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Yiannis N. Moschovakis
Yiannis Nicholas Moschovakis (Greek: Γιάννης Μοσχοβάκης; born January 18, 1938) is a set theorist, descriptive set theorist, and recursion (computability) theorist, at UCLA.
Yiannis N. Moschovakis
Born
Yiannis Nicholas Moschovakis
(1938-01-18) January 18, 1938
Athens, Greece
Alma materUniversity of Wisconsin–Madison
Known forEffective descriptive set theory
Scientific career
FieldsMathematics
InstitutionsUCLA
Doctoral advisorStephen Kleene
Doctoral studentsAlexander S. Kechris
Phokion G. Kolaitis
His book Descriptive Set Theory (North-Holland) is the primary reference for the subject. He is especially associated with the development of the effective, or lightface, version of descriptive set theory, and he is known for the Moschovakis coding lemma that is named after him.
Biography
Moschovakis earned his Ph.D. from the University of Wisconsin–Madison in 1963 under the direction of Stephen Kleene, with a dissertation entitled Recursive Analysis. In 2015, he was elected as a fellow of the American Mathematical Society "for contributions to mathematical logic, especially set theory and computability theory, and for exposition".[1]
For many years, he has split his time between UCLA and the University of Athens (he retired from the latter in July 2005).
Moschovakis is married to Joan Moschovakis, with whom he gave the 2014 Lindström Lectures at the University of Gothenburg.[2]
Publications
• Elementary induction on abstract structures. North-Holland. 1974.[3] 2nd edn. Dover. 2008. ISBN 9780486152011.
• Descriptive set theory. North-Holland. 1980. ISBN 9780444853059.[4] 2nd edn. 2005. Second edition available online
• Notes on set theory. North-Holland. 1994. ISBN 9783540941804. 2nd edn. 2005.
References
1. 2016 Class of the Fellows of the AMS, American Mathematical Society, retrieved 2015-11-16.
2. "The Lindström Lectures - Department of Philosophy, Linguistics and Theory of Science, University of Gothenburg, Sweden". flov.gu.se. Archived from the original on 2013-11-11.
3. Barwise, K. Jon (1975). "Review: Elementary induction on abstract structures, by Y. Moschovakis". Bull. Amer. Math. Soc. 81 (6): 1031–1035. doi:10.1090/s0002-9904-1975-13893-6.
4. Jech, Thomas (1981). "Review: Descriptive set theory, by Y. Moschovakis". Bull. Amer. Math. Soc. (N.S.). 5 (3): 339–349. doi:10.1090/s0273-0979-1981-14952-1.
External links
• Home page
• Yiannis N. Moschovakis at the Mathematics Genealogy Project
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• BnF data
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Yifeng Liu
Yifeng Liu (born July 19, 1985 in Shanghai, China) is a Chinese professor of mathematics at Zhejiang University specializing in number theory, automorphic forms and arithmetic geometry.[1]
Yifeng Liu
Born (1985-07-19) July 19, 1985
Shanghai, China
NationalityChinese
Alma materColumbia University
Peking University
Awards
• Sloan Research Fellowship (2017)
• SASTRA Ramanujan Prize (2018)
Scientific career
FieldsMathematics
Institutions
• Yale University
• Northwestern University
• Massachusetts Institute of Technology
• Zhejiang University
ThesisArithmetic inner product formula for unitary groups (2012)
Doctoral advisorShou-Wu Zhang
Career
Liu received his BS Degree from Peking University in 2007 and PhD degree from Columbia University, New York, in 2012 under the direction of Shou-Wu Zhang. He was a C.L.E. Moore Instructor at MIT from 2012 to 2015 and an assistant professor at Northwestern University from 2015 to 2018 before being appointed an associate professor at Yale University.[2][3] Liu returned to China in 2021 to join Zheijiang University became a full professor of mathematics.[1]
Liu has made important contributions to arithmetic geometry and number theory. His contributions span a wide spectrum of topics such as arithmetic theta lifts and derivatives of L-functions, the Gan–Gross–Prasad conjecture and its arithmetic counterpart, the Beilinson–Bloch–Kato conjecture, the geometric Langlands program, the p-adic Waldspurger theorem, and the study of étale cohomology on Artin stacks.[2]
Awards
He received a Sloan Research Fellowship in 2017.[2]
He was awarded the 2018 SASTRA Ramanujan Prize for his contributions to the field of mathematics. He shared the prize with Jack Thorne.[4][3]
References
1. "百度安全验证".
2. "Yifeng Liu and Jack Thorne to receive 2018 SASTRA Ramanujan Prize" (PDF). DPMMS News. Department of Pure Mathematics and Mathematical Statistics, Cambridge University. Retrieved 3 February 2019.
3. "Yale, Cambridge profs. get SASTRA-Ramanujan Award". The Hindu. December 22, 2018. Retrieved 3 February 2019.
4. Maeve Forti (25 October 2018). "Yifeng Liu wins prestigious award in mathematics". YaleNews. Yale University. Retrieved 3 February 2019.
External links
• Yifeng Liu's personal homepage
Recipients of SASTRA Ramanujan Prize
• Manjul Bhargava (2005)
• Kannan Soundararajan (2005)
• Terence Tao (2006)
• Ben Green (2007)
• Akshay Venkatesh (2008)
• Kathrin Bringmann (2009)
• Wei Zhang (2010)
• Roman Holowinsky (2011)
• Zhiwei Yun (2012)
• Peter Scholze (2013)
• James Maynard (2014)
• Jacob Tsimerman (2015)
• Kaisa Matomäki (2016)
• Maksym Radziwill (2016)
• Maryna Viazovska (2017)
• Yifeng Liu (2018)
• Jack Thorne (2018)
• Adam Harper (2019)
• Shai Evra (2020)
• Will Sawin (2021)
• Yunqing Tang (2022)
Authority control: Academics
• MathSciNet
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Yigu yanduan
Yigu yanduan (益古演段 Old Mathematics in Expanded Sections) is a 13th-century mathematical work by Yuan dynasty mathematician Li Zhi.
Overview
Yigu yanduan was based on Northern Song mathematician Jiang Zhou's (蒋周) Yigu Ji (益古集 Collection of Old Mathematics) which is not extant. However, from fragments quoted in Yang Hui's work The Complete Algorithms of Acreage (田亩比类算法大全), this lost mathematical treatise Yigu Ji was about solving area problems with geometry.
Li Zhi used the examples of Yigu Ji to introduce the art of Tian yuan shu to newcomers to this field. Although Li Zhi's previous monograph Ceyuan haijing also used Tian yuan shu, it is harder to understand than Yigu yanduan.
Yigu yanduan was later collected into Siku Quanshu.
Yigu yanduan consists of three volumes with 64 problems solved using Tian yuan sh] in parallel with the geometrical method. Li Zhi intended to introduce students to the art of Tian yuan shu through ancient geometry. Yigu yanduan together with Ceyuan haijing are considered major contributions to Tian yuan shu by Li Zhi. These two works are also considered as the earliest extant documents on Tian yuans shu.
All the 64 problems followed more or less the same format, starting with a question (问), followed by an answer (答曰), a diagram, then an algorithm (术), in which Li Zhi explained step by step how to set up algebra equation with Tian yuan shu, then followed by geometrical interpretation (Tiao duan shu). The order of arrangement of Tian yuan shu equation in Yigu yanduan is the reverse of that in Ceyuan haijing, i.e., here with the constant term at top, followed by first order tian yuan, second order tian yuan, third order tian yuan etc. This later arrangement conformed with contemporary convention of algebra equation( for instance, Qin Jiushao's Mathematical Treatise in Nine Sections), and later became a norm.
Yigu yanduan was first introduced to the English readers by the British Protestant Christian missionary to China, Alexander Wylie who wrote:
Yi koo yen t'wan...written in 1282 consists of 64 geometrical problem, illustrated the principle of Plane Measurement, Evolution and other rules, the whole being developed by means of T'een yuen.[1]
In 1913 Van Hée translated all 64 problems in Yigu yanduan into French.[2]
Volume I
Problem 1 to 22, all about the mathematics of a circle embedded in a square.
Example: problem 8
There is a square field, with a circular pool in the middle, given that the land is 13.75 mu, and the sum of the circumferences of the square field and the circular pool equals to 300 steps, what is the circumferences of the square and circle respective ?
Anwwer: The circumference of the square is 240 steps, the circumference of the circle is 60 steps.
Method: set up tian yuan one (celestial element 1) as the diameter of the circle, x
TAI
multiply it by 3 to get the circumference of the circle 3x (pi ~~3)
TAI
subtract this from the sum of circumferences to obtain the circumference of the square $300-3x$
TAI
The square of it equals to 16 times the area of the square $(300-3x)*(300-3x)=90000-1800x+9x^{2}$
TAI
Again set up tian yuan 1 as the diameter of circle, square it up and multiplied by 12 to get 16 times the area of circle as
TAI
subtract from 16 time square area we have 16 times area of land
TAI
put it at right hand side and put 16 times 13.75 mu = 16 * 13.75 *240 =52800 steps at left, after cancellation, we get $-3x^{2}-1800x+37200=0:$
TAI
Solve this equation to get diameter of circle = 20 steps, circumference of circle = 60 steps
Volume II
Problem 23 to 42, 20 problems in all solving geometry of rectangle embedded in circle with tian yuan shu
Example, problem 35
Suppose we have a circular field with a rectangular water pool in the center, and the distance of a corner to the circumference is 17.5 steps, and the sum of length and width of the pool is 85 steps, what is the diameter of the circle, the length and width of the pool ?
Answer: The diameter of the circle is one hundred steps, the length of pool is 60 steps, and the width 25 steps. Method: Let tian yuan one as the diagonal of rectangle, then the diameter of circle is tian yuan one plus 17.5*2
$x+35$
multiply the square of diameter with $\pi \approx 3$ equals to four times the area of the circle:
$3(x+35)^{2}=3x^{2}+210x+3675$
subtracting four times the area of land to obtain:
four times the area of pool = $3x^{2}+210x+3675-4x6000$= $3x^{2}+210x-20325$
now
The square of the sum of length and width of the pool =85*85 =7225 which is four times the pool area plus the square of the difference of its length and width ($(L-W)^{2}$)
Further double the pool area plus $(L-W)^{2}$ equals to $L^{2}+W^{2}$ = the square of the diagonal of the pool thus
( four time pool area + the square of its dimension difference) - (twice the pool area + square if its dimension difference) equals $7225-x^{2}$ = twice the pool area
so four times the area of pool = $2(7225-x^{2})$
equate this with the four times pool area obtained above
$2(7225-x^{2})$ =$3x^{2}+210x-20325$
we get a quadratic equation $5x^{2}+210x-34775$=0 Solve this equation to get
• diagonal of pool =65 steps
• diameter of circle =65 +2*17.5 =100 steps
• Length - width =35 steps
• Length + width =85 steps
• Length =60 steps
• Width =25 steps
Volume III
Problem 42 to 64, altogether 22 questions about the mathematics of more complex diagrams
Q: fifty-fourth. There is a square field, with a rectangular water pool lying on its diagonal. The area outside the pool is one thousand one hundred fifty paces. Given that from the corners of the field to the straight sides of the pool are fourteen paces and nineteen paces. What is the area of the square field, what is the length and width of the pool?
Answer: The area of the square field is 40 square paces, the length of the pool is thirty five paces, and the width is twenty five paces.
Let the width of the pool be Tianyuan 1.
TAI
Add the width of the pool to twice the distance from field corner to short long side of pool equals to the length of diagonal of the field x+38
TAI
Square it to obtain the area of square with the length of the pool diagonal as its sides
$x^{2}+76x+1444$
TAI
The length of pool minus the width of pool multiplied by 2 = 2 (19-14) = 10
Pool length = pool width +10: x+10
TAI
Pool area = pool with times pool length : x(x+10) =$x^{2}+10x$
TAI
Area of pool times 乘 1.96 ( the square root of 2) =1.4
one has $1.96x^{2}+19.6x$
tai
Area of diagonal square subtract area of pool multiplied 1.96 equals to area of land times 1.96:
$x^{2}+76x+1444$ - $1.96x^{2}+19.6x=$：$-0.96x^{2}+56.4x+1444$
TAI
Occupied plot times 1.96 =1150 * 1.96 =2254=$-0.96x^{2}+56.4x+1444$
hence =$-0.96x^{2}+56.4x-810$:
TAI
Solve this equation and we obtain
width of pool 25 paces therefore pool length =pool width +10 =35 paces length of pool =45 paces
References
1. Alexander Wylie, Notes on Chinese Literature, p117, Shanghai 1902, reprinted by kessinger Publishing
2. van Hée Li Yeh, Mathématicien Chinois du XIIIe siècle, TP,1913,14,537
Further reading
• Yoshio Mikami The Development of Mathematics in China and Japan, p. 81
• Annotated Yigu yanduan by Qing dynasty mathematician Li Rui.
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Yingying Fan
Yingying Fan is a Chinese-American statistician and Centennial Chair in Business Administration and Professor in Data Sciences and Operations Department of the Marshall School of Business at the University of Southern California.[1] She also holds joint appointments at the USC Dana and David Dornsife College of Letters, Arts and Sciences, and Keck Medicine of USC. Her contributions to statistics and data science were recognized by the Royal Statistical Society Guy Medal in Bronze in 2017[2] and the Institute of Mathematical Statistics Medallion Lecture in 2023.[3] She was elected Fellow of American Statistical Association in 2019[4] and Fellow of Institute of Mathematical Statistics "for seminal contributions to high-dimensional inference, variable selection, classification, networks, and nonparametric methodology, particularly in the field of financial econometrics, and for conscientious professional service" in 2020.[5]
References
1. "Yingying Fan | USC Marshall". www.marshall.usc.edu.
2. "RSS announces honours for 2017 | StatsLife". www.statslife.org.uk.
3. "Institute of Mathematical Statistics | Honored Special Awards & Lecturers Recipient List".
4. https://www.amstat.org/asa/files/pdfs/2019-ASAFellowAnnouncement.pdf
5. "Institute of Mathematical Statistics | Congratulations to the 2020 IMS Fellows!".
Guy Medallists
Gold Medallists
• Charles Booth (1892)
• Robert Giffen (1894)
• Jervoise Athelstane Baines (1900)
• Francis Ysidro Edgeworth (1907)
• Patrick G. Craigie (1908)
• G. Udny Yule (1911)
• T. H. C. Stevenson (1920)
• A. William Flux (1930)
• A. L. Bowley (1935)
• Major Greenwood (1945)
• R. A. Fisher (1946)
• A. Bradford Hill (1953)
• E. S. Pearson (1955)
• Frank Yates (1960)
• Harold Jeffreys (1962)
• Jerzy Neyman (1966)
• M. G. Kendall (1968)
• M. S. Bartlett (1969)
• Harald Cramér (1972)
• David Cox (1973)
• G. A. Barnard (1975)
• Roy Allen (1978)
• D. G. Kendall (1981)
• Henry Daniels (1984)
• Bernard Benjamin (1986)
• Robin Plackett (1987)
• Peter Armitage (1990)
• George E. P. Box (1993)
• Peter Whittle (1996)
• Michael Healy (1999)
• Dennis Lindley (2002)
• John Nelder (2005)
• James Durbin (2008)
• C. R. Rao (2011)
• John Kingman (2013)
• Bradley Efron (2014)
• Adrian Smith (2016)
• Stephen Buckland (2019)
• David Spiegelhalter (2020)
• Nancy Reid (2022)
Silver Medallists
• John Glover (1893)
• Augustus Sauerbeck (1894)
• A. L. Bowley (1895)
• F. J. Atkinson (1897)
• C. S. Loch (1899)
• Richard Crawford (1900)
• Thomas A. Welton (1901)
• R. H. Hooker (1902)
• Yves Guyot (1903)
• D. A. Thomas (1904)
• R. H. Rew (1905)
• W. H. Shaw (1906)
• N. A. Humphreys (1907)
• Edward Brabrook (1909)
• G. H. Wood (1910)
• R. Dudfield (1913)
• S. Rowson (1914)
• S. J. Chapman (1915)
• J. S. Nicholson (1918)
• J. C. Stamp (1919)
• A. William Flux (1921)
• H. W. Macrosty (1927)
• Ethel Newbold (1928)
• H. E. Soper (1930)
• J. H. Jones (1934)
• Ernest Charles Snow (1935)
• R. G. Hawtrey (1936)
• E. C. Ramsbottom (1938)
• L. Isserlis (1939)
• H. Leak (1940)
• M. G. Kendall (1945)
• Harry Campion (1950)
• F. A. A. Menzler (1951)
• M. S. Bartlett (1952)
• J. O. Irwin (1953)
• L. H. C. Tippett (1954)
• D. G. Kendall (1955)
• Henry Daniels (1957)
• G. A. Barnard (1958)
• E. C. Fieller (1960)
• D. R. Cox (1961)
• P. V. Sukhatme (1962)
• George E. P. Box (1964)
• C. R. Rao (1965)
• Peter Whittle (1966)
• Dennis Lindley (1968)
• Robin Plackett (1973)
• James Durbin (1976)
• John Nelder (1977)
• Peter Armitage (1978)
• Michael Healy (1979)
• M. Stone (1980)
• John Kingman (1981)
• Henry Wynn (1982)
• Julian Besag (1983)
• J. C. Gittins (1984)
• A. Bissell (1985)
• W. Pridmore (1985)
• Richard Peto (1986)
• John Copas (1987)
• John Aitchison (1988)
• F. P. Kelly (1989)
• David Clayton (1990)
• R. L. Smith (1991)
• Robert Nicholas Curnow (1992)
• A. F. M. Smith (1993)
• David Spiegelhalter (1994)
• B. W. Silverman (1995)
• Steffen Lauritzen (1996)
• Peter Diggle (1997)
• Harvey Goldstein (1998)
• Peter Green (1999)
• Walter Gilks (2000)
• Philip Dawid (2001)
• David Hand (2002)
• Kanti Mardia (2003)
• Peter Donnelly (2004)
• Peter McCullagh (2005)
• Michael Titterington (2006)
• Howell Tong (2007)
• Gareth Roberts (2008)
• Sylvia Richardson (2009)
• Iain M. Johnstone (2010)
• P. G. Hall (2011)
• David Firth (2012)
• Brian Ripley (2013)
• Jianqing Fan (2014)
• Anthony Davison (2015)
• Nancy Reid (2016)
• Neil Shephard (2017)
• Peter Bühlmann (2018)
• Susan Murphy (2019)
• Arnaud Doucet (2020)
• Håvard Rue (2021)
• Paul Fearnhead (2022)
Bronze Medallists
• William Gemmell Cochran (1936)
• R. F. George (1938)
• W. J. Jennett (1949)
• Peter Armitage (1962)
• James Durbin (1966)
• F. Downton (1967)
• Robin Plackett (1968)
• M. C. Pike (1969)
• P. G. Moore (1970)
• D. J. Bartholomew (1971)
• G. N. Wilkinson (1974)
• A. F. Bissell (1975)
• P. L. Goldsmith (1976)
• A. F. M. Smith (1977)
• Philip Dawid (1978)
• T. M. F. Smith (1979)
• A. J. Fox (1980)
• S. J. Pocock (1982)
• Peter McCullagh (1983)
• Bernard Silverman (1984)
• David Spiegelhalter (1985)
• D. F. Hendry (1986)
• Peter Green (1987)
• S. C. Darby (1988)
• S. M. Gore (1989)
• Valerie Isham (1990)
• M. G. Kenward (1991)
• C. Jennison (1992)
• Jonathan Tawn (1993)
• R. F. A. Poultney (1994)
• Iain M. Johnstone (1995)
• J. N. S. Matthews (1996)
• Gareth Roberts (1997)
• D. Firth (1998)
• P. W. F. Smith
• J. Forster (1999)
• J. Wakefield (2000)
• Guy Nason (2001)
• Geert Molenberghs (2002)
• Peter Lynn (2003)
• Nicola Best (2004)
• Steve Brooks (2005)
• Matthew Stephens (2006)
• Paul Fearnhead (2007)
• Fiona Steele (2008)
• Chris Holmes (2009)
• Omiros Papaspiliopoulos (2010)
• Nicolai Meinshausen (2011)
• Richard Samworth (2012)
• Piotr Fryzlewicz (2013)
• Ming Yuan (2014)
• Jinchi Lv (2015)
• Yingying Fan (2017)
• Peng Ding (2018)
• Jonas Peters (2019)
• Rachel McCrea (2020)
• Pierre E. Jacob (2021)
• Rajan Shah (2022)
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Yinyu Ye
Yinyu Ye (Chinese: 叶荫宇; pinyin: Yè Yīnyǔ; born 1948) is a Chinese American theoretical computer scientist working on mathematical optimization. He is a specialist in interior point methods, especially in convex minimization and linear programming. He is a professor of Management Science and Engineering and Kwoh-Ting Li Chair Professor of Engineering at Stanford University. He also holds a courtesy appointment in the Department of Electrical Engineering. Ye also is a co-founder of minMax Optimization Inc.
Education
Yinyu Ye was born in 1948 in Wuhan, Hubei, China. He attended Huazhong University of Science and Technology and graduated with a B.S. in Systems and Control in 1982. He received a Ph.D in Engineering Economic Systems from Stanford University in 1988, under the supervision of George B. Dantzig.
Research publications
Ye wrote Interior-Point Algorithms: Theory and Analysis. He joined David Luenberger for the third edition of Luenberger's Linear and Nonlinear Programming.[1]
In recent years, Ye has developed computational methods and theory using semidefinite programming for practical problems like the localization of network sensors. In computational economics, Ye has also established new complexity results for problems concerning the computation of an economic equilibrium.[2]
Awards
Ye was a 2009 co-recipient of the John von Neumann Theory Prize.[2] He was elected to the 2006 class of Fellows of the Institute for Operations Research and the Management Sciences.[3]
Positions
Before joining Stanford University, Ye was a Henry B. Tippie Research Professor at the University of Iowa. Ye is a co-founder of minMax Optimization, a technology company based in Palo Alto and Shanghai focused on creating optimization tools for geospatial and financial problems.
References
1. Luenberger, David G.; Ye, Yinyu (2008). Linear and nonlinear programming. International Series in Operations Research & Management Science. Vol. 116 (Third ed.). New York: Springer. pp. xiv+546. ISBN 978-0-387-74502-2. MR 2423726.
2. "John von Neumann Theory Prize / INFORMS Prizes & Awards / Recognize Excellence / IOL Home - INFORMS.org". www.informs.org. Archived from the original on 2010-03-12.
3. Fellows: Alphabetical List, Institute for Operations Research and the Management Sciences, retrieved 2019-10-09
• Luenberger, David G.; Ye, Yinyu (2008). Linear and nonlinear programming. International Series in Operations Research & Management Science. Vol. 116 (Third ed.). New York: Springer. pp. xiv+546. ISBN 978-0-387-74502-2. MR 2423726.
John von Neumann Theory Prize
1975–1999
• George Dantzig (1975)
• Richard Bellman (1976)
• Felix Pollaczek (1977)
• John F. Nash / Carlton E. Lemke (1978)
• David Blackwell (1979)
• David Gale / Harold W. Kuhn / Albert W. Tucker (1980)
• Lloyd Shapley (1981)
• Abraham Charnes / William W. Cooper / Richard J. Duffin (1982)
• Herbert Scarf (1983)
• Ralph Gomory (1984)
• Jack Edmonds (1985)
• Kenneth Arrow (1986)
• Samuel Karlin (1987)
• Herbert A. Simon (1988)
• Harry Markowitz (1989)
• Richard Karp (1990)
• Richard E. Barlow / Frank Proschan (1991)
• Alan J. Hoffman / Philip Wolfe (1992)
• Robert Herman (1993)
• Lajos Takacs (1994)
• Egon Balas (1995)
• Peter C. Fishburn (1996)
• Peter Whittle (1997)
• Fred W. Glover (1998)
• R. Tyrrell Rockafellar (1999)
2000–present
• Ellis L. Johnson / Manfred W. Padberg (2000)
• Ward Whitt (2001)
• Donald L. Iglehart / Cyrus Derman (2002)
• Arkadi Nemirovski / Michael J. Todd (2003)
• J. Michael Harrison (2004)
• Robert Aumann (2005)
• Martin Grötschel / László Lovász / Alexander Schrijver (2006)
• Arthur F. Veinott, Jr. (2007)
• Frank Kelly (2008)
• Yurii Nesterov / Yinyu Ye (2009)
• Søren Asmussen / Peter W. Glynn (2010)
• Gérard Cornuéjols (2011)
• George Nemhauser / Laurence Wolsey (2012)
• Michel Balinski (2013)
• Nimrod Megiddo (2014)
• Vašek Chvátal / Jean Bernard Lasserre (2015)
• Martin I. Reiman / Ruth J. Williams (2016)
• Donald Goldfarb / Jorge Nocedal (2017)
• Dimitri Bertsekas / John Tsitsiklis (2018)
• Dimitris Bertsimas / Jong-Shi Pang (2019)
• Adrian Lewis (2020)
• Alexander Shapiro (2021)
• Vijay Vazirani (2022)
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-yllion
-yllion (pronounced /aɪljən/)[1] is a proposal from Donald Knuth for the terminology and symbols of an alternate decimal superbase system. In it, he adapts the familiar English terms for large numbers to provide a systematic set of names for much larger numbers. In addition to providing an extended range, -yllion also dodges the long and short scale ambiguity of -illion.
Part of a series on
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• Āryabhaṭa
• Kaṭapayādi
• Coptic
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• Georgian
• Glagolitic
• Greek
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List of numeral systems
Knuth's digit grouping is exponential instead of linear; each division doubles the number of digits handled, whereas the familiar system only adds three or six more. His system is basically the same as one of the ancient and now-unused Chinese numeral systems, in which units stand for 104, 108, 1016, 1032, ..., 102n, and so on (with an exception that the -yllion proposal does not use a word for thousand which the original Chinese numeral system has). Today the corresponding Chinese characters are used for 104, 108, 1012, 1016, and so on.
Details and examples
Look up -yllion in Wiktionary, the free dictionary.
In Knuth's -yllion proposal:
• 1 to 999 have their usual names.
• 1000 to 9999 are divided before the 2nd-last digit and named "foo hundred bar." (e.g. 1234 is "twelve hundred thirty-four"; 7623 is "seventy-six hundred twenty-three")
• 104 to 108 − 1 are divided before the 4th-last digit and named "foo myriad bar". Knuth also introduces at this level a grouping symbol (comma) for the numeral. So 382,1902 is "three hundred eighty-two myriad nineteen hundred two."
• 108 to 1016 − 1 are divided before the 8th-last digit and named "foo myllion bar", and a semicolon separates the digits. So 1,0002;0003,0004 is "one myriad two myllion, three myriad four."
• 1016 to 1032 − 1 are divided before the 16th-last digit and named "foo byllion bar", and a colon separates the digits. So 12:0003,0004;0506,7089 is "twelve byllion, three myriad four myllion, five hundred six myriad seventy hundred eighty-nine."
• etc.
Each new number name is the square of the previous one — therefore, each new name covers twice as many digits. Knuth continues borrowing the traditional names changing "illion" to "yllion" on each one. Abstractly, then, "one n-yllion" is $10^{2^{n+2}}$. "One trigintyllion" ($10^{2^{32}}$) would have 232 + 1, or 42;9496,7297, or nearly forty-three myllion (4300 million) digits (by contrast, a conventional "trigintillion" has merely 94 digits — not even a hundred, let alone a thousand million, and still 7 digits short of a googol). Better yet, "one centyllion" ($10^{2^{102}}$) would have 2102 + 1, or 507,0602;4009,1291:7605,9868;1282,1505, or about 1/20 of a tryllion digits, whereas a conventional "centillion" has only 304 digits.
The corresponding Chinese "long scale" numerals are given, with the traditional form listed before the simplified form. Same numerals are used in the Chinese "short scale" (new number name every power of 10 after 1000 (or 103+n)), "myriad scale" (new number name every 104n), and "mid scale" (new number name every 108n). Today these numerals are still in use, but are used in their "myriad scale" values, which is also used in Japanese and in Korean. For a more extensive table, see Myriad system.
ValueNameNotation Standard English name (short scale)Chinese ("long scale")Pīnyīn (Mandarin)Jyutping (Cantonese)Pe̍h-ōe-jī (Hokkien)
100 One 1 One 一 yī jat1 it/chit
101 Ten 10 Ten 十 shí sap6 si̍p/cha̍p
102 One hundred 100 One hundred 百 bǎi baak3 pah
103 Ten hundred 1000 One thousand 千 qiān cin1 chhian
104 One myriad 1,0000 Ten thousand 萬, 万 wàn maan6 bān
105 Ten myriad 10,0000 One hundred thousand 十萬, 十万 shíwàn sap6 maan6 si̍p/cha̍p bān
106 One hundred myriad 100,0000 One million 百萬, 百万 bǎiwàn baak3 maan6 pah bān
107 Ten hundred myriad 1000,0000 Ten million 千萬, 千万 qiānwàn cin1 maan6 chhian bān
108 One myllion 1;0000,0000 One hundred million 億, 亿 yì jik1 ek
109 Ten myllion 10;0000,0000 One billion 十億, 十亿 shíyì sap6 jik1 si̍p/cha̍p ek
1012 One myriad myllion 1,0000;0000,0000 One trillion 萬億, 万亿 wànyì maan6 jik1 bān ek
1016 One byllion 1:0000,0000;0000,0000 Ten quadrillion 兆 zhào siu6 tiāu
1024 One myllion byllion 1;0000,0000:0000,0000;0000,0000 One septillion 億兆, 亿兆 yìzhào jik1 siu6 ek tiāu
1032 One tryllion 1'0000,0000;0000,0000:0000,0000;0000,0000 One hundred nonillion 京 jīng ging1 kiaⁿ
1064 One quadryllion Ten vigintillion 垓 gāi goi1 kai
10128 One quintyllion One hundred unquadragintillion 秭 zǐ zi2 chi
10256 One sextyllion Ten quattuoroctogintillion 穰 ráng joeng4 liōng
10512 One septyllion One hundred novensexagintacentillion 溝, 沟 gōu kau1 kau
101024 One octyllion Ten quadragintatrecentillion 澗, 涧 jiàn gaan3 kán
102048 One nonyllion One hundred unoctogintasescentillion 正 zhēng zing3 chiàⁿ
104096 One decyllion Ten milliquattuorsexagintatrecentillion 載, 载 zài zoi3 chài
Latin- prefix
In order to construct names of the form n-yllion for large values of n, Knuth appends the prefix "latin-" to the name of n without spaces and uses that as the prefix for n. For example, the number "latintwohundredyllion" corresponds to n = 200, and hence to the number $10^{2^{202}}$.
Negative powers
To refer to small quantities with this system, the suffix -th is used.
For instance, $10^{-4}$is a myriadth. $10^{-16777216}$ is a vigintyllionth.
See also
• Nicolas Chuquet – Mathematician
• Jacques Pelletier du Mans – Humanist, Poet, Mathematician
• Knuth's up-arrow notation – Method of notation of very large integers
• The Sand Reckoner – Work by Archimedes
References
1. "Large Numbers (Page 2) at MROB".
• Donald E. Knuth. Supernatural Numbers in The Mathematical Gardener (edited by David A. Klarner). Wadsworth, Belmont, CA, 1981. 310—325.
• Robert P. Munafo. The Knuth -yllion Notation ( Archived 2012-02-13 at the Wayback Machine 2012-02-25), 1996–2012.
Donald Knuth
Publications
• The Art of Computer Programming
• "The Complexity of Songs"
• Computers and Typesetting
• Concrete Mathematics
• Surreal Numbers
• Things a Computer Scientist Rarely Talks About
• Selected papers series
Software
• TeX
• Metafont
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• MMIX)
Fonts
• AMS Euler
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Literate programming
• WEB
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Algorithms
• Knuth's Algorithm X
• Knuth–Bendix completion algorithm
• Knuth–Morris–Pratt algorithm
• Knuth shuffle
• Robinson–Schensted–Knuth correspondence
• Trabb Pardo–Knuth algorithm
• Generalization of Dijkstra's algorithm
• Knuth's Simpath algorithm
Other
• Dancing Links
• Knuth reward check
• Knuth Prize
• Knuth's up-arrow notation
• Man or boy test
• Quater-imaginary base
• -yllion
• Potrzebie system of weights and measures
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Jean-Christophe Yoccoz
Jean-Christophe Yoccoz (29 May 1957 – 3 September 2016) was a French mathematician. He was awarded a Fields Medal in 1994, for his work on dynamical systems.[1][2] Yoccoz died on 3 September 2016 at the age of 59.
Jean-Christophe Yoccoz
Jean-Christophe Yoccoz in 2005
Born(1957-05-29)29 May 1957
Paris, France
Died3 September 2016(2016-09-03) (aged 59)
Paris, France
EducationLycée Louis-le-Grand
Alma materÉcole normale supérieure
École Polytechnique
Known forDynamical systems
Yoccoz puzzle
AwardsSalem Prize (1988)
Fields Medal (1994)
Scientific career
FieldsMathematics
InstitutionsCentre de mathématiques Laurent-Schwartz
Paris-Sud 11 University
Collège de France
Doctoral advisorMichael Herman
Doctoral studentsSylvain Crovisier
Ricardo Pérez-Marco
Websitewww.college-de-france.fr/site/jean-christophe-yoccoz/Hommage-a-Jean-Christophe-Yoccoz.htm
Wikimedia Commons has media related to Jean-Christophe Yoccoz.
Biography
Yoccoz attended the Lycée Louis-le-Grand,[3] during which time he was a silver medalist at the 1973 International Mathematical Olympiad and a gold medalist in 1974.[4][5] He entered the École Normale Supérieure in 1975, and completed an agrégation in mathematics in 1977.[6]
After completing military service in Brazil, he completed his PhD under Michael Herman in 1985 at Centre de mathématiques Laurent-Schwartz, which is a research unit jointly operated by the French National Center for Scientific Research (CNRS) and École Polytechnique.[6][7][8]
He took up a position at the University of Paris-Sud in 1987, and became a professor at the Collège de France in 1997, where he remained until his death.[2] He was a member of Bourbaki.[9]
Yoccoz won the Salem Prize in 1988. He was an invited speaker at the International Congress of Mathematicians in 1990 at Kyoto,[10] and was awarded the Fields Medal at the International Congress of Mathematicians in 1994 in Zürich.[11][10] He joined the French Academy of Sciences and Brazilian Academy of Sciences in 1994, became a chevalier in the French Legion of Honor in 1995, and was awarded the Grand Cross of the Brazilian National Order of Scientific Merit in 1998.[5]
Mathematical work
Yoccoz's worked on the theory of dynamical systems, his contributions include advances to KAM theory, and the introduction of the method of Yoccoz puzzles, a combinatorial technique which proved useful to the study of Julia sets.[5]
Notable publications
• Yoccoz, J.-C. Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne. Ann. Sci. École Norm. Sup. (4) 17 (1984), no. 3, 333–359. doi:10.24033/asens.1475
• Yoccoz, Jean-Christophe. Théorème de Siegel, nombres de Bruno et polynômes quadratiques. Petits diviseurs en dimension 1. Astérisque No. 231 (1995), 3–88. MR1367353
References
1. O'Connor, John J.; Robertson, Edmund F., "Jean-Christophe Yoccoz", MacTutor History of Mathematics Archive, University of St Andrews
2. "Le mathématicien Jean-Christophe Yoccoz est mort". 5 September 2016. Retrieved 6 September 2016.
3. Administrator. "Des mathématiciens". Retrieved 6 September 2016.
4. Jean-Christophe Yoccoz's results at International Mathematical Olympiad
5. "Jean-Christophe Yoccoz † › Heidelberg Laureate Forum". Archived from the original on 11 September 2016. Retrieved 6 September 2016.
6. "Yoccoz biography". Retrieved 6 September 2016.
7. "Ciência se despede de matemático francês Jean-Christophe Yoccoz". Archived from the original on 19 September 2016. Retrieved 6 September 2016.
8. Jean-Christophe Yoccoz at the Mathematics Genealogy Project
9. Mashaal, Maurice (2006), Bourbaki: a secret society of mathematicians, American Mathematical Society, p. 19, ISBN 978-0-8218-3967-6
10. "International Mathematical Union (IMU)". Archived from the original on 8 November 2017. Retrieved 6 September 2016.
11. "Fields Medals and Nevanlinna Prize 1994". Archived from the original on 29 July 2016. Retrieved 6 September 2016.
Fields Medalists
• 1936 Ahlfors
• Douglas
• 1950 Schwartz
• Selberg
• 1954 Kodaira
• Serre
• 1958 Roth
• Thom
• 1962 Hörmander
• Milnor
• 1966 Atiyah
• Cohen
• Grothendieck
• Smale
• 1970 Baker
• Hironaka
• Novikov
• Thompson
• 1974 Bombieri
• Mumford
• 1978 Deligne
• Fefferman
• Margulis
• Quillen
• 1982 Connes
• Thurston
• Yau
• 1986 Donaldson
• Faltings
• Freedman
• 1990 Drinfeld
• Jones
• Mori
• Witten
• 1994 Bourgain
• Lions
• Yoccoz
• Zelmanov
• 1998 Borcherds
• Gowers
• Kontsevich
• McMullen
• 2002 Lafforgue
• Voevodsky
• 2006 Okounkov
• Perelman
• Tao
• Werner
• 2010 Lindenstrauss
• Ngô
• Smirnov
• Villani
• 2014 Avila
• Bhargava
• Hairer
• Mirzakhani
• 2018 Birkar
• Figalli
• Scholze
• Venkatesh
• 2022 Duminil-Copin
• Huh
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Small retrosnub icosicosidodecahedron
In geometry, the small retrosnub icosicosidodecahedron (also known as a retrosnub disicosidodecahedron, small inverted retrosnub icosicosidodecahedron, or retroholosnub icosahedron) is a nonconvex uniform polyhedron, indexed as U72. It has 112 faces (100 triangles and 12 pentagrams), 180 edges, and 60 vertices.[1] It is given a Schläfli symbol sr{⁵/₃,³/₂}.
Small retrosnub icosicosidodecahedron
TypeUniform star polyhedron
ElementsF = 112, E = 180
V = 60 (χ = −8)
Faces by sides(40+60){3}+12{5/2}
Coxeter diagram
Wythoff symbol| 3/2 3/2 5/2
Symmetry groupIh, [5,3], *532
Index referencesU72, C91, W118
Dual polyhedronSmall hexagrammic hexecontahedron
Vertex figure
(35.5/3)/2
Bowers acronymSirsid
The 40 non-snub triangular faces form 20 coplanar pairs, forming star hexagons that are not quite regular. Unlike most snub polyhedra, it has reflection symmetries.
George Olshevsky nicknamed it the yog-sothoth (after the Cthulhu Mythos deity).[2][3]
Convex hull
Its convex hull is a nonuniform truncated dodecahedron.
Truncated dodecahedron
Convex hull
Small retrosnub icosicosidodecahedron
Cartesian coordinates
Cartesian coordinates for the vertices of a small retrosnub icosicosidodecahedron are all the even permutations of
(±(1-ϕ−α), 0, ±(3−ϕα))
(±(ϕ-1−α), ±2, ±(2ϕ-1−ϕα))
(±(ϕ+1−α), ±2(ϕ-1), ±(1−ϕα))
where ϕ = (1+√5)/2 is the golden ratio and α = √3ϕ−2.
See also
• List of uniform polyhedra
• Small snub icosicosidodecahedron
References
1. Maeder, Roman. "72: small retrosnub icosicosidodecahedron". MathConsult.{{cite web}}: CS1 maint: url-status (link)
2. Birrell, Robert J. (May 1992). The Yog-sothoth: analysis and construction of the small inverted retrosnub icosicosidodecahedron (M.S.). California State University.
3. Bowers, Jonathan (2000). "Uniform Polychora" (PDF). In Reza Sarhagi (ed.). Bridges 2000. Bridges Conference. pp. 239–246.
External links
• Weisstein, Eric W. "Small retrosnub icosicosidodecahedron". MathWorld.
• Klitzing, Richard. "3D star small retrosnub icosicosidodecahedron".
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Yoichi Miyaoka
Yoichi Miyaoka (宮岡 洋一, Miyaoka Yōichi) is a mathematician who works in algebraic geometry and who proved (independently of Shing-Tung Yau's work) the Bogomolov–Miyaoka–Yau inequality in an Inventiones Mathematicae paper.[1]
In 1984, Miyaoka extended the Bogomolov–Miyaoka–Yau inequality to surfaces with quotient singularities, and in 2008 to orbifold surfaces. Doing so, he obtains sharp bound on the number of quotient singularities on surfaces of general type. Moreover, the inequality for orbifold surfaces gives explicit values for the coefficients of the so-called Lang-Vojta conjecture relating the degree of a curve on a surface with its geometric genus.
References
1. Miyaoka, Yoichi (1977-12-01). "On the Chern numbers of surfaces of general type". Inventiones Mathematicae. 42 (1): 225–237. Bibcode:1977InMat..42..225M. doi:10.1007/BF01389789. ISSN 1432-1297. S2CID 120699065.
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Yom Tov Lipman Lipkin
Yom Tov Lipman Lipkin (Hebrew: יום טוב ליפמן ליפקין, Russian: Липман Израилевич Липкин; 1840 – February 21 [O.S. February 9] 1876) was a Lithuanian Jewish mathematician and inventor. He was the youngest son of Rabbi Yisroel Salanter, the father of the Musar movement.
Lipkin is best known for the Peaucellier–Lipkin linkage which was partly named after him.[1][2] The device is also known as the "Lipkin parallelogram".[3] Lipkin discovered the linkage independent from Peaucellier in 1871.[1] A model of Lipkin's invention was exhibited at the exposition at Vienna in 1873, and was later secured from the inventor by the Museum of the Institute of Engineers of Ways of Communication, St. Petersburg.
Biography
Lipkin was born in Salantai, department of Kovno, in 1846. He became interested in science and mathematics since childhood. Not knowing any non-Jewish languages, he had to derive his information from Hebrew books alone. He later learned German and French and went to study at University of Königsberg at the age of 17. He received a Ph.D. degree at Jena University with a thesis titled "Ueber die Räumlichen Strophoiden." He then moved to St. Petersburg, to work at University of St. Petersburg and continue his studies under Pafnuty Chebyshev. Soon afterwards he died in 1876 from smallpox.
Lipkin broke from traditional Jewish life, but kept interests in Jewish affairs and published in Ha-Tsefirah newspaper.
References
1. Mathematical tutorial of the Peaucellier–Lipkin linkage
2. How to draw a straight line by Daina Taimina
3. Lipkin, in 1906 Jewish Encyclopedia.
• Simona-Mariana Cretu, Gigi-Dragos Ciocioi-Troaca, Emil Soarece, and Eugen Marian Paun, Mechanical Models for Anti-Rhomb Linkage, in Explorations in the History of Machines and Mechanisms, Springer, 2012, pp. 421–430.
• Alan T. Levenson, Roger C. Klein, An Introduction to Modern Jewish Thinkers: From Spinoza to Soloveitchik, Rowman & Littlefield Publishers, 2006.
• I. Etkes, Rabbi Israel Salanter and the Mussar Movement: Seeking the Torah of Truth, Magness Press, Hebrew University of Jerusalem, 1982.
This article incorporates text from a publication now in the public domain: Singer, Isidore; et al., eds. (1901–1906). The Jewish Encyclopedia. New York: Funk & Wagnalls. {{cite encyclopedia}}: Missing or empty |title= (help)
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Yomtov Garti
Yomtov Bonjour Garti (2 September 1915 – 21 February 2011) was a Turkish mathematician and a teacher of mathematics, physics and cosmography in Istanbul, Turkey.
Life
Yomtov Garti was born in Kadıköy, at the Asian part of Istanbul (Ottoman Empire, later to become Turkey). His father Maer Garti was a veterinarian doctor, who died in the typhus epidemics during the World War I, while serving in the Ottoman Army. Yomtov Garti was educated at the French high school Lycée Saint-Joseph, Istanbul . After graduating from the Department of Mathematics and Physics of Istanbul University, he was approached by the famous mathematician Richard Edler von Mises, who was then located in Istanbul[1] and proposed him a PhD position. Garti’s PhD was on statistics and probability theory tutored by Richard Edler von Mises and William Prager.[2] Yomtov Garti received his PhD in 1939, as the first PhD student of von Mises and third PhD of Turkey. He later published his findings in an article (Garti, 1940),[3] which is a generalization of initial distributions to n dimensions published in[4](cited in[5]). Garti served as an assistant for a summer to Harry Dember, a professor in the Institute of Applied Physics in Istanbul. After receiving his doctoral degree in 1939, he started to teach at famous high schools in Istanbul. In 1954, he presented an article to Richard von Mises in a book published in his honor.[6] Yomtov Garti continued teaching until age 92. He died, at the age of 96 in 2011. All main newspapers of Turkey announced the loss of “The teacher of teachers”. At his funeral, the several generations of students overfilled the Hemdat Israel Synagogue at Kadıköy, Istanbul.
Career
Yomtov Garti became a renowned mathematics teacher in Istanbul. He thought mathematics and physics to thousands of students over six decades, in several famous schools, namely Galatasary High School (Galatasaray Lisesi), Haydarpaşa High School, Lycee Saint-Joseph Istanbul, and Notre Dame de Sion Lisesi. He also thought in Musevi Lisesi and Boğaziçi Üniversity. Yomtov Garti was rewarded the title of “Chevalier” of “Ordre des Palmes Academiques” by the French Government in recognition of his educational work in French schools in Turkey.
Publications
• Garti, Y., 1940. Les lois de probabilité pour les fonctions statistiques (cas de collectifs à plusieurs dimensions). Revue Mathématique de l’Union Interbalkanique 3, 21–39.
• Garti Y, Consoli T. 1954. Sur la densite de probabilite du produit de variables aleatoires de Pearson du type III. In: Studies in mathematics and mechanics presented to Richard von Mises. Academic Press, New York. pp 301–309.
References
1. Eden and Irzik, 2012. German mathematicians in exile in Turkey: Richard von Mises, William Prager, Hilda Geiringer, and their impact on Turkish mathematics. Historia Mathematica 2012, 29 (4): 432-459. https://doi.org/10.1016/j.hm.2012.07.002
2. "Yomtov Garti - The Mathematics Genealogy Project". www.genealogy.math.ndsu.nodak.edu. Genealogy of Mathematicians. Retrieved 29 May 2020.
3. Garti, 1940, Les lois de probabilité pour les fonctions statistiques (cas de collectifs à plusieurs dimensions). Revue Mathématique de l’Union Interbalkanique 3, 21–39.
4. Mises 1936 http://www.numdam.org/item?id=AIHP_1936__6_3-4_185_0
5. Mises, 1964. Mathematical theory of probability and statistics. Eited by Hilda Geiringer. Academic Press New York and London. Citing Y. Garti in chapter C.5. page 651.
6. Garti Y, Consoli T. 1954. Sur la densite de probabilite du produit de variables aleatoires de Pearson du type III. In: Studies in mathematics and mechanics presented to Richard von Mises. Academic Press, New York. pp 301-309
• Eden A, Irzik G. 2012. German mathematicians in exile in Turkey: Richard von Mises, William Prager, Hilda Geiringer, and their impact on Turkish mathematics. Historia Mathematica 2012, 29 (4): 432-459. https://doi.org/10.1016/j.hm.2012.07.002
• Garti, Y., 1940. Les lois de probabilité pour les fonctions statistiques (cas de collectifs à plusieurs dimensions). Revue Mathématique de l’Union Interbalkanique 3, 21–39.
• Garti Y, Consoli T. 1954. Sur la densite de probabilite du produit de variables aleatoires de Pearson du type III. In: Studies in mathematics and mechanics presented to Richard von Mises. Academic Press, New York. pp 301–309.
• Mises R. Les lois de probabilité pour les fonctions statistiques. Ann. Inst. Henri Poincare. 6, 185–212, 1936. http://www.numdam.org/item?id=AIHP_1936__6_3-4_185_0
• Mises R. 1964. Mathematical theory of probability and statistics. Eited by Hilda Geiringer. Academic Press New York and London. Citing Y. Garti in chapter C.5. page 651.
External links
• Misés, R. de (1936). "Les lois de probabilité pour les fonctions statistiques". Annales de l'institut Henri Poincaré. pp. 185–212. Retrieved 29 May 2020.
• http://www.matematikdunyasi.org/arsiv/PDF/11_01_59_60_garti.pdf
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Yoneda lemma
In mathematics, the Yoneda lemma is arguably the most important result in category theory.[1] It is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a miniature category with just one object and only isomorphisms). It allows the embedding of any locally small category into a category of functors (contravariant set-valued functors) defined on that category. It also clarifies how the embedded category, of representable functors and their natural transformations, relates to the other objects in the larger functor category. It is an important tool that underlies several modern developments in algebraic geometry and representation theory. It is named after Nobuo Yoneda.
Generalities
The Yoneda lemma suggests that instead of studying the locally small category ${\mathcal {C}}$, one should study the category of all functors of ${\mathcal {C}}$ into $\mathbf {Set} $ (the category of sets with functions as morphisms). $\mathbf {Set} $ is a category we think we understand well, and a functor of ${\mathcal {C}}$ into $\mathbf {Set} $ can be seen as a "representation" of ${\mathcal {C}}$ in terms of known structures. The original category ${\mathcal {C}}$ is contained in this functor category, but new objects appear in the functor category, which were absent and "hidden" in ${\mathcal {C}}$. Treating these new objects just like the old ones often unifies and simplifies the theory.
This approach is akin to (and in fact generalizes) the common method of studying a ring by investigating the modules over that ring. The ring takes the place of the category ${\mathcal {C}}$, and the category of modules over the ring is a category of functors defined on ${\mathcal {C}}$.
Formal statement
Yoneda's lemma concerns functors from a fixed category ${\mathcal {C}}$ to the category of sets, $\mathbf {Set} $. If ${\mathcal {C}}$ is a locally small category (i.e. the hom-sets are actual sets and not proper classes), then each object $A$ of ${\mathcal {C}}$ gives rise to a natural functor to $\mathbf {Set} $ called a hom-functor. This functor is denoted:
$h_{A}=\mathrm {Hom} (A,-)$.
The (covariant) hom-functor $h_{A}$ sends $X$ to the set of morphisms $\mathrm {Hom} (A,X)$ and sends a morphism $f\colon X\to Y$ (where $X$ and $Y$ are objects in ${\mathcal {C}}$) to the morphism $f\circ -$ (composition with $f$ on the left) that sends a morphism $g$ in $\mathrm {Hom} (A,X)$ to the morphism $f\circ g$ in $\mathrm {Hom} (A,Y)$. That is,
$h_{A}(f)=\mathrm {Hom} (A,f),{\text{ or}}$
$h_{A}(f)(g)=f\circ g$
Yoneda's lemma says that:
Lemma (Yoneda) — Let $F$ be a functor from a locally small category ${\mathcal {C}}$ to $\mathbf {Set} $. Then for each object $A$ of ${\mathcal {C}}$, the natural transformations $\mathrm {Nat} (h_{A},F)\equiv \mathrm {Hom} (\mathrm {Hom} (A,-),F)$ from $h_{A}$ to $F$ are in one-to-one correspondence with the elements of $F(A)$. That is,
$\mathrm {Nat} (h_{A},F)\cong F(A).$
Moreover, this isomorphism is natural in $A$ and $F$ when both sides are regarded as functors from ${\mathcal {C}}\times \mathbf {Set} ^{\mathcal {C}}$ to $\mathbf {Set} $.
Here the notation $\mathbf {Set} ^{\mathcal {C}}$ denotes the category of functors from ${\mathcal {C}}$ to $\mathbf {Set} $.
Given a natural transformation $\Phi $ from $h_{A}$ to $F$, the corresponding element of $F(A)$ is $u=\Phi _{A}(\mathrm {id} _{A})$;[lower-alpha 1] and given an element $u$ of $F(A)$, the corresponding natural transformation is given by $\Phi _{X}(f)=F(f)(u)$ which assigns to a morphism $f\colon A\to X$ a value of $F(X)$.
Contravariant version
There is a contravariant version of Yoneda's lemma, which concerns contravariant functors from ${\mathcal {C}}$ to $\mathbf {Set} $. This version involves the contravariant hom-functor
$h^{A}=\mathrm {Hom} (-,A),$
which sends $X$ to the hom-set $\mathrm {Hom} (X,A)$. Given an arbitrary contravariant functor $G$ from ${\mathcal {C}}$ to $\mathbf {Set} $, Yoneda's lemma asserts that
$\mathrm {Nat} (h^{A},G)\cong G(A).$
Naming conventions
The use of $h_{A}$ for the covariant hom-functor and $h^{A}$ for the contravariant hom-functor is not completely standard. Many texts and articles either use the opposite convention or completely unrelated symbols for these two functors. However, most modern algebraic geometry texts starting with Alexander Grothendieck's foundational EGA use the convention in this article.[lower-alpha 2]
The mnemonic "falling into something" can be helpful in remembering that $h_{A}$ is the covariant hom-functor. When the letter $A$ is falling (i.e. a subscript), $h_{A}$ assigns to an object $X$ the morphisms from $A$ into $X$.
Proof
Since $\Phi $ is a natural transformation, we have the following commutative diagram:
This diagram shows that the natural transformation $\Phi $ is completely determined by $\Phi _{A}(\mathrm {id} _{A})=u$ since for each morphism $f\colon A\to X$ one has
$\Phi _{X}(f)=(Ff)u.$
Moreover, any element $u\in F(A)$ defines a natural transformation in this way. The proof in the contravariant case is completely analogous.
The Yoneda embedding
An important special case of Yoneda's lemma is when the functor $F$ from ${\mathcal {C}}$ to $\mathbf {Set} $ is another hom-functor $h_{B}$. In this case, the covariant version of Yoneda's lemma states that
$\mathrm {Nat} (h_{A},h_{B})\cong \mathrm {Hom} (B,A).$
That is, natural transformations between hom-functors are in one-to-one correspondence with morphisms (in the reverse direction) between the associated objects. Given a morphism $f\colon B\to A$ the associated natural transformation is denoted $\mathrm {Hom} (f,-)$.
Mapping each object $A$ in ${\mathcal {C}}$ to its associated hom-functor $h_{A}=\mathrm {Hom} (A,-)$ and each morphism $f\colon B\to A$ to the corresponding natural transformation $\mathrm {Hom} (f,-)$ determines a contravariant functor $h_{\bullet }$ from ${\mathcal {C}}$ to $\mathbf {Set} ^{\mathcal {C}}$, the functor category of all (covariant) functors from ${\mathcal {C}}$ to $\mathbf {Set} $. One can interpret $h_{\bullet }$ as a covariant functor:
$h_{\bullet }\colon {\mathcal {C}}^{\text{op}}\to \mathbf {Set} ^{\mathcal {C}}.$
The meaning of Yoneda's lemma in this setting is that the functor $h_{\bullet }$ is fully faithful, and therefore gives an embedding of ${\mathcal {C}}^{\mathrm {op} }$ in the category of functors to $\mathbf {Set} $. The collection of all functors $\{h_{A}|A\in C\}$ is a subcategory of $\mathbf {Set} ^{\mathcal {C}}$. Therefore, Yoneda embedding implies that the category ${\mathcal {C}}^{\mathrm {op} }$ is isomorphic to the category $\{h_{A}|A\in C\}$.
The contravariant version of Yoneda's lemma states that
$\mathrm {Nat} (h^{A},h^{B})\cong \mathrm {Hom} (A,B).$
Therefore, $h^{\bullet }$ gives rise to a covariant functor from ${\mathcal {C}}$ to the category of contravariant functors to $\mathbf {Set} $:
$h^{\bullet }\colon {\mathcal {C}}\to \mathbf {Set} ^{{\mathcal {C}}^{\mathrm {op} }}.$
Yoneda's lemma then states that any locally small category ${\mathcal {C}}$ can be embedded in the category of contravariant functors from ${\mathcal {C}}$ to $\mathbf {Set} $ via $h^{\bullet }$. This is called the Yoneda embedding.
The Yoneda embedding is sometimes denoted by よ, the Hiragana kana Yo.[2]
Representable functor
The Yoneda embedding essentially states that for every (locally small) category, objects in that category can be represented by presheaves, in a full and faithful manner. That is,
$\mathrm {Nat} (h^{A},P)\cong P(A)$
for a presheaf P. Many common categories are, in fact, categories of pre-sheaves, and on closer inspection, prove to be categories of sheaves, and as such examples are commonly topological in nature, they can be seen to be topoi in general. The Yoneda lemma provides a point of leverage by which the topological structure of a category can be studied and understood.
In terms of (co)end calculus
Main article: End (category theory)
Given two categories $\mathbf {C} $ and $\mathbf {D} $ with two functors $F,G:\mathbf {C} \to \mathbf {D} $, natural transformations between them can be written as the following end.[3]
$\mathrm {Nat} (F,G)=\int _{c\in \mathbf {C} }\mathrm {Hom} _{\mathbf {D} }(Fc,Gc)$
For any functors $K\colon \mathbf {C} ^{op}\to \mathbf {Sets} $ and $H\colon \mathbf {C} \to \mathbf {Sets} $ the following formulas are all formulations of the Yoneda lemma.[4]
$K\cong \int ^{c\in \mathbf {C} }Kc\times \mathrm {Hom} _{\mathbf {C} }(-,c),\qquad K\cong \int _{c\in \mathbf {C} }(Kc)^{\mathrm {Hom} _{\mathbf {C} }(c,-)},$
$H\cong \int ^{c\in \mathbf {C} }Hc\times \mathrm {Hom} _{\mathbf {C} }(c,-),\qquad H\cong \int _{c\in \mathbf {C} }(Hc)^{\mathrm {Hom} _{\mathbf {C} }(-,c)}.$
Preadditive categories, rings and modules
A preadditive category is a category where the morphism sets form abelian groups and the composition of morphisms is bilinear; examples are categories of abelian groups or modules. In a preadditive category, there is both a "multiplication" and an "addition" of morphisms, which is why preadditive categories are viewed as generalizations of rings. Rings are preadditive categories with one object.
The Yoneda lemma remains true for preadditive categories if we choose as our extension the category of additive contravariant functors from the original category into the category of abelian groups; these are functors which are compatible with the addition of morphisms and should be thought of as forming a module category over the original category. The Yoneda lemma then yields the natural procedure to enlarge a preadditive category so that the enlarged version remains preadditive — in fact, the enlarged version is an abelian category, a much more powerful condition. In the case of a ring $R$, the extended category is the category of all right modules over $R$, and the statement of the Yoneda lemma reduces to the well-known isomorphism
$M\cong \mathrm {Hom} _{R}(R,M)$ for all right modules $M$ over $R$.
Relationship to Cayley's theorem
As stated above, the Yoneda lemma may be considered as a vast generalization of Cayley's theorem from group theory. To see this, let ${\mathcal {C}}$ be a category with a single object $*$ such that every morphism is an isomorphism (i.e. a groupoid with one object). Then $G=\mathrm {Hom} _{\mathcal {C}}(*,*)$ forms a group under the operation of composition, and any group can be realized as a category in this way.
In this context, a covariant functor ${\mathcal {C}}\to \mathbf {Set} $ consists of a set $X$ and a group homomorphism $G\to \mathrm {Perm} (X)$, where $\mathrm {Perm} (X)$ is the group of permutations of $X$; in other words, $X$ is a G-set. A natural transformation between such functors is the same thing as an equivariant map between $G$-sets: a set function $\alpha \colon X\to Y$ with the property that $\alpha (g\cdot x)=g\cdot \alpha (x)$ for all $g$ in $G$ and $x$ in $X$. (On the left side of this equation, the $\cdot $ denotes the action of $G$ on $X$, and on the right side the action on $Y$.)
Now the covariant hom-functor $\mathrm {Hom} _{\mathcal {C}}(*,-)$ corresponds to the action of $G$ on itself by left-multiplication (the contravariant version corresponds to right-multiplication). The Yoneda lemma with $F=\mathrm {Hom} _{\mathcal {C}}(*,-)$ states that
$\mathrm {Nat} (\mathrm {Hom} _{\mathcal {C}}(*,-),\mathrm {Hom} _{\mathcal {C}}(*,-))\cong \mathrm {Hom} _{\mathcal {C}}(*,*)$,
that is, the equivariant maps from this $G$-set to itself are in bijection with $G$. But it is easy to see that (1) these maps form a group under composition, which is a subgroup of $\mathrm {Perm} (G)$, and (2) the function which gives the bijection is a group homomorphism. (Going in the reverse direction, it associates to every $g$ in $G$ the equivariant map of right-multiplication by $g$.) Thus $G$ is isomorphic to a subgroup of $\mathrm {Perm} (G)$, which is the statement of Cayley's theorem.
History
Yoshiki Kinoshita stated in 1996 that the term "Yoneda lemma" was coined by Saunders Mac Lane following an interview he had with Yoneda in the Gare du Nord station.[5][6]
See also
• Representation theorem
Notes
1. Recall that $\Phi _{A}:\mathrm {Hom} (A,A)\to F(A)$ so the last expression is well-defined and sends a morphism from $A$ to $A$, to an element in $F(A)$.
2. A notable exception to modern algebraic geometry texts following the conventions of this article is Commutative algebra with a view toward algebraic geometry / David Eisenbud (1995), which uses $h_{A}$ to mean the covariant hom-functor. However, the later book The geometry of schemes / David Eisenbud, Joe Harris (1998) reverses this and uses $h_{A}$ to mean the contravariant hom-functor.
References
1. Riehl, Emily (2017). Category Theory in Context (PDF). Dover. ISBN 978-0-486-82080-4.
2. "Yoneda embedding". nLab. Retrieved 6 July 2019.
3. Loregian (2015), Theorem 1.4.1.
4. Loregian (2015), Proposition 2.2.1.
5. Kinoshita, Yoshiki (23 April 1996). "Prof. Nobuo Yoneda passed away". Retrieved 21 December 2013.
6. "le lemme de la Gare du Nord". neverendingbooks. 18 November 2016. Retrieved 2022-09-10.
• Freyd, Peter (1964), Abelian categories, Harper's Series in Modern Mathematics (2003 reprint ed.), Harper and Row, Zbl 0121.02103.
• Mac Lane, Saunders (1998), Categories for the Working Mathematician, Graduate Texts in Mathematics, vol. 5 (2nd ed.), New York, NY: Springer-Verlag, ISBN 0-387-98403-8, Zbl 0906.18001
• Loregian, Fosco (2015). "This is the (co)end, my only (co)friend". arXiv:1501.02503 [math.CT].
• Yoneda lemma at the nLab
External links
• Mizar system proof: Wojciechowski, M. (1997). "Yoneda Embedding". Formalized Mathematics journal. 6 (3): 377–380. CiteSeerX 10.1.1.73.7127.
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Presheaf (category theory)
In category theory, a branch of mathematics, a presheaf on a category $C$ is a functor $F\colon C^{\mathrm {op} }\to \mathbf {Set} $. If $C$ is the poset of open sets in a topological space, interpreted as a category, then one recovers the usual notion of presheaf on a topological space.
A morphism of presheaves is defined to be a natural transformation of functors. This makes the collection of all presheaves on $C$ into a category, and is an example of a functor category. It is often written as ${\widehat {C}}=\mathbf {Set} ^{C^{\mathrm {op} }}$. A functor into ${\widehat {C}}$ is sometimes called a profunctor.
A presheaf that is naturally isomorphic to the contravariant hom-functor Hom(–, A) for some object A of C is called a representable presheaf.
Some authors refer to a functor $F\colon C^{\mathrm {op} }\to \mathbf {V} $ as a $\mathbf {V} $-valued presheaf.[1]
Examples
• A simplicial set is a Set-valued presheaf on the simplex category $C=\Delta $.
Properties
• When $C$ is a small category, the functor category ${\widehat {C}}=\mathbf {Set} ^{C^{\mathrm {op} }}$ is cartesian closed.
• The poset of subobjects of $P$ form a Heyting algebra, whenever $P$ is an object of ${\widehat {C}}=\mathbf {Set} ^{C^{\mathrm {op} }}$ for small $C$.
• For any morphism $f:X\to Y$ of ${\widehat {C}}$, the pullback functor of subobjects $f^{*}:\mathrm {Sub} _{\widehat {C}}(Y)\to \mathrm {Sub} _{\widehat {C}}(X)$ has a right adjoint, denoted $\forall _{f}$, and a left adjoint, $\exists _{f}$. These are the universal and existential quantifiers.
• A locally small category $C$ embeds fully and faithfully into the category ${\widehat {C}}$ of set-valued presheaves via the Yoneda embedding which to every object $A$ of $C$ associates the hom functor $C(-,A)$.
• The category ${\widehat {C}}$ admits small limits and small colimits.[2] See limit and colimit of presheaves for further discussion.
• The density theorem states that every presheaf is a colimit of representable presheaves; in fact, ${\widehat {C}}$ is the colimit completion of $C$ (see #Universal property below.)
Universal property
The construction $C\mapsto {\widehat {C}}=\mathbf {Fct} (C^{\text{op}},\mathbf {Set} )$ is called the colimit completion of C because of the following universal property:
Proposition[3] — Let C, D be categories and assume D admits small colimits. Then each functor $\eta :C\to D$ factorizes as
$C{\overset {y}{\longrightarrow }}{\widehat {C}}{\overset {\widetilde {\eta }}{\longrightarrow }}D$
where y is the Yoneda embedding and ${\widetilde {\eta }}:{\widehat {C}}\to D$ is a, unique up to isomorphism, colimit-preserving functor called the Yoneda extension of $\eta $.
Proof: Given a presheaf F, by the density theorem, we can write $F=\varinjlim yU_{i}$ where $U_{i}$ are objects in C. Then let ${\widetilde {\eta }}F=\varinjlim \eta U_{i},$ which exists by assumption. Since $\varinjlim -$ is functorial, this determines the functor ${\widetilde {\eta }}:{\widehat {C}}\to D$. Succinctly, ${\widetilde {\eta }}$ is the left Kan extension of $\eta $ along y; hence, the name "Yoneda extension". To see ${\widetilde {\eta }}$ commutes with small colimits, we show ${\widetilde {\eta }}$ is a left-adjoint (to some functor). Define ${\mathcal {H}}om(\eta ,-):D\to {\widehat {C}}$ to be the functor given by: for each object M in D and each object U in C,
${\mathcal {H}}om(\eta ,M)(U)=\operatorname {Hom} _{D}(\eta U,M).$
Then, for each object M in D, since ${\mathcal {H}}om(\eta ,M)(U_{i})=\operatorname {Hom} (yU_{i},{\mathcal {H}}om(\eta ,M))$ by the Yoneda lemma, we have:
${\begin{aligned}\operatorname {Hom} _{D}({\widetilde {\eta }}F,M)&=\operatorname {Hom} _{D}(\varinjlim \eta U_{i},M)=\varprojlim \operatorname {Hom} _{D}(\eta U_{i},M)=\varprojlim {\mathcal {H}}om(\eta ,M)(U_{i})\\&=\operatorname {Hom} _{\widehat {C}}(F,{\mathcal {H}}om(\eta ,M)),\end{aligned}}$
which is to say ${\widetilde {\eta }}$ is a left-adjoint to ${\mathcal {H}}om(\eta ,-)$. $\square $
The proposition yields several corollaries. For example, the proposition implies that the construction $C\mapsto {\widehat {C}}$ is functorial: i.e., each functor $C\to D$ determines the functor ${\widehat {C}}\to {\widehat {D}}$.
Variants
A presheaf of spaces on an ∞-category C is a contravariant functor from C to the ∞-category of spaces (for example, the nerve of the category of CW-complexes.)[4] It is an ∞-category version of a presheaf of sets, as a "set" is replaced by a "space". The notion is used, among other things, in the ∞-category formulation of Yoneda's lemma that says: $C\to PShv(C)$ is fully faithful (here C can be just a simplicial set.)[5]
See also
• Topos
• Category of elements
• Simplicial presheaf (this notion is obtained by replacing "set" with "simplicial set")
• Presheaf with transfers
Notes
1. co-Yoneda lemma at the nLab
2. Kashiwara & Schapira 2005, Corollary 2.4.3.
3. Kashiwara & Schapira 2005, Proposition 2.7.1.
4. Lurie, Definition 1.2.16.1.
5. Lurie, Proposition 5.1.3.1.
References
• Kashiwara, Masaki; Schapira, Pierre (2005). Categories and sheaves. Grundlehren der mathematischen Wissenschaften. Vol. 332. Springer. ISBN 978-3-540-27950-1.
• Lurie, J. Higher Topos Theory.
• Mac Lane, Saunders; Moerdijk, Ieke (1992). Sheaves in Geometry and Logic. Springer. ISBN 0-387-97710-4.
Further reading
• Presheaf at the nLab
• Free cocompletion at the nLab
• Daniel Dugger, Sheaves and Homotopy Theory, the pdf file provided by nlab.
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Yoneda product
In algebra, the Yoneda product (named after Nobuo Yoneda) is the pairing between Ext groups of modules:
$\operatorname {Ext} ^{n}(M,N)\otimes \operatorname {Ext} ^{m}(L,M)\to \operatorname {Ext} ^{n+m}(L,N)$
induced by
$\operatorname {Hom} (N,M)\otimes \operatorname {Hom} (M,L)\to \operatorname {Hom} (N,L),\,f\otimes g\mapsto g\circ f.$
Specifically, for an element $\xi \in \operatorname {Ext} ^{n}(M,N)$, thought of as an extension
$\xi :0\rightarrow N\rightarrow E_{0}\rightarrow \cdots \rightarrow E_{n-1}\rightarrow M\rightarrow 0,$
and similarly
$\rho :0\rightarrow M\rightarrow F_{0}\rightarrow \cdots \rightarrow F_{m-1}\rightarrow L\rightarrow 0\in \operatorname {Ext} ^{m}(L,M),$
we form the Yoneda (cup) product
$\xi \smile \rho :0\rightarrow N\rightarrow E_{0}\rightarrow \cdots \rightarrow E_{n-1}\rightarrow F_{0}\rightarrow \cdots \rightarrow F_{m-1}\rightarrow L\rightarrow 0\in \operatorname {Ext} ^{m+n}(L,N).$
Note that the middle map $E_{n-1}\rightarrow F_{0}$ factors through the given maps to $M$.
We extend this definition to include $m,n=0$ using the usual functoriality of the $\operatorname {Ext} ^{*}(\cdot ,\cdot )$ groups.
Applications
Ext Algebras
Given a commutative ring $R$ and a module $M$, the Yoneda product defines a product structure on the groups ${\text{Ext}}^{\bullet }(M,M)$, where ${\text{Ext}}^{0}(M,M)={\text{Hom}}_{R}(M,M)$ is generally a non-commutative ring. This can be generalized to the case of sheaves of modules over a ringed space, or ringed topos.
Grothendieck duality
In Grothendieck's duality theory of coherent sheaves on a projective scheme $i:X\hookrightarrow \mathbb {P} _{k}^{n}$ of pure dimension $r$ over an algebraically closed field $k$, there is a pairing
${\text{Ext}}_{{\mathcal {O}}_{X}}^{p}({\mathcal {O}}_{X},{\mathcal {F}})\times {\text{Ext}}_{{\mathcal {O}}_{X}}^{r-p}({\mathcal {F}},\omega _{X}^{\bullet })\to k$
where $\omega _{X}$ is the dualizing complex $\omega _{X}={\mathcal {Ext}}_{{\mathcal {O}}_{\mathbb {P} }}^{n-r}(i_{*}{\mathcal {F}},\omega _{\mathbb {P} })$ and $\omega _{\mathbb {P} }={\mathcal {O}}_{\mathbb {P} }(-(n+1))$ given by the Yoneda pairing.[1]
Deformation theory
The Yoneda product is useful for understanding the obstructions to a deformation of maps of ringed topoi.[2] For example, given a composition of ringed topoi
$X\xrightarrow {f} Y\to S$
and an $S$-extension $j:Y\to Y'$ of $Y$ by an ${\mathcal {O}}_{Y}$-module $J$, there is an obstruction class
$\omega (f,j)\in {\text{Ext}}^{2}(\mathbf {L} _{X/Y},f^{*}J)$
which can be described as the yoneda product
$\omega (f,j)=f^{*}(e(j))\cdot K(X/Y/S)$
where
${\begin{aligned}K(X/Y/S)&\in {\text{Ext}}^{1}(\mathbf {L} _{X/Y},\mathbf {L} _{Y/S})\\f^{*}(e(j))&\in {\text{Ext}}^{1}(f^{*}\mathbf {L} _{Y/S},f^{*}J)\end{aligned}}$
and $\mathbf {L} _{X/Y}$ corresponds to the cotangent complex.
See also
• Ext functor
• Derived category
• Deformation theory
• Kodaira–Spencer map
References
1. Altman; Kleiman (1970). Grothendieck Duality. Lecture Notes in Mathematics. Vol. 146. p. 5. doi:10.1007/BFb0060932. ISBN 978-3-540-04935-7.
2. Illusie, Luc. "Complexe cotangent; application a la theorie des deformations" (PDF). p. 163.
• May, J. Peter. "Notes on Tor and Ext" (PDF).
External links
• Universality of Ext functor using Yoneda extensions
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Joram Lindenstrauss
Joram Lindenstrauss (Hebrew: יורם לינדנשטראוס) (October 28, 1936 – April 29, 2012) was an Israeli mathematician working in functional analysis. He was a professor of mathematics at the Einstein Institute of Mathematics.[1]
Joram Lindenstrauss
יורם לינדנשטראוס
Joram Lindenstrauss, 1975
Born(1936-10-28)October 28, 1936
Tel Aviv, Mandatory Palestine
DiedApril 29, 2012(2012-04-29) (aged 75)
Resting placeHar HaMenuchot
Alma materHebrew University of Jerusalem
AwardsIsrael Prize (1981)
Scientific career
InstitutionsEinstein Institute of Mathematics
Doctoral advisorsAryeh Dvoretzky
Branko Grünbaum
Doctoral studentsAssaf Naor, Gideon Schechtman
Biography
Joram Lindenstrauss was born in Tel Aviv.[2] He was the only child of a pair of lawyers who immigrated to Israel from Berlin. He began to study mathematics at the Hebrew University of Jerusalem in 1954 while serving in the army. He became a full-time student in 1956 and received his master's degree in 1959. In 1962 Lindenstrauss earned his Ph.D. from the Hebrew University (dissertation: Extension of Compact Operators, advisors: Aryeh Dvoretzky, Branko Grünbaum).[3] He worked as a postdoc at Yale University and the University of Washington in Seattle from 1962 - 1965. He was appointed senior lecturer at the Hebrew University in 1965, associate professor on 1967 and full professor in 1969. He became the Leon H. and Ada G. Miller Memorial Professor of Mathematics in 1985.[2] He retired in 2005.
Lindenstrauss was married to theoretical computer scientist Naomi Lindenstrauss. Two of their children, Ayelet Lindenstrauss and Fields Medallist Elon Lindenstrauss, are also mathematicians (providing a rare example of father, mother, son and daughter all having papers listed in Mathematical Reviews).[2] Joram was also the cousin of Micha Lindenstrauss.
Research
Lindenstrauss worked in various areas of functional analysis and geometry,[4] particularly Banach space theory, finite- and infinite-dimensional convexity, geometric nonlinear functional analysis and geometric measure theory.[2] He authored more than 100 papers as well as several books in Banach space theory.[5]
Among his results is the Johnson–Lindenstrauss lemma which concerns low-distortion embeddings of points from high-dimensional into low-dimensional Euclidean space. Another of his theorems states that in a Banach space with the Radon–Nikodym property, a closed and bounded set has an extreme point; compactness is not needed.[6]
Awards
In 1981 Lindenstrauss was awarded the Israel Prize, for mathematics.[7] In 1997, Lindenstrauss was the first mathematician from outside Poland to be awarded the Stefan Banach Medal of the Polish Academy of Sciences.[8]
Published works
• Classical Banach spaces I (with Lior Tzafriri). Springer-Verlag, 1977.
• Classical Banach spaces II (with Lior Tzafriri). Springer-Verlag, 1979.
• Banach spaces with a unique unconditional basis, up to permutation (with Jean Bourgain, Peter George Casazza, and Lior Tzafriri). Memoirs of the American Mathematical Society, vol 322. American Mathematical Society, 1985
• Geometric nonlinear functional analysis (with Yoav Benyamini). Colloquium publications, 48. American Mathematical Society, 2000.[9]
• Handbook of the geometry of Banach spaces (Edited, with William B. Johnson). Elsevier, Vol. 1 (2001), Vol. 2 (2003).
See also
• List of Israel Prize recipients
References
1. Professors emeriti, Einstein Institute of Mathematics, http://www.math.huji.ac.il/#news
2. "Joram Lindenstrauss CV" (PDF).
3. Joram Lindenstrauss at Mathematics Genealogy
4. A biographical sketch from the book "Classical Banach Spaces"
5. "MathSciNet author profile".
6. Artstein (1980, p. 173): Artstein, Zvi (1980). "Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points". SIAM Review. 22 (2): 172–185. doi:10.1137/1022026. JSTOR 2029960. MR 0564562.
7. "Israel Prize Official Site - Recipients in 1981 (in Hebrew)".
8. "Stefan Banach Medal". Polish Academy of Sciences. Archived from the original on 2020-11-05.
9. Virtual display of books written by members of the Einstein Institute of Mathematics
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Yoshie Katsurada
Yoshie Katsurada (Japanese: 桂田 芳枝, 3 September 1911 – 10 May 1980)[1][2] was a Japanese mathematician specializing in differential geometry.[3] She became the first Japanese woman to earn a doctorate in mathematics, in 1950, and the first to obtain an imperial university professorship in mathematics, in 1967.[4]
Life
Katsurada was born in Akaigawa, Hokkaido on 3 September 1911, a daughter of an elementary school principal. In high school in Otaru, she took special instruction in mathematics from a boys' mathematics instructor. Graduating from high school in 1929, she began auditing classes at the Tokyo Physics School, a predecessor to the Tokyo University of Science, in 1931.[1]
She began working as an administrative assistant in the Hokkaido University Department of Mathematics in 1936. In 1938 she began study in mathematics at Tokyo Woman's Christian University, withdrawing in 1940 to transfer to Hokkaido University. She graduated from Hokkaido University in 1942, and in the same year became an assistant professor there.[1]
In 1950, she completed a doctorate in mathematics at Hokkaido University, under the supervision of Shoji Kawaguchi,[1] becoming the first Japanese woman to earn a doctorate in mathematics,[1][4] and earning a promotion to associate professor.[1] She remained at Hokkaido University for the remainder of her career, with research visits to Sapienza University of Rome, ETH Zurich, and the University of California, Berkeley. She was promoted to full professor in 1967, the first female professor in mathematics at a former imperial university.[1]
She retired in 1975,[1] and died on 10 May 1980.[2]
Research
Katsurada's early research, from the beginning of her studies into the mid-1950s, primarily concerned line elements; this was the primary interest of her advisor Shoji Kawaguchi, with whom she continued to collaborate on this subject. After visiting Heinz Hopf at ETH Zurich in 1957–1958, she shifted interests to submanifolds and hypersurfaces in Riemannian manifolds, publishing well-regarded work in this area.[3]
Recognition
Several papers in the 1972 volume of the Hokkaido Mathematical Journal are dedicated to Katsurada in honor of her 60th birthday. Katsurada was given the Hokkaido Culture Award in 1973.[1]
References
1. 桂田芳枝 (Yoshie Katsurada, 1911-1980) (in Japanese), Department of Mathematics, Hokkaido University, retrieved 2021-08-30
2. "Deaths" (PDF), Notices of the American Mathematical Society, 28 (1): 98, January 1981
3. Kobayashi, Shoshichi (1996), "Differential Geometry in Japan in the 1940s and '50s", 総合講演, 企画特別講演アブストラクト: 9–18, doi:10.11429/emath1996.1996.1996_9; see in particular p. 13
4. Frédéric, Louis (2002), "Katsurada Yoshie", Japan Encyclopedia, translated by Roth, Käthe, The Belknap Press of Harvard University Press, p. 494
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Yoshiharu Kohayakawa
Yoshiharu Kohayakawa (Japanese: 小早川美晴; born 1963) is a Japanese-Brazilian mathematician working on discrete mathematics and probability theory.[1] He is known for his work on Szemerédi's regularity lemma, which he extended to sparser graphs.[2][3]
Biography
Kohayakawa was a student of Béla Bollobás at the University of Cambridge.[4]
According to Google Scholar, as of August 21, 2019, Kohayakawa's works have been cited over 3194 times, and his h-index is 33.[5]
He is a titular member of the Brazilian Academy of Sciences.[1]
In 2000, five American researchers received a USA NSF Research Grant in the value of $20,000 to go to Brazil to work in collaboration with him on mathematical problems.[6]
Kohayakawa has an Erdős number of 1.[7][8]
He was awarded the 2018 Fulkerson Prize.
References
1. Brazilian Academy of Sciences – Yoshiharu Kohayakawa Archived May 24, 2015, at the Wayback Machine
2. László Lovász – Large Networks and Graph Limits, p. 395
3. Bridget S. Webb – Surveys in Combinatorics 2005, p. 227
4. Mathematics Genealogy Project – Yoshiharu Kohayakawa
5. Google Scholar Profile – Yoshiharu Kohayakawa
6. U.S.-Brazil Cooperative Research: Problems on Random Graphs (Structures) and Set Systems: NSG GRANT 0072064
7. Celina Miraglia Herrera – My Erdős number
8. He wrote "The size of the largest bipartite subgraphs", on Discrete Mathematics with Erdős and Gyárfás
External links
• Home Page of Yoshiharu Kohayakawa at the University of São Paulo
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Yoshio Shimamoto
Yoshio Shimamoto was a nuclear physicist who also did work in mathematics and computer science. While at Brookhaven National Laboratory (1954-1987),[1] he designed the logic for the MERLIN digital computer in 1958, and served as chairman of the Applied Mathematics Department from 1964 to 1975. Shimamoto researched in combinatorial mathematics, the economics of outer continental shelf oil and gas lease sales (on behalf of the U.S. Geological Survey), the architecture of supercomputers, and the linking of computers for parallel processing.[2]
During the 1970s, he worked with Heinrich Heesch and Karl Durre on methods for a computer-aided proof of the four color theorem, using computer programs to apply Heesch's notion of "discharging" to eliminate 4-colorable cases. A proof of the Four Color Theorem, which he presented in 1971, was later shown to be flawed, but it served as the basis for further work.
Born in Hawaii in 1924, Shimamoto served with the U.S. Army Signal Corps and Strategic Bombing Survey in Japan, during World War II. He died in New Jersey on August 27, 2009.[2]
References
1. "Yoshio Shimamoto Obituary - New York, NY | New York Times", New York Times
2. "Paid notice: Deaths – Shimamoto , Yoshio", New York Times, August 31, 2009
Authority control: Academics
• MathSciNet
• zbMATH
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You-Dong Liang
You-Dong Liang (梁友栋) is a mathematician and educator, best known for his contributions in geometric modeling and the Liang-Barsky algorithm.
Education and teaching
You-Dong Liang was born on July 19, 1935, in Fuzhou, Fujian Province, China. Liang pursued his graduate degree in Fudan University, where he worked under the supervision of Professor Su Buqing and specialized in geometric theory. After graduating in 1960, he joined the mathematics teaching faculty at Zhejiang University, where he actively promoted the development of geometric design and graphics. From 1984–1990, he was the chairman of the mathematics department, and on several occasions, was a visiting scholar and visiting professor at the University of California at Berkeley, University of Utah, and University of Berlin. Liang helped form the Computational Geometry Collaborative Group in China. As the leader of this group, Liang supported the collaboration of scholars in geometric design and computational graphics. Liang was awarded the “Chinese Geometric Design and Calculation Contribution Award” in 2009.[1]
Contributions, papers, and awards
Liang has worked on computer-aided geometric design and computer graphics research. In 1984, Liang developed the Liang–Barsky algorithm, which has applications in computer graphics.[2] Liang made further improvements on this algorithm in 1992.[3] In the late 1980s and early 1990s, Liang proposed a series of theories and methodologies in geometric continuity. In 1991, Liang supervised the completion of "Generated Computer Graphics and Geometric Modeling Research" and was awarded the Chinese National Science Third Prize. During these years, Liang received other honors, including the Chinese Academy of Sciences Award and the “European Graphics Conference” Best Paper Award. Liang has published more than 50 papers, including "A New Concept and Method for Line Clipping", "Some Theorems on Geometrical Objects", "Curve and Surface Geometry Continuity", and "An Analysis and Algorithm for Polygon Clipping."
References
1. "CSIAM » 杰出贡献奖". cg.cs.tsinghua.edu.cn. Archived from the original on 2015-07-15.
2. Liang, YD, BA, Barsky, and M. Slater, Some Improvements to a Parametric Line Clipping Algorithm, CSD-92-688, Computer Science Division, University of California, Berkeley, 1992.
3. Liang, YD, and Barsky, B., "A New Concept and Method for Line Clipping", ACM Transactions on Graphics, 3 (1): 1–22, January 1984.
Authority control: Academics
• MathSciNet
• zbMATH
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Youden's J statistic
Youden's J statistic (also called Youden's index) is a single statistic that captures the performance of a dichotomous diagnostic test. (Bookmaker) Informedness is its generalization to the multiclass case and estimates the probability of an informed decision.
Definition
Youden's J statistic is
$J={\text{sensitivity}}+{\text{specificity}}-1={\text{recall}}_{1}+{\text{recall}}_{0}-1$
with the two right-hand quantities being sensitivity and specificity. Thus the expanded formula is:
$J={\frac {\text{true positives}}{{\text{true positives}}+{\text{false negatives}}}}+{\frac {\text{true negatives}}{{\text{true negatives}}+{\text{false positives}}}}-1$
The index was suggested by W. J. Youden in 1950[1] as a way of summarising the performance of a diagnostic test, however the formula was earlier published in Science by C. S. Pierce in 1884.[2] Its value ranges from -1 through 1 (inclusive),[1] and has a zero value when a diagnostic test gives the same proportion of positive results for groups with and without the disease, i.e the test is useless. A value of 1 indicates that there are no false positives or false negatives, i.e. the test is perfect. The index gives equal weight to false positive and false negative values, so all tests with the same value of the index give the same proportion of total misclassified results. While it is possible to obtain a value of less than zero from this equation, e.g. Classification yields only False Positives and False Negatives, a value of less than zero just indicates that the positive and negative labels have been switched. After correcting the labels the result will then be in the 0 through 1 range.
Youden's index is often used in conjunction with receiver operating characteristic (ROC) analysis.[3] The index is defined for all points of an ROC curve, and the maximum value of the index may be used as a criterion for selecting the optimum cut-off point when a diagnostic test gives a numeric rather than a dichotomous result. The index is represented graphically as the height above the chance line, and it is also equivalent to the area under the curve subtended by a single operating point.[4]
Youden's index is also known as deltaP' [5] and generalizes from the dichotomous to the multiclass case as informedness.[4]
The use of a single index is "not generally to be recommended",[6] but informedness or Youden's index is the probability of an informed decision (as opposed to a random guess) and takes into account all predictions.[4]
An unrelated but commonly used combination of basic statistics from information retrieval is the F-score, being a (possibly weighted) harmonic mean of recall and precision where recall = sensitivity = true positive rate, but specificity and precision are totally different measures. F-score, like recall and precision, only considers the so-called positive predictions, with recall being the probability of predicting just the positive class, precision being the probability of a positive prediction being correct, and F-score equating these probabilities under the effective assumption that the positive labels and the positive predictions should have the same distribution and prevalence,[4] similar to the assumption underlying of Fleiss' kappa. Youden's J, Informedness, Recall, Precision and F-score are intrinsically undirectional, aiming to assess the deductive effectiveness of predictions in the direction proposed by a rule, theory or classifier. Markedness (deltaP) is Youden's J used to assess the reverse or abductive direction,[4][7] and matches well human learning of associations; rules and, superstitions as we model possible causation;[5] while correlation and kappa evaluate bidirectionally.
Matthews correlation coefficient is the geometric mean of the regression coefficient of the problem and its dual, where the component regression coefficients of the Matthews correlation coefficient are Markedness (inverse of Youden's J or deltaP) and informedness (Youden's J or deltaP'). Kappa statistics such as Fleiss' kappa and Cohen's kappa are methods for calculating inter-rater reliability based on different assumptions about the marginal or prior distributions, and are increasingly used as chance corrected alternatives to accuracy in other contexts. Fleiss' kappa, like F-score, assumes that both variables are drawn from the same distribution and thus have the same expected prevalence, while Cohen's kappa assumes that the variables are drawn from distinct distributions and referenced to a model of expectation that assumes prevalences are independent.[7]
When the true prevalences for the two positive variables are equal as assumed in Fleiss kappa and F-score, that is the number of positive predictions matches the number of positive classes in the dichotomous (two class) case, the different kappa and correlation measure collapse to identity with Youden's J, and recall, precision and F-score are similarly identical with accuracy.[4][7]
References
1. Youden, W.J. (1950). "Index for rating diagnostic tests". Cancer. 3: 32–35. doi:10.1002/1097-0142(1950)3:1<32::aid-cncr2820030106>3.0.co;2-3. PMID 15405679.
2. Pierce, C.S. (1884). "The numerical measure of the success of predictions". Science. 4 (93): 453–454. doi:10.1126/science.ns-4.93.453.b.
3. Schisterman, E.F.; Perkins, N.J.; Liu, A.; Bondell, H. (2005). "Optimal cut-point and its corresponding Youden Index to discriminate individuals using pooled blood samples". Epidemiology. 16 (1): 73–81. doi:10.1097/01.ede.0000147512.81966.ba. PMID 15613948.
4. Powers, David M W (2011). "Evaluation: From Precision, Recall and F-Score to ROC, Informedness, Markedness & Correlation". Journal of Machine Learning Technologies. 2 (1): 37–63. hdl:2328/27165.
5. Perruchet, P.; Peereman, R. (2004). "The exploitation of distributional information in syllable processing". J. Neurolinguistics. 17 (2–3): 97–119. doi:10.1016/s0911-6044(03)00059-9.
6. Everitt B.S. (2002) The Cambridge Dictionary of Statistics. CUP ISBN 0-521-81099-X
7. Powers, David M W (2012). The Problem with Kappa. Conference of the European Chapter of the Association for Computational Linguistics. pp. 345–355. hdl:2328/27160.
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Hesse configuration
In geometry, the Hesse configuration is a configuration of 9 points and 12 lines with three points per line and four lines through each point. It can be realized in the complex projective plane as the set of inflection points of an elliptic curve, but it has no realization in the Euclidean plane. It was introduced by Colin Maclaurin and studied by Hesse (1844),[1] and is also known as Young's geometry,[2] named after the later work of John Wesley Young on finite geometry.[3][4]
Description
The Hesse configuration has the same incidence relations as the lines and points of the affine plane over the field of 3 elements. That is, the points of the Hesse configuration may be identified with ordered pairs of numbers modulo 3, and the lines of the configuration may correspondingly be identified with the triples of points (x, y) satisfying a linear equation ax + by = c (mod 3). Alternatively, the points of the configuration may be identified by the squares of a tic-tac-toe board, and the lines may be identified with the lines and broken diagonals of the board.
Each point belongs to four lines: in the tic tac toe interpretation of the configuration, one line is horizontal, one vertical, and two are diagonals or broken diagonals. Each line contains three points. In the language of configurations the Hesse configuration has the notation 94123, meaning that there are 9 points, 4 lines per point, 12 lines, and 3 points per line.
The Hesse configuration has 216 symmetries (its automorphism group has order 216). The group of its symmetries is known as the Hessian group.
Related configurations
Removing any one point and its four incident lines from the Hesse configuration produces another configuration of type 8383, the Möbius–Kantor configuration.[5][6][7]
In the Hesse configuration, the 12 lines may be grouped into four triples of parallel (non-intersecting) lines. Removing from the Hesse configuration the three lines belonging to a single triple produces a configuration of type 9393, the Pappus configuration.[6][7]
The Hesse configuration may in turn be augmented by adding four points, one for each triple of non-intersecting lines, and one line containing the four new points, to form a configuration of type 134134, the set of points and lines of the projective plane over the three-element field.
Realizability
The Hesse configuration can be realized in the complex projective plane as the 9 inflection points of an elliptic curve and the 12 lines through triples of inflection points.[3] If a given set of nine points in the complex plane is the set of inflections of an elliptic curve C, it is also the set of inflections of every curve in a pencil of curves generated by C and by the Hessian curve of C, the Hesse pencil.[8]
The Hessian polyhedron is a representation of the Hesse configuration in the complex plane.
The Hesse configuration shares with the Möbius–Kantor configuration the property of having a complex realization but not being realizable by points and straight lines in the Euclidean plane. In the Hesse configuration, every two points are connected by a line of the configuration (the defining property of the Sylvester–Gallai configurations) and therefore every line through two of its points contains a third point. But in the Euclidean plane, every finite set of points is either collinear, or includes a pair of points whose line does not contain any other points of the set; this is the Sylvester–Gallai theorem. Because the Hesse configuration disobeys the Sylvester–Gallai theorem, it has no Euclidean realization. This example also shows that the Sylvester–Gallai theorem cannot be generalized to the complex projective plane. However, in complex spaces, the Hesse configuration and all Sylvester–Gallai configurations must lie within a two-dimensional flat subspace.[9]
References
1. Hesse, O. (1844), "Über die Elimination der Variabeln aus drei algebraischen Gleichungen vom zweiten Grade mit zwei Variabeln" (PDF), Journal für die Reine und Angewandte Mathematik (in German), 28: 68–96, doi:10.1515/crll.1844.28.68, ISSN 0075-4102.
2. Wallace, Edward C.; West, Stephen F. (2015), Roads to Geometry (3rd ed.), Waveland Press, pp. 23–24, ISBN 9781478632047
3. MacNeish, H. F. (1942), "Four finite geometries", The American Mathematical Monthly, 49: 15–23, doi:10.2307/2303772, MR 0005625
4. Veblen, Oswald; Young, John Wesley (1910), Projective Geometry, vol. I, Ginn and Company, p. 249
5. Dolgachev, Igor V. (2004), "Abstract configurations in algebraic geometry", The Fano Conference, Univ. Torino, Turin, pp. 423–462, arXiv:math.AG/0304258, MR 2112585.
6. Coxeter, H. S. M. (1950), "Self-dual configurations and regular graphs", Bulletin of the American Mathematical Society, 56 (5): 413–455, doi:10.1090/S0002-9904-1950-09407-5.
7. Cullinane, Steven H. (2011), Configurations and squares.
8. Artebani, Michela; Dolgachev, Igor (2009), "The Hesse pencil of plane cubic curves", L'Enseignement Mathématique, 2e Série, 55 (3): 235–273, arXiv:math/0611590, doi:10.4171/lem/55-3-3, MR 2583779.
9. Elkies, Noam; Pretorius, Lou M.; Swanepoel, Konrad J. (2006), "Sylvester–Gallai theorems for complex numbers and quaternions", Discrete and Computational Geometry, 35 (3): 361–373, arXiv:math/0403023, doi:10.1007/s00454-005-1226-7, MR 2202107.
Incidence structures
Representation
• Incidence matrix
• Incidence graph
Fields
• Combinatorics
• Block design
• Steiner system
• Geometry
• Incidence
• Projective plane
• Graph theory
• Hypergraph
• Statistics
• Blocking
Configurations
• Complete quadrangle
• Fano plane
• Möbius–Kantor configuration
• Pappus configuration
• Hesse configuration
• Desargues configuration
• Reye configuration
• Schläfli double six
• Cremona–Richmond configuration
• Kummer configuration
• Grünbaum–Rigby configuration
• Klein configuration
• Dual
Theorems
• Sylvester–Gallai theorem
• De Bruijn–Erdős theorem
• Szemerédi–Trotter theorem
• Beck's theorem
• Bruck–Ryser–Chowla theorem
Applications
• Design of experiments
• Kirkman's schoolgirl problem
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Young tableau
In mathematics, a Young tableau (/tæˈbloʊ, ˈtæbloʊ/; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups and to study their properties. Young tableaux were introduced by Alfred Young, a mathematician at Cambridge University, in 1900.[1][2] They were then applied to the study of the symmetric group by Georg Frobenius in 1903. Their theory was further developed by many mathematicians, including Percy MacMahon, W. V. D. Hodge, G. de B. Robinson, Gian-Carlo Rota, Alain Lascoux, Marcel-Paul Schützenberger and Richard P. Stanley.
Definitions
Note: this article uses the English convention for displaying Young diagrams and tableaux.
Diagrams
A Young diagram (also called a Ferrers diagram, particularly when represented using dots) is a finite collection of boxes, or cells, arranged in left-justified rows, with the row lengths in non-increasing order. Listing the number of boxes in each row gives a partition λ of a non-negative integer n, the total number of boxes of the diagram. The Young diagram is said to be of shape λ, and it carries the same information as that partition. Containment of one Young diagram in another defines a partial ordering on the set of all partitions, which is in fact a lattice structure, known as Young's lattice. Listing the number of boxes of a Young diagram in each column gives another partition, the conjugate or transpose partition of λ; one obtains a Young diagram of that shape by reflecting the original diagram along its main diagonal.
There is almost universal agreement that in labeling boxes of Young diagrams by pairs of integers, the first index selects the row of the diagram, and the second index selects the box within the row. Nevertheless, two distinct conventions exist to display these diagrams, and consequently tableaux: the first places each row below the previous one, the second stacks each row on top of the previous one. Since the former convention is mainly used by Anglophones while the latter is often preferred by Francophones, it is customary to refer to these conventions respectively as the English notation and the French notation; for instance, in his book on symmetric functions, Macdonald advises readers preferring the French convention to "read this book upside down in a mirror" (Macdonald 1979, p. 2). This nomenclature probably started out as jocular. The English notation corresponds to the one universally used for matrices, while the French notation is closer to the convention of Cartesian coordinates; however, French notation differs from that convention by placing the vertical coordinate first. The figure on the right shows, using the English notation, the Young diagram corresponding to the partition (5, 4, 1) of the number 10. The conjugate partition, measuring the column lengths, is (3, 2, 2, 2, 1).
Arm and leg length
In many applications, for example when defining Jack functions, it is convenient to define the arm length aλ(s) of a box s as the number of boxes to the right of s in the diagram λ in English notation. Similarly, the leg length lλ(s) is the number of boxes below s. The hook length of a box s is the number of boxes to the right of s or below s in English notation, including the box s itself; in other words, the hook length is aλ(s) + lλ(s) + 1.
Tableaux
A Young tableau is obtained by filling in the boxes of the Young diagram with symbols taken from some alphabet, which is usually required to be a totally ordered set. Originally that alphabet was a set of indexed variables x1, x2, x3..., but now one usually uses a set of numbers for brevity. In their original application to representations of the symmetric group, Young tableaux have n distinct entries, arbitrarily assigned to boxes of the diagram. A tableau is called standard if the entries in each row and each column are increasing. The number of distinct standard Young tableaux on n entries is given by the involution numbers
1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, ... (sequence A000085 in the OEIS).
In other applications, it is natural to allow the same number to appear more than once (or not at all) in a tableau. A tableau is called semistandard, or column strict, if the entries weakly increase along each row and strictly increase down each column. Recording the number of times each number appears in a tableau gives a sequence known as the weight of the tableau. Thus the standard Young tableaux are precisely the semistandard tableaux of weight (1,1,...,1), which requires every integer up to n to occur exactly once.
In a standard Young tableau, the integer $k$ is a descent if $k+1$ appears in a row strictly below $k$. The sum of the descents is called the major index of the tableau.[3]
Variations
There are several variations of this definition: for example, in a row-strict tableau the entries strictly increase along the rows and weakly increase down the columns. Also, tableaux with decreasing entries have been considered, notably, in the theory of plane partitions. There are also generalizations such as domino tableaux or ribbon tableaux, in which several boxes may be grouped together before assigning entries to them.
Skew tableaux
A skew shape is a pair of partitions (λ, μ) such that the Young diagram of λ contains the Young diagram of μ; it is denoted by λ/μ. If λ = (λ1, λ2, ...) and μ = (μ1, μ2, ...), then the containment of diagrams means that μi ≤ λi for all i. The skew diagram of a skew shape λ/μ is the set-theoretic difference of the Young diagrams of λ and μ: the set of squares that belong to the diagram of λ but not to that of μ. A skew tableau of shape λ/μ is obtained by filling the squares of the corresponding skew diagram; such a tableau is semistandard if entries increase weakly along each row, and increase strictly down each column, and it is standard if moreover all numbers from 1 to the number of squares of the skew diagram occur exactly once. While the map from partitions to their Young diagrams is injective, this is not the case for the map from skew shapes to skew diagrams;[4] therefore the shape of a skew diagram cannot always be determined from the set of filled squares only. Although many properties of skew tableaux only depend on the filled squares, some operations defined on them do require explicit knowledge of λ and μ, so it is important that skew tableaux do record this information: two distinct skew tableaux may differ only in their shape, while they occupy the same set of squares, each filled with the same entries.[5] Young tableaux can be identified with skew tableaux in which μ is the empty partition (0) (the unique partition of 0).
Any skew semistandard tableau T of shape λ/μ with positive integer entries gives rise to a sequence of partitions (or Young diagrams), by starting with μ, and taking for the partition i places further in the sequence the one whose diagram is obtained from that of μ by adding all the boxes that contain a value ≤ i in T; this partition eventually becomes equal to λ. Any pair of successive shapes in such a sequence is a skew shape whose diagram contains at most one box in each column; such shapes are called horizontal strips. This sequence of partitions completely determines T, and it is in fact possible to define (skew) semistandard tableaux as such sequences, as is done by Macdonald (Macdonald 1979, p. 4). This definition incorporates the partitions λ and μ in the data comprising the skew tableau.
Overview of applications
Young tableaux have numerous applications in combinatorics, representation theory, and algebraic geometry. Various ways of counting Young tableaux have been explored and lead to the definition of and identities for Schur functions.
Many combinatorial algorithms on tableaux are known, including Schützenberger's jeu de taquin and the Robinson–Schensted–Knuth correspondence. Lascoux and Schützenberger studied an associative product on the set of all semistandard Young tableaux, giving it the structure called the plactic monoid (French: le monoïde plaxique).
In representation theory, standard Young tableaux of size k describe bases in irreducible representations of the symmetric group on k letters. The standard monomial basis in a finite-dimensional irreducible representation of the general linear group GLn are parametrized by the set of semistandard Young tableaux of a fixed shape over the alphabet {1, 2, ..., n}. This has important consequences for invariant theory, starting from the work of Hodge on the homogeneous coordinate ring of the Grassmannian and further explored by Gian-Carlo Rota with collaborators, de Concini and Procesi, and Eisenbud. The Littlewood–Richardson rule describing (among other things) the decomposition of tensor products of irreducible representations of GLn into irreducible components is formulated in terms of certain skew semistandard tableaux.
Applications to algebraic geometry center around Schubert calculus on Grassmannians and flag varieties. Certain important cohomology classes can be represented by Schubert polynomials and described in terms of Young tableaux.
Applications in representation theory
See also: Representation theory of the symmetric group
Young diagrams are in one-to-one correspondence with irreducible representations of the symmetric group over the complex numbers. They provide a convenient way of specifying the Young symmetrizers from which the irreducible representations are built. Many facts about a representation can be deduced from the corresponding diagram. Below, we describe two examples: determining the dimension of a representation and restricted representations. In both cases, we will see that some properties of a representation can be determined by using just its diagram. Young tableaux are involved in the use of the symmetric group in quantum chemistry studies of atoms, molecules and solids.[6][7]
Young diagrams also parametrize the irreducible polynomial representations of the general linear group GLn (when they have at most n nonempty rows), or the irreducible representations of the special linear group SLn (when they have at most n − 1 nonempty rows), or the irreducible complex representations of the special unitary group SUn (again when they have at most n − 1 nonempty rows). In these cases semistandard tableaux with entries up to n play a central role, rather than standard tableaux; in particular it is the number of those tableaux that determines the dimension of the representation.
Dimension of a representation
Main article: Hook length formula
Hook-lengths of the boxes for the partition 10 = 5 + 4 + 1
The dimension of the irreducible representation πλ of the symmetric group Sn corresponding to a partition λ of n is equal to the number of different standard Young tableaux that can be obtained from the diagram of the representation. This number can be calculated by the hook length formula.
A hook length hook(x) of a box x in Young diagram Y(λ) of shape λ is the number of boxes that are in the same row to the right of it plus those boxes in the same column below it, plus one (for the box itself). By the hook-length formula, the dimension of an irreducible representation is n! divided by the product of the hook lengths of all boxes in the diagram of the representation:
$\dim \pi _{\lambda }={\frac {n!}{\prod _{x\in Y(\lambda )}\operatorname {hook} (x)}}.$
The figure on the right shows hook-lengths for all boxes in the diagram of the partition 10 = 5 + 4 + 1. Thus
$\dim \pi _{\lambda }={\frac {10!}{7\cdot 5\cdot 4\cdot 3\cdot 1\cdot 5\cdot 3\cdot 2\cdot 1\cdot 1}}=288.$
Similarly, the dimension of the irreducible representation W(λ) of GLr corresponding to the partition λ of n (with at most r parts) is the number of semistandard Young tableaux of shape λ (containing only the entries from 1 to r), which is given by the hook-length formula:
$\dim W(\lambda )=\prod _{(i,j)\in Y(\lambda )}{\frac {r+j-i}{\operatorname {hook} (i,j)}},$
where the index i gives the row and j the column of a box.[8] For instance, for the partition (5,4,1) we get as dimension of the corresponding irreducible representation of GL7 (traversing the boxes by rows):
$\dim W(\lambda )={\frac {7\cdot 8\cdot 9\cdot 10\cdot 11\cdot 6\cdot 7\cdot 8\cdot 9\cdot 5}{7\cdot 5\cdot 4\cdot 3\cdot 1\cdot 5\cdot 3\cdot 2\cdot 1\cdot 1}}=66528.$
Restricted representations
A representation of the symmetric group on n elements, Sn is also a representation of the symmetric group on n − 1 elements, Sn−1. However, an irreducible representation of Sn may not be irreducible for Sn−1. Instead, it may be a direct sum of several representations that are irreducible for Sn−1. These representations are then called the factors of the restricted representation (see also induced representation).
The question of determining this decomposition of the restricted representation of a given irreducible representation of Sn, corresponding to a partition λ of n, is answered as follows. One forms the set of all Young diagrams that can be obtained from the diagram of shape λ by removing just one box (which must be at the end both of its row and of its column); the restricted representation then decomposes as a direct sum of the irreducible representations of Sn−1 corresponding to those diagrams, each occurring exactly once in the sum.
See also
• Robinson–Schensted correspondence
• Schur–Weyl duality
Notes
1. Knuth, Donald E. (1973), The Art of Computer Programming, Vol. III: Sorting and Searching (2nd ed.), Addison-Wesley, p. 48, Such arrangements were introduced by Alfred Young in 1900.
2. Young, A. (1900), "On quantitative substitutional analysis", Proceedings of the London Mathematical Society, Series 1, 33 (1): 97–145, doi:10.1112/plms/s1-33.1.97. See in particular p. 133.
3. Stembridge, John (1989-12-01). "On the eigenvalues of representations of reflection groups and wreath products". Pacific Journal of Mathematics. Mathematical Sciences Publishers. 140 (2): 353–396. doi:10.2140/pjm.1989.140.353. ISSN 0030-8730.
4. For instance the skew diagram consisting of a single square at position (2,4) can be obtained by removing the diagram of μ = (5,3,2,1) from the one of λ = (5,4,2,1), but also in (infinitely) many other ways. In general any skew diagram whose set of non-empty rows (or of non-empty columns) is not contiguous or does not contain the first row (respectively column) will be associated to more than one skew shape.
5. A somewhat similar situation arises for matrices: the 3-by-0 matrix A must be distinguished from the 0-by-3 matrix B, since AB is a 3-by-3 (zero) matrix while BA is the 0-by-0 matrix, but both A and B have the same (empty) set of entries; for skew tableaux however such distinction is necessary even in cases where the set of entries is not empty.
6. Philip R. Bunker and Per Jensen (1998) Molecular Symmetry and Spectroscopy, 2nd ed. NRC Research Press,Ottawa pp.198-202.ISBN 9780660196282
7. R.Pauncz (1995) The Symmetric Group in Quantum Chemistry, CRC Press, Boca Raton, Florida
8. Predrag Cvitanović (2008). Group Theory: Birdtracks, Lie's, and Exceptional Groups. Princeton University Press., eq. 9.28 and appendix B.4
References
• William Fulton. Young Tableaux, with Applications to Representation Theory and Geometry. Cambridge University Press, 1997, ISBN 0-521-56724-6.
• Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103. Lecture 4
• Howard Georgi, Lie Algebras in Particle Physics, 2nd Edition - Westview
• Macdonald, I. G. Symmetric functions and Hall polynomials. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 1979. viii+180 pp. ISBN 0-19-853530-9 MR553598
• Laurent Manivel. Symmetric Functions, Schubert Polynomials, and Degeneracy Loci. American Mathematical Society.
• Jean-Christophe Novelli, Igor Pak, Alexander V. Stoyanovskii, "A direct bijective proof of the Hook-length formula", Discrete Mathematics and Theoretical Computer Science 1 (1997), pp. 53–67.
• Bruce E. Sagan. The Symmetric Group. Springer, 2001, ISBN 0-387-95067-2
• Vinberg, E.B. (2001) [1994], "Young tableau", Encyclopedia of Mathematics, EMS Press
• Yong, Alexander (February 2007). "What is...a Young Tableau?" (PDF). Notices of the American Mathematical Society. 54 (2): 240–241. Retrieved 2008-01-16.
• Predrag Cvitanović, Group Theory: Birdtracks, Lie's, and Exceptional Groups. Princeton University Press, 2008.
External links
• Eric W. Weisstein. "Ferrers Diagram". From MathWorld—A Wolfram Web Resource.
• Eric W. Weisstein. "Young Tableau." From MathWorld—A Wolfram Web Resource.
• Semistandard tableaux entry in the FindStat database
• Standard tableaux entry in the FindStat database
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Young's convolution inequality
In mathematics, Young's convolution inequality is a mathematical inequality about the convolution of two functions,[1] named after William Henry Young.
Statement
Euclidean space
In real analysis, the following result is called Young's convolution inequality:[2]
Suppose $f$ is in the Lebesgue space $L^{p}(\mathbb {R} ^{d})$ and $g$ is in $L^{q}(\mathbb {R} ^{d})$ and
${\frac {1}{p}}+{\frac {1}{q}}={\frac {1}{r}}+1$
with $1\leq p,q,r\leq \infty .$ Then
$\|f*g\|_{r}\leq \|f\|_{p}\|g\|_{q}.$
Here the star denotes convolution, $L^{p}$ is Lebesgue space, and
$\|f\|_{p}={\Bigl (}\int _{\mathbb {R} ^{d}}|f(x)|^{p}\,dx{\Bigr )}^{1/p}$
denotes the usual $L^{p}$ norm.
Equivalently, if $p,q,r\geq 1$ and $ {\frac {1}{p}}+{\frac {1}{q}}+{\frac {1}{r}}=2$ then
$\left|\int _{\mathbb {R} ^{d}}\int _{\mathbb {R} ^{d}}f(x)g(x-y)h(y)\,\mathrm {d} x\,\mathrm {d} y\right|\leq \left(\int _{\mathbb {R} ^{d}}\vert f\vert ^{p}\right)^{\frac {1}{p}}\left(\int _{\mathbb {R} ^{d}}\vert g\vert ^{q}\right)^{\frac {1}{q}}\left(\int _{\mathbb {R} ^{d}}\vert h\vert ^{r}\right)^{\frac {1}{r}}$
Generalizations
Young's convolution inequality has a natural generalization in which we replace $\mathbb {R} ^{d}$ by a unimodular group $G.$ If we let $\mu $ be a bi-invariant Haar measure on $G$ and we let $f,g:G\to \mathbb {R} $ or $\mathbb {C} $ be integrable functions, then we define $f*g$ by
$f*g(x)=\int _{G}f(y)g(y^{-1}x)\,\mathrm {d} \mu (y).$
Then in this case, Young's inequality states that for $f\in L^{p}(G,\mu )$ and $g\in L^{q}(G,\mu )$ and $p,q,r\in [1,\infty ]$ such that
${\frac {1}{p}}+{\frac {1}{q}}={\frac {1}{r}}+1$
we have a bound
$\lVert f*g\rVert _{r}\leq \lVert f\rVert _{p}\lVert g\rVert _{q}.$
Equivalently, if $p,q,r\geq 1$ and $ {\frac {1}{p}}+{\frac {1}{q}}+{\frac {1}{r}}=2$ then
$\left|\int _{G}\int _{G}f(x)g(y^{-1}x)h(y)\,\mathrm {d} \mu (x)\,\mathrm {d} \mu (y)\right|\leq \left(\int _{G}\vert f\vert ^{p}\right)^{\frac {1}{p}}\left(\int _{G}\vert g\vert ^{q}\right)^{\frac {1}{q}}\left(\int _{G}\vert h\vert ^{r}\right)^{\frac {1}{r}}.$
Since $\mathbb {R} ^{d}$ is in fact a locally compact abelian group (and therefore unimodular) with the Lebesgue measure the desired Haar measure, this is in fact a generalization.
This generalization may be refined. Let $G$ and $\mu $ be as before and assume $1<p,q,r<\infty $ satisfy $ {\tfrac {1}{p}}+{\tfrac {1}{q}}={\tfrac {1}{r}}+1.$ Then there exists a constant $C$ such that for any $f\in L^{p}(G,\mu )$ and any measurable function $g$ on $G$ that belongs to the weak $L^{q}$ space $L^{q,w}(G,\mu ),$ which by definition means that the following supremum
$\|g\|_{q,w}^{q}~:=~\sup _{t>0}\,t^{q}\mu (|g|>t)$
is finite, we have $f*g\in L^{r}(G,\mu )$ and[3]
$\|f*g\|_{r}~\leq ~C\,\|f\|_{p}\,\|g\|_{q,w}.$
Applications
An example application is that Young's inequality can be used to show that the heat semigroup is a contracting semigroup using the $L^{2}$ norm (that is, the Weierstrass transform does not enlarge the $L^{2}$ norm).
Proof
Proof by Hölder's inequality
Young's inequality has an elementary proof with the non-optimal constant 1.[4]
We assume that the functions $f,g,h:G\to \mathbb {R} $ are nonnegative and integrable, where $G$ is a unimodular group endowed with a bi-invariant Haar measure $\mu .$ We use the fact that $\mu (S)=\mu (S^{-1})$ for any measurable $S\subseteq G.$ Since $ p(2-{\tfrac {1}{q}}-{\tfrac {1}{r}})=q(2-{\tfrac {1}{p}}-{\tfrac {1}{r}})=r(2-{\tfrac {1}{p}}-{\tfrac {1}{q}})=1$
${\begin{aligned}&\int _{G}\int _{G}f(x)g(y^{-1}x)h(y)\,\mathrm {d} \mu (x)\,\mathrm {d} \mu (y)\\={}&\int _{G}\int _{G}\left(f(x)^{p}g(y^{-1}x)^{q}\right)^{1-{\frac {1}{r}}}\left(f(x)^{p}h(y)^{r}\right)^{1-{\frac {1}{q}}}\left(g(y^{-1}x)^{q}h(y)^{r}\right)^{1-{\frac {1}{p}}}\,\mathrm {d} \mu (x)\,\mathrm {d} \mu (y)\end{aligned}}$
By the Hölder inequality for three functions we deduce that
${\begin{aligned}&\int _{G}\int _{G}f(x)g(y^{-1}x)h(y)\,\mathrm {d} \mu (x)\,\mathrm {d} \mu (y)\\&\leq \left(\int _{G}\int _{G}f(x)^{p}g(y^{-1}x)^{q}\,\mathrm {d} \mu (x)\,\mathrm {d} \mu (y)\right)^{1-{\frac {1}{r}}}\left(\int _{G}\int _{G}f(x)^{p}h(y)^{r}\,\mathrm {d} \mu (x)\,\mathrm {d} \mu (y)\right)^{1-{\frac {1}{q}}}\left(\int _{G}\int _{G}g(y^{-1}x)^{q}h(y)^{r}\,\mathrm {d} \mu (x)\,\mathrm {d} \mu (y)\right)^{1-{\frac {1}{p}}}.\end{aligned}}$
The conclusion follows then by left-invariance of the Haar measure, the fact that integrals are preserved by inversion of the domain, and by Fubini's theorem.
Proof by interpolation
Young's inequality can also be proved by interpolation; see the article on Riesz–Thorin interpolation for a proof.
Sharp constant
In case $p,q>1,$ Young's inequality can be strengthened to a sharp form, via
$\|f*g\|_{r}\leq c_{p,q}\|f\|_{p}\|g\|_{q}.$
where the constant $c_{p,q}<1.$[5][6][7] When this optimal constant is achieved, the function $f$ and $g$ are multidimensional Gaussian functions.
See also
• Minkowski inequality – Inequality that established Lp spaces are normed vector spaces
Notes
1. Young, W. H. (1912), "On the multiplication of successions of Fourier constants", Proceedings of the Royal Society A, 87 (596): 331–339, doi:10.1098/rspa.1912.0086, JFM 44.0298.02, JSTOR 93120
2. Bogachev, Vladimir I. (2007), Measure Theory, vol. I, Berlin, Heidelberg, New York: Springer-Verlag, ISBN 978-3-540-34513-8, MR 2267655, Zbl 1120.28001, Theorem 3.9.4
3. Bahouri, Chemin & Danchin 2011, pp. 5–6.
4. Lieb, Elliott H.; Loss, Michael (2001). Analysis. Graduate Studies in Mathematics (2nd ed.). Providence, R.I.: American Mathematical Society. p. 100. ISBN 978-0-8218-2783-3. OCLC 45799429.
5. Beckner, William (1975). "Inequalities in Fourier Analysis". Annals of Mathematics. 102 (1): 159–182. doi:10.2307/1970980. JSTOR 1970980.
6. Brascamp, Herm Jan; Lieb, Elliott H (1976-05-01). "Best constants in Young's inequality, its converse, and its generalization to more than three functions". Advances in Mathematics. 20 (2): 151–173. doi:10.1016/0001-8708(76)90184-5.
7. Fournier, John J. F. (1977), "Sharpness in Young's inequality for convolution", Pacific Journal of Mathematics, 72 (2): 383–397, doi:10.2140/pjm.1977.72.383, MR 0461034, Zbl 0357.43002
References
• Bahouri, Hajer; Chemin, Jean-Yves; Danchin, Raphaël (2011). Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften. Vol. 343. Berlin, Heidelberg: Springer. ISBN 978-3-642-16830-7. OCLC 704397128.
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Young's inequality for integral operators
In mathematical analysis, the Young's inequality for integral operators, is a bound on the $L^{p}\to L^{q}$ operator norm of an integral operator in terms of $L^{r}$ norms of the kernel itself.
Statement
Assume that $X$ and $Y$ are measurable spaces, $K:X\times Y\to \mathbb {R} $ is measurable and $q,p,r\geq 1$ are such that ${\frac {1}{q}}={\frac {1}{p}}+{\frac {1}{r}}-1$. If
$\int _{Y}|K(x,y)|^{r}\,\mathrm {d} y\leq C^{r}$ for all $x\in X$
and
$\int _{X}|K(x,y)|^{r}\,\mathrm {d} x\leq C^{r}$ for all $y\in Y$
then [1]
$\int _{X}\left|\int _{Y}K(x,y)f(y)\,\mathrm {d} y\right|^{q}\,\mathrm {d} x\leq C^{q}\left(\int _{Y}|f(y)|^{p}\,\mathrm {d} y\right)^{\frac {q}{p}}.$
Particular cases
Convolution kernel
If $X=Y=\mathbb {R} ^{d}$ and $K(x,y)=h(x-y)$, then the inequality becomes Young's convolution inequality.
See also
Young's inequality for products
Notes
1. Theorem 0.3.1 in: C. D. Sogge, Fourier integral in classical analysis, Cambridge University Press, 1993. ISBN 0-521-43464-5
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Young's inequality for products
In mathematics, Young's inequality for products is a mathematical inequality about the product of two numbers.[1] The inequality is named after William Henry Young and should not be confused with Young's convolution inequality.
Young's inequality for products can be used to prove Hölder's inequality. It is also widely used to estimate the norm of nonlinear terms in PDE theory, since it allows one to estimate a product of two terms by a sum of the same terms raised to a power and scaled.
Standard version for conjugate Hölder exponents
The standard form of the inequality is the following:
Theorem — If $a\geq 0$ and $b\geq 0$ are nonnegative real numbers and if $p>1$ and $q>1$ are real numbers such that ${\frac {1}{p}}+{\frac {1}{q}}=1,$ then
$ab~\leq ~{\frac {a^{p}}{p}}+{\frac {b^{q}}{q}}.$
Equality holds if and only if $a^{p}=b^{q}.$
It can be used to prove Hölder's inequality.
Proof[2]
Since ${\tfrac {1}{p}}+{\tfrac {1}{q}}=1,$ $p-1={\tfrac {1}{q-1}}.$ A graph $y=x^{p-1}$ on the $xy$-plane is thus also a graph $x=y^{q-1}.$ From sketching a visual representation of the integrals of the area between this curve and the axes, and the area in the rectangle bounded by the lines $x=0,x=a,y=0,y=b,$ and the fact that $y$ is always increasing for increasing $x$ and vice versa, we can see that $\int _{0}^{a}x^{p-1}\mathrm {d} x$ upper bounds the area of the rectangle below the curve (with equality when $b\geq a^{p-1}$) and $\int _{0}^{b}y^{q-1}\mathrm {d} y$ upper bounds the area of the rectangle above the curve (with equality when $b\leq a^{p-1}$). Thus, $\int _{0}^{a}x^{p-1}\mathrm {d} x+\int _{0}^{b}y^{q-1}\mathrm {d} y\geq ab,$ with equality when $b=a^{p-1}$ (or equivalently, $a^{p}=b^{q}$). Young's inequality follows from evaluating the integrals. (See below for a generalization.)
This form of Young's inequality can also be proved via Jensen's inequality.
Proof[3]
The claim is certainly true if $a=0$ or $b=0$ so henceforth assume that $a>0$ and $b>0.$ Put $t=1/p$ and $(1-t)=1/q.$ Because the logarithm function is concave,
$\ln \left(ta^{p}+(1-t)b^{q}\right)~\geq ~t\ln \left(a^{p}\right)+(1-t)\ln \left(b^{q}\right)=\ln(a)+\ln(b)=\ln(ab)$
with the equality holding if and only if $a^{p}=b^{q}.$ Young's inequality follows by exponentiating.
Young's inequality may equivalently be written as
$a^{\alpha }b^{\beta }\leq \alpha a+\beta b,\qquad \,0\leq \alpha ,\beta \leq 1,\quad \ \alpha +\beta =1.$
Where this is just the concavity of the logarithm function. Equality holds if and only if $a=b$ or $\{\alpha ,\beta \}=\{0,1\}.$
Generalizations
Theorem[4] — Suppose $a>0$ and $b>0.$ If $1<p<\infty $ and $q$ are such that ${\tfrac {1}{p}}+{\tfrac {1}{q}}=1$ then
$ab~=~\min _{0<t<\infty }\left({\frac {t^{p}a^{p}}{p}}+{\frac {t^{-q}b^{q}}{q}}\right).$
Using $t:=1$ and replacing $a$ with $a^{1/p}$ and $b$ with $b^{1/q}$ results in the inequality:
$a^{1/p}\,b^{1/q}~\leq ~{\frac {a}{p}}+{\frac {b}{q}},$
which is useful for proving Hölder's inequality.
Proof[4]
Define a real-valued function $f$ on the positive real numbers by
$f(t)~=~{\frac {t^{p}a^{p}}{p}}+{\frac {t^{-q}b^{q}}{q}}$
for every $t>0$ and then calculate its minimum.
Theorem — If $0\leq p_{i}\leq 1$ with $\sum _{i}p_{i}=1$ then
$\prod _{i}{a_{i}}^{p_{i}}~\leq ~\sum _{i}p_{i}a_{i}.$
Equality holds if and only if all the $a_{i}$s with non-zero $p_{i}$s are equal.
Elementary case
An elementary case of Young's inequality is the inequality with exponent $2,$
$ab\leq {\frac {a^{2}}{2}}+{\frac {b^{2}}{2}},$
which also gives rise to the so-called Young's inequality with $\varepsilon $ (valid for every $\varepsilon >0$), sometimes called the Peter–Paul inequality. [5] This name refers to the fact that tighter control of the second term is achieved at the cost of losing some control of the first term – one must "rob Peter to pay Paul"
$ab~\leq ~{\frac {a^{2}}{2\varepsilon }}+{\frac {\varepsilon b^{2}}{2}}.$
Proof: Young's inequality with exponent $2$ is the special case $p=q=2.$ However, it has a more elementary proof.
Start by observing that the square of every real number is zero or positive. Therefore, for every pair of real numbers $a$ and $b$ we can write:
$0\leq (a-b)^{2}$
Work out the square of the right hand side:
$0\leq a^{2}-2ab+b^{2}$
Add $2ab$ to both sides:
$2ab\leq a^{2}+b^{2}$
Divide both sides by 2 and we have Young's inequality with exponent $2:$
$ab\leq {\frac {a^{2}}{2}}+{\frac {b^{2}}{2}}$
Young's inequality with $\varepsilon $ follows by substituting $a'$ and $b'$ as below into Young's inequality with exponent $2:$
$a'=a/{\sqrt {\varepsilon }},\;b'={\sqrt {\varepsilon }}b.$
Matricial generalization
T. Ando proved a generalization of Young's inequality for complex matrices ordered by Loewner ordering.[6] It states that for any pair $A,B$ of complex matrices of order $n$ there exists a unitary matrix $U$ such that
$U^{*}|AB^{*}|U\preceq {\tfrac {1}{p}}|A|^{p}+{\tfrac {1}{q}}|B|^{q},$
where ${}^{*}$ denotes the conjugate transpose of the matrix and $|A|={\sqrt {A^{*}A}}.$
Standard version for increasing functions
For the standard version[7][8] of the inequality, let $f$ denote a real-valued, continuous and strictly increasing function on $[0,c]$ with $c>0$ and $f(0)=0.$ Let $f^{-1}$ denote the inverse function of $f.$ Then, for all $a\in [0,c]$ and $b\in [0,f(c)],$
$ab~\leq ~\int _{0}^{a}f(x)\,dx+\int _{0}^{b}f^{-1}(x)\,dx$
with equality if and only if $b=f(a).$
With $f(x)=x^{p-1}$ and $f^{-1}(y)=y^{q-1},$ this reduces to standard version for conjugate Hölder exponents.
For details and generalizations we refer to the paper of Mitroi & Niculescu.[9]
Generalization using Fenchel–Legendre transforms
By denoting the convex conjugate of a real function $f$ by $g,$ we obtain
$ab~\leq ~f(a)+g(b).$
This follows immediately from the definition of the convex conjugate. For a convex function $f$ this also follows from the Legendre transformation.
More generally, if $f$ is defined on a real vector space $X$ and its convex conjugate is denoted by $f^{\star }$ (and is defined on the dual space $X^{\star }$), then
$\langle u,v\rangle \leq f^{\star }(u)+f(v).$
where $\langle \cdot ,\cdot \rangle :X^{\star }\times X\to \mathbb {R} $ is the dual pairing.
Examples
The convex conjugate of $f(a)=a^{p}/p$ is $g(b)=b^{q}/q$ with $q$ such that ${\tfrac {1}{p}}+{\tfrac {1}{q}}=1,$ and thus Young's inequality for conjugate Hölder exponents mentioned above is a special case.
The Legendre transform of $f(a)=e^{a}-1$ is $g(b)=1-b+b\ln b$, hence $ab\leq e^{a}-b+b\ln b$ for all non-negative $a$ and $b.$ This estimate is useful in large deviations theory under exponential moment conditions, because $b\ln b$ appears in the definition of relative entropy, which is the rate function in Sanov's theorem.
See also
• Convex conjugate – the ("dual") lower-semicontinuous convex function resulting from the Legendre–Fenchel transformation of a "primal" functionPages displaying wikidata descriptions as a fallback
• Integral of inverse functions – Mathematical theorem, used in calculus
• Legendre transformation – Mathematical transformation
• Young's convolution inequality
Notes
1. Young, W. H. (1912), "On classes of summable functions and their Fourier series", Proceedings of the Royal Society A, 87 (594): 225–229, Bibcode:1912RSPSA..87..225Y, doi:10.1098/rspa.1912.0076, JFM 43.1114.12, JSTOR 93236
2. Pearse, Erin. "Math 209D - Real Analysis Summer Preparatory Seminar Lecture Notes" (PDF). Retrieved 17 September 2022.
3. Bahouri, Chemin & Danchin 2011.
4. Jarchow 1981, pp. 47–55.
5. Tisdell, Chris (2013), The Peter Paul Inequality, YouTube video on Dr Chris Tisdell's YouTube channel,
6. T. Ando (1995). "Matrix Young Inequalities". In Huijsmans, C. B.; Kaashoek, M. A.; Luxemburg, W. A. J.; et al. (eds.). Operator Theory in Function Spaces and Banach Lattices. Springer. pp. 33–38. ISBN 978-3-0348-9076-2.
7. Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1952) [1934], Inequalities, Cambridge Mathematical Library (2nd ed.), Cambridge: Cambridge University Press, ISBN 0-521-05206-8, MR 0046395, Zbl 0047.05302, Chapter 4.8
8. Henstock, Ralph (1988), Lectures on the Theory of Integration, Series in Real Analysis Volume I, Singapore, New Jersey: World Scientific, ISBN 9971-5-0450-2, MR 0963249, Zbl 0668.28001, Theorem 2.9
9. Mitroi, F. C., & Niculescu, C. P. (2011). An extension of Young's inequality. In Abstract and Applied Analysis (Vol. 2011). Hindawi.
References
• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
• Bahouri, Hajer; Chemin, Jean-Yves; Danchin, Raphaël (2011). Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften. Vol. 343. Berlin, Heidelberg: Springer. ISBN 978-3-642-16830-7. OCLC 704397128.
External links
• Young's Inequality at PlanetMath
• Weisstein, Eric W. "Young's Inequality". MathWorld.
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Young's inequality
Young's inequality may refer to:
• Young's inequality for products, bounding the product of two quantities
• Young's convolution inequality, bounding the convolution product of two functions
• Young's inequality for integral operators
See also
• William Henry Young, English mathematician (1863–1942)
• Hausdorff–Young inequality, bounding the coefficient of Fourier series
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Young's lattice
In mathematics, Young's lattice is a lattice that is formed by all integer partitions. It is named after Alfred Young, who, in a series of papers On quantitative substitutional analysis, developed the representation theory of the symmetric group. In Young's theory, the objects now called Young diagrams and the partial order on them played a key, even decisive, role. Young's lattice prominently figures in algebraic combinatorics, forming the simplest example of a differential poset in the sense of Stanley (1988). It is also closely connected with the crystal bases for affine Lie algebras.
Definition
Young's lattice is a lattice (and hence also a partially ordered set) Y formed by all integer partitions ordered by inclusion of their Young diagrams (or Ferrers diagrams).
Significance
The traditional application of Young's lattice is to the description of the irreducible representations of symmetric groups Sn for all n, together with their branching properties, in characteristic zero. The equivalence classes of irreducible representations may be parametrized by partitions or Young diagrams, the restriction from Sn +1 to Sn is multiplicity-free, and the representation of Sn with partition p is contained in the representation of Sn +1 with partition q if and only if q covers p in Young's lattice. Iterating this procedure, one arrives at Young's semicanonical basis in the irreducible representation of Sn with partition p, which is indexed by the standard Young tableaux of shape p.
Properties
• The poset Y is graded: the minimal element is ∅, the unique partition of zero, and the partitions of n have rank n. This means that given two partitions that are comparable in the lattice, their ranks are ordered in the same sense as the partitions, and there is at least one intermediate partition of each intermediate rank.
• The poset Y is a lattice. The meet and join of two partitions are given by the intersection and the union of the corresponding Young diagrams. Because it is a lattice in which the meet and join operations are represented by intersections and unions, it is a distributive lattice.
• If a partition p covers k elements of Young's lattice for some k then it is covered by k + 1 elements. All partitions covered by p can be found by removing one of the "corners" of its Young diagram (boxes at the end both of their row and of their column). All partitions covering p can be found by adding one of the "dual corners" to its Young diagram (boxes outside the diagram that are the first such box both in their row and in their column). There is always a dual corner in the first row, and for each other dual corner there is a corner in the previous row, whence the stated property.
• If distinct partitions p and q both cover k elements of Y then k is 0 or 1, and p and q are covered by k elements. In plain language: two partitions can have at most one (third) partition covered by both (their respective diagrams then each have one box not belonging to the other), in which case there is also one (fourth) partition covering them both (whose diagram is the union of their diagrams).
• Saturated chains between ∅ and p are in a natural bijection with the standard Young tableaux of shape p: the diagrams in the chain add the boxes of the diagram of the standard Young tableau in the order of their numbering. More generally, saturated chains between q and p are in a natural bijection with the skew standard tableaux of skew shape p/q.
• The Möbius function of Young's lattice takes values 0, ±1. It is given by the formula
$\mu (q,p)={\begin{cases}(-1)^{|p|-|q|}&{\text{if the skew diagram }}p/q{\text{ is a disconnected union of squares}}\\&{\text{(no common edges);}}\\[10pt]0&{\text{otherwise}}.\end{cases}}$
Dihedral symmetry
The portion of Young's lattice lying below 1 + 1 + 1 + 1, 2 + 2 + 2, 3 + 3, and 4
Conventional diagram with partitions of the same rank at the same height
Diagram showing dihedral symmetry
Conventionally, Young's lattice is depicted in a Hasse diagram with all elements of the same rank shown at the same height above the bottom. Suter (2002) has shown that a different way of depicting some subsets of Young's lattice shows some unexpected symmetries.
The partition
$n+\cdots +3+2+1$
of the nth triangular number has a Ferrers diagram that looks like a staircase. The largest elements whose Ferrers diagrams are rectangular that lie under the staircase are these:
${\begin{array}{c}\underbrace {1+\cdots \cdots \cdots +1} _{n{\text{ terms}}}\\\underbrace {2+\cdots \cdots +2} _{n-1{\text{ terms}}}\\\underbrace {3+\cdots +3} _{n-2{\text{ terms}}}\\\vdots \\\underbrace {{}\quad n\quad {}} _{1{\text{ term}}}\end{array}}$
Partitions of this form are the only ones that have only one element immediately below them in Young's lattice. Suter showed that the set of all elements less than or equal to these particular partitions has not only the bilateral symmetry that one expects of Young's lattice, but also rotational symmetry: the rotation group of order n + 1 acts on this poset. Since this set has both bilateral symmetry and rotational symmetry, it must have dihedral symmetry: the (n + 1)st dihedral group acts faithfully on this set. The size of this set is 2n.
For example, when n = 4, then the maximal element under the "staircase" that have rectangular Ferrers diagrams are
1 + 1 + 1 + 1
2 + 2 + 2
3 + 3
4
The subset of Young's lattice lying below these partitions has both bilateral symmetry and 5-fold rotational symmetry. Hence the dihedral group D5 acts faithfully on this subset of Young's lattice.
See also
• Young–Fibonacci lattice
• Bratteli diagram
References
• Misra, Kailash C.; Miwa, Tetsuji (1990). "Crystal base for the basic representation of $U_{q}({\widehat {\mathfrak {sl}}}(n))$". Communications in Mathematical Physics. 134 (1): 79–88. Bibcode:1990CMaPh.134...79M. doi:10.1007/BF02102090. S2CID 120298905.
• Sagan, Bruce (2000). The Symmetric Group. Berlin: Springer. ISBN 0-387-95067-2.
• Stanley, Richard P. (1988). "Differential posets". Journal of the American Mathematical Society. 1 (4): 919–961. doi:10.2307/1990995. JSTOR 1990995.
• Suter, Ruedi (2002). "Young's lattice and dihedral symmetries". European Journal of Combinatorics. 23 (2): 233–238. doi:10.1006/eujc.2001.0541.
Order theory
• Topics
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Young measure
In mathematical analysis, a Young measure is a parameterized measure that is associated with certain subsequences of a given bounded sequence of measurable functions. They are a quantification of the oscillation effect of the sequence in the limit. Young measures have applications in the calculus of variations, especially models from material science, and the study of nonlinear partial differential equations, as well as in various optimization (or optimal control problems). They are named after Laurence Chisholm Young who invented them, already in 1937 in one dimension (curves) and later in higher dimensions in 1942.[1]
Definition
We let $\{f_{k}\}_{k=1}^{\infty }$ be a bounded sequence in $L^{p}(U,\mathbb {R} ^{m})$, where $U$ denotes an open bounded subset of $\mathbb {R} ^{n}$. Then there exists a subsequence $\{f_{k_{j}}\}_{j=1}^{\infty }\subset \{f_{k}\}_{k=1}^{\infty }$ and for almost every $x\in U$ a Borel probability measure $\nu _{x}$ on $\mathbb {R} ^{m}$ such that for each $F\in C(\mathbb {R} ^{m})$ we have $F\circ f_{k_{j}}(x){\rightharpoonup }\int _{\mathbb {R} ^{m}}F(y)d\nu _{x}(y)$ weakly in $L^{p}(U)$ if the weak limit exists (or weak star in $L^{\infty }(U)$ in case of $p=+\infty $). The measures $\nu _{x}$ are called the Young measures generated by the sequence $\{f_{k_{j}}\}_{j=1}^{\infty }$. More generally, for any Caratheodory function $f(x,A):U\times R^{m}\to R,$ the limit of $\int _{U}f(x,f_{j}(x))\ dx,$ if it exists, will be given by $\int _{U}\int _{\mathbb {R} ^{m}}f(x,A)\ d\nu _{x}(A)\ dx$.[2]
Young's original idea in the case $f\in C_{0}(U\times \mathbb {R} ^{m})$ was to for each integer $j\geq 1$ consider the uniform measure, let's say $\Gamma _{j}:=(id,f_{j})_{\sharp }L^{d}\llcorner U,$ concentrated on graph of the function $f_{j}.$ (Here, $L^{d}\llcorner U$is the restriction of the Lebesgue measure on $U.$) By taking the weak-star limit of these measures as elements of $C_{0}(U\times \mathbb {R} ^{m})^{\star },$ we have $\langle \Gamma _{j},f\rangle =\int _{U}f(x,f_{j}(x))\ dx\to \langle \Gamma ,f\rangle ,$ where $\Gamma $ is the mentioned weak limit. After a disintegration of the measure $\Gamma $ on the product space $\Omega \times \mathbb {R} ^{m},$ we get the parameterized measure $\nu _{x}$.
Example
For every asymptotically minimizing sequence $u_{n}$ of $I(u)=\int _{0}^{1}(u'(x)^{2}-1)^{2}+u'(x)^{2}dx$ subject to $u(0)=u(1)=0$ (that is, the sequence satisfies $\lim _{n\to +\infty }I(u_{n})=\inf _{u\in C^{1}([0,1])}I(u)$), and perhaps after passing to a subsequence, the sequence of derivatives $u'_{n}$ generates Young measures of the form $\nu _{x}=\alpha (x)\delta _{-1}+(1-\alpha )(x)\delta _{1}$ with $\alpha \colon [0,1]\to [0,1]$ measurable. This captures the essential features of all minimizing sequences to this problem, namely, their derivatives $u'_{k}(x)$ will tend to concentrate along the minima $\{-1,1\}$ of the integrand $(u'(x)^{2}-1)^{2}+u'(x)^{2}$.
References
1. Young, L. C. (1942). "Generalized Surfaces in the Calculus of Variations". Annals of Mathematics. 43 (1): 84–103. doi:10.2307/1968882. ISSN 0003-486X. JSTOR 1968882.
2. Pedregal, Pablo (1997). Parametrized measures and variational principles. Basel: Birkhäuser Verlag. ISBN 978-3-0348-8886-8. OCLC 812613013.
• Ball, J. M. (1989). "A version of the fundamental theorem for Young measures". In Rascle, M.; Serre, D.; Slemrod, M. (eds.). PDEs and Continuum Models of Phase Transition. Lecture Notes in Physics. Vol. 344. Berlin: Springer. pp. 207–215.
• C.Castaing, P.Raynaud de Fitte, M.Valadier (2004). Young measures on topological spaces. Dordrecht: Kluwer.{{cite book}}: CS1 maint: multiple names: authors list (link)
• L.C. Evans (1990). Weak convergence methods for nonlinear partial differential equations. Regional conference series in mathematics. American Mathematical Society.
• S. Müller (1999). Variational models for microstructure and phase transitions. Lecture Notes in Mathematics. Springer.
• P. Pedregal (1997). Parametrized Measures and Variational Principles. Basel: Birkhäuser. ISBN 978-3-0348-9815-7.
• T. Roubíček (2020). Relaxation in Optimization Theory and Variational Calculus (2nd ed.). Berlin: W. de Gruyter. ISBN 978-3-11-014542-7.
• Valadier, M. (1990). "Young measures". Methods of Nonconvex Analysis. Lecture Notes in Mathematics. Vol. 1446. Berlin: Springer. pp. 152–188.
• Young, L. C. (1937), "Generalized curves and the existence of an attained absolute minimum in the Calculus of Variations", Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Classe III, XXX (7–9): 211–234, JFM 63.1064.01, Zbl 0019.21901, memoir presented by Stanisław Saks at the session of 16 December 1937 of the Warsaw Society of Sciences and Letters. The free PDF copy is made available by the RCIN –Digital Repository of the Scientifics Institutes.
• Young, L. C. (1969), Lectures on the Calculus of Variations and Optimal Control, Philadelphia–London–Toronto: W. B. Saunders, pp. xi+331, ISBN 9780721696409, MR 0259704, Zbl 0177.37801.
External links
• "Young measure", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
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Young symmetrizer
In mathematics, a Young symmetrizer is an element of the group algebra of the symmetric group, constructed in such a way that, for the homomorphism from the group algebra to the endomorphisms of a vector space $V^{\otimes n}$ obtained from the action of $S_{n}$ on $V^{\otimes n}$ by permutation of indices, the image of the endomorphism determined by that element corresponds to an irreducible representation of the symmetric group over the complex numbers. A similar construction works over any field, and the resulting representations are called Specht modules. The Young symmetrizer is named after British mathematician Alfred Young.
Definition
Given a finite symmetric group Sn and specific Young tableau λ corresponding to a numbered partition of n, and consider the action of $S_{n}$ given by permuting the boxes of $\lambda $. Define two permutation subgroups $P_{\lambda }$ and $Q_{\lambda }$ of Sn as follows:
$P_{\lambda }=\{g\in S_{n}:g{\text{ preserves each row of }}\lambda \}$
and
$Q_{\lambda }=\{g\in S_{n}:g{\text{ preserves each column of }}\lambda \}.$
Corresponding to these two subgroups, define two vectors in the group algebra $\mathbb {C} S_{n}$ as
$a_{\lambda }=\sum _{g\in P_{\lambda }}e_{g}$
and
$b_{\lambda }=\sum _{g\in Q_{\lambda }}\operatorname {sgn}(g)e_{g}$
where $e_{g}$ is the unit vector corresponding to g, and $\operatorname {sgn}(g)$ is the sign of the permutation. The product
$c_{\lambda }:=a_{\lambda }b_{\lambda }=\sum _{g\in P_{\lambda },h\in Q_{\lambda }}\operatorname {sgn}(h)e_{gh}$
is the Young symmetrizer corresponding to the Young tableau λ. Each Young symmetrizer corresponds to an irreducible representation of the symmetric group, and every irreducible representation can be obtained from a corresponding Young symmetrizer. (If we replace the complex numbers by more general fields the corresponding representations will not be irreducible in general.)
Construction
Let V be any vector space over the complex numbers. Consider then the tensor product vector space $V^{\otimes n}=V\otimes V\otimes \cdots \otimes V$ (n times). Let Sn act on this tensor product space by permuting the indices. One then has a natural group algebra representation $\mathbb {C} S_{n}\to \operatorname {End} (V^{\otimes n})$ on $V^{\otimes n}$ (i.e. $V^{\otimes n}$ is a right $\mathbb {C} S_{n}$ module).
Given a partition λ of n, so that $n=\lambda _{1}+\lambda _{2}+\cdots +\lambda _{j}$, then the image of $a_{\lambda }$ is
$\operatorname {Im} (a_{\lambda }):=V^{\otimes n}a_{\lambda }\cong \operatorname {Sym} ^{\lambda _{1}}V\otimes \operatorname {Sym} ^{\lambda _{2}}V\otimes \cdots \otimes \operatorname {Sym} ^{\lambda _{j}}V.$
For instance, if $n=4$, and $\lambda =(2,2)$, with the canonical Young tableau $\{\{1,2\},\{3,4\}\}$. Then the corresponding $a_{\lambda }$ is given by
$a_{\lambda }=e_{\text{id}}+e_{(1,2)}+e_{(3,4)}+e_{(1,2)(3,4)}.$
For any product vector $v_{1,2,3,4}:=v_{1}\otimes v_{2}\otimes v_{3}\otimes v_{4}$ of $V^{\otimes 4}$ we then have
$v_{1,2,3,4}a_{\lambda }=v_{1,2,3,4}+v_{2,1,3,4}+v_{1,2,4,3}+v_{2,1,4,3}=(v_{1}\otimes v_{2}+v_{2}\otimes v_{1})\otimes (v_{3}\otimes v_{4}+v_{4}\otimes v_{3}).$
Thus the set of all $a_{\lambda }v_{1,2,3,4}$ clearly spans $\operatorname {Sym} ^{2}V\otimes \operatorname {Sym} ^{2}V$ and since the $v_{1,2,3,4}$ span $V^{\otimes 4}$ we obtain $V^{\otimes 4}a_{\lambda }=\operatorname {Sym} ^{2}V\otimes \operatorname {Sym} ^{2}V$, where we wrote informally $V^{\otimes 4}a_{\lambda }\equiv \operatorname {Im} (a_{\lambda })$.
Notice also how this construction can be reduced to the construction for $n=2$. Let $\mathbb {1} \in \operatorname {End} (V^{\otimes 2})$ be the identity operator and $S\in \operatorname {End} (V^{\otimes 2})$ the swap operator defined by $S(v\otimes w)=w\otimes v$, thus $\mathbb {1} =e_{\text{id}}$ and $S=e_{(1,2)}$. We have that
$e_{\text{id}}+e_{(1,2)}=\mathbb {1} +S$
maps into $\operatorname {Sym} ^{2}V$, more precisely
${\frac {1}{2}}(\mathbb {1} +S)$
is the projector onto $\operatorname {Sym} ^{2}V$. Then
${\frac {1}{4}}a_{\lambda }={\frac {1}{4}}(e_{\text{id}}+e_{(1,2)}+e_{(3,4)}+e_{(1,2)(3,4)})={\frac {1}{4}}(\mathbb {1} \otimes \mathbb {1} +S\otimes \mathbb {1} +\mathbb {1} \otimes S+S\otimes S)={\frac {1}{2}}(\mathbb {1} +S)\otimes {\frac {1}{2}}(\mathbb {1} +S)$
which is the projector onto $\operatorname {Sym} ^{2}V\otimes \operatorname {Sym} ^{2}V$.
The image of $b_{\lambda }$ is
$\operatorname {Im} (b_{\lambda })\cong \bigwedge ^{\mu _{1}}V\otimes \bigwedge ^{\mu _{2}}V\otimes \cdots \otimes \bigwedge ^{\mu _{k}}V$
where μ is the conjugate partition to λ. Here, $\operatorname {Sym} ^{i}V$ and $\bigwedge ^{j}V$ are the symmetric and alternating tensor product spaces.
The image $\mathbb {C} S_{n}c_{\lambda }$ of $c_{\lambda }=a_{\lambda }\cdot b_{\lambda }$ in $\mathbb {C} S_{n}$ is an irreducible representation of Sn, called a Specht module. We write
$\operatorname {Im} (c_{\lambda })=V_{\lambda }$
for the irreducible representation.
Some scalar multiple of $c_{\lambda }$ is idempotent,[1] that is $c_{\lambda }^{2}=\alpha _{\lambda }c_{\lambda }$ for some rational number $\alpha _{\lambda }\in \mathbb {Q} .$ Specifically, one finds $\alpha _{\lambda }=n!/\dim V_{\lambda }$. In particular, this implies that representations of the symmetric group can be defined over the rational numbers; that is, over the rational group algebra $\mathbb {Q} S_{n}$.
Consider, for example, S3 and the partition (2,1). Then one has
$c_{(2,1)}=e_{123}+e_{213}-e_{321}-e_{312}.$
If V is a complex vector space, then the images of $c_{\lambda }$ on spaces $V^{\otimes d}$ provides essentially all the finite-dimensional irreducible representations of GL(V).
See also
• Representation theory of the symmetric group
Notes
1. See (Fulton & Harris 1991, Theorem 4.3, p. 46)
References
• William Fulton. Young Tableaux, with Applications to Representation Theory and Geometry. Cambridge University Press, 1997.
• Lecture 4 of Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
• Bruce E. Sagan. The Symmetric Group. Springer, 2001.
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YoungJu Choie
YoungJu Choie (Korean: 최영주, born June 15, 1959)[1] is a South Korean mathematician who works as a professor of mathematics at the Pohang University of Science and Technology (POSTECH). Her research interests include number theory and modular forms.[2]
YoungJu Choie
NationalitySouth Korean
Alma materTemple University
Ewha Womans University
Scientific career
FieldsMathematics
InstitutionsPohang University of Science and Technology
Doctoral advisorMarvin Knopp
Education and career
Choie graduated from Ewha Womans University in 1982,[2] and earned a doctorate in 1986 from Temple University under the supervision of Marvin Knopp.[3] After temporary positions at Ohio State University and the University of Maryland, she became an assistant professor at University of Colorado in 1989, and moved to POSTECH as a full professor in 1990. Choie became a Fellow of the American Mathematical Society in 2013.[4]
Mathematical work
Choie works on various aspects of Jacobi forms.[5][6] Together with Winfried Kohnen, she has proved upper bounds on the first sign change of Fourier coefficients of cusp forms,[7] generalizing the work of Siegel.
Selected works
• Y. Choie, Y. Park and D. Zagier, Periods of modular forms on $\Gamma _{0}(N)$ and Products of Jacobi Theta functions, Journal of the European Mathematical Society, Vol. 21, Issue 5, pp 1379–1410 (2019)
• R. Bruggeman, Y. Choie and N. Diamantis, Holomorphic automorphic forms and cohomology, Memoirs of the American Mathematical Society, 253 (2018), no. 1212, vii+167 pp. ISBN 978-1-4704-2855-6
• D. Bump and Y. Choie, “Schubert Eisenstein series”, American Journal of Mathematics Vol 136, No 6, Dec 2014, 1581-1608.
• Y. Choie and W. Kohnen, “The first sign change of Fourier coefficients of cusp forms”, American Journal of Mathematics 131 (2009), no. 2, 517-543.
• Y. Choie and D. Zagier, “Rational period functions for PSL(2, Z)”, Contemporary Mathematics, A tribute to Emil Grosswald: Number Theory and Related Analysis, 143, 89-108, 1993.
• (Book) Y. Choie and MH. Lee, "Jacobi-Like Forms, Pseudodifferential Operators, and Quasimodular Forms”, 318 pages, Springer Monographs in Mathematics on Springer Verlag 2019 (eBook ISBN 978-3-030-29123-5)
• (Book) M. Shi, Y. Choie, A. Sharma and P. Sole, “Codes and Modular forms”, World Scientific, ISBN 978-981-121-291-8 (hardcover) | December 2019 Pages: 232
Service
Choie has been an editor of International Journal of Number Theory since 2004. In 2010–2011 she was editor-in-chief of the Bulletin of the Korean Mathematical Society.[2] She became a president of the society of Korean Women in Mathematical Sciences in 2017. Selective Public Service:
• 2020-2022: NCsoft, Non-executive director
• 2019-2020: The Presidential Advisory Council on Science and Technology, Deliberative Member, Republic of Korea.
• 2019-2020: University Councilor, Representative of Faculty at POSTECH, Pohang, Korea
• 2019-2020: Academic vice president of Korean Mathematical Society, Korea
• 2018-2021: Non-standing member of Board of Trustees, UNIST(Ulsan Institute of Science and Technology), Korea
• 2018-2020: Non-standing member of Board of Trustees, NRF(National Research Foundation), Korea
• 2018–2020, 2009-2013: Director of Pohang Mathematical Institute, POSTECH, Korea
• 2017.12-2019.11: Member, General review committee for academic research supporting the field of education, Ministry of Education
• 2017-2018: Science and Technology Innovation Board, Ministry of Science and Technology Information and communication
• 2018-2019/2020-2021 : Vice President of KOFWST/Auditor, Korea
• 2018-2019: Chief of section committee of group activities of KOFWST, Korea
• 2018: University councilor of POSTECH , Representative of Faculty
• 2017: KWMS (Korea Women in Mathematical Sciences), President, 20170101-20171231
• 2016-2017: KOFWST(Korea Federation of Women’s Sciences and Technology Academics) Member of the board of trustee
• 2016-2018: IMU Committee for Women in Mathematics (CWM) ambassadors
• 2015-2016: CRB(Chief of Research Board), National Research Foundation
• 2009-2015: Organizing Committee, 2014 Seoul ICM, Seoul, Korea
• 2013-2015: Organizing committee, ICWM, 2014, Seoul.
• 2008-2009: ICM-2014 Seoul Bitteing committee.
• 2007-2009: Head of Department of Mathematics, POSTECH, Pohang, Korea.
• 2006:WISE Mentoring Fellow, 2006-
• 2004-2007, 2012: KWMS(Korean Women in Mathematical Sciences), Board of Trustees:
Recognition
Choie has received several awards such as "The best Journal Paper Award (2002)" from the Korean Mathematical Society, "Kwon, Kyungwhan" Chaired Professor (2004) at Pohang University of Science and Technology, "The best woman Scientist of the year" award (2005) from Ministry of Science and Technology, "Amore-Pacific The best Women in Science and Technology" (2007), KOFWST (Korea Federation of Woman's Science and Technology Association) and the "2014 Distinguished research" award from Ministry of Education of Korea.[2] In 2013, Choie became one of the inaugural fellows of the American Mathematical Society.[4] Choie became the first female mathematician as member of the Korean Academy of Science and Technology in 2018.,[1] retrieved 2018-12-22. She was the first female mathematician who received (2018) the academic award of Korean Mathematical Society.
References
1. Member profile, Korean Academy of Science and Technology
2. Faculty profile, POSTECH, retrieved 2015-02-19.
3. Young-Ju Choie at the Mathematics Genealogy Project
4. "List of Fellows of the American Mathematical Society". American Mathematical Society. Retrieved 2019-10-08.
5. Choie, YoungJu; Eholzer, Wolfgang (1998-02-01). "Rankin–Cohen Operators for Jacobi and Siegel Forms". Journal of Number Theory. 68 (2): 160–177. doi:10.1006/jnth.1997.2203. ISSN 0022-314X. S2CID 17316768.
6. Choie, YoungJu (1997-05-01). "Jacobi forms and the heat operator". Mathematische Zeitschrift. 225 (1): 95–101. doi:10.1007/PL00004603. ISSN 1432-1823. S2CID 117410236.
7. Choie, YoungJu; Kohnen, Winfried (2009-03-20). "The first sign change of Fourier coefficients of cusp forms". American Journal of Mathematics. 131 (2): 517–543. doi:10.1353/ajm.0.0050. ISSN 1080-6377. S2CID 14118329.
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Young–Deruyts development
In mathematics, the Young–Deruyts development is a method of writing invariants of an action of a group on an n-dimensional vector space V in terms of invariants depending on at most n–1 vectors (Dieudonné & Carrell 1970, 1971, p.36, 39).
References
• Dieudonné, Jean A.; Carrell, James B. (1970), "Invariant theory, old and new", Advances in Mathematics, 4: 1–80, doi:10.1016/0001-8708(70)90015-0, ISSN 0001-8708, MR 0255525
• Dieudonné, Jean A.; Carrell, James B. (1971), Invariant theory, old and new, Boston, MA: Academic Press, doi:10.1016/0001-8708(70)90015-0, ISBN 978-0-12-215540-6, MR 0279102
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Young–Fibonacci lattice
In mathematics, the Young–Fibonacci graph and Young–Fibonacci lattice, named after Alfred Young and Leonardo Fibonacci, are two closely related structures involving sequences of the digits 1 and 2. Any digit sequence of this type can be assigned a rank, the sum of its digits: for instance, the rank of 11212 is 1 + 1 + 2 + 1 + 2 = 7. As was already known in ancient India, the number of sequences with a given rank is a Fibonacci number. The Young–Fibonacci lattice is an infinite modular lattice having these digit sequences as its elements, compatible with this rank structure. The Young–Fibonacci graph is the graph of this lattice, and has a vertex for each digit sequence. As the graph of a modular lattice, it is a modular graph.
The Young–Fibonacci graph and the Young–Fibonacci lattice were both initially studied in two papers by Fomin (1988) and Stanley (1988). They are named after the closely related Young's lattice and after the Fibonacci number of their elements at any given rank.
Digit sequences with a given rank
A digit sequence with rank r may be formed either by adding the digit 2 to a sequence with rank r − 2, or by adding the digit 1 to a sequence with rank r − 1. If f is the function that maps r to the number of different digit sequences of that rank, therefore, f satisfies the recurrence relation f (r) = f (r − 2) + f (r − 1) defining the Fibonacci numbers, but with slightly different initial conditions: f (0) = f (1) = 1 (there is one rank-0 string, the empty string, and one rank-1 string, consisting of the single digit 1). These initial conditions cause the sequence of values of f to be shifted by one position from the Fibonacci numbers: f (r) = Fr +1.
In the ancient Indian study of prosody, the Fibonacci numbers were used to count the number of different sequences of short and long syllables with a given total length; if the digit 1 corresponds to a short syllable, and the digit 2 corresponds to a long syllable, the rank of a digit sequence measures the total length of the corresponding sequence of syllables. See the Fibonacci number article for details.
Graphs of digit sequences
The Young–Fibonacci graph is an infinite graph, with a vertex for each string of the digits "1" and "2" (including the empty string). The neighbors of a string s are the strings formed from s by one of the following operations:
1. Insert a "1" into s, prior to the leftmost "1" (or anywhere in s if it does not already contain a "1").
2. Change the leftmost "1" of s into a "2".
3. Remove the leftmost "1" from s.
4. Change a "2" that does not have a "1" to the left of it into a "1".
It is straightforward to verify that each operation can be inverted: operations 1 and 3 are inverse to each other, as are operations 2 and 4. Therefore, the resulting graph may be considered to be undirected. However, it is usually considered to be a directed acyclic graph in which each edge connects from a vertex of lower rank to a vertex of higher rank.
As both Fomin (1988) and Stanley (1988) observe, this graph has the following properties:
• It is connected: any nonempty string may have its rank reduced by some operation, so there is a sequence of operations leading from it to the empty string, reversing which gives a directed path in the graph from the empty string to every other vertex.
• It is compatible with the rank structure: every directed path has length equal to the difference in ranks of its endpoints.
• For every two distinct nodes u and v, the number of common immediate predecessors of u and v equals the number of common immediate successors of u and v; this number is either zero or one.
• The out-degree of every vertex equals one plus its in-degree.
Fomin (1988) calls a graph with these properties a Y-graph; Stanley (1988) calls a graph with a weaker version of these properties (in which the numbers of common predecessors and common successors of any pair of nodes must be equal but may be greater than one) the graph of a differential poset.
Partial order and lattice structure
The transitive closure of the Young–Fibonacci graph is a partial order. As Stanley (1988) shows, any two vertices x and y have a unique greatest common predecessor in this order (their meet) and a unique least common successor (their join); thus, this order is a lattice, called the Young–Fibonacci lattice.
To find the meet of x and y, one may first test whether one of x and y is a predecessor of the other. A string x is a predecessor of another string y in this order exactly when the number of "2" digits remaining in y, after removing the longest common suffix of x and y, is at least as large as the number of all digits remaining in x after removing the common suffix. If x is a predecessor of y according to this test, then their meet is x, and similarly if y is a predecessor of x then their meet is y. In a second case, if neither x nor y is the predecessor of the other, but one or both of them begins with a "1" digit, the meet is unchanged if these initial digits are removed. And finally, if both x and y begin with the digit "2", the meet of x and y may be found by removing this digit from both of them, finding the meet of the resulting suffixes, and adding the "2" back to the start.
A common successor of x and y (though not necessarily the least common successor) may be found by taking a string of "2" digits with length equal to the longer of x and y. The least common successor is then the meet of the finitely many strings that are common successors of x and y and predecessors of this string of "2"s.
As Stanley (1988) further observes, the Young–Fibonacci lattice is modular. Fomin (1988) incorrectly claims that it is distributive; however, the sublattice formed by the strings {21, 22, 121, 211, 221} forms a diamond sublattice, forbidden in distributive lattices.
References
• Fomin, S. V. (1988), "Generalized Robinson–Schensted–Knuth correspondence", Journal of Mathematical Sciences, 41 (2): 979–991, doi:10.1007/BF01247093, S2CID 120902883. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR 155: 156–175, 1986.
• Stanley, Richard P. (1988), "Differential posets", Journal of the American Mathematical Society, 1 (4): 919–961, doi:10.2307/1990995, JSTOR 1990995.
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Yozo Matsushima
Yozo Matsushima (松島 与三, Matsushima Yozō, February 11, 1921 – April 9, 1983) was a Japanese mathematician.
Yozo Matsushima
Born(1921-02-11)February 11, 1921
Sakai City, Osaka Prefecture, Japan
DiedApril 9, 1983(1983-04-09) (aged 62)
Osaka, Japan
NationalityJapanese
Alma materOsaka University
Known forresearch in Lie algebras and Lie groups
AwardsAsahi Prize
Scientific career
FieldsMathematics
InstitutionsOsaka University
Nagoya University
University of Notre Dame
Doctoral advisorKenjiro Shoda
Early life
Matsushima was born on February 11, 1921, in Sakai City, Osaka Prefecture, Japan. He studied at Osaka Imperial University (later named Osaka University) and graduated with a Bachelor of Science degree in mathematics in September 1942.[1] At Osaka, he was taught by mathematicians Kenjiro Shoda. After completing his degree, he was appointed as an assistant in the Mathematical Institute of Nagoya Imperial University (later named Nagoya University).[1] These were difficult years for Japanese students and researchers because of World War II.[2]
The first paper published by Matsushima contained a proof that a conjecture of Hans Zassenhaus was false. Zassenhaus had conjectured that every semisimple Lie algebra L over a field of prime characteristic, with [L, L] = L, is the direct sum of simple ideals. Matsushima constructed a counterexample. He then developed a proof that Cartan subalgebras of a complex Lie algebra are conjugate. However, Japanese researchers were out of touch with the research done in the West, and Matsushima was unaware that French mathematician Claude Chevalley had already published a proof. When he obtained details of another paper of Chevalley through a review in Mathematical Reviews, he was able to construct the proofs for himself.[2]
Matsushima published two papers in the 1947 volume of the Proceedings of the Japan Academy (which did not appear until 1950) and three papers in the first volume of Journal of the Mathematical Society of Japan.[2]
Professorship
Matsushima became a full professor at Nagoya University in 1953. Chevalley visited Matsushima in Nagoya in 1953 and invited him to spend the following year in France. He went to France in 1954 and returned to Nagoya in December 1955. He also spent time at the University of Strasbourg. He presented some of his results to Ehresmann's seminar in Strasbourg, extending Cartan's classification of complex irreducible Lie algebras to the case of real Lie algebras.[2]
In spring 1960, Matsushima became a professor of Osaka University as successor to the chair of Shoda.[1] His research took a somewhat different direction and he wrote a series of papers on cohomology of locally symmetric spaces, collaborating with Murakami. He went to the Institute for Advanced Study in September 1962 and returned to Osaka after one year. He jointly began to organize the United States-Japan Seminar in Differential Geometry, which was held in Kyoto in June 1965. After this, he went to France and spent the academic year 1965-66 as visiting professor at the University of Grenoble. He accepted a chair at the University of Notre Dame in Notre Dame, Indiana, in September 1966.[1] He continued to collaborate with Murakami. He introduced Matsushima's formula for the Betti numbers of quotients of symmetric spaces. In 1967, he became an editor of the Journal of Differential Geometry and remained on the editorial board for the rest of his life. After 14 years at Notre Dame, he returned to Japan in 1980. A conference was organized in his honor in May 1980 before he left Notre Dame.[2]
Later life
In February 1981, a volume of papers Manifolds and Lie groups, Papers in honour of Yozo Matsushima was published by his colleagues and former students at Osaka. It also contained some papers presented to the conference held in Notre Dame in the previous May. He died on April 9, 1983, in Osaka, Japan.[2]
Honours
Matsushima received the Asahi Prize for his research on continuous groups in 1962.[2]
References
1. Murakami, Shingo (1984). "Yozô Matsushima: 1921--1983". The Osaka Journal of Mathematics. 21 (1): i–ii. Retrieved 2008-07-09.
2. O'Connor, John J.; Robertson, Edmund F., "Yozo Matsushima", MacTutor History of Mathematics Archive, University of St Andrews
• Kobayashi, Shoshichi (1984). "The mathematical work of Y. Matsushima and its development". The Osaka Journal of Mathematics. 21 (1): iii–xix. Retrieved 2008-07-10.
• Matsushima, Yozô (1992), Murakami, Shingo; Kobayashi, Shoshichi (eds.), Collected papers of Yozô Matsushima, Series in Pure Mathematics, vol. 15, River Edge, NJ: World Scientific Publishing Co. Inc., doi:10.1142/9789814360067, ISBN 9789810208141, MR 1169467
• "List of publications of Yozô Matsushima". The Osaka Journal of Mathematics. 21 (1): xx–xxii. 1984. Retrieved 2008-07-10.
• Yozo Matsushima at the Mathematics Genealogy Project
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Yudell Luke
Yudell Leo Luke (26 June 1918 – 6 May 1983) was an American mathematician who made significant contributions to MRIGlobal, was awarded the N. T. Veatch award for Distinguished Research and Creative Activity in 1975, and appointed as Curator's Professor at the University of Missouri in 1978, a post he held until his death.
Yudell Luke
Born(1918-06-26)26 June 1918
Kansas City, Missouri, United States
Died(1983-05-06)6 May 1983
Moscow, Russia
CitizenshipAmerican
Known forSpecial functions
Hypergeometric functions
AwardsN T Veatch award for Distinguished Research and Creative Activity
Scientific career
FieldsMathematics
InstitutionsUniversity of Missouri
Luke published eight books and nearly 100 papers in a wide variety of mathematical areas, ranging from aeronautics to approximation theory. By his own estimation, Luke reviewed over 1,800 papers and books throughout his career.
Biography
Yudell Luke was born in Kansas City, Missouri, U.S. on 26 June 1918 to Jewish parents.[1] His father, David Luke, was sexton of Congregation Kerem Israel Beth Shalom. The young Luke attended the Kansas City Missouri Junior College, graduating in 1937. He read mathematics at the University of Illinois, receiving a bachelor's degree in 1939, and a Masters the following year. He then taught at the university for two years, but was called up for World War II military service in 1942.
Luke served in the United States Navy until 1946 and was stationed in Hawaii for the duration of the war. After his service, he returned to the university, where he met his future wife LaVerne (LaVerne B. (née Podolsky), 1922–2004) at the University of Illinois. They moved to Kansas City in 1946 and had four daughters, Molly, Janis, Linda, and Debra, and established the Yudell and LaVerne Luke Senior Adult Transportation Fund at the Kansas City Jewish Community Center.[1]
Soon after Luke moved to Kansas City, he was appointed to MRIGlobal (formerly Midwest Research Institute). His first position was as Head of the Mathematical Analysis Section, a position he held until his promotion to Senior Advisor for Mathematics in 1961. Luke also held posts at other universities. In 1955, he became a lecturer at the University of Missouri–Kansas City, and he also taught at the University of Kansas. After the mathematics group of MRIGlobal was disbanded in 1971, Luke was appointed professor at the University of Missouri, and in 1975, received the N T Veatch award for Distinguished Research and Creative Activity. He then became Curator's Professor at Missouri in 1978.
In 1982, an exchange programme between the University of Missouri and the University of Moscow was formed, and the following year, Luke travelled to Moscow to lecture on a series of topics as part of the programme, including special functions, asymptotic analysis and approximation theory. He died while in Russia on 6 May 1983.
Luke had a wide range of interests outside mathematics, including basketball, baseball, bridge, and cribbage. He wrote two books on the probabilities of winning at the latter.[2] He also expressed interest in opera and philosophy, and once gave a series of lectures on the history of philosophy, mainly focusing on Baruch Spinoza's ideas.
Selected bibliography
Papers
• "Rational approximations to the exponential function". Journal of the ACM, 4(1):24–29, January 1957.
Books
• "Integrals of Bessel functions". MacGraw-Hill. 1962
• "The Special Functions and Their Approximations: v. 1 (Mathematics in Science & Engineering)". Academic Press Inc. April 1969. ISBN 978-0-12-459901-7
• "The Special Functions and Their Approximations: v. 2 (Mathematics in Science & Engineering)". Academic Press Inc. 1969.
• "Cumulative Index to Mathematics of Computation 1943–1969". American Mathematical Society. December 1972. ISBN 978-0-8218-4000-9
• "Algorithms for the Computation of Mathematical Functions". Academic Press Inc. 1977. ISBN 978-0-12-459940-6
References
1. Kansas City Star (3 October 2004) Obits. Page B5.
2. O'Connor, 1998.
General
• O'Connor, John J.; Robertson, Edmund F., "Yudell Luke", MacTutor History of Mathematics Archive, University of St Andrews
• Gautschi, Walter; Wimp, Jet (1984). "In memoriam : Yudell L. Luke, 26 June 1918 – 6 May 1983". Math. Comp. 43 (168): 349–352. doi:10.1090/s0025-5718-1984-0758187-6. JSTOR 2008280.
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Shi Yuguang
Shi Yuguang (Chinese: 史宇光; born 1969, Yinxian, Zhejiang) is a Chinese mathematician at Peking University.[1] His areas of research are geometric analysis and differential geometry.[2]
He was awarded the ICTP Ramanujan Prize in 2010, for "outstanding contributions to the geometry of complete (noncompact) Riemannian manifolds, specifically the positivity of quasi-local mass and rigidity of asymptotically hyperbolic manifolds."[3]
He earned his Ph.D. from the Chinese Academy of Sciences in 1996 under the supervision of Ding Weiyue.[4]
Technical contributions
Shi is well-known for his foundational work with Luen-Fai Tam on compact and smooth Riemannian manifolds-with-boundary whose scalar curvature is nonnegative and whose boundary is mean-convex. In particular, if the manifold has a spin structure, and if each connected component of the boundary can be isometrically embedded as a strictly convex hypersurface in Euclidean space, then the average value of the mean curvature of each boundary component is less than or equal to the average value of the mean curvature of the corresponding hypersurface in Euclidean space.
This is particularly simple in three dimensions, where every manifold has a spin structure and a result of Louis Nirenberg shows that any positively-curved Riemannian metric on the two-dimensional sphere can be isometrically embedded in three-dimensional Euclidean space in a geometrically unique way.[5] Hence Shi and Tam's result gives a striking sense in which, given a compact and smooth three-dimensional Riemannian manifold-with-boundary of nonnegative scalar curvature, whose boundary components have positive intrinsic curvature and positive mean curvature, the extrinsic geometry of the boundary components are controlled by their intrinsic geometry. More precisely, the extrinsic geometry is controlled by the extrinsic geometry of the isometric embedding uniquely determined by the intrinsic geometry.
Shi and Tam's proof adopts a method, due to Robert Bartnik, of using parabolic partial differential equations to construct noncompact Riemannian manifolds-with-boundary of nonnegative scalar curvature and prescribed boundary behavior. By combining Bartnik's construction with the given compact manifold-with-boundary, one obtains a complete Riemannian manifold which is non-differentiable along a closed and smooth hypersurface. By using Bartnik's method to relate the geometry near infinity to the geometry of the hypersurface, and by proving a positive energy theorem in which certain singularities are allowed, Shi and Tam's result follows.
From the perspective of research literature in general relativity, Shi and Tam's result is notable in proving, in certain contexts, the nonnegativity of the Brown-York quasilocal energy of J. David Brown and James W. York.[6] The ideas of Shi−Tam and Brown−York have been further developed by Mu-Tao Wang and Shing-Tung Yau, among others.
Major publication
• Yuguang Shi and Luen-Fai Tam. Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature. J. Differential Geom. 62 (2002), no. 1, 79–125. doi:10.4310/jdg/1090425530
References
1. "News: Ramanujan prize awarded to Yuguang Shi". Archived from the original on 2015-02-13. Retrieved 2015-07-26.
2. http://eng.math.pku.edu.cn/en/view.php?uid=shiyg%5B%5D
3. http://www.ams.org/notices/201108/rtx110801131p.pdf
4. Shi Yuguang at the Mathematics Genealogy Project
5. Louis Nirenberg. The Weyl and Minkowski problems in differential geometry in the large. Comm. Pure Appl. Math. 6 (1953), 337–394.
6. J. David Brown and James W. York, Jr. Quasilocal energy and conserved charges derived from the gravitational action. Phys. Rev. D (3) 47 (1993), no. 4, 1407–1419.
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Yujiro Kawamata
Yujiro Kawamata (born 1952) is a Japanese mathematician working in algebraic geometry.
Yujiro Kawamata
Nationality Japanese
Alma materUniversity of Tokyo
Known forKawamata-Viehweg vanishing theorem
Kawamata log terminal (klt) singularities
Scientific career
FieldsMathematics
InstitutionsUniversity of Tokyo
Doctoral advisorShigeru Iitaka
Career
Kawamata completed the master's course at the University of Tokyo in 1977. He was an Assistant at the University of Mannheim from 1977 to 1979 and a Miller Fellow at the University of California, Berkeley from 1981 to 1983. Kawamata is now a professor at the University of Tokyo. He won the Mathematical Society of Japan Autumn award (1988) and the Japan Academy of Sciences award (1990) for his work in algebraic geometry.
Research
Kawamata was involved in the development of the minimal model program in the 1980s. The program aims to show that every algebraic variety is birational to one of an especially simple type: either a minimal model or a Fano fiber space. The Kawamata-Viehweg vanishing theorem, strengthening the Kodaira vanishing theorem, is a method. Building on that, Kawamata proved the basepoint-free theorem. The cone theorem and contraction theorem, central results in the theory, are the result of a joint effort by Kawamata, Kollár, Mori, Reid, and Shokurov.[1]
After Mori proved the existence of minimal models in dimension 3 in 1988, Kawamata and Miyaoka clarified the structure of minimal models by proving the abundance conjecture in dimension 3.[2] Kawamata used analytic methods in Hodge theory to prove the Iitaka conjecture over a base of dimension 1.[3]
More recently, a series of papers by Kawamata related the derived category of coherent sheaves on an algebraic variety to geometric properties in the spirit of minimal model theory.[4]
Notes
1. Y. Kawamata, K. Matsuda, and K. Matsuki. Introduction to the minimal model program. Algebraic Geometry, Sendai 1985. North-Holland (1987), 283-360.
2. Y. Kawamata. Abundance theorem for minimal threefolds. Invent. Math. 108 (1992), 229-246.
3. Y. Kawamata. Kodaira dimension of algebraic fiber spaces over curves. Invent. Math. 66 (1982), 57-71.
4. Y. Kawamata. D-equivalence and K-equivalence. J. Diff. Geom. 61 (2002), 147-171.
References
• Kawamata, Yujiro (1982), "Kodaira dimension of algebraic fiber spaces over curves", Inventiones Mathematicae, 66: 57–71, Bibcode:1982InMat..66...57K, doi:10.1007/BF01404756, MR 0652646, S2CID 123007245
• Kawamata, Yujiro; Matsuda, Katsumi; Matsuki, Kenji (1987), "Introduction to the minimal model program", Algebraic Geometry, Sendai 1985, Advanced Studies in Pure Mathematics, vol. 10, North-Holland, pp. 283–360, ISBN 0-444-70313-6, MR 0946243
• Kawamata, Yujiro (1992), "Abundance theorem for minimal threefolds", Inventiones Mathematicae, 108: 229–246, Bibcode:1992InMat.108..229K, doi:10.1007/BF02100604, MR 1161091, S2CID 121956975
• Kawamata, Yujiro (2002), "D-equivalence and K-equivalence", Journal of Differential Geometry, 61: 147–171, arXiv:math/0205287, Bibcode:2002math......5287K, doi:10.4310/jdg/1090351323, MR 1949787, S2CID 8778816
• Kawamata, Yujiro (2014), Kōjigen daisū tayōtairon / 高次元代数多様体論 (Higher Dimensional Algebraic Varieties), Iwanami Shoten, ISBN 978-4000075985
External links
• Homepage in Tokyo
• Page at KIAS
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Yukio Matsumoto
Yukio Matsumoto (松本 幸夫, Matsumoto Yukio; * 1944) is a japanese mathematician, who worked mostly in the field of geometric topology and low-dimensional topology. He was a former professor for mathematics at the university of Tokyo.[1]
He received his Ph.D in 1973 from the university of Tokyo and his supervisor was Ichiro Tamura.[2]
In 1984 he won the Iyanaga Prize of the Mathematical Society of Japan.[1][3][4]
Selected publications
Solo
• Matsumoto, Yukio (2001). An Introduction to Morse Theory. Translations of Mathematical Monographs. ISBN 978-0821810224.
Joint
• Kojima, Sadayoshi; Matsumoto, Yukio; Saito, Kyōji; Seppälä, Mika (1995). Topology and Teichmüller spaces. World Scientific Publishing. doi:10.1142/3122.
References
1. "Yukio Matsumoto". researchmap.jp. researchmap. Retrieved 2022-11-08.
2. Yukio Matsumoto at the Mathematics Genealogy Project
3. "List of Spring and Autumn Prizes Winners". www.mathsoc.jp.
4. https://www.gakushuin.ac.jp/univ/new/pamphlet-english.pdf
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Goodman and Kruskal's gamma
In statistics, Goodman and Kruskal's gamma is a measure of rank correlation, i.e., the similarity of the orderings of the data when ranked by each of the quantities. It measures the strength of association of the cross tabulated data when both variables are measured at the ordinal level. It makes no adjustment for either table size or ties. Values range from −1 (100% negative association, or perfect inversion) to +1 (100% positive association, or perfect agreement). A value of zero indicates the absence of association.
This statistic (which is distinct from Goodman and Kruskal's lambda) is named after Leo Goodman and William Kruskal, who proposed it in a series of papers from 1954 to 1972.[1][2][3][4]
Definition
The estimate of gamma, G, depends on two quantities:
• Ns, the number of pairs of cases ranked in the same order on both variables (number of concordant pairs),
• Nd, the number of pairs of cases ranked in reversed order on both variables (number of reversed pairs),
where "ties" (cases where either of the two variables in the pair are equal) are dropped. Then
$G={\frac {N_{s}-N_{d}}{N_{s}+N_{d}}}\ .$
This statistic can be regarded as the maximum likelihood estimator for the theoretical quantity $\gamma $, where
$\gamma ={\frac {P_{s}-P_{d}}{P_{s}+P_{d}}}\ ,$
and where Ps and Pd are the probabilities that a randomly selected pair of observations will place in the same or opposite order respectively, when ranked by both variables.
Critical values for the gamma statistic are sometimes found by using an approximation, whereby a transformed value, t of the statistic is referred to Student t distribution, where
$t\approx G{\sqrt {\frac {N_{s}+N_{d}}{n(1-G^{2})}}}\ ,$
and where n is the number of observations (not the number of pairs):
$n\neq N_{s}+N_{d}.\,$
Yule's Q
A special case of Goodman and Kruskal's gamma is Yule's Q, also known as the Yule coefficient of association,[5] which is specific to 2×2 matrices. Consider the following contingency table of events, where each value is a count of an event's frequency:
YesNoTotals
Positive aba+b
Negative cdc+d
Totals a+cb+dn
Yule's Q is given by:
$Q={\frac {ad-bc}{ad+bc}}\ .$
Although computed in the same fashion as Goodman and Kruskal's gamma, it has a slightly broader interpretation because the distinction between nominal and ordinal scales becomes a matter of arbitrary labeling for dichotomous distinctions. Thus, whether Q is positive or negative depends merely on which pairings the analyst considers to be concordant, but is otherwise symmetric.
Q varies from −1 to +1. −1 reflects total negative association, +1 reflects perfect positive association and 0 reflects no association at all. The sign depends on which pairings the analyst initially considered to be concordant, but this choice does not affect the magnitude.
In term of the odds ratio OR, Yule's Q is given by
$Q={\frac {{OR}-1}{{OR}+1}}\ .$
and so Yule's Q and Yule's Y are related by
$Q={\frac {2Y}{1+Y^{2}}}\ ,$
$Y={\frac {1-{\sqrt {1-Q^{2}}}}{Q}}\ .$
See also
• Kendall tau rank correlation coefficient
• Goodman and Kruskal's lambda
• Yule's Y, also known as the coefficient of colligation
References
1. Goodman, Leo A.; Kruskal, William H. (1954). "Measures of Association for Cross Classifications". Journal of the American Statistical Association. 49 (268): 732–764. doi:10.2307/2281536. JSTOR 2281536.
2. Goodman, Leo A.; Kruskal, William H. (1959). "Measures of Association for Cross Classifications. II: Further Discussion and References". Journal of the American Statistical Association. 54 (285): 123–163. doi:10.1080/01621459.1959.10501503. JSTOR 2282143.
3. Goodman, Leo A.; Kruskal, William H. (1963). "Measures of Association for Cross Classifications III: Approximate Sampling Theory". Journal of the American Statistical Association. 58 (302): 310–364. doi:10.1080/01621459.1963.10500850. JSTOR 2283271.
4. Goodman, Leo A.; Kruskal, William H. (1972). "Measures of Association for Cross Classifications, IV: Simplification of Asymptotic Variances". Journal of the American Statistical Association. 67 (338): 415–421. doi:10.1080/01621459.1972.10482401. JSTOR 2284396.
5. Yule, G U. (1912). "On the methods of measuring association between two attributes". Journal of the Royal Statistical Society. 49 (6): 579–652. doi:10.2307/2340126. JSTOR 2340126.
Further reading
• Sheskin, D.J. (2007) The Handbook of Parametric and Nonparametric Statistical Procedures. Chapman & Hall/CRC, ISBN 9781584888147
Statistics
• Outline
• Index
Descriptive statistics
Continuous data
Center
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Shape
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Count data
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Summary tables
• Contingency table
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Dependence
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Graphics
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Data collection
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Statistical inference
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Coefficient of colligation
In statistics, Yule's Y, also known as the coefficient of colligation, is a measure of association between two binary variables. The measure was developed by George Udny Yule in 1912,[1][2] and should not be confused with Yule's coefficient for measuring skewness based on quartiles.
Formula
For a 2×2 table for binary variables U and V with frequencies or proportions
V = 0V = 1
U = 0ab
U = 1cd
Yule's Y is given by
$Y={\frac {{\sqrt {ad}}-{\sqrt {bc}}}{{\sqrt {ad}}+{\sqrt {bc}}}}.$
Yule's Y is closely related to the odds ratio OR = ad/(bc) as is seen in following formula:
$Y={\frac {{\sqrt {OR}}-1}{{\sqrt {OR}}+1}}$
Yule's Y varies from −1 to +1. −1 reflects total negative correlation, +1 reflects perfect positive association while 0 reflects no association at all. These correspond to the values for the more common Pearson correlation.
Yule's Y is also related to the similar Yule's Q, which can also be expressed in terms of the odds ratio. Q and Y are related by:
$Q={\frac {2Y}{1+Y^{2}}}\ ,$
$Y={\frac {1-{\sqrt {1-Q^{2}}}}{Q}}\ .$
Interpretation
Yule's Y gives the fraction of perfect association in per unum (multiplied by 100 it represents this fraction in a more familiar percentage). Indeed, the formula transforms the original 2×2 table in a crosswise symmetric table wherein b = c = 1 and a = d = √OR.
For a crosswise symmetric table with frequencies or proportions a = d and b = c it is very easy to see that it can be split up in two tables. In such tables association can be measured in a perfectly clear way by dividing (a – b) by (a + b). In transformed tables b has to be substituted by 1 and a by √OR. The transformed table has the same degree of association (the same OR) as the original not-crosswise symmetric table. Therefore, the association in asymmetric tables can be measured by Yule's Y, interpreting it in just the same way as with symmetric tables. Of course, Yule's Y and (a − b)/(a + b) give the same result in crosswise symmetric tables, presenting the association as a fraction in both cases.
Yule's Y measures association in a substantial, intuitively understandable way and therefore it is the measure of preference to measure association.
Examples
The following crosswise symmetric table
V = 0V = 1
U = 04010
U = 11040
can be split up into two tables:
V = 0V = 1
U = 01010
U = 11010
and
V = 0V = 1
U = 0300
U = 1030
It is obvious that the degree of association equals 0.6 per unum (60%).
The following asymmetric table can be transformed in a table with an equal degree of association (the odds ratios of both tables are equal).
V = 0V = 1
U = 031
U = 139
Here follows the transformed table:
V = 0V = 1
U = 031
U = 113
The odds ratios of both tables are equal to 9. Y = (3 − 1)/(3 + 1) = 0.5 (50%)
References
1. Yule, G. Udny (1912). "On the Methods of Measuring Association Between Two Attributes". Journal of the Royal Statistical Society. 75 (6): 579–652. doi:10.2307/2340126. JSTOR 2340126.
2. Michel G. Soete. A new theory on the measurement of association between two binary variables in medical sciences: association can be expressed in a fraction (per unum, percentage, pro mille....) of perfect association (2013), e-article, BoekBoek.be
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Simpson's paradox
Simpson's paradox is a phenomenon in probability and statistics in which a trend appears in several groups of data but disappears or reverses when the groups are combined. This result is often encountered in social-science and medical-science statistics,[1][2][3] and is particularly problematic when frequency data are unduly given causal interpretations.[4] The paradox can be resolved when confounding variables and causal relations are appropriately addressed in the statistical modeling[4][5] (e.g., through cluster analysis[6]).
Simpson's paradox has been used to illustrate the kind of misleading results that the misuse of statistics can generate.[7][8]
Edward H. Simpson first described this phenomenon in a technical paper in 1951,[9] but the statisticians Karl Pearson (in 1899[10]) and Udny Yule (in 1903[11]) had mentioned similar effects earlier. The name Simpson's paradox was introduced by Colin R. Blyth in 1972.[12] It is also referred to as Simpson's reversal, the Yule–Simpson effect, the amalgamation paradox, or the reversal paradox.[13]
Mathematician Jordan Ellenberg argues that Simpson's paradox is misnamed as "there's no contradiction involved, just two different ways to think about the same data" and suggests that its lesson "isn't really to tell us which viewpoint to take but to insist that we keep both the parts and the whole in mind at once."[14]
Examples
UC Berkeley gender bias
One of the best-known examples of Simpson's paradox comes from a study of gender bias among graduate school admissions to University of California, Berkeley. The admission figures for the fall of 1973 showed that men applying were more likely than women to be admitted, and the difference was so large that it was unlikely to be due to chance.[15][16]
All Men Women
Applicants Admitted Applicants Admitted Applicants Admitted
Total 12,763 41% 8,442 44% 4,321 35%
However, when taking into account the information about departments being applied to, the different rejection percentages reveal the different difficulty of getting into the department, and at the same time it showed that women tended to apply to more competitive departments with lower rates of admission, even among qualified applicants (such as in the English department), whereas men tended to apply to less competitive departments with higher rates of admission (such as in the engineering department). The pooled and corrected data showed a "small but statistically significant bias in favor of women".[16]
The data from the six largest departments are listed below:
Department All Men Women
Applicants Admitted Applicants Admitted Applicants Admitted
A 933 64% 825 62% 108 82%
B 585 63% 560 63% 25 68%
C 918 35% 325 37% 593 34%
D 792 34% 417 33% 375 35%
E 584 25% 191 28% 393 24%
F 714 6% 373 6% 341 7%
Total 4526 39% 2691 45% 1835 30%
Legend:
greater percentage of successful applicants than the other gender
greater number of applicants than the other gender
bold - the two 'most applied for' departments for each gender
The entire data showed total of 4 out of 85 departments to be significantly biased against women, while 6 to be significantly biased against men (not all present in the 'six largest departments' table above). Notably, the numbers of biased departments were not the basis for the conclusion, but rather it was the gender admissions pooled across all departments, while weighing by each department's rejection rate across all of its applicants.[16]
Kidney stone treatment
Another example comes from a real-life medical study[17] comparing the success rates of two treatments for kidney stones.[18] The table below shows the success rates (the term success rate here actually means the success proportion) and numbers of treatments for treatments involving both small and large kidney stones, where Treatment A includes open surgical procedures and Treatment B includes closed surgical procedures. The numbers in parentheses indicate the number of success cases over the total size of the group.
Treatment
Stone size
Treatment A Treatment B
Small stones Group 1
93% (81/87)
Group 2
87% (234/270)
Large stones Group 3
73% (192/263)
Group 4
69% (55/80)
Both 78% (273/350)83% (289/350)
The paradoxical conclusion is that treatment A is more effective when used on small stones, and also when used on large stones, yet treatment B appears to be more effective when considering both sizes at the same time. In this example, the "lurking" variable (or confounding variable) causing the paradox is the size of the stones, which was not previously known to researchers to be important until its effects were included.
Which treatment is considered better is determined by which success ratio (successes/total) is larger. The reversal of the inequality between the two ratios when considering the combined data, which creates Simpson's paradox, happens because two effects occur together:
1. The sizes of the groups, which are combined when the lurking variable is ignored, are very different. Doctors tend to give cases with large stones the better treatment A, and the cases with small stones the inferior treatment B. Therefore, the totals are dominated by groups 3 and 2, and not by the two much smaller groups 1 and 4.
2. The lurking variable, stone size, has a large effect on the ratios; i.e., the success rate is more strongly influenced by the severity of the case than by the choice of treatment. Therefore, the group of patients with large stones using treatment A (group 3) does worse than the group with small stones, even if the latter used the inferior treatment B (group 2).
Based on these effects, the paradoxical result is seen to arise because the effect of the size of the stones overwhelms the benefits of the better treatment (A). In short, the less effective treatment B appeared to be more effective because it was applied more frequently to the small stones cases, which were easier to treat.[18]
Batting averages
A common example of Simpson's paradox involves the batting averages of players in professional baseball. It is possible for one player to have a higher batting average than another player each year for a number of years, but to have a lower batting average across all of those years. This phenomenon can occur when there are large differences in the number of at bats between the years. Mathematician Ken Ross demonstrated this using the batting average of two baseball players, Derek Jeter and David Justice, during the years 1995 and 1996:[19][20]
Year
Batter
1995 1996 Combined
Derek Jeter 12/48 .250 183/582 .314 195/630 .310
David Justice 104/411 .253 45/140 .321 149/551 .270
In both 1995 and 1996, Justice had a higher batting average (in bold type) than Jeter did. However, when the two baseball seasons are combined, Jeter shows a higher batting average than Justice. According to Ross, this phenomenon would be observed about once per year among the possible pairs of players.[19]
Vector interpretation
Simpson's paradox can also be illustrated using a 2-dimensional vector space.[21] A success rate of $ {\frac {p}{q}}$ (i.e., successes/attempts) can be represented by a vector ${\vec {A}}=(q,p)$, with a slope of $ {\frac {p}{q}}$. A steeper vector then represents a greater success rate. If two rates $ {\frac {p_{1}}{q_{1}}}$ and $ {\frac {p_{2}}{q_{2}}}$ are combined, as in the examples given above, the result can be represented by the sum of the vectors $(q_{1},p_{1})$ and $(q_{2},p_{2})$, which according to the parallelogram rule is the vector $(q_{1}+q_{2},p_{1}+p_{2})$, with slope $ {\frac {p_{1}+p_{2}}{q_{1}+q_{2}}}$.
Simpson's paradox says that even if a vector ${\vec {L}}_{1}$ (in orange in figure) has a smaller slope than another vector ${\vec {B}}_{1}$ (in blue), and ${\vec {L}}_{2}$ has a smaller slope than ${\vec {B}}_{2}$, the sum of the two vectors ${\vec {L}}_{1}+{\vec {L}}_{2}$ can potentially still have a larger slope than the sum of the two vectors ${\vec {B}}_{1}+{\vec {B}}_{2}$, as shown in the example. For this to occur one of the orange vectors must have a greater slope than one of the blue vectors (here ${\vec {L}}_{2}$ and ${\vec {B}}_{1}$), and these will generally be longer than the alternatively subscripted vectors – thereby dominating the overall comparison.
Correlation between variables
Simpson's reversal can also arise in correlations, in which two variables appear to have (say) a positive correlation towards one another, when in fact they have a negative correlation, the reversal having been brought about by a "lurking" confounder. Berman et al.[22] give an example from economics, where a dataset suggests overall demand is positively correlated with price (that is, higher prices lead to more demand), in contradiction of expectation. Analysis reveals time to be the confounding variable: plotting both price and demand against time reveals the expected negative correlation over various periods, which then reverses to become positive if the influence of time is ignored by simply plotting demand against price.
Psychology
Psychological interest in Simpson's paradox seeks to explain why people deem sign reversal to be impossible at first, offended by the idea that an action preferred both under one condition and under its negation should be rejected when the condition is unknown. The question is where people get this strong intuition from, and how it is encoded in the mind.
Simpson's paradox demonstrates that this intuition cannot be derived from either classical logic or probability calculus alone, and thus led philosophers to speculate that it is supported by an innate causal logic that guides people in reasoning about actions and their consequences.[4] Savage's sure-thing principle[12] is an example of what such logic may entail. A qualified version of Savage's sure thing principle can indeed be derived from Pearl's do-calculus[4] and reads: "An action A that increases the probability of an event B in each subpopulation Ci of C must also increase the probability of B in the population as a whole, provided that the action does not change the distribution of the subpopulations." This suggests that knowledge about actions and consequences is stored in a form resembling Causal Bayesian Networks.
Probability
A paper by Pavlides and Perlman presents a proof, due to Hadjicostas, that in a random 2 × 2 × 2 table with uniform distribution, Simpson's paradox will occur with a probability of exactly 1⁄60.[23] A study by Kock suggests that the probability that Simpson's paradox would occur at random in path models (i.e., models generated by path analysis) with two predictors and one criterion variable is approximately 12.8 percent; slightly higher than 1 occurrence per 8 path models.[24]
Simpson's second paradox
A second, less well-known paradox was also discussed in Simpson's 1951 paper. It can occur when the "sensible interpretation" is not necessarily found in the separated data, like in the Kidney Stone example, but can instead reside in the combined data. Whether the partitioned or combined form of the data should be used hinges on the process giving rise to the data, meaning the correct interpretation of the data cannot always be determined by simply observing the tables.[25]
Judea Pearl has shown that, in order for the partitioned data to represent the correct causal relationships between any two variables, $X$ and $Y$, the partitioning variables must satisfy a graphical condition called "back-door criterion":[26][27]
1. They must block all spurious paths between $X$ and $Y$
2. No variable can be affected by $X$
This criterion provides an algorithmic solution to Simpson's second paradox, and explains why the correct interpretation cannot be determined by data alone; two different graphs, both compatible with the data, may dictate two different back-door criteria.
When the back-door criterion is satisfied by a set Z of covariates, the adjustment formula (see Confounding) gives the correct causal effect of X on Y. If no such set exists, Pearl's do-calculus can be invoked to discover other ways of estimating the causal effect.[4][28] The completeness of do-calculus [29][28] can be viewed as offering a complete resolution of the Simpson's paradox.
Criticism
One criticism is that the paradox is not really a paradox at all, but rather a failure to properly account for confounding variables or to consider causal relationships between variables.[30]
Another criticism of the apparent Simpson's paradox is that it may be a result of the specific way that data is stratified or grouped. The phenomenon may disappear or even reverse if the data is stratified differently or if different confounding variables are considered. Simpson's example actually highlighted a phenomenon called noncollapsibility,[31] which occurs when subgroups with high proportions do not make simple averages when combined. This suggests that the paradox may not be a universal phenomenon, but rather a specific instance of a more general statistical issue.
Critics of the apparent Simpson's paradox also argue that the focus on the paradox may distract from more important statistical issues, such as the need for careful consideration of confounding variables and causal relationships when interpreting data.[32]
Despite these criticisms, the apparent Simpson's paradox remains a popular and intriguing topic in statistics and data analysis. It continues to be studied and debated by researchers and practitioners in a wide range of fields, and it serves as a valuable reminder of the importance of careful statistical analysis and the potential pitfalls of simplistic interpretations of data.
See also
• Aliasing – Signal processing effect
• Anscombe's quartet – Four data sets with the same descriptive statistics, yet very different distributions
• Berkson's paradox – Tendency to misinterpret statistical experiments involving conditional probabilities
• Cherry picking – Fallacy of incomplete evidence
• Condorcet paradox – Situation in social choice theory where collective preferences are cyclic
• Ecological fallacy – Logical fallacy that occurs when group characteristics are applied to individuals
• Gerrymandering – Form of political manipulation
• Low birth-weight paradox – Statistical quirk of babies' birth weights
• Modifiable areal unit problem – Source of statistical bias
• Prosecutor's fallacy – Error in thinking which involves under-valuing base rate informationPages displaying short descriptions of redirect targets
• Will Rogers phenomenon – phenomenon in which moving an element from one set to another set raises the average values of both setsPages displaying wikidata descriptions as a fallback
• Spurious correlation
• Omitted-variable bias
References
1. Clifford H. Wagner (February 1982). "Simpson's Paradox in Real Life". The American Statistician. 36 (1): 46–48. doi:10.2307/2684093. JSTOR 2684093.
2. Holt, G. B. (2016). Potential Simpson's paradox in multicenter study of intraperitoneal chemotherapy for ovarian cancer. Journal of Clinical Oncology, 34(9), 1016–1016.
3. Franks, Alexander; Airoldi, Edoardo; Slavov, Nikolai (2017). "Post-transcriptional regulation across human tissues". PLOS Computational Biology. 13 (5): e1005535. arXiv:1506.00219. Bibcode:2017PLSCB..13E5535F. doi:10.1371/journal.pcbi.1005535. ISSN 1553-7358. PMC 5440056. PMID 28481885.
4. Judea Pearl. Causality: Models, Reasoning, and Inference, Cambridge University Press (2000, 2nd edition 2009). ISBN 0-521-77362-8.
5. Kock, N., & Gaskins, L. (2016). Simpson's paradox, moderation and the emergence of quadratic relationships in path models: An information systems illustration. International Journal of Applied Nonlinear Science, 2(3), 200–234.
6. Rogier A. Kievit, Willem E. Frankenhuis, Lourens J. Waldorp and Denny Borsboom, Simpson's paradox in psychological science: a practical guide https://doi.org/10.3389/fpsyg.2013.00513
7. Robert L. Wardrop (February 1995). "Simpson's Paradox and the Hot Hand in Basketball". The American Statistician, 49 (1): pp. 24–28.
8. Alan Agresti (2002). "Categorical Data Analysis" (Second edition). John Wiley and Sons ISBN 0-471-36093-7
9. Simpson, Edward H. (1951). "The Interpretation of Interaction in Contingency Tables". Journal of the Royal Statistical Society, Series B. 13: 238–241.
10. Pearson, Karl; Lee, Alice; Bramley-Moore, Lesley (1899). "Genetic (reproductive) selection: Inheritance of fertility in man, and of fecundity in thoroughbred racehorses". Philosophical Transactions of the Royal Society A. 192: 257–330. doi:10.1098/rsta.1899.0006.
11. G. U. Yule (1903). "Notes on the Theory of Association of Attributes in Statistics". Biometrika. 2 (2): 121–134. doi:10.1093/biomet/2.2.121.
12. Colin R. Blyth (June 1972). "On Simpson's Paradox and the Sure-Thing Principle". Journal of the American Statistical Association. 67 (338): 364–366. doi:10.2307/2284382. JSTOR 2284382.
13. I. J. Good, Y. Mittal (June 1987). "The Amalgamation and Geometry of Two-by-Two Contingency Tables". The Annals of Statistics. 15 (2): 694–711. doi:10.1214/aos/1176350369. ISSN 0090-5364. JSTOR 2241334.
14. Ellenberg, Jordan (May 25, 2021). Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy and Everything Else. New York: Penguin Press. p. 228. ISBN 978-1-9848-7905-9. OCLC 1226171979.
15. David Freedman, Robert Pisani, and Roger Purves (2007), Statistics (4th edition), W. W. Norton. ISBN 0-393-92972-8.
16. P.J. Bickel, E.A. Hammel and J.W. O'Connell (1975). "Sex Bias in Graduate Admissions: Data From Berkeley" (PDF). Science. 187 (4175): 398–404. Bibcode:1975Sci...187..398B. doi:10.1126/science.187.4175.398. PMID 17835295. S2CID 15278703. Archived (PDF) from the original on 2016-06-04.
17. C. R. Charig; D. R. Webb; S. R. Payne; J. E. Wickham (29 March 1986). "Comparison of treatment of renal calculi by open surgery, percutaneous nephrolithotomy, and extracorporeal shockwave lithotripsy". Br Med J (Clin Res Ed). 292 (6524): 879–882. doi:10.1136/bmj.292.6524.879. PMC 1339981. PMID 3083922.
18. Steven A. Julious; Mark A. Mullee (3 December 1994). "Confounding and Simpson's paradox". BMJ. 309 (6967): 1480–1481. doi:10.1136/bmj.309.6967.1480. PMC 2541623. PMID 7804052.
19. Ken Ross. "A Mathematician at the Ballpark: Odds and Probabilities for Baseball Fans (Paperback)" Pi Press, 2004. ISBN 0-13-147990-3. 12–13
20. Statistics available from Baseball-Reference.com: Data for Derek Jeter; Data for David Justice.
21. Kocik Jerzy (2001). "Proofs without Words: Simpson's Paradox" (PDF). Mathematics Magazine. 74 (5): 399. doi:10.2307/2691038. JSTOR 2691038. Archived (PDF) from the original on 2010-06-12.
22. Berman, S. DalleMule, L. Greene, M., Lucker, J. (2012), "Simpson's Paradox: A Cautionary Tale in Advanced Analytics Archived 2020-05-10 at the Wayback Machine", Significance.
23. Marios G. Pavlides & Michael D. Perlman (August 2009). "How Likely is Simpson's Paradox?". The American Statistician. 63 (3): 226–233. doi:10.1198/tast.2009.09007. S2CID 17481510.
24. Kock, N. (2015). How likely is Simpson's paradox in path models? International Journal of e-Collaboration, 11(1), 1–7.
25. Norton, H. James; Divine, George (August 2015). "Simpson's paradox ... and how to avoid it". Significance. 12 (4): 40–43. doi:10.1111/j.1740-9713.2015.00844.x.
26. Pearl, Judea (2014). "Understanding Simpson's Paradox". The American Statistician. 68 (1): 8–13. doi:10.2139/ssrn.2343788. S2CID 2626833.
27. Pearl, Judea (1993). "Graphical Models, Causality, and Intervention". Statistical Science. 8 (3): 266–269. doi:10.1214/ss/1177010894.
28. Pearl, J.; Mackenzie, D. (2018). The Book of Why: The New Science of Cause and Effect. New York, NY: Basic Books.
29. Shpitser, I.; Pearl, J. (2006). Dechter, R.; Richardson, T.S. (eds.). "Identification of Conditional Interventional Distributions". Proceedings of the Twenty-Second Conference on Uncertainty in Artificial Intelligence. Corvallis, OR: AUAI Press: 437–444.
30. Blyth, Colin R. (June 1972). "On Simpson's Paradox and the Sure-Thing Principle". Journal of the American Statistical Association. 67 (338): 364–366. doi:10.1080/01621459.1972.10482387. ISSN 0162-1459.
31. Greenland, Sander (2021-11-01). "Noncollapsibility, confounding, and sparse-data bias. Part 2: What should researchers make of persistent controversies about the odds ratio?". Journal of Clinical Epidemiology. 139: 264–268. doi:10.1016/j.jclinepi.2021.06.004. ISSN 0895-4356. PMID 34119647.
32. Hernán, Miguel A.; Clayton, David; Keiding, Niels (June 2011). "The Simpson's paradox unraveled". International Journal of Epidemiology. 40 (3): 780–785. doi:10.1093/ije/dyr041. ISSN 1464-3685. PMC 3147074. PMID 21454324.
Bibliography
• Leila Schneps and Coralie Colmez, Math on trial. How numbers get used and abused in the courtroom, Basic Books, 2013. ISBN 978-0-465-03292-1. (Sixth chapter: "Math error number 6: Simpson's paradox. The Berkeley sex bias case: discrimination detection").
External links
Wikimedia Commons has media related to Simpson's paradox.
• Simpson's Paradox at the Stanford Encyclopedia of Philosophy, by Jan Sprenger and Naftali Weinberger.
• How statistics can be misleading – Mark Liddell – TED-Ed video and lesson.
• Pearl, Judea, "Understanding Simpson’s Paradox" (PDF)
• Simpson's Paradox, a short article by Alexander Bogomolny on the vector interpretation of Simpson's paradox
• The Wall Street Journal column "The Numbers Guy" for December 2, 2009 dealt with recent instances of Simpson's paradox in the news. Notably a Simpson's paradox in the comparison of unemployment rates of the 2009 recession with the 1983 recession.
• At the Plate, a Statistical Puzzler: Understanding Simpson's Paradox by Arthur Smith, August 20, 2010
• Simpson's Paradox, a video by Henry Reich of MinutePhysics
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Yule–Simon distribution
In probability and statistics, the Yule–Simon distribution is a discrete probability distribution named after Udny Yule and Herbert A. Simon. Simon originally called it the Yule distribution.[1]
Yule–Simon
Probability mass function
Yule–Simon PMF on a log-log scale. (Note that the function is only defined at integer values of k. The connecting lines do not indicate continuity.)
Cumulative distribution function
Yule–Simon CMF. (Note that the function is only defined at integer values of k. The connecting lines do not indicate continuity.)
Parameters $\rho >0\,$ shape (real)
Support $k\in \{1,2,\dotsc \}$
PMF $\rho \operatorname {B} (k,\rho +1)$
CDF $1-k\operatorname {B} (k,\rho +1)$
Mean ${\frac {\rho }{\rho -1}}$ for $\rho >1$
Mode $1$
Variance ${\frac {\rho ^{2}}{(\rho -1)^{2}(\rho -2)}}$ for $\rho >2$
Skewness ${\frac {(\rho +1)^{2}{\sqrt {\rho -2}}}{(\rho -3)\rho }}\,$ for $\rho >3$
Ex. kurtosis $\rho +3+{\frac {11\rho ^{3}-49\rho -22}{(\rho -4)(\rho -3)\rho }}$ for $\rho >4$
MGF does not exist
CF ${\frac {\rho }{\rho +1}}{}_{2}F_{1}(1,1;\rho +2;e^{i\,t})e^{i\,t}$
The probability mass function (pmf) of the Yule–Simon (ρ) distribution is
$f(k;\rho )=\rho \operatorname {B} (k,\rho +1),$
for integer $k\geq 1$ and real $\rho >0$, where $\operatorname {B} $ is the beta function. Equivalently the pmf can be written in terms of the rising factorial as
$f(k;\rho )={\frac {\rho \Gamma (\rho +1)}{(k+\rho )^{\underline {\rho +1}}}},$
where $\Gamma $ is the gamma function. Thus, if $\rho $ is an integer,
$f(k;\rho )={\frac {\rho \,\rho !\,(k-1)!}{(k+\rho )!}}.$ !\,(k-1)!}{(k+\rho )!}}.}
The parameter $\rho $ can be estimated using a fixed point algorithm.[2]
The probability mass function f has the property that for sufficiently large k we have
$f(k;\rho )\approx {\frac {\rho \Gamma (\rho +1)}{k^{\rho +1}}}\propto {\frac {1}{k^{\rho +1}}}.$
This means that the tail of the Yule–Simon distribution is a realization of Zipf's law: $f(k;\rho )$ can be used to model, for example, the relative frequency of the $k$th most frequent word in a large collection of text, which according to Zipf's law is inversely proportional to a (typically small) power of $k$.
Occurrence
The Yule–Simon distribution arose originally as the limiting distribution of a particular model studied by Udny Yule in 1925 to analyze the growth in the number of species per genus in some higher taxa of biotic organisms.[3] The Yule model makes use of two related Yule processes, where a Yule process is defined as a continuous time birth process which starts with one or more individuals. Yule proved that when time goes to infinity, the limit distribution of the number of species in a genus selected uniformly at random has a specific form and exhibits a power-law behavior in its tail. Thirty years later, the Nobel laureate Herbert A. Simon proposed a time-discrete preferential attachment model to describe the appearance of new words in a large piece of a text. Interestingly enough, the limit distribution of the number of occurrences of each word, when the number of words diverges, coincides with that of the number of species belonging to the randomly chosen genus in the Yule model, for a specific choice of the parameters. This fact explains the designation Yule–Simon distribution that is commonly assigned to that limit distribution. In the context of random graphs, the Barabási–Albert model also exhibits an asymptotic degree distribution that equals the Yule–Simon distribution in correspondence of a specific choice of the parameters and still presents power-law characteristics for more general choices of the parameters. The same happens also for other preferential attachment random graph models.[4]
The preferential attachment process can also be studied as an urn process in which balls are added to a growing number of urns, each ball being allocated to an urn with probability linear in the number (of balls) the urn already contains.
The distribution also arises as a compound distribution, in which the parameter of a geometric distribution is treated as a function of random variable having an exponential distribution. Specifically, assume that $W$ follows an exponential distribution with scale $1/\rho $ or rate $\rho $:
$W\sim \operatorname {Exponential} (\rho ),$
with density
$h(w;\rho )=\rho \exp(-\rho w).$
Then a Yule–Simon distributed variable K has the following geometric distribution conditional on W:
$K\sim \operatorname {Geometric} (\exp(-W)).$
The pmf of a geometric distribution is
$g(k;p)=p(1-p)^{k-1}$
for $k\in \{1,2,\dotsc \}$. The Yule–Simon pmf is then the following exponential-geometric compound distribution:
$f(k;\rho )=\int _{0}^{\infty }g(k;\exp(-w))h(w;\rho )\,dw.$
The maximum likelihood estimator for the parameter $\rho $ given the observations $k_{1},k_{2},k_{3},\dots ,k_{N}$ is the solution to the fixed point equation
$\rho ^{(t+1)}={\frac {N+a-1}{b+\sum _{i=1}^{N}\sum _{j=1}^{k_{i}}{\frac {1}{\rho ^{(t)}+j}}}},$
where $b=0,a=1$ are the rate and shape parameters of the gamma distribution prior on $\rho $.
This algorithm is derived by Garcia[2] by directly optimizing the likelihood. Roberts and Roberts[5]
generalize the algorithm to Bayesian settings with the compound geometric formulation described above. Additionally, Roberts and Roberts[5] are able to use the Expectation Maximisation (EM) framework to show convergence of the fixed point algorithm. Moreover, Roberts and Roberts[5] derive the sub-linearity of the convergence rate for the fixed point algorithm. Additionally, they use the EM formulation to give 2 alternate derivations of the standard error of the estimator from the fixed point equation. The variance of the $\lambda $ estimator is
$\operatorname {Var} ({\hat {\lambda }})={\frac {1}{{\frac {N}{{\hat {\lambda }}^{2}}}-\sum _{i=1}^{N}\sum _{j=1}^{k_{i}}{\frac {1}{({\hat {\lambda }}+j)^{2}}}}},$
the standard error is the square root of the quantity of this estimate divided by N.
Generalizations
The two-parameter generalization of the original Yule distribution replaces the beta function with an incomplete beta function. The probability mass function of the generalized Yule–Simon(ρ, α) distribution is defined as
$f(k;\rho ,\alpha )={\frac {\rho }{1-\alpha ^{\rho }}}\;\mathrm {B} _{1-\alpha }(k,\rho +1),\,$
with $0\leq \alpha <1$. For $\alpha =0$ the ordinary Yule–Simon(ρ) distribution is obtained as a special case. The use of the incomplete beta function has the effect of introducing an exponential cutoff in the upper tail.
See also
• Zeta distribution
• Scale-free network
• Beta negative binomial distribution
Bibliography
• Colin Rose and Murray D. Smith, Mathematical Statistics with Mathematica. New York: Springer, 2002, ISBN 0-387-95234-9. (See page 107, where it is called the "Yule distribution".)
References
1. Simon, H. A. (1955). "On a class of skew distribution functions". Biometrika. 42 (3–4): 425–440. doi:10.1093/biomet/42.3-4.425.
2. Garcia Garcia, Juan Manuel (2011). "A fixed-point algorithm to estimate the Yule-Simon distribution parameter". Applied Mathematics and Computation. 217 (21): 8560–8566. doi:10.1016/j.amc.2011.03.092.
3. Yule, G. U. (1924). "A Mathematical Theory of Evolution, based on the Conclusions of Dr. J. C. Willis, F.R.S". Philosophical Transactions of the Royal Society B. 213 (402–410): 21–87. doi:10.1098/rstb.1925.0002.
4. Pachon, Angelica; Polito, Federico; Sacerdote, Laura (2015). "Random Graphs Associated to Some Discrete and Continuous Time Preferential Attachment Models". Journal of Statistical Physics. 162 (6): 1608–1638. arXiv:1503.06150. doi:10.1007/s10955-016-1462-7. S2CID 119168040.
5. Roberts, Lucas; Roberts, Denisa (2017). "An Expectation Maximization Framework for Preferential Attachment Models". arXiv:1710.08511 [stat.CO].
Probability distributions (list)
Discrete
univariate
with finite
support
• Benford
• Bernoulli
• beta-binomial
• binomial
• categorical
• hypergeometric
• negative
• Poisson binomial
• Rademacher
• soliton
• discrete uniform
• Zipf
• Zipf–Mandelbrot
with infinite
support
• beta negative binomial
• Borel
• Conway–Maxwell–Poisson
• discrete phase-type
• Delaporte
• extended negative binomial
• Flory–Schulz
• Gauss–Kuzmin
• geometric
• logarithmic
• mixed Poisson
• negative binomial
• Panjer
• parabolic fractal
• Poisson
• Skellam
• Yule–Simon
• zeta
Continuous
univariate
supported on a
bounded interval
• arcsine
• ARGUS
• Balding–Nichols
• Bates
• beta
• beta rectangular
• continuous Bernoulli
• Irwin–Hall
• Kumaraswamy
• logit-normal
• noncentral beta
• PERT
• raised cosine
• reciprocal
• triangular
• U-quadratic
• uniform
• Wigner semicircle
supported on a
semi-infinite
interval
• Benini
• Benktander 1st kind
• Benktander 2nd kind
• beta prime
• Burr
• chi
• chi-squared
• noncentral
• inverse
• scaled
• Dagum
• Davis
• Erlang
• hyper
• exponential
• hyperexponential
• hypoexponential
• logarithmic
• F
• noncentral
• folded normal
• Fréchet
• gamma
• generalized
• inverse
• gamma/Gompertz
• Gompertz
• shifted
• half-logistic
• half-normal
• Hotelling's T-squared
• inverse Gaussian
• generalized
• Kolmogorov
• Lévy
• log-Cauchy
• log-Laplace
• log-logistic
• log-normal
• log-t
• Lomax
• matrix-exponential
• Maxwell–Boltzmann
• Maxwell–Jüttner
• Mittag-Leffler
• Nakagami
• Pareto
• phase-type
• Poly-Weibull
• Rayleigh
• relativistic Breit–Wigner
• Rice
• truncated normal
• type-2 Gumbel
• Weibull
• discrete
• Wilks's lambda
supported
on the whole
real line
• Cauchy
• exponential power
• Fisher's z
• Kaniadakis κ-Gaussian
• Gaussian q
• generalized normal
• generalized hyperbolic
• geometric stable
• Gumbel
• Holtsmark
• hyperbolic secant
• Johnson's SU
• Landau
• Laplace
• asymmetric
• logistic
• noncentral t
• normal (Gaussian)
• normal-inverse Gaussian
• skew normal
• slash
• stable
• Student's t
• Tracy–Widom
• variance-gamma
• Voigt
with support
whose type varies
• generalized chi-squared
• generalized extreme value
• generalized Pareto
• Marchenko–Pastur
• Kaniadakis κ-exponential
• Kaniadakis κ-Gamma
• Kaniadakis κ-Weibull
• Kaniadakis κ-Logistic
• Kaniadakis κ-Erlang
• q-exponential
• q-Gaussian
• q-Weibull
• shifted log-logistic
• Tukey lambda
Mixed
univariate
continuous-
discrete
• Rectified Gaussian
Multivariate
(joint)
• Discrete:
• Ewens
• multinomial
• Dirichlet
• negative
• Continuous:
• Dirichlet
• generalized
• multivariate Laplace
• multivariate normal
• multivariate stable
• multivariate t
• normal-gamma
• inverse
• Matrix-valued:
• LKJ
• matrix normal
• matrix t
• matrix gamma
• inverse
• Wishart
• normal
• inverse
• normal-inverse
• complex
Directional
Univariate (circular) directional
Circular uniform
univariate von Mises
wrapped normal
wrapped Cauchy
wrapped exponential
wrapped asymmetric Laplace
wrapped Lévy
Bivariate (spherical)
Kent
Bivariate (toroidal)
bivariate von Mises
Multivariate
von Mises–Fisher
Bingham
Degenerate
and singular
Degenerate
Dirac delta function
Singular
Cantor
Families
• Circular
• compound Poisson
• elliptical
• exponential
• natural exponential
• location–scale
• maximum entropy
• mixture
• Pearson
• Tweedie
• wrapped
• Category
• Commons
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Yuli Rudyak
Yuli B. Rudyak is a professor of mathematics at the University of Florida in Gainesville, Florida. He obtained his doctorate from Moscow State University under the supervision of M. M. Postnikov. His main research interests are geometry and topology and symplectic topology.
Books
• Rudyak, Yu. (1998), On Thom spectra, orientability, and cobordism. With a foreword by Haynes Miller, Springer Monographs in Mathematics, Berlin: Springer-Verlag
Reviewer Donald M. Davis (mathematician) for MathSciNet wrote: "This book provides an excellent and thorough treatment of various topics related to cobordism. It should become an indispensable tool for advanced graduate students and workers in algebraic topology."[1]
The book listed 118 cites at Google Scholar in 2011.[2]
Personal life
Rudyak is the father of Marina Rudyak, who is an Assistant Professor of Chinese Studies at the University of Heidelberg.
Notes
1. MR1627486
2. Google Scholar search results
External links
• Webpage at UFL
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Yulia Gel
Yulia R. Gel is a professor in the Department of Mathematical Sciences at the University of Texas at Dallas[1] and an adjunct professor in the Department of Statistics and Actuarial Science of the University of Waterloo.[2]
Yulia Gel
Education
• Saint Petersburg State University
• University of Washington
Known forTopological Data Analysis
Scientific career
FieldsStatistics
Institutions
• University of Texas at Dallas
• University of Waterloo
Academic advisorsVladimir N. Fomin
Early life and education
Gel earned her doctorate in mathematics at Saint Petersburg State University in Russia, under the supervision of Vladimir N. Fomin.[3] After postdoctoral research at the University of Washington, she joined the Waterloo faculty in 2004, and moved to Dallas in 2014.[4]
Research and career
Prior to joining the University of Texas at Dallas, Yulia Gel served as an Assistant/Associate Professor with tenure in the Department of Statistics and Actuarial Sciences at the University of Waterloo, Canada, from 2004 to 2014. She has also held visiting positions at prominent institutions such as NASA Jet Propulsion Lab (Caltech), the Isaac Newton Institute for Mathematical Sciences (Cambridge, UK), Johns Hopkins University, University of California at Berkeley, and George Washington University.
Yulia Gel has a diverse range of research interests that span statistical foundations of data science, machine learning, topological and geometric methods in statistics, and topological data analysis. Her work focuses on graph mining, inference for random graphs and complex networks, uncertainty quantification in network analysis, data depth on networks, time series analysis, spatio-temporal processes, and climate informatics. She is particularly interested in the application of statistical and data science techniques to domains such as healthcare predictive analytics and climate informatics.
Awards and honors
In 2014 Yulia was elected as a Fellow of the American Statistical Association" for theoretical contributions to nonparametric aspects of spatiotemporal processes; for promoting the application of modern statistical methodologies in law, public policy, and the environmental sciences; and for championing the advancement of women and other under-represented groups in the mathematical and physical sciences."[5]
References
1. Mathematics faculty and research, UT Dallas, retrieved 2016-07-12.
2. "Yulia Gel". Statistics and Actuarial Science. University of Waterloo. 23 February 2015. Retrieved 8 December 2017.
3. Yulia Gel at the Mathematics Genealogy Project
4. Golbeck, Amanda L.; Olkin, Ingram; Gel, Yulia R., eds. (2015), "About the Editors", Leadership and Women in Statistics, CRC Press, ISBN 9781482236453.
5. ASA Honors 63 New Fellows (PDF), American Statistical Association, June 11, 2014, retrieved 2016-07-11.
External links
• Google scholar profile
Authority control: Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
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Yuliya Mishura
Yuliya Stepanivna Mishura (Ukrainian: Юлія Степанівна Мішура) is a Ukrainian mathematician specializing in probability theory and mathematical finance. She is a professor at the Taras Shevchenko National University of Kyiv.[1]
Education and career
Mishura earned a Ph.D. in 1978 from the Taras Shevchenko National University of Kyiv with a dissertation on Limit Theorems for Functionals from Stochastic Fields supervised by Dmitrii Sergeevich Silvestrov. She earned a Dr. Sci. from the National Academy of Sciences of Ukraine in 1990 with a dissertation Martingale Methods in the Theory of Stochastic Fields.[1][2]
She became an assistant professor in the Faculty of Mechanics and Mathematics at National Taras Shevchenko University of Kyiv in 1976. She has been a full professor since 1991, and head of the Department of Probability, Statistics and Actuarial Mathematics since 2003.[1]
With Kęstutis Kubilius, she is the founding co-editor-in-chief of the journal Modern Stochastics: Theory and Applications.[3] She is the editor-in-chief of the journal Theory of Probability and Mathematical Statistics.
Books
Mishura is the author of many monographs and textbooks.[1] They include:
• Discrete-Time Approximations and Limit Theorems In Applications to Financial Markets (with Kostiantyn Ralchenko, De Gruyter Series in Probability and Stochastics, 2021)
• Asymptotic Analysis of Unstable Solutions of Stochastic Differential Equations (with G. Kulinich, S. Kushnirenko, Vol.9 Bocconi & Springer Series, Mathematics, Statistics, Finance and Economics, 2020)
• Fractional Brownian Motion. Approximations and Projections (with Oksana Banna, Kostiantyn Ralchenko, Sergiy Shklyar, Wiley-ISTE, 2019)
• Stochastic Analysis of Mixed Fractional Gaussian Processes (ISTE Press, 2018)[4]
• Theory and Statistical Applications of Stochastic Processes (with Georgiy Shevchenko, ISTE Press and John Wiley & Sons, 2017)
• Parameter Estimation in Fractional Diffusion Models (with Kęstutis Kubilius and Kostiantyn Ralchenko, Bocconi University Press and Springer, 2017)[5]
• Ruin Probabilities: Smoothness, Bounds, Supermartingale Approach (with Olena Ragulina, ISTE Press, 2016)
• Financial Mathematics: Optimization in Insurance and Finance Set (ISTE Press, 2016)[6]
• Theory of Stochastic Processes: With Applications to Financial Mathematics And Risk Theory (with Gusak, Kukush, Kulik, and Pilipenko, Problem Books in Mathematics, Springer, 2010)[7]
• Stochastic Calculus for Fractional Brownian Motion and Related Processes (Lecture Notes in Mathematics 1929, Springer, 2008)[8]
References
1. "Yuliya Mishura", Employee profile, Taras Shevchenko National University of Kyiv, retrieved 2020-03-29
2. Yuliya Mishura at the Mathematics Genealogy Project
3. "Editors-in-chief", Modern Stochastics: Theory and Applications, retrieved 2020-03-29
4. Review of Stochastic Analysis of Mixed Fractional Gaussian Processes:
• Aurzada, Frank, Mathematical Reviews, MR 3793191{{citation}}: CS1 maint: untitled periodical (link)
5. Reviews of Parameter Estimation in Fractional Diffusion Models:
• Kolnogorov, Alex V., zbMATH, Zbl 1388.60006{{citation}}: CS1 maint: untitled periodical (link)
• Lu, Fei, Mathematical Reviews, MR 3752152{{citation}}: CS1 maint: untitled periodical (link)
6. Review of Financial Mathematics:
• Vives, Josep, zbMATH, Zbl 1371.91001{{citation}}: CS1 maint: untitled periodical (link)
7. Reviews of Theory of Stochastic Processes:
• Hein, Claudia, zbMATH, Zbl 1189.60001{{citation}}: CS1 maint: untitled periodical (link)
• Hand, David J. (December 2010), International Statistical Review, 78 (3): 461, doi:10.1111/j.1751-5823.2010.00122_15.x, JSTOR 27919877{{citation}}: CS1 maint: untitled periodical (link)
• Castellacci, Giuseppe (2011), Mathematical Reviews, MR 2572942{{citation}}: CS1 maint: untitled periodical (link)
• Myers, Donald E. (August 2011), Technometrics, 53 (3): 324–325, JSTOR 23210411{{citation}}: CS1 maint: untitled periodical (link)
8. Reviews of Stochastic Calculus for Fractional Brownian Motion and Related Processes:
• Gapeev, Pavel, zbMATH, Zbl 1138.60006{{citation}}: CS1 maint: untitled periodical (link)
• Nourdin, Ivan (2008), Mathematical Reviews, MR 2378138{{citation}}: CS1 maint: untitled periodical (link)
External links
• Yuliya Mishura publications indexed by Google Scholar
Authority control
International
• VIAF
Academics
• DBLP
• Google Scholar
• MathSciNet
• Mathematics Genealogy Project
• Scopus
• zbMATH
Other
• IdRef
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Yunqing Tang
Yunqing Tang is a mathematician specialising in number theory and arithmetic geometry and an Assistant Professor at University of California, Berkeley. She was awarded the SASTRA Ramanujan Prize in 2022 for "having established, by herself and in collaboration, a number of striking results on some central problems in arithmetic geometry and number theory".[1][2]
Yunqing Tang was born in China and secured a BSc degree from Beijing University in 2011 and then moved to Harvard University for higher education from where she graduated with a Ph D degree in 2016 under the guidance of Mark Kisin. She was associated with Princeton University in several capacities. First she was with the IAS Princeton during 2016-2017, then as an instructor from July 2017 to Jan 2020 and then as an assistant professor from July 2021 to June 2022, In between, she worked as a researcher at CNRS from February 2020 to June 2021. She is with University of California, Berkeley since July 2022.[3] [4]
Work
The citation for SASTRA Ramnujan Prize summarizes Yunqing Tang's contributions to mathematics thus:[2]
"The prize notes that her works display a remarkable combination of sophisticated techniques, in which the arithmetic and geometry of modular curves and of Shimura varieties play a central role, and have strong links with the discoveries of Srinivasa Ramanujan in the area of modular equations. ... she established a new special case of the Ogus conjecture concerning cycles in de Rham cohomology of abelian varieties. She has shown that any abelian surface with real multiplication has infinitely many primes with split reduction. She resolved of the long-standing unbounded coefficient conjecture of Atkin and Swinnertin-Dyer that algebraic functions which are not invariant under any congruence subgroup of SL2(Z), must have unbounded denominators. The study of algebraic functions that are related to the moduli of elliptic integrals, stems from Ramanujan’s own investigations and the plethora of beautiful modular identities that he discovered."
Awards and recognition
The awards and recognition conferred on Yunqing Tang include:[4]
• SASTRA Ramanujan Prize, 2022.
• AWM Dissertation Prize, awarded for outstanding Ph.D dissertations by female students in the US, 2016.
• New World Mathematics Award, Gold Medal for Ph.D thesis awarded for outstanding Chinese mathematics students worldwide, 2016.
• Merit Research Fellowship, Graduate School of Arts and Sciences, Harvard University, 2015 – 2016.
References
1. The Hindu Bubeau (3 October 2022). "SASTRA Ramanujan Prize for 2022 goes to Yunqing Tang". The Hindu. Retrieved 8 November 2022.
2. "YUNQING TANG TO RECEIVE 2022 SASTRA RAMANUJAN PRIZE" (PDF). University of California, Berkeley. Retrieved 8 November 2022.
3. "Yunqing Tang". Princeton University. Retrieved 8 November 2022.
4. "Curriculun Vitae of Yunqing Tang" (PDF). Princeton University. Retrieved 8 November 2022.
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Yupana
A yupana (from Quechua: yupay 'count')[1] is a counting board used to perform arithmetic operations, dating back to the time of the Incas. Very little documentation exists concerning its precise physical form or how it was used.
See also: Mathematics of the Incas
Inca Empire
Inca society
• Education
• Religion
• Mythology
• Architecture
• Engineering
• Roads
• Army
• Agriculture
• Ayllu
• Cuisine
Inca history
• Kingdom of Cusco
• Inca Empire
• History of Cusco
• Chimor–Inca War
• Invasion of Chile
• Civil War
• Spanish conquest
• Neo-Inca State
Types
The term yupana refers to two distinct classes of objects:
• Table Yupana (or archaeological yupana): a system of geometric boxes of different sizes and materials. The first example of this type was found in 1869 in the Ecuadorian province of Azuay and prompted searches for more of these objects. All examples of the archaeological yupana vary greatly from each other.[2] Some archaeological yupanas found in Manchán (an archaeological site in Casma) and Huacones-Vilcahuasi (in Cañete) were embedded into the floor.
• Poma de Ayala Yupana: a picture on page 360 of El primer nueva corónica y buen gobierno, written by the Amerindian chronicler Felipe Guaman Poma de Ayala shows a 5x4 chessboard (shown right).[3] The chessboard, though resembling a table yupana, differs from this style in most notably in each of its rectangular trays have the same dimensions, while table yupanas have trays of other polygonal shapes of differing sizes.
Although very different from each other, most scholars who have dealt with table yupanas have extended reasoning and theories to the Poma de Ayala yupana and vice versa, perhaps in an attempt to find a unifying thread or a common method of creation. For example, the Nueva coronica (New Chronicle) discovered in 1916 in the library of Copenhagen contained evidence that a portion of the studies on the Poma de Ayala yupana were based on previous studies and theories regarding table yupanas.[2]
History
Several chroniclers of the Indies described, in brief, this Incan abacus and its operation.
Felipe Guaman Poma de Ayala
The first was Guaman Poma de Ayala around the year 1615 who wrote:
... They count using tables, numbered in increments one hundred thousand to ten thousand, one hundred to ten, and onward until they arrive at one. They keep records of everything that happens in this realm: holidays and Sundays, months and years. The accountants and treasurers of the kingdom are found in every city, town, or indigenous village...
— [3]
In addition to providing this brief description, Poma de Ayala drew a picture of the yupana: a board of five rows and four columns with each cell holding a series of black and white circles.
José de Acosta
Predating Pomo de Ayala's writings, in 1596 The Jesuit father José de Acosta wrote:
... Well, seeing another group which uses kernels of corn is an enchanting thing, as a very embarrassing account, which he will have a very good accountant do by pen and ink, to see how each contribution fits with so many people, taking so much from over there and adding so much from here, with another hundred small pieces, these Indians will take their kernels and put one here, three there, eight I don't know where; they will move a kernel from here, they will barter three from there, and, in fact, they leave with their account done punctually without missing a mark, and much more they know how to put into account and account for what each can pay or give, that we will know how to give to each of them as ascertained by pen and ink. If this is not ingenuity and these men are beasts, let whoever wishes to judge it so judge it, for what I judge to be true is that in what they apply they give us great advantages.
— [4]
Juan de Velasco
In 1841, Father Juan de Velasco wrote:
... these teachers were using something like a series of trays made of wood, stone, or clay, with different separations, in which they put stones of different shapes, colors and angularities...
— [5]
Table yupana
Various table yupana have been found across Ecuador and Peru.
The Chordeleg Yupana
The earliest known example of a table yupana was found in 1869 in Chordeleg, Azuay Province, Ecuador. A rectangular table (33x27 cm) of wood consisting of 17 compartments, 14 of which are square, 2 are rectangular, and one of which is octagonal. Two edges of the table contain other square compartments (12x12 cm) raised and arranged side by side, upon which two square platforms (7x7 cm), are superimposed. These structures are called "towers". The table's compartments are symmetrical with respect to the diagonal of the rectangular compartments. The four sides of the board are also engraved with images of human heads and a crocodile.[2] As a result of this discovery, Charles Wiener a systematic study of these objects in 1877. Wiener concluded that the table yupanas served to calculate the taxes that farmers paid to the Incan empire.[6]
The Caraz Yupana
Found at Caraz between 1878 and 1879, this table yupana differs from that of Chordeleg as the material of construction is the stone and the central octagonal compartment is replaced with a rectangular one; towers also have three shelves instead of two.[2]
The Callejón de Huaylas Yupana
A series of table yupanas much different from the first, was described by Erland Nordenskiöld in 1931. These yupana, made of stone, boast a series of rectangular and square compartments. The tower has two rectangular compartments. The compartments are arranged symmetrically with respect to the axis of the smaller side of the table.[2]
The Triangular Yupana
These yupana, made of stone, have 18 triangular compartments. On one side there is a rectangular tower with one level and three triangular compartments. In the central part there are four square compartments.[2]
The Chan Chan Yupana
Identical to the yupana of Chordeleg, both for the material and the arrangement of the compartments, this table yupana was found in the Chan Chan archaeological complex in Peru in 1967.[2]
The Cárhua de la Bahía Yupana
Discovered in the Peruvian province of Pisco, these are two table yupana in clay and bone. The first is rectangular (47x32 cm), has 22 square (5x5 cm) and three rectangular (16x18 cm) compartments, and has no towers. The second yupana is rectangular (32x23 cm) and has 22 square compartments, two L-shaped compartments and three rectangular compartments in the center. The compartments are arranged symmetrically with respect to the axis of the longer side.[2]
• Fig. A - Structure of a “Chordeleg” table-yupana. Colouring to differentiate the compartments.
• Fig. B - Identication of a stereotyped color
• Fig. C - Really existing tocapu catalogued by Victoria de la Jara
• Fig. D - Other tocapu pattern, possible stylization of the previous one
• Fig. E - Tocapu called “llave inca”, Inca key
The Huancarcuchu Yupana
Discovered in Northern Ecuador by Max Uhle in 1922, this yupana is made of stone and its compartments are drawn onto the surface of the tablet. It has the shape of a pyramid consisting of 10 overlapping rectangles: four on the first level, three on the second, two in the third and one in the fourth. This yupana is the one that is closest to the picture by Poma de Ayala in Nueva Coronica, while having a line fewer and being partially drawn.[2]
The Florio Yupana
C. Florio presents a study [7] which does not identify a yupana in these archaeological findings, but an object whose name has been forgotten and remains unknown. Instead, this object is used to connect to the tocapu (an ideogram already used by pre-Incas civilizations) called “llave inca” (i.e. Inca key) to the yanantin-masintin philosophy. The scholar justifies this based on from the lack of objective evidence that recognizes this object as a yupana, a belief that consolidated over years without repetition or demonstration of this hypothesis, and with the crossing of data from the Miccinelli Documents and the tocapu catalogued by Victoria de la Jara.
Supposing to color the different compartments of the table yupana (fig. A), C. Florio identifies a drawing (fig. B) very similar to an existing tocapu (fig. C) catalogued by Victoria de la Jara. In addition, in the tocapu reported in figure D, also catalogued by V. de la Jara, Florio identifies a stylization of tocapu C and the departure point for creating the tocapu “llave Inca” (Inca key). She finds the relation between the table yupana and the Inca key also similar in their connection with the concept of duality: the table yupana structure is clearly dual and Blas Valera in “Exsul Immeritus Blas Valera populo suo” (one of the two Miccinelli Documents) describes the "Inca key" tocapu as representing the concept of the “opposite forces” and the “number 2”, both strictly linked to the concept of duality.[8]
According to C. Florio, the real yupana used by the Incas is that of Guáman Poma, but with more columns and rows. The Poma de Ayala yupana would have represented just the part of the yupana useful for carrying out a specific calculation, which Florio identifies to be multiplication (see below).
Theories Based On the Poma de Ayala Yupana
Henry Wassen
In 1931, Henry Wassén studied the Poma de Ayala yupana, proposing for the first time a possible representation of the numbers on the board and the operations of addition and multiplication. He interpreted the white circles as gaps carved into yupana into which the seeds described by chroniclers would be inserted: so the white circles correspond to empty gaps, while the blacks circles correspond to the same gaps filled with a black seed.[2]
The numbering system at the base of the yupana was positional notation in base 10 (in line with the writings of the chroniclers of the Indies).
The representation of the numbers then followed a vertical progression such that the numbers 1-9 were positioned in the first row from the bottom, the second row contained the tens, the third contained the hundreds, and so on.
Wassen proposed a progression of values of the seeds that depends on their position in the table: 1, 5, 15, 30, respectively, depending on which seeds occupy a gap in the first, second, third and fourth columns (see the table below). Only a maximum of five seeds could be included in a box belonging to the first column, so that the maximum value of that box was 5, multiplied by the power of the corresponding row. These seeds could be replaced with one seed of the next column, useful during arithmetic operations. According to the theory of Wassen, therefore, the operations of sum and product were carried out horizontally.
This theory received a lot of criticism due to the high complexity of the calculations and was therefore considered inadequate and soon abandoned.
The following table shows the number 13457 as it would appear on Wassen's yupana:
Wassen's Yupana
Powers\Values151530
104 •◦◦◦◦ ◦◦◦ ◦◦ ◦
103 •••◦◦ ◦◦◦ ◦◦ ◦
102 ••••◦ ◦◦◦ ◦◦ ◦
101 ◦◦◦◦◦ •◦◦ ◦◦ ◦
100 ••◦◦◦ •◦◦ ◦◦ ◦
Representation of 13457
This first interpretation of the Poma de Ayala yupana was the starting point for the theories developed by subsequent authors, into the modern writing. No researcher moved away from the positional numbering system until 2008.
Emilio Mendizabal
Emilio Mendizabal was the first to propose in 1976 that the Inca a representation based on the progression 1, 2, 3, 5 in addition to the decimal representation.[9] In the same publication, Mendizabal pointed out that the series of numbers 1, 2, 3, and 5, appear in Poma de Ayala's drawing, and are part of the Fibonacci sequence, and stressed the importance of the "magic" that the number 5 contained for civilizations of Northern Peru, similar in significance to the number 8 for the civilizations of Southern Peru.[2]
Radicati di Primeglio
In 1979, Carlos Radicati di Primeglio emphasized the difference of table yupana from that of Poma de Ayala, describing the state-of-the-art research and advanced theories so far. He also proposed the algorithms for calculating the four basic arithmetic operations for the Poma de Ayala yupana, according to a new interpretation for which it was possible to have up to nine seeds in each box with a vertical progression of powers of ten.[2] Radicati associated each gap with a value of 1.
The following table shows the number 13457 as it would appear on Radicati's yupana:
Radicati's Yupana
Powers\Values1111
104 •◦◦◦◦
◦◦◦◦
◦◦◦◦◦
◦◦◦◦
◦◦◦◦◦
◦◦◦◦
◦◦◦◦◦
◦◦◦◦
103 •••◦◦
◦◦◦◦
◦◦◦◦◦
◦◦◦◦
◦◦◦◦◦
◦◦◦◦
◦◦◦◦◦
◦◦◦◦
102 ••••◦
◦◦◦◦
◦◦◦◦◦
◦◦◦◦
◦◦◦◦◦
◦◦◦◦
◦◦◦◦◦
◦◦◦◦
101 •••••
◦◦◦◦
◦◦◦◦◦
◦◦◦◦
◦◦◦◦◦
◦◦◦◦
◦◦◦◦◦
◦◦◦◦
100 •••••
••◦◦
◦◦◦◦◦
◦◦◦◦
◦◦◦◦◦
◦◦◦◦
◦◦◦◦◦
◦◦◦◦
Representation of 13457
William Burns Glynn
In 1981, the English textile engineer William Burns Glynn proposed a positional base 10 solution for the yupana of Poma de Ayala.[10]
Glynn, as Radicati, adopted Wassen's idea of full and empty gaps, as well as a vertical progression of the powers of ten, but proposed an architecture that allowed yupana users to greatly simplify the arithmetic operations themselves.
The horizontal progression of the values of the seeds in its representation is 1, 1, 1 for the first three columns, such that in each row is possible to deposit a maximum of ten seeds (5 + 3 + 2 seeds). Ten seeds in any row corresponds to a single seed in the line above it.
The last column in Glynn's yupana is dedicated to the "memory", a place that can hold up to ten seeds before they are moved to the upper line. According to the author, this is very useful during arithmetic operations in order to reduce the possibility of error.
Glynn's solution has been adopted in various teaching projects all over the world, and even today some of its variants are used in some schools of South America.[11][12]
The following table shows the number 13457 as it would appear on Glynn's yupana:
Glynn's Yupana
Potenze\Valori111Memoria
104 •◦◦◦◦ ◦◦◦ ◦◦ ◦
103 •••◦◦ ◦◦◦ ◦◦ ◦
102 ••••◦ ◦◦◦ ◦◦ ◦
101 ••••• ◦◦◦ ◦◦ ◦
100 ••••• ••◦ ◦◦ ◦
Nicolino de Pasquale
In 2001, the Italian engineer Nicolino de Pasquale proposed a positional solution in base 40 of the Poma de Ayala yupana, taking the representation theory of Fibonacci already proposed by Emilio Mendizabal and developing it for the four operations.
De Pasquale's yupana also adopts a vertical progression to represent numbers by powers of 40. The representation of the numbers is based on the fact that the sum of the values of the circles in each row is 39, if each circle takes the value 5 in the first column, 3 in the second column, 2 in the third and 1 in the fourth one; it is thus possible to represent 39 numbers, united to neutral element ( zero or "no seeds" in the table); this forms the basis of 40 symbols necessary for the numbering system.[13]
The following table shows one of the possible representations of the number 13457 in De Pasquale's yupana:
De Pasquale's Yupana
Powers\Values5321
404 ◦◦◦◦◦ ◦◦◦ ◦◦ ◦
403 ◦◦◦◦◦ ◦◦◦ ◦◦ ◦
402 •◦◦◦◦ ◦◦◦ •◦ •
401 ••◦◦◦ ••◦ ◦◦ ◦
400 ••◦◦◦ •◦◦ •• ◦
After its publication, De Pasquale's theory sparked great controversy among researchers who fell into two primary groups: a group supporting the base 10 theory and another supporting the base 40 theory. The Spanish chronicles written of the conquest of the Americas indicated that the Incas used a decimal system and since 2003 the base 10 theory has been proposed as the basis for calculating both with the abacus and the quipu[14]
De Pasquale has recently proposed the use of yupana as astronomical calendar running in mixed base 36/40[15] and provided his own interpretation of the Quechua word huno, translating it as "0.1".[16] This interpretation diverges from all chroniclers of the Indies, especially Domingo de Santo Tomas[1] who in 1560 translated huno into chunga guaranga (ten thousand).
Cinzia Florio
In 2008 Cinzia Florio proposed an alternative and revolutionary approach compared to all the theories proposed so far. Florio's newer theory deviated from the positional numbering system and adopted additive, or sign-value notation.[17]
Relying exclusively on Poma de Ayala's design, Florio explained the arrangement of white and black circles and interpreted the use of the yupana as a board for computing multiplications, in which the multiplicand is represented in the right column, the multiplier in the two central columns, and the product in the left column, illustrated in the following table:
Florio's Yupana
ProductMultiplierMultiplierMultiplicand
◦◦◦◦◦ ◦◦◦ ◦◦ ◦
◦◦◦◦◦ ◦◦◦ ◦◦ ◦
◦◦◦◦◦ ◦◦◦ ◦◦ ◦
◦◦◦◦◦ ◦◦◦ ◦◦ ◦
◦◦◦◦◦ ◦◦◦ ◦◦ ◦
The theory differs from all the previous in several aspects: first, the white and black circles would not be gaps that could be filled with a seed, but rather different colors of seeds, representing respectively tens and ones (this according to the chronicler Juan de Velasco).[5]
Secondly, the multiplicand is entered in the first column respecting the sign-value notation: so, the seeds can be entered in any order and the number is given by the sum of the values of these seeds.
The multiplier is represented as the sum of two factors, since the procedure for obtaining the product is based on the distributive property of multiplication over addition.
According to Florio, the multiplication table drawn by Poma de Ayala with provision of the seeds represented the calculation: 32 x 5, where the multiplier 5 is decomposed into 3 + 2. The sequence of numbers 1,2,3,5 would be causal, contingent to the calculation done and unrelated to the Fibonacci series.
Florio's Yupana
ProductMultiplicatorMultiplicatorMultiplicand
3X2X
◦◦◦•• ◦◦• •• ◦
◦◦◦◦• ◦◦• ◦◦ •
••••• ◦◦◦ ◦• ◦
◦◦◦◦• ◦◦• ◦• ◦
◦◦◦•• ••• ◦◦ •
151(160)966432
Key: ◦ = 10; • = 1; The operation represented is: 32 x 5 = 32 x (2 + 3) = (32 x 2) + (32 x 3) = 64 + 96 = 160
The numbers represented in the columns are, from left to right:
• 32 (the multiplicand),
• 64 = 32 x 2 and 32 x 3 = 96 (which together constitute the multiplicand, multiplied by the two factors in which the multiplier has been broken down)
• 151 (the product)
The final number in this computation (which is incorrect) is the basis for all possible criticisms of this interpretation, since 160, not 151, is the sum of 96 and 64. Florio notes, however, that the mistake could have been on the part of Poma de Ayala in the original drawing, in designing a space as being occupied by a black circle instead of a white one. In this case, changing just one black circle into a white one in the final column gives us the number 160, the correct product.
Poma de Ayala's yupana cannot represent every multiplicand either, it is necessary to extend the yupana vertically (adding rows) to represent numbers whose sum of digits exceeds 5. The case is the same for the multipliers: to represent all the numbers is necessary to extend the number of columns. Apart from the supposed erroneous calculation (or erroneous representation by the designer), this is the only theory that identifies in Poma de Ayala's yupana a mathematical and consistent message (multiplication) and not a series of random numbers as in other interpretations.
Andrés Chirinos (2010)
In October 2010, Peruvian researcher Andrés Chirinos with the support of the Spanish Agency for International Development Cooperation (ACEID), revised older drawings and descriptions chronicled by Poma de Ayala, and finally deciphered the use of the yupana: a table with eleven holes which Chirinos calls a "Pre-Columbian Calculator", capable of adding, subtracting, dividing, and multiplying, making him hopeful of applying this information to the investigation of how quipus were used and functioned.[18]
See also
• Quipu
• Inca Empire
• Numbering System
References
1. Santo Tomas, "Lexicon o Vocabulario de la lengua general del Peru", 1560
2. Radicati di Primeglio, "Il sistema contabile degli Inca: Yupana e Quipu", 1979
3. Guaman Poma de Ayala, "Primer Nueva Coronica y Buen Gobierno", 1615
4. José de Acosta - Historia Natural y Moral de las Indias - Libro VI cap XVIII (De los memoriales y cuentas que usaron los Indios del Perú)
5. Juan Velasco - “Historia del Reino de Quito” - 1841 44, Tomo II, 7
6. Wong Torres, Zelma (2014-03-16). "Origen de la Contabilidad a Traves del Tiempo". Quipukamayoc. 11 (21): 105. doi:10.15381/quipu.v11i21.5496. ISSN 1609-8196.
7. C. Florio, "Recovering memory - The Inca Key as Yanantin"
8. piruanorum., Laurencich Minelli, Laura, ed. lit. Valera, Blas. Exsul Immeritus Blas valera Populu Suo. Cumis, Joan Antonio. Historia et rudimenta linguae (2009). Exsul Immeritus Blas Valera populo suo e Historia et rudimenta linguae piruanorum : indios, gesuiti e spagnoli in due documenti segreti sul Perù del XVII secolo. Cooperativa Libraria Universitaria Editrice Bologna. OCLC 912444132.{{cite book}}: CS1 maint: multiple names: authors list (link)
9. Emilio., Mendizábal Losack (1976). La pasión racionalista andina. [Universidad Nacional Mayor de San Marcos]. OCLC 10567025.
10. William Burns Glynn, "Calculation table of the Incas", Bol. Lima No. 11, 1981, 1-15.
11. Mora & Valero "La Yupana come strumento pedagogico alle elementari"
12. Fiorentino, "La yupana elettronica: uno strumento per la didattica interculturale della matematica"
13. N. De Pasquale "Il volo del condor", Pescara Informa, 2001
14. Lorenzi, Incan counting system as easy as 1,2,3,5 (2004)
15. N. De Pasquale, "The Saved Kingdom"
16. N. De Pasquale, "Decimal Guaman Poma"
17. C. Florio, "Incontri e disincontri nella individuazione di una relazione matematica nella yupana in Guaman Poma de Ayala", Salerno, 14-15 maggio e 10-12 Dicembre 2008 - Oédipus Editore, 2009
18. Vega, Beatriz (2010-11-20). "Lo Relativo en la Matemática. El Caso de la Proporcionalidad en el 3° Ciclo de la EGB". Yupana (5): 41–52. doi:10.14409/yu.v1i5.260. ISSN 2362-5562.
External links
• Gilsdorf - Ethnomathematics of the Inkas
• Heliane Seline - Mathematics through cultures
• O'Connor & Robertson - Mathematics of the Incas
Chroniclers of the Indies
• (in Spanish) Poma de Ayala - El Primer Nueva Coronica y Buen Gobierno
• (in Spanish) José De Acosta - Historia Natural y Moral de las Indias
• (in Spanish) Velasco - Historia del reyno de Quito del America del Sur
Theory by Wassen and table-Yupana
• (in Spanish) Radicati di Primeglio - El sistema contable de los Incas: Yupana y Quipu
Theory by Glynn Burns and school projects
• (in Spanish) Mora & Valero - La Yupana come strumento pedagogico alle elementari
• Leonard & Shakiban - The Incan Abacus
• (in Italian) Fiorentino - La yupana elettronica: uno strumento per la didattica interculturale della matematica
Theory by De Pasquale
• (in Italian) Università Bocconi di Milano - La Matematica nelle civiltà pre-colombiane
• (in English)Incan counting system as easy as 1,2,3,5 - by Rossella Lorenzi
• (in Italian) Notizie sulla numerazione Inca e sulla yupana
• (in Italian) Un italiano scopre l'enigma della matematica inca
• (in Italian) Il Sole 24 Ore Domenica 10 Novembre 2002 – N. 308 – Pagina 35 - di Antonio Aimi - SCIENZA E FILOSOFIA Matematica precolombiana Scoperto il metodo di calcolo degli Inca
• (in Italian) L'unione Sarda - I numeri della natura nella scacchiera degli Inca - di Andrea Mameli
• (in English) "Guaman Poma Game, by N. De Pasquale, D. D'Ottavio
Theory by C. Florio
• (in Italian) Florio - Incontri e disincontri nella individuazione di una relazione matematica nella yupana in Guaman Poma de Ayala
• (in Spanish) Florio - Encuentros y Desencuentros en la identificación de unarelación matemática en la yupana de Guaman Poma de Ayala
Inca Empire
History
• Sapa Inca
• Kingdom of Cusco
• Inca Empire
• History of Cusco
• Chimor–Inca War
• Invasion of Chile
• Inca Civil War
• Spanish conquest
• Ransom Room
• Neo-Inca State
Inca society
• Inca education
• Aclla
• Amauta
• Ayllu
• Chasqui
• Mitma
• Ñusta
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Inca mathematics
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Yuri A. Kuznetsov
Yuri A. Kuznetsov is a Russian-American mathematician currently the M. D. Anderson Chair Professor of Mathematics at University of Houston and Editor-in-Chief of Journal of Numerical Mathematics.[1][2]
References
1. "Yuri A. Kuznetsov". uh.edu. Retrieved May 10, 2017.
2. "Yuri Kuznetsov". ras.ru. Retrieved May 10, 2017.
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Yuri Burago
Yuri Dmitrievich Burago (Russian: Ю́рий Дми́триевич Бура́го; born 1936) is a Russian mathematician. He works in differential and convex geometry.
Yuri Burago
Yuri D. Burago at Oberwolfach in 2006. Photo courtesy MFO.
Born1936
NationalityRussian
Alma materSt. Petersburg State University
AwardsLeroy P. Steele Prize (2014)[1]
Scientific career
FieldsMathematics
InstitutionsSt. Petersburg State University
Doctoral advisorVictor Zalgaller
Aleksandr Aleksandrov
Doctoral studentsSergei Ivanov
Grigori Perelman
Education and career
Burago studied at Leningrad University, where he obtained his Ph.D. and Habilitation degrees. His advisors were Victor Zalgaller and Aleksandr Aleksandrov.
Yuri is a creator (with his students Perelman and Petrunin, and M. Gromov) of what is known now as Alexandrov Geometry. Also brought geometric inequalities to the state of art.
Burago is the head of the Laboratory of Geometry and Topology that is part of the St. Petersburg Department of Steklov Institute of Mathematics.[2] He took part in a report for the United States Civilian Research and Development Foundation for the Independent States of the former Soviet Union.[3]
Works
• Burago, Dmitri; Yuri Burago; Sergei Ivanov (2001-06-12) [1984]. A Course in Metric Geometry. American Mathematical Society (publisher). pp. 417. ISBN 978-0-8218-2129-9.
• Burago, Yuri; Zalgaller, Victor (February 1988) [1980]. Geometric Inequalities. Transl. from Russian by A.B. Sossinsky. Springer Verlag. ISBN 3-540-13615-0.[4]
His other books and papers include:
• Geometry III: Theory of Surfaces (1992)[4]
• Potential Theory and Function Theory for Irregular Regions (1969)[4]
• Isoperimetric inequalities in the theory of surfaces of bounded external curvature (1970)[4]
Students
He has advised Grigori Perelman, who solved the Poincaré conjecture, one of the seven Millennium Prize Problems. Burago was an advisor to Perelman during the latter's post-graduate research at St. Petersburg Department of Steklov Institute of Mathematics.
Footnotes
1. "The Leroy P Steele Prize of the AMS". n.d. Retrieved 5 January 2023.
2. Laboratory of Geometry and Topology
3. U.S. Civilian Research and Development Foundation for the Independent States of the former Soviet Union Archived December 7, 2006, at the Wayback Machine. 2001 Program Report.
4. Bibliography
External links
• Burago's page on the site of Steklov Mathematical Institute
• Yuri Burago at the Mathematics Genealogy Project
• Yuri Dmitrievich Burago in the Oberwolfach Photo Collection
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Yuri Luchko
Yuri Luchko is a German professor of mathematics at the Berlin University of Applied Sciences and Technology. His 90 works were peer-reviewed and appeared in such journals as the Fractional Calculus and Applied Analysis and Journal of Mathematical Analysis and Applications, among others.[1]
References
1. "Yuri Luchko". Google Scholar. Retrieved December 11, 2013.
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Yuri Petunin
Yuri Ivanovich Petunin (Russian: Юрий Иванович Петунин) was a Soviet and Ukrainian mathematician. Petunin was born in the city of Michurinsk (USSR) on September 30, 1937. After graduating from the Tambov State Pedagogical Institute he began his studies at Voronezh State University under the supervision of S.G Krein. He completed his postgraduate studies in 1962, and in 1968 he received his Doctor of Science Degree, the highest scientific degree awarded in the Soviet Union. In 1970 he joined the faculty of the computational mathematics department at Kyiv State University.
Yuri Ivanovich Petunin
Born(1937-09-30)September 30, 1937
Michurinsk, Tambov Oblast, USSR
DiedJune 1, 2011(2011-06-01) (aged 73)
Kyiv, Ukraine
NationalitySoviet Union
Ukraine
Known forFunctional analysis, Mathematical statistics, Biology
Scientific career
FieldsMathematician
InstitutionsKyiv State University
Doctoral advisorSelim Krein
Yuri Petunin is highly regarded for his results in functional analysis. He developed the theory of Scales in Banach spaces,[1] the theory of characteristics of linear manifolds in conjugate Banach spaces,[2] and with S.G. Krein and E.M. Semenov contributed to the theory of interpolation of linear operators.[3] He solved Banach's problem of norming subspaces in conjugate Banach spaces[2] as well as a problem posted by Calderón and Lions concerning interpolation in factor spaces.[3]
In addition to his work in functional analysis, Professor Petunin made significant contributions to pattern recognition and mathematical statistics. He also worked on developing differential diagnostics for oncological disease.[4] The Vysochanskij–Petunin inequality that bears his name formally justifies the so-called 3-sigma rule for unimodal distributions, a rule that has been broadly used in statistics since the time of Gauss. In the area of pattern recognition he developed a theory of linear discriminant rules where he investigated the problems of linear separability of any number of sets in n-dimensional space.[5]
In his later years Yuri Petunin returned to the area of functional analysis where he had begun his scientific research. Together with his colleagues at the department of computational mathematics, he successfully worked toward a solution of Hilbert's 20th problem.[6]
See also
• Vysochanskii–Petunin inequality
References
1. S G Krein and Yu I Petunin, Scales of Banach spaces, 1966 Russ. Math. Surv. 21, 85–129
2. Yu. I. Petunin and A. N. Plichko, The Theory of the Characteristics of Subspaces and Its Applications [in Russian], Vishcha Shkola, Kyiv (1980)
3. S.G. Krein, Ju.I. Petunin, E.M. Semenov, Interpolation of linear operators, Providence, R.I. : American Mathematical Society, 1982. vii, 375 p.,ISBN 0821845047
4. R.I. Andrushkiw, N.V. Boroday, D.A. Klyushin, Yu.I. Petunin. Computer-aided cytogenetic method of cancer diagnosis. — New York: Nova Publishers, 2007.
5. Yu. I. Petunin and G.A Shuldeshov, The theory of linear discriminant functions I,II, Cybernetics (Russian) no1,2, pp. 34–44, 31–35, 1981.
6. D.A. Klyushin, S.I. Lyasko, D.A. Nomirovskii, Yu.I. Petunin, Vladimir Semenov, Generalized Solutions of Operator Equations and Extreme Elements (Springer Optimization and Its Applications, Vol. 55)
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Gradshteyn and Ryzhik
Gradshteyn and Ryzhik (GR) is the informal name of a comprehensive table of integrals originally compiled by the Russian mathematicians I. S. Gradshteyn and I. M. Ryzhik. Its full title today is Table of Integrals, Series, and Products.
Table of Integrals, Series, and Products
Gradshteyn and Ryzhik, seventh English edition, 2007
AuthorRyzhik, Gradshteyn, Geronimus, Tseytlin et al.
CountryRussia
LanguageRussian, German, Polish, English, Japanese, Chinese
GenreMath
PublisherAcademic Press
Publication date
1943, 2015
Since its first publication in 1943, it was considerably expanded and it soon became a "classic" and highly regarded reference for mathematicians, scientists and engineers. After the deaths of the original authors, the work was maintained and further expanded by other editors.
At some stage a German and English dual-language translation became available, followed by Polish, English-only and Japanese versions. After several further editions, the Russian and German-English versions went out of print and have not been updated after the fall of the Iron Curtain, but the English version is still being actively maintained and refined by new editors, and it has recently been retranslated back into Russian as well.
Overview
One of the valuable characteristics of Gradshteyn and Ryzhik compared to similar compilations is that most listed integrals are referenced. The literature list contains 92 main entries and 140 additional entries (in the eighth English edition). The integrals are classified by numbers, which haven't changed from the fourth Russian up to the seventh English edition (the numbering in older editions as well as in the eighth English edition is not fully compatible). The book does not only contain the integrals, but also lists additional properties and related special functions. It also includes tables for integral transforms. Another advantage of Gradshteyn and Ryzhik compared to computer algebra systems is the fact that all special functions and constants used in the evaluation of the integrals are listed in a registry as well, thereby allowing reverse lookup of integrals based on special functions or constants.
On the downsides, Gradshteyn and Ryzhik has become known to contain a relatively high number of typographical errors even in newer editions, which has repeatedly led to the publication of extensive errata lists. Earlier English editions were also criticized for their poor translation of mathematical terms[1][2][3] and mediocre print quality.[1][2][4][5]
History
The work was originally compiled by the Russian mathematicians Iosif Moiseevich Ryzhik (Russian: Иосиф Моисеевич Рыжик, German: Jossif Moissejewitsch Ryschik)[6][nb 1] and Izrail Solomonovich Gradshteyn (Russian: Израиль Соломонович Градштейн, German: Israil Solomonowitsch Gradstein).[6][nb 2] While some contents were original, significant portions were collected from other previously existing integral tables like David Bierens de Haan's Nouvelles tables d'intégrales définies (1867),[6][7] Václav Jan Láska's Sammlung von Formeln der reinen und angewandten Mathematik (1888–1894)[6][8] or Edwin Plimpton Adams' and Richard Lionel Hippisley's Smithsonian Mathematical Formulae and Tables of Elliptic Functions (1922).[6][9]
The first edition, which contained about 5 000 formulas,[10][11][nb 3] was authored by Ryzhik,[nb 1] who had already published a book on special functions in 1936[6][12] and died during World War II around 1941.[6] Not announcing this fact, his compilation was published posthumously[6][nb 1] in 1943, followed by a second corrected edition in his name in 1948.[nb 4]
The third edition (1951) was worked on by Gradshteyn,[13] who also introduced the chapter numbering system in decimal notation. Gradshteyn planned considerable expansion for the fourth edition, a work he could not finish due to his own death.[6][nb 2] Therefore, the fourth (1962/1963) and fifth (1971) editions were continued by Yuri Veniaminovich Geronimus (Russian: Юрий Вениаминович Геронимус, German: Juri Weniaminowitsch Geronimus)[6][nb 5] and Michail Yulyevich Tseytlin (Russian: Михаил Ю́льевич Цейтлин, German: Michael Juljewitsch Zeitlin).[nb 6] The fourth edition contained about 12 000 formulas already.[14][nb 3]
Based on the third Russian edition, the first German-English edition with 5 400 formulas[15][nb 3] was published in 1957 by the East-German Deutscher Verlag der Wissenschaften (DVW) with German translations by Christa[nb 7] and Lothar Berg[nb 8] and the English texts by Martin Strauss.[nb 9] In Zentralblatt für Mathematik Karl Prachar wrote:[16]
"Die sehr reichhaltigen Tafeln wurden nur aus dem Russischen ins Deutsche und Englische übersetzt."
(Translation: The very comprehensive tables were only translated into German and English language.)
In 1963, it was followed by the second edition, a reprint edition with a four-page inlet listing corrections compiled by Eldon Robert Hansen.
Derived from the 1963 edition, but considerably expanded and split into two volumes, the third German-English edition by Ludwig Boll[nb 10] was finally published by MIR Moscow in 1981 (with permission of DVW and distributed through Verlag Harri Deutsch in the Western world); it incorporated the material of the fifth Russian edition (1971) as well.[nb 11]
Pending this third German-English edition an English-only edition by Alan Jeffrey[nb 12] was published in 1965. Lacking a clear designation by itself it was variously known as first, third or fourth English edition, as it was based on the then-current fourth Russian edition. The formulas were photographically reproduced and the text translated. This still held true for the expanded fourth English edition in 1980, which added chapters 10 to 17.[17]
Both of these editions saw multiple print runs each incorporating newly found corrections. Starting with the third printing, updated table entries were marked by adding a small superscript number to the entry number indicating the corresponding print run ("3" etc.), a convention carried over into later editions by continuing to increase the superscript number as kind of a revision number (no longer directly corresponding with actual print runs).
The fifth edition (1994), which contained close to 20 000 formulas,[18][nb 3] was electronically reset[3] in preparation for a CD-ROM issue of the fifth edition (1996) and in anticipation of further editions. Since the sixth edition (2000), now corresponding with superscript number "10", Daniel Zwillinger[nb 13] started contributing as well. The last edition being edited by Jeffrey before his death[nb 12] was the seventh English edition published in 2007 (with superscript number "11").[19] This edition has been retranslated back into Russian as "seventh Russian edition" in 2011.[20][nb 11]
For the eighth edition (2014/2015, with superscript number "12") Zwillinger took over the role of the editor. He was assisted by Victor Hugo Moll.[21][nb 14] In order to make room for additional information without increasing the size of the book significantly, the former chapters 11 (on algebraic inequalities), chapters 13 to 16 (on matrices and related results, determinants, norms, ordinary differential equations) and chapter 18 (on z-transforms) worth about 50 pages in total were removed and some chapters renumbered (12 to 11, 17 to 12). This edition contains more than 10 000 entries.[21][nb 3]
Related projects
In 1995, Alan Jeffrey published his Handbook of Mathematical Formulas and Integrals.[22] It was partially based on the fifth English edition of Gradshteyn and Ryzhik's Table of Integrals, Series, and Products and meant as an companion, but written to be more accessible for students and practitioners.[22] It went through four editions up to 2008.[22][23][24][25] The fourth edition also took advantage of changes incorporated into the seventh English edition of Gradshteyn and Ryzhik.[25]
Inspired by a 1988 paper in which Ilan Vardi proved several integrals in Gradshteyn and Ryzhik,[26] Victor Hugo Moll with George Boros started a project to prove all integrals listed in Gradshteyn and Ryzhik and add additional commentary and references.[27] In the foreword of the book Irresistible Integrals (2004), they wrote:[28]
It took a short time to realize that this task was monumental.
Nevertheless, the efforts have meanwhile resulted in about 900 entries from Gradshteyn and Ryzhik discussed in a series of more than 30 articles[29][30][31] of which papers 1 to 28[lower-alpha 1] have been published in issues 14 to 26 of Scientia, Universidad Técnica Federico Santa María (UTFSM), between 2007 and 2015[60] and compiled into a two-volume book series Special Integrals of Gradshteyn and Ryzhik: the Proofs (2014–2015) already.[61][62]
Editions
Russian editions
• Рыжик, И. М. (1943). Таблицы интегралов, сумм, рядов и произведений [Tables of integrals, sums, series and products] (in Russian) (1 ed.). Moscow: Gosudarstvennoe Izdatel'stvo Tehniko-Teoretičeskoj Literatury (Государственное издательство технико-теоретической литературы) (GITTL / ГИТТЛ) (Tehteoretizdat / Техтеоретиздат). LCCN ltf89006085. 400 pages.[10][63]
• Рыжик, И. М. (1948). Таблицы интегралов, сумм, рядов и произведений (in Russian) (2 ed.). Moscow: Gosudarstvennoe Izdatel'stvo Tehniko-Teoretičeskoj Literatury (Государственное издательство технико-теоретической литературы) (GITTL / ГИТТЛ) (Tehteoretizdat / Техтеоретиздат). 400 pages.[11]
• Рыжик, И. М.; Градштейн, И. С. (1951). Таблицы интегралов, сумм, рядов и произведений (in Russian) (3 ed.). Moscow: Gosudarstvennoe Izdatel'stvo Tehniko-Teoretičeskoj Literatury (Государственное издательство технико-теоретической литературы) (GITTL / ГИТТЛ) (Tehteoretizdat / Техтеоретиздат). LCCN 52034158. 464 pages (+ errata inlet).[63][64][65]
• Градштейн, И. С.; Рыжик, И. М. (1963) [1962]. Геронимус, Ю. В.; Цейтлин, М. Ю́. (eds.). Tablitsy Integralov, Summ, Riadov I Proizvedenii Таблицы интегралов, сумм, рядов и произведений (in Russian) (4 ed.). Moscow: Gosudarstvennoe Izdatel'stvo Fiziko-Matematicheskoy Literatury (Государственное издательство физико-математической литературы) (Fizmatgiz / Физматгиз). LCCN 63027211. 1100 pages (+ 8 page errata inlet in later print runs).[14] Errata:[66]
• Градштейн, И. С.; Рыжик, И. М. (1971). Геронимус, Ю. В.; Цейтлин, М. Ю́. (eds.). Таблицы интегралов, сумм, рядов и произведений (in Russian) (5 ed.). Nauka (Наука). LCCN 78876185. Dark brown fake-leather, 1108 pages.[nb 11]
• Градштейн, И. С.; Рыжик, И. М.; Геронимус, Ю. В.; Цейтлин, М. Ю́. (2011). Jeffrey, Alan; Zwillinger, Daniel (eds.). Таблицы интегралов, ряда и продуктов [Table of Integrals, Series, and Products] (in Russian). Translated by Maximov, Vasily Vasilyevich [in Russian] (7 ed.). Saint-Petersburg, Russia: BHV (БХВ-Петербург). ISBN 978-5-9775-0360-0. GR:11. l+1182 pages.[20][nb 11]
German editions
• Ryshik, Jossif Moissejewitsch; Gradstein, Israil Solomonowitsch (1957). Tafeln / Tables: Summen-, Produkt- und Integral-Tafeln / Tables of Series, Products, and Integrals (in German and English). Translated by Berg, Christa; Berg, Lothar; Strauss, Martin. Foreword by Schröder, Kurt Erich (1 ed.). Berlin, Germany: Deutscher Verlag der Wissenschaften. LCCN 58028629. DNB-IDN 454242255, Lizenz-Nr. 206, 435/2/57. Retrieved 2016-02-21. Gray or green linen, xxiii+438 pages.[15][16] Errata:[65][67][68][69][70][71][72]
• Ryshik, Jossif Moissejewitsch; Gradstein, Israil Solomonowitsch (1963). Tafeln / Tables: Summen-, Produkt- und Integral-Tafeln / Tables of Series, Products, and Integrals (in German and English). Translated by Berg, Christa; Berg, Lothar; Strauss, Martin. Foreword by Schröder, Kurt Erich (2nd corrected ed.). Berlin, Germany: VEB Deutscher Verlag der Wissenschaften (DVW). LCCN 63025905. DNB-IDN 579497747, Lizenz-Nr. 206, 435/93/63. Retrieved 2016-02-21. (Printed by VEB Druckerei "Thomas Müntzer", Bad Langensalza. Distributed in the USA by Plenum Press, Inc., New York.) Green linen, xxiii+438 pages + 4 page errata inlet. Errata:[70]
• Gradstein, Israil Solomonowitsch; Ryshik, Jossif Moissejewitsch (1981). Geronimus, Juri Weneaminowitsch; Zeitlin, Michael Juljewitsch (eds.). Tafeln / Tables: Summen-, Produkt- und Integraltafeln / Tables of Series, Products, and Integrals (in German and English). Translated by Boll, Ludwig (3 ed.). Berlin / Thun / Frankfurt am Main / Moscow: Verlag Harri Deutsch / Verlag MIR Moscow. ISBN 3-87144-350-6. LCCN 82202345. DNB-IDN 551448512, 881086274, 881086282. Gray linen with gilded embossing by A. W. Schipow, 2 volumes, 677+3 & 504 pages.[73][74]
Polish edition
• Ryżyk, I. M.; Gradsztejn, I. S. (1964). Tablice całek, sum, szeregów i iloczynów (in Polish). Translated by Malesiński, Roman (1 ed.). Warsaw, Poland: Państwowe Wydawnictwo Naukowe (PWN). OCLC 749996828. VIAF 309184374. Retrieved 2016-02-16. Light grayish cover, 464 pages.
English editions
• Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich (February 1966) [1965]. Jeffrey, Alan (ed.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (3 ed.). Academic Press. ISBN 0-12-294750-9. LCCN 65029097. Retrieved 2016-02-21. Black cloth hardcover with gilt titles, white dust jacket, xiv+1086 pages.[1] Errata:[1][72][75][76][77][78][79][80][81][82][83][84][85][86][87][88][89][90][91][92]
• Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich (1980). Jeffrey, Alan (ed.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (4th corrected and enlarged ed.). Academic Press, Inc. ISBN 978-0-12-294760-5. GR:4,5,6,7. Retrieved 2016-02-21. xlvi+1160 pages.[2][17] Errata:[2][17][93][94][95][96][97][98][99][100][101]
• Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich (January 1994). Jeffrey, Alan (ed.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (5 ed.). Academic Press, Inc. MR 1243179. Retrieved 2016-02-21. Blue hardcover with green or blue rectangular and gilt titles, xlvii+1204 pages.[3][18][4][5] (A CD-ROM version with ISBN 0-12-294756-8 / ISBN 978-0-12-294756-8 and LCCN 96-801532 was prepared by Lightbinders, Inc. in July 1996.[102][103][4][5]) Errata:[3][104][105][106][107][4][5][108]
• Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich (July 2000). Jeffrey, Alan; Zwillinger, Daniel (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (6 ed.). Academic Press, Inc. ISBN 978-0-12-294757-5. MR 1773820. GR:10. Retrieved 2016-02-21. Red cover, xlvii+1163 pages.[109] (A reprint edition "积分, 级数和乘积表" by World Books Press became available in China under ISBN 7-5062-6546-X / ISBN 978-7-5062-6546-1 in April 2004.) Errata:[32][41][45][109][110][111][112][113]
• Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich (February 2007). Jeffrey, Alan; Zwillinger, Daniel (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (7 ed.). Academic Press, Inc. ISBN 978-0-12-373637-6. MR 2360010. GR:11. Retrieved 2016-02-21. xlviii+1171 pages, with CD-ROM.[19][114] (A reprint edition "积分, 级数和乘积表" by Beijing World Publishing Corporation (世界图书出版公司北京公司 / WPCBJ) became available in China under ISBN 7-5062-8235-6 / ISBN 978-7-5062-8235-2 in May 2007.) Errata:[41][45][47][51][53][57][59][115]
• Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica (8 ed.). Academic Press, Inc. ISBN 978-0-12-384933-5. GR:12. Retrieved 2016-02-21. xlvi+1133 pages.[21] Errata:[116][30][117]
Japanese edition
• Градштейн (Guradoshu グラドシュ), И. С.; Рыжик (Rijiku リジク), И. М. (December 1983). Sūgaku daikōshikishū 数学大公式集 [Large mathematics collection] (in Japanese). Translated by Otsuki, Yoshihiko (大槻 義彦) [in Japanese] (1 ed.). Tokyo, Japan: Maruzen (丸善). ISBN 978-4-621-02796-7. NCID BN00561932. JPNO JP84018271. Retrieved 2016-04-06. xv+1085 pages.
See also
• Prudnikov, Brychkov and Marichev (PBM)
• Bronshtein and Semendyayev (BS)
• Abramowitz and Stegun (AS)
• NIST Handbook of Mathematical Functions (DLMF)
• Jahnke and Emde (JE)
• Magnus and Oberhettinger (MO)
Notes
1. [32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59]
1. Iosif Moiseevich Ryzhik (Иосиф Моисеевич Рыжик)[nb 15] (1918?–1941?). VIAF 15286520. GND 107340518, 1087809320. (NB. Some sources identify him as a sergeant (сержантом) born in 1918, originally from Vitebsk (Витебска), who was drafted into the army in 1939 from Chkalovsk (Чкаловска), Orenburg (Оренбург), and got missing in December 1941. However, since a birth year 1918 would have made him a very young author (18), this could also have been a namesake. In the foreword of the first edition of the book, Ryzhik thanked three mathematicians of the Moscow Mathematical Society for their suggestions and advice: Vyacheslav Vassilievich Stepanov (Вячеслав Васильевич Степанов), Aleksei Ivanovich Markushevich (Алексей Иванович Маркушевич), and Ilya Nikolaevich Bronshtein (Илья Николаевич Бронштейн), suggesting that he must have been in some way associated with this group.)
2. Izrail Solomonovich Gradshteyn (Израиль Соломонович Градштейн) (1899, Odessa – 1958, Moscow). VIAF 20405466, VIAF 310677818, VIAF 270418384. ISNI 0000000116049405. GND 11526194X.
3. Following the sources, this article distinguishes between the documented number of formulas and the number of entries.
4. The fact that Ryzhik's death was not announced before the third edition of the book in 1951 might indicate that his status was unclear for a number of years, or, in the case of the first edition, that typesetting had already started, but actual production of the book had to be delayed and was then finalized in his absence as a consequence of the war.
5. Yuri Veniaminovich Geronimus (Юрий Венеаминович Геронимус) (1923–2013), GND 131451812.
6. Michail Yulyevich Tseytlin (Михаил Ю́льевич Цейтлин), also as M. Yu. Ceitlin, Michael Juljewitsch Zeitlin, Michael Juljewitsch Zeitlein, Michael Juljewitsch Tseitlin, Mikhail Juljewitsch Tseitlin (?–).
7. Christa Berg née Jahncke (?–), GND 122341597 (this entry contains an incorrect birth year and some incorrectly associated books).
8. Lothar Berg (1930-07-28 to 2015-07-27), GND 117708054.
9. Martin D. H. Strauss also as Martin D. H. Strauß (1907-03-18 Pillau, Baltijsk, Ostpreußen – 1978-05-17, East-Berlin, GDR), GND 139569200, German physicist and philosopher.
10. Ludwig Boll (1911-12-10 Gaulsheim, Germany – 1984-12-02), GND 1068090308, German mathematician.
11. The seventh Russian edition (2011) was named after the seventh English edition (2007), of which it was a retranslation. There was no sixth genuinely Russian edition. The English series of editions was originally (1965) based on the fourth Russian edition (1962/1963). It is unknown if any changes for the fifth Russian edition (1971) or the third German-English edition (1981), which did incorporate material from the fifth Russian edition, were reflected in any of the English editions in between (and thereby in the seventh Russian edition as well).
12. Alan Jeffrey (1929-07-16 to 2010-06-06), GND 113118120.
13. Daniel "Dan" Ian Zwillinger (1957–), GND 172475694.
14. Victor Hugo Moll (1956–), GND 173099572.
References
1. Shanks, Daniel (October 1966). "Reviews and Descriptions of Tables and Books 85: Table of Integrals, Series, and Products by I. S. Gradshteyn, I. M. Ryzhik" (PDF). Mathematics of Computation (review). 20 (96): 616–617. doi:10.2307/2003554. JSTOR 2003554. RMT:85. Retrieved 2016-03-04.
2. Luke, Yudell Leo (January 1981). "Reviews and Descriptions of Tables and Books 5: I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, Academic Press, New York, 1980" (PDF). Mathematics of Computation (review). 36 (153): 310–314. doi:10.2307/2007757. JSTOR 2007757. MSC:7.95,7.100. Retrieved 2016-03-04.
3. Kölbig, Kurt Siegfried (January 1995). "Reviews and Descriptions of Tables and Books 1: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Alan Jeffrey, ed.), Academic Press, Boston, 1994" (PDF). Mathematics of Computation. 64 (209): 439–441. doi:10.2307/2153347. JSTOR 2153347. MSC:00A22,33-00,44-00,65-00. Retrieved 2016-03-03.
4. Wimp, Jet (April 1997). "Tables of Integrals, Series and Products By I. S. Gradshteyn and I. M. Ryzhik, edited by Alan Jeffrey". American Mathematical Monthly.
5. Wimp, Jet (October 1997) [1997-09-30]. Koepf, Wolfram (ed.). "2. Tables of Integrals, Series and Products By I. S. Gradshteyn and I. M. Ryzhik, edited by Alan Jeffrey" (PDF). Books and Journals: Reviews. Orthogonal Polynomials and Special Functions. Vol. 8, no. 1. SIAM Activity Group on Orthogonal Polynomials and Special Functions. pp. 13–16. Archived (PDF) from the original on 2022-01-20. Retrieved 2022-01-23.
6. Wolfram, Stephen (2005-10-08). "The History and Future of Special Functions". Wolfram Technology Conference, Festschrift for Oleg Marichev, in honor of his 60th birthday (speech, blog post). Champaign, IL, USA: Stephen Wolfram, LLC. The story behind Gradshteyn-Ryzhik. Archived from the original on 2016-04-07. Retrieved 2016-04-06. […] In 1936 Iosif Moiseevich Ryzhik had a book entitled Special Functions published by the United Moscow-Leningrad Scientific-Technical Publisher. Ryzhik died in 1941, either during the siege of Leningrad, or fighting on the Russian front. In 1943, a table of formulas was published under Ryzhik's name by the Governmental Moscow-Leningrad Technical-Theoretical Publisher. The only thing the book seems to say about its origins is that it's responding to the shortage of books of formulas. It says that some integrals marked in it are original, but the others mostly come from three books—a French one from 1858, a German one from 1894, and an American one from 1922. It explains that effort went into the ordering of the integrals, and that some are simplified by using a new special function s equal to Gamma[x+y-1]/(Gamma[x]Gamma[y]). It then thanks three fairly prominent mathematicians from Moscow University. That's basically all we know about Ryzhik. […] Israil Solomonovitch Gradshteyn was born in 1899 in Odessa, and became a professor of mathematics at Moscow State University. But in 1948, he was fired as part of the Soviet attack on Jewish academics. To make money, he wanted to write a book. And so he decided to build on Ryzhik's tables. Apparently he never met Ryzhik. But he created a new edition, and by the third edition, the book was known as Gradshteyn-Ryzhik. […] Gradshteyn died of natural causes in Moscow in 1958. Though somehow there developed an urban legend that one of the authors of Gradshteyn-Ryzhik had been shot as a piece of anti-Semitism on the grounds that an error in their tables had caused an airplane crash. […] Meanwhile, starting around 1953, Yurii Geronimus, who had met Gradshteyn at Moscow State University, began helping with the editing of the tables, and actually added the appendices on special functions. Later on, several more people were involved. And when the tables were published in the West, there were arguments about royalties. But Geronimus [in 2005 was] still alive and well and living in Jerusalem, and Oleg phoned him […]
7. Bierens de Haan, David (1867). Nouvelles tables d'intégrales définies [New tables of definite integrals] (in French) (1 ed.). Leiden, Netherlands: P. Engels. Retrieved 2016-04-17. (NB. This book had a precursor in 1858 named Tables d'intégrales définies (published by C. G. van der Post in Amsterdam) with supplement Supplément aux tables d'intégrales définies in ca. 1864. Cited as BI (БХ) in GR.)
8. Láska, Václav Jan (1888–1894). Sammlung von Formeln der reinen und angewandten Mathematik [Compilation of formulae of pure and applied mathematics] (in German). Vol. 1–3 (1 ed.). Braunschweig, Germany: Friedrich Vieweg und Sohn. OCLC 24624148. Retrieved 2016-04-17. (NB. The book writes the author's name as Wenzel Láska. Cited as LA (Ла) in GR.)
9. Adams, Edwin Plimpton; Hippisley, Richard Lionel (1922). Greenhill, Alfred George (ed.). Smithsonian Mathematical Formulae and Tables of Elliptic Functions. Smithsonian Miscellaneous Collections. Vol. 74 (1 ed.). Washington D.C., USA: Smithsonian Institution. Retrieved 2016-04-17. (NB. Cited as AD (А) in GR.)
10. Archibald, Raymond Clare (October 1945). "Recent Mathematical Tables 219: I. M. Ryzhik, Tablitsy Integralov, Summ, Riadov i Proizvedeniĭ, Leningrad, OGIZ, 1943" (PDF). Mathematical Tables and Other Aids to Computation (MTAG). American Mathematical Society. 1 (12): 442–443. RMT:219. Retrieved 2016-03-04.
11. Hahn, Wolfgang (1950-07-05). "Rydzik, I. M.: Tabellen für Integrale, Summen, Reihen und Produkte. 2. Aufl. Moskau, Leningrad: Staatsverlag für techn.-theor. Lit., 1948". Zentralblatt für Mathematik (review) (in German). Berlin, Germany. 34 (1/3): 70. Zbl 0034.07001. Retrieved 2016-02-16.
12. Stepanov, Vyacheslav Vassilievich (1936). Preface. Специальные функции: Собрание формул и вспомогательные таблицы [Special functions: A collection of formulas and an auxiliary table]. By Ryzhik, Iosif Moiseevich (in Russian) (1 ed.). Moscow / Leningrad: Объединенное научно-техническое издательство, ONTI. Гострансизд-во. Глав. ред. общетехнич. лит-ры и номографии. Archived from the original on 2016-04-09. Retrieved 2016-04-09. (160 pages.)
13. Rosenfeld (Розенфельд), Boris Abramowitsch (Борис Абрамович) [in German] (2003). "Math.ru" Об Исааке Моисеевиче Ягломе [About Isaac Moiseevich Yaglom]. Мат. просвещение (Mat. enlightenment) (in Russian): 25–28. Archived from the original on 2022-01-12. Retrieved 2022-01-12 – via math.ru. […] во время антисемитской кампании, известной как «борьба с космополитизмом», был уволен вместе с И.М. Гельфандом и И. С. Градштейном […] [during the antisemitic campaign known as the "fight against cosmopolitanism", he was fired along with I. M. Gelfand and I. S. Gradstein.]
14. Bruins, Evert Marie [in German] (1964-01-02). "Gradšteĭn, I. S., und I. M. Ryžik: Integral-, Summen-, Reihen- und Produkttafeln. (Tablicy integralov, summ, rjadov i proizvedeniĭ) 4. Aufl. überarb. unter Mitwirkung von Ju. V. Geronimus und M. Ju. Ceĭtlin. Moskau: Staatsverlag für physikalisch-mathematische Literatur, 1962". Zentralblatt für Mathematik (list) (in German). 103 (1): 38. Zbl 0103.03801. Retrieved 2016-02-16.
15. Wrench, Jr., John William (October 1960). "Reviews and Descriptions of Tables and Books 69: I. M. Ryshik & I. S. Gradstein, Summen-, Produkt- und Integral-Tafeln: Tables of Series, Products, and Integrals, VEB Deutscher Verlag der Wissenschaften, Berlin" (PDF). Mathematics of Computation. 14 (72): 381–382. doi:10.2307/2003905. JSTOR 2003905. RMT:69. Archived (PDF) from the original on 2016-03-16. Retrieved 2016-03-16.
16. Prachar, Karl (1959-09-15). "Ryshik, I. M. und I. S. Gradstein: Summen-, Produkt- und Integraltafeln / Tables of series, products and integrals. Berlin: VEB Deutscher Verlag der Wissenschaften, 1957". Zentralblatt für Mathematik (review) (in German). 80 (2): 337–338. Zbl 0080.33703. Archived from the original on 2016-02-17. Retrieved 2016-02-12.
17. Papp, Frank J. "Gradshteyn, I. S.; Ryzhik, I. M.: Tables of integrals, series, and products. Corr. and enl. ed. by Alan Jeffrey. Incorporating the 4th ed. by Yu. V. Geronimus and M. Yu. Tseytlin (M. Yu. Tsejtlin). Transl. from the Russian – New York – London – Toronto. Volumes 1, 2. German and English Transl". Zentralblatt für Mathematik und ihre Grenzgebiete (review). 521: 193. MR 0582453. Zbl 0521.33001. Retrieved 2016-02-16.
18. "Gradshteyn, I. S.; Ryzhik, I. M. Table of integrals, series, and products. Transl. from the Russian by Scripta Technica, Inc. 5th ed. Boston, MA: Academic Press, Inc. (1994)". Zentralblatt MATH. 1994. ISBN 9780122947551. Zbl 0918.65002. Retrieved 2016-02-16.
19. Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich (February 2007). Jeffrey, Alan; Zwillinger, Daniel (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica (7 ed.). Academic Press, Inc. ISBN 978-0-12-373637-6. MR 2360010. GR:11. Retrieved 2016-02-21.
20. Таблицы интегралов, ряда и продуктов [Table of Integrals, Series, and Products] (PDF) (in Russian) (7 ed.). BHV (БХВ-Петербург). 2011. ISBN 978-5-9775-0360-0. Archived (PDF) from the original on 2016-03-16. Retrieved 2016-03-16.
21. Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica (8 ed.). Academic Press, Inc. ISBN 978-0-12-384933-5. GR:12. Retrieved 2016-02-21.
22. Jeffrey, Alan (1995-01-01). Handbook of Mathematical Formulas and Integrals (1 ed.). Academic Press, Inc. ISBN 978-0-12-382580-3.
23. Jeffrey, Alan (2000-08-01). Handbook of Mathematical Formulas and Integrals (2 ed.). Academic Press, Inc. ISBN 978-0-12-382251-2.
24. Jeffrey, Alan (2004). Handbook of Mathematical Formulas and Integrals (3 ed.). Academic Press, Inc. (published 2003-11-21). ISBN 978-0-12-382256-7. Archived from the original on 2022-01-01. Retrieved 2016-03-01.
25. Jeffrey, Alan; Dai, Hui-Hui (2008-02-01). Handbook of Mathematical Formulas and Integrals (4 ed.). Academic Press, Inc. ISBN 978-0-12-374288-9. Retrieved 2016-03-01. (NB. Contents of companion CD-ROM: )
26. Vardi, Ilan (April 1988). "Integrals: An Introduction to Analytic Number Theory" (PDF). American Mathematical Monthly. 95 (4): 308–315. doi:10.2307/2323562. JSTOR 2323562. Archived (PDF) from the original on 2016-03-15. Retrieved 2016-03-14.
27. Moll, Victor Hugo (April 2010) [2009-08-30]. "Seized Opportunities" (PDF). Notices of the American Mathematical Society. 57 (4): 476–484. Archived (PDF) from the original on 2016-04-08. Retrieved 2016-04-08.
28. Boros, George; Moll, Victor Hugo (2006) [September 2004]. Irresistible Integrals. Symbolics, Analysis and Experiments in the Evaluation of Integrals (reprinted 1st ed.). Cambridge University Press (CUP). p. xi. ISBN 978-0-521-79186-1. Retrieved 2016-02-22. (NB. This edition contains many typographical errors.)
29. Moll, Victor Hugo; Vignat, Christophe. "The integrals in Gradshteyn and Ryzhik. Part 29: Chebyshev polynomials" (PDF). Scientia. Series A: Mathematical Sciences. Archived from the original on 2016-03-13. Retrieved 2016-03-13.{{cite journal}}: CS1 maint: unfit URL (link) (NB. This paper discusses 19 GR entries: 1.14.2.1, 1.320, 2.18.1.9, 3.753.2, 3.771.8, 6.611, 7.341.1, 7.341.2, 7.342, 7.343.1, 7.344.1, 7.344.2, 7.346, 7.348, 7.349, 7.355.1, 7.355.2, 8.411.1, 8.921. )
30. Amdeberhan, Tewodros; Dixit, Atul; Guan, Xiao; Jiu, Lin; Kuznetsov, Alexey; Moll, Victor Hugo; Vignat, Christophe. "The integrals in Gradshteyn and Ryzhik. Part 30: Trigonometric functions" (PDF). Scientia. Series A: Mathematical Sciences. Archived from the original on 2016-03-13. Retrieved 2016-03-13.{{cite journal}}: CS1 maint: unfit URL (link) (NB. This paper discusses 51 GR entries: 1.320.1, 1.320.3, 1.320.5, 1.320.7, 2.01.5, 2.01.6, 2.01.7, 2.01.8, 2.01.9, 2.01.10, 2.01.11, 2.01.12, 2.01.13, 2.01.14, 2.513.1, 2.513.2, 2.513.3, 2.513.4, 3.231.5, 3.274.2, 3.541.8, 3.611.3, 3.621.3, 3.621.4, 3.624.6, 3.631.16, 3.661.3, 3.661.4, 3.675.1, 3.675.2, 3.688.1, 3.721.1, 3.747.7, 3.761.4, 4.381.1, 4.381.2, 4.381.3, 4.381.4, 4.521.1, 6.671.7, 6.671.8, 7.244.1, 7.244.2, 7.531.1, 7.531.2, 8.230.1, 8.230.2, 8.361.7, 8.370, 8.910.2, 8.911.1. It also contains 1 errata for GR entry 3.541.8. )
31. Gonzalez, Ivan; Kohl, Karen T.; Moll, Victor Hugo (2014-06-13) [2014-03-19]. "Evaluation of entries in Gradshteyn and Ryzhik employing the method of brackets" (PDF). Scientia. Series A: Mathematical Sciences (published 2014). 25: 65–84. Retrieved 2016-03-13. (NB. This paper is also incorporated into volume II.)
32. Moll, Victor Hugo (2006-11-06) [2006-07-21]. "The integrals in Gradshteyn and Ryzhik. Part 1: A family of logarithmic integrals" (PDF). Scientia. Series A: Mathematical Sciences (published 2007). 14: 1–6. Archived from the original (PDF) on 2017-02-02. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 10 GR entries: 3.419.2, 3.419.3, 3.419.4, 3.419.5, 3.419.6, 4.232.3, 4.261.4, 4.262.3, 4.263.1, 4.264.3. )
33. Moll, Victor Hugo (2006-11-06) [2006-06-27]. "The integrals in Gradshteyn and Ryzhik. Part 2: Elementary logarithmic integrals" (PDF). Scientia. Series A: Mathematical Sciences (published 2007). 14: 7–15. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 5 GR entries: 4.231.1, 4.231.5, 4.231.6, 4.232.1, 4.232.2. )
34. Moll, Victor Hugo (2007-01-16) [2006-12-27]. "The integrals in Gradshteyn and Ryzhik. Part 3: Combinations of logarithms and exponentials" (PDF). Scientia. Series A: Mathematical Sciences (published 2007). 15: 31–36. arXiv:0705.0175. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 8 GR entries: 4.331.1, 4.335.1, 4.335.3, 4.352.1, 4.352.2, 4.352.3, 4.352.4, 4.353.2. )
35. Moll, Victor Hugo (2007-01-16) [2006-12-27]. "The integrals in Gradshteyn and Ryzhik. Part 4: The gamma function" (PDF). Scientia. Series A: Mathematical Sciences (published 2007). 15: 37–46. arXiv:0705.0179. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 44 GR entries: 3.324.2, 3.326.1, 3.326.2, 3.328, 3.351.3, 3.371, 3.381.4, 3.382.2, 3.434.1, 3.434.2, 3.461.2, 3.461.3, 3.462.9, 3.471.3, 3.478.1, 3.478.2, 3.481.1, 3.481.2, 4.215.1, 4.215.2, 4.215.3, 4.215.4, 4.229.1, 4.229.3, 4.229.4, 4.269.3, 4.272.5, 4.272.6, 4.272.7, 4.325.8, 4.325.11, 4.325.12,, 4.331.1 4.333, 4.335.1, 4.335.3, 4.355.1, 4.355.3, 4.355.4, 4.358.2, 4.358.3, 4.358.4, 4.369.1, 4.369.2. )
36. Amdeberhan, Tewodros; Medina, Luis A.; Moll, Victor Hugo (2007-01-16) [2006-12-27]. "The integrals in Gradshteyn and Ryzhik. Part 5: Some trigonometric integrals" (PDF). Scientia. Series A: Mathematical Sciences (published 2007). 15: 47–60. arXiv:0705.2379. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 10 GR entries: 3.621.1, 3.621.3, 3.621.4, 3.761.11, 3.764.1, 3.764.2, 3.821.3, 3.821.14, 3.822.1, 3.822.2. )
37. Moll, Victor Hugo (2007-10-31) [2007-08-31]. "The integrals in Gradshteyn and Ryzhik. Part 6: The beta function" (PDF). Scientia. Series A: Mathematical Sciences (published 2008). 16: 9–24. arXiv:0707.2121. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 65 GR entries: 3.191.1, 3.191.2, 3.191.3, 3.192.1, 3.192.2, 3.193, 3.194.3, 3.194.4, 3.194.6, 3.194.7, 3.196.2, 3.196.3, 3.196.4, 3.196.5, 3.216.1, 3.216.2, 3.217, 3.218, 3.221.1, 3.221.2, 3.222.2, 3.223.1, 3.223.2, 3.223.3, 3.224, 3.225.1, 3.225.2, 3.225.3, 3.226.1, 3.226.2, 3.241.2, 3.241.4, 3.241.5, 3.248.1, 3.248.2, 3.248.3, 3.249.1, 3.249.2, 3.249.5, 3.249.7, 3.249.8, 3.251.1, 3.251.2, 3.251.3, 3.251.4, 3.251.5, 3.251.6, 3.251.8, 3.251.9, 3.251.10, 3.251.11, 3.267.1, 3.267.2, 3.267.3, 3.311.3, 3.311.9, 3.312.1, 3.313.1, 3.313.2, 3.314, 3.457.3, 4.251.1, 4.273, 4.275.1, 4.321.1. )
38. Amdeberhan, Tewodros; Moll, Victor Hugo (2007-10-31) [2007-08-21]. "The integrals in Gradshteyn and Ryzhik. Part 7: Elementary examples" (PDF). Scientia. Series A: Mathematical Sciences (published 2008). 16: 25–39. arXiv:0707.2122. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 30 GR entries: 2.321.1, 2.321.2, 2.322.1, 2.322.2, 2.322.3, 2.322.4, 3.195, 3.248.4, 3.248.6, 3.249.1, 3.249.6, 3.252.1, 3.252.2, 3.252.3, 3.268.1, 3.310, 3.311.1, 3.351.1, 3.351.2, 3.351.7, 3.351.8, 3.351.9, 3.353.4, 3.411.19, 3.411.20, 3.471.1, 3.622.3, 3.622.4, 4.212.7, 4.222.1. )
39. Moll, Victor Hugo; Rosenberg, Jason; Straub, Armin; Whitworth, Pat (2007-10-31) [2007-08-31]. "The integrals in Gradshteyn and Ryzhik. Part 8: Combinations of powers, exponentials and logarithms" (PDF). Scientia. Series A: Mathematical Sciences (published 2008). 16: 41–50. arXiv:0707.2123. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 7 GR entries: 3.351.1, 4.331.1, 4.351.1, 4.351.2, 4.353.3, 4.362.1, 8.350.1. )
40. Amdeberhan, Tewodros; Moll, Victor Hugo; Rosenberg, Jason; Straub, Armin; Whitworth, Pat (2008-11-18) [2007-11-29]. "The integrals in Gradshteyn and Ryzhik. Part 9: Combinations of logarithms, rational and trigonometric functions" (PDF). Scientia. Series A: Mathematical Sciences (published 2009). 17: 27–44. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 45 GR entries: 2.148.4, 3.747.7, 4.223.1, 4.223.2, 4.224.1, 4.224.2, 4.224.3, 4.224.4, 4.224.5, 4.224.6, 4.225.1, 4.225.2, 4.227.1, 4.227.2, 4.227.3, 4.227.9, 4.227.10, 4.227.11, 4.227.13, 4.227.14, 4.227.15, 4.231.1, 4.231.2, 4.231.3, 4.231.4, 4.231.8, 4.231.9, 4.231.11, 4.231.12, 4.231.13, 4.231.14, 4.231.15, 4.231.19, 4.231.20, 4.233.1, 4.261.8, 4.291.1, 4.291.2, 4.295.5, 4.295.6, 4.295.11, 4.521.1, 4.531.1, 8.366.3, 8.380.3. )
41. Medina, Luis A.; Moll, Victor Hugo (2008-11-18) [2007-11-29]. "The integrals in Gradshteyn and Ryzhik. Part 10: The digamma function" (PDF). Scientia. Series A: Mathematical Sciences (published 2009). 17: 45–66. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 76 GR entries: 3.219, 3.231.1, 3.231.3, 3.231.5, 3.231.6, 3.233, 3.234.1, 3.235, 3.244.2, 3.244.3, 3.265, 3.268.2, 3.269.1, 3.269.3, 3.311.5, 3.311.6, 3.311.7, 3.311.8, 3.311.10, 3.311.11, 3.311.12, 3.312.2, 3.316, 3.317.1, 3.317.2, 3.427.1, 3.427.2, 3.429, 3.434.2, 3.435.3, 3.435.4, 3.442.3, 3.457.1, 3.463, 3.467, 3.469.2, 3.469.3, 3.471.14, 3.475.1, 3.475.2, 3.475.3, 3.476.1, 3.476.2, 4.241.1, 4.241.2, 4.241.3, 4.241.4, 4.241.5, 4.241.7, 4.241.8, 4.241.9, 4.241.10, 4.241.11, 4.243, 4.244.1, 4.244.2, 4.244.3, 4.245.1, 4.245.2, 4.246, 4.247.1, 4.247.2, 4.251.4, 4.253.1, 4.254.1, 4.254.6, 4.256, 4.271.15, 4.275.2, 4.281.1, 4.281.4, 4.281.5, 4.293.8, 4.293.13, 4.331.1, 8.371.2. )
42. Boyadzhiev, Khristo N.; Medina, Luis A.; Moll, Victor Hugo (2009-03-16) [2008-07-02]. "The integrals in Gradshteyn and Ryzhik. Part 11: The incomplete beta function" (PDF). Scientia. Series A: Mathematical Sciences (published 2009). 18: 61–75. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 52 GR entries: 3.222.1, 3.231.2, 3.231.4, 3.241.1, 3.244.1, 3.249.4, 3.251.7, 3.269.2, 3.311.2, 3.311.13, 3.522.4, 3.541.6, 3.541.7, 3.541.8, 3.622.2, 3.623.2, 3.623.3, 3.624.1, 3.635.1, 3.651.1, 3.651.2, 3.656.1, 3.981.3, 4.231.1, 4.231.6, 4.231.11, 4.231.12, 4.231.14, 4.231.19, 4.231.20, 4.234.1, 4.234.2, 4.251.3, 4.254.4, 4.261.2, 4.261.6, 4.261.11, 4.262.1, 4.262.4, 4.263.2, 4.264.1, 4.265, 4.266.1, 4.271.1, 4.271.16, 8.361.7, 8.365.4, 8.366.3, 8.366.11, 8.366.12, 8.366.13, 8.370. )
43. Moll, Victor Hugo; Posey, Ronald A. (2009-03-16) [2008-07-02]. "The integrals in Gradshteyn and Ryzhik. Part 12: Some logarithmic integrals" (PDF). Scientia. Series A: Mathematical Sciences (published 2009). 18: 77–84. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 6 GR entries: 4.231.1, 4.233.1, 4.233.2, 4.233.3, 4.233.4, 4.261.8. )
44. Moll, Victor Hugo (2010-10-10) [2009-07-07]. "The integrals in Gradshteyn and Ryzhik. Part 13: Trigonometric forms of the beta function" (PDF). Scientia. Series A: Mathematical Sciences (published 2010). 19: 91–96. arXiv:1004.2439. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 21 GR entries: 3.621.1, 3.621.2, 3.621.3, 3.621.4, 3.621.5, 3.621.6, 3.621.7, 3.622.1, 3.623.1, 3.624.2, 3.624.3, 3.624.4, 3.624.5, 3.625.1, 3.625.2, 3.625.3, 3.625.4, 3.626.1, 3.626.2, 3.627, 3.628. )
45. Amdeberhan, Tewodros; Moll, Victor Hugo (2010-10-10) [2009-07-07]. "The integrals in Gradshteyn and Ryzhik. Part 14: An elementary evaluation of entry 3.411.5" (PDF). Scientia. Series A: Mathematical Sciences (published 2010). 19: 97–103. arXiv:1004.2440. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 1 GR entry: 3.411.5. )
46. Albano, Matthew; Amdeberhan, Tewodros; Beyerstedt, Erin; Moll, Victor Hugo (2010-07-18) [2010-04-20]. "The integrals in Gradshteyn and Ryzhik. Part 15: Frullani integrals" (PDF). Scientia. Series A: Mathematical Sciences (published 2010). 19: 113–119. arXiv:1005.2940. Retrieved 2016-03-14. (NB. This paper (from volume I) discusses 12 GR entries: 3.232, 3.329, 3.412.1, 3.434.2, 3.436, 3.476.1, 3.484, 4.267.8, 4.297.7, 4.319.3, 4.324.2, 4.536.2. )
47. Boettner, Stefan Thomas; Moll, Victor Hugo (2010-07-21) [2010-03-22]. "The integrals in Gradshteyn and Ryzhik. Part 16: Complete elliptic integrals" (PDF). Scientia. Series A: Mathematical Sciences (published 2010). 20: 45–59. arXiv:1005.2941. Retrieved 2016-03-14. (NB. This paper (from volume II) discusses 36 GR entries: 1.421.1, 3.166.16, 3.166.18, 3.721.1, 3.821.7, 3.841.1, 3.841.2, 3.841.3, 3.841.4, 3.842.3, 3.842.4, 3.844.1, 3.844.2, 3.844.3, 3.844.4, 3.844.5, 3.844.6, 3.844.7, 3.844.8, 3.846.1, 3.846.2, 3.846.3, 3.846.4, 3.846.5, 3.846.6, 3.846.7, 3.846.8, 4.242.1, 4.395.1, 4.414.1, 4.432.1, 4.522.4, 4.522.5, 4.522.6, 4.522.7, 8.129.1. )
48. Amdeberhan, Tewodros; Boyadzhiev, Khristo N.; Moll, Victor Hugo (2010-07-21) [2010-03-22]. "The integrals in Gradshteyn and Ryzhik. Part 17: The Riemann zeta function" (PDF). Scientia. Series A: Mathematical Sciences (published 2010). 20: 61–71. Retrieved 2016-03-14. (NB. This paper (from volume II) discusses 36 GR entries: 3.333.1, 3.333.2, 3.411.1, 3.411.2, 3.411.3, 3.411.4, 3.411.6, 3.411.7, 3.411.8, 3.411.9, 3.411.10, 3.411.11, 3.411.12, 3.411.13, 3.411.14, 3.411.15, 3.411.17, 3.411.18, 3.411.21, 3.411.22, 3.411.25, 3.411.26, 4.231.1, 4.231.2, 4.261.12, 4.261.13, 4.262.1, 4.262.2, 4.262.5, 4.262.6, 4.264.1, 4.264.2, 4.266.1, 4.266.2, 4.271.1, 4.271.2. )
49. Koutschan, Christoph; Moll, Victor Hugo (2010-08-21) [2010-04-26]. "The integrals in Gradshteyn and Ryzhik. Part 18: Some automatic proofs" (PDF). Scientia. Series A: Mathematical Sciences (published 2010). 20: 93–111. Archived (PDF) from the original on 2016-03-14. Retrieved 2016-03-14. (NB. This paper (from volume II) discusses 7 GR entries: 2.148.3, 3.953, 4.535.1, 6.512.1, 7.322, 7.349, 7.512.5. )
50. Albano, Matthew; Amdeberhan, Tewodros; Beyerstedt, Erin; Moll, Victor Hugo (2011-04-13) [2010-12-23]. "The integrals in Gradshteyn and Ryzhik. Part 19: The error function" (PDF). Scientia. Series A: Mathematical Sciences (published 2011). 21: 25–42. Retrieved 2016-03-14. (NB. This paper (from volume II) discusses 42 GR entries: 3.321.1, 3.321.2, 3.321.3, 3.321.4, 3.321.5, 3.321.6, 3.321.7, 3.322.1, 3.322.2, 3.323.1, 3.323.2, 3.325, 3.361.1, 3.361.2, 3.361.3, 3.362.1, 3.362.2, 3.363.1, 3.363.2, 3.369, 3.382.4, 3.461.5, 3.462.5, 3.462.6, 3.462.7, 3.462.8, 3.464, 3.466.1, 3.466.2, 3.471.15, 3.471.16, 3.472.1, 6.281.1, 6.282.1, 6.283.1, 6.283.2, 6.294.1, 8.258.1, 8.258.2, 8.258.3, 8.258.4, 8.258.5. )
51. Kohl, Karen T.; Moll, Victor Hugo (2011-04-13) [2010-12-23]. "The integrals in Gradshteyn and Ryzhik. Part 20: Hypergeometric functions" (PDF). Scientia. Series A: Mathematical Sciences (published 2011). 21: 43–54. Retrieved 2016-03-14. (NB. This paper (from volume II) discusses 26 GR entries: 3.194.1, 3.194.2, 3.194.5, 3.196.1, 3.197.1, 3.197.2, 3.197.3, 3.197.4, 3.197.5, 3.197.6, 3.197.7, 3.197.8, 3.197.9, 3.197.10, 3.197.11, 3.197.12, 3.198, 3.199, 3.227.1, 3.254.1, 3.254.2, 3.259.2, 3.312.3, 3.389.1, 9.121.4, 9.131.1. )
52. Boyadzhiev, Khristo N.; Moll, Victor Hugo (2012-10-20) [2012-05-15]. "The integrals in Gradshteyn and Ryzhik. Part 21: Hyperbolic functions" (PDF). Scientia. Series A: Mathematical Sciences (published 2012). 22: 109–127. Archived (PDF) from the original on 2016-03-14. Retrieved 2016-03-14. (NB. This paper (from volume II) discusses 75 GR entries: 2.423.9, 3.231.2, 3.231.5, 3.265, 3.511.1, 3.511.2, 3.511.4, 3.511.5, 3.511.8, 3.512.1, 3.512.2, 3.521.1, 3.521.2, 3.522.3, 3.522.6, 3.522.8, 3.522.10, 3.523.2, 3.523.3, 3.523.4, 3.523.5, 3.523.6, 3.523.7, 3.523.8, 3.523.9, 3.523.10, 3.523.11, 3.523.12, 3.524.2, 3.524.4, 3.524.6, 3.524.8, 3.524.9, 3.524.10, 3.524.11, 3.524.12, 3.524.13, 3.524.14, 3.524.15, 3.524.16, 3.524.17, 3.524.18, 3.524.19, 3.524.20, 3.524.21, 3.524.22, 3.524.23, 3.525.1, 3.525.2, 3.525.3, 3.525.4, 3.525.5, 3.525.6, 3.525.7, 3.525.8, 3.527.1, 3.527.2, 3.527.3, 3.527.4, 3.527.5, 3.527.6, 3.527.7, 3.527.8, 3.527.9, 3.527.10, 3.527.11, 3.527.12, 3.527.13, 3.527.14, 3.527.15, 3.527.16, 3.543.2, 4.113.3, 4.231.12, 8.365.9. )
53. Glasser, Larry; Kohl, Karen T.; Koutschan, Christoph; Moll, Victor Hugo; Straub, Armin (2012-10-20) [2012-05-15]. "The integrals in Gradshteyn and Ryzhik. Part 22: Bessel-K functions" (PDF). Scientia. Series A: Mathematical Sciences (published 2012). 22: 129–151. Archived (PDF) from the original on 2016-03-14. Retrieved 2016-03-14. (NB. This paper (from volume II) discusses 36 GR entries: 3.323.3, 3.324.1, 3.337.1, 3.364.3, 3.365.2, 3.366.2, 3.372, 3.383.3, 3.383.8, 3.387.1, 3.387.3, 3.387.6, 3.388.2, 3.389.4, 3.391, 3.395.1, 3.461.6, 3.461.7, 3.461.8, 3.461.9, 3.462.20, 3.462.21, 3.462.22, 3.462.23, 3.462.24, 3.462.25, 3.469.1, 3.471.4, 3.471.8, 3.471.9, 3.471.12, 3.478.4, 3.479.1, 3.547.2, 3.547.4, 8.432.6. )
54. Medina, Luis A.; Moll, Victor Hugo (2012-06-25) [2012-02-01]. "The integrals in Gradshteyn and Ryzhik. Part 23: Combination of logarithms and rational functions" (PDF). Scientia. Series A: Mathematical Sciences (published 2012). 23: 1–18. Retrieved 2016-03-14. (NB. This paper (from volume II) discusses 54 GR entries: 3.417.1, 3.417.2, 4.212.7, 4.224.5, 4.224.6, 4.225.1, 4.225.2, 4.231.1, 4.231.2, 4.231.8, 4.231.9, 4.231.10, 4.231.11, 4.231.16, 4.231.17, 4.231.18, 4.233.1, 4.234.3, 4.234.6, 4.234.7, 4.234.8, 4.262.7, 4.262.8, 4.262.9, 4.291.1, 4.291.2, 4.291.3, 4.291.4, 4.291.5, 4.291.6, 4.291.7, 4.291.8, 4.291.9, 4.291.10, 4.291.11, 4.291.12, 4.291.13, 4.291.14, 4.291.15, 4.291.16, 4.291.17, 4.291.18, 4.291.19, 4.291.20, 4.291.21, 4.291.22, 4.291.23, 4.291.24, 4.291.25, 4.291.26, 4.291.27, 4.291.28, 4.291.29, 4.291.30. )
55. McInturff, Kim; Moll, Victor Hugo (2012-07-28) [2012-02-01]. "The integrals in Gradshteyn and Ryzhik. Part 24: Polylogarithm functions" (PDF). Scientia. Series A: Mathematical Sciences (published 2012). 23: 45–51. Retrieved 2016-03-14. (NB. This paper (from volume II) discusses 10 GR entries: 3.418.1, 3.514.1, 3.531.1, 3.531.2, 3.531.3, 3.531.4, 3.531.5, 3.531.6, 3.531.7, 9.513.1. )
56. Moll, Victor Hugo (2012-07-28) [2012-02-01]. "The integrals in Gradshteyn and Ryzhik. Part 25: Evaluation by series" (PDF). Scientia. Series A: Mathematical Sciences (published 2012). 23: 53–65. Retrieved 2016-03-14. (NB. This paper (from volume II) discusses 18 GR entries: 3.194.8, 3.311.4, 3.342, 3.451.1, 3.451.2, 3.466.3, 3.747.7, 4.221.1, 4.221.2, 4.221.3, 4.251.5, 4.251.6, 4.269.1, 4.269.2, 8.339.1, 8.339.2, 8.365.4, 8.366.3. )
57. Boyadzhiev, Khristo N.; Moll, Victor Hugo (2015-01-30) [2014-09-19]. "The integrals in Gradshteyn and Ryzhik. Part 26: The exponential integral" (PDF). Scientia. Series A: Mathematical Sciences (published 2015). 26: 19–30. Retrieved 2016-03-14. (NB. This paper (from volume II) discusses 41 GR entries: 3.327, 3.351.4, 3.351.5, 3.351.6, 3.352.1, 3.352.2, 3.352.3, 3.352.4, 3.352.5, 3.352.6, 3.353.1, 3.353.2, 3.353.3, 3.353.4, 3.353.5, 3.354.1, 3.354.2, 3.354.3, 3.354.4, 3.355.1, 3.355.2, 3.355.3, 3.355.4, 3.356.1, 3.356.2, 3.356.3, 3.356.4, 3.357.1, 3.357.2, 3.357.3, 3.357.4, 3.357.5, 3.357.6, 3.358.1, 3.358.2, 3.358.3, 3.358.4, 4.211.1, 4.211.2, 4.212.1, 4.212.2. )
58. Medina, Luis A.; Moll, Victor Hugo (2015-01-30) [2014-09-16]. "The integrals in Gradshteyn and Ryzhik. Part 27: More logarithmic examples" (PDF). Scientia. Series A: Mathematical Sciences (published 2015). 26: 31–47. Retrieved 2016-03-14. (NB. This paper (from volume II) discusses 37 GR entries: 3.194.3, 3.231.1, 3.231.5, 3.231.6, 3.241.3, 3.621.3, 3.621.5, 4.224.2, 4.231.1, 4.231.2, 4.231.8, 4.233.5, 4.234.4, 4.234.5, 4.235.1, 4.235.2, 4.235.3, 4.235.4, 4.241.5, 4.241.6, 4.241.7, 4.241.11, 4.251.1, 4.251.2, 4.252.1, 4.252.2, 4.252.3, 4.252.4, 4.254.2, 4.254.3, 4.255.2, 4.255.3, 4.257.1, 4.261.9, 4.261.15, 4.261.16, 4.297.8. )
59. Dixit, Atul; Moll, Victor Hugo (2015-02-01) [2014-09-30]. "The integrals in Gradshteyn and Ryzhik. Part 28: The confluent hypergeometric function and Whittaker functions" (PDF). Scientia. Series A: Mathematical Sciences (published 2015). 26: 49–61. Retrieved 2016-03-14. (NB. This paper (from volume II) discusses 17 GR entries: 7.612.1, 7.612.2, 7.621.1, 7.621.2, 7.621.3, 7.621.4, 7.621.5, 7.621.6, 7.621.7, 7.621.8, 7.621.9, 7.621.10, 7.621.11, 7.621.12, 8.334.2, 8.703, 9.211.4. )
60. Moll, Victor Hugo (2012). "Index of the papers in Revista Scientia with formulas from GR". Retrieved 2016-02-17.
61. Moll, Victor Hugo (2014-10-01). Special Integrals of Gradshteyn and Ryzhik: the Proofs. Monographs and Research Notes in Mathematics. Vol. I (1 ed.). Chapman and Hall/CRC Press/Taylor & Francis Group, LLC (published 2014-11-12). ISBN 978-1-4822-5651-2. Retrieved 2016-02-12. (NB. This volume compiles Scientia papers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 from issues 14 to 19.)
62. Moll, Victor Hugo (2015-08-24). Special Integrals of Gradshteyn and Ryzhik: the Proofs. Monographs and Research Notes in Mathematics. Vol. II (1 ed.). Chapman and Hall/CRC Press/Taylor & Francis Group, LLC (published 2015-10-27). ISBN 978-1-4822-5653-6. Retrieved 2016-02-12. (NB. This volume compiles 14 Scientia articles from issues 20, 21, 22, 23, 25 and 26 including papers 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28 and one unnumbered paper.)
63. "Ryžik, I. M.: Tafeln von Integralen, Summen, Reihen und Produkten. Moskau-Leningrad: Staatsverlag für technisch-theoretische Literatur, 1943". Zentralblatt für Mathematik (list) (in German). 60 (1): 123. 1957-04-01. Zbl 0060.12305. Retrieved 2016-02-16.
64. Prachar, Karl (1952-09-01). "Ryžik, I. M. und I. S. Gradštejn: Tafeln von Integralen, Summen, Reihen und Produkten. 3. umgearb. Aufl. Moskau-Leningrad: Staatsverlag für technisch-theoretische Literatur, 1951". Zentralblatt für Mathematik (review) (in German). 44 (1/5): 133. Zbl 0044.13303. Retrieved 2016-02-16.
65. Wrench, Jr., John William (October 1960). "Table Errata 293: I. M. Ryshik & I. S. Gradstein, Summen-, Produkt- und Integral-Tafeln: Tables of Series, Products, and Integrals, Deutscher Verlag der Wissenschaften, Berlin, 1957" (PDF). Mathematics of Computation. 14 (72): 401–403. JSTOR 2003934. MTE:293. Archived (PDF) from the original on 2016-03-16. Retrieved 2016-03-16.
66. Градштейн, И. С.; Рыжик, И. М. (1971). "Errata in 4th edition". In Геронимус, Ю. В.; Цейтлин, М. Ю́. (eds.). Таблицы интегралов, сумм, рядов и произведений (in Russian) (5 ed.). Nauka (Наука). pp. 1101–1108. (NB. The 8-page errata list in later print runs of the fourth Russian edition affected 189 table entries.)
67. Ryshik-Gradstein: Summen-, Produkt- und Integral-Tafeln: Berichtigungen zur 1. Auflage (in German). Berlin, Germany: VEB Deutscher Verlag der Wissenschaften. 1962. MR 0175273. (NB. This brochure was available free of charge from the publisher on request.)
68. "Ryshik-Gradstein: Tafeln Summen Produkte Integrale: Berichtigungen zur 1. Auflage". L'Enseignement Mathématique. Bulletin Bibliographique: Livres Nouveaux (in French and German). 9: 5. 1963. Retrieved 2016-03-04. Die Berichtigung wird den Interessenten auf Anfrage kostenlos durch den Verlag geliefert.
69. Rodman, Richard B. (January 1963). "Table Errata 326: I. M. Ryshik & I. S. Gradstein, Summen-, Produkt- und Integral-Tafeln, Deutscher Verlag der Wissenschaften, Berlin, 1957" (PDF). Mathematics of Computation. 17 (81): 100–103. JSTOR 2003754. MTE:326. Archived (PDF) from the original on 2016-03-16. Retrieved 2016-03-16.
70. Schmieg, Glenn M. (July 1966). "Errata 392: I. M. Ryshik & I. S. Gradstein, Summen-, Produkt- und Integral-Tafeln: Tables of Series, Products, and Integrals, VEB Deutscher Verlag der Wissenschaften, Berlin, 1957" (PDF). Mathematics of Computation. 20 (95): 468–471. doi:10.2307/2003630. JSTOR 2003630. MTE:392. Archived (PDF) from the original on 2016-03-16. Retrieved 2016-03-16.
71. Filliben, James J. [at Wikidata] (January 1970). "Table Errata 456: I. M. Ryshik & I. S. Gradstein, Summen-, Produkt- und Integral-Tafeln, VEB Deutscher Verlag der Wissenschaften, Berlin, 1957" (PDF). Mathematics of Computation. 24 (109): 239–242. JSTOR 2004912. MTE:456. Archived (PDF) from the original on 2016-03-16. Retrieved 2016-03-16.
72. MacKinnon, Robert F. (January 1972). "Table Errata 486: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 26 (117): 305–307. doi:10.1090/s0025-5718-1972-0415970-6. JSTOR 2004754. MR 0415970. MTE:486. Retrieved 2016-03-04.
73. "Gradshtejn, I. S.; Ryzhik, I. M.: Summen-, Produkt- und Integraltafeln. Band 1, 2. Deutsch und englisch. Übers. aus dem Russischen auf der Basis der 5. russ. Aufl., überarb. von J. Geronimus und M. Zeitlin. Tables of series, products, and integrals. Volumes 1, 2. German and English Transl". Zentralblatt für Mathematik und ihre Grenzgebiete (list) (in German). 448: 395. Zbl 0448.65002. Retrieved 2016-02-16.
74. "Gradshtejn, I. S.; Ryzhik, I. M.: Summen-, Produkt- und Integraltafeln. Band 1, 2. Deutsch und englisch. Übers. aus dem Russischen auf der Basis der 5. russ. Aufl., überarb. von J. Geronimus und M. Zeitlin. Tables of series, products, and integrals. Volumes 1, 2. German and English Transl". Zentralblatt für Mathematik und ihre Grenzgebiete (list) (in German). 456: 355. Zbl 0456.65001. Retrieved 2016-02-16.
75. Fettis, Henry E. (April 1967). "Table Errata 408: I. S. Gradshteyn & I. M. Ryzhik, Tables of Integrals, Series and Products, Fourth Edition, Academic Press, New York, 1965" (PDF). Mathematics of Computation. 21 (98): 293–295. JSTOR 2004214. MTE:408. Retrieved 2016-03-04.
76. Blake, J. R.; Wood, Van E.; Glasser, M. Lawrence; Fettis, Henry E.; Hansen, Eldon Robert; Patrick, Merrell R. (October 1968). "Table Errata 428: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, fourth edition, prepared by Yu. V. Geronimus & M. Yu. Tseytlin, Academic Press, New York, 1965" (PDF). Mathematics of Computation. 22 (104): 903–907. JSTOR 2004606. MR 0239122. MTE:428. Retrieved 2016-03-04. (NB. See 1972 corrigenda by Fettis and 1979 corrigenda by Anderson.)
77. Corrington, Murlan S.; Fettis, Henry E. (April 1969). "Table Errata 437: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 23 (106): 467–472. JSTOR 2004471. MR 0415966. Retrieved 2016-03-04.
78. Bradley, Lee C. (October 1969). "Table Errata 446: I. S. Gradshteyn & I. M. Ryzhik, Tables of Integrals, Series and Products, 4th edition, Academic Press, New York, 1965" (PDF). Mathematics of Computation. 23 (108): 891–892, s15–s17. doi:10.1090/s0025-5718-1969-0415968-8. JSTOR 2004993. MR 0415968. MTE:446. Retrieved 2016-03-04.
79. Young, A. T. (January 1970). "Table Errata 451: A. Erdélyi, W. Magnus, F. Oberhettinger & F. G. Tricomi, Tables of Integral Transforms, McGraw-Hill Book Co., New York, 1954" (PDF). Mathematics of Computation. 24 (109): 239–242. doi:10.2307/2004614. JSTOR 2004912. MR 0257656. MTE:451. Archived (PDF) from the original on 2016-03-16. Retrieved 2016-03-16. (NB. The error also affects entry 3.252.10 on page 297 in GR.)
80. Muhlhausen, Carl W.; Konowalow, Daniel D. (January 1971). "Table Errata 473: I. S. Gradshteyn & I. M. Ryzhik, Tables of Integrals, Series and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 25 (113): 199–201. JSTOR 2005156. MR 0415969. MTE:473. Retrieved 2016-03-04.
81. Nash, John C. (April 1972). "Table Errata 492: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 26 (118): 597–599. JSTOR 2005198. MR 0415971. MTE:492. Retrieved 2016-03-04.
82. Fettis, Henry E. (April 1972). "Corrigendum: MTE 428, Math. Comp., v.22, 1968, pp. 903–907" (PDF). Mathematics of Computation. 26 (118): 601. doi:10.2307/2005199. JSTOR 2005199. MR 0415973. Retrieved 2016-03-04. (NB. This corrigenda applies to MTE 428.)
83. Ojo, Akin; Sadiku, J. (April 1973). "Table Errata 503: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 27 (122): 451–452. doi:10.1090/s0025-5718-1973-0415972-0. JSTOR 2005649. MR 0415972. MTE:503. Retrieved 2016-03-04. (NB. See 1982 corrigenda by Fettis.)
84. Fettis, Henry E. (April 1982). "Corrigenda: Ojo, Akin; Sadiku, J. (1973). Table Errata 503: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 38 (158): 657. doi:10.1090/S0025-5718-1982-0645681-X. JSTOR 2007312. MR 0645681. Retrieved 2016-03-14. (NB. This corrigenda applies to MTE 503.)
85. Scherzinger, Ann (October 1976). "Table Errata 528: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 30 (136): 899. doi:10.1090/s0025-5718-1976-0408192-x. JSTOR 2005420. MR 0408192. MTE:528. Retrieved 2016-03-04.
86. Carr, Alistair R. (April 1977). "Table Errata 534: I. S. Gradshteyn & I. M. Ryzhik, Tables of Integrals, Series and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 31 (138): 614–616. doi:10.1090/s0025-5718-1977-0428676-9. JSTOR 2006446. MR 0428676. MTE:534. Retrieved 2016-03-04.
87. Robinson, Neville I. (January 1978). "Table Errata 550: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 32 (141): 317–320. JSTOR 2006287. MR 0478539. MTE:550. Retrieved 2016-03-04.
88. Fettis, Henry E. (January 1979). "Table Errata 557: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 33 (145): 430–431. JSTOR 2006060. MTE:557. Retrieved 2016-03-04.
89. Anderson, Michael (January 1979). "Corrigenda: p. 906 of MTE 428. I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 33 (145): 432–433. doi:10.2307/2006061. JSTOR 2006061. Retrieved 2016-03-04. (NB. This corrigenda applies to MTE 428.)
90. McGregor, John Ross (April 1979). "Table Errata 564: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 33 (146): 845–846. JSTOR 2006322. MTE:564. Retrieved 2016-03-04.
91. Birtwistle, David T. (October 1979). "Table Errata 565: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 33 (148): 1377. JSTOR 2006476. MTE:565. Retrieved 2016-03-04.
92. Gallas, Jason A. Carlson (October 1980). "Table Errata 572: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 35 (152): 1444. doi:10.1090/S0025-5718-1980-0583522-8. JSTOR 2006418. MR 0583522. MTE:572. Retrieved 2016-03-04.
93. Fettis, Henry E. (January 1981). "Table Errata 577: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 36 (153): 317–318. doi:10.1090/S0025-5718-1981-0595074-8. JSTOR 2007758. MR 0595074. MTE:577. Retrieved 2016-03-04.
94. Fettis, Henry E. (January 1981). "Table Errata 582: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 36 (153): 315–320. JSTOR 2007758. MR 0595074. MTE:582. Archived (PDF) from the original on 2016-03-16. Retrieved 2016-03-04. (NB. See 1982 corrigenda by Fettis.)
95. Fettis, Henry E. (January 1982). "Corrigenda: Fettis, Henry E. (1981). Table Errata 582: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1965" (PDF). Mathematics of Computation. 38 (157): 337. doi:10.1090/S0025-5718-1982-0637313-1. JSTOR 2007492. MR 0637313. Archived (PDF) from the original on 2016-03-16. Retrieved 2016-03-16. (NB. This corrigenda applies to MTE 582.)
96. van Haeringen, Hendrik; Kok, Lambrecht P. [at Wikidata] (October 1982). "Table Errata 589: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, Corrected and enlarged edition, Academic Press, New York, First printing, 1980" (PDF). Mathematics of Computation. 39 (160): 747–757. doi:10.1090/S0025-5718-1982-0669666-2. JSTOR 2007357. MR 0669666. MTE:589. Retrieved 2016-02-22.
97. van Haeringen, Hendrik; Kok, Lambrecht P. [at Wikidata]. "I. S. Gradshteyn & I. M. Ryzhik, Tables of integrals, series, and products. Math. comput. 39, 747–757 (1982)". Zentralblatt für Mathematik und ihre Grenzgebiete (review). 521: 193. Zbl 0521.33002. Retrieved 2016-02-16.
98. Fettis, Henry E.; Deutsch, Emeric; Krupnikov, Ernst D. (October 1983). "Table Errata 601: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series and Products, Corrected and enlarged edition, Academic Press, New York, First Printing, 1980" (PDF). Mathematics of Computation. 41 (164): 780–783. doi:10.1090/S0025-5718-1983-0717727-2. JSTOR 2007718. MR 0717727. MTE:601. Retrieved 2016-03-04.
99. Solt, György (October 1986). "Table Errata 607: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, corrected and enlarged edition prepared by A. Jeffrey, Academic Press, New York, 1980" (PDF). Mathematics of Computation. 47 (176): 767–768. doi:10.1090/S0025-5718-1986-0856719-2. JSTOR 2008202. MR 0856719. MTE:607. Retrieved 2016-03-03.
100. "Errors in the Integral Tables of Gradshteyn and Ryzhik with Correct Results from Mathematica". Mathematica Information Centre / Wolfram Library Archive: Technical Notes. Champaign, IL, USA: Wolfram Research, Inc. 2004 [2003]. Archived from the original on 2003-04-25. Retrieved 2016-02-16.
101. "Errors in the Integral Tables of Gradshteyn and Ryzhik with Correct Results from Mathematica" (Technical note). Champaign, IL, USA: Wolfram Research, Inc. 2004 [2003]. Archived from the original on 2004-06-19. Retrieved 2016-03-04.
102. "Table of integrals, series, and products. Ed. by Alan Jeffrey. CD-ROM version 1.0 for PC, MAC, and UNIX computers. 5th ed. (English) San Diego, CA: Academic Press (1996)". Zentralblatt MATH. 1996. ISBN 9780122947568. Zbl 0918.65001. Retrieved 2016-02-16.
103. Rosenblum, Marvin (October 1996). Koepf, Wolfram (ed.). "4. Table of Integrals, Series, and Products, CD-ROM Version 1.0 Edited by Alan Jeffrey" (PDF). Books and Journals: Review. Orthogonal Polynomials and Special Functions. Vol. 7, no. 1. SIAM Activity Group on Orthogonal Polynomials and Special Functions. pp. 11–12. Archived (PDF) from the original on 2022-01-20. Retrieved 2022-01-23.
104. Kölbig, Kurt Siegfried (June 1996) [1995]. "Corrigenda: I. S. Gradshteyn & I. M. Ryzhik; Table of Integrals, Series, and Products, Fifth edition, Academic Press, Boston" (PDF). CERN Computing and Networks Division. CN/95/15. Retrieved 2016-02-12.
105. Kölbig, Kurt Siegfried (January 1995). "Table errata 617: I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Alan Jeffrey, ed.), Academic Press, Boston, 1994" (PDF). Mathematics of Computation. 64 (209): 449–460. doi:10.1090/S0025-5718-1995-1270626-0. JSTOR 2153354. MR 1270626. MTE:617. Retrieved 2016-03-03.
106. Lambert, Adeline (January 1997). "Table Errata 628". Mathematics of Computation. 66 (217): 463. JSTOR 2153671. MR 1388890. MTE:628.
107. Fikioris, George (October 1998). "Table Errata 634". Mathematics of Computation. 67 (224): 1753–1754. JSTOR 2584882. MR 1625064. MTE:634. MSC:00A22,33-00,65-00.
108. Ruderman, Dan L. (2001-01-22). "Errors in Gradshteyn and Ryzhik, 5th ed". Archived from the original on 2007-02-18. 3.381.3, 3.411.6, 3.721.3, 3.761.2, 3.761.9, 3.897.1, 6.561.13, 8.350.2
109. Rangarajan, Sarukkai Krishnamachari. "Gradshteyn, I. S.; Ryzhik, I. M. Table of integrals, series, and products. Translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. 6th ed. San Diego, CA: Academic Press". Zentralblatt MATH. ISBN 9780122947575. Zbl 0981.65001. Retrieved 2016-02-16.
110. "Table Errata 636". Mathematics of Computation. 71 (239): 1335–1336. July 2002. JSTOR 2698918. MTE:636.
111. "Table Errata 637". Mathematics of Computation. 71 (239): 1335–1336. July 2002. JSTOR 2698918. MTE:637.
112. Zwillinger, Daniel; Jeffrey, Alan (2005-11-10). "Errata for Tables of Integrals, Series, and Products, 6th edition by I. S. Gradshteyn and M. Ryzhik edited by Alan Jeffrey and Daniel Zwillinger, Academic Press, Orlando, Florida, 2000, ISBN 0-12-294757-6" (PDF). Archived (PDF) from the original on 2016-03-08. Retrieved 2016-03-08. (NB. This list of 64 pages has 398 entries. According to Daniel Zwillinger it is incomplete.)
113. De Vos, Alexis (2020-11-09) [2009-03-19]. "Alexis De Vos". Universiteit Gent, Belgium. Archived from the original on 2021-06-13. Retrieved 2022-01-12. […] Finally, he is the proud discoverer of an error in equation 3.454.1 of the Gradshteyn and Ryzhik "Tables of integrals, series, and products". See errata for 6th edition by Alan Jeffrey and Daniel Zwillinger, pages 1 and 19. The error is now corrected in the 7th edition page 363 (with acknowledgement in page xxvi). […]
114. "Gradshteyn, I. S.; Ryzhik, I. M. Table of integrals, series, and products. Translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. With one CD-ROM (Windows, Macintosh and UNIX). 7th ed. Amsterdam: Elsevier/Academic Press (ISBN 978-0-12-373637-6)". Zentralblatt MATH. ISBN 9780123736376. Zbl 1208.65001. Retrieved 2016-02-16.
115. Zwillinger, Daniel; Jeffrey, Alan (2008-04-11). "Errata for Tables of Integrals, Series, and Products (7th edition) by I. S. Gradshteyn and M. Ryzhik edited by Alan Jeffrey and Daniel Zwillinger, Academic Press, Orlando, Florida, 2007, ISBN 0-12-373637-4" (PDF). Archived (PDF) from the original on 2016-03-08. Retrieved 2016-03-08. (NB. This list of 7 pages has 42 entries. According to Daniel Zwillinger it is incomplete.)
116. Veestraeten, Dirk (2015-07-24) [2015-06-21]. Written at Amsterdam, Netherlands. "Some remarks, generalizations and misprints in the integrals in Gradshteyn and Ryzhik". Scientia. Series A: Mathematical Sciences. Valparaıso, Chile: Universidad Tecnica Federico Santa Marıa. 26: 115–131. ISSN 0716-8446. S2CID 124124467. Archived (PDF) from the original on 2021-12-26. Retrieved 2021-12-26. (18 pages)
117. Zwillinger, Daniel; Moll, Victor Hugo (2021-04-23) [2014-10-06]. "Errata for Tables of Integrals, Series, and Products (8th edition) by I. S. Gradshteyn and M. Ryzhik edited by Daniel Zwillinger and Victor Moll, Academic Press, 2014, ISBN 0-12-384933-0" (PDF) (6 ed.). Archived (PDF) from the original on 2021-04-25. Retrieved 2021-12-26. (NB. This list of 33 pages has 191 entries.)
External links
• Zwillinger, Daniel. "Gradshteyn and Ryzhik: Table of Integrals, Series, and Products (Home Page)". Archived from the original on 2016-03-08. Retrieved 2016-03-08.
• Moll, Victor Hugo. "List with the formulas and proofs in GR". Archived from the original on 2010-01-09. Retrieved 2016-03-08.
• "SCIENTIA, Series A: Mathematical Sciences – Official Journal of the Universidad Técnica Federico Santa María". Universidad Técnica Federico Santa María (UTFSM). Retrieved 2016-03-08.
• Blackley, Jonathan "Seamus" (2021-06-12) [2021-06-11]. "One of my most cherished things, a tool so useful and powerful that I honesty have come to look at it as a friend: Table of Integrals, Series, and Products". twitter. Archived from the original on 2021-09-11. Retrieved 2022-01-12.
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Yuri Zhuravlyov (mathematician)
Yuri Ivanovich Zhuravlyov (Russian: Юрий Иванович Журавлёв; 14 January 1935 – 14 January 2022) was a Soviet and Russian mathematician specializing in the algebraic theory of algorithms. His research in applied mathematics and computer science was foundational for a number of specialties within discrete mathematics, pattern recognition, and predictive analysis. Zhuravlyov was a full member of the Russian Academy of Sciences and the chairman of its "Applied Mathematics and Informatics" section. He was also the editor-in-chief of the international journal Pattern Recognition and Image Analysis.
Yuri Zhuravlyov
Born
Yuri Ivanovich Zhuravlyov
(1935-01-14)14 January 1935
Voronezh, RSFSR, Soviet Union
Died14 January 2022(2022-01-14) (aged 87)
Moscow, Russia
NationalityRussian
EducationFull Member RAS (1992)
Alma materMoscow State University
Scientific career
FieldsMathematics
InstitutionsDorodnitsyn Computing Centre, Moscow State University
Biography
Zhuravlyov was born on 14 January 1935 in Voronezh in the former Soviet Union. In 1952, after finishing high school, he applied and was accepted into the Mathematics Department at Moscow State University. Under the direction of Alexey Lyapunov, he completed his first serious work on the minimization of partially defined boolean functions. The work was published in 1955 and awarded first prize at the All-Soviet student research competition.
In 1957, Zhuravlyov completed his master's thesis on a solution to the problem of finding words in a finite set with consideration for its construction. In 1959, he completed his doctoral work which involved a proof for lack of local unsolvability for constructing the minimal disjunctive normal form.
In 1959, he moved to Novosibirsk, where he pursued government-sponsored research and taught algebra and mathematical logic at the Novosibirsk University. In 1966, he began research into pattern recognition. His first serious work in this field related to the identification of deposits in the area of gold mining. He then developed a model of algorithms for calculating estimates that became foundational for numerous subsequent research and works in the field.
In 1969, Zhuravlyov moved to Moscow to head the Pattern Recognition Lab at the Central Soviet Computing Center. In 1970, he also joined the faculty of the Moscow Institute of Physics and Technology as a full professor.
Throughout the 1970s and 1980s, Zhuravlyov published a series of seminal works in applied mathematics and informatics. In 1991, he founded the journal Pattern Recognition and Image Analysis. In 1992, he was invited to join the Russian Academy of Sciences. In 1997, he became a professor at Moscow State University.
Zhuravlyov died in Moscow on 14 January 2022, on his 87th birthday.[1]
References
1. Юрий Иванович Журавлев (14.01.1935 – 14.01.2022) (in Russian)
External links
• Maik journal page
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• Springer journal page
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Yurii Egorov
Yurii (or Yuri) Vladimirovich Egorov (Юрий Владимирович Егоров, born 14 July 1938 in Moscow, died October 2018 in Toulouse) was a Russian-Soviet mathematician who specialized in differential equations.
Biography
In 1960 he completed his undergraduate studies at the Mechanics and Mathematics Faculty of Moscow State University (MSU). In 1963 from MSU he received his Ph.D. with the thesis "Некоторые задачи теории оптимального управления в бесконечномерных пространствах" ("Some Problems of Optimal Control Theory in Infinite-Dimensional Spaces"). In 1970 from MSU he received his Russian doctorate of sciences (Doctor Nauk) with thesis: "О локальных свойствах псевдодифференциальных операторов главного типа" ("Local Properties of Pseudodifferential Operators of Principal Type"). He was employed at MSU from 1961 to 1992, and he was a full professor in the Department of Differential Equations of the Mechanics and Mathematics Faculty there from 1973 to 1992. Since 1992 he has been a professor of mathematics at Paul Sabatier University (Toulouse III).
Egorov's research deals with differential equations and applications in mathematical physics, spectral theory, and optimal control theory. In 1970 he was an Invited Speaker of the ICM in Nice.[1]
Awards
• 1981 — Lomonosov Memorial Prize (established in 1944) — for his series of publications on "Субэллиптические операторы и их применения к исследованию краевых задач" (Subelliptic operators and their applications to the study of boundary value problems)
• 1988 — USSR State Prize (with several co-authors) — for their series of publications (1958–1985) on "Исследования краевых задач для дифференциальных операторов и их приложения в математической физике" (Research on boundary value problems and their applications in mathematical physics)
• 1998 — Petrovsky Award (jointly with V. A. Kondratiev) for their series of publications on "Исследование спектра эллиптических операторов" (The study of the spectra of elliptic operators)
Selected publications
Articles
• "The canonical transformations of pseudodifferential operators." Uspekhi Matematicheskikh Nauk 24, no. 5 (1969): 235–236.
• "On the solubility of differential equations with simple characteristics." Russian Mathematical Surveys 26, no. 2 (1971): 113.
• with Mikhail Aleksandrovich Shubin: "Linear partial differential equations. Foundations of the classical theory." Itogi Nauki i Tekhniki. Seriya" Sovremennye Problemy Matematiki. Fundamental'nye Napravleniya" 30 (1988): 5–255.
• "A contribution to the theory of generalized functions." Russian Mathematical Surveys 45, no. 5 (1990): 1.
• with Vladimir Aleksandrovich Kondrat'ev and Olga Arsen'evna Oleynik: "Asymptotic behaviour of the solutions of non-linear elliptic and parabolic systems in tube domains." Sbornik: Mathematics 189, no. 3 (1998): 359–382.
• Victor A. Galaktionov, Vladimir A. Kondratiev, and Stanislav I. Pohozaev: "On the necessary conditions of global existence to a quasilinear inequality in the half-space." Comptes Rendus de l'Académie des Sciences-Series I-Mathematics 330, no. 2 (2000): 93–98.
Books
• with Vladimir A. Kondratiev: On spectral theory of elliptic operators. Operator theory, advances and applications ; vol. 89. Basel; Boston: Birkhäuser Verlag. 1996. ISBN 9783764353902; x+328 pages{{cite book}}: CS1 maint: postscript (link)
• with Bert-Wolfgang Schulze: Pseudo-differential operators, singularities, applications. Operator theory, advances and applications ; vol. 93. Basel; Boston: Birkhäuser Verlag. 1997. ISBN 9783764354848; xiii+349 pages{{cite book}}: CS1 maint: postscript (link)
References
1. Egorov, Yu V. "On the local solvability of pseudodifferential equations." In Actes du Congrès International des Mathématiciens, Tome 2, pp. 717–722. 1970.
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Yurii Nesterov
Yurii Nesterov is a Russian mathematician, an internationally recognized expert in convex optimization, especially in the development of efficient algorithms and numerical optimization analysis. He is currently a professor at the University of Louvain (UCLouvain).
Yurii Nesterov
2005 in Oberwolfach
Born (1956-01-25) January 25, 1956
Moscow, USSR
CitizenshipBelgium
Alma materMoscow State University (1977)
Awards
• Dantzig Prize, 2000
• John von Neumann Theory Prize, 2009
• EURO Gold Medal, 2016
Scientific career
Fields
• Convex optimization,
• Semidefinite programming,
• Nonlinear programming,
• Numerical analysis,
• Applied mathematics
Institutions
• UCLouvain
• National Research University
• Central Economic Mathematical Institute
Doctoral advisorBoris Polyak
Biography
In 1977, Yurii Nesterov graduated in applied mathematics at Moscow State University. From 1977 to 1992 he was a researcher at the Central Economic Mathematical Institute of the Russian Academy of Sciences. Since 1993, he has been working at UCLouvain, specifically in the Department of Mathematical Engineering from the Louvain School of Engineering, Center for Operations Research and Econometrics.
In 2000, Nesterov received the Dantzig Prize.[1]
In 2009, Nesterov won the John von Neumann Theory Prize.[2]
In 2016, Nesterov received the EURO Gold Medal.[3]
Academic work
Nesterov is most famous for his work in convex optimization, including his 2004 book, considered a canonical reference on the subject.[4] His main novel contribution is an accelerated version of gradient descent that converges considerably faster than ordinary gradient descent (commonly referred as Nesterov momentum, Nesterov Acceleration or Nesterov accelerated gradient, in short — NAG).[5][6][7][8][9] This method, sometimes called "FISTA", was further developed by Beck & Teboulle in their 2009 paper "A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems".[10]
His work with Arkadi Nemirovski in their 1994 book[11] is the first to point out that the interior point method can solve convex optimization problems, and the first to make a systematic study of semidefinite programming (SDP). Also in this book, they introduced the self-concordant functions which are useful in the analysis of Newton's method.[12]
References
1. "The George B. Dantzig Prize". 2000. Retrieved December 12, 2014.
2. "John Von Neumann Theory Prize". 2009. Retrieved June 4, 2014.
3. "EURO Gold Medal". 2016. Retrieved August 20, 2016.
4. Nesterov, Yurii (2004). Introductory lectures on convex optimization : A basic course. Kluwer Academic Publishers. CiteSeerX 10.1.1.693.855. ISBN 978-1402075537.
5. Nesterov, Y (1983). "A method for unconstrained convex minimization problem with the rate of convergence $O(1/k^{2})$". Doklady AN USSR. 269: 543–547.
6. Walkington, Noel J. (2023). "Nesterov's Method for Convex Optimization". SIAM Review. 65 (2): 539–562. doi:10.1137/21M1390037. ISSN 0036-1445.
7. Bubeck, Sebastien (April 1, 2013). "ORF523: Nesterov's Accelerated Gradient Descent". Retrieved June 4, 2014.
8. Bubeck, Sebastien (March 6, 2014). "Nesterov's Accelerated Gradient Descent for Smooth and Strongly Convex Optimization". Retrieved June 4, 2014.
9. "The zen of gradient descent". blog.mrtz.org. Retrieved 2023-05-13.
10. Beck, Amir; Teboulle, Marc (2009-01-01). "A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems". SIAM Journal on Imaging Sciences. 2 (1): 183–202. doi:10.1137/080716542.
11. Nesterov, Yurii; Arkadii, Nemirovskii (1995). Interior-Point Polynomial Algorithms in Convex Programming. Society for Industrial and Applied Mathematics. ISBN 978-0898715156.
12. Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (PDF). Cambridge University Press. ISBN 978-0-521-83378-3. Retrieved October 15, 2011.
External links
• Official website
John von Neumann Theory Prize
1975–1999
• George Dantzig (1975)
• Richard Bellman (1976)
• Felix Pollaczek (1977)
• John F. Nash / Carlton E. Lemke (1978)
• David Blackwell (1979)
• David Gale / Harold W. Kuhn / Albert W. Tucker (1980)
• Lloyd Shapley (1981)
• Abraham Charnes / William W. Cooper / Richard J. Duffin (1982)
• Herbert Scarf (1983)
• Ralph Gomory (1984)
• Jack Edmonds (1985)
• Kenneth Arrow (1986)
• Samuel Karlin (1987)
• Herbert A. Simon (1988)
• Harry Markowitz (1989)
• Richard Karp (1990)
• Richard E. Barlow / Frank Proschan (1991)
• Alan J. Hoffman / Philip Wolfe (1992)
• Robert Herman (1993)
• Lajos Takacs (1994)
• Egon Balas (1995)
• Peter C. Fishburn (1996)
• Peter Whittle (1997)
• Fred W. Glover (1998)
• R. Tyrrell Rockafellar (1999)
2000–present
• Ellis L. Johnson / Manfred W. Padberg (2000)
• Ward Whitt (2001)
• Donald L. Iglehart / Cyrus Derman (2002)
• Arkadi Nemirovski / Michael J. Todd (2003)
• J. Michael Harrison (2004)
• Robert Aumann (2005)
• Martin Grötschel / László Lovász / Alexander Schrijver (2006)
• Arthur F. Veinott, Jr. (2007)
• Frank Kelly (2008)
• Yurii Nesterov / Yinyu Ye (2009)
• Søren Asmussen / Peter W. Glynn (2010)
• Gérard Cornuéjols (2011)
• George Nemhauser / Laurence Wolsey (2012)
• Michel Balinski (2013)
• Nimrod Megiddo (2014)
• Vašek Chvátal / Jean Bernard Lasserre (2015)
• Martin I. Reiman / Ruth J. Williams (2016)
• Donald Goldfarb / Jorge Nocedal (2017)
• Dimitri Bertsekas / John Tsitsiklis (2018)
• Dimitris Bertsimas / Jong-Shi Pang (2019)
• Adrian Lewis (2020)
• Alexander Shapiro (2021)
• Vijay Vazirani (2022)
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Yurii Reshetnyak
Yurii Grigorievich Reshetnyak (Russian: Ю́рий Григо́рьевич Решетня́к, 26 September 1929 – 17 December 2021) was a Soviet and Russian mathematician and academician.[1]
Yurii Grigorievich Reshetnyak
Юрий Григорьевич Решетняк
Born(1929-09-26)26 September 1929
Leningrad,
RSFSR, USSR
Died17 December 2021(2021-12-17) (aged 92)
Novosibirsk, Russia
CitizenshipUSSR, Russia
Alma materLeningrad State University
Scientific career
FieldsMathematics
Doctoral advisorA. D. Aleksandrov
He worked in geometry and the theory of functions of a real variable. He was known for his work in the Reshetnyak gluing theorem. Reshetnyak received the 2000 Lobachevsky Prize from the Russian Academy of Sciences.[2]
Reshetnyak died on 17 December 2021, at the age of 92.[3]
Selected publications
• Space mappings with bounded distortion. Translations of Mathematical Monographs. Vol. 73. Providence, RI: American Mathematical Society. 1989. ISBN 0-8218-4526-8; 362 pp.{{cite book}}: CS1 maint: postscript (link)[4]
• with A. D. Aleksandrov: General theory of irregular curves [translated from the Russian by L. Ya. Yuzina]. Dordrecht & Boston: Kluwer Academic Publishers. 1989. ISBN 9027728119; x+288 pp.{{cite book}}: CS1 maint: postscript (link)
References
1. Решетняк Юрий Григорьевич
2. Lobachecvsky Prize, Russian Academy of Sciences. Accessed January 13, 2014
3. В Новосибирске скончался известный математик Юрий Решетняк (in Russian)
4. Vuorinen, Matti (1991). "Review: Space mappings with bounded distortion by Yu. G. Reshetnyak" (PDF). Bull. Amer. Math. Soc. (N.S.). 24 (2): 408–415. doi:10.1090/s0273-0979-1991-16051-9.
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Yuri Manin
Yuri Ivanovich Manin (Russian: Ю́рий Ива́нович Ма́нин; 16 February 1937 – 7 January 2023) was a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics.
Yuri Manin
Manin in 2006
Born
Yuri Ivanovich Manin
(1937-02-16)16 February 1937
Simferopol, Crimean ASSR, Russian SFSR, Soviet Union
Died7 January 2023(2023-01-07) (aged 85)
NationalityRussian
Alma mater
• Moscow State University
• Steklov Mathematics Institute (PhD)
Known forManin conjecture
Manin matrix
Manin obstruction
Manin triple
Manin–Drinfeld theorem
Manin–Mumford conjecture
ADHM construction
Gauss–Manin connection
Cartier–Manin operator
CH-quasigroup
Modular symbol
Quantum simulator
Awards
• Nemmers Prize in Mathematics (1994)
• Schock Prize (1999)
• Cantor Medal (2002)
• Bolyai Prize (2010)
• King Faisal International Prize (2002)
Scientific career
FieldsMathematics
Institutions
• Max-Planck-Institut für Mathematik
• Northwestern University
Doctoral advisorIgor Shafarevich
Doctoral students
• Alexander Beilinson
• Vladimir Berkovich
• Ivan Cherednik
• Mariusz Wodzicki
• Vladimir Drinfeld
• Ha Huy Khoai
• Vasilli Iskovskikh
• Mikhail Kapranov
• Victor Kolyvagin
• Alexander L. Rosenberg
• Vyacheslav Shokurov
• Alexei Skorobogatov
• Yuri Tschinkel
Life and career
Manin was born on 16 February 1937 in Simferopol, Crimean ASSR, Soviet Union.[1]
He received a doctorate in 1960 at the Steklov Mathematics Institute as a student of Igor Shafarevich. He became a professor at the Max-Planck-Institut für Mathematik in Bonn, where he was director from 1992 to 2005 and then director emeritus.[2][1] He was also a professor emeritus at Northwestern University.[3]
He had over the years more than 40 doctoral students, including Vladimir Berkovich, Mariusz Wodzicki, Alexander Beilinson, Ivan Cherednik, Alexei Skorobogatov, Vladimir Drinfeld, Mikhail Kapranov, Vyacheslav Shokurov, Ralph Kaufmann, Arend Bayer, Victor Kolyvagin and Hà Huy Khoái.[4]
Manin died on 7 January 2023.[1]
Research
Manin's early work included papers on the arithmetic and formal groups of abelian varieties, the Mordell conjecture in the function field case, and algebraic differential equations. The Gauss–Manin connection is a basic ingredient of the study of cohomology in families of algebraic varieties.[5][6]
He developed the Manin obstruction, indicating the role of the Brauer group in accounting for obstructions to the Hasse principle via Grothendieck's theory of global Azumaya algebras, setting off a generation of further work.[7][8]
Manin pioneered the field of arithmetic topology (along with John Tate, David Mumford, Michael Artin, and Barry Mazur).[9] He also formulated the Manin conjecture, which predicts the asymptotic behaviour of the number of rational points of bounded height on algebraic varieties.[10]
In mathematical physics, Manin wrote on Yang–Mills theory, quantum information, and mirror symmetry.[11][12] He was one of the first to propose the idea of a quantum computer in 1980 with his book Computable and Uncomputable.[13]
He wrote a book on cubic surfaces and cubic forms, showing how to apply both classical and contemporary methods of algebraic geometry, as well as nonassociative algebra.[14]
Awards
He was awarded the Brouwer Medal in 1987, the first Nemmers Prize in Mathematics in 1994, the Schock Prize of the Royal Swedish Academy of Sciences in 1999, the Cantor Medal of the German Mathematical Society in 2002, the King Faisal International Prize in 2002, and the Bolyai Prize of the Hungarian Academy of Sciences in 2010.[1]
In 1990, he became a foreign member of the Royal Netherlands Academy of Arts and Sciences.[15] He was a member of eight other academies of science and was also an honorary member of the London Mathematical Society.[1]
Selected works
• Mathematics as metaphor – selected essays. American Mathematical Society. 2009.
• Rational points of algebraic curves over function fields. {{cite book}}: |work= ignored (help)
• Manin, Yu I. (1965). "Algebraic topology of algebraic varieties". Russian Mathematical Surveys. 20 (6): 183–192. Bibcode:1965RuMaS..20..183M. doi:10.1070/RM1965v020n06ABEH001192. S2CID 250895773.
• Frobenius manifolds, quantum cohomology, and moduli spaces. American Mathematical Society. 1999.[16]
• Quantum groups and non commutative geometry. Montreal: Centre de Recherches Mathématiques. 1988.
• Topics in non-commutative geometry. Princeton University Press. 1991. ISBN 9780691635781.[17]
• Gauge field theory and complex geometry. Grundlehren der mathematischen Wissenschaften. Springer. 1988.[18]
• Cubic forms - algebra, geometry, arithmetics. North Holland. 1986.
• A course in mathematical logic. Springer. 1977.,[19] second expanded edition with new chapters by the author and Boris Zilber, Springer 2010.
• Computable and Uncomputable. Moscow. 1980.{{cite book}}: CS1 maint: location missing publisher (link)[13]
• Mathematics and physics. Birkhäuser. 1981.
• Manin, Yu. I. (1984). "New dimensions in geometry". Arbeitstagung. Lectures Notes in Mathematics. Vol. 1111. Bonn: Springer. pp. 59–101. doi:10.1007/BFb0084585. ISBN 978-3-540-15195-1.
• Manin, Yuri; Kostrikin, Alexei I. (1989). Linear algebra and geometry. London, England: Gordon and Breach. doi:10.1201/9781466593480. ISBN 9780429073816. S2CID 124713118.
• Manin, Yuri; Gelfand, Sergei (1994). Homological algebra. Encyclopedia of Mathematical Sciences. Springer.
• Manin, Yuri; Gelfand, Sergei Gelfand (1996). Methods of Homological algebra. Springer Monographs in Mathematics. Springer. doi:10.1007/978-3-662-12492-5. ISBN 978-3-642-07813-2.
• Manin, Yuri; Kobzarev, Igor (1989). Elementary Particles: mathematics, physics and philosophy. Dordrecht: Kluwer.
• Manin, Yuri; Panchishkin, Alexei A. (1995). Introduction to Number theory. Springer.
• Manin, Yuri I. (2000). "Moduli, Motives, Mirrors". European Congress of Mathematics. Progress in Mathematics. Barcelona. pp. 53–73. doi:10.1007/978-3-0348-8268-2_4. hdl:21.11116/0000-0004-357E-4. ISBN 978-3-0348-9497-5.{{cite book}}: CS1 maint: location missing publisher (link)
• Classical computing, quantum computing and Shor´s factoring algorithm (PDF). Bourbaki Seminar. 1999.{{cite book}}: CS1 maint: location missing publisher (link)
• Rademacher, Hans; Toeplitz, Otto (2002). Von Zahlen und Figuren [From Numbers and Figures] (in German). doi:10.1007/978-3-662-36239-6. ISBN 978-3-662-35411-7.
• Manin, Yuri; Marcolli, Matilde (2002). "Holography principle and arithmetic of algebraic curves". Advances in Theoretical and Mathematical Physics. Max-Planck-Institut für Mathematik, Bonn: International Press. 5 (3): 617–650. doi:10.4310/ATMP.2001.v5.n3.a6. S2CID 25731842.
• Manin, Yu. I. (December 1991). "Three-dimensional hyperbolic geometry as ∞-adic Arakelov geometry". Inventiones Mathematicae. 104 (1): 223–243. Bibcode:1991InMat.104..223M. doi:10.1007/BF01245074. S2CID 121350567.
• Mathematik, Kunst und Zivilisation [Mathematics, Art and Civilisation]. Die weltweit besten mathematischen Artikel im 21. Jahrhundert. Vol. 3. e-enterprise. 2014. ISBN 978-3-945059-15-9.
See also
• Arithmetic topology
• Noncommutative residue
References
1. "Max Planck Institute for Mathematics in Bonn Mourns Death of Yuri Manin". Max Planck Institute for Mathematics. 8 January 2023. Retrieved 8 January 2023.
2. "Yuri Manin | Max Planck Institute for Mathematics". www.mpim-bonn.mpg.de. Retrieved 6 August 2018.
3. "Emeriti Faculty: Department of Mathematics – Northwestern University". math.northwestern.edu. Retrieved 6 August 2018.
4. Yuri Manin at the Mathematics Genealogy Project
5. Manin, Ju. I. (1958), "Algebraic curves over fields with differentiation", Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya (in Russian), 22: 737–756, MR 0103889 English translation in Manin, Ju. I. (1964) [1958], "Algebraic curves over fields with differentiation", American Mathematical Society translations: 22 papers on algebra, number theory and differential geometry, vol. 37, Providence, R.I.: American Mathematical Society, pp. 59–78, ISBN 978-0-8218-1737-7, MR 0103889
6. "Gauss-Manin connection", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
7. Serge Lang (1997). Survey of Diophantine geometry. Springer-Verlag. pp. 250–258. ISBN 3-540-61223-8. Zbl 0869.11051.
8. Alexei N. Skorobogatov (1999). Appendix A by S. Siksek: 4-descent. "Beyond the Manin obstruction". Inventiones Mathematicae. 135 (2): 399–424. arXiv:alg-geom/9711006. Bibcode:1999InMat.135..399S. doi:10.1007/s002220050291. S2CID 14285244. Zbl 0951.14013.
9. Morishita, Masanori (2012). "Introduction". Knots and Primes. Universitext. London: Springer. pp. 1–7. doi:10.1007/978-1-4471-2158-9_1. ISBN 978-1-4471-2157-2.
10. Franke, J.; Manin, Y. I.; Tschinkel, Y. (1989). "Rational points of bounded height on Fano varieties". Inventiones Mathematicae. 95 (2): 421–435. Bibcode:1989InMat..95..421F. doi:10.1007/bf01393904. MR 0974910. S2CID 121044839. Zbl 0674.14012.
11. Atiyah, Michael; Drinfeld, Vladimir; Hitchin, Nigel; Manin, Yuri (1978). "Construction of instantons". Physics Letters A. 65 (3): 185–187. Bibcode:1978PhLA...65..185A. doi:10.1016/0375-9601(78)90141-X.
12. Devchand, Chandrashekar; Ogievetsky, Victor I. (1996). "Integrability of N=3 super Yang-Mills equations". Topics in statistical and theoretical physics. Amer. Math. Soc. Transl. Ser. 2. Vol. 177. Providence, RI: American Mathematical Society. pp. 51–58.
13. Manin, Yu. I. (1980). Vychislimoe i nevychislimoe [Computable and Noncomputable] (in Russian). Sov.Radio. pp. 13–15. Archived from the original on 10 May 2013. Retrieved 4 March 2013.
14. Manin: Cubic forms – algebra, geometry, arithmetics, North Holland 1986
15. "Y.I. Manin". Royal Netherlands Academy of Arts and Sciences. Retrieved 19 July 2015.
16. Getzler, Ezra (2001). "Review: Frobenius manifolds, quantum cohomology, and moduli spaces by Yuri I. Manin". Bull. Amer. Math. Soc. (N.S.). 38 (1): 101–108. doi:10.1090/S0273-0979-00-00888-0.
17. Penkov, Ivan (1993). "Review: Topics in non-commutative geometry by Yuri I. Manin". Bull. Amer. Math. Soc. (N.S.). 29 (1): 106–111. doi:10.1090/S0273-0979-1993-00391-4.
18. LeBrun, Claude (1989). "Review: Gauge field theory and complex geometry by Yuri I. Manin; trans. by N. Koblitz and J. R. King". Bull. Amer. Math. Soc. (N.S.). 21 (1): 192–196. doi:10.1090/S0273-0979-1989-15816-3.
19. Shoenfield, J. R. (1979). "Review: A course in mathematical logic by Yu. I Manin" (PDF). Bull. Amer. Math. Soc. (N.S.). 1 (3): 539–541. doi:10.1090/s0273-0979-1979-14613-5.
Further reading
• Némethi, A. (April 2011). "Yuri Ivanovich Manin" (PDF). Acta Mathematica Hungarica. 133 (1–2): 1–13. doi:10.1007/s10474-011-0151-x.
• Jean-Paul Pier (1 January 2000). Development of Mathematics 1950–2000. Springer Science & Business Media. p. 1116. ISBN 978-3-7643-6280-5.
External links
Wikiquote has quotations related to Yuri Manin.
• Manin's page at Max-Planck-Institut für Mathematik website
• Good Proofs are Proofs that Make us Wiser, interview by Martin Aigner and Vasco A. Schmidt
• Biography
• Interviewed by David Eisenbud for Simons Foundation "Science Lives"
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• Willard Van Orman Quine (1993)
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Mathematics
• Elias M. Stein (1993)
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Yuriy Drozd
Yuriy Drozd (Ukrainian: Юрій Анатолійович Дрозд; born October 15, 1944) is a Ukrainian mathematician working primarily in algebra. He is a Corresponding Member of the National Academy of Sciences of Ukraine and head of the Department of Algebra and Topology at the Institute of Mathematics of the National Academy of Sciences of Ukraine.
Yuriy Drozd
Born (1944-10-15) 15 October 1944
Kyiv, Ukrainian SSR
NationalityUkrainian
Alma materTaras Shevchenko National University of Kyiv,
Steklov Institute of Mathematics
AwardsState Prize of Ukraine in Science and Technology
Scientific career
Fieldsmathematics, algebra, representation theory, algebraic geometry
InstitutionsInstitute of Mathematics of NAS of Ukraine
Doctoral advisorIgor Shafarevich
Doctoral studentsVolodymyr Mazorchuk
Biography
Yiriy Drozd graduated from Kyiv University in 1966, pursuing a postgraduate degree at the Institute of Mathematics of the National Academy of Sciences of Ukraine in 1969. His PhD dissertation On Some Questions of the Theory of Integral Representations (1970) was supervised by Igor Shafarevich.
From 1969 to 2006 Drozd worked at the Faculty of Mechanics and Mathematics at Kyiv University (at first as lecturer, then as associate professor and full professor). From 1980 to 1998 he headed the Department of Algebra and Mathematical Logic. Since 2006 he has been the head of the Department of Algebra and Topology (until 2014 - the Department of Algebra) of the Institute of Mathematics of the National Academy of Sciences of Ukraine.
His doctoral students include Volodymyr Mazorchuk.
References
• Mathematics Genealogy Project.
• Institute of Mathematics of the National Academy of Sciences of Ukraine.
• Personal site.
• Oberwolfach Photo Collection.
External links
• Yuriy Drozd, Introduction to Algebraic Geometry (course lecture notes, University of Kaiserslautern).
• Yuriy Drozd, Vector Bundles over Projective Curves.
• Yuriy Drozd, General Properties of Surface Singularities.
• Drozd, Yuriy; Kirichenko, Vladimir (1994). Finite-Dimensional Algebras. Springer. ISBN 978-3-642-76244-4.
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Yuri Aleksandrovich Brychkov
Yury Aleksandrovich Brychkov (Russian: Юрий Александрович Брычков; born 29 February 1944 in Moscow, Russia) is a Russian mathematician.
Yury Aleksandrovich Brychkov
Юрий Александрович Брычков
Born (1944-02-29) 29 February 1944
Moscow, USSR
Alma materMoscow State University
Known forTables of Series, Special Functions, Integral Transforms
Scientific career
FieldsMathematics
Doctoral advisorYuri Mikhailovich Shirokov
He graduated from Moscow State University in 1966 and worked on quantum field theory at the Steklov Mathematical Institute of the Russian Academy of Sciences, under the supervision of Yuri Mikhailovich Shirokov. He received his PhD in 1971 and he has been with the Dorodnicyn Computing Centre of the Russian Academy of Sciences since 1969.
Yu. A. Brychkov has worked on various topics of pure mathematics, and he has made contributions to the fields of special functions and integral transforms. He has also worked on the computer implementation of special functions at the University of Waterloo,[1] Maplesoft, and Wolfram Research.[2] He is a founding editor of the Journal of Integral Transforms and Special Functions,[3] and has authored a number of handbooks, including the five volume Integrals and Series (Gordon and Breach Science Publishers, 1986–1992).[4]
Works
• Brychkov, Yu. A.; Prudnikov, A. P. (1989). Integral transformations of generalized functions. New York-London: Gordon & Breach Science Publishers / CRC Press. New York-London. ISBN 2-88124-705-9. (342 pages)
• Brychkov, Yu. A.; Marichev, O. I.; Prudnikov, A. P. (1989). Tables of indefinite integrals. New York-London: Gordon & Breach Science Publishers / CRC Press. New York-London. ISBN 2-88124-710-5. (192 pages)
• Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I. Integraly i ryady Интегралы и ряды [Integrals and series] (in Russian). Vol. Set 1-3 (1 ed.). Nauka (Наука). Moscow. 1981−1986.
• Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I. Integrals and series. Vol. Set 1-5. Gordon & Breach Science Publishers / CRC Press. New York-London. 1986−1992.
• Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I. (1986). Integrals and Series. Vol. 1: Elementary Functions. Gordon & Breach Science Publishers / CRC Press. New York-London. ISBN 978-2-88124-089-8. OCLC 916363878. (798 pages)
• Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I. (1986). Integrals and Series. Vol. 2: Special functions. Gordon & Breach Science Publishers / CRC Press. New York-London. ISBN 978-2-88124-090-4. OCLC 50653126. (750 pages.)
• Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I. (1990). Integrals and Series. Vol. 3: More special functions. Gordon & Breach Science Publishers / CRC Press. New York-London. ISBN 978-2-88124-682-1. OCLC 916363880. (800 pages.)
• Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I. (1992). Integrals and Series. Vol. 4: Direct Laplace Transforms. Gordon & Breach Science Publishers / CRC Press. New York-London. OCLC 63722509. (Second printing: 1998.) (xviii+618 pages.)
• Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I. (1992). Integrals and Series. Vol. 5: Inverse Laplace Transforms. Gordon & Breach Science Publishers / CRC Press. New York-London. ISBN 2-88124838-1. OCLC 489706146. (xx+595 pages.)
• Brychkov, Yu. A.; Glaeske, H.-Ju.; Prudnikov, A. P.; Vu Kim, Tuan (1992). Multidimensional integral transformations. Gordon & Breach Science Publishers / CRC Press. New York-London. ISBN 2-88124-839-X. (386 pages)
• Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I. (2003). Integraly i ryady Интегралы и ряды [Integrals and series] (in Russian). Vol. Set 1-3 (2nd revised ed.). Fizmatlit (Физматлит). ISBN 978-5-9221-0322-0.
• Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I. (2003). Integraly i ryady Интегралы и ряды [Integrals and series: Elementary functions] (in Russian). Vol. 1: Elementarnye funktsii (Элементарные функции) (2nd revised ed.). Fizmatlit (Физматлит). Moscow. ISBN 978-5-9221-0323-7. OCLC 937142305. (630 pages)
• Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I. (2003). Integraly i ryady Интегралы и ряды [Integrals and series: Special functions] (in Russian). Vol. 2: Spetsialnye funktsii (Специальные функции) (2nd revised ed.). Fizmatlit (Физматлит). Moscow. ISBN 978-5-9221-0324-4. (663 pages)
• Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I. (2003). Integraly i ryady Интегралы и ряды [Integrals and series: Special functions. Further chapters.] (in Russian). Vol. 3: Spetsialnye funktsii. Dopolnitelnye glavy (2nd revised ed.). Fizmatlit (Физматлит). Moscow. ISBN 978-5-9221-0325-1. (710 pages)
• Brychkov, Yu. A. (2008). Handbook of special functions. Derivatives, integrals, series and other formulas. CRC Press. Boca Raton. ISBN 978-1-58488-956-4. (xx+680 pages)
• Brychkov, Yu. A.; Marichev, O. I.; Savischenko., N. V. (2018). Handbook of Mellin Transforms. CRC Press. Boca Raton. ISBN 978-1-13835-335-0. (xx+587 pages)
References
1. "University of Waterloo: Telephone Directory, September 2002" (PDF). Information Systems & Technology, University of Waterloo. Retrieved 2018-05-10.
2. "Wolfram Blog: Author Index: Yury Brychkov". Wolfram. Retrieved 2018-05-10.
3. "Integral Transforms and Special Functions: Editorial Board". Integral Transforms and Special Functions. 26 (12): ebi. 2015. doi:10.1080/10652469.2015.1088624.
4. Integrals and series / A.P. Prudnikov, Yu. A. Brychkov, O.I. Marichev; translated from the Russian by N.M. Queen. 1986. ISBN 9782881240973. Retrieved 2018-05-10. {{cite book}}: |website= ignored (help)
External links
• New Derivatives of the Bessel Functions Have Been Discovered
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Yuri Gurevich
Yuri Gurevich, Professor Emeritus at the University of Michigan, is an American computer scientist and mathematician and the inventor of abstract state machines.
Gurevich was born and educated in the Soviet Union.[1] He taught mathematics there and then in Israel before moving to the United States in 1982. The best-known work of his Soviet period is on the classical decision problem.[2] In Israel, Gurevich worked with Saharon Shelah on monadic second-order theories.[3] The Forgetful Determinacy Theorem of Gurevich–Harrington is of that period as well.[4]
From 1982 to 1998, Gurevich taught computer science at the University of Michigan, where he started to work on various aspects of computational complexity theory[5] including average case complexity.[6] He became one of the founders of the emerging field of finite model theory.[7]
Most importantly, he became interested in the problem of what an algorithm is. This led him to the theory of abstract state machines (ASMs). The ASM Thesis says that, behaviorally, every algorithm is an ASM.[8] A few convincing axioms enabled derivation of the sequential ASM thesis[9] and the Church–Turing thesis.[10] The ASM thesis has also been proven for some other classes of algorithms.[11][12]
From 1998 to 2018, Gurevich was with Microsoft Research where he founded a group on Foundations of Software Engineering. The group built Spec Explorer based on the theory of abstract state machines. The tool was adopted by the Windows team; a modified version of the tool helped Microsoft meet the European Union demands for high-level executable specifications. Later, Gurevich worked with different Microsoft groups on various efficiency, safety, and security issues,[13] including access control,[14] differential compression,[15] and privacy.[16]
Since 1988, Gurevich has managed the column on Logic in Computer Science in the Bulletin of the European Association for Theoretical Computer Science.[17] Since 2013 Gurevich has worked primarily on quantum computing,[18] while continuing research in his traditional areas.
Gurevich is a 2020 AAAS Fellow,[19] a 1997 ACM Fellow,[20] a 1995 Guggenheim Fellow,[21] an inaugural fellow of the European Association for Theoretical Computer Science,[22] a member of Academia Europaea, and Dr. Honoris Causa of Hasselt University in Belgium and of Ural State University in Russia.
References
1. Blass, Andreas; Dershowitz, Nachum; Reisig, Wolfgang (2010), Blass, Andreas; Dershowitz, Nachum; Reisig, Wolfgang (eds.), "Yuri, Logic, and Computer Science", Fields of Logic and Computation, Berlin, Heidelberg: Springer Berlin Heidelberg, vol. 6300, pp. 1–48, doi:10.1007/978-3-642-15025-8_1, ISBN 978-3-642-15024-1, retrieved 2023-07-05
2. E. Börger, E. Grädel, and Y. Gurevich. The Classical Decision Problem. Springer, 1997.
3. Y. Gurevich. Monadic second-order theories. In J. Barwise and S. Feferman (eds.), Model-Theoretic Logics, Springer, 1985, 479-506.
4. Y. Gurevich and L. Harrington. Automata, Trees, and Games. STOC '82: Proceedings of the Fourteenth annual ACM Symposium on Theory of Computing, 1982, 60-65.
5. Y. Gurevich and S. Shelah. Expected computation time for Hamiltonian Path Problem. SIAM Journal on Computing 16:3, 1987, 486-502.
6. Y. Gurevich. Average case completeness. Journal of Computer and System Sciences 42:3, 1991, 346-398.
7. Y. Gurevich. Toward logic tailored for computational complexity. In M Richter et al. (eds.), Computation and Proof Theory. Springer Lecture Notes in Mathematics 1104, 1984, 175-216.
8. Y. Gurevich. Evolving Algebra 1993: Lipari Guide. In E. Börger (ed.), Specification and Validation Methods, Oxford University Press, 1995, 9–36. https://arxiv.org/abs/1808.06255
9. Y. Gurevich. Sequential Abstract State Machines capture sequential algorithms. ACM Transactions on Computational Logic 1(1), 2000.
10. N. Dershowitz and Y. Gurevich. A natural axiomatization of computability and proof of Church’s Thesis. Bulletin of Symbolic Logic 14:3, 2008, 299-350.
11. A. Blass and Y. Gurevich. Abstract State Machines Capture Parallel Algorithms. ACM Transactions on Computational Logic 4(4), 2003, 578–651, and 9(3), 2008, article 19.
12. A. Blass, Y. Gurevich, D. Rosenzweig, and B. Rossman. Interactive Small-Step Algorithms II: Abstract State Machines and the Characterization Theorem. Logical Methods in Computer Science 3(4), 2007, paper 4.
13. "Google Patents".
14. A. Blass, Y. Gurevich, M. Moskal, and I. Neeman. Evidential authorization. In S. Nanz (ed), The Future of Software Engineering, Springer 2011, 77–99.
15. N. Bjørner, A. Blass, and Y. Gurevich. Content-dependent chunking for differential compression: The local maximum approach. Journal of Computer Systems Science 76(3-4), 2010, 154-203.
16. Y. Gurevich, E. Hudis, and J.M. Wing. Inverse privacy. Communications of the ACM 59(7), 2016, 38-42.
17. https://eatcs.org/index.php/eatcs-bulletin
18. A. Bocharov, Y. Gurevich, and K.M. Svore. Efficient decomposition of single-qubit gates into V basis circuits. Physical Review A 88:1, 2013.
19. AAAS Fellows, retrieved on Jan 11, 2021.
20. ACM Fellows, Association for Computing Machinery. Accessed February 16, 2010.
21. Fellows List, Archived June 22, 2011, at the Wayback Machine John Simon Guggenheim Memorial Foundation. Accessed February 16, 2010.
22. "EATCS names 2014 fellows", Milestones: Computer Science Awards, Appointments, Communications of the ACM, 58 (1): 24, January 2015, doi:10.1145/2686734, S2CID 11485095
External links
• Gurevich's Homepage
• Yuri Gurevich, Mathematics Genealogy Project
Wikimedia Commons has media related to Yuri Gurevich.
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Yutaka Taniyama
Yutaka Taniyama (谷山 豊, Taniyama Yutaka[1], 12 November 1927 – 17 November 1958) was a Japanese mathematician known for the Taniyama–Shimura conjecture.
Yutaka Taniyama
谷山 豊
Born(1927-11-12)12 November 1927
Kisai near Tokyo, Japan
Died17 November 1958(1958-11-17) (aged 31)
Tokyo, Japan
Alma materUniversity of Tokyo
Known forContributions in Algebraic Number Theory, Taniyama–Shimura conjecture
Scientific career
FieldsMathematics
InstitutionsUniversity of Tokyo
Contribution
Taniyama was best known for conjecturing, in modern language, automorphic properties of L-functions of elliptic curves over any number field. A partial and refined case of this conjecture for elliptic curves over rationals is called the Taniyama–Shimura conjecture or the modularity theorem whose statement he subsequently refined in collaboration with Goro Shimura. The names Taniyama, Shimura and Weil have all been attached to this conjecture, but the idea is essentially due to Taniyama.
“Taniyama's interests were in algebraic number theory and his fame is mainly due to two problems posed by him at the symposium on Algebraic Number Theory held in Tokyo and Nikko in 1955. His meeting with André Weil at this symposium was to have a major influence on Taniyama's work. These problems form the basis of a conjecture: every elliptic curve defined over the rational field is a factor of the Jacobian of a modular function field. This conjecture proved to be a major component in the proof of Fermat's Last Theorem by Andrew Wiles.”[2]
In 1986 Ken Ribet proved that if the Taniyama–Shimura conjecture held, then so would Fermat's Last Theorem, which inspired Andrew Wiles to work for a number of years in secrecy on it, and to prove enough of it to prove Fermat's Last Theorem. Owing to the pioneering contribution of Wiles and the efforts of a number of mathematicians, the Taniyama–Shimura conjecture was finally proven in 1999. The original Taniyama conjecture for elliptic curves over arbitrary number fields remains open.
In an episode of Nova (American TV program) on the proof of Fermat's Last Theorem, reflecting on Taniyama's work, Goro Shimura stated:
Taniyama was not a very careful person as a mathematician. He made a lot of mistakes. But he made mistakes in a good direction and so eventually he got right answers. I tried to imitate him, but I found out that it is very difficult to make good mistakes.[3] [4]
Depression and death
In 1958, Taniyama worked for University of Tokyo as an assistant (joshu), was engaged, and was offered a position at the Institute for Advanced Study in Princeton, New Jersey. On 17 November 1958, Taniyama committed suicide. He left a note explaining how far he had progressed with his teaching duties, and apologizing to his colleagues for the trouble he was causing them. His suicide note read:
Until yesterday I had no definite intention of killing myself. But more than a few must have noticed that lately I have been tired both physically and mentally. As to the cause of my suicide, I don't quite understand it myself, but it is not the result of a particular incident, nor of a specific matter. Merely may I say, I am in the frame of mind that I lost confidence in my future. There may be someone to whom my suicide will be troubling or a blow to a certain degree. I sincerely hope that this incident will cast no dark shadow over the future of that person. At any rate, I cannot deny that this is a kind of betrayal, but please excuse it as my last act in my own way, as I have been doing my own way all my life.
Although his note is mostly enigmatic it does mention tiredness and a loss of confidence in his future. Taniyama's ideas had been criticized as unsubstantiated and his behavior had occasionally been deemed peculiar. Goro Shimura mentioned that he suffered from depression. Taniyama also mentioned in the note his concern that some might be harmed by his suicide and his hope that the act would not cast "a dark shadow over that person."
About a month later, Misako Suzuki, the woman whom he was planning to marry, also committed suicide by carbon monoxide poisoning, leaving a note reading: "We promised each other that no matter where we went, we would never be separated. Now that he is gone, I must go too in order to join him."
After Taniyama's death, Goro Shimura stated that:
He was always kind to his colleagues, especially to his juniors, and he genuinely cared about their welfare. He was the moral support of many of those who came into mathematical contact with him, including of course myself. Probably he was never conscious of this role he was playing. But I feel his noble generosity in this respect even more strongly now than when he was alive. And yet nobody was able to give him any support when he desperately needed it. Reflecting on this, I am overwhelmed by the bitterest grief.
See also
• Taniyama group
Notes
1. Taniyama's given name 豊 was intended to be read as Toyo, but was frequently misread as the more common form Yutaka, which he eventually adopted as his own name.
2. Yutaka Taniyama biography, University of St Andrews, Scotland: https://www-history.mcs.st-and.ac.uk/Biographies/Taniyama.html
3. "The Proof". Nova. Season 25. Episode 4. 28 October 1997. 14:21 minutes in. PBS. Transcript of episode.
4. "Fermat's Last Theorem". Horizon. 1995. 12:08 minutes in. BBC.
Publications
• Shimura, Goro; Taniyama, Yutaka (1961), Complex multiplication of abelian varieties and its applications to number theory, Publications of the Mathematical Society of Japan, vol. 6, Tokyo: The Mathematical Society of Japan, MR 0125113 This book is hard to find, but an expanded version was later published as Shimura, Goro (1997). Abelian Varieties with Complex Multiplication and Modular Functions (Hardcover ed.). Princeton University Press. ISBN 978-0-691-01656-6.
References
• Shimura, Goro (1989), "Yutaka Taniyama and his time. Very personal recollections", The Bulletin of the London Mathematical Society, 21 (2): 186–196, doi:10.1112/blms/21.2.186, ISSN 0024-6093, MR 0976064
• Singh, Simon (hardcover, 1998). Fermat's Enigma. Bantam Books. ISBN 0-8027-1331-9 (previously published under the title Fermat's Last Theorem).
• Weil, André, "Y. Taniyama", Sugaku-no Ayumi, 6 (4): 21–22, Reprinted in Weil's collected works, volume II
External links
• O'Connor, John J.; Robertson, Edmund F., "Yutaka Taniyama", MacTutor History of Mathematics Archive, University of St Andrews
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Yutaka Yamamoto (mathematician)
Yutaka Yamamoto (山本 裕, Yamamoto Yutaka, born 29 March 1950), is a Japanese mathematician working in systems theory, control theory, and signal processing.
References
• YYfest 2010 Symposium on Systems, Control, and Signal Processing In honor of Yutaka Yamamoto on the occasion of his 60th birthday Kyoto University. 29–31 March 2010
• Jan C. Willems; Shinji Hara; Yoshito Ohta; Hisaya Fujioka, eds. (2010). Perspectives in Mathematical System Theory, Control, and Signal Processing: A Festschrift in Honor of Yutaka Yamamoto on the Occasion of his 60th Birthday. Springer Science & Business Media. ISBN 978-3-540-93917-7.
External links
• Personal website at Kyoto University
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Yuval Flicker
Yuval Zvi Flicker (Hebrew: יוּבַל צְבִי פְלִיקֶר; born 1955 in Israel) is an American mathematician. His primary research interests include automorphic representations.[1]
Yuval Flicker
Born (1955-01-03) 3 January 1955
Kfar Saba, Israel
NationalityIsrael, United States
Alma materUniversity of Cambridge
Hebrew University of Jerusalem
Tel Aviv University
AwardsAlexander von Humboldt Fellow, Fulbright Award, Lady Davis Fellow, Simons Foundation Fellow, NUS Senior Fellow
Scientific career
FieldsMathematics
InstitutionsOhio State University
Ariel University
Doctoral advisorAlan Baker
He received his PhD degree from the University of Cambridge in 1978. His thesis advisor was Alan Baker, in the area of transcendental number theory.[1][2]
He taught at Princeton University, Columbia University, Harvard University and Ohio State University, where he now has the title of Faculty Emeritus.[3] He also worked with David Kazhdan[4] and Pierre Deligne.[1][5]
Education
Born 1955 in Kfar-Saba, raised in Ramat-Gan, Flicker studied Mathematics and Philosophy at Tel-Aviv University gaining a BA in 1973, then he studied Mathematics at the Hebrew University gaining an MA in 1974. After that he studied Part III of the Mathematical Tripos at DPMMS, Cambridge University in 1974-75, where he was awarded his PhD under the supervision of Fields Medalist Alan Baker in 1978. His dissertation was "Linear forms on Abelian Varieties over Local Fields". He was a Post Doctoral scholar at the Institute for Advanced Study Princeton 1978-79, at Columbia University 1979-81, at Princeton University 1981-85, and at Harvard University 1985-87. He worked as a member of the Mathematics Department at the Ohio State University from 1987 to 2015.
Research
Flicker's research interests include Automorphic and Admissible Representations, Automorphic forms over function fields, Arithmetic Geometry, Lifting of Representations, Hecke-Iwahori algebras, p-adic automorphic forms, Galois Cohomology, Local-Global Principles, Motives, Algebraic Groups, Covering Groups, Shimura Varieties. He coauthored works with David Kazhdan,[4] Pierre Deligne,[5] his students[6] and other scholars.[7] He acknowledges influence of Joseph Bernstein[8] and of Vladimir Drinfeld.[9] He is the author of several books.
Dissemination
Flicker visited and lectured at the Universities of Mannheim, Bielefeld, Münster, Essen, Köln, HU Berlin supported by a Humboldt Stiftung, DAAD and SFB; at MPIM in Bonn; at University of Tokyo; at TIFR Bombay (and later TIFR Mumbai); at University of Santiago, Chile; at University of Buenos Aires supported by a Fulbright award; at the Chinese Academy of Sciences; at National University of Singapore supported by an NUS Senior Fellowship; at the Hebrew University of Jerusalem supported by a Lady Davis Fellowship and Schonbrunn Professorship, and Simons Fellowship; at IMPA Rio de Janeiro; at Erzincan University supported by TÜBİTAK.
Flicker endorsed An Open Letter to Richard Riley, United States Secretary of Education.
Books
Yuval Flicker is the author of a number of books including:
• Arthur's Invariant Trace Formula and Comparison of Inner Forms (2016)[10]
• Drinfeld Moduli Schemes and Automorphic Forms (2013)[11]
• Automorphic Representations of Low Rank Groups (2006)[12]
• Automorphic Forms and Shimura Varieties of PGSp(2) (2005)[13]
• Matching of Orbital Integrals on GL(4) and GSp(2) (1999)[14]
External links
• Home Page at Ohio State
• Math Department Ohio State
• Personal Home Page
References
1. "Yuval Flicker OSU CV" (PDF).
2. Yuval Zvi Flicker at the Mathematics Genealogy Project.
3. "Yuval Flicker". Ohio State University. Retrieved 22 October 2021.
4. "Metaplectic correspondence". Publications Mathématiques de l'IHÉS.
5. "Counting local systems with principal unipotent local monodromy". Annals of Mathematics.
6. "Twister Character of a Small Representations of PGL(4)" (PDF). Moscow Mathematical Journal.
7. "Grothendieck's Theorem on Non-Abelian H2 and Local-Global Principles" (PDF). Journal of the American Mathematical Society.
8. "K-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras, Part 2". Proceedings of Symposia in Pure Mathematics.
9. "Eisenstein Series and the Trace Formula for GL(2) over a Function Field" (PDF). Documenta Mathematica.
10. Birkhäuser Basel, ISBN 978-3-319-31593-5.
11. Springer-Verlag New York, ISBN 978-1-4614-5888-3.
12. World Scientific, ISBN 978-981-256-803-8.
13. World Scientific, ISBN 978-981-256-403-0.
14. Memoirs of the American Mathematical Society 655, AMS, ISBN 978-0-8218-0959-4.
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Yves André
Yves André (born December 11, 1959) is a French mathematician, specializing in arithmetic geometry.
Yves André
André at Oberwolfach, 2007
Born (1959-12-11) December 11, 1959
NationalityFrench
Alma materPierre and Marie Curie University
AwardsPrix Paul Doistau–Émile Blutet (2011)
Member of the Academia Europaea (2015)
Scientific career
Doctoral advisorDaniel Bertrand
Biography
André received his doctorate in 1984 from Pierre and Marie Curie University (Paris VI) with thesis advisor Daniel Bertrand and thesis Structure de Hodge, équations différentielles p-adiques, et indépendance algébrique de périodes d'intégrales abéliennes.[1] He became at CNRS in 1985 a Researcher, in 2000 a Research Director 2nd Class, and in 2009 a Research Director 1st Class (at École Normale Supérieure and Institut de mathématiques de Jussieu – Paris Rive Gauche).[2]
Research
In 1989, he formulated the one-dimensional-subvariety case of what is now known as the André-Oort conjecture on special subvarieties of Shimura varieties.[3] Only partial results have been proven so far; by André himself and by Jonathan Pila in 2009. In 2016, André used Scholze's method of perfectoid spaces to prove Melvin Hochster's direct summand conjecture that any finite extension of a regular commutative ring splits as a module.[4][5]
Awards
In 2011, André received the Prix Paul Doistau–Émile Blutet of the Académie des Sciences. In 2015, he was elected as a Member of the Academia Europaea. He was an invited speaker at the 2018 International Congress of Mathematicians in Rio de Janeiro and gave a talk titled Perfectoid spaces and the homological conjectures.[6]
Selected publications
• André, Yves (1989). G-Functions and Geometry A Publication of the Max-Planck-Institut für Mathematik, Bonn. Wiesbaden. ISBN 978-3-663-14108-2. OCLC 860266118.{{cite book}}: CS1 maint: location missing publisher (link)
• André, Yves (18 October 2022). "Mumford-Tate groups of mixed Hodge structures and the theorem of the fixed part". Compositio Mathematica (in French). 82 (1): 1–24. ISSN 1570-5846. Retrieved 22 November 2022.
• Andre, Yves (1996). "On the Shafarevich and Tate conjectures for hyperkähler varieties". Mathematische Annalen. Springer Science and Business Media LLC. 305 (1): 205–248. doi:10.1007/bf01444219. ISSN 0025-5831. S2CID 122949797.
• André, Yves; Baldassarri, F. (2001). De Rham cohomology of differential modules on algebraic varieties. Basel, Switzerland: Birkhäuser. ISBN 978-3-0348-8336-8. OCLC 679321692.
• Period mappings and differential equations. From C to Cp: Tohoku-Hokkaido Lectures in Arithmetic Geometry, Tokyo, Memoirs Mathematical Society of Japan 2003 (with appendix by F. Kato, N. Tsuzuki)
• "Une introduction aux motifs (Motifs purs, motifs mixtes, périodes)". Société Mathématique de France (in French). Retrieved 22 November 2022.
• André, Yves (2009). "Galois theory, motives and transcendental numbers". Renormalization and Galois Theories. IRMA Lectures in Mathematics and Theoretical Physics. Vol. 15. Zuerich, Switzerland: European Mathematical Society Publishing House. pp. 165–177. doi:10.4171/073-1/4. ISBN 978-3-03719-073-9. S2CID 16880343.
• André, Yves (7 December 2017). "La conjecture du facteur direct". Publications mathématiques de l'IHÉS (in French). Springer Science and Business Media LLC. 127 (1): 71–93. arXiv:1609.00345. doi:10.1007/s10240-017-0097-9. ISSN 0073-8301. S2CID 254170253.
References
1. Yves André at the Mathematics Genealogy Project
2. "Yves André". Academia Europaea.
3. "G-functions and geometry", Vieweg 1989
4. André, Yves (2016). "La conjecture du facteur direct". arXiv:1609.00345 [math.AG].
5. Bhatt, Bhargav (2016). "On the direct summand conjecture and its derived variant". arXiv:1608.08882 [math.AG]..
6. André, Yves (2018). "Perfectoid spaces and the homological conjectures". arXiv:1801.10006 [math.AC].
External links
• "Yves André - Grothendieck et les équations différentielles". YouTube. 4 April 2016.
• "Yves André - Direct summand conjecture and perfectoid Abhyankar lemma: an overview". YouTube. 11 November 2016.
• "Yves André: What is... a motivic Galois group". YouTube. 18 January 2018.
• "Yves André: Periods of relative 1 motives". YouTube. 18 January 2018.
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Yves Colin de Verdière
Yves Colin de Verdière is a French mathematician.
Yves Colin de Verdière
NationalityFrench
Alma materParis Diderot University
Known forColin de Verdière graph invariant
AwardsPrize Ampère
Fellow of the United States National Academy of Sciences
Émile Picard Medal
Scientific career
FieldsMathematics
InstitutionsJoseph Fourier University
Doctoral advisorMarcel Berger
Life
He studied at the École Normale Supérieure in Paris in the late 1960s, obtained his Ph.D. in 1973, and then spent the bulk of his working life as faculty at Joseph Fourier University in Grenoble. He retired in December 2005.
Work
Colin de Verdière is known for work in spectral theory, in particular on the semiclassical limit of quantum mechanics (including quantum chaos); in graph theory where he introduced a new graph invariant, the Colin de Verdière graph invariant; and on a variety of other subjects within Riemannian geometry and number theory.
Honors and awards
His contributions have been recognized by several awards: senior member of the Institut Universitaire de France from 1991 to 2001; Prize Ampère of the French Academy of Sciences in 1999; Fellow of the American Academy of Arts and Sciences in 2004; Émile Picard Medal of the French Academy of Sciences in 2018. He was an invited speaker at the International Congress of Mathematicians, held in Berkeley, California in 1986.
External links
• Yves Colin de Verdière at the Mathematics Genealogy Project
• "Colin de Verdière's home page".
• "Conference in honour of his retirement". Archived from the original on 2006-05-06.
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Yves Meyer
Yves F. Meyer (French: [mɛjɛʁ]; born 19 July 1939) is a French mathematician. He is among the progenitors of wavelet theory, having proposed the Meyer wavelet. Meyer was awarded the Abel Prize in 2017.
Yves Meyer
Yves Meyer giving a lecture in 2016.
Born (1939-07-19) 19 July 1939
Paris, France
NationalityFrench
EducationÉcole Normale Supérieure
University of Strasbourg
Known forWavelet theory
AwardsSalem Prize
Carl Friedrich Gauss Prize
Abel Prize
Princess of Asturias Award
Scientific career
FieldsMathematics
ThesisIdéaux Fermés de L1 dans Lesquels une Suite Approche l'Identité (1966)
Doctoral advisorJean-Pierre Kahane
Doctoral students
• Pascal Auscher
• Aline Bonami
• Albert Cohen
Biography
Born in Paris in a Jewish family, Yves Meyer studied at the Lycée Carnot in Tunis;[1] he won the French General Student Competition (Concours Général) in Mathematics, and was placed first in the entrance examination for the École Normale Supérieure in 1957.[2] He obtained his Ph.D. in 1966, under the supervision of Jean-Pierre Kahane.[3][4] The Mexican historian Jean Meyer is his cousin.
Yves Meyer taught at the Prytanée national militaire during his military service (1960–1963), then was a teaching assistant at the Université de Strasbourg (1963–1966), a professor at Université Paris-Sud (1966–1980), a professor at École Polytechnique (1980–1986), a professor at Université Paris-Dauphine (1985–1995), a senior researcher at the Centre national de la recherche scientifique (CNRS) (1995–1999), an invited professor at the Conservatoire National des Arts et Métiers (2000), a professor at École Normale Supérieure de Cachan (1999–2003), and has been a professor emeritus at Ecole Normale Supérieure de Cachan since 2004.
He was awarded the 2010 Gauss Prize for fundamental contributions to number theory, operator theory and harmonic analysis, and his pivotal role in the development of wavelets and multiresolution analysis.[3] He also received the 2017 Abel Prize "for his pivotal role in the development of the mathematical theory of wavelets."[5]
Publications
• Meyer, Yves (1970). Nombres de Pisot, nombres de Salem, et analyse harmonique (in French). Berlin New York: Springer-Verlag. ISBN 978-3-540-36243-2. OCLC 295014081.
• Algebraic numbers and harmonic analysis. Burlington: Elsevier Science. 1972. ISBN 978-0-08-095412-7. OCLC 761646828.
• Meyer, Yves (1990). Ondelettes et opérateurs (in French). Paris: Hermann. ISBN 978-2-7056-6125-0. OCLC 945745937.
• Meyer, Yves (22 April 1993). Wavelets and Operators. D. H. Salinger. Cambridge University Press. doi:10.1017/cbo9780511623820. ISBN 978-0-521-42000-6.[6]
Awards and recognitions
• He is a member of the Académie des Sciences since 1993.[7]
• Meyer was an Invited Speaker at the ICM in 1970 in Nice, in 1983 in Warsaw,[8] and in 1990 in Kyoto.[9]
• In 2010, Yves Meyer was awarded the Carl Friedrich Gauss Prize.[3]
• In 2012 he became a fellow of the American Mathematical Society.[10]
• In 2017 he was awarded the Abel Prize for his pivotal role in developing the mathematical theory of wavelets.[11]
• In 2020 he received the Princess of Asturias Award for Technical and Scientific Research.[12]
See also
• Wavelet
• Alex Grossmann
• Meyer wavelet
• Compressed sensing
• Harmonious set
• JPEG 2000
• Meyer set
• Ingrid Daubechies
• Jean Morlet
References
1. "Home". lyceecarnotdetunis.com.
2. Société de Mathématiques Appliquées et Industrielles : Yves Meyer.
3. "Carl Friedrich Gauss Prize – Yves Meyer". International Congress of Mathematicians 2010, Hyderabad, India. Archived from the original on 23 September 2010.
4. Yves F. Meyer at the Mathematics Genealogy Project
5. "2017: Yves Meyer". www.abelprize.no. Retrieved 22 July 2022.{{cite web}}: CS1 maint: url-status (link)
6. Chui, Charles K. (1996). "Review: Wavelets and operators, by Yves Meyer; A friendly guide to wavelets, by Gerald Kaiser". Bull. Amer. Math. Soc. (N.S.). 33 (1): 131–134. doi:10.1090/s0273-0979-96-00635-0.
7. Académie des Sciences : Yves Meyer. Archived 9 August 2011 at the Wayback Machine
8. Meyer, Yves. "Intégrales singulières, opérateurs multilinéaires, analyse complexe et équations aux dérivées partielles." Proc. Intern. Cong. Math (1983): 1001–1010.
9. Meyer, Yves F. "Wavelets and applications." Proc. Intern. Cong. Math (1990): 1619–1626.
10. List of Fellows of the American Mathematical Society, retrieved 4 February 2013.
11. "Abel Prize 2017: Yves Meyer wins 'maths Nobel' for work on wavelets". The Guardian. 21 March 2017.
12. "Yves Meyer, Ingrid Daubechies, Terence Tao and Emmanuel Candès, Princess of Asturias Award for Technical and Scientific Research 2020". Princess of Asturias Foundation. Retrieved 23 June 2020.
External links
• Société Mathématiques de France : Lecture by Yves Meyer (2009)
• Yves Meyer at the Mathematics Genealogy Project
• Gauss prize 2010
Abel Prize laureates
• 2003 Jean-Pierre Serre
• 2004 Michael Atiyah
• Isadore Singer
• 2005 Peter Lax
• 2006 Lennart Carleson
• 2007 S. R. Srinivasa Varadhan
• 2008 John G. Thompson
• Jacques Tits
• 2009 Mikhail Gromov
• 2010 John Tate
• 2011 John Milnor
• 2012 Endre Szemerédi
• 2013 Pierre Deligne
• 2014 Yakov Sinai
• 2015 John Forbes Nash Jr.
• Louis Nirenberg
• 2016 Andrew Wiles
• 2017 Yves Meyer
• 2018 Robert Langlands
• 2019 Karen Uhlenbeck
• 2020 Hillel Furstenberg
• Grigory Margulis
• 2021 László Lovász
• Avi Wigderson
• 2022 Dennis Sullivan
• 2023 Luis Caffarelli
Laureates of the Prince or Princess of Asturias Award for Technical and Scientific Research
Prince of Asturias Award for Technical and Scientific Research
1980s
• 1981: Alberto Sols
• 1982: Manuel Ballester
• 1983: Luis Antonio Santaló Sors
• 1984: Antonio Garcia-Bellido
• 1985: David Vázquez Martínez and Emilio Rosenblueth
• 1986: Antonio González González
• 1987: Jacinto Convit and Pablo Rudomín
• 1988: Manuel Cardona and Marcos Moshinsky
• 1989: Guido Münch
1990s
• 1990: Santiago Grisolía and Salvador Moncada
• 1991: Francisco Bolívar Zapata
• 1992: Federico García Moliner
• 1993: Amable Liñán
• 1994: Manuel Patarroyo
• 1995: Manuel Losada Villasante and Instituto Nacional de Biodiversidad of Costa Rica
• 1996: Valentín Fuster
• 1997: Atapuerca research team
• 1998: Emilio Méndez Pérez and Pedro Miguel Echenique Landiríbar
• 1999: Ricardo Miledi and Enrique Moreno González
2000s
• 2000: Robert Gallo and Luc Montagnier
• 2001: Craig Venter, John Sulston, Francis Collins, Hamilton Smith and Jean Weissenbach
• 2002: Lawrence Roberts, Robert E. Kahn, Vinton Cerf and Tim Berners-Lee
• 2003: Jane Goodall
• 2004: Judah Folkman, Tony Hunter, Joan Massagué, Bert Vogelstein and Robert Weinberg
• 2005: Antonio Damasio
• 2006: Juan Ignacio Cirac
• 2007: Peter Lawrence and Ginés Morata
• 2008: Sumio Iijima, Shuji Nakamura, Robert Langer, George M. Whitesides and Tobin Marks
• 2009: Martin Cooper and Raymond Tomlinson
2010s
• 2010: David Julius, Baruch Minke and Linda Watkins
• 2011: Joseph Altman, Arturo Álvarez-Buylla and Giacomo Rizzolatti
• 2012: Gregory Winter and Richard A. Lerner
• 2013: Peter Higgs, François Englert and European Organization for Nuclear Research CERN
• 2014: Avelino Corma Canós, Mark E. Davis and Galen D. Stucky
Princess of Asturias Award for Technical and Scientific Research
2010s
• 2015: Emmanuelle Charpentier and Jennifer Doudna
• 2016: Hugh Herr
• 2017: Rainer Weiss, Kip S. Thorne, Barry C. Barish and the LIGO Scientific Collaboration
• 2018: Svante Pääbo
• 2019: Joanne Chory and Sandra Myrna Díaz
2020s
• 2020: Yves Meyer, Ingrid Daubechies, Terence Tao and Emmanuel Candès
• 2021: Katalin Karikó, Drew Weissman, Philip Felgner, Uğur Şahin, Özlem Türeci, Derrick Rossi and Sarah Gilbert
• 2022: Geoffrey Hinton, Yann LeCun, Yoshua Bengio and Demis Hassabis
• 2023: Jeffrey I. Gordon, Everett Peter Greenberg and Bonnie Bassler
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Yvette Amice
Yvette Amice (June 4, 1936 – July 4, 1993) was a French mathematician whose research concerned number theory and p-adic analysis.[1] She was president of the Société mathématique de France.[1]
Education
Amice studied mathematics at the École normale supérieure de jeunes filles in Sèvres, beginnining in 1956 and earning her agrégation in 1959.[1] She became an assistant at the Faculté des sciences de Paris until 1964, when she completed a state doctorate under the supervision of Charles Pisot. Her dissertation was Interpolation p-adique [p-adic interpolation].[1][2]
Career
On completing her doctorate, she became maître de conférences at the University of Poitiers and then, in 1966, professor at the University of Bordeaux. She returned to Poitiers in 1968 but then in 1970 became one of the founding professors of Paris Diderot University, where she was vice president from 1978 to 1981.
In 1975 she became president of the Société mathématique de France.[1]
Textbook
Amice was the author of a textbook on the p-adic number system, Les nombres p-adiques (Presses Universitaires de France, 1975).[3]
References
1. Barsky, Daniel; Kahane, Jean-Pierre (1994), "Yvette Amice (1936–1993)" (PDF), Gazette des Mathématiciens (61): 83–87, MR 1289341.
2. Yvette Amice at the Mathematics Genealogy Project
3. Review of Les nombres p-adiques by W. Bartenwerfer, MR0447195 (in German).
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Yvette Kosmann-Schwarzbach
Yvette Kosmann-Schwarzbach (born 30 April 1941)[1] is a French mathematician and professor.
Yvette Kosmann-Schwarzbach
Born (1941-04-30) 30 April 1941
NationalityFrench
Alma materUniversity of Paris
Known forKosmann lift
Scientific career
FieldsMathematics
InstitutionsÉcole polytechnique
University of Lille
ThesisDérivées de Lie des spineurs (1970)
Doctoral advisorAndré Lichnerowicz
Websitehttps://www.cmls.polytechnique.fr/perso/kosmann/
Education and career
Kosmann-Schwarzbach obtained her doctoral degree in 1970 at the University of Paris under supervision of André Lichnerowicz on a dissertation titled Dérivées de Lie des spineurs (Lie derivatives of spinors).[2]
She worked at Lille University of Science and Technology, and since 1993 at the École polytechnique.
Research
Kosmann-Schwarzbach is the author of over fifty articles on differential geometry, algebra and mathematical physics, of two books on Lie groups and on the Noether theorem, as well as the co-editor of several books concerning the theory of integrable systems. The Kosmann lift in differential geometry is named after her.[3][4]
Works
• Groups and Symmetries: From Finite Groups to Lie Groups. Translated by Stephanie Frank Singer. Springer 2010, ISBN 978-0387788654.[5]
• The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century. Translated by Bertram Schwarzbach. Springer 2011, ISBN 978-0387878676.[6]
References
1. Birth date from Library of Congress and French National Library, retrieved 2019-10-13
2. Yvette Kosmann-Schwarzbach at the Mathematics Genealogy Project
3. Fatibene, L.; Ferraris, M.; Francaviglia, M.; Godina, M. (28 August – 1 September 1995). Janyska, J.; Kolář, I.; Slovák, J. (eds.). "A geometric definition of Lie derivative for Spinor Fields". Proceedings of the 6th International Conference on Differential Geometry and Applications. Brno, Czech Republic: Masaryk University: 549–558.
4. Godina M. and Matteucci P. (2003), Reductive G-structures and Lie derivatives, Journal of Geometry and Physics, 47, pp. 66–86
5. Reviews of Groups and Symmetries: Aloysius Helminck (2011), MR2553682; Thomas R. Hagedorn (2010), MAA Reviews; Ilka Agricola, Zbl 1132.20001; Eugene Kryachko, Zbl 1201.20001.
6. Reviews of The Noether Theorems: Jeremy Gray (2008), Historia Mathematica, doi:10.1016/j.hm.2007.06.002; Narciso Román-Roy (2012), MR2761345; Michael Berg (2011), MAA Reviews; Teodora-Liliana Rădulescu, Zbl 1128.01024; Reinhard Siegmund-Schultze, Zbl 1216.01011.
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Yvonne Dold-Samplonius
Yvonne Dold-Samplonius (20 May 1937 – 16 June 2014) was a Dutch mathematician and historian who specialized in the history of Islamic mathematics during the Middle age. She was particularly interested in the mathematical methods used by Islamic architects and builders of the Middle Ages for measurements of volumes and measurements of religious buildings or in the design of muqarnas.
Biography
Born on 20 May 1937 in Haarlem, Yvonne Samplonius obtained her degree in mathematics and Arabic from the University of Amsterdam (Doktoratsexamen) in 1966.[1] Yvonne Dold-Samplonius married in 1965 the German mathematician Albrecht Dold. She studied from 1966 to 1967 at Harvard University under the direction of Professor John E. Murdoch.[1] She obtained in 1977 a PhD for her analysis of the treatise Kitāb al-mafrādāt li Aqāţun (Book of Assumptions of Aqātun) under the supervision of Prof. Evert Marie Bruins and Prof. Juan Vernet.[2][3]
She came into contact with the work of the Persian mathematician, physicist and astronomer Abū Sahl al-Qūhī, who worked in Baghdad in the 10th century and worked on the geometrical forms of buildings.[4] Through his work, she became interested in the geometrical calculations that helped building many domes of palaces and mosques, called muqarnas, in the Arab world and Persia.[5][6][7] She wrote articles on the Islamic mathematicians Jamshīd al-Kāshī and Abu-Abdullah Muhammad ibn Īsa Māhānī in the Dictionary of the Middle Ages and in the Dictionary of Scientific Biography.[8][9]
In her last years her interest shifted to mathematics in Islamic architecture from an historic point of view.[7][10] Since 1995, she has been an associate member of the Interdisciplinary Center for Scientific Computing (IWR) of the University of Heidelberg, with whom she has published several videos on Islamic geometrical art.[11][12] In 1985, she is visiting professor at the University of Siena. In 2000, she organized with Joseph Dauben the conference "2000 Years of Transmission of Mathematical Ideas".[13] In 2002, she became a Corresponding Member of the International Academy of the History of Sciences and was elected effective member in 2007. She was made honorary citizen of Kashan in Iran in 2000.[11]
Publications
• Yvonne Dold-Samplonius, Dissertation: Book of Assumptions by Aqatun (Kitab al-Mafrudat li-Aqatun), Amsterdam 1977.
• Yvonne Dold-Samplonius : Practical Arabic Mathematics: Measuring the Muqarnas by al-Kashi, Centaurus 35, 193–242, (1992/3).
• Yvonne Dold-Samplonius : How al-Kashi Measures the Muqarnas: A Second Look, M. Folkerts (Ed.), Mathematische Probleme im Mittelalter: Der lateinische und arabische Sprachbereich, Wolfenbütteler Mittelalter-Studien Vol. 10, 56 – 90, Wiesbaden, (1996).
• Yvonne Dold-Samplonius : Calculation of Arches and Domes in 15th Century Samarkand, Nexus Network Journal, Vol. 2(3), (2000).
• Yvonne Dold-Samplonius : Calculating Surface Areas and Volumes in Islamic Architecture, The Enterprise of Science in Islam, New Perspectives, Eds. Jan P. Hogendijk et Abdelhamid I. Sabra, MIT Press, Cambridge Mass. pp. 235–265, (2003).
• Yvonne Dold-Samplonius, Silvia L. Harmsen : The Muqarnas Plate Found at Takht-i Sulaiman, A New Interpretation, Muqarnas Vol. 22, Leiden, pp. 85–94, (2005).
Videos
• Yvonne Dold-Samplonius, Christoph Kindl, Norbert Quien : Qubba for al-Kashi, Video, Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, American Mathematical Society, (1996). Qubba for al-Kashi on YouTube
• Yvonne Dold-Samplonius, Silvia L. Harmsen, Susanne Krömker, Michael Winckler : Magic of Muqarnas, Video, Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, (2005).
References
1. de Waard, Peter (2014). "Yvonne Dold-Samplonius 1937–2014" (PDF). Obituary University of Heidelberg (in Dutch).
2. Dold-Samplonius, Yvonne (1977). Book of Assumptions by Aquatun: Text-critical Edition.
3. Dold-Samplonius, Yvonne (1978). "Some remarks on the Book of Assumptions by Aqatun" (PDF). Journal for the History of Arabic Science. 2 (2): 255–263.
4. Dold-Samplonius, Yvonne (2008) [1970–80]. "Al-Qūhī (or Al-Kūhī), Abū Sahl Wayjan Ibn Rustam". Complete Dictionary of Scientific Biography. Encyclopedia.com.
5. Dold-Samplonius, Yvonne (1992-10-01). "Practical Arabic Mathematics: Measuring the Muqarnas by al-K¯ash¯i" (PDF). Centaurus. 35 (3): 193–242. Bibcode:1992Cent...35..193D. doi:10.1111/j.1600-0498.1992.tb00699.x. ISSN 1600-0498.
6. Dold-Samplonius, Yvonne; Hermelink, Heinrich (1970). "Al-Jayyānī, Abū'Abd Allāh Muḥammad Ibn Mu'ādh". Complete Dictionary of Scientific Biography. Encyclopedia.com.
7. Dold-Samplonius, Yvonne; Harmsen, Silvia L. (2005). "The Muqarnas Plate Found at Takht-I Sulayman: A New Interpretation". Muqarnas. 22: 85–94. doi:10.1163/22118993_02201005. JSTOR 25482424.
8. Dold-Samplonius, Yvonne. "Al-Kāshī | Muslim astronomer and mathematician". Encyclopedia Britannica. Retrieved 2018-02-02.
9. Dold-Samplonius, Yvonne (2008) [1970-80]. "Al-Māhānī, Abū 'Abd Allāh Muḥammad Ibn 'Īsā". Complete Dictionary of Scientific Biography. Encyclopedia.com.
10. Dold-Samplonius, Yvonne (2003). "Calculating Surface Areas and Volumes in Islamic Architecture". In Hogendijk, J. P.; Sabra, A. I. (eds.). The Enterprise of Science in Islam: New Perspectives. MIT Press. pp. 235–265. ISBN 978-0-262-19482-2.
11. IWR - History of Islamic Mathematics (2014). "Curriculum Vitae – Dr. Yvonne Dold-Samplonius". www.iwr.uni-heidelberg.de. Retrieved 2018-02-04.
12. Yvonne, Dold-Samplonius; Silvia, Harmsen; Susanne, Krömker; Michael J., Winckler (2005). "Magic of Muqarnas" (in German). doi:10.11588/heidok.00017446. {{cite journal}}: Cite journal requires |journal= (help)
13. Dold-Samplonius, Yvonne; Dauben, Joseph W., eds. (2002). From China to Paris: 2000 Years Transmission of Mathematical Ideas. Franz Steiner Verlag. ISBN 978-3-515-08223-5.
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Z* theorem
In mathematics, George Glauberman's Z* theorem is stated as follows:
Z* theorem: Let G be a finite group, with O(G) being its maximal normal subgroup of odd order. If T is a Sylow 2-subgroup of G containing an involution not conjugate in G to any other element of T, then the involution lies in Z*(G), which is the inverse image in G of the center of G/O(G).
This generalizes the Brauer–Suzuki theorem (and the proof uses the Brauer–Suzuki theorem to deal with some small cases).
Details
The original paper Glauberman (1966) gave several criteria for an element to lie outside Z*(G). Its theorem 4 states:
For an element t in T, it is necessary and sufficient for t to lie outside Z*(G) that there is some g in G and abelian subgroup U of T satisfying the following properties:
1. g normalizes both U and the centralizer CT(U), that is g is contained in N = NG(U) ∩ NG(CT(U))
2. t is contained in U and tg ≠ gt
3. U is generated by the N-conjugates of t
4. the exponent of U is equal to the order of t
Moreover g may be chosen to have prime power order if t is in the center of T, and g may be chosen in T otherwise.
A simple corollary is that an element t in T is not in Z*(G) if and only if there is some s ≠ t such that s and t commute and s and t are G-conjugate.
A generalization to odd primes was recorded in Guralnick & Robinson (1993): if t is an element of prime order p and the commutator [t, g] has order coprime to p for all g, then t is central modulo the p′-core. This was also generalized to odd primes and to compact Lie groups in Mislin & Thévenaz (1991), which also contains several useful results in the finite case.
Henke & Semeraro (2015) have also studied an extension of the Z* theorem to pairs of groups (G, H) with H a normal subgroup of G.
Works cited
• Dade, Everett C. (1971), "Character theory pertaining to finite simple groups", in Powell, M. B.; Higman, Graham (eds.), Finite simple groups. Proceedings of an Instructional Conference organized by the London Mathematical Society (a NATO Advanced Study Institute), Oxford, September 1969, Boston, MA: Academic Press, pp. 249–327, ISBN 978-0-12-563850-0, MR 0360785 gives a detailed proof of the Brauer–Suzuki theorem.
• Glauberman, George (1966), "Central elements in core-free groups", Journal of Algebra, 4 (3): 403–420, doi:10.1016/0021-8693(66)90030-5, ISSN 0021-8693, MR 0202822, Zbl 0145.02802
• Guralnick, Robert M.; Robinson, Geoffrey R. (1993), "On extensions of the Baer-Suzuki theorem", Israel Journal of Mathematics, 82 (1): 281–297, doi:10.1007/BF02808114, ISSN 0021-2172, MR 1239051, Zbl 0794.20029
• Henke, Ellen; Semeraro, Jason (1 October 2015). "Centralizers of normal subgroups and the Z*-theorem". Journal of Algebra. 439: 511–514. arXiv:1411.1932. doi:10.1016/j.jalgebra.2015.06.027.
• Mislin, Guido; Thévenaz, Jacques (1991), "The Z*-theorem for compact Lie groups", Mathematische Annalen, 291 (1): 103–111, doi:10.1007/BF01445193, ISSN 0025-5831, MR 1125010
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Fisher's z-distribution
Fisher's z-distribution is the statistical distribution of half the logarithm of an F-distribution variate:
$z={\frac {1}{2}}\log F$
"z-distribution" redirects here. For the distribution related to z-scores, see Normal distribution § Standard normal distribution.
Fisher's z
Probability density function
Parameters $d_{1}>0,\ d_{2}>0$ deg. of freedom
Support $x\in (-\infty ;+\infty )\!$ ;+\infty )\!}
PDF ${\frac {2d_{1}^{d_{1}/2}d_{2}^{d_{2}/2}}{B(d_{1}/2,d_{2}/2)}}{\frac {e^{d_{1}x}}{\left(d_{1}e^{2x}+d_{2}\right)^{\left(d_{1}+d_{2}\right)/2}}}\!$
Mode $0$
It was first described by Ronald Fisher in a paper delivered at the International Mathematical Congress of 1924 in Toronto.[1] Nowadays one usually uses the F-distribution instead.
The probability density function and cumulative distribution function can be found by using the F-distribution at the value of $x'=e^{2x}\,$. However, the mean and variance do not follow the same transformation.
The probability density function is[2][3]
$f(x;d_{1},d_{2})={\frac {2d_{1}^{d_{1}/2}d_{2}^{d_{2}/2}}{B(d_{1}/2,d_{2}/2)}}{\frac {e^{d_{1}x}}{\left(d_{1}e^{2x}+d_{2}\right)^{(d_{1}+d_{2})/2}}},$
where B is the beta function.
When the degrees of freedom becomes large ($d_{1},d_{2}\rightarrow \infty $), the distribution approaches normality with mean[2]
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 \bar{x} = \frac 1 2 \left( \frac 1 {d_2} - \frac 1 {d_1} \right)}
and variance
$\sigma _{x}^{2}={\frac {1}{2}}\left({\frac {1}{d_{1}}}+{\frac {1}{d_{2}}}\right).$
Related distribution
• If $X\sim \operatorname {FisherZ} (n,m)$ then $e^{2X}\sim \operatorname {F} (n,m)\,$ (F-distribution)
• If $X\sim \operatorname {F} (n,m)$ then ${\tfrac {\log X}{2}}\sim \operatorname {FisherZ} (n,m)$
References
1. Fisher, R. A. (1924). "On a Distribution Yielding the Error Functions of Several Well Known Statistics" (PDF). Proceedings of the International Congress of Mathematics, Toronto. 2: 805–813. Archived from the original (PDF) on April 12, 2011.
2. Leo A. Aroian (December 1941). "A study of R. A. Fisher's z distribution and the related F distribution". The Annals of Mathematical Statistics. 12 (4): 429–448. doi:10.1214/aoms/1177731681. JSTOR 2235955.
3. Charles Ernest Weatherburn (1961). A first course in mathematical statistics.
External links
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Z-transform
In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation.[1][2]
It can be considered as a discrete-time equivalent of the Laplace transform (s-domain).[3] This similarity is explored in the theory of time-scale calculus.
Whereas the continuous-time Fourier transform is evaluated on the Laplace s-domain's imaginary line, the discrete-time Fourier transform is evaluated over the unit circle of the z-domain. What is roughly the s-domain's left half-plane, is now the inside of the complex unit circle; what is the z-domain's outside of the unit circle, roughly corresponds to the right half-plane of the s-domain.
One of the means of designing digital filters is to take analog designs, subject them to a bilinear transform which maps them from the s-domain to the z-domain, and then produce the digital filter by inspection, manipulation, or numerical approximation. Such methods tend not to be accurate except in the vicinity of the complex unity, i.e. at low frequencies.
History
The basic idea now known as the Z-transform was known to Laplace, and it was re-introduced in 1947 by W. Hurewicz[4][5] and others as a way to treat sampled-data control systems used with radar. It gives a tractable way to solve linear, constant-coefficient difference equations. It was later dubbed "the z-transform" by Ragazzini and Zadeh in the sampled-data control group at Columbia University in 1952.[6][7]
The modified or advanced Z-transform was later developed and popularized by E. I. Jury.[8][9]
The idea contained within the Z-transform is also known in mathematical literature as the method of generating functions which can be traced back as early as 1730 when it was introduced by de Moivre in conjunction with probability theory.[10] From a mathematical view the Z-transform can also be viewed as a Laurent series where one views the sequence of numbers under consideration as the (Laurent) expansion of an analytic function.
Definition
The Z-transform can be defined as either a one-sided or two-sided transform. (Just like we have the one-sided Laplace transform and the two-sided Laplace transform.) [11]
Bilateral Z-transform
The bilateral or two-sided Z-transform of a discrete-time signal $x[n]$ is the formal power series $X(z)$ defined as
$X(z)={\mathcal {Z}}\{x[n]\}=\sum _{n=-\infty }^{\infty }x[n]z^{-n}$
(Eq.1)
where $n$ is an integer and $z$ is, in general, a complex number:
$z=Ae^{j\phi }=A\cdot (\cos {\phi }+j\sin {\phi })$
where $A$ is the magnitude of $z$, $j$ is the imaginary unit, and $\phi $ is the complex argument (also referred to as angle or phase) in radians.
Unilateral Z-transform
Alternatively, in cases where $x[n]$ is defined only for $n\geq 0$, the single-sided or unilateral Z-transform is defined as
$X(z)={\mathcal {Z}}\{x[n]\}=\sum _{n=0}^{\infty }x[n]z^{-n}.$
(Eq.2)
In signal processing, this definition can be used to evaluate the Z-transform of the unit impulse response of a discrete-time causal system.
An important example of the unilateral Z-transform is the probability-generating function, where the component $x[n]$ is the probability that a discrete random variable takes the value $n$, and the function $X(z)$ is usually written as $X(s)$ in terms of $s=z^{-1}$. The properties of Z-transforms (below) have useful interpretations in the context of probability theory.
Inverse Z-transform
The inverse Z-transform is
$x[n]={\mathcal {Z}}^{-1}\{X(z)\}={\frac {1}{2\pi j}}\oint _{C}X(z)z^{n-1}dz$
(Eq.3)
where C is a counterclockwise closed path encircling the origin and entirely in the region of convergence (ROC). In the case where the ROC is causal (see Example 2), this means the path C must encircle all of the poles of $X(z)$.
A special case of this contour integral occurs when C is the unit circle. This contour can be used when the ROC includes the unit circle, which is always guaranteed when $X(z)$ is stable, that is, when all the poles are inside the unit circle. With this contour, the inverse Z-transform simplifies to the inverse discrete-time Fourier transform, or Fourier series, of the periodic values of the Z-transform around the unit circle:
$x[n]={\frac {1}{2\pi }}\int _{-\pi }^{+\pi }X(e^{j\omega })e^{j\omega n}d\omega .$
(Eq.4)
The Z-transform with a finite range of n and a finite number of uniformly spaced z values can be computed efficiently via Bluestein's FFT algorithm. The discrete-time Fourier transform (DTFT)—not to be confused with the discrete Fourier transform (DFT)—is a special case of such a Z-transform obtained by restricting z to lie on the unit circle.
Region of convergence
The region of convergence (ROC) is the set of points in the complex plane for which the Z-transform summation converges.
$\mathrm {ROC} =\left\{z:\left|\sum _{n=-\infty }^{\infty }x[n]z^{-n}\right|<\infty \right\}$
Example 1 (no ROC)
Let $x[n]=0.5^{n}\ $. Expanding x[n] on the interval (−∞, ∞) it becomes
$x[n]=\left\{\dots ,0.5^{-3},0.5^{-2},0.5^{-1},1,0.5,0.5^{2},0.5^{3},\dots \right\}=\left\{\dots ,2^{3},2^{2},2,1,0.5,0.5^{2},0.5^{3},\dots \right\}.$
Looking at the sum
$\sum _{n=-\infty }^{\infty }x[n]z^{-n}\to \infty .$
Therefore, there are no values of z that satisfy this condition.
Example 2 (causal ROC)
Let $x[n]=0.5^{n}u[n]\ $ (where u is the Heaviside step function). Expanding x[n] on the interval (−∞, ∞) it becomes
$x[n]=\left\{\dots ,0,0,0,1,0.5,0.5^{2},0.5^{3},\dots \right\}.$
Looking at the sum
$\sum _{n=-\infty }^{\infty }x[n]z^{-n}=\sum _{n=0}^{\infty }0.5^{n}z^{-n}=\sum _{n=0}^{\infty }\left({\frac {0.5}{z}}\right)^{n}={\frac {1}{1-0.5z^{-1}}}.$
The last equality arises from the infinite geometric series and the equality only holds if |0.5z−1| < 1, which can be rewritten in terms of z as |z| > 0.5. Thus, the ROC is |z| > 0.5. In this case the ROC is the complex plane with a disc of radius 0.5 at the origin "punched out".
Example 3 (anti causal ROC)
Let $x[n]=-(0.5)^{n}u[-n-1]\ $ (where u is the Heaviside step function). Expanding x[n] on the interval (−∞, ∞) it becomes
$x[n]=\left\{\dots ,-(0.5)^{-3},-(0.5)^{-2},-(0.5)^{-1},0,0,0,0,\dots \right\}.$
Looking at the sum
$\sum _{n=-\infty }^{\infty }x[n]z^{-n}=-\sum _{n=-\infty }^{-1}0.5^{n}z^{-n}=-\sum _{m=1}^{\infty }\left({\frac {z}{0.5}}\right)^{m}=-{\frac {0.5^{-1}z}{1-0.5^{-1}z}}=-{\frac {1}{0.5z^{-1}-1}}={\frac {1}{1-0.5z^{-1}}}.$
Using the infinite geometric series, again, the equality only holds if |0.5−1z| < 1 which can be rewritten in terms of z as |z| < 0.5. Thus, the ROC is |z| < 0.5. In this case the ROC is a disc centered at the origin and of radius 0.5.
What differentiates this example from the previous example is only the ROC. This is intentional to demonstrate that the transform result alone is insufficient.
Examples conclusion
Examples 2 & 3 clearly show that the Z-transform X(z) of x[n] is unique when and only when specifying the ROC. Creating the pole–zero plot for the causal and anticausal case show that the ROC for either case does not include the pole that is at 0.5. This extends to cases with multiple poles: the ROC will never contain poles.
In example 2, the causal system yields an ROC that includes |z| = ∞ while the anticausal system in example 3 yields an ROC that includes |z| = 0.
In systems with multiple poles it is possible to have a ROC that includes neither |z| = ∞ nor |z| = 0. The ROC creates a circular band. For example,
$x[n]=0.5^{n}u[n]-0.75^{n}u[-n-1]$
has poles at 0.5 and 0.75. The ROC will be 0.5 < |z| < 0.75, which includes neither the origin nor infinity. Such a system is called a mixed-causality system as it contains a causal term (0.5)nu[n] and an anticausal term −(0.75)nu[−n−1].
The stability of a system can also be determined by knowing the ROC alone. If the ROC contains the unit circle (i.e., |z| = 1) then the system is stable. In the above systems the causal system (Example 2) is stable because |z| > 0.5 contains the unit circle.
Let us assume we are provided a Z-transform of a system without a ROC (i.e., an ambiguous x[n]). We can determine a unique x[n] provided we desire the following:
• Stability
• Causality
For stability the ROC must contain the unit circle. If we need a causal system then the ROC must contain infinity and the system function will be a right-sided sequence. If we need an anticausal system then the ROC must contain the origin and the system function will be a left-sided sequence. If we need both stability and causality, all the poles of the system function must be inside the unit circle.
The unique x[n] can then be found.
Properties
Properties of the z-transform
Property
Time domain Z-domain Proof ROC
Definition of Z-transform $x[n]$ $X(z)$ $X(z)={\mathcal {Z}}\{x[n]\}$ (definition of the z-transform)
$x[n]={\mathcal {Z}}^{-1}\{X(z)\}$ (definition of the inverse z-transform)
$r_{2}<|z|<r_{1}$
Linearity $a_{1}x_{1}[n]+a_{2}x_{2}[n]$ $a_{1}X_{1}(z)+a_{2}X_{2}(z)$ ${\begin{aligned}X(z)&=\sum _{n=-\infty }^{\infty }(a_{1}x_{1}(n)+a_{2}x_{2}(n))z^{-n}\\&=a_{1}\sum _{n=-\infty }^{\infty }x_{1}(n)z^{-n}+a_{2}\sum _{n=-\infty }^{\infty }x_{2}(n)z^{-n}\\&=a_{1}X_{1}(z)+a_{2}X_{2}(z)\end{aligned}}$ Contains ROC1 ∩ ROC2
Time expansion $x_{K}[n]={\begin{cases}x[r],&n=Kr\\0,&n\notin K\mathbb {Z} \end{cases}}$
with $K\mathbb {Z} :=\{Kr:r\in \mathbb {Z} \}$ :=\{Kr:r\in \mathbb {Z} \}}
$X(z^{K})$ ${\begin{aligned}X_{K}(z)&=\sum _{n=-\infty }^{\infty }x_{K}(n)z^{-n}\\&=\sum _{r=-\infty }^{\infty }x(r)z^{-rK}\\&=\sum _{r=-\infty }^{\infty }x(r)(z^{K})^{-r}\\&=X(z^{K})\end{aligned}}$ $R^{\frac {1}{K}}$
Decimation $x[Kn]$ ${\frac {1}{K}}\sum _{p=0}^{K-1}X\left(z^{\tfrac {1}{K}}\cdot e^{-i{\tfrac {2\pi }{K}}p}\right)$ ohio-state.edu or ee.ic.ac.uk
Time delay $x[n-k]$
with $k>0$ and $x:x[n]=0\ \forall n<0$
$z^{-k}X(z)$ ${\begin{aligned}Z\{x[n-k]\}&=\sum _{n=0}^{\infty }x[n-k]z^{-n}\\&=\sum _{j=-k}^{\infty }x[j]z^{-(j+k)}&&j=n-k\\&=\sum _{j=-k}^{\infty }x[j]z^{-j}z^{-k}\\&=z^{-k}\sum _{j=-k}^{\infty }x[j]z^{-j}\\&=z^{-k}\sum _{j=0}^{\infty }x[j]z^{-j}&&x[\beta ]=0,\beta <0\\&=z^{-k}X(z)\end{aligned}}$ ROC, except z = 0 if k > 0 and z = ∞ if k < 0
Time advance $x[n+k]$
with $k>0$
Bilateral Z-transform:
$z^{k}X(z)$
Unilateral Z-transform:[12]
$z^{k}X(z)-z^{k}\sum _{n=0}^{k-1}x[n]z^{-n}$
First difference backward $x[n]-x[n-1]$
with $x[n]=0$ for $n<0$
$(1-z^{-1})X(z)$ Contains the intersection of ROC of X1(z) and z ≠ 0
First difference forward $x[n+1]-x[n]$ $(z-1)X(z)-zx[0]$
Time reversal $x[-n]$ $X(z^{-1})$ ${\begin{aligned}{\mathcal {Z}}\{x(-n)\}&=\sum _{n=-\infty }^{\infty }x(-n)z^{-n}\\&=\sum _{m=-\infty }^{\infty }x(m)z^{m}\\&=\sum _{m=-\infty }^{\infty }x(m){(z^{-1})}^{-m}\\&=X(z^{-1})\\\end{aligned}}$ ${\tfrac {1}{r_{1}}}<|z|<{\tfrac {1}{r_{2}}}$
Scaling in the z-domain $a^{n}x[n]$ $X(a^{-1}z)$ ${\begin{aligned}{\mathcal {Z}}\left\{a^{n}x[n]\right\}&=\sum _{n=-\infty }^{\infty }a^{n}x(n)z^{-n}\\&=\sum _{n=-\infty }^{\infty }x(n)(a^{-1}z)^{-n}\\&=X(a^{-1}z)\end{aligned}}$ $|a|r_{2}<|z|<|a|r_{1}$
Complex conjugation $x^{*}[n]$ $X^{*}(z^{*})$ ${\begin{aligned}{\mathcal {Z}}\{x^{*}(n)\}&=\sum _{n=-\infty }^{\infty }x^{*}(n)z^{-n}\\&=\sum _{n=-\infty }^{\infty }\left[x(n)(z^{*})^{-n}\right]^{*}\\&=\left[\sum _{n=-\infty }^{\infty }x(n)(z^{*})^{-n}\right]^{*}\\&=X^{*}(z^{*})\end{aligned}}$
Real part $\operatorname {Re} \{x[n]\}$ ${\tfrac {1}{2}}\left[X(z)+X^{*}(z^{*})\right]$
Imaginary part $\operatorname {Im} \{x[n]\}$ ${\tfrac {1}{2j}}\left[X(z)-X^{*}(z^{*})\right]$
Differentiation in the z-domain $nx[n]$ $-z{\frac {dX(z)}{dz}}$ ${\begin{aligned}{\mathcal {Z}}\{nx(n)\}&=\sum _{n=-\infty }^{\infty }nx(n)z^{-n}\\&=z\sum _{n=-\infty }^{\infty }nx(n)z^{-n-1}\\&=-z\sum _{n=-\infty }^{\infty }x(n)(-nz^{-n-1})\\&=-z\sum _{n=-\infty }^{\infty }x(n){\frac {d}{dz}}(z^{-n})\\&=-z{\frac {dX(z)}{dz}}\end{aligned}}$ ROC, if $X(z)$ is rational;
ROC possibly excluding the boundary, if $X(z)$ is irrational[13]
Convolution $x_{1}[n]*x_{2}[n]$ $X_{1}(z)X_{2}(z)$ ${\begin{aligned}{\mathcal {Z}}\{x_{1}(n)*x_{2}(n)\}&={\mathcal {Z}}\left\{\sum _{l=-\infty }^{\infty }x_{1}(l)x_{2}(n-l)\right\}\\&=\sum _{n=-\infty }^{\infty }\left[\sum _{l=-\infty }^{\infty }x_{1}(l)x_{2}(n-l)\right]z^{-n}\\&=\sum _{l=-\infty }^{\infty }x_{1}(l)\left[\sum _{n=-\infty }^{\infty }x_{2}(n-l)z^{-n}\right]\\&=\left[\sum _{l=-\infty }^{\infty }x_{1}(l)z^{-l}\right]\!\!\left[\sum _{n=-\infty }^{\infty }x_{2}(n)z^{-n}\right]\\&=X_{1}(z)X_{2}(z)\end{aligned}}$ Contains ROC1 ∩ ROC2
Cross-correlation $r_{x_{1},x_{2}}=x_{1}^{*}[-n]*x_{2}[n]$ $R_{x_{1},x_{2}}(z)=X_{1}^{*}({\tfrac {1}{z^{*}}})X_{2}(z)$ Contains the intersection of ROC of $X_{1}({\tfrac {1}{z^{*}}})$ and $X_{2}(z)$
Accumulation $\sum _{k=-\infty }^{n}x[k]$ ${\frac {1}{1-z^{-1}}}X(z)$ ${\begin{aligned}\sum _{n=-\infty }^{\infty }\sum _{k=-\infty }^{n}x[k]z^{-n}&=\sum _{n=-\infty }^{\infty }(x[n]+\cdots +x[-\infty ])z^{-n}\\&=X(z)\left(1+z^{-1}+z^{-2}+\cdots \right)\\&=X(z)\sum _{j=0}^{\infty }z^{-j}\\&=X(z){\frac {1}{1-z^{-1}}}\end{aligned}}$
Multiplication $x_{1}[n]x_{2}[n]$ ${\frac {1}{j2\pi }}\oint _{C}X_{1}(v)X_{2}({\tfrac {z}{v}})v^{-1}\mathrm {d} v$ -
Parseval's theorem
$\sum _{n=-\infty }^{\infty }x_{1}[n]x_{2}^{*}[n]\quad =\quad {\frac {1}{j2\pi }}\oint _{C}X_{1}(v)X_{2}^{*}({\tfrac {1}{v^{*}}})v^{-1}\mathrm {d} v$
Initial value theorem: If x[n] is causal, then
$x[0]=\lim _{z\to \infty }X(z).$
Final value theorem: If the poles of (z − 1)X(z) are inside the unit circle, then
$x[\infty ]=\lim _{z\to 1}(z-1)X(z).$
Table of common Z-transform pairs
Here:
$u:n\mapsto u[n]={\begin{cases}1,&n\geq 0\\0,&n<0\end{cases}}$
is the unit (or Heaviside) step function and
$\delta :n\mapsto \delta [n]={\begin{cases}1,&n=0\\0,&n\neq 0\end{cases}}$
is the discrete-time unit impulse function (cf Dirac delta function which is a continuous-time version). The two functions are chosen together so that the unit step function is the accumulation (running total) of the unit impulse function.
Signal, $x[n]$Z-transform, $X(z)$ROC
1$\delta [n]$1all z
2$\delta [n-n_{0}]$$z^{-n_{0}}$$z\neq 0$
3$u[n]\,$${\frac {1}{1-z^{-1}}}$$|z|>1$
4$-u[-n-1]$${\frac {1}{1-z^{-1}}}$$|z|<1$
5$nu[n]$${\frac {z^{-1}}{(1-z^{-1})^{2}}}$$|z|>1$
6$-nu[-n-1]\,$${\frac {z^{-1}}{(1-z^{-1})^{2}}}$$|z|<1$
7$n^{2}u[n]$${\frac {z^{-1}(1+z^{-1})}{(1-z^{-1})^{3}}}$$|z|>1\,$
8$-n^{2}u[-n-1]\,$${\frac {z^{-1}(1+z^{-1})}{(1-z^{-1})^{3}}}$$|z|<1\,$
9$n^{3}u[n]$${\frac {z^{-1}(1+4z^{-1}+z^{-2})}{(1-z^{-1})^{4}}}$$|z|>1\,$
10$-n^{3}u[-n-1]$${\frac {z^{-1}(1+4z^{-1}+z^{-2})}{(1-z^{-1})^{4}}}$$|z|<1\,$
11$a^{n}u[n]$${\frac {1}{1-az^{-1}}}$$|z|>|a|$
12$-a^{n}u[-n-1]$${\frac {1}{1-az^{-1}}}$$|z|<|a|$
13$na^{n}u[n]$${\frac {az^{-1}}{(1-az^{-1})^{2}}}$$|z|>|a|$
14$-na^{n}u[-n-1]$${\frac {az^{-1}}{(1-az^{-1})^{2}}}$$|z|<|a|$
15$n^{2}a^{n}u[n]$${\frac {az^{-1}(1+az^{-1})}{(1-az^{-1})^{3}}}$$|z|>|a|$
16$-n^{2}a^{n}u[-n-1]$${\frac {az^{-1}(1+az^{-1})}{(1-az^{-1})^{3}}}$$|z|<|a|$
17$\left({\begin{array}{c}n+m-1\\m-1\end{array}}\right)a^{n}u[n]$${\frac {1}{(1-az^{-1})^{m}}}$, for positive integer $m$[13]$|z|>|a|$
18$(-1)^{m}\left({\begin{array}{c}-n-1\\m-1\end{array}}\right)a^{n}u[-n-m]$${\frac {1}{(1-az^{-1})^{m}}}$, for positive integer $m$[13]$|z|<|a|$
19$\cos(\omega _{0}n)u[n]$${\frac {1-z^{-1}\cos(\omega _{0})}{1-2z^{-1}\cos(\omega _{0})+z^{-2}}}$$|z|>1$
20$\sin(\omega _{0}n)u[n]$${\frac {z^{-1}\sin(\omega _{0})}{1-2z^{-1}\cos(\omega _{0})+z^{-2}}}$$|z|>1$
21$a^{n}\cos(\omega _{0}n)u[n]$${\frac {1-az^{-1}\cos(\omega _{0})}{1-2az^{-1}\cos(\omega _{0})+a^{2}z^{-2}}}$$|z|>|a|$
22$a^{n}\sin(\omega _{0}n)u[n]$${\frac {az^{-1}\sin(\omega _{0})}{1-2az^{-1}\cos(\omega _{0})+a^{2}z^{-2}}}$$|z|>|a|$
Relationship to Fourier series and Fourier transform
Further information: Discrete-time Fourier transform § Relationship to the Z-transform
For values of $z$ in the region $|z|=1$, known as the unit circle, we can express the transform as a function of a single, real variable, ω, by defining $z=e^{j\omega }$. And the bi-lateral transform reduces to a Fourier series:
$\sum _{n=-\infty }^{\infty }x[n]\ z^{-n}=\sum _{n=-\infty }^{\infty }x[n]\ e^{-j\omega n},$
(Eq.4)
which is also known as the discrete-time Fourier transform (DTFT) of the $x[n]$ sequence. This 2π-periodic function is the periodic summation of a Fourier transform, which makes it a widely used analysis tool. To understand this, let $X(f)$ be the Fourier transform of any function, $x(t)$, whose samples at some interval, T, equal the x[n] sequence. Then the DTFT of the x[n] sequence can be written as follows.
$\underbrace {\sum _{n=-\infty }^{\infty }\overbrace {x(nT)} ^{x[n]}\ e^{-j2\pi fnT}} _{\text{DTFT}}={\frac {1}{T}}\sum _{k=-\infty }^{\infty }X(f-k/T).$
(Eq.5)
When T has units of seconds, $\scriptstyle f$ has units of hertz. Comparison of the two series reveals that $\omega =2\pi fT$ is a normalized frequency with unit of radian per sample. The value ω = 2π corresponds to $ f={\frac {1}{T}}$. And now, with the substitution $ f={\frac {\omega }{2\pi T}},$ Eq.4 can be expressed in terms of the Fourier transform, X(•):
$\sum _{n=-\infty }^{\infty }x[n]\ e^{-j\omega n}={\frac {1}{T}}\sum _{k=-\infty }^{\infty }\underbrace {X\left({\tfrac {\omega }{2\pi T}}-{\tfrac {k}{T}}\right)} _{X\left({\frac {\omega -2\pi k}{2\pi T}}\right)}.$
(Eq.6)
As parameter T changes, the individual terms of Eq.5 move farther apart or closer together along the f-axis. In Eq.6 however, the centers remain 2π apart, while their widths expand or contract. When sequence x(nT) represents the impulse response of an LTI system, these functions are also known as its frequency response. When the $x(nT)$ sequence is periodic, its DTFT is divergent at one or more harmonic frequencies, and zero at all other frequencies. This is often represented by the use of amplitude-variant Dirac delta functions at the harmonic frequencies. Due to periodicity, there are only a finite number of unique amplitudes, which are readily computed by the much simpler discrete Fourier transform (DFT). (See Discrete-time Fourier transform § Periodic data.)
Relationship to Laplace transform
Further information: Laplace transform § Z-transform
Bilinear transform
The bilinear transform can be used to convert continuous-time filters (represented in the Laplace domain) into discrete-time filters (represented in the Z-domain), and vice versa. The following substitution is used:
$s={\frac {2}{T}}{\frac {(z-1)}{(z+1)}}$
to convert some function $H(s)$ in the Laplace domain to a function $H(z)$ in the Z-domain (Tustin transformation), or
$z=e^{sT}\approx {\frac {1+sT/2}{1-sT/2}}$
from the Z-domain to the Laplace domain. Through the bilinear transformation, the complex s-plane (of the Laplace transform) is mapped to the complex z-plane (of the z-transform). While this mapping is (necessarily) nonlinear, it is useful in that it maps the entire $j\omega $ axis of the s-plane onto the unit circle in the z-plane. As such, the Fourier transform (which is the Laplace transform evaluated on the $j\omega $ axis) becomes the discrete-time Fourier transform. This assumes that the Fourier transform exists; i.e., that the $j\omega $ axis is in the region of convergence of the Laplace transform.
Starred transform
Given a one-sided Z-transform, X(z), of a time-sampled function, the corresponding starred transform produces a Laplace transform and restores the dependence on sampling parameter, T:
${\bigg .}X^{*}(s)=X(z){\bigg |}_z=e^{sT}}$
The inverse Laplace transform is a mathematical abstraction known as an impulse-sampled function.
Linear constant-coefficient difference equation
The linear constant-coefficient difference (LCCD) equation is a representation for a linear system based on the autoregressive moving-average equation.
$\sum _{p=0}^{N}y[n-p]\alpha _{p}=\sum _{q=0}^{M}x[n-q]\beta _{q}$
Both sides of the above equation can be divided by α0, if it is not zero, normalizing α0 = 1 and the LCCD equation can be written
$y[n]=\sum _{q=0}^{M}x[n-q]\beta _{q}-\sum _{p=1}^{N}y[n-p]\alpha _{p}.$
This form of the LCCD equation is favorable to make it more explicit that the "current" output y[n] is a function of past outputs y[n − p], current input x[n], and previous inputs x[n − q].
Transfer function
Taking the Z-transform of the above equation (using linearity and time-shifting laws) yields
$Y(z)\sum _{p=0}^{N}z^{-p}\alpha _{p}=X(z)\sum _{q=0}^{M}z^{-q}\beta _{q}$
and rearranging results in
$H(z)={\frac {Y(z)}{X(z)}}={\frac {\sum _{q=0}^{M}z^{-q}\beta _{q}}{\sum _{p=0}^{N}z^{-p}\alpha _{p}}}={\frac {\beta _{0}+z^{-1}\beta _{1}+z^{-2}\beta _{2}+\cdots +z^{-M}\beta _{M}}{\alpha _{0}+z^{-1}\alpha _{1}+z^{-2}\alpha _{2}+\cdots +z^{-N}\alpha _{N}}}.$
Zeros and poles
From the fundamental theorem of algebra the numerator has M roots (corresponding to zeros of H) and the denominator has N roots (corresponding to poles). Rewriting the transfer function in terms of zeros and poles
$H(z)={\frac {(1-q_{1}z^{-1})(1-q_{2}z^{-1})\cdots (1-q_{M}z^{-1})}{(1-p_{1}z^{-1})(1-p_{2}z^{-1})\cdots (1-p_{N}z^{-1})}},$
where qk is the kth zero and pk is the kth pole. The zeros and poles are commonly complex and when plotted on the complex plane (z-plane) it is called the pole–zero plot.
In addition, there may also exist zeros and poles at z = 0 and z = ∞. If we take these poles and zeros as well as multiple-order zeros and poles into consideration, the number of zeros and poles are always equal.
By factoring the denominator, partial fraction decomposition can be used, which can then be transformed back to the time domain. Doing so would result in the impulse response and the linear constant coefficient difference equation of the system.
Output response
If such a system H(z) is driven by a signal X(z) then the output is Y(z) = H(z)X(z). By performing partial fraction decomposition on Y(z) and then taking the inverse Z-transform the output y[n] can be found. In practice, it is often useful to fractionally decompose $\textstyle {\frac {Y(z)}{z}}$ before multiplying that quantity by z to generate a form of Y(z) which has terms with easily computable inverse Z-transforms.
See also
• Advanced Z-transform
• Bilinear transform
• Difference equation (recurrence relation)
• Discrete convolution
• Discrete-time Fourier transform
• Finite impulse response
• Formal power series
• Generating function
• Generating function transformation
• Laplace transform
• Laurent series
• Least-squares spectral analysis
• Probability-generating function
• Star transform
• Zak transform
• Zeta function regularization
References
1. Mandal, Jyotsna Kumar (2020). "Z-Transform-Based Reversible Encoding". Reversible Steganography and Authentication via Transform Encoding. Studies in Computational Intelligence. Vol. 901. Singapore: Springer Singapore. pp. 157–195. doi:10.1007/978-981-15-4397-5_7. ISBN 978-981-15-4396-8. ISSN 1860-949X. S2CID 226413693. Z is a complex variable. Z-transform converts the discrete spatial domain signal into complex frequency domain representation. Z-transform is derived from the Laplace transform.
2. Lynn, Paul A. (1986). "The Laplace Transform and the z-transform". Electronic Signals and Systems. London: Macmillan Education UK. pp. 225–272. doi:10.1007/978-1-349-18461-3_6. ISBN 978-0-333-39164-8. Laplace Transform and the z-transform are closely related to the Fourier Transform. z-transform is especially suitable for dealing with discrete signals and systems. It offers a more compact and convenient notation than the discrete-time Fourier Transform.
3. Palani, S. (2021-08-26). "The z-Transform Analysis of Discrete Time Signals and Systems". Signals and Systems. Cham: Springer International Publishing. pp. 921–1055. doi:10.1007/978-3-030-75742-7_9. ISBN 978-3-030-75741-0. S2CID 238692483. z-transform is the discrete counterpart of Laplace transform. z-transform converts difference equations of discrete time systems to algebraic equations which simplifies the discrete time system analysis. Laplace transform and z-transform are common except that Laplace transform deals with continuous time signals and systems.
4. E. R. Kanasewich (1981). Time Sequence Analysis in Geophysics. University of Alberta. pp. 186, 249. ISBN 978-0-88864-074-1.
5. E. R. Kanasewich (1981). Time sequence analysis in geophysics (3rd ed.). University of Alberta. pp. 185–186. ISBN 978-0-88864-074-1.
6. Ragazzini, J. R.; Zadeh, L. A. (1952). "The analysis of sampled-data systems". Transactions of the American Institute of Electrical Engineers, Part II: Applications and Industry. 71 (5): 225–234. doi:10.1109/TAI.1952.6371274. S2CID 51674188.
7. Cornelius T. Leondes (1996). Digital control systems implementation and computational techniques. Academic Press. p. 123. ISBN 978-0-12-012779-5.
8. Eliahu Ibrahim Jury (1958). Sampled-Data Control Systems. John Wiley & Sons.
9. Eliahu Ibrahim Jury (1973). Theory and Application of the Z-Transform Method. Krieger Pub Co. ISBN 0-88275-122-0.
10. Eliahu Ibrahim Jury (1964). Theory and Application of the Z-Transform Method. John Wiley & Sons. p. 1.
11. Jackson, Leland B. (1996). "The z Transform". Digital Filters and Signal Processing. Boston, MA: Springer US. pp. 29–54. doi:10.1007/978-1-4757-2458-5_3. ISBN 978-1-4419-5153-3. z transform is to discrete-time systems what the Laplace transform is to continuous-time systems. z is a complex variable. This is sometimes referred to as the two-sided z transform, with the one-sided z transform being the same except for a summation from n = 0 to infinity. The primary use of the one sided transform ... is for causal sequences, in which case the two transforms are the same anyway. We will not, therefore, make this distinction and will refer to ... as simply the z transform of x(n).
12. Bolzern, Paolo; Scattolini, Riccardo; Schiavoni, Nicola (2015). Fondamenti di Controlli Automatici (in Italian). MC Graw Hill Education. ISBN 978-88-386-6882-1.
13. A. R. Forouzan (2016). "Region of convergence of derivative of Z transform". Electronics Letters. 52 (8): 617–619. Bibcode:2016ElL....52..617F. doi:10.1049/el.2016.0189. S2CID 124802942.
Further reading
• Refaat El Attar, Lecture notes on Z-Transform, Lulu Press, Morrisville NC, 2005. ISBN 1-4116-1979-X.
• Ogata, Katsuhiko, Discrete Time Control Systems 2nd Ed, Prentice-Hall Inc, 1995, 1987. ISBN 0-13-034281-5.
• Alan V. Oppenheim and Ronald W. Schafer (1999). Discrete-Time Signal Processing, 2nd Edition, Prentice Hall Signal Processing Series. ISBN 0-13-754920-2.
External links
• "Z-transform", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Numerical inversion of the Z-transform
• Z-Transform table of some common Laplace transforms
• Mathworld's entry on the Z-transform
• Z-Transform threads in Comp.DSP
• A graphic of the relationship between Laplace transform s-plane to Z-plane of the Z transform
• A video-based explanation of the Z-Transform for engineers
• What is the z-Transform?
Digital signal processing
Theory
• Detection theory
• Discrete signal
• Estimation theory
• Nyquist–Shannon sampling theorem
Sub-fields
• Audio signal processing
• Digital image processing
• Speech processing
• Statistical signal processing
Techniques
• Z-transform
• Advanced z-transform
• Matched Z-transform method
• Bilinear transform
• Constant-Q transform
• Discrete cosine transform (DCT)
• Discrete Fourier transform (DFT)
• Discrete-time Fourier transform (DTFT)
• Impulse invariance
• Integral transform
• Laplace transform
• Post's inversion formula
• Starred transform
• Zak transform
Sampling
• Aliasing
• Anti-aliasing filter
• Downsampling
• Nyquist rate / frequency
• Oversampling
• Quantization
• Sampling rate
• Undersampling
• Upsampling
Authority control: National
• Israel
• United States
• Czech Republic
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Z-factor
The Z-factor is a measure of statistical effect size. It has been proposed for use in high-throughput screening (where it is also known as Z-prime[1]), and commonly written as Z' to judge whether the response in a particular assay is large enough to warrant further attention.
Background
In high-throughput screens, experimenters often compare a large number (hundreds of thousands to tens of millions) of single measurements of unknown samples to positive and negative control samples. The particular choice of experimental conditions and measurements is called an assay. Large screens are expensive in time and resources. Therefore, prior to starting a large screen, smaller test (or pilot) screens are used to assess the quality of an assay, in an attempt to predict if it would be useful in a high-throughput setting. The Z-factor is an attempt to quantify the suitability of a particular assay for use in a full-scale, high-throughput screen.
Definition
The Z-factor is defined in terms of four parameters: the means ($\mu $) and standard deviations ($\sigma $) of both the positive (p) and negative (n) controls ($\mu _{p}$, $\sigma _{p}$, and $\mu _{n}$, $\sigma _{n}$). Given these values, the Z-factor is defined as:
${\text{Z-factor}}=1-{3(\sigma _{p}+\sigma _{n}) \over |\mu _{p}-\mu _{n}|}$
In practice, the Z-factor is estimated from the sample means and sample standard deviations
${\text{Estimated Z-factor}}=1-{3({\hat {\sigma }}_{p}+{\hat {\sigma }}_{n}) \over |{\hat {\mu }}_{p}-{\hat {\mu }}_{n}|}$
Interpretation
The following interpretations for the Z-factor are taken from:[2]
Z-factorInterpretation
1.0Ideal. Z-factors can never exceed 1.
between 0.5 and 1.0An excellent assay. Note that if $\sigma _{p}=\sigma _{n}$, 0.5 is equivalent to a separation of 12 standard deviations between $\mu _{p}$ and $\mu _{n}$.
between 0 and 0.5A marginal assay.
less than 0There is too much overlap between the positive and negative controls for the assay to be useful.
Note that by the standards of many types of experiments, a zero Z-factor would suggest a large effect size, rather than a borderline useless result as suggested above. For example, if σp=σn=1, then μp=6 and μn=0 gives a zero Z-factor. But for normally-distributed data with these parameters, the probability that the positive control value would be less than the negative control value is less than 1 in 105. Extreme conservatism is used in high throughput screening due to the large number of tests performed.
Limitations
The constant factor 3 in the definition of the Z-factor is motivated by the normal distribution, for which more than 99% of values occur within 3 standard deviations of the mean. If the data follow a strongly non-normal distribution, the reference points (e.g. the meaning of a negative value) may be misleading. Another issue is that the usual estimates of the mean and standard deviation are not robust; accordingly many users in the high-throughput screening community prefer the "Robust Z-prime" which substitutes the median for the mean and the median absolute deviation for the standard deviation.[3] Extreme values (outliers) in either the positive or negative controls can adversely affect the Z-factor, potentially leading to an apparently unfavorable Z-factor even when the assay would perform well in actual screening .[4] In addition, the application of the single Z-factor-based criterion to two or more positive controls with different strengths in the same assay will lead to misleading results .[5] The absolute sign in the Z-factor makes it inconvenient to derive the statistical inference of Z-factor mathematically [6] . A recently proposed statistical parameter, strictly standardized mean difference (SSMD), can address these issues[5] [6] [7] . One estimate of SSMD is robust to outliers.
See also
• high-throughput screening
• SSMD
• Z-score or Standard score
References
1. "Orbitrap LC-MS - US". thermofisher.com.
2. Zhang, JH; Chung, TDY; Oldenburg, KR (1999). "A simple statistical parameter for use in evaluation and validation of high throughput screening assays". Journal of Biomolecular Screening. 4 (2): 67–73. doi:10.1177/108705719900400206. PMID 10838414. S2CID 36577200.
3. Birmingham, Amanda; et al. (August 2009). "Statistical Methods for Analysis of High-Throughput RNA Interference Screens". Nat Methods. 6 (8): 569–575. doi:10.1038/nmeth.1351. PMC 2789971. PMID 19644458.
4. Sui Y, Wu Z (2007). "Alternative Statistical Parameter for High-Throughput Screening Assay Quality Assessment". Journal of Biomolecular Screening. 12 (2): 229–34. doi:10.1177/1087057106296498. PMID 17218666.
5. Zhang XHD, Espeseth AS, Johnson E, Chin J, Gates A, Mitnaul L, Marine SD, Tian J, Stec EM, Kunapuli P, Holder DJ, Heyse JF, Stulovici B, Ferrer M (2008). "Integrating experimental and analytic approaches to improve data quality in genome-wide RNAi screens". Journal of Biomolecular Screening. 13 (5): 378–89. doi:10.1177/1087057108317145. PMID 18480473. S2CID 22679273.
6. Zhang, XHD (2007). "A pair of new statistical parameters for quality control in RNA interference high-throughput screening assays". Genomics. 89 (4): 552–61. doi:10.1016/j.ygeno.2006.12.014. PMID 17276655.
7. Zhang, XHD (2008). "Novel analytic criteria and effective plate designs for quality control in genome-wide RNAi screens". Journal of Biomolecular Screening. 13 (5): 363–77. doi:10.1177/1087057108317062. PMID 18567841. S2CID 12688742.
Further reading
• Kraybill, B. (2005) "Quantitative Assay Evaluation and Optimization" (unpublished note)
• Zhang XHD (2011) "Optimal High-Throughput Screening: Practical Experimental Design and Data Analysis for Genome-scale RNAi Research, Cambridge University Press"
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Z function
In mathematics, the Z function is a function used for studying the Riemann zeta function along the critical line where the argument is one-half. It is also called the Riemann–Siegel Z function, the Riemann–Siegel zeta function, the Hardy function, the Hardy Z function and the Hardy zeta function. It can be defined in terms of the Riemann–Siegel theta function and the Riemann zeta function by
$Z(t)=e^{i\theta (t)}\zeta \left({\frac {1}{2}}+it\right).$
It follows from the functional equation of the Riemann zeta function that the Z function is real for real values of t. It is an even function, and real analytic for real values. It follows from the fact that the Riemann-Siegel theta function and the Riemann zeta function are both holomorphic in the critical strip, where the imaginary part of t is between −1/2 and 1/2, that the Z function is holomorphic in the critical strip also. Moreover, the real zeros of Z(t) are precisely the zeros of the zeta function along the critical line, and complex zeros in the Z function critical strip correspond to zeros off the critical line of the Riemann zeta function in its critical strip.
The Riemann–Siegel formula
Calculation of the value of Z(t) for real t, and hence of the zeta function along the critical line, is greatly expedited by the Riemann–Siegel formula. This formula tells us
$Z(t)=2\sum _{n^{2}<t/2\pi }n^{-1/2}\cos(\theta (t)-t\log n)+R(t),$
where the error term R(t) has a complex asymptotic expression in terms of the function
$\Psi (z)={\frac {\cos 2\pi (z^{2}-z-1/16)}{\cos 2\pi z}}$
and its derivatives. If $u=\left({\frac {t}{2\pi }}\right)^{1/4}$, $N=\lfloor u^{2}\rfloor $ and $p=u^{2}-N$ then
$R(t)\sim (-1)^{N-1}\left(\Psi (p)u^{-1}-{\frac {1}{96\pi ^{2}}}\Psi ^{(3)}(p)u^{-3}+\cdots \right)$
where the ellipsis indicates we may continue on to higher and increasingly complex terms.
Other efficient series for Z(t) are known, in particular several using the incomplete gamma function. If
$Q(a,z)={\frac {\Gamma (a,z)}{\Gamma (a)}}={\frac {1}{\Gamma (a)}}\int _{z}^{\infty }u^{a-1}e^{-u}\,du$
then an especially nice example is
$Z(t)=2\Re \left(e^{i\theta (t)}\left(\sum _{n=1}^{\infty }Q\left({\frac {s}{2}},\pi in^{2}\right)-{\frac {\pi ^{s/2}e^{\pi is/4}}{s\Gamma \left({\frac {s}{2}}\right)}}\right)\right)$
Behavior of the Z function
From the critical line theorem, it follows that the density of the real zeros of the Z function is
${\frac {c}{2\pi }}\log {\frac {t}{2\pi }}$
for some constant c > 2/5. Hence, the number of zeros in an interval of a given size slowly increases. If the Riemann hypothesis is true, all of the zeros in the critical strip are real zeros, and the constant c is one. It is also postulated that all of these zeros are simple zeros.
An Omega theorem
Because of the zeros of the Z function, it exhibits oscillatory behavior. It also slowly grows both on average and in peak value. For instance, we have, even without the Riemann hypothesis, the Omega theorem that
$Z(t)=\Omega \left(\exp \left({\frac {3}{4}}{\sqrt {\frac {\log t}{\log \log t}}}\right)\right),$
where the notation means that $Z(t)$ divided by the function within the Ω does not tend to zero with increasing t.
Average growth
The average growth of the Z function has also been much studied. We can find the root mean square (abbreviated RMS) average from
${\frac {1}{T}}\int _{0}^{T}Z(t)^{2}dt\sim \log T$
or
${\frac {1}{T}}\int _{T}^{2T}Z(t)^{2}dt\sim \log T$
which tell us that the RMS size of Z(t) grows as ${\sqrt {\log t}}$.
This estimate can be improved to
${\frac {1}{T}}\int _{0}^{T}Z(t)^{2}dt=\log T+(2\gamma -2\log(2\pi )-1)+O(T^{-15/22})$
If we increase the exponent, we get an average value which depends more on the peak values of Z. For fourth powers, we have
${\frac {1}{T}}\int _{0}^{T}Z(t)^{4}dt\sim {\frac {1}{2\pi ^{2}}}(\log T)^{4}$
from which we may conclude that the fourth root of the mean fourth power grows as ${\frac {1}{2^{1/4}{\sqrt {\pi }}}}\log t.$
The Lindelöf hypothesis
Main article: Lindelöf hypothesis
Higher even powers have been much studied, but less is known about the corresponding average value. It is conjectured, and follows from the Riemann hypothesis, that
${\frac {1}{T}}\int _{0}^{T}Z(t)^{2k}\,dt=o(T^{\varepsilon })$
for every positive ε. Here the little "o" notation means that the left hand side divided by the right hand side does converge to zero; in other words little o is the negation of Ω. This conjecture is called the Lindelöf hypothesis, and is weaker than the Riemann hypothesis. It is normally stated in an important equivalent form, which is
$Z(t)=o(t^{\varepsilon });$
in either form it tells us the rate of growth of the peak values cannot be too high. The best known bound on this rate of growth is not strong, telling us that any $\epsilon >{\frac {89}{570}}\approx 0.156$ is suitable. It would be astonishing to find that the Z function grew anywhere close to as fast as this. Littlewood proved that on the Riemann hypothesis,
$Z(t)=o\left(\exp \left({\frac {10\log t}{\log \log t}}\right)\right),$
and this seems far more likely.
References
• Edwards, H.M. (1974). Riemann's zeta function. Pure and Applied Mathematics. Vol. 58. New York-London: Academic Press. ISBN 0-12-232750-0. Zbl 0315.10035.
• Ivić, Aleksandar (2013). The theory of Hardy's Z-function. Cambridge Tracts in Mathematics. Vol. 196. Cambridge: Cambridge University Press. ISBN 978-1-107-02883-8. Zbl 1269.11075.
• Paris, R. B.; Kaminski, D. (2001). Asymptotics and Mellin-Barnes Integrals. Encyclopedia of Mathematics and Its Applications. Vol. 85. Cambridge: Cambridge University Press. ISBN 0-521-79001-8. Zbl 0983.41019.
• Ramachandra, K. (February 1996). Lectures on the mean-value and Omega-theorems for the Riemann Zeta-function. Lectures on Mathematics and Physics. Mathematics. Tata Institute of Fundamental Research. Vol. 85. Berlin: Springer-Verlag. ISBN 3-540-58437-4. Zbl 0845.11003.
• Titchmarsh, E. C. (1986) [1951]. Heath-Brown, D.R. (ed.). The Theory of the Riemann Zeta-Function (second revised ed.). Oxford University Press.
External links
• Weisstein, Eric W. "Riemann–Siegel Functions". MathWorld.
• Wolfram Research – Riemann-Siegel function Z (includes function plotting and evaluation)
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Z-matrix (mathematics)
In mathematics, the class of Z-matrices are those matrices whose off-diagonal entries are less than or equal to zero; that is, the matrices of the form:
$Z=(z_{ij});\quad z_{ij}\leq 0,\quad i\neq j.$
Note that this definition coincides precisely with that of a negated Metzler matrix or quasipositive matrix, thus the term quasinegative matrix appears from time to time in the literature, though this is rare and usually only in contexts where references to quasipositive matrices are made.
The Jacobian of a competitive dynamical system is a Z-matrix by definition. Likewise, if the Jacobian of a cooperative dynamical system is J, then (−J) is a Z-matrix.
Related classes are L-matrices, M-matrices, P-matrices, Hurwitz matrices and Metzler matrices. L-matrices have the additional property that all diagonal entries are greater than zero. M-matrices have several equivalent definitions, one of which is as follows: a Z-matrix is an M-matrix if it is nonsingular and its inverse is nonnegative. All matrices that are both Z-matrices and P-matrices are nonsingular M-matrices.
In the context of quantum complexity theory, these are referred to as stoquastic operators.[1]
See also
• Hurwitz matrix
• M-matrix
• Metzler matrix
• P-matrix
References
1. Bravyi, Sergey; DiVincenzo, David P.; Oliveira, Roberto I.; Terhal, Barbara M. (2006). "The Complexity of Stoquastic Local Hamiltonian Problems". arXiv:quant-ph/0606140.
• Huan T.; Cheng G.; Cheng X. (1 April 2006). "Modified SOR-type iterative method for Z-matrices". Applied Mathematics and Computation. 175 (1): 258–268. doi:10.1016/j.amc.2005.07.050.
• Saad, Y. (1996). Iterative methods for sparse linear systems (2nd ed.). Philadelphia, PA.: Society for Industrial and Applied Mathematics. p. 28. ISBN 0-534-94776-X.
• Berman, Abraham; Plemmons, Robert J. (2014). Nonnegative Matrices in the Mathematical Sciences. Academic Press. ISBN 9781483260860.
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Mahler's 3/2 problem
In mathematics, Mahler's 3/2 problem concerns the existence of "Z-numbers".
A Z-number is a real number x such that the fractional parts of
$x\left({\frac {3}{2}}\right)^{n}$
are less than 1/2 for all positive integers n. Kurt Mahler conjectured in 1968 that there are no Z-numbers.
More generally, for a real number α, define Ω(α) as
$\Omega (\alpha )=\inf _{\theta >0}\left({\limsup _{n\rightarrow \infty }\left\lbrace {\theta \alpha ^{n}}\right\rbrace -\liminf _{n\rightarrow \infty }\left\lbrace {\theta \alpha ^{n}}\right\rbrace }\right).$
Mahler's conjecture would thus imply that Ω(3/2) exceeds 1/2. Flatto, Lagarias, and Pollington showed[1] that
$\Omega \left({\frac {p}{q}}\right)>{\frac {1}{p}}$
for rational p/q > 1 in lowest terms.
References
1. Flatto, Leopold; Lagarias, Jeffrey C.; Pollington, Andrew D. (1995). "On the range of fractional parts of ζ { (p/q)n }". Acta Arithmetica. LXX (2): 125–147. doi:10.4064/aa-70-2-125-147. ISSN 0065-1036. Zbl 0821.11038.
• Everest, Graham; van der Poorten, Alf; Shparlinski, Igor; Ward, Thomas (2003). Recurrence sequences. Mathematical Surveys and Monographs. Vol. 104. Providence, RI: American Mathematical Society. ISBN 0-8218-3387-1. Zbl 1033.11006.
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Integer points in convex polyhedra
The study of integer points in convex polyhedra[1] is motivated by questions such as "how many nonnegative integer-valued solutions does a system of linear equations with nonnegative coefficients have" or "how many solutions does an integer linear program have". Counting integer points in polyhedra or other questions about them arise in representation theory, commutative algebra, algebraic geometry, statistics, and computer science.[2]
The set of integer points, or, more generally, the set of points of an affine lattice, in a polyhedron is called Z-polyhedron,[3] from the mathematical notation $\mathbb {Z} $ or Z for the set of integer numbers.[4]
Properties
For a lattice Λ, Minkowski's theorem relates the number d(Λ) (the volume of a fundamental parallelepiped of the lattice) and the volume of a given symmetric convex set S to the number of lattice points contained in S.
The number of lattice points contained in a polytope all of whose vertices are elements of the lattice is described by the polytope's Ehrhart polynomial. Formulas for some of the coefficients of this polynomial involve d(Λ) as well.
Applications
Loop optimization
In certain approaches to loop optimization, the set of the executions of the loop body is viewed as the set of integer points in a polyhedron defined by loop constraints.
See also
• Convex lattice polytope
• Pick's theorem
References and notes
1. In some contexts convex polyhedra are called simply "polyhedra".
2. Integer points in polyhedra. Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics, ACM--SIAM Joint Summer Research Conference, 2006
3. The term "Z-polyhedron" is also used as a synonym to convex lattice polytope, the convex hull of finitely many points in an affine lattice.
4. "Computations on Iterated Spaces" in: The Compiler Design Handbook: Optimizations and Machine Code Generation, CRC Press 2007, 2nd edition, ISBN 1-4200-4382-X, p.15-7
Further reading
• Barvinok, Alexander; Beck, Matthias; Haase, Christian; Reznick, Bruce; Welker, Volkmar (2005), Integer Points In Polyhedra: Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference held in Snowbird, UT, July 13–17, 2003, Contemporary Mathematics, vol. 374, Providence, RI: American Mathematical Society, doi:10.1090/conm/374, ISBN 0-8218-3459-2, MR 2134757
• Barvinok, Alexander (2008), Integer Points In Polyhedra, Zurich Lectures in Advanced Mathematics, Zürich: European Mathematical Society, doi:10.4171/052, ISBN 978-3-03719-052-4, MR 2455889
• Beck, Matthias; Haase, Christian; Reznick, Bruce; Vergne, Michèle; Welker, Volkmar; Yoshida, Ruriko (2008), Integer Points In Polyhedra: Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics (PDF), Contemporary Mathematics, vol. 452, Providence, RI: American Mathematical Society, doi:10.1090/conm/452, ISBN 978-0-8218-4173-0, MR 2416261
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Z-test
A Z-test is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution. Z-tests test the mean of a distribution. For each significance level in the confidence interval, the Z-test has a single critical value (for example, 1.96 for 5% two tailed) which makes it more convenient than the Student's t-test whose critical values are defined by the sample size (through the corresponding degrees of freedom). Both the Z-test and Student's t-test have similarities in that they both help determine the significance of a set of data. However, the z-test is rarely used in practice because the population deviation is difficult to determine.
Applicability
Because of the central limit theorem, many test statistics are approximately normally distributed for large samples. Therefore, many statistical tests can be conveniently performed as approximate Z-tests if the sample size is large or the population variance is known. If the population variance is unknown (and therefore has to be estimated from the sample itself) and the sample size is not large (n < 30), the Student's t-test may be more appropriate (in some cases, n < 50, as described below).
Procedure
How to perform a Z test when T is a statistic that is approximately normally distributed under the null hypothesis is as follows:
First, estimate the expected value μ of T under the null hypothesis, and obtain an estimate s of the standard deviation of T.
Second, determine the properties of T : one tailed or two tailed.
For Null hypothesis H0: μ≥μ0 vs alternative hypothesis H1: μ<μ0 , it is lower/left-tailed (one tailed).
For Null hypothesis H0: μ≤μ0 vs alternative hypothesis H1: μ>μ0 , it is upper/right-tailed (one tailed).
For Null hypothesis H0: μ=μ0 vs alternative hypothesis H1: μ≠μ0 , it is two-tailed.
Third, calculate the standard score:
$Z={\frac {({\bar {X}}-\mu _{0})}{\sigma }},$
which one-tailed and two-tailed p-values can be calculated as Φ(Z)(for lower/left-tailed tests), Φ(−Z) (for upper/right-tailed tests) and 2Φ(−|Z|) (for two-tailed tests) where Φ is the standard normal cumulative distribution function.
Use in location testing
1. The term "Z-test" is often used to refer specifically to the one-sample location test comparing the mean of a set of measurements to a given constant when the sample variance is known. For example, if the observed data X1, ..., Xn are (i) independent, (ii) have a common mean μ, and (iii) have a common variance σ2, then the sample average X has mean μ and variance ${\frac {\sigma ^{2}}{n}}$.
2. The null hypothesis is that the mean value of X is a given number μ0. We can use X as a test-statistic, rejecting the null hypothesis if X − μ0 is large.
3. To calculate the standardized statistic $Z={\frac {({\bar {X}}-\mu _{0})}{s}}$, we need to either know or have an approximate value for σ2, from which we can calculate $s^{2}={\frac {\sigma ^{2}}{n}}$ . In some applications, σ2 is known, but this is uncommon.
4. If the sample size is moderate or large, we can substitute the sample variance for σ2, giving a plug-in test. The resulting test will not be an exact Z-test since the uncertainty in the sample variance is not accounted for—however, it will be a good approximation unless the sample size is small.
5. A t-test can be used to account for the uncertainty in the sample variance when the data are exactly normal.
6. Difference between Z-test and t-test: Z-test is used when sample size is large (n>50), or the population variance is known. t-test is used when sample size is small (n<50) and population variance is unknown.
7. There is no universal constant at which the sample size is generally considered large enough to justify use of the plug-in test. Typical rules of thumb: the sample size should be 50 observations or more.
8. For large sample sizes, the t-test procedure gives almost identical p-values as the Z-test procedure.
9. Other location tests that can be performed as Z-tests are the two-sample location test and the paired difference test.
Conditions
For the Z-test to be applicable, certain conditions must be met.
• Nuisance parameters should be known, or estimated with high accuracy (an example of a nuisance parameter would be the standard deviation in a one-sample location test). Z-tests focus on a single parameter, and treat all other unknown parameters as being fixed at their true values. In practice, due to Slutsky's theorem, "plugging in" consistent estimates of nuisance parameters can be justified. However if the sample size is not large enough for these estimates to be reasonably accurate, the Z-test may not perform well.
• The test statistic should follow a normal distribution. Generally, one appeals to the central limit theorem to justify assuming that a test statistic varies normally. There is a great deal of statistical research on the question of when a test statistic varies approximately normally. If the variation of the test statistic is strongly non-normal, a Z-test should not be used.
If estimates of nuisance parameters are plugged in as discussed above, it is important to use estimates appropriate for the way the data were sampled. In the special case of Z-tests for the one or two sample location problem, the usual sample standard deviation is only appropriate if the data were collected as an independent sample.
In some situations, it is possible to devise a test that properly accounts for the variation in plug-in estimates of nuisance parameters. In the case of one and two sample location problems, a t-test does this.
Example
Suppose that in a particular geographic region, the mean and standard deviation of scores on a reading test are 100 points, and 12 points, respectively. Our interest is in the scores of 55 students in a particular school who received a mean score of 96. We can ask whether this mean score is significantly lower than the regional mean—that is, are the students in this school comparable to a simple random sample of 55 students from the region as a whole, or are their scores surprisingly low?
First calculate the standard error of the mean:
$\mathrm {SE} ={\frac {\sigma }{\sqrt {n}}}={\frac {12}{\sqrt {55}}}={\frac {12}{7.42}}=1.62$
where ${\sigma }$ is the population standard deviation.
Next calculate the z-score, which is the distance from the sample mean to the population mean in units of the standard error:
$z={\frac {M-\mu }{\mathrm {SE} }}={\frac {96-100}{1.62}}=-2.47$
In this example, we treat the population mean and variance as known, which would be appropriate if all students in the region were tested. When population parameters are unknown, a Student's t-test should be conducted instead.
The classroom mean score is 96, which is −2.47 standard error units from the population mean of 100. Looking up the z-score in a table of the standard normal distribution cumulative probability, we find that the probability of observing a standard normal value below −2.47 is approximately 0.5 − 0.4932 = 0.0068. This is the one-sided p-value for the null hypothesis that the 55 students are comparable to a simple random sample from the population of all test-takers. The two-sided p-value is approximately 0.014 (twice the one-sided p-value).
Another way of stating things is that with probability 1 − 0.014 = 0.986, a simple random sample of 55 students would have a mean test score within 4 units of the population mean. We could also say that with 98.6% confidence we reject the null hypothesis that the 55 test takers are comparable to a simple random sample from the population of test-takers.
The Z-test tells us that the 55 students of interest have an unusually low mean test score compared to most simple random samples of similar size from the population of test-takers. A deficiency of this analysis is that it does not consider whether the effect size of 4 points is meaningful. If instead of a classroom, we considered a subregion containing 900 students whose mean score was 99, nearly the same z-score and p-value would be observed. This shows that if the sample size is large enough, very small differences from the null value can be highly statistically significant. See statistical hypothesis testing for further discussion of this issue.
Z-tests other than location tests
Location tests are the most familiar Z-tests. Another class of Z-tests arises in maximum likelihood estimation of the parameters in a parametric statistical model. Maximum likelihood estimates are approximately normal under certain conditions, and their asymptotic variance can be calculated in terms of the Fisher information. The maximum likelihood estimate divided by its standard error can be used as a test statistic for the null hypothesis that the population value of the parameter equals zero. More generally, if ${\hat {\theta }}$ is the maximum likelihood estimate of a parameter θ, and θ0 is the value of θ under the null hypothesis,
${\frac {{\hat {\theta }}-\theta _{0}}{{\rm {SE}}({\hat {\theta }})}}$
can be used as a Z-test statistic.
When using a Z-test for maximum likelihood estimates, it is important to be aware that the normal approximation may be poor if the sample size is not sufficiently large. Although there is no simple, universal rule stating how large the sample size must be to use a Z-test, simulation can give a good idea as to whether a Z-test is appropriate in a given situation.
Z-tests are employed whenever it can be argued that a test statistic follows a normal distribution under the null hypothesis of interest. Many non-parametric test statistics, such as U statistics, are approximately normal for large enough sample sizes, and hence are often performed as Z-tests.
See also
• Normal distribution
• Standard normal table
• Standard score
• Student's t-test
References
• Sprinthall, R. C. (2011). Basic Statistical Analysis (9th ed.). Pearson Education. ISBN 978-0-205-05217-2.
• Casella, G., Berger, R. L. (2002). Statistical Inference. Duxbury Press. ISBN 0-534-24312-6.
• Douglas C.Montgomery, George C.Runger.(2014). Applied Statistics And Probability For Engineers.(6th ed.). John Wiley & Sons, inc. ISBN 9781118539712, 9781118645062.
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G-module
In mathematics, given a group G, a G-module is an abelian group M on which G acts compatibly with the abelian group structure on M. This widely applicable notion generalizes that of a representation of G. Group (co)homology provides an important set of tools for studying general G-modules.
The term G-module is also used for the more general notion of an R-module on which G acts linearly (i.e. as a group of R-module automorphisms).
Definition and basics
Let $G$ be a group. A left $G$-module consists of[1] an abelian group $M$ together with a left group action $\rho :G\times M\to M$ such that
g·(a1 + a2) = g·a1 + g·a2
where g·a denotes ρ(g,a). A right G-module is defined similarly. Given a left G-module M, it can be turned into a right G-module by defining a·g = g−1·a.
A function f : M → N is called a morphism of G-modules (or a G-linear map, or a G-homomorphism) if f is both a group homomorphism and G-equivariant.
The collection of left (respectively right) G-modules and their morphisms form an abelian category G-Mod (resp. Mod-G). The category G-Mod (resp. Mod-G) can be identified with the category of left (resp. right) ZG-modules, i.e. with the modules over the group ring Z[G].
A submodule of a G-module M is a subgroup A ⊆ M that is stable under the action of G, i.e. g·a ∈ A for all g ∈ G and a ∈ A. Given a submodule A of M, the quotient module M/A is the quotient group with action g·(m + A) = g·m + A.
Examples
• Given a group G, the abelian group Z is a G-module with the trivial action g·a = a.
• Let M be the set of binary quadratic forms f(x, y) = ax2 + 2bxy + cy2 with a, b, c integers, and let G = SL(2, Z) (the 2×2 special linear group over Z). Define
$(g\cdot f)(x,y)=f((x,y)g^{t})=f\left((x,y)\cdot {\begin{bmatrix}\alpha &\gamma \\\beta &\delta \end{bmatrix}}\right)=f(\alpha x+\beta y,\gamma x+\delta y),$
where
$g={\begin{bmatrix}\alpha &\beta \\\gamma &\delta \end{bmatrix}}$
and (x, y)g is matrix multiplication. Then M is a G-module studied by Gauss.[2] Indeed, we have
$g(h(f(x,y)))=gf((x,y)h^{t})=f((x,y)h^{t}g^{t})=f((x,y)(gh)^{t})=(gh)f(x,y).$
• If V is a representation of G over a field K, then V is a G-module (it is an abelian group under addition).
Topological groups
If G is a topological group and M is an abelian topological group, then a topological G-module is a G-module where the action map G×M → M is continuous (where the product topology is taken on G×M).[3]
In other words, a topological G-module is an abelian topological group M together with a continuous map G×M → M satisfying the usual relations g(a + a′) = ga + ga′, (gg′)a = g(g′a), and 1a = a.
Notes
1. Curtis, Charles W.; Reiner, Irving (1962), Representation Theory of Finite Groups and Associative Algebras, John Wiley & Sons (Reedition 2006 by AMS Bookstore), ISBN 978-0-470-18975-7.
2. Kim, Myung-Hwan (1999), Integral Quadratic Forms and Lattices: Proceedings of the International Conference on Integral Quadratic Forms and Lattices, June 15–19, 1998, Seoul National University, Korea, American Mathematical Soc.
3. D. Wigner (1973). "Algebraic cohomology of topological groups". Trans. Amer. Math. Soc. 178: 83–93. doi:10.1090/s0002-9947-1973-0338132-7.
References
• Chapter 6 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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ZJ theorem
In mathematics, George Glauberman's ZJ theorem states that if a finite group G is p-constrained and p-stable and has a normal p-subgroup for some odd prime p, then Op′(G)Z(J(S)) is a normal subgroup of G, for any Sylow p-subgroup S.
Notation and definitions
• J(S) is the Thompson subgroup of a p-group S: the subgroup generated by the abelian subgroups of maximal order.
• Z(H) means the center of a group H.
• Op′ is the maximal normal subgroup of G of order coprime to p, the p′-core
• Op is the maximal normal p-subgroup of G, the p-core.
• Op′,p(G) is the maximal normal p-nilpotent subgroup of G, the p′,p-core, part of the upper p-series.
• For an odd prime p, a group G with Op(G) ≠ 1 is said to be p-stable if whenever P is a p-subgroup of G such that POp′(G) is normal in G, and [P,x,x] = 1, then the image of x in NG(P)/CG(P) is contained in a normal p-subgroup of NG(P)/CG(P).
• For an odd prime p, a group G with Op(G) ≠ 1 is said to be p-constrained if the centralizer CG(P) is contained in Op′,p(G) whenever P is a Sylow p-subgroup of Op′,p(G).
References
• Glauberman, George (1968), "A characteristic subgroup of a p-stable group", Canadian Journal of Mathematics, 20: 1101–1135, doi:10.4153/cjm-1968-107-2, ISSN 0008-414X, MR 0230807
• Gorenstein, D. (1980), Finite Groups, New York: Chelsea, ISBN 978-0-8284-0301-6, MR 0569209
• Thompson, John G. (1969), "A replacement theorem for p-groups and a conjecture", Journal of Algebra, 13 (2): 149–151, doi:10.1016/0021-8693(69)90068-4, ISSN 0021-8693, MR 0245683
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NL (complexity)
In computational complexity theory, NL (Nondeterministic Logarithmic-space) is the complexity class containing decision problems that can be solved by a nondeterministic Turing machine using a logarithmic amount of memory space.
Unsolved problem in computer science:
${\mathsf {L{\overset {?}{=}}NL}}$
(more unsolved problems in computer science)
NL is a generalization of L, the class for logspace problems on a deterministic Turing machine. Since any deterministic Turing machine is also a nondeterministic Turing machine, we have that L is contained in NL.
NL can be formally defined in terms of the computational resource nondeterministic space (or NSPACE) as NL = NSPACE(log n).
Important results in complexity theory allow us to relate this complexity class with other classes, telling us about the relative power of the resources involved. Results in the field of algorithms, on the other hand, tell us which problems can be solved with this resource. Like much of complexity theory, many important questions about NL are still open (see Unsolved problems in computer science).
Occasionally NL is referred to as RL due to its probabilistic definition below; however, this name is more frequently used to refer to randomized logarithmic space, which is not known to equal NL.
NL-complete problems
Several problems are known to be NL-complete under log-space reductions, including ST-connectivity and 2-satisfiability. ST-connectivity asks, for nodes S and T in a directed graph, whether T is reachable from S. 2-satisfiability asks, given a propositional formula of which each clause is the disjunction of two literals, if there is a variable assignment that makes the formula true. An example instance, where $\neg $ indicates not, might be:
$(x_{1}\vee \neg x_{3})\wedge (\neg x_{2}\vee x_{3})\wedge (\neg x_{1}\vee \neg x_{2})$
Containments
It is known that NL is contained in P, since there is a polynomial-time algorithm for 2-satisfiability, but it is not known whether NL = P or whether L = NL. It is known that NL = co-NL, where co-NL is the class of languages whose complements are in NL. This result (the Immerman–Szelepcsényi theorem) was independently discovered by Neil Immerman and Róbert Szelepcsényi in 1987; they received the 1995 Gödel Prize for this work.
In circuit complexity, NL can be placed within the NC hierarchy. In Papadimitriou 1994, Theorem 16.1, we have:
${\mathsf {NC_{1}\subseteq L\subseteq NL\subseteq NC_{2}}}$.
More precisely, NL is contained in AC1. It is known that NL is equal to ZPL, the class of problems solvable by randomized algorithms in logarithmic space and unbounded time, with no error. It is not, however, known or believed to be equal to RLP or ZPLP, the polynomial-time restrictions of RL and ZPL, which some authors refer to as RL and ZPL.
We can relate NL to deterministic space using Savitch's theorem, which tells us that any nondeterministic algorithm can be simulated by a deterministic machine in at most quadratically more space. From Savitch's theorem, we have directly that:
${\mathsf {NL\subseteq SPACE}}(\log ^{2}n)\ \ \ \ {\text{equivalently, }}{\mathsf {NL\subseteq L}}^{2}.$
This was the strongest deterministic-space inclusion known in 1994 (Papadimitriou 1994 Problem 16.4.10, "Symmetric space"). Since larger space classes are not affected by quadratic increases, the nondeterministic and deterministic classes are known to be equal, so that for example we have PSPACE = NPSPACE.
Alternative definitions
Probabilistic definition
Suppose C is the complexity class of decision problems solvable in logarithmithic space with probabilistic Turing machines that never accept incorrectly but are allowed to reject incorrectly less than 1/3 of the time; this is called one-sided error. The constant 1/3 is arbitrary; any x with 0 ≤ x < 1/2 would suffice.
It turns out that C = NL. Notice that C, unlike its deterministic counterpart L, is not limited to polynomial time, because although it has a polynomial number of configurations it can use randomness to escape an infinite loop. If we do limit it to polynomial time, we get the class RL, which is contained in but not known or believed to equal NL.
There is a simple algorithm that establishes that C = NL. Clearly C is contained in NL, since:
• If the string is not in the language, both reject along all computation paths.
• If the string is in the language, an NL algorithm accepts along at least one computation path and a C algorithm accepts along at least two-thirds of its computation paths.
To show that NL is contained in C, we simply take an NL algorithm and choose a random computation path of length n, and execute this 2n times. Because no computation path exceeds length n, and because there are 2n computation paths in all, we have a good chance of hitting the accepting one (bounded below by a constant).
The only problem is that we don't have room in log space for a binary counter that goes up to 2n. To get around this we replace it with a randomized counter, which simply flips n coins and stops and rejects if they all land on heads. Since this event has probability 2−n, we expect to take 2n steps on average before stopping. It only needs to keep a running total of the number of heads in a row it sees, which it can count in log space.
Because of the Immerman–Szelepcsényi theorem, according to which NL is closed under complements, the one-sided error in these probabilistic computations can be replaced by zero-sided error. That is, these problems can be solved by probabilistic Turing machines that use logarithmic space and never make errors. The corresponding complexity class that also requires the machine to use only polynomial time is called ZPLP.
Thus, when we only look at space alone, it seems that randomization and nondeterminism are equally powerful.
Certificate definition
NL can equivalently be characterised by certificates, analogous to classes such as NP. Consider a deterministic logarithmic-space bounded Turing machine that has an additional read-only read-once input tape. A language is in NL if and only if such a Turing machine accepts any word in the language for an appropriate choice of certificate in its additional input tape, and rejects any word not in the language regardless of the certificate.[1]
Cem Say and Abuzer Yakaryılmaz have proven that the deterministic logarithmic-space Turing machine in the statement above can be replaced by a bounded-error probabilistic constant-space Turing machine that is allowed to use only a constant number of random bits.[2]
Descriptive complexity
There is a simple logical characterization of NL: it contains precisely those languages expressible in first-order logic with an added transitive closure operator.
Closure properties
The class NL is closed under the operations complementation, union, and therefore intersection, concatenation, and Kleene star.
Notes
1. Arora, Sanjeev; Barak, Boaz (2009). "Definition 4.19". Complexity Theory: A Modern Approach. Cambridge University Press. ISBN 978-0-521-42426-4.
2. A. C. Cem Say, Abuzer Yakaryılmaz, "Finite state verifiers with constant randomness," Logical Methods in Computer Science, Vol. 10(3:6)2014, pp. 1-17.
References
• Complexity Zoo: NL
• Papadimitriou, C. (1994). "Chapter 16: Logarithmic Space". Computational Complexity. Addison-Wesley. ISBN 0-201-53082-1.
• Michael Sipser (27 June 1997). "Sections 8.4–8.6: The Classes L and NL, NL-completeness, NL equals coNL". Introduction to the Theory of Computation. PWS Publishing. pp. 294–302. ISBN 0-534-94728-X.
• Introduction to Complexity Theory: Lecture 7. Oded Goldreich. Proposition 6.1. Our C is what Goldreich calls badRSPACE(log n).
Important complexity classes
Considered feasible
• DLOGTIME
• AC0
• ACC0
• TC0
• L
• SL
• RL
• NL
• NL-complete
• NC
• SC
• CC
• P
• P-complete
• ZPP
• RP
• BPP
• BQP
• APX
• FP
Suspected infeasible
• UP
• NP
• NP-complete
• NP-hard
• co-NP
• co-NP-complete
• AM
• QMA
• PH
• ⊕P
• PP
• #P
• #P-complete
• IP
• PSPACE
• PSPACE-complete
Considered infeasible
• EXPTIME
• NEXPTIME
• EXPSPACE
• 2-EXPTIME
• ELEMENTARY
• PR
• R
• RE
• ALL
Class hierarchies
• Polynomial hierarchy
• Exponential hierarchy
• Grzegorczyk hierarchy
• Arithmetical hierarchy
• Boolean hierarchy
Families of classes
• DTIME
• NTIME
• DSPACE
• NSPACE
• Probabilistically checkable proof
• Interactive proof system
List of complexity classes
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Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is in A and b is in B.[1] In terms of set-builder notation, that is
$A\times B=\{(a,b)\mid a\in A\ {\mbox{ and }}\ b\in B\}.$[2][3]
"Cartesian square" redirects here. For Cartesian squares in category theory, see Cartesian square (category theory).
A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form (row value, column value).[4]
One can similarly define the Cartesian product of n sets, also known as an n-fold Cartesian product, which can be represented by an n-dimensional array, where each element is an n-tuple. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets.
The Cartesian product is named after René Descartes,[5] whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product.
Set-theoretic definition
A rigorous definition of the Cartesian product requires a domain to be specified in the set-builder notation. In this case the domain would have to contain the Cartesian product itself. For defining the Cartesian product of the sets $A$ and $B$, one such domain is the set ${\mathcal {P}}({\mathcal {P}}(A\cup B))$ where ${\mathcal {P}}$ denotes the power set. Then the Cartesian product of the sets $A$ and $B$ would be defined as[6]
$A\times B=\{x\in {\mathcal {P}}({\mathcal {P}}(A\cup B))\mid \exists a\in A\exists b\in B:x=(a,b)\}.$
Examples
A deck of cards
An illustrative example is the standard 52-card deck. The standard playing card ranks {A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2} form a 13-element set. The card suits {♠, ♥, ♦, ♣} form a four-element set. The Cartesian product of these sets returns a 52-element set consisting of 52 ordered pairs, which correspond to all 52 possible playing cards.
Ranks × Suits returns a set of the form {(A, ♠), (A, ♥), (A, ♦), (A, ♣), (K, ♠), …, (3, ♣), (2, ♠), (2, ♥), (2, ♦), (2, ♣)}.
Suits × Ranks returns a set of the form {(♠, A), (♠, K), (♠, Q), (♠, J), (♠, 10), …, (♣, 6), (♣, 5), (♣, 4), (♣, 3), (♣, 2)}.
These two sets are distinct, even disjoint, but there is a natural bijection between them, under which (3, ♣) corresponds to (♣, 3) and so on.
A two-dimensional coordinate system
The main historical example is the Cartesian plane in analytic geometry. In order to represent geometrical shapes in a numerical way, and extract numerical information from shapes' numerical representations, René Descartes assigned to each point in the plane a pair of real numbers, called its coordinates. Usually, such a pair's first and second components are called its x and y coordinates, respectively (see picture). The set of all such pairs (i.e., the Cartesian product ℝ×ℝ, with ℝ denoting the real numbers) is thus assigned to the set of all points in the plane.
Most common implementation (set theory)
Main article: Implementation of mathematics in set theory
A formal definition of the Cartesian product from set-theoretical principles follows from a definition of ordered pair. The most common definition of ordered pairs, Kuratowski's definition, is $(x,y)=\{\{x\},\{x,y\}\}$. Under this definition, $(x,y)$ is an element of ${\mathcal {P}}({\mathcal {P}}(X\cup Y))$, and $X\times Y$ is a subset of that set, where ${\mathcal {P}}$ represents the power set operator. Therefore, the existence of the Cartesian product of any two sets in ZFC follows from the axioms of pairing, union, power set, and specification. Since functions are usually defined as a special case of relations, and relations are usually defined as subsets of the Cartesian product, the definition of the two-set Cartesian product is necessarily prior to most other definitions.
Non-commutativity and non-associativity
Let A, B, C, and D be sets.
The Cartesian product A × B is not commutative,
$A\times B\neq B\times A,$[4]
because the ordered pairs are reversed unless at least one of the following conditions is satisfied:[7]
• A is equal to B, or
• A or B is the empty set.
For example:
A = {1,2}; B = {3,4}
A × B = {1,2} × {3,4} = {(1,3), (1,4), (2,3), (2,4)}
B × A = {3,4} × {1,2} = {(3,1), (3,2), (4,1), (4,2)}
A = B = {1,2}
A × B = B × A = {1,2} × {1,2} = {(1,1), (1,2), (2,1), (2,2)}
A = {1,2}; B = ∅
A × B = {1,2} × ∅ = ∅
B × A = ∅ × {1,2} = ∅
Strictly speaking, the Cartesian product is not associative (unless one of the involved sets is empty).
$(A\times B)\times C\neq A\times (B\times C)$
If for example A = {1}, then (A × A) × A = {((1, 1), 1)} ≠ {(1, (1, 1))} = A × (A × A).
Intersections, unions, and subsets
See also: List of set identities and relations
Example sets
A = {y ∈ ℝ : 1 ≤ y ≤ 4}, B = {x ∈ ℝ : 2 ≤ x ≤ 5},
and C = {x ∈ ℝ : 4 ≤ x ≤ 7}, demonstrating
A × (B∩C) = (A×B) ∩ (A×C),
A × (B∪C) = (A×B) ∪ (A×C), and
A × (B \ C) = (A×B) \ (A×C)
Example sets
A = {x ∈ ℝ : 2 ≤ x ≤ 5}, B = {x ∈ ℝ : 3 ≤ x ≤ 7},
C = {y ∈ ℝ : 1 ≤ y ≤ 3}, D = {y ∈ ℝ : 2 ≤ y ≤ 4}, demonstrating
(A∩B) × (C∩D) = (A×C) ∩ (B×D).
(A∪B) × (C∪D) ≠ (A×C) ∪ (B×D) can be seen from the same example.
The Cartesian product satisfies the following property with respect to intersections (see middle picture).
$(A\cap B)\times (C\cap D)=(A\times C)\cap (B\times D)$
In most cases, the above statement is not true if we replace intersection with union (see rightmost picture).
$(A\cup B)\times (C\cup D)\neq (A\times C)\cup (B\times D)$
In fact, we have that:
$(A\times C)\cup (B\times D)=[(A\setminus B)\times C]\cup [(A\cap B)\times (C\cup D)]\cup [(B\setminus A)\times D]$
For the set difference, we also have the following identity:
$(A\times C)\setminus (B\times D)=[A\times (C\setminus D)]\cup [(A\setminus B)\times C]$
Here are some rules demonstrating distributivity with other operators (see leftmost picture):[7]
${\begin{aligned}A\times (B\cap C)&=(A\times B)\cap (A\times C),\\A\times (B\cup C)&=(A\times B)\cup (A\times C),\\A\times (B\setminus C)&=(A\times B)\setminus (A\times C),\end{aligned}}$
$(A\times B)^{\complement }=\left(A^{\complement }\times B^{\complement }\right)\cup \left(A^{\complement }\times B\right)\cup \left(A\times B^{\complement }\right)\!,$
where $A^{\complement }$ denotes the absolute complement of A.
Other properties related with subsets are:
${\text{if }}A\subseteq B{\text{, then }}A\times C\subseteq B\times C;$
${\text{if both }}A,B\neq \emptyset {\text{, then }}A\times B\subseteq C\times D\!\iff \!A\subseteq C{\text{ and }}B\subseteq D.$[8]
Cardinality
See also: Cardinal arithmetic
The cardinality of a set is the number of elements of the set. For example, defining two sets: A = {a, b} and B = {5, 6}. Both set A and set B consist of two elements each. Their Cartesian product, written as A × B, results in a new set which has the following elements:
A × B = {(a,5), (a,6), (b,5), (b,6)}.
where each element of A is paired with each element of B, and where each pair makes up one element of the output set. The number of values in each element of the resulting set is equal to the number of sets whose Cartesian product is being taken; 2 in this case. The cardinality of the output set is equal to the product of the cardinalities of all the input sets. That is,
|A × B| = |A| · |B|.[4]
In this case, |A × B| = 4
Similarly
|A × B × C| = |A| · |B| · |C|
and so on.
The set A × B is infinite if either A or B is infinite, and the other set is not the empty set.[9]
Cartesian products of several sets
n-ary Cartesian product
The Cartesian product can be generalized to the n-ary Cartesian product over n sets X1, ..., Xn as the set
$X_{1}\times \cdots \times X_{n}=\{(x_{1},\ldots ,x_{n})\mid x_{i}\in X_{i}\ {\text{for every}}\ i\in \{1,\ldots ,n\}\}$
of n-tuples. If tuples are defined as nested ordered pairs, it can be identified with (X1 × ⋯ × Xn−1) × Xn. If a tuple is defined as a function on {1, 2, …, n} that takes its value at i to be the ith element of the tuple, then the Cartesian product X1×⋯×Xn is the set of functions
$\{x:\{1,\ldots ,n\}\to X_{1}\cup \cdots \cup X_{n}\ |\ x(i)\in X_{i}\ {\text{for every}}\ i\in \{1,\ldots ,n\}\}.$
n-ary Cartesian power
The Cartesian square of a set X is the Cartesian product X2 = X × X. An example is the 2-dimensional plane R2 = R × R where R is the set of real numbers:[1] R2 is the set of all points (x,y) where x and y are real numbers (see the Cartesian coordinate system).
The n-ary Cartesian power of a set X, denoted $X^{n}$, can be defined as
$X^{n}=\underbrace {X\times X\times \cdots \times X} _{n}=\{(x_{1},\ldots ,x_{n})\ |\ x_{i}\in X\ {\text{for every}}\ i\in \{1,\ldots ,n\}\}.$
An example of this is R3 = R × R × R, with R again the set of real numbers,[1] and more generally Rn.
The n-ary Cartesian power of a set X is isomorphic to the space of functions from an n-element set to X. As a special case, the 0-ary Cartesian power of X may be taken to be a singleton set, corresponding to the empty function with codomain X.
Infinite Cartesian products
It is possible to define the Cartesian product of an arbitrary (possibly infinite) indexed family of sets. If I is any index set, and $\{X_{i}\}_{i\in I}$ is a family of sets indexed by I, then the Cartesian product of the sets in $\{X_{i}\}_{i\in I}$ is defined to be
$\prod _{i\in I}X_{i}=\left\{\left.f:I\to \bigcup _{i\in I}X_{i}\ \right|\ \forall i\in I.\ f(i)\in X_{i}\right\},$
that is, the set of all functions defined on the index set I such that the value of the function at a particular index i is an element of Xi. Even if each of the Xi is nonempty, the Cartesian product may be empty if the axiom of choice, which is equivalent to the statement that every such product is nonempty, is not assumed. $\prod _{i\in I}X_{i}$ may also be denoted ${\mathsf {X}}$${}_{i\in I}X_{i}$.[10]
For each j in I, the function
$\pi _{j}:\prod _{i\in I}X_{i}\to X_{j},$
defined by $\pi _{j}(f)=f(j)$ is called the jth projection map.
Cartesian power is a Cartesian product where all the factors Xi are the same set X. In this case,
$\prod _{i\in I}X_{i}=\prod _{i\in I}X$
is the set of all functions from I to X, and is frequently denoted XI. This case is important in the study of cardinal exponentiation. An important special case is when the index set is $\mathbb {N} $, the natural numbers: this Cartesian product is the set of all infinite sequences with the ith term in its corresponding set Xi. For example, each element of
$\prod _{n=1}^{\infty }\mathbb {R} =\mathbb {R} \times \mathbb {R} \times \cdots $
can be visualized as a vector with countably infinite real number components. This set is frequently denoted $\mathbb {R} ^{\omega }$, or $\mathbb {R} ^{\mathbb {N} }$.
Other forms
Abbreviated form
If several sets are being multiplied together (e.g., X1, X2, X3, …), then some authors[11] choose to abbreviate the Cartesian product as simply ×Xi.
Cartesian product of functions
If f is a function from X to A and g is a function from Y to B, then their Cartesian product f × g is a function from X × Y to A × B with
$(f\times g)(x,y)=(f(x),g(y)).$
This can be extended to tuples and infinite collections of functions. This is different from the standard Cartesian product of functions considered as sets.
Cylinder
Let $A$ be a set and $B\subseteq A$. Then the cylinder of $B$ with respect to $A$ is the Cartesian product $B\times A$ of $B$ and $A$.
Normally, $A$ is considered to be the universe of the context and is left away. For example, if $B$ is a subset of the natural numbers $\mathbb {N} $, then the cylinder of $B$ is $B\times \mathbb {N} $.
Definitions outside set theory
Category theory
Although the Cartesian product is traditionally applied to sets, category theory provides a more general interpretation of the product of mathematical structures. This is distinct from, although related to, the notion of a Cartesian square in category theory, which is a generalization of the fiber product.
Exponentiation is the right adjoint of the Cartesian product; thus any category with a Cartesian product (and a final object) is a Cartesian closed category.
Graph theory
In graph theory, the Cartesian product of two graphs G and H is the graph denoted by G × H, whose vertex set is the (ordinary) Cartesian product V(G) × V(H) and such that two vertices (u,v) and (u′,v′) are adjacent in G × H, if and only if u = u′ and v is adjacent with v′ in H, or v = v′ and u is adjacent with u′ in G. The Cartesian product of graphs is not a product in the sense of category theory. Instead, the categorical product is known as the tensor product of graphs.
See also
• Binary relation
• Concatenation of sets of strings
• Coproduct
• Cross product
• Direct product of groups
• Empty product
• Euclidean space
• Exponential object
• Finitary relation
• Join (SQL) § Cross join
• Orders on the Cartesian product of totally ordered sets
• Axiom of power set (to prove the existence of the Cartesian product)
• Product (category theory)
• Product topology
• Product type
• Ultraproduct
References
1. Weisstein, Eric W. "Cartesian Product". mathworld.wolfram.com. Retrieved September 5, 2020.
2. Warner, S. (1990). Modern Algebra. Dover Publications. p. 6.
3. Nykamp, Duane. "Cartesian product definition". Math Insight. Retrieved September 5, 2020.
4. "Cartesian Product". web.mnstate.edu. Archived from the original on July 18, 2020. Retrieved September 5, 2020.
5. "Cartesian". Merriam-Webster.com. 2009. Retrieved December 1, 2009.
6. Corry, S. "A Sketch of the Rudiments of Set Theory" (PDF). Retrieved May 5, 2023.
7. Singh, S. (August 27, 2009). Cartesian product. Retrieved from the Connexions Web site: http://cnx.org/content/m15207/1.5/
8. Cartesian Product of Subsets. (February 15, 2011). ProofWiki. Retrieved 05:06, August 1, 2011 from https://proofwiki.org/w/index.php?title=Cartesian_Product_of_Subsets&oldid=45868
9. Peter S. (1998). A Crash Course in the Mathematics of Infinite Sets. St. John's Review, 44(2), 35–59. Retrieved August 1, 2011, from http://www.mathpath.org/concepts/infinity.htm
10. F. R. Drake, Set Theory: An Introduction to Large Cardinals, p.24. Studies in Logic and the Foundations of Mathematics, vol. 76 (1978). ISBN 0-7204-2200-0.
11. Osborne, M., and Rubinstein, A., 1994. A Course in Game Theory. MIT Press.
External links
• Cartesian Product at ProvenMath
• "Direct product", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• How to find the Cartesian Product, Education Portal Academy
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Zadeh's rule
In mathematical optimization, Zadeh's rule (also known as the least-entered rule) is an algorithmic refinement of the simplex method for linear optimization.
The rule was proposed around 1980 by Norman Zadeh (son of Lotfi A. Zadeh), and has entered the folklore of convex optimization since then.[1]
Zadeh offered a reward of $1,000 to anyone who can show that the rule admits polynomially many iterations or to prove that there is a family of linear programs on which the pivoting rule requires subexponentially many iterations to find the optimum.[2]
Algorithm
Zadeh's rule belongs to the family of history-based improvement rules which, during a run of the simplex algorithm, retain supplementary data in addition to the current basis of the linear program.
In particular, the rule chooses among all improving variables one which has entered the basis least often, intuitively ensuring that variables that might yield a substantive improvement in the long run but only a small improvement in a single step will be applied after a linear number of steps.
The supplementary data structure in Zadeh's algorithm can therefore be modeled as an occurrence record, mapping all variables to natural numbers, monitoring how often a particular variable has entered the basis. In every iteration, the algorithm then selects an improving variable that is minimal with respect to the retained occurrence record.
Note that the rule does not explicitly specify which particular improving variable should enter the basis in case of a tie.
Superpolynomial lower bound
Zadeh's rule has been shown to have at least super-polynomial time complexity in the worse-case by constructing a family of Markov decision processes on which the policy iteration algorithm requires a super-polynomial number of steps.[3][4]
Running the simplex algorithm with Zadeh's rule on the induced linear program then yields a super-polynomial lower bound.
The result was presented at the "Efficiency of the Simplex Method: Quo vadis Hirsch conjecture?" IPAM workshop in 2011 by Oliver Friedmann.[5] Zadeh, although not working in academia anymore at that time, attended the Workshop and honored his original proposal.[6]
Exponential lower bound
Friedmann's original result has since been strengthened by the construction of an exponential instance for Zadeh's rule.[7]
Notes
1. Zadeh, Norman (1980). "What is the worst case behaviour of the simplex algorithm?". Technical Report, Department of Operations Research, Stanford.
2. Ziegler, Günter (2004). "Typical and extremal linear programs". The Sharpest Cut (MPS-Siam Series on Optimization: 217–230. doi:10.1137/1.9780898718805.ch14. ISBN 978-0-89871-552-1.
3. Friedmann, Oliver (2011). "A subexponential lower bound for Zadeh's pivoting rule for solving linear programs and games". Proceedings of the 15th International Conference on Integer Programming and Combinatorial Optimization (IPCO). pp. 192–206.
4. Disser, Y.; Hopp, A.V. (2019). "On Friedmann's Subexponential Lower Bound for Zadeh's Pivot Rule". Proceedings of the 20th Conference on Integer Programming and Combinatorial Optimization (IPCO). pp. 168–180.
5. "Efficiency of the Simplex Method: Quo vadis Hirsch conjecture?".
6. "Günter Ziegler: 1000$ from Beverly Hills for a Math Problem. (IPAM remote blogging.)". 20 January 2011.
7. Disser, Yann; Friedmann, Oliver; Hopp, Alexander V. (2022). "An exponential lower bound for Zadeh's pivot rule". Mathematical Programming.
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Sara Zahedi
Sara Zahedi (born 1981 in Tehran)[1] is an Iranian-Swedish mathematician who works in computational fluid dynamics and holds an associate professorship in numerical analysis at the Royal Institute of Technology (KTH) in Sweden. She is one of ten winners and the only female winner of the European Mathematical Society Prize for 2016 "for her outstanding research regarding the development and analysis of numerical algorithms for partial differential equations with a focus on applications to problems with dynamically changing geometry". The topic of Zahedi's EMS Prize lecture was her recent research on the CutFEM method of solving fluid dynamics problems with changing boundary geometry, such as arise when simulating the dynamics of systems of two immiscible liquids.[2] This method combines level set methods to represent the domain boundaries as cuts through an underlying uniform grid, together with numerical simulation techniques that can adapt to the complex geometries of grid cells cut by these boundaries.[3]
When Zahedi was ten years old, with her father having been killed by the regime after the Iranian Revolution, her mother sent her on her own as a refugee to Sweden, and only rejoined her some years later.[1][4] She was drawn to mathematics in part because she understood mathematics better than the Swedish language,[4] and to fluid mechanics because of its real-world applications.[1] She earned a master's degree from KTH in 2006, and a doctorate in 2011;[2] her dissertation, Numerical Methods for Fluid Interface Problems, was supervised by Gunilla Kreiss.[5] After postdoctoral studies at Uppsala University, she returned to KTH as an assistant professor in 2014.[2]
Selected publications
• Olsson, Elin; Kreiss, Gunilla; Zahedi, Sara (2007), "A conservative level set method for two phase flow. II", Journal of Computational Physics, 225 (1): 785–807, Bibcode:2007JCoPh.225..785O, doi:10.1016/j.jcp.2006.12.027, MR 2346700.
• Hansbo, Peter; Larson, Mats G.; Zahedi, Sara (2014), "A cut finite element method for a Stokes interface problem", Applied Numerical Mathematics, 85: 90–114, arXiv:1205.5684, doi:10.1016/j.apnum.2014.06.009, MR 3239219, S2CID 119167194.
References
1. Timon, Ágata (July 20, 2016), "Me gusta resolver problemas matemáticos del mundo real: Sara Zahedi, del Royal Institute of Technology (Suecia), ganadora del premio EMS a matemáticos jóvenes", El Mundo (in Spanish).
2. Prize laureates, 7th Eur. Congress of Mathematics, July 18–22, 2016, retrieved 2016-07-26.
3. Burman, Erik; Claus, Susanne; Hansbo, Peter; Larson, Mats G.; Massing, André (December 2014), "CutFEM: Discretizing geometry and partial differential equations" (PDF), International Journal for Numerical Methods in Engineering, 104 (7): 472–501, Bibcode:2015IJNME.104..472B, doi:10.1002/nme.4823.
4. "A refugee's story: 'Math was a language I understood'", Don't call me a prodigy: the rising stars of European mathematics, Deutsche Welle, July 18, 2016, retrieved 2016-07-26.
5. Numerical Methods for Fluid Interface Problems, dissertations.se, retrieved 2016-07-26.
External links
• Home page
• Sara Zahedi publications indexed by Google Scholar
Mathematics in Iran
Mathematicians
Before
20th Century
• Abu al-Wafa' Buzjani
• Jamshīd al-Kāshī (al-Kashi's theorem)
• Omar Khayyam (Khayyam-Pascal's triangle, Khayyam-Saccheri quadrilateral, Khayyam's Solution of Cubic Equations)
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• Muhammad Baqir Yazdi
• Nizam al-Din al-Nisapuri
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• Najm al-Din al-Qazwini al-Katibi
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• Maryam Mirzakhani
• Caucher Birkar
• Sara Zahedi
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• S. L. Hakimi (Havel–Hakimi algorithm)
• Siamak Yassemi
• Freydoon Shahidi (Langlands–Shahidi method)
• Hamid Naderi Yeganeh
• Esmail Babolian
• Ramin Takloo-Bighash
• Lotfi A. Zadeh (Fuzzy mathematics, Fuzzy set, Fuzzy logic)
• Ebadollah S. Mahmoodian
• Reza Sarhangi (The Bridges Organization)
• Siavash Shahshahani
• Gholamhossein Mosaheb
• Amin Shokrollahi
• Reza Sadeghi
• Mohammad Mehdi Zahedi
• Mohsen Hashtroodi
• Hossein Zakeri
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Zahorski theorem
In mathematics, Zahorski's theorem is a theorem of real analysis. It states that a necessary and sufficient condition for a subset of the real line to be the set of points of non-differentiability of a continuous real-valued function, is that it be the union of a Gδ set and a ${G_{\delta }}_{\sigma }$ set of zero measure.
This result was proved by Zygmunt Zahorski in 1939 and first published in 1941.
References
• Zahorski, Zygmunt (1941), "Punktmengen, in welchen eine stetige Funktion nicht differenzierbar ist", Rec. Math. (Mat. Sbornik), Nouvelle Série (in Russian and German), 9 (51): 487–510, MR 0004869.
• Zahorski, Zygmunt (1946), "Sur l'ensemble des points de non-dérivabilité d'une fonction continue" (French translation of 1941 Russian paper), Bulletin de la Société Mathématique de France, 74: 147–178, doi:10.24033/bsmf.1381, MR 0022592.
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Porous set
In mathematics, a porous set is a concept in the study of metric spaces. Like the concepts of meagre and measure zero sets, a porous set can be considered "sparse" or "lacking bulk"; however, porous sets are not equivalent to either meagre sets or measure zero sets, as shown below.
Definition
Let (X, d) be a complete metric space and let E be a subset of X. Let B(x, r) denote the closed ball in (X, d) with centre x ∈ X and radius r > 0. E is said to be porous if there exist constants 0 < α < 1 and r0 > 0 such that, for every 0 < r ≤ r0 and every x ∈ X, there is some point y ∈ X with
$B(y,\alpha r)\subseteq B(x,r)\setminus E.$
A subset of X is called σ-porous if it is a countable union of porous subsets of X.
Properties
• Any porous set is nowhere dense. Hence, all σ-porous sets are meagre sets (or of the first category).
• If X is a finite-dimensional Euclidean space Rn, then porous subsets are sets of Lebesgue measure zero.
• However, there does exist a non-σ-porous subset P of Rn which is of the first category and of Lebesgue measure zero. This is known as Zajíček's theorem.
• The relationship between porosity and being nowhere dense can be illustrated as follows: if E is nowhere dense, then for x ∈ X and r > 0, there is a point y ∈ X and s > 0 such that
$B(y,s)\subseteq B(x,r)\setminus E.$
However, if E is also porous, then it is possible to take s = αr (at least for small enough r), where 0 < α < 1 is a constant that depends only on E.
References
• Reich, Simeon; Zaslavski, Alexander J. (2002). "Two convergence results for continuous descent methods". Electronic Journal of Differential Equations. 2002 (24): 1–11. ISSN 1072-6691.
• Zajíček, L. (1987–1988). "Porosity and σ-porosity". Real Anal. Exchange. 13 (2): 314–350. doi:10.2307/44151885. ISSN 0147-1937. JSTOR 44151885. MR943561
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Zakai equation
In filtering theory the Zakai equation is a linear stochastic partial differential equation for the un-normalized density of a hidden state. In contrast, the Kushner equation gives a non-linear stochastic partial differential equation for the normalized density of the hidden state. In principle either approach allows one to estimate a quantity function (the state of a dynamical system) from noisy measurements, even when the system is non-linear (thus generalizing the earlier results of Wiener and Kalman for linear systems and solving a central problem in estimation theory). The application of this approach to a specific engineering situation may be problematic however, as these equations are quite complex.[1][2] The Zakai equation is a bilinear stochastic partial differential equation. It was named after Moshe Zakai.[3]
Overview
Assume the state of the system evolves according to
$dx=f(x,t)dt+dw$
and a noisy measurement of the system state is available:
$dz=h(x,t)dt+dv$
where $w,v$ are independent Wiener processes. Then the unnormalized conditional probability density $p(x,t)$ of the state at time t is given by the Zakai equation:
$dp=L(p)dt+ph^{T}dz$
where the operator $L(p)=-\sum {\frac {\partial (f_{i}p)}{\partial x_{i}}}+{\frac {1}{2}}\sum {\frac {\partial ^{2}p}{\partial x_{i}\partial x_{j}}}$
As previously mentioned, $p$ is an unnormalized density and thus does not necessarily integrate to 1. After solving for $p$, integration and normalization can be done if desired (an extra step not required in the Kushner approach).
Note that if the last term on the right hand side is omitted (by choosing h identically zero), the result is a nonstochastic PDE: the familiar Fokker–Planck equation, which describes the evolution of the state when no measurement information is available.
See also
• Kushner equation
• Kalman filter
• Wiener filter
References
1. Sritharan, S. S. (1994). "Nonlinear filtering of stochastic Navier–Stokes equations". In Funaki, T.; Woyczynski, W. A. (eds.). Nonlinear Stochastic PDEs: Burgers Turbulence and Hydrodynamic Limit (PDF). Springer-Verlag. pp. 247–260. ISBN 0-387-94624-1.
2. Hobbs, S. L.; Sritharan, S. S. (1996). "Nonlinear filtering theory for stochastic reaction–diffusion equations". In Gretsky, N.; Goldstein, J.; Uhl, J. J. (eds.). Probability and Modern Analysis (PDF). Marcel Dekker. pp. 219–234.
3. Zakai, M. (1969). "On the optimal filtering of diffusion processes". Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete. 11 (3): 230–243. doi:10.1007/BF00536382. MR 0242552. S2CID 119763576. Zbl 0164.19201.
Further reading
• Grigelionis, B.; Mikulevičius, R. (1983). "Stochastic evolution equations and densities of the conditional distributions". Theory and Application of Random Fields. Berlin: Springer. pp. 49–88. doi:10.1007/BFb0044682.
• Schuss, Zeev (2012). "Nonlinear Filtering and Smoothing of Diffusions". Nonlinear Filtering and Optimal Phase Tracking. Boston: Springer. pp. 85–106. doi:10.1007/978-1-4614-0487-3_3. ISBN 978-1-4614-0486-6.
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Knizhnik–Zamolodchikov equations
In mathematical physics the Knizhnik–Zamolodchikov equations, or KZ equations, are linear differential equations satisfied by the correlation functions (on the Riemann sphere) of two-dimensional conformal field theories associated with an affine Lie algebra at a fixed level. They form a system of complex partial differential equations with regular singular points satisfied by the N-point functions of affine primary fields and can be derived using either the formalism of Lie algebras or that of vertex algebras.
The structure of the genus-zero part of the conformal field theory is encoded in the monodromy properties of these equations. In particular, the braiding and fusion of the primary fields (or their associated representations) can be deduced from the properties of the four-point functions, for which the equations reduce to a single matrix-valued first-order complex ordinary differential equation of Fuchsian type.
Originally the Russian physicists Vadim Knizhnik and Alexander Zamolodchikov derived the equations for the SU(2) Wess–Zumino–Witten model using the classical formulas of Gauss for the connection coefficients of the hypergeometric differential equation.
Definition
Let ${\hat {\mathfrak {g}}}_{k}$ denote the affine Lie algebra with level k and dual Coxeter number h. Let v be a vector from a zero mode representation of ${\hat {\mathfrak {g}}}_{k}$ and $\Phi (v,z)$ the primary field associated with it. Let $t^{a}$ be a basis of the underlying Lie algebra ${\mathfrak {g}}$, $t_{i}^{a}$ their representation on the primary field $\Phi (v_{i},z)$ and η the Killing form. Then for $i,j=1,2,\ldots ,N$ the Knizhnik–Zamolodchikov equations read
$\left((k+h)\partial _{z_{i}}+\sum _{j\neq i}{\frac {\sum _{a,b}\eta _{ab}t_{i}^{a}\otimes t_{j}^{b}}{z_{i}-z_{j}}}\right)\left\langle \Phi (v_{N},z_{N})\dots \Phi (v_{1},z_{1})\right\rangle =0.$
Informal derivation
The Knizhnik–Zamolodchikov equations result from the Sugawara construction of the Virasoro algebra from the affine Lie algebra. More specifically, they result from applying the identity
$L_{-1}={\frac {1}{2(k+h)}}\sum _{k\in \mathbf {Z} }\sum _{a,b}\eta _{ab}J_{-k}^{a}J_{k-1}^{b}$
to the affine primary field $\Phi (v_{i},z_{i})$ in a correlation function of affine primary fields. In this context, only the terms $k=0,1$ are non-vanishing. The action of $J_{-1}^{a}$ can then be rewritten using global Ward identities,
$\left(\left(J_{-1}^{a}\right)_{i}+\sum _{j\neq i}{\frac {t_{j}^{a}}{z_{i}-z_{j}}}\right)\left\langle \Phi (v_{N},z_{N})\dots \Phi (v_{1},z_{1})\right\rangle =0,$
and $L_{-1}$ can be identified with the infinitesimal translation operator ${\frac {\partial }{\partial z}}$.
Mathematical formulation
Since the treatment in Tsuchiya & Kanie (1988), the Knizhnik–Zamolodchikov equation has been formulated mathematically in the language of vertex algebras due to Borcherds (1986) and Frenkel, Lepowsky & Meurman (1988). This approach was popularized amongst theoretical physicists by Goddard (1988) harvtxt error: no target: CITEREFGoddard1988 (help) and amongst mathematicians by Kac (1996) harvtxt error: no target: CITEREFKac1996 (help).
The vacuum representation H0 of an affine Kac–Moody algebra at a fixed level can be encoded in a vertex algebra. The derivation d acts as the energy operator L0 on H0, which can be written as a direct sum of the non-negative integer eigenspaces of L0, the zero energy space being generated by the vacuum vector Ω. The eigenvalue of an eigenvector of L0 is called its energy. For every state a in L there is a vertex operator V(a,z) which creates a from the vacuum vector Ω, in the sense that
$V(a,0)\Omega =a.$
The vertex operators of energy 1 correspond to the generators of the affine algebra
$X(z)=\sum X(n)z^{-n-1}$
where X ranges over the elements of the underlying finite-dimensional simple complex Lie algebra ${\mathfrak {g}}$.
There is an energy 2 eigenvector L−2Ω which give the generators Ln of the Virasoro algebra associated to the Kac–Moody algebra by the Segal–Sugawara construction
$T(z)=\sum L_{n}z^{-n-2}.$
If a has energy α, then the corresponding vertex operator has the form
$V(a,z)=\sum V(a,n)z^{-n-\alpha }.$
The vertex operators satisfy
${\begin{aligned}{\frac {d}{dz}}V(a,z)&=\left[L_{-1},V(a,z)\right]=V\left(L_{-1}a,z\right)\\\left[L_{0},V(a,z)\right]&=\left(z^{-1}{\frac {d}{dz}}+\alpha \right)V(a,z)\end{aligned}}$
as well as the locality and associativity relations
$V(a,z)V(b,w)=V(b,w)V(a,z)=V(V(a,z-w)b,w).$
These last two relations are understood as analytic continuations: the inner products with finite energy vectors of the three expressions define the same polynomials in z±1, w±1 and (z − w)−1 in the domains |z| < |w|, |z| > |w| and |z – w| < |w|. All the structural relations of the Kac–Moody and Virasoro algebra can be recovered from these relations, including the Segal–Sugawara construction.
Every other integral representation Hi at the same level becomes a module for the vertex algebra, in the sense that for each a there is a vertex operator Vi(a, z) on Hi such that
$V_{i}(a,z)V_{i}(b,w)=V_{i}(b,w)V_{i}(a,z)=V_{i}(V(a,z-w)b,w).$
The most general vertex operators at a given level are intertwining operators Φ(v, z) between representations Hi and Hj where v lies in Hk. These operators can also be written as
$\Phi (v,z)=\sum \Phi (v,n)z^{-n-\delta }$
but δ can now be rational numbers. Again these intertwining operators are characterized by properties
$V_{j}(a,z)\Phi (v,w)=\Phi (v,w)V_{i}(a,w)=\Phi \left(V_{k}(a,z-w)v,w\right)$
and relations with L0 and L−1 similar to those above.
When v is in the lowest energy subspace for L0 on Hk, an irreducible representation of ${\mathfrak {g}}$, the operator Φ(v, w) is called a primary field of charge k.
Given a chain of n primary fields starting and ending at H0, their correlation or n-point function is defined by
$\left\langle \Phi (v_{1},z_{1})\Phi (v_{2},z_{2})\cdots \Phi (v_{n},z_{n})\right\rangle =\left(\Phi \left(v_{1},z_{1}\right)\Phi \left(v_{2},z_{2}\right)\cdots \Phi \left(v_{n},z_{n}\right)\Omega ,\Omega \right).$
In the physics literature the vi are often suppressed and the primary field written Φi(zi), with the understanding that it is labelled by the corresponding irreducible representation of ${\mathfrak {g}}$.
Vertex algebra derivation
If (Xs) is an orthonormal basis of ${\mathfrak {g}}$ for the Killing form, the Knizhnik–Zamolodchikov equations may be deduced by integrating the correlation function
$\sum _{s}\left\langle X_{s}(w)X_{s}(z)\Phi (v_{1},z_{1})\cdots \Phi (v_{n},z_{n})\right\rangle (w-z)^{-1}$
first in the w variable around a small circle centred at z; by Cauchy's theorem the result can be expressed as sum of integrals around n small circles centred at the zj's:
${1 \over 2}(k+h)\left\langle T(z)\Phi (v_{1},z_{1})\cdots \Phi (v_{n},z_{n})\right\rangle =-\sum _{j,s}\left\langle X_{s}(z)\Phi (v_{1},z_{1})\cdots \Phi (X_{s}v_{j},z_{j})\cdots \Phi (v_{n},z_{n})\right\rangle (z-z_{j})^{-1}.$
Integrating both sides in the z variable about a small circle centred on zi yields the ith Knizhnik–Zamolodchikov equation.
Lie algebra derivation
It is also possible to deduce the Knizhnik–Zamodchikov equations without explicit use of vertex algebras. The termΦ(vi, zi) may be replaced in the correlation function by its commutator with Lr where r = 0, ±1. The result can be expressed in terms of the derivative with respect to zi. On the other hand, Lr is also given by the Segal–Sugawara formula:
${\begin{aligned}L_{0}&=(k+h)^{-1}\sum _{s}\left[{\frac {1}{2}}X_{s}(0)^{2}+\sum _{m>0}X_{s}(-m)X_{s}(m)\right]\\L_{\pm 1}&=(k+h)^{-1}\sum _{s}\sum _{m\geq 0}X_{s}(-m\pm 1)X_{s}(m)\end{aligned}}$
After substituting these formulas for Lr, the resulting expressions can be simplified using the commutator formulas
$[X(m),\Phi (a,n)]=\Phi (Xa,m+n).$
Original derivation
The original proof of Knizhnik & Zamolodchikov (1984), reproduced in Tsuchiya & Kanie (1988), uses a combination of both of the above methods. First note that for X in ${\mathfrak {g}}$
$\left\langle X(z)\Phi (v_{1},z_{1})\cdots \Phi (v_{n},z_{n})\right\rangle =\sum _{j}\left\langle \Phi (v_{1},z_{1})\cdots \Phi (Xv_{j},z_{j})\cdots \Phi (v_{n},z_{n})\right\rangle (z-z_{j})^{-1}.$
Hence
$\sum _{s}\langle X_{s}(z)\Phi (z_{1},v_{1})\cdots \Phi (X_{s}v_{i},z_{i})\cdots \Phi (v_{n},z_{n})\rangle =\sum _{j}\sum _{s}\langle \cdots \Phi (X_{s}v_{j},z_{j})\cdots \Phi (X_{s}v_{i},z_{i})\cdots \rangle (z-z_{j})^{-1}.$
On the other hand,
$\sum _{s}X_{s}(z)\Phi \left(X_{s}v_{i},z_{i}\right)=(z-z_{i})^{-1}\Phi \left(\sum _{s}X_{s}^{2}v_{i},z_{i}\right)+(k+g){\partial \over \partial z_{i}}\Phi (v_{i},z_{i})+O(z-z_{i})$
so that
$(k+g){\frac {\partial }{\partial z_{i}}}\Phi (v_{i},z_{i})=\lim _{z\to z_{i}}\left[\sum _{s}X_{s}(z)\Phi \left(X_{s}v_{i},z_{i}\right)-(z-z_{i})^{-1}\Phi \left(\sum _{s}X_{s}^{2}v_{i},z_{i}\right)\right].$
The result follows by using this limit in the previous equality.
Monodromy representation of KZ equation
In conformal field theory along the above definition the n-point correlation function of the primary field satisfies KZ equation. In particular, for ${\mathfrak {sl}}_{2}$ and non negative integers k there are $k+1$ primary fields $\Phi _{j}(z_{j})$ 's corresponding to spin j representation ($j=0,1/2,1,3/2,\ldots ,k/2$). The correlation function $\Psi (z_{1},\dots ,z_{n})$ of the primary fields $\Phi _{j}(z_{j})$ 's for the representation $(\rho ,V_{i})$ takes values in the tensor product $V_{1}\otimes \cdots \otimes V_{n}$ and its KZ equation is
$(k+2){\frac {\partial }{\partial z_{i}}}\Psi =\sum _{i,j\neq i}{\frac {\Omega _{ij}}{z_{i}-z_{j}}}\Psi $,
where $\Omega _{ij}=\sum _{a}\rho _{i}(J^{a})\otimes \rho _{j}(J_{a})$ as the above informal derivation.
This n-point correlation function can be analytically continued as multi-valued holomorphic function to the domain $X_{n}\subset \mathbb {C} ^{n}$ with $z_{i}\neq z_{j}$ for $i\neq j$. Due to this analytic continuation, the holonomy of the KZ equation can be described by the braid group $B_{n}$ introduced by Emil Artin.[1] In general, A complex semi-simple Lie algebra ${\mathfrak {g}}$ and its representations $(\rho ,V_{i})$ give the linear representation of braid group
$\theta \colon B_{n}\rightarrow V_{1}\otimes \cdots \otimes V_{n}$
as the holonomy of KZ equation. Oppositely, a KZ equation gives the linear representation of braid groups as its holonomy.
The action on $V_{1}\otimes \dots \otimes V_{n}$ by the analytic continuation of KZ equation is called monodromy representation of KZ equation. In particular, if all $V_{i}$ 's have spin 1/2 representation then the linear representation obtained from KZ equation agrees with the representation constructed from operator algebra theory by Vaughan Jones. It is known that the monodromy representation of KZ equation with a general semi-simple Lie algebra agrees with the linear representation of braid group given by R-matrix of the corresponding quantum group.
Applications
• Representation theory of affine Lie algebra and quantum groups
• Braid groups
• Topology of hyperplane complements
• Knot theory and 3-folds
See also
• Quantum KZ equations
References
1. Kohno 2002
• Baik, Jinho; Deift, Percy; Johansson, Kurt (June 1999), "On the distribution of the length of the longest increasing subsequence of random permutations" (PDF), J. Amer. Math. Soc., 12 (4): 1119–78, doi:10.1090/S0894-0347-99-00307-0, S2CID 11355968
• Knizhnik, V.G.; Zamolodchikov, A.B. (1984), "Current Algebra and Wess–Zumino Model in Two-Dimensions", Nucl. Phys. B, 247 (1): 83–103, Bibcode:1984NuPhB.247...83K, doi:10.1016/0550-3213(84)90374-2
• Tsuchiya, A.; Kanie, Y. (1988), Vertex operators in conformal field theory on P(1) and monodromy representations of braid group, Adv. Stud. Pure Math., vol. 16, pp. 297–372 (Erratum in volume 19, pp. 675–682.)
• Borcherds, Richard (1986), "Vertex algebras, Kac–Moody algebras, and the Monster", Proc. Natl. Acad. Sci. USA, 83 (10): 3068–3071, Bibcode:1986PNAS...83.3068B, doi:10.1073/pnas.83.10.3068, PMC 323452, PMID 16593694
• Frenkel, Igor; Lepowsky, James; Meurman, Arne (1988), Vertex operator algebras and the Monster, Pure and Applied Mathematics, vol. 134, Academic Press, ISBN 0-12-267065-5
• Goddard, Peter (1989), "Meromorphic conformal field theory", in Kac, Victor G. (ed.), Infinite Dimensional Lie Algebras And Groups, Advanced Series In Mathematical Physics, vol. 7, World Scientific, pp. 556–587, ISBN 978-981-4663-17-5
• Kac, Victor (1998), Vertex algebras for beginners, University Lecture Series, vol. 10, American Mathematical Society, ISBN 0-8218-0643-2
• Etingof, Pavel I.; Frenkel, Igor; Kirillov, Alexander A. (1998), Lectures on Representation Theory and Knizhnik–Zamolodchikov Equations, Mathematical Surveys and Monographs, vol. 58, American Mathematical Society, ISBN 0821804960
• Frenkel, Edward; Ben-Zvi, David (2001), Vertex algebras and Algebraic Curves, Mathematical Surveys and Monographs, vol. 88, American Mathematical Society, ISBN 0-8218-2894-0
• Kohno, Toshitake (2002), Conformal Field Theory and Topology, Translation of Mathematical Monographs, vol. 210, American Mathematical Society, ISBN 978-0821821305
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Zappa–Szép product
In mathematics, especially group theory, the Zappa–Szép product (also known as the Zappa–Rédei–Szép product, general product, knit product, exact factorization or bicrossed product) describes a way in which a group can be constructed from two subgroups. It is a generalization of the direct and semidirect products. It is named after Guido Zappa (1940) and Jenő Szép (1950) although it was independently studied by others including B.H. Neumann (1935), G.A. Miller (1935), and J.A. de Séguier (1904).[1]
Internal Zappa–Szép products
Let G be a group with identity element e, and let H and K be subgroups of G. The following statements are equivalent:
• G = HK and H ∩ K = {e}
• For each g in G, there exists a unique h in H and a unique k in K such that g = hk.
If either (and hence both) of these statements hold, then G is said to be an internal Zappa–Szép product of H and K.
Examples
Let G = GL(n,C), the general linear group of invertible n × n matrices over the complex numbers. For each matrix A in G, the QR decomposition asserts that there exists a unique unitary matrix Q and a unique upper triangular matrix R with positive real entries on the main diagonal such that A = QR. Thus G is a Zappa–Szép product of the unitary group U(n) and the group (say) K of upper triangular matrices with positive diagonal entries.
One of the most important examples of this is Philip Hall's 1937 theorem on the existence of Sylow systems for soluble groups. This shows that every soluble group is a Zappa–Szép product of a Hall p'-subgroup and a Sylow p-subgroup, and in fact that the group is a (multiple factor) Zappa–Szép product of a certain set of representatives of its Sylow subgroups.
In 1935, George Miller showed that any non-regular transitive permutation group with a regular subgroup is a Zappa–Szép product of the regular subgroup and a point stabilizer. He gives PSL(2,11) and the alternating group of degree 5 as examples, and of course every alternating group of prime degree is an example. This same paper gives a number of examples of groups which cannot be realized as Zappa–Szép products of proper subgroups, such as the quaternion group and the alternating group of degree 6.
External Zappa–Szép products
As with the direct and semidirect products, there is an external version of the Zappa–Szép product for groups which are not known a priori to be subgroups of a given group. To motivate this, let G = HK be an internal Zappa–Szép product of subgroups H and K of the group G. For each k in K and each h in H, there exist α(k, h) in H and β(k, h) in K such that kh = α(k, h) β(k, h). This defines mappings α : K × H → H and β : K × H → K which turn out to have the following properties:
• α(e, h) = h and β(k, e) = k for all h in H and k in K.
• α(k1k2, h) = α(k1, α(k2, h))
• β(k, h1h2) = β(β(k, h1), h2)
• α(k, h1h2) = α(k, h1) α(β(k, h1), h2)
• β(k1k2, h) = β(k1, α(k2, h)) β(k2, h)
for all h1, h2 in H, k1, k2 in K. From these, it follows that
• For each k in K, the mapping h ↦ α(k, h) is a bijection of H.
• For each h in H, the mapping k ↦ β(k, h) is a bijection of K.
(Indeed, suppose α(k, h1) = α(k, h2). Then h1 = α(k−1k, h1) = α(k−1, α(k, h1)) = α(k−1, α(k, h2)) = h2. This establishes injectivity, and for surjectivity, use h = α(k, α(k−1, h)).)
More concisely, the first three properties above assert the mapping α : K × H → H is a left action of K on (the underlying set of) H and that β : K × H → K is a right action of H on (the underlying set of) K. If we denote the left action by h → kh and the right action by k → kh, then the last two properties amount to k(h1h2) = kh1 kh1h2 and (k1k2)h = k1k2h k2h.
Turning this around, suppose H and K are groups (and let e denote each group's identity element) and suppose there exist mappings α : K × H → H and β : K × H → K satisfying the properties above. On the cartesian product H × K, define a multiplication and an inversion mapping by, respectively,
• (h1, k1) (h2, k2) = (h1 α(k1, h2), β(k1, h2) k2)
• (h, k)−1 = (α(k−1, h−1), β(k−1, h−1))
Then H × K is a group called the external Zappa–Szép product of the groups H and K. The subsets H × {e} and {e} × K are subgroups isomorphic to H and K, respectively, and H × K is, in fact, an internal Zappa–Szép product of H × {e} and {e} × K.
Relation to semidirect and direct products
Let G = HK be an internal Zappa–Szép product of subgroups H and K. If H is normal in G, then the mappings α and β are given by, respectively, α(k,h) = k h k− 1 and β(k, h) = k. This is easy to see because $(h_{1}k_{1})(h_{2}k_{2})=(h_{1}k_{1}h_{2}k_{1}^{-1})(k_{1}k_{2})$ and $h_{1}k_{1}h_{2}k_{1}^{-1}\in H$ since by normality of $H$, $k_{1}h_{2}k_{1}^{-1}\in H$. In this case, G is an internal semidirect product of H and K.
If, in addition, K is normal in G, then α(k,h) = h. In this case, G is an internal direct product of H and K.
References
1. Martin W. Liebeck; Cheryl E. Praeger; Jan Saxl (2010). Regular Subgroups of Primitive Permutation Groups. American Mathematical Soc. pp. 1–2. ISBN 978-0-8218-4654-4.
• Huppert, B. (1967), Endliche Gruppen (in German), Berlin, New York: Springer-Verlag, ISBN 978-3-540-03825-2, MR 0224703, OCLC 527050, Kap. VI, §4.
• Michor, P. W. (1989), "Knit products of graded Lie algebras and groups", Proceedings of the Winter School on Geometry and Physics, Srni, Suppl. Rendiconti Circolo Matematico di Palermo, Ser. II, 22: 171–175, arXiv:math/9204220, Bibcode:1992math......4220M.
• Miller, G. A. (1935), "Groups which are the products of two permutable proper subgroups", Proceedings of the National Academy of Sciences, 21 (7): 469–472, Bibcode:1935PNAS...21..469M, doi:10.1073/pnas.21.7.469, PMC 1076628, PMID 16588002
• Szép, J. (1950), "On the structure of groups which can be represented as the product of two subgroups", Acta Sci. Math. Szeged, 12: 57–61.
• Takeuchi, M. (1981), "Matched pairs of groups and bismash products of Hopf algebras", Comm. Algebra, 9 (8): 841–882, doi:10.1080/00927878108822621.
• Zappa, G. (1940), "Sulla costruzione dei gruppi prodotto di due dati sottogruppi permutabili traloro", Atti Secondo Congresso Un. Mat. Ital., Bologna{{citation}}: CS1 maint: location missing publisher (link); Edizioni Cremonense, Rome, (1942) 119–125.
• Agore, A.L.; Chirvasitu, A.; Ion, B.; Militaru, G. (2007), Factorization problems for finite groups, arXiv:math/0703471, Bibcode:2007math......3471A, doi:10.1007/s10468-009-9145-6, S2CID 18024087.
• Brin, M. G. (2005). "On the Zappa-Szép Product". Communications in Algebra. 33 (2): 393–424. arXiv:math/0406044. doi:10.1081/AGB-200047404. S2CID 15169734.
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Turán's brick factory problem
Unsolved problem in mathematics:
Can any complete bipartite graph be drawn with fewer crossings than the number given by Zarankiewicz?
(more unsolved problems in mathematics)
In the mathematics of graph drawing, Turán's brick factory problem asks for the minimum number of crossings in a drawing of a complete bipartite graph. The problem is named after Pál Turán, who formulated it while being forced to work in a brick factory during World War II.[1]
A drawing method found by Kazimierz Zarankiewicz has been conjectured to give the correct answer for every complete bipartite graph, and the statement that this is true has come to be known as the Zarankiewicz crossing number conjecture. The conjecture remains open, with only some special cases solved.[2]
Origin and formulation
During World War II, Hungarian mathematician Pál Turán was forced to work in a brick factory, pushing wagon loads of bricks from kilns to storage sites. The factory had tracks from each kiln to each storage site, and the wagons were harder to push at the points where tracks crossed each other. Turán was inspired by this situation to ask how the factory might be redesigned to minimize the number of crossings between these tracks.[1]
Mathematically, this problem can be formalized as asking for a graph drawing of a complete bipartite graph, whose vertices represent kilns and storage sites, and whose edges represent the tracks from each kiln to each storage site. The graph should be drawn in the plane with each vertex as a point, each edge as a curve connecting its two endpoints, and no vertex placed on an edge that it is not incident to. A crossing is counted whenever two edges that are disjoint in the graph have a nonempty intersection in the plane. The question is then, what is the minimum number of crossings in such a drawing?[2][3]
Turán's formulation of this problem is often recognized as one of the first studies of the crossing numbers of graphs.[4] (Another independent formulation of the same concept occurred in sociology, in methods for drawing sociograms,[5] and a much older puzzle, the three utilities problem, can be seen as a special case of the brick factory problem with three kilns and three storage facilities.[6]) Crossing numbers have since gained greater importance, as a central object of study in graph drawing[7] and as an important tool in VLSI design[8] and discrete geometry.[9]
Upper bound
Both Zarankiewicz and Kazimierz Urbanik saw Turán speak about the brick factory problem in different talks in Poland in 1952,[3] and independently published attempted solutions of the problem, with equivalent formulas for the number of crossings.[10][11] As both of them showed, it is always possible to draw the complete bipartite graph Km,n (a graph with m vertices on one side, n vertices on the other side, and mn edges connecting the two sides) with a number of crossings equal to
$\operatorname {cr} (K_{m,n})\leq {\biggl \lfloor }{\frac {n}{2}}{\biggr \rfloor }{\biggl \lfloor }{\frac {n-1}{2}}{\biggr \rfloor }{\biggl \lfloor }{\frac {m}{2}}{\biggr \rfloor }{\biggl \lfloor }{\frac {m-1}{2}}{\biggr \rfloor }.$
The construction is simple: place m vertices on the x-axis of the plane, avoiding the origin, with equal or nearly-equal numbers of points to the left and right of the y-axis. Similarly, place n vertices on the y-axis of the plane, avoiding the origin, with equal or nearly-equal numbers of points above and below the x-axis. Then, connect every point on the x-axis by a straight line segment to every point on the y-axis.[3]
However, their proofs that this formula is optimal, that is, that there can be no drawings with fewer crossings, were erroneous. The gap was not discovered until eleven years after publication, nearly simultaneously by Gerhard Ringel and Paul Kainen.[12] Nevertheless, it is conjectured that Zarankiewicz's and Urbanik's formula is optimal. This has come to be known as the Zarankiewicz crossing number conjecture. Although some special cases of it are known to be true, the general case remains open.[2]
Lower bounds
Since Km,n and Kn,m are isomorphic, it is enough to consider the case where m ≤ n. In addition, for m ≤ 2 Zarankiewicz's construction gives no crossings, which of course cannot be bested. So the only nontrivial cases are those for which m and n are both ≥ 3.
Zarankiewicz's attempted proof of the conjecture, although invalid for the general case of Km,n, works for the case m = 3. It has since been extended to other small values of m, and the Zarankiewicz conjecture is known to be true for the complete bipartite graphs Km,n with m ≤ 6.[13] The conjecture is also known to be true for K7,7, K7,8, and K7,9.[14] If a counterexample exists, that is, a graph Km,n requiring fewer crossings than the Zarankiewicz bound, then in the smallest counterexample both m and n must be odd.[13]
For each fixed choice of m, the truth of the conjecture for all Km,n can be verified by testing only a finite number of choices of n.[15] More generally, it has been proven that every complete bipartite graph requires a number of crossings that is (for sufficiently large graphs) at least 83% of the number given by the Zarankiewicz bound. Closing the gap between this lower bound and the upper bound remains an open problem.[16]
Rectilinear crossing numbers
If edges are required to be drawn as straight line segments, rather than arbitrary curves, then some graphs need more crossings than they would when drawn with curved edges. However, the upper bound established by Zarankiewicz for the crossing numbers of complete bipartite graphs can be achieved using only straight edges. Therefore, if the Zarankiewicz conjecture is correct, then the complete bipartite graphs have rectilinear crossing numbers equal to their crossing numbers.[17]
References
1. Turán, P. (1977), "A note of welcome", Journal of Graph Theory, 1: 7–9, doi:10.1002/jgt.3190010105.
2. Pach, János; Sharir, Micha (2009), "5.1 Crossings—the Brick Factory Problem", Combinatorial Geometry and Its Algorithmic Applications: The Alcalá Lectures, Mathematical Surveys and Monographs, vol. 152, American Mathematical Society, pp. 126–127.
3. Beineke, Lowell; Wilson, Robin (2010), "The early history of the brick factory problem", The Mathematical Intelligencer, 32 (2): 41–48, doi:10.1007/s00283-009-9120-4, MR 2657999, S2CID 122588849.
4. Foulds, L. R. (1992), Graph Theory Applications, Universitext, Springer, p. 71, ISBN 9781461209331.
5. Bronfenbrenner, Urie (1944), "The graphic presentation of sociometric data", Sociometry, 7 (3): 283–289, doi:10.2307/2785096, JSTOR 2785096, The arrangement of subjects on the diagram, while haphazard in part, is determined largely by trial and error with the aim of minimizing the number of intersecting lines.
6. Bóna, Miklós (2011), A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory, World Scientific, pp. 275–277, ISBN 9789814335232. Bóna introduces the puzzle (in the form of three houses to be connected to three wells) on p. 275, and writes on p. 277 that it "is equivalent to the problem of drawing K3,3 on a plane surface without crossings".
7. Schaefer, Marcus (2014), "The graph crossing number and its variants: a survey", The Electronic Journal of Combinatorics: #DS21
8. Leighton, T. (1983), Complexity Issues in VLSI, Foundations of Computing Series, Cambridge, MA: MIT Press
9. Székely, L. A. (1997), "Crossing numbers and hard Erdős problems in discrete geometry", Combinatorics, Probability and Computing, 6 (3): 353–358, doi:10.1017/S0963548397002976, MR 1464571, S2CID 36602807
10. Zarankiewicz, K. (1954), "On a problem of P. Turan concerning graphs", Fundamenta Mathematicae, 41: 137–145, doi:10.4064/fm-41-1-137-145, MR 0063641.
11. Urbaník, K. (1955), "Solution du problème posé par P. Turán", Colloq. Math., 3: 200–201. As cited by Székely, László A. (2001) [1994], "Zarankiewicz crossing number conjecture", Encyclopedia of Mathematics, EMS Press
12. Guy, Richard K. (1969), "The decline and fall of Zarankiewicz's theorem", Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968), Academic Press, New York, pp. 63–69, MR 0253931.
13. Kleitman, Daniel J. (1970), "The crossing number of K5,n", Journal of Combinatorial Theory, 9 (4): 315–323, doi:10.1016/s0021-9800(70)80087-4, MR 0280403.
14. Woodall, D. R. (1993), "Cyclic-order graphs and Zarankiewicz's crossing-number conjecture", Journal of Graph Theory, 17 (6): 657–671, doi:10.1002/jgt.3190170602, MR 1244681.
15. Christian, Robin; Richter, R. Bruce; Salazar, Gelasio (2013), "Zarankiewicz's conjecture is finite for each fixed m", Journal of Combinatorial Theory, Series B, 103 (2): 237–247, doi:10.1016/j.jctb.2012.11.001, MR 3018068.
16. de Klerk, E.; Maharry, J.; Pasechnik, D. V.; Richter, R. B.; Salazar, G. (2006), "Improved bounds for the crossing numbers of Km,n and Kn", SIAM Journal on Discrete Mathematics, 20 (1): 189–202, arXiv:math/0404142, doi:10.1137/S0895480104442741, MR 2257255, S2CID 1509054.
17. Kainen, Paul C. (1968), "On a problem of P. Erdős", Journal of Combinatorial Theory, 5 (4): 374–377, doi:10.1016/s0021-9800(68)80013-4, MR 0231744
External links
• Weisstein, Eric W., "Zarankiewicz's Conjecture", MathWorld
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Zarhin trick
In mathematics, the Zarhin trick is a method for eliminating the polarization of abelian varieties A by observing that the abelian variety A4 × Â4 is principally polarized. The method was introduced by Zarhin (1974) in his proof of the Tate conjecture over global fields of positive characteristic.
References
• Zarhin, Ju. G. (1974), "A remark on endomorphisms of abelian varieties over function fields of finite characteristic", Mathematics of the USSR-Izvestiya, 8 (3): 477–480, doi:10.1070/IM1974v008n03ABEH002115, ISSN 0373-2436, MR 0354689
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Oscar Zariski
Oscar Zariski (April 24, 1899 – July 4, 1986) was a Russian-born American mathematician and one of the most influential algebraic geometers of the 20th century.
Oscar Zariski
Oscar Zariski (1899–1986)
Born
Russian: О́скар Зари́сский
(1899-04-24)April 24, 1899
Kobrin, Russian Empire
DiedJuly 4, 1986(1986-07-04) (aged 87)
Brookline, Massachusetts, United States
NationalityAmerican
Alma materUniversity of Kyiv
University of Rome
Known forContributions to algebraic geometry
AwardsCole Prize in Algebra (1944)
National Medal of Science (1965)
Wolf Prize (1981)
Steele Prize (1981)
Scientific career
FieldsMathematics
InstitutionsJohns Hopkins University
University of Illinois
Harvard University
Doctoral advisorGuido Castelnuovo
Doctoral studentsS. S. Abhyankar
Michael Artin
Iacopo Barsotti
Irvin Cohen
Daniel Gorenstein
Robin Hartshorne
Heisuke Hironaka
Steven Kleiman
Joseph Lipman
David Mumford
Maxwell Rosenlicht
Pierre Samuel
Abraham Seidenberg
Education
Zariski was born Oscher (also transliterated as Ascher or Osher) Zaritsky to a Jewish family (his parents were Bezalel Zaritsky and Hanna Tennenbaum) and in 1918 studied at the University of Kyiv. He left Kyiv in 1920 to study at the University of Rome where he became a disciple of the Italian school of algebraic geometry, studying with Guido Castelnuovo, Federigo Enriques and Francesco Severi.
Zariski wrote a doctoral dissertation in 1924 on a topic in Galois theory, which was proposed to him by Castelnuovo. At the time of his dissertation publication, he changed his name to Oscar Zariski.
Johns Hopkins University years
Zariski emigrated to the United States in 1927 supported by Solomon Lefschetz. He had a position at Johns Hopkins University where he became professor in 1937. During this period, he wrote Algebraic Surfaces as a summation of the work of the Italian school. The book was published in 1935 and reissued 36 years later, with detailed notes by Zariski's students that illustrated how the field of algebraic geometry had changed. It is still an important reference.
It seems to have been this work that set the seal of Zariski's discontent with the approach of the Italians to birational geometry. He addressed the question of rigour by recourse to commutative algebra. The Zariski topology, as it was later known, is adequate for biregular geometry, where varieties are mapped by polynomial functions. That theory is too limited for algebraic surfaces, and even for curves with singular points. A rational map is to a regular map as a rational function is to a polynomial: it may be indeterminate at some points. In geometric terms, one has to work with functions defined on some open, dense set of a given variety. The description of the behaviour on the complement may require infinitely near points to be introduced to account for limiting behaviour along different directions. This introduces a need, in the surface case, to use also valuation theory to describe the phenomena such as blowing up (balloon-style, rather than explosively).
Harvard University years
After spending a year 1946–1947 at the University of Illinois at Urbana–Champaign, Zariski became professor at Harvard University in 1947 where he remained until his retirement in 1969. In 1945, he fruitfully discussed foundational matters for algebraic geometry with André Weil. Weil's interest was in putting an abstract variety theory in place, to support the use of the Jacobian variety in his proof of the Riemann hypothesis for curves over finite fields, a direction rather oblique to Zariski's interests. The two sets of foundations weren't reconciled at that point.
At Harvard, Zariski's students included Shreeram Abhyankar, Heisuke Hironaka, David Mumford, Michael Artin and Steven Kleiman—thus spanning the main areas of advance in singularity theory, moduli theory and cohomology in the next generation. Zariski himself worked on equisingularity theory. Some of his major results, Zariski's main theorem and the Zariski theorem on holomorphic functions, were amongst the results generalized and included in the programme of Alexander Grothendieck that ultimately unified algebraic geometry.
Zariski proposed the first example of a Zariski surface in 1958.
Views
Zariski was a Jewish atheist.[1]
Awards and recognition
Zariski was elected to the United States National Academy of Sciences in 1944,[2] the American Academy of Arts and Sciences in 1948,[3] and the American Philosophical Society in 1951.[4] Zariski was awarded the Steele Prize in 1981, and in the same year the Wolf Prize in Mathematics with Lars Ahlfors. He wrote also Commutative Algebra in two volumes, with Pierre Samuel. His papers have been published by MIT Press, in four volumes. In 1997 a conference was held in his honor in Obergurgl, Austria.[5][6]
Publications
• Zariski, Oscar (2004) [1935], Abhyankar, Shreeram S.; Lipman, Joseph; Mumford, David (eds.), Algebraic surfaces, Classics in mathematics (second supplemented ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-58658-6, MR 0469915[7]
• Zariski, Oscar (1958), Introduction to the problem of minimal models in the theory of algebraic surfaces, Publications of the Mathematical Society of Japan, vol. 4, The Mathematical Society of Japan, Tokyo, MR 0097403
• Zariski, Oscar (1969) [1958], Cohn, James (ed.), An introduction to the theory of algebraic surfaces, Lecture notes in mathematics, vol. 83, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0082246, ISBN 978-3-540-04602-8, MR 0263819
• Zariski, Oscar; Samuel, Pierre (1975) [1958], Commutative algebra I, Graduate Texts in Mathematics, vol. 28, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90089-6, MR 0090581[8]
• Zariski, Oscar; Samuel, Pierre (1975) [1960], Commutative algebra. Vol. II, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90171-8, MR 0389876[9]
• Zariski, Oscar (2006) [1973], Kmety, François; Merle, Michel; Lichtin, Ben (eds.), The moduli problem for plane branches, University Lecture Series, vol. 39, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2983-7, MR 0414561(original title): Le problème des modules pour les branches planes{{citation}}: CS1 maint: postscript (link)[10]
• Zariski, Oscar (1972), Collected papers. Vol. I: Foundations of algebraic geometry and resolution of singularities, Cambridge, Massachusetts-London: MIT Press, ISBN 978-0-262-08049-1, MR 0505100
• Zariski, Oscar (1973), Collected papers. Vol. II: Holomorphic functions and linear systems, Mathematicians of Our Time, Cambridge, Massachusetts-London: MIT Press, ISBN 978-0-262-01038-2, MR 0505100
• Zariski, Oscar (1978), Artin, Michael; Mazur, Barry (eds.), Collected papers. Volume III. Topology of curves and surfaces, and special topics in the theory of algebraic varieties, Mathematicians of Our Time, Cambridge, Massachusetts-London: MIT Press, ISBN 978-0-262-24021-5, MR 0505104
• Zariski, Oscar (1979), Lipman, Joseph; Teissier, Bernard (eds.), Collected papers. Vol. IV. Equisingularity on algebraic varieties, Mathematicians of Our Time, vol. 16, MIT Press, ISBN 978-0-262-08049-1, MR 0545653
See also
• Zariski ring
• Zariski tangent space
• Zariski surface
• Zariski topology
• Zariski–Riemann surface
• Zariski space (disambiguation)
• Zariski's lemma
• Zariski's main theorem
Notes
1. Carol Parikh (2008). The Unreal Life of Oscar Zariski. Springer. p. 5. ISBN 9780387094298. And yet it did, even though since moving into the boarding house he had become an atheist and most of his friends, including his best friend, were Russians.
2. "Oscar Zariski". www.nasonline.org. Retrieved 2023-02-16.
3. "Oscar Zariski". American Academy of Arts & Sciences. 9 February 2023. Retrieved 2023-02-16.
4. "APS Member History". search.amphilsoc.org. Retrieved 2023-02-16.
5. Herwig Hauser; Joseph Lipman; Frans Oort; Adolfo Quirós (14 February 2000). Resolution of Singularities: A research textbook in tribute to Oscar Zariski Based on the courses given at the Working Week in Obergurgl, Austria, September 7–14, 1997. Springer Science & Business Media. ISBN 978-3-7643-6178-5.
6. Bogomolov, Fedor; Tschinkel, Yuri (2001). "Book Review: Alterations and resolution of singularities". Bulletin of the American Mathematical Society. 39 (1): 95–101. doi:10.1090/S0273-0979-01-00922-3. ISSN 0273-0979.
7. Lefschetz, Solomon (1936). "Review: Algebraic Surfaces, by Oscar Zariski" (PDF). Bulletin of the American Mathematical Society. 42 (1, Part 2): 13–14. doi:10.1090/s0002-9904-1936-06238-5.
8. Herstein, I. N. (1959). "Review: Commutative algebra, Vol. 1, by Oscar Zariski and Pierre Samuel" (PDF). Bull. Amer. Math. Soc. 6 (1): 26–30. doi:10.1090/S0002-9904-1959-10267-6.
9. Auslander, M. (1962). "Review: Commutative algebra, Vol. II, by O. Zariski and P. Samuel" (PDF). Bull. Amer. Math. Soc. 68 (1): 12–13. doi:10.1090/s0002-9904-1962-10674-0.
10. Washburn, Sherwood (1988). "Review: Le problème des modules pour les branches planes, by Oscar Zariski, with an appendix by Bernard Teissier" (PDF). Bull. Amer. Math. Soc. (N.S.). 18 (2): 209–214. doi:10.1090/s0273-0979-1988-15651-0.
References
• Blass, Piotr (2013), "The influence of Oscar Zariski on algebraic geometry" (PDF), Notices of the American Mathematical Society
• Mumford, David (1986), "Oscar Zariski: 1899–1986" (PDF), Notices of the American Mathematical Society, 33 (6): 891–894, ISSN 0002-9920, MR 0860889
• Parikh, Carol (2009) [1991], The Unreal Life of Oscar Zariski, Springer, ISBN 9780387094304, MR 1086628
• Gouvêa, Fernando Q. (1 January 2009). "Review of The Unreal Life of Oscar Zariski by Carol Parikh". MAA Reviews, Mathematical Association of America, maa.org.
External links
Wikiquote has quotations related to Oscar Zariski.
• O'Connor, John J.; Robertson, Edmund F., "Oscar Zariski", MacTutor History of Mathematics Archive, University of St Andrews
• Oscar Zariski at the Mathematics Genealogy Project
• Biography from United States Naval Academy.
Laureates of the Wolf Prize in Mathematics
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Zariski's finiteness theorem
In algebra, Zariski's finiteness theorem gives a positive answer to Hilbert's 14th problem for the polynomial ring in two variables, as a special case.[1] Precisely, it states:
Given a normal domain A, finitely generated as an algebra over a field k, if L is a subfield of the field of fractions of A containing k such that $\operatorname {tr.deg} _{k}(L)\leq 2$, then the k-subalgebra $L\cap A$ is finitely generated.
References
1. http://aix1.uottawa.ca/~ddaigle/articles/H14survey.pdf
• Zariski, O. (1954). "Interprétations algébrico-géométriques du quatorzième problème de Hilbert". Bull. Sci. Math. (2). 78: 155–168.
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Zariski's lemma
In algebra, Zariski's lemma, proved by Oscar Zariski (1947), states that, if a field K is finitely generated as an associative algebra over another field k, then K is a finite field extension of k (that is, it is also finitely generated as a vector space).
An important application of the lemma is a proof of the weak form of Hilbert's Nullstellensatz:[1] if I is a proper ideal of $k[t_{1},...,t_{n}]$ (k algebraically closed field), then I has a zero; i.e., there is a point x in $k^{n}$ such that $f(x)=0$ for all f in I. (Proof: replacing I by a maximal ideal ${\mathfrak {m}}$, we can assume $I={\mathfrak {m}}$ is maximal. Let $A=k[t_{1},...,t_{n}]$ and $\phi :A\to A/{\mathfrak {m}}$ be the natural surjection. By the lemma $A/{\mathfrak {m}}$ is a finite extension. Since k is algebraically closed that extension must be k. Then for any $f\in {\mathfrak {m}}$,
$f(\phi (t_{1}),\cdots ,\phi (t_{n}))=\phi (f(t_{1},\cdots ,t_{n}))=0$;
that is to say, $x=(\phi (t_{1}),\cdots ,\phi (t_{n}))$ is a zero of ${\mathfrak {m}}$.)
The lemma may also be understood from the following perspective. In general, a ring R is a Jacobson ring if and only if every finitely generated R-algebra that is a field is finite over R.[2] Thus, the lemma follows from the fact that a field is a Jacobson ring.
Proofs
Two direct proofs, one of which is due to Zariski, are given in Atiyah–MacDonald.[3][4] For Zariski's original proof, see the original paper.[5] Another direct proof in the language of Jacobson rings is given below. The lemma is also a consequence of the Noether normalization lemma. Indeed, by the normalization lemma, K is a finite module over the polynomial ring $k[x_{1},\ldots ,x_{d}]$ where $x_{1},\ldots ,x_{d}$ are elements of K that are algebraically independent over k. But since K has Krull dimension zero and since an integral ring extension (e.g., a finite ring extension) preserves Krull dimensions, the polynomial ring must have dimension zero; i.e., $d=0$.
The following characterization of a Jacobson ring contains Zariski's lemma as a special case. Recall that a ring is a Jacobson ring if every prime ideal is an intersection of maximal ideals. (When A is a field, A is a Jacobson ring and the theorem below is precisely Zariski's lemma.)
Theorem — [2] Let A be a ring. Then the following are equivalent.
1. A is a Jacobson ring.
2. Every finitely generated A-algebra B that is a field is finite over A.
Proof: 2. $\Rightarrow $ 1.: Let ${\mathfrak {p}}$ be a prime ideal of A and set $B=A/{\mathfrak {p}}$. We need to show the Jacobson radical of B is zero. For that end, let f be a nonzero element of B. Let ${\mathfrak {m}}$ be a maximal ideal of the localization $B[f^{-1}]$. Then $B[f^{-1}]/{\mathfrak {m}}$ is a field that is a finitely generated A-algebra and so is finite over A by assumption; thus it is finite over $B=A/{\mathfrak {p}}$ and so is finite over the subring $B/{\mathfrak {q}}$ where ${\mathfrak {q}}={\mathfrak {m}}\cap B$. By integrality, ${\mathfrak {q}}$ is a maximal ideal not containing f.
1. $\Rightarrow $ 2.: Since a factor ring of a Jacobson ring is Jacobson, we can assume B contains A as a subring. Then the assertion is a consequence of the next algebraic fact:
(*) Let $B\supset A$ be integral domains such that B is finitely generated as A-algebra. Then there exists a nonzero a in A such that every ring homomorphism $\phi :A\to K$, K an algebraically closed field, with $\phi (a)\neq 0$ extends to ${\widetilde {\phi }}:B\to K$.
Indeed, choose a maximal ideal ${\mathfrak {m}}$ of A not containing a. Writing K for some algebraic closure of $A/{\mathfrak {m}}$, the canonical map $\phi :A\to A/{\mathfrak {m}}\hookrightarrow K$ extends to ${\widetilde {\phi }}:B\to K$. Since B is a field, ${\widetilde {\phi }}$ is injective and so B is algebraic (thus finite algebraic) over $A/{\mathfrak {m}}$. We now prove (*). If B contains an element that is transcendental over A, then it contains a polynomial ring over A to which φ extends (without a requirement on a) and so we can assume B is algebraic over A (by Zorn's lemma, say). Let $x_{1},\dots ,x_{r}$ be the generators of B as A-algebra. Then each $x_{i}$ satisfies the relation
$a_{i0}x_{i}^{n}+a_{i1}x_{i}^{n-1}+\dots +a_{in}=0,\,\,a_{ij}\in A$
where n depends on i and $a_{i0}\neq 0$. Set $a=a_{10}a_{20}\dots a_{r0}$. Then $B[a^{-1}]$ is integral over $A[a^{-1}]$. Now given $\phi :A\to K$, we first extend it to ${\widetilde {\phi }}:A[a^{-1}]\to K$ by setting ${\widetilde {\phi }}(a^{-1})=\phi (a)^{-1}$. Next, let ${\mathfrak {m}}=\operatorname {ker} {\widetilde {\phi }}$. By integrality, ${\mathfrak {m}}={\mathfrak {n}}\cap A[a^{-1}]$ for some maximal ideal ${\mathfrak {n}}$ of $B[a^{-1}]$. Then ${\widetilde {\phi }}:A[a^{-1}]\to A[a^{-1}]/{\mathfrak {m}}\to K$ extends to $B[a^{-1}]\to B[a^{-1}]/{\mathfrak {n}}\to K$. Restrict the last map to B to finish the proof. $\square $
Notes
1. Milne 2017, Theorem 2.12.
2. Atiyah & MacDonald 1969, Ch 5. Exercise 25.
3. Atiyah & MacDonald 1969, Ch 5. Exercise 18.
4. Atiyah & MacDonald 1969, Proposition 7.9.
5. Zariski 1947, pp. 362–368.
Sources
• Atiyah, Michael; MacDonald, Ian G. (1969). Introduction to Commutative Algebra. Addison-Wesley Series in Mathematics. Addison–Wesley. ISBN 0-201-40751-5.
• Milne, James (19 March 2017). "Algebraic Geometry". Retrieved 1 February 2022.
• Zariski, Oscar (April 1947). "A new proof of Hilbert's Nullstellensatz". Bulletin of the American Mathematical Society. 53 (4): 362–368. doi:10.1090/s0002-9904-1947-08801-7. MR 0020075.
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Correspondence (algebraic geometry)
In algebraic geometry, a correspondence between algebraic varieties V and W is a subset R of V×W, that is closed in the Zariski topology. In set theory, a subset of a Cartesian product of two sets is called a binary relation or correspondence; thus, a correspondence here is a relation that is defined by algebraic equations. There are some important examples, even when V and W are algebraic curves: for example the Hecke operators of modular form theory may be considered as correspondences of modular curves.
However, the definition of a correspondence in algebraic geometry is not completely standard. For instance, Fulton, in his book on intersection theory,[1] uses the definition above. In literature, however, a correspondence from a variety X to a variety Y is often taken to be a subset Z of X×Y such that Z is finite and surjective over each component of X. Note the asymmetry in this latter definition; which talks about a correspondence from X to Y rather than a correspondence between X and Y. The typical example of the latter kind of correspondence is the graph of a function f:X→Y. Correspondences also play an important role in the construction of motives (cf. presheaf with transfers).[2]
See also
• Adequate equivalence relation
References
1. Fulton, William (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98549-7, MR 1644323
2. Mazza, Carlo; Voevodsky, Vladimir; Weibel, Charles (2006), Lecture notes on motivic cohomology, Clay Mathematics Monographs, vol. 2, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3847-1, MR 2242284
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Zariski geometry
In mathematics, a Zariski geometry consists of an abstract structure introduced by Ehud Hrushovski and Boris Zilber, in order to give a characterisation of the Zariski topology on an algebraic curve, and all its powers. The Zariski topology on a product of algebraic varieties is very rarely the product topology, but richer in closed sets defined by equations that mix two sets of variables. The result described gives that a very definite meaning, applying to projective curves and compact Riemann surfaces in particular.
Definition
A Zariski geometry consists of a set X and a topological structure on each of the sets
X, X2, X3, ...
satisfying certain axioms.
(N) Each of the Xn is a Noetherian topological space, of dimension at most n.
Some standard terminology for Noetherian spaces will now be assumed.
(A) In each Xn, the subsets defined by equality in an n-tuple are closed. The mappings
Xm → Xn
defined by projecting out certain coordinates and setting others as constants are all continuous.
(B) For a projection
p: Xm → Xn
and an irreducible closed subset Y of Xm, p(Y) lies between its closure Z and Z \ Z′ where Z′ is a proper closed subset of Z. (This is quantifier elimination, at an abstract level.)
(C) X is irreducible.
(D) There is a uniform bound on the number of elements of a fiber in a projection of any closed set in Xm, other than the cases where the fiber is X.
(E) A closed irreducible subset of Xm, of dimension r, when intersected with a diagonal subset in which s coordinates are set equal, has all components of dimension at least r − s + 1.
The further condition required is called very ample (cf. very ample line bundle). It is assumed there is an irreducible closed subset P of some Xm, and an irreducible closed subset Q of P× X2, with the following properties:
(I) Given pairs (x, y), (x′, y′) in X2, for some t in P, the set of (t, u, v) in Q includes (t, x, y) but not (t, x′, y′)
(J) For t outside a proper closed subset of P, the set of (x, y) in X2, (t, x, y) in Q is an irreducible closed set of dimension 1.
(K) For all pairs (x, y), (x′, y′) in X2, selected from outside a proper closed subset, there is some t in P such that the set of (t, u, v) in Q includes (t, x, y) and (t, x′, y′).
Geometrically this says there are enough curves to separate points (I), and to connect points (K); and that such curves can be taken from a single parametric family.
Then Hrushovski and Zilber prove that under these conditions there is an algebraically closed field K, and a non-singular algebraic curve C, such that its Zariski geometry of powers and their Zariski topology is isomorphic to the given one. In short, the geometry can be algebraized.
References
• Hrushovski, Ehud; Zilber, Boris (1996). "Zariski Geometries" (PDF). Journal of the American Mathematical Society. 9 (1): 1–56. doi:10.1090/S0894-0347-96-00180-4.
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Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in real or complex analysis; in particular, it is not Hausdorff.[1] This topology was introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring (called the spectrum of the ring) a topological space.
The Zariski topology allows tools from topology to be used to study algebraic varieties, even when the underlying field is not a topological field. This is one of the basic ideas of scheme theory, which allows one to build general algebraic varieties by gluing together affine varieties in a way similar to that in manifold theory, where manifolds are built by gluing together charts, which are open subsets of real affine spaces.
The Zariski topology of an algebraic variety is the topology whose closed sets are the algebraic subsets of the variety.[1] In the case of an algebraic variety over the complex numbers, the Zariski topology is thus coarser than the usual topology, as every algebraic set is closed for the usual topology.
The generalization of the Zariski topology to the set of prime ideals of a commutative ring follows from Hilbert's Nullstellensatz, that establishes a bijective correspondence between the points of an affine variety defined over an algebraically closed field and the maximal ideals of the ring of its regular functions. This suggests defining the Zariski topology on the set of the maximal ideals of a commutative ring as the topology such that a set of maximal ideals is closed if and only if it is the set of all maximal ideals that contain a given ideal. Another basic idea of Grothendieck's scheme theory is to consider as points, not only the usual points corresponding to maximal ideals, but also all (irreducible) algebraic varieties, which correspond to prime ideals. Thus the Zariski topology on the set of prime ideals (spectrum) of a commutative ring is the topology such that a set of prime ideals is closed if and only if it is the set of all prime ideals that contain a fixed ideal.
Zariski topology of varieties
In classical algebraic geometry (that is, the part of algebraic geometry in which one does not use schemes, which were introduced by Grothendieck around 1960), the Zariski topology is defined on algebraic varieties.[2] The Zariski topology, defined on the points of the variety, is the topology such that the closed sets are the algebraic subsets of the variety. As the most elementary algebraic varieties are affine and projective varieties, it is useful to make this definition more explicit in both cases. We assume that we are working over a fixed, algebraically closed field k (in classical algebraic geometry, k is usually the field of complex numbers).
Affine varieties
First, we define the topology on the affine space $\mathbb {A} ^{n},$ formed by the n-tuples of elements of k. The topology is defined by specifying its closed sets, rather than its open sets, and these are taken simply to be all the algebraic sets in $\mathbb {A} ^{n}.$ That is, the closed sets are those of the form
$V(S)=\{x\in \mathbb {A} ^{n}\mid f(x)=0,\forall f\in S\}$
where S is any set of polynomials in n variables over k. It is a straightforward verification to show that:
• V(S) = V((S)), where (S) is the ideal generated by the elements of S;
• For any two ideals of polynomials I, J, we have
1. $V(I)\cup V(J)\,=\,V(IJ);$
2. $V(I)\cap V(J)\,=\,V(I+J).$
It follows that finite unions and arbitrary intersections of the sets V(S) are also of this form, so that these sets form the closed sets of a topology (equivalently, their complements, denoted D(S) and called principal open sets, form the topology itself). This is the Zariski topology on $\mathbb {A} ^{n}.$
If X is an affine algebraic set (irreducible or not) then the Zariski topology on it is defined simply to be the subspace topology induced by its inclusion into some $\mathbb {A} ^{n}.$ Equivalently, it can be checked that:
• The elements of the affine coordinate ring
$A(X)\,=\,k[x_{1},\dots ,x_{n}]/I(X)$
act as functions on X just as the elements of $k[x_{1},\dots ,x_{n}]$ act as functions on $\mathbb {A} ^{n}$; here, I(X) is the ideal of all polynomials vanishing on X.
• For any set of polynomials S, let T be the set of their images in A(X). Then the subset of X
$V'(T)=\{x\in X\mid f(x)=0,\forall f\in T\}$
(these notations are not standard) is equal to the intersection with X of V(S).
This establishes that the above equation, clearly a generalization of the definition of the closed sets in $\mathbb {A} ^{n}$ above, defines the Zariski topology on any affine variety.
Projective varieties
Recall that n-dimensional projective space $\mathbb {P} ^{n}$ is defined to be the set of equivalence classes of non-zero points in $\mathbb {A} ^{n+1}$ by identifying two points that differ by a scalar multiple in k. The elements of the polynomial ring $k[x_{0},\dots ,x_{n}]$ are not functions on $\mathbb {P} ^{n}$ because any point has many representatives that yield different values in a polynomial; however, for homogeneous polynomials the condition of having zero or nonzero value on any given projective point is well-defined since the scalar multiple factors out of the polynomial. Therefore, if S is any set of homogeneous polynomials we may reasonably speak of
$V(S)=\{x\in \mathbb {P} ^{n}\mid f(x)=0,\forall f\in S\}.$
The same facts as above may be established for these sets, except that the word "ideal" must be replaced by the phrase "homogeneous ideal", so that the V(S), for sets S of homogeneous polynomials, define a topology on $\mathbb {P} ^{n}.$ As above the complements of these sets are denoted D(S), or, if confusion is likely to result, D′(S).
The projective Zariski topology is defined for projective algebraic sets just as the affine one is defined for affine algebraic sets, by taking the subspace topology. Similarly, it may be shown that this topology is defined intrinsically by sets of elements of the projective coordinate ring, by the same formula as above.
Properties
An important property of Zariski topologies is that they have a base consisting of simple elements, namely the D(f) for individual polynomials (or for projective varieties, homogeneous polynomials) f. That these form a basis follows from the formula for the intersection of two Zariski-closed sets given above (apply it repeatedly to the principal ideals generated by the generators of (S)). The open sets in this base are called distinguished or basic open sets. The importance of this property results in particular from its use in the definition of an affine scheme.
By Hilbert's basis theorem and some elementary properties of Noetherian rings, every affine or projective coordinate ring is Noetherian. As a consequence, affine or projective spaces with the Zariski topology are Noetherian topological spaces, which implies that any closed subset of these spaces is compact.
However, except for finite algebraic sets, no algebraic set is ever a Hausdorff space. In the old topological literature "compact" was taken to include the Hausdorff property, and this convention is still honored in algebraic geometry; therefore compactness in the modern sense is called "quasicompactness" in algebraic geometry. However, since every point (a1, ..., an) is the zero set of the polynomials x1 - a1, ..., xn - an, points are closed and so every variety satisfies the T1 axiom.
Every regular map of varieties is continuous in the Zariski topology. In fact, the Zariski topology is the weakest topology (with the fewest open sets) in which this is true and in which points are closed. This is easily verified by noting that the Zariski-closed sets are simply the intersections of the inverse images of 0 by the polynomial functions, considered as regular maps into $\mathbb {A} ^{1}.$
Spectrum of a ring
In modern algebraic geometry, an algebraic variety is often represented by its associated scheme, which is a topological space (equipped with additional structures) that is locally homeomorphic to the spectrum of a ring.[3] The spectrum of a commutative ring A, denoted Spec A, is the set of the prime ideals of A, equipped with the Zariski topology, for which the closed sets are the sets
$V(I)=\{P\in \operatorname {Spec} A\mid P\supset I\}$
where I is an ideal.
To see the connection with the classical picture, note that for any set S of polynomials (over an algebraically closed field), it follows from Hilbert's Nullstellensatz that the points of V(S) (in the old sense) are exactly the tuples (a1, ..., an) such that the ideal generated by the polynomials x1 − a1, ..., xn − an contains S; moreover, these are maximal ideals and by the "weak" Nullstellensatz, an ideal of any affine coordinate ring is maximal if and only if it is of this form. Thus, V(S) is "the same as" the maximal ideals containing S. Grothendieck's innovation in defining Spec was to replace maximal ideals with all prime ideals; in this formulation it is natural to simply generalize this observation to the definition of a closed set in the spectrum of a ring.
Another way, perhaps more similar to the original, to interpret the modern definition is to realize that the elements of A can actually be thought of as functions on the prime ideals of A; namely, as functions on Spec A. Simply, any prime ideal P has a corresponding residue field, which is the field of fractions of the quotient A/P, and any element of A has a reflection in this residue field. Furthermore, the elements that are actually in P are precisely those whose reflection vanishes at P. So if we think of the map, associated to any element a of A:
$e_{a}\colon {\bigl (}P\in \operatorname {Spec} A{\bigr )}\mapsto \left({\frac {a\;{\bmod {P}}}{1}}\in \operatorname {Frac} (A/P)\right)$
("evaluation of a"), which assigns to each point its reflection in the residue field there, as a function on Spec A (whose values, admittedly, lie in different fields at different points), then we have
$e_{a}(P)=0\Leftrightarrow P\in V(a)$
More generally, V(I) for any ideal I is the common set on which all the "functions" in I vanish, which is formally similar to the classical definition. In fact, they agree in the sense that when A is the ring of polynomials over some algebraically closed field k, the maximal ideals of A are (as discussed in the previous paragraph) identified with n-tuples of elements of k, their residue fields are just k, and the "evaluation" maps are actually evaluation of polynomials at the corresponding n-tuples. Since as shown above, the classical definition is essentially the modern definition with only maximal ideals considered, this shows that the interpretation of the modern definition as "zero sets of functions" agrees with the classical definition where they both make sense.
Just as Spec replaces affine varieties, the Proj construction replaces projective varieties in modern algebraic geometry. Just as in the classical case, to move from the affine to the projective definition we need only replace "ideal" by "homogeneous ideal", though there is a complication involving the "irrelevant maximal ideal," which is discussed in the cited article.
Examples
• Spec k, the spectrum of a field k is the topological space with one element.
• Spec ℤ, the spectrum of the integers has a closed point for every prime number p corresponding to the maximal ideal (p) ⊂ ℤ, and one non-closed generic point (i.e., whose closure is the whole space) corresponding to the zero ideal (0). So the closed subsets of Spec ℤ are precisely the whole space and the finite unions of closed points.
• Spec k[t], the spectrum of the polynomial ring over a field k: such a polynomial ring is known to be a principal ideal domain and the irreducible polynomials are the prime elements of k[t]. If k is algebraically closed, for example the field of complex numbers, a non-constant polynomial is irreducible if and only if it is linear, of the form t − a, for some element a of k. So, the spectrum consists of one closed point for every element a of k and a generic point, corresponding to the zero ideal, and the set of the closed points is homeomorphic with the affine line k equipped with its Zariski topology. Because of this homeomorphism, some authors call affine line the spectrum of k[t]. If k is not algebraically closed, for example the field of the real numbers, the picture becomes more complicated because of the existence of non-linear irreducible polynomials. In this case, the spectrum consists of one closed point for each monic irreducible polynomial, and a generic point corresponding to the zero ideal. For example, the spectrum of ℝ[t] consists of the closed points (x − a), for a in ℝ, the closed points (x2 + px + q) where p, q are in ℝ and with negative discriminant p2 − 4q < 0, and finally a generic point (0). For any field, the closed subsets of Spec k[t] are finite unions of closed points, and the whole space. (This results from the fact that k[t] is a principal ideal domain, and, in a principal ideal domain, the prime ideals that contain an ideal are the prime factors of the prime factorization of a generator of the ideal).
Further properties
The most dramatic change in the topology from the classical picture to the new is that points are no longer necessarily closed; by expanding the definition, Grothendieck introduced generic points, which are the points with maximal closure, that is the minimal prime ideals. The closed points correspond to maximal ideals of A. However, the spectrum and projective spectrum are still T0 spaces: given two points P, Q, which are prime ideals of A, at least one of them, say P, does not contain the other. Then D(Q) contains P but, of course, not Q.
Just as in classical algebraic geometry, any spectrum or projective spectrum is (quasi)compact, and if the ring in question is Noetherian then the space is a Noetherian space. However, these facts are counterintuitive: we do not normally expect open sets, other than connected components, to be compact, and for affine varieties (for example, Euclidean space) we do not even expect the space itself to be compact. This is one instance of the geometric unsuitability of the Zariski topology. Grothendieck solved this problem by defining the notion of properness of a scheme (actually, of a morphism of schemes), which recovers the intuitive idea of compactness: Proj is proper, but Spec is not.
See also
• Spectral space
Citations
1. Hulek 2003, p. 19, 1.1.1..
2. Mumford 1999.
3. Dummit & Foote 2004.
References
• Dummit, D. S.; Foote, R. (2004). Abstract Algebra (3 ed.). Wiley. pp. 71–72. ISBN 9780471433347.
• Hartshorne, Robin (1977), Algebraic Geometry, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157, OCLC 13348052
• Hulek, Klaus (2003). Elementary Algebraic Geometry. AMS. ISBN 978-0-8218-2952-3.
• Mumford, David (1999) [1967]. The Red Book of Varieties and Schemes. Lecture Notes in Mathematics. Vol. 1358 (expanded, Includes Michigan Lectures (1974) on Curves and their Jacobians ed.). Berlin, New York: Springer-Verlag. doi:10.1007/b62130. ISBN 978-3-540-63293-1. MR 1748380.
• Todd Rowland. "Zariski Topology". MathWorld.
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Zariski ring
In commutative algebra, a Zariski ring is a commutative Noetherian topological ring A whose topology is defined by an ideal ${\mathfrak {a}}$ contained in the Jacobson radical, the intersection of all maximal ideals. They were introduced by Oscar Zariski (1946) under the name "semi-local ring" which now means something different, and named "Zariski rings" by Pierre Samuel (1953). Examples of Zariski rings are noetherian local rings with the topology induced by the maximal ideal, and ${\mathfrak {a}}$-adic completions of Noetherian rings.
Let A be a Noetherian topological ring with the topology defined by an ideal ${\mathfrak {a}}$. Then the following are equivalent.
• A is a Zariski ring.
• The completion ${\widehat {A}}$ is faithfully flat over A (in general, it is only flat over A).
• Every maximal ideal is closed.
References
• Atiyah, Michael F.; Macdonald, Ian G. (1969), Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., MR 0242802
• Samuel, Pierre (1953), Algèbre locale, Mémor. Sci. Math., vol. 123, Paris: Gauthier-Villars, MR 0054995
• Zariski, Oscar (1946), "Generalized semi-local rings", Summa Brasil. Math., 1 (8): 169–195, MR 0022835
• Zariski, Oscar; Samuel, Pierre (1975), Commutative algebra. Vol. II, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90171-8, MR 0389876
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Zariski surface
In algebraic geometry, a branch of mathematics, a Zariski surface is a surface over a field of characteristic p > 0 such that there is a dominant inseparable map of degree p from the projective plane to the surface. In particular, all Zariski surfaces are unirational. They were named by Piotr Blass in 1977 after Oscar Zariski who used them in 1958 to give examples of unirational surfaces in characteristic p > 0 that are not rational. (In characteristic 0 by contrast, Castelnuovo's theorem implies that all unirational surfaces are rational.)
For spaces of valuations, see Zariski–Riemann surface.
Zariski surfaces are birational to surfaces in affine 3-space A3 defined by irreducible polynomials of the form
$z^{p}=f(x,y).\ $
The following problem was posed by Oscar Zariski in 1971: Let S be a Zariski surface with vanishing geometric genus. Is S necessarily a rational surface? For p = 2 and for p = 3 the answer to the above problem is negative as shown in 1977 by Piotr Blass in his University of Michigan Ph.D. thesis and by William E. Lang in his Harvard Ph.D. thesis in 1978. Kentaro Mitsui (2014) announced further examples giving a negative answer to Zariski's question in every characteristic p>0 . His method however is non constructive at the moment and we do not have explicit equations for p>3.
See also
• List of algebraic surfaces
References
• Blass, Piotr; Lang, Jeffrey (1987), Zariski surfaces and differential equations in characteristic p>0, Monographs and Textbooks in Pure and Applied Mathematics, vol. 106, New York: Marcel Dekker Inc., ISBN 978-0-8247-7637-4, MR 0879599
• Mitsui, Kentaro (2014), "On a question of Zariski on Zariski surfaces", Math. Z., 276 (1–2): 237–242, doi:10.1007/s00209-013-1195-0, MR 3150201
• Zariski, Oscar (1958), "On Castelnuovo's criterion of rationality pa=P2=0 of an algebraic surface", Illinois Journal of Mathematics, 2: 303–315, ISSN 0019-2082, MR 0099990
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Zariski tangent space
In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space at a point P on an algebraic variety V (and more generally). It does not use differential calculus, being based directly on abstract algebra, and in the most concrete cases just the theory of a system of linear equations.
Motivation
For example, suppose given a plane curve C defined by a polynomial equation
F(X,Y) = 0
and take P to be the origin (0,0). Erasing terms of higher order than 1 would produce a 'linearised' equation reading
L(X,Y) = 0
in which all terms XaYb have been discarded if a + b > 1.
We have two cases: L may be 0, or it may be the equation of a line. In the first case the (Zariski) tangent space to C at (0,0) is the whole plane, considered as a two-dimensional affine space. In the second case, the tangent space is that line, considered as affine space. (The question of the origin comes up, when we take P as a general point on C; it is better to say 'affine space' and then note that P is a natural origin, rather than insist directly that it is a vector space.)
It is easy to see that over the real field we can obtain L in terms of the first partial derivatives of F. When those both are 0 at P, we have a singular point (double point, cusp or something more complicated). The general definition is that singular points of C are the cases when the tangent space has dimension 2.
Definition
The cotangent space of a local ring R, with maximal ideal ${\mathfrak {m}}$ is defined to be
${\mathfrak {m}}/{\mathfrak {m}}^{2}$
where ${\mathfrak {m}}$2 is given by the product of ideals. It is a vector space over the residue field k:= R/${\mathfrak {m}}$. Its dual (as a k-vector space) is called tangent space of R.[1]
This definition is a generalization of the above example to higher dimensions: suppose given an affine algebraic variety V and a point v of V. Morally, modding out ${\mathfrak {m}}$2 corresponds to dropping the non-linear terms from the equations defining V inside some affine space, therefore giving a system of linear equations that define the tangent space.
The tangent space $T_{P}(X)$ and cotangent space $T_{P}^{*}(X)$ to a scheme X at a point P is the (co)tangent space of ${\mathcal {O}}_{X,P}$. Due to the functoriality of Spec, the natural quotient map $f:R\rightarrow R/I$ induces a homomorphism $g:{\mathcal {O}}_{X,f^{-1}(P)}\rightarrow {\mathcal {O}}_{Y,P}$ for X=Spec(R), P a point in Y=Spec(R/I). This is used to embed $T_{P}(Y)$ in $T_{f^{-1}P}(X)$.[2] Since morphisms of fields are injective, the surjection of the residue fields induced by g is an isomorphism. Then a morphism k of the cotangent spaces is induced by g, given by
${\mathfrak {m}}_{P}/{\mathfrak {m}}_{P}^{2}$
$\cong ({\mathfrak {m}}_{f^{-1}P}/I)/(({\mathfrak {m}}_{f^{-1}P}^{2}+I)/I)$
$\cong {\mathfrak {m}}_{f^{-1}P}/({\mathfrak {m}}_{f^{-1}P}^{2}+I)$
$\cong ({\mathfrak {m}}_{f^{-1}P}/{\mathfrak {m}}_{f^{-1}P}^{2})/\mathrm {Ker} (k).$
Since this is a surjection, the transpose $k^{*}:T_{P}(Y)\rightarrow T_{f^{-1}P}(X)$ is an injection.
(One often defines the tangent and cotangent spaces for a manifold in the analogous manner.)
Analytic functions
If V is a subvariety of an n-dimensional vector space, defined by an ideal I, then R = Fn / I, where Fn is the ring of smooth/analytic/holomorphic functions on this vector space. The Zariski tangent space at x is
mn / (I+mn2),
where mn is the maximal ideal consisting of those functions in Fn vanishing at x.
In the planar example above, I = (F(X,Y)), and I+m2 = (L(X,Y))+m2.
Properties
If R is a Noetherian local ring, the dimension of the tangent space is at least the dimension of R:
dim m/m2 ≧ dim R
R is called regular if equality holds. In a more geometric parlance, when R is the local ring of a variety V at a point v, one also says that v is a regular point. Otherwise it is called a singular point.
The tangent space has an interpretation in terms of K[t]/(t2), the dual numbers for K; in the parlance of schemes, morphisms from Spec K[t]/(t2) to a scheme X over K correspond to a choice of a rational point x ∈ X(k) and an element of the tangent space at x.[3] Therefore, one also talks about tangent vectors. See also: tangent space to a functor.
In general, the dimension of the Zariski tangent space can be extremely large. For example, let $C^{1}(\mathbf {R} )$ be the ring of continuously differentiable real-valued functions on $\mathbf {R} $. Define $R=C_{0}^{1}(\mathbf {R} )$ to be the ring of germs of such functions at the origin. Then R is a local ring, and its maximal ideal m consists of all germs which vanish at the origin. The functions $x^{\alpha }$ for $\alpha \in (1,2)$ define linearly independent vectors in the Zariski cotangent space $m/m^{2}$, so the dimension of $m/m^{2}$ is at least the ${\mathfrak {c}}$, the cardinality of the continuum. The dimension of the Zariski tangent space $(m/m^{2})^{*}$ is therefore at least $2^{\mathfrak {c}}$. On the other hand, the ring of germs of smooth functions at a point in an n-manifold has an n-dimensional Zariski cotangent space.[lower-alpha 1]
See also
• Tangent cone
• Jet (mathematics)
Notes
1. https://mathoverflow.net/questions/44705/cardinalities-larger-than-the-continuum-in-areas-besides-set-theory/44733#44733
Citations
1. Eisenbud & Harris 1998, I.2.2, pg. 26.
2. Smoothness and the Zariski Tangent Space, James McKernan, 18.726 Spring 2011 Lecture 5
3. Hartshorne 1977, Exercise II 2.8.
Sources
• Eisenbud, David; Harris, Joe (1998). The Geometry of Schemes. Springer-Verlag. ISBN 0-387-98637-5 – via Internet Archive.
• Hartshorne, Robin (1977). Algebraic Geometry. Graduate Texts in Mathematics. Vol. 52. New York: Springer-Verlag. ISBN 978-0-387-90244-9. MR 0463157.
• Zariski, Oscar (1947). "The concept of a simple point of an abstract algebraic variety". Transactions of the American Mathematical Society. 62: 1–52. doi:10.1090/S0002-9947-1947-0021694-1. MR 0021694. Zbl 0031.26101.
External links
• Zariski tangent space. V.I. Danilov (originator), Encyclopedia of Mathematics.
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Zariski's main theorem
In algebraic geometry, Zariski's main theorem, proved by Oscar Zariski (1943), is a statement about the structure of birational morphisms stating roughly that there is only one branch at any normal point of a variety. It is the special case of Zariski's connectedness theorem when the two varieties are birational.
Zariski's main theorem can be stated in several ways which at first sight seem to be quite different, but are in fact deeply related. Some of the variations that have been called Zariski's main theorem are as follows:
• The total transform of a normal fundamental point of a birational map has positive dimension. This is essentially Zariski's original form of his main theorem.
• A birational morphism with finite fibers to a normal variety is an isomorphism to an open subset.
• The total transform of a normal point under a proper birational morphism is connected.
• A closely related theorem of Grothendieck describes the structure of quasi-finite morphisms of schemes, which implies Zariski's original main theorem.
• Several results in commutative algebra that imply the geometric form of Zariski's main theorem.
• A normal local ring is unibranch, which is a variation of the statement that the transform of a normal point is connected.
• The local ring of a normal point of a variety is analytically normal. This is a strong form of the statement that it is unibranch.
The name "Zariski's main theorem" comes from the fact that Zariski labelled it as the "MAIN THEOREM" in Zariski (1943).
Zariski's main theorem for birational morphisms
Let f be a birational mapping of algebraic varieties V and W. Recall that f is defined by a closed subvariety $\Gamma \subset V\times W$ (a "graph" of f) such that the projection on the first factor $p_{1}$ induces an isomorphism between an open $U\subset V$ and $p_{1}^{-1}(U)$, and such that $p_{2}\circ p_{1}^{-1}$ is an isomorphism on U too. The complement of U in V is called a fundamental variety or indeterminacy locus, and the image of a subset of V under $p_{2}\circ p_{1}^{-1}$ is called a total transform of it.
The original statement of the theorem in (Zariski 1943, p. 522) reads:
MAIN THEOREM: If W is an irreducible fundamental variety on V of a birational correspondence T between V and V′ and if T has no fundamental elements on V′ then — under the assumption that V is locally normal at W — each irreducible component of the transform T[W] is of higher dimension than W.
Here T is essentially a morphism from V′ to V that is birational, W is a subvariety of the set where the inverse of T is not defined whose local ring is normal, and the transform T[W] means the inverse image of W under the morphism from V′ to V.
Here are some variants of this theorem stated using more recent terminology. Hartshorne (1977, Corollary III.11.4) calls the following connectedness statement "Zariski's Main theorem":
If f:X→Y is a birational projective morphism between noetherian integral schemes, then the inverse image of every normal point of Y is connected.
The following consequence of it (Theorem V.5.2,loc.cit.) also goes under this name:
If f:X→Y is a birational transformation of projective varieties with Y normal, then the total transform of a fundamental point of f is connected and of dimension at least 1.
Examples
• Suppose that V is a smooth variety of dimension greater than 1 and V′ is given by blowing up a point W on V. Then V is normal at W, and the component of the transform of W is a projective space, which has dimension greater than W as predicted by Zariski's original form of his main theorem.
• In the previous example the transform of W was irreducible. It is easy to find examples where the total transform is reducible by blowing up other points on the transform. For example, if V′ is given by blowing up a point W on V and then blowing up another point on this transform, the total transform of W has 2 irreducible components meeting at a point. As predicted by Hartshorne's form of the main theorem, the total transform is connected and of dimension at least 1.
• For an example where W is not normal and the conclusion of the main theorem fails, take V′ to be a smooth variety, and take V to be given by identifying two distinct points on V′, and take W to be the image of these two points. Then W is not normal, and the transform of W consists of two points, which is not connected and does not have positive dimension.
Zariski's main theorem for quasifinite morphisms
In EGA III, Grothendieck calls the following statement which does not involve connectedness a "Main theorem" of Zariski Grothendieck (1961, Théorème 4.4.3):
If f:X→Y is a quasi-projective morphism of Noetherian schemes then the set of points that are isolated in their fiber is open in X. Moreover the induced scheme of this set is isomorphic to an open subset of a scheme that is finite over Y.
In EGA IV, Grothendieck observed that the last statement could be deduced from a more general theorem about the structure of quasi-finite morphisms, and the latter is often referred to as the "Zariski's main theorem in the form of Grothendieck". It is well known that open immersions and finite morphisms are quasi-finite. Grothendieck proved that under the hypothesis of separatedness all quasi-finite morphisms are compositions of such Grothendieck (1966, Théorème 8.12.6):
if Y is a quasi-compact separated scheme and $f:X\to Y$ is a separated, quasi-finite, finitely presented morphism then there is a factorization into $X\to Z\to Y$, where the first map is an open immersion and the second one is finite.
The relation between this theorem about quasi-finite morphisms and Théorème 4.4.3 of EGA III quoted above is that if f:X→Y is a projective morphism of varieties, then the set of points that are isolated in their fiber is quasifinite over Y. Then structure theorem for quasi-finite morphisms applies and yields the desired result.
Zariski's main theorem for commutative rings
Zariski (1949) reformulated his main theorem in terms of commutative algebra as a statement about local rings. Grothendieck (1961, Théorème 4.4.7) generalized Zariski's formulation as follows:
If B is an algebra of finite type over a local Noetherian ring A, and n is a maximal ideal of B which is minimal among ideals of B whose inverse image in A is the maximal ideal m of A, then there is a finite A-algebra A′ with a maximal ideal m′ (whose inverse image in A is m) such that the localization Bn is isomorphic to the A-algebra A′m′.
If in addition A and B are integral and have the same field of fractions, and A is integrally closed, then this theorem implies that A and B are equal. This is essentially Zariski's formulation of his main theorem in terms of commutative rings.
Zariski's main theorem: topological form
A topological version of Zariski's main theorem says that if x is a (closed) point of a normal complex variety it is unibranch; in other words there are arbitrarily small neighborhoods U of x such that the set of non-singular points of U is connected (Mumford 1999, III.9).
The property of being normal is stronger than the property of being unibranch: for example, a cusp of a plane curve is unibranch but not normal.
Zariski's main theorem: power series form
A formal power series version of Zariski's main theorem says that if x is a normal point of a variety then it is analytically normal; in other words the completion of the local ring at x is a normal integral domain (Mumford 1999, III.9).
See also
• Deligne's connectedness theorem
• Fulton–Hansen connectedness theorem
• Grothendieck's connectedness theorem
• Stein factorization
• Theorem on formal functions
References
• Danilov, V.I. (2001) [1994], "Zariski theorem", Encyclopedia of Mathematics, EMS Press
• Grothendieck, Alexandre (1961), Eléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : III. Étude cohomologique des faisceaux cohérents, Première partie, Publications Mathématiques de l'IHÉS, vol. 11, pp. 5–167
• Grothendieck, Alexandre (1966), Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Troisième partie, Publications Mathématiques de l'IHÉS, vol. 28, pp. 43–48
• Hartshorne, Robin (1977), Algebraic Geometry, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
• Mumford, David (1999) [1988], The red book of varieties and schemes, Lecture Notes in Mathematics, vol. 1358 (expanded, Includes Michigan Lectures (1974) on Curves and their Jacobians ed.), Berlin, New York: Springer-Verlag, doi:10.1007/b62130, ISBN 978-3-540-63293-1, MR 1748380
• Peskine, Christian (1966), "Une généralisation du main theorem de Zariski", Bull. Sci. Math. (2), 90: 119–127
• Raynaud, Michel (1970), Anneaux locaux henséliens, Lecture Notes in Mathematics, vol. 169, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0069571, ISBN 978-3-540-05283-8, MR 0277519
• Zariski, Oscar (1943), "Foundations of a general theory of birational correspondences.", Trans. Amer. Math. Soc., 53 (3): 490–542, doi:10.2307/1990215, JSTOR 1990215, MR 0008468
• Zariski, Oscar (1949), "A simple analytical proof of a fundamental property of birational transformations.", Proc. Natl. Acad. Sci. U.S.A., 35 (1): 62–66, Bibcode:1949PNAS...35...62Z, doi:10.1073/pnas.35.1.62, JSTOR 88284, MR 0028056, PMC 1062959, PMID 16588856
External links
• Is there an intuitive reason for Zariski's main theorem?
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