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Zsolt Baranyai
Zsolt Baranyai (June 23, 1948 in Budapest – April 6, 1978) was a Hungarian mathematician known for his work in combinatorics.
He graduated from Fazekas High School where he was a classmate of László Lovász, Miklós Laczkovich, and Lajos Pósa. He studied mathematics at Eötvös Loránd University and went on to become a lecturer in the Analysis Department. He earned his Ph.D. in 1975 and was posthumously awarded the Candidate degree of the Hungarian Academy of Sciences in 1978.
Baranyai is best known for his theorem on the decompositions of complete hypergraphs, which solved a long-standing open problem.[1] In addition to his mathematical pursuits, Baranyai was also a professional musician who played the recorder. He died while touring Hungary with the Bakfark Consort in a car accident after a concert.[2]
References
1. van Lint & Wilson (2001). Chapter 38, "Baranyai's theorem", pp. 536–541.
2. van Lint, J. H.; Wilson, R. M. (2001), A Course in Combinatorics, Cambridge University Press, p. 540, ISBN 9780521006019.
External links
• A Panorama of Hungarian Mathematics in the Twentieth Century, p. 567.
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Zubov's method
Zubov's method is a technique for computing the basin of attraction for a set of ordinary differential equations (a dynamical system). The domain of attraction is the set $\{x:\,v(x)<1\}$, where $v(x)$ is the solution to a partial differential equation known as the Zubov equation.[1] Zubov's method can be used in a number of ways.
Statement
Zubov's theorem states that:
If $x'=f(x),t\in \mathbb {R} $ is an ordinary differential equation in $\mathbb {R} ^{n}$ with $f(0)=0$, a set $A$ containing 0 in its interior is the domain of attraction of zero if and only if there exist continuous functions $v,h$ such that:
• $v(0)=h(0)=0$, $0<v(x)<1$ for $x\in A\setminus \{0\}$, $h>0$ on $\mathbb {R} ^{n}\setminus \{0\}$
• for every $\gamma _{2}>0$ there exist $\gamma _{1}>0,\alpha _{1}>0$ such that $v(x)>\gamma _{1},h(x)>\alpha _{1}$ , if $||x||>\gamma _{2}$
• $v(x_{n})\rightarrow 1$ for $x_{n}\rightarrow \partial A$ or $||x_{n}||\rightarrow \infty $
• $\nabla v(x)\cdot f(x)=-h(x)(1-v(x)){\sqrt {1+||f(x)||^{2}}}$
If f is continuously differentiable, then the differential equation has at most one continuously differentiable solution satisfying $v(0)=0$.
References
1. Vladimir Ivanovich Zubov, Methods of A.M. Lyapunov and their application, Izdatel'stvo Leningradskogo Universiteta, 1961. (Translated by the United States Atomic Energy Commission, 1964.) ASIN B0007F2CDQ.
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Zuckerman functor
In mathematics, a Zuckerman functor is used to construct representations of real reductive Lie groups from representations of Levi subgroups. They were introduced by Gregg Zuckerman (1978). The Bernstein functor is closely related.
This article is about the Zuckerman induction functor, which is not the same as the (Zuckerman) translation functor.
Notation and terminology
• G is a connected reductive real affine algebraic group (for simplicity; the theory works for more general groups), and g is the Lie algebra of G. K is a maximal compact subgroup of G.
• L is a Levi subgroup of G, the centralizer of a compact connected abelian subgroup, and *l is the Lie algebra of L.
• A representation of K is called K-finite if every vector is contained in a finite-dimensional representation of K. Denote by WK the subspace of K-finite vectors of a representation W of K.
• A (g,K)-module is a vector space with compatible actions of g and K, on which the action of K is K-finite.
• R(g,K) is the Hecke algebra of G of all distributions on G with support in K that are left and right K finite. This is a ring which does not have an identity but has an approximate identity, and the approximately unital R(g,K)- modules are the same as (g,K) modules.
Definition
The Zuckerman functor Γ is defined by
$\Gamma _{g,L\cap K}^{g,K}(W)=\hom _{R(g,L\cap K)}(R(g,K),W)_{K}$
and the Bernstein functor Π is defined by
$\Pi _{g,L\cap K}^{g,K}(W)=R(g,K)\otimes _{R(g,L\cap K)}W.$
References
• David A. Vogan, Representations of real reductive Lie groups, ISBN 3-7643-3037-6
• Anthony W. Knapp, David A. Vogan, Cohomological induction and unitary representations, ISBN 0-691-03756-6 prefacereview by Dan BarbaschMR1330919
• David A. Vogan, Unitary Representations of Reductive Lie Groups. (AM-118) (Annals of Mathematics Studies) ISBN 0-691-08482-3
• Gregg J. Zuckerman, Construction of representations via derived functors, unpublished lecture series at the Institute for Advanced Study, 1978.
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Translation functor
In mathematical representation theory, a (Zuckerman) translation functor is a functor taking representations of a Lie algebra to representations with a possibly different central character. Translation functors were introduced independently by Zuckerman (1977) and Jantzen (1979). Roughly speaking, the functor is given by taking a tensor product with a finite-dimensional representation, and then taking a subspace with some central character.
Definition
By the Harish-Chandra isomorphism, the characters of the center Z of the universal enveloping algebra of a complex reductive Lie algebra can be identified with the points of L⊗C/W, where L is the weight lattice and W is the Weyl group. If λ is a point of L⊗C/W then write χλ for the corresponding character of Z.
A representation of the Lie algebra is said to have central character χλ if every vector v is a generalized eigenvector of the center Z with eigenvalue χλ; in other words if z∈Z and v∈V then (z − χλ(z))n(v)=0 for some n.
The translation functor ψμ
λ
takes representations V with central character χλ to representations with central character χμ. It is constructed in two steps:
• First take the tensor product of V with an irreducible finite dimensional representation with extremal weight λ−μ (if one exists).
• Then take the generalized eigenspace of this with eigenvalue χμ.
References
• Jantzen, Jens Carsten (1979), Moduln mit einem höchsten Gewicht, Lecture Notes in Mathematics, vol. 750, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0069521, ISBN 978-3-540-09558-3, MR 0552943
• Knapp, Anthony W.; Vogan, David A. (1995), Cohomological induction and unitary representations, Princeton Mathematical Series, vol. 45, Princeton University Press, doi:10.1515/9781400883936, ISBN 978-0-691-03756-1, MR 1330919
• Zuckerman, Gregg (1977), "Tensor products of finite and infinite dimensional representations of semisimple Lie groups", Ann. Math., 2, 106 (2): 295–308, doi:10.2307/1971097, JSTOR 1971097, MR 0457636
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Wadim Zudilin
Wadim Zudilin (Вадим Валентинович Зудилин) is a Russian mathematician and number theorist who is active in studying hypergeometric functions and zeta constants. He studied under Yuri V. Nesterenko and worked at Moscow State University, the Steklov Institute of Mathematics, the Max Planck Institute for Mathematics and the University of Newcastle, Australia. He now works at the Radboud University Nijmegen, the Netherlands.[1]
He has reproved Apéry's theorem that ζ(3) is irrational, and expanded it. Zudilin proved that at least one of the four numbers ζ(5), ζ(7), ζ(9), or ζ(11) is irrational.[2] For that accomplishment he won the Distinguished Award of the Hardy-Ramanujan Society in 2001.
With Doron Zeilberger, Zudilin[3] improved upper bound of irrationality measure for π, which as of November 2022 is the current best estimate.
References
1. "Wadim Zudilin appointed Professor of Pure Mathematics".
2. W. Zudilin (2001). "One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational". Russ. Math. Surv. 56 (4): 774–776. Bibcode:2001RuMaS..56..774Z. doi:10.1070/RM2001v056n04ABEH000427. S2CID 250734661.
3. Zeilberger, Doron; Zudilin, Wadim (2020-01-07). "The irrationality measure of π is at most 7.103205334137…". Moscow Journal of Combinatorics and Number Theory. 9 (4): 407–419. arXiv:1912.06345. doi:10.2140/moscow.2020.9.407. S2CID 209370638.
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External links
• Wadim Zudilin's homepage
• Wadim Zudilin's research profile
• Wadim Zudilin's list of published works
• Wadim Zudilin at the Mathematics Genealogy Project
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Zvezdelina Stankova
Zvezdelina Entcheva Stankova (Bulgarian: Звезделина Енчева Станкова; born 15 September 1969) is an American mathematician who is a professor of mathematics at Mills College and a teaching professor at the University of California, Berkeley, the founder of the Berkeley Math Circle, and an expert in the combinatorial enumeration of permutations with forbidden patterns.[1]
Zvezdelina Entcheva Stankova
Stankova in 2012
Born (1969-09-15) 15 September 1969
Ruse, People's Republic of Bulgaria
NationalityAmerican
Alma materBryn Mawr College
Harvard University
Known forSkew-merged permutation
Studying permutations with forbidden subsequences
Establishing math circles
AwardsAlice T. Schafer Prize (1992)
Deborah and Franklin Tepper Haimo Award (2011)
Scientific career
FieldsMathematics
InstitutionsMills College
University of California, Berkeley
Doctoral advisorJoe Harris
Websitehttps://math.berkeley.edu/~stankova/
Biography
Stankova was born in Ruse, Bulgaria.[2] She began attending the Ruse math circle as a fifth grader in Bulgaria, the same year she learned to solve the Rubik's Cube[3] and began winning regional mathematics competitions.[1] She later wrote of this experience that "if I was not a member of Ruse SMC I would not be able to make such profound achievements in mathematics".[4] She became a student at an elite English-language high school, and competed on the Bulgarian team in the International Mathematical Olympiads in 1987 and 1988, earning silver medals both times.[2][5] She entered Sofia University but in 1989, as the Iron Curtain was falling, became one of 15 Bulgarian students selected to travel to the US to complete their studies.[2]
Stankova studied at Bryn Mawr College, completing bachelor's and master's degrees there in 1992,[6] with Rhonda Hughes as a faculty mentor.[7] While an undergraduate, she participated in a summer research program with Joseph Gallian at the University of Minnesota Duluth, which began her interest in permutation patterns.[8] Next, she went to Harvard University for her doctoral studies, and earned a Ph.D. there in 1997; her dissertation, entitled Moduli of Trigonal Curves, was supervised by Joe Harris.[9]
She worked at the University of California, Berkeley as Morrey Assistant Professor of Mathematics[10] before joining the Mills College faculty in 1999,[6] and continues to teach one course per year as a visiting professor at Berkeley.[11][12] She also serves on the advisory board of the Proof School in San Francisco.[1]
Contributions
In the theory of permutation patterns, Stankova is known for proving that the permutations with the forbidden pattern 1342 are equinumerous with the permutations with forbidden pattern 2413, an important step in the enumeration of permutations avoiding a pattern of length 4.[8][13]
In 1998 she became the founder and director of the Berkeley Math Circle, an after-school mathematics enrichment program that Stankova modeled after her early experiences learning mathematics in Bulgaria.[3][7][14][15] The Berkeley circle was only the second math circle in the US (after one in Boston); following its success, over 100 other circles have been created,[3] and Stankova has assisted in the formation of many of them.[11]
Also in 1998, she founded the Bay Area Mathematical Olympiad.[10] For six years, she served as a coach of the US International Mathematical Olympiad team.[7][16]
Since 2013, she has featured in several videos on the mathematics-themed YouTube channel "Numberphile".[17]
Publications
With Tom Rike, she is co-editor of two books about her work with the Berkeley Math Circle, A Decade of the Berkeley Math Circle: The American Experience (Vol. I, 2008, Vol. II, 2014).[18]
Awards and honors
In 1992, Stankova won the Alice T. Schafer Prize of the Association for Women in Mathematics for her undergraduate research in permutation patterns.[8][11] In 2004 she became one of two inaugural winners of the Henry L. Alder Award for Distinguished Teaching by a Beginning College or University Mathematics Faculty Member.[19] In 2011 Stankova won the Deborah and Franklin Tepper Haimo Award for Distinguished College or University Teaching, given by the Mathematical Association of America, "for her outstanding work in teaching, mentoring, and inspiring students at all levels, and in leading the development of Math Circles, and promoting participation in mathematics competitions".[7][11][16] From 2009 to 2012 she was the Frederick A. Rice Professor of Mathematics at Mills.[20]
References
1. "Zvezdelina Stankova". Proof School. Archived from the original on 2018-12-09. Retrieved 2016-02-03.
2. Vigoda, Ralph (October 23, 1991), "Bulgarian Math Whiz Wows 'Em: Zvezdelina Stankova Already Is Joining The Ranks Of Top Mathematicians. "She's A Genius," Says Bryn Mawr's Head Of Math", The Philadelphia Inquirer, archived from the original on December 22, 2015.
3. Weld, Sarah (April 2014), "Proving Their Passion: The Berkeley Math Circle gives math kids a place to find solutions—together", The East Bay Monthly
4. Ruse Students Mathematical Circle (PDF), Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, retrieved 2016-02-02.
5. Participant record: Zvezdelina Stankova, International Mathematical Olympiad, retrieved 2016-02-02.
6. "Zvezdelina Stankova", Mathematics Faculty and Staff, Mills College, retrieved 2016-02-01.
7. Bakke, Katherine (April 6, 2011), "Zvezdelina Stankova '92 Carries on a Bryn Mawr Tradition: Excellent Teaching in Mathematics", Meaningful Contributions, Bryn Mawr College.
8. Third Annual Alice T. Schafer Prize, Association for Women in Mathematics, July 1992, retrieved 2016-02-02.
9. Zvezdelina Stankova at the Mathematics Genealogy Project
10. Keith, Tamara (February 10, 1999), UC Berkeley mathematicians feed the minds of young local math whizzes, University of California, Berkeley.
11. "Haimo Award Citation: Zvezdelina Stankova" (PDF), January 2011 Prizes and Awards, American Mathematical Society, p. 4, January 7, 2011.
12. Stankova's home page at Berkeley, retrieved 2016-02-02.
13. Bona, Miklos (2012), Combinatorics of Permutations, Discrete Mathematics and Its Applications (2nd ed.), CRC Press, pp. 154–155, ISBN 9781439850527.
14. Vandervelde, Sam (2009), Circle in a Box, MSRI mathematical circles library, vol. 2, Mathematical Sciences Research Institute and American Mathematical Society, pp. 4, 34, ISBN 9780821847527.
15. Bloom, Melanie (October 21, 2013), Moms Everyday: Making Math Fun, 10/11.
16. Melendez, Lyanne (February 25, 2011), Mills College professor wins highest math award, ABC7 News.
17. Numberphile (2013-12-19), Pebbling a Chessboard - Numberphile, retrieved 2016-08-21
18. MSRI Mathematical Circles Library, National Association of Math Circles, retrieved 2016-02-01.
19. Henry L. Alder Award, Mathematical Association of America, retrieved 2018-06-08
20. Curriculum vitae: Zvezdelina Stankova (PDF), retrieved 2016-02-01
External links
• Zvezdelina Stankova in the Oberwolfach photo collection
• Moduli of Trigonal Curves Paper based on PhD thesis.
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Zvi Arad
Zvi Arad (Hebrew: צבי ארד,16 April 1942, in Petah Tikva, Mandatory Palestine – 4 February 2018, in Petah Tikva, Israel) was an Israeli mathematician, acting president of Bar-Ilan University, and president of Netanya Academic College.[1][2]
Zvi Arad
Zvi Arad (2008)
Born
צבי ארד
(1942-04-16)April 16, 1942
Petah Tikva, Mandatory Palestine
DiedFebruary 4, 2018(2018-02-04) (aged 75)
Petah Tikva, Israel
NationalityIsraeli
Occupationmathematician
Known for
• Acting President of Bar-Ilan University
• President of Netanya Academic College
Biography
Zvi Arad began his academic studies in the Mathematics Department of Bar-Ilan University. He received his first degree in 1964 and after army service went on to complete a second and third degree in the Mathematics Department of Tel Aviv University.
Academic career
In 1968 Arad joined the academic staff at Bar-Ilan University as an assistant and in 1983 was appointed a full professor. During the years 1978/9 he held the position of visiting scientist at the University of Chicago, and from 1982 to 1983 held the position of visiting professor at the University of Toronto.
Arad held a variety of senior academic posts at Bar-Ilan University. He served as chairman of the Mathematics and Computer Science Department, dean of the Faculty of Natural Sciences and Mathematics, rector and president of the university (succeeding Ernest Krausz, and followed by Shlomo Eckstein).[3] Together with Professor Bernard Pinchuk he founded Gelbart Institute, an international research institute named after Abe Gelbart, and the Emmy Noether Institute (Minerva Center). Together with colleagues he established a journal, the Israel Mathematics Conference Proceedings, distributed by the American Mathematical Society (AMS). From 1984–1985 he served as a member of the Council for Higher Education of the State of Israel. In 1982 he was elected a member of Russia's Academy of Natural Sciences.
From 1994 he served on the editorial board of the Algebra Colloquium, a journal of the Chinese Academy of Sciences published by Springer-Verlag. He also serves on the editorial board of various international publications: South East Asian Bulletin of Mathematics of the Asian Mathematical Society, the IMCP of Contemporary Mathematics published by the American Mathematical Society, and the publication Cubo Matemática Educacional, Temuco, Chile.
He initiated numerous agreements of cooperation with universities and institutions throughout the world including academic institutes in the former Soviet Union, universities and research centers in America, Canada, Germany, the United Kingdom, Italy, Russia, China, South Africa, etc.
He was a member of Israel's first official delegation to the former USSR, under the leadership of President Ezer Weizman. In an official address, President Mikhail Gorbachev mentioned Professor Arad's contributions towards the establishment of scientific communications between Israel and the former USSR. In an effort to advance cooperation in research he has headed delegations of scientists to Russia, China, and East Germany.
Haaretz newspaper (January 21, 1998) described him as one of the pioneers of higher education reform in Israel. The Encyclopaedia Hebraica lists Zvi Arad as "fulfilling a key role in the development and advancement of Bar-Ilan University and in the establishment of the University's regional colleges in Safed, Ashkelon and the Jordan Valley)." For this achievement he was awarded a certificate of honor by the mayor of each city. The establishment of these colleges began in 1985 and went on to affect the whole of Israel. These colleges advanced the Galilee and Southern Israel and brought higher education to the peripheries of Israel.
Netanya Academic College
In 1994, at the request of the mayors of the city of Netanya, Yoel Elroi and Zvi Poleg, Arad established the Netanya Academic College. He served as president of the college for 24 years.[4] A partner in the initiation and establishment of the college was Miriam Feirberg, who at that time served as head of the Education Department of the City of Netanya. Today the college is an accredited institute of higher education that grant first and second academic degrees in a variety of fields.
Published works
Together with his colleague Professor Marcel Herzog, Arad wrote Products of Conjugacy Classes, published by Springer-Verlag. The book facilitated the basis of the establishment of mathematical theory and today forms part of the branch of abstract algebra known as Table Algebras, and is attached to central branches in mathematics: Graph theory, algebra combinations, and theory presentation. Arad coauthored two other books on the subject of table algebra. In 2000 his book was published in the series American Mathematical Society Memoirs and in January 2002 another book on table algebras was published in the international publication, Springer. Arad was the editor of Contemporary Mathematics, Volume 402.
See also
• Education in Israel
References
1. For 11-year-old Math Prodigy, 2 Plus 2 Equals 'Super-gifted'
2. "Bar-Ilan Presidents | Bar Ilan University". .biu.ac.il. Retrieved 2020-02-15.
3. "Bar-Ilan Presidents | Bar Ilan University". .biu.ac.il. Retrieved 2020-02-18.
4. "Professor Zvi Arad passes away". Israel National News. 4 February 2018. Retrieved 19 February 2018.
External links
• Profile Netanya Academic College
• Zvi Arad Dun's 100 (in Hebrew)
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Zvi Galil
Zvi Galil (Hebrew: צבי גליל; born June 26, 1947) is an Israeli-American computer scientist and mathematician. Galil served as the president of Tel Aviv University from 2007 through 2009. From 2010 to 2019, he was the dean of the Georgia Institute of Technology College of Computing.[3] His research interests include the design and analysis of algorithms, computational complexity and cryptography. He has been credited with coining the terms stringology and sparsification.[4][5] He has published over 200 scientific papers[6] and is listed as an ISI highly cited researcher.[7]
Zvi Galil
Galil in 2010
Born (1947-06-26) June 26, 1947[1]
Tel Aviv, Mandatory Palestine[1]
Alma mater
• Tel Aviv University (BSc, MSc)
• Cornell University (PhD)
Awards
• ACM Fellow
• NAE Member
• American Academy of Arts and Sciences Fellow
Scientific career
Fields
• Computer science
• Mathematics
Institutions
• IBM
• Tel Aviv University
• Columbia University
• Georgia Institute of Technology
Doctoral advisorJohn Hopcroft[2]
Doctoral students
• Mordechai Ben-Ari
• Moti Yung
• David Eppstein
• Giuseppe F. Italiano
• Matthew K. Franklin[2]
• Jonathan Katz
• Stuart Haber
Early life and education
Zvi Galil was born in Tel Aviv in Mandatory Palestine in 1947. He completed both his B.Sc. (1970) and his M.Sc. (1971) in applied mathematics, both summa cum laude, at Tel Aviv University before earning his Ph.D. in computer science at Cornell in 1975 under the supervision of John Hopcroft.[2] He then spent a year working as a post-doctorate researcher at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York.[8]
Career
From 1976 until 1995 he worked in the computer science department of Tel Aviv University, serving as its chair from 1979 to 1982. In 1982 he joined the faculty of Columbia University, serving as the chair of the computer science department from 1989-1994.[1][8] From 1995-2007, he served as the dean of the Fu Foundation School of Engineering & Applied Science.[9] In this position, he oversaw the naming of the school in honor of Chinese businessman Z. Y. Fu after a large donation was given in his name.[10] At Columbia, he was appointed the Julian Clarence Levi Professor of Mathematical Methods and Computer Science in 1987, and the Morris and Alma A. Schapiro Dean of Engineering in 1995.[1]
Galil served as the president of Tel Aviv University starting in 2007 (following Itamar Rabinovich),[11] but resigned and returned to the faculty in 2009, and was succeeded by Joseph Klafter.[12][13] He was named as the dean of Georgia Tech's college of computing on April 9, 2010.[3] At Georgia Tech, together with Udacity founder Sebastian Thrun, Galil conceived of the college of computing's online Master of Science in computer science (OMSCS) program, and he led the faculty creation of the program.[14] OMSCS went on to become the largest online master’s program in computer science in the United States.[15] OMSCS has been featured in hundreds of articles, including a 2013 front-page article in The New York Times and 2021 interviews in The Wall Street Journal and Forbes.[14][16][17] Inside Higher Education noted that OMSCS "suggests that institutions can successfully deliver high-quality, low-cost degrees to students at scale".[18] The Chronicle of Higher Education noted that OMSCS "may have the best chance of changing how much students pay for a traditional degree".[19] Galil stepped down as dean and returned to a regular faculty position in June 2019.[20][21] He now serves as the Frederick G. Storey Chair in Computing and Executive Advisor to Online Programs at Georgia Tech.
Professional service
In 1982, Galil founded the Columbia University Theory Day and organized the event for the first 15 years. It still exists as the New York Area Theory Day.[22] From 1983 to 1987, Galil served as the chairman of ACM SIGACT, an organization that promotes research in theoretical computer science.[23] He served as managing editor of SIAM Journal on Computing from 1991 to 1997 and editor in chief of Journal of Algorithms from 1988 to 2003.
Research
Galil's research is in the areas of algorithms (particularly string and graph algorithms) complexity, and cryptography. He has also conducted research in experimental design with Jack Kiefer.
Galil's real-time algorithms are the fastest possible for string matching and palindrome recognition, and they work even on the most basic computer model, the multi-tape Turing machine. More generally, he formulated a "predictability" condition that allows any complying online algorithm to be converted to a real-time algorithm.[24][25] With Joel Seiferas, Galil improved the time-optimal algorithms to be space optimal (logarithmic space) as well.[26]
Galil worked with Dany Breslauer to design a linear-work, O(loglogn) parallel algorithm for string matching,[27] and they later proved it to have the best possible time complexity among linear work algorithms.[28] With other computer scientists, he designed a constant-time linear-work randomized search algorithm to be used when the pattern preprocessing is given.[29]
With his students, Galil designed more than a dozen currently-fastest algorithms for exact or approximate, sequential or parallel, and one- or multi-dimensional string matching.
Galil worked with other computer scientists to develop several currently-fastest graph algorithms. Examples include: maximum weighted matching;[30] trivalent graph isomorphism;[31] and minimum weight spanning trees.[32]
With his students, Galil devised a technique he called "sparsification"[33] and a method he called "sparse dynamic programming".[34] The first was used to speed up dynamic graph algorithms. The second was used to speed up the computations of various edit distances between strings.
In 1979, together with Ofer Gabber, Galil solved the previously open problem of constructing a family of expander graphs with an explicit expansion ratio,[35] useful in the design of fast graph algorithms.
Awards and honors
In 1995, Galil was inducted as a Fellow of the Association for Computing Machinery for "fundamental contributions to the design and analysis of algorithms and outstanding service to the theoretical computer science community,"[36] and in 2004, he was elected to the National Academy of Engineering for "contributions to the design and analysis of algorithms and for leadership in computer science and engineering."[37][38] In 2005, he was selected as a Fellow of the American Academy of Arts and Sciences.[39] In 2008, Columbia University established the Zvi Galil award for student life.[40] In 2009, the Columbia Society of Graduates awarded him the Great Teacher Award.[41] In 2012, The University of Waterloo awarded Galil with an honorary Doctor of Mathematics degree for his "fundamental contributions in the areas of graph algorithms and string matching."[42] In 2020, Academic Influence included Galil in the list of the 10 most influential computer scientists of the last decade, and the advisory board of the College of Computing at Georgia Tech raised over $2 million from over 130 donors to establish an endowed chair named after Galil.[43][44]
References
1. Eppstein, David; Italiano, Giuseppe F. (March 1999). "PREFACE: Festschrift for Zvi Galil". Journal of Complexity. 15 (1): 1–3. doi:10.1006/jcom.1998.0492.
2. Zvi Galil at the Mathematics Genealogy Project
3. "Institute names next College of Computing Dean" (Press release). Georgia Institute of Technology. 2010-04-09. Retrieved 2010-04-09.
4. "Introduction to Stringology". The Prague Stringology Club. Czech Technical University in Prague. Retrieved May 14, 2012.
5. Zvi, Galil; David Eppstein; Giuseppe F. Italiano; Amnon Nissenzweig (September 1997). "Sparsification - a technique for speeding up dynamic graph algorithms". Journal of the ACM. 44 (5): 669–696. doi:10.1145/265910.265914. S2CID 340999.
6. "Zvi Galil". The DBLP Computer Science Bibliography. Digital Bibliography & Library Project. Retrieved 2016-03-24.
7. "ISI Highly Cited Researchers Version 1.1: Zvi Galil". ISI Web of Knowledge. Retrieved 2011-06-27.
8. "Zvi Galil Named Dean of Columbia's Engineering School" (Press release). Columbia University. July 14, 1995. Retrieved 2019-06-05.
9. McCaughey, Robert (2014). A Lever Long Enough: A History of Columbia's School of Engineering and Applied Science since 1864. Columbia University Press. p. 240. ISBN 9780231166881.
10. Arenson, Karen W. (1997-10-01). "Chinese Tycoon Gives Columbia $26 Million". The New York Times. Retrieved 2010-04-20.
11. "Computer expert nominated for TAU presidency". The Jerusalem Post. November 5, 2006.
12. Basch_Interactive (1980-01-01). "Presidents of Tel Aviv University | Tel Aviv University | Tel Aviv University". English.tau.ac.il. Retrieved 2020-02-18.
13. Ilani, Ofri; Kashti, Or (2009-07-02). "Tel Aviv University president quits / Sources: Galil was forced out of office". Haaretz. Retrieved 2011-06-27.
14. Lewin, Tamar (2013-08-18). "Master's Degree Is New Frontier of Study Online". The New York Times. ISSN 0362-4331. Retrieved 2023-01-02.
15. Galil, Zvi. "OMSCS: The Revolution Will Be Digitized". cacm.acm.org. Retrieved 2020-07-27.
16. Varadarajan, Tunku (2021-04-02). "Opinion | The Man Who Made Online College Work". Wall Street Journal. ISSN 0099-9660. Retrieved 2021-11-01.
17. Nietzel, Michael T. "Georgia Tech's Online MS In Computer Science Continues To Thrive. Why That's Important For The Future of MOOCs". Forbes. Retrieved 2022-03-25.
18. "Analysis shows Georgia Tech's online master's in computer science expanded access | Inside Higher Ed". www.insidehighered.com. 20 March 2018. Retrieved 2022-03-25.
19. "What Georgia Tech's Online Degree in Computer Science Means for Low-Cost Programs". www.chronicle.com. 6 November 2014. Retrieved 2022-03-25.
20. "College's Skyrocketing Stature, Global Impact Highlight Galil's Legacy". Georgia Tech College of Computing. April 16, 2019. Retrieved 2019-06-05.
21. "Georgia Tech Alumni Magazine, Vol. 95 No. 3, Fall 2019". Issuu. Retrieved 2020-04-21.
22. "New York Area Theory Day". www.cs.columbia.edu. Retrieved 2020-06-03.
23. "Front matter". ACM SIGACT News. 19 (1). Fall 1987.
24. Galil, Zvi (1981-01-01). "String Matching in Real Time". Journal of the ACM. 28 (1): 134–149. doi:10.1145/322234.322244. ISSN 0004-5411. S2CID 9164969.
25. Galil, Zvi (1978-04-01). "Palindrome recognition in real time by a multitape turing machine". Journal of Computer and System Sciences. 16 (2): 140–157. doi:10.1016/0022-0000(78)90042-9. ISSN 0022-0000.
26. Galil, Zvi; Seiferas, Joel (1983-06-01). "Time-space-optimal string matching". Journal of Computer and System Sciences. 26 (3): 280–294. doi:10.1016/0022-0000(83)90002-8. ISSN 0022-0000.
27. Breslauer, Dany; Galil, Zvi (1990-12-01). "An Optimal $O(\log\log n)$ Time Parallel String Matching Algorithm". SIAM Journal on Computing. 19 (6): 1051–1058. doi:10.1137/0219072. ISSN 0097-5397.
28. Breslauer, Dany; Galil, Zvi (1992-10-01). "A Lower Bound for Parallel String Matching". SIAM Journal on Computing. 21 (5): 856–862. doi:10.1137/0221050. ISSN 0097-5397.
29. Crochemore, Maxime; Galil, Zvi; Gasieniec, Leszek; Park, Kunsoo; Rytter, Wojciech (1997-08-01). "Constant-Time Randomized Parallel String Matching". SIAM Journal on Computing. 26 (4): 950–960. doi:10.1137/S009753979528007X. ISSN 0097-5397.
30. Galil, Zvi; Micali, Silvio; Gabow, Harold (1986-02-01). "An $O(EV\log V)$ Algorithm for Finding a Maximal Weighted Matching in General Graphs". SIAM Journal on Computing. 15 (1): 120–130. doi:10.1137/0215009. ISSN 0097-5397. S2CID 12854446.
31. Galil, Zvi; Hoffmann, Christoph M.; Luks, Eugene M.; Schnorr, Claus P.; Weber, Andreas (1987-07-01). "An O(n3log n) deterministic and an O(n3) Las Vegs isomorphism test for trivalent graphs". Journal of the ACM. 34 (3): 513–531. doi:10.1145/28869.28870. ISSN 0004-5411. S2CID 18031646.
32. Gabow, Harold N.; Galil, Zvi; Spencer, Thomas; Tarjan, Robert E. (1986-06-01). "Efficient algorithms for finding minimum spanning trees in undirected and directed graphs". Combinatorica. 6 (2): 109–122. doi:10.1007/BF02579168. ISSN 1439-6912. S2CID 35618095.
33. Eppstein, David; Galil, Zvi; Italiano, Giuseppe F.; Nissenzweig, Amnon (1997-09-01). "Sparsification—a technique for speeding up dynamic graph algorithms". Journal of the ACM. 44 (5): 669–696. doi:10.1145/265910.265914. ISSN 0004-5411. S2CID 340999.
34. Eppstein, David; Galil, Zvi; Giancarlo, Raffaele; Italiano, Giuseppe F. (1992-07-01). "Sparse dynamic programming I: linear cost functions". Journal of the ACM. 39 (3): 519–545. doi:10.1145/146637.146650. ISSN 0004-5411. S2CID 17060840.
35. Gabber, Ofer; Galil, Zvi (1981-06-01). "Explicit constructions of linear-sized superconcentrators". Journal of Computer and System Sciences. 22 (3): 407–420. doi:10.1016/0022-0000(81)90040-4. ISSN 0022-0000.
36. ACM Fellow Award / Zvi Galil
37. "Dr. Zvi Galil". NAE Members. National Academy of Engineering. Retrieved May 11, 2012.
38. "Zvi Galil Elected to National Academy of Engineering". Columbia News. Columbia University. Retrieved May 11, 2012.
39. Academy Elects 225th Class of Fellows and Foreign Honorary Members, American Association for the Advancement of Science, April 26, 2005
40. "Zvi Galil Award". Columbia College. Retrieved 2019-06-05.
41. "Quigley, Galil To Receive Great Teacher Awards". Columbia College Today. September 2009. Retrieved 2019-06-05.
42. Smyth, Pamela. "University of Waterloo to award eight honorary degrees at spring convocation". Waterloo Communications. University of Waterloo. Retrieved May 11, 2012.
43. Larson, Erik J.; PhD. "Top Influential Computer Scientists Today". academicinfluence.com. Retrieved 2021-05-05.
44. "New Endowed Chair Honors Inclusion and Diversity". College of Computing. 2021-06-02. Retrieved 2021-06-09.
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Zvika Brakerski
Zvika Brakerski is an Israeli mathematician, known for his work on homomorphic encryption, particularly in developing the foundations of the second generation FHE schema, for which he was awarded the 2022 Gödel Prize.[1][2] Brakerski is an Associate Professor in the Department of Computer Science and Applied Mathematics at the Weizmann Institute of Science.
Zvika Brakerski
OccupationAssociate Professor
Known forhomomorphic encryption
AwardsGödel Prize
Academic background
Doctoral advisorShafi Goldwasser
Other advisorsDan Boneh
Academic work
Disciplinecryptography
Research
In 2012 Brakerski published a paper at the Annual Cryptology Conference "Fully homomorphic encryption without modulus switching from classical GapSVP Authors",[3] this formed the basis of the Brakerski-Gentry-Vaikuntanathan (BGV)[4] - for which they were jointly awarded the Gödel Prize - and BFV Fully Homomorphic Encryption (FHE) schema. The two dominant second-generation FHE schema.
References
1. "ACM SIGACT - Gödel Prize". sigact.org. Archived from the original on 2022-11-24. Retrieved 2022-11-24.
2. "School of Engineering second quarter 2022 awards". MIT News | Massachusetts Institute of Technology. Archived from the original on 2022-11-24. Retrieved 2022-11-24.
3. Brakerski, Zvika (2012). "Fully Homomorphic Encryption without Modulus Switching from Classical GapSVP". In Safavi-Naini, Reihaneh; Canetti, Ran (eds.). Advances in Cryptology – CRYPTO 2012. Lecture Notes in Computer Science. Vol. 7417. Berlin, Heidelberg: Springer. pp. 868–886. doi:10.1007/978-3-642-32009-5_50. ISBN 978-3-642-32009-5.
4. Brakerski, Zvika; Gentry, Craig; Vaikuntanathan, Vinod (2011). "Fully Homomorphic Encryption without Bootstrapping". Cryptology ePrint Archive.
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Thomas von Randow
Thomas von Randow (26 December 1921 Breslau, Schlesien – 29 July 2009 Hamburg) was a German mathematician and journalist who published mathematical and logical puzzles under the pseudonym Zweistein in the "Logelei" column in Die Zeit. (After 2005 his column and pseudonym were continued by Bernhard Seckinger and Immanuel Halupczok.)
Publications
Many of his logic puzzles were published in the following books:
• 99 Logeleien von Zweistein. Christian Wegner, Hamburg 1968
• Neue Logeleien von Zweistein. Hoffmann und Campe, Hamburg 1976
• Logeleien für Kenner. Hoffmann und Campe, Hamburg 1975
• 88 neue Logeleien. Nymphenburger, München 1983
• 87 neue Logeleien. Rasch und Röhring, Hamburg 1985
• Weitere Logeleien von Zweistein. Deutscher Taschenbuchverlag (dtv), München 1985, ISBN 3-485-00446-4
• Zweisteins Zahlenmagie. Mathematisches und Mystisches über einen abstrakten Gebrauchsgegenstand. Von Eins bis Dreizehn. Illustrationen von Gerhard Gepp. Christian Brandstätter, Wien 1993, ISBN 3854474814
• Zweisteins Zahlen-Logeleien. Insel, Frankfurt am Main und Leipzig 1993, ISBN 3-458-33210-3
References
• Interview in Die Zeit, 15 November 2005
• Thomas von Randow – Visionär seines Fachs. Obituary in Die Zeit, 32/2009
• Thomas von Randow at the Mathematics Genealogy Project
External links
• Logelei puzzle by Zweistein in Die Zeit
• Collection of logical puzzles by Zweistein (in German)
• Index to articles by Thomas von Randow in Die Zeit
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Antoni Zygmund
Antoni Zygmund (December 25, 1900 – May 30, 1992) was a Polish mathematician. He worked mostly in the area of mathematical analysis, including especially harmonic analysis, and he is considered one of the greatest analysts of the 20th century.[1][2][3][4][5] Zygmund was responsible for creating the Chicago school of mathematical analysis together with his doctoral student Alberto Calderón, for which he was awarded the National Medal of Science in 1986.[1][2][3][4]
Antoni Zygmund
Antoni Zygmund
Born(1900-12-25)December 25, 1900
Warsaw, Congress Poland, Russian Empire
DiedMay 30, 1992(1992-05-30) (aged 91)
Chicago, Illinois, United States
NationalityPolish
CitizenshipPolish, American
Alma materUniversity of Warsaw (Ph.D., 1923)
Known forSingular integral operators
Calderón–Zygmund lemma
Marcinkiewicz–Zygmund inequality
Paley–Zygmund inequality
Calderón–Zygmund kernel
AwardsLeroy P. Steele Prize (1979)
National Medal of Science (1986)
Scientific career
FieldsMathematics
InstitutionsUniversity of Chicago
Stefan Batory University
Doctoral advisorAleksander Rajchman
Stefan Mazurkiewicz
Doctoral studentsAlberto Calderón
Elias M. Stein
Paul Cohen
Biography
Born in Warsaw, Zygmund obtained his Ph.D. from the University of Warsaw (1923) and was a professor at Stefan Batory University at Wilno from 1930 to 1939, when World War II broke out and Poland was occupied. In 1940 he managed to emigrate to the United States, where he became a professor at Mount Holyoke College in South Hadley, Massachusetts. In 1945–1947 he was a professor at the University of Pennsylvania, and from 1947, until his retirement, at the University of Chicago.
He was a member of several scientific societies. From 1930 until 1952 he was a member of the Warsaw Scientific Society (TNW), from 1946 of the Polish Academy of Learning (PAU), from 1959 of the Polish Academy of Sciences (PAN), and from 1961 of the National Academy of Sciences in the United States. In 1986 he received the National Medal of Science.
In 1935 Zygmund published in Polish the original edition of what has become, in its English translation, the two-volume Trigonometric Series. It was described by Robert A. Fefferman as "one of the most influential books in the history of mathematical analysis" and "an extraordinarily comprehensive and masterful presentation of a ... vast field".[6] Jean-Pierre Kahane called the book "The Bible" of a harmonic analyst. The theory of trigonometric series had remained the largest component of Zygmund's mathematical investigations.[5]
His work has had a pervasive influence in many fields of mathematics, mostly in mathematical analysis, and particularly in harmonic analysis. Among the most significant were the results he obtained with Calderón on singular integral operators.[7][6] George G. Lorentz called it Zygmund's crowning achievement, one that "stands somewhat apart from the rest of Zygmund's work".[5]
Zygmund's students included Alberto Calderón, Paul Cohen, Nathan Fine, Józef Marcinkiewicz, Victor L. Shapiro, Guido Weiss, Elias M. Stein and Mischa Cotlar. He died in Chicago.
Mathematical objects named after Zygmund
• Calderón–Zygmund lemma
• Marcinkiewicz–Zygmund inequality
• Paley–Zygmund inequality
• Calderón–Zygmund kernel
Books
• Trigonometric Series (Cambridge University Press 1959, 2002)
• Intégrales singulières (Springer-Verlag, 1971)
• Trigonometric Interpolation (University of Chicago, 1950)
• Measure and Integral: An Introduction to Real Analysis, With Richard L. Wheeden (Marcel Dekker, 1977)
• Analytic Functions, with Stanislaw Saks (Elsevier Science Ltd, 1971)
See also
• Calderón–Zygmund lemma
• Zygmunt Janiszewski
• Marcinkiewicz–Zygmund inequality
• Paley–Zygmund inequality
• List of Poles
• Centipede mathematics
References
1. Noble, Holcomb B. (1998-04-20). "Alberto Calderon, 77, Pioneer Of Mathematical Analysis". The New York Times. ISSN 0362-4331. Retrieved 2019-06-23.
2. Warnick, Mark S. (19 April 1998). "ALBERTO CALDERON, MATH GENIUS". chicagotribune.com. Retrieved 2019-06-23.
3. "Antoni Zygmund (1900-1992)". www-history.mcs.st-and.ac.uk. Retrieved 2019-06-23.
4. "PROFESSOR ALBERTO CALDERON, 77, DIES". Washington Post. ISSN 0190-8286. Retrieved 2019-06-22.
5. Lorentz, G. G. (1993). "Antoni Zygmund and His Work" (PDF). Journal of Approximation Theory. 75: 1–7. doi:10.1006/jath.1993.1084.
6. The 2nd edition of Zygmund's Trigonometric Series (Cambridge University Press, 1959) consists of 2 separate volumes. The 3rd edition (Cambridge University Press, 2002, ISBN 0 521 89053 5) consists of the two volumes combined with a foreword by Robert A. Fefferman. The nine pages in Fefferman's foreword (biographic and other information concerning Zygmund) are not numbered.
7. Carbery, Tony (17 July 1992). "Obituary: Professor Antoni Zygmund". The Independent. Archived from the original on 2022-05-07.
Further reading
• Kazimierz Kuratowski, A Half Century of Polish Mathematics: Remembrances and Reflections, Oxford, Pergamon Press, 1980, ISBN 0-08-023046-6.
• Gray, Jeremy (1970–1980). "Zygmund, Antoni". Dictionary of Scientific Biography. Vol. 25. New York: Charles Scribner's Sons. pp. 414–416. ISBN 978-0-684-10114-9.
External links
• Antoni Zygmund at the Mathematics Genealogy Project
• Mount Holyoke biography
• O'Connor, John J.; Robertson, Edmund F., "Antoni Zygmund", MacTutor History of Mathematics Archive, University of St Andrews
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1968
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1975
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1979
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1982
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1987
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1988
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1992
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1993
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1994
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1995
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1996
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1997
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Robert A. Weinberg
1998
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1999
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2001
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Mario R. Capecchi
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Harold Varmus
2002
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2003
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2004
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2005
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2006
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2007
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2008
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2009
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2011
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2012
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2013
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2014
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1986
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1987
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1988
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1989
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1993
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1997
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1998
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2013
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2014
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1964
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1965
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1967
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1975
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1979
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1986
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1987
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Zygmunt Janiszewski
Zygmunt Janiszewski (12 July 1888 – 3 January 1920)[1] was a Polish mathematician.
Zygmunt Janiszewski
Born(1888-07-12)12 July 1888
Warsaw, Vistula Land, Russian Empire
Died3 January 1920(1920-01-03) (aged 31)
Lwów, Poland
Resting placeLychakiv cemetery, Lwów
Alma materUniversity of Paris
Known forJaniszewski's theorem
Brouwer–Janiszewski–Knaster continuum
Scientific career
FieldsMathematics
InstitutionsUniversity of Warsaw
Doctoral advisorHenri Lebesgue
Doctoral studentsKazimierz Kuratowski
Early life and education
He was born to mother Julia Szulc-Chojnicka and father, Czeslaw Janiszewski who was a graduate of the University of Warsaw and served as the director of the Société du Crédit Municipal in Warsaw.
Janiszewski left Poland to study mathematics in Zurich, Munich and Göttingen, where he was taught by some of the most prominent mathematicians of the time, such as Heinrich Burkhardt, David Hilbert, Hermann Minkowski and Ernst Zermelo.[2] He then went to Paris and in 1911 received his doctorate in topology under the supervision of Henri Lebesgue. His thesis was titled Sur les continus irréductibles entre deux points (On the Irreducible Continuous Curves Between Two Points).[2] In 1913, he published a seminal work in the field of topology of surface entitled On Cutting the Plane by Continua.
Career
Janiszewski taught at the University of Lwów and was professor at the University of Warsaw. At the outbreak of World War I he was a soldier in the Polish Legions of Józef Piłsudski, and took part in operations around Volyn.[3] Along with other officers, he refused to swear an oath of allegiance to the Austrian government. He subsequently left the Legions and went into hiding under an assumed identity, Zygmunt Wicherkiewicz, in Boiska, near Zwoleń.[2] From Boiska he moved on to Ewin, near Włoszczowa, where he directed a shelter for homeless children.[2]
In 1917, he published an article O potrzebach matematyki w Polsce in the Nauka Polska journal, thus initiating the Polish School of Mathematics.[4] He also founded the journal Fundamenta Mathematicae.[5] Janiszewski proposed the name of the journal in 1919, though the first issue was published in 1920, after his death.
Janiszewski devoted the family property that he had inherited from his father to charity and education. He also donated all the prize money that he received from mathematical awards and competitions to the education and development of young Polish students.
Death
Z. Janiszewski's fiancée was Janina Kelles-Krauz, daughter of Kazimierz Kelles-Krauz. The date of the wedding was set, but due to Z. Janiszewski's death, it did not take place[6]. His life was cut short by the influenza pandemic of 1918–19,[5] which took his life at Lwów on 3 January 1920 at the age of 31. He willed his body for medical research, and his cranium for craniological study, desiring to be "useful after his death".
Samuel Dickstein wrote a commemorative address after Janiszewski's death, honoring his humility, kindness and dedication to his work:
Enthusiasm and strong will characterized Janiszewski not only in his scientific work, but in his life generally. His active participation in the Legions, his refusal to take an oath which was not compatible with his patriotic conscience, his work in the field of education, when at a most difficult time he entered that field as an enlightened and wise worker, free of any prejudice and partiality and ardently keen only to propagate light and truth - these facts prove that in the heart of a mathematician seemingly detached from active life there glowed the purest emotions of affection and self-denial. If we also mention that, having very moderate needs himself, he dispensed all the means at his disposal to educate young talents, and that he bequeathed the property that he had inherited from his parents for educational purposes, and in particular for the education of outstanding individuals, then we may indeed exclaim from the bottom of our hearts that the memory of that life, devoted to the cause and interrupted so early, lives on in its results and deeds and will remain treasured and living for us, the witnesses of his work, and for generations to come.[7]
While Janiszewski best remembered for his many contributions to topological mathematics in the early 20th century, for the founding of Fundamenta Mathematicae, and for his enthusiasm for teaching young minds, his loyalty to his homeland during World War I perhaps gives the greatest insight into his psyche. The orphans' shelter that he set up during the war doubtlessly saved many lives, and is perhaps his greatest contribution to the world.
On 3 January 2020, the 100th anniversary of his death, a researcher from Australia travelled to Lviv and met with the director of Lychakiv Cemetery. Restoration of the grave was arranged, and the stone was restored. Janiszewski is buried in field 58, plot 82 of Lychakiv Cemetery.
See also
• List of Polish mathematicians
Notes
1. "Janiszewski Zygmunt". Astro-Databank. Retrieved 13 November 2021.
2. "Zygmunt Janiszewski (1888 - 1920)". mathshistory.st-andrews.ac.uk. Retrieved 2020-02-11.
3. Domoradzki, Stanislaw; Stawiska, Malgorzata (6 April 2018). "Polish mathematicians and mathematics in World War I. Part II. Russian Empire". Studia Historiae Scientiarum (published 2019). 18: 55–92. arXiv:1804.02448. doi:10.4467/2543702XSHS.19.004.11010.
4. Iłowiecki, Maciej (1981). Dzieje nauki polskiej. Warsaw: Interpress. pp. 251–256.
5. "Placing World War I in the History of Mathematics". HAL archives-ouvertes (Sorbonne). 8 July 2014. Retrieved 11 February 2020.
6. Wilczkowska-Grabowska, Magdalena (1986). "Memories of Janina Kelles-Krauz. Librarian - Monthly of the Association of Polish Librarians, Number 1986 7-8, pages 37-39., 1986". {{cite journal}}: Cite journal requires |journal= (help)
7. Kuratowski 1980, pp. 162–163
References
• Kuratowski, Kazimierz (1980), A Half Century of Polish Mathematics: Remembrances and Reflections, Oxford: Pergamon Press, pp. 158–163, ISBN 0-08-023046-6 et passim.
External links
• Zygmunt Janiszewski at the Mathematics Genealogy Project
• O'Connor, John J.; Robertson, Edmund F., "Zygmunt Janiszewski", MacTutor History of Mathematics Archive, University of St Andrews
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Vertical bar
The vertical bar, |, is a glyph with various uses in mathematics, computing, and typography. It has many names, often related to particular meanings: Sheffer stroke (in logic), pipe, bar, or (literally the word "or"), vbar, and others.[1]
" ‖ " redirects here. For the use of a similar-looking character in African languages, see lateral clicks. For the parallelism symbol see parallel (geometry) and parallel (operator).
|
Vertical bar
In UnicodeU+007C | VERTICAL LINE (|, |, |)
Related
See alsoU+00A6 ¦ BROKEN BAR (¦)
U+2016 ‖ DOUBLE VERTICAL LINE (‖, ‖)
U+2223 ∣ DIVIDES
Usage
Mathematics
The vertical bar is used as a mathematical symbol in numerous ways:
• absolute value: $|x|$, read "the absolute value of x"[2]
• cardinality: $|S|$, read "the cardinality of the set S" or "the length of a string S".
• conditional probability: $P(X|Y)$, read "the probability of X given Y"
• determinant: $|A|$, read "the determinant of the matrix A".[2] When the matrix entries are written out, the determinant is denoted by surrounding the matrix entries by vertical bars instead of the usual brackets or parentheses of the matrix, as in ${\begin{vmatrix}a&b\\c&d\end{vmatrix}}$.
• distance: $P|ab$, denoting the shortest distance between point $P$ to line $ab$, so line $P|ab$ is perpendicular to line $ab$
• divisibility: $a\mid b$, read "a divides b" or "a is a factor of b", though Unicode also provides special 'divides' and 'does not divide' symbols (U+2223 and U+2224: ∣, ∤)[2]
• function evaluation: $f(x)|_{x=4}$, read "f of x, evaluated at x equals 4" (see subscripts at Wikibooks)
• order: $|G|$, read "the order of the group G", or $|g|$, "the order of the element $g\in G$"
• restriction: $f|_{A}$, denoting the restriction of the function $f$, with a domain that is a superset of $A$, to just $A$
• set-builder notation: $\{x|x<2\}$, read "the set of x such that x is less than two". Often, a colon ':' is used instead of a vertical bar
• the Sheffer stroke in logic: $a|b$, read "a nand b"
• subtraction: $f(x)\vert _{a}^{b}$, read "f(x) from a to b", denoting $f(b)-f(a)$. Used in the context of a definite integral with variable x.
• A vertical bar can be used to separate variables from fixed parameters in a function, for example $f(x|\mu ,\sigma )$, or in the notation for elliptic integrals.
The double vertical bar, $\|$, is also employed in mathematics.
• parallelism: $AB\parallel CD$, read "the line $AB$ is parallel to the line $CD$"
• norm: $\|A\|$, read "the norm (length, size, magnitude etc.) of the matrix $A$". The norm of a one-dimensional vector is the absolute value and single bars are used.[3]
• Propositional truncation (a type former that truncates a type down to a mere proposition in homotopy type theory): for any $a:A$ (read "term $a$ of type $A$") we have $|a|:\left\|A\right\|$[4] (here $|a|$ reads "image of $a:A$ in $\left\|A\right\|$" and $|a|:\left\|A\right\|$ reads "propositional truncation of $A$")[5]
In LaTeX mathematical mode, the ASCII vertical bar produces a vertical line, and \| creates a double vertical line (a | b \| c is set as $a|b\|c$). This has different spacing from \mid and \parallel, which are relational operators: a \mid b \parallel c is set as $a\mid b\parallel c$. See below about LaTeX in text mode.
Physics
The vertical bar is used in bra–ket notation in quantum physics. Examples:
• $|\psi \rangle $: the quantum physical state $\psi $
• $\langle \psi |$: the dual state corresponding to the state above
• $\langle \psi |\rho \rangle $: the inner product of states $\psi $ and $\rho $
• Supergroups in physics are denoted G(N|M), which reads "G, M vertical bar N"; here G denotes any supergroup, M denotes the bosonic dimensions, and N denotes the Grassmann dimensions.[6]
Pipe
A pipe is an inter-process communication mechanism originating in Unix, which directs the output (standard out and, optionally, standard error) of one process to the input (standard in) of another. In this way, a series of commands can be "piped" together, giving users the ability to quickly perform complex multi-stage processing from the command line or as part of a Unix shell script ("bash file"). In most Unix shells (command interpreters), this is represented by the vertical bar character. For example:
grep -i 'blair' filename.log | more
where the output from the grep process (all lines containing 'blair') is piped to the more process (which allows a command line user to read through results one page at a time).
The same "pipe" feature is also found in later versions of DOS and Microsoft Windows.
This usage has led to the character itself being called "pipe".
Disjunction
In many programming languages, the vertical bar is used to designate the logic operation or, either bitwise or or logical or.
Specifically, in C and other languages following C syntax conventions, such as C++, Perl, Java and C#, a | b denotes a bitwise or; whereas a double vertical bar a || b denotes a (short-circuited) logical or. Since the character was originally not available in all code pages and keyboard layouts, ANSI C can transcribe it in form of the trigraph ??!, which, outside string literals, is equivalent to the | character.
In regular expression syntax, the vertical bar again indicates logical or (alternation). For example: the Unix command grep -E 'fu|bar' matches lines containing 'fu' or 'bar'.
Concatenation
The double vertical bar operator "||" denotes string concatenation in PL/I, standard ANSI SQL, and theoretical computer science (particularly cryptography).
Delimiter
Although not as common as commas or tabs, the vertical bar can be used as a delimiter in a flat file. Examples of a pipe-delimited standard data format are LEDES 1998B and HL7. It is frequently used because vertical bars are typically uncommon in the data itself.
Similarly, the vertical bar may see use as a delimiter for regular expression operations (e.g. in sed). This is useful when the regular expression contains instances of the more common forward slash (/) delimiter; using a vertical bar eliminates the need to escape all instances of the forward slash. However, this makes the bar unusable as the regular expression "alternative" operator.
Backus–Naur form
In Backus–Naur form, an expression consists of sequences of symbols and/or sequences separated by '|', indicating a choice, the whole being a possible substitution for the symbol on the left.
<personal-name> ::= <name> | <initial>
Concurrency operator
In calculi of communicating processes (like pi-calculus), the vertical bar is used to indicate that processes execute in parallel.
APL
The pipe in APL is the modulo or residue function between two operands and the absolute value function next to one operand.
List comprehensions
The vertical bar is used for list comprehensions in some functional languages, e.g. Haskell and Erlang. Compare set-builder notation.
Text markup
The vertical bar is used as a special character in lightweight markup languages, notably MediaWiki's Wikitext (in the templates and internal links).
In LaTeX text mode, the vertical bar produces an em dash (—). The \textbar command can be used to produce a vertical bar.
Phonetics and orthography
In the Khoisan languages and the International Phonetic Alphabet, the vertical bar is used to write the dental click (ǀ). A double vertical bar is used to write the alveolar lateral click (ǁ). Since these are technically letters, they have their own Unicode code points in the Latin Extended-B range: U+01C0 for the single bar and U+01C1 for the double bar.
Some Northwest and Northeast Caucasian languages written in the Cyrillic script have a vertical bar called palochka (Russian: палочка, lit. 'little stick'), indicating the preceding consonant is an ejective.
Longer single and double vertical bars are used to mark prosodic boundaries in the IPA.
Literature
Punctuation
In medieval European manuscripts, a single vertical bar was a common variant of the virgula ⟨/⟩ used as a period, scratch comma,[7] and caesura mark.[7]
In Sanskrit and other Indian languages, a single vertical mark, a danda, has a similar function as a period (full stop). Two bars || (a 'double danda') is the equivalent of a pilcrow in marking the end of a stanza, paragraph or section. The danda has its own Unicode code point, U+0964.
Poetry
A double vertical bar ⟨||⟩ or ⟨ǁ⟩ is the standard caesura mark in English literary criticism and analysis. It marks the strong break or caesura common to many forms of poetry, particularly Old English verse. It is also traditionally used to mark the division between lines of verse printed as prose (the style preferred by Oxford University Press), though it is now often replaced by the forward slash.
Notation
In the Geneva Bible and early printings of the King James Version, a double vertical bar is used to mark margin notes that contain an alternative translation from the original text. These margin notes always begin with the conjunction "Or". In later printings of the King James Version, the double vertical bar is irregularly used to mark any comment in the margins.
Music scoring
In music, when writing chord sheets, single vertical bars associated with a colon (|: A / / / :|) represents the beginning and end of a section (e.g. Intro, Interlude, Verse, Chorus) of music. Single bars can also represent the beginning and end of measures (|: A / / / | D / / / | E / / / :|). A double vertical bar associated with a colon can represent the repeat of a given section (||: A / / / :|| - play twice).
Encoding
Solid vertical bar vs broken bar
Many early video terminals and dot-matrix printers rendered the vertical bar character as the allograph broken bar ¦. This may have been to distinguish the character from the lower-case 'L' and the upper-case 'I' on these limited-resolution devices, and to make a vertical line of them look more like a horizontal line of dashes. It was also (briefly) part of the ASCII standard.
An initial draft for a 7-bit character set that was published by the X3.2 subcommittee for Coded Character Sets and Data Format on June 8, 1961, was the first to include the vertical bar in a standard set. The bar was intended to be used as the representation for the logical OR symbol.[8] A subsequent draft on May 12, 1966, places the vertical bar in column 7 alongside regional entry codepoints, and formed the basis for the original draft proposal used by the International Standards Organisation.[8] This draft received opposition from the IBM user group SHARE, with its chairman, H. W. Nelson, writing a letter to the American Standards Association titled "The Proposed revised American Standard Code for Information Interchange does NOT meet the needs of computer programmers!"; in this letter, he argues that no characters within the international subset designated at columns 2-5 of the character set would be able to adequately represent logical OR and logical NOT in languages such as IBM's PL/I universally on all platforms.[9] As a compromise, a requirement was introduced where the exclamation mark (!) and circumflex (^) would display as logical OR (|) and logical NOT (¬) respectively in use cases such as programming, while outside of these use cases they would represent their original typographic symbols:
It may be desirable to employ distinctive styling to facilitate their use for specific purposes as, for example, to stylize the graphics in code positions 2/1 and 5/14 to those frequently associated with logical OR (|) and logical NOT (¬) respectively.
— X3.2 document X3.2/475[10]
The original vertical bar encoded at 0x7C in the original May 12, 1966 draft was then broken as ¦, so it could not be confused with the unbroken logical OR. In the 1967 revision of ASCII, along with the equivalent ISO 464 code published the same year, the code point was defined to be a broken vertical bar, and the exclamation mark character was allowed to be rendered as a solid vertical bar.[11][12] However, the 1977 revision (ANSI X.3-1977) undid the changes made in the 1967 revision, enforcing that the circumflex could no longer be stylised as a logical NOT symbol, the exclamation mark likewise no longer allowing stylisation as a vertical bar, and defining the code point originally set to the broken bar as a solid vertical bar instead;[11] the same changes were also reverted in ISO 646-1973 published four years prior.
Some variants of EBCDIC included both versions of the character as different code points. The broad implementation of the extended ASCII ISO/IEC 8859 series in the 1990s also made a distinction between the two forms. This was preserved in Unicode as a separate character at U+00A6 BROKEN BAR (the term "parted rule" is used sometimes in Unicode documentation). Some fonts draw the characters the same (both are solid vertical bars, or both are broken vertical bars).[13] The broken bar does not appear to have any clearly identified uses distinct from those of the vertical bar.[14] In non-computing use — for example in mathematics, physics and general typography — the broken bar is not an acceptable substitute for the vertical bar.
Many keyboards with US or US-International layout display the broken bar on a keycap even though the solid vertical bar character is produced in modern operating systems. This includes many German QWERTZ keyboards. This is a legacy of keyboards manufactured during the 1980s and 1990s for IBM PC compatible computers featuring the broken bar, as such computers used IBM's 8-bit Code page 437 character set based on ASCII, which continued to display the glyph for the broken bar at codepoint 7C on displays from MDA (1981) to VGA (1987) despite the changes made to ASCII in 1977. The UK/Ireland keyboard has both symbols engraved: the broken bar is given as an alternate graphic on the "grave" (backtick) key; the solid bar is on the backslash key.
The broken bar character can be typed (depending on the layout) as AltGr+` or AltGr+6 or AltGr+⇧ Shift+Right \ on Windows and Compose!^ on Linux. It can be inserted into HTML as ¦
In some dictionaries, the broken bar is used to mark stress that may be either primary or secondary. That is, [¦ba] covers the pronunciations [ˈba] and [ˌba].[15]
Unicode code points
These glyphs are encoded in Unicode as follows:
• U+007C | VERTICAL LINE (|, |, |) (single vertical line)
• U+00A6 ¦ BROKEN BAR (¦) (single broken line)
• U+2016 ‖ DOUBLE VERTICAL LINE (‖, ‖) (double vertical line ( $\|$ ): used in pairs to indicate norm)
• U+FF5C ｜ FULLWIDTH VERTICAL LINE (Fullwidth form)
• U+2225 ∥ PARALLEL TO (∥, ∥, ∥, ∥, ∥)
• U+01C0 ǀ LATIN LETTER DENTAL CLICK
• U+01C1 ǁ LATIN LETTER LATERAL CLICK
• U+2223 ∣ DIVIDES (∣, ∣, ∣, ∣)
• U+2502 │ BOX DRAWINGS LIGHT VERTICAL (│) (and various other box drawing characters in the range U+2500 to U+257F)
• U+0964 । DEVANAGARI DANDA
• U+0965 ॥ DEVANAGARI DOUBLE DANDA
Code pages and other historical encodings
Code pages, ASCII, ISO/IEC, EBCDIC, Shift-JIS, etc. Vertical bar (|) Broken bar (¦)
ASCII,
CP437, CP667, CP720, CP737, CP790, CP819, CP852, CP855, CP860, CP861, CP862, CP865, CP866, CP867, CP869, CP872, CP895, CP932, CP991
124 (7Ch) none
CP775 167 (A7h)
CP850, CP857, CP858 221 (DDh)
CP863 160 (A0h)
CP864 219 (DBh)
ISO/IEC 8859-1, -7, -8, -9, -13,
CP1250, CP1251, CP1252, CP1253, CP1254, CP1255, CP1256, CP1257, CP1258
166 (A6h)
ISO/IEC 8859-2, -3, -4, -5, -6, -10, -11, -14, -15, -16 none
EBCDIC CCSID 37 79 (4Fh) 106 (6Ah)
EBCDIC CCSID 500 187 (BBh)
JIS X 0208, JIS X 0213 Men-ku-ten 1-01-35 (7-bit: 2143h; Shift JIS: 8162h; EUC: A1C3h)[lower-alpha 1] none
See also
Look up vertical bar in Wiktionary, the free dictionary.
• Bar (diacritic) – Diacritic used in some languages
• Triple bar – Symbol with multiple meanings
Notes
1. The Shift JIS and EUC encoded forms also include the ASCII vertical bar in its usual encoding (see halfwidth and fullwidth forms). The same applies when the 7-bit form is used as part of ISO-2022-JP (allowing switching to and from ASCII).
References
1. Raymond, Eric S. "ASCII". The Jargon File.
2. Weisstein, Eric W. "Single Bar". mathworld.wolfram.com. Retrieved 2020-08-24.
3. Weisstein, Eric W. "Matrix Norm". mathworld.wolfram.com. Retrieved 2020-08-24.
4. Univalent Foundations Program (2013). Homotopy Type Theory: Univalent Foundations of Mathematics (GitHub version) (PDF). Institute for Advanced Study. p. 108. Archived from the original (PDF) on 2017-07-07. Retrieved 2017-07-01.
5. Univalent Foundations Program (2013). Homotopy Type Theory: Univalent Foundations of Mathematics (print version). Institute for Advanced Study. p. 450.
6. Larus Thorlacius, Thordur Jonsson (eds.), M-Theory and Quantum Geometry, Springer, 2012, p. 263.
7. "virgula, n.", Oxford English Dictionary, 1st ed., Oxford: Oxford University Press, 1917.
8. Fischer, Eric (2012). The Evolution of Character Codes, 1874-1968 (Thesis). Penn State University. CiteSeerX 10.1.1.96.678. Retrieved July 10, 2020.
9. H. W. Nelson, letter to Thomas B. Steel, June 8, 1966, Honeywell Inc. X3.2 Standards Subcommittee Records, 1961-1969 (CBI 67), Charles Babbage Institute, University of Minnesota, Minneapolis, box 1, folder 23.
10. X3.2 document X3.2/475, December 13, 1966, Honeywell Inc. X3.2 Standards Subcommittee Records, 1961-1969 (CBI 67), Charles Babbage Institute, University of Minnesota, Minneapolis, box 1, folder 22.
11. Salste, Tuomas (January 2016). "7-bit character sets: Revisions of ASCII". Aivosto Oy. urn:nbn:fi-fe201201011004. Archived from the original on 2016-06-13. Retrieved 2016-06-13.
12. Korpela, Jukka. "Character histories - notes on some Ascii code positions". Archived from the original on 2020-03-11. Retrieved 2020-05-31.
13. Jim Price (2010-05-24). "ASCII Chart: IBM PC Extended ASCII Display Characters". Retrieved 2012-02-23.
14. Jukka "Yucca" Korpela (2006-09-20). "Detailed descriptions of the characters". Retrieved 2012-02-23.
15. For example, "Balearic". Merriam-Webster Dictionary..
Common punctuation marks and other typographical marks or symbols
• space
• , comma
• : colon
• ; semicolon
• ‐ hyphen
• ’ ' apostrophe
• ′ ″ ‴ prime
• . full stop
• & ampersand
• @ at sign
• ^ caret
• / slash
• \ backslash
• … ellipsis
• * asterisk
• ⁂ asterism
• * * * dinkus
• - hyphen-minus
• ‒ – — dash
• = ⸗ double hyphen
• ? question mark
• ! exclamation mark
• ‽ interrobang
• ¡ ¿ inverted ! and ?
• ⸮ irony punctuation
• # number sign
• № numero sign
• º ª ordinal indicator
• % percent sign
• ‰ per mille
• ‱ basis point
• ° degree symbol
• ⌀ diameter sign
• + − plus and minus signs
• × multiplication sign
• ÷ division sign
• ~ tilde
• ± plus–minus sign
• ∓ minus-plus sign
• _ underscore
• ⁀ tie
• | ¦ ‖ vertical bar
• • bullet
• · interpunct
• © copyright symbol
• © copyleft
• ℗ sound recording copyright
• ® registered trademark
• SM service mark symbol
• TM trademark symbol
• ‘ ’ “ ” ' ' " " quotation mark
• ‹ › « » guillemet
• ( ) [ ] { } ⟨ ⟩ bracket
• ” 〃 ditto mark
• † ‡ dagger
• ❧ hedera/floral heart
• ☞ manicule
• ◊ lozenge
• ¶ ⸿ pilcrow (paragraph mark)
• ※ reference mark
• § section mark
• Version of this table as a sortable list
• Currency symbols
• Diacritics (accents)
• Logic symbols
• Math symbols
• Whitespace
• Chinese punctuation
• Hebrew punctuation
• Japanese punctuation
• Korean punctuation
|
Hexadecimal
In mathematics and computing, the hexadecimal (also base-16 or simply hex) numeral system is a positional numeral system that represents numbers using a radix (base) of sixteen. Unlike the decimal system representing numbers using ten symbols, hexadecimal uses sixteen distinct symbols, most often the symbols "0"–"9" to represent values 0 to 9, and "A"–"F" (or alternatively "a"–"f") to represent values from ten to fifteen.
"Sexadecimal" redirects here. For base 60, see Sexagesimal.
Part of a series on
Numeral systems
Place-value notation
Hindu-Arabic numerals
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• Bengali
• Devanagari
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• Tamil
• Malayalam
• Telugu
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• Balinese
• Burmese
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East Asian systems
Contemporary
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Historic
• Counting rods
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Other systems
• History
Ancient
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Post-classical
• Cistercian
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• Pentadic
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Contemporary
• Cherokee
• Kaktovik (Iñupiaq)
By radix/base
Common radices/bases
• 2
• 3
• 4
• 5
• 6
• 8
• 10
• 12
• 16
• 20
• 60
• (table)
Non-standard radices/bases
• Bijective (1)
• Signed-digit (balanced ternary)
• Mixed (factorial)
• Negative
• Complex (2i)
• Non-integer (φ)
• Asymmetric
Sign-value notation
Non-alphabetic
• Aegean
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Alphabetic
• Abjad
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• Glagolitic
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List of numeral systems
Software developers and system designers widely use hexadecimal numbers because they provide a human-friendly representation of binary-coded values. Each hexadecimal digit represents four bits (binary digits), also known as a nibble (or nybble).[1] For example, an 8-bit byte can have values ranging from 00000000 to 11111111 in binary form, which can be conveniently represented as 00 to FF in hexadecimal.
In mathematics, a subscript is typically used to specify the base. For example, the decimal value 38,967 would be expressed in hexadecimal as 983716. In programming, several notations denote hexadecimal numbers, usually involving a prefix. The prefix 0x is used in C, which would denote this value as 0x9837.
Hexadecimal is used in the transfer encoding Base16, in which each byte of the plaintext is broken into two 4-bit values and represented by two hexadecimal digits.
Representation
Written representation
In most current use cases, the letters A–F or a–f represent the values 10–15, while the numerals 0–9 are used to represent their decimal values.
There is no universal convention to use lowercase or uppercase, so each is prevalent or preferred in particular environments by community standards or convention; even mixed case is used. Seven-segment displays use mixed-case AbCdEF to make digits that can be distinguished from each other.
There is some standardization of using spaces (rather than commas or another punctuation mark) to separate hex values in a long list. For instance, in the following hex dump, each 8-bit byte is a 2-digit hex number, with spaces between them, while the 32-bit offset at the start is an 8-digit hex number.
00000000 57 69 6b 69 70 65 64 69 61 2c 20 74 68 65 20 66
00000010 72 65 65 20 65 6e 63 79 63 6c 6f 70 65 64 69 61
00000020 20 74 68 61 74 20 61 6e 79 6f 6e 65 20 63 61 6e
00000030 20 65 64 69 74 0a
Distinguishing from decimal
In contexts where the base is not clear, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously. A numerical subscript (itself written in decimal) can give the base explicitly: 15910 is decimal 159; 15916 is hexadecimal 159, which equals 34510. Some authors prefer a text subscript, such as 159decimal and 159hex, or 159d and 159h.
Donald Knuth introduced the use of a particular typeface to represent a particular radix in his book The TeXbook.[2] Hexadecimal representations are written there in a typewriter typeface: 5A3
In linear text systems, such as those used in most computer programming environments, a variety of methods have arisen:
• Unix (and related) shells, AT&T assembly language and likewise the C programming language (and its syntactic descendants such as C++, C#, Go, D, Java, JavaScript, Python and Windows PowerShell) use the prefix 0x for numeric constants represented in hex: 0x5A3. Character and string constants may express character codes in hexadecimal with the prefix \x followed by two hex digits: '\x1B' represents the Esc control character; "\x1B[0m\x1B[25;1H" is a string containing 11 characters with two embedded Esc characters.[3] To output an integer as hexadecimal with the printf function family, the format conversion code %X or %x is used.
• In XML and XHTML, characters can be expressed as hexadecimal numeric character references using the notation ode;, for instance ’ represents the character U+2019 (the right single quotation mark). If there is no x the number is decimal (thus ’ is the same character).[4]
• In Intel-derived assembly languages and Modula-2,[5] hexadecimal is denoted with a suffixed H or h: FFh or 05A3H. Some implementations require a leading zero when the first hexadecimal digit character is not a decimal digit, so one would write 0FFh instead of FFh. Some other implementations (such as NASM) allow C-style numbers (0x42).
• Other assembly languages (6502, Motorola), Pascal, Delphi, some versions of BASIC (Commodore), GameMaker Language, Godot and Forth use $ as a prefix: $5A3.
• Some assembly languages (Microchip) use the notation H'ABCD' (for ABCD16). Similarly, Fortran 95 uses Z'ABCD'.
• Ada and VHDL enclose hexadecimal numerals in based "numeric quotes": 16#5A3#. For bit vector constants VHDL uses the notation x"5A3".[6]
• Verilog represents hexadecimal constants in the form 8'hFF, where 8 is the number of bits in the value and FF is the hexadecimal constant.
• The Smalltalk language uses the prefix 16r: 16r5A3
• PostScript and the Bourne shell and its derivatives denote hex with prefix 16#: 16#5A3.
• Common Lisp uses the prefixes #x and #16r. Setting the variables *read-base*[7] and *print-base*[8] to 16 can also be used to switch the reader and printer of a Common Lisp system to Hexadecimal number representation for reading and printing numbers. Thus Hexadecimal numbers can be represented without the #x or #16r prefix code, when the input or output base has been changed to 16.
• MSX BASIC,[9] QuickBASIC, FreeBASIC and Visual Basic prefix hexadecimal numbers with &H: &H5A3
• BBC BASIC and Locomotive BASIC use & for hex.[10]
• TI-89 and 92 series uses a 0h prefix: 0h5A3
• ALGOL 68 uses the prefix 16r to denote hexadecimal numbers: 16r5a3. Binary, quaternary (base-4) and octal numbers can be specified similarly.
• The most common format for hexadecimal on IBM mainframes (zSeries) and midrange computers (IBM i) running the traditional OS's (zOS, zVSE, zVM, TPF, IBM i) is X'5A3', and is used in Assembler, PL/I, COBOL, JCL, scripts, commands and other places. This format was common on other (and now obsolete) IBM systems as well. Occasionally quotation marks were used instead of apostrophes.
Syntax that is always Hex
Sometimes the numbers are known to be Hex.
• In URIs (including URLs), character codes are written as hexadecimal pairs prefixed with %: http://www.example.com/name%20with%20spaces where %20 is the code for the space (blank) character, ASCII code point 20 in hex, 32 in decimal.
• In the Unicode standard, a character value is represented with U+ followed by the hex value, e.g. U+20AC is the Euro sign (€).
• Color references in HTML, CSS and X Window can be expressed with six hexadecimal digits (two each for the red, green and blue components, in that order) prefixed with #: white, for example, is represented as #FFFFFF.[11] CSS also allows 3-hexdigit abbreviations with one hexdigit per component: #FA3 abbreviates #FFAA33 (a golden orange: ).
• In MIME (e-mail extensions) quoted-printable encoding, character codes are written as hexadecimal pairs prefixed with =: Espa=F1a is "España" (F1 is the code for ñ in the ISO/IEC 8859-1 character set).[12])
• PostScript binary data (such as image pixels) can be expressed as unprefixed consecutive hexadecimal pairs: AA213FD51B3801043FBC...
• Any IPv6 address can be written as eight groups of four hexadecimal digits (sometimes called hextets), where each group is separated by a colon (:). This, for example, is a valid IPv6 address: 2001:0db8:85a3:0000:0000:8a2e:0370:7334 or abbreviated by removing zeros as 2001:db8:85a3::8a2e:370:7334 (IPv4 addresses are usually written in decimal).
• Globally unique identifiers are written as thirty-two hexadecimal digits, often in unequal hyphen-separated groupings, for example 3F2504E0-4F89-41D3-9A0C-0305E82C3301.
Other symbols for 10–15 and mostly different symbol sets
The use of the letters A through F to represent the digits above 9 was not universal in the early history of computers.
• During the 1950s, some installations, such as Bendix-14, favored using the digits 0 through 5 with an overline to denote the values 10–15 as 0, 1, 2, 3, 4 and 5.
• The SWAC (1950)[13] and Bendix G-15 (1956)[14][13] computers used the lowercase letters u, v, w, x, y and z for the values 10 to 15.
• The ORDVAC and ILLIAC I (1952) computers (and some derived designs, e.g. BRLESC) used the uppercase letters K, S, N, J, F and L for the values 10 to 15.[15][13]
• The Librascope LGP-30 (1956) used the letters F, G, J, K, Q and W for the values 10 to 15.[16][13]
• On the PERM (1956) computer, hexadecimal numbers were written as letters O for zero, A to N and P for 1 to 15. Many machine instructions had mnemonic hex-codes (A=add, M=multiply, L=load, F=fixed-point etc.); programs were written without instruction names.[17]
• The Honeywell Datamatic D-1000 (1957) used the lowercase letters b, c, d, e, f, and g whereas the Elbit 100 (1967) used the uppercase letters B, C, D, E, F and G for the values 10 to 15.[13]
• The Monrobot XI (1960) used the letters S, T, U, V, W and X for the values 10 to 15.[13]
• The NEC parametron computer NEAC 1103 (1960) used the letters D, G, H, J, K (and possibly V) for values 10–15.[18]
• The Pacific Data Systems 1020 (1964) used the letters L, C, A, S, M and D for the values 10 to 15.[13]
• New numeric symbols and names were introduced in the Bibi-binary notation by Boby Lapointe in 1968.
• Bruce Alan Martin of Brookhaven National Laboratory considered the choice of A–F "ridiculous". In a 1968 letter to the editor of the CACM, he proposed an entirely new set of symbols based on the bit locations.[19]
• Ronald O. Whitaker of Rowco Engineering Co., in 1972, proposed a triangular font that allows "direct binary reading" in order to "permit both input and output from computers without respect to encoding matrices."[20][21]
• Some seven-segment display decoder chips (i.e., 74LS47) show unexpected output due to logic designed only to produce 0–9 correctly.[22]
Verbal and digital representations
Since there were no traditional numerals to represent the quantities from ten to fifteen, alphabetic letters were re-employed as a substitute. Most European languages lack non-decimal-based words for some of the numerals eleven to fifteen. Some people read hexadecimal numbers digit by digit, like a phone number, or using the NATO phonetic alphabet, the Joint Army/Navy Phonetic Alphabet, or a similar ad-hoc system. In the wake of the adoption of hexadecimal among IBM System/360 programmers, Magnuson (1968)[23] suggested a pronunciation guide that gave short names to the letters of hexadecimal – for instance, "A" was pronounced "ann", B "bet", C "chris", etc.[23] Another naming-system was published online by Rogers (2007)[24] that tries to make the verbal representation distinguishable in any case, even when the actual number does not contain numbers A–F. Examples are listed in the tables below. Yet another naming system was elaborated by Babb (2015), based on a joke in Silicon Valley.[25]
Others have proposed using the verbal Morse Code conventions to express four-bit hexadecimal digits, with "dit" and "dah" representing zero and one, respectively, so that "0000" is voiced as "dit-dit-dit-dit" (....), dah-dit-dit-dah (-..-) voices the digit with a value of nine, and "dah-dah-dah-dah" (----) voices the hexadecimal digit for decimal 15.
Systems of counting on digits have been devised for both binary and hexadecimal. Arthur C. Clarke suggested using each finger as an on/off bit, allowing finger counting from zero to 102310 on ten fingers.[26] Another system for counting up to FF16 (25510) is illustrated on the right.
Magnuson (1968)[23]
naming method
NumberPronunciation
Aann
Bbet
Cchris
Ddot
Eernest
Ffrost
1Aannteen
A0annty
5Bfifty-bet
A01Cannty christeen
1AD0annteen dotty
3A7Dthirty-ann seventy-dot
Rogers (2007)[24]
naming method
NumberPronunciation
Aten
Beleven
Ctwelve
Ddraze
Eeptwin
Ffim
10tex
11oneteek
1Ffimteek
50fiftek
C0twelftek
100hundrek
1000thousek
3Ethirtek-eptwin
E1eptek-one
C4Atwelve-hundrek-fourtek-ten
1743one-thousek-seven-
-hundrek-fourtek-three
Signs
The hexadecimal system can express negative numbers the same way as in decimal: −2A to represent −4210 and so on.
Hexadecimal can also be used to express the exact bit patterns used in the processor, so a sequence of hexadecimal digits may represent a signed or even a floating-point value. This way, the negative number −4210 can be written as FFFF FFD6 in a 32-bit CPU register (in two's-complement), as C228 0000 in a 32-bit FPU register or C045 0000 0000 0000 in a 64-bit FPU register (in the IEEE floating-point standard).
Hexadecimal exponential notation
Just as decimal numbers can be represented in exponential notation, so too can hexadecimal numbers. P notation uses the letter P (or p, for "power"), whereas E (or e) serves a similar purpose in decimal E notation. The number after the P is decimal and represents the binary exponent. Increasing the exponent by 1 multiplies by 2, not 16: 20p0 = 10p1 = 8p2 = 4p3 = 2p4 = 1.0p5. Usually, the number is normalized so that the hexadecimal digits start with 1. (zero is usually 0 with no P).
Example: 1.3DEp42 represents 1.3DE16 × 24210.
P notation is required by the IEEE 754-2008 binary floating-point standard, and can be used for floating-point literals in the C99 edition of the C programming language.[27] Using the %a or %A conversion specifiers, this notation can be produced by implementations of the printf family of functions following the C99 specification[28] and Single Unix Specification (IEEE Std 1003.1) POSIX standard.[29]
Conversion
Binary conversion
Most computers manipulate binary data, but it is difficult for humans to work with a large number of digits for even a relatively small binary number. Although most humans are familiar with the base 10 system, it is much easier to map binary to hexadecimal than to decimal because each hexadecimal digit maps to a whole number of bits (410). This example converts 11112 to base ten. Since each position in a binary numeral can contain either a 1 or a 0, its value may be easily determined by its position from the right:
• 00012 = 110
• 00102 = 210
• 01002 = 410
• 10002 = 810
Therefore:
11112= 810 + 410 + 210 + 110
= 1510
With little practice, mapping 11112 to F16 in one step becomes easy: see table in written representation. The advantage of using hexadecimal rather than decimal increases rapidly with the size of the number. When the number becomes large, conversion to decimal is very tedious. However, when mapping to hexadecimal, it is trivial to regard the binary string as 4-digit groups and map each to a single hexadecimal digit.[30]
This example shows the conversion of a binary number to decimal, mapping each digit to the decimal value, and adding the results.
(01011110101101010010)2= 26214410 + 6553610 + 3276810 + 1638410 + 819210 + 204810 + 51210 + 25610 + 6410 + 1610 + 210
= 38792210
Compare this to the conversion to hexadecimal, where each group of four digits can be considered independently, and converted directly:
(01011110101101010010)2=0101 1110 1011 0101 00102
=5EB5216
=5EB5216
The conversion from hexadecimal to binary is equally direct.[30]
Other simple conversions
Although quaternary (base 4) is little used, it can easily be converted to and from hexadecimal or binary. Each hexadecimal digit corresponds to a pair of quaternary digits and each quaternary digit corresponds to a pair of binary digits. In the above example 5 E B 5 216 = 11 32 23 11 024.
The octal (base 8) system can also be converted with relative ease, although not quite as trivially as with bases 2 and 4. Each octal digit corresponds to three binary digits, rather than four. Therefore, we can convert between octal and hexadecimal via an intermediate conversion to binary followed by regrouping the binary digits in groups of either three or four.
Division-remainder in source base
As with all bases there is a simple algorithm for converting a representation of a number to hexadecimal by doing integer division and remainder operations in the source base. In theory, this is possible from any base, but for most humans only decimal and for most computers only binary (which can be converted by far more efficient methods) can be easily handled with this method.
Let d be the number to represent in hexadecimal, and the series hihi−1...h2h1 be the hexadecimal digits representing the number.
1. i ← 1
2. hi ← d mod 16
3. d ← (d − hi) / 16
4. If d = 0 (return series hi) else increment i and go to step 2
"16" may be replaced with any other base that may be desired.
The following is a JavaScript implementation of the above algorithm for converting any number to a hexadecimal in String representation. Its purpose is to illustrate the above algorithm. To work with data seriously, however, it is much more advisable to work with bitwise operators.
function toHex(d) {
var r = d % 16;
if (d - r == 0) {
return toChar(r);
}
return toHex((d - r) / 16) + toChar(r);
}
function toChar(n) {
const alpha = "0123456789ABCDEF";
return alpha.charAt(n);
}
Conversion through addition and multiplication
It is also possible to make the conversion by assigning each place in the source base the hexadecimal representation of its place value — before carrying out multiplication and addition to get the final representation. For example, to convert the number B3AD to decimal, one can split the hexadecimal number into its digits: B (1110), 3 (310), A (1010) and D (1310), and then get the final result by multiplying each decimal representation by 16p (p being the corresponding hex digit position, counting from right to left, beginning with 0). In this case, we have that:
B3AD = (11 × 163) + (3 × 162) + (10 × 161) + (13 × 160)
which is 45997 in base 10.
Tools for conversion
Many computer systems provide a calculator utility capable of performing conversions between the various radices frequently including hexadecimal.
In Microsoft Windows, the Calculator utility can be set to Programmer mode, which allows conversions between radix 16 (hexadecimal), 10 (decimal), 8 (octal) and 2 (binary), the bases most commonly used by programmers. In Programmer Mode, the on-screen numeric keypad includes the hexadecimal digits A through F, which are active when "Hex" is selected. In hex mode, however, the Windows Calculator supports only integers.
Elementary arithmetic
Elementary operations such as addition, subtraction, multiplication and division can be carried out indirectly through conversion to an alternate numeral system, such as the commonly-used decimal system or the binary system where each hex digit corresponds to four binary digits.
Alternatively, one can also perform elementary operations directly within the hex system itself — by relying on its addition/multiplication tables and its corresponding standard algorithms such as long division and the traditional subtraction algorithm.
Real numbers
Rational numbers
As with other numeral systems, the hexadecimal system can be used to represent rational numbers, although repeating expansions are common since sixteen (1016) has only a single prime factor; two.
For any base, 0.1 (or "1/10") is always equivalent to one divided by the representation of that base value in its own number system. Thus, whether dividing one by two for binary or dividing one by sixteen for hexadecimal, both of these fractions are written as 0.1. Because the radix 16 is a perfect square (42), fractions expressed in hexadecimal have an odd period much more often than decimal ones, and there are no cyclic numbers (other than trivial single digits). Recurring digits are exhibited when the denominator in lowest terms has a prime factor not found in the radix; thus, when using hexadecimal notation, all fractions with denominators that are not a power of two result in an infinite string of recurring digits (such as thirds and fifths). This makes hexadecimal (and binary) less convenient than decimal for representing rational numbers since a larger proportion lie outside its range of finite representation.
All rational numbers finitely representable in hexadecimal are also finitely representable in decimal, duodecimal and sexagesimal: that is, any hexadecimal number with a finite number of digits also has a finite number of digits when expressed in those other bases. Conversely, only a fraction of those finitely representable in the latter bases are finitely representable in hexadecimal. For example, decimal 0.1 corresponds to the infinite recurring representation 0.19 in hexadecimal. However, hexadecimal is more efficient than duodecimal and sexagesimal for representing fractions with powers of two in the denominator. For example, 0.062510 (one-sixteenth) is equivalent to 0.116, 0.0912, and 0;3,4560.
n Decimal
Prime factors of: base, b = 10: 2, 5;
b − 1 = 9: 3
Hexadecimal
Prime factors of: base, b = 1610 = 10: 2; b − 1 = 1510 = F: 3, 5
Reciprocal Prime factors Positional representation
(decimal)
Positional representation
(hexadecimal)
Prime factors Reciprocal
2 1/2 2 0.5 0.8 2 1/2
3 1/3 3 0.3333... = 0.3 0.5555... = 0.5 3 1/3
4 1/4 2 0.25 0.4 2 1/4
5 1/5 5 0.2 0.3 5 1/5
6 1/6 2, 3 0.16 0.2A 2, 3 1/6
7 1/7 7 0.142857 0.249 7 1/7
8 1/8 2 0.125 0.2 2 1/8
9 1/9 3 0.1 0.1C7 3 1/9
10 1/10 2, 5 0.1 0.19 2, 5 1/A
11 1/11 11 0.09 0.1745D B 1/B
12 1/12 2, 3 0.083 0.15 2, 3 1/C
13 1/13 13 0.076923 0.13B D 1/D
14 1/14 2, 7 0.0714285 0.1249 2, 7 1/E
15 1/15 3, 5 0.06 0.1 3, 5 1/F
16 1/16 2 0.0625 0.1 2 1/10
17 1/17 17 0.0588235294117647 0.0F 11 1/11
18 1/18 2, 3 0.05 0.0E38 2, 3 1/12
19 1/19 19 0.052631578947368421 0.0D79435E5 13 1/13
20 1/20 2, 5 0.05 0.0C 2, 5 1/14
21 1/21 3, 7 0.047619 0.0C3 3, 7 1/15
22 1/22 2, 11 0.045 0.0BA2E8 2, B 1/16
23 1/23 23 0.0434782608695652173913 0.0B21642C859 17 1/17
24 1/24 2, 3 0.0416 0.0A 2, 3 1/18
25 1/25 5 0.04 0.0A3D7 5 1/19
26 1/26 2, 13 0.0384615 0.09D8 2, D 1/1A
27 1/27 3 0.037 0.097B425ED 3 1/1B
28 1/28 2, 7 0.03571428 0.0924 2, 7 1/1C
29 1/29 29 0.0344827586206896551724137931 0.08D3DCB 1D 1/1D
30 1/30 2, 3, 5 0.03 0.08 2, 3, 5 1/1E
31 1/31 31 0.032258064516129 0.08421 1F 1/1F
32 1/32 2 0.03125 0.08 2 1/20
33 1/33 3, 11 0.03 0.07C1F 3, B 1/21
34 1/34 2, 17 0.02941176470588235 0.078 2, 11 1/22
35 1/35 5, 7 0.0285714 0.075 5, 7 1/23
36 1/36 2, 3 0.027 0.071C 2, 3 1/24
Irrational numbers
The table below gives the expansions of some common irrational numbers in decimal and hexadecimal.
Number Positional representation
Decimal Hexadecimal
√2 (the length of the diagonal of a unit square) 1.414213562373095048... 1.6A09E667F3BCD...
√3 (the length of the diagonal of a unit cube) 1.732050807568877293... 1.BB67AE8584CAA...
√5 (the length of the diagonal of a 1×2 rectangle) 2.236067977499789696... 2.3C6EF372FE95...
φ (phi, the golden ratio = (1+√5)/2) 1.618033988749894848... 1.9E3779B97F4A...
π (pi, the ratio of circumference to diameter of a circle) 3.141592653589793238462643
383279502884197169399375105...
3.243F6A8885A308D313198A2E0
3707344A4093822299F31D008...
e (the base of the natural logarithm) 2.718281828459045235... 2.B7E151628AED2A6B...
τ (the Thue–Morse constant) 0.412454033640107597... 0.6996 9669 9669 6996...
γ (the limiting difference between the harmonic series and the natural logarithm) 0.577215664901532860... 0.93C467E37DB0C7A4D1B...
Powers
Powers of two have very simple expansions in hexadecimal. The first sixteen powers of two are shown below.
2xValueValue (Decimal)
2011
2122
2244
2388
2410hex16dec
2520hex32dec
2640hex64dec
2780hex128dec
28100hex256dec
29200hex512dec
2A (210dec)400hex1024dec
2B (211dec)800hex2048dec
2C (212dec)1000hex4096dec
2D (213dec)2000hex8192dec
2E (214dec)4000hex16,384dec
2F (215dec)8000hex32,768dec
210 (216dec)10000hex65,536dec
Cultural history
The traditional Chinese units of measurement were base-16. For example, one jīn (斤) in the old system equals sixteen taels. The suanpan (Chinese abacus) can be used to perform hexadecimal calculations such as additions and subtractions.[31]
As with the duodecimal system, there have been occasional attempts to promote hexadecimal as the preferred numeral system. These attempts often propose specific pronunciation and symbols for the individual numerals.[32] Some proposals unify standard measures so that they are multiples of 16.[33][34] An early such proposal was put forward by John W. Nystrom in Project of a New System of Arithmetic, Weight, Measure and Coins: Proposed to be called the Tonal System, with Sixteen to the Base, published in 1862.[35] Nystrom among other things suggested hexadecimal time, which subdivides a day by 16, so that there are 16 "hours" (or "10 tims", pronounced tontim) in a day.[36]
Look up hexadecimal in Wiktionary, the free dictionary.
The word hexadecimal is first recorded in 1952.[37] It is macaronic in the sense that it combines Greek ἕξ (hex) "six" with Latinate -decimal. The all-Latin alternative sexadecimal (compare the word sexagesimal for base 60) is older, and sees at least occasional use from the late 19th century.[38] It is still in use in the 1950s in Bendix documentation. Schwartzman (1994) argues that use of sexadecimal may have been avoided because of its suggestive abbreviation to sex.[39] Many western languages since the 1960s have adopted terms equivalent in formation to hexadecimal (e.g. French hexadécimal, Italian esadecimale, Romanian hexazecimal, Serbian хексадецимални, etc.) but others have introduced terms which substitute native words for "sixteen" (e.g. Greek δεκαεξαδικός, Icelandic sextándakerfi, Russian шестнадцатеричной etc.)
Terminology and notation did not become settled until the end of the 1960s. Donald Knuth in 1969 argued that the etymologically correct term would be senidenary, or possibly sedenary, a Latinate term intended to convey "grouped by 16" modelled on binary, ternary and quaternary etc. According to Knuth's argument, the correct terms for decimal and octal arithmetic would be denary and octonary, respectively.[40] Alfred B. Taylor used senidenary in his mid-1800s work on alternative number bases, although he rejected base 16 because of its "incommodious number of digits".[41][42]
The now-current notation using the letters A to F establishes itself as the de facto standard beginning in 1966, in the wake of the publication of the Fortran IV manual for IBM System/360, which (unlike earlier variants of Fortran) recognizes a standard for entering hexadecimal constants.[43] As noted above, alternative notations were used by NEC (1960) and The Pacific Data Systems 1020 (1964). The standard adopted by IBM seems to have become widely adopted by 1968, when Bruce Alan Martin in his letter to the editor of the CACM complains that
With the ridiculous choice of letters A, B, C, D, E, F as hexadecimal number symbols adding to already troublesome problems of distinguishing octal (or hex) numbers from decimal numbers (or variable names), the time is overripe for reconsideration of our number symbols. This should have been done before poor choices gelled into a de facto standard!
Martin's argument was that use of numerals 0 to 9 in nondecimal numbers "imply to us a base-ten place-value scheme": "Why not use entirely new symbols (and names) for the seven or fifteen nonzero digits needed in octal or hex. Even use of the letters A through P would be an improvement, but entirely new symbols could reflect the binary nature of the system".[19] He also argued that "re-using alphabetic letters for numerical digits represents a gigantic backward step from the invention of distinct, non-alphabetic glyphs for numerals sixteen centuries ago" (as Brahmi numerals, and later in a Hindu–Arabic numeral system), and that the recent ASCII standards (ASA X3.4-1963 and USAS X3.4-1968) "should have preserved six code table positions following the ten decimal digits -- rather than needlessly filling these with punctuation characters" (":;<=>?") that might have been placed elsewhere among the 128 available positions.
Base16 (transfer encoding)
Base16 (as a proper name without a space) can also refer to a binary to text encoding belonging to the same family as Base32, Base58, and Base64.
In this case, data is broken into 4-bit sequences, and each value (between 0 and 15 inclusively) is encoded using one of 16 symbols from the ASCII character set. Although any 16 symbols from the ASCII character set can be used, in practice the ASCII digits '0'–'9' and the letters 'A'–'F' (or the lowercase 'a'–'f') are always chosen in order to align with standard written notation for hexadecimal numbers.
There are several advantages of Base16 encoding:
• Most programming languages already have facilities to parse ASCII-encoded hexadecimal
• Being exactly half a byte, 4-bits is easier to process than the 5 or 6 bits of Base32 and Base64 respectively
• The symbols 0–9 and A–F are universal in hexadecimal notation, so it is easily understood at a glance without needing to rely on a symbol lookup table
• Many CPU architectures have dedicated instructions that allow access to a half-byte (otherwise known as a "nibble"), making it more efficient in hardware than Base32 and Base64
The main disadvantages of Base16 encoding are:
• Space efficiency is only 50%, since each 4-bit value from the original data will be encoded as an 8-bit byte. In contrast, Base32 and Base64 encodings have a space efficiency of 63% and 75% respectively.
• Possible added complexity of having to accept both uppercase and lowercase letters
Support for Base16 encoding is ubiquitous in modern computing. It is the basis for the W3C standard for URL percent encoding, where a character is replaced with a percent sign "%" and its Base16-encoded form. Most modern programming languages directly include support for formatting and parsing Base16-encoded numbers.
See also
• Base32, Base64 (content encoding schemes)
• Hexadecimal time
• IBM hexadecimal floating-point
• Hex editor
• Hex dump
• Bailey–Borwein–Plouffe formula (BBP)
• Hexspeak
• P notation
References
1. "The hexadecimal system". Ionos Digital Guide. Archived from the original on 2022-08-26. Retrieved 2022-08-26.
2. Knuth, Donald Ervin (1986). The TeXbook. Duane Bibby. Reading, Mass. ISBN 0-201-13447-0. OCLC 12973034. Archived from the original on 2022-01-16. Retrieved 2022-03-15.{{cite book}}: CS1 maint: location missing publisher (link)
3. The string "\x1B[0m\x1B[25;1H" specifies the character sequence Esc [ 0 m Esc [ 2 5 ; 1 H Nul. These are the escape sequences used on an ANSI terminal that reset the character set and color, and then move the cursor to line 25.
4. "The Unicode Standard, Version 7" (PDF). Unicode. Archived (PDF) from the original on 2016-03-03. Retrieved 2018-10-28.
5. "Modula-2 – Vocabulary and representation". Modula −2. Archived from the original on 2015-12-13. Retrieved 2015-11-01.
6. "An Introduction to VHDL Data Types". FPGA Tutorial. 2020-05-10. Archived from the original on 2020-08-23. Retrieved 2020-08-21.
7. "*read-base* variable in Common Lisp". CLHS. Archived from the original on 2016-02-03. Retrieved 2015-01-10.
8. "*print-base* variable in Common Lisp". CLHS. Archived from the original on 2014-12-26. Retrieved 2015-01-10.
9. MSX is Coming — Part 2: Inside MSX Archived 2010-11-24 at the Wayback Machine Compute!, issue 56, January 1985, p. 52
10. BBC BASIC programs are not fully portable to Microsoft BASIC (without modification) since the latter takes & to prefix octal values. (Microsoft BASIC primarily uses &O to prefix octal, and it uses &H to prefix hexadecimal, but the ampersand alone yields a default interpretation as an octal prefix.
11. "Hexadecimal web colors explained". Archived from the original on 2006-04-22. Retrieved 2006-01-11.
12. "ISO-8859-1 (ISO Latin 1) Character Encoding". www.ic.unicamp.br. Archived from the original on 2019-06-29. Retrieved 2019-06-26.
13. Savard, John J. G. (2018) [2005]. "Computer Arithmetic". quadibloc. The Early Days of Hexadecimal. Archived from the original on 2018-07-16. Retrieved 2018-07-16.
14. "2.1.3 Sexadecimal notation". G15D Programmer's Reference Manual (PDF). Los Angeles, CA, USA: Bendix Computer, Division of Bendix Aviation Corporation. p. 4. Archived (PDF) from the original on 2017-06-01. Retrieved 2017-06-01. This base is used because a group of four bits can represent any one of sixteen different numbers (zero to fifteen). By assigning a symbol to each of these combinations we arrive at a notation called sexadecimal (usually hex in conversation because nobody wants to abbreviate sex). The symbols in the sexadecimal language are the ten decimal digits and, on the G-15 typewriter, the letters u, v, w, x, y and z. These are arbitrary markings; other computers may use different alphabet characters for these last six digits.
15. Gill, S.; Neagher, R. E.; Muller, D. E.; Nash, J. P.; Robertson, J. E.; Shapin, T.; Whesler, D. J. (1956-09-01). Nash, J. P. (ed.). "ILLIAC Programming – A Guide to the Preparation of Problems For Solution by the University of Illinois Digital Computer" (PDF). bitsavers.org (Fourth printing. Revised and corrected ed.). Urbana, Illinois, USA: Digital Computer Laboratory, Graduate College, University of Illinois. pp. 3–2. Archived (PDF) from the original on 2017-05-31. Retrieved 2014-12-18.
16. ROYAL PRECISION Electronic Computer LGP – 30 PROGRAMMING MANUAL. Port Chester, New York: Royal McBee Corporation. April 1957. Archived from the original on 2017-05-31. Retrieved 2017-05-31. (NB. This somewhat odd sequence was from the next six sequential numeric keyboard codes in the LGP-30's 6-bit character code.)
17. Manthey, Steffen; Leibrandt, Klaus (2002-07-02). "Die PERM und ALGOL" (PDF) (in German). Archived (PDF) from the original on 2018-10-03. Retrieved 2018-05-19.
18. NEC Parametron Digital Computer Type NEAC-1103 (PDF). Tokyo, Japan: Nippon Electric Company Ltd. 1960. Cat. No. 3405-C. Archived (PDF) from the original on 2017-05-31. Retrieved 2017-05-31.
19. Martin, Bruce Alan (October 1968). "Letters to the editor: On binary notation". Communications of the ACM. Associated Universities Inc. 11 (10): 658. doi:10.1145/364096.364107. S2CID 28248410.
20. Whitaker, Ronald O. (January 1972). Written at Indianapolis, Indiana, USA. "More on man/machine" (PDF). Letters. Datamation. Vol. 18, no. 1. Barrington, Illinois, USA: Technical Publishing Company. p. 103. Archived (PDF) from the original on 2022-12-05. Retrieved 2022-12-24. (1 page)
21. Whitaker, Ronald O. (1976-08-10) [1975-02-24]. "Combined display and range selector for use with digital instruments employing the binary numbering system" (PDF). Indianapolis, Indiana, USA. US Patent 3974444A. Archived (PDF) from the original on 2022-12-24. Retrieved 2022-12-24. (7 pages)
22. "SN5446A, '47A, '48, SN54LS47, 'LS48, 'LS49, SN7446A, '47A, '48, SN74LS47, 'LS48, 'LS49 BCD-to-Seven-Segment Decoders/Drivers". Dallas, Texas, USA: Texas Instruments Incorporated. March 1988 [1974]. SDLS111. Archived (PDF) from the original on 2021-10-20. Retrieved 2021-09-15. (29 pages)
23. Magnuson, Robert A. (January 1968). "A hexadecimal pronunciation guide". Datamation. Vol. 14, no. 1. p. 45.
24. Rogers, S.R. (2007). "Hexadecimal number words". Intuitor. Archived from the original on 2019-09-17. Retrieved 2019-08-26.
25. Babb, Tim (2015). "How to pronounce hexadecimal". Bzarg. Archived from the original on 2020-11-11. Retrieved 2021-01-01.
26. Clarke, Arthur; Pohl, Frederik (2008). The Last Theorem. Ballantine. p. 91. ISBN 978-0007289981.
27. "ISO/IEC 9899:1999 – Programming languages – C". ISO. Iso.org. 2011-12-08. Archived from the original on 2016-10-10. Retrieved 2014-04-08.
28. "Rationale for International Standard – Programming Languages – C" (PDF). Open Standards. 5.10. April 2003. pp. 52, 153–154, 159. Archived (PDF) from the original on 2016-06-06. Retrieved 2010-10-17.
29. The IEEE and The Open Group (2013) [2001]. "dprintf, fprintf, printf, snprintf, sprintf – print formatted output". The Open Group Base Specifications (Issue 7, IEEE Std 1003.1, 2013 ed.). Archived from the original on 2016-06-21. Retrieved 2016-06-21.
30. Mano, M. Morris; Ciletti, Michael D. (2013). Digital Design – With an Introduction to the Verilog HDL (Fifth ed.). Pearson Education. pp. 6, 8–10. ISBN 978-0-13-277420-8.
31. "算盤 Hexadecimal Addition & Subtraction on a Chinese Abacus". totton.idirect.com. Archived from the original on 2019-07-06. Retrieved 2019-06-26.
32. "Base 4^2 Hexadecimal Symbol Proposal". Hauptmech. Archived from the original on 2021-10-20. Retrieved 2008-09-04.
33. "Intuitor Hex Headquarters". Intuitor. Archived from the original on 2010-09-04. Retrieved 2018-10-28.
34. Niemietz, Ricardo Cancho (2003-10-21). "A proposal for addition of the six Hexadecimal digits (A-F) to Unicode". DKUUG Standardizing. Archived from the original on 2011-06-04. Retrieved 2018-10-28.
35. Nystrom, John William (1862). Project of a New System of Arithmetic, Weight, Measure and Coins: Proposed to be called the Tonal System, with Sixteen to the Base. Philadelphia: Lippincott.
36. Nystrom (1862), p. 33: "In expressing time, angle of a circle, or points on the compass, the unit tim should be noted as integer, and parts thereof as tonal fractions, as 5·86 tims is five times and metonby [*"sutim and metonby" John Nystrom accidentally gives part of the number in decimal names; in Nystrom's pronunciation scheme, 5=su, 8=me, 6=by, c.f. unifoundry.com Archived 2021-05-19 at the Wayback Machine ]."
37. C. E. Fröberg, Hexadecimal Conversion Tables, Lund (1952).
38. The Century Dictionary of 1895 has sexadecimal in the more general sense of "relating to sixteen". An early explicit use of sexadecimal in the sense of "using base 16" is found also in 1895, in the Journal of the American Geographical Society of New York, vols. 27–28, p. 197.
39. Schwartzman, Steven (1994). The Words of Mathematics: An etymological dictionary of mathematical terms used in English. The Mathematical Association of America. p. 105. ISBN 0-88385-511-9. s.v. hexadecimal
40. Knuth, Donald. (1969). The Art of Computer Programming, Volume 2. ISBN 0-201-03802-1. (Chapter 17.)
41. Alfred B. Taylor, Report on Weights and Measures, Pharmaceutical Association, 8th Annual Session, Boston, 15 September 1859. See pages and 33 and 41.
42. Alfred B. Taylor, "Octonary numeration and its application to a system of weights and measures", Proc Amer. Phil. Soc. Vol XXIV Archived 2016-06-24 at the Wayback Machine, Philadelphia, 1887; pages 296–366. See pages 317 and 322.
43. IBM System/360 FORTRAN IV Language Archived 2021-05-19 at the Wayback Machine (1966), p. 13.
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Easter
Easter,[nb 1] also called Pascha[nb 2] (Aramaic, Greek, Latin) or Resurrection Sunday,[nb 3] is a Christian festival and cultural holiday commemorating the resurrection of Jesus from the dead, described in the New Testament as having occurred on the third day of his burial following his crucifixion by the Romans at Calvary c. 30 AD.[10][11] It is the culmination of the Passion of Jesus Christ, preceded by Lent (or Great Lent), a 40-day period of fasting, prayer, and penance.
Easter
Icon of the Resurrection depicting Christ having destroyed the gates of hell and removing Adam and Eve from the grave. Christ is flanked by saints, and Satan, depicted as an old man, is bound and chained.
Observed byChristians
SignificanceCelebrates the resurrection of Jesus
CelebrationsChurch services, festive family meals, Easter egg decoration, and gift-giving
ObservancesPrayer, all-night vigil, sunrise service
DateVariable, determined by the Computus
2022 date
• April 17 (Western)
• April 24 (Eastern)
2023 date
• April 9 (Western)
• April 16 (Eastern)
2024 date
• March 31 (Western)
• May 5 (Eastern)
2025 date
• April 20 (Western)
• April 20 (Eastern)
Related toPassover, Septuagesima, Sexagesima, Quinquagesima, Shrove Tuesday, Ash Wednesday, Clean Monday, Lent, Great Lent, Palm Sunday, Holy Week, Maundy Thursday, Good Friday, and Holy Saturday which lead up to Easter; and Divine Mercy Sunday, Ascension, Pentecost, Trinity Sunday, Corpus Christi and Feast of the Sacred Heart which follow it.
Easter-observing Christians commonly refer to the week before Easter as Holy Week, which in Western Christianity begins on Palm Sunday (marking the entrance of Jesus in Jerusalem), includes Spy Wednesday (on which the betrayal of Jesus is mourned),[12] and contains the days of the Easter Triduum including Maundy Thursday, commemorating the Maundy and Last Supper,[13][14] as well as Good Friday, commemorating the crucifixion and death of Jesus.[15] In Eastern Christianity, the same days and events are commemorated with the names of days all starting with "Holy" or "Holy and Great"; and Easter itself might be called "Great and Holy Pascha", "Easter Sunday", "Pascha" or "Sunday of Pascha". In Western Christianity, Eastertide, or the Easter Season, begins on Easter Sunday and lasts seven weeks, ending with the coming of the 50th day, Pentecost Sunday. In Eastern Christianity, the Paschal season ends with Pentecost as well, but the leave-taking of the Great Feast of Pascha is on the 39th day, the day before the Feast of the Ascension.
Easter and its related holidays are moveable feasts, not falling on a fixed date; its date is computed based on a lunisolar calendar (solar year plus Moon phase) similar to the Hebrew calendar. The First Council of Nicaea (325) established only two rules, namely independence from the Hebrew calendar and worldwide uniformity. No details for the computation were specified; these were worked out in practice, a process that took centuries and generated a number of controversies. It has come to be the first Sunday after the ecclesiastical full moon that occurs on or soonest after 21 March.[16] Even if calculated on the basis of the Gregorian calendar, the date of that full moon sometimes differs from that of the astronomical first full moon after the March equinox.[17]
The English term is derived from the Saxon spring festival Ēostre;[18] Easter is linked to the Jewish Passover by its name (Hebrew: פֶּסַח pesach, Aramaic: פָּסחָא pascha are the basis of the term Pascha), by its origin (according to the synoptic Gospels, both the crucifixion and the resurrection took place during the week of Passover)[19][20] and by much of its symbolism, as well as by its position in the calendar. In most European languages, both the Christian Easter and the Jewish Passover are called by the same name; and in the older English versions of the Bible, as well, the term Easter was used to translate Passover.[21]
Easter traditions vary across the Christian world, and include sunrise services or late-night vigils, exclamations and exchanges of Paschal greetings, flowering the cross,[22] the wearing of Easter bonnets by women, clipping the church,[23] and the decoration and the communal breaking of Easter eggs (a symbol of the empty tomb).[24][25][26] The Easter lily, a symbol of the resurrection in Western Christianity,[27][28] traditionally decorates the chancel area of churches on this day and for the rest of Eastertide.[29] Additional customs that have become associated with Easter and are observed by both Christians and some non-Christians include Easter parades, communal dancing (Eastern Europe), the Easter Bunny and egg hunting.[30][31][32][33][34] There are also traditional Easter foods that vary by region and culture.
Etymology
The modern English term Easter, cognate with modern Dutch ooster and German Ostern, developed from an Old English word that usually appears in the form Ēastrun, Ēastron, or Ēastran; but also as Ēastru, Ēastro; and Ēastre or Ēostre.[nb 4] Bede provides the only documentary source for the etymology of the word, in his eighth-century The Reckoning of Time. He wrote that Ēosturmōnaþ (Old English for 'Month of Ēostre', translated in Bede's time as "Paschal month") was an English month, corresponding to April, which he says "was once called after a goddess of theirs named Ēostre, in whose honour feasts were celebrated in that month".[35]
In Latin and Greek, the Christian celebration was, and still is, called Pascha (Greek: Πάσχα), a word derived from Aramaic פסחא (Paskha), cognate to the Hebrew פֶּסַח (Pesach). The word originally denoted the Jewish festival known in English as Passover, commemorating the Jewish Exodus from slavery in Egypt.[36][37] As early as the 50s of the 1st century, Paul the Apostle, writing from Ephesus to the Christians in Corinth,[38] applied the term to Christ, and it is unlikely that the Ephesian and Corinthian Christians were the first to hear Exodus 12 interpreted as speaking about the death of Jesus, not just about the Jewish Passover ritual.[39] In most languages, Germanic languages such as English being exceptions, the feast is known by names derived from the Greek and Latin Pascha.[7][40] Pascha is also a name by which Jesus himself is remembered in the Orthodox Church, especially in connection with his resurrection and with the season of its celebration.[41] Others call the holiday "Resurrection Sunday" or "Resurrection Day", after the Greek Ἀνάστασις, Anastasis, 'Resurrection' day.[8][9][42][43]
Theological significance
Easter celebrates Jesus' supernatural resurrection from the dead, which is one of the chief tenets of the Christian faith.[44] Paul writes that, for those who trust in Jesus's death and resurrection, "death is swallowed up in victory." The First Epistle of Peter declares that God has given believers "a new birth into a living hope through the resurrection of Jesus Christ from the dead". Christian theology holds that, through faith in the working of God, those who follow Jesus are spiritually resurrected with him so that they may walk in a new way of life and receive eternal salvation, and can hope to be physically resurrected to dwell with him in the Kingdom of Heaven.[45]
Easter is linked to Passover and the Exodus from Egypt recorded in the Old Testament through the Last Supper, sufferings, and crucifixion of Jesus that preceded the resurrection.[40] According to the three Synoptic Gospels, Jesus gave the Passover meal a new meaning, as in the upper room during the Last Supper he prepared himself and his disciples for his death.[40] He identified the bread and cup of wine as his body, soon to be sacrificed, and his blood, soon to be shed. The Apostle Paul states, in his First Epistle to the Corinthians, "Get rid of the old yeast that you may be a new batch without yeast—as you really are. For Christ, our Passover lamb, has been sacrificed." This refers to the requirement in Jewish law that Jews eliminate all chametz, or leavening, from their homes in advance of Passover, and to the allegory of Jesus as the Paschal lamb.[46][47]
Early Christianity
As the Gospels assert that both the crucifixion and resurrection of Jesus during the week of Passover, the first Christians timed the observance of the annual celebration of the resurrections in relation to Passover.[48] Direct evidence for a more fully formed Christian festival of Pascha (Easter) begins to appear in the mid-2nd century. Perhaps the earliest extant primary source referring to Easter is a mid-2nd-century Paschal homily attributed to Melito of Sardis, which characterizes the celebration as a well-established one.[49] Evidence for another kind of annually recurring Christian festival, those commemorating the martyrs, began to appear at about the same time as the above homily.[50]
While martyrs' days (usually the individual dates of martyrdom) were celebrated on fixed dates in the local solar calendar, the date of Easter was fixed by means of the local Jewish[51] lunisolar calendar. This is consistent with the celebration of Easter having entered Christianity during its earliest, Jewish, period, but does not leave the question free of doubt.[52]
The ecclesiastical historian Socrates Scholasticus attributes the observance of Easter by the church to the perpetuation of pre-Christian custom, "just as many other customs have been established", stating that neither Jesus nor his Apostles enjoined the keeping of this or any other festival. Although he describes the details of the Easter celebration as deriving from local custom, he insists the feast itself is universally observed.[53]
Date
Easter and the holidays that are related to it are moveable feasts, in that they do not fall on a fixed date in the Gregorian or Julian calendars (both of which follow the cycle of the sun and the seasons). Instead, the date for Easter is determined on a lunisolar calendar similar to the Hebrew calendar. The First Council of Nicaea (325) established two rules, independence of the Jewish calendar and worldwide uniformity, which were the only rules for Easter explicitly laid down by the Council. No details for the computation were specified; these were worked out in practice, a process that took centuries and generated a number of controversies. (See also Computus and Reform of the date of Easter.) In particular, the Council did not decree that Easter must fall on Sunday, but this was already the practice almost everywhere.[55]
In Western Christianity, using the Gregorian calendar, Easter always falls on a Sunday between 22 March and 25 April,[56] within about seven days after the astronomical full moon.[57] The following day, Easter Monday, is a legal holiday in many countries with predominantly Christian traditions.[58]
Eastern Orthodox Christians base Paschal date calculations on the Julian calendar. Because of the thirteen-day difference between the calendars between 1900 and 2099, 21 March corresponds, during the 21st century, to 3 April in the Gregorian calendar. Since the Julian calendar is no longer used as the civil calendar of the countries where Eastern Christian traditions predominate, Easter varies between 4 April and 8 May in the Gregorian calendar. Because the Julian "full moon" is always several days after the astronomical full moon, the Eastern Easter is also often later, relative to the visible lunar phases, than Western Easter.[59]
Among the Oriental Orthodox, some churches have changed from the Julian to the Gregorian calendar and the date for Easter, as for other fixed and moveable feasts, is the same as in the Western church.[60]
Computations
In 725, Bede succinctly wrote, "The Sunday following the full Moon which falls on or after the equinox will give the lawful Easter."[61] However, this does not precisely reflect the ecclesiastical rules. The full moon referred to (called the Paschal full moon) is not an astronomical full moon, but the 14th day of a lunar month. Another difference is that the astronomical equinox is a natural astronomical phenomenon, which can fall on 19, 20 or 21 March,[62] while the ecclesiastical date is fixed by convention on 21 March.[63]
In addition, the lunar tables of the Julian calendar are currently five days behind those of the Gregorian calendar. Therefore, the Julian computation of the Paschal full moon is a full five days later than the astronomical full moon. The result of this combination of solar and lunar discrepancies is divergence in the date of Easter in most years (see table).[64]
Easter is determined on the basis of lunisolar cycles. The lunar year consists of 30-day and 29-day lunar months, generally alternating, with an embolismic month added periodically to bring the lunar cycle into line with the solar cycle. In each solar year (1 January to 31 December inclusive), the lunar month beginning with an ecclesiastical new moon falling in the 29-day period from 8 March to 5 April inclusive is designated as the paschal lunar month for that year.[65]
Easter is the third Sunday in the paschal lunar month, or, in other words, the Sunday after the paschal lunar month's 14th day. The 14th of the paschal lunar month is designated by convention as the Paschal full moon, although the 14th of the lunar month may differ from the date of the astronomical full moon by up to two days.[65] Since the ecclesiastical new moon falls on a date from 8 March to 5 April inclusive, the paschal full moon (the 14th of that lunar month) must fall on a date from 22 March to 18 April inclusive.[64]
The Gregorian calculation of Easter was based on a method devised by the Calabrian doctor Aloysius Lilius (or Lilio) for adjusting the epacts of the Moon,[66] and has been adopted by almost all Western Christians and by Western countries which celebrate national holidays at Easter. For the British Empire and colonies, a determination of the date of Easter Sunday using Golden Numbers and Sunday letters was defined by the Calendar (New Style) Act 1750 with its Annexe. This was designed to match exactly the Gregorian calculation.[67]
Controversies over the date
The precise date of Easter has at times been a matter of contention. By the later 2nd century, it was widely accepted that the celebration of the holiday was a practice of the disciples and an undisputed tradition. The Quartodeciman controversy, the first of several Easter controversies, arose concerning the date on which the holiday should be celebrated.[68]
The term "Quartodeciman" refers to the practice of ending the Lenten fast on Nisan 14 of the Hebrew calendar, "the LORD's passover".[69] According to the church historian Eusebius, the Quartodeciman Polycarp (bishop of Smyrna, by tradition a disciple of John the Apostle) debated the question with Anicetus (bishop of Rome). The Roman province of Asia was Quartodeciman, while the Roman and Alexandrian churches continued the fast until the Sunday following (the Sunday of Unleavened Bread), wishing to associate Easter with Sunday. Neither Polycarp nor Anicetus persuaded the other, but they did not consider the matter schismatic either, parting in peace and leaving the question unsettled.[70]
Controversy arose when Victor, bishop of Rome a generation after Anicetus, attempted to excommunicate Polycrates of Ephesus and all other bishops of Asia for their Quartodecimanism. According to Eusebius, a number of synods were convened to deal with the controversy, which he regarded as all ruling in support of Easter on Sunday.[71] Polycrates (c. 190), however, wrote to Victor defending the antiquity of Asian Quartodecimanism. Victor's attempted excommunication was apparently rescinded, and the two sides reconciled upon the intervention of bishop Irenaeus and others, who reminded Victor of the tolerant precedent of Anicetus.[72][73]
Quartodecimanism seems to have lingered into the 4th century, when Socrates of Constantinople recorded that some Quartodecimans were deprived of their churches by John Chrysostom[74] and that some were harassed by Nestorius.[75]
It is not known how long the Nisan 14 practice continued. But both those who followed the Nisan 14 custom, and those who set Easter to the following Sunday, had in common the custom of consulting their Jewish neighbors to learn when the month of Nisan would fall, and setting their festival accordingly. By the later 3rd century, however, some Christians began to express dissatisfaction with the custom of relying on the Jewish community to determine the date of Easter. The chief complaint was that the Jewish communities sometimes erred in setting Passover to fall before the Northern Hemisphere spring equinox.[76][77] The Sardica paschal table[78] confirms these complaints, for it indicates that the Jews of some eastern Mediterranean city (possibly Antioch) fixed Nisan 14 on dates well before the spring equinox on multiple occasions.[79]
Because of this dissatisfaction with reliance on the Jewish calendar, some Christians began to experiment with independent computations.[nb 5] Others, however, believed that the customary practice of consulting Jews should continue, even if the Jewish computations were in error.[82]
First Council of Nicaea (325 AD)
This controversy between those who advocated independent computations, and those who wished to continue the custom of relying on the Jewish calendar, was formally resolved by the First Council of Nicaea in 325, which endorsed changing to an independent computation by the Christian community in order to celebrate in common. This effectively required the abandonment of the old custom of consulting the Jewish community in those places where it was still used. Epiphanius of Salamis wrote in the mid-4th century:
[T]he emperor [...] convened a council of 318 bishops [...] in the city of Nicaea [...] They passed certain ecclesiastical canons at the council besides, and at the same time decreed in regard to the Passover [i.e., Easter] that there must be one unanimous concord on the celebration of God's holy and supremely excellent day. For it was variously observed by people; some kept it early, some between [the disputed dates], but others late. And in a word, there was a great deal of controversy at that time.[83]
Canons[84] and sermons[85] condemning the custom of computing Easter's date based on the Jewish calendar indicate that this custom (called "protopaschite" by historians) did not die out at once, but persisted for a time after the Council of Nicaea.[86]
Dionysius Exiguus, and others following him, maintained that the 318 bishops assembled at Nicaea had specified a particular method of determining the date of Easter; subsequent scholarship has refuted this tradition.[87] In any case, in the years following the council, the computational system that was worked out by the church of Alexandria came to be normative. The Alexandrian system, however, was not immediately adopted throughout Christian Europe. Following Augustalis' treatise De ratione Paschae (On the Measurement of Easter), Rome retired the earlier 8-year cycle in favor of Augustalis' 84-year lunisolar calendar cycle, which it used until 457. It then switched to Victorius of Aquitaine's adaptation of the Alexandrian system.[88][89]
Because this Victorian cycle differed from the unmodified Alexandrian cycle in the dates of some of the Paschal full moons, and because it tried to respect the Roman custom of fixing Easter to the Sunday in the week of the 16th to the 22nd of the lunar month (rather than the 15th to the 21st as at Alexandria), by providing alternative "Latin" and "Greek" dates in some years, occasional differences in the date of Easter as fixed by Alexandrian rules continued.[88][89] The Alexandrian rules were adopted in the West following the tables of Dionysius Exiguus in 525.[90]
Early Christians in Britain and Ireland also used an 84-year cycle. From the 5th century onward this cycle set its equinox to 25 March and fixed Easter to the Sunday falling in the 14th to the 20th of the lunar month inclusive.[91][92] This 84-year cycle was replaced by the Alexandrian method in the course of the 7th and 8th centuries. Churches in western continental Europe used a late Roman method until the late 8th century during the reign of Charlemagne, when they finally adopted the Alexandrian method. Since 1582, when the Roman Catholic Church adopted the Gregorian calendar while most of Europe used the Julian calendar, the date on which Easter is celebrated has again differed.[93]
The Greek island of Syros, whose population is divided almost equally between Catholics and Orthodox, is one of the few places where the two Churches share a common date for Easter, with the Catholics accepting the Orthodox date—a practice helping considerably in maintaining good relations between the two communities.[94] Conversely, Orthodox Christians in Finland celebrate Easter according to the Western Christian date.[95]
Proposed reforms of the date
In the 20th and 21st centuries, some individuals and institutions have propounded changing the method of calculating the date for Easter, the most prominent proposal being the Sunday after the second Saturday in April. Despite having some support, proposals to reform the date have not been implemented.[96] An Orthodox congress of Eastern Orthodox bishops, which included representatives mostly from the Patriarch of Constantinople and the Serbian Patriarch, met in Constantinople in 1923, where the bishops agreed to the Revised Julian calendar.[97]
The original form of this calendar would have determined Easter using precise astronomical calculations based on the meridian of Jerusalem.[98][99] However, all the Eastern Orthodox countries that subsequently adopted the Revised Julian calendar adopted only that part of the revised calendar that applied to festivals falling on fixed dates in the Julian calendar. The revised Easter computation that had been part of the original 1923 agreement was never permanently implemented in any Orthodox diocese.[97]
In the United Kingdom, Parliament passed the Easter Act 1928 to change the date of Easter to be the first Sunday after the second Saturday in April (or, in other words, the Sunday in the period from 9 to 15 April). However, the legislation has not been implemented, although it remains on the Statute book and could be implemented, subject to approval by the various Christian churches.[100]
At a summit in Aleppo, Syria, in 1997, the World Council of Churches (WCC) proposed a reform in the calculation of Easter which would have replaced the present divergent practices of calculating Easter with modern scientific knowledge taking into account actual astronomical instances of the spring equinox and full moon based on the meridian of Jerusalem, while also following the tradition of Easter being on the Sunday following the full moon.[101] The recommended World Council of Churches changes would have sidestepped the calendar issues and eliminated the difference in date between the Eastern and Western churches. The reform was proposed for implementation starting in 2001, and despite repeated calls for reform, it was not ultimately adopted by any member body.[102][103]
In January 2016, the Anglican Communion, Coptic Orthodox Church, Greek Orthodox Church, and Roman Catholic Church again considered agreeing on a common, universal date for Easter, while also simplifying the calculation of that date, with either the second or third Sunday in April being popular choices.[104]
In November 2022, the Patriarch of Constantinople said that conversations between the Roman Catholic Church and the Orthodox Churches had begun to determine a common date for the celebration of Easter. The agreement is expected to be reached for the 1700th anniversary of the Council of Nicaea in 2025.[105]
Table of the dates of Easter by Gregorian and Julian calendars
The WCC presented comparative data of the relationships:
Table of dates of Easter 2001–2025 (in Gregorian dates)[106]
Year Full Moon Jewish Passover [note 1] Astronomical Easter [note 2] Gregorian Easter Julian Easter
2001 8 April 15 April
2002 28 March 31 March5 May
2003 16 April17 April 20 April27 April
2004 5 April6 April 11 April
2005 25 March24 April 27 March1 May
2006 13 April 16 April23 April
2007 2 April3 April 8 April
2008 21 March20 April 23 March27 April
2009 9 April 12 April19 April
2010 30 March 4 April
2011 18 April19 April 24 April
2012 6 April7 April 8 April15 April
2013 27 March26 March 31 March5 May
2014 15 April 20 April
2015 4 April 5 April12 April
2016 23 March23 April 27 March1 May
2017 11 April 16 April
2018 31 March 1 April8 April
2019 21 March20 April 24 March21 April28 April
2020 8 April9 April 12 April19 April
2021 28 March 4 April2 May
2022 16 April 17 April24 April
2023 6 April 9 April16 April
2024 25 March23 April 31 March5 May
2025 13 April 20 April
1. Jewish Passover is on Nisan 15 of its calendar. It commences at sunset preceding the date indicated (as does Easter in many traditions).
2. Astronomical Easter is the first Sunday after the astronomical full moon after the astronomical March equinox as measured at the meridian of Jerusalem according to this WCC proposal.
Position in the church year
Western Christianity
In most branches of Western Christianity, Easter is preceded by Lent, a period of penitence that begins on Ash Wednesday, lasts 40 days (not counting Sundays), and is often marked with fasting. The week before Easter, known as Holy Week, is an important time for observers to commemorate the final week of Jesus' life on earth.[107] The Sunday before Easter is Palm Sunday, with the Wednesday before Easter being known as Spy Wednesday (or Holy Wednesday). The last three days before Easter are Maundy Thursday, Good Friday and Holy Saturday (sometimes referred to as Silent Saturday).[108]
Palm Sunday, Maundy Thursday and Good Friday respectively commemorate Jesus's entry in Jerusalem, the Last Supper and the crucifixion. Maundy Thursday, Good Friday, and Holy Saturday are sometimes referred to as the Easter Triduum (Latin for "Three Days"). Many churches begin celebrating Easter late in the evening of Holy Saturday at a service called the Easter Vigil.[109]
The week beginning with Easter Sunday is called Easter Week or the Octave of Easter, and each day is prefaced with "Easter", e.g. Easter Monday (a public holiday in many countries), Easter Tuesday (a much less widespread public holiday), etc. Easter Saturday is therefore the Saturday after Easter Sunday. The day before Easter is properly called Holy Saturday. Eastertide, or Paschaltide, the season of Easter, begins on Easter Sunday and lasts until the day of Pentecost, seven weeks later.[110][111][112]
Eastern Christianity
In Eastern Christianity, the spiritual preparation for Easter/Pascha begins with Great Lent, which starts on Clean Monday and lasts for 40 continuous days (including Sundays). Great Lent ends on a Friday, and the next day is Lazarus Saturday. The Vespers which begins Lazarus Saturday officially brings Great Lent to a close, although the fast continues through the following week.[113][114]
The Paschal Vigil begins with the Midnight Office, which is the last service of the Lenten Triodion and is timed so that it ends a little before midnight on Holy Saturday night. At the stroke of midnight the Paschal celebration itself begins, consisting of Paschal Matins, Paschal Hours, and Paschal Divine Liturgy.[115]
The liturgical season from Easter to the Sunday of All Saints (the Sunday after Pentecost) is known as the Pentecostarion (the "50 days"). The week which begins on Easter Sunday is called Bright Week, during which there is no fasting, even on Wednesday and Friday. The Afterfeast of Easter lasts 39 days, with its Apodosis (leave-taking) on the day before the Feast of the Ascension. Pentecost Sunday is the 50th day from Easter (counted inclusively).[116] In the Pentecostarion published by Apostoliki Diakonia of the Church of Greece, the Great Feast Pentecost is noted in the synaxarion portion of Matins to be the 8th Sunday of Pascha. However, the Paschal greeting of "Christ is risen!" is no longer exchanged among the faithful after the Apodosis of Pascha.[117][118]
Liturgical observance
Western Christianity
The Easter festival is kept in many different ways among Western Christians. The traditional, liturgical observation of Easter, as practised among Roman Catholics, Lutherans,[121] and some Anglicans begins on the night of Holy Saturday with the Easter Vigil which follows an ancient liturgy involving symbols of light, candles and water and numerous readings form the Old and New Testament.[122]
Services continue on Easter Sunday and in a number of countries on Easter Monday. In parishes of the Moravian Church, as well as some other denominations such as the Methodist Churches, there is a tradition of Easter Sunrise Services[123] often starting in cemeteries[124] in remembrance of the biblical narrative in the Gospels, or other places in the open where the sunrise is visible.[125]
In some traditions, Easter services typically begin with the Paschal greeting: "Christ is risen!" The response is: "He is risen indeed. Alleluia!"[126]
Eastern Christianity
Eastern Orthodox, Eastern Catholics and Byzantine Rite Lutherans have a similar emphasis on Easter in their calendars, and many of their liturgical customs are very similar.[127]
Preparation for Easter begins with the season of Great Lent, which begins on Clean Monday.[128] While the end of Lent is Lazarus Saturday, fasting does not end until Easter Sunday.[129] The Orthodox service begins late Saturday evening, observing the Jewish tradition that evening is the start of liturgical holy days.[129]
The church is darkened, then the priest lights a candle at midnight, representing the resurrection of Jesus Christ. Altar servers light additional candles, with a procession which moves three times around the church to represent the three days in the tomb.[129] The service continues early into Sunday morning, with a feast to end the fasting. An additional service is held later that day on Easter Sunday.[129]
Non-observing Christian groups
Many Puritans saw traditional feasts of the established Anglican Church, such as All Saints' Day and Easter, as abominations because the Bible does not mention them.[130][131] Conservative Reformed denominations such as the Free Presbyterian Church of Scotland and the Reformed Presbyterian Church of North America likewise reject the celebration of Easter as a violation of the regulative principle of worship and what they see as its non-Scriptural origin.[132][133]
Members of the Religious Society of Friends (Quakers), as part of their historic testimony against times and seasons, do not celebrate or observe Easter or any traditional feast days of the established Church, believing instead that "every day is the Lord's Day," and that elevation of one day above others suggests that it is acceptable to do un-Christian acts on other days.[134][135] During the 17th and 18th centuries, Quakers were persecuted for this non-observance of Holy Days.[136]
Groups such as the Restored Church of God reject the celebration of Easter, seeing it as originating in a pagan spring festival adopted by the Roman Catholic Church.[137][138][139]
Jehovah's Witnesses maintain a similar view, observing a yearly commemorative service of the Last Supper and the subsequent execution of Christ on the evening of Nisan 14 (as they calculate the dates derived from the lunar Hebrew calendar). It is commonly referred to by many Witnesses as simply "The Memorial". Jehovah's Witnesses believe that such verses as Luke 22:19–20 and 1 Corinthians 11:26 constitute a commandment to remember the death of Christ though not the resurrection.[140][141]
Easter celebrations around the world
In countries where Christianity is a state religion, or those with large Christian populations, Easter is often a public holiday.[142] As Easter always falls on a Sunday, many countries in the world also recognize Easter Monday as a public holiday.[143] Some retail stores, shopping malls, and restaurants are closed on Easter Sunday. Good Friday, which occurs two days before Easter Sunday, is also a public holiday in many countries, as well as in 12 U.S. states. Even in states where Good Friday is not a holiday, many financial institutions, stock markets, and public schools are closed – the few banks that are normally open on regular Sundays are closed on Easter.[144]
In the Nordic countries Good Friday, Easter Sunday, and Easter Monday are public holidays,[145] and Good Friday and Easter Monday are bank holidays.[146] In Denmark, Iceland and Norway Maundy Thursday is also a public holiday. It is a holiday for most workers, except those operating some shopping malls which keep open for a half-day. Many businesses give their employees almost a week off, called Easter break.[147] Schools are closed between Palm Sunday and Easter Monday. According to a 2014 poll, 6 of 10 Norwegians travel during Easter, often to a countryside cottage; 3 of 10 said their typical Easter included skiing.[148]
In the Netherlands both Easter Sunday and Easter Monday are national holidays. Like first and second Christmas Day, they are both considered Sundays, which results in a first and a second Easter Sunday, after which the week continues to a Tuesday.[149]
In Greece Good Friday and Saturday as well as Easter Sunday and Monday are traditionally observed public holidays. It is custom for employees of the public sector to receive Easter bonuses as a gift from the state.[150]
In Commonwealth nations Easter Day is rarely a public holiday, as is the case for celebrations which fall on a Sunday. In the United Kingdom both Good Friday and Easter Monday are bank holidays, except for Scotland, where only Good Friday is a bank holiday.[151] In Canada, Easter Monday is a statutory holiday for federal employees. In the Canadian province of Quebec, either Good Friday or Easter Monday are statutory holidays (although most companies give both).[152]
In Australia, Easter is associated with harvest time.[153] Good Friday and Easter Monday are public holidays across all states and territories. "Easter Saturday" (the Saturday before Easter Sunday) is a public holiday in every state except Tasmania and Western Australia, while Easter Sunday itself is a public holiday only in New South Wales. Easter Tuesday is additionally a conditional public holiday in Tasmania, varying between award, and was also a public holiday in Victoria until 1994.[154]
In the United States, because Easter falls on a Sunday, which is already a non-working day for federal and state employees, it has not been designated as a federal or state holiday.[155] Easter parades are held in many American cities, involving festive strolling processions.[30]
Traditional customs
The egg is an ancient symbol of new life and rebirth.[156] In Christianity it became associated with Jesus's crucifixion and resurrection.[157] The custom of the Easter egg originated in the early Christian community of Mesopotamia, who stained eggs red in memory of the blood of Christ, shed at his crucifixion.[158][159] As such, for Christians, the Easter egg is a symbol of the empty tomb.[25][26] The oldest tradition is to use dyed chicken eggs.
In the Eastern Orthodox Church Easter eggs are blessed by a priest[160] both in families' baskets together with other foods forbidden during Great Lent and alone for distribution or in church or elsewhere.
• Traditional red Easter eggs for blessing by a priest
• A priest blessing baskets with Easter eggs and other foods forbidden during Great Lent
• A priest distributing blessed Easter eggs after blessing the Soyuz rocket
Easter eggs are a widely popular symbol of new life among the Eastern Orthodox but also in folk traditions in Slavic countries and elsewhere. A batik-like decorating process known as pisanka produces intricate, brilliantly colored eggs. The celebrated House of Fabergé workshops created exquisite jewelled Easter eggs for the Russian Imperial family from 1885 to 1916.[161]
Modern customs
A modern custom in the Western world is to substitute decorated chocolate, or plastic eggs filled with candy such as jellybeans; as many people give up sweets as their Lenten sacrifice, individuals enjoy them at Easter after having abstained from them during the preceding forty days of Lent.[162]
• Easter eggs, a symbol of the empty tomb, are a popular cultural symbol of Easter.[24]
• Marshmallow rabbits, candy eggs and other treats in an Easter basket
• An Easter egg decorated with the Easter Bunny
Manufacturing their first Easter egg in 1875, British chocolate company Cadbury sponsors the annual Easter egg hunt which takes place in over 250 National Trust locations in the United Kingdom.[163][164] On Easter Monday, the President of the United States holds an annual Easter egg roll on the White House lawn for young children.[165]
Easter Bunny
In some traditions, the children put out their empty baskets for the Easter bunny to fill while they sleep. They wake to find their baskets filled with candy eggs and other treats.[166][31] A custom originating in Germany,[166] the Easter Bunny is a popular legendary anthropomorphic Easter gift-giving character analogous to Santa Claus in American culture. Many children around the world follow the tradition of coloring hard-boiled eggs and giving baskets of candy.[31] Historically, foxes, cranes and storks were also sometimes named as the mystical creatures.[166] Since the rabbit is a pest in Australia, the Easter Bilby is available as an alternative.[167]
Music
• Marc-Antoine Charpentier:[168][169]
• Messe pour le samedi de Pâques, for soloists, chorus and continuo, H.8 (1690).
• Prose pour le jour de Pâques, for 3 voices and continuo, H.13 (1670)
• Chant joyeux du temps de Pâques, for soloists, chorus, 2 treble viols, and continuo, H.339 (1685).
• O filii à 3 voix pareilles, for 3 voices, 2 flutes, and continuo, H.312 (1670).
• Pour Pâques, for 2 voices, 2 flutes, and continuo, H.308 (1670).
• O filii pour les voix, violons, flûtes et orgue, for soloists, chorus, flutes, strings, and continuo, H.356 (1685?).
• Louis-Nicolas Clérambault: Motet pour le Saint jour de Pâques, in F major, opus 73
• André Campra: Au Christ triomphant, cantata for Easter
• Dieterich Buxtehude: Cantatas BuxWV 15 and BuxWV 62
• Carl Heinrich Graun: Easter Oratorio
• Henrich Biber: Missa Christi resurgentis C.3 (1674)
• Michael Praetorius: Easter Mass
• Johann Sebastian Bach: Christ lag in Todesbanden, BWV 4; Der Himmel lacht! Die Erde jubilieret, BWV 31; Oster-Oratorium, BWV 249.
• Georg Philipp Telemann: more than 100 cantatas for Eastertide.
• Jacques-Nicolas Lemmens: Sonata n° 2 "O Filii", Sonata n° 3 "Pascale", for organ.
• Charles Gounod: Messe solennelle de Pâques (1883).
• Nikolai Rimsky-Korsakov: La Grande Pâque russe, symphonic overture (1888).
• Sergueï Vassilievitch Rachmaninov: Suite pour deux pianos n°1 – Pâques, op. 5, n° 4 (1893).
See also
• Divine Mercy Sunday
• Life of Jesus in the New Testament
• List of Easter films
• List of Easter hymns
• List of Easter television episodes
• Movable Eastern Christian Observances
• Regina Caeli
• Category:Film portrayals of Jesus' death and resurrection
Footnotes
1. Traditional names for the feast in English are "Easter Day", as in the Book of Common Prayer; "Easter Sunday", used by James Ussher (The Whole Works of the Most Rev. James Ussher, Volume 4[1]) and Samuel Pepys (The Diary of Samuel Pepys, Volume 2[2]), as well as the single word "Easter" in books printed in 1575,[3] 1584,[4] and 1586.[5]
2. In the Eastern Orthodox Church, the Greek word Pascha is used for the celebration; in English, the analogous word is Pasch.[6][7]
3. The term "Resurrection Sunday" is used particularly by Christian communities in the Middle East.[8][9]
4. Old English pronunciation: [ˈæːɑstre, ˈeːostre]
5. Eusebius reports that Dionysius, Bishop of Alexandria, proposed an 8-year Easter cycle, and quotes a letter from Anatolius, Bishop of Laodicea, that refers to a 19-year cycle.[80] An 8-year cycle has been found inscribed on a statue unearthed in Rome in the 17th century, and since dated to the 3rd century.[81]
References
1. Ussher, James; Elrington, Charles Richard (1631). The Whole Works of the Most Rev. James Ussher – James Ussher, Charles Richard Elrington – Google Books. Archived from the original on 1 August 2020. Retrieved 28 March 2023.
2. Pepys, Samuel (1665). The Diary of Samuel Pepys M.A. F.R.S. Archived from the original on 9 April 2023. Retrieved 7 April 2023.
3. Foxe, John (1575). A Sermon of Christ Crucified, Preached at Paules Crosse the Fridaie Before ... Archived from the original on 9 April 2023. Retrieved 20 June 2015.
4. Caradoc (St. of Llancarfan) (1584). The Historie of Cambria. Archived from the original on 9 April 2023. Retrieved 20 June 2015.
5. (de Granada), Luis (1586). "A Memoriall of a Christian Life: Wherein are Treated All Such Thinges, as ..." Archived from the original on 9 April 2023. Retrieved 20 June 2015.
6. Ferguson, Everett (2009). Baptism in the Early Church: History, Theology, and Liturgy in the First Five Centuries. Wm. B. Eerdmans Publishing. p. 351. ISBN 978-0802827487. Archived from the original on 1 August 2020. Retrieved 23 April 2014. The practices are usually interpreted in terms of baptism at the pasch (Easter), for which compare Tertullian, but the text does not specify this season, only that it was done on Sunday, and the instructions may apply to whenever the baptism was to be performed.
7. Davies, Norman (1998). Europe: A History. HarperCollins. p. 201. ISBN 978-0060974688. In most European languages Easter is called by some variant of the late Latin word Pascha, which in turn derives from the Hebrew pesach, meaning passover.
8. Gamman, Andrew; Bindon, Caroline (2014). Stations for Lent and Easter. Kereru Publishing Limited. p. 7. ISBN 978-0473276812. Easter Day, also known as Resurrection Sunday, marks the high point of the Christian year. It is the day that we celebrate the resurrection of Jesus Christ from the dead.
9. Boda, Mark J.; Smith, Gordon T. (2006). Repentance in Christian Theology. Liturgical Press. p. 316. ISBN 978-0814651759. Archived from the original on 4 August 2020. Retrieved 19 April 2014. Orthodox, Catholic, and all Reformed churches in the Middle East celebrate Easter according to the Eastern calendar, calling this holy day "Resurrection Sunday," not Easter.
10. Trawicky, Bernard; Gregory, Ruth Wilhelme (2000). Anniversaries and Holidays. American Library Association. ISBN 978-0838906958. Archived from the original on 12 October 2017. Retrieved 17 October 2020. Easter is the central celebration of the Christian liturgical year. It is the oldest and most important Christian feast, celebrating the Resurrection of Jesus Christ. The date of Easter determines the dates of all movable feasts except those of Advent.
11. Aveni, Anthony (2004). "The Easter/Passover Season: Connecting Time's Broken Circle", The Book of the Year: A Brief History of Our Seasonal Holidays. Oxford University Press. pp. 64–78. ISBN 0-19-517154-3. Archived from the original on 8 February 2021. Retrieved 17 October 2020.
12. Cooper, J.HB. (23 October 2013). Dictionary of Christianity. Routledge. p. 124. ISBN 9781134265466. Holy Week. The last week in LENT. It begins on PALM SUNDAY; the fourth day is called SPY WEDNESDAY; the fifth is MAUNDY THURSDAY or HOLY THURSDAY; the sixth is Good Friday; and the last 'Holy Saturday', or the 'Great Sabbath'.
13. Peter C. Bower (2003). The Companion to the Book of Common Worship. Geneva Press. ISBN 978-0664502324. Archived from the original on 8 June 2021. Retrieved 11 April 2009. Maundy Thursday (or le mandé; Thursday of the Mandatum, Latin, commandment). The name is taken from the first few words sung at the ceremony of the washing of the feet, "I give you a new commandment" (John 13:34); also from the commandment of Christ that we should imitate His loving humility in the washing of the feet (John 13:14–17). The term mandatum (maundy), therefore, was applied to the rite of foot-washing on this day.
14. Ramshaw, Gail (2004). Three Day Feast: Maundy Thursday, Good Friday, and Easter. Augsburg Fortress. ISBN 978-1451408164. Archived from the original on 5 November 2021. Retrieved 11 April 2009. In the liturgies of the Three Days, the service for Maundy Thursday includes both, telling the story of Jesus' last supper and enacting the footwashing.
15. Stuart, Leonard (1909). New century reference library of the world's most important knowledge: complete, thorough, practical, Volume 3. Syndicate Pub. Co. Archived from the original on 5 November 2021. Retrieved 11 April 2009. Holy Week, or Passion Week, the week which immediately precedes Easter, and is devoted especially to commemorating the passion of our Lord. The Days more especially solemnized during it are Holy Wednesday, Maundy Thursday, Good Friday, and Holy Saturday.
16. "Frequently asked questions about the date of Easter". Archived from the original on 22 April 2011. Retrieved 22 April 2009.
17. Woodman, Clarence E. (1923). "Clarence E. Woodman, "Easter and the Ecclesiastical Calendar" in Journal of the Royal Astronomical Society of Canada". Journal of the Royal Astronomical Society of Canada. 17: 141. Bibcode:1923JRASC..17..141W. Archived from the original on 12 May 2019. Retrieved 12 May 2019.
18. Gamber, Jenifer (September 2014). My Faith, My Life, Revised Edition: A Teen's Guide to the Episcopal Church. Church Publishing. p. 96. ISBN 978-0-8192-2962-5. The word "Easter" comes from the Anglo-Saxon spring festival called Eostre. Easter replaced the pagan festival of Eostre.
19. "5 April 2007: Mass of the Lord's Supper | BENEDICT XVI". www.vatican.va. Archived from the original on 5 April 2021. Retrieved 1 April 2021.
20. Reno, R. R. (14 April 2017). "The Profound Connection Between Easter and Passover". The Wall Street Journal. ISSN 0099-9660. Archived from the original on 17 December 2021. Retrieved 1 April 2021.
21. Weiser, Francis X. (1958). Handbook of Christian Feasts and Customs. New York: Harcourt, Brace and Company. p. 214. ISBN 0-15-138435-5.
22. Whitehouse, Bonnie Smith (15 November 2022). Seasons of Wonder: Making the Ordinary Sacred Through Projects, Prayers, Reflections, and Rituals: A 52-week devotional. Crown Publishing Group. p. 95-96. ISBN 978-0-593-44332-3.
23. Simpson, Jacqueline; Roud, Steve (2003). "clipping the church". Oxford Reference. Oxford University Press. doi:10.1093/acref/9780198607663.001.0001. ISBN 9780198607663. Archived from the original on 12 April 2020. Retrieved 31 March 2013.
24. Jordan, Anne (2000). Christianity. Nelson Thornes. ISBN 978-0748753208. Archived from the original on 8 February 2021. Retrieved 7 April 2012. Easter eggs are used as a Christian symbol to represent the empty tomb. The outside of the egg looks dead but inside there is new life, which is going to break out. The Easter egg is a reminder that Jesus will rise from His tomb and bring new life. Eastern Orthodox Christians dye boiled eggs red to represent the blood of Christ shed for the sins of the world.
25. The Guardian, Volume 29. H. Harbaugh. 1878. Archived from the original on 4 August 2020. Retrieved 7 April 2012. Just so, on that first Easter morning, Jesus came to life and walked out of the tomb, and left it, as it were, an empty shell. Just so, too, when the Christian dies, the body is left in the grave, an empty shell, but the soul takes wings and flies away to be with God. Thus you see that though an egg seems to be as dead as a stone, yet it really has life in it; and also it is like Christ's dead body, which was raised to life again. This is the reason we use eggs on Easter. (In olden times they used to color the eggs red, so as to show the kind of death by which Christ died, – a bloody death.)
26. Gordon Geddes, Jane Griffiths (2002). Christian belief and practice. Heinemann. ISBN 978-0435306915. Archived from the original on 29 July 2020. Retrieved 7 April 2012. Red eggs are given to Orthodox Christians after the Easter Liturgy. They crack their eggs against each other's. The cracking of the eggs symbolizes a wish to break away from the bonds of sin and misery and enter the new life issuing from Christ's resurrection.
27. Collins, Cynthia (19 April 2014). "Easter Lily Tradition and History". The Guardian. Archived from the original on 17 August 2020. Retrieved 20 April 2014. The Easter Lily is symbolic of the resurrection of Jesus Christ. Churches of all denominations, large and small, are filled with floral arrangements of these white flowers with their trumpet-like shape on Easter morning.
28. Schell, Stanley (1916). Easter Celebrations. Werner & Company. p. 84. We associate the lily with Easter, as pre-eminently the symbol of the Resurrection.
29. Luther League Review: 1936–1937. Luther League of America. 1936. Archived from the original on 3 August 2020. Retrieved 20 June 2015.
30. Duchak, Alicia (2002). An A–Z of Modern America. Rutledge. p. 372. ISBN 978-0415187558. Archived from the original on 27 December 2021. Retrieved 17 October 2020.
31. Sifferlin, Alexandra (21 February 2020) [2015]. "What's the Origin of the Easter Bunny?". Time. Archived from the original on 22 October 2021. Retrieved 4 April 2021.
32. Black, Vicki K. (2004). The Church Standard, Volume 74. Church Publishing, Inc. ISBN 978-0819225757. Archived from the original on 4 August 2020. Retrieved 7 April 2012. In parts of Europe, the eggs were dyed red and were then cracked together when people exchanged Easter greetings. Many congregations today continue to have Easter egg hunts for the children after the services on Easter Day.
33. The Church Standard, Volume 74. Walter N. Hering. 1897. Archived from the original on 30 August 2020. Retrieved 7 April 2012. When the custom was carried over into Christian practice the Easter eggs were usually sent to the priests to be blessed and sprinkled with holy water. In later times the coloring and decorating of eggs was introduced, and in a royal roll of the time of Edward I., which is preserved in the Tower of London, there is an entry of 18d. for 400 eggs, to be used for Easter gifts.
34. Brown, Eleanor Cooper (2010). From Preparation to Passion. ISBN 978-1609577650. Archived from the original on 4 August 2020. Retrieved 7 April 2012. So what preparations do most Christians and non-Christians make? Shopping for new clothing often signifies the belief that Spring has arrived, and it is a time of renewal. Preparations for the Easter Egg Hunts and the Easter Ham for the Sunday dinner are high on the list too.
35. Wallis, Faith (1999). Bede: The Reckoning of Time. Liverpool University Press. p. 54. ISBN 0853236933.
36. "History of Easter". The History Channel website. A&E Television Networks. Archived from the original on 31 May 2013. Retrieved 9 March 2013.
37. Karl Gerlach (1998). The Antenicene Pascha: A Rhetorical History. Peeters Publishers. p. xviii. ISBN 978-9042905702. Archived from the original on 8 August 2021. Retrieved 9 January 2020. The second century equivalent of easter and the paschal Triduum was called by both Greek and Latin writers "Pascha (πάσχα)", a Greek transliteration of the Aramaic form of the Hebrew פֶּסַח, the Passover feast of Ex. 12.
38. 1 Corinthians 5:7
39. Karl Gerlach (1998). The Antenicene Pascha: A Rhetorical History. Peters Publishers. p. 21. ISBN 978-9042905702. Archived from the original on 28 December 2021. Retrieved 17 October 2020. For while it is from Ephesus that Paul writes, "Christ our Pascha has been sacrificed for us," Ephesian Christians were not likely the first to hear that Ex 12 did not speak about the rituals of Pesach, but the death of Jesus of Nazareth.
40. Vicki K. Black (2004). Welcome to the Church Year: An Introduction to the Seasons of the Episcopal Church. Church Publishing, Inc. ISBN 978-0819219664. Archived from the original on 8 August 2021. Retrieved 9 January 2020. Easter is still called by its older Greek name, Pascha, which means "Passover", and it is this meaning as the Christian Passover-the celebration of Jesus's triumph over death and entrance into resurrected life-that is the heart of Easter in the church. For the early church, Jesus Christ was the fulfillment of the Jewish Passover feast: through Jesus, we have been freed from slavery of sin and granted to the Promised Land of everlasting life.
41. Orthros of Holy Pascha, Stichera: "Today the sacred Pascha is revealed to us. The new and holy Pascha, the mystical Pascha. The all-venerable Pascha. The Pascha which is Christ the Redeemer. The spotless Pascha. The great Pascha. The Pascha of the faithful. The Pascha which has opened unto us the gates of Paradise. The Pascha which sanctifies all faithful."
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Category:Easter
• Greek words (Wiktionary): Πάσχα (Easter) vs. πάσχα (Passover) vs. πάσχω (to suffer)
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• Roman Catholic View of Easter (from the Catholic Encyclopedia)
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• Orthodox Paschal Calculator Julian Easter and associated festivals in Gregorian calendar 1583–4099
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2E6 (mathematics)
In mathematics, 2E6 is the name of a family of Steinberg or twisted Chevalley groups. It is a quasi-split form of E6, depending on a quadratic extension of fields K⊂L. Unfortunately the notation for the group is not standardized, as some authors write it as 2E6(K) (thinking of 2E6 as an algebraic group taking values in K) and some as 2E6(L) (thinking of the group as a subgroup of E6(L) fixed by an outer involution).
Over finite fields these groups form one of the 18 infinite families of finite simple groups, and were introduced independently by Tits (1958) and Steinberg (1959).
Over finite fields
The group 2E6(q2) has order q36 (q12 − 1) (q9 + 1) (q8 − 1) (q6 − 1) (q5 + 1) (q2 − 1) /(3,q + 1).[1] This is similar to the order q36 (q12 − 1) (q9 − 1) (q8 − 1) (q6 − 1) (q5 − 1) (q2 − 1) /(3,q − 1) of E6(q).
Its Schur multiplier has order (3, q + 1) except for q=2, i. e. 2E6(22), when it has order 12 and is a product of cyclic groups of orders 2,2,3. One of the exceptional double covers of 2E6(22) is a subgroup of the baby monster group, and the exceptional central extension by the elementary abelian group of order 4 is a subgroup of the monster group.
The outer automorphism group has order (3, q + 1) · f where q2 = pf.
Over the real numbers
Over the real numbers, 2E6 is the quasisplit form of E6, and is one of the five real forms of E6 classified by Élie Cartan. Its maximal compact subgroup is of type F4.
Remarks
1. Reading example: If q2=22 in 2E6(q2) then q=2 in the order formula q36 (q12 − 1) (q9 + 1) (q8 − 1) (q6 − 1) (q5 + 1) (q2 − 1) /(3,q + 1). However, the group 2E6(22) is sometimes also written 2E6(2) (e. g. in Wilson's Atlas).
References
• Carter, Roger W. (1989) [1972], Simple groups of Lie type, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-50683-6, MR 0407163
• Steinberg, Robert (1959), "Variations on a theme of Chevalley", Pacific Journal of Mathematics, 9: 875–891, doi:10.2140/pjm.1959.9.875, ISSN 0030-8730, MR 0109191
• Steinberg, Robert (1968), Lectures on Chevalley groups, Yale University, New Haven, Conn., MR 0466335, archived from the original on 2012-09-10
• Tits, Jacques (1958), Les "formes réelles" des groupes de type E6, Séminaire Bourbaki; 10e année: 1957/1958. Textes des conférences; Exposés 152 à 168; 2e èd. corrigée, Exposé 162, vol. 15, Paris: Secrétariat math'ematique, MR 0106247
• Robert Wilson: Atlas of Finite Group Representations: Sporadic groups
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1/4 + 1/16 + 1/64 + 1/256 + ⋯
In mathematics, the infinite series 1/4 + 1/16 + 1/64 + 1/256 + ⋯ is an example of one of the first infinite series to be summed in the history of mathematics; it was used by Archimedes circa 250–200 BC.[1] As it is a geometric series with first term 1/4 and common ratio 1/4, its sum is
$\sum _{n=1}^{\infty }{\frac {1}{4^{n}}}={\frac {\frac {1}{4}}{1-{\frac {1}{4}}}}={\frac {1}{3}}.$
Visual demonstrations
The series 1/4 + 1/16 + 1/64 + 1/256 + ⋯ lends itself to some particularly simple visual demonstrations because a square and a triangle both divide into four similar pieces, each of which contains 1/4 the area of the original.
In the figure on the left,[2][3] if the large square is taken to have area 1, then the largest black square has area 1/2 × 1/2 = 1/4. Likewise, the second largest black square has area 1/16, and the third largest black square has area 1/64. The area taken up by all of the black squares together is therefore 1/4 + 1/16 + 1/64 + ⋯, and this is also the area taken up by the gray squares and the white squares. Since these three areas cover the unit square, the figure demonstrates that
$3\left({\frac {1}{4}}+{\frac {1}{4^{2}}}+{\frac {1}{4^{3}}}+{\frac {1}{4^{4}}}+\cdots \right)=1.$
Archimedes' own illustration, adapted at top,[4] was slightly different, being closer to the equation
$\sum _{n=1}^{\infty }{\frac {3}{4^{n}}}={\frac {3}{4}}+{\frac {3}{4^{2}}}+{\frac {3}{4^{3}}}+{\frac {3}{4^{4}}}+\cdots =1.$
See below for details on Archimedes' interpretation.
The same geometric strategy also works for triangles, as in the figure on the right:[2][5][6] if the large triangle has area 1, then the largest black triangle has area 1/4, and so on. The figure as a whole has a self-similarity between the large triangle and its upper sub-triangle. A related construction making the figure similar to all three of its corner pieces produces the Sierpiński triangle.[7]
Proof by Archimedes
Archimedes encounters the series in his work Quadrature of the Parabola. He is finding the area inside a parabola by the method of exhaustion, and he gets a series of triangles; each stage of the construction adds an area 1/4 times the area of the previous stage. His desired result is that the total area is 4/3 times the area of the first stage. To get there, he takes a break from parabolas to introduce an algebraic lemma:
Proposition 23. Given a series of areas A, B, C, D, ... , Z, of which A is the greatest, and each is equal to four times the next in order, then[8]
$A+B+C+D+\cdots +Z+{\frac {1}{3}}Z={\frac {4}{3}}A.$
Archimedes proves the proposition by first calculating
${\begin{array}{rcl}\displaystyle B+C+\cdots +Z+{\frac {B}{3}}+{\frac {C}{3}}+\cdots +{\frac {Z}{3}}&=&\displaystyle {\frac {4B}{3}}+{\frac {4C}{3}}+\cdots +{\frac {4Z}{3}}\\[1em]&=&\displaystyle {\frac {1}{3}}(A+B+\cdots +Y).\end{array}}$
On the other hand,
${\frac {B}{3}}+{\frac {C}{3}}+\cdots +{\frac {Y}{3}}={\frac {1}{3}}(B+C+\cdots +Y).$
Subtracting this equation from the previous equation yields
$B+C+\cdots +Z+{\frac {Z}{3}}={\frac {1}{3}}A$
and adding A to both sides gives the desired result.[9]
Today, a more standard phrasing of Archimedes' proposition is that the partial sums of the series 1 + 1/4 + 1/16 + ⋯ are:
$1+{\frac {1}{4}}+{\frac {1}{4^{2}}}+\cdots +{\frac {1}{4^{n}}}={\frac {1-\left({\frac {1}{4}}\right)^{n+1}}{1-{\frac {1}{4}}}}.$
This form can be proved by multiplying both sides by 1 − 1/4 and observing that all but the first and the last of the terms on the left-hand side of the equation cancel in pairs. The same strategy works for any finite geometric series.
The limit
Archimedes' Proposition 24 applies the finite (but indeterminate) sum in Proposition 23 to the area inside a parabola by a double reductio ad absurdum. He does not quite[10] take the limit of the above partial sums, but in modern calculus this step is easy enough:
$\lim _{n\to \infty }{\frac {1-\left({\frac {1}{4}}\right)^{n+1}}{1-{\frac {1}{4}}}}={\frac {1}{1-{\frac {1}{4}}}}={\frac {4}{3}}.$
Since the sum of an infinite series is defined as the limit of its partial sums,
$1+{\frac {1}{4}}+{\frac {1}{4^{2}}}+{\frac {1}{4^{3}}}+\cdots ={\frac {4}{3}}.$
Notes
1. Shawyer and Watson p. 3.
2. Nelsen and Alsina p. 74.
3. Ajose and Nelson. p. 230
4. Heath p. 250
5. Stein p. 46.
6. Mabry. p. 63
7. Nelson and Alsina p. 56
8. This is a quotation from Heath's English translation (p. 249).
9. This presentation is a shortened version of Heath p. 250.
10. Modern authors differ on how appropriate it is to say that Archimedes summed the infinite series. For example, Shawyer and Watson (p. 3) simply say he did; Swain and Dence say that "Archimedes applied an indirect limiting process"; and Stein (p. 45) stops short with the finite sums.
References
• Ajose, Sunday and Roger Nelsen (June 1994). "Proof without Words: Geometric Series". Mathematics Magazine. 67 (3): 230. doi:10.2307/2690617. JSTOR 2690617.
• Heath, T. L. (1953) [1897]. The Works of Archimedes. Cambridge UP. Page images at Casselman, Bill. "Archimedes' quadrature of the parabola". Archived from the original on 2012-03-20. Retrieved 2007-03-22. HTML with figures and commentary at Otero, Daniel E. (2002). "Archimedes of Syracuse". Archived from the original on 7 March 2007. Retrieved 2007-03-22.
• Mabry, Rick (February 1999). "Proof without Words: ${\frac {1}{4}}$ + $({\frac {1}{4}})^{2}$ + $({\frac {1}{4}})^{3}$ + ⋯ = ${\frac {1}{3}}$". Mathematics Magazine. 72 (1): 63. doi:10.1080/0025570X.1999.11996702. JSTOR 2691318.
• Nelsen, Roger B.; Alsina, Claudi (2006). Math Made Visual: Creating Images for Understanding Mathematics. MAA. ISBN 0-88385-746-4.
• Shawyer, Bruce; Watson, Bruce (1994). Borel's Methods of Summability: Theory and Applications. Oxford UP. ISBN 0-19-853585-6.
• Stein, Sherman K. (1999). Archimedes: What Did He Do Besides Cry Eureka?. MAA. ISBN 0-88385-718-9.
• Swain, Gordon; Dence, Thomas (April 1998). "Archimedes' Quadrature of the Parabola Revisited". Mathematics Magazine. 71 (2): 123–30. doi:10.2307/2691014. JSTOR 2691014.
Sequences and series
Integer sequences
Basic
• Arithmetic progression
• Geometric progression
• Harmonic progression
• Square number
• Cubic number
• Factorial
• Powers of two
• Powers of three
• Powers of 10
Advanced (list)
• Complete sequence
• Fibonacci sequence
• Figurate number
• Heptagonal number
• Hexagonal number
• Lucas number
• Pell number
• Pentagonal number
• Polygonal number
• Triangular number
Properties of sequences
• Cauchy sequence
• Monotonic function
• Periodic sequence
Properties of series
Series
• Alternating
• Convergent
• Divergent
• Telescoping
Convergence
• Absolute
• Conditional
• Uniform
Explicit series
Convergent
• 1/2 − 1/4 + 1/8 − 1/16 + ⋯
• 1/2 + 1/4 + 1/8 + 1/16 + ⋯
• 1/4 + 1/16 + 1/64 + 1/256 + ⋯
• 1 + 1/2s + 1/3s + ... (Riemann zeta function)
Divergent
• 1 + 1 + 1 + 1 + ⋯
• 1 − 1 + 1 − 1 + ⋯ (Grandi's series)
• 1 + 2 + 3 + 4 + ⋯
• 1 − 2 + 3 − 4 + ⋯
• 1 + 2 + 4 + 8 + ⋯
• 1 − 2 + 4 − 8 + ⋯
• Infinite arithmetic series
• 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
• 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
• 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
Kinds of series
• Taylor series
• Power series
• Formal power series
• Laurent series
• Puiseux series
• Dirichlet series
• Trigonometric series
• Fourier series
• Generating series
Hypergeometric series
• Generalized hypergeometric series
• Hypergeometric function of a matrix argument
• Lauricella hypergeometric series
• Modular hypergeometric series
• Riemann's differential equation
• Theta hypergeometric series
• Category
|
1/2 − 1/4 + 1/8 − 1/16 + ⋯
In mathematics, the infinite series 1/2 − 1/4 + 1/8 − 1/16 + ⋯ is a simple example of an alternating series that converges absolutely.
It is a geometric series whose first term is 1/2 and whose common ratio is −1/2, so its sum is
$\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{2^{n}}}={\frac {1}{2}}-{\frac {1}{4}}+{\frac {1}{8}}-{\frac {1}{16}}+\cdots ={\frac {\frac {1}{2}}{1-(-{\frac {1}{2}})}}={\frac {1}{3}}.$
Hackenbush and the surreals
A slight rearrangement of the series reads
$1-{\frac {1}{2}}-{\frac {1}{4}}+{\frac {1}{8}}-{\frac {1}{16}}+\cdots ={\frac {1}{3}}.$
The series has the form of a positive integer plus a series containing every negative power of two with either a positive or negative sign, so it can be translated into the infinite blue-red Hackenbush string that represents the surreal number 1/3:
LRRLRLR... = 1/3.[1]
A slightly simpler Hackenbush string eliminates the repeated R:
LRLRLRL... = 2/3.[2]
In terms of the Hackenbush game structure, this equation means that the board depicted on the right has a value of 0; whichever player moves second has a winning strategy.
Related series
• The statement that 1/2 − 1/4 + 1/8 − 1/16 + ⋯ is absolutely convergent means that the series 1/2 + 1/4 + 1/8 + 1/16 + ⋯ is convergent. In fact, the latter series converges to 1, and it proves that one of the binary expansions of 1 is 0.111....
• Pairing up the terms of the series 1/2 − 1/4 + 1/8 − 1/16 + ⋯ results in another geometric series with the same sum, 1/4 + 1/16 + 1/64 + 1/256 + ⋯. This series is one of the first to be summed in the history of mathematics; it was used by Archimedes circa 250–200 BC.[3]
• The Euler transform of the divergent series 1 − 2 + 4 − 8 + ⋯ is 1/2 − 1/4 + 1/8 − 1/16 + ⋯. Therefore, even though the former series does not have a sum in the usual sense, it is Euler summable to 1/3.[4]
Notes
1. Berkelamp et al. p. 79
2. Berkelamp et al. pp. 307–308
3. Shawyer and Watson p. 3
4. Korevaar p. 325
References
• Berlekamp, E. R.; Conway, J. H.; Guy, R. K. (1982). Winning Ways for your Mathematical Plays. Academic Press. ISBN 0-12-091101-9.
• Korevaar, Jacob (2004). Tauberian Theory: A Century of Developments. Springer. ISBN 3-540-21058-X.
• Shawyer, Bruce; Watson, Bruce (1994). Borel's Methods of Summability: Theory and Applications. Oxford UP. ISBN 0-19-853585-6.
Sequences and series
Integer sequences
Basic
• Arithmetic progression
• Geometric progression
• Harmonic progression
• Square number
• Cubic number
• Factorial
• Powers of two
• Powers of three
• Powers of 10
Advanced (list)
• Complete sequence
• Fibonacci sequence
• Figurate number
• Heptagonal number
• Hexagonal number
• Lucas number
• Pell number
• Pentagonal number
• Polygonal number
• Triangular number
Properties of sequences
• Cauchy sequence
• Monotonic function
• Periodic sequence
Properties of series
Series
• Alternating
• Convergent
• Divergent
• Telescoping
Convergence
• Absolute
• Conditional
• Uniform
Explicit series
Convergent
• 1/2 − 1/4 + 1/8 − 1/16 + ⋯
• 1/2 + 1/4 + 1/8 + 1/16 + ⋯
• 1/4 + 1/16 + 1/64 + 1/256 + ⋯
• 1 + 1/2s + 1/3s + ... (Riemann zeta function)
Divergent
• 1 + 1 + 1 + 1 + ⋯
• 1 − 1 + 1 − 1 + ⋯ (Grandi's series)
• 1 + 2 + 3 + 4 + ⋯
• 1 − 2 + 3 − 4 + ⋯
• 1 + 2 + 4 + 8 + ⋯
• 1 − 2 + 4 − 8 + ⋯
• Infinite arithmetic series
• 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
• 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
• 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
Kinds of series
• Taylor series
• Power series
• Formal power series
• Laurent series
• Puiseux series
• Dirichlet series
• Trigonometric series
• Fourier series
• Generating series
Hypergeometric series
• Generalized hypergeometric series
• Hypergeometric function of a matrix argument
• Lauricella hypergeometric series
• Modular hypergeometric series
• Riemann's differential equation
• Theta hypergeometric series
• Category
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Circulant graph
In graph theory, a circulant graph is an undirected graph acted on by a cyclic group of symmetries which takes any vertex to any other vertex. It is sometimes called a cyclic graph,[1] but this term has other meanings.
For the square matrices, see Circulant matrix.
Equivalent definitions
Circulant graphs can be described in several equivalent ways:[2]
• The automorphism group of the graph includes a cyclic subgroup that acts transitively on the graph's vertices. In other words, the graph has a graph automorphism, which is a cyclic permutation of its vertices.
• The graph has an adjacency matrix that is a circulant matrix.
• The n vertices of the graph can be numbered from 0 to n − 1 in such a way that, if some two vertices numbered x and (x + d) mod n are adjacent, then every two vertices numbered z and (z + d) mod n are adjacent.
• The graph can be drawn (possibly with crossings) so that its vertices lie on the corners of a regular polygon, and every rotational symmetry of the polygon is also a symmetry of the drawing.
• The graph is a Cayley graph of a cyclic group.[3]
Examples
Every cycle graph is a circulant graph, as is every crown graph with 2 modulo 4 vertices.
The Paley graphs of order n (where n is a prime number congruent to 1 modulo 4) is a graph in which the vertices are the numbers from 0 to n − 1 and two vertices are adjacent if their difference is a quadratic residue modulo n. Since the presence or absence of an edge depends only on the difference modulo n of two vertex numbers, any Paley graph is a circulant graph.
Every Möbius ladder is a circulant graph, as is every complete graph. A complete bipartite graph is a circulant graph if it has the same number of vertices on both sides of its bipartition.
If two numbers m and n are relatively prime, then the m × n rook's graph (a graph that has a vertex for each square of an m × n chessboard and an edge for each two squares that a chess rook can move between in a single move) is a circulant graph. This is because its symmetries include as a subgroup the cyclic group Cmn $\simeq $ Cm×Cn. More generally, in this case, the tensor product of graphs between any m- and n-vertex circulants is itself a circulant.[2]
Many of the known lower bounds on Ramsey numbers come from examples of circulant graphs that have small maximum cliques and small maximum independent sets.[1]
A specific example
The circulant graph $C_{n}^{s_{1},\ldots ,s_{k}}$ with jumps $s_{1},\ldots ,s_{k}$ is defined as the graph with $n$ nodes labeled $0,1,\ldots ,n-1$ where each node i is adjacent to 2k nodes $i\pm s_{1},\ldots ,i\pm s_{k}\mod n$.
• The graph $C_{n}^{s_{1},\ldots ,s_{k}}$ is connected if and only if $\gcd(n,s_{1},\ldots ,s_{k})=1$.
• If $1\leq s_{1}<\cdots <s_{k}$ are fixed integers then the number of spanning trees $t(C_{n}^{s_{1},\ldots ,s_{k}})=na_{n}^{2}$ where $a_{n}$ satisfies a recurrence relation of order $2^{s_{k}-1}$.
• In particular, $t(C_{n}^{1,2})=nF_{n}^{2}$ where $F_{n}$ is the n-th Fibonacci number.
Self-complementary circulants
A self-complementary graph is a graph in which replacing every edge by a non-edge and vice versa produces an isomorphic graph. For instance, a five-vertex cycle graph is self-complementary, and is also a circulant graph. More generally every Paley graph of prime order is a self-complementary circulant graph.[4] Horst Sachs showed that, if a number n has the property that every prime factor of n is congruent to 1 modulo 4, then there exists a self-complementary circulant with n vertices. He conjectured that this condition is also necessary: that no other values of n allow a self-complementary circulant to exist.[2][4] The conjecture was proven some 40 years later, by Vilfred.[2]
Ádám's conjecture
Define a circulant numbering of a circulant graph to be a labeling of the vertices of the graph by the numbers from 0 to n − 1 in such a way that, if some two vertices numbered x and y are adjacent, then every two vertices numbered z and (z − x + y) mod n are adjacent. Equivalently, a circulant numbering is a numbering of the vertices for which the adjacency matrix of the graph is a circulant matrix.
Let a be an integer that is relatively prime to n, and let b be any integer. Then the linear function that takes a number x to ax + b transforms a circulant numbering to another circulant numbering. András Ádám conjectured that these linear maps are the only ways of renumbering a circulant graph while preserving the circulant property: that is, if G and H are isomorphic circulant graphs, with different numberings, then there is a linear map that transforms the numbering for G into the numbering for H. However, Ádám's conjecture is now known to be false. A counterexample is given by graphs G and H with 16 vertices each; a vertex x in G is connected to the six neighbors x ± 1, x ± 2, and x ± 7 modulo 16, while in H the six neighbors are x ± 2, x ± 3, and x ± 5 modulo 16. These two graphs are isomorphic, but their isomorphism cannot be realized by a linear map.[2]
Toida's conjecture refines Ádám's conjecture by considering only a special class of circulant graphs, in which all of the differences between adjacent graph vertices are relatively prime to the number of vertices. According to this refined conjecture, these special circulant graphs should have the property that all of their symmetries come from symmetries of the underlying additive group of numbers modulo n. It was proven by two groups in 2001 and 2002.[5][6]
Algorithmic questions
There is a polynomial-time recognition algorithm for circulant graphs, and the isomorphism problem for circulant graphs can be solved in polynomial time.[7][8]
References
1. Small Ramsey Numbers, Stanisław P. Radziszowski, Electronic J. Combinatorics, dynamic survey 1, updated 2014.
2. Vilfred, V. (2004), "On circulant graphs", in Balakrishnan, R.; Sethuraman, G.; Wilson, Robin J. (eds.), Graph Theory and its Applications (Anna University, Chennai, March 14–16, 2001), Alpha Science, pp. 34–36.
3. Alspach, Brian (1997), "Isomorphism and Cayley graphs on abelian groups", Graph symmetry (Montreal, PQ, 1996), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 497, Dordrecht: Kluwer Acad. Publ., pp. 1–22, MR 1468786.
4. Sachs, Horst (1962). "Über selbstkomplementäre Graphen". Publicationes Mathematicae Debrecen. 9: 270–288. MR 0151953..
5. Muzychuk, Mikhail; Klin, Mikhail; Pöschel, Reinhard (2001), "The isomorphism problem for circulant graphs via Schur ring theory", Codes and association schemes (Piscataway, NJ, 1999), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 56, Providence, Rhode Island: American Mathematical Society, pp. 241–264, MR 1816402
6. Dobson, Edward; Morris, Joy (2002), "Toida's conjecture is true", Electronic Journal of Combinatorics, 9 (1): R35:1–R35:14, MR 1928787
7. Muzychuk, Mikhail (2004). "A Solution of the Isomorphism Problem for Circulant Graphs". Proc. London Math. Soc. 88: 1–41. doi:10.1112/s0024611503014412. MR 2018956.
8. Evdokimov, Sergei; Ponomarenko, Ilia (2004). "Recognition and verification of an isomorphism of circulant graphs in polynomial time". St. Petersburg Math. J. 15: 813–835. doi:10.1090/s1061-0022-04-00833-7. MR 2044629.
External links
• Weisstein, Eric W. "Circulant Graph". MathWorld.
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Ágnes Szendrei
Ágnes Szendrei is a Hungarian-American mathematician whose research concerns clones, the congruence lattice problem, and other topics in universal algebra. She is a professor of mathematics at the University of Colorado Boulder,[1] and the author of the well-cited book Clones in Universal Algebra (1986).[2] In May 2022[3][4], Dr. Szendrei was elected as an external member of the Hungarian Academy of Sciences[5]; such external memberships are for Hungarian scientists who live outside of Hungary and who have made exceptional contributions to scientific research.
Szendrei earned a doctorate from the Hungarian Academy of Sciences in 1982, and a habilitation in 1993.[6] Her 1982 dissertation was Clones of Linear Operations and Semi-Affine Algebras, supervised by Béla Csákány.[7] She was on the faculty of the University of Szeged from 1982 until 2003, when she moved to the University of Colorado.[6]
Szendrei is a Humboldt Fellow. She won the Kató Rényi Award for undergraduate research in 1975, the Géza Grünwald Commemorative Prize for young researchers of the János Bolyai Mathematical Society in 1978, and the Golden Ring of the Republic in 1979. She was the 1992 winner of the Paul Erdős Prize of the Hungarian Academy of Sciences, and the 2000 winner of the Academy's Farkas Bolyai Award.[6]
References
1. Agnes Szendrei, University of Colorado Boulder, 29 September 2016, retrieved 2019-10-11
2. Berman, Joel (1987), "Review of Clones in Universal Algebra", Mathematical Reviews, MR 0859550
3. University of Colorado Boulder Department of Mathematics News and Events, 28 September 2016, retrieved 2023-03-16
4. Hungarian Academy of Sciences announcement of new external members elected 2022, 13 May 2022, retrieved 2023-03-16
5. Department of Mathematical Sciences of the Hungarian Academy of Sciences, retrieved 2023-03-16
6. Curriculum vitae (PDF), January 31, 2019, retrieved 2019-10-11
7. Ágnes Szendrei at the Mathematics Genealogy Project
External links
• Home page
• Ágnes Szendrei publications indexed by Google Scholar
Authority control: Academics
• Google Scholar
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
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Árpád Varecza
Árpád Varecza (6 September 1941 – 26 September 2005), was a Hungarian mathematician, former lecturer at the College of Nyíregyháza, head of the Institute of Mathematics and Informatics, and deputy director general of the institution for three years.
Árpád Varecza
Born(1941-09-06)6 September 1941
Vác, Hungary
Died26 September 2005(2005-09-26) (aged 64)
Nyíregyháza, Hungary
NationalityHungarian
Alma materEötvös Loránd University, Budapest
Scientific career
FieldsMathematics
InstitutionsCollege of Nyíregyháza
Biography
He was born on September 6, 1941 in Vác. He graduated from the teacher training college in Szeged in 1963, then graduated from ELTE with a degree in mathematics, physics and technology. His first jobs were tied to his birthplace. He taught at the Primary School in Váchartyán and Verőce, then at the Géza Király Secondary School and Vocational Secondary School in Vác. He was admitted to the Department of Mathematics of the Teacher Training College in Nyíregyháza in 1969 as an assistant lecturer, in 1971 he was appointed an assistant professor, in 1977 he was appointed an associate professor and in 1983 he was appointed a college teacher. Between 1977 and 1980 he was an aspirant at the Mathematical Research Institute of the Hungarian Academy of Sciences. In 1975 he received his Ph.D. at Kossuth Lajos University, and in 1982 he defended his Ph.D. Following the era marked by the name of Gyula Bereznai, he was head of the Department of Mathematics in 1984, then Head of the Institute of Mathematics and Informatics in 2000, and served as Deputy Director General of the institution for three years.
Work
He specialized in combinatorics, including "sorting algorithms". He obtained his candidate's degree in his dissertation on "Optimal sorting algorithms". Honorary Heir President of the János Bolyai Mathematical Society, President of the Department of Mathematics, Physics and Astronomy of the DAB, Chairman of the College's Scientific Committee, Member of the MM Intensive Further Education Council and of the MM Computer Science Advisory Board, He was the editor of the Mathematics series. Under his guidance, in 1985, four colleagues earned their Ph.D.
Books
• Mathematics competitions for teacher training colleges[1]
• A tanárképző főiskolák Péter Rózsa matematikai versenyei[2]
Notes
1. Bereznai Gyula - Dr.Varecza Árpád - Dr.Rozgonyi Tibor: Tanárképző főiskolák matematika versenyei (1952-1970:ISBN 9789631736274; 1971-1979:ISBN 9789631761597; 1980-1985:ISBN 9789631816068)
Tanárképző főiskolák országos matematika versenyei: (ISBN 9789631736267)
2. Rozgonyi Tibor, Varecza Árpád: A tanárképző főiskolák Péter Rózsa matematikai versenyei; Typotex, 2003. - 160 p. ISBN 9639326798
External links
• Varecza Árpád
• On the smallest and largest elements
• In memoriam
Authority control
International
• ISNI
• VIAF
National
• Germany
Academics
• DBLP
• zbMATH
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Èlizbar Nadaraya
Èlizbar Nadaraya is a Georgian mathematician who is currently a Full Professor and the Chair of the Theory of Probability and Mathematical Statistics at the Tbilisi State University.[1] He developed the Nadaraya-Watson estimator along with Geoffrey Watson, which proposes estimating the conditional expectation of a random variable as a locally weighted average using a kernel as a weighting function.[2][3][4]
Nadaraya was born in 1936 in Khobi, Georgia. He received his doctoral degree from the V.I. Romanovski Institute of Mathematics, Tashkent in 1981. He has since co-authored over 120 publications including 5 textbooks in the area of probability and statistics.[5]
Most cited publications
Book
• E. A. Nadaraya, Nonparametric Estimation of Probability Densities and Regression Curves Springer, 1989 ISBN 978-90-277-2757-2 (Cited 319 times, according to Google Scholar.[6])
Journal articles
• Nadaraya EA. On estimating regression. Theory of Probability & Its Applications. 1964;9(1):141-2. (Cited 4408 times, according to Google Scholar [6])
• Nadaraya, E.A., 1965. On non-parametric estimates of density functions and regression curves. Theory of Probability & Its Applications, 10(1), pp. 186–190. (Cited 673 times, according to Google Scholar.[6])
• Nadaraya EA. Some new estimates for distribution functions. Theory of Probability & Its Applications. 1964;9(3):497-500. (Cited 329 times, according to Google Scholar.[6])
References
1. "Nadaraya Elizbar".
2. Nadaraya, E. A. (1964). "On Estimating Regression". Theory of Probability and Its Applications. 9 (1): 141–2. doi:10.1137/1109020.
3. Watson, G. S. (1964). "Smooth regression analysis". Sankhyā: The Indian Journal of Statistics, Series A. 26 (4): 359–372. JSTOR 25049340.
4. Bierens, Herman J. (1994). "The Nadaraya–Watson kernel regression function estimator". Topics in Advanced Econometrics. New York: Cambridge University Press. pp. 212–247. ISBN 0-521-41900-X.
5. http://science.org.ge/wp-content/cv/Nadaraya%20Elizbar.pdf
6. [] Google Scholar Author page, Accessed Dec. 2022
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ÉLECTRE
ÉLECTRE is a family of multi-criteria decision analysis (MCDA) methods that originated in Europe in the mid-1960s. The acronym ÉLECTRE stands for: ÉLimination Et Choix Traduisant la REalité ("Elimination and Choice Translating Reality").
The method was first proposed by Bernard Roy and his colleagues at SEMA consultancy company. A team at SEMA was working on the concrete, multiple criteria, real-world problem of how firms could decide on new activities and had encountered problems using a weighted sum technique. Roy was called in as a consultant and the group devised the ELECTRE method. As it was first applied in 1965, the ELECTRE method was to choose the best action(s) from a given set of actions, but it was soon applied to three main problems: choosing, ranking and sorting. The method became more widely known when a paper by B. Roy appeared in a French operations research journal.[1] It evolved into ELECTRE I (electre one) and the evolutions have continued with ELECTRE II, ELECTRE III, ELECTRE IV, ELECTRE IS and ELECTRE TRI (electre tree), to mention a few.[2] They are used in the fields of business, development,[3] design,[4] and small hydropower.[5]
Roy is widely recognized as the father of the ELECTRE method, which was one of the earliest approaches in what is sometimes known as the French school of decision making. It is usually classified as an "outranking method" of decision making.
There are two main parts to an ELECTRE application: first, the construction of one or several outranking relations, which aims at comparing in a comprehensive way each pair of actions; second, an exploitation procedure that elaborates on the recommendations obtained in the first phase. The nature of the recommendation depends on the problem being addressed: choosing, ranking or sorting.
Usually the ELECTRE methods are used to discard some alternatives to the problem, which are unacceptable. After that, another form of MCDA can be used to select the best one. The advantage of using the ELECTRE methods before is that another MCDA can be applied with a restricted set of alternatives, saving much time.
Criteria in ELECTRE methods have two distinct sets of parameters: the importance coefficients and the veto thresholds. ELECTRE method cannot determine the weights of the criteria. In this regard, it can be combined with other approaches such as Ordinal Priority Approach, Analytic Hierarchy Process, etc.
References
1. Roy, Bernard (1968). "Classement et choix en présence de points de vue multiples (la méthode ELECTRE)". La Revue d'Informatique et de Recherche Opérationelle (RIRO) (8): 57–75.
2. Figueira, José; Salvatore Greco; Matthias Ehrgott (2005). Multiple Criteria Decision Analysis: State of the Art Surveys. New York: Springer Science + Business Media, Inc. ISBN 0-387-23081-5.
3. Rangel, L. S. A. D.; Gomes, L. F. V. A. M.; Moreira, R. R. A. (2009). "Decision theory with multiple criteria: An aplication [sic] of ELECTRE IV and TODIM to SEBRAE/RJ". Pesquisa Operacional. 29 (3): 577. doi:10.1590/S0101-74382009000300007.
4. Shanian, A.; Savadogo, O. (2006). "A non-compensatory compromised solution for material selection of bipolar plates for polymer electrolyte membrane fuel cell (PEMFC) using ELECTRE IV". Electrochimica Acta. 51 (25): 5307. doi:10.1016/j.electacta.2006.01.055.
5. Saracoglu, B. O. (2015). "An Experimental Research Study on the Solution of a Private Small Hydropower Plant Investments Selection Problem by ELECTRE III/IV, Shannon's Entropy, and Saaty's Subjective Criteria Weighting". Advances in Decision Sciences. 2015: 1–20. doi:10.1155/2015/548460.
External links
• Decision Radar : A free online ELECTRE calculator written in Python.
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Élisabeth Bouscaren
Élisabeth Bouscaren (born 1956)[1] is a French mathematician who works on algebraic geometry, algebra and mathematical logic (model theory).[2]
Élisabeth Bouscaren
Bouscaren at Oberwolfach in 1988
Born1956 (age 66–67)
NationalityFrench
Academic background
Alma materUniversity of Paris VII
Academic work
DisciplineMathematics
InstitutionsUniversity of Paris XI,
French National Center for Scientific Research
Main interestsAlgebraic geometry,
Model theory
Education and career
Bouscaren received her doctorate in 1979 from the University of Paris VII and her habilitation in 1985. From 1981 she worked at the French National Center for Scientific Research (CNRS) until 2005, when she moved to the University of Paris XI. Since 2007, she has held the position of Research Director at CNRS.
She has been a visiting scholar at Yale University, the University of Notre Dame and MSRI, and has published a book on Ehud Hrushovski's proof of the Mordell-Lang conjecture. She was an invited speaker in the logic session of the 2002 International Congress of Mathematicians.[3]
In 2020, Bouscaren gave the Gödel Lecture, titled The ubiquity of configurations in Model Theory.
Selected publications
• Bouscaren, Elisabeth (2005), "Model theory and geometry", Logic Colloquium 2000, Lecture Notes in Logic, vol. 19, Urbana, IL: Association for Symbolic Logic, pp. 3–31, MR 2143876
• Bouscaren, E.; Delon, F. (2002), "Groups definable in separably closed fields", Transactions of the American Mathematical Society, 354 (3): 945–966, doi:10.1090/S0002-9947-01-02886-0, MR 1867366
• Bouscaren, E.; Delon, F. (2002), "Minimal groups in separably closed fields", Journal of Symbolic Logic, 67 (1): 239–259, doi:10.2178/jsl/1190150042, MR 1889549
• Bouscaren, Élisabeth (2002), "Théorie des modèles et conjecture de Manin-Mumford (d'après Ehud Hrushovski)" [Model theory and the Manin-Mumford conjecture (following Ehud Hrushovski)], Astérisque (in French) (276): 137–159, MR 1886759, Séminaire Bourbaki 1999/2000
• Bouscaren, Elisabeth, ed. (1998), Model Theory and Algebraic Geometry: An introduction to E. Hrushovski's proof of the Geometric Mordell–Lang Conjecture, Lecture Notes in Mathematics, vol. 1696, Berlin: Springer-Verlag, doi:10.1007/978-3-540-68521-0, ISBN 3-540-64863-1, MR 1678586
• Bouscaren, E.; Hrushovski, E. (1994), "On one-based theories", Journal of Symbolic Logic, 59 (2): 579–595, doi:10.2307/2275409, MR 1276634
References
1. Birth year from BNF catalog entry, accessed 2018-10-08
2. "Elisabeth Bouscaren". www.math.u-psud.fr. Retrieved 2018-10-08.
3. ICM Plenary and Invited Speakers, accessed 2018-10-08
External links
• Homepage
• CV (pdf)
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Éléments de géométrie algébrique
The Éléments de géométrie algébrique ("Elements of Algebraic Geometry") by Alexander Grothendieck (assisted by Jean Dieudonné), or EGA for short, is a rigorous treatise, in French, on algebraic geometry that was published (in eight parts or fascicles) from 1960 through 1967 by the Institut des Hautes Études Scientifiques. In it, Grothendieck established systematic foundations of algebraic geometry, building upon the concept of schemes, which he defined. The work is now considered the foundation stone and basic reference of modern algebraic geometry.
Éléments de géométrie algébrique
AuthorAlexander Grothendieck and Jean Dieudonné
LanguageFrench
SubjectAlgebraic geometry
PublisherInstitut des Hautes Études Scientifiques
Publication date
1960–1967
Editions
Initially thirteen chapters were planned, but only the first four (making a total of approximately 1500 pages) were published. Much of the material which would have been found in the following chapters can be found, in a less polished form, in the Séminaire de géométrie algébrique (known as SGA). Indeed, as explained by Grothendieck in the preface of the published version of SGA, by 1970 it had become clear that incorporating all of the planned material in EGA would require significant changes in the earlier chapters already published, and that therefore the prospects of completing EGA in the near term were limited. An obvious example is provided by derived categories, which became an indispensable tool in the later SGA volumes, but was not yet used in EGA III as the theory was not yet developed at the time. Considerable effort was therefore spent to bring the published SGA volumes to a high degree of completeness and rigour. Before work on the treatise was abandoned, there were plans in 1966–67 to expand the group of authors to include Grothendieck's students Pierre Deligne and Michel Raynaud, as evidenced by published correspondence between Grothendieck and David Mumford.[1] Grothendieck's letter of 4 November 1966 to Mumford also indicates that the second-edition revised structure was in place by that time, with Chapter VIII already intended to cover the Picard scheme. In that letter he estimated that at the pace of writing up to that point, the following four chapters (V to VIII) would have taken eight years to complete, indicating an intended length comparable to the first four chapters, which had been in preparation for about eight years at the time.
Grothendieck nevertheless wrote a revised version of EGA I which was published by Springer-Verlag. It updates the terminology, replacing "prescheme" by "scheme" and "scheme" by "separated scheme", and heavily emphasizes the use of representable functors. The new preface of the second edition also includes a slightly revised plan of the complete treatise, now divided into twelve chapters.
Grothendieck's EGA V which deals with Bertini type theorems is to some extent available from the Grothendieck Circle website. Monografie Matematyczne in Poland has accepted this volume for publication, but the editing process is quite slow (as of 2010). James Milne has preserved some of the original Grothendieck notes and a translation of them into English. They may be available from his websites connected with the University of Michigan in Ann Arbor.
Chapters
The following table lays out the original and revised plan of the treatise and indicates where (in SGA or elsewhere) the topics intended for the later, unpublished chapters were treated by Grothendieck and his collaborators.
# First edition Second edition Comments
I Le langage des schémas Le langage des schémas Second edition brings in certain schemes representing functors such as Grassmannians, presumably from intended Chapter V of the first edition. In addition, the contents of Section 1 of Chapter IV of first edition was moved to Chapter I in the second edition.
II Étude globale élémentaire de quelques classes de morphismes Étude globale élémentaire de quelques classes de morphismes First edition complete, second edition did not appear.
III Étude cohomologique des faisceaux cohérents Cohomologie des Faisceaux algébriques cohérents. Applications. First edition complete except for last four sections, intended for publication after Chapter IV: elementary projective duality, local cohomology and its relation to projective cohomology, and Picard groups (all but projective duality treated in SGA II).
IV Étude locale des schémas et des morphismes de schémas Étude locale des schémas et des morphismes de schémas First edition essentially complete; some changes made in last sections; the section on hyperplane sections made into the new Chapter V of second edition (draft exists)
V Procédés élémentaires de construction de schémas Complements sur les morphismes projectifs Did not appear. Some elementary constructions of schemes apparently intended for first edition appear in Chapter I of second edition. The existing draft of Chapter V corresponds to the second edition plan. It includes also expanded treatment of some material from SGA VII.
VI Technique de descente.
Méthode générale de construction des schémas
Techniques de construction de schémas Did not appear. Descent theory and related construction techniques summarised by Grothendieck in FGA. By 1968 the plan had evolved to treat algebraic spaces and algebraic stacks.
VII Schémas de groupes, espaces fibrés principaux Schémas en groupes, espaces fibrés principaux Did not appear. Treated in detail in SGA III.
VIII Étude différentielle des espaces fibrés Le schéma de Picard Did not appear. Material apparently intended for first edition can be found in SGA III, construction and results on Picard scheme are summarised in FGA.
IX Le groupe fondamental Le groupe fondamental Did not appear. Treated in detail in SGA I.
X Résidus et dualité Résidus et dualité Did not appear. Treated in detail in Hartshorne's edition of Grothendieck's notes "Residues and duality"
XI Théorie d'intersection, classes de Chern, théorème de Riemann-Roch Théorie d'intersection, classes de Chern, théorème de Riemann-Roch Did not appear. Treated in detail in SGA VI.
XII Schémas abéliens et schémas de Picard Cohomologie étale des schémas Did not appear. Étale cohomology treated in detail in SGA IV, SGA V.
XIII Cohomologie de Weil none Intended to cover étale cohomology in the first edition.
In addition to the actual chapters, an extensive "Chapter 0" on various preliminaries was divided between the volumes in which the treatise appeared. Topics treated range from category theory, sheaf theory and general topology to commutative algebra and homological algebra. The longest part of Chapter 0, attached to Chapter IV, is more than 200 pages.
Grothendieck never gave permission for the 2nd edition of EGA I to be republished, so copies are rare but found in many libraries. The work on EGA was finally disrupted by Grothendieck's departure first from IHÉS in 1970 and soon afterwards from the mathematical establishment altogether. Grothendieck's incomplete notes on EGA V can be found at Grothendieck Circle.
In historical terms, the development of the EGA approach set the seal on the application of sheaf theory to algebraic geometry, set in motion by Serre's basic paper FAC. It also contained the first complete exposition of the algebraic approach to differential calculus, via principal parts. The foundational unification it proposed (see for example unifying theories in mathematics) has stood the test of time.
EGA has been scanned by NUMDAM and is available at their website under "Publications mathématiques de l'IHÉS", volumes 4 (EGAI), 8 (EGAII), 11 (EGAIII.1re), 17 (EGAIII.2e), 20 (EGAIV.1re), 24 (EGAIV.2e), 28 (EGAIV.3e) and 32 (EGAIV.4e).
Bibliographic information
• Grothendieck, Alexandre; Dieudonné, Jean (1971). Éléments de géométrie algébrique: I. Le langage des schémas. Grundlehren der Mathematischen Wissenschaften (in French). Vol. 166 (2nd ed.). Berlin; New York: Springer-Verlag. ISBN 978-3-540-05113-8.
• Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas". Publications Mathématiques de l'IHÉS. 4: 5–228. doi:10.1007/bf02684778. MR 0217083.[2]
• Grothendieck, Alexandre; Dieudonné, Jean (1961). "Éléments de géométrie algébrique: II. Étude globale élémentaire de quelques classes de morphismes". Publications Mathématiques de l'IHÉS. 8: 5–222. doi:10.1007/bf02699291. MR 0217084.
• Grothendieck, Alexandre; Dieudonné, Jean (1961). "Eléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Première partie". Publications Mathématiques de l'IHÉS. 11: 5–167. doi:10.1007/bf02684274. MR 0217085.
• Grothendieck, Alexandre; Dieudonné, Jean (1963). "Éléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Seconde partie". Publications Mathématiques de l'IHÉS. 17: 5–91. doi:10.1007/bf02684890. MR 0163911.
• Grothendieck, Alexandre; Dieudonné, Jean (1964). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Première partie". Publications Mathématiques de l'IHÉS. 20: 5–259. doi:10.1007/bf02684747. MR 0173675.
• Grothendieck, Alexandre; Dieudonné, Jean (1965). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Seconde partie". Publications Mathématiques de l'IHÉS. 24: 5–231. doi:10.1007/bf02684322. MR 0199181.
• Grothendieck, Alexandre; Dieudonné, Jean (1966). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Troisième partie". Publications Mathématiques de l'IHÉS. 28: 5–255. doi:10.1007/bf02684343. MR 0217086.
• Grothendieck, Alexandre; Dieudonné, Jean (1967). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie". Publications Mathématiques de l'IHÉS. 32: 5–361. doi:10.1007/bf02732123. MR 0238860.
See also
• Fondements de la Géometrie Algébrique (FGA)
• Séminaire de Géométrie Algébrique du Bois Marie (SGA)
References
1. Mumford, David (2010). Ching-Li Chai; Amnon Neeman; Takahiro Shiota. (eds.). Selected papers, Volume II. On algebraic geometry, including correspondence with Grothendieck. Springer. pp. 720, 722. ISBN 978-0-387-72491-1.
2. Lang, S. (1961). "Review: Éléments de géométrie algébrique, par A. Grothendieck, rédigés avec la collaboration de J. Dieudonné" (PDF). Bull. Amer. Math. Soc. 67 (3): 239–246. doi:10.1090/S0002-9904-1961-10564-8.
External links
• Scanned copies and partial English translations: Mathematical Texts (published) Archived 2012-11-04 at the Wayback Machine
• Detailed table of contents: EGA
• SGA, EGA, FGA by Mateo Carmona
• The Grothendieck circle maintains copies of EGA, SGA, and other of Grothendieck's writings
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Émery topology
In martingale theory, Émery topology is a topology on the space of semimartingales. The topology is used in financial mathematics. The class of stochastic integrals with general predictable integrands coincides with the closure of the set of all simple integrals.[1]
The topology was introduced in 1979 by the french mathematician Michel Émery.[2]
Definition
Let $(\Omega ,{\mathcal {A}},\{{\mathcal {F_{t}}}\},P)$ be a filtred probability space, where the filtration satisfies the usual conditions and $T\in (0,\infty )$. Let ${\mathcal {S}}(P)$ be the space of real semimartingales and ${\mathcal {E}}(1)$ the space of simple predictable processes $H$ with $|H|=1$.
We define the quasinorm
$\|X\|_{{\mathcal {S}}(P)}:=\sup \limits _{H\in {\mathcal {E}}(1)}\mathbb {E} \left[1\wedge \left(\sup \limits _{t\in [0,T]}|(H\cdot X)_{t}|\right)\right].$
Then $({\mathcal {S}}(P),d)$ with the metric $d(X,Y):=\|X-Y\|_{{\mathcal {S}}(P)}$ is a complete space and the induced topology is called Émery topology.[3][1]
References
1. Kardaras, Constantinos (2013). "On the closure in the Emery topology of semimartingale wealth-process sets". Annals of Applied Probability. 23 (4): 1355–1376. doi:10.1214/12-AAP872.
2. Émery, Michel (1979). "Une topologie sur l'espace des semimartingales". Séminaire de probabilités de Strasbourg. 13: 260–280.
3. De Donno, M.; Pratelli, M. (2005). "A theory of stochastic integration for bond markets". Annals of Applied Probability. 15 (4): 2773–2791. arXiv:math/0602532. doi:10.1214/105051605000000548.
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Émile Léger
Émile Léger (1795–1838) was a French mathematician.
Émile Léger
Born(1795-08-15)15 August 1795
La Grange aux Bois, today Sainte-Menehould, France
Died15 December 1838(1838-12-15) (aged 43)
Paris, France
Alma materÉcole Polytechnique
Known forEuclidean algorithm
Scientific career
FieldsMathematics
Life and work
Leger studied at Lycée de Mayence (now Mainz in Germany, capital of the French department of Mont-Tonnerre during the French First Republic), where his father Claude was professor of rhetoric. In 1813 he entered the École Polytechnique. With other students, he helped defend Paris during the Hundred Days of Napoleon in March 1815, and was decorated for bravery.[1] In 1816, he left school to go to Montmorency where his father founded an institution to prepare young people for the entrance exams to Paris universities. After his father retired, he managed the institution.[2]
Léger only published four papers on mathematics,[1] but one of them seems to be the first to recognize the worst case in the euclidean algorithm: when the inputs are proportional to consecutive Fibonacci numbers.[2]
References
1. O'Connor & Robertson, MacTutor History of Mathematics.
2. Shallit, page 410.
Bibliography
• Shallit, Jeffrey (1994). "Origins of the analysis of the Euclidean algorithm". Historia Mathematica. 21 (4): 401–419. doi:10.1006/hmat.1994.1031. ISSN 0315-0860.
External links
• O'Connor, John J.; Robertson, Edmund F., "Émile Léger", MacTutor History of Mathematics Archive, University of St Andrews
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Émile Merlin
Émile Alphonse Louis Merlin (12 October 1875 in Mons – 29 July 1938 in Le Bourg d'Oisans) was a Belgian mathematician and astronomer.
Émile A. L. Merlin
Born(1875-10-12)12 October 1875
Mons, Belgium
Died29 July 1938(1938-07-29) (aged 62)
Le Bourg-d'Oisans, France
EducationPh.D.
Alma materUniversity of Liège
University of Ghent
AwardsA. de Potter Prize
Scientific career
FieldsMathematics, astronomy
InstitutionsUniversity of Ghent
Merlin attended the secondary school Athénée royal de Bruxelles. He then studied at the University of Liège and the University of Ghent, where in 1900 he received his doctorate in mathematics. This was followed by a stay abroad between 1901 and 1903 in Paris at the Sorbonne, at the Collège de France and in Göttingen. In 1904 he became an assistant at the observatory in Uccle. In 1909 he was promoted to astronomer adjoint. From 1912 he was a lecturer on astronomy and geodesy at the University of Ghent and in 1919 he became a full professor and director of the geographical station of the University of Ghent.[1]
He was an alpinist and died in a mountain accident in the French Alps in Le Bourg d'Oisans.[1]
Merlin was one of the editors for the French edition of Klein's encyclopedia.[2]
For his work on celestial mechanics he was awarded the A. de Potter Prize of the Royal Belgian Academy of Sciences. He was president of the Belgian Mathematical Society, honorary member of the Astronomical Society of Mexico and member of the Commission for Mathematics and Astronomy of the National Fund for Research in Spain.[1] He was Invited Speaker at the International Congress of Mathematicians in Toronto 1924 (Sur les lignes asymptotiques en géométrie infinitésimale) and Oslo 1936 (Sur certains mouvements des fluid parfaits).
Selected publications
• with Paul Stroobant, Jules Delvosal, Hector Philippot, and Eugène Delporte: Les observatoires astronomiques et les astronomes. Hayez. 1907.
• Merlin, E. (1907). "La Répartition des Taches Solaires en Latitudes Héliographiques". Bulletin de la Société Belge d'Astronomie. 12: A179. Bibcode:1907BSBA...12A.179M.
• Merlin, E. (1908). "Sur la détermination systématique des éléments de la figure elliptique d'une planète au moyen de mesures micrométriques de diamètres". Astronomische Nachrichten. 178 (24): 391–394. Bibcode:1908AN....178..391M. doi:10.1002/asna.19081782404.
• Merlin, Emile (1908). "Observations d'etoiles doubles effectuees a l'equatorial de 38 centime tres EN 1906 et 1907". Annales de l'Observatoire Royal de Belgique Nouvelle Serie. 11: 75. Bibcode:1908AnOBN..11...75M.
• Merlin, Émile (1930). "Sur la résolution graphique des triangles sphériques et la détermination rapide de la longitude et de la latitude d'un lieu". Bulletin Astronomique. 6: 119. Bibcode:1930BuAst...6..119M.
• Merlin, E. (1938). "Sur une formule usitée en astronomie". Ciel et Terre. 54: 1. Bibcode:1938C&T....54....1M.
References
1. Van Aerschodt, L. (1938). "Nécrologie: Emile Merlin (1875-1938)". Ciel et Terre. 54: 295. Bibcode:1938C&T....54..295V.
2. Merlin, É (1913). "Configurations". Encylopédie des Sciences Mathématiques, Edition française. Vol. Tome 3, volume 2. pp. 144–160.
External links
• "Emile Merlin (1875–1938)" by Henri Louis Vanderlinden (obituary in Flemish with bibliography of Merlin's publications)
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Émile Borel
Félix Édouard Justin Émile Borel (French: [bɔʁɛl]; 7 January 1871 – 3 February 1956)[1] was a French mathematician[2] and politician. As a mathematician, he was known for his founding work in the areas of measure theory and probability.
Not to be confused with Armand Borel.
Émile Borel
Émile Borel (1932)
Minister of the Navy
In office
17 April 1925 – 28 November 1925
Prime MinisterPaul Painlevé
Preceded byJacques-Louis Dumesnil
Succeeded byGeorges Leygues
Member of the Chamber of Deputies
In office
15 June 1924 – 4 June 1936
Personal details
Born
Félix Édouard Justin Émile Borel
(1871-01-07)7 January 1871
Saint-Affrique, France
Died3 February 1956(1956-02-03) (aged 85)
Paris, France
NationalityFrench
Alma materÉcole normale supérieure Paris
Known forMeasure theory
Probability theory
Heine–Borel theorem
Scientific career
FieldsMathematics
InstitutionsUniversity of Paris
ThesisSur quelques points de la théorie des fonctions (1893)
Doctoral advisorGaston Darboux
Doctoral students
• Paul Dienes
• Henri Lebesgue
• Paul Montel
• Georges Valiron
Biography
Borel was born in Saint-Affrique, Aveyron, the son of a Protestant pastor.[3] He studied at the Collège Sainte-Barbe and Lycée Louis-le-Grand before applying to both the École normale supérieure and the École Polytechnique. He qualified in the first position for both and chose to attend the former institution in 1889. That year he also won the concours général, an annual national mathematics competition. After graduating in 1892, he placed first in the agrégation, a competitive civil service examination leading to the position of professeur agrégé. His thesis, published in 1893, was titled Sur quelques points de la théorie des fonctions ("On some points in the theory of functions"). That year, Borel started a four-year stint as a lecturer at the University of Lille, during which time he published 22 research papers. He returned to the École normale supérieure in 1897, and was appointed to the chair of theory of functions, which he held until 1941.[4]
In 1901, Borel married 17-year-old Marguerite, the daughter of colleague Paul Émile Appel; she later wrote more than 30 novels under the pseudonym Camille Marbo. Émile Borel died in Paris on 3 February 1956.[4]
Work
Along with René-Louis Baire and Henri Lebesgue, Émile Borel was among the pioneers of measure theory and its application to probability theory. The concept of a Borel set is named in his honor. One of his books on probability introduced the amusing thought experiment that entered popular culture under the name infinite monkey theorem or the like. He also published a series of papers (1921–1927) that first defined games of strategy.[5] John von Neumann objected to this assignment of priority in a letter to Econometrica published in 1953 where he asserted that Borel could not have defined games of strategy because he rejected the minimax theorem.[6]
With the development of statistical hypothesis testing in the early 1900s various tests for randomness were proposed. Sometimes these were claimed to have some kind of general significance, but mostly they were just viewed as simple practical methods. In 1909, Borel formulated the notion that numbers picked randomly on the basis of their value are almost always normal, and with explicit constructions in terms of digits, it is quite straightforward to get numbers that are normal.[7]
In 1913 and 1914 he bridged the gap between hyperbolic geometry and special relativity with expository work. For instance, his book Introduction Géométrique à quelques Théories Physiques[8] described hyperbolic rotations as transformations that leave a hyperbola stable just as a circle around a rotational center is stable.
In 1922, he founded the Paris Institute of Statistics, the oldest French school for statistics; then in 1928 he co-founded the Institut Henri Poincaré in Paris.
Political career
In the 1920s, 1930s, and 1940s, he was active in politics. From 1924 to 1936, he was a member of the Chamber of Deputies.[9] In 1925, he was Minister of the Navy in the cabinet of fellow mathematician Paul Painlevé. During the Second World War, he was a member of the French Resistance.
Honors
Besides the Centre Émile Borel at the Institut Henri Poincaré in Paris and a crater on the Moon, the following mathematical notions are named after him:
• Borel algebra
• Borel's lemma
• Borel's law of large numbers
• Borel measure
• Borel–Kolmogorov paradox
• Borel–Cantelli lemma
• Borel–Carathéodory theorem
• Heine–Borel theorem
• Borel determinacy theorem
• Borel right process
• Borel set
• Borel summation
• Borel distribution
• Borel's conjecture about strong measure zero sets (not to be confused with Borel conjecture, named for Armand Borel).
Borel also described a poker model that he coins La Relance in his 1938 book Applications de la théorie des probabilités aux Jeux de Hasard.[10]
Borel was awarded the Resistance Medal in 1950.[4]
Works
• On a few points about the theory of functions (PhD thesis, 1894)
• Introduction to the study of number theory and superior algebra (1895)
• A course on the theory of functions (1898)
• A course on power series (1900)
• A course on divergent series (1901)
• A course on positive terms series (1902)
• A course on meromorphic functions (1903)
• A course on growth theory at the Paris faculty of sciences (1910)
• A course on functions of a real variable and polynomial serial developments (1905)
• Chance (1914)
• Geometrical introduction to some physical theories (1914)
• A course on complex variable uniform monogenic functions (1917)
• On the method in sciences (1919)
• Space and time (1921)
• Game theory and left symmetric core integral equations (1921)
• Methods and problems of the theory of functions (1922)
• Space and time (1922)
• A treatise on probability calculation and its applications (1924–1934)
• Application of probability theory to games of chance (1938)
• Principles and classical formulas for probability calculation (1925)
• Practical and philosophical values of probabilities (1939)
• Mathematical theory of contract bridge for everyone (1940)
• Game, luck and contemporary scientific theories (1941)
• Probabilities and life (1943)
• Evolution of mechanics (1943)
• Paradoxes of the infinite (1946)
• Elements of set theory (1949)
• Probability and certainty (1950)
• Inaccessible numbers (1952)
• Imaginary and real in mathematics and physics (1952)
• Emile Borel complete works (1972)
Articles
• (in French) "La science est-elle responsable de la crise mondiale?", Scientia : rivista internazionale di sintesi scientifica, 51, 1932, pp. 99–106.
• (in French) "La science dans une société socialiste", Scientia : rivista internazionale di sintesi scientifica, 31, 1922, pp. 223–228.
• (in French) "Le continu mathématique et le continu physique", Rivista di scienza, 6, 1909, pp. 21–35.
See also
• Borel right process
References
1. May, Kenneth (1970–1980). "Borel, Émile". Dictionary of Scientific Biography. Vol. 2. New York: Charles Scribner's Sons. pp. 302–305. ISBN 978-0-684-10114-9.
2. Émile Borel's biography – Université Lille Nord de France
3. McElroy, Tucker (2009). A to Z of Mathematicians. Infobase Publishing. p. 46. ISBN 978-1-4381-0921-3.
4. Chang, Sooyoung (2011). Academic Genealogy of Mathematicians. World Scientific. p. 107. ISBN 978-981-4282-29-1.
5. "Émile Borel," Encyclopædia Britannica
6. von Neumann, J.; Fréchet, M. (1953). "Communication on the Borel Notes". Econometrica. 21 (1): 124–127. doi:10.2307/1906950. ISSN 0012-9682. JSTOR 1906950.
7. Harman, Glyn (2002), "One hundred years of normal numbers", in Bennett, M. A.; Berndt, B. C.; Boston, N.; Diamond, H. G.; Hildebrand, A. J.; Philipp, W. (eds.), Surveys in Number Theory: Papers from the Millennial Conference on Number Theory, A K Peters, pp. 57–74, MR 1956249
8. Émile Borel (1914) Introduction Geometrique à quelques Théories Physiques, Gauthier-Villars, link from Cornell University Historical Math Monographs
9. "Émile Borel | French mathematician | Britannica". www.britannica.com. Retrieved 2023-03-12.
10. Émile Borel and Jean Ville. Applications de la théorie des probabilités aux jeux de hasard. Gauthier-Vilars, 1938
• Michel Pinault, Emile Borel, une carrière intellectuelle sous la 3ème République, Paris, L'Harmattan, 2017. Voir : michel-pinault.over-blog.com
External links
• Quotations related to Émile Borel at Wikiquote
• French Wikisource has original text related to this article: Auteur:Émile Borel
• Media related to Émile Borel (mathematician) at Wikimedia Commons
• Works by or about Émile Borel at Internet Archive
• O'Connor, John J.; Robertson, Edmund F., "Émile Borel", MacTutor History of Mathematics Archive, University of St Andrews
• Author profile in the database zbMATH
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Éric Leichtnam
Éric Leichtnam is director of research at the CNRS at the Institut de Mathématiques de Jussieu in Paris. His fields of interest are noncommutative geometry, ergodic theory, Dirichlet problem, non-commutative residue.
Selected publications
• Katz, Mikhail G.; Leichtnam, Eric (2013), "Commuting and noncommuting infinitesimals", American Mathematical Monthly, 120 (7): 631–641, arXiv:1304.0583, doi:10.4169/amer.math.monthly.120.07.631
• Gérard, Patrick; Leichtnam, Éric: Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Math. J. 71 (1993), no. 2, 559–607.
• Fedosov, Boris V.; Golse, François; Leichtnam, Eric; Schrohe, Elmar: The noncommutative residue for ----- (1996), no. 1, 1–31.
• Leichtnam, E.; Piazza, P.: Spectral sections and higher Atiyah–Patodi–Singer index theory on Galois coverings. Geometric and Functional Analysis 8 (1998), no. 1, 17–58.
• Leichtnam, Eric (2005), "An invitation to Deninger's work on arithmetic zeta functions", Geometry, spectral theory, groups, and dynamics, Contemp. Math., vol. 387, Providence, RI: Amer. Math. Soc., pp. 201–236, MR 2180209.
External links
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Eric Urban
Eric Jean-Paul Urban is a professor of mathematics at Columbia University working in number theory and automorphic forms, particularly Iwasawa theory.
Eric Urban
Urban at the Mathematical Research Institute of Oberwolfach in 2018
Alma materParis-Sud University
AwardsGuggenheim Fellowship (2007)
Scientific career
FieldsMathematics
InstitutionsColumbia University
ThesisArithmétique des formes automorphes pour GL(2) sur un corps imaginaire quadratique (1994)
Doctoral advisorJacques Tilouine
Career
Urban received his PhD in mathematics from Paris-Sud University in 1994 under the supervision of Jacques Tilouine.[1] He is a professor of mathematics at Columbia University.[2]
Research
Together with Christopher Skinner, Urban proved many cases of Iwasawa–Greenberg main conjectures for a large class of modular forms.[3] As a consequence, for a modular elliptic curve over the rational numbers, they prove that the vanishing of the Hasse–Weil L-function L(E, s) of E at s = 1 implies that the p-adic Selmer group of E is infinite. Combined with theorems of Gross-Zagier and Kolyvagin, this gave a conditional proof (on the Tate–Shafarevich conjecture) of the conjecture that E has infinitely many rational points if and only if L(E, 1) = 0, a (weak) form of the Birch–Swinnerton-Dyer conjecture. These results were used (in joint work with Manjul Bhargava and Wei Zhang) to prove that a positive proportion of elliptic curves satisfy the Birch–Swinnerton-Dyer conjecture.[4][5]
Awards
Urban was awarded a Guggenheim Fellowship in 2007.[6]
Selected publications
• Urban, Eric (2011). "Eigenvarieties for reductive groups". Annals of Mathematics. Second Series. 174 (3): 1685–1784. doi:10.4007/annals.2011.174.3.7. ISSN 0003-486X.
• Skinner, Christopher; Urban, Eric (2014). "The Iwasawa Main Conjectures for GL2". Inventiones Mathematicae. 195 (1): 1–277. Bibcode:2014InMat.195....1S. doi:10.1007/s00222-013-0448-1. ISSN 0020-9910. S2CID 120848645.
References
1. Eric Urban at the Mathematics Genealogy Project
2. "Eric Jean-Paul Urban » Department Directory". Columbia University. Retrieved 3 March 2020.
3. Skinner, Christopher; Urban, Eric (2014). "The Iwasawa Main Conjectures for GL2". Inventiones Mathematicae. 195 (1): 1–277. Bibcode:2014InMat.195....1S. doi:10.1007/s00222-013-0448-1. ISSN 0020-9910. S2CID 120848645.
4. Bhargava, Manjul; Skinner, Christopher; Zhang, Wei (2014-07-07). "A majority of elliptic curves over $\mathbb Q$ satisfy the Birch and Swinnerton-Dyer conjecture". arXiv:1407.1826 [math.NT].
5. Baker, Matt (2014-03-10). "The BSD conjecture is true for most elliptic curves". Matt Baker's Math Blog. Retrieved 2019-02-24.
6. "Eric Urban". John Simon Guggenheim Memorial Foundation. Retrieved 9 March 2021.
External links
• Eric Urban at the Mathematics Genealogy Project
Authority control: Academics
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• Mathematics Genealogy Project
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Algebraic space
In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin[1] for use in deformation theory. Intuitively, schemes are given by gluing together affine schemes using the Zariski topology, while algebraic spaces are given by gluing together affine schemes using the finer étale topology. Alternatively one can think of schemes as being locally isomorphic to affine schemes in the Zariski topology, while algebraic spaces are locally isomorphic to affine schemes in the étale topology.
The resulting category of algebraic spaces extends the category of schemes and allows one to carry out several natural constructions that are used in the construction of moduli spaces but are not always possible in the smaller category of schemes, such as taking the quotient of a free action by a finite group (cf. the Keel–Mori theorem).
Definition
There are two common ways to define algebraic spaces: they can be defined as either quotients of schemes by etale equivalence relations, or as sheaves on a big etale site that are locally isomorphic to schemes. These two definitions are essentially equivalent.
Algebraic spaces as quotients of schemes
An algebraic space X comprises a scheme U and a closed subscheme R ⊂ U × U satisfying the following two conditions:
1. R is an equivalence relation as a subset of U × U
2. The projections pi: R → U onto each factor are étale maps.
Some authors, such as Knutson, add an extra condition that an algebraic space has to be quasi-separated, meaning that the diagonal map is quasi-compact.
One can always assume that R and U are affine schemes. Doing so means that the theory of algebraic spaces is not dependent on the full theory of schemes, and can indeed be used as a (more general) replacement of that theory.
If R is the trivial equivalence relation over each connected component of U (i.e. for all x, y belonging to the same connected component of U, we have xRy if and only if x=y), then the algebraic space will be a scheme in the usual sense. Since a general algebraic space X does not satisfy this requirement, it allows a single connected component of U to cover X with many "sheets". The point set underlying the algebraic space X is then given by |U| / |R| as a set of equivalence classes.
Let Y be an algebraic space defined by an equivalence relation S ⊂ V × V. The set Hom(Y, X) of morphisms of algebraic spaces is then defined by the condition that it makes the descent sequence
$\mathrm {Hom} (Y,X)\rightarrow \mathrm {Hom} (V,X){{{} \atop \longrightarrow } \atop {\longrightarrow \atop {}}}\mathrm {Hom} (S,X)$
exact (this definition is motivated by a descent theorem of Grothendieck for surjective étale maps of affine schemes). With these definitions, the algebraic spaces form a category.
Let U be an affine scheme over a field k defined by a system of polynomials g(x), x = (x1, ..., xn), let
$k\{x_{1},\ldots ,x_{n}\}\ $
denote the ring of algebraic functions in x over k, and let X = {R ⊂ U × U} be an algebraic space.
The appropriate stalks ÕX, x on X are then defined to be the local rings of algebraic functions defined by ÕU, u, where u ∈ U is a point lying over x and ÕU, u is the local ring corresponding to u of the ring
k{x1, ..., xn} / (g)
of algebraic functions on U.
A point on an algebraic space is said to be smooth if ÕX, x ≅ k{z1, ..., zd} for some indeterminates z1, ..., zd. The dimension of X at x is then just defined to be d.
A morphism f: Y → X of algebraic spaces is said to be étale at y ∈ Y (where x = f(y)) if the induced map on stalks
ÕX, x → ÕY, y
is an isomorphism.
The structure sheaf OX on the algebraic space X is defined by associating the ring of functions O(V) on V (defined by étale maps from V to the affine line A1 in the sense just defined) to any algebraic space V which is étale over X.
Algebraic spaces as sheaves
An algebraic space ${\mathfrak {X}}$ can be defined as a sheaf of sets
${\mathfrak {X}}:({\text{Sch}}/S)_{\text{et}}^{op}\to {\text{Sets}}$
such that
1. There is a surjective etale morphism $h_{X}\to {\mathfrak {X}}$
2. the diagonal morphism $\Delta _{{\mathfrak {X}}/S}:{\mathfrak {X}}\to {\mathfrak {X}}\times {\mathfrak {X}}$ is representable.
The second condition is equivalent to the property that given any schemes $Y,Z$ and morphisms $h_{Y},h_{Z}\to {\mathfrak {X}}$, their fiber-product of sheaves
$h_{Y}\times _{\mathfrak {X}}h_{Z}$
is representable by a scheme over $S$. Note that some authors, such as Knutson, add an extra condition that an algebraic space has to be quasi-separated, meaning that the diagonal map is quasi-compact.
Algebraic spaces and schemes
Algebraic spaces are similar to schemes, and much of the theory of schemes extends to algebraic spaces. For example, most properties of morphisms of schemes also apply to algebraic spaces, one can define cohomology of quasicoherent sheaves, this has the usual finiteness properties for proper morphisms, and so on.
• Proper algebraic spaces over a field of dimension one (curves) are schemes.
• Non-singular proper algebraic spaces of dimension two over a field (smooth surfaces) are schemes.
• Quasi-separated group objects in the category of algebraic spaces over a field are schemes, though there are non quasi-separated group objects that are not schemes.
• Commutative-group objects in the category of algebraic spaces over an arbitrary scheme which are proper, locally finite presentation, flat, and cohomologically flat in dimension 0 are schemes.
• Not every singular algebraic surface is a scheme.
• Hironaka's example can be used to give a non-singular 3-dimensional proper algebraic space that is not a scheme, given by the quotient of a scheme by a group of order 2 acting freely. This illustrates one difference between schemes and algebraic spaces: the quotient of an algebraic space by a discrete group acting freely is an algebraic space, but the quotient of a scheme by a discrete group acting freely need not be a scheme (even if the group is finite).
• Every quasi-separated algebraic space contains a dense open affine subscheme, and the complement of such a subscheme always has codimension ≥ 1. Thus algebraic spaces are in a sense "close" to affine schemes.
• The quotient of the complex numbers by a lattice is an algebraic space, but is not an elliptic curve, even though the corresponding analytic space is an elliptic curve (or more precisely is the image of an elliptic curve under the functor from complex algebraic spaces to analytic spaces). In fact this algebraic space quotient is not a scheme, is not complete, and is not even quasi-separated. This shows that although the quotient of an algebraic space by an infinite discrete group is an algebraic space, it can have strange properties and might not be the algebraic space one was "expecting". Similar examples are given by the quotient of the complex affine line by the integers, or the quotient of the complex affine line minus the origin by the powers of some number: again the corresponding analytic space is a variety, but the algebraic space is not.
Algebraic spaces and analytic spaces
Algebraic spaces over the complex numbers are closely related to analytic spaces and Moishezon manifolds.
Roughly speaking, the difference between complex algebraic spaces and analytic spaces is that complex algebraic spaces are formed by gluing affine pieces together using the étale topology, while analytic spaces are formed by gluing with the classical topology. In particular there is a functor from complex algebraic spaces of finite type to analytic spaces. Hopf manifolds give examples of analytic surfaces that do not come from a proper algebraic space (though one can construct non-proper and non-separated algebraic spaces whose analytic space is the Hopf surface). It is also possible for different algebraic spaces to correspond to the same analytic space: for example, an elliptic curve and the quotient of C by the corresponding lattice are not isomorphic as algebraic spaces but the corresponding analytic spaces are isomorphic.
Artin showed that proper algebraic spaces over the complex numbers are more or less the same as Moishezon spaces.
Generalization
A far-reaching generalization of algebraic spaces is given by the algebraic stacks. In the category of stacks we can form even more quotients by group actions than in the category of algebraic spaces (the resulting quotient is called a quotient stack).
Citations
1. Artin 1969; Artin 1971.
References
• Artin, Michael (1969), "The implicit function theorem in algebraic geometry", in Abhyankar, Shreeram Shankar (ed.), Algebraic geometry: papers presented at the Bombay Colloquium, 1968, of Tata Institute of Fundamental Research studies in mathematics, vol. 4, Oxford University Press, pp. 13–34, MR 0262237
• Artin, Michael (1971), Algebraic spaces, Yale Mathematical Monographs, vol. 3, Yale University Press, ISBN 978-0-300-01396-2, MR 0407012
• Knutson, Donald (1971), Algebraic spaces, Lecture Notes in Mathematics, vol. 203, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0059750, ISBN 978-3-540-05496-2, MR 0302647
External links
• Danilov, V.I. (2001) [1994], "Algebraic space", Encyclopedia of Mathematics, EMS Press
• Algebraic space in the stacks project
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Étale fundamental group
The étale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces.
Topological analogue/informal discussion
In algebraic topology, the fundamental group $\pi _{1}(X,x)$ of a pointed topological space $(X,x)$ is defined as the group of homotopy classes of loops based at $x$. This definition works well for spaces such as real and complex manifolds, but gives undesirable results for an algebraic variety with the Zariski topology.
In the classification of covering spaces, it is shown that the fundamental group is exactly the group of deck transformations of the universal covering space. This is more promising: finite étale morphisms of algebraic varieties are the appropriate analogue of covering spaces of topological spaces. Unfortunately, an algebraic variety $X$ often fails to have a "universal cover" that is finite over $X$, so one must consider the entire category of finite étale coverings of $X$. One can then define the étale fundamental group as an inverse limit of finite automorphism groups.
Formal definition
Let $X$ be a connected and locally noetherian scheme, let $x$ be a geometric point of $X,$ and let $C$ be the category of pairs $(Y,f)$ such that $f\colon Y\to X$ is a finite étale morphism from a scheme $Y.$ Morphisms $(Y,f)\to (Y',f')$ in this category are morphisms $Y\to Y'$ as schemes over $X.$ This category has a natural functor to the category of sets, namely the functor
$F(Y)=\operatorname {Hom} _{X}(x,Y);$
geometrically this is the fiber of $Y\to X$ over $x,$ and abstractly it is the Yoneda functor represented by $x$ in the category of schemes over $X$. The functor $F$ is typically not representable in $C$; however, it is pro-representable in $C$, in fact by Galois covers of $X$. This means that we have a projective system $\{X_{j}\to X_{i}\mid i<j\in I\}$ in $C$, indexed by a directed set $I,$ where the $X_{i}$ are Galois covers of $X$, i.e., finite étale schemes over $X$ such that $\#\operatorname {Aut} _{X}(X_{i})=\operatorname {deg} (X_{i}/X)$.[1] It also means that we have given an isomorphism of functors
$F(Y)=\varinjlim _{i\in I}\operatorname {Hom} _{C}(X_{i},Y)$.
In particular, we have a marked point $P\in \varprojlim _{i\in I}F(X_{i})$ of the projective system.
For two such $X_{i},X_{j}$ the map $X_{j}\to X_{i}$ induces a group homomorphism $\operatorname {Aut} _{X}(X_{j})\to \operatorname {Aut} _{X}(X_{i})$ which produces a projective system of automorphism groups from the projective system $\{X_{i}\}$. We then make the following definition: the étale fundamental group $\pi _{1}(X,x)$ of $X$ at $x$ is the inverse limit
$\pi _{1}(X,x)=\varprojlim _{i\in I}{\operatorname {Aut} }_{X}(X_{i}),$
with the inverse limit topology.
The functor $F$ is now a functor from $C$ to the category of finite and continuous $\pi _{1}(X,x)$-sets, and establishes an equivalence of categories between $C$ and the category of finite and continuous $\pi _{1}(X,x)$-sets.[2]
Examples and theorems
The most basic example of is $\pi _{1}(\operatorname {Spec} k)$, the étale fundamental group of a field $k$. Essentially by definition, the fundamental group of $k$ can be shown to be isomorphic to the absolute Galois group $\operatorname {Gal} (k^{sep}/k)$. More precisely, the choice of a geometric point of $\operatorname {Spec} (k)$ is equivalent to giving a separably closed extension field $K$, and the étale fundamental group with respect to that base point identifies with the Galois group $\operatorname {Gal} (K/k)$. This interpretation of the Galois group is known as Grothendieck's Galois theory.
More generally, for any geometrically connected variety $X$ over a field $k$ (i.e., $X$ is such that $X^{sep}:=X\times _{k}k^{sep}$ is connected) there is an exact sequence of profinite groups
$1\to \pi _{1}(X^{sep},{\overline {x}})\to \pi _{1}(X,{\overline {x}})\to \operatorname {Gal} (k^{sep}/k)\to 1.$
Schemes over a field of characteristic zero
For a scheme $X$ that is of finite type over $\mathbb {C} $, the complex numbers, there is a close relation between the étale fundamental group of $X$ and the usual, topological, fundamental group of $X(\mathbb {C} )$, the complex analytic space attached to $X$. The algebraic fundamental group, as it is typically called in this case, is the profinite completion of $\pi _{1}(X)$. This is a consequence of the Riemann existence theorem, which says that all finite étale coverings of $X(\mathbb {C} )$ stem from ones of $X$. In particular, as the fundamental group of smooth curves over $\mathbb {C} $ (i.e., open Riemann surfaces) is well understood; this determines the algebraic fundamental group. More generally, the fundamental group of a proper scheme over any algebraically closed field of characteristic zero is known, because an extension of algebraically closed fields induces isomorphic fundamental groups.
Schemes over a field of positive characteristic and the tame fundamental group
For an algebraically closed field $k$ of positive characteristic, the results are different, since Artin–Schreier coverings exist in this situation. For example, the fundamental group of the affine line $\mathbf {A} _{k}^{1}$ is not topologically finitely generated. The tame fundamental group of some scheme U is a quotient of the usual fundamental group of $U$ which takes into account only covers that are tamely ramified along $D$, where $X$ is some compactification and $D$ is the complement of $U$ in $X$.[3][4] For example, the tame fundamental group of the affine line is zero.
Affine schemes over a field of characteristic p
It turns out that every affine scheme $X\subset \mathbf {A} _{k}^{n}$ is a $K(\pi ,1)$-space, in the sense that the etale homotopy type of $X$ is entirely determined by its etale homotopy group.[5] Note $\pi =\pi _{1}^{et}(X,{\overline {x}})$ where ${\overline {x}}$ is a geometric point.
Further topics
From a category-theoretic point of view, the fundamental group is a functor
{Pointed algebraic varieties} → {Profinite groups}.
The inverse Galois problem asks what groups can arise as fundamental groups (or Galois groups of field extensions). Anabelian geometry, for example Grothendieck's section conjecture, seeks to identify classes of varieties which are determined by their fundamental groups.[6]
Friedlander (1982) studies higher étale homotopy groups by means of the étale homotopy type of a scheme.
The pro-étale fundamental group
Bhatt & Scholze (2015, §7) have introduced a variant of the étale fundamental group called the pro-étale fundamental group. It is constructed by considering, instead of finite étale covers, maps which are both étale and satisfy the valuative criterion of properness. For geometrically unibranch schemes (e.g., normal schemes), the two approaches agree, but in general the pro-étale fundamental group is a finer invariant: its profinite completion is the étale fundamental group.
See also
• étale morphism
• Fundamental group
• Fundamental group scheme
Notes
1. J. S. Milne, Lectures on Étale Cohomology, version 2.21: 26-27
2. Grothendieck, Alexandre; Raynaud, Michèle (2003) [1971], Séminaire de Géométrie Algébrique du Bois Marie - 1960-61 - Revêtements étales et groupe fondamental - (SGA 1) (Documents Mathématiques 3), Paris: Société Mathématique de France, pp. xviii+327, see Exp. V, IX, X, arXiv:math.AG/0206203, ISBN 978-2-85629-141-2
3. Grothendieck, Alexander; Murre, Jacob P. (1971), The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme, Lecture Notes in Mathematics, Vol. 208, Berlin, New York: Springer-Verlag
4. Schmidt, Alexander (2002), "Tame coverings of arithmetic schemes", Mathematische Annalen, 322 (1): 1–18, arXiv:math/0005310, doi:10.1007/s002080100262, S2CID 29899627
5. Achinger, Piotr (November 2017). "Wild ramification and K(pi, 1) spaces". Inventiones Mathematicae. 210 (2): 453–499. arXiv:1701.03197. doi:10.1007/s00222-017-0733-5. ISSN 0020-9910. S2CID 119146164.
6. (Tamagawa 1997)
References
• Bhatt, Bhargav; Scholze, Peter (2015), "The pro-étale topology for schemes", Astérisque: 99–201, arXiv:1309.1198, Bibcode:2013arXiv1309.1198B, MR 3379634
• Friedlander, Eric M. (1982), Étale homotopy of simplicial schemes, Annals of Mathematics Studies, vol. 104, Princeton University Press, ISBN 978-0-691-08288-2
• Murre, J. P. (1967), Lectures on an introduction to Grothendieck's theory of the fundamental group, Bombay: Tata Institute of Fundamental Research, MR 0302650
• Tamagawa, Akio (1997), "The Grothendieck conjecture for affine curves", Compositio Mathematica, 109 (2): 135–194, doi:10.1023/A:1000114400142, MR 1478817
• This article incorporates material from étale fundamental group on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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Étale group scheme
In mathematics, more precisely in algebra, an étale group scheme is a certain kind of group scheme.
Definition
A finite group scheme $G$ over a field $K$ is called an étale group scheme if it is represented by an étale K-algebra ${\mathfrak {R}}$, i.e. if ${\mathfrak {R}}\otimes _{K}{\bar {K}}$ is isomorphic to ${\bar {K}}\times ...\times {\bar {K}}$.
References
• John Voight, Introduction to group schemes (PDF), Dartmouth College (lecture notes)
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Étale homotopy type
In mathematics, especially in algebraic geometry, the étale homotopy type is an analogue of the homotopy type of topological spaces for algebraic varieties.
Roughly speaking, for a variety or scheme X, the idea is to consider étale coverings $U\rightarrow X$ and to replace each connected component of U and the higher "intersections", i.e., fiber products, $U_{n}:=U\times _{X}U\times _{X}\dots \times _{X}U$ (n+1 copies of U, $n\geq 0$) by a single point. This gives a simplicial set which captures some information related to X and the étale topology of it.
Slightly more precisely, it is in general necessary to work with étale hypercovers $(U_{n})_{n\geq 0}$ instead of the above simplicial scheme determined by a usual étale cover. Taking finer and finer hypercoverings (which is technically accomplished by working with the pro-object in simplicial sets determined by taking all hypercoverings), the resulting object is the étale homotopy type of X. Similarly to classical topology, it is able to recover much of the usual data related to the étale topology, in particular the étale fundamental group of the scheme and the étale cohomology of locally constant étale sheaves.
References
• Artin, Michael; Mazur, Barry (1969). Etale homotopy. Springer.
• Friedlander, Eric (1982). Étale homotopy of simplicial schemes. Annals of Mathematics Studies, PUP.
External links
• http://ncatlab.org/nlab/show/étale+homotopy
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Étale spectrum
In algebraic geometry, a branch of mathematics, the étale spectrum of a commutative ring or an E∞-ring, denoted by Specét or Spét, is an analog of the prime spectrum Spec of a commutative ring that is obtained by replacing Zariski topology with étale topology. The precise definition depends on one's formalism. But the idea of the definition itself is simple. The usual prime spectrum Spec enjoys the relation: for a scheme (S, OS) and a commutative ring A,
$\operatorname {Hom} (S,\operatorname {Spec} (A))\simeq \operatorname {Hom} (A,\Gamma (S,{\mathcal {O}}_{S}))$
where Hom on the left is for morphisms of schemes and Hom on the right ring homomorphisms. This is to say Spec is the right adjoint to the global section functor $(S,{\mathcal {O}}_{S})\mapsto \Gamma (S,{\mathcal {O}}_{S})$. So, roughly, one can (and typically does) simply define the étale spectrum Spét to be the right adjoint to the global section functor on the category of "spaces" with étale topology.[1][2]
Over a field of characteristic zero, K. Behrend constructs the étale spectrum of a graded algebra called a perfect resolving algebra.[3] He then defines a differential graded scheme (a type of a derived scheme) as one that is étale-locally such an étale spectrum.
The notion makes sense in the usual algebraic geometry but appears more frequently in the context of derived algebraic geometry.
Notes
1. Lurie, Remark 1.2.3.6.
2. Lurie, Remark 1.4.2.7.
3. Behrend, Kai (2002-12-16). "Differential Graded Schemes II: The 2-category of Differential Graded Schemes". arXiv:math/0212226.
References
• Lurie, J. "Spectral Algebraic Geometry (under construction)" (PDF).
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Étale topology
In algebraic geometry, the étale topology is a Grothendieck topology on the category of schemes which has properties similar to the Euclidean topology, but unlike the Euclidean topology, it is also defined in positive characteristic. The étale topology was originally introduced by Alexander Grothendieck to define étale cohomology, and this is still the étale topology's most well-known use.
Definitions
For any scheme X, let Ét(X) be the category of all étale morphisms from a scheme to X. This is the analog of the category of open subsets of X (that is, the category whose objects are varieties and whose morphisms are open immersions). Its objects can be informally thought of as étale open subsets of X. The intersection of two objects corresponds to their fiber product over X. Ét(X) is a large category, meaning that its objects do not form a set.
An étale presheaf on X is a contravariant functor from Ét(X) to the category of sets. A presheaf F is called an étale sheaf if it satisfies the analog of the usual gluing condition for sheaves on topological spaces. That is, F is an étale sheaf if and only if the following condition is true. Suppose that U → X is an object of Ét(X) and that Ui → U is a jointly surjective family of étale morphisms over X. For each i, choose a section xi of F over Ui. The projection map Ui × Uj → Ui, which is loosely speaking the inclusion of the intersection of Ui and Uj in Ui, induces a restriction map F(Ui) → F(Ui × Uj). If for all i and j the restrictions of xi and xj to Ui × Uj are equal, then there must exist a unique section x of F over U which restricts to xi for all i.
Suppose that X is a Noetherian scheme. An abelian étale sheaf F on X is called finite locally constant if it is a representable functor which can be represented by an étale cover of X. It is called constructible if X can be covered by a finite family of subschemes on each of which the restriction of F is finite locally constant. It is called torsion if F(U) is a torsion group for all étale covers U of X. Finite locally constant sheaves are constructible, and constructible sheaves are torsion. Every torsion sheaf is a filtered inductive limit of constructible sheaves.
Grothendieck originally introduced the machinery of Grothendieck topologies and topoi to define the étale topology. In this language, the definition of the étale topology is succinct but abstract: It is the topology generated by the pretopology whose covering families are jointly surjective families of étale morphisms. The small étale site of X is the category O(Xét) whose objects are schemes U with a fixed étale morphism U → X. The morphisms are morphisms of schemes compatible with the fixed maps to X. The big étale site of X is the category Ét/X, that is, the category of schemes with a fixed map to X, considered with the étale topology.
The étale topology can be defined using slightly less data. First, notice that the étale topology is finer than the Zariski topology. Consequently, to define an étale cover of a scheme X, it suffices to first cover X by open affine subschemes, that is, to take a Zariski cover, and then to define an étale cover of an affine scheme. An étale cover of an affine scheme X can be defined as a jointly surjective family {uα : Xα → X} such that the set of all α is finite, each Xα is affine, and each uα is étale. Then an étale cover of X is a family {uα : Xα → X} which becomes an étale cover after base changing to any open affine subscheme of X.
Local rings
See also: Henselian ring
Let X be a scheme with its étale topology, and fix a point x of X. In the Zariski topology, the stalk of X at x is computed by taking a direct limit of the sections of the structure sheaf over all the Zariski open neighborhoods of x. In the étale topology, there are strictly more open neighborhoods of x, so the correct analog of the local ring at x is formed by taking the limit over a strictly larger family. The correct analog of the local ring at x for the étale topology turns out to be the strict henselization of the local ring ${\mathcal {O}}_{X,x}$. It is usually denoted ${\mathcal {O}}_{X,x}^{\text{sh}}$.
Examples
• For each étale morphism $U\to X$, let $\mathbb {G} _{m}(U)={\mathcal {O}}_{U}(U)^{\times }$. Then $U\mapsto \mathbb {G} _{m}(U)$ is a presheaf on X; it is a sheaf since it can be represented by the scheme $\operatorname {Spec} _{X}({\mathcal {O}}_{X}[t,t^{-1}])$.
See also
• Étale cohomology
• Nisnevich topology
• Smooth topology
• ℓ-adic sheaf
References
• Grothendieck, Alexandre; Dieudonné, Jean (1964). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Première partie". Publications Mathématiques de l'IHÉS. 20. doi:10.1007/bf02684747. MR 0173675.
• Grothendieck, Alexandre; Dieudonné, Jean (1967). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie". Publications Mathématiques de l'IHÉS. 32. doi:10.1007/bf02732123. MR 0238860.
• Artin, Michael (1972). Alexandre Grothendieck; Jean-Louis Verdier (eds.). Séminaire de Géométrie Algébrique du Bois Marie – 1963–64 – Théorie des topos et cohomologie étale des schémas – (SGA 4) – vol. 2. Lecture notes in mathematics (in French). Vol. 270. Berlin; New York: Springer-Verlag. pp. iv+418. doi:10.1007/BFb0061319. ISBN 978-3-540-06012-3.
• Artin, Michael (1972). Alexandre Grothendieck; Jean-Louis Verdier (eds.). Séminaire de Géométrie Algébrique du Bois Marie – 1963–64 – Théorie des topos et cohomologie étale des schémas – (SGA 4) – vol. 3. Lecture notes in mathematics (in French). Vol. 305. Berlin; New York: Springer-Verlag. pp. vi+640. doi:10.1007/BFb0070714. ISBN 978-3-540-06118-2.
• Deligne, Pierre (1977). Séminaire de Géométrie Algébrique du Bois Marie – Cohomologie étale – (SGA 4½). Lecture notes in mathematics (in French). Vol. 569. Berlin; New York: Springer-Verlag. pp. iv+312. doi:10.1007/BFb0091516. ISBN 978-3-540-08066-4.
• J. S. Milne (1980), Étale cohomology, Princeton, N.J: Princeton University Press, ISBN 0-691-08238-3
• J. S. Milne (2008). Lectures on Étale Cohomology
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Étale topos
In mathematics, the étale topos of a scheme X is the category of all étale sheaves on X. An étale sheaf is a sheaf on the étale site of X.
Definition
Let X be a scheme. An étale covering of X is a family $\{\varphi _{i}:U_{i}\to X\}_{i\in I}$, where each $\varphi _{i}$ is an étale morphism of schemes, such that the family is jointly surjective that is $X=\bigcup _{i\in I}\varphi _{i}(U_{i})$.
The category Ét(X) is the category of all étale schemes over X. The collection of all étale coverings of a étale scheme U over X i.e. an object in Ét(X) defines a Grothendieck pretopology on Ét(X) which in turn induces a Grothendieck topology, the étale topology on X. The category together with the étale topology on it is called the étale site on X.
The étale topos $X^{\text{ét}}$ of a scheme X is then the category of all sheaves of sets on the site Ét(X). Such sheaves are called étale sheaves on X. In other words, an étale sheaf ${\mathcal {F}}$ is a (contravariant) functor from the category Ét(X) to the category of sets satisfying the following sheaf axiom:
For each étale U over X and each étale covering $\{\varphi _{i}:U_{i}\to U\}$ of U the sequence
$0\to {\mathcal {F}}(U)\to \prod _{i\in I}{\mathcal {F}}(U_{i}){{{} \atop \longrightarrow } \atop {\longrightarrow \atop {}}}\prod _{i,j\in I}{\mathcal {F}}(U_{ij})$
is exact, where $U_{ij}=U_{i}\times _{U}U_{j}$.
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Étienne Fouvry
Étienne Fouvry is a French mathematician working primarily in analytic number theory.
Étienne Fouvry
Fouvry in 1986
Born1953
Nationality France
Alma materUniversity of Bordeaux
Scientific career
FieldsMathematics
InstitutionsUniversity of Paris-Sud
ThesisRepartitions des suites dans les progressions arithmetiques (1981)
Doctoral advisorsJean-Marc Deshouillers, Henryk Iwaniec
Websitewww.math.u-psud.fr/~fouvry/
In 1985, Fouvry showed that the first case of Fermat's Last Theorem is true for infinitely many primes.[1]
References
1. Fouvry, Étienne (1985). "Théorème de Brun-Titchmarsh: application au théorème de Fermat" [The Brun-Titchmarsh theorem: application to the Fermat theorem]. Invent. Math. (in French). 79 (2): 383–407. Bibcode:1985InMat..79..383F. doi:10.1007/BF01388980. MR 0778134. S2CID 122719070.
External links
• Videos of Étienne Fouvry in the AV-Portal of the German National Library of Science and Technology
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Étienne Pascal
Étienne Pascal (French: [etjɛn paskal]; 2 May 1588 – 24 September 1651) was a French chief tax officer and the father of Blaise Pascal (1623–1662).
Étienne Pascal
Born(1588-05-02)2 May 1588
Clermont-Ferrand, Puy-de-Dôme, France
Died24 September 1651(1651-09-24) (aged 63)
Paris
EducationParis (law degree in 1610)
Known forPascal's limaçon
SpouseAntoinette Begon
ChildrenGilberte Périer, Blaise Pascal, Jacqueline Pascal
Scientific career
FieldsTax officer, amateur mathematician
Biography
Pascal was born in Clermont to Martin Pascal, the treasurer of France, and Marguerite Pascal de Mons.[1] He had three daughters, two of whom survived past childhood: Gilberte (1620–1687) and Jacqueline (1625–1661). His wife Antoinette Begon died in 1626.
He was a tax official, lawyer, and a wealthy member of the petite noblesse, who also had an interest in science and mathematics. He was trained in the law at Paris and received his law degree in 1610. That year, he returned to Clermont and purchased the post of counsellor for Bas-Auvergne, the area surrounding Clermont.
In 1631, five years after his wife's death,[1] Pascal moved with his children to Paris. They hired Louise Delfault, a maid who eventually became an instrumental member of the family. Pascal, who never remarried, decided to home-educate his children, who showed extraordinary intellectual ability, particularly his son Blaise.
Pascal served on a scientific committee (whose members included Pierre Hérigone and Claude Mydorge) to determine whether Jean-Baptiste Morin's scheme for determining longitude from the Moon's motion was practical.
The limaçon was first studied and named by Pascal, and so this mathematical curve is often called Pascal's limaçon.
Pascal died in Paris.
Notes
1. O'Connor, J.J.; Robertson, E.F. (August 2006). "Étienne Pascal". University of St. Andrews, Scotland. Retrieved 5 February 2010.
External links
• O'Connor, John J.; Robertson, Edmund F., "Étienne Pascal", MacTutor History of Mathematics Archive, University of St Andrews
Blaise Pascal
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• Pascal's triangle
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Works
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Évariste Galois
Évariste Galois (/ɡælˈwɑː/;[1] French: [evaʁist ɡalwa]; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a problem that had been open for 350 years. His work laid the foundations for Galois theory and group theory,[2] two major branches of abstract algebra. He was a staunch republican and was heavily involved in the political turmoil that surrounded the French Revolution of 1830. As a result of his political activism, he was arrested repeatedly, serving one jail sentence of several months. For reasons that remain obscure, shortly after his release from prison he fought in a duel and died of the wounds he suffered.[3]
Évariste Galois
A portrait of Évariste Galois aged about 15
Born
Évariste Galois
(1811-10-25)25 October 1811
Bourg-la-Reine, French Empire
Died31 May 1832(1832-05-31) (aged 20)
Paris, Kingdom of France
Cause of deathGunshot wound to the abdomen
Alma materÉcole préparatoire
(no degree)
Known forWork on theory of equations, group theory and Galois theory
Scientific career
FieldsMathematics
InfluencesLouis Paul Émile Richard
Adrien-Marie Legendre
Joseph-Louis Lagrange
Signature
Life
Early life
Galois was born on 25 October 1811 to Nicolas-Gabriel Galois and Adélaïde-Marie (née Demante).[2][4] His father was a Republican and was head of Bourg-la-Reine's liberal party. His father became mayor of the village[2] after Louis XVIII returned to the throne in 1814. His mother, the daughter of a jurist, was a fluent reader of Latin and classical literature and was responsible for her son's education for his first twelve years.
In October 1823, he entered the Lycée Louis-le-Grand where his teacher Louis Paul Émile Richard recognized his brilliance.[5] At the age of 14, he began to take a serious interest in mathematics.[5]
He found a copy of Adrien-Marie Legendre's Éléments de Géométrie, which, it is said, he read "like a novel" and mastered at the first reading. At 15, he was reading the original papers of Joseph-Louis Lagrange, such as the Réflexions sur la résolution algébrique des équations which likely motivated his later work on equation theory,[6] and Leçons sur le calcul des fonctions, work intended for professional mathematicians, yet his classwork remained uninspired and his teachers accused him of putting on the airs of a genius.[4]
Budding mathematician
In 1828, he attempted the entrance examination for the École Polytechnique, the most prestigious institution for mathematics in France at the time, without the usual preparation in mathematics, and failed for lack of explanations on the oral examination. In that same year, he entered the École Normale (then known as l'École préparatoire), a far inferior institution for mathematical studies at that time, where he found some professors sympathetic to him.
In the following year Galois's first paper, on continued fractions,[7] was published. It was at around the same time that he began making fundamental discoveries in the theory of polynomial equations. He submitted two papers on this topic to the Academy of Sciences. Augustin-Louis Cauchy refereed these papers, but refused to accept them for publication for reasons that still remain unclear. However, in spite of many claims to the contrary, it is widely held that Cauchy recognized the importance of Galois's work, and that he merely suggested combining the two papers into one in order to enter it in the competition for the Academy's Grand Prize in Mathematics. Cauchy, an eminent mathematician of the time though with political views that were diametrically opposed to those of Galois, considered Galois's work to be a likely winner.[8]
On 28 July 1829, Galois's father died by suicide after a bitter political dispute with the village priest.[9] A couple of days later, Galois made his second and last attempt to enter the Polytechnique and failed yet again.[9] It is undisputed that Galois was more than qualified; accounts differ on why he failed. More plausible accounts state that Galois made too many logical leaps and baffled the incompetent examiner, which enraged Galois. The recent death of his father may have also influenced his behavior.[4]
Having been denied admission to the École polytechnique, Galois took the Baccalaureate examinations in order to enter the École normale.[9] He passed, receiving his degree on 29 December 1829.[9] His examiner in mathematics reported, "This pupil is sometimes obscure in expressing his ideas, but he is intelligent and shows a remarkable spirit of research."
He submitted his memoir on equation theory several times, but it was never published in his lifetime. Though his first attempt was refused by Cauchy, in February 1830 following Cauchy's suggestion he submitted it to the Academy's secretary Joseph Fourier,[9] to be considered for the Grand Prix of the Academy. Unfortunately, Fourier died soon after,[9] and the memoir was lost.[9] The prize would be awarded that year to Niels Henrik Abel posthumously and also to Carl Gustav Jacob Jacobi. Despite the lost memoir, Galois published three papers that year. One laid the foundations for Galois theory.[10] The second was about the numerical resolution of equations (root finding in modern terminology).[11] The third was an important one in number theory, in which the concept of a finite field was first articulated.[12]
Political firebrand
Galois lived during a time of political turmoil in France. Charles X had succeeded Louis XVIII in 1824, but in 1827 his party suffered a major electoral setback and by 1830 the opposition liberal party became the majority. Charles, faced with political opposition from the chambers, staged a coup d'état, and issued his notorious July Ordinances, touching off the July Revolution[9] which ended with Louis Philippe becoming king. While their counterparts at the Polytechnique were making history in the streets, Galois, at the École Normale, was locked in by the school's director. Galois was incensed and wrote a blistering letter criticizing the director, which he submitted to the Gazette des Écoles, signing the letter with his full name. Although the Gazette's editor omitted the signature for publication, Galois was expelled.[13]
Although his expulsion would have formally taken effect on 4 January 1831, Galois quit school immediately and joined the staunchly Republican artillery unit of the National Guard. He divided his time between his mathematical work and his political affiliations. Due to controversy surrounding the unit, soon after Galois became a member, on 31 December 1830, the artillery of the National Guard was disbanded out of fear that they might destabilize the government. At around the same time, nineteen officers of Galois's former unit were arrested and charged with conspiracy to overthrow the government.
In April 1831, the officers were acquitted of all charges, and on 9 May 1831, a banquet was held in their honor, with many illustrious people present, such as Alexandre Dumas. The proceedings grew riotous. At some point, Galois stood and proposed a toast in which he said, "To Louis Philippe," with a dagger above his cup. The republicans at the banquet interpreted Galois's toast as a threat against the king's life and cheered. He was arrested the following day at his mother's house and held in detention at Sainte-Pélagie prison until 15 June 1831, when he had his trial.[8] Galois's defense lawyer cleverly claimed that Galois actually said, "To Louis-Philippe, if he betrays," but that the qualifier was drowned out in the cheers. The prosecutor asked a few more questions, and perhaps influenced by Galois's youth, the jury acquitted him that same day.[8][9][13][14]
On the following Bastille Day (14 July 1831), Galois was at the head of a protest, wearing the uniform of the disbanded artillery, and came heavily armed with several pistols, a loaded rifle, and a dagger. He was again arrested.[9] During his stay in prison, Galois at one point drank alcohol for the first time at the goading of his fellow inmates. One of these inmates, François-Vincent Raspail, recorded what Galois said while drunk in a letter from 25 July. Excerpted from the letter:[8]
And I tell you, I will die in a duel on the occasion of some coquette de bas étage. Why? Because she will invite me to avenge her honor which another has compromised.
Do you know what I lack, my friend? I can confide it only to you: it is someone whom I can love and love only in spirit. I've lost my father and no one has ever replaced him, do you hear me...?
The first line is a haunting prophecy of how Galois would in fact die; the second shows how Galois was profoundly affected by the loss of his father. Raspail continues that Galois, still in a delirium, attempted suicide, and that he would have succeeded if his fellow inmates hadn't forcibly stopped him.[8] Months later, when Galois's trial occurred on 23 October, he was sentenced to six months in prison for illegally wearing a uniform.[9][15][16] While in prison, he continued to develop his mathematical ideas. He was released on 29 April 1832.
Final days
Galois returned to mathematics after his expulsion from the École Normale, although he continued to spend time in political activities. After his expulsion became official in January 1831, he attempted to start a private class in advanced algebra which attracted some interest, but this waned, as it seemed that his political activism had priority.[4][8] Siméon Denis Poisson asked him to submit his work on the theory of equations, which he did on 17 January 1831. Around 4 July 1831, Poisson declared Galois's work "incomprehensible", declaring that "[Galois's] argument is neither sufficiently clear nor sufficiently developed to allow us to judge its rigor"; however, the rejection report ends on an encouraging note: "We would then suggest that the author should publish the whole of his work in order to form a definitive opinion."[17] While Poisson's report was made before Galois's 14 July arrest, it took until October to reach Galois in prison. It is unsurprising, in the light of his character and situation at the time, that Galois reacted violently to the rejection letter, and decided to abandon publishing his papers through the Academy and instead publish them privately through his friend Auguste Chevalier. Apparently, however, Galois did not ignore Poisson's advice, as he began collecting all his mathematical manuscripts while still in prison, and continued polishing his ideas until his release on 29 April 1832,[13] after which he was somehow talked into a duel.[9]
Galois's fatal duel took place on 30 May.[18] The true motives behind the duel are obscure. There has been much speculation about them. What is known is that, five days before his death, he wrote a letter to Chevalier which clearly alludes to a broken love affair.[8]
Some archival investigation on the original letters suggests that the woman of romantic interest was Stéphanie-Félicie Poterin du Motel,[19] the daughter of the physician at the hostel where Galois stayed during the last months of his life. Fragments of letters from her, copied by Galois himself (with many portions, such as her name, either obliterated or deliberately omitted), are available.[20] The letters hint that du Motel had confided some of her troubles to Galois, and this might have prompted him to provoke the duel himself on her behalf. This conjecture is also supported by other letters Galois later wrote to his friends the night before he died. Galois's cousin, Gabriel Demante, when asked if he knew the cause of the duel, mentioned that Galois "found himself in the presence of a supposed uncle and a supposed fiancé, each of whom provoked the duel." Galois himself exclaimed: "I am the victim of an infamous coquette and her two dupes."[13]
Much more detailed speculation based on these scant historical details has been interpolated by many of Galois's biographers, such as the frequently repeated speculation that the entire incident was stage-managed by the police and royalist factions to eliminate a political enemy.
As to his opponent in the duel, Alexandre Dumas names Pescheux d'Herbinville,[14] who was actually one of the nineteen artillery officers whose acquittal was celebrated at the banquet that occasioned Galois's first arrest.[21] However, Dumas is alone in this assertion, and if he were correct it is unclear why d'Herbinville would have been involved. It has been speculated that he was du Motel's "supposed fiancé" at the time (she ultimately married someone else), but no clear evidence has been found supporting this conjecture. On the other hand, extant newspaper clippings from only a few days after the duel give a description of his opponent (identified by the initials "L.D.") that appear to more accurately apply to one of Galois's Republican friends, most probably Ernest Duchatelet, who was imprisoned with Galois on the same charges.[22] Given the conflicting information available, the true identity of his killer may well be lost to history.
Whatever the reasons behind the duel, Galois was so convinced of his impending death that he stayed up all night writing letters to his Republican friends and composing what would become his mathematical testament, the famous letter to Auguste Chevalier outlining his ideas, and three attached manuscripts.[23] Mathematician Hermann Weyl said of this testament, "This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind." However, the legend of Galois pouring his mathematical thoughts onto paper the night before he died seems to have been exaggerated.[8] In these final papers, he outlined the rough edges of some work he had been doing in analysis and annotated a copy of the manuscript submitted to the Academy and other papers.
Early in the morning of 30 May 1832, he was shot in the abdomen,[18] was abandoned by his opponents and his own seconds, and was found by a passing farmer. He died the following morning[18] at ten o'clock in the Hôpital Cochin (probably of peritonitis), after refusing the offices of a priest. His funeral ended in riots.[18] There were plans to initiate an uprising during his funeral, but during the same time the leaders heard of General Jean Maximilien Lamarque's death and the rising was postponed without any uprising occurring until 5 June. Only Galois's younger brother was notified of the events prior to Galois's death.[24] Galois was 20 years old. His last words to his younger brother Alfred were:
"Ne pleure pas, Alfred ! J'ai besoin de tout mon courage pour mourir à vingt ans !"
(Don't weep, Alfred! I need all my courage to die at twenty!)
On 2 June, Évariste Galois was buried in a common grave of the Montparnasse Cemetery whose exact location is unknown.[18][16] In the cemetery of his native town – Bourg-la-Reine – a cenotaph in his honour was erected beside the graves of his relatives.[25]
Évariste Galois died in 1832. Joseph Liouville began studying Galois' unpublished papers in 1842 and acknowledged their value in 1843. It is not clear what happened in the 10 years between 1832 and 1842 nor what eventually inspired Joseph Liouville to begin reading Galois' papers. Jesper Lützen explores this subject at some length in Chapter XIV Galois Theory of his book about Joseph Liouville without reaching any definitive conclusions.[26]
It is certainly possible that mathematicians (including Liouville) did not want to publicize Galois' papers because Galois was a republican political activist who died 5 days before the June Rebellion, an unsuccessful anti-monarchist insurrection of Parisian republicans. In Galois' obituary, his friend Auguste Chevalier almost accused academicians at the École Polytechnique of having killed Galois since, if they had not rejected his work, he would have become a mathematician and would not have devoted himself to the republican political activism for which some believed he was killed.[26]
Given that France was still living in the shadow of the Reign of Terror and the Napoleonic era, Liouville might have waited until the June Rebellion's political turmoil subsided before turning his attention to Galois' papers.[26]
Liouville finally published Galois' manuscripts in the October–November 1846 issue of the Journal de Mathématiques Pures et Appliquées.[27][28] Galois' most famous contribution was a novel proof that there is no quintic formula – that is, that fifth and higher degree equations are not generally solvable by radicals. Although Niels Henrik Abel had already proved the impossibility of a "quintic formula" by radicals in 1824 and Paolo Ruffini had published a solution in 1799 that turned out to be flawed, Galois's methods led to deeper research into what is now called Galois Theory, which can be used to determine, for any polynomial equation, whether it has a solution by radicals.
Contributions to mathematics
From the closing lines of a letter from Galois to his friend Auguste Chevalier, dated 29 May 1832, two days before Galois's death:[23]
Tu prieras publiquement Jacobi ou Gauss de donner leur avis, non sur la vérité, mais sur l'importance des théorèmes.
Après cela, il y aura, j'espère, des gens qui trouveront leur profit à déchiffrer tout ce gâchis.
(Ask Jacobi or Gauss publicly to give their opinion, not as to the truth, but as to the importance of these theorems. Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.)
Within the 60 or so pages of Galois's collected works are many important ideas that have had far-reaching consequences for nearly all branches of mathematics.[29][30] His work has been compared to that of Niels Henrik Abel (1802 – 1829), a contemporary mathematician who died at a very young age, and much of their work had significant overlap.
Algebra
While many mathematicians before Galois gave consideration to what are now known as groups, it was Galois who was the first to use the word group (in French groupe) in a sense close to the technical sense that is understood today, making him among the founders of the branch of algebra known as group theory. He called the decomposition of a group into its left and right cosets a proper decomposition if the left and right cosets coincide, which is what today is known as a normal subgroup.[23] He also introduced the concept of a finite field (also known as a Galois field in his honor) in essentially the same form as it is understood today.[12]
In his last letter to Chevalier[23] and attached manuscripts, the second of three, he made basic studies of linear groups over finite fields:
• He constructed the general linear group over a prime field, GL(ν, p) and computed its order, in studying the Galois group of the general equation of degree pν.[31]
• He constructed the projective special linear group PSL(2,p). Galois constructed them as fractional linear transforms, and observed that they were simple except if p was 2 or 3.[32] These were the second family of finite simple groups, after the alternating groups.[33]
• He noted the exceptional fact that PSL(2,p) is simple and acts on p points if and only if p is 5, 7, or 11.[34][35]
Galois theory
Main article: Galois theory
Galois's most significant contribution to mathematics is his development of Galois theory. He realized that the algebraic solution to a polynomial equation is related to the structure of a group of permutations associated with the roots of the polynomial, the Galois group of the polynomial. He found that an equation could be solved in radicals if one can find a series of subgroups of its Galois group, each one normal in its successor with abelian quotient, that is, its Galois group is solvable. This proved to be a fertile approach, which later mathematicians adapted to many other fields of mathematics besides the theory of equations to which Galois originally applied it.[29]
Analysis
Galois also made some contributions to the theory of Abelian integrals and continued fractions.
As written in his last letter,[23] Galois passed from the study of elliptic functions to consideration of the integrals of the most general algebraic differentials, today called Abelian integrals. He classified these integrals into three categories.
Continued fractions
In his first paper in 1828,[7] Galois proved that the regular continued fraction which represents a quadratic surd ζ is purely periodic if and only if ζ is a reduced surd, that is, $\zeta >1$ and its conjugate $\eta $ satisfies $-1<\eta <0$.
In fact, Galois showed more than this. He also proved that if ζ is a reduced quadratic surd and η is its conjugate, then the continued fractions for ζ and for (−1/η) are both purely periodic, and the repeating block in one of those continued fractions is the mirror image of the repeating block in the other. In symbols we have
${\begin{aligned}\zeta &=[\,{\overline {a_{0};a_{1},a_{2},\dots ,a_{m-1}}}\,]\\[3pt]{\frac {-1}{\eta }}&=[\,{\overline {a_{m-1};a_{m-2},a_{m-3},\dots ,a_{0}}}\,]\,\end{aligned}}$
where ζ is any reduced quadratic surd, and η is its conjugate.
From these two theorems of Galois a result already known to Lagrange can be deduced. If r > 1 is a rational number that is not a perfect square, then
${\sqrt {r}}=\left[\,a_{0};{\overline {a_{1},a_{2},\dots ,a_{2},a_{1},2a_{0}}}\,\right].$
In particular, if n is any non-square positive integer, the regular continued fraction expansion of √n contains a repeating block of length m, in which the first m − 1 partial denominators form a palindromic string.
See also
• List of things named after Évariste Galois
Notes
1. "Galois theory". Random House Webster's Unabridged Dictionary.
2. C., Bruno, Leonard (c. 2003) [1999]. Math and mathematicians : the history of math discoveries around the world. Baker, Lawrence W. Detroit, Mich.: U X L. p. 171. ISBN 978-0787638139. OCLC 41497065.{{cite book}}: CS1 maint: multiple names: authors list (link)
3. C., Bruno, Leonard (2003) [1999]. Math and mathematicians : the history of math discoveries around the world. Baker, Lawrence W. Detroit, Mich.: U X L. pp. 171, 174. ISBN 978-0787638139. OCLC 41497065.{{cite book}}: CS1 maint: multiple names: authors list (link)
4. Stewart, Ian (1973). Galois Theory. London: Chapman and Hall. pp. xvii–xxii. ISBN 978-0-412-10800-6.
5. C., Bruno, Leonard (2003) [1999]. Math and mathematicians : the history of math discoveries around the world. Baker, Lawrence W. Detroit, Mich.: U X L. p. 172. ISBN 978-0787638139. OCLC 41497065.{{cite book}}: CS1 maint: multiple names: authors list (link)
6. "Réflexions sur la résolution algébrique des équations". britannica encyclopedia.
7. Galois, Évariste (1828). "Démonstration d'un théorème sur les fractions continues périodiques". Annales de Mathématiques. XIX: 294.
8. Rothman, Tony (1982). "Genius and Biographers: The Fictionalization of Evariste Galois". The American Mathematical Monthly. 89 (2): 84–106. doi:10.2307/2320923. JSTOR 2320923. Retrieved 31 January 2015.
9. C., Bruno, Leonard (2003) [1999]. Math and mathematicians : the history of math discoveries around the world. Baker, Lawrence W. Detroit, Mich.: U X L. p. 173. ISBN 978-0787638139. OCLC 41497065.{{cite book}}: CS1 maint: multiple names: authors list (link)
10. Galois, Évariste (1830). "Analyse d'un Mémoire sur la résolution algébrique des équations". Bulletin des Sciences Mathématiques. XIII: 271.
11. Galois, Évariste (1830). "Note sur la résolution des équations numériques". Bulletin des Sciences Mathématiques. XIII: 413.
12. Galois, Évariste (1830). "Sur la théorie des nombres". Bulletin des Sciences Mathématiques. XIII: 428.
13. Dupuy, Paul (1896). "La vie d'Évariste Galois". Annales Scientifiques de l'École Normale Supérieure. 13: 197–266. doi:10.24033/asens.427.
14. Dumas (père), Alexandre. "CCIV". Mes Mémoires. ISBN 978-1-4371-5595-2. Retrieved 13 April 2010.
15. Bell, Eric Temple (1986). Men of Mathematics. New York: Simon and Schuster. ISBN 978-0-671-62818-5.
16. Escofier, Jean-Pierre (2001). Galois Theory. Springer. pp. 222–224. ISBN 978-0-387-98765-1.
17. Taton, R. (1947). "Les relations d'Évariste Galois avec les mathématiciens de son temps". Revue d'Histoire des Sciences et de Leurs Applications. 1 (2): 114–130. doi:10.3406/rhs.1947.2607.
18. C., Bruno, Leonard (2003) [1999]. Math and mathematicians : the history of math discoveries around the world. Baker, Lawrence W. Detroit, Mich.: U X L. p. 174. ISBN 978-0787638139. OCLC 41497065.{{cite book}}: CS1 maint: multiple names: authors list (link)
19. Infantozzi, Carlos Alberti (1968). "Sur la mort d'Évariste Galois". Revue d'Histoire des Sciences et de Leurs Applications. 21 (2): 157. doi:10.3406/rhs.1968.2554.
20. Bourgne, R.; J.-P. Azra (1962). Écrits et mémoires mathématiques d'Évariste Galois. Paris: Gauthier-Villars.
21. Blanc, Louis (1844). The History of Ten Years, 1830–1840, Volume 1. London: Chapman and Hall. p. 431.
22. Dalmas, Andre (1956). Évariste Galois: Révolutionnaire et Géomètre. Paris: Fasquelle.
23. Galois, Évariste (1846). "Lettre de Galois à M. Auguste Chevalier". Journal de Mathématiques Pures et Appliquées. XI: 408–415. Retrieved 4 February 2009.
24. Coutinho, S.C. (1999). The Mathematics of Ciphers. Natick: A K Peters, Ltd. pp. 127–128. ISBN 978-1-56881-082-9.
25. Toti Rigatelli, Laura (1996). Evariste Galois, 1811–1832 (Vita mathematica, 11). Birkhäuser. p. 114. ISBN 978-3-7643-5410-7.
26. Lützen, Jesper (1990). "Chapter XIV: Galois Theory". Joseph Liouville 1809–1882: Master of Pure and Applied Mathematics. Studies in the History of Mathematics and Physical Sciences. Vol. 15. Springer-Verlag. pp. 559–580. ISBN 3-540-97180-7.
27. Galois, Évariste (1846). "OEuvres mathématiques d'Évariste Galois". Journal de Mathématiques Pures et Appliquées. XI: 381–444. Retrieved 4 February 2009.
28. Pierpont, James (1899). "Review: Oeuvres mathématiques d'Evariste Galois; publiées sous les auspices de la Société Mathématique de France, avec une introduction par M. EMILE PICARD. Paris, Gauthier-Villars et Fils, 1897. 8vo, x + 63 pp" (PDF). Bull. Amer. Math. Soc. 5 (6): 296–300. doi:10.1090/S0002-9904-1899-00599-8. In 1897 the French Mathematical Society reprinted the 1846 publication.
29. Lie, Sophus (1895). "Influence de Galois sur le Développement des Mathématiques". Le centenaire de l'École Normale 1795–1895. Hachette.
30. See also: Sophus Lie, "Influence de Galois sur le développement des mathématiques" in: Évariste Galois, Oeuvres Mathématiques publiées en 1846 dans le Journal de Liouville (Sceaux, France: Éditions Jacques Gabay, 1989), appendix pages 1–9.
31. Letter, p. 410
32. Letter, p. 411
33. Wilson, Robert A. (2009). "Chapter 1: Introduction". The finite simple groups. Graduate Texts in Mathematics 251. Vol. 251. Berlin, New York: Springer-Verlag. doi:10.1007/978-1-84800-988-2. ISBN 978-1-84800-987-5. Zbl 1203.20012, 2007 preprint {{cite book}}: External link in |postscript= (help)CS1 maint: postscript (link)
34. Letter, pp. 411–412
35. "Galois's last letter, translated" (PDF).
References
• Artin, Emil (1998), Galois Theory, Dover Publications, Inc., ISBN 978-0-486-62342-9 – Reprinting of second revised edition of 1944, The University of Notre Dame Press.
• Astruc, Alexandre (1994), Évariste Galois, Grandes Biographies (in French), Flammarion, ISBN 978-2-08-066675-8
• Bell, E.T. (1937), "Galois", Men of Mathematics, vol. 2. Still in print.
• Désérable, François-Henri (2015), Évariste (in French), Gallimard, ISBN 9782070147045
• Edwards, Harold M. (May 1984), Galois Theory, Graduate Texts in Mathematics 101, Springer-Verlag, ISBN 978-0-387-90980-6 – This textbook explains Galois Theory with historical development and includes an English translation of Galois's memoir.
• Ehrhardt, Caroline (2011), Évariste Galois, la fabrication d'une icône mathématique, En temps et lieux (in French), Editions de l'Ecole Pratiques de Hautes Etudes en Sciences Sociales, ISBN 978-2-7132-2317-4
• Infeld, Leopold (1948), Whom the Gods Love: The Story of Evariste Galois, Classics in Mathematics Education Series, Reston, Va: National Council of Teachers of Mathematics, ISBN 978-0-87353-125-2 – Classic fictionalized biography by physicist Infeld.
• Livio, Mario (2006), "The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry", Physics Today, Souvenir Press, 59 (7): 50, Bibcode:2006PhT....59g..50L, doi:10.1063/1.2337831, ISBN 978-0-285-63743-6
• Toti Rigatelli, Laura (1996), Évariste Galois, Birkhauser, ISBN 978-3-7643-5410-7 – This biography challenges the common myth concerning Galois's duel and death.
• Stewart, Ian (1973), Galois Theory, Chapman and Hall, ISBN 978-0-412-10800-6 – This comprehensive text on Galois Theory includes a brief biography of Galois himself.
• Tignol, Jean-Pierre (2001), Galois' theory of algebraic equations, Singapore: World Scientific, ISBN 978-981-02-4541-2 – Historical development of Galois theory.
• Neumann, Peter (2011). The mathematical writings of Evariste Galois (PDF). Zürich, Switzerland: European Mathematical Society. ISBN 978-3-03719-104-0.
External links
Wikimedia Commons has media related to Évariste Galois.
Wikiquote has quotations related to Évariste Galois.
Wikisource has the text of the 1911 Encyclopædia Britannica article "Galois, Évariste".
• Works by Évariste Galois at Project Gutenberg
• Works by or about Évariste Galois at Internet Archive
• O'Connor, John J.; Robertson, Edmund F., "Évariste Galois", MacTutor History of Mathematics Archive, University of St Andrews
• The Galois Archive (biography, letters and texts in various languages)
• Two Galois articles, online and analyzed on BibNum : "Mémoire sur les conditions de résolubilité des équations par radicaux" (1830) (link)[for English analysis, click 'A télécharger']; "Démonstration d'un théorème sur les fractions continues périodiques" (1829) (link) [for English analysis, click 'A télécharger']
• Rothman, Tony (1982). "Genius and Biographers: The Fictionalization of Evariste Galois" (PDF). The American Mathematical Monthly. 89 (2): 84–106. doi:10.2307/2320923. JSTOR 2320923.
• La vie d'Évariste Galois by Paul Dupuy The first and still one of the most extensive biographies, referred to by every other serious biographer of Galois
• Œuvres Mathématiques published in 1846 in the Journal de Liouville, converted to Djvu format by Prof. Antoine Chambert-Loir at the University of Rennes.
• Alexandre Dumas, Mes Mémoires, the relevant chapter of Alexandre Dumas' memoires where he mentions Galois and the banquet.
• Évariste Galois at the Mathematics Genealogy Project
• Theatrical trailer of University College Utrecht's "Évariste – En Garde"
• A piece of music dedicated to Evariste Galois on YouTube
Évariste Galois
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Ólafur Daníelsson
Ólafur Dan Daníelsson (31 October 1877 – 10 December 1957) was an Icelandic mathematician.[1] He was the first Icelandic mathematician to complete a doctoral degree.[1] He was also the founder of the Icelandic Mathematical Society.[2]
Life
Early life and education
Danielsson was born in Viðvík in Viðvíkursveit in Skagafjördur.[2] In 1897, he finished his secondary education in Reykjavík, and in the same year, went to study mathematics in the University of Copenhagen.[3] Hieronymus Georg Zeuthen and Julius Petersen were his university tutors.[3] In 1900, his first scientific paper was published in the Danish journal Nyt Tidsskrift for Matematik B.[3] In 1901, he was awarded a gold medal for his mathematical treatise at the University of Copenhagen.[3] In 1904, he was awarded a master's degree, which enabled him to teach in Danish high schools.[3]
Returning to Iceland, he applied to be a mathematics teacher at the Reykjavik High School, where he had studied a few years previously.[2] However, he did not get the job.[2] The successful applicant was an engineer, Sigurður Thoroddsen.[2]
He started undertaking PhD research.[2] His thesis built upon the earlier works of Zeuthen and other scientists, such as Rudolph Clebsch, Guido Castelnuevo and Luigi Cremona.[3] In 1909, he submitted his thesis and graduated from the University of Copenhagen.[2] He was the first Icelandic mathematician to be awarded a doctorate.[2]
Career
He became a private tutor and began writing textbooks.[2] In 1906, his first textbook, Reikningsbók/Arithmetic, was published.[3] In 1908, he became the first mathematics teacher in the Iceland Teaher College when it was first established.[3] The students were experienced teachers, but had been lacking formal education themselves.[3] In 1914, his textbook Arithmetic was republished for the students' needs.[3]
In 1919, a mathematics stream at Reykjavík High School was founded in response to Danielsson's and his friends' initiative.[3] He was tasked with its development, with the goal of enabling students to attend the Polytechnic College in Copenhagen and to pursue university studies in sciences.[3] Prior to that, students needed to spend a preparatory year abroad.[3]
At the same time, Danielsson started writing high school mathematics textbooks.[3] In 1920s, his 4 textbooks were republished, including a rewritten version of the Arithmetic book.[3] Additionally, three new subjects were introduced in Icelandic: Um flatarmyndir/On plane geometry, Kenslubók í hornafræði/Trigonometry, and Kenslubók í algebru/A textbook in algebra.[3] These three textbooks were groundbreaking, being the first of their kind in Icelandic.[3] They were adopted for use at Reykjavík High School, along with the advanced Danish textbooks.[3] Later, when Akureyri High School was established in 1930, these textbooks were also incorporated into its curriculum.[3] The mathematician Sigurdur Helgason commented that, "The geometry textbooks by the remarkable mathematician Ólafur Daníelsson, the pioneering founder of mathematics education in Iceland, were written by a man with a real mission".[4]
In 1941, Daníelsson concluded his teaching career and retired.[3] His remarkable influence extended over almost seven decades, starting in 1906 when he published his initial textbook and continuing in 1908 when he commenced teaching at Iceland's Teacher College.[3] His significant impact on mathematics education persisted until 1976 when his textbooks were excluded from the reading list of the national entrance examination.[3] There is no doubt about his enduring legacy as a devoted mathematician, as his visionary approach helped shape mathematics education in Iceland.[3]
Research
In the 1920s, Daníelsson dedicated himself to advancing the field of algebraic geometry through his research. He actively participated in the Scandinavian Congress of Mathematicians held in 1925 and 1927.[5] His contributions were instrumental in fostering the development of mathematics in Iceland, which ultimately led to Iceland becoming a full member of the Nordic Congress of Mathematicians in the 1980s.[5]
He published several papers in the Danish Matematisk Tidsskrift, with notable contributions in the years 1926, 1940, 1945, and 1948.[3] His research work also appeared in esteemed journals such as Mathematische Annalen, specifically in volumes 102 (1930), 109 (1934), 113 (1937), and 114 (1937).[3]
In 1925, Daníelsson participated in the Sixth Scandinavian Congress of Mathematicians held in Copenhagen.[3] Two years later, in 1927, he also attended the seventh congress held in Oslo.[3] He delivered presentations at both congresses, accompanied by the publication of his papers. His first paper, titled "En Lösning af Malfattis problem" [A solution of Malfatti's Problem], was published in Matematisk Tidsskrift. Subsequently, he contributed to Matematische Annalen with a paper entitled "Überkorrespondierende Punkte der Steinerschen Fläche vierter Ordnung und die Hauptpunkte derselben" (Corresponding Points of Steiner's Surface of Fourth Order and their Principal Points). This journal featured the works of renowned mathematicians such as Einstein, van der Waerden, von Neumann, Landau, Ore, and Kolmogorov, among others, and Daníelsson's paper was among the 44 articles published.[3] It is worth noting that Danielsson was the only mathematician from Iceland contributing to Scandinavian Mathematicial journals before the second world war.[6]
Daníelsson's fascination with elementary geometry was evident, as he remarked that "it is difficult to find tasks simpler and more elegant than skillful mathematical problems." His final paper was published in both the Journal of the Icelandic Society of Engineers in 1946 and Matematisk Tidsskrift in 1948.[3]
The Icelandic Mathematical Society
On 31 October 1947, the Icelandic Mathematical Society was founded in Reykjavik when Daníelsson was 70. The society records:
“On Friday, 31 October 1947, which was the seventieth birthday of Ólafur Daníelsson, he gathered in his home several men and set up a Society. The purpose of the Society is to promote co-operation and promotion of people in Iceland who have completed a university degree in a mathematical subject. The Society holds meetings at which individual members explain their mathematical topics and, if desired, discussions on the topic will be conducted.”[7]
The first lecture was delivered by Ólafur Daníelsson himself.[2] He spoke "about the circle transcribed by the outer circumference of the triangle" and calculated its length relative to the radius of the inscribed circle and the circumference of the triangle. This result has been published in the Matematisk Tidsskrift.[2] However, this had been a longstanding interest of him, as the initial foundations of this subject could be traced back to an article he wrote in 1900, published in the same journal.[2] In this regard, the topic itself carried a sense of antiquity, yet it had recently witnessed a fresh comprehension shortly before his presentation.[2]
References
1. "Ólafur Daníelsson – Biography". Maths History. Retrieved 2023-07-09.
2. "Icelandic Mathematical Society". Maths History. Retrieved 2023-07-09.
3. Bjarnadóttir, Kristín (2013). "Mathematics Education in Twentieth Century Iceland–Ólafur Daníelsson's Impact". Dig Where You Stand. 3: 65–80.
4. Helgason, Sigurdur (2009). The Selected Works of Sigurdur Helgason. American Mathematical Society. pp. xiii.
5. Turner, Laura E (2023). "A Richer Gathering: On the History of the Nordic Congress of Mathematicians". European Mathematical Society Magazine (127): 39–44.
6. Siegmund-Schultze, Reinhard (1850–1950). "The Interplay of Various Scandinavian Mathematical Journals (1859–1953) and the Road towards Internationalization". Historia Mathematica. 45 (4): 354–75 – via Elsevier Science Direct.{{cite journal}}: CS1 maint: date format (link)
7. "Um félagið | stæ.is". www.stae.is. Retrieved 2023-07-09.
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Āryabhaṭa's sine table
Āryabhata's sine table is a set of twenty-four numbers given in the astronomical treatise Āryabhatiya composed by the fifth century Indian mathematician and astronomer Āryabhata (476–550 CE), for the computation of the half-chords of a certain set of arcs of a circle. The set of numbers appears in verse 12 in Chapter 1 Dasagitika of Aryabhatiya.[1] It is not a table in the modern sense of a mathematical table; that is, it is not a set of numbers arranged into rows and columns.[2] [3][4] Āryabhaṭa's table is also not a set of values of the trigonometric sine function in a conventional sense; it is a table of the first differences of the values of trigonometric sines expressed in arcminutes, and because of this the table is also referred to as Āryabhaṭa's table of sine-differences.[5][6]
Āryabhaṭa's table was the first sine table ever constructed in the history of mathematics.[7] The now lost tables of Hipparchus (c. 190 BC – c. 120 BC) and Menelaus (c. 70–140 CE) and those of Ptolemy (c. AD 90 – c. 168) were all tables of chords and not of half-chords.[7] Āryabhaṭa's table remained as the standard sine table of ancient India. There were continuous attempts to improve the accuracy of this table. These endeavors culminated in the eventual discovery of the power series expansions of the sine and cosine functions by Madhava of Sangamagrama (c. 1350 – c. 1425), the founder of the Kerala school of astronomy and mathematics, and the tabulation of a sine table by Madhava with values accurate to seven or eight decimal places.
Some historians of mathematics have argued that the sine table given in Āryabhaṭiya was an adaptation of earlier such tables constructed by mathematicians and astronomers of ancient Greece.[8] David Pingree, one of America's foremost historians of the exact sciences in antiquity, was an exponent of such a view. Assuming this hypothesis, G. J. Toomer[9][10][11] writes, "Hardly any documentation exists for the earliest arrival of Greek astronomical models in India, or for that matter what those models would have looked like. So it is very difficult to ascertain the extent to which what has come down to us represents transmitted knowledge, and what is original with Indian scientists. ... The truth is probably a tangled mixture of both."[12]
The table
In modern notations
The values encoded in Āryabhaṭa's Sanskrit verse can be decoded using the numerical scheme explained in Āryabhaṭīya, and the decoded numbers are listed in the table below. In the table, the angle measures relevant to Āryabhaṭa's sine table are listed in the second column. The third column contains the list the numbers contained in the Sanskrit verse given above in Devanagari script. For the convenience of users unable to read Devanagari, these word-numerals are reproduced in the fourth column in ISO 15919 transliteration. The next column contains these numbers in the Hindu-Arabic numerals. Āryabhaṭa's numbers are the first differences in the values of sines. The corresponding value of sine (or more precisely, of jya) can be obtained by summing up the differences up to that difference. Thus the value of jya corresponding to 18° 45′ is the sum 225 + 224 + 222 + 219 + 215 = 1105. For assessing the accuracy of Āryabhaṭa's computations, the modern values of jyas are given in the last column of the table.
In the Indian mathematical tradition, the sine ( or jya) of an angle is not a ratio of numbers. It is the length of a certain line segment, a certain half-chord. The radius of the base circle is basic parameter for the construction of such tables. Historically, several tables have been constructed using different values for this parameter. Āryabhaṭa has chosen the number 3438 as the value of radius of the base circle for the computation of his sine table. The rationale of the choice of this parameter is the idea of measuring the circumference of a circle in angle measures. In astronomical computations distances are measured in degrees, minutes, seconds, etc. In this measure, the circumference of a circle is 360° = (60 × 360) minutes = 21600 minutes. The radius of the circle, the measure of whose circumference is 21600 minutes, is 21600 / 2π minutes. Computing this using the value π = 3.1416 known to Aryabhata one gets the radius of the circle as 3438 minutes approximately. Āryabhaṭa's sine table is based on this value for the radius of the base circle. It has not yet been established who is the first ever to use this value for the base radius. But Aryabhatiya is the earliest surviving text containing a reference to this basic constant.[13]
Sl. No Angle ( A )
(in degrees,
arcminutes)
Value in Āryabhaṭa's
numerical notation
(in Devanagari)
Value in Āryabhaṭa's
numerical notation
(in ISO 15919 transliteration)
Value in
Hindu-Arabic numerals
Āryabhaṭa's
value of
jya (A)
Modern value
of jya (A)
(3438 × sin (A))
1
03° 45′
मखि
makhi
225
225′
224.8560
2
07° 30′
भखि
bhakhi
224
449′
448.7490
3
11° 15′
फखि
phakhi
222
671′
670.7205
4
15° 00′
धखि
dhakhi
219
890′
889.8199
5
18° 45′
णखि
ṇakhi
215
1105′
1105.1089
6
22° 30′
ञखि
ñakhi
210
1315′
1315.6656
7
26° 15′
ङखि
ṅakhi
205
1520′
1520.5885
8
30° 00′
हस्झ
hasjha
199
1719′
1719.0000
9
33° 45′
स्ककि
skaki
191
1910′
1910.0505
10
37° 30′
किष्ग
kiṣga
183
2093′
2092.9218
11
41° 15′
श्घकि
śghaki
174
2267′
2266.8309
12
45° 00′
किघ्व
kighva
164
2431′
2431.0331
13
48° 45′
घ्लकि
ghlaki
154
2585′
2584.8253
14
52° 30′
किग्र
kigra
143
2728′
2727.5488
15
56° 15′
हक्य
hakya
131
2859′
2858.5925
16
60° 00′
धकि
dhaki
119
2978′
2977.3953
17
63° 45′
किच
kica
106
3084′
3083.4485
18
67° 30′
स्ग
sga
93
3177′
3176.2978
19
71° 15′
झश
jhaśa
79
3256′
3255.5458
20
75° 00′
ङ्व
ṅva
65
3321′
3320.8530
21
78° 45′
क्ल
kla
51
3372′
3371.9398
22
82° 30′
प्त
pta
37
3409′
3408.5874
23
86° 15′
फ
pha
22
3431′
3430.6390
24
90° 00′
छ
cha
7
3438′
3438.0000
Āryabhaṭa's computational method
The second section of Āryabhaṭiya, titled Ganitapādd, a contains a stanza indicating a method for the computation of the sine table. There are several ambiguities in correctly interpreting the meaning of this verse. For example, the following is a translation of the verse given by Katz wherein the words in square brackets are insertions of the translator and not translations of texts in the verse.[13]
• "When the second half-[chord] partitioned is less than the first half-chord, which is [approximately equal to] the [corresponding] arc, by a certain amount, the remaining [sine-differences] are less [than the previous ones] each by that amount of that divided by the first half-chord."
This may be referring to the fact that the second derivative of the sine function is equal to the negative of the sine function.
See also
• Madhava's sine table
• Bhaskara I's sine approximation formula
References
1. Kripa Shankar Shukla and K V Sarma (1976). Aryabhatiya of Aryabhata (Critically ediited with Introduction, English Translation, Notes, Comments and Index). Dlehi: Indian national Science Academy. p. 29. Retrieved 25 January 2023.
2. Helaine Selin (Ed.) (2008). Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (2 ed.). Springer. pp. 986–988. ISBN 978-1-4020-4425-0.
3. Selin, Helaine, ed. (2008). Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (2 ed.). Springer. pp. 986–988. ISBN 978-1-4020-4425-0.
4. Eugene Clark (1930). Theastronomy. Chicago: The University of Chicago Press.
5. Takao Hayashi, T (November 1997). "Āryabhaṭa's rule and table for sine-differences". Historia Mathematica. 24 (4): 396–406. doi:10.1006/hmat.1997.2160.
6. B. L. van der Waerden, B. L. (March 1988). "Reconstruction of a Greek table of chords". Archive for History of Exact Sciences. 38 (1): 23–38. Bibcode:1988AHES...38...23V. doi:10.1007/BF00329978. S2CID 189793547.
7. J J O'Connor and E F Robertson (June 1996). "The trigonometric functions". Retrieved 4 March 2010.
8. "Hipparchus and Trigonometry". Retrieved 6 March 2010.
9. G. J. Toomer, G. J. (July 2007). "The Chord Table of Hipparchus and the Early History of Greek Trigonometry". Centaurus. 18 (1): 6–28. doi:10.1111/j.1600-0498.1974.tb00205.x.
10. B.N. Narahari Achar (2002). "Āryabhata and the table of Rsines" (PDF). Indian Journal of History of Science. 37 (2): 95–99. Retrieved 6 March 2010.
11. Glen Van Brummelen (March 2000). "[HM] Radian Measure". Historia Mathematica mailing List Archive. Retrieved 6 March 2010.
12. Glen Van Brummelen (25 January 2009). The mathematics of the heavens and the earth: the early 0. ISBN 9780691129730.
13. Katz, Victor J., ed. (2007). The mathematics of Egypt, Mesopotamia, China, India, and Islam: a sourcebook. Princeton: Princeton University Press. pp. 405–408. ISBN 978-0-691-11485-9.
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Čech-to-derived functor spectral sequence
In algebraic topology, a branch of mathematics, the Čech-to-derived functor spectral sequence is a spectral sequence that relates Čech cohomology of a sheaf and sheaf cohomology.[1]
Definition
Let ${\mathcal {F}}$ be a sheaf on a topological space X. Choose an open cover ${\mathfrak {U}}$ of X. That is, ${\mathfrak {U}}$ is a set of open subsets of X which together cover X. Let ${\mathcal {H}}^{q}(X,{\mathcal {F}})$ denote the presheaf which takes an open set U to the qth cohomology of ${\mathcal {F}}$ on U, that is, to $H^{q}(U,{\mathcal {F}})$. For any presheaf ${\mathcal {G}}$, let ${\check {H}}^{p}({\mathfrak {U}},{\mathcal {G}})$ denote the pth Čech cohomology of ${\mathcal {G}}$ with respect to the cover ${\mathfrak {U}}$. Then the Čech-to-derived functor spectral sequence is:[2]
$E_{2}^{p,q}={\check {H}}^{p}({\mathfrak {U}},{\mathcal {H}}^{q}(X,{\mathcal {F}}))\Rightarrow H^{p+q}(X,{\mathcal {F}}).$
Properties
If ${\mathfrak {U}}$ consists of only two open sets, then this spectral sequence degenerates to the Mayer–Vietoris sequence. See Spectral sequence#Long exact sequences.
If for all finite intersections of a covering the cohomology vanishes, the E2-term degenerates and the edge morphisms yield an isomorphism of Čech cohomology for this covering to sheaf cohomology. This provides a method of computing sheaf cohomology using Čech cohomology. For instance, this happens if ${\mathcal {F}}$ is a quasi-coherent sheaf on a scheme and each element of ${\mathfrak {U}}$ is an open affine subscheme such that all finite intersections are again affine (e.g. if the scheme is separated). This can be used to compute the cohomology of line bundles on projective space.[3]
See also
• Leray's theorem
Notes
1. Dimca 2004, 2.3.9.
2. Godement 1973, Théorème 5.4.1.
3. Hartshorne 1977, Theorem III.5.1.
References
• Dimca, Alexandru (2004), Sheaves in topology, Universitext, Berlin: Springer-Verlag, ISBN 978-3-540-20665-1, MR 2050072
• Godement, Roger (1973), Topologie algébrique et théorie des faisceaux, Paris: Hermann, MR 0345092
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
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Čech cohomology
In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech.
Motivation
Let X be a topological space, and let ${\mathcal {U}}$ be an open cover of X. Let $N({\mathcal {U}})$ denote the nerve of the covering. The idea of Čech cohomology is that, for an open cover ${\mathcal {U}}$ consisting of sufficiently small open sets, the resulting simplicial complex $N({\mathcal {U}})$ should be a good combinatorial model for the space X. For such a cover, the Čech cohomology of X is defined to be the simplicial cohomology of the nerve. This idea can be formalized by the notion of a good cover. However, a more general approach is to take the direct limit of the cohomology groups of the nerve over the system of all possible open covers of X, ordered by refinement. This is the approach adopted below.
Construction
Let X be a topological space, and let ${\mathcal {F}}$ be a presheaf of abelian groups on X. Let ${\mathcal {U}}$ be an open cover of X.
Simplex
A q-simplex σ of ${\mathcal {U}}$ is an ordered collection of q+1 sets chosen from ${\mathcal {U}}$, such that the intersection of all these sets is non-empty. This intersection is called the support of σ and is denoted |σ|.
Now let $\sigma =(U_{i})_{i\in \{0,\ldots ,q\}}$ be such a q-simplex. The j-th partial boundary of σ is defined to be the (q−1)-simplex obtained by removing the j-th set from σ, that is:
$\partial _{j}\sigma :=(U_{i})_{i\in \{0,\ldots ,q\}\setminus \{j\}}.$ :=(U_{i})_{i\in \{0,\ldots ,q\}\setminus \{j\}}.}
The boundary of σ is defined as the alternating sum of the partial boundaries:
$\partial \sigma :=\sum _{j=0}^{q}(-1)^{j+1}\partial _{j}\sigma $ :=\sum _{j=0}^{q}(-1)^{j+1}\partial _{j}\sigma }
viewed as an element of the free abelian group spanned by the simplices of ${\mathcal {U}}$.
Cochain
A q-cochain of ${\mathcal {U}}$ with coefficients in ${\mathcal {F}}$ is a map which associates with each q-simplex σ an element of ${\mathcal {F}}(|\sigma |)$, and we denote the set of all q-cochains of ${\mathcal {U}}$ with coefficients in ${\mathcal {F}}$ by $C^{q}({\mathcal {U}},{\mathcal {F}})$. $C^{q}({\mathcal {U}},{\mathcal {F}})$ is an abelian group by pointwise addition.
Differential
The cochain groups can be made into a cochain complex $(C^{\bullet }({\mathcal {U}},{\mathcal {F}}),\delta )$ by defining the coboundary operator $\delta _{q}:C^{q}({\mathcal {U}},{\mathcal {F}})\to C^{q+1}({\mathcal {U}},{\mathcal {F}})$ by:
$\quad (\delta _{q}f)(\sigma ):=\sum _{j=0}^{q+1}(-1)^{j}\mathrm {res} _{|\sigma |}^{|\partial _{j}\sigma |}f(\partial _{j}\sigma ),$
where $\mathrm {res} _{|\sigma |}^{|\partial _{j}\sigma |}$ is the restriction morphism from ${\mathcal {F}}(|\partial _{j}\sigma |)$ to ${\mathcal {F}}(|\sigma |).$ (Notice that ∂jσ ⊆ σ, but |σ| ⊆ |∂jσ|.)
A calculation shows that $\delta _{q+1}\circ \delta _{q}=0.$
The coboundary operator is analogous to the exterior derivative of De Rham cohomology, so it sometimes called the differential of the cochain complex.
Cocycle
A q-cochain is called a q-cocycle if it is in the kernel of $\delta $, hence $Z^{q}({\mathcal {U}},{\mathcal {F}}):=\ker(\delta _{q})\subseteq C^{q}({\mathcal {U}},{\mathcal {F}})$ is the set of all q-cocycles.
Thus a (q−1)-cochain $f$ is a cocycle if for all q-simplices $\sigma $ the cocycle condition
$\sum _{j=0}^{q}(-1)^{j}\mathrm {res} _{|\sigma |}^{|\partial _{j}\sigma |}f(\partial _{j}\sigma )=0$
holds.
A 0-cocycle $f$ is a collection of local sections of ${\mathcal {F}}$ satisfying a compatibility relation on every intersecting $A,B\in {\mathcal {U}}$
$f(A)|_{A\cap B}=f(B)|_{A\cap B}$
A 1-cocycle $f$ satisfies for every non-empty 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/":): {\displaystyle U = A\cap B \cap C} with $A,B,C\in {\mathcal {U}}$
$f(B\cap C)|_{U}-f(A\cap C)|_{U}+f(A\cap B)|_{U}=0$
Coboundary
A q-cochain is called a q-coboundary if it is in the image of $\delta $ and $B^{q}({\mathcal {U}},{\mathcal {F}}):=\mathrm {Im} (\delta _{q-1})\subseteq C^{q}({\mathcal {U}},{\mathcal {F}})$ is the set of all q-coboundaries.
For example, a 1-cochain $f$ is a 1-coboundary if there exists a 0-cochain $h$ such that for every intersecting $A,B\in {\mathcal {U}}$
$f(A\cap B)=h(A)|_{A\cap B}-h(B)|_{A\cap B}$
Cohomology
The Čech cohomology of ${\mathcal {U}}$ with values in ${\mathcal {F}}$ is defined to be the cohomology of the cochain complex $(C^{\bullet }({\mathcal {U}},{\mathcal {F}}),\delta )$. Thus the qth Čech cohomology is given by
${\check {H}}^{q}({\mathcal {U}},{\mathcal {F}}):=H^{q}((C^{\bullet }({\mathcal {U}},{\mathcal {F}}),\delta ))=Z^{q}({\mathcal {U}},{\mathcal {F}})/B^{q}({\mathcal {U}},{\mathcal {F}})$.
The Čech cohomology of X is defined by considering refinements of open covers. If ${\mathcal {V}}$ is a refinement of ${\mathcal {U}}$ then there is a map in cohomology ${\check {H}}^{*}({\mathcal {U}},{\mathcal {F}})\to {\check {H}}^{*}({\mathcal {V}},{\mathcal {F}}).$ The open covers of X form a directed set under refinement, so the above map leads to a direct system of abelian groups. The Čech cohomology of X with values in ${\mathcal {F}}$ is defined as the direct limit ${\check {H}}(X,{\mathcal {F}}):=\varinjlim _{\mathcal {U}}{\check {H}}({\mathcal {U}},{\mathcal {F}})$ of this system.
The Čech cohomology of X with coefficients in a fixed abelian group A, denoted ${\check {H}}(X;A)$, is defined as ${\check {H}}(X,{\mathcal {F}}_{A})$ where ${\mathcal {F}}_{A}$ is the constant sheaf on X determined by A.
A variant of Čech cohomology, called numerable Čech cohomology, is defined as above, except that all open covers considered are required to be numerable: that is, there is a partition of unity {ρi} such that each support $\{x\mid \rho _{i}(x)>0\}$ is contained in some element of the cover. If X is paracompact and Hausdorff, then numerable Čech cohomology agrees with the usual Čech cohomology.
Relation to other cohomology theories
If X is homotopy equivalent to a CW complex, then the Čech cohomology ${\check {H}}^{*}(X;A)$ is naturally isomorphic to the singular cohomology $H^{*}(X;A)\,$. If X is a differentiable manifold, then ${\check {H}}^{*}(X;\mathbb {R} )$ is also naturally isomorphic to the de Rham cohomology; the article on de Rham cohomology provides a brief review of this isomorphism. For less well-behaved spaces, Čech cohomology differs from singular cohomology. For example if X is the closed topologist's sine curve, then ${\check {H}}^{1}(X;\mathbb {Z} )=\mathbb {Z} ,$ whereas $H^{1}(X;\mathbb {Z} )=0.$
If X is a differentiable manifold and the cover ${\mathcal {U}}$ of X is a "good cover" (i.e. all the sets Uα are contractible to a point, and all finite intersections of sets in ${\mathcal {U}}$ are either empty or contractible to a point), then ${\check {H}}^{*}({\mathcal {U}};\mathbb {R} )$ is isomorphic to the de Rham cohomology.
If X is compact Hausdorff, then Čech cohomology (with coefficients in a discrete group) is isomorphic to Alexander-Spanier cohomology.
For a presheaf ${\mathcal {F}}$ on X, let ${\mathcal {F}}^{+}$ denote its sheafification. Then we have a natural comparison map
$\chi :{\check {H}}^{*}(X,{\mathcal {F}})\to H^{*}(X,{\mathcal {F}}^{+})$ :{\check {H}}^{*}(X,{\mathcal {F}})\to H^{*}(X,{\mathcal {F}}^{+})}
from Čech cohomology to sheaf cohomology. If X is paracompact Hausdorff, then $ \chi $ is an isomorphism. More generally, $ \chi $ is an isomorphism whenever the Čech cohomology of all presheaves on X with zero sheafification vanishes.[2]
In algebraic geometry
Čech cohomology can be defined more generally for objects in a site C endowed with a topology. This applies, for example, to the Zariski site or the etale site of a scheme X. The Čech cohomology with values in some sheaf ${\mathcal {F}}$ is defined as
${\check {H}}^{n}(X,{\mathcal {F}}):=\varinjlim _{\mathcal {U}}{\check {H}}^{n}({\mathcal {U}},{\mathcal {F}}).$
where the colimit runs over all coverings (with respect to the chosen topology) of X. Here ${\check {H}}^{n}({\mathcal {U}},{\mathcal {F}})$ is defined as above, except that the r-fold intersections of open subsets inside the ambient topological space are replaced by the r-fold fiber product
${\mathcal {U}}^{\times _{X}^{r}}:={\mathcal {U}}\times _{X}\dots \times _{X}{\mathcal {U}}.$
As in the classical situation of topological spaces, there is always a map
${\check {H}}^{n}(X,{\mathcal {F}})\rightarrow H^{n}(X,{\mathcal {F}})$
from Čech cohomology to sheaf cohomology. It is always an isomorphism in degrees n = 0 and 1, but may fail to be so in general. For the Zariski topology on a Noetherian separated scheme, Čech and sheaf cohomology agree for any quasi-coherent sheaf. For the étale topology, the two cohomologies agree for any étale sheaf on X, provided that any finite set of points of X are contained in some open affine subscheme. This is satisfied, for example, if X is quasi-projective over an affine scheme.[3]
The possible difference between Čech cohomology and sheaf cohomology is a motivation for the use of hypercoverings: these are more general objects than the Čech nerve
$N_{X}{\mathcal {U}}:\dots \to {\mathcal {U}}\times _{X}{\mathcal {U}}\times _{X}{\mathcal {U}}\to {\mathcal {U}}\times _{X}{\mathcal {U}}\to {\mathcal {U}}.$
A hypercovering K∗ of X is a certain simplicial object in C, i.e., a collection of objects Kn together with boundary and degeneracy maps. Applying a sheaf ${\mathcal {F}}$ to K∗ yields a simplicial abelian group $ {\mathcal {F}}(K_{\ast })$ whose n-th cohomology group is denoted $ H^{n}({\mathcal {F}}(K_{\ast }))$. (This group is the same as ${\check {H}}^{n}({\mathcal {U}},{\mathcal {F}})$ in case K∗ equals $N_{X}{\mathcal {U}}$.) Then, it can be shown that there is a canonical isomorphism
$H^{n}(X,{\mathcal {F}})\cong \varinjlim _{K_{*}}H^{n}({\mathcal {F}}(K_{*})),$
where the colimit now runs over all hypercoverings.[4]
Examples
For example, we can compute the coherent sheaf cohomology of $\Omega ^{1}$ on the projective line $\mathbb {P} _{\mathbb {C} }^{1}$ using the Čech complex. Using the cover
${\mathcal {U}}=\{U_{1}={\text{Spec}}(\mathbb {C} [y]),U_{2}={\text{Spec}}(\mathbb {C} [y^{-1}])\}$
we have the following modules from the cotangent sheaf
${\begin{aligned}&\Omega ^{1}(U_{1})=\mathbb {C} [y]dy\\&\Omega ^{1}(U_{2})=\mathbb {C} \left[y^{-1}\right]dy^{-1}\end{aligned}}$
If we take the conventions that $dy^{-1}=-(1/y^{2})dy$ then we get the Čech complex
$0\to \mathbb {C} [y]dy\oplus \mathbb {C} \left[y^{-1}\right]dy^{-1}{\xrightarrow {d^{0}}}\mathbb {C} \left[y,y^{-1}\right]dy\to 0$
Since $d^{0}$ is injective and the only element not in the image of $d^{0}$ is $y^{-1}dy$ we get that
${\begin{aligned}&H^{1}(\mathbb {P} _{\mathbb {C} }^{1},\Omega ^{1})\cong \mathbb {C} \\&H^{k}(\mathbb {P} _{\mathbb {C} }^{1},\Omega ^{1})\cong 0{\text{ for }}k\neq 1\end{aligned}}$
References
Citation footnotes
1. Penrose, Roger (1992), "On the Cohomology of Impossible Figures", Leonardo, 25 (3/4): 245–247, doi:10.2307/1575844, JSTOR 1575844, S2CID 125905129. Reprinted from Penrose, Roger (1991), "On the Cohomology of Impossible Figures / La Cohomologie des Figures Impossibles", Structural Topology, 17: 11–16, retrieved January 16, 2014
2. Brady, Zarathustra. "Notes on sheaf cohomology" (PDF). p. 11. Archived (PDF) from the original on 2022-06-17.
3. Milne, James S. (1980), "Section III.2, Theorem 2.17", Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, ISBN 978-0-691-08238-7, MR 0559531
4. Artin, Michael; Mazur, Barry (1969), "Lemma 8.6", Etale homotopy, Lecture Notes in Mathematics, vol. 100, Springer, p. 98, ISBN 978-3-540-36142-8
General references
• Bott, Raoul; Loring Tu (1982). Differential Forms in Algebraic Topology. Springer. ISBN 0-387-90613-4.
• Hatcher, Allen (2002). Algebraic Topology (PDF). Cambridge University Press. ISBN 0-521-79540-0.
• Wells, Raymond (1980). "2. Sheaf Theory: Appendix A. Cech Cohomology with Coefficients in a Sheaf". Differential Analysis on Complex Manifolds. Springer. pp. 63–64. doi:10.1007/978-1-4757-3946-6_2. ISBN 978-3-540-90419-9.
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Ultraproduct
The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factors need to have the same signature. The ultrapower is the special case of this construction in which all factors are equal.
For example, ultrapowers can be used to construct new fields from given ones. The hyperreal numbers, an ultrapower of the real numbers, are a special case of this.
Some striking applications of ultraproducts include very elegant proofs of the compactness theorem and the completeness theorem, Keisler's ultrapower theorem, which gives an algebraic characterization of the semantic notion of elementary equivalence, and the Robinson–Zakon presentation of the use of superstructures and their monomorphisms to construct nonstandard models of analysis, leading to the growth of the area of nonstandard analysis, which was pioneered (as an application of the compactness theorem) by Abraham Robinson.
Definition
The general method for getting ultraproducts uses an index set $I,$ a structure $M_{i}$ (assumed to be non-empty in this article) for each element $i\in I$ (all of the same signature), and an ultrafilter ${\mathcal {U}}$ on $I.$
For any two elements $a_{\bullet }=\left(a_{i}\right)_{i\in I}$ and $b_{\bullet }=\left(b_{i}\right)_{i\in I}$ of the Cartesian product $ \prod \limits _{i\in I}}M_{i},$ declare them to be ${\mathcal {U}}$-equivalent, written $a_{\bullet }\sim b_{\bullet }$ or $a_{\bullet }=_{\mathcal {U}}b_{\bullet },$ if and only if the set of indices $\left\{i\in I:a_{i}=b_{i}\right\}$ on which they agree is an element of ${\mathcal {U}};$ in symbols,
$a_{\bullet }\sim b_{\bullet }\;\iff \;\left\{i\in I:a_{i}=b_{i}\right\}\in {\mathcal {U}},$
which compares components only relative to the ultrafilter ${\mathcal {U}}.$ This binary relation $\,\sim \,$ is an equivalence relation[proof 1] on the Cartesian product $ \prod \limits _{i\in I}}M_{i}.$
The ultraproduct of $M_{\bullet }=\left(M_{i}\right)_{i\in I}$ modulo ${\mathcal {U}}$ is the quotient set of $ \prod \limits _{i\in I}}M_{i}$ with respect to $\sim $ and is therefore sometimes denoted by $ \prod \limits _{i\in I}}M_{i}\,/\,{\mathcal {U}}$ or $ \prod }_{\mathcal {U}}\,M_{\bullet }.$
Explicitly, if the ${\mathcal {U}}$-equivalence class of an element $a\in \prod \limits _{i\in I}}M_{i}$ is denoted by
$a_{\mathcal {U}}:={\big \{}x\in \prod \limits _{i\in I}}M_{i}\;:\;x\sim a{\big \}}$
then the ultraproduct is the set of all ${\mathcal {U}}$-equivalence classes
${\prod }_{\mathcal {U}}\,M_{\bullet }\;=\;\prod _{i\in I}M_{i}\,/\,{\mathcal {U}}\;:=\;\left\{a_{\mathcal {U}}\;:\;a\in \prod \limits _{i\in I}}M_{i}\right\}.$
Although ${\mathcal {U}}$ was assumed to be an ultrafilter, the construction above can be carried out more generally whenever ${\mathcal {U}}$ is merely a filter on $I,$ in which case the resulting quotient set $ \prod \limits _{i\in I}}M_{i}/\,{\mathcal {U}}$ is called a reduced product.
When ${\mathcal {U}}$ is a principal ultrafilter (which happens if and only if ${\mathcal {U}}$ contains its kernel $\cap \,{\mathcal {U}}$) then the ultraproduct is isomorphic to one of the factors. And so usually, ${\mathcal {U}}$ is not a principal ultrafilter, which happens if and only if ${\mathcal {U}}$ is free (meaning $\cap \,{\mathcal {U}}=\varnothing $), or equivalently, if every cofinite subsets of $I$ is an element of ${\mathcal {U}}.$ Since every ultrafilter on a finite set is principal, the index set $I$ is consequently also usually infinite.
The ultraproduct acts as a filter product space where elements are equal if they are equal only at the filtered components (non-filtered components are ignored under the equivalence). One may define a finitely additive measure $m$ on the index set $I$ by saying $m(A)=1$ if $A\in {\mathcal {U}}$ and $m(A)=0$ otherwise. Then two members of the Cartesian product are equivalent precisely if they are equal almost everywhere on the index set. The ultraproduct is the set of equivalence classes thus generated.
Finitary operations on the Cartesian product $ \prod \limits _{i\in I}}M_{i}$ are defined pointwise (for example, if $+$ is a binary function then $a_{i}+b_{i}=(a+b)_{i}$). Other relations can be extended the same way:
$R\left(a_{\mathcal {U}}^{1},\dots ,a_{\mathcal {U}}^{n}\right)~\iff ~\left\{i\in I:R^{M_{i}}\left(a_{i}^{1},\dots ,a_{i}^{n}\right)\right\}\in {\mathcal {U}},$
where $a_{\mathcal {U}}$ denotes the ${\mathcal {U}}$-equivalence class of $a$ with respect to $\sim .$ In particular, if every $M_{i}$ is an ordered field then so is the ultraproduct.
Ultrapower
An ultrapower is an ultraproduct for which all the factors $M_{i}$ are equal. Explicitly, the ultrapower of a set $M$ modulo ${\mathcal {U}}$ is the ultraproduct $ \prod \limits _{i\in I}}M_{i}\,/\,{\mathcal {U}}= \prod }_{\mathcal {U}}\,M_{\bullet }$ of the indexed family $M_{\bullet }:=\left(M_{i}\right)_{i\in I}$ defined by $M_{i}:=M$ for every index $i\in I.$ The ultrapower may be denoted by $ \prod }_{\mathcal {U}}\,M$ or (since $ \prod \limits _{i\in I}}M$ is often denoted by $M^{I}$) by
$M^{I}/{\mathcal {U}}~:=~\prod _{i\in I}M\,/\,{\mathcal {U}}\,$
For every $m\in M,$ let $(m)_{i\in I}$ denote the constant map $I\to M$ that is identically equal to $m.$ This constant map/tuple is an element of the Cartesian product $M^{I}= \prod \limits _{i\in I}}M$ and so the assignment $m\mapsto (m)_{i\in I}$ defines a map $M\to \prod \limits _{i\in I}}M.$ The natural embedding of $M$ into $ \prod }_{\mathcal {U}}\,M$ is the map $M\to \prod }_{\mathcal {U}}\,M$ that sends an element $m\in M$ to the ${\mathcal {U}}$-equivalence class of the constant tuple $(m)_{i\in I}.$
Examples
The hyperreal numbers are the ultraproduct of one copy of the real numbers for every natural number, with regard to an ultrafilter over the natural numbers containing all cofinite sets. Their order is the extension of the order of the real numbers. For example, the sequence $\omega $ given by $\omega _{i}=i$ defines an equivalence class representing a hyperreal number that is greater than any real number.
Analogously, one can define nonstandard integers, nonstandard complex numbers, etc., by taking the ultraproduct of copies of the corresponding structures.
As an example of the carrying over of relations into the ultraproduct, consider the sequence $\psi $ defined by $\psi _{i}=2i.$ Because $\psi _{i}>\omega _{i}=i$ for all $i,$ it follows that the equivalence class of $\psi _{i}=2i$ is greater than the equivalence class of $\omega _{i}=i,$ so that it can be interpreted as an infinite number which is greater than the one originally constructed. However, let $\chi _{i}=i$ for $i$ not equal to $7,$ but $\chi _{7}=8.$ The set of indices on which $\omega $ and $\chi $ agree is a member of any ultrafilter (because $\omega $ and $\chi $ agree almost everywhere), so $\omega $ and $\chi $ belong to the same equivalence class.
In the theory of large cardinals, a standard construction is to take the ultraproduct of the whole set-theoretic universe with respect to some carefully chosen ultrafilter ${\mathcal {U}}.$ Properties of this ultrafilter ${\mathcal {U}}$ have a strong influence on (higher order) properties of the ultraproduct; for example, if ${\mathcal {U}}$ is $\sigma $-complete, then the ultraproduct will again be well-founded. (See measurable cardinal for the prototypical example.)
Łoś's theorem
Łoś's theorem, also called the fundamental theorem of ultraproducts, is due to Jerzy Łoś (the surname is pronounced [ˈwɔɕ], approximately "wash"). It states that any first-order formula is true in the ultraproduct if and only if the set of indices $i$ such that the formula is true in $M_{i}$ is a member of ${\mathcal {U}}.$ More precisely:
Let $\sigma $ be a signature, ${\mathcal {U}}$ an ultrafilter over a set $I,$ and for each $i\in I$ let $M_{i}$ be a $\sigma $-structure. Let $ \prod }_{\mathcal {U}}\,M_{\bullet }$ or $ \prod \limits _{i\in I}}M_{i}/{\mathcal {U}}$ be the ultraproduct of the $M_{i}$ with respect to ${\mathcal {U}}.$ Then, for each $a^{1},\ldots ,a^{n}\in \prod \limits _{i\in I}}M_{i},$ where $a^{k}=\left(a_{i}^{k}\right)_{i\in I},$ and for every $\sigma $-formula $\phi ,$
${\prod }_{\mathcal {U}}\,M_{\bullet }\models \phi \left[a_{\mathcal {U}}^{1},\ldots ,a_{\mathcal {U}}^{n}\right]~\iff ~\{i\in I:M_{i}\models \phi [a_{i}^{1},\ldots ,a_{i}^{n}]\}\in {\mathcal {U}}.$
The theorem is proved by induction on the complexity of the formula $\phi .$ The fact that ${\mathcal {U}}$ is an ultrafilter (and not just a filter) is used in the negation clause, and the axiom of choice is needed at the existential quantifier step. As an application, one obtains the transfer theorem for hyperreal fields.
Examples
Let $R$ be a unary relation in the structure $M,$ and form the ultrapower of $M.$ Then the set $S=\{x\in M:Rx\}$ has an analog ${}^{*}S$ in the ultrapower, and first-order formulas involving $S$ are also valid for ${}^{*}S.$ For example, let $M$ be the reals, and let $Rx$ hold if $x$ is a rational number. Then in $M$ we can say that for any pair of rationals $x$ and $y,$ there exists another number $z$ such that $z$ is not rational, and $x<z<y.$ Since this can be translated into a first-order logical formula in the relevant formal language, Łoś's theorem implies that ${}^{*}S$ has the same property. That is, we can define a notion of the hyperrational numbers, which are a subset of the hyperreals, and they have the same first-order properties as the rationals.
Consider, however, the Archimedean property of the reals, which states that there is no real number $x$ such that $x>1,\;x>1+1,\;x>1+1+1,\ldots $ for every inequality in the infinite list. Łoś's theorem does not apply to the Archimedean property, because the Archimedean property cannot be stated in first-order logic. In fact, the Archimedean property is false for the hyperreals, as shown by the construction of the hyperreal number $\omega $ above.
Direct limits of ultrapowers (ultralimits)
For the ultraproduct of a sequence of metric spaces, see Ultralimit.
In model theory and set theory, the direct limit of a sequence of ultrapowers is often considered. In model theory, this construction can be referred to as an ultralimit or limiting ultrapower.
Beginning with a structure, $A_{0}$ and an ultrafilter, ${\mathcal {D}}_{0},$ form an ultrapower, $A_{1}.$ Then repeat the process to form $A_{2},$ and so forth. For each $n$ there is a canonical diagonal embedding $A_{n}\to A_{n+1}.$ At limit stages, such as $A_{\omega },$ form the direct limit of earlier stages. One may continue into the transfinite.
Ultraproduct monad
The ultrafilter monad is the codensity monad of the inclusion of the category of finite sets into the category of all sets.[1]
Similarly, the ultraproduct monad is the codensity monad of the inclusion of the category $\mathbf {FinFam} $ of finitely-indexed families of sets into the category $\mathbf {Fam} $ of all indexed families of sets. So in this sense, ultraproducts are categorically inevitable.[1] Explicitly, an object of $\mathbf {Fam} $ consists of a non-empty index set $I$ and an indexed family $\left(M_{i}\right)_{i\in I}$ of sets. A morphism $\left(N_{i}\right)_{j\in J}\to \left(M_{i}\right)_{i\in I}$ between two objects consists of a function $\phi :I\to J$ between the index sets and a $J$-indexed family $\left(\phi _{j}\right)_{j\in J}$ of function $\phi _{j}:M_{\phi (j)}\to N_{j}.$ The category $\mathbf {FinFam} $ is a full subcategory of this category of $\mathbf {Fam} $ consisting of all objects $\left(M_{i}\right)_{i\in I}$ whose index set $I$ is finite. The codensity monad of the inclusion map $\mathbf {FinFam} \hookrightarrow \mathbf {Fam} $ is then, in essence, given by
$\left(M_{i}\right)_{i\in I}~\mapsto ~\left(\prod _{i\in I}M_{i}\,/\,{\mathcal {U}}\right)_{{\mathcal {U}}\in U(I)}\,.$
See also
• Compactness theorem
• Extender (set theory) – in set theory, a system of ultrafilters representing an elementary embedding witnessing large cardinal propertiesPages displaying wikidata descriptions as a fallback
• Löwenheim–Skolem theorem – Existence and cardinality of models of logical theories
• Transfer principle – That all statements of some language that are true for some structure are true for another structure
• Ultrafilter – Maximal proper filter
Notes
1. Leinster, Tom (2013). "Codensity and the ultrafilter monad" (PDF). Theory and Applications of Categories. 28: 332–370. arXiv:1209.3606. Bibcode:2012arXiv1209.3606L.
Proofs
1. Although ${\mathcal {U}}$ is assumed to be an ultrafilter over $I,$ this proof only requires that ${\mathcal {U}}$ be a filter on $I.$ Throughout, let $a_{\bullet }=\left(a_{i}\right)_{i\in I},b_{\bullet }=\left(b_{i}\right)_{i\in I},$ and $c_{\bullet }=\left(c_{i}\right)_{i\in I}$ be elements of $ \prod \limits _{i\in I}}M_{i}.$ The relation $a_{\bullet }\,\sim \,a_{\bullet }$ always holds since $\{i\in I:a_{i}=a_{i}\}=I$ is an element of filter ${\mathcal {U}}.$ Thus the reflexivity of $\,\sim \,$ follows from that of equality $\,=.\,$ Similarly, $\,\sim \,$ is symmetric since equality is symmetric. For transitivity, assume that $R=\{i:a_{i}:=b_{i}\}$ and $S:=\{i:b_{i}=c_{i}\}$ are elements of ${\mathcal {U}};$ it remains to show that $T:=\{i:a_{i}=c_{i}\}$ also belongs to ${\mathcal {U}}.$ The transitivity of equality guarantees $R\cap S\subseteq T$ (since if $i\in R\cap S$ then $a_{i}=b_{i}$ and $b_{i}=c_{i}$). Because ${\mathcal {U}}$ is closed under binary intersections, $R\cap S\in {\mathcal {U}}.$ Since ${\mathcal {U}}$ is upward closed in $I,$ it contains every superset of $R\cap S$ (that consists of indices); in particular, ${\mathcal {U}}$ contains $T.$ $\blacksquare $
References
• Bell, John Lane; Slomson, Alan B. (2006) [1969]. Models and Ultraproducts: An Introduction (reprint of 1974 ed.). Dover Publications. ISBN 0-486-44979-3.
• Burris, Stanley N.; Sankappanavar, H.P. (2000) [1981]. A Course in Universal Algebra (Millennium ed.).
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Łukasiewicz logic
In mathematics and philosophy, Łukasiewicz logic (/ˌluːkəˈʃɛvɪtʃ/ LOO-kə-SHEV-itch, Polish: [wukaˈɕɛvitʂ]) is a non-classical, many-valued logic. It was originally defined in the early 20th century by Jan Łukasiewicz as a three-valued modal logic;[1] it was later generalized to n-valued (for all finite n) as well as infinitely-many-valued (ℵ0-valued) variants, both propositional and first order.[2] The ℵ0-valued version was published in 1930 by Łukasiewicz and Alfred Tarski; consequently it is sometimes called the Łukasiewicz–Tarski logic.[3] It belongs to the classes of t-norm fuzzy logics[4] and substructural logics.[5]
This article is about a system of logic. For the similarly named Łukasiewicz notation, see Polish notation.
Łukasiewicz logic was motivated by Aristotle's suggestion that bivalent logic was not applicable to future contingents, e.g. the statement "There will be a sea battle tomorrow". In other words, statements about the future were neither true nor false, but an intermediate value could be assigned to them, to represent their possibility of becoming true in the future.
This article presents the Łukasiewicz(–Tarski) logic in its full generality, i.e. as an infinite-valued logic. For an elementary introduction to the three-valued instantiation Ł3, see three-valued logic.
Language
The propositional connectives of Łukasiewicz logic are $\rightarrow $ ("implication"), and the constant $\bot $ ("false"). Additional connectives can be defined in terms of these:
${\begin{aligned}\neg A&=_{def}A\rightarrow \bot \\A\vee B&=_{def}(A\rightarrow B)\rightarrow B\\A\wedge B&=_{def}\neg (\neg A\vee \neg B)\\A\leftrightarrow B&=_{def}(A\rightarrow B)\wedge (B\rightarrow A)\end{aligned}}$
The $\vee $ and $\wedge $ connectives are called weak disjunction and conjunction, because they are non-classical, as the law of excluded middle does not hold for them. In the context of substructural logics, they are called additive connectives. They also correspond to lattice min/max connectives.
In terms of substructural logics, there are also strong or multiplicative disjunction and conjunction connectives, although these are not part of Łukasiewicz's original presentation:
${\begin{aligned}A\oplus B&=_{def}\neg A\rightarrow B\\A\otimes B&=_{def}\neg (A\rightarrow \neg B)\end{aligned}}$
There are also defined modal operators, using the Tarskian Möglichkeit:
${\begin{aligned}\Diamond A&=_{def}\neg A\rightarrow A\\\Box A&=_{def}\neg \Diamond \neg A\end{aligned}}$
Axioms
The original system of axioms for propositional infinite-valued Łukasiewicz logic used implication and negation as the primitive connectives, along with modus ponens:
${\begin{aligned}A&\rightarrow (B\rightarrow A)\\(A\rightarrow B)&\rightarrow ((B\rightarrow C)\rightarrow (A\rightarrow C))\\((A\rightarrow B)\rightarrow B)&\rightarrow ((B\rightarrow A)\rightarrow A)\\(\neg B\rightarrow \neg A)&\rightarrow (A\rightarrow B).\end{aligned}}$
Propositional infinite-valued Łukasiewicz logic can also be axiomatized by adding the following axioms to the axiomatic system of monoidal t-norm logic:
Divisibility
$(A\wedge B)\rightarrow (A\otimes (A\rightarrow B))$
Double negation
$\neg \neg A\rightarrow A.$
That is, infinite-valued Łukasiewicz logic arises by adding the axiom of double negation to basic fuzzy logic (BL), or by adding the axiom of divisibility to the logic IMTL.
Finite-valued Łukasiewicz logics require additional axioms.
Proof Theory
A hypersequent calculus for three-valued Łukasiewicz logic was introduced by Arnon Avron in 1991.[6]
Sequent calculi for finite and infinite-valued Łukasiewicz logics as an extension of linear logic were introduced by A. Prijatelj in 1994.[7] However, these are not cut-free systems.
Hypersequent calculi for Łukasiewicz logics were introduced by A. Ciabattoni et al in 1999.[8] However, these are not cut-free for $n>3$ finite-valued logics.
A labelled tableaux system was introduced by Nicola Olivetti in 2003.[9]
Real-valued semantics
Infinite-valued Łukasiewicz logic is a real-valued logic in which sentences from sentential calculus may be assigned a truth value of not only 0 or 1 but also any real number in between (e.g. 0.25). Valuations have a recursive definition where:
• $w(\theta \circ \phi )=F_{\circ }(w(\theta ),w(\phi ))$ for a binary connective $\circ ,$
• $w(\neg \theta )=F_{\neg }(w(\theta )),$
• $w\left({\overline {0}}\right)=0$ and $w\left({\overline {1}}\right)=1,$
and where the definitions of the operations hold as follows:
• Implication: $F_{\rightarrow }(x,y)=\min\{1,1-x+y\}$
• Equivalence: $F_{\leftrightarrow }(x,y)=1-|x-y|$
• Negation: $F_{\neg }(x)=1-x$
• Weak conjunction: $F_{\wedge }(x,y)=\min\{x,y\}$
• Weak disjunction: $F_{\vee }(x,y)=\max\{x,y\}$
• Strong conjunction: $F_{\otimes }(x,y)=\max\{0,x+y-1\}$
• Strong disjunction: $F_{\oplus }(x,y)=\min\{1,x+y\}.$
The truth function $F_{\otimes }$ of strong conjunction is the Łukasiewicz t-norm and the truth function $F_{\oplus }$ of strong disjunction is its dual t-conorm. Obviously, $F_{\otimes }(.5,.5)=0$ and $F_{\oplus }(.5,.5)=1$, so if $T(p)=.5$, then $T(p\wedge p)=T(\neg p\wedge \neg p)=0$ while the respective logically-equivalent propositions have $T(p\vee p)=T(\neg p\vee \neg p)=1$.
The truth function $F_{\rightarrow }$ is the residuum of the Łukasiewicz t-norm. All truth functions of the basic connectives are continuous.
By definition, a formula is a tautology of infinite-valued Łukasiewicz logic if it evaluates to 1 under each valuation of propositional variables by real numbers in the interval [0, 1].
Finite-valued and countable-valued semantics
Using exactly the same valuation formulas as for real-valued semantics Łukasiewicz (1922) also defined (up to isomorphism) semantics over
• any finite set of cardinality n ≥ 2 by choosing the domain as { 0, 1/(n − 1), 2/(n − 1), ..., 1 }
• any countable set by choosing the domain as { p/q | 0 ≤ p ≤ q where p is a non-negative integer and q is a positive integer }.
General algebraic semantics
The standard real-valued semantics determined by the Łukasiewicz t-norm is not the only possible semantics of Łukasiewicz logic. General algebraic semantics of propositional infinite-valued Łukasiewicz logic is formed by the class of all MV-algebras. The standard real-valued semantics is a special MV-algebra, called the standard MV-algebra.
Like other t-norm fuzzy logics, propositional infinite-valued Łukasiewicz logic enjoys completeness with respect to the class of all algebras for which the logic is sound (that is, MV-algebras) as well as with respect to only linear ones. This is expressed by the general, linear, and standard completeness theorems:[4]
The following conditions are equivalent:
• $A$ is provable in propositional infinite-valued Łukasiewicz logic
• $A$ is valid in all MV-algebras (general completeness)
• $A$ is valid in all linearly ordered MV-algebras (linear completeness)
• $A$ is valid in the standard MV-algebra (standard completeness).
Here valid means necessarily evaluates to 1.
Font, Rodriguez and Torrens introduced in 1984 the Wajsberg algebra as an alternative model for the infinite-valued Łukasiewicz logic.[10]
A 1940s attempt by Grigore Moisil to provide algebraic semantics for the n-valued Łukasiewicz logic by means of his Łukasiewicz–Moisil (LM) algebra (which Moisil called Łukasiewicz algebras) turned out to be an incorrect model for n ≥ 5. This issue was made public by Alan Rose in 1956. C. C. Chang's MV-algebra, which is a model for the ℵ0-valued (infinitely-many-valued) Łukasiewicz–Tarski logic, was published in 1958. For the axiomatically more complicated (finite) n-valued Łukasiewicz logics, suitable algebras were published in 1977 by Revaz Grigolia and called MVn-algebras.[11] MVn-algebras are a subclass of LMn-algebras, and the inclusion is strict for n ≥ 5.[12] In 1982 Roberto Cignoli published some additional constraints that added to LMn-algebras produce proper models for n-valued Łukasiewicz logic; Cignoli called his discovery proper Łukasiewicz algebras.[13]
Complexity
Łukasiewicz logics are co-NP complete.[14]
Modal Logic
Łukasiewicz logics can be seen as modal logics using the defined operators,
${\begin{aligned}\Diamond A&=_{def}\neg A\rightarrow A\\\Box A&=_{def}\neg \Diamond \neg A\\\end{aligned}}$
A third doubtful operator has been proposed, $\odot A=_{def}A\leftrightarrow \neg A$.[15]
From these we can prove the following theorems, which are common axioms in many modal logics:
${\begin{aligned}A&\rightarrow \Diamond A\\\Box A&\rightarrow A\\A&\rightarrow (A\rightarrow \Box A)\\\Box (A\rightarrow B)&\rightarrow (\Box A\rightarrow \Box B)\\\Box (A\rightarrow B)&\rightarrow (\Diamond A\rightarrow \Diamond B)\\\end{aligned}}$
We can also prove distribution theorems on the strong connectives:
${\begin{aligned}\Box (A\otimes B)&\leftrightarrow \Box A\otimes \Box B\\\Diamond (A\oplus B)&\leftrightarrow \Diamond A\oplus \Diamond B\\\Diamond (A\otimes B)&\rightarrow \Diamond A\otimes \Diamond B\\\Box A\oplus \Box B&\rightarrow \Box (A\oplus B)\end{aligned}}$
However, the following distribution theorems also hold:
${\begin{aligned}\Box A\vee \Box B&\leftrightarrow \Box (A\vee B)\\\Box A\wedge \Box B&\leftrightarrow \Box (A\wedge B)\\\Diamond A\vee \Diamond B&\leftrightarrow \Diamond (A\vee B)\\\Diamond A\wedge \Diamond B&\leftrightarrow \Diamond (A\wedge B)\end{aligned}}$
In other words, if $\Diamond A\wedge \Diamond \neg A$, then $\Diamond (A\wedge \neg A)$, which is counter-intuitive.[16] [17] However, these controversial theorems have been defended as a modal logic about future contingents by A. N. Prior.[18] Notably, $\Diamond A\wedge \Diamond \neg A\leftrightarrow \odot A$.
References
1. Łukasiewicz J., 1920, O logice trójwartościowej (in Polish). Ruch filozoficzny 5:170–171. English translation: On three-valued logic, in L. Borkowski (ed.), Selected works by Jan Łukasiewicz, North–Holland, Amsterdam, 1970, pp. 87–88. ISBN 0-7204-2252-3
2. Hay, L.S., 1963, Axiomatization of the infinite-valued predicate calculus. Journal of Symbolic Logic 28:77–86.
3. Lavinia Corina Ciungu (2013). Non-commutative Multiple-Valued Logic Algebras. Springer. p. vii. ISBN 978-3-319-01589-7. citing Łukasiewicz, J., Tarski, A.: Untersuchungen über den Aussagenkalkül. Comp. Rend. Soc. Sci. et Lettres Varsovie Cl. III 23, 30–50 (1930).
4. Hájek P., 1998, Metamathematics of Fuzzy Logic. Dordrecht: Kluwer.
5. Ono, H., 2003, "Substructural logics and residuated lattices — an introduction". In F.V. Hendricks, J. Malinowski (eds.): Trends in Logic: 50 Years of Studia Logica, Trends in Logic 20: 177–212.
6. A. Avron, "Natural 3-valued Logics– Characterization and Proof Theory", Journal of Symbolic Logic 56(1), doi:10.2307/2274919
7. A. Prijateli, "Bounded contraction and Gentzen-style formulation of Łukasiewicz logics", Studia Logica 57: 437-456, 1996
8. A. Ciabattoni, D.M. Gabbay, N. Olivetti, "Cut-free proof systems for logics of weak excluded middle" Soft Computing 2 (1999) 147—156
9. N. Olivetti, "Tableaux for Łukasiewicz Infinite-valued Logic", Studia Logica volume 73, pages 81–111 (2003)
10. http://journal.univagora.ro/download/pdf/28.pdf citing J. M. Font, A. J. Rodriguez, A. Torrens, Wajsberg Algebras, Stochastica, VIII, 1, 5-31, 1984
11. Lavinia Corina Ciungu (2013). Non-commutative Multiple-Valued Logic Algebras. Springer. pp. vii–viii. ISBN 978-3-319-01589-7. citing Grigolia, R.S.: "Algebraic analysis of Lukasiewicz-Tarski’s n-valued logical systems". In: Wójcicki, R., Malinkowski, G. (eds.) Selected Papers on Lukasiewicz Sentential Calculi, pp. 81–92. Polish Academy of Sciences, Wroclav (1977)
12. Iorgulescu, A.: Connections between MVn-algebras and n-valued Łukasiewicz–Moisil algebras Part I. Discrete Mathematics 181, 155–177 (1998) doi:10.1016/S0012-365X(97)00052-6
13. R. Cignoli, Proper n-Valued Łukasiewicz Algebras as S-Algebras of Łukasiewicz n-Valued Propositional Calculi, Studia Logica, 41, 1982, 3-16, doi:10.1007/BF00373490
14. A. Ciabattoni, M. Bongini and F. Montagna, Proof Search and Co-NP Completeness for Many-Valued Logics. Fuzzy Sets and Systems.
15. Clarence Irving Lewis and Cooper Harold Langford. Symbolic Logic. Dover, New York, second edition, 1959.
16. Robert Bull and Krister Segerberg. Basic modal logic. In Dov M. Gabbay and Franz Guenthner, editors, Handbook of Philosophical Logic, volume 2. D. Reidel Publishing Company, Lancaster, 1986
17. Alasdair Urquhart. An interpretation of many-valued logic. Zeitschr. f. math. Logik und Grundlagen d. Math., 19:111–114, 1973.
18. A.N. Prior. Three-valued logic and future contingents. 3(13):317–26, October 1953.
Further reading
• Rose, A.: 1956, Formalisation du Calcul Propositionnel Implicatif ℵ0 Valeurs de Łukasiewicz, C. R. Acad. Sci. Paris 243, 1183–1185.
• Rose, A.: 1978, Formalisations of Further ℵ0-Valued Łukasiewicz Propositional Calculi, Journal of Symbolic Logic 43(2), 207–210. doi:10.2307/2272818
• Cignoli, R., “The algebras of Lukasiewicz many-valued logic - A historical overview,” in S. Aguzzoli et al.(Eds.), Algebraic and Proof-theoretic Aspects of Non-classical Logics, LNAI 4460, Springer, 2007, 69-83. doi:10.1007/978-3-540-75939-3_5
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Šidák correction for t-test
One of the application of Student's t-test is to test the location of one sequence of independent and identically distributed random variables. If we want to test the locations of multiple sequences of such variables, Šidák correction should be applied in order to calibrate the level of the Student's t-test. Moreover, if we want to test the locations of nearly infinitely many sequences of variables, then Šidák correction should be used, but with caution. More specifically, the validity of Šidák correction depends on how fast the number of sequences goes to infinity.
Introduction
Suppose we are interested in m different hypotheses, $H_{1},...,H_{m}$, and would like to check if all of them are true. Now the hypothesis test scheme becomes
$H_{null}$: all of $H_{i}$ are true;
$H_{alternative}$: at least one of $H_{i}$ is false.
Let $\alpha $ be the level of this test (the type-I error), that is, the probability that we falsely reject $H_{null}$ when it is true.
We aim to design a test with certain level $\alpha $.
Suppose when testing each hypothesis $H_{i}$, the test statistic we use is $t_{i}$.
If these $t_{i}$'s are independent, then a test for $H_{null}$ can be developed by the following procedure, known as Šidák correction.
Step 1, we test each of m null hypotheses at level $1-(1-\alpha )^{\frac {1}{m}}$.
Step 2, if any of these m null hypotheses is rejected, we reject $H_{null}$.
Finite case
For finitely many t-tests, suppose $Y_{ij}=\mu _{i}+\epsilon _{ij},i=1,...,N,j=1,...,n,$ where for each i, $\epsilon _{i1},...,\epsilon _{in}$ are independently and identically distributed, for each j $\epsilon _{1j},...,\epsilon _{Nj}$ are independent but not necessarily identically distributed, and $\epsilon _{ij}$ has finite fourth moment.
Our goal is to design a test for $H_{null}:\mu _{i}=0,\forall i=1,...,N$ with level α. This test can be based on the t-statistic of each sequences, that is,
$t_{i}={\frac {{\bar {Y}}_{i}}{S_{i}/{\sqrt {n}}}},$
where:
${\bar {Y}}_{i}={\frac {1}{n}}\sum _{j=1}^{n}Y_{ij},\qquad S_{i}^{2}={\frac {1}{n}}\sum _{j=1}^{n}(Y_{ij}-{\bar {Y}}_{i})^{2}.$
Using Šidák correction, we reject $H_{null}$ if any of the t-tests based on the t-statistics above reject at level $1-(1-\alpha )^{\frac {1}{N}}.$ More specifically, we reject $H_{null}$ when
$\exists i\in \{1,\ldots ,N\}:|t_{i}|>\zeta _{\alpha ,N},$
where
$P(|Z|>\zeta _{\alpha ,N})=1-(1-\alpha )^{\frac {1}{N}},\qquad Z\sim N(0,1)$
The test defined above has asymptotic level α, because
${\begin{aligned}{\text{level}}&=P_{null}\left({\text{reject }}H_{null}\right)\\&=P_{null}\left(\exists i\in \{1,\ldots ,N\}:|t_{i}|>\zeta _{\alpha ,N}\right)\\&=1-P_{null}\left(\forall i\in \{1,\ldots ,N\}:|t_{i}|\leq \zeta _{\alpha ,N}\right)\\&=1-\prod _{i=1}^{N}P_{null}\left(|t_{i}|\leq \zeta _{\alpha ,N}\right)\\&\to 1-\prod _{i=1}^{N}P\left(|Z_{i}|\leq \zeta _{\alpha ,N}\right)&&Z_{i}\sim N(0,1)\\&=\alpha \end{aligned}}$
Infinite case
In some cases, the number of sequences, $N$, increase as the data size of each sequences, $n$, increase. In particular, suppose $N(n)\rightarrow \infty {\text{ as }}n\rightarrow \infty $. If this is true, then we will need to test a null including infinitely many hypotheses, that is
$H_{null}:{\text{ all of }}H_{i}{\text{ are true, }}i=1,2,....$
To design a test, Šidák correction may be applied, as in the case of finitely many t-test. However, when $N(n)\rightarrow \infty {\text{ as }}n\rightarrow \infty $, the Šidák correction for t-test may not achieve the level we want, that is, the true level of the test may not converges to the nominal level $\alpha $ as n goes to infinity. This result is related to high-dimensional statistics and is proven by Fan, Hall & Yao (2007).[1] Specifically, if we want the true level of the test converges to the nominal level $\alpha $, then we need a restraint on how fast $N(n)\rightarrow \infty $. Indeed,
• When all of $\epsilon _{ij}$ have distribution symmetric about zero, then it is sufficient to require $\log N=o(n^{1/3})$ to guarantee the true level converges to $\alpha $.
• When the distributions of $\epsilon _{ij}$ are asymmetric, then it is necessary to impose $\log N=o(n^{1/2})$ to ensure the true level converges to $\alpha $.
• Actually, if we apply bootstrapping method to the calibration of level, then we will only need $\log N=o(n^{1/3})$ even if $\epsilon _{ij}$ has asymmetric distribution.
The results above are based on Central Limit Theorem. According to Central Limit Theorem, each of our t-statistics $t_{i}$ possesses asymptotic standard normal distribution, and so the difference between the distribution of each $t_{i}$ and the standard normal distribution is asymptotically negligible. The question is, if we aggregate all the differences between the distribution of each $t_{i}$ and the standard normal distribution, is this aggregation of differences still asymptotically ignorable?
When we have finitely many $t_{i}$, the answer is yes. But when we have infinitely many $t_{i}$, the answer some time becomes no. This is because in the latter case we are summing up infinitely many infinitesimal terms. If the number of the terms goes to infinity too fast, that is, $N(n)\rightarrow \infty $ too fast, then the sum may not be zero, the distribution of the t-statistics can not be approximated by the standard normal distribution, the true level does not converges to the nominal level $\alpha $, and then the Šidák correction fails.
See also
• Šidák correction
• Multiple comparisons
• Bonferroni correction
• Family-wise error rate
• Closed testing procedure
References
1. Fan, Jianqing; Hall, Peter; Yao, Qiwei (2007). "To How Many Simultaneous Hypothesis Tests Can Normal, Student's t or Bootstrap Calibration Be Applied". Journal of the American Statistical Association. 102 (480): 1282–1288. arXiv:math/0701003. doi:10.1198/016214507000000969. S2CID 8622675.
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Švarc–Milnor lemma
In the mathematical subject of geometric group theory, the Švarc–Milnor lemma (sometimes also called Milnor–Švarc lemma, with both variants also sometimes spelling Švarc as Schwarz) is a statement which says that a group $G$, equipped with a "nice" discrete isometric action on a metric space $X$, is quasi-isometric to $X$.
This result goes back, in different form, before the notion of quasi-isometry was formally introduced, to the work of Albert S. Schwarz (1955)[1] and John Milnor (1968).[2] Pierre de la Harpe called the Švarc–Milnor lemma "the fundamental observation in geometric group theory"[3] because of its importance for the subject. Occasionally the name "fundamental observation in geometric group theory" is now used for this statement, instead of calling it the Švarc–Milnor lemma; see, for example, Theorem 8.2 in the book of Farb and Margalit.[4]
Precise statement
Several minor variations of the statement of the lemma exist in the literature (see the Notes section below). Here we follow the version given in the book of Bridson and Haefliger (see Proposition 8.19 on p. 140 there).[5]
Let $G$ be a group acting by isometries on a proper length space $X$ such that the action is properly discontinuous and cocompact.
Then the group $G$ is finitely generated and for every finite generating set $S$ of $G$ and every point $p\in X$ the orbit map
$f_{p}:(G,d_{S})\to X,\quad g\mapsto gp$
is a quasi-isometry.
Here $d_{S}$ is the word metric on $G$ corresponding to $S$.
Sometimes a properly discontinuous cocompact isometric action of a group $G$ on a proper geodesic metric space $X$ is called a geometric action.[6]
Explanation of the terms
Recall that a metric $X$ space is proper if every closed ball in $X$ is compact.
An action of $G$ on $X$ is properly discontinuous if for every compact $K\subseteq X$ the set
$\{g\in G\mid gK\cap K\neq \varnothing \}$
is finite.
The action of $G$ on $X$ is cocompact if the quotient space $X/G$, equipped with the quotient topology, is compact. Under the other assumptions of the Švarc–Milnor lemma, the cocompactness condition is equivalent to the existence of a closed ball $B$ in $X$ such that
$\bigcup _{g\in G}gB=X.$
Examples of applications of the Švarc–Milnor lemma
For Examples 1 through 5 below see pp. 89–90 in the book of de la Harpe.[3] Example 6 is the starting point of the part of the paper of Richard Schwartz.[7]
1. For every $n\geq 1$ the group $\mathbb {Z} ^{n}$ is quasi-isometric to the Euclidean space $\mathbb {R} ^{n}$.
2. If $\Sigma $ is a closed connected oriented surface of negative Euler characteristic then the fundamental group $\pi _{1}(\Sigma )$ is quasi-isometric to the hyperbolic plane $\mathbb {H} ^{2}$.
3. If $(M,g)$ is a closed connected smooth manifold with a smooth Riemannian metric $g$ then $\pi _{1}(M)$ is quasi-isometric to $({\tilde {M}},d_{\tilde {g}})$, where ${\tilde {M}}$ is the universal cover of $M$, where ${\tilde {g}}$ is the pull-back of $g$ to ${\tilde {M}}$, and where $d_{\tilde {g}}$ is the path metric on ${\tilde {M}}$ defined by the Riemannian metric ${\tilde {g}}$.
4. If $G$ is a connected finite-dimensional Lie group equipped with a left-invariant Riemannian metric and the corresponding path metric, and if $\Gamma \leq G$ is a uniform lattice then $\Gamma $ is quasi-isometric to $G$.
5. If $M$ is a closed hyperbolic 3-manifold, then $\pi _{1}(M)$ is quasi-isometric to $\mathbb {H} ^{3}$.
6. If $M$ is a complete finite volume hyperbolic 3-manifold with cusps, then $\Gamma =\pi _{1}(M)$ is quasi-isometric to $\Omega =\mathbb {H} ^{3}-{\mathcal {B}}$, where ${\mathcal {B}}$ is a certain $\Gamma $-invariant collection of horoballs, and where $\Omega $ is equipped with the induced path metric.
References
1. A. S. Švarc, A volume invariant of coverings (in Russian), Doklady Akademii Nauk SSSR, vol. 105, 1955, pp. 32–34.
2. J. Milnor, A note on curvature and fundamental group, Journal of Differential Geometry, vol. 2, 1968, pp. 1–7
3. Pierre de la Harpe, Topics in geometric group theory. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000. ISBN 0-226-31719-6; p. 87
4. Benson Farb, and Dan Margalit, A primer on mapping class groups. Princeton Mathematical Series, 49. Princeton University Press, Princeton, NJ, 2012. ISBN 978-0-691-14794-9; p. 224
5. M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer-Verlag, Berlin, 1999. ISBN 3-540-64324-9
6. I. Kapovich, and N. Benakli, Boundaries of hyperbolic groups. Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), pp. 39–93, Contemp. Math., 296, American Mathematical Society, Providence, RI, 2002, ISBN 0-8218-2822-3; Convention 2.22 on p. 46
7. Richard Schwartz, The quasi-isometry classification of rank one lattices, Publications Mathématiques de l'Institut des Hautes Études Scientifiques, vol. 82, 1995, pp. 133–168
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Ștefan Emilian
Ștefan Emilian (August 8, 1819 – November 1899) was an Imperial Austrian-born Romanian mathematician and architect.
Born in Bonchida, Kolozs County (now Bonțida, Cluj County), in the Principality of Transylvania, he was given the surname Kertész as a child, although his birth name was Emilian. He attended high school in Sibiu. Then, from 1841 to 1845, he studied at the Academy of Fine Arts Vienna, graduating with an architect's degree. Additionally, from 1841 to 1843, he took courses at the Vienna Polytechnic Institute. Emilian returned home shortly before 1848, in time for the Transylvanian Revolution. Pursued by the authorities, he sought refuge in Wallachia. By 1850, he was back in Transylvania, where he taught mathematics at Brașov's Greek Orthodox High School. He remained there until 1858, a period during which he designed the new school building. Additionally, he was the architect for the first paper factory in Zărnești.[1]
In 1858, he was invited to Iași, the capital of Moldavia, in order to teach drawing and geometry to the upper classes of Academia Mihăileană. Emilian remained there for two years, until the founding of the University of Iași. Additionally, he taught at the military officers' school and the technical school of arts and professions. At the new university, he was named full professor of descriptive geometry and linear perspective, remaining from October 1860 to October 1892, when he had to retire. Meanwhile, he designed the Iași anatomy institute, the Lipovan Church, and the church in Bosia. A single published book of his is known: the 1886 Curs practic de perspectivă liniară. Emilian's funeral eulogy was delivered by Alexandru Dimitrie Xenopol.[1]
He married Cornelia Ederlly de Medve.[2]
Notes
1. Ionel Maftei, Personalități ieșene, vol. I, pp. 229–30. Comitetul de cultură și educație socialistă al județului Iași, 1972
2. Ionela Băluță, "Apariția femeii ca actor social – a doua jumătate a secolului al XIX-lea", in Direcții și teme de cercetare în studiile de gen din România, p. 71. Bucharest: Editura Colegiul Noua Europă, 2003, ISBN 978-973-856975-1
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Overline
An overline, overscore, or overbar, is a typographical feature of a horizontal line drawn immediately above the text. In old mathematical notation, an overline was called a vinculum, a notation for grouping symbols which is expressed in modern notation by parentheses, though it persists for symbols under a radical sign. The original use in Ancient Greek was to indicate compositions of Greek letters as Greek numerals.[1] In Latin, it indicates Roman numerals multiplied by a thousand and it forms medieval abbreviations (sigla). Marking one or more words with a continuous line above the characters is sometimes called overstriking, though overstriking generally refers to printing one character on top of an already-printed character.
DescriptionSampleUnicodeCSS/HTML
Overline
(markup)
Xx— text-decoration: overline;
Overline
(character)
‾U+203E‾, ‾
X̅x̅ (combining)U+0305X̅
Double overline
(markup)
Xx— text-decoration: overline;
text-decoration-style: double;
Double overline
(character)
X̿x̿ (combining)U+033FX̿
Macron
(character)
¯U+00AF¯, ¯
X̄x̄ (combining)U+0304X̄
X̄x̄ (precomposed) varies
An overline, that is, a single line above a chunk of text, should not be confused with the macron, a diacritical mark placed above (or sometimes below) individual letters. The macron is narrower than the character box.[2]
Uses
Medicine
In most forms of Latin scribal abbreviation, an overline or macron indicates omitted letters similar to use of apostrophes in English contractions. Letters with macrons or overlines continue to be used in medical abbreviations in various European languages, particularly for prescriptions. Common examples include
• a, a̅, or ā for ante ("before")
• c, c̅, or c̄ for cum ("with")
• p, p̅, or p̄ for post ("after")[3]
• q, q̅, or q̄ for quisque and its inflections ("every", "each")
• s, s̅, or s̄ for sine ("without")
• x, x̅, or x̄ for exceptus and its inflections ("except")
Note, however, that abbreviations involving the letter h take their macron halfway up the ascending line rather than at the normal height for Unicode overlines and macrons: ħ. This is separately encoded in Unicode with the symbols using bar diacritics and appears shorter than other overlines in many fonts.
Decimal separator
Main article: Decimal separator
In the Middle Ages, from the original Indian decimal writing, before printing, an overline over the units digit was used to separate the integral part of a number from its fractional part, as in 9995 (meaning 99.95 in decimal point format). A similar notation remains in common use as an underbar to superscript digits, especially for monetary values without a decimal separator, as in 9995.
Vinculum
In mathematics, an overline can be used as a vinculum.
The vinculum can indicate a line segment:[4]
${\overline {\rm {AB}}}$
The vinculum can indicate a repeating decimal value:
${1 \over 7}=0.{\overline {142857}}=0.142857142857142857142857...$
When it is not possible to format the number so that the overline is over the digit(s) that repeat, one overline character is placed to the left of the digit(s) that repeat:
$3.{\overline {\phantom {I}}}3=3.{\overline {3}}=3.3333333333333333333333333...$
$3.12{\overline {\phantom {I}}}34=3.12{\overline {34}}=3.123434343434343434343434...$
Historically, the vinculum was used to group together symbols so that they could be treated as a unit. Today, parentheses are more commonly used for this purpose.
Statistics
The overline is used to indicate a sample mean:[5]
• ${\overline {x}}$ is the average value of $x_{i}$
Survival functions or complementary cumulative distribution functions are often denoted by placing an overline over the symbol for the cumulative: ${\overline {F}}(x)=1-F(x)$.
Negation
In set theory and some electrical engineering contexts, negation operators (also known as complement) can be written as an overline above the term or expression to be negated.[6] For example:
Common set theory notation:
${\begin{aligned}{\overline {A\cup B}}&\equiv {\overline {A}}\cap {\overline {B}}\\{\overline {A\cap B}}&\equiv {\overline {A}}\cup {\overline {B}}\end{aligned}}$
Electrical engineering notation:
${\begin{aligned}{\overline {A\cdot B}}&\equiv {\overline {A}}+{\overline {B}}\\{\overline {A+B}}&\equiv {\overline {A}}\cdot {\overline {B}}\end{aligned}}$
in which the times (cross) means multiplication, the dot means logical AND, and the plus sign means logical OR.
Both illustrate De Morgan's laws and its mnemonic, "break the line, change the sign".
Negative
In common logarithms, a bar over the characteristic indicates that it is negative—whilst the mantissa remains positive. This notation avoids the need for separate tables to convert positive and negative logarithms back to their original numbers.
$\log _{10}0.012\approx -2+0.07918={\bar {2}}.07918$
Complex numbers
The overline notation can indicate a complex conjugate and analogous operations.[7]
• if $x=a+ib$, then ${\overline {x}}=a-ib.$
Vector
In physics, an overline sometimes indicates a vector, although boldface and arrows are also commonly used:
• ${\overline {x}}=|x|{\hat {x}}$
Congruence classes
Congruence modulo n is an equivalence relation, and the equivalence class of the integer a, denoted by an, is the set {... , a − 2n, a − n, a, a + n, a + 2n, ...}. This set, consisting of all the integers congruent to a modulo n, is called the congruence class, residue class, or simply residue of the integer a modulo n. When the modulus n is known from the context, that residue may also be denoted [a] or a.
Topological closure
In topology, the closure of a subset S of a topological space is often denoted S or $\operatorname {cl} S$.
Improper rotation
In crystallography, an overline indicates an improper rotation or a negative number:
• ${\overline {3}}$ is the Hermann–Mauguin notation for a threefold rotoinversion, used in crystallography.
• $[{\overline {1}}1{\overline {2}}]$ is the direction with Miller indices $h=-1$, $k=1$, $l=-2$.
Maximal conductance
In computational neuroscience, an overline is used to indicate the "maximal" conductances in Hodgkin-Huxley models. This goes back to at least the landmark paper published by Nobel prize winners Alan Lloyd Hodgkin and Andrew Fielding Huxley around 1952.[8]
$I_{\mathrm {Na} }(t)={\bar {g}}_{\mathrm {Na} }m(V_{m})^{3}h(V_{m})(V_{m}-E_{\mathrm {Na} })$
Antiparticles
Overlines are used in subatomic particle physics to denote antiparticles for some particles (with the alternate being distinguishing based on electric charge). For example, the proton is denoted as
p
, and its corresponding antiparticle is denoted as
p
.
Engineering
An active low signal is designated by an overline, e.g. RESET, representing logical negation.
Morse (CW)
Some Morse code prosigns can be expressed as two or three characters run together, and an overline is often used to signify this. The most famous is the distress signal, SOS.
Writing
An overline-like symbol is traditionally used in Syriac text to mark abbreviations and numbers. It has dots at each end and the center. In German it is occasionally used to indicate a pair of letters which cannot both be fitted into the available space.[9][10]
When Morse code is written out as text, overlines are used to distinguish prosigns and other concatenated character groups from strings of individual characters.
In Arabic writing and printing, overlines are traditionally used instead of underlines for typographic emphasis,[11] although underlines are used more and more due to the rise of the internet.
Linguistics
X-bar theory makes use of overbar notation to indicate differing levels of syntactic structure. Certain structures are represented by adding an overbar to the unit, as in X. Due to difficulty in typesetting the overbar, the prime symbol is often used instead, as in X′. Contemporary typesetting software, such as LaTeX, has made typesetting overbars considerably simpler; both prime and overbar markers are accepted usages. Some variants of X-bar notation use a double-bar (or double-prime) to represent phrasal-level units.
X-bar theory derives its name from the overbar. One of the core proposals of the theory was the creation of an intermediate syntactic node between phrasal (XP) and unit (X) levels; rather than introduce a different label, the intermediate unit was marked with a bar.
Implementations
HTML with CSS
In HTML using CSS, overline is implemented via the text-decoration property; for example, <span style="text-decoration: overline">text</span> results in: text.
The text decoration property supports also other typographical features with horizontal lines: underline (a line below the text) and strikethrough (a line through the text).
Unicode
Unicode includes two graphic characters, U+00AF ¯ MACRON and U+203E ‾ OVERLINE. They are compatibility equivalent to the U+0020 SPACE with non-spacing diacritics U+0304 ◌̄ COMBINING MACRON and U+0305 ◌̅ COMBINING OVERLINE respectively; the latter allows an overline to be placed over any character. There is also U+033F ◌̿ COMBINING DOUBLE OVERLINE. As with any combining character, it appears in the same character box as the character that logically precedes it: for example, x̅, compared to x‾. A series of overlined characters, for example 1̅2̅3̅, may result either in a broken or an unbroken line, depending on the font.
In Unicode, character U+FE26 COMBINING CONJOINING MACRON is conjoining (bridging) two characters: ◌︦◌.
In East Asian (CJK) computing, U+FFE3 ￣ FULLWIDTH MACRON is available. Despite the name, Unicode maps this character to both U+203E and U+00AF.[12]
Unicode maps the overline-like character from ISO/IEC 8859-1 and code page 850 to the U+00AF ¯ MACRON symbol mentioned above. In a reversal of its official name (and compatibility decomposition), it is much wider than an actual macron diacritic over most letters, and actually wider than U+203E ‾ OVERLINE in most fonts. In Microsoft Windows, U+00AF can be entered with the keystrokes Alt+0175 (where numbers are entered from the numeric keypad). In GTK, the symbol can be added using the keystrokes Ctrl+⇧ Shift+U to activate Unicode input, then type "00AF" as the code for the character. On a Mac, with the ABC Extended keyboard, use ⌥ Option+a.
The Unicode character U+070F SYRIAC ABBREVIATION MARK is used to mark Syriac abbreviations and numbers. However, several computer environments do not render this line correctly or at all.
The Unicode character U+0B55 ୕ ORIYA SIGN OVERLINE is used as a length mark in Odia script.
Word processors
In Microsoft Word, overstriking of text can be accomplished with the EQ \O() field code. The field code {EQ \O(x,¯)} produces x and the field code {EQ \O(xyz,¯¯¯)} produces xyz. (Doesn't work in Word 2010; it is necessary to insert MS Equation object). Windows: Alt+0773 (once before character, one more time after character).
LibreOffice has direct support for several styles of overline in its "Format / Character / Font Effects" dialog.
Overstriking of longer sections of text, such as in 123, can also be produced in many text processors as text markup as a special form of understriking.
TeX
In LaTeX, a text <text> can be overlined with $\overline{\mbox{<text>}}$. The inner \mbox{} is necessary to override the math-mode (here invoked by the dollar signs) which the \overline{} demands.
See also
Wikimedia Commons has media related to Overlining.
• Ā
• Titlo, an overline used to indicate numerals or abbreviations in Cyrillic
• Underscore
References
1. Smith, T. P. (2013). How Big is Big and How Small is Small: The Sizes of Everything and Why.
2. Wells, J.C. (2001). "Orthographic diacritics and multilingual computing". University College London. Retrieved 23 March 2014.
3. Cappelli, Adriano (1961). Manuali Hoepli Lexicon Abbreviature Dizionario Di Abbreviature Latine ed Italiane. Milan: Editore Ulrico Hoepli Milano. p. 256.
4. "Line Segment Definition - Math Open Reference". www.mathopenref.com. Retrieved 2020-08-24.
5. "Sample Means". www.stat.yale.edu. Retrieved 2020-08-24.
6. "Set Operations | Union | Intersection | Complement | Difference | Mutually Exclusive | Partitions | De Morgan's Law | Distributive Law | Cartesian Product". www.probabilitycourse.com. Retrieved 2020-08-24.
7. Weisstein, Eric W. "Complex Conjugate". mathworld.wolfram.com. Retrieved 2020-08-24.
8. Hodgkin, A. L.; Huxley, A. F. (1952). "A quantitative description of membrane current and its application to conduction and excitation in nerve". The Journal of Physiology. 117 (4): 500–544. doi:10.1113/jphysiol.1952.sp004764. PMC 1392413. PMID 12991237.
9. Hardwig, Florian (2011-11-23). "Gräfinnen". Flickr. Retrieved 26 December 2017.
10. Hardwig, Florian (2015-12-26). "Lieder zur Weihnachtszeit (1940)". Fonts in Use. Retrieved 26 December 2017. It used to be common to mark omitted double letters with an overbar, especially for "mm" and "nn". These abbreviations come in handy when lyrics have to match the musical notes, see 'da kom[m]t er her'.
11. "Emphasis (typography)". Emphasis (typography). Retrieved 2020-09-02.
12. The Unicode Consortium (2012), "Halfwidth and Fullwidth Forms" (PDF), The Unicode Standard 6.1, Unicode Consortium, ISBN 978-1-936213-02-3, FULLWIDTH MACRON • sometimes treated as fullwidth overline
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Hippasus
Hippasus of Metapontum (/ˈhɪpəsəs/; Greek: Ἵππασος ὁ Μεταποντῖνος, Híppasos; c. 530 – c. 450 BC)[1] was a Greek philosopher and early follower of Pythagoras.[2][3] Little is known about his life or his beliefs, but he is sometimes credited with the discovery of the existence of irrational numbers. The discovery of irrational numbers is said to have been shocking to the Pythagoreans, and Hippasus is supposed to have drowned at sea, apparently as a punishment from the gods for divulging this. However, the few ancient sources which describe this story either do not mention Hippasus by name (e.g. Pappus)[4] or alternatively tell that Hippasus drowned because he revealed how to construct a dodecahedron inside a sphere. The discovery of irrationality is not specifically ascribed to Hippasus by any ancient writer.
Life
Little is known about the life of Hippasus. He may have lived in the late 5th century BC, about a century after the time of Pythagoras. Metapontum in Magna Graecia is usually referred to as his birthplace,[5][6][7][8][9] although according to Iamblichus some claim Metapontum to be his birthplace, while others the nearby city of Croton.[10] Hippasus is recorded under the city of Sybaris in Iamblichus list of each city's Pythagoreans.[11] He also states that Hippasus was the founder of a sect of the Pythagoreans called the Mathematici (μαθηματικοί) in opposition to the Acusmatici (ἀκουσματικοί);[12] but elsewhere he makes him the founder of the Acusmatici in opposition to the Mathematici.[13]
Iamblichus says about the death of Hippasus:
It is related to Hippasus that he was a Pythagorean, and that, owing to his being the first to publish and describe the sphere from the twelve pentagons, he perished at sea for his impiety, but he received credit for the discovery, though really it all belonged to HIM (for in this way they refer to Pythagoras, and they do not call him by his name).[14]
According to Iamblichus (ca. 245-325 AD, 1918 translation) in The life of Pythagoras, by Thomas Taylor[15]
There were also two forms of philosophy, for the two genera of those that pursued it: the Acusmatici and the Mathematici. The latter are acknowledged to be Pythagoreans by the rest but the Mathematici do not admit that the Acusmatici derived their instructions from Pythagoras but from Hippasus. The philosophy of the Acusmatici consisted in auditions unaccompanied with demonstrations and a reasoning process; because it merely ordered a thing to be done in a certain way and that they should endeavor to preserve such other things as were said by him, as divine dogmas. Memory was the most valued faculty. All these auditions were of three kinds; some signifying what a thing is; others what it especially is, others what ought or ought not to be done. (p. 61)
Doctrines
Aristotle speaks of Hippasus as holding the element of fire to be the cause of all things;[16] and Sextus Empiricus contrasts him with the Pythagoreans in this respect, that he believed the arche to be material, whereas they thought it was incorporeal, namely, number.[17] Diogenes Laërtius tells us that Hippasus believed that "there is a definite time which the changes in the universe take to complete, and that the universe is limited and ever in motion."[6] According to one statement, Hippasus left no writings,[6] according to another he was the author of the Mystic Discourse, written to bring Pythagoras into disrepute.[18]
A scholium on Plato's Phaedo notes him as an early experimenter in music theory, claiming that he made use of bronze disks to discover the fundamental musical ratios, 4:3, 3:2, and 2:1.[19]
Irrational numbers
Hippasus is sometimes credited with the discovery of the existence of irrational numbers, following which he was drowned at sea. Pythagoreans preached that all numbers could be expressed as the ratio of integers, and the discovery of irrational numbers is said to have shocked them. However, the evidence linking the discovery to Hippasus is unclear.
Pappus merely says that the knowledge of irrational numbers originated in the Pythagorean school, and that the member who first divulged the secret perished by drowning.[20] Iamblichus gives a series of inconsistent reports. In one story he explains how a Pythagorean was merely expelled for divulging the nature of the irrational; but he then cites the legend of the Pythagorean who drowned at sea for making known the construction of the regular dodecahedron in the sphere.[21] In another account he tells how it was Hippasus who drowned at sea for betraying the construction of the dodecahedron and taking credit for this construction himself;[22] but in another story this same punishment is meted out to the Pythagorean who divulged knowledge of the irrational.[23] Iamblichus clearly states that the drowning at sea was a punishment from the gods for impious behaviour.[21]
These stories are usually taken together to ascribe the discovery of irrationals to Hippasus, but whether he did or not is uncertain.[24] In principle, the stories can be combined, since it is possible to discover irrational numbers when constructing dodecahedra. Irrationality, by infinite reciprocal subtraction, can be easily seen in the golden ratio of the regular pentagon.[25]
Some scholars in the early 20th century credited Hippasus with the discovery of the irrationality of ${\sqrt {2}}$, the square root of 2. Plato in his Theaetetus,[26] describes how Theodorus of Cyrene (c. 400 BC) proved the irrationality of ${\sqrt {3}}$, ${\sqrt {5}}$, etc. up to ${\sqrt {17}}$, which implies that an earlier mathematician had already proved the irrationality of ${\sqrt {2}}$.[27] Aristotle referred to the method for a proof of the irrationality of ${\sqrt {2}}$,[28] and a full proof along these same lines is set out in the proposition interpolated at the end of Euclid's Book X,[29] which suggests that the proof was certainly ancient.[30] The method is a proof by contradiction, or reductio ad absurdum, which shows that if the diagonal of a square is assumed to be commensurable with the side, then the same number must be both odd and even.[30]
In the hands of modern writers this combination of vague ancient reports and modern guesswork has sometimes evolved into a much more emphatic and colourful tale. Some writers have Hippasus making his discovery while on board a ship, as a result of which his Pythagorean shipmates toss him overboard;[31] while one writer even has Pythagoras himself "to his eternal shame" sentencing Hippasus to death by drowning, for showing "that ${\sqrt {2}}$ is an irrational number".[32]
References
1. Huffman, Carl A. (1993). Philolaus of Croton: Pythagorean and Presocratic. Cambridge University Press. p. 8.
2. "Hippasus of Metapontum | Greek philosopher". Encyclopedia Britannica. Retrieved 2021-09-20.
3. Iamblichus (1918). The life of Pythagoras (1918 translation ed.). p. 327.
4. William Thompson. The Commentary of Pappus on Book X of Euclid's Elements (PDF).
5. Aristotle, Metaphysics I.3: 984a7
6. Diogenes Laertius, Lives of Eminent Philosophers VIII,84
7. Simplicius, Physica 23.33
8. Aetius I.5.5 (Dox. 292)
9. Clement of Alexandria, Protrepticus 64.2
10. Iamblichus, Vita Pythagorica, 18 (81)
11. Iamblichus, Vita Pythagorica, 34 (267)
12. Iamblichus, De Communi Mathematica Scientia, 76
13. Iamblichus, Vita Pythagorica, 18 (81); cf. Iamblichus, In Nic. 10.20; De anima ap. Stobaeus, i.49.32
14. Iamblichus, Thomas, ed. (1939). "18". On the Pythagorean Life. p. 88.
15. Iamblichus (1918). The life of Pythagoras.
16. Aristotle, Metaphysics (English translation)
17. Sextus Empiricus, ad Phys. i. 361
18. Diogenes Laertius, Lives of Eminent Philosophers, viii. 7
19. Scholium on Plato's Phaedo, 108d
20. Pappus, Commentary on Book X of Euclid's Elements. A similar story is quoted in a Greek scholium to the tenth book.
21. Iamblichus, Vita Pythagorica, 34 (246).
22. Iamblichus, Vita Pythagorica, 18 (88), De Communi Mathematica Scientia, 25.
23. Iamblichus, Vita Pythagorica, 34 (247).
24. Wilbur Richard Knorr (1975), The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and its Significance for Early Greek Geometry, pages 21–22, 50–51. Springer.
25. Walter Burkert (1972), Lore and Science in Ancient Pythagoreanism, page 459. Harvard University Press.
26. Plato, Theaetetus, 147d ff.
27. Thomas Heath (1921) A History of Greek Mathematics, Volume 1, From Thales to Euclid, p. 155.
28. Aristotle, Prior Analytics, I-23.
29. Thomas Heath (1921) A History of Greek Mathematics, Volume 1, From Thales to Euclid, p. 157.
30. Thomas Heath (1921) A History of Greek Mathematics, Volume 1, From Thales to Euclid, p. 168.
31. Morris Kline (1990), Mathematical Thought from Ancient to Modern Times, page 32. Oxford University Press.
32. Simon Singh (1998), Fermat's Enigma, p. 50.
External links
Wikisource has the text of the 1911 Encyclopædia Britannica article "Hippasus of Metapontum".
• Hippasus of Metapontum at scienceworld.wolfram.com
• Laërtius, Diogenes (1925). "Pythagoreans: Hippasus" . Lives of the Eminent Philosophers. Vol. 2:8. Translated by Hicks, Robert Drew (Two volume ed.). Loeb Classical Library.
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Alpha shape
In computational geometry, an alpha shape, or α-shape, is a family of piecewise linear simple curves in the Euclidean plane associated with the shape of a finite set of points. They were first defined by Edelsbrunner, Kirkpatrick & Seidel (1983). The alpha-shape associated with a set of points is a generalization of the concept of the convex hull, i.e. every convex hull is an alpha-shape but not every alpha shape is a convex hull.
Characterization
For each real number α, define the concept of a generalized disk of radius 1/α as follows:
• If α = 0, it is a closed half-plane;
• If α > 0, it is a closed disk of radius 1/α;
• If α < 0, it is the closure of the complement of a disk of radius −1/α.
Then an edge of the alpha-shape is drawn between two members of the finite point set whenever there exists a generalized disk of radius 1/α containing none of the point set and which has the property that the two points lie on its boundary.
If α = 0, then the alpha-shape associated with the finite point set is its ordinary convex hull.
Alpha complex
Alpha shapes are closely related to alpha complexes, subcomplexes of the Delaunay triangulation of the point set.
Each edge or triangle of the Delaunay triangulation may be associated with a characteristic radius, the radius of the smallest empty circle containing the edge or triangle. For each real number α, the α-complex of the given set of points is the simplicial complex formed by the set of edges and triangles whose radii are at most 1/α.
The union of the edges and triangles in the α-complex forms a shape closely resembling the α-shape; however it differs in that it has polygonal edges rather than edges formed from arcs of circles. More specifically, Edelsbrunner (1995) showed that the two shapes are homotopy equivalent. (In this later work, Edelsbrunner used the name "α-shape" to refer to the union of the cells in the α-complex, and instead called the related curvilinear shape an α-body.)
Examples
This technique can be employed to reconstruct a Fermi surface from the electronic Bloch spectral function evaluated at the Fermi level, as obtained from the Green's function in a generalised ab-initio study of the problem. The Fermi surface is then defined as the set of reciprocal space points within the first Brillouin zone, where the signal is highest. The definition has the advantage of covering also cases of various forms of disorder.
See also
• Beta skeleton
References
• N. Akkiraju, H. Edelsbrunner, M. Facello, P. Fu, E. P. Mucke, and C. Varela. "Alpha shapes: definition and software". In Proc. Internat. Comput. Geom. Software Workshop 1995, Minneapolis.
• Edelsbrunner, Herbert (1995), "Smooth surfaces for multi-scale shape representation", Foundations of software technology and theoretical computer science (Bangalore, 1995), Lecture Notes in Comput. Sci., vol. 1026, Berlin: Springer, pp. 391–412, MR 1458090.
• Edelsbrunner, Herbert; Kirkpatrick, David G.; Seidel, Raimund (1983), "On the shape of a set of points in the plane", IEEE Transactions on Information Theory, 29 (4): 551–559, doi:10.1109/TIT.1983.1056714.
External links
Wikimedia Commons has media related to Alpha shape.
• 2D Alpha Shapes and 3D Alpha Shapes in CGAL the Computational Geometry Algorithms Library
• Alpha Complex in the GUDHI library.
• Description and implementation by Duke University
• Everything You Always Wanted to Know About Alpha Shapes But Were Afraid to Ask – with illustrations and interactive demonstration
• Implementation of the 3D alpha-shape for the reconstruction of 3D sets from a point cloud in R
• Description of the implementation details for alpha shapes - lecture providing a description of the formal and intuitive aspects of alpha shape implementation
• Alpha Hulls, Shapes, and Weighted things - lecture slides by Robert Pless at the Washington University in St. Louis
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αΒΒ
αΒΒ is a second-order deterministic global optimization algorithm for finding the optima of general, twice continuously differentiable functions.[1][2] The algorithm is based around creating a relaxation for nonlinear functions of general form by superposing them with a quadratic of sufficient magnitude, called α, such that the resulting superposition is enough to overcome the worst-case scenario of non-convexity of the original function. Since a quadratic has a diagonal Hessian matrix, this superposition essentially adds a number to all diagonal elements of the original Hessian, such that the resulting Hessian is positive-semidefinite. Thus, the resulting relaxation is a convex function.
Theory
Let a function ${f({\boldsymbol {x}})\in C^{2}}$ be a function of general non-linear non-convex structure, defined in a finite box $X=\{{\boldsymbol {x}}\in \mathbb {R} ^{n}:{\boldsymbol {x}}^{L}\leq {\boldsymbol {x}}\leq {\boldsymbol {x}}^{U}\}$. Then, a convex underestimation (relaxation) $L({\boldsymbol {x}})$ of this function can be constructed over $X$ by superposing a sum of univariate quadratics, each of sufficient magnitude to overcome the non-convexity of ${f({\boldsymbol {x}})}$ everywhere in $X$, as follows:
$L({\boldsymbol {x}})=f({\boldsymbol {x}})+\sum _{i=1}^{i=n}\alpha _{i}(x_{i}^{L}-x_{i})(x_{i}^{U}-x_{i})$
$L({\boldsymbol {x}})$ is called the $\alpha {\text{BB}}$ underestimator for general functional forms. If all $\alpha _{i}$ are sufficiently large, the new function $L({\boldsymbol {x}})$ is convex everywhere in $X$. Thus, local minimization of $L({\boldsymbol {x}})$ yields a rigorous lower bound on the value of ${f({\boldsymbol {x}})}$ in that domain.
Calculation of ${\boldsymbol {\alpha }}$
There are numerous methods to calculate the values of the ${\boldsymbol {\alpha }}$ vector. It is proven that when $\alpha _{i}=\max\{0,-{\frac {1}{2}}\lambda _{i}^{\min }\}$, where $\lambda _{i}^{\min }$ is a valid lower bound on the $i$-th eigenvalue of the Hessian matrix of ${f({\boldsymbol {x}})}$, the $L({\boldsymbol {x}})$ underestimator is guaranteed to be convex.
One of the most popular methods to get these valid bounds on eigenvalues is by use of the Scaled Gerschgorin theorem. Let $H(X)$ be the interval Hessian matrix of ${f(X)}$ over the interval $X$. Then, $\forall d_{i}>0$ a valid lower bound on eigenvalue $\lambda _{i}$ may be derived from the $i$-th row of $H(X)$ as follows:
$\lambda _{i}^{\min }={\underline {h_{ii}}}-\sum _{i\neq j}(\max(|{\underline {h_{ij}}}|,|{\overline {h_{ij}}}|{\frac {d_{j}}{d_{i}}})$
References
1. "A global optimization approach for Lennard-Jones microclusters." Journal of Chemical Physics, 1992, 97(10), 7667-7677
2. "αBB: A global optimization method for general constrained nonconvex problems." Journal of Global Optimization, 1995, 7(4), 337-363
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Archimedes
Archimedes of Syracuse (/ˌɑːrkɪˈmiːdiːz/, ARK-ihm-EE-deez;[3][lower-alpha 1] c. 287 – c. 212 BC) was an Ancient Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily.[4] Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Considered the greatest mathematician of ancient history, and one of the greatest of all time,[5] Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical theorems.[6][7] These include the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral.[8][9]
Archimedes of Syracuse
Ἀρχιμήδης
Archimedes Thoughtful
by Domenico Fetti (1620)
Bornc. 287 BC
Syracuse, Sicily
Diedc. 212 BC (aged approximately 75)
Syracuse, Sicily
Known for
List
• Archimedes' principle
Archimedes' screw
Center of gravity
Statics
Hydrostatics
Law of the lever
Indivisibles
Neuseis constructions[1]
List of other things named after him
Scientific career
FieldsMathematics
Physics
Astronomy
Mechanics
Engineering
InfluencesEudoxus
InfluencedApollonius[2]
Hero
Pappus
Eutocius
Archimedes' other mathematical achievements include deriving an approximation of pi, defining and investigating the Archimedean spiral, and devising a system using exponentiation for expressing very large numbers. He was also one of the first to apply mathematics to physical phenomena, working on statics and hydrostatics. Archimedes' achievements in this area include a proof of the law of the lever,[10] the widespread use of the concept of center of gravity,[11] and the enunciation of the law of buoyancy or Archimedes' principle.[12] He is also credited with designing innovative machines, such as his screw pump, compound pulleys, and defensive war machines to protect his native Syracuse from invasion.
Archimedes died during the siege of Syracuse, when he was killed by a Roman soldier despite orders that he should not be harmed. Cicero describes visiting Archimedes' tomb, which was surmounted by a sphere and a cylinder that Archimedes requested be placed there to represent his mathematical discoveries.
Unlike his inventions, Archimedes' mathematical writings were little known in antiquity. Mathematicians from Alexandria read and quoted him, but the first comprehensive compilation was not made until c. 530 AD by Isidore of Miletus in Byzantine Constantinople, while commentaries on the works of Archimedes by Eutocius in the 6th century opened them to wider readership for the first time. The relatively few copies of Archimedes' written work that survived through the Middle Ages were an influential source of ideas for scientists during the Renaissance and again in the 17th century,[13][14] while the discovery in 1906 of previously lost works by Archimedes in the Archimedes Palimpsest has provided new insights into how he obtained mathematical results.[15][16][17][18]
Biography
Archimedes was born c. 287 BC in the seaport city of Syracuse, Sicily, at that time a self-governing colony in Magna Graecia. The date of birth is based on a statement by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years before his death in 212 BC.[9] In the Sand-Reckoner, Archimedes gives his father's name as Phidias, an astronomer about whom nothing else is known.[19] A biography of Archimedes was written by his friend Heracleides, but this work has been lost, leaving the details of his life obscure. It is unknown, for instance, whether he ever married or had children, or if he ever visited Alexandria, Egypt, during his youth.[20] From his surviving written works, it is clear that he maintained collegiate relations with scholars based there, including his friend Conon of Samos and the head librarian Eratosthenes of Cyrene.[lower-alpha 2]
The standard versions of Archimedes' life were written long after his death by Greek and Roman historians. The earliest reference to Archimedes occurs in The Histories by Polybius (c. 200–118 BC), written about 70 years after his death. It sheds little light on Archimedes as a person, and focuses on the war machines that he is said to have built in order to defend the city from the Romans.[21] Polybius remarks how, during the Second Punic War, Syracuse switched allegiances from Rome to Carthage, resulting in a military campaign under the command of Marcus Claudius Marcellus and Appius Claudius Pulcher, who besieged the city from 213 to 212 BC. He notes that the Romans underestimated Syracuse's defenses, and mentions several machines Archimedes designed, including improved catapults, crane-like machines that could be swung around in an arc, and other stone-throwers. Although the Romans ultimately captured the city, they suffered considerable losses due to Archimedes' inventiveness.[22]
Cicero (106–43 BC) mentions Archimedes in some of his works. While serving as a quaestor in Sicily, Cicero found what was presumed to be Archimedes' tomb near the Agrigentine gate in Syracuse, in a neglected condition and overgrown with bushes. Cicero had the tomb cleaned up and was able to see the carving and read some of the verses that had been added as an inscription. The tomb carried a sculpture illustrating Archimedes' favorite mathematical proof, that the volume and surface area of the sphere are two-thirds that of an enclosing cylinder including its bases.[23][24] He also mentions that Marcellus brought to Rome two planetariums Archimedes built.[25] The Roman historian Livy (59 BC–17 AD) retells Polybius' story of the capture of Syracuse and Archimedes' role in it.[21]
Plutarch (45–119 AD) wrote in his Parallel Lives that Archimedes was related to King Hiero II, the ruler of Syracuse.[27] He also provides at least two accounts on how Archimedes died after the city was taken. According to the most popular account, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet Marcellus, but he declined, saying that he had to finish working on the problem. This enraged the soldier, who killed Archimedes with his sword. Another story has Archimedes carrying mathematical instruments before being killed because a soldier thought they were valuable items. Marcellus was reportedly angered by Archimedes' death, as he considered him a valuable scientific asset (he called Archimedes "a geometrical Briareus") and had ordered that he should not be harmed.[28][29]
The last words attributed to Archimedes are "Do not disturb my circles" (Latin, "Noli turbare circulos meos"; Katharevousa Greek, "μὴ μου τοὺς κύκλους τάραττε"), a reference to the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier. There is no reliable evidence that Archimedes uttered these words and they do not appear in Plutarch's account. A similar quotation is found in the work of Valerius Maximus (fl. 30 AD), who wrote in Memorable Doings and Sayings, "... sed protecto manibus puluere 'noli' inquit, 'obsecro, istum disturbare'" ("... but protecting the dust with his hands, said 'I beg of you, do not disturb this'").[21]
Discoveries and inventions
Archimedes' principle
The most widely known anecdote about Archimedes tells of how he invented a method for determining the volume of an object with an irregular shape. According to Vitruvius, a crown for a temple had been made for King Hiero II of Syracuse, who supplied the pure gold to be used. The crown was likely made in the shape of a votive wreath.[30] Archimedes was asked to determine whether some silver had been substituted by the goldsmith without damaging the crown, so he could not melt it down into a regularly shaped body in order to calculate its density.[31]
In this account, Archimedes noticed while taking a bath that the level of the water in the tub rose as he got in, and realized that this effect could be used to determine the golden crown's volume. Archimedes was so excited by this discovery that he took to the streets naked, having forgotten to dress, crying "Eureka!" (Greek: "εὕρηκα, heúrēka!, lit. 'I have found [it]!'). For practical purposes water is incompressible,[32] so the submerged crown would displace an amount of water equal to its own volume. By dividing the mass of the crown by the volume of water displaced, its density could be obtained; if cheaper and less dense metals had been added, the density would be lower than that of gold. Archimedes found that this is what had happened, proving that silver had been mixed in.[30] [31]
The story of the golden crown does not appear anywhere in Archimedes' known works. The practicality of the method described has been called into question due to the extreme accuracy that would be required to measure water displacement.[33] Archimedes may have instead sought a solution that applied the hydrostatics principle known as Archimedes' principle, found in his treatise On Floating Bodies: a body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces.[34] Using this principle, it would have been possible to compare the density of the crown to that of pure gold by balancing it on a scale with a pure gold reference sample of the same weight, then immersing the apparatus in water. The difference in density between the two samples would cause the scale to tip accordingly.[12] Galileo Galilei, who invented a hydrostatic balance in 1586 inspired by Archimedes' work, considered it "probable that this method is the same that Archimedes followed, since, besides being very accurate, it is based on demonstrations found by Archimedes himself."[35][36]
Law of the lever
While Archimedes did not invent the lever, he gave a mathematical proof of the principle involved in his work On the Equilibrium of Planes.[37] Earlier descriptions of the principle of the lever are found in a work by Euclid and in the Mechanical Problems, belonging to the Peripatetic school of the followers of Aristotle, the authorship of which has been attributed by some to Archytas.[38][39]
There are several, often conflicting, reports regarding Archimedes' feats using the lever to lift very heavy objects. Plutarch describes how Archimedes designed block-and-tackle pulley systems, allowing sailors to use the principle of leverage to lift objects that would otherwise have been too heavy to move.[40] According to Pappus of Alexandria, Archimedes' work on levers and his understanding of mechanical advantage caused him to remark: "Give me a place to stand on, and I will move the Earth" (Greek: δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω).[41] Olympiodorus later attributed the same boast to Archimedes' invention of the baroulkos, a kind of windlass, rather than the lever.[42]
Astronomical instruments
Archimedes discusses astronomical measurements of the Earth, Sun, and Moon, as well as Aristarchus' heliocentric model of the universe, in the Sand-Reckoner. Without the use of either trigonometry or a table of chords, Archimedes describes the procedure and instrument used to make observations (a straight rod with pegs or grooves),[43][44] applies correction factors to these measurements, and finally gives the result in the form of upper and lower bounds to account for observational error.[19] Ptolemy, quoting Hipparchus, also references Archimedes' solstice observations in the Almagest. This would make Archimedes the first known Greek to have recorded multiple solstice dates and times in successive years.[20]
Cicero's De re publica portrays a fictional conversation taking place in 129 BC, after the Second Punic War. General Marcus Claudius Marcellus is said to have taken back to Rome two mechanisms after capturing Syracuse in 212 BC, which were constructed by Archimedes and which showed the motion of the Sun, Moon and five planets. Cicero also mentions similar mechanisms designed by Thales of Miletus and Eudoxus of Cnidus. The dialogue says that Marcellus kept one of the devices as his only personal loot from Syracuse, and donated the other to the Temple of Virtue in Rome. Marcellus' mechanism was demonstrated, according to Cicero, by Gaius Sulpicius Gallus to Lucius Furius Philus, who described it thus:[45][46]
Hanc sphaeram Gallus cum moveret, fiebat ut soli luna totidem conversionibus in aere illo quot diebus in ipso caelo succederet, ex quo et in caelo sphaera solis fieret eadem illa defectio, et incideret luna tum in eam metam quae esset umbra terrae, cum sol e regione.
When Gallus moved the globe, it happened that the Moon followed the Sun by as many turns on that bronze contrivance as in the sky itself, from which also in the sky the Sun's globe became to have that same eclipse, and the Moon came then to that position which was its shadow on the Earth when the Sun was in line.
This is a description of a small planetarium. Pappus of Alexandria reports on a treatise by Archimedes (now lost) dealing with the construction of these mechanisms entitled On Sphere-Making.[25][47] Modern research in this area has been focused on the Antikythera mechanism, another device built c. 100 BC that was probably designed for the same purpose.[48] Constructing mechanisms of this kind would have required a sophisticated knowledge of differential gearing.[49] This was once thought to have been beyond the range of the technology available in ancient times, but the discovery of the Antikythera mechanism in 1902 has confirmed that devices of this kind were known to the ancient Greeks.[50][51]
Archimedes' screw
A large part of Archimedes' work in engineering probably arose from fulfilling the needs of his home city of Syracuse. Athenaeus of Naucratis quotes a certain Moschion in a description on how King Hiero II commissioned the design of a huge ship, the Syracusia, which could be used for luxury travel, carrying supplies, and as a display of naval power.[52] The Syracusia is said to have been the largest ship built in classical antiquity and, according to Moschion's account, it was launched by Archimedes.[42] The ship presumably was capable of carrying 600 people and included garden decorations, a gymnasium, and a temple dedicated to the goddess Aphrodite among its facilities.[53] The account also mentions that, in order to remove any potential water leaking through the hull, a device with a revolving screw-shaped blade inside a cylinder was designed by Archimedes.
Archimedes' screw was turned by hand, and could also be used to transfer water from a low-lying body of water into irrigation canals. The screw is still in use today for pumping liquids and granulated solids such as coal and grain. Described by Vitruvius, Archimedes' device may have been an improvement on a screw pump that was used to irrigate the Hanging Gardens of Babylon.[54][55] The world's first seagoing steamship with a screw propeller was the SS Archimedes, which was launched in 1839 and named in honor of Archimedes and his work on the screw.[56]
Archimedes' claw
Archimedes is said to have designed a claw as a weapon to defend the city of Syracuse. Also known as "the ship shaker", the claw consisted of a crane-like arm from which a large metal grappling hook was suspended. When the claw was dropped onto an attacking ship the arm would swing upwards, lifting the ship out of the water and possibly sinking it.[57]
There have been modern experiments to test the feasibility of the claw, and in 2005 a television documentary entitled Superweapons of the Ancient World built a version of the claw and concluded that it was a workable device.[58] Archimedes has also been credited with improving the power and accuracy of the catapult, and with inventing the odometer during the First Punic War. The odometer was described as a cart with a gear mechanism that dropped a ball into a container after each mile traveled.[59]
Heat ray
Archimedes may have written a work on mirrors entitled Catoptrica,[lower-alpha 3] and later authors believed he might have used mirrors acting collectively as a parabolic reflector to burn ships attacking Syracuse. Lucian wrote, in the second century AD, that during the siege of Syracuse Archimedes destroyed enemy ships with fire. Almost four hundred years later, Anthemius of Tralles mentions, somewhat hesitantly, that Archimedes could have used burning-glasses as a weapon.[60]
Often called the "Archimedes heat ray", the purported mirror arrangement focused sunlight onto approaching ships, presumably causing them to catch fire. In the modern era, similar devices have been constructed and may be referred to as a heliostat or solar furnace.[61]
Archimedes' alleged heat ray has been the subject of an ongoing debate about its credibility since the Renaissance. René Descartes rejected it as false, while modern researchers have attempted to recreate the effect using only the means that would have been available to Archimedes, mostly with negative results.[62][63] It has been suggested that a large array of highly polished bronze or copper shields acting as mirrors could have been employed to focus sunlight onto a ship, but the overall effect would have been blinding, dazzling, or distracting the crew of the ship rather than fire.[64]
Mathematics
While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of mathematics. Plutarch wrote that Archimedes "placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life",[28] though some scholars believe this may be a mischaracterization.[65][66][67]
Method of exhaustion
Archimedes was able to use indivisibles (a precursor to infinitesimals) in a way that is similar to modern integral calculus.[6] Through proof by contradiction (reductio ad absurdum), he could give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the answer lay. This technique is known as the method of exhaustion, and he employed it to approximate the areas of figures and the value of π.
In Measurement of a Circle, he did this by drawing a larger regular hexagon outside a circle then a smaller regular hexagon inside the circle, and progressively doubling the number of sides of each regular polygon, calculating the length of a side of each polygon at each step. As the number of sides increases, it becomes a more accurate approximation of a circle. After four such steps, when the polygons had 96 sides each, he was able to determine that the value of π lay between 31/7 (approx. 3.1429) and 310/71 (approx. 3.1408), consistent with its actual value of approximately 3.1416.[68] He also proved that the area of a circle was equal to π multiplied by the square of the radius of the circle ($ \pi r^{2}$).
Archimedean property
In On the Sphere and Cylinder, Archimedes postulates that any magnitude when added to itself enough times will exceed any given magnitude. Today this is known as the Archimedean property of real numbers.[69]
Archimedes gives the value of the square root of 3 as lying between 265/153 (approximately 1.7320261) and 1351/780 (approximately 1.7320512) in Measurement of a Circle. The actual value is approximately 1.7320508, making this a very accurate estimate. He introduced this result without offering any explanation of how he had obtained it. This aspect of the work of Archimedes caused John Wallis to remark that he was: "as it were of set purpose to have covered up the traces of his investigation as if he had grudged posterity the secret of his method of inquiry while he wished to extort from them assent to his results."[70] It is possible that he used an iterative procedure to calculate these values.[71][72]
The infinite series
In Quadrature of the Parabola, Archimedes proved that the area enclosed by a parabola and a straight line is 4/3 times the area of a corresponding inscribed triangle as shown in the figure at right. He expressed the solution to the problem as an infinite geometric series with the common ratio 1/4:
$\sum _{n=0}^{\infty }4^{-n}=1+4^{-1}+4^{-2}+4^{-3}+\cdots ={4 \over 3}.\;$
If the first term in this series is the area of the triangle, then the second is the sum of the areas of two triangles whose bases are the two smaller secant lines, and whose third vertex is where the line that is parallel to the parabola's axis and that passes through the midpoint of the base intersects the parabola, and so on. This proof uses a variation of the series 1/4 + 1/16 + 1/64 + 1/256 + · · · which sums to 1/3.
Myriad of myriads
In The Sand Reckoner, Archimedes set out to calculate a number that was greater than the grains of sand needed to fill the universe. In doing so, he challenged the notion that the number of grains of sand was too large to be counted. He wrote:
There are some, King Gelo (Gelo II, son of Hiero II), who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited.
To solve the problem, Archimedes devised a system of counting based on the myriad. The word itself derives from the Greek μυριάς, murias, for the number 10,000. He proposed a number system using powers of a myriad of myriads (100 million, i.e., 10,000 x 10,000) and concluded that the number of grains of sand required to fill the universe would be 8 vigintillion, or 8×1063.[73]
Writings
The works of Archimedes were written in Doric Greek, the dialect of ancient Syracuse.[74] Many written works by Archimedes have not survived or are only extant in heavily edited fragments; at least seven of his treatises are known to have existed due to references made by other authors.[9] Pappus of Alexandria mentions On Sphere-Making and another work on polyhedra, while Theon of Alexandria quotes a remark about refraction from the now-lost Catoptrica.[lower-alpha 3]
Archimedes made his work known through correspondence with the mathematicians in Alexandria. The writings of Archimedes were first collected by the Byzantine Greek architect Isidore of Miletus (c. 530 AD), while commentaries on the works of Archimedes written by Eutocius in the sixth century AD helped to bring his work a wider audience. Archimedes' work was translated into Arabic by Thābit ibn Qurra (836–901 AD), and into Latin via Arabic by Gerard of Cremona (c. 1114–1187). Direct Greek to Latin translations were later done by William of Moerbeke (c. 1215–1286) and Iacobus Cremonensis (c. 1400–1453).[75][76]
During the Renaissance, the Editio princeps (First Edition) was published in Basel in 1544 by Johann Herwagen with the works of Archimedes in Greek and Latin.[77]
Surviving works
The following are ordered chronologically based on new terminological and historical criteria set by Knorr (1978) and Sato (1986).[78][79]
Measurement of a Circle
Main article: Measurement of a Circle
This is a short work consisting of three propositions. It is written in the form of a correspondence with Dositheus of Pelusium, who was a student of Conon of Samos. In Proposition II, Archimedes gives an approximation of the value of pi (π), showing that it is greater than 223/71 and less than 22/7.
The Sand Reckoner
Main article: The Sand Reckoner
In this treatise, also known as Psammites, Archimedes finds a number that is greater than the grains of sand needed to fill the universe. This book mentions the heliocentric theory of the solar system proposed by Aristarchus of Samos, as well as contemporary ideas about the size of the Earth and the distance between various celestial bodies. By using a system of numbers based on powers of the myriad, Archimedes concludes that the number of grains of sand required to fill the universe is 8×1063 in modern notation. The introductory letter states that Archimedes' father was an astronomer named Phidias. The Sand Reckoner is the only surviving work in which Archimedes discusses his views on astronomy.[80]
On the Equilibrium of Planes
There are two books to On the Equilibrium of Planes: the first contains seven postulates and fifteen propositions, while the second book contains ten propositions. In the first book, Archimedes proves the law of the lever, which states that:
Magnitudes are in equilibrium at distances reciprocally proportional to their weights.
Archimedes uses the principles derived to calculate the areas and centers of gravity of various geometric figures including triangles, parallelograms and parabolas.[81]
Quadrature of the Parabola
Main article: Quadrature of the Parabola
In this work of 24 propositions addressed to Dositheus, Archimedes proves by two methods that the area enclosed by a parabola and a straight line is 4/3 the area of a triangle with equal base and height. He achieves this in one of his proofs by calculating the value of a geometric series that sums to infinity with the ratio 1/4.
On the Sphere and Cylinder
Main article: On the Sphere and Cylinder
In this two-volume treatise addressed to Dositheus, Archimedes obtains the result of which he was most proud, namely the relationship between a sphere and a circumscribed cylinder of the same height and diameter. The volume is 4/3πr3 for the sphere, and 2πr3 for the cylinder. The surface area is 4πr2 for the sphere, and 6πr2 for the cylinder (including its two bases), where r is the radius of the sphere and cylinder.
On Spirals
Main article: On Spirals
This work of 28 propositions is also addressed to Dositheus. The treatise defines what is now called the Archimedean spiral. It is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity. Equivalently, in modern polar coordinates (r, θ), it can be described by the equation $\,r=a+b\theta $ with real numbers a and b.
This is an early example of a mechanical curve (a curve traced by a moving point) considered by a Greek mathematician.
On Conoids and Spheroids
This is a work in 32 propositions addressed to Dositheus. In this treatise Archimedes calculates the areas and volumes of sections of cones, spheres, and paraboloids.
On Floating Bodies
There are two books of On Floating Bodies. In the first book, Archimedes spells out the law of equilibrium of fluids and proves that water will adopt a spherical form around a center of gravity. This may have been an attempt at explaining the theory of contemporary Greek astronomers such as Eratosthenes that the Earth is round. The fluids described by Archimedes are not self-gravitating since he assumes the existence of a point towards which all things fall in order to derive the spherical shape. Archimedes' principle of buoyancy is given in this work, stated as follows:
Any body wholly or partially immersed in fluid experiences an upthrust equal to, but opposite in direction to, the weight of the fluid displaced.
In the second part, he calculates the equilibrium positions of sections of paraboloids. This was probably an idealization of the shapes of ships' hulls. Some of his sections float with the base under water and the summit above water, similar to the way that icebergs float.
Ostomachion
Main article: Ostomachion
Also known as Loculus of Archimedes or Archimedes' Box,[82] this is a dissection puzzle similar to a Tangram, and the treatise describing it was found in more complete form in the Archimedes Palimpsest. Archimedes calculates the areas of the 14 pieces which can be assembled to form a square. Reviel Netz of Stanford University argued in 2003 that Archimedes was attempting to determine how many ways the pieces could be assembled into the shape of a square. Netz calculates that the pieces can be made into a square 17,152 ways.[83] The number of arrangements is 536 when solutions that are equivalent by rotation and reflection are excluded.[84] The puzzle represents an example of an early problem in combinatorics.
The origin of the puzzle's name is unclear, and it has been suggested that it is taken from the Ancient Greek word for "throat" or "gullet", stomachos (στόμαχος).[85] Ausonius calls the puzzle Ostomachion, a Greek compound word formed from the roots of osteon (ὀστέον, 'bone') and machē (μάχη, 'fight').[82]
The cattle problem
Main article: Archimedes' cattle problem
Gotthold Ephraim Lessing discovered this work in a Greek manuscript consisting of a 44-line poem in the Herzog August Library in Wolfenbüttel, Germany in 1773. It is addressed to Eratosthenes and the mathematicians in Alexandria. Archimedes challenges them to count the numbers of cattle in the Herd of the Sun by solving a number of simultaneous Diophantine equations. There is a more difficult version of the problem in which some of the answers are required to be square numbers. A. Amthor first solved this version of the problem[86] in 1880, and the answer is a very large number, approximately 7.760271×10206544.[87]
The Method of Mechanical Theorems
Main article: The Method of Mechanical Theorems
This treatise was thought lost until the discovery of the Archimedes Palimpsest in 1906. In this work Archimedes uses indivisibles,[6][7] and shows how breaking up a figure into an infinite number of infinitely small parts can be used to determine its area or volume. He may have considered this method lacking in formal rigor, so he also used the method of exhaustion to derive the results. As with The Cattle Problem, The Method of Mechanical Theorems was written in the form of a letter to Eratosthenes in Alexandria.
Apocryphal works
Archimedes' Book of Lemmas or Liber Assumptorum is a treatise with 15 propositions on the nature of circles. The earliest known copy of the text is in Arabic. T. L. Heath and Marshall Clagett argued that it cannot have been written by Archimedes in its current form, since it quotes Archimedes, suggesting modification by another author. The Lemmas may be based on an earlier work by Archimedes that is now lost.[88]
It has also been claimed that the formula for calculating the area of a triangle from the length of its sides was known to Archimedes,[lower-alpha 4] though its first appearance is in the work of Heron of Alexandria in the 1st century AD.[89] Other questionable attributions to Archimedes' work include the Latin poem Carmen de ponderibus et mensuris (4th or 5th century), which describes the use of a hydrostatic balance to solve the problem of the crown, and the 12th-century text Mappae clavicula, which contains instructions on how to perform assaying of metals by calculating their specific gravities.[90][91]
Archimedes Palimpsest
Main article: Archimedes Palimpsest
The foremost document containing Archimedes' work is the Archimedes Palimpsest. In 1906, the Danish professor Johan Ludvig Heiberg visited Constantinople to examine a 174-page goatskin parchment of prayers, written in the 13th century, after reading a short transcription published seven years earlier by Papadopoulos-Kerameus.[92][93] He confirmed that it was indeed a palimpsest, a document with text that had been written over an erased older work. Palimpsests were created by scraping the ink from existing works and reusing them, a common practice in the Middle Ages, as vellum was expensive. The older works in the palimpsest were identified by scholars as 10th-century copies of previously lost treatises by Archimedes.[92][94] The parchment spent hundreds of years in a monastery library in Constantinople before being sold to a private collector in the 1920s. On 29 October 1998, it was sold at auction to an anonymous buyer for $2 million.[95]
The palimpsest holds seven treatises, including the only surviving copy of On Floating Bodies in the original Greek. It is the only known source of The Method of Mechanical Theorems, referred to by Suidas and thought to have been lost forever. Stomachion was also discovered in the palimpsest, with a more complete analysis of the puzzle than had been found in previous texts. The palimpsest was stored at the Walters Art Museum in Baltimore, Maryland, where it was subjected to a range of modern tests including the use of ultraviolet and X-ray light to read the overwritten text.[96] It has since returned to its anonymous owner.[97][98]
The treatises in the Archimedes Palimpsest include:
• On the Equilibrium of Planes
• On Spirals
• Measurement of a Circle
• On the Sphere and Cylinder
• On Floating Bodies
• The Method of Mechanical Theorems
• Stomachion
• Speeches by the 4th century BC politician Hypereides
• A commentary on Aristotle's Categories
• Other works
Legacy
Further information: List of things named after Archimedes and Eureka
Sometimes called the father of mathematics and mathematical physics, Archimedes had a wide influence on mathematics and science.[99]
Mathematics and physics
Historians of science and mathematics almost universally agree that Archimedes was the finest mathematician from antiquity. Eric Temple Bell, for instance, wrote:
Any list of the three “greatest” mathematicians of all history would include the name of Archimedes. The other two usually associated with him are Newton and Gauss. Some, considering the relative wealth—or poverty—of mathematics and physical science in the respective ages in which these giants lived, and estimating their achievements against the background of their times, would put Archimedes first.[100]
Likewise, Alfred North Whitehead and George F. Simmons said of Archimedes:
... in the year 1500 Europe knew less than Archimedes who died in the year 212 BC ...[101]
If we consider what all other men accomplished in mathematics and physics, on every continent and in every civilization, from the beginning of time down to the seventeenth century in Western Europe, the achievements of Archimedes outweighs it all. He was a great civilization all by himself.[102]
Reviel Netz, Suppes Professor in Greek Mathematics and Astronomy at Stanford University and an expert in Archimedes notes:
And so, since Archimedes led more than anyone else to the formation of the calculus and since he was the pioneer of the application of mathematics to the physical world, it turns out that Western science is but a series of footnotes to Archimedes. Thus, it turns out that Archimedes is the most important scientist who ever lived.[103]
Leonardo da Vinci repeatedly expressed admiration for Archimedes, and attributed his invention Architonnerre to Archimedes.[104][105][106] Galileo called him "superhuman" and "my master",[107][108] while Huygens said, "I think Archimedes is comparable to no one", consciously emulating him in his early work.[109] Leibniz said, "He who understands Archimedes and Apollonius will admire less the achievements of the foremost men of later times".[110] Gauss's heroes were Archimedes and Newton,[111] and Moritz Cantor, who studied under Gauss in the University of Göttingen, reported that he once remarked in conversation that "there had been only three epoch-making mathematicians: Archimedes, Newton, and Eisenstein".[112]
The inventor Nikola Tesla praised him, saying:
Archimedes was my ideal. I admired the works of artists, but to my mind, they were only shadows and semblances. The inventor, I thought, gives to the world creations which are palpable, which live and work.[113]
Honors and commemorations
There is a crater on the Moon named Archimedes (29.7°N 4.0°W / 29.7; -4.0) in his honor, as well as a lunar mountain range, the Montes Archimedes (25.3°N 4.6°W / 25.3; -4.6).[114]
The Fields Medal for outstanding achievement in mathematics carries a portrait of Archimedes, along with a carving illustrating his proof on the sphere and the cylinder. The inscription around the head of Archimedes is a quote attributed to 1st century AD poet Manilius, which reads in Latin: Transire suum pectus mundoque potiri ("Rise above oneself and grasp the world").[115][116][117]
Archimedes has appeared on postage stamps issued by East Germany (1973), Greece (1983), Italy (1983), Nicaragua (1971), San Marino (1982), and Spain (1963).[118]
The exclamation of Eureka! attributed to Archimedes is the state motto of California. In this instance, the word refers to the discovery of gold near Sutter's Mill in 1848 which sparked the California Gold Rush.[119]
See also
Concepts
• Arbelos
• Archimedean point
• Archimedes' axiom
• Archimedes number
• Archimedes paradox
• Archimedean solid
• Archimedes' twin circles
• Methods of computing square roots
• Salinon
• Steam cannon
• Trammel of Archimedes
People
• Diocles
• Pseudo-Archimedes
• Zhang Heng
References
Notes
1. Ancient Greek: Ἀρχιμήδης; Doric Greek: [ar.kʰi.mɛː.dɛ̂ːs]
2. In the preface to On Spirals addressed to Dositheus of Pelusium, Archimedes says that "many years have elapsed since Conon's death." Conon of Samos lived c. 280–220 BC, suggesting that Archimedes may have been an older man when writing some of his works.
3. The treatises by Archimedes known to exist only through references in the works of other authors are: On Sphere-Making and a work on polyhedra mentioned by Pappus of Alexandria; Catoptrica, a work on optics mentioned by Theon of Alexandria; Principles, addressed to Zeuxippus and explaining the number system used in The Sand Reckoner; On Balances or On Levers; On Centers of Gravity; On the Calendar.
4. Boyer, Carl Benjamin. 1991. A History of Mathematics. ISBN 978-0-471-54397-8: "Arabic scholars inform us that the familiar area formula for a triangle in terms of its three sides, usually known as Heron's formula — $k={\sqrt {s(s-a)(s-b)(s-c)}}$, where $s$ is the semiperimeter — was known to Archimedes several centuries before Heron lived. Arabic scholars also attribute to Archimedes the 'theorem on the broken chord' ... Archimedes is reported by the Arabs to have given several proofs of the theorem."
Citations
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42. Evans, James (1 August 1999). "The Material Culture of Greek Astronomy". Journal for the History of Astronomy. 30 (3): 238–307. Bibcode:1999JHA....30..237E. doi:10.1177/002182869903000305. ISSN 0021-8286. S2CID 120800329. Archived from the original on 14 July 2021. Retrieved 25 March 2021. But even before Hipparchus, Archimedes had described a similar instrument in his Sand-Reckoner. A fuller description of the same sort of instrument is given by Pappus of Alexandria ... Figure 30 is based on Archimedes and Pappus. Rod R has a groove that runs its whole length ... A cylinder or prism C is fixed to a small block that slides freely in the groove (p. 281).
43. Toomer, G. J.; Jones, Alexander (7 March 2016). "astronomical instruments". Oxford Research Encyclopedia of Classics. doi:10.1093/acrefore/9780199381135.013.886. ISBN 9780199381135. Archived from the original on 14 April 2021. Retrieved 25 March 2021. Perhaps the earliest instrument, apart from sundials, of which we have a detailed description is the device constructed by Archimedes (Sand-Reckoner 11-15) for measuring the sun's apparent diameter; this was a rod along which different coloured pegs could be moved.
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67. Heath, T.L. "Archimedes on measuring the circle". math.ubc.ca. Archived from the original on 3 July 2004. Retrieved 30 October 2012.
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72. Carroll, Bradley W. "The Sand Reckoner". Weber State University. Archived from the original on 13 August 2007. Retrieved 23 July 2007.
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74. Clagett, Marshall (1982). "William of Moerbeke: Translator of Archimedes". Proceedings of the American Philosophical Society. 126 (5): 356–366. ISSN 0003-049X. JSTOR 986212. Archived from the original on 8 March 2021. Retrieved 2 May 2021.
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77. Knorr, W. R. (1978). "Archimedes and the Elements: Proposal for a Revised Chronological Ordering of the Archimedean Corpus". Archive for History of Exact Sciences. 19 (3): 211–290. doi:10.1007/BF00357582. ISSN 0003-9519. JSTOR 41133526. S2CID 119774581. Archived from the original on 14 August 2021. Retrieved 14 August 2021.
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80. Heath, T.L. (1897). The Works of Archimedes (1897). The unabridged work in PDF form (19 MB). Cambridge University Press. Archived from the original on 6 October 2007. Retrieved 14 October 2007.
81. "Graeco Roman Puzzles". Gianni A. Sarcone and Marie J. Waeber. Archived from the original on 14 May 2008. Retrieved 9 May 2008.
82. Kolata, Gina (14 December 2003). "In Archimedes' Puzzle, a New Eureka Moment". The New York Times. Archived from the original on 14 July 2021. Retrieved 23 July 2007.
83. Ed Pegg Jr. (17 November 2003). "The Loculus of Archimedes, Solved". Mathematical Association of America. Archived from the original on 19 May 2008. Retrieved 18 May 2008.
84. Rorres, Chris. "Archimedes' Stomachion". Courant Institute of Mathematical Sciences. Archived from the original on 26 October 2007. Retrieved 14 September 2007.
85. Krumbiegel, B. and Amthor, A. Das Problema Bovinum des Archimedes, Historisch-literarische Abteilung der Zeitschrift für Mathematik und Physik 25 (1880) pp. 121–136, 153–171.
86. Calkins, Keith G. "Archimedes' Problema Bovinum". Andrews University. Archived from the original on 12 October 2007. Retrieved 14 September 2007.
87. "Archimedes' Book of Lemmas". cut-the-knot. Archived from the original on 11 July 2007. Retrieved 7 August 2007.
88. O'Connor, J.J.; Robertson, E.F. (April 1999). "Heron of Alexandria". University of St Andrews. Archived from the original on 9 May 2010. Retrieved 17 February 2010.
89. Dilke, Oswald A. W. 1990. [Untitled]. Gnomon 62(8):697–99. JSTOR 27690606.
90. Berthelot, Marcel. 1891. "Sur l histoire de la balance hydrostatique et de quelques autres appareils et procédés scientifiques." Annales de Chimie et de Physique 6(23):475–85.
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95. "X-rays reveal Archimedes' secrets". BBC News. 2 August 2006. Archived from the original on 25 August 2007. Retrieved 23 July 2007.
96. Piñar, G.; Sterflinger, K.; Ettenauer, J.; Quandt, A.; Pinzari, F. (2015). "A Combined Approach to Assess the Microbial Contamination of the Archimedes Palimpsest". Microbial Ecology. 69 (1): 118–134. doi:10.1007/s00248-014-0481-7. ISSN 1432-184X. PMC 4287661. PMID 25135817. Archived from the original on 22 April 2023. Retrieved 30 November 2021.
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• father of mathematics: Jane Muir, Of Men and Numbers: The Story of the Great Mathematicians, p 19.
• father of mathematical physics: James H. Williams Jr., Fundamentals of Applied Dynamics, p 30., Carl B. Boyer, Uta C. Merzbach, A History of Mathematics, p 111., Stuart Hollingdale, Makers of Mathematics, p 67., Igor Ushakov, In the Beginning, Was the Number (2), p 114.
98. E.T. Bell, Men of Mathematics, p 20.
99. Alfred North Whitehead. "The Influence of Western Medieval Culture Upon the Development of Modern Science". Archived from the original on 4 July 2022. Retrieved 4 April 2022.
100. George F. Simmons, Calculus Gems: Brief Lives and Memorable Mathematics, p 43.
101. Reviel Netz, William Noel, The Archimedes Codex: Revealing The Secrets Of The World's Greatest Palimpsest
102. "The Steam-Engine". Nelson Examiner and New Zealand Chronicle. Vol. I, no. 11. Nelson: National Library of New Zealand. 21 May 1842. p. 43. Archived from the original on 24 July 2011. Retrieved 14 February 2011.
103. The Steam Engine. The Penny Magazine. 1838. p. 104. Archived from the original on 7 May 2021. Retrieved 7 May 2021.
104. Robert Henry Thurston (1996). A History of the Growth of the Steam-Engine. Elibron. p. 12. ISBN 1-4021-6205-7. Archived from the original on 22 January 2021. Retrieved 7 May 2021.
105. Matthews, Michael. Time for Science Education: How Teaching the History and Philosophy of Pendulum Motion Can Contribute to Science Literacy. p. 96.
106. "Archimedes - Galileo Galilei and Archimedes". exhibits.museogalileo.it. Archived from the original on 17 April 2021. Retrieved 16 June 2021.
107. Yoder, J. (1996). "Following in the footsteps of geometry: the mathematical world of Christiaan Huygens". De Zeventiende Eeuw. Jaargang 12. Archived from the original on 12 May 2021.
108. Boyer, Carl B., and Uta C. Merzbach. 1968. A History of Mathematics. ch. 7.
109. Jay Goldman, The Queen of Mathematics: A Historically Motivated Guide to Number Theory, p 88.
110. E.T. Bell, Men of Mathematics, p 237
111. W. Bernard Carlson, Tesla: Inventor of the Electrical Age, p 57
112. Friedlander, Jay; Williams, Dave. "Oblique view of Archimedes crater on the Moon". NASA. Archived from the original on 19 August 2007. Retrieved 13 September 2007.
113. Riehm, C. (2002). "The early history of the Fields Medal" (PDF). Notices of the AMS. 49 (7): 778–782. Archived (PDF) from the original on 18 January 2021. Retrieved 28 April 2021. The Latin inscription from the Roman poet Manilius surrounding the image may be translated 'To pass beyond your understanding and make yourself master of the universe.' The phrase comes from Manilius's Astronomica 4.392 from the first century A.D. (p. 782).
114. "The Fields Medal". Fields Institute for Research in Mathematical Sciences. 5 February 2015. Archived from the original on 23 April 2021. Retrieved 23 April 2021.
115. "Fields Medal". International Mathematical Union. Archived from the original on 2 December 2017. Retrieved 23 April 2021.
116. Rorres, Chris. "Stamps of Archimedes". Courant Institute of Mathematical Sciences. Archived from the original on 2 October 2010. Retrieved 25 August 2007.
117. "California Symbols". California State Capitol Museum. Archived from the original on 12 October 2007. Retrieved 14 September 2007.
Further reading
Wikisource has the text of the 1911 Encyclopædia Britannica article "Archimedes".
• Boyer, Carl Benjamin. 1991. A History of Mathematics. New York: Wiley. ISBN 978-0-471-54397-8.
• Clagett, Marshall. 1964–1984. Archimedes in the Middle Ages 1–5. Madison, WI: University of Wisconsin Press.
• Dijksterhuis, Eduard J. [1938] 1987. Archimedes, translated. Princeton: Princeton University Press. ISBN 978-0-691-08421-3.
• Gow, Mary. 2005. Archimedes: Mathematical Genius of the Ancient World. Enslow Publishing. ISBN 978-0-7660-2502-8.
• Hasan, Heather. 2005. Archimedes: The Father of Mathematics. Rosen Central. ISBN 978-1-4042-0774-5.
• Heath, Thomas L. 1897. Works of Archimedes. Dover Publications. ISBN 978-0-486-42084-4. Complete works of Archimedes in English.
• Netz, Reviel, and William Noel. 2007. The Archimedes Codex. Orion Publishing Group. ISBN 978-0-297-64547-4.
• Pickover, Clifford A. 2008. Archimedes to Hawking: Laws of Science and the Great Minds Behind Them. Oxford University Press. ISBN 978-0-19-533611-5.
• Simms, Dennis L. 1995. Archimedes the Engineer. Continuum International Publishing Group. ISBN 978-0-7201-2284-8.
• Stein, Sherman. 1999. Archimedes: What Did He Do Besides Cry Eureka?. Mathematical Association of America. ISBN 978-0-88385-718-2.
External links
• Heiberg's Edition of Archimedes. Texts in Classical Greek, with some in English.
• Archimedes on In Our Time at the BBC
• Works by Archimedes at Project Gutenberg
• Works by or about Archimedes at Internet Archive
• Archimedes at the Indiana Philosophy Ontology Project
• Archimedes at PhilPapers
• The Archimedes Palimpsest project at The Walters Art Museum in Baltimore, Maryland
• "Archimedes and the Square Root of 3". MathPages.com.
• "Archimedes on Spheres and Cylinders". MathPages.com.
• Testing the Archimedes steam cannon Archived 29 March 2010 at the Wayback Machine
Archimedes
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|
Beta skeleton
In computational geometry and geometric graph theory, a β-skeleton or beta skeleton is an undirected graph defined from a set of points in the Euclidean plane. Two points p and q are connected by an edge whenever all the angles prq are sharper than a threshold determined from the numerical parameter β.
Circle-based definition
Let β be a positive real number, and calculate an angle θ using the formulas
$\theta ={\begin{cases}\sin ^{-1}{\frac {1}{\beta }},&{\text{if }}\beta \geq 1\\\pi -\sin ^{-1}{\beta },&{\text{if }}\beta \leq 1\end{cases}}$
For any two points p and q in the plane, let Rpq be the set of points for which angle prq is greater than θ. Then Rpq takes the form of a union of two open disks with diameter βd(p,q) for β ≥ 1 and θ ≤ π/2, and it takes the form of the intersection of two open disks with diameter d(p,q)/β for β ≤ 1 and θ ≥ π/2. When β = 1 the two formulas give the same value θ = π/2, and Rpq takes the form of a single open disk with pq as its diameter.
The β-skeleton of a discrete set S of points in the plane is the undirected graph that connects two points p and q with an edge pq whenever Rpq contains no points of S. That is, the β-skeleton is the empty region graph defined by the regions Rpq.[1] When S contains a point r for which angle prq is greater than θ, then pq is not an edge of the β-skeleton; the β-skeleton consists of those pairs pq for which no such point r exists.
Lune-based definition
Some authors use an alternative definition in which the empty regions Rpq for β > 1 are not unions of two disks but rather lenses (more often called in this context "lunes"), intersections of two congruent disks with diameter βd(pq), such that line segment pq lies on a radius of both disks and such that the points p and q both lie on the boundary of the intersection. As with the circle-based β-skeleton, the lune-based β-skeleton has an edge pq whenever region Rpq is empty of other input points. For this alternative definition, the relative neighborhood graph is a special case of a β-skeleton with β = 2. The two definitions coincide for β ≤ 1, and for larger values of β the circle-based skeleton is a subgraph of the lune-based skeleton.
One important difference between the circle-based and lune-based β-skeletons is that, for any point set that does not lie on a single line, there always exists a sufficiently large value of β such that the circle-based β-skeleton is the empty graph. In contrast, if a pair of points p and q has the property that, for all other points r, one of the two angles pqr and qpr is obtuse, then the lune-based β-skeleton will contain edge pq no matter how large β is.
History
β-skeletons were first defined by Kirkpatrick & Radke (1985) as a scale-invariant variation of the alpha shapes of Edelsbrunner, Kirkpatrick & Seidel (1983). The name, "β-skeleton", reflects the fact that in some sense the β-skeleton describes the shape of a set of points in the same way that a topological skeleton describes the shape of a two-dimensional region. Several generalizations of the β-skeleton to graphs defined by other empty regions have also been considered.[1][2]
Properties
If β varies continuously from 0 to ∞, the circle-based β-skeletons form a sequence of graphs extending from the complete graph to the empty graph. The special case β = 1 leads to the Gabriel graph, which is known to contain the Euclidean minimum spanning tree; therefore, the β-skeleton also contains the Gabriel graph and the minimum spanning tree whenever β ≤ 1.
For any constant β, a fractal construction resembling a flattened version of the Koch snowflake can be used to define a sequence of point sets whose β-skeletons are paths of arbitrarily large length within a unit square. Therefore, unlike the closely related Delaunay triangulation, β-skeletons have unbounded stretch factor and are not geometric spanners.[3]
Algorithms
A naïve algorithm that tests each triple p, q, and r for membership of r in the region Rpq can construct the β-skeleton of any set of n points in time O(n3).
When β ≥ 1, the β-skeleton (with either definition) is a subgraph of the Gabriel graph, which is a subgraph of the Delaunay triangulation. If pq is an edge of the Delaunay triangulation that is not an edge of the β-skeleton, then a point r that forms a large angle prq can be found as one of the at most two points forming a triangle with p and q in the Delaunay triangulation. Therefore, for these values of β, the circle-based β-skeleton for a set of n points can be constructed in time O(n log n) by computing the Delaunay triangulation and using this test to filter its edges.[2]
For β < 1, a different algorithm of Hurtado, Liotta & Meijer (2003) allows the construction of the β-skeleton in time O(n2). No better worst-case time bound is possible because, for any fixed value of β smaller than one, there exist point sets in general position (small perturbations of a regular polygon) for which the β-skeleton is a dense graph with a quadratic number of edges. In the same quadratic time bound, the entire β-spectrum (the sequence of circle-based β-skeletons formed by varying β) may also be calculated.
Applications
The circle-based β-skeleton may be used in image analysis to reconstruct the shape of a two-dimensional object, given a set of sample points on the boundary of the object (a computational form of the connect the dots puzzle where the sequence in which the dots are to be connected must be deduced by an algorithm rather than being given as part of the puzzle). Although, in general, this requires a choice of the value of the parameter β, it is possible to prove that the choice β = 1.7 will correctly reconstruct the entire boundary of any smooth surface, and not generate any edges that do not belong to the boundary, as long as the samples are generated sufficiently densely relative to the local curvature of the surface.[4] However in experimental testing a lower value, β = 1.2, was more effective for reconstructing street maps from sets of points marking the center lines of streets in a geographic information system.[5] For generalizations of the β-skeleton technique to reconstruction of surfaces in three dimensions, see Hiyoshi (2007).
Circle-based β-skeletons have been used to find subgraphs of the minimum weight triangulation of a point set: for sufficiently large values of β, every β-skeleton edge can be guaranteed to belong to the minimum weight triangulation. If the edges found in this way form a connected graph on all of the input points, then the remaining minimum weight triangulation edges may be found in polynomial time by dynamic programming. However, in general the minimum weight triangulation problem is NP-hard, and the subset of its edges found in this way may not be connected.[6]
β-skeletons have also been applied in machine learning to solve geometric classification problems[7] and in wireless ad hoc networks as a mechanism for controlling communication complexity by choosing a subset of the pairs of wireless stations that can communicate with each other.[8]
Notes
1. Cardinal, Collette & Langerman (2009).
2. Veltkamp (1992).
3. Eppstein (2002); Bose et al. (2002); Wang et al. (2003).
4. Amenta, Bern & Eppstein (1998); O'Rourke (2000).
5. Radke & Flodmark (1999).
6. Keil (1994); Cheng & Xu (2001).
7. Zhang & King (2002); Toussaint (2005).
8. Bhardwaj, Misra & Xue (2005).
References
• Amenta, Nina; Bern, Marshall; Eppstein, David (1998), "The crust and the beta-skeleton: combinatorial curve reconstruction", Graphical Models and Image Processing, 60/2 (2): 125–135, doi:10.1006/gmip.1998.0465, S2CID 6301659, archived from the original on 2006-03-22.
• Bhardwaj, Manvendu; Misra, Satyajayant; Xue, Guoliang (2005), "Distributed topology control in wireless ad hoc networks using ß-skeleton", Workshop on High Performance Switching and Routing (HPSR 2005), Hong Kong, China (PDF), archived from the original (PDF) on 2011-06-07.
• Bose, Prosenjit; Devroye, Luc; Evans, William; Kirkpatrick, David G. (2002), "On the spanning ratio of Gabriel graphs and β-skeletons", LATIN 2002: Theoretical Informatics, Lecture Notes in Computer Science, vol. 2286, Springer-Verlag, pp. 77–97, doi:10.1007/3-540-45995-2_42.
• Cardinal, Jean; Collette, Sébastian; Langerman, Stefan (2009), "Empty region graphs", Computational Geometry Theory & Applications, 42 (3): 183–195, doi:10.1016/j.comgeo.2008.09.003.
• Cheng, Siu-Wing; Xu, Yin-Feng (2001), "On β-skeleton as a subgraph of the minimum weight triangulation", Theoretical Computer Science, 262 (1–2): 459–471, doi:10.1016/S0304-3975(00)00318-2.
• Edelsbrunner, Herbert; Kirkpatrick, David G.; Seidel, Raimund (1983), "On the shape of a set of points in the plane", IEEE Transactions on Information Theory, 29 (4): 551–559, doi:10.1109/TIT.1983.1056714.
• Eppstein, David (2002), "Beta-skeletons have unbounded dilation", Computational Geometry Theory & Applications, 23 (1): 43–52, arXiv:cs.CG/9907031, doi:10.1016/S0925-7721(01)00055-4, S2CID 1617451.
• Hiyoshi, H. (2007), "Greedy beta-skeleton in three dimensions", Proc. 4th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2007), pp. 101–109, doi:10.1109/ISVD.2007.27, S2CID 23189942.
• Hurtado, Ferran; Liotta, Giuseppe; Meijer, Henk (2003), "Optimal and suboptimal robust algorithms for proximity graphs", Computational Geometry Theory & Applications, 25 (1–2): 35–49, doi:10.1016/S0925-7721(02)00129-3.
• Keil, J. Mark (1994), "Computing a subgraph of the minimum weight triangulation", Computational Geometry Theory & Applications, 4 (1): 18–26, doi:10.1016/0925-7721(94)90014-0.
• Kirkpatrick, David G.; Radke, John D. (1985), "A framework for computational morphology", Computational Geometry, Machine Intelligence and Pattern Recognition, vol. 2, Amsterdam: North-Holland, pp. 217–248.
• O'Rourke, Joseph (2000), "Computational Geometry Column 38", SIGACT News, 31 (1): 28–30, arXiv:cs.CG/0001025, doi:10.1145/346048.346050.
• Radke, John D.; Flodmark, Anders (1999), "The use of spatial decompositions for constructing street centerlines" (PDF), Geographic Information Sciences, 5 (1): 15–23.
• Toussaint, Godfried (2005), "Geometric proximity graphs for improving nearest neighbor methods in instance-based learning and data mining", International Journal of Computational Geometry and Applications, 15 (2): 101–150, doi:10.1142/S0218195905001622.
• Veltkamp, Remko C. (1992), "The γ-neighborhood graph" (PDF), Computational Geometry Theory & Applications, 1 (4): 227–246, doi:10.1016/0925-7721(92)90003-B.
• Wang, Weizhao; Li, Xiang-Yang; Moaveninejad, Kousha; Wang, Yu; Song, Wen-Zhan (2003), "The spanning ratio of β-skeletons", Proc. 15th Canadian Conference on Computational Geometry (CCCG 2003) (PDF), pp. 35–38.
• Zhang, Wan; King, Irwin (2002), "Locating support vectors via β-skeleton technique", Proc. Proceedings of the 9th International Conference on Neural Information Processing (ICONIP'02), Orchid Country Club, Singapore, November 18-22, 2002 (PDF), pp. 1423–1427.
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γ-space
In mathematics, a $\gamma $-space is a topological space that satisfies a certain a basic selection principle. An infinite cover of a topological space is an $\omega $-cover if every finite subset of this space is contained in some member of the cover, and the whole space is not a member the cover. A cover of a topological space is a $\gamma $-cover if every point of this space belongs to all but finitely many members of this cover. A $\gamma $-space is a space in which every open $\omega $-cover contains a $\gamma $-cover.
History
Gerlits and Nagy introduced the notion of γ-spaces.[1] They listed some topological properties and enumerated them by Greek letters. The above property was the third one on this list, and therefore it is called the γ-property.
Characterizations
Combinatorial characterization
Let $[\mathbb {N} ]^{\infty }$ be the set of all infinite subsets of the set of natural numbers. A set $A\subset [\mathbb {N} ]^{\infty }$is centered if the intersection of finitely many elements of $A$ is infinite. Every set $a\in [\mathbb {N} ]^{\infty }$we identify with its increasing enumeration, and thus the set $a$ we can treat as a member of the Baire space $\mathbb {N} ^{\mathbb {N} }$. Therefore, $[\mathbb {N} ]^{\infty }$is a topological space as a subspace of the Baire space $\mathbb {N} ^{\mathbb {N} }$. A zero-dimensional separable metric space is a γ-space if and only if every continuous image of that space into the space $[\mathbb {N} ]^{\infty }$that is centered has a pseudointersection.[2]
Topological game characterization
Let $X$ be a topological space. The $\gamma $-has a pseudo intersection if there is a set game played on $X$ is a game with two players Alice and Bob.
1st round: Alice chooses an open $\omega $-cover ${\mathcal {U}}_{1}$ of $X$. Bob chooses a set $U_{1}\in {\mathcal {U}}_{1}$.
2nd round: Alice chooses an open $\omega $-cover ${\mathcal {U}}_{2}$ of $X$. Bob chooses a set $U_{2}\in {\mathcal {U}}_{2}$.
etc.
If $\{U_{n}:n\in \mathbb {N} \}$ is a $\gamma $-cover of the space $X$, then Bob wins the game. Otherwise, Alice wins.
A player has a winning strategy if he knows how to play in order to win the game (formally, a winning strategy is a function).
A topological space is a $\gamma $-space iff Alice has no winning strategy in the $\gamma $-game played on this space.[1]
Properties
• A topological space is a γ-space if and only if it satisfies ${\text{S}}_{1}(\Omega ,\Gamma )$ selection principle.[1]
• Every Lindelöf space of cardinality less than the pseudointersection number ${\mathfrak {p}}$ is a $\gamma $-space.
• Every $\gamma $-space is a Rothberger space,[3] and thus it has strong measure zero.
• Let $X$ be a Tychonoff space, and $C(X)$ be the space of continuous functions $f\colon X\to \mathbb {R} $ with pointwise convergence topology. The space $X$ is a $\gamma $-space if and only if $C(X)$ is Fréchet–Urysohn if and only if $C(X)$ is strong Fréchet–Urysohn.[1]
• Let $A$ be a ${\binom {\mathbf {\Omega } }{\mathbf {\Gamma } }}$ subset of the real line, and $M$ be a meager subset of the real line. Then the set $A+M=\{a+x:a\in A,x\in M\}$ is meager.[4]
References
1. Gerlits, J.; Nagy, Zs. (1982). "Some properties of $C(X)$, I". Topology and Its Applications. 14 (2): 151–161. doi:10.1016/0166-8641(82)90065-7.
2. Recław, Ireneusz (1994). "Every Lusin set is undetermined in the point-open game". Fundamenta Mathematicae. 144: 43–54. doi:10.4064/fm-144-1-43-54.
3. Scheepers, Marion (1996). "Combinatorics of open covers I: Ramsey theory". Topology and Its Applications. 69: 31–62. doi:10.1016/0166-8641(95)00067-4.
4. Galvin, Fred; Miller, Arnold (1984). "$\gamma $-sets and other singular sets of real numbers". Topology and Its Applications. 17 (2): 145–155. doi:10.1016/0166-8641(84)90038-5.
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p-adic gamma function
In mathematics, the p-adic gamma function Γp is a function of a p-adic variable analogous to the gamma function. It was first explicitly defined by Morita (1975), though Boyarsky (1980) pointed out that Dwork (1964) implicitly used the same function. Diamond (1977) defined a p-adic analog Gp of log Γ. Overholtzer (1952) had previously given a definition of a different p-adic analogue of the gamma function, but his function does not have satisfactory properties and is not used much.
Definition
The p-adic gamma function is the unique continuous function of a p-adic integer x (with values in $\mathbb {Z} _{p}$) such that
$\Gamma _{p}(x)=(-1)^{x}\prod _{0<i<x,\ p\,\nmid \,i}i$
for positive integers x, where the product is restricted to integers i not divisible by p. As the positive integers are dense with respect to the p-adic topology in $\mathbb {Z} _{p}$, $\Gamma _{p}(x)$ can be extended uniquely to the whole of $\mathbb {Z} _{p}$. Here $\mathbb {Z} _{p}$ is the ring of p-adic integers. It follows from the definition that the values of $\Gamma _{p}(\mathbb {Z} )$ are invertible in $\mathbb {Z} _{p}$; this is because these values are products of integers not divisible by p, and this property holds after the continuous extension to $\mathbb {Z} _{p}$. Thus $\Gamma _{p}:\mathbb {Z} _{p}\to \mathbb {Z} _{p}^{\times }$. Here $\mathbb {Z} _{p}^{\times }$ is the set of invertible p-adic integers.
Basic properties of the p-adic gamma function
The classical gamma function satisfies the functional equation $\Gamma (x+1)=x\Gamma (x)$ for any $x\in \mathbb {C} \setminus \mathbb {Z} _{\leq 0}$. This has an analogue with respect to the Morita gamma function:
${\frac {\Gamma _{p}(x+1)}{\Gamma _{p}(x)}}={\begin{cases}-x,&{\mbox{if }}x\in \mathbb {Z} _{p}^{\times }\\-1,&{\mbox{if }}x\in p\mathbb {Z} _{p}.\end{cases}}$
The Euler's reflection formula $\Gamma (x)\Gamma (1-x)={\frac {\pi }{\sin {(\pi x)}}}$ has its following simple counterpart in the p-adic case:
$\Gamma _{p}(x)\Gamma _{p}(1-x)=(-1)^{x_{0}},$
where $x_{0}$ is the first digit in the p-adic expansion of x, unless $x\in p\mathbb {Z} _{p}$, in which case $x_{0}=p$ rather than 0.
Special values
$\Gamma _{p}(0)=1,$
$\Gamma _{p}(1)=-1,$
$\Gamma _{p}(2)=1,$
$\Gamma _{p}(3)=-2,$
and, in general,
$\Gamma _{p}(n+1)={\frac {(-1)^{n+1}n!}{[n/p]!p^{[n/p]}}}\quad (n\geq 2).$
At $x={\frac {1}{2}}$ the Morita gamma function is related to the Legendre symbol $\left({\frac {a}{p}}\right)$:
$\Gamma _{p}\left({\frac {1}{2}}\right)^{2}=-\left({\frac {-1}{p}}\right).$
It can also be seen, that $\Gamma _{p}(p^{n})\equiv 1{\pmod {p^{n}}},$ hence $\Gamma _{p}(p^{n})\to 1$ as $n\to \infty $.[1]: 369
Other interesting special values come from the Gross–Koblitz formula, which was first proved by cohomological tools, and later was proved using more elementary methods.[2] For example,
$\Gamma _{5}\left({\frac {1}{4}}\right)^{2}=-2+{\sqrt {-1}},$
$\Gamma _{7}\left({\frac {1}{3}}\right)^{3}={\frac {1-3{\sqrt {-3}}}{2}},$
where ${\sqrt {-1}}\in \mathbb {Z} _{5}$ denotes the square root with first digit 3, and ${\sqrt {-3}}\in \mathbb {Z} _{7}$ denotes the square root with first digit 2. (Such specifications must always be done if we talk about roots.)
Another example is
$\Gamma _{3}\left({\frac {1}{8}}\right)\Gamma _{3}\left({\frac {3}{8}}\right)=-(1+{\sqrt {-2}}),$
where ${\sqrt {-2}}$ is the square root of $-2$ in $\mathbb {Q} _{3}$ congruent to 1 modulo 3.[3]
p-adic Raabe formula
The Raabe-formula for the classical Gamma function says that
$\int _{0}^{1}\log \Gamma (x+t)dt={\frac {1}{2}}\log(2\pi )+x\log x-x.$
This has an analogue for the Iwasawa logarithm of the Morita gamma function:[4]
$\int _{\mathbb {Z} _{p}}\log \Gamma _{p}(x+t)dt=(x-1)(\log \Gamma _{p})'(x)-x+\left\lceil {\frac {x}{p}}\right\rceil \quad (x\in \mathbb {Z} _{p}).$
The ceiling function to be understood as the p-adic limit $\lim _{n\to \infty }\left\lceil {\frac {x_{n}}{p}}\right\rceil $ such that $x_{n}\to x$ through rational integers.
Mahler expansion
The Mahler expansion is similarly important for p-adic functions as the Taylor expansion in classical analysis. The Mahler expansion of the p-adic gamma function is the following:[1]: 374
$\Gamma _{p}(x+1)=\sum _{k=0}^{\infty }a_{k}{\binom {x}{k}},$
where the sequence $a_{k}$ is defined by the following identity:
$\sum _{k=0}^{\infty }(-1)^{k+1}a_{k}{\frac {x^{k}}{k!}}={\frac {1-x^{p}}{1-x}}\exp \left(x+{\frac {x^{p}}{p}}\right).$
See also
• Gross–Koblitz formula
References
• Boyarsky, Maurizio (1980), "p-adic gamma functions and Dwork cohomology", Transactions of the American Mathematical Society, 257 (2): 359–369, doi:10.2307/1998301, ISSN 0002-9947, JSTOR 1998301, MR 0552263
• Diamond, Jack (1977), "The p-adic log gamma function and p-adic Euler constants", Transactions of the American Mathematical Society, 233: 321–337, doi:10.2307/1997840, ISSN 0002-9947, JSTOR 1997840, MR 0498503
• Diamond, Jack (1984), "p-adic gamma functions and their applications", in Chudnovsky, David V.; Chudnovsky, Gregory V.; Cohn, Henry; et al. (eds.), Number theory (New York, 1982), Lecture Notes in Math., vol. 1052, Berlin, New York: Springer-Verlag, pp. 168–175, doi:10.1007/BFb0071542, ISBN 978-3-540-12909-7, MR 0750664
• Dwork, Bernard (1964), "On the zeta function of a hypersurface. II", Annals of Mathematics, Second Series, 80 (2): 227–299, doi:10.2307/1970392, ISSN 0003-486X, JSTOR 1970392, MR 0188215
• Morita, Yasuo (1975), "A p-adic analogue of the Γ-function", Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics, 22 (2): 255–266, hdl:2261/6494, ISSN 0040-8980, MR 0424762
• Overholtzer, Gordon (1952), "Sum functions in elementary p-adic analysis", American Journal of Mathematics, 74 (2): 332–346, doi:10.2307/2371998, ISSN 0002-9327, JSTOR 2371998, MR 0048493
1. Robert, Alain M. (2000). A course in p-adic analysis. New York: Springer-Verlag.
2. Robert, Alain M. (2001). "The Gross-Koblitz formula revisited". Rendiconti del Seminario Matematico della Università di Padova. The Mathematical Journal of the University of Padova. 105: 157–170. doi:10.1016/j.jnt.2009.08.005. hdl:2437/90539. ISSN 0041-8994. MR 1834987.
3. Cohen, H. (2007). Number Theory. Vol. 2. New York: Springer Science+Business Media. p. 406.
4. Cohen, Henri; Eduardo, Friedman (2008). "Raabe's formula for p-adic gamma and zeta functions". Annales de l'Institut Fourier. 88 (1): 363–376. doi:10.5802/aif.2353. hdl:10533/139530. MR 2401225.
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Delta-convergence
In mathematics, Delta-convergence, or Δ-convergence, is a mode of convergence in metric spaces, weaker than the usual metric convergence, and similar to (but distinct from) the weak convergence in Banach spaces. In Hilbert space, Delta-convergence and weak convergence coincide. For a general class of spaces, similarly to weak convergence, every bounded sequence has a Delta-convergent subsequence. Delta convergence was first introduced by Teck-Cheong Lim,[1] and, soon after, under the name of almost convergence, by Tadeusz Kuczumow.[2]
Definition
A sequence $(x_{k})$ in a metric space $(X,d)$ is said to be Δ-convergent to $x\in X$ if for every $y\in X$, $\limsup(d(x_{k},x)-d(x_{k},y))\leq 0$.
Characterization in Banach spaces
If $X$ is a uniformly convex and uniformly smooth Banach space, with the duality mapping $x\mapsto x^{*}$ given by $\|x\|=\|x^{*}\|$, $\langle x^{*},x\rangle =\|x\|^{2}$, then a sequence $(x_{k})\subset X$ is Delta-convergent to $x$ if and only if $(x_{k}-x)^{*}$ converges to zero weakly in the dual space $X^{*}$ (see [3]). In particular, Delta-convergence and weak convergence coincide if $X$ is a Hilbert space.
Opial property
Coincidence of weak convergence and Delta-convergence is equivalent, for uniformly convex Banach spaces, to the well-known Opial property[3]
Delta-compactness theorem
The Delta-compactness theorem of T. C. Lim[1] states that if $(X,d)$ is an asymptotically complete metric space, then every bounded sequence in $X$ has a Delta-convergent subsequence.
The Delta-compactness theorem is similar to the Banach–Alaoglu theorem for weak convergence but, unlike the Banach-Alaoglu theorem (in the non-separable case) its proof does not depend on the Axiom of Choice.
Asymptotic center and asymptotic completeness
An asymptotic center of a sequence $(x_{k})_{k\in \mathbb {N} }$, if it exists, is a limit of the Chebyshev centers $c_{n}$ for truncated sequences $(x_{k})_{k\geq n}$. A metric space is called asymptotically complete, if any bounded sequence in it has an asymptotic center.
Uniform convexity as sufficient condition of asymptotic completeness
Condition of asymptotic completeness in the Delta-compactness theorem is satisfied by uniformly convex Banach spaces, and more generally, by uniformly rotund metric spaces as defined by J. Staples.[4]
Further reading
• William Kirk, Naseer Shahzad, Fixed point theory in distance spaces. Springer, Cham, 2014. xii+173 pp.
• G. Devillanova, S. Solimini, C. Tintarev, On weak convergence in metric spaces, Nonlinear Analysis and Optimization (B. S. Mordukhovich, S. Reich, A. J. Zaslavski, Editors), 43–64, Contemporary Mathematics 659, AMS, Providence, RI, 2016.
References
1. T.C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976), 179–182.
2. T. Kuczumow, An almost convergence and its applications, Ann. Univ. Mariae Curie-Sklodowska Sect. A 32 (1978), 79–88.
3. S. Solimini, C. Tintarev, Concentration analysis in Banach spaces, Comm. Contemp. Math. 2015, DOI 10.1142/S0219199715500388
4. J. Staples, Fixed point theorems in uniformly rotund metric spaces, Bull. Austral. Math. Soc. 14 (1976), 181–192.
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Delta-functor
In homological algebra, a δ-functor between two abelian categories A and B is a collection of functors from A to B together with a collection of morphisms that satisfy properties generalising those of derived functors. A universal δ-functor is a δ-functor satisfying a specific universal property related to extending morphisms beyond "degree 0". These notions were introduced by Alexander Grothendieck in his "Tohoku paper" to provide an appropriate setting for derived functors.[1] In particular, derived functors are universal δ-functors.
Not to be confused with Delta-function.
The terms homological δ-functor and cohomological δ-functor are sometimes used to distinguish between the case where the morphisms "go down" (homological) and the case where they "go up" (cohomological). In particular, one of these modifiers is always implicit, although often left unstated.
Definition
Given two abelian categories A and B a covariant cohomological δ-functor between A and B is a family {Tn} of covariant additive functors Tn : A → B indexed by the non-negative integers, and for each short exact sequence
$0\rightarrow M^{\prime }\rightarrow M\rightarrow M^{\prime \prime }\rightarrow 0$
a family of morphisms
$\delta ^{n}:T^{n}(M^{\prime \prime })\rightarrow T^{n+1}(M^{\prime })$
indexed by the non-negative integers satisfying the following two properties:
1. For each short exact sequence as above, there is a long exact sequence
2. For each morphism of short exact sequences
and for each non-negative n, the induced square
is commutative (the δn on the top is that corresponding to the short exact sequence of M's whereas the one on the bottom corresponds to the short exact sequence of N's).
The second property expresses the functoriality of a δ-functor. The modifier "cohomological" indicates that the δn raise the index on the T. A covariant homological δ-functor between A and B is similarly defined (and generally uses subscripts), but with δn a morphism Tn(M '') → Tn-1(M'). The notions of contravariant cohomological δ-functor between A and B and contravariant homological δ-functor between A and B can also be defined by "reversing the arrows" accordingly.
Morphisms of δ-functors
A morphism of δ-functors is a family of natural transformations that, for each short exact sequence, commute with the morphisms δ. For example, in the case of two covariant cohomological δ-functors denoted S and T, a morphism from S to T is a family Fn : Sn → Tn of natural transformations such that for every short exact sequence
$0\rightarrow M^{\prime }\rightarrow M\rightarrow M^{\prime \prime }\rightarrow 0$
the following diagram commutes:
Universal δ-functor
A universal δ-functor is characterized by the (universal) property that giving a morphism from it to any other δ-functor (between A and B) is equivalent to giving just F0. If S denotes a covariant cohomological δ-functor between A and B, then S is universal if given any other (covariant cohomological) δ-functor T (between A and B), and given any natural transformation
$F_{0}:S^{0}\rightarrow T^{0}$
there is a unique sequence Fn indexed by the positive integers such that the family { Fn }n ≥ 0 is a morphism of δ-functors.
See also
• Effaceable functor
Notes
1. Grothendieck 1957
References
• Grothendieck, Alexander (1957), "Sur quelques points d'algèbre homologique", The Tohoku Mathematical Journal, Second Series, 9 (2–3), MR 0102537
• Section XX.7 of Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001
• Section 2.1 of Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press. ISBN 978-0-521-55987-4. MR 1269324. OCLC 36131259.
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ΔP
ΔP (Delta P) is a mathematical term symbolizing a change (Δ) in pressure (P).
Uses
• Young–Laplace equation
Darcy–Weisbach equation
Given that the head loss hf expresses the pressure loss Δp as the height of a column of fluid,
$\Delta p=\rho \cdot g\cdot h_{f}$
where ρ is the density of the fluid. The Darcy–Weisbach equation can also be written in terms of pressure loss:
$\Delta p=f\cdot {\frac {L}{D}}\cdot {\frac {\rho V^{2}}{2}}$
Lung compliance
In general, compliance is defined by the change in volume (ΔV) versus the associated change in pressure (ΔP), or ΔV/ΔP:
$Compliance={\frac {\Delta V}{\Delta P}}$
During mechanical ventilation, compliance is influenced by three main physiologic factors:
1. Lung compliance
2. Chest wall compliance
3. Airway resistance
Lung compliance is influenced by a variety of primary abnormalities of lung parenchyma, both chronic and acute. Airway resistance is typically increased by bronchospasm and airway secretions. Chest wall compliance can be decreased by fixed abnormalities (e.g. kyphoscoliosis, morbid obesity) or more variable problems driven by patient agitation while intubated.[1]
Calculating compliance on minute volume (VE: ΔV is always defined by tidal volume (VT), but ΔP is different for the measurement of dynamic vs. static compliance.
Dynamic compliance (Cdyn)
$C_{dyn}={\frac {V_{T}}{\mathrm {PIP-PEEP} }}$
where PIP = peak inspiratory pressure (the maximum pressure during inspiration), and PEEP = positive end expiratory pressure. Alterations in airway resistance, lung compliance and chest wall compliance influence Cdyn.
Static compliance (Cstat)
$C_{stat}={\frac {V_{T}}{P_{plat}-PEEP}}$
where Pplat = plateau pressure. Pplat is measured at the end of inhalation and prior to exhalation using an inspiratory hold maneuver. During this maneuver, airflow is transiently (~0.5 sec) discontinued, which eliminates the effects of airway resistance. Pplat is never > PIP and is typically < 3-5 cmH2O lower than PIP when airway resistance is normal.
See also
• Pressure measurement
• Pressure drop
• Head loss
References
1. Dellamonica J, Lerolle N, Sargentini C, Beduneau G, Di Marco F, Mercat A, et al. (2011). "PEEP-induced changes in lung volume in acute respiratory distress syndrome. Two methods to estimate alveolar recruitment". Intensive Care Med. 37 (10): 1595–604. doi:10.1007/s00134-011-2333-y. PMID 21866369. S2CID 36231036.
External links
• Delta P, Diving Pressure Hazard
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ε-quadratic form
In mathematics, specifically the theory of quadratic forms, an ε-quadratic form is a generalization of quadratic forms to skew-symmetric settings and to *-rings; ε = ±1, accordingly for symmetric or skew-symmetric. They are also called $(-)^{n}$-quadratic forms, particularly in the context of surgery theory.
There is the related notion of ε-symmetric forms, which generalizes symmetric forms, skew-symmetric forms (= symplectic forms), Hermitian forms, and skew-Hermitian forms. More briefly, one may refer to quadratic, skew-quadratic, symmetric, and skew-symmetric forms, where "skew" means (−) and the * (involution) is implied.
The theory is 2-local: away from 2, ε-quadratic forms are equivalent to ε-symmetric forms: half the symmetrization map (below) gives an explicit isomorphism.
Definition
ε-symmetric forms and ε-quadratic forms are defined as follows.[1]
Given a module M over a *-ring R, let B(M) be the space of bilinear forms on M, and let T : B(M) → B(M) be the "conjugate transpose" involution B(u, v) ↦ B(v, u)*. Since multiplication by −1 is also an involution and commutes with linear maps, −T is also an involution. Thus we can write ε = ±1 and εT is an involution, either T or −T (ε can be more general than ±1; see below). Define the ε-symmetric forms as the invariants of εT, and the ε-quadratic forms are the coinvariants.
As an exact sequence,
$0\to Q^{\varepsilon }(M)\to B(M){\stackrel {1-\varepsilon T}{\longrightarrow }}B(M)\to Q_{\varepsilon }(M)\to 0$
As kernel and cokernel,
$Q^{\varepsilon }(M):={\mbox{ker}}\,(1-\varepsilon T)$
$Q_{\varepsilon }(M):={\mbox{coker}}\,(1-\varepsilon T)$
The notation Qε(M), Qε(M) follows the standard notation MG, MG for the invariants and coinvariants for a group action, here of the order 2 group (an involution).
Composition of the inclusion and quotient maps (but not 1 − εT) as $Q^{\varepsilon }(M)\to B(M)\to Q_{\varepsilon }(M)$ yields a map Qε(M) → Qε(M): every ε-symmetric form determines an ε-quadratic form.
Symmetrization
Conversely, one can define a reverse homomorphism "1 + εT": Qε(M) → Qε(M), called the symmetrization map (since it yields a symmetric form) by taking any lift of a quadratic form and multiplying it by 1 + εT. This is a symmetric form because (1 − εT)(1 + εT) = 1 − T2 = 0, so it is in the kernel. More precisely, $(1+\varepsilon T)B(M)<Q^{\varepsilon }(M)$. The map is well-defined by the same equation: choosing a different lift corresponds to adding a multiple of (1 − εT), but this vanishes after multiplying by 1 + εT. Thus every ε-quadratic form determines an ε-symmetric form.
Composing these two maps either way: Qε(M) → Qε(M) → Qε(M) or Qε(M) → Qε(M) → Qε(M) yields multiplication by 2, and thus these maps are bijective if 2 is invertible in R, with the inverse given by multiplication with 1/2.
An ε-quadratic form ψ ∈ Qε(M) is called non-degenerate if the associated ε-symmetric form (1 + εT)(ψ) is non-degenerate.
Generalization from *
If the * is trivial, then ε = ±1, and "away from 2" means that 2 is invertible: 1/2 ∈ R.
More generally, one can take for ε ∈ R any element such that ε*ε = 1. ε = ±1 always satisfy this, but so does any element of norm 1, such as complex numbers of unit norm.
Similarly, in the presence of a non-trivial *, ε-symmetric forms are equivalent to ε-quadratic forms if there is an element λ ∈ R such that λ* + λ = 1. If * is trivial, this is equivalent to 2λ = 1 or λ = 1/2, while if * is non-trivial there can be multiple possible λ; for example, over the complex numbers any number with real part 1/2 is such a λ.
For instance, in the ring $R=\mathbf {Z} \left[\textstyle {\frac {1+i}{2}}\right]$ (the integral lattice for the quadratic form 2x2 − 2x + 1), with complex conjugation, $\lambda =\textstyle {\frac {1\pm i}{2}}$ are two such elements, though 1/2 ∉ R.
Intuition
In terms of matrices (we take V to be 2-dimensional), if * is trivial:
• matrices ${\begin{pmatrix}a&b\\c&d\end{pmatrix}}$ correspond to bilinear forms
• the subspace of symmetric matrices ${\begin{pmatrix}a&b\\b&c\end{pmatrix}}$ correspond to symmetric forms
• the subspace of (−1)-symmetric matrices ${\begin{pmatrix}0&b\\-b&0\end{pmatrix}}$ correspond to symplectic forms
• the bilinear form ${\begin{pmatrix}a&b\\c&d\end{pmatrix}}$ yields the quadratic form
$ax^{2}+bxy+cyx+dy^{2}=ax^{2}+(b+c)xy+dy^{2}\,$,
• the map 1 + T from quadratic forms to symmetric forms maps $ex^{2}+fxy+gy^{2}$
to ${\begin{pmatrix}2e&f\\f&2g\end{pmatrix}}$, for example by lifting to ${\begin{pmatrix}e&f\\0&g\end{pmatrix}}$ and then adding to transpose. Mapping back to quadratic forms yields double the original: $2ex^{2}+2fxy+2gy^{2}=2(ex^{2}+fxy+gy^{2})$.
If ${\bar {\cdot }}$ is complex conjugation, then
• the subspace of symmetric matrices are the Hermitian matrices ${\begin{pmatrix}a&z\\{\bar {z}}&c\end{pmatrix}}$
• the subspace of skew-symmetric matrices are the skew-Hermitian matrices ${\begin{pmatrix}bi&z\\-{\bar {z}}&di\end{pmatrix}}$
Refinements
An intuitive way to understand an ε-quadratic form is to think of it as a quadratic refinement of its associated ε-symmetric form.
For instance, in defining a Clifford algebra over a general field or ring, one quotients the tensor algebra by relations coming from the symmetric form and the quadratic form: vw + wv = 2B(v, w) and $v^{2}=Q(v)$. If 2 is invertible, this second relation follows from the first (as the quadratic form can be recovered from the associated bilinear form), but at 2 this additional refinement is necessary.
Examples
An easy example for an ε-quadratic form is the standard hyperbolic ε-quadratic form $H_{\varepsilon }(R)\in Q_{\varepsilon }(R\oplus R^{*})$. (Here, R* := HomR(R, R) denotes the dual of the R-module R.) It is given by the bilinear form $((v_{1},f_{1}),(v_{2},f_{2}))\mapsto f_{2}(v_{1})$. The standard hyperbolic ε-quadratic form is needed for the definition of L-theory.
For the field of two elements R = F2 there is no difference between (+1)-quadratic and (−1)-quadratic forms, which are just called quadratic forms. The Arf invariant of a nonsingular quadratic form over F2 is an F2-valued invariant with important applications in both algebra and topology, and plays a role similar to that played by the discriminant of a quadratic form in characteristic not equal to two.
Manifolds
Further information: Intersection product
The free part of the middle homology group (with integer coefficients) of an oriented even-dimensional manifold has an ε-symmetric form, via Poincaré duality, the intersection form. In the case of singly even dimension 4k + 2, this is skew-symmetric, while for doubly even dimension 4k, this is symmetric. Geometrically this corresponds to intersection, where two n/2-dimensional submanifolds in an n-dimensional manifold generically intersect in a 0-dimensional submanifold (a set of points), by adding codimension. For singly even dimension the order switches sign, while for doubly even dimension order does not change sign, hence the ε-symmetry. The simplest cases are for the product of spheres, where the product S2k × S2k and S2k+1 × S2k+1 respectively give the symmetric form $\left({\begin{smallmatrix}0&1\\1&0\end{smallmatrix}}\right)$ and skew-symmetric form $\left({\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}}\right).$ In dimension two, this yields a torus, and taking the connected sum of g tori yields the surface of genus g, whose middle homology has the standard hyperbolic form.
With additional structure, this ε-symmetric form can be refined to an ε-quadratic form. For doubly even dimension this is integer valued, while for singly even dimension this is only defined up to parity, and takes values in Z/2. For example, given a framed manifold, one can produce such a refinement. For singly even dimension, the Arf invariant of this skew-quadratic form is the Kervaire invariant.
Given an oriented surface Σ embedded in R3, the middle homology group H1(Σ) carries not only a skew-symmetric form (via intersection), but also a skew-quadratic form, which can be seen as a quadratic refinement, via self-linking. The skew-symmetric form is an invariant of the surface Σ, whereas the skew-quadratic form is an invariant of the embedding Σ ⊂ R3, e.g. for the Seifert surface of a knot. The Arf invariant of the skew-quadratic form is a framed cobordism invariant generating the first stable homotopy group $\pi _{1}^{s}$.
For the standard embedded torus, the skew-symmetric form is given by $\left({\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}}\right)$ (with respect to the standard symplectic basis), and the skew-quadratic refinement is given by xy with respect to this basis: Q(1, 0) = Q(0, 1) = 0: the basis curves don't self-link; and Q(1, 1) = 1: a (1, 1) self-links, as in the Hopf fibration. (This form has Arf invariant 0, and thus this embedded torus has Kervaire invariant 0.)
Applications
A key application is in algebraic surgery theory, where even L-groups are defined as Witt groups of ε-quadratic forms, by C.T.C.Wall
References
1. Ranicki, Andrew (2001). "Foundations of algebraic surgery". arXiv:math/0111315.
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Topologies on spaces of linear maps
In mathematics, particularly functional analysis, spaces of linear maps between two vector spaces can be endowed with a variety of topologies. Studying space of linear maps and these topologies can give insight into the spaces themselves.
The article operator topologies discusses topologies on spaces of linear maps between normed spaces, whereas this article discusses topologies on such spaces in the more general setting of topological vector spaces (TVSs).
Topologies of uniform convergence on arbitrary spaces of maps
Throughout, the following is assumed:
1. $T$ is any non-empty set and ${\mathcal {G}}$ is a non-empty collection of subsets of $T$ directed by subset inclusion (i.e. for any $G,H\in {\mathcal {G}}$ there exists some $K\in {\mathcal {G}}$ such that $G\cup H\subseteq K$).
2. $Y$ is a topological vector space (not necessarily Hausdorff or locally convex).
3. ${\mathcal {N}}$ is a basis of neighborhoods of 0 in $Y.$
4. $F$ is a vector subspace of $Y^{T}=\prod _{t\in T}Y,$[note 1] which denotes the set of all $Y$-valued functions $f:T\to Y$ with domain $T.$
𝒢-topology
The following sets will constitute the basic open subsets of topologies on spaces of linear maps. For any subsets $G\subseteq T$ and $N\subseteq Y,$ let
${\mathcal {U}}(G,N):=\{f\in F:f(G)\subseteq N\}.$
The family
$\{{\mathcal {U}}(G,N):G\in {\mathcal {G}},N\in {\mathcal {N}}\}$
forms a neighborhood basis[1] at the origin for a unique translation-invariant topology on $F,$ where this topology is not necessarily a vector topology (that is, it might not make $F$ into a TVS). This topology does not depend on the neighborhood basis ${\mathcal {N}}$ that was chosen and it is known as the topology of uniform convergence on the sets in ${\mathcal {G}}$ or as the ${\mathcal {G}}$-topology.[2] However, this name is frequently changed according to the types of sets that make up ${\mathcal {G}}$ (e.g. the "topology of uniform convergence on compact sets" or the "topology of compact convergence", see the footnote for more details[3]).
A subset ${\mathcal {G}}_{1}$ of ${\mathcal {G}}$ is said to be fundamental with respect to ${\mathcal {G}}$ if each $G\in {\mathcal {G}}$ is a subset of some element in ${\mathcal {G}}_{1}.$ In this case, the collection ${\mathcal {G}}$ can be replaced by ${\mathcal {G}}_{1}$ without changing the topology on $F.$[2] One may also replace ${\mathcal {G}}$ with the collection of all subsets of all finite unions of elements of ${\mathcal {G}}$ without changing the resulting ${\mathcal {G}}$-topology on $F.$[4]
Call a subset $B$ of $T$ $F$-bounded if $f(B)$ is a bounded subset of $Y$ for every $f\in F.$[5]
Theorem[2][5] — The ${\mathcal {G}}$-topology on $F$ is compatible with the vector space structure of $F$ if and only if every $G\in {\mathcal {G}}$ is $F$-bounded; that is, if and only if for every $G\in {\mathcal {G}}$ and every $f\in F,$ $f(G)$ is bounded in $Y.$
Properties
Properties of the basic open sets will now be described, so assume that $G\in {\mathcal {G}}$ and $N\in {\mathcal {N}}.$ Then ${\mathcal {U}}(G,N)$ is an absorbing subset of $F$ if and only if for all $f\in F,$ $N$ absorbs $f(G)$.[6] If $N$ is balanced[6] (respectively, convex) then so is ${\mathcal {U}}(G,N).$
The equality ${\mathcal {U}}(\varnothing ,N)=F$ always holds. If $s$ is a scalar then $s{\mathcal {U}}(G,N)={\mathcal {U}}(G,sN),$ so that in particular, $-{\mathcal {U}}(G,N)={\mathcal {U}}(G,-N).$[6] Moreover,[4]
${\mathcal {U}}(G,N)-{\mathcal {U}}(G,N)\subseteq {\mathcal {U}}(G,N-N)$
and similarly[5]
${\mathcal {U}}(G,M)+{\mathcal {U}}(G,N)\subseteq {\mathcal {U}}(G,M+N).$
For any subsets $G,H\subseteq X$ and any non-empty subsets $M,N\subseteq Y,$[5]
${\mathcal {U}}(G\cup H,M\cap N)\subseteq {\mathcal {U}}(G,M)\cap {\mathcal {U}}(H,N)$
which implies:
• if $M\subseteq N$ then ${\mathcal {U}}(G,M)\subseteq {\mathcal {U}}(G,N).$[6]
• if $G\subseteq H$ then ${\mathcal {U}}(H,N)\subseteq {\mathcal {U}}(G,N).$
• For any $M,N\in {\mathcal {N}}$ and subsets $G,H,K$ of $T,$ if $G\cup H\subseteq K$ then ${\mathcal {U}}(K,M\cap N)\subseteq {\mathcal {U}}(G,M)\cap {\mathcal {U}}(H,N).$
For any family ${\mathcal {S}}$ of subsets of $T$ and any family ${\mathcal {M}}$ of neighborhoods of the origin in $Y,$[4]
${\mathcal {U}}\left(\bigcup _{S\in {\mathcal {S}}}S,N\right)=\bigcap _{S\in {\mathcal {S}}}{\mathcal {U}}(S,N)\qquad {\text{ and }}\qquad {\mathcal {U}}\left(G,\bigcap _{M\in {\mathcal {M}}}M\right)=\bigcap _{M\in {\mathcal {M}}}{\mathcal {U}}(G,M).$
Uniform structure
See also: Uniform space
For any $G\subseteq T$ and $U\subseteq Y\times Y$ be any entourage of $Y$ (where $Y$ is endowed with its canonical uniformity), let
${\mathcal {W}}(G,U)~:=~\left\{(u,v)\in Y^{T}\times Y^{T}~:~(u(g),v(g))\in U\;{\text{ for every }}g\in G\right\}.$
Given $G\subseteq T,$ the family of all sets ${\mathcal {W}}(G,U)$ as $U$ ranges over any fundamental system of entourages of $Y$ forms a fundamental system of entourages for a uniform structure on $Y^{T}$ called the uniformity of uniform converges on $G$ or simply the $G$-convergence uniform structure.[7] The ${\mathcal {G}}$-convergence uniform structure is the least upper bound of all $G$-convergence uniform structures as $G\in {\mathcal {G}}$ ranges over ${\mathcal {G}}.$[7]
Nets and uniform convergence
Let $f\in F$ and let $f_{\bullet }=\left(f_{i}\right)_{i\in I}$ be a net in $F.$ Then for any subset $G$ of $T,$ say that $f_{\bullet }$ converges uniformly to $f$ on $G$ if for every $N\in {\mathcal {N}}$ there exists some $i_{0}\in I$ such that for every $i\in I$ satisfying $i\geq i_{0},I$ $f_{i}-f\in {\mathcal {U}}(G,N)$ (or equivalently, $f_{i}(g)-f(g)\in N$ for every $g\in G$).[5]
Theorem[5] — If $f\in F$ and if $f_{\bullet }=\left(f_{i}\right)_{i\in I}$ is a net in $F,$ then $f_{\bullet }\to f$ in the ${\mathcal {G}}$-topology on $F$ if and only if for every $G\in {\mathcal {G}},$ $f_{\bullet }$ converges uniformly to $f$ on $G.$
Inherited properties
Local convexity
If $Y$ is locally convex then so is the ${\mathcal {G}}$-topology on $F$ and if $\left(p_{i}\right)_{i\in I}$ is a family of continuous seminorms generating this topology on $Y$ then the ${\mathcal {G}}$-topology is induced by the following family of seminorms:
$p_{G,i}(f):=\sup _{x\in G}p_{i}(f(x)),$
as $G$ varies over ${\mathcal {G}}$ and $i$ varies over $I$.[8]
Hausdorffness
If $Y$ is Hausdorff and $T=\bigcup _{G\in {\mathcal {G}}}G$ then the ${\mathcal {G}}$-topology on $F$ is Hausdorff.[5]
Suppose that $T$ is a topological space. If $Y$ is Hausdorff and $F$ is the vector subspace of $Y^{T}$ consisting of all continuous maps that are bounded on every $G\in {\mathcal {G}}$ and if $\bigcup _{G\in {\mathcal {G}}}G$ is dense in $T$ then the ${\mathcal {G}}$-topology on $F$ is Hausdorff.
Boundedness
A subset $H$ of $F$ is bounded in the ${\mathcal {G}}$-topology if and only if for every $G\in {\mathcal {G}},$ $H(G)=\bigcup _{h\in H}h(G)$ is bounded in $Y.$[8]
Examples of 𝒢-topologies
Pointwise convergence
If we let ${\mathcal {G}}$ be the set of all finite subsets of $T$ then the ${\mathcal {G}}$-topology on $F$ is called the topology of pointwise convergence. The topology of pointwise convergence on $F$ is identical to the subspace topology that $F$ inherits from $Y^{T}$ when $Y^{T}$ is endowed with the usual product topology.
If $X$ is a non-trivial completely regular Hausdorff topological space and $C(X)$ is the space of all real (or complex) valued continuous functions on $X,$ the topology of pointwise convergence on $C(X)$ is metrizable if and only if $X$ is countable.[5]
𝒢-topologies on spaces of continuous linear maps
Throughout this section we will assume that $X$ and $Y$ are topological vector spaces. ${\mathcal {G}}$ will be a non-empty collection of subsets of $X$ directed by inclusion. $L(X;Y)$ will denote the vector space of all continuous linear maps from $X$ into $Y.$ If $L(X;Y)$ is given the ${\mathcal {G}}$-topology inherited from $Y^{X}$ then this space with this topology is denoted by $L_{\mathcal {G}}(X;Y)$. The continuous dual space of a topological vector space $X$ over the field $\mathbb {F} $ (which we will assume to be real or complex numbers) is the vector space $L(X;\mathbb {F} )$ and is denoted by $X^{\prime }$.
The ${\mathcal {G}}$-topology on $L(X;Y)$ is compatible with the vector space structure of $L(X;Y)$ if and only if for all $G\in {\mathcal {G}}$ and all $f\in L(X;Y)$ the set $f(G)$ is bounded in $Y,$ which we will assume to be the case for the rest of the article. Note in particular that this is the case if ${\mathcal {G}}$ consists of (von-Neumann) bounded subsets of $X.$
Assumptions on 𝒢
Assumptions that guarantee a vector topology
• (${\mathcal {G}}$ is directed): ${\mathcal {G}}$ will be a non-empty collection of subsets of $X$ directed by (subset) inclusion. That is, for any $G,H\in {\mathcal {G}},$ there exists $K\in {\mathcal {G}}$ such that $G\cup H\subseteq K$.
The above assumption guarantees that the collection of sets ${\mathcal {U}}(G,N)$ forms a filter base. The next assumption will guarantee that the sets ${\mathcal {U}}(G,N)$ are balanced. Every TVS has a neighborhood basis at 0 consisting of balanced sets so this assumption isn't burdensome.
• ($N\in {\mathcal {N}}$ are balanced): ${\mathcal {N}}$ is a neighborhoods basis of the origin in $Y$ that consists entirely of balanced sets.
The following assumption is very commonly made because it will guarantee that each set ${\mathcal {U}}(G,N)$ is absorbing in $L(X;Y).$
• ($G\in {\mathcal {G}}$ are bounded): ${\mathcal {G}}$ is assumed to consist entirely of bounded subsets of $X.$
The next theorem gives ways in which ${\mathcal {G}}$ can be modified without changing the resulting ${\mathcal {G}}$-topology on $Y.$
Theorem[6] — Let ${\mathcal {G}}$ be a non-empty collection of bounded subsets of $X.$ Then the ${\mathcal {G}}$-topology on $L(X;Y)$ is not altered if ${\mathcal {G}}$ is replaced by any of the following collections of (also bounded) subsets of $X$:
1. all subsets of all finite unions of sets in ${\mathcal {G}}$;
2. all scalar multiples of all sets in ${\mathcal {G}}$;
3. all finite Minkowski sums of sets in ${\mathcal {G}}$;
4. the balanced hull of every set in ${\mathcal {G}}$;
5. the closure of every set in ${\mathcal {G}}$;
and if $X$ and $Y$ are locally convex, then we may add to this list:
1. the closed convex balanced hull of every set in ${\mathcal {G}}.$
Common assumptions
Some authors (e.g. Narici) require that ${\mathcal {G}}$ satisfy the following condition, which implies, in particular, that ${\mathcal {G}}$ is directed by subset inclusion:
${\mathcal {G}}$ is assumed to be closed with respect to the formation of subsets of finite unions of sets in ${\mathcal {G}}$ (i.e. every subset of every finite union of sets in ${\mathcal {G}}$ belongs to ${\mathcal {G}}$).
Some authors (e.g. Trèves [9]) require that ${\mathcal {G}}$ be directed under subset inclusion and that it satisfy the following condition:
If $G\in {\mathcal {G}}$ and $s$ is a scalar then there exists a $H\in {\mathcal {G}}$ such that $sG\subseteq H.$
If ${\mathcal {G}}$ is a bornology on $X,$ which is often the case, then these axioms are satisfied. If ${\mathcal {G}}$ is a saturated family of bounded subsets of $X$ then these axioms are also satisfied.
Properties
Hausdorffness
A subset of a TVS $X$ whose linear span is a dense subset of $X$ is said to be a total subset of $X.$ If ${\mathcal {G}}$ is a family of subsets of a TVS $T$ then ${\mathcal {G}}$ is said to be total in $T$ if the linear span of $\bigcup _{G\in {\mathcal {G}}}G$ is dense in $T.$[10]
If $F$ is the vector subspace of $Y^{T}$ consisting of all continuous linear maps that are bounded on every $G\in {\mathcal {G}},$ then the ${\mathcal {G}}$-topology on $F$ is Hausdorff if $Y$ is Hausdorff and ${\mathcal {G}}$ is total in $T.$[6]
Completeness
For the following theorems, suppose that $X$ is a topological vector space and $Y$ is a locally convex Hausdorff spaces and ${\mathcal {G}}$ is a collection of bounded subsets of $X$ that covers $X,$ is directed by subset inclusion, and satisfies the following condition: if $G\in {\mathcal {G}}$ and $s$ is a scalar then there exists a $H\in {\mathcal {G}}$ such that $sG\subseteq H.$
• $L_{\mathcal {G}}(X;Y)$ is complete if
1. $X$ is locally convex and Hausdorff,
2. $Y$ is complete, and
3. whenever $u:X\to Y$ is a linear map then $u$ restricted to every set $G\in {\mathcal {G}}$ is continuous implies that $u$ is continuous,
• If $X$ is a Mackey space then $L_{\mathcal {G}}(X;Y)$is complete if and only if both $X_{\mathcal {G}}^{\prime }$ and $Y$ are complete.
• If $X$ is barrelled then $L_{\mathcal {G}}(X;Y)$ is Hausdorff and quasi-complete.
• Let $X$ and $Y$ be TVSs with $Y$ quasi-complete and assume that (1) $X$ is barreled, or else (2) $X$ is a Baire space and $X$ and $Y$ are locally convex. If ${\mathcal {G}}$ covers $X$ then every closed equicontinuous subset of $L(X;Y)$ is complete in $L_{\mathcal {G}}(X;Y)$ and $L_{\mathcal {G}}(X;Y)$ is quasi-complete.[11]
• Let $X$ be a bornological space, $Y$ a locally convex space, and ${\mathcal {G}}$ a family of bounded subsets of $X$ such that the range of every null sequence in $X$ is contained in some $G\in {\mathcal {G}}.$ If $Y$ is quasi-complete (respectively, complete) then so is $L_{\mathcal {G}}(X;Y)$.[12]
Boundedness
Let $X$ and $Y$ be topological vector spaces and $H$ be a subset of $L(X;Y).$ Then the following are equivalent:[8]
1. $H$ is bounded in $L_{\mathcal {G}}(X;Y)$;
2. For every $G\in {\mathcal {G}},$ $H(G):=\bigcup _{h\in H}h(G)$ is bounded in $Y$;[8]
3. For every neighborhood $V$ of the origin in $Y$ the set $\bigcap _{h\in H}h^{-1}(V)$ absorbs every $G\in {\mathcal {G}}.$
If ${\mathcal {G}}$ is a collection of bounded subsets of $X$ whose union is total in $X$ then every equicontinuous subset of $L(X;Y)$ is bounded in the ${\mathcal {G}}$-topology.[11] Furthermore, if $X$ and $Y$ are locally convex Hausdorff spaces then
• if $H$ is bounded in $L_{\sigma }(X;Y)$ (that is, pointwise bounded or simply bounded) then it is bounded in the topology of uniform convergence on the convex, balanced, bounded, complete subsets of $X.$[13]
• if $X$ is quasi-complete (meaning that closed and bounded subsets are complete), then the bounded subsets of $L(X;Y)$ are identical for all ${\mathcal {G}}$-topologies where ${\mathcal {G}}$ is any family of bounded subsets of $X$ covering $X.$[13]
Examples
${\mathcal {G}}\subseteq \wp (X)$ ("topology of uniform convergence on ...") Notation Name ("topology of...") Alternative name
finite subsets of $X$ $L_{\sigma }(X;Y)$ pointwise/simple convergence topology of simple convergence
precompact subsets of $X$ precompact convergence
compact convex subsets of $X$ $L_{\gamma }(X;Y)$ compact convex convergence
compact subsets of $X$ $L_{c}(X;Y)$ compact convergence
bounded subsets of $X$ $L_{b}(X;Y)$ bounded convergence strong topology
The topology of pointwise convergence
By letting ${\mathcal {G}}$ be the set of all finite subsets of $X,$ $L(X;Y)$ will have the weak topology on $L(X;Y)$ or the topology of pointwise convergence or the topology of simple convergence and $L(X;Y)$ with this topology is denoted by $L_{\sigma }(X;Y)$. Unfortunately, this topology is also sometimes called the strong operator topology, which may lead to ambiguity;[6] for this reason, this article will avoid referring to this topology by this name.
A subset of $L(X;Y)$ is called simply bounded or weakly bounded if it is bounded in $L_{\sigma }(X;Y)$.
The weak-topology on $L(X;Y)$ has the following properties:
• If $X$ is separable (that is, it has a countable dense subset) and if $Y$ is a metrizable topological vector space then every equicontinuous subset $H$ of $L_{\sigma }(X;Y)$ is metrizable; if in addition $Y$ is separable then so is $H.$[14]
• So in particular, on every equicontinuous subset of $L(X;Y),$ the topology of pointwise convergence is metrizable.
• Let $Y^{X}$ denote the space of all functions from $X$ into $Y.$ If $L(X;Y)$ is given the topology of pointwise convergence then space of all linear maps (continuous or not) $X$ into $Y$ is closed in $Y^{X}$.
• In addition, $L(X;Y)$ is dense in the space of all linear maps (continuous or not) $X$ into $Y.$
• Suppose $X$ and $Y$ are locally convex. Any simply bounded subset of $L(X;Y)$ is bounded when $L(X;Y)$ has the topology of uniform convergence on convex, balanced, bounded, complete subsets of $X.$ If in addition $X$ is quasi-complete then the families of bounded subsets of $L(X;Y)$ are identical for all ${\mathcal {G}}$-topologies on $L(X;Y)$ such that ${\mathcal {G}}$ is a family of bounded sets covering $X.$[13]
Equicontinuous subsets
• The weak-closure of an equicontinuous subset of $L(X;Y)$ is equicontinuous.
• If $Y$ is locally convex, then the convex balanced hull of an equicontinuous subset of $L(X;Y)$ is equicontinuous.
• Let $X$ and $Y$ be TVSs and assume that (1) $X$ is barreled, or else (2) $X$ is a Baire space and $X$ and $Y$ are locally convex. Then every simply bounded subset of $L(X;Y)$ is equicontinuous.[11]
• On an equicontinuous subset $H$ of $L(X;Y),$ the following topologies are identical: (1) topology of pointwise convergence on a total subset of $X$; (2) the topology of pointwise convergence; (3) the topology of precompact convergence.[11]
Compact convergence
By letting ${\mathcal {G}}$ be the set of all compact subsets of $X,$ $L(X;Y)$ will have the topology of compact convergence or the topology of uniform convergence on compact sets and $L(X;Y)$ with this topology is denoted by $L_{c}(X;Y)$.
The topology of compact convergence on $L(X;Y)$ has the following properties:
• If $X$ is a Fréchet space or a LF-space and if $Y$ is a complete locally convex Hausdorff space then $L_{c}(X;Y)$ is complete.
• On equicontinuous subsets of $L(X;Y),$ the following topologies coincide:
• The topology of pointwise convergence on a dense subset of $X,$
• The topology of pointwise convergence on $X,$
• The topology of compact convergence.
• The topology of precompact convergence.
• If $X$ is a Montel space and $Y$ is a topological vector space, then $L_{c}(X;Y)$ and $L_{b}(X;Y)$ have identical topologies.
Topology of bounded convergence
By letting ${\mathcal {G}}$ be the set of all bounded subsets of $X,$ $L(X;Y)$ will have the topology of bounded convergence on $X$ or the topology of uniform convergence on bounded sets and $L(X;Y)$ with this topology is denoted by $L_{b}(X;Y)$.[6]
The topology of bounded convergence on $L(X;Y)$ has the following properties:
• If $X$ is a bornological space and if $Y$ is a complete locally convex Hausdorff space then $L_{b}(X;Y)$ is complete.
• If $X$ and $Y$ are both normed spaces then the topology on $L(X;Y)$ induced by the usual operator norm is identical to the topology on $L_{b}(X;Y)$.[6]
• In particular, if $X$ is a normed space then the usual norm topology on the continuous dual space $X^{\prime }$ is identical to the topology of bounded convergence on $X^{\prime }$.
• Every equicontinuous subset of $L(X;Y)$ is bounded in $L_{b}(X;Y)$.
Polar topologies
Throughout, we assume that $X$ is a TVS.
𝒢-topologies versus polar topologies
If $X$ is a TVS whose bounded subsets are exactly the same as its weakly bounded subsets (e.g. if $X$ is a Hausdorff locally convex space), then a ${\mathcal {G}}$-topology on $X^{\prime }$ (as defined in this article) is a polar topology and conversely, every polar topology if a ${\mathcal {G}}$-topology. Consequently, in this case the results mentioned in this article can be applied to polar topologies.
However, if $X$ is a TVS whose bounded subsets are not exactly the same as its weakly bounded subsets, then the notion of "bounded in $X$" is stronger than the notion of "$\sigma \left(X,X^{\prime }\right)$-bounded in $X$" (i.e. bounded in $X$ implies $\sigma \left(X,X^{\prime }\right)$-bounded in $X$) so that a ${\mathcal {G}}$-topology on $X^{\prime }$ (as defined in this article) is not necessarily a polar topology. One important difference is that polar topologies are always locally convex while ${\mathcal {G}}$-topologies need not be.
Polar topologies have stronger results than the more general topologies of uniform convergence described in this article and we refer the read to the main article: polar topology. We list here some of the most common polar topologies.
List of polar topologies
Suppose that $X$ is a TVS whose bounded subsets are the same as its weakly bounded subsets.
Notation: If $\Delta (Y,X)$ denotes a polar topology on $Y$ then $Y$ endowed with this topology will be denoted by $Y_{\Delta (Y,X)}$ or simply $Y_{\Delta }$ (e.g. for $\sigma (Y,X)$ we would have $\Delta =\sigma $ so that $Y_{\sigma (Y,X)}$ and $Y_{\sigma }$ all denote $Y$ with endowed with $\sigma (Y,X)$).
>${\mathcal {G}}\subseteq \wp (X)$
("topology of uniform convergence on ...")
Notation Name ("topology of...") Alternative name
finite subsets of $X$ $\sigma (Y,X)$
$s(Y,X)$
pointwise/simple convergence weak/weak* topology
$\sigma (X,Y)$-compact disks $\tau (Y,X)$ Mackey topology
$\sigma (X,Y)$-compact convex subsets $\gamma (Y,X)$ compact convex convergence
$\sigma (X,Y)$-compact subsets
(or balanced $\sigma (X,Y)$-compact subsets)
$c(Y,X)$ compact convergence
$\sigma (X,Y)$-bounded subsets $b(Y,X)$
$\beta (Y,X)$
bounded convergence strong topology
𝒢-ℋ topologies on spaces of bilinear maps
We will let ${\mathcal {B}}(X,Y;Z)$ denote the space of separately continuous bilinear maps and $B(X,Y;Z)$denote the space of continuous bilinear maps, where $X,Y,$ and $Z$ are topological vector space over the same field (either the real or complex numbers). In an analogous way to how we placed a topology on $L(X;Y)$ we can place a topology on ${\mathcal {B}}(X,Y;Z)$ and $B(X,Y;Z)$.
Let ${\mathcal {G}}$ (respectively, ${\mathcal {H}}$) be a family of subsets of $X$ (respectively, $Y$) containing at least one non-empty set. Let ${\mathcal {G}}\times {\mathcal {H}}$ denote the collection of all sets $G\times H$ where $G\in {\mathcal {G}},$ $H\in {\mathcal {H}}.$ We can place on $Z^{X\times Y}$ the ${\mathcal {G}}\times {\mathcal {H}}$-topology, and consequently on any of its subsets, in particular on $B(X,Y;Z)$and on ${\mathcal {B}}(X,Y;Z)$. This topology is known as the ${\mathcal {G}}-{\mathcal {H}}$-topology or as the topology of uniform convergence on the products $G\times H$ of ${\mathcal {G}}\times {\mathcal {H}}$.
However, as before, this topology is not necessarily compatible with the vector space structure of ${\mathcal {B}}(X,Y;Z)$ or of $B(X,Y;Z)$without the additional requirement that for all bilinear maps, $b$ in this space (that is, in ${\mathcal {B}}(X,Y;Z)$ or in $B(X,Y;Z)$) and for all $G\in {\mathcal {G}}$ and $H\in {\mathcal {H}},$ the set $b(G,H)$ is bounded in $X.$ If both ${\mathcal {G}}$ and ${\mathcal {H}}$ consist of bounded sets then this requirement is automatically satisfied if we are topologizing $B(X,Y;Z)$but this may not be the case if we are trying to topologize ${\mathcal {B}}(X,Y;Z)$. The ${\mathcal {G}}-{\mathcal {H}}$-topology on ${\mathcal {B}}(X,Y;Z)$ will be compatible with the vector space structure of ${\mathcal {B}}(X,Y;Z)$ if both ${\mathcal {G}}$ and ${\mathcal {H}}$ consists of bounded sets and any of the following conditions hold:
• $X$ and $Y$ are barrelled spaces and $Z$ is locally convex.
• $X$ is a F-space, $Y$ is metrizable, and $Z$ is Hausdorff, in which case ${\mathcal {B}}(X,Y;Z)=B(X,Y;Z).$
• $X,Y,$ and $Z$ are the strong duals of reflexive Fréchet spaces.
• $X$ is normed and $Y$ and $Z$ the strong duals of reflexive Fréchet spaces.
The ε-topology
Main article: Injective tensor product
Suppose that $X,Y,$ and $Z$ are locally convex spaces and let ${\mathcal {G}}^{\prime }$ and ${\mathcal {H}}^{\prime }$ be the collections of equicontinuous subsets of $X^{\prime }$ and $X^{\prime }$, respectively. Then the ${\mathcal {G}}^{\prime }-{\mathcal {H}}^{\prime }$-topology on ${\mathcal {B}}\left(X_{b\left(X^{\prime },X\right)}^{\prime },Y_{b\left(X^{\prime },X\right)}^{\prime };Z\right)$ will be a topological vector space topology. This topology is called the ε-topology and ${\mathcal {B}}\left(X_{b\left(X^{\prime },X\right)}^{\prime },Y_{b\left(X^{\prime },X\right)};Z\right)$ with this topology it is denoted by ${\mathcal {B}}_{\epsilon }\left(X_{b\left(X^{\prime },X\right)}^{\prime },Y_{b\left(X^{\prime },X\right)}^{\prime };Z\right)$ or simply by ${\mathcal {B}}_{\epsilon }\left(X_{b}^{\prime },Y_{b}^{\prime };Z\right).$
Part of the importance of this vector space and this topology is that it contains many subspace, such as ${\mathcal {B}}\left(X_{\sigma \left(X^{\prime },X\right)}^{\prime },Y_{\sigma \left(X^{\prime },X\right)}^{\prime };Z\right),$ which we denote by ${\mathcal {B}}\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime };Z\right).$ When this subspace is given the subspace topology of ${\mathcal {B}}_{\epsilon }\left(X_{b}^{\prime },Y_{b}^{\prime };Z\right)$ it is denoted by ${\mathcal {B}}_{\epsilon }\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime };Z\right).$
In the instance where $Z$ is the field of these vector spaces, ${\mathcal {B}}\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)$ is a tensor product of $X$ and $Y.$ In fact, if $X$ and $Y$ are locally convex Hausdorff spaces then ${\mathcal {B}}\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)$ is vector space-isomorphic to $L\left(X_{\sigma \left(X^{\prime },X\right)}^{\prime };Y_{\sigma (Y^{\prime },Y)}\right),$ which is in turn is equal to $L\left(X_{\tau \left(X^{\prime },X\right)}^{\prime };Y\right).$
These spaces have the following properties:
• If $X$ and $Y$ are locally convex Hausdorff spaces then ${\mathcal {B}}_{\varepsilon }\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)$ is complete if and only if both $X$ and $Y$ are complete.
• If $X$ and $Y$ are both normed (respectively, both Banach) then so is ${\mathcal {B}}_{\epsilon }\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)$
See also
• Bornological space – Space where bounded operators are continuous
• Bounded linear operator – Linear transformation between topological vector spacesPages displaying short descriptions of redirect targets
• Dual system
• Dual topology
• List of topologies – List of concrete topologies and topological spaces
• Modes of convergence – Property of a sequence or series
• Operator norm – Measure of the "size" of linear operators
• Polar topology – Dual space topology of uniform convergence on some sub-collection of bounded subsets
• Strong dual space – Continuous dual space endowed with the topology of uniform convergence on bounded sets
• Topologies on the set of operators on a Hilbert space
• Uniform convergence – Mode of convergence of a function sequence
• Uniform space – Topological space with a notion of uniform properties
• Weak topology – Mathematical term
• Vague topology
References
1. Because $T$ is just a set that is not yet assumed to be endowed with any vector space structure, $F\subseteq Y^{T}$ should not yet be assumed to consist of linear maps, which is a notation that currently can not be defined.
1. Note that each set ${\mathcal {U}}(G,N)$ is a neighborhood of the origin for this topology, but it is not necessarily an open neighborhood of the origin.
2. Schaefer & Wolff 1999, pp. 79–88.
3. In practice, ${\mathcal {G}}$ usually consists of a collection of sets with certain properties and this name is changed appropriately to reflect this set so that if, for instance, ${\mathcal {G}}$ is the collection of compact subsets of $T$ (and $T$ is a topological space), then this topology is called the topology of uniform convergence on the compact subsets of $T.$
4. Narici & Beckenstein 2011, pp. 19–45.
5. Jarchow 1981, pp. 43–55.
6. Narici & Beckenstein 2011, pp. 371–423.
7. Grothendieck 1973, pp. 1–13.
8. Schaefer & Wolff 1999, p. 81.
9. Trèves 2006, Chapter 32.
10. Schaefer & Wolff 1999, p. 80.
11. Schaefer & Wolff 1999, p. 83.
12. Schaefer & Wolff 1999, p. 117.
13. Schaefer & Wolff 1999, p. 82.
14. Schaefer & Wolff 1999, p. 87.
Bibliography
• 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.
• Hogbe-Nlend, Henri (1977). Bornologies and Functional Analysis: Introductory Course on the Theory of Duality Topology-Bornology and its use in Functional Analysis. North-Holland Mathematics Studies. Vol. 26. Amsterdam New York New York: North Holland. ISBN 978-0-08-087137-0. MR 0500064. OCLC 316549583.
• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
Functional analysis (topics – glossary)
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• Dual space
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Eta invariant
In mathematics, the eta invariant of a self-adjoint elliptic differential operator on a compact manifold is formally the number of positive eigenvalues minus the number of negative eigenvalues. In practice both numbers are often infinite so are defined using zeta function regularization. It was introduced by Atiyah, Patodi, and Singer (1973, 1975) who used it to extend the Hirzebruch signature theorem to manifolds with boundary. The name comes from the fact that it is a generalization of the Dirichlet eta function.
They also later used the eta invariant of a self-adjoint operator to define the eta invariant of a compact odd-dimensional smooth manifold.
Michael Francis Atiyah, H. Donnelly, and I. M. Singer (1983) defined the signature defect of the boundary of a manifold as the eta invariant, and used this to show that Hirzebruch's signature defect of a cusp of a Hilbert modular surface can be expressed in terms of the value at s=0 or 1 of a Shimizu L-function.
Definition
The eta invariant of self-adjoint operator A is given by ηA(0), where η is the analytic continuation of
$\eta (s)=\sum _{\lambda \neq 0}{\frac {\operatorname {sign} (\lambda )}{|\lambda |^{s}}}$
and the sum is over the nonzero eigenvalues λ of A.
References
• Atiyah, Michael Francis; Patodi, V. K.; Singer, I. M. (1973), "Spectral asymmetry and Riemannian geometry", The Bulletin of the London Mathematical Society, 5 (2): 229–234, CiteSeerX 10.1.1.597.6432, doi:10.1112/blms/5.2.229, ISSN 0024-6093, MR 0331443
• Atiyah, Michael Francis; Patodi, V. K.; Singer, I. M. (1975), "Spectral asymmetry and Riemannian geometry. I", Mathematical Proceedings of the Cambridge Philosophical Society, 77 (1): 43–69, Bibcode:1975MPCPS..77...43A, doi:10.1017/S0305004100049410, ISSN 0305-0041, MR 0397797, S2CID 17638224
• Atiyah, Michael Francis; Donnelly, H.; Singer, I. M. (1983), "Eta invariants, signature defects of cusps, and values of L-functions", Annals of Mathematics, Second Series, 118 (1): 131–177, doi:10.2307/2006957, ISSN 0003-486X, JSTOR 2006957, MR 0707164
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η set
In mathematics, an η set (eta set) is a type of totally ordered set introduced by Hausdorff (1907, p. 126, 1914, chapter 6 section 8) that generalizes the order type η of the rational numbers.
Definition
If $\alpha $ is an ordinal then an $\eta _{\alpha }$ set is a totally ordered set in which for any two subsets $X$ and $Y$ of cardinality less than $\aleph _{\alpha }$, if every element of $X$ is less than every element of $Y$ then there is some element greater than all elements of $X$ and less than all elements of $Y$.
Examples
The only non-empty countable η0 set (up to isomorphism) is the ordered set of rational numbers.
Suppose that κ = ℵα is a regular cardinal and let X be the set of all functions f from κ to {−1,0,1} such that if f(α) = 0 then f(β) = 0 for all β > α, ordered lexicographically. Then X is a ηα set. The union of all these sets is the class of surreal numbers.
A dense totally ordered set without endpoints is an ηα set if and only if it is ℵα saturated.
Properties
Any ηα set X is universal for totally ordered sets of cardinality at most ℵα, meaning that any such set can be embedded into X.
For any given ordinal α, any two ηα sets of cardinality ℵα are isomorphic (as ordered sets). An ηα set of cardinality ℵα exists if ℵα is regular and Σβ<α 2ℵβ ≤ ℵα.
References
• Alling, Norman L. (1962), "On the existence of real-closed fields that are ηα-sets of power ℵα.", Trans. Amer. Math. Soc., 103: 341–352, doi:10.1090/S0002-9947-1962-0146089-X, MR 0146089
• Chang, Chen Chung; Keisler, H. Jerome (1990) [1973]. Model Theory. Studies in Logic and the Foundations of Mathematics (3rd ed.). Elsevier. ISBN 978-0-444-88054-3.
• Felgner, U. (2002), "Die Hausdorffsche Theorie der ηα-Mengen und ihre Wirkungsgeschichte" (PDF), Hausdorff Gesammelte Werke, vol. II, Berlin, Heidelberg: Springer-Verlag, pp. 645–674
• Hausdorff (1907), "Untersuchungen über Ordnungstypen V", Ber. über die Verhandlungen der Königl. Sächs. Ges. Der Wiss. Zu Leipzig. Math.-phys. Klasse, 59: 105–159 English translation in Hausdorff (2005)
• Hausdorff, F. (1914), Grundzüge der Mengenlehre, Leipzig: Veit & Co
• Hausdorff, Felix (2005), Plotkin, J. M. (ed.), Hausdorff on ordered sets, History of Mathematics, vol. 25, Providence, RI: American Mathematical Society, ISBN 0-8218-3788-5, MR 2187098
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Theta correspondence
In mathematics, the theta correspondence or Howe correspondence is a mathematical relation between representations of two groups of a reductive dual pair. The local theta correspondence relates irreducible admissible representations over a local field, while the global theta correspondence relates irreducible automorphic representations over a global field.
The theta correspondence was introduced by Roger Howe in Howe (1979). Its name arose due to its origin in André Weil's representation theoretical formulation of the theory of theta series in Weil (1964). The Shimura correspondence as constructed by Jean-Loup Waldspurger in Waldspurger (1980) and Waldspurger (1991) may be viewed as an instance of the theta correspondence.
Statement
Setup
Let $F$ be a local or a global field, not of characteristic $2$. Let $W$ be a symplectic vector space over $F$, and $Sp(W)$ the symplectic group.
Fix a reductive dual pair $(G,H)$ in $Sp(W)$. There is a classification of reductive dual pairs.[1] [2]
Local theta correspondence
$F$ is now a local field. Fix a non-trivial additive character $\psi $ of $F$. There exists a Weil representation of the metaplectic group $Mp(W)$ associated to $\psi $, which we write as $\omega _{\psi }$.
Given the reductive dual pair $(G,H)$ in $Sp(W)$, one obtains a pair of commuting subgroups $({\widetilde {G}},{\widetilde {H}})$ in $Mp(W)$ by pulling back the projection map from $Mp(W)$ to $Sp(W)$.
The local theta correspondence is a 1-1 correspondence between certain irreducible admissible representations of ${\widetilde {G}}$ and certain irreducible admissible representations of ${\widetilde {H}}$, obtained by restricting the Weil representation $\omega _{\psi }$ of $Mp(W)$ to the subgroup ${\widetilde {G}}\cdot {\widetilde {H}}$. The correspondence was defined by Roger Howe in Howe (1979). The assertion that this is a 1-1 correspondence is called the Howe duality conjecture.
Key properties of local theta correspondence include its compatibility with Bernstein-Zelevinsky induction [3] and conservation relations concerning the first occurrence indices along Witt towers .[4]
Global theta correspondence
Stephen Rallis showed a version of the global Howe duality conjecture for cuspidal automorphic representations over a global field, assuming the validity of the Howe duality conjecture for all local places. [5]
Howe duality conjecture
Define ${\mathcal {R}}({\widetilde {G}},\omega _{\psi })$ the set of irreducible admissible representations of ${\widetilde {G}}$, which can be realized as quotients of $\omega _{\psi }$. Define ${\mathcal {R}}({\widetilde {H}},\omega _{\psi })$ and ${\mathcal {R}}({\widetilde {G}}\cdot {\widetilde {H}},\omega _{\psi })$, likewise.
The Howe duality conjecture asserts that ${\mathcal {R}}({\widetilde {G}}\cdot {\widetilde {H}},\omega _{\psi })$ is the graph of a bijection between ${\mathcal {R}}({\widetilde {G}},\omega _{\psi })$ and ${\mathcal {R}}({\widetilde {H}},\omega _{\psi })$.
The Howe duality conjecture for archimedean local fields was proved by Roger Howe.[6] For $p$-adic local fields with $p$ odd it was proved by Jean-Loup Waldspurger.[7] Alberto Mínguez later gave a proof for dual pairs of general linear groups, that works for arbitrary residue characteristic. [8] For orthogonal-symplectic or unitary dual pairs, it was proved by Wee Teck Gan and Shuichiro Takeda. [9] The final case of quaternionic dual pairs was completed by Wee Teck Gan and Binyong Sun.[10]
See also
• Reductive dual pair
• Metaplectic group
References
1. Howe 1979.
2. Mœglin, Vignéras & Waldspurger 1987.
3. Kudla 1986.
4. Sun & Zhu 2015.
5. Rallis 1984.
6. Howe 1989.
7. Waldspurger 1990.
8. Mínguez 2008.
9. Gan & Takeda 2016.
10. Gan & Sun 2017.
Bibliography
• Gan, Wee Teck; Takeda, Shuichiro (2016), "A proof of the Howe duality conjecture", J. Amer. Math. Soc., 29 (2): 473–493, arXiv:1407.1995, doi:10.1090/jams/839, S2CID 942882
• Gan, Wee Teck; Sun, Binyong (2017), "The Howe duality conjecture: quaternionic case", in Cogdell, J.; Kim, J.-L.; Zhu, C.-B. (eds.), Representation Theory, Number Theory, and Invariant Theory, Progr. Math., 323, Birkhäuser/Springer, pp. 175–192
• Howe, Roger E. (1979), "θ-series and invariant theory", in Borel, A.; Casselman, W. (eds.), Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, Providence, R.I.: American Mathematical Society, pp. 275–285, ISBN 978-0-8218-1435-2, MR 0546602
• Howe, Roger E. (1989), "Transcending classical invariant theory", J. Amer. Math. Soc., 2 (3): 535–552, doi:10.2307/1990942, JSTOR 1990942
• Kudla, Stephen S. (1986), "On the local theta-correspondence", Invent. Math., 83 (2): 229–255, doi:10.1007/BF01388961, S2CID 122106772
• Mínguez, Alberto (2008), "Correspondance de Howe explicite: paires duales de type II", Ann. Sci. Éc. Norm. Supér., 4, 41 (5): 717–741, doi:10.24033/asens.2080
• Mœglin, Colette; Vignéras, Marie-France; Waldspurger, Jean-Loup (1987), Correspondances de Howe sur un corps p-adique, Lecture Notes in Mathematics, vol. 1291, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0082712, ISBN 978-3-540-18699-1, MR 1041060
• Rallis, Stephen (1984), "On the Howe duality conjecture", Compositio Math., 51 (3): 333–399
• Sun, Binyong; Zhu, Chen-Bo (2015), "Conservation relations for local theta correspondence", J. Amer. Math. Soc., 28 (4): 939–983, arXiv:1204.2969, doi:10.1090/S0894-0347-2014-00817-1, S2CID 5936119
• Waldspurger, Jean-Loup (1980), "Correspondance de Shimura", J. Math. Pures Appl., 59 (9): 1–132
• Waldspurger, Jean-Loup (1990), "Démonstration d'une conjecture de dualité de Howe dans le cas p-adique, p ≠ 2", Festschrift in Honor of I. I. Piatetski-Shapiro on the Occasion of His Sixtieth Birthday, Part I, Israel Math. Conf. Proc., 2: 267–324
• Waldspurger, Jean-Loup (1991), "Correspondances de Shimura et quaternions", Forum Math., 3 (3): 219–307, doi:10.1515/form.1991.3.219, S2CID 123512840
• Weil, André (1964), "Sur certains groupes d'opérateurs unitaires", Acta Math., 111: 143–211, doi:10.1007/BF02391012
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K-Poincaré group
In physics and mathematics, the κ-Poincaré group, named after Henri Poincaré, is a quantum group, obtained by deformation of the Poincaré group into a Hopf algebra. It is generated by the elements $a^{\mu }$ and ${\Lambda ^{\mu }}_{\nu }$ with the usual constraint:
$\eta ^{\rho \sigma }{\Lambda ^{\mu }}_{\rho }{\Lambda ^{\nu }}_{\sigma }=\eta ^{\mu \nu }~,$
where $\eta ^{\mu \nu }$ is the Minkowskian metric:
$\eta ^{\mu \nu }=\left({\begin{array}{cccc}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{array}}\right)~.$
The commutation rules reads:
• $[a_{j},a_{0}]=i\lambda a_{j}~,\;[a_{j},a_{k}]=0$
• $[a^{\mu },{\Lambda ^{\rho }}_{\sigma }]=i\lambda \left\{\left({\Lambda ^{\rho }}_{0}-{\delta ^{\rho }}_{0}\right){\Lambda ^{\mu }}_{\sigma }-\left({\Lambda ^{\alpha }}_{\sigma }\eta _{\alpha 0}+\eta _{\sigma 0}\right)\eta ^{\rho \mu }\right\}$
In the (1 + 1)-dimensional case the commutation rules between $a^{\mu }$ and ${\Lambda ^{\mu }}_{\nu }$ are particularly simple. The Lorentz generator in this case is:
${\Lambda ^{\mu }}_{\nu }=\left({\begin{array}{cc}\cosh \tau &\sinh \tau \\\sinh \tau &\cosh \tau \end{array}}\right)$
and the commutation rules reads:
• $[a_{0},\left({\begin{array}{c}\cosh \tau \\\sinh \tau \end{array}}\right)]=i\lambda ~\sinh \tau \left({\begin{array}{c}\sinh \tau \\\cosh \tau \end{array}}\right)$
• $[a_{1},\left({\begin{array}{c}\cosh \tau \\\sinh \tau \end{array}}\right)]=i\lambda \left(1-\cosh \tau \right)\left({\begin{array}{c}\sinh \tau \\\cosh \tau \end{array}}\right)$
The coproducts are classical, and encode the group composition law:
• $\Delta a^{\mu }={\Lambda ^{\mu }}_{\nu }\otimes a^{\nu }+a^{\mu }\otimes 1$
• $\Delta {\Lambda ^{\mu }}_{\nu }={\Lambda ^{\mu }}_{\rho }\otimes {\Lambda ^{\rho }}_{\nu }$
Also the antipodes and the counits are classical, and represent the group inversion law and the map to the identity:
• $S(a^{\mu })=-{(\Lambda ^{-1})^{\mu }}_{\nu }a^{\nu }$
• $S({\Lambda ^{\mu }}_{\nu })={(\Lambda ^{-1})^{\mu }}_{\nu }={\Lambda _{\nu }}^{\mu }$
• $\varepsilon (a^{\mu })=0$
• $\varepsilon ({\Lambda ^{\mu }}_{\nu })={\delta ^{\mu }}_{\nu }$
The κ-Poincaré group is the dual Hopf algebra to the K-Poincaré algebra, and can be interpreted as its “finite” version.
References
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Konstantin Adolfovic Semendyayev
Konstantin Adolfovic Semendyayev or Semendyaev (Russian: Константин Адольфович Семендяев, German: Konstantin Adolfowitsch Semendjajew); born 9 December 1908 in Simferopol, died 15 November 1988) was a Russian engineer and applied mathematician. He worked in the department of applied mathematics of the Steklov Institute in Moscow. He carried out pioneering work in the area of numerical weather forecasting in Russia.
Work and life
Semendyayev studied at the Lomonosov University with the degree in 1929 and was then at various higher schools. From 1931 to 1936 he was in the Faculty of Mathematics and Mechanics at Lomonosov University. He habilitated in 1940 (Russian doctorate). From 1936 he headed the Department of Mathematical Instruments of the USSR Academy of Sciences. He was evacuated to Kazan with the institute during World War II. After World War II, he headed a department for numerical calculations at the Steklov Institute in Moscow and, when the Institute for Applied Mathematics at the Steklov Institute was founded in 1953, his group became the Department of Gas Dynamics. In 1961, he became deputy head of the Institute for Applied Mathematics. In 1963, he went to the Hydrometeorological Center of the USSR, where he led the programming work. He also supported the teaching of applied mathematics at various Moscow educational institutions.
Semendyayev is known as the co-author of a handbook of mathematics for engineers and students of technical universities,[1] which he wrote together with Ilya Nikolaevich Bronshtein around the 1939/1940 timeframe. Hot lead typesetting for the work had already started when the Siege of Leningrad prohibited further development and the print matrices were relocated.[1] After the war, they were first considered lost, but could be found again years later, so that the first edition of Справочник по математике для инженеров и учащихся втузов could finally be published in 1945.[1][2] This was a major success and went through eleven editions in Russia and was translated into various languages, including German and English, until the publisher Nauka planned to replace it with a translation of the American Mathematical Handbook for Scientists and Engineers by Granino and Theresa M. Korn in 1968.[1][2] However, in a parallel development starting in 1970, the so called "Bronshtein and Semendyayev" (BS), which had been translated into German in 1958, underwent a major overhaul by a team of East-German authors around Günter Grosche, Viktor & Dorothea Ziegler (of University of Leipzig), to which Semendyayev contributed as well (a section on computer systems and numerical harmonic analysis).[1] This was published in 1979 and spawned translations into many other languages as well, including a retranslation into Russian and an English edition. In 1986, the 13th Russian edition was published. The German 'Wende' and the later reunification led to considerable changes in the publishing environment in Germany between 1989 and 1991, which eventually resulted in two independent German publishing branches by Eberhard Zeidler (published 1995–2013) and by Gerhard Musiol & Heiner Mühlig (published 1992–2020) to expand and maintain the work up to the present, again with translations into many other languages including English.
Semendyayev has been on the editorial board of the Russian journal Journal of Numerical Mathematics and Mathematical Physics (Журнал вычислительной математики и математической физики) since its inception.
He received the Order of Lenin, the USSR State Prize and the Order of the Red Banner of Labor.
Publications
• With Bronshtein: "Handbook of Mathematics for Engineers and Students of Technical Universities" (Справочник по математике для инженеров и учащихся втузов), Moscow, 1945
See also
• Bronshtein and Semendyayev (BS)
• Ilya Nikolaevich Bronshtein
References
1. Ziegler, Dorothea (2002-02-21). "Der "Bronstein"". Archiv der Stiftung Benedictus Gotthelf Teubner, Leipzig (in German). Frauwalde, Germany. Archived from the original on 2016-03-25. Retrieved 2016-03-25.
2. Girlich, Hans-Joachim [in German] (March 2014). "Von Pascals Repertorium zum Springer-Taschenbuch der Mathematik – über eine mathematische Bestsellerserie" [From Pascal's finding aid to Springer's pocketbook of mathematics – about a bestseller series in mathematics] (PDF) (in German) (preprint ed.). Leipzig, Germany: University of Leipzig, Mathematisches Institut. DNB-IDN 1052022731. Archived (PDF) from the original on 2016-04-06. Retrieved 2016-04-06.
Further reading
• "Кафедра «Высшая математика» История кафедры" [Department of Higher Mathematics - Department history] (in Russian). Moscow State University of Mechanical Engineering (MAMI). 2016 [2010]. Archived from the original on 2016-04-03. Retrieved 2022-01-23.
• Volume 29, 1989, pp. 474–475, http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=zvmmf&paperid=3490&option_lang=rus
• https://web.archive.org/web/20200705112735/http://www.mathnet.ru/links/feea4ab25995d6344cb609e4dcfc8c88/zvmmf3490.pdf
• https://web.archive.org/web/20211019135410/https://keldysh.ru/memory/index1.htm Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences, 2010
External links
• "Семендяев Константин Адольфович" [Semendyaev, Konstantin Adol'fovich]. Math-Net.Ru (in Russian and English). 2021 [2016]. Archived from the original on 2022-01-23. Retrieved 2022-01-23.
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Suslin representation
In mathematics, a Suslin representation of a set of reals (more precisely, elements of Baire space) is a tree whose projection is that set of reals. More generally, a subset A of κω is λ-Suslin if there is a tree T on κ × λ such that A = p[T].
By a tree on κ × λ we mean here a subset T of the union of κi × λi for all i ∈ N (or i < ω in set-theoretical notation).
Here, p[T] = { f | ∃g : (f,g) ∈ [T] } is the projection of T, where [T] = { (f, g ) | ∀n ∈ ω : (f(n), g(n)) ∈ T } is the set of branches through T.
Since [T] is a closed set for the product topology on κω × λω where κ and λ are equipped with the discrete topology (and all closed sets in κω × λω come in this way from some tree on κ × λ), λ-Suslin subsets of κω are projections of closed subsets in κω × λω.
When one talks of Suslin sets without specifying the space, then one usually means Suslin subsets of R, which descriptive set theorists usually take to be the set ωω.
See also
• Suslin cardinal
• Suslin operation
External links
• R. Ketchersid, The strength of an ω1-dense ideal on ω1 under CH, 2004.
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Lambda cube
In mathematical logic and type theory, the λ-cube (also written lambda cube) is a framework introduced by Henk Barendregt[1] to investigate the different dimensions in which the calculus of constructions is a generalization of the simply typed λ-calculus. Each dimension of the cube corresponds to a new kind of dependency between terms and types. Here, "dependency" refers to the capacity of a term or type to bind a term or type. The respective dimensions of the λ-cube correspond to:
• x-axis ($\rightarrow $): types that can bind terms, corresponding to dependent types.
• y-axis ($\uparrow $): terms that can bind types, corresponding to polymorphism.
• z-axis ($\nearrow $): types that can bind types, corresponding to (binding) type operators.
The different ways to combine these three dimensions yield the 8 vertices of the cube, each corresponding to a different kind of typed system. The λ-cube can be generalized into the concept of a pure type system.
Examples of Systems
(λ→) Simply typed lambda calculus
The simplest system found in the λ-cube is the simply typed lambda calculus, also called λ→. In this system, the only way to construct an abstraction is by making a term depend on a term, with the typing rule
${\frac {\Gamma ,x:\sigma \;\vdash \;t:\tau }{\Gamma \;\vdash \;\lambda x.t:\sigma \to \tau }}$
(λ2) System F
In System F (also named λ2 for the "second-order typed lambda calculus")[2] there is another type of abstraction, written with a $\Lambda $, that allows terms to depend on types, with the following rule:
${\frac {\Gamma \;\vdash \;t:\sigma }{\Gamma \;\vdash \;\Lambda \alpha .t:\Pi \alpha .\sigma }}\;{\text{ if }}\alpha {\text{ does not occur free in }}\Gamma $
The terms beginning with a $\Lambda $ are called polymorphic, as they can be applied to different types to get different functions, similarly to polymorphic functions in ML-like languages. For instance, the polymorphic identity
fun x -> x
of OCaml has type
'a -> 'a
meaning it can take an argument of any type 'a and return an element of that type. This type corresponds in λ2 to the type $\Pi \alpha .\alpha \to \alpha $.
(λω) System Fω
In System F${\underline {\omega }}$ a construction is introduced to supply types that depend on other types. This is called a type constructor and provides a way to build "a function with a type as a value".[3] An example of such a type constructor is the type of binary trees with leaves labeled by data of a given type $A$: ${\mathsf {TREE}}:=\lambda A:*.\Pi B.(A\to B)\to (B\to B\to B)\to B$, where "$A:*$" informally means "$A$ is a type". This is a function that takes a type parameter $A$ as an argument and returns the type of ${\mathsf {TREE}}$s of values of type $A$. In concrete programming, this feature corresponds to the ability to define type constructors inside the language, rather than considering them as primitives. The previous type constructor roughly corresponds to the following definition of a tree with labeled leaves in OCaml:
type 'a tree = | Leaf of 'a | Node of 'a tree * 'a tree
This type constructor can be applied to other types to obtain new types. E.g., to obtain type of trees of integers:
type int_tree = int tree
System F${\underline {\omega }}$ is generally not used on its own, but is useful to isolate the independent feature of type constructors.[4]
(λP) Lambda-P
In the λP system, also named ΛΠ, and closely related to the LF Logical Framework, one has so called dependent types. These are types that are allowed to depend on terms. The crucial introduction rule of the system is
${\frac {\Gamma ,x:A\;\vdash \;B:*}{\Gamma \;\vdash \;(\Pi x:A.B):*}}$
where $*$ represents valid types. The new type constructor $\Pi $ corresponds via the Curry-Howard isomorphism to a universal quantifier, and the system λP as a whole corresponds to first-order logic with implication as only connective. An example of these dependent types in concrete programming is the type of vectors on a certain length: the length is a term, on which the type depends.
(Fω) System Fω
System Fω combines both the $\Lambda $ constructor of System F and the type constructors from System F${\underline {\omega }}$. Thus System Fω provides both terms that depend on types and types that depend on types.
(λC) Calculus of constructions
In the calculus of constructions, denoted as λC in the cube or as λPω,[1]: 130 these four features cohabit, so that both types and terms can depend on types and terms. The clear border that exists in λ→ between terms and types is somewhat abolished, as all types except the universal $\square $ are themselves terms with a type.
Formal definition
As for all systems based upon the simply typed lambda calculus, all systems in the cube are given in two steps: first, raw terms, together with a notion of β-reduction, and then typing rules that allow to type those terms.
The set of sorts is defined as $S:=\{*,\square \}$, sorts are represented with the letter $s$. There is also a set $V$ of variables, represented by the letters $x,y,\dots $. The raw terms of the eight systems of the cube are given by the following syntax:
$A:=x\mid s\mid A~A\mid \lambda x:A.A\mid \Pi x:A.A$
and $A\to B$ denoting $\Pi x:A.B$ when $x$ does not occur free in $B$.
The environment, as is usual in typed systems, are given by $\Gamma :=\emptyset \mid \Gamma ,x:T$ :=\emptyset \mid \Gamma ,x:T}
The notion of β-reduction is common to all systems in the cube. It is written $\to _{\beta }$ and given by the rules
${\frac {}{(\lambda x:A.B)~C\to _{\beta }B[C/x]}}$
${\frac {B\to _{\beta }B'}{\lambda x:A.B\to _{\beta }\lambda x:A.B'}}$
${\frac {A\to _{\beta }A'}{\lambda x:A.B\to _{\beta }\lambda x:A'.B}}$
${\frac {B\to _{\beta }B'}{\Pi x:A.B\to _{\beta }\Pi x:A.B'}}$
${\frac {A\to _{\beta }A'}{\Pi x:A.B\to _{\beta }\Pi x:A'.B}}$
Its reflexive, transitive closure is written $=_{\beta }$. The following typing rules are also common to all systems in the cube:
${\frac {}{\vdash *:\square }}\quad {\text{(Axiom)}}$
${\frac {\Gamma \vdash A:s\quad x{\text{ does not occur in }}\Gamma }{\Gamma ,x:A\vdash x:A}}\quad {\text{(Start)}}$
${\frac {\Gamma \vdash A:B\quad \Gamma \vdash C:s}{\Gamma ,x:C\vdash A:B}}\quad {\text{(Weakening)}}$
${\frac {\Gamma \vdash C:\Pi x:A.B\quad \Gamma \vdash a:A}{\Gamma \vdash Ca:B[a/x]}}\quad {\text{(Application)}}$
${\frac {\Gamma \vdash A:B\quad B=_{\beta }B'\quad \Gamma \vdash B':s}{\Gamma \vdash A:B'}}\quad {\text{(Conversion)}}$
The difference between the systems is in the pairs of sorts $ (s_{1},s_{2})$ that are allowed in the following two typing rules:
${\frac {\Gamma \vdash A:s_{1}\quad \Gamma ,x:A\vdash B:s_{2}}{\Gamma \vdash \Pi x:A.B:s_{2}}}\quad {\text{(Product)}}$
${\frac {\Gamma \vdash A:s_{1}\quad \Gamma ,x:A\vdash b:B\quad \Gamma ,x:A\vdash B:s_{2}}{\Gamma \vdash \lambda x:A.b:\Pi x:A.B}}\quad {\text{(Abstraction)}}$
The correspondence between the systems and the pairs $ (s_{1},s_{2})$ allowed in the rules is the following:
$(s_{1},s_{2})$ $(*,*)$ $(*,\square )$ $(\square ,*)$ $(\square ,\square )$
λ→
λP
λ2
λω
λP2
λPω
λω
λC
Each direction of the cube corresponds to one pair (excluding the pair $ (*,*)$ shared by all systems), and in turn each pair corresponds to one possibility of dependency between terms and types:
• $ (*,*)$ allows terms to depend on terms.
• $ (*,\square )$ allows types to depend on terms.
• $ (\square ,*)$ allows terms to depend on types.
• $ (\square ,\square )$ allows types to depend on types.
Comparison between the systems
λ→
A typical derivation that can be obtained is
$\alpha :*\vdash \lambda x:\alpha .x:\Pi x:\alpha .\alpha $ :*\vdash \lambda x:\alpha .x:\Pi x:\alpha .\alpha }
or with the arrow shortcut
$\alpha :*\vdash \lambda x:\alpha .x:\alpha \to \alpha $ :*\vdash \lambda x:\alpha .x:\alpha \to \alpha }
closely resembling the identity (of type $ \alpha $) of the usual λ→. Note that all types used must appear in the context, because the only derivation that can be done in an empty context is $ \vdash *:\square $.
The computing power is quite weak, it corresponds to the extended polynomials (polynomials together with a conditional operator).[5]
λ2
In λ2, such terms can be obtained as
$\vdash (\lambda \beta :*.\lambda x:\bot .x\beta ):\Pi \beta :*.\bot \to \beta $ :*.\lambda x:\bot .x\beta ):\Pi \beta :*.\bot \to \beta }
with $ \bot =\Pi \alpha :*.\alpha $ :*.\alpha } . If one reads $ \Pi $ as a universal quantification, via the Curry-Howard isomorphism, this can be seen as a proof of the principle of explosion. In general, λ2 adds the possibility to have impredicative types such as $ \bot $, that is terms quantifying over all types including themselves.
The polymorphism also allows the construction of functions that were not constructible in λ→. More precisely, the functions definable in λ2 are those provably total in second-order Peano arithmetic.[6] In particular, all primitive recursive functions are definable.
λP
In λP, the ability to have types depending on terms means one can express logical predicates. For instance, the following is derivable:
$\alpha :*,a_{0}:\alpha ,p:\alpha \to *,q:*\vdash \lambda z:(\Pi x:\alpha .px\to q).\lambda y:(\Pi x:\alpha .px).(za_{0})(ya_{0}):(\Pi x:\alpha .px\to q)\to (\Pi x:\alpha .px)\to q$ :*,a_{0}:\alpha ,p:\alpha \to *,q:*\vdash \lambda z:(\Pi x:\alpha .px\to q).\lambda y:(\Pi x:\alpha .px).(za_{0})(ya_{0}):(\Pi x:\alpha .px\to q)\to (\Pi x:\alpha .px)\to q}
which corresponds, via the Curry-Howard isomorphism, to a proof of $(\forall x:A,Px\to Q)\to (\forall x:A,Px)\to Q$.
From the computational point of view, however, having dependent types does not enhance computational power, only the possibility to express more precise type properties.[7]
The conversion rule is strongly needed when dealing with dependent types, because it allows to perform computation on the terms in the type. For instance, if you have $\Gamma \vdash A:P((\lambda x.x)y)$ and $\Gamma \vdash B:\Pi x:P(y).C$, you need to apply the conversion rule to obtain $\Gamma \vdash A:P(y)$ to be able to type $\Gamma \vdash BA:C$.
λω
In λω, the following operator
$AND:=\lambda \alpha :*.\lambda \beta :*.\Pi \gamma :*.(\alpha \to \beta \to \gamma )\to \gamma $ :*.\lambda \beta :*.\Pi \gamma :*.(\alpha \to \beta \to \gamma )\to \gamma }
is definable, that is $\vdash AND:*\to *\to *$. The derivation
$\alpha :*,\beta :*\vdash \Pi \gamma :*.(\alpha \to \beta \to \gamma )\to \gamma :*$ :*,\beta :*\vdash \Pi \gamma :*.(\alpha \to \beta \to \gamma )\to \gamma :*}
can be obtained already in λ2, however the polymorphic $ AND$ can only be defined if the rule $ (\square ,*)$ is also present.
From a computing point of view, λω is extremely strong, and has been considered as a basis for programming languages.[8]
λC
The calculus of constructions has both the predicate expressiveness of λP and the computational power of λω, hence why λC is also called λPω,[1]: 130 so it is very powerful, both on the logical side and on the computational side.
Relation to other systems
The system Automath is similar to λ2 from a logical point of view. The ML-like languages, from a typing point of view, lie somewhere between λ→ and λ2, as they admit a restricted kind of polymorphic types, that is the types in prenex normal form. However, because they feature some recursion operators, their computing power is greater than that of λ2.[7] The Coq system is based on an extension of λC with a linear hierarchy of universes, rather than only one untypable $ \square $, and the ability to construct inductive types.
Pure type systems can be seen as a generalization of the cube, with an arbitrary set of sorts, axiom, product and abstraction rules. Conversely, the systems of the lambda cube can be expressed as pure type systems with two sorts $\{*,\square \}$, the only axiom $ \{*,\square \}$, and a set of rules $ R$ such that $\{(*,*,*)\}\subseteq R\subseteq \{(*,*,*),(*,\square ,\square ),(\square ,*,*),(\square ,\square ,\square )\}$.[1]
Via the Curry-Howard isomorphism, there is a one-to-one correspondence between the systems in the lambda cube and logical systems,[1] namely:
System of the cube Logical System
λ→ (Zeroth-order) Propositional Calculus
λ2 Second-order Propositional Calculus
λω Weakly Higher Order Propositional Calculus
λω Higher Order Propositional Calculus
λP (First order) Predicate Logic
λP2 Second-order Predicate Calculus
λPω Weak Higher Order Predicate Calculus
λC Calculus of Constructions
All the logics are implicative (i.e. the only connectives are $ \to $ and $ \forall $), however one can define other connectives such as $\wedge $ or $\neg $ in an impredicative way in second and higher order logics. In the weak higher order logics, there are variables for higher order predicates, but no quantification on those can be done.
Common properties
All systems in the cube enjoy
• the Church-Rosser property: if $M\to _{\beta }N$ and $M\to _{\beta }N'$ then there exists $N''$ such that $N\to _{\beta }^{*}N''$ and $N'\to _{\beta }^{*}N''$;
• the subject reduction property: if $\Gamma \vdash M:T$ and $M\to _{\beta }M'$ then $\Gamma \vdash M':T$;
• the uniqueness of types: if $\Gamma \vdash A:B$ and $\Gamma \vdash A:B'$ then $B=_{\beta }B'$.
All of these can be proven on generic pure type systems.[9]
Any term well-typed in a system of the cube is strongly normalizing,[1] although this property is not common to all pure type systems. No system in the cube is Turing complete.[7]
Subtyping
Subtyping however is not represented in the cube, even though systems like $F_{<:}^{\omega }$, known as higher-order bounded quantification, which combines subtyping and polymorphism are of practical interest, and can be further generalized to bounded type operators. Further extensions to $F_{<:}^{\omega }$ allow the definition of purely functional objects; these systems were generally developed after the lambda cube paper was published.[10]
The idea of the cube is due to the mathematician Henk Barendregt (1991). The framework of pure type systems generalizes the lambda cube in the sense that all corners of the cube, as well as many other systems can be represented as instances of this general framework.[11] This framework predates the lambda cube by a couple of years. In his 1991 paper, Barendregt also defines the corners of the cube in this framework.
See also
• In his Habilitation à diriger les recherches,[12] Olivier Ridoux gives a cut-out template of the lambda cube and also a dual representation of the cube as an octahedron, where the 8 vertices are replaced by faces, as well as a dual representation as a dodecahedron, where the 12 edges are replaced by faces.
• Homotopy type theory
Notes
1. Barendregt, Henk (1991). "Introduction to generalized type systems". Journal of Functional Programming. 1 (2): 125–154. doi:10.1017/s0956796800020025. hdl:2066/17240. ISSN 0956-7968. S2CID 44757552.
2. Nederpelt, Rob; Geuvers, Herman (2014). Type Theory and Formal Proof. Cambridge University Press. p. 69. ISBN 9781107036505.
3. Nederpelt & Geuvers 2014, p. 85
4. Nederpelt & Geuvers 2014, p. 100
5. Schwichtenberg, Helmut (1975). "Definierbare Funktionen imλ-Kalkül mit Typen". Archiv für Mathematische Logik und Grundlagenforschung (in German). 17 (3–4): 113–114. doi:10.1007/bf02276799. ISSN 0933-5846. S2CID 11598130.
6. Girard, Jean-Yves; Lafont, Yves; Taylor, Paul (1989). Proofs and Types. Cambridge Tracts in Theoretical Computer Science. Vol. 7. Cambridge University Press. ISBN 9780521371810.
7. Ridoux, Olivier (1998). Lambda-Prolog de A à Z ... ou presque (PDF). [s.n.] OCLC 494473554.
8. Pierce, Benjamin; Dietzen, Scott; Michaylov, Spiro (1989). Programming in higher-order typed lambda-calculi. Computer Science Department, Carnegie Mellon University. OCLC 20442222. CMU-CS-89-111 ERGO-89-075.
9. Sørensen, Morten Heine; Urzyczyin, Pawel (2006). "Pure type systems and the λ-cube". Lectures on the Curry-Howard Isomorphism. Elsevier. pp. 343–359. doi:10.1016/s0049-237x(06)80015-7. ISBN 9780444520777.
10. Pierce, Benjamin (2002). Types and programming languages. MIT Press. pp. 467–490. ISBN 978-0262162098. OCLC 300712077.
11. Pierce 2002, p. 466
12. Ridoux 1998, p. 70
Further reading
• Peyton Jones, Simon; Meijer, Erik (1997). "Henk: A Typed Intermediate Language" (PDF). Microsoft. Henk is based directly on the lambda cube, an expressive family of typed lambda calculi.
External links
• Barendregt's Lambda Cube in the context of pure type systems by Roger Bishop Jones
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Polynomial matrix
In mathematics, a polynomial matrix or matrix of polynomials is a matrix whose elements are univariate or multivariate polynomials. Equivalently, a polynomial matrix is a polynomial whose coefficients are matrices.
A univariate polynomial matrix P of degree p is defined as:
$P=\sum _{n=0}^{p}A(n)x^{n}=A(0)+A(1)x+A(2)x^{2}+\cdots +A(p)x^{p}$
where $A(i)$ denotes a matrix of constant coefficients, and $A(p)$ is non-zero. An example 3×3 polynomial matrix, degree 2:
$P={\begin{pmatrix}1&x^{2}&x\\0&2x&2\\3x+2&x^{2}-1&0\end{pmatrix}}={\begin{pmatrix}1&0&0\\0&0&2\\2&-1&0\end{pmatrix}}+{\begin{pmatrix}0&0&1\\0&2&0\\3&0&0\end{pmatrix}}x+{\begin{pmatrix}0&1&0\\0&0&0\\0&1&0\end{pmatrix}}x^{2}.$
We can express this by saying that for a ring R, the rings $M_{n}(R[X])$ and $(M_{n}(R))[X]$ are isomorphic.
Properties
• A polynomial matrix over a field with determinant equal to a non-zero element of that field is called unimodular, and has an inverse that is also a polynomial matrix. Note that the only scalar unimodular polynomials are polynomials of degree 0 – nonzero constants, because an inverse of an arbitrary polynomial of higher degree is a rational function.
• The roots of a polynomial matrix over the complex numbers are the points in the complex plane where the matrix loses rank.
• The determinant of a matrix polynomial with Hermitian positive-definite (semidefinite) coefficients is a polynomial with positive (nonnegative) coefficients.[1]
Note that polynomial matrices are not to be confused with monomial matrices, which are simply matrices with exactly one non-zero entry in each row and column.
If by λ we denote any element of the field over which we constructed the matrix, by I the identity matrix, and we let A be a polynomial matrix, then the matrix λI − A is the characteristic matrix of the matrix A. Its determinant, |λI − A| is the characteristic polynomial of the matrix A.
References
1. Friedland, S.; Melman, A. (2020). "A note on Hermitian positive semidefinite matrix polynomials". Linear Algebra and Its Applications. 598: 105–109. doi:10.1016/j.laa.2020.03.038.
• E.V.Krishnamurthy, Error-free Polynomial Matrix computations, Springer Verlag, New York, 1985
Matrix classes
Explicitly constrained entries
• Alternant
• Anti-diagonal
• Anti-Hermitian
• Anti-symmetric
• Arrowhead
• Band
• Bidiagonal
• Bisymmetric
• Block-diagonal
• Block
• Block tridiagonal
• Boolean
• Cauchy
• Centrosymmetric
• Conference
• Complex Hadamard
• Copositive
• Diagonally dominant
• Diagonal
• Discrete Fourier Transform
• Elementary
• Equivalent
• Frobenius
• Generalized permutation
• Hadamard
• Hankel
• Hermitian
• Hessenberg
• Hollow
• Integer
• Logical
• Matrix unit
• Metzler
• Moore
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λ-ring
In algebra, a λ-ring or lambda ring is a commutative ring together with some operations λn on it that behave like the exterior powers of vector spaces. Many rings considered in K-theory carry a natural λ-ring structure. λ-rings also provide a powerful formalism for studying an action of the symmetric functions on the ring of polynomials, recovering and extending many classical results (Lascoux (2003)).
λ-rings were introduced by Grothendieck (1957, 1958, p.148). For more about λ-rings see Atiyah & Tall (1969), Knutson (1973), Hazewinkel (2009) and Yau (2010).
Motivation
If V and W are finite-dimensional vector spaces over a field k, then we can form the direct sum V ⊕ W, the tensor product V ⊗ W, and the n-th exterior power of V, Λn(V). All of these are again finite-dimensional vector spaces over k. The same three operations of direct sum, tensor product and exterior power are also available when working with k-linear representations of a finite group, when working with vector bundles over some topological space, and in more general situations.
λ-rings are designed to abstract the common algebraic properties of these three operations, where we also allow for formal inverses with respect to the direct sum operation. (These formal inverses also appear in Grothendieck groups, which is why the underlying additive groups of most λ-rings are Grothendieck groups.) The addition in the ring corresponds to the direct sum, the multiplication in the ring corresponds to the tensor product, and the λ-operations to the exterior powers. For example, the isomorphism
$\Lambda ^{2}(V\oplus W)\cong \Lambda ^{2}(V)\oplus \left(\Lambda ^{1}(V)\otimes \Lambda ^{1}(W)\right)\oplus \Lambda ^{2}(W)$
corresponds to the formula
$\lambda ^{2}(x+y)=\lambda ^{2}(x)+\lambda ^{1}(x)\lambda ^{1}(y)+\lambda ^{2}(y)$
valid in all λ-rings, and the isomorphism
$\Lambda ^{1}(V\otimes W)\cong \Lambda ^{1}(V)\otimes \Lambda ^{1}(W)$
corresponds to the formula
$\lambda ^{1}(xy)=\lambda ^{1}(x)\lambda ^{1}(y)$
valid in all λ-rings. Analogous but (much) more complicated formulas govern the higher order λ-operators.
Motivation with Vector Bundles
If we have a short exact sequence of vector bundles over a smooth scheme $X$
$0\to {\mathcal {E}}''\to {\mathcal {E}}\to {\mathcal {E}}'\to 0,$
then locally, for a small enough open neighborhood $U$ we have the isomorphism
$\bigwedge ^{n}{\mathcal {E}}|_{U}\cong \bigoplus _{i+j=n}\bigwedge ^{i}{\mathcal {E}}'|_{U}\otimes \bigwedge ^{j}{\mathcal {E}}''|_{U}$
Now, in the Grothendieck group $K(X)$ of $X$ (which is actually a ring), we get this local equation globally for free, from the defining equivalence relations. So
${\begin{aligned}\left[\bigwedge ^{n}{\mathcal {E}}\right]&=\left[\bigoplus _{i+j=n}\bigwedge ^{i}{\mathcal {E}}'\otimes \bigwedge ^{j}{\mathcal {E}}''\right]\\&=\sum _{i+j=n}\left[\bigwedge ^{i}{\mathcal {E}}'\right]\cdot \left[\bigwedge ^{j}{\mathcal {E}}''\right]\end{aligned}}$
demonstrating the basic relation in a λ-ring,[1] that
$\lambda ^{n}(x+y)=\sum _{i+j=n}\lambda ^{i}(x)\lambda ^{j}(y).$
Definition
A λ-ring is a commutative ring R together with operations λn : R → R for every non-negative integer n. These operations are required to have the following properties valid for all x, y in R and all n, m ≥ 0:
• λ0(x) = 1
• λ1(x) = x
• λn(1) = 0 if n ≥ 2
• λn(x + y) = Σi+j=n λi(x) λj(y)
• λn(xy) = Pn(λ1(x), ..., λn(x), λ1(y), ..., λn(y))
• λn(λm(x)) = Pn,m(λ1(x), ..., λmn(x))
where Pn and Pn,m are certain universal polynomials with integer coefficients that describe the behavior of exterior powers on tensor products and under composition. These polynomials can be defined as follows.
Let e1, ..., emn be the elementary symmetric polynomials in the variables X1, ..., Xmn. Then Pn,m is the unique polynomial in nm variables with integer coefficients such that Pn,m(e1, ..., emn) is the coefficient of tn in the expression
$\prod _{1\leq i_{1}<i_{2}<\cdots <i_{m}\leq mn}(1+tX_{i_{1}}X_{i_{2}}\cdots X_{i_{m}})$
(Such a polynomial exists, because the expression is symmetric in the Xi and the elementary symmetric polynomials generate all symmetric polynomials.)
Now let e1, ..., en be the elementary symmetric polynomials in the variables X1, ..., Xn and f1, ..., fn be the elementary symmetric polynomials in the variables Y1, ..., Yn. Then Pn is the unique polynomial in 2n variables with integer coefficients such that Pn(e1, ..., en, f1, ..., fn) is the coefficient of tn in the expression
$\prod _{i,j=1}^{n}(1+tX_{i}Y_{j})$
Variations
The λ-rings defined above are called "special λ-rings" by some authors, who use the term "λ-ring" for a more general concept where the conditions on λn(1), λn(xy) and λm(λn(x)) are dropped.
Examples
• The ring Z of integers, with the binomial coefficients $\lambda ^{n}(x)={x \choose n}$ as operations (which are also defined for negative x) is a λ-ring. In fact, this is the only λ-structure on Z. This example is closely related to the case of finite-dimensional vector spaces mentioned in the Motivation section above, identifying each vector space with its dimension and remembering that $\dim(\Lambda ^{n}(k^{x}))={x \choose n}$.
• More generally, any binomial ring becomes a λ-ring if we define the λ-operations to be the binomial coefficients, λn(x) = (x
n
). In these λ-rings, all Adams operations are the identity.
• The K-theory K(X) of a topological space X is a λ-ring, with the lambda operations induced by taking exterior powers of a vector bundle.
• Given a group G and a base field k, the representation ring R(G) is a λ-ring; the λ-operations are induced by the exterior powers of k-linear representations of the group G.
• The ring ΛZ of symmetric functions is a λ-ring. On the integer coefficients the λ-operations are defined by binomial coefficients as above, and if e1, e2, ... denote the elementary symmetric functions, we set λn(e1) = en. Using the axioms for the λ-operations, and the fact that the functions ek are algebraically independent and generate the ring ΛZ, this definition can be extended in a unique fashion so as to turn ΛZ into a λ-ring. In fact, this is the free λ-ring on one generator, the generator being e1. (Yau (2010, p.14)).
Further properties and definitions
Every λ-ring has characteristic 0 and contains the λ-ring Z as a λ-subring.
Many notions of commutative algebra can be extended to λ-rings. For example, a λ-homomorphism between λ-rings R and S is a ring homomorphism f : R → S such that f(λn(x)) = λn(f(x)) for all x in R and all n ≥ 0. A λ-ideal in the λ-ring R is an ideal I in R such that λn(x) ϵ I for all x in R and all n ≥ 1.
If x is an element of a λ-ring and m a non-negative integer such that λm(x) ≠ 0 and λn(x) = 0 for all n > m, we write dim(x) = m and call the element x finite-dimensional. Not all elements need to be finite-dimensional. We have dim(x+y) ≤ dim(x) + dim(y) and the product of 1-dimensional elements is 1-dimensional.
See also
• Chern class
• Symmetric Function
• K-theory
• Adams operation
• Plethystic exponential
References
1. Pieter Belmans (23 October 2014). "Three filtrations on the grothendieck ring of a scheme".
• Atiyah, M. F.; Tall, D. O. (1969), "Group representations, λ-rings and the J-homomorphism.", Topology, 8: 253–297, doi:10.1016/0040-9383(69)90015-9, MR 0244387
• Expo 0 and V of Berthelot, Pierre; Alexandre Grothendieck; Luc Illusie, eds. (1971). Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics 225) (in French). Berlin; New York: Springer-Verlag. xii+700. doi:10.1007/BFb0066283. ISBN 978-3-540-05647-8. MR 0354655.
• Grothendieck, Alexander (1957), "Special λ-rings", Unpublished
• Grothendieck, Alexander (1958), "La théorie des classes de Chern", Bull. Soc. Math. France, 86: 137–154, MR 0116023
• Hazewinkel, Michiel (2009), "Witt vectors. I.", Handbook of algebra. Vol. 6, Amsterdam: Elsevier/North-Holland, pp. 319–472, arXiv:0804.3888, doi:10.1016/S1570-7954(08)00207-6, ISBN 978-0-444-53257-2, MR 2553661
• Knutson, Donald (1973), λ-rings and the representation theory of the symmetric group, Lecture Notes in Mathematics, vol. 308, Berlin-New York: Springer-Verlag, doi:10.1007/BFb0069217, MR 0364425
• Lascoux, Alain (2003), Symmetric functions and combinatorial operators on polynomials (PDF), CBMS Reg. Conf. Ser. in Math. 99, American Mathematical Society
• Soulé, C.; Abramovich, Dan; Burnol, J.-F.; Kramer, Jürg (1992). Lectures on Arakelov geometry. Cambridge Studies in Advanced Mathematics. Vol. 33. Joint work with H. Gillet. Cambridge: Cambridge University Press. ISBN 0-521-47709-3. Zbl 0812.14015.
• Yau, Donald (2010), Lambda-rings, Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., doi:10.1142/7664, ISBN 978-981-4299-09-1, MR 2649360
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Lambda-mu calculus
In mathematical logic and computer science, the lambda-mu calculus is an extension of the lambda calculus introduced by M. Parigot.[1] It introduces two new operators: the μ operator (which is completely different both from the μ operator found in computability theory and from the μ operator of modal μ-calculus) and the bracket operator. Proof-theoretically, it provides a well-behaved formulation of classical natural deduction.
One of the main goals of this extended calculus is to be able to describe expressions corresponding to theorems in classical logic. According to the Curry–Howard isomorphism, lambda calculus on its own can express theorems in intuitionistic logic only, and several classical logical theorems can't be written at all. However with these new operators one is able to write terms that have the type of, for example, Peirce's law.
Semantically these operators correspond to continuations, found in some functional programming languages.
Formal definition
We can augment the definition of a lambda expression to gain one in the context of lambda-mu calculus. The three main expressions found in lambda calculus are as follows:
1. V, a variable, where V is any identifier.
2. λV.E, an abstraction, where V is any identifier and E is any lambda expression.
3. (E E′), an application, where E and E'; are any lambda expressions.
For details, see the corresponding article.
In addition to the traditional λ-variables, the lambda-mu calculus includes a distinct set of μ-variables. These μ-variables can be used to name or freeze arbitrary subterms, allowing us to later abstract on those names. The set of terms contains unnamed (all traditional lambda expressions are of this kind) and named terms. The terms that are added by the lambda-mu calculus are of the form:
1. [α]t is a named term, where α is a μ-variable and t is an unnamed term.
2. (μ α. E) is an unnamed term, where α is a μ-variable and E is a named term.
Reduction
The basic reduction rules used in the lambda-mu calculus are the following:
logical reduction
$(\lambda x.u)v\;\triangleright _{c}\;u[v/x]$
structural reduction
$(\mu \beta .u)v\;\triangleright _{c}\;\mu \beta .u\left[[\beta ](wv)/[\beta ]w\right]$
renaming
$[\alpha ]\mu \beta .u\;\triangleright _{c}\;u[\alpha /\beta ]$
the equivalent of η-reduction
$\mu \alpha .[\alpha ]u\;\triangleright _{c}\;u$, for α not freely occurring in u
These rules cause the calculus to be confluent. Further reduction rules could be added to provide us with a stronger notion of normal form, though this would be at the expense of confluence.
See also
• Classical pure type systems for typed generalizations of lambda calculi with control
References
1. Michel Parigot (1992). λμ-Calculus: An algorithmic interpretation of classical natural deduction. Lecture Notes in Computer Science. Vol. 624. pp. 190–201. doi:10.1007/BFb0013061. ISBN 3-540-55727-X.
External links
• Lambda-mu relevant discussion on Lambda the Ultimate.
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Explicit substitution
In computer science, lambda calculi are said to have explicit substitutions if they pay special attention to the formalization of the process of substitution. This is in contrast to the standard lambda calculus where substitutions are performed by beta reductions in an implicit manner which is not expressed within the calculus; the "freshness" conditions in such implicit calculi are a notorious source of errors.[1] The concept has appeared in a large number of published papers in quite different fields, such as in abstract machines, predicate logic, and symbolic computation.
Overview
A simple example of a lambda calculus with explicit substitution is "λx", which adds one new form of term to the lambda calculus, namely the form M⟨x:=N⟩, which reads "M where x will be substituted by N". (The meaning of the new term is the same as the common idiom let x:=N in M from many programming languages.) λx can be written with the following rewriting rules:
1. (λx.M) N → M⟨x:=N⟩
2. x⟨x:=N⟩ → N
3. x⟨y:=N⟩ → x (x≠y)
4. (M1M2) ⟨x:=N⟩ → (M1⟨x:=N⟩) (M2⟨x:=N⟩)
5. (λx.M) ⟨y:=N⟩ → λx.(M⟨y:=N⟩) (x≠y and x not free in N)
While making substitution explicit, this formulation still retains the complexity of the lambda calculus "variable convention", requiring arbitrary renaming of variables during reduction to ensure that the "(x≠y and x not free in N)" condition on the last rule is always satisfied before applying the rule. Therefore many calculi of explicit substitution avoid variable names altogether by using a so-called "name-free" De Bruijn index notation.
History
Explicit substitutions were sketched in the preface of Curry's book on Combinatory logic[2] and grew out of an ‘implementation trick’ used, for example, by AUTOMATH, and became a respectable syntactic theory in lambda calculus and rewriting theory. Though it actually originated with de Bruijn,[3] the idea of a specific calculus where substitutions are part of the object language, and not of the informal meta-theory, is traditionally credited to Abadi, Cardelli, Curien, and Lévy. Their seminal paper[4] on the λσ calculus explains that implementations of lambda calculus need to be very careful when dealing with substitutions. Without sophisticated mechanisms for structure-sharing, substitutions can cause a size explosion, and therefore, in practice, substitutions are delayed and explicitly recorded. This makes the correspondence between the theory and the implementation highly non-trivial and correctness of implementations can be hard to establish. One solution is to make the substitutions part of the calculus, that is, to have a calculus of explicit substitutions.
Once substitution has been made explicit, however, the basic properties of substitution change from being semantic to syntactic properties. One most important example is the "substitution lemma", which with the notation of λx becomes
• (M⟨x:=N⟩)⟨y:=P⟩ = (M⟨y:=P⟩)⟨x:=(N⟨y:=P⟩)⟩ (where x≠y and x not free in P)
A surprising counterexample, due to Melliès,[5] shows that the way this rule is encoded in the original calculus of explicit substitutions is not strongly normalizing. Following this, a multitude of calculi were described trying to offer the best compromise between syntactic properties of explicit substitution calculi.[6][7][8]
See also
• Combinatory logic
• Substitution instance
References
1. Clouston, Ranald; Bizjak, Aleš; Grathwohl, Hans; Birkedal, Lars (27 April 2017). "The Guarded Lambda-Calculus: Programming and Reasoning with Guarded Recursion for Coinductive Types". Logical Methods in Computer Science. 12 (3): 36. arXiv:1606.09455. doi:10.2168/LMCS-12(3:7)2016.
2. Curry, Haskell; Feys, Robert (1958). Combinatory Logic Volume I. Amsterdam: North-Holland Publishing Company.
3. N. G. de Bruijn: A namefree lambda calculus with facilities for internal definition of expressions and segments. Technological University Eindhoven, Netherlands, Department of Mathematics, (1978), (TH-Report), Number 78-WSK-03.
4. M. Abadi, L. Cardelli, P-L. Curien and J-J. Levy, Explicit Substitutions, Journal of Functional Programming 1, 4 (October 1991), 375–416.
5. P-A. Melliès: Typed lambda-calculi with explicit substitutions may not terminate. TLCA 1995: 328–334
6. P. Lescanne, From λσ to λυ: a journey through calculi of explicit substitutions, POPL 1994, pp. 60–69.
7. K. H. Rose, Explicit Substitution – Tutorial & Survey, BRICS LS-96-3, September 1996 (ps.gz).
8. Delia Kesner: A Theory of Explicit Substitutions with Safe and Full Composition. Logical Methods in Computer Science 5(3) (2009)
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Marinus of Tyre
Marinus of Tyre (Greek: Μαρῖνος ὁ Τύριος, Marînos ho Týrios; c. AD 70–130) was a Greek-speaking Roman geographer, cartographer and mathematician, who founded mathematical geography and provided the underpinnings of Claudius Ptolemy's influential Geography.
Life
Marinus was originally from Tyre in the Roman province of Syria.[1] His work was a precursor to that of the great geographer Claudius Ptolemy, who used Marinus' work as a source for his Geography and acknowledges his great obligations to him.[2][3] Ptolemy said, "Marinus says of the merchant class generally that they are only intent on their business, and have little interest in exploration, and that often through their love of boasting they magnify distances."[4] Later, Marinus was also cited by the Arab geographer al-Masʿūdī. Beyond this, little is known of his life.
Legacy
Marinus' geographical treatise is lost and known only from Ptolemy's remarks. He introduced improvements to the construction of maps and developed a system of nautical charts. His chief legacy is that he was the first to assign to each place a proper latitude and longitude. His zero meridian ran through the westernmost land known during his time, the Isles of the Blessed, around the location of the present-day Canary or Cape Verde Islands. He used the parallel of Rhodes for measurements of latitude.
Ptolemy mentions several revisions of Marinus' geographical work, which is often dated to AD 114, although this is uncertain. Marinus estimated a length of 180,000 stadia for the equator, roughly corresponding[5] to a circumference of the Earth of 33,300 kilometres (20,700 mi), about 17% less than the actual value.
Marinus also carefully studied the works of his predecessors and the diaries of travelers. His maps were the first in the Roman Empire to show China. He invented equirectangular projection, which is still used in map creation today. A few of Marinus' opinions are also reported by Ptolemy. Marinus was of the opinion that the World Ocean was separated into an eastern and a western part by the continents of Europe, Asia and Africa. He thought that the inhabited world stretched in latitude from Thule (Norway) to Agisymba (around the Tropic of Capricorn) and in longitude from the Isles of the Blessed (around the Canaries) to Shera (China). Marinus also coined the term Antarctic, referring to the opposite of the Arctic.
In 1935, an impact crater on the Moon was named after Marinus.
See also
• 1st century in Lebanon
References
1. George Sarton (1936). "The Unity and Diversity of the Mediterranean World", Osiris 2, p. 406-463 [430].
2. Chisholm 1911.
3. Harley, J. B. (John Brian); Woodward, David (1987). The History of cartography. Humana Press. pp. 178–. ISBN 978-0-226-31633-8. Retrieved 4 June 2010.
4. Ptolemy, "33".
5. For a value of a 185 m or 607 ft per stadion.
Attribution
Chisholm, Hugh, ed. (1911). "Marinus of Tyre" . Encyclopædia Britannica (11th ed.). Cambridge University Press.
• A. Forbiger, Handbuch der alten Geographie, vol. i. (1842);
• E. H. Bunbury, Hist. of Ancient Geography (1879), ii. p. 519;
• E. H. Berger, Geschichte der wissenschaftlichen Erdkunde der Griechen (1903).
• "Marinus" in Brill's New Pauly (Brill, 2010)
External links
• Jones, Alexander (2008) [1970-80]. "Marinus of Tyre". Complete Dictionary of Scientific Biography. Encyclopedia.com.
• https://web.archive.org/web/20080314171517/http://www.tmth.edu.gr/en/aet/3/66.html
• http://www.dioi.org/gad.htm
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Ξ function
In mathematics, the Ξ function (named for the Greek letter Ξ or Xi) may refer to:
• Riemann Xi function, a variant of the Riemann zeta function with a simpler functional equation
• Harish-Chandra's Ξ function, a special spherical function on a semisimple Lie group
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Prime-counting function
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number x.[1][2] It is denoted by π(x) (unrelated to the number π).
"Π(x)" redirects here. For the variant of the gamma function, see Gamma function § Pi function.
Growth rate
Main article: Prime number theorem
Of great interest in number theory is the growth rate of the prime-counting function.[3][4] It was conjectured in the end of the 18th century by Gauss and by Legendre to be approximately
${\frac {x}{\log(x)}}$
where log is the natural logarithm, in the sense that
$\lim _{x\rightarrow \infty }{\frac {\pi (x)}{x/\log(x)}}=1.$
This statement is the prime number theorem. An equivalent statement is
$\lim _{x\rightarrow \infty }\pi (x)/\operatorname {li} (x)=1$
where li is the logarithmic integral function. The prime number theorem was first proved in 1896 by Jacques Hadamard and by Charles de la Vallée Poussin independently, using properties of the Riemann zeta function introduced by Riemann in 1859. Proofs of the prime number theorem not using the zeta function or complex analysis were found around 1948 by Atle Selberg and by Paul Erdős (for the most part independently).[5]
More precise estimates
In 1899, de la Vallée Poussin proved that [6]
$\pi (x)=\operatorname {li} (x)+O\left(xe^{-a{\sqrt {\log x}}}\right)\quad {\text{as }}x\to \infty $
for some positive constant a. Here, O(...) is the big O notation.
More precise estimates of $\pi (x)\!$ are now known. For example, in 2002, Kevin Ford proved that[7]
$\pi (x)=\operatorname {li} (x)+O\left(x\exp \left(-0.2098(\log x)^{\frac {3}{5}}(\log \log x)^{-{\frac {1}{5}}}\right)\right).$
Mossinghoff and Trudgian proved[8] an explicit upper bound for the difference between $\pi (x)$ and $\operatorname {li} (x)$:
${\big |}\pi (x)-\operatorname {li} (x){\big |}\leq 0.2593{\frac {x}{(\log x)^{3/4}}}\exp \left(-{\sqrt {\frac {\log x}{6.315}}}\right)$
for $x\geq 229$.
For values of $x$ that are not unreasonably large, $\operatorname {li} (x)$ is greater than $\pi (x)$. However, $\pi (x)-\operatorname {li} (x)$ is known to change sign infinitely many times. For a discussion of this, see Skewes' number.
Exact form
For $x>1$ let $\pi _{0}(x)=\pi (x)-1/2$ when $x$ is a prime number, and $\pi _{0}(x)=\pi (x)$ otherwise. Bernhard Riemann, in his work On the Number of Primes Less Than a Given Magnitude, proved that $\pi _{0}(x)$ is equal to[9]
$\pi _{0}(x)=\operatorname {R} (x)-\sum _{\rho }\operatorname {R} (x^{\rho }),$
where
$\operatorname {R} (x)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\operatorname {li} (x^{1/n}),$
μ(n) is the Möbius function, li(x) is the logarithmic integral function, ρ indexes every zero of the Riemann zeta function, and li(xρ/n) is not evaluated with a branch cut but instead considered as Ei(ρ/n log x) where Ei(x) is the exponential integral. If the trivial zeros are collected and the sum is taken only over the non-trivial zeros ρ of the Riemann zeta function, then $\pi _{0}(x)$ may be approximated by[10]
$\pi _{0}(x)\approx \operatorname {R} (x)-\sum _{\rho }\operatorname {R} (x^{\rho })-{\frac {1}{\log {x}}}+{\frac {1}{\pi }}\arctan {\frac {\pi }{\log {x}}}.$
The Riemann hypothesis suggests that every such non-trivial zero lies along Re(s) = 1/2.
Table of π(x), x / log x, and li(x)
The table shows how the three functions π(x), x / log x and li(x) compare at powers of 10. See also,[3][11] and[12]
x π(x) π(x) − x / log x li(x) − π(x) x / π(x) x / log x % Error
10 4 0 2 2.500 -8.57%
102 25 3 5 4.000 13.14%
103 168 23 10 5.952 13.83%
104 1,229 143 17 8.137 11.66%
105 9,592 906 38 10.425 9.45%
106 78,498 6,116 130 12.739 7.79%
107 664,579 44,158 339 15.047 6.64%
108 5,761,455 332,774 754 17.357 5.78%
109 50,847,534 2,592,592 1,701 19.667 5.10%
1010 455,052,511 20,758,029 3,104 21.975 4.56%
1011 4,118,054,813 169,923,159 11,588 24.283 4.13%
1012 37,607,912,018 1,416,705,193 38,263 26.590 3.77%
1013 346,065,536,839 11,992,858,452 108,971 28.896 3.47%
1014 3,204,941,750,802 102,838,308,636 314,890 31.202 3.21%
1015 29,844,570,422,669 891,604,962,452 1,052,619 33.507 2.99%
1016 279,238,341,033,925 7,804,289,844,393 3,214,632 35.812 2.79%
1017 2,623,557,157,654,233 68,883,734,693,928 7,956,589 38.116 2.63%
1018 24,739,954,287,740,860 612,483,070,893,536 21,949,555 40.420 2.48%
1019 234,057,667,276,344,607 5,481,624,169,369,961 99,877,775 42.725 2.34%
1020 2,220,819,602,560,918,840 49,347,193,044,659,702 222,744,644 45.028 2.22%
1021 21,127,269,486,018,731,928 446,579,871,578,168,707 597,394,254 47.332 2.11%
1022 201,467,286,689,315,906,290 4,060,704,006,019,620,994 1,932,355,208 49.636 2.02%
1023 1,925,320,391,606,803,968,923 37,083,513,766,578,631,309 7,250,186,216 51.939 1.93%
1024 18,435,599,767,349,200,867,866 339,996,354,713,708,049,069 17,146,907,278 54.243 1.84%
1025 176,846,309,399,143,769,411,680 3,128,516,637,843,038,351,228 55,160,980,939 56.546 1.77%
1026 1,699,246,750,872,437,141,327,603 28,883,358,936,853,188,823,261 155,891,678,121 58.850 1.70%
1027 16,352,460,426,841,680,446,427,399 267,479,615,610,131,274,163,365 508,666,658,006 61.153 1.64%
1028 157,589,269,275,973,410,412,739,598 2,484,097,167,669,186,251,622,127 1,427,745,660,374 63.456 1.58%
1029 1,520,698,109,714,272,166,094,258,063 23,130,930,737,541,725,917,951,446 4,551,193,622,464 65.759 1.52%
In the On-Line Encyclopedia of Integer Sequences, the π(x) column is sequence OEIS: A006880, π(x) − x/log x is sequence OEIS: A057835, and li(x) − π(x) is sequence OEIS: A057752.
The value for π(1024) was originally computed by J. Buethe, J. Franke, A. Jost, and T. Kleinjung assuming the Riemann hypothesis.[13] It was later verified unconditionally in a computation by D. J. Platt.[14] The value for π(1025) is due to J. Buethe, J. Franke, A. Jost, and T. Kleinjung.[15] The value for π(1026) was computed by D. B. Staple.[16] All other prior entries in this table were also verified as part of that work.
The value for 1027 was announced in 2015 by David Baugh and Kim Walisch.[17]
The value for 1028 was announced in 2020 by David Baugh and Kim Walisch.[18]
The value for 1029 was announced in 2022 by David Baugh and Kim Walisch.[19]
Algorithms for evaluating π(x)
A simple way to find $\pi (x)$, if $x$ is not too large, is to use the sieve of Eratosthenes to produce the primes less than or equal to $x$ and then to count them.
A more elaborate way of finding $\pi (x)$ is due to Legendre (using the inclusion–exclusion principle): given $x$, if $p_{1},p_{2},\ldots ,p_{n}$ are distinct prime numbers, then the number of integers less than or equal to $x$ which are divisible by no $p_{i}$ is
$\lfloor x\rfloor -\sum _{i}\left\lfloor {\frac {x}{p_{i}}}\right\rfloor +\sum _{i<j}\left\lfloor {\frac {x}{p_{i}p_{j}}}\right\rfloor -\sum _{i<j<k}\left\lfloor {\frac {x}{p_{i}p_{j}p_{k}}}\right\rfloor +\cdots $
(where $\lfloor {x}\rfloor $ denotes the floor function). This number is therefore equal to
$\pi (x)-\pi \left({\sqrt {x}}\right)+1$
when the numbers $p_{1},p_{2},\ldots ,p_{n}$ are the prime numbers less than or equal to the square root of $x$.
The Meissel–Lehmer algorithm
Main article: Meissel–Lehmer algorithm
In a series of articles published between 1870 and 1885, Ernst Meissel described (and used) a practical combinatorial way of evaluating $\ \pi (x)\ :$ :} Let $\ p_{1},p_{2},\ldots ,p_{n}\ $ be the first $\ n\ $ primes and denote by $\Phi (m,n)$ the number of natural numbers not greater than $\ m\ $ which are divisible by none of the $\ p_{i}\ $ for any $\ i\leq n\ .$ Then
$\Phi (m,n)=\Phi (m,n-1)-\Phi \left({\frac {m}{p_{n}}},n-1\right).$
Given a natural number $\ m\ ,$ if $\ n=\pi \left({\sqrt[{3}]{m}}\right)\ $ and if $\ \mu =\pi \left({\sqrt {m}}\right)-n\ ,$ then
$\pi (m)=\Phi (m,n)+n(\mu +1)+{\frac {\mu ^{2}-\mu }{2}}-1-\sum _{k=1}^{\mu }\pi \left({\frac {m}{p_{n+k}}}\right)\ .$
Using this approach, Meissel computed $\ \pi (x)\ ,$ for $\ x\ $ equal to 5×105, 106, 107, and 108.
In 1959, Derrick Henry Lehmer extended and simplified Meissel's method. Define, for real $\ m\ $ and for natural numbers $\ n\ $ and $\ k\ ,$ $\ P_{k}(m,n)\ $ as the number of numbers not greater than m with exactly k prime factors, all greater than $\ p_{n}\ .$ Furthermore, set $\ P_{0}(m,n)=1\ .$ Then
$\ \Phi (m,n)=\sum _{k=0}^{+\infty }P_{k}(m,n)\ $
where the sum actually has only finitely many nonzero terms. Let $\ y\ $ denote an integer such that $\ {\sqrt[{3}]{m\ }}\leq y\leq {\sqrt {m\ }}\ ,$ and set $\ n=\pi (y)\ .$ Then $\ P_{1}(m,n)=\pi (m)-n\ $ and $\ P_{k}(m,n)=0\ $ when $\ k\geq 3\ .$ Therefore,
$\ \pi (m)=\Phi (m,n)+n-1-P_{2}(m,n)\ $
The computation of $\ P_{2}(m,n)\ $ can be obtained this way:
$P_{2}(m,n)=\sum _{y<p\leq {\sqrt {m\ }}}\left(\pi \left({\frac {m}{p}}\right)-\pi (p)+1\right)\ ,$
where the sum is over prime numbers.
On the other hand, the computation of $\ \Phi (m,n)\ $ can be done using the following rules:
1. $\ \Phi (m,0)=\lfloor m\rfloor \ $
2. $\ \Phi (m,b)=\Phi (m,b-1)-\Phi \left({\frac {m}{p_{b}}},b-1\right)\ $
Using his method and an IBM 701, Lehmer was able to compute the correct value of $\ \pi \left(10^{9}\right)\ $ and missed the correct value of $\pi \left(10^{10}\right)$ by 1.[20]
Further improvements to this method were made by Lagarias, Miller, Odlyzko, Deléglise, and Rivat.[21]
Other prime-counting functions
Other prime-counting functions are also used because they are more convenient to work with.
Riemann's prime-power counting function
Riemann's prime-power counting function is usually denoted as $\ \Pi _{0}(x)\ $ or $\ J_{0}(x)\ .$ It has jumps of $\ {\tfrac {1}{\ n\ }}\ $ at prime powers $\ p^{n}\ ,$ and it takes a value halfway between the two sides at the discontinuities of $\ \pi (x)\ .$ That added detail is used because the function may then be defined by an inverse Mellin transform.
Formally, we may define $\ \Pi _{0}(x)\ $ by
$\ \Pi _{0}(x)={\frac {1}{2}}\left(\sum _{p^{n}<x}{\frac {1}{n}}~+~\sum _{p^{n}\leq x}{\frac {1}{n}}\right)\ $
where the variable p in each sum ranges over all primes within the specified limits.
We may also write
$\ \Pi _{0}(x)=\sum _{n=2}^{x}{\frac {\Lambda (n)}{\log n}}-{\frac {\Lambda (x)}{2\log x}}=\sum _{n=1}^{\infty }{\frac {1}{n}}\pi _{0}{\bigl (}x^{1/n}{\bigr )}\ $
where $\ \Lambda (n)\ $ is the von Mangoldt function and
$\pi _{0}(x)=\lim _{\varepsilon \to 0}{\frac {\ \pi (x-\varepsilon )+\pi (x+\varepsilon )\ }{2}}\ .$
The Möbius inversion formula then gives
$\pi _{0}(x)=\sum _{n=1}^{\infty }{\frac {\ \mu (n)\ }{n}}\ \Pi _{0}{\bigl (}x^{1/n}{\bigr )}\ ,$
where $\ \mu (n)\ $ is the Möbius function.
Knowing the relationship between the logarithm of the Riemann zeta function and the von Mangoldt function $\Lambda $, and using the Perron formula we have
$\ \log \zeta (s)=s\int _{0}^{\infty }\Pi _{0}(x)x^{-s-1}\ \mathrm {d} x\ $
Chebyshev's function
The Chebyshev function weights primes or prime powers pn by log(p):
$\ \theta (x)=\sum _{p\leq x}\log p\ $
$\ \psi (x)\ =\ \sum _{p^{n}\leq x}\log p\ =\ \sum _{n=1}^{\infty }\theta {\bigl (}x^{1/n}{\bigr )}\ =\ \sum _{n\leq x}\Lambda (n)\ .$
For $x\geq 2$,
$\ \vartheta (x)=\pi (x)\ \log x\ -\ \int _{2}^{x}{\frac {\ \pi (t)\ }{t}}\ \mathrm {d} t\ $
and
$\ \pi (x)={\frac {\vartheta (x)}{\ \log x\ }}+\int _{2}^{x}{\frac {\vartheta (t)}{\ t\ \log ^{2}(t)\ }}\ \mathrm {d} t\ .$[22]
Formulas for prime-counting functions
Formulas for prime-counting functions come in two kinds: arithmetic formulas and analytic formulas. Analytic formulas for prime-counting were the first used to prove the prime number theorem. They stem from the work of Riemann and von Mangoldt, and are generally known as explicit formulas.[23]
We have the following expression for the second Chebyshev function ψ:
$\psi _{0}(x)=x-\sum _{\rho }{\frac {x^{\rho }}{\rho }}-\log 2\pi -{\frac {1}{2}}\log \left(1-x^{-2}\right),$
where
$\psi _{0}(x)=\lim _{\varepsilon \to 0}{\frac {\psi (x-\varepsilon )+\psi (x+\varepsilon )}{2}}.$
Here ρ are the zeros of the Riemann zeta function in the critical strip, where the real part of ρ is between zero and one. The formula is valid for values of x greater than one, which is the region of interest. The sum over the roots is conditionally convergent, and should be taken in order of increasing absolute value of the imaginary part. Note that the same sum over the trivial roots gives the last subtrahend in the formula.
For $\Pi _{0}(x)$ we have a more complicated formula
$\Pi _{0}(x)=\operatorname {li} (x)-\sum _{\rho }\operatorname {li} (x^{\rho })-\log 2+\int _{x}^{\infty }{\frac {dt}{t\left(t^{2}-1\right)\log t}}.$
Again, the formula is valid for x > 1, while ρ are the nontrivial zeros of the zeta function ordered according to their absolute value. The integral is equal to the series over the trivial zeros:
$\int _{x}^{\infty }{\frac {\mathrm {d} t}{t\left(t^{2}-1\right)\log t}}=\int _{x}^{\infty }{\frac {1}{t\log t}}\left(\sum _{m}t^{-2m}\right)\,\mathrm {d} t=\sum _{m}\int _{x}^{\infty }{\frac {t^{-2m}}{t\log t}}\,\mathrm {d} t\,\,{\overset {(u=t^{-2m})}{=}}-\sum _{m}\operatorname {li} (x^{-2m})$
The first term li(x) is the usual logarithmic integral function; the expression li(xρ) in the second term should be considered as Ei(ρ log x), where Ei is the analytic continuation of the exponential integral function from negative reals to the complex plane with branch cut along the positive reals.
Thus, Möbius inversion formula gives us[10]
$\pi _{0}(x)=\operatorname {R} (x)-\sum _{\rho }\operatorname {R} (x^{\rho })-\sum _{m}\operatorname {R} (x^{-2m})$
valid for x > 1, where
$\operatorname {R} (x)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\operatorname {li} (x^{1/n})=1+\sum _{k=1}^{\infty }{\frac {(\log x)^{k}}{k!k\zeta (k+1)}}$
is Riemann's R-function[24] and μ(n) is the Möbius function. The latter series for it is known as Gram series.[25][26] Because $\log(x)<x$ for all $x>0$, this series converges for all positive x by comparison with the series for $e^{x}$. The logarithm in the Gram series of the sum over the non-trivial zero contribution should be evaluated as $\rho \log x$ and not $\log x^{\rho }$.
Folkmar Bornemann proved,[27] when assuming the conjecture that all zeros of the Riemann zeta function are simple,[note 1] that
$\operatorname {R} (e^{-2\pi t})={\frac {1}{\pi }}\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}t^{-2k-1}}{(2k+1)\zeta (2k+1)}}+{\frac {1}{2}}\sum _{\rho }{\frac {t^{-\rho }}{\rho \cos(\pi \rho /2)\zeta '(\rho )}}$
where $\rho $ runs over the non-trivial zeros of the Riemann zeta function and $t>0$.
The sum over non-trivial zeta zeros in the formula for $\pi _{0}(x)$ describes the fluctuations of $\pi _{0}(x),$ while the remaining terms give the "smooth" part of prime-counting function,[28] so one can use
$\operatorname {R} (x)-\sum _{m=1}^{\infty }\operatorname {R} (x^{-2m})$
as a good estimator of $\pi (x)$ for x > 1. In fact, since the second term approaches 0 as $x\to \infty $, while the amplitude of the "noisy" part is heuristically about ${\sqrt {x}}/\log x,$ estimating $\pi (x)$ by $\operatorname {R} (x)$ alone is just as good, and fluctuations of the distribution of primes may be clearly represented with the function
${\bigl (}\pi _{0}(x)-\operatorname {R} (x){\bigr )}{\frac {\log x}{\sqrt {x}}}.$
Inequalities
Here are some useful inequalities for π(x).
${\frac {x}{\log x}}<\pi (x)<1.25506{\frac {x}{\log x}}$
for x ≥ 17.
The left inequality holds for x ≥ 17 and the right inequality holds for x > 1. The constant 1.25506 is $ {\frac {30\log 113}{113}}$ to 5 decimal places, as $ {\frac {\pi (x)\log x}{x}}$ has its maximum value at x = 113.[29]
Pierre Dusart proved in 2010:
${\frac {x}{\log x-1}}<\pi (x)$ for $x\geq 5393$, and
$\pi (x)<{\frac {x}{\log x-1.1}}$ for $x\geq 60184$.[30]
Here are some inequalities for the nth prime, pn. The upper bound is due to Rosser (1941),[31] the lower one to Dusart (1999):[32]
$n(\log(n\log n)-1)<p_{n}<n{\log(n\log n)}$ for n ≥ 6.
The left inequality holds for n ≥ 2 and the right inequality holds for n ≥ 6.
An approximation for the nth prime number is
$p_{n}=n(\log(n\log n)-1)+{\frac {n(\log \log n-2)}{\log n}}+O\left({\frac {n(\log \log n)^{2}}{(\log n)^{2}}}\right).$
Ramanujan[33] proved that the inequality
$\pi (x)^{2}<{\frac {ex}{\log x}}\pi \left({\frac {x}{e}}\right)$
holds for all sufficiently large values of $x$.
In [30] Dusart proved (Proposition 6.6) that, for $n\geq 688383$,
$p_{n}\leq n\left(\log n+\log \log n-1+{\frac {\log \log n-2}{\log n}}\right),$
and (Proposition 6.7) that, for $n\geq 3$,
$p_{n}\geq n\left(\log n+\log \log n-1+{\frac {\log \log n-2.1}{\log n}}\right).$
More recently, Dusart[34] has proved (Theorem 5.1) that, for $x>1$,
$\pi (x)\leq {\frac {x}{\log x}}\left(1+{\frac {1}{\log x}}+{\frac {2}{\log ^{2}x}}+{\frac {7.59}{\log ^{3}x}}\right)$ ,
and that, for $x\geq 88789$,
$\pi (x)>{\frac {x}{\log x}}\left(1+{\frac {1}{\log x}}+{\frac {2}{\log ^{2}x}}\right).$
The Riemann hypothesis
The Riemann hypothesis implies a much tighter bound on the error in the estimate for $\pi (x)$, and hence to a more regular distribution of prime numbers,
$\pi (x)=\operatorname {li} (x)+O({\sqrt {x}}\log {x}).$
Specifically,[35]
$|\pi (x)-\operatorname {li} (x)|<{\frac {\sqrt {x}}{8\pi }}\,\log {x},\qquad {\text{for all }}x\geq 2657.$
See also
• Foias constant
• Bertrand's postulate
• Oppermann's conjecture
• Ramanujan prime
References
1. Bach, Eric; Shallit, Jeffrey (1996). Algorithmic Number Theory. MIT Press. volume 1 page 234 section 8.8. ISBN 0-262-02405-5.
2. Weisstein, Eric W. "Prime Counting Function". MathWorld.
3. "How many primes are there?". Chris K. Caldwell. Archived from the original on 2012-10-15. Retrieved 2008-12-02.
4. Dickson, Leonard Eugene (2005). History of the Theory of Numbers, Vol. I: Divisibility and Primality. Dover Publications. ISBN 0-486-44232-2.
5. Ireland, Kenneth; Rosen, Michael (1998). A Classical Introduction to Modern Number Theory (Second ed.). Springer. ISBN 0-387-97329-X.
6. See also Theorem 23 of A. E. Ingham (2000). The Distribution of Prime Numbers. Cambridge University Press. ISBN 0-521-39789-8.
7. Kevin Ford (November 2002). "Vinogradov's Integral and Bounds for the Riemann Zeta Function" (PDF). Proc. London Math. Soc. 85 (3): 565–633. arXiv:1910.08209. doi:10.1112/S0024611502013655. S2CID 121144007.
8. Mossinghoff, Michael J.; Trudgian, Timothy S. (2015). "Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function". J. Number Theory. 157: 329–349. arXiv:1410.3926. doi:10.1016/J.JNT.2015.05.010. S2CID 117968965.
9. Hutama, Daniel (2017). "Implementation of Riemann's Explicit Formula for Rational and Gaussian Primes in Sage" (PDF). Institut des sciences mathématiques.
10. Riesel, Hans; Göhl, Gunnar (1970). "Some calculations related to Riemann's prime number formula" (PDF). Mathematics of Computation. American Mathematical Society. 24 (112): 969–983. doi:10.2307/2004630. ISSN 0025-5718. JSTOR 2004630. MR 0277489.
11. "Tables of values of pi(x) and of pi2(x)". Tomás Oliveira e Silva. Retrieved 2008-09-14.
12. "A table of values of pi(x)". Xavier Gourdon, Pascal Sebah, Patrick Demichel. Retrieved 2008-09-14.
13. "Conditional Calculation of pi(1024)". Chris K. Caldwell. Retrieved 2010-08-03.
14. Platt, David J. (2012). "Computing π(x) Analytically)". arXiv:1203.5712 [math.NT].
15. "How Many Primes Are There?". J. Buethe. Retrieved 2015-09-01.
16. Staple, Douglas (19 August 2015). The combinatorial algorithm for computing pi(x) (Thesis). Dalhousie University. Retrieved 2015-09-01.
17. Walisch, Kim (September 6, 2015). "New confirmed pi(10^27) prime counting function record". Mersenne Forum.
18. Baugh, David (Oct 26, 2020). "New confirmed pi(10^28) prime counting function record". OEIS.
19. Baugh, David (Feb 28, 2022). "New confirmed pi(10^29) prime counting function record". OEIS.
20. Lehmer, Derrick Henry (1 April 1958). "On the exact number of primes less than a given limit". Illinois J. Math. 3 (3): 381–388. Retrieved 1 February 2017.
21. Deléglise, Marc; Rivat, Joel (January 1996). "Computing π(x): The Meissel, Lehmer, Lagarias, Miller, Odlyzko method" (PDF). Mathematics of Computation. 65 (213): 235–245. doi:10.1090/S0025-5718-96-00674-6.
22. Apostol, Tom M. (2010). Introduction to Analytic Number Theory. Springer.
23. Titchmarsh, E.C. (1960). The Theory of Functions, 2nd ed. Oxford University Press.
24. Weisstein, Eric W. "Riemann Prime Counting Function". MathWorld.
25. Riesel, Hans (1994). Prime Numbers and Computer Methods for Factorization. Progress in Mathematics. Vol. 126 (2nd ed.). Birkhäuser. pp. 50–51. ISBN 0-8176-3743-5.
26. Weisstein, Eric W. "Gram Series". MathWorld.
27. Bornemann, Folkmar. "Solution of a Problem Posed by Jörg Waldvogel" (PDF).
28. "The encoding of the prime distribution by the zeta zeros". Matthew Watkins. Retrieved 2008-09-14.
29. Rosser, J. Barkley; Schoenfeld, Lowell (1962). "Approximate formulas for some functions of prime numbers". Illinois J. Math. 6: 64–94. doi:10.1215/ijm/1255631807. ISSN 0019-2082. Zbl 0122.05001.
30. Dusart, Pierre (2 Feb 2010). "Estimates of Some Functions Over Primes without R.H.". arXiv:1002.0442v1 [math.NT].
31. Rosser, Barkley (1941). "Explicit bounds for some functions of prime numbers". American Journal of Mathematics. 63 (1): 211–232. doi:10.2307/2371291. JSTOR 2371291.
32. Dusart, Pierre (1999). "The $k$th prime is greater than $k(\ln k+\ln \ln k-1)$ for $k\geq 2$". Mathematics of Computation. 68 (225): 411–415. doi:10.1090/S0025-5718-99-01037-6.
33. Berndt, Bruce C. (2012-12-06). Ramanujan's Notebooks, Part IV. Springer Science & Business Media. pp. 112–113. ISBN 9781461269328.
34. Dusart, Pierre (January 2018). "Explicit estimates of some functions over primes". Ramanujan Journal. 45 (1): 225–234. doi:10.1007/s11139-016-9839-4. S2CID 125120533.
35. Schoenfeld, Lowell (1976). "Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II". Mathematics of Computation. American Mathematical Society. 30 (134): 337–360. doi:10.2307/2005976. ISSN 0025-5718. JSTOR 2005976. MR 0457374.
Notes
1. Montgomery showed that (assuming the Riemann hypothesis) at least 2/3 of all zeros are simple.
External links
• Chris Caldwell, The Nth Prime Page at The Prime Pages.
• Tomás Oliveira e Silva, Tables of prime-counting functions.
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Bridgeland stability condition
In mathematics, and especially algebraic geometry, a Bridgeland stability condition, defined by Tom Bridgeland, is an algebro-geometric stability condition defined on elements of a triangulated category. The case of original interest and particular importance is when this triangulated category is the derived category of coherent sheaves on a Calabi–Yau manifold, and this situation has fundamental links to string theory and the study of D-branes.
Such stability conditions were introduced in a rudimentary form by Michael Douglas called $\Pi $-stability and used to study BPS B-branes in string theory.[1] This concept was made precise by Bridgeland, who phrased these stability conditions categorically, and initiated their study mathematically.[2]
Definition
The definitions in this section are presented as in the original paper of Bridgeland, for arbitrary triangulated categories.[2] Let ${\mathcal {D}}$ be a triangulated category.
Slicing of triangulated categories
A slicing ${\mathcal {P}}$ of ${\mathcal {D}}$ is a collection of full additive subcategories ${\mathcal {P}}(\varphi )$ for each $\varphi \in \mathbb {R} $ such that
• ${\mathcal {P}}(\varphi )[1]={\mathcal {P}}(\varphi +1)$ for all $\varphi $, where $[1]$ is the shift functor on the triangulated category,
• if $\varphi _{1}>\varphi _{2}$ and $A\in {\mathcal {P}}(\varphi _{1})$ and $B\in {\mathcal {P}}(\varphi _{2})$, then $\operatorname {Hom} (A,B)=0$, and
• for every object $E\in {\mathcal {D}}$ there exists a finite sequence of real numbers $\varphi _{1}>\varphi _{2}>\cdots >\varphi _{n}$ and a collection of triangles
with $A_{i}\in {\mathcal {P}}(\varphi _{i})$ for all $i$.
The last property should be viewed as axiomatically imposing the existence of Harder–Narasimhan filtrations on elements of the category ${\mathcal {D}}$.
Stability conditions
A Bridgeland stability condition on a triangulated category ${\mathcal {D}}$ is a pair $(Z,{\mathcal {P}})$ consisting of a slicing ${\mathcal {P}}$ and a group homomorphism $Z:K({\mathcal {D}})\to \mathbb {C} $, where $K({\mathcal {D}})$ is the Grothendieck group of ${\mathcal {D}}$, called a central charge, satisfying
• if $0\neq E\in {\mathcal {P}}(\varphi )$ then $Z(E)=m(E)\exp(i\pi \varphi )$ for some strictly positive real number $m(E)\in \mathbb {R} _{>0}$.
It is convention to assume the category ${\mathcal {D}}$ is essentially small, so that the collection of all stability conditions on ${\mathcal {D}}$ forms a set $\operatorname {Stab} ({\mathcal {D}})$. In good circumstances, for example when ${\mathcal {D}}={\mathcal {D}}^{b}\operatorname {Coh} (X)$ is the derived category of coherent sheaves on a complex manifold $X$, this set actually has the structure of a complex manifold itself.
Technical remarks about stability condition
It is shown by Bridgeland that the data of a Bridgeland stability condition is equivalent to specifying a bounded t-structure ${\mathcal {P}}(>0)$ on the category ${\mathcal {D}}$ and a central charge $Z:K({\mathcal {A}})\to \mathbb {C} $ on the heart ${\mathcal {A}}={\mathcal {P}}((0,1])$ of this t-structure which satisfies the Harder–Narasimhan property above.[2]
An element $E\in {\mathcal {A}}$ is semi-stable (resp. stable) with respect to the stability condition $(Z,{\mathcal {P}})$ if for every surjection $E\to F$ for $F\in {\mathcal {A}}$, we have $\varphi (E)\leq ({\text{resp.}}<)\,\varphi (F)$ where $Z(E)=m(E)\exp(i\pi \varphi (E))$ and similarly for $F$.
Examples
From the Harder–Narasimhan filtration
Recall the Harder–Narasimhan filtration for a smooth projective curve $X$ implies for any coherent sheaf $E$ there is a filtration
$0=E_{0}\subset E_{1}\subset \cdots \subset E_{n}=E$
such that the factors $E_{j}/E_{j-1}$ have slope $\mu _{i}={\text{deg}}/{\text{rank}}$. We can extend this filtration to a bounded complex of sheaves $E^{\bullet }$ by considering the filtration on the cohomology sheaves $E^{i}=H^{i}(E^{\bullet })[+i]$ and defining the slope of $E_{j}^{i}=\mu _{i}+j$, giving a function
$\phi :K(X)\to \mathbb {R} $
for the central charge.
Elliptic curves
There is an analysis by Bridgeland for the case of Elliptic curves. He finds[2][3] there is an equivalence
${\text{Stab}}(X)/{\text{Aut}}(X)\cong {\text{GL}}^{+}(2,\mathbb {R} )/{\text{SL}}(2,\mathbb {Z} )$
where ${\text{Stab}}(X)$ is the set of stability conditions and ${\text{Aut}}(X)$ is the set of autoequivalences of the derived category $D^{b}(X)$.
References
1. Douglas, M.R., Fiol, B. and Römelsberger, C., 2005. Stability and BPS branes. Journal of High Energy Physics, 2005(09), p. 006.
2. Bridgeland, Tom (2006-02-08). "Stability conditions on triangulated categories". arXiv:math/0212237.
3. Uehara, Hokuto (2015-11-18). "Autoequivalences of derived categories of elliptic surfaces with non-zero Kodaira dimension". pp. 10–12. arXiv:1501.06657 [math.AG].
Papers
• Stability conditions on $A_{n}$ singularities
• Interactions between autoequivalences, stability conditions, and moduli problems
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Π01 class
In computability theory, a Π01 class is a subset of 2ω of a certain form. These classes are of interest as technical tools within recursion theory and effective descriptive set theory. They are also used in the application of recursion theory to other branches of mathematics (Cenzer 1999, p. 39).
Definition
The set 2<ω consists of all finite sequences of 0s and 1s, while the set 2ω consists of all infinite sequences of 0s and 1s (that is, functions from ω = {0, 1, 2, ...} to the set {0,1}).
A tree on 2<ω is a subset of 2<ω that is closed under taking initial segments. An element f of 2ω is a path through a tree T on 2<ω if every finite initial segment of f is in T.
A (lightface) Π01 class is a subset C of 2ω for which there is a computable tree T such that C consists of exactly the paths through T. A boldface Π01 class is a subset D of 2ω for which there is an oracle f in 2ω and a subtree tree T of 2< ω from computable from f such that D is the set of paths through T.
As effectively closed sets
The boldface Π01 classes are exactly the same as the closed sets of 2ω and thus the same as the boldface Π01 subsets of 2ω in the Borel hierarchy.
Lightface Π01 classes in 2ω (that is, Π01 classes whose tree is computable with no oracle) correspond to effectively closed sets. A subset B of 2ω is effectively closed if there is a recursively enumerable sequence ⟨σi : i ∈ ω⟩ of elements of 2< ω such that each g ∈ 2ω is in B if and only if there exists some i such that σi is an initial segment of B.
Relationship with effective theories
For each effectively axiomatized theory T of first-order logic, the set of all completions of T is a $\Pi _{1}^{0}$ class. Moreover, for each $\Pi _{1}^{0}$ subset S of $2^{\omega }$ there is an effectively axiomatized theory T such that each element of S computes a completion of T, and each completion of T computes an element of S (Jockusch and Soare 1972b).
See also
• Arithmetical hierarchy
• Basis theorem (computability)
References
• Cenzer, Douglas (1999), "$\Pi _{1}^{0}$ classes in computability theory", Handbook of computability theory, Stud. Logic Found. Math., vol. 140, Amsterdam: North-Holland, pp. 37 85, MR 1720779
• Jockush, Carl G.; Soare, Robert I. (1972a), "Degrees of members of $\Pi _{1}^{0}$ classes." (PDF), Pacific Journal of Mathematics, 40 (3): 605–616, doi:10.2140/pjm.1972.40.605
• Jockush, Carl G.; Soare, Robert I. (1972b), "$\Pi _{1}^{0}$ Classes and Degrees of Theories", Transactions of the American Mathematical Society, 173: 33–56, doi:10.1090/s0002-9947-1972-0316227-0
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Countably barrelled space
In functional analysis, a topological vector space (TVS) is said to be countably barrelled if every weakly bounded countable union of equicontinuous subsets of its continuous dual space is again equicontinuous. This property is a generalization of barrelled spaces.
Definition
A TVS X with continuous dual space $X^{\prime }$ is said to be countably barrelled if $B^{\prime }\subseteq X^{\prime }$ is a weak-* bounded subset of $X^{\prime }$ that is equal to a countable union of equicontinuous subsets of $X^{\prime }$, then $B^{\prime }$ is itself equicontinuous.[1] A Hausdorff locally convex TVS is countably barrelled if and only if each barrel in X that is equal to the countable intersection of closed convex balanced neighborhoods of 0 is itself a neighborhood of 0.[1]
σ-barrelled space
A TVS with continuous dual space $X^{\prime }$ is said to be σ-barrelled if every weak-* bounded (countable) sequence in $X^{\prime }$ is equicontinuous.[1]
Sequentially barrelled space
A TVS with continuous dual space $X^{\prime }$ is said to be sequentially barrelled if every weak-* convergent sequence in $X^{\prime }$ is equicontinuous.[1]
Properties
Every countably barrelled space is a countably quasibarrelled space, a σ-barrelled space, a σ-quasi-barrelled space, and a sequentially barrelled space.[1] An H-space is a TVS whose strong dual space is countably barrelled.[1]
Every countably barrelled space is a σ-barrelled space and every σ-barrelled space is sequentially barrelled.[1] Every σ-barrelled space is a σ-quasi-barrelled space.[1]
A locally convex quasi-barrelled space that is also a 𝜎-barrelled space is a barrelled space.[1]
Examples and sufficient conditions
Every barrelled space is countably barrelled.[1] However, there exist semi-reflexive countably barrelled spaces that are not barrelled.[1] The strong dual of a distinguished space and of a metrizable locally convex space is countably barrelled.[1]
Counter-examples
There exist σ-barrelled spaces that are not countably barrelled.[1] There exist normed DF-spaces that are not countably barrelled.[1] There exists a quasi-barrelled space that is not a 𝜎-barrelled space.[1] There exist σ-barrelled spaces that are not Mackey spaces.[1] There exist σ-barrelled spaces that are not countably quasi-barrelled spaces and thus not countably barrelled.[1] There exist sequentially barrelled spaces that are not σ-quasi-barrelled.[1] There exist quasi-complete locally convex TVSs that are not sequentially barrelled.[1]
See also
• Barrelled space
• H-space
• Quasibarrelled space
References
1. Khaleelulla 1982, pp. 28–63.
• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
• Wong (1979). Schwartz spaces, nuclear spaces, and tensor products. Berlin New York: Springer-Verlag. ISBN 3-540-09513-6. OCLC 5126158.
Topological vector spaces (TVSs)
Basic concepts
• Banach space
• Completeness
• Continuous linear operator
• Linear functional
• Fréchet space
• Linear map
• Locally convex space
• Metrizability
• Operator topologies
• Topological vector space
• Vector space
Main results
• Anderson–Kadec
• Banach–Alaoglu
• Closed graph theorem
• F. Riesz's
• Hahn–Banach (hyperplane separation
• Vector-valued Hahn–Banach)
• Open mapping (Banach–Schauder)
• Bounded inverse
• Uniform boundedness (Banach–Steinhaus)
Maps
• Bilinear operator
• form
• Linear map
• Almost open
• Bounded
• Continuous
• Closed
• Compact
• Densely defined
• Discontinuous
• Topological homomorphism
• Functional
• Linear
• Bilinear
• Sesquilinear
• Norm
• Seminorm
• Sublinear function
• Transpose
Types of sets
• Absolutely convex/disk
• Absorbing/Radial
• Affine
• Balanced/Circled
• Banach disks
• Bounding points
• Bounded
• Complemented subspace
• Convex
• Convex cone (subset)
• Linear cone (subset)
• Extreme point
• Pre-compact/Totally bounded
• Prevalent/Shy
• Radial
• Radially convex/Star-shaped
• Symmetric
Set operations
• Affine hull
• (Relative) Algebraic interior (core)
• Convex hull
• Linear span
• Minkowski addition
• Polar
• (Quasi) Relative interior
Types of TVSs
• Asplund
• B-complete/Ptak
• Banach
• (Countably) Barrelled
• BK-space
• (Ultra-) Bornological
• Brauner
• Complete
• Convenient
• (DF)-space
• Distinguished
• F-space
• FK-AK space
• FK-space
• Fréchet
• tame Fréchet
• Grothendieck
• Hilbert
• Infrabarreled
• Interpolation space
• K-space
• LB-space
• LF-space
• Locally convex space
• Mackey
• (Pseudo)Metrizable
• Montel
• Quasibarrelled
• Quasi-complete
• Quasinormed
• (Polynomially
• Semi-) Reflexive
• Riesz
• Schwartz
• Semi-complete
• Smith
• Stereotype
• (B
• Strictly
• Uniformly) convex
• (Quasi-) Ultrabarrelled
• Uniformly smooth
• Webbed
• With the approximation property
• Mathematics portal
• Category
• Commons
|
Countably quasi-barrelled space
In functional analysis, a topological vector space (TVS) is said to be countably quasi-barrelled if every strongly bounded countable union of equicontinuous subsets of its continuous dual space is again equicontinuous. This property is a generalization of quasibarrelled spaces.
Definition
A TVS X with continuous dual space $X^{\prime }$ is said to be countably quasi-barrelled if $B^{\prime }\subseteq X^{\prime }$ is a strongly bounded subset of $X^{\prime }$ that is equal to a countable union of equicontinuous subsets of $X^{\prime }$, then $B^{\prime }$ is itself equicontinuous.[1] A Hausdorff locally convex TVS is countably quasi-barrelled if and only if each bornivorous barrel in X that is equal to the countable intersection of closed convex balanced neighborhoods of 0 is itself a neighborhood of 0.[1]
σ-quasi-barrelled space
A TVS with continuous dual space $X^{\prime }$ is said to be σ-quasi-barrelled if every strongly bounded (countable) sequence in $X^{\prime }$ is equicontinuous.[1]
Sequentially quasi-barrelled space
A TVS with continuous dual space $X^{\prime }$ is said to be sequentially quasi-barrelled if every strongly convergent sequence in $X^{\prime }$ is equicontinuous.
Properties
Every countably quasi-barrelled space is a σ-quasi-barrelled space.
Examples and sufficient conditions
Every barrelled space, every countably barrelled space, and every quasi-barrelled space is countably quasi-barrelled and thus also σ-quasi-barrelled space.[1] The strong dual of a distinguished space and of a metrizable locally convex space is countably quasi-barrelled.[1]
Every σ-barrelled space is a σ-quasi-barrelled space.[1] Every DF-space is countably quasi-barrelled.[1] A σ-quasi-barrelled space that is sequentially complete is a σ-barrelled space.[1]
There exist σ-barrelled spaces that are not Mackey spaces.[1] There exist σ-barrelled spaces (which are consequently σ-quasi-barrelled spaces) that are not countably quasi-barrelled spaces.[1] There exist sequentially complete Mackey spaces that are not σ-quasi-barrelled.[1] There exist sequentially barrelled spaces that are not σ-quasi-barrelled.[1] There exist quasi-complete locally convex TVSs that are not sequentially barrelled.[1]
See also
• Barrelled space
• Countably barrelled space
• DF-space
• H-space
• Quasibarrelled space
References
1. Khaleelulla 1982, pp. 28–63.
• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
• Wong (1979). Schwartz spaces, nuclear spaces, and tensor products. Berlin New York: Springer-Verlag. ISBN 3-540-09513-6. OCLC 5126158.
Topological vector spaces (TVSs)
Basic concepts
• Banach space
• Completeness
• Continuous linear operator
• Linear functional
• Fréchet space
• Linear map
• Locally convex space
• Metrizability
• Operator topologies
• Topological vector space
• Vector space
Main results
• Anderson–Kadec
• Banach–Alaoglu
• Closed graph theorem
• F. Riesz's
• Hahn–Banach (hyperplane separation
• Vector-valued Hahn–Banach)
• Open mapping (Banach–Schauder)
• Bounded inverse
• Uniform boundedness (Banach–Steinhaus)
Maps
• Bilinear operator
• form
• Linear map
• Almost open
• Bounded
• Continuous
• Closed
• Compact
• Densely defined
• Discontinuous
• Topological homomorphism
• Functional
• Linear
• Bilinear
• Sesquilinear
• Norm
• Seminorm
• Sublinear function
• Transpose
Types of sets
• Absolutely convex/disk
• Absorbing/Radial
• Affine
• Balanced/Circled
• Banach disks
• Bounding points
• Bounded
• Complemented subspace
• Convex
• Convex cone (subset)
• Linear cone (subset)
• Extreme point
• Pre-compact/Totally bounded
• Prevalent/Shy
• Radial
• Radially convex/Star-shaped
• Symmetric
Set operations
• Affine hull
• (Relative) Algebraic interior (core)
• Convex hull
• Linear span
• Minkowski addition
• Polar
• (Quasi) Relative interior
Types of TVSs
• Asplund
• B-complete/Ptak
• Banach
• (Countably) Barrelled
• BK-space
• (Ultra-) Bornological
• Brauner
• Complete
• Convenient
• (DF)-space
• Distinguished
• F-space
• FK-AK space
• FK-space
• Fréchet
• tame Fréchet
• Grothendieck
• Hilbert
• Infrabarreled
• Interpolation space
• K-space
• LB-space
• LF-space
• Locally convex space
• Mackey
• (Pseudo)Metrizable
• Montel
• Quasibarrelled
• Quasi-complete
• Quasinormed
• (Polynomially
• Semi-) Reflexive
• Riesz
• Schwartz
• Semi-complete
• Smith
• Stereotype
• (B
• Strictly
• Uniformly) convex
• (Quasi-) Ultrabarrelled
• Uniformly smooth
• Webbed
• With the approximation property
• Mathematics portal
• Category
• Commons
|
Sigma-ring
In mathematics, a nonempty collection of sets is called a 𝜎-ring (pronounced sigma-ring) if it is closed under countable union and relative complementation.
Formal definition
Let ${\mathcal {R}}$ be a nonempty collection of sets. Then ${\mathcal {R}}$ is a 𝜎-ring if:
1. Closed under countable unions: $\bigcup _{n=1}^{\infty }A_{n}\in {\mathcal {R}}$ if $A_{n}\in {\mathcal {R}}$ for all $n\in \mathbb {N} $
2. Closed under relative complementation: $A\setminus B\in {\mathcal {R}}$ if $A,B\in {\mathcal {R}}$
Properties
These two properties imply:
$\bigcap _{n=1}^{\infty }A_{n}\in {\mathcal {R}}$
whenever $A_{1},A_{2},\ldots $ are elements of ${\mathcal {R}}.$
This is because
$\bigcap _{n=1}^{\infty }A_{n}=A_{1}\setminus \bigcup _{n=2}^{\infty }\left(A_{1}\setminus A_{n}\right).$
Every 𝜎-ring is a δ-ring but there exist δ-rings that are not 𝜎-rings.
Similar concepts
If the first property is weakened to closure under finite union (that is, $A\cup B\in {\mathcal {R}}$ whenever $A,B\in {\mathcal {R}}$) but not countable union, then ${\mathcal {R}}$ is a ring but not a 𝜎-ring.
Uses
𝜎-rings can be used instead of 𝜎-fields (𝜎-algebras) in the development of measure and integration theory, if one does not wish to require that the universal set be measurable. Every 𝜎-field is also a 𝜎-ring, but a 𝜎-ring need not be a 𝜎-field.
A 𝜎-ring ${\mathcal {R}}$ that is a collection of subsets of $X$ induces a 𝜎-field for $X.$ Define ${\mathcal {A}}=\{E\subseteq X:E\in {\mathcal {R}}\ {\text{or}}\ E^{c}\in {\mathcal {R}}\}.$ Then ${\mathcal {A}}$ is a 𝜎-field over the set $X$ - to check closure under countable union, recall a $\sigma $-ring is closed under countable intersections. In fact ${\mathcal {A}}$ is the minimal 𝜎-field containing ${\mathcal {R}}$ since it must be contained in every 𝜎-field containing ${\mathcal {R}}.$
See also
• δ-ring – Ring closed under countable intersections
• Field of sets – Algebraic concept in measure theory, also referred to as an algebra of sets
• Join (sigma algebra) – Algebric structure of set algebraPages displaying short descriptions of redirect targets
• 𝜆-system (Dynkin system) – Family closed under complements and countable disjoint unions
• Measurable function – Function for which the preimage of a measurable set is measurable
• Monotone class – theoremPages displaying wikidata descriptions as a fallbackPages displaying short descriptions with no spaces
• π-system – Family of sets closed under intersection
• Ring of sets – Family closed under unions and relative complements
• Sample space – Set of all possible outcomes or results of a statistical trial or experiment
• 𝜎 additivity – Mapping function
• σ-algebra – Algebric structure of set algebra
• 𝜎-ideal – Family closed under subsets and countable unions
References
• Walter Rudin, 1976. Principles of Mathematical Analysis, 3rd. ed. McGraw-Hill. Final chapter uses 𝜎-rings in development of Lebesgue theory.
Families ${\mathcal {F}}$ of sets over $\Omega $
Is necessarily true of ${\mathcal {F}}\colon $
or, is ${\mathcal {F}}$ closed under:
Directed
by $\,\supseteq $
$A\cap B$ $A\cup B$ $B\setminus A$ $\Omega \setminus A$ $A_{1}\cap A_{2}\cap \cdots $ $A_{1}\cup A_{2}\cup \cdots $ $\Omega \in {\mathcal {F}}$ $\varnothing \in {\mathcal {F}}$ F.I.P.
π-system
Semiring Never
Semialgebra (Semifield) Never
Monotone class only if $A_{i}\searrow $only if $A_{i}\nearrow $
𝜆-system (Dynkin System) only if
$A\subseteq B$
only if $A_{i}\nearrow $ or
they are disjoint
Never
Ring (Order theory)
Ring (Measure theory) Never
δ-Ring Never
𝜎-Ring Never
Algebra (Field) Never
𝜎-Algebra (𝜎-Field) Never
Dual ideal
Filter NeverNever$\varnothing \not \in {\mathcal {F}}$
Prefilter (Filter base) NeverNever$\varnothing \not \in {\mathcal {F}}$
Filter subbase NeverNever$\varnothing \not \in {\mathcal {F}}$
Open Topology
(even arbitrary $\cup $)
Never
Closed Topology
(even arbitrary $\cap $)
Never
Is necessarily true of ${\mathcal {F}}\colon $
or, is ${\mathcal {F}}$ closed under:
directed
downward
finite
intersections
finite
unions
relative
complements
complements
in $\Omega $
countable
intersections
countable
unions
contains $\Omega $ contains $\varnothing $ Finite
Intersection
Property
Additionally, a semiring is a π-system where every complement $B\setminus A$ is equal to a finite disjoint union of sets in ${\mathcal {F}}.$
A semialgebra is a semiring that contains $\Omega .$
$A,B,A_{1},A_{2},\ldots $ are arbitrary elements of ${\mathcal {F}}$ and it is assumed that ${\mathcal {F}}\neq \varnothing .$
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Phi-hiding assumption
The phi-hiding assumption or Φ-hiding assumption is an assumption about the difficulty of finding small factors of φ(m) where m is a number whose factorization is unknown, and φ is Euler's totient function. The security of many modern cryptosystems comes from the perceived difficulty of certain problems. Since P vs. NP problem is still unresolved, cryptographers cannot be sure computationally intractable problems exist. Cryptographers thus make assumptions as to which problems are hard. It is commonly believed that if m is the product of two large primes, then calculating φ(m) is currently computationally infeasible; this assumption is required for the security of the RSA Cryptosystem. The Φ-Hiding assumption is a stronger assumption, namely that if p1 and p2 are small primes exactly one of which divides φ(m), there is no polynomial-time algorithm which can distinguish which of the primes p1 and p2 divides φ(m) with probability significantly greater than one-half.
This assumption was first stated in the 1999 paper Computationally Private Information Retrieval with Polylogarithmic Communication,[1] where it was used in a Private Information Retrieval scheme.
Applications
The Phi-hiding assumption has found applications in the construction of a few cryptographic primitives. Some of the constructions include:
• Computationally Private Information Retrieval with Polylogarithmic Communication (1999)
• Efficient Private Bidding and Auctions with an Oblivious Third Party (1999)
• Single-Database Private Information Retrieval with Constant Communication Rate (2005)
• Password authenticated key exchange using hidden smooth subgroups (2005)
References
1. Cachin, Christian; Micali, Silvio; Stadler, Markus (1999). "Computationally Private Information Retrieval with Polylogarithmic Communication". In Stern, Jacques (ed.). Advances in Cryptology — EUROCRYPT '99. Lecture Notes in Computer Science. Vol. 1592. Springer. pp. 402–414. doi:10.1007/3-540-48910-X_28. ISBN 978-3-540-65889-4. S2CID 29690672.
Computational hardness assumptions
Number theoretic
• Integer factorization
• Phi-hiding
• RSA problem
• Strong RSA
• Quadratic residuosity
• Decisional composite residuosity
• Higher residuosity
Group theoretic
• Discrete logarithm
• Diffie-Hellman
• Decisional Diffie–Hellman
• Computational Diffie–Hellman
Pairings
• External Diffie–Hellman
• Sub-group hiding
• Decision linear
Lattices
• Shortest vector problem (gap)
• Closest vector problem (gap)
• Learning with errors
• Ring learning with errors
• Short integer solution
Non-cryptographic
• Exponential time hypothesis
• Unique games conjecture
• Planted clique conjecture
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χ-bounded
In graph theory, a $\chi $-bounded family ${\mathcal {F}}$ of graphs is one for which there is some function $f$ such that, for every integer $t$ the graphs in ${\mathcal {F}}$ with $t=\omega (G)$ (clique number) can be colored with at most $f(t)$ colors. This concept and its notation were formulated by András Gyárfás.[1] The use of the Greek letter chi in the term $\chi $-bounded is based on the fact that the chromatic number of a graph $G$ is commonly denoted $\chi (G)$.
Nontriviality
It is not true that the family of all graphs is $\chi $-bounded. As Zykov (1949) and Mycielski (1955) showed, there exist triangle-free graphs of arbitrarily large chromatic number,[2][3] so for these graphs it is not possible to define a finite value of $f(2)$. Thus, $\chi $-boundedness is a nontrivial concept, true for some graph families and false for others.[4]
Specific classes
Every class of graphs of bounded chromatic number is (trivially) $\chi $-bounded, with $f(t)$ equal to the bound on the chromatic number. This includes, for instance, the planar graphs, the bipartite graphs, and the graphs of bounded degeneracy. Complementarily, the graphs in which the independence number is bounded are also $\chi $-bounded, as Ramsey's theorem implies that they have large cliques.
Vizing's theorem can be interpreted as stating that the line graphs are $\chi $-bounded, with $f(t)=t+1$.[5][6] The claw-free graphs more generally are also $\chi $-bounded with $f(t)\leq t^{2}$. This can be seen by using Ramsey's theorem to show that, in these graphs, a vertex with many neighbors must be part of a large clique. This bound is nearly tight in the worst case, but connected claw-free graphs that include three mutually-nonadjacent vertices have even smaller chromatic number, $f(t)=2t$.[5]
Other $\chi $-bounded graph families include:
• The perfect graphs, with $f(t)=t$
• The graphs of boxicity two[7], which is the intersection graphs of axis-parallel rectangles, with $f(t)\in O(t\log(t))$(big O notation)[8]
• The graphs of bounded clique-width[9]
• The intersection graphs of scaled and translated copies of any compact convex shape in the plane[10]
• The polygon-circle graphs, with $f(t)=2^{t}$
• The circle graphs, with $f(t)=7t^{2}$[11] and (generalizing circle graphs) the "outerstring graphs", intersection graphs of bounded curves in the plane that all touch the unbounded face of the arrangement of the curves[12]
• The outerstring graph is an intersection graph of curves that lie in a common half-plane and have one endpoint on the boundary of that half-plane[13]
• The intersection graphs of curves that cross a fixed curve between 1 and $n\in \mathbb {N} $ times[14]
However, although intersection graphs of convex shapes, circle graphs, and outerstring graphs are all special cases of string graphs, the string graphs themselves are not $\chi $-bounded. They include as a special case the intersection graphs of line segments, which are also not $\chi $-bounded.[4]
Unsolved problems
Unsolved problem in mathematics:
Are all tree-free graph classes $\chi $-bounded?
(more unsolved problems in mathematics)
According to the Gyárfás–Sumner conjecture, for every tree $T$, the graphs that do not contain $T$ as an induced subgraph are $\chi $-bounded. For instance, this would include the case of claw-free graphs, as a claw is a special kind of tree. However, the conjecture is known to be true only for certain special trees, including paths[1] and radius-two trees.[15]
Another problem on $\chi $-boundedness was posed by Louis Esperet, who asked whether every hereditary class of graphs that is $\chi $-bounded has a function $f(t)$ that grows at most polynomially as a function of $t$.[6] A strong counterexample to Esperet's conjecture was announced in 2022 by Briański, Davies, and Walczak, who proved that there exist $\chi $-bounded hereditary classes whose function $f(t)$ can be chosen arbitrarily as long as it grows more quickly than a certain cubic polynomial.[16]
References
1. Gyárfás, A. (1987), "Problems from the world surrounding perfect graphs" (PDF), Proceedings of the International Conference on Combinatorial Analysis and its Applications (Pokrzywna, 1985), Zastosowania Matematyki, 19 (3–4): 413–441 (1988), MR 0951359
2. Zykov, A. A. (1949), "О некоторых свойствах линейных комплексов" [On some properties of linear complexes], Mat. Sbornik, New Series (in Russian), 24 (66): 163–188, MR 0035428. Translated into English in Amer. Math. Soc. Translation, 1952, MR0051516. As cited by Pawlik et al. (2014)
3. Mycielski, Jan (1955), "Sur le coloriage des graphs", Colloq. Math. (in French), 3 (2): 161–162, doi:10.4064/cm-3-2-161-162, MR 0069494
4. Pawlik, Arkadiusz; Kozik, Jakub; Krawczyk, Tomasz; Lasoń, Michał; Micek, Piotr; Trotter, William T.; Walczak, Bartosz (2014), "Triangle-free intersection graphs of line segments with large chromatic number", Journal of Combinatorial Theory, Series B, 105: 6–10, arXiv:1209.1595, doi:10.1016/j.jctb.2013.11.001, MR 3171778, S2CID 9705484
5. Chudnovsky, Maria; Seymour, Paul (2010), "Claw-free graphs VI. Colouring", Journal of Combinatorial Theory, Series B, 100 (6): 560–572, doi:10.1016/j.jctb.2010.04.005, MR 2718677
6. Karthick, T.; Maffray, Frédéric (2016), "Vizing bound for the chromatic number on some graph classes", Graphs and Combinatorics, 32 (4): 1447–1460, doi:10.1007/s00373-015-1651-1, MR 3514976, S2CID 41279514
7. Asplund, E.; Grünbaum, B. (1960), "On a coloring problem", Mathematica Scandinavica, 8: 181–188, doi:10.7146/math.scand.a-10607, MR 0144334
8. Chalermsook; Walczak (2020), Coloring and Maximum Weight Independent Set of Rectangles, arXiv:2007.07880
9. Dvořák, Zdeněk; Král', Daniel (2012), "Classes of graphs with small rank decompositions are $\chi $-bounded", Electronic Journal of Combinatorics, 33 (4): 679–683, arXiv:1107.2161, doi:10.1016/j.ejc.2011.12.005, MR 3350076, S2CID 5530520
10. Kim, Seog-Jin; Kostochka, Alexandr; Nakprasit, Kittikorn (2004), "On the chromatic number of intersection graphs of convex sets in the plane", Electronic Journal of Combinatorics, 11 (1), R52, doi:10.37236/1805, MR 2097318
11. Davies; McCarty (2020), "Circle graphs are quadratically χ‐bounded", Bulletin of the London Mathematical Society, 53 (3): 673–679, arXiv:1905.11578v1, doi:10.1112/blms.12447, S2CID 167217723
12. Rok, Alexandre; Walczak, Bartosz (2014), "Outerstring graphs are $\chi $-bounded", Proceedings of the Thirtieth Annual Symposium on Computational Geometry (SoCG'14), New York: ACM, pp. 136–143, doi:10.1145/2582112.2582115, MR 3382292, S2CID 33362942
13. Rok; Walczak (2019), "Outerstring Graphs are $\chi$-Bounded", SIAM Journal on Discrete Mathematics, 33 (4): 2181–2199, arXiv:1312.1559, doi:10.1137/17M1157374, S2CID 9474387
14. Rok; Walczak (2019), "Coloring Curves that Cross a Fixed Curve", Discrete & Computational Geometry, 61 (4): 830–851, doi:10.1007/s00454-018-0031-z, S2CID 124706442
15. Kierstead, H. A.; Penrice, S. G. (1994), "Radius two trees specify $\chi $-bounded classes", Journal of Graph Theory, 18 (2): 119–129, doi:10.1002/jgt.3190180203, MR 1258244
16. Briański, Marcin; Davies, James; Walczak, Bartosz (2022), Separating polynomial $\chi $-boundedness from $\chi $-boundedness, arXiv:2201.08814
External links
• Chi-bounded, Open Problem Garden
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Buchholz's ordinal
In mathematics, ψ0(Ωω), widely known as Buchholz's ordinal, is a large countable ordinal that is used to measure the proof-theoretic strength of some mathematical systems. In particular, it is the proof theoretic ordinal of the subsystem $\Pi _{1}^{1}$-CA0 of second-order arithmetic;[1][2] this is one of the "big five" subsystems studied in reverse mathematics (Simpson 1999). It is also the proof-theoretic ordinal of ${\mathsf {ID_{<\omega }}}$, the theory of finitely iterated inductive definitions, and of $KP\ell _{0}$,[3] a fragment of Kripke-Platek set theory extended by an axiom stating every set is contained in an admissible set. Buchholz's ordinal is also the order type of the segment bounded by $D_{0}D_{\omega }0$ in Buchholz's ordinal notation ${\mathsf {(OT,<)}}$.[1] Lastly, it can be expressed as the limit of the sequence: $\varepsilon _{0}=\psi _{0}(\Omega )$, ${\mathsf {BHO}}=\psi _{0}(\Omega _{2})$, $\psi _{0}(\Omega _{3})$, ...
Definition
Main article: Buchholz psi functions
• $\Omega _{0}=1$, and $\Omega _{n}=\aleph _{n}$ for n > 0.
• $C_{i}(\alpha )$ is the closure of $\Omega _{i}$ under addition and the $\psi _{\eta }(\mu )$ function itself (the latter of which only for $\mu <\alpha $ and $\eta \leq \omega $).
• $\psi _{i}(\alpha )$ is the smallest ordinal not in $C_{i}(\alpha )$.
• Thus, ψ0(Ωω) is the smallest ordinal not in the closure of $1$ under addition and the $\psi _{\eta }(\mu )$ function itself (the latter of which only for $\mu <\Omega _{\omega }$ and $\eta \leq \omega $).
References
• G. Takeuti, Proof theory, 2nd edition 1987 ISBN 0-444-10492-5
• K. Schütte, Proof theory, Springer 1977 ISBN 0-387-07911-4
1. Buchholz, W. (1986-01-01). "A new system of proof-theoretic ordinal functions". Annals of Pure and Applied Logic. 32: 195–207. doi:10.1016/0168-0072(86)90052-7. ISSN 0168-0072.
2. Simpson, Stephen G. (2009). Subsystems of Second Order Arithmetic. Perspectives in Logic (2 ed.). Cambridge: Cambridge University Press. ISBN 978-0-521-88439-6.
3. T. Carlson, Elementary Patterns of Resemblance (1999). Accessed 12 August 2022.
Large countable ordinals
• First infinite ordinal ω
• Epsilon numbers ε0
• Feferman–Schütte ordinal Γ0
• Ackermann ordinal θ(Ω2)
• small Veblen ordinal θ(Ωω)
• large Veblen ordinal θ(ΩΩ)
• Bachmann–Howard ordinal ψ(εΩ+1)
• Buchholz's ordinal ψ0(Ωω)
• Takeuti–Feferman–Buchholz ordinal ψ(εΩω+1)
• Proof-theoretic ordinals of the theories of iterated inductive definitions
• Nonrecursive ordinal ≥ ωCK
1
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ω-automaton
In automata theory, a branch of theoretical computer science, an ω-automaton (or stream automaton) is a variation of finite automata that runs on infinite, rather than finite, strings as input. Since ω-automata do not stop, they have a variety of acceptance conditions rather than simply a set of accepting states.
ω-automata are useful for specifying behavior of systems that are not expected to terminate, such as hardware, operating systems and control systems. For such systems, one may want to specify a property such as "for every request, an acknowledge eventually follows", or its negation "there is a request that is not followed by an acknowledge". The former is a property of infinite words: one cannot say of a finite sequence that it satisfies this property.
Classes of ω-automata include the Büchi automata, Rabin automata, Streett automata, parity automata and Muller automata, each deterministic or non-deterministic. These classes of ω-automata differ only in terms of acceptance condition. They all recognize precisely the regular ω-languages except for the deterministic Büchi automata, which is strictly weaker than all the others. Although all these types of automata recognize the same set of ω-languages, they nonetheless differ in succinctness of representation for a given ω-language.
Deterministic ω-automata
Formally, a deterministic ω-automaton is a tuple A = (Q,Σ,δ,Q0,Acc) that consists of the following components:
• Q is a finite set. The elements of Q are called the states of A.
• Σ is a finite set called the alphabet of A.
• δ: Q × Σ → Q is a function, called the transition function of A.
• Q0 is an element of Q, called the initial state.
• Acc is the acceptance condition, formally a subset of Qω.
An input for A is an infinite string over the alphabet Σ, i.e. it is an infinite sequence α = (a1,a2,a3,...). The run of A on such an input is an infinite sequence ρ = (r0,r1,r2,...) of states, defined as follows:
• r0 = q0.
• r1 = δ(r0,a1).
• r2 = δ(r1,a2).
...
• rn = δ(rn-1,an).
The main purpose of an ω-automaton is to define a subset of the set of all inputs: The set of accepted inputs. Whereas in the case of an ordinary finite automaton every run ends with a state rn and the input is accepted if and only if rn is an accepting state, the definition of the set of accepted inputs is more complicated for ω-automata. Here we must look at the entire run ρ. The input is accepted if the corresponding run is in Acc. The set of accepted input ω-words is called the recognized ω-language by the automaton, which is denoted as L(A).
The definition of Acc as a subset of Qω is purely formal and not suitable for practice because normally such sets are infinite. The difference between various types of ω-automata (Büchi, Rabin etc.) consists in how they encode certain subsets Acc of Qω as finite sets, and therefore in which such subsets they can encode.
Nondeterministic ω-automata
Formally, a nondeterministic ω-automaton is a tuple A = (Q,Σ,Δ,Q0,Acc) that consists of the following components:
• Q is a finite set. The elements of Q are called the states of A.
• Σ is a finite set called the alphabet of A.
• Δ is a subset of Q × Σ × Q and is called the transition relation of A.
• Q0 is a subset of Q, called the initial set of states.
• Acc is the acceptance condition, a subset of Qω.
Unlike a deterministic ω-automaton, which has a transition function δ, the non-deterministic version has a transition relation Δ. Note that Δ can be regarded as a function : Q × Σ → P(Q) from Q × Σ to the power set P(Q). Thus, given a state qn and a symbol an, the next state qn+1 is not necessarily determined uniquely, rather there is a set of possible next states.
A run of A on the input α = (a1,a2,a3,...) is any infinite sequence ρ = (r0,r1,r2,...) of states that satisfies the following conditions:
• r0 is an element of Q0.
• r1 is an element of Δ(r0,a1).
• r2 is an element of Δ(r1,a2).
...
• rn is an element of Δ(rn-1,an).
A nondeterministic ω-automaton may admit many different runs on any given input, or none at all. The input is accepted if at least one of the possible runs is accepting. Whether a run is accepting depends only on Acc, as for deterministic ω-automata. Every deterministic ω-automaton can be regarded as a nondeterministic ω-automaton by taking Δ to be the graph of δ. The definitions of runs and acceptance for deterministic ω-automata are then special cases of the nondeterministic cases.
Acceptance conditions
Acceptance conditions may be infinite sets of ω-words. However, people mostly study acceptance conditions that are finitely representable. The following lists a variety of popular acceptance conditions.
Before discussing the list, let's make the following observation. In the case of infinitely running systems, one is often interested in whether certain behavior is repeated infinitely often. For example, if a network card receives infinitely many ping requests, then it may fail to respond to some of the requests but should respond to an infinite subset of received ping requests. This motivates the following definition: For any run ρ, let Inf(ρ) be the set of states that occur infinitely often in ρ. This notion of certain states being visited infinitely often will be helpful in defining the following acceptance conditions.
• A Büchi automaton is an ω-automaton A that uses the following acceptance condition, for some subset F of Q:
Büchi condition
A accepts exactly those runs ρ for which Inf(ρ) ∩ F is not empty, i.e. there is an accepting state that occurs infinitely often in ρ.
• A Rabin automaton is an ω-automaton A that uses the following acceptance condition, for some set Ω of pairs (Bi,Gi) of sets of states:
Rabin condition
A accepts exactly those runs ρ for which there exists a pair (Bi,Gi) in Ω such that Bi ∩ Inf(ρ) is empty and Gi ∩ Inf(ρ) is not empty.
• A Streett automaton is an ω-automaton A that uses the following acceptance condition, for some set Ω of pairs (Bi,Gi) of sets of states:
Streett condition
A accepts exactly those runs ρ such that for all pairs (Bi,Gi) in Ω, Bi ∩ Inf(ρ) is empty or Gi ∩ Inf(ρ) is not empty.
• A parity automaton is an automaton A whose set of states is Q = {0,1,2,...,k} for some natural number k, and that has the following acceptance condition:
Parity condition
A accepts ρ if and only if the smallest number in Inf(ρ) is even.
• A Muller automaton is an ω-automaton A that uses the following acceptance condition, for a subset F of P(Q) (the power set of Q):
Muller condition
A accepts exactly those runs ρ for which Inf(ρ) is an element of F.
Every Büchi automaton can be regarded as a Muller automaton. It suffices to replace F by F' consisting of all subsets of Q that contain at least one element of F. Similarly every Rabin, Streett or parity automaton can also be regarded as a Muller automaton.
Example
The following ω-language L over the alphabet Σ = {0,1}, which can be recognized by a nondeterministic Büchi automaton: L consists of all ω-words in Σω in which 1 occurs only finitely many times. A non-deterministic Büchi automaton recognizing L needs only two states q0 (the initial state) and q1. Δ consists of the triples (q0,0,q0), (q0,1,q0), (q0,0,q1) and (q1,0,q1). F = {q1}. For any input α in which 1 occurs only finitely many times, there is a run that stays in state q0 as long as there are 1s to read, and goes to state q1 afterwards. This run is successful. If there are infinitely many 1s, then there is only one possible run: the one that always stays in state q0. (Once the machine has left q0 and reached q1, it cannot return. If another 1 is read, there is no successor state.)
Notice that above language cannot be recognized by a deterministic Büchi automaton, which is strictly less expressive than its non-deterministic counterpart.
Expressive power of ω-automata
An ω-language over a finite alphabet Σ is a set of ω-words over Σ, i.e. it is a subset of Σω. An ω-language over Σ is said to be recognized by an ω-automaton A (with the same alphabet) if it is the set of all ω-words accepted by A. The expressive power of a class of ω-automata is measured by the class of all ω-languages that can be recognized by some automaton in the class.
The nondeterministic Büchi, parity, Rabin, Streett, and Muller automata, respectively, all recognize exactly the same class of ω-languages.[1] These are known as the ω-Kleene closure of the regular languages or as the regular ω-languages. Using different proofs it can also be shown that the deterministic parity, Rabin, Streett, and Muller automata all recognize the regular ω-languages. It follows from this that the class of regular ω-languages is closed under complementation. However, the example above shows that the class of deterministic Büchi automata is strictly weaker.
Conversion between ω-automata
Because nondeterministic Muller, Rabin, Streett, parity, and Büchi automata are equally expressive, they can be translated to each other. Let us use the following abbreviation $\{N,D\}\times \{M,R,S,P,B\}$: for example, NB stands for nondeterministic Büchi ω-automaton, while DP stands for deterministic parity ω-automaton. Then the following holds.
• Clearly, any deterministic automaton can be viewed as a nondeterministic one.
• $NB\rightarrow NR/NS/NP$ with no blow-up in the state space.
• $NR\rightarrow NB$ with a polynomial blow-up in the state space, i.e., the number of states in the resulting NB is $2nm+1$, where $n$ is the number of states in the NB and $m$ is the number of Rabin acceptance pairs (see, for example, [2]).
• $NS/NM/NP\rightarrow NB$ with exponential blow-up in the state space.
• $NB\rightarrow DR/DP$ with exponential blow-up in the state space. This determinization result uses Safra's construction.
A comprehensive overview of translations can be found on the referenced web source. [3]
Applications to decidability
ω-automata can be used to prove decidability of S1S, the monadic second-order (MSO) theory of natural numbers under successor. Infinite-tree automata extend ω-automata to infinite trees and can be used to prove decidability of S2S, the MSO theory with two successors, and this can be extended to the MSO theory of graphs with bounded (given a fixed bound) treewidth.
Further reading
• Farwer, Berndt (2002), "ω-Automata", in Grädel, Erich; Thomas, Wolfgang; Wilke, Thomas (eds.), Automata, Logics, and Infinite Games, Lecture Notes in Computer Science, Springer, pp. 3–21, ISBN 978-3-540-00388-5.
• Perrin, Dominique; Pin, Jean-Éric (2004), Infinite Words: Automata, Semigroups, Logic and Games, Elsevier, ISBN 978-0-12-532111-2
• Thomas, Wolfgang (1990), "Automata on infinite objects", in van Leeuwen, Jan (ed.), Handbook of Theoretical Computer Science, vol. B, MIT Press, pp. 133–191, ISBN 978-0-262-22039-2
• Bakhadyr Khoussainov; Anil Nerode (6 December 2012). Automata Theory and its Applications. Springer Science & Business Media. ISBN 978-1-4612-0171-7.
References
1. Safra, S. (1988), "On the complexity of ω-automata", Proceedings of the 29th Annual Symposium on Foundations of Computer Science (FOCS '88), Washington, DC, USA: IEEE Computer Society, pp. 319–327, doi:10.1109/SFCS.1988.21948.
2. Esparza, Javier (2017), Automata Theory: An Algorithmic Approach (PDF)
3. Boker, Udi (18 April 2018). "Word-Automata Translations". Udi Boker's webpage. Retrieved 30 March 2019.
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ω-bounded space
In mathematics, an ω-bounded space is a topological space in which the closure of every countable subset is compact. More generally, if P is some property of subspaces, then a P-bounded space is one in which every subspace with property P has compact closure.
Every compact space is ω-bounded, and every ω-bounded space is countably compact. The long line is ω-bounded but not compact.
The bagpipe theorem describes the ω-bounded surfaces.
References
• Juhász, Istvan; van Mill, Jan; Weiss, William (2013), "Variations on ω-boundedness", Israel Journal of Mathematics, 194 (2): 745–766, doi:10.1007/s11856-012-0062-8, MR 3047090
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Omega-categorical theory
In mathematical logic, an omega-categorical theory is a theory that has exactly one countably infinite model up to isomorphism. Omega-categoricity is the special case κ = $\aleph _{0}$ = ω of κ-categoricity, and omega-categorical theories are also referred to as ω-categorical. The notion is most important for countable first-order theories.
Equivalent conditions for omega-categoricity
Many conditions on a theory are equivalent to the property of omega-categoricity. In 1959 Erwin Engeler, Czesław Ryll-Nardzewski and Lars Svenonius, proved several independently.[1] Despite this, the literature still widely refers to the Ryll-Nardzewski theorem as a name for these conditions. The conditions included with the theorem vary between authors.[2][3]
Given a countable complete first-order theory T with infinite models, the following are equivalent:
• The theory T is omega-categorical.
• Every countable model of T has an oligomorphic automorphism group (that is, there are finitely many orbits on Mn for every n).
• Some countable model of T has an oligomorphic automorphism group.[4]
• The theory T has a model which, for every natural number n, realizes only finitely many n-types, that is, the Stone space Sn(T) is finite.
• For every natural number n, T has only finitely many n-types.
• For every natural number n, every n-type is isolated.
• For every natural number n, up to equivalence modulo T there are only finitely many formulas with n free variables, in other words, for every n, the nth Lindenbaum–Tarski algebra of T is finite.
• Every model of T is atomic.
• Every countable model of T is atomic.
• The theory T has a countable atomic and saturated model.
• The theory T has a saturated prime model.
Examples
The theory of any countably infinite structure which is homogeneous over a finite relational language is omega-categorical.[5] Hence, the following theories are omega-categorical:
• The theory of dense linear orders without endpoints (Cantor's isomorphism theorem)
• The theory of the Rado graph
• The theory of infinite linear spaces over any finite field
Notes
1. Rami Grossberg, José Iovino and Olivier Lessmann, A primer of simple theories
2. Hodges, Model Theory, p. 341.
3. Rothmaler, p. 200.
4. Cameron (1990) p.30
5. Macpherson, p. 1607.
References
• Cameron, Peter J. (1990), Oligomorphic permutation groups, London Mathematical Society Lecture Note Series, vol. 152, Cambridge: Cambridge University Press, ISBN 0-521-38836-8, Zbl 0813.20002
• Chang, Chen Chung; Keisler, H. Jerome (1989) [1973], Model Theory, Elsevier, ISBN 978-0-7204-0692-4
• Hodges, Wilfrid (1993), Model theory, Cambridge: Cambridge University Press, ISBN 978-0-521-30442-9
• Hodges, Wilfrid (1997), A shorter model theory, Cambridge: Cambridge University Press, ISBN 978-0-521-58713-6
• Macpherson, Dugald (2011), "A survey of homogeneous structures", Discrete Mathematics, 311 (15): 1599–1634, doi:10.1016/j.disc.2011.01.024, MR 2800979
• Poizat, Bruno (2000), A Course in Model Theory: An Introduction to Contemporary Mathematical Logic, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98655-5
• Rothmaler, Philipp (2000), Introduction to Model Theory, New York: Taylor & Francis, ISBN 978-90-5699-313-9
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Ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend enumeration to infinite sets.[1]
A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used. To extend this process to various infinite sets, ordinal numbers are defined more generally as linearly ordered labels that include the natural numbers and have the property that every set of ordinals has a least element (this is needed for giving a meaning to "the least unused element").[2] This more general definition allows us to define an ordinal number $\omega $ (omega) that is greater than every natural number, along with ordinal numbers $\omega +1$, $\omega +2$, etc., which are even greater than $\omega $.
A linear order such that every non-empty subset has a least element is called a well-order. The axiom of choice implies that every set can be well-ordered, and given two well-ordered sets, one is isomorphic to an initial segment of the other. So ordinal numbers exist and are essentially unique.
Ordinal numbers are distinct from cardinal numbers, which measure the size of sets. Although the distinction between ordinals and cardinals is not always apparent on finite sets (one can go from one to the other just by counting labels), they are very different in the infinite case, where different infinite ordinals can correspond to sets having the same cardinal. Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated, although none of these operations are commutative.
Ordinals were introduced by Georg Cantor in 1883[3] in order to accommodate infinite sequences and classify derived sets, which he had previously introduced in 1872 while studying the uniqueness of trigonometric series.[4]
Ordinals extend the natural numbers
A natural number (which, in this context, includes the number 0) can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence. When restricted to finite sets, these two concepts coincide, since all linear orders of a finite set are isomorphic.
When dealing with infinite sets, however, one has to distinguish between the notion of size, which leads to cardinal numbers, and the notion of position, which leads to the ordinal numbers described here. This is because while any set has only one size (its cardinality), there are many nonisomorphic well-orderings of any infinite set, as explained below.
Whereas the notion of cardinal number is associated with a set with no particular structure on it, the ordinals are intimately linked with the special kind of sets that are called well-ordered. A well-ordered set is a totally ordered set in which every non-empty subset has a least element (a totally ordered set is an ordered set such that, given two distinct elements, one is less than the other). Equivalently, assuming the axiom of dependent choice, it is a totally ordered set without any infinite decreasing sequence — though there may be infinite increasing sequences. Ordinals may be used to label the elements of any given well-ordered set (the smallest element being labelled 0, the one after that 1, the next one 2, "and so on"), and to measure the "length" of the whole set by the least ordinal that is not a label for an element of the set. This "length" is called the order type of the set.
Any ordinal is defined by the set of ordinals that precede it. In fact, the most common definition of ordinals identifies each ordinal as the set of ordinals that precede it. For example, the ordinal 42 is generally identified as the set {0, 1, 2, ..., 41}. Conversely, any set S of ordinals that is downward closed — meaning that for any ordinal α in S and any ordinal β < α, β is also in S — is (or can be identified with) an ordinal.
This definition of ordinals in terms of sets allows for infinite ordinals. The smallest infinite ordinal is $\omega $, which can be identified with the set of natural numbers (so that the ordinal associated with every natural number precedes $\omega $). Indeed, the set of natural numbers is well-ordered—as is any set of ordinals—and since it is downward closed, it can be identified with the ordinal associated with it.
Perhaps a clearer intuition of ordinals can be formed by examining a first few of them: as mentioned above, they start with the natural numbers, 0, 1, 2, 3, 4, 5, ... After all natural numbers comes the first infinite ordinal, ω, and after that come ω+1, ω+2, ω+3, and so on. (Exactly what addition means will be defined later on: just consider them as names.) After all of these come ω·2 (which is ω+ω), ω·2+1, ω·2+2, and so on, then ω·3, and then later on ω·4. Now the set of ordinals formed in this way (the ω·m+n, where m and n are natural numbers) must itself have an ordinal associated with it: and that is ω2. Further on, there will be ω3, then ω4, and so on, and ωω, then ωωω, then later ωωωω, and even later ε0 (epsilon nought) (to give a few examples of relatively small—countable—ordinals). This can be continued indefinitely (as every time one says "and so on" when enumerating ordinals, it defines a larger ordinal). The smallest uncountable ordinal is the set of all countable ordinals, expressed as ω1 or $\Omega $.[5][6]
Definitions
Well-ordered sets
In a well-ordered set, every non-empty subset contains a distinct smallest element. Given the axiom of dependent choice, this is equivalent to saying that the set is totally ordered and there is no infinite decreasing sequence (the latter being easier to visualize). In practice, the importance of well-ordering is justified by the possibility of applying transfinite induction, which says, essentially, that any property that passes on from the predecessors of an element to that element itself must be true of all elements (of the given well-ordered set). If the states of a computation (computer program or game) can be well-ordered—in such a way that each step is followed by a "lower" step—then the computation will terminate.
It is inappropriate to distinguish between two well-ordered sets if they only differ in the "labeling of their elements", or more formally: if the elements of the first set can be paired off with the elements of the second set such that if one element is smaller than another in the first set, then the partner of the first element is smaller than the partner of the second element in the second set, and vice versa. Such a one-to-one correspondence is called an order isomorphism, and the two well-ordered sets are said to be order-isomorphic or similar (with the understanding that this is an equivalence relation).
Formally, if a partial order ≤ is defined on the set S, and a partial order ≤' is defined on the set S' , then the posets (S,≤) and (S' ,≤') are order isomorphic if there is a bijection f that preserves the ordering. That is, f(a) ≤' f(b) if and only if a ≤ b. Provided there exists an order isomorphism between two well-ordered sets, the order isomorphism is unique: this makes it quite justifiable to consider the two sets as essentially identical, and to seek a "canonical" representative of the isomorphism type (class). This is exactly what the ordinals provide, and it also provides a canonical labeling of the elements of any well-ordered set. Every well-ordered set (S,<) is order-isomorphic to the set of ordinals less than one specific ordinal number under their natural ordering. This canonical set is the order type of (S,<).
Essentially, an ordinal is intended to be defined as an isomorphism class of well-ordered sets: that is, as an equivalence class for the equivalence relation of "being order-isomorphic". There is a technical difficulty involved, however, in the fact that the equivalence class is too large to be a set in the usual Zermelo–Fraenkel (ZF) formalization of set theory. But this is not a serious difficulty. The ordinal can be said to be the order type of any set in the class.
Definition of an ordinal as an equivalence class
The original definition of ordinal numbers, found for example in the Principia Mathematica, defines the order type of a well-ordering as the set of all well-orderings similar (order-isomorphic) to that well-ordering: in other words, an ordinal number is genuinely an equivalence class of well-ordered sets. This definition must be abandoned in ZF and related systems of axiomatic set theory because these equivalence classes are too large to form a set. However, this definition still can be used in type theory and in Quine's axiomatic set theory New Foundations and related systems (where it affords a rather surprising alternative solution to the Burali-Forti paradox of the largest ordinal).
Von Neumann definition of ordinals
See also: Set-theoretic definition of natural numbers and Zermelo ordinals
First several von Neumann ordinals
0 ={} =∅
1 ={0} ={∅}
2 ={0,1} ={∅,{∅}}
3 ={0,1,2} ={∅,{∅},{∅,{∅}}}
4 ={0,1,2,3} ={∅,{∅},{∅,{∅}},{∅,{∅},{∅,{∅}}}}
Rather than defining an ordinal as an equivalence class of well-ordered sets, it will be defined as a particular well-ordered set that (canonically) represents the class. Thus, an ordinal number will be a well-ordered set; and every well-ordered set will be order-isomorphic to exactly one ordinal number.
For each well-ordered set T, $a\mapsto T_{<a}$ defines an order isomorphism between T and the set of all subsets of T having the form $T_{<a}:=\{x\in T\mid x<a\}$ ordered by inclusion. This motivates the standard definition, suggested by John von Neumann at the age of 19, now called definition of von Neumann ordinals: "each ordinal is the well-ordered set of all smaller ordinals". In symbols, $\lambda =[0,\lambda )$.[7][8] Formally:
A set S is an ordinal if and only if S is strictly well-ordered with respect to set membership and every element of S is also a subset of S.
The natural numbers are thus ordinals by this definition. For instance, 2 is an element of 4 = {0, 1, 2, 3}, and 2 is equal to {0, 1} and so it is a subset of {0, 1, 2, 3}.
It can be shown by transfinite induction that every well-ordered set is order-isomorphic to exactly one of these ordinals, that is, there is an order preserving bijective function between them.
Furthermore, the elements of every ordinal are ordinals themselves. Given two ordinals S and T, S is an element of T if and only if S is a proper subset of T. Moreover, either S is an element of T, or T is an element of S, or they are equal. So every set of ordinals is totally ordered. Further, every set of ordinals is well-ordered. This generalizes the fact that every set of natural numbers is well-ordered.
Consequently, every ordinal S is a set having as elements precisely the ordinals smaller than S. For example, every set of ordinals has a supremum, the ordinal obtained by taking the union of all the ordinals in the set. This union exists regardless of the set's size, by the axiom of union.
The class of all ordinals is not a set. If it were a set, one could show that it was an ordinal and thus a member of itself, which would contradict its strict ordering by membership. This is the Burali-Forti paradox. The class of all ordinals is variously called "Ord", "ON", or "∞".
An ordinal is finite if and only if the opposite order is also well-ordered, which is the case if and only if each of its non-empty subsets has a maximum.
Other definitions
There are other modern formulations of the definition of ordinal. For example, assuming the axiom of regularity, the following are equivalent for a set x:
• x is a (von Neumann) ordinal,
• x is a transitive set, and set membership is trichotomous on x,
• x is a transitive set totally ordered by set inclusion,
• x is a transitive set of transitive sets.
These definitions cannot be used in non-well-founded set theories. In set theories with urelements, one has to further make sure that the definition excludes urelements from appearing in ordinals.
Transfinite sequence
If α is any ordinal and X is a set, an α-indexed sequence of elements of X is a function from α to X. This concept, a transfinite sequence (if α is infinite) or ordinal-indexed sequence, is a generalization of the concept of a sequence. An ordinary sequence corresponds to the case α = ω, while a finite α corresponds to a tuple, a.k.a. string.
Transfinite induction
Main article: Transfinite induction
Transfinite induction holds in any well-ordered set, but it is so important in relation to ordinals that it is worth restating here.
Any property that passes from the set of ordinals smaller than a given ordinal α to α itself, is true of all ordinals.
That is, if P(α) is true whenever P(β) is true for all β < α, then P(α) is true for all α. Or, more practically: in order to prove a property P for all ordinals α, one can assume that it is already known for all smaller β < α.
Transfinite recursion
Transfinite induction can be used not only to prove things, but also to define them. Such a definition is normally said to be by transfinite recursion – the proof that the result is well-defined uses transfinite induction. Let F denote a (class) function F to be defined on the ordinals. The idea now is that, in defining F(α) for an unspecified ordinal α, one may assume that F(β) is already defined for all β < α and thus give a formula for F(α) in terms of these F(β). It then follows by transfinite induction that there is one and only one function satisfying the recursion formula up to and including α.
Here is an example of definition by transfinite recursion on the ordinals (more will be given later): define function F by letting F(α) be the smallest ordinal not in the set {F(β) | β < α}, that is, the set consisting of all F(β) for β < α. This definition assumes the F(β) known in the very process of defining F; this apparent vicious circle is exactly what definition by transfinite recursion permits. In fact, F(0) makes sense since there is no ordinal β < 0, and the set {F(β) | β < 0} is empty. So F(0) is equal to 0 (the smallest ordinal of all). Now that F(0) is known, the definition applied to F(1) makes sense (it is the smallest ordinal not in the singleton set {F(0)} = {0}), and so on (the and so on is exactly transfinite induction). It turns out that this example is not very exciting, since provably F(α) = α for all ordinals α, which can be shown, precisely, by transfinite induction.
Successor and limit ordinals
Any nonzero ordinal has the minimum element, zero. It may or may not have a maximum element. For example, 42 has maximum 41 and ω+6 has maximum ω+5. On the other hand, ω does not have a maximum since there is no largest natural number. If an ordinal has a maximum α, then it is the next ordinal after α, and it is called a successor ordinal, namely the successor of α, written α+1. In the von Neumann definition of ordinals, the successor of α is $\alpha \cup \{\alpha \}$ since its elements are those of α and α itself.[7]
A nonzero ordinal that is not a successor is called a limit ordinal. One justification for this term is that a limit ordinal is the limit in a topological sense of all smaller ordinals (under the order topology).
When $\langle \alpha _{\iota }|\iota <\gamma \rangle $ is an ordinal-indexed sequence, indexed by a limit $\gamma $ and the sequence is increasing, i.e. $\alpha _{\iota }<\alpha _{\rho }$ whenever $\iota <\rho ,$ its limit is defined as the least upper bound of the set $\{\alpha _{\iota }|\iota <\gamma \},$ that is, the smallest ordinal (it always exists) greater than any term of the sequence. In this sense, a limit ordinal is the limit of all smaller ordinals (indexed by itself). Put more directly, it is the supremum of the set of smaller ordinals.
Another way of defining a limit ordinal is to say that α is a limit ordinal if and only if:
There is an ordinal less than α and whenever ζ is an ordinal less than α, then there exists an ordinal ξ such that ζ < ξ < α.
So in the following sequence:
0, 1, 2, ..., ω, ω+1
ω is a limit ordinal because for any smaller ordinal (in this example, a natural number) there is another ordinal (natural number) larger than it, but still less than ω.
Thus, every ordinal is either zero, or a successor (of a well-defined predecessor), or a limit. This distinction is important, because many definitions by transfinite recursion rely upon it. Very often, when defining a function F by transfinite recursion on all ordinals, one defines F(0), and F(α+1) assuming F(α) is defined, and then, for limit ordinals δ one defines F(δ) as the limit of the F(β) for all β<δ (either in the sense of ordinal limits, as previously explained, or for some other notion of limit if F does not take ordinal values). Thus, the interesting step in the definition is the successor step, not the limit ordinals. Such functions (especially for F nondecreasing and taking ordinal values) are called continuous. Ordinal addition, multiplication and exponentiation are continuous as functions of their second argument (but can be defined non-recursively).
Indexing classes of ordinals
Any well-ordered set is similar (order-isomorphic) to a unique ordinal number $\alpha $; in other words, its elements can be indexed in increasing fashion by the ordinals less than $\alpha $. This applies, in particular, to any set of ordinals: any set of ordinals is naturally indexed by the ordinals less than some $\alpha $. The same holds, with a slight modification, for classes of ordinals (a collection of ordinals, possibly too large to form a set, defined by some property): any class of ordinals can be indexed by ordinals (and, when the class is unbounded in the class of all ordinals, this puts it in class-bijection with the class of all ordinals). So the $\gamma $-th element in the class (with the convention that the "0-th" is the smallest, the "1-st" is the next smallest, and so on) can be freely spoken of. Formally, the definition is by transfinite induction: the $\gamma $-th element of the class is defined (provided it has already been defined for all $\beta <\gamma $), as the smallest element greater than the $\beta $-th element for all $\beta <\gamma $.
This could be applied, for example, to the class of limit ordinals: the $\gamma $-th ordinal, which is either a limit or zero is $\omega \cdot \gamma $ (see ordinal arithmetic for the definition of multiplication of ordinals). Similarly, one can consider additively indecomposable ordinals (meaning a nonzero ordinal that is not the sum of two strictly smaller ordinals): the $\gamma $-th additively indecomposable ordinal is indexed as $\omega ^{\gamma }$. The technique of indexing classes of ordinals is often useful in the context of fixed points: for example, the $\gamma $-th ordinal $\alpha $ such that $\omega ^{\alpha }=\alpha $ is written $\varepsilon _{\gamma }$. These are called the "epsilon numbers".
Closed unbounded sets and classes
A class $C$ of ordinals is said to be unbounded, or cofinal, when given any ordinal $\alpha $, there is a $\beta $ in $C$ such that $\alpha <\beta $ (then the class must be a proper class, i.e., it cannot be a set). It is said to be closed when the limit of a sequence of ordinals in the class is again in the class: or, equivalently, when the indexing (class-)function $F$ is continuous in the sense that, for $\delta $ a limit ordinal, $F(\delta )$ (the $\delta $-th ordinal in the class) is the limit of all $F(\gamma )$ for $\gamma <\delta $; this is also the same as being closed, in the topological sense, for the order topology (to avoid talking of topology on proper classes, one can demand that the intersection of the class with any given ordinal is closed for the order topology on that ordinal, this is again equivalent).
Of particular importance are those classes of ordinals that are closed and unbounded, sometimes called clubs. For example, the class of all limit ordinals is closed and unbounded: this translates the fact that there is always a limit ordinal greater than a given ordinal, and that a limit of limit ordinals is a limit ordinal (a fortunate fact if the terminology is to make any sense at all!). The class of additively indecomposable ordinals, or the class of $\varepsilon _{\cdot }$ ordinals, or the class of cardinals, are all closed unbounded; the set of regular cardinals, however, is unbounded but not closed, and any finite set of ordinals is closed but not unbounded.
A class is stationary if it has a nonempty intersection with every closed unbounded class. All superclasses of closed unbounded classes are stationary, and stationary classes are unbounded, but there are stationary classes that are not closed and stationary classes that have no closed unbounded subclass (such as the class of all limit ordinals with countable cofinality). Since the intersection of two closed unbounded classes is closed and unbounded, the intersection of a stationary class and a closed unbounded class is stationary. But the intersection of two stationary classes may be empty, e.g. the class of ordinals with cofinality ω with the class of ordinals with uncountable cofinality.
Rather than formulating these definitions for (proper) classes of ordinals, one can formulate them for sets of ordinals below a given ordinal $\alpha $: A subset of a limit ordinal $\alpha $ is said to be unbounded (or cofinal) under $\alpha $ provided any ordinal less than $\alpha $ is less than some ordinal in the set. More generally, one can call a subset of any ordinal $\alpha $ cofinal in $\alpha $ provided every ordinal less than $\alpha $ is less than or equal to some ordinal in the set. The subset is said to be closed under $\alpha $ provided it is closed for the order topology in $\alpha $, i.e. a limit of ordinals in the set is either in the set or equal to $\alpha $ itself.
Arithmetic of ordinals
Main article: Ordinal arithmetic
There are three usual operations on ordinals: addition, multiplication, and (ordinal) exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set that represents the operation or by using transfinite recursion. The Cantor normal form provides a standardized way of writing ordinals. It uniquely represents each ordinal as a finite sum of ordinal powers of ω. However, this cannot form the basis of a universal ordinal notation due to such self-referential representations as ε0 = ωε0. The so-called "natural" arithmetical operations retain commutativity at the expense of continuity.
Interpreted as nimbers (a game-theoretic variant of numbers), ordinals are also subject to nimber arithmetic operations.
Ordinals and cardinals
Initial ordinal of a cardinal
Each ordinal associates with one cardinal, its cardinality. If there is a bijection between two ordinals (e.g. ω = 1 + ω and ω + 1 > ω), then they associate with the same cardinal. Any well-ordered set having an ordinal as its order-type has the same cardinality as that ordinal. The least ordinal associated with a given cardinal is called the initial ordinal of that cardinal. Every finite ordinal (natural number) is initial, and no other ordinal associates with its cardinal. But most infinite ordinals are not initial, as many infinite ordinals associate with the same cardinal. 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 theories with the axiom of choice, the cardinal number of any set has an initial ordinal, and one may employ the Von Neumann cardinal assignment as the cardinal's representation. (However, we must then be careful to distinguish between cardinal arithmetic and ordinal arithmetic.) In set theories without the axiom of choice, a cardinal may be represented by the set of sets with that cardinality having minimal rank (see Scott's trick).
One issue with Scott's trick is that it identifies the cardinal number $0$ with $\{\emptyset \}$, which in some formulations is the ordinal number $1$. It may be clearer to apply Von Neumann cardinal assignment to finite cases and to use Scott's trick for sets which are infinite or do not admit well orderings. Note that cardinal and ordinal arithmetic agree for finite numbers.
The α-th infinite initial ordinal is written $\omega _{\alpha }$, it is always a limit ordinal. Its cardinality is written $\aleph _{\alpha }$. For example, the cardinality of ω0 = ω is $\aleph _{0}$, which is also the cardinality of ω2 or ε0 (all are countable ordinals). So ω can be identified with $\aleph _{0}$, except that the notation $\aleph _{0}$ is used when writing cardinals, and ω when writing ordinals (this is important since, for example, $\aleph _{0}^{2}$ = $\aleph _{0}$ whereas $\omega ^{2}>\omega $). 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 the $\omega _{n}$ for natural numbers n (any limit of cardinals is a cardinal, so this limit is indeed the first cardinal after all the $\omega _{n}$).
Cofinality
The cofinality of an ordinal $\alpha $ is the smallest ordinal $\delta $ that is the order type of a cofinal subset of $\alpha $. Notice that a number of authors define cofinality or use it only for limit ordinals. The cofinality of a set of ordinals or any other well-ordered set is the cofinality of the order type of that set.
Thus for a limit ordinal, there exists a $\delta $-indexed strictly increasing sequence with limit $\alpha $. For example, the cofinality of ω2 is ω, because the sequence ω·m (where m ranges over the natural numbers) tends to ω2; but, more generally, any countable limit ordinal has cofinality ω. An uncountable limit ordinal may have either cofinality ω as does $\omega _{\omega }$ or an uncountable cofinality.
The cofinality of 0 is 0. And the cofinality of any successor ordinal is 1. The cofinality of any limit ordinal is at least $\omega $.
An ordinal that is equal to its cofinality is called regular and it is always an initial ordinal. Any limit of regular ordinals is a limit of initial ordinals and thus is also initial even if it is not regular, which it usually is not. If the Axiom of Choice, then $\omega _{\alpha +1}$ is regular for each α. In this case, the ordinals 0, 1, $\omega $, $\omega _{1}$, and $\omega _{2}$ are regular, whereas 2, 3, $\omega _{\omega }$, and ωω·2 are initial ordinals that are not regular.
The cofinality of any ordinal α is a regular ordinal, i.e. the cofinality of the cofinality of α is the same as the cofinality of α. So the cofinality operation is idempotent.
Some "large" countable ordinals
Further information: Large countable ordinal
As mentioned above (see Cantor normal form), the ordinal ε0 is the smallest satisfying the equation $\omega ^{\alpha }=\alpha $, so it is the limit of the sequence 0, 1, $\omega $, $\omega ^{\omega }$, $\omega ^{\omega ^{\omega }}$, etc. Many ordinals can be defined in such a manner as fixed points of certain ordinal functions (the $\iota $-th ordinal such that $\omega ^{\alpha }=\alpha $ is called $\varepsilon _{\iota }$, then one could go on trying to find the $\iota $-th ordinal such that $\varepsilon _{\alpha }=\alpha $, "and so on", but all the subtlety lies in the "and so on"). One could try to do this systematically, but no matter what system is used to define and construct ordinals, there is always an ordinal that lies just above all the ordinals constructed by the system. Perhaps the most important ordinal that limits a system of construction in this manner is the Church–Kleene ordinal, $\omega _{1}^{\mathrm {CK} }$ (despite the $\omega _{1}$ in the name, this ordinal is countable), which is the smallest ordinal that cannot in any way be represented by a computable function (this can be made rigorous, of course). Considerably large ordinals can be defined below $\omega _{1}^{\mathrm {CK} }$, however, which measure the "proof-theoretic strength" of certain formal systems (for example, $\varepsilon _{0}$ measures the strength of Peano arithmetic). Large countable ordinals such as countable admissible ordinals can also be defined above the Church-Kleene ordinal, which are of interest in various parts of logic.
Topology and ordinals
Further information: Order topology
Any ordinal number can be made into a topological space by endowing it with the order topology; this topology is discrete if and only if the ordinal is a countable cardinal, i.e. at most ω. A subset of ω + 1 is open in the order topology if and only if either it is cofinite or it does not contain ω as an element.
See the Topology and ordinals section of the "Order topology" article.
History
The transfinite ordinal numbers, which first appeared in 1883,[9] originated in Cantor's work with derived sets. If P is a set of real numbers, the derived set P′ is the set of limit points of P. In 1872, Cantor generated the sets P(n) by applying the derived set operation n times to P. In 1880, he pointed out that these sets form the sequence P' ⊇ ··· ⊇ P(n) ⊇ P(n + 1) ⊇ ···, and he continued the derivation process by defining P(∞) as the intersection of these sets. Then he iterated the derived set operation and intersections to extend his sequence of sets into the infinite: P(∞) ⊇ P(∞ + 1) ⊇ P(∞ + 2) ⊇ ··· ⊇ P(2∞) ⊇ ··· ⊇ P(∞2) ⊇ ···.[10] The superscripts containing ∞ are just indices defined by the derivation process.[11]
Cantor used these sets in the theorems:
1. If P(α) = ∅ for some index α, then P′ is countable;
2. Conversely, if P′ is countable, then there is an index α such that P(α) = ∅.
These theorems are proved by partitioning P′ into pairwise disjoint sets: P′ = (P′\ P(2)) ∪ (P(2) \ P(3)) ∪ ··· ∪ (P(∞) \ P(∞ + 1)) ∪ ··· ∪ P(α). For β < α: since P(β + 1) contains the limit points of P(β), the sets P(β) \ P(β + 1) have no limit points. Hence, they are discrete sets, so they are countable. Proof of first theorem: If P(α) = ∅ for some index α, then P′ is the countable union of countable sets. Therefore, P′ is countable.[12]
The second theorem requires proving the existence of an α such that P(α) = ∅. To prove this, Cantor considered the set of all α having countably many predecessors. To define this set, he defined the transfinite ordinal numbers and transformed the infinite indices into ordinals by replacing ∞ with ω, the first transfinite ordinal number. Cantor called the set of finite ordinals the first number class. The second number class is the set of ordinals whose predecessors form a countably infinite set. The set of all α having countably many predecessors—that is, the set of countable ordinals—is the union of these two number classes. Cantor proved that the cardinality of the second number class is the first uncountable cardinality.[13]
Cantor's second theorem becomes: If P′ is countable, then there is a countable ordinal α such that P(α) = ∅. Its proof uses proof by contradiction. Let P′ be countable, and assume there is no such α. This assumption produces two cases.
• Case 1: P(β) \ P(β + 1) is non-empty for all countable β. Since there are uncountably many of these pairwise disjoint sets, their union is uncountable. This union is a subset of P′, so P' is uncountable.
• Case 2: P(β) \ P(β + 1) is empty for some countable β. Since P(β + 1) ⊆ P(β), this implies P(β + 1) = P(β). Thus, P(β) is a perfect set, so it is uncountable.[14] Since P(β) ⊆ P′, the set P′ is uncountable.
In both cases, P′ is uncountable, which contradicts P′ being countable. Therefore, there is a countable ordinal α such that P(α) = ∅. Cantor's work with derived sets and ordinal numbers led to the Cantor-Bendixson theorem.[15]
Using successors, limits, and cardinality, Cantor generated an unbounded sequence of ordinal numbers and number classes.[16] The (α + 1)-th number class is the set of ordinals whose predecessors form a set of the same cardinality as the α-th number class. The cardinality of the (α + 1)-th number class is the cardinality immediately following that of the α-th number class.[17] For a limit ordinal α, the α-th number class is the union of the β-th number classes for β < α.[18] Its cardinality is the limit of the cardinalities of these number classes.
If n is finite, the n-th number class has cardinality $\aleph _{n-1}$. If α ≥ ω, the α-th number class has cardinality $\aleph _{\alpha }$.[19] Therefore, the cardinalities of the number classes correspond one-to-one with the aleph numbers. Also, the α-th number class consists of ordinals different from those in the preceding number classes if and only if α is a non-limit ordinal. Therefore, the non-limit number classes partition the ordinals into pairwise disjoint sets.
See also
• Counting
• Even and odd ordinals
• First uncountable ordinal
• Ordinal space
• Surreal number, a generalization of ordinals which includes negatives
Notes
1. "Ordinal Number - Examples and Definition of Ordinal Number". Literary Devices. 2017-05-21. Retrieved 2021-08-31.
2. Sterling, Kristin (2007-09-01). Ordinal Numbers. LernerClassroom. ISBN 978-0-8225-8846-7.
3. Thorough introductions are given by (Levy 1979) and (Jech 2003).
4. Hallett, Michael (1979), "Towards a theory of mathematical research programmes. I", The British Journal for the Philosophy of Science, 30 (1): 1–25, doi:10.1093/bjps/30.1.1, MR 0532548. See the footnote on p. 12.
5. "Ordinal Numbers | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2020-08-12.
6. Weisstein, Eric W. "Ordinal Number". mathworld.wolfram.com. Retrieved 2020-08-12.
7. von Neumann 1923
8. (Levy 1979, p. 52) attributes the idea to unpublished work of Zermelo in 1916 and several papers by von Neumann the 1920s.
9. Cantor 1883. English translation: Ewald 1996, pp. 881–920
10. Ferreirós 1995, pp. 34–35; Ferreirós 2007, pp. 159, 204–5
11. Ferreirós 2007, p. 269
12. Ferreirós 1995, pp. 35–36; Ferreirós 2007, p. 207
13. Ferreirós 1995, pp. 36–37; Ferreirós 2007, p. 271
14. Dauben 1979, p. 111
15. Ferreirós 2007, pp. 207–8
16. Dauben 1979, pp. 97–98
17. Hallett 1986, pp. 61–62
18. Tait 1997, p. 5 footnote
19. The first number class has cardinality $\aleph _{0}$. Mathematical induction proves that the n-th number class has cardinality $\aleph _{n-1}$. Since the ω-th number class is the union of the n-th number classes, its cardinality is $\aleph _{\omega }$, the limit of the $\aleph _{n-1}$. Transfinite induction proves that if α ≥ ω, the α-th number class has cardinality $\aleph _{\alpha }$.
References
• Cantor, Georg (1883), "Ueber unendliche, lineare Punktmannichfaltigkeiten. 5.", Mathematische Annalen, 21 (4): 545–591, doi:10.1007/bf01446819, S2CID 121930608. Published separately as: Grundlagen einer allgemeinen Mannigfaltigkeitslehre.
• Cantor, Georg (1897), "Beitrage zur Begrundung der transfiniten Mengenlehre. II", Mathematische Annalen, vol. 49, no. 2, pp. 207–246, doi:10.1007/BF01444205, S2CID 121665994 English translation: Contributions to the Founding of the Theory of Transfinite Numbers II.
• Conway, John H.; Guy, Richard (2012) [1996], "Cantor's Ordinal Numbers", The Book of Numbers, Springer, pp. 266–7, 274, ISBN 978-1-4612-4072-3
• Dauben, Joseph (1979), Georg Cantor: His Mathematics and Philosophy of the Infinite, Harvard University Press, ISBN 0-674-34871-0.
• Ewald, William B., ed. (1996), From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, Volume 2, Oxford University Press, ISBN 0-19-850536-1.
• Ferreirós, José (1995), "'What fermented in me for years': Cantor's discovery of transfinite numbers" (PDF), Historia Mathematica, 22: 33–42, doi:10.1006/hmat.1995.1003.
• Ferreirós, José (2007), Labyrinth of Thought: A History of Set Theory and Its Role in Mathematical Thought (2nd revised ed.), Birkhäuser, ISBN 978-3-7643-8349-7.
• Hallett, Michael (1986), Cantorian Set Theory and Limitation of Size, Oxford University Press, ISBN 0-19-853283-0.
• Hamilton, A. G. (1982), "6. Ordinal and cardinal numbers", Numbers, Sets, and Axioms : the Apparatus of Mathematics, New York: Cambridge University Press, ISBN 0-521-24509-5.
• Kanamori, Akihiro (2012), "Set Theory from Cantor to Cohen" (PDF), in Gabbay, Dov M.; Kanamori, Akihiro; Woods, John H. (eds.), Sets and Extensions in the Twentieth Century, Cambridge University Press, pp. 1–71, ISBN 978-0-444-51621-3.
• Levy, A. (2002) [1979], Basic Set Theory, Springer-Verlag, ISBN 0-486-42079-5.
• Jech, Thomas (2013), Set Theory (2nd ed.), Springer, ISBN 978-3-662-22400-7.
• Sierpiński, W. (1965), Cardinal and Ordinal Numbers (2nd ed.), Warszawa: Państwowe Wydawnictwo Naukowe Also defines ordinal operations in terms of the Cantor Normal Form.
• Suppes, Patrick (1960), Axiomatic Set Theory, D.Van Nostrand, ISBN 0-486-61630-4.
• Tait, William W. (1997), "Frege versus Cantor and Dedekind: On the Concept of Number" (PDF), in William W. Tait (ed.), Early Analytic Philosophy: Frege, Russell, Wittgenstein, Open Court, pp. 213–248, ISBN 0-8126-9344-2.
• von Neumann, John (1923), "Zur Einführung der transfiniten Zahlen", Acta litterarum ac scientiarum Ragiae Universitatis Hungaricae Francisco-Josephinae, Sectio scientiarum mathematicarum, vol. 1, pp. 199–208, archived from the original on 2014-12-18, retrieved 2013-09-15
• von Neumann, John (January 2002) [1923], "On the introduction of transfinite numbers", in Jean van Heijenoort (ed.), From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931 (3rd ed.), Harvard University Press, pp. 346–354, ISBN 0-674-32449-8 - English translation of von Neumann 1923.
External links
Look up ordinal in Wiktionary, the free dictionary.
Wikimedia Commons has media related to Ordinal numbers.
• "Ordinal number", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Ordinals at ProvenMath
• Ordinal calculator GPL'd free software for computing with ordinals and ordinal notations
• Chapter 4 of Don Monk's lecture notes on set theory is an introduction to ordinals.
Large countable ordinals
• First infinite ordinal ω
• First uncountable ordinal Ω
• Epsilon numbers ε0
• Feferman–Schütte ordinal Γ0
• Ackermann ordinal θ(Ω2)
• small Veblen ordinal θ(Ωω)
• large Veblen ordinal θ(ΩΩ)
• Bachmann–Howard ordinal ψ(εΩ+1)
• Buchholz's ordinal ψ0(Ωω)
• Takeuti–Feferman–Buchholz ordinal ψ(εΩω+1)
• Proof-theoretic ordinals of the theories of iterated inductive definitions
• Nonrecursive ordinal ≥ ωCK
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Ω-logic
In set theory, Ω-logic is an infinitary logic and deductive system proposed by W. Hugh Woodin (1999) as part of an attempt to generalize the theory of determinacy of pointclasses to cover the structure $H_{\aleph _{2}}$. Just as the axiom of projective determinacy yields a canonical theory of $H_{\aleph _{1}}$, he sought to find axioms that would give a canonical theory for the larger structure. The theory he developed involves a controversial argument that the continuum hypothesis is false.
Not to be confused with ω-logic.
Analysis
Woodin's Ω-conjecture asserts that if there is a proper class of Woodin cardinals (for technical reasons, most results in the theory are most easily stated under this assumption), then Ω-logic satisfies an analogue of the completeness theorem. From this conjecture, it can be shown that, if there is any single axiom which is comprehensive over $H_{\aleph _{2}}$ (in Ω-logic), it must imply that the continuum is not $\aleph _{1}$. Woodin also isolated a specific axiom, a variation of Martin's maximum, which states that any Ω-consistent $\Pi _{2}$ (over $H_{\aleph _{2}}$) sentence is true; this axiom implies that the continuum is $\aleph _{2}$.
Woodin also related his Ω-conjecture to a proposed abstract definition of large cardinals: he took a "large cardinal property" to be a $\Sigma _{2}$ property $P(\alpha )$ of ordinals which implies that α is a strong inaccessible, and which is invariant under forcing by sets of cardinal less than α. Then the Ω-conjecture implies that if there are arbitrarily large models containing a large cardinal, this fact will be provable in Ω-logic.
The theory involves a definition of Ω-validity: a statement is an Ω-valid consequence of a set theory T if it holds in every model of T having the form $V_{\alpha }^{\mathbb {B} }$ for some ordinal $\alpha $ and some forcing notion $\mathbb {B} $. This notion is clearly preserved under forcing, and in the presence of a proper class of Woodin cardinals it will also be invariant under forcing (in other words, Ω-satisfiability is preserved under forcing as well). There is also a notion of Ω-provability;[1] here the "proofs" consist of universally Baire sets and are checked by verifying that for every countable transitive model of the theory, and every forcing notion in the model, the generic extension of the model (as calculated in V) contains the "proof", restricted its own reals. For a proof-set A the condition to be checked here is called "A-closed". A complexity measure can be given on the proofs by their ranks in the Wadge hierarchy. Woodin showed that this notion of "provability" implies Ω-validity for sentences which are $\Pi _{2}$ over V. The Ω-conjecture states that the converse of this result also holds. In all currently known core models, it is known to be true; moreover the consistency strength of the large cardinals corresponds to the least proof-rank required to "prove" the existence of the cardinals.
Notes
1. Bhatia, Rajendra, ed. (2010), Proceedings of the International Congress of Mathematicians: Hyderabad, 2010, vol. 1, World Scientific, p. 519
References
• Bagaria, Joan; Castells, Neus; Larson, Paul (2006), "An Ω-logic primer", Set theory (PDF), Trends Math., Basel, Boston, Berlin: Birkhäuser, pp. 1–28, doi:10.1007/3-7643-7692-9_1, ISBN 978-3-7643-7691-8, MR 2267144
• Koellner, Peter (2013), "The Continuum Hypothesis", The Stanford Encyclopedia of Philosophy, Edward N. Zalta (Ed.)
• Woodin, W. Hugh (1999), The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, Walter de Gruyter, doi:10.1515/9783110804737, ISBN 3-11-015708-X, MR 1713438
• Woodin, W. Hugh (2001), "The continuum hypothesis. I" (PDF), Notices of the American Mathematical Society, 48 (6): 567–576, ISSN 0002-9920, MR 1834351
• Woodin, W. Hugh (2001b), "The Continuum Hypothesis, Part II" (PDF), Notices of the AMS, 48 (7): 681–690
• Woodin, W. Hugh (2005), "The continuum hypothesis", in Cori, Rene; Razborov, Alexander; Todorčević, Stevo; et al. (eds.), Logic Colloquium 2000, Lect. Notes Log., vol. 19, Urbana, IL: Assoc. Symbol. Logic, pp. 143–197, MR 2143878
External links
• W. H. Woodin, Slides for 3 talks
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ω-consistent theory
In mathematical logic, an ω-consistent (or omega-consistent, also called numerically segregative)[1] theory is a theory (collection of sentences) that is not only (syntactically) consistent[2] (that is, does not prove a contradiction), but also avoids proving certain infinite combinations of sentences that are intuitively contradictory. The name is due to Kurt Gödel, who introduced the concept in the course of proving the incompleteness theorem.[3]
Definition
A theory T is said to interpret the language of arithmetic if there is a translation of formulas of arithmetic into the language of T so that T is able to prove the basic axioms of the natural numbers under this translation.
A T that interprets arithmetic is ω-inconsistent if, for some property P of natural numbers (defined by a formula in the language of T), T proves P(0), P(1), P(2), and so on (that is, for every standard natural number n, T proves that P(n) holds), but T also proves that there is some natural number n such that P(n) fails.[2] This may not generate a contradiction within T because T may not be able to prove for any specific value of n that P(n) fails, only that there is such an n. In particular, such n is necessarily a nonstandard integer in any model for T (Quine has thus called such theories "numerically insegregative").[4]
T is ω-consistent if it is not ω-inconsistent.
There is a weaker but closely related property of Σ1-soundness. A theory T is Σ1-sound (or 1-consistent, in another terminology) if every Σ01-sentence[5] provable in T is true in the standard model of arithmetic N (i.e., the structure of the usual natural numbers with addition and multiplication). If T is strong enough to formalize a reasonable model of computation, Σ1-soundness is equivalent to demanding that whenever T proves that a Turing machine C halts, then C actually halts. Every ω-consistent theory is Σ1-sound, but not vice versa.
More generally, we can define an analogous concept for higher levels of the arithmetical hierarchy. If Γ is a set of arithmetical sentences (typically Σ0n for some n), a theory T is Γ-sound if every Γ-sentence provable in T is true in the standard model. When Γ is the set of all arithmetical formulas, Γ-soundness is called just (arithmetical) soundness. If the language of T consists only of the language of arithmetic (as opposed to, for example, set theory), then a sound system is one whose model can be thought of as the set ω, the usual set of mathematical natural numbers. The case of general T is different, see ω-logic below.
Σn-soundness has the following computational interpretation: if the theory proves that a program C using a Σn−1-oracle halts, then C actually halts.
Examples
Consistent, ω-inconsistent theories
Write PA for the theory Peano arithmetic, and Con(PA) for the statement of arithmetic that formalizes the claim "PA is consistent". Con(PA) could be of the form "For every natural number n, n is not the Gödel number of a proof from PA that 0=1". (This formulation uses 0=1 instead of a direct contradiction; that gives the same result, because PA certainly proves ¬0=1, so if it proved 0=1 as well we would have a contradiction, and on the other hand, if PA proves a contradiction, then it proves anything, including 0=1.)
Now, assuming PA is really consistent, it follows that PA + ¬Con(PA) is also consistent, for if it were not, then PA would prove Con(PA) (reductio), contradicting Gödel's second incompleteness theorem. However, PA + ¬Con(PA) is not ω-consistent. This is because, for any particular natural number n, PA + ¬Con(PA) proves that n is not the Gödel number of a proof that 0=1 (PA itself proves that fact; the extra assumption ¬Con(PA) is not needed). However, PA + ¬Con(PA) proves that, for some natural number n, n is the Gödel number of such a proof (this is just a direct restatement of the claim ¬Con(PA)).
In this example, the axiom ¬Con(PA) is Σ1, hence the system PA + ¬Con(PA) is in fact Σ1-unsound, not just ω-inconsistent.
Arithmetically sound, ω-inconsistent theories
Let T be PA together with the axioms c ≠ n for each natural number n, where c is a new constant added to the language. Then T is arithmetically sound (as any nonstandard model of PA can be expanded to a model of T), but ω-inconsistent (as it proves $\exists x\,c=x$, and c ≠ n for every number n).
Σ1-sound ω-inconsistent theories using only the language of arithmetic can be constructed as follows. Let IΣn be the subtheory of PA with the induction schema restricted to Σn-formulas, for any n > 0. The theory IΣn + 1 is finitely axiomatizable, let thus A be its single axiom, and consider the theory T = IΣn + ¬A. We can assume that A is an instance of the induction schema, which has the form
$\forall w\,[B(0,w)\land \forall x\,(B(x,w)\to B(x+1,w))\to \forall x\,B(x,w)].$
If we denote the formula
$\forall w\,[B(0,w)\land \forall x\,(B(x,w)\to B(x+1,w))\to B(n,w)]$
by P(n), then for every natural number n, the theory T (actually, even the pure predicate calculus) proves P(n). On the other hand, T proves the formula $\exists x\,\neg P(x)$, because it is logically equivalent to the axiom ¬A. Therefore, T is ω-inconsistent.
It is possible to show that T is Πn + 3-sound. In fact, it is Πn + 3-conservative over the (obviously sound) theory IΣn. The argument is more complicated (it relies on the provability of the Σn + 2-reflection principle for IΣn in IΣn + 1).
Arithmetically unsound, ω-consistent theories
Let ω-Con(PA) be the arithmetical sentence formalizing the statement "PA is ω-consistent". Then the theory PA + ¬ω-Con(PA) is unsound (Σ3-unsound, to be precise), but ω-consistent. The argument is similar to the first example: a suitable version of the Hilbert–Bernays–Löb derivability conditions holds for the "provability predicate" ω-Prov(A) = ¬ω-Con(PA + ¬A), hence it satisfies an analogue of Gödel's second incompleteness theorem.
ω-logic
Not to be confused with Ω-logic.
The concept of theories of arithmetic whose integers are the true mathematical integers is captured by ω-logic.[6] Let T be a theory in a countable language that includes a unary predicate symbol N intended to hold just of the natural numbers, as well as specified names 0, 1, 2, ..., one for each (standard) natural number (which may be separate constants, or constant terms such as 0, 1, 1+1, 1+1+1, ..., etc.). Note that T itself could be referring to more general objects, such as real numbers or sets; thus in a model of T the objects satisfying N(x) are those that T interprets as natural numbers, not all of which need be named by one of the specified names.
The system of ω-logic includes all axioms and rules of the usual first-order predicate logic, together with, for each T-formula P(x) with a specified free variable x, an infinitary ω-rule of the form:
From $P(0),P(1),P(2),\ldots $ infer $\forall x\,(N(x)\to P(x))$.
That is, if the theory asserts (i.e. proves) P(n) separately for each natural number n given by its specified name, then it also asserts P collectively for all natural numbers at once via the evident finite universally quantified counterpart of the infinitely many antecedents of the rule. For a theory of arithmetic, meaning one with intended domain the natural numbers such as Peano arithmetic, the predicate N is redundant and may be omitted from the language, with the consequent of the rule for each P simplifying to $\forall x\,P(x)$.
An ω-model of T is a model of T whose domain includes the natural numbers and whose specified names and symbol N are standardly interpreted, respectively as those numbers and the predicate having just those numbers as its domain (whence there are no nonstandard numbers). If N is absent from the language then what would have been the domain of N is required to be that of the model, i.e. the model contains only the natural numbers. (Other models of T may interpret these symbols nonstandardly; the domain of N need not even be countable, for example.) These requirements make the ω-rule sound in every ω-model. As a corollary to the omitting types theorem, the converse also holds: the theory T has an ω-model if and only if it is consistent in ω-logic.
There is a close connection of ω-logic to ω-consistency. A theory consistent in ω-logic is also ω-consistent (and arithmetically sound). The converse is false, as consistency in ω-logic is a much stronger notion than ω-consistency. However, the following characterization holds: a theory is ω-consistent if and only if its closure under unnested applications of the ω-rule is consistent.
Relation to other consistency principles
If the theory T is recursively axiomatizable, ω-consistency has the following characterization, due to Craig Smoryński:[7]
T is ω-consistent if and only if $T+\mathrm {RFN} _{T}+\mathrm {Th} _{\Pi _{2}^{0}}(\mathbb {N} )$ is consistent.
Here, $\mathrm {Th} _{\Pi _{2}^{0}}(\mathbb {N} )$ is the set of all Π02-sentences valid in the standard model of arithmetic, and $\mathrm {RFN} _{T}$ is the uniform reflection principle for T, which consists of the axioms
$\forall x\,(\mathrm {Prov} _{T}(\ulcorner \varphi ({\dot {x}})\urcorner )\to \varphi (x))$
for every formula $\varphi $ with one free variable. In particular, a finitely axiomatizable theory T in the language of arithmetic is ω-consistent if and only if T + PA is $\Sigma _{2}^{0}$-sound.
Notes
1. W. V. O. Quine (1971), Set Theory and Its Logic.
2. S. C. Kleene, Introduction to Metamathematics (1971), p.207. Bibliotheca Mathematica: A Series of Monographs on Pure and Applied Mathematics Vol. I, Wolters-Noordhoff, North-Holland 0-7204-2103-9, Elsevier 0-444-10088-1.
3. Smorynski, "The incompleteness theorems", Handbook of Mathematical Logic, 1977, p. 851.
4. Floyd, Putnam, A Note on Wittgenstein's "Notorious Paragraph" about the Gödel Theorem (2000)
5. The definition of this symbolism can be found at arithmetical hierarchy.
6. J. Barwise (ed.), Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977.
7. Smoryński, Craig (1985). Self-reference and modal logic. Berlin: Springer. ISBN 978-0-387-96209-2. Reviewed in Boolos, G.; Smorynski, C. (1988). "Self-Reference and Modal Logic". The Journal of Symbolic Logic. 53: 306. doi:10.2307/2274450. JSTOR 2274450.
Bibliography
• Kurt Gödel (1931). 'Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I'. In Monatshefte für Mathematik. Translated into English as On Formally Undecidable Propositions of Principia Mathematica and Related Systems.
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Stable theory
In the mathematical field of model theory, a theory is called stable if it satisfies certain combinatorial restrictions on its complexity. Stable theories are rooted in the proof of Morley's categoricity theorem and were extensively studied as part of Saharon Shelah's classification theory, which showed a dichotomy that either the models of a theory admit a nice classification or the models are too numerous to have any hope of a reasonable classification. A first step of this program was showing that if a theory is not stable then its models are too numerous to classify.
For differential equations, see Stability theory.
Stable theories were the predominant subject of pure model theory from the 1970s through the 1990s, so their study shaped modern model theory[1] and there is a rich framework and set of tools to analyze them. A major direction in model theory is "neostability theory," which tries to generalize the concepts of stability theory to broader contexts, such as simple and NIP theories.
Motivation and history
A common goal in model theory is to study a first-order theory by analyzing the complexity of the Boolean algebras of (parameter) definable sets in its models. One can equivalently analyze the complexity of the Stone duals of these Boolean algebras, which are type spaces. Stability restricts the complexity of these type spaces by restricting their cardinalities. Since types represent the possible behaviors of elements in a theory's models, restricting the number of types restricts the complexity of these models.[2]
Stability theory has its roots in Michael Morley's 1965 proof of Łoś's conjecture on categorical theories. In this proof, the key notion was that of a totally transcendental theory, defined by restricting the topological complexity of the type spaces. However, Morley showed that (for countable theories) this topological restriction is equivalent to a cardinality restriction, a strong form of stability now called $\omega $-stability, and he made significant use of this equivalence. In the course of generalizing Morley's categoricity theorem to uncountable theories, Frederick Rowbottom generalized $\omega $-stability by introducing $\kappa $-stable theories for some cardinal $\kappa $, and finally Shelah introduced stable theories.[3]
Stability theory was much further developed in the course of Shelah's classification theory program. The main goal of this program was to show a dichotomy that either the models of a first-order theory can be nicely classified up to isomorphism using a tree of cardinal-invariants (generalizing, for example, the classification of vector spaces over a fixed field by their dimension), or are so complicated that no reasonable classification is possible.[4] Among the concrete results from this classification theory were theorems on the possible spectrum functions of a theory, counting the number of models of cardinality $\kappa $ as a function of $\kappa $.[lower-alpha 1] Shelah's approach was to identify a series of "dividing lines" for theories. A dividing line is a property of a theory such that both it and its negation have strong structural consequences; one should imply the models of the theory are chaotic, while the other should yield a positive structure theory. Stability was the first such dividing line in the classification theory program, and since its failure was shown to rule out any reasonable classification, all further work could assume the theory to be stable. Thus much of classification theory was concerned with analyzing stable theories and various subsets of stable theories given by further dividing lines, such as superstable theories.[3]
One of the key features of stable theories developed by Shelah is that they admit a general notion of independence called non-forking independence, generalizing linear independence from vector spaces and algebraic independence from field theory. Although non-forking independence makes sense in arbitrary theories, and remains a key tool beyond stable theories, it has particularly good geometric and combinatorial properties in stable theories. As with linear independence, this allows the definition of independent sets and of local dimensions as the cardinalities of maximal instances of these independent sets, which are well-defined under additional hypotheses. These local dimensions then give rise to the cardinal-invariants classifying models up to isomorphism.[4]
Definition and alternate characterizations
Let T be a complete first-order theory.
For a given infinite cardinal $\kappa $, T is $\kappa $-stable if for every set A of cardinality $\kappa $ in a model of T, the set S(A) of complete types over A also has cardinality $\kappa $. This is the smallest the cardinality of S(A) can be, while it can be as large as $2^{\kappa }$. For the case $\kappa =\aleph _{0}$, it is common to say T is $\omega $-stable rather than $\aleph _{0}$-stable.[5]
T is stable if it is $\kappa $-stable for some infinite cardinal $\kappa $.[6]
Restrictions on the cardinals $\kappa $ for which a theory can simultaneously by $\kappa $-stable are described by the stability spectrum,[7] which singles out the even tamer subset of superstable theories.
A common alternate definition of stable theories is that they do not have the order property. A theory has the order property if there is a formula $\phi ({\bar {x}},{\bar {y}})$ and two infinite sequences of tuples $A=({\bar {a}}_{i}:i\in \mathbb {N} )$, $B=({\bar {b}}_{j}:j\in \mathbb {N} )$ in some model M such that $\phi $ defines an infinite half graph on $A\times B$, i.e. $\phi ({\bar {a}}_{i},{\bar {b}}_{j})$ is true in M $\iff i\leq j$.[8] This is equivalent to there being a formula $\psi ({\bar {x}},{\bar {y}})$ and an infinite sequence of tuples $A=({\bar {a}}_{i}:i\in \mathbb {N} )$ in some model M such that $\psi $ defines an infinite linear order on A, i.e. $\psi ({\bar {a}}_{i},{\bar {a}}_{j})$ is true in M $\iff i\leq j$.[9][lower-alpha 2][lower-alpha 3]
There are numerous further characterizations of stability. As with Morley's totally transcendental theories, the cardinality restrictions of stability are equivalent to bounding the topological complexity of type spaces in terms of Cantor-Bendixson rank.[12] Another characterization is via the properties that non-forking independence has in stable theories, such as being symmetric. This characterizes stability in the sense that any theory with an abstract independence relation satisfying certain of these properties must be stable and the independence relation must be non-forking independence.[13]
Any of these definitions, except via an abstract independence relation, can instead be used to define what it means for a single formula to be stable in a given theory T. Then T can be defined to be stable if every formula is stable in T.[14] Localizing results to stable formulas allows these results to be applied to stable formulas in unstable theories, and this localization to single formulas is often useful even in the case of stable theories.[15]
Examples and non-examples
For an unstable theory, consider the theory DLO of dense linear orders without endpoints. Then the atomic order relation has the order property. Alternatively, unrealized 1-types over a set A correspond to cuts (generalized Dedekind cuts, without the requirements that the two sets be non-empty and that the lower set have no greatest element) in the ordering of A,[16] and there exist dense orders of any cardinality $\kappa $ with $2^{\kappa }$-many cuts.[17]
Another unstable theory is the theory of the Rado graph, where the atomic edge relation has the order property.[18]
For a stable theory, consider the theory $ACF_{p}$ of algebraically closed fields of characteristic p, allowing $p=0$. Then if K is a model of $ACF_{p}$, counting types over a set $A\subset K$ is equivalent to counting types over the field k generated by A in K. There is a (continuous) bijection from the space of n-types over k to the space of prime ideals in the polynomial ring $k[X_{1},\dots ,X_{n}]$. Since such ideals are finitely generated, there are only $|k|+\aleph _{0}$ many, so $ACF_{p}$ is $\kappa $-stable for all infinite $\kappa $.[19]
Some further examples of stable theories are listed below.
• The theory of any module over a ring (in particular, any theory of vector spaces or abelian groups).[20]
• The theory of non-abelian free groups.[21]
• The theory of differentially closed fields of characteristic p. When $p=0$, the theory is $\omega $-stable.[22]
• The theory of any nowhere dense graph class.[23] These include graph classes with bounded expansion, which in turn include planar graphs and any graph class of bounded degree.
Geometric stability theory
Geometric stability theory is concerned with the fine analysis of local geometries in models and how their properties influence global structure. This line of results was later key in various applications of stability theory, for example to Diophantine geometry. It is usually taken to start in the late 1970s with Boris Zilber's analysis of totally categorical theories, eventually showing that they are not finitely axiomatizble. Every model of a totally categorical theory is controlled by (i.e. is prime and minimal over) a strongly minimal set, which carries a matroid structure[lower-alpha 4] determined by (model-theoretic) algebraic closure that gives notions of independence and dimension. In this setting, geometric stability theory then asks the local question of what the possibilities are for the structure of the strongly minimal set, and the local-to-global question of how the strongly minimal set controls the whole model.[24]
The second question is answered by Zilber's Ladder Theorem, showing every model of a totally categorical theory is built up by a finite sequence of something like "definable fiber bundles" over the strongly minimal set.[25] For the first question, Zilber's Trichotomy Conjecture was that the geometry of a strongly minimal set must be either like that of a set with no structure, or the set must essentially carry the structure of a vector space, or the structure of an algebraically closed field, with the first two cases called locally modular.[26] This conjecture illustrates two central themes. First, that (local) modularity serves to divide combinatorial or linear behavior from nonlinear, geometric complexity as in algebraic geometry.[27] Second, that complicated combinatorial geometry necessarily comes from algebraic objects;[28] this is akin to the classical problem of finding a coordinate ring for an abstract projective plane defined by incidences, and further examples are the group configuration theorems showing certain combinatorial dependencies among elements must arise from multiplication in a definable group.[29] By developing analogues of parts of algebraic geometry in strongly minimal sets, such as intersection theory, Zilber proved a weak form of the Trichotomy Conjecture for uncountably categorical theories.[30] Although Ehud Hrushovski developed the Hrushovski construction to disprove the full conjecture, it was later proved with additional hypotheses in the setting of "Zariski geometries".[31]
Notions from Shelah's classification program, such as regular types, forking, and orthogonality, allowed these ideas to be carried to greater generality, especially in superstable theories. Here, sets defined by regular types play the role of strongly minimal sets, with their local geometry determined by forking dependence rather than algebraic dependence. In place of the single strongly minimal set controlling models of a totally categorical theory, there may be many such local geometries defined by regular types, and orthogonality describes when these types have no interaction.[32]
Applications
While stable theories are fundamental in model theory, this section lists applications of stable theories to other areas of mathematics. This list does not aim for completeness, but rather a sense of breadth.
• Since the theory of differentially closed fields of characteristic 0 is $\omega $-stable, there are many applications of stability theory in differential algebra. For example, the existence and uniqueness of the differential closure of such a field (an analogue of the algebraic closure) were proved by Lenore Blum and Shelah respectively, using general results on prime models in $\omega $-stable theories.[33]
• In Diophantine geometry, Ehud Hrushovski used geometric stability theory to prove the Mordell-Lang conjecture for function fields in all characteristics, which generalizes Faltings's theorem about counting rational points on curves and the Manin-Mumford conjecture about counting torsion points on curves.[34] The key point in the proof was using Zilber's Trichotomy in differential fields to show certain arithmetically defined groups are locally modular.[35]
• In online machine learning, the Littlestone dimension of a concept class is a complexity measure characterizing learnability, analogous to the VC-dimension in PAC learning. Bounding the Littlestone dimension of a concept class is equivalent to a combinatorial characterization of stability involving binary trees.[36] This equivlanece has been used, for example, to prove that online learnability of a concept class is equivalent to differentially private PAC learnability.[37]
• In functional analysis, Jean-Louis Krivine and Bernard Maurey defined a notion of stability for Banach spaces, equivalent to stating that no quantifier-free formula has the order property (in continuous logic, rather than first-order logic). They then showed that every stable Banach space admits an almost-isometric embedding of ℓp for some $p\in [1,\infty )$.[38] This is part of a broader interplay between functional analysis and stability in continuous logic; for example, early results of Alexander Grothendieck in functional analysis can be interpreted as equivalent to fundamental results of stability theory.[39]
• A countable (possibly finite) structure is ultrahomogeneous if every finite partial automorphism extends to an automorphism of the full structure. Gregory Cherlin and Alistair Lachlan provided a general classification theory for stable ultrahomogeneous structures, including all finite ones. In particular, their results show that for any fixed finite relational language, the finite homogeneous structures fall into finitely many infinite families with members parametrized by numerical invariants and finitely many sporadic examples. Furthermore, every sporadic example becomes part of an infinite family in some richer language, and new sporadic examples always appear in suitably richer languages.[40]
• In arithmetic combinatorics, Hrushovski proved results on the structure of approximate subgroups, for example implying a strengthened version of Gromov's theorem on groups of polynomial growth. Although this did not directly use stable theories, the key insight was that fundamental results from stable group theory could be generalized and applied in this setting.[41] This directly led to the Breuillard-Green-Tao theorem classifying approximate subgroups.[42]
Generalizations
For about twenty years after its introduction, stability was the main subject of pure model theory.[43] A central direction of modern pure model theory, sometimes called "neostability" or "classification theory,"[lower-alpha 5]consists of generalizing the concepts and techniques developed for stable theories to broader classes of theories, and this has fed into many of the more recent applications of model theory.[44]
Two notable examples of such broader classes are simple and NIP theories. These are orthogonal generalizations of stable theories, since a theory is both simple and NIP if and only if it is stable.[43] Roughly, NIP theories keep the good combinatorial behavior from stable theories, while simple theories keep the good geometric behavior of non-forking independence.[45] In particular, simple theories can be characterized by non-forking independence being symmetric,[46] while NIP can be characterized by bounding the number of types realized over either finite[47] or infinite[48] sets.
Another direction of generalization is to recapitulate classification theory beyond the setting of complete first-order theories, such as in abstract elementary classes.[49]
See also
• Stability spectrum
• Spectrum of a theory
• Morley's categoricity theorem
• NIP theories
Notes
1. One such result is Shelah's proof of Morley's conjecture for countable theories, stating that the number of models of cardinality $\kappa $ is non-decreasing for uncountable $\kappa $.[4]
2. In work on Łoś's conjecture preceding Morley's proof, Andrzej Ehrenfeucht introduced a property slightly stronger than the order property, which Shelah later called property (E). This was another precursor of (uns)stable theories.[10]
3. One benefit of the definition of stability via the order property is that it is more clearly set-theoretically absolute.[11]
4. The term "pregeometry" is often used instead of "matroid" in this setting.
5. The term "classification theory" has two uses. The narrow use described earlier refers to Shelah's program of identifying classifiable theories, and takes place almost entirely within stable theories. The broader use described here refers to the larger program of classifying theories by dividing lines possibly more general than stability.[11]
References
1. Baldwin, John (2021). "The dividing line methodology: Model theory motivating set theory" (PDF). Theoria. 87 (2): 1. doi:10.1111/theo.12297. S2CID 211239082.
2. van den Dries, Lou (2005). "Introduction to model-theoretic stability" (PDF). Introduction. Retrieved 9 January 2023.
3. Pillay, Anand (1983). "Preface". An Introduction to Stability Theory.
4. Baldwin, John (2021). "The dividing line methodology: Model theory motivating set theory" (PDF). Theoria. 87 (2). Section 1.1. doi:10.1111/theo.12297. S2CID 211239082.
5. Marker, David (2006). Model Theory: An Introduction. Definition 4.2.17.
6. Marker, David (2006). Model Theory: An Introduction. Definition 5.3.1.
7. Tent, Katrin; Ziegler, Martin (2012). A Course in Model Theory. Theorem 8.6.5.
8. Tent, Katrin; Ziegler, Martin (2012). A Course in Model Theory. Definition 8.2.1.
9. Tent, Katrin; Ziegler, Martin (2012). A Course in Model Theory. Exercise 8.2.1.
10. Shelah, Saharon (1974). "Categoricity of uncountable theories" (PDF). Proceedings of the Tarski symposium.
11. Hodges, Wilfrid. "First-order Model Theory". Stanford Encyclopedia of Philosophy. Section 5.1. Retrieved 9 January 2023.
12. Casanovas, Enrique. "Stable and simple theories (Lecture Notes)" (PDF). Proposition 6.6. Retrieved 11 January 2023.
13. Tent, Katrin; Ziegler, Martin (2012). A Course in Model Theory. Theorem 8.5.10.
14. Tent, Katrin; Ziegler, Martin (2012). A Course in Model Theory. Chapter 8.2.
15. Baldwin, John (2017). Fundamentals of Stability Theory. Chapter 3.1.
16. Marker, David (2006). Model Theory: An Introduction. Example 4.1.12.
17. Marker, David (2006). Model Theory: An Introduction. Lemma 5.2.12.
18. Tent, Katrin; Ziegler, Martin (2012). A Course in Model Theory. Exercise 8.2.3.
19. Marker, David (2006). Model Theory: An Introduction. Example 4.1.14.
20. Tent, Katrin; Ziegler, Martin (2012). A Course in Model Theory. Example 8.6.6.
21. Sela, Zlil (2013). "Diophantine geometry over groups VIII: Stability" (PDF). Annals of Mathematics. 177 (3): 787–868. doi:10.4007/annals.2013.177.3.1. S2CID 119143329.
22. Shelah, Saharon (1973). "Differentially closed fields" (PDF). Israel Journal of Mathematics. 16 (3): 314–328. doi:10.1007/BF02756711. S2CID 119906669.
23. Adler, Hans; Adler, Isolde (2014). "Interpreting nowhere dense graph classes as a classical notion of model theory". European Journal of Combinatorics. 36: 322–330. doi:10.1016/j.ejc.2013.06.048.
24. Pillay, Anand (2001). "Aspects of geometric model theory". Logic Colloquium ’99.
25. Pillay, Anand (1996). Geometric Stability Theory. p. 343.
26. Scanlon, Thomas. "Zilber's Trichotomy Conjecture". Retrieved 27 January 2023.
27. Hrushovski, Ehud (1998). "Geometric model theory". Proceedings of the International Congress of Mathematicians. Vol. 1.
28. Scanlon, Thomas. "Combinatorial geometric stability". Retrieved 27 January 2023.
29. Ben-Yaacov, Itaï; Tomašić, Ivan; Wagner, Frank (2002). "The Group Configuration in Simple Theories and Its Applications" (PDF). 8. 2.
30. Scanlon, Thomas. "Zilber's trichotomy theorem". Retrieved 27 January 2023.
31. Scanlon, Thomas. "Combinatorial geometric stability". Retrieved 27 January 2023.
32. Pillay, Anand (2001). "Aspects of geometric model theory". Logic Colloquium ’99.
33. Sacks, Gerald (1972). "The differential closure of a differential field" (PDF). Bulletin of the American Mathematical Society. 78 (5): 629–634. doi:10.1090/S0002-9904-1972-12969-0. S2CID 17860378.
34. Hrushovski, Ehud (1996). "The Mordell-Lang conjecture for function fields" (PDF). Journal of the American Mathematical Society. 9 (3): 667–690. doi:10.1090/S0894-0347-96-00202-0.
35. Scanlon, Thomas. "Mordell-Lang and variants". Retrieved 27 January 2023.
36. Chase, Hunter; Freitag, James (2019). "Model theory and machine learning". Bulletin of Symbolic Logic. 25 (3): 319–332. arXiv:1801.06566. doi:10.1017/bsl.2018.71. S2CID 119689419.
37. Alon, Noga; Bun, Mark; Livni, Roi; Malliaris, Maryanthe; Moran, Shay (2022). "Private and Online Learnability are Equivalent" (PDF). Journal of the ACM. 69 (4): 1–34. doi:10.1145/3526074. S2CID 247186721.
38. Iovino, José (2014). Applications of model theory to functional analysis (PDF). Chapters 13,15.
39. Ben Yaacov, Itaï (2014). "Model theoretic stability and definability of types, after A. Grothendieck". Bulletin of Symbolic Logic. 20 (4). arXiv:1306.5852.
40. Cherlin, Gregory (2000). "Sporadic homogeneous structures" (PDF). The Gelfand mathematical seminars, 1996--1999.
41. Hrushovski, Ehud (2012). "Stable group theory and approximate subgroups" (PDF). Journal of the American Mathematical Society. 25 (1).
42. Breuillard, Emmanuel; Green, Ben; Tao, Terence (2012). "The structure of approximate groups" (PDF). Publications mathématiques de l'IHÉS. 116. Acknowledgments. arXiv:1110.5008. doi:10.1007/s10240-012-0043-9. S2CID 254166823.
43. Simon, Pierre (2015). "Introduction". A Guide to NIP Theories (PDF).
44. Hart, Bradd; Hrushovski, Ehud; Onshuus, Alf; Pillay, Anand; Scanlon, Thomas; Wagner, Frank. "Neostability Theory" (PDF).
45. Adler, Hans (2008). "An introduction to theories without the independence property" (PDF). Archive for Mathematical Logic. 5: 21.
46. Kim, Byunghan (2001). "Simplicity, and stability in there". The Journal of Symbolic Logic. 66 (2): 822–836. doi:10.2307/2695047. JSTOR 2695047. S2CID 7033889.
47. Chernikov, Artem; Simon, Pierre (2015). "Externally definable sets and dependent pairs II" (PDF). Transactions of the American Mathematical Society. 367 (7). Fact 3. doi:10.1090/S0002-9947-2015-06210-2. S2CID 53968137.
48. Simon, Pierre (2015). A Guide to NIP Theories (PDF). Proposition 2.69.
49. Shelah, Saharon (2009). Classification Theory for Abstract Elementary Classes Volume 1 (PDF).
External links
• A map of the model-theoretic classification of theories, highlighting stability
• Two book reviews discussing stability and classification theory for non-model theorists: Fundamentals of Stability Theory and Classification Theory
• An overview of (geometric) stability theory for non-model theorists
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First uncountable ordinal
In mathematics, the first uncountable ordinal, traditionally denoted by $\omega _{1}$ or sometimes by $\Omega $, is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum (least upper bound) of all countable ordinals. When considered as a set, the elements of $\omega _{1}$ are the countable ordinals (including finite ordinals),[1] of which there are uncountably many.
Like any ordinal number (in von Neumann's approach), $\omega _{1}$ is a well-ordered set, with set membership serving as the order relation. $\omega _{1}$ is a limit ordinal, i.e. there is no ordinal $\alpha $ such that $\omega _{1}=\alpha +1$.
The cardinality of the set $\omega _{1}$ is the first uncountable cardinal number, $\aleph _{1}$ (aleph-one). The ordinal $\omega _{1}$ is thus the initial ordinal of $\aleph _{1}$. Under the continuum hypothesis, the cardinality of $\omega _{1}$ is $\beth _{1}$, the same as that of $\mathbb {R} $—the set of real numbers.[2]
In most constructions, $\omega _{1}$ and $\aleph _{1}$ are considered equal as sets. To generalize: if $\alpha $ is an arbitrary ordinal, we define $\omega _{\alpha }$ as the initial ordinal of the cardinal $\aleph _{\alpha }$.
The existence of $\omega _{1}$ can be proven without the axiom of choice. For more, see Hartogs number.
Topological properties
Any ordinal number can be turned into a topological space by using the order topology. When viewed as a topological space, $\omega _{1}$ is often written as $[0,\omega _{1})$, to emphasize that it is the space consisting of all ordinals smaller than $\omega _{1}$.
If the axiom of countable choice holds, every increasing ω-sequence of elements of $[0,\omega _{1})$ converges to a limit in $[0,\omega _{1})$. The reason is that the union (i.e., supremum) of every countable set of countable ordinals is another countable ordinal.
The topological space $[0,\omega _{1})$ is sequentially compact, but not compact. As a consequence, it is not metrizable. It is, however, countably compact and thus not Lindelöf (a countably compact space is compact if and only if it is Lindelöf). In terms of axioms of countability, $[0,\omega _{1})$ is first-countable, but neither separable nor second-countable.
The space $[0,\omega _{1}]=\omega _{1}+1$ is compact and not first-countable. $\omega _{1}$ is used to define the long line and the Tychonoff plank—two important counterexamples in topology.
See also
• Epsilon numbers (mathematics)
• Large countable ordinal
• Ordinal arithmetic
References
1. "Set Theory > Basic Set Theory (Stanford Encyclopedia of Philosophy)". plato.stanford.edu. Retrieved 2020-08-12.
2. "first uncountable ordinal in nLab". ncatlab.org. Retrieved 2020-08-12.
Bibliography
• Thomas Jech, Set Theory, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, ISBN 3-540-44085-2.
• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).
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Epsilon
Epsilon (/ˈɛpsɪlɒn/,[1] UK also /ɛpˈsaɪlən/;[2] uppercase Ε, lowercase ε or lunate ϵ; Greek: έψιλον) is the fifth letter of the Greek alphabet, corresponding phonetically to a mid front unrounded vowel IPA: [e̞] or IPA: [ɛ̝]. In the system of Greek numerals it also has the value five. It was derived from the Phoenician letter He . Letters that arose from epsilon include the Roman E, Ë and Ɛ, and Cyrillic Е, È, Ё, Є and Э.
Not to be confused with Upsilon.
Greek alphabet
Αα Alpha Νν Nu
Ββ Beta Ξξ Xi
Γγ Gamma Οο Omicron
Δδ Delta Ππ Pi
Εε Epsilon Ρρ Rho
Ζζ Zeta Σσς Sigma
Ηη Eta Ττ Tau
Θθ Theta Υυ Upsilon
Ιι Iota Φφ Phi
Κκ Kappa Χχ Chi
Λλ Lambda Ψψ Psi
Μμ Mu Ωω Omega
History
Archaic local variants
• Ϝ
• Ͱ
• Ϻ
• Ϙ
• Ͳ
• Ͷ
• Diacritics
• Ligatures
Numerals
• ϛ (6)
• ϟ (90)
• ϡ (900)
Use in other languages
• Bactrian
• Coptic
• Albanian
Related topics
• Use as scientific symbols
• Category
The name of the letter was originally εἶ (Ancient Greek: [êː]), but it was later changed to ἒ ψιλόν (e psilon 'simple e') in the Middle Ages to distinguish the letter from the digraph αι, a former diphthong that had come to be pronounced the same as epsilon.
The uppercase form of epsilon is identical to Latin E but has its own code point in Unicode: U+0395 Ε GREEK CAPITAL LETTER EPSILON. The lowercase version has two typographical variants, both inherited from medieval Greek handwriting. One, the most common in modern typography and inherited from medieval minuscule, looks like a reversed number "3" and is encoded U+03B5 ε GREEK SMALL LETTER EPSILON. The other, also known as lunate or uncial epsilon and inherited from earlier uncial writing,[3][4] looks like a semicircle crossed by a horizontal bar: it is encoded U+03F5 ϵ GREEK LUNATE EPSILON SYMBOL. While in normal typography these are just alternative font variants, they may have different meanings as mathematical symbols: computer systems therefore offer distinct encodings for them.[3] In TeX, \epsilon ( $\epsilon \!$ ) denotes the lunate form, while \varepsilon ( $\varepsilon \!$ ) denotes the reversed-3 form. Unicode versions 2.0.0 and onwards use ɛ as the lowercase Greek epsilon letter,[5] but in version 1.0.0, ϵ was used.[6]
There is also a 'Latin epsilon', ɛ or "open e", which looks similar to the Greek lowercase epsilon. It is encoded in Unicode as U+025B ɛ LATIN SMALL LETTER OPEN E and U+0190 Ɛ LATIN CAPITAL LETTER OPEN E and is used as an IPA phonetic symbol. The lunate or uncial epsilon provided inspiration for the euro sign, €.[7]
The lunate epsilon, ϵ, is not to be confused with the set membership symbol ∈; nor should the Latin uppercase epsilon, Ɛ, be confused with the Greek uppercase Σ (sigma). The symbol $\in $, first used in set theory and logic by Giuseppe Peano and now used in mathematics in general for set membership ("belongs to") evolved from the letter epsilon, since the symbol was originally used as an abbreviation for the Latin word est. In addition, mathematicians often read the symbol ∈ as "element of", as in "1 is an element of the natural numbers" for $1\in \mathbb {N} $, for example. As late as 1960, ε itself was used for set membership, while its negation "does not belong to" (now ∉) was denoted by ε' (epsilon prime).[8] Only gradually did a fully separate, stylized symbol take the place of epsilon in this role. In a related context, Peano also introduced the use of a backwards epsilon, ϶, for the phrase "such that", although the abbreviation s.t. is occasionally used in place of ϶ in informal cardinals.
History
Origin
The letter Ε was adopted from the Phoenician letter He () when Greeks first adopted alphabetic writing. In archaic Greek writing, its shape is often still identical to that of the Phoenician letter. Like other Greek letters, it could face either leftward or rightward (), depending on the current writing direction, but, just as in Phoenician, the horizontal bars always faced in the direction of writing. Archaic writing often preserves the Phoenician form with a vertical stem extending slightly below the lowest horizontal bar. In the classical era, through the influence of more cursive writing styles, the shape was simplified to the current E glyph.[9]
Sound value
While the original pronunciation of the Phoenician letter He was [h], the earliest Greek sound value of Ε was determined by the vowel occurring in the Phoenician letter name, which made it a natural choice for being reinterpreted from a consonant symbol to a vowel symbol denoting an [e] sound.[10] Besides its classical Greek sound value, the short /e/ phoneme, it could initially also be used for other [e]-like sounds. For instance, in early Attic before c. 500 BC, it was used also both for the long, open /ɛː/, and for the long close /eː/. In the former role, it was later replaced in the classic Greek alphabet by Eta (Η), which was taken over from eastern Ionic alphabets, while in the latter role it was replaced by the digraph spelling ΕΙ.
Epichoric alphabets
Some dialects used yet other ways of distinguishing between various e-like sounds.
In Corinth, the normal function of Ε to denote /e/ and /ɛː/ was taken by a glyph resembling a pointed B (), while Ε was used only for long close /eː/.[11] The letter Beta, in turn, took the deviant shape .
In Sicyon, a variant glyph resembling an X () was used in the same function as Corinthian .[12]
In Thespiai (Boeotia), a special letter form consisting of a vertical stem with a single rightward-pointing horizontal bar () was used for what was probably a raised variant of /e/ in pre-vocalic environments.[13][14] This tack glyph was used elsewhere also as a form of "Heta", i.e. for the sound /h/.
Glyph variants
After the establishment of the canonical classical Ionian (Euclidean) Greek alphabet, new glyph variants for Ε were introduced through handwriting. In the uncial script (used for literary papyrus manuscripts in late antiquity and then in early medieval vellum codices), the "lunate" shape () became predominant. In cursive handwriting, a large number of shorthand glyphs came to be used, where the cross-bar and the curved stroke were linked in various ways.[15] Some of them resembled a modern lowercase Latin "e", some a "6" with a connecting stroke to the next letter starting from the middle, and some a combination of two small "c"-like curves. Several of these shapes were later taken over into minuscule book hand. Of the various minuscule letter shapes, the inverted-3 form became the basis for lower-case Epsilon in Greek typography during the modern era.
Uncial Uncial variants Cursive variants Minuscule Minuscule with ligatures
Uses
International Phonetic Alphabet
Despite its pronunciation as mid, in the International Phonetic Alphabet, the Latin epsilon /ɛ/ represents open-mid front unrounded vowel, as in the English word pet /pɛt/.
Symbol
The uppercase Epsilon is not commonly used outside of the Greek language because of its similarity to the Latin letter E. However, it is commonly used in structural mechanics with Young's Modulus equations for calculating tensile, compressive and areal strain.
The Greek lowercase epsilon ε, the lunate epsilon symbol ϵ, and the Latin lowercase epsilon ɛ (see above) are used in a variety of places:
• In engineering mechanics, strain calculations ϵ = increase of length / original length. Usually this relates to extensometer testing of metallic materials.
• In mathematics
• (particularly calculus), an infinitesimally small positive quantity is commonly denoted ε; see (ε, δ)-definition of limit.
• Hilbert introduced epsilon terms $\epsilon x.\phi $ as an extension to first-order logic; see epsilon calculus.
• it is used to represent the Levi-Civita symbol.
• it is used to represent dual numbers: $a+b\varepsilon $, with $\varepsilon ^{2}=0$ and $\varepsilon \neq 0$.
• it is sometimes used to denote the Heaviside step function.[16]
• in set theory, the epsilon numbers are ordinal numbers that satisfy the fixed point ε = ωε. The first epsilon number, ε0, is the limit ordinal of the set {ω, ωω, ωωω, ...}.
• in numerical analysis and statistics it is used as the error term
• in group theory it is used as the idempotent group when e is in use as a variable name
• In computer science
• it often represents the empty string, though different writers use a variety of other symbols for the empty string as well; usually the lower-case Greek letter lambda (λ).
• the machine epsilon indicates the upper bound on the relative error due to rounding in floating point arithmetic.
• In physics,
• it indicates the permittivity of a medium; with the subscript 0 (ε0) it is the permittivity of free space.
• it can also indicate the strain of a material (a ratio of extensions).
• In automata theory, it shows a transition that involves no shifting of an input symbol.
• In astronomy,
• it stands for the fifth-brightest star in a constellation (see Bayer designation).
• Epsilon is the name for the most distant and most visible ring of Uranus.
• In planetary science, ε denotes the axial tilt.
• In chemistry, it represents the molar extinction coefficient of a chromophore.
• In economics, ε refers to elasticity.
• In statistics,
• it is used to refer to error terms.
• it also can to refer to the degree of sphericity in repeated measures ANOVAs.
• In agronomy, it is used to represent the "photosynthetic efficiency" of a particular plant or crop.
Unicode
• Greek Epsilon
Character information
PreviewΕεϵ϶
Unicode name GREEK CAPITAL LETTER EPSILON GREEK SMALL LETTER EPSILON GREEK LUNATE EPSILON SYMBOL GREEK REVERSED LUNATE EPSILON SYMBOL
Encodingsdecimalhexdechexdechexdechex
Unicode917U+0395949U+03B51013U+03F51014U+03F6
UTF-8206 149CE 95206 181CE B5207 181CF B5207 182CF B6
Numeric character referenceΕΕεεϵϵ϶϶
Named character referenceΕε, εϵ, ϵ, ϵ϶, ϶
DOS Greek132841569C
DOS Greek-2168A8222DE
Windows 1253197C5229E5
TeX\varepsilon\epsilon
• Coptic Eie
Character information
PreviewⲈⲉ
Unicode name COPTIC CAPITAL LETTER EIE COPTIC SMALL LETTER EIE
Encodingsdecimalhexdechex
Unicode11400U+2C8811401U+2C89
UTF-8226 178 136E2 B2 88226 178 137E2 B2 89
Numeric character referenceⲈⲈⲉⲉ
• Latin Open E
Character information
PreviewƐɛᶓᵋ
Unicode name LATIN CAPITAL LETTER
OPEN E
LATIN SMALL LETTER
OPEN E
LATIN SMALL LETTER
OPEN E WITH RETROFLEX HOOK
MODIFIER LETTER
SMALL OPEN E
Encodingsdecimalhexdechexdechexdechex
Unicode400U+0190603U+025B7571U+1D937499U+1D4B
UTF-8198 144C6 90201 155C9 9B225 182 147E1 B6 93225 181 139E1 B5 8B
Numeric character referenceƐƐɛɛᶓᶓᵋᵋ
Character information
Previewɜɝᶔᶟ
Unicode name LATIN SMALL LETTER
REVERSED OPEN E
LATIN SMALL LETTER
REVERSED OPEN E WITH HOOK
LATIN SMALL LETTER REVERSED
OPEN E WITH RETROFLEX HOOK
MODIFIER LETTER
SMALL REVERSED OPEN E
Encodingsdecimalhexdechexdechexdechex
Unicode604U+025C605U+025D7572U+1D947583U+1D9F
UTF-8201 156C9 9C201 157C9 9D225 182 148E1 B6 94225 182 159E1 B6 9F
Numeric character referenceɜɜɝɝᶔᶔᶟᶟ
Character information
Previewᴈᵌʚɞ
Unicode name LATIN SMALL LETTER
TURNED OPEN E
MODIFIER LETTER SMALL
TURNED OPEN E
LATIN SMALL LETTER
CLOSED OPEN E
LATIN SMALL LETTER
CLOSED REVERSED OPEN E
Encodingsdecimalhexdechexdechexdechex
Unicode7432U+1D087500U+1D4C666U+029A606U+025E
UTF-8225 180 136E1 B4 88225 181 140E1 B5 8C202 154CA 9A201 158C9 9E
Numeric character referenceᴈᴈᵌᵌʚʚɞɞ
• Mathematical Epsilon
Character information
Preview𝚬𝛆𝛦𝜀𝜠𝜺
Unicode name MATHEMATICAL BOLD
CAPITAL EPSILON
MATHEMATICAL BOLD
SMALL EPSILON
MATHEMATICAL ITALIC
CAPITAL EPSILON
MATHEMATICAL ITALIC
SMALL EPSILON
MATHEMATICAL BOLD ITALIC
CAPITAL EPSILON
MATHEMATICAL BOLD ITALIC
SMALL EPSILON
Encodingsdecimalhexdechexdechexdechexdechexdechex
Unicode120492U+1D6AC120518U+1D6C6120550U+1D6E6120576U+1D700120608U+1D720120634U+1D73A
UTF-8240 157 154 172F0 9D 9A AC240 157 155 134F0 9D 9B 86240 157 155 166F0 9D 9B A6240 157 156 128F0 9D 9C 80240 157 156 160F0 9D 9C A0240 157 156 186F0 9D 9C BA
UTF-1655349 57004D835 DEAC55349 57030D835 DEC655349 57062D835 DEE655349 57088D835 DF0055349 57120D835 DF2055349 57146D835 DF3A
Numeric character reference𝚬𝚬𝛆𝛆𝛦𝛦𝜀𝜀𝜠𝜠𝜺𝜺
Character information
Preview𝛜𝜖𝝐
Unicode name MATHEMATICAL BOLD
EPSILON SYMBOL
MATHEMATICAL ITALIC
EPSILON SYMBOL
MATHEMATICAL BOLD ITALIC
EPSILON SYMBOL
Encodingsdecimalhexdechexdechex
Unicode120540U+1D6DC120598U+1D716120656U+1D750
UTF-8240 157 155 156F0 9D 9B 9C240 157 156 150F0 9D 9C 96240 157 157 144F0 9D 9D 90
UTF-1655349 57052D835 DEDC55349 57110D835 DF1655349 57168D835 DF50
Numeric character reference𝛜𝛜𝜖𝜖𝝐𝝐
Character information
Preview𝝚𝝴𝞔𝞮
Unicode name MATHEMATICAL SANS-SERIF
BOLD CAPITAL EPSILON
MATHEMATICAL SANS-SERIF
BOLD SMALL EPSILON
MATHEMATICAL SANS-SERIF
BOLD ITALIC CAPITAL EPSILON
MATHEMATICAL SANS-SERIF
BOLD ITALIC SMALL EPSILON
Encodingsdecimalhexdechexdechexdechex
Unicode120666U+1D75A120692U+1D774120724U+1D794120750U+1D7AE
UTF-8240 157 157 154F0 9D 9D 9A240 157 157 180F0 9D 9D B4240 157 158 148F0 9D 9E 94240 157 158 174F0 9D 9E AE
UTF-1655349 57178D835 DF5A55349 57204D835 DF7455349 57236D835 DF9455349 57262D835 DFAE
Numeric character reference𝝚𝝚𝝴𝝴𝞔𝞔𝞮𝞮
Character information
Preview𝞊𝟄
Unicode name MATHEMATICAL SANS-SERIF
BOLD EPSILON SYMBOL
MATHEMATICAL SANS-SERIF
BOLD ITALIC EPSILON SYMBOL
Encodingsdecimalhexdechex
Unicode120714U+1D78A120772U+1D7C4
UTF-8240 157 158 138F0 9D 9E 8A240 157 159 132F0 9D 9F 84
UTF-1655349 57226D835 DF8A55349 57284D835 DFC4
Numeric character reference𝞊𝞊𝟄𝟄
These characters are used only as mathematical symbols. Stylized Greek text should be encoded using the normal Greek letters, with markup and formatting to indicate text style.
Initial
• Initial epsilon in Lectionary 226, folio 20 verso
• folio 64 verso
• folio 125 verso
See also
• Е and е, the letter Ye of the Cyrillic alphabet
• Є є, Ukrainian Ye
• Ԑ ԑ, Reversed Ze
• E (disambiguation)
References
1. Wells, John C. (1990). "epsilon". Longman Pronunciation Dictionary. Harlow, England: Longman. p. 250. ISBN 0582053838.
2. "epsilon". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
3. Nick Nicholas: Letters Archived 2012-12-15 at archive.today, 2003–2008. (Greek Unicode Issues)
4. Colwell, Ernest C. (1969). "A chronology for the letters Ε, Η, Λ, Π in the Byzantine minuscule book hand". Studies in methodology in textual criticism of the New Testament. Leiden: Brill. p. 127.
5. "Code Charts" (PDF). The Unicode Standard, Version 2.0. p. 130. ISBN 0-201-48345-9.
6. "Code Charts" (PDF). The Unicode Standard, Version 1.0. Vol. 1. p. 130. ISBN 0-201-56788-1.
7. "European Commission – Economic and Financial Affairs – How to use the euro name and symbol". Ec.europa.eu. Retrieved 7 April 2010. Inspiration for the € symbol itself came from the Greek epsilon, ϵ – a reference to the cradle of European civilization – and the first letter of the word Europe, crossed by two parallel lines to 'certify' the stability of the euro.
8. Halmos, Paul R. (1960). Naive Set Theory. New York: Van Nostrand. pp. 5–6. ISBN 978-1614271314.
9. Jeffery, Lilian H. (1961). The local scripts of archaic Greece. Oxford: Clarendon. pp. 63–64.
10. Jeffery, Local scripts, p. 24.
11. Jeffery, Local scripts, p. 114.
12. Jeffery, Local scripts, p. 138.
13. Nicholas, Nick (2005). "Proposal to add Greek epigraphical letters to the UCS" (PDF). Archived from the original (PDF) on 2006-05-05. Retrieved 2010-08-12.
14. Jeffery, Local scripts, p. 89.
15. Thompson, Edward M. (1911). An introduction to Greek and Latin palaeography. Oxford: Clarendon. pp. 191–194.
16. Weisstein, Eric W. "Delta Function". mathworld.wolfram.com. Retrieved 2019-02-19.
Further reading
Look up Ε or ɛ in Wiktionary, the free dictionary.
• Hoffman, Paul; The Man Who Loved Only Numbers. Hyperion, 1998. ISBN 0-7868-6362-5.
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Sigma
Sigma (/ˈsɪɡmə/;[1] uppercase Σ, lowercase σ, lowercase in word-final position ς; Greek: σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used as an operator for summation. When used at the end of a letter-case word (one that does not use all caps), the final form (ς) is used. In Ὀδυσσεύς (Odysseus), for example, the two lowercase sigmas (σ) in the center of the name are distinct from the word-final sigma (ς) at the end. The Latin letter S derives from sigma while the Cyrillic letter Es derives from a lunate form of this letter.
Greek alphabet
Αα Alpha Νν Nu
Ββ Beta Ξξ Xi
Γγ Gamma Οο Omicron
Δδ Delta Ππ Pi
Εε Epsilon Ρρ Rho
Ζζ Zeta Σσς Sigma
Ηη Eta Ττ Tau
Θθ Theta Υυ Upsilon
Ιι Iota Φφ Phi
Κκ Kappa Χχ Chi
Λλ Lambda Ψψ Psi
Μμ Mu Ωω Omega
History
Archaic local variants
• Ϝ
• Ͱ
• Ϻ
• Ϙ
• Ͳ
• Ͷ
• Diacritics
• Ligatures
Numerals
• ϛ (6)
• ϟ (90)
• ϡ (900)
Use in other languages
• Bactrian
• Coptic
• Albanian
Related topics
• Use as scientific symbols
• Category
History
The shape (Σς) and alphabetic position of sigma is derived from the Phoenician letter (shin).
Sigma's original name may have been san, but due to the complicated early history of the Greek epichoric alphabets, san came to be identified as a separate letter in the Greek alphabet, represented as Ϻ.[2] Herodotus reports that "san" was the name given by the Dorians to the same letter called "sigma" by the Ionians.[lower-roman 1][3]
According to one hypothesis,[4] the name "sigma" may continue that of Phoenician samekh (), the letter continued through Greek xi, represented as Ξ. Alternatively, the name may have been a Greek innovation that simply meant 'hissing', from the root of σίζω (sízō, from Proto-Greek *sig-jō 'I hiss').[2]
Lunate sigma
In handwritten Greek during the Hellenistic period (4th–3rd century BC), the epigraphic form of Σ was simplified into a C-like shape,[5] which has also been found on coins from the 4th century BC onward.[6] This became the universal standard form of sigma during late antiquity and the Middle Ages.
Today, it is known as lunate sigma (uppercase Ϲ, lowercase ϲ), because of its crescent-like shape, and is still widely used in decorative typefaces in Greece, especially in religious and church contexts, as well as in some modern print editions of classical Greek texts.
A dotted lunate sigma (sigma periestigmenon, Ͼ) was used by Aristarchus of Samothrace (220–143 BC) as an editorial sign indicating that the line marked as such is at an incorrect position. Similarly, a reversed sigma (antisigma, Ͻ), may mark a line that is out of place. A dotted antisigma (antisigma periestigmenon, Ͽ) may indicate a line after which rearrangements should be made, or to variant readings of uncertain priority.
In Greek inscriptions from the late first century BC onwards, Ͻ was an abbreviation indicating that a man's father's name is the same as his own name, thus Dionysodoros son of Dionysodoros would be written Διονυσόδωρος Ͻ (Dionysodoros Dionysodorou).[7][8]
In Unicode, the above variations of lunate sigma are encoded as U+03F9 Ϲ ; U+03FD Ͻ , U+03FE Ͼ , and U+03FF Ͽ .
Derived alphabets
Sigma was adopted in the Old Italic alphabets beginning in the 8th century BC. At that time a simplified three-stroke version, omitting the lowermost stroke, was already found in Western Greek alphabets, and was incorporated into classical Etruscan and Oscan, as well as in the earliest Latin epigraphy (early Latin S), such as the Duenos inscription. The alternation between three and four (and occasionally more than four) strokes was also adopted into the early runic alphabet (early form of the s-rune). Both the Anglo-Saxon runes and the Younger Futhark consistently use the simplified three-stroke version.
The letter С of Cyrillic script originates in the lunate form of Sigma.
Uses
Language and linguistics
• In both Ancient and Modern Greek, the sigma represents the voiceless alveolar fricative [s]. In Modern Greek, this sound is voiced to the voiced alveolar fricative [z] when occurring before [m], [n], [v], [ð] or [ɣ].
• The uppercase form of sigma (Σ) was re-borrowed into the Latin alphabet—more precisely, the International African Alphabet—to serve as the uppercase of modern esh (lowercase: ʃ).
• In phonology, σ is used to represent syllables.
• In linguistics, Σ represents the set of symbols that form an alphabet (see also computer science).
• In historical linguistics, Σ is used to represent a Common Brittonic consonant with a sound between [s] and [h]; perhaps an aspirated [[Voiceless postalveolar fricative|[ʃʰ]]].[9]
Mathematics
• In general mathematics, lowercase σ is commonly used to represent unknown angles, as well as serving as a shorthand for "countably", whereas Σ is regularly used as the operator for summation, e.g.:
$\sum _{k=0}^{5}k=0+1+2+3+4+5=15$
• In mathematical logic, $\Sigma _{n}^{0}$ is used to denote the set of formulae with bounded quantifiers beginning with existential quantifiers, alternating $n-1$ times between existential and universal quantifiers. This notation reflects an indirect analogy between the relationship of summation and products on one hand, and existential and universal quantifiers on the other. See the article on the arithmetic hierarchy.
• In statistics, σ represents the standard deviation of population or probability distribution (where mu or μ is used for the mean).
• In topology, σ-compact topological space is one that can be written as a countable union of compact subsets.
• In mathematical analysis and in probability theory, there is a type of algebra of sets known as σ-algebra (aka σ-field). Sigma algebra also includes terms such as:
• σ(A), denoting the generated sigma-algebra of a set A
• Σ-finite measure (see measure theory)
• In number theory, σ is included in various divisor functions, especially the sigma function or sum-of-divisors function.
• In applied mathematics, σ(T) denotes the spectrum of a linear map T.
• In complex analysis, σ is used in the Weierstrass sigma-function.
• In probability theory and statistics, Σ denotes the covariance matrix of a set of random variables, sometimes in the form $\;|\!\!\!\Sigma $ to distinguish it from the summation operator.
• Theoretical spectral analysis uses σ as standard deviation opposed to lowercase mu as the absolute mean value.
Biology, physiology, and medicine
• In biology, the sigma receptor (σ–receptors) is a type of cell surface receptor.
• In biochemistry, the σ factor (or specificity factor) is a protein found in RNA polymerase.
• In bone physiology, the bone remodeling period—i.e., the life span of a basic multicellular unit—has historically been referred to as the sigma period
• In early 20th-century physiology literature, σ had been used to represent milliseconds[10]
Business, finance, and economics
• In finance, σ is the symbol used to represent volatility of stocks, usually measured by the standard deviation of logarithmic returns.
• In accounting, Σ indicates the balance of invoice classes and the overall amount of debts and demands.
• In macroeconomics, σ is used in equations to represent the elasticity of substitution between two inputs.
• In the machine industry, Six Sigma (6σ) is a quality model based on the standard deviation.
Chemistry
• Sigma bonds (σ bonds) are the strongest type of covalent chemical bond.
• In organic chemistry, σ symbolizes the sigma constant of Hammett equation.
• In alchemy, Σ was sometimes used to represent sugar.
Engineering and computer science
• In computer science, Σ represents the set of symbols that form an alphabet (see also linguistics)
• Relational algebra uses the values $\sigma _{a\theta b}(R)$ and $\sigma _{a\theta v}(R)$ to denote selections, which are a type of unary operation.
• In machine learning, σ is used in the formula that derives the Sigmoid function.
• In radar jamming or electronic warfare, radar cross-sections (RCS) are commonly represented as σ when measuring the size of a target's image on radar.
• In signal processing, σ denotes the damping ratio of a system parameter.
• In theoretical computer science, Σ serves as the busy beaver function.
• In civil engineering, σ refers to the normal stress applied on a material or structure.
Physics
• In nuclear and particle physics, σ is used to denote cross sections in general (see also RCS), while Σ represents macroscopic cross sections [1/length].
• The symbol is to denote the Stefan–Boltzmann constant.
• In relation to fundamental properties of material, σ is often used to signify electrical conductivity.
• In electrostatics, σ represents surface charge density.
• In continuum mechanics, σ is used to signify stress.
• In condensed matter physics, Σ denotes self-energy.
• The symbol can be used to signify surface tension (alternatively, γ or T are also used instead).
• In quantum mechanics, σ is used to indicate Pauli matrices.
• In astronomy, σ represents velocity dispersion.
• In astronomy, the prefix Σ is used to designate double stars of the Catalogus Novus Stellarum Duplicium by Friedrich Georg Wilhelm von Struve.
• In particle physics, Σ represents a class of baryons.
Organizations
• During the 1930s, an uppercase Σ was in use as the symbol of the Ação Integralista Brasileira, a fascist political party in Brazil.
• Sigma Corporation uses the name of the letter but not the letter itself, but in many Internet forums, photographers refer to the company or its lenses using the letter.
• Sigma Aldrich incorporate both the name and the character in their logo.
Character encoding
Greek sigma
Character information
PreviewΣσςϹϲ
Unicode name GREEK CAPITAL LETTER SIGMA GREEK SMALL LETTER SIGMA GREEK SMALL LETTER FINAL SIGMA GREEK CAPITAL LUNATE SIGMA SYMBOL GREEK LUNATE SIGMA SYMBOL
Encodingsdecimalhexdechexdechexdechexdechex
Unicode931U+03A3963U+03C3962U+03C21017U+03F91010U+03F2
UTF-8206 163CE A3207 131CF 83207 130CF 82207 185CF B9207 178CF B2
Numeric character referenceΣΣσσςςϹϹϲϲ
Named character referenceΣσς, ς, ς
DOS Greek14591169A9170AA
DOS Greek-2207CF236EC237ED
Windows 1253211D3243F3242F2
TeX\Sigma\sigma\varsigma
[11]
Character information
PreviewϽͻϾͼϿͽ
Unicode name GREEK CAPITAL REVERSED LUNATE SIGMA SYMBOL GREEK SMALL REVERSED LUNATE SIGMA SYMBOL GREEK CAPITAL DOTTED LUNATE SIGMA SYMBOL GREEK SMALL DOTTED LUNATE SIGMA SYMBOL GREEK CAPITAL REVERSED DOTTED LUNATE SIGMA SYMBOL GREEK SMALL REVERSED DOTTED LUNATE SIGMA SYMBOL
Encodingsdecimalhexdechexdechexdechexdechexdechex
Unicode1021U+03FD891U+037B1022U+03FE892U+037C1023U+03FF893U+037D
UTF-8207 189CF BD205 187CD BB207 190CF BE205 188CD BC207 191CF BF205 189CD BD
Numeric character referenceϽϽͻͻϾϾͼͼϿϿͽͽ
Coptic sima
Character information
PreviewⲤⲥ
Unicode name COPTIC CAPITAL LETTER SIMA COPTIC SMALL LETTER SIMA
Encodingsdecimalhexdechex
Unicode11428U+2CA411429U+2CA5
UTF-8226 178 164E2 B2 A4226 178 165E2 B2 A5
Numeric character referenceⲤⲤⲥⲥ
Mathematical sigma
These characters are used only as mathematical symbols. Stylized Greek text should be encoded using the normal Greek letters, with markup and formatting to indicate text style.
Character information
Preview∑𝚺𝛔𝛓𝛴𝜎
Unicode name N-ARY SUMMATION MATHEMATICAL BOLD
CAPITAL SIGMA
MATHEMATICAL BOLD
SMALL SIGMA
MATHEMATICAL BOLD
SMALL FINAL SIGMA
MATHEMATICAL ITALIC
CAPITAL SIGMA
MATHEMATICAL ITALIC
SMALL SIGMA
Encodingsdecimalhexdechexdechexdechexdechexdechex
Unicode8721U+2211120506U+1D6BA120532U+1D6D4120531U+1D6D3120564U+1D6F4120590U+1D70E
UTF-8226 136 145E2 88 91240 157 154 186F0 9D 9A BA240 157 155 148F0 9D 9B 94240 157 155 147F0 9D 9B 93240 157 155 180F0 9D 9B B4240 157 156 142F0 9D 9C 8E
UTF-168721221155349 57018D835 DEBA55349 57044D835 DED455349 57043D835 DED355349 57076D835 DEF455349 57102D835 DF0E
Numeric character reference∑∑𝚺𝚺𝛔𝛔𝛓𝛓𝛴𝛴𝜎𝜎
Named character reference∑, ∑
Character information
Preview𝜍𝜮𝝈𝝇𝝨
Unicode name MATHEMATICAL ITALIC
SMALL FINAL SIGMA
MATHEMATICAL BOLD ITALIC
CAPITAL SIGMA
MATHEMATICAL BOLD ITALIC
SMALL SIGMA
MATHEMATICAL BOLD ITALIC
SMALL FINAL SIGMA
MATHEMATICAL SANS-SERIF
BOLD CAPITAL SIGMA
Encodingsdecimalhexdechexdechexdechexdechex
Unicode120589U+1D70D120622U+1D72E120648U+1D748120647U+1D747120680U+1D768
UTF-8240 157 156 141F0 9D 9C 8D240 157 156 174F0 9D 9C AE240 157 157 136F0 9D 9D 88240 157 157 135F0 9D 9D 87240 157 157 168F0 9D 9D A8
UTF-1655349 57101D835 DF0D55349 57134D835 DF2E55349 57160D835 DF4855349 57159D835 DF4755349 57192D835 DF68
Numeric character reference𝜍𝜍𝜮𝜮𝝈𝝈𝝇𝝇𝝨𝝨
Character information
Preview𝞂𝞁𝞢𝞼𝞻
Unicode name MATHEMATICAL SANS-SERIF
BOLD SMALL SIGMA
MATHEMATICAL SANS-SERIF
BOLD SMALL FINAL SIGMA
MATHEMATICAL SANS-SERIF
BOLD ITALIC CAPITAL SIGMA
MATHEMATICAL SANS-SERIF
BOLD ITALIC SMALL SIGMA
MATHEMATICAL SANS-SERIF
BOLD ITALIC SMALL FINAL SIGMA
Encodingsdecimalhexdechexdechexdechexdechex
Unicode120706U+1D782120705U+1D781120738U+1D7A2120764U+1D7BC120763U+1D7BB
UTF-8240 157 158 130F0 9D 9E 82240 157 158 129F0 9D 9E 81240 157 158 162F0 9D 9E A2240 157 158 188F0 9D 9E BC240 157 158 187F0 9D 9E BB
UTF-1655349 57218D835 DF8255349 57217D835 DF8155349 57250D835 DFA255349 57276D835 DFBC55349 57275D835 DFBB
Numeric character reference𝞂𝞂𝞁𝞁𝞢𝞢𝞼𝞼𝞻𝞻
See also
Wikimedia Commons has media related to the letter sigma.
Look up Σ, σ, or ς in Wiktionary, the free dictionary.
• Antisigma
• Greek letters used in mathematics, science, and engineering
• Sampi
• Sho (letter)
• Stigma (letter)
• Sibilant consonant
• Summation (Σ)
• Combining form "sigm-" (e.g. sigmodon, sigmurethra, etc.)
• Derivative "sigmoid" (e.g. sigmoid sinus, sigmoid colon, sigmoidoscopy, etc.)
References
Notes
1. "the same letter, which the Dorians call "san", but the Ionians 'sigma'..." [translated from Ancient Greek: "τὠυτὸ γράμμα, τὸ Δωριέες μὲν σὰν καλέουσι ,Ἴωνες δὲ σίγμα"] (Herodotus 1.139)
Citations
1. "sigma". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
2. Woodard, Roger D. (2006). "Alphabet". In Wilson, Nigel Guy (ed.). Encyclopedia of Ancient Greece. London: Routledge. p. 38.
3. Herodotus, Histories 1.139 — Everson, Michael and Nicholas Sims-Williams. 2002. "Non-Attic letters," transcribed by N. Nicholas. Archived from the original 2020-06-28.
4. Jeffery, Lilian H. (1961). The Local Scripts of Archaic Greece. Oxford: Clarendon. pp. 25–7.
5. Thompson, Edward M. (1912). Introduction to Greek and Latin Paleography. Oxford: Clarendon. p. 108, 144.
6. Hopkins, Edward C. D. (2004). "Letterform Usage | Numismatica Font Projects" Parthia.
7. de Lisle, Christopher (2020). "Attic Inscriptions in UK Collections: Ashmolean Museum, Oxford". AIUK. 11: 11. ISSN 2054-6769. Retrieved 2 June 2022.
8. Follet, Simone (2000). "Les deux archontes Pamménès du Ier siècle a.c. à Athènes". Revue des Études Grecques. 113: 188–192. doi:10.3406/reg.2000.4402.
9. Conroy, Kevin M. (21 February 2008). "Celtic initial consonant mutations - nghath and bhfuil?" – via dlib.bc.edu.
10. Hill, A. V. (1935). "Units and Symbols". Nature. 136 (3432): 222. Bibcode:1935Natur.136..222H. doi:10.1038/136222a0. S2CID 4087300.
11. Unicode Code Charts: Greek and Coptic (Range: 0370-03FF)
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Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov (Russian: Андре́й Никола́евич Колмого́ров, IPA: [ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf] (listen), 25 April 1903 – 20 October 1987)[4][5] was a Soviet mathematician who contributed to the mathematics of probability theory, topology, intuitionistic logic, turbulence, classical mechanics, algorithmic information theory and computational complexity.[6][2][7]
Andrey Kolmogorov
Born
Andrey Nikolaevich Kolmogorov
(1903-04-25)25 April 1903
Tambov, Russian Empire
Died20 October 1987(1987-10-20) (aged 84)
Moscow, Russian SFSR, Soviet Union
CitizenshipSoviet Union
Alma materMoscow State University (Ph.D.)
Known for
• Probability theory
• Probability space
• Topology
• Intuitionistic logic
• Turbulence studies
• Classical mechanics
• Mathematical analysis
• Kolmogorov complexity
• KAM theorem
• KPP equation
Spouse
Anna Dmitrievna Egorova
(m. 1942–1987)
Awards
• Member of the Russian Academy of Sciences[1]
• Stalin Prize (1941)
• Balzan Prize (1962)
• ForMemRS (1964)[2]
• Lenin Prize (1965)
• Wolf Prize (1980)
• Lobachevsky Prize (1986)
Scientific career
FieldsMathematics
InstitutionsMoscow State University
Doctoral advisorNikolai Luzin[3]
Doctoral students
• Vladimir Alekseev
• Vladimir Arnold
• Sergei N. Artemov
• Grigory Barenblatt
• Roland Dobrushin
• Eugene Dynkin
• Israil Gelfand
• Boris Gnedenko
• Leonid Levin
• Valerii Kozlov
• Per Martin-Löf
• Robert Minlos
• Andrei Monin
• Sergey Nikolsky
• Alexander Obukhov
• Yuri Prokhorov
• Yakov Sinai
• Albert Shiryaev
• Anatoli Vitushkin
• Vladimir Uspensky
• Akiva Yaglom
• Vladimir Vovk[3]
Biography
Early life
Andrey Kolmogorov was born in Tambov, about 500 kilometers south-southeast of Moscow, in 1903. His unmarried mother, Maria Yakovlevna Kolmogorova, died giving birth to him.[8] Andrey was raised by two of his aunts in Tunoshna (near Yaroslavl) at the estate of his grandfather, a well-to-do nobleman.
Little is known about Andrey's father. He was supposedly named Nikolai Matveyevich Katayev and had been an agronomist. Katayev had been exiled from Saint Petersburg to the Yaroslavl province after his participation in the revolutionary movement against the tsars. He disappeared in 1919 and was presumed to have been killed in the Russian Civil War.
Andrey Kolmogorov was educated in his aunt Vera's village school, and his earliest literary efforts and mathematical papers were printed in the school journal "The Swallow of Spring". Andrey (at the age of five) was the "editor" of the mathematical section of this journal. Kolmogorov's first mathematical discovery was published in this journal: at the age of five he noticed the regularity in the sum of the series of odd numbers: $1=1^{2};1+3=2^{2};1+3+5=3^{2},$ etc.[9]
In 1910, his aunt adopted him, and they moved to Moscow, where he graduated from high school in 1920. Later that same year, Kolmogorov began to study at Moscow State University and at the same time Mendeleev Moscow Institute of Chemistry and Technology.[10] Kolmogorov writes about this time: "I arrived at Moscow University with a fair knowledge of mathematics. I knew in particular the beginning of set theory. I studied many questions in articles in the Encyclopedia of Brockhaus and Efron, filling out for myself what was presented too concisely in these articles."[11]
Kolmogorov gained a reputation for his wide-ranging erudition. While an undergraduate student in college, he attended the seminars of the Russian historian S. V. Bakhrushin, and he published his first research paper on the fifteenth and sixteenth centuries' landholding practices in the Novgorod Republic.[12] During the same period (1921–22), Kolmogorov worked out and proved several results in set theory and in the theory of Fourier series.
Adulthood
In 1922, Kolmogorov gained international recognition for constructing a Fourier series that diverges almost everywhere.[13][14] Around this time, he decided to devote his life to mathematics.
In 1925, Kolmogorov graduated from Moscow State University and began to study under the supervision of Nikolai Luzin.[3] He formed a lifelong close friendship with Pavel Alexandrov, a fellow student of Luzin; indeed, several researchers have concluded that the two friends were involved in a homosexual relationship,[15][16][17][18] although neither acknowledged this openly during their lifetimes. Kolmogorov (together with Aleksandr Khinchin) became interested in probability theory. Also in 1925, he published his work in intuitionistic logic, "On the principle of the excluded middle," in which he proved that under a certain interpretation all statements of classical formal logic can be formulated as those of intuitionistic logic. In 1929, Kolmogorov earned his Doctor of Philosophy (Ph.D.) degree from Moscow State University.
In 1930, Kolmogorov went on his first long trip abroad, traveling to Göttingen and Munich and then to Paris. He had various scientific contacts in Göttingen, first with Richard Courant and his students working on limit theorems, where diffusion processes proved to be the limits of discrete random processes, then with Hermann Weyl in intuitionistic logic, and lastly with Edmund Landau in function theory. His pioneering work About the Analytical Methods of Probability Theory was published (in German) in 1931. Also in 1931, he became a professor at Moscow State University.
In 1933, Kolmogorov published his book, Foundations of the Theory of Probability, laying the modern axiomatic foundations of probability theory and establishing his reputation as the world's leading expert in this field. In 1935, Kolmogorov became the first chairman of the department of probability theory at Moscow State University. Around the same years (1936) Kolmogorov contributed to the field of ecology and generalized the Lotka–Volterra model of predator–prey systems.
During the Great Purge in 1936, Kolmogorov's doctoral advisor Nikolai Luzin became a high-profile target of Stalin's regime in what is now called the "Luzin Affair." Kolmogorov and several other students of Luzin testified against Luzin, accusing him of plagiarism, nepotism, and other forms of misconduct; the hearings eventually concluded that he was a servant to "fascistoid science" and thus an enemy of the Soviet people. Luzin lost his academic positions, but curiously he was neither arrested nor expelled from the Academy of Sciences of the Soviet Union.[19][20] The question of whether Kolmogorov and others were coerced into testifying against their teacher remains a topic of considerable speculation among historians; all parties involved refused to publicly discuss the case for the rest of their lives. Soviet-Russian mathematician Semën Samsonovich Kutateladze concluded in 2013, after reviewing archival documents made available during the 1990s and other surviving testimonies, that the students of Luzin had initiated the accusations against Luzin out of personal acrimony; there was no definitive evidence that the students were coerced by the state, nor was there any definitive evidence to support their allegations of academic misconduct.[21] Soviet historian of mathematics A.P. Yushkevich surmised that, unlike many of the other high-profile persecutions of the era, Stalin did not personally initiate the persecution of Luzin and instead eventually concluded that he was not a threat to the regime, which would explain the unusually mild punishment relative to other contemporaries.[22]
In a 1938 paper, Kolmogorov "established the basic theorems for smoothing and predicting stationary stochastic processes"—a paper that had major military applications during the Cold War.[23] In 1939, he was elected a full member (academician) of the USSR Academy of Sciences.
During World War II Kolmogorov contributed to the Soviet war effort by applying statistical theory to artillery fire, developing a scheme of stochastic distribution of barrage balloons intended to help protect Moscow from German bombers during the Battle of Moscow.[24]
In his study of stochastic processes, especially Markov processes, Kolmogorov and the British mathematician Sydney Chapman independently developed a pivotal set of equations in the field that have been given the name of the Chapman–Kolmogorov equations.
Later, Kolmogorov focused his research on turbulence, beginning his publications in 1941. In classical mechanics, he is best known for the Kolmogorov–Arnold–Moser theorem, first presented in 1954 at the International Congress of Mathematicians.[6] In 1957, working jointly with his student Vladimir Arnold, he solved a particular interpretation of Hilbert's thirteenth problem. Around this time he also began to develop, and has since been considered a founder of, algorithmic complexity theory – often referred to as Kolmogorov complexity theory.
Kolmogorov married Anna Dmitrievna Egorova in 1942. He pursued a vigorous teaching routine throughout his life both at the university level and also with younger children, as he was actively involved in developing a pedagogy for gifted children in literature, music, and mathematics. At Moscow State University, Kolmogorov occupied different positions including the heads of several departments: probability, statistics, and random processes; mathematical logic. He also served as the Dean of the Moscow State University Department of Mechanics and Mathematics.
In 1971, Kolmogorov joined an oceanographic expedition aboard the research vessel Dmitri Mendeleev. He wrote a number of articles for the Great Soviet Encyclopedia. In his later years, he devoted much of his effort to the mathematical and philosophical relationship between probability theory in abstract and applied areas.[25]
Kolmogorov died in Moscow in 1987 and his remains were buried in the Novodevichy cemetery.
A quotation attributed to Kolmogorov is [translated into English]: "Every mathematician believes that he is ahead of the others. The reason none state this belief in public is because they are intelligent people."
Vladimir Arnold once said: "Kolmogorov – Poincaré – Gauss – Euler – Newton, are only five lives separating us from the source of our science."
Awards and honours
Kolmogorov received numerous awards and honours both during and after his lifetime:
• Member of the Russian Academy of Sciences[1]
• Awarded the Stalin Prize in 1941
• Elected an Honorary Member of the American Academy of Arts and Sciences in 1959[26]
• Elected member of the American Philosophical Society in 1961[27]
• Awarded the Balzan Prize in 1962
• Elected a Foreign Member of the Royal Netherlands Academy of Arts and Sciences in 1963[28]
• Elected a Foreign Member of the Royal Society (ForMemRS) in 1964.[2]
• Awarded the Lenin Prize in 1965
• Elected member of the United States National Academy of Sciences in 1967[29]
• Awarded the Wolf Prize in 1980
• Awarded the Lobachevsky Prize in 1986
The following are named in Kolmogorov's honour:
• Fisher–Kolmogorov equation
• Johnson–Mehl–Avrami–Kolmogorov equation
• Kolmogorov axioms
• Kolmogorov equations (also known as the Fokker–Planck equations in the context of diffusion and in the forward case)
• Kolmogorov dimension (upper box dimension)
• Kolmogorov–Arnold theorem
• Kolmogorov–Arnold–Moser theorem
• Kolmogorov continuity theorem
• Kolmogorov's criterion
• Kolmogorov extension theorem
• Kolmogorov's three-series theorem
• Convergence of Fourier series
• Gnedenko-Kolmogorov central limit theorem
• Quasi-arithmetic mean (it is also called Kolmogorov mean)
• Kolmogorov homology
• Kolmogorov's inequality
• Landau–Kolmogorov inequality
• Kolmogorov integral
• Brouwer–Heyting–Kolmogorov interpretation
• Kolmogorov microscales
• Kolmogorov's normability criterion
• Fréchet–Kolmogorov theorem
• Kolmogorov space
• Kolmogorov complexity
• Kolmogorov–Smirnov test
• Wiener filter (also known as Wiener–Kolmogorov filtering theory)
• Wiener–Kolmogorov prediction
• Kolmogorov automorphism
• Kolmogorov's characterization of reversible diffusions
• Borel–Kolmogorov paradox
• Chapman–Kolmogorov equation
• Hahn–Kolmogorov theorem
• Johnson–Mehl–Avrami–Kolmogorov equation
• Kolmogorov–Sinai entropy
• Astronomical seeing described by Kolmogorov's turbulence law
• Kolmogorov structure function
• Kolmogorov–Uspenskii machine model
• Kolmogorov's zero–one law
• Kolmogorov–Zurbenko filter
• Kolmogorov's two-series theorem
• Rao–Blackwell–Kolmogorov theorem
• Khinchin–Kolmogorov theorem
• Kolmogorov's Strong Law of Large Numbers
Bibliography
A bibliography of his works appeared in "Publications of A. N. Kolmogorov". Annals of Probability. 17 (3): 945–964. July 1989. doi:10.1214/aop/1176991252.
• Kolmogorov, Andrey (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung (in German). Berlin: Julius Springer.[30]
• Translation: Kolmogorov, Andrey (1956). Foundations of the Theory of Probability (2nd ed.). New York: Chelsea. ISBN 978-0-8284-0023-7. Archived from the original on 14 September 2018. Retrieved 17 February 2016.
• 1991–93. Selected works of A.N. Kolmogorov, 3 vols. Tikhomirov, V. M., ed., Volosov, V. M., trans. Dordrecht:Kluwer Academic Publishers. ISBN 90-277-2796-1
• 1925. "On the principle of the excluded middle" in Jean van Heijenoort, 1967. A Source Book in Mathematical Logic, 1879–1931. Harvard Univ. Press: 414–37.
• Kolmogorov, Andrei N. (1963). "On Tables of Random Numbers". Sankhyā Ser. A. 25: 369–375. MR 0178484.
• Kolmogorov, Andrei N. (1998) [1963]. "On Tables of Random Numbers". Theoretical Computer Science. 207 (2): 387–395. doi:10.1016/S0304-3975(98)00075-9. MR 1643414.
• Kolmogorov, Andrei N. (2005) Selected works. In 6 volumes. Moscow (in Russian)
Textbooks:
• A. N. Kolmogorov and B. V. Gnedenko. "Limit distributions for sums of independent random variables", 1954.
• A. N. Kolmogorov and S. V. Fomin. "Elements of the Theory of Functions and Functional Analysis", Publication 1999, Publication 2012
• Kolmogorov, Andrey Nikolaevich; Fomin, Sergei Vasilyevich (1975) [1970]. Introductory real analysis. New York: Dover Publications. ISBN 978-0-486-61226-3..
References
1. Youschkevitch, A. P. (1983), "A. N. Kolmogorov: Historian and philosopher of mathematics on the occasion of his 80th birfhday", Historia Mathematica, 10 (4): 383–395, doi:10.1016/0315-0860(83)90001-0
2. Kendall, D. G. (1991). "Andrei Nikolaevich Kolmogorov. 25 April 1903-20 October 1987". Biographical Memoirs of Fellows of the Royal Society. 37: 300–326. doi:10.1098/rsbm.1991.0015. S2CID 58080873.
3. Andrey Kolmogorov at the Mathematics Genealogy Project
4. "Academician Andrei Nikolaevich Kolmogorov (obituary)". Russian Mathematical Surveys. 43 (1): 1–9. 1988. Bibcode:1988RuMaS..43....1.. doi:10.1070/RM1988v043n01ABEH001555. S2CID 250857950.
5. Parthasarathy, K. R. (1988). "Obituary: Andrei Nikolaevich Kolmogorov". Journal of Applied Probability. 25 (2): 445–450. doi:10.1017/S0021900200041115. JSTOR 3214455.
6. Yaglom, A M (January 1994). "A. N. Kolmogorov as a Fluid Mechanician and Founder of a School in Turbulence Research". Annual Review of Fluid Mechanics. 26 (1): 1–23. doi:10.1146/annurev.fl.26.010194.000245. ISSN 0066-4189. Retrieved 23 February 2023.
7. O'Connor, John J.; Robertson, Edmund F., "Andrey Kolmogorov", MacTutor History of Mathematics Archive, University of St Andrews
8. Encyclopædia Britannica Online, s. v. "Andrey Nikolayevich Kolmogorov", accessed February 22, 2013.
9. "Andrei N Kolmogorov prepared by V M Tikhomirov". Wolf Prize in Mathematics, v.2. World Scientific. 2001. pp. 119–141. ISBN 9789812811769.
10. "Андрей Николаевич КОЛМОГОРОВ. Curriculum Vitae". Retrieved 19 June 2023.
11. Society, American Mathematical (2000). Kolmogorov in Perspective (History of Mathematics). American Mathematical Soc. p. 6. ISBN 978-0821829189.
12. Salsburg, David (2001). The Lady Tasting Tea: How Statistics Revolutionized Science in the Twentieth Century. New York: W. H. Freeman. pp. 137–50. ISBN 978-0-7167-4106-0.
13. Kolmogorov, A. (1923). "Une série de Fourier–Lebesgue divergente presque partout" [A Fourier–Lebesgue series that diverges almost everywhere] (PDF). Fundamenta Mathematicae (in French). 4 (1): 324–328. doi:10.4064/fm-4-1-324-328.
14. V. I. Arnold-Max Dresden. "In Brief". Archived from the original on 5 October 2013.
15. Graham, Loren R.; Kantor, Jean-Michel (2009). Naming infinity: a true story of religious mysticism and mathematical creativity. Harvard University Press. p. 185. ISBN 978-0-674-03293-4. The police soon learned of Kolmogorov and Alexandrov's homosexual bond, and they used that knowledge to obtain the behavior that they wished.
16. Gessen, Masha (2011). Perfect Rigour: A Genius and the Mathematical Breakthrough of a Lifetime. Icon Books Ltd. p. 17. Kolmogorov alone among the top Soviet mathematicians avoided being drafted into the postwar military effort. His students always wondered why-and the only likely explanation seems to be Kolmogorov's homosexuality. His lifelong partner, with whom he shared a home starting in 1929, was the topologist Pavel Alexandrov.
17. Graham, Loren; Kantor, Jean-Michel (2009), Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity, Harvard University Press, p. 185, ISBN 9780674032934
18. Szpiro, George (2011), Pricing the Future: Finance, Physics, and the 300-year Journey to the Black-Scholes Equation, Basic Books, p. 152, ISBN 9780465022489, It was generally known that they had a homosexual relationship, although they never acknowledged their liaison
19. Lorentz, G. G. (2001). "Who discovered analytic sets?". The Mathematical Intelligencer. 23 (4): 28–32. doi:10.1007/BF03024600. S2CID 121273798.
20. O'Connor, John J.; Robertson, Edmund F., "The 1936 Luzin affair", MacTutor History of Mathematics Archive, University of St Andrews
21. "СИБИРСКИЕ ЭЛЕКТРОННЫЕ МАТЕМАТИЧЕСКИЕ ИЗВЕСТИ" (PDF). semr.math.nsc.ru (in Russian). Retrieved 19 June 2023.
22. A.P. Yushkevich, The Lusin Affair (in Russian).
23. Salsburg, p. 139.
24. Gleick, James (2012). The Information: a history, a theory, a flood. New York: Vintage Books. p. 334. ISBN 978-1-4000-9623-7.
25. Salsburg, pp. 145–7.
26. "Andrei Nikolayevich Kolmogorov". American Academy of Arts & Sciences. Retrieved 21 November 2022.
27. "APS Member History". search.amphilsoc.org. Retrieved 21 November 2022.
28. "A.N. Kolmogorov (1903–1987)". Royal Netherlands Academy of Arts and Sciences. Retrieved 22 July 2015.
29. "A. Kolmogorov". www.nasonline.org. Retrieved 21 November 2022.
30. Rietz, H. L. (1934). "Review: Grundbegriffe der Wahrscheinlichkeitsrechnung by A. Kolmogoroff" (PDF). Bull. Amer. Math. Soc. 40 (7): 522–523. doi:10.1090/s0002-9904-1934-05895-6. Archived (PDF) from the original on 9 October 2022.
External links
Wikimedia Commons has media related to Andrey Kolmogorov.
• Portal dedicated to AN Kolmogorov (his scientific and popular publications, articles about him).(in Russian)
• The Legacy of Andrei Nikolaevich Kolmogorov
• Biography at Scholarpedia
• Derzhavin Tambov State University - Institute of Mathematics, Physics and Information Technology Archived 2019-10-05 at the Wayback Machine
• The origins and legacy of Kolmogorov's Grundbegriffe
• Vitanyi, P.M.B., Andrey Nikolaevich Kolmogorov. Scholarpedia, 2(2):2798; 2007
• Collection of links to Kolmogorov resources
• Interview with Professor A. M. Yaglom about Kolmogorov, Gelfand and other (1988, Ithaca, New York)
• Kolmogorov School at Moscow University
• Annual Kolmogorov Lecture at the Computer Learning Research Centre at Royal Holloway, University of London
• Lorentz G. G., Mathematics and Politics in the Soviet Union from 1928 to 1953
• Kutateladze S. S., Sic Transit... or Heroes, Villains, and Rights of Memory.
• Kutateladze S. S., The Tragedy of Mathematics in Russia
• Video recording of the G. Falkovich's lecture: "Andrey Nikolaevich Kolmogorov (1903–1987) and the Russian school"
• Andrey Kolmogorov at the Mathematics Genealogy Project
Laureates of the Wolf Prize in Mathematics
1970s
• Israel Gelfand / Carl L. Siegel (1978)
• Jean Leray / André Weil (1979)
1980s
• Henri Cartan / Andrey Kolmogorov (1980)
• Lars Ahlfors / Oscar Zariski (1981)
• Hassler Whitney / Mark Krein (1982)
• Shiing-Shen Chern / Paul Erdős (1983/84)
• Kunihiko Kodaira / Hans Lewy (1984/85)
• Samuel Eilenberg / Atle Selberg (1986)
• Kiyosi Itô / Peter Lax (1987)
• Friedrich Hirzebruch / Lars Hörmander (1988)
• Alberto Calderón / John Milnor (1989)
1990s
• Ennio de Giorgi / Ilya Piatetski-Shapiro (1990)
• Lennart Carleson / John G. Thompson (1992)
• Mikhail Gromov / Jacques Tits (1993)
• Jürgen Moser (1994/95)
• Robert Langlands / Andrew Wiles (1995/96)
• Joseph Keller / Yakov G. Sinai (1996/97)
• László Lovász / Elias M. Stein (1999)
2000s
• Raoul Bott / Jean-Pierre Serre (2000)
• Vladimir Arnold / Saharon Shelah (2001)
• Mikio Sato / John Tate (2002/03)
• Grigory Margulis / Sergei Novikov (2005)
• Stephen Smale / Hillel Furstenberg (2006/07)
• Pierre Deligne / Phillip A. Griffiths / David B. Mumford (2008)
2010s
• Dennis Sullivan / Shing-Tung Yau (2010)
• Michael Aschbacher / Luis Caffarelli (2012)
• George Mostow / Michael Artin (2013)
• Peter Sarnak (2014)
• James G. Arthur (2015)
• Richard Schoen / Charles Fefferman (2017)
• Alexander Beilinson / Vladimir Drinfeld (2018)
• Jean-François Le Gall / Gregory Lawler (2019)
2020s
• Simon K. Donaldson / Yakov Eliashberg (2020)
• George Lusztig (2022)
• Ingrid Daubechies (2023)
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Akiva Yaglom
Akiva Moiseevich Yaglom (Russian: Аки́ва Моисе́евич Ягло́м; 6 March 1921 – 13 December 2007) was a Soviet and Russian physicist, mathematician, statistician, and meteorologist. He was known for his contributions to the statistical theory of turbulence and theory of random processes. Yaglom spent most of his career in Russia working in various institutions, including the Institute of Theoretical Geophysics.
Akiva Moiseevich Yaglom
Yaglom in 1976
Born(1921-03-06)6 March 1921
Kharkiv, Ukrainian SSR
Died13 December 2007(2007-12-13) (aged 86)
Boston, Massachusetts, United States
NationalityRussian
Alma materLomonosov Moscow State University
Steklov Institute of Mathematics
AwardsOtto Laporte Award (1988)
Lewis Fry Richardson Medal
Scientific career
FieldsProbability theory, Turbulence
InstitutionsInstitute of Theoretical Geophysics
A.M. Obukhov Institute of Atmospheric Physics
Massachusetts Institute of Technology
Doctoral advisorAndrey Kolmogorov
From 1992 until his death, Yaglom worked at the Massachusetts Institute of Technology as a research fellow in the Department of Aeronautics and Astronautics.[1] He authored several popular books in mathematics and probability, some of them with his twin brother and mathematician Isaak Yaglom.[2]
Education and career
Akiva Yaglom was born on 6 March 1921 in Kharkiv, Ukraine to the family of an engineer. He had a twin brother Isaak. The family moved to Moscow when the Yaglom brothers were five years old. During their school years they were keen on mathematics. In 1938 they shared the first prize at the Moscow mathematical competition for schoolchildren.[2] Yaglom joined Moscow State University in 1938, where he studied physics and mathematics. He completed his fourth year of diploma at the Sverdlovsk State University and received the masters in science degree in 1942. After a short period of work at the Main Geophysical Observatory, Yaglom joined the Steklov Institute of Mathematics of the USSR Academy of Sciences and completed his postgraduate studies in 1946 under the mentorship of Andrey Kolmogorov. His dissertation was "On the Statistical Reversibility of Brownian Motion".[3]
After he received his Ph.D, Yaglom was offered a job at the Lebedev Physical Institute by the future Nobel laureates Igor Tamm and Vitaly Ginzburg, but he declined the offer because he knew that the job would have required him to deal with applied problems related to the development of nuclear weapons.[4] He joined in the Institute of Atmospheric Physics of the USSR Academy of Sciences and worked at the Laboratory of Atmospheric Turbulence and worked there for more than 45 years. In 1955, he defended his second doctoral thesis "The Theory of Correlation between Continuous Processes and Fields with Applications to the Problems of Statistical Exploration of Time Series and to Turbulence Theory".[4]
Yaglom was also a full professor in the Faculty of Probability Theory at the Mathematics and Mechanics Department of Moscow State University. In 1992, Yaglom went to the United States and joined the Massachusetts Institute of Technology. He died in Boston, Massachusetts on 13 December 2007.[2]
Principal works
Yaglom worked in many fields in applied mathematics and statistics, including the theory of random processes and the statistical theory of fluid mechanics. His initial studies on the theory of random functions were published in the lengthy 1952-article "Introduction to the Theory of Random Functions" which appeared in the journal Uspekhi Fizicheskikh Nauk. Later, this work was published in United States. His study on local structure of the acceleration field in a turbulent flow established the fact that the frequency spectrum of Lagrangian acceleration of a fluid particle in a turbulent flow is constant. This work was later independently repeated by Werner Heisenberg.
Awards and honors
In 1955, Yaglom received a Doctor of Science degree, the highest scientific degree in the Soviet Union, for his work on theories of stochastic processes and their application to turbulence theory.[1] He received the American Physical Society's Otto Laporte Award in 1988 for his "fundamental contribution to the statistical theory of turbulence and the study of its underlying mathematical structure."[5]
Yaglom received the European Geosciences Union's 2008 Lewis Fry Richardson Medal, posthumously, for his "eminent and pioneering contributions to the development of statistical theories of turbulence, atmospheric dynamics and diffusion, including spectral techniques, stochastic and cascade models."[6]
Books authored
Yaglom authored six books and about 120 research papers. Most of his materials have been published in English and many other languages.[1] The monograph titled Statistical Fluid Mechanics, co-authored with Andrei Monin, is regarded as an encyclopedic work in the subject field.[3]
• A. M. Yaglom, An Introduction to the Theory of Stationary Random Functions, Dover Publications, 1962.[7]
• A. M. Yaglom and I. M. Yaglom, Challenging Mathematical Problems With Elementary Solutions, Volume 1, Dover Publications, 1987.
• A. M. Yaglom and I. M. Yaglom, Challenging Mathematical Problems With Elementary Solutions, Volume 2, Dover Publications, 1987.[8]
• A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence, Dover Publications, 2007.[9]
References
1. "Akiva Yaglom, research fellow, dies at 86". Massachusetts Institute of Technology. Retrieved 10 March 2018.
2. Bradshaw, Peter (March 2008). "Prof. A.M. Yaglom". Flow, Turbulence and Combustion. Springer Netherlands. 80 (3): 287–289. doi:10.1007/s10494-008-9141-7. ISSN 1573-1987. S2CID 121550164.
3. "Akiva Moiseevich Yaglom (on his 85th birthday)". Izvestiya, Atmospheric and Oceanic Physics. MAIK Nauka. 42 (1): 127–128. January 2006. Bibcode:2006IzAOP..42..127.. doi:10.1134/S0001433806010130. ISSN 1555-628X. S2CID 195301245.
4. Golitsyn, G.S.; B.A. Kader; B.M. Koprov; M.I. Fortus (December 2008). "In memory of A. M. Yaglom". Izvestiya, Atmospheric and Oceanic Physics. MAIK Nauka. 44 (6): 796–798. Bibcode:2008IzAOP..44..796G. doi:10.1134/S0001433808060157. ISSN 1555-628X. S2CID 122297950.
5. "Otto Laporte Award". American Physical Society. Archived from the original on 2 December 2008. Retrieved 31 July 2010.
6. "EGU Lewis Fry Richardson Medal 2008". European Geosciences Union. Archived from the original on 18 July 2011. Retrieved 31 July 2010.
7. Yaglom, A. M. (January 2004). An Introduction to the Theory of Stationary Random Functions. ISBN 9780486495712. Retrieved 31 July 2010.
8. Yaglom, Akiva Moiseevich; Yaglom, Isaak Moiseevich (January 1987). Challenging Mathematical Problems with Elementary Solutions. ISBN 9780486655376. Retrieved 31 July 2010.
9. Statistical fluid mechanics: mechanics of turbulence. Retrieved 31 July 2010.
External links
• Akiva Yaglom at the Mathematics Genealogy Project
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Anatoli Prudnikov
Anatolii Platonovich Prudnikov (Анатолий Платонович Прудников; 14 January 1927 in Ulyanovsk, Russia – 10 January 1999) was a Russian mathematician.
Anatolii Platonovich Prudnikov
Born(1927-01-14)14 January 1927
Died10 January 1999(1999-01-10) (aged 71)
NationalityRussian
OccupationMathematician
In 1930 the Prudnikov family moved to Samara, where Anatolii passed his Abitur in 1944. He then studied at the Kuibyshev Aviation Institute for three years and at the Kuibyshev Pedagogical Institute for one year before completing his degree qualifying him as a teacher. In 1968 he received his doctorate under the direction of professor Vitalii Arsenievich Ditkin with a thesis entitled On a class of integral transforms of Volterra type and some generalizations of operational calculus.[1] With Ditkin, he published several handbooks on integral transforms and operational calculus. Prudnikov's fame derives mainly from the five-volume work "Integrals and Series" (1981–1992), written with Yuri Aleksandrovich Brychkov and Oleg Igorevich Marichev.[2]
Works
• Prudnikov, Anatolii Platonovich (Прудников, Анатолий Платонович); Brychkov, Yury Aleksandrovich (Брычков, Юрий Александрович); Marichev, Oleg Igorevich (Маричев, Олег Игоревич) (1986). Tables of Indefinite Integrals (in Russian). Moscow: Nauka.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Prudnikov, Anatolii Platonovich (Прудников, Анатолий Платонович); Brychkov, Yury Aleksandrovich (Брычков, Юрий Александрович); Marichev, Oleg Igorevich (Маричев, Олег Игоревич). Integrals and Series (in Russian). Vol. 1–5 (1 ed.). Nauka.{{cite book}}: CS1 maint: multiple names: authors list (link) 1981−1986.[3] (English, translated from the Russian by N. M. Queen), volumes 1–5, Gordon & Breach Science Publishers / CRC Press, 1988–1992, ISBN 2-88124-097-6. Second revised edition (Russian), volumes 1–3, Fiziko-Matematicheskaya Literatura, 2003.
References
1. "Obituary: A. P. Prudnikov". Integral Transforms and Special Functions. 8 (1–2): 1–2. 1999. doi:10.1080/10652469908819211. eISSN 1476-8291. ISSN 1065-2469.
2. Marichev, Oleg Igorevich (1999-03-15). "Memorial note about A. P. Prudnikov (Topic #10)". OP-SF Net. 6 (2). Archived from the original on 2014-08-26.
3. Kölbig, Kurt Siegfried (1988). "Reviews and Descriptions of Tables and Books (pp. 349–352 subsection on "Integrals and Series")". Mathematics of Computation. 50 (181): 343–357. doi:10.1090/S0025-5718-88-99807-9. Retrieved 2016-04-15.
External links
• Anatoli Prudnikov at the Mathematics Genealogy Project
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Andrey Markov Jr.
Andrey Andreyevich Markov (Russian: Андре́й Андре́евич Ма́рков; 22 September 1903, Saint Petersburg – 11 October 1979, Moscow) was a Soviet mathematician, the son of the Russian mathematician Andrey Markov Sr, and one of the key founders of the Russian school of constructive mathematics and logic. He made outstanding contributions to various areas of mathematics, including differential equations, topology, mathematical logic and the foundations of mathematics.[1][2]
His name is in particular associated with Markov's principle and Markov's rule in mathematical logic, Markov's theorem in knot theory and Markov algorithm in theoretical computer science. An important result that he proved in 1947 was that the word problem for semigroups was unsolvable; Emil Leon Post obtained the same result independently at about the same time. In 1953 he became a member of the Communist Party.
In 1960, Markov obtained fundamental results showing that the classification of four-dimensional manifolds is undecidable: no general algorithm exists for distinguishing two arbitrary manifolds with four or more dimensions. This is because four-dimensional manifolds have sufficient flexibility to allow us to embed any algorithm within their structure. Hence, classifying all four-manifolds would imply a solution to Turing's halting problem. Embedding implies failure to create a correspondence between algorithms and indexing (naturally uncountably infinite, but even larger) of the four-manifolds structure. Failure is in Cantor's sense. Indexing is in Godel's sense. This result has profound implications for the limitations of mathematical analysis.
His doctoral students include Boris Kushner, Gennady Makanin, and Nikolai Shanin.
Awards and honors
• Medal "For Valiant Labour in the Great Patriotic War 1941–1945" (1945)
• Order of the Badge of Honour (1945)
• Medal "For the Defence of Leningrad" (1946)
• Order of Lenin (1954)
• Order of the Red Banner of Labour (1963)
Notes
1. Kushner, Boris A (2006). "The constructive mathematics of A. A. Markov". Amer. Math. Monthly. 113 (6): 559–566. doi:10.2307/27641983. JSTOR 27641983. MR 2231143.
2. Glukhov, M. M.; Nagornyĭ, N. M. (2004). "Andreĭ Andreevich Markov (on the centenary of his birth)". Diskrete Math. Appl. 14 (1): 1–6. doi:10.1515/156939204774148776. MR 2069985. S2CID 120486293.
External links
• Andrey Markov Jr. at the Mathematics Genealogy Project
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Arithmetic (book)
Arithmetic (Russian: Арифметика, romanized: Arifmetika) is a 1703 mathematics textbook by the Russian educator and mathematician Leonty Magnitsky. The book served as the standard Russian mathematics textbook until the mid-18th century. Mikhail Lomonosov was educated on this book, and referred to it as the "gates of my own erudition".[1] It was the first mathematics textbook written in the Russian language that was not a translated edition of a foreign work.[2] It consisted essentially of Magnitsky's own lecture notes, and offered an encyclopedic overview of arithmetic at the time, with sections on navigational astronomy, geodesy, algebra, geometry, and trigonometry.[2]
It was organized in instructive question and answer format, and rooted not in the abstract but in practical and demonstrable applications of theories and axioms. The book also contained astronomical tables and coordinate maps for various Russian locales.[2]
The origins of the book lie in Peter the Great's establishment of the School of Navigation in Moscow, and the subsequent appointment of Magnitsky at the school's helm. He needed a text to teach from, and so formulated the book around his lectures and the prevailing European mathematics texts of the age.[3]
The full title and subtitle reads: "Arithmetic, that is the science of numbering. Translated from different languages into Russian, put together and divided into two parts". The book runs 600 pages. Its publication was extensively researched in 1914 by Dmitrii Galanin in his book Leonty Filippovich Magnitsky and His Arithmetic. Original copies are preserved in the Moscow State University library.[4]
Gallery
Folios from Arithmetic
References
1. Billington, James (2010). Icon and Axe: An Interpretative History of Russian Culture. Random House. pp. 289–290. ISBN 9780307765284.
2. O'Connor, JJ (December 2008). "Leonty Filippovich Magnitsky". St. Andrews. Retrieved 20 February 2022.
3. Swetz, Frank J. (April 2018). "Mathematical Treasure: 18th-Century Russian Arithmetic by Magnitsky". MAA. Retrieved 20 February 2022.
4. "Arithmetic by Magnitsky". Mathematical Etudes. Etudes. Retrieved 20 February 2022.
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