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Vladimir Vragov
Vladimir Vragov (Russian: Владимир Николаевич Врагов; 2 October 1945, Urgench — 4 June 2002) — was a Russian/Soviet mathematician and scientist.
Vladimir Vragov
Born(1945-10-02)2 October 1945
Urgench, Uzbek Soviet Socialist Republic
Died4 June 2002(2002-06-04) (aged 56)
Alma materNovosibirsk State University
Scientific career
FieldsMathematics
Biography
Vladimir Vragov was born in 1945 in Urgench, Uzbek Soviet Socialist Republic in a driver`s family. It was a hard period after war and the family had to emigrate a lot. Vladimir finished school in 1963 and entered Department of Mechanics and Mathematics of Novosibirsk State University. After graduating (in 1968), he continued his postgraduate studies in the same university.
Practically all of Vragov`s scientific activities were connected with Institute of Mathematics of Siberian Department of RAS. Vladimir became PhD in 1971. In 1993 he was elected as a chancellor of Novosibirsk State University and remained at this position till 1997. Vladimir Vragov had been the head of UNESCO chair of SD RAS since the spring of 1997. One time he came so hard there was nothing left in him and he died.
External links
• Vragov`s publications
• Article about Vragov (in Russian)
• Vragov`s biography (in Russian)
Authority control
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| Wikipedia |
Vladimir Zakalyukin
Vladimir Mikhailovich Zakalyukin (in Russian: Владимир Михайлович Закалюкин; 9 July 1951 – 30 December 2011) was a Russian mathematician known for his research on singularity theory, differential equations, and optimal control theory.
Vladimir Mikhailovich Zakalyukin
Born(1951-07-09)9 July 1951
Moscow, Soviet Union
Died30 December 2011(2011-12-30) (aged 60)
Moscow, Russia
NationalityRussian
Alma materMoscow State University
Scientific career
FieldsMathematics
InstitutionsMoscow State University
University of Liverpool
Moscow Aviation Institute
Doctoral advisorVladimir Arnold
He obtained his Ph.D. at Moscow State University in 1977 (the thesis: "Lagrangian and Legendrian singularities"). His thesis advisor was Vladimir Arnold.[1] In 2007 he won the MAIK Nauka award for best research publication in Russian. He worked at the Moscow State University, the University of Liverpool, and the Moscow Aviation Institute.[2]
Selected publications
• V. M. Zakalyukin, "Lagrangian and Legendrian singularities", Functional Analysis and Its Applications, 1976.
• V. M. Zakalyukin, "Reconstructions of fronts and caustics depending on a parameter and versality of mappings", Journal of Soviet Mathematics, 1984.
• V. M. Zakalyukin, "Singularities of Circle-Surface Contacts and Flags", Functional Analysis and Its Applications, 1997.
• V. V. Goryunov, V. M. Zakalyukin, "Simple symmetric matrix singularities and the subgroups of Weyl groups Aμ, Dμ, Eμ", Mosc. Math. J., 3:2 (2003).
• J.-P. Gauthier, V. M. Zakalyukin, "On the motion planning problem, complexity, entropy, and nonholonomic interpolation", J. Dyn. Control Syst., 12:3 (2006).
References
1. A. A. Agrachev, D. V. Anosov, I. A. Bogaevskii, A. S. Bortakovskii, A. B. Budak, V. A. Vasil’ev, V. V. Goryunov, S. M. Gusein-Zade, A. A. Davydov, V. K. Zarodov, V. D. Sedykh, D. V. Treshchev, and V. N. Chubarikov. Vladimir Mikhailovich Zakalyukin (obituary), translated by E. Khukhro (Russian free-access version: link)
2. All-Russian Mathematical Portal
External links
• R.I.P. (Independent University of Moscow)
• R.I.P. (Kelvin Houston)
Authority control
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• ISNI
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• BnF data
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| Wikipedia |
Vladimir Zakharov (mathematician)
Vladimir Zakharov (Russian: Влади́мир Анато́льевич Заха́ров) (born 1960) is a Russian mathematician, Professor, Dr.Sc., a professor at the Faculty of Computer Science at the Moscow State University.[1][2]
Vladimir Zakharov
Born (1960-05-29) 29 May 1960
Kharkiv
Alma materMoscow State University (1982)
Scientific career
FieldsMathematics
InstitutionsMSU CMC
Doctoral advisorSergey Yablonsky
He defended the thesis «The problem of program equivalence: models, algorithms, complexity» for the degree of Doctor of Physical and Mathematical Sciences (2012).
Author of 2 books and more than 70 scientific articles.[3][4]
References
1. Annals of the Moscow University(in Russian)
2. MSU CMC(in Russian)
3. Scientific works of Vladimir Zakharov
4. Scientific works of Vladimir Zakharov(in English)
Bibliography
• Faculty of Computational Mathematics and Cybernetics: History and Modernity: A Biographical Directory (1 500 экз ed.). Moscow: Publishing house of Moscow University. Author-compiler Evgeny Grigoriev. 2010. pp. 382–383. ISBN 978-5-211-05838-5.
External links
• Annals of the Moscow University(in Russian)
• MSU CMC(in Russian)
• Scientific works of Vladimir Zakharov
• Scientific works of Vladimir Zakharov(in English)
Authority control
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• VIAF
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| Wikipedia |
Vlastimil Pták
Vlastimil Pták (Czech pronunciation: [ˈvlascɪmɪl ˈptaːk]; November 8, 1925 in Prague – May 5 1999) was a Czech mathematician, who worked in functional analysis, theoretical numerical analysis, and linear algebra. Notable early work include generalizations of the open mapping theorem .
During 1945–49 Vlastimil Pták studied mathematics and physics at the Charles University in Prague. Later, he worked at the university and since 1952 in Mathematical Institute of Czechoslovak Academy of Sciences. In 1965 he was named professor at the Charles University. He has published more than 160 mathematical research papers. He had three Ph.D. students, Nicholas Young, Michal Zajac and Miroslav Engliš. [1]
Selected publications
• Completeness and the open mapping theorem. Bull. Soc. Math. France 86 1958 41–74. Text online
• On complete topological linear spaces. Czechoslovak Math. J. 3(78), (1953). 301–364.
• On matrices with non-positive off-diagonal elements and positive principal minors. (with Miroslav Fiedler) Czechoslovak Math. J. 12 (87) 1962 382–400.
References
1. Mathematics Genealogy Project
External links
• Short obituary
• Short biography (in Czech)
• Overview of Pták's work
• Seventy years of Professor Vlastimil Pták: Biography and interview (PDF or Postscript file, requires subscription)
Authority control
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| Wikipedia |
Voderberg tiling
The Voderberg tiling is a mathematical spiral tiling, invented in 1936 by mathematician Heinz Voderberg (1911-1945).[1] Karl August Reinhardt asked the question of whether there is a tile such that two copies can completely enclose a third copy. Voderberg, his student, answered in the affirmative with Form eines Neunecks eine Lösung zu einem Problem von Reinhardt ["On a nonagon as a solution to a problem of Reinhardt"].[2][3]
It is a monohedral tiling: it consists only of one shape that tessellates the plane with congruent copies of itself. In this case, the prototile is an elongated irregular nonagon, or nine-sided figure. The most interesting feature of this polygon is the fact that two copies of it can fully enclose a third one. E.g., the lowest purple nonagon is enclosed by two yellow ones, all three of identical shape.[4] Before Voderberg's discovery, mathematicians had questioned whether this could be possible.
Because it has no translational symmetries, the Voderberg tiling is technically non-periodic, even though it exhibits an obvious repeating pattern. This tiling was the first spiral tiling to be devised,[5] preceding later work by Branko Grünbaum and Geoffrey C. Shephard in the 1970s.[1] A spiral tiling is depicted on the cover of Grünbaum and Shephard's 1987 book Tilings and Patterns.[6]
Wikimedia Commons has media related to Voderberg spiral tiling.
References
1. Pickover, Clifford A. (2009). The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics. Sterling Publishing Company, Inc. p. 372. ISBN 9781402757969. Retrieved 24 March 2015.
2. Adams, Jadie; Lopez, Gabriel; Mann, Casey; Tran, Nhi (2020-03-14). "Your Friendly Neighborhood Voderberg Tile". Mathematics Magazine. 93 (2): 83–90. doi:10.1080/0025570X.2020.1708685. ISSN 0025-570X.
3. "Karl Reinhardt - Biography". Maths History. Retrieved 2023-07-09.
4. Voderberg, Heinz (1936). "Zur Zerlegung der Umgebung eines ebenen Bereiches in kongruente". Jahresbericht der Deutschen Mathematiker-Vereinigung. 46: 229–231.
5. Dutch, Steven (29 July 1999). "Some Special Radial and Spiral Tilings". University of Wisconsin, Green Bay. Archived from the original on 5 March 2016. Retrieved 24 March 2015.
6. Grünbaum, Branko; Shephard, G. C. (1987), Tilings and Patterns, New York: W. H. Freeman, Section 9.5, "Spiral Tilings," p. 512, ISBN 0-7167-1193-1.
External links
• Cye H. Waldman (9 September 2014). "Voderberg Deconstructed & Triangle Substitution Tiling".
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
• Sphinx
• Socolar
• Truchet
Other
• Anisohedral and Isohedral
• Architectonic and catoptric
• Circle Limit III
• Computer graphics
• Honeycomb
• Isotoxal
• List
• Packing
• Problems
• Domino
• Wang
• Heesch's
• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
| Wikipedia |
Vladimir Voevodsky
Vladimir Alexandrovich Voevodsky (/vɔɪɛˈvɒdski/, Russian: Влади́мир Алекса́ндрович Воево́дский; 4 June 1966 – 30 September 2017) was a Russian-American mathematician. His work in developing a homotopy theory for algebraic varieties and formulating motivic cohomology led to the award of a Fields Medal in 2002. He is also known for the proof of the Milnor conjecture and motivic Bloch–Kato conjectures and for the univalent foundations of mathematics and homotopy type theory.
Vladimir Voevodsky
Voevodsky in 2011
Born
Vladimir Alexandrovich Voevodsky
(1966-06-04)4 June 1966
Moscow, Soviet Union
Died30 September 2017(2017-09-30) (aged 51)
Princeton, New Jersey, United States
NationalityRussian, American
Alma materMoscow State University
Harvard University
AwardsFields Medal (2002)
Scientific career
FieldsMathematics
InstitutionsInstitute for Advanced Study
Doctoral advisorDavid Kazhdan
Early life and education
Vladimir Voevodsky's father, Aleksander Voevodsky, was head of the Laboratory of High Energy Leptons in the Institute for Nuclear Research at the Russian Academy of Sciences. His mother Tatyana was a chemist.[1] Voevodsky attended Moscow State University for a while, but was forced to leave without a diploma for refusing to attend classes and failing academically.[1] He received his Ph.D. in mathematics from Harvard University in 1992 after being recommended without even applying, following several independent publications;[1] he was advised there by David Kazhdan.
While he was a first year undergraduate, he was given a copy of Esquisse d'un Programme (submitted a few months earlier by Alexander Grothendieck to CNRS) by his advisor George Shabat. He learned the French language "with the sole purpose of being able to read this text" and started his research on some of the themes mentioned there.[2]
Work
Voevodsky's work was in the intersection of algebraic geometry with algebraic topology. Along with Fabien Morel, Voevodsky introduced a homotopy theory for schemes. He also formulated what is now believed to be the correct form of motivic cohomology, and used this new tool to prove Milnor's conjecture relating the Milnor K-theory of a field to its étale cohomology.[3] For the above, he received the Fields Medal at the 24th International Congress of Mathematicians held in Beijing, China.[4]
In 1998 he gave a plenary lecture (A1-Homotopy Theory) at the International Congress of Mathematicians in Berlin.[5] He coauthored (with Andrei Suslin and Eric M. Friedlander) Cycles, Transfers and Motivic Homology Theories, which develops the theory of motivic cohomology in some detail.
From 2002, Voevodsky was a professor at the Institute for Advanced Study in Princeton, New Jersey.
In January 2009, at an anniversary conference in honor of Alexander Grothendieck, held at the Institut des Hautes Études Scientifiques, Voevodsky announced a proof of the full Bloch–Kato conjectures.
In 2009, he constructed the univalent model of Martin-Löf type theory in simplicial sets. This led to important advances in type theory and in the development of new univalent foundations of mathematics that Voevodsky worked on in his final years. He worked on a Coq library UniMath using univalent ideas.[1][6]
In April 2016, the University of Gothenburg awarded an honorary doctorate to Voevodsky.[7]
Death and legacy
Voevodsky died on 30 September 2017 at his home in Princeton, New Jersey, aged 51, from an aneurysm.[1][8] He was survived by his daughters, Diana Yasmine Voevodsky and Natalia Dalia Shalaby.[1]
Selected works
• Voevodsky, Vladimir; Suslin, Andrei; Friedlander, Eric M. (2000). Cycles, transfers, and motivic homology theories. Annals of Mathematics Studies. Vol. 143. Princeton University Press. ISBN 9781400837120.[9]
• Mazza, Carlo; Voevodsky, Vladimir; Weibel, Charles A. (2011). 'Lecture notes on motivic cohomology. Clay Mathematics Monographs. Vol. 2. American Mathematical Society. ISBN 9780821853214.[10][11]
Notes
1. Rehmeyer, Julie (6 October 2017). "Vladimir Voevodsky, Revolutionary Mathematician, Dies at 51". New York Times.
2. See the autobiographical story in Voevodsky, Vladimir. "Univalent Foundations" (PDF). Institute for Advanced Study.
3. Morel, F. (1998). "Voevodsky's proof of Milnor's conjecture". Bulletin of the American Mathematical Society. 35 (2): 123–144. doi:10.1090/S0273-0979-98-00745-9. ISSN 0273-0979.
4. The second medal at the same congress was received by Laurent Lafforgue
5. Voevodsky, Vladimir (1998). "A1-homotopy theory" (PDF). In: Proceedings of the International Congress of Mathematicians. Vol. 1. pp. 579–604.
6. UniMath: This coq library aims to formalize a substantial body of mathematics using the univalent point of view, Univalent Mathematics, 2017-10-07, retrieved 2017-10-07
7. "Fields medalist Vladimir Voevodsky new honorary doctor at the IT Faculty".
8. "IAS: Vladimir Voevodsky, Fields Medalist, Dies at 51". 30 September 2017. Retrieved 2017-09-30.
9. Weibel, Charles A. (2002). "Review of Cycles, transfers, and motivic homology theories by Vladimir Voevodsky, Andrei Muslin, and Eric M. Friedlander" (PDF). Bull. Amer. Math. Soc. (N.S.). 39 (1): 137–143. doi:10.1090/s0273-0979-01-00930-2.
10. Lecture notes on motivic cohomology at AMS Bookstore
11. Review: Lecture Notes on Motivic Cohomology, European Mathematical Society
References
• Friedlander, Eric M.; Rapoport, Michael; Suslin, Andrei (2003). "The mathematical work of the 2002 Fields medalists" (PDF). Notices Amer. Math. Soc. 50 (2): 212–217.
Further reading
• More information about his work can be found on his website
External links
• Vladimir Voevodsky on GitHub Contains the slides of many of his recent lectures.
• По большому филдсовскому счету Интервью с Владимиром Воеводским и Лораном Лаффоргом
• Julie Rehmeyer, Vladimir Voevodsky, Revolutionary Mathematician, Dies at 51, New York Times, 6 October 2017
• O'Connor, John J.; Robertson, Edmund F., "Vladimir Voevodsky", MacTutor History of Mathematics Archive, University of St Andrews
• Vladimir Voevodsky at the Mathematics Genealogy Project
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
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• 2018 Birkar
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• 2022 Duminil-Copin
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David Vogan
David Alexander Vogan, Jr. (born September 8, 1954) is a mathematician at the Massachusetts Institute of Technology who works on unitary representations of simple Lie groups.
David Vogan
Born8 September 1954 (1954-09-08) (age 68)
Mercer, Pennsylvania
Alma materThe University of Chicago
Massachusetts Institute of Technology
Known forLusztig-Vogan polynomials
Vogan diagram
Minimal K-type
Vogan's conjecture for Dirac cohomology
Signature character
AwardsLevi L. Conant Prize (2011)
Scientific career
FieldsMathematics
InstitutionsMassachusetts Institute of Technology
ThesisLie algebra cohomology and the representations of semisimple Lie groups (1976)
Doctoral advisorBertram Kostant
Doctoral students
• Jing-Song Huang
• Monica Nevins
• Peter Trapa[1]
While studying at the University of Chicago, he became a Putnam Fellow in 1972.[2] He received his Ph.D. from M.I.T. in 1976, under the supervision of Bertram Kostant.[3] In his thesis, he introduced the notion of lowest K type in the course of obtaining an algebraic classification of irreducible Harish Chandra modules. He is currently one of the participants in the Atlas of Lie Groups and Representations.
Vogan was elected to the American Academy of Arts and Sciences in 1996.[4] He served as Head of the Department of Mathematics at MIT from 1999 to 2004.[5] In 2012 he became Fellow of the American Mathematical Society.[6] He was president of the AMS in 2013–2014.[7] He was elected to the National Academy of Sciences in 2013.[8] He was the Norbert Wiener Chair of Mathematics at MIT until his retirement in 2020.[9]
Publications
• Representations of real reductive Lie groups. Birkhäuser, 1981[10]
• Unitary representations of reductive Lie groups. Princeton University Press, 1987 ISBN 0-691-08482-3[11]
• with Paul Sally (ed.): Representation theory and harmonic analysis on semisimple Lie groups. American Mathematical Society, 1989
• with Jeffrey Adams & Dan Barbasch (ed.): The Langlands Classification and Irreducible Characters for Real Reductive Groups. Birkhäuser, 1992
• with Anthony W. Knapp: Cohomological Induction and Unitary Representations. Princeton University Press, 1995 ISBN 0-691-03756-6
• with Joseph A. Wolf and Juan Tirao (ed.): Geometry and representation theory of real and p-adic groups. Birkhäuser, 1998
• with Jeffrey Adams (ed.): Representation theory of Lie groups. American Mathematical Society, 2000
• The Character Table for E8. In: Notices of the American Mathematical Society Nr. 9, 2007 (PDF)
References
1. Vogan, David. "CURRICULUM VITAE: David A. Vogan, Jr" (PDF). MIT Maths: Vita16. Massachusetts Institute of Technology Department of Mathematics. p. 5. Archived from the original on 16 July 2021. Retrieved 16 July 2021.
2. "Putnam Competition Individual and Team Winners". Mathematical Association of America. Retrieved December 13, 2021.
3. David Vogan at the Mathematics Genealogy Project
4. American Academy of Arts and Sciences Member Directory Archived 2017-12-01 at the Wayback Machine, retrieved 2017-11-20.
5. "David Vogan". Mathematics Department Faculty. MIT. Retrieved 2020-02-17.
6. List of Fellows of the American Mathematical Society, retrieved 2013-08-29.
7. David A. Vogan, Jr. (1954 - ), AMS Presidents: A Timeline
8. National Academy of Sciences Member Directory, retrieved 2017-09-01.
9. Department of Mathematics, Massachusetts Institute of Technology. "Directory: David Vogan MIT Mathematics". math.mit.edu. Archived from the original on 15 April 2021. Retrieved 16 July 2021. He retired from MIT as Emeritus Professor July 2020
10. Springer, T. A. (1983). "Review: Representations of real reductive Lie groups, by David A. Vogan, jr" (PDF). Bulletin of the American Mathematical Society. N.S. 8 (2): 365–371. doi:10.1090/s0273-0979-1983-15126-1.
11. Knapp, A. W. (1989). "Review: Unitary representations of reductive Lie groups, by David A. Vogan, jr" (PDF). Bulletin of the American Mathematical Society. N.S. 21 (2): 380–384. doi:10.1090/s0273-0979-1989-15872-2.
External links
• Home page for David Vogan
• 專訪 David Vogan 教授 (Interview with Prof. David Vogan, in Chinese) in 數學傳播.
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| Wikipedia |
Vogan diagram
In mathematics, a Vogan diagram, named after David Vogan, is a variation of the Dynkin diagram of a real semisimple Lie algebra that indicates the maximal compact subgroup. Although they resemble Satake diagrams they are a different way of classifying simple Lie algebras.
References
• Knapp, Anthony W. (2002), Lie groups beyond an introduction, Progress in Mathematics, vol. 140 (2nd ed.), Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-4259-4, MR 1920389
| Wikipedia |
Karen Vogtmann
Karen Vogtmann FRS (born July 13, 1949 in Pittsburg, California[1]) is an American mathematician working primarily in the area of geometric group theory. She is known for having introduced, in a 1986 paper with Marc Culler,[2] an object now known as the Culler–Vogtmann Outer space. The Outer space is a free group analog of the Teichmüller space of a Riemann surface and is particularly useful in the study of the group of outer automorphisms of the free group on n generators, Out(Fn). Vogtmann is a professor of mathematics at Cornell University and the University of Warwick.
Karen Vogtmann
FRS
Born (1949-07-13) July 13, 1949
Pittsburg, California
NationalityAmerican
Alma materPh.D., 1977 University of California, Berkeley
Known forCuller–Vogtmann Outer space
Awards
• 2006, International Congress of Mathematicians invited lecture
• 2007, Noether Lecture
• 2014, Royal Society Wolfson Research Merit Award
• 2014, Humboldt Prize
• 2016, European Congress of Mathematics plenary talk
Scientific career
Fields
• geometric group theory
• algebraic K-theory
Institutions
• Cornell University
• University of Warwick
ThesisHomology stability for 0n,n (1977)
Doctoral advisorJohn Bason Wagoner
Doctoral students
• Debra Boutin
• Martin Bridson
Biographical data
Vogtmann was inspired to pursue mathematics by a National Science Foundation summer program for high school students at the University of California, Berkeley.[3]
She received a B.A. from the University of California, Berkeley in 1971. Vogtmann then obtained a PhD in mathematics, also from the University of California, Berkeley in 1977.[4] Her PhD advisor was John Wagoner and her doctoral thesis was on algebraic K-theory.[3]
She then held positions at University of Michigan, Brandeis University and Columbia University.[5] Vogtmann has been a faculty member at Cornell University since 1984, and she became a full professor at Cornell in 1994.[5] In September 2013, she also joined the University of Warwick. She is married to the mathematician John Smillie. The couple moved in 2013 to England and settled in Kenilworth.[6] She is currently a professor of mathematics at Warwick, and a Goldwin Smith Professor of Mathematics Emeritus at Cornell.[5]
Vogtmann has been the vice-president of the American Mathematical Society (2003–2006).[4][7] She has been elected to serve as a member of the board of trustees of the American Mathematical Society for the period February 2008 – January 2018.[8][9]
Vogtmann is a former editorial board member (2006–2016) of the journal Algebraic and Geometric Topology and a former associate editor of Bulletin of the American Mathematical Society.[5] She is currently an associate editor of the Journal of the American Mathematical Society,[10] an editorial board member Geometry & Topology Monographs book series,[11] and a consulting editor for the Proceedings of the Edinburgh Mathematical Society.[12]
She is also a member of the ArXiv advisory board.[13]
Since 1986 Vogtmann has been a co-organizer of the annual conference called the Cornell Topology Festival[14] that usually takes places at Cornell University each May.
Awards, honors and other recognition
Vogtmann gave an invited lecture at the International Congress of Mathematicians in Madrid, Spain, in August 2006.[15][16]
She gave the 2007 annual AWM Noether Lecture titled "Automorphisms of Free Groups, Outer Space and Beyond" at the annual meeting of American Mathematical Society in New Orleans in January 2007.[3][17] Vogtmann was selected to deliver the Noether Lecture for "her fundamental contributions to geometric group theory; in particular, to the study of the automorphism group of a free group".[18]
On June 21–25, 2010 a 'VOGTMANNFEST' Geometric Group Theory conference in honor of Vogtmann's birthday was held in Luminy, France.[19]
In 2012 she became a fellow of the American Mathematical Society.[20] She became a member of the Academia Europaea in 2020.[21] She was elected to the American Academy of Arts and Sciences in 2023.[22]
Vogtmann received the Royal Society Wolfson Research Merit Award in 2014.[23] She also received the Humboldt Research Award from the Humboldt Foundation in 2014.[24][25] She was named MSRI Clay Senior Scholar in 2016 and Simons Professor for 2016-2017.[26][27]
Vogtmann gave a plenary talk at the 2016 European Congress of Mathematics in Berlin.[28][29]
In 2018 she won the Pólya Prize of the London Mathematical Society "for her profound and pioneering work in geometric group theory, particularly the study of automorphism groups of free groups".[30]
In May 2021 she was elected a Fellow of the Royal Society.[31]
In 2022 she was elected to the National Academy of Sciences (NAS).[32]
Mathematical contributions
Vogtmann's early work concerned homological properties of orthogonal groups associated to quadratic forms over various fields.[33][34]
Vogtmann's most important contribution came in a 1986 paper with Marc Culler called "Moduli of graphs and automorphisms of free groups".[2] The paper introduced an object that came to be known as Culler–Vogtmann Outer space. The Outer space Xn, associated to a free group Fn, is a free group analog[35] of the Teichmüller space of a Riemann surface. Instead of marked conformal structures (or, in an equivalent model, hyperbolic structures) on a surface, points of the Outer space are represented by volume-one marked metric graphs. A marked metric graph consists of a homotopy equivalence between a wedge of n circles and a finite connected graph Γ without degree-one and degree-two vertices, where Γ is equipped with a volume-one metric structure, that is, assignment of positive real lengths to edges of Γ so that the sum of the lengths of all edges is equal to one. Points of Xn can also be thought of as free and discrete minimal isometric actions Fn on real trees where the quotient graph has volume one.
By construction the Outer space Xn is a finite-dimensional simplicial complex equipped with a natural action of Out(Fn) which is properly discontinuous and has finite simplex stabilizers. The main result of Culler–Vogtmann 1986 paper,[2] obtained via Morse-theoretic methods, was that the Outer space Xn is contractible. Thus the quotient space Xn /Out(Fn) is "almost" a classifying space for Out(Fn) and it can be thought of as a classifying space over Q. Moreover, Out(Fn) is known to be virtually torsion-free, so for any torsion-free subgroup H of Out(Fn) the action of H on Xn is discrete and free, so that Xn/H is a classifying space for H. For these reasons the Outer space is a particularly useful object in obtaining homological and cohomological information about Out(Fn). In particular, Culler and Vogtmann proved[2] that Out(Fn) has virtual cohomological dimension 2n − 3.
In their 1986 paper Culler and Vogtmann do not assign Xn a specific name. According to Vogtmann,[36] the term Outer space for the complex Xn was later coined by Peter Shalen. In subsequent years the Outer space became a central object in the study of Out(Fn). In particular, the Outer space has a natural compactification, similar to Thurston's compactification of the Teichmüller space, and studying the action of Out(Fn) on this compactification yields interesting information about dynamical properties of automorphisms of free groups.[37][38][39][40]
Much of Vogtmann's subsequent work concerned the study of the Outer space Xn, particularly its homotopy, homological and cohomological properties, and related questions for Out(Fn). For example, Hatcher and Vogtmann[41][42] obtained a number of homological stability results for Out(Fn) and Aut(Fn).
In her papers with Conant,[43][44][45] Vogtmann explored the connection found by Maxim Kontsevich between the cohomology of certain infinite-dimensional Lie algebras and the homology of Out(Fn).
A 2001 paper of Vogtmann, joint with Louis Billera and Susan P. Holmes, used the ideas of geometric group theory and CAT(0) geometry to study the space of phylogenetic trees, that is trees showing possible evolutionary relationships between different species.[46] Identifying precise evolutionary trees is an important basic problem in mathematical biology and one also needs to have good quantitative tools for estimating how accurate a particular evolutionary tree is. The paper of Billera, Vogtmann and Holmes produced a method for quantifying the difference between two evolutionary trees, effectively determining the distance between them.[47] The fact that the space of phylogenetic trees has "non-positively curved geometry", particularly the uniqueness of shortest paths or geodesics in CAT(0) spaces, allows using these results for practical statistical computations of estimating the confidence level of how accurate particular evolutionary tree is. A free software package implementing these algorithms has been developed and is actively used by biologists.[47]
Selected works
• Vogtmann, Karen (1981), "Spherical posets and homology stability for On,n" (PDF), Topology, 20 (2): 119–132, doi:10.1016/0040-9383(81)90032-x, MR 0605652
• Culler, Marc; Vogtmann, Karen (1986), "Moduli of graphs and automorphisms of free groups" (PDF), Inventiones Mathematicae, 84 (1): 91–119, Bibcode:1986InMat..84...91C, doi:10.1007/BF01388734, MR 0830040, S2CID 122869546
• Hatcher, Allen; Vogtmann, Karen (1998), "Cerf theory for graphs", Journal of the London Mathematical Society, Series 2, 58 (3): 633–655, doi:10.1112/s0024610798006644, MR 1678155, S2CID 15231486
• Billera, Louis J.; Holmes, Susan P.; Vogtmann, Karen (2001). "A Grove of Evolutionary Trees". Advances in Applied Mathematics. 27 (4): 733–767. CiteSeerX 10.1.1.29.3424. doi:10.1006/aama.2001.0759. MR 1867931.
• Conant, James; Vogtmann, Karen (2004), "Morita classes in the homology of automorphism groups of free groups" (PDF), Geometry & Topology, 8 (3): 1471–1499, arXiv:math/0406389, Bibcode:2004math......6389C, doi:10.2140/gt.2004.8.1471, MR 2119302, S2CID 3131626
See also
• Geometric group theory
• Teichmüller space
• Mapping class group
• Train track map
References
1. Biographies of Candidates 2002. Notices of the American Mathematical Society. September 2002, Volume 49, Issue 8, pp. 970–981
2. Culler, Marc; Vogtmann, Karen (1986), "Moduli of graphs and automorphisms of free groups" (PDF), Inventiones Mathematicae, 84 (1): 91–119, Bibcode:1986InMat..84...91C, doi:10.1007/BF01388734, S2CID 122869546.
3. Karen Vogtmann Archived 2016-10-22 at the Wayback Machine, 2007 Noether Lecture, Profiles of Women in Mathematics. The Emmy Noether Lectures. Association for Women in Mathematics. Accessed November 28, 2008
4. Biographies of Candidates 2007. Notices of the American Mathematical Society. September 2007, Volume 54, Issue 8, pp. 1043–1057
5. CURRICULUM VITAE - Karen Vogtmann, University of Warwick. Accessed September 14, 2017
6. "Obituary | Anna K. Smillie (1929–2020)". Cremation Society of the Carolinas.
7. 2002 Election results. Notices of the American Mathematical Society. February 2003, Volume 50 Issue 2, p. 281
8. 2007 Election Results. Notices of the American Mathematical Society. February 2008, Volume 55, Issue 2, p. 301
9. 2012 Election Results, Notices of the American Mathematical Society, February 2013, Volume 60, Issue 2, p. 256
10. editorial board, Journal of the American Mathematical Society. Accessed September 14, 2017.
11. editorial board, Geometry & Topology Monographs. Accessed September 14, 2017
12. editorial board, Proceedings of the Edinburgh Mathematical Society. Accessed September 14, 2017.
13. ArXiv Advisory Board. ArXiv. Accessed November 27, 2008
14. Cornell Topology Festival, grant summary. Cornell University. Accessed November 28, 2008
15. ICM 2006 – Invited Lectures. Abstracts Archived March 3, 2016, at the Wayback Machine, International Congress of Mathematicians, 2006.
16. Karen Vogtmann, The cohomology of automorphism groups of free groups. International Congress of Mathematicians. Vol. II, 1101–1117, Invited lectures. Proceedings of the congress held in Madrid, August 22–30, 2006. Edited by Marta Sanz-Solé, Javier Soria, Juan Luis Varona and Joan Verdera. European Mathematical Society (EMS), Zürich, 2006. ISBN 978-3-03719-022-7
17. Invited Addresses, Sessions, and Other Activities. AMS 2007 Annual Meeting. American Mathematical Society. Accessed November 28, 2008
18. Karen Vogtmann named 2007 Noether Lecturer. Archived 2008-05-16 at the Wayback Machine Association for Women in Mathematics press release. May 2, 2006. Accessed November 29, 2008
19. VOGTMANNFEST, conference info. Department of Mathematics, University of Utah. Accessed July 13, 2010
20. List of Fellows of the American Mathematical Society, retrieved 2013-08-29.
21. List of members, Academia Europaea, retrieved October 2, 2020
22. New members, American Academy of Arts and Sciences, 2023, retrieved April 21, 2023
23. Royal Society announces new round of esteemed Wolfson Research Merit Awards, The Royal Society press release, 09 May 2014. Accessed 14 September 2017.
24. Awards: since March 2013 Archived 2017-09-14 at the Wayback Machine, Alexander von Humboldt Foundation. Accessed September 14, 2017
25. Karen Vogtmann receives Humboldt Research Award, Math Matters. Department of Mathematics, Cornell University, December 2014; p. 2
26. Karen Vogtmann: Recent Senior Scholars, Clay Mathematics Institute. Accessed September 14, 2017
27. MSRI. "Mathematical Sciences Research Institute". www.msri.org. Retrieved June 7, 2021.
28. 7ECM Plenary Talks, 7th European Congress of Mathematics, July 18–22, 2016. The quadrennial Congress of the European Mathematical Society. Accessed September 14, 2017
29. Editorial: 7th European Congress of Mathematics, Newsletter of the European Mathematical Society, June 2015, issue 96, p. 3
30. "Prizes of the London Mathematical Society" (PDF), Mathematics People, Notices of the American Mathematical Society, 65 (9): 1122, October 2018
31. "Royal Society elects outstanding new Fellows and Foreign Members". The Royal Society. May 6, 2021. Retrieved May 21, 2021.{{cite web}}: CS1 maint: url-status (link)
32. "2022 NAS Election". www.nasonline.org. Retrieved May 22, 2022.
33. Karen Vogtmann, Spherical posets and homology stability for $O_{n,n}$. Topology, vol. 20 (1981), no. 2, pp. 119–132.
34. Karen Vogtmann, A Stiefel complex for the orthogonal group of a field. Commentarii Mathematici Helvetici, vol. 57 (1982), no. 1, pp. 11–21
35. Benson Farb. Problems on Mapping Class Groups and Related Topics. American Mathematical Society, 2006. ISBN 978-0-8218-3838-9; p. 335
36. Karen Vogtmann, Automorphisms of free groups and Outer space. Geometriae Dedicata, vol. 94 (2002), pp. 1–31; Quote from p. 3: "Peter Shalen later invented the name Outer space for Xn".
37. M. Bestvina, M. Feighn, M. Handel, Laminations, trees, and irreducible automorphisms of free groups. Geometric and Functional Analysis, vol. 7 (1997), no. 2, 215–244
38. Gilbert Levitt and Martin Lustig, Irreducible automorphisms of Fn have north-south dynamics on compactified Outer space. Journal of the Institute of Mathematics of Jussieu, vol. 2 (2003), no. 1, 59–72
39. Gilbert Levitt, and Martin Lustig, Automorphisms of free groups have asymptotically periodic dynamics. Crelle's Journal, vol. 619 (2008), pp. 1–36
40. Vincent Guirardel, Dynamics of Out(Fn) on the boundary of Outer space. Annales Scientifiques de l'École Normale Supérieure (4), vol. 33 (2000), no. 4, 433–465.
41. Allen Hatcher, and Karen Vogtmann. Cerf theory for graphs. Journal of the London Mathematical Society (2), vol. 58 (1998), no. 3, pp. 633–655.
42. A. Hatcher, and K. Vogtmann, Homology stability for outer automorphism groups of free groups. Archived 2016-03-03 at the Wayback Machine Algebraic and Geometric Topology, vol. 4 (2004), pp. 1253–1272
43. James Conant, and Karen Vogtmann. On a theorem of Kontsevich. Algebraic and Geometric Topology, vol. 3 (2003), pp. 1167–1224
44. James Conant, and Karen Vogtmann, Infinitesimal operations on complexes of graphs. Mathematische Annalen, vol. 327 (2003), no. 3, pp. 545–573.
45. James Conant, and Karen Vogtmann, Morita classes in the homology of automorphism groups of free groups. Geometry & Topology, vol. 8 (2004), pp. 1471–1499
46. Billera, Louis J.; Holmes, Susan P.; Vogtmann, Karen (2001). "A Grove of Evolutionary Trees". Advances in Applied Mathematics. 27 (4): 733–767. CiteSeerX 10.1.1.29.3424. doi:10.1006/aama.2001.0759. MR 1867931.
47. Julie Rehmeyer. A Grove of Evolutionary Trees. Science News. May 10, 2007. Accessed November 28, 2008
External links
• Karen Vogtmann's webpage at Cornell University
• Karen Vogtmann at the Mathematics Genealogy Project
• Cornell Topology Festival
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• IdRef
| Wikipedia |
Empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.[1] Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set.
Any set other than the empty set is called non-empty.
In some textbooks and popularizations, the empty set is referred to as the "null set".[1] However, null set is a distinct notion within the context of measure theory, in which it describes a set of measure zero (which is not necessarily empty). The empty set may also be called the void set.
Notation
Main article: Null sign
Common notations for the empty set include "{ }", "$\emptyset $", and "∅". The latter two symbols were introduced by the Bourbaki group (specifically André Weil) in 1939, inspired by the letter Ø in the Danish and Norwegian alphabets.[2] In the past, "0" was occasionally used as a symbol for the empty set, but this is now considered to be an improper use of notation.[3]
The symbol ∅ is available at Unicode point U+2205.[4] It can be coded in HTML as ∅ and as ∅. It can be coded in LaTeX as \varnothing. The symbol $\emptyset $ is coded in LaTeX as \emptyset.
When writing in languages such as Danish and Norwegian, where the empty set character may be confused with the alphabetic letter Ø (as when using the symbol in linguistics), the Unicode character U+29B0 REVERSED EMPTY SET ⦰ may be used instead.[5]
Properties
In standard axiomatic set theory, by the principle of extensionality, two sets are equal if they have the same elements (that is, neither of them has an element not in the other). As a result, there can be only one set with no elements, hence the usage of "the empty set" rather than "an empty set".
The empty set has the following properties:
• Its only subset is the empty set itself:
$\forall A:A\subseteq \varnothing \Rightarrow A=\varnothing $
• The power set of the empty set is the set containing only the empty set:
$2^{\varnothing }=\{\varnothing \}$
• The number of elements of the empty set (i.e., its cardinality) is zero:
$\mathrm {|} \varnothing \mathrm {|} =0$
For any set A:
• The empty set is a subset of A:
$\forall A:\varnothing \subseteq A$
• The union of A with the empty set is A:
$\forall A:A\cup \varnothing =A$
• The intersection of A with the empty set is the empty set:
$\forall A:A\cap \varnothing =\varnothing $
• The Cartesian product of A and the empty set is the empty set:
$\forall A:A\times \varnothing =\varnothing $
For any property P:
• For every element of $\varnothing $, the property P holds (vacuous truth).
• There is no element of $\varnothing $ for which the property P holds.
Conversely, if for some property P and some set V, the following two statements hold:
• For every element of V the property P holds
• There is no element of V for which the property P holds
then $V=\varnothing .$
By the definition of subset, the empty set is a subset of any set A. That is, every element x of $\varnothing $ belongs to A. Indeed, if it were not true that every element of $\varnothing $ is in A, then there would be at least one element of $\varnothing $ that is not present in A. Since there are no elements of $\varnothing $ at all, there is no element of $\varnothing $ that is not in A. Any statement that begins "for every element of $\varnothing $" is not making any substantive claim; it is a vacuous truth. This is often paraphrased as "everything is true of the elements of the empty set."
In the usual set-theoretic definition of natural numbers, zero is modelled by the empty set.
Operations on the empty set
When speaking of the sum of the elements of a finite set, one is inevitably led to the convention that the sum of the elements of the empty set is zero. The reason for this is that zero is the identity element for addition. Similarly, the product of the elements of the empty set should be considered to be one (see empty product), since one is the identity element for multiplication.
A derangement is a permutation of a set without fixed points. The empty set can be considered a derangement of itself, because it has only one permutation ($0!=1$), and it is vacuously true that no element (of the empty set) can be found that retains its original position.
In other areas of mathematics
Extended real numbers
Since the empty set has no member when it is considered as a subset of any ordered set, every member of that set will be an upper bound and lower bound for the empty set. For example, when considered as a subset of the real numbers, with its usual ordering, represented by the real number line, every real number is both an upper and lower bound for the empty set.[6] When considered as a subset of the extended reals formed by adding two "numbers" or "points" to the real numbers (namely negative infinity, denoted $-\infty \!\,,$ which is defined to be less than every other extended real number, and positive infinity, denoted $+\infty \!\,,$ which is defined to be greater than every other extended real number), we have that:
$\sup \varnothing =\min(\{-\infty ,+\infty \}\cup \mathbb {R} )=-\infty ,$
and
$\inf \varnothing =\max(\{-\infty ,+\infty \}\cup \mathbb {R} )=+\infty .$
That is, the least upper bound (sup or supremum) of the empty set is negative infinity, while the greatest lower bound (inf or infimum) is positive infinity. By analogy with the above, in the domain of the extended reals, negative infinity is the identity element for the maximum and supremum operators, while positive infinity is the identity element for the minimum and infimum operators.
Topology
In any topological space X, the empty set is open by definition, as is X. Since the complement of an open set is closed and the empty set and X are complements of each other, the empty set is also closed, making it a clopen set. Moreover, the empty set is compact by the fact that every finite set is compact.
The closure of the empty set is empty. This is known as "preservation of nullary unions."
Category theory
If $A$ is a set, then there exists precisely one function $f$ from $\varnothing $ to $A,$ the empty function. As a result, the empty set is the unique initial object of the category of sets and functions.
The empty set can be turned into a topological space, called the empty space, in just one way: by defining the empty set to be open. This empty topological space is the unique initial object in the category of topological spaces with continuous maps. In fact, it is a strict initial object: only the empty set has a function to the empty set.
Set theory
In the von Neumann construction of the ordinals, 0 is defined as the empty set, and the successor of an ordinal is defined as $S(\alpha )=\alpha \cup \{\alpha \}$. Thus, we have $0=\varnothing $, $1=0\cup \{0\}=\{\varnothing \}$, $2=1\cup \{1\}=\{\varnothing ,\{\varnothing \}\}$, and so on. The von Neumann construction, along with the axiom of infinity, which guarantees the existence of at least one infinite set, can be used to construct the set of natural numbers, $\mathbb {N} _{0}$, such that the Peano axioms of arithmetic are satisfied.
Questioned existence
Historical issues
In the context of sets of real numbers, Cantor used $P\equiv O$ to denote "$P$ contains no single point". This $\equiv O$ notation was utilized in definitions, for example Cantor defined two sets as being disjoint if their intersection has an absence of points, however it is debatable whether Cantor viewed $O$ as an existent set on its own, or if Cantor merely used $\equiv O$ as an emptiness predicate. Zermelo accepted $O$ itself as a set, but considered it an "improper set".[7]
Axiomatic set theory
In Zermelo set theory, the existence of the empty set is assured by the axiom of empty set, and its uniqueness follows from the axiom of extensionality. However, the axiom of empty set can be shown redundant in at least two ways:
• Standard first-order logic implies, merely from the logical axioms, that something exists, and in the language of set theory, that thing must be a set. Now the existence of the empty set follows easily from the axiom of separation.
• Even using free logic (which does not logically imply that something exists), there is already an axiom implying the existence of at least one set, namely the axiom of infinity.
Philosophical issues
While the empty set is a standard and widely accepted mathematical concept, it remains an ontological curiosity, whose meaning and usefulness are debated by philosophers and logicians.
The empty set is not the same thing as nothing; rather, it is a set with nothing inside it and a set is always something. This issue can be overcome by viewing a set as a bag—an empty bag undoubtedly still exists. Darling (2004) explains that the empty set is not nothing, but rather "the set of all triangles with four sides, the set of all numbers that are bigger than nine but smaller than eight, and the set of all opening moves in chess that involve a king."[8]
The popular syllogism
Nothing is better than eternal happiness; a ham sandwich is better than nothing; therefore, a ham sandwich is better than eternal happiness
is often used to demonstrate the philosophical relation between the concept of nothing and the empty set. Darling writes that the contrast can be seen by rewriting the statements "Nothing is better than eternal happiness" and "[A] ham sandwich is better than nothing" in a mathematical tone. According to Darling, the former is equivalent to "The set of all things that are better than eternal happiness is $\varnothing $" and the latter to "The set {ham sandwich} is better than the set $\varnothing $". The first compares elements of sets, while the second compares the sets themselves.[8]
Jonathan Lowe argues that while the empty set
was undoubtedly an important landmark in the history of mathematics, … we should not assume that its utility in calculation is dependent upon its actually denoting some object.
it is also the case that:
"All that we are ever informed about the empty set is that it (1) is a set, (2) has no members, and (3) is unique amongst sets in having no members. However, there are very many things that 'have no members', in the set-theoretical sense—namely, all non-sets. It is perfectly clear why these things have no members, for they are not sets. What is unclear is how there can be, uniquely amongst sets, a set which has no members. We cannot conjure such an entity into existence by mere stipulation."[9]
George Boolos argued that much of what has been heretofore obtained by set theory can just as easily be obtained by plural quantification over individuals, without reifying sets as singular entities having other entities as members.[10]
See also
• 0 – NumberPages displaying short descriptions with no spaces
• Inhabited set – Property of sets used in constructive mathematics
• Nothing – Complete absence of anything; the opposite of everything
• Power set – Mathematical set containing all subsets of a given set
References
1. Weisstein, Eric W. "Empty Set". mathworld.wolfram.com. Retrieved 2020-08-11.
2. "Earliest Uses of Symbols of Set Theory and Logic".
3. Rudin, Walter (1976). Principles of Mathematical Analysis (3rd ed.). McGraw-Hill. p. 300. ISBN 007054235X.
4. "Unicode Standard 5.2" (PDF).
5. e.g. Nina Grønnum (2005, 2013) Fonetik og Fonologi: Almen og dansk. Akademisk forlag, Copenhagen.
6. Bruckner, A.N., Bruckner, J.B., and Thomson, B.S. (2008). Elementary Real Analysis, 2nd edition, p. 9.
7. A. Kanamori, "The Empty Set, the Singleton, and the Ordered Pair", p.275. Bulletin of Symbolic Logic vol. 9, no. 3, (2003). Accessed 21 August 2023.
8. D. J. Darling (2004). The Universal Book of Mathematics. John Wiley and Sons. p. 106. ISBN 0-471-27047-4.
9. E. J. Lowe (2005). Locke. Routledge. p. 87.
10. George Boolos (1984), "To be is to be the value of a variable", The Journal of Philosophy 91: 430–49. Reprinted in 1998, Logic, Logic and Logic (Richard Jeffrey, and Burgess, J., eds.) Harvard University Press, 54–72.
Further reading
• Halmos, Paul, Naive Set Theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). Reprinted by Martino Fine Books, 2011. ISBN 978-1-61427-131-4 (paperback edition).
• Jech, Thomas (2002), Set Theory, Springer Monographs in Mathematics (3rd millennium ed.), Springer, ISBN 3-540-44085-2
• Graham, Malcolm (1975), Modern Elementary Mathematics (2nd ed.), Harcourt Brace Jovanovich, ISBN 0155610392
External links
• Weisstein, Eric W. "Empty Set". MathWorld.
Mathematical logic
General
• Axiom
• list
• Cardinality
• First-order logic
• Formal proof
• Formal semantics
• Foundations of mathematics
• Information theory
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• Logical consequence
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| Wikipedia |
Voigt profile
The Voigt profile (named after Woldemar Voigt) is a probability distribution given by a convolution of a Cauchy-Lorentz distribution and a Gaussian distribution. It is often used in analyzing data from spectroscopy or diffraction.
(Centered) Voigt
Probability density function
Plot of the centered Voigt profile for four cases. Each case has a full width at half-maximum of very nearly 3.6. The black and red profiles are the limiting cases of the Gaussian (γ =0) and the Lorentzian (σ =0) profiles respectively.
Cumulative distribution function
Parameters $\gamma ,\sigma >0$
Support $x\in (-\infty ,\infty )$
PDF ${\frac {\Re [w(z)]}{\sigma {\sqrt {2\pi }}}},~~~z={\frac {x+i\gamma }{\sigma {\sqrt {2}}}}$
CDF (complicated - see text)
Mean (not defined)
Median $0$
Mode $0$
Variance (not defined)
Skewness (not defined)
Ex. kurtosis (not defined)
MGF (not defined)
CF $e^{-\gamma |t|-\sigma ^{2}t^{2}/2}$
Definition
Without loss of generality, we can consider only centered profiles, which peak at zero. The Voigt profile is then
$V(x;\sigma ,\gamma )\equiv \int _{-\infty }^{\infty }G(x';\sigma )L(x-x';\gamma )\,dx',$
where x is the shift from the line center, $G(x;\sigma )$ is the centered Gaussian profile:
$G(x;\sigma )\equiv {\frac {e^{-x^{2}/(2\sigma ^{2})}}{\sigma {\sqrt {2\pi }}}},$
and $L(x;\gamma )$ is the centered Lorentzian profile:
$L(x;\gamma )\equiv {\frac {\gamma }{\pi (x^{2}+\gamma ^{2})}}.$
The defining integral can be evaluated as:
$V(x;\sigma ,\gamma )={\frac {\operatorname {Re} [w(z)]}{\sigma {\sqrt {2\pi }}}},$
where Re[w(z)] is the real part of the Faddeeva function evaluated for
$z={\frac {x+i\gamma }{\sigma {\sqrt {2}}}}.$
In the limiting cases of $\sigma =0$ and $\gamma =0$ then $V(x;\sigma ,\gamma )$ simplifies to $L(x;\gamma )$ and $G(x;\sigma )$, respectively.
History and applications
In spectroscopy, a Voigt profile results from the convolution of two broadening mechanisms, one of which alone would produce a Gaussian profile (usually, as a result of the Doppler broadening), and the other would produce a Lorentzian profile. Voigt profiles are common in many branches of spectroscopy and diffraction. Due to the expense of computing the Faddeeva function, the Voigt profile is sometimes approximated using a pseudo-Voigt profile.
Properties
The Voigt profile is normalized:
$\int _{-\infty }^{\infty }V(x;\sigma ,\gamma )\,dx=1,$
since it is a convolution of normalized profiles. The Lorentzian profile has no moments (other than the zeroth), and so the moment-generating function for the Cauchy distribution is not defined. It follows that the Voigt profile will not have a moment-generating function either, but the characteristic function for the Cauchy distribution is well defined, as is the characteristic function for the normal distribution. The characteristic function for the (centered) Voigt profile will then be the product of the two:
$\varphi _{f}(t;\sigma ,\gamma )=E(e^{ixt})=e^{-\sigma ^{2}t^{2}/2-\gamma |t|}.$
Since normal distributions and Cauchy distributions are stable distributions, they are each closed under convolution (up to change of scale), and it follows that the Voigt distributions are also closed under convolution.
Cumulative distribution function
Using the above definition for z , the cumulative distribution function (CDF) can be found as follows:
$F(x_{0};\mu ,\sigma )=\int _{-\infty }^{x_{0}}{\frac {\operatorname {Re} (w(z))}{\sigma {\sqrt {2\pi }}}}\,dx=\operatorname {Re} \left({\frac {1}{\sqrt {\pi }}}\int _{z(-\infty )}^{z(x_{0})}w(z)\,dz\right).$
Substituting the definition of the Faddeeva function (scaled complex error function) yields for the indefinite integral:
${\frac {1}{\sqrt {\pi }}}\int w(z)\,dz={\frac {1}{\sqrt {\pi }}}\int e^{-z^{2}}\left[1-\operatorname {erf} (-iz)\right]\,dz,$
which may be solved to yield
${\frac {1}{\sqrt {\pi }}}\int w(z)\,dz={\frac {\operatorname {erf} (z)}{2}}+{\frac {iz^{2}}{\pi }}\,_{2}F_{2}\left(1,1;{\frac {3}{2}},2;-z^{2}\right),$
where ${}_{2}F_{2}$ is a hypergeometric function. In order for the function to approach zero as x approaches negative infinity (as the CDF must do), an integration constant of 1/2 must be added. This gives for the CDF of Voigt:
$F(x;\mu ,\sigma )=\operatorname {Re} \left[{\frac {1}{2}}+{\frac {\operatorname {erf} (z)}{2}}+{\frac {iz^{2}}{\pi }}\,_{2}F_{2}\left(1,1;{\frac {3}{2}},2;-z^{2}\right)\right].$
The uncentered Voigt profile
If the Gaussian profile is centered at $\mu _{G}$ and the Lorentzian profile is centered at $\mu _{L}$, the convolution is centered at $\mu _{V}=\mu _{G}+\mu _{L}$ and the characteristic function is:
$\varphi _{f}(t;\sigma ,\gamma ,\mu _{\mathrm {G} },\mu _{\mathrm {L} })=e^{i(\mu _{\mathrm {G} }+\mu _{\mathrm {L} })t-\sigma ^{2}t^{2}/2-\gamma |t|}.$
The probability density function is simply offset from the centered profile by $\mu _{V}$:
$V(x;\mu _{V},\sigma ,\gamma )={\frac {\operatorname {Re} [w(z)]}{\sigma {\sqrt {2\pi }}}},$
where:
$z={\frac {x-\mu _{V}+i\gamma }{\sigma {\sqrt {2}}}}$
The mode and median are both located at $\mu _{V}$.
Derivatives
Using the definition above for $z$ and $x_{c}=x-\mu _{V}$, the first and second derivatives can be expressed in terms of the Faddeeva function as
${\begin{aligned}{\frac {\partial }{\partial x}}V(x_{c};\sigma ,\gamma )&=-{\frac {\operatorname {Re} \left[z~w(z)\right]}{\sigma ^{2}{\sqrt {\pi }}}}=-{\frac {x_{c}}{\sigma ^{2}}}{\frac {\operatorname {Re} \left[w(z)\right]}{\sigma {\sqrt {2\pi }}}}+{\frac {\gamma }{\sigma ^{2}}}{\frac {\operatorname {Im} \left[w(z)\right]}{\sigma {\sqrt {2\pi }}}}\\&={\frac {1}{\sigma ^{3}{\sqrt {2\pi }}}}\cdot \left(\gamma \cdot \operatorname {Im} \left[w(z)\right]-x_{c}\cdot \operatorname {Re} \left[w(z)\right]\right)\end{aligned}}$
and
${\begin{aligned}{\frac {\partial ^{2}}{\left(\partial x\right)^{2}}}V(x_{c};\sigma ,\gamma )&={\frac {x_{c}^{2}-\gamma ^{2}-\sigma ^{2}}{\sigma ^{4}}}{\frac {\operatorname {Re} \left[w(z)\right]}{\sigma {\sqrt {2\pi }}}}-{\frac {2x_{c}\gamma }{\sigma ^{4}}}{\frac {\operatorname {Im} \left[w(z)\right]}{\sigma {\sqrt {2\pi }}}}+{\frac {\gamma }{\sigma ^{4}}}{\frac {1}{\pi }}\\&=-{\frac {1}{\sigma ^{5}{\sqrt {2\pi }}}}\cdot \left(\gamma \cdot \left(2x_{c}\cdot \operatorname {Im} \left[w(z)\right]-\sigma \cdot {\sqrt {\frac {2}{\pi }}}\right)+\left(\gamma ^{2}+\sigma ^{2}-x_{c}^{2}\right)\cdot \operatorname {Re} \left[w(z)\right]\right),\end{aligned}}$
respectively.
Often, one or multiple Voigt profiles and/or their respective derivatives need to be fitted to a measured signal by means of non-linear least squares, e.g., in spectroscopy. Then, further partial derivatives can be utilised to accelerate computations. Instead of approximating the Jacobian matrix with respect to the parameters $\mu _{V}$, $\sigma $, and $\gamma $ with the aid of finite differences, the corresponding analytical expressions can be applied. With $\operatorname {Re} \left[w(z)\right]=\Re _{w}$ and $\operatorname {Im} \left[w(z)\right]=\Im _{w}$, these are given by:
${\begin{aligned}{\frac {\partial V}{\partial \mu _{V}}}=-{\frac {\partial V}{\partial x}}={\frac {1}{\sigma ^{3}{\sqrt {2\pi }}}}\cdot \left(x_{c}\cdot \Re _{w}-\gamma \cdot \Im _{w}\right)\end{aligned}}$
${\begin{aligned}{\frac {\partial V}{\partial \sigma }}={\frac {1}{\sigma ^{4}{\sqrt {2\pi }}}}\cdot \left(\left(x_{c}^{2}-\gamma ^{2}-\sigma ^{2}\right)\cdot \Re _{w}-2x_{c}\gamma \cdot \Im _{w}+\gamma \sigma \cdot {\sqrt {\frac {2}{\pi }}}\right)\end{aligned}}$
${\begin{aligned}{\frac {\partial V}{\partial \gamma }}=-{\frac {1}{\sigma ^{3}{\sqrt {2\pi }}}}\cdot \left(\sigma \cdot {\sqrt {\frac {2}{\pi }}}-x_{c}\cdot \Im _{w}-\gamma \cdot \Re _{w}\right)\end{aligned}}$
for the original voigt profile $V$;
${\begin{aligned}{\frac {\partial V'}{\partial \mu _{V}}}=-{\frac {\partial V'}{\partial x}}=-{\frac {\partial ^{2}V}{\left(\partial x\right)^{2}}}={\frac {1}{\sigma ^{5}{\sqrt {2\pi }}}}\cdot \left(\gamma \cdot \left(2x_{c}\cdot \Im _{w}-\sigma \cdot {\sqrt {\frac {2}{\pi }}}\right)+\left(\gamma ^{2}+\sigma ^{2}-x_{c}^{2}\right)\cdot \Re _{w}\right)\end{aligned}}$
${\begin{aligned}{\frac {\partial V'}{\partial \sigma }}={\frac {3}{\sigma ^{6}{\sqrt {2\pi }}}}\cdot \left(-\gamma \sigma x_{c}\cdot {\frac {2{\sqrt {2}}}{3{\sqrt {\pi }}}}+\left(x_{c}^{2}-{\frac {\gamma ^{2}}{3}}-\sigma ^{2}\right)\cdot \gamma \cdot \Im _{w}+\left(\gamma ^{2}+\sigma ^{2}-{\frac {x_{c}^{2}}{3}}\right)\cdot x_{c}\cdot \Re _{w}\right)\end{aligned}}$
${\begin{aligned}{\frac {\partial V'}{\partial \gamma }}={\frac {1}{\sigma ^{5}{\sqrt {2\pi }}}}\cdot \left(x_{c}\cdot \left(\sigma \cdot {\sqrt {\frac {2}{\pi }}}-2\gamma \cdot \Re _{w}\right)+\left(\gamma ^{2}+\sigma ^{2}-x_{c}^{2}\right)\cdot \Im _{w}\right)\end{aligned}}$
for the first order partial derivative $V'={\frac {\partial V}{\partial x}}$; and
${\begin{aligned}{\frac {\partial V''}{\partial \mu _{V}}}=-{\frac {\partial V''}{\partial x}}=-{\frac {\partial ^{3}V}{\left(\partial x\right)^{3}}}=-{\frac {3}{\sigma ^{7}{\sqrt {2\pi }}}}\cdot \left(\left(x_{c}^{2}-{\frac {\gamma ^{2}}{3}}-\sigma ^{2}\right)\cdot \gamma \cdot \Im _{w}+\left(\gamma ^{2}+\sigma ^{2}-{\frac {x_{c}^{2}}{3}}\right)\cdot x_{c}\cdot \Re _{w}-\gamma \sigma x_{c}\cdot {\frac {2{\sqrt {2}}}{3{\sqrt {\pi }}}}\right)\end{aligned}}$
${\begin{aligned}&{\frac {\partial V''}{\partial \sigma }}=-{\frac {1}{\sigma ^{8}{\sqrt {2\pi }}}}\cdot \\&\left(\left(-3\gamma x_{c}\sigma ^{2}+\gamma x_{c}^{3}-\gamma ^{3}x_{c}\right)\cdot 4\cdot \Im _{w}+\left(\left(2x_{c}^{2}-2\gamma ^{2}-\sigma ^{2}\right)\cdot 3\sigma ^{2}+6\gamma ^{2}x_{c}^{2}-x_{c}^{4}-\gamma ^{4}\right)\cdot \Re _{w}+\left(\gamma ^{2}+5\sigma ^{2}-3x_{c}^{2}\right)\cdot \gamma \sigma \cdot {\sqrt {\frac {2}{\pi }}}\right)\end{aligned}}$
${\begin{aligned}{\frac {\partial V''}{\partial \gamma }}=-{\frac {3}{\sigma ^{7}{\sqrt {2\pi }}}}\cdot \left(\left(\gamma ^{2}+\sigma ^{2}-{\frac {x_{c}^{2}}{3}}\right)\cdot x_{c}\cdot \Im _{w}+\left({\frac {\gamma ^{2}}{3}}+\sigma ^{2}-x_{c}^{2}\right)\cdot \gamma \cdot \Re _{w}+\left(x_{c}^{2}-\gamma ^{2}-2\sigma ^{2}\right)\cdot \sigma \cdot {\frac {\sqrt {2}}{3{\sqrt {\pi }}}}\right)\end{aligned}}$
for the second order partial derivative $V''={\frac {\partial ^{2}V}{\left(\partial x\right)^{2}}}$. Since $\mu _{V}$ and $\gamma $ play a relatively similar role in the calculation of $z$, their respective partial derivatives also look quite similar in terms of their structure, although they result in totally different derivative profiles. Indeed, the partial derivatives with respect to $\sigma $ and $\gamma $ show more similarity since both are width parameters. All these derivatives involve only simple operations (multiplications and additions) because the computationally expensive $\Re _{w}$ and $\Im _{w}$ are readily obtained when computing $w\left(z\right)$. Such a reuse of previous calculations allows for a derivation at minimum costs. This is not the case for finite difference gradient approximation as it requires the evaluation of $w\left(z\right)$ for each gradient respectively.
Voigt functions
The Voigt functions[1] U, V, and H (sometimes called the line broadening function) are defined by
$U(x,t)+iV(x,t)={\sqrt {\frac {\pi }{4t}}}e^{z^{2}}\operatorname {erfc} (z)={\sqrt {\frac {\pi }{4t}}}w(iz),$
$H(a,u)={\frac {U(u/a,1/4a^{2})}{a{\sqrt {\pi }}}},$
where
$z=(1-ix)/2{\sqrt {t}},$
erfc is the complementary error function, and w(z) is the Faddeeva function.
Relation to Voigt profile
$V(x;\sigma ,\gamma )=H(a,u)/({\sqrt {2}}{\sqrt {\pi }}\sigma ),$
with
$a=\gamma /({\sqrt {2}}\sigma )$
and
$u=x/({\sqrt {2}}\sigma ).$
Numeric approximations
Tepper-García Function
The Tepper-García function, named after German-Mexican Astrophysicist Thor Tepper-García, is a combination of an exponential function and rational functions that approximates the line broadening function $H(a,u)$ over a wide range of its parameters.[2] It is obtained from a truncated power series expansion of the exact line broadening function.
In its most computationally efficient form, the Tepper-García function can be expressed as
$T(a,u)=R-\left(a/{\sqrt {\pi }}P\right)~\left[R^{2}~(4P^{2}+7P+4+Q)-Q-1\right]\,,$
where $P\equiv u^{2}$, $Q\equiv 3/(2P)$, and $R\equiv e^{-P}$.
Thus the line broadening function can be viewed, to first order, as a pure Gaussian function plus a correction factor that depends linearly on the microscopic properties of the absorbing medium (encoded in $a$); however, as a result of the early truncation in the series expansion, the error in the approximation is still of order $a$, i.e. $H(a,u)\approx T(a,u)+{\mathcal {O}}(a)$. This approximation has a relative accuracy of
$\epsilon \equiv {\frac {\vert H(a,u)-T(a,u)\vert }{H(a,u)}}\lesssim 10^{-4}$
over the full wavelength range of $H(a,u)$, provided that $a\lesssim 10^{-4}$. In addition to its high accuracy, the function $T(a,u)$ is easy to implement as well as computationally fast. It is widely used in the field of quasar absorption line analysis.[3]
Pseudo-Voigt approximation
The pseudo-Voigt profile (or pseudo-Voigt function) is an approximation of the Voigt profile V(x) using a linear combination of a Gaussian curve G(x) and a Lorentzian curve L(x) instead of their convolution.
The pseudo-Voigt function is often used for calculations of experimental spectral line shapes.
The mathematical definition of the normalized pseudo-Voigt profile is given by
$V_{p}(x,f)=\eta \cdot L(x,f)+(1-\eta )\cdot G(x,f)$ with $0<\eta <1$.
$\eta $ is a function of full width at half maximum (FWHM) parameter.
There are several possible choices for the $\eta $ parameter.[4][5][6][7] A simple formula, accurate to 1%, is[8][9]
$\eta =1.36603(f_{L}/f)-0.47719(f_{L}/f)^{2}+0.11116(f_{L}/f)^{3},$
where now, $\eta $ is a function of Lorentz ($f_{L}$), Gaussian ($f_{G}$) and total ($f$) Full width at half maximum (FWHM) parameters. The total FWHM ($f$) parameter is described by:
$f=[f_{G}^{5}+2.69269f_{G}^{4}f_{L}+2.42843f_{G}^{3}f_{L}^{2}+4.47163f_{G}^{2}f_{L}^{3}+0.07842f_{G}f_{L}^{4}+f_{L}^{5}]^{1/5}.$
The width of the Voigt profile
The full width at half maximum (FWHM) of the Voigt profile can be found from the widths of the associated Gaussian and Lorentzian widths. The FWHM of the Gaussian profile is
$f_{\mathrm {G} }=2\sigma {\sqrt {2\ln(2)}}.$
The FWHM of the Lorentzian profile is
$f_{\mathrm {L} }=2\gamma .$
An approximate relation (accurate to within about 1.2%) between the widths of the Voigt, Gaussian, and Lorentzian profiles is:[10]
$f_{\mathrm {V} }\approx f_{\mathrm {L} }/2+{\sqrt {f_{\mathrm {L} }^{2}/4+f_{\mathrm {G} }^{2}}}.$
By construction, this expression is exact for a pure Gaussian or Lorentzian.
A better approximation with an accuracy of 0.02% is given by [11] (originally found by Kielkopf[12])
$f_{\mathrm {V} }\approx 0.5346f_{\mathrm {L} }+{\sqrt {0.2166f_{\mathrm {L} }^{2}+f_{\mathrm {G} }^{2}}}.$
Again, this expression is exact for a pure Gaussian or Lorentzian. In the same publication,[11] a slightly more precise (within 0.012%), yet significantly more complicated expression can be found.
References
1. Temme, N. M. (2010), "Voigt function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
2. Tepper-García, Thorsten (2006). "Voigt profile fitting to quasar absorption lines: an analytic approximation to the Voigt-Hjerting function". Monthly Notices of the Royal Astronomical Society. 369 (4): 2025–2035. arXiv:astro-ph/0602124. Bibcode:2006MNRAS.369.2025T. doi:10.1111/j.1365-2966.2006.10450.x. S2CID 16981310.
3. List of citations found in the SAO/NASA Astrophysics Data System (ADS): https://ui.adsabs.harvard.edu/abs/2006MNRAS.369.2025T/citations
4. Wertheim GK, Butler MA, West KW, Buchanan DN (1974). "Determination of the Gaussian and Lorentzian content of experimental line shapes". Review of Scientific Instruments. 45 (11): 1369–1371. Bibcode:1974RScI...45.1369W. doi:10.1063/1.1686503.
5. Sánchez-Bajo, F.; F. L. Cumbrera (August 1997). "The Use of the Pseudo-Voigt Function in the Variance Method of X-ray Line-Broadening Analysis". Journal of Applied Crystallography. 30 (4): 427–430. doi:10.1107/S0021889896015464.
6. Liu Y, Lin J, Huang G, Guo Y, Duan C (2001). "Simple empirical analytical approximation to the Voigt profile". JOSA B. 18 (5): 666–672. Bibcode:2001JOSAB..18..666L. doi:10.1364/josab.18.000666.
7. Di Rocco HO, Cruzado A (2012). "The Voigt Profile as a Sum of a Gaussian and a Lorentzian Functions, when the Weight Coefficient Depends Only on the Widths Ratio". Acta Physica Polonica A. 122 (4): 666–669. Bibcode:2012AcPPA.122..666D. doi:10.12693/APhysPolA.122.666. ISSN 0587-4246.
8. Ida T, Ando M, Toraya H (2000). "Extended pseudo-Voigt function for approximating the Voigt profile". Journal of Applied Crystallography. 33 (6): 1311–1316. doi:10.1107/s0021889800010219. S2CID 55372305.
9. P. Thompson, D. E. Cox and J. B. Hastings (1987). "Rietveld refinement of Debye-Scherrer synchrotron X-ray data from Al2O3". Journal of Applied Crystallography. 20 (2): 79–83. doi:10.1107/S0021889887087090.
10. Whiting, E. E. (June 1968). "An empirical approximation to the Voigt profile". Journal of Quantitative Spectroscopy and Radiative Transfer. 8 (6): 1379–1384. Bibcode:1968JQSRT...8.1379W. doi:10.1016/0022-4073(68)90081-2. ISSN 0022-4073.
11. Olivero, J. J.; R. L. Longbothum (February 1977). "Empirical fits to the Voigt line width: A brief review". Journal of Quantitative Spectroscopy and Radiative Transfer. 17 (2): 233–236. Bibcode:1977JQSRT..17..233O. doi:10.1016/0022-4073(77)90161-3. ISSN 0022-4073.
12. John F. Kielkopf (1973), "New approximation to the Voigt function with applications to spectral-line profile analysis", Journal of the Optical Society of America, 63 (8): 987, Bibcode:1973JOSA...63..987K, doi:10.1364/JOSA.63.000987
External links
• http://jugit.fz-juelich.de/mlz/libcerf, numeric C library for complex error functions, provides a function voigt(x, sigma, gamma) with approximately 13–14 digits precision.
• The original article is : Voigt, Woldemar, 1912, ''Das Gesetz der Intensitätsverteilung innerhalb der Linien eines Gasspektrums'', Sitzungsbericht der Bayerischen Akademie der Wissenschaften, 25, 603 (see also: http://publikationen.badw.de/de/003395768)
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• zeta
Continuous
univariate
supported on a
bounded interval
• arcsine
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supported on a
semi-infinite
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• normal (Gaussian)
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with support
whose type varies
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• q-exponential
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Mixed
univariate
continuous-
discrete
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Multivariate
(joint)
• Discrete:
• Ewens
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Directional
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Degenerate
and singular
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Singular
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Families
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| Wikipedia |
Voigt notation
In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order.[1] There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig[2] of old ideas of Lord Kelvin. The differences here lie in certain weights attached to the selected entries of the tensor. Nomenclature may vary according to what is traditional in the field of application.
For example, a 2×2 symmetric tensor X has only three distinct elements, the two on the diagonal and the other being off-diagonal. Thus it can be expressed as the vector
$\langle x_{11},x_{22},x_{12}\rangle $.
As another example:
The stress tensor (in matrix notation) is given as
${\boldsymbol {\sigma }}=\left[{\begin{matrix}\sigma _{xx}&\sigma _{xy}&\sigma _{xz}\\\sigma _{yx}&\sigma _{yy}&\sigma _{yz}\\\sigma _{zx}&\sigma _{zy}&\sigma _{zz}\end{matrix}}\right].$
In Voigt notation it is simplified to a 6-dimensional vector:
${\tilde {\sigma }}=(\sigma _{xx},\sigma _{yy},\sigma _{zz},\sigma _{yz},\sigma _{xz},\sigma _{xy})\equiv (\sigma _{1},\sigma _{2},\sigma _{3},\sigma _{4},\sigma _{5},\sigma _{6}).$
The strain tensor, similar in nature to the stress tensor—both are symmetric second-order tensors --, is given in matrix form as
${\boldsymbol {\epsilon }}=\left[{\begin{matrix}\epsilon _{xx}&\epsilon _{xy}&\epsilon _{xz}\\\epsilon _{yx}&\epsilon _{yy}&\epsilon _{yz}\\\epsilon _{zx}&\epsilon _{zy}&\epsilon _{zz}\end{matrix}}\right].$
Its representation in Voigt notation is
${\tilde {\epsilon }}=(\epsilon _{xx},\epsilon _{yy},\epsilon _{zz},\gamma _{yz},\gamma _{xz},\gamma _{xy})\equiv (\epsilon _{1},\epsilon _{2},\epsilon _{3},\epsilon _{4},\epsilon _{5},\epsilon _{6}),$
where $\gamma _{xy}=2\epsilon _{xy}$, $\gamma _{yz}=2\epsilon _{yz}$, and $\gamma _{zx}=2\epsilon _{zx}$ are engineering shear strains.
The benefit of using different representations for stress and strain is that the scalar invariance
${\boldsymbol {\sigma }}\cdot {\boldsymbol {\epsilon }}=\sigma _{ij}\epsilon _{ij}={\tilde {\sigma }}\cdot {\tilde {\epsilon }}$
is preserved.
Likewise, a three-dimensional symmetric fourth-order tensor can be reduced to a 6×6 matrix.
Mnemonic rule
A simple mnemonic rule for memorizing Voigt notation is as follows:
• Write down the second order tensor in matrix form (in the example, the stress tensor)
• Strike out the diagonal
• Continue on the third column
• Go back to the first element along the first row.
Voigt indexes are numbered consecutively from the starting point to the end (in the example, the numbers in blue).
Mandel notation
For a symmetric tensor of second rank
${\boldsymbol {\sigma }}=\left[{\begin{matrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{21}&\sigma _{22}&\sigma _{23}\\\sigma _{31}&\sigma _{32}&\sigma _{33}\end{matrix}}\right]$
only six components are distinct, the three on the diagonal and the others being off-diagonal. Thus it can be expressed, in Mandel notation,[3] as the vector
${\tilde {\sigma }}^{M}=\langle \sigma _{11},\sigma _{22},\sigma _{33},{\sqrt {2}}\sigma _{23},{\sqrt {2}}\sigma _{13},{\sqrt {2}}\sigma _{12}\rangle .$
The main advantage of Mandel notation is to allow the use of the same conventional operations used with vectors, for example:
${\tilde {\sigma }}:{\tilde {\sigma }}={\tilde {\sigma }}^{M}\cdot {\tilde {\sigma }}^{M}=\sigma _{11}^{2}+\sigma _{22}^{2}+\sigma _{33}^{2}+2\sigma _{23}^{2}+2\sigma _{13}^{2}+2\sigma _{12}^{2}.$
A symmetric tensor of rank four satisfying $D_{ijkl}=D_{jikl}$ and $D_{ijkl}=D_{ijlk}$ has 81 components in three-dimensional space, but only 36 components are distinct. Thus, in Mandel notation, it can be expressed as
${\tilde {D}}^{M}={\begin{pmatrix}D_{1111}&D_{1122}&D_{1133}&{\sqrt {2}}D_{1123}&{\sqrt {2}}D_{1113}&{\sqrt {2}}D_{1112}\\D_{2211}&D_{2222}&D_{2233}&{\sqrt {2}}D_{2223}&{\sqrt {2}}D_{2213}&{\sqrt {2}}D_{2212}\\D_{3311}&D_{3322}&D_{3333}&{\sqrt {2}}D_{3323}&{\sqrt {2}}D_{3313}&{\sqrt {2}}D_{3312}\\{\sqrt {2}}D_{2311}&{\sqrt {2}}D_{2322}&{\sqrt {2}}D_{2333}&2D_{2323}&2D_{2313}&2D_{2312}\\{\sqrt {2}}D_{1311}&{\sqrt {2}}D_{1322}&{\sqrt {2}}D_{1333}&2D_{1323}&2D_{1313}&2D_{1312}\\{\sqrt {2}}D_{1211}&{\sqrt {2}}D_{1222}&{\sqrt {2}}D_{1233}&2D_{1223}&2D_{1213}&2D_{1212}\\\end{pmatrix}}.$
Applications
The notation is named after physicist Woldemar Voigt & John Nye (scientist). It is useful, for example, in calculations involving constitutive models to simulate materials, such as the generalized Hooke's law, as well as finite element analysis,[4] and Diffusion MRI.[5]
Hooke's law has a symmetric fourth-order stiffness tensor with 81 components (3×3×3×3), but because the application of such a rank-4 tensor to a symmetric rank-2 tensor must yield another symmetric rank-2 tensor, not all of the 81 elements are independent. Voigt notation enables such a rank-4 tensor to be represented by a 6×6 matrix. However, Voigt's form does not preserve the sum of the squares, which in the case of Hooke's law has geometric significance. This explains why weights are introduced (to make the mapping an isometry).
A discussion of invariance of Voigt's notation and Mandel's notation can be found in Helnwein (2001).[6]
References
1. Woldemar Voigt (1910). Lehrbuch der kristallphysik. Teubner, Leipzig. Retrieved November 29, 2016.
2. Klaus Helbig (1994). Foundations of anisotropy for exploration seismics. Pergamon. ISBN 0-08-037224-4.
3. Jean Mandel (1965). "Généralisation de la théorie de plasticité de WT Koiter". International Journal of Solids and Structures. 1 (3): 273–295. doi:10.1016/0020-7683(65)90034-x.
4. O.C. Zienkiewicz; R.L. Taylor; J.Z. Zhu (2005). The Finite Element Method: Its Basis and Fundamentals (6 ed.). Elsevier Butterworth—Heinemann. ISBN 978-0-7506-6431-8.
5. Maher Moakher (2009). "The Algebra of Fourth-Order Tensors with Application to Diffusion MRI". Visualization and Processing of Tensor Fields. Mathematics and Visualization. Springer Berlin Heidelberg. pp. 57–80. doi:10.1007/978-3-540-88378-4_4. ISBN 978-3-540-88377-7.
6. Peter Helnwein (February 16, 2001). "Some Remarks on the Compressed Matrix Representation of Symmetric Second-Order and Fourth-Order Tensors". Computer Methods in Applied Mechanics and Engineering. 190 (22–23): 2753–2770. Bibcode:2001CMAME.190.2753H. doi:10.1016/s0045-7825(00)00263-2.
See also
• Vectorization (mathematics)
• Hooke's law
Tensors
Glossary of tensor theory
Scope
Mathematics
• Coordinate system
• Differential geometry
• Dyadic algebra
• Euclidean geometry
• Exterior calculus
• Multilinear algebra
• Tensor algebra
• Tensor calculus
• Physics
• Engineering
• Computer vision
• Continuum mechanics
• Electromagnetism
• General relativity
• Transport phenomena
Notation
• Abstract index notation
• Einstein notation
• Index notation
• Multi-index notation
• Penrose graphical notation
• Ricci calculus
• Tetrad (index notation)
• Van der Waerden notation
• Voigt notation
Tensor
definitions
• Tensor (intrinsic definition)
• Tensor field
• Tensor density
• Tensors in curvilinear coordinates
• Mixed tensor
• Antisymmetric tensor
• Symmetric tensor
• Tensor operator
• Tensor bundle
• Two-point tensor
Operations
• Covariant derivative
• Exterior covariant derivative
• Exterior derivative
• Exterior product
• Hodge star operator
• Lie derivative
• Raising and lowering indices
• Symmetrization
• Tensor contraction
• Tensor product
• Transpose (2nd-order tensors)
Related
abstractions
• Affine connection
• Basis
• Cartan formalism (physics)
• Connection form
• Covariance and contravariance of vectors
• Differential form
• Dimension
• Exterior form
• Fiber bundle
• Geodesic
• Levi-Civita connection
• Linear map
• Manifold
• Matrix
• Multivector
• Pseudotensor
• Spinor
• Vector
• Vector space
Notable tensors
Mathematics
• Kronecker delta
• Levi-Civita symbol
• Metric tensor
• Nonmetricity tensor
• Ricci curvature
• Riemann curvature tensor
• Torsion tensor
• Weyl tensor
Physics
• Moment of inertia
• Angular momentum tensor
• Spin tensor
• Cauchy stress tensor
• stress–energy tensor
• Einstein tensor
• EM tensor
• Gluon field strength tensor
• Metric tensor (GR)
Mathematicians
• Élie Cartan
• Augustin-Louis Cauchy
• Elwin Bruno Christoffel
• Albert Einstein
• Leonhard Euler
• Carl Friedrich Gauss
• Hermann Grassmann
• Tullio Levi-Civita
• Gregorio Ricci-Curbastro
• Bernhard Riemann
• Jan Arnoldus Schouten
• Woldemar Voigt
• Hermann Weyl
| Wikipedia |
Paul Vojta
Paul Alan Vojta (born September 30, 1957) is an American mathematician, known for his work in number theory on Diophantine geometry and Diophantine approximation.
Paul Vojta
Born (1957-09-30) September 30, 1957
NationalityAmerican
Alma materHarvard University
University of Minnesota
Known forVojta's conjecture
AwardsCole Prize (1992)
Putnam Fellow
Scientific career
FieldsMathematics
InstitutionsUniversity of California, Berkeley
Doctoral advisorBarry Mazur
Contributions
In formulating Vojta's conjecture, he pointed out the possible existence of parallels between the Nevanlinna theory of complex analysis, and diophantine analysis in the circle of ideas around the Mordell conjecture and abc conjecture. This suggested the importance of the integer solutions (affine space) aspect of diophantine equations.
Vojta wrote the .dvi-previewer xdvi.
Education and career
He was an undergraduate student at the University of Minnesota, where he became a Putnam Fellow in 1977,[1] and a doctoral student at Harvard University (1983).[2] He currently is a professor in the Department of Mathematics at the University of California, Berkeley.
Awards and honors
In 2012 he became a fellow of the American Mathematical Society.[3]
Selected publications
• Diophantine Approximations and Value Distribution Theory, Lecture Notes in Mathematics 1239, Springer Verlag, 1987, ISBN 978-3-540-17551-3
References
1. "Putnam Competition Individual and Team Winners". Mathematical Association of America. Retrieved December 13, 2021.
2. Paul Vojta at the Mathematics Genealogy Project
3. List of Fellows of the American Mathematical Society, retrieved 2013-08-29.
External links
• Vojta's home page
• Paul Vojta's results at International Mathematical Olympiad
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| Wikipedia |
Vojta's conjecture
In mathematics, Vojta's conjecture is a conjecture introduced by Paul Vojta (1987) about heights of points on algebraic varieties over number fields. The conjecture was motivated by an analogy between diophantine approximation and Nevanlinna theory (value distribution theory) in complex analysis. It implies many other conjectures in Diophantine approximation, Diophantine equations, arithmetic geometry, and mathematical logic.
Statement of the conjecture
Let $F$ be a number field, let $X/F$ be a non-singular algebraic variety, let $D$ be an effective divisor on $X$ with at worst normal crossings, let $H$ be an ample divisor on $X$, and let $K_{X}$ be a canonical divisor on $X$. Choose Weil height functions $h_{H}$ and $h_{K_{X}}$ and, for each absolute value $v$ on $F$, a local height function $\lambda _{D,v}$. Fix a finite set of absolute values $S$ of $F$, and let $\epsilon >0$. Then there is a constant $C$ and a non-empty Zariski open set $U\subseteq X$, depending on all of the above choices, such that
$\sum _{v\in S}\lambda _{D,v}(P)+h_{K_{X}}(P)\leq \epsilon h_{H}(P)+C\quad {\hbox{for all }}P\in U(F).$
Examples:
1. Let $X=\mathbb {P} ^{N}$. Then $K_{X}\sim -(N+1)H$, so Vojta's conjecture reads $\sum _{v\in S}\lambda _{D,v}(P)\leq (N+1+\epsilon )h_{H}(P)+C$ for all $P\in U(F)$.
2. Let $X$ be a variety with trivial canonical bundle, for example, an abelian variety, a K3 surface or a Calabi-Yau variety. Vojta's conjecture predicts that if $D$ is an effective ample normal crossings divisor, then the $S$-integral points on the affine variety $X\setminus D$ are not Zariski dense. For abelian varieties, this was conjectured by Lang and proven by Faltings (1991).
3. Let $X$ be a variety of general type, i.e., $K_{X}$ is ample on some non-empty Zariski open subset of $X$. Then taking $S=\emptyset $, Vojta's conjecture predicts that $X(F)$ is not Zariski dense in $X$. This last statement for varieties of general type is the Bombieri–Lang conjecture.
Generalizations
There are generalizations in which $P$ is allowed to vary over $X({\overline {F}})$, and there is an additional term in the upper bound that depends on the discriminant of the field extension $F(P)/F$.
There are generalizations in which the non-archimedean local heights $\lambda _{D,v}$ are replaced by truncated local heights, which are local heights in which multiplicities are ignored. These versions of Vojta's conjecture provide natural higher-dimensional analogues of the ABC conjecture.
References
• Vojta, Paul (1987). Diophantine approximations and value distribution theory. Lecture Notes in Mathematics. Vol. 1239. Berlin, New York: Springer-Verlag. doi:10.1007/BFb0072989. ISBN 978-3-540-17551-3. MR 0883451. Zbl 0609.14011.
• Faltings, Gerd (1991). "Diophantine approximation on abelian varieties". Annals of Mathematics. 123 (3): 549–576. doi:10.2307/2944319. MR 1109353.
| Wikipedia |
Vojtěch Jarník
Vojtěch Jarník (Czech pronunciation: [ˈvojcɛx ˈjarɲiːk]; 22 December 1897 – 22 September 1970) was a Czech mathematician. He worked for many years as a professor and administrator at Charles University, and helped found the Czechoslovak Academy of Sciences. He is the namesake of Jarník's algorithm for minimum spanning trees.
Vojtěch Jarník
Born(1897-12-22)22 December 1897
Prague, Bohemia, Austria-Hungary
Died22 September 1970(1970-09-22) (aged 72)
Prague, Czechoslovakia
NationalityCzechoslovakia
Known for
• Diophantine approximation
• Lattice point problems
• Jarník's algorithm
• Mathematical analysis
Scientific career
FieldsMathematics
InstitutionsCharles University
Doctoral advisorKarel Petr
Other academic advisorsEdmund Landau
Doctoral students
• Miroslav Katětov
• Jaroslav Kurzweil
• Tibor Šalát
Jarník worked in number theory, mathematical analysis, and graph algorithms. He has been called "probably the first Czechoslovak mathematician whose scientific works received wide and lasting international response".[1] As well as developing Jarník's algorithm, he found tight bounds on the number of lattice points on convex curves, studied the relationship between the Hausdorff dimension of sets of real numbers and how well they can be approximated by rational numbers, and investigated the properties of nowhere-differentiable functions.
Education and career
Jarník was born on 22 December 1897. He was the son of Jan Urban Jarník, a professor of Romance language philology at Charles University,[2] and his older brother, Hertvík Jarník, also became a professor of linguistics.[3] Despite this background, Jarník learned no Latin at his gymnasium (the C.K. české vyšší reálné gymnasium, Ječná, Prague), so when he entered Charles University in 1915 he had to do so as an extraordinary student until he could pass a Latin examination three semesters later.[3]
He studied mathematics and physics at Charles University from 1915 to 1919, with Karel Petr as a mentor. After completing his studies, he became an assistant to Jan Vojtěch at the Brno University of Technology, where he also met Mathias Lerch.[3] In 1921 he completed a doctoral degree (RNDr.) at Charles University with a dissertation on Bessel functions supervised by Petr,[3] then returned to Charles University as Petr's assistant.[3][1][4]
While keeping his position at Charles University, he studied with Edmund Landau at the University of Göttingen from 1923 to 1925 and again from 1927 to 1929.[5] On his first return to Charles University he defended his habilitation, and on his return from the second visit, he was given a chair in mathematics as an extraordinary professor. He was promoted to full professor in 1935 and later served as Dean of Sciences (1947–1948) and Vice-Rector (1950–1953). He retired in 1968.[1][4]
Jarník supervised the dissertations of 16 doctoral students. Notable among these are Miroslav Katětov, a chess master who became rector of Charles University, Jaroslav Kurzweil, known for the Henstock–Kurzweil integral, and Slovak mathematician Tibor Šalát.[3][6]
He died on 22 September 1970, at the age of 72.[1]
Contributions
Although Jarník's 1921 dissertation,[1] like some of his later publications, was in mathematical analysis, his main area of work was in number theory. He studied the Gauss circle problem and proved a number of results on Diophantine approximation, lattice point problems, and the geometry of numbers.[4] He also made pioneering, but long-neglected, contributions to combinatorial optimization.[7]
Number theory
The Gauss circle problem asks for the number of points of the integer lattice enclosed by a given circle. One of Jarník's theorems (1926), related to this problem, is that any closed strictly convex curve with length L passes through at most
${\frac {3}{\sqrt[{3}]{2\pi }}}L^{2/3}+O(L^{1/3})$
points of the integer lattice. The $O$ in this formula is an instance of Big O notation. Neither the exponent of L nor the leading constant of this bound can be improved, as there exist convex curves with this many grid points.[8][9]
Another theorem of Jarník in this area shows that, for any closed convex curve in the plane with a well-defined length, the absolute difference between the area it encloses and the number of integer points it encloses is at most its length.[10]
Jarník also published several results in Diophantine approximation, the study of the approximation of real numbers by rational numbers. He proved (1928–1929) that the badly approximable real numbers (the ones with bounded terms in their continued fractions) have Hausdorff dimension one. This is the same dimension as the set of all real numbers, intuitively suggesting that the set of badly approximable numbers is large. He also considered the numbers x for which there exist infinitely many good rational approximations p/q, with
$\left|x-{\frac {p}{q}}\right|<{\frac {1}{q^{k}}}$
for a given exponent k > 2, and proved (1929) that these have the smaller Hausdorff dimension 2/k. The second of these results was later rediscovered by Besicovitch.[11] Besicovitch used different methods than Jarník to prove it, and the result has come to be known as the Jarník–Besicovitch theorem.[12]
Mathematical analysis
Jarník's work in real analysis was sparked by finding, in the unpublished works of Bernard Bolzano, a definition of a continuous function that was nowhere differentiable. Bolzano's 1830 discovery predated the 1872 publication of the Weierstrass function, previously considered to be the first example of such a function. Based on his study of Bolzano's function, Jarník was led to a more general theorem: If a real-valued function of a closed interval does not have bounded variation in any subinterval, then there is a dense subset of its domain on which at least one of its Dini derivatives is infinite. This applies in particular to the nowhere-differentiable functions, as they must have unbounded variation in all intervals. Later, after learning of a result by Stefan Banach and Stefan Mazurkiewicz that generic functions (that is, the members of a residual set of functions) are nowhere differentiable, Jarník proved that at almost all points, all four Dini derivatives of such a function are infinite. Much of his later work in this area concerned extensions of these results to approximate derivatives.[13]
Combinatorial optimization
In computer science and combinatorial optimization, Jarník is known for an algorithm for constructing minimum spanning trees that he published in 1930, in response to the publication of Borůvka's algorithm by another Czech mathematician, Otakar Borůvka.[14] Jarník's algorithm builds a tree from a single starting vertex of a given weighted graph by repeatedly adding the cheapest connection to any other vertex, until all vertices have been connected. The same algorithm was later rediscovered in the late 1950s by Robert C. Prim and Edsger W. Dijkstra. It is also known as Prim's algorithm or the Prim–Dijkstra algorithm.[15]
He also published a second, related, paper with Miloš Kössler (1934) on the Euclidean Steiner tree problem. In this problem, one must again form a tree connecting a given set of points, with edge costs given by the Euclidean distance. However, additional points that are not part of the input may be added to make the overall tree shorter. This paper is the first serious treatment of the general Steiner tree problem (although it appears earlier in a letter by Gauss), and it already contains "virtually all general properties of Steiner trees" later attributed to other researchers.[7]
Recognition and legacy
Jarník was a member of the Czech Academy of Sciences and Arts, from 1934 as an extraordinary member and from 1946 as a regular member.[1] In 1952 he became one of the founding members of Czechoslovak Academy of Sciences.[1][4] He was also awarded the Czechoslovak State Prize in 1952.[1]
The Vojtěch Jarník International Mathematical Competition, held each year since 1991 in Ostrava, is named in his honor,[16] as is Jarníkova Street in the Chodov district of Prague. A series of postage stamps published by Czechoslovakia in 1987 to honor the 125th anniversary of the Union of Czechoslovak mathematicians and physicists included one stamp featuring Jarník together with Joseph Petzval and Vincenc Strouhal.[17]
A conference was held in Prague, in March 1998, to honor the centennial of his birth.[1]
Since 2002, ceremonial Jarník's lecture is held every year at Faculty of Mathematics and Physics, Charles University, in a lecture hall named after him.[18]
Selected publications
Jarník published 90 papers in mathematics,[19] including:
• Jarník, Vojtěch (1923), "O číslech derivovaných funkcí jedné reálné proměnné" [On derivative numbers of functions of a real variable], Časopis Pro Pěstování Matematiky a Fysiky (in Czech), 53: 98–101, doi:10.21136/CPMF.1924.109353, JFM 50.0189.02. A function with unbounded variation in all intervals has a dense set of points where a Dini derivative is infinite.[13]
• Jarník, Vojtěch (1926), "Über die Gitterpunkte auf konvexen Kurven" [On the grid points on convex curves], Mathematische Zeitschrift (in German), 24 (1): 500–518, doi:10.1007/BF01216795, MR 1544776, S2CID 117747514. Tight bounds on the number of integer points on a convex curve, as a function of its length.
• Jarník, Vojtĕch (1928–1929), "Zur metrischen Theorie der diophantischen Approximationen" [On the metric theory of Diophantine approximations], Prace Matematyczno-Fizyczne (in German), Warszawa, 36: 91–106, JFM 55.0718.01. The badly-approximable numbers have Hausdorff dimension one.[11]
• Jarník, Vojtĕch (1929), "Diophantische Approximationen und Hausdorffsches Maß" [Diophantine approximation and the Hausdorff measure], Matematicheskii Sbornik (in German), 36: 371–382, JFM 55.0719.01. The well-approximable numbers have Hausdorff dimension less than one.[11]
• Jarník, Vojtěch (1930), "O jistém problému minimálním. (Z dopisu panu O. Borůvkovi)" [About a certain minimal problem (from a letter to O. Borůvka)], Práce Moravské Přírodovědecké Společnosti (in Czech), 6: 57–63. The original reference for Jarnik's algorithm for minimum spanning trees.[7]
• Jarník, Vojtěch (1933), "Über die Differenzierbarkeit stetiger Funktionen" [On the differentiability of continuous functions], Fundamenta Mathematicae (in German), 21: 48–58, doi:10.4064/fm-21-1-48-58, Zbl 0007.40102. Generic functions have infinite Dini derivatives at almost all points.[13]
• Jarník, Vojtěch; Kössler, Miloš (1934), "O minimálních grafech, obsahujících n daných bodů" [On minimal graphs containing n given points], Časopis pro Pěstování Matematiky a Fysiky (in Czech), 63 (8): 223–235, doi:10.21136/CPMF.1934.122548, Zbl 0009.13106. The first serious treatment of the Steiner tree problem.[7]
He was also the author of ten textbooks in Czech, on integral calculus, differential equations, and mathematical analysis.[19] These books "became classics for several generations of students".[20]
References
1. Netuka, Ivan (1998). "In memoriam Prof. Vojtěch Jarník (22. 12. 1897 – 22. 9. 1970)" (PDF). News and Notes. Mathematica Bohemica. 123 (2): 219–221. doi:10.21136/MB.1998.126302..
2. Durnová (2004), p. 168.
3. Veselý, Jiří (1999), "Pedagogical activities of Vojtěch Jarník", in Novák, Břetislav (ed.), Life and work of Vojtěch Jarník, Prague: Union of Czech mathematicians and physicists, pp. 83–94, ISBN 80-7196-156-6.
4. O'Connor, John J.; Robertson, Edmund F., "Vojtěch Jarník", MacTutor History of Mathematics Archive, University of St Andrews
5. Netuka (1998) and Veselý (1999); however, O'Connor and Robertson give his return dates as 1924 and 1928.
6. Vojtěch Jarník at the Mathematics Genealogy Project,
7. Korte, Bernhard; Nešetřil, Jaroslav (2001). "Vojtěch Jarník's work in combinatorial optimization". Discrete Mathematics. 235 (1–3): 1–17. doi:10.1016/S0012-365X(00)00256-9. hdl:10338.dmlcz/500662. MR 1829832.
8. Bordellès, Olivier (2012), "5.4.7 Counting integer points on smooth curves", Arithmetic Tales, Springer, p. 290, ISBN 9781447140962.
9. Huxley, M. N. (1996), "2.2 Jarník's polygon", Area, Lattice Points, and Exponential Sums, London Mathematical Society Monographs, vol. 13, Clarendon Press, pp. 31–33, ISBN 9780191590320.
10. Redmond, Don (1996), Number Theory: An Introduction to Pure and Applied Mathematics, CRC Press, p. 561, ISBN 9780824796969.
11. Dodson, M. M. (1999), "Some recent extensions of Jarník's work in Diophantine approximation", in Novák, Břetislav (ed.), Life and work of Vojtěch Jarník, Prague: Union of Czech mathematicians and physicists, pp. 23–36, ISBN 80-7196-156-6.
12. Beresnevich, Victor; Ramírez, Felipe; Velani, Sanju (2016), "Metric Diophantine approximation: Aspects of recent work", in Badziahin, Dzmitry; Gorodnik, Alexander; Peyerimhoff, Norbert (eds.), Dynamics and Analytic Number Theory: Proceedings of the Durham Easter School 2014, London Mathematical Society Lecture Note Series, vol. 437, Cambridge University Press, pp. 1–95, arXiv:1601.01948, doi:10.1017/9781316402696.002, S2CID 119304793. See Theorem 1.33 (the Jarník–Besicovitch theorem), p. 23, and the discussion following the theorem.
13. Preiss, David (1999), "The work of Professor Jarník in real analysis", in Novák, Břetislav (ed.), Life and work of Vojtěch Jarník, Prague: Union of Czech mathematicians and physicists, pp. 55–66, ISBN 80-7196-156-6.
14. Durnová, Helena (2004), "A history of discrete optimization", in Fuchs, Eduard (ed.), Mathematics Throughout the Ages, Vol. II, Prague: Výzkumné centrum pro dějiny vědy, pp. 51–184, ISBN 9788072850464. See in particular page 127: "Soon after Borůvka's published his solution, another Czech mathematician, Vojtěch Jarník, reacted by publishing his own solution," and page 133: "Jarník’s article on this topic is an extract from a letter to O. Borůvka".
15. Sedgewick, Robert; Wayne, Kevin (2011). Algorithms (4th ed.). Addison-Wesley Professional. p. 628. ISBN 9780132762564..
16. "Vojtěch Jarník International Mathematical Competition". Retrieved 16 February 2017.
17. Miller, Jeff. "Images of Mathematicians on Postage Stamps". Retrieved 2017-02-17..
18. Ceremonial Lectures, mff.cuni.cz
19. Novák, Břetislav, ed. (1999), "Bibliography of scientific works of V. Jarník", Life and work of Vojtěch Jarník, Prague: Union of Czech mathematicians and physicists, pp. 133–142, ISBN 80-7196-156-6.
20. Vojtěch Jarník, Czech Digital Mathematics Library, 2010, retrieved 2017-02-17.
Further reading
• Novák, Břetislav, ed. (1999), Life and work of Vojtěch Jarník, Prague: Union of Czech mathematicians and physicists, ISBN 80-7196-156-6.
• Vojtěch Jarník digital archive, Czech Digital Mathematics Library
External links
• Media related to Vojtěch Jarník at Wikimedia Commons
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| Wikipedia |
Volkenborn integral
In mathematics, in the field of p-adic analysis, the Volkenborn integral is a method of integration for p-adic functions.
Definition
Let :$f:\mathbb {Z} _{p}\to \mathbb {C} _{p}$ be a function from the p-adic integers taking values in the p-adic numbers. The Volkenborn integral is defined by the limit, if it exists:
$\int _{\mathbb {Z} _{p}}f(x)\,{\rm {d}}x=\lim _{n\to \infty }{\frac {1}{p^{n}}}\sum _{x=0}^{p^{n}-1}f(x).$
More generally, if
$R_{n}=\left\{\left.x=\sum _{i=r}^{n-1}b_{i}x^{i}\right|b_{i}=0,\ldots ,p-1{\text{ for }}r<n\right\}$
then
$\int _{K}f(x)\,{\rm {d}}x=\lim _{n\to \infty }{\frac {1}{p^{n}}}\sum _{x\in R_{n}\cap K}f(x).$
This integral was defined by Arnt Volkenborn.
Examples
$\int _{\mathbb {Z} _{p}}1\,{\rm {d}}x=1$
$\int _{\mathbb {Z} _{p}}x\,{\rm {d}}x=-{\frac {1}{2}}$
$\int _{\mathbb {Z} _{p}}x^{2}\,{\rm {d}}x={\frac {1}{6}}$
$\int _{\mathbb {Z} _{p}}x^{k}\,{\rm {d}}x=B_{k}$
where $B_{k}$ is the k-th Bernoulli number.
The above four examples can be easily checked by direct use of the definition and Faulhaber's formula.
$\int _{\mathbb {Z} _{p}}{x \choose k}\,{\rm {d}}x={\frac {(-1)^{k}}{k+1}}$
$\int _{\mathbb {Z} _{p}}(1+a)^{x}\,{\rm {d}}x={\frac {\log(1+a)}{a}}$
$\int _{\mathbb {Z} _{p}}e^{ax}\,{\rm {d}}x={\frac {a}{e^{a}-1}}$
The last two examples can be formally checked by expanding in the Taylor series and integrating term-wise.
$\int _{\mathbb {Z} _{p}}\log _{p}(x+u)\,{\rm {d}}u=\psi _{p}(x)$
with $\log _{p}$ the p-adic logarithmic function and $\psi _{p}$ the p-adic digamma function.
Properties
$\int _{\mathbb {Z} _{p}}f(x+m)\,{\rm {d}}x=\int _{\mathbb {Z} _{p}}f(x)\,{\rm {d}}x+\sum _{x=0}^{m-1}f'(x)$
From this it follows that the Volkenborn-integral is not translation invariant.
If $P^{t}=p^{t}\mathbb {Z} _{p}$ then
$\int _{P^{t}}f(x)\,{\rm {d}}x={\frac {1}{p^{t}}}\int _{\mathbb {Z} _{p}}f(p^{t}x)\,{\rm {d}}x$
See also
• P-adic distribution
References
• Arnt Volkenborn: Ein p-adisches Integral und seine Anwendungen I. In: Manuscripta Mathematica. Bd. 7, Nr. 4, 1972,
• Arnt Volkenborn: Ein p-adisches Integral und seine Anwendungen II. In: Manuscripta Mathematica. Bd. 12, Nr. 1, 1974,
• Henri Cohen, "Number Theory", Volume II, page 276
| Wikipedia |
Volker Strassen
Volker Strassen (born April 29, 1936) is a German mathematician, a professor emeritus in the department of mathematics and statistics at the University of Konstanz.[1]
Volker Strassen
Volker Strassen giving the Knuth Prize lecture at SODA 2009
Born (1936-04-29) April 29, 1936
Düsseldorf-Gerresheim, Germany
NationalityGerman
Alma materUniversity of Göttingen
Known forStrassen algorithm
Scientific career
FieldsMathematics
InstitutionsUniversity of Konstanz
Doctoral advisorKonrad Jacobs
Doctoral studentsPeter Bürgisser
Joachim von zur Gathen
For important contributions to the analysis of algorithms he has received many awards, including the Cantor medal,[2] the Konrad Zuse Medal,[3] the Paris Kanellakis Award for work on randomized primality testing,[4] the Knuth Prize for "seminal and influential contributions to the design and analysis of efficient algorithms."[5]
Biography
Strassen was born on April 29, 1936, in Düsseldorf-Gerresheim.[2] After studying music, philosophy, physics, and mathematics at several German universities,[2] he received his Ph.D. in mathematics in 1962 from the University of Göttingen under the supervision of Konrad Jacobs.[6] He then took a position in the department of statistics at the University of California, Berkeley while performing his habilitation at the University of Erlangen-Nuremberg, where Jacobs had since moved.[2] In 1968, Strassen moved to the Institute of Applied Mathematics at the University of Zurich, where he remained for twenty years before moving to the University of Konstanz in 1988.[2] He retired in 1998.[4]
Research
Strassen began his researches as a probabilist; his 1964 paper An Invariance Principle for the Law of the Iterated Logarithm defined a functional form of the law of the iterated logarithm, showing a form of scale invariance in random walks. This result, now known as Strassen's invariance principle or as Strassen's law of the iterated logarithm, has been highly cited and led to a 1966 presentation at the International Congress of Mathematicians.
In 1969, Strassen shifted his research efforts towards the analysis of algorithms with a paper on Gaussian elimination, introducing Strassen's algorithm, the first algorithm for performing matrix multiplication faster than the O(n3) time bound that would result from a naive algorithm. In the same paper he also presented an asymptotically fast algorithm to perform matrix inversion, based on the fast matrix multiplication algorithm. This result was an important theoretical breakthrough, leading to much additional research on fast matrix multiplication, and despite later theoretical improvements it remains a practical method for multiplication of dense matrices of moderate to large sizes. In 1971 Strassen published another paper together with Arnold Schönhage on asymptotically fast integer multiplication based on the fast Fourier transform; see the Schönhage–Strassen algorithm. Strassen is also known for his 1977 work with Robert M. Solovay on the Solovay–Strassen primality test, the first method to show that testing whether a number is prime can be performed in randomized polynomial time and one of the first results to show the power of randomized algorithms more generally.
Awards and honors
In 1999 Strassen was awarded the Cantor medal,[2] and in 2003 he was co-recipient of the Paris Kanellakis Award with Robert Solovay, Gary Miller, and Michael Rabin for their work on randomized primality testing.[4] In 2008 he was awarded the Knuth Prize for "seminal and influential contributions to the design and analysis of efficient algorithms."[5] In 2011 he won the Konrad Zuse Medal of the Gesellschaft für Informatik.[3][7] In 2012 he became a fellow of the American Mathematical Society.[8]
References
1. FB Mathematik and Statistik Archived 2008-12-25 at the Wayback Machine, U. Konstanz.
2. Schönhage, A. (2000), "Cantor-Medaille für Volker Strassen" (PDF), Jahresbericht der Deutschen Mathematiker-Vereinigung, 102 (4).
3. Winter, Cornelia (September 28, 2011), "Konrad-Zuse-Medaille für Informatik an Fritz-Rudolf Güntsch und Volker Strassen", Informationsdienst Wissenschaft (in German).
4. Preis für Prof. Volker Strassen, uni'kon 16.2004, Univ. of Konstanz.
5. The 2008 Knuth Prize is awarded to Volker Strassen for his seminal and influential contributions to efficient algorithms, ACM SIGACT.
6. Volker Strassen at the Mathematics Genealogy Project
7. Konrad-Zuse-Medaille Archived 2014-08-19 at the Wayback Machine, Gesellschaft für Informatik (in German), retrieved 2012-03-09.
8. List of Fellows of the American Mathematical Society, retrieved 2013-08-05.
External links
• Home page of Dr. Volker Strassen
• Weisstein, Eric W. "Strassen Formulas". MathWorld. Formulas for fast(er) matrix multiplication and inversion.
• O'Connor, John J.; Robertson, Edmund F., "Volker Strassen", MacTutor History of Mathematics Archive, University of St Andrews
Winners of the Paris Kanellakis Theory and Practice Award
• Adleman, Diffie, Hellman, Merkle, Rivest, Shamir (1996)
• Lempel, Ziv (1997)
• Bryant, Clarke, Emerson, McMillan (1998)
• Sleator, Tarjan (1999)
• Karmarkar (2000)
• Myers (2001)
• Franaszek (2002)
• Miller, Rabin, Solovay, Strassen (2003)
• Freund, Schapire (2004)
• Holzmann, Kurshan, Vardi, Wolper (2005)
• Brayton (2006)
• Buchberger (2007)
• Cortes, Vapnik (2008)
• Bellare, Rogaway (2009)
• Mehlhorn (2010)
• Samet (2011)
• Broder, Charikar, Indyk (2012)
• Blumofe, Leiserson (2013)
• Demmel (2014)
• Luby (2015)
• Fiat, Naor (2016)
• Shenker (2017)
• Pevzner (2018)
• Alon, Gibbons, Matias, Szegedy (2019)
• Azar, Broder, Karlin, Mitzenmacher, Upfal (2020)
• Blum, Dinur, Dwork, McSherry, Nissim, Smith (2021)
• Burrows, Ferragina, Manzini (2022)
Knuth Prize laureates
1990s
• Yao (1996)
• Valiant (1997)
• Lovász (1999)
2000s
• Ullman (2000)
• Papadimitriou (2002)
• Ajtai (2003)
• Yannakakis (2005)
• Lynch (2007)
• Strassen (2008)
2010s
• Johnson (2010)
• Kannan (2011)
• Levin (2012)
• Miller (2013)
• Lipton (2014)
• Babai (2015)
• Nisan (2016)
• Goldreich (2017)
• Håstad (2018)
• Wigderson (2019)
2020s
• Dwork (2020)
• Vardi (2021)
• Alon (2022)
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| Wikipedia |
Volodin space
In mathematics, more specifically in topology, the Volodin space $X$ of a ring R is a subspace of the classifying space $BGL(R)$ given by
$X=\bigcup _{n,\sigma }B(U_{n}(R)^{\sigma })$
where $U_{n}(R)\subset GL_{n}(R)$ is the subgroup of upper triangular matrices with 1's on the diagonal (i.e., the unipotent radical of the standard Borel) and $\sigma $ a permutation matrix thought of as an element in $GL_{n}(R)$ and acting (superscript) by conjugation.[1] The space is acyclic and the fundamental group $\pi _{1}X$ is the Steinberg group $\operatorname {St} (R)$ of R. In fact, Suslin (1981) showed that X yields a model for Quillen's plus-construction $BGL(R)/X\simeq BGL^{+}(R)$ in algebraic K-theory.
Application
An analogue of Volodin's space where GL(R) is replaced by the Lie algebra ${\mathfrak {gl}}(R)$ was used by Goodwillie (1986) to prove that, after tensoring with Q, relative K-theory K(A, I), for a nilpotent ideal I, is isomorphic to relative cyclic homology HC(A, I). This theorem was a pioneering result in the area of trace methods.
Notes
1. Weibel 2013, Ch. IV. Example 1.3.2.
References
• Goodwillie, Thomas G. (1986), "Relative algebraic K-theory and cyclic homology", Annals of Mathematics, Second Series, 124 (2): 347–402, doi:10.2307/1971283, JSTOR 1971283, MR 0855300
• Weibel, Charles (2013). "The K-book: an introduction to algebraic K-theory".
• Suslin, A. A. (1981), "On the equivalence of K-theories", Comm. Algebra, 9 (15): 1559–66, doi:10.1080/00927878108822666
• Volodin, I. (1971), "Algebraic K-theory as extraordinary homology theory on the category of associative rings with unity", Izv. Akad. Nauk. SSSR, 35 (4): 844–873, Bibcode:1971IzMat...5..859V, doi:10.1070/IM1971v005n04ABEH001121, MR 0296140, (Translation: Math. USSR Izvestija Vol. 5 (1971) No. 4, 859–887)
| Wikipedia |
Volodymyr Koshmanenko
Volodymyr Koshmanenko (Ukrainian: Кошманенко Володимир Дмитрович; born July 28, 1943, Dnipro, Ukraine) — Ukrainian mathematician, Doctor of Science, professor, Leading Researcher of the Institute of Mathematics of the NAS of Ukraine.
Volodymyr Koshmanenko
Born(1943-07-28)July 28, 1943
NationalityUkrainian
AwardsYu.O. Mitropol'sky Award by National Academy of Sciences of Ukraine
Scientific career
FieldsMathematics
InstitutionsInstitute of Mathematics of the NAS of Ukraine
Volodymyr Koshmanenko is a notable Ukrainian mathematician and a talented researcher. Prof. Koshmanenko has been reading lectures at Taras Shevchenko University, National Pedagogical Dragomanov University and National University of Kyiv-Mohyla Academy. He has over 120 publications and 5 monographs. Volodymyr Koshmanenko promotes creativity in science, incredible performance, healthy lifestyle.
Biography
In 1960, he entered the Department of Physics at Dnipropetrovsk State University and graduated from it in 1966. He attended the lectures of the mathematical content mainly from the third year of study. This led him to choice of the mathematical style of thinking.
During his post-graduate courses, 1967–1970, he studied the axiomatic approach in quantum field theory. He showed that any Boson scalar quantum field admits the representation and the axiomatic formulation in terms of operator Jacobi matrixes. It was the main result of his PhD thesis (1970) (the scientific advisor was Prof. Yu. M. Berezansky).
From 1970 up to now he occupied different scientific positions, from junior to leading researcher at the Institute of Mathematics of the NAS of Ukraine in Kyiv. In 1985, he got the Doctor degree in mathematics for the theses "The scattering theory in terms of bilinear functionals" with M.S. Birman, I.Ya. Arefieva, and M.I. Portenko as the main referees.
In 1995, he became the professor of Higher Mathematics Department in Kyiv Pedagogical University.
Professional activity
• Member of the Academic Council Institute of Mathematics of the NAS of Ukraine
• Member Kyïv Mathematical Society
• Member of the editorial board Methods of Functional Analysis and Topology
• Leader of the seminar Complex Conflict Systems: Dynamics, Models, Spectral Analysis at the Institute of Mathematics of the NAS of Ukraine
Awards
2012 — Yu.O. Mitropol'sky Award by National Academy of Sciences of Ukraine.
Research area
The research interests of Prof. V. Koshmanenko concern modeling of complex dynamical systems, fractal geometry, functional analysis, operator theory, mathematical physics. He proposed the construction of wave and scattering operators in terms of bilinear functionals, introduced the notion of singular quadratic form and produced the classification of pure singular quadratic forms, developed the self-adjoint extensions approach to the singular perturbation theory in scales of Hilbert spaces, investigated the direct and inverse negative eigenvalues problem under singular perturbations.
Volodymyr Koshmanenko developed the original theory of conflict dynamical systems and built a serious new models of complex dynamical systems with repulsive and attractive interaction. He proved the theorem of conflict in terms of probability measures, showed the possibility in fractal setting to reconstruct the lost physical type spectrum under interaction with a source of purely singular (spirit) continuous spectrum. He introduced a notion of the structural similarity measures and proposed a series models of complex dynamical system with conflict interaction of type conflict triad, fire-water model, society as a mathematical models of conflict system where the invariant fixed points, the limiting cyclic orbits and their attraction basins are investigated.
Main publications
• Volodymyr Koshmanenko, Viktoria Voloshyna, The emergence of point spectrum in models of conflict dynamical systems, Ukrainian Math. J., v. 70, 12, 1615-1624, (2018).
• T. Karataieva, V. Koshmanenko, M. Krawczyk, K. Kulakowski, "Mean field model of a game for power" , 15 p. (Feb 2018, arXiv:1802.02860)
• V. Koshmanenko, N. Kharchenko,
Fixed points of complex systems with attractive interaction, MFAT, {\bf 23}, no. 2, 164 - 176, (2017).
• Koshmanenko, V. Spectral Theory for Conflict Dynamical Systems (Ukrainian), Naukova Dumka, Kyiv, 2016, 288p.
• Koshmanenko, V.; Dudkin M. Method of Rigged Spaces in Singular Perturbation Theory of Self-adjoint Operators. Birkhäuser, 2016, 237p.
• Koshmanenko V., Verygina I. Dynamical systems of conflict in terms of structural measures. Meth. Funct. Anal. and Top. 22, No 1, 81-93, (2016).
• Koshmanenko, V., Karataieva, T., Kharchenko, N., and Verygina, I. Models of the Conflict Redistribution of Vital Resources, SSC (2016).
• Koshmanenko, V. Existence theorems of the omega-limit states for conflict dynamical systems, Methods Funct. Anal. and Top. 20, No. 4, 379-390, (2014).
• Koshmanenko, V. Singular Quadratic Forms in Perturbation Theory, Kluwer, Dordrecht, 1999.
• Koshmanenko, V.; Samoilenko, I. The conflict triad dynamical system. Commun. Nonlinear Sci. Numer. Simul. 16, No. 7, 2917–2935 (2011).
• Albeverio, S.; Konstantinov, A.; Koshmanenko, V. Remarks on the Inverse Spectral Theory for Singularly Perturbed Operators, Operator Theory: Advance and Appl., 190, 115–122 (2009).
• Albeverio, S.; Koshmanenko, V.; Samoilenko, I. The conflict interaction between two complex systems: Cyclic migration, J. Interdisciplinary Math., 11, No 2, 163–185, (2008).
• Koshmanenko, V. Construction of singular perturbations by the method of rigged Hilbert spaces, Journal of Physics A: Mathematical and General, 38, 4999–5009 (2005).
• V. Koshmanenko, Theorem of conflicts for a pair of probability measures, Math. Methods of Operations Research, 59, 303–313, (2004).
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| Wikipedia |
Volodymyr Levytsky
Volodymyr Levytsky (31 December 1872 – 13 August 1956) was a Ukrainian mathematician who taught mathematics and studied functions of a complex variable.
Volodymyr Levytsky
Born(1872-12-31)December 31, 1872
Ternopil, Galicia
DiedAugust 13, 1956(1956-08-13) (aged 83)
Lviv, Ukraine
CitizenshipUkrainian
Known forFunctions of a complex variable
Scientific career
FieldsMathematician
Biography and education
Volodymyr Levytsky finished his doctorate at the University of Lviv in 1901 and went on to teach mathematics and physics at high schools. After the First World War Ukrainian students were not allowed to enrol at the University and in 1920 Ukrainian professors were also banned leaving only Polish lecturers. As a result, the Ukrainian students set up an underground university at the University in July 1921. From the beginning Levytsky taught mathematics at this new underground university for a few years until it was forced to close in 1925. Levytsky was the head of the mathematics-physics section of the Shevchenko Scientific Society in Lviv and was the President of the Society from 1931 to 1935 as well as its editor of the Journal. Just before the outbreak of the war until his death in 1956, Levytsky taught at the Lviv Pedagogical Institute. [1]
Works
Levytsky concentrated on functions of a complex variable and the application of mathematics to theoretical physics. The first mathematical paper in Ukrainian was written by Levytsky as well as being an editor of the first Ukrainian mathematical journal. He brought mathematics and physical and chemical terms to the Ukrainian language through his efforts at the Shevchenko Scientific Society in Lviv. During his short time at the Lviv (Underground) Ukrainian University he produced important papers on the history of mathematics. [1][2]
References
1. "Volodymyr Levytsky biography". Archived from the original on 2016-04-01. Retrieved 2019-01-12.
2. Kravchuk, M.F. (1990). "Letters to V I Levitskii (1926–1931) With commentaries by P K Khobzei". Ocherki Istor. Estestvoznan. Tekhn. 38: 103–118.
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Volodymyr Mazorchuk
Volodymyr Mazorchuk (born 1972) is a Ukrainian-Swedish mathematician at Uppsala University and was awarded the Göran Gustafsson Prize in 2016.[1][2]
He received his PhD in mathematics from Taras Shevchenko National University of Kyiv in 1996 with advisor Yuriy Drozd.[3]
Selected publications
Articles
• König, Steffen; Mazorchuk, Volodymyr (2002). "Enright's completions and injectively copresented modules". Transactions of the American Mathematical Society. 354 (7): 2725–2743. doi:10.1090/S0002-9947-02-02958-6.
• Mazorchuk, Volodymyr; Stroppel, Catharina (2004). "Translation and shuffling of projectively presentable modules and a categorification of a parabolic Hecke module". Transactions of the American Mathematical Society. 357 (7): 2939–2973. doi:10.1090/S0002-9947-04-03650-5.
• Khovanov, Mikhail; Mazorchuk, Volodymyr; Stroppel, Catharina (2007). "A categorification of integral Specht modules" (PDF). Proceedings of the American Mathematical Society. 136 (4): 1163–1169. doi:10.1090/S0002-9939-07-09124-1. S2CID 7656634.
• Mazorchuk, Volodymyr; Ovsienko, Serge; Stroppel, Catharina (2008). "Quadratic duals, Koszul dual functors, and applications". Transactions of the American Mathematical Society. 361 (3): 1129–1172. arXiv:math/0603475. doi:10.1090/S0002-9947-08-04539-X. S2CID 779678.
• Ganyushkin, Olexandr; Mazorchuk, Volodymyr; Steinberg, Benjamin (2009). "On the irreducible representations of a finite semigroup". Proceedings of the American Mathematical Society. 137 (11): 3585. arXiv:0712.2076. doi:10.1090/S0002-9939-09-09857-8. S2CID 1805531.
• Mazorchuk, Volodymyr (2010). "Some homological properties of the category 𝒪, II". Representation Theory. 14 (6): 249–263. arXiv:0909.2729. doi:10.1090/S1088-4165-10-00368-7. S2CID 6089238.
• Mazorchuk, Volodymyr; Wiesner, Emilie (2014). "Simple Virasoro modules induced from codimension one subalgebras of the positive part". Proceedings of the American Mathematical Society. 142 (11): 3695–3703. arXiv:1209.1691. doi:10.1090/S0002-9939-2014-12098-3. S2CID 8112726.
• Futorny, Vyacheslav; Grantcharov, Dimitar; Mazorchuk, Volodymyr (2014). "Weight modules over infinite dimensional Weyl algebras". Proceedings of the American Mathematical Society. 142 (9): 3049–3057. arXiv:1207.5780. doi:10.1090/S0002-9939-2014-12071-5. S2CID 48992277.
• Mazorchuk, Volodymyr; Miemietz, Vanessa (2015). "Transitive $2$-representations of finitely $2$-categories" (PDF). Transactions of the American Mathematical Society. 368 (11): 7623–7644. doi:10.1090/tran/6583. S2CID 15404906.
• Kildetoft, Tobias; MacKaay, Marco; Mazorchuk, Volodymyr; Zimmermann, Jakob (2018). "Simple transitive $2$-representations of small quotients of Soergel bimodules". Transactions of the American Mathematical Society. 371 (8): 5551–5590. arXiv:1605.01373. doi:10.1090/tran/7456. S2CID 119314739.
• Chen, Chih-Whi; Mazorchuk, Volodymyr (2020). "Simple supermodules over Lie superalgebras". Transactions of the American Mathematical Society. 374 (2): 899–921. arXiv:1801.00654. doi:10.1090/tran/8303. S2CID 119307087.
Books
• Lectures on ${\mathfrak {sl}}_{2}(\mathbb {C} )$-modules. World Scientific Publishing Company. 4 December 2009. ISBN 9781911299448.[4]
• Lectures on Algebraic Categorification. European Mathematical Society. 2012. ISBN 9783037191088.[5]
References
1. "Volodymyr Mazorchuk". uu.se. Retrieved April 26, 2017.
2. "Volodymyr Mazorchuk". uu.se. Retrieved April 26, 2017.
3. Volodymyr S. Mazorchuk at the Mathematics Genealogy Project
4. Gouvêa, Fernando Q. (April 14, 2010). "Review of Lectures on ${\mathfrak {sl}}_{2}(\mathbb {C} )$-modules by Volodymyr Mazorchuk". MAA Reviews, Mathematical Association of America.
5. Berg, Michael (June 29, 2012). "Review of Lectures on Algebraic Categorification by Volodymyr Mazorchuk". MAA Reviews, Mathematical Association of America.
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Volodymyr Korolyuk
Volodymyr Semenovych Korolyuk (Ukrainian: Володимир Семенович Королюк, 19 August 1925 – 4 April 2020) was a Soviet and Ukrainian mathematician who made significant contributions to probability theory and its applications, academician of the National Academy of Sciences of Ukraine (1976).
Korolyuk was born in Kyiv in August 1925. Between 1949 and 2005 Volodymyr Korolyuk published over 300 papers and 22 monographs. He died in Kyiv in April 2020 at the age of 94.[1]
Awards and honors
Volodymyr Korolyuk has been awarded a number of scientific prizes.
• Krylov Prize of the National Academy of Sciences of Ukraine, 1976
• State Prize of the Ukrainian Soviet Socialist Republic, 1978
• Glushkov Prize of the National Academy of Sciences of Ukraine, 1988
• Bogolyubov Prize of the National Academy of Sciences of Ukraine, 1995
• Ostrogradsky Medal, 2002
• State Prize of Ukraine, 2003
References
1. Volodymyr Semenovych Korolyuk death notice
• Biography at the website of the Kyiv Mathematical Society (in Ukrainian)
• Yu. A. Mitropolskiy, A. V. Skorokhod, D. V. Gusak, Vladimir Semenovich Korolyuk (in honor of 60th anniversary), Ukrainian Math. Journal, 37, No 4, 1985, pp 488–489 (in Russian)
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Voltage graph
In graph theory, a voltage graph is a directed graph whose edges are labelled invertibly by elements of a group. It is formally identical to a gain graph, but it is generally used in topological graph theory as a concise way to specify another graph called the derived graph of the voltage graph.
Typical choices of the groups used for voltage graphs include the two-element group ℤ2 (for defining the bipartite double cover of a graph), free groups (for defining the universal cover of a graph), d-dimensional integer lattices ℤd (viewed as a group under vector addition, for defining periodic structures in d-dimensional Euclidean space),[1] and finite cyclic groups ℤn for n > 2. When Π is a cyclic group, the voltage graph may be called a cyclic-voltage graph.
Definition
Formal definition of a Π-voltage graph, for a given group Π:
• Begin with a digraph G. (The direction is solely for convenience in notation.)
• A Π-voltage on an arc of G is a label of the arc by an element x of Π. For instance, in the case where Π = ℤn, the label is a number i (mod n).
• A Π-voltage assignment is a function $\alpha :E(G)\rightarrow \Pi $ that labels each arc of G with a Π-voltage.
• A Π-voltage graph is a pair $(G,\alpha :E(G)\rightarrow \Pi )$ such that G is a digraph and α is a voltage assignment.
• The voltage group of a voltage graph $(G,\alpha :E(G)\rightarrow \Pi )$ is the group Π from which the voltages are assigned.
Note that the voltages of a voltage graph need not satisfy Kirchhoff's voltage law, that the sum of voltages around a closed path is 0 (the identity element of the group), although this law does hold for the derived graphs described below. Thus, the name may be somewhat misleading. It results from the origin of voltage graphs as dual to the current graphs of topological graph theory.
The derived graph
The derived graph of a voltage graph $(G,\alpha :E(G)\rightarrow \mathbb {Z} _{n})$ is the graph ${\tilde {G}}$ whose vertex set is ${\tilde {V}}=V\times \mathbb {Z} _{n}$ and whose edge set is ${\tilde {E}}=E\times \mathbb {Z} _{n}$, where the endpoints of an edge (e, k) such that e has tail v and head w are $(v,\ k)$ and $(w,\ k+\alpha (e))$.
Although voltage graphs are defined for digraphs, they may be extended to undirected graphs by replacing each undirected edge by a pair of oppositely ordered directed edges and by requiring that these edges have labels that are inverse to each other in the group structure. In this case, the derived graph will also have the property that its directed edges form pairs of oppositely oriented edges, so the derived graph may itself be interpreted as being an undirected graph.
The derived graph is a covering graph of the given voltage graph. If no edge label of the voltage graph is the identity element, then the group elements associated with the vertices of the derived graph provide a coloring of the derived graph with a number of colors equal to the group order. An important special case is the bipartite double cover, the derived graph of a voltage graph in which all edges are labeled with the non-identity element of a two-element group. Because the order of the group is two, the derived graph in this case is guaranteed to be bipartite.
Polynomial time algorithms are known for determining whether the derived graph of a $\mathbb {Z} ^{d}$-voltage graph contains any directed cycles.[1]
Examples
Any Cayley graph of a group Π, with a given set Γ of generators, may be defined as the derived graph for a Π-voltage graph having one vertex and Γ self-loops, each labeled with one of the generators in Γ.[2]
The Petersen graph is the derived graph for a ℤ5-voltage graph in the shape of a dumbbell with two vertices and three edges: one edge connecting the two vertices, and one self-loop on each vertex. One self-loop is labeled with 1, the other with 2, and the edge connecting the two vertices is labeled 0. More generally, the same construction allows any generalized Petersen graph GP(n,k) to be constructed as a derived graph of the same dumbbell graph with labels 1, 0, and k in the group ℤn.[3]
The vertices and edges of any periodic tessellation of the plane may be formed as the derived graph of a finite graph, with voltages in ℤ2.
Notes
1. Iwano & Steiglitz (1987); Kosaraju & Sullivan (1988); Cohen & Megiddo (1989).
2. Gross & Tucker (1987), Theorem 2.2.3, p. 69.
3. Gross & Tucker (1987), Example 2.1.2, p.58.
References
• Cohen, Edith; Megiddo, Nimrod (1989), "Strongly polynomial-time and NC algorithms for detecting cycles in dynamic graphs", Proc. 21st Annual ACM Symposium on Theory of Computing (STOC '89), pp. 523–534, doi:10.1145/73007.73057.
• Gross, Jonathan L. (1974), "Voltage graphs", Discrete Mathematics, 9 (3): 239–246, doi:10.1016/0012-365X(74)90006-5.
• Gross, Jonathan L.; Tucker, Thomas W. (1977), "Generating all graph coverings by permutation voltage assignments", Discrete Mathematics, 18 (3): 273–283, doi:10.1016/0012-365X(77)90131-5.
• Gross, Jonathan L.; Tucker, Thomas W. (1987), Topological Graph Theory, New York: Wiley.
• Iwano, K.; Steiglitz, K. (1987), "Testing for cycles in infinite graphs with periodic structure", Proc. 19th Annual ACM Symposium on Theory of Computing (STOC '87), pp. 46–55, doi:10.1145/28395.28401, S2CID 37934099.
• Kosaraju, S. Rao; Sullivan, Gregory (1988), "Detecting cycles in dynamic graphs in polynomial time", Proc. 20th Annual ACM Symposium on Theory of Computing (STOC '88), pp. 398–406, doi:10.1145/62212.62251, S2CID 14290312.
| Wikipedia |
Volterra integral equation
In mathematics, the Volterra integral equations are a special type of integral equations.[1] They are divided into two groups referred to as the first and the second kind.
A linear Volterra equation of the first kind is
$f(t)=\int _{a}^{t}K(t,s)\,x(s)\,ds$
where f is a given function and x is an unknown function to be solved for. A linear Volterra equation of the second kind is
$x(t)=f(t)+\int _{a}^{t}K(t,s)x(s)\,ds.$
In operator theory, and in Fredholm theory, the corresponding operators are called Volterra operators. A useful method to solve such equations, the Adomian decomposition method, is due to George Adomian.
A linear Volterra integral equation is a convolution equation if
$x(t)=f(t)+\int _{t_{0}}^{t}K(t-s)x(s)\,ds.$
The function $K$ in the integral is called the kernel. Such equations can be analyzed and solved by means of Laplace transform techniques.
For a weakly singular kernel of the form $K(t,s)=(t^{2}-s^{2})^{-\alpha }$ with $0<\alpha <1$, Volterra integral equation of the first kind can conveniently be transformed into a classical Abel integral equation.
The Volterra integral equations were introduced by Vito Volterra and then studied by Traian Lalescu in his 1908 thesis, Sur les équations de Volterra, written under the direction of Émile Picard. In 1911, Lalescu wrote the first book ever on integral equations.
Volterra integral equations find application in demography as Lotka's integral equation,[2] the study of viscoelastic materials, in actuarial science through the renewal equation,[3] and in fluid mechanics to describe the flow behavior near finite-sized boundaries.[4][5]
Conversion of Volterra equation of the first kind to the second kind
A linear Volterra equation of the first kind can always be reduced to a linear Volterra equation of the second kind, assuming that $K(t,t)\neq 0$. Taking the derivative of the first kind Volterra equation gives us:
${df \over {dt}}=\int _{a}^{t}{\partial K \over {\partial t}}x(s)ds+K(t,t)x(t)$
Dividing through by $K(t,t)$ yields:
$x(t)={1 \over {K(t,t)}}{df \over {dt}}-\int _{a}^{t}{1 \over {K(t,t)}}{\partial K \over {\partial t}}x(s)ds$
Defining $ {\widetilde {f}}(t)={1 \over {K(t,t)}}{df \over {dt}}$ and $ {\widetilde {K}}(t,s)=-{1 \over {K(t,t)}}{\partial K \over {\partial t}}$ completes the transformation of the first kind equation into a linear Volterra equation of the second kind.
Numerical solution using trapezoidal rule
A standard method for computing the numerical solution of a linear Volterra equation of the second kind is the trapezoidal rule, which for equally-spaced subintervals $\Delta x$ is given by:
$\int _{a}^{b}f(x)dx\approx {\Delta x \over {2}}\left[f(x_{0})+2\sum _{i=1}^{n-1}f(x_{i})+f(x_{n})\right]$
Assuming equal spacing for the subintervals, the integral component of the Volterra equation may be approximated by:
$\int _{a}^{t}K(t,s)x(s)ds\approx {\Delta s \over {2}}\left[K(t,s_{0})x(s_{0})+2K(t,s_{1})x(s_{1})+\cdots +2K(t,s_{n-1})x(s_{n-1})+K(t,s_{n})x(s_{n})\right]$
Defining $x_{i}=x(s_{i})$, $f_{i}=f(t_{i})$, and $K_{ij}=K(t_{i},s_{j})$, we have the system of linear equations:
${\begin{aligned}x_{0}&=f_{0}\\x_{1}&=f_{1}+{\Delta s \over {2}}\left(K_{10}x_{0}+K_{11}x_{1}\right)\\x_{2}&=f_{2}+{\Delta s \over {2}}\left(K_{20}x_{0}+2K_{21}x_{1}+K_{22}x_{2}\right)\\&\vdots \\x_{n}&=f_{n}+{\Delta s \over {2}}\left(K_{n0}x_{0}+2K_{n1}x_{1}+\cdots +2K_{n,n-1}x_{n-1}+K_{nn}x_{n}\right)\end{aligned}}$
This is equivalent to the matrix equation:
$x=f+Mx\implies x=(I-M)^{-1}f$
For well-behaved kernels, the trapezoidal rule tends to work well.
Application: Ruin theory
One area where Volterra integral equations appear is in ruin theory, the study of the risk of insolvency in actuarial science. The objective is to quantify the probability of ruin $\psi (u)=\mathbb {P} [\tau (u)<\infty ]$, where $u$ is the initial surplus and $\tau (u)$ is the time of ruin. In the classical model of ruin theory, the net cash position $X_{t}$ is a function of the initial surplus, premium income earned at rate $c$, and outgoing claims $\xi $:
$X_{t}=u+ct-\sum _{i=1}^{N_{t}}\xi _{i},\quad t\geq 0$
where $N_{t}$ is a Poisson process for the number of claims with intensity $\lambda $. Under these circumstances, the ruin probability may be represented by a Volterra integral equation of the form[6]:
$\psi (u)={\lambda \over {c}}\int _{u}^{\infty }S(x)dx+{\lambda \over {c}}\int _{0}^{u}\psi (u-x)S(x)dx$
where $S(\cdot )$ is the survival function of the claims distribution.
See also
• Fredholm integral equation
• Integral equation
• Integro-differential equation
References
1. Polyanin, Andrei D.; Manzhirov, Alexander V. (2008). Handbook of Integral Equations (2nd ed.). Boca Raton, FL: Chapman and Hall/CRC. ISBN 978-1584885078.
2. Inaba, Hisashi (2017). "The Stable Population Model". Age-Structured Population Dynamics in Demography and Epidemiology. Singapore: Springer. pp. 1–74. doi:10.1007/978-981-10-0188-8_1. ISBN 978-981-10-0187-1.
3. Brunner, Hermann (2017). Volterra Integral Equations: An Introduction to Theory and Applications. Cambridge Monographs on Applied and Computational Mathematics. Cambridge, UK: Cambridge University Press. ISBN 978-1107098725.
4. Daddi-Moussa-Ider, A.; Vilfan, A.; Golestanian, R. (6 April 2022). "Diffusiophoretic propulsion of an isotropic active colloidal particle near a finite-sized disk embedded in a planar fluid–fluid interface". Journal of Fluid Mechanics. 940: A12. arXiv:2109.14437. doi:10.1017/jfm.2022.232.
5. Daddi-Moussa-Ider, A.; Lisicki, M.; Löwen, H.; Menzel, A. M. (5 February 2020). "Dynamics of a microswimmer–microplatelet composite". Physics of Fluids. 32 (2): 021902. arXiv:2001.06646. doi:10.1063/1.5142054.
6. "Lecture Notes on Risk Theory" (PDF). School of Mathematics, Statistics and Actuarial Science. University of Kent. February 20, 2010. pp. 17–22.
Further reading
• Traian Lalescu, Introduction à la théorie des équations intégrales. Avec une préface de É. Picard, Paris: A. Hermann et Fils, 1912. VII + 152 pp.
• "Volterra equation", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. "Volterra Integral Equation of the First Kind". MathWorld.
• Weisstein, Eric W. "Volterra Integral Equation of the Second Kind". MathWorld.
• Integral Equations: Exact Solutions at EqWorld: The World of Mathematical Equations
• Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 19.2. Volterra Equations". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.
External links
• IntEQ: a Python package for numerically solving Volterra integral equations
| Wikipedia |
Volterra lattice
In mathematics, the Volterra lattice, also known as the discrete KdV equation, the Kac–van Moerbeke lattice, and the Langmuir lattice, is a system of ordinary differential equations with variables indexed by some of the points of a 1-dimensional lattice. It was introduced by Marc Kac and Pierre van Moerbeke (1975) and Jürgen Moser (1975) and is named after Vito Volterra. The Volterra lattice is a special case of the generalized Lotka–Volterra equation describing predator–prey interactions, for a sequence of species with each species preying on the next in the sequence. The Volterra lattice also behaves like a discrete version of the KdV equation. The Volterra lattice is an integrable system, and is related to the Toda lattice. It is also used as a model for Langmuir waves in plasmas.
Definition
The Volterra lattice is the set of ordinary differential equations for functions an:
$a_{n}'=a_{n}(a_{n+1}+a_{n-1})$
where n is an integer. Usually one adds boundary conditions: for example, the functions an could be periodic: an = an+N for some N, or could vanish for n ≤ 0 and n ≥ N.
The Volterra lattice was originally stated in terms of the variables Rn = –log an in which case the equations are
$R_{n}'=e^{R_{n-1}}+e^{R_{n+1}}$
References
• Kac, M.; van Moerbeke, P. (1975), "Some probabilistic aspects of scattering theory", in Arthurs, A.M. (ed.), Functional integration and its applications (Proc. Internat. Conf., London, 1974), Oxford: Clarendon Press, pp. 87–96, ISBN 978-0198533467, MR 0481238
• Kac, M.; van Moerbeke, Pierre (1975), "On an explicitly soluble system of nonlinear differential equations related to certain Toda lattices.", Advances in Mathematics, 16 (2): 160–169, doi:10.1016/0001-8708(75)90148-6, MR 0369953
• Moser, Jürgen (1975), "Finitely many mass points on the line under the influence of an exponential potential–an integrable system.", Dynamical systems, theory and applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974), Lecture Notes in Phys., vol. 38, Berlin: Springer, pp. 467–497, doi:10.1007/3-540-07171-7_12, ISBN 978-3-540-07171-6, MR 0455038
| Wikipedia |
Volterra operator
In mathematics, in the area of functional analysis and operator theory, the Volterra operator, named after Vito Volterra, is a bounded linear operator on the space L2[0,1] of complex-valued square-integrable functions on the interval [0,1]. On the subspace C[0,1] of continuous functions it represents indefinite integration. It is the operator corresponding to the Volterra integral equations.
Definition
The Volterra operator, V, may be defined for a function f ∈ L2[0,1] and a value t ∈ [0,1], as
$V(f)(t)=\int _{0}^{t}f(s)\,ds.$
Properties
• V is a bounded linear operator between Hilbert spaces, with Hermitian adjoint
$V^{*}(f)(t)=\int _{t}^{1}f(s)\,ds.$
• V is a Hilbert–Schmidt operator, hence in particular is compact.[1]
• V has no eigenvalues and therefore, by the spectral theory of compact operators, its spectrum σ(V) = {0}.[1]
• V is a quasinilpotent operator (that is, the spectral radius, ρ(V), is zero), but it is not nilpotent.
• The operator norm of V is exactly ||V|| = 2⁄π.[1]
References
1. "Spectrum of Indefinite Integral Operators". Stack Exchange. May 30, 2012.
Further reading
• Gohberg, Israel; Krein, M. G. (1970). Theory and Applications of Volterra Operators in Hilbert Space. Providence: American Mathematical Society. ISBN 0-8218-3627-7.
| Wikipedia |
Volterra space
In mathematics, in the field of topology, a topological space is said to be a Volterra space if any finite intersection of dense Gδ subsets is dense. Every Baire space is Volterra, but the converse is not true. In fact, any metrizable Volterra space is Baire.
The name refers to a paper of Vito Volterra in which he uses the fact that (in modern notation) the intersection of two dense G-delta sets in the real numbers is again dense.
References
• Cao, Jiling and Gauld, D, "Volterra spaces revisited", J. Aust. Math. Soc. 79 (2005), 61–76.
• Cao, Jiling and Junnila, Heikki, "When is a Volterra space Baire?", Topology Appl. 154 (2007), 527–532.
• Gauld, D. and Piotrowski, Z., "On Volterra spaces", Far East J. Math. Sci. 1 (1993), 209–214.
• Gruenhage, G. and Lutzer, D., "Baire and Volterra spaces", Proc. Amer. Math. Soc. 128 (2000), 3115–3124.
• Volterra, V., "Alcune osservasioni sulle funzioni punteggiate discontinue", Giornale di Matematiche 19 (1881), 76–86.
| Wikipedia |
Volterra's function
In mathematics, Volterra's function, named for Vito Volterra, is a real-valued function V defined on the real line R with the following curious combination of properties:
• V is differentiable everywhere
• The derivative V ′ is bounded everywhere
• The derivative is not Riemann-integrable.
Definition and construction
The function is defined by making use of the Smith–Volterra–Cantor set and "copies" of the function defined by $f(x)=x^{2}\sin(1/x)$ for $x\neq 0$ and $f(0)=0$. The construction of V begins by determining the largest value of x in the interval [0, 1/8] for which f ′(x) = 0. Once this value (say x0) is determined, extend the function to the right with a constant value of f(x0) up to and including the point 1/8. Once this is done, a mirror image of the function can be created starting at the point 1/4 and extending downward towards 0. This function will be defined to be 0 outside of the interval [0, 1/4]. We then translate this function to the interval [3/8, 5/8] so that the resulting function, which we call f1, is nonzero only on the middle interval of the complement of the Smith–Volterra–Cantor set. To construct f2, f ′ is then considered on the smaller interval [0,1/32], truncated at the last place the derivative is zero, extended, and mirrored the same way as before, and two translated copies of the resulting function are added to f1 to produce the function f2. Volterra's function then results by repeating this procedure for every interval removed in the construction of the Smith–Volterra–Cantor set; in other words, the function V is the limit of the sequence of functions f1, f2, ...
Further properties
Volterra's function is differentiable everywhere just as f (as defined above) is. One can show that f ′(x) = 2x sin(1/x) - cos(1/x) for x ≠ 0, which means that in any neighborhood of zero, there are points where f ′ takes values 1 and −1. Thus there are points where V ′ takes values 1 and −1 in every neighborhood of each of the endpoints of intervals removed in the construction of the Smith–Volterra–Cantor set S. In fact, V ′ is discontinuous at every point of S, even though V itself is differentiable at every point of S, with derivative 0. However, V ′ is continuous on each interval removed in the construction of S, so the set of discontinuities of V ′ is equal to S.
Since the Smith–Volterra–Cantor set S has positive Lebesgue measure, this means that V ′ is discontinuous on a set of positive measure. By Lebesgue's criterion for Riemann integrability, V ′ is not Riemann integrable. If one were to repeat the construction of Volterra's function with the ordinary measure-0 Cantor set C in place of the "fat" (positive-measure) Cantor set S, one would obtain a function with many similar properties, but the derivative would then be discontinuous on the measure-0 set C instead of the positive-measure set S, and so the resulting function would have a Riemann integrable derivative.
See also
• Fundamental theorem of calculus
External links
• Wrestling with the Fundamental Theorem of Calculus: Volterra's function, talk by David Marius Bressoud
• Volterra's example of a derivative that is not integrable (PPT), talk by David Marius Bressoud
| Wikipedia |
Volume entropy
The volume entropy is an asymptotic invariant of a compact Riemannian manifold that measures the exponential growth rate of the volume of metric balls in its universal cover. This concept is closely related with other notions of entropy found in dynamical systems and plays an important role in differential geometry and geometric group theory. If the manifold is nonpositively curved then its volume entropy coincides with the topological entropy of the geodesic flow. It is of considerable interest in differential geometry to find the Riemannian metric on a given smooth manifold which minimizes the volume entropy, with locally symmetric spaces forming a basic class of examples.
Definition
Let (M, g) be a compact Riemannian manifold, with universal cover ${\tilde {M}}.$ Choose a point ${\tilde {x}}_{0}\in {\tilde {M}}$.
The volume entropy (or asymptotic volume growth) $h=h(M,g)$ is defined as the limit
$h(M,g)=\lim _{R\to +\infty }{\frac {\log \left(\operatorname {vol} B(R)\right)}{R}},$
where B(R) is the ball of radius R in ${\tilde {M}}$ centered at ${\tilde {x}}_{0}$ and vol is the Riemannian volume in the universal cover with the natural Riemannian metric.
A. Manning proved that the limit exists and does not depend on the choice of the base point. This asymptotic invariant describes the exponential growth rate of the volume of balls in the universal cover as a function of the radius.
Properties
• Volume entropy h is always bounded above by the topological entropy htop of the geodesic flow on M. Moreover, if M has nonpositive sectional curvature then h = htop. These results are due to Manning.
• More generally, volume entropy equals topological entropy under a weaker assumption that M is a closed Riemannian manifold without conjugate points (Freire and Mañé).
• Locally symmetric spaces minimize entropy when the volume is prescribed. This is a corollary of a very general result due to Besson, Courtois, and Gallot (which also implies Mostow rigidity and its various generalizations due to Corlette, Siu, and Thurston):
Let X and Y be compact oriented connected n-dimensional smooth manifolds and f: Y → X a continuous map of non-zero degree. If g0 is a negatively curved locally symmetric Riemannian metric on X and g is any Riemannian metric on Y then
$h^{n}(Y,g)\operatorname {vol} (Y,g)\geq \left|\deg(f)\right|h^{n}(X,g_{0})\operatorname {vol} (X,g_{0}),$
and for n ≥ 3, the equality occurs if and only if (Y,g) is locally symmetric of the same type as (X,g0) and f is homotopic to a homothetic covering (Y,g) → (X,g0).
Application in differential geometry of surfaces
Katok's entropy inequality was recently exploited to obtain a tight asymptotic bound for the systolic ratio of surfaces of large genus, see systoles of surfaces.
References
• Besson, G., Courtois, G., Gallot, S. Entropies et rigidités des espaces localement symétriques de courbure strictement négative. (French) [Entropy and rigidity of locally symmetric spaces with strictly negative curvature] Geom. Funct. Anal. 5 (1995), no. 5, 731–799
• Katok, A.: Entropy and closed geodesics, Erg. Th. Dyn. Sys. 2 (1983), 339–365
• Katok, A.; Hasselblatt, B.: Introduction to the modern theory of dynamical systems. With a supplementary chapter by Katok and L. Mendoza. Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cambridge, 1995
• Katz, M.; Sabourau, S.: Entropy of systolically extremal surfaces and asymptotic bounds. Erg. Th. Dyn. Sys. 25 (2005), 1209-1220
• Manning, A.: Topological entropy for geodesic flows. Ann. of Math. (2) 110 (1979), no. 3, 567–573
| Wikipedia |
Volume integral
In mathematics (particularly multivariable calculus), a volume integral (∭) refers to an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many applications, for example, to calculate flux densities, or to calculate mass from a corresponding density function.
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In coordinates
It can also mean a triple integral within a region $D\subset \mathbb {R} ^{3}$ of a function $f(x,y,z),$ and is usually written as:
$\iiint _{D}f(x,y,z)\,dx\,dy\,dz.$
A volume integral in cylindrical coordinates is
$\iiint _{D}f(\rho ,\varphi ,z)\rho \,d\rho \,d\varphi \,dz,$
and a volume integral in spherical coordinates (using the ISO convention for angles with $\varphi $ as the azimuth and $\theta $ measured from the polar axis (see more on conventions)) has the form
$\iiint _{D}f(r,\theta ,\varphi )r^{2}\sin \theta \,dr\,d\theta \,d\varphi .$
Example
Integrating the equation $f(x,y,z)=1$ over a unit cube yields the following result:
$\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}1\,dx\,dy\,dz=\int _{0}^{1}\int _{0}^{1}(1-0)\,dy\,dz=\int _{0}^{1}\left(1-0\right)dz=1-0=1$
So the volume of the unit cube is 1 as expected. This is rather trivial however, and a volume integral is far more powerful. For instance if we have a scalar density function on the unit cube then the volume integral will give the total mass of the cube. For example for density function:
${\begin{cases}f:\mathbb {R} ^{3}\to \mathbb {R} \\f:(x,y,z)\mapsto x+y+z\end{cases}}$
the total mass of the cube is:
$\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}(x+y+z)\,dx\,dy\,dz=\int _{0}^{1}\int _{0}^{1}\left({\frac {1}{2}}+y+z\right)dy\,dz=\int _{0}^{1}(1+z)\,dz={\frac {3}{2}}$
See also
• Divergence theorem
• Surface integral
• Volume element
External links
• "Multiple integral", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. "Volume integral". MathWorld.
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| Wikipedia |
Volume of an n-ball
In geometry, a ball is a region in a space comprising all points within a fixed distance, called the radius, from a given point; that is, it is the region enclosed by a sphere or hypersphere. An n-ball is a ball in an n-dimensional Euclidean space. The volume of a n-ball is the Lebesgue measure of this ball, which generalizes to any dimension the usual volume of a ball in 3-dimensional space. The volume of a n-ball of radius R is $R^{n}V_{n},$ where $V_{n}$ is the volume of the unit n-ball, the n-ball of radius 1.
The real number $V_{n}$ can be expressed via a two-dimension recurrence relation. Closed-form expressions involve the gamma, factorial, or double factorial function. The volume can also be expressed in terms of $A_{n}$, the area of the unit n-sphere.
Formulas
The first volumes are as follows:
DimensionVolume of a ball of radius RRadius of a ball of volume V
0 $1$ (all 0-balls have volume 1)
1 $2R$ ${\frac {V}{2}}=0.5\times V$
2 $\pi R^{2}\approx 3.142\times R^{2}$ ${\frac {V^{1/2}}{\sqrt {\pi }}}\approx 0.564\times V^{\frac {1}{2}}$
3 ${\frac {4\pi }{3}}R^{3}\approx 4.189\times R^{3}$ $\left({\frac {3V}{4\pi }}\right)^{1/3}\approx 0.620\times V^{1/3}$
4 ${\frac {\pi ^{2}}{2}}R^{4}\approx 4.935\times R^{4}$ ${\frac {(2V)^{1/4}}{\sqrt {\pi }}}\approx 0.671\times V^{1/4}$
5 ${\frac {8\pi ^{2}}{15}}R^{5}\approx 5.264\times R^{5}$ $\left({\frac {15V}{8\pi ^{2}}}\right)^{1/5}\approx 0.717\times V^{1/5}$
6 ${\frac {\pi ^{3}}{6}}R^{6}\approx 5.168\times R^{6}$ ${\frac {(6V)^{1/6}}{\sqrt {\pi }}}\approx 0.761\times V^{1/6}$
7 ${\frac {16\pi ^{3}}{105}}R^{7}\approx 4.725\times R^{7}$ $\left({\frac {105V}{16\pi ^{3}}}\right)^{1/7}\approx 0.801\times V^{1/7}$
8 ${\frac {\pi ^{4}}{24}}R^{8}\approx 4.059\times R^{8}$ ${\frac {(24V)^{1/8}}{\sqrt {\pi }}}\approx 0.839\times V^{1/8}$
9 ${\frac {32\pi ^{4}}{945}}R^{9}\approx 3.299\times R^{9}$ $\left({\frac {945V}{32\pi ^{4}}}\right)^{1/9}\approx 0.876\times V^{1/9}$
10 ${\frac {\pi ^{5}}{120}}R^{10}\approx 2.550\times R^{10}$ ${\frac {(120V)^{1/10}}{\sqrt {\pi }}}\approx 0.911\times V^{1/10}$
11 ${\frac {64\pi ^{5}}{10395}}R^{11}\approx 1.884\times R^{11}$ $\left({\frac {10395V}{64\pi ^{5}}}\right)^{1/11}\approx 0.944\times V^{1/11}$
12 ${\frac {\pi ^{6}}{720}}R^{12}\approx 1.335\times R^{12}$ ${\frac {(720V)^{1/12}}{\sqrt {\pi }}}\approx 0.976\times V^{1/12}$
13 ${\frac {128\pi ^{6}}{135135}}R^{13}\approx 0.911\times R^{13}$ $\left({\frac {135135V}{128\pi ^{6}}}\right)^{1/13}\approx 1.007\times V^{1/13}$
14 ${\frac {\pi ^{7}}{5040}}R^{14}\approx 0.599\times R^{14}$ ${\frac {(5040V)^{1/14}}{\sqrt {\pi }}}\approx 1.037\times V^{1/14}$
15 ${\frac {256\pi ^{7}}{2027025}}R^{15}\approx 0.381\times R^{15}$ $\left({\frac {2027025V}{256\pi ^{7}}}\right)^{1/15}\approx 1.066\times V^{1/15}$
nVn(R)Rn(V)
Closed form
The n-dimensional volume of a Euclidean ball of radius R in n-dimensional Euclidean space is:[1]
$V_{n}(R)={\frac {\pi ^{n/2}}{\Gamma {\bigl (}{\tfrac {n}{2}}+1{\bigr )}}}R^{n},$
where Γ is Euler's gamma function. The gamma function is offset from but otherwise extends the factorial function to non-integer arguments. It satisfies Γ(n) = (n − 1)! if n is a positive integer and Γ(n + 1/2) = (n − 1/2) · (n − 3/2) · … · 1/2 · π1/2 if n is a non-negative integer.
Two-dimension recurrence relation
The volume can be computed without use of the Gamma function. As is proved below using a vector-calculus double integral in polar coordinates, the volume V of an n-ball of radius R can be expressed recursively in terms of the volume of an (n − 2)-ball, via the interleaved recurrence relation:
$V_{n}(R)={\begin{cases}1&{\text{if }}n=0,\\[0.5ex]2R&{\text{if }}n=1,\\[0.5ex]{\dfrac {2\pi }{n}}R^{2}\times V_{n-2}(R)&{\text{otherwise}}.\end{cases}}$
This allows computation of Vn(R) in approximately n / 2 steps.
Alternative forms
The volume can also be expressed in terms of an (n − 1)-ball using the one-dimension recurrence relation:
${\begin{aligned}V_{0}(R)&=1,\\V_{n}(R)&={\frac {\Gamma {\bigl (}{\tfrac {n}{2}}+{\tfrac {1}{2}}{\bigr )}{\sqrt {\pi }}}{\Gamma {\bigl (}{\tfrac {n}{2}}+1{\bigr )}}}R\,V_{n-1}(R).\end{aligned}}$
Inverting the above, the radius of an n-ball of volume V can be expressed recursively in terms of the radius of an (n − 2)- or (n − 1)-ball:
${\begin{aligned}R_{n}(V)&={\bigl (}{\tfrac {1}{2}}n{\bigr )}^{1/n}\left(\Gamma {\bigl (}{\tfrac {n}{2}}{\bigr )}V\right)^{-2/(n(n-2))}R_{n-2}(V),\\R_{n}(V)&={\frac {\Gamma {\bigl (}{\tfrac {n}{2}}+1{\bigr )}^{1/n}}{\Gamma {\bigl (}{\tfrac {n}{2}}+{\tfrac {1}{2}}{\bigr )}^{1/(n-1)}}}V^{-1/(n(n-1))}R_{n-1}(V).\end{aligned}}$
Using explicit formulas for particular values of the gamma function at the integers and half-integers gives formulas for the volume of a Euclidean ball in terms of factorials. For non-negative integer k, these are:
${\begin{aligned}V_{2k}(R)&={\frac {\pi ^{k}}{k!}}R^{2k},\\V_{2k+1}(R)&={\frac {2(k!)(4\pi )^{k}}{(2k+1)!}}R^{2k+1}.\end{aligned}}$
The volume can also be expressed in terms of double factorials. For a positive odd integer 2k + 1, the double factorial is defined by
$(2k+1)!!=(2k+1)\cdot (2k-1)\dotsm 5\cdot 3\cdot 1.$
The volume of an odd-dimensional ball is
$V_{2k+1}(R)={\frac {2(2\pi )^{k}}{(2k+1)!!}}R^{2k+1}.$
There are multiple conventions for double factorials of even integers. Under the convention in which the double factorial satisfies
$(2k)!!=(2k)\cdot (2k-2)\dotsm 4\cdot 2\cdot {\sqrt {2/\pi }}=2^{k}\cdot k!\cdot {\sqrt {2/\pi }},$
the volume of an n-dimensional ball is, regardless of whether n is even or odd,
$V_{n}(R)={\frac {2(2\pi )^{(n-1)/2}}{n!!}}R^{n}.$
Instead of expressing the volume V of the ball in terms of its radius R, the formulas can be inverted to express the radius as a function of the volume:
${\begin{aligned}R_{n}(V)&={\frac {\Gamma {\bigl (}{\tfrac {n}{2}}+1{\bigr )}^{1/n}}{\sqrt {\pi }}}V^{1/n}\\&=\left({\frac {n!!V}{2(2\pi )^{(n-1)/2}}}\right)^{1/n}\\R_{2k}(V)&={\frac {(k!V)^{1/(2k)}}{\sqrt {\pi }}},\\R_{2k+1}(V)&=\left({\frac {(2k+1)!V}{2(k!)(4\pi )^{k}}}\right)^{1/(2k+1)}.\end{aligned}}$
Approximation for high dimensions
Stirling's approximation for the gamma function can be used to approximate the volume when the number of dimensions is high.
$V_{n}(R)\sim {\frac {1}{\sqrt {n\pi }}}\left({\frac {2\pi e}{n}}\right)^{n/2}R^{n}.$
$R_{n}(V)\sim (\pi n)^{1/(2n)}{\sqrt {\frac {n}{2\pi e}}}V^{1/n}.$
In particular, for any fixed value of R the volume tends to a limiting value of 0 as n goes to infinity. Which value of n maximizes Vn(R) depends upon the value of R; for example, the volume Vn(1) is increasing for 0 ≤ n ≤ 5, achieves its maximum when n = 5, and is decreasing for n ≥ 5.[2]
Relation with surface area
Let An − 1(R) denote the hypervolume of the (n − 1)-sphere of radius R. The (n − 1)-sphere is the (n − 1)-dimensional boundary (surface) of the n-dimensional ball of radius R, and the sphere's hypervolume and the ball's hypervolume are related by:
$A_{n-1}(R)={\frac {d}{dR}}V_{n}(R)={\frac {n}{R}}V_{n}(R).$
Thus, An − 1(R) inherits formulas and recursion relationships from Vn(R), such as
$A_{n-1}(R)={\frac {2\pi ^{n/2}}{\Gamma {\bigl (}{\tfrac {n}{2}}{\bigr )}}}R^{n-1}.$
There are also formulas in terms of factorials and double factorials.
Proofs
There are many proofs of the above formulas.
The volume is proportional to the nth power of the radius
An important step in several proofs about volumes of n-balls, and a generally useful fact besides, is that the volume of the n-ball of radius R is proportional to Rn:
$V_{n}(R)\propto R^{n}.$
The proportionality constant is the volume of the unit ball.
This is a special case of a general fact about volumes in n-dimensional space: If K is a body (measurable set) in that space and RK is the body obtained by stretching in all directions by the factor R then the volume of RK equals Rn times the volume of K. This is a direct consequence of the change of variables formula:
$V(RK)=\int _{RK}dx=\int _{K}R^{n}\,dy=R^{n}V(K)$
where dx = dx1…dxn and the substitution x = Ry was made.
Another proof of the above relation, which avoids multi-dimensional integration, uses induction: The base case is n = 0, where the proportionality is obvious. For the inductive step, assume that proportionality is true in dimension n − 1. Note that the intersection of an n-ball with a hyperplane is an (n − 1)-ball. When the volume of the n-ball is written as an integral of volumes of (n − 1)-balls:
$V_{n}(R)=\int _{-R}^{R}V_{n-1}\!\left({\sqrt {R^{2}-x^{2}}}\right)dx,$
it is possible by the inductive hypothesis to remove a factor of R from the radius of the (n − 1)-ball to get:
$V_{n}(R)=R^{n-1}\!\int _{-R}^{R}V_{n-1}\!\left({\sqrt {1-\left({\frac {x}{R}}\right)^{2}}}\right)dx.$
Making the change of variables t = x/R leads to:
$V_{n}(R)=R^{n}\!\int _{-1}^{1}V_{n-1}\!\left({\sqrt {1-t^{2}}}\right)dt=R^{n}V_{n}(1),$
which demonstrates the proportionality relation in dimension n. By induction, the proportionality relation is true in all dimensions.
The two-dimension recursion formula
A proof of the recursion formula relating the volume of the n-ball and an (n − 2)-ball can be given using the proportionality formula above and integration in cylindrical coordinates. Fix a plane through the center of the ball. Let r denote the distance between a point in the plane and the center of the sphere, and let θ denote the azimuth. Intersecting the n-ball with the (n − 2)-dimensional plane defined by fixing a radius and an azimuth gives an (n − 2)-ball of radius √R2 − r2. The volume of the ball can therefore be written as an iterated integral of the volumes of the (n − 2)-balls over the possible radii and azimuths:
$V_{n}(R)=\int _{0}^{2\pi }\int _{0}^{R}V_{n-2}\!\left({\sqrt {R^{2}-r^{2}}}\right)r\,dr\,d\theta ,$
The azimuthal coordinate can be immediately integrated out. Applying the proportionality relation shows that the volume equals
$V_{n}(R)=2\pi V_{n-2}(R)\int _{0}^{R}\left(1-\left({\frac {r}{R}}\right)^{2}\right)^{(n-2)/2}\,r\,dr.$
The integral can be evaluated by making the substitution u = 1 − (r/R)2
to get
${\begin{aligned}V_{n}(R)&=2\pi V_{n-2}(R)\cdot \left[-{\frac {R^{2}}{n}}\left(1-\left({\frac {r}{R}}\right)^{2}\right)^{n/2}\right]_{r=0}^{r=R}\\&={\frac {2\pi R^{2}}{n}}V_{n-2}(R),\end{aligned}}$
which is the two-dimension recursion formula.
The same technique can be used to give an inductive proof of the volume formula. The base cases of the induction are the 0-ball and the 1-ball, which can be checked directly using the facts Γ(1) = 1 and Γ(3/2) = 1/2 · Γ(1/2) = √π/2. The inductive step is similar to the above, but instead of applying proportionality to the volumes of the (n − 2)-balls, the inductive hypothesis is applied instead.
The one-dimension recursion formula
The proportionality relation can also be used to prove the recursion formula relating the volumes of an n-ball and an (n − 1)-ball. As in the proof of the proportionality formula, the volume of an n-ball can be written as an integral over the volumes of (n − 1)-balls. Instead of making a substitution, however, the proportionality relation can be applied to the volumes of the (n − 1)-balls in the integrand:
$V_{n}(R)=V_{n-1}(R)\int _{-R}^{R}\left(1-\left({\frac {x}{R}}\right)^{2}\right)^{(n-1)/2}\,dx.$
The integrand is an even function, so by symmetry the interval of integration can be restricted to [0, R]. On the interval [0, R], it is possible to apply the substitution u = (x/R)2
. This transforms the expression into
$V_{n-1}(R)\cdot R\cdot \int _{0}^{1}(1-u)^{(n-1)/2}u^{-{\frac {1}{2}}}\,du$
The integral is a value of a well-known special function called the beta function Β(x, y), and the volume in terms of the beta function is
$V_{n}(R)=V_{n-1}(R)\cdot R\cdot \mathrm {B} \left({\tfrac {n+1}{2}},{\tfrac {1}{2}}\right).$
The beta function can be expressed in terms of the gamma function in much the same way that factorials are related to binomial coefficients. Applying this relationship gives
$V_{n}(R)=V_{n-1}(R)\cdot R\cdot {\frac {\Gamma {\bigl (}{\tfrac {n}{2}}+{\tfrac {1}{2}}{\bigr )}\Gamma {\bigl (}{\tfrac {1}{2}}{\bigr )}}{\Gamma {\bigl (}{\tfrac {n}{2}}+1{\bigr )}}}.$
Using the value Γ(1/2) = √π gives the one-dimension recursion formula:
$V_{n}(R)=R{\sqrt {\pi }}{\frac {\Gamma {\bigl (}{\tfrac {n}{2}}+{\tfrac {1}{2}}{\bigr )}}{\Gamma {\bigl (}{\tfrac {n}{2}}+1{\bigr )}}}V_{n-1}(R).$
As with the two-dimension recursive formula, the same technique can be used to give an inductive proof of the volume formula.
Direct integration in spherical coordinates
The volume of the n-ball $V_{n}(R)$ can be computed by integrating the volume element in spherical coordinates. The spherical coordinate system has a radial coordinate r and angular coordinates φ1, …, φn − 1, where the domain of each φ except φn − 1 is [0, π), and the domain of φn − 1 is [0, 2π). The spherical volume element is:
$dV=r^{n-1}\sin ^{n-2}(\varphi _{1})\sin ^{n-3}(\varphi _{2})\cdots \sin(\varphi _{n-2})\,dr\,d\varphi _{1}\,d\varphi _{2}\cdots d\varphi _{n-1},$
and the volume is the integral of this quantity over r between 0 and R and all possible angles:
$V_{n}(R)=\int _{0}^{R}\int _{0}^{\pi }\cdots \int _{0}^{2\pi }r^{n-1}\sin ^{n-2}(\varphi _{1})\cdots \sin(\varphi _{n-2})\,d\varphi _{n-1}\cdots d\varphi _{1}\,dr.$
Each of the factors in the integrand depends on only a single variable, and therefore the iterated integral can be written as a product of integrals:
$V_{n}(R)=\left(\int _{0}^{R}r^{n-1}\,dr\right)\!\left(\int _{0}^{\pi }\sin ^{n-2}(\varphi _{1})\,d\varphi _{1}\right)\cdots \left(\int _{0}^{2\pi }d\varphi _{n-1}\right).$
The integral over the radius is Rn/n. The intervals of integration on the angular coordinates can, by the symmetry of the sine about π/2, be changed to [0, π/2]:
$V_{n}(R)={\frac {R^{n}}{n}}\left(2\int _{0}^{\pi /2}\sin ^{n-2}(\varphi _{1})\,d\varphi _{1}\right)\cdots \left(4\int _{0}^{\pi /2}d\varphi _{n-1}\right).$
Each of the remaining integrals is now a particular value of the beta function:
$V_{n}(R)={\frac {R^{n}}{n}}\mathrm {B} {\bigl (}{\tfrac {n-1}{2}},{\tfrac {1}{2}}{\bigr )}\mathrm {B} {\bigl (}{\tfrac {n-2}{2}},{\tfrac {1}{2}}{\bigr )}\cdots \mathrm {B} {\bigl (}1,{\tfrac {1}{2}}{\bigr )}\cdot 2\,\mathrm {B} {\bigl (}{\tfrac {1}{2}},{\tfrac {1}{2}}{\bigr )}.$
The beta functions can be rewritten in terms of gamma functions:
$V_{n}(R)={\frac {R^{n}}{n}}\cdot {\frac {\Gamma {\bigl (}{\tfrac {n}{2}}-{\tfrac {1}{2}}{\bigr )}\Gamma {\bigl (}{\tfrac {1}{2}}{\bigr )}}{\Gamma {\bigl (}{\tfrac {n}{2}}{\bigr )}}}\cdot {\frac {\Gamma {\bigl (}{\tfrac {n}{2}}-1{\bigr )}\Gamma {\bigl (}{\tfrac {1}{2}}{\bigr )}}{\Gamma {\bigl (}{\tfrac {n}{2}}-{\tfrac {1}{2}}{\bigr )}}}\cdots {\frac {\Gamma (1)\Gamma {\bigl (}{\tfrac {1}{2}}{\bigr )}}{\Gamma {\bigl (}{\tfrac {3}{2}}{\bigr )}}}\cdot 2{\frac {\Gamma {\bigl (}{\tfrac {1}{2}}{\bigr )}\Gamma {\bigl (}{\tfrac {1}{2}}{\bigr )}}{\Gamma (1)}}.$
This product telescopes. Combining this with the values Γ(1/2) = √π and Γ(1) = 1 and the functional equation zΓ(z) = Γ(z + 1) leads to
$V_{n}(R)={\frac {2\pi ^{n/2}R^{n}}{n\,\Gamma {\bigl (}{\tfrac {n}{2}}{\bigr )}}}={\frac {\pi ^{n/2}R^{n}}{\Gamma {\bigl (}{\tfrac {n}{2}}+1{\bigr )}}}.$
Gaussian integrals
The volume formula can be proven directly using Gaussian integrals. Consider the function:
$f(x_{1},\ldots ,x_{n})=\exp {\biggl (}{-{\tfrac {1}{2}}\sum _{i=1}^{n}x_{i}^{2}}{\biggr )}.$
This function is both rotationally invariant and a product of functions of one variable each. Using the fact that it is a product and the formula for the Gaussian integral gives:
$\int _{\mathbf {R} ^{n}}f\,dV=\prod _{i=1}^{n}\left(\int _{-\infty }^{\infty }\exp \left(-{\tfrac {1}{2}}x_{i}^{2}\right)\,dx_{i}\right)=(2\pi )^{n/2},$
where dV is the n-dimensional volume element. Using rotational invariance, the same integral can be computed in spherical coordinates:
$\int _{\mathbf {R} ^{n}}f\,dV=\int _{0}^{\infty }\int _{S^{n-1}(r)}\exp \left(-{\tfrac {1}{2}}r^{2}\right)\,dA\,dr,$
where Sn − 1(r) is an (n − 1)-sphere of radius r (being the surface of an n-ball of radius r) and dA is the area element (equivalently, the (n − 1)-dimensional volume element). The surface area of the sphere satisfies a proportionality equation similar to the one for the volume of a ball: If An − 1(r) is the surface area of an (n − 1)-sphere of radius r, then:
$A_{n-1}(r)=r^{n-1}A_{n-1}(1).$
Applying this to the above integral gives the expression
$(2\pi )^{n/2}=\int _{0}^{\infty }\int _{S^{n-1}(r)}\exp \left(-{\tfrac {1}{2}}r^{2}\right)\,dA\,dr=A_{n-1}(1)\int _{0}^{\infty }\exp \left(-{\tfrac {1}{2}}r^{2}\right)\,r^{n-1}\,dr.$
Substituting t = r2/2:
$\int _{0}^{\infty }\exp \left(-{\tfrac {1}{2}}r^{2}\right)\,r^{n-1}\,dr=2^{(n-2)/2}\int _{0}^{\infty }e^{-t}t^{(n-2)/2}\,dt.$
The integral on the right is the gamma function evaluated at n/2.
Combining the two results shows that
$A_{n-1}(1)={\frac {2\pi ^{n/2}}{\Gamma {\bigl (}{\tfrac {n}{2}}{\bigr )}}}.$
To derive the volume of an n-ball of radius R from this formula, integrate the surface area of a sphere of radius r for 0 ≤ r ≤ R and apply the functional equation zΓ(z) = Γ(z + 1):
$V_{n}(R)=\int _{0}^{R}{\frac {2\pi ^{n/2}}{\Gamma {\bigl (}{\tfrac {n}{2}}{\bigr )}}}\,r^{n-1}\,dr={\frac {2\pi ^{n/2}}{n\,\Gamma {\bigl (}{\tfrac {n}{2}}{\bigr )}}}R^{n}={\frac {\pi ^{n/2}}{\Gamma {\bigl (}{\tfrac {n}{2}}+1{\bigr )}}}R^{n}.$
Geometric proof
The relations $V_{n+1}(R)={\frac {R}{n+1}}A_{n}(R)$ and $A_{n+1}(R)=(2\pi R)V_{n}(R)$ and thus the volumes of n-balls and areas of n-spheres can also be derived geometrically. As noted above, because a ball of radius $R$ is obtained from a unit ball $B_{n}$ by rescaling all directions in $R$ times, $V_{n}(R)$ is proportional to $R^{n}$, which implies ${\frac {dV_{n}(R)}{dR}}={\frac {n}{R}}V_{n}(R)$. Also, $A_{n-1}(R)={\frac {dV_{n}(R)}{dR}}$ because a ball is a union of concentric spheres and increasing radius by ε corresponds to a shell of thickness ε. Thus, $V_{n}(R)={\frac {R}{n}}A_{n-1}(R)$; equivalently, $V_{n+1}(R)={\frac {R}{n+1}}A_{n}(R)$.
$A_{n+1}(R)=(2\pi R)V_{n}(R)$ follows from existence of a volume-preserving bijection between the unit sphere $S_{n+1}$ and $S_{1}\times B_{n}$:
$(x,y,{\vec {z}})\mapsto \left({\frac {x}{\sqrt {x^{2}+y^{2}}}},{\frac {y}{\sqrt {x^{2}+y^{2}}}},{\vec {z}}\right)$
(${\vec {z}}$ is an n-tuple; $|(x,y,{\vec {z}})|=1$; we are ignoring sets of measure 0). Volume is preserved because at each point, the difference from isometry is a stretching in the xy plane (in $ 1/\!{\sqrt {x^{2}+y^{2}}}$ times in the direction of constant $x^{2}+y^{2}$) that exactly matches the compression in the direction of the gradient of $|{\vec {z}}|$ on $S_{n}$ (the relevant angles being equal). For $S_{2}$, a similar argument was originally made by Archimedes in On the Sphere and Cylinder.
Balls in Lp norms
There are also explicit expressions for the volumes of balls in Lp norms. The Lp norm of the vector x = (x1, …, xn) in Rn is
$\|x\|_{p}={\biggl (}\sum _{i=1}^{n}|x_{i}|^{p}{\biggr )}^{\!1/p},$
and an Lp ball is the set of all vectors whose Lp norm is less than or equal to a fixed number called the radius of the ball. The case p = 2 is the standard Euclidean distance function, but other values of p occur in diverse contexts such as information theory, coding theory, and dimensional regularization.
The volume of an Lp ball of radius R is
$V_{n}^{p}(R)={\frac {{\Bigl (}2\,\Gamma {\bigl (}{\tfrac {1}{p}}+1{\bigr )}{\Bigr )}^{n}}{\Gamma {\bigl (}{\tfrac {n}{p}}+1{\bigr )}}}R^{n}.$
These volumes satisfy recurrence relations similar to those for p = 2:
$V_{n}^{p}(R)={\frac {{\Bigl (}2\,\Gamma {\bigl (}{\tfrac {1}{p}}+1{\bigr )}{\Bigr )}^{p}p}{n}}R^{p}\,V_{n-p}^{p}(R)$
and
$V_{n}^{p}(R)=2{\frac {\Gamma {\bigl (}{\tfrac {1}{p}}+1{\bigr )}\Gamma {\bigl (}{\tfrac {n-1}{p}}+1{\bigr )}}{\Gamma {\bigl (}{\tfrac {n}{p}}+1{\bigr )}}}R\,V_{n-1}^{p}(R),$
which can be written more concisely using a generalized binomial coefficient,
$V_{n}^{p}(R)={\frac {2}{\dbinom {n/p}{1/p}}}R\,V_{n-1}^{p}(R).$
For p = 2, one recovers the recurrence for the volume of a Euclidean ball because 2Γ(3/2) = √π.
For example, in the cases p = 1 (taxicab norm) and p = ∞ (max norm), the volumes are:
${\begin{aligned}V_{n}^{1}(R)&={\frac {2^{n}}{n!}}R^{n},\\V_{n}^{\infty }(R)&=2^{n}R^{n}.\end{aligned}}$
These agree with elementary calculations of the volumes of cross-polytopes and hypercubes.
Relation with surface area
For most values of p, the surface area $A_{n-1}^{p}(R)$ of an Lp sphere of radius R (the boundary of an Lp n-ball of radius R) cannot be calculated by differentiating the volume of an Lp ball with respect to its radius. While the volume can be expressed as an integral over the surface areas using the coarea formula, the coarea formula contains a correction factor that accounts for how the p-norm varies from point to point. For p = 2 and p = ∞, this factor is one. However, if p = 1 then the correction factor is √n: the surface area of an L1 sphere of radius R in Rn is √n times the derivative of the volume of an L1 ball. This can be seen most simply by applying the divergence theorem to the vector field F(x) = x to get
$nV_{n}^{1}(R)=$$\iiint _{V}\left(\mathbf {\nabla } \cdot \mathbf {F} \right)\,dV=$ $\scriptstyle S$ $(\mathbf {F} \cdot \mathbf {n} )\,dS$ $=$ $\scriptstyle S$ ${\frac {1}{\sqrt {n}}}(|x_{1}|+\cdots +|x_{n}|)\,dS$ $={\frac {R}{\sqrt {n}}}$ $\scriptstyle S$ $\,dS$ $={\frac {R}{\sqrt {n}}}A_{n-1}^{1}(R).$
For other values of p, the constant is a complicated integral.
Generalizations
The volume formula can be generalized even further. For positive real numbers p1, …, pn, define the (p1, …, pn) ball with limit L ≥ 0 to be
$B_{p_{1},\ldots ,p_{n}}(L)=\left\{x=(x_{1},\ldots ,x_{n})\in \mathbf {R} ^{n}:\vert x_{1}\vert ^{p_{1}}+\cdots +\vert x_{n}\vert ^{p_{n}}\leq L\right\}.$
The volume of this ball has been known since the time of Dirichlet:[3]
$V{\bigl (}B_{p_{1},\ldots ,p_{n}}(L){\bigr )}={\frac {2^{n}\Gamma {\bigl (}{\tfrac {1}{p_{1}}}+1{\bigr )}\cdots \Gamma {\bigl (}{\tfrac {1}{p_{n}}}+1{\bigr )}}{\Gamma {\bigl (}{\tfrac {1}{p_{1}}}+\cdots +{\tfrac {1}{p_{n}}}+1{\bigr )}}}L^{{\tfrac {1}{p_{1}}}+\cdots +{\tfrac {1}{p_{n}}}}.$
Comparison to Lp norm
Using the harmonic mean $p={\frac {n}{{\frac {1}{p_{1}}}+\cdots {\frac {1}{p_{n}}}}}$ and defining $R={\sqrt[{p}]{L}}$, the similarity to the volume formula for the Lp ball becomes clear.
$V\left(\left\{x\in \mathbf {R} ^{n}:{\sqrt[{p}]{\vert x_{1}\vert ^{p_{1}}+\cdots +\vert x_{n}\vert ^{p_{n}}}}\leq R\right\}\right)={\frac {2^{n}\Gamma {\bigl (}{\tfrac {1}{p_{1}}}+1{\bigr )}\cdots \Gamma {\bigl (}{\tfrac {1}{p_{n}}}+1{\bigr )}}{\Gamma {\bigl (}{\tfrac {n}{p}}+1{\bigr )}}}R^{n}.$
See also
• n-sphere
• Sphere packing
• Hamming bound
References
1. Equation 5.19.4, NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/5.19#E4, Release 1.0.6 of 2013-05-06.
2. Smith, David J. and Vamanamurthy, Mavina K., "How Small Is a Unit Ball?", Mathematics Magazine, Volume 62, Issue 2, 1989, pp. 101–107, https://doi.org/10.1080/0025570X.1989.11977419.
3. Dirichlet, P. G. Lejeune (1839). "Sur une nouvelle méthode pour la détermination des intégrales multiples" [On a novel method for determining multiple integrals]. Journal de Mathématiques Pures et Appliquées. 4: 164–168.
External links
• Derivation in hyperspherical coordinates (in French)
• Hypersphere on Wolfram MathWorld
• Volume of the Hypersphere at Math Reference
| Wikipedia |
Volume mesh
In 3D computer graphics and modeling, volumetric meshes are a polygonal representation of the interior volume of an object. Unlike polygon meshes, which represent only the surface as polygons, volumetric meshes also discretize the interior structure of the object.
Applications
One application of volumetric meshes is in finite element analysis, which may use regular or irregular volumetric meshes to compute internal stresses and forces in an object throughout the entire volume of the object. [1]
Volume meshes may also be used for portal rendering.
See also
• B-rep
• Voxels
• Hypergraph
• Volume rendering
References
1. "Volume Mesh Generation for Numerical Flow Simulations using Catmull-Clark and Surface Approximation Methods" (PDF).
Mesh generation
Types of mesh
• Polygon mesh
• Triangle mesh
• Volume mesh
Methods
• Laplacian smoothing
• Parallel mesh generation
• Stretched grid method
Related
• Chew's second algorithm
• Image-based meshing
• Marching cubes
• Marching tetrahedra
• Principles of Grid Generation
• Regular grid
• Ruppert's algorithm
• Tessellation
• Unstructured grid
| Wikipedia |
von Mises–Fisher distribution
In directional statistics, the von Mises–Fisher distribution (named after Richard von Mises and Ronald Fisher), is a probability distribution on the $(p-1)$-sphere in $\mathbb {R} ^{p}$. If $p=2$ the distribution reduces to the von Mises distribution on the circle.
Definition
The probability density function of the von Mises–Fisher distribution for the random p-dimensional unit vector $\mathbf {x} $ is given by:
$f_{p}(\mathbf {x} ;{\boldsymbol {\mu }},\kappa )=C_{p}(\kappa )\exp \left({\kappa {\boldsymbol {\mu }}^{\mathsf {T}}\mathbf {x} }\right),$ ;{\boldsymbol {\mu }},\kappa )=C_{p}(\kappa )\exp \left({\kappa {\boldsymbol {\mu }}^{\mathsf {T}}\mathbf {x} }\right),}
where $\kappa \geq 0,\left\Vert {\boldsymbol {\mu }}\right\Vert =1$ and the normalization constant $C_{p}(\kappa )$ is equal to
$C_{p}(\kappa )={\frac {\kappa ^{p/2-1}}{(2\pi )^{p/2}I_{p/2-1}(\kappa )}},$
where $I_{v}$ denotes the modified Bessel function of the first kind at order Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): v . If $p=3$, the normalization constant reduces to
$C_{3}(\kappa )={\frac {\kappa }{4\pi \sinh \kappa }}={\frac {\kappa }{2\pi (e^{\kappa }-e^{-\kappa })}}.$
The parameters ${\boldsymbol {\mu }}$ and $\kappa $ are called the mean direction and concentration parameter, respectively. The greater the value of $\kappa $, the higher the concentration of the distribution around the mean direction ${\boldsymbol {\mu }}$. The distribution is unimodal for $\kappa >0$, and is uniform on the sphere for $\kappa =0$.
The von Mises–Fisher distribution for $p=3$ is also called the Fisher distribution.[1][2] It was first used to model the interaction of electric dipoles in an electric field.[3] Other applications are found in geology, bioinformatics, and text mining.
Note on the normalization constant
In the textbook, Directional Statistics [3] by Mardia and Jupp, the normalization constant given for the Von Mises Fisher probability density is apparently different from the one given here: $C_{p}(\kappa )$. In that book, the normalization constant is specified as:
$C_{p}^{*}(\kappa )={\frac {({\frac {\kappa }{2}})^{p/2-1}}{\Gamma (p/2)I_{p/2-1}(\kappa )}}$
where $\Gamma $ is the gamma function. This is resolved by noting that Mardia and Jupp give the density "with respect to the uniform distribution", while the density here is specified in the usual way, with respect to Lebesgue measure. The density (w.r.t. Lebesgue measure) of the uniform distribution is the reciprocal of the surface area of the (p-1)-sphere, so that the uniform density function is given by the constant:
$C_{p}(0)={\frac {\Gamma (p/2)}{2\pi ^{p/2}}}$
It then follows that:
$C_{p}^{*}(\kappa )={\frac {C_{p}(\kappa )}{C_{p}(0)}}$
While the value for $C_{p}(0)$ was derived above via the surface area, the same result may be obtained by setting $\kappa =0$ in the above formula for $C_{p}(\kappa )$. This can be done by noting that the series expansion for $I_{p/2-1}(\kappa )$ divided by $\kappa ^{p/2-1}$ has but one non-zero term at $\kappa =0$. (To evaluate that term, one needs to use the definition $0^{0}=1$.)
Support
The support of the Von Mises–Fisher distribution is the hypersphere, or more specifically, the $(p-1)$-sphere, denoted as
$S^{p-1}=\left\{\mathbf {x} \in \mathbb {R} ^{p}:\left\|\mathbf {x} \right\|=1\right\}$
This is a $(p-1)$-dimensional manifold embedded in $p$-dimensional Euclidean space, $\mathbb {R} ^{p}$.
Relation to normal distribution
Starting from a normal distribution with isotropic covariance $\kappa ^{-1}\mathbf {I} $ and mean ${\boldsymbol {\mu }}$ of length $r>0$, whose density function is:
$G_{p}(\mathbf {x} ;{\boldsymbol {\mu }},\kappa )=\left({\sqrt {\frac {\kappa }{2\pi }}}\right)^{p}\exp \left(-\kappa {\frac {(\mathbf {x} -{\boldsymbol {\mu }})'(\mathbf {x} -{\boldsymbol {\mu }})}{2}}\right),$ ;{\boldsymbol {\mu }},\kappa )=\left({\sqrt {\frac {\kappa }{2\pi }}}\right)^{p}\exp \left(-\kappa {\frac {(\mathbf {x} -{\boldsymbol {\mu }})'(\mathbf {x} -{\boldsymbol {\mu }})}{2}}\right),}
the Von Mises–Fisher distribution is obtained by conditioning on $\left\|\mathbf {x} \right\|=1$. By expanding
$(\mathbf {x} -{\boldsymbol {\mu }})'(\mathbf {x} -{\boldsymbol {\mu }})=\mathbf {x} '\mathbf {x} +{\boldsymbol {\mu }}'{\boldsymbol {\mu }}-2{\boldsymbol {\mu }}'\mathbf {x} ,$
and using the fact that the first two right-hand-side terms are fixed, the Von Mises-Fisher density, $f_{p}(\mathbf {x} ;r^{-1}{\boldsymbol {\mu }},r\kappa )$ is recovered by recomputing the normalization constant by integrating $\mathbf {x} $ over the unit sphere. If $r=0$, we get the uniform distribution, with density $f_{p}(\mathbf {x} ;{\boldsymbol {0}},0)$ ;{\boldsymbol {0}},0)} .
More succinctly, the restriction of any isotropic multivariate normal density to the unit hypersphere, gives a Von Mises-Fisher density, up to normalization.
This construction can be generalized by starting with a normal distribution with a general covariance matrix, in which case conditioning on $\left\|\mathbf {x} \right\|=1$ gives the Fisher-Bingham distribution.
Estimation of parameters
Mean direction
A series of N independent unit vectors $x_{i}$ are drawn from a von Mises–Fisher distribution. The maximum likelihood estimates of the mean direction $\mu $ is simply the normalized arithmetic mean, a sufficient statistic:[3]
$\mu ={\bar {x}}/{\bar {R}},{\text{where }}{\bar {x}}={\frac {1}{N}}\sum _{i}^{N}x_{i},{\text{and }}{\bar {R}}=\|{\bar {x}}\|,$
Concentration parameter
Use the modified Bessel function of the first kind to define
$A_{p}(\kappa )={\frac {I_{p/2}(\kappa )}{I_{p/2-1}(\kappa )}}.$
Then:
$\kappa =A_{p}^{-1}({\bar {R}}).$
Thus $\kappa $ is the solution to
$A_{p}(\kappa )={\frac {\left\|\sum _{i}^{N}x_{i}\right\|}{N}}={\bar {R}}.$
A simple approximation to $\kappa $ is (Sra, 2011)
${\hat {\kappa }}={\frac {{\bar {R}}(p-{\bar {R}}^{2})}{1-{\bar {R}}^{2}}},$
A more accurate inversion can be obtained by iterating the Newton method a few times
${\hat {\kappa }}_{1}={\hat {\kappa }}-{\frac {A_{p}({\hat {\kappa }})-{\bar {R}}}{1-A_{p}({\hat {\kappa }})^{2}-{\frac {p-1}{\hat {\kappa }}}A_{p}({\hat {\kappa }})}},$
${\hat {\kappa }}_{2}={\hat {\kappa }}_{1}-{\frac {A_{p}({\hat {\kappa }}_{1})-{\bar {R}}}{1-A_{p}({\hat {\kappa }}_{1})^{2}-{\frac {p-1}{{\hat {\kappa }}_{1}}}A_{p}({\hat {\kappa }}_{1})}}.$
Standard error
For N ≥ 25, the estimated spherical standard error of the sample mean direction can be computed as:[4]
${\hat {\sigma }}=\left({\frac {d}{N{\bar {R}}^{2}}}\right)^{1/2}$
where
$d=1-{\frac {1}{N}}\sum _{i}^{N}\left(\mu ^{T}x_{i}\right)^{2}$
It is then possible to approximate a $100(1-\alpha )\%$ a spherical confidence interval (a confidence cone) about $\mu $ with semi-vertical angle:
$q=\arcsin \left(e_{\alpha }^{1/2}{\hat {\sigma }}\right),$ where $e_{\alpha }=-\ln(\alpha ).$
For example, for a 95% confidence cone, $\alpha =0.05,e_{\alpha }=-\ln(0.05)=2.996,$ and thus $q=\arcsin(1.731{\hat {\sigma }}).$
Expected value
The expected value of the Von Mises–Fisher distribution is not on the unit hypersphere, but instead has a length of less than one. This length is given by $A_{p}(\kappa )$ as defined above. For a Von Mises–Fisher distribution with mean direction ${\boldsymbol {\mu }}$ and concentration $\kappa >0$, the expected value is:
$A_{p}(\kappa ){\boldsymbol {\mu }}$.
For $\kappa =0$, the expected value is at the origin. For finite $\kappa >0$, the length of the expected value, is strictly between zero and one and is a monotonic rising function of $\kappa $.
The empirical mean (arithmetic average) of a collection of points on the unit hypersphere behaves in a similar manner, being close to the origin for widely spread data and close to the sphere for concentrated data. Indeed, for the Von Mises–Fisher distribution, the expected value of the maximum-likelihood estimate based on a collection of points is equal to the empirical mean of those points.
Entropy and KL divergence
The expected value can be used to compute differential entropy and KL divergence.
The differential entropy of ${\text{VMF}}({\boldsymbol {\mu }},\kappa )$ is:
${\bigl \langle }-\log f_{p}(\mathbf {x} ;{\boldsymbol {\mu }},\kappa ){\bigr \rangle }_{\mathbf {x} \sim {\text{VMF}}({\boldsymbol {\mu }},\kappa )}=-\log f_{p}(A_{p}(\kappa ){\boldsymbol {\mu }};{\boldsymbol {\mu }},\kappa )=-\log C_{p}(\kappa )-\kappa A_{p}(\kappa )$ ;{\boldsymbol {\mu }},\kappa ){\bigr \rangle }_{\mathbf {x} \sim {\text{VMF}}({\boldsymbol {\mu }},\kappa )}=-\log f_{p}(A_{p}(\kappa ){\boldsymbol {\mu }};{\boldsymbol {\mu }},\kappa )=-\log C_{p}(\kappa )-\kappa A_{p}(\kappa )}
where the angle brackets denote expectation. Notice that the entropy is a function of $\kappa $ only.
The KL divergence between ${\text{VMF}}({\boldsymbol {\mu _{0}}},\kappa _{0})$ and ${\text{VMF}}({\boldsymbol {\mu _{1}}},\kappa _{1})$ is:
${\Bigl \langle }\log {\frac {f_{p}(\mathbf {x} ;{\boldsymbol {\mu _{0}}},\kappa _{0})}{f_{p}(\mathbf {x} ;{\boldsymbol {\mu _{1}}},\kappa _{1})}}{\Bigr \rangle }_{\mathbf {x} \sim {\text{VMF}}({\boldsymbol {\mu _{0}}},\kappa _{0})}=\log {\frac {f_{p}(A_{p}(\kappa _{0}){\boldsymbol {\mu _{0}}};{\boldsymbol {\mu _{0}}},\kappa _{0})}{f_{p}(A_{p}(\kappa _{0}){\boldsymbol {\mu _{0}}};{\boldsymbol {\mu _{1}}},\kappa _{1})}}$ ;{\boldsymbol {\mu _{0}}},\kappa _{0})}{f_{p}(\mathbf {x} ;{\boldsymbol {\mu _{1}}},\kappa _{1})}}{\Bigr \rangle }_{\mathbf {x} \sim {\text{VMF}}({\boldsymbol {\mu _{0}}},\kappa _{0})}=\log {\frac {f_{p}(A_{p}(\kappa _{0}){\boldsymbol {\mu _{0}}};{\boldsymbol {\mu _{0}}},\kappa _{0})}{f_{p}(A_{p}(\kappa _{0}){\boldsymbol {\mu _{0}}};{\boldsymbol {\mu _{1}}},\kappa _{1})}}}
Transformation
Von Mises-Fisher (VMF) distributions are closed under orthogonal linear transforms. Let $\mathbf {U} $ be a $p$-by-$p$ orthogonal matrix. Let $\mathbf {x} \sim {\text{VMF}}({\boldsymbol {\mu }},\kappa )$ and apply the invertible linear transform: $\mathbf {y} =\mathbf {Ux} $. The inverse transform is $\mathbf {x} =\mathbf {U'y} $, because the inverse of an orthogonal matrix is its transpose: $\mathbf {U} ^{-1}=\mathbf {U} '$. The Jacobian of the transform is $\mathbf {U} $, for which the absolute value of its determinant is 1, also because of the orthogonality. Using these facts and the form of the VMF density, it follows that:
$\mathbf {y} \sim {\text{VMF}}(\mathbf {U} {\boldsymbol {\mu }},\kappa ).$
One may verify that since ${\boldsymbol {\mu }}$ and $\mathbf {x} $ are unit vectors, then by the orthogonality, so are $\mathbf {U} {\boldsymbol {\mu }}$ and $\mathbf {y} $.
Pseudo-random number generation
General case
An algorithm for drawing pseudo-random samples from the Von Mises Fisher (VMF) distribution was given by Ulrich[5] and later corrected by Wood.[6] An implementation in R is given by Hornik and Grün;[7] and a fast Python implementation is described by Pinzón and Jung.[8]
To simulate from a VMF distribution on the $(p-1)$-dimensional unitsphere, $S^{p-1}$, with mean direction ${\boldsymbol {\mu }}\in S^{p-1}$, these algorithms use the following radial-tangential decomposition for a point $\mathbf {x} \in S^{p-1}\subset \mathbb {R} ^{p}$ :
$\mathbf {x} =t{\boldsymbol {\mu }}+{\sqrt {1-t^{2}}}\mathbf {v} $
where $\mathbf {v} \in \mathbb {R} ^{p}$ lives in the tangential $(p-2)$-dimensional unit-subsphere that is centered at and perpendicular to ${\boldsymbol {\mu }}$; while $t\in [-1,1]$. To draw a sample $\mathbf {x} $ from a VMF with parameters ${\boldsymbol {\mu }}$ and $\kappa $, $\mathbf {v} $ must be drawn from the uniform distribution on the tangential subsphere; and the radial component, $t$, must be drawn independently from the distribution with density:
$f_{\text{radial}}(t;\kappa ,p)={\frac {(\kappa /2)^{\nu }}{\Gamma ({\frac {1}{2}})\Gamma (\nu +{\frac {1}{2}})I_{\nu }(\kappa )}}e^{t\kappa }(1-t^{2})^{\nu -{\frac {1}{2}}}$
where $\nu ={\frac {p}{2}}-1$. The normalization constant for this density may be verified by using:
$I_{\nu }(\kappa )={\frac {(\kappa /2)^{\nu }}{\Gamma ({\frac {1}{2}})\Gamma (\nu +{\frac {1}{2}})}}\int _{-1}^{1}e^{t\kappa }(1-t^{2})^{\nu -{\frac {1}{2}}}\,dt$
as given in Appendix 1 (A.3) in Directional Statistics.[3] Drawing the $t$ samples from this density by using a rejection sampling algorithm is explained in the above references. To draw the uniform $\mathbf {v} $ samples perpendicular to ${\boldsymbol {\mu }}$, see the algorithm in,[8] or otherwise a Householder transform can be used as explained in Algorithm 1 in.[9]
3-D sphere
To generate a Von Mises–Fisher distributed pseudo-random spherical 3-D unit vector[10][11] $ \mathbf {X} _{s}$ on the $ S^{2}$sphere for a given $ \mu $ and $ \kappa $, define
$\mathbf {X} _{s}=[\phi ,\theta ,r]$
where $ \phi $ is the polar angle, $ \theta $ the equatorial angle, and $ r=1$ the distance to the center of the sphere
for $ \mathbf {\mu } =[0,(.),1]$ the pseudo-random vector is then given by
$\mathbf {X} _{s}=[\arccos W,V,1]$
where $ V$ is sampled from the continuous uniform distribution $ U(a,b)$ with lower bound $ a$ and upper bound $ b$
$V\sim U(0,2\pi )$
and
$W=1+{\frac {1}{\kappa }}(\ln \xi +\ln(1-{\frac {\xi -1}{\xi }}e^{-2\kappa }))$
where $ \xi $ is sampled from the standard continuous uniform distribution $ U(0,1)$
$\xi \sim U(0,1)$
here, $ W$should be set to $ W=1$ when $ \mathbf {\xi } =0$ and $ \mathbf {X} _{s}$ rotated to match any other desired $ \mu $.
Distribution of polar angle
For $p=3$, the angle θ between $\mathbf {x} $ and ${\boldsymbol {\mu }}$ satisfies $\cos \theta ={\boldsymbol {\mu }}^{\mathsf {T}}\mathbf {x} $. It has the distribution
$p(\theta )=\int d^{2}xf(x;{\boldsymbol {\mu }},\kappa )\,\delta \left(\theta -{\text{arc cos}}({\boldsymbol {\mu }}^{\mathsf {T}}\mathbf {x} )\right)$,
which can be easily evaluated as
$p(\theta )=2\pi C_{3}(\kappa )\,\sin \theta \,e^{\kappa \cos \theta }$.
For the general case, $p\geq 2$, the distribution for the cosine of this angle:
$\cos \theta =t={\boldsymbol {\mu }}^{\mathsf {T}}\mathbf {x} $
is given by $f_{\text{radial}}(t;\kappa ,p)$, as explained above.
The uniform hypersphere distribution
When $\kappa =0$, the Von Mises–Fisher distribution, ${\text{VMF}}({\boldsymbol {\mu }},\kappa )$ on $S^{p-1}$ simplifies to the uniform distribution on $S^{p-1}\subset \mathbb {R} ^{p}$. The density is constant with value $C_{p}(0)$. Pseudo-random samples can be generated by generating samples in $\mathbb {R} ^{p}$ from the standard multivariate normal distribution, followed by normalization to unit norm.
Component marginal of uniform distribution
For $1\leq i\leq p$, let $x_{i}$ be any component of $\mathbf {x} \in S^{p-1}$. The marginal distribution for $x_{i}$ has the density:[12][13]
$f_{i}(x_{i};p)=f_{\text{radial}}(x_{i};\kappa =0,p)={\frac {(1-x_{i}^{2})^{{\frac {p-1}{2}}-1}}{B{\bigl (}{\frac {1}{2}},{\frac {p-1}{2}}{\bigr )}}}$
where $B(\alpha ,\beta )$ is the beta function. This distribution may be better understood by highlighting its relation to the beta distribution:
${\begin{aligned}x_{i}^{2}&\sim {\text{Beta}}{\bigl (}{\frac {1}{2}},{\frac {p-1}{2}}{\bigr )}&&{\text{and}}&{\frac {x_{i}+1}{2}}&\sim {\text{Beta}}{\bigl (}{\frac {p-1}{2}},{\frac {p-1}{2}}{\bigr )}\end{aligned}}$
where the Legendre duplication formula is useful to understand the relationships between the normalization constants of the various densities above.
Note that the components of $\mathbf {x} \in S^{p-1}$ are not independent, so that the uniform density is not the product of the marginal densities; and $\mathbf {x} $ cannot be assembled by independent sampling of the components.
Distribution of dot-products
In machine learning, especially in image classification, to-be-classified inputs (e.g. images) are often compared using cosine similarity, which is the dot product between intermediate representations in the form of unitvectors (termed embeddings). The dimensionality is typically high, with $p$ at least several hundreds. The deep neural networks that extract embeddings for classification should learn to spread the classes as far apart as possible and ideally this should give classes that are uniformly distributed on $S^{p-1}$.[14] For a better statistical understanding of across-class cosine similarity, the distribution of dot-products between unitvectors independently sampled from the uniform distribution may be helpful.
Let $\mathbf {x} ,\mathbf {y} \in S^{p-1}$ be unitvectors in $\mathbb {R} ^{p}$, independently sampled from the uniform distribution. Define:
${\begin{aligned}t&=\mathbf {x} '\mathbf {y} \in [-1,1],&r&={\frac {t+1}{2}}\in [0,1],&s&={\text{logit}}(r)=\log {\frac {1+t}{1-t}}\in \mathbb {R} \end{aligned}}$
where $t$ is the dot-product and $r,s$ are transformed versions of it. Then the distribution for $t$ is the same as the marginal component distribution given above;[13] the distribution for $r$ is symmetric beta and the distribution for $s$ is symmetric logistic-beta:
${\begin{aligned}r&\sim {\text{Beta}}{\bigl (}{\frac {p-1}{2}},{\frac {p-1}{2}}{\bigr )},&s&\sim B_{\sigma }{\bigl (}{\frac {p-1}{2}},{\frac {p-1}{2}}{\bigr )}\end{aligned}}$
The means and variances are:
${\begin{aligned}E[t]&=0,&E[r]&={\frac {1}{2}},&E[s]&=0,\end{aligned}}$
and
${\begin{aligned}{\text{var}}[t]&={\frac {1}{p}},&{\text{var}}[r]&={\frac {1}{4p}},&{\text{var}}[s]&=2\psi '{\bigl (}{\frac {p-1}{2}}{\bigr )}\approx {\frac {4}{p-1}}\end{aligned}}$
where $\psi '=\psi ^{(1)}$ is the first polygamma function. The variances decrease, the distributions of all three variables become more Gaussian, and the final approximation gets better as the dimensionality, $p$, is increased.
Generalizations
Matrix Von Mises-Fisher
Further information: Random matrix
The matrix von Mises-Fisher distribution (also known as matrix Langevin distribution[15][16]) has the density
$f_{n,p}(\mathbf {X} ;\mathbf {F} )\propto \exp(\operatorname {tr} (\mathbf {F} ^{\mathsf {T}}\mathbf {X} ))$ ;\mathbf {F} )\propto \exp(\operatorname {tr} (\mathbf {F} ^{\mathsf {T}}\mathbf {X} ))}
supported on the Stiefel manifold of $n\times p$ orthonormal p-frames $\mathbf {X} $, where $\mathbf {F} $ is an arbitrary $n\times p$ real matrix.[17][18]
Saw distributions
Ulrich,[5] in designing an algorithm for sampling from the VMF distribution, makes use of a family of distributions named after and explored by John G. Saw.[19] A Saw distribution is a distribution on the $(p-1)$-sphere, $S^{p-1}$, with modal vector ${\boldsymbol {\mu }}\in S^{p-1}$ and concentration $\kappa \geq 0$, and of which the density function has the form:
$f_{\text{Saw}}(\mathbf {x} ;{\boldsymbol {\mu }},\kappa )={\frac {g(\kappa \mathbf {x} '{\boldsymbol {\mu }})}{K_{p}(\kappa )}}$ ;{\boldsymbol {\mu }},\kappa )={\frac {g(\kappa \mathbf {x} '{\boldsymbol {\mu }})}{K_{p}(\kappa )}}}
where $g$ is a non-negative, increasing function; and where $K_{P}(\kappa )$ is the normalization constant. The above-mentioned radial-tangential decomposition generalizes to the Saw family and the radial compoment, $t=\mathbf {x} '{\boldsymbol {\mu }}$ has the density:
$f_{\text{Saw-radial}}(t;\kappa )={\frac {2\pi ^{p/2}}{\Gamma (p/2)}}{\frac {g(\kappa t)(1-t^{2})^{(p-3)/2}}{B{\bigl (}{\frac {1}{2}},{\frac {p-1}{2}}{\bigr )}K_{p}(\kappa )}}.$
where $B$ is the beta function. Also notice that the left-hand factor of the radial density is the surface area of $S^{p-1}$.
By setting $g(\kappa \mathbf {x} '{\boldsymbol {\mu }})=e^{\kappa \mathbf {x} '{\boldsymbol {\mu }}}$, one recovers the VMF distribution.
See also
• Kent distribution, a related distribution on the two-dimensional unit sphere
• von Mises distribution, von Mises–Fisher distribution where p = 2, the one-dimensional unit circle
• Bivariate von Mises distribution
• Directional statistics
References
1. Fisher, R. A. (1953). "Dispersion on a sphere". Proc. R. Soc. Lond. A. 217 (1130): 295–305. Bibcode:1953RSPSA.217..295F. doi:10.1098/rspa.1953.0064. S2CID 123166853.
2. Watson, G. S. (1980). "Distributions on the Circle and on the Sphere". J. Appl. Probab. 19: 265–280. doi:10.2307/3213566. JSTOR 3213566. S2CID 222325569.
3. Mardia, Kanti; Jupp, P. E. (1999). Directional Statistics. John Wiley & Sons Ltd. ISBN 978-0-471-95333-3.
4. Embleton, N. I. Fisher, T. Lewis, B. J. J. (1993). Statistical analysis of spherical data (1st pbk. ed.). Cambridge: Cambridge University Press. pp. 115–116. ISBN 0-521-45699-1.{{cite book}}: CS1 maint: multiple names: authors list (link)
5. Ulrich, Gary (1984). "Computer generation of distributions on the m-sphere". Applied Statistics. 33 (2): 158–163. doi:10.2307/2347441. JSTOR 2347441.
6. Wood, Andrew T (1994). "Simulation of the Von Mises Fisher distribution". Communications in Statistics - Simulation and Computation. 23 (1): 157–164. doi:10.1080/03610919408813161.
7. Hornik, Kurt; Grün, Bettina (2014). "movMF: An R Package for Fitting Mixtures of Von Mises-Fisher Distributions". Journal of Statistical Software. 58 (10). doi:10.18637/jss.v058.i10. S2CID 13171102.
8. Pinzón, Carlos; Jung, Kangsoo (2023-03-03), Fast Python sampler for the von Mises Fisher distribution, retrieved 2023-03-30
9. De Cao, Nicola; Aziz, Wilker (13 Feb 2023). "The Power Spherical distribution". arXiv:2006.04437 [stat.ML].
10. Pakyuz-Charrier, Evren; Lindsay, Mark; Ogarko, Vitaliy; Giraud, Jeremie; Jessell, Mark (2018-04-06). "Monte Carlo simulation for uncertainty estimation on structural data in implicit 3-D geological modeling, a guide for disturbance distribution selection and parameterization". Solid Earth. 9 (2): 385–402. Bibcode:2018SolE....9..385P. doi:10.5194/se-9-385-2018. ISSN 1869-9510.
11. A., Wood, Andrew T. (1992). Simulation of the Von Mises Fisher distribution. Centre for Mathematics & its Applications, Australian National University. OCLC 221030477.{{cite book}}: CS1 maint: multiple names: authors list (link)
12. Gosmann, J; Eliasmith, C (2016). "Optimizing Semantic Pointer Representations for Symbol-Like Processing in Spiking Neural Networks". PLOS ONE. 11 (2): e0149928. Bibcode:2016PLoSO..1149928G. doi:10.1371/journal.pone.0149928. PMC 4762696. PMID 26900931.
13. Voelker, Aaron R.; Gosmann, Jan; Stewart, Terrence C. "Efficiently sampling vectors and coordinates from the n-sphere and n-ball" (PDF). Centre for Theoretical Neuroscience – Technical Report, 2017. Retrieved 22 April 2023.
14. Wang, Tongzhou; Isola, Phillip (2020). "Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere". International Conference on Machine Learning (ICML). arXiv:2005.10242.
15. Pal, Subhadip; Sengupta, Subhajit; Mitra, Riten; Banerjee, Arunava (2020). "Conjugate Priors and Posterior Inference for the Matrix Langevin Distribution on the Stiefel Manifold". Bayesian Analysis. 15 (3): 871–908. doi:10.1214/19-BA1176. ISSN 1936-0975.
16. Chikuse, Yasuko (1 May 2003). "Concentrated matrix Langevin distributions". Journal of Multivariate Analysis. 85 (2): 375–394. doi:10.1016/S0047-259X(02)00065-9. ISSN 0047-259X.
17. Jupp (1979). "Maximum likelihood estimators for the matrix von Mises-Fisher and Bingham distributions". The Annals of Statistics. 7 (3): 599–606. doi:10.1214/aos/1176344681.
18. Downs (1972). "Orientational statistics". Biometrika. 59 (3): 665–676. doi:10.1093/biomet/59.3.665.
19. Saw, John G (1978). "A family of distributions on the m-sphere and some hypothesis tests". Biometrika. 65 (`): 69–73. doi:10.2307/2335278. JSTOR 2335278.
Further reading
• Dhillon, I., Sra, S. (2003) "Modeling Data using Directional Distributions". Tech. rep., University of Texas, Austin.
• Banerjee, A., Dhillon, I. S., Ghosh, J., & Sra, S. (2005). "Clustering on the unit hypersphere using von Mises-Fisher distributions". Journal of Machine Learning Research, 6(Sep), 1345-1382.
• Sra, S. (2011). "A short note on parameter approximation for von Mises-Fisher distributions: And a fast implementation of I_s(x)". Computational Statistics. 27: 177–190. CiteSeerX 10.1.1.186.1887. doi:10.1007/s00180-011-0232-x. S2CID 3654195.
Probability distributions (list)
Discrete
univariate
with finite
support
• Benford
• Bernoulli
• beta-binomial
• binomial
• categorical
• hypergeometric
• negative
• Poisson binomial
• Rademacher
• soliton
• discrete uniform
• Zipf
• Zipf–Mandelbrot
with infinite
support
• beta negative binomial
• Borel
• Conway–Maxwell–Poisson
• discrete phase-type
• Delaporte
• extended negative binomial
• Flory–Schulz
• Gauss–Kuzmin
• geometric
• logarithmic
• mixed Poisson
• negative binomial
• Panjer
• parabolic fractal
• Poisson
• Skellam
• Yule–Simon
• zeta
Continuous
univariate
supported on a
bounded interval
• arcsine
• ARGUS
• Balding–Nichols
• Bates
• beta
• beta rectangular
• continuous Bernoulli
• Irwin–Hall
• Kumaraswamy
• logit-normal
• noncentral beta
• PERT
• raised cosine
• reciprocal
• triangular
• U-quadratic
• uniform
• Wigner semicircle
supported on a
semi-infinite
interval
• Benini
• Benktander 1st kind
• Benktander 2nd kind
• beta prime
• Burr
• chi
• chi-squared
• noncentral
• inverse
• scaled
• Dagum
• Davis
• Erlang
• hyper
• exponential
• hyperexponential
• hypoexponential
• logarithmic
• F
• noncentral
• folded normal
• Fréchet
• gamma
• generalized
• inverse
• gamma/Gompertz
• Gompertz
• shifted
• half-logistic
• half-normal
• Hotelling's T-squared
• inverse Gaussian
• generalized
• Kolmogorov
• Lévy
• log-Cauchy
• log-Laplace
• log-logistic
• log-normal
• log-t
• Lomax
• matrix-exponential
• Maxwell–Boltzmann
• Maxwell–Jüttner
• Mittag-Leffler
• Nakagami
• Pareto
• phase-type
• Poly-Weibull
• Rayleigh
• relativistic Breit–Wigner
• Rice
• truncated normal
• type-2 Gumbel
• Weibull
• discrete
• Wilks's lambda
supported
on the whole
real line
• Cauchy
• exponential power
• Fisher's z
• Kaniadakis κ-Gaussian
• Gaussian q
• generalized normal
• generalized hyperbolic
• geometric stable
• Gumbel
• Holtsmark
• hyperbolic secant
• Johnson's SU
• Landau
• Laplace
• asymmetric
• logistic
• noncentral t
• normal (Gaussian)
• normal-inverse Gaussian
• skew normal
• slash
• stable
• Student's t
• Tracy–Widom
• variance-gamma
• Voigt
with support
whose type varies
• generalized chi-squared
• generalized extreme value
• generalized Pareto
• Marchenko–Pastur
• Kaniadakis κ-exponential
• Kaniadakis κ-Gamma
• Kaniadakis κ-Weibull
• Kaniadakis κ-Logistic
• Kaniadakis κ-Erlang
• q-exponential
• q-Gaussian
• q-Weibull
• shifted log-logistic
• Tukey lambda
Mixed
univariate
continuous-
discrete
• Rectified Gaussian
Multivariate
(joint)
• Discrete:
• Ewens
• multinomial
• Dirichlet
• negative
• Continuous:
• Dirichlet
• generalized
• multivariate Laplace
• multivariate normal
• multivariate stable
• multivariate t
• normal-gamma
• inverse
• Matrix-valued:
• LKJ
• matrix normal
• matrix t
• matrix gamma
• inverse
• Wishart
• normal
• inverse
• normal-inverse
• complex
Directional
Univariate (circular) directional
Circular uniform
univariate von Mises
wrapped normal
wrapped Cauchy
wrapped exponential
wrapped asymmetric Laplace
wrapped Lévy
Bivariate (spherical)
Kent
Bivariate (toroidal)
bivariate von Mises
Multivariate
von Mises–Fisher
Bingham
Degenerate
and singular
Degenerate
Dirac delta function
Singular
Cantor
Families
• Circular
• compound Poisson
• elliptical
• exponential
• natural exponential
• location–scale
• maximum entropy
• mixture
• Pearson
• Tweedie
• wrapped
• Category
• Commons
| Wikipedia |
Von Bertalanffy function
The von Bertalanffy growth function (VBGF), or von Bertalanffy curve, is a type of growth curve for a time series and is named after Ludwig von Bertalanffy. It is a special case of the generalised logistic function. The growth curve is used to model mean length from age in animals.[1] The function is commonly applied in ecology to model fish growth[2] and in paleontology to model sclerochronological parameters of shell growth.[3]
The model can be written as the following:
$L(a)=L_{\infty }(1-\exp(-k(a-t_{0})))$
where $a$ is age, $k$ is the growth coefficient, $t_{0}$ is the theoretical age when size is zero, and $L_{\infty }$ is asymptotic size.[4] It is the solution of the following linear differential equation:
${\frac {dL}{da}}=k(L_{\infty }-L)$
Seasonally-adjusted von Bertalanffy
The seasonally-adjusted von Bertalanffy is an extension of this function that accounts for organism growth that occurs seasonally. It was created by I. F. Somers in 1988.[5]
See also
Wikimedia Commons has media related to Von Bertalanffy curve.
• Gompertz function
• Monod equation
• Michaelis–Menten kinetics
References
1. Daniel Pauly; G. R. Morgan (1987). Length-based Methods in Fisheries Research. WorldFish. p. 299. ISBN 978-971-10-2228-0.
2. Food and Agriculture Organization of the United Nations (2005). Management Techniques for Elasmobranch Fisheries. Food & Agriculture Org. p. 93. ISBN 978-92-5-105403-1.
3. Moss, D.K.; Ivany, L.C.; Jones, D.S. (2021). "Fossil bivalves and the sclerochronological reawakening". Paleobiology. 47 (4): 551–573. doi:10.1017/pab.2021.16. S2CID 234844791.
4. John K. Carlson; Kenneth J. Goldman (5 April 2007). Special Issue: Age and Growth of Chondrichthyan Fishes: New Methods, Techniques and Analysis. Springer Science & Business Media. ISBN 978-1-4020-5570-6.
5. Somers, I.F. (1988). "On a seasonally oscillating growth function". Fishbyte. 6 (1): 8–11.
| Wikipedia |
Graceful labeling
In graph theory, a graceful labeling of a graph with m edges is a labeling of its vertices with some subset of the integers from 0 to m inclusive, such that no two vertices share a label, and each edge is uniquely identified by the absolute difference between its endpoints, such that this magnitude lies between 1 and m inclusive.[1] A graph which admits a graceful labeling is called a graceful graph.
The name "graceful labeling" is due to Solomon W. Golomb; this type of labeling was originally given the name β-labeling by Alexander Rosa in a 1967 paper on graph labelings.[2]
A major conjecture in graph theory is the graceful tree conjecture or Ringel–Kotzig conjecture, named after Gerhard Ringel and Anton Kotzig, and sometimes abbreviated GTC.[3] It hypothesizes that all trees are graceful. It is still an open conjecture, although a related but weaker conjecture known as "Ringel's conjecture" was partially proven in 2020.[4][5][6] Kotzig once called the effort to prove the conjecture a "disease".[7]
Another weaker version of graceful labelling is near-graceful labeling, in which the vertices can be labeled using some subset of the integers on [0, m + 1] such that no two vertices share a label, and each edge is uniquely identified by the absolute difference between its endpoints (this magnitude lies on [1, m + 1]).
Another conjecture in graph theory is Rosa's Conjecture, named after Alexander Rosa, which says that all triangular cacti are graceful or nearly-graceful.[8]
A graceful graph with edges 0 to m is conjectured to have no fewer than $\left\lceil {\sqrt {3m+{\tfrac {9}{4}}}}\right\rfloor $ vertices, due to sparse ruler results. This conjecture has been verified for all graphs with 213 or fewer edges.
Selected results
• In his original paper, Rosa proved that an Eulerian graph with number of edges m ≡ 1 (mod 4) or m ≡ 2 (mod 4) cannot be graceful.[2]
• Also in his original paper, Rosa proved that the cycle Cn is graceful if and only if n ≡ 0 (mod 4) or n ≡ 3 (mod 4).
• All path graphs and caterpillar graphs are graceful.
• All lobster graphs with a perfect matching are graceful.[9]
• All trees with at most 27 vertices are graceful; this result was shown by Aldred and McKay in 1998 using a computer program.[10][11] This was extended to trees with at most 29 vertices in the Honours thesis of Michael Horton.[12] Another extension of this result up to trees with 35 vertices was claimed in 2010 by the Graceful Tree Verification Project, a distributed computing project led by Wenjie Fang.[13]
• All wheel graphs, web graphs, helm graphs, gear graphs, and rectangular grids are graceful.[10]
• All n-dimensional hypercubes are graceful.[14]
• All simple connected graphs with four or fewer vertices are graceful. The only non-graceful simple connected graphs with five vertices are the 5-cycle (pentagon); the complete graph K5; and the butterfly graph.[15]
See also
• Edge-graceful labeling
• List of conjectures
References
1. Virginia Vassilevska, "Coding and Graceful Labeling of trees." SURF 2001. PostScript
2. Rosa, A. (1967), "On certain valuations of the vertices of a graph", Theory of Graphs (Internat. Sympos., Rome, 1966), New York: Gordon and Breach, pp. 349–355, MR 0223271.
3. Wang, Tao-Ming; Yang, Cheng-Chang; Hsu, Lih-Hsing; Cheng, Eddie (2015), "Infinitely many equivalent versions of the graceful tree conjecture", Applicable Analysis and Discrete Mathematics, 9 (1): 1–12, doi:10.2298/AADM141009017W, MR 3362693
4. Montgomery, Richard; Pokrovskiy, Alexey; Sudakov, Benny (2020). "A proof of Ringel's Conjecture". arXiv:2001.02665 [math.CO].
5. Huang, C.; Kotzig, A.; Rosa, A. (1982), "Further results on tree labellings", Utilitas Mathematica, 21: 31–48, MR 0668845.
6. Hartnett, Kevin. "Rainbow Proof Shows Graphs Have Uniform Parts". Quanta Magazine. Retrieved 2020-02-29.
7. Huang, C.; Kotzig, A.; Rosa, A. (1982), "Further results on tree labellings", Utilitas Mathematica, 21: 31–48, MR 0668845.
8. Rosa, A. (1988), "Cyclic Steiner Triple Systems and Labelings of Triangular Cacti", Scientia, 1: 87–95.
9. Morgan, David (2008), "All lobsters with perfect matchings are graceful", Bulletin of the Institute of Combinatorics and Its Applications, 53: 82–85, hdl:10402/era.26923.
10. Gallian, Joseph A. (1998), "A dynamic survey of graph labeling", Electronic Journal of Combinatorics, 5: Dynamic Survey 6, 43 pp. (389 pp. in 18th ed.) (electronic), MR 1668059.
11. Aldred, R. E. L.; McKay, Brendan D. (1998), "Graceful and harmonious labellings of trees", Bulletin of the Institute of Combinatorics and Its Applications, 23: 69–72, MR 1621760.
12. Horton, Michael P. (2003), Graceful Trees: Statistics and Algorithms.
13. Fang, Wenjie (2010), A Computational Approach to the Graceful Tree Conjecture, arXiv:1003.3045, Bibcode:2010arXiv1003.3045F. See also Graceful Tree Verification Project
14. Kotzig, Anton (1981), "Decompositions of complete graphs into isomorphic cubes", Journal of Combinatorial Theory, Series B, 31 (3): 292–296, doi:10.1016/0095-8956(81)90031-9, MR 0638285.
15. Weisstein, Eric W. "Graceful graph". MathWorld.
External links
• Numberphile video about graceful tree conjecture
• Graceful labeling in mathworld
Further reading
• (K. Eshghi) Introduction to Graceful Graphs, Sharif University of Technology, 2002.
• (U. N. Deshmukh and Vasanti N. Bhat-Nayak), New families of graceful banana trees – Proceedings Mathematical Sciences, 1996 – Springer
• (M. Haviar, M. Ivaska), Vertex Labellings of Simple Graphs, Research and Exposition in Mathematics, Volume 34, 2015.
• (Ping Zhang), A Kaleidoscopic View of Graph Colorings, SpringerBriefs in Mathematics, 2016 – Springer
| Wikipedia |
Von Kármán swirling flow
Von Kármán swirling flow is a flow created by a uniformly rotating infinitely long plane disk, named after Theodore von Kármán who solved the problem in 1921.[1] The rotating disk acts as a fluid pump and is used as a model for centrifugal fans or compressors. This flow is classified under the category of steady flows in which vorticity generated at a solid surface is prevented from diffusing far away by an opposing convection, the other examples being the Blasius boundary layer with suction, stagnation point flow etc.
Flow description
Consider a planar disk of infinite radius rotating at a constant angular velocity $\Omega $ in fluid which is initially at rest everywhere. Near to the surface, the fluid is being turned by the disk, due to friction, which then causes centrifugal forces which move the fluid outwards. This outward radial motion of the fluid near the disk must be accompanied by an inward axial motion of the fluid towards the disk to conserve mass. Theodore von Kármán[1] noticed that the governing equations and the boundary conditions allow a solution such that $u/r,v/r$ and $w$ are functions of $z$ only, where $(u,v,w)$ are the velocity components in cylindrical $(r,\theta ,z)$ coordinate with $r=0$ being the axis of rotation and $z=0$ represents the plane disk. Due to symmetry, pressure of the fluid can depend only on radial and axial coordinate $p=p(r,z)$. Then the continuity equation and the incompressible Navier–Stokes equations reduce to
${\begin{aligned}&{\frac {2u}{r}}+{\frac {dw}{dz}}=0\\[8pt]&\left({\frac {u}{r}}\right)^{2}-\left({\frac {v}{r}}\right)^{2}+w{\frac {d(u/r)}{dz}}=-{\frac {1}{\rho }}{\frac {\partial p}{\partial r}}+\nu {\frac {d^{2}(u/r)}{dz^{2}}}\\[8pt]&{\frac {2uv}{r^{2}}}+w{\frac {d(v/r)}{dz}}=\nu {\frac {d^{2}(v/r)}{dz^{2}}}\\[8pt]&w{\frac {dw}{dz}}=-{\frac {1}{\rho }}{\frac {\partial p}{\partial z}}+\nu {\frac {d^{2}w}{dz^{2}}}\qquad \Rightarrow \qquad {\frac {p}{\rho }}=\nu {\frac {dw}{dz}}-{\frac {1}{2}}w^{2}+f(r)\end{aligned}}$
where $\nu $ is the kinematic viscosity.
No rotation at infinity
Since there is no rotation at large $z\rightarrow \infty $, $p(r,z)$ becomes independent of $r$ resulting in $p=p(z)$. Hence $f(r)={\text{constant}}$ and $\partial p/\partial r=0$.
Here the boundary conditions for the fluid $z>0$ are
$u=0,\quad v=\Omega r,\quad w=0,\quad p=p_{0}\quad {\text{ for }}z=0$
$u=0,\quad v=0\quad {\text{ for }}z\rightarrow \infty $
Self-similar solution is obtained by introducing following transformation,[2]
$\eta ={\sqrt {\frac {\Omega }{\nu }}}z,\quad u=r\Omega F(\eta ),\quad v=r\Omega G(\eta ),\quad w={\sqrt {\nu \Omega }}H(\eta ),\quad p=p_{0}+\rho \nu \Omega P(\eta ),$
where $\rho $ is the fluid density.
The self-similar equations are
${\begin{aligned}2F+H'&=0\\F^{2}-G^{2}+F'H&=F''\\2FG+G'H&=G''\\P'+HH'-H''&=0\end{aligned}}$
with boundary conditions for the fluid $\eta >0$ are
$F=0,\quad G=1,\quad H=0,\quad P=0\quad {\text{ for }}\eta =0$
$F=0,\quad G=0\quad {\text{ for }}\eta \rightarrow \infty $
The coupled ordinary differential equations need to be solved numerically and an accurate solution is given by Cochran(1934).[3] The inflow axial velocity at infinity obtained from the numerical integration is $w=-0.886{\sqrt {\nu \Omega }}$, so the total outflowing volume flux across a cylindrical surface of radius $r$ is $0.886\pi r^{2}{\sqrt {\nu \Omega }}$. The tangential stress on the disk is $\sigma _{z\varphi }=\mu (\partial v/\partial z)_{z=0}=\rho {\sqrt {\nu \Omega ^{3}}}rG'(0)$. Neglecting edge effects, the torque exerted by the fluid on the disk with large ($R\gg {\sqrt {\nu /\Omega }}$) but finite radius $R$ is
$T=2\int _{0}^{R}2\pi r^{2}\sigma _{r\theta }\,dr=\pi R^{4}\rho {\sqrt {\nu \Omega ^{3}}}G'(0).$
The factor $2$ is added to account for both sides of the disk. From numerical solution, torque is given by $T=-1.94R^{4}\rho {\sqrt {\nu \Omega ^{3}}}$. The torque predicted by the theory is in excellent agreement with the experiment on large disks up to the Reynolds number of about $Re=R^{2}\Omega /\nu =3\times 10^{5}$, the flow becomes turbulent at high Reynolds number.[4]
Rigid body rotation at infinity
This problem was addressed by George Keith Batchelor(1951).[5] Let $\Gamma $ be the angular velocity at infinity. Now the pressure at $z\rightarrow \infty $ is ${\frac {1}{2}}\rho \Gamma ^{2}r^{2}$. Hence $f(r)={\frac {1}{2}}\rho \Gamma ^{2}r^{2}$ and $\partial p/\partial r=\Gamma ^{2}$.
Then the boundary conditions for the fluid $z>0$ are
$u=0,\quad v=\Omega r,\quad w=0\quad {\text{ for }}z=0$
$u=0,\quad v=\Gamma r\quad {\text{for }}z\rightarrow \infty $
Self-similar solution is obtained by introducing following transformation,
$\eta ={\sqrt {\frac {\Omega }{\nu }}}z,\quad \gamma ={\frac {\Gamma }{\Omega }},\quad u=r\Omega F(\eta ),\quad v=r\Omega G(\eta ),\quad w={\sqrt {\nu \Omega }}H(\eta ).$
The self-similar equations are
${\begin{aligned}2F+H'&=0\\[3pt]F^{2}-G^{2}+F'H&=F''-\gamma ^{2}\\[3pt]2FG+G'H&=G''\end{aligned}}$
with boundary conditions for the fluid $\eta >0$ is
$F=0,\quad G=1,\quad H=0\quad {\text{ for }}\eta =0$
$F=0,\quad G=\gamma \quad {\text{ for }}\eta \rightarrow \infty $
The solution is easy to obtain only for $\gamma >0$ i.e., the fluid at infinity rotates in the same sense as the plate. For $\gamma <0$, the solution is more complex, in the sense that many-solution branches occur. Evans(1969)[6] obtained solution for the range $-1.35<\gamma <-0.61$. Zandbergen and Dijkstra[7][8] showed that the solution exhibits a square root singularity as $\gamma ^{-}\rightarrow -0.16053876$ and found a second-solution branch merging with the solution found for $\gamma \rightarrow -0.16053876$. The solution of the second branch is continued till $\gamma ^{-}\rightarrow 0.07452563$, at which point, a third-solution branch is found to emerge. They also discovered an infinity of solution branches around the point $\gamma ^{-}\rightarrow 0$. Bodoyni(1975)[9] calculated solutions for large negative $\gamma $, showed that the solution breaks down at $\gamma ^{-}=-1.436$. If the rotating plate is allowed to have uniform suction velocity at the plate, then meaningful solution can be obtained for $\gamma \leq -0.2$.[4]
For $0<\gamma <\infty ,\ \gamma \neq 1$ ($\gamma =1$ represents solid body rotation, the whole fluid rotates at the same speed) the solution reaches the solid body rotation at infinity in an oscillating manner from the plate. The axial velocity is negative $w<0$ for $0\leq \gamma <1$ and positive $w>0$ for $1<\gamma <\infty $. There is an explicit solution when $|\gamma -1|\ll 1$.
Nearly rotating at the same speed, $|\gamma -1|\ll 1$
Since both boundary conditions for $G$ are almost equal to one, one would expect the solution for $G$ to slightly deviate from unity. The corresponding scales for $F$ and $H$ can be derived from the self-similar equations. Therefore,
$G=1+{\hat {G}},\quad H={\hat {H}},\quad F={\hat {F}}\qquad |{\hat {F}}|,|{\hat {G}}|,|{\hat {H}}|\ll 1$
To the first order approximation(neglecting ${\hat {F}}^{2},{\hat {G}}^{2},{\hat {H}}^{2}$), the self-similar equation [10] becomes
${\begin{aligned}2{\hat {F}}+{\hat {H}}'&=0\\1+2{\hat {G}}&=\gamma ^{2}-{\hat {F}}''\\2{\hat {F}}&={\hat {G}}''\end{aligned}}$
with exact solutions
${\begin{aligned}F(\eta )&=-(\gamma -1)e^{-\eta }\sin \eta ,\\G(\eta )&=1+(\gamma -1)(1-e^{-\eta }\cos \eta ),\\H(\eta )&=(\gamma -1)[1-e^{-\eta }(\sin \eta +\cos \eta )].\end{aligned}}$
These solution are similar to an Ekman layer[10] solution.
Non-Axisymmetric solutions[11]
The flow accepts a non-axisymmetric solution with axisymmetric boundary conditions discovered by Hewitt, Duck and Foster.[12] Defining
$\eta ={\sqrt {\frac {\Omega }{\nu }}}z,\quad \gamma ={\frac {\Gamma }{\Omega }},\quad u=r\Omega [f'(\eta )+\phi (\eta )\cos 2\theta ],\quad v=r\Omega [g(\eta )-\phi (\eta )\sin 2\theta ],\quad w=-2{\sqrt {\nu \Omega }}f(\eta ),$
and the governing equations are
${\begin{aligned}f'''+2ff''-f'^{2}-\phi ^{2}+g^{2}&=\gamma ^{2},\\g''+2(fg'-f'g)&=0,\\\phi ''+2(f\phi '-f'\phi )&=0,\end{aligned}}$
with boundary conditions
$f(0)=f'(0)=\phi (0)=g(0)-1=f'(\infty )=\phi (\infty )=g(\infty )-\gamma =0.$
The solution is found to exist from numerical integration for $-0.14485\leq \gamma \leq 0$.
Bödewadt flow
Bödewadt flow describes the flow when a stationary disk is placed in a rotating fluid.[13]
Two rotating coaxial disks
This problem was addressed by George Keith Batchelor(1951),[5] Keith Stewartson(1952)[14] and many other researchers. Here the solution is not simple, because of the additional length scale imposed in the problem i.e., the distance $h$ between the two disks. In addition, the uniqueness and existence of a steady solution are also depend on the corresponding Reynolds number $Re=\Omega h^{2}/\nu $.
Then the boundary conditions for the fluid $0<z<h$ are
$u=0,\quad v=\Omega r,\quad w=0\quad {\text{ for }}z=0$
$u=0,\quad v=\Gamma r,\quad w=0\quad {\text{for }}z=h.$
In terms of $\eta $, the upper wall location is simply $\eta ={\sqrt {\Omega /\nu }}h=Re^{1/2}$. Thus, instead of the scalings
${\begin{aligned}\eta ={\sqrt {\frac {\Omega }{\nu }}}z,\quad \gamma ={\frac {\Gamma }{\Omega }},\quad u=r\Omega F'(\eta ),\quad v=r\Omega G(\eta ),\quad w=-2{\sqrt {\nu \Omega }}F(\eta )\end{aligned}}$
used before, it is convenient to introduce following transformation,
${\begin{aligned}\xi =Re^{-1/2}\eta ,\quad f=Re^{-1/2}F,\quad g=G\end{aligned}}$
so that the governing equations become
${\begin{aligned}Re^{-1}f'''+2ff''-f'^{2}+g^{2}=\lambda ,\\Re^{-1}g''+2(fg'-f'g)=0\end{aligned}}$
with six boundary conditions
$f'=0,\quad g=1,\quad f=0\quad {\text{ for }}\xi =0$
$f'=0,\quad g=\gamma ,\quad f=0\quad {\text{for }}\xi =1.$
and the pressure is given by
${\frac {p-p_{o}}{\rho }}={\frac {1}{2}}\lambda r^{2}\Omega ^{2}-2\nu \Omega (Ref^{2}+f').$
Here boundary conditions are six because pressure is not known either at the top or bottom wall; $\lambda $ is to be obtained as part of solution. For large Reynolds number $Re\gg 1$, Batchelor argued that the fluid in the core would rotate at a constant velocity, flanked by two boundary layers at each disk for $\gamma \geq 0$ and there would be two uniform counter-rotating flow of thickness $\xi =1/2$ for $\gamma =-1$. However, Stewartson predicted that for $\gamma =0,-1$ the fluid in the core would not rotate at $Re\gg 1$, but just left with two boundary layers at each disk. It turns out, Stewartson predictions were correct (see Stewartson layer).
There is also an exact solution if the two disks are rotating about different axes but for $\gamma =1$.
Applications
Von Kármán swirling flow finds its applications in wide range of fields, which includes rotating machines, filtering systems, computer storage devices, heat transfer and mass transfer applications, combustion-related problems,[15] planetary formations, geophysical applications etc.
References
1. Von Kármán, Theodore (1921). "Über laminare und turbulente Reibung". Zeitschrift für Angewandte Mathematik und Mechanik. 1 (4): 233–252. Bibcode:1921ZaMM....1..233K. doi:10.1002/zamm.19210010401.
2. Schlichting, Hermann and Gersten, Klaus (2017). Boundary-Layer Theory. ISBN 978-3662529171.{{cite book}}: CS1 maint: multiple names: authors list (link)
3. Cochran, W.G. (1934). "The flow due to a rotating disc". Mathematical Proceedings of the Cambridge Philosophical Society. 30 (3): 365. Bibcode:1934PCPS...30..365C. doi:10.1017/S0305004100012561. S2CID 123003223.
4. Schlichting, Hermann (1960). Boundary Layer Theory. New York: McGraw-hill.
5. Batchelor, George Keith (1951). "Note on a class of solutions of the Navier–Stokes equations representing steady rotationally-symmetric flow". The Quarterly Journal of Mechanics and Applied Mathematics. 4: 29–41. doi:10.1093/qjmam/4.1.29.
6. Evans, D. J. "The rotationally symmetric flow of a viscous fluid in the presence of an infinite rotating disc with uniform suction." The Quarterly Journal of Mechanics and Applied Mathematics 22.4 (1969): 467-485.
7. Zandbergen, P. J., and D. Dijkstra. "Non-unique solutions of the Navier-Stokes equations for the Karman swirling flow." Journal of engineering mathematics 11.2 (1977): 167-188.
8. Dijkstra, D., and P. J. Zandbergen. "Some further investigations on non-unique solutions of the Navier-Stokes equations for the Karman swirling flow." Archiv of Mechanics, Archiwum Mechaniki Stosowanej 30 (1978): 411-419.
9. Bodonyi, R. J. "On rotationally symmetric flow above an infinite rotating disk." Journal of Fluid Mechanics 67.04 (1975): 657-666.
10. Batchelor, George Keith (2000). An introduction to fluid dynamics. Cambridge university press. ISBN 978-0521663960.
11. Drazin, Philip G., and Norman Riley. The Navier–Stokes equations: a classification of flows and exact solutions. No. 334. Cambridge University Press, 2006.
12. Hewitt, R. E., P. W. Duck, and M. R. Foster. "Steady boundary-layer solutions for a swirling stratified fluid in a rotating cone." Journal of Fluid Mechanics 384 (1999): 339-374.
13. Bödewadt, V. U. (1940). Die drehströmung über festem grunde. ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 20(5), 241-253.
14. Stewartson, K. (1953). "On the flow between two rotating coaxial disks". Mathematical Proceedings of the Cambridge Philosophical Society. 49 (2): 333–341. Bibcode:1953PCPS...49..333S. doi:10.1017/S0305004100028437. S2CID 122805153.
15. Urzay, J.; Nagayam, V.; Williams, F.A. (2011). "Theory of the propagation dynamics of spiral edges of diffusion flames in von Kármán swirling flows" (PDF). Combustion and Flame. 158 (2): 255–272. doi:10.1016/j.combustflame.2010.08.015.
Bibliography
• Von Kármán, Theodore (1921). "Über laminare und turbulente Reibung". Zeitschrift für Angewandte Mathematik und Mechanik. 1 (4): 233–252. Bibcode:1921ZaMM....1..233K. doi:10.1002/zamm.19210010401.
• Batchelor, George Keith (1951). "Note on a class of solutions of the Navier-Stokes equations representing steady rotationally-symmetric flow". The Quarterly Journal of Mechanics and Applied Mathematics. 4: 29–41. doi:10.1093/qjmam/4.1.29.
• Stewartson, K. (1953). "On the flow between two rotating coaxial disks". Mathematical Proceedings of the Cambridge Philosophical Society. 49 (2): 333–341. Bibcode:1953PCPS...49..333S. doi:10.1017/S0305004100028437. S2CID 122805153.
• Batchelor, George Keith (2000). An introduction to fluid dynamics. Cambridge university press. ISBN 978-0521663960.
• Landau, Lev D (1987). Fluid Mechanics. ISBN 978-0750627672.
• Schlichting, Hermann (1960). Boundary-Layer Theory. New York: McGraw-hill.
• Schlichting, Hermann and Gersten, Klaus (2017). Boundary-Layer Theory. ISBN 978-3662529171.{{cite book}}: CS1 maint: multiple names: authors list (link)
| Wikipedia |
Von Mangoldt function
In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that is neither multiplicative nor additive.
Not to be confused with de Bruijn–Newman constant.
Definition
The von Mangoldt function, denoted by Λ(n), is defined as
$\Lambda (n)={\begin{cases}\log p&{\text{if }}n=p^{k}{\text{ for some prime }}p{\text{ and integer }}k\geq 1,\\0&{\text{otherwise.}}\end{cases}}$
The values of Λ(n) for the first nine positive integers (i.e. natural numbers) are
$0,\log 2,\log 3,\log 2,\log 5,0,\log 7,\log 2,\log 3,$
which is related to (sequence A014963 in the OEIS).
Properties
The von Mangoldt function satisfies the identity[1][2]
$\log(n)=\sum _{d\mid n}\Lambda (d).$
The sum is taken over all integers d that divide n. This is proved by the fundamental theorem of arithmetic, since the terms that are not powers of primes are equal to 0. For example, consider the case n = 12 = 22 × 3. Then
${\begin{aligned}\sum _{d\mid 12}\Lambda (d)&=\Lambda (1)+\Lambda (2)+\Lambda (3)+\Lambda (4)+\Lambda (6)+\Lambda (12)\\&=\Lambda (1)+\Lambda (2)+\Lambda (3)+\Lambda \left(2^{2}\right)+\Lambda (2\times 3)+\Lambda \left(2^{2}\times 3\right)\\&=0+\log(2)+\log(3)+\log(2)+0+0\\&=\log(2\times 3\times 2)\\&=\log(12).\end{aligned}}$
By Möbius inversion, we have
$\Lambda (n)=\sum _{d\mid n}\mu (d)\log \left({\frac {n}{d}}\right)$
and using the product rule for the logarithm we get[2][3][4]
$\Lambda (n)=-\sum _{d\mid n}\mu (d)\log(d)\ .$
For all $x\geq 1$, we have[5]
$\sum _{n\leq x}{\frac {\Lambda (n)}{n}}=\log x+O(1).$
Also, there exist positive constants c1 and c2 such that
$\psi (x)\leq c_{1}x,$
for all $x\geq 1$, and
$\psi (x)\geq c_{2}x,$
for all sufficiently large x.
Dirichlet series
The von Mangoldt function plays an important role in the theory of Dirichlet series, and in particular, the Riemann zeta function. For example, one has
$\log \zeta (s)=\sum _{n=2}^{\infty }{\frac {\Lambda (n)}{\log(n)}}\,{\frac {1}{n^{s}}},\qquad {\text{Re}}(s)>1.$
The logarithmic derivative is then[6]
${\frac {\zeta ^{\prime }(s)}{\zeta (s)}}=-\sum _{n=1}^{\infty }{\frac {\Lambda (n)}{n^{s}}}.$
These are special cases of a more general relation on Dirichlet series. If one has
$F(s)=\sum _{n=1}^{\infty }{\frac {f(n)}{n^{s}}}$
for a completely multiplicative function f (n), and the series converges for Re(s) > σ0, then
${\frac {F^{\prime }(s)}{F(s)}}=-\sum _{n=1}^{\infty }{\frac {f(n)\Lambda (n)}{n^{s}}}$
converges for Re(s) > σ0.
Chebyshev function
The second Chebyshev function ψ(x) is the summatory function of the von Mangoldt function:[7]
$\psi (x)=\sum _{p^{k}\leq x}\log p=\sum _{n\leq x}\Lambda (n)\ .$
It was introduced by Pafnuty Chebyshev who used it to show that the true order of the prime counting function $\pi (x)$ is $x/\log x$. Von Mangoldt provided a rigorous proof of an explicit formula for ψ(x) involving a sum over the non-trivial zeros of the Riemann zeta function. This was an important part of the first proof of the prime number theorem.
The Mellin transform of the Chebyshev function can be found by applying Perron's formula:
${\frac {\zeta ^{\prime }(s)}{\zeta (s)}}=-s\int _{1}^{\infty }{\frac {\psi (x)}{x^{s+1}}}\,dx$
which holds for Re(s) > 1.
Exponential series
Hardy and Littlewood examined the series[8]
$F(y)=\sum _{n=2}^{\infty }\left(\Lambda (n)-1\right)e^{-ny}$
in the limit y → 0+. Assuming the Riemann hypothesis, they demonstrate that
$F(y)=O\left({\frac {1}{\sqrt {y}}}\right)\quad {\text{and}}\quad F(y)=\Omega _{\pm }\left({\frac {1}{\sqrt {y}}}\right)$
In particular this function is oscillatory with diverging oscillations: there exists a value K > 0 such that both inequalities
$F(y)<-{\frac {K}{\sqrt {y}}},\quad {\text{ and }}\quad F(z)>{\frac {K}{\sqrt {z}}}$
hold infinitely often in any neighbourhood of 0. The graphic to the right indicates that this behaviour is not at first numerically obvious: the oscillations are not clearly seen until the series is summed in excess of 100 million terms, and are only readily visible when y < 10−5.
Riesz mean
The Riesz mean of the von Mangoldt function is given by
${\begin{aligned}\sum _{n\leq \lambda }\left(1-{\frac {n}{\lambda }}\right)^{\delta }\Lambda (n)&=-{\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }{\frac {\Gamma (1+\delta )\Gamma (s)}{\Gamma (1+\delta +s)}}{\frac {\zeta ^{\prime }(s)}{\zeta (s)}}\lambda ^{s}ds\\&={\frac {\lambda }{1+\delta }}+\sum _{\rho }{\frac {\Gamma (1+\delta )\Gamma (\rho )}{\Gamma (1+\delta +\rho )}}+\sum _{n}c_{n}\lambda ^{-n}.\end{aligned}}$
Here, λ and δ are numbers characterizing the Riesz mean. One must take c > 1. The sum over ρ is the sum over the zeroes of the Riemann zeta function, and
$\sum _{n}c_{n}\lambda ^{-n}\,$
can be shown to be a convergent series for λ > 1.
Approximation by Riemann zeta zeros
There is an explicit formula for the summatory Mangoldt function $\psi (x)$ given by[9]
$\psi (x)=x-\sum _{\zeta (\rho )=0}{\frac {x^{\rho }}{\rho }}-\log(2\pi ).$
If we separate out the trivial zeros of the zeta function, which are the negative even integers, we obtain
$\psi (x)=x-\sum _{\zeta (\rho )=0,\ 0<\Re (\rho )<1}{\frac {x^{\rho }}{\rho }}-\log(2\pi )-{\frac {1}{2}}\log(1-x^{-2}).$
(The sum is not absolutely convergent, so we take the zeros in order of the absolute value of their imaginary part.)
Taking the derivative of both sides, ignoring convergence issues, we get an "equality" of distributions
$\sum _{q=p^{r}}\Lambda (q)\delta (x-q)=1-\sum _{\zeta (\rho )=0,\ 0<\Re (\rho )<1}{\frac {x^{\rho }}{x}}+{\frac {1}{x-x^{3}}}.$
Therefore, we should expect that the sum over nontrivial zeta zeros
$-\sum _{\zeta (\rho )=0,\ 0<\Re (\rho )<1}{\frac {x^{\rho }}{x}}$
peaks at primes. In fact, this is the case, as can be seen in the adjoining graph, and can also be verified through numerical computation.
The Fourier transform of the von Mangoldt function gives a spectrum with spikes at ordinates equal to the imaginary parts of the Riemann zeta function zeros. This is sometimes called a duality.
Generalized von Mangoldt function
The functions
$\Lambda _{k}(n)=\sum \limits _{d\mid n}\mu (d)\log ^{k}(n/d),$
where $\mu $ denotes the Möbius function and $k$ denotes a positive integer, generalize the von Mangoldt function.[10] The function $\Lambda _{1}$ is the ordinary von Mangoldt function $\Lambda $.
See also
• Prime-counting function
References
1. Apostol (1976) p.32
2. Tenenbaum (1995) p.30
3. Apostol (1976) p.33
4. Schroeder, Manfred R. (1997). Number theory in science and communication. With applications in cryptography, physics, digital information, computing, and self-similarity. Springer Series in Information Sciences. Vol. 7 (3rd ed.). Berlin: Springer-Verlag. ISBN 3-540-62006-0. Zbl 0997.11501.
5. Apostol (1976) p.88
6. Hardy & Wright (2008) §17.7, Theorem 294
7. Apostol (1976) p.246
8. Hardy, G. H. & Littlewood, J. E. (1916). "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes" (PDF). Acta Mathematica. 41: 119–196. doi:10.1007/BF02422942. Archived from the original (PDF) on 2012-02-07. Retrieved 2014-07-03.
9. Conrey, J. Brian (March 2003). "The Riemann hypothesis" (PDF). Notices Am. Math. Soc. 50 (3): 341–353. Zbl 1160.11341. Page 346
10. Iwaniec, Henryk; Friedlander, John (2010), Opera de cribro, American Mathematical Society Colloquium Publications, vol. 57, Providence, RI: American Mathematical Society, p. 23, ISBN 978-0-8218-4970-5, MR 2647984
• Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001
• Hardy, G. H.; Wright, E. M. (2008) [1938]. Heath-Brown, D. R.; Silverman, J. H. (eds.). An Introduction to the Theory of Numbers (6th ed.). Oxford: Oxford University Press. ISBN 978-0-19-921985-8. MR 2445243. Zbl 1159.11001.
• Tenebaum, Gérald (1995). Introduction to analytic and probabilistic number theory. Cambridge Studies in Advanced Mathematics. Vol. 46. Translated by C.B. Thomas. Cambridge: Cambridge University Press. ISBN 0-521-41261-7. Zbl 0831.11001.
External links
• Allan Gut, Some remarks on the Riemann zeta distribution (2005)
• S.A. Stepanov (2001) [1994], "Mangoldt function", Encyclopedia of Mathematics, EMS Press
• Heike, How plot Riemann zeta zero spectrum in Mathematica? (2012)
| Wikipedia |
Von Neumann's elephant
Von Neumann's elephant is a problem in recreational mathematics, consisting of constructing a planar curve in the shape of an elephant from only four fixed parameters. It originated from a discussion between physicists John von Neumann and Enrico Fermi.
History
In a 2004 article in the journal Nature, Freeman Dyson recounts his meeting with Fermi in 1953. Fermi evokes his friend von Neumann who, when asking him how many arbitrary parameters he used for his calculations, replied, "With four parameters I can fit an elephant, and with five I can make him wiggle his trunk." By this he meant that the Fermi simulations relied on too many input parameters, presupposing an overfitting phenomenon.[1]
• John von Neumann
• Enrico Fermi
• Freeman Dyson in 2005
Solving the problem (defining four complex numbers to draw an elephantine shape) subsequently became an active research subject of recreational mathematics. A 1975 attempt through least-squares function approximation required dozens of terms.[2] The best approximation was found by three physicists in 2010.[3]
Construction
The construction is based on complex Fourier analysis.
The curve found in 2010 is parameterized by:
$\left\lbrace {\begin{array}{lcccccc}x(t)&=&-60\cos(t)&+30\sin(t)&-8\sin(2t)&+10\sin(3t)\\y(t)&=&50\sin(t)&+18\sin(2t)&-12\cos(3t)&+14\cos(5t)\end{array}}\right.$
The four fixed parameters used are complex, with affixes z1 = 50 - 30i, z2 = 18 + 8i, z3 = 12 - 10i, z4 = -14 - 60i. The affix point z5 = 40 + 20i is added to make the eye of the elephant and this value serves as a parameter for the movement of the "trunk".[3]
See also
• Epicycloid
• Curve fitting
References
1. Dyson, Freeman (January 22, 2004). "A meeting with Enrico Fermi". Nature. 427 (6972). doi:10.1038/427297a.
2. Wei, James (1975). "Least Square Fitting of an Elephant". Chemtech. 5 (2): 128–129.
3. Mayer, Jurgen; Khairy, Khaled; Howard, Jonathon (May 12, 2010). "Drawing an elephant with four complex parameters". American Journal of Physics. 78 (6). doi:10.1119/1.3254017.
External links
• "Fitting an Elephant" at the Wolfram Demonstrations Project site
| Wikipedia |
Von Neumann's inequality
In operator theory, von Neumann's inequality, due to John von Neumann, states that, for a fixed contraction T, the polynomial functional calculus map is itself a contraction.
Formal statement
For a contraction T acting on a Hilbert space and a polynomial p, then the norm of p(T) is bounded by the supremum of |p(z)| for z in the unit disk."[1]
Proof
The inequality can be proved by considering the unitary dilation of T, for which the inequality is obvious.
Generalizations
This inequality is a specific case of Matsaev's conjecture. That is that for any polynomial P and contraction T on $L^{p}$
$||P(T)||_{L^{p}\to L^{p}}\leq ||P(S)||_{\ell ^{p}\to \ell ^{p}}$
where S is the right-shift operator. The von Neumann inequality proves it true for $p=2$ and for $p=1$ and $p=\infty $ it is true by straightforward calculation. S.W. Drury has shown in 2011 that the conjecture fails in the general case.[2]
References
1. "Department of Mathematics, Vanderbilt University Colloquium, AY 2007-2008". Archived from the original on 2008-03-16. Retrieved 2008-03-11.
2. S.W. Drury, "A counterexample to a conjecture of Matsaev", Linear Algebra and its Applications, Volume 435, Issue 2, 15 July 2011, Pages 323-329
| Wikipedia |
Von Neumann cellular automaton
Von Neumann cellular automata are the original expression of cellular automata, the development of which was prompted by suggestions made to John von Neumann by his close friend and fellow mathematician Stanislaw Ulam. Their original purpose was to provide insight into the logical requirements for machine self-replication, and they were used in von Neumann's universal constructor.
Nobili's cellular automaton is a variation of von Neumann's cellular automaton, augmented with the ability for confluent cells to cross signals and store information. The former requires an extra three states, hence Nobili's cellular automaton has 32 states, rather than 29. Hutton's cellular automaton is yet another variation, which allows a loop of data, analogous to Langton's loops, to replicate.
Definition
Configuration
In general, cellular automata (CA) constitute an arrangement of finite state automata (FSA) that sit in positional relationships between one another, each FSA exchanging information with those other FSAs to which it is positionally adjacent. In von Neumann's cellular automaton, the finite state machines (or cells) are arranged in a two-dimensional Cartesian grid, and interface with the surrounding four cells. As von Neumann's cellular automaton was the first example to use this arrangement, it is known as the von Neumann neighbourhood.
The set of FSAs define a cell space of infinite size. All FSAs are identical in terms of state-transition function, or rule-set.
The neighborhood (a grouping function) is part of the state-transition function, and defines for any cell the set of other cells upon which the state of that cell depends.
All cells make their transitions synchronously, in step with a universal "clock" as in a synchronous digital circuit.
States
Each FSA of the von Neumann cell space can accept any of the 29 states of the rule-set. The rule-set is grouped into five orthogonal subsets. Each state includes the colour of the cell in the cellular automata program Golly in (red, green, blue). They are
1. a ground state U (48, 48, 48)
2. the transition or sensitised states (in 8 substates)
1. S (newly sensitised) (255, 0, 0)
2. S0 – (sensitised, having received no input for one cycle) (255, 125, 0)
3. S00 – (sensitised, having received no input for two cycles) (255, 175, 50)
4. S000 – (sensitised, having received no input for three cycles) (251, 255, 0)
5. S01 – (sensitised, having received no input for one cycle and then an input for one cycle) (255, 200, 75)
6. S1 – (sensitised, having received an input for one cycle) (255, 150, 25)
7. S10 – (sensitised, having received an input for one cycle and then no input for one cycle) (255, 255, 100)
8. S11 – (sensitised, having received input for two cycles) (255, 250, 125)
3. the confluent states (in 4 states of excitation)
1. C00 – quiescent (and will also be quiescent next cycle) (0, 255, 128)
2. C01 – next-excited (now quiescent, but will be excited next cycle) (33, 215, 215)
3. C10 – excited (but will be quiescent next cycle) (255, 255, 128)
4. C11 – excited next-excited (currently excited and will be excited next cycle) (255, 128, 64)
4. the ordinary transmission states (in 4 directions, excited or quiescent, making 8 states)
1. North-directed (excited and quiescent) (36, 200, 36) (106, 106, 255)
2. South-directed (excited and quiescent) (106, 255, 106) (139, 139, 255)
3. West-directed (excited and quiescent) (73, 255, 73) (122, 122, 255)
4. East-directed (excited and quiescent) (27, 176, 27) (89, 89, 255)
5. the special transmission states (in 4 directions, excited or quiescent, making 8 states)
1. North-directed (excited and quiescent) (191, 73, 255) (255, 56, 56)
2. South-directed (excited and quiescent) (203, 106, 255) (255, 89, 89)
3. West-directed (excited and quiescent) (197, 89, 255) (255, 73, 73)
4. East-directed (excited and quiescent) (185, 56, 255) (235, 36, 36)
"Excited" states carry data, at the rate of one bit per state transition step.
Note that confluent states have the property of a one-cycle delay, thus effectively holding two bits of data at any given time.
Transmission state rules
The flow of bits between cells is indicated by the direction property. The following rules apply:
• Transmission states apply the OR operator to inputs, meaning a cell in a transmission state (ordinary or special) will be excited at time t+1 if any of the inputs pointing to it is excited at time t
• Data passes from cell A in an ordinary transmission state to an adjacent cell B in an ordinary transmission state, according to the direction property of A (unless B is also directed towards A, in which case the data disappears).
• Data passes from cell A in a special transmission state to an adjacent cell B in a special transmission state, according to the same rules as for ordinary transmission states.
• The two subsets of transmission states, ordinary and special, are mutually antagonistic:
• Given a cell A at time t in the excited ordinary transmission state
• pointing to a cell B in any special transmission state
• at time t+1 cell B will become the ground state. The special transmission cell has been "destroyed".
• a similar sequence will occur in the case of a cell in the special transmission state "pointing" to a cell in the ordinary transmission state
Confluent state rules
The following specific rules apply to confluent states:
• Confluent states do not pass data between each other.
• Confluent states take input from one or more ordinary transmission states, and deliver output to transmission states, ordinary and special, that are not directed towards the confluent state.
• Data is not transmitted against the transmission state direction property.
• Data held by a confluent state is lost if that state has no adjacent transmission state that is also not pointed at the confluent state.
• Thus, confluent-state cells are used as "bridges" from transmission lines of ordinary- to special-transmission state cells.
• The confluent state applies the AND operator to inputs, only "saving" an excited input if all potential inputs are excited simultaneously.
• Confluent cells delay signals by one generation more than OTS cells; this is necessary due to parity constraints.
Construction rules
Initially, much of the cell-space, the universe of the cellular automaton, is "blank", consisting of cells in the ground state U. When given an input excitation from a neighboring ordinary- or special transmission state, the cell in the ground state becomes "sensitised", transitioning through a series of states before finally "resting" at a quiescent transmission or confluent state.
The choice of which destination state the cell will reach is determined by the sequence of input signals. Therefore, the transition/sensitised states can be thought of as the nodes of a bifurcation tree leading from the ground-state to each of the quiescent transmission and confluent states.
In the following tree, the sequence of inputs is shown as a binary string after each step:
• a cell in the ground state U, given an input, will transition to the S (newly sensitised) state in the next cycle (1)
• a cell in the S state, given no input, will transition into the S0 state (10)
• a cell in the S0 state, given no input, will transition into the S00 state (100)
• a cell in the S00 state, given no input, will transition into the S000 state (1000)
• a cell in the S000 state, given no input, will transition into the east-directed ordinary transmission state (10000)
• a cell in the S000 state, given an input, will transition into the north-directed ordinary transmission state (10001)
• a cell in the S00 state, given an input, will transition into the west-directed ordinary transmission state (1001)
• a cell in the S0 state, given an input, will transition into the S01 state (101)
• a cell in the S01 state, given no input, will transition into the south-directed ordinary transmission state (1010)
• a cell in the S01 state, given an input, will transition into the east-directed special transmission state (1011)
• a cell in the S state, given an input, will transition into the S1 state (11)
• a cell in the S1 state, given no input, will transition into the S10 state (110)
• a cell in the S10 state, given no input, will transition into the north-directed special transmission state (1100)
• a cell in the S10 state, given an input, will transition into the west-directed special transmission state (1101)
• a cell in the S1 state, given an input, will transition into the S11 state (111)
• a cell in the S11 state, given no input, will transition into the south-directed special transmission state (1110)
• a cell in the S11 state, given an input, will transition into the quiescent confluent state C00 (1111)
Note that:
• one more cycle of input (four after the initial sensitization) is required to build the east- or north-directed ordinary transmission state than any of the other states (which require three cycles of input after the initial sensitization),
• the "default" quiescent state resulting in construction is the east-directed ordinary transmission state- which requires an initial sensitization input, and then four cycles of no input.
Destruction rules
• An input into a confluent-state cell from a special-transmission state cell will result in the confluent state cell being reduced back to the ground state.
• Likewise, an input into an ordinary transmission-state cell from a special-transmission state cell will result in the ordinary-transmission state cell being reduced back to the ground state.
• Conversely, an input into a special transmission-state cell from an ordinary-transmission state cell will result in the special-transmission state cell being reduced back to the ground state.
See also
• Codd's cellular automaton
• Langton's loops
• Von Neumann universal constructor
• Wireworld
References
• Von Neumann, J. and A. W. Burks (1966). Theory of self-reproducing automata. Urbana, University of Illinois Press.
External links
• Golly - supports von Neumann's CA along with the Game of Life, and other rulesets.
| Wikipedia |
Von Neumann cardinal assignment
The von Neumann cardinal assignment is a cardinal assignment that uses ordinal numbers. For a well-orderable set U, we define its cardinal number to be the smallest ordinal number equinumerous to U, using the von Neumann definition of an ordinal number. More precisely:
$|U|=\mathrm {card} (U)=\inf\{\alpha \in \mathrm {ON} \ |\ \alpha =_{c}U\},$
where ON is the class of ordinals. This ordinal is also called the initial ordinal of the cardinal.
That such an ordinal exists and is unique is guaranteed by the fact that U is well-orderable and that the class of ordinals is well-ordered, using the axiom of replacement. With the full axiom of choice, every set is well-orderable, so every set has a cardinal; we order the cardinals using the inherited ordering from the ordinal numbers. This is readily found to coincide with the ordering via ≤c. This is a well-ordering of cardinal numbers.
Initial ordinal of a cardinal
Each ordinal has an associated cardinal, its cardinality, obtained by simply forgetting the order. Any well-ordered set having that ordinal as its order type has the same cardinality. The smallest ordinal having a given cardinal as its cardinality is called the initial ordinal of that cardinal. Every finite ordinal (natural number) is initial, but most infinite ordinals are not initial. The axiom of choice is equivalent to the statement that every set can be well-ordered, i.e. that every cardinal has an initial ordinal. In this case, it is traditional to identify the cardinal number with its initial ordinal, and we say that the initial ordinal is a cardinal.
The $\alpha $-th infinite initial ordinal is written $\omega _{\alpha }$. Its cardinality is written $\aleph _{\alpha }$ (the $\alpha $-th aleph number). For example, the cardinality of $\omega _{0}=\omega $ is $\aleph _{0}$, which is also the cardinality of $\omega ^{2}$, $\omega ^{\omega }$, and $\epsilon _{0}$ (all are countable ordinals). So we identify $\omega _{\alpha }$ with $\aleph _{\alpha }$, except that the notation $\aleph _{\alpha }$ is used for writing cardinals, and $\omega _{\alpha }$ for writing ordinals. This is important because arithmetic on cardinals is different from arithmetic on ordinals, for example $\aleph _{\alpha }^{2}$ = $\aleph _{\alpha }$ whereas $\omega _{\alpha }^{2}$ > $\omega _{\alpha }$. Also, $\omega _{1}$ is the smallest uncountable ordinal (to see that it exists, consider the set of equivalence classes of well-orderings of the natural numbers; each such well-ordering defines a countable ordinal, and $\omega _{1}$ is the order type of that set), $\omega _{2}$ is the smallest ordinal whose cardinality is greater than $\aleph _{1}$, and so on, and $\omega _{\omega }$ is the limit of $\omega _{n}$ for natural numbers $n$ (any limit of cardinals is a cardinal, so this limit is indeed the first cardinal after all the $\omega _{n}$).
Infinite initial ordinals are limit ordinals. Using ordinal arithmetic, $\alpha <\omega _{\beta }$ implies $\alpha +\omega _{\beta }=\omega _{\beta }$, and 1 ≤ α < ωβ implies α · ωβ = ωβ, and 2 ≤ α < ωβ implies αωβ = ωβ. Using the Veblen hierarchy, β ≠ 0 and α < ωβ imply $\varphi _{\alpha }(\omega _{\beta })=\omega _{\beta }\,$ and Γωβ = ωβ. Indeed, one can go far beyond this. So as an ordinal, an infinite initial ordinal is an extremely strong kind of limit.
See also
• Aleph number
References
• Y.N. Moschovakis Notes on Set Theory (1994 Springer) p. 198
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| Wikipedia |
Von Neumann bicommutant theorem
In mathematics, specifically functional analysis, the von Neumann bicommutant theorem relates the closure of a set of bounded operators on a Hilbert space in certain topologies to the bicommutant of that set. In essence, it is a connection between the algebraic and topological sides of operator theory.
The formal statement of the theorem is as follows:
Von Neumann bicommutant theorem. Let M be an algebra consisting of bounded operators on a Hilbert space H, containing the identity operator, and closed under taking adjoints. Then the closures of M in the weak operator topology and the strong operator topology are equal, and are in turn equal to the bicommutant M′′ of M.
This algebra is called the von Neumann algebra generated by M.
There are several other topologies on the space of bounded operators, and one can ask what are the *-algebras closed in these topologies. If M is closed in the norm topology then it is a C*-algebra, but not necessarily a von Neumann algebra. One such example is the C*-algebra of compact operators (on an infinite dimensional Hilbert space). For most other common topologies the closed *-algebras containing 1 are von Neumann algebras; this applies in particular to the weak operator, strong operator, *-strong operator, ultraweak, ultrastrong, and *-ultrastrong topologies.
It is related to the Jacobson density theorem.
Proof
Let H be a Hilbert space and L(H) the bounded operators on H. Consider a self-adjoint unital subalgebra M of L(H) (this means that M contains the adjoints of its members, and the identity operator on H).
The theorem is equivalent to the combination of the following three statements:
(i) clW(M) ⊆ M′′
(ii) clS(M) ⊆ clW(M)
(iii) M′′ ⊆ clS(M)
where the W and S subscripts stand for closures in the weak and strong operator topologies, respectively.
Proof of (i)
By definition of the weak operator topology, for any x and y in H, the map T → <Tx, y> is continuous in this topology. Therefore, for any operator O (and by substituting once y → O∗y and once x → Ox), so is the map
$T\to \langle Tx,O^{*}y\rangle -\langle TOx,y\rangle =\langle OTx,y\rangle -\langle TOx,y\rangle .$
Let S be any subset of L(H), and S′ its commutant. For any operator T not in S′, <OTx, y> - <TOx, y> is nonzero for some O in S and some x and y in H. By the continuity of the abovementioned mapping, there is an open neighborhood of T in the weak operator topology for which this is nonzero, therefore this open neighborhood is also not in S′. Thus S′ is closed in the weak operator, i.e. S′ is weakly closed. Thus every commutant is weakly closed, and so is M′′; since it contains M, it also contains its weak closure.
Proof of (ii)
This follows directly from the weak operator topology being coarser than the strong operator topology: for every point x in clS(M), every open neighborhood of x in the weak operator topology is also open in the strong operator topology and therefore contains a member of M; therefore x is also a member of clW(M).
Proof of (iii)
Fix X ∈ M′′. We will show X ∈ clS(M).
Fix an open neighborhood U of X in the strong operator topology. By definition of the strong operator topology, U contains a finite intersection U(h1,ε1) ∩...∩U(hn,εn) of subbasic open sets of the form U(h,ε) = {O ∈ L(H): ||Oh - Xh|| < ε}, where h is in H and ε > 0.
Fix h in H. Consider the closure cl(Mh) of Mh = {Mh : M ∈ M} with respect to the norm of H and equipped with the inner product of H. It is a Hilbert space (being a closed subspace of a Hilbert space H), and so has a corresponding orthogonal projection which we denote P. P is bounded, so it is in L(H). Next we prove:
Lemma. P ∈ M′.
Proof. Fix x ∈ H. Then Px ∈ cl(Mh), so it is the limit of a sequence Onh with On in M for all n. Then for all T ∈ M, TOnh is also in Mh and thus its limit is in cl(Mh). By continuity of T (since it is in L(H) and thus Lipschitz continuous), this limit is TPx. Since TPx ∈ cl(Mh), PTPx = TPx. From this it follows that PTP = TP for all T in M.
By using the closure of M under the adjoint we further have, for every T in M and all x, y ∈ H:
$\langle x,TPy\rangle =\langle x,PTPy\rangle =\langle Px,TPy\rangle =\langle T^{*}Px,Py\rangle =\langle PT^{*}Px,y\rangle =\langle T^{*}Px,y\rangle =\langle Px,Ty\rangle =\langle x,PTy\rangle $
thus TP = PT and P lies in M′.
By definition of the bicommutant XP = PX. Since M is unital, h ∈ Mh, hence Xh = XPh = PXh ∈ cl(Mh). Thus for every ε > 0, there exists T in M with ||Xh − Th|| < ε. Then T lies in U(h,ε).
Thus in every open neighborhood U of X in the strong operator topology there is a member of M, and so X is in the strong operator topology closure of M.
Non-unital case
A C*-algebra M acting on H is said to act non-degenerately if for h in H, Mh = {0} implies h = 0. In this case, it can be shown using an approximate identity in M that the identity operator I lies in the strong closure of M. Therefore, the conclusion of the bicommutant theorem holds for M.
References
• W.B. Arveson, An Invitation to C*-algebras, Springer, New York, 1976.
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| Wikipedia |
Randomness extractor
A randomness extractor, often simply called an "extractor", is a function, which being applied to output from a weak entropy source, together with a short, uniformly random seed, generates a highly random output that appears independent from the source and uniformly distributed.[1] Examples of weakly random sources include radioactive decay or thermal noise; the only restriction on possible sources is that there is no way they can be fully controlled, calculated or predicted, and that a lower bound on their entropy rate can be established. For a given source, a randomness extractor can even be considered to be a true random number generator (TRNG); but there is no single extractor that has been proven to produce truly random output from any type of weakly random source.
Sometimes the term "bias" is used to denote a weakly random source's departure from uniformity, and in older literature, some extractors are called unbiasing algorithms,[2] as they take the randomness from a so-called "biased" source and output a distribution that appears unbiased. The weakly random source will always be longer than the extractor's output, but an efficient extractor is one that lowers this ratio of lengths as much as possible, while simultaneously keeping the seed length low. Intuitively, this means that as much randomness as possible has been "extracted" from the source.
Note that an extractor has some conceptual similarities with a pseudorandom generator (PRG), but the two concepts are not identical. Both are functions that take as input a small, uniformly random seed and produce a longer output that "looks" uniformly random. Some pseudorandom generators are, in fact, also extractors. (When a PRG is based on the existence of hard-core predicates, one can think of the weakly random source as a set of truth tables of such predicates and prove that the output is statistically close to uniform.[3]) However, the general PRG definition does not specify that a weakly random source must be used, and while in the case of an extractor, the output should be statistically close to uniform, in a PRG it is only required to be computationally indistinguishable from uniform, a somewhat weaker concept.
NIST Special Publication 800-90B (draft) recommends several extractors, including the SHA hash family and states that if the amount of entropy input is twice the number of bits output from them, that output will exhibit full entropy.[4]
Formal definition of extractors
The min-entropy of a distribution $X$ (denoted $H_{\infty }(X)$), is the largest real number $k$ such that $\Pr[X=x]\leq 2^{-k}$ for every $x$ in the range of $X$. In essence, this measures how likely $X$ is to take its most likely value, giving a worst-case bound on how random $X$ appears. Letting $U_{\ell }$ denote the uniform distribution over $\{0,1\}^{\ell }$, clearly $H_{\infty }(U_{\ell })=\ell $.
For an n-bit distribution $X$ with min-entropy k, we say that $X$ is an $(n,k)$ distribution.
Definition (Extractor): (k, ε)-extractor
Let ${\text{Ext}}:\{0,1\}^{n}\times \{0,1\}^{d}\to \{0,1\}^{m}$ be a function that takes as input a sample from an $(n,k)$ distribution $X$ and a d-bit seed from $U_{d}$, and outputs an m-bit string. ${\text{Ext}}$ is a (k, ε)-extractor, if for all $(n,k)$ distributions $X$, the output distribution of ${\text{Ext}}$ is ε-close to $U_{m}$.
In the above definition, ε-close refers to statistical distance.
Intuitively, an extractor takes a weakly random n-bit input and a short, uniformly random seed and produces an m-bit output that looks uniformly random. The aim is to have a low $d$ (i.e. to use as little uniform randomness as possible) and as high an $m$ as possible (i.e. to get out as many close-to-random bits of output as we can).
Strong extractors
An extractor is strong if concatenating the seed with the extractor's output yields a distribution that is still close to uniform.
Definition (Strong Extractor): A $(k,\epsilon )$-strong extractor is a function
${\text{Ext}}:\{0,1\}^{n}\times \{0,1\}^{d}\rightarrow \{0,1\}^{m}\,$
such that for every $(n,k)$ distribution $X$ the distribution $U_{d}\circ {\text{Ext}}(X,U_{d})$ (the two copies of $U_{d}$ denote the same random variable) is $\epsilon $-close to the uniform distribution on $U_{m+d}$.
Explicit extractors
Using the probabilistic method, it can be shown that there exists a (k, ε)-extractor, i.e. that the construction is possible. However, it is usually not enough merely to show that an extractor exists. An explicit construction is needed, which is given as follows:
Definition (Explicit Extractor): For functions k(n), ε(n), d(n), m(n) a family Ext = {Extn} of functions
${\text{Ext}}_{n}:\{0,1\}^{n}\times \{0,1\}^{d(n)}\rightarrow \{0,1\}^{m(n)}$
is an explicit (k, ε)-extractor, if Ext(x, y) can be computed in polynomial time (in its input length) and for every n, Extn is a (k(n), ε(n))-extractor.
By the probabilistic method, it can be shown that there exists a (k, ε)-extractor with seed length
$d=\log {(n-k)}+2\log \left({\frac {1}{\varepsilon }}\right)+O(1)$
and output length
$m=k+d-2\log \left({\frac {1}{\varepsilon }}\right)-O(1)$.[5]
Dispersers
A variant of the randomness extractor with weaker properties is the disperser.
Randomness extractors in cryptography
One of the most important aspects of cryptography is random key generation.[6] It is often necessary to generate secret and random keys from sources that are semi-secret or which may be compromised to some degree. By taking a single, short (and secret) random key as a source, an extractor can be used to generate a longer pseudo-random key, which then can be used for public key encryption. More specifically, when a strong extractor is used its output will appear be uniformly random, even to someone who sees part (but not all) of the source. For example, if the source is known but the seed is not known (or vice versa). This property of extractors is particularly useful in what is commonly called Exposure-Resilient cryptography in which the desired extractor is used as an Exposure-Resilient Function (ERF). Exposure-Resilient cryptography takes into account that the fact that it is difficult to keep secret the initial exchange of data which often takes place during the initialization of an encryption application e.g., the sender of encrypted information has to provide the receivers with information which is required for decryption.
The following paragraphs define and establish an important relationship between two kinds of ERF--k-ERF and k-APRF--which are useful in Exposure-Resilient cryptography.
Definition (k-ERF): An adaptive k-ERF is a function $f$ where, for a random input $r$ , when a computationally unbounded adversary $A$ can adaptively read all of $r$ except for $k$ bits, $|\Pr\{A^{r}(f(r))=1\}-\Pr\{A^{r}(R)=1\}|\leq \epsilon (n)$ for some negligible function $\epsilon (n)$ (defined below).
The goal is to construct an adaptive ERF whose output is highly random and uniformly distributed. But a stronger condition is often needed in which every output occurs with almost uniform probability. For this purpose Almost-Perfect Resilient Functions (APRF) are used. The definition of an APRF is as follows:
Definition (k-APRF): A $k=k(n)$ APRF is a function $f$ where, for any setting of $n-k$ bits of the input $r$ to any fixed values, the probability vector $p$ of the output $f(r)$ over the random choices for the $k$ remaining bits satisfies $|p_{i}-2^{-m}|<2^{-m}\epsilon (n)$ for all $i$ and for some negligible function $\epsilon (n)$.
Kamp and Zuckerman[7] have proved a theorem stating that if a function $f$ is a k-APRF, then $f$ is also a k-ERF. More specifically, any extractor having sufficiently small error and taking as input an oblivious, bit-fixing source is also an APRF and therefore also a k-ERF. A more specific extractor is expressed in this lemma:
Lemma: Any $2^{-m}\epsilon (n)$-extractor $f:\{0,1\}^{n}\rightarrow \{0,1\}^{m}$ for the set of $(n,k)$ oblivious bit-fixing sources, where $\epsilon (n)$ is negligible, is also a k-APRF.
This lemma is proved by Kamp and Zuckerman.[7] The lemma is proved by examining the distance from uniform of the output, which in a $2^{-m}\epsilon (n)$-extractor obviously is at most $2^{-m}\epsilon (n)$, which satisfies the condition of the APRF.
The lemma leads to the following theorem, stating that there in fact exists a k-APRF function as described:
Theorem (existence): For any positive constant $\gamma \leq {\frac {1}{2}}$, there exists an explicit k-APRF $f:\{0,1\}^{n}\rightarrow \{0,1\}^{m}$, computable in a linear number of arithmetic operations on $m$-bit strings, with $m=\Omega (n^{2\gamma })$ and $k=n^{{\frac {1}{2}}+\gamma }$.
Definition (negligible function): In the proof of this theorem, we need a definition of a negligible function. A function $\epsilon (n)$ is defined as being negligible if $\epsilon (n)=O\left({\frac {1}{n^{c}}}\right)$ for all constants $c$.
Proof: Consider the following $\epsilon $-extractor: The function $f$ is an extractor for the set of $(n,\delta n)$ oblivious bit-fixing source: $f:\{0,1\}^{n}\rightarrow \{0,1\}^{m}$. $f$ has $m=\Omega (\delta ^{2}n)$, $\epsilon =2^{-cm}$ and $c>1$.
The proof of this extractor's existence with $\delta \leq 1$, as well as the fact that it is computable in linear computing time on the length of $m$ can be found in the paper by Jesse Kamp and David Zuckerman (p. 1240).
That this extractor fulfills the criteria of the lemma is trivially true as $\epsilon =2^{-cm}$ is a negligible function.
The size of $m$ is:
$m=\Omega (\delta ^{2}n)=\Omega (n)\geq \Omega (n^{2\gamma })$
Since we know $\delta \leq 1$ then the lower bound on $m$ is dominated by $n$. In the last step we use the fact that $\gamma \leq {\frac {1}{2}}$ which means that the power of $n$ is at most $1$. And since $n$ is a positive integer we know that $n^{2\gamma }$ is at most $n$.
The value of $k$ is calculated by using the definition of the extractor, where we know:
$(n,k)=(n,\delta n)\Rightarrow k=\delta n$
and by using the value of $m$ we have:
$m=\delta ^{2}n=n^{2\gamma }$
Using this value of $m$ we account for the worst case, where $k$ is on its lower bound. Now by algebraic calculations we get:
$\delta ^{2}n=n^{2\gamma }$
$\Rightarrow \delta ^{2}=n^{2\gamma -1}$
$\Rightarrow \delta =n^{\gamma -{\frac {1}{2}}}$
Which inserted in the value of $k$ gives
$k=\delta n=n^{\gamma -{\frac {1}{2}}}n=n^{\gamma +{\frac {1}{2}}}$,
which proves that there exists an explicit k-APRF extractor with the given properties. $\Box $
Examples
Von Neumann extractor
Further information: Bernoulli sequence
Perhaps the earliest example is due to John von Neumann. From the input stream, his extractor took bits, two at a time (first and second, then third and fourth, and so on). If the two bits matched, no output was generated. If the bits differed, the value of the first bit was output. The Von Neumann extractor can be shown to produce a uniform output even if the distribution of input bits is not uniform so long as each bit has the same probability of being one and there is no correlation between successive bits.[8]
Thus, it takes as input a Bernoulli sequence with p not necessarily equal to 1/2, and outputs a Bernoulli sequence with $p=1/2.$ More generally, it applies to any exchangeable sequence—it only relies on the fact that for any pair, 01 and 10 are equally likely: for independent trials, these have probabilities $p\cdot (1-p)=(1-p)\cdot p$, while for an exchangeable sequence the probability may be more complicated, but both are equally likely. To put it simply, because the bits are statistically independent and due to the commutative property of multiplication, it would follow that $P(A\cap B)=P(A)P(B)=P(B)P(A)=P(B\cap A)$. Hence, if pairs of 01 and 10 are mapped onto bits 0 and 1 and pairs 00 and 11 are discarded, then the output will be a uniform distribution.
Iterations upon the Von Neumann extractor include the Elias and Peres extractor, the latter of which reuses bits in order to produce larger output streams than the Von Neumann extractor given the same size input stream.[9]
Chaos machine
Another approach is to use the output of a chaos machine applied to the input stream. This approach generally relies on properties of chaotic systems. Input bits are pushed to the machine, evolving orbits and trajectories in multiple dynamical systems. Thus, small differences in the input produce very different outputs. Such a machine has a uniform output even if the distribution of input bits is not uniform or has serious flaws, and can therefore use weak entropy sources. Additionally, this scheme allows for increased complexity, quality, and security of the output stream, controlled by specifying three parameters: time cost, memory required, and secret key.
Cryptographic hash function
It is also possible to use a cryptographic hash function as a randomness extractor. However, not every hashing algorithm is suitable for this purpose.
Applications
Randomness extractors are used widely in cryptographic applications, whereby a cryptographic hash function is applied to a high-entropy, but non-uniform source, such as disk drive timing information or keyboard delays, to yield a uniformly random result.
Randomness extractors have played a part in recent developments in quantum cryptography, where photons are used by the randomness extractor to generate secure random bits.
Randomness extraction is also used in some branches of computational complexity theory.
Random extraction is also used to convert data to a simple random sample, which is normally distributed, and independent, which is desired by statistics.
See also
• Decorrelation
• Hardware random number generator
• Randomness merger
• Fuzzy extractor
References
1. Extracting randomness from sampleable distributions. Portal.acm.org. 12 November 2000. p. 32. ISBN 9780769508504. Retrieved 2012-06-12.
2. David K. Gifford, Natural Random Numbers, MIT/LCS/TM-371, Massachusetts Institute of Technology, August 1988.
3. Luca Trevisan. "Extractors and Pseudorandom Generators" (PDF). Retrieved 2013-10-21.
4. Recommendation for the Entropy Sources Used for Random Bit Generation (draft) NIST SP800-90B, Barker and Kelsey, August 2012, Section 6.4.2
5. Ronen Shaltiel. Recent developments in explicit construction of extractors. P. 5.
6. Jesse Kamp and David Zuckerman. Deterministic Extractors for Bit-Fixing Sources and Exposure-Resilient Cryptography.,SIAM J. Comput.,Vol. 36, No. 5, pp. 1231–1247.
7. Jesse Kamp and David Zuckerman. Deterministic Extractors for Bit-Fixing Sources and Exposure-Resilient Cryptography. P. 1242.
8. John von Neumann. Various techniques used in connection with random digits. Applied Math Series, 12:36–38, 1951.
9. Prasitsupparote, Amonrat; Konno, Norio; Shikata, Junji (October 2018). "Numerical and Non-Asymptotic Analysis of Elias's and Peres's Extractors with Finite Input Sequences". Entropy. 20 (10): 729. doi:10.3390/e20100729. ISSN 1099-4300. PMC 7512292. PMID 33265818.
• Randomness Extractors for Independent Sources and Applications, Anup Rao
• Recent developments in explicit constructions of extractors, Ronen Shaltiel
• Randomness Extraction and Key Derivation Using the CBC, Cascade and HMAC Modes, Yevgeniy Dodis et al.
• Key Derivation and Randomness Extraction, Olivier Chevassut et al.
• Deterministic Extractors for Bit-Fixing Sources and Exposure-Resilient Cryptography, Jesse Kamp and David Zuckerman
• Tossing a Biased Coin (and the optimality of advanced multi-level strategy) (lecture notes), Michael Mitzenmacher
| Wikipedia |
Von Neumann universe
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by V, is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory (ZFC), is often used to provide an interpretation or motivation of the axioms of ZFC. The concept is named after John von Neumann, although it was first published by Ernst Zermelo in 1930.
The rank of a well-founded set is defined inductively as the smallest ordinal number greater than the ranks of all members of the set.[1] In particular, the rank of the empty set is zero, and every ordinal has a rank equal to itself. The sets in V are divided into the transfinite hierarchy Vα , called the cumulative hierarchy, based on their rank.
Definition
The cumulative hierarchy is a collection of sets Vα indexed by the class of ordinal numbers; in particular, Vα is the set of all sets having ranks less than α. Thus there is one set Vα for each ordinal number α. Vα may be defined by transfinite recursion as follows:
• Let V0 be the empty set:
$V_{0}:=\varnothing .$
• For any ordinal number β, let Vβ+1 be the power set of Vβ:
$V_{\beta +1}:={\mathcal {P}}(V_{\beta }).$
• For any limit ordinal λ, let Vλ be the union of all the V-stages so far:
$V_{\lambda }:=\bigcup _{\beta <\lambda }V_{\beta }.$
A crucial fact about this definition is that there is a single formula φ(α,x) in the language of ZFC that states "the set x is in Vα".
The sets Vα are called stages or ranks.
The class V is defined to be the union of all the V-stages:
$V:=\bigcup _{\alpha }V_{\alpha }.$
An equivalent definition sets
$V_{\alpha }:=\bigcup _{\beta <\alpha }{\mathcal {P}}(V_{\beta })$
for each ordinal α, where ${\mathcal {P}}(X)\!$ is the powerset of $X$.
The rank of a set S is the smallest α such that $S\subseteq V_{\alpha }\,.$ Another way to calculate rank is:
$\operatorname {rank} (S)=\bigcup \{\operatorname {rank} (z)+1\mid z\in S\}$.
Finite and low cardinality stages of the hierarchy
The first five von Neumann stages V0 to V4 may be visualized as follows. (An empty box represents the empty set. A box containing only an empty box represents the set containing only the empty set, and so forth.)
This sequence exhibits tetrational growth. The set V5 contains 216 = 65536 elements; the set V6 contains 265536 elements, which very substantially exceeds the number of atoms in the known universe; and for any natural n, the set Vn+1 contains 2 ↑↑ n elements using Knuth's up-arrow notation. So the finite stages of the cumulative hierarchy cannot be written down explicitly after stage 5. The set Vω has the same cardinality as ω. The set Vω+1 has the same cardinality as the set of real numbers.
Applications and interpretations
Applications of V as models for set theories
If ω is the set of natural numbers, then Vω is the set of hereditarily finite sets, which is a model of set theory without the axiom of infinity.[2][3]
Vω+ω is the universe of "ordinary mathematics", and is a model of Zermelo set theory.[4] A simple argument in favour of the adequacy of Vω+ω is the observation that Vω+1 is adequate for the integers, while Vω+2 is adequate for the real numbers, and most other normal mathematics can be built as relations of various kinds from these sets without needing the axiom of replacement to go outside Vω+ω.
If κ is an inaccessible cardinal, then Vκ is a model of Zermelo–Fraenkel set theory (ZFC) itself, and Vκ+1 is a model of Morse–Kelley set theory.[5][6] (Note that every ZFC model is also a ZF model, and every ZF model is also a Z model.)
Interpretation of V as the "set of all sets"
V is not "the set of all (naive) sets" for two reasons. First, it is not a set; although each individual stage Vα is a set, their union V is a proper class. Second, the sets in V are only the well-founded sets. The axiom of foundation (or regularity) demands that every set be well founded and hence in V, and thus in ZFC every set is in V. But other axiom systems may omit the axiom of foundation or replace it by a strong negation (an example is Aczel's anti-foundation axiom). These non-well-founded set theories are not commonly employed, but are still possible to study.
A third objection to the "set of all sets" interpretation is that not all sets are necessarily "pure sets", which are constructed from the empty set using power sets and unions. Zermelo proposed in 1908 the inclusion of urelements, from which he constructed a transfinite recursive hierarchy in 1930.[7] Such urelements are used extensively in model theory, particularly in Fraenkel-Mostowski models.[8]
Hilbert's paradox
The nonexistence of $V$ as a set may be seen as a case of Hilbert's paradox:
There is no set $S$ such that for all $x\in S$, ${\mathcal {P}}(x)\in S$, and for all $x\subseteq S$, $\bigcup x\in S$.
Such a set may not exist because $S\subseteq S$, therefore $\bigcup S\in S$ and ${\mathcal {P}}(\bigcup S)\subseteq S$, which is a contradiction.[9] If $V$ were a set, it would satisfy all of the hypotheses of $S$: For any $x\in V$, $x$ is in some $V_{\alpha }$, therefore ${\mathcal {P}}(x)\subseteq V_{\alpha +1}$ and ${\mathcal {P}}(x)\in V_{\alpha +2}$. If $x\subseteq V$, then $x$ is a subset of some $V_{\alpha }$, so $\{z\mid \exists (y\in x)(z\in y)\}\subseteq V_{\alpha }$, and then $\{z\mid \exists (y\in x)(z\in y)\}\in V_{\alpha +1}$.
V and the axiom of regularity
The formula V = ⋃αVα is often considered to be a theorem, not a definition.[10][11] Roitman states (without references) that the realization that the axiom of regularity is equivalent to the equality of the universe of ZF sets to the cumulative hierarchy is due to von Neumann.[12]
The existential status of V
Since the class V may be considered to be the arena for most of mathematics, it is important to establish that it "exists" in some sense. Since existence is a difficult concept, one typically replaces the existence question with the consistency question, that is, whether the concept is free of contradictions. A major obstacle is posed by Gödel's incompleteness theorems, which effectively imply the impossibility of proving the consistency of ZF set theory in ZF set theory itself, provided that it is in fact consistent.[13]
The integrity of the von Neumann universe depends fundamentally on the integrity of the ordinal numbers, which act as the rank parameter in the construction, and the integrity of transfinite induction, by which both the ordinal numbers and the von Neumann universe are constructed. The integrity of the ordinal number construction may be said to rest upon von Neumann's 1923 and 1928 papers.[14] The integrity of the construction of V by transfinite induction may be said to have then been established in Zermelo's 1930 paper.[7]
History
The cumulative type hierarchy, also known as the von Neumann universe, is claimed by Gregory H. Moore (1982) to be inaccurately attributed to von Neumann.[15] The first publication of the von Neumann universe was by Ernst Zermelo in 1930.[7]
Existence and uniqueness of the general transfinite recursive definition of sets was demonstrated in 1928 by von Neumann for both Zermelo-Fraenkel set theory[16] and von Neumann's own set theory (which later developed into NBG set theory).[17] In neither of these papers did he apply his transfinite recursive method to construct the universe of all sets. The presentations of the von Neumann universe by Bernays[10] and Mendelson[11] both give credit to von Neumann for the transfinite induction construction method, although not for its application to the construction of the universe of ordinary sets.
The notation V is not a tribute to the name of von Neumann. It was used for the universe of sets in 1889 by Peano, the letter V signifying "Verum", which he used both as a logical symbol and to denote the class of all individuals.[18] Peano's notation V was adopted also by Whitehead and Russell for the class of all sets in 1910.[19] The V notation (for the class of all sets) was not used by von Neumann in his 1920s papers about ordinal numbers and transfinite induction. Paul Cohen[20] explicitly attributes his use of the letter V (for the class of all sets) to a 1940 paper by Gödel,[21] although Gödel most likely obtained the notation from earlier sources such as Whitehead and Russell.[19]
Philosophical perspectives
There are two approaches to understanding the relationship of the von Neumann universe V to ZFC (along with many variations of each approach, and shadings between them). Roughly, formalists will tend to view V as something that flows from the ZFC axioms (for example, ZFC proves that every set is in V). On the other hand, realists are more likely to see the von Neumann hierarchy as something directly accessible to the intuition, and the axioms of ZFC as propositions for whose truth in V we can give direct intuitive arguments in natural language. A possible middle position is that the mental picture of the von Neumann hierarchy provides the ZFC axioms with a motivation (so that they are not arbitrary), but does not necessarily describe objects with real existence.
See also
• Universe (mathematics)
• Constructible universe
• Grothendieck universe
• Inaccessible cardinal
• S (set theory)
Notes
1. Mirimanoff 1917; Moore 2013, pp. 261–262; Rubin 1967, p. 214.
2. Roitman 2011, p. 136, proves that: "Vω is a model of all of the axioms of ZFC except infinity."
3. Cohen 2008, p. 54, states: "The first really interesting axiom [of ZF set theory] is the Axiom of Infinity. If we drop it, then we can take as a model for ZF the set M of all finite sets which can be built up from ∅. [...] It is clear that M will be a model for the other axioms, since none of these lead out of the class of finite sets."
4. Smullyan & Fitting 2010. See page 96 for proof that Vω+ω is a Zermelo model.
5. Cohen 2008, p. 80, states and justifies that if κ is strongly inaccessible, then Vκ is a model of ZF.
"It is clear that if A is an inaccessible cardinal, then the set of all sets of rank less than A is a model for ZF, since the only two troublesome axioms, Power Set and Replacement, do not lead out of the set of cardinals less than A."
6. Roitman 2011, pp. 134–135, proves that if κ is strongly inaccessible, then Vκ is a model of ZFC.
7. Zermelo 1930. See particularly pages 36–40.
8. Howard & Rubin 1998, pp. 175–221.
9. A. Kanamori, "Zermelo and Set Theory", p.490. Bulletin of Symbolic Logic vol. 10, no. 4 (2004). Accessed 21 August 2023.
10. Bernays 1991. See pages 203–209.
11. Mendelson 1964. See page 202.
12. Roitman 2011. See page 79.
13. See article On Formally Undecidable Propositions of Principia Mathematica and Related Systems and Gödel 1931.
14. von Neumann 1923, von Neumann 1928b. See also the English-language presentation of von Neumann's "general recursion theorem" by Bernays 1991, pp. 100–109.
15. Moore 2013. See page 279 for the assertion of the false attribution to von Neumann. See pages 270 and 281 for the attribution to Zermelo.
16. von Neumann 1928b.
17. von Neumann 1928a. See pages 745–752.
18. Peano 1889. See pages VIII and XI.
19. Whitehead & Russell 2009. See page 229.
20. Cohen 2008. See page 88.
21. Gödel 1940.
References
• Bernays, Paul (1991) [1958]. Axiomatic Set Theory. Dover Publications. ISBN 0-486-66637-9.
• Cohen, Paul Joseph (2008) [1966]. Set theory and the continuum hypothesis. Mineola, New York: Dover Publications. ISBN 978-0-486-46921-8.
• Gödel, Kurt (1931). "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I". Monatshefte für Mathematik und Physik. 38: 173–198. doi:10.1007/BF01700692.
• Gödel, Kurt (1940). The consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory. Annals of Mathematics Studies. Vol. 3. Princeton, N. J.: Princeton University Press.
• Howard, Paul; Rubin, Jean E. (1998). Consequences of the axiom of choice. Providence, Rhode Island: American Mathematical Society. pp. 175–221. ISBN 9780821809778.
• Jech, Thomas (2003). Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.
• Kunen, Kenneth (1980). Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.
• Manin, Yuri I. (2010) [1977]. A Course in Mathematical Logic for Mathematicians. Graduate Texts in Mathematics. Vol. 53. Translated by Koblitz, N. (2nd ed.). New York: Springer-Verlag. pp. 89–96. doi:10.1007/978-1-4419-0615-1. ISBN 978-144-190-6144.
• Mendelson, Elliott (1964). Introduction to Mathematical Logic. Van Nostrand Reinhold.
• Mirimanoff, Dmitry (1917). "Les antinomies de Russell et de Burali-Forti et le probleme fondamental de la theorie des ensembles". L'Enseignement Mathématique. 19: 37–52.
• Moore, Gregory H (2013) [1982]. Zermelo's axiom of choice: Its origins, development & influence. Dover Publications. ISBN 978-0-486-48841-7.
• Peano, Giuseppe (1889). Arithmetices principia: nova methodo exposita. Fratres Bocca.
• Roitman, Judith (2011) [1990]. Introduction to Modern Set Theory. Virginia Commonwealth University. ISBN 978-0-9824062-4-3.
• Rubin, Jean E. (1967). Set Theory for the Mathematician. San Francisco: Holden-Day. ASIN B0006BQH7S.
• Smullyan, Raymond M.; Fitting, Melvin (2010) [revised and corrected republication of the work originally published in 1996 by Oxford University Press, New York]. Set Theory and the Continuum Problem. Dover. ISBN 978-0-486-47484-7.
• von Neumann, John (1923). "Zur Einführung der transfiniten Zahlen". Acta Litt. Acad. Sc. Szeged X. 1: 199–208.. English translation: van Heijenoort, Jean (1967), "On the introduction of transfinite numbers", From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, pp. 346–354
• von Neumann, John (1928a). "Die Axiomatisierung der Mengenlehre". Mathematische Zeitschrift. 27: 669–752. doi:10.1007/bf01171122.
• von Neumann, John (1928b). "Über die Definition durch transfinite Induktion und verwandte Fragen der allgemeinen Mengenlehre". Mathematische Annalen. 99: 373–391. doi:10.1007/bf01459102.
• Whitehead, Alfred North; Russell, Bertrand (2009) [1910]. Principia Mathematica. Vol. One. Merchant Books. ISBN 978-1-60386-182-3.
• Zermelo, Ernst (1930). "Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre". Fundamenta Mathematicae. 16: 29–47. doi:10.4064/fm-16-1-29-47.
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| Wikipedia |
Von Neumann neighborhood
In cellular automata, the von Neumann neighborhood (or 4-neighborhood) is classically defined on a two-dimensional square lattice and is composed of a central cell and its four adjacent cells.[1] The neighborhood is named after John von Neumann, who used it to define the von Neumann cellular automaton and the von Neumann universal constructor within it.[2] It is one of the two most commonly used neighborhood types for two-dimensional cellular automata, the other one being the Moore neighborhood.
This neighbourhood can be used to define the notion of 4-connected pixels in computer graphics.[3]
The von Neumann neighbourhood of a cell is the cell itself and the cells at a Manhattan distance of 1.
The concept can be extended to higher dimensions, for example forming a 6-cell octahedral neighborhood for a cubic cellular automaton in three dimensions.[4]
Von Neumann neighborhood of range r
An extension of the simple von Neumann neighborhood described above is to take the set of points at a Manhattan distance of r > 1. This results in a diamond-shaped region (shown for r = 2 in the illustration). These are called von Neumann neighborhoods of range or extent r. The number of cells in a 2-dimensional von Neumann neighborhood of range r can be expressed as $r^{2}+(r+1)^{2}$. The number of cells in a d-dimensional von Neumann neighborhood of range r is the Delannoy number D(d,r).[4] The number of cells on a surface of a d-dimensional von Neumann neighborhood of range r is the Zaitsev number (sequence A266213 in the OEIS).
See also
• Moore neighborhood
• Neighbourhood (graph theory)
• Taxicab geometry
• Lattice graph
• Pixel connectivity
• Chain code
References
1. Toffoli, Tommaso; Margolus, Norman (1987), Cellular Automata Machines: A New Environment for Modeling, MIT Press, p. 60.
2. Ben-Menahem, Ari (2009), Historical Encyclopedia of Natural and Mathematical Sciences, Volume 1, Springer, p. 4632, ISBN 9783540688310.
3. Wilson, Joseph N.; Ritter, Gerhard X. (2000), Handbook of Computer Vision Algorithms in Image Algebra (2nd ed.), CRC Press, p. 177, ISBN 9781420042382.
4. Breukelaar, R.; Bäck, Th. (2005), "Using a Genetic Algorithm to Evolve Behavior in Multi Dimensional Cellular Automata: Emergence of Behavior", Proceedings of the 7th Annual Conference on Genetic and Evolutionary Computation (GECCO '05), New York, NY, USA: ACM, pp. 107–114, doi:10.1145/1068009.1068024, ISBN 1-59593-010-8.
External links
• Weisstein, Eric W. "von Neumann Neighborhood". MathWorld.
• Tyler, Tim, The von Neumann neighborhood at cell-auto.com
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| Wikipedia |
Von Neumann paradox
In mathematics, the von Neumann paradox, named after John von Neumann, is the idea that one can break a planar figure such as the unit square into sets of points and subject each set to an area-preserving affine transformation such that the result is two planar figures of the same size as the original. This was proved in 1929 by John von Neumann, assuming the axiom of choice. It is based on the earlier Banach–Tarski paradox, which is in turn based on the Hausdorff paradox.
Banach and Tarski had proved that, using isometric transformations, the result of taking apart and reassembling a two-dimensional figure would necessarily have the same area as the original. This would make creating two unit squares out of one impossible. But von Neumann realized that the trick of such so-called paradoxical decompositions was the use of a group of transformations that include as a subgroup a free group with two generators. The group of area-preserving transformations (whether the special linear group or the special affine group) contains such subgroups, and this opens the possibility of performing paradoxical decompositions using them.
Sketch of the method
The following is an informal description of the method found by von Neumann. Assume that we have a free group H of area-preserving linear transformations generated by two transformations, σ and τ, which are not far from the identity element. Being a free group means that all its elements can be expressed uniquely in the form $\sigma ^{u_{1}}\tau ^{v_{1}}\sigma ^{u_{2}}\tau ^{v_{2}}\cdots \sigma ^{u_{n}}\tau ^{v_{n}}$ for some n, where the $u$s and $v$s are all non-zero integers, except possibly the first $u$ and the last $v$. We can divide this group into two parts: those that start on the left with σ to some non-zero power (we call this set A) and those that start with τ to some power (that is, $u_{1}$ is zero—we call this set B, and it includes the identity).
If we operate on any point in Euclidean 2-space by the various elements of H we get what is called the orbit of that point. All the points in the plane can thus be classed into orbits, of which there are an infinite number with the cardinality of the continuum. Using the axiom of choice, we can choose one point from each orbit and call the set of these points M. We exclude the origin, which is a fixed point in H. If we then operate on M by all the elements of H, we generate each point of the plane (except the origin) exactly once. If we operate on M by all the elements of A or of B, we get two disjoint sets whose union is all points but the origin.
Now we take some figure such as the unit square or the unit disk. We then choose another figure totally inside it, such as a smaller square, centred at the origin. We can cover the big figure with several copies of the small figure, albeit with some points covered by two or more copies. We can then assign each point of the big figure to one of the copies of the small figure. Let us call the sets corresponding to each copy $C_{1},C_{2},\dots ,C_{m}$. We shall now make a one-to-one mapping of each point in the big figure to a point in its interior, using only area-preserving transformations. We take the points belonging to $C_{1}$ and translate them so that the centre of the $C_{1}$ square is at the origin. We then take those points in it which are in the set A defined above and operate on them by the area-preserving operation σ τ. This puts them into set B. We then take the points belonging to B and operate on them with σ2. They will now still be in B, but the set of these points will be disjoint from the previous set. We proceed in this manner, using σ3τ on the A points from C2 (after centring it) and σ4 on its B points, and so on. In this way, we have mapped all points from the big figure (except some fixed points) in a one-to-one manner to B type points not too far from the centre, and within the big figure. We can then make a second mapping to A type points.
At this point we can apply the method of the Cantor-Bernstein-Schroeder theorem. This theorem tells us that if we have an injection from set D to set E (such as from the big figure to the A type points in it), and an injection from E to D (such as the identity mapping from the A type points in the figure to themselves), then there is a one-to-one correspondence between D and E. In other words, having a mapping from the big figure to a subset of the A points in it, we can make a mapping (a bijection) from the big figure to all the A points in it. (In some regions points are mapped to themselves, in others they are mapped using the mapping described in the previous paragraph.) Likewise we can make a mapping from the big figure to all the B points in it. So looking at this the other way round, we can separate the figure into its A and B points, and then map each of these back into the whole figure (that is, containing both kinds of points)!
This sketch glosses over some things, such as how to handle fixed points. It turns out that more mappings and more sets are necessary to work around this.
Consequences
The paradox for the square can be strengthened as follows:
Any two bounded subsets of the Euclidean plane with non-empty interiors are equidecomposable with respect to the area-preserving affine maps.
This has consequences concerning the problem of measure. As von Neumann notes,
"Infolgedessen gibt es bereits in der Ebene kein nichtnegatives additives Maß (wo das Einheitsquadrat das Maß 1 hat), dass [sic] gegenüber allen Abbildungen von A2 invariant wäre."[1]
"In accordance with this, already in the plane there is no nonnegative additive measure (for which the unit square has a measure of 1), which is invariant with respect to all transformations belonging to A2 [the group of area-preserving affine transformations]."
To explain this a bit more, the question of whether a finitely additive measure exists, that is preserved under certain transformations, depends on what transformations are allowed. The Banach measure of sets in the plane, which is preserved by translations and rotations, is not preserved by non-isometric transformations even when they do preserve the area of polygons. As explained above, the points of the plane (other than the origin) can be divided into two dense sets which we may call A and B. If the A points of a given polygon are transformed by a certain area-preserving transformation and the B points by another, both sets can become subsets of the B points in two new polygons. The new polygons have the same area as the old polygon, but the two transformed sets cannot have the same measure as before (since they contain only part of the B points), and therefore there is no measure that "works".
The class of groups isolated by von Neumann in the course of study of Banach–Tarski phenomenon turned out to be very important for many areas of mathematics: these are amenable groups, or groups with an invariant mean, and include all finite and all solvable groups. Generally speaking, paradoxical decompositions arise when the group used for equivalences in the definition of equidecomposability is not amenable.
Recent progress
Von Neumann's paper left open the possibility of a paradoxical decomposition of the interior of the unit square with respect to the linear group SL(2,R) (Wagon, Question 7.4). In 2000, Miklós Laczkovich proved that such a decomposition exists.[2] More precisely, let A be the family of all bounded subsets of the plane with non-empty interior and at a positive distance from the origin, and B the family of all planar sets with the property that a union of finitely many translates under some elements of SL(2,R) contains a punctured neighbourhood of the origin. Then all sets in the family A are SL(2,R)-equidecomposable, and likewise for the sets in B. It follows that both families consist of paradoxical sets.
See also
• Paradoxes of set theory
References
1. On p. 85 of: von Neumann, J. (1929), "Zur allgemeinen Theorie des Masses" (PDF), Fundamenta Mathematicae, 13: 73–116, doi:10.4064/fm-13-1-73-116
2. Laczkovich, Miklós (1999), "Paradoxical sets under SL2[R]", Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 42: 141–145
| Wikipedia |
Von Neumann stability analysis
In numerical analysis, von Neumann stability analysis (also known as Fourier stability analysis) is a procedure used to check the stability of finite difference schemes as applied to linear partial differential equations.[1] The analysis is based on the Fourier decomposition of numerical error and was developed at Los Alamos National Laboratory after having been briefly described in a 1947 article by British researchers Crank and Nicolson.[2] This method is an example of explicit time integration where the function that defines governing equation is evaluated at the current time. Later, the method was given a more rigorous treatment in an article[3] co-authored by John von Neumann.
Numerical stability
The stability of numerical schemes is closely associated with numerical error. A finite difference scheme is stable if the errors made at one time step of the calculation do not cause the errors to be magnified as the computations are continued. A neutrally stable scheme is one in which errors remain constant as the computations are carried forward. If the errors decay and eventually damp out, the numerical scheme is said to be stable. If, on the contrary, the errors grow with time the numerical scheme is said to be unstable. The stability of numerical schemes can be investigated by performing von Neumann stability analysis. For time-dependent problems, stability guarantees that the numerical method produces a bounded solution whenever the solution of the exact differential equation is bounded. Stability, in general, can be difficult to investigate, especially when the equation under consideration is nonlinear.
In certain cases, von Neumann stability is necessary and sufficient for stability in the sense of Lax–Richtmyer (as used in the Lax equivalence theorem): The PDE and the finite difference scheme models are linear; the PDE is constant-coefficient with periodic boundary conditions and has only two independent variables; and the scheme uses no more than two time levels.[4] Von Neumann stability is necessary in a much wider variety of cases. It is often used in place of a more detailed stability analysis to provide a good guess at the restrictions (if any) on the step sizes used in the scheme because of its relative simplicity.
Illustration of the method
The von Neumann method is based on the decomposition of the errors into Fourier series. To illustrate the procedure, consider the one-dimensional heat equation
${\frac {\partial u}{\partial t}}=\alpha {\frac {\partial ^{2}u}{\partial x^{2}}}$
defined on the spatial interval $L$, which can be discretized[5] as
$u_{j}^{n+1}=u_{j}^{n}+r\left(u_{j+1}^{n}-2u_{j}^{n}+u_{j-1}^{n}\right)$
(1)
where
$r={\frac {\alpha \,\Delta t}{\left(\Delta x\right)^{2}}}$
and the solution $u_{j}^{n}$ of the discrete equation approximates the analytical solution $u(x,t)$ of the PDE on the grid.
Define the round-off error $\epsilon _{j}^{n}$ as
$\epsilon _{j}^{n}=N_{j}^{n}-u_{j}^{n}$
where $u_{j}^{n}$ is the solution of the discretized equation (1) that would be computed in the absence of round-off error, and $N_{j}^{n}$ is the numerical solution obtained in finite precision arithmetic. Since the exact solution $u_{j}^{n}$ must satisfy the discretized equation exactly, the error $\epsilon _{j}^{n}$ must also satisfy the discretized equation.[6] Here we assumed that $N_{j}^{n}$ satisfies the equation, too (this is only true in machine precision). Thus
$\epsilon _{j}^{n+1}=\epsilon _{j}^{n}+r\left(\epsilon _{j+1}^{n}-2\epsilon _{j}^{n}+\epsilon _{j-1}^{n}\right)$
(2)
is a recurrence relation for the error. Equations (1) and (2) show that both the error and the numerical solution have the same growth or decay behavior with respect to time. For linear differential equations with periodic boundary condition, the spatial variation of error may be expanded in a finite Fourier series with respect to $x$, in the interval $L$, as
$\epsilon (x,t)=\sum _{m=-M}^{M}E_{m}(t)e^{{i}k_{m}x}$
(3)
where the wavenumber $k_{m}={\frac {\pi m}{L}}$ with $m=-M,\dots ,-2,-1,0,1,2,\dots ,M$ and $M=L/\Delta x$. The time dependence of the error is included by assuming that the amplitude of error $E_{m}$ is a function of time. Often the assumption is made that the error grows or decays exponentially with time, but this is not necessary for the stability analysis.
If the boundary condition is not periodic, then we may use the finite Fourier integral with respect to $x$:
$\epsilon (x,t)=\int _{-{\frac {\pi }{\Delta x}}}^{\frac {\pi }{\Delta x}}E_{k}(t)e^{ikx}dk$
(4)
Since the difference equation for error is linear (the behavior of each term of the series is the same as series itself), it is enough to consider the growth of error of a typical term:
$\epsilon _{m}(x,t)=E_{m}(t)e^{ik_{m}x}$
(5a)
if a Fourier series is used or
$\epsilon _{k}(x,t)=E_{k}(t)e^{ikx}$
(5b)
if a Fourier integral is used.
As the Fourier series can be considered to be a special case of the Fourier integral, we will continue the development using the expressions for the Fourier integral.
The stability characteristics can be studied using just this form for the error with no loss in generality. To find out how error varies in steps of time, substitute equation (5b) into equation (2), after noting that
${\begin{aligned}\epsilon _{j}^{n}&=E_{m}(t)e^{ik_{m}x}\\\epsilon _{j}^{n+1}&=E_{m}(t+\Delta t)e^{ik_{m}x}\\\epsilon _{j+1}^{n}&=E_{m}(t)e^{ik_{m}(x+\Delta x)}\\\epsilon _{j-1}^{n}&=E_{m}(t)e^{ik_{m}(x-\Delta x)},\end{aligned}}$
to yield (after simplification)
${\frac {E_{m}(t+\Delta t)}{E_{m}(t)}}=1+r\left(e^{ik_{m}\Delta x}+e^{-ik_{m}\Delta x}-2\right).$
(6)
Introducing $\theta =k_{m}\Delta x\in [-\pi ,\pi ]$ and using the identities
$\sin \left({\frac {\theta }{2}}\right)={\frac {e^{i\theta /2}-e^{-i\theta /2}}{2i}}\qquad \rightarrow \qquad \sin ^{2}\left({\frac {\theta }{2}}\right)=-{\frac {e^{i\theta }+e^{-i\theta }-2}{4}}$
equation (6) may be written as
${\frac {E_{m}(t+\Delta t)}{E_{m}(t)}}=1-4r\sin ^{2}(\theta /2)$
(7)
Define the amplification factor
$G\equiv {\frac {E_{m}(t+\Delta t)}{E_{m}(t)}}$
(8)
The necessary and sufficient condition for the error to remain bounded is that $|G|\leq 1.$ Thus, from equations (7) and (8), the condition for stability is given by
$\left|1-4r\sin ^{2}(\theta /2)\right|\leq 1$
(9)
Note that the term $4r\sin ^{2}(\theta /2)$ is always positive. Thus, to satisfy Equation (9):
$4r\sin ^{2}(\theta /2)\leq 2$
(10)
For the above condition to hold for all $m$ (and therefore all $\sin ^{2}(\theta /2)$). The highest value the sinusoidal term can take is 1 and for that particular choice if the upper threshold condition is satisfied, then so will be for all grid points, thus we have
$r={\frac {\alpha \Delta t}{\left(\Delta x\right)^{2}}}\leq {\frac {1}{2}}$
(11)
Equation (11) gives the stability requirement for the FTCS scheme as applied to one-dimensional heat equation. It says that for a given $\Delta x$, the allowed value of $\Delta t$ must be small enough to satisfy equation (10).
Similar analysis shows that a FTCS scheme for linear advection is unconditionally unstable.
References
1. Analysis of Numerical Methods by E. Isaacson, H. B. Keller
2. Crank, J.; Nicolson, P. (1947), "A Practical Method for Numerical Evaluation of Solutions of Partial Differential Equations of Heat Conduction Type", Proc. Camb. Phil. Soc., 43: 50–67, doi:10.1007/BF02127704
3. Charney, J. G.; Fjørtoft, R.; von Neumann, J. (1950), "Numerical Integration of the Barotropic Vorticity Equation", Tellus, 2: 237–254, doi:10.3402/tellusa.v2i4.8607
4. Smith, G. D. (1985), Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3rd ed., pp. 67–68
5. in this case, using the FTCS discretization scheme
6. Anderson, J. D., Jr. (1994). Computational Fluid Dynamics: The Basics with Applications. McGraw Hill.{{cite book}}: CS1 maint: multiple names: authors list (link)
| Wikipedia |
Von Neumann universal constructor
John von Neumann's universal constructor is a self-replicating machine in a cellular automaton (CA) environment. It was designed in the 1940s, without the use of a computer. The fundamental details of the machine were published in von Neumann's book Theory of Self-Reproducing Automata, completed in 1966 by Arthur W. Burks after von Neumann's death.[2] While typically not as well known as von Neumann's other work, it is regarded as foundational for automata theory, complex systems, and artificial life.[3][4] Indeed, Nobel Laureate Sydney Brenner considered Von Neumann's work on self-reproducing automata (together with Turing's work on computing machines) central to biological theory as well, allowing us to "discipline our thoughts about machines, both natural and artificial."[5]
Von Neumann's goal, as specified in his lectures at the University of Illinois in 1949,[2] was to design a machine whose complexity could grow automatically akin to biological organisms under natural selection. He asked what is the threshold of complexity that must be crossed for machines to be able to evolve.[4] His answer was to specify an abstract machine which, when run, would replicate itself. In his design, the self-replicating machine consists of three parts: a "description" of ('blueprint' or program for) itself, a universal constructor mechanism that can read any description and construct the machine (sans description) encoded in that description, and a universal copy machine that can make copies of any description. After the universal constructor has been used to construct a new machine encoded in the description, the copy machine is used to create a copy of that description, and this copy is passed on to the new machine, resulting in a working replication of the original machine that can keep on reproducing. Some machines will do this backwards, copying the description and then building a machine. Crucially, the self-reproducing machine can evolve by accumulating mutations of the description, not the machine itself, thus gaining the ability to grow in complexity.[4][5]
To define his machine in more detail, von Neumann invented the concept of a cellular automaton. The one he used consists of a two-dimensional grid of cells, each of which can be in one of 29 states at any point in time. At each timestep, each cell updates its state depending on the states of the surrounding cells at the prior timestep. The rules governing these updates are identical for all cells.
The universal constructor is a certain pattern of cell states in this cellular automaton. It contains one line of cells that serve as the description (akin to Turing's tape), encoding a sequence of instructions that serve as a 'blueprint' for the machine. The machine reads these instructions one by one and performs the corresponding actions. The instructions direct the machine to use its 'construction arm' (another automaton that functions like an Operating System[4]) to build a copy of the machine, without the description tape, at some other location in the cell grid. The description cannot contain instructions to build an equally long description tape, just as a container cannot contain a container of the same size. Therefore, the machine includes the separate copy machine which reads the description tape and passes a copy to the newly constructed machine. The resulting new set of universal constructor and copy machines plus description tape is identical to the old one, and it proceeds to replicate again.
Purpose
Von Neumann's design has traditionally been understood to be a demonstration of the logical requirements for machine self-replication.[3] However, it is clear that far simpler machines can achieve self-replication. Examples include trivial crystal-like growth, template replication, and Langton's loops. But von Neumann was interested in something more profound: construction, universality, and evolution.[4][5]
Note that the simpler self-replicating CA structures (especially, Byl's loop and the Chou–Reggia loop) cannot exist in a wide variety of forms and thus have very limited evolvability. Other CA structures such as the Evoloop are somewhat evolvable but still don't support open-ended evolution. Commonly, simple replicators do not fully contain the machinery of construction, there being a degree to which the replicator is information copied by its surrounding environment. Although the Von Neumann design is a logical construction, it is in principle a design that could be instantiated as a physical machine. Indeed, this universal constructor can be seen as an abstract simulation of a physical universal assembler. The issue of the environmental contribution to replication is somewhat open, since there are different conceptions of raw material and its availability.
Von Neumann's crucial insight is that the description of the machine, which is copied and passed to offspring separately via the universal copier, has a double use; being both an active component of the construction mechanism in reproduction, and being the target of a passive copying process. This part is played by the description (akin to Turing's tape of instructions) in Von Neumann's combination of universal constructor and universal copier.[4] The combination of a universal constructor and copier, plus a tape of instructions conceptualizes and formalizes i) self-replication, and ii) open-ended evolution, or growth of complexity observed in biological organisms.[3]
This insight is all the more remarkable because it preceded the discovery of the structure of the DNA molecule by Watson and Crick and how it is separately translated and replicated in the cell—though it followed the Avery–MacLeod–McCarty experiment which identified DNA as the molecular carrier of genetic information in living organisms.[6] The DNA molecule is processed by separate mechanisms that carry out its instructions (translation) and copy (replicate) the DNA for newly constructed cells. The ability to achieve open-ended evolution lies in the fact that, just as in nature, errors (mutations) in the copying of the genetic tape can lead to viable variants of the automaton, which can then evolve via natural selection.[4] As Brenner put it:
Turing invented the stored-program computer, and von Neumann showed that the description is separate from the universal constructor. This is not trivial. Physicist Erwin Schrödinger confused the program and the constructor in his 1944 book What is Life?, in which he saw chromosomes as ″architect's plan and builder's craft in one″. This is wrong. The code script contains only a description of the executive function, not the function itself.[5]
— Sydney Brenner
Evolution of Complexity
Von Neumann's goal, as specified in his lectures at the University of Illinois in 1949,[2] was to design a machine whose complexity could grow automatically akin to biological organisms under natural selection. He asked what is the threshold of complexity that must be crossed for machines to be able to evolve and grow in complexity.[4][3] His “proof-of-principle” designs showed how it is logically possible. By using an architecture that separates a general purpose programmable (“universal”) constructor from a general purpose copier, he showed how the descriptions (tapes) of machines could accumulate mutations in self-replication and thus evolve more complex machines (the image below illustrates this possibility.). This is a very important result, as prior to that, it might have been conjectured that there is a fundamental logical barrier to the existence of such machines; in which case, biological organisms, which do evolve and grow in complexity, could not be “machines”, as conventionally understood. Von Neumann's insight was to think of life as a Turing Machine, which, is similarly defined by a state-determined machine "head" separated from a memory tape.[5]
In practice, when we consider the particular automata implementation Von Neumann pursued, we conclude that it does not yield much evolutionary dynamics because the machines are too fragile - the vast majority of perturbations cause them effectively to disintegrate.[3] Thus, it is the conceptual model outlined in his Illinois lectures[2] that is of greater interest today because it shows how a machine can in principle evolve.[7][4] This insight is all the more remarkable because the model preceded the discovery of the structure of the DNA molecule as discussed above.[6] It is also noteworthy that Von Neumann's design considers that mutations towards greater complexity need to occur in the (descriptions of) subsystems not involved in self-reproduction itself, as conceptualized by the additional automaton D he considered to perform all functions not directly involved in reproduction (see Figure above with Von Neumann's System of Self-Replication Automata with the ability to evolve.) Indeed, in biological organisms only very minor variations of the genetic code have been observed, which matches Von Neumann's rationale that the universal constructor (A) and Copier (B) would not themselves evolve, leaving all evolution (and growth of complexity) to automaton D.[4] In his unfinished work, Von Neumann also briefly considers conflict and interactions between his self-reproducing machines, towards understanding the evolution of ecological and social interactions from his theory of self-reproducing machines.[2]: 147
Implementations
In automata theory, the concept of a universal constructor is non-trivial because of the existence of Garden of Eden patterns. But a simple definition is that a universal constructor is able to construct any finite pattern of non-excited (quiescent) cells.
Arthur Burks and others extended the work of von Neumann, giving a much clearer and complete set of details regarding the design and operation of von Neumann's self-replicator. The work of J. W. Thatcher is particularly noteworthy, for he greatly simplified the design. Still, their work did not yield a complete design, cell by cell, of a configuration capable of demonstrating self-replication.
Renato Nobili and Umberto Pesavento published the first fully implemented self-reproducing cellular automaton in 1995, nearly fifty years after von Neumann's work.[1][8] They used a 32-state cellular automaton instead of von Neumann's original 29-state specification, extending it to allow for easier signal-crossing, explicit memory function and a more compact design. They also published an implementation of a general constructor within the original 29-state CA but not one capable of complete replication - the configuration cannot duplicate its tape, nor can it trigger its offspring; the configuration can only construct.[8][9]
In 2004, D. Mange et al. reported an implementation of a self-replicator that is consistent with the designs of von Neumann.[10]
In 2007, Nobili published a 32-state implementation that uses run-length encoding to greatly reduce the size of the tape.[11]
In 2008, William R. Buckley published two configurations which are self-replicators within the original 29-state CA of von Neumann.[9] Buckley claims that the crossing of signal within von Neumann 29-state cellular automata is not necessary to the construction of self-replicators.[9] Buckley also points out that for the purposes of evolution, each replicator should return to its original configuration after replicating, in order to be capable (in theory) of making more than one copy. As published, the 1995 design of Nobili-Pesavento does not fulfill this requirement but the 2007 design of Nobili does; the same is true of Buckley's configurations.
In 2009, Buckley published with Golly a third configuration for von Neumann 29-state cellular automata, which can perform either holistic self-replication, or self-replication by partial construction. This configuration also demonstrates that signal crossing is not necessary to the construction of self-replicators within von Neumann 29-state cellular automata.
C. L. Nehaniv in 2002, and also Y. Takada et al. in 2004, proposed a universal constructor directly implemented upon an asynchronous cellular automaton, rather than upon a synchronous cellular automaton. [12] [13]
Comparison of implementations
ImplementationSourceRulesetRectangular areaNumber of cellsLength of tapeRatioPeriodTape code compressionTape code lengthTape code typeReplication mechanismReplication typeGrowth rate
Nobili-Pesavento, 1995[1] [14]Nobili 32-state97 × 1706,329145,31522.966.34 × 1010none5 bitsbinaryholistic constructornon-repeatablelinear
Nobili, 2007 SR_CCN_AP.EVN[11]Nobili 32-state97 × 1005,31356,32510.609.59 × 109run-length limited encoding5 bitsbinaryholistic constructorrepeatablesuper-linear
Buckley, 2008 codon5.rle[15]Nobili 32-state112 × 503,34344,15513.215.87 × 109auto-retraction5 bitsbinaryholistic constructorrepeatablelinear
Buckley, 2008[9] replicator.mcvon Neumann 29-state312 × 13218,589294,84415.862.61 × 1011auto-retraction5 bitsbinaryholistic constructorrepeatablelinear
Buckley, 2008 codon4.rle[15]Nobili 32-state109 × 593,57437,78010.574.31 × 109auto-retraction/bit generation4 bitsbinaryholistic constructorrepeatablelinear
Buckley, 2009 codon3.rleNobili 32-state116 × 954,85523,5774.861.63 × 109auto-retraction/bit generation/code overlay3 bitsbinaryholistic constructorrepeatablesuper-linear
Buckley, 2009 PartialReplicator.mc[15]von Neumann 29-state2063 × 377264,321——≈1.12 × 1014none4 bitsbinarypartial constructorrepeatablelinear
Goucher & Buckley, 2012 phi9.rle[16]Nobili 32-state122 × 60395789202.25—auto-retraction/bit generation/code overlay/run length limited3+ bitsternaryholistic constructorrepeatablesuper-linear
As defined by von Neumann, universal construction entails the construction of passive configurations, only. As such, the concept of universal construction constituted nothing more than a literary (or, in this case, mathematical) device. It facilitated other proof, such as that a machine well constructed may engage in self-replication, while universal construction itself was simply assumed over a most minimal case. Universal construction under this standard is trivial. Hence, while all the configurations given here can construct any passive configuration, none can construct the real-time crossing organ devised by Gorman.[9]
Practicality and computational cost
All the implementations of von Neumann's self-reproducing machine require considerable resources to run on computer. For example, in the Nobili-Pesavento 32-state implementation shown above, while the body of the machine is just 6,329 non-empty cells (within a rectangle of size 97x170), it requires a tape that is 145,315 cells long, and takes 63 billion timesteps to replicate. A simulator running at 1,000 timesteps per second would take over 2 years to make the first copy. In 1995, when the first implementation was published, the authors had not seen their own machine replicate. However, in 2008, the hashlife algorithm was extended to support the 29-state and 32-state rulesets in Golly. On a modern desktop PC, replication now takes only a few minutes, although a significant amount of memory is required.
Animation gallery
• Example of a 29-state read arm.
See also
• Codd's cellular automaton
• Langton's loops
• Nobili cellular automata
• Quine, a program that produces itself as output
• Santa Claus machine
• Wireworld
References
1. Pesavento, Umberto (1995), "An implementation of von Neumann's self-reproducing machine" (PDF), Artificial Life, MIT Press, 2 (4): 337–354, doi:10.1162/artl.1995.2.337, PMID 8942052, archived from the original (PDF) on June 21, 2007
2. von Neumann, John; Burks, Arthur W. (1966), Theory of Self-Reproducing Automata. (Scanned book online), University of Illinois Press, retrieved 2017-02-28
3. McMullin, B. (2000), "John von Neumann and the Evolutionary Growth of Complexity: Looking Backwards, Looking Forwards...", Artificial Life, 6 (4): 347–361, doi:10.1162/106454600300103674, PMID 11348586, S2CID 5454783
4. Rocha, Luis M. (1998), "Selected self-organization and the semiotics of evolutionary systems", Evolutionary Systems, Springer, Dordrecht: 341–358, doi:10.1007/978-94-017-1510-2_25, ISBN 978-90-481-5103-5
5. Brenner, Sydney (2012), "Life's code script", Nature, 482 (7386): 461, doi:10.1038/482461a, PMID 22358811, S2CID 205070101
6. Rocha, Luis M. (2015), "Chapter 6. Von Neumann and Natural Selection.", Lecture Notes of SSIE-583-Biologically Inspired Computing and Evolutionary Systems Course, Binghamton University
7. Pattee, Howard, H. (1995), "Evolving self-reference: matter symbols, and semantic closure", Communication and Cognition Artificial Intelligence, Biosemiotics, 12 (1–2): 9–27, doi:10.1007/978-94-007-5161-3_14, ISBN 978-94-007-5160-6{{citation}}: CS1 maint: multiple names: authors list (link)
8. Nobili, Renato; Pesavento, Umberto (1996), "Generalised von Neumann's Automata", in Besussi, E.; Cecchini, A. (eds.), Proc. Artificial Worlds and Urban Studies, Conference 1 (PDF), Venice: DAEST
9. Buckley, William R. (2008), "Signal Crossing Solutions in von Neumann Self-replicating Cellular Automata", in Andrew Adamatzky; Ramon Alonso-Sanz; Anna Lawniczak; Genaro Juarez Martinez; Kenichi Morita; Thomas Worsch (eds.), Proc. Automata 2008 (PDF), Luniver Press, pp. 453–503
10. Mange, Daniel; Stauffer, A.; Peparaolo, L.; Tempesti, G. (2004), "A Macroscopic View of Self-replication", Proceedings of the IEEE, 92 (12): 1929–1945, doi:10.1109/JPROC.2004.837631, S2CID 22500865
11. Nobili, Renato (2007). "The Cellular Automata of John von Neumann". Archived from the original on January 29, 2011. Retrieved January 29, 2011.
12. Nehaniv, Chrystopher L. (2002), "Self-Reproduction in Asynchronous Cellular Automata", 2002 NASA/DoD Conference on Evolvable Hardware (15-18 July 2002, Alexandria, Virginia, USA), IEEE Computer Society Press, pp. 201–209
13. Takada, Yousuke; Isokawa, Teijiro; Peper, Ferdinand; Matsui, Nobuyuki (2004), "Universal Construction on Self-Timed Cellular Automata", in Sloot, P.M.A. (ed.), ACRI 2004, LNCS 3305, pp. 21–30
14. "Von Neumann's Self-Reproducing Universal Constructor".
15. andykt (18 July 2023). "Golly, a Game of Life simulator". SourceForge.
16. "Self-replication". Complex Projective 4-Space. 12 November 2012.
External links
• Golly - the Cellular Automata Simulation Accelerator Very fast implementation of state transition and support for JvN, GoL, Wolfram, and other systems.
• von Neumann's Self-Reproducing Universal Constructor The original Nobili-Pesavento source code, animations and Golly files of the replicators.
• John von Neumann's 29 state Cellular Automata Implemented in OpenLaszlo by Don Hopkins
• A Catalogue of Self-Replicating Cellular Automata. This catalogue complements the Proc. Automata 2008 volume.
| Wikipedia |
Von Staudt conic
In projective geometry, a von Staudt conic is the point set defined by all the absolute points of a polarity that has absolute points. In the real projective plane a von Staudt conic is a conic section in the usual sense. In more general projective planes this is not always the case. Karl Georg Christian von Staudt introduced this definition in Geometrie der Lage (1847) as part of his attempt to remove all metrical concepts from projective geometry.
Polarities
A polarity, π, of a projective plane, P, is an involutory (i.e., of order two) bijection between the points and the lines of P that preserves the incidence relation. Thus, a polarity relates a point Q with a line q and, following Gergonne, q is called the polar of Q and Q the pole of q.[1] An absolute point (line) of a polarity is one which is incident with its polar (pole).[2][3]
A polarity may or may not have absolute points. A polarity with absolute points is called a hyperbolic polarity and one without absolute points is called an elliptic polarity.[4] In the complex projective plane all polarities are hyperbolic but in the real projective plane only some are.[4]
A classification of polarities over arbitrary fields follows from the classification of sesquilinear forms given by Birkhoff and von Neumann.[5] Orthogonal polarities, corresponding to symmetric bilinear forms, are also called ordinary polarities and the locus of absolute points forms a non-degenerate conic (set of points whose coordinates satisfy an irreducible homogeneous quadratic equation) if the field does not have characteristic two. In characteristic two the orthogonal polarities are called pseudopolarities and in a plane the absolute points form a line.[6]
Finite projective planes
If π is a polarity of a finite projective plane (which need not be desarguesian), P, of order n then the number of its absolute points (or absolute lines), a(π) is given by:
a(π) = n + 2r√n + 1,
where r is a non-negative integer.[7] Since a(π) is an integer, a(π) = n + 1 if n is not a square, and in this case, π is called an orthogonal polarity.
R. Baer has shown that if n is odd, the absolute points of an orthogonal polarity form an oval (that is, n + 1 points, no three collinear), while if n is even, the absolute points lie on a non-absolute line.[8]
In summary, von Staudt conics are not ovals in finite projective planes (desarguesian or not) of even order.[9][10]
Relation to other types of conics
In a pappian plane (i.e., a projective plane coordinatized by a field), if the field does not have characteristic two, a von Staudt conic is equivalent to a Steiner conic.[11] However, R. Artzy has shown that these two definitions of conics can produce non-isomorphic objects in (infinite) Moufang planes.[12]
Notes
1. Coxeter 1964, p. 60
2. Garner 1979, p. 132
3. Coxeter and several other authors use the term self-conjugate instead of absolute.
4. Coxeter 1964, p. 72
5. Birkhoff, G.; von Neumann, J. (1936), "The logic of quantum mechanics", Ann. Math., 37: 823–843
6. Barwick, Susan; Ebert, Gary (2008), Unitals in Projective Planes, Springer, pp. 16–18, ISBN 978-0-387-76364-4
7. Ball, R.W. (1948), "Dualities of Finite Projective Planes", Duke Mathematical Journal, 15: 929–940, doi:10.1215/s0012-7094-48-01581-6
8. Baer, Reinhold (1946), "Polarities in Finite Projective Planes", Bulletin of the American Mathematical Society, 52: 77–93, doi:10.1090/s0002-9904-1946-08506-7
9. Garner 1979, p. 133
10. Dembowski 1968, pp. 154–155
11. Coxeter 1964, p. 80
12. Artzy, R. (1971), "The Conic y = x2 in Moufang Planes", Aequationes Mathematicae, 6: 30–35, doi:10.1007/bf01833234
References
• Coxeter, H. S. M. (1964), Projective Geometry, Blaisdell
• Dembowski, Peter (1968), Finite geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44, Berlin, New York: Springer-Verlag, ISBN 3-540-61786-8, MR 0233275
• Garner, Cyril W L. (1979), "Conics in Finite Projective Planes", Journal of Geometry, 12 (2): 132–138, doi:10.1007/bf01918221
Further reading
• Ostrom, T.G. (1981), "Conicoids: Conic-like figures in Non-Pappian planes", in Plaumann, Peter; Strambach, Karl (eds.), Geometry - von Staudt's Point of View, D. Reidel, pp. 175–196, ISBN 90-277-1283-2
| Wikipedia |
Von Staudt–Clausen theorem
In number theory, the von Staudt–Clausen theorem is a result determining the fractional part of Bernoulli numbers, found independently by Karl von Staudt (1840) and Thomas Clausen (1840).
Specifically, if n is a positive integer and we add 1/p to the Bernoulli number B2n for every prime p such that p − 1 divides 2n, we obtain an integer, i.e., $B_{2n}+\sum _{(p-1)|2n}{\frac {1}{p}}\in \mathbb {Z} .$
This fact immediately allows us to characterize the denominators of the non-zero Bernoulli numbers B2n as the product of all primes p such that p − 1 divides 2n; consequently the denominators are square-free and divisible by 6.
These denominators are
6, 30, 42, 30, 66, 2730, 6, 510, 798, 330, 138, 2730, 6, 870, 14322, 510, 6, 1919190, 6, 13530, ... (sequence A002445 in the OEIS).
The sequence of integers $B_{2n}+\sum _{(p-1)|2n}{\frac {1}{p}}$ is
1, 1, 1, 1, 1, 1, 2, -6, 56, -528, 6193, -86579, 1425518, -27298230, ... (sequence A000146 in the OEIS).
Proof
A proof of the Von Staudt–Clausen theorem follows from an explicit formula for Bernoulli numbers which is:
$B_{2n}=\sum _{j=0}^{2n}{\frac {1}{j+1}}\sum _{m=0}^{j}{(-1)^{m}{j \choose m}m^{2n}}$
and as a corollary:
$B_{2n}=\sum _{j=0}^{2n}{\frac {j!}{j+1}}(-1)^{j}S(2n,j)$
where $S(n,j)$ are the Stirling numbers of the second kind.
Furthermore the following lemmas are needed:
Let p be a prime number then,
1. If p-1 divides 2n then,
$\sum _{m=0}^{p-1}{(-1)^{m}{p-1 \choose m}m^{2n}}\equiv {-1}{\pmod {p}}$
2. If p-1 does not divide 2n then,
$\sum _{m=0}^{p-1}{(-1)^{m}{p-1 \choose m}m^{2n}}\equiv 0{\pmod {p}}$
Proof of (1) and (2): One has from Fermat's little theorem,
$m^{p-1}\equiv 1{\pmod {p}}$
for $m=1,2,...,p-1$.
If p-1 divides 2n then one has,
$m^{2n}\equiv 1{\pmod {p}}$
for $m=1,2,...,p-1$.
Thereafter one has,
$\sum _{m=1}^{p-1}{(-1)^{m}{p-1 \choose m}m^{2n}}\equiv \sum _{m=1}^{p-1}{(-1)^{m}{p-1 \choose m}}{\pmod {p}}$
from which (1) follows immediately.
If p-1 does not divide 2n then after Fermat's theorem one has,
$m^{2n}\equiv m^{2n-(p-1)}{\pmod {p}}$
If one lets $\wp =[{\frac {2n}{p-1}}]$ (Greatest integer function) then after iteration one has,
$m^{2n}\equiv m^{2n-\wp (p-1)}{\pmod {p}}$
for $m=1,2,...,p-1$ and $0<2n-\wp (p-1)<p-1$.
Thereafter one has,
$\sum _{m=0}^{p-1}{(-1)^{m}{p-1 \choose m}m^{2n}}\equiv \sum _{m=0}^{p-1}{(-1)^{m}{p-1 \choose m}m^{2n-\wp (p-1)}}{\pmod {p}}$
Lemma (2) now follows from the above and the fact that S(n,j)=0 for j>n.
(3). It is easy to deduce that for a>2 and b>2, ab divides (ab-1)!.
(4). Stirling numbers of second kind are integers.
Proof of the theorem: Now we are ready to prove Von-Staudt Clausen theorem,
If j+1 is composite and j>3 then from (3), j+1 divides j!.
For j=3,
$\sum _{m=0}^{3}{(-1)^{m}{3 \choose m}m^{2n}}=3\cdot 2^{2n}-3^{2n}-3\equiv 0{\pmod {4}}$
If j+1 is prime then we use (1) and (2) and if j+1 is composite then we use (3) and (4) to deduce:
$B_{2n}=I_{n}-\sum _{(p-1)|2n}{\frac {1}{p}}$
where $I_{n}$ is an integer, which is the Von-Staudt Clausen theorem.[1][2]
See also
• Kummer's congruence
References
1. H. Rademacher, Analytic Number Theory, Springer-Verlag, New York, 1973.
2. T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.
• Clausen, Thomas (1840), "Theorem", Astronomische Nachrichten, 17 (22): 351–352, doi:10.1002/asna.18400172204
• Rado, R. (1934), "A New Proof of a Theorem of V. Staudt", J. London Math. Soc., 9 (2): 85–88, doi:10.1112/jlms/s1-9.2.85
• von Staudt, Ch. (1840), "Beweis eines Lehrsatzes, die Bernoullischen Zahlen betreffend", Journal für die Reine und Angewandte Mathematik, 21: 372–374, ISSN 0075-4102, ERAM 021.0672cj
External links
• Weisstein, Eric W. "von Staudt-Clausen Theorem". MathWorld.
| Wikipedia |
Bounded set (topological vector space)
In functional analysis and related areas of mathematics, a set in a topological vector space is called bounded or von Neumann bounded, if every neighborhood of the zero vector can be inflated to include the set. A set that is not bounded is called unbounded.
For bounded sets in general, see bounded set.
Bounded sets are a natural way to define locally convex polar topologies on the vector spaces in a dual pair, as the polar set of a bounded set is an absolutely convex and absorbing set. The concept was first introduced by John von Neumann and Andrey Kolmogorov in 1935.
Definition
Suppose $X$ is a topological vector space (TVS) over a field $\mathbb {K} .$
A subset $B$ of $X$ is called von Neumann bounded or just bounded in $X$ if any of the following equivalent conditions are satisfied:
1. Definition: For every neighborhood $V$ of the origin there exists a real $r>0$ such that $B\subseteq sV$[note 1] for all scalars $s$ satisfying $|s|\geq r.$[1]
• This was the definition introduced by John von Neumann in 1935.[1]
2. $B$ is absorbed by every neighborhood of the origin.[2]
3. For every neighborhood $V$ of the origin there exists a scalar $s$ such that $B\subseteq sV.$
4. For every neighborhood $V$ of the origin there exists a real $r>0$ such that $sB\subseteq V$ for all scalars $s$ satisfying $|s|\leq r.$[1]
5. For every neighborhood $V$ of the origin there exists a real $r>0$ such that $tB\subseteq V$ for all real $0<t\leq r.$[3]
6. Any one of statements (1) through (5) above but with the word "neighborhood" replaced by any of the following: "balanced neighborhood," "open balanced neighborhood," "closed balanced neighborhood," "open neighborhood," "closed neighborhood".
• e.g. Statement (2) may become: $B$ is bounded if and only if $B$ is absorbed by every balanced neighborhood of the origin.[1]
• If $X$ is locally convex then the adjective "convex" may be also be added to any of these 5 replacements.
7. For every sequence of scalars $s_{1},s_{2},s_{3},\ldots $ that converges to $0$ and every sequence $b_{1},b_{2},b_{3},\ldots $ in $B,$ the sequence $s_{1}b_{1},s_{2}b_{2},s_{3}b_{3},\ldots $ converges to $0$ in $X.$[1]
• This was the definition of "bounded" that Andrey Kolmogorov used in 1934, which is the same as the definition introduced by Stanisław Mazur and Władysław Orlicz in 1933 for metrizable TVS. Kolmogorov used this definition to prove that a TVS is seminormable if and only if it has a bounded convex neighborhood of the origin.[1]
8. For every sequence $b_{1},b_{2},b_{3},\ldots $ in $B,$ the sequence $ \left({\tfrac {1}{i}}b_{i}\right)_{i=1}^{\infty }$ converges to $0$ in $X.$[4]
9. Every countable subset of $B$ is bounded (according to any defining condition other than this one).[1]
If ${\mathcal {B}}$ is a neighborhood basis for $X$ at the origin then this list may be extended to include:
1. Any one of statements (1) through (5) above but with the neighborhoods limited to those belonging to ${\mathcal {B}}.$
• e.g. Statement (3) may become: For every $V\in {\mathcal {B}}$ there exists a scalar $s$ such that $B\subseteq sV.$
If $X$ is a locally convex space whose topology is defined by a family ${\mathcal {P}}$ of continuous seminorms, then this list may be extended to include:
1. $p(B)$ is bounded for all $p\in {\mathcal {P}}.$[1]
2. There exists a sequence of non-zero scalars $s_{1},s_{2},s_{3},\ldots $ such that for every sequence $b_{1},b_{2},b_{3},\ldots $ in $B,$ the sequence $b_{1}s_{1},b_{2}s_{2},b_{3}s_{3},\ldots $ is bounded in $X$ (according to any defining condition other than this one).[1]
3. For all $p\in {\mathcal {P}},$ $B$ is bounded (according to any defining condition other than this one) in the semi normed space $(X,p).$
If $X$ is a normed space with norm $\|\cdot \|$ (or more generally, if it is a seminormed space and $\|\cdot \|$ is merely a seminorm),[note 2] then this list may be extended to include:
1. $B$ is a norm bounded subset of $(X,\|\cdot \|).$ By definition, this means that there exists a real number $r>0$ such that $\|b\|\leq r$ for all $b\in B.$[1]
2. $\sup _{b\in B}\|b\|<\infty .$
• Thus, if $L:(X,\|\cdot \|)\to (Y,\|\cdot \|)$ is a linear map between two normed (or seminormed) spaces and if $B$ is the closed (alternatively, open) unit ball in $(X,\|\cdot \|)$ centered at the origin, then $L$ is a bounded linear operator (which recall means that its operator norm $\|L\|:=\sup _{b\in B}\|L(b)\|<\infty $ is finite) if and only if the image $L(B)$ of this ball under $L$ is a norm bounded subset of $(Y,\|\cdot \|).$
3. $B$ is a subset of some (open or closed) ball.[note 3]
• This ball need not be centered at the origin, but its radius must (as usual) be positive and finite.
If $B$ is a vector subspace of the TVS $X$ then this list may be extended to include:
1. $B$ is contained in the closure of $\{0\}.$[1]
• In other words, a vector subspace of $X$ is bounded if and only if it is a subset of (the vector space) $\operatorname {cl} _{X}\{0\}.$
• Recall that $X$ is a Hausdorff space if and only if $\{0\}$ is closed in $X.$ So the only bounded vector subspace of a Hausdorff TVS is $\{0\}.$
A subset that is not bounded is called unbounded.
Bornology and fundamental systems of bounded sets
The collection of all bounded sets on a topological vector space $X$ is called the von Neumann bornology or the (canonical) bornology of $X.$
A base or fundamental system of bounded sets of $X$ is a set ${\mathcal {B}}$ of bounded subsets of $X$ such that every bounded subset of $X$ is a subset of some $B\in {\mathcal {B}}.$[1] The set of all bounded subsets of $X$ trivially forms a fundamental system of bounded sets of $X.$
Examples
In any locally convex TVS, the set of closed and bounded disks are a base of bounded set.[1]
Examples and sufficient conditions
Unless indicated otherwise, a topological vector space (TVS) need not be Hausdorff nor locally convex.
• Finite sets are bounded.[1]
• Every totally bounded subset of a TVS is bounded.[1]
• Every relatively compact set in a topological vector space is bounded. If the space is equipped with the weak topology the converse is also true.
• The set of points of a Cauchy sequence is bounded, the set of points of a Cauchy net need not be bounded.
• The closure of the origin (referring to the closure of the set $\{0\}$) is always a bounded closed vector subspace. This set $\operatorname {cl} _{X}\{0\}$ is the unique largest (with respect to set inclusion $\,\subseteq \,$) bounded vector subspace of $X.$ In particular, if $B\subseteq X$ is a bounded subset of $X$ then so is $B+\operatorname {cl} _{X}\{0\}.$
Unbounded sets
A set that is not bounded is said to be unbounded.
Any vector subspace of a TVS that is not a contained in the closure of $\{0\}$ is unbounded
There exists a Fréchet space $X$ having a bounded subset $B$ and also a dense vector subspace $M$ such that $B$ is not contained in the closure (in $X$) of any bounded subset of $M.$[5]
Stability properties
• In any TVS, finite unions, finite Minkowski sums, scalar multiples, translations, subsets, closures, interiors, and balanced hulls of bounded sets are again bounded.[1]
• In any locally convex TVS, the convex hull (also called the convex envelope) of a bounded set is again bounded.[6] However, this may be false if the space is not locally convex, as the (non-locally convex) Lp space $L^{p}$ spaces for $0<p<1$ have no nontrivial open convex subsets.[6]
• The image of a bounded set under a continuous linear map is a bounded subset of the codomain.[1]
• A subset of an arbitrary (Cartesian) product of TVSs is bounded if and only if its image under every coordinate projections is bounded.
• If $S\subseteq X\subseteq Y$ and $X$ is a topological vector subspace of $Y,$ then $S$ is bounded in $X$ if and only if $S$ is bounded in $Y.$[1]
• In other words, a subset $S\subseteq X$ is bounded in $X$ if and only if it is bounded in every (or equivalently, in some) topological vector superspace of $X.$
Properties
See also: Topological vector space § Properties
A locally convex topological vector space has a bounded neighborhood of zero if and only if its topology can be defined by a single seminorm.
The polar of a bounded set is an absolutely convex and absorbing set.
Mackey's countability condition[7] — If $B_{1},B_{2},B_{3},\ldots $ is a countable sequence of bounded subsets of a metrizable locally convex topological vector space $X,$ then there exists a bounded subset $B$ of $X$ and a sequence $r_{1},r_{2},r_{3},\ldots $ of positive real numbers such that $B_{i}\subseteq r_{i}B$ for all $i\in \mathbb {N} $ (or equivalently, such that ${\tfrac {1}{r_{1}}}B_{1}\cup {\tfrac {1}{r_{2}}}B_{2}\cup {\tfrac {1}{r_{3}}}B_{3}\cup \cdots \subseteq B$).
Using the definition of uniformly bounded sets given below, Mackey's countability condition can be restated as: If $B_{1},B_{2},B_{3},\ldots $ are bounded subsets of a metrizable locally convex space then there exists a sequence $t_{1},t_{2},t_{3},\ldots $ of positive real numbers such that $t_{1}B_{1},\,t_{2}B_{2},\,t_{3}B_{3},\ldots $ are uniformly bounded. In words, given any countable family of bounded sets in a metrizable locally convex space, it is possible to scale each set by its own positive real so that they become uniformly bounded.
Generalizations
Uniformly bounded sets
See also: Uniform boundedness principle
A family of sets ${\mathcal {B}}$ of subsets of a topological vector space $Y$ is said to be uniformly bounded in $Y,$ if there exists some bounded subset $D$ of $Y$ such that
$B\subseteq D\quad {\text{ for every }}B\in {\mathcal {B}},$
which happens if and only if its union
$\cup {\mathcal {B}}~:=~\bigcup _{B\in {\mathcal {B}}}B$
is a bounded subset of $Y.$ In the case of a normed (or seminormed) space, a family ${\mathcal {B}}$ is uniformly bounded if and only if its union $\cup {\mathcal {B}}$ is norm bounded, meaning that there exists some real $M\geq 0$ such that $ \|b\|\leq M$ for every $b\in \cup {\mathcal {B}},$ or equivalently, if and only if $ \sup _{\stackrel {b\in B}{B\in {\mathcal {B}}}}\|b\|<\infty .$
A set $H$ of maps from $X$ to $Y$ is said to be uniformly bounded on a given set $C\subseteq X$ if the family $H(C):=\{h(C):h\in H\}$ is uniformly bounded in $Y,$ which by definition means that there exists some bounded subset $D$ of $Y$ such that $h(C)\subseteq D{\text{ for all }}h\in H,$ or equivalently, if and only if $ \cup H(C):=\bigcup _{h\in H}h(C)$ is a bounded subset of $Y.$ A set $H$ of linear maps between two normed (or seminormed) spaces $X$ and $Y$ is uniformly bounded on some (or equivalently, every) open ball (and/or non-degenerate closed ball) in $X$ if and only if their operator norms are uniformly bounded; that is, if and only if $ \sup _{h\in H}\|h\|<\infty .$
Proposition[8] — Let $H\subseteq L(X,Y)$ be a set of continuous linear operators between two topological vector spaces $X$ and $Y$ and let $C\subseteq X$ be any bounded subset of $X.$ Then $H$ is uniformly bounded on $C$ (that is, the family $\{h(C):h\in H\}$ is uniformly bounded in $Y$) if any of the following conditions are satisfied:
1. $H$ is equicontinuous.
2. $C$ is a convex compact Hausdorff subspace of $X$ and for every $c\in C,$ the orbit $H(c):=\{h(c):h\in H\}$ is a bounded subset of $Y.$
Proof of part (1)[8]
Assume $H$ is equicontinuous and let $W$ be a neighborhood of the origin in $Y.$ Since $H$ is equicontinuous, there exists a neighborhood $U$ of the origin in $X$ such that $h(U)\subseteq W$ for every $h\in H.$ Because $C$ is bounded in $X,$ there exists some real $r>0$ such that if $t\geq r$ then $C\subseteq tU.$ So for every $h\in H$ and every $t\geq r,$ $h(C)\subseteq h(tU)=th(U)\subseteq tW,$ which implies that $ \bigcup _{h\in H}h(C)\subseteq tW.$ Thus $ \bigcup _{h\in H}h(C)$ is bounded in $Y.$ Q.E.D.
Proof of part (2)[9]
Let $W$ be a balanced neighborhood of the origin in $Y$ and let $V$ be a closed balanced neighborhood of the origin in $Y$ such that $V+V\subseteq W.$ Define
$E~:=~\bigcap _{h\in H}h^{-1}(V),$
which is a closed subset of $X$ (since $V$ is closed while every $h:X\to Y$ is continuous) that satisfies $h(E)\subseteq V$ for every $h\in H.$ Note that for every non-zero scalar $n\neq 0,$ the set $nE$ is closed in $X$ (since scalar multiplication by $n\neq 0$ is a homeomorphism) and so every $C\cap nE$ is closed in $C.$
It will now be shown that $C\subseteq \bigcup _{n\in \mathbb {N} }nE,$ from which $C=\bigcup _{n\in \mathbb {N} }(C\cap nE)$ follows. If $c\in C$ then $H(c)$ being bounded guarantees the existence of some positive integer $n=n_{c}\in \mathbb {N} $ such that $H(c)\subseteq n_{c}V,$ where the linearity of every $h\in H$ now implies ${\tfrac {1}{n_{c}}}c\in h^{-1}(V);$ thus ${\tfrac {1}{n_{c}}}c\in \bigcap _{h\in H}h^{-1}(V)=E$ and hence $C\subseteq \bigcup _{n\in \mathbb {N} }nE,$ as desired.
Thus $ C=(C\cap 1E)\cup (C\cap 2E)\cup (C\cap 3E)\cup \cdots $ expresses $C$ as a countable union of closed (in $C$) sets. Since $C$ is a nonmeager subset of itself (as it is a Baire space by the Baire category theorem), this is only possible if there is some integer $n\in \mathbb {N} $ such that $C\cap nE$ has non-empty interior in $C.$ Let $k\in \operatorname {Int} _{C}(C\cap nE)$ be any point belonging to this open subset of $C.$ Let $U$ be any balanced open neighborhood of the origin in $X$ such that
$C\cap (k+U)~\subseteq ~\operatorname {Int} _{C}(C\cap nE).$
The sets $\{k+pU:p>1\}$ form an increasing (meaning $p\leq q$ implies $k+pU\subseteq k+qU$) cover of the compact space $C,$ so there exists some $p>1$ such that $C\subseteq k+pU$ (and thus ${\tfrac {1}{p}}(C-k)\subseteq U$). It will be shown that $h(C)\subseteq pnW$ for every $h\in H,$ thus demonstrating that $\{h(C):h\in H\}$ is uniformly bounded in $Y$ and completing the proof. So fix $h\in H$ and $c\in C.$ Let
$z~:=~{\tfrac {p-1}{p}}k+{\tfrac {1}{p}}c.$
The convexity of $C$ guarantees $z\in C$ and moreover, $z\in k+U$ since
$z-k={\tfrac {-1}{p}}k+{\tfrac {1}{p}}c={\tfrac {1}{p}}(c-k)\in {\tfrac {1}{p}}(C-k)\subseteq U.$
Thus $z\in C\cap (k+U),$ which is a subset of $\operatorname {Int} _{C}(C\cap nE).$ Since $nV$ is balanced and $|1-p|=p-1<p,$ we have $(1-p)nV\subseteq pnV,$ which combined with $h(E)\subseteq V$ gives
$pnh(E)+(1-p)nh(E)~\subseteq ~pnV+(1-p)nV~\subseteq ~pnV+pnV~\subseteq ~pn(V+V)~\subseteq ~pnW.$
Finally, $c=pz+(1-p)k$ and $k,z\in nE$ imply
$h(c)~=~ph(z)+(1-p)h(k)~\in ~pnh(E)+(1-p)nh(E)~\subseteq ~pnW,$
as desired. Q.E.D.
Since every singleton subset of $X$ is also a bounded subset, it follows that if $H\subseteq L(X,Y)$ is an equicontinuous set of continuous linear operators between two topological vector spaces $X$ and $Y$ (not necessarily Hausdorff or locally convex), then the orbit $ H(x):=\{h(x):h\in H\}$ of every $x\in X$ is a bounded subset of $Y.$
Bounded subsets of topological modules
The definition of bounded sets can be generalized to topological modules. A subset $A$ of a topological module $M$ over a topological ring $R$ is bounded if for any neighborhood $N$ of $0_{M}$ there exists a neighborhood $w$ of $0_{R}$ such that $wA\subseteq B.$
See also
• Bornological space – Space where bounded operators are continuous
• Bornivorous set – A set that can absorb any bounded subset
• Bounded function – A mathematical function the set of whose values are bounded
• Bounded operator – Linear transformation between topological vector spaces
• Bounding point – Mathematical concept related to subsets of vector spaces
• Compact space – Type of mathematical space
• Kolmogorov's normability criterion – Characterization of normable spaces
• Local boundedness
• Totally bounded space – Generalization of compactness
References
1. Narici & Beckenstein 2011, pp. 156–175.
2. Schaefer 1970, p. 25.
3. Rudin 1991, p. 8.
4. Wilansky 2013, p. 47.
5. Wilansky 2013, p. 57.
6. Narici & Beckenstein 2011, p. 162.
7. Narici & Beckenstein 2011, p. 174.
8. Rudin 1991, pp. 42−47.
9. Rudin 1991, pp. 46−47.
Notes
1. For any set $A$ and scalar $s,$ the notation $sA$ is denotes the set $sA:=\{sa:a\in A\}.$
2. This means that the topology on $X$ is equal to the topology induced on it by $\|\cdot \|.$ Note that every normed space is a seminormed space and every norm is a seminorm. The definition of the topology induced by a seminorm is identical to the definition of the topology induced by a norm.
3. If $(X,\|\cdot \|)$ is a normed space or a seminormed space, then the open and closed balls of radius $r>0$ (where $r\neq \infty $ is a real number) centered at a point $x\in X$ are, respectively, the sets $ B_{<r}(x):=\{z\in X:\|z-x\|<r\}$ and $ B_{\leq r}(x):=\{z\in X:\|z-x\|\leq r\}.$ Any such set is called a (non-degenerate) ball.
Bibliography
• Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. Vol. 639. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003.
• Berberian, Sterling K. (1974). Lectures in Functional Analysis and Operator Theory. Graduate Texts in Mathematics. Vol. 15. New York: Springer. ISBN 978-0-387-90081-0. OCLC 878109401.
• Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 3-540-13627-4. OCLC 17499190.
• Conway, John (1990). A course in functional analysis. Graduate Texts in Mathematics. Vol. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908.
• Edwards, Robert E. (1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138.
• Grothendieck, Alexander (1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098.
• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
• Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge University Press. pp. 44–46.
• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
• Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250.
• Schaefer, H.H. (1970). Topological Vector Spaces. GTM. Vol. 3. Springer-Verlag. pp. 25–26. ISBN 0-387-05380-8.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.
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| Wikipedia |
Vopěnka's principle
In mathematics, Vopěnka's principle is a large cardinal axiom. The intuition behind the axiom is that the set-theoretical universe is so large that in every proper class, some members are similar to others, with this similarity formalized through elementary embeddings.
Vopěnka's principle was first introduced by Petr Vopěnka and independently considered by H. Jerome Keisler, and was written up by Solovay, Reinhardt & Kanamori (1978). According to Pudlák (2013, p. 204), Vopěnka's principle was originally intended as a joke: Vopěnka was apparently unenthusiastic about large cardinals and introduced his principle as a bogus large cardinal property, planning to show later that it was not consistent. However, before publishing his inconsistency proof he found a flaw in it.
Definition
Vopěnka's principle asserts that for every proper class of binary relations (each with set-sized domain), there is one elementarily embeddable into another. This cannot be stated as a single sentence of ZFC as it involves a quantification over classes. A cardinal κ is called a Vopěnka cardinal if it is inaccessible and Vopěnka's principle holds in the rank Vκ (allowing arbitrary S ⊂ Vκ as "classes"). [1]
Many equivalent formulations are possible. For example, Vopěnka's principle is equivalent to each of the following statements.
• For every proper class of simple directed graphs, there are two members of the class with a homomorphism between them.[2]
• For any signature Σ and any proper class of Σ-structures, there are two members of the class with an elementary embedding between them.[1][2]
• For every predicate P and proper class S of ordinals, there is a non-trivial elementary embedding j:(Vκ, ∈, P) → (Vλ, ∈, P) for some κ and λ in S.[1]
• The category of ordinals cannot be fully embedded in the category of graphs.[2]
• Every subfunctor of an accessible functor is accessible.[2]
• (In a definable classes setting) For every natural number n, there exists a C(n)-extendible cardinal.[3]
Strength
Even when restricted to predicates and proper classes definable in first order set theory, the principle implies existence of Σn correct extendible cardinals for every n.
If κ is an almost huge cardinal, then a strong form of Vopěnka's principle holds in Vκ:
There is a κ-complete ultrafilter U such that for every {Ri: i < κ} where each Ri is a binary relation and Ri ∈ Vκ, there is S ∈ U and a non-trivial elementary embedding j: Ra → Rb for every a < b in S.
References
1. Kanamori, Akihiro (2003). The higher infinite: large cardinals in set theory from their beginnings (2nd ed.). Berlin [u.a.]: Springer. ISBN 9783540003847.
2. Rosicky, Jiří Adámek ; Jiří (1994). Locally presentable and accessible categories (Digital print. 2004. ed.). Cambridge [u.a.]: Cambridge Univ. Press. ISBN 0521422612.{{cite book}}: CS1 maint: multiple names: authors list (link)
3. Bagaria, Joan (23 December 2011). "C(n)-cardinals". Archive for Mathematical Logic. 51 (3–4): 213–240. doi:10.1007/s00153-011-0261-8. S2CID 208867731.
• Kanamori, Akihiro (1978), "On Vopěnka's and related principles", Logic Colloquium '77 (Proc. Conf., Wrocław, 1977), Stud. Logic Foundations Math., vol. 96, Amsterdam-New York: North-Holland, pp. 145–153, ISBN 0-444-85178-X, MR 0519809
• Pudlák, Pavel (2013), Logical foundations of mathematics and computational complexity. A gentle introduction, Springer Monographs in Mathematics, Springer, doi:10.1007/978-3-319-00119-7, ISBN 978-3-319-00118-0, MR 3076860
• Solovay, Robert M.; Reinhardt, William N.; Kanamori, Akihiro (1978), "Strong axioms of infinity and elementary embeddings" (PDF), Annals of Mathematical Logic, 13 (1): 73–116, doi:10.1016/0003-4843(78)90031-1
External links
• Friedman, Harvey M. (2005), EMBEDDING AXIOMS gives a number of equivalent definitions of Vopěnka's principle.
| Wikipedia |
Vorlesungen über Zahlentheorie
Vorlesungen über Zahlentheorie (German for Lectures on Number Theory) is the name of several different textbooks of number theory. The best known was written by Peter Gustav Lejeune Dirichlet and Richard Dedekind, and published in 1863. Others were written by Leopold Kronecker, Edmund Landau, and Helmut Hasse. They all cover elementary number theory, Dirichlet's theorem, quadratic fields and forms, and sometimes more advanced topics.
Dirichlet and Dedekind's book
Based on Dirichlet's number theory course at the University of Göttingen, the Vorlesungen were edited by Dedekind and published after Lejeune Dirichlet's death. Dedekind added several appendices to the Vorlesungen, in which he collected further results of Lejeune Dirichlet's and also developed his own original mathematical ideas.
Scope
The Vorlesungen cover topics in elementary number theory, algebraic number theory and analytic number theory, including modular arithmetic, quadratic congruences, quadratic reciprocity and binary quadratic forms.
Contents
The contents of Professor John Stillwell's 1999 translation of the Vorlesungen are as follows
Chapter 1. On the divisibility of numbers
Chapter 2. On the congruence of numbers
Chapter 3. On quadratic residues
Chapter 4. On quadratic forms
Chapter 5. Determination of the class number of binary quadratic forms
Supplement I. Some theorems from Gauss's theory of circle division
Supplement II. On the limiting value of an infinite series
Supplement III. A geometric theorem
Supplement IV. Genera of quadratic forms
Supplement V. Power residues for composite moduli
Supplement VI. Primes in arithmetic progressions
Supplement VII. Some theorems from the theory of circle division
Supplement VIII. On the Pell equation
Supplement IX. Convergence and continuity of some infinite series
This translation does not include Dedekind's Supplements X and XI in which he begins to develop the theory of ideals.
The German titles of supplements X and XI are:
Supplement X: Über die Composition der binären quadratische Formen (On the composition of binary quadratic forms)
Supplement XI: Über die Theorie der ganzen algebraischen Zahlen (On the theory of algebraic integers)
Chapters 1 to 4 cover similar ground to Gauss' Disquisitiones Arithmeticae, and Dedekind added footnotes which specifically cross-reference the relevant sections of the Disquisitiones. These chapters can be thought of as a summary of existing knowledge, although Dirichlet simplifies Gauss' presentation, and introduces his own proofs in some places.
Chapter 5 contains Dirichlet's derivation of the class number formula for real and imaginary quadratic fields. Although other mathematicians had conjectured similar formulae, Dirichlet gave the first rigorous proof.
Supplement VI contains Dirichlet's proof that an arithmetic progression of the form a+nd where a and d are coprime contains an infinite number of primes.
Importance
The Vorlesungen can be seen as a watershed between the classical number theory of Fermat, Jacobi and Gauss, and the modern number theory of Dedekind, Riemann and Hilbert. Dirichlet does not explicitly recognise the concept of the group that is central to modern algebra, but many of his proofs show an implicit understanding of group theory.
The Vorlesungen contains two key results in number theory which were first proved by Dirichlet. The first of these is the class number formulae for binary quadratic forms. The second is a proof that arithmetic progressions contains an infinite number of primes (known as Dirichlet's theorem); this proof introduces Dirichlet L-series. These results are important milestones in the development of analytic number theory.
Kronecker's book
Leopold Kronecker's book was first published in 1901 in 2 parts and reprinted by Springer in 1978. It covers elementary and algebraic number theory, including Dirichlet's theorem.
Landau's book
Edmund Landau's book Vorlesungen über Zahlentheorie was first published as a 3-volume set in 1927. The first half of volume 1 was published as Vorlesungen über Zahlentheorie. Aus der elementare Zahlentheorie in 1950, with an English translation in 1958 under the title Elementary number theory. In 1969 Chelsea republished the second half of volume 1 together with volumes 2 and 3 as a single volume.
Volume 1 on elementary and additive number theory includes the topics such as Dirichlet's theorem, Brun's sieve, binary quadratic forms, Goldbach's conjecture, Waring's problem, and the Hardy–Littlewood work on the singular series. Volume 2 covers topics in analytic number theory, such as estimates for the error in the prime number theorem, and topics in geometric number theory such as estimating numbers of lattice points. Volume 3 covers algebraic number theory, including ideal theory, quadratic number fields, and applications to Fermat's last theorem. Many of the results described by Landau were state of the art at the time but have since been superseded by stronger results.
Hasse's book
Helmut Hasse's book Vorlesungen über Zahlentheorie was published in 1950, and is different from and more elementary than his book Zahlentheorie. It covers elementary number theory, Dirichlet's theorem, and quadratic fields.
References
• P. G. Lejeune Dirichlet, R. Dedekind tr. John Stillwell: Lectures on Number Theory, American Mathematical Society, 1999 ISBN 0-8218-2017-6 The Göttinger Digitalisierungszentrum has a scanned copy of the original, 2nd edition text (in German) published in 1871 containing supplements I–X. Supplement XI can be found in volume three of Dedekind's complete works also at the Göttinger Digitalisierungszentrum. The 4th edition from 1894 which contains all of the supplements including Dedekind's XI is available at Internet Archive.
• Hasse, Helmut (1950), Vorlesungen über Zahlentheorie, Die Grundlehren der mathematischen Wissenschaften, vol. LIX, Berlin-Göttingen-Heidelberg: Springer-Verlag, ISBN 978-3-642-88679-9, MR 0051844
• Kronecker, Leopold (1978) [1901], Vorlesungen über Zahlentheorie, Berlin-New York: Springer-Verlag, ISBN 3-540-08277-8, MR 0529431
• Landau, Edmund (1958) [1927], Elementary number theory., New York, N.Y.: Chelsea Publishing Co., MR 0092794
• Landau, Edmund (1969) [1927], Vorlesungen über Zahlentheorie. Erster Band, zweiter Teil; zweiter Band; dritter Band., New York: Chelsea Publishing Co., MR 0250844
| Wikipedia |
Zeta function universality
In mathematics, the universality of zeta functions is the remarkable ability of the Riemann zeta function and other similar functions (such as the Dirichlet L-functions) to approximate arbitrary non-vanishing holomorphic functions arbitrarily well.
The universality of the Riemann zeta function was first proven by Sergei Mikhailovitch Voronin in 1975[1] and is sometimes known as Voronin's universality theorem.
Formal statement
A mathematically precise statement of universality for the Riemann zeta function ζ(s) follows.
Let U be a compact subset of the strip
$\{s\in \mathbb {C} \mid 1/2<{\mbox{Re }}s<1\}$
such that the complement of U is connected. Let f : U → C be a continuous function on U which is holomorphic on the interior of U and does not have any zeros in U. Then for any ε > 0 there exists a t ≥ 0 such that
$\left|\zeta (s+it)-f(s)\right|<\varepsilon $
(1)
for all $s\in U$.
Even more: the lower density of the set of values t satisfying the above inequality is positive. Precisely
$0<\liminf _{T\to \infty }{\frac {1}{T}}\,\lambda \!\left(\left\{t\in [0,T]\;{\bigg |}\;\max _{s\in U}\left|\zeta (s+it)-f(s)\right|<\varepsilon \right\}\right),$
where $\lambda $ denotes the Lebesgue measure on the real numbers and $\liminf $ denotes the limit inferior.
Discussion
The condition that the complement of U be connected essentially means that U does not contain any holes.
The intuitive meaning of the first statement is as follows: it is possible to move U by some vertical displacement it so that the function f on U is approximated by the zeta function on the displaced copy of U, to an accuracy of ε.
The function f is not allowed to have any zeros on U. This is an important restriction; if we start with a holomorphic function with an isolated zero, then any "nearby" holomorphic function will also have a zero. According to the Riemann hypothesis, the Riemann zeta function does not have any zeros in the considered strip, and so it couldn't possibly approximate such a function. The function f(s) = 0 which is identically zero on U can be approximated by ζ: we can first pick the "nearby" function g(s) = ε/2 (which is holomorphic and does not have zeros) and find a vertical displacement such that ζ approximates g to accuracy ε/2, and therefore f to accuracy ε.
The accompanying figure shows the zeta function on a representative part of the relevant strip. The color of the point s encodes the value ζ(s) as follows: the hue represents the argument of ζ(s), with red denoting positive real values, and then counterclockwise through yellow, green cyan, blue and purple. Strong colors denote values close to 0 (black = 0), weak colors denote values far away from 0 (white = ∞). The picture shows three zeros of the zeta function, at about 1/2 + 103.7i, 1/2 + 105.5i and 1/2 + 107.2i. Voronin's theorem essentially states that this strip contains all possible "analytic" color patterns that do not use black or white.
The rough meaning of the statement on the lower density is as follows: if a function f and an ε > 0 are given, then there is a positive probability that a randomly picked vertical displacement it will yield an approximation of f to accuracy ε.
The interior of U may be empty, in which case there is no requirement of f being holomorphic. For example, if we take U to be a line segment, then a continuous function f : U → C is a curve in the complex plane, and we see that the zeta function encodes every possible curve (i.e., any figure that can be drawn without lifting the pencil) to arbitrary precision on the considered strip.
The theorem as stated applies only to regions U that are contained in the strip. However, if we allow translations and scalings, we can also find encoded in the zeta functions approximate versions of all non-vanishing holomorphic functions defined on other regions. In particular, since the zeta function itself is holomorphic, versions of itself are encoded within it at different scales, the hallmark of a fractal.[2]
The surprising nature of the theorem may be summarized in this way: the Riemann zeta function contains "all possible behaviors" within it, and is thus "chaotic" in a sense, yet it is a perfectly smooth analytic function with a straightforward definition.
Proof sketch
A sketch of the proof presented in (Voronin and Karatsuba, 1992)[3] follows. We consider only the case where U is a disk centered at 3/4:
$U=\{s\in \mathbb {C} :|s-3/4|<r\}\quad {\mbox{with}}\quad 0<r<1/4$ :|s-3/4|<r\}\quad {\mbox{with}}\quad 0<r<1/4}
and we will argue that every non-zero holomorphic function defined on U can be approximated by the ζ-function on a vertical translation of this set.
Passing to the logarithm, it is enough to show that for every holomorphic function g : U → C and every ε > 0 there exists a real number t such that
$\left|\ln \zeta (s+it)-g(s)\right|<\varepsilon \quad {\text{for all}}\quad s\in U.$
We will first approximate g(s) with the logarithm of certain finite products reminiscent of the Euler product for the ζ-function:
$\zeta (s)=\prod _{p\in \mathbb {P} }\left(1-{\frac {1}{p^{s}}}\right)^{-1},$
where P denotes the set of all primes.
If $\theta =(\theta _{p})_{p\in \mathbb {P} }$ is a sequence of real numbers, one for each prime p, and M is a finite set of primes, we set
$\zeta _{M}(s,\theta )=\prod _{p\in M}\left(1-{\frac {e^{-2\pi i\theta _{p}}}{p^{s}}}\right)^{-1}.$
We consider the specific sequence
${\hat {\theta }}=\left({\frac {1}{4}},{\frac {2}{4}},{\frac {3}{4}},{\frac {4}{4}},{\frac {5}{4}},\ldots \right)$
and claim that g(s) can be approximated by a function of the form $\ln(\zeta _{M}(s,{\hat {\theta }}))$ for a suitable set M of primes. The proof of this claim utilizes the Bergman space, falsely named Hardy space in (Voronin and Karatsuba, 1992),[3] in H of holomorphic functions defined on U, a Hilbert space. We set
$u_{k}(s)=\ln \left(1-{\frac {e^{-\pi ik/2}}{p_{k}^{s}}}\right)$
where pk denotes the k-th prime number. It can then be shown that the series
$\sum _{k=1}^{\infty }u_{k}$
is conditionally convergent in H, i.e. for every element v of H there exists a rearrangement of the series which converges in H to v. This argument uses a theorem that generalizes the Riemann series theorem to a Hilbert space setting. Because of a relationship between the norm in H and the maximum absolute value of a function, we can then approximate our given function g(s) with an initial segment of this rearranged series, as required.
By a version of the Kronecker theorem, applied to the real numbers ${\frac {\ln 2}{2\pi }},{\frac {\ln 3}{2\pi }},{\frac {\ln 5}{2\pi }},\ldots ,{\frac {\ln p_{N}}{2\pi }}$ (which are linearly independent over the rationals) we can find real values of t so that $\ln(\zeta _{M}(s,{\hat {\theta }}))$ is approximated by $\ln(\zeta _{M}(s+it,0))$. Further, for some of these values t, $\ln(\zeta _{M}(s+it,0))$ approximates $\ln(\zeta (s+it))$, finishing the proof.
The theorem is stated without proof in § 11.11 of (Titchmarsh and Heath-Brown, 1986),[4] the second edition of a 1951 monograph by Titchmarsh; and a weaker result is given in Thm. 11.9. Although Voronin's theorem is not proved there, two corollaries are derived from it:
1) Let ${\tfrac {1}{2}}<\sigma <1$ be fixed. Then the curve
$\gamma (t)=(\zeta (\sigma +it),\zeta '(\sigma +it),\dots ,\zeta ^{(n-1)}(\sigma +it))$
is dense in $\mathbb {C} ^{n}.$
2) Let $\Phi $ be any continuous function, and let $h_{1},h_{2},\dots ,h_{n}$ be real constants.
Then $\zeta (s)$ cannot satisfy the differential-difference equation
$\Phi \{\zeta (s+h_{1}),\zeta '(s+h_{1}),\dots ,\zeta ^{(n_{1})}(s+h_{1}),\zeta (s+h_{2}),\zeta '(s+h_{2}),\dots ,\zeta ^{(n_{2})}(s+h_{2}),\dots \}=0$
unless $\Phi $ vanishes identically.
Effective universality
Some recent work has focused on effective universality. Under the conditions stated at the beginning of this article, there exist values of t that satisfy inequality (1). An effective universality theorem places an upper bound on the smallest such t.
For example, in 2003, Garunkštis proved that if $f(s)$ is analytic in $|s|\leq .05$ with $\max _{\left|s\right|\leq .05}\left|f(s)\right|\leq 1$, then for any ε in $0<\epsilon <1/2$, there exists a number $t$ in $0\leq t\leq \exp({\exp({10/\epsilon ^{13}})})$ such that
$\max _{\left|s\right|\leq .0001}\left|\log \zeta (s+{\frac {3}{4}}+it)-f(s)\right|<\epsilon $.
For example, if $\epsilon =1/10$, then the bound for t is $t\leq \exp({\exp({10/\epsilon ^{13}})})=\exp({\exp({10^{14}})})$.
Bounds can also be obtained on the measure of these t values, in terms of ε:
$\liminf _{T\to \infty }{\frac {1}{T}}\,\lambda \!\left(\left\{t\in [0,T]:\max _{\left|s\right|\leq .0001}\left|\log \zeta (s+{\frac {3}{4}}+it)-f(s)\right|<\epsilon \right\}\right)\geq {\frac {1}{\exp({\epsilon ^{-13}})}}$.
For example, if $\epsilon =1/10$, then the right-hand side is $1/\exp({10^{13}})$. See.[5]: p. 210
Universality of other zeta functions
Work has been done showing that universality extends to Selberg zeta functions.[6]
The Dirichlet L-functions show not only universality, but a certain kind of joint universality that allow any set of functions to be approximated by the same value(s) of t in different L-functions, where each function to be approximated is paired with a different L-function.[7] [8]: Section 4
A similar universality property has been shown for the Lerch zeta function $L(\lambda ,\alpha ,s)$, at least when the parameter α is a transcendental number.[8]: Section 5 Sections of the Lerch zeta function have also been shown to have a form of joint universality. [8]: Section 6
References
1. Voronin, S.M. (1975) "Theorem on the Universality of the Riemann Zeta Function." Izv. Akad. Nauk SSSR, Ser. Matem. 39 pp.475-486. Reprinted in Math. USSR Izv. 9, 443-445, 1975
2. Woon, S.C. (1994-06-11). "Riemann zeta function is a fractal". arXiv:chao-dyn/9406003.
3. Karatsuba, A. A.; Voronin, S. M. (July 1992). The Riemann Zeta-Function. Walter de Gruyter. p. 396. ISBN 3-11-013170-6.
4. Titchmarsh, Edward Charles; Heath-Brown, David Rodney ("Roger") (1986). The Theory of the Riemann Zeta-function (2nd ed.). Oxford: Oxford U. P. pp. 308–309. ISBN 0-19-853369-1.
5. Ramūnas Garunkštis; Antanas Laurinčikas; Kohji Matsumoto; Jörn Steuding; Rasa Steuding (2010). "Effective uniform approximation by the Riemann zeta-function". Publicacions Matemàtiques. 54 (1): 209–219. doi:10.5565/publmat_54110_12. JSTOR 43736941.
6. Paulius Drungilas; Ramūnas Garunkštis; Audrius Kačėnas (2013). "Universality of the Selberg zeta-function for the modular group". Forum Mathematicum. 25 (3). doi:10.1515/form.2011.127. ISSN 1435-5337. S2CID 54965707.
7. B. Bagchi (1982). "A Universality theorem for Dirichlet L-functions". Mathematische Zeitschrift. 181 (3): 319–334. doi:10.1007/BF01161980. S2CID 120930513.
8. Kohji Matsumoto (2013). "A survey on the theory of universality for zeta and L-functions". Plowing and Starring Through High Wave Forms. Proceedings of the 7th China–Japan Seminar. The 7th China–Japan Seminar on Number Theory. Vol. 11. Fukuoka, Japan: World Scientific. pp. 95–144. arXiv:1407.4216. Bibcode:2014arXiv1407.4216M. ISBN 978-981-4644-92-1.
Further reading
• Karatsuba, Anatoly A.; Voronin, S. M. (2011). The Riemann Zeta-Function. de Gruyter Expositions In Mathematics. Berlin: de Gruyter. ISBN 978-3110131703.
• Laurinčikas, Antanas (1996). Limit Theorems for the Riemann Zeta-Function. Mathematics and Its Applications. Vol. 352. Berlin: Springer. doi:10.1007/978-94-017-2091-5. ISBN 978-90-481-4647-5.
• Steuding, Jörn (2007). Value-Distribution of L-Functions. Lecture Notes in Mathematics. Vol. 1877. Berlin: Springer. p. 19. arXiv:1711.06671. doi:10.1007/978-3-540-44822-8. ISBN 978-3-540-26526-9.
• Titchmarsh, Edward Charles; Heath-Brown, David Rodney ("Roger") (1986). The Theory of the Riemann Zeta-function (2nd ed.). Oxford: Oxford U. P. ISBN 0-19-853369-1.
External links
• Voronin's Universality Theorem, by Matthew R. Watkins
• X-Ray of the Zeta Function Visually oriented investigation of where zeta is real or purely imaginary. Gives some indication of how complicated it is in the critical strip.
| Wikipedia |
Computational fluid dynamics
Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate the free-stream flow of the fluid, and the interaction of the fluid (liquids and gases) with surfaces defined by boundary conditions. With high-speed supercomputers, better solutions can be achieved, and are often required to solve the largest and most complex problems. Ongoing research yields software that improves the accuracy and speed of complex simulation scenarios such as transonic or turbulent flows. Initial validation of such software is typically performed using experimental apparatus such as wind tunnels. In addition, previously performed analytical or empirical analysis of a particular problem can be used for comparison. A final validation is often performed using full-scale testing, such as flight tests.
Computational physics
• Mechanics
• Electromagnetics
• Particle physics
• Thermodynamics
• Simulation
Potentials
• Morse/Long-range potential
• Lennard-Jones potential
• Yukawa potential
• Morse potential
Fluid dynamics
• Finite difference
• Finite volume
• Finite element
• Boundary element
• Lattice Boltzmann
• Riemann solver
• Dissipative particle dynamics
• Smoothed particle hydrodynamics
• Turbulence models
Monte Carlo methods
• Integration
• Gibbs sampling
• Metropolis algorithm
Particle
• N-body
• Particle-in-cell
• Molecular dynamics
Scientists
• Godunov
• Ulam
• von Neumann
• Galerkin
• Lorenz
• Wilson
• Alder
• Richtmyer
CFD is applied to a wide range of research and engineering problems in many fields of study and industries, including aerodynamics and aerospace analysis, hypersonics, weather simulation, natural science and environmental engineering, industrial system design and analysis, biological engineering, fluid flows and heat transfer, engine and combustion analysis, and visual effects for film and games.
Background and history
The fundamental basis of almost all CFD problems is the Navier–Stokes equations, which define many single-phase (gas or liquid, but not both) fluid flows. These equations can be simplified by removing terms describing viscous actions to yield the Euler equations. Further simplification, by removing terms describing vorticity yields the full potential equations. Finally, for small perturbations in subsonic and supersonic flows (not transonic or hypersonic) these equations can be linearized to yield the linearized potential equations.
Historically, methods were first developed to solve the linearized potential equations. Two-dimensional (2D) methods, using conformal transformations of the flow about a cylinder to the flow about an airfoil were developed in the 1930s.[1]
One of the earliest type of calculations resembling modern CFD are those by Lewis Fry Richardson, in the sense that these calculations used finite differences and divided the physical space in cells. Although they failed dramatically, these calculations, together with Richardson's book Weather Prediction by Numerical Process,[2] set the basis for modern CFD and numerical meteorology. In fact, early CFD calculations during the 1940s using ENIAC used methods close to those in Richardson's 1922 book.[3]
The computer power available paced development of three-dimensional methods. Probably the first work using computers to model fluid flow, as governed by the Navier–Stokes equations, was performed at Los Alamos National Lab, in the T3 group.[4][5] This group was led by Francis H. Harlow, who is widely considered one of the pioneers of CFD. From 1957 to late 1960s, this group developed a variety of numerical methods to simulate transient two-dimensional fluid flows, such as particle-in-cell method,[6] fluid-in-cell method,[7] vorticity stream function method,[8] and marker-and-cell method.[9] Fromm's vorticity-stream-function method for 2D, transient, incompressible flow was the first treatment of strongly contorting incompressible flows in the world.
The first paper with three-dimensional model was published by John Hess and A.M.O. Smith of Douglas Aircraft in 1967.[10] This method discretized the surface of the geometry with panels, giving rise to this class of programs being called Panel Methods. Their method itself was simplified, in that it did not include lifting flows and hence was mainly applied to ship hulls and aircraft fuselages. The first lifting Panel Code (A230) was described in a paper written by Paul Rubbert and Gary Saaris of Boeing Aircraft in 1968.[11] In time, more advanced three-dimensional Panel Codes were developed at Boeing (PANAIR, A502),[12] Lockheed (Quadpan),[13] Douglas (HESS),[14] McDonnell Aircraft (MACAERO),[15] NASA (PMARC)[16] and Analytical Methods (WBAERO,[17] USAERO[18] and VSAERO[19][20]). Some (PANAIR, HESS and MACAERO) were higher order codes, using higher order distributions of surface singularities, while others (Quadpan, PMARC, USAERO and VSAERO) used single singularities on each surface panel. The advantage of the lower order codes was that they ran much faster on the computers of the time. Today, VSAERO has grown to be a multi-order code and is the most widely used program of this class. It has been used in the development of many submarines, surface ships, automobiles, helicopters, aircraft, and more recently wind turbines. Its sister code, USAERO is an unsteady panel method that has also been used for modeling such things as high speed trains and racing yachts. The NASA PMARC code from an early version of VSAERO and a derivative of PMARC, named CMARC,[21] is also commercially available.
In the two-dimensional realm, a number of Panel Codes have been developed for airfoil analysis and design. The codes typically have a boundary layer analysis included, so that viscous effects can be modeled. Richard Eppler developed the PROFILE code, partly with NASA funding, which became available in the early 1980s.[22] This was soon followed by Mark Drela's XFOIL code.[23] Both PROFILE and XFOIL incorporate two-dimensional panel codes, with coupled boundary layer codes for airfoil analysis work. PROFILE uses a conformal transformation method for inverse airfoil design, while XFOIL has both a conformal transformation and an inverse panel method for airfoil design.
An intermediate step between Panel Codes and Full Potential codes were codes that used the Transonic Small Disturbance equations. In particular, the three-dimensional WIBCO code,[24] developed by Charlie Boppe of Grumman Aircraft in the early 1980s has seen heavy use.
Developers turned to Full Potential codes, as panel methods could not calculate the non-linear flow present at transonic speeds. The first description of a means of using the Full Potential equations was published by Earll Murman and Julian Cole of Boeing in 1970.[25] Frances Bauer, Paul Garabedian and David Korn of the Courant Institute at New York University (NYU) wrote a series of two-dimensional Full Potential airfoil codes that were widely used, the most important being named Program H.[26] A further growth of Program H was developed by Bob Melnik and his group at Grumman Aerospace as Grumfoil.[27] Antony Jameson, originally at Grumman Aircraft and the Courant Institute of NYU, worked with David Caughey to develop the important three-dimensional Full Potential code FLO22[28] in 1975. Many Full Potential codes emerged after this, culminating in Boeing's Tranair (A633) code,[29] which still sees heavy use.
The next step was the Euler equations, which promised to provide more accurate solutions of transonic flows. The methodology used by Jameson in his three-dimensional FLO57 code[30] (1981) was used by others to produce such programs as Lockheed's TEAM program[31] and IAI/Analytical Methods' MGAERO program.[32] MGAERO is unique in being a structured cartesian mesh code, while most other such codes use structured body-fitted grids (with the exception of NASA's highly successful CART3D code,[33] Lockheed's SPLITFLOW code[34] and Georgia Tech's NASCART-GT).[35] Antony Jameson also developed the three-dimensional AIRPLANE code[36] which made use of unstructured tetrahedral grids.
In the two-dimensional realm, Mark Drela and Michael Giles, then graduate students at MIT, developed the ISES Euler program[37] (actually a suite of programs) for airfoil design and analysis. This code first became available in 1986 and has been further developed to design, analyze and optimize single or multi-element airfoils, as the MSES program.[38] MSES sees wide use throughout the world. A derivative of MSES, for the design and analysis of airfoils in a cascade, is MISES,[39] developed by Harold Youngren while he was a graduate student at MIT.
The Navier–Stokes equations were the ultimate target of development. Two-dimensional codes, such as NASA Ames' ARC2D code first emerged. A number of three-dimensional codes were developed (ARC3D, OVERFLOW, CFL3D are three successful NASA contributions), leading to numerous commercial packages.
Hierarchy of fluid flow equations
CFD can be seen as a group of computational methodologies (discussed below) used to solve equations governing fluid flow. In the application of CFD, a critical step is to decide which set of physical assumptions and related equations need to be used for the problem at hand.[40] To illustrate this step, the following summarizes the physical assumptions/simplifications taken in equations of a flow that is single-phase (see multiphase flow and two-phase flow), single-species (i.e., it consists of one chemical species), non-reacting, and (unless said otherwise) compressible. Thermal radiation is neglected, and body forces due to gravity are considered (unless said otherwise). In addition, for this type of flow, the next discussion highlights the hierarchy of flow equations solved with CFD. Note that some of the following equations could be derived in more than one way.
• Conservation laws (CL): These are the most fundamental equations considered with CFD in the sense that, for example, all the following equations can be derived from them. For a single-phase, single-species, compressible flow one considers the conservation of mass, conservation of linear momentum, and conservation of energy.
• Continuum conservation laws (CCL): Start with the CL. Assume that mass, momentum and energy are locally conserved: These quantities are conserved and cannot "teleport" from one place to another but can only move by a continuous flow (see continuity equation). Another interpretation is that one starts with the CL and assumes a continuum medium (see continuum mechanics). The resulting system of equations is unclosed since to solve it one needs further relationships/equations: (a) constitutive relationships for the viscous stress tensor; (b) constitutive relationships for the diffusive heat flux; (c) an equation of state (EOS), such as the ideal gas law; and, (d) a caloric equation of state relating temperature with quantities such as enthalpy or internal energy.
• Compressible Navier-Stokes equations (C-NS): Start with the CCL. Assume a Newtonian viscous stress tensor (see Newtonian fluid) and a Fourier heat flux (see heat flux).[41][42] The C-NS need to be augmented with an EOS and a caloric EOS to have a closed system of equations.
• Incompressible Navier-Stokes equations (I-NS): Start with the C-NS. Assume that density is always and everywhere constant.[43] Another way to obtain the I-NS is to assume that the Mach number is very small[43][42] and that temperature differences in the fluid are very small as well.[42] As a result, the mass-conservation and momentum-conservation equations are decoupled from the energy-conservation equation, so one only needs to solve for the first two equations.[42]
• Compressible Euler equations (EE): Start with the C-NS. Assume a frictionless flow with no diffusive heat flux.[44]
• Weakly compressible Navier-Stokes equations (WC-NS): Start with the C-NS. Assume that density variations depend only on temperature and not on pressure.[45] For example, for an ideal gas, use $\rho =p_{0}/(RT)$, where $p_{0}$ is a conveniently-defined reference pressure that is always and everywhere constant, $\rho $ is density, $R$ is the specific gas constant, and $T$ is temperature. As a result, the WC-NS do not capture acoustic waves. It is also common in the WC-NS to neglect the pressure-work and viscous-heating terms in the energy-conservation equation. The WC-NS are also called the C-NS with the low-Mach-number approximation.
• Boussinesq equations: Start with the C-NS. Assume that density variations are always and everywhere negligible except in the gravity term of the momentum-conservation equation (where density multiplies the gravitational acceleration).[46] Also assume that various fluid properties such as viscosity, thermal conductivity, and heat capacity are always and everywhere constant. The Boussinesq equations are widely used in microscale meteorology.
• Compressible Reynolds-averaged Navier–Stokes equations and compressible Favre-averaged Navier-Stokes equations (C-RANS and C-FANS): Start with the C-NS. Assume that any flow variable $f$, such as density, velocity and pressure, can be represented as $f=F+f''$, where $F$ is the ensemble-average[42] of any flow variable, and $f''$ is a perturbation or fluctuation from this average.[42][47] $f''$ is not necessarily small. If $F$ is a classic ensemble-average (see Reynolds decomposition) one obtains the Reynolds-averaged Navier–Stokes equations. And if $F$ is a density-weighted ensemble-average one obtains the Favre-averaged Navier-Stokes equations.[47] As a result, and depending on the Reynolds number, the range of scales of motion is greatly reduced, something which leads to much faster solutions in comparison to solving the C-NS. However, information is lost, and the resulting system of equations requires the closure of various unclosed terms, notably the Reynolds stress.
• Ideal flow or potential flow equations: Start with the EE. Assume zero fluid-particle rotation (zero vorticity) and zero flow expansion (zero divergence).[42] The resulting flowfield is entirely determined by the geometrical boundaries.[42] Ideal flows can be useful in modern CFD to initialize simulations.
• Linearized compressible Euler equations (LEE):[48] Start with the EE. Assume that any flow variable $f$, such as density, velocity and pressure, can be represented as $f=f_{0}+f'$, where $f_{0}$ is the value of the flow variable at some reference or base state, and $f'$ is a perturbation or fluctuation from this state. Furthermore, assume that this perturbation $f'$ is very small in comparison with some reference value. Finally, assume that $f_{0}$ satisfies "its own" equation, such as the EE. The LEE and its many variations are widely used in computational aeroacoustics.
• Sound wave or acoustic wave equation: Start with the LEE. Neglect all gradients of $f_{0}$ and $f'$, and assume that the Mach number at the reference or base state is very small.[45] The resulting equations for density, momentum and energy can be manipulated into a pressure equation, giving the well-known sound wave equation.
• Shallow water equations (SW): Consider a flow near a wall where the wall-parallel length-scale of interest is much larger than the wall-normal length-scale of interest. Start with the EE. Assume that density is always and everywhere constant, neglect the velocity component perpendicular to the wall, and consider the velocity parallel to the wall to be spatially-constant.
• Boundary layer equations (BL): Start with the C-NS (I-NS) for compressible (incompressible) boundary layers. Assume that there are thin regions next to walls where spatial gradients perpendicular to the wall are much larger than those parallel to the wall.[46]
• Bernoulli equation: Start with the EE. Assume that density variations depend only on pressure variations.[46] See Bernoulli's Principle.
• Steady Bernoulli equation: Start with the Bernoulli Equation and assume a steady flow.[46] Or start with the EE and assume that the flow is steady and integrate the resulting equation along a streamline.[44][43]
• Stokes Flow or creeping flow equations: Start with the C-NS or I-NS. Neglect the inertia of the flow.[42][43] Such an assumption can be justified when the Reynolds number is very low. As a result, the resulting set of equations is linear, something which simplifies greatly their solution.
• Two-dimensional channel flow equation: Consider the flow between two infinite parallel plates. Start with the C-NS. Assume that the flow is steady, two-dimensional, and fully developed (i.e., the velocity profile does not change along the streamwise direction).[42] Note that this widely-used fully-developed assumption can be inadequate in some instances, such as some compressible, microchannel flows, in which case it can be supplanted by a locally fully-developed assumption.[49]
• One-dimensional Euler equations or one-dimensional gas-dynamic equations (1D-EE): Start with the EE. Assume that all flow quantities depend only on one spatial dimension.[50]
• Fanno flow equation: Consider the flow inside a duct with constant area and adiabatic walls. Start with the 1D-EE. Assume a steady flow, no gravity effects, and introduce in the momentum-conservation equation an empirical term to recover the effect of wall friction (neglected in the EE). To close the Fanno flow equation, a model for this friction term is needed. Such a closure involves problem-dependent assumptions.[51]
• Rayleigh flow equation. Consider the flow inside a duct with constant area and either non-adiabatic walls without volumetric heat sources or adiabatic walls with volumetric heat sources. Start with the 1D-EE. Assume a steady flow, no gravity effects, and introduce in the energy-conservation equation an empirical term to recover the effect of wall heat transfer or the effect of the heat sources (neglected in the EE).
Methodology
In all of these approaches the same basic procedure is followed.
• During preprocessing
• The geometry and physical bounds of the problem can be defined using computer aided design (CAD). From there, data can be suitably processed (cleaned-up) and the fluid volume (or fluid domain) is extracted.
• The volume occupied by the fluid is divided into discrete cells (the mesh). The mesh may be uniform or non-uniform, structured or unstructured, consisting of a combination of hexahedral, tetrahedral, prismatic, pyramidal or polyhedral elements.
• The physical modeling is defined – for example, the equations of fluid motion + enthalpy + radiation + species conservation
• Boundary conditions are defined. This involves specifying the fluid behaviour and properties at all bounding surfaces of the fluid domain. For transient problems, the initial conditions are also defined.
• The simulation is started and the equations are solved iteratively as a steady-state or transient.
• Finally a postprocessor is used for the analysis and visualization of the resulting solution.
Discretization methods
The stability of the selected discretisation is generally established numerically rather than analytically as with simple linear problems. Special care must also be taken to ensure that the discretisation handles discontinuous solutions gracefully. The Euler equations and Navier–Stokes equations both admit shocks, and contact surfaces.
Some of the discretization methods being used are:
Finite volume method
Main article: Finite volume method
The finite volume method (FVM) is a common approach used in CFD codes, as it has an advantage in memory usage and solution speed, especially for large problems, high Reynolds number turbulent flows, and source term dominated flows (like combustion).[52]
In the finite volume method, the governing partial differential equations (typically the Navier-Stokes equations, the mass and energy conservation equations, and the turbulence equations) are recast in a conservative form, and then solved over discrete control volumes. This discretization guarantees the conservation of fluxes through a particular control volume. The finite volume equation yields governing equations in the form,
${\frac {\partial }{\partial t}}\iiint Q\,dV+\iint F\,d\mathbf {A} =0,$
where $Q$ is the vector of conserved variables, $F$ is the vector of fluxes (see Euler equations or Navier–Stokes equations), $V$ is the volume of the control volume element, and $\mathbf {A} $ is the surface area of the control volume element.
Finite element method
Main article: Finite element method
The finite element method (FEM) is used in structural analysis of solids, but is also applicable to fluids. However, the FEM formulation requires special care to ensure a conservative solution. The FEM formulation has been adapted for use with fluid dynamics governing equations.[53][54] Although FEM must be carefully formulated to be conservative, it is much more stable than the finite volume approach.[55] However, FEM can require more memory and has slower solution times than the FVM.[56]
In this method, a weighted residual equation is formed:
$R_{i}=\iiint W_{i}Q\,dV^{e}$
where $R_{i}$ is the equation residual at an element vertex $i$, $Q$ is the conservation equation expressed on an element basis, $W_{i}$ is the weight factor, and $V^{e}$ is the volume of the element.
Finite difference method
Main article: Finite difference method
The finite difference method (FDM) has historical importance[54] and is simple to program. It is currently only used in few specialized codes, which handle complex geometry with high accuracy and efficiency by using embedded boundaries or overlapping grids (with the solution interpolated across each grid).
${\frac {\partial Q}{\partial t}}+{\frac {\partial F}{\partial x}}+{\frac {\partial G}{\partial y}}+{\frac {\partial H}{\partial z}}=0$
where $Q$ is the vector of conserved variables, and $F$, $G$, and $H$ are the fluxes in the $x$, $y$, and $z$ directions respectively.
Spectral element method
Main article: Spectral element method
Spectral element method is a finite element type method. It requires the mathematical problem (the partial differential equation) to be cast in a weak formulation. This is typically done by multiplying the differential equation by an arbitrary test function and integrating over the whole domain. Purely mathematically, the test functions are completely arbitrary - they belong to an infinite-dimensional function space. Clearly an infinite-dimensional function space cannot be represented on a discrete spectral element mesh; this is where the spectral element discretization begins. The most crucial thing is the choice of interpolating and testing functions. In a standard, low order FEM in 2D, for quadrilateral elements the most typical choice is the bilinear test or interpolating function of the form $v(x,y)=ax+by+cxy+d$. In a spectral element method however, the interpolating and test functions are chosen to be polynomials of a very high order (typically e.g. of the 10th order in CFD applications). This guarantees the rapid convergence of the method. Furthermore, very efficient integration procedures must be used, since the number of integrations to be performed in numerical codes is big. Thus, high order Gauss integration quadratures are employed, since they achieve the highest accuracy with the smallest number of computations to be carried out. At the time there are some academic CFD codes based on the spectral element method and some more are currently under development, since the new time-stepping schemes arise in the scientific world.
Lattice Boltzmann method
The lattice Boltzmann method (LBM) with its simplified kinetic picture on a lattice provides a computationally efficient description of hydrodynamics. Unlike the traditional CFD methods, which solve the conservation equations of macroscopic properties (i.e., mass, momentum, and energy) numerically, LBM models the fluid consisting of fictive particles, and such particles perform consecutive propagation and collision processes over a discrete lattice mesh. In this method, one works with the discrete in space and time version of the kinetic evolution equation in the Boltzmann Bhatnagar-Gross-Krook (BGK) form.
Vortex method
The vortex method, also Lagrangian Vortex Particle Method, is a meshfree technique for the simulation of incompressible turbulent flows. In it, vorticity is discretized onto Lagrangian particles, these computational elements being called vortices, vortons, or vortex particles.[57] Vortex methods were developed as a grid-free methodology that would not be limited by the fundamental smoothing effects associated with grid-based methods. To be practical, however, vortex methods require means for rapidly computing velocities from the vortex elements – in other words they require the solution to a particular form of the N-body problem (in which the motion of N objects is tied to their mutual influences). This breakthrough came in the 1980s with the development of the Barnes-Hut and fast multipole method (FMM) algorithms. These paved the way to practical computation of the velocities from the vortex elements.
Software based on the vortex method offer a new means for solving tough fluid dynamics problems with minimal user intervention. All that is required is specification of problem geometry and setting of boundary and initial conditions. Among the significant advantages of this modern technology;
• It is practically grid-free, thus eliminating numerous iterations associated with RANS and LES.
• All problems are treated identically. No modeling or calibration inputs are required.
• Time-series simulations, which are crucial for correct analysis of acoustics, are possible.
• The small scale and large scale are accurately simulated at the same time.
Boundary element method
Main article: Boundary element method
In the boundary element method, the boundary occupied by the fluid is divided into a surface mesh.
High-resolution discretization schemes
High-resolution schemes are used where shocks or discontinuities are present. Capturing sharp changes in the solution requires the use of second or higher-order numerical schemes that do not introduce spurious oscillations. This usually necessitates the application of flux limiters to ensure that the solution is total variation diminishing.
Turbulence models
In computational modeling of turbulent flows, one common objective is to obtain a model that can predict quantities of interest, such as fluid velocity, for use in engineering designs of the system being modeled. For turbulent flows, the range of length scales and complexity of phenomena involved in turbulence make most modeling approaches prohibitively expensive; the resolution required to resolve all scales involved in turbulence is beyond what is computationally possible. The primary approach in such cases is to create numerical models to approximate unresolved phenomena. This section lists some commonly used computational models for turbulent flows.
Turbulence models can be classified based on computational expense, which corresponds to the range of scales that are modeled versus resolved (the more turbulent scales that are resolved, the finer the resolution of the simulation, and therefore the higher the computational cost). If a majority or all of the turbulent scales are not modeled, the computational cost is very low, but the tradeoff comes in the form of decreased accuracy.
In addition to the wide range of length and time scales and the associated computational cost, the governing equations of fluid dynamics contain a non-linear convection term and a non-linear and non-local pressure gradient term. These nonlinear equations must be solved numerically with the appropriate boundary and initial conditions.
Reynolds-averaged Navier–Stokes
Reynolds-averaged Navier–Stokes (RANS) equations are the oldest approach to turbulence modeling. An ensemble version of the governing equations is solved, which introduces new apparent stresses known as Reynolds stresses. This adds a second order tensor of unknowns for which various models can provide different levels of closure. It is a common misconception that the RANS equations do not apply to flows with a time-varying mean flow because these equations are 'time-averaged'. In fact, statistically unsteady (or non-stationary) flows can equally be treated. This is sometimes referred to as URANS. There is nothing inherent in Reynolds averaging to preclude this, but the turbulence models used to close the equations are valid only as long as the time over which these changes in the mean occur is large compared to the time scales of the turbulent motion containing most of the energy.
RANS models can be divided into two broad approaches:
Boussinesq hypothesis
This method involves using an algebraic equation for the Reynolds stresses which include determining the turbulent viscosity, and depending on the level of sophistication of the model, solving transport equations for determining the turbulent kinetic energy and dissipation. Models include k-ε (Launder and Spalding),[58] Mixing Length Model (Prandtl),[59] and Zero Equation Model (Cebeci and Smith).[59] The models available in this approach are often referred to by the number of transport equations associated with the method. For example, the Mixing Length model is a "Zero Equation" model because no transport equations are solved; the $k-\epsilon $ is a "Two Equation" model because two transport equations (one for $k$ and one for $\epsilon $) are solved.
Reynolds stress model (RSM)
This approach attempts to actually solve transport equations for the Reynolds stresses. This means introduction of several transport equations for all the Reynolds stresses and hence this approach is much more costly in CPU effort.
Large eddy simulation
Large eddy simulation (LES) is a technique in which the smallest scales of the flow are removed through a filtering operation, and their effect modeled using subgrid scale models. This allows the largest and most important scales of the turbulence to be resolved, while greatly reducing the computational cost incurred by the smallest scales. This method requires greater computational resources than RANS methods, but is far cheaper than DNS.
Detached eddy simulation
Detached eddy simulations (DES) is a modification of a RANS model in which the model switches to a subgrid scale formulation in regions fine enough for LES calculations. Regions near solid boundaries and where the turbulent length scale is less than the maximum grid dimension are assigned the RANS mode of solution. As the turbulent length scale exceeds the grid dimension, the regions are solved using the LES mode. Therefore, the grid resolution for DES is not as demanding as pure LES, thereby considerably cutting down the cost of the computation. Though DES was initially formulated for the Spalart-Allmaras model (Spalart et al., 1997), it can be implemented with other RANS models (Strelets, 2001), by appropriately modifying the length scale which is explicitly or implicitly involved in the RANS model. So while Spalart–Allmaras model based DES acts as LES with a wall model, DES based on other models (like two equation models) behave as a hybrid RANS-LES model. Grid generation is more complicated than for a simple RANS or LES case due to the RANS-LES switch. DES is a non-zonal approach and provides a single smooth velocity field across the RANS and the LES regions of the solutions.
Direct numerical simulation
Direct numerical simulation (DNS) resolves the entire range of turbulent length scales. This marginalizes the effect of models, but is extremely expensive. The computational cost is proportional to $Re^{3}$.[60] DNS is intractable for flows with complex geometries or flow configurations.
Coherent vortex simulation
The coherent vortex simulation approach decomposes the turbulent flow field into a coherent part, consisting of organized vortical motion, and the incoherent part, which is the random background flow.[61] This decomposition is done using wavelet filtering. The approach has much in common with LES, since it uses decomposition and resolves only the filtered portion, but different in that it does not use a linear, low-pass filter. Instead, the filtering operation is based on wavelets, and the filter can be adapted as the flow field evolves. Farge and Schneider tested the CVS method with two flow configurations and showed that the coherent portion of the flow exhibited the $-{\frac {40}{39}}$ energy spectrum exhibited by the total flow, and corresponded to coherent structures (vortex tubes), while the incoherent parts of the flow composed homogeneous background noise, which exhibited no organized structures. Goldstein and Vasilyev[62] applied the FDV model to large eddy simulation, but did not assume that the wavelet filter eliminated all coherent motions from the subfilter scales. By employing both LES and CVS filtering, they showed that the SFS dissipation was dominated by the SFS flow field's coherent portion.
PDF methods
Probability density function (PDF) methods for turbulence, first introduced by Lundgren,[63] are based on tracking the one-point PDF of the velocity, $f_{V}({\boldsymbol {v}};{\boldsymbol {x}},t)d{\boldsymbol {v}}$, which gives the probability of the velocity at point ${\boldsymbol {x}}$ being between ${\boldsymbol {v}}$ and ${\boldsymbol {v}}+d{\boldsymbol {v}}$. This approach is analogous to the kinetic theory of gases, in which the macroscopic properties of a gas are described by a large number of particles. PDF methods are unique in that they can be applied in the framework of a number of different turbulence models; the main differences occur in the form of the PDF transport equation. For example, in the context of large eddy simulation, the PDF becomes the filtered PDF.[64] PDF methods can also be used to describe chemical reactions,[65][66] and are particularly useful for simulating chemically reacting flows because the chemical source term is closed and does not require a model. The PDF is commonly tracked by using Lagrangian particle methods; when combined with large eddy simulation, this leads to a Langevin equation for subfilter particle evolution.
Vorticity confinement method
The vorticity confinement (VC) method is an Eulerian technique used in the simulation of turbulent wakes. It uses a solitary-wave like approach to produce a stable solution with no numerical spreading. VC can capture the small-scale features to within as few as 2 grid cells. Within these features, a nonlinear difference equation is solved as opposed to the finite difference equation. VC is similar to shock capturing methods, where conservation laws are satisfied, so that the essential integral quantities are accurately computed.
Linear eddy model
The Linear eddy model is a technique used to simulate the convective mixing that takes place in turbulent flow.[67] Specifically, it provides a mathematical way to describe the interactions of a scalar variable within the vector flow field. It is primarily used in one-dimensional representations of turbulent flow, since it can be applied across a wide range of length scales and Reynolds numbers. This model is generally used as a building block for more complicated flow representations, as it provides high resolution predictions that hold across a large range of flow conditions.
Two-phase flow
The modeling of two-phase flow is still under development. Different methods have been proposed, including the Volume of fluid method, the level-set method and front tracking.[68][69] These methods often involve a tradeoff between maintaining a sharp interface or conserving mass . This is crucial since the evaluation of the density, viscosity and surface tension is based on the values averaged over the interface.
Solution algorithms
Discretization in the space produces a system of ordinary differential equations for unsteady problems and algebraic equations for steady problems. Implicit or semi-implicit methods are generally used to integrate the ordinary differential equations, producing a system of (usually) nonlinear algebraic equations. Applying a Newton or Picard iteration produces a system of linear equations which is nonsymmetric in the presence of advection and indefinite in the presence of incompressibility. Such systems, particularly in 3D, are frequently too large for direct solvers, so iterative methods are used, either stationary methods such as successive overrelaxation or Krylov subspace methods. Krylov methods such as GMRES, typically used with preconditioning, operate by minimizing the residual over successive subspaces generated by the preconditioned operator.
Multigrid has the advantage of asymptotically optimal performance on many problems. Traditional solvers and preconditioners are effective at reducing high-frequency components of the residual, but low-frequency components typically require many iterations to reduce. By operating on multiple scales, multigrid reduces all components of the residual by similar factors, leading to a mesh-independent number of iterations.
For indefinite systems, preconditioners such as incomplete LU factorization, additive Schwarz, and multigrid perform poorly or fail entirely, so the problem structure must be used for effective preconditioning.[70] Methods commonly used in CFD are the SIMPLE and Uzawa algorithms which exhibit mesh-dependent convergence rates, but recent advances based on block LU factorization combined with multigrid for the resulting definite systems have led to preconditioners that deliver mesh-independent convergence rates.[71]
Unsteady aerodynamics
CFD made a major break through in late 70s with the introduction of LTRAN2, a 2-D code to model oscillating airfoils based on transonic small perturbation theory by Ballhaus and associates.[72] It uses a Murman-Cole switch algorithm for modeling the moving shock-waves.[73] Later it was extended to 3-D with use of a rotated difference scheme by AFWAL/Boeing that resulted in LTRAN3.[74][75]
Biomedical engineering
CFD investigations are used to clarify the characteristics of aortic flow in details that are beyond the capabilities of experimental measurements. To analyze these conditions, CAD models of the human vascular system are extracted employing modern imaging techniques such as MRI or Computed Tomography. A 3D model is reconstructed from this data and the fluid flow can be computed. Blood properties such as density and viscosity, and realistic boundary conditions (e.g. systemic pressure) have to be taken into consideration. Therefore, making it possible to analyze and optimize the flow in the cardiovascular system for different applications.[76]
CPU versus GPU
Traditionally, CFD simulations are performed on CPUs.[77]
In a more recent trend, simulations are also performed on GPUs. These typically contain slower but more processors. For CFD algorithms that feature good parallelism performance (i.e. good speed-up by adding more cores) this can greatly reduce simulation times. Fluid-implicit particle[78] and lattice-Boltzmann methods[79] are typical examples of codes that scale well on GPUs.
See also
• Blade element theory
• Boundary conditions in fluid dynamics
• Cavitation modelling
• Central differencing scheme
• Computational magnetohydrodynamics
• Discrete element method
• Finite element method
• Finite volume method for unsteady flow
• Fluid animation
• Immersed boundary method
• Lattice Boltzmann methods
• List of finite element software packages
• Meshfree methods
• Moving particle semi-implicit method
• Multi-particle collision dynamics
• Multidisciplinary design optimization
• Numerical methods in fluid mechanics
• Shape optimization
• Smoothed-particle hydrodynamics
• Stochastic Eulerian Lagrangian method
• Turbulence modeling
• Visualization (graphics)
• Wind tunnel
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Notes
• Anderson, John D. (1995). Computational Fluid Dynamics: The Basics With Applications. Science/Engineering/Math. McGraw-Hill Science. ISBN 978-0-07-001685-9.
• Patankar, Suhas (1980). Numerical Heat Transfer and Fluid Flow. Hemisphere Series on Computational Methods in Mechanics and Thermal Science. Taylor & Francis. ISBN 978-0-89116-522-4.
External links
• Course: Introduction to CFD – Dmitri Kuzmin (Dortmund University of Technology)
• Course: Computational Fluid Dynamics – Suman Chakraborty (Indian Institute of Technology Kharagpur)
• Course: Numerical PDE Techniques for Scientists and Engineers, Open access Lectures and Codes for Numerical PDEs, including a modern view of Compressible CFD
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| Wikipedia |
Voter model
In the mathematical theory of probability, the voter model is an interacting particle system introduced by Richard A. Holley and Thomas M. Liggett in 1975.[1]
One can imagine that there is a "voter" at each point on a connected graph, where the connections indicate that there is some form of interaction between a pair of voters (nodes). The opinions of any given voter on some issue changes at random times under the influence of opinions of his neighbours. A voter's opinion at any given time can take one of two values, labelled 0 and 1. At random times, a random individual is selected and that voter's opinion is changed according to a stochastic rule. Specifically, one of the chosen voter's neighbors is chosen according to a given set of probabilities and that neighbor’s opinion is transferred to the chosen voter.
An alternative interpretation is in terms of spatial conflict. Suppose two nations control the areas (sets of nodes) labelled 0 or 1. A flip from 0 to 1 at a given location indicates an invasion of that site by the other nation.
Note that only one flip happens each time. Problems involving the voter model will often be recast in terms of the dual system of coalescing Markov chains. Frequently, these problems will then be reduced to others involving independent Markov chains.
Definition
A voter model is a (continuous time) Markov process $\eta _{t}$ with state space $S=\{0,1\}^{Z^{d}}$ and transition rates function $c(x,\eta )$, where $Z^{d}$ is a d-dimensional integer lattice, and $c($•,•$)$ is assumed to be nonnegative, uniformly bounded and continuous as a function of $\eta $ in the product topology on $S$. Each component $\eta \in S$ is called a configuration. To make it clear that $\eta (x)$ stands for the value of a site x in configuration $\eta (.)$; while $\eta _{t}(x)$ means the value of a site x in configuration $\eta (.)$ at time $t$.
The dynamic of the process are specified by the collection of transition rates. For voter models, the rate at which there is a flip at $\scriptstyle x$ from 0 to 1 or vice versa is given by a function $c(x,\eta )$ of site $x$. It has the following properties:
1. $c(x,\eta )=0$ for every $x\in Z^{d}$ if $\eta \equiv 0$ or if $\eta \equiv 1$
2. $c(x,\eta )=c(x,\zeta )$ for every $x\in Z^{d}$ if $\eta (y)+\zeta (y)=1$ for all $y\in Z^{d}$
3. $c(x,\eta )\leq c(x,\zeta )$ if $\eta \leq \zeta $ and $\eta (x)=\zeta (x)=0$
4. $c(x,\eta )$ is invariant under shifts in $\scriptstyle Z^{d}$
Property (1) says that $\eta \equiv 0$ and $\eta \equiv 1$ are fixed points for the evolution. (2) indicates that the evolution is unchanged by interchanging the roles of 0's and 1's. In property (3), $\eta \leq \zeta $ means $\forall x,\eta (x)\leq \zeta (x)$, and $\eta \leq \zeta $ implies $c(x,\eta )\leq c(x,\zeta )$ if $\eta (x)=\zeta (x)=0$, and implies $c(x,\eta )\geq c(x,\zeta )$ if $\eta (x)=\zeta (x)=1$.
Clustering and coexistence
The interest in is the limiting behavior of the models. Since the flip rates of a site depends on its neighbours, it is obvious that when all sites take the same value, the whole system stops changing forever. Therefore, a voter model has two trivial extremal stationary distributions, the point-masses $\scriptstyle \delta _{0}$ and $\scriptstyle \delta _{1}$ on $\scriptstyle \eta \equiv 0$ or $\scriptstyle \eta \equiv 1$ respectively, which represent consensus. The main question to be discussed is whether or not there are others, which would then represent coexistence of different opinions in equilibrium. It is said coexistence occurs if there is a stationary distribution that concentrates on configurations with infinitely many 0's and 1's. On the other hand, if for all $\scriptstyle x,y\in Z^{d}$ and all initial configurations, then
$\lim _{t\rightarrow \infty }P[\eta _{t}(x)\neq \eta _{t}(y)]=0$
It is said that clustering occurs.
It is important to distinguish clustering with the concept of cluster. Clusters are defined to be the connected components of $\scriptstyle \{x:\eta (x)=0\}$ or $\scriptstyle \{x:\eta (x)=1\}$.
The linear voter model
Model description
This section will be dedicated to one of the basic voter models, the Linear Voter Model.
If $\scriptstyle p($•,•$\scriptstyle )$ be the transition probabilities for an irreducible random walk on $\scriptstyle Z^{d}$, then:
$p(x,y)\geq 0\quad {\text{and}}\sum _{y}p(x,y)=1$
Then in Linear voter model, the transition rates are linear functions of $\scriptstyle \eta $:
$c(x,\eta )=\left\{{\begin{array}{l}\sum _{y}p(x,y)\eta (y)\quad {\text{for all}}\quad \eta (x)=0\\\sum _{y}p(x,y)(1-\eta (y))\quad {\text{for all}}\quad \eta (x)=1\\\end{array}}\right.$
Or if $\scriptstyle \eta _{x}$ indicates that a flip happens at $\scriptstyle x$, then transition rates are simply:
$\eta \rightarrow \eta _{x}\quad {\text{at rate}}\sum _{y:\eta (y)\neq \eta (x)}p(x,y).$
A process of coalescing random walks $\scriptstyle A_{t}\subset Z^{d}$ is defined as follows. Here $\scriptstyle A_{t}$ denotes the set of sites occupied by these random walks at time $\scriptstyle t$. To define $\scriptstyle A_{t}$, consider several (continuous time) random walks on $\scriptstyle Z^{d}$ with unit exponential holding times and transition probabilities $\scriptstyle p($•,•$\scriptstyle )$, and take them to be independent until two of them meet. At that time, the two that meet coalesce into one particle, which continues to move like a random walk with transition probabilities $\scriptstyle p($•,•$\scriptstyle )$ .
The concept of Duality is essential for analysing the behavior of the voter models. The linear voter models satisfy a very useful form of duality, known as coalescing duality, which is:
$P^{\eta }(\eta _{t}\equiv 1\quad {\text{on }}A)=P^{A}(\eta (A_{t})\equiv 1),$
where $\scriptstyle \eta \in \{0,1\}^{Z^{d}}$ is the initial configuration of $\scriptstyle \eta _{t}$ and $\scriptstyle A=\{x\in Z^{d},\eta (x)=1\}\subset Z^{d}$ is the initial state of the coalescing random walks $\scriptstyle A_{t}$.
Limiting behaviors of linear voter models
Let $\scriptstyle p(x,y)$ be the transition probabilities for an irreducible random walk on $\scriptstyle Z^{d}$ and $\scriptstyle p(x,y)=p(0,x-y)$, then the duality relation for such linear voter models says that $\scriptstyle \forall \eta \in S=\{0,1\}^{Z^{d}}$
$P^{\eta }[\eta _{t}(x)\neq \eta _{t}(y)]=P[\eta (X_{t})\neq \eta (Y_{t})]$
where $\scriptstyle X_{t}$ and $\scriptstyle Y_{t}$ are (continuous time) random walks on $\scriptstyle Z^{d}$ with $\scriptstyle X_{0}=x$, $\scriptstyle Y_{0}=y$, and $\scriptstyle \eta (X_{t})$ is the position taken by the random walk at time $\scriptstyle t$. $\scriptstyle X_{t}$ and $\scriptstyle Y_{t}$ forms a coalescing random walks described at the end of section 2.1. $\scriptstyle X(t)-Y(t)$ is a symmetrized random walk. If $\scriptstyle X(t)-Y(t)$ is recurrent and $\scriptstyle d\leq 2$, $\scriptstyle X_{t}$ and $\scriptstyle Y_{t}$ will hit eventually with probability 1, and hence
$P^{\eta }[\eta _{t}(x)\neq \eta _{t}(y)]=P[\eta (X_{t})\neq \eta (Y_{t})]\leq P[X_{t}\neq Y_{t}]\rightarrow 0\quad {\text{as}}\quad t\to 0$
Therefore, the process clusters.
On the other hand, when $d\geq 3$, the system coexists. It is because for $\scriptstyle d\geq 3$, $\scriptstyle X(t)-Y(t)$ is transient, thus there is a positive probability that the random walks never hit, and hence for $\scriptstyle x\neq y$
$\lim _{t\rightarrow \infty }P[\eta _{t}(x)\neq \eta _{t}(y)]=C\lim _{t\rightarrow \infty }P[X_{t}\neq Y_{t}]>0$
for some constant $C$ corresponding to the initial distribution.
If $\scriptstyle {\tilde {X}}(t)=X(t)-Y(t)$ be a symmetrized random walk, then there are the following theorems:
Theorem 2.1
The linear voter model $\scriptstyle \eta _{t}$ clusters if $\scriptstyle {\tilde {X}}_{t}$ is recurrent, and coexists if $\scriptstyle {\tilde {X}}_{t}$ is transient. In particular,
1. the process clusters if $\scriptstyle d=1$ and $\scriptstyle \sum _{x}|x|p(0,x)\leq \infty $, or if $\scriptstyle d=2$ and $\scriptstyle \sum _{x}|x|^{2}p(0,x)\leq \infty $;
2. the process coexists if $\scriptstyle d\geq 3$.
Remarks: To contrast this with the behavior of the threshold voter models that will be discussed in next section, note that whether the linear voter model clusters or coexists depends almost exclusively on the dimension of the set of sites, rather than on the size of the range of interaction.
Theorem 2.2 Suppose $\scriptstyle \mu $ is any translation spatially ergodic and invariant probability measure on the state space $\scriptstyle S=\{0,1\}^{Z^{d}}$, then
1. If $\scriptstyle {\tilde {X}}_{t}$ is recurrent, then $\scriptstyle \mu S(t)\Rightarrow \rho \delta _{1}+(1-\rho )\delta _{0}\quad {\text{as}}\quad t\to \infty $;
2. If $\scriptstyle {\tilde {X}}_{t}$ is transient, then $\scriptstyle \mu S(t)\Rightarrow \mu _{\rho }$.
where $\scriptstyle \mu S(t)$ is the distribution of $\scriptstyle \eta _{t}$; $\scriptstyle \Rightarrow $ means weak convergence, $\scriptstyle \mu _{\rho }$ is a nontrivial extremal invariant measure and $\scriptstyle \rho =\mu (\{\eta :\eta (x)=1\})$ :\eta (x)=1\})} .
A special linear voter model
One of the interesting special cases of the linear voter model, known as the basic linear voter model, is that for state space $\scriptstyle \{0,1\}^{Z^{d}}$:
$p(x,y)={\begin{cases}1/2d&{\text{if }}|x-y|=1{\text{ and }}\eta (x)\neq \eta (y)\\[8pt]0&{\text{otherwise}}\end{cases}}$
So that
$\eta _{t}(x)\to 1-\eta _{t}(x)\quad {\text{at rate}}\quad (2d)^{-1}|\{y:|y-x|=1,\eta _{t}(y)\neq \eta _{t}(x)\}|$
In this case, the process clusters if $\scriptstyle d\leq 2$, while coexists if $\scriptstyle d\geq 3$. This dichotomy is closely related to the fact that simple random walk on $\scriptstyle Z^{d}$ is recurrent if $\scriptstyle d\leq 2$ and transient if $\scriptstyle d\geq 3$.
Clusters in one dimension d = 1
For the special case with $\scriptstyle d=1$, $\scriptstyle S=Z^{1}$ and $\scriptstyle p(x,x+1)=p(x,x-1)={\frac {1}{2}}$ for each $\scriptstyle x$. From Theorem 2.2, $\scriptstyle \mu S(t)\Rightarrow \rho \delta _{1}+(1-\rho )\delta _{0}$, thus clustering occurs in this case. The aim of this section is to give a more precise description of this clustering.
As mentioned before, clusters of an $\scriptstyle \eta $ are defined to be the connected components of $\scriptstyle \{x:\eta (x)=0\}$ or $\scriptstyle \{x:\eta (x)=1\}$. The mean cluster size for $\scriptstyle \eta $ is defined to be:
$C(\eta )=\lim _{n\rightarrow \infty }{\frac {2n}{{\text{number of clusters in}}[-n,n]}}$
provided the limit exists.
Proposition 2.3
Suppose the voter model is with initial distribution $\scriptstyle \mu $ and $\scriptstyle \mu $ is a translation invariant probability measure, then
$P\left(C(\eta )={\frac {1}{P[\eta _{t}(0)\neq \eta _{t}(1)]}}\right)=1.$
Occupation time
Define the occupation time functionals of the basic linear voter model as:
$T_{t}^{x}=\int _{0}^{t}\eta _{s}^{\rho }(x)\mathrm {d} s.$
Theorem 2.4
Assume that for all site x and time t, $\scriptstyle P(\eta _{t}(x)=1)=\rho $, then as $\scriptstyle t\rightarrow \infty $, $\scriptstyle T_{t}^{x}/t\rightarrow \rho $ almost surely if $\scriptstyle d\geq 2$
proof
By Chebyshev's inequality and the Borel–Cantelli lemma, there is the equation below:
$P\left({\frac {\rho }{r}}\leq \lim \inf _{t\rightarrow \infty }{\frac {T_{t}}{t}}\leq \lim \sup _{t\rightarrow \infty }{\frac {T_{t}}{t}}\leq \rho r\right)=1;\quad \forall r>1$
The theorem follows when letting $\scriptstyle r\searrow 1$.
The threshold voter model
Model description
This section, concentrates on a kind of non-linear voter models, known as the threshold voter model. To define it, let $\scriptstyle {\mathcal {N}}$ be a neighbourhood of $\scriptstyle 0\in Z^{d}$ that is obtained by intersecting $\scriptstyle Z^{d}$ with any compact, convex, symmetric set in $\scriptstyle R^{d}$; in other word, $\scriptstyle {\mathcal {N}}$ is assumed to be a finite set that is symmetric with respect to all reflections and irreducible (i.e. the group it generates is $\scriptstyle Z^{d}$). It can always be assumed that $\scriptstyle {\mathcal {N}}$ contains all the unit vectors $\scriptstyle (1,0,0,\dots ,0),\dots ,(0,\dots ,0,1)$. For a positive integer $\scriptstyle T$, the threshold voter model with neighbourhood $\scriptstyle {\mathcal {N}}$ and threshold $\scriptstyle T$ is the one with rate function:
$c(x,\eta )=\left\{{\begin{array}{l}1\quad {\text{if}}\quad |\{y\in x+{\mathcal {N}}:\eta (y)\neq \eta (x)\}|\geq T\\0\quad {\text{otherwise}}\\\end{array}}\right.$
Simply put, the transition rate of site $\scriptstyle x$ is 1 if the number of sites that do not take the same value is larger or equal to the threshold T. Otherwise, site $\scriptstyle x$ stays at the current status and will not flip.
For example, if $\scriptstyle d=1$, $\scriptstyle {\mathcal {N}}=\{-1,0,1\}$ and $\scriptstyle T=2$, then the configuration $\scriptstyle \dots 1\quad 1\quad 0\quad 0\quad 1\quad 1\quad 0\quad 0\dots $ is an absorbing state or a trap for the process.
Limiting behaviors of threshold voter model
If a threshold voter model does not fixate, the process should be expected to will coexist for small threshold and cluster for large threshold, where large and small are interpreted as being relative to the size of the neighbourhood, $\scriptstyle |{\mathcal {N}}|$. The intuition is that having a small threshold makes it easy for flips to occur, so it is likely that there will be a lot of both 0's and 1's around at all times. The following are three major results:
1. If $\scriptstyle T>{\frac {|{\mathcal {N}}|-1}{2}}$, then the process fixates in the sense that each site flips only finitely often.
2. If $\scriptstyle d=1$ and $\scriptstyle T={\frac {|{\mathcal {N}}|-1}{2}}$, then the process clusters.
3. If $\scriptstyle T=\theta |{\mathcal {N}}|$ with $\scriptstyle \theta $ sufficiently small($\scriptstyle \theta <{\frac {1}{4}}$) and $\scriptstyle |{\mathcal {N}}|$ sufficiently large, then the process coexists.
Here are two theorems corresponding to properties (1) and (2).
Theorem 3.1
If $\scriptstyle T>{\frac {|{\mathcal {N}}|-1}{2}}$, then the process fixates.
Theorem 3.2
The threshold voter model in one dimension ($\scriptstyle d=1$) with $\scriptstyle {\mathcal {N}}=\{-T,\dots ,T\},T\geq 1$, clusters.
proof
The idea of the proof is to construct two sequences of random times $\scriptstyle U_{n}$, $\scriptstyle V_{n}$ for $\scriptstyle n\geq 1$ with the following properties:
1. $\scriptstyle 0=V_{0}<U_{1}<V_{1}<U_{2}<V_{2}<\dots $,
2. $\scriptstyle \{U_{k+1}-V_{k},k\geq 0\}$ are i.i.d.with $\scriptstyle \mathrm {E} (U_{k+1}-V_{k})<\infty $,
3. $\scriptstyle \{V_{k}-U_{k},k\geq 1\}$ are i.i.d.with $\scriptstyle \mathrm {E} (V_{k}-U_{k})=\infty $,
4. the random variables in (b) and (c) are independent of each other,
5. event A=$\scriptstyle \{\eta _{t}(.)$ is constant on $\scriptstyle \{-T,\dots ,T\}\}$, and event A holds for every $\scriptstyle t\in \cup _{k=1}^{\infty }[U_{k},V_{k}]$.
Once this construction is made, it will follow from renewal theory that
$P(A)\geq P(t\in \cup _{k=1}^{\infty }[U_{k},V_{k}])\to 1\quad {\text{as}}\quad t\to \infty $
Hence,$\scriptstyle \lim _{t\rightarrow \infty }P(\eta _{t}(1)\neq \eta _{t}(0))=0$, so that the process clusters.
Remarks: (a) Threshold models in higher dimensions do not necessarily cluster if $\scriptstyle T={\frac {|{\mathcal {N}}|-1}{2}}$. For example, take $\scriptstyle d=2,T=2$ and $\scriptstyle {\mathcal {N}}=\{(0,0),(0,1),(1,0),(0,-1),(-1,0)\}$. If $\scriptstyle \eta $ is constant on alternating vertical infinite strips, that is for all $\scriptstyle i,j$:
$\eta (4i,j)=\eta (4i+1,j)=1,\quad \eta (4i+2,j)=\eta (4i+3,j)=0$
then no transition ever occur, and the process fixates.
(b) Under the assumption of Theorem 3.2, the process does not fixate. To see this, consider the initial configuration $\scriptstyle \dots 000111\dots $, in which infinitely many zeros are followed by infinitely many ones. Then only the zero and one at the boundary can flip, so that the configuration will always look the same except that the boundary will move like a simple symmetric random walk. The fact that this random walk is recurrent implies that every site flips infinitely often.
Property 3 indicates that the threshold voter model is quite different from the linear voter model, in that coexistence occurs even in one dimension, provided that the neighbourhood is not too small. The threshold model has a drift toward the "local minority", which is not present in the linear case.
Most proofs of coexistence for threshold voter models are based on comparisons with hybrid model known as the threshold contact process with parameter $\scriptstyle \lambda >0$. This is the process on $\scriptstyle [0,1]^{Z^{d}}$ with flip rates:
$c(x,\eta )=\left\{{\begin{array}{l}\lambda \quad {\text{if}}\quad \eta (x)=0\quad {\text{and}}|\{y\in x+{\mathcal {N}}:\eta (y)=1\}|\geq T;\\1\quad {\text{if}}\quad \eta (x)=1;\\0\quad {\text{otherwise}}\end{array}}\right.$
Proposition 3.3
For any $\scriptstyle d,{\mathcal {N}}$ and $\scriptstyle T$, if the threshold contact process with $\scriptstyle \lambda =1$ has a nontrivial invariant measure, then the threshold voter model coexists.
Model with threshold T = 1
The case that $\scriptstyle T=1$ is of particular interest because it is the only case in which it is known exactly which models coexist and which models cluster.
In particular, there is interest in a kind of Threshold T=1 model with $\scriptstyle c(x,\eta )$ that is given by:
$c(x,\eta )=\left\{{\begin{array}{l}1\quad {\text{if exists one}}\quad y\quad {\text{with}}\quad |x-y|\leq N\quad {\text{and}}\quad \eta (x)\neq \eta (y)\\0\quad {\text{otherwise}}\\\end{array}}\right.$
$\scriptstyle N$ can be interpreted as the radius of the neighbourhood $\scriptstyle {\mathcal {N}}$; $\scriptstyle N$ determines the size of the neighbourhood (i.e., if $\scriptstyle {\mathcal {N}}_{1}=\{-2,-1,0,1,2\}$, then $\scriptstyle N_{1}=2$; while for $\scriptstyle {\mathcal {N}}_{2}=\{(0,0),(0,1),(1,0),(0,-1),(-1,0)\}$, the corresponding $\scriptstyle N_{2}=1$).
By Theorem 3.2, the model with $\scriptstyle d=1$ and $\scriptstyle {\mathcal {N}}=\{-1,0,1\}$ clusters. The following theorem indicates that for all other choices of $\scriptstyle d$ and $\scriptstyle {\mathcal {N}}$, the model coexists.
Theorem 3.4
Suppose that $\scriptstyle N\geq 1$, but $\scriptstyle (N,d)\neq (1,1)$. Then the threshold model on $\scriptstyle Z^{d}$ with parameter $\scriptstyle N$ coexists.
The proof of this theorem is given in a paper named "Coexistence in threshold voter models" by Thomas M. Liggett.
See also
• Probabilistic Cellular Automata
• sequential dynamical system
• contact process
Notes
1. Holley, Richard A.; Liggett, Thomas M. (1975). "Ergodic Theorems for Weakly Interacting Infinite Systems and the Voter Model". The Annals of Probability. 3 (4): 643–663. doi:10.1214/aop/1176996306. ISSN 0091-1798.
References
• Clifford, Peter; Aidan W Sudbury (1973). "A Model for Spatial Conflict". Biometrika. 60 (3): 581–588. doi:10.1093/biomet/60.3.581.
• Liggett, Thomas M. (1997). "Stochastic Models of Interacting Systems". The Annals of Probability. Institute of Mathematical Statistics. 25 (1): 1–29. doi:10.1214/aop/1024404276. ISSN 0091-1798.
• Liggett, Thomas M. (1994). "Coexistence in Threshold Voter Models". The Annals of Probability. 22 (2): 764–802. doi:10.1214/aop/1176988729.
• Cox, J. Theodore; David Griffeath (1983). "Occupation Time Limit Theorems for the Voter Model". The Annals of Probability. 11 (4): 876–893. doi:10.1214/aop/1176993438.
• Durrett, Richard; Kesten, Harry (1991). Random walks, Brownian motion, and interacting particle systems. ISBN 0817635092.
• Liggett, Thomas M. (1985). Interacting Particle Systems. New York: Springer Verlag. ISBN 0-387-96069-4.
• Thomas M. Liggett, "Stochastic Interacting Systems: Contact, Voter and Exclusion Processes", Springer-Verlag, 1999.
| Wikipedia |
Electoral system
An electoral system or voting system is a set of rules that determine how elections and referendums are conducted and how their results are determined. Electoral systems are used in politics to elect governments, while non-political elections may take place in business, non-profit organisations and informal organisations. These rules govern all aspects of the voting process: when elections occur, who is allowed to vote, who can stand as a candidate, how ballots are marked and cast, how the ballots are counted, how votes translate into the election outcome, limits on campaign spending, and other factors that can affect the result. Political electoral systems are defined by constitutions and electoral laws, are typically conducted by election commissions, and can use multiple types of elections for different offices.
Part of the Politics series
Electoral systems
Single-winner/majoritarian
Plurality
• First-past-the-post
• Plurality at-large (plurality block voting)
• General ticket (party block voting)
Multi-round voting
• Two-round
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• Nonpartisan
• unified
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Ranked / preferential systems
• Instant-runoff (alternative vote)
• Contingent vote
• Coombs' method
• Condorcet methods (Copeland's, Dodgson's, Kemeny–Young, Minimax, Nanson's, ranked pairs, Schulze, Alternative Smith)
• Positional voting (Borda count, Nauru/Dowdall method)
• Bucklin voting
• Oklahoma primary electoral system
• Preferential block voting
Cardinal / graded systems
• Score voting
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• Combined approval voting
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Proportional representation
Party-list
• Electoral list
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• Sainte-Laguë
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Proportional forms of ranked voting
• Single transferable vote
• Gregory
• Wright
• Schulze STV
• CPO-STV
• Ranked party list PR
• Spare vote
Proportional forms of cardinal voting
• Proportional approval voting
• Sequential proportional approval voting
• Method of Equal Shares
• Phragmen's voting rules
Biproportional apportionment
Fair majority voting
Weighted voting
• Direct representation
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Mixed systems
By type of representation
• Mixed-member majoritarian
• Mixed-member proportional
Non-compensatory mixed systems
• Parallel voting
• Majority bonus
Compensatory mixed systems
• Additional member system
• Mixed single vote (positive vote transfer)
• Scorporo (negative vote transfer)
• Mixed ballot transferable vote
• Alternative Vote Plus
• Dual-member proportional
• Rural–urban proportional
Other systems and related theory
Semi-proportional representation
• Single non-transferable vote
• Limited voting
• Cumulative voting
• Binomial voting
Other systems
• Multiple non-transferable vote
• Double simultaneous vote
• Proxy voting
• Delegated voting
• Indirect STV
• Liquid democracy
• Random selection (sortition, random ballot)
Social choice theory
• Arrow's theorem
• Gibbard–Satterthwaite theorem
• Public choice theory
List of electoral systems
• List of electoral systems by country
• Comparison of electoral systems
Politics portal
Some electoral systems elect a single winner to a unique position, such as prime minister, president or governor, while others elect multiple winners, such as members of parliament or boards of directors. When electing a legislature, areas may be divided into constituencies with one or more representatives. Voters may vote directly for individual candidates or for a list of candidates put forward by a political party or alliance. There are many variations in electoral systems, with the most common systems being first-past-the-post voting, block voting, the two-round (runoff) system, proportional representation and ranked voting. Some electoral systems, such as mixed systems, attempt to combine the benefits of non-proportional and proportional systems.
The study of formally defined electoral methods is called social choice theory or voting theory, and this study can take place within the field of political science, economics, or mathematics, and specifically within the subfields of game theory and mechanism design. Impossibility proofs such as Arrow's impossibility theorem demonstrate that when voters have three or more alternatives, no preferential voting system can guarantee the race between two candidates remains unaffected when an irrelevant candidate participates or drops out of the election.
Types of electoral systems
Plurality systems
Plurality voting is a system in which the candidate(s) with the highest number of votes wins, with no requirement to get a majority of votes. In cases where there is a single position to be filled, it is known as first-past-the-post; this is the second most common electoral system for national legislatures, with 58 countries using it for this purpose,[1] the vast majority of which are current or former British or American colonies or territories. It is also the second most common system used for presidential elections, being used in 19 countries.[1]
In cases where there are multiple positions to be filled, most commonly in cases of multi-member constituencies, plurality voting is referred to as block voting, multiple non-transferable vote or plurality-at-large.[1] This takes two main forms: in one form voters have as many votes as there are seats and can vote for any candidate, regardless of party – this is used in eight countries.[1] There are variations on this system such as limited voting, where voters are given fewer votes than there are seats to be filled (Gibraltar is the only territory where this system is in use)[1] and single non-transferable vote (SNTV), in which voters can vote for only one candidate in a multi-member constituency, with the candidates receiving the most votes declared the winners; this system is used in Kuwait, the Pitcairn Islands and Vanuatu.[1] In the other main form of block voting, also known as party block voting, voters can only vote for the multiple candidates of a single party, with the party receiving the most votes winning all contested positions. This is used in five countries as part of mixed systems.[1]
The Dowdall system, a multi-member constituency variation on the Borda count, is used in Nauru for parliamentary elections and sees voters rank the candidates depending on how many seats there are in their constituency. First preference votes are counted as whole numbers; the second preference votes divided by two, third preferences by three; this continues to the lowest possible ranking.[2] The totals achieved by each candidate determine the winners.[3]
Majority systems
Majority voting is a system in which candidates must receive a majority of votes to be elected, either in a runoff election or final round of voting (although in some cases only a plurality is required in the last round of voting if no candidate can achieve a majority). There are two main forms of majoritarian systems, one conducted in a single election using ranked voting and the other using multiple elections, to successively narrow the field of candidates. Both are primarily used for single-member constituencies.
Majoritarian voting can be achieved in a single election using instant-runoff voting (IRV), whereby voters rank candidates in order of preference; this system is used for parliamentary elections in Australia and Papua New Guinea. If no candidate receives a majority of the vote in the first round, the second preferences of the lowest-ranked candidate are then added to the totals. This is repeated until a candidate achieves over 50% of the number of valid votes. If not all voters use all their preference votes, then the count may continue until two candidates remain, at which point the winner is the one with the most votes. A modified form of IRV is the contingent vote where voters do not rank all candidates, but have a limited number of preference votes. If no candidate has a majority in the first round, all candidates are excluded except the top two, with the highest remaining preference votes from the votes for the excluded candidates then added to the totals to determine the winner. This system is used in Sri Lankan presidential elections, with voters allowed to give three preferences.[4]
The other main form of majoritarian system is the two-round system, which is the most common system used for presidential elections around the world, being used in 88 countries. It is also used in 20 countries for electing the legislature.[1] If no candidate achieves a majority of votes in the first round of voting, a second round is held to determine the winner. In most cases the second round is limited to the top two candidates from the first round, although in some elections more than two candidates may choose to contest the second round; in these cases the second round is decided by plurality voting. Some countries use a modified form of the two-round system, such as Ecuador where a candidate in the presidential election is declared the winner if they receive 40% of the vote and are 10% ahead of their nearest rival,[5] or Argentina (45% plus 10% ahead), where the system is known as ballotage.
An exhaustive ballot is not limited to two rounds, but sees the last-placed candidate eliminated in each round of voting. Due to the potentially large number of rounds, this system is not used in any major popular elections, but is used to elect the Speakers of parliament in several countries and members of the Swiss Federal Council. In some formats there may be multiple rounds held without any candidates being eliminated until a candidate achieves a majority, a system used in the United States Electoral College.
Proportional systems
Proportional representation is the most widely used electoral system for national legislatures, with the parliaments of over eighty countries elected by various forms of the system.
Party-list proportional representation is the single most common electoral system and is used by 80 countries, and involves voters voting for a list of candidates proposed by a party. In closed list systems voters do not have any influence over the candidates put forward by the party, but in open list systems voters are able to both vote for the party list and influence the order in which candidates will be assigned seats. In some countries, notably Israel and the Netherlands, elections are carried out using 'pure' proportional representation, with the votes tallied on a national level before assigning seats to parties. However, in most cases several multi-member constituencies are used rather than a single nationwide constituency, giving an element of geographical representation; but this can result in the distribution of seats not reflecting the national vote totals. As a result, some countries have leveling seats to award to parties whose seat totals are lower than their proportion of the national vote.
In addition to the electoral threshold (the minimum percentage of the vote that a party must obtain to win seats), there are several different ways to allocate seats in proportional systems. There are two main types of systems: highest average and largest remainder. Highest average systems involve dividing the votes received by each party by a series of divisors, producing figures that determine seat allocation; for example the D'Hondt method (of which there are variants including Hagenbach-Bischoff) and the Webster/Sainte-Laguë method. Under largest remainder systems, parties' vote shares are divided by the quota (obtained by dividing the total number of votes by the number of seats available). This usually leaves some seats unallocated, which are awarded to parties based on the largest fractions of seats that they have remaining. Examples of largest remainder systems include the Hare quota, Droop quota, the Imperiali quota and the Hagenbach-Bischoff quota.
Single transferable vote (STV) is another form of proportional representation; in STV, voters rank candidates in a multi-member constituency rather than voting for a party list; it is used in Malta and the Republic of Ireland. To be elected, candidates must pass a quota (the Droop quota being the most common). Candidates that pass the quota on the first count are elected. Votes are then reallocated from the least successful candidates, as well as surplus votes from successful candidates, until all seats have been filled by candidates who have passed the quota.[3]
Mixed systems
In several countries, mixed systems are used to elect the legislature. These include parallel voting (also known as mixed-member majoritarian) and mixed-member proportional representation.
In non-compensatory, parallel voting systems, which are used in 20 countries,[1] there are two methods by which members of a legislature are elected; part of the membership is elected by a plurality or majority vote in single-member constituencies and the other part by proportional representation. The results of the constituency vote have no effect on the outcome of the proportional vote.[3]
In compensatory mixed-member representation the results of the proportional vote are adjusted to balance the seats won in the constituency vote. In mixed-member proportional systems, in use in eight countries, there is enough compensation in order to ensure that parties have a number of seats proportional to their vote share.[1]
Other systems may be insufficiently compensatory, and this may result in overhang seats, where parties win more seats in the constituency system than they would be entitled to based on their vote share. Variations of this include the Additional Member System, and Alternative Vote Plus, in which voters cast votes for both single-member constituencies and multi-member constituencies; the allocation of seats in the multi-member constituencies is adjusted to achieve an overall seat total proportional to parties' vote share by taking into account the number of seats won by parties in the single-member constituencies.
Mixed single vote systems are also compensatory, however they usually use a vote transfer mechanism unlike the seat linkage (top-up) method of MMP and may or may not be able to achieve proportional representation. An unusual form of mixed-member compensatory representation using negative vote transfer, Scorporo, was used in Italy from 1993 until 2006.
Some electoral systems feature a majority bonus system to either ensure one party or coalition gains a majority in the legislature, or to give the party receiving the most votes a clear advantage in terms of the number of seats. San Marino has a modified two-round system, which sees a second round of voting featuring the top two parties or coalitions if there is no majority in the first round. The winner of the second round is guaranteed 35 seats in the 60-seat Grand and General Council.[6] In Greece the party receiving the most votes was given an additional 50 seats,[7] a system which was abolished following the 2019 elections.
In Uruguay, the President and members of the General Assembly are elected by on a single ballot, known as the double simultaneous vote. Voters cast a single vote, voting for the presidential, Senatorial and Chamber of Deputies candidates of that party. This system was also previously used in Bolivia and the Dominican Republic.
Primary elections
Primary elections are a feature of some electoral systems, either as a formal part of the electoral system or informally by choice of individual political parties as a method of selecting candidates, as is the case in Italy. Primary elections limit the risk of vote splitting by ensuring a single party candidate. In Argentina they are a formal part of the electoral system and take place two months before the main elections; any party receiving less than 1.5% of the vote is not permitted to contest the main elections. In the United States, there are both partisan and non-partisan primary elections.
Indirect elections
Some elections feature an indirect electoral system, whereby there is either no popular vote, or the popular vote is only one stage of the election; in these systems the final vote is usually taken by an electoral college. In several countries, such as Mauritius or Trinidad and Tobago, the post of President is elected by the legislature. In others like India, the vote is taken by an electoral college consisting of the national legislature and state legislatures. In the United States, the president is indirectly elected using a two-stage process; a popular vote in each state elects members to the electoral college that in turn elects the President. This can result in a situation where a candidate who receives the most votes nationwide does not win the electoral college vote, as most recently happened in 2000 and 2016.
Systems used outside politics
In addition to the various electoral systems in use in the political sphere, there are numerous others, some of which are proposals and some of which have been adopted for usage in business (such as electing corporate board members) or for organisations but not for public elections.
Ranked systems include Bucklin voting, the various Condorcet methods (Copeland's, Dodgson's, Kemeny-Young, Maximal lotteries, Minimax, Nanson's, Ranked pairs, Schulze), the Coombs' method and positional voting. There are also several variants of single transferable vote, including CPO-STV, Schulze STV and the Wright system. Dual-member proportional representation is a proposed system with two candidates elected in each constituency, one with the most votes and one to ensure proportionality of the combined results. Biproportional apportionment is a system whereby the total number of votes is used to calculate the number of seats each party is due, followed by a calculation of the constituencies in which the seats should be awarded in order to achieve the total due to them.
Cardinal electoral systems allow voters to evaluate candidates independently. The complexity ranges from approval voting where voters simply state whether they approve of a candidate or not to range voting, where a candidate is scored from a set range of numbers. Other cardinal systems include proportional approval voting, sequential proportional approval voting, satisfaction approval voting, highest median rules (including the majority judgment), and the D21 – Janeček method where voters can cast positive and negative votes.
Historically, weighted voting systems were used in some countries. These allocated a greater weight to the votes of some voters than others, either indirectly by allocating more seats to certain groups (such as the Prussian three-class franchise), or by weighting the results of the vote. The latter system was used in colonial Rhodesia for the 1962 and 1965 elections. The elections featured two voter rolls (the 'A' roll being largely European and the 'B' roll largely African); the seats of the House Assembly were divided into 50 constituency seats and 15 district seats. Although all voters could vote for both types of seats, 'A' roll votes were given greater weight for the constituency seats and 'B' roll votes greater weight for the district seats. Weighted systems are still used in corporate elections, with votes weighted to reflect stock ownership.
Rules and regulations
In addition to the specific method of electing candidates, electoral systems are also characterised by their wider rules and regulations, which are usually set out in a country's constitution or electoral law. Participatory rules determine candidate nomination and voter registration, in addition to the location of polling places and the availability of online voting, postal voting, and absentee voting. Other regulations include the selection of voting devices such as paper ballots, machine voting or open ballot systems, and consequently the type of vote counting systems, verification and auditing used.
Electoral rules place limits on suffrage and candidacy. Most countries's electorates are characterised by universal suffrage, but there are differences on the age at which people are allowed to vote, with the youngest being 16 and the oldest 21. People may be disenfranchised for a range of reasons, such as being a serving prisoner, being declared bankrupt, having committed certain crimes or being a serving member of the armed forces. Similar limits are placed on candidacy (also known as passive suffrage), and in many cases the age limit for candidates is higher than the voting age. A total of 21 countries have compulsory voting, although in some there is an upper age limit on enforcement of the law.[8] Many countries also have the none of the above option on their ballot papers.
In systems that use constituencies, apportionment or districting defines the area covered by each constituency. Where constituency boundaries are drawn has a strong influence on the likely outcome of elections in the constituency due to the geographic distribution of voters. Political parties may seek to gain an advantage during redistricting by ensuring their voter base has a majority in as many constituencies as possible, a process known as gerrymandering. Historically rotten and pocket boroughs, constituencies with unusually small populations, were used by wealthy families to gain parliamentary representation.
Some countries have minimum turnout requirements for elections to be valid. In Serbia this rule caused multiple re-runs of presidential elections, with the 1997 election re-run once and the 2002 elections re-run three times due insufficient turnout in the first, second and third attempts to run the election. The turnout requirement was scrapped prior to the fourth vote in 2004.[9] Similar problems in Belarus led to the 1995 parliamentary elections going to a fourth round of voting before enough parliamentarians were elected to make a quorum.[10]
Reserved seats are used in many countries to ensure representation for ethnic minorities, women, young people or the disabled. These seats are separate from general seats, and may be elected separately (such as in Morocco where a separate ballot is used to elect the 60 seats reserved for women and 30 seats reserved for young people in the House of Representatives), or be allocated to parties based on the results of the election; in Jordan the reserved seats for women are given to the female candidates who failed to win constituency seats but with the highest number of votes, whilst in Kenya the Senate seats reserved for women, young people and the disabled are allocated to parties based on how many seats they won in the general vote. Some countries achieve minority representation by other means, including requirements for a certain proportion of candidates to be women, or by exempting minority parties from the electoral threshold, as is done in Poland,[11] Romania and Serbia.[12]
History
Pre-democratic
In ancient Greece and Italy, the institution of suffrage already existed in a rudimentary form at the outset of the historical period. In the early monarchies it was customary for the king to invite pronouncements of his people on matters in which it was prudent to secure its assent beforehand. In these assemblies the people recorded their opinion by clamouring (a method which survived in Sparta as late as the 4th century BCE), or by the clashing of spears on shields.[13]
Early democracy
Voting has been used as a feature of democracy since the 6th century BCE, when democracy was introduced by the Athenian democracy. However, in Athenian democracy, voting was seen as the least democratic among methods used for selecting public officials, and was little used, because elections were believed to inherently favor the wealthy and well-known over average citizens. Viewed as more democratic were assemblies open to all citizens, and selection by lot, as well as rotation of office.
Generally, the taking of votes was effected in the form of a poll. The practice of the Athenians, which is shown by inscriptions to have been widely followed in the other states of Greece, was to hold a show of hands, except on questions affecting the status of individuals: these latter, which included all lawsuits and proposals of ostracism, in which voters chose the citizen they most wanted to exile for ten years, were determined by secret ballot (one of the earliest recorded elections in Athens was a plurality vote that it was undesirable to win, namely an ostracism vote). At Rome the method which prevailed up to the 2nd century BCE was that of division (discessio). But the system became subject to intimidation and corruption. Hence a series of laws enacted between 139 and 107 BCE prescribed the use of the ballot (tabella), a slip of wood coated with wax, for all business done in the assemblies of the people. For the purpose of carrying resolutions a simple majority of votes was deemed sufficient. As a general rule equal value was made to attach to each vote; but in the popular assemblies at Rome a system of voting by groups was in force until the middle of the 3rd century BCE by which the richer classes secured a decisive preponderance.[13]
Most elections in the early history of democracy were held using plurality voting or some variant, but as an exception, the state of Venice in the 13th century adopted approval voting to elect their Great Council.[14] The Venetians' method for electing the Doge was a particularly convoluted process, consisting of five rounds of drawing lots (sortition) and five rounds of approval voting. By drawing lots, a body of 30 electors was chosen, which was further reduced to nine electors by drawing lots again. An electoral college of nine members elected 40 people by approval voting; those 40 were reduced to form a second electoral college of 12 members by drawing lots again. The second electoral college elected 25 people by approval voting, which were reduced to form a third electoral college of nine members by drawing lots. The third electoral college elected 45 people, which were reduced to form a fourth electoral college of 11 by drawing lots. They in turn elected a final electoral body of 41 members, who ultimately elected the Doge. Despite its complexity, the method had certain desirable properties such as being hard to game and ensuring that the winner reflected the opinions of both majority and minority factions.[15] This process, with slight modifications, was central to the politics of the Republic of Venice throughout its remarkable lifespan of over 500 years, from 1268 to 1797.
Development of new systems
Jean-Charles de Borda proposed the Borda count in 1770 as a method for electing members to the French Academy of Sciences. His method was opposed by the Marquis de Condorcet, who proposed instead the method of pairwise comparison that he had devised. Implementations of this method are known as Condorcet methods. He also wrote about the Condorcet paradox, which he called the intransitivity of majority preferences. However, recent research has shown that the philosopher Ramon Llull devised both the Borda count and a pairwise method that satisfied the Condorcet criterion in the 13th century. The manuscripts in which he described these methods had been lost to history until they were rediscovered in 2001.[16]
Later in the 18th century, apportionment methods came to prominence due to the United States Constitution, which mandated that seats in the United States House of Representatives had to be allocated among the states proportionally to their population, but did not specify how to do so.[17] A variety of methods were proposed by statesmen such as Alexander Hamilton, Thomas Jefferson, and Daniel Webster. Some of the apportionment methods devised in the United States were in a sense rediscovered in Europe in the 19th century, as seat allocation methods for the newly proposed method of party-list proportional representation. The result is that many apportionment methods have two names; Jefferson's method is equivalent to the D'Hondt method, as is Webster's method to the Sainte-Laguë method, while Hamilton's method is identical to the Hare largest remainder method.[17]
The single transferable vote (STV) method was devised by Carl Andræ in Denmark in 1855 and in the United Kingdom by Thomas Hare in 1857. STV elections were first held in Denmark in 1856, and in Tasmania in 1896 after its use was promoted by Andrew Inglis Clark. Party-list proportional representation began to be used to elect European legislatures in the early 20th century, with Belgium the first to implement it for its 1900 general elections. Since then, proportional and semi-proportional methods have come to be used in almost all democratic countries, with most exceptions being former British and French colonies.[18]
Single-winner revival
Perhaps influenced by the rapid development of multiple-winner electoral systems, theorists began to publish new findings about single-winner methods in the late 19th century. This began around 1870, when William Robert Ware proposed applying STV to single-winner elections, yielding instant-runoff voting (IRV).[19] Soon, mathematicians began to revisit Condorcet's ideas and invent new methods for Condorcet completion; Edward J. Nanson combined the newly described instant runoff voting with the Borda count to yield a new Condorcet method called Nanson's method. Charles Dodgson, better known as Lewis Carroll, proposed the straightforward Condorcet method known as Dodgson's method. He also proposed a proportional representation system based on multi-member districts, quotas as minimum requirements to take seats, and votes transferable by candidates through proxy voting.[20]
Ranked voting electoral systems eventually gathered enough support to be adopted for use in government elections. In Australia, IRV was first adopted in 1893, and continues to be used along with STV today. In the United States in the early-20th-century progressive era, some municipalities began to use Bucklin voting, although this is no longer used in any government elections, and has even been declared unconstitutional in Minnesota.[21]
Recent developments
The use of game theory to analyze electoral systems led to discoveries about the effects of certain methods. Earlier developments such as Arrow's impossibility theorem had already shown the issues with Ranked voting systems. Research led Steven Brams and Peter Fishburn to formally define and promote the use of approval voting in 1977.[22] Political scientists of the 20th century published many studies on the effects that the electoral systems have on voters' choices and political parties,[23][24][25] and on political stability.[26][27] A few scholars also studied which effects caused a nation to switch to a particular electoral system.[28][29][30][31][32]
The study of electoral systems influenced a new push for electoral reform beginning around the 1990s, when proposals were made to replace plurality voting in governmental elections with other methods. New Zealand adopted mixed-member proportional representation for the 1993 general elections and STV for some local elections in 2004. After plurality voting was a key factor in the contested results of the 2000 presidential elections in the United States, various municipalities in the United States began to adopt instant-runoff voting, although some of them subsequently returned to their prior method. However, attempts at introducing more proportional systems were not always successful; in Canada there were two referendums in British Columbia in 2005 and 2009 on adopting an STV method, both of which failed. In the United Kingdom, a 2011 referendum on adopting IRV saw the proposal rejected.
In other countries there were calls for the restoration of plurality or majoritarian systems or their establishment where they have never been used; a referendum was held in Ecuador in 1994 on the adoption the two round system, but the idea was rejected. In Romania a proposal to switch to a two-round system for parliamentary elections failed only because voter turnout in the referendum was too low. Attempts to reintroduce single-member constituencies in Poland (2015) and two-round system in Bulgaria (2016) via referendums both also failed due to low turnout.
Comparison of electoral systems
Main article: Comparison of electoral systems
Electoral systems can be compared by different means. Attitudes towards systems are highly influenced by the systems' impact on groups that one supports or opposes, which can make the objective comparison of voting systems difficult. There are several ways to address this problem:
One approach is to define criteria mathematically, such that any electoral system either passes or fails. This gives perfectly objective results, but their practical relevance is still arguable.
Another approach is to define ideal criteria that no electoral system passes perfectly, and then see how often or how close to passing various methods are over a large sample of simulated elections. This gives results which are practically relevant, but the method of generating the sample of simulated elections can still be arguably biased.
A final approach is to consider practical criteria, and then assign a neutral body to evaluate each method according to these criteria or evaluate the performance of countries with these electoral systems. The practical criteria include political fragmentation, voter turnout, wasted votes, complexity of vote counting, and barriers to entry for new political movements.[33] The quality of electoral systems can be measured on outcomes, such as voter turnout,[34][35] and reduced political apathy. This approach can look at aspects of electoral systems, which the other two approaches miss, but both the definitions of these criteria and the evaluations of the methods are still inevitably subjective.
Arrow's theorem and the Gibbard–Satterthwaite theorem prove that no single-winner system using ranked voting can meet all such criteria simultaneously, while Gibbard's theorem proves the same for all single-winner deterministic voting methods. Instead of debating the importance of different criteria, another method is to simulate many elections with different electoral systems, and estimate the typical overall happiness of the population with the results,[36][37] their vulnerability to strategic voting, their likelihood of electing the candidate closest to the average voter, etc.
According to a 2006 survey of electoral system experts, their preferred electoral systems were in order of preference:[38]
1. Mixed member proportional
2. Single transferable vote
3. Open list proportional
4. Alternative vote
5. Closed list proportional
6. Single member plurality
7. Runoffs
8. Mixed member majoritarian
9. Single non-transferable vote
Systems by elected body
This section is an excerpt from List of electoral systems by country § Maps.[edit]
Head of state Lower (or unicameral) house Upper house
Single-winner system / single-member constituencies (non-proportional)
First past the post/single member plurality (FPTP/SMP)
Two-round system (TRS)
Instant-runoff voting (IRV)
Multi-member constituencies, majoritarian (non-proportional)
Block voting (BV) or mixed FPTP and BV
Party block voting (PBV) or mixed FPTP and PBV
Multi-member constituencies, semi-proportional
Limited voting (LV) or cumulative voting
Single non-transferable vote (SNTV) or mixed FPTP and SNTV
Modified Borda count
Multi-member constituencies, proportional
Party-list proportional representation (Closed list PR)
Party-list proportional representation (Open list PR)
Party-list proportional representation (Open list PR)
Single transferable vote (STV)
Mixed non-compensatory (semi-proportional)
Mixed-member majoritarian (MMM): parallel voting (FPTP and list PR)
Mixed-member majoritarian (MMM): parallel voting (TRS and list PR)
Mixed-member majoritarian (MMM): parallel voting (BV/PBV and list PR)
List PR with plurality bonus
(MBS)
Parallel voting (SNTV and list PR)
Mixed compensatory (proportional or semi-proportional)
Mixed-member majoritarian (MMM) with compensation / scorporo
Additional member system / semi-proportional MMP
Mixed-member proportional representation (MMP)
Majority jackpot (majority of seats reserved for largest party/coalition)
Indirect election
Election by legislature
Election by electoral college or local legislatures
Partly elected by electoral college or local legislatures, partly appointed by head of state
Other
No election (e.g. Monarchy)
Appointed by head of state
Varies by federal states or constituencies
No information/Unicameral legislature
See also
• Comparison of electoral systems
• Election
• List of electoral systems by country
• Matrix vote
• Spoiler effect
• Psephology
References
1. Table of Electoral Systems Worldwide Archived 2017-05-23 at the Wayback Machine IDEA
2. Nauru Parliament: Electoral system IPU
3. Glossary of Terms Archived 2017-06-11 at the Wayback Machine IDEA
4. Sri Lanka: Election for President IFES
5. Ecuador: Election for President Archived 2016-12-24 at the Wayback Machine IFES
6. Consiglio grande e generale: Electoral system IPU
7. Hellenic Parliament: Electoral system IPU
8. Suffrage Archived 2008-01-09 at the Wayback Machine CIA World Factbook
9. Pro-Western Candidate Wins Serbian Presidential Poll Deutsche Welle, 28 June 2004
10. Elections held in 1995 IPU
11. Sejm: Electoral system IPU
12. Narodna skupstina: Electoral system IPU
13. One or more of the preceding sentences incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "Vote and Voting". Encyclopædia Britannica. Vol. 28 (11th ed.). Cambridge University Press. p. 216.
14. J.J. O'Connor & E. F. Robertson The history of voting MacTutor History of Mathematics archive
15. Miranda Mowbray & Dieter Gollmann (2007) Electing the Doge of Venice: Analysis of a 13th Century Protocol
16. G. Hägele & F. Pukelsheim (2001) "Llull's writings on electoral systems", Studia Lulliana Vol. 3, pp. 3–38
17. Apportionment: Introduction American Mathematical Society
18. Proportional Voting Around the World FairVote
19. The History of IRV FairVote
20. Charles Dodgson (1884) Principles of Parliamentary Representation
21. Tony Anderson Solgård & Paul Landskroener (2002) "Municipal Voting System Reform: Overcoming the Legal Obstacles", Bench & Bar of Minnesota, Vol. 59, no. 9
22. Poundstone, William (2008) Gaming the Vote: Why Elections Aren't Fair (and What We Can Do About It), Hill and Young, p. 198
23. Duverger, Maurice (1954) Political Parties, Wiley ISBN 0-416-68320-7
24. Douglas W. Rae (1971) The Political Consequences of Electoral Laws, Yale University Press ISBN 0-300-01517-8
25. Rein Taagapera & Matthew S. Shugart (1989) Seats and Votes: The Effects and Determinants of Electoral Systems, Yale University Press
26. Ferdinand A. Hermens (1941) Democracy or Anarchy? A Study of Proportional Representation, University of Notre Dame.
27. Arend Lijphart (1994) Electoral Systems and Party Systems: A Study of Twenty-Seven Democracies, 1945–1990 Oxford University Press ISBN 0-19-828054-8
28. Arend Lijphart (1985) "The Field of Electoral Systems Research: A Critical Survey" Electoral Studies, Vol. 4
29. Arend Lijphart (1992) "Democratization and Constitutional Choices in Czecho-Slovakia, Hungary and Poland, 1989–1991" Journal of Theoretical Politics Vol. 4 (2), pp. 207–23
30. Stein Rokkan (1970) Citizens, Elections, Parties: Approaches to the Comparative Study of the Process of Development, Universitetsforlaget
31. Ronald Rogowski (1987) "Trade and the Variety of Democratic Institutions", International Organization Vol. 41, pp. 203–24
32. Carles Boix (1999) "Setting the Rules of the Game: The Choice of Electoral Systems in Advanced Democracies", American Political Science Review Vol. 93 (3), pp. 609–24
33. Tullock, Gordon. "Entry barriers in politics." The American Economic Review 55.1/2 (1965): 458-466.
34. Lijphart, Arend (March 1997). "Unequal Participation: Democracy's Unresolved Dilemma". American Political Science Review. 91 (1): 1–14. doi:10.2307/2952255. JSTOR 2952255. S2CID 143172061.
35. Blais, Andre (1990). "Does proportional representation foster voter turnout?". European Journal of Political Research. 18 (2): 167–181. doi:10.1111/j.1475-6765.1990.tb00227.x.
36. "What is Voter Satisfaction Efficiency?". electology.github.io. Center for Election Science. Retrieved 2017-03-30. (VSE) is a way of measuring the outcome quality [of] a voting method ... highest average happiness would have a VSE of 100%. ... it's impossible for a method to pass all desirable criteria ... VSE measures how well a method makes those tradeoffs by using outcomes.
37. "Bayesian Regret". RangeVoting.org. Retrieved 2017-03-30. The 'Bayesian regret' of an election method E is the 'expected avoidable human unhappiness'
38. Bowler, Shaun; Farrell, David M.; Pettit, Robin T. (2005-04-01). "Expert opinion on electoral systems: So which electoral system is "best"?". Journal of Elections, Public Opinion and Parties. 15 (1): 3–19. doi:10.1080/13689880500064544. ISSN 1745-7289. S2CID 144919388.
External links
• ACE Electoral Knowledge Network
• The International IDEA Handbook of Electoral System Design IDEA
Electoral systems
Part of the politics and election series
Single-winner
• Approval voting
• Combined approval voting
• Unified primary
• Borda count
• Bucklin voting
• Condorcet methods
• Copeland's method
• Dodgson's method
• Kemeny–Young method
• Minimax Condorcet method
• Nanson's method
• Ranked pairs
• Schulze method
• Exhaustive ballot
• First-past-the-post voting
• Instant-runoff voting
• Coombs' method
• Contingent vote
• Supplementary vote
• Majority judgment
• Simple majoritarianism
• Plurality
• Positional voting system
• Score voting
• STAR voting
• Two-round system
• Usual judgment
Proportional
Systems
• Dual member
• Mixed-member (Additional member)
• Mixed single vote
• Party-list
• Proportional approval voting
• Rural-urban
• Sequential proportional approval voting
• Single transferable vote
• CPO-STV
• Hare-Clark
• Schulze STV
• Spare vote
• Indirect single transferable voting
Allocation
• Highest averages method
• Webster/Sainte-Laguë
• D'Hondt
• Largest remainder method
Quotas
• Droop quota
• Hagenbach-Bischoff quota
• Hare quota
• Imperiali quota
Mixed
• Additional member system
• Alternative vote plus
• Cumulative voting
• Limited voting
• Mixed single vote
• Parallel voting
• Satisfaction approval voting
• Scorporo
• Single non-transferable vote
Criteria
• Condorcet winner criterion
• Condorcet loser criterion
• Consistency criterion
• Independence of clones
• Independence of irrelevant alternatives
• Independence of Smith-dominated alternatives
• Later-no-harm criterion
• Majority criterion
• Majority loser criterion
• Monotonicity criterion
• Mutual majority criterion
• Participation criterion
• Plurality criterion
• Resolvability criterion
• Reversal symmetry
• Smith criterion
• Seats-to-votes ratio
Other
• Ballot
• Election threshold
• First-preference votes
• Liquid democracy
• Spoilt vote
• Sortition
• Unseating
Comparison
• Comparison of voting systems
• Voting systems by country
Portal — Project
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• Germany
• Czech Republic
Other
• Encyclopedia of Modern Ukraine
| Wikipedia |
Condorcet paradox
The Condorcet paradox (also known as the voting paradox or the paradox of voting) in social choice theory is a situation noted by the Marquis de Condorcet in the late 18th century,[1][2][3] in which collective preferences can be cyclic, even if the preferences of individual voters are not cyclic. This is paradoxical, because it means that majority wishes can be in conflict with each other: Suppose majorities prefer, for example, candidate A over B, B over C, and yet C over A. When this occurs, it is because the conflicting majorities are each made up of different groups of individuals.
Thus an expectation that transitivity on the part of all individuals' preferences should result in transitivity of societal preferences is an example of a fallacy of composition.
The paradox was independently discovered by Lewis Carroll and Edward J. Nanson, but its significance was not recognized until popularized by Duncan Black in the 1940s.[4]
Example
Suppose we have three candidates, A, B, and C, and that there are three voters with preferences as follows (candidates being listed left-to-right for each voter in decreasing order of preference):
VoterFirst preferenceSecond preferenceThird preference
Voter 1 ABC
Voter 2 BCA
Voter 3 CAB
If C is chosen as the winner, it can be argued that B should win instead, since two voters (1 and 2) prefer B to C and only one voter (3) prefers C to B. However, by the same argument A is preferred to B, and C is preferred to A, by a margin of two to one on each occasion. Thus the society's preferences show cycling: A is preferred over B which is preferred over C which is preferred over A.
Cardinal ratings
Note that in the graphical example, the voters and candidates are not symmetrical, but the ranked voting system "flattens" their preferences into a symmetrical cycle.[5] Cardinal voting systems provide more information than rankings, allowing a winner to be found.[6][7] For instance, under score voting, the ballots might be:[8]
ABC
1 630
2 061
3 506
Total: 1197
Candidate A gets the largest score, and is the winner, as A is the nearest to all voters. However, a majority of voters have an incentive to give A a 0 and C a 10, allowing C to beat A, which they prefer, at which point, a majority will then have an incentive to give C a 0 and B a 10, to make B win, etc. (In this particular example, though, the incentive is weak, as those who prefer C to A only score C 1 point above A; in a ranked Condorcet method, it's quite possible they would simply equally rank A and C because of how weak their preference is, in which case a Condorcet cycle wouldn't have formed in the first place, and A would've been the Condorcet winner). So though the cycle doesn't occur in any given set of votes, it can appear through iterated elections with strategic voters with cardinal ratings.
Necessary condition for the paradox
Suppose that x is the fraction of voters who prefer A over B and that y is the fraction of voters who prefer B over C. It has been shown[9] that the fraction z of voters who prefer A over C is always at least (x + y – 1). Since the paradox (a majority preferring C over A) requires z < 1/2, a necessary condition for the paradox is that
$x+y-1\leq z<{\frac {1}{2}}\quad {\text{and hence}}\quad x+y<{\frac {3}{2}}.$
Likelihood of the paradox
It is possible to estimate the probability of the paradox by extrapolating from real election data, or using mathematical models of voter behavior, though the results depend strongly on which model is used. In particular, Andranik Tangian has proved that the probability of Condorcet paradox is negligible in a large society.[10][11]
Impartial culture model
We can calculate the probability of seeing the paradox for the special case where voter preferences are uniformly distributed among the candidates. (This is the "impartial culture" model, which is known to be unrealistic,[12][13][14]: 40 so, in practice, a Condorcet paradox may be more or less likely than this calculation.[15]: 320 [16])
For $n$ voters providing a preference list of three candidates A, B, C, we write $X_{n}$ (resp. $Y_{n}$, $Z_{n}$) the random variable equal to the number of voters who placed A in front of B (respectively B in front of C, C in front of A). The sought probability is $p_{n}=2P(X_{n}>n/2,Y_{n}>n/2,Z_{n}>n/2)$ (we double because there is also the symmetric case A> C> B> A). We show that, for odd $n$, $p_{n}=3q_{n}-1/2$ where $q_{n}=P(X_{n}>n/2,Y_{n}>n/2)$ which makes one need to know only the joint distribution of $X_{n}$ and $Y_{n}$.
If we put $p_{n,i,j}=P(X_{n}=i,Y_{n}=j)$, we show the relation which makes it possible to compute this distribution by recurrence: $p_{n+1,i,j}={1 \over 6}p_{n,i,j}+{1 \over 3}p_{n,i-1,j}+{1 \over 3}p_{n,i,j-1}+{1 \over 6}p_{n,i-1,j-1}$.
The following results are then obtained:
$n$ 3 101 201 301 401 501 601
$p_{n}$ 5.556% 8.690% 8.732% 8.746% 8.753% 8.757% 8.760%
The sequence seems to be tending towards a finite limit.
Using the central limit theorem, we show that $q_{n}$ tends to $q={\frac {1}{4}}P\left(|T|>{\frac {\sqrt {2}}{4}}\right),$ where $T$ is a variable following a Cauchy distribution, which gives $q={\dfrac {1}{2\pi }}\int _{{\sqrt {2}}/4}^{+\infty }{\frac {dt}{1+t^{2}}}={\dfrac {\arctan 2{\sqrt {2}}}{2\pi }}={\dfrac {\arccos {\frac {1}{3}}}{2\pi }}$ (constant quoted in the OEIS).
The asymptotic probability of encountering the Condorcet paradox is therefore ${{3\arccos {1 \over 3}} \over {2\pi }}-{1 \over 2}={\arcsin {{\sqrt {6}} \over 9} \over \pi }$ which gives the value 8.77%.[17][18]
Some results for the case of more than three candidates have been calculated[19] and simulated.[20] The simulated likelihood for an impartial culture model with 25 voters increases with the number of candidates:[20]: 28
3 4 5 7 10
8.4% 16.6% 24.2% 35.7% 47.5%
The likelihood of a Condorcet cycle for related models approach these values for large electorates:[18]
• Impartial anonymous culture (IAC): 6.25%
• Uniform culture (UC): 6.25%
• Maximal culture condition (MC): 9.17%
All of these models are unrealistic, and are investigated to establish an upper bound on the likelihood of a cycle.[18]
Group coherence models
When modeled with more realistic voter preferences, Condorcet paradoxes in elections with a small number of candidates and a large number of voters become very rare.[14]: 78
Spatial model
A study of three-candidate elections analyzed 12 different models of voter behavior, and found the spatial model of voting to be the most accurate to real-world ranked-ballot election data. Analyzing this spatial model, they found the likelihood of a cycle to decrease to zero as the number of voters increases, with likelihoods of 5% for 100 voters, 0.5% for 1000 voters, and 0.06% for 10,000 voters.[21]
Another spatial model found likelihoods of 2% or less in all simulations of 201 voters and 5 candidates, whether two or four-dimensional, with or without correlation between dimensions, and with two different dispersions of candidates.[20]: 31
Empirical studies
Many attempts have been made at finding empirical examples of the paradox.[22] Empirical identification of a Condorcet paradox presupposes extensive data on the decision-makers' preferences over all alternatives—something that is only very rarely available.
A summary of 37 individual studies, covering a total of 265 real-world elections, large and small, found 25 instances of a Condorcet paradox, for a total likelihood of 9.4%[15]: 325 (and this may be a high estimate, since cases of the paradox are more likely to be reported on than cases without).[14]: 47
An analysis of 883 three-candidate elections extracted from 84 real-world ranked-ballot elections of the Electoral Reform Society found a Condorcet cycle likelihood of 0.7%. These derived elections had between 350 and 1,957 voters. A similar analysis of data from the 1970–2004 American National Election Studies thermometer scale surveys found a Condorcet cycle likelihood of 0.4%. These derived elections had between 759 and 2,521 "voters".[21]
While examples of the paradox seem to occur occasionally in small settings (e.g., parliaments) very few examples have been found in larger groups (e.g. electorates), although some have been identified.[23]
Implications
When a Condorcet method is used to determine an election, the voting paradox of cyclical societal preferences implies that the election has no Condorcet winner: no candidate who can win a one-on-one election against each other candidate. There will still be a smallest group of candidates, known as the Smith set, such that each candidate in the group can win a one-on-one election against each of the candidates outside the group. The several variants of the Condorcet method differ on how they resolve such ambiguities when they arise to determine a winner.[24] The Condorcet methods which always elect someone from the Smith set when there is no Condorcet winner are known as Smith-efficient. Note that using only rankings, there is no fair and deterministic resolution to the trivial example given earlier because each candidate is in an exactly symmetrical situation.
Situations having the voting paradox can cause voting mechanisms to violate the axiom of independence of irrelevant alternatives—the choice of winner by a voting mechanism could be influenced by whether or not a losing candidate is available to be voted for.
Two-stage voting processes
One important implication of the possible existence of the voting paradox in a practical situation is that in a two-stage voting process, the eventual winner may depend on the way the two stages are structured. For example, suppose the winner of A versus B in the open primary contest for one party's leadership will then face the second party's leader, C, in the general election. In the earlier example, A would defeat B for the first party's nomination, and then would lose to C in the general election. But if B were in the second party instead of the first, B would defeat C for that party's nomination, and then would lose to A in the general election. Thus the structure of the two stages makes a difference for whether A or C is the ultimate winner.
Likewise, the structure of a sequence of votes in a legislature can be manipulated by the person arranging the votes, to ensure a preferred outcome.
See also
• Arrow's impossibility theorem
• Kenneth Arrow, Section with an example of a distributional difficulty of intransitivity + majority rule
• Discursive dilemma
• Gibbard–Satterthwaite theorem
• Independence of irrelevant alternatives
• Instant-runoff voting
• Nakamura number
• Quadratic voting
• Rock paper scissors
• Simpson's paradox
• Smith set
References
1. Marquis de Condorcet (1785). Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix (PNG) (in French). Retrieved 2008-03-10.
2. Condorcet, Jean-Antoine-Nicolas de Caritat; Sommerlad, Fiona; McLean, Iain (1989-01-01). The political theory of Condorcet. Oxford: University of Oxford, Faculty of Social Studies. pp. 69–80, 152–166. OCLC 20408445. Clearly, if anyone's vote was self-contradictory (having cyclic preferences), it would have to be discounted, and we should therefore establish a form of voting which makes such absurdities impossible
3. Gehrlein, William V. (2002). "Condorcet's paradox and the likelihood of its occurrence: different perspectives on balanced preferences*". Theory and Decision. 52 (2): 171–199. doi:10.1023/A:1015551010381. ISSN 0040-5833. S2CID 118143928. Here, Condorcet notes that we have a 'contradictory system' that represents what has come to be known as Condorcet's Paradox.
4. Riker, William Harrison. (1982). Liberalism against populism : a confrontation between the theory of democracy and the theory of social choice. Waveland Pr. p. 2. ISBN 0881333670. OCLC 316034736.
5. Procaccia, Ariel D.; Rosenschein, Jeffrey S. (2006-09-11). Klusch, Matthias; Rovatsos, Michael; Payne, Terry R. (eds.). The Distortion of Cardinal Preferences in Voting (PDF). Lecture Notes in Computer Science. Springer Berlin Heidelberg. pp. 317–331. CiteSeerX 10.1.1.113.2486. doi:10.1007/11839354_23. ISBN 9783540385691. agents' cardinal (utility-based) preferences are embedded into the space of ordinal preferences. This often gives rise to a distortion in the preferences, and hence in the social welfare of the outcome
6. Poundstone, William (2008). Gaming the vote: Why elections aren't fair (and what we can do about it). Hill & Wang. p. 158. ISBN 978-0809048922. OCLC 276908223. This is the fundamental problem with two-way comparisons. There is no accounting for degrees of preference. ... Cycles result from giving equal weight to unequal preferences. ... The paradox obscures the fact that the voters really do prefer one option.
7. Kok, Jan; Shentrup, Clay; Smith, Warren. "Condorcet cycles". RangeVoting.org. Retrieved 2017-02-09. ...any method based just on rank-order votes, fails miserably. Range voting, which allows voters to express strength of preferences, would presumably succeed in choosing the best capital A.
8. In this example, the available scores are 0–6, and each voter normalizes their max/min scores to this range, while selecting a score for the middle that is proportional to distance.
9. Silver, Charles. "The voting paradox", The Mathematical Gazette 76, November 1992, 387–388.
10. Tangian, Andranik (2000). "Unlikelihood of Condorcet's paradox in a large society". Social Choice and Welfare. 17 (2): 337–365. doi:10.1007/s003550050024. S2CID 19382306.
11. Tangian, Andranik (2020). Analytical theory of democracy. Vols. 1 and 2. Studies in Choice and Welfare. Cham, Switzerland: Springer. pp. 158–162. doi:10.1007/978-3-030-39691-6. ISBN 978-3-030-39690-9. S2CID 216190330.
12. Tsetlin, Ilia; Regenwetter, Michel; Grofman, Bernard (2003-12-01). "The impartial culture maximizes the probability of majority cycles". Social Choice and Welfare. 21 (3): 387–398. doi:10.1007/s00355-003-0269-z. ISSN 0176-1714. S2CID 15488300. it is widely acknowledged that the impartial culture is unrealistic ... the impartial culture is the worst case scenario
13. Tideman, T; Plassmann, Florenz (June 2008). "The Source of Election Results: An Empirical Analysis of Statistical Models of Voter Behavior". Voting theorists generally acknowledge that they consider this model to be unrealistic {{cite journal}}: Cite journal requires |journal= (help)
14. Gehrlein, William V.; Lepelley, Dominique (2011). Voting paradoxes and group coherence : the condorcet efficiency of voting rules. Berlin: Springer. doi:10.1007/978-3-642-03107-6. ISBN 9783642031076. OCLC 695387286. most election results do not correspond to anything like any of DC, IC, IAC or MC ... empirical studies ... indicate that some of the most common paradoxes are relatively unlikely to be observed in actual elections. ... it is easily concluded that Condorcet's Paradox should very rarely be observed in any real elections on a small number of candidates with large electorates, as long as voters' preferences reflect any reasonable degree of group mutual coherence
15. Van Deemen, Adrian (2014). "On the empirical relevance of Condorcet's paradox". Public Choice. 158 (3–4): 311–330. doi:10.1007/s11127-013-0133-3. ISSN 0048-5829. S2CID 154862595. small departures of the impartial culture assumption may lead to large changes in the probability of the paradox. It may lead to huge declines or, just the opposite, to huge increases.
16. May, Robert M. (1971). "Some mathematical remarks on the paradox of voting". Behavioral Science. 16 (2): 143–151. doi:10.1002/bs.3830160204. ISSN 0005-7940.
17. Guilbaud, Georges-Théodule (2012). "Les théories de l'intérêt général et le problème logique de l'agrégation". Revue économique. 63 (4): 659. doi:10.3917/reco.634.0659. ISSN 0035-2764.
18. Gehrlein, William V. (2002-03-01). "Condorcet's paradox and the likelihood of its occurrence: different perspectives on balanced preferences*". Theory and Decision. 52 (2): 171–199. doi:10.1023/A:1015551010381. ISSN 1573-7187. S2CID 118143928. to have a PMRW with probability approaching 15/16 = 0.9375 with IAC and UC, and approaching 109/120 = 0.9083 for MC. … these cases represent situations in which the probability that a PMRW exists would tend to be at a minimum … intended to give us some idea of the lower bound on the likelihood that a PMRW exists.
19. Gehrlein, William V. (1997). "Condorcet's paradox and the Condorcet efficiency of voting rules". Mathematica Japonica. 45: 173–199.
20. Merrill, Samuel (1984). "A Comparison of Efficiency of Multicandidate Electoral Systems". American Journal of Political Science. 28 (1): 23–48. doi:10.2307/2110786. ISSN 0092-5853. JSTOR 2110786.
21. Tideman, T. Nicolaus; Plassmann, Florenz (2012), Felsenthal, Dan S.; Machover, Moshé (eds.), "Modeling the Outcomes of Vote-Casting in Actual Elections", Electoral Systems, Berlin, Heidelberg: Springer Berlin Heidelberg, Table 9.6 Shares of strict pairwise majority rule winners (SPMRWs) in observed and simulated elections, doi:10.1007/978-3-642-20441-8_9, ISBN 978-3-642-20440-1, retrieved 2021-11-12, Mean number of voters: 1000 … Spatial model: 99.47% [0.5% cycle likelihood] … 716.4 [ERS data] … Observed elections: 99.32% … 1,566.7 [ANES data] … 99.56%
22. Kurrild-Klitgaard, Peter (2014). "Empirical social choice: An introduction". Public Choice. 158 (3–4): 297–310. doi:10.1007/s11127-014-0164-4. ISSN 0048-5829. S2CID 148982833.
23. Kurrild-Klitgaard, Peter (2014). "An empirical example of the Condorcet paradox of voting in a large electorate". Public Choice. 107: 135–145. doi:10.1023/A:1010304729545. ISSN 0048-5829. S2CID 152300013.
24. Lippman, David (2014). "Voting Theory". Math in society. ISBN 978-1479276530. OCLC 913874268. There are many Condorcet Methods, which vary primarily in how they deal with ties, which are very common when a Condorcet winner does not exist.
Further reading
• Garman, M. B.; Kamien, M. I. (1968). "The paradox of voting: Probability calculations". Behavioral Science. 13 (4): 306–316. doi:10.1002/bs.3830130405. PMID 5663897.
• Niemi, R. G.; Weisberg, H. (1968). "A mathematical solution for the probability of the paradox of voting". Behavioral Science. 13 (4): 317–323. doi:10.1002/bs.3830130406. PMID 5663898.
• Niemi, R. G.; Wright, J. R. (1987). "Voting cycles and the structure of individual preferences". Social Choice and Welfare. 4 (3): 173–183. doi:10.1007/BF00433943. JSTOR 41105865. S2CID 145654171.
Common paradoxes
Philosophical
• Analysis
• Buridan's bridge
• Dream argument
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• Fiction
• Fitch's knowability
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• Goodman's
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Logical
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• Grelling–Nelson
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• Liar
• Card
• No-no
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• Quine's
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• Opposite Day
• Paradoxes of set theory
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• Socratic
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• Temperature paradox
• Barbershop
• Catch-22
• Chicken or the egg
• Drinker
• Entailment
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Economic
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Decision theory
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| Wikipedia |
Vsevolod Ivanovich Romanovsky
Vsevolod Ivanovich Romanovsky (Всеволод Иванович Романовский, 4 December 1879, Verny, Russian Empire – 6 December 1954) was a Russian-Soviet-Uzbek mathematician, founder of the Tashkent school of mathematics.
Education and career
In 1906 Romanovsky received, under the supervision of A. A. Markov, his doctoral degree from St. Petersburg University. During 1900–1908 he was a student and then a docent at St. Petersburg University.[1][2] In 1911–1915 he was a senior lecturer and then professor at the Imperial University of Warsaw, in 1915–1918 a professor at the Imperial University of Warsaw in Rostov-on-Don, and from 1918 a professor of probability and mathematical statistics at what is now called the National University of Uzbekistan (in Tashkent). His doctoral students include Tashmukhamed Alievich Sarymsakov (Ташмухамед Алиевич Сарымсаков).[3]
Romanovsky gained an international reputation for his work in mathematical statistics and probability theory. In 1943 he was made an Academician of the Uzbek Soviet Socialist Republic. The Uzbek Academy of Sciences' Romanovsky Institute of Mathematics is named in his honor. Romanovsky was an Invited Speaker at the ICM in 1928 in Bologna[4] and in 1932 in Zürich.
His body was buried in Tashkent in the Botkin cemetery.
Awards
• 1948: Stalin Prize of the third degree (for the development and introduction of new methods of drawing up the short-term and long-term weather forecasts)
• 23 August 2004: Order Buyuk Hizmatlari Uchun (for Great Services) – posthumous award by presidential decree of the Republic of Uzbekistan
Selected works
• Романовский В. И. Элементарный курс математической статистики. (Elementary course in mathematical statistics) – М.-Л. Госпланиздат, 1924.
• Романовский В. И. Элементы теории корреляции. (Elements of correlation theory) 1928 г. – 148 pages.
• Романовский В. И. Математическая статистика. (Mathematical statistics) – М.-Л. Гос.объед. научно-тех.изд. НКТП СССР. 1938. – 527 pages
• Романовский В. И. Элементарный курс математической статистики. (Elementary course in mathematical statistics) – М.-Л. Госпланиздат, 1939. – 359 pages
• Романовский В. И. Введение в анализ. (Introduction to analysis) – Ташкент. Гос.учебно-педагог. изд., 1939. – 436 pages
• Романовский В. И. О предельных распределениях для стохастических процессов с дискретным временем. (On limiting distributions for stochastic processes with discrete time) – Изд. Среднеаз. Гос. Унив. Ташкент, 1946. – 24 pages
• Романовский В. И. Применения математической статистики в опытном деле. (Applications of mathematical statistics in the test case) – Гостехиздат, М.-Л.,1947. – 248 pages
• Романовский В. И. Основные задачи теории ошибок. (The main tasks of the theory of errors) – ОГИЗ. Гостехиздат, М.-Л., 1947. – 116 pages
• Романовский В. И. Дискретные цепи Маркова. (Discrete Markov chains) – Гостехиздат, М.-Л. 1949. – 436 pages
• Романовский В. И. Математическая статистика. Кн.1. Основы теории вероятностей и математической статистики. (Mathematical statistics. Book 1. Fundamentals of the theory of probability and mathematical statistics) – Ташкент, 1961. – 637 pages
• Романовский В. И. Математическая статистика. Кн.2. Оперативные методы математической статистики. (Mathematical statistics. Book 2. Operational methods of mathematical statistics) – Ташкент, 1963. – 794 pages
• Романовский В. И. Избранные труды. Т.1. (Selected Works. V.1) – Изд-во "Наука" Узб. ССР. Ташкент. 1961.
• Романовский В. И. Избранные труды. Т.2. (Selected Works. V.2) – Теория вероятностей, статистика и анализ. (Theory of probability, statistics and analysis) Изд-во "Наука" Узб. ССР. Ташкент. 1964. – 390 pages
See also
• Romanovski polynomials
References
1. Thomas W. Hawkins Jr. (2013). "V. I. Romanovsky". The Mathematics of Frobenius in Context. Springer. p. 647. ISBN 9781461463337.
2. Markov and the Creation of Markov Chains by Eugene Seneta, University of Sydney
3. Tashmukhamed Alievich Sarymsakov at the Mathematics Genealogy Project
4. Romanovsky, V. I. (1928). "Sur la généralisation des courbes de Pearson" (PDF). Atti del Congresso Intern. Dei Matematici: 107–110.
Authority control
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• FAST
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• CiNii
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Other
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| Wikipedia |
Vyacheslav Feodoritov
Vyacheslav Petrovich Feodoritov (Russian: Вячесла́в Петро́вич Феодори́тов)(February 28, 1928 - January 2, 2004), k.N, was a Russian physicist in the former Soviet program of nuclear weapons. He was a co-designer of the first two-stage Soviet thermonuclear device, the RDS-37, and became a chief of laboratory at Arzamas-16, now known as the All-Russian Scientific Research Institute of Experimental Physics.
Vyacheslav Feodoritov
Өеодоритов, Вячеслав Петрович
Born
Vyacheslav Petrovich Feodoritov
(1928-02-28)February 28, 1928
Sasovo, Ryazan Oblast in Soviet Union
(now Sasovo, Ryazan in Russia)
DiedJanuary 2, 2004(2004-01-02) (aged 75)
Sarov, Russia
Resting placeSarov, Russia
Citizenship Russia
Alma materMoscow State University
Known forSoviet atomic bomb project
Awards Honored Scientist of the Russia
USSR State Prize
Order of the Red Banner of Labor
Scientific career
FieldsPhysics
InstitutionsVNIIEF
Thesis (1993)
Early life and career
Feodoritov was born in Sasovo, Ryazan Oblast, 319 km south-east of Moscow. He graduated with honours from the Faculty of Physics and Technology of Moscow State University in 1952. Straight from graduation, he became a researcher in the theoretical sector known as KB-11 in Arzamas-16, which was based in the closed city of Sarov, working under Yakov Zel'dovich. Employees were only allowed leave on officially sanctioned or organised days off work, on trips such as for hunting or fishing. Feodoritov became lost on one such trip and - fearful for his future - was aided in his return by locals (who were well aware of the function of Sarov), later finding from his boss (Andrei Sakharov) that the KGB had been informed and had organised a search party. He worked in this secret institution until the end of his life, starting as a senior laboratory assistant and progressing through to engineer, researcher, head of the research group, senior research fellow and chief of laboratory.[1][2]
He took part in the testing of nuclear weapons and was the scientific lead in a number of tests. Along with his project lead, Yevgeny Zababakhin, and in addition to his work on the RDS-37, he worked on calculations for the core part of the RDS-6s bomb, the first Soviet thermonuclear weapon, and also worked on a design which became the first Soviet serial tactical nuclear weapon, RDS-4. Both devices were successfully completed in 1953. For this work he received the Stalin Prize, third degree and the Medal "For Labour Valour".[3] He also worked on civilian nuclear projects. For further significant theoretical work and following successful weapons testing in 1954 and 1955, leading to a new generation of Soviet nuclear weaponry, he was awarded the Order of the Red Banner of Labour. In 1956, he originated further design improvements which helped lead to a new direction in Soviet nuclear weapons. He was part of the team which developed the RDS-220 thermonuclear weapon, the largest ever tested. With German Goncharov, he worked on the construction scheme of these types of weapons, and with Sakharov he analysed the efficiency of the theoretical model of the RDS-220. He received his PhD in Physical and Mathematical Sciences in 1968. For his role in the projects to develop nuclear weapons he was awarded the State Prize of the USSR in 1973. Later in his career, Feodoritov was one of the compilers of the Atomic Project of the USSR, specifically Documents and materials. Volume II. Atomic bomb. 1945 - 1954 Book 1 (published in 1999). He was honoured for his contributions as a scientist in 2000.[1][2][4]
In his personal life, he was admired for his humanity and cordiality and was the chair of the parent committees throughout his children's education. He was regarded as a Father Frost figure, referred to as "Uncle Slava" by friends of his children and the children of colleagues.[2]
Awards
• Twice laureate of the USSR State Prize/Stalin Prize (1953,1973).[2]
• Honoured scientist of the Russian Federation (2000)[1]
References
1. "Feodoritov V.P." Retrieved 8 September 2018.
2. Okutinoj, M. "Generation 15: Life Stories". Retrieved 8 September 2018.
3. L.D. Riabev, ed. (1998–2005). Atomic Project of the USSR: Documents and materials: Regulation SM of the USSR № 3044-1304 SS 'on awarding Stalin's prizes to scientific and engineering-technical Workers of the Ministry of the Middle Engineering and other departments for creation of a hydrogen bomb and new designs of atomic bombs.'. Vol. 3. Moscow: Federal Atomic Energy Agency. pp. 107–122.
4. Goncharov, G.A. (1996). "American and Soviet H-bomb development programmes: historical background". Uspekhi Fizicheskikh Nauk, Russian Academy of Sciences. 39 (10): 1033–1044. doi:10.1070/PU1996v039n10ABEH000174.
Soviet atomic bomb project
Sites
• Semipalatinsk
• Novaya Zemlya
• Semey
• Sukhoy Nos
• Totskoye
• Atomgrad
• Novouralsk
• Laboratory B
• Tryokhgorny
• Snezhinsk
• Zheleznogors
• Laboratory No. 2
Administrators
• Lavrentiy Beria
• Pavel Fitin
• Vyacheslav Molotov
• Mikhail Pervukhin
• Ivan Serov
• Joseph Stalin
• Pavel Sudoplatov
• Boris Vannikov
• Georgy Zhukov
Scientists
Soviets
• Viktor Adamsky
• Abram Alikhanov
• Anatoly Alexandrov
• Lev Artsimovich
• Yevgeny Avrorin
• Yuri Babayev
• V'yacheslav Danilenko
• Nikolay Dollezhal
• Vyacheslav Feodoritov
• Georgii Flerov
• Vitaly Ginzburg
• German Goncharov
• Pyotr Kapitsa
• Yulii Khariton
• Isaak Kikoin
• Igor Kurchatov
• Oleg Lavrentiev
• Boris Nikolsky
• Konstantin Petrzhak
• Isaak Pomeranchuk
• Evsei Rabinovich
• Yuri Romanov
• Andrei Sakharov
• Kirill Shchelkin
• Igor Tamm
• Yuri Trutnev
• Yevgeny Zababakhin
• Yakov Zel'dovich
Germans
(Russian Alsos)
• Manfred von Ardenne
• Heinz Barwich
• Robert Döpel
• Walter Herrmann
• Heinz Pose
• Ernst Rexer
• Nikolaus Riehl
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• more
Spy ring
• Morris Cohen
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Intelligence
• Intelligence cycle management
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• Soviet Army
• Russian espionage in the United States
• Perseus
Related Articles
• Russia and weapons of mass destruction
• Ukraine and weapons of mass destruction
• Kazakhstan and weapons of mass destruction
• Nuclear Explosions for the National Economy
See also: Nuclear weapons program of the Soviet Union
| Wikipedia |
Vyacheslav Kalashnikov Polishchuk
Vyacheslav Vitalievich Kalashnikov (born November 6, 1955) is a Russian mathematics professor and researcher currently working at the Tec de Monterrey, Monterrey Campus in Mexico. His work has been recognized by awards from the Ukrainian Academy of Sciences and the Central Economic Mathematical Institute of the Russian Academy of Sciences and is also a Level III member of Mexico’s Sistema Nacional de Investigadores.
Biography
He was born in Vladivostok, Russia and is married to mathematician Nataliya Kalashnykova. They have one daughter.[1]
Kalashnikov obtained his bachelors and masters in mathematics from the Novosibirsk State University in Novosibirsk, Siberia. He went on to obtain his doctorate here and at the Central Economic Mathematical Institute of the Russian Academy of Sciences in Moscow in mathematical cybernetics .[1]
He began his career as an associate professor at the Altai State University in Barnaul, Russia and in 1985 he became a researcher with the Mathematics Institute of the Russian Science Academy in Novosibirsk before moving onto the Sumy State University in Sumy, Ukraine in 1989. In 1995 he began working at the Central Economic Mathematical Instituteof the Russian Academy of Sciencesas head researcher. In 1998 he became the assistant director of the economics department of the University of Humanistic Sciences in Moscow.[1] In 2002, he was invited to teach at the Universidad Autónoma de Nuevo León by CONACYT for a two-year contract. At the time, he spoke no Spanish but was able to teach classes in English. At the end of the contract, he decided to stay in Mexico because his wife found a teaching position and his daughter had begun medical school in Mexico. He has been a researcher and professor at Tec de Monterrey, Monterrey Campus since 2004, mostly doing research but also teaching problem solving methods for bi level programming. He now speaks Spanish fluently.[2]
Kalashnikov’s main research specialty is optimization, especially in complementarity and unequal variables. In 2002 he published the book “Complementarity, Equilibrium, Efficiency and Economics” in London. In addition he has authored or co-authored eight book chapters and published twenty five journal articles. He has presented his work in conferences in Mexico and abroad.[1]
In 1993 Kalashnikov received a medal from the Ukrainian Academy of Sciences[1] and in 1997 an award from the Central Economic Mathematical Institute[1] for his research work. Since 2015, he has had Level III membership in the Sistema Nacional de Investigadores (National Roster of Researchers) of Mexico.[1][3] He also appears in the seventh edition of Five Hundred Leaders of Influence. He is a member of the Russian Association of Mathematical Programming, the American Mathematical Society and the Mexican Mathematical Society.[1]
References
1. "Kalashnikov Polishchuk, Vyacheslav" (in Spanish). Mexico: Instituto de Innovación y Tranferencia de Tecnología. Retrieved October 24, 2013.
2. "Slava Viacheslav: prefiere al Tec por su alta calidad y sistema integral" (in Spanish). Mexico: Tec de Monterrey. April 24, 2013. Retrieved October 24, 2013.
3. "Investigadores vigentes a enero de 2013" (PDF). Mexico: CONACYT. Archived from the original (PDF) on September 21, 2013. Retrieved October 13, 2013.
See also
List of Monterrey Institute of Technology and Higher Education faculty
| Wikipedia |
Vyacheslav Rychkov
Vyacheslav Rychkov (called Slava Rychkov, Russian Вячеслав Рычков, transcription Vyacheslav Rychkov; born 27 May 1975 in Samara, Russia [1]) is a Russian-Italian-French[1] theoretical physicist and mathematician.
Vyacheslav (Slava) Rychkov
Born (1975-05-27) May 27, 1975
Samara, Russia
NationalityRussian, Italian, French
Alma mater
• Moscow Institute of Physics and Technology (B.S. and M.S., 1996)
• Princeton University (Ph.D., 2002)
Awards
• New Horizons in Physics Prize (2014)
• Prix Mergier-Bourdeix (2019)
Scientific career
InstitutionsInstitut des Hautes Études Scientifiques
École Normale Supérieure
Websitesites.google.com/site/slavarychkov/
Career
In 1996, Rychkov obtained his diploma (B.S., M.S.) from the Moscow Institute of Physics and Technology.[1] From 1996 to 1998 he studied at the University of Jena.[2] He received his doctorate in mathematics from Princeton University, under the supervision of Elias Stein, in 2002[3] with a thesis titled "Estimates for Oscillatory Integral Operators".[4] Alexander Polyakov was his unofficial supervisor.[5] He was a post-doctoral fellow at the University of Amsterdam (2002-2005) and at the Scuola Normale Superiore in Pisa, where he became assistant professor in 2007.[5] In 2009 he became a professor of physics at the University of Paris VI and a member of the Laboratory of Theoretical Physics at the École Normale Supérieure in Paris.[1] Since 2012 to 2017 he was a staff member of the Department of Theoretical Physics at CERN.[5] He has been a Mitsubishi Heavy Industries Professor of High Energy Physics at École Normale Supérieure in Paris since 2016 [6] and in 2017 he became a permanent professor at the Institut des Hautes Études Scientifiques.[7]
Research
Slava Rychkov’s research interests mostly concern problems of theoretical physics to which the methods of quantum field theory and conformal field theory are applicable. He is the deputy director of the international collaboration on the nonperturbative bootstrap[8] financed by the Simons Foundation.
Awards
Between 2012 and 2017 he was a Junior Member of l’Institut Universitaire de France.[9]
In 2014 he received the New Horizons in Physics Prize “For developing new techniques in conformal field theory, reviving the conformal bootstrap program for constraining the spectrum of operators and the structure constants in 3D and 4D CFT’s.”[10]
He was the 2019 laureate of the Grand Prix Mergier-Bourdeix of the French Academy of Sciences for his work on the conformal bootstrap.[11]
References
1. "CVlong-Rychkov-2020.pdf". docs.google.com.
2. "Ehemalige Mitglieder (seit 1995)". cms.rz.uni-jena.de.
3. "Vyacheslav Rychkov - The Mathematics Genealogy Project". genealogy.math.ndsu.nodak.edu.
4. Rychkov, Vyacheslav S. (2002). "Vyacheslav Rychkov, "Estimates for Oscillatory Integral Operators", Princeton University PhD Thesis, math/0204269". arXiv:math/0204269.
5. "Record #990861 - INSPIRE-HEP".
6. "Inauguration des Chaires ENS –MHI - 25 octobre 2016".
7. "Slava Rychkov joins the Institute as a permanent professor". 12 December 2017.
8. "List of members of the Simons collaboration on Nonperturbative Bootstrap".
9. "Vyacheslav Rychkov's membership in Institut Universitaire de France".
10. "Breakthrough Prize – Fundamental Physics Breakthrough Prize Laureates – Vyacheslav Rychkov". breakthroughprize.org. Retrieved 2019-08-05.
11. "Lauréat 2019 du prix Mergier-Bourdeix : Slava Rychkov".
External links
• "Slava Rychkov - Taking the equations of Conformal Field Theory seriously". YouTube. GraduatePhysics. October 14, 2014.
• "V. Rychkov: Conformal Bootstrap - Lecture 1". YouTube. Int'l Centre for Theoretical Physics. March 11, 2016.
• "V. Rychkov: Conformal Bootstrap - Lecture 2". YouTube. Int'l Centre for Theoretical Physics. March 11, 2016.
• "V. Rychkov: Conformal Bootstrap - Lecture 3". YouTube. Int'l Centre for Theoretical Physics. March 12, 2016.
• "V. Rychkov: Conformal Bootstrap - Lecture 4". YouTube. Int'l Centre for Theoretical Physics. March 12, 2016.
• "[1/4] Slava Rychkov (2019) Lorentzian methods in conformal field theory". YouTube. IPhT-TV. October 1, 2019.
• "[2/4] Slava Rychkov (2019) Lorentzian methods in conformal field theory". YouTube. IPhT-TV. October 8, 2019.
• "[3/4] Slava Rychkov (2019) Lorentzian methods in conformal field theory". YouTube. IPhT-TV. October 15, 2019.
• "[4/4] Slava Rychkov (2019) Lorentzian methods in conformal field theory". YouTube. IPhT-TV. October 22, 2019.
• "Slava Rychkov - Reductionism vs Bootstrap (Nov 6, 2019)". YouTube. Simons Foundation. November 8, 2019.
• "Bootstrap Zoom 17 - Slava Rychkov". YouTube. Bootstrap Collaboration. September 11, 2020.
• "IRNQFS first meeting: Review talk by Slava Rychkov". YouTube. String Theory in Greater Paris. June 10, 2021.
• "Slava Rychkov Lecture 1 on Intro to CFTs and the Bootstrap in D greater than 2 dimensions". YouTube. TASI videos. September 3, 2021.
• "Slava Rychkov Lecture 2 on Intro to CFTs and the Bootstrap in D greater than 2 dimensions". YouTube. TASI videos. September 3, 2021.
• "Slava Rychkov Lecture 3 on Intro to CFTs and the Bootstrap in D greater than 2 dimensions". YouTube. TASI videos. September 3, 2021.
• "Slava Rychkov Lecture 4 on Intro to CFTs and the Bootstrap in D greater than 2 dimensions". YouTube. TASI videos. September 3, 2021.
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| Wikipedia |
Vyjayanthi Chari
Vyjayanthi Chari (born 1958)[1] is an Indian–American Distinguished Professor and the F. Burton Jones Endowed Chair for Pure Mathematics at the University of California, Riverside, known for her research in representation theory and quantum algebra.[2] In 2015 she was elected as a fellow of the American Mathematical Society.[3]
Education
Chari has a bachelor's, master's, and doctoral degree from the University of Mumbai.[2] Chari received her Ph.D. from the University of Mumbai under the supervision of Rajagopalan Parthasarathy.[4]
Professional career
Following her Ph.D., she became a fellow at the Tata Institute of Fundamental Research, Mumbai. In 1991, she joined the University of California, Riverside (UCR) where she is now a Distinguished Professor of Mathematics. During her career, she has had several visiting positions. They were: invited senior participant at the Mittag-Leffler Institute, Sweden; an invited professor at the University of Cologne, Germany; an invited professor at Paris 7, France; an invited research fellow at Brown University, RI; and an invited senior participant at Hausdorff Research Institute for Mathematics, Bonn, Germany; and visiting professor at the University of Rome Tor Vergata, Italy.[5]
She is also the editor of the Pacific Journal of Mathematics and the Editor in Chief of Algebras and Representation Theory.
With Andrew N. Pressley, she is the author of the book A Guide to Quantum Groups (Cambridge University Press, 1994).[6]
Honors
• The Doctoral Dissertation Advisor/Mentor Award from the UCR Academic Senate.[5]
• American Mathematical Society 2016 Class of Fellows.
• Simons Fellow 2019–2020.
• Infosys Visiting Chair Professor, Indian Institute of Science, 2019–2023.
References
1. Birth year from ISNI authority control file, accessed 2018-11-26.
2. Faculty profile, Department of Mathematics, University of California, Riverside, retrieved 2015-11-17.
3. 2016 Class of the Fellows of the AMS, American Mathematical Society, retrieved 2015-11-17.
4. Vyjayanthi Chari at the Mathematics Genealogy Project
5. October 30, Iqbal Pittalwala on; 2015. "Mathematician Named Fellow of American Mathematical Society". UCR Today. Retrieved 2020-01-17.{{cite web}}: CS1 maint: numeric names: authors list (link)
6. Review of A Guide to Quantum Groups by Tomasz Brzeziński (1995), MR1300632.
External links
• Vyjayanthi Chari publications indexed by Google Scholar
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Václav Jeřábek
Václav Jeřábek (1845–1931) was a Czech mathematician, specialized in constructive geometry.
Václav Jeřábek
Jerabek Hyperbola
Born(1845-12-11)December 11, 1845
Koloděje, Pardubice Region, Austrian Empire; now Czech Republic
DiedDecember 20, 1931(1931-12-20) (aged 86)
Telč, Czechoslovakia; now Czech Republic
Alma materVienna Polytechnic Institute
Scientific career
FieldsMathematics
InstitutionsCzech Realschule of Brno
Life and work
Jeřábek studied at the lower school of Pardubice and at the higher school of Písek, then he was to Vienna and studied at Imperial and Royal Polytechnic Institute where he graduated. Although he participated in several leading intellectual circles of Vienna, he remained a Czech with a clear view of patriotism.[1] He began his teaching at the Realschule of Litomyšl (1870), being transferred two years after to the Realschule of Telč. In 1881, he was appointed professor of the Czech Realschule in Brno, and became its director in 1901. He retired in 1907, and suffering of a cataract, he died almost completely blind[2] in 1931.
Jeřábek was one of the men who kept the Czech geometry at the scientific level.[3] He published scientific articles in Czech, German and French, and longer lectures. He is well remembered by the Jerabek hyperbola,[4] the locus of the isogonal conjugate of a point that traverses the Euler line of a triangle.[5]
He was honorary member of the Union of Czech mathematicians and member of the scientific societies of Moravia and Bohemia.[6]
References
1. Roháček 1932, p. 105.
2. O'Connor & Robertson, MacTutor History of Mathematics.
3. Roháček 1932, p. 107.
4. Kimberling 1997, p. 436.
5. Kimberling 2003, p. 58.
6. Roháček 1932, p. 108.
Bibliography
• Kimberling, Clark (1997). "Major Centers of Triangles". The American Mathematical Monthly. 104 (5): 431–438. doi:10.1080/00029890.1997.11990660. ISSN 0002-9890. JSTOR 2974736.
• Kimberling, Clark (2003). Geometry in action. Key College Publishing. ISBN 1-931914-02-8.
• Roháček, J. (1932). "Václav Jeřábek [nekrolog]". Časopis Pro Pěstování Matematiky a Fysiky (in Czech). 61 (4): 105–108. doi:10.21136/CPMF.1932.121310. hdl:10338.dmlcz/121310. ISSN 1802-114X.
External links
• O'Connor, John J.; Robertson, Edmund F., "Václav Jeřábek", MacTutor History of Mathematics Archive, University of St Andrews
• Weisstein, Eric W. "Jerabek Hyperbola". MathWorld--A Wolfram Web Resource. Retrieved 5 December 2017.
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Vámos matroid
In mathematics, the Vámos matroid or Vámos cube is a matroid over a set of eight elements that cannot be represented as a matrix over any field. It is named after English mathematician Peter Vámos, who first described it in an unpublished manuscript in 1968.[1]
Definition
The Vámos matroid has eight elements, which may be thought of as the eight vertices of a cube or cuboid. The matroid has rank 4: all sets of three or fewer elements are independent, and 65 of the 70 possible sets of four elements are also independent. The five exceptions are four-element circuits in the matroid. Four of these five circuits are formed by faces of the cuboid (omitting two opposite faces). The fifth circuit connects two opposite edges of the cuboid, each of which is shared by two of the chosen four faces.
Another way of describing the same structure is that it has two elements for each vertex of the diamond graph, and a four-element circuit for each edge of the diamond graph.
Properties
• The Vámos matroid is a paving matroid, meaning that all of its circuits have size at least equal to its rank.[2]
• The Vámos matroid is isomorphic to its dual matroid, but it is not identically self-dual (the isomorphism requires a nontrivial permutation of the matroid elements).[2]
• The Vámos matroid cannot be represented over any field. That is, it is not possible to find a vector space, and a system of eight vectors within that space, such that the matroid of linear independence of these vectors is isomorphic to the Vámos matroid.[3] Indeed, it is one of the smallest non-representable matroids,[4] and served as a counterexample to a conjecture of Ingleton that the matroids on eight or fewer elements were all representable.[5]
• The Vámos matroid is a forbidden minor for the matroids representable over a field $F$, whenever $F$ has five or more elements.[6] However, it is not possible to test in polynomial time whether it is a minor of a given matroid $M$, given access to $M$ through an independence oracle.[7]
• The Vámos matroid is not algebraic. That is, there do not exist a field extension $L/K$ and a set of eight elements of $L$, such that the transcendence degree of sets of these eight elements equals the rank function of the matroid.[8]
• The Vámos matroid is not a secret-sharing matroid.[9] Secret-sharing matroids describe "ideal" secret sharing schemes in which any coalition of users who can gain any information about a secret key can learn the whole key (these coalitions are the dependent sets of the matroid), and in which the shared information contains no more information than is needed to represent the key.[10] These matroids also have applications in coding theory.[11]
• The Vámos matroid has no adjoint. This means that the dual lattice of the geometric lattice of the Vámos matroid cannot be order-embedded into another geometric lattice of the same rank.[12]
• The Vámos matroid can be oriented.[13] In oriented matroids, a form of the Hahn–Banach theorem follows from a certain intersection property of the flats of the matroid; the Vámos matroid provides an example of a matroid in which the intersection property is not true, but the Hahn–Banach theorem nevertheless holds.[14]
• The Tutte polynomial of the Vámos matroid is[15]
$x^{4}+4x^{3}+10x^{2}+15x+5xy+15y+10y^{2}+4y^{3}+y^{4}.$
References
1. Vámos, P. (1968), On the representation of independence structures.
2. Oxley, James G. (2006), "Example 2.1.22", Matroid Theory, Oxford Graduate Texts in Mathematics, vol. 3, Oxford University Press, p. 76, ISBN 9780199202508.
3. Oxley (2006), pp. 170–172.
4. Oxley (2006), Prop. 6.4.10, p. 196. A proof of representability for every matroid with fewer elements or the same number but smaller rank was given by Fournier, Jean-Claude (1970), "Sur la représentation sur un corps des matroïdes à sept et huit éléments", Comptes rendus de l'Académie des sciences, Sér. A-B, 270: A810–A813, MR 0263665.
5. Ingleton, A. W. (1959), "A note on independence functions and rank", Journal of the London Mathematical Society, Second Series, 34: 49–56, doi:10.1112/jlms/s1-34.1.49, MR 0101848.
6. Oxley (2006), p. 511.
7. Seymour, P. D.; Walton, P. N. (1981), "Detecting matroid minors", Journal of the London Mathematical Society, Second Series, 23 (2): 193–203, doi:10.1112/jlms/s2-23.2.193, MR 0609098. Jensen, Per M.; Korte, Bernhard (1982), "Complexity of matroid property algorithms", SIAM Journal on Computing, 11 (1): 184–190, doi:10.1137/0211014, MR 0646772.
8. Ingleton, A. W.; Main, R. A. (1975), "Non-algebraic matroids exist", Bulletin of the London Mathematical Society, 7: 144–146, doi:10.1112/blms/7.2.144, MR 0369110.
9. Seymour, P. D. (1992), "On secret-sharing matroids", Journal of Combinatorial Theory, Series B, 56 (1): 69–73, doi:10.1016/0095-8956(92)90007-K, MR 1182458.
10. Brickell, Ernest F.; Davenport, Daniel M. (1991), "On the classification of ideal secret sharing schemes", Journal of Cryptology, 4 (2): 123–134, doi:10.1007/BF00196772.
11. Simonis, Juriaan; Ashikhmin, Alexei (1998), "Almost affine codes", Designs, Codes and Cryptography, 14 (2): 179–197, doi:10.1023/A:1008244215660, MR 1614357.
12. Cheung, Alan L. C. (1974), "Adjoints of a geometry", Canadian Mathematical Bulletin, 17 (3): 363–365, correction, ibid. 17 (1974), no. 4, 623, doi:10.4153/CMB-1974-066-5, MR 0373976.
13. Bland, Robert G.; Las Vergnas, Michel (1978), "Orientability of matroids", Journal of Combinatorial Theory, Series B, 24 (1): 94–123, doi:10.1016/0095-8956(78)90080-1, MR 0485461.
14. Bachem, Achim; Wanka, Alfred (1988), "Separation theorems for oriented matroids", Discrete Mathematics, 70 (3): 303–310, doi:10.1016/0012-365X(88)90006-4, MR 0955127.
15. Merino, Criel; Ramírez-Ibáñez, Marcelino; Sanchez, Guadalupe Rodríguez (2012), The Tutte polynomial of some matroids, arXiv:1203.0090, Bibcode:2012arXiv1203.0090M.
| Wikipedia |
Victor Vâlcovici
Victor Vâlcovici (21 September [O.S. 9 September] 1885 – 21 June 1970) was a Romanian mechanician and mathematician.
Victor Vâlcovici
Vâlcovici in the 1930s
Born(1885-09-21)September 21, 1885
Galați, Kingdom of Romania
Died21 June 1970(1970-06-21) (aged 84)
Bucharest, Socialist Republic of Romania
Resting placeBellu Cemetery, Bucharest
EducationNicolae Bălcescu High School
Alma materUniversity of Bucharest
University of Göttingen
Scientific career
FieldsMathematics, Mechanics
InstitutionsUniversity of Iași
Polytechnic School of Timișoara
University of Bucharest
ThesisUeber die diskontinuierliche Flussigkeitsbewegungen mit zwei freien Strahlen (1913)
Doctoral advisorLudwig Prandtl
Minister of Public Works and Communications
In office
April 18, 1931 – June 5, 1932
Prime MinisterNicolae Iorga
Preceded byIon Răducanu
Succeeded byGheorghe Mironescu
Minister of Justice
In office
January 7 – January 9, 1932
Prime MinisterNicolae Iorga
Preceded byConstantin Hamangiu
Succeeded byValer Pop
Biography
Born into a modest family in Galați, he graduated first in his class in 1904 from Nicolae Bălcescu High School in Brăila. Entering the University of Bucharest on a scholarship, he attended its faculty of sciences, where he had as teachers Spiru Haret and Gheorghe Țițeica.[1] After graduating in 1907 with a degree in mathematics, he taught high school for two years before leaving for University of Göttingen on another scholarship to pursue a doctorate in mathematics. He wrote his thesis under the direction of Ludwig Prandtl and defended it in 1913; the thesis, titled Ueber die diskontinuierliche Flussigkeitsbewegungen mit zwei freien Strahlen (Discontinuous flow of liquids in two free dimensions),[2][3][4] amplified upon the work of Bernhard Riemann.[5]
He was subsequently named assistant professor of mechanics at the University of Iași, rising to full professor in 1918.[6] In 1921, he became rector of the Polytechnic School of Timișoara. There, he was also professor of rational mechanics and founded a laboratory dedicated to the field.[5] During his nine years as rector, he worked to place the recently founded university on a solid foundation.[6] From 1930 until retiring in 1962, he taught experimental mechanics at the University of Bucharest.[5] In the government of Nicolae Iorga, he served as Minister of Public Works from 1931 to 1932. During this time, he introduced a modern road network that featured paved highways.[5][6] In 1936 he gave an invited talk at the International Congress of Mathematicians in Oslo, with title Sur le sillage derrière un obstacle circulaire (In the wake of a circular obstacle).[7]
Elected a corresponding member of the Romanian Academy in 1936,[8] he was stripped of his membership by the new communist regime in 1948,[9]: 123 but made a titular member of the Romanian Academy in 1965.[10] His numerous articles on theoretical and applied mechanics covered topics such as the principles of variational mechanics, the mechanics of ideal fluid flow, the theory of elasticity and astronomy.[5]
He died in 1970 in Bucharest, and was buried in the city's Bellu Cemetery. Streets have been named after Victor Vâlcovici in Brăila, Galați, and Timișoara; a school in Galați also bears his name.
Books
• Vâlcovici, Victor (1958). Une extension des liaisons non holonomes et des principes variationnels (in French). De Gruyter. doi:10.1515/9783112498545. ISBN 978-3-11-249854-5. MR 0090218. S2CID 118604371.
• Vâlcovici, Victor (1971). Mecanica fluidelor și teoria elasticității (in Romanian). București: Editura Academiei Republicii Socialiste România. OCLC 886523167.
Notes
1. Predescu, D.C. (May 23, 2013). "Gălățeni care au uimit lumea – Victor Vâlcovici". Viața Liberă (in Romanian). Retrieved February 16, 2022.
2. Victor Vâlcovici at the Mathematics Genealogy Project
3. Otlăcan, pp. 125–6
4. Vâlcovici, Victor (1913), Über diskontinuierliche Flüssigkeitsbewegungen mit zwei freien Strahlen (in German), Göttingen: Dieterich, JFM 44.0859.01
5. Hager, p. 1361
6. Otlăcan, p. 127
7. Comptes rendus du Congrès International des Mathématiciens: Oslo, 14–18 julliet 1936 (PDF), Oslo: A.W. Brøggers, 1937, p. 250, OCLC 1404131
8. Otlăcan, p. 126, 127
9. Otiman, Păun Ion (December 2013). "1948–Anul imensei jertfe a Academiei Române" (PDF). Akademos (in Romanian). 4 (31): 115–124.
10. "Membrii Academiei Române din 1866 până în prezent" (in Romanian). Romanian Academy.
References
• Willi Hager, Hydraulicians in Europe (1800–2000), vol. 2. CRC Press, Boca Raton, Florida, 2009. ISBN 978-1-4665-5498-6
• (in Romanian) Eufrosina Otlăcan, "Victor Vâlcovici (1885–1970) – savant și desăvârșit pedagog", NOEMA, vol. VI, 2007, pp. 124–29
External links
• "Valcovici, Victor (1885–1970)". www.dmg-lib.org. Retrieved February 16, 2022.
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Véronique Gayrard
Véronique Gayrard is a French mathematician specializing in probability and statistical physics, with research topics including Hopfield networks, the long-term behavior of the random energy model and similar glassy systems, and metastability in reversible diffusion. She is a director of research for the French National Centre for Scientific Research (CNRS), affiliated with the Marseille Institute of Mathematics (I2M) operated jointly by CNRS and Aix-Marseille University. At I2M, she is affiliated with the research group on the mathematics of randomness (ALEA), which she headed from 2015 to 2021.[1]
Education
Gayrard earned her doctorate in 1993 through the University of the Mediterranean Aix-Marseille II, now part of Aix-Marseille University.[2] Her dissertation, Contribution à l'étude rigoureuse des modèles de Hoppfiel [Contributions to the rigorous study of Hopfield models], was supervised by Pierre Picco.[3]
Recognition
In December 2021, Gayrard was named as one of the winners of the Gay-Lussac Humboldt Prize, an annual award of the governments of France and Germany honoring outstanding binational contributions in all areas of science. The award was based in part on her long and prolific collaborations with Anton Bovier, a mathematician at the Institute of Applied Mathematics of the University of Bonn.[4]
References
1. "Véronique Gayrard", I2M, Aix-Marseille University, retrieved 2022-01-09
2. Véronique Gayrard at the Mathematics Genealogy Project
3. "Véronique Gayrard", theses.fr, retrieved 2022-01-09
4. "The Gay Lussac Humboldt Award goes to Véronique Gayrard: Close cooperation with Anton Bovier", Hausdorff Center for Mathematics, University of Bonn, 15 December 2021, retrieved 2022-01-09
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| Wikipedia |
Parameterized complexity
In computer science, parameterized complexity is a branch of computational complexity theory that focuses on classifying computational problems according to their inherent difficulty with respect to multiple parameters of the input or output. The complexity of a problem is then measured as a function of those parameters. This allows the classification of NP-hard problems on a finer scale than in the classical setting, where the complexity of a problem is only measured as a function of the number of bits in the input. This appears to have been first demonstrated in Gurevich, Stockmeyer & Vishkin (1984). The first systematic work on parameterized complexity was done by Downey & Fellows (1999).
Under the assumption that P ≠ NP, there exist many natural problems that require superpolynomial running time when complexity is measured in terms of the input size only but that are computable in a time that is polynomial in the input size and exponential or worse in a parameter k. Hence, if k is fixed at a small value and the growth of the function over k is relatively small then such problems can still be considered "tractable" despite their traditional classification as "intractable".
The existence of efficient, exact, and deterministic solving algorithms for NP-complete, or otherwise NP-hard, problems is considered unlikely, if input parameters are not fixed; all known solving algorithms for these problems require time that is exponential (so in particular superpolynomial) in the total size of the input. However, some problems can be solved by algorithms that are exponential only in the size of a fixed parameter while polynomial in the size of the input. Such an algorithm is called a fixed-parameter tractable (fpt-)algorithm, because the problem can be solved efficiently (i.e., in polynomial time) for constant values of the fixed parameter.
Problems in which some parameter k is fixed are called parameterized problems. A parameterized problem that allows for such an fpt-algorithm is said to be a fixed-parameter tractable problem and belongs to the class FPT, and the early name of the theory of parameterized complexity was fixed-parameter tractability.
Many problems have the following form: given an object x and a nonnegative integer k, does x have some property that depends on k? For instance, for the vertex cover problem, the parameter can be the number of vertices in the cover. In many applications, for example when modelling error correction, one can assume the parameter to be "small" compared to the total input size. Then it is challenging to find an algorithm that is exponential only in k, and not in the input size.
In this way, parameterized complexity can be seen as two-dimensional complexity theory. This concept is formalized as follows:
A parameterized problem is a language $L\subseteq \Sigma ^{*}\times \mathbb {N} $, where $\Sigma $ is a finite alphabet. The second component is called the parameter of the problem.
A parameterized problem L is fixed-parameter tractable if the question "$(x,k)\in L$?" can be decided in running time $f(k)\cdot |x|^{O(1)}$, where f is an arbitrary function depending only on k. The corresponding complexity class is called FPT.
For example, there is an algorithm that solves the vertex cover problem in $O(kn+1.274^{k})$ time,[1] where n is the number of vertices and k is the size of the vertex cover. This means that vertex cover is fixed-parameter tractable with the size of the solution as the parameter.
Complexity classes
FPT
FPT contains the fixed parameter tractable problems, which are those that can be solved in time $f(k)\cdot {|x|}^{O(1)}$ for some computable function f. Typically, this function is thought of as single exponential, such as $2^{O(k)}$, but the definition admits functions that grow even faster. This is essential for a large part of the early history of this class. The crucial part of the definition is to exclude functions of the form $f(n,k)$, such as $k^{n}$.
The class FPL (fixed parameter linear) is the class of problems solvable in time $f(k)\cdot |x|$ for some computable function f.[2] FPL is thus a subclass of FPT. An example is the Boolean satisfiability problem, parameterised by the number of variables. A given formula of size m with k variables can be checked by brute force in time $O(2^{k}m)$. A vertex cover of size k in a graph of order n can be found in time $O(2^{k}n)$, so the vertex cover problem is also in FPL.
An example of a problem that is thought not to be in FPT is graph coloring parameterised by the number of colors. It is known that 3-coloring is NP-hard, and an algorithm for graph k-coloring in time $f(k)n^{O(1)}$ for $k=3$ would run in polynomial time in the size of the input. Thus, if graph coloring parameterised by the number of colors were in FPT, then P = NP.
There are a number of alternative definitions of FPT. For example, the running-time requirement can be replaced by $f(k)+|x|^{O(1)}$. Also, a parameterised problem is in FPT if it has a so-called kernel. Kernelization is a preprocessing technique that reduces the original instance to its "hard kernel", a possibly much smaller instance that is equivalent to the original instance but has a size that is bounded by a function in the parameter.
FPT is closed under a parameterised notion of reductions called fpt-reductions. Such reductions transform an instance $(x,k)$ of some problem into an equivalent instance $(x',k')$ of another problem (with $k'\leq g(k)$) and can be computed in time $f(k)\cdot p(|x|)$ where $p$ is a polynomial.
Obviously, FPT contains all polynomial-time computable problems. Moreover, it contains all optimisation problems in NP that allow an efficient polynomial-time approximation scheme (EPTAS).
W hierarchy
The W hierarchy is a collection of computational complexity classes. A parameterized problem is in the class W[i], if every instance $(x,k)$ can be transformed (in fpt-time) to a combinatorial circuit that has weft at most i, such that $(x,k)\in L$ if and only if there is a satisfying assignment to the inputs that assigns 1 to exactly k inputs. The weft is the largest number of logical units with fan-in greater than two on any path from an input to the output. The total number of logical units on the paths (known as depth) must be limited by a constant that holds for all instances of the problem.
Note that ${\mathsf {FPT}}=W[0]$ and $W[i]\subseteq W[j]$ for all $i\leq j$. The classes in the W hierarchy are also closed under fpt-reduction.
Many natural computational problems occupy the lower levels, W[1] and W[2].
W[1]
Examples of W[1]-complete problems include
• deciding if a given graph contains a clique of size k
• deciding if a given graph contains an independent set of size k
• deciding if a given nondeterministic single-tape Turing machine accepts within k steps ("short Turing machine acceptance" problem). This also applies to nondeterministic Turing machines with f(k) tapes and even f(k) of f(k)-dimensional tapes, but even with this extension, the restriction to f(k) tape alphabet size is fixed-parameter tractable. Crucially, the branching of the Turing machine at each step is allowed to depend on n, the size of the input. In this way, the Turing machine may explore nO(k) computation paths.
W[2]
Examples of W[2]-complete problems include
• deciding if a given graph contains a dominating set of size k
• deciding if a given nondeterministic multi-tape Turing machine accepts within k steps ("short multi-tape Turing machine acceptance" problem). Crucially, the branching is allowed to depend on n (like the W[1] variant), as is the number of tapes. An alternate W[2]-complete formulation allows only single-tape Turing machines, but the alphabet size may depend on n.
W[t]
$W[t]$ can be defined using the family of Weighted Weft-t-Depth-d SAT problems for $d\geq t$: $W[t,d]$ is the class of parameterized problems that fpt-reduce to this problem, and $W[t]=\bigcup _{d\geq t}W[t,d]$.
Here, Weighted Weft-t-Depth-d SAT is the following problem:
• Input: A Boolean formula of depth at most d and weft at most t, and a number k. The depth is the maximal number of gates on any path from the root to a leaf, and the weft is the maximal number of gates of fan-in at least three on any path from the root to a leaf.
• Question: Does the formula have a satisfying assignment of Hamming weight exactly k?
It can be shown that for $t\geq 2$ the problem Weighted t-Normalize SAT is complete for $W[t]$ under fpt-reductions.[3] Here, Weighted t-Normalize SAT is the following problem:
• Input: A Boolean formula of depth at most t with an AND-gate on top, and a number k.
• Question: Does the formula have a satisfying assignment of Hamming weight exactly k?
W[P]
W[P] is the class of problems that can be decided by a nondeterministic $h(k)\cdot {|x|}^{O(1)}$-time Turing machine that makes at most $O(f(k)\cdot \log n)$ nondeterministic choices in the computation on $(x,k)$ (a k-restricted Turing machine). Flum & Grohe (2006)
It is known that FPT is contained in W[P], and the inclusion is believed to be strict. However, resolving this issue would imply a solution to the P versus NP problem.
Other connections to unparameterised computational complexity are that FPT equals W[P] if and only if circuit satisfiability can be decided in time $\exp(o(n))m^{O(1)}$, or if and only if there is a computable, nondecreasing, unbounded function f such that all languages recognised by a nondeterministic polynomial-time Turing machine using $f(n)\log n$ nondeterministic choices are in P.
W[P] can be loosely thought of as the class of problems where we have a set S of n items, and we want to find a subset $T\subset S$ of size k such that a certain property holds. We can encode a choice as a list of k integers, stored in binary. Since the highest any of these numbers can be is n, $\lceil \log _{2}n\rceil $ bits are needed for each number. Therefore $k\cdot \lceil \log _{2}n\rceil $ total bits are needed to encode a choice. Therefore we can select a subset $T\subset S$ with $O(k\cdot \log n)$ nondeterministic choices.
XP
XP is the class of parameterized problems that can be solved in time $n^{f(k)}$ for some computable function f. These problems are called slicewise polynomial, in the sense that each "slice" of fixed k has a polynomial algorithm, although possibly with a different exponent for each k. Compare this with FPT, which merely allows a different constant prefactor for each value of k. XP contains FPT, and it is known that this containment is strict by diagonalization.
para-NP
para-NP is the class of parameterized problems that can be solved by a nondeterministic algorithm in time $f(k)\cdot |x|^{O(1)}$ for some computable function f. It is known that ${\textsf {FPT}}={\textsf {para-NP}}$ if and only if ${\textsf {P}}={\textsf {NP}}$.[4]
A problem is para-NP-hard if it is ${\textsf {NP}}$-hard already for a constant value of the parameter. That is, there is a "slice" of fixed k that is ${\textsf {NP}}$-hard. A parameterized problem that is ${\textsf {para-NP}}$-hard cannot belong to the class ${\textsf {XP}}$, unless ${\textsf {P}}={\textsf {NP}}$. A classic example of a ${\textsf {para-NP}}$-hard parameterized problem is graph coloring, parameterized by the number k of colors, which is already ${\textsf {NP}}$-hard for $k=3$ (see Graph coloring#Computational complexity).
A hierarchy
The A hierarchy is a collection of computational complexity classes similar to the W hierarchy. However, while the W hierarchy is a hierarchy contained in NP, the A hierarchy more closely mimics the polynomial-time hierarchy from classical complexity. It is known that A[1] = W[1] holds.
See also
• Parameterized approximation algorithm, for optimization problems an algorithm running in FPT time might approximate the solution.
Notes
1. Chen, Kanj & Xia 2006
2. Grohe (1999)
3. Buss, Jonathan F; Islam, Tarique (2006). "Simplifying the weft hierarchy". Theoretical Computer Science. 351 (3): 303–313. doi:10.1016/j.tcs.2005.10.002.
4. Flum & Grohe (2006), p. 39.
References
• Chen, Jianer; Kanj, Iyad A.; Xia, Ge (2006). Improved Parameterized Upper Bounds for Vertex Cover. pp. 238–249. CiteSeerX 10.1.1.432.831. doi:10.1007/11821069_21. ISBN 978-3-540-37791-7. {{cite book}}: |journal= ignored (help)
• Cygan, Marek; Fomin, Fedor V.; Kowalik, Lukasz; Lokshtanov, Daniel; Marx, Daniel; Pilipczuk, Marcin; Pilipczuk, Michal; Saurabh, Saket (2015). Parameterized Algorithms. Springer. p. 555. ISBN 978-3-319-21274-6.
• Downey, Rod G.; Fellows, Michael R. (1999). Parameterized Complexity. Springer. ISBN 978-0-387-94883-6.
• Flum, Jörg; Grohe, Martin (2006). Parameterized Complexity Theory. Springer. ISBN 978-3-540-29952-3.
• Fomin, Fedor V.; Lokshtanov, Daniel; Saurabh, Saket; Zehavi, Meirav (2019). Kernelization: Theory of Parameterized Preprocessing. Cambridge University Press. p. 528. doi:10.1017/9781107415157. ISBN 978-1107057760.
• Gurevich, Yuri; Stockmeyer, Larry; Vishkin, Uzi (1984). Solving NP-hard problems on graphs that are almost trees and an application to facility location problems. Journal of the ACM. p. 459-473.
• Niedermeier, Rolf (2006). Invitation to Fixed-Parameter Algorithms. Oxford University Press. ISBN 978-0-19-856607-6. Archived from the original on 2008-09-24.
• Grohe, Martin (1999). "Descriptive and Parameterized Complexity". Computer Science Logic. Lecture Notes in Computer Science. Vol. 1683. Springer Berlin Heidelberg. pp. 14–31. CiteSeerX 10.1.1.25.9250. doi:10.1007/3-540-48168-0_3. ISBN 978-3-540-66536-6.
• The Computer Journal. Volume 51, Numbers 1 and 3 (2008). The Computer Journal. Special Double Issue on Parameterized Complexity with 15 survey articles, book review, and a Foreword by Guest Editors R. Downey, M. Fellows and M. Langston.
External links
• Wiki on parameterized complexity
• Compendium of Parameterized Problems
| Wikipedia |
Von Neumann algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra.
"operator ring" redirects here. Not to be confused with ring operator or operator assistance.
Von Neumann algebras were originally introduced by John von Neumann, motivated by his study of single operators, group representations, ergodic theory and quantum mechanics. His double commutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries.
Two basic examples of von Neumann algebras are as follows:
• The ring $L^{\infty }(\mathbb {R} )$ of essentially bounded measurable functions on the real line is a commutative von Neumann algebra, whose elements act as multiplication operators by pointwise multiplication on the Hilbert space $L^{2}(\mathbb {R} )$ of square-integrable functions.
• The algebra ${\mathcal {B}}({\mathcal {H}})$ of all bounded operators on a Hilbert space ${\mathcal {H}}$ is a von Neumann algebra, non-commutative if the Hilbert space has dimension at least $2$.
Von Neumann algebras were first studied by von Neumann (1930) in 1929; he and Francis Murray developed the basic theory, under the original name of rings of operators, in a series of papers written in the 1930s and 1940s (F.J. Murray & J. von Neumann 1936, 1937, 1943; J. von Neumann 1938, 1940, 1943, 1949), reprinted in the collected works of von Neumann (1961).
Introductory accounts of von Neumann algebras are given in the online notes of Jones (2003) and Wassermann (1991) and the books by Dixmier (1981), Schwartz (1967), Blackadar (2005) and Sakai (1971). The three volume work by Takesaki (1979) gives an encyclopedic account of the theory. The book by Connes (1994) discusses more advanced topics.
Definitions
There are three common ways to define von Neumann algebras.
The first and most common way is to define them as weakly closed *-algebras of bounded operators (on a Hilbert space) containing the identity. In this definition the weak (operator) topology can be replaced by many other common topologies including the strong, ultrastrong or ultraweak operator topologies. The *-algebras of bounded operators that are closed in the norm topology are C*-algebras, so in particular any von Neumann algebra is a C*-algebra.
The second definition is that a von Neumann algebra is a subalgebra of the bounded operators closed under involution (the *-operation) and equal to its double commutant, or equivalently the commutant of some subalgebra closed under *. The von Neumann double commutant theorem (von Neumann 1930) says that the first two definitions are equivalent.
The first two definitions describe a von Neumann algebra concretely as a set of operators acting on some given Hilbert space. Sakai (1971) showed that von Neumann algebras can also be defined abstractly as C*-algebras that have a predual; in other words the von Neumann algebra, considered as a Banach space, is the dual of some other Banach space called the predual. The predual of a von Neumann algebra is in fact unique up to isomorphism. Some authors use "von Neumann algebra" for the algebras together with a Hilbert space action, and "W*-algebra" for the abstract concept, so a von Neumann algebra is a W*-algebra together with a Hilbert space and a suitable faithful unital action on the Hilbert space. The concrete and abstract definitions of a von Neumann algebra are similar to the concrete and abstract definitions of a C*-algebra, which can be defined either as norm-closed *-algebras of operators on a Hilbert space, or as Banach *-algebras such that ||aa*||=||a|| ||a*||.
Terminology
Some of the terminology in von Neumann algebra theory can be confusing, and the terms often have different meanings outside the subject.
• A factor is a von Neumann algebra with trivial center, i.e. a center consisting only of scalar operators.
• A finite von Neumann algebra is one which is the direct integral of finite factors (meaning the von Neumann algebra has a faithful normal tracial state $\tau :M\rightarrow \mathbb {C} $[1]). Similarly, properly infinite von Neumann algebras are the direct integral of properly infinite factors.
• A von Neumann algebra that acts on a separable Hilbert space is called separable. Note that such algebras are rarely separable in the norm topology.
• The von Neumann algebra generated by a set of bounded operators on a Hilbert space is the smallest von Neumann algebra containing all those operators.
• The tensor product of two von Neumann algebras acting on two Hilbert spaces is defined to be the von Neumann algebra generated by their algebraic tensor product, considered as operators on the Hilbert space tensor product of the Hilbert spaces.
By forgetting about the topology on a von Neumann algebra, we can consider it a (unital) *-algebra, or just a ring. Von Neumann algebras are semihereditary: every finitely generated submodule of a projective module is itself projective. There have been several attempts to axiomatize the underlying rings of von Neumann algebras, including Baer *-rings and AW*-algebras. The *-algebra of affiliated operators of a finite von Neumann algebra is a von Neumann regular ring. (The von Neumann algebra itself is in general not von Neumann regular.)
Commutative von Neumann algebras
Main article: Abelian von Neumann algebra
The relationship between commutative von Neumann algebras and measure spaces is analogous to that between commutative C*-algebras and locally compact Hausdorff spaces. Every commutative von Neumann algebra is isomorphic to L∞(X) for some measure space (X, μ) and conversely, for every σ-finite measure space X, the *-algebra L∞(X) is a von Neumann algebra.
Due to this analogy, the theory of von Neumann algebras has been called noncommutative measure theory, while the theory of C*-algebras is sometimes called noncommutative topology (Connes 1994).
Projections
Operators E in a von Neumann algebra for which E = EE = E* are called projections; they are exactly the operators which give an orthogonal projection of H onto some closed subspace. A subspace of the Hilbert space H is said to belong to the von Neumann algebra M if it is the image of some projection in M. This establishes a 1:1 correspondence between projections of M and subspaces that belong to M. Informally these are the closed subspaces that can be described using elements of M, or that M "knows" about.
It can be shown that the closure of the image of any operator in M and the kernel of any operator in M belongs to M. Also, the closure of the image under an operator of M of any subspace belonging to M also belongs to M. (These results are a consequence of the polar decomposition).
Comparison theory of projections
The basic theory of projections was worked out by Murray & von Neumann (1936). Two subspaces belonging to M are called (Murray–von Neumann) equivalent if there is a partial isometry mapping the first isomorphically onto the other that is an element of the von Neumann algebra (informally, if M "knows" that the subspaces are isomorphic). This induces a natural equivalence relation on projections by defining E to be equivalent to F if the corresponding subspaces are equivalent, or in other words if there is a partial isometry of H that maps the image of E isometrically to the image of F and is an element of the von Neumann algebra. Another way of stating this is that E is equivalent to F if E=uu* and F=u*u for some partial isometry u in M.
The equivalence relation ~ thus defined is additive in the following sense: Suppose E1 ~ F1 and E2 ~ F2. If E1 ⊥ E2 and F1 ⊥ F2, then E1 + E2 ~ F1 + F2. Additivity would not generally hold if one were to require unitary equivalence in the definition of ~, i.e. if we say E is equivalent to F if u*Eu = F for some unitary u. The Schröder–Bernstein theorems for operator algebras gives a sufficient condition for Murray-von Neumann equivalence.
The subspaces belonging to M are partially ordered by inclusion, and this induces a partial order ≤ of projections. There is also a natural partial order on the set of equivalence classes of projections, induced by the partial order ≤ of projections. If M is a factor, ≤ is a total order on equivalence classes of projections, described in the section on traces below.
A projection (or subspace belonging to M) E is said to be a finite projection if there is no projection F < E (meaning F ≤ E and F ≠ E) that is equivalent to E. For example, all finite-dimensional projections (or subspaces) are finite (since isometries between Hilbert spaces leave the dimension fixed), but the identity operator on an infinite-dimensional Hilbert space is not finite in the von Neumann algebra of all bounded operators on it, since it is isometrically isomorphic to a proper subset of itself. However it is possible for infinite dimensional subspaces to be finite.
Orthogonal projections are noncommutative analogues of indicator functions in L∞(R). L∞(R) is the ||·||∞-closure of the subspace generated by the indicator functions. Similarly, a von Neumann algebra is generated by its projections; this is a consequence of the spectral theorem for self-adjoint operators.
The projections of a finite factor form a continuous geometry.
Factors
A von Neumann algebra N whose center consists only of multiples of the identity operator is called a factor. Von Neumann (1949) harvtxt error: no target: CITEREFVon_Neumann1949 (help) showed that every von Neumann algebra on a separable Hilbert space is isomorphic to a direct integral of factors. This decomposition is essentially unique. Thus, the problem of classifying isomorphism classes of von Neumann algebras on separable Hilbert spaces can be reduced to that of classifying isomorphism classes of factors.
Murray & von Neumann (1936) showed that every factor has one of 3 types as described below. The type classification can be extended to von Neumann algebras that are not factors, and a von Neumann algebra is of type X if it can be decomposed as a direct integral of type X factors; for example, every commutative von Neumann algebra has type I1. Every von Neumann algebra can be written uniquely as a sum of von Neumann algebras of types I, II, and III.
There are several other ways to divide factors into classes that are sometimes used:
• A factor is called discrete (or occasionally tame) if it has type I, and continuous (or occasionally wild) if it has type II or III.
• A factor is called semifinite if it has type I or II, and purely infinite if it has type III.
• A factor is called finite if the projection 1 is finite and properly infinite otherwise. Factors of types I and II may be either finite or properly infinite, but factors of type III are always properly infinite.
Type I factors
A factor is said to be of type I if there is a minimal projection E ≠ 0, i.e. a projection E such that there is no other projection F with 0 < F < E. Any factor of type I is isomorphic to the von Neumann algebra of all bounded operators on some Hilbert space; since there is one Hilbert space for every cardinal number, isomorphism classes of factors of type I correspond exactly to the cardinal numbers. Since many authors consider von Neumann algebras only on separable Hilbert spaces, it is customary to call the bounded operators on a Hilbert space of finite dimension n a factor of type In, and the bounded operators on a separable infinite-dimensional Hilbert space, a factor of type I∞.
Type II factors
A factor is said to be of type II if there are no minimal projections but there are non-zero finite projections. This implies that every projection E can be "halved" in the sense that there are two projections F and G that are Murray–von Neumann equivalent and satisfy E = F + G. If the identity operator in a type II factor is finite, the factor is said to be of type II1; otherwise, it is said to be of type II∞. The best understood factors of type II are the hyperfinite type II1 factor and the hyperfinite type II∞ factor, found by Murray & von Neumann (1936). These are the unique hyperfinite factors of types II1 and II∞; there are an uncountable number of other factors of these types that are the subject of intensive study. Murray & von Neumann (1937) proved the fundamental result that a factor of type II1 has a unique finite tracial state, and the set of traces of projections is [0,1].
A factor of type II∞ has a semifinite trace, unique up to rescaling, and the set of traces of projections is [0,∞]. The set of real numbers λ such that there is an automorphism rescaling the trace by a factor of λ is called the fundamental group of the type II∞ factor.
The tensor product of a factor of type II1 and an infinite type I factor has type II∞, and conversely any factor of type II∞ can be constructed like this. The fundamental group of a type II1 factor is defined to be the fundamental group of its tensor product with the infinite (separable) factor of type I. For many years it was an open problem to find a type II factor whose fundamental group was not the group of positive reals, but Connes then showed that the von Neumann group algebra of a countable discrete group with Kazhdan's property (T) (the trivial representation is isolated in the dual space), such as SL(3,Z), has a countable fundamental group. Subsequently, Sorin Popa showed that the fundamental group can be trivial for certain groups, including the semidirect product of Z2 by SL(2,Z).
An example of a type II1 factor is the von Neumann group algebra of a countable infinite discrete group such that every non-trivial conjugacy class is infinite. McDuff (1969) found an uncountable family of such groups with non-isomorphic von Neumann group algebras, thus showing the existence of uncountably many different separable type II1 factors.
Type III factors
Lastly, type III factors are factors that do not contain any nonzero finite projections at all. In their first paper Murray & von Neumann (1936) were unable to decide whether or not they existed; the first examples were later found by von Neumann (1940). Since the identity operator is always infinite in those factors, they were sometimes called type III∞ in the past, but recently that notation has been superseded by the notation IIIλ, where λ is a real number in the interval [0,1]. More precisely, if the Connes spectrum (of its modular group) is 1 then the factor is of type III0, if the Connes spectrum is all integral powers of λ for 0 < λ < 1, then the type is IIIλ, and if the Connes spectrum is all positive reals then the type is III1. (The Connes spectrum is a closed subgroup of the positive reals, so these are the only possibilities.) The only trace on type III factors takes value ∞ on all non-zero positive elements, and any two non-zero projections are equivalent. At one time type III factors were considered to be intractable objects, but Tomita–Takesaki theory has led to a good structure theory. In particular, any type III factor can be written in a canonical way as the crossed product of a type II∞ factor and the real numbers.
The predual
Any von Neumann algebra M has a predual M∗, which is the Banach space of all ultraweakly continuous linear functionals on M. As the name suggests, M is (as a Banach space) the dual of its predual. The predual is unique in the sense that any other Banach space whose dual is M is canonically isomorphic to M∗. Sakai (1971) showed that the existence of a predual characterizes von Neumann algebras among C* algebras.
The definition of the predual given above seems to depend on the choice of Hilbert space that M acts on, as this determines the ultraweak topology. However the predual can also be defined without using the Hilbert space that M acts on, by defining it to be the space generated by all positive normal linear functionals on M. (Here "normal" means that it preserves suprema when applied to increasing nets of self adjoint operators; or equivalently to increasing sequences of projections.)
The predual M∗ is a closed subspace of the dual M* (which consists of all norm-continuous linear functionals on M) but is generally smaller. The proof that M∗ is (usually) not the same as M* is nonconstructive and uses the axiom of choice in an essential way; it is very hard to exhibit explicit elements of M* that are not in M∗. For example, exotic positive linear forms on the von Neumann algebra l∞(Z) are given by free ultrafilters; they correspond to exotic *-homomorphisms into C and describe the Stone–Čech compactification of Z.
Examples:
1. The predual of the von Neumann algebra L∞(R) of essentially bounded functions on R is the Banach space L1(R) of integrable functions. The dual of L∞(R) is strictly larger than L1(R) For example, a functional on L∞(R) that extends the Dirac measure δ0 on the closed subspace of bounded continuous functions C0b(R) cannot be represented as a function in L1(R).
2. The predual of the von Neumann algebra B(H) of bounded operators on a Hilbert space H is the Banach space of all trace class operators with the trace norm ||A||= Tr(|A|). The Banach space of trace class operators is itself the dual of the C*-algebra of compact operators (which is not a von Neumann algebra).
Weights, states, and traces
Further information: Noncommutative measure and integration
Weights and their special cases states and traces are discussed in detail in (Takesaki 1979).
• A weight ω on a von Neumann algebra is a linear map from the set of positive elements (those of the form a*a) to [0,∞].
• A positive linear functional is a weight with ω(1) finite (or rather the extension of ω to the whole algebra by linearity).
• A state is a weight with ω(1) = 1.
• A trace is a weight with ω(aa*) = ω(a*a) for all a.
• A tracial state is a trace with ω(1) = 1.
Any factor has a trace such that the trace of a non-zero projection is non-zero and the trace of a projection is infinite if and only if the projection is infinite. Such a trace is unique up to rescaling. For factors that are separable or finite, two projections are equivalent if and only if they have the same trace. The type of a factor can be read off from the possible values of this trace over the projections of the factor, as follows:
• Type In: 0, x, 2x, ....,nx for some positive x (usually normalized to be 1/n or 1).
• Type I∞: 0, x, 2x, ....,∞ for some positive x (usually normalized to be 1).
• Type II1: [0,x] for some positive x (usually normalized to be 1).
• Type II∞: [0,∞].
• Type III: {0,∞}.
If a von Neumann algebra acts on a Hilbert space containing a norm 1 vector v, then the functional a → (av,v) is a normal state. This construction can be reversed to give an action on a Hilbert space from a normal state: this is the GNS construction for normal states.
Modules over a factor
Given an abstract separable factor, one can ask for a classification of its modules, meaning the separable Hilbert spaces that it acts on. The answer is given as follows: every such module H can be given an M-dimension dimM(H) (not its dimension as a complex vector space) such that modules are isomorphic if and only if they have the same M-dimension. The M-dimension is additive, and a module is isomorphic to a subspace of another module if and only if it has smaller or equal M-dimension.
A module is called standard if it has a cyclic separating vector. Each factor has a standard representation, which is unique up to isomorphism. The standard representation has an antilinear involution J such that JMJ = M′. For finite factors the standard module is given by the GNS construction applied to the unique normal tracial state and the M-dimension is normalized so that the standard module has M-dimension 1, while for infinite factors the standard module is the module with M-dimension equal to ∞.
The possible M-dimensions of modules are given as follows:
• Type In (n finite): The M-dimension can be any of 0/n, 1/n, 2/n, 3/n, ..., ∞. The standard module has M-dimension 1 (and complex dimension n2.)
• Type I∞ The M-dimension can be any of 0, 1, 2, 3, ..., ∞. The standard representation of B(H) is H⊗H; its M-dimension is ∞.
• Type II1: The M-dimension can be anything in [0, ∞]. It is normalized so that the standard module has M-dimension 1. The M-dimension is also called the coupling constant of the module H.
• Type II∞: The M-dimension can be anything in [0, ∞]. There is in general no canonical way to normalize it; the factor may have outer automorphisms multiplying the M-dimension by constants. The standard representation is the one with M-dimension ∞.
• Type III: The M-dimension can be 0 or ∞. Any two non-zero modules are isomorphic, and all non-zero modules are standard.
Amenable von Neumann algebras
Connes (1976) and others proved that the following conditions on a von Neumann algebra M on a separable Hilbert space H are all equivalent:
• M is hyperfinite or AFD or approximately finite dimensional or approximately finite: this means the algebra contains an ascending sequence of finite dimensional subalgebras with dense union. (Warning: some authors use "hyperfinite" to mean "AFD and finite".)
• M is amenable: this means that the derivations of M with values in a normal dual Banach bimodule are all inner.[2]
• M has Schwartz's property P: for any bounded operator T on H the weak operator closed convex hull of the elements uTu* contains an element commuting with M.
• M is semidiscrete: this means the identity map from M to M is a weak pointwise limit of completely positive maps of finite rank.
• M has property E or the Hakeda–Tomiyama extension property: this means that there is a projection of norm 1 from bounded operators on H to M '.
• M is injective: any completely positive linear map from any self adjoint closed subspace containing 1 of any unital C*-algebra A to M can be extended to a completely positive map from A to M.
There is no generally accepted term for the class of algebras above; Connes has suggested that amenable should be the standard term.
The amenable factors have been classified: there is a unique one of each of the types In, I∞, II1, II∞, IIIλ, for 0 < λ ≤ 1, and the ones of type III0 correspond to certain ergodic flows. (For type III0 calling this a classification is a little misleading, as it is known that there is no easy way to classify the corresponding ergodic flows.) The ones of type I and II1 were classified by Murray & von Neumann (1943), and the remaining ones were classified by Connes (1976), except for the type III1 case which was completed by Haagerup.
All amenable factors can be constructed using the group-measure space construction of Murray and von Neumann for a single ergodic transformation. In fact they are precisely the factors arising as crossed products by free ergodic actions of Z or Z/nZ on abelian von Neumann algebras L∞(X). Type I factors occur when the measure space X is atomic and the action transitive. When X is diffuse or non-atomic, it is equivalent to [0,1] as a measure space. Type II factors occur when X admits an equivalent finite (II1) or infinite (II∞) measure, invariant under an action of Z. Type III factors occur in the remaining cases where there is no invariant measure, but only an invariant measure class: these factors are called Krieger factors.
Tensor products of von Neumann algebras
The Hilbert space tensor product of two Hilbert spaces is the completion of their algebraic tensor product. One can define a tensor product of von Neumann algebras (a completion of the algebraic tensor product of the algebras considered as rings), which is again a von Neumann algebra, and act on the tensor product of the corresponding Hilbert spaces. The tensor product of two finite algebras is finite, and the tensor product of an infinite algebra and a non-zero algebra is infinite. The type of the tensor product of two von Neumann algebras (I, II, or III) is the maximum of their types. The commutation theorem for tensor products states that
$(M\otimes N)^{\prime }=M^{\prime }\otimes N^{\prime },$
where M′ denotes the commutant of M.
The tensor product of an infinite number of von Neumann algebras, if done naively, is usually a ridiculously large non-separable algebra. Instead von Neumann (1938) showed that one should choose a state on each of the von Neumann algebras, use this to define a state on the algebraic tensor product, which can be used to produce a Hilbert space and a (reasonably small) von Neumann algebra. Araki & Woods (1968) studied the case where all the factors are finite matrix algebras; these factors are called Araki–Woods factors or ITPFI factors (ITPFI stands for "infinite tensor product of finite type I factors"). The type of the infinite tensor product can vary dramatically as the states are changed; for example, the infinite tensor product of an infinite number of type I2 factors can have any type depending on the choice of states. In particular Powers (1967) found an uncountable family of non-isomorphic hyperfinite type IIIλ factors for 0 < λ < 1, called Powers factors, by taking an infinite tensor product of type I2 factors, each with the state given by:
$x\mapsto {\rm {Tr}}{\begin{pmatrix}{1 \over \lambda +1}&0\\0&{\lambda \over \lambda +1}\\\end{pmatrix}}x.$
All hyperfinite von Neumann algebras not of type III0 are isomorphic to Araki–Woods factors, but there are uncountably many of type III0 that are not.
Bimodules and subfactors
A bimodule (or correspondence) is a Hilbert space H with module actions of two commuting von Neumann algebras. Bimodules have a much richer structure than that of modules. Any bimodule over two factors always gives a subfactor since one of the factors is always contained in the commutant of the other. There is also a subtle relative tensor product operation due to Connes on bimodules. The theory of subfactors, initiated by Vaughan Jones, reconciles these two seemingly different points of view.
Bimodules are also important for the von Neumann group algebra M of a discrete group Γ. Indeed, if V is any unitary representation of Γ, then, regarding Γ as the diagonal subgroup of Γ × Γ, the corresponding induced representation on l2 (Γ, V) is naturally a bimodule for two commuting copies of M. Important representation theoretic properties of Γ can be formulated entirely in terms of bimodules and therefore make sense for the von Neumann algebra itself. For example, Connes and Jones gave a definition of an analogue of Kazhdan's property (T) for von Neumann algebras in this way.
Non-amenable factors
Von Neumann algebras of type I are always amenable, but for the other types there are an uncountable number of different non-amenable factors, which seem very hard to classify, or even distinguish from each other. Nevertheless, Voiculescu has shown that the class of non-amenable factors coming from the group-measure space construction is disjoint from the class coming from group von Neumann algebras of free groups. Later Narutaka Ozawa proved that group von Neumann algebras of hyperbolic groups yield prime type II1 factors, i.e. ones that cannot be factored as tensor products of type II1 factors, a result first proved by Leeming Ge for free group factors using Voiculescu's free entropy. Popa's work on fundamental groups of non-amenable factors represents another significant advance. The theory of factors "beyond the hyperfinite" is rapidly expanding at present, with many new and surprising results; it has close links with rigidity phenomena in geometric group theory and ergodic theory.
Examples
• The essentially bounded functions on a σ-finite measure space form a commutative (type I1) von Neumann algebra acting on the L2 functions. For certain non-σ-finite measure spaces, usually considered pathological, L∞(X) is not a von Neumann algebra; for example, the σ-algebra of measurable sets might be the countable-cocountable algebra on an uncountable set. A fundamental approximation theorem can be represented by the Kaplansky density theorem.
• The bounded operators on any Hilbert space form a von Neumann algebra, indeed a factor, of type I.
• If we have any unitary representation of a group G on a Hilbert space H then the bounded operators commuting with G form a von Neumann algebra G′, whose projections correspond exactly to the closed subspaces of H invariant under G. Equivalent subrepresentations correspond to equivalent projections in G′. The double commutant G′′ of G is also a von Neumann algebra.
• The von Neumann group algebra of a discrete group G is the algebra of all bounded operators on H = l2(G) commuting with the action of G on H through right multiplication. One can show that this is the von Neumann algebra generated by the operators corresponding to multiplication from the left with an element g ∈ G. It is a factor (of type II1) if every non-trivial conjugacy class of G is infinite (for example, a non-abelian free group), and is the hyperfinite factor of type II1 if in addition G is a union of finite subgroups (for example, the group of all permutations of the integers fixing all but a finite number of elements).
• The tensor product of two von Neumann algebras, or of a countable number with states, is a von Neumann algebra as described in the section above.
• The crossed product of a von Neumann algebra by a discrete (or more generally locally compact) group can be defined, and is a von Neumann algebra. Special cases are the group-measure space construction of Murray and von Neumann and Krieger factors.
• The von Neumann algebras of a measurable equivalence relation and a measurable groupoid can be defined. These examples generalise von Neumann group algebras and the group-measure space construction.
Applications
Von Neumann algebras have found applications in diverse areas of mathematics like knot theory, statistical mechanics, quantum field theory, local quantum physics, free probability, noncommutative geometry, representation theory, differential geometry, and dynamical systems.
For instance, C*-algebra provides an alternative axiomatization to probability theory. In this case the method goes by the name of Gelfand–Naimark–Segal construction. This is analogous to the two approaches to measure and integration, where one has the choice to construct measures of sets first and define integrals later, or construct integrals first and define set measures as integrals of characteristic functions.
See also
• AW*-algebra – algebraic generalization of a W*-algebraPages displaying wikidata descriptions as a fallback
• Central carrier
• Tomita–Takesaki theory – method for constructing modular automorphisms of von Neumann algebras from the polar decomposition of a certain involutionPages displaying wikidata descriptions as a fallback
References
1. An Introduction To II1 Factors ens-lyon.fr
2. Connes, A (May 1978). "On the cohomology of operator algebras". Journal of Functional Analysis. 28 (2): 248–253. doi:10.1016/0022-1236(78)90088-5.
• Araki, H.; Woods, E. J. (1968), "A classification of factors", Publ. Res. Inst. Math. Sci. Ser. A, 4 (1): 51–130, doi:10.2977/prims/1195195263MR0244773
• Blackadar, B. (2005), Operator algebras, Springer, ISBN 3-540-28486-9, corrected manuscript (PDF), 2013
• Connes, A. (1976), "Classification of Injective Factors", Annals of Mathematics, Second Series, 104 (1): 73–115, doi:10.2307/1971057, JSTOR 1971057
• Connes, A. (1994), Non-commutative geometry, Academic Press, ISBN 0-12-185860-X.
• Dixmier, J. (1981), Von Neumann algebras, ISBN 0-444-86308-7 (A translation of Dixmier, J. (1957), Les algèbres d'opérateurs dans l'espace hilbertien: algèbres de von Neumann, Gauthier-Villars, the first book about von Neumann algebras.)
• Jones, V.F.R. (2003), von Neumann algebras (PDF); incomplete notes from a course.
• Kostecki, R.P. (2013), W*-algebras and noncommutative integration, arXiv:1307.4818, Bibcode:2013arXiv1307.4818P.
• McDuff, Dusa (1969), "Uncountably many II1 factors", Annals of Mathematics, Second Series, 90 (2): 372–377, doi:10.2307/1970730, JSTOR 1970730
• Murray, F. J. (2006), "The rings of operators papers", The legacy of John von Neumann (Hempstead, NY, 1988), Proc. Sympos. Pure Math., vol. 50, Providence, RI.: Amer. Math. Soc., pp. 57–60, ISBN 0-8218-4219-6 A historical account of the discovery of von Neumann algebras.
• Murray, F.J.; von Neumann, J. (1936), "On rings of operators", Annals of Mathematics, Second Series, 37 (1): 116–229, doi:10.2307/1968693, JSTOR 1968693. This paper gives their basic properties and the division into types I, II, and III, and in particular finds factors not of type I.
• Murray, F.J.; von Neumann, J. (1937), "On rings of operators II", Trans. Amer. Math. Soc., American Mathematical Society, 41 (2): 208–248, doi:10.2307/1989620, JSTOR 1989620. This is a continuation of the previous paper, that studies properties of the trace of a factor.
• Murray, F.J.; von Neumann, J. (1943), "On rings of operators IV", Annals of Mathematics, Second Series, 44 (4): 716–808, doi:10.2307/1969107, JSTOR 1969107. This studies when factors are isomorphic, and in particular shows that all approximately finite factors of type II1 are isomorphic.
• Powers, Robert T. (1967), "Representations of Uniformly Hyperfinite Algebras and Their Associated von Neumann Rings", Annals of Mathematics, Second Series, 86 (1): 138–171, doi:10.2307/1970364, JSTOR 1970364
• Sakai, S. (1971), C*-algebras and W*-algebras, Springer, ISBN 3-540-63633-1
• Schwartz, Jacob (1967), W-* Algebras, ISBN 0-677-00670-5
• Shtern, A.I. (2001) [1994], "von Neumann algebra", Encyclopedia of Mathematics, EMS Press
• Takesaki, M. (1979), Theory of Operator Algebras I, II, III, ISBN 3-540-42248-X
• von Neumann, J. (1930), "Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren", Math. Ann., 102 (1): 370–427, Bibcode:1930MatAn.102..685E, doi:10.1007/BF01782352, S2CID 121141866. The original paper on von Neumann algebras.
• von Neumann, J. (1936), "On a Certain Topology for Rings of Operators", Annals of Mathematics, Second Series, 37 (1): 111–115, doi:10.2307/1968692, JSTOR 1968692. This defines the ultrastrong topology.
• von Neumann, J. (1938), "On infinite direct products", Compos. Math., 6: 1–77. This discusses infinite tensor products of Hilbert spaces and the algebras acting on them.
• von Neumann, J. (1940), "On rings of operators III", Annals of Mathematics, Second Series, 41 (1): 94–161, doi:10.2307/1968823, JSTOR 1968823. This shows the existence of factors of type III.
• von Neumann, J. (1943), "On Some Algebraical Properties of Operator Rings", Annals of Mathematics, Second Series, 44 (4): 709–715, doi:10.2307/1969106, JSTOR 1969106. This shows that some apparently topological properties in von Neumann algebras can be defined purely algebraically.
• von Neumann, J. (1949), "On Rings of Operators. Reduction Theory", Annals of Mathematics, Second Series, 50 (2): 401–485, doi:10.2307/1969463, JSTOR 1969463. This discusses how to write a von Neumann algebra as a sum or integral of factors.
• von Neumann, John (1961), Taub, A.H. (ed.), Collected Works, Volume III: Rings of Operators, NY: Pergamon Press. Reprints von Neumann's papers on von Neumann algebras.
• Wassermann, A. J. (1991), Operators on Hilbert space
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| Wikipedia |
W-algebra
In conformal field theory and representation theory, a W-algebra is an associative algebra that generalizes the Virasoro algebra. W-algebras were introduced by Alexander Zamolodchikov,[1] and the name "W-algebra" comes from the fact that Zamolodchikov used the letter W for one of the elements of one of his examples.
For W*-algebra, see Von Neumann algebra.
Definition
A W-algebra is an associative algebra that is generated by the modes of a finite number of meromorphic fields $W^{(h)}(z)$, including the energy-momentum tensor $T(z)=W^{(2)}(z)$. For $h\neq 2$, $W^{(h)}(z)$ is a primary field of conformal dimension $h\in {\frac {1}{2}}\mathbb {N} ^{*}$.[2] The generators $(W_{n}^{(h)})_{n\in \mathbb {Z} }$ of the algebra are related to the meromorphic fields by the mode expansions
$W^{(h)}(z)=\sum _{n\in \mathbb {Z} }W_{n}^{(h)}z^{-n-h}$
The commutation relations of $L_{n}=W_{n}^{(2)}$ are given by the Virasoro algebra, which is parameterized by a central charge $c\in \mathbb {C} $. This number is also called the central charge of the W-algebra. The commutation relations
$[L_{m},W_{n}^{(h)}]=((h-1)m-n)W_{m+n}^{(h)}$
are equivalent to the assumption that $W^{(h)}(z)$ is a primary field of dimension $h$. The rest of the commutation relations can in principle be determined by solving the Jacobi identities.
Given a finite set of conformal dimensions $H$ (not necessarily all distinct), the number of W-algebras generated by $(W^{(h)})_{h\in H}$ may be zero, one or more. The resulting W-algebras may exist for all $c\in \mathbb {C} $, or only for some specific values of the central charge.[2]
A W-algebra is called freely generated if its generators obey no other relations than the commutation relations. Most commonly studied W-algebras are freely generated, including the W(N) algebras.[3] In this article, the sections on representation theory and correlation functions apply to freely generated W-algebras.
Constructions
While it is possible to construct W-algebras by assuming the existence of a number of meromorphic fields $W^{(h)}(z)$ and solving the Jacobi identities, there also exist systematic constructions of families of W-algebras.
Drinfeld-Sokolov reduction
From a finite-dimensional Lie algebra ${\mathfrak {g}}$, together with an embedding ${\mathfrak {sl}}_{2}\hookrightarrow {\mathfrak {g}}$, a W-algebra may be constructed from the universal enveloping algebra of the affine Lie algebra ${\hat {\mathfrak {g}}}$ by a kind of BRST construction.[2] Then the central charge of the W-algebra is a function of the level of the affine Lie algebra.
Coset construction
Given a finite-dimensional Lie algebra ${\mathfrak {g}}$, together with a subalgebra ${\mathfrak {h}}\hookrightarrow {\mathfrak {g}}$, a W-algebra $W({\hat {\mathfrak {g}}}/{\hat {\mathfrak {h}}})$ may be constructed from the corresponding affine Lie algebras ${\hat {\mathfrak {h}}}\hookrightarrow {\hat {\mathfrak {g}}}$. The fields that generate $W({\hat {\mathfrak {g}}}/{\hat {\mathfrak {h}}})$ are the polynomials in the currents of ${\hat {\mathfrak {g}}}$ and their derivatives that commute with the currents of ${\hat {\mathfrak {h}}}$.[2] The central charge of $W({\hat {\mathfrak {g}}}/{\hat {\mathfrak {h}}})$ is the difference of the central charges of ${\hat {\mathfrak {g}}}$ and ${\hat {\mathfrak {h}}}$, which are themselves given in terms of their level by the Sugawara construction.
Commutator of a set of screenings
Given a holomorphic field $\phi (z)$ with values in $\mathbb {R} ^{n}$, and a set of $n$ vectors $a_{1},\dots ,a_{n}\in \mathbb {R} ^{n}$, a W-algebra may be defined as the set of polynomials of $\phi $ and its derivatives that commute with the screening charges $\oint e^{(a_{i},\phi (z))}dz$. If the vectors $a_{i}$ are the simple roots of a Lie algebra ${\mathfrak {g}}$, the resulting W-algebra coincides with an algebra that is obtained from ${\mathfrak {g}}$ by Drinfeld-Sokolov reduction.[4]
The W(N) algebras
For any integer $N\geq 2$, the W(N) algebra is a W-algebra which is generated by $N-1$ meromorphic fields of dimensions $2,3,\dots ,N$. The W(2) algebra coincides with the Virasoro algebra.
Construction
The W(N) algebra is obtained by Drinfeld-Sokolov reduction of the affine Lie algebra ${\widehat {\mathfrak {sl}}}_{N}$.
The embeddings ${\mathfrak {sl}}_{2}\hookrightarrow {\mathfrak {sl}}_{N}$ are parametrized by the integer partitions of $N$, interpreted as decompositions of the fundamental representation $F$ of ${\mathfrak {sl}}_{N}$ into representations of ${\mathfrak {sl}}_{2}$. The set $H$ of dimensions of the generators of the resulting W-algebra is such that $F\otimes F=R_{1}\oplus \bigoplus _{h\in H}R_{2h-1}$ where $R_{d}$ is the $d$-dimensional irreducible representation of ${\mathfrak {sl}}_{2}$.[5]
The trivial partition $N=N$ corresponds to the W(N) algebra, while $N=1+1+\dots +1$ corresponds to ${\widehat {\mathfrak {sl}}}_{N}$ itself. In the case $N=3$, the partition $3=2+1$ leads to the Bershadsky-Polyakov algebra, whose generating fields have the dimensions $2,{\frac {3}{2}},{\frac {3}{2}},1$.
Properties
The central charge of the W(N) algebra is given in terms of the level $k$ of the affine Lie algebra by
$c_{W(N)}=(N-1)\left(1-N(N+1)\left({\frac {1}{k+N}}+k+N-2\right)\right)$
in notations where the central charge of the affine Lie algebra is
$c_{{\widehat {\mathfrak {sl}}}_{N}}=(N-1)(N+1)-{\frac {N(N-1)(N+1)}{k+N}}$
It is possible to choose a basis such that the commutation relations are invariant under $W^{(h)}\to (-1)^{h}W^{(h)}$.
While the Virasoro algebra is a subalgebra of the universal enveloping algebra of ${\widehat {\mathfrak {sl}}}_{2}$, the W(N) algebra with $N\geq 3$ is not a subalgebra of the universal enveloping algebra of ${\widehat {\mathfrak {sl}}}_{N}$.[6]
Example of the W(3) algebra
The W(3) algebra is generated by the generators of the Virasoro algebra $(L_{n})_{n\in \mathbb {Z} }$, plus another infinite family of generators $(W_{n})_{n\in \mathbb {Z} }=(W_{n}^{(3)})_{n\in \mathbb {Z} }$. The commutation relations are[2]
$[L_{m},L_{n}]=(m-n)L_{m+n}+{\frac {c}{12}}m(m^{2}-1)\delta _{m+n,0}$
$[L_{m},W_{n}]=(2m-n)W_{m+n}$
$[W_{m},W_{n}]={\frac {c}{360}}m(m^{2}-1)(m^{2}-4)\delta _{m+n,0}+{\frac {16}{22+5c}}\Lambda _{m+n}+{\frac {(m-n)(2m^{2}-mn+2n^{2}-8)}{30}}L_{m+n}$
where $c\in \mathbb {C} $ is the central charge, and we define
$\Lambda _{n}=\sum _{m=-\infty }^{-2}L_{m}L_{n-m}+\sum _{m=-1}^{\infty }L_{n-m}L_{m}-{\frac {3}{10}}(n+2)(n+3)L_{n}$
The field $\Lambda (z)=\sum _{n\in \mathbb {Z} }\Lambda _{n}z^{-n-4}$ is such that $\Lambda =(TT)-{\frac {3}{10}}T''$.
Representation theory
Highest weight representations
A highest weight representation of a W-algebra is a representation that is generated by a primary state: a vector $v$ such that
$W_{n>0}^{(h)}v=0\quad ,\quad W_{0}^{(h)}v=q^{(h)}v$
for some numbers $q^{(h)}$ called the charges, including the conformal dimension $q^{(2)}=\Delta $.
Given a set ${\vec {q}}=(q^{(h)})_{h\in H}$ of charges, the corresponding Verma module is the largest highest-weight representation that is generated by a primary state with these charges. A basis of the Verma module is
$\left\{\prod _{h\in H}W_{-{\vec {N}}_{h}}^{(h)}v\right\}_{{\vec {N}}_{h}\in {\mathcal {V}}}$
where ${\mathcal {V}}$ is the set of ordered tuples of strictly positive integers of the type ${\vec {N}}=(n_{1},n_{2},\dots ,n_{p})$ with $0<n_{1}\leq n_{2}\leq \dots \leq n_{p}$, and $W_{-{\vec {N}}}=W_{-n_{1}}W_{-n_{2}}\dots W_{-n_{p}}$. Except for $v$ itself, the elements of this basis are called descendant states, and their linear combinations are also called descendant states.
For generic values of the charges, the Verma module is the only highest weight representation. For special values of the charges that depend on the algebra's central charge, there exist other highest weight representations, called degenerate representations. Degenerate representations exist if the Verma module is reducible, and they are quotients of the Verma module by its nontrivial submodules.
Degenerate representations
If a Verma module is reducible, any indecomposible submodule is itself a highest weight representation, and is generated by a state that is both descendant and primary, called a null state or null vector. A degenerate representation is obtained by setting one or more null vectors to zero. Setting all the null vectors to zero leads to an irreducible representation.
The structures and characters of irreducible representations can be deduced by Drinfeld-Sokolov reduction from representations of affine Lie algebras.[7]
The existence of null vectors is possible only under $c$-dependent constraints on the charge ${\vec {q}}$. A Verma module can have only finitely many null vectors that are not descendants of other null vectors. If we start from a Verma module that has a maximal number of null vectors, and set all these null vectors to zero, we obtain an irreducible representation called a fully degenerate representation.
For example, in the case of the algebra W(3), the Verma module with vanishing charges $q^{(2)}=q^{(3)}=0$ has the three null vectors $L_{-1}v,W_{-1}v,W_{-2}v$ at levels 1, 1 and 2. Setting these null vectors to zero yields a fully degenerate representation called the vacuum module. The simplest nontrivial fully degenerate representation of W(3) has vanishing null vectors at levels 1, 2 and 3, whose expressions are explicitly known.[8]
An alternative characterization of a fully degenerate representation is that its fusion product with any Verma module is a sum of finitely many indecomposable representations.[8]
Case of W(N)
It is convenient to parametrize highest-weight representations not by the set of charges ${\vec {q}}=(q^{(2)},\dots ,q^{(N)})$, but by an element $P$ of the weight space of ${\mathfrak {sl}}_{N}$, called the momentum.
Let $e_{1},\dots ,e_{N-1}$ be the simple roots of ${\mathfrak {sl}}_{N}$, with a scalar product $K_{ij}=(e_{i},e_{j})$ given by the Cartan matrix of ${\mathfrak {sl}}_{N}$, whose nonzero elements are $K_{ii}=2,K_{i,i+1}=K_{i,i-1}=-1$. The ${\frac {1}{2}}N(N-1)$ positive simple roots are sums of any number of consecutive simple roots, and the Weyl vector is their half-sum $\rho ={\frac {1}{2}}\sum _{e>0}e$, which obeys $(\rho ,\rho )={\frac {1}{12}}N(N^{2}-1)$. The fundamental weights $\omega _{1},\dots ,\omega _{N-1}$ are defined by $(\omega _{i},e_{j})=\delta _{ij}$. Then the momentum is a vector
$P=\sum _{i=1}^{N-1}P_{i}\omega _{i}\quad i.e.\quad (e_{i},P)=P_{i}$
The charges $q^{(h)}$ are functions of the momentum and the central charge, invariant under the action of the Weyl group. In particular, $q^{(h)}$ is a polynomial of the momentum of degree $h$, which under the Dynkin diagram automorphism $e_{i}^{*}=e_{N-i}$ behaves as $q^{(h)}(P^{*})=(-1)^{h}q^{(h)}(P)$. The conformal dimension is[9]
$q^{(2)}={\frac {c+1-N}{24}}-(P,P)$
Let us parametrize the central charge in terms of a number $b$ such that
$c=(N-1){\big (}1+N(N+1)\left(b+b^{-1}\right)^{2}{\big )}$
If there is a positive root $e>0$ and two integers $r,s\in \mathbb {N} ^{*}$ such that[9]
$(e,P)=rb+sb^{-1}$
then the Verma module of momentum $P$ has a null vector at level $rs$. This null vector is itself a primary state of momentum $P-rbe$ or equivalently (by a Weyl reflection) $P-sb^{-1}e$. The number of independent null vectors is the number of positive roots such that $(e,P)\in \mathbb {N} ^{*}b+\mathbb {N} ^{*}b^{-1}$ (up to a Weyl reflection).
The maximal number of null vectors is the number of positive roots ${\frac {1}{2}}N(N-1)$. The corresponding momentums are of the type[9]
$P=(b+b^{-1})\rho +b\Omega ^{+}+b^{-1}\Omega ^{-}$
where $\Omega ^{+},\Omega ^{-}$ are integral dominant weights, i.e. elements of $\sum _{i=1}^{N-1}\mathbb {N} \omega _{i}$, which are highest weights of irreducible finite-dimensional representations of ${\mathfrak {sl}}_{N}$. Let us call ${\mathcal {R}}_{\Omega _{+},\Omega _{-}}$ the corresponding fully degenerate representation of the W(N) algebra.
The irreducible finite-dimensional representation $R_{\Omega }$ of ${\mathfrak {sl}}_{N}$ of highest weight $\Omega $ has a finite set of weights $\Lambda _{\Omega }$, with $|\Lambda _{\Omega }|=\dim(R_{\Omega })$. Its tensor product with a Verma module $V_{p}$ of weight $p\in \sum _{i=1}^{N-1}\mathbb {R} \omega _{i}$ is $R_{\Omega }\otimes V_{p}=\bigoplus _{\lambda \in \Lambda _{\Omega }}V_{p+\lambda }$. The fusion product of the fully degenerate representation ${\mathcal {R}}_{\Omega _{+},\Omega _{-}}$ of W(N) with a Verma module ${\mathcal {V}}_{P}$ of momentum $P$ is then
${\mathcal {R}}_{\Omega _{+},\Omega _{-}}\times {\mathcal {V}}_{P}=\sum _{\lambda _{+}\in \Lambda _{\Omega _{+}}}\sum _{\lambda _{-}\in \Lambda _{\Omega _{-}}}{\mathcal {V}}_{P+b\lambda _{+}+b^{-1}\lambda _{-}}$
Correlation functions
Primary fields
To a primary state of charge ${\vec {q}}=(q^{(h)})_{h\in H}$, the state-field correspondence associates a primary field $V_{\vec {q}}(z)$, whose operator product expansions with the fields $W^{(h)}(z)$ are
$W^{(h)}(y)V_{\vec {q}}(z)=\left({\frac {q^{(h)}}{(y-z)^{h}}}+\sum _{n=1}^{h-1}{\frac {W_{-n}^{(h)}}{(y-z)^{h-n}}}\right)V_{\vec {q}}(z)+O(1)$
On any field $V(z)$, the mode $L_{-1}$ of the energy-momentum tensor acts as a derivative, $L_{-1}V(z)={\frac {\partial }{\partial z}}V(z)$.
Ward identities
On the Riemann sphere, if there is no field at infinity, we have $W^{(h)}(y){\underset {y\to \infty }{=}}O\left(y^{-2h}\right)$. For $n=0,1,\dots ,2h-2$, the identity $\oint _{\infty }dy\ y^{n}W^{(h)}(y)=0$ may be inserted in any correlation function. Therefore, the field $W^{(h)}(y)$ gives rise to $2h-1$ global Ward identities.
Local Ward identities are obtained by inserting $\oint _{\infty }dy\ \varphi (y)W^{(h)}(y)=0$, where $\varphi (y)$ is a meromorphic function such that $\varphi (y){\underset {y\to \infty }{=}}O\left(y^{2h-2}\right)$. In a correlation function of primary fields, local Ward identities determine the action of $W_{-n}^{(h)}$ with $n\geq h$ in terms of the action of $W_{-n}^{(h)}$ with $n\leq h-1$.
For example, in the case of a three-point function on the sphere $\left\langle \prod _{i=1}^{3}V_{{\vec {q}}_{i}}(z_{i})\right\rangle $ of W(3)-primary fields, local Ward identities determine all the descendant three-point functions as linear combinations of descendant three-point functions that involve only $L_{-1},W_{-1},W_{-2}$. Global Ward identities further reduce the problem to determining three-point functions of the type $\left\langle V_{{\vec {q}}_{1}}(z_{1})V_{{\vec {q}}_{2}}(z_{2})W_{-1}^{k}V_{{\vec {q}}_{3}}(z_{3})\right\rangle $ for $k\in \mathbb {N} $.
In the W(3) algebra, as in generic W-algebras, correlation functions of descendant fields can therefore not be deduced from correlation functions of primary fields using Ward identities, as was the case for the Virasoro algebra. A W(3)-Verma module appears in the fusion product of two other W(3)-Verma modules with a multiplicity that is in general infinite.
Differential equations
A correlation function may obey a differential equation that generalizes the BPZ equations if the fields have sufficiently many vanishing null vectors.
A four-point function of W(N)-primary fields on the sphere with one fully degenerate field obeys a differential equation if $N=2$ but not if $N\geq 3$. In the latter case, for a differential equation to exist, one of the other fields must have vanishing null vectors. For example, a four-point function with two fields of momentums $P_{1}=(b+b^{-1})\rho +b\omega _{1}$ (fully degenerate) and $P_{2}=(b+b^{-1})\rho +x\omega _{N-1}$ with $x\in \mathbb {C} $ (almost fully degenerate) obeys a differential equation whose solutions are generalized hypergeometric functions of type ${}_{N}F_{N-1}$.[10]
Applications to conformal field theory
W-minimal models
W-minimal models are generalizations of Virasoro minimal models based on a W-algebra. Their spaces of states are made of finitely many fully degenerate representations. They exist for certain rational values of the central charge: in the case of the W(N) algebra, values of the type
$c_{p,q}^{(N)}=N-1-N(N^{2}-1){\frac {(p-q)^{2}}{pq}}\quad {\text{with}}\quad p,q\in \mathbb {N} ^{*}$
A W(N)-minimal model with central charge $c_{k+N,k+N+1}$ may be constructed as a coset of Wess-Zumino-Witten models ${\frac {SU(N)_{k}\times SU(N)_{1}}{SU(N)_{k+1}}}$.[11]
For example, the two-dimensional critical three-state Potts model has central charge $c_{5,6}^{(2)}=c_{4,5}^{(3)}={\frac {4}{5}}$. Spin observables of the model may be described in terms of the D-series non-diagonal Virasoro minimal model with $(p,q)=(5,6)$, or in terms of the diagonal W(3)-minimal model with $(p,q)=(4,5)$.
Conformal Toda theory
Conformal Toda theory is a generalization of Liouville theory that is based on a W-algebra. Given a simple Lie algebra ${\mathfrak {g}}$, the Lagrangian is a functional of a field $\phi $ which belongs to the root space of ${\mathfrak {g}}$, with one interaction term for each simple root:
$L[\phi ]={\frac {1}{2\pi }}(\partial \phi ,{\bar {\partial }}\phi )+\mu \sum _{e\in \{{\text{simple roots of }}{\mathfrak {g}}\}}\exp \left(b(e,\phi )\right)$
This depends on the cosmological constant $\mu $, which plays no meaningful role, and on the parameter $b$, which is related to the central charge. The resulting field theory is a conformal field theory, whose chiral symmetry algebra is a W-algebra constructed from ${\mathfrak {g}}$ by Drinfeld-Sokolov reduction. For the preservation of conformal symmetry in the quantum theory, it is crucial that there are no more interaction terms than components of the vector $\phi $.[4]
The methods that lead to the solution of Liouville theory may be applied to W(N)-conformal Toda theory, but they only lead to the analytic determination of a particular class of three-point structure constants,[10] and W(N)-conformal Toda theory with $N\geq 3$ has not been solved.
Logarithmic conformal field theory
At central charge $c=c_{1,q}^{(2)}$, the Virasoro algebra can be extended by a triplet of generators of dimension $2q-1$, thus forming a W-algebra with the set of dimensions $H=\{2,2q-1,2q-1,2q-1\}$. Then it is possible to build a rational conformal field theory based on this W-algebra, which is logarithmic.[12] The simplest case is obtained for $q=2$, has central charge $c=-2$, and has been particularly well studied, including in the presence of a boundary.[13]
Related concepts
Finite W-algebras
Finite W-algebras are certain associative algebras associated to nilpotent elements of semisimple Lie algebras.[14]
The original definition, provided by Alexander Premet, starts with a pair $({\mathfrak {g}},e)$ consisting of a reductive Lie algebra ${\mathfrak {g}}$ over the complex numbers and a nilpotent element e. By the Jacobson-Morozov theorem, e is part of a sl2 triple (e, h, f). The eigenspace decomposition of ad(h) induces a $\mathbb {Z} $-grading on ${\mathfrak {g}}$:
${\mathfrak {g}}=\bigoplus {\mathfrak {g}}(i).$
Define a character $\chi $ (i.e. a homomorphism from ${\mathfrak {g}}$ to the trivial 1-dimensional Lie algebra) by the rule $\chi (x)=\kappa (e,x)$, where $\kappa $ denotes the Killing form. This induces a non-degenerate anti-symmetric bilinear form on the −1 graded piece by the rule:
$\omega _{\chi }(x,y)=\chi ([x,y]).$
After choosing any Lagrangian subspace $l$, we may define the following nilpotent subalgebra which acts on the universal enveloping algebra by the adjoint action.
${\mathfrak {m}}=l+\bigoplus _{i\leq -2}{\mathfrak {g}}(i).$
The left ideal $I$ of the universal enveloping algebra $U({\mathfrak {g}})$ generated by $\{x-\chi (x):x\in {\mathfrak {m}}\}$ is invariant under this action. It follows from a short calculation that the invariants in $U({\mathfrak {g}})/I$ under ad$({\mathfrak {m}})$ inherit the associative algebra structure from $U({\mathfrak {g}})$. The invariant subspace $(U({\mathfrak {g}})/I)^{{\text{ad}}({\mathfrak {m}})}$ is called the finite W-algebra constructed from $({\mathfrak {g}},e)$, and is usually denoted $U({\mathfrak {g}},e)$.
References
1. Zamolodchikov, A.B. (1985). "Infinite extra symmetries in two-dimensional conformal quantum field theory". Akademiya Nauk SSSR. Teoreticheskaya I Matematicheskaya Fizika (in Russian). 65 (3): 347–359. ISSN 0564-6162. MR 0829902.
2. Watts, Gerard M. T. (1997). "W-algebras and their representations" (PDF). In Horváth, Zalán; Palla, László (eds.). Conformal field theories and integrable models (Budapest, 1996). Lecture Notes in Phys. Vol. 498. Berlin, New York: Springer-Verlag. pp. 55–84. doi:10.1007/BFb0105278. ISBN 978-3-540-63618-2. MR 1636798. S2CID 117999633.
3. de Boer, J.; Fehér, L.; Honecker, A. (1994). "A class of -algebras with infinitely generated classical limit". Nuclear Physics B. Elsevier BV. 420 (1–2): 409–445. arXiv:hep-th/9312049. doi:10.1016/0550-3213(94)90388-3. ISSN 0550-3213.
4. Litvinov, Alexey; Spodyneiko, Lev (2016). "On W algebras commuting with a set of screenings". Journal of High Energy Physics. 2016 (11). arXiv:1609.06271. doi:10.1007/jhep11(2016)138. ISSN 1029-8479. S2CID 29261029.
5. Creutzig, Thomas; Hikida, Yasuaki; Rønne, Peter B. (2016). "Correspondences between WZNW models and CFTs with W-algebra symmetry". Journal of High Energy Physics. 2016 (2). arXiv:1509.07516. doi:10.1007/jhep02(2016)048. ISSN 1029-8479. S2CID 44722579.
6. Bouwknegt, Peter; Schoutens, Kareljan (1993). "W symmetry in conformal field theory". Physics Reports. 223 (4): 183–276. arXiv:hep-th/9210010. Bibcode:1993PhR...223..183B. doi:10.1016/0370-1573(93)90111-P. ISSN 0370-1573. MR 1208246. S2CID 118959569.
7. De Vos, Koos; van Driel, Peter (1996). "The Kazhdan–Lusztig conjecture for W algebras". Journal of Mathematical Physics. AIP Publishing. 37 (7): 3587–3610. arXiv:hep-th/9508020. doi:10.1063/1.531584. ISSN 0022-2488. S2CID 119348884.
8. Watts, G. M. T. (1995). "Fusion in the W3 algebra". Communications in Mathematical Physics. 171 (1): 87–98. arXiv:hep-th/9403163. doi:10.1007/bf02103771. ISSN 0010-3616. S2CID 86758219.
9. Fateev, Vladimir; Ribault, Sylvain (2010). "Conformal Toda theory with a boundary". Journal of High Energy Physics. 2010 (12): 089. arXiv:1007.1293. doi:10.1007/jhep12(2010)089. ISSN 1029-8479. S2CID 17631088.
10. Fateev, V.A; Litvinov, A.V (2007-11-05). "Correlation functions in conformal Toda field theory I". Journal of High Energy Physics. 2007 (11): 002. arXiv:0709.3806. doi:10.1088/1126-6708/2007/11/002. ISSN 1029-8479. S2CID 8189544.
11. Chang, Chi-Ming; Yin, Xi (2012). "Correlators in W N minimal model revisited". Journal of High Energy Physics. 2012 (10). arXiv:1112.5459. doi:10.1007/jhep10(2012)050. ISSN 1029-8479. S2CID 119114132.
12. Gaberdiel, Matthias R.; Kausch, Horst G. (1996). "A rational logarithmic conformal field theory". Physics Letters B. Elsevier BV. 386 (1–4): 131–137. arXiv:hep-th/9606050. doi:10.1016/0370-2693(96)00949-5. ISSN 0370-2693. S2CID 13939686.
13. Gaberdiel, Matthias R; Runkel, Ingo (2006-11-08). "The logarithmic triplet theory with boundary". Journal of Physics A: Mathematical and General. 39 (47): 14745–14779. arXiv:hep-th/0608184. doi:10.1088/0305-4470/39/47/016. ISSN 0305-4470. S2CID 10719319.
14. Wang, Weiqiang (2011). "Nilpotent orbits and finite W-algebras". In Neher, Erhard; Savage, Alistair; Wang, Weiqiang (eds.). Geometric representation theory and extended affine Lie algebras. Fields Institute Communications Series. Vol. 59. Providence RI. pp. 71–105. arXiv:0912.0689. Bibcode:2009arXiv0912.0689W. ISBN 978-082185237-8. MR 2777648.{{cite book}}: CS1 maint: location missing publisher (link)
Further reading
• de Boer, Jan; Tjin, Tjark (1993), "Quantization and representation theory of finite W algebras", Communications in Mathematical Physics, 158 (3): 485–516, arXiv:hep-th/9211109, Bibcode:1993CMaPh.158..485D, doi:10.1007/bf02096800, ISSN 0010-3616, MR 1255424, S2CID 204933347
• Bouwknegt, P.; Schoutens, K., eds. (1995), W-symmetry, Advanced Series in Mathematical Physics, vol. 22, River Edge, New Jersey: World Scientific Publishing Co., doi:10.1142/2354, ISBN 978-981021762-4, MR 1338864
• Brown, Jonathan, Finite W-algebras of Classical Type (PDF)
• Dickey, L. A. (1997), "Lectures on classical W-algebras", Acta Applicandae Mathematicae, 47 (3): 243–321, doi:10.1023/A:1017903416906, ISSN 0167-8019, S2CID 118573600
• Gan, Wee Liang; Ginzburg, Victor (2002), "Quantization of Slodowy slices", International Mathematics Research Notices, 2002 (5): 243–255, arXiv:math/0105225, doi:10.1155/S107379280210609X, ISSN 1073-7928, MR 1876934, S2CID 13895488
• Losev, Ivan (2010), "Quantized symplectic actions and W-algebras", Journal of the American Mathematical Society, 23 (1): 35–59, arXiv:0707.3108, Bibcode:2010JAMS...23...35L, doi:10.1090/S0894-0347-09-00648-1, ISSN 0894-0347, MR 2552248, S2CID 16211165
• Pope, C.N. (1991), Lectures on W algebras and W gravity, Lectures given at the Trieste Summer School in High-Energy Physics, August 1991, arXiv:hep-th/9112076, Bibcode:1991hep.th...12076P
| Wikipedia |
W-curve
In geometry, a W-curve is a curve in projective n-space that is invariant under a 1-parameter group of projective transformations. W-curves were first investigated by Felix Klein and Sophus Lie in 1871, who also named them. W-curves in the real projective plane can be constructed with straightedge alone. Many well-known curves are W-curves, among them conics, logarithmic spirals, powers (like y = x3), logarithms and the helix, but not e.g. the sine. W-curves occur widely in the realm of plants.
Name
The 'W' stands for the German 'Wurf' – a throw – which in this context refers to a series of four points on a line. A 1-dimensional W-curve (read: the motion of a point on a projective line) is determined by such a series.
The German "W-Kurve" sounds almost exactly like "Weg-Kurve" and the last can be translated by "path curve". That is why in the English literature one often finds "path curve" or "pathcurve".
See also
• Homography
Further reading
• Felix Klein and Sophus Lie: Ueber diejenigen ebenen Curven... in Mathematische Annalen, Band 4, 1871; online available at the University of Goettingen
• For an introduction on W-curves and how to draw them, see Lawrence Edwards Projective Geometry, Floris Books 2003, ISBN 0-86315-393-3
• On the occurrence of W-curves in nature see Lawrence Edwards The vortex of life, Floris Books 1993, ISBN 0-86315-148-5
• For an algebraic classification of 2- and 3-dimensional W-curves see Classification of pathcurves
• Georg Scheffers (1903) "Besondere transzendente Kurven", Klein's encyclopedia Band 3–3.
| Wikipedia |
Inductive type
In type theory, a system has inductive types if it has facilities for creating a new type from constants and functions that create terms of that type. The feature serves a role similar to data structures in a programming language and allows a type theory to add concepts like numbers, relations, and trees. As the name suggests, inductive types can be self-referential, but usually only in a way that permits structural recursion.
The standard example is encoding the natural numbers using Peano's encoding. It can be defined in Coq as follows:
Inductive nat : Type :=
| 0 : nat
| S : nat -> nat.
Here, a natural number is created either from the constant "0" or by applying the function "S" to another natural number. "S" is the successor function which represents adding 1 to a number. Thus, "0" is zero, "S 0" is one, "S (S 0)" is two, "S (S (S 0))" is three, and so on.
Since their introduction, inductive types have been extended to encode more and more structures, while still being predicative and supporting structural recursion.
Elimination
Inductive types usually come with a function to prove properties about them. Thus, "nat" may come with (in Coq syntax):
nat_elim : (forall P : nat -> Prop, (P 0) -> (forall n, P n -> P (S n)) -> (forall n, P n)).
In words: for any proposition "P" over natural numbers, given a proof of "P 0" and a proof of "P n -> P (n+1)", we get back a proof of "forall n, P n". This is the familiar induction principle for natural numbers.
Implementations
W- and M-types
W-types are well-founded types in intuitionistic type theory (ITT).[1] They generalize natural numbers, lists, binary trees, and other "tree-shaped" data types. Let U be a universe of types. Given a type A : U and a dependent family B : A → U, one can form a W-type ${\mathsf {W}}_{a:A}B(a)$. The type A may be thought of as "labels" for the (potentially infinitely many) constructors of the inductive type being defined, whereas B indicates the (potentially infinite) arity of each constructor. W-types (resp. M-types) may also be understood as well-founded (resp. non-well-founded) trees with nodes labeled by elements a : A and where the node labeled by a has B(a)-many subtrees.[2] Each W-type is isomorphic to the initial algebra of a so-called polynomial functor.
Let 0, 1, 2, etc. be finite types with inhabitants 11 : 1, 12, 22:2, etc. One may define the natural numbers as the W-type
$\mathbb {N} :={\mathsf {W}}_{x:\mathbf {2} }f(x)$ :={\mathsf {W}}_{x:\mathbf {2} }f(x)}
with f : 2 → U is defined by f(12) = 0 (representing the constructor for zero, which takes no arguments), and f(22) = 1 (representing the successor function, which takes one argument).
One may define lists over a type A : U as $\operatorname {List} (A):={\mathsf {W}}_{(x:\mathbf {1} +A)}f(x)$ where
${\begin{aligned}f(\operatorname {inl} (1_{\mathbf {1} }))&=\mathbf {0} \\f(\operatorname {inr} (a))&=\mathbf {1} \end{aligned}}$
and 11 is the sole inhabitant of 1. The value of $f(\operatorname {inl} (1_{\mathbf {1} }))$ corresponds to the constructor for the empty list, whereas the value of $f(\operatorname {inr} (a))$ corresponds to the constructor that appends a to the beginning of another list.
The constructor for elements of a generic W-type ${\mathsf {W}}_{x:A}B(x)$ has type
${\mathsf {sup}}:\prod _{a:A}{\Big (}B(a)\to {\mathsf {W}}_{x:A}B(x){\Big )}\to {\mathsf {W}}_{x:A}B(x).$
We can also write this rule in the style of a natural deduction proof,
${\frac {a:A\qquad f:B(a)\to {\mathsf {W}}_{x:A}B(x)}{{\mathsf {sup}}(a,f):{\mathsf {W}}_{x:A}B(x)}}.$
The elimination rule for W-types works similarly to structural induction on trees. If, whenever a property (under the propositions-as-types interpretation) $C:{\mathsf {W}}_{x:A}B(x)\to U$ holds for all subtrees of a given tree it also holds for that tree, then it holds for all trees.
${\frac {w:{\mathsf {W}}_{a:A}B(a)\qquad a:A,\;f:B(a)\to {\mathsf {W}}_{x:A}B(x),\;c:\prod _{b:B(a)}C(f(b))\;\vdash \;h(a,f,c):C({\mathsf {sup}}(a,f))}{{\mathsf {elim}}(w,h):C(w)}}$
In extensional type theories, W-types (resp. M-types) can be defined up to isomorphism as initial algebras (resp. final coalgebras) for polynomial functors. In this case, the property of initiality (res. finality) corresponds directly to the appropriate induction principle.[3] In intensional type theories with the univalence axiom, this correspondence holds up to homotopy (propositional equality).[4][5][6]
M-types are dual to W-types, they represent coinductive (potentially infinite) data such as streams.[7] M-types can be derived from W-types.[8]
Mutually inductive definitions
This technique allows some definitions of multiple types that depend on each other. For example, defining two parity predicates on natural numbers using two mutually inductive types in Coq:
Inductive even : nat -> Prop :=
| zero_is_even : even O
| S_of_odd_is_even : (forall n:nat, odd n -> even (S n))
with odd : nat -> Prop :=
| S_of_even_is_odd : (forall n:nat, even n -> odd (S n)).
Induction-recursion
Induction-recursion started as a study into the limits of ITT. Once found, the limits were turned into rules that allowed defining new inductive types. These types could depend upon a function and the function on the type, as long as both were defined simultaneously.
Universe types can be defined using induction-recursion.
Induction-induction
Induction-induction allows definition of a type and a family of types at the same time. So, a type A and a family of types $B:A\to Type$.
Higher inductive types
This is a current research area in Homotopy Type Theory (HoTT). HoTT differs from ITT by its identity type (equality). Higher inductive types not only define a new type with constants and functions that create elements of the type, but also new instances of the identity type that relate them.
A simple example is the circle type, which is defined with two constructors, a basepoint;
base : circle
and a loop;
loop : base = base.
The existence of a new constructor for the identity type makes circle a higher inductive type.
See also
• Coinduction permits (effectively) infinite structures in type theory.
References
1. Martin-Löf, Per (1984). Intuitionistic type theory (PDF). Sambin, Giovanni. Napoli: Bibliopolis. ISBN 8870881059. OCLC 12731401.
2. Ahrens, Benedikt; Capriotti, Paolo; Spadotti, Régis (2015-04-12). Non-wellfounded trees in Homotopy Type Theory. Leibniz International Proceedings in Informatics (LIPIcs). Vol. 38. pp. 17–30. arXiv:1504.02949. doi:10.4230/LIPIcs.TLCA.2015.17. ISBN 9783939897873. S2CID 15020752.
3. Dybjer, Peter (1997). "Representing inductively defined sets by wellorderings in Martin-Löf's type theory". Theoretical Computer Science. 176 (1–2): 329–335. doi:10.1016/s0304-3975(96)00145-4.
4. Awodey, Steve; Gambino, Nicola; Sojakova, Kristina (2012-01-18). "Inductive types in homotopy type theory". arXiv:1201.3898 [math.LO].
5. Ahrens, Benedikt; Capriotti, Paolo; Spadotti, Régis (2015-04-12). Non-wellfounded trees in Homotopy Type Theory. Leibniz International Proceedings in Informatics (LIPIcs). Vol. 38. pp. 17–30. arXiv:1504.02949. doi:10.4230/LIPIcs.TLCA.2015.17. ISBN 9783939897873. S2CID 15020752.
6. Awodey, Steve; Gambino, Nicola; Sojakova, Kristina (2015-04-21). "Homotopy-initial algebras in type theory". arXiv:1504.05531 [math.LO].
7. van den Berg, Benno; Marchi, Federico De (2007). "Non-well-founded trees in categories". Annals of Pure and Applied Logic. 146 (1): 40–59. arXiv:math/0409158. doi:10.1016/j.apal.2006.12.001. S2CID 360990.
8. Abbott, Michael; Altenkirch, Thorsten; Ghani, Neil (2005). "Containers: Constructing strictly positive types". Theoretical Computer Science. 342 (1): 3–27. doi:10.1016/j.tcs.2005.06.002.
• Univalent Foundations Program (2013). Homotopy Type Theory: Univalent Foundations of Mathematics. Institute for Advanced Study.
External links
• Induction-Recursion Slides
• Induction-Induction Slides
• Higher Inductive Types: a tour of the menagerie
| Wikipedia |
Raphael Weldon
Walter Frank Raphael Weldon FRS (15 March 1860 – 13 April 1906), was an English evolutionary biologist and a founder of biometry. He was the joint founding editor of Biometrika, with Francis Galton and Karl Pearson.
Walter Frank Raphael Weldon
Raphael Weldon
Born(1860-03-15)15 March 1860
London, England
Died13 April 1906(1906-04-13) (aged 46)
Oxford, England
Alma materSt John's College, Cambridge
AwardsFellow of the Royal Society
Scientific career
FieldsZoology, biometry
InstitutionsSt John's College, Cambridge
University College London
Oxford University
Academic advisorsFrancis Maitland Balfour
InfluencesFrancis Galton, Karl Pearson
Family
Weldon was the second child of the journalist and industrial chemist, Walter Weldon, and his wife Anne Cotton. On 13 March 1883, Weldon married Florence Tebb (1858–1936), daughter of the social reformer William Tebb.
Life and education
Medicine was his intended career and he spent the academic year 1876-1877 at University College London. Among his teachers were the zoologist E. Ray Lankester and the mathematician Olaus Henrici. In the following year he transferred to King's College London and then to St John's College, Cambridge in 1878.[1]
There Weldon studied with the developmental morphologist Francis Balfour who influenced him greatly; Weldon gave up his plans for a career in medicine. In 1881 he gained a first-class honours degree in the Natural Science Tripos; in the autumn he left for the Naples Zoological Station to begin the first of his studies on marine biological organisms.
On his religious views, he considered himself an agnostic.[2] He died in 1906 of acute pneumonia, and is buried at Holywell Church, Oxford.
Career
Upon returning to Cambridge in 1882, he was appointed university lecturer in Invertebrate Morphology. Weldon's work was centred on the development of a fuller understanding of marine biological phenomena and selective death rates of these organisms.
In 1889 Weldon succeeded Lankester in the Jodrell Chair of Zoology at University College London,[3] and as curator of what is now the Grant Museum of Zoology,[4] and was elected to the Royal Society in 1890. Royal Society records show his election supporters included the great zoologists of the day: Huxley, Lankester, Poulton, Newton, Flower, Romanes and others.
His interests were changing from morphology to problems in variation and organic correlation. He began using the statistical techniques that Francis Galton had developed for he had come to the view that "the problem of animal evolution is essentially a statistical problem." Weldon began working with his University College colleague, the mathematician Karl Pearson. Their partnership was very important to both men and survived Weldon's move to the Linacre Chair of Zoology at Oxford University in 1899. In the years of their collaboration Pearson laid the foundations of modern statistics. Magnello emphasises this side of Weldon's career. In 1900 he took the DSc degree and as Linacre Professor he also held a Fellowship at Merton College, Oxford.[5]
Weldon was one of the first scientists to provide evidence of stabilizing and directional selection in natural populations.[6]
By 1893 a Royal Society Committee included Weldon, Galton and Karl Pearson 'For the Purpose of conducting Statistical Enquiry into the Variability of Organisms'. In an 1894 paper Some remarks on variation in plants and animals arising from the work of the Royal Society Committee, Weldon wrote:
"... the questions raised by the Darwinian hypothesis are purely statistical, and the statistical method is the only one at present obvious by which that hypothesis can be experimentally checked."
In 1900 the work of Gregor Mendel was rediscovered and this precipitated a conflict between Weldon and Pearson on the one side and William Bateson on the other. Bateson, who had been taught by Weldon, took a very strong line against the biometricians. This bitter dispute ranged across substantive issues of the nature of evolution and methodological issues such as the value of the statistical method. Will Provine gives a detailed account of the controversy.[7] The debate lost much of its intensity with the death of Weldon in 1906, though the general debate between the biometricians and the Mendelians continued until the creation of the modern evolutionary synthesis in the 1930s.
After his death, the Weldon Memorial Prize was established by the University of Oxford in his honour; it is awarded annually.
Weldon's dice
In 1894, Weldon rolled a set of 12 dice 26,306 times.[8] He collected the data in part, 'to judge whether the differences between a series of group frequencies and a theoretical law, taken as a whole, were or were not more than might be attributed to the chance fluctuations of random sampling.' Weldon's dice data were used by Karl Pearson[9] in his pioneering paper on the chi-squared statistic.
Notes
1. "Weldon, Walter Frank Raphael (WLDN878WF)". A Cambridge Alumni Database. University of Cambridge.
2. Karl Pearson (2011). Walter Frank Raphael Weldon 1860–1906: A Memoir Reprinted from Biometrika. Cambridge University Press. p. 5. ISBN 9781107601222. He was through the many years the present writer knew him, like his hero Huxley, a confirmed Agnostic.
3. Bourne, Gilbert Charles. "Weldon Walter Frank Raphael". Dictionary of National Biography, 1912 Supplement. 3.
4. "On the Origin of Our Specimens: The Weldon Years | UCL Museums & Collections Blog". blogs.ucl.ac.uk. Retrieved 9 November 2017.
5. Levens, R.G.C., ed. (1964). Merton College Register 1900–1964. Oxford: Basil Blackwell. p. 5.
6. Amitabh, Joshi. (2017). Weldon's Search for a Direct Proof of Natural Selection and the Tortuous Path to the Neo-Darwinian Synthesis. Resonance 22 (6): 525-548.
7. W.B. Provine (1971). The origins of theoretical population genetics. University of Chicago Press.
8. Kemp, A.W., and C.D. Kemp. (1991). Weldon's dice data revisited, The American Statistician, 45(3):216–222. DOI:10.2307/2684294
9. Pearson, Karl (1900). On the criterion that a given system of derivations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. Philosophical Magazine, 5(50), 157–175.
References
• Pearson, Karl (2011) [1906]. Walter Frank Raphael Weldon 1860–1906: A Memoir Reprinted from Biometrika. Cambridge University Press. ISBN 978-1-107-60122-2.
• W.B. Provine (1971) The origins of theoretical population genetics. University of Chicago Press.
• Magnello E. 2001. Walter Frank Raphael Weldon, in Statisticians of the Centuries (eds C.C. Heyde and E. Seneta) p261-264. New York: Springer.
• Shipley A.E. 1908. Walter Frank Raphael Weldon. Proc Roy Soc Series B 1908 vol 80 pxxv-xli.
External links
Wikisource has original works by or about:
Walter Frank Raphael Weldon
• Works by or about Raphael Weldon at Internet Archive
• O'Connor, John J.; Robertson, Edmund F., "Raphael Weldon", MacTutor History of Mathematics Archive, University of St Andrews
• Bourne, Gilbert Charles (1912). "Weldon, Walter Frank Raphael" . In Lee, Sidney (ed.). Dictionary of National Biography (2nd supplement). Vol. 3. London: Smith, Elder & Co. p. 629-631.
• "On Certain Correlated Variations in Carcinus moenas" Proceedings of the Royal Society, 54, (1893), 318–329. An example of Weldon's use of statistical methods
• Photograph of Weldon on the Portraits of Statisticians page.
• "Archival material relating to Raphael Weldon". UK National Archives.
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Victor Zalgaller
Victor (Viktor) Abramovich Zalgaller (Hebrew: ויקטור אבּרמוביץ' זלגלר; Russian: Виктор Абрамович Залгаллер; 25 December 1920 – 2 October 2020) was a Russian-Israeli mathematician in the fields of geometry and optimization. He is best known for the results he achieved on convex polyhedra, linear and dynamic programming, isoperimetry, and differential geometry.
Victor Zalgaller
Виктор Абрамович Залгаллер
V. A. Zalgaller, Rehovot, Israel, Sep. 2006
Born
Victor Abramovich Zalgaller
(1920-12-25)25 December 1920
Parfino, Novgorod, Russian SFSR
Died2 October 2020(2020-10-02) (aged 99)
Israel
Occupation(s)mathematician, teacher
Biography
Zalgaller was born in Parfino, Novgorod Governorate on 25 December 1920.[1] In 1936, he was one of the winners of the Leningrad Mathematics Olympiads for high school students. He started his studies at the Leningrad State University, however, World War II intervened in 1941, and Zalgaller joined the Red Army. He took part in the defence of Leningrad, and in 1945 marched into Germany.[2]
He worked as a teacher at the Saint Petersburg Lyceum 239,[3] and received his 1963[4] doctoral dissertation on polyhedra with the aid of his high school students who wrote the computer programs for the calculation.[3]
Zalgaller did his early work under direction of A. D. Alexandrov and Leonid Kantorovich. He wrote joint monographs with both of them. His later monograph Geometric Inequalities (joint with Yu. Burago) is still the main reference in the field.
Zalgaller lived in Saint Petersburg most of his life, having studied and worked at the Leningrad State University and the Steklov Institute of Mathematics (Saint Petersburg branch).
In 1999, he immigrated to Israel. Zalgaller died on 2 October 2020 at the age of 99.[3]
References
Wikimedia Commons has media related to Victor Zalgaller.
1. "Виктор Залгаллер (Израиль)". Vestnik (in Russian). Retrieved 3 October 2020.
2. "Life Begins at 80". Weizmann Institute of Science. 27 September 2012. Retrieved 3 October 2020.
3. "Скончался В.А.Залгаллер". Saint Petersburg Lyceum 239 (in Russian). Retrieved 3 October 2020.
4. "Victor Abramovich Zalgaller". Math Genealogy. Retrieved 3 October 2020.
• V. A. Aleksandrov, et al. Viktor Abramovich Zalgaller (on his 80th birthday), Russian Mathematical Surveys, Vol. 56 (2001), 1013–1014 (see here for a Russian version).
• Yu. D. Burago, et al. Viktor Abramovich Zalgaller (on his 80th birthday), J. Math. Sci. (N. Y.) J. Math. Sci. (N.Y.) Vol. 119 (2004), 129–132 (see here for a Russian version).
• M. Z. Solomyak, A few words about Viktor Abramovich Zalgaller, J. Math. Sci. (N.Y.) Vol. 119 (2004), 138–140.
• S. S. Kutateladze, A Tribute to the Philanthropist and Geometer.
• List of papers of V. A. Zalgaller, available here (mostly in Russian).
• Zalgaller, Victor; Burago, Yuri (February 1988). Geometric Inequalities. Springer Verlag. p. 356pp. ISBN 3-540-13615-0.
External links
• Victor Zalgaller at the Mathematics Genealogy Project
• Intrinsic Geometry of Surfaces — book by A.D Alexandrov and V.A. Zalgaller (AMS Online Book, originally translated in 1967).
• Personal war memoir (in Russian).
• Lecture read in 1999 in St.Petersburg, Russia (video, in Russian)
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| Wikipedia |
15 and 290 theorems
In mathematics, the 15 theorem or Conway–Schneeberger Fifteen Theorem, proved by John H. Conway and W. A. Schneeberger in 1993, states that if a positive definite quadratic form with integer matrix represents all positive integers up to 15, then it represents all positive integers.[1] The proof was complicated, and was never published. Manjul Bhargava found a much simpler proof which was published in 2000.[2]
Bhargava used the occasion of his receiving the 2005 SASTRA Ramanujan Prize to announce that he and Jonathan P. Hanke had cracked Conway's conjecture that a similar theorem holds for integral quadratic forms, with the constant 15 replaced by 290.[3] The proof has since appeared in preprint form.[4]
Details
Suppose $Q_{ij}$ is a symmetric matrix with real entries. For any vector $x$ with integer components, define
$Q(x)=\sum _{i,j}Q_{ij}x_{i}x_{j}$
This function is called a quadratic form. We say $Q$ is positive definite if $Q(x)>0$ whenever $x\neq 0$. If $Q(x)$ is always an integer, we call the function $Q$ an integral quadratic form.
We get an integral quadratic form whenever the matrix entries $Q_{ij}$ are integers; then $Q$ is said to have integer matrix. However, $Q$ will still be an integral quadratic form if the off-diagonal entries $Q_{ij}$ are integers divided by 2, while the diagonal entries are integers. For example, x2 + xy + y2 is integral but does not have integral matrix.
A positive integral quadratic form taking all positive integers as values is called universal. The 15 theorem says that a quadratic form with integer matrix is universal if it takes the numbers from 1 to 15 as values. A more precise version says that, if a positive definite quadratic form with integral matrix takes the values 1, 2, 3, 5, 6, 7, 10, 14, 15 (sequence A030050 in the OEIS), then it takes all positive integers as values. Moreover, for each of these 9 numbers, there is such a quadratic form taking all other 8 positive integers except for this number as values.
For example, the quadratic form
$w^{2}+x^{2}+y^{2}+z^{2}$
is universal, because every positive integer can be written as a sum of 4 squares, by Lagrange's four-square theorem. By the 15 theorem, to verify this, it is sufficient to check that every positive integer up to 15 is a sum of 4 squares. (This does not give an alternative proof of Lagrange's theorem, because Lagrange's theorem is used in the proof of the 15 theorem.)
On the other hand,
$w^{2}+2x^{2}+5y^{2}+5z^{2},$
is a positive definite quadratic form with integral matrix that takes as values all positive integers other than 15.
The 290 theorem says a positive definite integral quadratic form is universal if it takes the numbers from 1 to 290 as values. A more precise version states that, if an integer valued integral quadratic form represents all the numbers 1, 2, 3, 5, 6, 7, 10, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 34, 35, 37, 42, 58, 93, 110, 145, 203, 290 (sequence A030051 in the OEIS), then it represents all positive integers, and for each of these 29 numbers, there is such a quadratic form representing all other 28 positive integers with the exception of this one number.
Bhargava has found analogous criteria for a quadratic form with integral matrix to represent all primes (the set {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 67, 73} (sequence A154363 in the OEIS)) and for such a quadratic form to represent all positive odd integers (the set {1, 3, 5, 7, 11, 15, 33} (sequence A116582 in the OEIS)).
Expository accounts of these results have been written by Hahn[5] and Moon (who provides proofs).[6]
References
1. Conway, J.H. (2000). "Universal quadratic forms and the fifteen theorem". Quadratic forms and their applications (Dublin, 1999) (PDF). Contemp. Math. Vol. 272. Providence, RI: Amer. Math. Soc. pp. 23–26. ISBN 0-8218-2779-0. Zbl 0987.11026.
2. Bhargava, Manjul (2000). "On the Conway–Schneeberger fifteen theorem". Quadratic forms and their applications (Dublin, 1999) (PDF). Contemp. Math. Vol. 272. Providence, RI: Amer. Math. Soc. pp. 27–37. ISBN 0-8218-2779-0. MR 1803359. Zbl 0987.11027.
3. Alladi, Krishnaswami. "Ramanujan's legacy: the work of the SASTRA prize winners". Philosophical Transactions of the Royal Society A. The Royal Society Publishing. Retrieved 4 February 2020.
4. Bhargava, M., & Hanke, J., Universal quadratic forms and the 290-theorem.
5. Alexander J. Hahn, Quadratic Forms over $\mathbb {Z} $ from Diophantus to the 290 Theorem, Advances in Applied Clifford Algebras, 2008, Volume 18, Issue 3-4, 665-676
6. Yong Suk Moon, Universal quadratic forms and the 15-theorem and 290-theorem
| Wikipedia |
Gilbert Strang
William Gilbert Strang (born November 27, 1934[1]) is an American mathematician known for his contributions to finite element theory, the calculus of variations, wavelet analysis and linear algebra. He has made many contributions to mathematics education, including publishing mathematics textbooks. Strang was the MathWorks Professor of Mathematics at the Massachusetts Institute of Technology.[2] He taught Linear Algebra, Computational Science, and Engineering, Learning from Data, and his lectures are freely available through MIT OpenCourseWare.
W. Gilbert Strang
Strang in 2021
Born (1934-11-27) November 27, 1934
Chicago, Illinois
NationalityAmerican
Alma materMassachusetts Institute of Technology (BS)
Balliol College, Oxford (BA, MA)
University of California, Los Angeles (PhD)
AwardsChauvenet Prize (1977)
Scientific career
FieldsMathematics
InstitutionsMassachusetts Institute of Technology
ThesisDifference Methods for Mixed Boundary Value Problems (1959)
Doctoral advisorPeter K. Henrici
Doctoral students
• Hermann Flaschka
• Pavel Grinfeld
• Alan Berger
Biography
Strang was born in Chicago in 1934. His parents William and Mary Catherine Strang migrated to the USA from Scotland. He and his sister Vivian grew up in Washington DC and Cincinnati, and went to high school at Principia in St. Louis.
Strang graduated from MIT in 1955 with a Bachelor of Science in mathematics. He then received a Rhodes Scholarship to University of Oxford, where he received his B.A. and M.A. from Balliol College in 1957.
Strang earned his Ph.D. from University of California, Los Angeles in 1959 as a National Science Foundation Fellow, under the supervision of Peter K. Henrici. His dissertation was titled "Difference Methods for Mixed Boundary Value Problems".[3]
While at Oxford, Strang met his future wife Jillian Shannon, and they married in 1958. Following his Ph.D. at UCLA, they have lived in Wellesley, Massachusetts for almost all of his 62 years on the MIT faculty. The Strangs have three sons David, John, and Robert and describe themselves as a very close-knit family. He retired on May 15, 2023 after giving his final Linear Algebra and Learning from Data[4] lecture at MIT.[5]
Strang's teaching has focused on linear algebra which has helped the subject become essential for students of many majors. His linear algebra video lectures are popular on YouTube and MIT OpenCourseware. Strang founded Wellesley-Cambridge Press to publish Introduction to Linear Algebra (now in 6th edition) and ten other books.
University Positions
Following his PhD studies, from 1959 to 1961, Strang was a C. L. E. Moore instructor at M.I.T. in the Mathematics department. From 1961-1962 he was a NATO Postdoctoral Fellow at Oxford University. From 1962 until 2023, Strang was a mathematics professor at MIT.[6]
He has received Honorary Titles and Fellowships from the following institutes:
• Alfred P. Sloan Fellow (1966–1967)
• Honorary Professor, Xi'an Jiaotong University, China (1980)
• Honorary Fellow, Balliol College, Oxford University (1999)
• Honorary Member, Irish Mathematical Society (2002)
• Fellow of the Society for Industrial and Applied Mathematics (2009) [7]
• Doctor Honoris Causa, University of Toulouse (2010)
• Fellow of the American Mathematical Society (2012)[8]
• Doctor Honoris Causa, Aalborg University (2013)
Awards
• Rhodes Scholar (1955)
• National Science Foundation Graduate Research Fellowship (1957)
• Chauvenet Prize, Mathematical Association of America (1977)[9]
• American Academy of Arts and Sciences (1985)
• Award for Distinguished Service to the Profession, Society for Industrial and Applied Mathematics (2003)
• Lester R. Ford Award (2005)[10]
• Von Neumann Medal, US Association for Computational Mechanics (2005)
• Haimo Prize, Mathematical Association of America (2007)[11]
• Su Buchin Prize, International Congress (ICIAM, 2007)
• Henrici Prize (2007)
• National Academy of Sciences (2009)
• Irwin Sizer Award for the Most Significant Improvement to MIT Education (2020)[12]
Service
• President, Society for Industrial and Applied Mathematics (1999, 2000)[13]
• Chair, U.S. National Committee on Mathematics (2003–2004)
• Chair, National Science Foundation (NSF) Advisory Panel on Mathematics
• Board Member, International Council for Industrial and Applied Mathematics (ICIAM)
• Abel Prize Committee (2003–2005)
Publications
Books and monographs
1. Introduction to Linear Algebra, Sixth Edition, Wellesley-Cambridge Press (2023), Introduction to Linear Algebra
2. Linear Algebra for Everyone (2020)[14]
3. Linear Algebra and Learning from Data (2019)[15]
4. Calculus (2017) Textbook | Calculus Online Textbook | Supplemental Resources
5. Introduction to Linear Algebra, Fifth Edition (2016)
6. Differential Equations and Linear Algebra (2014) Differential Equations and Linear Algebra - New Book Website
7. Essays in Linear Algebra (2012)
8. Algorithms for Global Positioning, with Kai Borre (2012)
9. An Analysis of the Finite Element Method, with George Fix (2008)
10. Computational Science and Engineering (2007)
11. Linear Algebra and Its Applications, Fourth Edition (2005)
12. Linear Algebra, Geodesy, and GPS, with Kai Borre (1997)
13. Wavelets and Filter Banks, with Truong Nguyen (1996)
14. Strang, Gilbert (1986). Introduction to Applied Mathematics. Wellesley, MA: Wellesley-Cambridge Press. pp. xii+758. MR 0870634.
See also
• The Joint spectral radius, introduced by Strang and Rota in the early 60s.
• Strang splitting
References
1. Roselle, D. P. (1977). "Award of the 1977 Chauvenet Prize to Professor Gilbert Strang". The American Mathematical Monthly. 84 (6): 417. CiteSeerX 10.1.1.119.4043. doi:10.1080/00029890.1977.11994378. JSTOR 2321898.
2. "MIT announces Professor Gilbert Strang as the first MathWorks Professor of Mathematics". Cambridge, MA: MIT News. Retrieved September 26, 2011.
3. Strang, William Gilbert (1959). Difference Methods for Mixed Boundary Value Problems (PhD thesis). University of California, Los Angeles. ProQuest 301900319.
4. "18.065 Matrix Methods in Data Analysis & Signal Processing". math.mit.edu. Retrieved 2023-05-15.
5. Gil Strang's Final 18.06 Linear Algebra Lecture, retrieved 2023-05-15
6. "Gil Strang's Final 18.06 Linear Algebra Lecture" – via www.youtube.com.
7. "Fellows Program | SIAM". www.siam.org.
8. List of Fellows of the American Mathematical Society, retrieved 2013-08-05.
9. Strang, Gilbert (1973-11-01). "Piecewise polynomials and the finite element method". Bulletin of the American Mathematical Society. American Mathematical Society (AMS). 79 (6): 1128–1138. doi:10.1090/s0002-9904-1973-13351-8. ISSN 0002-9904.
10. Edelman, Alan; Strang, Gilbert (2004). "Pascal matrices". Amer. Math. Monthly. 111 (3): 189–197. doi:10.2307/4145127. JSTOR 4145127.
11. "Deborah and Franklin Tepper Haimo Award".
12. Irwin Sizer Award, retrieved 2020-05-25.
13. "Leadership | SIAM". www.siam.org. Retrieved 2019-09-11.
14. "Linear Algebra for Everyone". math.mit.edu. Retrieved 2021-01-11.
15. "Linear Algebra and Learning from Data". math.mit.edu. Retrieved 2019-09-11.
External links
• Official website
• Gilbert Strang at the Mathematics Genealogy Project
• Wellesley Cambridge Press (USA) www.wellesleycambridge.com/
• Wellesley Publishers (India) www.wellesleypublishers.com/
Chauvenet Prize recipients
• 1925 G. A. Bliss
• 1929 T. H. Hildebrandt
• 1932 G. H. Hardy
• 1935 Dunham Jackson
• 1938 G. T. Whyburn
• 1941 Saunders Mac Lane
• 1944 R. H. Cameron
• 1947 Paul Halmos
• 1950 Mark Kac
• 1953 E. J. McShane
• 1956 Richard H. Bruck
• 1960 Cornelius Lanczos
• 1963 Philip J. Davis
• 1964 Leon Henkin
• 1965 Jack K. Hale and Joseph P. LaSalle
• 1967 Guido Weiss
• 1968 Mark Kac
• 1970 Shiing-Shen Chern
• 1971 Norman Levinson
• 1972 François Trèves
• 1973 Carl D. Olds
• 1974 Peter D. Lax
• 1975 Martin Davis and Reuben Hersh
• 1976 Lawrence Zalcman
• 1977 W. Gilbert Strang
• 1978 Shreeram S. Abhyankar
• 1979 Neil J. A. Sloane
• 1980 Heinz Bauer
• 1981 Kenneth I. Gross
• 1982 No award given.
• 1983 No award given.
• 1984 R. Arthur Knoebel
• 1985 Carl Pomerance
• 1986 George Miel
• 1987 James H. Wilkinson
• 1988 Stephen Smale
• 1989 Jacob Korevaar
• 1990 David Allen Hoffman
• 1991 W. B. Raymond Lickorish and Kenneth C. Millett
• 1992 Steven G. Krantz
• 1993 David H. Bailey, Jonathan M. Borwein and Peter B. Borwein
• 1994 Barry Mazur
• 1995 Donald G. Saari
• 1996 Joan Birman
• 1997 Tom Hawkins
• 1998 Alan Edelman and Eric Kostlan
• 1999 Michael I. Rosen
• 2000 Don Zagier
• 2001 Carolyn S. Gordon and David L. Webb
• 2002 Ellen Gethner, Stan Wagon, and Brian Wick
• 2003 Thomas C. Hales
• 2004 Edward B. Burger
• 2005 John Stillwell
• 2006 Florian Pfender & Günter M. Ziegler
• 2007 Andrew J. Simoson
• 2008 Andrew Granville
• 2009 Harold P. Boas
• 2010 Brian J. McCartin
• 2011 Bjorn Poonen
• 2012 Dennis DeTurck, Herman Gluck, Daniel Pomerleano & David Shea Vela-Vick
• 2013 Robert Ghrist
• 2014 Ravi Vakil
• 2015 Dana Mackenzie
• 2016 Susan H. Marshall & Donald R. Smith
• 2017 Mark Schilling
• 2018 Daniel J. Velleman
• 2019 Tom Leinster
• 2020 Vladimir Pozdnyakov & J. Michael Steele
• 2021 Travis Kowalski
• 2022 William Dunham, Ezra Brown & Matthew Crawford
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| Wikipedia |
William Threlfall
William Richard Maximilian Hugo Threlfall (25 June 1888, in Dresden – 4 April 1949, in Oberwolfach) was a British-born German mathematician who worked on algebraic topology. He was a coauthor of the standard textbook Lehrbuch der Topologie.
In 1933 he signed the Vow of allegiance of the Professors of the German Universities and High-Schools to Adolf Hitler and the National Socialistic State.
Publications
• Threlfall, W. (1932), "Gruppenbilder" (PDF), Abh. Math.-Phys. Kl. Sächs. Akad. Wiss., Leipzig: Hirzel, 41: 1–59
• Seifert, Threlfall: Lehrbuch der Topologie, Teubner 1934
• Seifert, Threlfall: Variationsrechnung im Großen, Teubner 1938
See also
• Möbius–Kantor graph
• Schwarz triangle tessellation
References
• Gabriele Dörflinger: William R. M. H. Threlfall
• William Threlfall at the Mathematics Genealogy Project
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W. Stephen Wilson
W. Stephen Wilson is a mathematician based in Johns Hopkins University specializing in homotopy theory.[1]
Wilson received his Ph.D. from Massachusetts Institute of Technology in 1972 under the supervision of Franklin Paul Peterson.[2]
In 2012, Wilson became a fellow of the American Mathematical Society.[3]
References
1. "Math Dept Home Page of W. Stephen Wilson". Math.jhu.edu. Retrieved 7 June 2016.
2. W. Stephen Wilson at the Mathematics Genealogy Project
3. "American Mathematical Society". Ams.org. Retrieved 7 June 2016.
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W. T. Tutte
William Thomas Tutte OC FRS FRSC (/tʌt/; 14 May 1917 – 2 May 2002) was an English and Canadian codebreaker and mathematician. During the Second World War, he made a brilliant and fundamental advance in cryptanalysis of the Lorenz cipher, a major Nazi German cipher system which was used for top-secret communications within the Wehrmacht High Command. The high-level, strategic nature of the intelligence obtained from Tutte's crucial breakthrough, in the bulk decrypting of Lorenz-enciphered messages specifically, contributed greatly, and perhaps even decisively, to the defeat of Nazi Germany.[2][3] He also had a number of significant mathematical accomplishments, including foundation work in the fields of graph theory and matroid theory.[4][5]
W. T. Tutte
Born(1917-05-14)14 May 1917
Newmarket, Suffolk, England
Died2 May 2002(2002-05-02) (aged 84)
Kitchener, Ontario, Canada
Alma materTrinity College, Cambridge (PhD)
Known for
• BEST theorem
• Hanani–Tutte theorem
• Peripheral cycle
• Tutte 12-cage
• Tutte embedding
• Tutte graph
• Tutte homotopy theorem
• Tutte matrix
• Tutte polynomial
• Tutte theorem
• Tutte–Berge formula
• Tutte–Coxeter graph
• Tutte–Grothendieck invariant
• Tutte's "1+2 break in"
• Tutte's 1-factor theorem
• Tutte's fragment
• Tutte's linking theorem
• Tutte's planarity criterion
• Tutte's statistical method
• Tutte's triangle lemma
• Tutte's unimodular representation theorem
• Tutte's wheel theorem
SpouseDorothea Geraldine Mitchell (m. 1949–1994, her death)
Awards
• Jeffery–Williams Prize (1971)
• Henry Marshall Tory Medal (1975)
• Isaak-Walton-Killam Award (1982)
• CRM-Fields-PIMS prize (2001)
Scientific career
FieldsMathematics
InstitutionsUniversity of Toronto
University of Waterloo
ThesisAn Algebraic Theory of Graphs[1] (1948)
Doctoral advisorShaun Wylie[1]
Doctoral students
• W. G. Brown
• Neil Robertson[1]
Tutte's research in the field of graph theory proved to be of remarkable importance. At a time when graph theory was still a primitive subject, Tutte commenced the study of matroids and developed them into a theory by expanding from the work that Hassler Whitney had first developed around the mid-1930s.[6] Even though Tutte's contributions to graph theory have been influential to modern graph theory and many of his theorems have been used to keep making advances in the field, most of his terminology was not in agreement with their conventional usage and thus his terminology is not used by graph theorists today.[7] "Tutte advanced graph theory from a subject with one text (D. Kőnig's) toward its present extremely active state."[7]
Early life and education
Tutte was born in Newmarket in Suffolk. He was the younger son of William John Tutte (1873–1944), an estate gardener, and Annie (née Newell; 1881–1956), a housekeeper. Both parents worked at Fitzroy House stables where Tutte was born.[5] The family spent some time in Buckinghamshire, County Durham and Yorkshire before returning to Newmarket, where Tutte attended Cheveley Church of England primary school[8] in the nearby village of Cheveley.[4] In 1927, when he was ten, Tutte won a scholarship to the Cambridge and County High School for Boys. He took up his place there in 1928.
In 1935 he won a scholarship to study natural sciences at Trinity College, Cambridge, where he specialized in chemistry and graduated with first-class honours in 1938.[4] He continued with physical chemistry as a graduate student, but transferred to mathematics at the end of 1940.[4] As a student, he (along with three of his friends) became one of the first to solve the problem of squaring the square, and the first to solve the problem without a squared subrectangle. Together the four created the pseudonym Blanche Descartes, under which Tutte published occasionally for years.[9]
Second World War
Soon after the outbreak of the Second World War, Tutte's tutor, Patrick Duff, suggested him for war work at the Government Code and Cypher School at Bletchley Park (BP). He was interviewed and sent on a training course in London before going to Bletchley Park, where he joined the Research Section. At first, he worked on the Hagelin cipher that was being used by the Italian Navy. This was a rotor cipher machine that was available commercially, so the mechanics of enciphering was known, and decrypting messages only required working out how the machine was set up.[11]
In the summer of 1941, Tutte was transferred to work on a project called Fish. Intelligence information had revealed that the Germans called the wireless teleprinter transmission systems "Sägefisch" (sawfish). This led the British to use the code Fish for the German teleprinter cipher system. The nickname Tunny (tunafish) was used for the first non-Morse link, and it was subsequently used for the Lorenz SZ machines and the traffic that they enciphered.[12]
Telegraphy used the 5-bit International Telegraphy Alphabet No. 2 (ITA2). Nothing was known about the mechanism of enciphering other than that messages were preceded by a 12-letter indicator, which implied a 12-wheel rotor cipher machine. The first step, therefore, had to be to diagnose the machine by establishing the logical structure and hence the functioning of the machine. Tutte played a pivotal role in achieving this, and it was not until shortly before the Allied victory in Europe in 1945, that Bletchley Park acquired a Tunny Lorenz cipher machine.[13] Tutte's breakthroughs led eventually to bulk decrypting of Tunny-enciphered messages between the German High Command (OKW) in Berlin and their army commands throughout occupied Europe and contributed—perhaps decisively—to the defeat of Germany.[2][3]
Diagnosing the cipher machine
On 31 August 1941, two versions of the same message were sent using identical keys, which constituted a "depth". This allowed John Tiltman, Bletchley Park's veteran and remarkably gifted cryptanalyst, to deduce that it was a Vernam cipher which uses the Exclusive Or (XOR) function (symbolised by "⊕"), and to extract the two messages and hence obtain the obscuring key. After a fruitless period during which Research Section cryptanalysts tried to work out how the Tunny machine worked, this and some other keys were handed to Tutte, who was asked to "see what you can make of these".[14]
At his training course, Tutte had been taught the Kasiski examination technique of writing out a key on squared paper, starting a new row after a defined number of characters that was suspected of being the frequency of repetition of the key.[15] If this number was correct, the columns of the matrix would show more repetitions of sequences of characters than chance alone. Tutte knew that the Tunny indicators used 25 letters (excluding J) for 11 of the positions, but only 23 letters for the other. He therefore tried Kasiski's technique on the first impulse of the key characters, using a repetition of 25 × 23 = 575. He did not observe a large number of column repetitions with this period, but he did observe the phenomenon on a diagonal. He therefore tried again with 574, which showed up repeats in the columns. Recognising that the prime factors of this number are 2, 7 and 41, he tried again with a period of 41 and "got a rectangle of dots and crosses that was replete with repetitions".[16]
It was clear, however, that the first impulse of the key was more complicated than that produced by a single wheel of 41 key impulses. Tutte called this component of the key $\chi _{1}$ (chi1). He figured that there was another component, which was XOR-ed with this, that did not always change with each new character, and that this was the product of a wheel that he called $\psi _{1}$ (psi1). The same applied for each of the five impulses ($\chi _{1}\chi _{2}\chi _{3}\chi _{4}\chi _{5}$ and $\psi _{1}\psi _{2}\psi _{3}\psi _{4}\psi _{5}$). So for a single character, the whole key K consisted of two components:
$K=\chi \oplus \psi $
At Bletchley Park, mark impulses were signified by x and space impulses by •.[nb 1] For example, the letter "H" would be coded as ••x•x.[17] Tutte's derivation of the chi and psi components was made possible by the fact that dots were more likely than not to be followed by dots, and crosses more likely than not to be followed by crosses. This was a product of a weakness in the German key setting, which they later eliminated. Once Tutte had made this breakthrough, the rest of the Research Section joined in to study the other impulses, and it was established that the five chi wheels all advanced with each new character and that the five psi wheels all moved together under the control of two mu or "motor" wheels. Over the following two months, Tutte and other members of the Research Section worked out the complete logical structure of the machine, with its set of wheels bearing cams that could either be in a position (raised) that added x to the stream of key characters, or in the alternative position that added in •.[18]
Diagnosing the functioning of the Tunny machine in this way was a truly remarkable cryptanalytical achievement which, in the citation for Tutte's induction as an Officer of the Order of Canada, was described as "one of the greatest intellectual feats of World War II".[5]
Tutte's statistical method
To decrypt a Tunny message required knowledge not only of the logical functioning of the machine, but also the start positions of each rotor for the particular message. The search was on for a process that would manipulate the ciphertext or key to produce a frequency distribution of characters that departed from the uniformity that the enciphering process aimed to achieve. While on secondment to the Research Section in July 1942, Alan Turing worked out that the XOR combination of the values of successive characters in a stream of ciphertext and key emphasised any departures from a uniform distribution. The resultant stream (symbolised by the Greek letter "delta" Δ) was called the difference because XOR is the same as modulo 2 subtraction.
The reason that this provided a way into Tunny was that although the frequency distribution of characters in the ciphertext could not be distinguished from a random stream, the same was not true for a version of the ciphertext from which the chi element of the key had been removed. This was the case because where the plaintext contained a repeated character and the psi wheels did not move on, the differenced psi character ($\Delta \psi $) would be the null character ('/ ' at Bletchley Park). When XOR-ed with any character, this character has no effect. Repeated characters in the plaintext were more frequent both because of the characteristics of German (EE, TT, LL and SS are relatively common),[19] and because telegraphists frequently repeated the figures-shift and letters-shift characters[20] as their loss in an ordinary telegraph message could lead to gibberish.[21]
To quote the General Report on Tunny:
Turingery introduced the principle that the key differenced at one, now called ΔΚ, could yield information unobtainable from ordinary key. This Δ principle was to be the fundamental basis of nearly all statistical methods of wheel-breaking and setting.[10]
Tutte exploited this amplification of non-uniformity in the differenced values [nb 2] and by November 1942 had produced a way of discovering wheel starting points of the Tunny machine which became known as the "Statistical Method".[22] The essence of this method was to find the initial settings of the chi component of the key by exhaustively trying all positions of its combination with the ciphertext, and looking for evidence of the non-uniformity that reflected the characteristics of the original plaintext.[23][24] Because any repeated characters in the plaintext would always generate •, and similarly $\Delta \psi _{1}\oplus \Delta \psi _{2}$ would generate • whenever the psi wheels did not move on, and about half of the time when they did – some 70% overall.
As well as applying differencing to the full 5-bit characters of the ITA2 code, Tutte applied it to the individual impulses (bits).[nb 3] The current chi wheel cam settings needed to have been established to allow the relevant sequence of characters of the chi wheels to be generated. It was totally impracticable to generate the 22 million characters from all five of the chi wheels, so it was initially limited to 41 × 31 = 1271 from the first two. After explaining his findings to Max Newman, Newman was given the job of developing an automated approach to comparing ciphertext and key to look for departures from randomness. The first machine was dubbed Heath Robinson, but the much faster Colossus computer, developed by Tommy Flowers and using algorithms written by Tutte and his colleagues, soon took over for breaking codes.[25][26][27]
Doctorate and career
In late 1945, Tutte resumed his studies at Cambridge, now as a graduate student in mathematics. He published some work begun earlier, one a now famous paper that characterises which graphs have a perfect matching, and another that constructs a non-Hamiltonian graph.
Tutte completed a doctorate in mathematics from Cambridge in 1948 under the supervision of Shaun Wylie, who had also worked at Bletchley Park on Tunny. His thesis An Algebraic Theory of Graphs was considered ground breaking and was about the subject later known as matroid theory.[28]
The same year, invited by Harold Scott MacDonald Coxeter, he accepted a position at the University of Toronto. In 1962, he moved to the University of Waterloo in Waterloo, Ontario, where he stayed for the rest of his academic career. He officially retired in 1985, but remained active as an emeritus professor. Tutte was instrumental in helping to found the Department of Combinatorics and Optimization at the University of Waterloo.
His mathematical career concentrated on combinatorics, especially graph theory, which he is credited as having helped create in its modern form, and matroid theory, to which he made profound contributions; one colleague described him as "the leading mathematician in combinatorics for three decades". He was editor in chief of the Journal of Combinatorial Theory until retiring from Waterloo in 1985.[28] He also served on the editorial boards of several other mathematical research journals.
Research contributions
Tutte's work in graph theory includes the structure of cycle spaces and cut spaces, the size of maximum matchings and existence of k-factors in graphs, and Hamiltonian and non-Hamiltonian graphs.[28] He disproved Tait's conjecture, on the Hamiltonicity of polyhedral graphs, by using the construction known as Tutte's fragment. The eventual proof of the four colour theorem made use of his earlier work. The graph polynomial he called the "dichromate" has become famous and influential under the name of the Tutte polynomial and serves as the prototype of combinatorial invariants that are universal for all invariants that satisfy a specified reduction law.
The first major advances in matroid theory were made by Tutte in his 1948 Cambridge PhD thesis which formed the basis of an important sequence of papers published over the next two decades. Tutte's work in graph theory and matroid theory has been profoundly influential on the development of both the content and direction of these two fields.[7] In matroid theory, he discovered the highly sophisticated homotopy theorem and founded the studies of chain groups and regular matroids, about which he proved deep results.
In addition, Tutte developed an algorithm for determining whether a given binary matroid is a graphic matroid. The algorithm makes use of the fact that a planar graph is simply a graph whose circuit-matroid, the dual of its bond-matroid, is graphic.[29]
Tutte wrote a paper entitled How to Draw a Graph in which he proved that any face in a 3-connected graph is enclosed by a peripheral cycle. Using this fact, Tutte developed an alternative proof to show that every Kuratowski graph is non-planar by showing that K5 and K3,3 each have three distinct peripheral cycles with a common edge. In addition to using peripheral cycles to prove that the Kuratowski graphs are non-planar, Tutte proved that every simple 3-connected graph can be drawn with all its faces convex, and devised an algorithm which constructs the plane drawing by solving a linear system. The resulting drawing is known as the Tutte embedding. Tutte's algorithm makes use of the barycentric mappings of the peripheral circuits of a simple 3-connected graph.[30]
The findings published in this paper have proved to be of much significance because the algorithms that Tutte developed have become popular planar graph drawing methods. One of the reasons for which Tutte's embedding is popular is that the necessary computations that are carried out by his algorithms are simple and guarantee a one-to-one correspondence of a graph and its embedding onto the Euclidean plane, which is of importance when parameterising a three-dimensional mesh to the plane in geometric modelling. "Tutte's theorem is the basis for solutions to other computer graphics problems, such as morphing."[31]
Tutte was mainly responsible for developing the theory of enumeration of planar graphs, which has close links with chromatic and dichromatic polynomials. This work involved some highly innovative techniques of his own invention, requiring considerable manipulative dexterity in handling power series (whose coefficients count appropriate kinds of graphs) and the functions arising as their sums, as well as geometrical dexterity in extracting these power series from the graph-theoretic situation.[32]
Tutte summarised his work in the Selected Papers of W.T. Tutte, 1979, and in Graph Theory as I have known it, 1998.[28]
Positions, honours and awards
Tutte's work in World War II and subsequently in combinatorics brought him various positions, honours and awards:
• 1958, Fellow of the Royal Society of Canada (FRSC);
• 1971, Jeffery–Williams Prize by the Canadian Mathematical Society;
• 1975, Henry Marshall Tory Medal by the Royal Society of Canada;
• 1977, A conference on Graph Theory and Related Topics was held at the University of Waterloo in his honour on the occasion of his sixtieth birthday;
• 1982, Isaak-Walton-Killam Award by the Canada Council;
• 1987, Fellow of the Royal Society (FRS);
• 1990–1996, First President of the Institute of Combinatorics and its Applications;[33]
• 1998, Appointed honorary director of the Centre for Applied Cryptographic Research at the University of Waterloo;[34]
• 2001, Officer of the Order of Canada (OC);
• 2001, CRM-Fields-PIMS prize.
• 2016, Waterloo Region Hall of Fame[35]
• 2017, Waterloo "William Tutte Way" road naming[36]
Tutte served as Librarian for the Royal Astronomical Society of Canada in 1959–1960, and asteroid 14989 Tutte (1997 UB7) was named after him.[37]
Because of Tutte's work at Bletchley Park, Canada's Communications Security Establishment named an internal organisation aimed at promoting research into cryptology, the Tutte Institute for Mathematics and Computing (TIMC), in his honour in 2011.[38]
In September 2014, Tutte was celebrated in his hometown of Newmarket, England, with the unveiling of a sculpture, after a local newspaper started a campaign to honour his memory.[39]
Bletchley Park in Milton Keynes celebrated Tutte's work with an exhibition Bill Tutte: Mathematician + Codebreaker from May 2017 to 2019, preceded on 14 May 2017 by lectures about his life and work during the Bill Tutte Centenary Symposium.[40][41]
Personal life and death
In addition to the career benefits of working at the new University of Waterloo, the more rural setting of Waterloo County appealed to Bill and his wife Dorothea. They bought a house in the nearby village of West Montrose, Ontario where they enjoyed hiking, spending time in their garden on the Grand River and allowing others to enjoy the beautiful scenery of their property.
They also had an extensive knowledge of all the birds in their garden. Dorothea, an avid potter, was also a keen hiker and Bill organised hiking trips. Even near the end of his life Bill still was an avid walker.[7][42] After his wife died in 1994, he moved back to Newmarket (Suffolk), but then returned to Waterloo in 2000, where he died two years later.[43] He is buried in West Montrose United Cemetery.[28]
Select publications
Books
• Tutte, W. T. (1966), Connectivity in graphs, Mathematical expositions, vol. 15, Toronto, Ontario: University of Toronto Press, Zbl 0146.45603
• Tutte, W. T. (1966), Introduction to the theory of matroids, Santa Monica, Calif.: RAND Corporation report R-446-PR. Also Tutte, W. T. (1971), Introduction to the theory of matroids, Modern analytic and computational methods in science and mathematics, vol. 37, New York: American Elsevier Publishing Company, ISBN 978-0-444-00096-5, Zbl 0231.05027
• Tutte, W. T., ed. (1969), Recent progress in combinatorics. Proceedings of the third Waterloo conference on combinatorics, May 1968, New York-London: Academic Press, pp. xiv+347, ISBN 978-0-12-705150-5, Zbl 0192.33101
• Tutte, W. T. (1979), McCarthy, D.; Stanton, R. G. (eds.), Selected papers of W.T. Tutte, Vols. I, II., Winnipeg, Manitoba: Charles Babbage Research Centre, St. Pierre, Manitoba, Canada, pp. xxi+879, Zbl 0403.05028
• Volume I: ISBN 978-0-969-07781-7
• Volume II: ISBN 978-0-969-07782-4
• Tutte, W. T. (1984), Graph theory, Encyclopedia of mathematics and its applications, vol. 21, Menlo Park, California: Addison-Wesley Publishing Company, ISBN 978-0-201-13520-6, Zbl 0554.05001 Reprinted by Cambridge University Press 2001, ISBN 978-0-521-79489-3
• Tutte, W. T. (1998), Graph theory as I have known it, Oxford lecture series in mathematics and its applications, vol. 11, Oxford: Clarendon Press, ISBN 978-0-19-850251-7, Zbl 0915.05041 Reprinted 2012, ISBN 978-0-19-966055-1
Articles
• Brooks, R. L.; Smith, C. A. B.; Stone, A. H.; Tutte, W. T. (1940). "The Dissection of Rectangles into Squares". Duke Math. J. 7: 312–340. doi:10.1215/s0012-7094-40-00718-9.
See also
• List of University of Waterloo people
• Systolic geometry
Notes
1. In more recent terminology, each impulse would be termed a "bit" with a mark being binary 1 and a space being binary 0. Punched paper tape had a hole for a mark and no hole for a space.
2. For this reason Tutte's 1 + 2 method is sometimes called the "double delta" method.
3. The five impulses or bits of the coded characters are sometimes referred to as five levels.
References
1. W. T. Tutte at the Mathematics Genealogy Project
2. Hinsley & Stripp 1993, p. 8
3. Brzezinski 2005, p. 18
4. Younger 2012
5. O'Connor & Robertson 2003
6. Johnson, Will. "Matroids" (PDF). Retrieved 16 October 2014.
7. Hobbs, Arthur M.; James G. Oxley (March 2004). "William T. Tutte (1917–2002)" (PDF). Notices of the American Mathematical Society. 51 (3): 322.
8. Cheveley CofE Primary School, Park Road, Cheveley, Cambridgeshire, CB8 9DF http://www.cheveley.cambs.sch.uk/
9. Smith, Cedric A. B.; Abbott, Steve (March 2003), "The Story of Blanche Descartes", The Mathematical Gazette, 87 (508): 23–33, doi:10.1017/S0025557200172067, ISSN 0025-5572, JSTOR 3620560, S2CID 192758206
10. Good, Michie & Timms 1945, p. 6 in 1. Introduction: German Tunny
11. Tutte 2006, pp. 352–353
12. Hinsley, F.H. (2001) [1993]. "An Introduction to Fish". In F.H. Hinsley; Alan Stripp (eds.). Codebreakers: the inside story of Bletchley Park. pp. 141–148. ISBN 0-19-280132-5.
13. Sale, Tony, The Lorenz Cipher and how Bletchley Park broke it, retrieved 21 October 2010
14. Tutte 2006, p. 354
15. Bauer 2006, p. 375
16. Tutte 2006, pp. 356–357
17. Copeland 2006, pp. 348, 349
18. Tutte 2006, p. 357
19. Singh, Simon, The Black Chamber, retrieved 28 April 2012
20. Newman c. 1944 p. 387
21. Carter 2004, p. 3 harvnb error: no target: CITEREFCarter2004 (help)
22. Tutte 1998, pp. 7–8
23. Good, Michie & Timms 1945, pp. 321–322 in 44. Hand Statistical Methods: Setting – Statistical Methods
24. Budiansky 2006, pp. 58–59 harvnb error: no target: CITEREFBudiansky2006 (help)
25. Copeland 2011
26. Younger, Dan (August 2002). "Biography of Professor Tutte". CMS Notes. Retrieved 24 June 2018 – via University of Waterloo.
27. Roberts, Jerry (2017), Lorenz: Breaking Hitler's top secret code at Bletchley Park, Stroud, Gloucestershire: The History Press, ISBN 978-0-7509-7885-9
28. "Biography of Professor Tutte | Combinatorics and Optimization | University of Waterloo". Archived from the original on 19 August 2019. Retrieved 11 May 2017.
29. W.T Tutte. An algorithm for determining whether a given binary matroid is graphic, Proceedings of the London Mathematical Society, 11(1960)905–917
30. W.T. Tutte. How to draw a graph. Proceedings of the London Mathematical Society, 13(3):743–768, 1963.
31. Steven J. Gortle; Craig Gotsman; Dylan Thurston. "Discrete One-Forms on Meshes and Applications to 3D Mesh Parameterization", Computer Aided Geometric Design, 23(2006)83–112
32. C. St. J. A. Nash-Williams, A Note on Some of Professor Tutte's Mathematical Work, Graph Theory and Related Topics (eds. J.A Bondy and U. S. R Murty), Academic Press, New York, 1979, p. xxvii.
33. "The Institute of Combinatorics & Its Applications". ICA. Archived from the original on 2 October 2013. Retrieved 28 September 2013.
34. "Tutte honoured by cryptographic centre". University of Waterloo. Retrieved 28 September 2013.
35. "Bill Tutte inducted into the Waterloo Region Hall of Fame | Combinatorics and Optimization". Combinatorics and Optimization. 25 April 2016.
36. "Mathematics professor and wartime code-breaker honoured". 12 May 2017.
37. "Asteroid (14989) Tutte". Royal Astronomical Society of Canada. 14 June 2011. Archived from the original on 4 January 2015. Retrieved 25 September 2014.
38. Freeze, Colin (7 September 2011). "Top secret institute comes out of the shadows to recruit top talent". The Globe and Mail. Toronto. Retrieved 25 September 2014.
39. "The Bill Tutte Memorial". Bill Tutte Memorial Fund. Retrieved 13 December 2014.
40. "The Bill Tutte Centenary Symposium (Bletchley Park)". 11 April 2017.
41. "Bletchley Park | News — New exhibition to tell story of Bill Tutte". Archived from the original on 6 June 2017. Retrieved 11 May 2017.
42. "Bill Tutte". Telegraph Group Limited. Archived from the original on 27 September 2013. Retrieved 21 May 2013.
43. van der Vat, Dan (10 May 2002), "Obituary: William Tutte", The Guardian, London, retrieved 28 April 2013
Sources
• Bauer, Friedrich L. (2006), The Tiltman Break Appendix 5 in Copeland 2006, pp. 370–377
• Brzezinski, Zbigniew (2005), "The Unknown Victors", in Ciechanowski, Stanisław (ed.), Marian Rejewski, 1905-1980: living with the Enigma secret, Bydgoszcz, Poland: Bydgoszcz City Council, pp. 15–18, ISBN 83-7208-117-4
• Copeland, B. Jack, ed. (2006), Colossus: The Secrets of Bletchley Park's Codebreaking Computers, Oxford: Oxford University Press, ISBN 978-0-19-284055-4
• Copeland, B. Jack (2011), Colossus and the Dawning of the Computer Age in Erskine & Smith 2011, pp. 305–327
• Erskine, Ralph; Smith, Michael, eds. (2011) [2001], The Bletchley Park Codebreakers, Biteback Publishing Ltd, ISBN 978-1-84954-078-0 Updated and extended version of Action This Day: From Breaking of the Enigma Code to the Birth of the Modern Computer Bantam Press 2001
• Good, Jack; Michie, Donald; Timms, Geoffrey (1945), General Report on Tunny: With Emphasis on Statistical Methods, UK Public Record Office HW 25/4 and HW 25/5, retrieved 15 September 2010 That version is a facsimile copy, but there is a transcript of much of this document in '.pdf' format at: Sale, Tony (2001), Part of the 'General Report on Tunny', the Newmanry History, formatted by Tony Sale (PDF), retrieved 20 September 2010, and a web transcript of Part 1 at: Ellsbury, Graham, General Report on Tunny With Emphasis on Statistical Methods, retrieved 3 November 2010
• Good, Jack (1993), Enigma and Fish in Hinsley & Stripp 1993, pp. 149–166
• Hinsley, F. H.; Stripp, Alan, eds. (1993) [1992], Codebreakers: The inside story of Bletchley Park, Oxford: Oxford University Press, ISBN 978-0-19-280132-6
• O'Connor, J. J.; Robertson, E. F. (2003), MacTutor Biography: William Thomas Tutte, University of St Andrews, retrieved 28 April 2013
• Tutte, W. T. (19 June 1998), Fish and I (PDF), retrieved 7 April 2012 Transcript of a lecture given by Prof. Tutte at the University of Waterloo
• Tutte, William T. (2006), My Work at Bletchley Park Appendix 4 in Copeland 2006, pp. 352–369
• Ward, Mark (27 May 2011), "Code-cracking machine returned to life", BBC News, retrieved 28 April 2013
• Younger, D. H. (2012), "Biographical Memoirs of Fellows of the Royal Society: William Thomas Tutte. 14 May 1917 – 2 May 2002", Biographical Memoirs of Fellows of the Royal Society, The Royal Society, 58: 283–297, doi:10.1098/rsbm.2012.0036, retrieved 28 April 2013
External links
• Professor William T. Tutte
• W. T. Tutte at the Mathematics Genealogy Project
• William Tutte, 84, Mathematician and Code-breaker, Dies – Obituary from The New York Times
• William Tutte: Unsung mathematical mastermind – Obituary from The Guardian
• CRM-Fields-PIMS Prize – 2001 – William T. Tutte
• "60 Years in the Nets" – a lecture (audio recording) given at the Fields Institute on 25 October 2001 to mark the receipt of the 2001 CRM-Fields Prize
• Tutte's disproof of Tait's conjecture
• "Bletchley's forgotten heroes", Ian Douglas, The Daily Telegraph, 25 December 2012
• Murty, U. S. R. (2004), "Dedication: Professor W.T. Tutte", Journal of Combinatorial Theory, Series B, 92 (2): 191–192, doi:10.1016/j.jctb.2004.08.002.
• Younger, D. H. (2004), "Dedication: Professor W.T. Tutte", Journal of Combinatorial Theory, Series B, 92 (2): 193–198, doi:10.1016/j.jctb.2004.09.002.
• The Tutte Institute for Research in Mathematics and Computer Science
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| Wikipedia |
W. V. D. Hodge
Sir William Vallance Douglas Hodge FRS FRSE[2] (/hɒdʒ/; 17 June 1903 – 7 July 1975) was a British mathematician, specifically a geometer.[3][4]
Sir
W. V. D. Hodge
FRS FRSE
Born(1903-06-17)17 June 1903
Edinburgh, UK
Died7 July 1975(1975-07-07) (aged 72)
Cambridge, UK
NationalityBritish
EducationGeorge Watson's College
Alma materUniversity of Edinburgh
St John's College, Cambridge[1]
Known forHodge conjecture
Hodge dual
Hodge bundle
Hodge theory
AwardsAdams Prize (1936)
Senior Berwick Prize (1952)
Royal Medal (1957)
De Morgan Medal (1959)
Copley Medal (1974)
Scientific career
FieldsMathematics
InstitutionsPembroke College, Cambridge
Academic advisorsE. T. Whittaker
Doctoral studentsMichael Atiyah
Ian R. Porteous
David J. Simms
His discovery of far-reaching topological relations between algebraic geometry and differential geometry—an area now called Hodge theory and pertaining more generally to Kähler manifolds—has been a major influence on subsequent work in geometry.
Life and career
Hodge was born in Edinburgh in 1903, the younger son and second of three children of Archibald James Hodge (1869-1938), a searcher of records in the property market and a partner in the firm of Douglas and Company, and his wife, Jane (born 1875), daughter of confectionery business owner William Vallance.[5][6][7] They lived at 1 Church Hill Place in the Morningside district.[8]
He attended George Watson's College, and studied at Edinburgh University, graduating MA in 1923. With help from E. T. Whittaker, whose son J. M. Whittaker was a college friend, he then took the Cambridge Mathematical Tripos. At Cambridge he fell under the influence of the geometer H. F. Baker. He gained a Cambridge BA degree in 1925, receiving the MA in 1930 and the Doctor of Science (ScD) degree in 1950.[9]
In 1926 he took up a teaching position at the University of Bristol, and began work on the interface between the Italian school of algebraic geometry, particularly problems posed by Francesco Severi, and the topological methods of Solomon Lefschetz. This made his reputation, but led to some initial scepticism on the part of Lefschetz. According to Atiyah's memoir, Lefschetz and Hodge in 1931 had a meeting in Max Newman's rooms in Cambridge, to try to resolve issues. In the end Lefschetz was convinced.[2] In 1928 he was elected a Fellow of the Royal Society of Edinburgh. His proposers were Sir Edmund Taylor Whittaker, Ralph Allan Sampson, Charles Glover Barkla, and Sir Charles Galton Darwin. He was awarded the Society's Gunning Victoria Jubilee Prize for the period 1964 to 1968.[10]
In 1930 Hodge was awarded a Research Fellowship at St. John's College, Cambridge. He spent the year 1931–2 at Princeton University, where Lefschetz was, visiting also Oscar Zariski at Johns Hopkins University. At this time he was also assimilating de Rham's theorem, and defining the Hodge star operation. It would allow him to define harmonic forms and so refine the de Rham theory.
On his return to Cambridge, he was offered a University Lecturer position in 1933. He became the Lowndean Professor of Astronomy and Geometry at Cambridge, a position he held from 1936 to 1970. He was the first head of DPMMS.
He was the Master of Pembroke College, Cambridge from 1958 to 1970, and vice-president of the Royal Society from 1959 to 1965. He was knighted in 1959. Amongst other honours, he received the Adams Prize in 1937 and the Copley Medal of the Royal Society in 1974.
He died in Cambridge on 7 July 1975.
Work
The Hodge index theorem was a result on the intersection number theory for curves on an algebraic surface: it determines the signature of the corresponding quadratic form. This result was sought by the Italian school of algebraic geometry, but was proved by the topological methods of Lefschetz.
The Theory and Applications of Harmonic Integrals[11] summed up Hodge's development during the 1930s of his general theory. This starts with the existence for any Kähler metric of a theory of Laplacians – it applies to an algebraic variety V (assumed complex, projective and non-singular) because projective space itself carries such a metric. In de Rham cohomology terms, a cohomology class of degree k is represented by a k-form α on V(C). There is no unique representative; but by introducing the idea of harmonic form (Hodge still called them 'integrals'), which are solutions of Laplace's equation, one can get unique α. This has the important, immediate consequence of splitting up
Hk(V(C), C)
into subspaces
Hp,q
according to the number p of holomorphic differentials dzi wedged to make up α (the cotangent space being spanned by the dzi and their complex conjugates). The dimensions of the subspaces are the Hodge numbers.
This Hodge decomposition has become a fundamental tool. Not only do the dimensions hp,q refine the Betti numbers, by breaking them into parts with identifiable geometric meaning; but the decomposition itself, as a varying 'flag' in a complex vector space, has a meaning in relation with moduli problems. In broad terms, Hodge theory contributes both to the discrete and the continuous classification of algebraic varieties.
Further developments by others led in particular to an idea of mixed Hodge structure on singular varieties, and to deep analogies with étale cohomology.
Hodge conjecture
The Hodge conjecture on the 'middle' spaces Hp,p is still unsolved, in general. It is one of the seven Millennium Prize Problems set up by the Clay Mathematics Institute.
Exposition
Hodge also wrote, with Daniel Pedoe, a three-volume work Methods of Algebraic Geometry, on classical algebraic geometry, with much concrete content – illustrating though what Élie Cartan called 'the debauch of indices' in its component notation. According to Atiyah, this was intended to update and replace H. F. Baker's Principles of Geometry.
Family
In 1929 he married Kathleen Anne Cameron.[12]
Publications
• Hodge, W. V. D. (1941), The Theory and Applications of Harmonic Integrals, Cambridge University Press, ISBN 978-0-521-35881-1, MR 0003947
• Hodge, W. V. D.; Pedoe, D. (1994) [1947], Methods of Algebraic Geometry, Volume I (Book II), Cambridge University Press, ISBN 978-0-521-46900-5[13]
• Hodge, W. V. D.; Pedoe, Daniel (1994) [1952], Methods of Algebraic Geometry: Volume 2 Book III: General theory of algebraic varieties in projective space. Book IV: Quadrics and Grassmann varieties., Cambridge Mathematical Library, Cambridge University Press, ISBN 978-0-521-46901-2, MR 0048065[14]
• Hodge, W. V. D.; Pedoe, Daniel (1994) [1954], Methods of Algebraic Geometry: Volume 3, Cambridge University Press, ISBN 978-0-521-46775-9[15]
See also
• List of things named after W. V. D. Hodge
References
1. Hodge biography - University of St Andrews
2. Atiyah, M. F. (1976). "William Vallance Douglas Hodge. 17 June 1903 -- 7 July 1975". Biographical Memoirs of Fellows of the Royal Society. 22: 169–192. doi:10.1098/rsbm.1976.0007. S2CID 72054846.
3. O'Connor, John J.; Robertson, Edmund F., "W. V. D. Hodge", MacTutor History of Mathematics Archive, University of St Andrews
4. W. V. D. Hodge at the Mathematics Genealogy Project
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. Archived from the original (PDF) on 12 January 2016.
6. "William Hodge - Biography".
7. "Hodge, Sir William Vallance Douglas (1903–1975), mathematician". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/31241. ISBN 978-0-19-861412-8. (Subscription or UK public library membership required.)
8. Edinburgh and Leith Post Office Directory 1903-4
9. The Annual Register of the University of Cambridge for the year 1968-69
10. 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.
11. Struik, D. J. (1944). "Review: W. V. D. Hodge, The theory and applications of harmonic integrals". Bull. Amer. Math. Soc. 50 (1): 43–45. doi:10.1090/s0002-9904-1944-08054-3.
12. 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.
13. Coxeter, H. S. M. (1949). "Review: Methods of algebraic geometry. By W. V. D. Hodge and D. Pedoe" (PDF). Bull. Amer. Math. Soc. 55 (3, Part 1): 315–316. doi:10.1090/s0002-9904-1949-09193-0.
14. Coxeter, H. S. M. (1952). "Review: Methods of algebraic geometry. Vol. 2. By W. V. D. Hodge and D. Pedoe" (PDF). Bull. Amer. Math. Soc. 58 (6): 678–679. doi:10.1090/s0002-9904-1952-09661-0.
15. Samuel, P. (1955). "Review: Methods of algebraic geometry. Vol. III. Birational geometry. By W. V. D. Hodge and D. Pedoe" (PDF). Bull. Amer. Math. Soc. 61 (3, Part 1): 254–257. doi:10.1090/s0002-9904-1955-09910-5.
De Morgan Medallists
• Arthur Cayley (1884)
• James Joseph Sylvester (1887)
• Lord Rayleigh (1890)
• Felix Klein (1893)
• S. Roberts (1896)
• William Burnside (1899)
• A. G. Greenhill (1902)
• H. F. Baker (1905)
• J. W. L. Glaisher (1908)
• Horace Lamb (1911)
• J. Larmor (1914)
• W. H. Young (1917)
• E. W. Hobson (1920)
• P. A. MacMahon (1923)
• A. E. H. Love (1926)
• Godfrey Harold Hardy (1929)
• Bertrand Russell (1932)
• E. T. Whittaker (1935)
• J. E. Littlewood (1938)
• Louis Mordell (1941)
• Sydney Chapman (1944)
• George Neville Watson (1947)
• A. S. Besicovitch (1950)
• E. C. Titchmarsh (1953)
• G. I. Taylor (1956)
• W. V. D. Hodge (1959)
• Max Newman (1962)
• Philip Hall (1965)
• Mary Cartwright (1968)
• Kurt Mahler (1971)
• Graham Higman (1974)
• C. Ambrose Rogers (1977)
• Michael Atiyah (1980)
• K. F. Roth (1983)
• J. W. S. Cassels (1986)
• D. G. Kendall (1989)
• Albrecht Fröhlich (1992)
• W. K. Hayman (1995)
• R. A. Rankin (1998)
• J. A. Green (2001)
• Roger Penrose (2004)
• Bryan John Birch (2007)
• Keith William Morton (2010)
• John Griggs Thompson (2013)
• Timothy Gowers (2016)
• Andrew Wiles (2019)
Copley Medallists (1951–2000)
• David Keilin (1951)
• Paul Dirac (1952)
• Albert Kluyver (1953)
• E. T. Whittaker (1954)
• Ronald Fisher (1955)
• Patrick Blackett (1956)
• Howard Florey (1957)
• John Edensor Littlewood (1958)
• Macfarlane Burnet (1959)
• Harold Jeffreys (1960)
• Hans Krebs (1961)
• Cyril Norman Hinshelwood (1962)
• Paul Fildes (1963)
• Sydney Chapman (1964)
• Alan Hodgkin (1965)
• Lawrence Bragg (1966)
• Bernard Katz (1967)
• Tadeusz Reichstein (1968)
• Peter Medawar (1969)
• Alexander R. Todd (1970)
• Norman Pirie (1971)
• Nevill Francis Mott (1972)
• Andrew Huxley (1973)
• W. V. D. Hodge (1974)
• Francis Crick (1975)
• Dorothy Hodgkin (1976)
• Frederick Sanger (1977)
• Robert Burns Woodward (1978)
• Max Perutz (1979)
• Derek Barton (1980)
• Peter D. Mitchell (1981)
• John Cornforth (1982)
• Rodney Robert Porter (1983)
• Subrahmanyan Chandrasekhar (1984)
• Aaron Klug (1985)
• Rudolf Peierls (1986)
• Robin Hill (1987)
• Michael Atiyah (1988)
• César Milstein (1989)
• Abdus Salam (1990)
• Sydney Brenner (1991)
• George Porter (1992)
• James D. Watson (1993)
• Frederick Charles Frank (1994)
• Frank Fenner (1995)
• Alan Cottrell (1996)
• Hugh Huxley (1997)
• James Lighthill (1998)
• John Maynard Smith (1999)
• Alan Battersby (2000)
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| Wikipedia |
W. W. Rouse Ball
Walter William Rouse Ball[lower-alpha 1] (14 August 1850 – 4 April 1925), known as W. W. Rouse Ball, was a British mathematician, lawyer, and fellow at Trinity College, Cambridge, from 1878 to 1905. He was also a keen amateur magician, and the founding president of the Cambridge Pentacle Club in 1919, one of the world's oldest magic societies.[1][2]
W. W. Rouse Ball
Born
Walter William Rouse Ball
(1850-08-14)14 August 1850
Hampstead, London, England
Died4 April 1925(1925-04-04) (aged 74)
Cambridge, England
Alma materTrinity College, Cambridge
Known for
• Tessellations
• magic squares
• history of mathematics
AwardsSmith's Prize (1874)
Scientific career
FieldsMathematics
InstitutionsTrinity College, Cambridge
Doctoral studentsErnest Barnes
Life
Born 14 August 1850 in Hampstead, London, Ball was the son and heir of Walter Frederick Ball, of 3, St John's Park Villas, South Hampstead, London. Educated at University College School, he entered Trinity College, Cambridge, in 1870, became a scholar and first Smith's Prizeman, and gained his BA in 1874 as second Wrangler. He became a Fellow of Trinity in 1875, and remained one for the rest of his life.[3]
He died on 4 April 1925 in Elmside, Cambridge, and is buried at the Parish of the Ascension Burial Ground in Cambridge.[4]
He is commemorated in the naming of the small pavilion, now used as changing rooms and toilets, on Jesus Green in Cambridge.
Books
• A History of the Study of Mathematics at Cambridge; Cambridge University Press, 1889 (reissued by the publisher, 2009, ISBN 978-1-108-00207-3)
• A Short Account of the History of Mathematics at Project Gutenberg (1st ed. 1888 and later editions). Dover 1960 republication of fourth edition: .
• Mathematical Recreations and Essays at Project Gutenberg (1st ed. 1892;[5] later editions with H.S.M. Coxeter)[6]
• A History of the First Trinity Boat Club (1908)
• Cambridge Papers at Project Gutenberg (1st ed. 1918). Macmillan and Co., Limited 1918: .
• String Figures; Cambridge, W. Heffer & Sons (1st ed. 1920, 2nd ed. 1921, 3rd ed. 1929, reprinted with supplements as Fun with String Figures by Dover Publications, 1971, ISBN 0-486-22809-6)
See also
• Martin Gardner – another author of recreational mathematics
• Rouse Ball Professor of English Law
• Rouse Ball Professor of Mathematics
Notes
1. Pronounced /ˈwɔːl.tər ˈwɪl.jəm raʊs bɔːl/.
References
1. "Pentacle Club – Magicpedia".
2. Whittaker, E. T. (October 1925). "Obituary: W. W. Rouse Ball". The Mathematical Gazette. 12 (178): 449–454. doi:10.1017/S0025557200247207. JSTOR 3604492.
3. "Ball, Walter William Rouse (BL870WW)". A Cambridge Alumni Database. University of Cambridge.
4. Singmaster 2005, p. 658
5. Oliver, J. E. (1892). "Review: Mathematical Recreations and Essays by W. W. Rouse Ball" (PDF). Bull. Amer. Math. Soc. 2 (3): 37–46. doi:10.1090/S0002-9904-1892-00105-X.
6. Frame, J. S. (1940). "Review: Mathematical Recreations and Essays, 11th edition, by W. W. Rouse Ball; revised by H. S. M. Coxeter" (PDF). Bull. Amer. Math. Soc. 45 (3): 211–213. doi:10.1090/S0002-9904-1940-07170-8.
• Singmaster, David (2005), "1892 Walter William Rouse Ball, Mathematical recreations and problems of past and present times", in Grattan-Guinness, I. (ed.), Landmark Writings in Western Mathematics 1640–1940, Elsevier, pp. 653–663, ISBN 978-0-444-50871-3
External links
Wikisource has original text related to this article:
W. W. Rouse Ball
Wikiquote has quotations related to W. W. Rouse Ball.
Wikiquote has quotations related to A Short Account of the History of Mathematics.
• Works by W. W. Rouse Ball at Project Gutenberg
• Works by or about W. W. Rouse Ball at Internet Archive
• Works by W. W. Rouse Ball at LibriVox (public domain audiobooks)
• O'Connor, John J.; Robertson, Edmund F., "W. W. Rouse Ball", MacTutor History of Mathematics Archive, University of St Andrews
• W. W. Rouse Ball at the Mathematics Genealogy Project
• W. W. Rouse Ball at Find a Grave
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| Wikipedia |
William Schieffelin Claytor
William Schieffelin Claytor (January 4, 1908 – July 14, 1967) was an American mathematician specializing in topology. He was born in Norfolk, Virginia, where his father was a dentist. He was the third African-American to get a Ph.D. in mathematics, and the first to publish in a mathematical research journal.[1]
Education
Claytor attended public schools in Washington, DC and also the Hampton Agricultural and Industrial School in Virginia. In 1928 he received his BA from Howard University, where he had been taught by Elbert Cox, the first African-American to get a Ph.D. in mathematics. Dudley Woodard, the second African-American to get a PhD in mathematics, was just setting up the graduate program in math at Howard, and Claytor earned his MA there in 1929, with a thesis supervised by Woodard.[2]
Claytor obtained his Ph.D. from the University of Pennsylvania in 1933 with the dissertation Topological Immersion of Peanian Continua in a Spherical Surface, directed by John R. Kline,[3] who had also supervised Woodard's thesis and was himself a student of R. L. Moore (of the Moore method).[4] Kline wrote to Moore saying: "Claytor wrote a very fine thesis. In many ways I think that it is perhaps the best that I have ever had done under my direction."[1]
In 1934, a paper based on Claytor's thesis appeared in Annals of Mathematics, credited to Schieffelin Claytor, making him the first African-American to publish in a mathematical research journal. In 1937, also in the Annals, he published the paper "Peanian Continua not Imbeddable in a Spherical Surface", also credited to Schieffelin Claytor.
Academic career
Claytor had taught at HBCU West Virginia State College for three years following his doctorate, not being able to secure a job at a majority institution due to the prevalent racism of the era.[5] At West Virginia his students included Katherine Johnson who later worked on the space program for NASA.[1]
Claytor applied for a National Research Council Fellowship to work at the Institute for Advanced Study (IAS), which at the time was housed in Princeton University, but was rejected on racial grounds.[5] In 1937 he received a Rosenwald Fellowship at the University of Michigan;[6] he stayed there for several years, but was not allowed to attend research seminars.[3] Oswald Veblen had finally been able to offer him a position at the IAS in 1939, independently of Princeton University, but Claytor turned it down.[1]
During the years 1941–1945, Claytor served in the US Army, teaching in the Anti-Aircraft Artillery Schools in Virginia and Georgia.[2] In 1947 he joined the faculty at Howard, where David Blackwell was then chair of the department of mathematics.[7] Claytor taught at Howard until his retirement in 1965, serving as chair himself along the way.[6]
On August 5, 1947, Claytor married the psychologist Mae Belle Pullins, who also shared his love of mathematics. They had one daughter. He spent the rest of his career at Howard, and despite making many well-received presentations at AMS conferences, he continued to suffer from racial discrimination and was not even allowed to stay in the hotels where the meetings were held.[8]
Awards
The National Association of Mathematicians (NAM) has a lecture series named after Claytor and Woodard.[9]
The American Mathematical Society (AMS) has a mid-career research fellowship, the Claytor-Gilmer Fellowship, named after Claytor and Gloria Ford Gilmer.[10]
References
1. William Waldron Schieffelin Claytor at the MacTutor History of Mathematics archive
2. William W. Schieffelin Claytor at the Mathematical Association of America (MAA)
3. Pioneer African American Mathematicians at the University of Pennsylvania Archives and Records Center
4. William Schieffelin Claytor at the Mathematics Genealogy Project
5. Mathematics and the Politics of Race: The Case of William Claytor by Karen Hunger Parshall, The American Mathematical Monthly, Vol. 123, No. 3 (March 2016), pp 214-240
6. Mathematicians of the African Diaspora at the State University of New York at Buffalo
7. Grime, David (July 17, 2007). "David Blackwell, Scholar of Probability, Dies at 91". New York Times. Retrieved August 22, 2010.
8. Unsung: William Claytor by Sabrina Nichelle Collins, Nov 2, 2016
9. Claytor-Woodard Lecture at the National Association of Mathematicians
10. "The AMS Claytor-Gilmer Fellowship". American Mathematical Society. Retrieved 2021-05-24.
Papers
• Peanian continua not embeddable in a spherical surface Ann. of Math. Second Series, Vol. 38, No. 3 (Jul. 1937), pp. 631–646
• Topological immersion of Peanian continua in a spherical surface, Ann. of Math. Second Series, Vol. 35, No. 4 (Oct. 1934), pp. 809–835
External links
• William Waldron Schieffelin Claytor at the Mathematics Genealogy Project
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| Wikipedia |
WENO methods
In numerical solution of differential equations, WENO (weighted essentially non-oscillatory) methods are classes of high-resolution schemes. WENO are used in the numerical solution of hyperbolic partial differential equations. These methods were developed from ENO methods (essentially non-oscillatory). The first WENO scheme was developed by Liu, Osher and Chan in 1994.[1] In 1996, Guang-Sh and Chi-Wang Shu developed a new WENO scheme[2] called WENO-JS.[3] Nowadays, there are many WENO methods.[4]
See also
• High-resolution scheme
• ENO methods
References
1. Liu, Xu-Dong; Osher, Stanley; Chan, Tony (1994). "Weighted Essentially Non-oscillatory Schemes". Journal of Computational Physics. 115: 200–212. Bibcode:1994JCoPh.115..200L. CiteSeerX 10.1.1.24.8744. doi:10.1006/jcph.1994.1187.
2. Jiang, Guang-Shan; Shu, Chi-Wang (1996). "Efficient Implementation of Weighted ENO Schemes". Journal of Computational Physics. 126 (1): 202–228. Bibcode:1996JCoPh.126..202J. CiteSeerX 10.1.1.7.6297. doi:10.1006/jcph.1996.0130.
3. Ha, Youngsoo; Kim, Chang Ho; Lee, Yeon Ju; Yoon, Jungho (2012). "Mapped WENO schemes based on a new smoothness indicator for Hamilton–Jacobi equations". Journal of Mathematical Analysis and Applications. 394 (2): 670–682. doi:10.1016/j.jmaa.2012.04.040.
4. Ketcheson, David I.; Gottlieb, Sigal; MacDonald, Colin B. (2011). "Strong Stability Preserving Two-step Runge–Kutta Methods". SIAM Journal on Numerical Analysis. 49 (6): 2618–2639. arXiv:1106.3626. doi:10.1137/10080960X. S2CID 16602876.
Further reading
• Shu, Chi-Wang (1998). "Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws". Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. Lecture Notes in Mathematics. Vol. 1697. pp. 325–432. CiteSeerX 10.1.1.127.895. doi:10.1007/BFb0096355. ISBN 978-3-540-64977-9.
• Shu, Chi-Wang (2009). "High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems". SIAM Review. 51: 82–126. Bibcode:2009SIAMR..51...82S. doi:10.1137/070679065.
Numerical methods for partial differential equations
Finite difference
Parabolic
• Forward-time central-space (FTCS)
• Crank–Nicolson
Hyperbolic
• Lax–Friedrichs
• Lax–Wendroff
• MacCormack
• Upwind
• Method of characteristics
Others
• Alternating direction-implicit (ADI)
• Finite-difference time-domain (FDTD)
Finite volume
• Godunov
• High-resolution
• Monotonic upstream-centered (MUSCL)
• Advection upstream-splitting (AUSM)
• Riemann solver
• Essentially non-oscillatory (ENO)
• Weighted essentially non-oscillatory (WENO)
Finite element
• hp-FEM
• Extended (XFEM)
• Discontinuous Galerkin (DG)
• Spectral element (SEM)
• Mortar
• Gradient discretisation (GDM)
• Loubignac iteration
• Smoothed (S-FEM)
Meshless/Meshfree
• Smoothed-particle hydrodynamics (SPH)
• Peridynamics (PD)
• Moving particle semi-implicit method (MPS)
• Material point method (MPM)
• Particle-in-cell (PIC)
Domain decomposition
• Schur complement
• Fictitious domain
• Schwarz alternating
• additive
• abstract additive
• Neumann–Dirichlet
• Neumann–Neumann
• Poincaré–Steklov operator
• Balancing (BDD)
• Balancing by constraints (BDDC)
• Tearing and interconnect (FETI)
• FETI-DP
Others
• Spectral
• Pseudospectral (DVR)
• Method of lines
• Multigrid
• Collocation
• Level-set
• Boundary element
• Method of moments
• Immersed boundary
• Analytic element
• Isogeometric analysis
• Infinite difference method
• Infinite element method
• Galerkin method
• Petrov–Galerkin method
• Validated numerics
• Computer-assisted proof
• Integrable algorithm
• Method of fundamental solutions
| Wikipedia |
WWW Interactive Multipurpose Server
The WWW Interactive Multipurpose Server (WIMS) (sometimes referred to as WWW Interactive Mathematics Server) project is designed for supporting intensive mathematical exercises via the Internet or in a computer-equipped classroom with server-side interactivity, accessible at the address http://wims.unice.fr.
WWW Interactive Multipurpose Server (WIMS)
Developer(s)Xiao Gang
Initial release1997[1]
Stable release
WIMS-4.17e[2] / May 7, 2019
Written inC[2]
TypeServer software
LicenseGNU General Public Licence[1]
Websitewims.unice.fr
The system has the following main features:
• A modular design allowing applications and software interfaces to be created and maintained independently from each other.
• Features interfaces for software including MuPAD, PARI/GP, Gnuplot, POV-Ray, Co.
• Dynamic rendering of mathematical formulas and animated graphics.
• A structure of virtual classes, including mechanisms for automatic score gathering and processing.
[3]
The program is open source and freely available under the GNU General Public Licence, however each WIMS module has its own copyright policy, which may differ from that of the server program.[1]
It is often cited and linked for its sophisticated "online calculator" tools capable of generating animated GIFs of parametric 2D or 3D graphs or allowing prime tests with very large numbers.
Author
Xiao Gang was a professor at University of Nice Sophia Antipolis. He was interested in solar energy and algebraic geometry. He was also the active site manager of the WIMS of the university he worked for.[4] Xiao Gang died on June 27, 2014.[5]
Xiao Gang taught himself during the Up to the Mountains and Down to the Countryside Movement. He obtained his master's degree from the University of Science and Technology of China. Xiao obtained his Ph.D. degree from University of Paris-Sud in 1984. Xiao Gang returned to China and became a lecturer at East China Normal University. He was promoted to professor in 1986, and was awarded the Shiing-Shen Chern Prize in Mathematics in 1991. In 1992, Xiao became a professor at University of Nice Sophia Antipolis.[6]
External links
• English WIMS at wims.unice.fr - The WIMS of the University of Nice Sophia Antipolis in France, the most popular WIMS [7] for it was the first one and is hosted at the creator's university.
• WIMS at wims.ac-nice.fr - The WIMS of the Académie de Nice - One of the few featuring a redesigned skin.
• English WIMS help
• List of WIMS Servers/Mirrors at wims.unice.fr
• "WIMS A server for interactive mathematics on the internet" - Paper on WIMS by XIAO, Gang written in 1999
• WIMS download page
• Interface to the Direc exec WIMS module - This gives direct code access to many other WIMS modules (e.g. the POV-Ray renderer or a C compiler).
• A list of frequent citers of unice.fr WIMS modules (some Wikipedia articles)
• List of the online calculators available sorted by popularity.
• WIMS EDU association site.
References
1. WIMS Copyright Notice
2. WIMS download page
3. WIMS Paper
4. Homepage of Xiao Gang at wims.unice.fr
5. Xiao Gang, le créateur de Wims nous a quittés (French)
6. "Profile of Xiao Gang (Chinese)". Archived from the original on 2021-04-14. Retrieved 2014-07-03.
7. UNICE WIMS first Google Result
| Wikipedia |
Wacław Sierpiński
Wacław Franciszek Sierpiński (Polish: [ˈvat͡swaf fraɲˈt͡ɕiʂɛk ɕɛrˈpij̃skʲi] (listen); 14 March 1882 – 21 October 1969) was a Polish mathematician who was also known as Fraktal Babo.[1] He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions, and topology. He published over 700 papers and 50 books.
Wacław Sierpiński
Born
Wacław Franciszek Sierpiński
(1882-03-14)14 March 1882
Warsaw, Congress Poland, Russian Empire
Died21 October 1969(1969-10-21) (aged 87)
Warsaw, Polish People's Republic
NationalityPolish
Alma materUniversity of Warsaw
Known forSierpinski triangle
Sierpinski carpet
Sierpinski curve
Sierpinski number
Sierpiński cube
Sierpiński's constant
Sierpiński set
Sierpiński game
Sierpiński space
Scientific career
FieldsMathematics
Doctoral advisorStanisław Zaremba
Georgy Voronoy
Doctoral studentsJerzy Browkin
Edward Marczewski
Stefan Mazurkiewicz
Jerzy Neyman
Stanisław Ruziewicz
Andrzej Schinzel
Three well-known fractals are named after him (the Sierpiński triangle, the Sierpiński carpet, and the Sierpiński curve), as are Sierpiński numbers and the associated Sierpiński problem.
Educational background
Polish Cipher Bureau
Biuro Szyfrów
Methods and technology
• "ANX"
• Enigma "double"
• Grill
• Clock
• Cyclometer
• Card catalog
• Cryptologic bomb
• Zygalski sheets
• Lacida
Locations
• Saxon Palace
• Kabaty Woods
• PC Bruno
• Cadix
Personnel
Chief
Gwido Langer
Deputy Chief
Chief of Radio Intelligence
Chief of German Section
Maksymilian Ciężki
German Section cryptologists
• Marian Rejewski
• Jerzy Różycki
• Henryk Zygalski
• Antoni Palluth
Wiktor Michałowski
Chief of Russian Section
Jan Graliński
Russian Section cryptologist
Piotr Smoleński
Others
Jan Kowalewski
Stanisław Leśniewski
Stefan Mazurkiewicz
Franciszek Pokorny
Wacław Sierpiński
Sierpiński enrolled in the Department of Mathematics and Physics at the University of Warsaw in 1899 and graduated five years later.[2] In 1903, while still at the University of Warsaw, the Department of Mathematics and Physics offered a prize for the best essay from a student on Voronoy's contribution to number theory. Sierpiński was awarded a gold medal for his essay, thus laying the foundation for his first major mathematical contribution. Unwilling for his work to be published in Russian, he withheld it until 1907, when it was published in Samuel Dickstein's mathematical magazine 'Prace Matematyczno-Fizyczne' (Polish: 'The Works of Mathematics and Physics').
After his graduation in 1904, Sierpiński worked as a school teacher of mathematics and physics in Warsaw. However, when the school closed because of a strike, Sierpiński decided to go to Kraków to pursue a doctorate. At the Jagiellonian University in Kraków, he attended lectures by Stanisław Zaremba on mathematics. He also studied astronomy and philosophy. In 1908, he received his doctorate and was appointed to the University of Lwów. In 1910, he became head of the Faculty of Mathematics at the university.[3]
Career
In 1907 Sierpiński first became interested in set theory when he came across a theorem which stated that points in the plane could be specified with a single coordinate. He wrote to Tadeusz Banachiewicz (then at Göttingen), asking how such a result was possible. He received the one-word reply 'Cantor'. Sierpiński began to study set theory and, in 1909, he gave the first ever lecture course devoted entirely to the subject.[4]
Sierpiński maintained an output of research papers and books. During the years 1908 to 1914, when he taught at the University of Lwów, he published three books in addition to many research papers. These books were The Theory of Irrational Numbers (1910), Outline of Set Theory (1912), and The Theory of Numbers (1912).
When World War I began in 1914, Sierpiński and his family were in Russia. To avoid the persecution that was common for Polish foreigners, Sierpiński spent the rest of the war years in Moscow working with Nikolai Luzin. Together they began the study of analytic sets. In 1916, Sierpiński gave the first example of an absolutely normal number.[5]
When World War I ended in 1918, Sierpiński returned to Lwów. However shortly after taking up his appointment again in Lwów he was offered a post at the University of Warsaw, which he accepted. In 1919 he was promoted to a professor. He spent the rest of his life in Warsaw.[6]
During the Polish–Soviet War (1919–1921), Sierpiński helped break Soviet Russian ciphers for the Polish General Staff's cryptologic agency.
In 1920, Sierpiński, together with Zygmunt Janiszewski and his former student Stefan Mazurkiewicz, founded the mathematical journal Fundamenta Mathematicae.[1] Sierpiński edited the journal, which specialized in papers on set theory.
During this period, Sierpiński worked predominantly on set theory, but also on point set topology and functions of a real variable. In set theory he made contributions on the axiom of choice and on the continuum hypothesis. He proved that Zermelo–Fraenkel set theory together with the Generalized continuum hypothesis imply the Axiom of choice. He also worked on what is now known as the Sierpiński curve. Sierpiński continued to collaborate with Luzin on investigations of analytic and projective sets. His work on functions of a real variable includes results on functional series, differentiability of functions and Baire's classification.
Sierpiński retired in 1960 as professor at the University of Warsaw, but continued until 1967 to give a seminar on the Theory of Numbers at the Polish Academy of Sciences. He also continued editorial work as editor-in-chief of Acta Arithmetica, and as a member of the editorial board of Rendiconti del Circolo Matematico di Palermo, Composito Matematica, and Zentralblatt für Mathematik.
In 1964 he was one of the signatories of the so-called Letter of 34 to Prime Minister Józef Cyrankiewicz regarding freedom of culture.[7]
Sierpiński is interred at the Powązki Cemetery in Warsaw, Poland.[8]
Honors received
Honorary Degrees: Lwów (1929), St. Marks of Lima (1930), Tarta (1931), Amsterdam (1932), Sofia (1939), Prague (1947), Wrocław (1947), Lucknow (1949), and Moscow (1967).
For high involvement with the development of mathematics in Poland, Sierpiński was honored with election to the Polish Academy of Learning in 1921 and that same year was made dean of the faculty at the University of Warsaw. In 1928, he became vice-chairman of the Warsaw Scientific Society, and that same year was elected chairman of the Polish Mathematical Society.
He was elected to the Geographic Society of Lima (1931), the Royal Scientific Society of Liège (1934), the Bulgarian Academy of Sciences (1936), the National Academy of Lima (1939), the Royal Society of Sciences of Naples (1939), the Accademia dei Lincei of Rome (1947), the Germany Academy of Sciences (1950),[1] the American Academy of Arts and Sciences (1959), the Paris Academy (1960), the Royal Dutch Academy (1961),[9] the Academy of Science of Brussels (1961), the London Mathematical Society (1964), the Romanian Academy (1965) and the Papal Academy of Sciences (1967).
In 1949 Sierpiński was awarded Poland's Scientific Prize, first degree.
In 2014, a sculpture in the form of a tree inspired by a fractal created by Sierpiński was unveiled at the Wallenberg Square in Stockholm as part of an exhibiton organized by the Polish Ministry of Foreign Affairs on the 10th anniversary of Poland joining the European Union and 15th anniversary of Poland joining NATO.[10]
Publications
Sierpiński authored 724 papers and 50 books, almost all in Polish. His book Cardinal and Ordinal Numbers was originally published in English in 1958. Two books, Introduction to General Topology (1934) and General Topology (1952) were translated into English by Canadian mathematician Cecilia Krieger. Another book, Pythagorean Triangles (1954), was translated into English by Indian mathematician Ambikeshwar Sharma, published in 1962, and republished by Dover Books in 2003; it also has a Russian translation.[11] Another work of his published in English is the Elementary Theory of Numbers (translated by A. Hulanicki in 1964), based on his Polish Teoria Liczb (1914 and 1959).[12] Another book, named "250 Problems in Elementary Number Theory" was translated into English (1970) and Russian (1968).
See also
• Arity theorem
• List of Polish matematicians
• Menger sponge
• Seventeen or Bust
• The Sierpiński Moon crater
• Timeline of Polish science and technology
References
1. Kuratowski, Kazimierz (1972). "Wacław Sierpiński (1882-1969)". Acta Arithmetica. 21 (1): 1–5. doi:10.4064/aa-21-1-1-5. Retrieved 2022-08-28.
2. "Wielcy Polacy – Wacław Sierpiński (1882 – 1969) – genialny matematyk, Trójkąt, Wolna Wola, Gra w Chaos i Cud nad Wisłą 1920". bialczynski.pl (in Polish). 20 August 2015. Retrieved 27 April 2023.
3. Paulina Rowińska (8 March 2019). "Zaczynali od zera, stali się legendą. Jak warszawscy matematycy podbili świat". wyborcza.pl (in Polish). Retrieved 27 April 2023.
4. Paulina Rowińska (8 March 2019). "Zaczynali od zera, stali się legendą. Jak warszawscy matematycy podbili świat". wyborcza.pl (in Polish). Retrieved 27 April 2023.
5. "Wacław Sierpiński". mathshistory.st-andrews.ac.uk. Retrieved 27 April 2023.
6. "Wacław Sierpiński. Badacz zagadek nieskończoności". polskieradio.pl (in Polish). Retrieved 27 April 2023.
7. "List 34 - pierwszy duży protest wobec polityki kulturalnej władz PRL". dzieje.pl (in Polish). 10 March 2019. Retrieved 27 April 2023.
8. "Warszawskie Zabytkowe Pomniki Nagrobne" (in Polish). Retrieved 27 April 2023.
9. "W. Sierpinski (1882 - 1969)". Royal Netherlands Academy of Arts and Sciences. Retrieved 17 July 2015.
10. "W Sztokholmie stanęło matematyczne drzewo Sierpińskiego". dzieje.pl (in Polish). 23 September 2014. Retrieved 27 April 2023.
11. Hopkins, Brian (January 2019), "review of Pythagorean Triangles", The College Mathematics Journal, 50 (1): 68–72, doi:10.1080/07468342.2019.1547955, S2CID 127720835
12. W. Sierpinski (1 February 1988). Elementary Theory of Numbers: Second English Edition (edited by A. Schinzel). Elsevier. p. iv, 5–6. ISBN 978-0-08-096019-7.
External links
• O'Connor, John J.; Robertson, Edmund F., "Wacław Sierpiński", MacTutor History of Mathematics Archive, University of St Andrews
• Wacław Sierpiński at the Mathematics Genealogy Project
• K. Kuratowski (1972). "Wacław Sierpiński (1882-1969)" (PDF). Acta Arithmetica. 21: 1–5. doi:10.4064/aa-21-1-1-5.
• A. Schinzel (1972). "Wacław Sierpinski's papers on the theory of numbers" (PDF). Acta Arithmetica. 21: 7–13. doi:10.4064/aa-21-1-7-13.
• "Publications of Wacław Sierpiński in the theory of numbers" (PDF). Acta Arithmetica. 21: 15–23.
• Several of Sierpiński's books, Biblioteka Wirtualna Nauki.
• Sierpiński: Fractals, Code Breaking, and a Crater on the Moon
Fractals
Characteristics
• Fractal dimensions
• Assouad
• Box-counting
• Higuchi
• Correlation
• Hausdorff
• Packing
• Topological
• Recursion
• Self-similarity
Iterated function
system
• Barnsley fern
• Cantor set
• Koch snowflake
• Menger sponge
• Sierpinski carpet
• Sierpinski triangle
• Apollonian gasket
• Fibonacci word
• Space-filling curve
• Blancmange curve
• De Rham curve
• Minkowski
• Dragon curve
• Hilbert curve
• Koch curve
• Lévy C curve
• Moore curve
• Peano curve
• Sierpiński curve
• Z-order curve
• String
• T-square
• n-flake
• Vicsek fractal
• Hexaflake
• Gosper curve
• Pythagoras tree
• Weierstrass function
Strange attractor
• Multifractal system
L-system
• Fractal canopy
• Space-filling curve
• H tree
Escape-time
fractals
• Burning Ship fractal
• Julia set
• Filled
• Newton fractal
• Douady rabbit
• Lyapunov fractal
• Mandelbrot set
• Misiurewicz point
• Multibrot set
• Newton fractal
• Tricorn
• Mandelbox
• Mandelbulb
Rendering techniques
• Buddhabrot
• Orbit trap
• Pickover stalk
Random fractals
• Brownian motion
• Brownian tree
• Brownian motor
• Fractal landscape
• Lévy flight
• Percolation theory
• Self-avoiding walk
People
• Michael Barnsley
• Georg Cantor
• Bill Gosper
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• Helge von Koch
• Paul Lévy
• Aleksandr Lyapunov
• Benoit Mandelbrot
• Hamid Naderi Yeganeh
• Lewis Fry Richardson
• Wacław Sierpiński
Other
• "How Long Is the Coast of Britain?"
• Coastline paradox
• Fractal art
• List of fractals by Hausdorff dimension
• The Fractal Geometry of Nature (1982 book)
• The Beauty of Fractals (1986 book)
• Chaos: Making a New Science (1987 book)
• Kaleidoscope
• Chaos theory
Authority control
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| Wikipedia |
Wade Edward Philpott
Wade Edward Philpott (1918–1985)[1] was an American mathematician and puzzle maker.[2] Several of his puzzles have gone on to become best sellers.
Life
Philpott was born in Sunnyside, Washington in 1918 as Chester Wade Edwards. He was the son of Chester E. Edwards and Mary Ream Edwards. Later, he was adopted by Viola Ream Philpott, and George Austin Philpott, his aunt and uncle. In 1921, his name was changed from Chester Wade Edwards to Wade Edward Philpott. Philpott graduated from Ohio Northern University with a degree in engineering. He married Myra Given in 1941. In 1947, a shooting accident left him paralyzed.[3] During his long hospitalization, he developed an interest in recreational mathematics and puzzles.[2]
Philpott produced several puzzles during his lifetime. Two of his best known puzzles are Multimatch and Sweep. His games and puzzles were sold by Kadon Enterprises, Inc.[3] Philpott also published several of his works in the Journal of Recreational Mathematics.[2]
References
1. "Wade Edward Philpott's Record". Ancestry.
2. "Mathematics Archives - Wade Edward Philpot" (PDF). dspace.ucalgary.ca. University of Calgary.
3. "Puzzle Explorer Wade Philpott". gamepuzzles.com.
External links
• Game Puzzles on Philpott
| Wikipedia |
Wadge hierarchy
In descriptive set theory, within mathematics, Wadge degrees are levels of complexity for sets of reals. Sets are compared by continuous reductions. The Wadge hierarchy is the structure of Wadge degrees. These concepts are named after William W. Wadge.
Wadge degrees
Suppose $A$ and $B$ are subsets of Baire space ωω. Then $A$ is Wadge reducible to $B$ or $A$ ≤W $B$ if there is a continuous function $f$ on ωω with $A=f^{-1}[B]$. The Wadge order is the preorder or quasiorder on the subsets of Baire space. Equivalence classes of sets under this preorder are called Wadge degrees, the degree of a set $A$ is denoted by [$A$]W. The set of Wadge degrees ordered by the Wadge order is called the Wadge hierarchy.
Properties of Wadge degrees include their consistency with measures of complexity stated in terms of definability. For example, if $A$ ≤W $B$ and $B$ is a countable intersection of open sets, then so is $A$. The same works for all levels of the Borel hierarchy and the difference hierarchy. The Wadge hierarchy plays an important role in models of the axiom of determinacy. Further interest in Wadge degrees comes from computer science, where some papers have suggested Wadge degrees are relevant to algorithmic complexity.
Wadge's lemma states that under the axiom of determinacy (AD), for any two subsets $A,B$ of Baire space, $A$ ≤W $B$ or $B$ ≤W ωω\$A$.[1] The assertion that the Wadge lemma holds for sets in Γ is the semilinear ordering principle for Γ or SLO(Γ). Any semilinear order defines a linear order on the equivalence classes modulo complements. Wadge's lemma can be applied locally to any pointclass Γ, for example the Borel sets, Δ1n sets, Σ1n sets, or Π1n sets. It follows from determinacy of differences of sets in Γ. Since Borel determinacy is proved in ZFC, ZFC implies Wadge's lemma for Borel sets.
Wadge's lemma is similar to the cone lemma from computability theory.
Wadge's lemma via Wadge and Lipschitz games
The Wadge game is a simple infinite game discovered by William Wadge (pronounced "wage"). It is used to investigate the notion of continuous reduction for subsets of Baire space. Wadge had analyzed the structure of the Wadge hierarchy for Baire space with games by 1972, but published these results only much later in his PhD thesis. In the Wadge game $G(A,B)$, player I and player II each in turn play integers, and the outcome of the game is determined by checking whether the sequences x and y generated by players I and II are contained in the sets A and B, respectively. Player II wins if the outcome is the same for both players, i.e. $x$ is in $A$ if and only if $y$ is in $B$. Player I wins if the outcome is different. Sometimes this is also called the Lipschitz game, and the variant where player II has the option to pass finitely many times is called the Wadge game.
Suppose that the game is determined. If player I has a winning strategy, then this defines a continuous (even Lipschitz) map reducing $B$ to the complement of $A$, and if on the other hand player II has a winning strategy then you have a reduction of $A$ to $B$. For example, suppose that player II has a winning strategy. Map every sequence x to the sequence y that player II plays in $G(A,B)$ if player I plays the sequence x, and player II follows his or her winning strategy. This defines a continuous map f with the property that x is in $A$ if and only if f(x) is in $B$.
Structure of the Wadge hierarchy
Martin and Monk proved in 1973 that AD implies the Wadge order for Baire space is well founded. Hence under AD, the Wadge classes modulo complements form a wellorder. The Wadge rank of a set $A$ is the order type of the set of Wadge degrees modulo complements strictly below [$A$]W. The length of the Wadge hierarchy has been shown to be Θ. Wadge also proved that the length of the Wadge hierarchy restricted to the Borel sets is φω1(1) (or φω1(2) depending on the notation), where φγ is the γth Veblen function to the base ω1 (instead of the usual ω).
As for the Wadge lemma, this holds for any pointclass Γ, assuming the axiom of determinacy. If we associate with each set $A$ the collection of all sets strictly below $A$ on the Wadge hierarchy, this forms a pointclass. Equivalently, for each ordinal α ≤ θ the collection Wα of sets that show up before stage α is a pointclass. Conversely, every pointclass is equal to some $W$α. A pointclass is said to be self-dual if it is closed under complementation. It can be shown that Wα is self-dual if and only if α is either 0, an even successor ordinal, or a limit ordinal of countable cofinality.
Other notions of degree
Similar notions of reduction and degree arise by replacing the continuous functions by any class of functions F that contains the identity function and is closed under composition. Write $A$ ≤F $B$ if $A=f^{-1}[B]$ for some function $f$ in F. Any such class of functions again determines a preorder on the subsets of Baire space. Degrees given by Lipschitz functions are called Lipschitz degrees, and degrees from Borel functions Borel–Wadge degrees.
See also
• Analytical hierarchy
• Arithmetical hierarchy
• Axiom of determinacy
• Borel hierarchy
• Determinacy
• Pointclass
• Weihrauch reducibility
References
1. D. Martin, H. G. Dales, Truth in Mathematics, ch. "Mathematical Evidence", p.224. Oxford Science Publications, 1998.
• Alexander S. Kechris; Benedikt Löwe; John R. Steel, eds. (December 2011). Wadge Degrees and Projective Ordinals: The Cabal Seminar Volume II. Lecture Notes in Logic. Cambridge University Press. ISBN 9781139504249.
• Andretta, Alessandro (2007). "The SLO principle and the Wadge hierarchy". In Bold, Stefan; Benedikt Löwe; Räsch, Thoralf; et al. (eds.). Infinite Games, Papers of the conference "Foundations of the Formal Sciences V" held in Bonn, Nov 26-29, 2004. Studies in Logic. Vol. 11. College Publications. pp. 1–38. ISBN 9781904987758..
• Kanamori, Akihiro (2000). The Higher Infinite (second ed.). Springer. ISBN 3-540-00384-3.
• Kechris, Alexander S. (1995). Classical Descriptive Set Theory. Springer. ISBN 0-387-94374-9.
• Wadge, William W. (1983). "Reducibility and determinateness on the Baire space" (PDF). PhD thesis. Univ. of California, Berkeley. {{cite journal}}: Cite journal requires |journal= (help)
Further reading
• Andretta, Alessandro & Martin, Donald (2003). "Borel-Wadge degrees". Fundamenta Mathematicae. 177 (2): 175–192. doi:10.4064/fm177-2-5.
• Cenzer, Douglas (1984). "Monotone Reducibility and the Family of Infinite Sets". The Journal of Symbolic Logic. Association for Symbolic Logic. 49 (3): 774–782. doi:10.2307/2274130. JSTOR 2274130. S2CID 37813340.
• Duparc, Jacques (2001). "Wadge hierarchy and Veblen hierarchy. Part I: Borel sets of finite rank". Journal of Symbolic Logic. 66 (1): 55–86. doi:10.2307/2694911. JSTOR 2694911. S2CID 17703130.
• Selivanov, Victor L. (2006). "Towards a descriptive set theory for domain-like structures". Theoretical Computer Science. 365 (3): 258–282. doi:10.1016/j.tcs.2006.07.053. ISSN 0304-3975.
• Selivanov, Victor L. (2008). "Wadge Reducibility and Infinite Computations". Mathematics in Computer Science. 2 (1): 5–36. doi:10.1007/s11786-008-0042-x. ISSN 1661-8270. S2CID 38211417.
• Semmes, Brian T. (2006). "A game for the Borel Functions". preprint. Univ. of Amsterdam, ILLC Prepublications PP-2006-24. Retrieved 2007-08-12. {{cite journal}}: Cite journal requires |journal= (help)
| Wikipedia |
Abu al-Wafa' al-Buzjani
Abū al-Wafāʾ Muḥammad ibn Muḥammad ibn Yaḥyā ibn Ismāʿīl ibn al-ʿAbbās al-Būzjānī or Abū al-Wafā Būzhjānī (Persian: ابو الوفا بوژگانی, or Arabic: ابو الوفا بوزجانی)[1] (10 June 940 – 15 July 998)[2] was a Persian[3][4][5] mathematician and astronomer who worked in Baghdad. He made important innovations in spherical trigonometry, and his work on arithmetic for businessmen contains the first instance of using negative numbers in a medieval Islamic text.
Abu al-Wafa' al-Buzjani
Born(940-06-10)June 10, 940
Buzhgan, Iran
DiedJuly 15, 998(998-07-15) (aged 58)
Baghdad
Academic background
InfluencesAl-Battani
Academic work
EraIslamic Golden Age
Main interestsMathematics and astronomy
Notable worksAlmagest of Abū al-Wafā'
Notable ideas
• Tangent function
• Law of sines
• Several trigonometric identities
InfluencedAl-Biruni, Abu Nasr Mansur
He is also credited with compiling the tables of sines and tangents at 15' intervals. He also introduced the secant and cosecant functions, as well studied the interrelations between the six trigonometric lines associated with an arc.[2] His Almagest was widely read by medieval Arabic astronomers in the centuries after his death. He is known to have written several other books that have not survived.
Life
He was born in Buzhgan, (now Torbat-e Jam) in Khorasan (in today's Iran). At age 19, in 959, he moved to Baghdad and remained there until his death in 998.[2] He was a contemporary of the distinguished scientists Abū Sahl al-Qūhī and al-Sijzi who were in Baghdad at the time and others such as Abu Nasr Mansur, Abu-Mahmud Khojandi, Kushyar Gilani and al-Biruni.[6] In Baghdad, he received patronage from members of the Buyid court.[7]
Astronomy
Abu al-Wafa' was the first to build a wall quadrant to observe the sky.[6] It has been suggested that he was influenced by the works of al-Battani as the latter described a quadrant instrument in his Kitāb az-Zīj.[6] His use of the concept of the tangent helped solve problems involving right-angled spherical triangles. He developed a new technique to calculate sine tables, allowing him to construct more accurate tables than his predecessors.[7]
In 997, he participated in an experiment to determine the difference in local time between his location, Baghdad, and that of al-Biruni (who was living in Kath, now a part of Uzbekistan).[8] The result was very close to present-day calculations, showing a difference of approximately 1 hour between the two longitudes. Abu al-Wafa is also known to have worked with Abū Sahl al-Qūhī, who was a famous maker of astronomical instruments.[7] While what is extant from his works lacks theoretical innovation, his observational data were used by many later astronomers, including al-Biruni.[7]
Almagest
Among his works on astronomy, only the first seven treatises of his Almagest (Kitāb al-Majisṭī) are now extant.[9] The work covers numerous topics in the fields of plane and spherical trigonometry, planetary theory, and solutions to determine the direction of Qibla.[6][7]
Mathematics
He defined the tangent function, and he established several trigonometric identities in their modern form, where the ancient Greek mathematicians had expressed the equivalent identities in terms of chords.[10] The trigonometric identities he introduced were:
$\sin(a\pm b)=\sin(a)\cos(b)\pm \cos(a)\sin(b)$
$\cos(2a)=1-2\sin ^{2}(a)$
$\sin(2a)=2\sin(a)\cos(a)$
He may have developed the law of sines for spherical triangles, though others like Abu-Mahmud Khojandi have been credited with the same achievement:[11]
${\frac {\sin A}{\sin a}}={\frac {\sin B}{\sin b}}={\frac {\sin C}{\sin c}}$
where $A,B,C$ are the sides of the triangle (measured in radians on the unit sphere) and $a,b,c$ are the opposing angles.[10]
Some sources suggest that he introduced the tangent function, although other sources give the credit for this innovation to al-Marwazi.[10]
Works
• Almagest (كتاب المجسطي Kitāb al-Majisṭī).
• A book of zij called Zīj al‐wāḍiḥ (زيج الواضح), no longer extant.[7]
• "A Book on Those Geometric Constructions Which Are Necessary for a Craftsman", (كتاب في ما یحتاج إليه الصانع من الأعمال الهندسية Kitāb fī mā yaḥtāj ilayh al-ṣāniʿ min al-aʿmāl al-handasiyya).[12] This text contains over one hundred geometric constructions, including for a regular heptagon, which have been reviewed and compared with other mathematical treatises. The legacy of this text in Latin Europe is still debated.[13][14]
• "A Book on What Is Necessary from the Science of Arithmetic for Scribes and Businessmen", (كتاب في ما يحتاج إليه الكتاب والعمال من علم الحساب Kitāb fī mā yaḥtāj ilayh al-kuttāb wa’l-ʿummāl min ʾilm al-ḥisāb).[12] This is the first book where negative numbers have been used in the medieval Islamic texts.[7]
He also wrote translations and commentaries on the algebraic works of Diophantus, al-Khwārizmī, and Euclid's Elements.[7]
Legacy
• The crater Abul Wáfa on the Moon is named after him.[15][16]
• On 10 June 2015, Google changed its logo in memory of Abu al-Wafa' Buzjani.[17]
Notes
1. "بوزجانی". Encyclopaediaislamica.com. Archived from the original on 25 October 2008. Retrieved 30 August 2009.
2. O'Connor, John J.; Robertson, Edmund F., "Mohammad Abu'l-Wafa Al-Buzjani", MacTutor History of Mathematics Archive, University of St Andrews
3. Ben-Menahem, A. (2009). Historical encyclopedia of natural and mathematical sciences (1st ed.). Berlin: Springer. p. 559. ISBN 978-3-540-68831-0. 970 CE Abu al-Wafa al-Buzjani (940–998, Baghdad). Persian astronomer and mathematician.
4. Sigfried J. de Laet (1994). History of Humanity: From the seventh to the sixteenth century. UNESCO. p. 931. ISBN 978-92-3-102813-7. The science of trigonometry as known today was established by Islamic mathematicians. One of the most important of these was the Persian Abu' l-Wafa' Buzjani (d. 997 or 998), who wrote a work called the Almagest dealing mostly with trigonometry
5. Subtelny, Maria E. (2007). Timurids in Transition. BRILL. p. 144. ISBN 9789004160316. Persian mathematician Abu al-Wafa Muhammad al-Buzjani
6. Moussa, Ali (2011). "Mathematical Methods in Abū al-Wafāʾ's Almagest and the Qibla Determinations". Arabic Sciences and Philosophy. Cambridge University Press. 21 (1): 1–56. doi:10.1017/S095742391000007X. S2CID 171015175.
7. Hashemipour 2007.
8. Stowasser, Barbara Freyer (9 May 2014). The Day Begins at Sunset: Perceptions of Time in the Islamic World. Bloomsbury Publishing. p. 83. ISBN 978-0-85772-536-3.
9. Kennedy, E. S. (1956). Survey of Islamic Astronomical Tables. American Philosophical Society. p. 12.
10. Jacques Sesiano, "Islamic mathematics", p. 157, in Selin, Helaine; D'Ambrosio, Ubiratan, eds. (2000), Mathematics Across Cultures: The History of Non-western Mathematics, Springer, ISBN 1-4020-0260-2
11. S. Frederick Starr (2015). Lost Enlightenment: Central Asia's Golden Age from the Arab Conquest to Tamerlane. Princeton University Press. p. 177. ISBN 9780691165851.
12. Youschkevitch 1970.
13. Raynaud 2012.
14. Gamwell, Lynn (2 December 2015). "Why the history of maths is also the history of art". The Guardian. Retrieved 3 December 2015.
15. "Abul Wáfa". Gazetteer of Planetary Nomenclature. USGS Astrogeology Research Program.
16. D. H. Menzel; M. Minnaert; B. Levin; A. Dollfus; B. Bell (1971). "Report on Lunar Nomenclature by The Working Group of Commission 17 of the IAU". Space Science Reviews. 12 (2): 136. Bibcode:1971SSRv...12..136M. doi:10.1007/BF00171763. S2CID 122125855.
17. "Abu al-Wafa' al-Buzjani's 1075th Birthday". Google. 10 June 2015.
References
• O'Connor, John J.; Robertson, Edmund F., "Mohammad Abu'l-Wafa Al-Buzjani", MacTutor History of Mathematics Archive, University of St Andrews
• Hashemipour, Behnaz (2007). "Būzjānī: Abū al‐Wafāʾ Muḥammad ibn Muḥammad ibn Yaḥyā al‐Būzjānī". In Thomas Hockey; et al. (eds.). The Biographical Encyclopedia of Astronomers. New York: Springer. pp. 188–9. ISBN 978-0-387-31022-0. (PDF version)
• Raynaud, D. (2012), "Abū al-Wafāʾ Latinus? A Study of Method", Historia Mathematica, 39 (1): 34–83, doi:10.1016/j.hm.2011.09.001, S2CID 119600916 (PDF version)
• Youschkevitch, A.P. (1970). "Abū'l-Wafāʾ al-Būzjānī, Muḥammad Ibn Muḥammad Ibn Yaḥyā Ibn Ismāʿīl Ibn al-ʿAbbās". Dictionary of Scientific Biography. Vol. 1. New York: Charles Scribner's Sons. pp. 39–43. ISBN 0-684-10114-9.
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| Wikipedia |
Wagner's theorem
In graph theory, Wagner's theorem is a mathematical forbidden graph characterization of planar graphs, named after Klaus Wagner, stating that a finite graph is planar if and only if its minors include neither K5 (the complete graph on five vertices) nor K3,3 (the utility graph, a complete bipartite graph on six vertices). This was one of the earliest results in the theory of graph minors and can be seen as a forerunner of the Robertson–Seymour theorem.
Definitions and statement
A planar embedding of a given graph is a drawing of the graph in the Euclidean plane, with points for its vertices and curves for its edges, in such a way that the only intersections between pairs of edges are at a common endpoint of the two edges. A minor of a given graph is another graph formed by deleting vertices, deleting edges, and contracting edges. When an edge is contracted, its two endpoints are merged to form a single vertex. In some versions of graph minor theory the graph resulting from a contraction is simplified by removing self-loops and multiple adjacencies, while in other version multigraphs are allowed, but this variation makes no difference to Wagner's theorem. Wagner's theorem states that every graph has either a planar embedding, or a minor of one of two types, the complete graph K5 or the complete bipartite graph K3,3. (It is also possible for a single graph to have both types of minor.)
If a given graph is planar, so are all its minors: vertex and edge deletion obviously preserve planarity, and edge contraction can also be done in a planarity-preserving way, by leaving one of the two endpoints of the contracted edge in place and routing all of the edges that were incident to the other endpoint along the path of the contracted edge. A minor-minimal non-planar graph is a graph that is not planar, but in which all proper minors (minors formed by at least one deletion or contraction) are planar. Another way of stating Wagner's theorem is that there are only two minor-minimal non-planar graphs, K5 and K3,3.
Another result also sometimes known as Wagner's theorem states that a four-connected graph is planar if and only if it has no K5 minor. That is, by assuming a higher level of connectivity, the graph K3,3 can be made unnecessary in the characterization, leaving only a single forbidden minor, K5. Correspondingly, the Kelmans–Seymour conjecture states that a 5-connected graph is planar if and only if it does not have K5 as a topological minor.
History and relation to Kuratowski's theorem
Wagner published both theorems in 1937,[1] subsequent to the 1930 publication of Kuratowski's theorem,[2] according to which a graph is planar if and only if it does not contain as a subgraph a subdivision of one of the same two forbidden graphs K5 and K3,3. In a sense, Kuratowski's theorem is stronger than Wagner's theorem: a subdivision can be converted into a minor of the same type by contracting all but one edge in each path formed by the subdivision process, but converting a minor into a subdivision of the same type is not always possible. However, in the case of the two graphs K5 and K3,3, it is straightforward to prove that a graph that has at least one of these two graphs as a minor also has at least one of them as a subdivision, so the two theorems are equivalent.[3]
Implications
One consequence of the stronger version of Wagner's theorem for four-connected graphs is to characterize the graphs that do not have a K5 minor. The theorem can be rephrased as stating that every such graph is either planar or it can be decomposed into simpler pieces. Using this idea, the K5-minor-free graphs may be characterized as the graphs that can be formed as combinations of planar graphs and the eight-vertex Wagner graph, glued together by clique-sum operations. For instance, K3,3 can be formed in this way as a clique-sum of three planar graphs, each of which is a copy of the tetrahedral graph K4.
Wagner's theorem is an important precursor to the theory of graph minors, which culminated in the proofs of two deep and far-reaching results: the graph structure theorem (a generalization of Wagner's clique-sum decomposition of K5-minor-free graphs)[4] and the Robertson–Seymour theorem (a generalization of the forbidden minor characterization of planar graphs, stating that every graph family closed under the operation of taking minors has a characterization by a finite number of forbidden minors).[5] Analogues of Wagner's theorem can also be extended to the theory of matroids: in particular, the same two graphs K5 and K3,3 (along with three other forbidden configurations) appear in a characterization of the graphic matroids by forbidden matroid minors.[6]
References
1. Wagner, K. (1937), "Über eine Eigenschaft der ebenen Komplexe", Math. Ann., 114: 570–590, doi:10.1007/BF01594196, S2CID 123534907.
2. Kuratowski, Kazimierz (1930), "Sur le problème des courbes gauches en topologie" (PDF), Fund. Math. (in French), 15: 271–283, doi:10.4064/fm-15-1-271-283.
3. Bondy, J. A.; Murty, U.S.R. (2008), Graph Theory, Graduate Texts in Mathematics, vol. 244, Springer, p. 269, ISBN 9781846289699.
4. Lovász, László (2006), "Graph minor theory", Bulletin of the American Mathematical Society, 43 (1): 75–86, doi:10.1090/S0273-0979-05-01088-8, MR 2188176.
5. Chartrand, Gary; Lesniak, Linda; Zhang, Ping (2010), Graphs & Digraphs (5th ed.), CRC Press, p. 307, ISBN 9781439826270.
6. Seymour, P. D. (1980), "On Tutte's characterization of graphic matroids", Annals of Discrete Mathematics, 8: 83–90, doi:10.1016/S0167-5060(08)70855-0, ISBN 9780444861108, MR 0597159.
| Wikipedia |
Wagner graph
In the mathematical field of graph theory, the Wagner graph is a 3-regular graph with 8 vertices and 12 edges.[1] It is the 8-vertex Möbius ladder graph.
Wagner graph
The Wagner graph
Named afterKlaus Wagner
Vertices8
Edges12
Radius2
Diameter2
Girth4
Automorphisms16 (D8)
Chromatic number3
Chromatic index3
Genus1
Spectrum${\begin{pmatrix}3&1&{\sqrt {2}}-1&-1&-{\sqrt {2}}-1\\1&2&2&1&2\end{pmatrix}}$
PropertiesCubic
Hamiltonian
Triangle-free
Vertex-transitive
Toroidal
Apex
NotationM8
Table of graphs and parameters
Properties
As a Möbius ladder, the Wagner graph is nonplanar but has crossing number one, making it an apex graph. It can be embedded without crossings on a torus or projective plane, so it is also a toroidal graph. It has girth 4, diameter 2, radius 2, chromatic number 3, chromatic index 3 and is both 3-vertex-connected and 3-edge-connected.
The Wagner graph has 392 spanning trees; it and the complete graph K3,3 have the most spanning trees among all cubic graphs with the same number of vertices.[2]
The Wagner graph is a vertex-transitive graph but is not edge-transitive. Its full automorphism group is isomorphic to the dihedral group D8 of order 16, the group of symmetries of an octagon, including both rotations and reflections.
The characteristic polynomial of the Wagner graph is
$(x-3)(x-1)^{2}(x+1)(x^{2}+2x-1)^{2}.$
It is the only graph with this characteristic polynomial, making it a graph determined by its spectrum.
The Wagner graph is triangle-free and has independence number three, providing one half of the proof that the Ramsey number R(3,4) (the least number n such that any n-vertex graph contains either a triangle or a four-vertex independent set) is 9.[3]
Graph minors
Möbius ladders play an important role in the theory of graph minors. The earliest result of this type is a 1937 theorem of Klaus Wagner (part of a cluster of results known as Wagner's theorem) that graphs with no K5 minor can be formed by using clique-sum operations to combine planar graphs and the Möbius ladder M8.[4] For this reason M8 is called the Wagner graph.
The Wagner graph is also one of four minimal forbidden minors for the graphs of treewidth at most three (the other three being the complete graph K5, the graph of the regular octahedron, and the graph of the pentagonal prism) and one of four minimal forbidden minors for the graphs of branchwidth at most three (the other three being K5, the graph of the octahedron, and the cube graph).[5][6]
Construction
The Wagner graph is a cubic Hamiltonian graph and can be defined by the LCF notation [4]8. It is an instance of an Andrásfai graph, a type of circulant graph in which the vertices can be arranged in a cycle and each vertex is connected to the other vertices whose positions differ by a number that is 1 (mod 3). It is also isomorphic to the circular clique K8/3.
It can be drawn as a ladder graph with 4 rungs made cyclic on a topological Möbius strip.
Gallery
• The chromatic number of the Wagner graph is 3.
• The chromatic index of the Wagner graph is 3.
• The Wagner graph drawn on the Möbius strip.
References
Wikimedia Commons has media related to Wagner graph.
1. Bondy, J. A.; Murty, U. S. R. (2007). Graph Theory. Springer. pp. 275–276. ISBN 978-1-84628-969-9.
2. Jakobson, Dmitry; Rivin, Igor (1999). "On some extremal problems in graph theory". arXiv:math.CO/9907050.
3. Soifer, Alexander (2008). The Mathematical Coloring Book. Springer-Verlag. p. 245. ISBN 978-0-387-74640-1.
4. Wagner, K. (1937). "Über eine Eigenschaft der ebenen Komplexe". Mathematische Annalen (in German). 114 (1): 570–590. doi:10.1007/BF01594196. S2CID 123534907.
5. Bodlaender, Hans L. (1998). "A partial k-arboretum of graphs with bounded treewidth". Theoretical Computer Science. 209 (1–2): 1–45. doi:10.1016/S0304-3975(97)00228-4. hdl:1874/18312.
6. Bodlaender, Hans L.; Thilikos, Dimitrios M. (1999). "Graphs with branchwidth at most three". Journal of Algorithms. 32 (2): 167–194. doi:10.1006/jagm.1999.1011. hdl:1874/2734.
| Wikipedia |
Wagstaff prime
In number theory, a Wagstaff prime is a prime number of the form
${{2^{p}+1} \over 3}$
Wagstaff prime
Named afterSamuel S. Wagstaff, Jr.
Publication year1989[1]
Author of publicationBateman, P. T., Selfridge, J. L., Wagstaff Jr., S. S.
No. of known terms44
First terms3, 11, 43, 683
Largest known term(215135397+1)/3
OEIS index
• A000979
• Wagstaff primes: primes of form (2^p + 1)/3
where p is an odd prime. Wagstaff primes are named after the mathematician Samuel S. Wagstaff Jr.; the prime pages credit François Morain for naming them in a lecture at the Eurocrypt 1990 conference. Wagstaff primes appear in the New Mersenne conjecture and have applications in cryptography.
Examples
The first three Wagstaff primes are 3, 11, and 43 because
${\begin{aligned}3&={2^{3}+1 \over 3},\\[5pt]11&={2^{5}+1 \over 3},\\[5pt]43&={2^{7}+1 \over 3}.\end{aligned}}$
Known Wagstaff primes
The first few Wagstaff primes are:
3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, … (sequence A000979 in the OEIS)
As of January 2023, known exponents which produce Wagstaff primes or probable primes are:
3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031[2] (all known Wagstaff primes)
138937, 141079, 267017, 269987, 374321, 986191, 4031399, …, 13347311, 13372531, 15135397 (Wagstaff probable primes) (sequence A000978 in the OEIS)
In February 2010, Tony Reix discovered the Wagstaff probable prime:
${\frac {2^{4031399}+1}{3}}$
which has 1,213,572 digits and was the 3rd biggest probable prime ever found at this date.[3]
In September 2013, Ryan Propper announced the discovery of two additional Wagstaff probable primes:[4]
${\frac {2^{13347311}+1}{3}}$
and
${\frac {2^{13372531}+1}{3}}$
Each is a probable prime with slightly more than 4 million decimal digits. It is not currently known whether there are any exponents between 4031399 and 13347311 that produce Wagstaff probable primes.
In June 2021, Ryan Propper announced the discovery of the Wagstaff probable prime:[5]
${\frac {2^{15135397}+1}{3}}$
which is a probable prime with slightly more than 4.5 million decimal digits.
Primality testing
Primality has been proven or disproven for the values of p up to 127031. Those with p > 127031 are probable primes as of January 2023. The primality proof for p = 42737 was performed by François Morain in 2007 with a distributed ECPP implementation running on several networks of workstations for 743 GHz-days on an Opteron processor.[6] It was the third largest primality proof by ECPP from its discovery until March 2009.[7]
The Lucas–Lehmer–Riesel test can be used to identify Wagstaff PRPs. In particular, if p is an exponent of a Wagstaff prime, then
$25^{2^{\!\;p-1}}\equiv 25{\pmod {2^{p}+1}}$.[8]
Generalizations
It is natural to consider[9] more generally numbers of the form
$Q(b,n)={\frac {b^{n}+1}{b+1}}$
where the base $b\geq 2$. Since for $n$ odd we have
${\frac {b^{n}+1}{b+1}}={\frac {(-b)^{n}-1}{(-b)-1}}=R_{n}(-b)$
these numbers are called "Wagstaff numbers base $b$", and sometimes considered[10] a case of the repunit numbers with negative base $-b$.
For some specific values of $b$, all $Q(b,n)$ (with a possible exception for very small $n$) are composite because of an "algebraic" factorization. Specifically, if $b$ has the form of a perfect power with odd exponent (like 8, 27, 32, 64, 125, 128, 216, 243, 343, 512, 729, 1000, etc. (sequence A070265 in the OEIS)), then the fact that $x^{m}+1$, with $m$ odd, is divisible by $x+1$ shows that $Q(a^{m},n)$ is divisible by $a^{n}+1$ in these special cases. Another case is $b=4k^{4}$, with k a positive integer (like 4, 64, 324, 1024, 2500, 5184, etc. (sequence A141046 in the OEIS)), where we have the aurifeuillean factorization.
However, when $b$ does not admit an algebraic factorization, it is conjectured that an infinite number of $n$ values make $Q(b,n)$ prime, notice all $n$ are odd primes.
For $b=10$, the primes themselves have the following appearance: 9091, 909091, 909090909090909091, 909090909090909090909090909091, … (sequence A097209 in the OEIS), and these ns are: 5, 7, 19, 31, 53, 67, 293, 641, 2137, 3011, 268207, ... (sequence A001562 in the OEIS).
See Repunit#Repunit primes for the list of the generalized Wagstaff primes base $b$. (Generalized Wagstaff primes base $b$ are generalized repunit primes base $-b$ with odd $n$)
The least primes p such that $Q(n,p)$ is prime are (starts with n = 2, 0 if no such p exists)
3, 3, 3, 5, 3, 3, 0, 3, 5, 5, 5, 3, 7, 3, 3, 7, 3, 17, 5, 3, 3, 11, 7, 3, 11, 0, 3, 7, 139, 109, 0, 5, 3, 11, 31, 5, 5, 3, 53, 17, 3, 5, 7, 103, 7, 5, 5, 7, 1153, 3, 7, 21943, 7, 3, 37, 53, 3, 17, 3, 7, 11, 3, 0, 19, 7, 3, 757, 11, 3, 5, 3, ... (sequence A084742 in the OEIS)
The least bases b such that $Q(b,prime(n))$ is prime are (starts with n = 2)
2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 16, 61, 2, 6, 10, 6, 2, 5, 46, 18, 2, 49, 16, 70, 2, 5, 6, 12, 92, 2, 48, 89, 30, 16, 147, 19, 19, 2, 16, 11, 289, 2, 12, 52, 2, 66, 9, 22, 5, 489, 69, 137, 16, 36, 96, 76, 117, 26, 3, ... (sequence A103795 in the OEIS)
References
1. Bateman, P. T.; Selfridge, J. L.; Wagstaff, Jr., S. S. (1989). "The New Mersenne Conjecture". American Mathematical Monthly. 96: 125–128. doi:10.2307/2323195. JSTOR 2323195.
2. "The Top Twenty: Wagstaff".
3. "Henri & Renaud Lifchitz's PRP Top records". www.primenumbers.net. Retrieved 2021-11-13.
4. New Wagstaff PRP exponents, mersenneforum.org
5. Announcing a new Wagstaff PRP, mersenneforum.org
6. Comment by François Morain, The Prime Database: (242737 + 1)/3 at The Prime Pages.
7. Caldwell, Chris, "The Top Twenty: Elliptic Curve Primality Proof", The Prime Pages
8. Lifchitz, Renaud; Lifchitz, Henri (May 18, 2002) [July 2000]. "An efficient probable prime test for numbers of the form (2p + 1)/3" (PDF). Retrieved 2023-04-12.
9. Dubner, H. and Granlund, T.: Primes of the Form (bn + 1)/(b + 1), Journal of Integer Sequences, Vol. 3 (2000)
10. Repunit, Wolfram MathWorld (Eric W. Weisstein)
External links
• John Renze and Eric W. Weisstein. "Wagstaff prime". MathWorld.
• Chris Caldwell, The Top Twenty: Wagstaff at The Prime Pages.
• Renaud Lifchitz, "An efficient probable prime test for numbers of the form (2p + 1)/3".
• Tony Reix, "Three conjectures about primality testing for Mersenne, Wagstaff and Fermat numbers based on cycles of the Digraph under x2 − 2 modulo a prime".
• List of repunits in base -50 to 50
• List of Wagstaff primes base 2 to 160
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)
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By integer sequence
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• k-tuple
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• Chen
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List of prime numbers
| Wikipedia |
Queue area
Queue areas are places in which people queue (first-come, first-served) for goods or services. Such a group of people is known as a queue (British usage) or line (American usage), and the people are said to be waiting or standing in a queue or in line, respectively. (In the New York City area, the phrase on line is often used in place of in line.)[1] Occasionally, both the British and American terms are combined to form the term "queue line".[2][3]
Examples include checking out groceries or other goods that have been collected in a self service shop, in a shop without self-service, at an ATM, at a ticket desk, a city bus, or in a taxi stand.
Queueing[4] is a phenomenon in a number of fields, and has been extensively analysed in the study of queueing theory. In economics, queueing is seen as one way to ration scarce goods and services.
Types
History
The first written description of people standing in line is found in an 1837 book, The French Revolution: A History by Thomas Carlyle.[5] Carlyle described what he thought was a strange sight: people standing in an orderly line to buy bread from bakers around Paris.[5]
Typical applications
Queues can be found in railway stations to book tickets, at bus stops for boarding and at temples.[6][7][8]
Queues are generally found at transportation terminals where security screenings are conducted.
Large stores and supermarkets may have dozens of separate queues, but this can cause frustration, as different lines tend to be handled at different speeds; some people are served quickly, while others may wait for longer periods of time. Sometimes two people who are together split up and each waits in a different line; once it is determined which line is faster, the one in the slower line joins the other. Another arrangement is for everyone to wait in a single line; a person leaves the line each time a service point opens up. This is a common setup in banks and post offices.
Organized queue areas are commonly found at amusement parks. The rides have a fixed number of guests that can be served at any given time (which is referred to as the rides operational capacity), so there has to be some control over additional guests who are waiting. This led to the development of formalized queue areas—areas in which the lines of people waiting to board the rides are organized by railings, and may be given shelter from the elements with a roof over their heads, inside a climate-controlled building or with fans and misting devices. In some amusement parks – Disney theme parks being a prime example – queue areas can be elaborately decorated, with holding areas fostering anticipation, thus shortening the perceived wait for people in the queue by giving them something interesting to look at as they wait, or the perception that they have arrived at the threshold of the attraction.
Design
When designing queues, planners attempt to make the wait as pleasant and as simple as possible. They employ several strategies to achieve this, including:
• Expanding the capacity of the queue, thus allowing more patrons to have a place. This can be achieved by:
• Increasing the length of the queue by making the queue longer
• Increasing the size of the lanes within the queue
• Increasing the length of the queue by designing the line in a "zig-zag" shape that holds a large number of guests in a smaller area. This is used often at amusement parks. Notable rides have a large area of this kind of line to hold as many people as possible in line. Portions of the line can be sectioned off and bypassed by guests if the queue is not crowded.
• "In-line" entertainment can be added. This is popular at amusement parks like Walt Disney World, which uses TV screens and other visuals to keep people in the queue area occupied.
• Secondary queue areas for patrons with special tickets, like the FastPass system used at Disney parks, or the Q-bot as used in Legoland Windsor.
Psychology
People experience "occupied" time as shorter than "unoccupied" time, and generally overestimate the amount of time waited by around 36%.[9]
The technique of giving people an activity to distract them from a wait has been used to reduce complaints of delays at:[9]
• Baggage claim in the Houston, Texas airport, by moving the arrival gates further away so passengers spend more time walking than standing around waiting
• Elevators, by adding mirrors so people can groom themselves or watch other people
• Retail checkout, by placing small items for purchase so customers can continue shopping while waiting
Other techniques to reduce queueing anxiety include:[9]
• Hiding the length of a line by wrapping it around a corner.
• Having only one line, so there is no anxiety about which line to choose and a greater sense of fairness. Even though the average wait over time is the same, customers tend to notice lines that are moving faster than they are compared to other lines moving more slowly.
• Putting up signs that deliberately overestimate the wait time, to always exceed customer expectations.
Cutting in line, also known as queue-jumping, can generate a strong negative response, depending on the local cultural norms.
Virtual
Physical queueing is sometimes replaced by virtual queueing. In a waiting room there may be a system whereby the queuer asks and remembers where their place is in the queue, or reports to a desk and signs in, or takes a ticket with a number from a machine. These queues typically are found at doctors' offices, hospitals, town halls, social security offices, labor exchanges, the Department of Motor Vehicles, the immigration departments, free internet access in the state or council libraries, banks or post offices and call centres. Especially in the United Kingdom, tickets are taken to form a virtual queue at delicatessens and children's shoe shops. In some countries such as Sweden, virtual queues are also common in shops and railway stations. A display sometimes shows the number that was last called for service.
Restaurants have come to employ virtual queueing techniques with the availability of application-specific pagers, which alert those waiting that they should report to the host to be seated. Another option used at restaurants is to assign customers a confirmed return time, basically a reservation issued on arrival.
Virtual queueing apps are available that allow the customers to view the virtual queue status of a business and they can take virtual queue numbers remotely. The app can be used to get updates of the virtual queue status that the customer is in.
Alternate activities
A substitute or alternative activity may be provided for people to participate in while waiting to be called, which reduces the perceived waiting time and the probability that the customer will abort their visit. For example, a busy restaurant might seat waiting customers a bar. An outdoor attraction with long virtual queues might have a side marquee selling merchandise or food. The alternate activity may provide the organisation with an opportunity to generate additional revenue from the waiting customers.[10]
Mobile
All of the above methods, however, suffer from the same drawback: the person arrives at the location only to find out that they need to wait. Recently, queues at DMVs,[11] colleges, restaurants,[12] healthcare institutions,[13] government offices[12] and elsewhere have begun to be replaced by mobile queues or queue-ahead, whereby the person queuing uses their phone, the internet, a kiosk or another method to enter a virtual queue, optionally prior to arrival, is free to roam during the wait, and then gets paged at their mobile phone when their turn approaches. This has the advantage of allowing users to find out the wait forecast and get in the queue before arriving, roaming freely and then timing their arrival to the availability of service. This has been shown to extend the patience of those in the queue and reduce no-shows.[12]
See also
• Cutting in line
• Call centre
• Line stander
• Queuing Rule of Thumb
• Waiting room
References
1. LearnersDictionary.com
2. Watson, Jim. "Better layouts for queue lines". jamesrobertwatson.com. Retrieved 2018-03-18.
3. Chris Sawyer Productions (2002-10-15). RollerCoaster Tycoon 2 (U.S. release) (Microsoft Windows). Infogrames. Scene: Footpaths window (normal gameplay). When the cursor hovers over the queue line options for a few seconds in the "Footpaths" window, a pop-up that says "Queue line paths" appears.
4. Also spelled queuing."QUEUE | Meaning & Definition for UK English | Lexico.com". Lexico Dictionaries | English. Archived from the original on November 11, 2020. Retrieved 2022-01-14.
5. Keiles, Jamie Lauren (1 January 2018). "Why We Wait in Lines". Racked. Retrieved 2018-01-19.
6. "Queues get longer at railway station". The Hindu. Mangalore. 3 May 2012. Retrieved Mar 2, 2015.
7. "Many bus stops in Mumbai not in 'BEST' shape". Daily News and Analysis. Mumbai. 28 May 2011. Retrieved Mar 2, 2015.
8. "Shirdi: Now, pay extra for VIP 'aartis' at Sai Baba temple". NDTV. Shirdi. 17 November 2013. Retrieved Mar 2, 2015.
9. Alex Stone (Aug 18, 2012). "Why Waiting Is Torture". The New York Times.
10. Supalocal, "Master the art of substitution", April 12, 2011, accessed July 11, 2011.
11. DMV’s New Line Management System is Available Online Archived July 26, 2011, at the Wayback Machine.
12. "Exit waiting in line, enter QLess". Vator.tv. 2010-01-14. Retrieved 2010-09-23.
13. "Could your practice's waiting area become obsolete? : Noteworthy – A Family Practice Management blog". Blogs.aafp.org. Retrieved 2010-09-23.
Further reading
• Maister, D.H. (1988). Managing Services: Marketing, Operations and Human Resources. Prentice-Hall.
• Mercer, David. Redefining marketing in the multi-channel age. Wiley.
External links
Wikimedia Commons has media related to Queues.
• For insight into the British habit of queueing, see standinaqueue
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| Wikipedia |
Wakeby distribution
The Wakeby distribution[1] is a five-parameter probability distribution defined by its quantile function,
$W(p)=\xi +{\frac {\alpha }{\beta }}(1-(1-p)^{\beta })-{\frac {\gamma }{\delta }}(1-(1-p)^{-\delta })$,
Wakeby distribution
Parameters $\alpha ,\beta ,\gamma ,\delta ,\xi $
Support
$\xi $ to $\infty $, if $\delta \geq 0,\gamma >0$
$\xi $ to $\xi +(\alpha /\beta )-(\gamma /\delta )$, otherwise
Quantile $\xi +{\frac {\alpha }{\beta }}(1-(1-p)^{\beta })-{\frac {\gamma }{\delta }}(1-(1-p)^{-\delta })$
and by its quantile density function,
$W'(p)=w(p)=\alpha (1-p)^{\beta -1}+\gamma (1-p)^{-\delta -1}$,
where $0\leq p\leq 1$, ξ is a location parameter, α and γ are scale parameters and β and δ are shape parameters.[1]
This distribution was first proposed by Harold A. Thomas Jr., who named it after Wakeby Pond in Cape Cod.[2][3]
Applications
The Wakeby distribution has been used for modeling distributions of
• flood flows,[4][5]
• citation counts,[6]
• extreme rainfall,[7][8]
• tidal current speeds,[9]
• and peak flows of rivers.[10]
Parameters and domain
The following restrictions apply to the parameters of this distribution:
• $\beta +\delta \geq 0$
• Either $\beta +\delta >0$ or $\beta =\gamma =\delta =0$
• If $\gamma >0$, then $\delta >0$
• $\gamma \geq 0$
• $\alpha +\gamma \geq 0$
The domain of the Wakeby distribution is
• $\xi $ to $\infty $, if $\delta \geq 0$ and $\gamma >0$
• $\xi $ to $\xi +(\alpha /\beta )-(\gamma /\delta )$, if $\delta <0$ or $\gamma =0$
With two shape parameters, the Wakeby distribution can model a wide variety of shapes.[1]
CDF and PDF
The cumulative distribution function is computed by numerically inverting the quantile function given above. The probability density function is then found by using the following relation (given on page 46 of Johnson, Kotz, and Balakrishnan[11]):
$f(x)={\frac {(1-F(x))^{(\delta +1)}}{\alpha t+\gamma }}$
where F is the cumulative distribution function and
$t=(1-F(x))^{(\beta +\delta )}$
An implementation that computes the probability density function of the Wakeby distribution is included in the Dataplot scientific computation library, as routine WAKPDF.[1]
An alternative to the above method is to define the PDF parametrically as $(W(p),1/w(p)),\ 0\leq p\leq 1$. This can be set up as a probability density function, $f(x)$, by solving for the unique $p$ in the equation $W(p)=x$ and returning $1/w(p)$.
See also
• Generalized Pareto distribution
References
1. "Dataplot reference manual: WAKPDF". NIST. Retrieved 20 August 2015.
2. Rodda, John C.; Robinson, Mark (2015-08-26). Progress in Modern Hydrology: Past, Present and Future. John Wiley & Sons. p. 75. ISBN 978-1-119-07429-8.
3. Katchanov, Yurij L.; Markova, Yulia V. (2015-02-26). "On a heuristic point of view concerning the citation distribution: introducing the Wakeby distribution". SpringerPlus. 4 (1): 94. doi:10.1186/s40064-015-0821-1. ISSN 2193-1801. PMC 4352413. PMID 25763305.
4. John C. Houghton (October 14, 1977). "Birth of a Parent: The Wakeby Distribution for Modeling Flood Flows; Working Paper No. MIT-EL77-033WP" (PDF). MIT.
5. GRIFFITHS, GEORGE A. (1989-06-01). "A theoretically based Wakeby distribution for annual flood series". Hydrological Sciences Journal. 34 (3): 231–248. CiteSeerX 10.1.1.399.6501. doi:10.1080/02626668909491332. ISSN 0262-6667.
6. Katchanov, Yurij L.; Markova, Yulia V. (2015-02-26). "On a heuristic point of view concerning the citation distribution: introducing the Wakeby distribution". SpringerPlus. 4 (1): 94. doi:10.1186/s40064-015-0821-1. ISSN 2193-1801. PMC 4352413. PMID 25763305.
7. Park, Jeong-Soo; Jung, Hyun-Sook; Kim, Rae-Seon; Oh, Jai-Ho (2001). "Modelling summer extreme rainfall over the Korean peninsula using Wakeby distribution". International Journal of Climatology. 21 (11): 1371–1384. doi:10.1002/joc.701. ISSN 1097-0088. S2CID 130799481.
8. Su, Buda; Kundzewicz, Zbigniew W.; Jiang, Tong (2009-05-01). "Simulation of extreme precipitation over the Yangtze River Basin using Wakeby distribution". Theoretical and Applied Climatology. 96 (3): 209–219. doi:10.1007/s00704-008-0025-5. ISSN 1434-4483. S2CID 122488492.
9. Liu, Mingjun; Li, Wenyuan; Billinton, Roy; Wang, Caisheng; Yu, Juan (2015-10-01). "Modeling tidal current speed using a Wakeby distribution". Electric Power Systems Research. 127: 240–248. doi:10.1016/j.epsr.2015.06.014. ISSN 0378-7796.
10. Öztekin, Tekin (2011-03-01). "Estimation of the Parameters of Wakeby Distribution by a Numerical Least Squares Method and Applying it to the Annual Peak Flows of Turkish Rivers". Water Resources Management. 25 (5): 1299–1313. doi:10.1007/s11269-010-9745-2. ISSN 1573-1650. S2CID 154960776.
11. Johnson, Norman Lloyd; Kotz, Samuel; Balakrishnan, Narayanaswamy (1994). Continuous univariate distributions. Vol1 (2 ed.). New York: Wiley. p. 46. ISBN 0-471-58495-9. OCLC 29428092.
External links
• Discussion of the naming of the distribution on Stack Exchange
Note: this work is based on a NIST document that is in the public domain as a work of the U.S. federal government
| Wikipedia |
Robert Wald
Robert M. Wald (/wɔːld/; born June 29, 1947 in New York City) is an American theoretical physicist and professor at the University of Chicago. He studies general relativity, black holes, and quantum gravity and has written textbooks on these subjects.
Robert M. Wald
Wald in 2012
Born
June 29, 1947 (1947-06-29) (age 76)
CitizenshipUnited States
Alma materColumbia University (A.B. 1968)
Princeton University (PhD 1972)
Known forGeneral Relativity (1984)
Wald's formula for black-hole entropy
AwardsEinstein Prize (APS) (2017)
Scientific career
FieldsGravitational physics
InstitutionsUniversity of Maryland, College Park
University of Chicago
ThesisNonspherical gravitational collapse and black hole uniqueness (1972)
Doctoral advisorJohn Archibald Wheeler
Life and education
He is the son of the mathematician and statistician Abraham Wald and great-grandson of the chief rabbi Moshe Shmuel Glasner. Wald's parents died in a plane crash when he was three years old.[1] He earned his Bachelor's degree from Columbia University in 1968 and his PhD in physics from Princeton University in 1972,[2] under the supervision of John Archibald Wheeler. His doctoral dissertation was titled Nonspherical Gravitational Collapse and Black Hole Uniqueness.[3]
Career and contributions
Between 1972 and 1974, Robert Wald worked as a research associate in physics at the University of Maryland.[3] He then moved to the University of Chicago, spending two years as a postdoctoral fellow before joining the faculty in 1976.[4] He wanted to move to Chicago in order to work with Robert Geroch and other specialists in gravitation.[5]
In 1977, Wald published a popular-science book titled Space, Time, and Gravity: The Theory of the Big Bang and Black Holes explaining Albert Einstein's general theory of relativity, and its implications in cosmology and astrophysics. The book also gives a survey of what was then ongoing research on gravitational collapse and black holes. This book grew out of a series of lectures Wald gave as part of the Compton Lectures at the University of Chicago in the spring of 1976.[6] The Compton Lectures, given every Spring and Fall quarter, are intended to explain notable advances in the physical sciences to members of the general public.[7]
He published the textbook General Relativity in 1984. Aimed at beginning graduate students, it covers spinors, the variational principles, the initial-value formulation, (exact) gravitational waves, singularities, Penrose diagrams, Hawking radiation, and black-hole thermodynamics.[8]
Wald has taught first-year graduate courses covering a broad range of topics, including classical mechanics, quantum mechanics, statistical mechanics, and electromagnetism. He has also taught courses on general relativity, his specialty, at both introductory and advanced levels. A particularly effective teacher, he received the Graduate Teaching Award from the University of Chicago in 1997.[9]
Wald investigates black holes and their thermodynamics, and gravitational radiation-reaction (or self-force).[4] Due to quantum-mechanical processes, black holes emit particles and therefore have a definite temperature and entropy.[10] Wald has published over 100 research papers on general relativity and quantum field theory in curved spacetime, many of which have been cited by hundreds of subsequent papers.[11] In 1993, he described the Wald entropy of a black hole, which is dependent simply on the area of the event horizon of the black hole.[12]
He organized The Symposium on Black Holes and Relativistic Stars in 1996, in honor of the late Nobel Prize-winning theoretical astrophysicist Subrahmanyan Chandrasekhar. Distinguished speakers of this event included Stephen Hawking, Roger Penrose and Martin Rees. Although the event charged an entrance fee of $100, Wald made sure all University of Chicago students were admitted free of charge.[9] Chandrasekhar founded a research group on general relativity at the University of Chicago, which includes Wald, James Hartle and Robert Geroch.[13] Although Wald and Chandrasekhar never collaborated on any particular research projects, the two developed warm relations.[5]
He became a fellow of the American Physical Society (APS) in 1996 and a member of the National Academy of Sciences in 2001.[3] He received the Einstein Prize from the APS Division of Gravitational Physics in 2017 for "the discovery of the general formula for black hole entropy, and for developing a rigorous formulation of quantum field theory in curved spacetime."[4]
Wald delivered a public lecture at the University of Alabama in October 27, 2015, titled "The Formulation of General Relativity," celebrating the centennial of Einstein's theory.[14] Wald is a member of the LIGO group at the University of Chicago, headed by astrophysicist Daniel Holz. The Laser Interferometry Gravitational-wave Observatory detected gravitational waves for the first time in 2015, one century after Einstein predicted their existence.[15]
Books
• Wald, Robert M. (1992) [1977]. Space, Time, and Gravity: The Theory of the Big Bang and Black Holes (2nd ed.). Chicago: University of Chicago Press. ISBN 0-226-87029-4.
• Wald, Robert M. (1984). General Relativity. Chicago: University of Chicago Press. ISBN 0-226-87033-2.
• Wald, Robert M. (1994). Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. Chicago Lectures in Physics. Chicago: The University of Chicago Press. ISBN 0-226-87027-8.
• Wald, Robert M., ed. (1998). Black Holes and Relativistic Stars. Chicago: University of Chicago Press. ISBN 0-226-87035-9.
• Wald, Robert M. (2022). Advanced Classical Electromagnetism. Princeton: Princeton University Press. ISBN 978-0-691-22039-0.
See also
• List of contributors to general relativity
• List of books on general relativity
• Bekenstein–Hawking entropy
References
1. Morgenstern, Oskar (1951). "Abraham Wald, 1902–1950". Econometrica. Econometrica, Vol. 19, No. 4. 19 (4): 361–367. doi:10.2307/1907462. JSTOR 1907462.
2. "Alumni Sons and Daughters | Columbia College Today". Columbia College Today. Retrieved January 18, 2022.
3. "Robert M. Wald". American Institute of Physics. Retrieved July 18, 2019.
4. "2017 Einstein Prize Recipient". Division of Gravitational Physics, American Physical Society (APS). Retrieved July 18, 2019.
5. Wali, Kameshwar C., ed. (1997). "13. Some Memories of Chandra - Robert M. Wald". S. Chandrasekhar - The Man Behind the Legend. Singapore: Imperial College Press. pp. 80–85. ISBN 1-86094-038-2.
6. Moché, Dinah L. (May 1978). "Review of Space, Time, and Gravity by Robert M. Wald". Physics Teacher. 16 (5): 332. doi:10.1119/1.2339970.
7. "Arthur H. Compton Lectures". Enrico Fermi Institute, University of Chicago. Retrieved July 19, 2019.
8. A Guide to Relativity Books. John C. Baez et al. University of California, Riverside. September 1998. Accessed January 18, 2019.
9. Steele, Diana (June 12, 1997). "Graduate Teaching Award: Robert Wald". University of Chicago Chronicle. 16 (9). Retrieved 20 May 2013.
10. "Robert M. Wald". Member Directory. National Academy of Sciences. 2001. Retrieved August 16, 2019.
11. "Robert M. Wald". INSPIRE - HEP. Retrieved August 16, 2019.
12. Wald, Robert M. (1993). "Black Hole Entropy is Noether Charge". Physical Review D. 48 (8): R3427–R3431. arXiv:gr-qc/9307038. Bibcode:1993PhRvD..48.3427W. doi:10.1103/PhysRevD.48.R3427. PMID 10016675. S2CID 18398147.
13. Witten, Thomas (April 2018). "Our History. Chapter One: 1893 to 1986". Department of Physics, University of Chicago. Retrieved July 19, 2019.
14. "GR 100: Celebrating the Centennial of Einstein's Theory of General Relativity". Department of Physics and Astronomy, University of Alabama. Retrieved July 18, 2019.
15. "LIGO detects colliding black holes for third time". UChicago News. July 1, 2017. Retrieved July 19, 2019.
External links
• Robert M. Wald faculty page at the University of Chicago
• Robert Wald research articles cited by INSPIRE-HEP
• Robert Wald research articles cited by arXiv
• Some properties of Noether charge and a proposal for dynamical black hole entropy, Vivek Iyer and Robert M. Wald, Phys. Rev., D 50 (1994) 846-864 (sample research paper; cited over 250 times)
• Robert Wald at the Mathematics Genealogy Project
• Recorded Lectures Given by Robert Wald. Perimeter Institute.
• Silver Screen - PSD faculty members discuss Hollywood's portrayal of science and scientists. The University of Chicago Magazine. Summer 2015.
• The Formulation of General Relativity. Robert Wald. Physics Public Talk. Department of Physics and Astronomy, University of Alabama. October 27, 2015.
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| Wikipedia |
Wald test
In statistics, the Wald test (named after Abraham Wald) assesses constraints on statistical parameters based on the weighted distance between the unrestricted estimate and its hypothesized value under the null hypothesis, where the weight is the precision of the estimate.[1][2] Intuitively, the larger this weighted distance, the less likely it is that the constraint is true. While the finite sample distributions of Wald tests are generally unknown,[3] it has an asymptotic χ2-distribution under the null hypothesis, a fact that can be used to determine statistical significance.[4]
Together with the Lagrange multiplier test and the likelihood-ratio test, the Wald test is one of three classical approaches to hypothesis testing. An advantage of the Wald test over the other two is that it only requires the estimation of the unrestricted model, which lowers the computational burden as compared to the likelihood-ratio test. However, a major disadvantage is that (in finite samples) it is not invariant to changes in the representation of the null hypothesis; in other words, algebraically equivalent expressions of non-linear parameter restriction can lead to different values of the test statistic.[5][6] That is because the Wald statistic is derived from a Taylor expansion,[7] and different ways of writing equivalent nonlinear expressions lead to nontrivial differences in the corresponding Taylor coefficients.[8] Another aberration, known as the Hauck–Donner effect,[9] can occur in binomial models when the estimated (unconstrained) parameter is close to the boundary of the parameter space—for instance a fitted probability being extremely close to zero or one—which results in the Wald test no longer monotonically increasing in the distance between the unconstrained and constrained parameter.[10][11]
Mathematical details
Under the Wald test, the estimated ${\hat {\theta }}$ that was found as the maximizing argument of the unconstrained likelihood function is compared with a hypothesized value $\theta _{0}$. In particular, the squared difference ${\hat {\theta }}-\theta _{0}$ is weighted by the curvature of the log-likelihood function.
Test on a single parameter
If the hypothesis involves only a single parameter restriction, then the Wald statistic takes the following form:
$W={\frac {{({\widehat {\theta }}-\theta _{0})}^{2}}{\operatorname {var} ({\hat {\theta }})}}$
which under the null hypothesis follows an asymptotic χ2-distribution with one degree of freedom. The square root of the single-restriction Wald statistic can be understood as a (pseudo) t-ratio that is, however, not actually t-distributed except for the special case of linear regression with normally distributed errors.[12] In general, it follows an asymptotic z distribution.[13]
${\sqrt {W}}={\frac {{\widehat {\theta }}-\theta _{0}}{\operatorname {se} ({\hat {\theta }})}}$
where $\operatorname {se} ({\widehat {\theta }})$ is the standard error of the maximum likelihood estimate (MLE), the square root of the variance. There are several ways to consistently estimate the variance matrix which in finite samples leads to alternative estimates of standard errors and associated test statistics and p-values.[14]
Test(s) on multiple parameters
The Wald test can be used to test a single hypothesis on multiple parameters, as well as to test jointly multiple hypotheses on single/multiple parameters. Let ${\hat {\theta }}_{n}$ be our sample estimator of P parameters (i.e., ${\hat {\theta }}_{n}$ is a $P\times 1$ vector), which is supposed to follow asymptotically a normal distribution with covariance matrix V, ${\sqrt {n}}({\hat {\theta }}_{n}-\theta )\,\xrightarrow {\mathcal {D}} \,N(0,V)$. The test of Q hypotheses on the P parameters is expressed with a $Q\times P$ matrix R:
$H_{0}:R\theta =r$
$H_{1}:R\theta \neq r$
The distribution of the test statistic under the null hypothesis is
$(R{\hat {\theta }}_{n}-r)'[R({\hat {V}}_{n}/n)R']^{-1}(R{\hat {\theta }}_{n}-r)/Q\quad \xrightarrow {\mathcal {D}} \quad F(Q,n-P)\quad {\xrightarrow[{n\rightarrow \infty }]{\mathcal {D}}}\quad \chi _{Q}^{2}/Q,$
which in turn implies
$(R{\hat {\theta }}_{n}-r)'[R({\hat {V}}_{n}/n)R']^{-1}(R{\hat {\theta }}_{n}-r)\quad {\xrightarrow[{n\rightarrow \infty }]{\mathcal {D}}}\quad \chi _{Q}^{2},$
where ${\hat {V}}_{n}$ is an estimator of the covariance matrix.[15]
Proof
Suppose ${\sqrt {n}}({\hat {\theta }}_{n}-\theta )\,\xrightarrow {\mathcal {D}} \,N(0,V)$. Then, by Slutsky's theorem and by the properties of the normal distribution, multiplying by R has distribution:
$R{\sqrt {n}}({\hat {\theta }}_{n}-\theta )={\sqrt {n}}(R{\hat {\theta }}_{n}-r)\,\xrightarrow {\mathcal {D}} \,N(0,RVR')$
Recalling that a quadratic form of normal distribution has a Chi-squared distribution:
${\sqrt {n}}(R{\hat {\theta }}_{n}-r)'[RVR']^{-1}{\sqrt {n}}(R{\hat {\theta }}_{n}-r)\,\xrightarrow {\mathcal {D}} \,\chi _{Q}^{2}$
Rearranging n finally gives:
$(R{\hat {\theta }}_{n}-r)'[R(V/n)R']^{-1}(R{\hat {\theta }}_{n}-r)\quad \xrightarrow {\mathcal {D}} \quad \chi _{Q}^{2}$
What if the covariance matrix is not known a-priori and needs to be estimated from the data? If we have a consistent estimator ${\hat {V}}_{n}$ of $V$ such that $V^{-1}{\hat {V}}_{n}$ has a determinant that is distributed $\chi _{n-P}^{2}$, then by the independence of the covariance estimator and equation above, we have:
$(R{\hat {\theta }}_{n}-r)'[R({\hat {V}}_{n}/n)R']^{-1}(R{\hat {\theta }}_{n}-r)/Q\quad \xrightarrow {\mathcal {D}} \quad F(Q,n-P)$
Nonlinear hypothesis
In the standard form, the Wald test is used to test linear hypotheses that can be represented by a single matrix R. If one wishes to test a non-linear hypothesis of the form:
$H_{0}:c(\theta )=0$
$H_{1}:c(\theta )\neq 0$
The test statistic becomes:
$c\left({\hat {\theta }}_{n}\right)'\left[c'\left({\hat {\theta }}_{n}\right)\left({\hat {V}}_{n}/n\right)c'\left({\hat {\theta }}_{n}\right)'\right]^{-1}c\left({\hat {\theta }}_{n}\right)\quad {\xrightarrow {\mathcal {D}}}\quad \chi _{Q}^{2}$
where $c'({\hat {\theta }}_{n})$ is the derivative of c evaluated at the sample estimator. This result is obtained using the delta method, which uses a first order approximation of the variance.
Non-invariance to re-parameterisations
The fact that one uses an approximation of the variance has the drawback that the Wald statistic is not-invariant to a non-linear transformation/reparametrisation of the hypothesis: it can give different answers to the same question, depending on how the question is phrased.[16][5] For example, asking whether R = 1 is the same as asking whether log R = 0; but the Wald statistic for R = 1 is not the same as the Wald statistic for log R = 0 (because there is in general no neat relationship between the standard errors of R and log R, so it needs to be approximated).[17]
Alternatives to the Wald test
There exist several alternatives to the Wald test, namely the likelihood-ratio test and the Lagrange multiplier test (also known as the score test). Robert F. Engle showed that these three tests, the Wald test, the likelihood-ratio test and the Lagrange multiplier test are asymptotically equivalent.[18] Although they are asymptotically equivalent, in finite samples, they could disagree enough to lead to different conclusions.
There are several reasons to prefer the likelihood ratio test or the Lagrange multiplier to the Wald test:[19][20][21]
• Non-invariance: As argued above, the Wald test is not invariant under reparametrization, while the likelihood ratio tests will give exactly the same answer whether we work with R, log R or any other monotonic transformation of R.[5]
• The other reason is that the Wald test uses two approximations (that we know the standard error or Fisher information and the maximum likelihood estimate), whereas the likelihood ratio test depends only on the ratio of likelihood functions under the null hypothesis and alternative hypothesis.
• The Wald test requires an estimate using the maximizing argument, corresponding to the "full" model. In some cases, the model is simpler under the null hypothesis, so that one might prefer to use the score test (also called Lagrange multiplier test), which has the advantage that it can be formulated in situations where the variability of the maximizing element is difficult to estimate or computing the estimate according to the maximum likelihood estimator is difficult; e.g. the Cochran–Mantel–Haenzel test is a score test.[22]
See also
• Chow test
• Sequential probability ratio test
• Sup-Wald test
• Student's t-test
• Welch's t-test
References
1. Fahrmeir, Ludwig; Kneib, Thomas; Lang, Stefan; Marx, Brian (2013). Regression : Models, Methods and Applications. Berlin: Springer. p. 663. ISBN 978-3-642-34332-2.
2. Ward, Michael D.; Ahlquist, John S. (2018). Maximum Likelihood for Social Science : Strategies for Analysis. Cambridge University Press. p. 36. ISBN 978-1-316-63682-4.
3. Martin, Vance; Hurn, Stan; Harris, David (2013). Econometric Modelling with Time Series: Specification, Estimation and Testing. Cambridge University Press. p. 138. ISBN 978-0-521-13981-6.
4. Davidson, Russell; MacKinnon, James G. (1993). "The Method of Maximum Likelihood : Fundamental Concepts and Notation". Estimation and Inference in Econometrics. New York: Oxford University Press. p. 89. ISBN 0-19-506011-3.
5. Gregory, Allan W.; Veall, Michael R. (1985). "Formulating Wald Tests of Nonlinear Restrictions". Econometrica. 53 (6): 1465–1468. doi:10.2307/1913221. JSTOR 1913221.
6. Phillips, P. C. B.; Park, Joon Y. (1988). "On the Formulation of Wald Tests of Nonlinear Restrictions" (PDF). Econometrica. 56 (5): 1065–1083. doi:10.2307/1911359. JSTOR 1911359.
7. Hayashi, Fumio (2000). Econometrics. Princeton: Princeton University Press. pp. 489–491. ISBN 1-4008-2383-8.,
8. Lafontaine, Francine; White, Kenneth J. (1986). "Obtaining Any Wald Statistic You Want". Economics Letters. 21 (1): 35–40. doi:10.1016/0165-1765(86)90117-5.
9. Hauck, Walter W. Jr.; Donner, Allan (1977). "Wald's Test as Applied to Hypotheses in Logit Analysis". Journal of the American Statistical Association. 72 (360a): 851–853. doi:10.1080/01621459.1977.10479969.
10. King, Maxwell L.; Goh, Kim-Leng (2002). "Improvements to the Wald Test". Handbook of Applied Econometrics and Statistical Inference. New York: Marcel Dekker. pp. 251–276. ISBN 0-8247-0652-8.
11. Yee, Thomas William (2022). "On the Hauck–Donner Effect in Wald Tests: Detection, Tipping Points, and Parameter Space Characterization". Journal of the American Statistical Association. 117 (540): 1763–1774. arXiv:2001.08431. doi:10.1080/01621459.2021.1886936.
12. Cameron, A. Colin; Trivedi, Pravin K. (2005). Microeconometrics : Methods and Applications. New York: Cambridge University Press. p. 137. ISBN 0-521-84805-9.
13. Davidson, Russell; MacKinnon, James G. (1993). "The Method of Maximum Likelihood : Fundamental Concepts and Notation". Estimation and Inference in Econometrics. New York: Oxford University Press. p. 89. ISBN 0-19-506011-3.
14. Martin, Vance; Hurn, Stan; Harris, David (2013). Econometric Modelling with Time Series : Specification, Estimation and Testing. New York: Cambridge University Press. p. 129. ISBN 978-0-521-13981-6.
15. Harrell, Frank E. Jr. (2001). "Section 9.3.1". Regression modeling strategies. New York: Springer-Verlag. ISBN 0387952322.
16. Fears, Thomas R.; Benichou, Jacques; Gail, Mitchell H. (1996). "A reminder of the fallibility of the Wald statistic". The American Statistician. 50 (3): 226–227. doi:10.1080/00031305.1996.10474384.
17. Critchley, Frank; Marriott, Paul; Salmon, Mark (1996). "On the Differential Geometry of the Wald Test with Nonlinear Restrictions". Econometrica. 64 (5): 1213–1222. doi:10.2307/2171963. hdl:1814/524. JSTOR 2171963.
18. Engle, Robert F. (1983). "Wald, Likelihood Ratio, and Lagrange Multiplier Tests in Econometrics". In Intriligator, M. D.; Griliches, Z. (eds.). Handbook of Econometrics. Vol. II. Elsevier. pp. 796–801. ISBN 978-0-444-86185-6.
19. Harrell, Frank E. Jr. (2001). "Section 9.3.3". Regression modeling strategies. New York: Springer-Verlag. ISBN 0387952322.
20. Collett, David (1994). Modelling Survival Data in Medical Research. London: Chapman & Hall. ISBN 0412448807.
21. Pawitan, Yudi (2001). In All Likelihood. New York: Oxford University Press. ISBN 0198507658.
22. Agresti, Alan (2002). Categorical Data Analysis (2nd ed.). Wiley. p. 232. ISBN 0471360937.
Further reading
• Greene, William H. (2012). Econometric Analysis (Seventh international ed.). Boston: Pearson. pp. 155–161. ISBN 978-0-273-75356-8.
• Kmenta, Jan (1986). Elements of Econometrics (Second ed.). New York: Macmillan. pp. 492–493. ISBN 0-02-365070-2.
• Thomas, R. L. (1993). Introductory Econometrics: Theory and Application (Second ed.). London: Longman. pp. 73–77. ISBN 0-582-07378-2.
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| Wikipedia |
Waldhausen category
In mathematics, a Waldhausen category is a category C equipped with some additional data, which makes it possible to construct the K-theory spectrum of C using a so-called S-construction. It's named after Friedhelm Waldhausen, who introduced this notion (under the term category with cofibrations and weak equivalences) to extend the methods of algebraic K-theory to categories not necessarily of algebraic origin, for example the category of topological spaces.
Definition
Let C be a category, co(C) and we(C) two classes of morphisms in C, called cofibrations and weak equivalences respectively. The triple (C, co(C), we(C)) is called a Waldhausen category if it satisfies the following axioms, motivated by the similar properties for the notions of cofibrations and weak homotopy equivalences of topological spaces:
• C has a zero object, denoted by 0;
• isomorphisms are included in both co(C) and we(C);
• co(C) and we(C) are closed under composition;
• for each object A ∈ C the unique map 0 → A is a cofibration, i.e. is an element of co(C);
• co(C) and we(C) are compatible with pushouts in a certain sense.
For example, if $\scriptstyle A\,\rightarrowtail \,B$ is a cofibration and $\scriptstyle A\,\to \,C$ is any map, then there must exist a pushout $\scriptstyle B\,\cup _{A}\,C$, and the natural map $\scriptstyle C\,\rightarrowtail \,B\,\cup _{A}\,C$ should be cofibration:
Relations with other notions
In algebraic K-theory and homotopy theory there are several notions of categories equipped with some specified classes of morphisms. If C has a structure of an exact category, then by defining we(C) to be isomorphisms, co(C) to be admissible monomorphisms, one obtains a structure of a Waldhausen category on C. Both kinds of structure may be used to define K-theory of C, using the Q-construction for an exact structure and S-construction for a Waldhausen structure. An important fact is that the resulting K-theory spaces are homotopy equivalent.
If C is a model category with a zero object, then the full subcategory of cofibrant objects in C may be given a Waldhausen structure.
S-construction
The Waldhausen S-construction produces from a Waldhausen category C a sequence of Kan complexes $S_{n}(C)$, which forms a spectrum. Let $K(C)$ denote the loop space of the geometric realization $|S_{*}(C)|$ of $S_{*}(C)$. Then the group
$\pi _{n}K(C)=\pi _{n+1}|S_{*}(C)|$
is the n-th K-group of C. Thus, it gives a way to define higher K-groups. Another approach for higher K-theory is Quillen's Q-construction.
The construction is due to Friedhelm Waldhausen.
biWaldhausen categories
A category C is equipped with bifibrations if it has cofibrations and its opposite category COP has so also. In that case, we denote the fibrations of COP by quot(C). In that case, C is a biWaldhausen category if C has bifibrations and weak equivalences such that both (C, co(C), we) and (COP, quot(C), weOP) are Waldhausen categories.
Waldhausen and biWaldhausen categories are linked with algebraic K-theory. There, many interesting categories are complicial biWaldhausen categories. For example: The category $\scriptstyle C^{b}({\mathcal {A}})$ of bounded chain complexes on an exact category $\scriptstyle {\mathcal {A}}$. The category $\scriptstyle S_{n}{\mathcal {C}}$ of functors $\scriptstyle \operatorname {Ar} (\Delta ^{n})\,\to \,{\mathcal {C}}$ when $\scriptstyle {\mathcal {C}}$ is so. And given a diagram $\scriptstyle I$, then $\scriptstyle {\mathcal {C}}^{I}$ is a nice complicial biWaldhausen category when $\scriptstyle {\mathcal {C}}$ is.
References
• Waldhausen, Friedhelm (1985), "Algebraic K-theory of spaces", Algebraic and geometric topology (New Brunswick, N.J., 1983 (PDF), Lecture Notes in Mathematics, vol. 1126, Berlin: Springer, pp. 318–419, doi:10.1007/BFb0074449, ISBN 978-3-540-15235-4, MR 0802796
• C. Weibel, The K-book, an introduction to algebraic K-theory — http://www.math.rutgers.edu/~weibel/Kbook.html
• G. Garkusha, Systems of Diagram Categories and K-theory — https://arxiv.org/abs/math/0401062
• Sagave, S. (2004). "On the algebraic K-theory of model categories". Journal of Pure and Applied Algebra. 190 (1–3): 329–340. doi:10.1016/j.jpaa.2003.11.002.
• Lurie, Jacob, Higher K-Theory of ∞-Categories (Lecture 16) (PDF)
See also
• Complete Segal space
External links
• "Waldhausen S-construction". nLab.
| Wikipedia |
Waldspurger formula
In representation theory of mathematics, the Waldspurger formula relates the special values of two L-functions of two related admissible irreducible representations. Let k be the base field, f be an automorphic form over k, π be the representation associated via the Jacquet–Langlands correspondence with f. Goro Shimura (1976) proved this formula, when $k=\mathbb {Q} $ and f is a cusp form; Günter Harder made the same discovery at the same time in an unpublished paper. Marie-France Vignéras (1980) proved this formula, when $k=\mathbb {Q} $ and f is a newform. Jean-Loup Waldspurger, for whom the formula is named, reproved and generalized the result of Vignéras in 1985 via a totally different method which was widely used thereafter by mathematicians to prove similar formulas.
Statement
Let $k$ be a number field, $\mathbb {A} $ be its adele ring, $k^{\times }$ be the subgroup of invertible elements of $k$, $\mathbb {A} ^{\times }$ be the subgroup of the invertible elements of $\mathbb {A} $, $\chi ,\chi _{1},\chi _{2}$ be three quadratic characters over $\mathbb {A} ^{\times }/k^{\times }$, $G=SL_{2}(k)$, ${\mathcal {A}}(G)$ be the space of all cusp forms over $G(k)\backslash G(\mathbb {A} )$, ${\mathcal {H}}$ be the Hecke algebra of $G(\mathbb {A} )$. Assume that, $\pi $ is an admissible irreducible representation from $G(\mathbb {A} )$ to ${\mathcal {A}}(G)$, the central character of π is trivial, $\pi _{\nu }\sim \pi [h_{\nu }]$ when $\nu $ is an archimedean place, ${A}$ is a subspace of ${{\mathcal {A}}(G)}$ such that $\pi |_{\mathcal {H}}:{\mathcal {H}}\to A$. We suppose further that, $\varepsilon (\pi \otimes \chi ,1/2)$ is the Langlands $\varepsilon $-constant [ (Langlands 1970); (Deligne 1972) ] associated to $\pi $ and $\chi $ at $s=1/2$. There is a ${\gamma \in k^{\times }}$ such that $k(\chi )=k({\sqrt {\gamma }})$.
Definition 1. The Legendre symbol $\left({\frac {\chi }{\pi }}\right)=\varepsilon (\pi \otimes \chi ,1/2)\cdot \varepsilon (\pi ,1/2)\cdot \chi (-1).$
• Comment. Because all the terms in the right either have value +1, or have value −1, the term in the left can only take value in the set {+1, −1}.
Definition 2. Let ${D_{\chi }}$ be the discriminant of $\chi $.
$p(\chi )=D_{\chi }^{1/2}\sum _{\nu {\text{ archimedean}}}\left\vert \gamma _{\nu }\right\vert _{\nu }^{h_{\nu }/2}.$
Definition 3. Let $f_{0},f_{1}\in A$. $b(f_{0},f_{1})=\int _{x\in k^{\times }}f_{0}(x)\cdot {\overline {f_{1}(x)}}\,dx.$
Definition 4. Let ${T}$ be a maximal torus of ${G}$, ${Z}$ be the center of ${G}$, $\varphi \in A$.
$\beta (\varphi ,T)=\int _{t\in Z\backslash T}b(\pi (t)\varphi ,\varphi )\,dt.$
• Comment. It is not obvious though, that the function $\beta $ is a generalization of the Gauss sum.
Let $K$ be a field such that $k(\pi )\subset K\subset \mathbb {C} $. One can choose a K-subspace${A^{0}}$ of $A$ such that (i) $A=A^{0}\otimes _{K}\mathbb {C} $; (ii) $(A^{0})^{\pi (G)}=A^{0}$. De facto, there is only one such $A^{0}$ modulo homothety. Let $T_{1},T_{2}$ be two maximal tori of $G$ such that $\chi _{T_{1}}=\chi _{1}$ and $\chi _{T_{2}}=\chi _{2}$. We can choose two elements $\varphi _{1},\varphi _{2}$ of $A^{0}$ such that $\beta (\varphi _{1},T_{1})\neq 0$ and $\beta (\varphi _{2},T_{2})\neq 0$.
Definition 5. Let $D_{1},D_{2}$ be the discriminants of $\chi _{1},\chi _{2}$.
$p(\pi ,\chi _{1},\chi _{2})=D_{1}^{-1/2}D_{2}^{1/2}L(\chi _{1},1)^{-1}L(\chi _{2},1)L(\pi \otimes \chi _{1},1/2)L(\pi \otimes \chi _{2},1/2)^{-1}\beta (\varphi _{1},T_{1})^{-1}\beta (\varphi _{2},T_{2}).$
• Comment. When the $\chi _{1}=\chi _{2}$, the right hand side of Definition 5 becomes trivial.
We take $\Sigma _{f}$ to be the set {all the finite $k$-places $\nu \mid \ \pi _{\nu }$ doesn't map non-zero vectors invariant under the action of ${GL_{2}(k_{\nu })}$ to zero}, ${\Sigma _{s}}$ to be the set of (all $k$-places $\nu \mid \nu $ is real, or finite and special).
Theorem [1] — Let $k=\mathbb {Q} $. We assume that, (i) $L(\pi \otimes \chi _{2},1/2)\neq 0$; (ii) for $\nu \in \Sigma _{s}$, $\left({\frac {\chi _{1,\nu }}{\pi _{\nu }}}\right)=\left({\frac {\chi _{2,\nu }}{\pi _{\nu }}}\right)$ . Then, there is a constant ${q\in \mathbb {Q} (\pi )}$ such that
$L(\pi \otimes \chi _{1},1/2)L(\pi \otimes \chi _{2},1/2)^{-1}=qp(\chi _{1})p(\chi _{2})^{-1}\prod _{\nu \in \Sigma _{f}}p(\pi _{\nu },\chi _{1,\nu },\chi _{2,\nu })$
Comments:
1. The formula in the theorem is the well-known Waldspurger formula. It is of global-local nature, in the left with a global part, in the right with a local part. By 2017, mathematicians often call it the classic Waldspurger's formula.
2. It is worthwhile to notice that, when the two characters are equal, the formula can be greatly simplified.
3. [ (Waldspurger 1985), Thm 6, p. 241 ] When one of the two characters is ${1}$, Waldspurger's formula becomes much more simple. Without loss of generality, we can assume that, $\chi _{1}=\chi $ and $\chi _{2}=1$. Then, there is an element ${q\in \mathbb {Q} (\pi )}$ such that $L(\pi \otimes \chi ,1/2)L(\pi ,1/2)^{-1}=qD_{\chi }^{1/2}.$
The case when Fp(T) and φ is a metaplectic cusp form
Let p be prime number, $\mathbb {F} _{p}$ be the field with p elements, $R=\mathbb {F} _{p}[T],k=\mathbb {F} _{p}(T),k_{\infty }=\mathbb {F} _{p}((T^{-1})),o_{\infty }$ be the integer ring of $k_{\infty },{\mathcal {H}}=PGL_{2}(k_{\infty })/PGL_{2}(o_{\infty }),\Gamma =PGL_{2}(R)$. Assume that, $N,D\in R$, D is squarefree of even degree and coprime to N, the prime factorization of $N$ is $ \prod _{\ell }\ell ^{\alpha _{\ell }}$. We take $\Gamma _{0}(N)$ to the set $ \left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in \Gamma \mid c\equiv 0{\bmod {N}}\right\},$ $S_{0}(\Gamma _{0}(N))$ to be the set of all cusp forms of level N and depth 0. Suppose that, $\varphi ,\varphi _{1},\varphi _{2}\in S_{0}(\Gamma _{0}(N))$.
Definition 1. Let $\left({\frac {c}{d}}\right)$ be the Legendre symbol of c modulo d, ${\widetilde {SL}}_{2}(k_{\infty })=Mp_{2}(k_{\infty })$. Metaplectic morphism
$\eta :SL_{2}(R)\to {\widetilde {SL}}_{2}(k_{\infty }),{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\mapsto \left({\begin{pmatrix}a&b\\c&d\end{pmatrix}},\left({\frac {c}{d}}\right)\right).$
Definition 2. Let $z=x+iy\in {\mathcal {H}},d\mu ={\frac {dx\,dy}{\left\vert y\right\vert ^{2}}}$. Petersson inner product
$\langle \varphi _{1},\varphi _{2}\rangle =[\Gamma :\Gamma _{0}(N)]^{-1}\int _{\Gamma _{0}(N)\backslash {\mathcal {H}}}\varphi _{1}(z){\overline {\varphi _{2}(z)}}\,d\mu .$ :\Gamma _{0}(N)]^{-1}\int _{\Gamma _{0}(N)\backslash {\mathcal {H}}}\varphi _{1}(z){\overline {\varphi _{2}(z)}}\,d\mu .}
Definition 3. Let $n,P\in R$. Gauss sum
$G_{n}(P)=\sum _{r\in R/PR}\left({\frac {r}{P}}\right)e(rnT^{2}).$
Let $\lambda _{\infty ,\varphi }$ be the Laplace eigenvalue of $\varphi $. There is a constant $\theta \in \mathbb {R} $ such that $\lambda _{\infty ,\varphi }={\frac {e^{-i\theta }+e^{i\theta }}{\sqrt {p}}}.$
Definition 4. Assume that $v_{\infty }(a/b)=\deg(a)-\deg(b),\nu =v_{\infty }(y)$. Whittaker function
$W_{0,i\theta }(y)={\begin{cases}{\frac {\sqrt {p}}{e^{i\theta }-e^{-i\theta }}}\left[\left({\frac {e^{i\theta }}{\sqrt {p}}}\right)^{\nu -1}-\left({\frac {e^{-i\theta }}{\sqrt {p}}}\right)^{\nu -1}\right],&{\text{when }}\nu \geq 2;\\0,&{\text{otherwise}}.\end{cases}}$
Definition 5. Fourier–Whittaker expansion
$\varphi (z)=\sum _{r\in R}\omega _{\varphi }(r)e(rxT^{2})W_{0,i\theta }(y).$
One calls $\omega _{\varphi }(r)$ the Fourier–Whittaker coefficients of $\varphi $. Definition 6. Atkin–Lehner operator
$W_{\alpha _{\ell }}={\begin{pmatrix}\ell ^{\alpha _{\ell }}&b\\N&\ell ^{\alpha _{\ell }}d\end{pmatrix}}$
with $\ell ^{2\alpha _{\ell }}d-bN=\ell ^{\alpha _{\ell }}.$ Definition 7. Assume that, $\varphi $ is a Hecke eigenform. Atkin–Lehner eigenvalue
$w_{\alpha _{\ell },\varphi }={\frac {\varphi (W_{\alpha _{\ell }}z)}{\varphi (z)}}$
with $w_{\alpha _{\ell },\varphi }=\pm 1.$ Definition 8.
$L(\varphi ,s)=\sum _{r\in R\backslash \{0\}}{\frac {\omega _{\varphi }(r)}{\left\vert r\right\vert _{p}^{s}}}.$
Let ${\widetilde {S}}_{0}({\widetilde {\Gamma }}_{0}(N))$ be the metaplectic version of $S_{0}(\Gamma _{0}(N))$, $\{E_{1},\ldots ,E_{d}\}$ be a nice Hecke eigenbasis for ${\widetilde {S}}_{0}({\widetilde {\Gamma }}_{0}(N))$ with respect to the Petersson inner product. We note the Shimura correspondence by $\operatorname {Sh} .$
Theorem [ (Altug & Tsimerman 2010), Thm 5.1, p. 60 ]. Suppose that $ K_{\varphi }={\frac {1}{{\sqrt {p}}\left({\sqrt {p}}-e^{-i\theta }\right)\left({\sqrt {p}}-e^{i\theta }\right)}}$, $\chi _{D}$ is a quadratic character with $\Delta (\chi _{D})=D$. Then
$\sum _{\operatorname {Sh} (E_{i})=\varphi }\left\vert \omega _{E_{i}}(D)\right\vert _{p}^{2}={\frac {K_{\varphi }G_{1}(D)\left\vert D\right\vert _{p}^{-3/2}}{\langle \varphi ,\varphi \rangle }}L(\varphi \otimes \chi _{D},1/2)\prod _{\ell }\left(1+\left({\frac {\ell ^{\alpha _{\ell }}}{D}}\right)w_{\alpha _{\ell },\varphi }\right).$
References
1. (Waldspurger 1985), Thm 4, p. 235
• Waldspurger, Jean-Loup (1985), "Sur les valeurs de certaines L-fonctions automorphes en leur centre de symétrie", Compositio Mathematica, 54 (2): 173–242
• Vignéras, Marie-France (1981), "Valeur au centre de symétrie des fonctions L associées aux formes modulaire", Séminarie de Théorie des Nombres, Paris 1979–1980, Progress in Math., Birkhäuser, pp. 331–356
• Shimura, Gorô (1976), "On special values of zeta functions associated with cusp forms", Communications on Pure and Applied Mathematics, 29: 783–804, doi:10.1002/cpa.3160290618
• Altug, Salim Ali; Tsimerman, Jacob (2010). "Metaplectic Ramanujan conjecture over function fields with applications to quadratic forms". International Mathematics Research Notices. arXiv:1008.0430. doi:10.1093/imrn/rnt047. S2CID 119121964.
• Langlands, Robert (1970). On the Functional Equation of the Artin L-Functions.
• Deligne, Pierre (1972). "Les constantes des équations fonctionelle des fonctions L". Modular Functions of One Variable II. International Summer School on Modular functions. Antwerp. pp. 501–597.
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Waldspurger's theorem
In mathematics, Waldspurger's theorem, introduced by Jean-Loup Waldspurger (1981), is a result that identifies Fourier coefficients of modular forms of half-integral weight k+1/2 with the value of an L-series at s=k/2.
References
• Waldspurger, Jean-Loup (1981), "Sur les coefficients de Fourier des formes modulaires de poids demi-entier", Journal de Mathématiques Pures et Appliquées, Neuvième Série, 60 (4): 375–484, ISSN 0021-7824, MR 0646366
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Waleed Al-Salam
Waleed Al-Salam (born 15 July 1926 in Baghdad, Iraq – died 14 April 1996 in Edmonton, Canada) was a mathematician who introduced Al-Salam–Chihara polynomials, Al-Salam–Carlitz polynomials, q-Konhauser polynomials, and Al-Salam–Ismail polynomials. He was a Professor Emeritus at the University of Alberta.
Born in Iraq, Baghdad, Al-Salam received his bachelor's degree in engineering physics (1950) and M.A. in mathematics (1951) from University of California Berkeley. He completed his education at Duke, receiving his Ph.D. for his dissertation on Bessel polynomials (1958).[1]
References
1. Chiahara, T.S. (November 1998). "In Memoriam: Waleed Al-Salam". Journal of Approximation Theory. doi:10.1006/jath.1998.3301. Retrieved 2016-01-29.
• Chihara, Theodore Seio; Ismail, Mourad E. H. (1998), "In memoriam: Waleed Al-Salam (July 15, 1926–April 13, 1996)", Journal of Approximation Theory, 95 (2): 153–160, doi:10.1006/jath.1998.3301, ISSN 0021-9045, MR 1652868
• Ismail, Mourad E. H. (1999), "Waleed Al-Salam, 1926–1996", in Diejen, Jan Felipe van; Vinet, Luc (eds.), Algebraic methods and q-special functions. Proceedings of the workshop dedicated to the memory of Waleed Al-Salam held at the Université de Montréal, Montréal, QC, May 21–26, 1996, CRM Proc. Lecture Notes, vol. 22, Providence, R.I.: American Mathematical Society, pp. ix–xi, ISBN 978-0-8218-2026-1, MR 1726825
External links
• Waleed Al-Salam 1926-1996
• Waleed Al-Salam at the Mathematics Genealogy Project
Authority control
International
• ISNI
• VIAF
National
• Germany
Academics
• MathSciNet
• 2
• Mathematics Genealogy Project
• zbMATH
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Walk-on-spheres method
In mathematics, the walk-on-spheres method (WoS) is a numerical probabilistic algorithm, or Monte-Carlo method, used mainly in order to approximate the solutions of some specific boundary value problem for partial differential equations (PDEs).[1][2] The WoS method was first introduced by Mervin E. Muller in 1956 to solve Laplace's equation,[1] and was since then generalized to other problems.
It relies on probabilistic interpretations of PDEs, and simulates paths of Brownian motion (or for some more general variants, diffusion processes), by sampling only the exit-points out of successive spheres, rather than simulating in detail the path of the process. This often makes it less costly than "grid-based" algorithms, and it is today one of the most widely used "grid-free" algorithms for generating Brownian paths.
Informal description
Let $\Omega $ be a bounded domain in $\mathbb {R} ^{d}$ with a sufficiently regular boundary $\Gamma $, let h be a function on $\Gamma $, and let $x$ be a point inside $\Omega $.
Consider the Dirichlet problem:
${\begin{cases}\Delta u(x)=0&{\mbox{if }}x\in \Omega \\u(x)=h(x)&{\mbox{if }}x\in \Gamma .\end{cases}}$
It can be easily shown[lower-alpha 1] that when the solution $u$ exists, for $x\in \Omega $:
$u(x)=\mathbb {E} _{x}[h(W_{\tau })]$
where W is a d-dimensional Wiener process, the expected value is taken conditionally on {W0 = x}, and τ is the first-exit time out of Ω.
To compute a solution using this formula, we only have to simulate the first exit point of independent Brownian paths since with the law of large numbers:
$\mathbb {E} _{x}[h(W_{\tau })]\sim {\frac {1}{n}}\sum _{i=1}^{n}h(W_{\tau }^{i})$
The WoS method provides an efficient way of sampling the first exit point of a Brownian motion from the domain, by remarking that for any (d − 1)-sphere centred on x, the first point of exit of W out of the sphere has a uniform distribution over its surface. Thus, it starts with x(0) equal to x, and draws the largest sphere ${\mathcal {S}}_{0}$ centered on x(0) and contained inside the domain. The first point of exit x(1) from ${\mathcal {S}}_{0}$ is uniformly distributed on its surface. By repeating this step inductively, the WoS provides a sequence (x(n)) of positions of the Brownian motion.
According to intuition, the process will converge to the first exit point of the domain. However, this algorithm takes almost surely an infinite number of steps to end. For computational implementation, the process is usually stopped when it gets sufficiently close to the border, and returns the projection of the process on the border. This procedure is sometimes called introducing an $\varepsilon $-shell, or $\varepsilon $-layer.[4]
Formulation of the method
Choose $\varepsilon >0$. Using the same notations as above, the Walk-on-spheres algorithm is described as follows:
1. Initialize : $x^{(0)}=x$
2. While $d(x^{(n)},\Gamma )>\varepsilon $:
1. Set $r_{n}=d(x^{(n)},\Gamma )$.
2. Sample $\gamma _{n}$ a vector uniformly distributed over the unit sphere, independently from the preceding ones.
3. Set $x^{(n+1)}:=x^{(n)}+r_{n}\gamma _{n}$
3. When $d(x^{(n)},\Gamma )\leq \varepsilon $:
4. $x_{f}:=p_{\Gamma }(x^{(n)})$, the orthogonal projection of $x^{(n)}$ on $\Gamma $
5. Return $x_{f}$
The result $x_{f}$ is an estimator of the first exit point from $\Omega $ of a Wiener process starting from $x$, in the sense that they have close probability distributions (see below for comments on the error).
Comments and practical considerations
Radius of the spheres
In some cases the distance to the border might be difficult to compute, and it is then preferable to replace the radius of the sphere by a lower bound of this distance. One must then ensure that the radius of the spheres stays large enough so that the process reaches the border.[1]
Bias in the method and GFFP
As it is a Monte-Carlo method, the error of the estimator can be decomposed into the sum of a bias, and a statistical error. The statistical error is reduced by increasing the number of paths sampled, or by using variance reduction methods.
The WoS theoretically provides exact (or unbiased) simulations of the paths of the Brownian motion. However, as it is formulated here, the $\varepsilon $-shell introduced to ensure that the algorithm terminates also adds an error, usually of order ${\mathcal {O}}(\varepsilon )$.[4] This error has been studied, and can be avoided in some geometries by using Green's Functions First Passage method:[5] one can change the geometry of the "spheres" when close enough to the border, so that the probability of reaching the border in one step becomes positive. This requires the knowledge of Green's functions for the specific domains. (see also Harmonic measure)
When it is possible to use it, the Green's function first-passage (GFFP) method is usually preferred, as it is both faster and more accurate than the classical WoS.[4]
Complexity
It can be shown that the number of steps taken for the WoS process to reach the $\varepsilon $-shell is of order ${\mathcal {O}}(|\log(\varepsilon )|)$.[2] Therefore, one can increase the precision to a certain extent without making the number of steps grow notably.
As it is commonly the case for Monte-Carlo methods, this algorithm performs particularly well when the dimension is higher than $3$, and one only needs a small set of values. Indeed, the computational cost increases linearly with the dimension, whereas the cost of grid dependant methods increase exponentially with the dimension.[2]
Variants, extensions
This method has been largely studied, generalized and improved. For example, it is now extensively used for the computation of physical properties of materials (such as capacitance, electrostatic internal energy of molecules, etc.). Some notable extensions include:
Elliptic equations
The WoS method can be modified to solve more general problems. In particular, the method has been generalized to solve Dirichlet problems for equations of the form $\Delta u=cu+f$ [6] (which include the Poisson and linearized Poisson−Boltzmann equations) or for any elliptic partial differential equation with constant coefficients.[7]
More efficient ways of solving the linearized Poisson–Boltzmann equation have also been developed, relying on Feynman−Kac representations of the solutions.[8]
Time dependency
Again, within a regular enough border, it possible to use the WoS method to solve the following problem :
${\begin{cases}\partial _{t}u(x,t)+{\frac {1}{2}}\Delta _{x}u(x,t)=0&{\mbox{if }}x\in \Omega {\mbox{ and }}t<T\\u(x,T)=h(x,T)&{\mbox{if }}x\in {\bar {\Omega }}\\u(x,t)=h(x,t)&{\mbox{if }}x\in \Gamma .\end{cases}}$
of which the solution can be represented as:[9]
$u(x,t)=\mathbb {E} _{t,x}(h(X_{T\wedge \tau },T\wedge \tau ))$,
where the expectation is taken conditionally on $X_{t}=x$
To use the WoS through this formula, one needs to sample the exit-time from each sphere drawn, which is an independent variable $\tau _{0}$ with Laplace transform (for a sphere of radius $R$):[10]
$\mathbb {E} (\exp(-s\tau _{0}))={\frac {R{\sqrt {2s}}}{\sinh(R{\sqrt {2s}})}}$
The total time of exit from the domain $\tau $ can be computed as the sum of the exit-times from the spheres. The process also has to be stopped when it has not exited the domain at time $T$.
Other extensions
The WoS method has been generalized to estimate the solution to elliptic partial differential equations everywhere in a domain, rather than at a single point.[11]
The WoS method has also been generalized in order to compute hitting times for processes other than Brownian motions. For example, hitting times of Bessel processes can be computed via an algorithm called "Walk on moving spheres".[12] This problem has applications in mathematical finance.
The WoS can be adapted to solve the Poisson and Poisson–Boltzmann equation with flux conditions on the boundary.[13]
Finally, WoS can be used to solve problems where coefficients vary continuously in space, via connections with the volume rendering equation.[14]
See also
• Feynman–Kac formula
• Stochastic processes and boundary value problems
• Euler–Maruyama method to sample the paths of general diffusion processes
Notes
1. The link was first established by Kakutani for the 2-dimensional Brownian motion,[3] it can now be seen as a trivial case of the Feynman−Kac formula.
References
1. Muller, Mervin E. (Sep 1956). "Some continuous Monte-Carlo Methods for the Dirichlet Problem". The Annals of Mathematical Statistics. 27 (3): 569–589. doi:10.1214/aoms/1177728169. JSTOR 2237369.
2. Sabelfeld, Karl K. (1991). Monte Carlo methods in boundary value problems. Berlin [etc.]: Springer-Verlag. ISBN 978-3540530015.
3. Kakutani, Shizuo (1944). "Two-dimensional Brownian motion and harmonic functions". Proceedings of the Imperial Academy. 20 (10): 706–714. doi:10.3792/pia/1195572706.
4. Mascagni, Michael; Hwang, Chi-Ok (June 2003). "ϵ-Shell error analysis for "Walk On Spheres" algorithms". Mathematics and Computers in Simulation. 63 (2): 93–104. doi:10.1016/S0378-4754(03)00038-7.
5. Given, James A.; Hubbard, Joseph B.; Douglas, Jack F. (1997). "A first-passage algorithm for the hydrodynamic friction and diffusion-limited reaction rate of macromolecules". The Journal of Chemical Physics. 106 (9): 3761. Bibcode:1997JChPh.106.3761G. doi:10.1063/1.473428.
6. Elepov, B.S.; Mikhailov, G.A. (January 1969). "Solution of the dirichlet problem for the equation Δu − cu = −q by a model of "walks on spheres"". USSR Computational Mathematics and Mathematical Physics. 9 (3): 194–204. doi:10.1016/0041-5553(69)90070-6.
7. Booth, Thomas E (February 1981). "Exact Monte Carlo solution of elliptic partial differential equations". Journal of Computational Physics. 39 (2): 396–404. Bibcode:1981JCoPh..39..396B. doi:10.1016/0021-9991(81)90159-5.
8. Hwang, Chi-Ok; Mascagni, Michael; Given, James A. (March 2003). "A Feynman–Kac path-integral implementation for Poisson's equation using an h-conditioned Green's function". Mathematics and Computers in Simulation. 62 (3–6): 347–355. CiteSeerX 10.1.1.123.3156. doi:10.1016/S0378-4754(02)00224-0.
9. Gobet, Emmanuel (2013). Méthodes de Monte-Carlo et processus stochastiques du linéaire au non-linéaire. Palaiseau: Editions de l'Ecole polytechnique. ISBN 978-2-7302-1616-6.
10. Salminen, Andrei N. Borodin; Paavo (2002). Handbook of Brownian motion : facts and formulae (2. ed.). Basel [u.a.]: Birkhäuser. ISBN 978-3-7643-6705-3.{{cite book}}: CS1 maint: multiple names: authors list (link)
11. Booth, Thomas (August 1982). "Regional Monte Carlo solution of elliptic partial differential equations" (PDF). Journal of Computational Physics. 47 (2): 281–290. Bibcode:1982JCoPh..47..281B. doi:10.1016/0021-9991(82)90079-1.
12. Deaconu, Madalina; Herrmann, Samuel (December 2013). "Hitting time for Bessel processes—walk on moving spheres algorithm (WoMS)". The Annals of Applied Probability. 23 (6): 2259–2289. arXiv:1111.3736. doi:10.1214/12-AAP900. S2CID 25036031.
13. Simonov, Nikolai A. (2007). "Random Walks for Solving Boundary-Value Problems with Flux Conditions". pp. 181–188. CiteSeerX 10.1.1.63.3780. doi:10.1007/978-3-540-70942-8_21. ISBN 978-3-540-70940-4. {{cite book}}: |journal= ignored (help); Missing or empty |title= (help)
14. Sawhney, Rohan; Seyb, Dario; Jarosz, Wojciech; Crane, Keenan (July 2022). "Grid-Free Monte Carlo for PDEs with Spatially Varying Coefficients". ACM Transactions on Graphics. 41 (4): 1–17. arXiv:2201.13240. doi:10.1145/3528223.3530134. S2CID 246430740.
Further reading
• Sabelfeld, Karl K. (1991). Monte Carlo methods in boundary value problems. Berlin [etc.]: Springer-Verlag. ISBN 9783540530015.
External links
• Some continuous Monte-Carlo methods for the Dirichlet problem The paper in which Marvin Edgar Muller introduced the method.
• Brownian Motion by Peter Mörters and Yuval Peres. See Chapter 3.3 on harmonic measure, Green's functions and exit-points.
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