Nuance announces the Winograd Schemas Challenge to Advance Artificial Intelligence Innovation". Business Wire. 28 July 2014. Retrieved 9 November 2014. 4. Turing, Alan (October 1950). "Computing Machinery and Intelligence" (PDF). Mind. LIX (236): 433–460. doi:10.1093/mind/LIX.236.433. Retrieved 28 October 2014. 5. Levesque, Hector; Davis, Ernest; Morgenstern, Leora (2012). The Winograd Schema Challenge. Proceedings of the Thirteenth International Conference on Principles of Knowledge Representation and Reasoning. 6. Michael, Julian (18 May 2015). The Theory of Correlation Formulas and Their Application to Discourse Coherence (Thesis). UT Digital Repository. p. 6. hdl:2152/29979. 7. Winograd, Terry (January 1972). "Understanding Natural Language" (PDF). Cognitive Psychology. 3 (1): 1–191. doi:10.1016/0010-0285(72)90002-3. Retrieved 4 November 2014. 8. Davis, Ernest. "A Collection of Winograd Schemas". cs.nyu.edu. NYU. Retrieved 30 October 2014. 9. Sakaguchi, Keisuke; Le Bras, Ronan; Bhagavatula, Chandra; Choi, Yejin (2019). "WinoGrande: An Adversarial Winograd Schema Challenge at Scale". arXiv:1907.10641 [cs.CL]. 10. Boguslavsky, I.M.; Frolova, T.I.; Iomdin, L.L.; Lazursky, A.V.; Rygaev, I.P.; Timoshenko, S.P. (2019). "Knowledge-based approach to Winograd Schema Challenge" (PDF). Proceedings of the International Conference of Computational Linguistics and Intellectual Technologies. Moscow. The prize could not be awarded to anybody. Most of the participants showed a result close to the random choice or even worse. The second competition scheduled for 2018 was canceled due to the lack of prospective participants. 11. "Winograd Schema Challenge". CommonsenseReasoning.org. Retrieved 24 January 2020. 12. "AAAI 2015 Spring Symposia". Association for the Advancement of Artificial Intelligence. Retrieved 1 January 2015. 13. Davis, Ernest; Morgenstern, Leora; Ortiz, Charles (Fall 2017). "The First Winograd Schema Challenge at IJCAI-16". AI Magazine. 14. Liu, Quan; Jiang, Hui; Ling, Zhen-Hua; Zhu, Xiaodan; Wei, Si; Hu, Yu (2016). "Commonsense Knowledge Enhanced Embeddings for Solving Pronoun Disambiguation Problems in Winograd Schema Challenge". arXiv:1611.04146 [cs.AI]. 15. Morgenstern, Leora; Davis, Ernest; Ortiz, Charles L. (March 2016). "Planning, Executing, and Evaluating the Winograd Schema Challenge". AI Magazine. 37 (1): 50–54. doi:10.1609/aimag.v37i1.2639. ISSN 0738-4602. 16. Liu, Quan; Jiang, Hui; Evdokimov, Andrew; Ling, Zhen-Hua; Zhu, Xiaodan; Wei, Si; Hu, Yu (2017). "Cause-Effect Knowledge Acquisition and Neural Association Model for Solving A Set of Winograd Schema Problems". Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence: 2344–2350. doi:10.24963/ijcai.2017/326. ISBN 9780999241103. 17. Trinh, Trieu H.; Le, Quoc V. (26 September 2019). "A Simple Method for Commonsense Reasoning". arXiv:1806.02847 [cs.AI]. 18. Brown, Tom B.; Mann, Benjamin; Ryder, Nick; Subbiah, Melanie; Kaplan, Jared; Dhariwal, Prafulla; Neelakantan, Arvind; Shyam, Pranav; Sastry, Girish; Askell, Amanda; Agarwal, Sandhini; Herbert-Voss, Ariel; Krueger, Gretchen; Henighan, Tom; Child, Rewon; Ramesh, Aditya; Ziegler, Daniel M.; Wu, Jeffrey; Winter, Clemens; Hesse, Christopher; Chen, Mark; Sigler, Eric; Litwin, Mateusz; Gray, Scott; Chess, Benjamin; Clark, Jack; Berner, Christopher; McCandlish, Sam; Radford, Alec; et al. (2020). "Language Models are Few-Shot Learners". arXiv:2005.14165 [cs.CL]. 19. "GLUE Benchmark". GlueBenchmark.com. Retrieved 30 July 2019. External links • Website for the contest sponsored by Nuance Communications • https://arxiv.org/abs/2201.02387
Hash chain A hash chain is the successive application of a cryptographic hash function to a piece of data. In computer security, a hash chain is a method used to produce many one-time keys from a single key or password. For non-repudiation, a hash function can be applied successively to additional pieces of data in order to record the chronology of data's existence. Definition A hash chain is a successive application of a cryptographic hash function $h$ to a string $x$. For example, $h(h(h(h(x))))$ gives a hash chain of length 4, often denoted $h^{4}(x)$ Applications Leslie Lamport[1] suggested the use of hash chains as a password protection scheme in an insecure environment. A server which needs to provide authentication may store a hash chain rather than a plain text password and prevent theft of the password in transmission or theft from the server. For example, a server begins by storing $h^{1000}(\mathrm {password} )$ which is provided by the user. When the user wishes to authenticate, they supply $h^{999}(\mathrm {password} )$ to the server. The server computes $h(h^{999}(\mathrm {password} ))=h^{1000}(\mathrm {password} )$ and verifies this matches the hash chain it has stored. It then stores $h^{999}(\mathrm {password} )$ for the next time the user wishes to authenticate. An eavesdropper seeing $h^{999}(\mathrm {password} )$ communicated to the server will be unable to re-transmit the same hash chain to the server for authentication since the server now expects $h^{998}(\mathrm {password} )$. Due to the one-way property of cryptographically secure hash functions, it is infeasible for the eavesdropper to reverse the hash function and obtain an earlier piece of the hash chain. In this example, the user could authenticate 1000 times before the hash chain were exhausted. Each time the hash value is different, and thus cannot be duplicated by an attacker. Binary hash chains Binary hash chains are commonly used in association with a hash tree. A binary hash chain takes two hash values as inputs, concatenates them and applies a hash function to the result, thereby producing a third hash value. The above diagram shows a hash tree consisting of eight leaf nodes and the hash chain for the third leaf node. In addition to the hash values themselves the order of concatenation (right or left 1,0) or "order bits" are necessary to complete the hash chain. Winternitz chains Winternitz chains (also known as function chains[2]) are used in hash-based cryptography. The chain is parameterized by the Winternitz parameter w (number of bits in a "digit" d) and security parameter n (number of bits in the hash value, typically double the security strength,[3] 256 or 512). The chain consists of $2^{w}$ values that are results of repeated application of a one-way "chain" function F to a secret key sk: $sk,F(sk),F(F(sk)),...,F^{2^{w-1}}(sk)$. The chain function is typically based on a standard cryptographic hash, but needs to be parameterized ("randomized"[4]), so it involves few invocations of the underlying hash.[5] In the Winternitz signature scheme a chain is used to encode one digit of the m-bit message, so the Winternitz signature uses approximately $mn/w$ bits, its calculation takes about $2^{w}m/w$ applications of the function F.[3] Note that some signature standards (like Extended Merkle signature scheme, XMSS) define w as the number of possible values in a digit, so $w=16$ in XMSS corresponds to $w=4$ in standards (like Leighton-Micali Signature, LMS) that define w in the same way as above - as a number of bits in the digit.[6] Hash chain vs. blockchain A hash chain is similar to a blockchain, as they both utilize a cryptographic hash function for creating a link between two nodes. However, a blockchain (as used by Bitcoin and related systems) is generally intended to support distributed agreement around a public ledger (data), and incorporates a set of rules for encapsulation of data and associated data permissions. See also • Challenge–response authentication • Hash list – In contrast to the recursive structure of hash chains, the elements of a hash list are independent of each other. • One-time password • Key stretching • Linked timestamping – Binary hash chains are a key component in linked timestamping. • X.509 References 1. L. Lamport, “Password Authentication with Insecure Communication”, Communications of the ACM 24.11 (November 1981), pp 770-772. 2. Hülsing 2013b, pp. 18–20. 3. Buchmann et al. 2011, p. 2. 4. Hülsing 2013b. 5. RFC 8391 6. NIST SP 800-208, Recommendation for Stateful Hash-Based Signature Schemes, p. 5 Sources • Buchmann, Johannes; Dahmen, Erik; Ereth, Sarah; Hülsing, Andreas; Rückert, Markus (2011). "On the Security of the Winternitz One-Time Signature Scheme" (PDF). Lecture Notes in Computer Science. Vol. 6737. Springer Berlin Heidelberg. pp. 363–378. doi:10.1007/978-3-642-21969-6_23. eISSN 1611-3349. ISBN 978-3-642-21968-9. ISSN 0302-9743. • Hülsing, Andreas (2013b). Practical Forward Secure Signatures using Minimal Security Assumptions (PDF) (PhD). TU Darmstadt. • Hülsing, Andreas (2013a). "W-OTS+ – Shorter Signatures for Hash-Based Signature Schemes" (PDF). Progress in Cryptology – AFRICACRYPT 2013. Lecture Notes in Computer Science. Vol. 7918. Springer Berlin Heidelberg. pp. 173–188. doi:10.1007/978-3-642-38553-7_10. eISSN 1611-3349. ISBN 978-3-642-38552-0. ISSN 0302-9743.
Wireworld Wireworld, alternatively WireWorld, is a cellular automaton first proposed by Brian Silverman in 1987, as part of his program Phantom Fish Tank. It subsequently became more widely known as a result of an article in the "Computer Recreations" column of Scientific American.[1] Wireworld is particularly suited to simulating transistors, and is Turing-complete. Rules A Wireworld cell can be in one of four different states, usually numbered 0–3 in software, modeled by colors in the examples here: 1. empty (black), 2. electron head (blue), 3. electron tail (red), 4. conductor (yellow). As in all cellular automata, time proceeds in discrete steps called generations (sometimes "gens" or "ticks"). Cells behave as follows: • empty → empty, • electron head → electron tail, • electron tail → conductor, • conductor → electron head if exactly one or two of the neighbouring cells are electron heads, otherwise remains conductor. Wireworld uses what is called the Moore neighborhood, which means that in the rules above, neighbouring means one cell away (range value of one) in any direction, both orthogonal and diagonal. These simple rules can be used to construct logic gates (see below). Applications Entities built within Wireworld universes include Langton's Ant (allowing any Langton's Ant pattern to be built within Wireworld)[2] and the Wireworld computer, a Turing-complete computer implemented as a cellular automaton.[3] See also • von Neumann's cellular automaton References 1. Dewdney, A K (January 1990). "Computer recreations: The cellular automata programs that create Wireworld, Rugworld and other diversions". Scientific American. 262 (1): 146–149. JSTOR 24996654. Retrieved 2 December 2018. 2. Nyles Heise. "Wireworld". Archived from the original on 2011-02-04. 3. Mark Owen. "The Wireworld Computer". External links • Wireworld on Rosetta Code • The Wireworld computer in Java • No Wires (contains an interactive Wireworld widget)
Wirtinger's inequality for functions In the mathematical field of analysis, the Wirtinger inequality is an important inequality for functions of a single variable, named after Wilhelm Wirtinger. It was used by Adolf Hurwitz in 1901 to give a new proof of the isoperimetric inequality for curves in the plane. A variety of closely related results are today known as Wirtinger's inequality, all of which can be viewed as certain forms of the Poincaré inequality. For other inequalities named after Wirtinger, see Wirtinger's inequality. Theorem There are several inequivalent versions of the Wirtinger inequality: • Let y be a continuous and differentiable function on the interval [0, L] with average value zero and with y(0) = y(L). Then $\int _{0}^{L}y(x)^{2}\,\mathrm {d} x\leq {\frac {L^{2}}{4\pi ^{2}}}\int _{0}^{L}y'(x)^{2}\,\mathrm {d} x,$ and equality holds if and only if y(x) = c sin 2π(x − α)/L for some numbers c and α.[1] • Let y be a continuous and differentiable function on the interval [0, L] with y(0) = y(L) = 0. Then $\int _{0}^{L}y(x)^{2}\,\mathrm {d} x\leq {\frac {L^{2}}{\pi ^{2}}}\int _{0}^{L}y'(x)^{2}\,\mathrm {d} x,$ and equality holds if and only if y(x) = c sin πx/L for some number c.[1] • Let y be a continuous and differentiable function on the interval [0, L] with average value zero. Then $\int _{0}^{L}y(x)^{2}\,\mathrm {d} x\leq {\frac {L^{2}}{\pi ^{2}}}\int _{0}^{L}y'(x)^{2}\,\mathrm {d} x.$ and equality holds if and only if y(x) = c cos πx/L for some number c.[2] Despite their differences, these are closely related to one another, as can be seen from the account given below in terms of spectral geometry. They can also all be regarded as special cases of various forms of the Poincaré inequality, with the optimal Poincaré constant identified explicitly. The middle version is also a special case of the Friedrichs inequality, again with the optimal constant identified. Proofs The three versions of the Wirtinger inequality can all be proved by various means. This is illustrated in the following by a different kind of proof for each of the three Wirtinger inequalities given above. In each case, by a linear change of variables in the integrals involved, there is no loss of generality in only proving the theorem for one particular choice of L. Fourier series Consider the first Wirtinger inequality given above. Take L to be 2π. Since Dirichlet's conditions are met, we can write $y(x)={\frac {1}{2}}a_{0}+\sum _{n\geq 1}\left(a_{n}{\frac {\sin nx}{\sqrt {\pi }}}+b_{n}{\frac {\cos nx}{\sqrt {\pi }}}\right),$ and the fact that the average value of y is zero means that a0 = 0. By Parseval's identity, $\int _{0}^{2\pi }y(x)^{2}\,\mathrm {d} x=\sum _{n=1}^{\infty }(a_{n}^{2}+b_{n}^{2})$ and $\int _{0}^{2\pi }y'(x)^{2}\,\mathrm {d} x=\sum _{n=1}^{\infty }n^{2}(a_{n}^{2}+b_{n}^{2})$ and since the summands are all nonnegative, the Wirtinger inequality is proved. Furthermore it is seen that equality holds if and only if an = bn = 0 for all n ≥ 2, which is to say that y(x) = a1 sin x + b1 cos x. This is equivalent to the stated condition by use of the trigonometric addition formulas. Integration by parts Consider the second Wirtinger inequality given above.[1] Take L to be π. Any differentiable function y(x) satisfies the identity $y(x)^{2}+{\big (}y'(x)-y(x)\cot x{\big )}^{2}=y'(x)^{2}-{\frac {d}{dx}}{\big (}y(x)^{2}\cot x{\big )}.$ Integration using the fundamental theorem of calculus and the boundary conditions y(0) = y(π) = 0 then shows $\int _{0}^{\pi }y(x)^{2}\,\mathrm {d} x+\int _{0}^{\pi }{\big (}y'(x)-y(x)\cot x{\big )}^{2}\,\mathrm {d} x=\int _{0}^{\pi }y'(x)^{2}\,\mathrm {d} x.$ This proves the Wirtinger inequality, since the second integral is clearly nonnegative. Furthermore, equality in the Wirtinger inequality is seen to be equivalent to y′(x) = y(x) cot x, the general solution of which (as computed by separation of variables) is y(x) = c sin x for an arbitrary number c. There is a subtlety in the above application of the fundamental theorem of calculus, since it is not the case that y(x)2 cot x extends continuously to x = 0 and x = π for every function y(x). This is resolved as follows. It follows from the Hölder inequality and y(0) = 0 that $\|y(x)\|=\left\|\int _{0}^{x}y'(x)\,\mathrm {d} x\right\|\leq \int _{0}^{x}\|y'(x)\|\,\mathrm {d} x\leq {\sqrt {x}}\left(\int _{0}^{x}y'(x)^{2}\,\mathrm {d} x\right)^{1/2},$ which shows that as long as $\int _{0}^{\pi }y'(x)^{2}\,\mathrm {d} x$ is finite, the limit of 1/x y(x)2 as x converges to zero is zero. Since cot x < 1/x for small positive values of x, it follows from the squeeze theorem that y(x)2 cot x converges to zero as x converges to zero. In exactly the same way, it can be proved that y(x)2 cot x converges to zero as x converges to π. Functional analysis Consider the third Wirtinger inequality given above. Take L to be 1. Given a continuous function f on [0, 1] of average value zero, let Tf) denote the function u on [0, 1] which is of average value zero, and with u′′ + f = 0 and u′(0) = u′(1) = 0. From basic analysis of ordinary differential equations with constant coefficients, the eigenvalues of T are (kπ)−2 for nonzero integers k, the largest of which is then π−2. Because T is a bounded and self-adjoint operator, it follows that $\int _{0}^{1}Tf(x)^{2}\,\mathrm {d} x\leq \pi ^{-2}\int _{0}^{1}f(x)Tf(x)\,\mathrm {d} x={\frac {1}{\pi ^{2}}}\int _{0}^{1}(Tf)'(x)^{2}\,\mathrm {d} x$ for all f of average value zero, where the equality is due to integration by parts. Finally, for any continuously differentiable function y on [0, 1] of average value zero, let gn be a sequence of compactly supported continuously differentiable functions on (0, 1) which converge in L2 to y′. Then define $y_{n}(x)=\int _{0}^{x}g_{n}(z)\,\mathrm {d} z-\int _{0}^{1}\int _{0}^{w}g_{n}(z)\,\mathrm {d} z\,\mathrm {d} w.$ Then each yn has average value zero with yn′(0) = yn′(1) = 0, which in turn implies that −yn′′ has average value zero. So application of the above inequality to f = −yn′′ is legitimate and shows that $\int _{0}^{1}y_{n}(x)^{2}\,\mathrm {d} x\leq {\frac {1}{\pi ^{2}}}\int _{0}^{1}y_{n}'(x)^{2}\,\mathrm {d} x.$ It is possible to replace yn by y, and thereby prove the Wirtinger inequality, as soon as it is verified that yn converges in L2 to y. This is verified in a standard way, by writing $y(x)-y_{n}(x)=\int _{0}^{x}{\big (}y_{n}'(z)-g_{n}(z){\big )}\,\mathrm {d} z-\int _{0}^{1}\int _{0}^{w}(y_{n}'(z)-g_{n}(z){\big )}\,\mathrm {d} z\,\mathrm {d} w$ and applying the Hölder or Jensen inequalities. This proves the Wirtinger inequality. In the case that y(x) is a function for which equality in the Wirtinger inequality holds, then a standard argument in the calculus of variations says that y must be a weak solution of the Euler–Lagrange equation y′′(x) + y(x) = 0 with y′(0) = y′(1) = 0, and the regularity theory of such equations, followed by the usual analysis of ordinary differential equations with constant coefficients, shows that y(x) = c cos πx for some number c. To make this argument fully formal and precise, it is necessary to be more careful about the function spaces in question.[2] Spectral geometry In the language of spectral geometry, the three versions of the Wirtinger inequality above can be rephrased as theorems about the first eigenvalue and corresponding eigenfunctions of the Laplace–Beltrami operator on various one-dimensional Riemannian manifolds:[3] • the first eigenvalue of the Laplace–Beltrami operator on the Riemannian circle of length L is 4π2/L2, and the corresponding eigenfunctions are the linear combinations of the two coordinate functions. • the first Dirichlet eigenvalue of the Laplace–Beltrami operator on the interval [0, L] is π2/L2 and the corresponding eigenfunctions are given by c sin πx/L for arbitrary nonzero numbers c. • the first Neumann eigenvalue of the Laplace–Beltrami operator on the interval [0, L] is π2/L2 and the corresponding eigenfunctions are given by c cos πx/L for arbitrary nonzero numbers c. These can also be extended to statements about higher-dimensional spaces. For example, the Riemannian circle may be viewed as the one-dimensional version of either a sphere, real projective space, or torus (of arbitrary dimension). The Wirtinger inequality, in the first version given here, can then be seen as the n = 1 case of any of the following: • the first eigenvalue of the Laplace–Beltrami operator on the unit-radius n-dimensional sphere is n, and the corresponding eigenfunctions are the linear combinations of the n + 1 coordinate functions.[4] • the first eigenvalue of the Laplace–Beltrami operator on the n-dimensional real projective space (with normalization given by the covering map from the unit-radius sphere) is 2n + 2, and the corresponding eigenfunctions are the restrictions of the homogeneous quadratic polynomials on Rn + 1 to the unit sphere (and then to the real projective space).[5] • the first eigenvalue of the Laplace–Beltrami operator on the n-dimensional torus (given as the n-fold product of the circle of length 2π with itself) is 1, and the corresponding eigenfunctions are arbitrary linear combinations of n-fold products of the eigenfunctions on the circles.[6] The second and third versions of the Wirtinger inequality can be extended to statements about first Dirichlet and Neumann eigenvalues of the Laplace−Beltrami operator on metric balls in Euclidean space: • the first Dirichlet eigenvalue of the Laplace−Beltrami operator on the unit ball in Rn is the square of the smallest positive zero of the Bessel function of the first kind J(n − 2)/2.[7] • the first Neumann eigenvalue of the Laplace−Beltrami operator on the unit ball in Rn is the square of the smallest positive zero of the first derivative of the Bessel function of the first kind Jn/2.[7] Application to the isoperimetric inequality In the first form given above, the Wirtinger inequality can be used to prove the isoperimetric inequality for curves in the plane, as found by Adolf Hurwitz in 1901.[8] Let (x, y) be a differentiable embedding of the circle in the plane. Parametrizing the circle by [0, 2π] so that (x, y) has constant speed, the length L of the curve is given by $\int _{0}^{2\pi }{\sqrt {x'(t)^{2}+y'(t)^{2}}}\,\mathrm {d} t$ and the area A enclosed by the curve is given (due to Stokes theorem) by $-\int _{0}^{2\pi }y(t)x'(t)\,\mathrm {d} t.$ Since the integrand of the integral defining L is assumed constant, there is ${\frac {L^{2}}{2\pi }}-2A=\int _{0}^{2\pi }{\big (}x'(t)^{2}+y'(t)^{2}+2y(t)x'(t){\big )}\,\mathrm {d} t$ which can be rewritten as $\int _{0}^{2\pi }{\big (}x'(t)+y(t){\big )}^{2}\,\mathrm {d} t+\int _{0}^{2\pi }{\big (}y'(t)^{2}-y(t)^{2}{\big )}\,\mathrm {d} t.$ The first integral is clearly nonnegative. Without changing the area or length of the curve, (x, y) can be replaced by (x, y + z) for some number z, so as to make y have average value zero. Then the Wirtinger inequality can be applied to see that the second integral is also nonnegative, and therefore ${\frac {L^{2}}{4\pi }}\geq A,$ which is the isoperimetric inequality. Furthermore, equality in the isoperimetric inequality implies both equality in the Wirtinger inequality and also the equality x′(t) + y(t) = 0, which amounts to y(t) = c1 sin(t – α) and then x(t) = c1 cos(t – α) + c2 for arbitrary numbers c1 and c2. These equations mean that the image of (x, y) is a round circle in the plane. References 1. Hardy, Littlewood & Pólya 1952, Section 7.7. 2. Brezis 2011, pp. 511–513, 576–578. 3. Chavel 1984, Sections I.3 and I.5. 4. Stein & Weiss 1971, Chapter IV.2. 5. Chavel 1984, p. 36. 6. Chavel 1984, Section II.2. 7. Chavel 1984, Theorem II.5.4. 8. Hardy, Littlewood & Pólya 1952, Section 7.7; Hurwitz 1901. • Brezis, Haim (2011). Functional analysis, Sobolev spaces and partial differential equations. Universitext. New York: Springer. doi:10.1007/978-0-387-70914-7. ISBN 978-0-387-70913-0. MR 2759829. Zbl 1220.46002. • Chavel, Isaac (1984). Eigenvalues in Riemannian geometry. Pure and Applied Mathematics. Vol. 115. Orlando, FL: Academic Press. doi:10.1016/s0079-8169(08)x6051-9. ISBN 0-12-170640-0. MR 0768584. Zbl 0551.53001. • Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1952). Inequalities (Second edition of 1934 original ed.). Cambridge University Press. MR 0046395. Zbl 0047.05302. • Hurwitz, A. (1901). "Sur le problème des isopérimètres". Comptes Rendus des Séances de l'Académie des Sciences. 132: 401–403. JFM 32.0386.01. • Stein, Elias M.; Weiss, Guido (1971). Introduction to Fourier analysis on Euclidean spaces. Princeton Mathematical Series. Vol. 32. Princeton, NJ: Princeton University Press. MR 0304972. Zbl 0232.42007.
Wirtinger's representation and projection theorem In mathematics, Wirtinger's representation and projection theorem is a theorem proved by Wilhelm Wirtinger in 1932 in connection with some problems of approximation theory. This theorem gives the representation formula for the holomorphic subspace $\left.\right.H_{2}$ of the simple, unweighted holomorphic Hilbert space $\left.\right.L^{2}$ of functions square-integrable over the surface of the unit disc $\left.\right.\{z:\|z\|<1\}$ of the complex plane, along with a form of the orthogonal projection from $\left.\right.L^{2}$ to $\left.\right.H_{2}$. Wirtinger's paper [1] contains the following theorem presented also in Joseph L. Walsh's well-known monograph [2] (p. 150) with a different proof. If $\left.\right.\left.F(z)\right.$ is of the class $\left.\right.L^{2}$ on $\left.\right.\|z\|<1$, i.e. $\iint _{\|z\|<1}\|F(z)\|^{2}\,dS<+\infty ,$ where $\left.\right.dS$ is the area element, then the unique function $\left.\right.f(z)$ of the holomorphic subclass $H_{2}\subset L^{2}$, such that $\iint _{\|z\|<1}\|F(z)-f(z)\|^{2}\,dS$ is least, is given by $f(z)={\frac {1}{\pi }}\iint _{\|\zeta \|<1}F(\zeta ){\frac {dS}{(1-{\overline {\zeta }}z)^{2}}},\quad \|z\|<1.$ The last formula gives a form for the orthogonal projection from $\left.\right.L^{2}$ to $\left.\right.H_{2}$. Besides, replacement of $\left.\right.F(\zeta )$ by $\left.\right.f(\zeta )$ makes it Wirtinger's representation for all $f(z)\in H_{2}$. This is an analog of the well-known Cauchy integral formula with the square of the Cauchy kernel. Later, after the 1950s, a degree of the Cauchy kernel was called reproducing kernel, and the notation $\left.\right.A_{0}^{2}$ became common for the class $\left.\right.H_{2}$. In 1948 Mkhitar Djrbashian[3] extended Wirtinger's representation and projection to the wider, weighted Hilbert spaces $\left.\right.A_{\alpha }^{2}$ of functions $\left.\right.f(z)$ holomorphic in $\left.\right.\|z\|<1$, which satisfy the condition $\\|f\\|_{A_{\alpha }^{2}}=\left\{{\frac {1}{\pi }}\iint _{\|z\|<1}\|f(z)\|^{2}(1-\|z\|^{2})^{\alpha -1}\,dS\right\}^{1/2}<+\infty {\text{ for some }}\alpha \in (0,+\infty ),$ and also to some Hilbert spaces of entire functions. The extensions of these results to some weighted $\left.\right.A_{\omega }^{2}$ spaces of functions holomorphic in $\left.\right.\|z\|<1$ and similar spaces of entire functions, the unions of which respectively coincide with all functions holomorphic in $\left.\right.\|z\|<1$ and the whole set of entire functions can be seen in.[4] See also • Jerbashian, A. M.; V. S. Zakaryan (2009). "The Contemporary Development in M. M. Djrbashian Factorization Theory and Related Problems of Analysis". Izv. NAN of Armenia, Matematika (English translation: Journal of Contemporary Mathematical Analysis). 44 (6). References 1. Wirtinger, W. (1932). "Uber eine Minimumaufgabe im Gebiet der analytischen Functionen". Monatshefte für Mathematik und Physik. 39: 377–384. doi:10.1007/bf01699078. S2CID 120529823. 2. Walsh, J. L. (1956). "Interpolation and Approximation by Rational Functions in the Complex Domain". Amer. Math. Soc. Coll. Publ. XX. Ann Arbor, Michigan: Edwards Brothers, Inc. 3. Djrbashian, M. M. (1948). "On the Representability Problem of Analytic Functions" (PDF). Soobsch. Inst. Matem. I Mekh. Akad. Nauk Arm. SSR. 2: 3–40. 4. Jerbashian, A. M. (2005). "On the Theory of Weighted Classes of Area Integrable Regular Functions". Complex Variables. 50 (3): 155–183. doi:10.1080/02781070500032846. S2CID 218556016. Functional analysis (topics – glossary) Spaces • Banach • Besov • Fréchet • Hilbert • Hölder • Nuclear • Orlicz • Schwartz • Sobolev • Topological vector Properties • Barrelled • Complete • Dual (Algebraic/Topological) • Locally convex • Reflexive • Reparable Theorems • Hahn–Banach • Riesz representation • Closed graph • Uniform boundedness principle • Kakutani fixed-point • Krein–Milman • Min–max • Gelfand–Naimark • Banach–Alaoglu Operators • Adjoint • Bounded • Compact • Hilbert–Schmidt • Normal • Nuclear • Trace class • Transpose • Unbounded • Unitary Algebras • Banach algebra • C-algebra • Spectrum of a C-algebra • Operator algebra • Group algebra of a locally compact group • Von Neumann algebra Open problems • Invariant subspace problem • Mahler's conjecture Applications • Hardy space • Spectral theory of ordinary differential equations • Heat kernel • Index theorem • Calculus of variations • Functional calculus • Integral operator • Jones polynomial • Topological quantum field theory • Noncommutative geometry • Riemann hypothesis • Distribution (or Generalized functions) Advanced topics • Approximation property • Balanced set • Choquet theory • Weak topology • Banach–Mazur distance • Tomita–Takesaki theory • Mathematics portal • Category • Commons
Wirtinger derivatives In complex analysis of one and several complex variables, Wirtinger derivatives (sometimes also called Wirtinger operators[1]), named after Wilhelm Wirtinger who introduced them in 1927 in the course of his studies on the theory of functions of several complex variables, are partial differential operators of the first order which behave in a very similar manner to the ordinary derivatives with respect to one real variable, when applied to holomorphic functions, antiholomorphic functions or simply differentiable functions on complex domains. These operators permit the construction of a differential calculus for such functions that is entirely analogous to the ordinary differential calculus for functions of real variables.[2] Historical notes Early days (1899–1911): the work of Henri Poincaré Wirtinger derivatives were used in complex analysis at least as early as in the paper (Poincaré 1899), as briefly noted by Cherry & Ye (2001, p. 31) and by Remmert (1991, pp. 66–67).[3] As a matter of fact, in the third paragraph of his 1899 paper,[4] Henri Poincaré first defines the complex variable in $\mathbb {C} ^{n}$ and its complex conjugate as follows ${\begin{cases}x_{k}+iy_{k}=z_{k}\\x_{k}-iy_{k}=u_{k}\end{cases}}\qquad 1\leqslant k\leqslant n.$ Then he writes the equation defining the functions $V$ he calls biharmonique,[5] previously written using partial derivatives with respect to the real variables $x_{k},y_{q}$ with $k,q$ ranging from 1 to $n$, exactly in the following way[6] ${\frac {d^{2}V}{dz_{k}\,du_{q}}}=0$ This implies that he implicitly used definition 2 below: to see this it is sufficient to compare equations 2 and 2' of (Poincaré 1899, p. 112). Apparently, this paper was not noticed by early researchers in the theory of functions of several complex variables: in the papers of Levi-Civita (1905), Levi (1910) (and Levi 1911) and of Amoroso (1912) all fundamental partial differential operators of the theory are expressed directly by using partial derivatives respect to the real and imaginary parts of the complex variables involved. In the long survey paper by Osgood (1966) (first published in 1913),[7] partial derivatives with respect to each complex variable of a holomorphic function of several complex variables seem to be meant as formal derivatives: as a matter of fact when Osgood expresses the pluriharmonic operator[8] and the Levi operator, he follows the established practice of Amoroso, Levi and Levi-Civita. The work of Dimitrie Pompeiu in 1912 and 1913: a new formulation According to Henrici (1993, p. 294), a new step in the definition of the concept was taken by Dimitrie Pompeiu: in the paper (Pompeiu 1912), given a complex valued differentiable function (in the sense of real analysis) of one complex variable $g(z)$ defined in the neighbourhood of a given point $z_{0}\in \mathbb {C} ,$ he defines the areolar derivative as the following limit ${{\frac {\partial g}{\partial {\bar {z}}}}(z_{0})}\mathrel {\overset {\mathrm {def} }{=}} \lim _{r\to 0}{\frac {1}{2\pi ir^{2}}}\oint _{\Gamma (z_{0},r)}g(z)\mathrm {d} z,$ where $\Gamma (z_{0},r)=\partial D(z_{0},r)$ is the boundary of a disk of radius $r$ entirely contained in the domain of definition of $g(z),$ i.e. his bounding circle.[9] This is evidently an alternative definition of Wirtinger derivative respect to the complex conjugate variable:[10] it is a more general one, since, as noted a by Henrici (1993, p. 294), the limit may exist for functions that are not even differentiable at $z=z_{0}.$[11] According to Fichera (1969, p. 28), the first to identify the areolar derivative as a weak derivative in the sense of Sobolev was Ilia Vekua.[12] In his following paper, Pompeiu (1913) uses this newly defined concept in order to introduce his generalization of Cauchy's integral formula, the now called Cauchy–Pompeiu formula. The work of Wilhelm Wirtinger The first systematic introduction of Wirtinger derivatives seems due to Wilhelm Wirtinger in the paper Wirtinger 1927 in order to simplify the calculations of quantities occurring in the theory of functions of several complex variables: as a result of the introduction of these differential operators, the form of all the differential operators commonly used in the theory, like the Levi operator and the Cauchy–Riemann operator, is considerably simplified and consequently easier to handle. The paper is deliberately written from a formal point of view, i.e. without giving a rigorous derivation of the properties deduced. Formal definition Despite their ubiquitous use,[13] it seems that there is no text listing all the properties of Wirtinger derivatives: however, fairly complete references are the short course on multidimensional complex analysis by Andreotti (1976, pp. 3–5),[14] the monograph of Gunning & Rossi (1965, pp. 3–6),[15] and the monograph of Kaup & Kaup (1983, p. 2,4)[16] which are used as general references in this and the following sections. Functions of one complex variable Definition 1. Consider the complex plane $\mathbb {C} \equiv \mathbb {R} ^{2}=\{(x,y)\mid x,y\in \mathbb {R} \}$ (in a sense of expressing a complex number $z=x+iy$ for real numbers $x$ and $y$). The Wirtinger derivatives are defined as the following linear partial differential operators of first order: ${\begin{aligned}{\frac {\partial }{\partial z}}&={\frac {1}{2}}\left({\frac {\partial }{\partial x}}-i{\frac {\partial }{\partial y}}\right)\\{\frac {\partial }{\partial {\bar {z}}}}&={\frac {1}{2}}\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right)\end{aligned}}$ Clearly, the natural domain of definition of these partial differential operators is the space of $C^{1}$ functions on a domain $\Omega \subseteq \mathbb {R} ^{2},$ but, since these operators are linear and have constant coefficients, they can be readily extended to every space of generalized functions. Functions of n > 1 complex variables Definition 2. Consider the Euclidean space on the complex field $\mathbb {C} ^{n}=\mathbb {R} ^{2n}=\left\{\left(\mathbf {x} ,\mathbf {y} \right)=\left(x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n}\right)\mid \mathbf {x} ,\mathbf {y} \in \mathbb {R} ^{n}\right\}.$ The Wirtinger derivatives are defined as the following linear partial differential operators of first order: ${\begin{cases}{\frac {\partial }{\partial z_{1}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x_{1}}}-i{\frac {\partial }{\partial y_{1}}}\right)\\\qquad \vdots \\{\frac {\partial }{\partial z_{n}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x_{n}}}-i{\frac {\partial }{\partial y_{n}}}\right)\\\end{cases}},\qquad {\begin{cases}{\frac {\partial }{\partial {\bar {z}}_{1}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x_{1}}}+i{\frac {\partial }{\partial y_{1}}}\right)\\\qquad \vdots \\{\frac {\partial }{\partial {\bar {z}}_{n}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x_{n}}}+i{\frac {\partial }{\partial y_{n}}}\right)\\\end{cases}}.$ As for Wirtinger derivatives for functions of one complex variable, the natural domain of definition of these partial differential operators is again the space of $C^{1}$ functions on a domain $\Omega \subset \mathbb {R} ^{2n},$ and again, since these operators are linear and have constant coefficients, they can be readily extended to every space of generalized functions. Relation with complex differentiation Wirtinger derivatives are closely related with complex differentiation (differentiation with respect to a complex variable $z=x+iy$ where $x$ and $y$ are real variables). The first Wirtinger derivative in the definition 1 is really differentiation with respect to $z$. For a complex function $f(z)=u(z)+iv(z)$ which is complex differentiable (equivalent to satisfying the Cauchy-Riemann equations ${\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}},{\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}$), ${\begin{aligned}{\frac {\partial f}{\partial z}}&={\frac {1}{2}}\left({\frac {\partial f}{\partial x}}-i{\frac {\partial f}{\partial y}}\right)\\&={\frac {1}{2}}\left({\frac {\partial u}{\partial x}}+i{\frac {\partial v}{\partial x}}-i{\frac {\partial u}{\partial y}}+{\frac {\partial v}{\partial y}}\right)\\&={\frac {\partial u}{\partial x}}+i{\frac {\partial v}{\partial x}}={\frac {\partial f}{\partial x}}\end{aligned}}$ where the 3rd equality uses the Cauchy-Riemann equations. Because the complex derivative is independent of the choice of a path in differentiation, the first Wirtinger derivative is the complex derivative. The second Wirtinger derivative is also related with complex differentiation; ${\frac {\partial f}{\partial {\bar {z}}}}=0$ is equivalent to the Cauchy-Riemann equations in a complex form. Basic properties In the present section and in the following ones it is assumed that $z\in \mathbb {C} ^{n}$ is a complex vector and that $z\equiv (x,y)=(x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n})$ where $x,y$ are real vectors, with n ≥ 1: also it is assumed that the subset $\Omega $ can be thought of as a domain in the real euclidean space $\mathbb {R} ^{2n}$ or in its isomorphic complex counterpart $\mathbb {C} ^{n}.$ All the proofs are easy consequences of definition 1 and definition 2 and of the corresponding properties of the derivatives (ordinary or partial). Linearity Lemma 1. If $f,g\in C^{1}(\Omega )$ and $\alpha ,\beta $ are complex numbers, then for $i=1,\dots ,n$ the following equalities hold ${\begin{aligned}{\frac {\partial }{\partial z_{i}}}\left(\alpha f+\beta g\right)&=\alpha {\frac {\partial f}{\partial z_{i}}}+\beta {\frac {\partial g}{\partial z_{i}}}\\{\frac {\partial }{\partial {\bar {z}}_{i}}}\left(\alpha f+\beta g\right)&=\alpha {\frac {\partial f}{\partial {\bar {z}}_{i}}}+\beta {\frac {\partial g}{\partial {\bar {z}}_{i}}}\end{aligned}}$ Product rule Lemma 2. If $f,g\in C^{1}(\Omega ),$ then for $i=1,\dots ,n$ the product rule holds ${\begin{aligned}{\frac {\partial }{\partial z_{i}}}(f\cdot g)&={\frac {\partial f}{\partial z_{i}}}\cdot g+f\cdot {\frac {\partial g}{\partial z_{i}}}\\{\frac {\partial }{\partial {\bar {z}}_{i}}}(f\cdot g)&={\frac {\partial f}{\partial {\bar {z}}_{i}}}\cdot g+f\cdot {\frac {\partial g}{\partial {\bar {z}}_{i}}}\end{aligned}}$ This property implies that Wirtinger derivatives are derivations from the abstract algebra point of view, exactly like ordinary derivatives are. Chain rule This property takes two different forms respectively for functions of one and several complex variables: for the n > 1 case, to express the chain rule in its full generality it is necessary to consider two domains $\Omega '\subseteq \mathbb {C} ^{m}$ and $\Omega ''\subseteq \mathbb {C} ^{p}$ and two maps $g:\Omega '\to \Omega $ and $f:\Omega \to \Omega ''$ having natural smoothness requirements.[17] Functions of one complex variable Lemma 3.1 If $f,g\in C^{1}(\Omega ),$ and $g(\Omega )\subseteq \Omega ,$ then the chain rule holds ${\begin{aligned}{\frac {\partial }{\partial z}}(f\circ g)&=\left({\frac {\partial f}{\partial z}}\circ g\right){\frac {\partial g}{\partial z}}+\left({\frac {\partial f}{\partial {\bar {z}}}}\circ g\right){\frac {\partial {\bar {g}}}{\partial z}}\\{\frac {\partial }{\partial {\bar {z}}}}(f\circ g)&=\left({\frac {\partial f}{\partial z}}\circ g\right){\frac {\partial g}{\partial {\bar {z}}}}+\left({\frac {\partial f}{\partial {\bar {z}}}}\circ g\right){\frac {\partial {\bar {g}}}{\partial {\bar {z}}}}\end{aligned}}$ Functions of n > 1 complex variables Lemma 3.2 If $g\in C^{1}(\Omega ',\Omega )$ and $f\in C^{1}(\Omega ,\Omega ''),$ then for $i=1,\dots ,m$ the following form of the chain rule holds ${\begin{aligned}{\frac {\partial }{\partial z_{i}}}\left(f\circ g\right)&=\sum _{j=1}^{n}\left({\frac {\partial f}{\partial z_{j}}}\circ g\right){\frac {\partial g_{j}}{\partial z_{i}}}+\sum _{j=1}^{n}\left({\frac {\partial f}{\partial {\bar {z}}_{j}}}\circ g\right){\frac {\partial {\bar {g}}_{j}}{\partial z_{i}}}\\{\frac {\partial }{\partial {\bar {z}}_{i}}}\left(f\circ g\right)&=\sum _{j=1}^{n}\left({\frac {\partial f}{\partial z_{j}}}\circ g\right){\frac {\partial g_{j}}{\partial {\bar {z}}_{i}}}+\sum _{j=1}^{n}\left({\frac {\partial f}{\partial {\bar {z}}_{j}}}\circ g\right){\frac {\partial {\bar {g}}_{j}}{\partial {\bar {z}}_{i}}}\end{aligned}}$ Conjugation Lemma 4. If $f\in C^{1}(\Omega ),$ then for $i=1,\dots ,n$ the following equalities hold ${\begin{aligned}{\overline {\left({\frac {\partial f}{\partial z_{i}}}\right)}}&={\frac {\partial {\bar {f}}}{\partial {\bar {z}}_{i}}}\\{\overline {\left({\frac {\partial f}{\partial {\bar {z}}_{i}}}\right)}}&={\frac {\partial {\bar {f}}}{\partial z_{i}}}\end{aligned}}$ See also • CR–function • Dolbeault complex • Dolbeault operator • Pluriharmonic function Notes 1. See references Fichera 1986, p. 62 and Kracht & Kreyszig 1988, p. 10. 2. Some of the basic properties of Wirtinger derivatives are the same ones as the properties characterizing the ordinary (or partial) derivatives and used for the construction of the usual differential calculus. 3. Reference to the work Poincaré 1899 of Henri Poincaré is precisely stated by Cherry & Ye (2001), while Reinhold Remmert does not cite any reference to support his assertion. 4. See reference (Poincaré 1899, pp. 111–114) 5. These functions are precisely pluriharmonic functions, and the linear differential operator defining them, i.e. the operator in equation 2 of (Poincaré 1899, p. 112), is exactly the n-dimensional pluriharmonic operator. 6. See (Poincaré 1899, p. 112), equation 2': note that, throughout the paper, the symbol $d$ is used to signify partial differentiation respect to a given variable, instead of the now commonplace symbol ∂. 7. The corrected Dover edition of the paper (Osgood 1913) harv error: no target: CITEREFOsgood1913 (help) contains much important historical information on the early development of the theory of functions of several complex variables, and is therefore a useful source. 8. See Osgood (1966, pp. 23–24): curiously, he calls Cauchy–Riemann equations this set of equations. 9. This is the definition given by Henrici (1993, p. 294) in his approach to Pompeiu's work: as Fichera (1969, p. 27) remarks, the original definition of Pompeiu (1912) does not require the domain of integration to be a circle. See the entry areolar derivative for further information. 10. See the section "Formal definition" of this entry. 11. See problem 2 in Henrici 1993, p. 294 for one example of such a function. 12. See also the excellent book by Vekua (1962, p. 55), Theorem 1.31: If the generalized derivative $\partial _{\bar {z}}w\in $$L_{p}(\Omega )$, p > 1, then the function $w(z)$ has almost everywhere in $G$ a derivative in the sense of Pompeiu, the latter being equal to the Generalized derivative in the sense of Sobolev $\partial _{\bar {z}}w$. 13. With or without the attribution of the concept to Wilhelm Wirtinger: see, for example, the well known monograph Hörmander 1990, p. 1,23. 14. In this course lectures, Aldo Andreotti uses the properties of Wirtinger derivatives in order to prove the closure of the algebra of holomorphic functions under certain operations: this purpose is common to all references cited in this section. 15. This is a classical work on the theory of functions of several complex variables dealing mainly with its sheaf theoretic aspects: however, in the introductory sections, Wirtinger derivatives and a few other analytical tools are introduced and their application to the theory is described. 16. In this work, the authors prove some of the properties of Wirtinger derivatives also for the general case of $C^{1}$ functions: in this single aspect, their approach is different from the one adopted by the other authors cited in this section, and perhaps more complete. 17. See Kaup & Kaup 1983, p. 4 and also Gunning 1990, p. 5: Gunning considers the general case of $C^{1}$ functions but only for p = 1. References Andreotti 1976, p. 5 and Gunning & Rossi 1965, p. 6, as already pointed out, consider only holomorphic maps with p = 1: however, the resulting formulas are formally very similar. References Historical references • Amoroso, Luigi (1912), "Sopra un problema al contorno", Rendiconti del Circolo Matematico di Palermo (in Italian), 33 (1): 75–85, doi:10.1007/BF03015289, JFM 43.0453.03, S2CID 122956910. "On a boundary value problem" (free translation of the title) is the first paper where a set of (fairly complicate) necessary and sufficient conditions for the solvability of the Dirichlet problem for holomorphic functions of several variables is given. • Cherry, W.; Ye, Z. (2001), Nevanlinna's theory of value distribution: the second main theorem and its error terms, Springer Monographs in Mathematics, Berlin: Springer Verlag, pp. XII+202, ISBN 978-3-540-66416-1, MR 1831783, Zbl 0981.30001. • Fichera, Gaetano (1969), "Derivata areolare e funzioni a variazione limitata", Revue Roumaine de Mathématiques Pures et Appliquées (in Italian), XIV (1): 27–37, MR 0265616, Zbl 0201.10002. "Areolar derivative and functions of bounded variation" (free English translation of the title) is an important reference paper in the theory of areolar derivatives. • Levi, Eugenio Elia (1910), "Studii sui punti singolari essenziali delle funzioni analitiche di due o più variabili complesse", Annali di Matematica Pura ed Applicata, s. III (in Italian), XVII (1): 61–87, doi:10.1007/BF02419336, JFM 41.0487.01, S2CID 122678686. "Studies on essential singular points of analytic functions of two or more complex variables" (English translation of the title) is an important paper in the theory of functions of several complex variables, where the problem of determining what kind of hypersurface can be the boundary of a domain of holomorphy. • Levi, Eugenio Elia (1911), "Sulle ipersuperficie dello spazio a 4 dimensioni che possono essere frontiera del campo di esistenza di una funzione analitica di due variabili complesse", Annali di Matematica Pura ed Applicata, s. III (in Italian), XVIII (1): 69–79, doi:10.1007/BF02420535, JFM 42.0449.02, S2CID 120133326. "On the hypersurfaces of the 4-dimensional space that can be the boundary of the domain of existence of an analytic function of two complex variables" (English translation of the title) is another important paper in the theory of functions of several complex variables, investigating further the theory started in (Levi 1910). • Levi-Civita, Tullio (1905), "Sulle funzioni di due o più variabili complesse", Rendiconti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, 5 (in Italian), XIV (2): 492–499, JFM 36.0482.01. "On the functions of two or more complex variables" (free English translation of the title) is the first paper where a sufficient condition for the solvability of the Cauchy problem for holomorphic functions of several complex variables is given. • Osgood, William Fogg (1966) [1913], Topics in the theory of functions of several complex variables (unabridged and corrected ed.), New York: Dover, pp. IV+120, JFM 45.0661.02, MR 0201668, Zbl 0138.30901. • Peschl, Ernst (1932), "Über die Krümmung von Niveaukurven bei der konformen Abbildung einfachzusammenhängender Gebiete auf das Innere eines Kreises. Eine Verallgemeinerung eines Satzes von E. Study.", Mathematische Annalen (in German), 106: 574–594, doi:10.1007/BF01455902, JFM 58.1096.05, MR 1512774, S2CID 127138808, Zbl 0004.30001, available at DigiZeitschriften. • Poincaré, H. (1899), "Sur les propriétés du potentiel et sur les fonctions Abéliennes", Acta Mathematica (in French), 22 (1): 89–178, doi:10.1007/BF02417872, JFM 29.0370.02. • Pompeiu, D. (1912), "Sur une classe de fonctions d'une variable complexe", Rendiconti del Circolo Matematico di Palermo (in French), 33 (1): 108–113, doi:10.1007/BF03015292, JFM 43.0481.01, S2CID 120717465. • Pompeiu, D. (1913), "Sur une classe de fonctions d'une variable complexe et sur certaines équations intégrales", Rendiconti del Circolo Matematico di Palermo (in French), 35 (1): 277–281, doi:10.1007/BF03015607, S2CID 121616964. • Vekua, I. N. (1962), Generalized Analytic Functions, International Series of Monographs in Pure and Applied Mathematics, vol. 25, London–Paris–Frankfurt: Pergamon Press, pp. xxx+668, MR 0150320, Zbl 0100.07603 • Wirtinger, Wilhelm (1927), "Zur formalen Theorie der Funktionen von mehr komplexen Veränderlichen", Mathematische Annalen (in German), 97: 357–375, doi:10.1007/BF01447872, JFM 52.0342.03, S2CID 121149132, available at DigiZeitschriften. In this important paper, Wirtinger introduces several important concepts in the theory of functions of several complex variables, namely Wirtinger's derivatives and the tangential Cauchy-Riemann condition. Scientific references • Andreotti, Aldo (1976), Introduzione all'analisi complessa (Lezioni tenute nel febbraio 1972), Contributi del Centro Linceo Interdisciplinare di Scienze Matematiche e Loro Applicazioni (in Italian), vol. 24, Rome: Accademia Nazionale dei Lincei, p. 34, archived from the original on 2012-03-07, retrieved 2010-08-28. Introduction to complex analysis is a short course in the theory of functions of several complex variables, held in February 1972 at the Centro Linceo Interdisciplinare di Scienze Matematiche e Loro Applicazioni "Beniamino Segre". • Fichera, Gaetano (1986), "Unification of global and local existence theorems for holomorphic functions of several complex variables", Memorie della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, 8, 18 (3): 61–83, MR 0917525, Zbl 0705.32006. • Gunning, Robert C.; Rossi, Hugo (1965), Analytic Functions of Several Complex Variables, Prentice-Hall series in Modern Analysis, Englewood Cliffs, N.J.: Prentice-Hall, pp. xiv+317, ISBN 9780821869536, MR 0180696, Zbl 0141.08601. • Gunning, Robert C. (1990), Introduction to Holomorphic Functions of Several Variables. Volume I: Function Theory, Wadsworth & Brooks/Cole Mathematics Series, Belmont, California: Wadsworth & Brooks/Cole, pp. xx+203, ISBN 0-534-13308-8, MR 1052649, Zbl 0699.32001. • Henrici, Peter (1993) [1986], Applied and Computational Complex Analysis Volume 3, Wiley Classics Library (Reprint ed.), New York–Chichester–Brisbane–Toronto–Singapore: John Wiley & Sons, pp. X+637, ISBN 0-471-58986-1, MR 0822470, Zbl 1107.30300. • Hörmander, Lars (1990) [1966], An Introduction to Complex Analysis in Several Variables, North–Holland Mathematical Library, vol. 7 (3rd (Revised) ed.), Amsterdam–London–New York–Tokyo: North-Holland, ISBN 0-444-88446-7, MR 1045639, Zbl 0685.32001. • Kaup, Ludger; Kaup, Burchard (1983), Holomorphic functions of several variables, de Gruyter Studies in Mathematics, vol. 3, Berlin–New York: Walter de Gruyter, pp. XV+349, ISBN 978-3-11-004150-7, MR 0716497, Zbl 0528.32001. • Kracht, Manfred; Kreyszig, Erwin (1988), Methods of Complex Analysis in Partial Differential Equations and Applications, Canadian Mathematical Society Series of Monographs and Advanced Texts, New York–Chichester–Brisbane–Toronto–Singapore: John Wiley & Sons, pp. xiv+394, ISBN 0-471-83091-7, MR 0941372, Zbl 0644.35005. • Martinelli, Enzo (1984), Introduzione elementare alla teoria delle funzioni di variabili complesse con particolare riguardo alle rappresentazioni integrali, Contributi del Centro Linceo Interdisciplinare di Scienze Matematiche e Loro Applicazioni (in Italian), vol. 67, Rome: Accademia Nazionale dei Lincei, pp. 236+II, archived from the original on 2011-09-27, retrieved 2010-08-24. "Elementary introduction to the theory of functions of complex variables with particular regard to integral representations" (English translation of the title) are the notes form a course, published by the Accademia Nazionale dei Lincei, held by Martinelli when he was "Professore Linceo". • Remmert, Reinhold (1991), Theory of Complex Functions, Graduate Texts in Mathematics, vol. 122 (Fourth corrected 1998 printing ed.), New York–Berlin–Heidelberg–Barcelona–Hong Kong–London–Milan–Paris–Singapore–Tokyo: Springer Verlag, pp. xx+453, ISBN 0-387-97195-5, MR 1084167, Zbl 0780.30001 ISBN 978-0-387-97195-7. A textbook on complex analysis including many historical notes on the subject. • Severi, Francesco (1958), Lezioni sulle funzioni analitiche di più variabili complesse – Tenute nel 1956–57 all'Istituto Nazionale di Alta Matematica in Roma (in Italian), Padova: CEDAM – Casa Editrice Dott. Antonio Milani, pp. XIV+255, Zbl 0094.28002. Notes from a course held by Francesco Severi at the Istituto Nazionale di Alta Matematica (which at present bears his name), containing appendices of Enzo Martinelli, Giovanni Battista Rizza and Mario Benedicty. An English translation of the title reads as:-"Lectures on analytic functions of several complex variables – Lectured in 1956–57 at the Istituto Nazionale di Alta Matematica in Rome".
Wirtinger inequality (2-forms) In mathematics, the Wirtinger inequality, named after Wilhelm Wirtinger, is a fundamental result in complex linear algebra which relates the symplectic and volume forms of a hermitian inner product. It has important consequences in complex geometry, such as showing that the normalized exterior powers of the Kähler form of a Kähler manifold are calibrations. For other inequalities named after Wirtinger, see Wirtinger's inequality. Statement Consider a real vector space with positive-definite inner product g, symplectic form ω, and almost-complex structure J, linked by ω(u, v) = g(J(u), v) for any vectors u and v. Then for any orthonormal vectors v1, ..., v2k there is $(\underbrace {\omega \wedge \cdots \wedge \omega } _{k{\text{ times}}})(v_{1},\ldots ,v_{2k})\leq k!.$ There is equality if and only if the span of v1, ..., v2k is closed under the operation of J.[1] In the language of the comass of a form, the Wirtinger theorem (although without precision about when equality is achieved) can also be phrased as saying that the comass of the form ω ∧ ⋅⋅⋅ ∧ ω is equal to k!.[1] Proof k = 1 In the special case k = 1, the Wirtinger inequality is a special case of the Cauchy–Schwarz inequality: $\omega (v_{1},v_{2})=g(J(v_{1}),v_{2})\leq \\|J(v_{1})\\|_{g}\\|v_{2}\\|_{g}=1.$ According to the equality case of the Cauchy–Schwarz inequality, equality occurs if and only if J(v1) and v2 are collinear, which is equivalent to the span of v1, v2 being closed under J. k > 1 Let v1, ..., v2k be fixed, and let T denote their span. Then there is an orthonormal basis e1, ..., e2k of T with dual basis w1, ..., w2k such that $\iota ^{\ast }\omega =\sum _{j=1}^{k}\omega (e_{2j-1},e_{2j})w_{2j-1}\wedge w_{2j},$ where ι denotes the inclusion map from T into V.[2] This implies $\underbrace {\iota ^{\ast }\omega \wedge \cdots \wedge \iota ^{\ast }\omega } _{k{\text{ times}}}=k!\prod _{i=1}^{k}\omega (e_{2i-1},e_{2i})w_{1}\wedge \cdots \wedge w_{2k},$ which in turn implies $(\underbrace {\omega \wedge \cdots \wedge \omega } _{k{\text{ times}}})(e_{1},\ldots ,e_{2k})=k!\prod _{i=1}^{k}\omega (e_{2i-1},e_{2i})\leq k!,$ where the inequality follows from the previously-established k = 1 case. If equality holds, then according to the k = 1 equality case, it must be the case that ω(e2i − 1, e2i) = ±1 for each i. This is equivalent to either ω(e2i − 1, e2i) = 1 or ω(e2i, e2i − 1) = 1, which in either case (from the k = 1 case) implies that the span of e2i − 1, e2i is closed under J, and hence that the span of e1, ..., e2k is closed under J. Finally, the dependence of the quantity $(\underbrace {\omega \wedge \cdots \wedge \omega } _{k{\text{ times}}})(v_{1},\ldots ,v_{2k})$ on v1, ..., v2k is only on the quantity v1 ∧ ⋅⋅⋅ ∧ v2k, and from the orthonormality condition on v1, ..., v2k, this wedge product is well-determined up to a sign. This relates the above work with e1, ..., e2k to the desired statement in terms of v1, ..., v2k. Consequences Given a complex manifold with hermitian metric, the Wirtinger theorem immediately implies that for any 2k-dimensional embedded submanifold M, there is $\operatorname {vol} (M)\geq {\frac {1}{k!}}\int _{M}\omega ^{k},$ where ω is the Kähler form of the metric. Furthermore, equality is achieved if and only if M is a complex submanifold.[3] In the special case that the hermitian metric satisfies the Kähler condition, this says that 1/k!ωk is a calibration for the underlying Riemannian metric, and that the corresponding calibrated submanifolds are the complex submanifolds of complex dimension k.[4] This says in particular that every complex submanifold of a Kähler manifold is a minimal submanifold, and is even volume-minimizing among all submanifolds in its homology class. Using the Wirtinger inequality, these facts even extend to the more sophisticated context of currents in Kähler manifolds.[5] See also • Gromov's inequality for complex projective space • Systolic geometry Notes 1. Federer 1969, Section 1.8.2. 2. McDuff & Salamon 2017, Lemma 2.4.5. 3. Griffiths & Harris 1978, Section 0.2. 4. Harvey & Lawson 1982. 5. Federer 1969, Section 5.4.19. References • Federer, Herbert (1969). Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften. Vol. 153. Berlin–Heidelberg–New York: Springer-Verlag. doi:10.1007/978-3-642-62010-2. ISBN 978-3-540-60656-7. MR 0257325. Zbl 0176.00801. • Griffiths, Phillip; Harris, Joseph (1978). Principles of algebraic geometry. Pure and Applied Mathematics. New York: John Wiley & Sons. ISBN 0-471-32792-1. MR 0507725. Zbl 0408.14001. • Harvey, Reese; Lawson, H. Blaine, Jr. (1982). "Calibrated geometries". Acta Mathematica. 148: 47–157. doi:10.1007/BF02392726. MR 0666108. Zbl 0584.53021.{{cite journal}}: CS1 maint: multiple names: authors list (link) • McDuff, Dusa; Salamon, Dietmar (2017). Introduction to symplectic topology. Oxford Graduate Texts in Mathematics (Third edition of 1995 original ed.). Oxford: Oxford University Press. doi:10.1093/oso/9780198794899.001.0001. ISBN 978-0-19-879490-5. MR 3674984. Zbl 1380.53003. • Wirtinger, W. (1936). "Eine Determinantenidentität und ihre Anwendung auf analytische Gebilde in euklidischer und Hermitescher Maßbestimmung". Monatshefte für Mathematik und Physik. 44: 343–365. doi:10.1007/BF01699328. MR 1550581. Zbl 0015.07602.
Wirtinger presentation In mathematics, especially in group theory, a Wirtinger presentation is a finite presentation where the relations are of the form $wg_{i}w^{-1}=g_{j}$ where $w$ is a word in the generators, $\{g_{1},g_{2},\ldots ,g_{k}\}.$ Wilhelm Wirtinger observed that the complements of knots in 3-space have fundamental groups with presentations of this form. Preliminaries and definition A knot K is an embedding of the one-sphere S1 in three-dimensional space R3. (Alternatively, the ambient space can also be taken to be the three-sphere S3, which does not make a difference for the purposes of the Wirtinger presentation.) The open subspace which is the complement of the knot, $S^{3}\setminus K$ is the knot complement. Its fundamental group $\pi _{1}(S^{3}\setminus K)$ is an invariant of the knot in the sense that equivalent knots have isomorphic knot groups. It is therefore interesting to understand this group in an accessible way. A Wirtinger presentation is derived from a regular projection of an oriented knot. Such a projection can be pictured as a finite number of (oriented) arcs in the plane, separated by the crossings of the projection. The fundamental group is generated by loops winding around each arc. Each crossing gives rise to a certain relation among the generators corresponding to the arcs meeting at the crossing. Wirtinger presentations of high-dimensional knots More generally, co-dimension two knots in spheres are known to have Wirtinger presentations. Michel Kervaire proved that an abstract group is the fundamental group of a knot exterior (in a perhaps high-dimensional sphere) if and only if all the following conditions are satisfied: 1. The abelianization of the group is the integers. 2. The 2nd homology of the group is trivial. 3. The group is finitely presented. 4. The group is the normal closure of a single generator. Conditions (3) and (4) are essentially the Wirtinger presentation condition, restated. Kervaire proved in dimensions 5 and larger that the above conditions are necessary and sufficient. Characterizing knot groups in dimension four is an open problem. Examples For the trefoil knot, a Wirtinger presentation can be shown to be $\pi _{1}(\mathbb {R} ^{3}\backslash {\text{trefoil}})=\langle x,y\mid yxy=xyx\rangle .$ See also • Knot group Further reading • Rolfsen, Dale (1990), Knots and links, Mathematics Lecture Series, vol. 7, Houston, TX: Publish or Perish, ISBN 978-0-914098-16-4, section 3D • Kawauchi, Akio (1996), A survey of knot theory, Birkhäuser, doi:10.1007/978-3-0348-9227-8, ISBN 978-3-0348-9953-6 • Hillman, Jonathan (2012), Algebraic invariants of links, Series on Knots and Everything, vol. 52, World Scientific, doi:10.1142/9789814407397, ISBN 9789814407397 • Livingston, Charles (1993), Knot Theory, The Mathematical Association of America
Wirtinger sextic In mathematics, the Wirtinger plane sextic curve, studied by Wirtinger, is a degree 6 genus 4 plane curve with double points at the 6 vertices of a complete quadrilateral. References • Coble, Arthur B. (1929), Algebraic geometry and theta functions, American Mathematical Society Colloquium Publications, vol. 10, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1010-1, MR 0733252 • Dolgachev, Igor. (2012), Classical Algebraic Geometry:a modern view, Cambridge.: Cambridge University Press, ISBN 978-1-107-01765-8, MR 2964027
Witness (mathematics) In mathematical logic, a witness is a specific value t to be substituted for variable x of an existential statement of the form ∃x φ(x) such that φ(t) is true. Examples For example, a theory T of arithmetic is said to be inconsistent if there exists a proof in T of the formula "0 = 1". The formula I(T), which says that T is inconsistent, is thus an existential formula. A witness for the inconsistency of T is a particular proof of "0 = 1" in T. Boolos, Burgess, and Jeffrey (2002:81) define the notion of a witness with the example, in which S is an n-place relation on natural numbers, R is an (n+1)-place recursive relation, and ↔ indicates logical equivalence (if and only if): S(x1, ..., xn) ↔ ∃y R(x1, . . ., xn, y) "A y such that R holds of the xi may be called a 'witness' to the relation S holding of the xi (provided we understand that when the witness is a number rather than a person, a witness only testifies to what is true)." In this particular example, the authors defined s to be (positively) recursively semidecidable, or simply semirecursive. Henkin witnesses In predicate calculus, a Henkin witness for a sentence $\exists x\,\varphi (x)$ in a theory T is a term c such that T proves φ(c) (Hinman 2005:196). The use of such witnesses is a key technique in the proof of Gödel's completeness theorem presented by Leon Henkin in 1949. Relation to game semantics The notion of witness leads to the more general idea of game semantics. In the case of sentence $\exists x\,\varphi (x)$ the winning strategy for the verifier is to pick a witness for $\varphi $. For more complex formulas involving universal quantifiers, the existence of a winning strategy for the verifier depends on the existence of appropriate Skolem functions. For example, if S denotes $\forall x\,\exists y\,\varphi (x,y)$ then an equisatisfiable statement for S is $\exists f\,\forall x\,\varphi (x,f(x))$. The Skolem function f (if it exists) actually codifies a winning strategy for the verifier of S by returning a witness for the existential sub-formula for every choice of x the falsifier might make. See also • Certificate (complexity), an analogous concept in computational complexity theory References • George S. Boolos, John P. Burgess, and Richard C. Jeffrey, 2002, Computability and Logic: Fourth Edition, Cambridge University Press, ISBN 0-521-00758-5. • Leon Henkin, 1949, "The completeness of the first-order functional calculus", Journal of Symbolic Logic v. 14 n. 3, pp. 159–166. • Peter G. Hinman, 2005, Fundamentals of mathematical logic, A.K. Peters, ISBN 1-56881-262-0. • J. Hintikka and G. Sandu, 2009, "Game-Theoretical Semantics" in Keith Allan (ed.) Concise Encyclopedia of Semantics, Elsevier, ISBN 0-08095-968-7, pp. 341–343
Disjunction and existence properties In mathematical logic, the disjunction and existence properties are the "hallmarks" of constructive theories such as Heyting arithmetic and constructive set theories (Rathjen 2005). Definitions • The disjunction property is satisfied by a theory if, whenever a sentence A ∨ B is a theorem, then either A is a theorem, or B is a theorem. • The existence property or witness property is satisfied by a theory if, whenever a sentence (∃x)A(x) is a theorem, where A(x) has no other free variables, then there is some term t such that the theory proves A(t). Related properties Rathjen (2005) lists five properties that a theory may possess. These include the disjunction property (DP), the existence property (EP), and three additional properties: • The numerical existence property (NEP) states that if the theory proves $(\exists x\in \mathbb {N} )\varphi (x)$, where φ has no other free variables, then the theory proves $\varphi ({\bar {n}})$ for some $n\in \mathbb {N} {\text{.}}$ Here ${\bar {n}}$ is a term in $T$ representing the number n. • Church's rule (CR) states that if the theory proves $(\forall x\in \mathbb {N} )(\exists y\in \mathbb {N} )\varphi (x,y)$ then there is a natural number e such that, letting $f_{e}$ be the computable function with index e, the theory proves $(\forall x)\varphi (x,f_{e}(x))$. • A variant of Church's rule, CR1, states that if the theory proves $(\exists f\colon \mathbb {N} \to \mathbb {N} )\psi (f)$ then there is a natural number e such that the theory proves $f_{e}$ is total and proves $\psi (f_{e})$. These properties can only be directly expressed for theories that have the ability to quantify over natural numbers and, for CR1, quantify over functions from $\mathbb {N} $ to $\mathbb {N} $. In practice, one may say that a theory has one of these properties if a definitional extension of the theory has the property stated above (Rathjen 2005). Results Non-examples and examples Almost by definition, a theory that accepts excluded middle while having independent statements does not have the disjunction property. So all classical theories expressing Robinson arithmetic do not have it. Most classical theories, such as Peano arithmetic and ZFC in turn do not validate the existence property either, e.g. because they validate the least number principle existence claim. But some classical theories, such as ZFC plus the axiom of constructibility, do have a weaker form of the existence property (Rathjen 2005). Heyting arithmetic is well known for having the disjunction property and the (numerical) existence property. While the earliest results were for constructive theories of arithmetic, many results are also known for constructive set theories (Rathjen 2005). John Myhill (1973) showed that IZF with the axiom of replacement eliminated in favor of the axiom of collection has the disjunction property, the numerical existence property, and the existence property. Michael Rathjen (2005) proved that CZF has the disjunction property and the numerical existence property. Freyd and Scedrov (1990) observed that the disjunction property holds in free Heyting algebras and free topoi. In categorical terms, in the free topos, that corresponds to the fact that the terminal object, $\mathbf {1} $, is not the join of two proper subobjects. Together with the existence property it translates to the assertion that $\mathbf {1} $ is an indecomposable projective object—the functor it represents (the global-section functor) preserves epimorphisms and coproducts. Relationship between properties There are several relationship between the five properties discussed above. In the setting of arithmetic, the numerical existence property implies the disjunction property. The proof uses the fact that a disjunction can be rewritten as an existential formula quantifying over natural numbers: $A\vee B\equiv (\exists n)[(n=0\to A)\wedge (n\neq 0\to B)]$. Therefore, if $A\vee B$ is a theorem of $T$, so is $\exists n\colon (n=0\to A)\wedge (n\neq 0\to B)$. Thus, assuming the numerical existence property, there exists some $s$ such that $({\bar {s}}=0\to A)\wedge ({\bar {s}}\neq 0\to B)$ is a theorem. Since ${\bar {s}}$ is a numeral, one may concretely check the value of $s$: if $s=0$ then $A$ is a theorem and if $s\neq 0$ then $B$ is a theorem. Harvey Friedman (1974) proved that in any recursively enumerable extension of intuitionistic arithmetic, the disjunction property implies the numerical existence property. The proof uses self-referential sentences in way similar to the proof of Gödel's incompleteness theorems. The key step is to find a bound on the existential quantifier in a formula (∃x)A(x), producing a bounded existential formula (∃x<n)A(x). The bounded formula may then be written as a finite disjunction A(1)∨A(2)∨...∨A(n). Finally, disjunction elimination may be used to show that one of the disjuncts is provable. History Kurt Gödel (1932) stated without proof that intuitionistic propositional logic (with no additional axioms) has the disjunction property; this result was proven and extended to intuitionistic predicate logic by Gerhard Gentzen (1934, 1935). Stephen Cole Kleene (1945) proved that Heyting arithmetic has the disjunction property and the existence property. Kleene's method introduced the technique of realizability, which is now one of the main methods in the study of constructive theories (Kohlenbach 2008; Troelstra 1973). See also • Constructive set theory • Heyting arithmetic • Law of excluded middle • Realizability • Existential quantifier References • Peter J. Freyd and Andre Scedrov, 1990, Categories, Allegories. North-Holland. • Harvey Friedman, 1975, The disjunction property implies the numerical existence property, State University of New York at Buffalo. • Gerhard Gentzen, 1934, "Untersuchungen über das logische Schließen. I", Mathematische Zeitschrift v. 39 n. 2, pp. 176–210. • Gerhard Gentzen, 1935, "Untersuchungen über das logische Schließen. II", Mathematische Zeitschrift v. 39 n. 3, pp. 405–431. • Kurt Gödel, 1932, "Zum intuitionistischen Aussagenkalkül", Anzeiger der Akademie der Wissenschaftischen in Wien, v. 69, pp. 65–66. • Stephen Cole Kleene, 1945, "On the interpretation of intuitionistic number theory," Journal of Symbolic Logic, v. 10, pp. 109–124. • Ulrich Kohlenbach, 2008, Applied proof theory, Springer. • John Myhill, 1973, "Some properties of Intuitionistic Zermelo-Fraenkel set theory", in A. Mathias and H. Rogers, Cambridge Summer School in Mathematical Logic, Lectures Notes in Mathematics v. 337, pp. 206–231, Springer. • Michael Rathjen, 2005, "The Disjunction and Related Properties for Constructive Zermelo-Fraenkel Set Theory", Journal of Symbolic Logic, v. 70 n. 4, pp. 1233–1254. • Anne S. Troelstra, ed. (1973), Metamathematical investigation of intuitionistic arithmetic and analysis, Springer. External links • Intuitionistic Logic by Joan Moschovakis, Stanford Encyclopedia of Philosophy
Witt algebra In mathematics, the complex Witt algebra, named after Ernst Witt, is the Lie algebra of meromorphic vector fields defined on the Riemann sphere that are holomorphic except at two fixed points. It is also the complexification of the Lie algebra of polynomial vector fields on a circle, and the Lie algebra of derivations of the ring C[z,z−1]. The Witt algebra is not directly related to the Witt ring of quadratic forms, or to the algebra of Witt vectors. There are some related Lie algebras defined over finite fields, that are also called Witt algebras. The complex Witt algebra was first defined by Élie Cartan (1909), and its analogues over finite fields were studied by Witt in the 1930s. Basis A basis for the Witt algebra is given by the vector fields $L_{n}=-z^{n+1}{\frac {\partial }{\partial z}}$, for n in $\mathbb {Z} $. The Lie bracket of two basis vector fields is given by $[L_{m},L_{n}]=(m-n)L_{m+n}.$ This algebra has a central extension called the Virasoro algebra that is important in two-dimensional conformal field theory and string theory. Note that by restricting n to 1,0,-1, one gets a subalgebra. Taken over the field of complex numbers, this is just the Lie algebra ${\mathfrak {sl}}(2,\mathbb {C} )$ of the Lorentz group $\mathrm {SO} (3,1)$. Over the reals, it is the algebra sl(2,R) = su(1,1). Conversely, su(1,1) suffices to reconstruct the original algebra in a presentation.[1] Over finite fields Over a field k of characteristic p>0, the Witt algebra is defined to be the Lie algebra of derivations of the ring k[z]/zp The Witt algebra is spanned by Lm for −1≤ m ≤ p−2. Images n = -1 Witt vector field n = 0 Witt vector field n = 1 Witt vector field n = -2 Witt vector field n = 2 Witt vector field n = -3 Witt vector field See also • Virasoro algebra • Heisenberg algebra References 1. D Fairlie, J Nuyts, and C Zachos (1988). Phys Lett B202 320-324. doi:10.1016/0370-2693(88)90478-9 • Élie Cartan, Les groupes de transformations continus, infinis, simples. Ann. Sci. Ecole Norm. Sup. 26, 93-161 (1909). • "Witt algebra", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Witt vector cohomology In mathematics, Witt vector cohomology was an early p-adic cohomology theory for algebraic varieties introduced by Serre (1958). Serre constructed it by defining a sheaf of truncated Witt rings Wn over a variety V and then taking the inverse limit of the sheaf cohomology groups Hi(V, Wn) of these sheaves. Serre observed that though it gives cohomology groups over a field of characteristic 0, it cannot be a Weil cohomology theory because the cohomology groups vanish when i > dim(V). For Abelian varieties Serre (1958b) showed that one could obtain a reasonable first cohomology group by taking the direct sum of the Witt vector cohomology and the Tate module of the Picard variety. References • Serre, J.P. (1958), "Sur la topologie des variétés algébriques en caractéristique p", 1958 Symposium internacional de topología algebraica, Mexico City: Universidad Nacional Autónoma de México and UNESCO, pp. 24–53, MR 0098097 • Serre, Jean-Pierre (1958b), "Quelques propriétés des variétés abéliennes en caractéristique p", Amer. J. Math., 80: 715–739, doi:10.2307/2372780, MR 0098100
Witten conjecture In algebraic geometry, the Witten conjecture is a conjecture about intersection numbers of stable classes on the moduli space of curves, introduced by Edward Witten in the paper Witten (1991), and generalized in Witten (1993). Witten's original conjecture was proved by Maxim Kontsevich in the paper Kontsevich (1992). Witten's motivation for the conjecture was that two different models of 2-dimensional quantum gravity should have the same partition function. The partition function for one of these models can be described in terms of intersection numbers on the moduli stack of algebraic curves, and the partition function for the other is the logarithm of the τ-function of the KdV hierarchy. Identifying these partition functions gives Witten's conjecture that a certain generating function formed from intersection numbers should satisfy the differential equations of the KdV hierarchy. Statement Suppose that Mg,n is the moduli stack of compact Riemann surfaces of genus g with n distinct marked points x1,...,xn, and Mg,n is its Deligne–Mumford compactification. There are n line bundles Li on Mg,n, whose fiber at a point of the moduli stack is given by the cotangent space of a Riemann surface at the marked point xi. The intersection index 〈τd1, ..., τdn〉 is the intersection index of Π c1(Li)di on Mg,n where Σdi = dimMg,n = 3g – 3 + n, and 0 if no such g exists, where c1 is the first Chern class of a line bundle. Witten's generating function $F(t_{0},t_{1},\ldots )=\sum \langle \tau _{0}^{k_{0}}\tau _{1}^{k_{1}}\cdots \rangle \prod _{i\geq 0}{\frac {t_{i}^{k_{i}}}{k_{i}!}}={\frac {t_{0}^{3}}{6}}+{\frac {t_{1}}{24}}+{\frac {t_{0}t_{2}}{24}}+{\frac {t_{1}^{2}}{24}}+{\frac {t_{0}^{2}t_{3}}{48}}+\cdots $ encodes all the intersection indices as its coefficients. Witten's conjecture states that the partition function Z = exp F is a τ-function for the KdV hierarchy, in other words it satisfies a certain series of partial differential equations corresponding to the basis $\{L_{-1},L_{0},L_{1},\ldots \}$ of the Virasoro algebra. Proof Kontsevich used a combinatorial description of the moduli spaces in terms of ribbon graphs to show that $\sum _{d_{1}+\cdots +d_{n}=3g-3+n}\langle \tau _{d_{1}},\ldots ,\tau _{d_{n}}\rangle \prod _{1\leq i\leq n}{\frac {(2d_{i}-1)!!}{\lambda _{i}^{2d_{i}+1}}}=\sum _{\Gamma \in G_{g,n}}{\frac {2^{-\|X_{0}\|}}{\|{\text{Aut}}\Gamma \|}}\prod _{e\in X_{1}}{\frac {2}{\lambda (e)}}$ Here the sum on the right is over the set Gg,n of ribbon graphs X of compact Riemann surfaces of genus g with n marked points. The set of edges e and points of X are denoted by X 0 and X1. The function λ is thought of as a function from the marked points to the reals, and extended to edges of the ribbon graph by setting λ of an edge equal to the sum of λ at the two marked points corresponding to each side of the edge. By Feynman diagram techniques, this implies that F(t0,...) is an asymptotic expansion of $\log \int \exp(i{\text{tr}}X^{3}/6)d\mu $ as Λ lends to infinity, where Λ and Χ are positive definite N by N hermitian matrices, and ti is given by $t_{i}={\frac {-{\text{tr }}\Lambda ^{-1-2i}}{1\times 3\times 5\times \cdots \times (2i-1)}}$ and the probability measure μ on the positive definite hermitian matrices is given by $d\mu =c_{\Lambda }\exp(-{\text{tr}}X^{2}\Lambda /2)dX$ where cΛ is a normalizing constant. This measure has the property that $\int X_{ij}X_{kl}d\mu =\delta _{il}\delta _{jk}{\frac {2}{\Lambda _{i}+\Lambda _{j}}}$ which implies that its expansion in terms of Feynman diagrams is the expression for F in terms of ribbon graphs. From this he deduced that exp F is a τ-function for the KdV hierarchy, thus proving Witten's conjecture. Generalizations The Witten conjecture is a special case of a more general relation between integrable systems of Hamiltonian PDEs and the geometry of certain families of 2D topological field theories (axiomatized in the form of the so-called cohomological field theories by Kontsevich and Manin), which was explored and studied systematically by B. Dubrovin and Y. Zhang, A. Givental, C. Teleman and others. The Virasoro conjecture is a generalization of the Witten conjecture. References • Cornalba, Maurizio; Arbarello, Enrico; Griffiths, Phillip A. (2011), Geometry of algebraic curves. Volume II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 268, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-540-69392-5, ISBN 978-3-540-42688-2, MR 2807457 • Kazarian, M. E.; Lando, Sergei K. (2007), "An algebro-geometric proof of Witten's conjecture", Journal of the American Mathematical Society, 20 (4): 1079–1089, arXiv:math/0601760, Bibcode:2007JAMS...20.1079K, doi:10.1090/S0894-0347-07-00566-8, ISSN 0894-0347, MR 2328716 • Kontsevich, Maxim (1992), "Intersection theory on the moduli space of curves and the matrix Airy function", Communications in Mathematical Physics, 147 (1): 1–23, Bibcode:1992CMaPh.147....1K, doi:10.1007/BF02099526, ISSN 0010-3616, MR 1171758 • Lando, Sergei K.; Zvonkin, Alexander K. (2004), Graphs on surfaces and their applications (PDF), Encyclopaedia of Mathematical Sciences, vol. 141, Berlin, New York: Springer-Verlag, ISBN 978-3-540-00203-1, MR 2036721 • Witten, Edward (1991), "Two-dimensional gravity and intersection theory on moduli space", Surveys in differential geometry (Cambridge, MA, 1990), vol. 1, Bethlehem, PA: Lehigh Univ., pp. 243–310, ISBN 978-0-8218-0168-0, MR 1144529 • Witten, Edward (1993), "Algebraic geometry associated with matrix models of two-dimensional gravity", in Goldberg, Lisa R.; Phillips, Anthony V. (eds.), Topological methods in modern mathematics (Stony Brook, NY, 1991), Proceedings of the symposium in honor of John Milnor's sixtieth birthday held at the State University of New York, Stony Brook, New York, June 14–21, 1991., Houston, TX: Publish or Perish, pp. 235–269, ISBN 978-0-914098-26-3, MR 1215968 Topics in algebraic curves Rational curves • Five points determine a conic • Projective line • Rational normal curve • Riemann sphere • Twisted cubic Elliptic curves Analytic theory • Elliptic function • Elliptic integral • Fundamental pair of periods • Modular form Arithmetic theory • Counting points on elliptic curves • Division polynomials • Hasse's theorem on elliptic curves • Mazur's torsion theorem • Modular elliptic curve • Modularity theorem • Mordell–Weil theorem • Nagell–Lutz theorem • Supersingular elliptic curve • Schoof's algorithm • Schoof–Elkies–Atkin algorithm Applications • Elliptic curve cryptography • Elliptic curve primality Higher genus • De Franchis theorem • Faltings's theorem • Hurwitz's automorphisms theorem • Hurwitz surface • Hyperelliptic curve Plane curves • AF+BG theorem • Bézout's theorem • Bitangent • Cayley–Bacharach theorem • Conic section • Cramer's paradox • Cubic plane curve • Fermat curve • Genus–degree formula • Hilbert's sixteenth problem • Nagata's conjecture on curves • Plücker formula • Quartic plane curve • Real plane curve Riemann surfaces • Belyi's theorem • Bring's curve • Bolza surface • Compact Riemann surface • Dessin d'enfant • Differential of the first kind • Klein quartic • Riemann's existence theorem • Riemann–Roch theorem • Teichmüller space • Torelli theorem Constructions • Dual curve • Polar curve • Smooth completion Structure of curves Divisors on curves • Abel–Jacobi map • Brill–Noether theory • Clifford's theorem on special divisors • Gonality of an algebraic curve • Jacobian variety • Riemann–Roch theorem • Weierstrass point • Weil reciprocity law Moduli • ELSV formula • Gromov–Witten invariant • Hodge bundle • Moduli of algebraic curves • Stable curve Morphisms • Hasse–Witt matrix • Riemann–Hurwitz formula • Prym variety • Weber's theorem (Algebraic curves) Singularities • Acnode • Crunode • Cusp • Delta invariant • Tacnode Vector bundles • Birkhoff–Grothendieck theorem • Stable vector bundle • Vector bundles on algebraic curves
Witten zeta function In mathematics, the Witten zeta function, is a function associated to a root system that encodes the degrees of the irreducible representations of the corresponding Lie group. These zeta functions were introduced by Don Zagier who named them after Edward Witten's study of their special values (among other things).[1][2] Note that in,[2] Witten zeta functions do not appear as explicit objects in their own right. Definition If $G$ is a compact semisimple Lie group, the associated Witten zeta function is (the meromorphic continuation of) the series $\zeta _{G}(s)=\sum _{\rho }{\frac {1}{(\dim \rho )^{s}}},$ where the sum is over equivalence classes of irreducible representations of $G$. In the case where $G$ is connected and simply connected, the correspondence between representations of $G$ and of its Lie algebra, together with the Weyl dimension formula, implies that $\zeta _{G}(s)$ can be written as $\sum _{m_{1},\dots ,m_{r}>0}\prod _{\alpha \in \Phi ^{+}}{\frac {1}{\langle \alpha ^{\lor },m_{1}\lambda _{1}+\cdots +m_{r}\lambda _{r}\rangle ^{s}}},$ where $\Phi ^{+}$ denotes the set of positive roots, $\{\lambda _{i}\}$ is a set of simple roots and $r$ is the rank. Examples • $\zeta _{SU(2)}(s)=\zeta (s)$, the Riemann zeta function. • $\zeta _{SU(3)}(s)=\sum _{x=1}^{\infty }\sum _{y=1}^{\infty }{\frac {1}{(xy(x+y)/2)^{s}}}.$ Abscissa of convergence If $G$ is simple and simply connected, the abscissa of convergence of $\zeta _{G}(s)$ is $r/\kappa $, where $r$ is the rank and $\kappa =\|\Phi ^{+}\|$. This is a theorem due to Alex Lubotzky and Michael Larsen.[3] A new proof is given by Jokke Häsä and Alexander Stasinski [4] which yields a more general result, namely it gives an explicit value (in terms of simple combinatorics) of the abscissa of convergence of any "Mellin zeta function" of the form $\sum _{x_{1},\dots ,x_{r}=1}^{\infty }{\frac {1}{P(x_{1},\dots ,x_{r})^{s}}},$ where $P(x_{1},\dots ,x_{r})$ is a product of linear polynomials with non-negative real coefficients. Singularities and values of the Witten zeta function associated to SU(3) $\zeta _{SU(3)}$ is absolutely convergent in $\{s\in \mathbb {C} ,\Re (s)>2/3\}$, and it can be extended meromorphicaly in $\mathbb {C} $. Its singularities are in ${\Bigl \{}{\frac {2}{3}}{\Bigr \}}\cup {\Bigl \{}{\frac {1}{2}}-k,k\in \mathbb {N} {\Bigr \}},$ and all of those singularities are simple poles.[5] In particular, the values of $\zeta _{SU(3)}(s)$ are well defined at all integers, and have been computed by Kazuhiro Onodera.[6] At $s=0$, we have $\zeta _{SU(3)}(0)={\frac {1}{3}},$ and $\zeta _{SU(3)}'(0)=\log(2^{4/3}\pi ).$ Let $a\in \mathbb {N} ^{*}$ be a positive integer. We have $\zeta _{SU(3)}(a)={\frac {2^{a+2}}{1+(-1)^{a}2}}\sum _{k=0}^{[a/2]}{2a-2k-1 \choose a-1}\zeta (2k)\zeta (3a-k).$ If a is odd, then $\zeta _{SU(3)}$ has a simple zero at $s=-a,$ and $\zeta _{SU(3)}'(-a)={\frac {2^{-a+1}(a!)^{2}}{(2a+1)!}}\zeta '(-3a-1)+2^{-a+2}\sum _{k=0}^{(a-1)/2}{a \choose 2k}\zeta (-a-2k)\zeta '(-2a+2k).$ If a is even, then $\zeta _{SU(3)}$ has a zero of order $2$ at $s=-a,$ and $\zeta _{SU(3)}''(-a)=2^{-a+2}\sum _{k=0}^{a/2}{a \choose 2k}\zeta '(-a-2k)\zeta '(-2a+2k).$ References 1. Zagier, Don (1994), "Values of Zeta Functions and Their Applications", First European Congress of Mathematics Paris, July 6–10, 1992, Birkhäuser Basel, pp. 497–512, doi:10.1007/978-3-0348-9112-7_23, ISBN 9783034899123 2. Witten, Edward (October 1991). "On quantum gauge theories in two dimensions". Communications in Mathematical Physics. 141 (1): 153–209. doi:10.1007/bf02100009. ISSN 0010-3616. S2CID 121994550. 3. Larsen, Michael; Lubotzky, Alexander (2008). "Representation growth of linear groups". Journal of the European Mathematical Society. 10 (2): 351–390. arXiv:math/0607369. doi:10.4171/JEMS/113. ISSN 1435-9855. S2CID 9322647. 4. Häsä, Jokke; Stasinski, Alexander (2019). "Representation growth of compact linear groups". Transactions of the American Mathematical Society. 372 (2): 925–980. doi:10.1090/tran/7618. 5. Romik, Dan (2017). "On the number of $n$-dimensional representations of $\operatorname{SU}(3)$, the Bernoulli numbers, and the Witten zeta function". Acta Arithmetica. 180 (2): 111–159. doi:10.4064/aa8455-3-2017. ISSN 0065-1036. 6. Onodera, Kazuhiro (2014). "A functional relation for Tornheim's double zeta functions". Acta Arithmetica. 162 (4): 337–354. arXiv:1211.1480. doi:10.4064/aa162-4-2. ISSN 0065-1036. S2CID 119636956.
Witting polytope In 4-dimensional complex geometry, the Witting polytope is a regular complex polytope, named as: 3{3}3{3}3{3}3, and Coxeter diagram . It has 240 vertices, 2160 3{} edges, 2160 3{3}3 faces, and 240 3{3}3{3}3 cells. It is self-dual. Each vertex belongs to 27 edges, 72 faces, and 27 cells, corresponding to the Hessian polyhedron vertex figure. Witting polytope Schläfli symbol3{3}3{3}3{3}3 Coxeter diagram Cells240 3{3}3{3}3 Faces2160 3{3}3 Edges2160 3{} Vertices240 Petrie polygon30-gon van Oss polygon90 3{4}3 Shephard groupL4 = 3[3]3[3]3[3]3, order 155,520 Dual polyhedronSelf-dual PropertiesRegular Symmetry Its symmetry by 3[3]3[3]3[3]3 or , order 155,520.[1] It has 240 copies of , order 648 at each cell.[2] Structure The configuration matrix is:[3] $\left[{\begin{smallmatrix}240&27&72&27\\3&2160&8&8\\8&8&2160&3\\27&72&27&240\end{smallmatrix}}\right]$ The number of vertices, edges, faces, and cells are seen in the diagonal of the matrix. These are computed by the order of the group divided by the order of the subgroup, by removing certain complex reflections, shown with X below. The number of elements of the k-faces are seen in rows below the diagonal. The number of elements in the vertex figure, etc., are given in rows above the digonal. L4 k-facefkf0f1f2f3k-figure Notes L3( ) f0 2402772273{3}3{3}3L4/L3 = 2166!/27/4! = 240 L2L13{ } f1 32160883{3}3L4/L2L1 = 2166!/4!/3 = 2160 3{3}3 f2 88216033{ } L33{3}3{3}3 f3 277227240( )L4/L3 = 216*6!/27/4! = 240 Coordinates Its 240 vertices are given coordinates in $\mathbb {C} ^{4}$: (0, ±ωμ, -±ων, ±ωλ) (-±ωμ, 0, ±ων, ±ωλ) (±ωμ, -±ων, 0, ±ωλ) (-±ωλ, -±ωμ, -±ων, 0) (±iωλ√3, 0, 0, 0) (0, ±iωλ√3, 0, 0) (0, 0, ±iωλ√3, 0) (0, 0, 0, ±iωλ√3) where $\omega ={\tfrac {-1+i{\sqrt {3}}}{2}},\lambda ,\nu ,\mu =0,1,2$. The last 6 points form hexagonal holes on one of its 40 diameters. There are 40 hyperplanes contain central 3{3}3{4}2, figures, with 72 vertices. Witting configuration Coxeter named it after Alexander Witting for being a Witting configuration in complex projective 3-space:[4] $\left[{\begin{smallmatrix}40&12&12\\2&240&2\\12&12&40\end{smallmatrix}}\right]$ or $\left[{\begin{smallmatrix}40&9&12\\4&90&4\\12&9&40\end{smallmatrix}}\right]$ The Witting configuration is related to the finite space PG(3,22), consisting of 85 points, 357 lines, and 85 planes.[5] Related real polytope Its 240 vertices are shared with the real 8-dimensional polytope 421, . Its 2160 3-edges are sometimes drawn as 6480 simple edges, slightly less than the 6720 edges of 421. The 240 difference is accounted by 40 central hexagons in 421 whose edges are not included in 3{3}3{3}3{3}3.[6] The honeycomb of Witting polytopes The regular Witting polytope has one further stage as a 4-dimensional honeycomb, . It has the Witting polytope as both its facets, and vertex figure. It is self-dual, and its dual coincides with itself.[7] Hyperplane sections of this honeycomb include 3-dimensional honeycombs . The honeycomb of Witting polytopes has a real representation as the 8-dimensional polytope 521, . Its f-vector element counts are in proportion: 1, 80, 270, 80, 1.[8] The configuration matrix for the honeycomb is: L5 k-facefkf0f1f2f3f4k-figure Notes L4( ) f0 N240216021602403{3}3{3}3{3}3L5/L4 = N L3L13{ } f1 380N2772273{3}3{3}3L5/L3L1 = 80N L2L23{3}3 f2 88270N883{3}3L5/L2L2 = 270N L3L13{3}3{3}3 f3 27722780N33{}L5/L3L1 = 80N L43{3}3{3}3{3}3 f4 24021602160240N( )L5/L4 = N Notes 1. Coxeter Regular Convex Polytopes, 12.5 The Witting polytope 2. Coxeter, Complex Regular Polytopes, p.134 3. Coxeter, Complex Regular polytopes, p.132 4. Alexander Witting, Ueber Jacobi'sche Functionen kter Ordnung Zweier Variabler, Mathemematische Annalen 29 (1887), 157-70, see especially p.169 5. Coxeter, Complex regular polytopes, p.133 6. Coxeter, Complex Regular Polytopes, p.134 7. Coxeter, Complex Regular Polytopes, p.135 8. Coxeter Regular Convex Polytopes, 12.5 The Witting polytope References • Coxeter, H. S. M. and Moser, W. O. J.; Generators and Relations for Discrete Groups (1965), esp pp 67–80. • Coxeter, H. S. M.; Regular Complex Polytopes, Cambridge University Press, second edition (1991). pp. 132–5, 143, 146, 152. • Coxeter, H. S. M. and Shephard, G.C.; Portraits of a family of complex polytopes, Leonardo Vol 25, No 3/4, (1992), pp 239–244
Witt group In mathematics, a Witt group of a field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear forms over the field. For Witt groups in the theory of algebraic groups, see Witt vector. Definition Fix a field k of characteristic not equal to two. All vector spaces will be assumed to be finite-dimensional. We say that two spaces equipped with symmetric bilinear forms are equivalent if one can be obtained from the other by adding a metabolic quadratic space, that is, zero or more copies of a hyperbolic plane, the non-degenerate two-dimensional symmetric bilinear form with a norm 0 vector.[1] Each class is represented by the core form of a Witt decomposition.[2] The Witt group of k is the abelian group W(k) of equivalence classes of non-degenerate symmetric bilinear forms, with the group operation corresponding to the orthogonal direct sum of forms. It is additively generated by the classes of one-dimensional forms.[3] Although classes may contain spaces of different dimension, the parity of the dimension is constant across a class and so rk : W(k) → Z/2Z is a homomorphism.[4] The elements of finite order in the Witt group have order a power of 2;[5][6] the torsion subgroup is the kernel of the functorial map from W(k) to W(kpy), where kpy is the Pythagorean closure of k;[7] it is generated by the Pfister forms $\langle \!\langle w\rangle \!\rangle =\langle 1,-w\rangle $ with $w$ a non-zero sum of squares.[8] If k is not formally real, then the Witt group is torsion, with exponent a power of 2.[9] The height of the field k is the exponent of the torsion in the Witt group, if this is finite, or ∞ otherwise.[8] Ring structure The Witt group of k can be given a commutative ring structure, by using the tensor product of quadratic forms to define the ring product. This is sometimes called the Witt ring W(k), though the term "Witt ring" is often also used for a completely different ring of Witt vectors. To discuss the structure of this ring we assume that k is of characteristic not equal to 2, so that we may identify symmetric bilinear forms and quadratic forms. The kernel of the rank mod 2 homomorphism is a prime ideal, I, of the Witt ring[4] termed the fundamental ideal.[10] The ring homomorphisms from W(k) to Z correspond to the field orderings of k, by taking signature with respective to the ordering.[10] The Witt ring is a Jacobson ring.[9] It is a Noetherian ring if and only if there are finitely many square classes; that is, if the squares in k form a subgroup of finite index in the multiplicative group of k.[11] If k is not formally real, the fundamental ideal is the only prime ideal of W[12] and consists precisely of the nilpotent elements;[9] W is a local ring and has Krull dimension 0.[13] If k is real, then the nilpotent elements are precisely those of finite additive order, and these in turn are the forms all of whose signatures are zero;[14] W has Krull dimension 1.[13] If k is a real Pythagorean field then the zero-divisors of W are the elements for which some signature is zero; otherwise, the zero-divisors are exactly the fundamental ideal.[5][15] If k is an ordered field with positive cone P then Sylvester's law of inertia holds for quadratic forms over k and the signature defines a ring homomorphism from W(k) to Z, with kernel a prime ideal KP. These prime ideals are in bijection with the orderings Xk of k and constitute the minimal prime ideal spectrum MinSpec W(k) of W(k). The bijection is a homeomorphism between MinSpec W(k) with the Zariski topology and the set of orderings Xk with the Harrison topology.[16] The n-th power of the fundamental ideal is additively generated by the n-fold Pfister forms.[17] Examples • The Witt ring of C, and indeed any algebraically closed field or quadratically closed field, is Z/2Z.[18] • The Witt ring of R is Z.[18] • The Witt ring of a finite field Fq with q odd is Z/4Z if q ≡ 3 mod 4 and isomorphic to the group ring (Z/2Z)[F/F2] if q ≡ 1 mod 4.[19] • The Witt ring of a local field with maximal ideal of norm congruent to 1 modulo 4 is isomorphic to the group ring (Z/2Z)[V] where V is the Klein 4-group.[20] • The Witt ring of a local field with maximal ideal of norm congruent to 3 modulo 4 is (Z/4Z)[C2] where C2 is a cyclic group of order 2.[20] • The Witt ring of Q2 is of order 32 and is given by[21] $\mathbf {Z} _{8}[s,t]/\langle 2s,2t,s^{2},t^{2},st-4\rangle .$ Invariants Certain invariants of a quadratic form can be regarded as functions on Witt classes. We have seen that dimension mod 2 is a function on classes: the discriminant is also well-defined. The Hasse invariant of a quadratic form is again a well-defined function on Witt classes with values in the Brauer group of the field of definition.[22] Rank and discriminant We define a ring over K, Q(K), as a set of pairs (d, e) with d in K/K 2 and e in Z/2Z. Addition and multiplication are defined by: $(d_{1},e_{1})+(d_{2},e_{2})=((-1)^{e_{1}e_{2}}d_{1}d_{2},e_{1}+e_{2})$ $(d_{1},e_{1})\cdot (d_{2},e_{2})=(d_{1}^{e_{2}}d_{2}^{e_{1}},e_{1}e_{2}).$ Then there is a surjective ring homomorphism from W(K) to this obtained by mapping a class to discriminant and rank mod 2. The kernel is I2.[23] The elements of Q may be regarded as classifying graded quadratic extensions of K.[24] Brauer–Wall group The triple of discriminant, rank mod 2 and Hasse invariant defines a map from W(K) to the Brauer–Wall group BW(K).[25] Witt ring of a local field Let K be a complete local field with valuation v, uniformiser π and residue field k of characteristic not equal to 2. There is an injection W(k) → W(K) which lifts the diagonal form ⟨a1,...an⟩ to ⟨u1,...un⟩ where ui is a unit of K with image ai in k. This yields $W(K)=W(k)\oplus \langle \pi \rangle \cdot W(k)$ identifying W(k) with its image in W(K).[26] Witt ring of a number field Let K be a number field. For quadratic forms over K, there is a Hasse invariant ±1 for every finite place corresponding to the Hilbert symbols. The invariants of a form over a number field are precisely the dimension, discriminant, all local Hasse invariants and the signatures coming from real embeddings.[27] We define the symbol ring over K, Sym(K), as a set of triples (d, e, f ) with d in K/K 2, e in Z/2 and f a sequence of elements ±1 indexed by the places of K, subject to the condition that all but finitely many terms of f are +1, that the value on acomplex places is +1 and that the product of all the terms in f in +1. Let [a, b] be the sequence of Hilbert symbols: it satisfies the conditions on f just stated.[28] We define addition and multiplication as follows: $(d_{1},e_{1},f_{1})+(d_{2},e_{2},f_{2})=((-1)^{e_{1}e_{2}}d_{1}d_{2},e_{1}+e_{2},[d_{1},d_{2}][-d_{1}d_{2},(-1)^{e_{1}e_{2}}]f_{1}f_{2})$ $(d_{1},e_{1},f_{1})\cdot (d_{2},e_{2},f_{2})=(d_{1}^{e_{2}}d_{2}^{e_{1}},e_{1}e_{2},[d_{1},d_{2}]^{1+e_{1}e_{2}}f_{1}^{e_{2}}f_{2}^{e_{1}})\ .$ Then there is a surjective ring homomorphism from W(K) to Sym(K) obtained by mapping a class to discriminant, rank mod 2, and the sequence of Hasse invariants. The kernel is I3.[29] The symbol ring is a realisation of the Brauer-Wall group.[30] Witt ring of the rationals The Hasse–Minkowski theorem implies that there is an injection[31] $W(\mathbf {Q} )\rightarrow W(\mathbf {R} )\oplus \prod _{p}W(\mathbf {Q} _{p})\ .$ We make this concrete, and compute the image, by using the "second residue homomorphism" W(Qp) → W(Fp). Composed with the map W(Q) → W(Qp) we obtain a group homomorphism ∂p: W(Q) → W(Fp) (for p = 2 we define ∂2 to be the 2-adic valuation of the discriminant, taken mod 2). We then have a split exact sequence[32] $0\rightarrow \mathbf {Z} \rightarrow W(\mathbf {Q} )\rightarrow \mathbf {Z} /2\oplus \bigoplus _{p\neq 2}W(\mathbf {F} _{p})\rightarrow 0\ $ which can be written as an isomorphism $W(\mathbf {Q} )\cong \mathbf {Z} \oplus \mathbf {Z} /2\oplus \bigoplus _{p\neq 2}W(\mathbf {F} _{p})\ $ where the first component is the signature.[33] Witt ring and Milnor's K-theory Let k be a field of characteristic not equal to 2. The powers of the ideal I of forms of even dimension ("fundamental ideal") in $W(k)$ form a descending filtration and one may consider the associated graded ring, that is the direct sum of quotients $I^{n}/I^{n+1}$. Let $\langle a\rangle $ be the quadratic form $ax^{2}$ considered as an element of the Witt ring. Then $\langle a\rangle -\langle 1\rangle $ is an element of I and correspondingly a product of the form $\langle \langle a_{1},\ldots ,a_{n}\rangle \rangle =(\langle a_{1}\rangle -\langle 1\rangle )\cdots (\langle a_{n}\rangle -\langle 1\rangle )$ is an element of $I^{n}.$ John Milnor in a 1970 paper [34] proved that the mapping from $(k^{})^{n}$ to $I^{n}/I^{n+1}$ that sends $(a_{1},\ldots ,a_{n})$ to $\langle \langle a_{1},\ldots ,a_{n}\rangle \rangle $ is multilinear and maps Steinberg elements (elements such that for some $i$ and $j$ such that $i\neq j$ one has $a_{i}+a_{j}=1$) to zero. This means that this mapping defines a homomorphism from the Milnor ring of k to the graded Witt ring. Milnor showed also that this homomorphism sends elements divisible by 2 to zero and that it is surjective. In the same paper he made a conjecture that this homomorphism is an isomorphism for all fields k (of characteristic different from 2). This became known as the Milnor conjecture on quadratic forms. The conjecture was proved by Dmitry Orlov, Alexander Vishik and Vladimir Voevodsky[35] in 1996 (published in 2007) for the case ${\textrm {char}}(k)=0$, leading to increased understanding of the structure of quadratic forms over arbitrary fields. Grothendieck-Witt ring The Grothendieck-Witt ring GW is a related construction generated by isometry classes of nonsingular quadratic spaces with addition given by orthogonal sum and multiplication given by tensor product. Since two spaces that differ by a hyperbolic plane are not identified in GW, the inverse for the addition needs to be introduced formally through the construction that was discovered by Grothendieck (see Grothendieck group). There is a natural homomorphism GW → Z given by dimension: a field is quadratically closed if and only if this is an isomorphism.[18] The hyperbolic spaces generate an ideal in GW and the Witt ring W is the quotient.[36] The exterior power gives the Grothendieck-Witt ring the additional structure of a λ-ring.[37] Examples • The Grothendieck-Witt ring of C, and indeed any algebraically closed field or quadratically closed field, is Z.[18] • The Grothendieck-Witt ring of R is isomorphic to the group ring Z[C2], where C2 is a cyclic group of order 2.[18] • The Grothendieck-Witt ring of any finite field of odd characteristic is Z ⊕ Z/2Z with trivial multiplication in the second component.[38] The element (1, 0) corresponds to the quadratic form ⟨a⟩ where a is not a square in the finite field. • The Grothendieck-Witt ring of a local field with maximal ideal of norm congruent to 1 modulo 4 is isomorphic to Z ⊕ (Z/2Z)3.[20] • The Grothendieck-Witt ring of a local field with maximal ideal of norm congruent to 3 modulo 4 it is Z' ⊕ Z/4Z ⊕ Z/2Z.[20] Grothendieck-Witt ring and motivic stable homotopy groups of spheres Fabien Morel[39][40] showed that the Grothendieck-Witt ring of a perfect field is isomorphic to the motivic stable homotopy group of spheres π0,0(S0,0) (see "A¹ homotopy theory"). Witt equivalence Two fields are said to be Witt equivalent if their Witt rings are isomorphic. For global fields there is a local-to-global principle: two global fields are Witt equivalent if and only if there is a bijection between their places such that the corresponding local fields are Witt equivalent.[41] In particular, two number fields K and L are Witt equivalent if and only if there is a bijection T between the places of K and the places of L and a group isomorphism t between their square-class groups, preserving degree 2 Hilbert symbols. In this case the pair (T, t) is called a reciprocity equivalence or a degree 2 Hilbert symbol equivalence.[42] Some variations and extensions of this condition, such as "tame degree l Hilbert symbol equivalence", have also been studied.[43] Generalizations Main article: L-theory Witt groups can also be defined in the same way for skew-symmetric forms, and for quadratic forms, and more generally ε-quadratic forms, over any -ring R. The resulting groups (and generalizations thereof) are known as the even-dimensional symmetric L-groups L2k(R) and even-dimensional quadratic L-groups L2k(R). The quadratic L-groups are 4-periodic, with L0(R) being the Witt group of (1)-quadratic forms (symmetric), and L2(R) being the Witt group of (−1)-quadratic forms (skew-symmetric); symmetric L-groups are not 4-periodic for all rings, hence they provide a less exact generalization. L-groups are central objects in surgery theory, forming one of the three terms of the surgery exact sequence. See also • Reduced height of a field Notes 1. Milnor & Husemoller (1973) p. 14 2. Lorenz (2008) p. 30 3. Milnor & Husemoller (1973) p. 65 4. Milnor & Husemoller (1973) p. 66 5. Lorenz (2008) p. 37 6. Milnor & Husemoller (1973) p. 72 7. Lam (2005) p. 260 8. Lam (2005) p. 395 9. Lorenz (2008) p. 35 10. Lorenz (2008) p. 31 11. Lam (2005) p. 32 12. Lorenz (2008) p. 33 13. Lam (2005) p. 280 14. Lorenz (2008) p. 36 15. Lam (2005) p. 282 16. Lam (2005) pp. 277–280 17. Lam (2005) p.316 18. Lam (2005) p. 34 19. Lam (2005) p.37 20. Lam (2005) p.152 21. Lam (2005) p.166 22. Lam (2005) p.119 23. Conner & Perlis (1984) p.12 24. Lam (2005) p.113 25. Lam (2005) p.117 26. Garibaldi, Merkurjev & Serre (2003) p.64 27. Conner & Perlis (1984) p.16 28. Conner & Perlis (1984) p.16-17 29. Conner & Perlis (1984) p.18 30. Lam (2005) p.116 31. Lam (2005) p.174 32. Lam (2005) p.175 33. Lam (2005) p.178 34. Milnor, John Willard (1970), "Algebraic K-theory and quadratic forms", Inventiones Mathematicae, 9 (4): 318–344, doi:10.1007/BF01425486, ISSN 0020-9910, MR 0260844 35. Orlov, Dmitry; Vishik, Alexander; Voevodsky, Vladimir (2007), "An exact sequence for K*M/2 with applications to quadratic forms", Annals of Mathematics, 165 (1): 1–13, arXiv:math/0101023, doi:10.4007/annals.2007.165.1 36. Lam (2005) p. 28 37. Garibaldi, Merkurjev & Serre (2003) p.63 38. Lam (2005) p.36, Theorem 3.5 39. , On the motivic stable π0 of the sphere spectrum, In: Axiomatic, Enriched and Motivic Homotopy Theory, pp. 219–260, J.P.C. Greenlees (ed.), 2004 Kluwer Academic Publishers. 40. Fabien Morel, A1-Algebraic topology over a field. Lecture Notes in Mathematics 2052, Springer Verlag, 2012. 41. Perlis, R.; Szymiczek, K.; Conner, P.E.; Litherland, R. (1994). "Matching Witts with global fields". In Jacob, William B.; et al. (eds.). Recent advances in real algebraic geometry and quadratic forms. Contemp. Math. Vol. 155. Providence, RI: American Mathematical Society. pp. 365–387. ISBN 0-8218-5154-3. Zbl 0807.11024. 42. Szymiczek, Kazimierz (1997). "Hilbert-symbol equivalence of number fields". Tatra Mt. Math. Publ. 11: 7–16. Zbl 0978.11012. 43. Czogała, A. (1999). "Higher degree tame Hilbert-symbol equivalence of number fields". Abh. Math. Sem. Univ. Hamburg. 69: 175–185. doi:10.1007/bf02940871. Zbl 0968.11038. References • Conner, Pierre E.; Perlis, Robert (1984). A Survey of Trace Forms of Algebraic Number Fields. Series in Pure Mathematics. Vol. 2. World Scientific. ISBN 9971-966-05-0. Zbl 0551.10017. • Garibaldi, Skip; Merkurjev, Alexander; Serre, Jean-Pierre (2003). Cohomological invariants in Galois cohomology. University Lecture Series. Vol. 28. Providence, RI: American Mathematical Society. ISBN 0-8218-3287-5. Zbl 1159.12311. • Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. Vol. 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023. • Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001 • Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. ISBN 978-0-387-72487-4. Zbl 1130.12001. • Milnor, John; Husemoller, Dale (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 73. Springer-Verlag. ISBN 3-540-06009-X. Zbl 0292.10016. • Witt, Ernst (1936), "Theorie der quadratischen Formen in beliebigen Korpern", Journal für die reine und angewandte Mathematik, 176 (3): 31–44, Zbl 0015.05701 Further reading • Balmer, Paul (2005). "Witt groups". In Friedlander, Eric M.; Grayson, D. R. (eds.). Handbook of K-theory. Vol. 2. Springer-Verlag. pp. 539–579. ISBN 3-540-23019-X. Zbl 1115.19004. External links • Witt rings in the Springer encyclopedia of mathematics
William Kingdon Clifford William Kingdon Clifford FRS (4 May 1845 – 3 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his honour. The operations of geometric algebra have the effect of mirroring, rotating, translating, and mapping the geometric objects that are being modelled to new positions. Clifford algebras in general and geometric algebra in particular have been of ever increasing importance to mathematical physics,[1] geometry,[2] and computing.[3] Clifford was the first to suggest that gravitation might be a manifestation of an underlying geometry. In his philosophical writings he coined the expression mind-stuff. William Clifford William Kingdon Clifford (1845–1879) Born4 May 1845 (1845-05-04) Exeter, Devon, England Died3 March 1879 (1879-03-04) (aged 33) Madeira, Portugal NationalityEnglish Alma materKing's College London Trinity College, Cambridge Known forClifford algebra Clifford's circle theorems Clifford's theorem Clifford torus Clifford–Klein form Clifford parallel Bessel–Clifford function Dual quaternion Elements of Dynamic SpouseLucy Clifford (1875–1879) Scientific career FieldsMathematics Philosophy InstitutionsUniversity College London Doctoral studentsArthur Black InfluencesGeorg Friedrich Bernhard Riemann Nikolai Ivanovich Lobachevsky Biography Born at Exeter, William Clifford showed great promise at school. He went on to King's College London (at age 15) and Trinity College, Cambridge, where he was elected fellow in 1868, after being second wrangler in 1867 and second Smith's prizeman.[4][5] Being second was a fate he shared with others who became famous scientists, including William Thomson (Lord Kelvin) and James Clerk Maxwell. In 1870, he was part of an expedition to Italy to observe the solar eclipse of 22 December 1870. During that voyage he survived a shipwreck along the Sicilian coast.[6] In 1871, he was appointed professor of mathematics and mechanics at University College London, and in 1874 became a fellow of the Royal Society.[4] He was also a member of the London Mathematical Society and the Metaphysical Society. Clifford married Lucy Lane on 7 April 1875, with whom he had two children.[7] Clifford enjoyed entertaining children and wrote a collection of fairy stories, The Little People.[8] Death and legacy In 1876, Clifford suffered a breakdown, probably brought on by overwork. He taught and administered by day, and wrote by night. A half-year holiday in Algeria and Spain allowed him to resume his duties for 18 months, after which he collapsed again. He went to the island of Madeira to recover, but died there of tuberculosis after a few months, leaving a widow with two children. Clifford and his wife are buried in London's Highgate Cemetery, near the graves of George Eliot and Herbert Spencer, just north of the grave of Karl Marx. The academic journal Advances in Applied Clifford Algebras publishes on Clifford's legacy in kinematics and abstract algebra. Mathematics "Clifford was above all and before all a geometer." — Henry John Stephen Smith[4] The discovery of non-Euclidean geometry opened new possibilities in geometry in Clifford's era. The field of intrinsic differential geometry was born, with the concept of curvature broadly applied to space itself as well as to curved lines and surfaces. Clifford was very much impressed by Bernhard Riemann’s 1854 essay "On the hypotheses which lie at the bases of geometry".[9] In 1870, he reported to the Cambridge Philosophical Society on the curved space concepts of Riemann, and included speculation on the bending of space by gravity. Clifford's translation[10][11] of Riemann's paper was published in Nature in 1873. His report at Cambridge, "On the Space-Theory of Matter", was published in 1876, anticipating Albert Einstein's general relativity by 40 years. Clifford elaborated elliptic space geometry as a non-Euclidean metric space. Equidistant curves in elliptic space are now said to be Clifford parallels. Clifford's contemporaries considered him acute and original, witty and warm. He often worked late into the night, which may have hastened his death. He published papers on a range of topics including algebraic forms and projective geometry and the textbook Elements of Dynamic. His application of graph theory to invariant theory was followed up by William Spottiswoode and Alfred Kempe.[12] Algebras In 1878, Clifford published a seminal work, building on Grassmann's extensive algebra.[13] He had succeeded in unifying the quaternions, developed by William Rowan Hamilton, with Grassmann's outer product (aka the exterior product). He understood the geometric nature of Grassmann's creation, and that the quaternions fit cleanly into the algebra Grassmann had developed. The versors in quaternions facilitate representation of rotation. Clifford laid the foundation for a geometric product, composed of the sum of the inner product and Grassmann's outer product. The geometric product was eventually formalized by the Hungarian mathematician Marcel Riesz. The inner product equips geometric algebra with a metric, fully incorporating distance and angle relationships for lines, planes, and volumes, while the outer product gives those planes and volumes vector-like properties, including a directional bias. Combining the two brought the operation of division into play. This greatly expanded our qualitative understanding of how objects interact in space. Crucially, it also provided the means for quantitatively calculating the spatial consequences of those interactions. The resulting geometric algebra, as he called it, eventually realized the long sought goal[lower-roman 1] of creating an algebra that mirrors the movements and projections of objects in 3-dimensional space.[14] Moreover, Clifford's algebraic schema extends to higher dimensions. The algebraic operations have the same symbolic form as they do in 2 or 3-dimensions. The importance of general Clifford algebras has grown over time, while their isomorphism classes - as real algebras - have been identified in other mathematical systems beyond simply the quaternions.[15] The realms of real analysis and complex analysis have been expanded through the algebra H of quaternions, thanks to its notion of a three-dimensional sphere embedded in a four-dimensional space. Quaternion versors, which inhabit this 3-sphere, provide a representation of the rotation group SO(3). Clifford noted that Hamilton's biquaternions were a tensor product $H\otimes C$ of known algebras, and proposed instead two other tensor products of H: Clifford argued that the "scalars" taken from the complex numbers C might instead be taken from split-complex numbers D or from the dual numbers N. In terms of tensor products, $H\otimes D$ produces split-biquaternions, while $H\otimes N$ forms dual quaternions. The algebra of dual quaternions is used to express screw displacement, a common mapping in kinematics. Philosophy As a philosopher, Clifford's name is chiefly associated with two phrases of his coining, mind-stuff and the tribal self. The former symbolizes his metaphysical conception, suggested to him by his reading of Baruch Spinoza,[4] which Clifford (1878) defined as follows:[17] That element of which, as we have seen, even the simplest feeling is a complex, I shall call Mind-stuff. A moving molecule of inorganic matter does not possess mind or consciousness ; but it possesses a small piece of mind-stuff. When molecules are so combined together as to form the film on the under side of a jelly-fish, the elements of mind-stuff which go along with them are so combined as to form the faint beginnings of Sentience. When the molecules are so combined as to form the brain and nervous system of a vertebrate, the corresponding elements of mind-stuff are so combined as to form some kind of consciousness; that is to say, changes in the complex which take place at the same time get so linked together that the repetition of one implies the repetition of the other. When matter takes the complex form of a living human brain, the corresponding mind-stuff takes the form of a human consciousness, having intelligence and volition. — "On the Nature of Things-in-Themselves" (1878) Regarding Clifford's concept, Sir Frederick Pollock wrote: Briefly put, the conception is that mind is the one ultimate reality; not mind as we know it in the complex forms of conscious feeling and thought, but the simpler elements out of which thought and feeling are built up. The hypothetical ultimate element of mind, or atom of mind-stuff, precisely corresponds to the hypothetical atom of matter, being the ultimate fact of which the material atom is the phenomenon. Matter and the sensible universe are the relations between particular organisms, that is, mind organized into consciousness, and the rest of the world. This leads to results which would in a loose and popular sense be called materialist. But the theory must, as a metaphysical theory, be reckoned on the idealist side. To speak technically, it is an idealist monism.[4] Tribal self, on the other hand, gives the key to Clifford's ethical view, which explains conscience and the moral law by the development in each individual of a 'self,' which prescribes the conduct conducive to the welfare of the 'tribe.' Much of Clifford's contemporary prominence was due to his attitude toward religion. Animated by an intense love of his conception of truth and devotion to public duty, he waged war on such ecclesiastical systems as seemed to him to favour obscurantism, and to put the claims of sect above those of human society. The alarm was greater, as theology was still unreconciled with Darwinism; and Clifford was regarded as a dangerous champion of the anti-spiritual tendencies then imputed to modern science.[4] There has also been debate on the extent to which Clifford's doctrine of 'concomitance' or 'psychophysical parallelism' influenced John Hughlings Jackson's model of the nervous system and, through him, the work of Janet, Freud, Ribot, and Ey.[18] Ethics In his 1877 essay, The Ethics of Belief, Clifford argues that it is immoral to believe things for which one lacks evidence.[19] He describes a ship-owner who planned to send to sea an old and not well built ship full of passengers. The ship-owner had doubts suggested to him that the ship might not be seaworthy: "These doubts preyed upon his mind, and made him unhappy." He considered having the ship refitted even though it would be expensive. At last, "he succeeded in overcoming these melancholy reflections." He watched the ship depart, "with a light heart…and he got his insurance money when she went down in mid-ocean and told no tales."[19] Clifford argues that the ship-owner was guilty of the deaths of the passengers even though he sincerely believed the ship was sound: "[H]e had no right to believe on such evidence as was before him."[lower-roman 2] Moreover, he contends that even in the case where the ship successfully reaches the destination, the decision remains immoral, because the morality of the choice is defined forever once the choice is made, and actual outcome, defined by blind chance, doesn't matter. The ship-owner would be no less guilty: his wrongdoing would never be discovered, but he still had no right to make that decision given the information available to him at the time. Clifford famously concludes with what has come to be known as Clifford's principle: "it is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence."[19] As such, he is arguing in direct opposition to religious thinkers for whom 'blind faith' (i.e. belief in things in spite of the lack of evidence for them) was a virtue. This paper was famously attacked by pragmatist philosopher William James in his "Will to Believe" lecture. Often these two works are read and published together as touchstones for the debate over evidentialism, faith, and overbelief. Premonition of relativity Though Clifford never constructed a full theory of spacetime and relativity, there are some remarkable observations he made in print that foreshadowed these modern concepts: In his book Elements of Dynamic (1878), he introduced "quasi-harmonic motion in a hyperbola". He wrote an expression for a parametrized unit hyperbola, which other authors later used as a model for relativistic velocity. Elsewhere he states:[20] The geometry of rotors and motors…forms the basis of the whole modern theory of the relative rest (Static) and the relative motion (Kinematic and Kinetic) of invariable systems.[lower-roman 3] This passage makes reference to biquaternions, though Clifford made these into split-biquaternions as his independent development. The book continues with a chapter "On the bending of space", the substance of general relativity. Clifford also discussed his views in On the Space-Theory of Matter in 1876. In 1910, William Barrett Frankland quoted the Space-Theory of Matter in his book on parallelism: "The boldness of this speculation is surely unexcelled in the history of thought. Up to the present, however, it presents the appearance of an Icarian flight."[21] Years later, after general relativity had been advanced by Albert Einstein, various authors noted that Clifford had anticipated Einstein. Hermann Weyl (1923), for instance, mentioned Clifford as one of those who, like Bernhard Riemann, anticipated the geometric ideas of relativity.[22] In 1940, Eric Temple Bell published The Development of Mathematics, in which he discusses the prescience of Clifford on relativity:[23] Bolder even than Riemann, Clifford confessed his belief (1870) that matter is only a manifestation of curvature in a space-time manifold. This embryonic divination has been acclaimed as an anticipation of Einstein's (1915–16) relativistic theory of the gravitational field. The actual theory, however, bears but slight resemblance to Clifford's rather detailed creed. As a rule, those mathematical prophets who never descend to particulars make the top scores. Almost anyone can hit the side of a barn at forty yards with a charge of buckshot. John Archibald Wheeler, during the 1960 International Congress for Logic, Methodology, and Philosophy of Science (CLMPS) at Stanford, introduced his geometrodynamics formulation of general relativity by crediting Clifford as the initiator.[24] In The Natural Philosophy of Time (1961), Gerald James Whitrow recalls Clifford's prescience, quoting him in order to describe the Friedmann–Lemaître–Robertson–Walker metric in cosmology.[25] Cornelius Lanczos (1970) summarizes Clifford's premonitions:[26] [He] with great ingenuity foresaw in a qualitative fashion that physical matter might be conceived as a curved ripple on a generally flat plane. Many of his ingenious hunches were later realized in Einstein's gravitational theory. Such speculations were automatically premature and could not lead to anything constructive without an intermediate link which demanded the extension of 3-dimensional geometry to the inclusion of time. The theory of curved spaces had to be preceded by the realization that space and time form a single four-dimensional entity. Likewise, Banesh Hoffmann (1973) writes:[27] Riemann, and more specifically Clifford, conjectured that forces and matter might be local irregularities in the curvature of space, and in this they were strikingly prophetic, though for their pains they were dismissed at the time as visionaries. In 1990, Ruth Farwell and Christopher Knee examined the record on acknowledgement of Clifford's foresight.[28] They conclude that "it was Clifford, not Riemann, who anticipated some of the conceptual ideas of General Relativity." To explain the lack of recognition of Clifford's prescience, they point out that he was an expert in metric geometry, and "metric geometry was too challenging to orthodox epistemology to be pursued."[28] In 1992, Farwell and Knee continued their study of Clifford and Riemann:[29] [They] hold that once tensors had been used in the theory of general relativity, the framework existed in which a geometrical perspective in physics could be developed and allowed the challenging geometrical conceptions of Riemann and Clifford to be rediscovered. Selected writings • 1872. On the aims and instruments of scientific thought, 524–41. • 1876 [1870]. On the Space-Theory of Matter.[30][31] • 1877. "The Ethics of Belief." Contemporary Review 29:289.[19][32] • 1878. Elements of Dynamic: An Introduction to the Study of Motion And Rest In Solid And Fluid Bodies.[33] • Book I: "Translations" • Book II: "Rotations" • Book III: "Strains" • 1878. "Applications of Grassmann's Extensive Algebra." American Journal of Mathematics 1(4):353.[34] • 1879: Seeing and Thinking[35]—includes four popular science lectures:[4] • "The Eye and the Brain" • "The Eye and Seeing" • "The Brain and Thinking" • "Of Boundaries in General" • 1879. Lectures and Essays I & II, with an introduction by Sir Frederick Pollock.[36] • 1881. "Mathematical fragments" (facsimiles).[37] • 1882. Mathematical Papers, edited by Robert Tucker, with an introduction by Henry J. S. Smith.[38] • 1885. The Common Sense of the Exact Sciences, completed by Karl Pearson.[39][4] • 1887. Elements of Dynamic 2.[40] • 1885 copy of "The Common Sense of the Exact Sciences" • Title page of an 1885 copy of "The Common Sense of the Exact Sciences" • Table of contents page for an 1885 copy of "The Common Sense of the Exact Sciences" • First page of an 1885 copy of "The Common Sense of the Exact Sciences" Quotations "I…hold that in the physical world nothing else takes place but this variation [of the curvature of space]." — Mathematical Papers (1882) "There is no scientific discoverer, no poet, no painter, no musician, who will not tell you that he found ready made his discovery or poem or picture—that it came to him from outside, and that he did not consciously create it from within." — "Some of the conditions of mental development" (1882), lecture to the Royal Institution "It is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence." — The Ethics of Belief (1879) [1877] "If a man, holding a belief which he was taught in childhood or persuaded of afterwards, keeps down and pushes away any doubts which arise about it in his mind, purposely avoids the reading of books and the company of men that call in question or discuss it, and regards as impious those questions which cannot easily be asked without disturbing it—the life of that man is one long sin against mankind." — The Ethics of Belief (1879) [1877] "I was not, and was conceived. I loved and did a little work. I am not and grieve not." — Epitaph See also • Bessel–Clifford function • Clifford's principle • Clifford analysis • Clifford gates • Clifford bundle • Clifford module • Clifford number • Motor • Rotor • Simplex • Split-biquaternion • Will to Believe Doctrine References Notes 1. "I believe that, so far as geometry is concerned, we need still another analysis which is distinctly geometrical or linear and which will express situation directly as algebra expresses magnitude directly." Leibniz, Gottfried. 1976 [1679]. "Letter to Christian Huygens (8 September 1679)." In Philosophical Papers and Letters (2nd ed.). Springer. 2. The italics are in the original. 3. This passage is immediately followed by a section on "The bending of space." However, according to the preface (p.vii), this section was written by Karl Pearson Citations 1. Doran, Chris; Lasenby, Anthony (2007). Geometric Algebra for Physicists. Cambridge, England: Cambridge University Press. p. 592. ISBN 9780521715959. 2. Hestenes, David (2011). "Grassmann's legacy". Grassmann's Legacy in From Past to Future: Graßmann's Work in Context, Petsche, Hans-Joachim, Lewis, Albert C., Liesen, Jörg, Russ, Steve (ed). Basel, Germany: Springer. pp. 243–260. doi:10.1007/978-3-0346-0405-5_22. ISBN 978-3-0346-0404-8. 3. Dorst, Leo (2009). Geometric Algebra for Computer Scientists. Amsterdam: Morgan Kaufmann. p. 664. ISBN 9780123749420. 4. Chisholm 1911, p. 506. 5. "Clifford, William Kingdon (CLFT863WK)". A Cambridge Alumni Database. University of Cambridge. 6. Chisholm, M. (2002). Such Silver Currents. Cambridge: The Lutterworth Press. p. 26. ISBN 978-0-7188-3017-5. 7. Stephen, Leslie; Pollock, Frederick (1901). Lectures and Essays by the Late William Kingdon Clifford, F.R.S. Vol. 1. New York: Macmillan and Company. p. 20. 8. Eves, Howard W. (1969). In Mathematical Circles: A Selection of Mathematical Stories and Anecdotes. Vol. 3–4. Prindle, Weber and Schmidt. pp. 91–92. 9. Riemann, Bernhard. 1867 [1854]. "On the hypotheses which lie at the bases of geometry" (Habilitationsschrift), translated by W. K. Clifford. – via School of Mathematics, Trinity College Dublin. 10. Clifford, William K. 1873. "On the hypotheses which lie at the bases of geometry." Nature 8:14–17, 36–37. 11. Clifford, William K. 1882. "Paper #9." P. 55–71 in Mathematical Papers. 12. Biggs, Norman L.; Lloyd, Edward Keith; Wilson, Robin James (1976). Graph Theory: 1736-1936. Oxford University Press. p. 67. ISBN 978-0-19-853916-2. 13. Clifford, William (1878). "Applications of Grassmann's extensive algebra". American Journal of Mathematics. 1 (4): 350–358. doi:10.2307/2369379. JSTOR 2369379. 14. Hestenes, David. "On the Evolution of Geometric Algebra and Geometric Calculus". 15. Dechant, Pierre-Philippe (March 2014). "A Clifford algebraic framework for Coxeter group theoretic computations". Advances in Applied Clifford Algebras. 14 (1): 89–108. arXiv:1207.5005. Bibcode:2012arXiv1207.5005D. doi:10.1007/s00006-013-0422-4. S2CID 54035515. 16. Frontispiece of Lectures and Essays by the Late William Kingdon Clifford, F.R.S., vol 2. 17. Clifford, William K. 1878. "On the Nature of Things-in-Themselves." Mind 3(9):57–67. doi:10.1093/mind/os-3.9.57. JSTOR 2246617. 18. Clifford, C. K., and G. E. Berrios. 2000. "Body and Mind." History of Psychiatry 11(43):311–38. doi:10.1177/0957154x0001104305. PMID 11640231. 19. Clifford, William K. 1877. "The Ethics of Belief." Contemporary Review 29:289. 20. Clifford, William K. 1885. Common Sense of the Exact Sciences. London: Kegan Paul, Trench and Co. p. 214. 21. Frankland, William Barrett. 1910. Theories of Parallelism. Cambridge: Cambridge University Press. pp. 48–49. 22. Weyl, Hermann. 1923. Raum Zeit Materie. Berlin: Springer-Verlag. p. 101 23. Bell, Eric Temple. 1940. The Development of Mathematics. pp. 359–60. 24. Wheeler, John Archibald. 1962 [1960]. "Curved empty space as the building material of the physical world: an assessment." In Logic, Methodology, and Philosophy of Science, edited by E. Nagel. Stanford University Press. 25. Whitrow, Gerald James. 1961. The Natural Philosophy of Time (1st ed.). pp. 246–47.—1980 [1961]. The Natural Philosophy of Time (2nd ed.). pp. 291. 26. Lanczos, Cornelius. 1970. Space Through the Ages: The Evolution of Geometrical Ideas from Pythagoras to Hilbert and Einstein. Academic Press. p. 222. 27. Hoffmann, Banesh. 1973. "Relativity." Dictionary of the History of Ideas 4:80. Charles Scribner's Sons. 28. Farwell, Ruth, and Christopher Knee. 1990. Studies in History and Philosophy of Science 21:91–121. 29. Farwell, Ruth, and Christopher Knee. 1992. "The Geometric Challenge of Riemann and Clifford." Pp. 98–106 in 1830–1930: A Century of Geometry, edited by L. Boi, D. Flament, and J. Salanskis. Lecture Notes in Physics 402. Springer Berlin Heidelberg. ISBN 978-3-540-47058-8. doi:10.1007/3-540-55408-4_56. 30. Clifford, William K. 1876 [1870]. "On the Space-Theory of Matter." Proceedings of the Cambridge Philosophical Society 2:157–58. OCLC 6084206. OL 20550270M. proceedingscamb06socigoog at the Internet Archive 31. Clifford, William K. 2007 [1870]. "On the Space-Theory of Matter." P. 71 in Beyond Geometry: Classic Papers from Riemann to Einstein, edited by P. Pesic. Mineola: Dover Publications. Bibcode:2007bgcp.book...71K. 32. Clifford, William K. 1886 [1877]. "The Ethics of Belief" (full text). Lectures and Essays (2nd ed.), edited by L. Stephen and F. Pollock. Macmillan and Co. – via A. J. Burger (2008). 33. Clifford, William K. 1878. Elements of Dynamic: An Introduction to the Study of Motion And Rest In Solid And Fluid Bodies I, II, & III. London: MacMillan and Co. – via Internet Archive. 34. Clifford, William K. 1878. "Applications of Grassmann's Extensive Algebra." American Journal of Mathematics 1(4):353. doi:10.2307/2369379. 35. Clifford, William K. 1879. Seeing and Thinking. London: Macmillan and Co. 36. Clifford, William K. 1901 [1879]. Lectures and Essays I (3rd ed.), edited by L. Stephen and F. Pollock. New York: The Macmillan Company. 37. Clifford, William K. 1881. "Mathematical Fragments" (facsimile). London: Macmillan Company. Located at University of Bordeaux. Science and Technology Library. FR 14652. 38. Clifford, William K. 1882. Mathematical Papers, edited by R. Tucker, introduction by H. J. S. Smith. London: MacMillan and Co. – via Internet Archive. 39. Clifford, William K. 1885. The Common Sense of the Exact Sciences, completed by K. Pearson. London: Kegan, Paul, Trench, and Co. 40. Clifford, William K. 1996 [1887]. "Elements of Dynamic" 2. In From Kant to Hilbert: A Source Book in the Foundations of Mathematics, edited by W. B. Ewald. Oxford. Oxford University Press. • This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "Clifford, William Kingdon". Encyclopædia Britannica. Vol. 6 (11th ed.). Cambridge University Press. p. 506. Further reading • Chisholm, M. (1997). "William Kingdon Clifford (1845-1879) and his wife Lucy (1846-1929)". Advances in Applied Clifford Algebras. 7S: 27–41. (The on-line version lacks the article's photographs.) • Chisholm, M. (2002). Such Silver Currents - The Story of William and Lucy Clifford, 1845-1929. Cambridge, UK: The Lutterworth Press. ISBN 978-0-7188-3017-5. • Farwell, Ruth; Knee, Christopher (1990). "The End of the Absolute: a nineteenth century contribution to General Relativity". Studies in History and Philosophy of Science. 21 (1): 91–121. Bibcode:1990SHPSA..21...91F. doi:10.1016/0039-3681(90)90016-2. • Macfarlane, Alexander (1916). Lectures on Ten British Mathematicians of the Nineteenth Century. New York: John Wiley and Sons. Lectures on Ten British Mathematicians of the Nineteenth Century. (See especially pages 78–91) • Madigan, Timothy J. (2010). W.K. Clifford and "The Ethics of Belief Cambridge Scholars Press, Cambridge, UK 978-1847-18503-7. • Penrose, Roger (2004). The Road to Reality: A Complete Guide to the Laws of the Universe. Alfred A. Knopf. ISBN 9780679454434. (See especially Chapter 11) • Stephen, Leslie; Pollock, Frederick (1879). Lectures and Essays by the Late William Kingdon Clifford, F.R.S. Vol. 1. New York: Macmillan and Company. • Stephen, Leslie; Pollock, Frederick (1879). Lectures and Essays by the Late William Kingdon Clifford, F.R.S. Vol. 2. New York: Macmillan and Company. External links • Works by William Kingdon Clifford at Project Gutenberg • William and Lucy Clifford (with pictures) • O'Connor, John J.; Robertson, Edmund F., "William Kingdon Clifford", MacTutor History of Mathematics Archive, University of St Andrews • Works by or about William Kingdon Clifford at Internet Archive • Works by William Kingdon Clifford at LibriVox (public domain audiobooks) • Clifford, William Kingdon, William James, and A.J. Burger (Ed.), The Ethics of Belief. • Joe Rooney William Kingdon Clifford, Department of Design and Innovation, the Open University, London. Philosophy of religion Concepts in religion • Afterlife • Euthyphro dilemma • Faith • or religious belief • Intelligent design • Miracle • Problem of evil • Soul • Spirit • Theodicy • Theological veto Conceptions of God • Brahman • Demiurge • Divine simplicity • Egoism • Holy Spirit • Misotheism • Pandeism • Personal god • Process theology • Supreme Being • Unmoved mover God in • Abrahamic religions • Buddhism • Christianity • Hinduism • Islam • Jainism • Judaism • Mormonism • Sikhism • Baháʼí Faith • Wicca Existence of God For • Beauty • Christological • Consciousness • Cosmological • Kalam • Contingency • Degree • Desire • Experience • Fine-tuning of the universe • Love • Miracles • Morality • Necessary existent • Ontological • Pascal's wager • Proper basis and Reformed epistemology • Reason • Teleological • Natural law • Watchmaker analogy • Transcendental Against • 747 gambit • Atheist's Wager • Evil • Free will • Hell • Inconsistent revelations • Nonbelief • Noncognitivism • Occam's razor • Omnipotence • Poor design • Russell's teapot Theology • Acosmism • Agnosticism • Animism • Antireligion • Atheism • Creationism • Dharmism • Deism • Demonology • Divine command theory • Dualism • Esotericism • Exclusivism • Existentialism • Christian • Atheistic • Feminist theology • Thealogy • Womanist theology • Fideism • Fundamentalism • Gnosticism • Henotheism • Humanism • Religious • Secular • Christian • Inclusivism • Theories about religions • Monism • Monotheism • Mysticism • Naturalism • Metaphysical • Religious • Humanistic • New Age • Nonduality • Nontheism • Pandeism • Panentheism • Pantheism • Perennialism • Polytheism • Possibilianism • Process theology • Religious skepticism • Spiritualism • Shamanism • Taoic • Theism • Transcendentalism • more... Religious language • Eschatological verification • Language game • Logical positivism • Apophatic theology • Verificationism Problem of evil • Augustinian theodicy • Best of all possible worlds • Euthyphro dilemma • Inconsistent triad • Irenaean theodicy • Natural evil • Theodicy Philosophers of religion (by date active) Ancient and medieval • Anselm of Canterbury • Augustine of Hippo • Avicenna • Averroes • Boethius • Gaudapada • Gaunilo of Marmoutiers • Pico della Mirandola • Heraclitus • King James VI and I • Marcion of Sinope • Maimonides • Adi Shankara • Thomas Aquinas • William of Ockham Early modern • Augustin Calmet • René Descartes • Blaise Pascal • Desiderius Erasmus • Baruch Spinoza • Nicolas Malebranche • Gottfried W Leibniz • William Wollaston • Thomas Chubb • David Hume • Baron d'Holbach • Immanuel Kant • Johann G Herder 1800 1850 • Friedrich Schleiermacher • Karl C F Krause • Georg W F Hegel • Thomas Carlyle • William Whewell • Ludwig Feuerbach • Søren Kierkegaard • Karl Marx • Albrecht Ritschl • Afrikan Spir 1880 1900 • Ernst Haeckel • W K Clifford • Friedrich Nietzsche • Harald Høffding • William James • Vladimir Solovyov • Ernst Troeltsch • Rudolf Otto • Lev Shestov • Sergei Bulgakov • Pavel Florensky • Ernst Cassirer • Joseph Maréchal 1920 postwar • George Santayana • Bertrand Russell • Martin Buber • René Guénon • Paul Tillich • Karl Barth • Emil Brunner • Rudolf Bultmann • Gabriel Marcel • Reinhold Niebuhr • Charles Hartshorne • Mircea Eliade • Frithjof Schuon • J L Mackie • Walter Kaufmann • Martin Lings • Peter Geach • George I Mavrodes • William Alston • Antony Flew 1970 1990 2010 • William L Rowe • Dewi Z Phillips • Alvin Plantinga • Anthony Kenny • Nicholas Wolterstorff • Richard Swinburne • Robert Merrihew Adams • Ravi Zacharias • Peter van Inwagen • Daniel Dennett • Loyal Rue • Jean-Luc Marion • William Lane Craig • Ali Akbar Rashad • Alexander Pruss Related topics • Criticism of religion • Desacralization of knowledge • Ethics in religion • Exegesis • History of religion • Religion • Religious language • Religious philosophy • Relationship between religion and science • Faith and rationality • more... • Portal • Category Authority control International • FAST • ISNI • VIAF National • Spain • France • BnF data • Germany • Israel • United States • Japan • Czech Republic • Australia • Greece • Netherlands • Poland • Portugal • Vatican Academics • CiNii • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie • Trove Other • SNAC • IdRef
Without loss of generality Without loss of generality (often abbreviated to WOLOG, WLOG[1] or w.l.o.g.; less commonly stated as without any loss of generality or with no loss of generality) is a frequently used expression in mathematics. The term is used to indicate the assumption that follows is chosen arbitrarily, narrowing the premise to a particular case, but does not affect the validity of the proof in general. The other cases are sufficiently similar to the one presented that proving them follows by essentially the same logic.[2] As a result, once a proof is given for the particular case, it is trivial to adapt it to prove the conclusion in all other cases. In many scenarios, the use of "without loss of generality" is made possible by the presence of symmetry.[3] For example, if some property P(x,y) of real numbers is known to be symmetric in x and y, namely that P(x,y) is equivalent to P(y,x), then in proving that P(x,y) holds for every x and y, one may assume "without loss of generality" that x ≤ y. There is no loss of generality in this assumption, since once the case x ≤ y ⇒ P(x,y) has been proved, the other case follows by interchanging x and y : y ≤ x ⇒ P(y,x), and by symmetry of P, this implies P(x,y), thereby showing that P(x,y) holds for all cases. On the other hand, if neither such a symmetry nor another form of equivalence can be established, then the use of "without loss of generality" is incorrect and can amount to an instance of proof by example – a logical fallacy of proving a claim by proving a non-representative example.[4] Example Consider the following theorem (which is a case of the pigeonhole principle): If three objects are each painted either red or blue, then there must be at least two objects of the same color. A proof: Assume, without loss of generality, that the first object is red. If either of the other two objects is red, then we are finished; if not, then the other two objects must both be blue and we are still finished. The above argument works because the exact same reasoning could be applied if the alternative assumption, namely, that the first object is blue, were made, or, similarly, that the words 'red' and 'blue' can be freely exchanged in the wording of the proof. As a result, the use of "without loss of generality" is valid in this case. See also • Up to • Mathematical jargon References 1. "Without Loss of Generality". Art of Problem Solving. Retrieved 2019-10-21. 2. Chartrand, Gary; Polimeni, Albert D.; Zhang, Ping (2008). Mathematical Proofs / A Transition to Advanced Mathematics (2nd ed.). Pearson/Addison Wesley. pp. 80–81. ISBN 978-0-321-39053-0. 3. Dijkstra, Edsger W. (1997). "WLOG, or the misery of the unordered pair (EWD1223)". In Broy, Manfred; Schieder, Birgit (eds.). Mathematical Methods in Program Development (PDF). NATO ASI Series F: Computer and Systems Sciences. Vol. 158. Springer. pp. 33–34. doi:10.1007/978-3-642-60858-2_9. 4. "An Acyclic Inequality in Three Variables". www.cut-the-knot.org. Retrieved 2019-10-21. External links • WLOG at PlanetMath. • "Without Loss of Generality" by John Harrison - discussion of formalizing "WLOG" arguments in an automated theorem prover.
Wold's theorem In statistics, Wold's decomposition or the Wold representation theorem (not to be confused with the Wold theorem that is the discrete-time analog of the Wiener–Khinchin theorem), named after Herman Wold, says that every covariance-stationary time series $Y_{t}$ can be written as the sum of two time series, one deterministic and one stochastic. This article is about the theorem as used in time series analysis. For an abstract mathematical statement, see Wold decomposition. Formally $Y_{t}=\sum _{j=0}^{\infty }b_{j}\varepsilon _{t-j}+\eta _{t},$ where: • $Y_{t}$ is the time series being considered, • $\varepsilon _{t}$ is an uncorrelated sequence which is the innovation process to the process $Y_{t}$ – that is, a white noise process that is input to the linear filter $\{b_{j}\}$. • $b$ is the possibly infinite vector of moving average weights (coefficients or parameters) • $\eta _{t}$ is a deterministic time series, such as one represented by a sine wave. The moving average coefficients have these properties: 1. Stable, that is square summable $\sum _{j=1}^{\infty }\|b_{j}\|^{2}$ < $\infty $ 2. Causal (i.e. there are no terms with j < 0) 3. Minimum delay 4. Constant ($b_{j}$ independent of t) 5. It is conventional to define $b_{0}=1$ This theorem can be considered as an existence theorem: any stationary process has this seemingly special representation. Not only is the existence of such a simple linear and exact representation remarkable, but even more so is the special nature of the moving average model. Imagine creating a process that is a moving average but not satisfying these properties 1–4. For example, the coefficients $b_{j}$ could define an acausal and non-minimum delay model. Nevertheless the theorem assures the existence of a causal minimum delay moving average that exactly represents this process. How this all works for the case of causality and the minimum delay property is discussed in Scargle (1981), where an extension of the Wold Decomposition is discussed. The usefulness of the Wold Theorem is that it allows the dynamic evolution of a variable $Y_{t}$ to be approximated by a linear model. If the innovations $\varepsilon _{t}$ are independent, then the linear model is the only possible representation relating the observed value of $Y_{t}$ to its past evolution. However, when $\varepsilon _{t}$ is merely an uncorrelated but not independent sequence, then the linear model exists but it is not the only representation of the dynamic dependence of the series. In this latter case, it is possible that the linear model may not be very useful, and there would be a nonlinear model relating the observed value of $Y_{t}$ to its past evolution. However, in practical time series analysis, it is often the case that only linear predictors are considered, partly on the grounds of simplicity, in which case the Wold decomposition is directly relevant. The Wold representation depends on an infinite number of parameters, although in practice they usually decay rapidly. The autoregressive model is an alternative that may have only a few coefficients if the corresponding moving average has many. These two models can be combined into an autoregressive-moving average (ARMA) model, or an autoregressive-integrated-moving average (ARIMA) model if non-stationarity is involved. See Scargle (1981) and references there; in addition this paper gives an extension of the Wold Theorem that allows more generality for the moving average (not necessarily stable, causal, or minimum delay) accompanied by a sharper characterization of the innovation (identically and independently distributed, not just uncorrelated). This extension allows the possibility of models that are more faithful to physical or astrophysical processes, and in particular can sense ″the arrow of time.″ References • Anderson, T. W. (1971). The Statistical Analysis of Time Series. Wiley. • Nerlove, M.; Grether, David M.; Carvalho, José L. (1995). Analysis of Economic Time Series (Revised ed.). San Diego: Academic Press. pp. 30–36. ISBN 0-12-515751-7. • Scargle, J. D. (1981). Studies in astronomical time series analysis. I – Modeling random processes in the time domain. Astrophysical Journal Supplement Series. Vol. 45. pp. 1–71. • Wold, H. (1954) A Study in the Analysis of Stationary Time Series, Second revised edition, with an Appendix on "Recent Developments in Time Series Analysis" by Peter Whittle. Almqvist and Wiksell Book Co., Uppsala. Statistics • Outline • Index Descriptive statistics Continuous data Center • Mean • Arithmetic • Arithmetic-Geometric • Cubic • Generalized/power • Geometric • Harmonic • Heronian • Heinz • Lehmer • Median • Mode Dispersion • Average absolute deviation • Coefficient of variation • Interquartile range • Percentile • Range • Standard deviation • Variance Shape • Central limit theorem • Moments • Kurtosis • L-moments • Skewness Count data • Index of dispersion Summary tables • Contingency table • Frequency distribution • Grouped data Dependence • Partial correlation • Pearson product-moment correlation • Rank correlation • Kendall's τ • Spearman's ρ • Scatter plot Graphics • Bar chart • Biplot • Box plot • Control chart • Correlogram • Fan chart • Forest plot • Histogram • Pie chart • Q–Q plot • Radar chart • Run chart • Scatter plot • Stem-and-leaf display • Violin plot Data collection Study design • Effect size • Missing data • Optimal design • Population • Replication • Sample size determination • Statistic • Statistical power Survey methodology • Sampling • Cluster • Stratified • Opinion poll • Questionnaire • Standard error Controlled experiments • Blocking • Factorial experiment • Interaction • Random assignment • Randomized controlled trial • Randomized experiment • Scientific control Adaptive designs • Adaptive clinical trial • Stochastic approximation • Up-and-down designs Observational studies • Cohort study • Cross-sectional study • Natural experiment • Quasi-experiment Statistical inference Statistical theory • Population • Statistic • Probability distribution • Sampling distribution • Order statistic • Empirical distribution • Density estimation • Statistical model • Model specification • Lp space • Parameter • location • scale • shape • Parametric family • Likelihood (monotone) • Location–scale family • Exponential family • Completeness • Sufficiency • Statistical functional • Bootstrap • U • V • Optimal decision • loss function • Efficiency • Statistical distance • divergence • Asymptotics • Robustness Frequentist inference Point estimation • Estimating equations • Maximum likelihood • Method of moments • M-estimator • Minimum distance • Unbiased estimators • Mean-unbiased minimum-variance • Rao–Blackwellization • Lehmann–Scheffé theorem • Median unbiased • Plug-in Interval estimation • Confidence interval • Pivot • Likelihood interval • Prediction interval • Tolerance interval • Resampling • Bootstrap • Jackknife Testing hypotheses • 1- & 2-tails • Power • Uniformly most powerful test • Permutation test • Randomization test • Multiple comparisons Parametric tests • Likelihood-ratio • Score/Lagrange multiplier • Wald Specific tests • Z-test (normal) • Student's t-test • F-test Goodness of fit • Chi-squared • G-test • Kolmogorov–Smirnov • Anderson–Darling • Lilliefors • Jarque–Bera • Normality (Shapiro–Wilk) • Likelihood-ratio test • Model selection • Cross validation • AIC • BIC Rank statistics • Sign • Sample median • Signed rank (Wilcoxon) • Hodges–Lehmann estimator • Rank sum (Mann–Whitney) • Nonparametric anova • 1-way (Kruskal–Wallis) • 2-way (Friedman) • Ordered alternative (Jonckheere–Terpstra) • Van der Waerden test Bayesian inference • Bayesian probability • prior • posterior • Credible interval • Bayes factor • Bayesian estimator • Maximum posterior estimator • Correlation • Regression analysis Correlation • Pearson product-moment • Partial correlation • Confounding variable • Coefficient of determination Regression analysis • Errors and residuals • Regression validation • Mixed effects models • Simultaneous equations models • Multivariate adaptive regression splines (MARS) Linear regression • Simple linear regression • Ordinary least squares • General linear model • Bayesian regression Non-standard predictors • Nonlinear regression • Nonparametric • Semiparametric • Isotonic • Robust • Heteroscedasticity • Homoscedasticity Generalized linear model • Exponential families • Logistic (Bernoulli) / Binomial / Poisson regressions Partition of variance • Analysis of variance (ANOVA, anova) • Analysis of covariance • Multivariate ANOVA • Degrees of freedom Categorical / Multivariate / Time-series / Survival analysis Categorical • Cohen's kappa • Contingency table • Graphical model • Log-linear model • McNemar's test • Cochran–Mantel–Haenszel statistics Multivariate • Regression • Manova • Principal components • Canonical correlation • Discriminant analysis • Cluster analysis • Classification • Structural equation model • Factor analysis • Multivariate distributions • Elliptical distributions • Normal Time-series General • Decomposition • Trend • Stationarity • Seasonal adjustment • Exponential smoothing • Cointegration • Structural break • Granger causality Specific tests • Dickey–Fuller • Johansen • Q-statistic (Ljung–Box) • Durbin–Watson • Breusch–Godfrey Time domain • Autocorrelation (ACF) • partial (PACF) • Cross-correlation (XCF) • ARMA model • ARIMA model (Box–Jenkins) • Autoregressive conditional heteroskedasticity (ARCH) • Vector autoregression (VAR) Frequency domain • Spectral density estimation • Fourier analysis • Least-squares spectral analysis • Wavelet • Whittle likelihood Survival Survival function • Kaplan–Meier estimator (product limit) • Proportional hazards models • Accelerated failure time (AFT) model • First hitting time Hazard function • Nelson–Aalen estimator Test • Log-rank test Applications Biostatistics • Bioinformatics • Clinical trials / studies • Epidemiology • Medical statistics Engineering statistics • Chemometrics • Methods engineering • Probabilistic design • Process / quality control • Reliability • System identification Social statistics • Actuarial science • Census • Crime statistics • Demography • Econometrics • Jurimetrics • National accounts • Official statistics • Population statistics • Psychometrics Spatial statistics • Cartography • Environmental statistics • Geographic information system • Geostatistics • Kriging • Category • Mathematics portal • Commons • WikiProject
Wold's decomposition In mathematics, particularly in operator theory, Wold decomposition or Wold–von Neumann decomposition, named after Herman Wold and John von Neumann, is a classification theorem for isometric linear operators on a given Hilbert space. It states that every isometry is a direct sum of copies of the unilateral shift and a unitary operator. This article is about the general mathematical result. For the application to time series analysis, see Wold's theorem. In time series analysis, the theorem implies that any stationary discrete-time stochastic process can be decomposed into a pair of uncorrelated processes, one deterministic, and the other being a moving average process. Details Let H be a Hilbert space, L(H) be the bounded operators on H, and V ∈ L(H) be an isometry. The Wold decomposition states that every isometry V takes the form $V=(\oplus _{\alpha \in A}S)\oplus U$ for some index set A, where S is the unilateral shift on a Hilbert space Hα, and U is a unitary operator (possible vacuous). The family {Hα} consists of isomorphic Hilbert spaces. A proof can be sketched as follows. Successive applications of V give a descending sequences of copies of H isomorphically embedded in itself: $H=H\supset V(H)\supset V^{2}(H)\supset \cdots =H_{0}\supset H_{1}\supset H_{2}\supset \cdots ,$ where V(H) denotes the range of V. The above defined Hi = Vi(H). If one defines $M_{i}=H_{i}\ominus H_{i+1}=V^{i}(H\ominus V(H))\quad {\text{for}}\quad i\geq 0\;,$ then $H=(\oplus _{i\geq 0}M_{i})\oplus (\cap _{i\geq 0}H_{i})=K_{1}\oplus K_{2}.$ It is clear that K1 and K2 are invariant subspaces of V. So V(K2) = K2. In other words, V restricted to K2 is a surjective isometry, i.e., a unitary operator U. Furthermore, each Mi is isomorphic to another, with V being an isomorphism between Mi and Mi+1: V "shifts" Mi to Mi+1. Suppose the dimension of each Mi is some cardinal number α. We see that K1 can be written as a direct sum Hilbert spaces $K_{1}=\oplus H_{\alpha }$ where each Hα is an invariant subspaces of V and V restricted to each Hα is the unilateral shift S. Therefore $V=V\vert _{K_{1}}\oplus V\vert _{K_{2}}=(\oplus _{\alpha \in A}S)\oplus U,$ which is a Wold decomposition of V. Remarks It is immediate from the Wold decomposition that the spectrum of any proper, i.e. non-unitary, isometry is the unit disk in the complex plane. An isometry V is said to be pure if, in the notation of the above proof, ∩i≥0 Hi = {0}. The multiplicity of a pure isometry V is the dimension of the kernel of V, i.e. the cardinality of the index set A in the Wold decomposition of V. In other words, a pure isometry of multiplicity N takes the form $V=\oplus _{1\leq \alpha \leq N}S.$ In this terminology, the Wold decomposition expresses an isometry as a direct sum of a pure isometry and a unitary operator. A subspace M is called a wandering subspace of V if Vn(M) ⊥ Vm(M) for all n ≠ m. In particular, each Mi defined above is a wandering subspace of V. A sequence of isometries The decomposition above can be generalized slightly to a sequence of isometries, indexed by the integers. The C-algebra generated by an isometry Consider an isometry V ∈ L(H). Denote by C(V) the C-algebra generated by V, i.e. C(V) is the norm closure of polynomials in V and V. The Wold decomposition can be applied to characterize C(V). Let C(T) be the continuous functions on the unit circle T. We recall that the C-algebra C(S) generated by the unilateral shift S takes the following form C(S) = {Tf + K \| Tf is a Toeplitz operator with continuous symbol f ∈ C(T) and K is a compact operator}. In this identification, S = Tz where z is the identity function in C(T). The algebra C(S) is called the Toeplitz algebra. Theorem (Coburn) C(V) is isomorphic to the Toeplitz algebra and V is the isomorphic image of Tz. The proof hinges on the connections with C(T), in the description of the Toeplitz algebra and that the spectrum of a unitary operator is contained in the circle T. The following properties of the Toeplitz algebra will be needed: 1. $T_{f}+T_{g}=T_{f+g}.\,$ 2. $T_{f}^{}=T_{\bar {f}}.$ 3. The semicommutator $T_{f}T_{g}-T_{fg}\,$ is compact. The Wold decomposition says that V is the direct sum of copies of Tz and then some unitary U: $V=(\oplus _{\alpha \in A}T_{z})\oplus U.$ So we invoke the continuous functional calculus f → f(U), and define $\Phi :C^{}(S)\rightarrow C^{}(V)\quad {\text{by}}\quad \Phi (T_{f}+K)=\oplus _{\alpha \in A}(T_{f}+K)\oplus f(U).$ One can now verify Φ is an isomorphism that maps the unilateral shift to V: By property 1 above, Φ is linear. The map Φ is injective because Tf is not compact for any non-zero f ∈ C(T) and thus Tf + K = 0 implies f = 0. Since the range of Φ is a C-algebra, Φ is surjective by the minimality of C(V). Property 2 and the continuous functional calculus ensure that Φ preserves the -operation. Finally, the semicommutator property shows that Φ is multiplicative. Therefore the theorem holds. References • Coburn, L. (1967). "The C-algebra of an isometry". Bull. Amer. Math. Soc. 73 (5): 722–726. doi:10.1090/S0002-9904-1967-11845-7. • Constantinescu, T. (1996). Schur Parameters, Factorization and Dilation Problems. Operator Theory, Advances and Applications. Vol. 82. Birkhäuser. ISBN 3-7643-5285-X. • Douglas, R. G. (1972). Banach Algebra Techniques in Operator Theory. Academic Press. ISBN 0-12-221350-5. • Rosenblum, Marvin; Rovnyak, James (1985). Hardy Classes and Operator Theory. Oxford University Press. ISBN 0-19-503591-7. Functional analysis (topics – glossary) Spaces • Banach • Besov • Fréchet • Hilbert • Hölder • Nuclear • Orlicz • Schwartz • Sobolev • Topological vector Properties • Barrelled • Complete • Dual (Algebraic/Topological) • Locally convex • Reflexive • Reparable Theorems • Hahn–Banach • Riesz representation • Closed graph • Uniform boundedness principle • Kakutani fixed-point • Krein–Milman • Min–max • Gelfand–Naimark • Banach–Alaoglu Operators • Adjoint • Bounded • Compact • Hilbert–Schmidt • Normal • Nuclear • Trace class • Transpose • Unbounded • Unitary Algebras • Banach algebra • C-algebra • Spectrum of a C*-algebra • Operator algebra • Group algebra of a locally compact group • Von Neumann algebra Open problems • Invariant subspace problem • Mahler's conjecture Applications • Hardy space • Spectral theory of ordinary differential equations • Heat kernel • Index theorem • Calculus of variations • Functional calculus • Integral operator • Jones polynomial • Topological quantum field theory • Noncommutative geometry • Riemann hypothesis • Distribution (or Generalized functions) Advanced topics • Approximation property • Balanced set • Choquet theory • Weak topology • Banach–Mazur distance • Tomita–Takesaki theory • Mathematics portal • Category • Commons Hilbert spaces Basic concepts • Adjoint • Inner product and L-semi-inner product • Hilbert space and Prehilbert space • Orthogonal complement • Orthonormal basis Main results • Bessel's inequality • Cauchy–Schwarz inequality • Riesz representation Other results • Hilbert projection theorem • Parseval's identity • Polarization identity (Parallelogram law) Maps • Compact operator on Hilbert space • Densely defined • Hermitian form • Hilbert–Schmidt • Normal • Self-adjoint • Sesquilinear form • Trace class • Unitary Examples • Cn(K) with K compact & n<∞ • Segal–Bargmann F
Wolf Prize in Mathematics The Wolf Prize in Mathematics is awarded almost annually[lower-alpha 1] by the Wolf Foundation in Israel. It is one of the six Wolf Prizes established by the Foundation and awarded since 1978; the others are in Agriculture, Chemistry, Medicine, Physics and Arts. According to a reputation survey conducted in 2013 and 2014, the Wolf Prize in Mathematics is the third most prestigious international academic award in mathematics, after the Abel Prize and the Fields Medal.[1][2] Until the establishment of the Abel Prize, it was probably the closest equivalent of a "Nobel Prize in Mathematics", since the Fields Medal is awarded every four years only to mathematicians under the age of 40. The Wolf Prize includes a monetary award of $100,000. Laureates Year Name Nationality Citation 1978 Israel Gelfand Soviet Union for his work in functional analysis, group representation, and for his seminal contributions to many areas of mathematics and its applications. Carl L. Siegel Germany for his contributions to the theory of numbers, theory of several complex variables, and celestial mechanics. 1979 Jean Leray France for pioneering work on the development and application of topological methods to the study of differential equations. André Weil France for his inspired introduction of algebraic-geometric methods to the theory of numbers. 1980 Henri Cartan France for pioneering work in algebraic topology, complex variables, homological algebra and inspired leadership of a generation of mathematicians. Andrey Kolmogorov Soviet Union for deep and original discoveries in Fourier analysis, probability theory, ergodic theory and dynamical systems. 1981 Lars Ahlfors Finland for seminal discoveries and the creation of powerful new methods in geometric function theory. Oscar Zariski United States creator of the modern approach to algebraic geometry, by its fusion with commutative algebra. 1982 Hassler Whitney United States for his fundamental work in algebraic topology, differential geometry and differential topology. Mark Krein Soviet Union for his fundamental contributions to functional analysis and its applications. 1983/84 Shiing-Shen Chern China United States for outstanding contributions to global differential geometry, which have profoundly influenced all mathematics. Paul Erdős Hungary for his numerous contributions to number theory, combinatorics, probability, set theory and mathematical analysis, and for personally stimulating mathematicians the world over. 1984/85 Kunihiko Kodaira Japan for his outstanding contributions to the study of complex manifolds and algebraic varieties. Hans Lewy United States for initiating many, now classic and essential, developments in partial differential equations. 1986 Samuel Eilenberg Poland United States for his fundamental work in algebraic topology and homological algebra. Atle Selberg Norway for his profound and original work on number theory and on discrete groups and automorphic forms. 1987 Kiyoshi Itō Japan for his fundamental contributions to pure and applied probability theory, especially the creation of the stochastic differential and integral calculus. Peter Lax Hungary United States for his outstanding contributions to many areas of analysis and applied mathematics. 1988 Friedrich Hirzebruch Germany for outstanding work combining topology, algebraic geometry and differential geometry, and algebraic number theory; and for his stimulation of mathematical cooperation and research. Lars Hörmander Sweden for fundamental work in modern analysis, in particular, the application of pseudo-differential operators and Fourier integral operators to linear partial differential equations. 1989 Alberto Calderón Argentina for his groundbreaking work on singular integral operators and their application to important problems in partial differential equations. John Milnor United States for ingenious and highly original discoveries in geometry, which have opened important new vistas in topology from the algebraic, combinatorial, and differentiable viewpoint. 1990 Ennio de Giorgi Italy for his innovating ideas and fundamental achievements in partial differential equations and calculus of variations. Ilya Piatetski-Shapiro Israel for his fundamental contributions in the fields of homogeneous complex domains, discrete groups, representation theory and automorphic forms. 1991 No award 1992 Lennart Carleson Sweden for his fundamental contributions to Fourier analysis, complex analysis, quasi-conformal mappings and dynamical systems. John G. Thompson United States for his profound contributions to all aspects of finite group theory and connections with other branches of mathematics. 1993 Mikhail Gromov Russia France for his revolutionary contributions to global Riemannian and symplectic geometry, algebraic topology, geometric group theory and the theory of partial differential equations; Jacques Tits Belgium France for his pioneering and fundamental contributions to the theory of the structure of algebraic and other classes of groups and in particular for the theory of buildings. 1994/95 Jürgen Moser Switzerland United States for his fundamental work on stability in Hamiltonian mechanics and his profound and influential contributions to nonlinear differential equations. 1995/96 Robert Langlands Canada for his path-blazing work and extraordinary insight in the fields of number theory, automorphic forms and group representation. Andrew Wiles United Kingdom for spectacular contributions to number theory and related fields, major advances on fundamental conjectures, and for settling Fermat's Last Theorem. 1996/97 Joseph B. Keller United States for his profound and innovative contributions, in particular to electromagnetic, optical, and acoustic wave propagation and to fluid, solid, quantum and statistical mechanics. Yakov G. Sinai Russia for his fundamental contributions to mathematically rigorous methods in statistical mechanics and the ergodic theory of dynamical systems and their applications in physics. 1998 No award 1999 László Lovász Hungary United States for his outstanding contributions to combinatorics, theoretical computer science and combinatorial optimization. Elias M. Stein United States for his contributions to classical and Euclidean Fourier analysis and for his exceptional impact on a new generation of analysts through his eloquent teaching and writing. 2000 Raoul Bott Hungary for his deep discoveries in topology and differential geometry and their applications to Lie groups, differential operators and mathematical physics. Jean-Pierre Serre France for his many fundamental contributions to topology, algebraic geometry, algebra, and number theory and for his inspirational lectures and writing. 2001 Vladimir Arnold Ukraine for his deep and influential work in a multitude of areas of mathematics, including dynamical systems, differential equations, and singularity theory. Saharon Shelah Israel for his many fundamental contributions to mathematical logic and set theory, and their applications within other parts of mathematics. 2002/03 Mikio Sato Japan for his creation of algebraic analysis, including hyperfunction theory and microfunction theory, holonomic quantum field theory, and a unified theory of soliton equations. John Tate United States for his creation of fundamental concepts in algebraic number theory. 2004 No award 2005 Gregory Margulis Russia for his monumental contributions to algebra, in particular to the theory of lattices in semi-simple Lie groups, and striking applications of this to ergodic theory, representation theory, number theory, combinatorics, and measure theory. Sergei Novikov Russia for his fundamental and pioneering contributions to algebraic and differential topology, and to mathematical physics, notably the introduction of algebraic-geometric methods. 2006/07 Stephen Smale United States for his groundbreaking contributions that have played a fundamental role in shaping differential topology, dynamical systems, mathematical economics, and other subjects in mathematics. Hillel Furstenberg United States Israel for his profound contributions to ergodic theory, probability, topological dynamics, analysis on symmetric spaces and homogeneous flows. 2008 Pierre Deligne Belgium for his work on mixed Hodge theory; the Weil conjectures; the Riemann-Hilbert correspondence; and for his contributions to arithmetic. Phillip A. Griffiths United States for his work on variations of Hodge structures; the theory of periods of abelian integrals; and for his contributions to complex differential geometry. David B. Mumford United States for his work on algebraic surfaces; on geometric invariant theory; and for laying the foundations of the modern algebraic theory of moduli of curves and theta functions. 2009 No award 2010 Shing-Tung Yau United States for his work in geometric analysis that has had a profound and dramatic impact on many areas of geometry and physics. Dennis P. Sullivan United States for his innovative contributions to algebraic topology and conformal dynamics. 2011 No award 2012 Michael Aschbacher United States for his work on the theory of finite groups. Luis Caffarelli Argentina for his work on partial differential equations. 2013 George D. Mostow United States for his fundamental and pioneering contribution to geometry and Lie group theory. Michael Artin United States for his fundamental contributions to algebraic geometry, both in commutative and noncommutative. 2014 Peter Sarnak South Africa United States for his deep contributions in analysis, number theory, geometry, and combinatorics. 2015 James G. Arthur Canada for his monumental work on the trace formula and his fundamental contributions to the theory of automorphic representations of reductive groups. 2016 No award 2017 Richard Schoen United States for his contributions to geometric analysis and the understanding of the interconnectedness of partial differential equations and differential geometry. Charles Fefferman United States for his contributions in a number of mathematical areas including complex multivariate analysis, partial differential equations and sub-elliptical problems. 2018 Alexander Beilinson United States for their work that has made significant progress at the interface of geometry and mathematical physics. Vladimir Drinfeld Russia United States 2019 Jean-Francois Le Gall France for his several deep and elegant contributions to the theory of stochastic processes. Gregory Lawler United States for his comprehensive and pioneering research on erased loops and random walks.[3] 2020 Simon K. Donaldson United Kingdom for their contributions to differential geometry and topology.[4] Yakov Eliashberg United States 2021 No award 2022 George Lusztig Romania United States for his groundbreaking contributions to representation theory and related areas.[5] 2023 Ingrid Daubechies Belgium United States for her work in wavelet theory and applied harmonic analysis.[6] Laureates per country Below is a chart of all laureates per country (updated to 2023 laureates). Some laureates are counted more than once if have multiple citizenship. Country Number of laureates United States 30 Soviet Union / Russia 9 France 7 Hungary 5 Israel 3 Japan 3 Belgium 3 Germany 2 United Kingdom 2 Canada 2 Argentina 2 Sweden 2 South Africa 1 Poland 1 Italy 1 China 1 Norway 1 Finland 1 Romania 1 Ukraine 1 Notes 1. The Wolf Foundation website describes the prize as annual; however, some prizes are split across years, while in some years no prize is awarded. See also • List of mathematics awards References 1. IREG Observatory on Academic Ranking and Excellence. IREG List of International Academic Awards (PDF). Brussels: IREG Observatory on Academic Ranking and Excellence. Retrieved 3 March 2018. 2. Zheng, Juntao; Liu, Niancai (2015). "Mapping of important international academic awards". Scientometrics. 104: 763–791. doi:10.1007/s11192-015-1613-7. 3. Wolf Prize 2019 - Mathematics 4. Wolf Prize 2020 - Mathematics 5. Wolf Prize 2022 - Mathematics 6. Wolf Prize 2023 - Mathematics External links • "The Wolf Foundation Prize in Mathematics". • "Huffingtonpost Israel-Wolf-Prizes 2012". Huffington Post. 10 January 2012. • "Jerusalempost Israel-Wolf-Prizes 2013". • Israel-Wolf-Prizes 2015 • Jerusalempost Wolf Prizes 2017 • Jerusalempost Wolf Prizes 2018 • Wolf Prize 2019 Wolf Foundation Prizes • Agriculture • Arts • Chemistry • Mathematics • Medicine • Physics 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
Wolfe conditions In the unconstrained minimization problem, the Wolfe conditions are a set of inequalities for performing inexact line search, especially in quasi-Newton methods, first published by Philip Wolfe in 1969.[1][2] In these methods the idea is to find $\min _{x}f(\mathbf {x} )$ for some smooth $f\colon \mathbb {R} ^{n}\to \mathbb {R} $. Each step often involves approximately solving the subproblem $\min _{\alpha }f(\mathbf {x} _{k}+\alpha \mathbf {p} _{k})$ where $\mathbf {x} _{k}$ is the current best guess, $\mathbf {p} _{k}\in \mathbb {R} ^{n}$ is a search direction, and $\alpha \in \mathbb {R} $ is the step length. The inexact line searches provide an efficient way of computing an acceptable step length $\alpha $ that reduces the objective function 'sufficiently', rather than minimizing the objective function over $\alpha \in \mathbb {R} ^{+}$ exactly. A line search algorithm can use Wolfe conditions as a requirement for any guessed $\alpha $, before finding a new search direction $\mathbf {p} _{k}$. Armijo rule and curvature A step length $\alpha _{k}$ is said to satisfy the Wolfe conditions, restricted to the direction $\mathbf {p} _{k}$, if the following two inequalities hold: ${\begin{aligned}{\textbf {i)}}&\quad f(\mathbf {x} _{k}+\alpha _{k}\mathbf {p} _{k})\leq f(\mathbf {x} _{k})+c_{1}\alpha _{k}\mathbf {p} _{k}^{\mathrm {T} }\nabla f(\mathbf {x} _{k}),\\[6pt]{\textbf {ii)}}&\quad {-\mathbf {p} }_{k}^{\mathrm {T} }\nabla f(\mathbf {x} _{k}+\alpha _{k}\mathbf {p} _{k})\leq -c_{2}\mathbf {p} _{k}^{\mathrm {T} }\nabla f(\mathbf {x} _{k}),\end{aligned}}$ with $0<c_{1}<c_{2}<1$. (In examining condition (ii), recall that to ensure that $\mathbf {p} _{k}$ is a descent direction, we have $\mathbf {p} _{k}^{\mathrm {T} }\nabla f(\mathbf {x} _{k})<0$, as in the case of gradient descent, where $\mathbf {p} _{k}=-\nabla f(\mathbf {x} _{k})$, or Newton–Raphson, where $\mathbf {p} _{k}=-\mathbf {H} ^{-1}\nabla f(\mathbf {x} _{k})$ with $\mathbf {H} $ positive definite.) $c_{1}$ is usually chosen to be quite small while $c_{2}$ is much larger; Nocedal and Wright give example values of $c_{1}=10^{-4}$ and $c_{2}=0.9$ for Newton or quasi-Newton methods and $c_{2}=0.1$ for the nonlinear conjugate gradient method.[3] Inequality i) is known as the Armijo rule[4] and ii) as the curvature condition; i) ensures that the step length $\alpha _{k}$ decreases $f$ 'sufficiently', and ii) ensures that the slope has been reduced sufficiently. Conditions i) and ii) can be interpreted as respectively providing an upper and lower bound on the admissible step length values. Strong Wolfe condition on curvature Denote a univariate function $\varphi $ restricted to the direction $\mathbf {p} _{k}$ as $\varphi (\alpha )=f(\mathbf {x} _{k}+\alpha \mathbf {p} _{k})$. The Wolfe conditions can result in a value for the step length that is not close to a minimizer of $\varphi $. If we modify the curvature condition to the following, ${\textbf {iii)}}\quad {\big \|}\mathbf {p} _{k}^{\mathrm {T} }\nabla f(\mathbf {x} _{k}+\alpha _{k}\mathbf {p} _{k}){\big \|}\leq c_{2}{\big \|}\mathbf {p} _{k}^{\mathrm {T} }\nabla f(\mathbf {x} _{k}){\big \|}$ then i) and iii) together form the so-called strong Wolfe conditions, and force $\alpha _{k}$ to lie close to a critical point of $\varphi $. Rationale The principal reason for imposing the Wolfe conditions in an optimization algorithm where $\mathbf {x} _{k+1}=\mathbf {x} _{k}+\alpha \mathbf {p} _{k}$ is to ensure convergence of the gradient to zero. In particular, if the cosine of the angle between $\mathbf {p} _{k}$ and the gradient, $\cos \theta _{k}={\frac {\nabla f(\mathbf {x} _{k})^{\mathrm {T} }\mathbf {p} _{k}}{\\|\nabla f(\mathbf {x} _{k})\\|\\|\mathbf {p} _{k}\\|}}$ is bounded away from zero and the i) and ii) conditions hold, then $\nabla f(\mathbf {x} _{k})\rightarrow 0$. An additional motivation, in the case of a quasi-Newton method, is that if $\mathbf {p} _{k}=-B_{k}^{-1}\nabla f(\mathbf {x} _{k})$, where the matrix $B_{k}$ is updated by the BFGS or DFP formula, then if $B_{k}$ is positive definite ii) implies $B_{k+1}$ is also positive definite. Comments Wolfe's conditions are more complicated than Armijo's condition, and a gradient descent algorithm based on Armijo's condition has a better theoretical guarantee than one based on Wolfe conditions (see the sections on "Upper bound for learning rates" and "Theoretical guarantee" in the Backtracking line search article). See also • Backtracking line search References 1. Wolfe, P. (1969). "Convergence Conditions for Ascent Methods". SIAM Review. 11 (2): 226–235. doi:10.1137/1011036. JSTOR 2028111. 2. Wolfe, P. (1971). "Convergence Conditions for Ascent Methods. II: Some Corrections". SIAM Review. 13 (2): 185–188. doi:10.1137/1013035. JSTOR 2028821. 3. Nocedal, Jorge; Wright, Stephen (1999). Numerical Optimization. p. 38. 4. Armijo, Larry (1966). "Minimization of functions having Lipschitz continuous first partial derivatives". Pacific J. Math. 16 (1): 1–3. doi:10.2140/pjm.1966.16.1. Further reading • "Line Search Methods". Numerical Optimization. Springer Series in Operations Research and Financial Engineering. 2006. pp. 30–32. doi:10.1007/978-0-387-40065-5_3. ISBN 978-0-387-30303-1. • "Quasi-Newton Methods". Numerical Optimization. Springer Series in Operations Research and Financial Engineering. 2006. pp. 135–163. doi:10.1007/978-0-387-40065-5_6. ISBN 978-0-387-30303-1. Optimization: Algorithms, methods, and heuristics Unconstrained nonlinear Functions • Golden-section search • Interpolation methods • Line search • Nelder–Mead method • Successive parabolic interpolation Gradients Convergence • Trust region • Wolfe conditions Quasi–Newton • Berndt–Hall–Hall–Hausman • Broyden–Fletcher–Goldfarb–Shanno and L-BFGS • Davidon–Fletcher–Powell • Symmetric rank-one (SR1) Other methods • Conjugate gradient • Gauss–Newton • Gradient • Mirror • Levenberg–Marquardt • Powell's dog leg method • Truncated Newton Hessians • Newton's method Constrained nonlinear General • Barrier methods • Penalty methods Differentiable • Augmented Lagrangian methods • Sequential quadratic programming • Successive linear programming Convex optimization Convex minimization • Cutting-plane method • Reduced gradient (Frank–Wolfe) • Subgradient method Linear and quadratic Interior point • Affine scaling • Ellipsoid algorithm of Khachiyan • Projective algorithm of Karmarkar Basis-exchange • Simplex algorithm of Dantzig • Revised simplex algorithm • Criss-cross algorithm • Principal pivoting algorithm of Lemke Combinatorial Paradigms • Approximation algorithm • Dynamic programming • Greedy algorithm • Integer programming • Branch and bound/cut Graph algorithms Minimum spanning tree • Borůvka • Prim • Kruskal Shortest path • Bellman–Ford • SPFA • Dijkstra • Floyd–Warshall Network flows • Dinic • Edmonds–Karp • Ford–Fulkerson • Push–relabel maximum flow Metaheuristics • Evolutionary algorithm • Hill climbing • Local search • Parallel metaheuristics • Simulated annealing • Spiral optimization algorithm • Tabu search • Software
Wolfe duality In mathematical optimization, Wolfe duality, named after Philip Wolfe, is type of dual problem in which the objective function and constraints are all differentiable functions. Using this concept a lower bound for a minimization problem can be found because of the weak duality principle.[1] Mathematical formulation For a minimization problem with inequality constraints, ${\begin{aligned}&{\underset {x}{\operatorname {minimize} }}&&f(x)\\&\operatorname {subject\;to} &&g_{i}(x)\leq 0,\quad i=1,\dots ,m\end{aligned}}$ the Lagrangian dual problem is ${\begin{aligned}&{\underset {u}{\operatorname {maximize} }}&&\inf _{x}\left(f(x)+\sum _{j=1}^{m}u_{j}g_{j}(x)\right)\\&\operatorname {subject\;to} &&u_{i}\geq 0,\quad i=1,\dots ,m\end{aligned}}$ where the objective function is the Lagrange dual function. Provided that the functions $f$ and $g_{1},\ldots ,g_{m}$ are convex and continuously differentiable, the infimum occurs where the gradient is equal to zero. The problem ${\begin{aligned}&{\underset {x,u}{\operatorname {maximize} }}&&f(x)+\sum _{j=1}^{m}u_{j}g_{j}(x)\\&\operatorname {subject\;to} &&\nabla f(x)+\sum _{j=1}^{m}u_{j}\nabla g_{j}(x)=0\\&&&u_{i}\geq 0,\quad i=1,\dots ,m\end{aligned}}$ is called the Wolfe dual problem.[2] This problem employs the KKT conditions as a constraint. Also, the equality constraint $\nabla f(x)+\sum _{j=1}^{m}u_{j}\nabla g_{j}(x)$ is nonlinear in general, so the Wolfe dual problem may be a nonconvex optimization problem. In any case, weak duality holds.[3] See also • Lagrangian duality • Fenchel duality References 1. Philip Wolfe (1961). "A duality theorem for non-linear programming". Quarterly of Applied Mathematics. 19 (3): 239–244. doi:10.1090/qam/135625. 2. "Chapter 3. Duality in convex optimization" (PDF). October 30, 2011. Retrieved May 20, 2012. 3. Geoffrion, Arthur M. (1971). "Duality in Nonlinear Programming: A Simplified Applications-Oriented Development". SIAM Review. 13 (1): 1–37. doi:10.1137/1013001. JSTOR 2028848.
Wolfgang Haken Wolfgang Haken (June 21, 1928 – October 2, 2022) was a German American mathematician who specialized in topology, in particular 3-manifolds. Wolfgang Haken Haken in 2008 Born(1928-06-21)June 21, 1928 Berlin, Germany DiedOctober 2, 2022(2022-10-02) (aged 94) Champaign, Illinois Alma materKiel University Occupation(s)Mathematician, professor Known forSolving the four-color theorem Biography Haken was born on June 21, 1928, in Berlin, Germany. His father was Werner Haken, a physicist who had Max Planck as a doctoral thesis advisor.[1] In 1953, Haken earned a Ph.D. degree in mathematics from Christian-Albrechts-Universität zu Kiel (Kiel University) and married Anna-Irmgard von Bredow, who earned a Ph.D. degree in mathematics from the same university in 1959. In 1962, they left Germany so he could accept a position as visiting professor at the University of Illinois at Urbana-Champaign. He became a full professor in 1965, retiring in 1998. In 1976, together with colleague Kenneth Appel at the University of Illinois at Urbana-Champaign, Haken solved the four-color problem: they proved that any planar graph can be properly colored using at most four colors. Haken has introduced several ideas, including Haken manifolds, Kneser-Haken finiteness, and an expansion of the work of Kneser into a theory of normal surfaces. Much of his work has an algorithmic aspect, and he is a figure in algorithmic topology. One of his key contributions to this field is an algorithm to detect whether a knot is unknotted. In 1978, Haken delivered an invited address at the International Congress of Mathematicians in Helsinki.[2] He was a recipient of the 1979 Fulkerson Prize of the American Mathematical Society for his proof with Appel of the four-color theorem.[3] Haken died in Champaign, Illinois, on October 2, 2022, aged 94.[4] Family Haken's eldest son, Armin, proved that there exist propositional tautologies that require resolution proofs of exponential size.[5] Haken's eldest daughter, Dorothea Blostein, is a professor of computer science, known for her discovery of the master theorem for divide-and-conquer recurrences. Haken’s second son, Lippold, is the inventor of the Continuum Fingerboard. Haken’s youngest son, Rudolf, is a professor of music, who established the world's first Electric Strings university degree program at the University of Illinois at Urbana-Champaign.[6] Wolfgang is the cousin of Hermann Haken, a physicist known for laser theory and synergetics. See also • Unknotting problem References 1. Werner Haken, Beitrag zur Kenntnis der thermoelektrischen Eigenschaften der Metallegierungen. Accessed May 6, 2019 2. International Congress of Mathematicians 1978. International Mathematical Union. Accessed May 29, 2011 3. Delbert Ray Fulkerson Prize, American Mathematical Society. Accessed May 29, 2011 4. "Wolfgang Haken's obituary". news-gazette.com. October 13, 2022. Archived from the original on October 14, 2022. Retrieved October 14, 2022. 5. Avi Wigderson, Mathematics and Computation, March 27 2018, footnote at Theorem 6.11 6. University of Illinois Electric Strings Degree Program Accessed November 15, 2022 • Haken, W. "Theorie der Normalflachen." Acta Math. 105, 245–375, 1961. • Ilya Kapovich (2016). "Wolfgang Haken: A biographical sketch". Illinois Journal of Mathematics. 60 (1): iii–ix. External links • Wolfgang Haken memorial website • Wolfgang Haken at the Mathematics Genealogy Project • Haken's faculty page at the University of Illinois at Urbana-Champaign • Wolfgang Haken biography from World of Mathematics • Lippold Haken's life story • Haken, Armin (1985), "The intractability of resolution", Theoretical Computer Science, 39: 297–308, doi:10.1016/0304-3975(85)90144-6 • Appel, Kenneth; Haken, Wolfgang (1989), Every Planar Map is Four Colorable, AMS, p. xv, ISBN 0-8218-5103-9 Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • United States • Netherlands Academics • DBLP • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie Other • IdRef
Wolfgang Krieger Wolfgang Krieger (born 3 June 3, 1940, Garmisch-Partenkirchen) is a German mathematician, specializing in analysis. Krieger studied mathematics and physics at the Ludwig-Maximilians-Universität München from 1959, where he obtained his doctorate in 1968 under Elmar Thoma with the thesis Über Maßklassen.[1] Krieger studied at Harvard University from 1962 to 1965, earning a master's degree in 1964. From 1966 to 1968 he was a research assistant in Munich; and from 1968 assistant professor, from 1970 associate professor and from 1972 full professor at Ohio State University. For the academic year 1973–1974 he was a visiting professor at the University of Göttingen. In 1974, until his retirement in 2006, he was a professor at Heidelberg University. From 1985 to 1987 he was Dean of the Faculty of Mathematics.[2] His research deals with ergodic theory, dynamical systems, and operator algebras. Cuntz-Krieger algebras, introduced in 1980, are named after him and Joachim Cuntz.[3] Krieger was a visiting scholar at IHES and Paris VI University in 1977–1978, at the University of Ottawa in 1982–1983, at the Almaden Research Center of IBM in 1988–1989, and at the Hebrew University of Jerusalem in 2005–2006. In 1997 he was a Fellow of the Japan Society for the Promotion of Science. He was elected a Fellow of the American Mathematical Society in 2012. He was an invited speaker with the talk On Generators in Ergodic Theory at the ICM in Vancouver in 1974. Selected publications • On non-singular transformations of a measure space. I, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, Vol. 11, No. 2, 1969, pp. 83–97. doi:10.1007/BF00531811 • On entropy and generators of measure-preserving transformations, Transactions of the American Mathematical Society, Vol. 149, 1970, pp. 453–464. doi:10.1090/S0002-9947-1970-0259068-3 • On ergodic flows and the isomorphism of factors, Mathematical Annals, Vol. 223, 1976, pp. 19–70 doi:10.1007/BF01360278 • with Alain Connes: Measure space automorphisms, the normalizers of their full groups, and approximate finiteness, Journal of Functional Analysis, Vol. 24, 1977, pp. 336–352 doi:10.1016/0022-1236(77)90062-3 • with Joachim Cuntz: A class of C*-algebras and topological Markov chains, Inventiones Mathematicae, Vol. 56, 1980, pp. 251–268. doi:10.1007/BF01390048 • On the Subsystems of Topological Markov Chains, Ergodic Theory and Dynamical Systems, Vol. 2, 1982, pp. 195–202 doi:10.1017/S0143385700001516 • with Mike Boyle: Periodic points and automorphisms of the shift, Trans. Amer. Math. Soc., Vol. 302, 1987, pp. 125–149 doi:10.1090/S0002-9947-1987-0887501-5 • with Brian Marcus and Selim Tuncel: On automorphisms of Markov chains, Trans. Amer. Math. Soc., Vol. 333, 1992, pp. 531–565 doi:10.1090/S0002-9947-1992-1099353-3 References 1. Wolfgang Krieger at the Mathematics Genealogy Project 2. Heidelberg Scholarly Lexicon 3. Nasr-Isfahani, Alireza (2017). "Algebraic Cuntz-Krieger algebras". arXiv:1708.01780 [math.RA]. External links • Prof. Dr. Wolfgang Krieger, math.uni-heidelberg.de Authority control International • ISNI • VIAF National • Germany Academics • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie
Wolfgang Walter Wolfgang Ludwig Walter (2 May 1927 – 26 June 2010) was a German mathematician, who specialized in the theory of differential equations. His textbook[2] on ordinary differential equations became a standard graduate text on the subject at many institutions. Wolfgang Ludwig Walter Born(1927-05-02)2 May 1927 Schwäbisch Gmünd, Weimar Republic Died26 June 2010(2010-06-26) (aged 83) Karlsruhe, Germany NationalityGerman Alma materUniversity of Tübingen Scientific career FieldsMathematics InstitutionsUniversity of Karlsruhe Doctoral advisorErich Kamke, Hellmuth Kneser[1] Biography Wolfgang Walter was born in 1927 in Schwäbisch Gmünd, Baden-Württemberg. His school studies were interrupted in 1943 when he was drafted into the army. He served as a soldier on the Eastern Front, was subsequently wounded and later interned as a prisoner of war by US troops. In 1946 after his release he completed his school education and in the years 1947–1952 studied mathematics and physics at the University of Tübingen, where he stayed on to study for his PhD under Erich Kamke and Hellmuth Kneser, defending his thesis in 1956.[3] In 1986–1992 Walter held the post of president of GAMM, the German society of applied mathematics and mechanics.[4] He died in Karlsruhe in 2010 at the age of 83. Works • Walter, Wolfgang (1998). Ordinary Differential Equations. Springer. ISBN 978-0387984599. • Walter, Wolfgang (2012). Differential and Integral Inequalities. Springer; Softcover reprint of the original 1st ed. 1970 edition. ISBN 978-3642864070. References 1. Wolfgang Walter. Mathematical Genealogy Project 2. Walter, Wolfgang (1998). Ordinary Differential Equations. Springer. ISBN 978-0387984599. 3. Wolfgang Reichel. In Memoriam Wolfgang Walter (1927–2010), Jahresber Dtsch Math-Ver, 2011. (in German) 4. R.M. Redheffer's 66th birthday tribute to Wolfgang Walter, World Scientific Publishing Company, 1994. Authority control International • ISNI • VIAF National • Norway • France • BnF data • Germany • Israel • Belgium • United States • Czech Republic • Netherlands • Poland Academics • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie Other • IdRef
WolframAlpha WolframAlpha (/ˈwʊlf.rəm-/ WUULf-rəm-) is an answer engine developed by Wolfram Research.[3] It answers factual queries by computing answers from externally sourced data.[4][5] WolframAlpha Type of site Answer engine OwnerWolframAlpha LLC Created byWolfram Research URLwww.wolframalpha.com CommercialYes RegistrationOptional LaunchedMay 18, 2009 (2009-05-18)[1] (official launch) May 15, 2009 (2009-05-15)[2] (public launch) Current statusActive Written inWolfram Language WolframAlpha was released on May 18, 2009, and is based on Wolfram's earlier product Wolfram Mathematica, a technical computing platform.[1] WolframAlpha gathers data from academic and commercial websites such as the CIA's The World Factbook, the United States Geological Survey, a Cornell University Library publication called All About Birds, Chambers Biographical Dictionary, Dow Jones, the Catalogue of Life,[3] CrunchBase,[6] Best Buy,[7] and the FAA to answer queries.[8] A Spanish language version was launched in 2022.[9] Technology Overview Users submit queries and computation requests via a text field. WolframAlpha then computes answers and relevant visualizations from a knowledge base of curated, structured data that come from other sites and books. It can respond to particularly phrased natural language fact-based questions. It displays its "Input interpretation" of such a question, using standardized phrases. It can also parse mathematical symbolism and respond with numerical and statistical results. Development WolframAlpha is written in the Wolfram Language, a general multi-paradigm programming language, and implemented in Mathematica. Wolfram language is proprietary and not commonly used by developers.[10] Usage WolframAlpha was used to power some searches in the Microsoft Bing and DuckDuckGo search engines but is no longer used to provide search results.[11][12] For factual question answering, WolframAlpha was used by Apple's Siri and Amazon Alexa for math and science queries but is no longer operational within those services.[13][14] WolframAlpha data types became available in July 2020 within Microsoft Excel, but the Microsoft-Wolfram partnership ended nearly two years later, in 2022, in favor of Microsoft Power Query data types.[15] WolframAlpha functionality in Microsoft Excel will end in June 2023.[16] History Launch preparations for WolframAlpha began on May 15, 2009 at 7 p.m. CDT and were broadcast live on Justin.tv. The plan was to publicly launch the service a few hours later. However, there were issues due to extreme load. The service officially launched on May 18, 2009,[17] receiving mixed reviews.[18][19] In 2009, WolframAlpha advocates pointed to its potential, some stating that how it determines results is more important than current usefulness.[18] WolframAlpha was free at launch, but later Wolfram Research attempted to monetize the service by launching an iOS application with a cost of $50, while the website itself was free.[20] That plan was abandoned after criticism.[21] On February 8, 2012, WolframAlpha Pro was released,[22] offering users additional features for a monthly subscription fee.[22][23] Some high-school and college students use WolframAlpha to cheat on math homework, though Wolfram Research says the service helps students understand math with its problem-solving capabilities.[24] Copyright claims InfoWorld published an article warning readers of the potential implications of giving an automated website proprietary rights to the data it generates.[25] Free software advocate Richard Stallman also opposes recognizing the site as a copyright holder and suspects that Wolfram Research would not be able to make this case under existing copyright law.[26] See also • Commonsense knowledge problem • Strong AI • Watson (computer) References 1. The Wolfram\|Alpha Launch Team (May 8, 2009). "So Much for A Quiet Launch". Wolfram\|Alpha Blog. Wolfram Alpha. Retrieved February 9, 2013.{{cite web}}: CS1 maint: multiple names: authors list (link) 2. The Wolfram\|Alpha Launch Team (May 12, 2009). "Going Live—and Webcasting It". Wolfram\|Alpha Blog. Wolfram Alpha. Retrieved February 9, 2013.{{cite web}}: CS1 maint: multiple names: authors list (link) 3. Bobbie Johnson (May 21, 2009). "Where does Wolfram Alpha get its information?". The Guardian. Retrieved March 8, 2013. 4. "About Wolfram\|Alpha: Making the World's Knowledge Computable". wolframalpha.com. Retrieved November 25, 2015. 5. Johnson, Bobbie (March 9, 2009). "British search engine 'could rival Google'". The Guardian. UK: Guardian News and Media. Retrieved February 9, 2013. 6. Dillet, Romain (September 7, 2012). "Wolfram Alpha Makes CrunchBase Data Computable Just In Time For Disrupt SF". TechCrunch. Retrieved February 9, 2013. 7. Golson, Jordan (December 16, 2011). "Wolfram Delivers Siri-Enabled Shopping Results From Best Buy". MacRumors. Retrieved February 9, 2013. 8. Barylick, Chris (November 19, 2011). "Wolfram Alpha search engine now tracks flight paths, trajectory information". Engadget. Retrieved February 9, 2013. 9. "Wolfram Alpha Spanish Announcement". Wolfram Alpha. Wolfram Research. July 22, 2022. Retrieved July 22, 2022. 10. "TIOBE Index". TIOBE. Retrieved October 6, 2022. 11. Krazit, Tom (August 21, 2009). "Bing strikes licensing deal with Wolfram Alpha". CNET. Archived from the original on October 23, 2013. Retrieved February 9, 2013. 12. The Wolfram\|Alpha Team (April 18, 2011). "Wolfram\|Alpha and DuckDuckGo Partner on API Binding and Search Integration". Wolfram\|Alpha Blog. Wolfram Alpha. Retrieved February 9, 2013.{{cite web}}: CS1 maint: multiple names: authors list (link) 13. "Alexa gets access to Wolfram Alpha's knowledge engine". TechCrunch. December 20, 2018. Retrieved March 17, 2021. 14. "Alexa Can Now Answer Those Tricky Math Questions". News18. December 26, 2018. 15. "Excel Data Types with Wolfram End of Support FAQ". support.microsoft.com. Retrieved August 15, 2022. 16. "Microsoft is killing Money in Excel along with Wolfram Alpha data types". XDA. May 31, 2022. Retrieved August 15, 2022. 17. "Wolfram 'search engine' goes live". BBC News. May 18, 2009. Retrieved February 9, 2013. 18. Spivack, Nova (March 7, 2009). "Wolfram Alpha is Coming – and It Could be as Important as Google". Nova Spivack – Minding the Planet. Retrieved February 9, 2013. 19. Singel, Ryan (May 18, 2009). "Wolfram\|Alpha Fails the Cool Test". Wired. Retrieved February 9, 2013. 20. "Nice Try, Wolfram Alpha. Still Not Paying $50 For Your App". TechCrunch. December 3, 2009. Retrieved August 15, 2022. 21. "Nice Try, Wolfram Alpha. Still Not Paying $50 For Your App". TechCrunch. December 3, 2009. Retrieved August 15, 2022. 22. Wolfram, Stephen (February 8, 2012). "Announcing Wolfram\|Alpha Pro". Wolfram\|Alpha Blog. Wolfram Alpha. Retrieved February 9, 2013. 23. "Step-by-Step Math". 24. Biddle, Pippa. "AI Is Making It Extremely Easy for Students to Cheat \| Backchannel". Wired. ISSN 1059-1028. Retrieved October 6, 2022. 25. McAllister, Neil (July 29, 2009). "How Wolfram Alpha could change software". InfoWorld. Retrieved February 28, 2012. 26. Stallman, Richard (August 4, 2009). "How Wolfram Alpha's Copyright Claims Could Change Software". Access 2 Knowledge (Mailing list). Archived from the original on April 28, 2013. Retrieved February 17, 2012. External links • Official website Wolfram Research Products • Computable Document Format • Mathematica • GridMathematica • MathWorld • WolframAlpha • Wolfram Demonstrations Project • Wolfram Language • Wolfram SystemModeler People • Stephen Wolfram • Conrad Wolfram • Theodore Gray • Eric Weisstein • Ed Pegg Jr. Virtual assistants Active • AliGenie • Alexa • Alice • Bixby • Viv • Braina • Celia • Clova • Cortana • Google Assistant • Maluuba • Mycroft • Siri • Voice Mate • Watson • WolframAlpha • Xiaoice Discontinued • BlackBerry Assistant • Google Now • M • Microsoft Agent • Microsoft Bob • Microsoft Voice Command • Ms. Dewey • Mya • Office Assistant (Clippy) • S Voice • Speaktoit Assistant • Tafiti • Vlingo
Wolfram Demonstrations Project The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-sized) interactive programmes called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hosted by Wolfram Research, whose stated goal is to bring computational exploration to a large population. At its launch, it contained 1300 demonstrations but has grown to over 10,000. The site won a Parents' Choice Award in 2008. Technology The Demonstrations run in Mathematica 6 or above and in Wolfram CDF Player which is a free modified version of Wolfram's Mathematica [1] and available for Windows, Linux and macOS[2] and can operate as a web browser plugin. They typically consist of a very direct user interface to a graphic or visualization, which dynamically recomputes in response to user actions such as moving a slider, clicking a button, or dragging a piece of graphics. Each Demonstration also has a brief description of the concept. Demonstrations are now easily embeddable into any website or blog.[3] Each Demonstration page includes a snippet of JavaScript code in the Share section of the sidebar. Topics The website is organized by topic: for example, science,[4] mathematics, computer science, art, biology, and finance. They cover a variety of levels, from elementary school mathematics to much more advanced topics such as quantum mechanics and models of biological organisms. The site is aimed at both educators and students, as well as researchers who wish to present their ideas to the broadest possible audience. Process Wolfram Research's staff organizes and edits the Demonstrations, which may be created by any user of Mathematica, then freely published[5] and freely downloaded. The Demonstrations are open-source, which means that they not only demonstrate the concept itself but also show how to implement it. Alternatives The use of the web to transmit small interactive programs is reminiscent of Sun's Java applets, Adobe's Flash, and the open-source Processing. However, those creating Demonstrations have access to the algorithmic and visualization capabilities of Mathematica making it more suitable for technical demonstrations. The Demonstrations Project also has similarities to user-generated content websites like Wikipedia and Flickr. Its business model is similar to Adobe's Acrobat and Flash strategy of charging for development tools but providing a free reader. References 1. Math Games MAA Online, May 1, 2007. 2. Adventures with the Wolfram Demonstrations Project John Wass, Scientific Computing 3. Kaurov, Vitaliy. "Add a Wolfram Demonstration to Your Site in One Easy Step". 4. Molecular Wolfram Demonstrations ScienceBase 5. Throwing beanbags in Mathematica 6, Scientific Computing, May 17, 2007. External links • Official site Wolfram Research Products • Computable Document Format • Mathematica • GridMathematica • MathWorld • WolframAlpha • Wolfram Demonstrations Project • Wolfram Language • Wolfram SystemModeler People • Stephen Wolfram • Conrad Wolfram • Theodore Gray • Eric Weisstein • Ed Pegg Jr.
MathWorld MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein.[2][3] It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana–Champaign.[3] MathWorld Type of businessPrivate Type of site Internet encyclopedia project Available inEnglish OwnerWolfram Research, Inc. Created byEric W. Weisstein[1] URLmathworld.wolfram.com LaunchedNovember 1999 (1999-11) Current statusActive History Eric W. Weisstein, the creator of the site, was a physics and astronomy student who got into the habit of writing notes on his mathematical readings. In 1995 he put his notes online and called it "Eric's Treasure Trove of Mathematics." It contained hundreds of pages/articles, covering a wide range of mathematical topics. The site became popular as an extensive single resource on mathematics on the web. In 1998, he made a contract with CRC Press and the contents of the site were published in print and CD-ROM form, titled "CRC Concise Encyclopedia of Mathematics." The free online version became only partially accessible to the public. In 1999 Weisstein went to work for Wolfram Research, Inc. (WRI), and WRI renamed the Math Treasure Trove to MathWorld and hosted it on the company's website without access restrictions. CRC lawsuit In 2000, CRC Press sued Wolfram Research Inc. (WRI), WRI president Stephen Wolfram, and author Eric W. Weisstein, due to what they considered a breach of contract: that the MathWorld content was to remain in print only. The site was taken down by a court injunction.[4] The case was later settled out of court, with WRI paying an unspecified amount and complying with other stipulations. Among these stipulations is the inclusion of a copyright notice at the bottom of the website and broad rights for the CRC Press to produce MathWorld in printed book form. The site then became once again available free to the public. This case made a wave of headlines in online publishing circles. The PlanetMath project was a result of MathWorld's being unavailable.[5] See also • List of online encyclopedias • Mathematica References 1. Eric Weisstein (2007). "Making MathWorld". Mathematica Journal. 10 (3). Archived from the original on 2012-07-09. Retrieved 2010-08-22. 2. "Wolfram MathWorld : the web's most extensive mathematics resource \| WorldCat.org". www.worldcat.org. Retrieved 2023-03-28. 3. "Making MathWorld « The Mathematica Journal". Retrieved 2023-03-28. 4. "CRC Press, LLC v. Wolfram Research, Inc., 149 F. Supp. 2d 500 \| Casetext Search + Citator". casetext.com. Retrieved 2023-03-28. 5. Corneli, Joseph (2011). "The PlanetMath Encyclopedia" (PDF). ITP 2011 Workshop on Mathematical Wikis (MathWikis 2011) Nijmegen, Netherlands, August 27, 2011. External links • Official website Wolfram Research Products • Computable Document Format • Mathematica • GridMathematica • MathWorld • WolframAlpha • Wolfram Demonstrations Project • Wolfram Language • Wolfram SystemModeler People • Stephen Wolfram • Conrad Wolfram • Theodore Gray • Eric Weisstein • Ed Pegg Jr.
Wolstenholme number A Wolstenholme number is a number that is the numerator of the generalized harmonic number Hn,2. Not to be confused with Wolstenholme prime. The first such numbers are 1, 5, 49, 205, 5269, 5369, 266681, 1077749, ... (sequence A007406 in the OEIS). These numbers are named after Joseph Wolstenholme, who proved Wolstenholme's theorem on modular relations of the generalized harmonic numbers. References • Weisstein, Eric W. "WolstenholmeNumber". MathWorld. Classes of natural numbers Powers and related numbers • Achilles • Power of 2 • Power of 3 • Power of 10 • Square • Cube • Fourth power • Fifth power • Sixth power • Seventh power • Eighth power • Perfect power • Powerful • Prime power Of the form a × 2b ± 1 • Cullen • Double Mersenne • Fermat • Mersenne • Proth • Thabit • Woodall Other polynomial numbers • Hilbert • Idoneal • Leyland • Loeschian • Lucky numbers of Euler Recursively defined numbers • Fibonacci • Jacobsthal • Leonardo • Lucas • Padovan • Pell • Perrin Possessing a specific set of other numbers • Amenable • Congruent • Knödel • Riesel • Sierpiński Expressible via specific sums • Nonhypotenuse • Polite • Practical • Primary pseudoperfect • Ulam • Wolstenholme Figurate numbers 2-dimensional centered • Centered triangular • Centered square • Centered pentagonal • Centered hexagonal • Centered heptagonal • Centered octagonal • Centered nonagonal • Centered decagonal • Star non-centered • Triangular • Square • Square triangular • Pentagonal • Hexagonal • Heptagonal • Octagonal • Nonagonal • Decagonal • Dodecagonal 3-dimensional centered • Centered tetrahedral • Centered cube • Centered octahedral • Centered dodecahedral • Centered icosahedral non-centered • Tetrahedral • Cubic • Octahedral • Dodecahedral • Icosahedral • Stella octangula pyramidal • Square pyramidal 4-dimensional non-centered • Pentatope • Squared triangular • Tesseractic Combinatorial numbers • Bell • Cake • Catalan • Dedekind • Delannoy • Euler • Eulerian • Fuss–Catalan • Lah • Lazy caterer's sequence • Lobb • Motzkin • Narayana • Ordered Bell • Schröder • Schröder–Hipparchus • Stirling first • Stirling second • Telephone number • Wedderburn–Etherington Primes • Wieferich • Wall–Sun–Sun • Wolstenholme prime • Wilson Pseudoprimes • Carmichael number • Catalan pseudoprime • Elliptic pseudoprime • Euler pseudoprime • Euler–Jacobi pseudoprime • Fermat pseudoprime • Frobenius pseudoprime • Lucas pseudoprime • Lucas–Carmichael number • Somer–Lucas pseudoprime • Strong pseudoprime Arithmetic functions and dynamics Divisor functions • Abundant • Almost perfect • Arithmetic • Betrothed • Colossally abundant • Deficient • Descartes • Hemiperfect • Highly abundant • Highly composite • Hyperperfect • Multiply perfect • Perfect • Practical • Primitive abundant • Quasiperfect • Refactorable • Semiperfect • Sublime • Superabundant • Superior highly composite • Superperfect Prime omega functions • Almost prime • Semiprime Euler's totient function • Highly cototient • Highly totient • Noncototient • Nontotient • Perfect totient • Sparsely totient Aliquot sequences • Amicable • Perfect • Sociable • Untouchable Primorial • Euclid • Fortunate Other prime factor or divisor related numbers • Blum • Cyclic • Erdős–Nicolas • Erdős–Woods • Friendly • Giuga • Harmonic divisor • Jordan–Pólya • Lucas–Carmichael • Pronic • Regular • Rough • Smooth • Sphenic • Størmer • Super-Poulet • Zeisel Numeral system-dependent numbers Arithmetic functions and dynamics • Persistence • Additive • Multiplicative Digit sum • Digit sum • Digital root • Self • Sum-product Digit product • Multiplicative digital root • Sum-product Coding-related • Meertens Other • Dudeney • Factorion • Kaprekar • Kaprekar's constant • Keith • Lychrel • Narcissistic • Perfect digit-to-digit invariant • Perfect digital invariant • Happy P-adic numbers-related • Automorphic • Trimorphic Digit-composition related • Palindromic • Pandigital • Repdigit • Repunit • Self-descriptive • Smarandache–Wellin • Undulating Digit-permutation related • Cyclic • Digit-reassembly • Parasitic • Primeval • Transposable Divisor-related • Equidigital • Extravagant • Frugal • Harshad • Polydivisible • Smith • Vampire Other • Friedman Binary numbers • Evil • Odious • Pernicious Generated via a sieve • Lucky • Prime Sorting related • Pancake number • Sorting number Natural language related • Aronson's sequence • Ban Graphemics related • Strobogrammatic • Mathematics portal
Wolstenholme prime In number theory, a Wolstenholme prime is a special type of prime number satisfying a stronger version of Wolstenholme's theorem. Wolstenholme's theorem is a congruence relation satisfied by all prime numbers greater than 3. Wolstenholme primes are named after mathematician Joseph Wolstenholme, who first described this theorem in the 19th century. Not to be confused with Wolstenholme number. Wolstenholme prime Named afterJoseph Wolstenholme Publication year1995[1] Author of publicationMcIntosh, R. J. No. of known terms2 Conjectured no. of termsInfinite Subsequence ofIrregular primes First terms16843, 2124679 Largest known term2124679 OEIS index • A088164 • Wolstenholme primes: primes p such that binomial(2p-1,p-1) == 1 (mod p^4) Interest in these primes first arose due to their connection with Fermat's Last Theorem. Wolstenholme primes are also related to other special classes of numbers, studied in the hope to be able to generalize a proof for the truth of the theorem to all positive integers greater than two. The only two known Wolstenholme primes are 16843 and 2124679 (sequence A088164 in the OEIS). There are no other Wolstenholme primes less than 109.[2] Definition Unsolved problem in mathematics: Are there any Wolstenholme primes other than 16843 and 2124679? (more unsolved problems in mathematics) Wolstenholme prime can be defined in a number of equivalent ways. Definition via binomial coefficients A Wolstenholme prime is a prime number p > 7 that satisfies the congruence ${2p-1 \choose p-1}\equiv 1{\pmod {p^{4}}},$ where the expression in left-hand side denotes a binomial coefficient.[3] In comparison, Wolstenholme's theorem states that for every prime p > 3 the following congruence holds: ${2p-1 \choose p-1}\equiv 1{\pmod {p^{3}}}.$ Definition via Bernoulli numbers A Wolstenholme prime is a prime p that divides the numerator of the Bernoulli number Bp−3.[4][5][6] The Wolstenholme primes therefore form a subset of the irregular primes. Definition via irregular pairs Main article: Irregular prime A Wolstenholme prime is a prime p such that (p, p–3) is an irregular pair.[7][8] Definition via harmonic numbers A Wolstenholme prime is a prime p such that[9] $H_{p-1}\equiv 0{\pmod {p^{3}}}\,,$ i.e. the numerator of the harmonic number $H_{p-1}$ expressed in lowest terms is divisible by p3. Search and current status The search for Wolstenholme primes began in the 1960s and continued over the following decades, with the latest results published in 2007. The first Wolstenholme prime 16843 was found in 1964, although it was not explicitly reported at that time.[10] The 1964 discovery was later independently confirmed in the 1970s. This remained the only known example of such a prime for almost 20 years, until the discovery announcement of the second Wolstenholme prime 2124679 in 1993.[11] Up to 1.2×107, no further Wolstenholme primes were found.[12] This was later extended to 2×108 by McIntosh in 1995 [5] and Trevisan & Weber were able to reach 2.5×108.[13] The latest result as of 2007 is that there are only those two Wolstenholme primes up to 109.[14] Expected number of Wolstenholme primes It is conjectured that infinitely many Wolstenholme primes exist. It is conjectured that the number of Wolstenholme primes ≤ x is about ln ln x, where ln denotes the natural logarithm. For each prime p ≥ 5, the Wolstenholme quotient is defined as $W_{p}{=}{\frac {{2p-1 \choose p-1}-1}{p^{3}}}.$ Clearly, p is a Wolstenholme prime if and only if Wp ≡ 0 (mod p). Empirically one may assume that the remainders of Wp modulo p are uniformly distributed in the set {0, 1, ..., p–1}. By this reasoning, the probability that the remainder takes on a particular value (e.g., 0) is about 1/p.[5] See also • Wieferich prime • Wall–Sun–Sun prime • Wilson prime • Table of congruences Notes 1. Wolstenholme primes were first described by McIntosh in McIntosh 1995, p. 385 2. Weisstein, Eric W., "Wolstenholme prime", MathWorld 3. Cook, J. D., Binomial coefficients, retrieved 21 December 2010 4. Clarke & Jones 2004, p. 553. 5. McIntosh 1995, p. 387. 6. Zhao 2008, p. 25. 7. Johnson 1975, p. 114. 8. Buhler et al. 1993, p. 152. 9. Zhao 2007, p. 18. 10. Selfridge and Pollack published the first Wolstenholme prime in Selfridge & Pollack 1964, p. 97 (see McIntosh & Roettger 2007, p. 2092). 11. Ribenboim 2004, p. 23. 12. Zhao 2007, p. 25. 13. Trevisan & Weber 2001, p. 283–284. 14. McIntosh & Roettger 2007, p. 2092. References • Buhler, J.; Crandall, R.; Ernvall, R.; Metsänkylä, T. (1993), "Irregular Primes and Cyclotomic Invariants to Four Million" (PDF), Mathematics of Computation, 61 (203): 151–153, Bibcode:1993MaCom..61..151B, doi:10.2307/2152942, JSTOR 2152942 Archived 12 November 2010 at WebCite • Clarke, F.; Jones, C. (2004), "A Congruence for Factorials" (PDF), Bulletin of the London Mathematical Society, 36 (4): 553–558, doi:10.1112/S0024609304003194, S2CID 120202453 Archived 2 January 2011 at WebCite • Johnson, W. (1975), "Irregular Primes and Cyclotomic Invariants" (PDF), Mathematics of Computation, 29 (129): 113–120, doi:10.2307/2005468, JSTOR 2005468 Archived 20 December 2010 at WebCite • McIntosh, R. J. (1995), "On the converse of Wolstenholme's Theorem" (PDF), Acta Arithmetica, 71 (4): 381–389, doi:10.4064/aa-71-4-381-389 Archived 8 November 2010 at WebCite • McIntosh, R. J.; Roettger, E. L. (2007), "A search for Fibonacci-Wieferich and Wolstenholme primes" (PDF), Mathematics of Computation, 76 (260): 2087–2094, Bibcode:2007MaCom..76.2087M, doi:10.1090/S0025-5718-07-01955-2 Archived 10 December 2010 at WebCite • Ribenboim, P. (2004), "Chapter 2. How to Recognize Whether a Natural Number is a Prime", The Little Book of Bigger Primes, New York: Springer-Verlag New York, Inc., ISBN 978-0-387-20169-6 Archived 24 November 2010 at WebCite • Selfridge, J. L.; Pollack, B. W. (1964), "Fermat's last theorem is true for any exponent up to 25,000", Notices of the American Mathematical Society, 11: 97 • Trevisan, V.; Weber, K. E. (2001), "Testing the Converse of Wolstenholme's Theorem" (PDF), Matemática Contemporânea, 21 (16): 275–286, doi:10.21711/231766362001/rmc2116 Archived 10 December 2010 at WebCite • Zhao, J. (2007), "Bernoulli numbers, Wolstenholme's theorem, and p5 variations of Lucas' theorem" (PDF), Journal of Number Theory, 123: 18–26, doi:10.1016/j.jnt.2006.05.005, S2CID 937685Archived 12 November 2010 at WebCite • Zhao, J. (2008), "Wolstenholme Type Theorem for Multiple Harmonic Sums" (PDF), International Journal of Number Theory, 4 (1): 73–106, doi:10.1142/s1793042108001146 Archived 27 November 2010 at WebCite Further reading • Babbage, C. (1819), "Demonstration of a theorem relating to prime numbers", The Edinburgh Philosophical Journal, 1: 46–49 • Krattenthaler, C.; Rivoal, T. (2009), "On the integrality of the Taylor coefficients of mirror maps, II", Communications in Number Theory and Physics, 3 (3): 555–591, arXiv:0907.2578, Bibcode:2009arXiv0907.2578K, doi:10.4310/CNTP.2009.v3.n3.a5 • Wolstenholme, J. (1862), "On Certain Properties of Prime Numbers", The Quarterly Journal of Pure and Applied Mathematics, 5: 35–39 External links • Caldwell, Chris K. Wolstenholme prime from The Prime Glossary • McIntosh, R. J. Wolstenholme Search Status as of March 2004 e-mail to Paul Zimmermann • Bruck, R. Wolstenholme's Theorem, Stirling Numbers, and Binomial Coefficients • Conrad, K. The p-adic Growth of Harmonic Sums interesting observation involving the two Wolstenholme primes Prime number classes By formula • Fermat (22n + 1) • Mersenne (2p − 1) • Double Mersenne (22p−1 − 1) • Wagstaff (2p + 1)/3 • Proth (k·2n + 1) • Factorial (n! ± 1) • Primorial (pn# ± 1) • Euclid (pn# + 1) • Pythagorean (4n + 1) • Pierpont (2m·3n + 1) • Quartan (x4 + y4) • Solinas (2m ± 2n ± 1) • Cullen (n·2n + 1) • Woodall (n·2n − 1) • Cuban (x3 − y3)/(x − y) • Leyland (xy + yx) • Thabit (3·2n − 1) • Williams ((b−1)·bn − 1) • Mills (⌊A3n⌋) By integer sequence • Fibonacci • Lucas • Pell • Newman–Shanks–Williams • Perrin • Partitions • Bell • Motzkin By property • Wieferich (pair) • Wall–Sun–Sun • Wolstenholme • Wilson • Lucky • Fortunate • Ramanujan • Pillai • Regular • Strong • Stern • Supersingular (elliptic curve) • Supersingular (moonshine theory) • Good • Super • Higgs • Highly cototient • Unique Base-dependent • Palindromic • Emirp • Repunit (10n − 1)/9 • Permutable • Circular • Truncatable • Minimal • Delicate • Primeval • Full reptend • Unique • Happy • Self • Smarandache–Wellin • Strobogrammatic • Dihedral • Tetradic Patterns • Twin (p, p + 2) • Bi-twin chain (n ± 1, 2n ± 1, 4n ± 1, …) • Triplet (p, p + 2 or p + 4, p + 6) • Quadruplet (p, p + 2, p + 6, p + 8) • k-tuple • Cousin (p, p + 4) • Sexy (p, p + 6) • Chen • Sophie Germain/Safe (p, 2p + 1) • Cunningham (p, 2p ± 1, 4p ± 3, 8p ± 7, ...) • Arithmetic progression (p + a·n, n = 0, 1, 2, 3, ...) • Balanced (consecutive p − n, p, p + n) By size • Mega (1,000,000+ digits) • Largest known • list Complex numbers • Eisenstein prime • Gaussian prime Composite numbers • Pseudoprime • Catalan • Elliptic • Euler • Euler–Jacobi • Fermat • Frobenius • Lucas • Somer–Lucas • Strong • Carmichael number • Almost prime • Semiprime • Sphenic number • Interprime • Pernicious Related topics • Probable prime • Industrial-grade prime • Illegal prime • Formula for primes • Prime gap First 60 primes • 2 • 3 • 5 • 7 • 11 • 13 • 17 • 19 • 23 • 29 • 31 • 37 • 41 • 43 • 47 • 53 • 59 • 61 • 67 • 71 • 73 • 79 • 83 • 89 • 97 • 101 • 103 • 107 • 109 • 113 • 127 • 131 • 137 • 139 • 149 • 151 • 157 • 163 • 167 • 173 • 179 • 181 • 191 • 193 • 197 • 199 • 211 • 223 • 227 • 229 • 233 • 239 • 241 • 251 • 257 • 263 • 269 • 271 • 277 • 281 List of prime numbers
Timeline of women in mathematics This is a timeline of women in mathematics. Timeline Early Common Era Before 350: Pandrosion, a Greek Alexandrine mathematician known for an approximate solution to doubling the cube and a simplified exact solution to the construction of the geometric mean.[1] c. 350–370 until 415: The lifetime of Hypatia, a Greek Alexandrine Neoplatonist philosopher in Egypt who was the first well-documented woman in mathematics.[2] 18th Century 1748: Italian mathematician Maria Agnesi published the first book discussing both differential and integral calculus, called Instituzioni analitiche ad uso della gioventù italiana.[3][4] 1759: French mathematician Émilie du Châtelet's translation and commentary on Isaac Newton’s work Principia Mathematica was published posthumously; it is still considered the standard French translation.[5] c. 1787 – 1797: Self-taught Chinese astronomer Wang Zhenyi published at least twelve books and multiple articles on astronomy and mathematics.[6] 19th Century 1827: French mathematician Sophie Germain saw her theorem, known as Germain's Theorem, published in a footnote of a book by the mathematician Adrien-Marie Legendre.[7][8] In this theorem Germain proved that if x, y, and z are integers and if x5 + y5 = z5 then either x, y, or z must be divisible by 5. Germain's theorem was a major step toward proving Fermat's Last Theorem for the case where n equals 5.[7] 1829: The first public examination of an American girl in geometry was held.[9] 1858: Florence Nightengale became the first female member of the Royal Statistical Society.[10] 1873: Sarah Woodhead of Britain became the first woman to take, and to pass, the Cambridge Mathematical Tripos Exam.[11] 1874: Russian mathematician Sofia Kovalevskaya became the first woman in modern Europe to gain a doctorate in mathematics, which she earned from the University of Göttingen in Germany.[12] 1880: Charlotte Angas Scott of Britain obtained special permission to take the Cambridge Mathematical Tripos Exam, as women were not normally allowed to sit for the exam. She came eighth on the Tripos of all students taking them, but due to her sex, the title of "eighth wrangler," a high honour, went officially to a male student.[13] At the ceremony, however, after the seventh wrangler had been announced, all the students in the audience shouted her name. Because she could not attend the award ceremony, Scott celebrated her accomplishment at Girton College where there were cheers and clapping at dinner, and a special evening ceremony where the students sang "See the Conquering Hero Comes", and she received an ode written by a staff member, and was crowned with laurels.[13] 1885: Charlotte Angas Scott became the first British woman to receive a doctorate in mathematics, which she received from the University of London.[14] 1886: Winifred Edgerton Merrill became the first American woman to earn a PhD in mathematics, which she earned from Columbia University.[15] 1888: The Kovalevskaya top, one of a brief list of known examples of integrable rigid body motion, was discovered by Sofia Kovalevskaya.[16][17] 1889: Sofia Kovalevskaya was appointed as the first female professor in Northern Europe, at the University of Stockholm.[12][18] 1890: Philippa Fawcett of Britain[19] became the first woman to obtain the top score in the Cambridge Mathematical Tripos Exam. Her score was 13 per cent higher than the second highest score. When the women's list was announced, Fawcett was described as "above the senior wrangler", but she did not receive the title of senior wrangler, as at that time only men could receive degrees and therefore only men were eligible for the Senior Wrangler title.[20][21] 1891: Charlotte Angas Scott of Britain became the first woman to join the American Mathematical Society, then called the New York Mathematical Society.[22] 1894: Charlotte Angas Scott of Britain became the first woman on the first Council of the American Mathematical Society.[23] 1897: Four women attended the inaugural International Congress of Mathematicians in Zurich in 1897 - Charlotte Angas Scott, Iginia Massarini, Vera von Schiff, and Charlotte Wedell.[24] 20th Century 1911: Swedish mathematician Louise Petrén-Overton became the first woman in Sweden with a doctorate in mathematics.[25] 1913: American mathematician Mildred Sanderson earned her PhD for a thesis that included an important theorem about modular invariants.[26] 1918: German mathematician Emmy Noether published Noether's (first) theorem, which states that any differentiable symmetry of the action of a physical system has a corresponding conservation law.[27] 1927: American mathematician Anna Pell-Wheeler became the first woman to present a lecture at the American Mathematical Society Colloquium.[28][29] 1930: Cecilia Kreiger became the first woman to earn a PhD in mathematics in Canada, at the University of Toronto.[30] 1930s: British mathematician Mary Cartwright proved her theorem, now known as Cartwright's theorem, which gives an estimate for the maximum modulus of an analytic function that takes the same value no more than p times in the unit disc. To prove the theorem she used a new approach, applying a technique introduced by Lars Ahlfors for conformal mappings.[31] 1943: Euphemia Haynes became the first African-American woman to earn a Ph.D. in mathematics, which she earned from Catholic University of America.[32] 1944: Helen Walker became the first female president of the American Statistical Association.[33] 1949: American mathematician Gertrude Mary Cox became the first woman elected into the International Statistical Institute.[34] Also, Maria Laura Lopes obtained her PhD in Mathematics, being the first woman to obtain the title in Brazil. 1951: Mary Cartwright of Britain became the first female president of the Mathematical Association.[35][31] 1956: American mathematician Gladys West began collecting data from satellites at the Naval Surface Warfare Center Dahlgren Division. Her calculations directly impacted the development of accurate GPS systems.[36] 1960s 1960 and 1966: British mathematician Lucy Joan Slater published two books about the hypergeometric functions from the Cambridge University Press.[37][38] 1961: Mary Cartwright of Britain became the first woman to be President of the London Mathematical Society.[39] 1962: American mathematician Mina Rees became the first person to receive the Award for Distinguished Service to Mathematics from the Mathematical Association of America.[40] 1963: Grace Alele-Williams became the first Nigerian woman to earn a Ph.D when she defended her thesis in Mathematics Education at the University of Chicago (U.S.)[41][42] 1964: Mary Cartwright of Britain became the first woman to be given the Sylvester Medal of the Royal Society.[39][43] 1965: Scottish mathematician Elizabeth McHarg became the first female president of the Edinburgh Mathematical Society.[44][45] 1966: American mathematician Mary L. Boas published Mathematical Methods in the Physical Sciences, which was still widely used in college classrooms as of 1999.[46][47][48] 1968: Mary Cartwright of Britain became the first woman to be given the De Morgan Medal, the London Mathematical Society’s premier award.[49][43] 1970s 1970: American mathematician Mina Rees became the first female president of the American Association for the Advancement of Science.[50] 1971: American mathematician Mary Ellen Rudin constructed the first Dowker space.[51][52] 1971: The Association for Women in Mathematics (AWM) was founded. It is a professional society whose mission is to encourage women and girls to study and to have active careers in the mathematical sciences, and to promote equal opportunity for and the equal treatment of women and girls in the mathematical sciences. It is incorporated in the state of Massachusetts.[53] 1971: The American Mathematical Society established its Joint Committee on Women in the Mathematical Sciences (JCW), which later became a joint committee of multiple scholarly societies.[54] 1973: American mathematician Jean Taylor published her dissertation on "Regularity of the Singular Set of Two-Dimensional Area-Minimizing Flat Chains Modulo 3 in R3" which solved a long-standing problem about length and smoothness of soap-film triple function curves.[55] 1974: American mathematician Joan Birman published the book Braids, Links, and Mapping Class Groups. It has become a standard introduction, with many of today's researchers having learned the subject through it.[56] 1975: American mathematician Julia Robinson became the first female mathematician elected to the National Academy of Sciences.[57][58] 1975: Stella Cunliffe became the first female president of the Royal Statistical Society.[10] 1976-1977: Marjorie Rice, an amateur American mathematician, discovered four new types of tessellating pentagons in 1976 and 1977.[59][60][61] 1979: American mathematician Dorothy Lewis Bernstein became the first female president of the Mathematical Association of America.[62] 1979: American mathematician Mary Ellen Rudin became the first woman to present the Mathematical Association of America’s Earle Raymond Hedrick Lectures, intended to showcase skilled expositors and enrich the understanding of instructors of college-level mathematics.[52][28] 1980s 1980: Joséphine Guidy Wandja, from the Ivory Coast, became the first African woman to earn a doctorate in mathematics.[63][64] 1981: Canadian-American mathematician Cathleen Morawetz became the first woman to give the Gibbs Lecture of the American Mathematical Society.[65] 1981: American mathematician Doris Schattschneider became the first female editor of Mathematics Magazine, a refereed bimonthly publication of the Mathematical Association of America.[66][67] 1982: Rebecca Walo Omana became the first female mathematics professor in the Democratic Republic of the Congo.[68][69] 1983: American mathematician Julia Robinson was elected the first female president of the American Mathematical Society for the term of 1983-1984 (but was unable to complete her term as she was suffering from leukemia),[58][70] and became the first female mathematician to be awarded a MacArthur Fellowship.[28] 1986: European Women in Mathematics (EWM) was founded as an organization in 1986 by Bodil Branner, Caroline Series, Gudrun Kalmbach, Marie-Françoise Roy, and Dona Strauss, inspired by the activities of the Association for Women in Mathematics in the USA.[71] It is the "first and best known" of several organizations devoted to women in mathematics in Europe.[72] 1987: Eileen Poiani became the first female president of Pi Mu Epsilon.[73] 1988: American mathematician Doris Schattschneider became the first woman to present the Mathematical Association of America’s J. Sutherland Frame Lectures.[28][74] 1990s 1992: Australian mathematician Cheryl Praeger became the first female President of the Australian Mathematical Society.[75] 1992: American mathematician Gloria Gilmer became the first woman to deliver a major National Association of Mathematicians lecture (it was the Cox-Talbot address).[76] 1995: American mathematician Margaret Wright became the first female president of the Society for Industrial and Applied Mathematics.[28][77] 1995: Israeli-Canadian mathematician Leah Edelstein-Keshet became the first female president of the Society for Mathematical Biology.[78] 1995: Ina Kersten became the president of the German Mathematical Society, which meant she was the first woman to head the society.[79][80] 1996: American mathematician Joan Birman became the first woman to receive the Mathematical Association of America’s Chauvenet Prize.[81][28] 1996: Katherine Heinrich became the first female President of the Canadian Mathematical Society.[82] 1996: Ioana Dumitriu, a New York University sophomore from Romania, became the first woman to be named a Putnam Fellow.[83] Putnam Fellows are the top five (or six, in case of a tie) scorers on The William Lowell Putnam Mathematical Competition.[84][85] 1998: Melanie Wood became the first female American to make the U.S. International Math Olympiad Team. She won silver medals in the 1998 and 1999 International Mathematical Olympiads.[86] 2000s 2002: Susan Howson became the first woman to be given the Adams Prize, given annually by the University of Cambridge to a British mathematician under the age of 40.[87] 2002: Melanie Wood became the first American woman and second woman overall to be named a Putnam Fellow in 2002. Putnam Fellows are the top five (or six, in case of a tie) scorers on William Lowell Putnam Mathematical Competition.[84][85] 2004: American Melanie Wood became the first woman to win the Frank and Brennie Morgan Prize for Outstanding Research in Mathematics by an Undergraduate Student. It is an annual award given to an undergraduate student in the US, Canada, or Mexico who demonstrates superior mathematics research.[88][86] 2004: American Alison Miller became the first female gold medal winner on the U.S. International Mathematical Olympiad Team.[89][90] 2006: Polish-Canadian mathematician Nicole Tomczak-Jaegermann became the first woman to win the CRM-Fields-PIMS prize.[91][92][93] 2006: Stefanie Petermichl, a German mathematical analyst then at the University of Texas at Austin, became the first woman to win the Salem Prize, an annual award given to young mathematicians who have worked in Raphael Salem's field of interest, chiefly topics in analysis related to Fourier series.[94][28] She shared the prize with Artur Avila.[95][28] 2006: When Olga Gil Medrano became president of the Royal Spanish Mathematical Society in 2006, she was the first woman elected to that position.[96] 2010s 2011: Belgian mathematician Ingrid Daubechies became the first female president of the International Mathematical Union.[97] 2012: Latvian mathematician Daina Taimina became the first woman to win the Euler Book Prize, for her 2009 book Crocheting Adventures with Hyperbolic Planes.[98][99] 2012: The Working Committee for Women in Mathematics, Chinese Mathematical Society (WCWM-CMS) was founded; it is a national non-profit academic organization in which female mathematicians who are engaged in research, teaching, and applications of mathematics can share their scientific research through academic exchanges both in China and abroad.[100] It is one of the branches of the Chinese Mathematical Society (CMS).[100] 2013: The African Women in Mathematics Association was founded. This professional organization with over 300 members promotes mathematics to African women and girls and supports female mathematicians.[101][102] 2014: Maryam Mirzakhani became the first woman as well as the first Iranian to be awarded the Fields Medal, which she was awarded for "her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces."[103][104][105] That year the Fields Medal was also awarded to Martin Hairer, Manjul Bhargava, and Artur Avila.[106] It is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union, and is often viewed as the greatest honor a mathematician can receive.[107][108] 2016: French mathematician Claire Voisin received the CNRS Gold medal, the highest scientific research award in France.[109] 2016: The London Mathematical Society's Women in Mathematics Committee was awarded the Royal Society's inaugural Athena Prize.[110] 2017: Nouzha El Yacoubi became the first female president of the African Mathematical Union.[111] 2019: American mathematician Karen Uhlenbeck became the first woman to win the Abel Prize, with the award committee citing "the fundamental impact of her work on analysis, geometry and mathematical physics."[112] 2019: Marissa Kawehi Loving became the first Native Hawaiian woman to earn a PhD in mathematics when she graduated from the University of Illinois Urbana-Champaign in 2019. In addition to being Native Hawaiian, she is also black, Japanese, and Puerto Rican.[113] 2020s 2022: Maryna Viazovska was awarded the Fields Medal in July 2022, making her the second woman (after Maryam Mirzakhani) and the first Ukrainian to be awarded it.[114][115] That year the Fields Medal was also awarded to Hugo Duminil-Copin, June Huh, and James Maynard.[115] The Fields Medal is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union, and is often viewed as the greatest honor a mathematician can receive.[107][108] 2023: Ingrid Daubechies was awarded the Wolf Prize in Mathematics in February 2023, becoming the first woman to receive this award.[116] See also • List of women in mathematics • Timeline of women in mathematics in the United States • Timeline of mathematics • Timeline of mathematical innovation in South and West Asia References 1. O'Connor, John J.; Robertson, Edmund F., "Pandrosion of Alexandria", MacTutor History of Mathematics Archive, University of St Andrews 2. Scholasticus, Socrates. Ecclesiastical History. Archived from the original on 2009-04-18. 3. According to Dirk Jan Struik, Agnesi is "the first important woman mathematician since Hypatia (fifth century A.D.)". 4. "Epigenesys - Maria Gaetana Agnesi \| Women in science". epigenesys.eu. Retrieved 2014-01-25. 5. "Brooklyn Museum: Elizabeth A. Sackler Center for Feminist Art: The Dinner Party: Heritage Floor: Emilie du Chatelet". brooklynmuseum.org. Retrieved 2014-01-25. 6. Bennett Peterson, Barbara (2016-09-16). Notable Women of China. doi:10.4324/9781315702063. ISBN 9781315702063. 7. "Sophie Germain". agnesscott.edu. Retrieved 2014-01-25. 8. "Sophie Germain page". math.rochester.edu. Retrieved 2014-01-25. 9. Elizabeth Cady Stanton; Susan B. Anthony; Matilda Joslyn Gage; Ida Husted Harper, eds. (1889). History of Woman Suffrage: 1848–1861, Volume 1. Susan B. Anthony. p. 36. Retrieved 2011-04-18. 10. "History". RSS. 11. Jensen-Vallin, Jacqueline A.; Beery, Janet L.; Mast, Maura B.; Greenwald, Sarah J., eds. (2018). Women in Mathematics: Celebrating the Centennial of the Mathematical Association of America. Springer. p. "Sarah+woodhead"+tripos+1873&pg=PA8 8. ISBN 978-3-319-88303-8. 12. "Sofya Vasilyevna Kovalevskaya (Russian mathematician) -- Encyclopædia Britannica". britannica.com. Retrieved 2014-01-25. 13. Patricia Clark Kenschaft (1987). "Charlotte Angas Scott (1858–1931)" in Women of Mathematics: A Biobibliographic Sourcebook. New York: Greenwood Press. pp. 193–203. ISBN 0-313-24849-4. 14. 🖉"Charlotte Angas Scott". mathwomen.agnesscott.org. 15. Susan E. Kelly & Sarah A. Rozner (28 February 2012). "Winifred Edgerton Merrill:"She Opened the Door"" (PDF). Notices of the AMS. 59 (4). Retrieved 25 January 2014. 16. S. Kovalevskaya, Sur Le Probleme De La Rotation D'Un Corps Solide Autour D'Un Point Fixe, Acta Mathematica 12 (1889) 177–232. 17. E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge University Press (1952). 18. "COOL, CREATIEF, HIP met ICT - Innovative women". chai-x.nl. Retrieved 2014-01-25. 19. "Philippa Fawcett Internship Programme \| Philippa Fawcett Internship Programme". www.maths.cam.ac.uk. 20. "Philippa Garrett Fawcett". agnesscott.edu. Retrieved 2014-01-25. 21. "The Woman Who Bested the Men at Math \| History \| Smithsonian". smithsonianmag.com. Retrieved 2014-01-25. 22. Oakes, Elizabeth (2007). Encyclopedia of World Scientists, Revised Edition. Infobase Publishing. p. 655. ISBN 9781438118826. 23. Chaplin, Stephanie (1997). "Biographies of Women Mathematicians: Charlotte Angas Scott". Agnes Scott College. Retrieved 22 October 2012. 24. Curbera, Guillermo (2009), Mathematicians of the World, Unite!: The International Congress of Mathematicians—A Human Endeavor, CRC Press, p. 16, ISBN 9781439865125 25. Larsson, Lisbeth, "Hedvig Louise Beata Petrén-Overton", Svenskt kvinnobiografiskt lexikon [Biographical Dictionary of Swedish Women] (in Swedish), retrieved 2019-01-13 26. "Mildred Leonora Sanderson". agnesscott.edu. Retrieved 2014-01-25. 27. Noether E (1918). "Invariante Variationsprobleme". Nachr. D. König. Gesellsch. D. Wiss. Zu Göttingen, Math-phys. Klasse. 1918: 235–257. 28. "Prizes, Awards, and Honors for Women Mathematicians". agnesscott.edu. Retrieved 2014-01-25. 29. "Anna Johnson Pell Wheeler". mathwomen.agnesscott.org. 30. Zuschlag, Anna. "Cecilia Krieger". The Canadian Encyclopedia. Retrieved 2018-08-22. 31. "Cartwright biography". -history.mcs.st-and.ac.uk. Retrieved 2014-01-25. 32. "Euphemia Lofton Haynes, first African American woman mathematician". math.buffalo.edu. Retrieved 2014-01-25. 33. "Helen Walker, 91, First Woman To Head U.S. Statistical Group". The New York Times. 18 January 1983. Retrieved 1 December 2014. 34. "Gertrude Mary Cox". agnesscott.edu. Retrieved 2014-01-25. 35. Williams, Mrs. E. M. (October 1966), "Presidential Address: The Changing Role of Mathematics in Education", The Mathematical Gazette, 50 (373): 243–254, doi:10.2307/3614669, JSTOR 3614669, S2CID 186846165 36. "How Gladys West uncovered the 'Hidden Figures' of GPS". GPS World. 2018-03-19. Retrieved 2018-09-22. 37. Slater, Lucy Joan (1960), Confluent hypergeometric functions, Cambridge, UK: Cambridge University Press, 38. Slater, Lucy Joan (1966), Generalized hypergeometric functions, Cambridge, UK: Cambridge University Press. 39. O'Connor, J. J.; Robertson, E. F. "Dame Mary Lucy Cartwright". School of Mathematics and Statistics, University of St Andrews. Retrieved 3 April 2019. 40. "Mina Rees". mathwomen.agnesscott.org. 41. "Grace Alele Williams". Retrieved 2021-01-18. 42. "5 women who have made their marks in education". Pulse Nigeria. 0100. Retrieved 2021-01-18. 43. "Mary Lucy Cartwright". mathwomen.agnesscott.org. 44. "Edinburgh Mathematical Society – Presidents", MacTutor History of Mathematics archive, School of Mathematics and Statistics, University of St Andrews, retrieved 2018-10-12 45. Hoyles, Celia (December 2017), "Female Presidents for Three Maths Societies", Mathematics Today, Institute of Mathematics and its Applications 46. Mary L. Boas (1966). Mathematical methods in the physical sciences. Wiley. ISBN 9780471084174. 47. Spector, Donald (1999). "Book Reviews". American Journal of Physics. 67 (2): 165–169. doi:10.1119/1.19216. 48. "DePaul Department of Physics". Archived from the original on June 19, 2010. 49. 🖉"Prizes, Awards, and Honors for Women Mathematicians". mathwomen.agnesscott.org. 50. "Mina Rees". agnesscott.edu. Retrieved 2014-01-25. 51. "New Zealand Mathematical Societu Newsletter Number 84, April 2002". Massey.ac.nz. Retrieved 2017-06-20. 52. "Mary Ellen Rudin - Biography". Maths History. 53. "About AWM - AWM Association for Women in Mathematics". Retrieved 2014-01-25. 54. "JCW-Math \| Joint Committee on Women in the Mathematical Sciences". jcwmath.wordpress.com. Retrieved 2014-01-25. 55. "Jean Taylor". agnesscott.edu. Retrieved 2014-01-25. 56. "Interview with Joan Birman" (PDF). Notices of the AMS. 54 (1). 4 December 2006. Retrieved 25 January 2014. 57. "Profiles of Women in Mathematics: Julia Robinson". awm-math.org. Retrieved 2014-01-25. 58. "Julia Bowman Robinson". mathwomen.agnesscott.org. 59. Schattschneider, Doris (1978), "Tiling the plane with congruent pentagons", Mathematics Magazine, 51 (1): 29–44, doi:10.2307/2689644, ISSN 0025-570X, JSTOR 2689644, MR 0493766 60. Marjorie Rice, "Tessellations", Intriguing Tessellations, retrieved 22 August 2015 – via Google Sites 61. Wolchover, Natalie (July 11, 2017). "Marjorie Rice's Secret Pentagons". Quanta Magazine. 62. Oakes, E.H. (2007). Encyclopedia of World Scientists. Facts On File, Incorporated. ISBN 9781438118826. 63. Cassiau-Haurie, Christophe (2008-02-20). "Les femmes peinent à percer les bulles". Africultures. Retrieved 2021-01-18. 64. "Joséphine Guidy-Wandja - Biography". Maths History. Retrieved 2021-01-18. 65. "Cathleen Morawetz". agnesscott.edu. Retrieved 2014-01-25. 66. "2005 Parson Lecturer - Dr. Doris Schattschneider". University of North Carolina at Asheville, Department of Mathematics. Archived from the original on 2014-01-11. Retrieved 2013-07-13.. 67. Riddle, Larry (April 5, 2013). "Biographies of Women Mathematicians \| Doris Schattschneider". Agnes Scott College. Retrieved 2013-07-13. 68. "Rebecca Walo OMANA \| African Women in Mathematics Association". Retrieved 2021-01-15. 69. 2019_AUR Conf_ConceptNote-Bios-Abstract.pdf (PDF), retrieved 2021-01-16 70. "Julia Bowman Robinson". Encyclopedia.com. 71. "European Women in Mathematics". MacTutor History of Mathematics archive. February 2018. Retrieved 2018-08-30. 72. Series, Caroline (December 2013), "European Level Organisations for Women Mathematicians" (PDF), EMS Newsletter, European Mathematical Society, vol. 90, p. 11 73. C.C. MacDuffee Award 1995 – Eileen L. Poiani, Pi Mu Epsilon, retrieved 2019-11-08 74. "Doris Schattschneider". mathwomen.agnesscott.org. 75. "Prof Cheryl Praeger - first female mathematician awarded George Szekeres Medal". Alliance of Girls’ Schools Australasia. 76. "Gloria Ford Gilmer". math.buffalo.edu. Retrieved 2014-01-25. 77. "Margaret Wright". mathwomen.agnesscott.org. 78. "Leah Edelstein-Keshet". math.ubc.ca. Retrieved 2014-01-25. 79. Abele, Andrea E.; Neunzert, Helmut; Tobies, Renate (2013), Traumjob Mathematik!: Berufswege von Frauen und Männern in der Mathematik (in German), Springer-Verlag, p. 5, ISBN 978-3-0348-7963-7 80. von Randow, Thomas (20 January 1995), "Ästhetik der Algebra", Die Zeit (in German) 81. "Joan S. Birman". mathwomen.agnesscott.org. 82. Women in Mathematics (PDF), Canadian Mathematical Society, archived from the original (PDF) on 2020-10-19, retrieved 2018-02-09 83. Karen W. Arenson (1997-05-01). "Q: How Many Women Have Won the Top Math Contest? - New York Times". The New York Times. Retrieved 2014-03-04. 84. "Duke Magazine-Where Are They Now?-January/February 2010". dukemagazine.duke.edu. Retrieved 2014-01-25. 85. "Melanie Wood: The Making of a Mathematician - Cogito". cogito.cty.jhu.edu. Retrieved 2014-01-25. 86. Rimer, Sara (10 October 2008). "Math Skills Suffer in U.S., Study Finds". The New York Times. Retrieved 2019-11-20. 87. "Prizes, Awards, and Honors for Women Mathematicians". mathwomen.agnesscott.org. 88. "2003 Morgan Prize" (PDF). Notices of the AMS. 51 (4). 26 February 2004. Retrieved 25 January 2014. 89. "Math Forum @ Drexel: Congratulations, Alison!". mathforum.org. Retrieved 2014-01-25. 90. "2004 IMO US Team Results in Athens, Greece \| Mathematical Association of America". www.maa.org. 91. Canada Research Chair in Geometric Analysis, retrieved 2010-12-03. 92. "Prizes, Awards, and Honors for Women Mathematicians". mathwomen.agnesscott.org. 93. "Fields Institute - CRM-Fields Prize Recipients". fields.utoronto.ca. Retrieved 2014-01-25. 94. Short vita, retrieved 2016-07-04. 95. "UZH - Fields Medal Winner Artur Avila Appointed Full Professor at the University of Zurich". Media.uzh.ch. 2018-07-24. Retrieved 2018-10-09. 96. Claramunt Vallespí, Rosa M.a; Claramunt Vallespí, Teresa (2012), Mujeres en ciencia y tecnología, UNED, ISBN 9788436265255 97. "Math professor Ingrid Daubechies awarded $1.5 million grant". The Chronicle. Retrieved 2018-02-03. 98. "Daina Taimina \| The Guardian". the Guardian. 99. "Prizes, Awards, and Honors for Women Mathematicians". mathwomen.agnesscott.org. 100. "Women Mathematicians, Sponsored by Agnes Scott College". agnesscott.edu. Retrieved 2014-01-25. 101. Ouedraogo, Pr Marie Françoise (2015-05-30). AWMA: une association au service des femmes mathématiciennes africaines (PDF) (Speech). Femmes et Mathematiques: Mathématiciennes africaines (in French). Institut Henri Poincaré. 102. "Organization \| African Women in Mathematics Association". africanwomeninmath.org. Retrieved 2021-01-15. 103. "Maryam Mirzakhani Becomes First Woman to Earn Fields Medal for Mathematics in Its 78 Year History \| The Mary Sue". themarysue.com. 12 August 2014. Retrieved 2014-09-13. 104. "IMU Prizes 2014 citations". International Mathematical Union. Retrieved 2014-08-12. 105. "IMU Prizes 2014". International Mathematical Union. Retrieved 2014-08-12. 106. "Fields Medals 2014 \| International Mathematical Union (IMU)". Mathunion.org. 2014-08-13. Retrieved 2018-10-09. 107. "2006 Fields Medals awarded" (PDF). Notices of the American Mathematical Society. American Mathematical Society. 53 (9): 1037–1044. October 2006. 108. "Reclusive Russian turns down math world's highest honour". Canadian Broadcasting Corporation. 22 August 2006. Retrieved 2006-08-26. 109. "Mathematician Claire Voisin awarded the CNRS 2016 gold medal". 110. "Royal Society Athena Prize \| Royal Society". April 19, 2022. Archived from the original on 19 April 2022. 111. "African women 1". Maths History. 112. Change, Kenneth (March 19, 2019). "Karen Uhlenbeck Is First Woman to Receive Abel Prize in Mathematics". New York Times. Retrieved 19 March 2019. 113. Communications, Brown Office of University. "Horizons Seminar: Marissa Kawehi Loving". events.brown.edu. 114. "Ukrainian Viazovska wins Fields Medal 2022". www.ukrinform.net. 115. "Fields Medal \| International Mathematical Union (IMU)". www.mathunion.org. 116. "Ingrid Daubechies". Wolf Foundation.
Female education in STEM Female education in STEM refers to child and adult female representation in the educational fields of science, technology, engineering, and mathematics (STEM). In 2017, 33% of students in STEM fields were women. The organization UNESCO has stated that this gender disparity is due to discrimination, biases, social norms and expectations that influence the quality of education women receive and the subjects they study.[1] UNESCO also believes that having more women in STEM fields is desirable because it would help bring about sustainable development.[1] Current status of girls and women in STEM education Overall trends in STEM education Gender differences in STEM education participation are already visible in early childhood care and education in science- and math-related play, and become more pronounced at higher levels of education. Girls appear to lose interest in STEM subjects with age, particularly between early and late adolescence.[1] This decreased interest affects participation in advanced studies at the secondary level and in higher education.[1] Female students represent 35% of all students enrolled in STEM-related fields of study at this level globally. Differences are also observed by disciplines, with female enrollment lowest in engineering, manufacturing and construction, natural science, mathematics and statistics and ICT fields. Significant regional and country differences in female representation in STEM studies can be observed, though, suggesting the presence of contextual factors affecting girls’ and women's engagement in these fields. Women leave STEM disciplines in disproportionate numbers during their higher education studies, in their transition to the world of work and even in their career cycle.[1][3][4][5][6][7] Learning achievement in STEM education Data on gender differences in learning achievement present a complex picture, depending on what is measured (subject, knowledge acquisition against knowledge application), the level of education/age of students, and geographic location. Overall, women's participation has been increasing, but significant regional variations exist. For example, where data are available in Africa, Latin America and the Caribbean, the gender gap is largely in favor of boys in mathematics achievement in secondary education. In contrast, in the Arab States, girls perform better than boys in both subjects in primary and secondary education. As with the data on participation, national and regional variations in data on learning achievement suggest the presence of contextual factors affecting girls’ and women's engagement in these fields. Girls’ achievement seems to be stronger in science than mathematics and where girls do better than boys, the score differential is up to three times higher than where boys do better.[8] Girls tend to outperform boys in certain sub-topics such as biology and chemistry but do less well in physics and earth science. The gender gap has fallen significantly in science in secondary education among TIMSS trend countries: 14 out of 17 participating countries had no gender gap in science in 2015, compared to only one in 1995. However, the data are less well known outside of these 17 countries. The gender gap in boys' favor is slightly bigger in mathematics but improvements over time in girls’ favor are also observed in certain countries, despite the important regional variations. Gender differences are observed within mathematical sub-topics with girls outperforming boys in topics such as algebra and geometry but doing less well in "number". Girls’ performance is stronger in assessments that measure knowledge acquisition than those measuring knowledge application. Country coverage in terms of data availability is quite limited while data are collected at a different frequency and against different variables in the existing studies. There are large gaps in our knowledge of the situation in low- and middle-income countries in sub-Saharan Africa, Central Asia, and South and West Asia, particularly at the secondary level.[1][4][5][9][10][11][12][13] Factors influencing girls' and women's participation and achievement in STEM education According to UNESCO, there are multiple and overlapping factors which influence girls' and women's participation, achievement and progression in STEM studies and careers, all of which interact in complex ways, including: • Individual level: biological factors that may influence individuals’ abilities, skills, and behaviour such as brain structure and function, hormones, genetics, and cognitive traits like spatial and linguistic skills. It also considers psychological factors, including self-efficacy, interest and motivation. • Family and peer level: parental beliefs and expectations, parental education and socioeconomic status, and other household factors, as well as peer influences. • School level: factors within the learning environment, including teachers’ profile, experience, beliefs and expectations, curricula, learning materials and resources, teaching strategies and student teacher interactions, assessment practices, and the overall school environment. • Societal level: social and cultural norms related to gender equality, and gender stereotypes in the media.[1] Individual level Individual level The question of whether there are differences in cognitive ability between men and women has long been a topic of debate among researchers and scholars. Some studies have found no differences in the neural mechanism of learning based on sex. Loss of interest has been the major reason cited for girls opting out of STEM. However, some have stated that this choice is influenced heavily by the socialisation process and stereotyped ideas about gender roles, including stereotypes about gender and STEM. Gender stereotypes that communicate the idea that STEM studies and careers are male domains can negatively affect girls' interest, engagement, and achievement in STEM and discourage them from pursuing STEM careers. Girls who assimilate such stereotypes have lower levels of self-efficacy and confidence in their ability than boys.[15] Self-efficacy affects both STEM education outcomes and aspirations for STEM careers to a considerable extent. In recent years, more women have been majoring in STEM, although we still continue to witness vast imbalances between men and women studying math, engineering, or science.[16] Family and peer level Parents, including their beliefs and expectations, play an important role in shaping girls' attitudes towards, and interest in, STEM studies. Parents with traditional beliefs about gender roles and who treat girls and boys unequally can reinforce stereotypes about gender and ability in STEM. Parents can also have a strong influence on girls' STEM participation and learning achievement through the family values, environment, experiences, and encouragement that they provide. Some research finds that parents’ expectations, particularly the mother's expectations, have more influence on the higher education and career choices of girls than those of boys.[1] Higher socio-economic status and parental educational qualifications are associated with higher scores in mathematics and science for both girls and boys. Girls' science performance appears to be more strongly associated with mothers' higher educational qualifications, and boys' with their fathers'. Family members with STEM careers can also influence girls’ STEM engagement. The broader socio-cultural context of the family can also play a role. Factors such as ethnicity, language used at home, immigrant status, and family structure may also have an influence on girls' participation and performance in STEM. Peers can also impact on girls’ motivation and feeling of belonging in STEM education. Influence of female peers is a significant predictor of girls' interest and confidence in mathematics and science.[9] School level Qualified teachers with specialisation in STEM can positively influence girls' performance and engagement with STEM education and their interest in pursuing STEM careers. Female STEM teachers often have stronger benefits for girls, possibly by acting as role models and by helping to dispel stereotypes about sex-based STEM ability. Teachers' beliefs, attitudes, behaviours, and interactions with students, as well as curricula and learning materials, can all play a role as well. Opportunities for real-life experiences with STEM, including hands-on practice, apprenticeships, career counselling, and mentoring can expand girls' understanding of STEM studies and professions and maintain interest. Assessment processes and tools that are gender-biased or include gender stereotypes may negatively affect girls' performance in STEM. Girls' learning outcomes in STEM can also be compromised by psychological factors such as mathematics or test anxiety.[1][4][6][9] The confidence of a female teacher in STEM subjects also has a strong impact on how well female students will perform in those subjects in the elementary school classroom. For example, female elementary teachers with anxiety around math will negatively affect the achievement of their female students in math.[17] Correlations have been found between gender bias in female elementary students and their achievement in mathematics. Those who had lower achievement over time have also been found to believe that boys are inherently better at mathematics than girls.[17] Societal level Cultural and social norms influence girls’ perceptions about their abilities, roles in society and career and life aspirations. The degree of gender equality in wider society influences girls' participation and performance in STEM. In countries with greater gender equality, girls tend to have more positive attitudes and confidence about mathematics, and the gender gap in achievement in the subject is smaller. Additionally, in some countries there were more women receiving computer science degrees than men.[18] That was primarily because a computer science degree was seen as indoor work. When the job title was adjusted to sound less masculine and more geared towards relationship building, females appeared to be more likely to enter the STEM field. Gender stereotypes portrayed in the media are internalised by children and adults and affect the way they view themselves and others. Media can perpetuate or challenge gender stereotypes about STEM abilities and careers.[19] Effects of gender disparities The prolonged consequence of consistent gendered stereotypes relating to women's inability to become successful in the field of STEM is the development of a fixed mindset that they are not sufficiently equipped to think critically or contribute valuable ideas in careers in fields that currently employ predominantly male workers. Stepping into a workplace where men outnumber women, knowing that male co-workers expect lower capabilities from a woman, significantly undermines women's skills and performance in their jobs. This in part is due to the heuristic representativeness – when people do not look the part, others are more critical of them. In a heavily male populated environment, men are more critical of women because they do not appear how the abstract representation in STEM fields typically appear. A study demonstrating the effects of construal level priming conditions between men and women, concluded that high construal levels facilitate the use of representativeness heuristic. In contrast, low construal conditions portrayed a decrease in the use of representativeness heuristic.[15] Possible solutions to reduce gender gap • Inclusive STEM approaches such as Problem-Based Learning (PBL) and personalization of learning could generate solutions to lower gender disparities in STEM.[20] • Students' intellectual engagement and success can develop and improve as a result of the instructor's gender. Gender disparities decrease when a course is taught by a female instructor.[21] • Increasing awareness about gender biases in STEM careers can also reduce the gender gap.[22] Sources This article incorporates text from a free content work. (license statement/permission). Text taken from Cracking the code: girls' and women's education in science, technology, engineering and mathematics (STEM), 23, 37, 46, 49, 56, 58, UNESCO, UNESCO. To learn how to add open license text to Wikipedia articles, please see this how-to page. For information on reusing text from Wikipedia, please see the terms of use. References 1. Cracking the code: Girls' and women's education in science, technology, engineering and mathematics (STEM) (PDF). UNESCO. 2017. ISBN 978-92-3-100233-5. 2. Mullis, I. V. S., Martin, M. O., Foy, P. and Hooper, M. (2016). "TIMSS Advanced 2015 International Results in Advanced Mathematics and Physics". TIMSS & PIRLS International Study Center website. Archived from the original on 2017-02-15. Retrieved 2 June 2017.{{cite web}}: CS1 maint: multiple names: authors list (link) 3. "STEM and Gender Advancement (SAGA) \| United Nations Educational, Scientific and Cultural Organization". www.unesco.org. Retrieved 2017-10-12. 4. PISA 2015 Results (Volume I): Excellence and Equity in Education. Paris: OECD. 2016. 5. "TIMSS ADVANCED 2015 INTERNATIONAL RESULTS REPORT – TIMSS 2015 INTERNATIONAL RESULTS REPORT". timssandpirls.bc.edu. Retrieved 2017-10-12. 6. UIS. "UIS Statistics". data.uis.unesco.org. Retrieved 2017-10-12. 7. Science and Engineering Indicators 2014. Arlington: National Science Board. 2014. 8. UNESCO (2017). Cracking the code girls' and women's education in science, technology, engineering and mathematics (STEM). Paris: Unesco. ISBN 978-92-3-100233-5. OCLC 1113762987. 9. Mullis, I. V. S., Martin, M. O. and Loveless, T. (2016). International Trends in Mathematics and Science Achievement, Curriculum and Instruction. Boston: 20 Years of TIMSS.{{cite book}}: CS1 maint: multiple names: authors list (link) 10. Gender Inequality in Learning Achievement in Primary Education. What can TERCE Tell us?. Santiago: UNESCO. 2016. 11. PASEC 2014: Education System Performance in Francophone Sub-Saharan Africa. Dakar: PASEC. 2015. 12. Salto, M. (2011). Trends in the Magnitude and Direction of Gender Differences in Learning Outcomes. SACMEQ. 13. Fraillon, J., Ainley, J., Schulz, W., Friedman, T. and Gebhardt, E. (2014). Preparing for Life in a Digital Age. The IEA International Computer and Information Literacy Study (ICILS) Report. Melbourne: ICILS and Springer Open.{{cite book}}: CS1 maint: multiple names: authors list (link) 14. Catherine André/VoxEurop/EDJNet; Marzia Bona/OBC Transeuropa/EDJNet (19 April 2018). "The ICT sector is booming. But are women missing out?". Retrieved 27 August 2018. 15. "Supplemental Material for The Effects of Construal Level on Heuristic Reasoning: The Case of Representativeness and Availability". Decision. 2014-12-22. doi:10.1037/dec0000021.supp. ISSN 2325-9965. 16. Sonnert, Gerhard; Fox, Mary Frank; Adkins, Kristen (December 2007). "Undergraduate Women in Science and Engineering: Effects of Faculty, Fields, and Institutions Over Time". Social Science Quarterly. 88 (5): 1333–1356. doi:10.1111/j.1540-6237.2007.00505.x. ISSN 0038-4941. 17. Beilock, Sian L.; Gunderson, Elizabeth A.; Ramirez, Gerardo; Levine, Susan C.; Smith, Edward E. (February 5, 2010). "Female Teachers' Math Anxiety Affects Girls' Math Achievement". Proceedings of the National Academy of Sciences. 107 (5): 1860–1863. Bibcode:2010PNAS..107.1860B. doi:10.1073/pnas.0910967107. JSTOR 40536499. PMC 2836676. PMID 20133834. 18. El-Hout, Mona; Garr-Schultz, Alexandra; Cheryan, Sapna (January 2021). "Beyond biology: The importance of cultural factors in explaining gender disparities in STEM preferences". European Journal of Personality. 35 (1): 45–50. doi:10.1177/0890207020980934. ISSN 0890-2070. S2CID 231606736. 19. Beasley, Maya (Summer 2012). "Why they leave: the impact of stereotype threat on the attrition of women and minorities from science, math and engineering majors". Social Psychology of Education. 15 (4): 427–448. doi:10.1007/s11218-012-9185-3. S2CID 2470487. 20. Zuo, Huifang; LaForce, Melanie; Ferris, Kaitlyn; Noble, Elizabeth (2019-07-18). "Revisiting Race and Gender Differences in STEM: Can Inclusive STEM High Schools Reduce Gaps?". European Journal of STEM Education. 4 (1). doi:10.20897/ejsteme/5840. ISSN 2468-4368. S2CID 199136661. 21. Solanki, Sabrina M.; Xu, Di (August 2018). "Looking Beyond Academic Performance: The Influence of Instructor Gender on Student Motivation in STEM Fields". American Educational Research Journal. 55 (4): 801–835. doi:10.3102/0002831218759034. ISSN 0002-8312. S2CID 149379494. 22. "Introduction: Women in science: Why so few?", Athena Unbound, Cambridge University Press, pp. 1–4, 2000-10-19, doi:10.1017/cbo9780511541414.001, ISBN 9780521563802, retrieved 2021-11-28
Woo circles In geometry, the Woo circles, introduced by Peter Y. Woo, are a set of infinitely many Archimedean circles. Construction Form an arbelos with the two inner semicircles tangent at point C. Let m denote any nonnegative real number. Draw two circles, with radii m times the radii of the smaller two arbelos semicircles, centered on the arbelos ground line, also tangent to each other at point C and with radius m times the radius of the corresponding small arbelos arc. Any circle centered on the Schoch line and externally tangent to the circles is a Woo circle.[1] See also • Schoch circles References 1. Thomas Schoch (2007). "Arbelos - The Woo Circles". Retrieved 2008-06-05.
Woodall's conjecture In the mathematics of directed graphs, Woodall's conjecture is an unproven relationship between dicuts and dijoins. It was posed by Douglas Woodall in 1976.[1] Unsolved problem in mathematics: Does the minimum number of edges in a dicut of a directed graph always equal the maximum number of disjoint dijoins? (more unsolved problems in mathematics) Statement A dicut is a partition of the vertices into two subsets such that all edges that cross the partition do so in the same direction. A dijoin is a subset of edges that, when contracted, produces a strongly connected graph; equivalently, it is a subset of edges that includes at least one edge from each dicut.[2] If the minimum number of edges in a dicut is $k$, then there can be at most $k$ disjoint dijoins in the graph, because each one must include a different edge from the smallest dicut. Woodall's conjecture states that, in this case, it is always possible to find $k$ disjoint dijoins. That is, any directed graph the minimum number of edges in a dicut equals the maximum number of disjoint dijoins that can be found in the graph (a packing of dijoins).[2][1] Partial results It is a folklore result that the theorem is true for directed graphs whose minimum dicut has two edges.[2] Any instance of the problem can be reduced to a directed acyclic graph by taking the condensation of the instance, a graph formed by contracting each strongly connected component to a single vertex. Another class of graphs for which the theorem has been proven true are the directed acyclic graphs in which every source vertex (a vertex without incoming edges) has a path to every sink vertex (a vertex without outgoing edges).[3][4] Related results A fractional weighted version of the conjecture, posed by Jack Edmonds and Rick Giles, was refuted by Alexander Schrijver.[5][6][2] In the other direction, the Lucchesi–Younger theorem states that the minimum size of a dijoin equals the maximum number of disjoint dicuts that can be found in a given graph.[7][8] References 1. Woodall, D. R. (1978), "Menger and König systems", in Alavi, Yousef; Lick, Don R. (eds.), Theory and applications of graphs (Proc. Internat. Conf., Western Mich. Univ., Kalamazoo, Mich., 1976), Lecture Notes in Mathematics, vol. 642, Berlin: Springer, pp. 620–635, doi:10.1007/BFb0070416, MR 0499529 2. Abdi, Ahmad; Cornuéjols, Gérard; Zlatin, Michael (2022), On packing dijoins in digraphs and weighted digraphs, arXiv:2202.00392 3. Schrijver, A. (1982), "Min-max relations for directed graphs", Bonn Workshop on Combinatorial Optimization (Bonn, 1980), Annals of Discrete Mathematics, vol. 16, North-Holland, pp. 261–280, MR 0686312 4. Feofiloff, P.; Younger, D. H. (1987), "Directed cut transversal packing for source-sink connected graphs", Combinatorica, 7 (3): 255–263, doi:10.1007/BF02579302, MR 0918396 5. Edmonds, Jack; Giles, Rick (1977), "A min-max relation for submodular functions on graphs", Studies in integer programming (Proc. Workshop, Bonn, 1975), Annals of Discrete Mathematics, vol. 1, North-Holland, Amsterdam, pp. 185–204, MR 0460169 6. Schrijver, A. (1980), Bachem, Achim; Grötschel, Martin; Korte, Bernhard (eds.), "A counterexample to a conjecture of Edmonds and Giles", Discrete Mathematics, 32 (2): 213–215, doi:10.1016/0012-365X(80)90057-6, MR 0592858 7. Lovász, László (1976), "On two minimax theorems in graph", Journal of Combinatorial Theory, Series B, 21 (2): 96–103, doi:10.1016/0095-8956(76)90049-6, MR 0427138 8. Lucchesi, C. L.; Younger, D. H. (1978), "A minimax theorem for directed graphs", Journal of the London Mathematical Society, Second Series, 17 (3): 369–374, doi:10.1112/jlms/s2-17.3.369, MR 0500618 External links • Feofiloff, Paulo (November 30, 2005), Woodall’s conjecture on Packing Dijoins: a survey (PDF) • "Woodall's conjecture", Open Problem Garden, April 5, 2007
Woodall number In number theory, a Woodall number (Wn) is any natural number of the form $W_{n}=n\cdot 2^{n}-1$ for some natural number n. The first few Woodall numbers are: 1, 7, 23, 63, 159, 383, 895, … (sequence A003261 in the OEIS). History Woodall numbers were first studied by Allan J. C. Cunningham and H. J. Woodall in 1917,[1] inspired by James Cullen's earlier study of the similarly defined Cullen numbers. Woodall primes Unsolved problem in mathematics: Are there infinitely many Woodall primes? (more unsolved problems in mathematics) Woodall numbers that are also prime numbers are called Woodall primes; the first few exponents n for which the corresponding Woodall numbers Wn are prime are 2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, ... (sequence A002234 in the OEIS); the Woodall primes themselves begin with 7, 23, 383, 32212254719, ... (sequence A050918 in the OEIS). In 1976 Christopher Hooley showed that almost all Cullen numbers are composite.[2] In October 1995, Wilfred Keller published a paper discussing several new Cullen primes and the efforts made to factorise other Cullen and Woodall numbers. Included in that paper is a personal communication to Keller from Hiromi Suyama, asserting that Hooley's method can be reformulated to show that it works for any sequence of numbers n · 2n + a + b, where a and b are integers, and in particular, that almost all Woodall numbers are composite.[3] It is an open problem whether there are infinitely many Woodall primes. As of October 2018, the largest known Woodall prime is 17016602 × 217016602 − 1.[4] It has 5,122,515 digits and was found by Diego Bertolotti in March 2018 in the distributed computing project PrimeGrid.[5] Restrictions Starting with W4 = 63 and W5 = 159, every sixth Woodall number is divisible by 3; thus, in order for Wn to be prime, the index n cannot be congruent to 4 or 5 (modulo 6). Also, for a positive integer m, the Woodall number W2m may be prime only if 2m + m is prime. As of January 2019, the only known primes that are both Woodall primes and Mersenne primes are W2 = M3 = 7, and W512 = M521. Divisibility properties Like Cullen numbers, Woodall numbers have many divisibility properties. For example, if p is a prime number, then p divides W(p + 1) / 2 if the Jacobi symbol $\left({\frac {2}{p}}\right)$ is +1 and W(3p − 1) / 2 if the Jacobi symbol $\left({\frac {2}{p}}\right)$ is −1. Generalization A generalized Woodall number base b is defined to be a number of the form n × bn − 1, where n + 2 > b; if a prime can be written in this form, it is then called a generalized Woodall prime. The smallest value of n such that n × bn − 1 is prime for b = 1, 2, 3, ... are[6] 3, 2, 1, 1, 8, 1, 2, 1, 10, 2, 2, 1, 2, 1, 2, 167, 2, 1, 12, 1, 2, 2, 29028, 1, 2, 3, 10, 2, 26850, 1, 8, 1, 42, 2, 6, 2, 24, 1, 2, 3, 2, 1, 2, 1, 2, 2, 140, 1, 2, 2, 22, 2, 8, 1, 2064, 2, 468, 6, 2, 1, 362, 1, 2, 2, 6, 3, 26, 1, 2, 3, 20, 1, 2, 1, 28, 2, 38, 5, 3024, 1, 2, 81, 858, 1, 2, 3, 2, 8, 60, 1, 2, 2, 10, 5, 2, 7, 182, 1, 17782, 3, ... (sequence A240235 in the OEIS) As of November 2021, the largest known generalized Woodall prime with base greater than 2 is 2740879 × 322740879 − 1.[7] b Numbers n such that n × bn − 1 is prime[6] OEIS sequence 3 1, 2, 6, 10, 18, 40, 46, 86, 118, 170, 1172, 1698, 1810, 2268, 4338, 18362, 72662, 88392, 94110, 161538, 168660, 292340, 401208, 560750, 1035092, ... A006553 4 1, 2, 3, 5, 8, 14, 23, 63, 107, 132, 428, 530, 1137, 1973, 2000, 7064, 20747, 79574, 113570, 293912, ..., 1993191, ... A086661 5 8, 14, 42, 384, 564, 4256, 6368, 21132, 27180, 96584, 349656, 545082, ... A059676 6 1, 2, 3, 19, 20, 24, 34, 77, 107, 114, 122, 165, 530, 1999, 4359, 11842, 12059, 13802, 22855, 41679, 58185, 145359, 249987, ... A059675 7 2, 18, 68, 84, 3812, 14838, 51582, ... A242200 8 1, 2, 7, 12, 25, 44, 219, 252, 507, 1155, 2259, 2972, 4584, 12422, 13905, 75606, ... A242201 9 10, 58, 264, 1568, 4198, 24500, ... A242202 10 2, 3, 8, 11, 15, 39, 60, 72, 77, 117, 183, 252, 396, 1745, 2843, 4665, 5364, 524427, ... A059671 11 2, 8, 252, 1184, 1308, 1182072, ... A299374 12 1, 6, 43, 175, 821, 910, 1157, 13748, 27032, 71761, 229918, 549721, 866981, 1405486, ... A299375 13 2, 6, 563528, ... A299376 14 1, 3, 7, 98, 104, 128, 180, 834, 1633, 8000, 28538, 46605, 131941, 147684, 433734, 1167708, ... A299377 15 2, 10, 14, 2312, 16718, 26906, 27512, 41260, 45432, 162454, 217606, 1527090, ... A299378 16 167, 189, 639, ... A299379 17 2, 18, 20, 38, 68, 3122, 3488, 39500, ... A299380 18 1, 2, 6, 8, 10, 28, 30, 39, 45, 112, 348, 380, 458, 585, 17559, 38751, 43346, 46984, 92711, ... A299381 19 12, 410, 33890, 91850, 146478, 189620, 280524, ... A299382 20 1, 18, 44, 60, 80, 123, 429, 1166, 2065, 8774, 35340, 42968, 50312, 210129, 663703, ... A299383 See also • Mersenne prime - Prime numbers of the form 2n − 1. References 1. Cunningham, A. J. C; Woodall, H. J. (1917), "Factorisation of $Q=(2^{q}\mp q)$ and $(q\cdot {2^{q}}\mp 1)$", Messenger of Mathematics, 47: 1–38. 2. Everest, Graham; van der Poorten, Alf; Shparlinski, Igor; Ward, Thomas (2003). Recurrence sequences. Mathematical Surveys and Monographs. Vol. 104. Providence, RI: American Mathematical Society. p. 94. ISBN 0-8218-3387-1. Zbl 1033.11006. 3. Keller, Wilfrid (January 1995). "New Cullen primes". Mathematics of Computation. 64 (212): 1739. doi:10.1090/S0025-5718-1995-1308456-3. ISSN 0025-5718. Keller, Wilfrid (December 2013). "Wilfrid Keller". www.fermatsearch.org. Hamburg. Archived from the original on February 28, 2020. Retrieved October 1, 2020. 4. "The Prime Database: 85083012^17016603-1", Chris Caldwell's The Largest Known Primes Database, retrieved March 24, 2018 5. PrimeGrid, Announcement of 170166022^17016602 - 1 (PDF), retrieved April 1, 2018 6. List of generalized Woodall primes base 3 to 10000 7. "The Top Twenty: Generalized Woodall". primes.utm.edu. Retrieved 20 November 2021. Further reading • Guy, Richard K. (2004), Unsolved Problems in Number Theory (3rd ed.), New York: Springer Verlag, pp. section B20, ISBN 0-387-20860-7. • Keller, Wilfrid (1995), "New Cullen Primes" (PDF), Mathematics of Computation, 64 (212): 1733–1741, doi:10.2307/2153382, JSTOR 2153382. • Caldwell, Chris, "The Top Twenty: Woodall Primes", The Prime Pages, retrieved December 29, 2007. External links • Chris Caldwell, The Prime Glossary: Woodall number, and The Top Twenty: Woodall, and The Top Twenty: Generalized Woodall, at The Prime Pages. • Weisstein, Eric W. "Woodall number". MathWorld. • Steven Harvey, List of Generalized Woodall primes. • Paul Leyland, Generalized Cullen and Woodall Numbers Prime number classes By formula • Fermat (22n + 1) • Mersenne (2p − 1) • Double Mersenne (22p−1 − 1) • Wagstaff (2p + 1)/3 • Proth (k·2n + 1) • Factorial (n! ± 1) • Primorial (pn# ± 1) • Euclid (pn# + 1) • Pythagorean (4n + 1) • Pierpont (2m·3n + 1) • Quartan (x4 + y4) • Solinas (2m ± 2n ± 1) • Cullen (n·2n + 1) • Woodall (n·2n − 1) • Cuban (x3 − y3)/(x − y) • Leyland (xy + yx) • Thabit (3·2n − 1) • Williams ((b−1)·bn − 1) • Mills (⌊A3n⌋) By integer sequence • Fibonacci • Lucas • Pell • Newman–Shanks–Williams • Perrin • Partitions • Bell • Motzkin By property • Wieferich (pair) • Wall–Sun–Sun • Wolstenholme • Wilson • Lucky • Fortunate • Ramanujan • Pillai • Regular • Strong • Stern • Supersingular (elliptic curve) • Supersingular (moonshine theory) • Good • Super • Higgs • Highly cototient • Unique Base-dependent • Palindromic • Emirp • Repunit (10n − 1)/9 • Permutable • Circular • Truncatable • Minimal • Delicate • Primeval • Full reptend • Unique • Happy • Self • Smarandache–Wellin • Strobogrammatic • Dihedral • Tetradic Patterns • Twin (p, p + 2) • Bi-twin chain (n ± 1, 2n ± 1, 4n ± 1, …) • Triplet (p, p + 2 or p + 4, p + 6) • Quadruplet (p, p + 2, p + 6, p + 8) • k-tuple • Cousin (p, p + 4) • Sexy (p, p + 6) • Chen • Sophie Germain/Safe (p, 2p + 1) • Cunningham (p, 2p ± 1, 4p ± 3, 8p ± 7, ...) • Arithmetic progression (p + a·n, n = 0, 1, 2, 3, ...) • Balanced (consecutive p − n, p, p + n) By size • Mega (1,000,000+ digits) • Largest known • list Complex numbers • Eisenstein prime • Gaussian prime Composite numbers • Pseudoprime • Catalan • Elliptic • Euler • Euler–Jacobi • Fermat • Frobenius • Lucas • Somer–Lucas • Strong • Carmichael number • Almost prime • Semiprime • Sphenic number • Interprime • Pernicious Related topics • Probable prime • Industrial-grade prime • Illegal prime • Formula for primes • Prime gap First 60 primes • 2 • 3 • 5 • 7 • 11 • 13 • 17 • 19 • 23 • 29 • 31 • 37 • 41 • 43 • 47 • 53 • 59 • 61 • 67 • 71 • 73 • 79 • 83 • 89 • 97 • 101 • 103 • 107 • 109 • 113 • 127 • 131 • 137 • 139 • 149 • 151 • 157 • 163 • 167 • 173 • 179 • 181 • 191 • 193 • 197 • 199 • 211 • 223 • 227 • 229 • 233 • 239 • 241 • 251 • 257 • 263 • 269 • 271 • 277 • 281 List of prime numbers Classes of natural numbers Powers and related numbers • Achilles • Power of 2 • Power of 3 • Power of 10 • Square • Cube • Fourth power • Fifth power • Sixth power • Seventh power • Eighth power • Perfect power • Powerful • Prime power Of the form a × 2b ± 1 • Cullen • Double Mersenne • Fermat • Mersenne • Proth • Thabit • Woodall Other polynomial numbers • Hilbert • Idoneal • Leyland • Loeschian • Lucky numbers of Euler Recursively defined numbers • Fibonacci • Jacobsthal • Leonardo • Lucas • Padovan • Pell • Perrin Possessing a specific set of other numbers • Amenable • Congruent • Knödel • Riesel • Sierpiński Expressible via specific sums • Nonhypotenuse • Polite • Practical • Primary pseudoperfect • Ulam • Wolstenholme Figurate numbers 2-dimensional centered • Centered triangular • Centered square • Centered pentagonal • Centered hexagonal • Centered heptagonal • Centered octagonal • Centered nonagonal • Centered decagonal • Star non-centered • Triangular • Square • Square triangular • Pentagonal • Hexagonal • Heptagonal • Octagonal • Nonagonal • Decagonal • Dodecagonal 3-dimensional centered • Centered tetrahedral • Centered cube • Centered octahedral • Centered dodecahedral • Centered icosahedral non-centered • Tetrahedral • Cubic • Octahedral • Dodecahedral • Icosahedral • Stella octangula pyramidal • Square pyramidal 4-dimensional non-centered • Pentatope • Squared triangular • Tesseractic Combinatorial numbers • Bell • Cake • Catalan • Dedekind • Delannoy • Euler • Eulerian • Fuss–Catalan • Lah • Lazy caterer's sequence • Lobb • Motzkin • Narayana • Ordered Bell • Schröder • Schröder–Hipparchus • Stirling first • Stirling second • Telephone number • Wedderburn–Etherington Primes • Wieferich • Wall–Sun–Sun • Wolstenholme prime • Wilson Pseudoprimes • Carmichael number • Catalan pseudoprime • Elliptic pseudoprime • Euler pseudoprime • Euler–Jacobi pseudoprime • Fermat pseudoprime • Frobenius pseudoprime • Lucas pseudoprime • Lucas–Carmichael number • Somer–Lucas pseudoprime • Strong pseudoprime Arithmetic functions and dynamics Divisor functions • Abundant • Almost perfect • Arithmetic • Betrothed • Colossally abundant • Deficient • Descartes • Hemiperfect • Highly abundant • Highly composite • Hyperperfect • Multiply perfect • Perfect • Practical • Primitive abundant • Quasiperfect • Refactorable • Semiperfect • Sublime • Superabundant • Superior highly composite • Superperfect Prime omega functions • Almost prime • Semiprime Euler's totient function • Highly cototient • Highly totient • Noncototient • Nontotient • Perfect totient • Sparsely totient Aliquot sequences • Amicable • Perfect • Sociable • Untouchable Primorial • Euclid • Fortunate Other prime factor or divisor related numbers • Blum • Cyclic • Erdős–Nicolas • Erdős–Woods • Friendly • Giuga • Harmonic divisor • Jordan–Pólya • Lucas–Carmichael • Pronic • Regular • Rough • Smooth • Sphenic • Størmer • Super-Poulet • Zeisel Numeral system-dependent numbers Arithmetic functions and dynamics • Persistence • Additive • Multiplicative Digit sum • Digit sum • Digital root • Self • Sum-product Digit product • Multiplicative digital root • Sum-product Coding-related • Meertens Other • Dudeney • Factorion • Kaprekar • Kaprekar's constant • Keith • Lychrel • Narcissistic • Perfect digit-to-digit invariant • Perfect digital invariant • Happy P-adic numbers-related • Automorphic • Trimorphic Digit-composition related • Palindromic • Pandigital • Repdigit • Repunit • Self-descriptive • Smarandache–Wellin • Undulating Digit-permutation related • Cyclic • Digit-reassembly • Parasitic • Primeval • Transposable Divisor-related • Equidigital • Extravagant • Frugal • Harshad • Polydivisible • Smith • Vampire Other • Friedman Binary numbers • Evil • Odious • Pernicious Generated via a sieve • Lucky • Prime Sorting related • Pancake number • Sorting number Natural language related • Aronson's sequence • Ban Graphemics related • Strobogrammatic • Mathematics portal
Sarah Woodhead Sarah Woodhead (1851–1912) was the first woman to take and pass a Tripos examination. In particular, she was the first woman to take, and to pass, the Mathematical Tripos exam, which she did in 1873.[1] Education Woodhead’s family had long belonged to the Society of Friends, so she was able to attend Ackworth School, a Quaker school that accepted daughters of Friends as well as their sons.[2] Woodhead later studied at Girton College, the first women's college to be founded at Oxford or Cambridge. As the physical college had yet to be built, she attended courses set up by Girton founder Emily Davies at Benslow House, Hitchin. In 1873, Woodhead took the same Mathematical Tripos examination as the male students, having already gained a first at Part I, and was classed equivalent to Senior Optime in mathematics. She was the first woman to take, and to pass, the Mathematical Tripos exam.[3] This also made her the very first of the first three women to complete any Tripos at Girton College.[4] The three "honorary" (rather than actual) graduates became known as "Woodhead, Cook and Lumsden, the Girton Pioneers".[5] Later life and death Woodhead married architect Christopher Corbett, after which she ran her own school in Bolton. She then became the second headmistress of Bolton School, known then as Bolton High School for Girls.[6] After her husband moved the family back to Manchester to take over his family firm, she found employment as an inspector of schools. Widowed in her fifties, she moved to Harrogate and died there in July, 1908, aged fifty-seven.[7] See also • Philippa Fawcett, the first woman to obtain the top score on the Mathematical Tripos. References 1. Jensen-Vallin, Jacqueline A.; Beery, Janet L.; Mast, Maura B.; Greenwald, Sarah J., eds. (2018). Women in Mathematics: Celebrating the Centennial of the Mathematical Association of America. Springer. p. "Sarah+woodhead"+tripos+1873&pg=PA8 8. ISBN 978-3-319-88303-8. 2. Woodhead, David L., Sarah Woodhead: Trail-Blazer - Quaker Girl & Pioneer, self (see Girton College Library) 3. Jensen-Vallin, Jacqueline A.; Beery, Janet L.; Mast, Maura B.; Greenwald, Sarah J., eds. (2018). Women in Mathematics: Celebrating the Centennial of the Mathematical Association of America. Springer. p. "Sarah+woodhead"+tripos+1873&pg=PA8 8. ISBN 978-3-319-88303-8. 4. Tuker, Mildred Anna Rosalie; Matthison, William (1907), Cambridge, A. - and C. Black, p. 321 5. Megson, Barbara; Lindsay, Jean Olivia (1961), Girton College, 1869–1959: an informal history, W. Heffer for the Girton Historical and Political Society, p. 19 6. Stephen, Barbara (1932), Girton College 1869–1932, Cambridge University Press, p. 194. 7. Campion, Val (2008), Pioneering Women: The Origins of Girton College in Hitchin, Hitchin Historical Society Publication, p. 49
Robert Woodhouse Robert Woodhouse FRS (28 April 1773 – 23 December 1827) was a British mathematician and astronomer. Robert Woodhouse FRS Born(1773-04-28)28 April 1773 Norwich, Norfolk, England Died23 December 1827(1827-12-23) (aged 54) Cambridge, England NationalityBritish Alma materCambridge University AwardsFirst Smith's Prize (1795) Scientific career FieldsMathematics and astronomy Biography Early life and education Robert Woodhouse was born on 28 April 1773 in Norwich, Norfolk, the son of Robert Woodhouse, linen draper, and Judith Alderson, the daughter of a Unitarian minister from Lowestoft.[1] Robert junior was baptised at St George's Church, Colegate, Norwich, on 19 May, 1773.[2] A younger son, John Thomas Woodhouse, was born in 1780. The brothers were educated at the Paston School in North Walsham, 24 kilometres (15 mi) north of Norwich.[1] In May 1790 Woodhouse was admitted to Gonville and Caius College, Cambridge,[1] the college where Paston pupils were traditionally sent.[3] In 1795 he graduated as the Senior Wrangler (ranked first among the mathematics undergraduates at the university), and took the First Smith's Prize.[1] He obtained his Master's degree at Cambridge in 1798.[4] Marriage and career at Cambridge Woodhouse was a fellow of the college from 1798 to 1823,[note 1] after which he resigned so as to be able to marry Harriet, the daughter of William Wilkin, a Norwich architect.[1][4] They were married on 20 February 1823; the marriage produced a son, also named Robert.[1][4] Harriet Woodhouse died at Cambridge on 31 March 1826.[6] Woodhouse was elected a Fellow of the Royal Society on 16 December 1802.[7] His earliest work, entitled the Principles of Analytical Calculation, was published at Cambridge in 1803.[8] In this he explained the differential notation and strongly pressed the employment of it; but he severely criticised the methods used by continental writers, and their constant assumption of non-evident principles.[9] In 1809 Woodhouse published a textbook covering planar trigonometry and spherical trigonometry and the next year a historical treatise on the calculus of variations and isoperimetrical problems. He next produced an astronomy; of which the first book (usually bound in two volumes), on practical and descriptive astronomy, was issued in 1812, and the second book, containing an account of the treatment of physical astronomy by Pierre-Simon Laplace and other continental writers, was issued in 1818.[8] Woodhouse became the Lucasian Professor of Mathematics in 1820, but the small income caused him to resign the professorship in 1822 and instead accept the better paid post as the Plumian professor in the university.[1][10] As Plumian Professor he was responsible for installing and adjusting the transit instruments and clocks at the Cambridge Observatory.[11] Woodhouse did not exercise much influence on the majority of his contemporaries, and the movement might have died away for the time being if it had not been for the advocacy of George Peacock, Charles Babbage, and John Herschel, who formed the Analytical Society, with the object of advocating the general use in the university of analytical methods and of the differential notation.[12] Woodhouse was the first director of the newly built observatory at Cambridge, a post he held until his death in 1827.[1] On his death in Cambridge he was buried in Caius College chapel.[7] Notes 1. Woodhouse's younger brother John was also a fellow at the college.[5] References 1. O'Connor, John J.; Robertson, Edmund F. (2005). "Robert Woodhouse". MacTutor. University of St Andrews. Retrieved 24 September 2021. 2. Robert Woodhouse in "the Norfolk, England, Transcripts of Church of England Baptism, Marriage and Burial Registers, 1600-1935, FamilySearch (Robert Woodhouse). Citing Archdeacon Transcripts 1600-1812. (registration required) 3. "The History of Paston College". Paston College. Retrieved 24 September 2021. 4. "Robert Woodhouse (WDHS790R)". A Cambridge Alumni Database. University of Cambridge. Retrieved 24 September 2021. 5. Becher 2004. 6. "Biographies T - Z". Institute of Astronomy, Cambridge. Retrieved 24 September 2021. 7. "Record: Woodhouse; Robert (1773 - 1827)". The Royal Society. Retrieved 24 September 2021. 8. Rouse Ball 1912, p. 440. 9. Becher 1980, pp. 389–400. 10. Guicciardini 1989, p. 126. 11. Woodhouse 1825, pp. 418–428. 12. Rouse Ball 1912, pp. 440–441. Sources Wikimedia Commons has media related to Robert Woodhouse (mathematician). Wikiquote has quotations related to Robert Woodhouse. • Becher, Harvey W. (1980). "Woodhouse, Babbage, Peacock and Modern Algebra". Historia Mathematica. 7 (4): 389–400. doi:10.1016/0315-0860(80)90003-8. • Becher, Harvey W. (2004). "Woodhouse, Robert". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/29926. OCLC 56568095. Retrieved 24 September 2021. (Subscription or UK public library membership required.) (subscription may be required or content may be available in libraries that are in the UK) • Guicciardini, Niccolò (1989). "Robert Woodhouse". The Development of Newtonian Calculus in Britain 1700–1800. New York: Cambridge University Press. ISBN 0-521-36466-3. • Rouse Ball, W.W. (1912). A Short Account Of The History Of Mathematics. London: MacMillan and Co. Ltd. OCLC 844389098. • Woodhouse, Robert (1825). "Some account of the transit instrument made by Mr. Dollond, and lately put up at the Cambridge Observatory". Philosophical Transactions of the Royal Society of London: 418–428. Further reading • Harman, P. M. (1988). "Newton to Maxwell: The 'Principia' and British Physics". Notes and Records of the Royal Society of London. 42 (1: Newton's 'Principia' and Its Legacy): 75–96. Bibcode:1988npl..conf...75H. doi:10.1098/rsnr.1988.0008. JSTOR 531370. S2CID 122622492 – via JSTOR. • Johnson, W. (1995). "Contributors to Improving the Teaching of Calculus in Early 19th-Century England". Notes and Records of the Royal Society of London. 49 (1): 93–103. doi:10.1098/rsnr.1995.0006. JSTOR 531886. S2CID 145534544 – via JSTOR. External links • Facsimile of Woodhouse's certificate of election to the Royal Society Works • 1803: Principles of Analytical Calculation • 1809: A Treatise on Plane and Spherical Trigonometry (5th edition 1827) • 1810: A Treatise on Isoperimetric Problems and the Calculus of Variations • 1818: An Elementary Treatise on Physical Astronomy, volume 1 • 1818: An Elementary Treatise on Astronomy, volume 2 • 1821: A Treatise on Astronomy, Theoretical and Practical Lucasian Professors of Mathematics • Isaac Barrow (1664) • Isaac Newton (1669) • William Whiston (1702) • Nicholas Saunderson (1711) • John Colson (1739) • Edward Waring (1760) • Isaac Milner (1798) • Robert Woodhouse (1820) • Thomas Turton (1822) • George Biddell Airy (1826) • Charles Babbage (1828) • Joshua King (1839) • George Stokes (1849) • Joseph Larmor (1903) • Paul Dirac (1932) • James Lighthill (1969) • Stephen Hawking (1979) • Michael Green (2009) • Michael Cates (2015) Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • United States • Czech Republic • Netherlands Other • IdRef
Woodin cardinal In set theory, a Woodin cardinal (named for W. Hugh Woodin) is a cardinal number $\lambda $ such that for all functions $f:\lambda \to \lambda $ there exists a cardinal $\kappa <\lambda $ with $\{f(\beta )\mid \beta <\kappa \}\subseteq \kappa $ and an elementary embedding $j:V\to M$ from the Von Neumann universe $V$ into a transitive inner model $M$ with critical point $\kappa $ and $V_{j(f)(\kappa )}\subseteq M.$ An equivalent definition is this: $\lambda $ is Woodin if and only if $\lambda $ is strongly inaccessible and for all $A\subseteq V_{\lambda }$ there exists a $\lambda _{A}<\lambda $ which is $<\lambda $-$A$-strong. $\lambda _{A}$ being $<\lambda $-$A$-strong means that for all ordinals $\alpha <\lambda $, there exist a $j:V\to M$ which is an elementary embedding with critical point $\lambda _{A}$, $j(\lambda _{A})>\alpha $, $V_{\alpha }\subseteq M$ and $j(A)\cap V_{\alpha }=A\cap V_{\alpha }$. (See also strong cardinal.) A Woodin cardinal is preceded by a stationary set of measurable cardinals, and thus it is a Mahlo cardinal. However, the first Woodin cardinal is not even weakly compact. Consequences Woodin cardinals are important in descriptive set theory. By a result[1] of Martin and Steel, existence of infinitely many Woodin cardinals implies projective determinacy, which in turn implies that every projective set is Lebesgue measurable, has the Baire property (differs from an open set by a meager set, that is, a set which is a countable union of nowhere dense sets), and the perfect set property (is either countable or contains a perfect subset). The consistency of the existence of Woodin cardinals can be proved using determinacy hypotheses. Working in ZF+AD+DC one can prove that $\Theta _{0}$ is Woodin in the class of hereditarily ordinal-definable sets. $\Theta _{0}$ is the first ordinal onto which the continuum cannot be mapped by an ordinal-definable surjection (see Θ (set theory)). Mitchell and Steel showed that assuming a Woodin cardinal exists, there is an inner model containing a Woodin cardinal in which there is a $\Delta _{4}^{1}$-well-ordering of the reals, ◊ holds, and the generalized continuum hypothesis holds.[2] Shelah proved that if the existence of a Woodin cardinal is consistent then it is consistent that the nonstationary ideal on $\omega _{1}$ is $\aleph _{2}$-saturated. Woodin also proved the equiconsistency of the existence of infinitely many Woodin cardinals and the existence of an $\aleph _{1}$-dense ideal over $\aleph _{1}$. Hyper-Woodin cardinals A cardinal $\kappa $ is called hyper-Woodin if there exists a normal measure $U$ on $\kappa $ such that for every set $S$, the set $\{\lambda <\kappa \mid \lambda $ is $<\kappa $-$S$-strong$\}$ is in $U$. $\lambda $ is $<\kappa $-$S$-strong if and only if for each $\delta <\kappa $ there is a transitive class $N$ and an elementary embedding $j:V\to N$ with $\lambda ={\text{crit}}(j),$ $j(\lambda )\geq \delta $, and $j(S)\cap H_{\delta }=S\cap H_{\delta }$. The name alludes to the classical result that a cardinal is Woodin if and only if for every set $S$, the set $\{\lambda <\kappa \mid \lambda $ is $<\kappa $-$S$-strong$\}$ is a stationary set. The measure $U$ will contain the set of all Shelah cardinals below $\kappa $. Weakly hyper-Woodin cardinals A cardinal $\kappa $ is called weakly hyper-Woodin if for every set $S$ there exists a normal measure $U$ on $\kappa $ such that the set $\{\lambda <\kappa \mid \lambda $ is $<\kappa $-$S$-strong$\}$ is in $U$. $\lambda $ is $<\kappa $-$S$-strong if and only if for each $\delta <\kappa $ there is a transitive class $N$ and an elementary embedding $j:V\to N$ with $\lambda ={\text{crit}}(j)$, $j(\lambda )\geq \delta $, and $j(S)\cap H_{\delta }=S\cap H_{\delta }.$ The name alludes to the classic result that a cardinal is Woodin if for every set $S$, the set $\{\lambda <\kappa \mid \lambda $ is $<\kappa $-$S$-strong$\}$ is stationary. The difference between hyper-Woodin cardinals and weakly hyper-Woodin cardinals is that the choice of $U$ does not depend on the choice of the set $S$ for hyper-Woodin cardinals. Notes and references 1. A Proof of Projective Determinacy 2. W. Mitchell, Inner models for large cardinals (2012, p.32). Accessed 2022-12-08. Further reading • Kanamori, Akihiro (2003). The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. ISBN 3-540-00384-3. • For proofs of the two results listed in consequences see Handbook of Set Theory (Eds. Foreman, Kanamori, Magidor) (to appear). Drafts of some chapters are available. • Ernest Schimmerling, Woodin cardinals, Shelah cardinals and the Mitchell-Steel core model, Proceedings of the American Mathematical Society 130/11, pp. 3385–3391, 2002, online • Steel, John R. (October 2007). "What is a Woodin Cardinal?" (PDF). Notices of the American Mathematical Society. 54 (9): 1146–7. Retrieved 2008-01-15.
Word-representable graph In the mathematical field of graph theory, a word-representable graph is a graph that can be characterized by a word (or sequence) whose entries alternate in a prescribed way. In particular, if the vertex set of the graph is V, one should be able to choose a word w over the alphabet V such that letters a and b alternate in w if and only if the pair ab is an edge in the graph. (Letters a and b alternate in w if, after removing from w all letters but the copies of a and b, one obtains a word abab... or a word baba....) For example, the cycle graph labeled by a, b, c and d in clock-wise direction is word-representable because it can be represented by abdacdbc: the pairs ab, bc, cd and ad alternate, but the pairs ac and bd do not. The word w is G's word-representant, and one says that that w represents G. The smallest (by the number of vertices) non-word-representable graph is the wheel graph W5, which is the only non-word-representable graph on 6 vertices. The definition of a word-representable graph works both in labelled and unlabelled cases since any labelling of a graph is equivalent to any other labelling. Also, the class of word-representable graphs is hereditary. Word-representable graphs generalise several important classes of graphs such as circle graphs, 3-colorable graphs and comparability graphs. Various generalisations of the theory of word-representable graphs accommodate representation of any graph. History Word-representable graphs were introduced by Sergey Kitaev in 2004 based on joint research with Steven Seif[1] on the Perkins semigroup, which has played an important role in semigroup theory since 1960.[2] The first systematic study of word-representable graphs was undertaken in a 2008 paper by Kitaev and Artem Pyatkin,[3] starting development of the theory. One of key contributors to the area is Magnús M. Halldórsson.[4][5][6] Up to date, 35+ papers have been written on the subject, and the core of the book[2] by Sergey Kitaev and Vadim Lozin is devoted to the theory of word-representable graphs. A quick way to get familiar with the area is to read one of the survey articles.[7][8][9] Motivation to study the graphs According to,[2] word-representable graphs are relevant to various fields, thus motivating to study the graphs. These fields are algebra, graph theory, computer science, combinatorics on words, and scheduling. Word-representable graphs are especially important in graph theory, since they generalise several important classes of graphs, e.g. circle graphs, 3-colorable graphs and comparability graphs. Early results It was shown in [3] that a graph G is word-representable if it is k-representable for some k, that is, G can be represented by a word having k copies of each letter. Moreover, if a graph is k-representable then it is also (k + 1)-representable. Thus, the notion of the representation number of a graph, as the minimum k such that a graph is word-representable, is well-defined. Non-word-representable graphs have the representation number ∞. Graphs with representation number 1 are precisely the set of complete graphs, while graphs with representation number 2 are precisely the class of circle non-complete graphs. In particular, forests (except for single trees on at most 2 vertices), ladder graphs and cycle graphs have representation number 2. No classification for graphs with representation number 3 is known. However, there are examples of such graphs, e.g. Petersen's graph and prisms. Moreover, the 3-subdivision of any graph is 3-representable. In particular, for every graph G there exists a 3-representable graph H that contains G as a minor.[3] A graph G is permutationally representable if it can be represented by a word of the form p1p2...pk, where pi is a permutation. On can also say that G is permutationally k-representable. A graph is permutationally representable iff it is a comparability graph.[1] A graph is word-representable implies that the neighbourhood of each vertex is permutationally representable (i.e. is a comparability graph).[1] Converse to the last statement is not true.[4] However, the fact that the neighbourhood of each vertex is a comparability graph implies that the Maximum Clique problem is polynomially solvable on word-representable graphs.[5][6] Semi-transitive orientations Semi-transitive orientations provide a powerful tool to study word-representable graphs. A directed graph is semi-transitively oriented iff it is acyclic and for any directed path u1→u2→ ...→ut, t ≥ 2, either there is no edge from u1 to ut or all edges ui → uj exist for 1 ≤ i < j ≤ t. A key theorem in the theory of word-representable graphs states that a graph is word-representable iff it admits a semi-transitive orientation.[6] As a corollary to the proof of the key theorem one obtain an upper bound on word-representants: Each non-complete word-representable graph G is 2(n − κ(G))-representable, where κ(G) is the size of a maximal clique in G.[6] As an immediate corollary of the last statement, one has that the recognition problem of word-representability is in NP. In 2014, Vincent Limouzy observed that it is an NP-complete problem to recognise whether a given graph is word-representable.[2][7] Another important corollary to the key theorem is that any 3-colorable graph is word-representable. The last fact implies that many classical graph problems are NP-hard on word-representable graphs. Overview of selected results Non-word-representable graphs Wheel graphs W2n+1, for n ≥ 2, are not word-representable and W5 is the minimum (by the number of vertices) non-word-representable graph. Taking any non-comparability graph and adding an apex (a vertex connected to any other vertex), we obtain a non-word-representable graph, which then can produce infinitely many non-word-representable graphs.[2] Any graph produced in this way will necessarily have a triangle (a cycle of length 3), and a vertex of degree at least 5. Non-word-representable graphs of maximum degree 4 exist [10] and non-word-representable triangle-free graphs exist.[5] Regular non-word representable graphs also exist.[2] Non-isomorphic non-word-representable connected graphs on at most eight vertices were first enumerated by Heman Z.Q. Chen. His calculations were extended in,[11] where it was shown that the numbers of non-isomorphic non-word-representable connected graphs on 5−11 vertices are given, respectively, by 0, 1, 25, 929, 54957, 4880093, 650856040. This is the sequence A290814 in the Online Encyclopaedia of Integer Sequences (OEIS). Operations on graphs and word-representability Operations preserving word-representability are removing a vertex, replacing a vertex with a module, Cartesian product, rooted product, subdivision of a graph, connecting two graphs by an edge and gluing two graphs in a vertex.[2] The operations not necessarily preserving word-representability are taking the complement, taking the line graph, edge contraction,[2] gluing two graphs in a clique of size 2 or more,[12] tensor product, lexicographic product and strong product.[13] Edge-deletion, edge-addition and edge-lifting with respect to word-representability (equivalently, semi-transitive orientability) are studied in.[13] Graphs with high representation number While each non-complete word-representable graph G is 2(n − κ(G))-representable, where κ(G) is the size of a maximal clique in G,[6] the highest known representation number is floor(n/2) given by crown graphs with an all-adjacent vertex.[6] Interestingly, such graphs are not the only graphs that require long representations.[14] Crown graphs themselves are shown to require long (possibly longest) representations among bipartite graphs.[15] Computational complexity Known computational complexities for problems on word-representable graphs can be summarised as follows:[2][7] PROBLEM COMPLEXITY deciding word-representability NP-complete Dominating Set NP-hard Clique Covering NP-hard Maximum Independent Set NP-hard Maximum Clique in P approximating the graph representation number within a factor n1−ε for any ε > 0 NP-hard Representation of planar graphs Triangle-free planar graphs are word-representable.[6] A K4-free near-triangulation is 3-colourable if and only if it is word-representable;[16] this result generalises studies in.[17][18] Word-representability of face subdivisions of triangular grid graphs is studied in [19] and word-representability of triangulations of grid-covered cylinder graphs is studied in.[20] Representation of split graphs Word-representation of split graphs is studied in.[21][12] In particular,[21] offers a characterisation in terms of forbidden induced subgraphs of word-representable split graphs in which vertices in the independent set are of degree at most 2, or the size of the clique is 4, while a computational characterisation of word-representable split graphs with the clique of size 5 is given in.[12] Also, necessary and sufficient conditions for an orientation of a split graph to be semi-transitive are given in,[21] while in [12] threshold graphs are shown to be word-representable and the split graphs are used to show that gluing two word-representable graphs in any clique of size at least 2 may, or may not result in a word-representable graph, which solved a long-standing open problem. Graphs representable by pattern avoiding words A graph is p-representable if it can be represented by a word avoiding a pattern p. For example, 132-representable graphs are those that can be represented by words w1w2...wn where there are no 1 ≤ a < b < c ≤ n such that wa < wc < wb. In [22] it is shown that any 132-representable graph is necessarily a circle graph, and any tree and any cycle graph, as well as any graph on at most 5 vertices, are 132-representable. It was shown in [23] that not all circle graphs are 132-representable, and that 123-representable graphs are also a proper subclass of the class of circle graphs. Generalisations A number of generalisations [24][25][26] of the notion of a word-representable graph are based on the observation by Jeff Remmel that non-edges are defined by occurrences of the pattern 11 (two consecutive equal letters) in a word representing a graph, while edges are defined by avoidance of this pattern. For example, instead of the pattern 11, one can use the pattern 112, or the pattern, 1212, or any other binary pattern where the assumption that the alphabet is ordered can be made so that a letter in a word corresponding to 1 in the pattern is less than that corresponding to 2 in the pattern. Letting u be an ordered binary pattern we thus get the notion of a u-representable graph. So, word-representable graphs are just the class of 11-representable graphs. Intriguingly, any graph can be u-represented assuming u is of length at least 3.[27] Another way to generalise the notion of a word-representable graph, again suggested by Jeff Remmel, is to introduce the "degree of tolerance" k for occurrences of a pattern p defining edges/non-edges. That is, we can say that if there are up to k occurrence of p formed by letters a and b, then there is an edge between a and b. This gives the notion of a k-p-representable graph, and k-11-representable graphs are studied in.[28] Note that 0-11-representable graphs are precisely word-representable graphs. The key results in [28] are that any graph is 2-11-representable and that word-representable graphs are a proper subclass of 1-11-representable graphs. Whether or not any graph is 1-11-representable is a challenging open problem. For yet another type of relevant generalisation, Hans Zantema suggested the notion of a k-semi-transitive orientation refining the notion of a semi-transitive orientation.[14] The idea here is restricting ourselves to considering only directed paths of length not exceeding k while allowing violations of semi-transitivity on longer paths. Open problems Open problems on word-representable graphs can be found in,[2][7][8][9] and they include: • Characterise (non-)word-representable planar graphs. • Characterise word-representable near-triangulations containing the complete graph K4 (such a characterisation is known for K4-free planar graphs [16]). • Classify graphs with representation number 3. (See [29] for the state-of-the-art in this direction.) • Is the line graph of a non-word-representable graph always non-word-representable? • Are there any graphs on n vertices whose representation requires more than floor(n/2) copies of each letter? • Is it true that out of all bipartite graphs crown graphs require longest word-representants? (See [15] for relevant discussion.) • Characterise word-representable graphs in terms of (induced) forbidden subgraphs. • Which (hard) problems on graphs can be translated to words representing them and solved on words (efficiently)? Literature The list of publications to study representation of graphs by words contains, but is not limited to 1. Ö. Akgün, I.P. Gent, S. Kitaev, H. Zantema. Solving computational problems in the theory of word-representable graphs. Journal of Integer Sequences 22 (2019), Article 19.2.5. 2. P. Akrobotu, S. Kitaev, and Z. Masárová. On word-representability of polyomino triangulations. Siberian Adv. Math. 25 (2015), 1−10. 3. B. Broere. Word representable graphs, 2018. Master thesis at Radboud University, Nijmegen. 4. B. Broere and H. Zantema. "The k-cube is k-representable," J. Autom., Lang., and Combin. 24 (2019) 1, 3-12. 5. J. N. Chen and S. Kitaev. On the 12-representability of induced subgraphs of a grid graph, Discussiones Mathematicae Graph Theory, to appear 6. T. Z. Q. Chen, S. Kitaev, and A. Saito. Representing split graphs by words, arXiv:1909.09471 7. T. Z. Q. Chen, S. Kitaev, and B. Y. Sun. Word-representability of face subdivisions of triangular grid graphs, Graphs and Combin. 32(5) (2016), 1749−1761. 8. T. Z. Q. Chen, S. Kitaev, and B. Y. Sun. Word-representability of triangulations of grid-covered cylinder graphs, Discr. Appl. Math. 213 (2016), 60−70. 9. G.-S. Cheon, J. Kim, M. Kim, and S. Kitaev. Word-representability of Toeplitz graphs, Discr. Appl. Math., to appear. 10. G.-S. Cheon, J. Kim, M. Kim, and A. Pyatkin. On k-11-representable graphs. J. Combin. 10 (2019) 3, 491−513. 11. I. Choi, J. Kim, and M. Kim. On operations preserving semi-transitive orient ability of graphs, Journal of Combinatorial Optimization 37 (2019) 4, 1351−1366. 12. A. Collins, S. Kitaev, and V. Lozin. New results on word-representable graphs, Discr. Appl. Math. 216 (2017), 136−141. 13. A. Daigavane, M. Singh, B.K. George. 2-uniform words: cycle graphs, and an algorithm to verify specific word-representations of graphs. arXiv:1806.04673 (2018). 14. M. Gaetz and C. Ji. Enumeration and extensions of word-representable graphs. Lecture Notes in Computer Science 11682 (2019) 180−192. In R. Mercas, D. Reidenbach (Eds) Combinatorics on Words. WORDS 2019. 15. M. Gaetz and C. Ji. Enumeration and Extensions of Word-representants, arXiv:1909.00019. 16. M. Gaetz and C. Ji. Enumeration and Extensions of Word-representants, Combinatorics on words, 180-192, Lecture Notes in Comput. Sci., 11682, Springer, Cham, 2019. 17. A. Gao, S. Kitaev, and P. Zhang. On 132-representable graphs. Australasian J. Combin. 69 (2017), 105−118. 18. M. Glen. Colourability and word-representability of near-triangulations, Pure and Applied Mathematics, to appear, 2019. 19. M. Glen. On word-representability of polyomino triangulations & crown graphs, 2019. PhD thesis, University of Strathclyde. 20. M. Glen and S. Kitaev. Word-Representability of Triangulations of Rectangular Polyomino with a Single Domino Tile, J. Combin.Math. Combin. Comput. 100, 131−144, 2017. 21. M. Glen, S. Kitaev, and A. Pyatkin. On the representation number of a crown graph, Discr. Appl. Math. 244, 2018, 89−93. 22. M.M. Halldórsson, S. Kitaev, A. Pyatkin On representable graphs, semi-transitive orientations, and the representation numbers, arXiv:0810.0310 (2008). 23. M.M. Halldórsson, S. Kitaev, A. Pyatkin (2010) Graphs capturing alternations in words. In: Y. Gao, H. Lu, S. Seki, S. Yu (eds), Developments in Language Theory. DLT 2010. Lecture Notes Comp. Sci. 6224, Springer, 436−437. 24. M.M. Halldórsson, S. Kitaev, A. Pyatkin (2011) Alternation graphs. In: P. Kolman, J. Kratochvíl (eds), Graph-Theoretic Concepts in Computer Science. WG 2011. Lecture Notes Comp. Sci. 6986, Springer, 191−202. 25. M. Halldórsson, S. Kitaev and A. Pyatkin. Semi-transitive orientations and word-representable graphs, Discr. Appl. Math. 201 (2016), 164−171. 26. M. Jones, S. Kitaev, A. Pyatkin, and J. Remmel. Representing Graphs via Pattern Avoiding Words, Electron. J. Combin. 22 (2), Res. Pap. P2.53, 1−20 (2015). 27. S. Kitaev. On graphs with representation number 3, J. Autom., Lang. and Combin. 18 (2013), 97−112. 28. S. Kitaev. A comprehensive introduction to the theory of word-representable graphs. In: É. Charlier, J. Leroy, M. Rigo (eds), Developments in Language Theory. DLT 2017. Lecture Notes Comp. Sci. 10396, Springer, 36−67. 29. S. Kitaev. Existence of u-representation of graphs, Journal of Graph Theory 85 (2017) 3, 661−668. 30. S. Kitaev, Y. Long, J. Ma, H. Wu. Word-representability of split graphs, arXiv:1709.09725 (2017). 31. S. Kitaev and V. Lozin. Words and Graphs, Springer, 2015. ISBN 978-3-319-25859-1. 32. S. Kitaev and A. Pyatkin. On representable graphs, J. Autom., Lang. and Combin. 13 (2008), 45−54. 33. S. Kitaev and A. Pyatkin. Word-representable graphs: a Survey, Journal of Applied and Industrial Mathematics 12(2) (2018) 278−296. 34. S. Kitaev and A. Pyatkin. On semi-transitive orientability of triangle-free graphs, arXiv:2003.06204v1. 35. S. Kitaev and A. Saito. On semi-transitive orientability of Kneser graphs and their complements, Discrete Math., to appear. 36. S. Kitaev, P. Salimov, C. Severs, and H. Úlfarsson (2011) On the representability of line graphs. In: G. Mauri, A. Leporati (eds), Developments in Language Theory. DLT 2011. Lecture Notes Comp. Sci. 6795, Springer, 478−479. 37. S. Kitaev and S. Seif. Word problem of the Perkins semigroup via directed acyclic graphs, Order 25 (2008), 177−194. 38. E. Leloup. Graphes représentables par mot. Master Thesis, University of Liège, 2019 39. Mandelshtam. On graphs representable by pattern-avoiding words, Discussiones Mathematicae Graph Theory 39 (2019) 375−389. 40. С. В. Китаев, А. В. Пяткин. Графы, представимые в виде слов. Обзор результатов, Дискретн. анализ и исслед. опер., 2018, том 25,номер 2, 19−53. Software Software to study word-representable graphs can be found here: 1. M. Glen. Software to deal with word-representable graphs, 2017. Available at https://personal.cis.strath.ac.uk/sergey.kitaev/word-representable-graphs.html. 2. H. Zantema. Software REPRNR to compute the representation number of a graph, 2018. Available at https://www.win.tue.nl/~hzantema/reprnr.html. References 1. S. Kitaev and S. Seif. Word problem of the Perkins semigroup via directed acyclic graphs, Order 25 (2008), 177−194. 2. S. Kitaev and V. Lozin. Words and Graphs, Springer, 2015. ISBN 978-3-319-25859-1 3. S. Kitaev and A. Pyatkin. On representable graphs, J. Autom., Lang. and Combin. 13 (2008), 45−54. 4. M.M. Halldórsson, S. Kitaev, A. Pyatkin (2010) Graphs capturing alternations in words. In: Y. Gao, H. Lu, S. Seki, S. Yu (eds), Developments in Language Theory. DLT 2010. Lecture Notes Comp. Sci. 6224, Springer, 436−437. 5. M.M. Halldórsson, S. Kitaev, A. Pyatkin (2011) Alternation graphs. In: P. Kolman, J. Kratochvíl (eds), Graph-Theoretic Concepts in Computer Science. WG 2011. Lecture Notes Comp. Sci. 6986, Springer, 191−202. 6. M. Halldórsson, S. Kitaev and A. Pyatkin. Semi-transitive orientations and word-representable graphs, Discr. Appl. Math. 201 (2016), 164−171. 7. S. Kitaev. A comprehensive introduction to the theory of word-representable graphs. In: É. Charlier, J. Leroy, M. Rigo (eds), Developments in Language Theory. DLT 2017. Lecture Notes Comp. Sci. 10396, Springer, 36−67. 8. S. Kitaev and A. Pyatkin. Word-representable graphs: a Survey, Journal of Applied and Industrial Mathematics 12(2) (2018) 278−296. 9. С. В. Китаев, А. В. Пяткин. Графы, представимые в виде слов. Обзор результатов, Дискретн. анализ и исслед. опер., 2018, том 25,номер 2, 19−53 10. A. Collins, S. Kitaev, and V. Lozin. New results on word-representable graphs, Discr. Appl. Math. 216 (2017), 136–141. 11. Ö. Akgün, I.P. Gent, S. Kitaev, H. Zantema. Solving computational problems in the theory of word-representable graphs. Journal of Integer Sequences 22 (2019), Article 19.2.5. 12. T. Z. Q. Chen, S. Kitaev, and A. Saito. Representing split graphs by words, arXiv:1909.09471 13. I. Choi, J. Kim, and M. Kim. On operations preserving semi-transitive orient ability of graphs, Journal of Combinatorial Optimization 37 (2019) 4, 1351−1366. 14. Ö. Akgün, I.P. Gent, S. Kitaev, H. Zantema. Solving computational problems in the theory of word-representable graphs. Journal of Integer Sequences 22 (2019), Article 19.2.5. 15. M. Glen, S. Kitaev, and A. Pyatkin. On the representation number of a crown graph, Discr. Appl. Math. 244, 2018, 89–93. 16. M. Glen. Colourability and word-representability of near-triangulations, Pure and Applied Mathematics, to appear, 2019. 17. P. Akrobotu, S. Kitaev, and Z. Masárová. On word-representability of polyomino triangulations. Siberian Adv. Math. 25 (2015), 1−10. 18. M. Glen and S. Kitaev. Word-Representability of Triangulations of Rectangular Polyomino with a Single Domino Tile, J. Combin.Math. Combin. Comput. 100, 131−144, 2017. 19. T. Z. Q. Chen, S. Kitaev, and B. Y. Sun. Word-representability of face subdivisions of triangular grid graphs, Graphs and Combin. 32(5) (2016), 1749−1761. 20. T. Z. Q. Chen, S. Kitaev, and B. Y. Sun. Word-representability of triangulations of grid-covered cylinder graphs, Discr. Appl. Math. 213 (2016), 60−70. 21. S. Kitaev, Y. Long, J. Ma, H. Wu. Word-representability of split graphs, arXiv:1709.09725 (2017). 22. A. Gao, S. Kitaev, and P. Zhang. On 132-representable graphs. Australasian J. Combin. 69 (2017), 105−118. 23. Mandelshtam. On graphs representable by pattern-avoiding words, Discussiones Mathematicae Graph Theory 39 (2019) 375−389. 24. M. Jones, S. Kitaev, A. Pyatkin, and J. Remmel. Representing Graphs via Pattern Avoiding Words, Electron. J. Combin. 22 (2), Res. Pap. P2.53, 1−20 (2015). 25. M. Gaetz and C. Ji. Enumeration and extensions of word-representable graphs. Lecture Notes in Computer Science 11682 (2019) 180−192. In R. Mercas, D. Reidenbach (Eds) Combinatorics on Words. WORDS 2019. 26. M. Gaetz and C. Ji. Enumeration and Extensions of Word-representants, arXiv:1909.00019. 27. S. Kitaev. Existence of u-representation of graphs, Journal of Graph Theory 85 (2017) 3, 661−668. 28. G.-S. Cheon, J. Kim, M. Kim, and A. Pyatkin. On k-11-representable graphs. J. Combin. 10 (2019) 3, 491−513. 29. S. Kitaev. On graphs with representation number 3, J. Autom., Lang. and Combin. 18 (2013), 97−112.

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.