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abc conjecture
The abc conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory that arose out of a discussion of Joseph Oesterlé and David Masser in 1985.[1][2] It is stated in terms of three positive integers $a,b$ and $c$ (hence the name) that are relatively prime and satisfy $a+b=c$. The conjecture essentially states that the product of the distinct prime factors of $abc$ is usually not much smaller than $c$. A number of famous conjectures and theorems in number theory would follow immediately from the abc conjecture or its versions. Mathematician Dorian Goldfeld described the abc conjecture as "The most important unsolved problem in Diophantine analysis".[3]
abc conjecture
FieldNumber theory
Conjectured by
• Joseph Oesterlé
• David Masser
Conjectured in1985
Equivalent toModified Szpiro conjecture
Consequences
• Beal conjecture
• Erdős–Ulam problem
• Faltings's theorem
• Fermat's Last Theorem
• Fermat–Catalan conjecture
• Roth's theorem
• Tijdeman's theorem
The abc conjecture originated as the outcome of attempts by Oesterlé and Masser to understand the Szpiro conjecture about elliptic curves,[4] which involves more geometric structures in its statement than the abc conjecture. The abc conjecture was shown to be equivalent to the modified Szpiro's conjecture.[1]
Various attempts to prove the abc conjecture have been made, but none are currently accepted by the mainstream mathematical community, and, as of 2020, the conjecture is still regarded as unproven.[5][6]
Formulations
Before stating the conjecture, the notion of the radical of an integer must be introduced: for a positive integer $n$, the radical of $n$, denoted ${\text{rad}}(n)$, is the product of the distinct prime factors of $n$. For example,
${\text{rad}}(16)={\text{rad}}(2^{4})={\text{rad}}(2)=2$
${\text{rad}}(17)=17$
${\text{rad}}(18)={\text{rad}}(2\cdot 3^{2})=2\cdot 3=6$
${\text{rad}}(1000000)={\text{rad}}(2^{6}\cdot 5^{6})=2\cdot 5=10$
If a, b, and c are coprime[notes 1] positive integers such that a + b = c, it turns out that "usually" $c<{\text{rad}}(abc)$. The abc conjecture deals with the exceptions. Specifically, it states that:
For every positive real number ε, there exist only finitely many triples (a, b, c) of coprime positive integers, with a + b = c, such that[7]
$c>\operatorname {rad} (abc)^{1+\varepsilon }.$
An equivalent formulation is:
For every positive real number ε, there exists a constant Kε such that for all triples (a, b, c) of coprime positive integers, with a + b = c:[7]
$c<K_{\varepsilon }\cdot \operatorname {rad} (abc)^{1+\varepsilon }.$
Equivalently (using the little o notation):
For all triples (a, b, c) of coprime positive integers with a + b = c, rad(abc) is at least c1-o(1).
A fourth equivalent formulation of the conjecture involves the quality q(a, b, c) of the triple (a, b, c), which is defined as
$q(a,b,c)={\frac {\log(c)}{\log {\big (}\operatorname {rad} (abc){\big )}}}.$
For example:
q(4, 127, 131) = log(131) / log(rad(4·127·131)) = log(131) / log(2·127·131) = 0.46820...
q(3, 125, 128) = log(128) / log(rad(3·125·128)) = log(128) / log(30) = 1.426565...
A typical triple (a, b, c) of coprime positive integers with a + b = c will have c < rad(abc), i.e. q(a, b, c) < 1. Triples with q > 1 such as in the second example are rather special, they consist of numbers divisible by high powers of small prime numbers. The fourth formulation is:
For every positive real number ε, there exist only finitely many triples (a, b, c) of coprime positive integers with a + b = c such that q(a, b, c) > 1 + ε.
Whereas it is known that there are infinitely many triples (a, b, c) of coprime positive integers with a + b = c such that q(a, b, c) > 1, the conjecture predicts that only finitely many of those have q > 1.01 or q > 1.001 or even q > 1.0001, etc. In particular, if the conjecture is true, then there must exist a triple (a, b, c) that achieves the maximal possible quality q(a, b, c).
Examples of triples with small radical
The condition that ε > 0 is necessary as there exist infinitely many triples a, b, c with c > rad(abc). For example, let
$a=1,\quad b=2^{6n}-1,\quad c=2^{6n},\qquad n>1.$
The integer b is divisible by 9:
$b=2^{6n}-1=64^{n}-1=(64-1)(\cdots )=9\cdot 7\cdot (\cdots ).$
Using this fact, the following calculation is made:
${\begin{aligned}\operatorname {rad} (abc)&=\operatorname {rad} (a)\operatorname {rad} (b)\operatorname {rad} (c)\\&=\operatorname {rad} (1)\operatorname {rad} \left(2^{6n}-1\right)\operatorname {rad} \left(2^{6n}\right)\\&=2\operatorname {rad} \left(2^{6n}-1\right)\\&=2\operatorname {rad} \left(9\cdot {\tfrac {b}{9}}\right)\\&\leqslant 2\cdot 3\cdot {\tfrac {b}{9}}\\&=2{\tfrac {b}{3}}\\&<{\tfrac {2}{3}}c.\end{aligned}}$
By replacing the exponent 6n by other exponents forcing b to have larger square factors, the ratio between the radical and c can be made arbitrarily small. Specifically, let p > 2 be a prime and consider
$a=1,\quad b=2^{p(p-1)n}-1,\quad c=2^{p(p-1)n},\qquad n>1.$
Now it may be plausibly claimed that b is divisible by p2:
${\begin{aligned}b&=2^{p(p-1)n}-1\\&=\left(2^{p(p-1)}\right)^{n}-1\\&=\left(2^{p(p-1)}-1\right)(\cdots )\\&=p^{2}\cdot r(\cdots ).\end{aligned}}$
The last step uses the fact that p2 divides 2p(p−1) − 1. This follows from Fermat's little theorem, which shows that, for p > 2, 2p−1 = pk + 1 for some integer k. Raising both sides to the power of p then shows that 2p(p−1) = p2(...) + 1.
And now with a similar calculation as above, the following results:
$\operatorname {rad} (abc)<{\tfrac {2}{p}}c.$
A list of the highest-quality triples (triples with a particularly small radical relative to c) is given below; the highest quality, 1.6299, was found by Eric Reyssat (Lando & Zvonkin 2004, p. 137) for
a = 2,
b = 310·109 = 6436341,
c = 235 = 6436343,
rad(abc) = 15042.
Some consequences
The abc conjecture has a large number of consequences. These include both known results (some of which have been proven separately only since the conjecture has been stated) and conjectures for which it gives a conditional proof. The consequences include:
• Roth's theorem on Diophantine approximation of algebraic numbers.[8][7]
• The Mordell conjecture (already proven in general by Gerd Faltings).[9]
• As equivalent, Vojta's conjecture in dimension 1.[10]
• The Erdős–Woods conjecture allowing for a finite number of counterexamples.[11]
• The existence of infinitely many non-Wieferich primes in every base b > 1.[12]
• The weak form of Marshall Hall's conjecture on the separation between squares and cubes of integers.[13]
• Fermat's Last Theorem has a famously difficult proof by Andrew Wiles. However it follows easily, at least for $n\geq 6$, from an effective form of a weak version of the abc conjecture. The abc conjecture says the lim sup of the set of all qualities (defined above) is 1, which implies the much weaker assertion that there is a finite upper bound for qualities. The conjecture that 2 is such an upper bound suffices for a very short proof of Fermat's Last Theorem for $n\geq 6$.[14]
• The Fermat–Catalan conjecture, a generalization of Fermat's Last Theorem concerning powers that are sums of powers.[15]
• The L-function L(s, χd) formed with the Legendre symbol, has no Siegel zero, given a uniform version of the abc conjecture in number fields, not just the abc conjecture as formulated above for rational integers.[16]
• A polynomial P(x) has only finitely many perfect powers for all integers x if P has at least three simple zeros.[17]
• A generalization of Tijdeman's theorem concerning the number of solutions of ym = xn + k (Tijdeman's theorem answers the case k = 1), and Pillai's conjecture (1931) concerning the number of solutions of Aym = Bxn + k.
• As equivalent, the Granville–Langevin conjecture, that if f is a square-free binary form of degree n > 2, then for every real β > 2 there is a constant C(f, β) such that for all coprime integers x, y, the radical of f(x, y) exceeds C · max{|x|, |y|}n−β.[18]
• As equivalent, the modified Szpiro conjecture, which would yield a bound of rad(abc)1.2+ε.[1]
• Dąbrowski (1996) has shown that the abc conjecture implies that the Diophantine equation n! + A = k2 has only finitely many solutions for any given integer A.
• There are ~cfN positive integers n ≤ N for which f(n)/B' is square-free, with cf > 0 a positive constant defined as:[19]
$c_{f}=\prod _{{\text{prime }}p}x_{i}\left(1-{\frac {\omega \,\!_{f}(p)}{p^{2+q_{p}}}}\right).$
• The Beal conjecture, a generalization of Fermat's Last Theorem proposing that if A, B, C, x, y, and z are positive integers with Ax + By = Cz and x, y, z > 2, then A, B, and C have a common prime factor. The abc conjecture would imply that there are only finitely many counterexamples.
• Lang's conjecture, a lower bound for the height of a non-torsion rational point of an elliptic curve.
• A negative solution to the Erdős–Ulam problem on dense sets of Euclidean points with rational distances.[20]
• An effective version of Siegel's theorem about integral points on algebraic curves.[21]
Theoretical results
The abc conjecture implies that c can be bounded above by a near-linear function of the radical of abc. Bounds are known that are exponential. Specifically, the following bounds have been proven:
$c<\exp {\left(K_{1}\operatorname {rad} (abc)^{15}\right)}$ (Stewart & Tijdeman 1986),
$c<\exp {\left(K_{2}\operatorname {rad} (abc)^{{\frac {2}{3}}+\varepsilon }\right)}$ (Stewart & Yu 1991), and
$c<\exp {\left(K_{3}\operatorname {rad} (abc)^{\frac {1}{3}}\left(\log(\operatorname {rad} (abc)\right)^{3}\right)}$ (Stewart & Yu 2001).
In these bounds, K1 and K3 are constants that do not depend on a, b, or c, and K2 is a constant that depends on ε (in an effectively computable way) but not on a, b, or c. The bounds apply to any triple for which c > 2.
There are also theoretical results that provide a lower bound on the best possible form of the abc conjecture. In particular, Stewart & Tijdeman (1986) showed that there are infinitely many triples (a, b, c) of coprime integers with a + b = c and
$c>\operatorname {rad} (abc)\exp {\left(k{\sqrt {\log c}}/\log \log c\right)}$
for all k < 4. The constant k was improved to k = 6.068 by van Frankenhuysen (2000).
Computational results
In 2006, the Mathematics Department of Leiden University in the Netherlands, together with the Dutch Kennislink science institute, launched the ABC@Home project, a grid computing system, which aims to discover additional triples a, b, c with rad(abc) < c. Although no finite set of examples or counterexamples can resolve the abc conjecture, it is hoped that patterns in the triples discovered by this project will lead to insights about the conjecture and about number theory more generally.
Distribution of triples with q > 1[22]
q
c
q > 1 q > 1.05 q > 1.1 q > 1.2 q > 1.3 q > 1.4
c < 102 644200
c < 103 311714831
c < 104 12074502283
c < 105 41824015251136
c < 106 1,2686673791022911
c < 107 3,4991,6698562106017
c < 108 8,9873,8691,8013849825
c < 109 22,3168,7423,69370614434
c < 1010 51,67718,2337,0351,15921851
c < 1011 116,97837,61213,2661,94732764
c < 1012 252,85673,71423,7733,02845574
c < 1013 528,275139,76241,4384,51959984
c < 1014 1,075,319258,16870,0476,66576998
c < 1015 2,131,671463,446115,0419,497998112
c < 1016 4,119,410812,499184,72713,1181,232126
c < 1017 7,801,3341,396,909290,96517,8901,530143
c < 1018 14,482,0652,352,105449,19424,0131,843160
As of May 2014, ABC@Home had found 23.8 million triples.[23]
Highest-quality triples[24]
Rank q a b c Discovered by
1 1.62992310·109235Eric Reyssat
2 1.626011232·56·73221·23Benne de Weger
3 1.623519·13077·292·31828·322·54Jerzy Browkin, Juliusz Brzezinski
4 1.5808283511·13228·38·173Jerzy Browkin, Juliusz Brzezinski, Abderrahmane Nitaj
5 1.567912·3754·7Benne de Weger
Note: the quality q(a, b, c) of the triple (a, b, c) is defined above.
Refined forms, generalizations and related statements
The abc conjecture is an integer analogue of the Mason–Stothers theorem for polynomials.
A strengthening, proposed by Baker (1998), states that in the abc conjecture one can replace rad(abc) by
ε−ω rad(abc),
where ω is the total number of distinct primes dividing a, b and c.[25]
Andrew Granville noticed that the minimum of the function ${\big (}\varepsilon ^{-\omega }\operatorname {rad} (abc){\big )}^{1+\varepsilon }$ over $\varepsilon >0$ occurs when $\varepsilon ={\frac {\omega }{\log {\big (}\operatorname {rad} (abc){\big )}}}.$
This inspired Baker (2004) to propose a sharper form of the abc conjecture, namely:
$c<\kappa \operatorname {rad} (abc){\frac {{\Big (}\log {\big (}\operatorname {rad} (abc){\big )}{\Big )}^{\omega }}{\omega !}}$ !}}}
with κ an absolute constant. After some computational experiments he found that a value of $6/5$ was admissible for κ. This version is called the "explicit abc conjecture".
Baker (1998) also describes related conjectures of Andrew Granville that would give upper bounds on c of the form
$K^{\Omega (abc)}\operatorname {rad} (abc),$
where Ω(n) is the total number of prime factors of n, and
$O{\big (}\operatorname {rad} (abc)\Theta (abc){\big )},$
where Θ(n) is the number of integers up to n divisible only by primes dividing n.
Robert, Stewart & Tenenbaum (2014) proposed a more precise inequality based on Robert & Tenenbaum (2013). Let k = rad(abc). They conjectured there is a constant C1 such that
$c<k\exp \left(4{\sqrt {\frac {3\log k}{\log \log k}}}\left(1+{\frac {\log \log \log k}{2\log \log k}}+{\frac {C_{1}}{\log \log k}}\right)\right)$
holds whereas there is a constant C2 such that
$c>k\exp \left(4{\sqrt {\frac {3\log k}{\log \log k}}}\left(1+{\frac {\log \log \log k}{2\log \log k}}+{\frac {C_{2}}{\log \log k}}\right)\right)$
holds infinitely often.
Browkin & Brzeziński (1994) formulated the n conjecture—a version of the abc conjecture involving n > 2 integers.
Claimed proofs
Lucien Szpiro proposed a solution in 2007, but it was found to be incorrect shortly afterwards.[26]
Since August 2012, Shinichi Mochizuki has claimed a proof of Szpiro's conjecture and therefore the abc conjecture.[27] He released a series of four preprints developing a new theory he called inter-universal Teichmüller theory (IUTT), which is then applied to prove the abc conjecture.[28] The papers have not been widely accepted by the mathematical community as providing a proof of abc.[29] This is not only because of their length and the difficulty of understanding them,[30] but also because at least one specific point in the argument has been identified as a gap by some other experts.[31] Although a few mathematicians have vouched for the correctness of the proof[32] and have attempted to communicate their understanding via workshops on IUTT, they have failed to convince the number theory community at large.[33][34]
In March 2018, Peter Scholze and Jakob Stix visited Kyoto for discussions with Mochizuki.[35][36] While they did not resolve the differences, they brought them into clearer focus. Scholze and Stix wrote a report asserting and explaining an error in the logic of the proof and claiming that the resulting gap was "so severe that ... small modifications will not rescue the proof strategy";[31] Mochizuki claimed that they misunderstood vital aspects of the theory and made invalid simplifications.[37][38][39]
On April 3, 2020, two mathematicians from the Kyoto research institute where Mochizuki works announced that his claimed proof would be published in Publications of the Research Institute for Mathematical Sciences, the institute's journal. Mochizuki is chief editor of the journal but recused himself from the review of the paper.[5] The announcement was received with skepticism by Kiran Kedlaya and Edward Frenkel, as well as being described by Nature as "unlikely to move many researchers over to Mochizuki's camp".[5] In March 2021, Mochizuki's proof was published in RIMS.[40]
See also
• List of unsolved problems in mathematics
Notes
1. When a + b = c, coprimality of a, b, c implies pairwise coprimality of a, b, c. So in this case, it does not matter which concept we use.
References
1. Oesterlé 1988.
2. Masser 1985.
3. Goldfeld 1996.
4. Fesenko, Ivan (September 2015). "Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta-functions, notes on the work of Shinichi Mochizuki". European Journal of Mathematics. 1 (3): 405–440. doi:10.1007/s40879-015-0066-0.
5. Castelvecchi, Davide (9 April 2020). "Mathematical proof that rocked number theory will be published". Nature. 580 (7802): 177. Bibcode:2020Natur.580..177C. doi:10.1038/d41586-020-00998-2. PMID 32246118.
6. Further comment by P. Scholze at Not Even Wrong math.columbia.edu
7. Waldschmidt 2015.
8. Bombieri (1994), p. .
9. Elkies (1991).
10. Van Frankenhuijsen (2002).
11. Langevin (1993).
12. Silverman (1988).
13. Nitaj (1996).
14. Granville, Andrew; Tucker, Thomas (2002). "It's As Easy As abc" (PDF). Notices of the AMS. 49 (10): 1224–1231.
15. Pomerance (2008).
16. Granville & Stark (2000).
17. The ABC-conjecture, Frits Beukers, ABC-DAY, Leiden, Utrecht University, 9 September 2005.
18. Mollin (2009); Mollin (2010, p. 297)
19. Granville (1998).
20. Pasten, Hector (2017), "Definability of Frobenius orbits and a result on rational distance sets", Monatshefte für Mathematik, 182 (1): 99–126, doi:10.1007/s00605-016-0973-2, MR 3592123, S2CID 7805117
21. arXiv:math/0408168 Andrea Surroca, Siegel’s theorem and the abc conjecture, Riv. Mat. Univ. Parma (7) 3, 2004, S. 323–332
22. "Synthese resultaten", RekenMeeMetABC.nl (in Dutch), archived from the original on December 22, 2008, retrieved October 3, 2012.
23. "Data collected sofar", ABC@Home, archived from the original on May 15, 2014, retrieved April 30, 2014
24. "100 unbeaten triples". Reken mee met ABC. 2010-11-07.
25. Bombieri & Gubler (2006), p. 404.
26. "Finiteness Theorems for Dynamical Systems", Lucien Szpiro, talk at Conference on L-functions and Automorphic Forms (on the occasion of Dorian Goldfeld's 60th Birthday), Columbia University, May 2007. See Woit, Peter (May 26, 2007), "Proof of the abc Conjecture?", Not Even Wrong.
27. Ball, Peter (10 September 2012). "Proof claimed for deep connection between primes". Nature. doi:10.1038/nature.2012.11378. Retrieved 19 March 2018.
28. Mochizuki, Shinichi (4 March 2021). "Inter-universal Teichmüller Theory IV: Log-Volume Computations and Set-Theoretic Foundations". Publications of the Research Institute for Mathematical Sciences. 57 (1): 627–723. doi:10.4171/PRIMS/57-1-4. S2CID 3135393.
29. Calegari, Frank (December 17, 2017). "The ABC conjecture has (still) not been proved". Retrieved March 17, 2018.
30. Revell, Timothy (September 7, 2017). "Baffling ABC maths proof now has impenetrable 300-page 'summary'". New Scientist.
31. Scholze, Peter; Stix, Jakob. "Why abc is still a conjecture" (PDF). Archived from the original (PDF) on February 8, 2020. Retrieved September 23, 2018. (updated version of their May report Archived 2020-02-08 at the Wayback Machine)
32. Fesenko, Ivan (28 September 2016). "Fukugen". Inference. 2 (3). Retrieved 30 October 2021.
33. Conrad, Brian (December 15, 2015). "Notes on the Oxford IUT workshop by Brian Conrad". Retrieved March 18, 2018.
34. Castelvecchi, Davide (8 October 2015). "The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof". Nature. 526 (7572): 178–181. Bibcode:2015Natur.526..178C. doi:10.1038/526178a. PMID 26450038.
35. Klarreich, Erica (September 20, 2018). "Titans of Mathematics Clash Over Epic Proof of ABC Conjecture". Quanta Magazine.
36. "March 2018 Discussions on IUTeich". Retrieved October 2, 2018. Web-page by Mochizuki describing discussions and linking consequent publications and supplementary material
37. Mochizuki, Shinichi. "Report on Discussions, Held during the Period March 15 – 20, 2018, Concerning Inter-Universal Teichmüller Theory" (PDF). Retrieved February 1, 2019. the ... discussions ... constitute the first detailed, ... substantive discussions concerning negative positions ... IUTch.
38. Mochizuki, Shinichi (July 2018). "Comments on the manuscript by Scholze-Stix concerning Inter-Universal Teichmüller Theory" (PDF). S2CID 174791744. Retrieved October 2, 2018.
39. Mochizuki, Shinichi. "Comments on the manuscript (2018-08 version) by Scholze-Stix concerning Inter-Universal Teichmüller Theory" (PDF). Retrieved October 2, 2018.
40. Mochizuki, Shinichi. "Mochizuki's proof of ABC conjecture". Retrieved July 13, 2021.
Sources
• Baker, Alan (1998). "Logarithmic forms and the abc-conjecture". In Győry, Kálmán (ed.). Number theory. Diophantine, computational and algebraic aspects. Proceedings of the international conference, Eger, Hungary, July 29-August 2, 1996. Berlin: de Gruyter. pp. 37–44. ISBN 3-11-015364-5. Zbl 0973.11047.
• Baker, Alan (2004). "Experiments on the abc-conjecture". Publicationes Mathematicae Debrecen. 65 (3–4): 253–260. doi:10.5486/PMD.2004.3348. S2CID 253834357.
• Bombieri, Enrico (1994). "Roth's theorem and the abc-conjecture" (Preprint). ETH Zürich.
• Bombieri, Enrico; Gubler, Walter (2006). Heights in Diophantine Geometry. New Mathematical Monographs. Vol. 4. Cambridge University Press. ISBN 978-0-521-71229-3. Zbl 1130.11034.
• Browkin, Jerzy; Brzeziński, Juliusz (1994). "Some remarks on the abc-conjecture". Math. Comp. 62 (206): 931–939. Bibcode:1994MaCom..62..931B. doi:10.2307/2153551. JSTOR 2153551.
• Browkin, Jerzy (2000). "The abc-conjecture". In Bambah, R. P.; Dumir, V. C.; Hans-Gill, R. J. (eds.). Number Theory. Trends in Mathematics. Basel: Birkhäuser. pp. 75–106. ISBN 3-7643-6259-6.
• Dąbrowski, Andrzej (1996). "On the diophantine equation x! + A = y2". Nieuw Archief voor Wiskunde, IV. 14: 321–324. Zbl 0876.11015.
• Elkies, N. D. (1991). "ABC implies Mordell". International Mathematics Research Notices. 1991 (7): 99–109. doi:10.1155/S1073792891000144.
• Frey, Gerhard (1997). "On Ternary Equations of Fermat Type and Relations with Elliptic Curves". Modular Forms and Fermat's Last Theorem. New York: Springer. pp. 527–548. ISBN 0-387-94609-8.
• Goldfeld, Dorian (1996). "Beyond the last theorem". Math Horizons. 4 (September): 26–34. doi:10.1080/10724117.1996.11974985. JSTOR 25678079.
• Goldfeld, Dorian (2002). "Modular forms, elliptic curves and the abc-conjecture". In Wüstholz, Gisbert (ed.). A panorama in number theory or The view from Baker's garden. Based on a conference in honor of Alan Baker's 60th birthday, Zürich, Switzerland, 1999. Cambridge: Cambridge University Press. pp. 128–147. ISBN 0-521-80799-9. Zbl 1046.11035.
• Gowers, Timothy; Barrow-Green, June; Leader, Imre, eds. (2008). The Princeton Companion to Mathematics. Princeton: Princeton University Press. pp. 361–362, 681. ISBN 978-0-691-11880-2.
• Granville, A. (1998). "ABC Allows Us to Count Squarefrees" (PDF). International Mathematics Research Notices. 1998 (19): 991–1009. doi:10.1155/S1073792898000592.
• Granville, Andrew; Stark, H. (2000). "ABC implies no "Siegel zeros" for L-functions of characters with negative exponent" (PDF). Inventiones Mathematicae. 139 (3): 509–523. Bibcode:2000InMat.139..509G. doi:10.1007/s002229900036. S2CID 6901166.
• Granville, Andrew; Tucker, Thomas (2002). "It's As Easy As abc" (PDF). Notices of the AMS. 49 (10): 1224–1231. CiteSeerX 10.1.1.146.610.
• Guy, Richard K. (2004). Unsolved Problems in Number Theory. Berlin: Springer-Verlag. ISBN 0-387-20860-7.
• Lando, Sergei K.; Zvonkin, Alexander K. (2004). "Graphs on Surfaces and Their Applications". Encyclopaedia of Mathematical Sciences: Lower-Dimensional Topology II. Vol. 141. Springer-Verlag. ISBN 3-540-00203-0.
• Langevin, M. (1993). "Cas d'égalité pour le théorème de Mason et applications de la conjecture abc". Comptes rendus de l'Académie des sciences (in French). 317 (5): 441–444.
• Masser, D. W. (1985). "Open problems". In Chen, W. W. L. (ed.). Proceedings of the Symposium on Analytic Number Theory. London: Imperial College.
• Mollin, R.A. (2009). "A note on the ABC-conjecture" (PDF). Far East Journal of Mathematical Sciences. 33 (3): 267–275. ISSN 0972-0871. Zbl 1241.11034. Archived from the original (PDF) on 2016-03-04. Retrieved 2013-06-14.
• Mollin, Richard A. (2010). Advanced number theory with applications. Boca Raton, FL: CRC Press. ISBN 978-1-4200-8328-6. Zbl 1200.11002.
• Nitaj, Abderrahmane (1996). "La conjecture abc". Enseign. Math. (in French). 42 (1–2): 3–24.
• Oesterlé, Joseph (1988), "Nouvelles approches du "théorème" de Fermat", Astérisque, Séminaire Bourbaki exp 694 (161): 165–186, ISSN 0303-1179, MR 0992208
• Pomerance, Carl (2008). "Computational Number Theory". The Princeton Companion to Mathematics. Princeton University Press. pp. 361–362.
• Silverman, Joseph H. (1988). "Wieferich's criterion and the abc-conjecture". Journal of Number Theory. 30 (2): 226–237. doi:10.1016/0022-314X(88)90019-4. Zbl 0654.10019.
• Robert, Olivier; Stewart, Cameron L.; Tenenbaum, Gérald (2014). "A refinement of the abc conjecture" (PDF). Bulletin of the London Mathematical Society. 46 (6): 1156–1166. doi:10.1112/blms/bdu069. S2CID 123460044.
• Robert, Olivier; Tenenbaum, Gérald (November 2013). "Sur la répartition du noyau d'un entier" [On the distribution of the kernel of an integer]. Indagationes Mathematicae (in French). 24 (4): 802–914. doi:10.1016/j.indag.2013.07.007.
• Stewart, C. L.; Tijdeman, R. (1986). "On the Oesterlé-Masser conjecture". Monatshefte für Mathematik. 102 (3): 251–257. doi:10.1007/BF01294603. S2CID 123621917.
• Stewart, C. L.; Yu, Kunrui (1991). "On the abc conjecture". Mathematische Annalen. 291 (1): 225–230. doi:10.1007/BF01445201. S2CID 123894587.
• Stewart, C. L.; Yu, Kunrui (2001). "On the abc conjecture, II". Duke Mathematical Journal. 108 (1): 169–181. doi:10.1215/S0012-7094-01-10815-6.
• van Frankenhuysen, Machiel (2000). "A Lower Bound in the abc Conjecture". J. Number Theory. 82 (1): 91–95. doi:10.1006/jnth.1999.2484. MR 1755155.
• Van Frankenhuijsen, Machiel (2002). "The ABC conjecture implies Vojta's height inequality for curves". J. Number Theory. 95 (2): 289–302. doi:10.1006/jnth.2001.2769. MR 1924103.
• Waldschmidt, Michel (2015). "Lecture on the abc Conjecture and Some of Its Consequences" (PDF). Mathematics in the 21st Century. Springer Proceedings in Mathematics & Statistics. Vol. 98. pp. 211–230. doi:10.1007/978-3-0348-0859-0_13. ISBN 978-3-0348-0858-3.
External links
• ABC@home Distributed computing project called ABC@Home.
• Easy as ABC: Easy to follow, detailed explanation by Brian Hayes.
• Weisstein, Eric W. "abc Conjecture". MathWorld.
• Abderrahmane Nitaj's ABC conjecture home page
• Bart de Smit's ABC Triples webpage
• http://www.math.columbia.edu/~goldfeld/ABC-Conjecture.pdf
• The ABC's of Number Theory by Noam D. Elkies
• Questions about Number by Barry Mazur
• Philosophy behind Mochizuki’s work on the ABC conjecture on MathOverflow
• ABC Conjecture Polymath project wiki page linking to various sources of commentary on Mochizuki's papers.
• abc Conjecture Numberphile video
• News about IUT by Mochizuki
| Wikipedia |
ABS methods
ABS methods, where the acronym contains the initials of Jozsef Abaffy, Charles G. Broyden and Emilio Spedicato, have been developed since 1981 to generate a large class of algorithms for the following applications:
• solution of general linear algebraic systems, determined or underdetermined,
• full or deficient rank;
• solution of linear Diophantine systems, i.e. equation systems where the coefficient matrix and the right hand side are integer valued and an integer solution is sought; this is a special but important case of Hilbert's tenth problem, the only one in practice soluble;
• solution of nonlinear algebraic equations;
• solution of continuous unconstrained or constrained optimization.
At the beginning of 2007 ABS literature consisted of over 400 papers and reports and two monographs, one due to Abaffy and Spedicato and published in 1989, one due to Xia and Zhang and published, in Chinese, in 1998. Moreover, three conferences had been organized in China.
Research on ABS methods has been the outcome of an international collaboration coordinated by Spedicato of university of Bergamo, Italy. It has involved over forty mathematicians from Hungary, UK, China, Iran and other countries.
The central element in such methods is the use of a special matrix transformation due essentially to the Hungarian mathematician Jenő Egerváry, who investigated its main properties in some papers that went unnoticed. For the basic problem of solving a linear system of m equations in n variables, where $\scriptstyle m\,\leq \,n$, ABS methods use the following simple geometric idea:
1. Given an arbitrary initial estimate of the solution, find one of the infinite solutions, defining a linear variety of dimension n − 1, of the first equation.
2. Find a solution of the second equation that is also a solution of the first, i.e. find a solution lying in the intersection of the linear varieties of the solutions of the first two equations considered separately.
3. By iteration of the above approach after m' steps one gets a solution of the last equation that is also a solution of the previous equations, hence of the full system. Moreover, it is possible to detect equations that are either redundant or incompatible.
Among the main results obtained so far:
• unification of algorithms for linear, nonlinear algebraic equations and for linearly constrained nonlinear optimization, including the LP problem as a special case;
• the method of Gauss has been improved by reducing the required memory and eliminating the need for pivoting;
• new methods for nonlinear systems with convergence properties better than for Newton method;
• derivation of a general algorithm for Hilbert tenth problem, linear case, with the extension of a classic Euler theorem from one equation to a system;
• solvers have been obtained that are more stable than classical ones, especially for the problem arising in primal-dual interior point method;
• ABS methods are usually faster on vector or parallel machines;
• ABS methods provide a simpler approach for teaching for a variety of classes of problems, since particular methods are obtained just by specific parameter choices.
Knowledge of ABS methods is still quite limited among mathematicians, but they have great potential for improving the methods currently in use.
Bibliography
• Jozsef Abaffy, Emilio Spedicato (1989): ABS Projection Algorithms: Mathematical Techniques for Linear and Nonlinear Algebraic Equations, Ellis Horwood, Chichester. The first monograph on the subject
• Jozsef Abaffy, Charles G. Broyden, Emilio Spedicato (1984): A class of direct methods for linear equations, Numerische Mathematik 45, 361–376. Paper introducing ABS methods for continuous linear systems.
• H. Esmaeili, N. Mahdavi-Amiri, Emilio Spedicato: A class of ABS algorithms for Diophantine linear systems, Numerische Mathematik 90, 101–115. Paper introducing ABS methods for integer linear systems.
| Wikipedia |
AC0
AC0 is a complexity class used in circuit complexity. It is the smallest class in the AC hierarchy, and consists of all families of circuits of depth O(1) and polynomial size, with unlimited-fanin AND gates and OR gates (we allow NOT gates only at the inputs).[1] It thus contains NC0, which has only bounded-fanin AND and OR gates.[1]
Example problems
Integer addition and subtraction are computable in AC0,[2] but multiplication is not (specifically, when the inputs are two integers under the usual binary[3] or base-10 representations of integers).
Since it is a circuit class, like P/poly, AC0 also contains every unary language.
Descriptive complexity
From a descriptive complexity viewpoint, DLOGTIME-uniform AC0 is equal to the descriptive class FO+BIT of all languages describable in first-order logic with the addition of the BIT predicate, or alternatively by FO(+, ×), or by Turing machine in the logarithmic hierarchy.[4]
Separations
In 1984 Furst, Saxe, and Sipser showed that calculating the parity of the input bits (unlike the aforementioned addition/subtraction problems above which had two inputs) cannot be decided by any AC0 circuits, even with non-uniformity.[5][1] It follows that AC0 is not equal to NC1, because a family of circuits in the latter class can compute parity.[1] More precise bounds follow from switching lemma. Using them, it has been shown that there is an oracle separation between the polynomial hierarchy and PSPACE.
References
1. Arora, Sanjeev; Barak, Boaz (2009). Computational complexity. A modern approach. Cambridge University Press. pp. 117–118, 287. ISBN 978-0-521-42426-4. Zbl 1193.68112.
2. Barrington, David Mix; Maciel, Alexis (July 18, 2000). "Lecture 2: The Complexity of Some Problems" (PDF). IAS/PCMI Summer Session 2000, Clay Mathematics Undergraduate Program: Basic Course on Computational Complexity.
3. Kayal, Neeraj; Hegde, Sumant (2015). "Lecture 5: Feb 4, 2015" (PDF). E0 309: Topics in Complexity Theory. Archived (PDF) from the original on 2021-10-16. Retrieved 2021-10-16.
4. Immerman, N. (1999). Descriptive Complexity. Springer. p. 85.
5. Furst, Merrick; Saxe, James B.; Sipser, Michael (1984). "Parity, circuits, and the polynomial-time hierarchy". Mathematical Systems Theory. 17 (1): 13–27. doi:10.1007/BF01744431. MR 0738749. Zbl 0534.94008.
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
| Wikipedia |
ACC0
ACC0, sometimes called ACC, is a class of computational models and problems defined in circuit complexity, a field of theoretical computer science. The class is defined by augmenting the class AC0 of constant-depth "alternating circuits" with the ability to count; the acronym ACC stands for "AC with counters".[1] Specifically, a problem belongs to ACC0 if it can be solved by polynomial-size, constant-depth circuits of unbounded fan-in gates, including gates that count modulo a fixed integer. ACC0 corresponds to computation in any solvable monoid. The class is very well studied in theoretical computer science because of the algebraic connections and because it is one of the largest concrete computational models for which computational impossibility results, so-called circuit lower bounds, can be proved.
Definitions
Informally, ACC0 models the class of computations realised by Boolean circuits of constant depth and polynomial size, where the circuit gates includes "modular counting gates" that compute the number of true inputs modulo some fixed constant.
More formally, a language belongs to AC0[m] if it can be computed by a family of circuits C1, C2, ..., where Cn takes n inputs, the depth of every circuit is constant, the size of Cn is a polynomial function of n, and the circuit uses the following gates: AND gates and OR gates of unbounded fan-in, computing the conjunction and disjunction of their inputs; NOT gates computing the negation of their single input; and unbounded fan-in MOD-m gates, which compute 1 if the number of input 1s is a multiple of m. A language belongs to ACC0 if it belongs to AC0[m] for some m.
In some texts, ACCi refers to a hierarchy of circuit classes with ACC0 at its lowest level, where the circuits in ACCi have depth O(login) and polynomial size.[1]
The class ACC0 can also be defined in terms of computations of nonuniform deterministic finite automata (NUDFA) over monoids. In this framework, the input is interpreted as elements from a fixed monoid, and the input is accepted if the product of the input elements belongs to a given list of monoid elements. The class ACC0 is the family of languages accepted by a NUDFA over some monoid that does not contain an unsolvable group as a subsemigroup.[2]
Computational power
The class ACC0 includes AC0. This inclusion is strict, because a single MOD-2 gate computes the parity function, which is known to be impossible to compute in AC0. More generally, the function MODm cannot be computed in AC0[p] for prime p unless m is a power of p.[3]
The class ACC0 is included in TC0. It is conjectured that ACC0 is unable to compute the majority function of its inputs (i.e. the inclusion in TC0 is strict), but this remains unresolved as of July 2018.
Every problem in ACC0 can be solved by circuits of depth 2, with AND gates of polylogarithmic fan-in at the inputs, connected to a single gate computing a symmetric function.[4] These circuits are called SYM+-circuits. The proof follows ideas of the proof of Toda's theorem.
Williams (2011) proves that ACC0 does not contain NEXPTIME. The proof uses many results in complexity theory, including the time hierarchy theorem, IP = PSPACE, derandomization, and the representation of ACC0 via SYM+ circuits.[5] Murray & Williams (2018) improves this bound and proves that ACC0 does not contain NQP (nondeterministic quasipolynomial time).
It is known that computing the permanent is impossible for LOGTIME-uniform ACC0 circuits, which implies that the complexity class PP is not contained in LOGTIME-uniform ACC0.[6]
Notes
1. Vollmer (1999), p. 126
2. Thérien (1981), Barrington & Thérien (1988)
3. Razborov (1987), Smolensky (1987)
4. Beigel & Tarui (1994)
5. Addendum to Arora, Barak textbook
6. Allender & Gore (1994)
References
• Allender, Eric (1996), "Circuit complexity before the dawn of the new millennium", 16th Conference on Foundations of Software Technology and Theoretical Computer Science,Hyderabad, India, December 18–20, 1996 (PDF), Lecture Notes in Computer Science, vol. 1180, Springer, pp. 1–18, doi:10.1007/3-540-62034-6_33
• Allender, Eric; Gore, Vivec (1994), "A uniform circuit lower bound for the permanent" (PDF), SIAM Journal on Computing, 23 (5): 1026–1049, doi:10.1137/S0097539792233907, archived from the original (PDF) on 2016-03-03, retrieved 2012-07-02
• Barrington, D.A. (1989), "Bounded-width polynomial-size branching programs recognize exactly those languages in NC1" (PDF), Journal of Computer and System Sciences, 38 (1): 150–164, doi:10.1016/0022-0000(89)90037-8.
• Barrington, David A. Mix (1992), "Some problems involving Razborov-Smolensky polynomials", in Paterson, M.S. (ed.), Boolean function complexity, Sel. Pap. Symp., Durham/UK 1990., London Mathematical Society Lecture Notes Series, vol. 169, pp. 109–128, ISBN 0-521-40826-1, Zbl 0769.68041.
• Barrington, D.A.; Thérien, D. (1988), "Finite monoids and the fine structure of NC1", Journal of the ACM, 35 (4): 941–952, doi:10.1145/48014.63138, S2CID 52148641
• Beigel, Richard; Tarui, Jun (1994), "On ACC", Computational Complexity, 4 (4): 350–366, doi:10.1007/BF01263423, S2CID 2582220.
• Clote, Peter; Kranakis, Evangelos (2002), Boolean functions and computation models, Texts in Theoretical Computer Science. An EATCS Series, Berlin: Springer-Verlag, ISBN 3-540-59436-1, Zbl 1016.94046
• Razborov, A. A. (1987), "Lower bounds for the size of circuits of bounded depth with basis {⊕,∨}", Math. Notes of the Academy of Science of the USSR, 41 (4): 333–338.
• Smolensky, R. (1987), "Algebraic methods in the theory of lower bounds for Boolean circuit complexity", Proc. 19th ACM Symposium on Theory of Computing, pp. 77–82, doi:10.1145/28395.28404.
• Murray, Cody D.; Williams, Ryan (2018), "Circuit Lower Bounds for Nondeterministic Quasi-Polytime: An Easy Witness Lemma for NP and NQP", Proc. 50th ACM Symposium on Theory of Computing, pp. 890–901, doi:10.1145/3188745.3188910, hdl:1721.1/130542, S2CID 3685013
• Thérien, D. (1981), "Classification of finite monoids: The language approach", Theoretical Computer Science, 14 (2): 195–208, doi:10.1016/0304-3975(81)90057-8.
• Vollmer, Heribert (1999), Introduction to Circuit Complexity, Berlin: Springer, ISBN 3-540-64310-9.
• Williams, Ryan (2011). "Non-uniform ACC Circuit Lower Bounds". 2011 IEEE 26th Annual Conference on Computational Complexity (PDF). pp. 115–125. doi:10.1109/CCC.2011.36. ISBN 978-1-4577-0179-5..
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
| Wikipedia |
ACORN (random number generator)
The ACORN or ″Additive Congruential Random Number″ generators are a robust family of pseudorandom number generators (PRNGs) for sequences of uniformly distributed pseudo-random numbers, introduced in 1989 and still valid in 2019, thirty years later.
Introduced by R.S.Wikramaratna,[1] ACORN was originally designed for use in geostatistical and geophysical Monte Carlo simulations, and later extended for use on parallel computers.[2]
Over the ensuing decades, theoretical analysis (formal proof of convergence and statistical results), empirical testing (using standard test suites), and practical application work have continued, despite the appearance and promotion of other better-known [but not necessarily better performing] PRNGs.
Benefits
The main advantages of ACORN are simplicity of concept and coding, speed of execution, long period length, and mathematically proven convergence.[3]
The algorithm can be extended, if future applications require “better quality” pseudo random numbers and longer period, by increasing the order and the modulus as required. In addition, recent research has shown that the ACORN generators pass all the tests in the TestU01 test suite, current version 1.2.3, with an appropriate choice of parameters and with a few very straightforward constraints on the choice of initialisation; it is worth noting, as pointed out by the authors of TestU01, that some widely-used pseudo-random number generators fail badly on some of the tests .
ACORN is particularly simple to implement in exact integer arithmetic, in various computer languages, using only a few lines of code.[4] Integer arithmetic is preferred to the real arithmetic modulo 1 in the original presentation, as the algorithm is then reproducible, producing exactly the same sequence on any machine and in any language,[2] and its periodicity is mathematically provable.
The ACORN generator has not seen the wide adoption of some other PRNGs, although it is included in the Fortran and C library routines of NAG Numerical Library.[5] Various reasons have been put forward for this.[6] Nevertheless, theoretical and empirical research is ongoing to further justify the continuing use of ACORN as a robust and effective PRNG.
Provisos
In testing, ACORN performs extremely well, for appropriate parameters.[6] However in its present form, ACORN has not been shown to be suitable for cryptography.
There have been few critical appraisals regarding ACORN. One of these,[7] warns of an unsatisfactory configuration of the acorni() routine when using GSLIB GeoStatistical modelling and simulation library,[8] and proposes a simple solution for this issue. Essentially the modulus parameter should be increased in order to avoid this issue.[9][6]
Another brief reference to ACORN simply states that the "... ACORN generator proposed recently […] is in fact equivalent to a MLCG with matrix A such that a~ = 1 for i 2 j, aq = 0 otherwise"[10] but the analysis is not taken further.
ACORN is not the same as ACG (Additive Congruential Generator) and should not be confused with it - ACG appears to have been used for a variant of the LCG (Linear Congruential Generator) described by Knuth (1997).
History and development
Initially, ACORN was implemented in real arithmetic in FORTRAN77,[1] and shown to give better speed of execution and statistical performance than Linear Congruential Generators and Chebyshev Generators.
In 1992, further results were published,[11] implementing the ACORN Pseudo-Random Number Generator in exact integer arithmetic which ensures reproducibility across different platforms and languages, and stating that for arbitrary real-precision arithmetic it is possible to prove convergence of the ACORN sequence to k-distributed as the precision increases.
In 2000, ACORN was stated to be a special case of a Multiple Recursive Generator (and, therefore, of a Matrix Generator),[2] and this was formally demonstrated in 2008[12] in a paper which also published results of empirical Diehard tests and comparisons with the NAG LCG (Linear Congruential Generator).
In 2009, formal proof was given[4] of theoretical convergence of ACORN to k-distributed for modulus M=2m as m tends to infinity (as previously alluded to in 1992[11]), together with empirical results supporting this, which showed that ACORN generators are able to pass all the tests in the standard TESTU01[13] suite for testing of PRNGs (when appropriate order and modulus parameters are selected).
Since 2009 ACORN has been included in the NAG (Numerical Algorithms Group) FORTRAN and C library routines,[14][5] together with other well-known PRNG routines. This implementation of ACORN works for arbitrarily large modulus and order, and is available for researchers to download.[5]
ACORN was also implemented in GSLIB GeoStatistical modelling and simulation library.[8]
More recently, ACORN was presented in April 2019 at a poster session at a conference on Numerical algorithms for high-performance computational science[15] at the Royal Society in London, and in June 2019 at a Seminar of the Numerical Analysis Group at the Mathematical Institute of the University of Oxford.[16] where it was stated that statistical performance is better than some very widely used generators (including the Mersenne Twister MT19937) and comparable to the best currently available methods" and that ACORN generators have been shown to reliably pass all the tests in the TestU01, while some other generators including Mersenne Twister do not pass all these tests. The poster and the presentation can be found at.[9]
Code example
The following example in Fortran77 was published in 2008[12] which includes a discussion of how to initialise :
DOUBLE PRECISION FUNCTION ACORNJ(XDUMMY)
C
C Fortran implementation of ACORN random number generator
C of order less than or equal to 120 (higher orders can be
C obtained by increasing the parameter value MAXORD) and
C modulus less than or equal to 2^60.
C
C After appropriate initialization of the common block /IACO2/
C each call to ACORNJ generates a single variate drawn from
C a uniform distribution over the unit interval.
C
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
PARAMETER (MAXORD=120,MAXOP1=MAXORD+1)
COMMON /IACO2/ KORDEJ,MAXJNT,IXV1(MAXOP1),IXV2(MAXOP1)
DO 7 I=1,KORDEJ
IXV1(I+1)=(IXV1(I+1)+IXV1(I))
IXV2(I+1)=(IXV2(I+1)+IXV2(I))
IF (IXV2(I+1).GE.MAXJNT) THEN
IXV2(I+1)=IXV2(I+1)-MAXJNT
IXV1(I+1)=IXV1(I+1)+1
ENDIF
IF (IXV1(I+1).GE.MAXJNT) IXV1(I+1)=IXV1(I+1)-MAXJNT
7 CONTINUE
ACORNJ=(DBLE(IXV1(KORDEJ+1))
1 +DBLE(IXV2(KORDEJ+1))/MAXJNT)/MAXJNT
RETURN
END
External links
• The ACORN website (ACORN.wikramaratna.org): contains information regarding the ACORN concept and algorithm, its author, a complete list of references, and information about current research directions.
References
1. Wikramaratna, R.S. (1989). ACORN — A new method for generating sequences of uniformly distributed Pseudo-random Numbers. Journal of Computational Physics. 83. 16-31.
2. R.S. Wikramaratna, Pseudo-random number generation for parallel processing — A splitting approach, SIAM News 33 (9) (2000).
3. "ACORN concept and algorithm". acorn.wikramaratna.org/concept.html.
4. R.S. Wikramaratna, Theoretical and empirical convergence results for additive congruential random number generators, Journal of Computational and Applied Mathematics (2009), doi:10.1016/j.cam.2009.10.015
5. "g05 Chapter Introduction : NAG Library, Mark 26". www.nag.co.uk.
6. "ACORN initialisation and critique". acorn.wikramaratna.org/critique.html.
7. Ortiz, Julián & V. Deutsch, Clayton. (2014). Random Number Generation with acorni: A Warning Note.
8. GsLib An open-source package dedicated to geostatistics, source code in Fortran 77 and 90.
9. "ACORN references and links". acorn.wikramaratna.org/references.html.
10. L’Ecuyer, Pierre. (1990). Random Numbers for Simulation.. Commun. ACM. 33. 85-97. 10.1145/84537.84555.
11. R.S. Wikramaratna, Theoretical background for the ACORN random number generator, Report AEA-APS-0244, AEA Technology, Winfrith, Dorset, UK, 1992.
12. Wikramaratna, Roy (2008). "The additive congruential random number generator – a special case of a multiple recursive generator". J. Comput. Appl. Math. 216 (2): 371–387. Bibcode:2008JCoAM.216..371W. doi:10.1016/j.cam.2007.05.018.
13. P. L'Ecuyer, R. Simard, TestU01: A C library for empirical testing of random number generators, ACM Trans. on Math. Software 33 (4) (2007) Article 22.
14. NAG, Numerical Algorithms Group (NAG) Fortran Library Mark 22, Numerical Algorithms Group Ltd., Oxford, UK, 2009.
15. "Numerical algorithms for high-performance computational science". The Royal Society.
16. "The Additive Congruential Random Number (ACORN) Generator - pseudo-random sequences that are well distributed in k-dimensions". Oxford University Mathematical Institute.
| Wikipedia |
AC (complexity)
In circuit complexity, AC is a complexity class hierarchy. Each class, ACi, consists of the languages recognized by Boolean circuits with depth $O(\log ^{i}n)$ and a polynomial number of unlimited fan-in AND and OR gates.
The name "AC" was chosen by analogy to NC, with the "A" in the name standing for "alternating" and referring both to the alternation between the AND and OR gates in the circuits and to alternating Turing machines.[1]
The smallest AC class is AC0, consisting of constant-depth unlimited fan-in circuits.
The total hierarchy of AC classes is defined as
${\mbox{AC}}=\bigcup _{i\geq 0}{\mbox{AC}}^{i}$
Relation to NC
The AC classes are related to the NC classes, which are defined similarly, but with gates having only constant fanin. For each i, we have[2][3]
${\mbox{NC}}^{i}\subseteq {\mbox{AC}}^{i}\subseteq {\mbox{NC}}^{i+1}.$
As an immediate consequence of this, we have that NC = AC.[4]
It is known that inclusion is strict for i = 0.[3]
Variations
The power of the AC classes can be affected by adding additional gates. If we add gates which calculate the modulo operation for some modulus m, we have the classes ACCi[m].[4]
Notes
1. Regan (1999, p. 27-18)
2. Clote & Kranakis (2002, p. 437)
3. Arora & Barak (2009, p. 118)
4. Clote & Kranakis (2002, p. 12)
References
• Arora, Sanjeev; Barak, Boaz (2009), Computational complexity. A modern approach, Cambridge University Press, ISBN 978-0-521-42426-4, Zbl 1193.68112
• Clote, Peter; Kranakis, Evangelos (2002), Boolean functions and computation models, Texts in Theoretical Computer Science. An EATCS Series, Berlin: Springer-Verlag, ISBN 3-540-59436-1, Zbl 1016.94046
• Regan, Kenneth W. (1999), "Complexity classes", Algorithms and Theory of Computation Handbook, CRC Press.
• Vollmer, Heribert (1998), Introduction to circuit complexity. A uniform approach, Texts in Theoretical Computer Science, Berlin: Springer-Verlag, ISBN 3-540-64310-9, Zbl 0931.68055
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
| Wikipedia |
AD+
In set theory, AD+ is an extension, proposed by W. Hugh Woodin, to the axiom of determinacy. The axiom, which is to be understood in the context of ZF plus DC$\mathbb {R} $ (the axiom of dependent choice for real numbers), states two things:
1. Every set of reals is ∞-Borel.
2. For any ordinal λ less than Θ, any subset A of ωω, and any continuous function π:λω→ωω, the preimage π−1[A] is determined. (Here λω is to be given the product topology, starting with the discrete topology on λ.)
The second clause by itself is referred to as ordinal determinacy.
See also
• Axiom of projective determinacy
• Axiom of real determinacy
• Suslin's problem
• Topological game
References
• Woodin, W. Hugh (1999). The axiom of determinacy, forcing axioms, and the nonstationary ideal (1st ed.). Berlin: W. de Gruyter. p. 618. ISBN 311015708X.
| Wikipedia |
Advanced Encryption Standard
The Advanced Encryption Standard (AES), also known by its original name Rijndael (Dutch pronunciation: [ˈrɛindaːl]),[5] is a specification for the encryption of electronic data established by the U.S. National Institute of Standards and Technology (NIST) in 2001.[6]
Advanced Encryption Standard
(Rijndael)
Visualization of the AES round function
General
DesignersJoan Daemen, Vincent Rijmen
First published1998
Derived fromSquare
SuccessorsAnubis, Grand Cru, Kalyna
CertificationAES winner, CRYPTREC, NESSIE, NSA
Cipher detail
Key sizes128, 192 or 256 bits[note 1]
Block sizes128 bits[note 2]
StructureSubstitution–permutation network
Rounds10, 12 or 14 (depending on key size)
Best public cryptanalysis
Attacks have been published that are computationally faster than a full brute-force attack, though none as of 2013 are computationally feasible.[1]
For AES-128, the key can be recovered with a computational complexity of 2126.1 using the biclique attack. For biclique attacks on AES-192 and AES-256, the computational complexities of 2189.7 and 2254.4 respectively apply. Related-key attacks can break AES-256 and AES-192 with complexities 299.5 and 2176 in both time and data, respectively.[2]
Another attack was blogged[3] and released as a preprint[4] in 2009. This attack is against AES-256 that uses only two related keys and 239 time to recover the complete 256-bit key of a 9-round version, or 245 time for a 10-round version with a stronger type of related subkey attack, or 270 time for an 11-round version.
AES is a variant of the Rijndael block cipher[5] developed by two Belgian cryptographers, Joan Daemen and Vincent Rijmen, who submitted a proposal[7] to NIST during the AES selection process.[8] Rijndael is a family of ciphers with different key and block sizes. For AES, NIST selected three members of the Rijndael family, each with a block size of 128 bits, but three different key lengths: 128, 192 and 256 bits.
AES has been adopted by the U.S. government. It supersedes the Data Encryption Standard (DES),[9] which was published in 1977. The algorithm described by AES is a symmetric-key algorithm, meaning the same key is used for both encrypting and decrypting the data.
In the United States, AES was announced by the NIST as U.S. FIPS PUB 197 (FIPS 197) on November 26, 2001.[6] This announcement followed a five-year standardization process in which fifteen competing designs were presented and evaluated, before the Rijndael cipher was selected as the most suitable.[note 3]
AES is included in the ISO/IEC 18033-3 standard. AES became effective as a U.S. federal government standard on May 26, 2002, after approval by the U.S. Secretary of Commerce. AES is available in many different encryption packages, and is the first (and only) publicly accessible cipher approved by the U.S. National Security Agency (NSA) for top secret information when used in an NSA approved cryptographic module.[note 4]
Definitive standards
The Advanced Encryption Standard (AES) is defined in each of:
• FIPS PUB 197: Advanced Encryption Standard (AES)[6]
• ISO/IEC 18033-3: Block ciphers[10]
Description of the ciphers
AES is based on a design principle known as a substitution–permutation network, and is efficient in both software and hardware.[11] Unlike its predecessor DES, AES does not use a Feistel network. AES is a variant of Rijndael, with a fixed block size of 128 bits, and a key size of 128, 192, or 256 bits. By contrast, Rijndael per se is specified with block and key sizes that may be any multiple of 32 bits, with a minimum of 128 and a maximum of 256 bits. Most AES calculations are done in a particular finite field.
AES operates on a 4 × 4 column-major order array of 16 bytes b0, b1, ..., b15 termed the state:[note 5]
${\begin{bmatrix}b_{0}&b_{4}&b_{8}&b_{12}\\b_{1}&b_{5}&b_{9}&b_{13}\\b_{2}&b_{6}&b_{10}&b_{14}\\b_{3}&b_{7}&b_{11}&b_{15}\end{bmatrix}}$
The key size used for an AES cipher specifies the number of transformation rounds that convert the input, called the plaintext, into the final output, called the ciphertext. The number of rounds are as follows:
• 10 rounds for 128-bit keys.
• 12 rounds for 192-bit keys.
• 14 rounds for 256-bit keys.
Each round consists of several processing steps, including one that depends on the encryption key itself. A set of reverse rounds are applied to transform ciphertext back into the original plaintext using the same encryption key.
High-level description of the algorithm
1. KeyExpansion – round keys are derived from the cipher key using the AES key schedule. AES requires a separate 128-bit round key block for each round plus one more.
2. Initial round key addition:
1. AddRoundKey – each byte of the state is combined with a byte of the round key using bitwise xor.
3. 9, 11 or 13 rounds:
1. SubBytes – a non-linear substitution step where each byte is replaced with another according to a lookup table.
2. ShiftRows – a transposition step where the last three rows of the state are shifted cyclically a certain number of steps.
3. MixColumns – a linear mixing operation which operates on the columns of the state, combining the four bytes in each column.
4. AddRoundKey
4. Final round (making 10, 12 or 14 rounds in total):
1. SubBytes
2. ShiftRows
3. AddRoundKey
The SubBytes step
In the SubBytes step, each byte $a_{i,j}$ in the state array is replaced with a SubByte $S(a_{i,j})$ using an 8-bit substitution box. Note that before round 0, the state array is simply the plaintext/input. This operation provides the non-linearity in the cipher. The S-box used is derived from the multiplicative inverse over GF(28), known to have good non-linearity properties. To avoid attacks based on simple algebraic properties, the S-box is constructed by combining the inverse function with an invertible affine transformation. The S-box is also chosen to avoid any fixed points (and so is a derangement), i.e., $S(a_{i,j})\neq a_{i,j}$, and also any opposite fixed points, i.e., $S(a_{i,j})\oplus a_{i,j}\neq {\text{FF}}_{16}$. While performing the decryption, the InvSubBytes step (the inverse of SubBytes) is used, which requires first taking the inverse of the affine transformation and then finding the multiplicative inverse.
The ShiftRows step
The ShiftRows step operates on the rows of the state; it cyclically shifts the bytes in each row by a certain offset. For AES, the first row is left unchanged. Each byte of the second row is shifted one to the left. Similarly, the third and fourth rows are shifted by offsets of two and three respectively.[note 6] In this way, each column of the output state of the ShiftRows step is composed of bytes from each column of the input state. The importance of this step is to avoid the columns being encrypted independently, in which case AES would degenerate into four independent block ciphers.
The MixColumns step
Main article: Rijndael MixColumns
In the MixColumns step, the four bytes of each column of the state are combined using an invertible linear transformation. The MixColumns function takes four bytes as input and outputs four bytes, where each input byte affects all four output bytes. Together with ShiftRows, MixColumns provides diffusion in the cipher.
During this operation, each column is transformed using a fixed matrix (matrix left-multiplied by column gives new value of column in the state):
${\begin{bmatrix}b_{0,j}\\b_{1,j}\\b_{2,j}\\b_{3,j}\end{bmatrix}}={\begin{bmatrix}2&3&1&1\\1&2&3&1\\1&1&2&3\\3&1&1&2\end{bmatrix}}{\begin{bmatrix}a_{0,j}\\a_{1,j}\\a_{2,j}\\a_{3,j}\end{bmatrix}}\qquad 0\leq j\leq 3$
Matrix multiplication is composed of multiplication and addition of the entries. Entries are bytes treated as coefficients of polynomial of order $x^{7}$. Addition is simply XOR. Multiplication is modulo irreducible polynomial $x^{8}+x^{4}+x^{3}+x+1$. If processed bit by bit, then, after shifting, a conditional XOR with 1B16 should be performed if the shifted value is larger than FF16 (overflow must be corrected by subtraction of generating polynomial). These are special cases of the usual multiplication in $\operatorname {GF} (2^{8})$.
In more general sense, each column is treated as a polynomial over $\operatorname {GF} (2^{8})$ and is then multiplied modulo ${01}_{16}\cdot z^{4}+{01}_{16}$ with a fixed polynomial $c(z)={03}_{16}\cdot z^{3}+{01}_{16}\cdot z^{2}+{01}_{16}\cdot z+{02}_{16}$. The coefficients are displayed in their hexadecimal equivalent of the binary representation of bit polynomials from $\operatorname {GF} (2)[x]$. The MixColumns step can also be viewed as a multiplication by the shown particular MDS matrix in the finite field $\operatorname {GF} (2^{8})$. This process is described further in the article Rijndael MixColumns.
The AddRoundKey
In the AddRoundKey step, the subkey is combined with the state. For each round, a subkey is derived from the main key using Rijndael's key schedule; each subkey is the same size as the state. The subkey is added by combining of the state with the corresponding byte of the subkey using bitwise XOR.
Optimization of the cipher
On systems with 32-bit or larger words, it is possible to speed up execution of this cipher by combining the SubBytes and ShiftRows steps with the MixColumns step by transforming them into a sequence of table lookups. This requires four 256-entry 32-bit tables (together occupying 4096 bytes). A round can then be performed with 16 table lookup operations and 12 32-bit exclusive-or operations, followed by four 32-bit exclusive-or operations in the AddRoundKey step.[12] Alternatively, the table lookup operation can be performed with a single 256-entry 32-bit table (occupying 1024 bytes) followed by circular rotation operations.
Using a byte-oriented approach, it is possible to combine the SubBytes, ShiftRows, and MixColumns steps into a single round operation.[13]
Security
The National Security Agency (NSA) reviewed all the AES finalists, including Rijndael, and stated that all of them were secure enough for U.S. Government non-classified data. In June 2003, the U.S. Government announced that AES could be used to protect classified information:
The design and strength of all key lengths of the AES algorithm (i.e., 128, 192 and 256) are sufficient to protect classified information up to the SECRET level. TOP SECRET information will require use of either the 192 or 256 key lengths. The implementation of AES in products intended to protect national security systems and/or information must be reviewed and certified by NSA prior to their acquisition and use.[14]
AES has 10 rounds for 128-bit keys, 12 rounds for 192-bit keys, and 14 rounds for 256-bit keys.
By 2006, the best known attacks were on 7 rounds for 128-bit keys, 8 rounds for 192-bit keys, and 9 rounds for 256-bit keys.[15]
Known attacks
For cryptographers, a cryptographic "break" is anything faster than a brute-force attack – i.e., performing one trial decryption for each possible key in sequence.[note 7] A break can thus include results that are infeasible with current technology. Despite being impractical, theoretical breaks can sometimes provide insight into vulnerability patterns. The largest successful publicly known brute-force attack against a widely implemented block-cipher encryption algorithm was against a 64-bit RC5 key by distributed.net in 2006.[16]
The key space increases by a factor of 2 for each additional bit of key length, and if every possible value of the key is equiprobable, this translates into a doubling of the average brute-force key search time. This implies that the effort of a brute-force search increases exponentially with key length. Key length in itself does not imply security against attacks, since there are ciphers with very long keys that have been found to be vulnerable.
AES has a fairly simple algebraic framework.[17] In 2002, a theoretical attack, named the "XSL attack", was announced by Nicolas Courtois and Josef Pieprzyk, purporting to show a weakness in the AES algorithm, partially due to the low complexity of its nonlinear components.[18] Since then, other papers have shown that the attack, as originally presented, is unworkable; see XSL attack on block ciphers.
During the AES selection process, developers of competing algorithms wrote of Rijndael's algorithm "we are concerned about [its] use ... in security-critical applications."[19] In October 2000, however, at the end of the AES selection process, Bruce Schneier, a developer of the competing algorithm Twofish, wrote that while he thought successful academic attacks on Rijndael would be developed someday, he "did not believe that anyone will ever discover an attack that will allow someone to read Rijndael traffic."[20]
Until May 2009, the only successful published attacks against the full AES were side-channel attacks on some specific implementations. In 2009, a new related-key attack was discovered that exploits the simplicity of AES's key schedule and has a complexity of 2119. In December 2009 it was improved to 299.5.[2] This is a follow-up to an attack discovered earlier in 2009 by Alex Biryukov, Dmitry Khovratovich, and Ivica Nikolić, with a complexity of 296 for one out of every 235 keys.[21] However, related-key attacks are not of concern in any properly designed cryptographic protocol, as a properly designed protocol (i.e., implementational software) will take care not to allow related keys, essentially by constraining an attacker's means of selecting keys for relatedness.
Another attack was blogged by Bruce Schneier[22] on July 30, 2009, and released as a preprint[23] on August 3, 2009. This new attack, by Alex Biryukov, Orr Dunkelman, Nathan Keller, Dmitry Khovratovich, and Adi Shamir, is against AES-256 that uses only two related keys and 239 time to recover the complete 256-bit key of a 9-round version, or 245 time for a 10-round version with a stronger type of related subkey attack, or 270 time for an 11-round version. 256-bit AES uses 14 rounds, so these attacks are not effective against full AES.
The practicality of these attacks with stronger related keys has been criticized,[24] for instance, by the paper on chosen-key-relations-in-the-middle attacks on AES-128 authored by Vincent Rijmen in 2010.[25]
In November 2009, the first known-key distinguishing attack against a reduced 8-round version of AES-128 was released as a preprint.[26] This known-key distinguishing attack is an improvement of the rebound, or the start-from-the-middle attack, against AES-like permutations, which view two consecutive rounds of permutation as the application of a so-called Super-S-box. It works on the 8-round version of AES-128, with a time complexity of 248, and a memory complexity of 232. 128-bit AES uses 10 rounds, so this attack is not effective against full AES-128.
The first key-recovery attacks on full AES were by Andrey Bogdanov, Dmitry Khovratovich, and Christian Rechberger, and were published in 2011.[27] The attack is a biclique attack and is faster than brute force by a factor of about four. It requires 2126.2 operations to recover an AES-128 key. For AES-192 and AES-256, 2190.2 and 2254.6 operations are needed, respectively. This result has been further improved to 2126.0 for AES-128, 2189.9 for AES-192 and 2254.3 for AES-256,[28] which are the current best results in key recovery attack against AES.
This is a very small gain, as a 126-bit key (instead of 128-bits) would still take billions of years to brute force on current and foreseeable hardware. Also, the authors calculate the best attack using their technique on AES with a 128-bit key requires storing 288 bits of data. That works out to about 38 trillion terabytes of data, which is more than all the data stored on all the computers on the planet in 2016. As such, there are no practical implications on AES security.[29] The space complexity has later been improved to 256 bits,[28] which is 9007 terabytes (while still keeping a time complexity of 2126.2).
According to the Snowden documents, the NSA is doing research on whether a cryptographic attack based on tau statistic may help to break AES.[30]
At present, there is no known practical attack that would allow someone without knowledge of the key to read data encrypted by AES when correctly implemented.
Side-channel attacks
Side-channel attacks do not attack the cipher as a black box, and thus are not related to cipher security as defined in the classical context, but are important in practice. They attack implementations of the cipher on hardware or software systems that inadvertently leak data. There are several such known attacks on various implementations of AES.
In April 2005, D. J. Bernstein announced a cache-timing attack that he used to break a custom server that used OpenSSL's AES encryption.[31] The attack required over 200 million chosen plaintexts.[32] The custom server was designed to give out as much timing information as possible (the server reports back the number of machine cycles taken by the encryption operation). However, as Bernstein pointed out, "reducing the precision of the server's timestamps, or eliminating them from the server's responses, does not stop the attack: the client simply uses round-trip timings based on its local clock, and compensates for the increased noise by averaging over a larger number of samples."[31]
In October 2005, Dag Arne Osvik, Adi Shamir and Eran Tromer presented a paper demonstrating several cache-timing attacks against the implementations in AES found in OpenSSL and Linux's dm-crypt partition encryption function.[33] One attack was able to obtain an entire AES key after only 800 operations triggering encryptions, in a total of 65 milliseconds. This attack requires the attacker to be able to run programs on the same system or platform that is performing AES.
In December 2009 an attack on some hardware implementations was published that used differential fault analysis and allows recovery of a key with a complexity of 232.[34]
In November 2010 Endre Bangerter, David Gullasch and Stephan Krenn published a paper which described a practical approach to a "near real time" recovery of secret keys from AES-128 without the need for either cipher text or plaintext. The approach also works on AES-128 implementations that use compression tables, such as OpenSSL.[35] Like some earlier attacks, this one requires the ability to run unprivileged code on the system performing the AES encryption, which may be achieved by malware infection far more easily than commandeering the root account.[36]
In March 2016, Ashokkumar C., Ravi Prakash Giri and Bernard Menezes presented a side-channel attack on AES implementations that can recover the complete 128-bit AES key in just 6–7 blocks of plaintext/ciphertext, which is a substantial improvement over previous works that require between 100 and a million encryptions.[37] The proposed attack requires standard user privilege and key-retrieval algorithms run under a minute.
Many modern CPUs have built-in hardware instructions for AES, which protect against timing-related side-channel attacks.[38][39]
NIST/CSEC validation
The Cryptographic Module Validation Program (CMVP) is operated jointly by the United States Government's National Institute of Standards and Technology (NIST) Computer Security Division and the Communications Security Establishment (CSE) of the Government of Canada. The use of cryptographic modules validated to NIST FIPS 140-2 is required by the United States Government for encryption of all data that has a classification of Sensitive but Unclassified (SBU) or above. From NSTISSP #11, National Policy Governing the Acquisition of Information Assurance: “Encryption products for protecting classified information will be certified by NSA, and encryption products intended for protecting sensitive information will be certified in accordance with NIST FIPS 140-2.”[40]
The Government of Canada also recommends the use of FIPS 140 validated cryptographic modules in unclassified applications of its departments.
Although NIST publication 197 (“FIPS 197”) is the unique document that covers the AES algorithm, vendors typically approach the CMVP under FIPS 140 and ask to have several algorithms (such as Triple DES or SHA1) validated at the same time. Therefore, it is rare to find cryptographic modules that are uniquely FIPS 197 validated and NIST itself does not generally take the time to list FIPS 197 validated modules separately on its public web site. Instead, FIPS 197 validation is typically just listed as an "FIPS approved: AES" notation (with a specific FIPS 197 certificate number) in the current list of FIPS 140 validated cryptographic modules.
The Cryptographic Algorithm Validation Program (CAVP)[41] allows for independent validation of the correct implementation of the AES algorithm. Successful validation results in being listed on the NIST validations page.[42] This testing is a pre-requisite for the FIPS 140-2 module validation. However, successful CAVP validation in no way implies that the cryptographic module implementing the algorithm is secure. A cryptographic module lacking FIPS 140-2 validation or specific approval by the NSA is not deemed secure by the US Government and cannot be used to protect government data.[40]
FIPS 140-2 validation is challenging to achieve both technically and fiscally.[43] There is a standardized battery of tests as well as an element of source code review that must be passed over a period of a few weeks. The cost to perform these tests through an approved laboratory can be significant (e.g., well over $30,000 US)[43] and does not include the time it takes to write, test, document and prepare a module for validation. After validation, modules must be re-submitted and re-evaluated if they are changed in any way. This can vary from simple paperwork updates if the security functionality did not change to a more substantial set of re-testing if the security functionality was impacted by the change.
Test vectors
Test vectors are a set of known ciphers for a given input and key. NIST distributes the reference of AES test vectors as AES Known Answer Test (KAT) Vectors.[note 8]
Performance
High speed and low RAM requirements were some of the criteria of the AES selection process. As the chosen algorithm, AES performed well on a wide variety of hardware, from 8-bit smart cards to high-performance computers.
On a Pentium Pro, AES encryption requires 18 clock cycles per byte,[44] equivalent to a throughput of about 11 MiB/s for a 200 MHz processor.
On Intel Core and AMD Ryzen CPUs supporting AES-NI instruction set extensions, throughput can be multiple GiB/s (even over 15 GiB/s on an i7-12700k).[45]
See also
• AES modes of operation
• Disk encryption
• Encryption
• Whirlpool – hash function created by Vincent Rijmen and Paulo S. L. M. Barreto
• List of free and open-source software packages
Notes
1. Key sizes of 128, 160, 192, 224, and 256 bits are supported by the Rijndael algorithm, but only the 128, 192, and 256-bit key sizes are specified in the AES standard.
2. Block sizes of 128, 160, 192, 224, and 256 bits are supported by the Rijndael algorithm for each key size, but only the 128-bit block size is specified in the AES standard.
3. See Advanced Encryption Standard process for more details.
4. See Security of AES below.
5. Large-block variants of Rijndael use an array with additional columns, but always four rows.
6. Rijndael variants with a larger block size have slightly different offsets. For blocks of sizes 128 bits and 192 bits, the shifting pattern is the same. Row $n$ is shifted left circular by $n-1$ bytes. For a 256-bit block, the first row is unchanged and the shifting for the second, third and fourth row is 1 byte, 3 bytes and 4 bytes respectively—this change only applies for the Rijndael cipher when used with a 256-bit block, as AES does not use 256-bit blocks.
7. See Cryptanalysis.
8. The AES Known Answer Test (KAT) Vectors are available in Zip format within the NIST site here Archived 2009-10-23 at the Wayback Machine
References
1. "Biclique Cryptanalysis of the Full AES" (PDF). Archived from the original (PDF) on March 6, 2016. Retrieved May 1, 2019.
2. Alex Biryukov and Dmitry Khovratovich, Related-key Cryptanalysis of the Full AES-192 and AES-256, "Archived copy". Table 1. Archived from the original on 2009-09-28. Retrieved 2010-02-16.{{cite web}}: CS1 maint: archived copy as title (link)
3. Bruce Schneier (2009-07-30). "Another New AES Attack". Schneier on Security, A blog covering security and security technology. Archived from the original on 2009-10-05. Retrieved 2010-03-11.
4. Alex Biryukov; Orr Dunkelman; Nathan Keller; Dmitry Khovratovich; Adi Shamir (2009-08-19). "Key Recovery Attacks of Practical Complexity on AES Variants With Up To 10 Rounds". Archived from the original on 28 January 2010. Retrieved 2010-03-11.
5. Daemen, Joan; Rijmen, Vincent (March 9, 2003). "AES Proposal: Rijndael" (PDF). National Institute of Standards and Technology. p. 1. Archived (PDF) from the original on 5 March 2013. Retrieved 21 February 2013.
6. "Announcing the ADVANCED ENCRYPTION STANDARD (AES)" (PDF). Federal Information Processing Standards Publication 197. United States National Institute of Standards and Technology (NIST). November 26, 2001. Archived (PDF) from the original on March 12, 2017. Retrieved October 2, 2012.
7. Joan Daemen and Vincent Rijmen (September 3, 1999). "AES Proposal: Rijndael" (PDF). Archived from the original (PDF) on February 3, 2007.
8. John Schwartz (October 3, 2000). "U.S. Selects a New Encryption Technique". New York Times. Archived from the original on March 28, 2017.
9. Westlund, Harold B. (2002). "NIST reports measurable success of Advanced Encryption Standard". Journal of Research of the National Institute of Standards and Technology. Archived from the original on 2007-11-03.
10. "ISO/IEC 18033-3: Information technology – Security techniques – Encryption algorithms – Part 3: Block ciphers". Archived from the original on 2013-12-03.
11. Bruce Schneier; John Kelsey; Doug Whiting; David Wagner; Chris Hall; Niels Ferguson; Tadayoshi Kohno; et al. (May 2000). "The Twofish Team's Final Comments on AES Selection" (PDF). Archived (PDF) from the original on 2010-01-02.
12. Bertoni, Guido; Breveglieri, Luca; Fragneto, Pasqualina; MacChetti, Marco; Marchesin, Stefano (2003). "Efficient Software Implementation of AES on 32-Bit Platforms". Cryptographic Hardware and Embedded Systems - CHES 2002. Lecture Notes in Computer Science. Vol. 2523. pp. 159–171. doi:10.1007/3-540-36400-5_13. ISBN 978-3-540-00409-7.
13. "byte-oriented-aes – A public domain byte-oriented implementation of AES in C – Google Project Hosting". Archived from the original on 2013-07-20. Retrieved 2012-12-23.
14. Lynn Hathaway (June 2003). "National Policy on the Use of the Advanced Encryption Standard (AES) to Protect National Security Systems and National Security Information" (PDF). Archived (PDF) from the original on 2010-11-06. Retrieved 2011-02-15.
15. John Kelsey, Stefan Lucks, Bruce Schneier, Mike Stay, David Wagner, and Doug Whiting, Improved Cryptanalysis of Rijndael, Fast Software Encryption, 2000 pp213–230 "Academic: Improved Cryptanalysis of Rijndael - Schneier on Security". Archived from the original on 2007-02-23. Retrieved 2007-03-06.
16. Ou, George (April 30, 2006). "Is encryption really crackable?". Ziff-Davis. Archived from the original on August 8, 2010. Retrieved August 7, 2010.
17. "Sean Murphy". University of London. Archived from the original on 2009-01-31. Retrieved 2008-11-02.
18. Bruce Schneier. "AES News, Crypto-Gram Newsletter, September 15, 2002". Archived from the original on 7 July 2007. Retrieved 2007-07-27.
19. Niels Ferguson; Richard Schroeppel; Doug Whiting (2001). "A simple algebraic representation of Rijndael". Proceedings of Selected Areas in Cryptography, 2001, Lecture Notes in Computer Science. Springer-Verlag. pp. 103–111. CiteSeerX 10.1.1.28.4921. Archived from the original (PDF/PostScript) on 4 November 2006. Retrieved 2006-10-06.
20. Bruce Schneier, AES Announced Archived 2009-02-01 at the Wayback Machine, October 15, 2000
21. Nikolić, Ivica (2009). "Distinguisher and Related-Key Attack on the Full AES-256". Advances in Cryptology – CRYPTO 2009. Lecture Notes in Computer Science. Vol. 5677. Springer Berlin / Heidelberg. pp. 231–249. doi:10.1007/978-3-642-03356-8_14. ISBN 978-3-642-03355-1.
22. Bruce Schneier (2009-07-30). "Another New AES Attack". Schneier on Security, A blog covering security and security technology. Archived from the original on 2009-10-05. Retrieved 2010-03-11.
23. Alex Biryukov; Orr Dunkelman; Nathan Keller; Dmitry Khovratovich; Adi Shamir (2009-08-19). "Key Recovery Attacks of Practical Complexity on AES Variants With Up To 10 Rounds". Archived from the original on 28 January 2010. Retrieved 2010-03-11.
24. Agren, Martin (2012). On Some Symmetric Lightweight Cryptographic Designs. Dissertation, Lund University. pp. 38–39.
25. Vincent Rijmen (2010). "Practical-Titled Attack on AES-128 Using Chosen-Text Relations" (PDF). Archived (PDF) from the original on 2010-07-02. {{cite journal}}: Cite journal requires |journal= (help)
26. Henri Gilbert; Thomas Peyrin (2009-11-09). "Super-Sbox Cryptanalysis: Improved Attacks for AES-like permutations". Archived from the original on 2010-06-04. Retrieved 2010-03-11.
27. Andrey Bogdanov; Dmitry Khovratovich & Christian Rechberger (2011). "Biclique Cryptanalysis of the Full AES" (PDF). Archived from the original (PDF) on 2012-09-05.
28. Biaoshuai Tao & Hongjun Wu (2015). Information Security and Privacy. Lecture Notes in Computer Science. Vol. 9144. pp. 39–56. doi:10.1007/978-3-319-19962-7_3. ISBN 978-3-319-19961-0.
29. Jeffrey Goldberg (2011-08-18). "AES Encryption isn't Cracked". Archived from the original on 8 January 2015. Retrieved 30 December 2014.
30. SPIEGEL ONLINE, Hamburg, Germany (28 December 2014). "Inside the NSA's War on Internet Security". SPIEGEL ONLINE. Archived from the original on 24 January 2015. Retrieved 4 September 2015.
31. "Index of formal scientific papers". Cr.yp.to. Archived from the original on 2008-09-17. Retrieved 2008-11-02.
32. Bruce Schneier. "AES Timing Attack". Archived from the original on 12 February 2007. Retrieved 2007-03-17.
33. Dag Arne Osvik; Adi Shamir; Eran Tromer (2005-11-20). "Cache Attacks and Countermeasures: the Case of AES" (PDF). Archived (PDF) from the original on 2006-06-19. Retrieved 2008-11-02. {{cite journal}}: Cite journal requires |journal= (help)
34. Dhiman Saha; Debdeep Mukhopadhyay; Dipanwita RoyChowdhury. "A Diagonal Fault Attack on the Advanced Encryption Standard" (PDF). Archived (PDF) from the original on 22 December 2009. Retrieved 2009-12-08. {{cite journal}}: Cite journal requires |journal= (help)
35. Endre Bangerter; David Gullasch & Stephan Krenn (2010). "Cache Games – Bringing Access-Based Cache Attacks on AES to Practice" (PDF). Archived (PDF) from the original on 2010-12-14. {{cite journal}}: Cite journal requires |journal= (help)
36. "Breaking AES-128 in realtime, no ciphertext required". Hacker News. Archived from the original on 2011-10-03. Retrieved 2012-12-23.
37. Ashokkumar C.; Ravi Prakash Giri; Bernard Menezes (2016). 2016 IEEE European Symposium on Security and Privacy (EuroS&P). pp. 261–275. doi:10.1109/EuroSP.2016.29. ISBN 978-1-5090-1751-5. S2CID 11251391.
38. "Are AES x86 Cache Timing Attacks Still Feasible?" (PDF). cseweb.ucsd.edu. Archived (PDF) from the original on 2017-08-09.
39. "Archived copy" (PDF). Archived (PDF) from the original on 2013-03-31. Retrieved 2017-07-26.{{cite web}}: CS1 maint: archived copy as title (link) Securing the Enterprise with Intel AES-NI.
40. "Archived copy" (PDF). Archived from the original (PDF) on 2012-04-21. Retrieved 2012-05-29.{{cite web}}: CS1 maint: archived copy as title (link)
41. "NIST.gov – Computer Security Division – Computer Security Resource Center". Csrc.nist.gov. Archived from the original on 2013-01-02. Retrieved 2012-12-23.
42. "Validated FIPS 140-1 and FIPS 140-2 Cryptographic Modules". Archived from the original on 2014-12-26. Retrieved 2014-06-26.
43. OpenSSL, [email protected]. "OpenSSL's Notes about FIPS certification". Openssl.org. Archived from the original on 2013-01-02. Retrieved 2012-12-23.
44. Schneier, Bruce; Kelsey, John; Whiting, Doug; Wagner, David; Hall, Chris; Ferguson, Niels (1999-02-01). "Performance Comparisons of the AES submissions" (PDF). Archived (PDF) from the original on 2011-06-22. Retrieved 2010-12-28.
45. "AMD Ryzen 7 1700X Review".
• Courtois, Nicolas; Pieprzyk, Josef (2003). "Cryptanalysis of Block Ciphers with Overdefined Systems of Equations". In Zheng, Yuliang (ed.). Advances in Cryptology – ASIACRYPT 2002: 8th International Conference on the Theory and Application of Cryptology and Information Security, Queenstown, New Zealand, December 1–5, 2002, Proceedings. Springer. pp. 268–287. ISBN 978-3-540-36178-7.
• Daemen, Joan; Rijmen, Vincent (2002). The Design of Rijndael: AES – The Advanced Encryption Standard. Springer. ISBN 978-3-540-42580-9.
• Paar, Christof; Pelzl, Jan (2009). Understanding Cryptography: A Textbook for Students and Practitioners. Springer. pp. 87–122. ISBN 978-3-642-04101-3. alternate link (companion web site contains online lectures on AES)
External links
• "256bit key – 128bit block – AES". Cryptography – 256 bit Ciphers: Reference source code and submissions to international cryptographic designs contests. EmbeddedSW.
• "Advanced Encryption Standard (AES)" (PDF). Federal Information Processing Standards. 26 November 2001. doi:10.6028/NIST.FIPS.197. 197.
• AES algorithm archive information – (old, unmaintained)
• "Part 3: Block ciphers" (PDF). Information technology – Security techniques – Encryption algorithms (2nd ed.). ISO. 2010-12-15. ISO/IEC 18033-3:2010(E). Archived (PDF) from the original on 2022-10-09.
• Animation of Rijndael – AES deeply explained and animated using Flash (by Enrique Zabala / University ORT / Montevideo / Uruguay). This animation (in English, Spanish, and German) is also part of CrypTool 1 (menu Indiv. Procedures → Visualization of Algorithms → AES).
• HTML5 Animation of Rijndael – Same Animation as above made in HTML5.
Block ciphers (security summary)
Common
algorithms
• AES
• Blowfish
• DES (internal mechanics, Triple DES)
• Serpent
• Twofish
Less common
algorithms
• ARIA
• Camellia
• CAST-128
• GOST
• IDEA
• LEA
• RC2
• RC5
• RC6
• SEED
• Skipjack
• TEA
• XTEA
Other
algorithms
• 3-Way
• Akelarre
• Anubis
• BaseKing
• BassOmatic
• BATON
• BEAR and LION
• CAST-256
• Chiasmus
• CIKS-1
• CIPHERUNICORN-A
• CIPHERUNICORN-E
• CLEFIA
• CMEA
• Cobra
• COCONUT98
• Crab
• Cryptomeria/C2
• CRYPTON
• CS-Cipher
• DEAL
• DES-X
• DFC
• E2
• FEAL
• FEA-M
• FROG
• G-DES
• Grand Cru
• Hasty Pudding cipher
• Hierocrypt
• ICE
• IDEA NXT
• Intel Cascade Cipher
• Iraqi
• Kalyna
• KASUMI
• KeeLoq
• KHAZAD
• Khufu and Khafre
• KN-Cipher
• Kuznyechik
• Ladder-DES
• LOKI (97, 89/91)
• Lucifer
• M6
• M8
• MacGuffin
• Madryga
• MAGENTA
• MARS
• Mercy
• MESH
• MISTY1
• MMB
• MULTI2
• MultiSwap
• New Data Seal
• NewDES
• Nimbus
• NOEKEON
• NUSH
• PRESENT
• Prince
• Q
• REDOC
• Red Pike
• S-1
• SAFER
• SAVILLE
• SC2000
• SHACAL
• SHARK
• Simon
• SM4
• Speck
• Spectr-H64
• Square
• SXAL/MBAL
• Threefish
• Treyfer
• UES
• xmx
• XXTEA
• Zodiac
Design
• Feistel network
• Key schedule
• Lai–Massey scheme
• Product cipher
• S-box
• P-box
• SPN
• Confusion and diffusion
• Round
• Avalanche effect
• Block size
• Key size
• Key whitening (Whitening transformation)
Attack
(cryptanalysis)
• Brute-force (EFF DES cracker)
• MITM
• Biclique attack
• 3-subset MITM attack
• Linear (Piling-up lemma)
• Differential
• Impossible
• Truncated
• Higher-order
• Differential-linear
• Distinguishing (Known-key)
• Integral/Square
• Boomerang
• Mod n
• Related-key
• Slide
• Rotational
• Side-channel
• Timing
• Power-monitoring
• Electromagnetic
• Acoustic
• Differential-fault
• XSL
• Interpolation
• Partitioning
• Rubber-hose
• Black-bag
• Davies
• Rebound
• Weak key
• Tau
• Chi-square
• Time/memory/data tradeoff
Standardization
• AES process
• CRYPTREC
• NESSIE
Utilization
• Initialization vector
• Mode of operation
• Padding
Cryptography
General
• History of cryptography
• Outline of cryptography
• Cryptographic protocol
• Authentication protocol
• Cryptographic primitive
• Cryptanalysis
• Cryptocurrency
• Cryptosystem
• Cryptographic nonce
• Cryptovirology
• Hash function
• Cryptographic hash function
• Key derivation function
• Digital signature
• Kleptography
• Key (cryptography)
• Key exchange
• Key generator
• Key schedule
• Key stretching
• Keygen
• Cryptojacking malware
• Ransomware
• Random number generation
• Cryptographically secure pseudorandom number generator (CSPRNG)
• Pseudorandom noise (PRN)
• Secure channel
• Insecure channel
• Subliminal channel
• Encryption
• Decryption
• End-to-end encryption
• Harvest now, decrypt later
• Information-theoretic security
• Plaintext
• Codetext
• Ciphertext
• Shared secret
• Trapdoor function
• Trusted timestamping
• Key-based routing
• Onion routing
• Garlic routing
• Kademlia
• Mix network
Mathematics
• Cryptographic hash function
• Block cipher
• Stream cipher
• Symmetric-key algorithm
• Authenticated encryption
• Public-key cryptography
• Quantum key distribution
• Quantum cryptography
• Post-quantum cryptography
• Message authentication code
• Random numbers
• Steganography
• Category
| Wikipedia |
AES-GCM-SIV
AES-GCM-SIV is a mode of operation for the Advanced Encryption Standard which provides similar performance to Galois/Counter Mode as well as misuse resistance in the event of the reuse of a cryptographic nonce. The construction is defined in RFC 8452.[1]
About
AES-GCM-SIV is designed to preserve both privacy and integrity even if nonces are repeated. To accomplish this, encryption is a function of a nonce, the plaintext message, and optional additional associated data (AAD). In the event a nonce is misused (i.e. used more than once), nothing is revealed except in the case that same message is encrypted multiple times with the same nonce. When that happens, an attacker is able to observe repeat encryptions, since encryption is a deterministic function of the nonce and message. However, beyond that, no additional information is revealed to the attacker. For this reason, AES-GCM-SIV is an ideal choice in cases that unique nonces cannot be guaranteed, such as multiple servers or network devices encrypting messages under the same key without coordination.
Operation
Like Galois/Counter Mode, AES-GCM-SIV combines the well-known counter mode of encryption with the Galois mode of authentication. The key feature is the use of a synthetic initialization vector which is computed with Galois field multiplication using a construction called POLYVAL (a little-endian variant of Galois/Counter Mode's GHASH). POLYVAL is run over the combination of nonce, plaintext, and additional data, so that the IV is different for each combination.
POLYVAL is defined over GF(2128) by the polynomial:
$x^{128}+x^{127}+x^{126}+x^{121}+1$
Note that GHASH is defined over the "reverse" polynomial:
$x^{128}+x^{7}+x^{2}+x+1$
This change provides efficiency benefits on little-endian architectures.
See also
• Authenticated encryption
• Stream cipher
References
1. Gueron, S. (April 2019). AES-GCM-SIV: Nonce Misuse-Resistant Authenticated Encryption. IETF. doi:10.17487/RFC8452. RFC 8452. Retrieved August 14, 2019.
External links
• RFC 8452: AES-GCM-SIV: Nonce Misuse-Resistant Authenticated Encryption
• BIU: Webpage for the AES-GCM-SIV Mode of Operation
Implementations
Implementations of AES-GCM-SIV are available, among others, in the following languages:
• C
• C#
• Go
• Go
• Java
• PHP
• Python
• Rust
Block ciphers (security summary)
Common
algorithms
• AES
• Blowfish
• DES (internal mechanics, Triple DES)
• Serpent
• Twofish
Less common
algorithms
• ARIA
• Camellia
• CAST-128
• GOST
• IDEA
• LEA
• RC2
• RC5
• RC6
• SEED
• Skipjack
• TEA
• XTEA
Other
algorithms
• 3-Way
• Akelarre
• Anubis
• BaseKing
• BassOmatic
• BATON
• BEAR and LION
• CAST-256
• Chiasmus
• CIKS-1
• CIPHERUNICORN-A
• CIPHERUNICORN-E
• CLEFIA
• CMEA
• Cobra
• COCONUT98
• Crab
• Cryptomeria/C2
• CRYPTON
• CS-Cipher
• DEAL
• DES-X
• DFC
• E2
• FEAL
• FEA-M
• FROG
• G-DES
• Grand Cru
• Hasty Pudding cipher
• Hierocrypt
• ICE
• IDEA NXT
• Intel Cascade Cipher
• Iraqi
• Kalyna
• KASUMI
• KeeLoq
• KHAZAD
• Khufu and Khafre
• KN-Cipher
• Kuznyechik
• Ladder-DES
• LOKI (97, 89/91)
• Lucifer
• M6
• M8
• MacGuffin
• Madryga
• MAGENTA
• MARS
• Mercy
• MESH
• MISTY1
• MMB
• MULTI2
• MultiSwap
• New Data Seal
• NewDES
• Nimbus
• NOEKEON
• NUSH
• PRESENT
• Prince
• Q
• REDOC
• Red Pike
• S-1
• SAFER
• SAVILLE
• SC2000
• SHACAL
• SHARK
• Simon
• SM4
• Speck
• Spectr-H64
• Square
• SXAL/MBAL
• Threefish
• Treyfer
• UES
• xmx
• XXTEA
• Zodiac
Design
• Feistel network
• Key schedule
• Lai–Massey scheme
• Product cipher
• S-box
• P-box
• SPN
• Confusion and diffusion
• Round
• Avalanche effect
• Block size
• Key size
• Key whitening (Whitening transformation)
Attack
(cryptanalysis)
• Brute-force (EFF DES cracker)
• MITM
• Biclique attack
• 3-subset MITM attack
• Linear (Piling-up lemma)
• Differential
• Impossible
• Truncated
• Higher-order
• Differential-linear
• Distinguishing (Known-key)
• Integral/Square
• Boomerang
• Mod n
• Related-key
• Slide
• Rotational
• Side-channel
• Timing
• Power-monitoring
• Electromagnetic
• Acoustic
• Differential-fault
• XSL
• Interpolation
• Partitioning
• Rubber-hose
• Black-bag
• Davies
• Rebound
• Weak key
• Tau
• Chi-square
• Time/memory/data tradeoff
Standardization
• AES process
• CRYPTREC
• NESSIE
Utilization
• Initialization vector
• Mode of operation
• Padding
Cryptographic hash functions and message authentication codes
• List
• Comparison
• Known attacks
Common functions
• MD5 (compromised)
• SHA-1 (compromised)
• SHA-2
• SHA-3
• BLAKE2
SHA-3 finalists
• BLAKE
• Grøstl
• JH
• Skein
• Keccak (winner)
Other functions
• BLAKE3
• CubeHash
• ECOH
• FSB
• Fugue
• GOST
• HAS-160
• HAVAL
• Kupyna
• LSH
• Lane
• MASH-1
• MASH-2
• MD2
• MD4
• MD6
• MDC-2
• N-hash
• RIPEMD
• RadioGatún
• SIMD
• SM3
• SWIFFT
• Shabal
• Snefru
• Streebog
• Tiger
• VSH
• Whirlpool
Password hashing/
key stretching functions
• Argon2
• Balloon
• bcrypt
• Catena
• crypt
• LM hash
• Lyra2
• Makwa
• PBKDF2
• scrypt
• yescrypt
General purpose
key derivation functions
• HKDF
• KDF1/KDF2
MAC functions
• CBC-MAC
• DAA
• GMAC
• HMAC
• NMAC
• OMAC/CMAC
• PMAC
• Poly1305
• SipHash
• UMAC
• VMAC
Authenticated
encryption modes
• CCM
• ChaCha20-Poly1305
• CWC
• EAX
• GCM
• IAPM
• OCB
Attacks
• Collision attack
• Preimage attack
• Birthday attack
• Brute-force attack
• Rainbow table
• Side-channel attack
• Length extension attack
Design
• Avalanche effect
• Hash collision
• Merkle–Damgård construction
• Sponge function
• HAIFA construction
Standardization
• CAESAR Competition
• CRYPTREC
• NESSIE
• NIST hash function competition
• Password Hashing Competition
Utilization
• Hash-based cryptography
• Merkle tree
• Message authentication
• Proof of work
• Salt
• Pepper
Cryptography
General
• History of cryptography
• Outline of cryptography
• Cryptographic protocol
• Authentication protocol
• Cryptographic primitive
• Cryptanalysis
• Cryptocurrency
• Cryptosystem
• Cryptographic nonce
• Cryptovirology
• Hash function
• Cryptographic hash function
• Key derivation function
• Digital signature
• Kleptography
• Key (cryptography)
• Key exchange
• Key generator
• Key schedule
• Key stretching
• Keygen
• Cryptojacking malware
• Ransomware
• Random number generation
• Cryptographically secure pseudorandom number generator (CSPRNG)
• Pseudorandom noise (PRN)
• Secure channel
• Insecure channel
• Subliminal channel
• Encryption
• Decryption
• End-to-end encryption
• Harvest now, decrypt later
• Information-theoretic security
• Plaintext
• Codetext
• Ciphertext
• Shared secret
• Trapdoor function
• Trusted timestamping
• Key-based routing
• Onion routing
• Garlic routing
• Kademlia
• Mix network
Mathematics
• Cryptographic hash function
• Block cipher
• Stream cipher
• Symmetric-key algorithm
• Authenticated encryption
• Public-key cryptography
• Quantum key distribution
• Quantum cryptography
• Post-quantum cryptography
• Message authentication code
• Random numbers
• Steganography
• Category
| Wikipedia |
AF+BG theorem
In algebraic geometry the AF+BG theorem (also known as Max Noether's fundamental theorem) is a result of Max Noether that asserts that, if the equation of an algebraic curve in the complex projective plane belongs locally (at each intersection point) to the ideal generated by the equations of two other algebraic curves, then it belongs globally to this ideal.
Statement
Let F, G, and H be homogeneous polynomials in three variables, with H having higher degree than F and G; let a = deg H − deg F and b = deg H − deg G (both positive integers) be the differences of the degrees of the polynomials. Suppose that the greatest common divisor of F and G is a constant, which means that the projective curves that they define in the projective plane $\mathbb {P} ^{2}$ have an intersection consisting in a finite number of points. For each point P of this intersection, the polynomials F and G generate an ideal (F, G)P of the local ring of $\mathbb {P} ^{2}$ at P (this local ring is the ring of the fractions ${\tfrac {n}{d}},$ where n and d are polynomials in three variables and d(P) ≠ 0). The theorem asserts that, if H lies in (F, G)P for every intersection point P, then H lies in the ideal (F, G); that is, there are homogeneous polynomials A and B of degrees a and b, respectively, such that H = AF + BG. Furthermore, any two choices of A differ by a multiple of G, and similarly any two choices of B differ by a multiple of F.
Related results
This theorem may be viewed as a generalization of Bézout's identity, which provides a condition under which an integer or a univariate polynomial h may be expressed as an element of the ideal generated by two other integers or univariate polynomials f and g: such a representation exists exactly when h is a multiple of the greatest common divisor of f and g. The AF+BG condition expresses, in terms of divisors (sets of points, with multiplicities), a similar condition under which a homogeneous polynomial H in three variables can be written as an element of the ideal generated by two other polynomials F and G.
This theorem is also a refinement, for this particular case, of Hilbert's Nullstellensatz, which provides a condition expressing that some power of a polynomial h (in any number of variables) belongs to the ideal generated by a finite set of polynomials.
References
• Fulton, William (2008), "5.5 Max Noether's Fundamental Theorem and 5.6 Applications of Noether's Theorem", Algebraic Curves: An Introduction to Algebraic Geometry (PDF), pp. 60–65.
• Griffiths, Phillip; Harris, Joseph (1978), Principles of Algebraic Geometry, John Wiley & Sons, ISBN 978-0-471-05059-9.
External links
• Weisstein, Eric W., "Noether's Fundamental Theorem", MathWorld
Topics in algebraic curves
Rational curves
• Five points determine a conic
• Projective line
• Rational normal curve
• Riemann sphere
• Twisted cubic
Elliptic curves
Analytic theory
• Elliptic function
• Elliptic integral
• Fundamental pair of periods
• Modular form
Arithmetic theory
• Counting points on elliptic curves
• Division polynomials
• Hasse's theorem on elliptic curves
• Mazur's torsion theorem
• Modular elliptic curve
• Modularity theorem
• Mordell–Weil theorem
• Nagell–Lutz theorem
• Supersingular elliptic curve
• Schoof's algorithm
• Schoof–Elkies–Atkin algorithm
Applications
• Elliptic curve cryptography
• Elliptic curve primality
Higher genus
• De Franchis theorem
• Faltings's theorem
• Hurwitz's automorphisms theorem
• Hurwitz surface
• Hyperelliptic curve
Plane curves
• AF+BG theorem
• Bézout's theorem
• Bitangent
• Cayley–Bacharach theorem
• Conic section
• Cramer's paradox
• Cubic plane curve
• Fermat curve
• Genus–degree formula
• Hilbert's sixteenth problem
• Nagata's conjecture on curves
• Plücker formula
• Quartic plane curve
• Real plane curve
Riemann surfaces
• Belyi's theorem
• Bring's curve
• Bolza surface
• Compact Riemann surface
• Dessin d'enfant
• Differential of the first kind
• Klein quartic
• Riemann's existence theorem
• Riemann–Roch theorem
• Teichmüller space
• Torelli theorem
Constructions
• Dual curve
• Polar curve
• Smooth completion
Structure of curves
Divisors on curves
• Abel–Jacobi map
• Brill–Noether theory
• Clifford's theorem on special divisors
• Gonality of an algebraic curve
• Jacobian variety
• Riemann–Roch theorem
• Weierstrass point
• Weil reciprocity law
Moduli
• ELSV formula
• Gromov–Witten invariant
• Hodge bundle
• Moduli of algebraic curves
• Stable curve
Morphisms
• Hasse–Witt matrix
• Riemann–Hurwitz formula
• Prym variety
• Weber's theorem (Algebraic curves)
Singularities
• Acnode
• Crunode
• Cusp
• Delta invariant
• Tacnode
Vector bundles
• Birkhoff–Grothendieck theorem
• Stable vector bundle
• Vector bundles on algebraic curves
| Wikipedia |
Approximately finite-dimensional C*-algebra
In mathematics, an approximately finite-dimensional (AF) C*-algebra is a C*-algebra that is the inductive limit of a sequence of finite-dimensional C*-algebras. Approximate finite-dimensionality was first defined and described combinatorially by Ola Bratteli. Later, George A. Elliott gave a complete classification of AF algebras using the K0 functor whose range consists of ordered abelian groups with sufficiently nice order structure.
The classification theorem for AF-algebras serves as a prototype for classification results for larger classes of separable simple amenable stably finite C*-algebras. Its proof divides into two parts. The invariant here is K0 with its natural order structure; this is a functor. First, one proves existence: a homomorphism between invariants must lift to a *-homomorphism of algebras. Second, one shows uniqueness: the lift must be unique up to approximate unitary equivalence. Classification then follows from what is known as the intertwining argument. For unital AF algebras, both existence and uniqueness follow from the fact the Murray-von Neumann semigroup of projections in an AF algebra is cancellative.
The counterpart of simple AF C*-algebras in the von Neumann algebra world are the hyperfinite factors, which were classified by Connes and Haagerup.
In the context of noncommutative geometry and topology, AF C*-algebras are noncommutative generalizations of C0(X), where X is a totally disconnected metrizable space.
Definition and basic properties
Finite-dimensional C*-algebras
An arbitrary finite-dimensional C*-algebra A takes the following form, up to isomorphism:
$\oplus _{k}M_{n_{k}},$
where Mi denotes the full matrix algebra of i × i matrices.
Up to unitary equivalence, a unital *-homomorphism Φ : Mi → Mj is necessarily of the form
$\Phi (a)=a\otimes I_{r},$
where r·i = j. The number r is said to be the multiplicity of Φ. In general, a unital homomorphism between finite-dimensional C*-algebras
$\Phi :\oplus _{1}^{s}M_{n_{k}}\rightarrow \oplus _{1}^{t}M_{m_{l}}$ :\oplus _{1}^{s}M_{n_{k}}\rightarrow \oplus _{1}^{t}M_{m_{l}}}
is specified, up to unitary equivalence, by a t × s matrix of partial multiplicities (rl k) satisfying, for all l
$\sum _{k}r_{lk}n_{k}=m_{l}.\;$
In the non-unital case, the equality is replaced by ≤. Graphically, Φ, equivalently (rl k), can be represented by its Bratteli diagram. The Bratteli diagram is a directed graph with nodes corresponding to each nk and ml and the number of arrows from nk to ml is the partial multiplicity rlk.
Consider the category whose objects are isomorphism classes of finite-dimensional C*-algebras and whose morphisms are *-homomorphisms modulo unitary equivalence. By the above discussion, the objects can be viewed as vectors with entries in N and morphisms are the partial multiplicity matrices.
AF algebras
A C*-algebra is AF if it is the direct limit of a sequence of finite-dimensional C*-algebras:
$A=\varinjlim \cdots \rightarrow A_{i}\,{\stackrel {\alpha _{i}}{\rightarrow }}A_{i+1}\rightarrow \cdots ,$
where each Ai is a finite-dimensional C*-algebra and the connecting maps αi are *-homomorphisms. We will assume that each αi is unital. The inductive system specifying an AF algebra is not unique. One can always drop to a subsequence. Suppressing the connecting maps, A can also be written as
$A={\overline {\cup _{n}A_{n}}}.$
The Bratteli diagram of A is formed by the Bratteli diagrams of {αi} in the obvious way. For instance, the Pascal triangle, with the nodes connected by appropriate downward arrows, is the Bratteli diagram of an AF algebra. A Bratteli diagram of the CAR algebra is given on the right. The two arrows between nodes means each connecting map is an embedding of multiplicity 2.
$1\rightrightarrows 2\rightrightarrows 4\rightrightarrows 8\rightrightarrows \dots $
(A Bratteli diagram of the CAR algebra)
If an AF algebra A = (∪nAn)−, then an ideal J in A takes the form ∪n (J ∩ An)−. In particular, J is itself an AF algebra. Given a Bratteli diagram of A and some subset S of nodes, the subdiagram generated by S gives inductive system that specifies an ideal of A. In fact, every ideal arises in this way.
Due to the presence of matrix units in the inductive sequence, AF algebras have the following local characterization: a C*-algebra A is AF if and only if A is separable and any finite subset of A is "almost contained" in some finite-dimensional C*-subalgebra.
The projections in ∪nAn in fact form an approximate unit of A.
It is clear that the extension of a finite-dimensional C*-algebra by another finite-dimensional C*-algebra is again finite-dimensional. More generally, the extension of an AF algebra by another AF algebra is again AF.[1]
Classification
K0
The K-theoretic group K0 is an invariant of C*-algebras. It has its origins in topological K-theory and serves as the range of a kind of "dimension function." For an AF algebra A, K0(A) can be defined as follows. Let Mn(A) be the C*-algebra of n × n matrices whose entries are elements of A. Mn(A) can be embedded into Mn + 1(A) canonically, into the "upper left corner". Consider the algebraic direct limit
$M_{\infty }(A)=\varinjlim \cdots \rightarrow M_{n}(A)\rightarrow M_{n+1}(A)\rightarrow \cdots .$
Denote the projections (self-adjoint idempotents) in this algebra by P(A). Two elements p and q are said to be Murray-von Neumann equivalent, denoted by p ~ q, if p = vv* and q = v*v for some partial isometry v in M∞(A). It is clear that ~ is an equivalence relation. Define a binary operation + on the set of equivalences P(A)/~ by
$[p]+[q]=[p\oplus q]$
where ⊕ yields the orthogonal direct sum of two finite-dimensional matrices corresponding to p and q. While we could choose matrices of arbitrarily large dimension to stand in for p and q, our result will be equivalent regardless. This makes P(A)/~ a semigroup that has the cancellation property. We denote this semigroup by K0(A)+. Performing the Grothendieck group construction gives an abelian group, which is K0(A).
K0(A) carries a natural order structure: we say [p] ≤ [q] if p is Murray-von Neumann equivalent to a subprojection of q. This makes K0(A) an ordered group whose positive cone is K0(A)+.
For example, for a finite-dimensional C*-algebra
$A=\oplus _{k=1}^{m}M_{n_{k}},$
one has
$(K_{0}(A),K_{0}(A)^{+})=(\mathbb {Z} ^{m},\mathbb {Z} _{+}^{m}).$
Two essential features of the mapping A ↦ K0(A) are:
1. K0 is a (covariant) functor. A *-homomorphism α : A → B between AF algebras induces a group homomorphism α* : K0(A) → K0(B). In particular, when A and B are both finite-dimensional, α* can be identified with the partial multiplicities matrix of α.
2. K0 respects direct limits. If A = ∪nαn(An)−, then K0(A) is the direct limit ∪nαn*(K0(An)).
The dimension group
Since M∞(M∞(A)) is isomorphic to M∞(A), K0 can only distinguish AF algebras up to stable isomorphism. For example, M2 and M4 are not isomorphic but stably isomorphic; K0(M2) = K0(M4) = Z.
A finer invariant is needed to detect isomorphism classes. For an AF algebra A, we define the scale of K0(A), denoted by Γ(A), to be the subset whose elements are represented by projections in A:
$\Gamma (A)=\{[p]\,|\,p^{*}=p^{2}=p\in A\}.$
When A is unital with unit 1A, the K0 element [1A] is the maximal element of Γ(A) and in fact,
$\Gamma (A)=\{x\in K_{0}(A)\,|\,0\leq x\leq [1_{A}]\}.$
The triple (K0, K0+, Γ(A)) is called the dimension group of A. If A = Ms, its dimension group is (Z, Z+, {1, 2,..., s}).
A group homomorphism between dimension group is said to be contractive if it is scale-preserving. Two dimension group are said to be isomorphic if there exists a contractive group isomorphism between them.
The dimension group retains the essential properties of K0:
1. A *-homomorphism α : A → B between AF algebras in fact induces a contractive group homomorphism α* on the dimension groups. When A and B are both finite-dimensional, corresponding to each partial multiplicities matrix ψ, there is a unique, up to unitary equivalence, *-homomorphism α : A → B such that α* = ψ.
2. If A = ∪nαn(An)−, then the dimension group of A is the direct limit of those of An.
Elliott's theorem
Elliott's theorem says that the dimension group is a complete invariant of AF algebras: two AF algebras A and B are isomorphic if and only if their dimension groups are isomorphic.
Two preliminary facts are needed before one can sketch a proof of Elliott's theorem. The first one summarizes the above discussion on finite-dimensional C*-algebras.
Lemma For two finite-dimensional C*-algebras A and B, and a contractive homomorphism ψ: K0(A) → K0(B), there exists a *-homomorphism φ: A → B such that φ* = ψ, and φ is unique up to unitary equivalence.
The lemma can be extended to the case where B is AF. A map ψ on the level of K0 can be "moved back", on the level of algebras, to some finite stage in the inductive system.
Lemma Let A be finite-dimensional and B AF, B = (∪nBn)−. Let βm be the canonical homomorphism of Bm into B. Then for any a contractive homomorphism ψ: K0(A) → K0(B), there exists a *-homomorphism φ: A → Bm such that βm* φ* = ψ, and φ is unique up to unitary equivalence in B.
The proof of the lemma is based on the simple observation that K0(A) is finitely generated and, since K0 respects direct limits, K0(B) = ∪n βn* K0 (Bn).
Theorem (Elliott) Two AF algebras A and B are isomorphic if and only if their dimension groups (K0(A), K0+(A), Γ(A)) and (K0(B), K0+(B), Γ(B)) are isomorphic.
The crux of the proof has become known as Elliott's intertwining argument. Given an isomorphism between dimension groups, one constructs a diagram of commuting triangles between the direct systems of A and B by applying the second lemma.
We sketch the proof for the non-trivial part of the theorem, corresponding to the sequence of commutative diagrams on the right.
Let Φ: (K0(A), K0+(A), Γ(A)) → (K0(B), K0+(B), Γ(B)) be a dimension group isomorphism.
1. Consider the composition of maps Φ α1* : K0(A1) → K0(B). By the previous lemma, there exists B1 and a *-homomorphism φ1: A1 → B1 such that the first diagram on the right commutes.
2. Same argument applied to β1* Φ−1 shows that the second diagram commutes for some A2.
3. Comparing diagrams 1 and 2 gives diagram 3.
4. Using the property of the direct limit and moving A2 further down if necessary, we obtain diagram 4, a commutative triangle on the level of K0.
5. For finite-dimensional algebras, two *-homomorphisms induces the same map on K0 if and only if they are unitary equivalent. So, by composing ψ1 with a unitary conjugation if needed, we have a commutative triangle on the level of algebras.
6. By induction, we have a diagram of commuting triangles as indicated in the last diagram. The map φ: A → B is the direct limit of the sequence {φn}. Let ψ: B → A is the direct limit of the sequence {ψn}. It is clear that φ and ψ are mutual inverses. Therefore, A and B are isomorphic.
Furthermore, on the level of K0, the adjacent diagram commutates for each k. By uniqueness of direct limit of maps, φ* = Φ.
The Effros-Handelman-Shen theorem
The dimension group of an AF algebra is a Riesz group. The Effros-Handelman-Shen theorem says the converse is true. Every Riesz group, with a given scale, arises as the dimension group of some AF algebra. This specifies the range of the classifying functor K0 for AF-algebras and completes the classification.
Riesz groups
A group G with a partial order is called an ordered group. The set G+ of elements ≥ 0 is called the positive cone of G. One says that G is unperforated if k·g ∈ G+ implies g ∈ G+.
The following property is called the Riesz decomposition property: if x, yi ≥ 0 and x ≤ Σ yi, then there exists xi ≥ 0 such that x = Σ xi, and xi ≤ yi for each i.
A Riesz group (G, G+) is an ordered group that is unperforated and has the Riesz decomposition property.
It is clear that if A is finite-dimensional, (K0, K0+) is a Riesz group, where Zk is given entrywise order. The two properties of Riesz groups are preserved by direct limits, assuming the order structure on the direct limit comes from those in the inductive system. So (K0, K0+) is a Riesz group for an AF algebra A.
A key step towards the Effros-Handelman-Shen theorem is the fact that every Riesz group is the direct limit of Zk 's, each with the canonical order structure. This hinges on the following technical lemma, sometimes referred to as the Shen criterion in the literature.
Lemma Let (G, G+) be a Riesz group, ϕ: (Zk, Zk+) → (G, G+) be a positive homomorphism. Then there exists maps σ and ψ, as indicated in the adjacent diagram, such that ker(σ) = ker(ϕ).
Corollary Every Riesz group (G, G+) can be expressed as a direct limit
$(G,G^{+})=\varinjlim (\mathbb {Z} ^{n_{k}},\mathbb {Z} _{+}^{n_{k}}),$
where all the connecting homomorphisms in the directed system on the right hand side are positive.
The theorem
Theorem If (G, G+) is a countable Riesz group with scale Γ(G), then there exists an AF algebra A such that (K0, K0+, Γ(A)) = (G, G+, Γ(G)). In particular, if Γ(G) = [0, uG] with maximal element uG, then A is unital with [1A] = [uG].
Consider first the special case where Γ(G) = [0, uG] with maximal element uG. Suppose
$(G,G^{+})=\varinjlim (H_{k},H_{k}^{+}),\quad {\mbox{where}}\quad (H,H_{k}^{+})=(\mathbb {Z} ^{n_{k}},\mathbb {Z} _{+}^{n_{k}}).$
Dropping to a subsequence if necessary, let
$\Gamma (H_{1})=\{v\in H_{1}^{+}|\phi _{1}(v)\in \Gamma (G)\},$
where φ1(u1) = uG for some element u1. Now consider the order ideal G1 generated by u1. Because each H1 has the canonical order structure, G1 is a direct sum of Z 's (with the number of copies possible less than that in H1). So this gives a finite-dimensional algebra A1 whose dimension group is (G1 G1+, [0, u1]). Next move u1 forward by defining u2 = φ12(u1). Again u2 determines a finite-dimensional algebra A2. There is a corresponding homomorphism α12 such that α12* = φ12. Induction gives a directed system
$A=\varinjlim A_{k},$
whose K0 is
$\varinjlim (G_{k},G_{k}^{+}),$
with scale
$\cup _{k}\phi _{k}[0,u_{k}]=[0,u_{G}].$
This proves the special case.
A similar argument applies in general. Observe that the scale is by definition a directed set. If Γ(G) = {vk}, one can choose uk ∈ Γ(G) such that uk ≥ v1 ... vk. The same argument as above proves the theorem.
Examples
By definition, uniformly hyperfinite algebras are AF and unital. Their dimension groups are the subgroups of Q. For example, for the 2 × 2 matrices M2, K0(M2) is the group of rational numbers of the form a/2 for a in Z. The scale is Γ(M2) = {0, 1/2, 1}. For the CAR algebra A, K0(A) is the group of dyadic rationals with scale K0(A) ∩ [0, 1], with 1 = [1A]. All such groups are simple, in a sense appropriate for ordered groups. Thus UHF algebras are simple C*-algebras. In general, the groups which are not dense in Q are the dimension groups of Mk for some k.
Commutative C*-algebras, which were characterized by Gelfand, are AF precisely when the spectrum is totally disconnected.[2] The continuous functions C(X) on the Cantor set X is one such example.
Elliott's classification program
It was proposed by Elliott that other classes of C*-algebras may be classifiable by K-theoretic invariants. For a C*-algebra A, the Elliott invariant is defined to be
${\mbox{Ell}}(A)\;{\stackrel {\mbox{def}}{=}}\;(\;(K_{0}(A),K_{0}(A)^{+},\Gamma (A)),K_{1}(A),T^{+}(A),\rho _{A}\;),$
where T+(A) is the tracial positive linear functionals in the weak-* topology, and ρA is the natural pairing between T+(A) and K0(A).
The original conjecture by Elliott stated that the Elliott invariant classifies simple unital separable amenable C*-algebras.
In the literature, one can find several conjectures of this kind with corresponding modified/refined Elliott invariants.
Von Neumann algebras
In a related context, an approximately finite-dimensional, or hyperfinite, von Neumann algebra is one with a separable predual and contains a weakly dense AF C*-algebra. Murray and von Neumann showed that, up to isomorphism, there exists a unique hyperfinite type II1 factor. Connes obtained the analogous result for the II∞ factor. Powers exhibited a family of non-isomorphic type III hyperfinite factors with cardinality of the continuum. Today we have a complete classification of hyperfinite factors.
Notes
1. Lawrence G. Brown. Extensions of AF Algebras: The Projection Lifting Problem. Operator Algebras and Applications, Proceedings of symposia in pure mathematics, vol. 38, Part 1, pp. 175-176, American Mathematical Soc., 1982
2. Davidson 1996, p. 77.
References
• Bratteli, Ola. (1972), Inductive limits of finite dimensional C*-algebras, Trans. Amer. Math. Soc. 171, 195-234.
• Davidson, K.R. (1996), C*-algebras by Example, Field Institute Monographs 6, American Mathematical Society.
• Effros, E.G., Handelman, D.E., and Shen C.L. (1980), Dimension groups and their affine representations, Amer. J. Math. 102, 385-402.
• Elliott, G.A. (1976), On the Classification of Inductive Limits of Sequences of Semisimple Finite-Dimensional Algebras, J. Algebra 38, 29-44.
• Elliott, G.A. and Toms, A.S. (2008), Regularity properties in the classification program for separable amenable C-algebras, Bull. Amer. Math. Soc. 45, 229-245.
• Fillmore, P.A.(1996), A User's Guide for Operator Algebras, Wiley-Interscience.
• Rørdam, M. (2002), Classification of Nuclear C*-Algebras, Encyclopaedia of Mathematical Sciences 126, Springer-Verlag.
External links
• "AF-algebra", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
| Wikipedia |
AKNS system
In mathematics, the AKNS system is an integrable system of partial differential equations, introduced by and named after Mark J. Ablowitz, David J. Kaup, Alan C. Newell, and Harvey Segur from their publication in Studies in Applied Mathematics: Ablowitz, Kaup, and Newell et al. (1974).
Definition
The AKNS system is a pair of two partial differential equations for two complex-valued functions p and q of 2 variables t and x:
$p_{t}=+ip^{2}q-{\frac {i}{2}}p_{xx}$
$q_{t}=-iq^{2}p+{\frac {i}{2}}q_{xx}$
If p and q are complex conjugates this reduces to the nonlinear Schrödinger equation.
Huygens' principle applied to the Dirac operator gives rise to the AKNS hierarchy.[1]
Applications to General Relativity
In 2021, the dynamics of three-dimensional (extremal) black holes on General Relativity with negative cosmological constant were showed equivalent to two independent copies of the AKNS system.[2] This duality was addressed through the imposition of suitable boundary conditions to the Chern-Simons action. In this scheme, the involution of conserved charges of the AKNS system yields an infinite-dimensional commuting asymptotic symmetry algebra of gravitational charges.
See also
• Huygens principle
References
1. Fabio A. C. C. Chalub and Jorge P. Zubelli, "Huygens’ Principle for Hyperbolic Operators and Integrable Hierarchies" ""
2. Cárdenas, Marcela; Correa, Francisco; Lara, Kristiansen; Pino, Miguel (2021-10-12). "Integrable Systems and Spacetime Dynamics". Physical Review Letters. 127 (16): 161601. arXiv:2104.09676. doi:10.1103/PhysRevLett.127.161601.
• Ablowitz, Mark J.; Kaup, David J.; Newell, Alan C.; Segur, Harvey (1974), "The inverse scattering transform-Fourier analysis for nonlinear problems", Studies in Appl. Math., 53 (4): 249–315, doi:10.1002/sapm1974534249, MR 0450815
| Wikipedia |
ALL (complexity)
In computability and complexity theory, ALL is the class of all decision problems.
Relations to other classes
ALL contains all of the complex classes of decision problems, including RE and co-RE.
External links
• Complexity Zoo: Class ALL
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
| Wikipedia |
AM-GM Inequality
In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same (in which case they are both that number).
The simplest non-trivial case – i.e., with more than one variable – for two non-negative numbers x and y, is the statement that
${\frac {x+y}{2}}\geq {\sqrt {xy}}$
with equality if and only if x = y. This case can be seen from the fact that the square of a real number is always non-negative (greater than or equal to zero) and from the elementary case (a ± b)2 = a2 ± 2ab + b2 of the binomial formula:
${\begin{aligned}0&\leq (x-y)^{2}\\&=x^{2}-2xy+y^{2}\\&=x^{2}+2xy+y^{2}-4xy\\&=(x+y)^{2}-4xy.\end{aligned}}$
Hence (x + y)2 ≥ 4xy, with equality precisely when (x − y)2 = 0, i.e. x = y. The AM–GM inequality then follows from taking the positive square root of both sides and then dividing both sides by 2.
For a geometrical interpretation, consider a rectangle with sides of length x and y, hence it has perimeter 2x + 2y and area xy. Similarly, a square with all sides of length √xy has the perimeter 4√xy and the same area as the rectangle. The simplest non-trivial case of the AM–GM inequality implies for the perimeters that 2x + 2y ≥ 4√xy and that only the square has the smallest perimeter amongst all rectangles of equal area.
Extensions of the AM–GM inequality are available to include weights or generalized means.
Background
The arithmetic mean, or less precisely the average, of a list of n numbers x1, x2, . . . , xn is the sum of the numbers divided by n:
${\frac {x_{1}+x_{2}+\cdots +x_{n}}{n}}.$
The geometric mean is similar, except that it is only defined for a list of nonnegative real numbers, and uses multiplication and a root in place of addition and division:
${\sqrt[{n}]{x_{1}\cdot x_{2}\cdots x_{n}}}.$
If x1, x2, . . . , xn > 0, this is equal to the exponential of the arithmetic mean of the natural logarithms of the numbers:
$\exp \left({\frac {\ln {x_{1}}+\ln {x_{2}}+\cdots +\ln {x_{n}}}{n}}\right).$
The inequality
Restating the inequality using mathematical notation, we have that for any list of n nonnegative real numbers x1, x2, . . . , xn,
${\frac {x_{1}+x_{2}+\cdots +x_{n}}{n}}\geq {\sqrt[{n}]{x_{1}\cdot x_{2}\cdots x_{n}}}\,,$
and that equality holds if and only if x1 = x2 = · · · = xn.
Geometric interpretation
In two dimensions, 2x1 + 2x2 is the perimeter of a rectangle with sides of length x1 and x2. Similarly, 4√x1x2 is the perimeter of a square with the same area, x1x2, as that rectangle. Thus for n = 2 the AM–GM inequality states that a rectangle of a given area has the smallest perimeter if that rectangle is also a square.
The full inequality is an extension of this idea to n dimensions. Every vertex of an n-dimensional box is connected to n edges. If these edges' lengths are x1, x2, . . . , xn, then x1 + x2 + · · · + xn is the total length of edges incident to the vertex. There are 2n vertices, so we multiply this by 2n; since each edge, however, meets two vertices, every edge is counted twice. Therefore, we divide by 2 and conclude that there are 2n−1n edges. There are equally many edges of each length and n lengths; hence there are 2n−1 edges of each length and the total of all edge lengths is 2n−1(x1 + x2 + · · · + xn). On the other hand,
$2^{n-1}(x_{1}+\ldots +x_{n})=2^{n-1}n{\sqrt[{n}]{x_{1}x_{2}\cdots x_{n}}}$
is the total length of edges connected to a vertex on an n-dimensional cube of equal volume, since in this case x1=...=xn. Since the inequality says
${x_{1}+x_{2}+\cdots +x_{n} \over n}\geq {\sqrt[{n}]{x_{1}x_{2}\cdots x_{n}}},$
it can be restated by multiplying through by n2n–1 to obtain
$2^{n-1}(x_{1}+x_{2}+\cdots +x_{n})\geq 2^{n-1}n{\sqrt[{n}]{x_{1}x_{2}\cdots x_{n}}}$
with equality if and only if x1 = x2 = · · · = xn.
Thus the AM–GM inequality states that only the n-cube has the smallest sum of lengths of edges connected to each vertex amongst all n-dimensional boxes with the same volume.[2]
Examples
Example 1
If $a,b,c>0$, then the A.M.-G.M. tells us that
$(1+a)(1+b)(1+c)\geq 8{\sqrt {abc}}$
Example 2
A simple upper bound for $n!$ can be found. AM-GM tells us
$1+2+\dots +n\geq n{\sqrt[{n}]{n!}}$
${\frac {n(n+1)}{2}}\geq n{\sqrt[{n}]{n!}}$
and so
$\left({\frac {n+1}{2}}\right)^{n}\geq n!$
with equality at $n=1$.
Equivalently,
$(n+1)^{n}\geq 2^{n}n!$
Example 3
Consider the function
$f(x,y,z)={\frac {x}{y}}+{\sqrt {\frac {y}{z}}}+{\sqrt[{3}]{\frac {z}{x}}}$
for all positive real numbers x, y and z. Suppose we wish to find the minimal value of this function. It can be rewritten as:
${\begin{aligned}f(x,y,z)&=6\cdot {\frac {{\frac {x}{y}}+{\frac {1}{2}}{\sqrt {\frac {y}{z}}}+{\frac {1}{2}}{\sqrt {\frac {y}{z}}}+{\frac {1}{3}}{\sqrt[{3}]{\frac {z}{x}}}+{\frac {1}{3}}{\sqrt[{3}]{\frac {z}{x}}}+{\frac {1}{3}}{\sqrt[{3}]{\frac {z}{x}}}}{6}}\\&=6\cdot {\frac {x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}}{6}}\end{aligned}}$
with
$x_{1}={\frac {x}{y}},\qquad x_{2}=x_{3}={\frac {1}{2}}{\sqrt {\frac {y}{z}}},\qquad x_{4}=x_{5}=x_{6}={\frac {1}{3}}{\sqrt[{3}]{\frac {z}{x}}}.$
Applying the AM–GM inequality for n = 6, we get
${\begin{aligned}f(x,y,z)&\geq 6\cdot {\sqrt[{6}]{{\frac {x}{y}}\cdot {\frac {1}{2}}{\sqrt {\frac {y}{z}}}\cdot {\frac {1}{2}}{\sqrt {\frac {y}{z}}}\cdot {\frac {1}{3}}{\sqrt[{3}]{\frac {z}{x}}}\cdot {\frac {1}{3}}{\sqrt[{3}]{\frac {z}{x}}}\cdot {\frac {1}{3}}{\sqrt[{3}]{\frac {z}{x}}}}}\\&=6\cdot {\sqrt[{6}]{{\frac {1}{2\cdot 2\cdot 3\cdot 3\cdot 3}}{\frac {x}{y}}{\frac {y}{z}}{\frac {z}{x}}}}\\&=2^{2/3}\cdot 3^{1/2}.\end{aligned}}$
Further, we know that the two sides are equal exactly when all the terms of the mean are equal:
$f(x,y,z)=2^{2/3}\cdot 3^{1/2}\quad {\mbox{when}}\quad {\frac {x}{y}}={\frac {1}{2}}{\sqrt {\frac {y}{z}}}={\frac {1}{3}}{\sqrt[{3}]{\frac {z}{x}}}.$
All the points (x, y, z) satisfying these conditions lie on a half-line starting at the origin and are given by
$(x,y,z)={\biggr (}t,{\sqrt[{3}]{2}}{\sqrt {3}}\,t,{\frac {3{\sqrt {3}}}{2}}\,t{\biggr )}\quad {\mbox{with}}\quad t>0.$
Applications
An important practical application in financial mathematics is to computing the rate of return: the annualized return, computed via the geometric mean, is less than the average annual return, computed by the arithmetic mean (or equal if all returns are equal). This is important in analyzing investments, as the average return overstates the cumulative effect.
Proofs of the AM–GM inequality
Proof using Jensen's inequality
Jensen's inequality states that the value of a concave function of an arithmetic mean is greater than or equal to the arithmetic mean of the function's values. Since the logarithm function is concave, we have
$\log \left({\frac {\sum _{i}x_{i}}{n}}\right)\geq \sum {\frac {1}{n}}\log x_{i}=\sum \left(\log x_{i}^{1/n}\right)=\log \left(\prod x_{i}^{1/n}\right).$
Taking antilogs of the far left and far right sides, we have the AM–GM inequality.
Proof by successive replacement of elements
We have to show that
$\alpha ={\frac {x_{1}+x_{2}+\cdots +x_{n}}{n}}\geq {\sqrt[{n}]{x_{1}x_{2}\cdots x_{n}}}=\beta $
with equality only when all numbers are equal.
If not all numbers are equal, then there exist $x_{i},x_{j}$ such that $x_{i}<\alpha <x_{j}$. Replacing xi by $\alpha $ and xj by $(x_{i}+x_{j}-\alpha )$ will leave the arithmetic mean of the numbers unchanged, but will increase the geometric mean because
$\alpha (x_{j}+x_{i}-\alpha )-x_{i}x_{j}=(\alpha -x_{i})(x_{j}-\alpha )>0$
If the numbers are still not equal, we continue replacing numbers as above. After at most $(n-1)$ such replacement steps all the numbers will have been replaced with $\alpha $ while the geometric mean strictly increases at each step. After the last step, the geometric mean will be ${\sqrt[{n}]{\alpha \alpha \cdots \alpha }}=\alpha $, proving the inequality.
It may be noted that the replacement strategy works just as well from the right hand side. If any of the numbers is 0 then so will the geometric mean thus proving the inequality trivially. Therefore we may suppose that all the numbers are positive. If they are not all equal, then there exist $x_{i},x_{j}$ such that $0<x_{i}<\beta <x_{j}$. Replacing $x_{i}$ by $\beta $ and $x_{j}$ by ${\frac {x_{i}x_{j}}{\beta }}$leaves the geometric mean unchanged but strictly decreases the arithmetic mean since
$x_{i}+x_{j}-\beta -{\frac {x_{i}x_{j}}{\beta }}={\frac {(\beta -x_{i})(x_{j}-\beta )}{\beta }}>0$. The proof then follows along similar lines as in the earlier replacement.
Proof by induction #1
Of the non-negative real numbers x1, . . . , xn, the AM–GM statement is equivalent to
$\alpha ^{n}\geq x_{1}x_{2}\cdots x_{n}$
with equality if and only if α = xi for all i ∈ {1, . . . , n}.
For the following proof we apply mathematical induction and only well-known rules of arithmetic.
Induction basis: For n = 1 the statement is true with equality.
Induction hypothesis: Suppose that the AM–GM statement holds for all choices of n non-negative real numbers.
Induction step: Consider n + 1 non-negative real numbers x1, . . . , xn+1, . Their arithmetic mean α satisfies
$(n+1)\alpha =\ x_{1}+\cdots +x_{n}+x_{n+1}.$
If all the xi are equal to α, then we have equality in the AM–GM statement and we are done. In the case where some are not equal to α, there must exist one number that is greater than the arithmetic mean α, and one that is smaller than α. Without loss of generality, we can reorder our xi in order to place these two particular elements at the end: xn > α and xn+1 < α. Then
$x_{n}-\alpha >0\qquad \alpha -x_{n+1}>0$
$\implies (x_{n}-\alpha )(\alpha -x_{n+1})>0\,.\qquad (*)$
Now define y with
$y:=x_{n}+x_{n+1}-\alpha \geq x_{n}-\alpha >0\,,$
and consider the n numbers x1, . . . , xn–1, y which are all non-negative. Since
$(n+1)\alpha =x_{1}+\cdots +x_{n-1}+x_{n}+x_{n+1}$
$n\alpha =x_{1}+\cdots +x_{n-1}+\underbrace {x_{n}+x_{n+1}-\alpha } _{=\,y},$
Thus, α is also the arithmetic mean of n numbers x1, . . . , xn–1, y and the induction hypothesis implies
$\alpha ^{n+1}=\alpha ^{n}\cdot \alpha \geq x_{1}x_{2}\cdots x_{n-1}y\cdot \alpha .\qquad (**)$
Due to (*) we know that
$(\underbrace {x_{n}+x_{n+1}-\alpha } _{=\,y})\alpha -x_{n}x_{n+1}=(x_{n}-\alpha )(\alpha -x_{n+1})>0,$
hence
$y\alpha >x_{n}x_{n+1}\,,\qquad ({*}{*}{*})$
in particular α > 0. Therefore, if at least one of the numbers x1, . . . , xn–1 is zero, then we already have strict inequality in (**). Otherwise the right-hand side of (**) is positive and strict inequality is obtained by using the estimate (***) to get a lower bound of the right-hand side of (**). Thus, in both cases we can substitute (***) into (**) to get
$\alpha ^{n+1}>x_{1}x_{2}\cdots x_{n-1}x_{n}x_{n+1}\,,$
which completes the proof.
Proof by induction #2
First of all we shall prove that for real numbers x1 < 1 and x2 > 1 there follows
$x_{1}+x_{2}>x_{1}x_{2}+1.$
Indeed, multiplying both sides of the inequality x2 > 1 by 1 – x1, gives
$x_{2}-x_{1}x_{2}>1-x_{1},$
whence the required inequality is obtained immediately.
Now, we are going to prove that for positive real numbers x1, . . . , xn satisfying x1 . . . xn = 1, there holds
$x_{1}+\cdots +x_{n}\geq n.$
The equality holds only if x1 = ... = xn = 1.
Induction basis: For n = 2 the statement is true because of the above property.
Induction hypothesis: Suppose that the statement is true for all natural numbers up to n – 1.
Induction step: Consider natural number n, i.e. for positive real numbers x1, . . . , xn, there holds x1 . . . xn = 1. There exists at least one xk < 1, so there must be at least one xj > 1. Without loss of generality, we let k =n – 1 and j = n.
Further, the equality x1 . . . xn = 1 we shall write in the form of (x1 . . . xn–2) (xn–1 xn) = 1. Then, the induction hypothesis implies
$(x_{1}+\cdots +x_{n-2})+(x_{n-1}x_{n})>n-1.$
However, taking into account the induction basis, we have
${\begin{aligned}x_{1}+\cdots +x_{n-2}+x_{n-1}+x_{n}&=(x_{1}+\cdots +x_{n-2})+(x_{n-1}+x_{n})\\&>(x_{1}+\cdots +x_{n-2})+x_{n-1}x_{n}+1\\&>n,\end{aligned}}$
which completes the proof.
For positive real numbers a1, . . . , an, let's denote
$x_{1}={\frac {a_{1}}{\sqrt[{n}]{a_{1}\cdots a_{n}}}},...,x_{n}={\frac {a_{n}}{\sqrt[{n}]{a_{1}\cdots a_{n}}}}.$
The numbers x1, . . . , xn satisfy the condition x1 . . . xn = 1. So we have
${\frac {a_{1}}{\sqrt[{n}]{a_{1}\cdots a_{n}}}}+\cdots +{\frac {a_{n}}{\sqrt[{n}]{a_{1}\cdots a_{n}}}}\geq n,$
whence we obtain
${\frac {a_{1}+\cdots +a_{n}}{n}}\geq {\sqrt[{n}]{a_{1}\cdots a_{n}}},$
with the equality holding only for a1 = ... = an.
Proof by Cauchy using forward–backward induction
The following proof by cases relies directly on well-known rules of arithmetic but employs the rarely used technique of forward-backward-induction. It is essentially from Augustin Louis Cauchy and can be found in his Cours d'analyse.[3]
The case where all the terms are equal
If all the terms are equal:
$x_{1}=x_{2}=\cdots =x_{n},$
then their sum is nx1, so their arithmetic mean is x1; and their product is x1n, so their geometric mean is x1; therefore, the arithmetic mean and geometric mean are equal, as desired.
The case where not all the terms are equal
It remains to show that if not all the terms are equal, then the arithmetic mean is greater than the geometric mean. Clearly, this is only possible when n > 1.
This case is significantly more complex, and we divide it into subcases.
The subcase where n = 2
If n = 2, then we have two terms, x1 and x2, and since (by our assumption) not all terms are equal, we have:
${\begin{aligned}{\Bigl (}{\frac {x_{1}+x_{2}}{2}}{\Bigr )}^{2}-x_{1}x_{2}&={\frac {1}{4}}(x_{1}^{2}+2x_{1}x_{2}+x_{2}^{2})-x_{1}x_{2}\\&={\frac {1}{4}}(x_{1}^{2}-2x_{1}x_{2}+x_{2}^{2})\\&={\Bigl (}{\frac {x_{1}-x_{2}}{2}}{\Bigr )}^{2}>0,\end{aligned}}$
hence
${\frac {x_{1}+x_{2}}{2}}\geq {\sqrt {x_{1}x_{2}}}$
as desired.
The subcase where n = 2k
Consider the case where n = 2k, where k is a positive integer. We proceed by mathematical induction.
In the base case, k = 1, so n = 2. We have already shown that the inequality holds when n = 2, so we are done.
Now, suppose that for a given k > 1, we have already shown that the inequality holds for n = 2k−1, and we wish to show that it holds for n = 2k. To do so, we apply the inequality twice for 2k-1 numbers and once for 2 numbers to obtain:
${\begin{aligned}{\frac {x_{1}+x_{2}+\cdots +x_{2^{k}}}{2^{k}}}&{}={\frac {{\frac {x_{1}+x_{2}+\cdots +x_{2^{k-1}}}{2^{k-1}}}+{\frac {x_{2^{k-1}+1}+x_{2^{k-1}+2}+\cdots +x_{2^{k}}}{2^{k-1}}}}{2}}\\[7pt]&\geq {\frac {{\sqrt[{2^{k-1}}]{x_{1}x_{2}\cdots x_{2^{k-1}}}}+{\sqrt[{2^{k-1}}]{x_{2^{k-1}+1}x_{2^{k-1}+2}\cdots x_{2^{k}}}}}{2}}\\[7pt]&\geq {\sqrt {{\sqrt[{2^{k-1}}]{x_{1}x_{2}\cdots x_{2^{k-1}}}}{\sqrt[{2^{k-1}}]{x_{2^{k-1}+1}x_{2^{k-1}+2}\cdots x_{2^{k}}}}}}\\[7pt]&={\sqrt[{2^{k}}]{x_{1}x_{2}\cdots x_{2^{k}}}}\end{aligned}}$
where in the first inequality, the two sides are equal only if
$x_{1}=x_{2}=\cdots =x_{2^{k-1}}$
and
$x_{2^{k-1}+1}=x_{2^{k-1}+2}=\cdots =x_{2^{k}}$
(in which case the first arithmetic mean and first geometric mean are both equal to x1, and similarly with the second arithmetic mean and second geometric mean); and in the second inequality, the two sides are only equal if the two geometric means are equal. Since not all 2k numbers are equal, it is not possible for both inequalities to be equalities, so we know that:
${\frac {x_{1}+x_{2}+\cdots +x_{2^{k}}}{2^{k}}}\geq {\sqrt[{2^{k}}]{x_{1}x_{2}\cdots x_{2^{k}}}}$
as desired.
The subcase where n < 2k
If n is not a natural power of 2, then it is certainly less than some natural power of 2, since the sequence 2, 4, 8, . . . , 2k, . . . is unbounded above. Therefore, without loss of generality, let m be some natural power of 2 that is greater than n.
So, if we have n terms, then let us denote their arithmetic mean by α, and expand our list of terms thus:
$x_{n+1}=x_{n+2}=\cdots =x_{m}=\alpha .$
We then have:
${\begin{aligned}\alpha &={\frac {x_{1}+x_{2}+\cdots +x_{n}}{n}}\\[6pt]&={\frac {{\frac {m}{n}}\left(x_{1}+x_{2}+\cdots +x_{n}\right)}{m}}\\[6pt]&={\frac {x_{1}+x_{2}+\cdots +x_{n}+{\frac {(m-n)}{n}}\left(x_{1}+x_{2}+\cdots +x_{n}\right)}{m}}\\[6pt]&={\frac {x_{1}+x_{2}+\cdots +x_{n}+\left(m-n\right)\alpha }{m}}\\[6pt]&={\frac {x_{1}+x_{2}+\cdots +x_{n}+x_{n+1}+\cdots +x_{m}}{m}}\\[6pt]&\geq {\sqrt[{m}]{x_{1}x_{2}\cdots x_{n}x_{n+1}\cdots x_{m}}}\\[6pt]&={\sqrt[{m}]{x_{1}x_{2}\cdots x_{n}\alpha ^{m-n}}}\,,\end{aligned}}$
so
$\alpha ^{m}\geq x_{1}x_{2}\cdots x_{n}\alpha ^{m-n}$
and
$\alpha \geq {\sqrt[{n}]{x_{1}x_{2}\cdots x_{n}}}$
as desired.
Proof by induction using basic calculus
The following proof uses mathematical induction and some basic differential calculus.
Induction basis: For n = 1 the statement is true with equality.
Induction hypothesis: Suppose that the AM–GM statement holds for all choices of n non-negative real numbers.
Induction step: In order to prove the statement for n + 1 non-negative real numbers x1, . . . , xn, xn+1, we need to prove that
${\frac {x_{1}+\cdots +x_{n}+x_{n+1}}{n+1}}-({x_{1}\cdots x_{n}x_{n+1}})^{\frac {1}{n+1}}\geq 0$
with equality only if all the n + 1 numbers are equal.
If all numbers are zero, the inequality holds with equality. If some but not all numbers are zero, we have strict inequality. Therefore, we may assume in the following, that all n + 1 numbers are positive.
We consider the last number xn+1 as a variable and define the function
$f(t)={\frac {x_{1}+\cdots +x_{n}+t}{n+1}}-({x_{1}\cdots x_{n}t})^{\frac {1}{n+1}},\qquad t>0.$
Proving the induction step is equivalent to showing that f(t) ≥ 0 for all t > 0, with f(t) = 0 only if x1, . . . , xn and t are all equal. This can be done by analyzing the critical points of f using some basic calculus.
The first derivative of f is given by
$f'(t)={\frac {1}{n+1}}-{\frac {1}{n+1}}({x_{1}\cdots x_{n}})^{\frac {1}{n+1}}t^{-{\frac {n}{n+1}}},\qquad t>0.$
A critical point t0 has to satisfy f′(t0) = 0, which means
$({x_{1}\cdots x_{n}})^{\frac {1}{n+1}}t_{0}^{-{\frac {n}{n+1}}}=1.$
After a small rearrangement we get
$t_{0}^{\frac {n}{n+1}}=({x_{1}\cdots x_{n}})^{\frac {1}{n+1}},$
and finally
$t_{0}=({x_{1}\cdots x_{n}})^{\frac {1}{n}},$
which is the geometric mean of x1, . . . , xn. This is the only critical point of f. Since f′′(t) > 0 for all t > 0, the function f is strictly convex and has a strict global minimum at t0. Next we compute the value of the function at this global minimum:
${\begin{aligned}f(t_{0})&={\frac {x_{1}+\cdots +x_{n}+({x_{1}\cdots x_{n}})^{1/n}}{n+1}}-({x_{1}\cdots x_{n}})^{\frac {1}{n+1}}({x_{1}\cdots x_{n}})^{\frac {1}{n(n+1)}}\\&={\frac {x_{1}+\cdots +x_{n}}{n+1}}+{\frac {1}{n+1}}({x_{1}\cdots x_{n}})^{\frac {1}{n}}-({x_{1}\cdots x_{n}})^{\frac {1}{n}}\\&={\frac {x_{1}+\cdots +x_{n}}{n+1}}-{\frac {n}{n+1}}({x_{1}\cdots x_{n}})^{\frac {1}{n}}\\&={\frac {n}{n+1}}{\Bigl (}{\frac {x_{1}+\cdots +x_{n}}{n}}-({x_{1}\cdots x_{n}})^{\frac {1}{n}}{\Bigr )}\\&\geq 0,\end{aligned}}$
where the final inequality holds due to the induction hypothesis. The hypothesis also says that we can have equality only when x1, . . . , xn are all equal. In this case, their geometric mean t0 has the same value, Hence, unless x1, . . . , xn, xn+1 are all equal, we have f(xn+1) > 0. This completes the proof.
This technique can be used in the same manner to prove the generalized AM–GM inequality and Cauchy–Schwarz inequality in Euclidean space Rn.
Proof by Pólya using the exponential function
George Pólya provided a proof similar to what follows. Let f(x) = ex–1 – x for all real x, with first derivative f′(x) = ex–1 – 1 and second derivative f′′(x) = ex–1. Observe that f(1) = 0, f′(1) = 0 and f′′(x) > 0 for all real x, hence f is strictly convex with the absolute minimum at x = 1. Hence x ≤ ex–1 for all real x with equality only for x = 1.
Consider a list of non-negative real numbers x1, x2, . . . , xn. If they are all zero, then the AM–GM inequality holds with equality. Hence we may assume in the following for their arithmetic mean α > 0. By n-fold application of the above inequality, we obtain that
${\begin{aligned}{{\frac {x_{1}}{\alpha }}{\frac {x_{2}}{\alpha }}\cdots {\frac {x_{n}}{\alpha }}}&\leq {e^{{\frac {x_{1}}{\alpha }}-1}e^{{\frac {x_{2}}{\alpha }}-1}\cdots e^{{\frac {x_{n}}{\alpha }}-1}}\\&=\exp {\Bigl (}{\frac {x_{1}}{\alpha }}-1+{\frac {x_{2}}{\alpha }}-1+\cdots +{\frac {x_{n}}{\alpha }}-1{\Bigr )},\qquad (*)\end{aligned}}$
with equality if and only if xi = α for every i ∈ {1, . . . , n}. The argument of the exponential function can be simplified:
${\begin{aligned}{\frac {x_{1}}{\alpha }}-1+{\frac {x_{2}}{\alpha }}-1+\cdots +{\frac {x_{n}}{\alpha }}-1&={\frac {x_{1}+x_{2}+\cdots +x_{n}}{\alpha }}-n\\&={\frac {n\alpha }{\alpha }}-n\\&=0.\end{aligned}}$
Returning to (*),
${\frac {x_{1}x_{2}\cdots x_{n}}{\alpha ^{n}}}\leq e^{0}=1,$
which produces x1 x2 · · · xn ≤ αn, hence the result[4]
${\sqrt[{n}]{x_{1}x_{2}\cdots x_{n}}}\leq \alpha .$
Proof by Lagrangian multipliers
If any of the $x_{i}$ are $0$, then there is nothing to prove. So we may assume all the $x_{i}$ are strictly positive.
Because the arithmetic and geometric means are homogeneous of degree 1, without loss of generality assume that $\prod _{i=1}^{n}x_{i}=1$. Set $G(x_{1},x_{2},\ldots ,x_{n})=\prod _{i=1}^{n}x_{i}$, and $F(x_{1},x_{2},\ldots ,x_{n})={\frac {1}{n}}\sum _{i=1}^{n}x_{i}$. The inequality will be proved (together with the equality case) if we can show that the minimum of $F(x_{1},x_{2},...,x_{n}),$ subject to the constraint $G(x_{1},x_{2},\ldots ,x_{n})=1,$ is equal to $1$, and the minimum is only achieved when $x_{1}=x_{2}=\cdots =x_{n}=1$. Let us first show that the constrained minimization problem has a global minimum.
Set $K=\{(x_{1},x_{2},\ldots ,x_{n})\colon 0\leq x_{1},x_{2},\ldots ,x_{n}\leq n\}$. Since the intersection $K\cap \{G=1\}$ is compact, the extreme value theorem guarantees that the minimum of $F(x_{1},x_{2},...,x_{n})$ subject to the constraints $G(x_{1},x_{2},\ldots ,x_{n})=1$ and $(x_{1},x_{2},\ldots ,x_{n})\in K$ is attained at some point inside $K$. On the other hand, observe that if any of the $x_{i}>n$, then $F(x_{1},x_{2},\ldots ,x_{n})>1$, while $F(1,1,\ldots ,1)=1$, and $(1,1,\ldots ,1)\in K\cap \{G=1\}$. This means that the minimum inside $K\cap \{G=1\}$ is in fact a global minimum, since the value of $F$ at any point inside $K\cap \{G=1\}$ is certainly no smaller than the minimum, and the value of $F$ at any point $(y_{1},y_{2},\ldots ,y_{n})$ not inside $K$ is strictly bigger than the value at $(1,1,\ldots ,1)$, which is no smaller than the minimum.
The method of Lagrange multipliers says that the global minimum is attained at a point $(x_{1},x_{2},\ldots ,x_{n})$ where the gradient of $F(x_{1},x_{2},\ldots ,x_{n})$ is $\lambda $ times the gradient of $G(x_{1},x_{2},\ldots ,x_{n})$, for some $\lambda $. We will show that the only point at which this happens is when $x_{1}=x_{2}=\cdots =x_{n}=1$ and $F(x_{1},x_{2},...,x_{n})=1.$
Compute ${\frac {\partial F}{\partial x_{i}}}={\frac {1}{n}}$ and
${\frac {\partial G}{\partial x_{i}}}=\prod _{j\neq i}x_{j}={\frac {G(x_{1},x_{2},\ldots ,x_{n})}{x_{i}}}={\frac {1}{x_{i}}}$
along the constraint. Setting the gradients proportional to one another therefore gives for each $i$ that ${\frac {1}{n}}={\frac {\lambda }{x_{i}}},$ and so $n\lambda =x_{i}.$ Since the left-hand side does not depend on $i$, it follows that $x_{1}=x_{2}=\cdots =x_{n}$, and since $G(x_{1},x_{2},\ldots ,x_{n})=1$, it follows that $x_{1}=x_{2}=\cdots =x_{n}=1$ and $F(x_{1},x_{2},\ldots ,x_{n})=1$, as desired.
Generalizations
Weighted AM–GM inequality
There is a similar inequality for the weighted arithmetic mean and weighted geometric mean. Specifically, let the nonnegative numbers x1, x2, . . . , xn and the nonnegative weights w1, w2, . . . , wn be given. Set w = w1 + w2 + · · · + wn. If w > 0, then the inequality
${\frac {w_{1}x_{1}+w_{2}x_{2}+\cdots +w_{n}x_{n}}{w}}\geq {\sqrt[{w}]{x_{1}^{w_{1}}x_{2}^{w_{2}}\cdots x_{n}^{w_{n}}}}$
holds with equality if and only if all the xk with wk > 0 are equal. Here the convention 00 = 1 is used.
If all wk = 1, this reduces to the above inequality of arithmetic and geometric means.
One stronger version of this, which also gives strengthened version of the unweighted version, is due to Aldaz. In particular, There is a similar inequality for the weighted arithmetic mean and weighted geometric mean. Specifically, let the nonnegative numbers x1, x2, . . . , xn and the nonnegative weights w1, w2, . . . , wn be given. Assume further that the sum of the weights is 1. Then $\sum _{i=1}^{n}w_{i}x_{i}\geq \prod _{i=1}^{n}x_{i}^{w_{i}}+\sum _{i=1}^{n}w_{i}\left(x_{i}^{\frac {1}{2}}-\sum _{i=1}^{n}w_{i}x_{i}^{\frac {1}{2}}\right)^{2}$. [5]
Proof using Jensen's inequality
Using the finite form of Jensen's inequality for the natural logarithm, we can prove the inequality between the weighted arithmetic mean and the weighted geometric mean stated above.
Since an xk with weight wk = 0 has no influence on the inequality, we may assume in the following that all weights are positive. If all xk are equal, then equality holds. Therefore, it remains to prove strict inequality if they are not all equal, which we will assume in the following, too. If at least one xk is zero (but not all), then the weighted geometric mean is zero, while the weighted arithmetic mean is positive, hence strict inequality holds. Therefore, we may assume also that all xk are positive.
Since the natural logarithm is strictly concave, the finite form of Jensen's inequality and the functional equations of the natural logarithm imply
${\begin{aligned}\ln {\Bigl (}{\frac {w_{1}x_{1}+\cdots +w_{n}x_{n}}{w}}{\Bigr )}&>{\frac {w_{1}}{w}}\ln x_{1}+\cdots +{\frac {w_{n}}{w}}\ln x_{n}\\&=\ln {\sqrt[{w}]{x_{1}^{w_{1}}x_{2}^{w_{2}}\cdots x_{n}^{w_{n}}}}.\end{aligned}}$
Since the natural logarithm is strictly increasing,
${\frac {w_{1}x_{1}+\cdots +w_{n}x_{n}}{w}}>{\sqrt[{w}]{x_{1}^{w_{1}}x_{2}^{w_{2}}\cdots x_{n}^{w_{n}}}}.$
Matrix arithmetic–geometric mean inequality
Most matrix generalizations of the arithmetic geometric mean inequality apply on the level of unitarily invariant norms, owing to the fact that even if the matrices $A$ and $B$ are positive semi-definite the matrix $AB$ may not be positive semi-definite and hence may not have a canonical square root. In [6] Bhatia and Kittaneh proved that for any unitarily invariant norm $|||\cdot |||$ and positive semi-definite matrices $A$ and $B$ it is the case that
$|||AB|||\leq {\frac {1}{2}}|||A^{2}+B^{2}|||$
Later, in [7] the same authors proved the stronger inequality that
$|||AB|||\leq {\frac {1}{4}}|||(A+B)^{2}|||$
Finally, it is known for dimension $n=2$ that the following strongest possible matrix generalization of the arithmetic-geometric mean inequality holds, and it is conjectured to hold for all $n$
$|||(AB)^{\frac {1}{2}}|||\leq {\frac {1}{2}}|||A+B|||$
This conjectured inequality was shown by Stephen Drury in 2012. Indeed, he proved[8]
${\sqrt {\sigma _{j}(AB)}}\leq {\frac {1}{2}}\lambda _{j}(A+B),\ j=1,\ldots ,n.$
Other generalizations
Other generalizations of the inequality of arithmetic and geometric means include:
• Muirhead's inequality,
• Maclaurin's inequality,
• Generalized mean inequality,
• Means of complex numbers.[9]
See also
• Hoffman's packing puzzle
• Ky Fan inequality
• Young's inequality for products
Notes
1. If AC = a and BC = b. OC = AM of a and b, and radius r = QO = OG.
Using Pythagoras' theorem, QC² = QO² + OC² ∴ QC = √QO² + OC² = QM.
Using Pythagoras' theorem, OC² = OG² + GC² ∴ GC = √OC² − OG² = GM.
Using similar triangles, HC/GC = GC/OC ∴ HC = GC²/OC = HM.
References
1. Hoffman, D. G. (1981), "Packing problems and inequalities", in Klarner, David A. (ed.), The Mathematical Gardner, Springer, pp. 212–225, doi:10.1007/978-1-4684-6686-7_19
2. Steele, J. Michael (2004). The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities. MAA Problem Books Series. Cambridge University Press. ISBN 978-0-521-54677-5. OCLC 54079548.
3. Cauchy, Augustin-Louis (1821). Cours d'analyse de l'École Royale Polytechnique, première partie, Analyse algébrique, Paris. The proof of the inequality of arithmetic and geometric means can be found on pages 457ff.
4. Arnold, Denise; Arnold, Graham (1993). Four unit mathematics. Hodder Arnold H&S. p. 242. ISBN 978-0-340-54335-1. OCLC 38328013.
5. Aldaz, J.M. (2009). "Self-Improvement of the Inequality Between Arithmetic and Geometric Means". Journal of Mathematical Inequalities. 3 (2): 213-216. doi:10.7153/jmi-03-21. Retrieved 11 January 2023.
6. Bhatia, Rajendra; Kittaneh, Fuad (1990). "On the singular values of a product of operators". SIAM Journal on Matrix Analysis and Applications. 11 (2): 272–277. doi:10.1137/0611018.
7. Bhatia, Rajendra; Kittaneh, Fuad (2000). "Notes on matrix arithmetic-geometric mean inequalities". Linear Algebra and Its Applications. 308 (1–3): 203–211. doi:10.1016/S0024-3795(00)00048-3.
8. S.W. Drury, On a question of Bhatia and Kittaneh, Linear Algebra Appl. 437 (2012) 1955–1960.
9. cf. Iordanescu, R.; Nichita, F.F.; Pasarescu, O. Unification Theories: Means and Generalized Euler Formulas. Axioms 2020, 9, 144.
External links
• Arthur Lohwater (1982). "Introduction to Inequalities". Online e-book in PDF format.
| Wikipedia |
Pythagorean means
In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians[1] because of their importance in geometry and music.
Definition
They are defined by:
${\begin{aligned}\operatorname {AM} \left(x_{1},\;\ldots ,\;x_{n}\right)&={\frac {x_{1}+\;\cdots \;+x_{n}}{n}}\\[9pt]\operatorname {GM} \left(x_{1},\;\ldots ,\;x_{n}\right)&={\sqrt[{n}]{\left\vert x_{1}\times \,\cdots \,\times x_{n}\right\vert }}\\[9pt]\operatorname {HM} \left(x_{1},\;\ldots ,\;x_{n}\right)&={\frac {n}{\frac {1}{x_{1}}}+\;\cdots \;+{\frac {1}{x_{n}}}}}\end{aligned}}$
Properties
Each mean, $ \operatorname {M} $, has the following properties:
First order homogeneity
$\operatorname {M} (bx_{1},\,\ldots ,\,bx_{n})=b\operatorname {M} (x_{1},\,\ldots ,\,x_{n})$
Invariance under exchange
$\operatorname {M} (\ldots ,\,x_{i},\,\ldots ,\,x_{j},\,\ldots )=\operatorname {M} (\ldots ,\,x_{j},\,\ldots ,\,x_{i},\,\ldots )$
for any $i$ and $j$.
Monotonicity
$a<b\rightarrow \operatorname {M} (a,x_{1},x_{2},\ldots x_{n})<\operatorname {M} (b,x_{1},x_{2},\ldots x_{n})$
Idempotence
$\forall x,\;M(x,x,\ldots x)=x$
Monotonicity and idempotence together imply that a mean of a set always lies between the extremes of the set.
$\min(x_{1},\,\ldots ,\,x_{n})\leq \operatorname {M} (x_{1},\,\ldots ,\,x_{n})\leq \max(x_{1},\,\ldots ,\,x_{n})$
The harmonic and arithmetic means are reciprocal duals of each other for positive arguments:
$\operatorname {HM} \left({\frac {1}{x_{1}}},\,\ldots ,\,{\frac {1}{x_{n}}}\right)={\frac {1}{\operatorname {AM} \left(x_{1},\,\ldots ,\,x_{n}\right)}}$
while the geometric mean is its own reciprocal dual:
$\operatorname {GM} \left({\frac {1}{x_{1}}},\,\ldots ,\,{\frac {1}{x_{n}}}\right)={\frac {1}{\operatorname {GM} \left(x_{1},\,\ldots ,\,x_{n}\right)}}$
Inequalities among means
There is an ordering to these means (if all of the $x_{i}$ are positive)
$\min \leq \operatorname {HM} \leq \operatorname {GM} \leq \operatorname {AM} \leq \max $
with equality holding if and only if the $x_{i}$ are all equal.
This is a generalization of the inequality of arithmetic and geometric means and a special case of an inequality for generalized means. The proof follows from the arithmetic-geometric mean inequality, $\operatorname {AM} \leq \max $, and reciprocal duality ($\min $ and $\max $ are also reciprocal dual to each other).
The study of the Pythagorean means is closely related to the study of majorization and Schur-convex functions. The harmonic and geometric means are concave symmetric functions of their arguments, and hence Schur-concave, while the arithmetic mean is a linear function of its arguments, so both concave and convex.
History
Almost everything that we know about the Pythagorean means came from arithmetic handbooks written in the first and second century. Nicomachus of Gerasa says that they were “acknowledged by all the ancients, Pythagoras, Plato and Aristotle.” Their earliest known use is a fragment of the Pythagorean philosopher Archytas of Tarentum:
There are three means in music: one is arithmetic, second is the geometric, third is sub-contrary, which they call harmonic. The mean is arithmetic when three terms are in proportion such that the excess by which the first exceeds the second is that by which the second exceeds the third. In this proportion it turns out that the interval of the greater terms is less, but that of the lesser terms greater. The mean is the geometric when they are such that as the first is to the second, so the second is to the third. Of these terms the greater and the lesser have the interval between them equal. Subcontrary, which we call harmonic, is the mean when they are such that, by whatever part of itself the first term exceeds the second, by that part of the third the middle term exceeds the third. It turns out that in this proportion the interval between the greater terms is greater and that between the lesser terms is less. [2]
The name "harmonic mean", according to Iamblichus, was coined by Archytas and Hippasus. The Pythagorean means also appear in Plato's Timaeus. Another evidence of their early use is a commentary by Pappus.
It was […] Theaetetus who distinguished the powers which are commensurable in length from those which are incommensurable, and who divided the more generally known irrational lines according to the different means, assigning the medial lines to geometry, the binomial to arithmetic, and the apotome to harmony, as is stated by Eudemus, the Peripatetic.[3]
The term "mean" (μεσότης, mesótēs in Ancient Greek) appears in the Neopythagorean arithmetic handbooks in connection with the term "proportion" (ἀναλογία, analogía in Ancient Greek).
Trivia
The smallest pairs of different natural numbers for which the arithmetic, geometric and harmonic means are all also natural numbers are (5,45) and (10,40).
See also
• Arithmetic–geometric mean
• Average
• Golden ratio
• Kepler triangle
Notes
1. If AC = a and BC = b. OC = AM of a and b, and radius r = QO = OG.
Using Pythagoras' theorem, QC² = QO² + OC² ∴ QC = √QO² + OC² = QM.
Using Pythagoras' theorem, OC² = OG² + GC² ∴ GC = √OC² − OG² = GM.
Using similar triangles, HC/GC = GC/OC ∴ HC = GC²/OC = HM.
References
1. Heath, Thomas. History of Ancient Greek Mathematics.
2. Huffman, Carl (2005). Archytas of Tarentum: Pythagorean, philosopher and mathematician king. Cambridge University Press. p. 163. ISBN 1139444077.
3. Huffman, Carl (2014). A History of Pythagoreanism. Cambridge University Press. p. 168. ISBN 1139915983.
External links
• Cantrell, David W. "Pythagorean Means". MathWorld.
| Wikipedia |
American Mathematics Competitions
The American Mathematics Competitions (AMC) are the first of a series of competitions in secondary school mathematics that determine the United States of America's team for the International Mathematical Olympiad (IMO). The selection process takes place over the course of roughly five stages. At the last stage, the US selects six members to form the IMO team. The 1994 US IMO Team is the first of the only two teams ever to achieve a perfect score (all six members earned perfect marks), and is colloquially known as the "dream team".[1][2]
There are three levels of AMCs:
• the AMC 8, for students under the age of 14.5 and in grades 8 and below[3]
• the AMC 10, for students under the age of 17.5 and in grades 10 and below
• the AMC 12, for students under the age of 19.5 and in grades 12 and below[4]
Students who perform well on the AMC 10 or AMC 12 competitions are invited to participate in the American Invitational Mathematics Examination (AIME). Students who perform exceptionally well on the AMC 12 and AIME are invited to the United States of America Mathematical Olympiad (USAMO), while students who perform exceptionally well on the AMC 10 and AIME are invited to United States of America Junior Mathematical Olympiad (USAJMO). Students who do exceptionally well on the USAMO (typically around 45 based on score and grade level) and USAJMO (typically around the top 15 students) are invited to attend the Mathematical Olympiad Program (MOP). See here for more details about MOP invitations and here for more information about the subsequent selection process for next year's USA IMO team.
American Mathematics Competitions is also the name of the organization, based in Washington, D.C., responsible for creating, distributing and coordinating the American Mathematics Competitions contests, which include the American Mathematics Contest, AIME, and USA(J)MO. The American Mathematics Competitions organization also conducts outreach to identify talent and strengthen problem-solving in middle and high school students.[5]
History
The AMC contest series includes the American Mathematics Contest 8 (AMC 8) (formerly the American Junior High School Mathematics Examination) for students in grades 8 and below, begun in 1985; the American Mathematics Contest 10 (AMC 10), for students in grades 9 and 10, begun in 2000; the American Mathematics Contest 12 (AMC 12) (formerly the American High School Mathematics Examination) for students in grades 11 and 12, begun in 1950; the American Invitational Mathematics Examination (AIME), begun in 1983; and the USA Mathematical Olympiad (USAMO), begun in 1972.[6]
Years Name No. of questions Comments
1950–1951 Annual High School Contest 50New York state only
1952–1959Nationwide
1960–196740
1968–197235
1973 Annual High School Mathematics Examination 35
1974–198230
1983–1999 American High School Mathematics Examination 30 AIME introduced in 1983, now is a middle step between AHSME and USAMO
AJHSME, now AMC 8, introduced in 1985
2000–present American Mathematics Competition 25 AHSME split into AMC10 and AMC12
A&B versions introduced in 2002.
USAMO split into USAJMO and USAMO in 2010. AMC 10 qualifiers who pass AIME go to USAJMO, instead of USAMO. USAJMO is supposed to be easier than USAMO.
Benefits of Participating
There are certain rewards for doing well on the AMC tests. For the AMC 8, a perfect score may earn a book prize or a plaque (as it did for the students who achieved perfect scores in 2002); a list of high scoring students is also available to colleges, institutions, and programs who want to attract students strong in mathematics. This may earn a high scorer an invitation to apply to places like MathPath, a summer program for middle schoolers. The top-scoring student in each school is also awarded a special pin.
For the AMC 10 and AMC 12, a high score earns recognition (in particular, perfect scorers' names and pictures are published in a special awards book); as with the AMC 8, a list of high-scoring students is also available to colleges, institutions, etc. The top-scoring student in each school is awarded a special pin, or a bronze, silver, or gold medal, depending on how many times he or she was the top scorer.
In addition, high scorers on the AMC 10 and AMC 12 qualify to take the next round of competition, the 3-hour long American Invitational Mathematics Examination (AIME), typically held in February. Cutoffs vary from year-to-year, but the total number of individuals who qualify for the AIME is typically around 6,000.
The combined scores of the AMC 12 and the AIME are used to determine approximately 270 individuals that will be invited back to take a 9-hour, 2-day, 6-problem proof-based contest known as the United States of America Mathematical Olympiad (USAMO), while the combined scores of the AMC 10 and the AIME are used to determine approximately 230 individuals that will be invited to take the United States of America Junior Mathematical Olympiad (USAJMO), which is easier but follows the same format. Students who do exceptionally well on the USAMO (typically around 45 based on score and grade level) and USAJMO (typically around the top 15 students) are invited to attend the Mathematical Olympiad Program (MOP). See here for more details about MOP invitations and here for more information about the subsequent selection process for next year's USA IMO team. In particular, High school seniors are admitted only if they are members of that year's IMO team, as the US IMO team selection process takes two years from start to finish.
During this summer camp, a 3-day competition called the Team Selection Test Selection Test (TSTST) is held to determine the approximately 30 individuals who will form the USA IMO Team Selection Test (TST) Group for the following school year. These individuals take a series of contests throughout the year, such as the Romanian Master of Mathematics and Asian Pacific Mathematics Olympiad, to finally pick the 6 member US Mathematics Team that will represent the US at the International Math Olympiad. The current director of MOP and team leader of the US IMO team is Po-Shen Loh from Carnegie Mellon University.[7]
Rules and scoring
AMC 8
The AMC 8 is a 25 multiple-choice question, 40-minute competition designed for middle schoolers.[6] No problems require the use of a calculator, and their use has been banned since 2008. The competition was previously held on a Thursday in November. However, starting in 2022, the competition will be held in January.
The AMC 8 is scored based on the number of questions answered correctly only. There is no penalty for getting a question wrong, and each question has equal value. Thus, a student who answers 23 questions correctly and 2 questions incorrectly receives a score of 23.
Rankings and awards
Ranking[8]
Based on questions correct:
• Distinguished Honor Roll: Top 1% (has ranged from 19–25)
• Honor Roll: Top 5% (has ranged from 15–19)
Awards
• A Certificate of Distinction is given to all students who receive a perfect score.
• An AMC 8 Winner Pin is given to the student(s) in each school with the highest score.
• The top three students for each school section will receive respectively a gold, silver, or bronze Certificate for Outstanding Achievement.
• An AMC 8 Honor Roll Certificate is given to all high scoring students.
• An AMC 8 Merit Certificate is given to high scoring students who are in 6th grade or below.
AMC 10 and AMC 12
The AMC 10 and AMC 12 are 25 question, 75-minute multiple choice competitions in secondary school mathematics containing problems which can be understood and solved with pre-calculus concepts. Calculators have not been allowed on the AMC 10/12 since 2008.[9]
High scores on the AMC 10 or 12 can qualify the participant for the American Invitational Mathematics Examination (AIME).[10] On the AMC 10, the top 2.5% make it, typically around 100 to 115 points. On the AMC 12, the top 5% make it, typically around 85 to 95 points.
The competitions are scored based on the number of questions answered correctly and the number of questions left blank. A student receives 6 points for each question answered correctly, 1.5 points for each question left blank, and 0 points for incorrect answers. Thus, a student who answers 24 correctly, leaves 1 blank, and misses 0 gets $24\times 6+1.5\times 1=145.5$ points. The maximum possible score is $25\times 6=150$ points; in 2020, the AMC 12 had a total of 18 perfect scores between its two administrations, and the AMC 10 also had 18.
From 1974 until 1999, the competition (then known as the American High School Math Examination, or AHSME) had 30 questions and was 90 minutes long, scoring 5 points for correct answers. Originally during this time, 1 point was awarded for leaving an answer blank, however, it was changed in the late 1980s to 2 points. When the competition was shortened as part of the 2000 rebranding from AHSME to AMC, the value of a correct answer was increased to 6 points and the number of questions reduced to 25 (keeping 150 as a perfect score). In 2001, the score of a blank was increased to 2.5 to penalize guessing. The 2007 competitions were the first with only 1.5 points awarded for a blank, to discourage students from leaving a large number of questions blank in order to assure qualification for the AIME. For example, prior to this change, on the AMC 12, a student could advance with only 11 correct answers, presuming the remaining questions were left blank. After the change, a student must answer 14 questions correctly to reach 100 points.
The competitions have historically overlapped to an extent, with the medium-hard AMC 10 questions usually being the same as the medium-easy ones on the AMC 12. However, this trend has diverged recently, and questions that are in both the AMC 10 and 12 are in increasingly similar positions. Problem 18 on the 2022 AMC 10A was the same as problem 18 on the 2022 AMC 12A. [11] Since 2002, two administrations have been scheduled, so as to avoid conflicts with school breaks. Students are eligible to compete in an A competition and a B competition, and may even take the AMC 10-A and the AMC 12-B, though they may not take both the AMC 10 and AMC 12 from the same date.[4] If a student participates in both competitions, they may use either score towards qualification to the AIME or USAMO/USAJMO.
See also
• American Invitational Mathematics Examination (AIME)
• United States of America Mathematical Olympiad (USAMO)
• Mathematical Olympiad Program (MOP)
• International Mathematical Olympiad (IMO)
• List of mathematics competitions
References
1. "United States of America". International Mathematical Olympiad. Retrieved 28 December 2020.
2. www.imo-official.org https://www.imo-official.org/year_country_r.aspx?year=2022. Retrieved 2022-09-08. {{cite web}}: Missing or empty |title= (help)
3. "AMC 8". Mathematical Association of America. Retrieved 29 December 2020.
4. "AMC 10/12". Mathematical Association of America. Retrieved 28 December 2020.
5. About AMC | MAA AMC. Maa-amc.org. Retrieved on 2020-06-24.
6. American Mathematics Competitions | Mathematical Association of America. Amc-reg.maa.org. Retrieved on 2013-08-14.
7. "United States of America". International Mathematical Olympiad. Retrieved 28 December 2020.
8. "American Mathematics Contest 8". Mathematical Association of America. Retrieved 28 December 2020.
9. "2021 AMC 10/12 A The Official Teacher's Manual" (PDF). Mathematical Association of America. Archived (PDF) from the original on 2020-09-21. Retrieved 3 February 2021.
10. American Mathematics Competitions | Mathematical Association of America. Amc.maa.org. Retrieved on 2013-08-14.
11. "Art of Problem Solving".
External links
• Official website
• Problems and Solutions from past AMC exams Archived 2015-02-09 at the Wayback Machine
• The Art of Problem Solving: AMC Forum
• About AMC
American mathematics
Organizations
• AMS
• MAA
• SIAM
• AMATYC
• AWM
Institutions
• AIM
• CIMS
• IAS
• ICERM
• IMA
• IPAM
• MBI
• SLMath
• SAMSI
• Geometry Center
Competitions
• MATHCOUNTS
• AMC
• AIME
• USAMO
• MOP
• Putnam Competition
• Integration Bee
| Wikipedia |
Analysis of variance
Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statistician Ronald Fisher. ANOVA is based on the law of total variance, where the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether two or more population means are equal, and therefore generalizes the t-test beyond two means. In other words, the ANOVA is used to test the difference between two or more means.
History
While the analysis of variance reached fruition in the 20th century, antecedents extend centuries into the past according to Stigler.[1] These include hypothesis testing, the partitioning of sums of squares, experimental techniques and the additive model. Laplace was performing hypothesis testing in the 1770s.[2] Around 1800, Laplace and Gauss developed the least-squares method for combining observations, which improved upon methods then used in astronomy and geodesy. It also initiated much study of the contributions to sums of squares. Laplace knew how to estimate a variance from a residual (rather than a total) sum of squares.[3] By 1827, Laplace was using least squares methods to address ANOVA problems regarding measurements of atmospheric tides.[4] Before 1800, astronomers had isolated observational errors resulting from reaction times (the "personal equation") and had developed methods of reducing the errors.[5] The experimental methods used in the study of the personal equation were later accepted by the emerging field of psychology [6] which developed strong (full factorial) experimental methods to which randomization and blinding were soon added.[7] An eloquent non-mathematical explanation of the additive effects model was available in 1885.[8]
Ronald Fisher introduced the term variance and proposed its formal analysis in a 1918 article on theoretical population genetics,The Correlation Between Relatives on the Supposition of Mendelian Inheritance.[9] His first application of the analysis of variance to data analysis was published in 1921, Studies in Crop Variation I, [10] This divided the variation of a time series into components representing annual causes and slow deterioration. Fisher's next piece, Studies in Crop Variation II, written with Winifred Mackenzie and published in 1923, studied the variation in yield across plots sown with different varieties and subjected to different fertiliser treatments. [11] Analysis of variance became widely known after being included in Fisher's 1925 book Statistical Methods for Research Workers.
Randomization models were developed by several researchers. The first was published in Polish by Jerzy Neyman in 1923.[12]
Example
The analysis of variance can be used to describe otherwise complex relations among variables. A dog show provides an example. A dog show is not a random sampling of the breed: it is typically limited to dogs that are adult, pure-bred, and exemplary. A histogram of dog weights from a show might plausibly be rather complex, like the yellow-orange distribution shown in the illustrations. Suppose we wanted to predict the weight of a dog based on a certain set of characteristics of each dog. One way to do that is to explain the distribution of weights by dividing the dog population into groups based on those characteristics. A successful grouping will split dogs such that (a) each group has a low variance of dog weights (meaning the group is relatively homogeneous) and (b) the mean of each group is distinct (if two groups have the same mean, then it isn't reasonable to conclude that the groups are, in fact, separate in any meaningful way).
In the illustrations to the right, groups are identified as X1, X2, etc. In the first illustration, the dogs are divided according to the product (interaction) of two binary groupings: young vs old, and short-haired vs long-haired (e.g., group 1 is young, short-haired dogs, group 2 is young, long-haired dogs, etc.). Since the distributions of dog weight within each of the groups (shown in blue) has a relatively large variance, and since the means are very similar across groups, grouping dogs by these characteristics does not produce an effective way to explain the variation in dog weights: knowing which group a dog is in doesn't allow us to predict its weight much better than simply knowing the dog is in a dog show. Thus, this grouping fails to explain the variation in the overall distribution (yellow-orange).
An attempt to explain the weight distribution by grouping dogs as pet vs working breed and less athletic vs more athletic would probably be somewhat more successful (fair fit). The heaviest show dogs are likely to be big, strong, working breeds, while breeds kept as pets tend to be smaller and thus lighter. As shown by the second illustration, the distributions have variances that are considerably smaller than in the first case, and the means are more distinguishable. However, the significant overlap of distributions, for example, means that we cannot distinguish X1 and X2 reliably. Grouping dogs according to a coin flip might produce distributions that look similar.
An attempt to explain weight by breed is likely to produce a very good fit. All Chihuahuas are light and all St Bernards are heavy. The difference in weights between Setters and Pointers does not justify separate breeds. The analysis of variance provides the formal tools to justify these intuitive judgments. A common use of the method is the analysis of experimental data or the development of models. The method has some advantages over correlation: not all of the data must be numeric and one result of the method is a judgment in the confidence in an explanatory relationship.
Classes of models
There are three classes of models used in the analysis of variance, and these are outlined here.
Fixed-effects models
The fixed-effects model (class I) of analysis of variance applies to situations in which the experimenter applies one or more treatments to the subjects of the experiment to see whether the response variable values change. This allows the experimenter to estimate the ranges of response variable values that the treatment would generate in the population as a whole.
Random-effects models
Random-effects model (class II) is used when the treatments are not fixed. This occurs when the various factor levels are sampled from a larger population. Because the levels themselves are random variables, some assumptions and the method of contrasting the treatments (a multi-variable generalization of simple differences) differ from the fixed-effects model.[13]
Mixed-effects models
Main article: Mixed model
A mixed-effects model (class III) contains experimental factors of both fixed and random-effects types, with appropriately different interpretations and analysis for the two types.
Example
Teaching experiments could be performed by a college or university department to find a good introductory textbook, with each text considered a treatment. The fixed-effects model would compare a list of candidate texts. The random-effects model would determine whether important differences exist among a list of randomly selected texts. The mixed-effects model would compare the (fixed) incumbent texts to randomly selected alternatives.
Defining fixed and random effects has proven elusive, with multiple competing definitions.[14]
Assumptions
The analysis of variance has been studied from several approaches, the most common of which uses a linear model that relates the response to the treatments and blocks. Note that the model is linear in parameters but may be nonlinear across factor levels. Interpretation is easy when data is balanced across factors but much deeper understanding is needed for unbalanced data.
Textbook analysis using a normal distribution
The analysis of variance can be presented in terms of a linear model, which makes the following assumptions about the probability distribution of the responses:[15][16][17][18]
• Independence of observations – this is an assumption of the model that simplifies the statistical analysis.
• Normality – the distributions of the residuals are normal.
• Equality (or "homogeneity") of variances, called homoscedasticity—the variance of data in groups should be the same.
The separate assumptions of the textbook model imply that the errors are independently, identically, and normally distributed for fixed effects models, that is, that the errors ($\varepsilon $) are independent and
$\varepsilon \thicksim N(0,\sigma ^{2}).$
Randomization-based analysis
In a randomized controlled experiment, the treatments are randomly assigned to experimental units, following the experimental protocol. This randomization is objective and declared before the experiment is carried out. The objective random-assignment is used to test the significance of the null hypothesis, following the ideas of C. S. Peirce and Ronald Fisher. This design-based analysis was discussed and developed by Francis J. Anscombe at Rothamsted Experimental Station and by Oscar Kempthorne at Iowa State University.[19] Kempthorne and his students make an assumption of unit treatment additivity, which is discussed in the books of Kempthorne and David R. Cox.[20][21]
Unit-treatment additivity
In its simplest form, the assumption of unit-treatment additivity[nb 1] states that the observed response $y_{i,j}$ from experimental unit $i$ when receiving treatment $j$ can be written as the sum of the unit's response $y_{i}$ and the treatment-effect $t_{j}$, that is [22][23][24]
$y_{i,j}=y_{i}+t_{j}.$
The assumption of unit-treatment additivity implies that, for every treatment $j$, the $j$th treatment has exactly the same effect $t_{j}$ on every experiment unit.
The assumption of unit treatment additivity usually cannot be directly falsified, according to Cox and Kempthorne. However, many consequences of treatment-unit additivity can be falsified. For a randomized experiment, the assumption of unit-treatment additivity implies that the variance is constant for all treatments. Therefore, by contraposition, a necessary condition for unit-treatment additivity is that the variance is constant.
The use of unit treatment additivity and randomization is similar to the design-based inference that is standard in finite-population survey sampling.
Derived linear model
Kempthorne uses the randomization-distribution and the assumption of unit treatment additivity to produce a derived linear model, very similar to the textbook model discussed previously.[25] The test statistics of this derived linear model are closely approximated by the test statistics of an appropriate normal linear model, according to approximation theorems and simulation studies.[26] However, there are differences. For example, the randomization-based analysis results in a small but (strictly) negative correlation between the observations.[27][28] In the randomization-based analysis, there is no assumption of a normal distribution and certainly no assumption of independence. On the contrary, the observations are dependent!
The randomization-based analysis has the disadvantage that its exposition involves tedious algebra and extensive time. Since the randomization-based analysis is complicated and is closely approximated by the approach using a normal linear model, most teachers emphasize the normal linear model approach. Few statisticians object to model-based analysis of balanced randomized experiments.
Statistical models for observational data
However, when applied to data from non-randomized experiments or observational studies, model-based analysis lacks the warrant of randomization.[29] For observational data, the derivation of confidence intervals must use subjective models, as emphasized by Ronald Fisher and his followers. In practice, the estimates of treatment-effects from observational studies generally are often inconsistent. In practice, "statistical models" and observational data are useful for suggesting hypotheses that should be treated very cautiously by the public.[30]
Summary of assumptions
See also: Shapiro–Wilk test, Bartlett's test, and Levene's test
The normal-model based ANOVA analysis assumes the independence, normality, and homogeneity of variances of the residuals. The randomization-based analysis assumes only the homogeneity of the variances of the residuals (as a consequence of unit-treatment additivity) and uses the randomization procedure of the experiment. Both these analyses require homoscedasticity, as an assumption for the normal-model analysis and as a consequence of randomization and additivity for the randomization-based analysis.
However, studies of processes that change variances rather than means (called dispersion effects) have been successfully conducted using ANOVA.[31] There are no necessary assumptions for ANOVA in its full generality, but the F-test used for ANOVA hypothesis testing has assumptions and practical limitations which are of continuing interest.
Problems which do not satisfy the assumptions of ANOVA can often be transformed to satisfy the assumptions. The property of unit-treatment additivity is not invariant under a "change of scale", so statisticians often use transformations to achieve unit-treatment additivity. If the response variable is expected to follow a parametric family of probability distributions, then the statistician may specify (in the protocol for the experiment or observational study) that the responses be transformed to stabilize the variance.[32] Also, a statistician may specify that logarithmic transforms be applied to the responses which are believed to follow a multiplicative model.[23][33] According to Cauchy's functional equation theorem, the logarithm is the only continuous transformation that transforms real multiplication to addition.
Characteristics
ANOVA is used in the analysis of comparative experiments, those in which only the difference in outcomes is of interest. The statistical significance of the experiment is determined by a ratio of two variances. This ratio is independent of several possible alterations to the experimental observations: Adding a constant to all observations does not alter significance. Multiplying all observations by a constant does not alter significance. So ANOVA statistical significance result is independent of constant bias and scaling errors as well as the units used in expressing observations. In the era of mechanical calculation it was common to subtract a constant from all observations (when equivalent to dropping leading digits) to simplify data entry.[34][35] This is an example of data coding.
Algorithm
The calculations of ANOVA can be characterized as computing a number of means and variances, dividing two variances and comparing the ratio to a handbook value to determine statistical significance. Calculating a treatment effect is then trivial: "the effect of any treatment is estimated by taking the difference between the mean of the observations which receive the treatment and the general mean".[36]
Partitioning of the sum of squares
Main article: Partition of sums of squares
See also: Lack-of-fit sum of squares
ANOVA uses traditional standardized terminology. The definitional equation of sample variance is $ s^{2}={\frac {1}{n-1}}\sum _{i}(y_{i}-{\bar {y}})^{2}$, where the divisor is called the degrees of freedom (DF), the summation is called the sum of squares (SS), the result is called the mean square (MS) and the squared terms are deviations from the sample mean. ANOVA estimates 3 sample variances: a total variance based on all the observation deviations from the grand mean, an error variance based on all the observation deviations from their appropriate treatment means, and a treatment variance. The treatment variance is based on the deviations of treatment means from the grand mean, the result being multiplied by the number of observations in each treatment to account for the difference between the variance of observations and the variance of means.
The fundamental technique is a partitioning of the total sum of squares SS into components related to the effects used in the model. For example, the model for a simplified ANOVA with one type of treatment at different levels.
$SS_{\text{Total}}=SS_{\text{Error}}+SS_{\text{Treatments}}$
The number of degrees of freedom DF can be partitioned in a similar way: one of these components (that for error) specifies a chi-squared distribution which describes the associated sum of squares, while the same is true for "treatments" if there is no treatment effect.
$DF_{\text{Total}}=DF_{\text{Error}}+DF_{\text{Treatments}}$
The F-test
Main article: F-test
The F-test is used for comparing the factors of the total deviation. For example, in one-way, or single-factor ANOVA, statistical significance is tested for by comparing the F test statistic
$F={\frac {\text{variance between treatments}}{\text{variance within treatments}}}$
$F={\frac {MS_{\text{Treatments}}}{MS_{\text{Error}}}}={{SS_{\text{Treatments}}/(I-1)} \over {SS_{\text{Error}}/(n_{T}-I)}}$
where MS is mean square, $I$ is the number of treatments and $n_{T}$ is the total number of cases
to the F-distribution with $I-1$ being the numerator degrees of freedom and $n_{T}-I$ the denominator degrees of freedom. Using the F-distribution is a natural candidate because the test statistic is the ratio of two scaled sums of squares each of which follows a scaled chi-squared distribution.
The expected value of F is $1+{n\sigma _{\text{Treatment}}^{2}}/{\sigma _{\text{Error}}^{2}}$ (where $n$ is the treatment sample size) which is 1 for no treatment effect. As values of F increase above 1, the evidence is increasingly inconsistent with the null hypothesis. Two apparent experimental methods of increasing F are increasing the sample size and reducing the error variance by tight experimental controls.
There are two methods of concluding the ANOVA hypothesis test, both of which produce the same result:
• The textbook method is to compare the observed value of F with the critical value of F determined from tables. The critical value of F is a function of the degrees of freedom of the numerator and the denominator and the significance level (α). If F ≥ FCritical, the null hypothesis is rejected.
• The computer method calculates the probability (p-value) of a value of F greater than or equal to the observed value. The null hypothesis is rejected if this probability is less than or equal to the significance level (α).
The ANOVA F-test is known to be nearly optimal in the sense of minimizing false negative errors for a fixed rate of false positive errors (i.e. maximizing power for a fixed significance level). For example, to test the hypothesis that various medical treatments have exactly the same effect, the F-test's p-values closely approximate the permutation test's p-values: The approximation is particularly close when the design is balanced.[26][37] Such permutation tests characterize tests with maximum power against all alternative hypotheses, as observed by Rosenbaum.[nb 2] The ANOVA F-test (of the null-hypothesis that all treatments have exactly the same effect) is recommended as a practical test, because of its robustness against many alternative distributions.[38][nb 3]
Extended algorithm
ANOVA consists of separable parts; partitioning sources of variance and hypothesis testing can be used individually. ANOVA is used to support other statistical tools. Regression is first used to fit more complex models to data, then ANOVA is used to compare models with the objective of selecting simple(r) models that adequately describe the data. "Such models could be fit without any reference to ANOVA, but ANOVA tools could then be used to make some sense of the fitted models, and to test hypotheses about batches of coefficients."[39] "[W]e think of the analysis of variance as a way of understanding and structuring multilevel models—not as an alternative to regression but as a tool for summarizing complex high-dimensional inferences ..."[39]
For a single factor
Main article: One-way analysis of variance
The simplest experiment suitable for ANOVA analysis is the completely randomized experiment with a single factor. More complex experiments with a single factor involve constraints on randomization and include completely randomized blocks and Latin squares (and variants: Graeco-Latin squares, etc.). The more complex experiments share many of the complexities of multiple factors. A relatively complete discussion of the analysis (models, data summaries, ANOVA table) of the completely randomized experiment is available.
There are some alternatives to conventional one-way analysis of variance, e.g.: Welch's heteroscedastic F test, Welch's heteroscedastic F test with trimmed means and Winsorized variances, Brown-Forsythe test, Alexander-Govern test, James second order test and Kruskal-Wallis test, available in onewaytests R
It is useful to represent each data point in the following form, called a statistical model:
$Y_{ij}=\mu +\tau _{j}+\varepsilon _{ij}$
where
• i = 1, 2, 3, ..., R
• j = 1, 2, 3, ..., C
• μ = overall average (mean)
• τj = differential effect (response) associated with the j level of X;
this assumes that overall the values of τj add to zero (that is, $ \sum _{j=1}^{C}\tau _{j}=0$)
• εij = noise or error associated with the particular ij data value
That is, we envision an additive model that says every data point can be represented by summing three quantities: the true mean, averaged over all factor levels being investigated, plus an incremental component associated with the particular column (factor level), plus a final component associated with everything else affecting that specific data value.
For multiple factors
ANOVA generalizes to the study of the effects of multiple factors. When the experiment includes observations at all combinations of levels of each factor, it is termed factorial. Factorial experiments are more efficient than a series of single factor experiments and the efficiency grows as the number of factors increases.[40] Consequently, factorial designs are heavily used.
The use of ANOVA to study the effects of multiple factors has a complication. In a 3-way ANOVA with factors x, y and z, the ANOVA model includes terms for the main effects (x, y, z) and terms for interactions (xy, xz, yz, xyz). All terms require hypothesis tests. The proliferation of interaction terms increases the risk that some hypothesis test will produce a false positive by chance. Fortunately, experience says that high order interactions are rare.[41] The ability to detect interactions is a major advantage of multiple factor ANOVA. Testing one factor at a time hides interactions, but produces apparently inconsistent experimental results.[40]
Caution is advised when encountering interactions; Test interaction terms first and expand the analysis beyond ANOVA if interactions are found. Texts vary in their recommendations regarding the continuation of the ANOVA procedure after encountering an interaction. Interactions complicate the interpretation of experimental data. Neither the calculations of significance nor the estimated treatment effects can be taken at face value. "A significant interaction will often mask the significance of main effects."[42] Graphical methods are recommended to enhance understanding. Regression is often useful. A lengthy discussion of interactions is available in Cox (1958).[43] Some interactions can be removed (by transformations) while others cannot.
A variety of techniques are used with multiple factor ANOVA to reduce expense. One technique used in factorial designs is to minimize replication (possibly no replication with support of analytical trickery) and to combine groups when effects are found to be statistically (or practically) insignificant. An experiment with many insignificant factors may collapse into one with a few factors supported by many replications.[44]
Associated analysis
Some analysis is required in support of the design of the experiment while other analysis is performed after changes in the factors are formally found to produce statistically significant changes in the responses. Because experimentation is iterative, the results of one experiment alter plans for following experiments.
The number of experimental units
In the design of an experiment, the number of experimental units is planned to satisfy the goals of the experiment. Experimentation is often sequential.
Early experiments are often designed to provide mean-unbiased estimates of treatment effects and of experimental error. Later experiments are often designed to test a hypothesis that a treatment effect has an important magnitude; in this case, the number of experimental units is chosen so that the experiment is within budget and has adequate power, among other goals.
Reporting sample size analysis is generally required in psychology. "Provide information on sample size and the process that led to sample size decisions."[45] The analysis, which is written in the experimental protocol before the experiment is conducted, is examined in grant applications and administrative review boards.
Besides the power analysis, there are less formal methods for selecting the number of experimental units. These include graphical methods based on limiting the probability of false negative errors, graphical methods based on an expected variation increase (above the residuals) and methods based on achieving a desired confidence interval.[46]
Power analysis
Power analysis is often applied in the context of ANOVA in order to assess the probability of successfully rejecting the null hypothesis if we assume a certain ANOVA design, effect size in the population, sample size and significance level. Power analysis can assist in study design by determining what sample size would be required in order to have a reasonable chance of rejecting the null hypothesis when the alternative hypothesis is true.[47][48][49][50]
Effect size
Main article: Effect size
Several standardized measures of effect have been proposed for ANOVA to summarize the strength of the association between a predictor(s) and the dependent variable or the overall standardized difference of the complete model. Standardized effect-size estimates facilitate comparison of findings across studies and disciplines. However, while standardized effect sizes are commonly used in much of the professional literature, a non-standardized measure of effect size that has immediately "meaningful" units may be preferable for reporting purposes.[51]
Model confirmation
Sometimes tests are conducted to determine whether the assumptions of ANOVA appear to be violated. Residuals are examined or analyzed to confirm homoscedasticity and gross normality.[52] Residuals should have the appearance of (zero mean normal distribution) noise when plotted as a function of anything including time and modeled data values. Trends hint at interactions among factors or among observations.
Follow-up tests
A statistically significant effect in ANOVA is often followed by additional tests. This can be done in order to assess which groups are different from which other groups or to test various other focused hypotheses. Follow-up tests are often distinguished in terms of whether they are "planned" (a priori) or "post hoc." Planned tests are determined before looking at the data, and post hoc tests are conceived only after looking at the data (though the term "post hoc" is inconsistently used).
The follow-up tests may be "simple" pairwise comparisons of individual group means or may be "compound" comparisons (e.g., comparing the mean pooling across groups A, B and C to the mean of group D). Comparisons can also look at tests of trend, such as linear and quadratic relationships, when the independent variable involves ordered levels. Often the follow-up tests incorporate a method of adjusting for the multiple comparisons problem.
Follow-up tests to identify which specific groups, variables, or factors have statistically different means include the Tukey's range test, and Duncan's new multiple range test. In turn, these tests are often followed with a Compact Letter Display (CLD) methodology in order to render the output of the mentioned tests more transparent to a non-statistician audience.
Study designs
There are several types of ANOVA. Many statisticians base ANOVA on the design of the experiment,[53] especially on the protocol that specifies the random assignment of treatments to subjects; the protocol's description of the assignment mechanism should include a specification of the structure of the treatments and of any blocking. It is also common to apply ANOVA to observational data using an appropriate statistical model.[54]
Some popular designs use the following types of ANOVA:
• One-way ANOVA is used to test for differences among two or more independent groups (means), e.g. different levels of urea application in a crop, or different levels of antibiotic action on several different bacterial species,[55] or different levels of effect of some medicine on groups of patients. However, should these groups not be independent, and there is an order in the groups (such as mild, moderate and severe disease), or in the dose of a drug (such as 5 mg/mL, 10 mg/mL, 20 mg/mL) given to the same group of patients, then a linear trend estimation should be used. Typically, however, the one-way ANOVA is used to test for differences among at least three groups, since the two-group case can be covered by a t-test.[56] When there are only two means to compare, the t-test and the ANOVA F-test are equivalent; the relation between ANOVA and t is given by F = t2.
• Factorial ANOVA is used when there is more than one factor.
• Repeated measures ANOVA is used when the same subjects are used for each factor (e.g., in a longitudinal study).
• Multivariate analysis of variance (MANOVA) is used when there is more than one response variable.
Cautions
Balanced experiments (those with an equal sample size for each treatment) are relatively easy to interpret; unbalanced experiments offer more complexity. For single-factor (one-way) ANOVA, the adjustment for unbalanced data is easy, but the unbalanced analysis lacks both robustness and power.[57] For more complex designs the lack of balance leads to further complications. "The orthogonality property of main effects and interactions present in balanced data does not carry over to the unbalanced case. This means that the usual analysis of variance techniques do not apply. Consequently, the analysis of unbalanced factorials is much more difficult than that for balanced designs."[58] In the general case, "The analysis of variance can also be applied to unbalanced data, but then the sums of squares, mean squares, and F-ratios will depend on the order in which the sources of variation are considered."[39]
ANOVA is (in part) a test of statistical significance. The American Psychological Association (and many other organisations) holds the view that simply reporting statistical significance is insufficient and that reporting confidence bounds is preferred.[51]
Generalizations
ANOVA is considered to be a special case of linear regression[59][60] which in turn is a special case of the general linear model.[61] All consider the observations to be the sum of a model (fit) and a residual (error) to be minimized.
The Kruskal–Wallis test and the Friedman test are nonparametric tests which do not rely on an assumption of normality.[62][63]
Connection to linear regression
Below we make clear the connection between multi-way ANOVA and linear regression.
Linearly re-order the data so that $k$-th observation is associated with a response $y_{k}$ and factors $Z_{k,b}$ where $b\in \{1,2,\ldots ,B\}$ denotes the different factors and $B$ is the total number of factors. In one-way ANOVA $B=1$ and in two-way ANOVA $B=2$. Furthermore, we assume the $b$-th factor has $I_{b}$ levels, namely $\{1,2,\ldots ,I_{b}\}$. Now, we can one-hot encode the factors into the $ \sum _{b=1}^{B}I_{b}$ dimensional vector $v_{k}$.
The one-hot encoding function $g_{b}:\{1,2,\ldots ,I_{b}\}\mapsto \{0,1\}^{I_{b}}$ is defined such that the $i$-th entry of $g_{b}(Z_{k,b})$ is
$g_{b}(Z_{k,b})_{i}={\begin{cases}1&{\text{if }}i=Z_{k,b}\\0&{\text{otherwise}}\end{cases}}$
The vector $v_{k}$ is the concatenation of all of the above vectors for all $b$. Thus, $v_{k}=[g_{1}(Z_{k,1}),g_{2}(Z_{k,2}),\ldots ,g_{B}(Z_{k,B})]$. In order to obtain a fully general $B$-way interaction ANOVA we must also concatenate every additional interaction term in the vector $v_{k}$ and then add an intercept term. Let that vector be $X_{k}$.
With this notation in place, we now have the exact connection with linear regression. We simply regress response $y_{k}$ against the vector $X_{k}$. However, there is a concern about identifiability. In order to overcome such issues we assume that the sum of the parameters within each set of interactions is equal to zero. From here, one can use F-statistics or other methods to determine the relevance of the individual factors.
Example
We can consider the 2-way interaction example where we assume that the first factor has 2 levels and the second factor has 3 levels.
Define $a_{i}=1$ if $Z_{k,1}=i$ and $b_{i}=1$ if $Z_{k,2}=i$, i.e. $a$ is the one-hot encoding of the first factor and $b$ is the one-hot encoding of the second factor.
With that,
$X_{k}=[a_{1},a_{2},b_{1},b_{2},b_{3},a_{1}\times b_{1},a_{1}\times b_{2},a_{1}\times b_{3},a_{2}\times b_{1},a_{2}\times b_{2},a_{2}\times b_{3},1]$
where the last term is an intercept term. For a more concrete example suppose that
${\begin{aligned}Z_{k,1}&=2\\Z_{k,2}&=1\end{aligned}}$
Then,
$X_{k}=[0,1,1,0,0,0,0,0,1,0,0,1]$
See also
• ANOVA on ranks
• ANOVA-simultaneous component analysis
• Analysis of covariance (ANCOVA)
• Analysis of molecular variance (AMOVA)
• Analysis of rhythmic variance (ANORVA)
• Expected mean squares
• Explained variation
• Linear trend estimation
• Mixed-design analysis of variance
• Multivariate analysis of covariance (MANCOVA)
• Permutational analysis of variance
• Variance decomposition
Footnotes
1. Unit-treatment additivity is simply termed additivity in most texts. Hinkelmann and Kempthorne add adjectives and distinguish between additivity in the strict and broad senses. This allows a detailed consideration of multiple error sources (treatment, state, selection, measurement and sampling) on page 161.
2. Rosenbaum (2002, page 40) cites Section 5.7 (Permutation Tests), Theorem 2.3 (actually Theorem 3, page 184) of Lehmann's Testing Statistical Hypotheses (1959).
3. The F-test for the comparison of variances has a mixed reputation. It is not recommended as a hypothesis test to determine whether two different samples have the same variance. It is recommended for ANOVA where two estimates of the variance of the same sample are compared. While the F-test is not generally robust against departures from normality, it has been found to be robust in the special case of ANOVA. Citations from Moore & McCabe (2003): "Analysis of variance uses F statistics, but these are not the same as the F statistic for comparing two population standard deviations." (page 554) "The F test and other procedures for inference about variances are so lacking in robustness as to be of little use in practice." (page 556) "[The ANOVA F-test] is relatively insensitive to moderate nonnormality and unequal variances, especially when the sample sizes are similar." (page 763) ANOVA assumes homoscedasticity, but it is robust. The statistical test for homoscedasticity (the F-test) is not robust. Moore & McCabe recommend a rule of thumb.
Notes
1. Stigler (1986)
2. Stigler (1986, p 134)
3. Stigler (1986, p 153)
4. Stigler (1986, pp 154–155)
5. Stigler (1986, pp 240–242)
6. Stigler (1986, Chapter 7 – Psychophysics as a Counterpoint)
7. Stigler (1986, p 253)
8. Stigler (1986, pp 314–315)
9. The Correlation Between Relatives on the Supposition of Mendelian Inheritance. Ronald A. Fisher. Philosophical Transactions of the Royal Society of Edinburgh. 1918. (volume 52, pages 399–433)
10. Fisher, Ronald A. (1921). ") Studies in Crop Variation. I. An Examination of the Yield of Dressed Grain from Broadbalk". Journal of Agricultural Science. 11 (2): 107–135. doi:10.1017/S0021859600003750. hdl:2440/15170. S2CID 86029217.
11. Fisher, Ronald A. (1923). ") Studies in Crop Variation. II. The Manurial Response of Different Potato Varieties". Journal of Agricultural Science. 13 (3): 311–320. doi:10.1017/S0021859600003592. hdl:2440/15179. S2CID 85985907.
12. Scheffé (1959, p 291, "Randomization models were first formulated by Neyman (1923) for the completely randomized design, by Neyman (1935) for randomized blocks, by Welch (1937) and Pitman (1937) for the Latin square under a certain null hypothesis, and by Kempthorne (1952, 1955) and Wilk (1955) for many other designs.")
13. Montgomery (2001, Chapter 12: Experiments with random factors)
14. Gelman (2005, pp. 20–21)
15. Snedecor, George W.; Cochran, William G. (1967). Statistical Methods (6th ed.). p. 321.
16. Cochran & Cox (1992, p 48)
17. Howell (2002, p 323)
18. Anderson, David R.; Sweeney, Dennis J.; Williams, Thomas A. (1996). Statistics for business and economics (6th ed.). Minneapolis/St. Paul: West Pub. Co. pp. 452–453. ISBN 978-0-314-06378-6.
19. Anscombe (1948)
20. Hinkelmann, Klaus; Kempthorne, Oscar (2005). Design and Analysis of Experiments, Volume 2: Advanced Experimental Design. John Wiley. p. 213. ISBN 978-0-471-70993-0.
21. Cox, D. R. (1992). Planning of Experiments. Wiley. ISBN 978-0-471-57429-3.
22. Kempthorne (1979, p 30)
23. Cox (1958, Chapter 2: Some Key Assumptions)
24. Hinkelmann and Kempthorne (2008, Volume 1, Throughout. Introduced in Section 2.3.3: Principles of experimental design; The linear model; Outline of a model)
25. Hinkelmann and Kempthorne (2008, Volume 1, Section 6.3: Completely Randomized Design; Derived Linear Model)
26. Hinkelmann and Kempthorne (2008, Volume 1, Section 6.6: Completely randomized design; Approximating the randomization test)
27. Bailey (2008, Chapter 2.14 "A More General Model" in Bailey, pp. 38–40)
28. Hinkelmann and Kempthorne (2008, Volume 1, Chapter 7: Comparison of Treatments)
29. Kempthorne (1979, pp 125–126, "The experimenter must decide which of the various causes that he feels will produce variations in his results must be controlled experimentally. Those causes that he does not control experimentally, because he is not cognizant of them, he must control by the device of randomization." "[O]nly when the treatments in the experiment are applied by the experimenter using the full randomization procedure is the chain of inductive inference sound. It is only under these circumstances that the experimenter can attribute whatever effects he observes to the treatment and the treatment only. Under these circumstances his conclusions are reliable in the statistical sense.")
30. Freedman
31. Montgomery (2001, Section 3.8: Discovering dispersion effects)
32. Hinkelmann and Kempthorne (2008, Volume 1, Section 6.10: Completely randomized design; Transformations)
33. Bailey (2008)
34. Montgomery (2001, Section 3-3: Experiments with a single factor: The analysis of variance; Analysis of the fixed effects model)
35. Cochran & Cox (1992, p 2 example)
36. Cochran & Cox (1992, p 49)
37. Hinkelmann and Kempthorne (2008, Volume 1, Section 6.7: Completely randomized design; CRD with unequal numbers of replications)
38. Moore and McCabe (2003, page 763)
39. Gelman (2008)
40. Montgomery (2001, Section 5-2: Introduction to factorial designs; The advantages of factorials)
41. Belle (2008, Section 8.4: High-order interactions occur rarely)
42. Montgomery (2001, Section 5-1: Introduction to factorial designs; Basic definitions and principles)
43. Cox (1958, Chapter 6: Basic ideas about factorial experiments)
44. Montgomery (2001, Section 5-3.7: Introduction to factorial designs; The two-factor factorial design; One observation per cell)
45. Wilkinson (1999, p 596)
46. Montgomery (2001, Section 3-7: Determining sample size)
47. Howell (2002, Chapter 8: Power)
48. Howell (2002, Section 11.12: Power (in ANOVA))
49. Howell (2002, Section 13.7: Power analysis for factorial experiments)
50. Moore and McCabe (2003, pp 778–780)
51. Wilkinson (1999, p 599)
52. Montgomery (2001, Section 3-4: Model adequacy checking)
53. Cochran & Cox (1957, p 9, "The general rule [is] that the way in which the experiment is conducted determines not only whether inferences can be made, but also the calculations required to make them.")
54. "ANOVA Design". bluebox.creighton.edu. Retrieved 23 January 2023.
55. "One-way/single factor ANOVA". Archived from the original on 7 November 2014.
56. "The Probable Error of a Mean" (PDF). Biometrika. 6: 1–25. 1908. doi:10.1093/biomet/6.1.1. hdl:10338.dmlcz/143545.
57. Montgomery (2001, Section 3-3.4: Unbalanced data)
58. Montgomery (2001, Section 14-2: Unbalanced data in factorial design)
59. Gelman (2005, p.1) (with qualification in the later text)
60. Montgomery (2001, Section 3.9: The Regression Approach to the Analysis of Variance)
61. Howell (2002, p 604)
62. Howell (2002, Chapter 18: Resampling and nonparametric approaches to data)
63. Montgomery (2001, Section 3-10: Nonparametric methods in the analysis of variance)
References
• Anscombe, F. J. (1948). "The Validity of Comparative Experiments". Journal of the Royal Statistical Society. Series A (General). 111 (3): 181–211. doi:10.2307/2984159. JSTOR 2984159. MR 0030181.
• Bailey, R. A. (2008). Design of Comparative Experiments. Cambridge University Press. ISBN 978-0-521-68357-9. Pre-publication chapters are available on-line.
• Belle, Gerald van (2008). Statistical rules of thumb (2nd ed.). Hoboken, N.J: Wiley. ISBN 978-0-470-14448-0.
• Cochran, William G.; Cox, Gertrude M. (1992). Experimental designs (2nd ed.). New York: Wiley. ISBN 978-0-471-54567-5.
• Cohen, Jacob (1988). Statistical power analysis for the behavior sciences (2nd ed.). Routledge ISBN 978-0-8058-0283-2
• Cohen, Jacob (1992). "Statistics a power primer". Psychological Bulletin. 112 (1): 155–159. doi:10.1037/0033-2909.112.1.155. PMID 19565683. S2CID 14411587.
• Cox, David R. (1958). Planning of experiments. Reprinted as ISBN 978-0-471-57429-3
• Cox, David R. (2006). Principles of statistical inference. Cambridge New York: Cambridge University Press. ISBN 978-0-521-68567-2.
• Freedman, David A.(2005). Statistical Models: Theory and Practice, Cambridge University Press. ISBN 978-0-521-67105-7
• Gelman, Andrew (2005). "Analysis of variance? Why it is more important than ever". The Annals of Statistics. 33: 1–53. arXiv:math/0504499. doi:10.1214/009053604000001048. S2CID 13529149.
• Gelman, Andrew (2008). "Variance, analysis of". The new Palgrave dictionary of economics (2nd ed.). Basingstoke, Hampshire New York: Palgrave Macmillan. ISBN 978-0-333-78676-5.
• Hinkelmann, Klaus & Kempthorne, Oscar (2008). Design and Analysis of Experiments. Vol. I and II (Second ed.). Wiley. ISBN 978-0-470-38551-7.
• Howell, David C. (2002). Statistical methods for psychology (5th ed.). Pacific Grove, CA: Duxbury/Thomson Learning. ISBN 978-0-534-37770-0.
• Kempthorne, Oscar (1979). The Design and Analysis of Experiments (Corrected reprint of (1952) Wiley ed.). Robert E. Krieger. ISBN 978-0-88275-105-4.
• Lehmann, E.L. (1959) Testing Statistical Hypotheses. John Wiley & Sons.
• Montgomery, Douglas C. (2001). Design and Analysis of Experiments (5th ed.). New York: Wiley. ISBN 978-0-471-31649-7.
• Moore, David S. & McCabe, George P. (2003). Introduction to the Practice of Statistics (4e). W H Freeman & Co. ISBN 0-7167-9657-0
• Rosenbaum, Paul R. (2002). Observational Studies (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-98967-9
• Scheffé, Henry (1959). The Analysis of Variance. New York: Wiley.
• Stigler, Stephen M. (1986). The history of statistics : the measurement of uncertainty before 1900. Cambridge, Mass: Belknap Press of Harvard University Press. ISBN 978-0-674-40340-6.
• Wilkinson, Leland (1999). "Statistical Methods in Psychology Journals; Guidelines and Explanations". American Psychologist. 5 (8): 594–604. CiteSeerX 10.1.1.120.4818. doi:10.1037/0003-066X.54.8.594. S2CID 428023.
Further reading
• Box, G. e. p. (1953). "Non-Normality and Tests on Variances". Biometrika. 40 (3/4): 318–335. doi:10.1093/biomet/40.3-4.318. JSTOR 2333350.
• Box, G. E. P. (1954). "Some Theorems on Quadratic Forms Applied in the Study of Analysis of Variance Problems, I. Effect of Inequality of Variance in the One-Way Classification". The Annals of Mathematical Statistics. 25 (2): 290. doi:10.1214/aoms/1177728786.
• Box, G. E. P. (1954). "Some Theorems on Quadratic Forms Applied in the Study of Analysis of Variance Problems, II. Effects of Inequality of Variance and of Correlation Between Errors in the Two-Way Classification". The Annals of Mathematical Statistics. 25 (3): 484. doi:10.1214/aoms/1177728717.
• Caliński, Tadeusz; Kageyama, Sanpei (2000). Block designs: A Randomization approach, Volume I: Analysis. Lecture Notes in Statistics. Vol. 150. New York: Springer-Verlag. ISBN 978-0-387-98578-7.
• Christensen, Ronald (2002). Plane Answers to Complex Questions: The Theory of Linear Models (Third ed.). New York: Springer. ISBN 978-0-387-95361-8.
• Cox, David R. & Reid, Nancy M. (2000). The theory of design of experiments. (Chapman & Hall/CRC). ISBN 978-1-58488-195-7
• Fisher, Ronald (1918). "Studies in Crop Variation. I. An examination of the yield of dressed grain from Broadbalk" (PDF). Journal of Agricultural Science. 11 (2): 107–135. doi:10.1017/S0021859600003750. hdl:2440/15170. S2CID 86029217. Archived from the original (PDF) on 12 June 2001.
• Freedman, David A.; Pisani, Robert; Purves, Roger (2007) Statistics, 4th edition. W.W. Norton & Company ISBN 978-0-393-92972-0
• Hettmansperger, T. P.; McKean, J. W. (1998). Edward Arnold (ed.). Robust nonparametric statistical methods. Kendall's Library of Statistics. Vol. 5 (First ed.). New York: John Wiley & Sons, Inc. pp. xiv+467 pp. ISBN 978-0-340-54937-7. MR 1604954.
• Lentner, Marvin; Thomas Bishop (1993). Experimental design and analysis (Second ed.). Blacksburg, VA: Valley Book Company. ISBN 978-0-9616255-2-8.
• Tabachnick, Barbara G. & Fidell, Linda S. (2007). Using Multivariate Statistics (5th ed.). Boston: Pearson International Edition. ISBN 978-0-205-45938-4
• Wichura, Michael J. (2006). The coordinate-free approach to linear models. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge: Cambridge University Press. pp. xiv+199. ISBN 978-0-521-86842-6. MR 2283455.
• Phadke, Madhav S. (1989). Quality Engineering using Robust Design. New Jersey: Prentice Hall PTR. ISBN 978-0-13-745167-8.
External links
Wikimedia Commons has media related to Analysis of variance.
Wikiversity has learning resources about Analysis of variance
• SOCR: ANOVA Activity
• Examples of all ANOVA and ANCOVA models with up to three treatment factors, including randomized block, split plot, repeated measures, and Latin squares, and their analysis in R (University of Southampton)
• NIST/SEMATECH e-Handbook of Statistical Methods, section 7.4.3: "Are the means equal?"
• Analysis of variance: Introduction
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| Wikipedia |
Adleman–Pomerance–Rumely primality test
In computational number theory, the Adleman–Pomerance–Rumely primality test is an algorithm for determining whether a number is prime. Unlike other, more efficient algorithms for this purpose, it avoids the use of random numbers, so it is a deterministic primality test. It is named after its discoverers, Leonard Adleman, Carl Pomerance, and Robert Rumely. The test involves arithmetic in cyclotomic fields.
It was later improved by Henri Cohen and Hendrik Willem Lenstra, commonly referred to as APR-CL. It can test primality of an integer n in time:
$(\log n)^{O(\log \,\log \,\log n)}.$
Software implementations
• UBASIC provides an implementation under the name APRT-CLE (APR Test CL extended)
• A factoring applet that uses APR-CL on certain conditions (source code included)
• Pari/GP uses APR-CL conditionally in its implementation of isprime().
• mpz_aprcl is an open source implementation using C and GMP.
• Jean Penné's LLR uses APR-CL on certain types of prime tests as a fallback option.
References
• Adleman, Leonard M.; Pomerance, Carl; Rumely, Robert S. (1983). "On distinguishing prime numbers from composite numbers". Annals of Mathematics. 117 (1): 173–206. doi:10.2307/2006975. JSTOR 2006975.
• Cohen, Henri; Lenstra, Hendrik W., Jr. (1984). "Primality testing and Jacobi sums". Mathematics of Computation. 42 (165): 297–330. doi:10.2307/2007581. JSTOR 2007581.{{cite journal}}: CS1 maint: multiple names: authors list (link)
• Riesel, Hans (1994). Prime Numbers and Computer Methods for Factorization. Birkhäuser. pp. 131–136. ISBN 978-0-8176-3743-9.
• APR and APR-CL
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• Italics indicate that algorithm is for numbers of special forms
| Wikipedia |
ASUI
The Average Service Unavailability Index (ASUI)[1] is a reliability index commonly used by electric power utilities. It is calculated as
${\mbox{ASUI}}={\frac {\sum {U_{i}N_{i}}}{\sum {N_{i}}\times 8760}}=1-{\mbox{ASAI}}$
where $N_{i}$ is the number of customers and $U_{i}$ is the annual outage time (in hours) for location $i$. ASUI can be represented in relation to SAIDI (where annual SAIDI is given in hours) as
${\mbox{ASUI}}={\frac {\mbox{SAIDI}}{8760}}$
References
1. Pham, Hoang (8 April 2003). Handbook of reliability engineering. Birkhäuser. p. 514. ISBN 978-1-85233-453-6. Retrieved 21 July 2011.
Reliability indices
Generic
• MTBF
• MTTR
• FOR
• EFOR
Electricity distribution
• ASUI
• ASAI
• CAIDI/SAIDI
• CAIFI/SAIFI
• MAIFI
• CTAIDI
Electricity generation
• LOLP
• LOLE
• LOLEV
• LOLF
• LOLH
| Wikipedia |
ATS theorem
In mathematics, the ATS theorem is the theorem on the approximation of a trigonometric sum by a shorter one. The application of the ATS theorem in certain problems of mathematical and theoretical physics can be very helpful.
History of the problem
In some fields of mathematics and mathematical physics, sums of the form
$S=\sum _{a<k\leq b}\varphi (k)e^{2\pi if(k)}\qquad (1)$
are under study.
Here $\varphi (x)$ and $f(x)$ are real valued functions of a real argument, and $i^{2}=-1.$ Such sums appear, for example, in number theory in the analysis of the Riemann zeta function, in the solution of problems connected with integer points in the domains on plane and in space, in the study of the Fourier series, and in the solution of such differential equations as the wave equation, the potential equation, the heat conductivity equation.
The problem of approximation of the series (1) by a suitable function was studied already by Euler and Poisson.
We shall define the length of the sum $S$ to be the number $b-a$ (for the integers $a$ and $b,$ this is the number of the summands in $S$).
Under certain conditions on $\varphi (x)$ and $f(x)$ the sum $S$ can be substituted with good accuracy by another sum $S_{1},$
$S_{1}=\sum _{\alpha <k\leq \beta }\Phi (k)e^{2\pi iF(k)},\ \ \ (2)$
where the length $\beta -\alpha $ is far less than $b-a.$
First relations of the form
$S=S_{1}+R,\qquad (3)$
where $S,$ $S_{1}$ are the sums (1) and (2) respectively, $R$ is a remainder term, with concrete functions $\varphi (x)$ and $f(x),$ were obtained by G. H. Hardy and J. E. Littlewood,[1][2][3] when they deduced approximate functional equation for the Riemann zeta function $\zeta (s)$ and by I. M. Vinogradov,[4] in the study of the amounts of integer points in the domains on plane. In general form the theorem was proved by J. Van der Corput,[5][6] (on the recent results connected with the Van der Corput theorem one can read at [7]).
In every one of the above-mentioned works, some restrictions on the functions $\varphi (x)$ and $f(x)$ were imposed. With convenient (for applications) restrictions on $\varphi (x)$ and $f(x),$ the theorem was proved by A. A. Karatsuba in [8] (see also,[9][10]).
Certain notations
[1]. For $B>0,B\to +\infty ,$ or $B\to 0,$ the record
$1\ll {\frac {A}{B}}\ll 1$
means that there are the constants $C_{1}>0$
and $C_{2}>0,$
such that
$C_{1}\leq {\frac {|A|}{B}}\leq C_{2}.$
[2]. For a real number $\alpha ,$ the record $\|\alpha \|$ means that
$\|\alpha \|=\min(\{\alpha \},1-\{\alpha \}),$
where
$\{\alpha \}$
is the fractional part of $\alpha .$
ATS theorem
Let the real functions ƒ(x) and $\varphi (x)$ satisfy on the segment [a, b] the following conditions:
1) $f''''(x)$ and $\varphi ''(x)$ are continuous;
2) there exist numbers $H,$ $U$ and $V$ such that
$H>0,\qquad 1\ll U\ll V,\qquad 0<b-a\leq V$
and
${\begin{array}{rc}{\frac {1}{U}}\ll f''(x)\ll {\frac {1}{U}}\ ,&\varphi (x)\ll H,\\\\f'''(x)\ll {\frac {1}{UV}}\ ,&\varphi '(x)\ll {\frac {H}{V}},\\\\f''''(x)\ll {\frac {1}{UV^{2}}}\ ,&\varphi ''(x)\ll {\frac {H}{V^{2}}}.\\\\\end{array}}$
Then, if we define the numbers $x_{\mu }$ from the equation
$f'(x_{\mu })=\mu ,$
we have
$\sum _{a<\mu \leq b}\varphi (\mu )e^{2\pi if(\mu )}=\sum _{f'(a)\leq \mu \leq f'(b)}C(\mu )Z(\mu )+R,$
where
$R=O\left({\frac {HU}{b-a}}+HT_{a}+HT_{b}+H\log \left(f'(b)-f'(a)+2\right)\right);$
$T_{j}={\begin{cases}0,&{\text{if }}f'(j){\text{ is an integer}};\\\min \left({\frac {1}{\|f'(j)\|}},{\sqrt {U}}\right),&{\text{if }}\|f'(j)\|\neq 0;\\\end{cases}}$
$j=a,b;$
$C(\mu )={\begin{cases}1,&{\text{if }}f'(a)<\mu <f'(b);\\{\frac {1}{2}},&{\text{if }}\mu =f'(a){\text{ or }}\mu =f'(b);\\\end{cases}}$
$Z(\mu )={\frac {1+i}{\sqrt {2}}}{\frac {\varphi (x_{\mu })}{\sqrt {f''(x_{\mu })}}}e^{2\pi i(f(x_{\mu })-\mu x_{\mu })}\ .$
The most simple variant of the formulated theorem is the statement, which is called in the literature the Van der Corput lemma.
Van der Corput lemma
Let $f(x)$ be a real differentiable function in the interval $a<x\leq b,$ moreover, inside of this interval, its derivative $f'(x)$ is a monotonic and a sign-preserving function, and for the constant $\delta $ such that $0<\delta <1$ satisfies the inequality $|f'(x)|\leq \delta .$ Then
$\sum _{a<k\leq b}e^{2\pi if(k)}=\int _{a}^{b}e^{2\pi if(x)}dx+\theta \left(3+{\frac {2\delta }{1-\delta }}\right),$
where $|\theta |\leq 1.$
Remark
If the parameters $a$ and $b$ are integers, then it is possible to substitute the last relation by the following ones:
$\sum _{a<k\leq b}e^{2\pi if(k)}=\int _{a}^{b}e^{2\pi if(x)}\,dx+{\frac {1}{2}}e^{2\pi if(b)}-{\frac {1}{2}}e^{2\pi if(a)}+\theta {\frac {2\delta }{1-\delta }},$
where $|\theta |\leq 1.$
On the applications of ATS to the problems of physics see,;[11][12] see also,.[13][14]
Notes
1. Hardy, G. H.; Littlewood, J. E. (1914). "Some problems of diophantine approximation: Part II. The trigonometrical series associated with the elliptic θ-functions". Acta Mathematica. International Press of Boston. 37: 193–239. doi:10.1007/bf02401834. ISSN 0001-5962.
2. Hardy, G. H.; Littlewood, J. E. (1916). "Contributions to the theory of the riemann zeta-function and the theory of the distribution of primes". Acta Mathematica. International Press of Boston. 41: 119–196. doi:10.1007/bf02422942. ISSN 0001-5962.
3. Hardy, G. H.; Littlewood, J. E. (1921). "The zeros of Riemann's zeta-function on the critical line". Mathematische Zeitschrift. Springer Science and Business Media LLC. 10 (3–4): 283–317. doi:10.1007/bf01211614. ISSN 0025-5874. S2CID 126338046.
4. I. M. Vinogradov. On the average value of the number of classes of purely root form of the negative determinant Communic. of Khar. Math. Soc., 16, 10–38 (1917).
5. van der Corput, J. G. (1921). "Zahlentheoretische Abschätzungen". Mathematische Annalen (in German). Springer Science and Business Media LLC. 84 (1–2): 53–79. doi:10.1007/bf01458693. ISSN 0025-5831. S2CID 179178113.
6. van der Corput, J. G. (1922). "Verschärfung der Abschätzung beim Teilerproblem". Mathematische Annalen (in German). Springer Science and Business Media LLC. 87 (1–2): 39–65. doi:10.1007/bf01458035. ISSN 0025-5831. S2CID 177789678.
7. Montgomery, Hugh (1994). Ten lectures on the interface between analytic number theory and harmonic analysis. Providence, R.I: Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society. ISBN 978-0-8218-0737-8. OCLC 30811108.
8. Karatsuba, A. A. (1987). "Approximation of exponential sums by shorter ones". Proceedings of the Indian Academy of Sciences, Section A. Springer Science and Business Media LLC. 97 (1–3): 167–178. doi:10.1007/bf02837821. ISSN 0370-0089. S2CID 120389154.
9. A. A. Karatsuba, S. M. Voronin. The Riemann Zeta-Function. (W. de Gruyter, Verlag: Berlin, 1992).
10. A. A. Karatsuba, M. A. Korolev. The theorem on the approximation of a trigonometric sum by a shorter one. Izv. Ross. Akad. Nauk, Ser. Mat. 71:3, pp. 63—84 (2007).
11. Karatsuba, Ekatherina A. (2004). "Approximation of sums of oscillating summands in certain physical problems". Journal of Mathematical Physics. AIP Publishing. 45 (11): 4310–4321. doi:10.1063/1.1797552. ISSN 0022-2488.
12. Karatsuba, Ekatherina A. (2007-07-20). "On an approach to the study of the Jaynes–Cummings sum in quantum optics". Numerical Algorithms. Springer Science and Business Media LLC. 45 (1–4): 127–137. doi:10.1007/s11075-007-9070-x. ISSN 1017-1398. S2CID 13485016.
13. Chassande-Mottin, Éric; Pai, Archana (2006-02-27). "Best chirplet chain: Near-optimal detection of gravitational wave chirps". Physical Review D. American Physical Society (APS). 73 (4): 042003. arXiv:gr-qc/0512137. doi:10.1103/physrevd.73.042003. hdl:11858/00-001M-0000-0013-4BBD-B. ISSN 1550-7998. S2CID 56344234.
14. Fleischhauer, M.; Schleich, W. P. (1993-05-01). "Revivals made simple: Poisson summation formula as a key to the revivals in the Jaynes-Cummings model". Physical Review A. American Physical Society (APS). 47 (5): 4258–4269. doi:10.1103/physreva.47.4258. ISSN 1050-2947. PMID 9909432.
| Wikipedia |
AW*-algebra
In mathematics, an AW*-algebra is an algebraic generalization of a W*-algebra. They were introduced by Irving Kaplansky in 1951.[1] As operator algebras, von Neumann algebras, among all C*-algebras, are typically handled using one of two means: they are the dual space of some Banach space, and they are determined to a large extent by their projections. The idea behind AW*-algebras is to forgo the former, topological, condition, and use only the latter, algebraic, condition.
Definition
Recall that a projection of a C*-algebra is a self-adjoint idempotent element. A C*-algebra A is an AW*-algebra if for every subset S of A, the left annihilator
$\mathrm {Ann} _{L}(S)=\{a\in A\mid \forall s\in S,as=0\}\,$
is generated as a left ideal by some projection p of A, and similarly the right annihilator is generated as a right ideal by some projection q:
$\forall S\subseteq A\,\exists p,q\in \mathrm {Proj} (A)\colon \mathrm {Ann} _{L}(S)=Ap,\quad \mathrm {Ann} _{R}(S)=qA$.
Hence an AW*-algebra is a C*-algebras that is at the same time a Baer *-ring.
The original definition of Kaplansky states that an AW*-algebra is a C*-algebra such that (1) any set of orthogonal projections has a least upper bound, and (2) that each maximal commutative C*-subalgebra is generated by its projections. The first condition states that the projections have an interesting structure, while the second condition ensures that there are enough projections for it to be interesting.[1] Note that the second condition is equivalent to the condition that each maximal commutative C*-subalgebra is monotone complete.
Structure theory
Many results concerning von Neumann algebras carry over to AW*-algebras. For example, AW*-algebras can be classified according to the behavior of their projections, and decompose into types.[2] For another example, normal matrices with entries in an AW*-algebra can always be diagonalized.[3] AW*-algebras also always have polar decomposition.[4]
However, there are also ways in which AW*-algebras behave differently from von Neumann algebras.[5] For example, AW*-algebras of type I can exhibit pathological properties,[6] even though Kaplansky already showed that such algebras with trivial center are automatically von Neumann algebras.
The commutative case
A commutative C*-algebra is an AW*-algebra if and only if its spectrum is a Stonean space. Via Stone duality, commutative AW*-algebras therefore correspond to complete Boolean algebras. The projections of a commutative AW*-algebra form a complete Boolean algebra, and conversely, any complete Boolean algebra is isomorphic to the projections of some commutative AW*-algebra.
References
1. Kaplansky, Irving (1951). "Projections in Banach algebras". Annals of Mathematics. 53 (2): 235–249. doi:10.2307/1969540.
2. Berberian, Sterling (1972). Baer *-rings. Springer.
3. Heunen, Chris; Reyes, Manuel L. (2013). "Diagonalizing matrices over AW*-algebras". Journal of Functional Analysis. 264 (8): 1873–1898. arXiv:1208.5120. doi:10.1016/j.jfa.2013.01.022.
4. Ara, Pere (1989). "Left and right projections are equivalent in Rickart C*-algebras". Journal of Algebra. 120 (2): 433–448. doi:10.1016/0021-8693(89)90209-3.
5. Wright, J. D. Maitland. "AW*-algebra". Springer.
6. Ozawa, Masanao (1984). "Nonuniqueness of the cardinality attached to homogeneous AW*-algebras". Proceedings of the American Mathematical Society. 93: 681–684. doi:10.2307/2045544.
| Wikipedia |
AWM-SIAM Sonia Kovalevsky Lecture
The AWM-SIAM Sonia Kovalevsky Lecture is an award and lecture series that "highlights significant contributions of women to applied or computational mathematics."[1] The Association for Women in Mathematics (AWM) and the Society for Industrial and Applied Mathematics (SIAM) planned the award[2] and lecture series in 2002 and first awarded it in 2003. The lecture is normally given each year at the SIAM Annual Meeting.[3] Award winners receive a signed certificate from the AWM and SIAM presidents.[4]
The lectures are named after Sonia Kovalevsky (1850–1891), a well-known Russian mathematician of the late 19th century.[5] Karl Weierstrass regarded Kovalevsky as his most talented student. In 1874, she received her Doctor of Philosophy degree from the University of Göttingen under the supervision of Weierstrass. She was granted privatdozentin status and taught at the University of Stockholm in 1883; she became an ordinary professor (the equivalent of full professor) at this institution in 1889. She was also an editor of the journal Acta Mathematica. Kovalevsky did her important work in the theory of partial differential equations and the rotation of a solid around a fixed point.[1]
Recipients
The Kovalevky Lecturers have been:[6]
• 2003 Linda R. Petzold, University of California, Santa Barbara, “Towards the Multiscale Simulation of Biochemical Networks”
• 2004 Joyce R. McLaughlin, Rensselaer Polytechnic Institute, “Interior Elastodynamics Inverse Problems: Creating Shear Wave Speed Images of Tissue”
• 2005 Ingrid Daubechies, Princeton University, “Superfast and (Super)sparse Algorithms”
• 2006 Irene Fonseca, Carnegie Mellon University, “New Challenges in the Calculus of Variations”
• 2007 Lai-Sang Young, Courant Institute, “Shear-Induced Chaos”[7]
• 2008 Dianne P. O'Leary, University of Maryland, “A Noisy Adiabatic Theorem: Wilkinson Meets Schrödinger’s Cat”[8]
• 2009 Andrea Bertozzi, University of California, Los Angeles
• 2010 Suzanne Lenhart, University of Tennessee at Knoxville, “Mixing it up: Discrete and Continuous Optimal Control for Biological Models”
• 2011 Susanne C. Brenner, Louisiana State University, “A Cautionary Tale in Numerical PDEs”[9]
• 2012 Barbara Keyfitz, Ohio State University, “The Role of Characteristics in Conservation Laws”
• 2013 Margaret Cheney, Colorado State University, “Introduction to Radar Imaging”
• 2014 Irene M. Gamba, University of Texas at Austin, “The evolution of complex interactions in non-linear kinetic systems”
• 2015 Linda J. S. Allen, Texas Tech University, “Predicting Population Extinction”[2]
• 2016 Lisa J. Fauci, Tulane University, “Biofluids of Reproduction: Oscillators, Viscoelastic Networks and Sticky Situations”[5]
• 2017 Liliana Borcea, University of Michigan, “Mitigating Uncertainty in Inverse Wave Scattering”[4]
• 2018 Eva Tardos, Cornell University, “Learning and Efficiency of Outcomes in Games”[10]
• 2019 Catherine Sulem, University of Toronto, “The Dynamics of Ocean Waves”
• 2020 Bonnie Berger, MIT, “Compressive genomics: leveraging the geometry of biological data”
• 2021 Vivette Girault, Université Pierre et Marie Curie, "From linear poroelasticity to nonlinear implicit elastic and related models"[11]
• 2022 Anne Greenbaum, University of Washington, "Two of my Favorite Problems”[12]
• 2023 Annalisa Buffa, Ecole Polytechnique Fédérale de Lausanne (EPFL), TBA [13]
See also
• Falconer Lecture
• Noether Lecture
• List of mathematics awards
References
1. Koblitz, Ann Hibner (1993). A Convergence of Lives: Sofia Kovalevskaia: Scientist, Writer, Revolutionary. New Brunswick, New Jersey: Rutgers University Press. p. 305. ISBN 0-8135-1963-2.
2. Young, Glenys (31 July 2015). "Professor Awarded Distinguished Lecture for Contributions to Mathematics". Texas Tech Today. Retrieved 2021-01-06.
3. "AWM-SIAM Sonia Kovalevsky Lecture". Society of Industrial and Applied Mathematics. Retrieved 6 January 2021.
4. Duraisamy, Karthik (2017). "2017 Kovalevsky and Reid Prizes". SIAM News: 12.
5. Bronston, Barri (6 July 2016). "Honoring women in mathematics". Tulane News. Retrieved 2021-01-06.
6. "Sonia Kovalevsky Lectures". Association for Women in Mathematics. Retrieved 1 January 2021.
7. "Mathematics People: SIAM Prizes Awarded" (PDF). Notices of the American Mathematical Society. 54 (9): 1164. October 2007. Retrieved 7 January 2021.
8. "Join SIAM in San Diego: 2008 SIAM Annual Meeting" (PDF). University of Maryland. Archived from the original (PDF) on 9 January 2021. Retrieved 6 January 2021.
9. "Susanne Brenner to be AWM-SIAM Sonia Kovalevky Lecturer". LSU Center for Computation & Technology. Retrieved 6 January 2021.
10. "Mathematicians receive high honors for global contributions". EurekAlert!. Retrieved 2021-01-06.
11. "2021 Lecturer: Vivette Girault". Association for Women in Mathematics. Retrieved 17 February 2021.
12. "Anne Greenbaum Named AWM-SIAM Sonia Kovalevsky Lecturer". SIAM News. Retrieved 2022-03-06.
13. "PR Kovalevsky 2023 - Buffa" (PDF). Association for Women in Mathematics. Retrieved 2023-06-16.
External links
• "Kovalevsky Lectures – Association for Women in Mathematics (AWM)". awm-math.org. Retrieved 1 January 2021.
• "Prizes, Awards, and Honors for Women Mathematicians". agnesscott.edu. Retrieved 1 January 2021.
| Wikipedia |
A Beautiful Mind (film)
A Beautiful Mind is a 2001 American biographical drama film directed by Ron Howard. Written by Akiva Goldsman, its screenplay was inspired by Sylvia Nasar's 1998 biography of the mathematician John Nash, a Nobel Laureate in Economics. A Beautiful Mind stars Russell Crowe as Nash, along with Ed Harris, Jennifer Connelly, Paul Bettany, Adam Goldberg, Judd Hirsch, Josh Lucas, Anthony Rapp, and Christopher Plummer in supporting roles. The story begins in Nash's days as a brilliant but asocial mathematics graduate student at Princeton University. After Nash accepts secret work in cryptography, his life takes a turn for the nightmarish.
A Beautiful Mind
Theatrical release poster
Directed byRon Howard
Written byAkiva Goldsman
Based onA Beautiful Mind
by Sylvia Nasar
Produced by
• Brian Grazer
• Ron Howard
Starring
• Russell Crowe
• Ed Harris
• Jennifer Connelly
• Paul Bettany
• Adam Goldberg
• Judd Hirsch
• Josh Lucas
• Anthony Rapp
• Christopher Plummer
CinematographyRoger Deakins
Edited by
• Daniel P. Hanley
• Mike Hill
Music byJames Horner
Production
companies
• Universal Searchlight Pictures[1]
• DreamWorks Pictures[1]
• Imagine Entertainment[1]
Distributed by
• Twentieth Century Universal Film Company[1]
(North America)
• DreamWorks Pictures LLC[1]
(International)
Release dates
• December 13, 2001 (2001-12-13) (Beverly Hills premiere)
• December 21, 2001 (2001-12-21) (United States)
Running time
135 minutes
CountryUnited States
LanguageEnglish
Budget$58 million[2]
Box office$316.8 million[2]
A Beautiful Mind was released theatrically in the United States on December 21, 2001. It went on to gross over $313 million worldwide and won four Academy Awards, for Best Picture, Best Director, Best Adapted Screenplay and Best Supporting Actress for Connelly. It was also nominated for Best Actor, Best Film Editing, Best Makeup, and Best Original Score.
Plot
In 1947, John Nash arrives at Princeton University as a co-recipient, with Martin Hansen, of the Carnegie Scholarship for Mathematics. He meets fellow math and science graduate students Sol, Ainsley, and Bender, as well as his roommate Charles Herman, a literature student.
Determined to publish his own original idea, Nash is inspired when he and his classmates discuss how to approach a group of women at a bar. Hansen quotes Adam Smith advocating "every man for himself", but Nash argues that a cooperative approach would lead to better chances of success in developing a new concept of governing dynamics. Publishing an article on his theory, he earns an appointment at MIT where he chooses Sol and Bender over Hansen to join him.
In 1953, Nash is invited to the Pentagon to study encrypted enemy telecommunications, which he deciphers mentally. Bored with his regular duties at MIT, including teaching, he is recruited by the mysterious William Parcher of the United States Department of Defense with a classified assignment: to look for hidden patterns in magazines and newspapers to thwart a Soviet plot. Nash becomes increasingly obsessive in his search for these patterns, delivering his results to a secret mailbox, and comes to believe he is being followed.
One of his students, Alicia Larde, asks him to dinner, and they fall in love. On a return visit to Princeton, Nash runs into Charles and his niece, Marcee. With Charles' encouragement, he proposes to Alicia and they marry. Nash fears for his life after surviving a shootout between Parcher and Soviet agents, and learns Alicia is pregnant, but he is forced to continue his assignment. While delivering a guest lecture at Harvard University, Nash tries to flee from people he thinks are Soviet agents, led by a psychiatrist named Dr. Rosen, but is forcibly sedated and committed to a psychiatric facility.
Dr. Rosen tells Alicia that Nash has schizophrenia and that Charles, Marcee, and Parcher exist only in his imagination. Alicia backs up the doctor, telling Nash that no "William Parcher" is in the Defense Department and takes out the unopened documents he delivered to the secret mailbox. Nash is given a course of insulin shock therapy and eventually released. Frustrated with the side effects of his antipsychotic medication, he secretly stops taking it and starts seeing Parcher and Charles again.
In 1956, Alicia discovers Nash has resumed his "assignment" in a shed near their home. Realizing he has relapsed, Alicia rushes to the house to find Nash had left their infant son in the running bathtub, believing "Charles" was watching the baby. Alicia calls Dr. Rosen, but Nash accidentally knocks her and the baby to the ground, believing he's fighting Parcher.
As Alicia flees with the baby, Nash fights with his visions and realizes that all of them have looked the same ever since he first saw them. He stops Alicia's car and tells her he realizes that "Marcee" isn't real because she doesn't age, finally accepting that Parcher and other figures are hallucinations. Against Dr. Rosen's advice, Nash chooses not to restart his medication, believing he can deal with his symptoms himself, and Alicia decides to stay and support him.
Nash returns to Princeton, approaching his old rival Hansen, now head of the mathematics department, who allows him to work out of the library and audit classes. Over the next two decades, Nash learns to ignore his hallucinations and, by the late 1970s, is allowed to teach again.
In 1994, Nash is awarded the Nobel Memorial Prize in Economic Sciences for his revolutionary work on game theory, and is honored by his fellow professors. At the ceremony, he dedicates the prize to his wife. As Nash, Alicia, and their son leave the auditorium in Stockholm, Nash sees Charles, Marcee, and Parcher watching him, but merely glances at them before departing.
Cast
• Russell Crowe as John Nash
• Jennifer Connelly as Alicia Nash
• Ed Harris as William Parcher
• Christopher Plummer as Dr. Rosen
• Paul Bettany as Charles Herman
• Adam Goldberg as Richard Sol
• Josh Lucas as Martin Hansen
• Anthony Rapp as Bender
• Jason Gray-Stanford as Ainsley Neilson
• Judd Hirsch as Hellinger
• Austin Pendleton as Thomas King
• Vivien Cardone as Marcee Herman
• Killian, Christian, and Daniel Coffinet-Crean as Baby
Production
Development
A Beautiful Mind was the second schizophrenia-themed film that Ron Howard had planned to direct. The first, Laws of Madness, would have been based on the true story of schizophrenic Michael Laudor, who overcame difficult odds to successfully graduate from Yale Law School. Howard purchased the rights to Laudor's life story for $1.5 million in 1995 and had Brad Pitt slated to play the lead role. However, after Laudor killed his fiancée in 1998 in the midst of a psychotic episode, plans for the movie were cancelled.[3][4]
After producer Brian Grazer first read an excerpt of Sylvia Nasar's 1998 book A Beautiful Mind in Vanity Fair magazine, he immediately purchased the rights to the film. Grazer later said that many A-list directors were calling with their point of view on the project. He eventually brought the project to Ron Howard, his long time partner.[5]
Grazer met with a number of screenwriters, mostly consisting of "serious dramatists", but he chose Akiva Goldsman because of his strong passion and desire for the project. Goldsman's creative take on the project was to avoid having viewers understand they are viewing an alternative reality until a specific point in the film. This was done to rob the viewers of their understanding, to mimic how Nash comprehended his experiences. Howard agreed to direct the film based on the first draft. He asked Goldsman to emphasize the love story of Nash and his wife; she was critical to his being able to continue living at home.[6]
Dave Bayer, a professor of mathematics at Barnard College, Columbia University,[7] was consulted on the mathematical equations that appear in the film. For the scene where Nash has to teach a calculus class and gives them a complicated problem to keep them busy, Bayer chose a problem physically unrealistic but mathematically very rich, in keeping with Nash as "someone who really doesn't want to teach the mundane details, who will home in on what's really interesting". Bayer received a cameo role in the film as a professor who lays his pen down for Nash in the pen ceremony near the end of the film.[8]
Greg Cannom was chosen to create the makeup effects for A Beautiful Mind, specifically the age progression of the characters. Crowe had previously worked with Cannom on The Insider. Howard had also worked with Cannom on Cocoon. Each character's stages of makeup were broken down by the number of years that would pass between levels. Cannom stressed subtlety between the stages, but worked toward the ultimate stage of "Older Nash". The production team originally decided that the makeup department would age Russell Crowe throughout the film; however, at Crowe's request, the makeup was used to push his look to resemble the facial features of John Nash. Cannom developed a new silicone-type makeup that could simulate skin and be used for overlapping applications; this shortened make-up application time from eight to four hours. Crowe was also fitted with a number of dentures to give him a slight overbite in the film.[9]
Howard and Grazer chose frequent collaborator James Horner to score the film because they knew of his ability to communicate. Howard said, regarding Horner, "it's like having a conversation with a writer or an actor or another director". A running discussion between the director and the composer was the concept of high-level mathematics being less about numbers and solutions, and more akin to a kaleidoscope, in that the ideas evolve and change. After the first screening of the film, Horner told Howard: "I see changes occurring like fast-moving weather systems". He chose it as another theme to connect to Nash's ever-changing character. Horner chose Welsh singer Charlotte Church to sing the soprano vocals after deciding that he needed a balance between a child and adult singing voice. He wanted a "purity, clarity and brightness of an instrument" but also a vibrato to maintain the humanity of the voice.[10]
The film was shot 90% chronologically. Three separate trips were made to the Princeton University campus. During filming, Howard decided that Nash's delusions should always be introduced first audibly and then visually. This provides a clue for the audience and establishes the delusions from Nash's point of view. The historic John Nash had only auditory delusions. The filmmakers developed a technique to represent Nash's mental epiphanies. Mathematicians described to them such moments as a sense of "the smoke clearing", "flashes of light" and "everything coming together", so the filmmakers used a flash of light appearing over an object or person to signify Nash's creativity at work.[11] Two night shots were done at Fairleigh Dickinson University's campus in Florham Park, New Jersey, in the Vanderbilt Mansion ballroom.[12] Portions of the film set at Harvard were filmed at Manhattan College.[13] (Harvard has turned down most requests for on-location filming ever since the filming of Love Story (1970), which caused significant physical damage to trees on campus.)[14]
Tom Cruise was considered for the lead role.[15][16] Howard ultimately cast Russell Crowe. For the role of Alicia Nash, Rachel Weisz was offered the role but turned it down. Charlize Theron and Julia Ormond auditioned for the role. According to Ron Howard, the four finalists for the role of Alicia were Ashley Judd, Claire Forlani, Mary McCormack and Jennifer Connelly, with Connelly winning the role. Before the casting of Connelly, Hilary Swank and Salma Hayek were also candidates for the part.
Writing
The narrative of the film differs considerably from the events of Nash's life, as filmmakers made choices for the sense of the story. The film has been criticized for this aspect, but the filmmakers said they never intended a literal representation of his life.[17]
One difficulty was the portrayal of his mental illness and trying to find a visual film language for this.[18] As a matter of fact, Nash never had visual hallucinations: Charles Herman (the "roommate"), Marcee Herman and William Parcher (the Defense agent) are a scriptwriter's invention. Sylvia Nasar said that the filmmakers "invented a narrative that, while far from a literal telling, is true to the spirit of Nash's story".[19] Nash spent his years between Princeton and MIT as a consultant for the RAND Corporation in California, but in the film he is portrayed as having worked for the Department of Defense at the Pentagon instead. His handlers, both from faculty and administration, had to introduce him to assistants and strangers.[11] The PBS documentary A Brilliant Madness tried to portray his life more accurately.[20]
Few of the characters in the film, besides John and Alicia Nash, correspond directly to actual people.[21] The discussion of the Nash equilibrium was criticized as over-simplified. In the film, Nash has schizophrenic hallucinations while he is in graduate school, but in his life he did not have this experience until some years later. No mention is made of Nash's homosexual experiences at RAND,[19] which are noted in the biography,[22] though both Nash and his wife deny this occurred.[23] Nash fathered a son, John David Stier (born June 19, 1953), by Eleanor Agnes Stier (1921–2005), a nurse whom he abandoned when she told him of her pregnancy.[24] The film did not include Alicia's divorce of John in 1963. It was not until after Nash won the Nobel Memorial Prize in 1994 that they renewed their relationship. Beginning in 1970, Alicia allowed him to live with her as a boarder. They remarried in 2001.[22]
Nash is shown to join Wheeler Laboratory at MIT, but there is no such lab. Instead, he was appointed as C. L. E. Moore instructor at MIT, and later as a professor.[25] The film furthermore does not touch on the revolutionary work of John Nash in differential geometry and partial differential equations, such as the Nash embedding theorem or his proof of Hilbert's nineteenth problem, work which he did in his time at MIT and for which he was given the Abel Prize in 2015. The so-called pen ceremony tradition at Princeton shown in the film is completely fictitious.[11][26] The film has Nash saying in 1994: "I take the newer medications", but in fact, he did not take any medication from 1970 onwards, something highlighted in Nasar's biography. Howard later stated that they added the line of dialogue because they worried that the film would be criticized for suggesting that all people with schizophrenia can overcome their illness without medication.[11] In addition, Nash never gave an acceptance speech for his Nobel prize.
Release and response
A Beautiful Mind received a limited release on December 21, 2001, receiving positive reviews, with Crowe receiving wide acclaim for his performance. It was later released in the United States on January 4, 2002.
Critical response
On Rotten Tomatoes, A Beautiful Mind holds an approval rating of 74% based on 214 reviews and an average score of 7.20/10. The website's critical consensus states: "The well-acted A Beautiful Mind is both a moving love story and a revealing look at mental illness."[27] On Metacritic, the film has a weighted average score of 72 out of 100 based on 33 reviews, indicating "generally favorable reviews".[28] Audiences polled by CinemaScore gave the film an average grade of "A-" on an A+ to F scale.[29]
Roger Ebert of Chicago Sun-Times gave the film four out of four stars.[30] Mike Clark of USA Today gave three-and-a-half out of four stars and also praised Crowe's performance, calling it a welcome follow-up to Howard's previous film, 2001's How the Grinch Stole Christmas.[31] Desson Thomson of The Washington Post found the film to be "one of those formulaically rendered Important Subject movies".[32] The portrayal of mathematics in the film was praised by the mathematics community, including John Nash himself.[8]
John Sutherland of The Guardian noted the film's biopic distortions, but said:
Howard pulls off an extraordinary trick in A Beautiful Mind by seducing the audience into Nash's paranoid world. We may not leave the cinema with A-level competence in game theory, but we do get a glimpse into what it feels like to be mad - and not know it.[33]
Some writers such as Shailee Koranne argue that the film presents an unrealistic or inappropriate depiction of the disorder schizophrenia, which the protagonist John Nash suffers from, stating that places too much emphasis on “fixing” the disorder.[34]
Writing in the Los Angeles Times, Lisa Navarrette criticized the casting of Jennifer Connelly as Alicia Nash as an example of whitewashing. Alicia Nash was born in El Salvador and had an accent not portrayed in the film.[35]
Box office
During the five-day weekend of the limited release, A Beautiful Mind opened at the #12 spot at the box office,[36] peaking at the #2 spot following the wide release.[37] The film went on to gross $170,742,341 in the United States and Canada and $313,542,341 worldwide.[2]
Awards and nominations
Award Category Recipient Result
Academy Awards[38] Best Picture Brian Grazer and Ron Howard Won
Best Director Ron Howard Won
Best Actor Russell Crowe Nominated
Best Supporting Actress Jennifer Connelly Won
Best Screenplay – Based on Material Previously Produced or Published Akiva Goldsman Won
Best Film Editing Mike Hill and Daniel P. Hanley Nominated
Best Makeup Greg Cannom and Colleen Callaghan Nominated
Best Original Score James Horner Nominated
Amanda Awards Best Foreign Feature Film Ron Howard Nominated
American Cinema Editors Awards Best Edited Feature Film – Dramatic Mike Hill and Daniel P. Hanley Nominated
American Film Institute Awards[39] Movie of the Year Nominated
Actor of the Year – Male – Movies Russell Crowe Nominated
Featured Actor of the Year – Female – Movies Jennifer Connelly Won
Screenwriter of the Year Akiva Goldsman Nominated
Artios Awards[40] Outstanding Achievement in Feature Film Casting – Drama Jane Jenkins and Janet Hirshenson Nominated
ASCAP Film and Television Music Awards Top Box Office Films James Horner Won
Australian Film Institute Awards[41] Best Foreign Film Brian Grazer and Ron Howard Nominated
Awards Circuit Community Awards Best Actor in a Leading Role Russell Crowe Won
Best Actress in a Supporting Role Jennifer Connelly Won
Best Adapted Screenplay Akiva Goldsman Nominated
Best Original Score James Horner Nominated
Best Cast Ensemble Paul Bettany, Jennifer Connelly, Russell Crowe, Adam Goldberg,
Jason Gray-Stanford, Ed Harris, Judd Hirsch, Josh Lucas,
Austin Pendleton, Christopher Plummer, and Anthony Rapp
Nominated
British Academy Film Awards[42] Best Film Brian Grazer and Ron Howard Nominated
Best Direction Ron Howard Nominated
Best Actor in a Leading Role Russell Crowe Won
Best Actress in a Supporting Role Jennifer Connelly Won
Best Adapted Screenplay Akiva Goldsman Nominated
Chicago Film Critics Association Awards[43] Best Film Nominated
Best Director Ron Howard Nominated
Best Actor Russell Crowe Nominated
Best Supporting Actress Jennifer Connelly Nominated
Best Screenplay Akiva Goldsman Nominated
Best Original Score James Horner Nominated
Christopher Awards Feature Film Won
Critics' Choice Awards[44] Best Picture Won
Best Director Ron Howard Won[lower-alpha 1]
Best Actor Russell Crowe Won
Best Supporting Actress Jennifer Connelly Won
Best Screenplay Akiva Goldsman Nominated
Czech Lion Awards Best Foreign Film Nominated
Dallas–Fort Worth Film Critics Association Awards Best Picture Won
Best Director Ron Howard Won
Best Actor Russell Crowe Won
Best Supporting Actress Jennifer Connelly Nominated
Best Screenplay Akiva Goldsman Won
Directors Guild of America Awards[45] Outstanding Directorial Achievement in Motion Pictures Ron Howard Won
DVD Exclusive Awards Best Audio Commentary – New Release Nominated
Original Retrospective Documentary – New Release Colleen A. Benn and Marian Mansi Nominated
Empire Awards Best Actress Jennifer Connelly Nominated
Golden Eagle Awards[46] Best Foreign Language Film Ron Howard Nominated
Golden Globe Awards[47] Best Motion Picture – Drama Won
Best Actor in a Motion Picture – Drama Russell Crowe Won
Best Supporting Actress – Motion Picture Jennifer Connelly Won
Best Director – Motion Picture Ron Howard Nominated
Best Screenplay – Motion Picture Akiva Goldsman Won
Best Original Score – Motion Picture James Horner Nominated
Golden Reel Awards[48] Best Sound Editing – Dialogue & ADR, Domestic Feature Film Anthony J. Ciccolini III, Deborah Wallach,
Stan Bochner, Louis Cerborino, and Marc Laub
Nominated
Best Sound Editing – Music (Foreign & Domestic) Jim Henrikson Nominated
Golden Schmoes Awards Best Actor of the Year Russell Crowe Nominated
Best Supporting Actress of the Year Jennifer Connelly Won
GoldSpirit Awards Best Soundtrack James Horner Nominated
Best Drama Soundtrack Nominated
Grammy Awards[49] Best Score Soundtrack Album for a Motion Picture, Television or Other Visual Media A Beautiful Mind: Original Motion Picture Soundtrack – James Horner Nominated
Humanitas Prize[50] Feature Film Category Akiva Goldsman Nominated
Kansas City Film Critics Circle Awards[51] Best Supporting Actress Jennifer Connelly Won[lower-alpha 2]
Las Vegas Film Critics Society Awards[52] Best Supporting Actress Nominated
London Film Critics Circle Awards British Supporting Actor of the Year Paul Bettany Nominated
MTV Movie Awards[53] Best Male Performance Russell Crowe Nominated
Online Film & Television Association Awards[54] Best Picture Brian Grazer and Ron Howard Nominated
Best Actor Russell Crowe Nominated
Best Supporting Actress Jennifer Connelly Won
Best Adapted Screenplay Akiva Goldsman Nominated
Best Original Score James Horner Nominated
Online Film Critics Society Awards[55] Best Actor Russell Crowe Nominated
Best Supporting Actress Jennifer Connelly Won
Producers Guild of America Awards[56] Outstanding Producer of Theatrical Motion Pictures Brian Grazer and Ron Howard Nominated
Phoenix Film Critics Society Awards Best Picture Nominated
Best Director Ron Howard Nominated
Best Actor in a Leading Role Russell Crowe Won
Best Actress in a Supporting Role Jennifer Connelly Won
Best Screenplay – Adaptation Akiva Goldsman Nominated
Best Original Score James Horner Nominated
Russian Guild of Film Critics Awards Best Foreign Actor Russell Crowe Nominated
San Diego Film Critics Society Awards Best Actor Nominated
Satellite Awards[57] Best Actor in a Motion Picture – Drama Nominated
Best Supporting Actor in a Motion Picture – Drama Ed Harris Nominated
Best Supporting Actress in a Motion Picture – Drama Jennifer Connelly Won
Best Adapted Screenplay Akiva Goldsman Nominated
Best Editing Mike Hill and Daniel P. Hanley Nominated
Best Original Score James Horner Nominated
Best Original Song "All Love Can Be"
Music by James Horner;
Lyrics by Will Jennings
Won
Screen Actors Guild Awards[58] Outstanding Performance by a Cast in a Motion Picture Paul Bettany, Jennifer Connelly, Russell Crowe, Adam Goldberg,
Jason Gray-Stanford, Ed Harris, Judd Hirsch, Josh Lucas,
Austin Pendleton, Christopher Plummer, and Anthony Rapp
Nominated
Outstanding Performance by a Male Actor in a Leading Role Russell Crowe Won
Outstanding Performance by a Female Actor in a Leading Role Jennifer Connelly Nominated
Southeastern Film Critics Association Awards[59] Best Picture 7th Place[lower-alpha 3]
Best Supporting Actress Jennifer Connelly Won[lower-alpha 4]
Teen Choice Awards Choice Movie – Drama/Action Adventure Nominated
Turkish Film Critics Association Awards Best Foreign Film 12th Place
USC Scripter Awards[60] Akiva Goldsman (screenwriter); Sylvia Nasar (author) Won
Vancouver Film Critics Circle Awards[61] Best Actor Russell Crowe Nominated
Voices in the Shadow Dubbing Festival Best Male Voice Fabrizio Pucci (for the dubbing of Russell Crowe) Nominated
World Soundtrack Awards[62] Soundtrack Composer of the Year James Horner Nominated
Writers Guild of America Awards[63] Best Screenplay – Based on Material Previously Produced or Published Akiva Goldsman Won
Yoga Awards Worst Foreign Director Ron Howard Won
• In 2006, it was named No. 93 in AFI's 100 Years... 100 Cheers. In the following year, it was nominated for AFI's 100 Years...100 Movies (10th Anniversary Edition).[64]
Home media
A Beautiful Mind was released on VHS and DVD, in wide- and full-screen editions, in North America on June 25, 2002.[65] The DVD set includes audio commentaries, deleted scenes, and documentaries.[66] The film was also released on Blu-ray in North America on January 25, 2011.[67]
Soundtrack
See also
• List of American films of 2001
• List of films about mathematicians
• Mental illness in films
Notes
1. Tied with Baz Luhrmann for Moulin Rouge!.
2. Tied with Maggie Smith for Gosford Park.
3. Tied with Mulholland Drive.
4. Tied with Maggie Smith for Gosford Park and Marisa Tomei for In the Bedroom.
References
1. "A Beautiful Mind (2002)". AFI Catalog of Feature Films. Retrieved December 20, 2020.
2. "A Beautiful Mind (2001)". Box Office Mojo. IMDb. Archived from the original on January 2, 2012. Retrieved November 8, 2010.
3. Abramowitz, Rachel (March 25, 2002). "In a Crisis, It Was a 'Beautiful' Job". Los Angeles Times. Retrieved April 24, 2023.
4. Friedman, Roger (February 15, 2002). "Exclusive: Ron Howard Changed His Mind; and Screenwriter Admits to 'Semi-Fictional Movie'". Fox News. Retrieved April 24, 2023.
5. "A Beautiful Partnership: Ron Howard and Brian Grazer", from A Beautiful Mind DVD, 2002.
6. "Development of the Screenplay", from A Beautiful Mind DVD, 2002.
7. "Dave Bayer: Professor of Mathematics". Barnard College, Columbia University. Archived from the original on May 11, 2012. Retrieved May 8, 2011.
8. "Beautiful Math" (PDF). June 2, 2002. Archived (PDF) from the original on November 19, 2012. Retrieved October 20, 2015.
9. "The Process of Age Progression", from A Beautiful Mind DVD. 2002.
10. "Scoring the Film", from A Beautiful Mind DVD, 2002.
11. A Beautiful Mind DVD commentary featuring Ron Howard, 2002.
12. "The Vanderbilt-Twombly Florham Estate / Fairleigh Dickinson University". Retrieved April 18, 2022.
13. "10 Movies Filmed in Manhattan College's Backyard". Archived from the original on October 25, 2015.
14. Schwartz, Nathaniel L. (September 21, 1999). "University, Hollywood Relationship Not Always a 'Love Story'". The Harvard Crimson. Archived from the original on March 3, 2012. Retrieved October 6, 2020.
15. "A Beautiful Mind Preview". Entertainment Weekly. October 24, 2001. Archived from the original on October 14, 2014.
16. Lyndall Bell (May 23, 2013). "Tales from A Beautiful Mind". Australian Broadcasting Corporation. it was potentially going to be a Robert Redford/Tom Cruise film.
17. "Ron Howard Interview". About.com. Archived from the original on January 9, 2012. Retrieved September 27, 2012.
18. "A Beautiful Mind". Mathematical Association of America. Archived from the original on October 14, 2014. Retrieved October 13, 2013.
19. "A Real Number". Slate Magazine. December 21, 2001. Archived from the original on August 24, 2007. Retrieved August 16, 2007.
20. "A Brilliant Madness". PBS.org. Archived from the original on July 14, 2007. Retrieved August 16, 2007.
21. Sylvia Nasar, A Beautiful Mind, Touchstone 1998.
22. Nasar, Sylvia (1998). A Beautiful Mind: A Biography of John Forbes Nash, Jr. Simon & Schuster. ISBN 0-684-81906-6.
23. "Nash: Film No Whitewash". CBS News: 60 Minutes. March 14, 2002. Archived from the original on August 7, 2007. Retrieved August 16, 2007.
24. Goldstein, Scott (April 10, 2005). "Eleanor Stier, 84". The Boston Globe. Archived from the original on May 8, 2008. Retrieved December 5, 2007.
25. "MIT facts meet fiction in 'A Beautiful Mind'". Massachusetts Institute of Technology. February 13, 2002. Archived from the original on July 12, 2007. Retrieved August 16, 2007.
26. "FAQ John Nash". Seeley G. Mudd Library at Princeton University. Archived from the original on July 16, 2007. Retrieved August 16, 2007.
27. "A Beautiful Mind (2001)". Rotten Tomatoes. Fandango Media. Archived from the original on August 24, 2007. Retrieved April 6, 2019.
28. "A Beautiful Mind Reviews". Metacritic. Archived from the original on March 6, 2018. Retrieved February 27, 2018.
29. "Home". CinemaScore. Retrieved February 25, 2022.
30. Ebert, Roger. "A Beautiful Mind". Chicago Sun-Times. RogerEbert.com. Archived from the original on January 11, 2012.
31. Clark, Mike (December 20, 2001). "Crowe brings to 'Mind' a great performance". USA Today. Archived from the original on July 13, 2007. Retrieved August 27, 2007.
32. Howe, Desson (December 21, 2001). "'Beautiful Mind': A Terrible Thing to Waste". Archived from the original on December 10, 2017 – via www.washingtonpost.com.
33. John Sutherland, 'Beautiful mind, lousy character' Archived October 28, 2016, at the Wayback Machine, The Guardian, 17 March 2002
34. Shailee Koranne, 'How schizophrenia is misrepresented in TV and film'
35. Navarrette, Lisa (April 1, 2002). "Why the Whitewashing of Alicia Nash?". Los Angeles Times. Retrieved January 6, 2021.
36. "Weekend Box Office Results for December 21–25, 2001". Box Office Mojo. IMDb. Archived from the original on January 12, 2012. Retrieved May 22, 2008.
37. "Weekend Box Office Results for January 4–6, 2002". Box Office Mojo. IMDb. Archived from the original on January 13, 2012. Retrieved May 22, 2008.
38. "The 74th Academy Awards (2002) Nominees and Winners". Academy of Motion Picture Arts and Sciences. AMPAS. Archived from the original on November 9, 2014. Retrieved November 19, 2011.
39. "AFI AWARDS 2001". American Film Institute. Retrieved April 19, 2016.
40. "Nominees/Winners". Casting Society of America. Retrieved July 10, 2019.
41. "AFI Past Winners - 2002 Winners & Nominees". AFI-AACTA. Archived from the original on January 4, 2015. Retrieved January 24, 2016.
42. "BAFTA Awards: Film in 2002". BAFTA. 2002. Retrieved September 16, 2016.
43. "1988-2013 Award Winner Archives". Chicago Film Critics Association. January 2013. Retrieved August 24, 2021.
44. "The BFCA Critics' Choice Awards :: 2001". Broadcast Film Critics Association. January 11, 2002. Archived from the original on January 7, 2013. Retrieved March 16, 2011.
45. "54th DGA Awards". Directors Guild of America Awards. Retrieved July 5, 2021.
46. Золотой Орел 2002 [Golden Eagle 2002] (in Russian). Ruskino.ru. Retrieved March 6, 2017.
47. "A Beautiful Mind – Golden Globes". HFPA. Retrieved July 5, 2021.
48. "Sound editors tap noms for Golden Reel Awards". Variety. Retrieved June 27, 2019.
49. "2002 Grammy Award Winners". Grammy.com. Retrieved May 1, 2011.
50. "Past Winners & Nominees". Humanitas Prize. Retrieved June 11, 2022.
51. "KCFCC Award Winners – 2000-09". December 14, 2013. Retrieved July 10, 2021.
52. "2001 Sierra Award Winners". December 13, 2021. Retrieved January 31, 2022.
53. "Pop stars claim victories at MTV Movie Awards". CNN. Associated Press. June 2, 2002. Archived from the original on March 16, 2016. Retrieved September 2, 2015.
54. "6th Annual Film Awards (2001)". Online Film & Television Association. Retrieved May 15, 2021.
55. "The Annual 5th Online Film Critics Society Awards". Online Film Critics Society. January 3, 2012. Retrieved August 24, 2021.
56. McNary, Dave (January 10, 2002). "Studio pix dominate PGA noms". Variety. Archived from the original on September 23, 2017. Retrieved September 22, 2017.
57. "International Press Academy website – 2002 6th Annual SATELLITE Awards". Archived from the original on February 13, 2008.
58. "The 8th Annual Screen Actors Guild Awards". Screen Actors Guild Awards. Archived from the original on November 1, 2011. Retrieved May 21, 2016.
59. "2001 SEFA Awards". sefca.net. Retrieved May 15, 2021.
60. "Past Scripter Awards". USC Scripter Award. Retrieved November 8, 2021.
61. "2nd Annual VFCC Award Winners". Vancouver Film Critics Circle. January 31, 2002. Retrieved January 31, 2002.
62. "World Soundtrack Awards". World Soundtrack Awards. Retrieved December 18, 2021.
63. "Awards Winners". wga.org. Writers Guild of America. Archived from the original on December 5, 2012. Retrieved June 6, 2010.
64. "AFI's 100 Years...100 Movies (10th Anniversary Edition) Ballot" (PDF). Archived (PDF) from the original on March 28, 2014.
65. "A Beautiful Mind". FilmCritic.com. Archived from the original on November 16, 2011. Retrieved July 24, 2011.
66. Rivero, Enrique (December 14, 2001). "DVD Preview: Howard Has Plans for Beautiful Mind DVD". hive4media.com. Archived from the original on January 10, 2002. Retrieved September 9, 2019.
67. "A Beautiful Mind (2001)". ReleasedOn.com. Archived from the original on March 15, 2011. Retrieved July 24, 2011.
Further reading
• Akiva Goldsman. A Beautiful Mind: Screenplay and Introduction. New York, New York: Newmarket Press, 2002. ISBN 1-55704-526-7.
External links
Wikiquote has quotations related to A Beautiful Mind (film).
• Official website
• A Beautiful Mind at IMDb
• A Beautiful Mind at the TCM Movie Database
• A Beautiful Mind at AllMovie
• A Beautiful Mind at Rotten Tomatoes
• A Beautiful Mind at Box Office Mojo
• A Beautiful Mind at MSN Movies
• A Beautiful Mind at Film Insight
Ron Howard
Feature films
Directed
• Grand Theft Auto (1977, also wrote)
• Cotton Candy (1978, also wrote)
• Skyward (1980)
• Night Shift (1982)
• Splash (1984)
• Cocoon (1985)
• Gung Ho (1986)
• Willow (1988)
• Parenthood (1989, also wrote)
• Backdraft (1991)
• Far and Away (1992, also wrote)
• The Paper (1994)
• Apollo 13 (1995)
• Ransom (1996)
• EDtv (1999)
• How the Grinch Stole Christmas (2000)
• A Beautiful Mind (2001)
• The Missing (2003)
• Cinderella Man (2005)
• The Da Vinci Code (2006)
• Frost/Nixon (2008)
• Angels & Demons (2009)
• The Dilemma (2011)
• Rush (2013)
• In the Heart of the Sea (2015)
• Inferno (2016)
• Solo: A Star Wars Story (2018)
• Hillbilly Elegy (2020)
• Thirteen Lives (2022)
• The Shrinking of Treehorn (TBA)
Produced
• Clean and Sober (1988)
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• The Alamo (2004)
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Documentaries
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• Pavarotti (2019)
• Rebuilding Paradise (2020)
• We Feed People (2022)
Produced
• Beyond the Mat (1999)
• Katy Perry: Part of Me (2012)
See also
• Imagine Entertainment
• Brian Grazer
Brian Grazer
Films produced
• Night Shift (1982)
• Real Genius (1985)
• Spies Like Us (1985)
• Like Father Like Son (1987)
• Parenthood (1989)
• Kindergarten Cop (1990)
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• My Girl (1991)
• Far and Away (1992)
• Boomerang (1992)
• For Love or Money (1993)
• My Girl 2 (1994)
• Greedy (1994)
• The Paper (1994)
• The Cowboy Way (1994)
• Apollo 13 (1995)
• Sgt. Bilko (1996)
• Fear (1996)
• The Nutty Professor (1996)
• The Chamber (1996)
• Ransom (1996)
• Liar Liar (1997)
• Inventing the Abbotts (1997)
• Mercury Rising (1998)
• Psycho (1998)
• EDtv (1999)
• Life (1999)
• Bowfinger (1999)
• Beyond the Mat (1999)
• Nutty Professor II: The Klumps (2000)
• How the Grinch Stole Christmas (2000)
• A Beautiful Mind (2001)
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• 8 Mile (2002)
• Intolerable Cruelty (2003)
• The Cat in the Hat (2003)
• The Missing (2003)
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• Cinderella Man (2005)
• Flightplan (2005)
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• Inside Man (2006)
• The Da Vinci Code (2006)
• American Gangster (2007)
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• Frost/Nixon (2008)
• Angels & Demons (2009)
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• The Dilemma (2011)
• Restless (2011)
• Cowboys & Aliens (2011)
• J. Edgar (2011)
• Tower Heist (2011)
• Katy Perry: Part of Me (2012)
• Rush (2013)
• Get on Up (2014)
• The Good Lie (2014)
• In the Heart of the Sea (2015)
• Lowriders (2016)
• Inferno (2016)
• American Made (2017)
• The Spy Who Dumped Me (2018)
• Pavarotti (2019)
• Dads (2019)
• Rebuilding Paradise (2020)
• Hillbilly Elegy (2020)
• Tick, Tick... Boom! (2021)
• Thirteen Lives (2022)
• Candy Cane Lane (TBA)
Films produced
and wrote
• Splash (1984)
• Armed and Dangerous (1986)
• Housesitter (1992)
TV series created
• Shadow Chasers (1985)
Related
• Imagine Entertainment
• Ron Howard
Akiva Goldsman
Films directed
• Winter's Tale (2014; also wrote and produced)
• Stephanie (2017)
Films written
• The Client (1994)
• Silent Fall (1994)
• Batman Forever (1995)
• A Time to Kill (1996)
• Batman & Robin (1997)
• Lost in Space (1998; also produced)
• Practical Magic (1998)
• A Beautiful Mind (2001)
• I, Robot (2004)
• Cinderella Man (2005)
• The Da Vinci Code (2006)
• I Am Legend (2007; also produced)
• Angels & Demons (2009)
• The Divergent Series: Insurgent (2015)
• The 5th Wave (2016)
• Rings (2017)
• Transformers: The Last Knight (2017)
• The Dark Tower (2017; also produced)
Films produced
• Deep Blue Sea (1999)
• Starsky & Hutch (2004)
• Mindhunters (2004)
• Constantine (2005)
• Mr. & Mrs. Smith (2005)
• Poseidon (2006)
• Hancock (2008)
• The Losers (2010)
• Fair Game (2010)
• Jonah Hex (2010)
• Lone Survivor (2013)
• King Arthur: Legend of the Sword (2017)
• The Map of Tiny Perfect Things (2021)
• Without Remorse (2021)
• Firestarter (2022)
• Meet Cute (2022)
TV series created
• Titans (2018–2023)
• Star Trek: Picard (2020–2023)
• Star Trek: Strange New Worlds (2022)
• The Crowded Room (2023)
Category
Awards for A Beautiful Mind
Academy Award for Best Picture
1927–1950
• Wings (1927–1928)
• The Broadway Melody (1928–1929)
• All Quiet on the Western Front (1929–1930)
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• Mrs. Miniver (1942)
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• The Lost Weekend (1945)
• The Best Years of Our Lives (1946)
• Gentleman's Agreement (1947)
• Hamlet (1948)
• All the King's Men (1949)
• All About Eve (1950)
1951–1975
• An American in Paris (1951)
• The Greatest Show on Earth (1952)
• From Here to Eternity (1953)
• On the Waterfront (1954)
• Marty (1955)
• Around the World in 80 Days (1956)
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• Gigi (1958)
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• In the Heat of the Night (1967)
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• Midnight Cowboy (1969)
• Patton (1970)
• The French Connection (1971)
• The Godfather (1972)
• The Sting (1973)
• The Godfather Part II (1974)
• One Flew Over the Cuckoo's Nest (1975)
1976–2000
• Rocky (1976)
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• The Deer Hunter (1978)
• Kramer vs. Kramer (1979)
• Ordinary People (1980)
• Chariots of Fire (1981)
• Gandhi (1982)
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• Amadeus (1984)
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• Forrest Gump (1994)
• Braveheart (1995)
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• Titanic (1997)
• Shakespeare in Love (1998)
• American Beauty (1999)
• Gladiator (2000)
2001–present
• A Beautiful Mind (2001)
• Chicago (2002)
• The Lord of the Rings: The Return of the King (2003)
• Million Dollar Baby (2004)
• Crash (2005)
• The Departed (2006)
• No Country for Old Men (2007)
• Slumdog Millionaire (2008)
• The Hurt Locker (2009)
• The King's Speech (2010)
• The Artist (2011)
• Argo (2012)
• 12 Years a Slave (2013)
• Birdman or (The Unexpected Virtue of Ignorance) (2014)
• Spotlight (2015)
• Moonlight (2016)
• The Shape of Water (2017)
• Green Book (2018)
• Parasite (2019)
• Nomadland (2020)
• CODA (2021)
• Everything Everywhere All at Once (2022)
Critics' Choice Movie Award for Best Picture
• Sense and Sensibility (1995)
• Fargo (1996)
• L.A. Confidential (1997)
• Saving Private Ryan (1998)
• American Beauty (1999)
• Gladiator (2000)
• A Beautiful Mind (2001)
• Chicago (2002)
• The Lord of the Rings: The Return of the King (2003)
• Sideways (2004)
• Brokeback Mountain (2005)
• The Departed (2006)
• No Country for Old Men (2007)
• Slumdog Millionaire (2008)
• The Hurt Locker (2009)
• The Social Network (2010)
• The Artist (2011)
• Argo (2012)
• 12 Years a Slave (2013)
• Boyhood (2014)
• Spotlight (2015)
• La La Land (2016)
• The Shape of Water (2017)
• Roma (2018)
• Once Upon a Time in Hollywood (2019)
• Nomadland (2020)
• The Power of the Dog (2021)
• Everything Everywhere All at Once (2022)
Dallas–Fort Worth Film Critics Association Award for Best Film
• Dances with Wolves (1990)
• JFK (1991)
• Unforgiven (1992)
• Schindler's List (1993)
• Pulp Fiction (1994)
• Leaving Las Vegas (1995)
• Fargo (1996)
• L.A. Confidential (1997)
• Saving Private Ryan (1998)
• American Beauty (1999)
• Traffic (2000)
• A Beautiful Mind (2001)
• Chicago (2002)
• The Lord of the Rings: The Return of the King (2003)
• Million Dollar Baby (2004)
• Brokeback Mountain (2005)
• United 93 (2006)
• No Country for Old Men (2007)
• Slumdog Millionaire (2008)
• Up in the Air (2009)
• The Social Network (2010)
• The Descendants (2011)
• Lincoln (2012)
• 12 Years a Slave (2013)
• Birdman or (The Unexpected Virtue of Ignorance) (2014)
• Spotlight (2015)
• Moonlight (2016)
• The Shape of Water (2017)
• A Star Is Born (2018)
• 1917 (2019)
• Nomadland (2020)
• The Power of the Dog (2021)
• Everything Everywhere All at Once (2022)
Golden Globe Award for Best Motion Picture – Drama
1943–1975
• The Song of Bernadette (1943)
• Going My Way (1944)
• The Lost Weekend (1945)
• The Best Years of Our Lives (1946)
• Gentleman's Agreement (1947)
• Johnny Belinda / The Treasure of the Sierra Madre (1948)
• All the King's Men (1949)
• Sunset Boulevard (1950)
• A Place in the Sun (1951)
• The Greatest Show on Earth (1952)
• The Robe (1953)
• On the Waterfront (1954)
• East of Eden (1955)
• Around the World in 80 Days (1956)
• The Bridge on the River Kwai (1957)
• The Defiant Ones (1958)
• Ben-Hur (1959)
• Spartacus (1960)
• The Guns of Navarone (1961)
• Lawrence of Arabia (1962)
• The Cardinal (1963)
• Becket (1964)
• Doctor Zhivago (1965)
• A Man for All Seasons (1966)
• In the Heat of the Night (1967)
• The Lion in Winter (1968)
• Anne of the Thousand Days (1969)
• Love Story (1970)
• The French Connection (1971)
• The Godfather (1972)
• The Exorcist (1973)
• Chinatown (1974)
• One Flew Over the Cuckoo's Nest (1975)
1976–2000
• Rocky (1976)
• The Turning Point (1977)
• Midnight Express (1978)
• Kramer vs. Kramer (1979)
• Ordinary People (1980)
• On Golden Pond (1981)
• E.T. the Extra-Terrestrial (1982)
• Terms of Endearment (1983)
• Amadeus (1984)
• Out of Africa (1985)
• Platoon (1986)
• The Last Emperor (1987)
• Rain Man (1988)
• Born on the Fourth of July (1989)
• Dances with Wolves (1990)
• Bugsy (1991)
• Scent of a Woman (1992)
• Schindler's List (1993)
• Forrest Gump (1994)
• Sense and Sensibility (1995)
• The English Patient (1996)
• Titanic (1997)
• Saving Private Ryan (1998)
• American Beauty (1999)
• Gladiator (2000)
2001–present
• A Beautiful Mind (2001)
• The Hours (2002)
• The Lord of the Rings: The Return of the King (2003)
• The Aviator (2004)
• Brokeback Mountain (2005)
• Babel (2006)
• Atonement (2007)
• Slumdog Millionaire (2008)
• Avatar (2009)
• The Social Network (2010)
• The Descendants (2011)
• Argo (2012)
• 12 Years a Slave (2013)
• Boyhood (2014)
• The Revenant (2015)
• Moonlight (2016)
• Three Billboards Outside Ebbing, Missouri (2017)
• Bohemian Rhapsody (2018)
• 1917 (2019)
• Nomadland (2020)
• The Power of the Dog (2021)
• The Fabelmans (2022)
Authority control
International
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National
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| Wikipedia |
A Biography of Maria Gaetana Agnesi
A Biography of Maria Gaetana Agnesi, an Eighteenth-Century Woman Mathematician: With Translations of Some of Her Work from Italian into English is a biography of Italian mathematician and philosopher Maria Gaetana Agnesi (1718–1799). It was written and translated by Antonella Cupillari, with a foreword by Patricia R. Allaire, and published in 2008 by the Edwin Mellen Press.
Topics
The main part of the book, over 100 pages, is a translation into English of an Italian-language biography of Agnesi, Elogio storico di Donna Maria Gaetana Agnesi, which was written in the year of her death by historian Antonio Francesco Frisi and republished in 1965.[1] It covers the cultural background that allowed her to become a mathematician, and her brief mathematical career from her teens to her thirties, as well as her work caring for the needy in the remaining fifty years of her life.[2]
Frisi was a family friend of Agnesi. He was the first to write a biography about her. To balance this material with a more objective view of Agnesi,[2] Cupillari has added over 50 pages of notes,[1] derived from two more Italian-language biographies of Agnesi, Maria Gaetana Àgnesi (Luisa Anzoletti, 1900) and Maria Gaetana Agnesi (Giovanna Tilche, 1984).[3] Another large section includes translations and explanations of excerpts from Agnesi's mathematical textbook, Institutioni Analitiche (1748),[1][3] which was "the first textbook to provide a unified treatment of algebra, Cartesian geometry and calculus", and by being written in vernacular Italian rather than Latin was aimed at a wider audience than the educated scholars of her day.[2] Cupillari concludes her biography with a bibliography of material about Agnesi.[1]
Audience and reception
Reviewers Luigi Pepe and Franka Bruckler recommend the book as a "useful introduction" and "unique, comprehensive source" on Agnesi and her work, particularly for people who read English but not Italian.[1][3] Bruckler includes among its potential readers historians of mathematics, mathematics educators, and members of the public.[3] Reviewer Edith Mendez describes the book as "an easy read", and its mathematics as accessible to undergraduate mathematics students,[4] but this is contradicted by Peter Ruane, who found the "fragmented" and "eulogistic" first part difficult to follow and to stomach.[2] Mendez also criticizes the book for being inadequately copyedited,[4] and Ruane suggests that the book would have been improved by more context of what was happening in mathematics in Europe at the time.[2]
References
1. Pepe, Luigi (2011), "Review of A Biography of Maria Gaetana Agnesi, an Eighteenth-Century Woman Mathematician", MathSciNet, MR 2675954
2. Ruane, P. N. (January 2009), "Review of A Biography of Maria Gaetana Agnesi, an Eighteenth-Century Woman Mathematician", MAA Reviews
3. Bruckler, Franka Miriam, "Review of A Biography of Maria Gaetana Agnesi, an Eighteenth-Century Woman Mathematician", zbMATH, Zbl 1228.01037
4. Mendez, Edith Prentice (June 2008), "Review of A Biography of Maria Gaetana Agnesi, an Eighteenth-Century Woman Mathematician", Convergence
| Wikipedia |
A Guide to the Classification Theorem for Compact Surfaces
A Guide to the Classification Theorem for Compact Surfaces is a textbook in topology, on the classification of two-dimensional surfaces. It was written by Jean Gallier and Dianna Xu, and published in 2013 by Springer-Verlag as volume 9 of their Geometry and Computing series (doi:10.1007/978-3-642-34364-3, ISBN 978-3-642-34363-6). The Basic Library List Committee of the Mathematical Association of America has recommended its inclusion in undergraduate mathematics libraries.[1]
Topics
The classification of surfaces (more formally, compact two-dimensional manifolds without boundary) can be stated very simply, as it depends only on the Euler characteristic and orientability of the surface. An orientable surface of this type must be topologically equivalent (homeomorphic) to a sphere, torus, or more general handlebody, classified by its number of handles. A non-orientable surface must be equivalent to a projective plane, Klein bottle, or more general surface characterized by an analogous number, its number of cross-caps. For compact surfaces with boundary, the only extra information needed is the number of boundary components.[1] This result is presented informally at the start of the book, as the first of its six chapters. The rest of the book presents a more rigorous formulation of the problem, a presentation of the topological tools needed to prove the result, and a formal proof of the classification.[2][3]
Other topics in topology discussed as part of this presentation include simplicial complexes, fundamental groups, simplicial homology and singular homology, and the Poincaré conjecture. Appendices include additional material on embeddings and self-intersecting mappings of surfaces into three-dimensional space such as the Roman surface, the structure of finitely generated abelian groups, general topology, the history of the classification theorem, and the Hauptvermutung (the theorem that every surface can be triangulated).[2]
Audience and reception
This is a textbook aimed at the level of advanced undergraduates or beginning graduate students in mathematics,[2] perhaps after having already completed a first course in topology. Readers of the book are expected to already be familiar with general topology, linear algebra, and group theory.[1] However, as a textbook, it lacks exercises, and reviewer Bill Wood suggests its use for a student project rather than for a formal course.[1]
Many other graduate algebraic topology textbooks include coverage of the same topic.[4] However, by focusing on a single topic, the classification theorem, the book is able to prove the result rigorously while remaining at a lower overall level,[4][5] provide a greater amount of intuition and history,[4] and serve as "a motivating tour of the discipline’s fundamental techniques".[1]
Reviewer Clara Löh complains that parts of the book are redundant, and in particular that the classification theorem can be proven either with the fundamental group or with homology (not needing both), that on the other hand several important tools from topology including the Jordan–Schoenflies theorem are not proven, and that several related classification results are omitted.[3] Nevertheless, reviewer D. V. Feldman highly recommends the book,[5] Wood writes "This is a book I wish I’d had in graduate school",[1] and reviewer Werner Kleinert calls it "an introductory text of remarkable didactic value".[2]
References
1. Wood, Bill (March 2014), "Review of A Guide to the Classification Theorem for Compact Surfaces", MAA Reviews, Mathematical Association of America
2. Kleinert, Werner, "Review of A Guide to the Classification Theorem for Compact Surfaces", zbMATH, Zbl 1270.57001
3. Löh, Clara, "Review of A Guide to the Classification Theorem for Compact Surfaces", Mathematical Reviews, 9, MR 3026641
4. Castrillon Lopez, Marco (January 2018), "Review of A Guide to the Classification Theorem for Compact Surfaces", EMS Reviews, European Mathematical Society
5. Feldman, D. V. (August 2013), "Review of A Guide to the Classification Theorem for Compact Surfaces" (PDF), Choice Reviews, 51 (1), Review 51-0331, doi:10.5860/choice.51-0331
External links
• Author's web site for A Guide to the Classification Theorem for Compact Surfaces including a PDF version of Chapter 1
| Wikipedia |
A History of Folding in Mathematics
A History of Folding in Mathematics: Mathematizing the Margins is a book in the history of mathematics on the mathematics of paper folding. It was written by Michael Friedman and published in 2018 by Birkhäuser as volume 59 of their Historical Studies series.
Topics
The book consists of six chapters, the first of which introduces the problem, sets it in the context of the investigation of the mathematical strength of straightedge and compass constructions, and introduces one of the major themes of the book, the relegation of paper folding to recreational mathematics as this sort of investigation fell out of favor among professional mathematicians, and its more recent resurrection as a serious topic of investigation.[1][2][3] As a work of history, the book follows Hans-Jörg Rheinberger in making a distinction between epistemic objects, the not-yet-fully-defined subjects of scientific investigation, and technical objects, the tools used in these investigations, and it links the perceived technicality of folding with its fall from mathematical favor.[3][4]
The remaining chapters are organized chronologically, beginning in the 16th century and the second chapter. This chapter includes the work of Albrecht Dürer on polyhedral nets, arrangements of polygons in the plane that can be folded to form a given polyhedron, and of Luca Pacioli on the use of folding to replace the compass and straightedge in geometric constructions; it also discusses the history of paper, and paper folding in the context of bookbinding.[1][5] The third chapter discusses the confluence of Arabic and European mathematics, into the 18th century, with topics including the symmetries of folded objects and the attempted use of folding to prove the parallel postulate.[1][5] Although Eugenio Beltrami continued to use folded models to investigate non-Euclidean geometry into the 19th century,[2] the fourth chapter of the book argues that other 19th-century uses of folding were more pedagogical, including the use of folded models to demonstrate mathematical concepts, their applications in chemistry, and the introduction of folding into kindergarten programs by Friedrich Fröbel.[1][5] The late 19th century also saw the publication in India and then the west of the book Geometric Exercises in Paper Folding, by T. Sundara Row.[2]
The final two chapters concern the 20th century and current topics of research in this area. They include work on formalizing paper folding as a form of axiomatic geometry beginning with Margherita Piazzola Beloch, the work of Wilhelm Ahrens in recreational mathematics, and the community of mathematical researchers coming together through the series of International Meetings of Origami Science and Technology (now known as the International Conference on Origami in Science, Math, and Education),[1][5] and through works popularizing this area within mathematics such as the book Geometric Folding Algorithms by Erik Demaine and Joseph O'Rourke.[2] Appendices include a translation of Beloch's work in this area, and a response to the book The Fold: Leibniz and the Baroque by Gilles Deleuze.[1][5]
Audience and reception
In reviewing the book, mathematicians Thomas Sonar and James J. Tattersall recommend the book to a general audience interested in mathematics and its history,[1][5] and reviewer James J. Tattersall writes that it contains "a wealth of mathematical and historical information on a wide selection of topics".[5]
Reviewer William J. Satzer, also a mathematician, disagrees on the target audience, writing that although the book would be of interest to historians of mathematics, it would make difficult reading for others because its topics are too loosely connected. Satzer also faults the book for its omission of Japanese and Chinese threads in its tapestry.[2] On the other hand, Argentine origami book author Laura Rozenberg, despite admitting to skipping over the more mathematical parts of the story, says it "can be read by the non-mathematician without pausing", writing that it felt that "Friedman had read our minds and had decided to indulge us with answers to problems that have beset paperfolding aficionados for years".[3] And historian Anne Por, reviewing the book, writes that "this work is not only highly informative but also particularly pleasant to read."[4]
References
1. Sonar, Thomas, "Review of A History of Folding in Mathematics", zbMATH, Zbl 1401.01003
2. Satzer, William J. (September 2018), "Review of A History of Folding in Mathematics", MAA Reviews, Mathematical Association of America
3. Rozenberg, Laura (January–February 2020), translated by Buschman, James, "Review of A History of Folding in Mathematics", The Fold, 56
4. Por, Anne (September 2019), "Review of A History of Folding in Mathematics", Isis, 110 (3): 577–578, doi:10.1086/704936, S2CID 203074266
5. Tattersall, James J., "Review of A History of Folding in Mathematics", Mathematical Reviews, MR 3793627
Mathematics of paper folding
Flat folding
• Big-little-big lemma
• Crease pattern
• Huzita–Hatori axioms
• Kawasaki's theorem
• Maekawa's theorem
• Map folding
• Napkin folding problem
• Pureland origami
• Yoshizawa–Randlett system
Strip folding
• Dragon curve
• Flexagon
• Möbius strip
• Regular paperfolding sequence
3d structures
• Miura fold
• Modular origami
• Paper bag problem
• Rigid origami
• Schwarz lantern
• Sonobe
• Yoshimura buckling
Polyhedra
• Alexandrov's uniqueness theorem
• Blooming
• Flexible polyhedron (Bricard octahedron, Steffen's polyhedron)
• Net
• Source unfolding
• Star unfolding
Miscellaneous
• Fold-and-cut theorem
• Lill's method
Publications
• Geometric Exercises in Paper Folding
• Geometric Folding Algorithms
• Geometric Origami
• A History of Folding in Mathematics
• Origami Polyhedra Design
• Origamics
People
• Roger C. Alperin
• Margherita Piazzola Beloch
• Robert Connelly
• Erik Demaine
• Martin Demaine
• Rona Gurkewitz
• David A. Huffman
• Tom Hull
• Kôdi Husimi
• Humiaki Huzita
• Toshikazu Kawasaki
• Robert J. Lang
• Anna Lubiw
• Jun Maekawa
• Kōryō Miura
• Joseph O'Rourke
• Tomohiro Tachi
• Eve Torrence
| Wikipedia |
A History of Mathematical Notations
A History of Mathematical Notations is a book on the history of mathematics and of mathematical notation. It was written by Swiss-American historian of mathematics Florian Cajori (1859–1930), and originally published as a two-volume set by the Open Court Publishing Company in 1928 and 1929, with the subtitles Volume I: Notations in Elementary Mathematics (1928) and Volume II: Notations Mainly in Higher Mathematics (1929).[1] Although Open Court republished it in a second edition in 1974, it was unchanged from the first edition.[2] In 1993, it was published as an 820-page single volume edition by Dover Publications, with its original pagination unchanged.[1]
The Basic Library List Committee of the Mathematical Association of America has listed this book as essential for inclusion in undergraduate mathematics libraries.[1] It was already described as long-awaited at the time of its publication,[3] and by 2013, when the Dover edition was reviewed by Fernando Q. Gouvêa, he wrote that it was "one of those books so well known that it doesn’t need a review".[1] However, some of its claims on the history of the notations it describes have been subsumed by more recent research, and its coverage of modern mathematics is limited, so it should be used with care as a reference.[1][2]
Topics
The first volume of the book concerns elementary mathematics. It has 400 pages of material on arithmetic. This includes the history of notation for numbers from many ancient cultures, arranged by culture,[3][4] with the Hindu–Arabic numeral system treated separately.[1] Following this, it covers notation for arithmetic operations, arranged separately by operation and by the mathematicians who used those notations (although not in a strict chronological ordering).[3][4][5] The first volume concludes with 30 pages on elementary geometry,[3][5] including also the struggle between symbolists and rhetoricians in the 18th and 19th centuries on whether to express mathematics in notation or words, respectively.[6]
The second volume is divided more evenly into four parts. The first part, on arithmetic and algebra, also includes mathematical constants and Special functionss that would nowadays be considered part of mathematical analysis, as well as notations for binomial coefficients and other topics in combinatorics,[7][8] and even the history of the dollar sign.[9] The second part is entitled "modern analysis", but its topics are primarily trigonometry, calculus, and mathematical logic,[7][8] including the conflicting calculus notations of Isaac Newton and Gottfried Wilhelm Leibniz.[9] The third part concerns geometry, while the fourth concerns scholarship in the history of mathematics as well as the movement for international standardization.[7][8]
Audience and reception
This book is mainly a reference work and sourcebook, containing excerpts from many texts illustrating their use of notation.[10]
Among reviewers from the time of the book's original publication, George Sarton took as the main lesson from this book "the slowness and timidity of human advance",[3] while some other reviewers saw differently that the confusing multiplicity of notations documented by the book should lead to a greater push for standardization.[11][12] Although praising the book's "richness of explanation" and "familiarity with the ground", Lao Genevra Simons expressed a wish that Cajori had access to a greater number of original sources,[10] and pointed to some historical inaccuracies in the work.[10][11] Sarton concluded, accurately, that the book "will remain a standard work for many years to come".[7] Although one reviewer found the treatment of dollar signs appropriate for an American book,[9] reviewer G. Feigl disagreed, finding this part off-topic.[13]
By 1974, and echoing Feigl,[13] reviewer Herbert Meschkowski complained that the book's coverage of mathematics from after the beginning of the 19th century was inadequate.[2] In a review published in 2013, Fernando Q. Gouvêa wrote that the book remained useful, especially for its photographic reproductions of samples of old notation. He added that it was still the only comprehensive text in this area, although other works cover more specialized subtopics. However, Gouvêa wrote that modern scholarship on the numbering systems of past civilizations and on the first uses of some symbols has changed since Cajori's work, so such claims need to be checked against more recent publications instead of taking Cajori's word for them. In the case of ancient number systems, Gouvêa recommends instead Numerical Notation: A Comparative History by Stephen Chrisomalis (Cambridge University Press, 2010).[1]
References
1. Gouvêa, Fernando Q. (August 2013), "Review of A History of Mathematical Notations (Dover edition)", MAA Reviews, Mathematical Association of America
2. Meschkowski, H., "Review of A History of Mathematical Notations, 2nd ed.", zbMATH (in German), Zbl 0334.01003
3. Sarton, George (May 1929), "Review of A History of Mathematical Notations, Vol. I", Isis, 12 (2): 332–336, doi:10.1086/346417, JSTOR 224794
4. Jervis, S. D. (July 1930), "Review of A History of Mathematical Notations, Vol. I", Science Progress in the Twentieth Century, 25 (97): 134–136, JSTOR 43429258
5. Feigl, G., "Review of A History of Mathematical Notations, Vol. I", Jahrbuch über die Fortschritte der Mathematik (in German), JFM 54.0001.04
6. J. B. (1928–1929), "Review of A History of Mathematical Notations, Vol. I", Transactions of the Faculty of Actuaries, 12 (113): 241–247, JSTOR 41218127
7. Sarton, George (September 1929), "Review of A History of Mathematical Notations, Vol. II", Isis, 13 (1): 129–130, doi:10.1086/346448, JSTOR 224613
8. Jervis, S. D. (January 1932), "Review of A History of Mathematical Notations, Vol. II", Science Progress in the Twentieth Century, 26 (103): 518, JSTOR 43429174
9. B., J. (1928–1929), "Review of A History of Mathematical Notations, Vol. II", Transactions of the Faculty of Actuaries, 12 (114): 283–285, JSTOR 41218131
10. Simons, Lao G. (1929), "Review of A History of Mathematical Notations, Vol. I", American Mathematical Monthly, 36 (4): 230–232, doi:10.2307/2299309, JSTOR 2299309, MR 1521716
11. Simons, Lao G. (1930), "Review of A History of Mathematical Notations, Vol. II", American Mathematical Monthly, 37 (4): 193–195, doi:10.2307/2299795, JSTOR 2299795, MR 1521979
12. "Review of A History of Mathematical Notations, Vol. I", The Mathematical Gazette, 15 (208): 170–171, July 1930, doi:10.2307/3607176, JSTOR 3607176, S2CID 119432905
13. Feigl, G., "Review of A History of Mathematical Notations, Vol. II", Jahrbuch über die Fortschritte der Mathematik (in German), JFM 55.0002.02
External links
• Works related to A History Of Mathematical Notations, Vol. I at Wikisource
• A History of Mathematical Notations, Vol. I and A History of Mathematical Notations, Vol. II on the Internet Archive
| Wikipedia |
A History of Pi
A History of Pi (also titled A History of π) is a 1970 non-fiction book by Petr Beckmann that presents a layman's introduction to the concept of the mathematical constant pi (π).[1]
A History of Pi
Book cover of A History of Pi (3rd ed.)
AuthorPetr Beckmann
CountryUnited States
LanguageEnglish
SubjectMathematics
General Sciences
History of mathematics
PublisherGolem Press (1st, 2nd ed.)
St. Martin's Press (3rd ed.)
Hippocrene Books (Reprint ed.)
Publication date
1970
Pages190 pages
ISBN978-0-911762-07-5
OCLC99082
Part of a series of articles on the
mathematical constant π
3.1415926535897932384626433...
Uses
• Area of a circle
• Circumference
• Use in other formulae
Properties
• Irrationality
• Transcendence
Value
• Less than 22/7
• Approximations
• Madhava's correction term
• Memorization
People
• Archimedes
• Liu Hui
• Zu Chongzhi
• Aryabhata
• Madhava
• Jamshīd al-Kāshī
• Ludolph van Ceulen
• François Viète
• Seki Takakazu
• Takebe Kenko
• William Jones
• John Machin
• William Shanks
• Srinivasa Ramanujan
• John Wrench
• Chudnovsky brothers
• Yasumasa Kanada
History
• Chronology
• A History of Pi
In culture
• Indiana Pi Bill
• Pi Day
Related topics
• Squaring the circle
• Basel problem
• Six nines in π
• Other topics related to π
Author
Beckmann was a Czechoslovakian who fled the Communist regime to go to the United States. His dislike of authority gives A History of Pi a style that belies its dry title. For example, his chapter on the era following the classical age of ancient Greece is titled "The Roman Pest";[2] he calls the Catholic Inquisition the act of "insane religious fanatic"; and he says that people who question public spending on scientific research are "intellectual cripples who drivel about 'too much technology' because technology has wounded them with the ultimate insult: 'They can't understand it any more.'"
Beckmann was a prolific scientific author who wrote several electrical engineering textbooks and non-technical works, founded Golem Press, which published most of his books, and published his own monthly newsletter, Access to Energy. In his self-published book Einstein Plus Two and in Internet flame wars, he claimed that the theory of relativity is incorrect.[3]
Bibliography
A History of Pi was originally published as A History of π in 1970 by Golem Press. This edition did not cover any approximations of π calculated after 1946. A second edition, printed in 1971, added material on the calculation of π by electronic computers, but still contained historical and mathematical errors, such as an incorrect proof that there exist infinitely many prime numbers.[4] A third edition was published as A History of Pi in 1976 by St. Martin's Press. It was published as A History of Pi by Hippocrene Books in 1990.[5] The title is given as A History of Pi by both Amazon[6] and by WorldCat.[7]
1. Beckmann, Petr (1970), A History of π (1st ed.), Golem Press, p. 190, ISBN 0-911762-07-8
2. Beckmann, Petr (1971-01-01), A History of π (2nd ed.), Golem Press, p. 196, ISBN 0-911762-12-4
3. Beckmann, Petr (1976-07-15), A History of Pi (3rd ed.), St. Martin's Press, p. 208, ISBN 0-312-38185-9
4. Beckmann, Petr (1977), A History of π (4th ed.), Golem Press, p. 202, ISBN 0-911762-18-3
5. Beckmann, Petr (1982), A History of π (5th ed.), Golem Press, p. 202, ISBN 0-911762-18-3
6. Beckmann, Petr (1990-06-01), A History of Pi (Reprint ed.), Hippocrene Books, p. 200, ISBN 0-88029-418-3
See also
• History of Pi
References
1. Drum, Kevin (December 2, 1996). "A History of Pi, by Petr Beckman". Archived from the original on July 4, 2007. Retrieved April 13, 2014.
2. Thoreau, Book Recommendation: A History of Pi Archived 2011-07-16 at the Wayback Machine
3. Farrell, John (2000-07-06). "Did Einstein cheat?". Salon. Retrieved 2022-03-25.
4. Gould, Henry W. (1974). "Review of A History of π". Mathematics of Computation. 28 (125): 325–327. doi:10.2307/2005843. ISSN 0025-5718. JSTOR 2005843.
5. "A History of PI by Petr Beckmann ", GoodReads
6. ASIN 0312381859
7. OCLC 472118858
| Wikipedia |
A History of Greek Mathematics
A History of Greek Mathematics is a book by English historian of mathematics Thomas Heath about history of Greek mathematics. It was published in Oxford in 1921, in two volumes titled Volume I, From Thales to Euclid and Volume II, From Aristarchus to Diophantus. It got positive reviews and is still used today. Ten years later, in 1931, Heath published A Manual of Greek Mathematics, a concise version of the two-volume History.
Background
Thomas Heath was a British civil servant, whose hobby was Greek mathematics (he called it a "hobby" himself). He published a number of translations of major works of Euclid, Archimedes, Apollonius of Perga and others; most are still used today.[1]
Heath wrote in the preface to the book:[2]
The work was begun in 1913, but the bulk of it was written, as a distraction, during the first three years of the war, the hideous course of which seemed day by day to enforce the profound truth conveyed in the answer of Plato to the Delians. When they consulted him on the problem set them by the Oracle, namely, that of duplicating the cube, he replied, 'It must be supposed, not that the god specially wished this problem solved, but that he would have the Greeks desist from war and wickedness and cultivate the Muses, so that, their passion being assuaged by philosophy and mathematics, they might live in innocent and mutually helpful intercourse with one another.'
Ten years later, in 1931, Heath published A Manual of Greek Mathematics, a concise version of the two-volume History. In a preface Heath wrote that the Manual is for "the general reader who has not lost interest in the studies of his youth", while History was written for scholars.[3] The Manual contains some discoveries made in ten years after the publication of History, for example the new edition of Rhind Papyrus (published in 1923), some parts of then unpublished Moscow Papyrus,[3][4][5] and decipherment of Babylonian tablets and "the newest studies" of Babylonian astronomy.[5]
Contents
I. Introductory
II. Greek numerical notation and arithmetical operations (logistiké)
III. Pythagorean arithmetic (arithmetiké)
IV. The earliest Greek geometry (Thales)
V. Pythagorean geometry (Pythagoras)
VI. Progress in the Elements down to Plato's time ("the formative stage in which proofs were discovered and the logical bases of the science were beginning to be sought"[6])
VII. Special problems ("three famous problems" of antiquity[6])
VIII. Zeno of Elea
IX. Plato
X. From Plato to Euclid (Eudoxus and Aristotle)
XI. Euclid
XII. Aristarchus of Samos
XIII. Archimedes
XIV. Conic Sections: Apollonius of Perga
XV. The successors of the great geometers (Nicomedes, Diocles, Perseus, Zenodorus, Hypsicles, Dionysodorus, Posidonius, Geminus)
XVI. Some handbooks (Cleomedes, Nicomachus, and Theon of Smyrna)
XVII. Trigonometry: Hipparchus, Menelaus, Ptolemy
XVIII. Mensuration: Heron of Alexandria
XIX. Pappus of Alexandria
XX. Algebra: Diophantus of Alexandria
XXI. Commentators and Byzantines (Serenus, Theon of Alexandria, Proclus, Hypatia, Porphyry, Iamblichus, Marinus of Neapolis, Domninus of Larissa, Simplicius, Eutocius, Anthemius of Tralles, Hero the Younger, Michael Psellus, Georgius Pachymeres, Maximus Planudes, Manuel Moschopoulos, Nicholas Rhabdas, John Pediasimos, Barlaam of Seminara, Isaac Argyrus)
Reception
The book got positive reviews. Mathematician David Eugene Smith praised the book, writing in 1923 that "no man now living is more capable than he of interpreting the Greek mathematical mind to the scholar of today; indeed, there is no one who ranks even in the same class with Sir Thomas Heath in this particular". He also noted that Heath wrote in length about "five of the greatest names in the field of ancient mathematical research" (Euclid, Archimedes, Apollonius, Pappus, and Diophantus), given "each approximately a hundred pages". He called the book "destined to be the standard work".[6]
Philosopher John Alexander Smith wrote in 1923 that the book "has the eminent merit of being readable", and that "for most scholars the work is full and detailed enough to form almost a library of reference".[7]
Another reviewer from 1923 wrote that "covering as it does so much ground, it is not surprising that the book shows signs of ruthless compression".[8] The author was praised for the book, with one reviewer writing "In Sir Thomas Heath we have, as Erasmus said of Tunstall, a scholar who is dictus ad unguem".[2]
Historian of science George Sarton also praised the book in his 1922 review, writing that "it seems hardly necessary to speak at great length of a book of which most scholars knew long before it appeared, for few books have been awaited with greater impatience". He also noted careful explanation of solutions written in modern language, and "perfect clearness of the exposition, its excellent order, its thoroughness".[9]
The Manual, concise version of History, also received positive reviews. It was called a "fascinating little book", "a mine of information, a delight to read".[10] Sarton criticized the book because of the absence of chapters devoted to Egyptian and Mesopotamian mathematics.[5] Herbert Turnbull praised the book, especially its treatment of new discoveries of Egyptian and Babylonian mathematics.[11]
Mathematician Howard Eves praised the book in his 1984 review, writing that "the English-speaking population is particularly fortunate in having available the extraordinary treatise ... one finds one of the most scholarly, most complete, and most charmingly written treatments of the subject, a treatment certain to kindle a deep appreciation of that early period of mathematical development and a genuine admiration of those who played leading roles in it."[12]
Fernando Q. Gouvêa, writing in 2006, criticizes Heath's books as outdated and old-fashioned.[4][13]
Benjamin Wardhaugh, writing in 2016, finds that Heath's approach to Greek mathematics is to "made them look like works of classic literature", and that "what Heath constructed might be characterized today as a history of the contents of Greek theoretical mathematics."[1] Reviel Netz in his 2022 book calls Heath's History "a reliable guide to many generations of scholars and curious readers". He writes that "Historiographies went in and out of fashion, but Heath still stands, providing a clear and readable survey of the contents of most of the works of pure mathematics attested from Greek antiquity." He has also noted that there was no other book on the subject written in a hundred years.[14]
Publication history
• A History of Greek Mathematics, Oxford, Clarendon Press. 1921. Volume I, From Thales to Euclid, Volume II, From Aristarchus to Diophantus
• A History of Greek Mathematics. New York: Dover Publications. 1981. ISBN 978-0-486-24073-2. Volume I, From Thales to Euclid, Volume II, From Aristarchus to Diophantus
• A History of Greek Mathematics. Cambridge University Press. 2013. ISBN 978-1-108-06306-7.
• A Manual of Greek Mathematics, Oxford, Clarendon Press. 1931.
• A Manual of Greek Mathematics. Mineola, NY: Dover Publications. 2003. ISBN 978-0486432311.
References
1. Wardhaugh, Benjamin (2016). "Greek Mathematics in English: The Work of Sir Thomas L. Heath (1861–1940)". Historiography of Mathematics in the 19th and 20th Centuries: 109–122. doi:10.1007/978-3-319-39649-1_6.
2. "Review of A History of Greek Mathematics". The Mathematical Gazette. 11 (165): 348–351. 1923. doi:10.2307/3602335. ISSN 0025-5572. Retrieved 30 May 2023.
3. Sanford, Vera (November 1931). "Review: Thomas L. Heath, A Manual of Greek Mathematics". Bulletin of the American Mathematical Society. 37 (11): 805–805. ISSN 0002-9904. Retrieved 2 June 2023.
4. Gouvêa, Fernando Q. "A Manual of Greek Mathematics". www.maa.org. Mathematical Association of America. Retrieved 2 June 2023.
5. Sarton, George (November 1931). "A Manual of Greek Mathematics. Thomas L. Heath". Isis. 16 (2): 450–451. doi:10.1086/346620. Retrieved 2 June 2023.
6. Smith, David Eugene (1923). "Heath on Greek Mathematics". Bull. Amer. Math. Soc. 29 (2): 79–84. doi:10.1090/s0002-9904-1923-03668-9.
7. Smith, J. A. (May 1923). "A History of Greek Mathematics - A History of Greek Mathematics. By Sir Thomas Heath. Clarendon Press, Oxford, 1921. Two vols. 50s. net". The Classical Review. 37 (3–4): 69–71. doi:10.1017/S0009840X0004169X. ISSN 1464-3561.
8. "A History of Greek Mathematics. By Sir Thomas Heath. 2 Vols., pp. xv + 446, xi + 586. Oxford: The Clarendon Press, 1921. £2 10s". The Journal of Hellenic Studies. 43 (1): 81–82. January 1923. doi:10.2307/625884. ISSN 2041-4099. Retrieved 30 May 2023.
9. Sarton, George (1922). "A History of Greek Mathematics by Thomas Heath". Isis. 4: 532–535. doi:10.1086/358094. Retrieved 30 May 2023.
10. B, T. a. A. (October 1931). "A Manual of Greek Mathematics. By Sir Thomas Heath, K.C.B., K.C.V.O., F.R.S. Pp. xvi+552. 15s. 1931. (Clarendon Press.)". The Mathematical Gazette. 15 (215): 476–476. doi:10.2307/3606228. ISSN 0025-5572. Retrieved 2 June 2023.
11. Turnbull, H. W. (October 1931). "A Manual of Greek Mathematics". Nature. 128 (3235): 739–740. doi:10.1038/128739a0. Retrieved 2 June 2023.
12. Eves, Howard (January 1984). "A History of Greek Mathematics (2 vols.). By Sir Thomas Heath". The American Mathematical Monthly. 91 (1): 62–64. doi:10.1080/00029890.1984.11971341. ISSN 0002-9890. Retrieved 30 May 2023.
13. Gouvêa, Fernando Q. "Ancient Mathematics". www.maa.org. Mathematical Association of America. Retrieved 2 June 2023.
14. Netz, Reviel (2022). A new history of Greek mathematics. Cambridge, United Kingdom: Cambridge University Press. pp. Preface. ISBN 978-1-108-83384-4.
Ancient Greek mathematics
Mathematicians
(timeline)
• Anaxagoras
• Anthemius
• Archytas
• Aristaeus the Elder
• Aristarchus
• Aristotle
• Apollonius
• Archimedes
• Autolycus
• Bion
• Bryson
• Callippus
• Carpus
• Chrysippus
• Cleomedes
• Conon
• Ctesibius
• Democritus
• Dicaearchus
• Diocles
• Diophantus
• Dinostratus
• Dionysodorus
• Domninus
• Eratosthenes
• Eudemus
• Euclid
• Eudoxus
• Eutocius
• Geminus
• Heliodorus
• Heron
• Hipparchus
• Hippasus
• Hippias
• Hippocrates
• Hypatia
• Hypsicles
• Isidore of Miletus
• Leon
• Marinus
• Menaechmus
• Menelaus
• Metrodorus
• Nicomachus
• Nicomedes
• Nicoteles
• Oenopides
• Pappus
• Perseus
• Philolaus
• Philon
• Philonides
• Plato
• Porphyry
• Posidonius
• Proclus
• Ptolemy
• Pythagoras
• Serenus
• Simplicius
• Sosigenes
• Sporus
• Thales
• Theaetetus
• Theano
• Theodorus
• Theodosius
• Theon of Alexandria
• Theon of Smyrna
• Thymaridas
• Xenocrates
• Zeno of Elea
• Zeno of Sidon
• Zenodorus
Treatises
• Almagest
• Archimedes Palimpsest
• Arithmetica
• Conics (Apollonius)
• Catoptrics
• Data (Euclid)
• Elements (Euclid)
• Measurement of a Circle
• On Conoids and Spheroids
• On the Sizes and Distances (Aristarchus)
• On Sizes and Distances (Hipparchus)
• On the Moving Sphere (Autolycus)
• Optics (Euclid)
• On Spirals
• On the Sphere and Cylinder
• Ostomachion
• Planisphaerium
• Sphaerics
• The Quadrature of the Parabola
• The Sand Reckoner
Problems
• Constructible numbers
• Angle trisection
• Doubling the cube
• Squaring the circle
• Problem of Apollonius
Concepts
and definitions
• Angle
• Central
• Inscribed
• Axiomatic system
• Axiom
• Chord
• Circles of Apollonius
• Apollonian circles
• Apollonian gasket
• Circumscribed circle
• Commensurability
• Diophantine equation
• Doctrine of proportionality
• Euclidean geometry
• Golden ratio
• Greek numerals
• Incircle and excircles of a triangle
• Method of exhaustion
• Parallel postulate
• Platonic solid
• Lune of Hippocrates
• Quadratrix of Hippias
• Regular polygon
• Straightedge and compass construction
• Triangle center
Results
In Elements
• Angle bisector theorem
• Exterior angle theorem
• Euclidean algorithm
• Euclid's theorem
• Geometric mean theorem
• Greek geometric algebra
• Hinge theorem
• Inscribed angle theorem
• Intercept theorem
• Intersecting chords theorem
• Intersecting secants theorem
• Law of cosines
• Pons asinorum
• Pythagorean theorem
• Tangent-secant theorem
• Thales's theorem
• Theorem of the gnomon
Apollonius
• Apollonius's theorem
Other
• Aristarchus's inequality
• Crossbar theorem
• Heron's formula
• Irrational numbers
• Law of sines
• Menelaus's theorem
• Pappus's area theorem
• Problem II.8 of Arithmetica
• Ptolemy's inequality
• Ptolemy's table of chords
• Ptolemy's theorem
• Spiral of Theodorus
Centers
• Cyrene
• Mouseion of Alexandria
• Platonic Academy
Related
• Ancient Greek astronomy
• Attic numerals
• Greek numerals
• Latin translations of the 12th century
• Non-Euclidean geometry
• Philosophy of mathematics
• Neusis construction
History of
• A History of Greek Mathematics
• by Thomas Heath
• algebra
• timeline
• arithmetic
• timeline
• calculus
• timeline
• geometry
• timeline
• logic
• timeline
• mathematics
• timeline
• numbers
• prehistoric counting
• numeral systems
• list
Other cultures
• Arabian/Islamic
• Babylonian
• Chinese
• Egyptian
• Incan
• Indian
• Japanese
Ancient Greece portal • Mathematics portal
| Wikipedia |
A Passage to Infinity
A Passage to Infinity: Medieval Indian Mathematics from Kerala and Its Impact [1][2][3] is a 2009 book by George Gheverghese Joseph chronicling the social and mathematical origins of the Kerala school of astronomy and mathematics. The book discusses the highlights of the achievements of Kerala school and also analyses the hypotheses and conjectures on the possible transmission of Kerala mathematics to Europe.
A Passage to Infinity: Medieval Indian Mathematics from Kerala and Its Impact
AuthorGeorge Gheverghese Joseph
LanguageEnglish
SubjectHistory of mathematics
PublisherSAGE India
Publication date
November 2009
Pages323
ISBN978-8132101680
An outline of the contents
1. Introduction
2. The Social Origins of the Kerala School
3. The Mathematical Origins of the Kerala School
4. The Highlights of Kerala Mathematics and Astronomy
5. Indian Trigonometry: From Ancient Beginnings to Nilakantha
6. Squaring the Circle: The Kerala Answer
7. Reaching for the Stars: The Power Series for Sines and Cosines
8. Changing Perspectives on Indian Mathematics
9. Exploring Transmissions: A Case Study of Kerala Mathematics
10. A Final Assessment
See also
• Indian astronomy
• Indian mathematics
• History of mathematics
References
1. Joseph, George Gheverghese (2009). A Passage to Infinity : Medieval Indian Mathematics from Kerala and Its Impact. Delhi: Sage Publications (Inda) Pvt. Ltd. p. 236. ISBN 978-81-321-0168-0.
2. Plofker, Kim (21 December 2015). "A Passage to Infinity: Medieval Indian Mathematics from Kerala and Its Impact". Aestimatio: Critical Reviews in the History of Science. 10: 56–62. doi:10.33137/aestimatio.v10i0.26020. ISSN 1549-4497.
3. Sriram, M.S. (2011). "Book Review: A Passage to Infinity—Medieval Indian Mathematics and Its Impact". Indian Historical Review. 38 (2): 247–250. doi:10.1177/037698361103800207. ISSN 0376-9836. S2CID 149427309.
Further references
• In association with the Royal Society's 350th anniversary celebrations in 2010, Asia House presented a talk based on A Passage to Infinity. See : "A Passage to Infinity: Indian Mathematics in World Mathematics". Retrieved 3 May 2010.
• For an audio-visual presentation of George Gheverghese Joseph's views on the ideas presented in the book, see : Joseph, George Gheverghese (16 September 2008). "George Gheverghese Joseph on the Transmission to Europe of Non-European Mathematics". The Mathematical Association of America. Archived from the original on 15 April 2010. Retrieved 3 May 2010.
• The Economic Times talks to George Gheverghese Joseph on The Passage to Infinity. See : Lal, Amrith (23 April 2010). "Indian mathematics loved numbers". The Economic Times.
• Review of "A PASSAGE TO INFINITY: Medieval Indian Mathematics from Kerala and its impact" by M. Ram Murty in Hardy-Ramanujan Journal, 36 (2013), 43–46.
• Nair, R. Madhavan (3 February 2011). "In search of the roots of mathematics". The Hindu. Retrieved 15 October 2014.
Scientific Research in Kerala
Pre 19th Century
• Achyuta Pisharati
• Candravakyas
• Citrabhanu
• Damodara
• Ganita-yukti-bhasa
• Govinda Bhattathiri
• Haridatta
• Jyā, koti-jyā and utkrama-jyā
• Jyeṣṭhadeva
• Jyotirmimamsa
• Karanapaddhati
• Katapayadi system
• Kriyakramakari
• Madhava of Sangamagrama
• Madhava series
• Madhava's sine table
• Melpathur Narayana Bhattathiri
• Nilakantha Somayaji
• Parameshvara
• Sadratnamala
• Śaṅkaranārāyaṇa
• Tantrasamgraha
Organizations
• C-DAC Thiruvananthapuram
• Centre for Development Studies
• Centre for Earth Science Studies
• Centre for Mathematical Sciences (Kerala)
• Centre for Rural Management
• Centre of Science and Technology for Rural Development
• Cyberpark
• Kerala Mathematical Association
• Kerala Science and Technology Museum
• Kerala Science Congress
• Kerala Soil Museum
• Kerala State Council for Science, Technology and Environment
• Krishi Vigyan Kendra Kannur
• Liquid Propulsion Systems Centre
• Rajiv Gandhi Centre for Biotechnology
• Regional Agricultural Research Station, Pattambi
• Rice Research Station, Moncombu
• Vikram Sarabhai Space Centre
Institutions
• Agricultural Research Station, Anakkayam
• Agricultural Research Station, Mannuthy
• Agronomic Research Station, Chalakudy
• Amrita Institute of Medical Sciences and Research Centre
• Aromatic and Medicinal Plants Research Station, Odakkali
• Banana Research Station, Kannara
• Cashew Research Station, Madakkathara
• Central Institute of Fisheries Technology
• Central Marine Fisheries Research Institute
• Central Tuber Crops Research Institute
• Centre for Marine Living Resources & Ecology
• College of Horticulture
• Crocodile Rehabilitation and Research Centre
• Institute of Handloom and Textile Technology
• Indian Institute of Science Education and Research, Thiruvananthapuram
• Indian Institute of Space Science and Technology
• Indian Institute of Spices Research
• Jubilee Mission Medical College and Research Institute
• Kerala Forest Research Institute
• Kerala School of Mathematics, Kozhikode
• Mar Athanasios College for Advanced Studies, Tiruvalla
• National Institute for Interdisciplinary Science and Technology
• National Institute of Speech and Hearing
• National Research Institute for Panchakarma
• Naval Physical and Oceanographic Laboratory
• Oriental Research Institute & Manuscripts Library
• Sree Chitra Tirunal Institute for Medical Sciences and Technology
• Srinivasa Ramanujan Institute of Basic Sciences
• St. Ephrem Ecumenical Research Institute
• Tropical Botanical Garden and Research Institute
• Vadakke Madham Brahmaswam Vedic Research Centre
Scientists
• Ajit Varki
• C. V. Subramanian
• Dileep George
• E. C. George Sudarshan
• G. Madhavan Nair
• George Varghese
• K. R. Ramanathan
• K. Radhakrishnan
• M. G. K. Menon
• Matthew Pothen Thekaekara
• P. N. Vinayachandran
• Pisharoth Rama Pisharoty
• Pulickel Ajayan
• R. S. Krishnan
• Sainudeen Pattazhy
• T. A. Venkitasubramanian
• Thanu Padmanabhan
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| Wikipedia |
A Primer of Real Functions
A Primer of Real Functions is a revised edition of a classic Carus Monograph on the theory of functions of a real variable. It is authored by R. P. Boas, Jr and updated by his son Harold P. Boas.[1]
A Primer of Real Functions
AuthorR. P. Boas, Jr, Harold P. Boas
CountryUnited States
LanguageEnglish
SeriesMathematical Association of America Textbooks
SubjectMathematics
PublisherAmerican Mathematical Society
Publication date
1960
Pages319
ISBN9780883850442
References
1. GáL, I. S. (1962). "Review: R. P. Boas, Jr., A primer of real functions". Bulletin of the American Mathematical Society. 68 (1): 10–12. doi:10.1090/s0002-9904-1962-10672-7. ISSN 0002-9904. Retrieved 28 March 2018.
| Wikipedia |
A Topological Picturebook
A Topological Picturebook is a book on mathematical visualization in low-dimensional topology by George K. Francis. It was originally published by Springer in 1987, and reprinted in paperback in 2007. The Basic Library List Committee of the Mathematical Association of America has recommended its inclusion in undergraduate mathematics libraries.[1]
A Topological Picturebook
AuthorGeorge K. Francis
PublisherSpringer
Publication date
1987
ISBN0387964266
Topics
Although the book includes some computer-generated images,[2] most of it is centered on hand drawing techniques.[1] After an introductory chapter on topological surfaces, the cusps in the outlines of surfaces formed when viewing them from certain angles, and the self-intersections of immersed surfaces, the next two chapters are centered on drawing techniques: chapter two concerns ink, paper, cross-hatching, and shading techniques for indicating the curvature of surfaces, while chapter three provides some basic techniques of graphical perspective.[3]
The remaining five chapters of the book provide case studies of different visualization problems in mathematics, called by the book "picture stories".[4][5] The mathematical topics visualized in these chapters include the Penrose triangle and related optical illusions; the Roman surface and Boy's surface, two different immersions of the projective plane, and deformations between them; sphere eversion and the Morin surface; group theory, the mapping class groups of surfaces, and the braid groups; and knot theory, Seifert surfaces, the Hopf fibration of space by linked circles, and the construction of knot complements by gluing polyhedra.[3][4]
Audience and reception
Reviewer Athanase Papadopoulos calls the book "a drawing manual for mathematicians".[3] However, reviewer Dave Auckly disagrees, writing that, although the book explains the principles of Francis's own visualizations, it is not really a practical guide to constructing visualizations more generally. Auckly also calls the chapter on perspective "a bizarre mix of mathematical formulas and artistic constructions". Nevertheless, he reviews it positively as "mathematics book loaded with pictures", aimed at undergraduates interested in mathematics.[4]
More generally, Bill Satzer suggests that the book can provide inspiration for other mathematical illustrators, and for how mathematics is taught and imagined,[1] and Dušan Repovš sees the book as an encouragement to professional mathematicians to more heavily illustrate their work.[6] Jeffrey Weeks sees the book as an embodiment of the principle that abstract mathematical results can often be best appreciated through concrete examples.[5] Thomas Banchoff writes that most readers from a general audience will be "captivated" by the intricate artworks of the book, and professional mathematicians will find sufficient depth in its explanation of these works.[2] However, Weeks writes that the book fails at another stated purpose, allowing artists to appreciate the mathematics behind the artworks it presents, because the mathematics is too advanced for easy understanding by a general audience.[5]
References
1. Satzer, William J. (December 2006), "Review of A Topological Picturebook (reprint)", MAA Reviews, Mathematical Association of America
2. Banchoff, Thomas (January–February 1991), "Review of A Topological Picturebook", American Scientist, 79 (1): 85–86, JSTOR 29774302
3. Papadopoulos, Athanase, "Review of A Topological Picturebook (reprint)", zbMATH, Zbl 1105.57001
4. Auckly, Dave (1988), "Review of A Topological Picturebook", Mathematical Reviews, MR 0880519
5. Weeks, Jeffrey R. (December 1988), American Mathematical Monthly, 95 (10): 970–974, doi:10.2307/2322408, JSTOR 2322408{{citation}}: CS1 maint: untitled periodical (link)
6. Repovš, D., "Review of A Topological Picturebook", zbMATH, Zbl 0612.57001
| Wikipedia |
A Treatise on the Circle and the Sphere
A Treatise on the Circle and the Sphere is a mathematics book on circles, spheres, and inversive geometry. It was written by Julian Coolidge, and published by the Clarendon Press in 1916.[1][2][3][4] The Chelsea Publishing Company published a corrected reprint in 1971,[5][6] and after the American Mathematical Society acquired Chelsea Publishing it was reprinted again in 1997.[7]
Topics
As is now standard in inversive geometry, the book extends the Euclidean plane to its one-point compactification, and considers Euclidean lines to be a degenerate case of circles, passing through the point at infinity. It identifies every circle with the inversion through it, and studies circle inversions as a group, the group of Möbius transformations of the extended plane. Another key tool used by the book are the "tetracyclic coordinates" of a circle, quadruples of complex numbers $a,b,c,d$ describing the circle in the complex plane as the solutions to the equation $az{\bar {z}}+bz+c{\bar {z}}+d=0$. It applies similar methods in three dimensions to identify spheres (and planes as degenerate spheres) with the inversions through them, and to coordinatize spheres by "pentacyclic coordinates".[7]
Other topics described in the book include:
• Tangent circles[2][3] and pencils of circles[3]
• Steiner chains, rings of circles tangent to two given circles[4]
• Ptolemy's theorem on the sides and diagonals of quadrilaterals inscribed in circles[4]
• Triangle geometry, and circles associated with triangles, including the nine-point circle, Brocard circle, and Lemoine circle[1][2][3]
• The Problem of Apollonius on constructing a circle tangent to three given circles, and the Malfatti problem of constructing three mutually-tangent circles, each tangent to two sides of a given triangle[1][3]
• The work of Wilhelm Fiedler on "cyclography", constructions involving circles and spheres[1][3]
• The Mohr–Mascheroni theorem, that in straightedge and compass constructions, it is possible to use only the compass[1]
• Laguerre transformations, analogues of Möbius transformations for oriented projective geometry[1][3]
• Dupin cyclides, shapes obtained from cylinders and tori by inversion[3]
Legacy
At the time of its original publication this book was called encyclopedic,[2][3] and "likely to become and remain the standard for a long period".[2] It has since been called a classic,[5][7] in part because of its unification of aspects of the subject previously studied separately in synthetic geometry, analytic geometry, projective geometry, and differential geometry.[5] At the time of its 1971 reprint, it was still considered "one of the most complete publications on the circle and the sphere", and "an excellent reference".[6]
References
1. Bieberbach, Ludwig, "Review of A Treatise on the Circle and the Sphere (1916 edition)", Jahrbuch über die Fortschritte der Mathematik, JFM 46.0921.02
2. H. P. H. (December 1916), "Review of A Treatise on the Circle and the Sphere (1916 edition)", The Mathematical Gazette, 8 (126): 338–339, doi:10.2307/3602790, hdl:2027/coo1.ark:/13960/t39z9q113, JSTOR 3602790
3. Emch, Arnold (June 1917), "Review of A Treatise on the Circle and the Sphere (1916 edition)", The American Mathematical Monthly, 24 (6): 276–279, doi:10.1080/00029890.1917.11998325, JSTOR 2973184
4. White, H. S. (July 1919), "Circle and sphere geometry (Review of A Treatise on the Circle and the Sphere)", Bulletin of the American Mathematical Society, American Mathematical Society ({AMS}), 25 (10): 464–468, doi:10.1090/s0002-9904-1919-03230-3
5. "Review of A Treatise on the Circle and the Sphere (1971 reprint)", Mathematical Reviews, MR 0389515
6. Peak, Philip (May 1974), "Review of A Treatise on the Circle and the Sphere (1971 reprint)", The Mathematics Teacher, 67 (5): 445, JSTOR 27959760
7. Steinke, G. F., "Review of A Treatise on the Circle and the Sphere (1997 reprint)", zbMATH, Zbl 0913.51004
External links
• A Treatise on the Circle and the Sphere (1916 edition) at the Internet Archive
| Wikipedia |
A-group
In mathematics, in the area of abstract algebra known as group theory, an A-group is a type of group that is similar to abelian groups. The groups were first studied in the 1940s by Philip Hall, and are still studied today. A great deal is known about their structure.
Definition
An A-group is a finite group with the property that all of its Sylow subgroups are abelian.
History
The term A-group was probably first used in (Hall 1940, Sec. 9), where attention was restricted to soluble A-groups. Hall's presentation was rather brief without proofs, but his remarks were soon expanded with proofs in (Taunt 1949). The representation theory of A-groups was studied in (Itô 1952). Carter then published an important relationship between Carter subgroups and Hall's work in (Carter 1962). The work of Hall, Taunt, and Carter was presented in textbook form in (Huppert 1967). The focus on soluble A-groups broadened, with the classification of finite simple A-groups in (Walter 1969) which allowed generalizing Taunt's work to finite groups in (Broshi 1971). Interest in A-groups also broadened due to an important relationship to varieties of groups discussed in (Ol'šanskiĭ 1969). Modern interest in A-groups was renewed when new enumeration techniques enabled tight asymptotic bounds on the number of distinct isomorphism classes of A-groups in (Venkataraman 1997).
Properties
The following can be said about A-groups:
• Every subgroup, quotient group, and direct product of A-groups are A-groups.
• Every finite abelian group is an A-group.
• A finite nilpotent group is an A-group if and only if it is abelian.
• The symmetric group on three points is an A-group that is not abelian.
• Every group of cube-free order is an A-group.
• The derived length of an A-group can be arbitrarily large, but no larger than the number of distinct prime divisors of the order, stated in (Hall 1940), and presented in textbook form as (Huppert 1967, Kap. VI, Satz 14.16).
• The lower nilpotent series coincides with the derived series (Hall 1940).
• A soluble A-group has a unique maximal abelian normal subgroup (Hall 1940).
• The Fitting subgroup of a solvable A-group is equal to the direct product of the centers of the terms of the derived series, first stated in (Hall 1940), then proven in (Taunt 1949), and presented in textbook form in (Huppert 1967, Kap. VI, Satz 14.8).
• A non-abelian finite simple group is an A-group if and only if it is isomorphic to the first Janko group or to PSL(2,q) where q > 3 and either q = 2n or q ≡ 3,5 mod 8, as shown in (Walter 1969).
• All the groups in the variety generated by a finite group are finitely approximable if and only if that group is an A-group, as shown in (Ol'šanskiĭ 1969).
• Like Z-groups, whose Sylow subgroups are cyclic, A-groups can be easier to study than general finite groups because of the restrictions on the local structure. For instance, a more precise enumeration of soluble A-groups was found after an enumeration of soluble groups with fixed, but arbitrary Sylow subgroups (Venkataraman 1997). A more leisurely exposition is given in (Blackburn, Neumann & Venkataraman 2007, Ch. 12).
References
• Blackburn, Simon R.; Neumann, Peter M.; Venkataraman, Geetha (2007), Enumeration of finite groups, Cambridge Tracts in Mathematics no 173 (1st ed.), Cambridge University Press, ISBN 978-0-521-88217-0, OCLC 154682311
• Broshi, Aviad M. (1971), "Finite groups whose Sylow subgroups are abelian", Journal of Algebra, 17: 74–82, doi:10.1016/0021-8693(71)90044-5, ISSN 0021-8693, MR 0269741
• Carter, Roger W. (1962), "Nilpotent self-normalizing subgroups and system normalizers", Proceedings of the London Mathematical Society, Third Series, 12: 535–563, doi:10.1112/plms/s3-12.1.535, MR 0140570
• Hall, Philip (1940), "The construction of soluble groups", Journal für die reine und angewandte Mathematik, 182: 206–214, ISSN 0075-4102, MR 0002877
• Huppert, B. (1967), Endliche Gruppen (in German), Berlin, New York: Springer-Verlag, ISBN 978-3-540-03825-2, MR 0224703, OCLC 527050, especially Kap. VI, §14, p751–760
• Itô, Noboru (1952), "Note on A-groups", Nagoya Mathematical Journal, 4: 79–81, doi:10.1017/S0027763000023023, ISSN 0027-7630, MR 0047656
• Ol'šanskiĭ, A. Ju. (1969), "Varieties of finitely approximable groups", Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya (in Russian), 33 (4): 915–927, Bibcode:1969IzMat...3..867O, doi:10.1070/IM1969v003n04ABEH000807, ISSN 0373-2436, MR 0258927
• Taunt, D. R. (1949), "On A-groups", Proc. Cambridge Philos. Soc., 45 (1): 24–42, Bibcode:1949PCPS...45...24T, doi:10.1017/S0305004100000414, MR 0027759
• Venkataraman, Geetha (1997), "Enumeration of finite soluble groups with abelian Sylow subgroups", The Quarterly Journal of Mathematics, Second Series, 48 (189): 107–125, doi:10.1093/qmath/48.1.107, MR 1439702
• Walter, John H. (1969), "The characterization of finite groups with abelian Sylow 2-subgroups.", Annals of Mathematics, Second Series, 89 (3): 405–514, doi:10.2307/1970648, JSTOR 1970648, MR 0249504
| Wikipedia |
Ak singularity
In mathematics, and in particular singularity theory, an Ak singularity, where k ≥ 0 is an integer, describes a level of degeneracy of a function. The notation was introduced by V. I. Arnold.
Let $f:\mathbb {R} ^{n}\to \mathbb {R} $ be a smooth function. We denote by $\Omega (\mathbb {R} ^{n},\mathbb {R} )$ the infinite-dimensional space of all such functions. Let $\operatorname {diff} (\mathbb {R} ^{n})$ denote the infinite-dimensional Lie group of diffeomorphisms $\mathbb {R} ^{n}\to \mathbb {R} ^{n},$ and $\operatorname {diff} (\mathbb {R} )$ the infinite-dimensional Lie group of diffeomorphisms $\mathbb {R} \to \mathbb {R} .$ The product group $\operatorname {diff} (\mathbb {R} ^{n})\times \operatorname {diff} (\mathbb {R} )$ acts on $\Omega (\mathbb {R} ^{n},\mathbb {R} )$ in the following way: let $\varphi :\mathbb {R} ^{n}\to \mathbb {R} ^{n}$ :\mathbb {R} ^{n}\to \mathbb {R} ^{n}} and $\psi :\mathbb {R} \to \mathbb {R} $ :\mathbb {R} \to \mathbb {R} } be diffeomorphisms and $f:\mathbb {R} ^{n}\to \mathbb {R} $ any smooth function. We define the group action as follows:
$(\varphi ,\psi )\cdot f:=\psi \circ f\circ \varphi ^{-1}$
The orbit of f , denoted orb(f), of this group action is given by
${\mbox{orb}}(f)=\{\psi \circ f\circ \varphi ^{-1}:\varphi \in {\mbox{diff}}(\mathbb {R} ^{n}),\psi \in {\mbox{diff}}(\mathbb {R} )\}\ .$
The members of a given orbit of this action have the following fact in common: we can find a diffeomorphic change of coordinate in $\mathbb {R} ^{n}$ and a diffeomorphic change of coordinate in $\mathbb {R} $ such that one member of the orbit is carried to any other. A function f is said to have a type Ak-singularity if it lies in the orbit of
$f(x_{1},\ldots ,x_{n})=1+\varepsilon _{1}x_{1}^{2}+\cdots +\varepsilon _{n-1}x_{n-1}^{2}\pm x_{n}^{k+1}$
where $\varepsilon _{i}=\pm 1$ and k ≥ 0 is an integer.
By a normal form we mean a particularly simple representative of any given orbit. The above expressions for f give normal forms for the type Ak-singularities. The type Ak-singularities are special because they are amongst the simple singularities, this means that there are only a finite number of other orbits in a sufficiently small neighbourhood of the orbit of f.
This idea extends over the complex numbers where the normal forms are much simpler; for example: there is no need to distinguish εi = +1 from εi = −1.
References
• Arnold, V. I.; Varchenko, A. N.; Gusein-Zade, S. M. (1985), The Classification of Critical Points, Caustics and Wave Fronts: Singularities of Differentiable Maps, Vol 1, Birkhäuser, ISBN 0-8176-3187-9
| Wikipedia |
Proof that 22/7 exceeds π
Proofs of the mathematical result that the rational number 22/7 is greater than π (pi) date back to antiquity. One of these proofs, more recently developed but requiring only elementary techniques from calculus, has attracted attention in modern mathematics due to its mathematical elegance and its connections to the theory of Diophantine approximations. Stephen Lucas calls this proof "one of the more beautiful results related to approximating π".[1] Julian Havil ends a discussion of continued fraction approximations of π with the result, describing it as "impossible to resist mentioning" in that context.[2]
Part of a series of articles on the
mathematical constant π
3.1415926535897932384626433...
Uses
• Area of a circle
• Circumference
• Use in other formulae
Properties
• Irrationality
• Transcendence
Value
• Less than 22/7
• Approximations
• Madhava's correction term
• Memorization
People
• Archimedes
• Liu Hui
• Zu Chongzhi
• Aryabhata
• Madhava
• Jamshīd al-Kāshī
• Ludolph van Ceulen
• François Viète
• Seki Takakazu
• Takebe Kenko
• William Jones
• John Machin
• William Shanks
• Srinivasa Ramanujan
• John Wrench
• Chudnovsky brothers
• Yasumasa Kanada
History
• Chronology
• A History of Pi
In culture
• Indiana Pi Bill
• Pi Day
Related topics
• Squaring the circle
• Basel problem
• Six nines in π
• Other topics related to π
The purpose of the proof is not primarily to convince its readers that 22/7 (or 3+1/7) is indeed bigger than π; systematic methods of computing the value of π exist. If one knows that π is approximately 3.14159, then it trivially follows that π < 22/7, which is approximately 3.142857. But it takes much less work to show that π < 22/7 by the method used in this proof than to show that π is approximately 3.14159.
Background
22/7 is a widely used Diophantine approximation of π. It is a convergent in the simple continued fraction expansion of π. It is greater than π, as can be readily seen in the decimal expansions of these values:
${\begin{aligned}{\frac {22}{7}}&=3.{\overline {142\,857}},\\\pi \,&=3.141\,592\,65\ldots \end{aligned}}$
The approximation has been known since antiquity. Archimedes wrote the first known proof that 22/7 is an overestimate in the 3rd century BCE, although he may not have been the first to use that approximation. His proof proceeds by showing that 22/7 is greater than the ratio of the perimeter of a regular polygon with 96 sides to the diameter of a circle it circumscribes.[note 1]
The proof
The proof can be expressed very succinctly:
$0<\int _{0}^{1}{\frac {x^{4}\left(1-x\right)^{4}}{1+x^{2}}}\,dx={\frac {22}{7}}-\pi .$
Therefore, 22/7 > π.
The evaluation of this integral was the first problem in the 1968 Putnam Competition.[4] It is easier than most Putnam Competition problems, but the competition often features seemingly obscure problems that turn out to refer to something very familiar. This integral has also been used in the entrance examinations for the Indian Institutes of Technology.[5]
Details of evaluation of the integral
That the integral is positive follows from the fact that the integrand is non-negative; the denominator is positive and the numerator is a product of nonnegative numbers. One can also easily check that the integrand is strictly positive for at least one point in the range of integration, say at 1/2. Since the integrand is continuous at that point and nonnegative elsewhere, the integral from 0 to 1 must be strictly positive.
It remains to show that the integral in fact evaluates to the desired quantity:
${\begin{aligned}0&<\int _{0}^{1}{\frac {x^{4}\left(1-x\right)^{4}}{1+x^{2}}}\,dx\\[8pt]&=\int _{0}^{1}{\frac {x^{4}-4x^{5}+6x^{6}-4x^{7}+x^{8}}{1+x^{2}}}\,dx&{\text{expansion of terms in the numerator}}\\[8pt]&=\int _{0}^{1}\left(x^{6}-4x^{5}+5x^{4}-4x^{2}+4-{\frac {4}{1+x^{2}}}\right)\,dx&{\text{ using polynomial long division}}&\\[8pt]&=\left.\left({\frac {x^{7}}{7}}-{\frac {2x^{6}}{3}}+x^{5}-{\frac {4x^{3}}{3}}+4x-4\arctan {x}\right)\,\right|_{0}^{1}&{\text{definite integration}}\\[6pt]&={\frac {1}{7}}-{\frac {2}{3}}+1-{\frac {4}{3}}+4-\pi \quad &{\text{with }}\arctan(1)={\frac {\pi }{4}}{\text{ and }}\arctan(0)=0\\[8pt]&={\frac {22}{7}}-\pi .&{\text{addition}}\end{aligned}}$
(See polynomial long division.)
Quick upper and lower bounds
In Dalzell (1944), it is pointed out that if 1 is substituted for x in the denominator, one gets a lower bound on the integral, and if 0 is substituted for x in the denominator, one gets an upper bound:[6]
${\frac {1}{1260}}=\int _{0}^{1}{\frac {x^{4}\left(1-x\right)^{4}}{2}}\,dx<\int _{0}^{1}{\frac {x^{4}\left(1-x\right)^{4}}{1+x^{2}}}\,dx<\int _{0}^{1}{\frac {x^{4}\left(1-x\right)^{4}}{1}}\,dx={1 \over 630}.$
Thus we have
${\frac {22}{7}}-{\frac {1}{630}}<\pi <{\frac {22}{7}}-{\frac {1}{1260}},$
hence 3.1412 < π < 3.1421 in decimal expansion. The bounds deviate by less than 0.015% from π. See also Dalzell (1971).[7]
Proof that 355/113 exceeds π
As discussed in Lucas (2005), the well-known Diophantine approximation and far better upper estimate 355/113 for π follows from the relation
$0<\int _{0}^{1}{\frac {x^{8}\left(1-x\right)^{8}\left(25+816x^{2}\right)}{3164\left(1+x^{2}\right)}}\,dx={\frac {355}{113}}-\pi .$
${\frac {355}{113}}=3.141\,592\,92\ldots ,$
where the first six digits after the decimal point agree with those of π. Substituting 1 for x in the denominator, we get the lower bound
$\int _{0}^{1}{\frac {x^{8}\left(1-x\right)^{8}\left(25+816x^{2}\right)}{6328}}\,dx={\frac {911}{5\,261\,111\,856}}=0.000\,000\,173\ldots ,$
substituting 0 for x in the denominator, we get twice this value as an upper bound, hence
${\frac {355}{113}}-{\frac {911}{2\,630\,555\,928}}<\pi <{\frac {355}{113}}-{\frac {911}{5\,261\,111\,856}}\,.$
In decimal expansion, this means 3.141 592 57 < π < 3.141 592 74, where the bold digits of the lower and upper bound are those of π.
Extensions
The above ideas can be generalized to get better approximations of π; see also Backhouse (1995)[8] and Lucas (2005) (in both references, however, no calculations are given). For explicit calculations, consider, for every integer n ≥ 1,
${\frac {1}{2^{2n-1}}}\int _{0}^{1}x^{4n}(1-x)^{4n}\,dx<{\frac {1}{2^{2n-2}}}\int _{0}^{1}{\frac {x^{4n}(1-x)^{4n}}{1+x^{2}}}\,dx<{\frac {1}{2^{2n-2}}}\int _{0}^{1}x^{4n}(1-x)^{4n}\,dx,$
where the middle integral evaluates to
${\begin{aligned}{\frac {1}{2^{2n-2}}}&\int _{0}^{1}{\frac {x^{4n}(1-x)^{4n}}{1+x^{2}}}\,dx\\[6pt]={}&\sum _{j=0}^{2n-1}{\frac {(-1)^{j}}{2^{2n-j-2}(8n-j-1){\binom {8n-j-2}{4n+j}}}}+(-1)^{n}\left(\pi -4\sum _{j=0}^{3n-1}{\frac {(-1)^{j}}{2j+1}}\right)\end{aligned}}$
involving π. The last sum also appears in Leibniz' formula for π. The correction term and error bound is given by
${\begin{aligned}{\frac {1}{2^{2n-1}}}\int _{0}^{1}x^{4n}(1-x)^{4n}\,dx&={\frac {1}{2^{2n-1}(8n+1){\binom {8n}{4n}}}}\\[6pt]&\sim {\frac {\sqrt {\pi n}}{2^{10n-2}(8n+1)}},\end{aligned}}$
where the approximation (the tilde means that the quotient of both sides tends to one for large n) of the central binomial coefficient follows from Stirling's formula and shows the fast convergence of the integrals to π.
Calculation of these integrals: For all integers k ≥ 0 and ℓ ≥ 2 we have
${\begin{aligned}x^{k}(1-x)^{\ell }&=(1-2x+x^{2})x^{k}(1-x)^{\ell -2}\\[6pt]&=(1+x^{2})\,x^{k}(1-x)^{\ell -2}-2x^{k+1}(1-x)^{\ell -2}.\end{aligned}}$
Applying this formula recursively 2n times yields
$x^{4n}(1-x)^{4n}=\left(1+x^{2}\right)\sum _{j=0}^{2n-1}(-2)^{j}x^{4n+j}(1-x)^{4n-2(j+1)}+(-2)^{2n}x^{6n}.$
Furthermore,
${\begin{aligned}x^{6n}-(-1)^{3n}&=\sum _{j=1}^{3n}(-1)^{3n-j}x^{2j}-\sum _{j=0}^{3n-1}(-1)^{3n-j}x^{2j}\\[6pt]&=\sum _{j=0}^{3n-1}\left((-1)^{3n-(j+1)}x^{2(j+1)}-(-1)^{3n-j}x^{2j}\right)\\[6pt]&=-(1+x^{2})\sum _{j=0}^{3n-1}(-1)^{3n-j}x^{2j},\end{aligned}}$
where the first equality holds, because the terms for 1 ≤ j ≤ 3n – 1 cancel, and the second equality arises from the index shift j → j + 1 in the first sum.
Application of these two results gives
${\begin{aligned}{\frac {x^{4n}(1-x)^{4n}}{2^{2n-2}(1+x^{2})}}=\sum _{j=0}^{2n-1}&{\frac {(-1)^{j}}{2^{2n-j-2}}}x^{4n+j}(1-x)^{4n-2j-2}\\[6pt]&{}-4\sum _{j=0}^{3n-1}(-1)^{3n-j}x^{2j}+(-1)^{3n}{\frac {4}{1+x^{2}}}.\qquad (1)\end{aligned}}$
For integers k, ℓ ≥ 0, using integration by parts ℓ times, we obtain
${\begin{aligned}\int _{0}^{1}x^{k}(1-x)^{\ell }\,dx&={\frac {\ell }{k+1}}\int _{0}^{1}x^{k+1}(1-x)^{\ell -1}\,dx\\[6pt]&\,\,\,\vdots \\[6pt]&={\frac {\ell }{k+1}}{\frac {\ell -1}{k+2}}\cdots {\frac {1}{k+\ell }}\int _{0}^{1}x^{k+\ell }\,dx\\[6pt]&={\frac {1}{(k+\ell +1){\binom {k+\ell }{k}}}}.\qquad (2)\end{aligned}}$
Setting k = ℓ = 4n, we obtain
$\int _{0}^{1}x^{4n}(1-x)^{4n}\,dx={\frac {1}{(8n+1){\binom {8n}{4n}}}}.$
Integrating equation (1) from 0 to 1 using equation (2) and arctan(1) = π/4, we get the claimed equation involving π.
The results for n = 1 are given above. For n = 2 we get
${\frac {1}{4}}\int _{0}^{1}{\frac {x^{8}(1-x)^{8}}{1+x^{2}}}\,dx=\pi -{\frac {47\,171}{15\,015}}$
and
${\frac {1}{8}}\int _{0}^{1}x^{8}(1-x)^{8}\,dx={\frac {1}{1\,750\,320}},$
hence 3.141 592 31 < π < 3.141 592 89, where the bold digits of the lower and upper bound are those of π. Similarly for n = 3,
${\frac {1}{16}}\int _{0}^{1}{\frac {x^{12}\left(1-x\right)^{12}}{1+x^{2}}}\,dx={\frac {431\,302\,721}{137\,287\,920}}-\pi $
with correction term and error bound
${\frac {1}{32}}\int _{0}^{1}x^{12}(1-x)^{12}\,dx={\frac {1}{2\,163\,324\,800}},$
hence 3.141 592 653 40 < π < 3.141 592 653 87. The next step for n = 4 is
${\frac {1}{64}}\int _{0}^{1}{\frac {x^{16}(1-x)^{16}}{1+x^{2}}}\,dx=\pi -{\frac {741\,269\,838\,109}{235\,953\,517\,800}}$
with
${\frac {1}{128}}\int _{0}^{1}x^{16}(1-x)^{16}\,dx={\frac {1}{2\,538\,963\,567\,360}},$
which gives 3.141 592 653 589 55 < π < 3.141 592 653 589 96.
See also
• Approximations of π
• Chronology of computation of π
• Lindemann–Weierstrass theorem (proof that π is transcendental)
• List of topics related to π
• Proof that π is irrational
Footnotes
Notes
1. Proposition 3: The ratio of the circumference of any circle to its diameter is less than 3+1/7 but greater than 3+10/71.[3]
Citations
1. Lucas, Stephen (2005), "Integral proofs that 355/113 > π" (PDF), Australian Mathematical Society Gazette, 32 (4): 263–266, MR 2176249, Zbl 1181.11077
2. Havil, Julian (2003), Gamma. Exploring Euler's Constant, Princeton, NJ: Princeton University Press, p. 96, ISBN 0-691-09983-9, MR 1968276, Zbl 1023.11001
3. Archimedes (2002) [1897], "Measurement of a circle", in Heath, T.L. (ed.), The Works of Archimedes, Dover Publications, pp. 93–96, ISBN 0-486-42084-1
4. Alexanderson, Gerald L.; Klosinski, Leonard F.; Larson, Loren C., eds. (1985), The William Lowell Putnam Mathematical Competition: Problems and Solutions: 1965–1984, Washington, DC: The Mathematical Association of America, ISBN 0-88385-463-5, Zbl 0584.00003
5. 2010 IIT Joint Entrance Exam, question 41 on page 12 of the mathematics section.
6. Dalzell, D. P. (1944), "On 22/7", Journal of the London Mathematical Society, 19 (75 Part 3): 133–134, doi:10.1112/jlms/19.75_part_3.133, MR 0013425, Zbl 0060.15306.
7. Dalzell, D. P. (1971), "On 22/7 and 355/113", Eureka; the Archimedeans' Journal, 34: 10–13, ISSN 0071-2248.
8. Backhouse, Nigel (July 1995), "Note 79.36, Pancake functions and approximations to π", The Mathematical Gazette, 79 (485): 371–374, doi:10.2307/3618318, JSTOR 3618318, S2CID 126397479
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• The problems of the 1968 Putnam competition, with this proof listed as question A1.
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| Wikipedia |
Adriaan Cornelis Zaanen
Adriaan Cornelis "Aad" Zaanen (14 June 1913 in Rotterdam – 1 April 2003 in Wassenaar) was a Dutch mathematician working in analysis. He is known for his books on Riesz spaces (together with Wim Luxemburg).
Aad Zaanen
Zaanen in 1967
Born
Adriaan Cornelis Zaanen
14 June 1913
Rotterdam, Netherlands
Died1 April 2003 (2003-05) (aged 89)
Wassenaar, Netherlands
NationalityDutch
Alma materLeiden University
Known forContributions to the theory of Riesz spaces
SpouseAda van der Woude
AwardsMember of the Royal Netherlands Academy of Arts and Sciences (1960)
Knight of the Order of the Netherlands Lion (1982)
Honorary member of the Dutch Mathematical Society (1988)
Scientific career
FieldsFunctional analysis
InstitutionsBandung Institute of Technology
Delft University of Technology
Leiden University
Doctoral advisorJohannes Droste
Doctoral studentsW.A.J. Luxemburg, B.C. Strydom, M.A. Kaashoek, A.C. van Eijnsbergen, J.J. Grobler, N.A. van Arkel, C.B. Huijsmans, E. de Jonge, P. Maritz, W.J. Claas, A.R. Schep, W.K. Vietsch, B. de Pagter
Biography
Zaanen was born in Rotterdam, where he attended the Hogere Burgerschool. He graduated in 1930 with excellent marks, and started his studies in mathematics at Leiden University. Having obtained his master's degree in 1935, he did research under the guidance of his doctoral advisor Johannes Droste,[1] and was awarded a Ph.D. in 1938. His doctoral thesis dealt with the convergence of series of eigenvalues of boundary value problems of the Sturm–Liouville type.[2] The same year he was appointed a mathematics teacher at the Hogere Burgerschool in Rotterdam, a profession that he continued until 1947.[3]
In the next years and also in the period of the German occupation of the Netherlands, Zaanen continued to do mathematical research in his spare time. He studied Stefan Banach's Théorie des Opérations Linéaires, the book that laid the foundations of functional analysis, and Marshall H. Stone's Linear Transformations in Hilbert Space. During this period he wrote nine scientific papers on integral equations with symmetrisable kernels that were published in the Proceedings of the Royal Netherlands Academy of Arts and Sciences in 1946-47.[4]
In parallel to his job as a secondary-school teacher, Zaanen was appointed in 1946 as a mathematics teacher for three hours per week at the Technische Hogeschool Delft, and as an unpaid privaatdocent at Leiden University where he taught a course on Lebesgue integration.[4]
In 1947 Zaanen accepted the position of Professor of Mathematics at the Technische Hogeschool Bandoeng. In 1950 he returned to the Netherlands where he was appointed Professor of Mathematics at the Technische Hogeschool Delft. In these years he continued his work on the book Linear Analysis, which was published in 1953 and for years was a prominent work on functional analysis and the theory of integral equations.[5]
In 1956 Zaanen was appointed Professor of Mathematics at Leiden University. There he started a large research programme into the theory of Riesz spaces, together with his first doctoral student Wim Luxemburg, Professor of Mathematics at the California Institute of Technology. Most of their results were published in a series of papers in the Proceedings of the Royal Netherlands Academy of Arts and Sciences. Unusually for mathematics research in the Netherlands at the time, Zaanen pursued a long-term research programme involving a number of collaborators and doctoral students. Eight doctoral theses on various topics in the theory of Riesz spaces were produced in this school.[6]
Zaanen took retirement in 1982.
Publications
Zaanen published almost 70 papers in scientific journals and reviewed conference proceedings.[7] He is however best known for four large books that each took a prominent place in scientific literature:
• A.C. Zaanen, Linear Analysis, North-Holland Publishing Company, Amsterdam and P. Noordhoff, Groningen (1953, 1957, 1960), 600 pages
• Adriaan Cornelis Zaanen, Integration, North-Holland Publishing Company, Amsterdam (1967) 604 pages. Revised and enlarged edition of An Introduction to the Theory of Integration (1958, 1961, 1965).
• W.A.J. Luxemburg and A.C. Zaanen, Riesz Spaces Volume I, North-Holland Publishing Company, Amsterdam London (1971), 514 pages
• A.C. Zaanen, Riesz Spaces II, North-Holland Publishing Company, Amsterdam New York Oxford (1983), 720 pages
Other functions and honours
Zaanen was elected a member of the Royal Netherlands Academy of Arts and Sciences in 1960.[3]
He served as President of the Dutch Mathematical Society from 1970 until 1972. He was an editor of the Society's journal Nieuw Archief voor Wiskunde from 1953 until 1982. In 1988 he was appointed an honorary member of the Society.
He served as a member of the Curatorium of the Mathematisch Centrum from 1965 until 1979.
On his retirement in 1982 Zaanen was appointed Knight of the Order of the Netherlands Lion.[4]
References
1. Gerrit van Dijk (2011). Leidse hoogleraren Wiskunde 1575-1975. Mathematisch Instituut, Universiteit Leiden. pp. 56–57. ISBN 978-90-817201-1-3.
2. Adriaan Cornelis Zaanen in the Mathematics Genealogy Project. Retrieved 12 September 2013.
3. C.B. Huijsmans; M.A. Kaashoek; W.A.J. Luxemburg & W.K. Vietsch (1982). From A to Z. Proceedings of a Symposium in honour of A.C. Zaanen. Mathematisch Centrum. pp. 123–124. ISBN 90-6196-241-2.
4. Gerrit van Dijk (2011). Leidse hoogleraren Wiskunde 1575-1975. Mathematisch Instituut, Universiteit Leiden. pp. 63–64. ISBN 978-90-817201-1-3.
5. John J. O'Connor & Edmund F. Robertson. Adriaan Cornelis Zaanen in the MacTutor History of Mathematics archive. University of St. Andrews. Retrieved 12 September 2013.
6. C.B. Huijsmans; M.A. Kaashoek; W.A.J. Luxemburg & W.K. Vietsch (1982). From A to Z. Proceedings of a Symposium in honour of A.C. Zaanen. Mathematisch Centrum. p. 130. ISBN 90-6196-241-2.
7. C.B. Huijsmans; M.A. Kaashoek; W.A.J. Luxemburg & W.K. Vietsch (1982). From A to Z. Proceedings of a Symposium in honour of A.C. Zaanen. Mathematisch Centrum. pp. 125–129. ISBN 90-6196-241-2.
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| Wikipedia |
Aaron Galuten
Aaron Galuten (March 2, 1917–September 23, 1994)[1] was an American mathematician, known mainly as the founder and principal operator of the Chelsea Publishing Company.[2]
References
1. "Aaron Galuten (1917-1994)".
2. Halmos, Paul R. (1985). I Want to be a Mathematician: An Automathography. Springer Verlag. p. 159. ISBN 9781461210849.
| Wikipedia |
Aaron Robertson (mathematician)
Aaron Robertson (born November 8, 1971) is an American mathematician who specializes in Ramsey theory. He is a professor at Colgate University.[1]
Aaron Robertson
Born
Aaron Robertson
(1971-11-08) November 8, 1971
Torrance, California, U.S.
Alma mater
• University of Michigan (BS)
• Temple University (PhD)
Scientific career
Fields
• Mathematics
• Ramsey Theory
InstitutionsColgate University
ThesisSome New Results in Ramsey Theory (1999)
Doctoral advisorDoron Zeilberger
Life and education
Aaron Robertson was born in Torrance, California, and moved with his parents to Midland, Michigan at the age of 4. He studied actuarial science as an undergraduate at the University of Michigan, and went on to graduate school in mathematics at Temple University in Philadelphia, where he was supervised by Doron Zeilberger. Robertson received his Ph.D. in 1999 with his thesis titled Some New Results in Ramsey Theory.[2]
Following his Ph.D., Robertson became an assistant professor of mathematics at Colgate University, where he is currently a full professor.
Mathematical work
Robertson's work in mathematics since 1998 has consisted predominantly of topics related to Ramsey theory.
One of Robertson's earliest publications is a paper, co-authored with his supervisor Doron Zeilberger, which came out of his Ph.D. work. The authors prove that "the minimum number (asymptotically) of monochromatic Schur Triples that a 2-colouring of $[1,n]$ can have $ n^{2}/22+O(n)$".[2]
After completing his dissertation, Robertson worked with 3-term arithmetic progressions where he found the best-known values that were close to each other and titled this piece New Lower Bounds for Some Multicolored Ramsey Numbers.[3]
Another notable piece of Robertson's research is a paper co-authored with Doron Zeilberger and Herbert Wilf titled Permutation Patterns and Continued Fractions.[4] In the paper, they "find a generating function for the number of (132)-avoiding permutations that have a given number of (123) patterns"[4] with the result being "in the form of a continued fraction".[4] Robertson's contribution to this specific paper includes discussion on permutations that avoid a certain pattern but contain others.
A notable paper Robertson wrote titled A Probalistic Threshold For Monochromatic Arithmetic Progressions[5] explores the function $f_{r}(k)={\sqrt {k}}\cdot r^{k/2}$ (where $r\geq 2$ is fixed) and the r-colourings of $[1,n_{k}]=\{1,2,\ldots ,n_{k}\}$. Robertson analyzes the threshold function for $k$-term arithmetic progressions and improves the bounds found previously.
In 2004, Robertson and Bruce M. Landman published the book Ramsey Theory on the Integers, of which a second expanded edition appeared in 2014.[6] The book introduced new topics such as rainbow Ramsey theory, an “inequality” version of Schur's theorem, monochromatic solutions of recurrence relations, Ramsey results involving both sums and products, monochromatic sets avoiding certain differences, Ramsey properties for polynomial progressions, generalizations of the Erdős–Ginzberg–Ziv theorem, and the number of arithmetic progressions under arbitrary colourings.
More recently, in 2021, Robertson published a book titled Fundamentals of Ramsey Theory.[7] Robertson's goal in writing this book was to "help give an overview of Ramsey theory from several points of view, adding intuition and detailed proofs as we go, while being, hopefully, a bit gentler than most of the other books on Ramsey theory".[7] Throughout the book, Robertson discusses several theorems including Ramsey's Theorem, Van der Waerden's Theorem, Rado's Theorem, and Hales–Jewett Theorem.
References
1. "Aaron Robertson | Colgate University". www.colgate.edu. Retrieved 2021-10-17.
2. Robertson, Aaron; Zeilberger, Doron (1998-03-25). "A 2-Coloring of $[1,N]$ can have $(1/22)N^2+O(N)$ Monochromatic Schur Triples, but not less!". The Electronic Journal of Combinatorics. 5: R19. doi:10.37236/1357. ISSN 1077-8926.
3. Robertson, Aaron (1999). "New Lower Bounds for Some Multicolored Ramsey Numbers". The Electronic Journal of Combinatorics. 6: R3. doi:10.37236/1435. ISSN 1077-8926.
4. Robertson, Aaron; Wilf, Herbert S.; Zeilberger, Doron (1999-10-01). "Permutation Patterns and Continued Fractions". The Electronic Journal of Combinatorics. 6: R38. doi:10.37236/1470. ISSN 1077-8926.
5. Robertson, Aaron (2016-01-01). "A probabilistic threshold for monochromatic arithmetic progressions". Journal of Combinatorial Theory. Series A. 137: 79–87. doi:10.1016/j.jcta.2015.08.003. ISSN 0097-3165.
6. Landman, Bruce M.; Robertson, Aaron (2014). Ramsey Theory on the Integers. Vol. 73 (2nd ed.). The Student Mathematical Library. doi:10.1090/stml/073. ISBN 978-1-4704-2000-0.
7. Robertson, Aaron (2021-06-18). Fundamentals of Ramsey Theory. Boca Raton: Chapman and Hall/CRC. doi:10.1201/9780429431418. ISBN 978-0-429-43141-8. S2CID 234866085.
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| Wikipedia |
Mental abacus
The abacus system of mental calculation is a system where users mentally visualize an abacus to carry out arithmetical calculations.[1] No physical abacus is used; only the answers are written down. Calculations can be made at great speed in this way. For example, in the Flash Anzan event at the All Japan Soroban Championship, champion Takeo Sasano was able to add fifteen three-digit numbers in just 1.7 seconds.[2]
This system is being propagated in China,[3] Singapore, South Korea, Thailand, Malaysia, and Japan. Mental calculation is said to improve mental capability, increases speed of response, memory power, and concentration power.
Many veteran and prolific abacus users in China, Japan, South Korea, and others who use the abacus daily, naturally tend to not use the abacus any more, but perform calculations by visualizing the abacus. This was verified when the right brain of visualisers showed heightened EEG activity when calculating, compared with others using an actual abacus to perform calculations.
The abacus can be used routinely to perform addition, subtraction, multiplication, and division; it can also be used to extract square and cube[4] roots.
See also
• Abacus logic
• Abacus
References
1. "Research on the benefits of mental abacus for development". Retrieved March 12, 2012.
2. Alex Bellos (2012), "World's fastest number game wows spectators and scientists", The Guardian
3. "(Chinese)Teaching Kids Visit to use abacus for mental calculation". Archived from the original on February 13, 2017. Retrieved March 12, 2012.
4. Feynman, Richard (1985). "Lucky Numbers". Surely you're joking, Mr. Feynman!. New York: W.W. Norton. ISBN 0-393-31604-1. OCLC 10925248.
External links
• Mental abacus does away with words, New Scientist, August 9, 2011
• Ku Y, Hong B, Zhou W, Bodner M, Zhou YD (2012). "Sequential neural processes in abacus mental addition: an EEG and FMRI case study". PLOS ONE. 7 (5): e36410. Bibcode:2012PLoSO...736410K. doi:10.1371/journal.pone.0036410. PMC 3344852. PMID 22574155.
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| Wikipedia |
Saeid Abbasbandy
Saeid Abbasbandy is an Iranian mathematician[2] and university professor at Imam Khomeini International University. Abbasbandy was born on March 17, 1967, in Tehran. He finished his high school course in Shariati High school and attended in the university entrance exam, then could enter University of Tehran.
Saeid Abbasbandy
Born (1967-03-17) March 17, 1967
Tehran
NationalityIranian
Alma materUniversity for Teacher Education
Known forComputational Mathematics, Fuzzy Numerical Analysis,
AwardsCNSNS, top cited paper
AMM, top cited paper[1]
Scientific career
FieldsMathematician
InstitutionsImam Khomeini International University
Doctoral advisorEsmail Babolian
His paper "Homotopy analysis method for quadratic Riccati differential equation" was singled out by Science Watch as a "Hot Paper in Mathematics" in March 2009.[3] Abbasbandy is editor-in-chief of a non-profit journal, Communications in Numerical Analysis.
Education
He received a Master of Science degree in September 1991 and a PhD in February 1996 from Kharazmi University. The title of his thesis was Numerical Galerkin Methods for Integral Equations of the First kind.[4]
References
1. "Working Reports". Abbasbandy.com. Retrieved 2020-01-03.
2. Selected works
3. "Saeid Abbasbandy". ScienceWatch.com. 2009-02-27. Retrieved 2012-09-04.
4. "Education". Abbasbandy.com. Retrieved 2012-09-04.
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| Wikipedia |
Abdias Treu
Abdias Treu (sometimes spelled Trew) (29 July 1597 – 12 April 1669) was a German mathematician and academic. He was the professor of mathematics and physical science at the University of Altdorf from 1636-1669. He is best known for his contributions to the field of astronomy.[1] He also contributed writings on the mathematical nature of music theory.[2] He is the grandfather of physician and botanist Christoph Jacob Treu.[3]
References
1. Library, British (1975). The British Library Journal. Vol. 1–3. p. 62.
2. Paolo Gozza (2013). Number to Sound: The Musical Way to the Scientific Revolution. Springer Science & Business Media. p. 175. ISBN 9789401595780.
3. Martin Bircher (1997). Die Fruchtbringende Gesellschaft unter Herzog August von Sachsen-Weissenfels. p. 57. {{cite book}}: |work= ignored (help)
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| Wikipedia |
Abdul–Aziz Yakubu (mathematician)
Abdul–Aziz Yakubu (???? – August 2022)[1] was a mathematical biologist.[2] Yakubu was a professor at Howard University for over 20 years and served as chair of the mathematics department from 2004 to 2014.[3]
Abdul–Aziz Yakubu
Born????
Accra, Ghana
DiedAugust 13 or 14, 2022
Academic background
EducationUniversity of Ghana (BS 1982)
University of Toledo (MS 1985)
North Carolina State University (PhD 1990)
Academic work
DisciplineMathematics
Sub-disciplineMathematical biology
InstitutionsHoward University (home institution)
University of Minnesota
Cornell University
Early life and education
Abdul Aziz-Yakubu was born in the capital city of Ghana, Accra. He became interested in math while studying at the University of Ghana - Legon, where he earned his B.S in mathematics and computer science in 1982.[4] In 1985, Yakubu earned his master's degree in applied mathematics from the University of Toledo in Ohio.[4] Before continuing to Howard University, he attended North Carolina State University and received his doctoral degree in applied mathematics in 1990.[4][3] For his Ph.D., he wrote his dissertation, "Discrete time competitive systems" under the advisement of John Franke.[5]
Career
Yakubu’s first research experience was at North Carolina State University, and his first faculty position was held at Howard University, a historically black research university (HBCU) in Washington, D.C. While at Howard he spent a year as a long-term visitor by participating in “Mathematics in Biology” at the University of Minnesota. The program at the University of Minnesota allowed him to make connections to further his positive contributions to the world, especially his research on infectious diseases in Africa. Two years after his long-term visit, he took a two-year leave from Howard University in 2002 to visit the Department of Statistics and Computational Biology at Cornell University, where he collaborated with Carlos Castillo-Chavez.[4]
After his Cornell experience, he returned to Howard University to focus on mathematical biology. Yakubu had a successful collaboration in scholarly work on exploited fisheries with scientists at the Northeast Fisheries Science Center of Woods Hole, Massachusetts. He worked on projects that investigated biodiversity and infectious diseases with Avner Friedman of the Mathematical Bioscience Institute students at Ohio State University and his graduate students at Howard University. His research on mathematical biology helped him connect with students and researchers internationally. Yakubu has attended and presented his contributions at several research conferences and workshops in Europe and Asia. The National Science Foundation funded DIMACS-MBI Africa initiative, led by Avner Friedmann, Marty Golubitsky, Fred Roberts as well as the NSF-funded Masamu project of Overton Jenda. It made it possible for him to give several lectures in Cameroon, Ghana, Morocco, South Africa, Uganda, and Zambia. Yakubu left on sabbatical leave to MBI.DIMACS in Piscataway, New Jersey, MBI in Columbus, Ohio, NIMBioS in Knoxville, Tennessee, and similar mathematical biology institutes.[6]
Yakubu published his research in several academic journals like the Bulletin of Mathematical Biology, Journal of Mathematical Biology, Mathematical Biosciences, SIAM and Journal of Applied Mathematics.[2] He was a member and held leadership roles in professional mathematics organizations, including the Society for Mathematical Biology, Mathematical Bioscience Institute of the Ohio State and DIMACS of Rutgers University. From 2007 to 2016, he served as Chair of the World Outreach Committee of the Society for Mathematical Biology.[2]
Yakubu died on August 13 or 14, 2022.[3] As of November 2022, the editors of the Journal of Biological Dynamics are working on a special issue to be published in remembrance of Yakubu. The special edition will cover topics inspired by or related to Yakubu's work in mathematics and population biology.[1]
Building the next generation of scientists
Professor Yakubu has advised several students, among which a notable number of PhD students
Watch Yakubu as he advises prospective international students who aspire to study at the highest level of excellence.
Selected publications
• Mathematical Approaches For Emerging and Reemerging Infectious Diseases: Models, Methods and Theory, Springer-Verlag, Volume 125, Edited by Carlos Castillo-Chavez with Sally Blower, Pauline van den Driessche, Denise Kirschner and Abdul-Aziz Yakubu (2002)[7]
• Mathematical Approaches For Emerging and Reemerging Infectious Diseases: Models, Methods and Theory, Springer-Verlag, Volume 126, Edited by Carlos Castillo-Chavez with Sally Blower, Pauline van den Driessche, Denise Kirschner and Abdul-Aziz Yakubu (2002)[7]
References
1. Gumel, Abba; Henson, Shandelle (eds.). "Journal of Biological Dynamics: In memory of Abdul-Aziz Yakubu". Taylor and Francis. Archived from the original on 2022-11-28. Retrieved 2022-11-28.
2. "Abdul-Aziz Yakubu". Mathematically Gifted & Black. The Network of Minorities in Mathematical Sciences. Retrieved 2023-05-09.
3. Howard University Mourns the Loss of Abdul-Aziz Yakubu. National Association of Mathematicians Newsletter. 2022. Volume 53 (issue 2):15.
4. Parks, Clinton (2 Feb 2004). "Seeing the Forest for the Trees". Science. AAAS. Retrieved 2023-04-26.
5. "Abdul-Aziz Yakubu - Mathematician of the African Diaspora". www.math.buffalo.edu. Retrieved 2023-05-09.
6. "My Career in Mathematical Biology" (PDF).
7. "Abdul-Aziz Yakubu". scholar.google.com. Retrieved 2023-05-09.
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| Wikipedia |
Abdus Salam School of Mathematical Sciences
The Abdus Salam School of Mathematical Sciences (AS-SMS), (درس گاہ عبدالسلام برائے علوم ریاضی) is an autonomous institute affiliated with the Government College University Lahore, Pakistan.[1] The school is named after the theoretical physicist and Nobel Laureate Abdus Salam.
Abdus Salam School of Mathematical Sciences
TypeResearch Institute
Established2003 (2003)
AffiliationGovernment College University Lahore (GCU Lahore)
Address
68-B
,
New Muslim Town, Lahore
,
Punjab
,
Pakistan
31.513142°N 74.314912°E / 31.513142; 74.314912
Websitewww.sms.edu.pk
Location in Pakistan
History
Abdus Salam played an influential role in formulating policies for the development of science in Pakistan in the late 1960s and early 1970s. He helped establish a number of research institutes in Pakistan including the Pakistan Institute of Nuclear Science and Technology (PINSTECH) and the Atomic Energy Commission.
In 2002, the Government of Punjab established a center of excellence in mathematical sciences, the School of Mathematical Sciences, affiliated with Government College University Lahore.[2] The idea behind the school is to create a world-class doctoral research institute in mathematics, but rooted in a developing country like Pakistan.[3][4] The school was later named after Abdus Salam who was professor of mathematics at the Government College Lahore from 1951 to 1954. The school starts operating in late 2003. The school is ruled by a board of governors (BoG), with the Vice Chancellor of GCU Lahore as its chairman.[4]
In 2003, Alla Ditta Raza Choudary, an emeritus professor from Central Washington University, was appointed as the founding director general of the school. Using the financial support from the Higher Education Commission of Pakistan that at that time implemented the Foreign Faculty Hiring Program, as well as from the Punjab Government, he made the important decision to look for Faculty entirely in the international market.[5] Jürgen Herzog, Josip Pečarić,[6] Amer Iqbal, Gerhard Pfister, Ioan Tomescu, Tudor Zamfirescu, and Alexandru Dimca are among Foreign Faculty that have supervised Ph.D. students at AS-SMS in the tenure of A.D.R. Choudary.
A.D.R. Choudary held the position of director general from August 2003 to August 2014. The next director general of the school was Shahid Saeed Siddiqi, a retired professor of mathematics at the Punjab University, who held the position from March 2015 to March 2017. From March 2017 until early 2020, the caretaker director general of the school was Ghulam Murtaza who also held the Abdus Salam Chair in Physics at the Government College University Lahore.
Program overview
Initially, AS-SMS required an intensive two-year course for all the students admitted. The first year is dedicated to basic university mathematics courses, and in the second year, students take more advanced and optional courses in diverse areas of mathematics. At the end of their second year, students choose one of the research groups at AS-SMS, which are geometry, algebra, analysis, discrete mathematics, applied mathematics, and stochastic processes.[5] The graduation rate at the end of the fifth year was high.[7] Students that admitted after 2010 are required to write an MPhil thesis, and the school program formally became an MPhil leading to Ph.D. program.
On the recommendation of the HEC, starting in 2016 the school discontinue new admission for its MPhil leading to Ph.D. program, and admission to MPhil and Ph.D. programs now requires separate admission examinations. [8]
The institution provides free-of-cost education to its students. It also provides financial assistance to its students.[3]
Achievements
Since its inception until early 2014, AS-SMS has produced more than 100 Ph.D. This is more than twice the total number of Ph.D. in mathematics produced by all other universities in Pakistan in the same period.[9] It has produced more than 800 publications in Web of Science-indexed journals.[3]
AS-SMS has organized several international level workshops and conferences. Some of workshops at AS-SMS has been chosen to have funding from the ICTP, UNESCO, and CIMPA (International Centre for Pure and Applied Mathematics). The signature conferences of AS-SMS are its “World/International Conferences of Mathematics in the 21st Century” series, which has been held in 2004,[10] 2005,[11] 2007,[12] 2009,[13] 2011,[14] and 2013;[15] all were held in the tenure of A.D.R. Choudary. Some notable mathematicians that have visited and given talks in the World Conference in 21st Century Mathematics series are Ari Laptev (President of the EMS 2007-2010),[13] Marta Sanz-Solé (President of the EMS 2011-2014),[14] Pierre Cartier (an associate of the Bourbaki Group),[15] Arnfinn Laudal,[14] Michel Waldschmidt (President of Société mathématique de France (SMF) 2001-2004),[15] János Pach,[14] Alan Huckleberry,[14] Aline Bonami (President of SMF 2012-2013),[15] and Ragni Piene (Chair of the Abel Prize Committee 2010-2014 and the first-ever woman in the International Mathematics Union Executive Committee).[15] The last conference has its proceedings published by the prestigious Springer Publishing by the title "Mathematics in the 21st Century"; with Pierre Cartier, A.D.R. Choudary, and Michel Waldschmidt as the editors.[16]
AS-SMS was chosen as a Post-Doctoral research destination from Ph.D. holders from prestigious institutions around the world. Young researchers from Russia, Italy, Romania, Indonesia, and other countries have become Post Doctoral Fellows in ASSMS. This is something that not used to happen in any mathematics institution in Pakistan before the existence of AS-SMS.[3][4]
On the recommendation of the Committee for Developing Countries (CDC) of the European Mathematical Society, AS-SMS was labeled “Emerging Regional Centre of Excellence of the European Mathematical Society” (ERCE) for the period of 2011–2015.[17][18] Furthermore, AS-SMS and some other ERCE have form Network of International Mathematics Centres in 2013, where AS-SMS was supposed to be its initial secretariat.[19]
At the high school level, the school initiated the training camps for the preparation and selection of the national team of Pakistan to compete at the International Mathematical Olympiad (IMO). The school continued to organize these training camps until 2014, in the end of thenure of A.D.R. Choudary. The national team of Pakistan competed at IMO for the very first time in 2005, won its first medal in 2007 and its first silver medal in 2013.[20]
References
1. (GCU), Government College University. "Science Research at GCU: The Abdus Salam School of Mathematics". The Government College University Official website. Archived from the original on 2012-10-16.
2. ASSMS, The Abdus Salam School of Mathematical Sciences. "Achievements". The Abdus Salam School of Mathematical Sciences Official website. Retrieved 30 September 2016.
3. "Abdus Salam School of Mathematical Sciences" (PDF). GC University Lahore. Archived from the original (PDF) on 24 September 2015. Retrieved 18 August 2015.
4. "A Report of ASSMS" (PDF). Abdus Salam School of Mathematical Sciences. Archived from the original (PDF) on May 4, 2015. Retrieved 18 August 2015.
5. Laudal, Arnfinn (June 2011). "Editorial: Abdus Salam School of Mathematical Sciences" (PDF). Newsletter of the European Mathematical Society. EMS Publishing House (80): 3–4. Retrieved September 26, 2016.
6. "Josip Pecaric's activities in Pakistan". Croatian World Network. Retrieved 29 June 2021.
7. Tu, Loring (August 2011). "The Abdus Salam School of Mathematical Sciences in Lahore" (PDF). Notices of the American Mathematical Society. American Mathematical Society. 58 (7): 938–943. Retrieved September 26, 2016.
8. "ASSMS". GCU Lahore. Archived from the original on 9 May 2021. Retrieved 16 June 2021.
9. Laudal, Arnfinn; Sanz-Solé, Marta; Waldschmidt, Michel (March 2014). "A Mathematical Anniversary in Pakistan" (PDF). Newsletter of the European Mathematical Society. EMS Publishing House (91): 8–9. Retrieved September 26, 2016.
10. "1st World Conference in 21st Century Mathematics". Abdus Salam School of Mathematical Sciences. Archived from the original on May 6, 2004. Retrieved 18 August 2015.
11. "2nd World Conference in 21st Century Mathematics". Abdus Salam School of Mathematical Sciences. Archived from the original on February 9, 2005. Retrieved 18 August 2015.
12. "3rd International Conference in 21st Century Mathematics". Abdus Salam School of Mathematical Sciences. Archived from the original on August 13, 2006. Retrieved 18 August 2015.
13. "4th World Conference in 21st Century Mathematics". Abdus Salam School of Mathematical Sciences. Archived from the original on November 7, 2008. Retrieved 18 August 2015.
14. "5th World Conference in 21st Century Mathematics". Abdus Salam School of Mathematical Sciences. Archived from the original on July 28, 2011. Retrieved 18 August 2015.
15. "6th World Conference in 21st Century Mathematics". Abdus Salam School of Mathematical Sciences. Archived from the original on March 20, 2013. Retrieved 18 November 2015.
16. Mathematics in the 21st Century. Springer Proceedings in Mathematics & Statistics. Vol. 98. Springer. 2015. doi:10.1007/978-3-0348-0859-0. ISBN 978-3-0348-0858-3. Retrieved 18 August 2015.
17. "Emerging Regional Centre of Excellence". European Mathematical Society. Archived from the original on 6 September 2015. Retrieved 18 August 2015.
18. "Foreign Faculty Hiring Program: ASSMS". Higher Education Commission of Pakistan. Archived from the original on March 4, 2016. Retrieved 18 August 2015.
19. "Network of International Mathematic Centres" (PDF). The International Mathematical Union. Retrieved 18 August 2015.
20. "Pakistan Results in IMO". International Mathematics Olympiad. Retrieved 17 August 2015.
| Wikipedia |
Abdusalam Abubakar
Abdusalam Abubakar (born 1989/1990)[1] is a Somali-born Irish scientist from Dublin. He was the winner of the 43rd Young Scientist and Technology Exhibition in 2007 at the age of seventeen. He went on to be named EU Young Scientist of the Year in September 2007.
Abdusalam Abubakar
Born1989 or 1990 (age 33–34)[1]
Somalia
NationalityIrish
CitizenshipIreland
Alma materDublin City University
Known forAn Extension of Wiener's Attack on RSA
AwardsBT Young Scientist of the Year (2007)
EU Young Scientist of the Year (2007)
Scientific career
FieldsMathematical Sciences
Biography
Abubakar was born in Somalia to an Irish father[2] of Somali descent. He is an only child.[2] He moved to Ireland in May 2005,[2] joining Synge Street CBS in central Dublin.[1] He first entered the Young Scientist and Technology Exhibition alongside two fellow students who invited him along and taught him to research and solve properly.[2] They won an award for mathematics at the event.[2] He was mentored by Jim Cooke.
Abubakar then re-entered the Young Scientist and Technology Exhibition for the 2007 event as a third-year student at Synge Street CBS.[1] His project at the exhibition was titled "An Extension of Wiener's Attack on RSA".[3] His project was based on the topic of cryptography.[1]
Abubakar won the Young Scientist and Technology Exhibition at the RDS, Dublin on 12 January 2007.[1] He defeated runner-up Beara Community School in County Cork's Ciara Murphy and her study on hearing loss in teenagers.[3] He admitted afterwards that he had never used a computer before coming to Ireland twenty months earlier.[2]
An interview with Abubakar in Xclusive Magazine called him "the hottest name in Ireland right now" and said achievement was "obviously a landmark in science" after his win.[2] He appeared on the front cover of that edition of the magazine, under the headline "GENIUS! How Abdusalam Abubakar, a sixteen-year-old Somali, broke a 13-year-old Irish record".[2] Abubakar appeared on Dustin's Daily News on 19 January 2007.[4]
He went on to represent Ireland at the 19th European Union Contest for Young Scientists in Valencia, Spain in September 2007, claiming first prize in the field of mathematics for Ireland.[5]
Abubakar studied financial mathematics at Dublin City University.[6]
References
1. "Dublin student wins Young Scientist prize". Raidió Teilifís Éireann. 12 January 2007. Retrieved 15 January 2010. 17-year-old student ... Abu Salam Abu Bakar
2. "ABDUSALAM ABUBAKAR". Xclusive Magazine. Archived from the original on 3 July 2007. Retrieved 15 January 2010.
3. "Special awards". Irish Independent. 13 January 2007. Retrieved 15 January 2010.
4. "January 19th – Abdusalam Abubakar". Dustin's Daily News. 12 January 2007. Archived from the original on 16 February 2008. Retrieved 15 January 2010. In Turkey Towers there's Young Scientist winner Abdusalam Abubakar. Plus there's more Beak Beak Brother and Niamh reports on more of Ireland's top young scientists.
5. "The 19th European Union Contest for Young Scientists". European Union Contest for Young Scientists. 14–19 September 2007. Archived from the original on 21 May 2008. Retrieved 15 January 2010.
6. "Previous Young Scientist winners: where are they now?". The Irish Examiner. 9 January 2014. Retrieved 18 June 2016.
Young Scientist and Technology Exhibition
Winners
1990s
• Jane Feehan (1994)
• Brian Fitzpatrick and Shane Markey (1995)
• Elsie O'Sullivan, Rowens Mooney and Patricia Lyle (1996)
• Michael Flynn (1997)
• Raphael Hurley (1998)
• Sarah Flannery (1999)
2000s
• Thomas Gernon (2000)
• Peter Taylor, Shane Browne and Michael O'Toole (2001)
• David Michael O'Doherty (2002)
• Adnan Osmani (2003)
• Ronan Larkin (2004)
• Patrick Collison (2005)
• Aisling Judge (2006)
• Abdusalam Abubakar (2007)
• Emer Jones (2008)
• Liam McCarthy and John D. O'Callaghan (2009)
2010s
• Richard O'Shea (2010)
• Alexander Amini (2011)
• Mark Kelly and Eric Doyle (2012)
• Emer Hickey, Sophie Healy-Thow and Ciara Judge (2013)
• Paul Clarke (2014)
• Ian O’Sullivan and Eimear Murphy (2015)
• Maria Louise Fufezan and Diana Bura (2016)
• Shane Curran (2017)
• Simon Meehan (2018)
• Adam Kelly (2019)
Related articles
• BT Ireland
• European Union Contest for Young Scientists
• Royal Dublin Society
• Synge Street CBS
| Wikipedia |
Abel's binomial theorem
Abel's binomial theorem, named after Niels Henrik Abel, is a mathematical identity involving sums of binomial coefficients. It states the following:
$\sum _{k=0}^{m}{\binom {m}{k}}(w+m-k)^{m-k-1}(z+k)^{k}=w^{-1}(z+w+m)^{m}.$
Example
The case m = 2
${\begin{aligned}&{}\quad {\binom {2}{0}}(w+2)^{1}(z+0)^{0}+{\binom {2}{1}}(w+1)^{0}(z+1)^{1}+{\binom {2}{2}}(w+0)^{-1}(z+2)^{2}\\&=(w+2)+2(z+1)+{\frac {(z+2)^{2}}{w}}\\&={\frac {(z+w+2)^{2}}{w}}.\end{aligned}}$
See also
• Binomial theorem
• Binomial type
References
• Weisstein, Eric W. "Abel's binomial theorem". MathWorld.
| Wikipedia |
Abel's test
In mathematics, Abel's test (also known as Abel's criterion) is a method of testing for the convergence of an infinite series. The test is named after mathematician Niels Henrik Abel. There are two slightly different versions of Abel's test – one is used with series of real numbers, and the other is used with power series in complex analysis. Abel's uniform convergence test is a criterion for the uniform convergence of a series of functions dependent on parameters.
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Abel's test in real analysis
Suppose the following statements are true:
1. $\sum a_{n}$ is a convergent series,
2. {bn} is a monotone sequence, and
3. {bn} is bounded.
Then $\sum a_{n}b_{n}$ is also convergent.
It is important to understand that this test is mainly pertinent and useful in the context of non absolutely convergent series $\sum a_{n}$. For absolutely convergent series, this theorem, albeit true, is almost self evident.
This theorem can be proved directly using summation by parts.
Abel's test in complex analysis
A closely related convergence test, also known as Abel's test, can often be used to establish the convergence of a power series on the boundary of its circle of convergence. Specifically, Abel's test states that if a sequence of positive real numbers $(a_{n})$ is decreasing monotonically (or at least that for all n greater than some natural number m, we have $a_{n}\geq a_{n+1}$) with
$\lim _{n\rightarrow \infty }a_{n}=0$
then the power series
$f(z)=\sum _{n=0}^{\infty }a_{n}z^{n}$
converges everywhere on the closed unit circle, except when z = 1. Abel's test cannot be applied when z = 1, so convergence at that single point must be investigated separately. Notice that Abel's test implies in particular that the radius of convergence is at least 1. It can also be applied to a power series with radius of convergence R ≠ 1 by a simple change of variables ζ = z/R.[1] Notice that Abel's test is a generalization of the Leibniz Criterion by taking z = −1.
Proof of Abel's test: Suppose that z is a point on the unit circle, z ≠ 1. For each $n\geq 1$, we define
$f_{n}(z):=\sum _{k=0}^{n}a_{k}z^{k}.$
By multiplying this function by (1 − z), we obtain
${\begin{aligned}(1-z)f_{n}(z)&=\sum _{k=0}^{n}a_{k}(1-z)z^{k}=\sum _{k=0}^{n}a_{k}z^{k}-\sum _{k=0}^{n}a_{k}z^{k+1}=a_{0}+\sum _{k=1}^{n}a_{k}z^{k}-\sum _{k=1}^{n+1}a_{k-1}z^{k}\\&=a_{0}-a_{n}z^{n+1}+\sum _{k=1}^{n}(a_{k}-a_{k-1})z^{k}.\end{aligned}}$
The first summand is constant, the second converges uniformly to zero (since by assumption the sequence $(a_{n})$ converges to zero). It only remains to show that the series converges. We will show this by showing that it even converges absolutely: $\sum _{k=1}^{\infty }\left|(a_{k}-a_{k-1})z^{k}\right|=\sum _{k=1}^{\infty }|a_{k}-a_{k-1}|\cdot |z|^{k}\leq \sum _{k=1}^{\infty }(a_{k-1}-a_{k})$ where the last sum is a converging telescoping sum. The absolute value vanished because the sequence $(a_{n})$ is decreasing by assumption.
Hence, the sequence $(1-z)f_{n}(z)$ converges (even uniformly) on the closed unit disc. If $z\not =1$, we may divide by (1 − z) and obtain the result.
Another way to obtain the result is to apply the Dirichlet's test. Indeed, for $z\neq 1,\ |z|=1$ holds $\left|\sum _{k=0}^{n}z^{k}\right|=\left|{\frac {z^{n+1}-1}{z-1}}\right|\leq {\frac {2}{|z-1|}}$, hence the assumptions of the Dirichlet's test are fulfilled.
Abel's uniform convergence test
Abel's uniform convergence test is a criterion for the uniform convergence of a series of functions or an improper integration of functions dependent on parameters. It is related to Abel's test for the convergence of an ordinary series of real numbers, and the proof relies on the same technique of summation by parts.
The test is as follows. Let {gn} be a uniformly bounded sequence of real-valued continuous functions on a set E such that gn+1(x) ≤ gn(x) for all x ∈ E and positive integers n, and let {fn} be a sequence of real-valued functions such that the series Σfn(x) converges uniformly on E. Then Σfn(x)gn(x) converges uniformly on E.
Notes
1. (Moretti, 1964, p. 91)
References
• Gino Moretti, Functions of a Complex Variable, Prentice-Hall, Inc., 1964
• Apostol, Tom M. (1974), Mathematical analysis (2nd ed.), Addison-Wesley, ISBN 978-0-201-00288-1
• Weisstein, Eric W. "Abel's uniform convergence test". MathWorld.
External links
• Proof (for real series) at PlanetMath.org
Calculus
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| Wikipedia |
Abel's identity
In mathematics, Abel's identity (also called Abel's formula[1] or Abel's differential equation identity) is an equation that expresses the Wronskian of two solutions of a homogeneous second-order linear ordinary differential equation in terms of a coefficient of the original differential equation. The relation can be generalised to nth-order linear ordinary differential equations. The identity is named after the Norwegian mathematician Niels Henrik Abel.
Since Abel's identity relates the different linearly independent solutions of the differential equation, it can be used to find one solution from the other. It provides useful identities relating the solutions, and is also useful as a part of other techniques such as the method of variation of parameters. It is especially useful for equations such as Bessel's equation where the solutions do not have a simple analytical form, because in such cases the Wronskian is difficult to compute directly.
A generalisation to first-order systems of homogeneous linear differential equations is given by Liouville's formula.
Statement
Consider a homogeneous linear second-order ordinary differential equation
$y''+p(x)y'+q(x)\,y=0$
on an interval I of the real line with real- or complex-valued continuous functions p and q. Abel's identity states that the Wronskian $W=(y_{1},y_{2})$ of two real- or complex-valued solutions $y_{1}$ and $y_{2}$ of this differential equation, that is the function defined by the determinant
$W(y_{1},y_{2})(x)={\begin{vmatrix}y_{1}(x)&y_{2}(x)\\y'_{1}(x)&y'_{2}(x)\end{vmatrix}}=y_{1}(x)\,y'_{2}(x)-y'_{1}(x)\,y_{2}(x),\qquad x\in I,$
satisfies the relation
$W(y_{1},y_{2})(x)=W(y_{1},y_{2})(x_{0})\cdot e^{-\int _{x_{0}}^{x}p(t)\,dt},\qquad x\in I,$
for each point $x_{0}\in I$.
Remarks
• In particular, when the differential equation is real-valued, the Wronskian $W(y_{1},y_{2})$ is always either identically zero, always positive, or always negative at every point $x$ in $I$ (see proof below). The latter cases imply the two solutions $y_{1}$ and $y_{2}$ are linearly independent (see Wronskian for a proof).
• It is not necessary to assume that the second derivatives of the solutions $y_{1}$ and $y_{2}$ are continuous.
• Abel's theorem is particularly useful if $p(x)=0$, because it implies that $W$ is constant.
Proof
Differentiating the Wronskian using the product rule gives (writing $W$ for $W(y_{1},y_{2})$ and omitting the argument $x$ for brevity)
${\begin{aligned}W'&=y_{1}'y_{2}'+y_{1}y_{2}''-y_{1}''y_{2}-y_{1}'y_{2}'\\&=y_{1}y_{2}''-y_{1}''y_{2}.\end{aligned}}$
Solving for $y''$ in the original differential equation yields
$y''=-(py'+qy).$
Substituting this result into the derivative of the Wronskian function to replace the second derivatives of $y_{1}$ and $y_{2}$ gives
${\begin{aligned}W'&=-y_{1}(py_{2}'+qy_{2})+(py_{1}'+qy_{1})y_{2}\\&=-p(y_{1}y_{2}'-y_{1}'y_{2})\\&=-pW.\end{aligned}}$
This is a first-order linear differential equation, and it remains to show that Abel's identity gives the unique solution, which attains the value $W(x_{0})$ at $x_{0}$. Since the function $p$ is continuous on $I$, it is bounded on every closed and bounded subinterval of $I$ and therefore integrable, hence
$V(x)=W(x)\,\exp \!\left(\int _{x_{0}}^{x}p(\xi )\,{\textrm {d}}\xi \right),\qquad x\in I,$
is a well-defined function. Differentiating both sides, using the product rule, the chain rule, the derivative of the exponential function and the fundamental theorem of calculus, one obtains
$V'(x)={\bigl (}W'(x)+W(x)p(x){\bigr )}\,\exp \!{\biggl (}\int _{x_{0}}^{x}p(\xi )\,{\textrm {d}}\xi {\biggr )}=0,\qquad x\in I,$
due to the differential equation for $W$. Therefore, $V$ has to be constant on $I$, because otherwise we would obtain a contradiction to the mean value theorem (applied separately to the real and imaginary part in the complex-valued case). Since $V(x_{0})=W(x_{0})$, Abel's identity follows by solving the definition of $V$ for $W(x)$.
Proof that the Wronskian never changes sign
For all $x\in I$, the Wronskian $W(y_{1},y_{2})(x)$ is either identically zero, always positive, or always negative, given that $y_{1}$, $y_{2}$, and $p(x)$ are real-valued. This is demonstrated as follows.
Abel's identity states that $W(y_{1},y_{2})(x)=W(y_{1},y_{2})(x_{0})\cdot e^{-\int _{x_{0}}^{x}p(x')\,{\textrm {d}}x'}\forall x\in I,$
Let $c=W(y_{1},y_{2})(x_{0})$. Then $c$ must be a real-valued constant because $y_{1}$ and $y_{2}$ are real-valued.
Let $f(x)=-\int _{x_{0}}^{x}p(x')\,{\textrm {d}}x'$. As $p(x)$ is real-valued, so is $f(x)$, so $e^{f(x)}$ is strictly positive.
Thus, $W(y_{1},y_{2})(x)=c\cdot e^{f(x)}$ is identically zero when $c=0$, always positive when $c$ is positive, and always negative when $c$ is negative.
Furthermore, when $y_{1}$, $y_{2}$, and $p(x)$, one can similarly show that $W(y_{1},y_{2})(x)$ is either identically $0$ or non-zero for all values of x.
Generalization
The Wronskian $W(y_{1},\ldots ,y_{n})$ of $n$ functions $y_{1},\ldots ,y_{n}$ on an interval $I$ is the function defined by the determinant
$W(y_{1},\ldots ,y_{n})(x)={\begin{vmatrix}y_{1}(x)&y_{2}(x)&\cdots &y_{n}(x)\\y'_{1}(x)&y'_{2}(x)&\cdots &y'_{n}(x)\\\vdots &\vdots &\ddots &\vdots \\y_{1}^{(n-1)}(x)&y_{2}^{(n-1)}(x)&\cdots &y_{n}^{(n-1)}(x)\end{vmatrix}},\qquad x\in I,$
Consider a homogeneous linear ordinary differential equation of order $n\geq 1$:
$y^{(n)}+p_{n-1}(x)\,y^{(n-1)}+\cdots +p_{1}(x)\,y'+p_{0}(x)\,y=0,$
on an interval $I$ of the real line with a real- or complex-valued continuous function $p_{n-1}$. Let $y_{1},\ldots ,y_{n}$ by solutions of this nth order differential equation. Then the generalisation of Abel's identity states that this Wronskian satisfies the relation:
$W(y_{1},\ldots ,y_{n})(x)=W(y_{1},\ldots ,y_{n})(x_{0})\exp {\biggl (}-\int _{x_{0}}^{x}p_{n-1}(\xi )\,{\textrm {d}}\xi {\biggr )},\qquad x\in I,$
for each point $x_{0}\in I$.
Direct proof
For brevity, we write $W$ for $W(y_{1},\ldots ,y_{n})$ and omit the argument $x$. It suffices to show that the Wronskian solves the first-order linear differential equation
$W'=-p_{n-1}\,W,$
because the remaining part of the proof then coincides with the one for the case $n=2$.
In the case $n=1$ we have $W=y_{1}$ and the differential equation for $W$ coincides with the one for $y_{1}$. Therefore, assume $n\geq 2$ in the following.
The derivative of the Wronskian $W$ is the derivative of the defining determinant. It follows from the Leibniz formula for determinants that this derivative can be calculated by differentiating every row separately, hence
${\begin{aligned}W'&={\begin{vmatrix}y'_{1}&y'_{2}&\cdots &y'_{n}\\y'_{1}&y'_{2}&\cdots &y'_{n}\\y''_{1}&y''_{2}&\cdots &y''_{n}\\y'''_{1}&y'''_{2}&\cdots &y'''_{n}\\\vdots &\vdots &\ddots &\vdots \\y_{1}^{(n-1)}&y_{2}^{(n-1)}&\cdots &y_{n}^{(n-1)}\end{vmatrix}}+{\begin{vmatrix}y_{1}&y_{2}&\cdots &y_{n}\\y''_{1}&y''_{2}&\cdots &y''_{n}\\y''_{1}&y''_{2}&\cdots &y''_{n}\\y'''_{1}&y'''_{2}&\cdots &y'''_{n}\\\vdots &\vdots &\ddots &\vdots \\y_{1}^{(n-1)}&y_{2}^{(n-1)}&\cdots &y_{n}^{(n-1)}\end{vmatrix}}\\&\qquad +\ \cdots \ +{\begin{vmatrix}y_{1}&y_{2}&\cdots &y_{n}\\y'_{1}&y'_{2}&\cdots &y'_{n}\\\vdots &\vdots &\ddots &\vdots \\y_{1}^{(n-3)}&y_{2}^{(n-3)}&\cdots &y_{n}^{(n-3)}\\y_{1}^{(n-2)}&y_{2}^{(n-2)}&\cdots &y_{n}^{(n-2)}\\y_{1}^{(n)}&y_{2}^{(n)}&\cdots &y_{n}^{(n)}\end{vmatrix}}.\end{aligned}}$
However, note that every determinant from the expansion contains a pair of identical rows, except the last one. Since determinants with linearly dependent rows are equal to 0, one is only left with the last one:
$W'={\begin{vmatrix}y_{1}&y_{2}&\cdots &y_{n}\\y'_{1}&y'_{2}&\cdots &y'_{n}\\\vdots &\vdots &\ddots &\vdots \\y_{1}^{(n-2)}&y_{2}^{(n-2)}&\cdots &y_{n}^{(n-2)}\\y_{1}^{(n)}&y_{2}^{(n)}&\cdots &y_{n}^{(n)}\end{vmatrix}}.$
Since every $y_{i}$ solves the ordinary differential equation, we have
$y_{i}^{(n)}+p_{n-2}\,y_{i}^{(n-2)}+\cdots +p_{1}\,y'_{i}+p_{0}\,y_{i}=-p_{n-1}\,y_{i}^{(n-1)}$
for every $i\in \lbrace 1,\ldots ,n\rbrace $. Hence, adding to the last row of the above determinant $p_{0}$ times its first row, $p_{1}$ times its second row, and so on until $p_{n-2}$ times its next to last row, the value of the determinant for the derivative of $W$ is unchanged and we get
$W'={\begin{vmatrix}y_{1}&y_{2}&\cdots &y_{n}\\y'_{1}&y'_{2}&\cdots &y'_{n}\\\vdots &\vdots &\ddots &\vdots \\y_{1}^{(n-2)}&y_{2}^{(n-2)}&\cdots &y_{n}^{(n-2)}\\-p_{n-1}\,y_{1}^{(n-1)}&-p_{n-1}\,y_{2}^{(n-1)}&\cdots &-p_{n-1}\,y_{n}^{(n-1)}\end{vmatrix}}=-p_{n-1}W.$
Proof using Liouville's formula
The solutions $y_{1},\ldots ,y_{n}$ form the square-matrix valued solution
$\Phi (x)={\begin{pmatrix}y_{1}(x)&y_{2}(x)&\cdots &y_{n}(x)\\y'_{1}(x)&y'_{2}(x)&\cdots &y'_{n}(x)\\\vdots &\vdots &\ddots &\vdots \\y_{1}^{(n-2)}(x)&y_{2}^{(n-2)}(x)&\cdots &y_{n}^{(n-2)}(x)\\y_{1}^{(n-1)}(x)&y_{2}^{(n-1)}(x)&\cdots &y_{n}^{(n-1)}(x)\end{pmatrix}},\qquad x\in I,$
of the $n$-dimensional first-order system of homogeneous linear differential equations
${\begin{pmatrix}y'\\y''\\\vdots \\y^{(n-1)}\\y^{(n)}\end{pmatrix}}={\begin{pmatrix}0&1&0&\cdots &0\\0&0&1&\cdots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\cdots &1\\-p_{0}(x)&-p_{1}(x)&-p_{2}(x)&\cdots &-p_{n-1}(x)\end{pmatrix}}{\begin{pmatrix}y\\y'\\\vdots \\y^{(n-2)}\\y^{(n-1)}\end{pmatrix}}.$
The trace of this matrix is $-p_{n-1}(x)$, hence Abel's identity follows directly from Liouville's formula.
References
1. Rainville, Earl David; Bedient, Phillip Edward (1969). Elementary Differential Equations. Collier-Macmillan International Editions.
• Abel, N. H., "Précis d'une théorie des fonctions elliptiques" J. Reine Angew. Math., 4 (1829) pp. 309–348.
• Boyce, W. E. and DiPrima, R. C. (1986). Elementary Differential Equations and Boundary Value Problems, 4th ed. New York: Wiley.
• Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.
• Weisstein, Eric W. "Abel's Differential Equation Identity". MathWorld.
| Wikipedia |
Functional equation
In mathematics, a functional equation [1][2] is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning is often used, where a functional equation is an equation that relates several values of the same function. For example, the logarithm functions are essentially characterized by the logarithmic functional equation $\log(xy)=\log(x)+\log(y).$
If the domain of the unknown function is supposed to be the natural numbers, the function is generally viewed as a sequence, and, in this case, a functional equation (in the narrower meaning) is called a recurrence relation. Thus the term functional equation is used mainly for real functions and complex functions. Moreover a smoothness condition is often assumed for the solutions, since without such a condition, most functional equations have very irregular solutions. For example, the gamma function is a function that satisfies the functional equation $f(x+1)=xf(x)$ and the initial value $f(1)=1.$ There are many functions that satisfy these conditions, but the gamma function is the unique one that is meromorphic in the whole complex plane, and logarithmically convex for x real and positive (Bohr–Mollerup theorem).
Examples
• Recurrence relations can be seen as functional equations in functions over the integers or natural numbers, in which the differences between terms' indexes can be seen as an application of the shift operator. For example, the recurrence relation defining the Fibonacci numbers, $F_{n}=F_{n-1}+F_{n-2}$, where $F_{0}=0$ and $F_{1}=1$
• $f(x+P)=f(x)$, which characterizes the periodic functions
• $f(x)=f(-x)$, which characterizes the even functions, and likewise $f(x)=-f(-x)$, which characterizes the odd functions
• $f(f(x))=g(x)$, which characterizes the functional square roots of the function g
• $f(x+y)=f(x)+f(y)\,\!$ (Cauchy's functional equation), satisfied by linear maps. The equation may, contingent on the axiom of choice, also have other pathological nonlinear solutions, whose existence can be proven with a Hamel basis for the real numbers
• $f(x+y)=f(x)f(y),\,\!$ satisfied by all exponential functions. Like Cauchy's additive functional equation, this too may have pathological, discontinuous solutions
• $f(xy)=f(x)+f(y)\,\!$, satisfied by all logarithmic functions and, over coprime integer arguments, additive functions
• $f(xy)=f(x)f(y)\,\!$, satisfied by all power functions and, over coprime integer arguments, multiplicative functions
• $f(x+y)+f(x-y)=2[f(x)+f(y)]\,\!$ (quadratic equation or parallelogram law)
• $f((x+y)/2)=(f(x)+f(y))/2\,\!$ (Jensen's functional equation)
• $g(x+y)+g(x-y)=2[g(x)g(y)]\,\!$ (d'Alembert's functional equation)
• $f(h(x))=h(x+1)\,\!$ (Abel equation)
• $f(h(x))=cf(x)\,\!$ (Schröder's equation).
• $f(h(x))=(f(x))^{c}\,\!$ (Böttcher's equation).
• $f(h(x))=h'(x)f(x)\,\!$ (Julia's equation).
• $f(xy)=\sum g_{l}(x)h_{l}(y)\,\!$ (Levi-Civita),
• $f(x+y)=f(x)g(y)+f(y)g(x)\,\!$ (sine addition formula and hyperbolic sine addition formula),
• $g(x+y)=g(x)g(y)-f(y)f(x)\,\!$ (cosine addition formula),
• $g(x+y)=g(x)g(y)+f(y)f(x)\,\!$ (hyperbolic cosine addition formula).
• The commutative and associative laws are functional equations. In its familiar form, the associative law is expressed by writing the binary operation in infix notation,
$(a\circ b)\circ c=a\circ (b\circ c)~,$
but if we write f(a, b) instead of a ○ b then the associative law looks more like a conventional functional equation,
$f(f(a,b),c)=f(a,f(b,c)).\,\!$
• The functional equation
$f(s)=2^{s}\pi ^{s-1}\sin \left({\frac {\pi s}{2}}\right)\Gamma (1-s)f(1-s)$
is satisfied by the Riemann zeta function, as proved here. The capital Γ denotes the gamma function.
• The gamma function is the unique solution of the following system of three equations:
• $f(x)={f(x+1) \over x}$
• $f(y)f\left(y+{\frac {1}{2}}\right)={\frac {\sqrt {\pi }}{2^{2y-1}}}f(2y)$
• $f(z)f(1-z)={\pi \over \sin(\pi z)}$ (Euler's reflection formula)
• The functional equation
$f\left({az+b \over cz+d}\right)=(cz+d)^{k}f(z)$
where a, b, c, d are integers satisfying $ad-bc=1$, i.e. ${\begin{vmatrix}a&b\\c&d\end{vmatrix}}$ = 1, defines f to be a modular form of order k.
One feature that all of the examples listed above share in common is that, in each case, two or more known functions (sometimes multiplication by a constant, sometimes addition of two variables, sometimes the identity function) are inside the argument of the unknown functions to be solved for.
When it comes to asking for all solutions, it may be the case that conditions from mathematical analysis should be applied; for example, in the case of the Cauchy equation mentioned above, the solutions that are continuous functions are the 'reasonable' ones, while other solutions that are not likely to have practical application can be constructed (by using a Hamel basis for the real numbers as vector space over the rational numbers). The Bohr–Mollerup theorem is another well-known example.
Involutions
The involutions are characterized by the functional equation $f(f(x))=x$. These appear in Babbage's functional equation (1820),[3]
$f(f(x))=1-(1-x)=x\,.$
Other involutions, and solutions of the equation, include
• $f(x)=a-x\,,$
• $f(x)={\frac {a}{x}}\,,$ and
• $f(x)={\frac {b-x}{1+cx}}~,$
which includes the previous three as special cases or limits.
Solution
One method of solving elementary functional equations is substitution.
Some solutions to functional equations have exploited surjectivity, injectivity, oddness, and evenness.
Some functional equations have been solved with the use of ansatzes, mathematical induction.
Some classes of functional equations can be solved by computer-assisted techniques.[4]
In dynamic programming a variety of successive approximation methods[5][6] are used to solve Bellman's functional equation, including methods based on fixed point iterations.
See also
• Functional equation (L-function)
• Bellman equation
• Dynamic programming
• Implicit function
• Functional differential equation
Notes
1. Rassias, Themistocles M. (2000). Functional Equations and Inequalities. 3300 AA Dordrecht, The Netherlands: Kluwer Academic Publishers. p. 335. ISBN 0-7923-6484-8.{{cite book}}: CS1 maint: location (link)
2. Czerwik, Stephan (2002). Functional Equations and Inequalities in Several Variables. P O Box 128, Farrer Road, Singapore 912805: World Scientific Publishing Co. p. 410. ISBN 981-02-4837-7.{{cite book}}: CS1 maint: location (link)
3. Ritt, J. F. (1916). "On Certain Real Solutions of Babbage's Functional Equation". The Annals of Mathematics. 17 (3): 113–122. doi:10.2307/2007270. JSTOR 2007270.
4. Házy, Attila (2004-03-01). "Solving linear two variable functional equations with computer". Aequationes Mathematicae. 67 (1): 47–62. doi:10.1007/s00010-003-2703-9. ISSN 1420-8903. S2CID 118563768.
5. Bellman, R. (1957). Dynamic Programming, Princeton University Press.
6. Sniedovich, M. (2010). Dynamic Programming: Foundations and Principles, Taylor & Francis.
References
• János Aczél, Lectures on Functional Equations and Their Applications, Academic Press, 1966, reprinted by Dover Publications, ISBN 0486445232.
• János Aczél & J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, 1989.
• C. Efthimiou, Introduction to Functional Equations, AMS, 2011, ISBN 978-0-8218-5314-6 ; online.
• Pl. Kannappan, Functional Equations and Inequalities with Applications, Springer, 2009.
• Marek Kuczma, Introduction to the Theory of Functional Equations and Inequalities, second edition, Birkhäuser, 2009.
• Henrik Stetkær, Functional Equations on Groups, first edition, World Scientific Publishing, 2013.
• Christopher G. Small (3 April 2007). Functional Equations and How to Solve Them. Springer Science & Business Media. ISBN 978-0-387-48901-8.
External links
• Functional Equations: Exact Solutions at EqWorld: The World of Mathematical Equations.
• Functional Equations: Index at EqWorld: The World of Mathematical Equations.
• IMO Compendium text (archived) on functional equations in problem solving.
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| Wikipedia |
Abel's inequality
In mathematics, Abel's inequality, named after Niels Henrik Abel, supplies a simple bound on the absolute value of the inner product of two vectors in an important special case.
Mathematical description
Let {a1, a2,...} be a sequence of real numbers that is either nonincreasing or nondecreasing, and let {b1, b2,...} be a sequence of real or complex numbers. If {an} is nondecreasing, it holds that
$\left|\sum _{k=1}^{n}a_{k}b_{k}\right|\leq \operatorname {max} _{k=1,\dots ,n}|B_{k}|(|a_{n}|+a_{n}-a_{1}),$
and if {an} is nonincreasing, it holds that
$\left|\sum _{k=1}^{n}a_{k}b_{k}\right|\leq \operatorname {max} _{k=1,\dots ,n}|B_{k}|(|a_{n}|-a_{n}+a_{1}),$
where
$B_{k}=b_{1}+\cdots +b_{k}.$
In particular, if the sequence {an} is nonincreasing and nonnegative, it follows that
$\left|\sum _{k=1}^{n}a_{k}b_{k}\right|\leq \operatorname {max} _{k=1,\dots ,n}|B_{k}|a_{1},$
Relation to Abel's transformation
Abel's inequality follows easily from Abel's transformation, which is the discrete version of integration by parts: If {a1, a2, ...} and {b1, b2, ...} are sequences of real or complex numbers, it holds that
$\sum _{k=1}^{n}a_{k}b_{k}=a_{n}B_{n}-\sum _{k=1}^{n-1}B_{k}(a_{k+1}-a_{k}).$
References
• Weisstein, Eric W. "Abel's inequality". MathWorld.
• Abel's inequality in Encyclopedia of Mathematics.
| Wikipedia |
Abel's irreducibility theorem
In mathematics, Abel's irreducibility theorem, a field theory result described in 1829 by Niels Henrik Abel,[1] asserts that if ƒ(x) is a polynomial over a field F that shares a root with a polynomial g(x) that is irreducible over F, then every root of g(x) is a root of ƒ(x). Equivalently, if ƒ(x) shares at least one root with g(x) then ƒ is divisible evenly by g(x), meaning that ƒ(x) can be factored as g(x)h(x) with h(x) also having coefficients in F.[2][3]
Corollaries of the theorem include:[2]
• If ƒ(x) is irreducible, there is no lower-degree polynomial (other than the zero polynomial) that shares any root with it. For example, x2 − 2 is irreducible over the rational numbers and has ${\sqrt {2}}$ as a root; hence there is no linear or constant polynomial over the rationals having ${\sqrt {2}}$ as a root. Furthermore, there is no same-degree polynomial that shares any roots with ƒ(x), other than constant multiples of ƒ(x).
• If ƒ(x) ≠ g(x) are two different irreducible monic polynomials, then they share no roots.
References
1. Abel, N. H. (1829), "Mémoire sur une classe particulière d'équations résolubles algébriquement" [Note on a particular class of algebraically solvable equations], Journal für die reine und angewandte Mathematik, 1829 (4): 131–156, doi:10.1515/crll.1829.4.131, S2CID 121388045.
2. Dörrie, Heinrich (1965), 100 Great Problems of Elementary Mathematics: Their History and Solution, Courier Dover Publications, p. 120, ISBN 9780486613482.
3. This theorem, for minimal polynomials rather than irreducible polynomials more generally, is Lemma 4.1.3 of Cox (2012). Irreducible polynomials, divided by their leading coefficient, are minimal for their roots (Cox Proposition 4.1.5), and all minimal polynomials are irreducible, so Cox's formulation is equivalent to Abel's. Cox, David A. (2012), Galois Theory, Pure and Applied Mathematics (2nd ed.), John Wiley & Sons, doi:10.1002/9781118218457, ISBN 978-1-118-07205-9.
External links
• Larry Freeman. Fermat's Last Theorem blog: Abel's Lemmas on Irreducibility. September 4, 2008.
• Weisstein, Eric W. "Abel's Irreducibility Theorem". MathWorld.
| Wikipedia |
Abel's summation formula
In mathematics, Abel's summation formula, introduced by Niels Henrik Abel, is intensively used in analytic number theory and the study of special functions to compute series.
Other concepts sometimes known by this name are summation by parts and Abel–Plana formula.
Formula
Wikibooks has a book on the topic of: Analytic Number Theory/Useful summation formulas
Let $(a_{n})_{n=0}^{\infty }$ be a sequence of real or complex numbers. Define the partial sum function $A$ by
$A(t)=\sum _{0\leq n\leq t}a_{n}$
for any real number $t$. Fix real numbers $x<y$, and let $\phi $ be a continuously differentiable function on $[x,y]$. Then:
$\sum _{x<n\leq y}a_{n}\phi (n)=A(y)\phi (y)-A(x)\phi (x)-\int _{x}^{y}A(u)\phi '(u)\,du.$
The formula is derived by applying integration by parts for a Riemann–Stieltjes integral to the functions $A$ and $\phi $.
Variations
Taking the left endpoint to be $-1$ gives the formula
$\sum _{0\leq n\leq x}a_{n}\phi (n)=A(x)\phi (x)-\int _{0}^{x}A(u)\phi '(u)\,du.$
If the sequence $(a_{n})$ is indexed starting at $n=1$, then we may formally define $a_{0}=0$. The previous formula becomes
$\sum _{1\leq n\leq x}a_{n}\phi (n)=A(x)\phi (x)-\int _{1}^{x}A(u)\phi '(u)\,du.$
A common way to apply Abel's summation formula is to take the limit of one of these formulas as $x\to \infty $. The resulting formulas are
${\begin{aligned}\sum _{n=0}^{\infty }a_{n}\phi (n)&=\lim _{x\to \infty }{\bigl (}A(x)\phi (x){\bigr )}-\int _{0}^{\infty }A(u)\phi '(u)\,du,\\\sum _{n=1}^{\infty }a_{n}\phi (n)&=\lim _{x\to \infty }{\bigl (}A(x)\phi (x){\bigr )}-\int _{1}^{\infty }A(u)\phi '(u)\,du.\end{aligned}}$
These equations hold whenever both limits on the right-hand side exist and are finite.
A particularly useful case is the sequence $a_{n}=1$ for all $n\geq 0$. In this case, $A(x)=\lfloor x+1\rfloor $. For this sequence, Abel's summation formula simplifies to
$\sum _{0\leq n\leq x}\phi (n)=\lfloor x+1\rfloor \phi (x)-\int _{0}^{x}\lfloor u+1\rfloor \phi '(u)\,du.$
Similarly, for the sequence $a_{0}=0$ and $a_{n}=1$ for all $n\geq 1$, the formula becomes
$\sum _{1\leq n\leq x}\phi (n)=\lfloor x\rfloor \phi (x)-\int _{1}^{x}\lfloor u\rfloor \phi '(u)\,du.$
Upon taking the limit as $x\to \infty $, we find
${\begin{aligned}\sum _{n=0}^{\infty }\phi (n)&=\lim _{x\to \infty }{\bigl (}\lfloor x+1\rfloor \phi (x){\bigr )}-\int _{0}^{\infty }\lfloor u+1\rfloor \phi '(u)\,du,\\\sum _{n=1}^{\infty }\phi (n)&=\lim _{x\to \infty }{\bigl (}\lfloor x\rfloor \phi (x){\bigr )}-\int _{1}^{\infty }\lfloor u\rfloor \phi '(u)\,du,\end{aligned}}$
assuming that both terms on the right-hand side exist and are finite.
Abel's summation formula can be generalized to the case where $\phi $ is only assumed to be continuous if the integral is interpreted as a Riemann–Stieltjes integral:
$\sum _{x<n\leq y}a_{n}\phi (n)=A(y)\phi (y)-A(x)\phi (x)-\int _{x}^{y}A(u)\,d\phi (u).$
By taking $\phi $ to be the partial sum function associated to some sequence, this leads to the summation by parts formula.
Examples
Harmonic numbers
If $a_{n}=1$ for $n\geq 1$ and $\phi (x)=1/x,$ then $A(x)=\lfloor x\rfloor $ and the formula yields
$\sum _{n=1}^{\lfloor x\rfloor }{\frac {1}{n}}={\frac {\lfloor x\rfloor }{x}}+\int _{1}^{x}{\frac {\lfloor u\rfloor }{u^{2}}}\,du.$
The left-hand side is the harmonic number $H_{\lfloor x\rfloor }$.
Representation of Riemann's zeta function
Fix a complex number $s$. If $a_{n}=1$ for $n\geq 1$ and $\phi (x)=x^{-s},$ then $A(x)=\lfloor x\rfloor $ and the formula becomes
$\sum _{n=1}^{\lfloor x\rfloor }{\frac {1}{n^{s}}}={\frac {\lfloor x\rfloor }{x^{s}}}+s\int _{1}^{x}{\frac {\lfloor u\rfloor }{u^{1+s}}}\,du.$
If $\Re (s)>1$, then the limit as $x\to \infty $ exists and yields the formula
$\zeta (s)=s\int _{1}^{\infty }{\frac {\lfloor u\rfloor }{u^{1+s}}}\,du.$
where $\zeta (s)$ is the Riemann zeta function. This may be used to derive Dirichlet's theorem that $\zeta (s)$ has a simple pole with residue 1 at s = 1.
Reciprocal of Riemann zeta function
The technique of the previous example may also be applied to other Dirichlet series. If $a_{n}=\mu (n)$ is the Möbius function and $\phi (x)=x^{-s}$, then $A(x)=M(x)=\sum _{n\leq x}\mu (n)$ is Mertens function and
${\frac {1}{\zeta (s)}}=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}=s\int _{1}^{\infty }{\frac {M(u)}{u^{1+s}}}\,du.$
This formula holds for $\Re (s)>1$.
See also
• Summation by parts
• Integration by parts
References
• Apostol, Tom (1976), Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics, Springer-Verlag.
| Wikipedia |
Abel–Jacobi map
In mathematics, the Abel–Jacobi map is a construction of algebraic geometry which relates an algebraic curve to its Jacobian variety. In Riemannian geometry, it is a more general construction mapping a manifold to its Jacobi torus. The name derives from the theorem of Abel and Jacobi that two effective divisors are linearly equivalent if and only if they are indistinguishable under the Abel–Jacobi map.
Construction of the map
In complex algebraic geometry, the Jacobian of a curve C is constructed using path integration. Namely, suppose C has genus g, which means topologically that
$H_{1}(C,\mathbb {Z} )\cong \mathbb {Z} ^{2g}.$
Geometrically, this homology group consists of (homology classes of) cycles in C, or in other words, closed loops. Therefore, we can choose 2g loops $\gamma _{1},\ldots ,\gamma _{2g}$ generating it. On the other hand, another more algebro-geometric way of saying that the genus of C is g is that
$H^{0}(C,K)\cong \mathbb {C} ^{g},$
where K is the canonical bundle on C.
By definition, this is the space of globally defined holomorphic differential forms on C, so we can choose g linearly independent forms $\omega _{1},\ldots ,\omega _{g}$. Given forms and closed loops we can integrate, and we define 2g vectors
$\Omega _{j}=\left(\int _{\gamma _{j}}\omega _{1},\ldots ,\int _{\gamma _{j}}\omega _{g}\right)\in \mathbb {C} ^{g}.$
It follows from the Riemann bilinear relations that the $\Omega _{j}$ generate a nondegenerate lattice $\Lambda $ (that is, they are a real basis for $\mathbb {C} ^{g}\cong \mathbb {R} ^{2g}$), and the Jacobian is defined by
$J(C)=\mathbb {C} ^{g}/\Lambda .$
The Abel–Jacobi map is then defined as follows. We pick some base point $p_{0}\in C$ and, nearly mimicking the definition of $\Lambda ,$ define the map
${\begin{cases}u:C\to J(C)\\u(p)=\left(\int _{p_{0}}^{p}\omega _{1},\dots ,\int _{p_{0}}^{p}\omega _{g}\right){\bmod {\Lambda }}\end{cases}}$
Although this is seemingly dependent on a path from $p_{0}$ to $p,$ any two such paths define a closed loop in $C$ and, therefore, an element of $H_{1}(C,\mathbb {Z} ),$ so integration over it gives an element of $\Lambda .$ Thus the difference is erased in the passage to the quotient by $\Lambda $. Changing base-point $p_{0}$ does change the map, but only by a translation of the torus.
The Abel–Jacobi map of a Riemannian manifold
Let $M$ be a smooth compact manifold. Let $\pi =\pi _{1}(M)$ be its fundamental group. Let $f:\pi \to \pi ^{ab}$ be its abelianisation map. Let $\operatorname {tor} =\operatorname {tor} (\pi ^{ab})$ be the torsion subgroup of $\pi ^{ab}$. Let $g:\pi ^{ab}\to \pi ^{ab}/\operatorname {tor} $ be the quotient by torsion. If $M$ is a surface, $\pi ^{ab}/\operatorname {tor} $ is non-canonically isomorphic to $\mathbb {Z} ^{2g}$, where $g$ is the genus; more generally, $\pi ^{ab}/\operatorname {tor} $ is non-canonically isomorphic to $\mathbb {Z} ^{b}$, where $b$ is the first Betti number. Let $\varphi =g\circ f:\pi \to \mathbb {Z} ^{b}$ be the composite homomorphism.
Definition. The cover ${\bar {M}}$ of the manifold $M$ corresponding to the subgroup $\ker(\varphi )\subset \pi $ is called the universal (or maximal) free abelian cover.
Now assume M has a Riemannian metric. Let $E$ be the space of harmonic 1-forms on $M$, with dual $E^{*}$ canonically identified with $H_{1}(M,\mathbb {R} )$. By integrating an integral harmonic 1-form along paths from a basepoint $x_{0}\in M$, we obtain a map to the circle $\mathbb {R} /\mathbb {Z} =S^{1}$.
Similarly, in order to define a map $M\to H_{1}(M,\mathbb {R} )/H_{1}(M,\mathbb {Z} )_{\mathbb {R} }$ without choosing a basis for cohomology, we argue as follows. Let $x$ be a point in the universal cover ${\tilde {M}}$ of $M$. Thus $x$ is represented by a point of $M$ together with a path $c$ from $x_{0}$ to it. By integrating along the path $c$, we obtain a linear form on $E$:
$h\to \int _{c}h.$
This gives rise a map
${\tilde {M}}\to E^{*}=H_{1}(M,\mathbb {R} ),$
which, furthermore, descends to a map
${\begin{cases}{\overline {A}}_{M}:{\overline {M}}\to E^{*}\\c\mapsto \left(h\mapsto \int _{c}h\right)\end{cases}}$
where ${\overline {M}}$ is the universal free abelian cover.
Definition. The Jacobi variety (Jacobi torus) of $M$ is the torus
$J_{1}(M)=H_{1}(M,\mathbb {R} )/H_{1}(M,\mathbb {Z} )_{\mathbb {R} }.$
Definition. The Abel–Jacobi map
$A_{M}:M\to J_{1}(M),$
is obtained from the map above by passing to quotients.
The Abel–Jacobi map is unique up to translations of the Jacobi torus. The map has applications in Systolic geometry. The Abel–Jacobi map of a Riemannian manifold shows up in the large time asymptotics of the heat kernel on a periodic manifold (Kotani & Sunada (2000) and Sunada (2012)).
In much the same way, one can define a graph-theoretic analogue of Abel–Jacobi map as a piecewise-linear map from a finite graph into a flat torus (or a Cayley graph associated with a finite abelian group), which is closely related to asymptotic behaviors of random walks on crystal lattices, and can be used for design of crystal structures.
The Abel-Jacobi map of a compact Riemann surface
We provide an analytic construction of the Abel-Jacobi map on compact Riemann surfaces.
Let $M$ denotes a compact Riemann surface of genus $g>0$. Let $\{a_{1},...,a_{g},b_{1},...,b_{g}\}$ be a canonical homology basis on $M$, and $\{\zeta _{1},...,\zeta _{g}\}$ the dual basis for ${\mathcal {H}}^{1}(M)$, which is a $g$ dimensional complex vector space consists of holomorphic differential forms. Dual basis we mean $\int _{a_{k}}\zeta _{j}=\delta _{jk}$, for $j,k=1,...,g$. We can form a symmetric matrix whose entries are $\int _{b_{k}}\zeta _{j}$, for $j,k=1,...,g$. Let $L$ be the lattice generated by the $2g$-columns of the $g\times 2g$ matrix whose entries consists of $\int _{c_{k}}\zeta _{j}$ for $j,k=1,...,g$ where $c_{k}\in \{a_{k},b_{k}\}$. We call $J(M)={\mathbb {C}}^{g}/L(M)$ the Jacobian variety of $M$ which is a compact, commutative $g$-dimensional complex Lie group.
We can define a map $\varphi :M\to J(M)$ by choosing a point $P_{0}\in M$ and setting $\varphi (P)=\left(\int _{P_{0}}^{P}\zeta _{1},...,\int _{P_{0}}^{P}\zeta _{g}\right).$ which is a well-defined holomorphic mapping with rank 1 (maximal rank). Then we can naturally extend this to a mapping of divisor classes;
If we denote $\mathrm {Div} (M)$ the divisor class group of $M$ then define a map $\varphi :\mathrm {Div} (M)\to J(M)$ :\mathrm {Div} (M)\to J(M)} by setting $\varphi (D)=\sum _{j=1}^{r}\varphi (P_{j})-\sum _{j=1}^{s}\varphi (Q_{j}),\quad D=P_{1}\cdots P_{r}/Q_{1}\cdots Q_{s}.$
Note that if $r=s$ then this map is independent of the choice of the base point so we can define the base point independent map $\varphi _{0}:\mathrm {Div} ^{(0)}(M)\to J(M)$ where $\mathrm {Div} ^{(0)}(M)$ denotes the divisors of degree zero of $M$.
The below Abel's theorem show that the kernel of the map $\varphi _{0}$ is precisely the subgroup of principal divisors. Together with the Jacobi inversion problem, we can say that As a group, $J(M)$ is isomorphic to the group of divisors of degree zero modulo its subgroup of principal divisors.
Abel–Jacobi theorem
The following theorem was proved by Abel (known as Abel's theorem): Suppose that
$D=\sum \nolimits _{i}n_{i}p_{i}$
is a divisor (meaning a formal integer-linear combination of points of C). We can define
$u(D)=\sum \nolimits _{i}n_{i}u(p_{i})$
and therefore speak of the value of the Abel–Jacobi map on divisors. The theorem is then that if D and E are two effective divisors, meaning that the $n_{i}$ are all positive integers, then
$u(D)=u(E)$ if and only if $D$ is linearly equivalent to $E.$ This implies that the Abel-Jacobi map induces an injective map (of abelian groups) from the space of divisor classes of degree zero to the Jacobian.
Jacobi proved that this map is also surjective (known as Jacobi inversion problem), so the two groups are naturally isomorphic.
The Abel–Jacobi theorem implies that the Albanese variety of a compact complex curve (dual of holomorphic 1-forms modulo periods) is isomorphic to its Jacobian variety (divisors of degree 0 modulo equivalence). For higher-dimensional compact projective varieties the Albanese variety and the Picard variety are dual but need not be isomorphic.
References
• E. Arbarello; M. Cornalba; P. Griffiths; J. Harris (1985). "1.3, Abel's Theorem". Geometry of Algebraic Curves, Vol. 1. Grundlehren der Mathematischen Wissenschaften. Springer-Verlag. ISBN 978-0-387-90997-4.
• Kotani, Motoko; Sunada, Toshikazu (2000), "Albanese maps and an off diagonal long time asymptotic for the heat kernel", Comm. Math. Phys., 209: 633–670, Bibcode:2000CMaPh.209..633K, doi:10.1007/s002200050033
• Sunada, Toshikazu (2012), "Lecture on topological crystallography", Japan. J. Math., 7: 1–39, doi:10.1007/s11537-012-1144-4
• Farkas, Hershel M; Kra, Irwin (23 December 1991), Riemann surfaces, New York: Springer, ISBN 978-0387977034
Topics in algebraic curves
Rational curves
• Five points determine a conic
• Projective line
• Rational normal curve
• Riemann sphere
• Twisted cubic
Elliptic curves
Analytic theory
• Elliptic function
• Elliptic integral
• Fundamental pair of periods
• Modular form
Arithmetic theory
• Counting points on elliptic curves
• Division polynomials
• Hasse's theorem on elliptic curves
• Mazur's torsion theorem
• Modular elliptic curve
• Modularity theorem
• Mordell–Weil theorem
• Nagell–Lutz theorem
• Supersingular elliptic curve
• Schoof's algorithm
• Schoof–Elkies–Atkin algorithm
Applications
• Elliptic curve cryptography
• Elliptic curve primality
Higher genus
• De Franchis theorem
• Faltings's theorem
• Hurwitz's automorphisms theorem
• Hurwitz surface
• Hyperelliptic curve
Plane curves
• AF+BG theorem
• Bézout's theorem
• Bitangent
• Cayley–Bacharach theorem
• Conic section
• Cramer's paradox
• Cubic plane curve
• Fermat curve
• Genus–degree formula
• Hilbert's sixteenth problem
• Nagata's conjecture on curves
• Plücker formula
• Quartic plane curve
• Real plane curve
Riemann surfaces
• Belyi's theorem
• Bring's curve
• Bolza surface
• Compact Riemann surface
• Dessin d'enfant
• Differential of the first kind
• Klein quartic
• Riemann's existence theorem
• Riemann–Roch theorem
• Teichmüller space
• Torelli theorem
Constructions
• Dual curve
• Polar curve
• Smooth completion
Structure of curves
Divisors on curves
• Abel–Jacobi map
• Brill–Noether theory
• Clifford's theorem on special divisors
• Gonality of an algebraic curve
• Jacobian variety
• Riemann–Roch theorem
• Weierstrass point
• Weil reciprocity law
Moduli
• ELSV formula
• Gromov–Witten invariant
• Hodge bundle
• Moduli of algebraic curves
• Stable curve
Morphisms
• Hasse–Witt matrix
• Riemann–Hurwitz formula
• Prym variety
• Weber's theorem (Algebraic curves)
Singularities
• Acnode
• Crunode
• Cusp
• Delta invariant
• Tacnode
Vector bundles
• Birkhoff–Grothendieck theorem
• Stable vector bundle
• Vector bundles on algebraic curves
| Wikipedia |
Abel elliptic functions
In mathematics Abel elliptic functions are a special kind of elliptic functions, that were established by the Norwegian mathematician Niels Henrik Abel. He published his paper "Recherches sur les Fonctions elliptiques" in Crelle's Journal in 1827.[1] It was the first work on elliptic functions that was actually published.[2] Abel's work on elliptic functions also influenced Jacobi's studies of elliptic functions, whose 1829 published book "Fundamenta nova theoriae functionum ellipticarum" became the standard work on elliptic functions.[3]
History
Abels starting point were the elliptic integrals which had been studied in great detail by Adrien-Marie Legendre. He began his research in 1823 when he still was a student. In particular he viewed them as complex functions which at that time were still in their infancy. In the following years Abel continued to explore these functions. He also tried to generalize them to functions with even more periods, but seemed to be in no hurry to publish his results.
But in the beginning of the year 1827 he wrote together his first, long presentation Recherches sur les fonctions elliptiques of his discoveries.[4] At the end of the same year he became aware of Carl Gustav Jacobi and his works on new transformations of elliptic integrals. Abel finishes then a second part of his article on elliptic functions and shows in an appendix how the transformation results of Jacobi would easily follow.[5][3] When he then sees the next publication by Jacobi where he makes use of elliptic functions to prove his results without referring to Abel, the Norwegian mathematician finds himself to be in a struggle with Jacobi over priority. He finishes several new articles about related issues, now for the first time dating them, but dies less than a year later in 1829.[6] In the meantime Jacobi completes his great work Fundamenta nova theoriae functionum ellipticarum on elliptic functions which appears the same year as a book. It ended up defining what would be the standard form of elliptic functions in the years that followed.[6]
Derivation from elliptic Integrals
Consider the elliptic integral of the first kind in the following symmetric form:[7]
$\alpha (x):=\int _{0}^{x}{\frac {\mathrm {d} t}{\sqrt {(1-c^{2}t^{2})(1+e^{2}t^{2})}}}$ with $c,e\in \mathbb {R} $.
$\alpha $ is an odd increasing function on the interval ${\bigl [}{-{\tfrac {1}{c}}},{\tfrac {1}{c}}{\bigr ]}$ with the maximum:[2]
${\omega \over 2}:=\int _{0}^{1/c}{\frac {dt}{\sqrt {(1-c^{2}t^{2})(1+e^{2}t^{2})}}}.$
That means $\alpha $ is invertible: There exists a function $\varphi $ such that $x=\varphi (\alpha (x))$, which is well-defined on the interval ${\bigl [}{-{\tfrac {\omega }{2}}},{\tfrac {\omega }{2}}{\bigr ]}$.
Like the function $\alpha $, it depends on the parameters $c$ and $e$ which can be expressed by writing $\varphi (u;e,c)$.
Since $\alpha $ is an odd function, $\varphi $ is also an odd function which means $\varphi (-u)=-\varphi (u)$.
By taking the derivative with respect to $u$ one gets:
$\varphi '(u)={\sqrt {(1-c^{2}\varphi ^{2}(u))(1+e^{2}\varphi ^{2}(u))}}$
which is an even function, i.e., $\varphi (-u)=\varphi (u)$.
Abel introduced the new functions
$f(u)={\sqrt {1-c^{2}\varphi ^{2}(u)}},\;\;\;F(u)={\sqrt {1+e^{2}\varphi ^{2}(u)}}$.
Thereby it holds that[2] $\varphi '(u)=f(u)F(u)$.
$\varphi $, $f$ and $F$ are the functions known as Abel elliptic functions. They can be continued using the addition theorems.
For example adding $\pm {\tfrac {1}{2}}\omega $ one gets:
$\varphi {\big (}u\pm {\omega \over 2}{\big )}=\pm {1 \over c}{f(u) \over F(u)},\quad $$f{\big (}u\pm {\omega \over 2}{\big )}=\mp {\sqrt {c^{2}+e^{2}}}{\varphi (u) \over F(u)},\;\;F{\big (}u\pm {\omega \over 2}{\big )}={{\sqrt {c^{2}+e^{2}}} \over c}{1 \over F(u)}$.
Complex extension
$\varphi $ can be continued onto purely imaginary numbers by introducing the substitution $t\rightarrow it$. One gets $xi=\varphi (\beta i)$, where
$\beta (x)=\int _{0}^{x}{\frac {\mathrm {d} t}{\sqrt {(1+c^{2}t^{2})(1-e^{2}t^{2})}}}$.
$\beta $ is an increasing function on the interval ${\bigl [}{-{\tfrac {1}{e}}},{\tfrac {1}{e}}{\bigr ]}$ with the maximum[8]
${\frac {\tilde {\omega }}{2}}:=\int _{0}^{\frac {1}{e}}{\frac {\mathrm {d} t}{\sqrt {(1+c^{2}t^{2})(1-e^{2}t^{2})}}}$.
That means $\varphi $, $f$ and $F$ are known along the real and imaginary axes. Using the addition theorems again they can be extended onto the complex plane.
For example for $\alpha \in {\bigl [}{-{\tfrac {\omega }{2}}},{\tfrac {\omega }{2}}{\bigr ]}$ yields to
$\varphi (\alpha +{\tfrac {1}{2}}{\tilde {\omega }}i)={\frac {\varphi (\alpha )f({\tfrac {1}{2}}{\tilde {\omega }}i)F({\tfrac {1}{2}}{\tilde {\omega }}i)+\varphi ({\tfrac {1}{2}}{\tilde {\omega }}i)f(\alpha )F(\alpha )}{1+c^{2}e^{2}\varphi ^{2}(\alpha )\varphi ^{2}({\tfrac {1}{2}}{\tilde {\omega }}i)}}={\frac {{\frac {i}{e}}f(\alpha )F(\alpha )}{1+c^{2}e^{2}\varphi ^{2}(\alpha ){\frac {i^{2}}{e^{2}}}}}={\frac {i}{e}}{\frac {f(\alpha )F(\alpha )}{1-c^{2}\varphi ^{2}(\alpha )}}={\frac {i}{e}}{\frac {f(\alpha )F(\alpha )}{f^{2}(\alpha )}}={\frac {i}{e}}{\frac {F(\alpha )}{f(\alpha )}}$.
Double periodicity and poles
The periodicity of $\varphi $, $f$ and $F$ can be shown by applying the addition theorems multiple times. All three functions are doubly periodic which means they have two $\mathbb {R} $-linear independent periods in the complex plane:[9]
$\varphi (\alpha +2\omega )=\varphi (\alpha )=\varphi (\alpha +2{\tilde {\omega }}i)=\varphi (\alpha +\omega +{\tilde {\omega }}i)$
$f(\alpha +2\omega )=f(\alpha )=f(\alpha +{\tilde {\omega }}i)$
$F(\alpha +\omega )=F(\alpha )=F(\alpha +2{\tilde {\omega }}i)$.
The poles of the functions $\varphi (\alpha )$,$f(\alpha )$ and $F(\alpha )$ are at[10]
$\alpha =(m+{\tfrac {1}{2}})\omega +(n+{\tfrac {1}{2}}){\tilde {\omega i}},\quad $ for $m,n\in \mathbb {Z} $.
Relation to Jacobi elliptic functions
Abel's elliptic functions can be expressed by the Jacobi elliptic functions, which do not depend on the parameters $c$ and 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/":): e but on a modulus $k$:
$\varphi (u;c,e)={\frac {1}{c}}\operatorname {sn} (cu,k)$
$f(u;c,e)=\operatorname {cn} (cu,k)$
$F(u;c,e)=\operatorname {dn} (cu,k)$,
where $k={\frac {ie}{c}}$.
Addition Theorems
For the functions $\varphi $, $f$ and $F$ the following addition theorems hold:[8]
$\varphi (\alpha +\beta )={\frac {\varphi (\alpha )f(\beta )F(\beta )+\varphi (\beta )f(\alpha )F(\alpha )}{R}}$
$f(\alpha +\beta )={\frac {f(\alpha )f(\beta )-c^{2}\varphi (\alpha )\varphi (\beta )F(\alpha )F(\beta )}{R}}$
$F(\alpha +\beta )={\frac {F(\alpha )F(\beta )+e^{2}\varphi (\alpha )\varphi (\beta )f(\alpha )f(\beta )}{R}}$,
where $R=1+c^{2}e^{2}\varphi ^{2}(\alpha )\varphi ^{2}(\beta )$.
These follow from the addition theorems for elliptic integrals that Euler already had proven.[8]
References
1. Gray, Jeremy, Real and the complex: a history of analysis in the 19th century, Springer Cham, p. 73, ISBN 978-3-319-23715-2
2. Gray, Jeremy, Real and the complex: a history of analysis in the 19th century, Springer Cham, pp. 74f, ISBN 978-3-319-23715-2
3. Gray, Jeremy, Real and the complex: a history of analysis in the 19th century, Springer Cham, pp. 84f, ISBN 978-3-319-23715-2
4. N.H. Abel, Recherches sur les fonctions elliptiques, Journal für die reine und angewandte Mathematik, 2, 101–181 (1827).
5. N.H. Abel, Recherches sur les fonctions elliptiques, Journal für die reine und angewandte Mathematik, 3, 160–190 (1828).
6. Gray, Jeremy (2015), Real and the complex: a history of analysis in the 19th century (in German), Cham, p. 85, ISBN 978-3-319-23715-2{{citation}}: CS1 maint: location missing publisher (link)
7. Abel, Niels Henrik; Laudal, Olav Arnfinn; Piene, Ragni (2004). The legacy of Niels Henrik Abel: the Abel bicentennial, Oslo, 2002. Berlin: Springer. p. 106. ISBN 3-540-43826-2. OCLC 53919054.
8. Houzel, Christian; Laudal, Olav Arnfinn; Piene, Ragni (2004), The legacy of Niels Henrik Abel: the Abel bicentennial, Oslo, 2002 (in German), Berlin: Springer, p. 107, ISBN 3-540-43826-2
9. Houzel, Christian; Laudal, Olav Arnfinn; Piene, Ragni (2004), The legacy of Niels Henrik Abel: the Abel bicentennial, Oslo, 2002 (in German), Berlin: Springer, p. 108, ISBN 3-540-43826-2
10. Houzel, Christian; Laudal, Olav Arnfinn; Piene, Ragni (2004), The legacy of Niels Henrik Abel: the Abel bicentennial, Oslo, 2002 (in German), Berlin: Springer, p. 109, ISBN 3-540-43826-2
Literature
• Niels Henrik Abel, Recherches sur le fonctions elliptiques Archived 2016-09-13 at the Wayback Machine, first and second part in Sophus Lie and Ludwig Sylow (eds.) Collected Works, Oslo (1881).
• Christian Houzel, The Work of Niels Henrik Abel, in O.A. Laudal and R. Piene, The Legacy of Niels Henrik Abel – The Abel Bicentennial, Oslo 2002, Springer Verlag, Berlin (2004). ISBN 3-540-43826-2.
| Wikipedia |
Abel equation
The Abel equation, named after Niels Henrik Abel, is a type of functional equation of the form
$f(h(x))=h(x+1)$
This article is about certain functional equations. For ordinary differential equations which are cubic in the unknown function, see Abel equation of the first kind.
or
$\alpha (f(x))=\alpha (x)+1$.
The forms are equivalent when α is invertible. h or α control the iteration of f.
Equivalence
The second equation can be written
$\alpha ^{-1}(\alpha (f(x)))=\alpha ^{-1}(\alpha (x)+1)\,.$
Taking x = α−1(y), the equation can be written
$f(\alpha ^{-1}(y))=\alpha ^{-1}(y+1)\,.$
For a known function f(x) , a problem is to solve the functional equation for the function α−1 ≡ h, possibly satisfying additional requirements, such as α−1(0) = 1.
The change of variables sα(x) = Ψ(x), for a real parameter s, brings Abel's equation into the celebrated Schröder's equation, Ψ(f(x)) = s Ψ(x) .
The further change F(x) = exp(sα(x)) into Böttcher's equation, F(f(x)) = F(x)s.
The Abel equation is a special case of (and easily generalizes to) the translation equation,[1]
$\omega (\omega (x,u),v)=\omega (x,u+v)~,$
e.g., for $\omega (x,1)=f(x)$,
$\omega (x,u)=\alpha ^{-1}(\alpha (x)+u)$. (Observe ω(x,0) = x.)
The Abel function α(x) further provides the canonical coordinate for Lie advective flows (one parameter Lie groups).
See also: Iterated function § Abelian property and Iteration sequences
History
Initially, the equation in the more general form [2] [3] was reported. Even in the case of a single variable, the equation is non-trivial, and admits special analysis.[4][5][6]
In the case of a linear transfer function, the solution is expressible compactly.[7]
Special cases
The equation of tetration is a special case of Abel's equation, with f = exp.
In the case of an integer argument, the equation encodes a recurrent procedure, e.g.,
$\alpha (f(f(x)))=\alpha (x)+2~,$
and so on,
$\alpha (f_{n}(x))=\alpha (x)+n~.$
Solutions
The Abel equation has at least one solution on $E$ if and only if for all $x\in E$ and all $n\in \mathbb {N} $, $f^{n}(x)\neq x$, where $f^{n}=f\circ f\circ ...\circ f$, is the function f iterated n times.[8]
Analytic solutions (Fatou coordinates) can be approximated by asymptotic expansion of a function defined by power series in the sectors around a parabolic fixed point.[9] The analytic solution is unique up to a constant.[10]
See also
• Functional equation
• Schröder's equation
• Böttcher's equation
• Infinite compositions of analytic functions
• Iterated function
• Shift operator
• Superfunction
References
1. Aczél, János, (1966): Lectures on Functional Equations and Their Applications, Academic Press, reprinted by Dover Publications, ISBN 0486445232 .
2. Abel, N.H. (1826). "Untersuchung der Functionen zweier unabhängig veränderlichen Größen x und y, wie f(x, y), welche die Eigenschaft haben, ..." Journal für die reine und angewandte Mathematik. 1: 11–15.
3. A. R. Schweitzer (1912). "Theorems on functional equations". Bull. Amer. Math. Soc. 19 (2): 51–106. doi:10.1090/S0002-9904-1912-02281-4.
4. Korkine, A (1882). "Sur un problème d'interpolation", Bull Sci Math & Astron 6(1) 228—242. online
5. G. Belitskii; Yu. Lubish (1999). "The real-analytic solutions of the Abel functional equations" (PDF). Studia Mathematica. 134 (2): 135–141.
6. Jitka Laitochová (2007). "Group iteration for Abel's functional equation". Nonlinear Analysis: Hybrid Systems. 1 (1): 95–102. doi:10.1016/j.nahs.2006.04.002.
7. G. Belitskii; Yu. Lubish (1998). "The Abel equation and total solvability of linear functional equations" (PDF). Studia Mathematica. 127: 81–89.
8. R. Tambs Lyche,Sur l'équation fonctionnelle d'Abel, University of Trondlyim, Norvege
9. Dudko, Artem (2012). Dynamics of holomorphic maps: Resurgence of Fatou coordinates, and Poly-time computability of Julia sets Ph.D. Thesis
10. Classifications of parabolic germs and fractal properties of orbits by Maja Resman, University of Zagreb, Croatia
| Wikipedia |
Abel equation of the first kind
In mathematics, an Abel equation of the first kind, named after Niels Henrik Abel, is any ordinary differential equation that is cubic in the unknown function. In other words, it is an equation of the form
$y'=f_{3}(x)y^{3}+f_{2}(x)y^{2}+f_{1}(x)y+f_{0}(x)\,$
This article is about certain differential equations. For certain functional equations named after Abel, see Abel equation.
where $f_{3}(x)\neq 0$. If $f_{3}(x)=0$ and $f_{0}(x)=0$, or $f_{2}(x)=0$ and $f_{0}(x)=0$, the equation reduces to a Bernoulli equation, while if $f_{3}(x)=0$ the equation reduces to a Riccati equation.
Properties
The substitution $y={\dfrac {1}{u}}$ brings the Abel equation of the first kind to the "Abel equation of the second kind" of the form
$uu'=-f_{0}(x)u^{3}-f_{1}(x)u^{2}-f_{2}(x)u-f_{3}(x).\,$
The substitution
${\begin{aligned}\xi &=\int f_{3}(x)E^{2}~dx,\\[6pt]u&=\left(y+{\dfrac {f_{2}(x)}{3f_{3}(x)}}\right)E^{-1},\\[6pt]E&=\exp \left(\int \left(f_{1}(x)-{\frac {f_{2}^{2}(x)}{3f_{3}(x)}}\right)~dx\right)\end{aligned}}$
brings the Abel equation of the first kind to the canonical form
$u'=u^{3}+\phi (\xi ).\,$
Dimitrios E. Panayotounakos and Theodoros I. Zarmpoutis discovered an analytic method to solve the above equation in an implicit form.[1]
Notes
1. Panayotounakos, Dimitrios E.; Zarmpoutis, Theodoros I. (2011). "Construction of Exact Parametric or Closed Form Solutions of Some Unsolvable Classes of Nonlinear ODEs (Abel's Nonlinear ODEs of the First Kind and Relative Degenerate Equations)". International Journal of Mathematics and Mathematical Sciences. Hindawi Publishing Corporation. 2011: 1–13. doi:10.1155/2011/387429.
References
• Panayotounakos, D.E.; Panayotounakou, N.D.; Vakakis, A.F.A (2002). "On the Solution of the Unforced Damped Duffing Oscillator with No Linear Stiffness Term". Nonlinear Dynamics. 28: 1–16. doi:10.1023/A:1014925032022. S2CID 117115358.. (Old link: On the Solution of the Unforced Damped Duffing Oscillator with No Linear Stiffness Term)
• Construction of Exact Parametric or Closed Form Solutions of Some Unsolvable Classes of Nonlinear ODEs (Abel's Nonlinear ODEs of the First Kind and Relative Degenerate Equations)
• Mancas, Stefan C., Rosu, Haret C., Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations. Physics Letters A 377 (2013) 1434–1438. [arXiv.org:1212.3636v3]
| Wikipedia |
Abel's theorem
In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel.
This article is about Abel's theorem on power series. For Abel's theorem on algebraic curves, see Abel–Jacobi map. For Abel's theorem on the insolubility of the quintic equation, see Abel–Ruffini theorem. For Abel's theorem on linear differential equations, see Abel's identity. For Abel's theorem on irreducible polynomials, see Abel's irreducibility theorem. For Abel's formula for summation of a series, using an integral, see Abel's summation formula.
Theorem
Let the Taylor series
$G(x)=\sum _{k=0}^{\infty }a_{k}x^{k}$
be a power series with real coefficients $a_{k}$ with radius of convergence $1.$ Suppose that the series
$\sum _{k=0}^{\infty }a_{k}$
converges. Then $G(x)$ is continuous from the left at $x=1,$ that is,
$\lim _{x\to 1^{-}}G(x)=\sum _{k=0}^{\infty }a_{k}.$
The same theorem holds for complex power series
$G(z)=\sum _{k=0}^{\infty }a_{k}z^{k},$
provided that $z\to 1$ entirely within a single Stolz sector, that is, a region of the open unit disk where
$|1-z|\leq M(1-|z|)$
for some fixed finite $M>1$. Without this restriction, the limit may fail to exist: for example, the power series
$\sum _{n>0}{\frac {z^{3^{n}}-z^{2\cdot 3^{n}}}{n}}$
converges to $0$ at $z=1,$ but is unbounded near any point of the form $e^{\pi i/3^{n}},$ so the value at $z=1$ is not the limit as $z$ tends to 1 in the whole open disk.
Note that $G(z)$ is continuous on the real closed interval $[0,t]$ for $t<1,$ by virtue of the uniform convergence of the series on compact subsets of the disk of convergence. Abel's theorem allows us to say more, namely that $G(z)$ is continuous on $[0,1].$
Stolz sector
The Stolz sector $|1-z|\leq M(1-|z|)$ has explicit formula
$y^{2}=-{\frac {M^{4}(x^{2}-1)-2M^{2}((x-1)x+1)+2{\sqrt {M^{4}(-2M^{2}(x-1)+2x-1)}}+(x-1)^{2}}{(M^{2}-1)^{2}}}$
and is plotted on the right for various values.
The left end of the sector is $x={\frac {1-M}{1+M}}$, and the right end is $x=1$. On the right end, it becomes a cone with angle $2\theta $, where $\cos \theta ={\frac {1}{M}}$.
Remarks
As an immediate consequence of this theorem, if $z$ is any nonzero complex number for which the series
$\sum _{k=0}^{\infty }a_{k}z^{k}$
converges, then it follows that
$\lim _{t\to 1^{-}}G(tz)=\sum _{k=0}^{\infty }a_{k}z^{k}$
in which the limit is taken from below.
The theorem can also be generalized to account for sums which diverge to infinity. If
$\sum _{k=0}^{\infty }a_{k}=\infty $
then
$\lim _{z\to 1^{-}}G(z)\to \infty .$
However, if the series is only known to be divergent, but for reasons other than diverging to infinity, then the claim of the theorem may fail: take, for example, the power series for
${\frac {1}{1+z}}.$
At $z=1$ the series is equal to $1-1+1-1+\cdots ,$ but ${\tfrac {1}{1+1}}={\tfrac {1}{2}}.$
We also remark the theorem holds for radii of convergence other than $R=1$: let
$G(x)=\sum _{k=0}^{\infty }a_{k}x^{k}$
be a power series with radius of convergence $R,$ and suppose the series converges at $x=R.$ Then $G(x)$ is continuous from the left at $x=R,$ that is,
$\lim _{x\to R^{-}}G(x)=G(R).$
Applications
The utility of Abel's theorem is that it allows us to find the limit of a power series as its argument (that is, $z$) approaches $1$ from below, even in cases where the radius of convergence, $R,$ of the power series is equal to $1$ and we cannot be sure whether the limit should be finite or not. See for example, the binomial series. Abel's theorem allows us to evaluate many series in closed form. For example, when
$a_{k}={\frac {(-1)^{k}}{k+1}},$
we obtain
$G_{a}(z)={\frac {\ln(1+z)}{z}},\qquad 0<z<1,$
by integrating the uniformly convergent geometric power series term by term on $[-z,0]$; thus the series
$\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{k+1}}$
converges to $\ln(2)$ by Abel's theorem. Similarly,
$\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}$
converges to $\arctan(1)={\tfrac {\pi }{4}}.$
$G_{a}(z)$ is called the generating function of the sequence $a.$ Abel's theorem is frequently useful in dealing with generating functions of real-valued and non-negative sequences, such as probability-generating functions. In particular, it is useful in the theory of Galton–Watson processes.
Outline of proof
After subtracting a constant from $a_{0},$ we may assume that $\sum _{k=0}^{\infty }a_{k}=0.$ Let $s_{n}=\sum _{k=0}^{n}a_{k}\!.$ Then substituting $a_{k}=s_{k}-s_{k-1}$ and performing a simple manipulation of the series (summation by parts) results in
$G_{a}(z)=(1-z)\sum _{k=0}^{\infty }s_{k}z^{k}.$
Given $\varepsilon >0,$ pick $n$ large enough so that $|s_{k}|<\varepsilon $ for all $k\geq n$ and note that
$\left|(1-z)\sum _{k=n}^{\infty }s_{k}z^{k}\right|\leq \varepsilon |1-z|\sum _{k=n}^{\infty }|z|^{k}=\varepsilon |1-z|{\frac {|z|^{n}}{1-|z|}}<\varepsilon M$
when $z$ lies within the given Stolz angle. Whenever $z$ is sufficiently close to $1$ we have
$\left|(1-z)\sum _{k=0}^{n-1}s_{k}z^{k}\right|<\varepsilon ,$
so that $\left|G_{a}(z)\right|<(M+1)\varepsilon $ when $z$ is both sufficiently close to $1$ and within the Stolz angle.
Related concepts
Converses to a theorem like Abel's are called Tauberian theorems: There is no exact converse, but results conditional on some hypothesis. The field of divergent series, and their summation methods, contains many theorems of abelian type and of tauberian type.
See also
• Abel's summation formula – Integration by parts version of Abel's method for summation by parts
• Nachbin resummation – Theorem bounding the growth rate of analytic functionsPages displaying short descriptions of redirect targets
• Summation by parts – Theorem to simplify sums of products of sequences
Further reading
• Ahlfors, Lars Valerian (September 1, 1980). Complex Analysis (Third ed.). McGraw Hill Higher Education. pp. 41–42. ISBN 0-07-085008-9. - Ahlfors called it Abel's limit theorem.
External links
• Abel summability at PlanetMath. (a more general look at Abelian theorems of this type)
• A.A. Zakharov (2001) [1994], "Abel summation method", Encyclopedia of Mathematics, EMS Press
• Weisstein, Eric W. "Abel's Convergence Theorem". MathWorld.
| Wikipedia |
Abel polynomials
The Abel polynomials in mathematics form a polynomial sequence, the nth term of which is of the form
$p_{n}(x)=x(x-an)^{n-1}.$
The sequence is named after Niels Henrik Abel (1802-1829), the Norwegian mathematician.
This polynomial sequence is of binomial type: conversely, every polynomial sequence of binomial type may be obtained from the Abel sequence in the umbral calculus.
Examples
For a = 1, the polynomials are (sequence A137452 in the OEIS)
$p_{0}(x)=1;$
$p_{1}(x)=x;$
$p_{2}(x)=-2x+x^{2};$
$p_{3}(x)=9x-6x^{2}+x^{3};$
$p_{4}(x)=-64x+48x^{2}-12x^{3}+x^{4};$
For a = 2, the polynomials are
$p_{0}(x)=1;$
$p_{1}(x)=x;$
$p_{2}(x)=-4x+x^{2};$
$p_{3}(x)=36x-12x^{2}+x^{3};$
$p_{4}(x)=-512x+192x^{2}-24x^{3}+x^{4};$
$p_{5}(x)=10000x-4000x^{2}+600x^{3}-40x^{4}+x^{5};$
$p_{6}(x)=-248832x+103680x^{2}-17280x^{3}+1440x^{4}-60x^{5}+x^{6};$
References
• Rota, Gian-Carlo; Shen, Jianhong; Taylor, Brian D. (1997). "All Polynomials of Binomial Type Are Represented by Abel Polynomials". Annali della Scuola Normale Superiore di Pisa - Classe di Scienze. Series 4. 25 (3–4): 731–738. MR 1655539. Zbl 1003.05011.
External links
• Weisstein, Eric W. "Abel Polynomial". MathWorld.
| Wikipedia |
Abel Prize
The Abel Prize (/ˈɑːbəl/ AH-bəl; Norwegian: Abelprisen [ˈɑ̀ːbl̩ˌpriːsn̩]) is awarded annually by the King of Norway to one or more outstanding mathematicians.[1] It is named after the Norwegian mathematician Niels Henrik Abel (1802–1829) and directly modeled after the Nobel Prizes.[2][3][4][5][6][7][8] It comes with a monetary award of 7.5 million Norwegian kroner (NOK; increased from 6 million NOK in 2019).
Abel Prize
Awarded forOutstanding scientific work in the field of mathematics
CountryNorway
Presented byGovernment of Norway
First awarded2003
Websitewww.abelprize.no
The Abel Prize's history dates back to 1899, when its establishment was proposed by the Norwegian mathematician Sophus Lie when he learned that Alfred Nobel's plans for annual prizes would not include a prize in mathematics. In 1902, King Oscar II of Sweden and Norway indicated his willingness to finance the creation of a mathematics prize to complement the Nobel Prizes, but the establishment of the prize was prevented by the dissolution of the union between Norway and Sweden in 1905. It took almost a century before the prize was finally established by the Government of Norway in 2001, and it was specifically intended "to give the mathematicians their own equivalent of a Nobel Prize."[7] The laureates are selected by the Abel Committee, the members of whom are appointed by the Norwegian Academy of Science and Letters.
The award ceremony takes place in the aula of the University of Oslo, where the Nobel Peace Prize was awarded between 1947 and 1989.[9] The Abel Prize board has also established an Abel symposium, administered by the Norwegian Mathematical Society, which takes place twice a year.[10]
History
The prize was first proposed in 1899, to be part of the celebration of the 100th anniversary of Niels Henrik Abel's birth in 1802.[11] The Norwegian mathematician Sophus Lie proposed establishing an Abel Prize when he learned that Alfred Nobel's plans for annual prizes would not include a prize in mathematics. King Oscar II was willing to finance a mathematics prize in 1902, and the mathematicians Ludwig Sylow and Carl Størmer drew up statutes and rules for the proposed prize. However, Lie's influence decreased after his death, and the dissolution of the union between Sweden and Norway in 1905 ended the first attempt to create an Abel Prize.[11]
After interest in the concept of the prize had risen in 2001, a working group was formed to develop a proposal, which was presented to the Prime Minister of Norway in May. In August 2001, the Norwegian government announced that the prize would be awarded beginning in 2002, the two-hundredth anniversary of Abel's birth. Atle Selberg received an honorary Abel Prize in 2002, but the first actual Abel Prize was awarded in 2003.[11][12]
A book series presenting Abel Prize laureates and their research was commenced in 2010. The first three volumes cover the years 2003–2007, 2008–2012, and 2013-2017 respectively.[13][14][15]
In 2019, Karen Uhlenbeck became the first woman to win the Abel Prize, with the award committee citing "the fundamental impact of her work on analysis, geometry and mathematical physics.[16] The Bernt Michael Holmboe Memorial Prize was created in 2005. Named after Abel's teacher, it promotes excellence in teaching.[17]
Selection criteria and funding
Anyone may submit a nomination for the Abel Prize, although self-nominations are not permitted. The nominee must be alive. If the awardee dies after being declared the winner, the prize will be awarded posthumously.
The Norwegian Academy of Science and Letters declares the winner of the Abel Prize each March after recommendation by the Abel Committee, which consists of five leading mathematicians. Both Norwegians and non-Norwegians may serve on the Committee. They are elected by the Norwegian Academy of Science and Letters and nominated by the International Mathematical Union and the European Mathematical Society.[11][18] As of 2022, the committee is chaired by Norwegian mathematician Helge Holden[19] and before then was headed by Hans Munthe Kaas, John Rognes, Ragni Piene, Kristian Seip, and Erling Stormer.[20]
Funding
The Norwegian Government gave the prize an initial funding of NOK 200 million (about €21.7 million[21]) in 2001. Previously, the funding came from the Abel foundation, but today the prize is financed directly through the national budget.
The funding is controlled by the Board, which consists of members elected by the Norwegian Academy of Science and Letters.[18] The current board consists of Ingrid K. Glad (chair), Aslak Bakke Buan, Helge K. Dahle, Kristin Vinje, Cordian Riener and Gunn Elisabeth Birkelund.[22]
Laureates
Year Laureate(s) Image Institution(s) Citation
2003 Jean-Pierre Serre Collège de France "For playing a key role in shaping the modern form of many parts of mathematics, including topology, algebraic geometry and number theory."[23]
2004 Michael Atiyah University of Edinburgh
University of Cambridge
"For their discovery and proof of the index theorem, bringing together topology, geometry and analysis, and their outstanding role in building new bridges between mathematics and theoretical physics."[24]
Isadore Singer Massachusetts Institute of Technology
University of California, Berkeley
2005 Peter Lax Courant Institute (NYU) "For his groundbreaking contributions to the theory and application of partial differential equations and to the computation of their solutions."[25]
2006 Lennart Carleson Royal Institute of Technology "For his profound and seminal contributions to harmonic analysis and the theory of smooth dynamical systems."[26]
2007 S. R. Srinivasa Varadhan Courant Institute (NYU) "For his fundamental contributions to probability theory and in particular for creating a unified theory of large deviation."[27]
2008 John G. Thompson University of Florida "For their profound achievements in algebra and in particular for shaping modern group theory."[28]
Jacques Tits Collège de France
2009 Mikhail Gromov Institut des Hautes Études Scientifiques[29] and Courant Institute[30] "For his revolutionary contributions to geometry."[31]
2010 John Tate University of Texas at Austin "For his vast and lasting impact on the theory of numbers."[32]
2011 John Milnor Stony Brook University "For pioneering discoveries in topology, geometry, and algebra."[33]
2012 Endre Szemerédi Alfréd Rényi Institute
and Rutgers University
"For his fundamental contributions to discrete mathematics and theoretical computer science, and in recognition of the profound and lasting impact of these contributions on additive number theory and ergodic theory."[34]
2013 Pierre Deligne Institute for Advanced Study "For seminal contributions to algebraic geometry and for their transformative impact on number theory, representation theory, and related fields."[35]
2014 Yakov Sinai Princeton University and Landau Institute for Theoretical Physics[36] "For his fundamental contributions to dynamical systems, ergodic theory, and mathematical physics."[37]
2015 John F. Nash Jr. Princeton University "For striking and seminal contributions to the theory of nonlinear partial differential equations and its applications to geometric analysis."[38]
Louis Nirenberg Courant Institute (NYU)
2016 Andrew Wiles University of Oxford[39][40] "For his stunning proof of Fermat's Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory."[41]
2017 Yves Meyer École normale supérieure Paris-Saclay "For his pivotal role in the development of the mathematical theory of wavelets."[42]
2018 Robert Langlands Institute for Advanced Study "For his visionary program connecting representation theory to number theory."[43]
2019 Karen Uhlenbeck University of Texas at Austin "For her pioneering achievements in geometric partial differential equations, gauge theory and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics."[44]
2020 Hillel Furstenberg Hebrew University of Jerusalem "For pioneering the use of methods from probability and dynamics in group theory, number theory and combinatorics."[45]
Grigory Margulis Yale University
2021 László Lovász Eötvös Loránd University "For their foundational contributions to theoretical computer science and discrete mathematics, and their leading role in shaping them into central fields of modern mathematics".[46]
Avi Wigderson Institute for Advanced Study
2022 Dennis Sullivan Stony Brook University and The Graduate Center, CUNY "For his groundbreaking contributions to topology in its broadest sense, and in particular its algebraic, geometric and dynamical aspects."[47]
2023 Luis Caffarelli University of Texas at Austin "For his seminal contributions to regularity theory for nonlinear partial differential equations including free-boundary problems and the Monge–Ampère equation."[48]
See also
• Fields Medal
• List of prizes known as the Nobel of a field
• List of mathematics awards
References
1. "Statutes for Niels Henrik Abel's Prize in Mathematics (The Abel Prize)". Retrieved 21 July 2022.
2. Chang, Kenneth (20 March 2018). "Robert P. Langlands Is Awarded the Abel Prize, a Top Math Honor". The New York Times. Archived from the original on 3 April 2019. Retrieved 23 March 2018.
3. Dreifus, Claudia (29 March 2005). "From Budapest to Los Alamos, a Life in Mathematics". The New York Times. Archived from the original on 29 May 2015. Retrieved 18 February 2017.
4. Cipra, Barry A. (26 March 2009). "Russian Mathematician Wins Abel Prize". ScienceNOW. Archived from the original on 29 March 2009. Retrieved 29 March 2009.
5. Laursen, Lucas (26 March 2009). "Geometer wins maths 'Nobel'". Nature. doi:10.1038/news.2009.196. Archived from the original on 22 March 2019. Retrieved 17 October 2012.
6. Foderaro, Lisa W. (31 May 2009). "In N.Y.U.'s Tally of Abel Prizes for Mathematics, Gromov Makes Three". The New York Times. Archived from the original on 2 April 2019. Retrieved 17 October 2012.
7. Devlin, Keith (April 2004). "Abel Prize Awarded: The Mathematicians' Nobel". Mathematical Association of America. Archived from the original on 27 August 2012. Retrieved 4 November 2012.
8. Piergiorgio Odifreddi; Arturo Sangalli (2006). The Mathematical Century: The 30 Greatest Problems of the Last 100 Years. Princeton University Press. p. 6. ISBN 0-691-12805-7. Archived from the original on 29 December 2016. Retrieved 23 March 2016.
9. "University of Oslo". Oslo Opera House. Archived from the original on 3 August 2018. Retrieved 22 December 2012.
10. "Main Page". The Norwegian Academy of Science and Letters. Archived from the original on 18 June 2019. Retrieved 26 July 2012.
11. "The History of the Abel Prize". www.abelprize.no. Retrieved 21 July 2022.
12. O'Connor, John J.; Robertson, Edmund F., "Atle Selberg", MacTutor History of Mathematics Archive, University of St Andrews
13. H. Holden; R. Piene, eds. (2010). The Abel Prize 2003–2007. The Abel Prize. Heidelberg: Springer. doi:10.1007/978-3-642-01373-7. ISBN 978-3-642-01372-0. Archived from the original on 1 November 2009. Retrieved 28 August 2017.
14. H. Holden; R. Piene, eds. (2014). The Abel Prize 2008–2012. The Abel Prize. Heidelberg: Springer. doi:10.1007/978-3-642-39449-2. ISBN 978-3-642-39449-2. Archived from the original on 21 February 2015. Retrieved 28 August 2017.
15. H. Holden; R. Piene, eds. (2019). The Abel Prize 2013-2017. The Abel Prize. Heidelberg: Springer. doi:10.1007/978-3-319-99028-6. ISBN 978-3-319-99027-9. S2CID 239378974. Archived from the original on 6 August 2019. Retrieved 6 August 2019.
16. Change, Kenneth (19 March 2019). "Karen Uhlenbeck Is First Woman to Receive Abel Prize in Mathematics". The New York Times. Archived from the original on 4 May 2019. Retrieved 19 March 2019.
17. "Abel Prize | mathematics award". Encyclopedia Britannica. Archived from the original on 26 January 2020. Retrieved 19 May 2020.
18. "Nomination Guidelines". The Norwegian Academy of Science and Letters. Archived from the original on 2 August 2018. Retrieved 26 July 2012.
19. "The Abel Committee". www.abelprize.no. Retrieved 30 December 2022.
20. "Former Abel committees". www.abelprize.no. Retrieved 30 December 2022.
21. "Google Currency Converter". Archived from the original on 28 March 2017. Retrieved 27 March 2017.
22. "The Abel Board". www.abelprize.no. Retrieved 30 December 2022.
23. "2003: Jean-Pierre Serre". The Norwegian Academy of Science and Letters. Retrieved 21 July 2022.
24. "2004: Sir Michael Francis Atiyah and Isadore M. Singer". The Norwegian Academy of Science and Letters. Retrieved 21 July 2022.
25. "2005: Peter D. Lax". The Norwegian Academy of Science and Letters. Retrieved 21 July 2022.
26. "2006: Lennart Carleson". The Norwegian Academy of Science and Letters. Retrieved 21 July 2022.
27. "2007: Srinivasa S. R. Varadhan". The Norwegian Academy of Science and Letters. Retrieved 21 July 2022.
28. "2008: John Griggs Thompson and Jacques Tits". The Norwegian Academy of Science and Letters. Retrieved 21 July 2022.
29. "The Abel Committee's Citation 2009". The Norwegian Academy of Science and Letters. Archived from the original on 22 August 2016. Retrieved 9 August 2016.
30. Foderaro, Lisa W. (31 May 2009). "In N.Y.U.'s Tally of Abel Prizes for Mathematics, Gromov Makes Three". The New York Times. Archived from the original on 2 April 2019. Retrieved 17 October 2012.
31. "2009: Mikhail Leonidovich Gromov". The Norwegian Academy of Science and Letters. Retrieved 21 July 2022.
32. "2010: John Torrence Tate". The Norwegian Academy of Science and Letters. Retrieved 21 July 2022.
33. "2011: John Milnor". The Norwegian Academy of Science and Letters. Retrieved 21 July 2022.
34. "2012: Endre Szemerédi". The Norwegian Academy of Science and Letters. Retrieved 21 July 2022.
35. "The Abel Prize Laureate 2013". The Norwegian Academy of Science and Letters. Retrieved 21 July 2022.
36. "The Abel Committee's Citation 2014". The Norwegian Academy of Science and Letters. Archived from the original on 22 August 2016. Retrieved 9 August 2016.
37. "2014: Yakov G. Sinai". The Norwegian Academy of Science and Letters. Retrieved 21 July 2022.
38. "2015: John F. Nash and Louis Nirenberg". The Norwegian Academy of Science and Letters. Retrieved 21 July 2022.
39. "The Abel Committee's Citation 2016". The Norwegian Academy of Science and Letters. Archived from the original on 2 August 2016. Retrieved 9 August 2016.
40. "Sir Andrew J. Wiles receives the Abel Prize" (Press release). The Norwegian Academy of Science and Letters. Archived from the original on 22 August 2016. Retrieved 9 August 2016.
41. "2016: Sir Andrew J. Wiles". The Norwegian Academy of Science and Letters. Retrieved 21 July 2022.
42. "2017: Yves Meyer". The Norwegian Academy of Science and Letters. Retrieved 21 July 2022.
43. "2018: Robert P. Langlands". The Norwegian Academy of Science and Letters. Retrieved 21 July 2022.
44. "2019: Karen Keskulla Uhlenbeck". The Norwegian Academy of Science and Letters. Retrieved 21 July 2022.
45. "2020: Hillel Furstenberg and Gregory Margulis". The Norwegian Academy of Science and Letters. Retrieved 21 July 2022.
46. "2021: László Lovász and Avi Wigderson". abelprize.no. Retrieved 21 July 2022.
47. "Prize winner 2022". The Norwegian Academy of Science and Letters. Retrieved 25 March 2022.
48. "Prize winner 2023". The Norwegian Academy of Science and Letters. Retrieved 22 March 2023.
External links
Wikimedia Commons has media related to Abel Prize.
• Official website
• Official website of the Abel Symposium
• Barile, Margherita & Weisstein, Eric W. "Abel Prize". MathWorld.
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
International mathematical activities
Organizations and Projects
• International Mathematical Union
• International Association for Mathematics and Computers in Simulation
• International Association of Mathematical Physics
• International Commission on the History of Mathematics
• International Congress of Chinese Mathematicians
• International Council for Industrial and Applied Mathematics
• International Linear Algebra Society
• International Society for Mathematical Sciences
• International Mathematical Knowledge Trust
• International Society for the Interaction of Mechanics and Mathematics
• International Workshop on Operator Theory and its Applications
• The Bridges Organization
• International Congress of Mathematicians
Competitions
• International Mathematical Olympiad
• International Mathematical Olympiad selection process
• International Mathematical Modeling Challenge
• Mathematical Kangaroo
• International Mathematics Competition for University Students
Awards
• Fields Medal
• Abel Prize
• International Giovanni Sacchi Landriani Prize
• Brouwer Medal
• David Hilbert Award
• Kolmogorov Medal
• Lobachevsky Prize
• Hans Schneider Prize in Linear Algebra
| Wikipedia |
Abel transform
In mathematics, the Abel transform,[1] named for Niels Henrik Abel, is an integral transform often used in the analysis of spherically symmetric or axially symmetric functions. The Abel transform of a function f(r) is given by
$F(y)=2\int _{y}^{\infty }{\frac {f(r)r}{\sqrt {r^{2}-y^{2}}}}\,dr.$
Assuming that f(r) drops to zero more quickly than 1/r, the inverse Abel transform is given by
$f(r)=-{\frac {1}{\pi }}\int _{r}^{\infty }{\frac {dF}{dy}}\,{\frac {dy}{\sqrt {y^{2}-r^{2}}}}.$
In image analysis, the forward Abel transform is used to project an optically thin, axially symmetric emission function onto a plane, and the inverse Abel transform is used to calculate the emission function given a projection (i.e. a scan or a photograph) of that emission function.
In absorption spectroscopy of cylindrical flames or plumes, the forward Abel transform is the integrated absorbance along a ray with closest distance y from the center of the flame, while the inverse Abel transform gives the local absorption coefficient at a distance r from the center. Abel transform is limited to applications with axially symmetric geometries. For more general asymmetrical cases, more general-oriented reconstruction algorithms such as algebraic reconstruction technique (ART), maximum likelihood expectation maximization (MLEM), filtered back-projection (FBP) algorithms should be employed.
In recent years, the inverse Abel transform (and its variants) has become the cornerstone of data analysis in photofragment-ion imaging and photoelectron imaging. Among recent most notable extensions of inverse Abel transform are the "onion peeling" and "basis set expansion" (BASEX) methods of photoelectron and photoion image analysis.
Geometrical interpretation
In two dimensions, the Abel transform F(y) can be interpreted as the projection of a circularly symmetric function f(r) along a set of parallel lines of sight at a distance y from the origin. Referring to the figure on the right, the observer (I) will see
$F(y)=\int _{-\infty }^{\infty }f\left({\sqrt {x^{2}+y^{2}}}\right)\,dx,$
where f(r) is the circularly symmetric function represented by the gray color in the figure. It is assumed that the observer is actually at x = ∞, so that the limits of integration are ±∞, and all lines of sight are parallel to the x axis. Realizing that the radius r is related to x and y as r2 = x2 + y2, it follows that
$dx={\frac {r\,dr}{\sqrt {r^{2}-y^{2}}}}$
for x > 0. Since f(r) is an even function in x, we may write
$F(y)=2\int _{0}^{\infty }f\left({\sqrt {x^{2}+y^{2}}}\right)\,dx=2\int _{|y|}^{\infty }f(r)\,{\frac {r\,dr}{\sqrt {r^{2}-y^{2}}}},$
which yields the Abel transform of f(r).
The Abel transform may be extended to higher dimensions. Of particular interest is the extension to three dimensions. If we have an axially symmetric function f(ρ, z), where ρ2 = x2 + y2 is the cylindrical radius, then we may want to know the projection of that function onto a plane parallel to the z axis. Without loss of generality, we can take that plane to be the yz plane, so that
$F(y,z)=\int _{-\infty }^{\infty }f(\rho ,z)\,dx=2\int _{y}^{\infty }{\frac {f(\rho ,z)\rho \,d\rho }{\sqrt {\rho ^{2}-y^{2}}}},$
which is just the Abel transform of f(ρ, z) in ρ and y.
A particular type of axial symmetry is spherical symmetry. In this case, we have a function f(r), where r2 = x2 + y2 + z2. The projection onto, say, the yz plane will then be circularly symmetric and expressible as F(s), where s2 = y2 + z2. Carrying out the integration, we have
$F(s)=\int _{-\infty }^{\infty }f(r)\,dx=2\int _{s}^{\infty }{\frac {f(r)r\,dr}{\sqrt {r^{2}-s^{2}}}},$
which is again, the Abel transform of f(r) in r and s.
Verification of the inverse Abel transform
Assuming $f$ is continuously differentiable and $f$, $f'$ drop to zero faster than $1/r$, we can set $u=f(r)$ and $v={\sqrt {r^{2}-y^{2}}}$. Integration by parts then yields
$F(y)=-2\int _{y}^{\infty }f'(r){\sqrt {r^{2}-y^{2}}}\,dr.$
Differentiating formally,
$F'(y)=2y\int _{y}^{\infty }{\frac {f'(r)}{\sqrt {r^{2}-y^{2}}}}\,dr.$
Now substitute this into the inverse Abel transform formula:
$-{\frac {1}{\pi }}\int _{r}^{\infty }{\frac {F'(y)}{\sqrt {y^{2}-r^{2}}}}\,dy=\int _{r}^{\infty }\int _{y}^{\infty }{\frac {-2y}{\pi {\sqrt {(y^{2}-r^{2})(s^{2}-y^{2})}}}}f'(s)\,dsdy.$
By Fubini's theorem, the last integral equals
$\int _{r}^{\infty }\int _{r}^{s}{\frac {-2y}{\pi {\sqrt {(y^{2}-r^{2})(s^{2}-y^{2})}}}}\,dyf'(s)\,ds=\int _{r}^{\infty }(-1)f'(s)\,ds=f(r).$
Generalization of the Abel transform to discontinuous F(y)
Consider the case where $F(y)$ is discontinuous at $y=y_{\Delta }$, where it abruptly changes its value by a finite amount $\Delta F$. That is, $y_{\Delta }$ and $\Delta F$ are defined by $\Delta F\equiv \lim _{\epsilon \rightarrow 0}[F(y_{\Delta }-\epsilon )-F(y_{\Delta }+\epsilon )]$. Such a situation is encountered in tethered polymers (Polymer brush) exhibiting a vertical phase separation, where $F(y)$ stands for the polymer density profile and $f(r)$ is related to the spatial distribution of terminal, non-tethered monomers of the polymers.
The Abel transform of a function f(r) is under these circumstances again given by:
$F(y)=2\int _{y}^{\infty }{\frac {f(r)r\,dr}{\sqrt {r^{2}-y^{2}}}}.$
Assuming f(r) drops to zero more quickly than 1/r, the inverse Abel transform is however given by
$f(r)=\left[{\frac {1}{2}}\delta (r-y_{\Delta }){\sqrt {1-(y_{\Delta }/r)^{2}}}-{\frac {1}{\pi }}{\frac {H(y_{\Delta }-r)}{\sqrt {y_{\Delta }^{2}-r^{2}}}}\right]\Delta F-{\frac {1}{\pi }}\int _{r}^{\infty }{\frac {dF}{dy}}{\frac {dy}{\sqrt {y^{2}-r^{2}}}}.$
where $\delta $ is the Dirac delta function and $H(x)$ the Heaviside step function. The extended version of the Abel transform for discontinuous F is proven upon applying the Abel transform to shifted, continuous $F(y)$, and it reduces to the classical Abel transform when $\Delta F=0$. If $F(y)$ has more than a single discontinuity, one has to introduce shifts for any of them to come up with a generalized version of the inverse Abel transform which contains n additional terms, each of them corresponding to one of the n discontinuities.
Relationship to other integral transforms
Relationship to the Fourier and Hankel transforms
The Abel transform is one member of the FHA cycle of integral operators. For example, in two dimensions, if we define A as the Abel transform operator, F as the Fourier transform operator and H as the zeroth-order Hankel transform operator, then the special case of the projection-slice theorem for circularly symmetric functions states that
$FA=H.$
In other words, applying the Abel transform to a 1-dimensional function and then applying the Fourier transform to that result is the same as applying the Hankel transform to that function. This concept can be extended to higher dimensions.
Relationship to the Radon transform
Abel transform can be viewed as the Radon transform of an isotropic 2D function f(r). As f(r) is isotropic, its Radon transform is the same at different angles of the viewing axis. Thus, the Abel transform is a function of the distance along the viewing axis only.
See also
• GPS radio occultation
References
1. N. H. Abel, Journal für die reine und angewandte Mathematik, 1, pp. 153–157 (1826).
• Bracewell, R. (1965). The Fourier Transform and its Applications. New York: McGraw-Hill. ISBN 0-07-007016-4.
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Abelian integral
In mathematics, an abelian integral, named after the Norwegian mathematician Niels Henrik Abel, is an integral in the complex plane of the form
$\int _{z_{0}}^{z}R(x,w)\,dx,$
where $R(x,w)$ is an arbitrary rational function of the two variables $x$ and $w$, which are related by the equation
$F(x,w)=0,$
where $F(x,w)$ is an irreducible polynomial in $w$,
$F(x,w)\equiv \varphi _{n}(x)w^{n}+\cdots +\varphi _{1}(x)w+\varphi _{0}\left(x\right),$
whose coefficients $\varphi _{j}(x)$, $j=0,1,\ldots ,n$ are rational functions of $x$. The value of an abelian integral depends not only on the integration limits, but also on the path along which the integral is taken; it is thus a multivalued function of $z$.
Abelian integrals are natural generalizations of elliptic integrals, which arise when
$F(x,w)=w^{2}-P(x),\,$
where $P\left(x\right)$ is a polynomial of degree 3 or 4. Another special case of an abelian integral is a hyperelliptic integral, where $P(x)$, in the formula above, is a polynomial of degree greater than 4.
History
The theory of abelian integrals originated with a paper by Abel[1] published in 1841. This paper was written during his stay in Paris in 1826 and presented to Augustin-Louis Cauchy in October of the same year. This theory, later fully developed by others,[2] was one of the crowning achievements of nineteenth century mathematics and has had a major impact on the development of modern mathematics. In more abstract and geometric language, it is contained in the concept of abelian variety, or more precisely in the way an algebraic curve can be mapped into abelian varieties. Abelian integrals were later connected to the prominent mathematician David Hilbert's 16th Problem, and they continue to be considered one of the foremost challenges in contemporary mathematics.
Modern view
In the theory of Riemann surfaces, an abelian integral is a function related to the indefinite integral of a differential of the first kind. Suppose we are given a Riemann surface $S$ and on it a differential 1-form $\omega $ that is everywhere holomorphic on $S$, and fix a point $P_{0}$ on $S$, from which to integrate. We can regard
$\int _{P_{0}}^{P}\omega $
as a multi-valued function $f\left(P\right)$, or (better) an honest function of the chosen path $C$ drawn on $S$ from $P_{0}$ to $P$. Since $S$ will in general be multiply connected, one should specify $C$, but the value will in fact only depend on the homology class of $C$.
In the case of $S$ a compact Riemann surface of genus 1, i.e. an elliptic curve, such functions are the elliptic integrals. Logically speaking, therefore, an abelian integral should be a function such as $f$.
Such functions were first introduced to study hyperelliptic integrals, i.e., for the case where $S$ is a hyperelliptic curve. This is a natural step in the theory of integration to the case of integrals involving algebraic functions ${\sqrt {A}}$, where $A$ is a polynomial of degree $>4$. The first major insights of the theory were given by Abel; it was later formulated in terms of the Jacobian variety $J\left(S\right)$. Choice of $P_{0}$ gives rise to a standard holomorphic function
$S\to J(S)$
of complex manifolds. It has the defining property that the holomorphic 1-forms on $S\to J(S)$, of which there are g independent ones if g is the genus of S, pull back to a basis for the differentials of the first kind on S.
Notes
1. Abel 1841.
2. Appell & Goursat 1895, p. 248.
References
• Abel, Niels H. (1841). "Mémoire sur une propriété générale d'une classe très étendue de fonctions transcendantes". Mémoires présentés par divers savants à l’Académie Royale des Sciences de l’Institut de France (in French). Paris. pp. 176–264.
• Appell, Paul; Goursat, Édouard (1895). Théorie des fonctions algébriques et de leurs intégrales (in French). Paris: Gauthier-Villars.
• Bliss, Gilbert A. (1933). Algebraic Functions. Providence: American Mathematical Society.
• Forsyth, Andrew R. (1893). Theory of Functions of a Complex Variable. Providence: Cambridge University Press.
• Griffiths, Phillip; Harris, Joseph (1978). Principles of Algebraic Geometry. New York: John Wiley & Sons.
• Neumann, Carl (1884). Vorlesungen über Riemann's Theorie der Abel'schen Integrale (2nd ed.). Leipzig: B. G. Teubner.
| Wikipedia |
Abelian Lie group
In geometry, an abelian Lie group is a Lie group that is an abelian group.
A connected abelian real Lie group is isomorphic to $\mathbb {R} ^{k}\times (S^{1})^{h}$.[1] In particular, a connected abelian (real) compact Lie group is a torus; i.e., a Lie group isomorphic to $(S^{1})^{h}$. A connected complex Lie group that is a compact group is abelian and a connected compact complex Lie group is a complex torus; i.e., a quotient of $\mathbb {\mathbb {C} } ^{n}$ by a lattice.
Let A be a compact abelian Lie group with the identity component $A_{0}$. If $A/A_{0}$ is a cyclic group, then $A$ is topologically cyclic; i.e., has an element that generates a dense subgroup.[2] (In particular, a torus is topologically cyclic.)
See also
• Cartan subgroup
Citations
1. Procesi 2007, Ch. 4. § 2..
2. Knapp 2001, Ch. IV, § 6, Lemma 4.20..
Works cited
• Knapp, Anthony W. (2001). Representation theory of semisimple groups. An overview based on examples. Princeton Landmarks in Mathematics. Princeton University Press. ISBN 0-691-09089-0.
• Procesi, Claudio (2007). Lie Groups: an approach through invariants and representation. Springer. ISBN 978-0387260402.
| Wikipedia |
Abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of abelian groups, Ab. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum. Abelian categories are very stable categories; for example they are regular and they satisfy the snake lemma. The class of abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory.
Definitions
A category is abelian if it is preadditive and
• it has a zero object,
• it has all binary biproducts,
• it has all kernels and cokernels, and
• all monomorphisms and epimorphisms are normal.
This definition is equivalent[1] to the following "piecemeal" definition:
• A category is preadditive if it is enriched over the monoidal category Ab of abelian groups. This means that all hom-sets are abelian groups and the composition of morphisms is bilinear.
• A preadditive category is additive if every finite set of objects has a biproduct. This means that we can form finite direct sums and direct products. In [2] Def. 1.2.6, it is required that an additive category have a zero object (empty biproduct).
• An additive category is preabelian if every morphism has both a kernel and a cokernel.
• Finally, a preabelian category is abelian if every monomorphism and every epimorphism is normal. This means that every monomorphism is a kernel of some morphism, and every epimorphism is a cokernel of some morphism.
Note that the enriched structure on hom-sets is a consequence of the first three axioms of the first definition. This highlights the foundational relevance of the category of Abelian groups in the theory and its canonical nature.
The concept of exact sequence arises naturally in this setting, and it turns out that exact functors, i.e. the functors preserving exact sequences in various senses, are the relevant functors between abelian categories. This exactness concept has been axiomatized in the theory of exact categories, forming a very special case of regular categories.
Examples
• As mentioned above, the category of all abelian groups is an abelian category. The category of all finitely generated abelian groups is also an abelian category, as is the category of all finite abelian groups.
• If R is a ring, then the category of all left (or right) modules over R is an abelian category. In fact, it can be shown that any small abelian category is equivalent to a full subcategory of such a category of modules (Mitchell's embedding theorem).
• If R is a left-noetherian ring, then the category of finitely generated left modules over R is abelian. In particular, the category of finitely generated modules over a noetherian commutative ring is abelian; in this way, abelian categories show up in commutative algebra.
• As special cases of the two previous examples: the category of vector spaces over a fixed field k is abelian, as is the category of finite-dimensional vector spaces over k.
• If X is a topological space, then the category of all (real or complex) vector bundles on X is not usually an abelian category, as there can be monomorphisms that are not kernels.
• If X is a topological space, then the category of all sheaves of abelian groups on X is an abelian category. More generally, the category of sheaves of abelian groups on a Grothendieck site is an abelian category. In this way, abelian categories show up in algebraic topology and algebraic geometry.
• If C is a small category and A is an abelian category, then the category of all functors from C to A forms an abelian category. If C is small and preadditive, then the category of all additive functors from C to A also forms an abelian category. The latter is a generalization of the R-module example, since a ring can be understood as a preadditive category with a single object.
Grothendieck's axioms
In his Tōhoku article, Grothendieck listed four additional axioms (and their duals) that an abelian category A might satisfy. These axioms are still in common use to this day. They are the following:
• AB3) For every indexed family (Ai) of objects of A, the coproduct *Ai exists in A (i.e. A is cocomplete).
• AB4) A satisfies AB3), and the coproduct of a family of monomorphisms is a monomorphism.
• AB5) A satisfies AB3), and filtered colimits of exact sequences are exact.
and their duals
• AB3*) For every indexed family (Ai) of objects of A, the product PAi exists in A (i.e. A is complete).
• AB4*) A satisfies AB3*), and the product of a family of epimorphisms is an epimorphism.
• AB5*) A satisfies AB3*), and filtered limits of exact sequences are exact.
Axioms AB1) and AB2) were also given. They are what make an additive category abelian. Specifically:
• AB1) Every morphism has a kernel and a cokernel.
• AB2) For every morphism f, the canonical morphism from coim f to im f is an isomorphism.
Grothendieck also gave axioms AB6) and AB6*).
• AB6) A satisfies AB3), and given a family of filtered categories $I_{j},j\in J$ and maps $A_{j}:I_{j}\to A$, we have $\prod _{j\in J}\lim _{I_{j}}A_{j}=\lim _{I_{j},\forall j\in J}\prod _{j\in J}A_{j}$, where lim denotes the filtered colimit.
• AB6*) A satisfies AB3*), and given a family of cofiltered categories $I_{j},j\in J$ and maps $A_{j}:I_{j}\to A$, we have $\sum _{j\in J}\lim _{I_{j}}A_{j}=\lim _{I_{j},\forall j\in J}\sum _{j\in J}A_{j}$, where lim denotes the cofiltered limit.
Elementary properties
Given any pair A, B of objects in an abelian category, there is a special zero morphism from A to B. This can be defined as the zero element of the hom-set Hom(A,B), since this is an abelian group. Alternatively, it can be defined as the unique composition A → 0 → B, where 0 is the zero object of the abelian category.
In an abelian category, every morphism f can be written as the composition of an epimorphism followed by a monomorphism. This epimorphism is called the coimage of f, while the monomorphism is called the image of f.
Subobjects and quotient objects are well-behaved in abelian categories. For example, the poset of subobjects of any given object A is a bounded lattice.
Every abelian category A is a module over the monoidal category of finitely generated abelian groups; that is, we can form a tensor product of a finitely generated abelian group G and any object A of A. The abelian category is also a comodule; Hom(G,A) can be interpreted as an object of A. If A is complete, then we can remove the requirement that G be finitely generated; most generally, we can form finitary enriched limits in A.
Related concepts
Abelian categories are the most general setting for homological algebra. All of the constructions used in that field are relevant, such as exact sequences, and especially short exact sequences, and derived functors. Important theorems that apply in all abelian categories include the five lemma (and the short five lemma as a special case), as well as the snake lemma (and the nine lemma as a special case).
Semi-simple Abelian categories
Main article: Semi-simplicity
An abelian category $\mathbf {A} $ is called semi-simple if there is a collection of objects $\{X_{i}\}_{i\in I}\in {\text{Ob}}(\mathbf {A} )$ called simple objects (meaning the only sub-objects of any $X_{i}$ are the zero object $0$ and itself) such that an object $X\in {\text{Ob}}(\mathbf {A} )$ can be decomposed as a direct sum (denoting the coproduct of the abelian category)
$X\cong \bigoplus _{i\in I}X_{i}$
This technical condition is rather strong and excludes many natural examples of abelian categories found in nature. For example, most module categories over a ring $R$ are not semi-simple; in fact, this is the case if and only if $R$ is a semisimple ring.
Examples
Some Abelian categories found in nature are semi-simple, such as
• Category of finite-dimensional vector spaces ${\text{FinVect}}(k)$ over a fixed field $k$
• By Maschke's theorem the category of representations ${\text{Rep}}_{k}(G)$ of a finite group $G$ over a field $k$ whose characteristic does not divide $|G|$ is a semi-simple abelian category.
• The category of coherent sheaves on a Noetherian scheme is semi-simple if and only if $X$ is a finite disjoint union of irreducible points. This is equivalent to a finite coproduct of categories of vector spaces over different fields. Showing this is true in the forward direction is equivalent to showing all ${\text{Ext}}^{1}$ groups vanish, meaning the cohomological dimension is 0. This only happens when the skyscraper sheaves $k_{x}$ at a point $x\in X$ have Zariski tangent space equal to zero, which is isomorphic to ${\text{Ext}}^{1}(k_{x},k_{x})$ using local algebra for such a scheme.[3]
Non-examples
There do exist some natural counter-examples of abelian categories which are not semi-simple, such as certain categories of representations. For example, the category of representations of the Lie group $(\mathbb {R} ,+)$ has the representation
$a\mapsto {\begin{bmatrix}1&a\\0&1\end{bmatrix}}$
which only has one subrepresentation of dimension $1$. In fact, this is true for any unipotent group[4]pg 112.
Subcategories of abelian categories
There are numerous types of (full, additive) subcategories of abelian categories that occur in nature, as well as some conflicting terminology.
Let A be an abelian category, C a full, additive subcategory, and I the inclusion functor.
• C is an exact subcategory if it is itself an exact category and the inclusion I is an exact functor. This occurs if and only if C is closed under pullbacks of epimorphisms and pushouts of monomorphisms. The exact sequences in C are thus the exact sequences in A for which all objects lie in C.
• C is an abelian subcategory if it is itself an abelian category and the inclusion I is an exact functor. This occurs if and only if C is closed under taking kernels and cokernels. Note that there are examples of full subcategories of an abelian category that are themselves abelian but where the inclusion functor is not exact, so they are not abelian subcategories (see below).
• C is a thick subcategory if it is closed under taking direct summands and satisfies the 2-out-of-3 property on short exact sequences; that is, if $0\to M'\to M\to M''\to 0$ is a short exact sequence in A such that two of $M',M,M''$ lie in C, then so does the third. In other words, C is closed under kernels of epimorphisms, cokernels of monomorphisms, and extensions. Note that P. Gabriel used the term thick subcategory to describe what we here call a Serre subcategory.
• C is a topologizing subcategory if it is closed under subquotients.
• C is a Serre subcategory if, for all short exact sequences $0\to M'\to M\to M''\to 0$ in A we have M in C if and only if both $M',M''$ are in C. In other words, C is closed under extensions and subquotients. These subcategories are precisely the kernels of exact functors from A to another abelian category.
• C is a localizing subcategory if it is a Serre subcategory such that the quotient functor $Q\colon \mathbf {A} \to \mathbf {A} /\mathbf {C} $ admits a right adjoint.
• There are two competing notions of a wide subcategory. One version is that C contains every object of A (up to isomorphism); for a full subcategory this is obviously not interesting. (This is also called a lluf subcategory.) The other version is that C is closed under extensions.
Here is an explicit example of a full, additive subcategory of an abelian category that is itself abelian but the inclusion functor is not exact. Let k be a field, $T_{n}$ the algebra of upper-triangular $n\times n$ matrices over k, and $\mathbf {A} _{n}$ the category of finite-dimensional $T_{n}$-modules. Then each $\mathbf {A} _{n}$ is an abelian category and we have an inclusion functor $I\colon \mathbf {A} _{2}\to \mathbf {A} _{3}$ identifying the simple projective, simple injective and indecomposable projective-injective modules. The essential image of I is a full, additive subcategory, but I is not exact.
History
Abelian categories were introduced by Buchsbaum (1955) (under the name of "exact category") and Grothendieck (1957) in order to unify various cohomology theories. At the time, there was a cohomology theory for sheaves, and a cohomology theory for groups. The two were defined differently, but they had similar properties. In fact, much of category theory was developed as a language to study these similarities. Grothendieck unified the two theories: they both arise as derived functors on abelian categories; the abelian category of sheaves of abelian groups on a topological space, and the abelian category of G-modules for a given group G.
See also
• Triangulated category
References
1. Peter Freyd, Abelian Categories
2. Handbook of categorical algebra, vol. 2, F. Borceux
3. "algebraic geometry - Tangent space in a point and First Ext group". Mathematics Stack Exchange. Retrieved 2020-08-23.
4. Humphreys, James E. (2004). Linear algebraic groups. Springer. ISBN 0-387-90108-6. OCLC 77625833.
• Buchsbaum, David A. (1955), "Exact categories and duality", Transactions of the American Mathematical Society, 80 (1): 1–34, doi:10.1090/S0002-9947-1955-0074407-6, ISSN 0002-9947, JSTOR 1993003, MR 0074407
• Freyd, Peter (1964), Abelian Categories, New York: Harper and Row
• Grothendieck, Alexander (1957), "Sur quelques points d'algèbre homologique", Tohoku Mathematical Journal, Second Series, 9: 119–221, doi:10.2748/tmj/1178244839, ISSN 0040-8735, MR 0102537
• Mitchell, Barry (1965), Theory of Categories, Boston, MA: Academic Press
• Popescu, Nicolae (1973), Abelian categories with applications to rings and modules, Boston, MA: Academic Press
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Abelian extension
In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvable if its Galois group is solvable, i.e., if the group can be decomposed into a series of normal extensions of an abelian group. Every finite extension of a finite field is a cyclic extension.
Description
Class field theory provides detailed information about the abelian extensions of number fields, function fields of algebraic curves over finite fields, and local fields.
There are two slightly different definitions of the term cyclotomic extension. It can mean either an extension formed by adjoining roots of unity to a field, or a subextension of such an extension. The cyclotomic fields are examples. A cyclotomic extension, under either definition, is always abelian.
If a field K contains a primitive n-th root of unity and the n-th root of an element of K is adjoined, the resulting Kummer extension is an abelian extension (if K has characteristic p we should say that p doesn't divide n, since otherwise this can fail even to be a separable extension). In general, however, the Galois groups of n-th roots of elements operate both on the n-th roots and on the roots of unity, giving a non-abelian Galois group as semi-direct product. The Kummer theory gives a complete description of the abelian extension case, and the Kronecker–Weber theorem tells us that if K is the field of rational numbers, an extension is abelian if and only if it is a subfield of a field obtained by adjoining a root of unity.
There is an important analogy with the fundamental group in topology, which classifies all covering spaces of a space: abelian covers are classified by its abelianisation which relates directly to the first homology group.
Further information: Ring class field
References
• Kuz'min, L.V. (2001) [1994], "cyclotomic extension", Encyclopedia of Mathematics, EMS Press
• Weisstein, Eric W. "Abelian Extension". MathWorld.
| Wikipedia |
Mathematical joke
A mathematical joke is a form of humor which relies on aspects of mathematics or a stereotype of mathematicians. The humor may come from a pun, or from a double meaning of a mathematical term, or from a lay person's misunderstanding of a mathematical concept. Mathematician and author John Allen Paulos in his book Mathematics and Humor described several ways that mathematics, generally considered a dry, formal activity, overlaps with humor, a loose, irreverent activity: both are forms of "intellectual play"; both have "logic, pattern, rules, structure"; and both are "economical and explicit".[2]
Some performers combine mathematics and jokes to entertain and/or teach math.[3][4][5]
Humor of mathematicians may be classified into the esoteric and exoteric categories. Esoteric jokes rely on the intrinsic knowledge of mathematics and its terminology. Exoteric jokes are intelligible to the outsiders, and most of them compare mathematicians with representatives of other disciplines or with common folk.[6]
Pun-based jokes
${\sqrt {-1}}\;2^{3}\;\Sigma \;\pi $
Rebus for "I ate some pie."
Some jokes use a mathematical term with a second non-technical meaning as the punchline of a joke.
Q. What's purple and commutes?
A. An abelian grape. (A pun on abelian group.)
Occasionally, multiple mathematical puns appear in the same jest:
When Noah sends his animals to go forth and multiply, a pair of snakes replies "We can't multiply, we're adders" – so Noah builds them a log table.[7]
This invokes four double meanings: adder (snake) vs. addition (algebraic operation); multiplication (biological reproduction) vs. multiplication (algebraic operation); log (a cut tree trunk) vs. log (logarithm); and table (set of facts) vs. table (piece of furniture).[8]
Other jokes create a double meaning from a direct calculation involving facetious variable names, such as this retold from Gravity's Rainbow:[9]
Person 1: What's the integral of 1/cabin with respect to cabin?
Person 2: A log cabin.
Person 1: No, a houseboat; you forgot to add the C![10]
The first part of this joke relies on the fact that the primitive (formed when finding the antiderivative) of the function 1/x is log(x). The second part is then based on the fact that the antiderivative is actually a class of functions, requiring the inclusion of a constant of integration, usually denoted as C—something which calculus students may forget. Thus, the indefinite integral of 1/cabin is "log(cabin) + C", or "A log cabin plus the sea", i.e., "A houseboat".
Jokes with numeral bases
Some jokes depend on ambiguity of numeral bases.
There are only 10 types of people in the world: those who understand binary, and those who don't.
This joke subverts the trope of phrases that begin with "there are two types of people in the world..." and relies on an ambiguous meaning of the expression 10, which in the binary numeral system is equal to the decimal number 2.[11] There are many alternative versions of the joke, such as "There are two types of people in this world. Those who can extrapolate from incomplete information."[12]
Another pun using different radices, asks:
Q. Why do mathematicians confuse Halloween and Christmas?
A. Because 31 Oct = 25 Dec.[13][14]
The play on words lies in the similarity of the abbreviation for October/Octal and December/Decimal, and the coincidence that both equal the same amount ($31_{8}=25_{10}$).
Imaginary numbers
Some jokes are based on imaginary number i, treating it as if it is a real number. A telephone intercept message of "you have dialed an imaginary number, please rotate your handset ninety degrees and try again" is a typical example.[15] Another popular example is: "What did π say to i? Get real. What did i say to π? Be rational."[16]
Stereotypes of mathematicians
Some jokes are based on stereotypes of mathematicians tending to think in complicated, abstract terms, causing them to lose touch with the "real world". These compare mathematicians to physicists, engineers, or the "soft" sciences in a form similar to an Englishman, an Irishman and a Scotsman, showing the other scientists doing something practical, while the mathematician proposes a theoretically valid but physically nonsensical solution.
A physicist, a biologist and a mathematician are sitting in a street café watching people entering and leaving a nearby house. First they see two people entering the house. Time passes. After a while they notice three people leaving the house. The physicist says, "The measurement wasn't accurate." The biologist says, "They must have reproduced." The mathematician says, "If one more person enters the house, then it will be empty."[17]
Mathematicians are also shown as averse to making hasty generalizations from a small amount of data, even if some form of generalization seems plausible:
An astronomer, a physicist and a mathematician are on a train in Scotland. The astronomer looks out of the window, sees a black sheep standing in a field, and remarks, "How odd. All the sheep in Scotland are black!" "No, no, no!" says the physicist. "Only some Scottish sheep are black." The mathematician rolls his eyes at his companions' muddled thinking and says, "In Scotland, there is at least one sheep, at least one side of which appears to be black from here some of the time."[18]
A classic joke involving stereotypes is the "Dictionary of Definitions of Terms Commonly Used in Math Lectures".[19] Examples include "Trivial: If I have to show you how to do this, you're in the wrong class" and "Similarly: At least one line of the proof of this case is the same as before."
Non-mathematician's math
This category of jokes comprises those that exploit common misunderstandings of mathematics, or the expectation that most people have only a basic mathematical education, if any.
A museum visitor was admiring a Tyrannosaurus fossil, and asked a nearby museum employee how old it was. "That skeleton's sixty-five million and three years, two months and eighteen days old," the employee replied. "How can you be so precise?" she asked. "Well, when I started working here, I asked a scientist the exact same question, and he said it was sixty-five million years old—and that was three years, two months and eighteen days ago."[20]
The joke is that the employee fails to understand the scientist's implication of the uncertainty in the age of the fossil and uses false precision.
Mock mathematics
A form of mathematical humor comes from using mathematical tools (both abstract symbols and physical objects such as calculators) in various ways which transgress their intended scope. These constructions are generally devoid of any substantial mathematical content, besides some basic arithmetic.
Mock mathematical reasoning
A set of jokes applies mathematical reasoning to situations where it is not entirely valid. Many are based on a combination of well-known quotes and basic logical constructs such as syllogisms:
Premise I:Knowledge is power.
Premise II:Power corrupts.
Conclusion:Therefore, knowledge corrupts.[21]
Another set of jokes relates to the absence of mathematical reasoning, or misinterpretation of conventional notation:
$\left(\lim _{x\to 8^{+}}{\frac {1}{x-8}}=\infty \right)\Rightarrow \left(\lim _{x\to 3^{+}}{\frac {1}{x-3}}=\omega \right)$
That is, the limit as x goes to 8 from above is a sideways 8 or the infinity sign, in the same way that the limit as x goes to three from above is a sideways 3 or the Greek letter omega (conventionally used to notate the smallest infinite ordinal number).[22]
An anomalous cancellation is a kind of arithmetic procedural error that gives a numerically correct answer:
• ${\frac {64}{16}}={\frac {{\cancel {6}}4}{\,\,1{\cancel {6}}}}={\frac {4}{1}}=4$
• ${\sqrt[{6}]{64}}={\sqrt[{\cancel {6}}]{{\cancel {6}}4}}={\sqrt {4}}=2$
• ${\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {1}{x}}={\frac {\cancel {\mathrm {d} }}{{\cancel {\mathrm {d} }}x}}{\frac {1}{x}}={\frac {}{x}}{\frac {1}{x}}=-{\frac {1}{x^{2}}}$
Mathematical fallacies
A number of mathematical fallacies are part of mathematical humorous folklore. For example:
${\begin{aligned}b&=a\\ab&=a^{2}\\ab-b^{2}&=a^{2}-b^{2}\\b(a-b)&=(a+b)(a-b)\\b&=a+b\\b&=b+b\\b&=2b\\1&=2\end{aligned}}$
This appears to prove that 1 = 2, but uses division by zero to produce the result.[23]
Some jokes attempt a seemingly plausible, but in fact impossible, mathematical operation. For example:
Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed?[24]
To reverse the digits of a number's decimal expansion, we have to start at the last digit and work backwards. However, that is not possible if the expansion never ends, which is true in the case of $\pi $ and $e$.
Humorous numbers
Many numbers have been given humorous names, either as pure numbers or as units of measurement. Some examples:
Sagan has been defined as "billions and billions", a metric of the number of stars in the observable universe.[25][26]
Jenny's constant has been defined as $J=(7^{e-1/e}-9)\cdot \pi ^{2}=867.5309\ldots .$ (sequence A182369 in the OEIS), from the pop song 867-5309/Jenny, which concerns the telephone number 867-5309.[27]
The number 42 appears prominently in the Douglas Adams trilogy The Hitchhiker's Guide to the Galaxy, where it is portrayed as "the answer to the ultimate question of life, the universe and everything".[28] This number appears as a fixed value in the TIFF image file format and its derivatives (including for example the ISO standard TIFF/EP) where the content of bytes 2–3 is defined as 42: "An arbitrary but carefully chosen number that further identifies the file as a TIFF file".[29]
The number 69 is commonly used in reference to a group of sex positions in which two people align to perform oral sex, thus becoming mutually inverted like the numerals 6 and 9. Because of this association, "69" has become an internet meme and is known as "the sex number" in certain communities.[30]
In the context of numerical humor, one classic example is the joke, "Why was six afraid of seven? Because seven eight (ate) nine!" The humor in this statement originates from a linguistic play on numbers and fundamental arithmetic.[31]
Calculator spelling
Calculator spelling is the formation of words and phrases by displaying a number and turning the calculator upside down.[32] The jest may be formulated as a mathematical problem where the result, when read upside down, appears to be an identifiable phrase like "ShELL OIL" or "Esso" using seven-segment display character representations where the open-top "4" is an inverted 'h' and '5' looks like 'S'. Other letters can be used as numbers too with 8 and 9 representing B and G, respectively.
An attributed example of calculator spelling, which dates from the 1970s,[33] is 5318008, which when turned over spells "BOOBIES".
Limericks
A mathematical limerick is an expression which, when read aloud, matches the form of a limerick. The following example is attributed to Leigh Mercer:[34]
${\frac {12+144+20+3{\sqrt {4}}}{7}}+(5\times 11)=9^{2}+0$
This is read as follows:
A dozen, a gross, and a score
Plus three times the square root of four
Divided by seven
Plus five times eleven
Is nine squared and not a bit more.
Another example using calculus is[lower-alpha 1]:
$\int _{1}^{\sqrt[{3}]{3}}z^{2}dz\cdot \cos \left({\frac {3\pi }{9}}\right)=\log({\sqrt[{3}]{e}})$
which may be read:
Integral z-squared dz
From one to the cube root of three
Times the cosine
of three pi over nine
Equals log of the cube root of e
The limerick is true if $\log $ is interpreted as the natural logarithm.
Doughnut and coffee mug topology joke
An oft-repeated joke is that topologists cannot tell a coffee cup from a doughnut,[35] since they are topologically equivalent: a sufficiently pliable doughnut could be reshaped (by a homeomorphism) to the form of a cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.
See also
• New Math (song)
• Spherical cow
• All horses are the same color
Notes
1. The variable's name 'z' should be pronounced as zee to make a proper rhyme to 'three' and 'e'.
References
1. "13 Jokes That Every Math Geek Will Find Hilarious". Business Insider.
2. John Allen Paulos (1980). Mathematics and Humor. ISBN 9780226650241.
3. "Matt Parker, math stand-up comedian". Mathscareers.org.uk. 2011-08-04. Retrieved 2013-07-01.
4. "Dara O'Briain: School of hard sums". Comedy.co.uk. Retrieved 2013-07-01.
5. Schimmrich, Steven (2011-05-17). "Dave Gorman – stand-up math comedy". Retrieved 2013-07-01.
6. Paul Renteln, Alan Dundes, Foolproof: A Sampling of Mathematical Folk Humor, NOTICES OF THE AMS, VOLUME 52, NUMBER 1, 2005, pp. 24-34.
7. Simanek, Donald E.; Holden, John C. (2001-10-01). Science Askew : a light-hearted look at the scientific world. ISBN 9780750307147.
8. Ermida, Isabel (2008-12-10). The Language of Comic Narratives. ISBN 9783110208337.
9. polyb (April 6, 2005). "Quick and Witty!". Physics Forums. Retrieved February 28, 2013.
10. Bloom, Harold (2009-01-01). Thomas Pynchon. ISBN 9781438116112.
11. Ritchie, Graeme (2002-06-01). The Linguistic Analysis of Jokes. ISBN 9780203406953.
12. Olicity8. "Two Types Of People". Wattpad. Retrieved 29 April 2019.
13. Larman, Craig (2002). Applying Uml and Patterns. ISBN 9780130925695.
14. Collins, Tim (2006-08-29). Are You a Geek?. ISBN 9780440336280.
15. Elizabeth Longmier (2007-05-01). In the Lab. ISBN 9781430322160. Retrieved 2013-06-19.
16. Joke number 11 on Page 9, 36 Teacher Jokes, 17 pages, October 17, 2013. Simple K12, InfoSource Inc
17. Krawcewicz, Wiesław; Rai, B. (2003-01-01). Calculus with Maple Labs : early transcendentals. ISBN 9781842650745.
18. Stewart, Ian (1995). Concepts of Modern Mathematics. ISBN 9780486134956.
19. "Dictionary of Definitions of Terms Commonly Used in Math lectures."
20. Seife, Charles (2010-09-23). Proofiness. ISBN 9781101443507.
21. Lawless, Andrew (2005). Plato's Sun. ISBN 9780802038098.
22. Xu, Chao (2008-02-21). "A mathematical look into the limit joke". Archived from the original on 2008-02-24. Retrieved 2008-04-19.
23. Harro Heuser: Lehrbuch der Analysis – Teil 1, 6th edition, Teubner 1989, ISBN 978-3-8351-0131-9, page 51 (German).
24. "Pi goes on and on and on …". JUST FOR FUN. Math Majors Matter. Retrieved 12 August 2018.
25. William Safire, ON LANGUAGE; Footprints on the Infobahn, New York Times, April 17, 1994
26. Sizing up the Universe – Stars, Sand and Nucleons – Numericana
27. "Jenny's Constant". Wolfram MathWorld. 2012-04-26. Retrieved 2013-06-19.
28. Gill, Peter (2011-02-03). "42: Douglas Adams' Amazingly Accurate Answer to Life the Universe and Everything". London: Guardian. Retrieved 2011-04-03.
29. Aldus/Microsoft (1999-08-09). "1) Structure". TIFF. Revision 5.0. Aldus Corporation and Microsoft Corporation. Archived from the original on 2012-02-11. Retrieved 2009-06-29. The number 42 was chosen for its deep philosophical significance.
30. Feldman, Brian (2016-06-09). "Why 69 Is the Internet's Coolest Number (Sex)". Intelligencer. Retrieved 2020-09-04.
31. "Adding Laughter to Numbers". Harvard.edu.
32. Bolt, Brian (1984-09-27). The Amazing Mathematical Amusement Arcade. Cambridge University Press. p. 48. ISBN 9780521269803.
33. Tom Dalzell; Terry Victor (27 November 2014). The Concise New Partridge Dictionary of Slang and Unconventional English. Taylor & Francis. p. 2060. ISBN 978-1-317-62511-7.
34. "Math Mayhem". Lhup.edu. Retrieved 2011-06-29.
35. West, Beverly H (1995-03-30). Differential Equations: A Dynamical Systems Approach : Higher-Dimensional Systems. ISBN 9780387943770. Retrieved 2011-06-29.
External links
• Mathematical Humor – from Mathworld
• Paul Renteln and Alan Dundes (2004-12-08). "Foolproof: A Sampling of Mathematical Folk Humor" (PDF). Notices of the AMS. 52 (1).
• 13 Jokes That Every Math Geek Will Find Hilarious
| Wikipedia |
Abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.[1]
For the group described by the archaic use of the related term "Abelian linear group", see Symplectic group.
The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified.
Algebraic structure → Group theory
Group theory
Basic notions
• Subgroup
• Normal subgroup
• Quotient group
• (Semi-)direct product
Group homomorphisms
• kernel
• image
• direct sum
• wreath product
• simple
• finite
• infinite
• continuous
• multiplicative
• additive
• cyclic
• abelian
• dihedral
• nilpotent
• solvable
• action
• Glossary of group theory
• List of group theory topics
Finite groups
• Cyclic group Zn
• Symmetric group Sn
• Alternating group An
• Dihedral group Dn
• Quaternion group Q
• Cauchy's theorem
• Lagrange's theorem
• Sylow theorems
• Hall's theorem
• p-group
• Elementary abelian group
• Frobenius group
• Schur multiplier
Classification of finite simple groups
• cyclic
• alternating
• Lie type
• sporadic
• Discrete groups
• Lattices
• Integers ($\mathbb {Z} $)
• Free group
Modular groups
• PSL(2, $\mathbb {Z} $)
• SL(2, $\mathbb {Z} $)
• Arithmetic group
• Lattice
• Hyperbolic group
Topological and Lie groups
• Solenoid
• Circle
• General linear GL(n)
• Special linear SL(n)
• Orthogonal O(n)
• Euclidean E(n)
• Special orthogonal SO(n)
• Unitary U(n)
• Special unitary SU(n)
• Symplectic Sp(n)
• G2
• F4
• E6
• E7
• E8
• Lorentz
• Poincaré
• Conformal
• Diffeomorphism
• Loop
Infinite dimensional Lie group
• O(∞)
• SU(∞)
• Sp(∞)
Algebraic groups
• Linear algebraic group
• Reductive group
• Abelian variety
• Elliptic curve
Definition
An abelian group is a set $A$, together with an operation $\cdot $ that combines any two elements $a$ and $b$ of $A$ to form another element of $A,$ denoted $a\cdot b$. The symbol $\cdot $ is a general placeholder for a concretely given operation. To qualify as an abelian group, the set and operation, $(A,\cdot )$, must satisfy four requirements known as the abelian group axioms (some authors include in the axioms some properties that belong to the definition of an operation: namely that the operation is defined for any ordered pair of elements of A, that the result is well-defined, and that the result belongs to A):
Associativity
For all $a$, $b$, and $c$ in $A$, the equation $(a\cdot b)\cdot c=a\cdot (b\cdot c)$ holds.
Identity element
There exists an element $e$ in $A$, such that for all elements $a$ in $A$, the equation $e\cdot a=a\cdot e=a$ holds.
Inverse element
For each $a$ in $A$ there exists an element $b$ in $A$ such that $a\cdot b=b\cdot a=e$, where $e$ is the identity element.
Commutativity
For all $a$, $b$ in $A$, $a\cdot b=b\cdot a$.
A group in which the group operation is not commutative is called a "non-abelian group" or "non-commutative group".[2]: 11
Facts
Notation
See also: Additive group and Multiplicative group
There are two main notational conventions for abelian groups – additive and multiplicative.
Convention Operation Identity Powers Inverse
Addition $x+y$0$nx$$-x$
Multiplication $x\cdot y$ or $xy$1 $x^{n}$ $x^{-1}$
Generally, the multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for modules and rings. The additive notation may also be used to emphasize that a particular group is abelian, whenever both abelian and non-abelian groups are considered, some notable exceptions being near-rings and partially ordered groups, where an operation is written additively even when non-abelian.[3]: 28–29
Multiplication table
To verify that a finite group is abelian, a table (matrix) – known as a Cayley table – can be constructed in a similar fashion to a multiplication table.[4]: 10 If the group is $G=\{g_{1}=e,g_{2},\dots ,g_{n}\}$ under the operation $\cdot $, the $(i,j)$-th entry of this table contains the product $g_{i}\cdot g_{j}$.
The group is abelian if and only if this table is symmetric about the main diagonal. This is true since the group is abelian iff $g_{i}\cdot g_{j}=g_{j}\cdot g_{i}$ for all $i,j=1,...,n$, which is iff the $(i,j)$ entry of the table equals the $(j,i)$ entry for all $i,j=1,...,n$, i.e. the table is symmetric about the main diagonal.
Examples
• For the integers and the operation addition $+$, denoted $(\mathbb {Z} ,+)$, the operation + combines any two integers to form a third integer, addition is associative, zero is the additive identity, every integer $n$ has an additive inverse, $-n$, and the addition operation is commutative since $n+m=m+n$ for any two integers $m$ and $n$.
• Every cyclic group $G$ is abelian, because if $x$, $y$ are in $G$, then $xy=a^{m}a^{n}=a^{m+n}=a^{n}a^{m}=yx$. Thus the integers, $\mathbb {Z} $, form an abelian group under addition, as do the integers modulo $n$, $\mathbb {Z} /n\mathbb {Z} $.
• Every ring is an abelian group with respect to its addition operation. In a commutative ring the invertible elements, or units, form an abelian multiplicative group. In particular, the real numbers are an abelian group under addition, and the nonzero real numbers are an abelian group under multiplication.
• Every subgroup of an abelian group is normal, so each subgroup gives rise to a quotient group. Subgroups, quotients, and direct sums of abelian groups are again abelian. The finite simple abelian groups are exactly the cyclic groups of prime order.[5]: 32
• The concepts of abelian group and $\mathbb {Z} $-module agree. More specifically, every $\mathbb {Z} $-module is an abelian group with its operation of addition, and every abelian group is a module over the ring of integers $\mathbb {Z} $ in a unique way.
In general, matrices, even invertible matrices, do not form an abelian group under multiplication because matrix multiplication is generally not commutative. However, some groups of matrices are abelian groups under matrix multiplication – one example is the group of $2\times 2$ rotation matrices.
Historical remarks
Camille Jordan named abelian groups after Norwegian mathematician Niels Henrik Abel, as Abel had found that the commutativity of the group of a polynomial implies that the roots of the polynomial can be calculated by using radicals.[6]: 144–145
Properties
If $n$ is a natural number and $x$ is an element of an abelian group $G$ written additively, then $nx$ can be defined as $x+x+\cdots +x$ ($n$ summands) and $(-n)x=-(nx)$. In this way, $G$ becomes a module over the ring $\mathbb {Z} $ of integers. In fact, the modules over $\mathbb {Z} $ can be identified with the abelian groups.[7]: 94–97
Theorems about abelian groups (i.e. modules over the principal ideal domain $\mathbb {Z} $) can often be generalized to theorems about modules over an arbitrary principal ideal domain. A typical example is the classification of finitely generated abelian groups which is a specialization of the structure theorem for finitely generated modules over a principal ideal domain. In the case of finitely generated abelian groups, this theorem guarantees that an abelian group splits as a direct sum of a torsion group and a free abelian group. The former may be written as a direct sum of finitely many groups of the form $\mathbb {Z} /p^{k}\mathbb {Z} $ for $p$ prime, and the latter is a direct sum of finitely many copies of $\mathbb {Z} $.
If $f,g:G\to H$ are two group homomorphisms between abelian groups, then their sum $f+g$, defined by $(f+g)(x)=f(x)+g(x)$, is again a homomorphism. (This is not true if $H$ is a non-abelian group.) The set ${\text{Hom}}(G,H)$ of all group homomorphisms from $G$ to $H$ is therefore an abelian group in its own right.
Somewhat akin to the dimension of vector spaces, every abelian group has a rank. It is defined as the maximal cardinality of a set of linearly independent (over the integers) elements of the group.[8]: 49–50 Finite abelian groups and torsion groups have rank zero, and every abelian group of rank zero is a torsion group. The integers and the rational numbers have rank one, as well as every nonzero additive subgroup of the rationals. On the other hand, the multiplicative group of the nonzero rationals has an infinite rank, as it is a free abelian group with the set of the prime numbers as a basis (this results from the fundamental theorem of arithmetic).
The center $Z(G)$ of a group $G$ is the set of elements that commute with every element of $G$. A group $G$ is abelian if and only if it is equal to its center $Z(G)$. The center of a group $G$ is always a characteristic abelian subgroup of $G$. If the quotient group $G/Z(G)$ of a group by its center is cyclic then $G$ is abelian.[9]
Finite abelian groups
Cyclic groups of integers modulo $n$, $\mathbb {Z} /n\mathbb {Z} $, were among the first examples of groups. It turns out that an arbitrary finite abelian group is isomorphic to a direct sum of finite cyclic groups of prime power order, and these orders are uniquely determined, forming a complete system of invariants. The automorphism group of a finite abelian group can be described directly in terms of these invariants. The theory had been first developed in the 1879 paper of Georg Frobenius and Ludwig Stickelberger and later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter of linear algebra.
Any group of prime order is isomorphic to a cyclic group and therefore abelian. Any group whose order is a square of a prime number is also abelian.[10] In fact, for every prime number $p$ there are (up to isomorphism) exactly two groups of order $p^{2}$, namely $\mathbb {Z} _{p^{2}}$ and $\mathbb {Z} _{p}\times \mathbb {Z} _{p}$.
Classification
The fundamental theorem of finite abelian groups states that every finite abelian group $G$ can be expressed as the direct sum of cyclic subgroups of prime-power order; it is also known as the basis theorem for finite abelian groups. Moreover, automorphism groups of cyclic groups are examples of abelian groups.[11] This is generalized by the fundamental theorem of finitely generated abelian groups, with finite groups being the special case when G has zero rank; this in turn admits numerous further generalizations.
The classification was proven by Leopold Kronecker in 1870, though it was not stated in modern group-theoretic terms until later, and was preceded by a similar classification of quadratic forms by Carl Friedrich Gauss in 1801; see history for details.
The cyclic group $\mathbb {Z} _{mn}$ of order $mn$ is isomorphic to the direct sum of $\mathbb {Z} _{m}$ and $\mathbb {Z} _{n}$ if and only if $m$ and $n$ are coprime. It follows that any finite abelian group $G$ is isomorphic to a direct sum of the form
$\bigoplus _{i=1}^{u}\ \mathbb {Z} _{k_{i}}$
in either of the following canonical ways:
• the numbers $k_{1},k_{2},\dots ,k_{u}$ are powers of (not necessarily distinct) primes,
• or $k_{1}$ divides $k_{2}$, which divides $k_{3}$, and so on up to $k_{u}$.
For example, $\mathbb {Z} _{15}$ can be expressed as the direct sum of two cyclic subgroups of order 3 and 5: $\mathbb {Z} _{15}\cong \{0,5,10\}\oplus \{0,3,6,9,12\}$. The same can be said for any abelian group of order 15, leading to the remarkable conclusion that all abelian groups of order 15 are isomorphic.
For another example, every abelian group of order 8 is isomorphic to either $\mathbb {Z} _{8}$ (the integers 0 to 7 under addition modulo 8), $\mathbb {Z} _{4}\oplus \mathbb {Z} _{2}$ (the odd integers 1 to 15 under multiplication modulo 16), or $\mathbb {Z} _{2}\oplus \mathbb {Z} _{2}\oplus \mathbb {Z} _{2}$.
See also list of small groups for finite abelian groups of order 30 or less.
Automorphisms
One can apply the fundamental theorem to count (and sometimes determine) the automorphisms of a given finite abelian group $G$. To do this, one uses the fact that if $G$ splits as a direct sum $H\oplus K$ of subgroups of coprime order, then
$\operatorname {Aut} (H\oplus K)\cong \operatorname {Aut} (H)\oplus \operatorname {Aut} (K).$
Given this, the fundamental theorem shows that to compute the automorphism group of $G$ it suffices to compute the automorphism groups of the Sylow $p$-subgroups separately (that is, all direct sums of cyclic subgroups, each with order a power of $p$). Fix a prime $p$ and suppose the exponents $e_{i}$ of the cyclic factors of the Sylow $p$-subgroup are arranged in increasing order:
$e_{1}\leq e_{2}\leq \cdots \leq e_{n}$
for some $n>0$. One needs to find the automorphisms of
$\mathbf {Z} _{p^{e_{1}}}\oplus \cdots \oplus \mathbf {Z} _{p^{e_{n}}}.$
One special case is when $n=1$, so that there is only one cyclic prime-power factor in the Sylow $p$-subgroup $P$. In this case the theory of automorphisms of a finite cyclic group can be used. Another special case is when $n$ is arbitrary but $e_{i}=1$ for $1\leq i\leq n$. Here, one is considering $P$ to be of the form
$\mathbf {Z} _{p}\oplus \cdots \oplus \mathbf {Z} _{p},$
so elements of this subgroup can be viewed as comprising a vector space of dimension $n$ over the finite field of $p$ elements $\mathbb {F} _{p}$. The automorphisms of this subgroup are therefore given by the invertible linear transformations, so
$\operatorname {Aut} (P)\cong \mathrm {GL} (n,\mathbf {F} _{p}),$
where $\mathrm {GL} $ is the appropriate general linear group. This is easily shown to have order
$\left|\operatorname {Aut} (P)\right|=(p^{n}-1)\cdots (p^{n}-p^{n-1}).$
In the most general case, where the $e_{i}$ and $n$ are arbitrary, the automorphism group is more difficult to determine. It is known, however, that if one defines
$d_{k}=\max\{r\mid e_{r}=e_{k}\}$
and
$c_{k}=\min\{r\mid e_{r}=e_{k}\}$
then one has in particular $k\leq d_{k}$, $c_{k}\leq k$, and
$\left|\operatorname {Aut} (P)\right|=\prod _{k=1}^{n}(p^{d_{k}}-p^{k-1})\prod _{j=1}^{n}(p^{e_{j}})^{n-d_{j}}\prod _{i=1}^{n}(p^{e_{i}-1})^{n-c_{i}+1}.$
One can check that this yields the orders in the previous examples as special cases (see Hillar, C., & Rhea, D.).
Finitely generated abelian groups
Main article: Finitely generated abelian group
An abelian group A is finitely generated if it contains a finite set of elements (called generators) $G=\{x_{1},\ldots ,x_{n}\}$ such that every element of the group is a linear combination with integer coefficients of elements of G.
Let L be a free abelian group with basis $B=\{b_{1},\ldots ,b_{n}\}.$ There is a unique group homomorphism $p\colon L\to A,$ such that
$p(b_{i})=x_{i}\quad {\text{for }}i=1,\ldots ,n.$
This homomorphism is surjective, and its kernel is finitely generated (since integers form a Noetherian ring). Consider the matrix M with integer entries, such that the entries of its jth column are the coefficients of the jth generator of the kernel. Then, the abelian group is isomorphic to the cokernel of linear map defined by M. Conversely every integer matrix defines a finitely generated abelian group.
It follows that the study of finitely generated abelian groups is totally equivalent with the study of integer matrices. In particular, changing the generating set of A is equivalent with multiplying M on the left by a unimodular matrix (that is, an invertible integer matrix whose inverse is also an integer matrix). Changing the generating set of the kernel of M is equivalent with multiplying M on the right by a unimodular matrix.
The Smith normal form of M is a matrix
$S=UMV,$
where U and V are unimodular, and S is a matrix such that all non-diagonal entries are zero, the non-zero diagonal entries $d_{1,1},\ldots ,d_{k,k}$ are the first ones, and $d_{j,j}$ is a divisor of $d_{i,i}$ for i > j. The existence and the shape of the Smith normal proves that the finitely generated abelian group A is the direct sum
$\mathbb {Z} ^{r}\oplus \mathbb {Z} /d_{1,1}\mathbb {Z} \oplus \cdots \oplus \mathbb {Z} /d_{k,k}\mathbb {Z} ,$
where r is the number of zero rows at the bottom of r (and also the rank of the group). This is the fundamental theorem of finitely generated abelian groups.
The existence of algorithms for Smith normal form shows that the fundamental theorem of finitely generated abelian groups is not only a theorem of abstract existence, but provides a way for computing expression of finitely generated abelian groups as direct sums.[12]: 26–27
Infinite abelian groups
The simplest infinite abelian group is the infinite cyclic group $\mathbb {Z} $. Any finitely generated abelian group $A$ is isomorphic to the direct sum of $r$ copies of $\mathbb {Z} $ and a finite abelian group, which in turn is decomposable into a direct sum of finitely many cyclic groups of prime power orders. Even though the decomposition is not unique, the number $r$, called the rank of $A$, and the prime powers giving the orders of finite cyclic summands are uniquely determined.
By contrast, classification of general infinitely generated abelian groups is far from complete. Divisible groups, i.e. abelian groups $A$ in which the equation $nx=a$ admits a solution $x\in A$ for any natural number $n$ and element $a$ of $A$, constitute one important class of infinite abelian groups that can be completely characterized. Every divisible group is isomorphic to a direct sum, with summands isomorphic to $\mathbb {Q} $ and Prüfer groups $\mathbb {Q} _{p}/Z_{p}$ for various prime numbers $p$, and the cardinality of the set of summands of each type is uniquely determined.[13] Moreover, if a divisible group $A$ is a subgroup of an abelian group $G$ then $A$ admits a direct complement: a subgroup $C$ of $G$ such that $G=A\oplus C$. Thus divisible groups are injective modules in the category of abelian groups, and conversely, every injective abelian group is divisible (Baer's criterion). An abelian group without non-zero divisible subgroups is called reduced.
Two important special classes of infinite abelian groups with diametrically opposite properties are torsion groups and torsion-free groups, exemplified by the groups $\mathbb {Q} /\mathbb {Z} $ (periodic) and $\mathbb {Q} $ (torsion-free).
Torsion groups
An abelian group is called periodic or torsion, if every element has finite order. A direct sum of finite cyclic groups is periodic. Although the converse statement is not true in general, some special cases are known. The first and second Prüfer theorems state that if $A$ is a periodic group, and it either has a bounded exponent, i.e., $nA=0$ for some natural number $n$, or is countable and the $p$-heights of the elements of $A$ are finite for each $p$, then $A$ is isomorphic to a direct sum of finite cyclic groups.[14] The cardinality of the set of direct summands isomorphic to $\mathbb {Z} /p^{m}\mathbb {Z} $ in such a decomposition is an invariant of $A$.[15]: 6 These theorems were later subsumed in the Kulikov criterion. In a different direction, Helmut Ulm found an extension of the second Prüfer theorem to countable abelian $p$-groups with elements of infinite height: those groups are completely classified by means of their Ulm invariants.
Torsion-free and mixed groups
An abelian group is called torsion-free if every non-zero element has infinite order. Several classes of torsion-free abelian groups have been studied extensively:
• Free abelian groups, i.e. arbitrary direct sums of $\mathbb {Z} $
• Cotorsion and algebraically compact torsion-free groups such as the $p$-adic integers
• Slender groups[16]: 259–274
An abelian group that is neither periodic nor torsion-free is called mixed. If $A$ is an abelian group and $T(A)$ is its torsion subgroup, then the factor group $A/T(A)$ is torsion-free. However, in general the torsion subgroup is not a direct summand of $A$, so $A$ is not isomorphic to $T(A)\oplus A/T(A)$. Thus the theory of mixed groups involves more than simply combining the results about periodic and torsion-free groups. The additive group $\mathbb {Z} $ of integers is torsion-free $\mathbb {Z} $-module.[17]: 206
Invariants and classification
One of the most basic invariants of an infinite abelian group $A$ is its rank: the cardinality of the maximal linearly independent subset of $A$. Abelian groups of rank 0 are precisely the periodic groups, while torsion-free abelian groups of rank 1 are necessarily subgroups of $\mathbb {Q} $ and can be completely described. More generally, a torsion-free abelian group of finite rank $r$ is a subgroup of $\mathbb {Q} _{r}$. On the other hand, the group of $p$-adic integers $\mathbb {Z} _{p}$ is a torsion-free abelian group of infinite $\mathbb {Z} $-rank and the groups $\mathbb {Z} _{p}^{n}$ with different $n$ are non-isomorphic, so this invariant does not even fully capture properties of some familiar groups.
The classification theorems for finitely generated, divisible, countable periodic, and rank 1 torsion-free abelian groups explained above were all obtained before 1950 and form a foundation of the classification of more general infinite abelian groups. Important technical tools used in classification of infinite abelian groups are pure and basic subgroups. Introduction of various invariants of torsion-free abelian groups has been one avenue of further progress. See the books by Irving Kaplansky, László Fuchs, Phillip Griffith, and David Arnold, as well as the proceedings of the conferences on Abelian Group Theory published in Lecture Notes in Mathematics for more recent findings.
Additive groups of rings
The additive group of a ring is an abelian group, but not all abelian groups are additive groups of rings (with nontrivial multiplication). Some important topics in this area of study are:
• Tensor product
• A.L.S. Corner's results on countable torsion-free groups
• Shelah's work to remove cardinality restrictions
• Burnside ring
Relation to other mathematical topics
Many large abelian groups possess a natural topology, which turns them into topological groups.
The collection of all abelian groups, together with the homomorphisms between them, forms the category ${\textbf {Ab}}$, the prototype of an abelian category.
Wanda Szmielew (1955) proved that the first-order theory of abelian groups, unlike its non-abelian counterpart, is decidable. Most algebraic structures other than Boolean algebras are undecidable.
There are still many areas of current research:
• Amongst torsion-free abelian groups of finite rank, only the finitely generated case and the rank 1 case are well understood;
• There are many unsolved problems in the theory of infinite-rank torsion-free abelian groups;
• While countable torsion abelian groups are well understood through simple presentations and Ulm invariants, the case of countable mixed groups is much less mature.
• Many mild extensions of the first-order theory of abelian groups are known to be undecidable.
• Finite abelian groups remain a topic of research in computational group theory.
Moreover, abelian groups of infinite order lead, quite surprisingly, to deep questions about the set theory commonly assumed to underlie all of mathematics. Take the Whitehead problem: are all Whitehead groups of infinite order also free abelian groups? In the 1970s, Saharon Shelah proved that the Whitehead problem is:
• Undecidable in ZFC (Zermelo–Fraenkel axioms), the conventional axiomatic set theory from which nearly all of present-day mathematics can be derived. The Whitehead problem is also the first question in ordinary mathematics proved undecidable in ZFC;
• Undecidable even if ZFC is augmented by taking the generalized continuum hypothesis as an axiom;
• Positively answered if ZFC is augmented with the axiom of constructibility (see statements true in L).
A note on typography
Among mathematical adjectives derived from the proper name of a mathematician, the word "abelian" is rare in that it is often spelled with a lowercase a, rather than an uppercase A, the lack of capitalization being a tacit acknowledgment not only of the degree to which Abel's name has been institutionalized but also of how ubiquitous in modern mathematics are the concepts introduced by him.[18]
See also
Algebraic structures
Group-like
• Group
• Semigroup / Monoid
• Rack and quandle
• Quasigroup and loop
• Abelian group
• Magma
• Lie group
Group theory
Ring-like
• Ring
• Rng
• Semiring
• Near-ring
• Commutative ring
• Domain
• Integral domain
• Field
• Division ring
• Lie ring
Ring theory
Lattice-like
• Lattice
• Semilattice
• Complemented lattice
• Total order
• Heyting algebra
• Boolean algebra
• Map of lattices
• Lattice theory
Module-like
• Module
• Group with operators
• Vector space
• Linear algebra
Algebra-like
• Algebra
• Associative
• Non-associative
• Composition algebra
• Lie algebra
• Graded
• Bialgebra
• Hopf algebra
• Commutator subgroup – Smallest normal subgroup by which the quotient is commutative
• Abelianization – Quotienting a group by its commutator subgroup
• Dihedral group of order 6 – Non-commutative group with 6 elements, the smallest non-abelian group
• Elementary abelian group – Commutative group in which all nonzero elements have the same order
• Grothendieck group – Abelian group extending a commutative monoid
• Pontryagin duality – Duality for locally compact abelian groups
Notes
1. Jacobson (2009) p. 41
2. Ramík, J., Pairwise Comparisons Method: Theory and Applications in Decision Making (Cham: Springer Nature Switzerland, 2020), p. 11.
3. Auslander, M., & Buchsbaum, D., Groups, Rings, Modules (Mineola, NY: Dover Publications, 1974), pp. 28–29.
4. Isaev, A. P., & Rubakov, V. A., Theory of Groups and Symmetries: Finite Groups, Lie Groups, and Lie Algebras (Singapore: World Scientific, 2018), p. 10.
5. Rose 2012, p. 32.
6. Cox, D. A., Galois Theory (Hoboken, New Jersey: John Wiley & Sons, 2004), pp. 144–145.
7. Eklof, Paul C., & Göbel, Rüdiger, eds., Abelian Groups and Modules: International Conference in Dublin, August 10–14, 1998 (Basel: Springer Basel AG, 1999), pp. 94–97.
8. Dixon, M. R., Kurdachenko, L. A., & Subbotin, I. Y., Linear Groups: The Accent on Infinite Dimensionality (Milton Park, Abingdon-on-Thames & Oxfordshire: Taylor & Francis, 2020), pp. 49–50.
9. Rose 2012, p. 48.
10. Rose 2012, p. 79.
11. Kurzweil, H., & Stellmacher, B., The Theory of Finite Groups: An Introduction (New York, Berlin, Heidelberg: Springer Verlag, 2004), pp. 43–54.
12. Finkelstein, L., & Kantor, W. M., eds., Groups and Computation II: Workshop on Groups and Computation, June 7–10, 1995 (Providence: AMS, 1997), pp. 26–27.
13. For example, $\mathbb {Q} /\mathbb {Z} \cong \sum _{p}\mathbb {Q} _{p}/\mathbb {Z} _{p}$.
14. Countability assumption in the second Prüfer theorem cannot be removed: the torsion subgroup of the direct product of the cyclic groups $\mathbb {Z} /p^{m}\mathbb {Z} $ for all natural $m$ is not a direct sum of cyclic groups.
15. Faith, C. C., Rings and Things and a Fine Array of Twentieth Century Associative Algebra (Providence: AMS, 2004), p. 6.
16. Albrecht, U., "Products of Slender Abelian Groups", in Göbel, R., & Walker, E., eds., Abelian Group Theory: Proceedings of the Third Conference Held on Abelian Group Theory at Oberwolfach, August 11-17, 1985 (New York: Gordon & Breach, 1987), pp. 259–274.
17. Lal, R., Algebra 2: Linear Algebra, Galois Theory, Representation Theory, Group Extensions and Schur Multiplier (Berlin, Heidelberg: Springer, 2017), p. 206.
18. "Abel Prize Awarded: The Mathematicians' Nobel". Archived from the original on 31 December 2012. Retrieved 3 July 2016.
References
• Cox, David (2004). Galois Theory. Wiley-Interscience. ISBN 9781118031339. MR 2119052.
• Fuchs, László (1970). Infinite Abelian Groups. Pure and Applied Mathematics. Vol. 36-I. Academic Press. MR 0255673.
• Fuchs, László (1973). Infinite Abelian Groups. Pure and Applied Mathematics. Vol. 36-II. Academic Press. MR 0349869.
• Griffith, Phillip A. (1970). Infinite Abelian group theory. Chicago Lectures in Mathematics. University of Chicago Press. ISBN 0-226-30870-7.
• Herstein, I. N. (1975). Topics in Algebra (2nd ed.). John Wiley & Sons. ISBN 0-471-02371-X.
• Hillar, Christopher; Rhea, Darren (2007). "Automorphisms of finite abelian groups" (PDF). American Mathematical Monthly. 114 (10): 917–923. arXiv:math/0605185. Bibcode:2006math......5185H. doi:10.1080/00029890.2007.11920485. JSTOR 27642365. S2CID 1038507.
• Jacobson, Nathan (2009). Basic Algebra I (2nd ed.). Dover Publications. ISBN 978-0-486-47189-1.
• Rose, John S. (2012). A Course on Group Theory. Dover Publications. ISBN 978-0-486-68194-8. Unabridged and unaltered republication of a work first published by the Cambridge University Press, Cambridge, England, in 1978.
• Szmielew, Wanda (1955). "Elementary Properties of Abelian Groups" (PDF). Fundamenta Mathematicae. 41 (2): 203–271. doi:10.4064/fm-41-2-203-271. MR 0072131. Zbl 0248.02049.
• Robinson, Abraham; Zakon, Elias (1960). "Elementary Properties of Ordered Abelian Groups" (PDF). Transactions of the American Mathematical Society. 96 (2): 222–236. doi:10.2307/1993461. JSTOR 1993461. Archived (PDF) from the original on 2022-10-09.
External links
• "Abelian group". Encyclopedia of Mathematics. EMS Press. 2001 [1994].
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| Wikipedia |
Abelian surface
In mathematics, an abelian surface is a 2-dimensional abelian variety.
One-dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most complex tori of dimension 2 are not algebraic via the Riemann bilinear relations. Essentially, these are conditions on the parameter space of period matrices for complex tori which define an algebraic subvariety. This subvariety contains all of the points whose period matrices correspond to a period matrix of an abelian variety.
The algebraic ones are called abelian surfaces and are exactly the 2-dimensional abelian varieties. Most of their theory is a special case of the theory of higher-dimensional tori or abelian varieties. Finding criteria for a complex torus of dimension 2 to be a product of two elliptic curves (up to isogeny) was a popular subject of study in the nineteenth century.
Invariants: The plurigenera are all 1. The surface is diffeomorphic to S1×S1×S1×S1 so the fundamental group is Z4.
Hodge diamond:
1
22
141
22
1
Examples: A product of two elliptic curves. The Jacobian variety of a genus 2 curve.
See also
• Hodge theory
• Complex torus
References
• Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004), Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 4, Springer-Verlag, Berlin, ISBN 978-3-540-00832-3, MR 2030225
• Beauville, Arnaud (1996), Complex algebraic surfaces, London Mathematical Society Student Texts, vol. 34 (2nd ed.), Cambridge University Press, ISBN 978-0-521-49510-3, MR 1406314
• Birkenhake, Ch. (2001) [1994], "Abelian surface", Encyclopedia of Mathematics, EMS Press
| Wikipedia |
Abelian von Neumann algebra
In functional analysis, an abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute.
The prototypical example of an abelian von Neumann algebra is the algebra L∞(X, μ) for μ a σ-finite measure on X realized as an algebra of operators on the Hilbert space L2(X, μ) as follows: Each f ∈ L∞(X, μ) is identified with the multiplication operator
$\psi \mapsto f\psi .$
Of particular importance are the abelian von Neumann algebras on separable Hilbert spaces, particularly since they are completely classifiable by simple invariants.
Though there is a theory for von Neumann algebras on non-separable Hilbert spaces (and indeed much of the general theory still holds in that case) the theory is considerably simpler for algebras on separable spaces and most applications to other areas of mathematics or physics only use separable Hilbert spaces. Note that if the measure spaces (X, μ) is a standard measure space (that is X − N is a standard Borel space for some null set N and μ is a σ-finite measure) then L2(X, μ) is separable.
Classification
The relationship between commutative von Neumann algebras and measure spaces is analogous to that between commutative C*-algebras and locally compact Hausdorff spaces. Every commutative von Neumann algebra on a separable Hilbert space is isomorphic to L∞(X) for some standard measure space (X, μ) and conversely, for every standard measure space X, L∞(X) is a von Neumann algebra. This isomorphism as stated is an algebraic isomorphism. In fact we can state this more precisely as follows:
Theorem. Any abelian von Neumann algebra of operators on a separable Hilbert space is *-isomorphic to exactly one of the following
• $\ell ^{\infty }(\{1,2,\ldots ,n\}),\quad n\geq 1$
• $\ell ^{\infty }(\mathbf {N} )$
• $L^{\infty }([0,1])$
• $L^{\infty }([0,1]\cup \{1,2,\ldots ,n\}),\quad n\geq 1$
• $L^{\infty }([0,1]\cup \mathbf {N} ).$
The isomorphism can be chosen to preserve the weak operator topology.
In the above list, the interval [0,1] has Lebesgue measure and the sets {1, 2, ..., n} and N have counting measure. The unions are disjoint unions. This classification is essentially a variant of Maharam's classification theorem for separable measure algebras. The version of Maharam's classification theorem that is most useful involves a point realization of the equivalence, and is somewhat of a folk theorem.
Although every standard measure space is isomorphic to one of the above and the list is exhaustive in this sense, there is a more canonical choice for the measure space in the case of abelian von Neumann algebras A: The set of all projectors is a $\sigma $-complete Boolean algebra, that is a pointfree $\sigma $-algebra. In the special case $A=L^{\infty }(X,{\mathfrak {A}},\mu )$ one recovers the abstract $\sigma $-algebra ${\mathfrak {A}}/\{A\mid \mu (A)=0\}$. This pointfree approach can be turned into a duality theorem analogue to Gelfand-duality between the category of abelian von Neumann algebras and the category of abstract $\sigma $-algebras.
Let μ and ν be non-atomic probability measures on standard Borel spaces X and Y respectively. Then there is a μ null subset N of X, a ν null subset M of Y and a Borel isomorphism
$\phi :X\setminus N\rightarrow Y\setminus M,\quad $
which carries μ into ν.[1]
Notice that in the above result, it is necessary to clip away sets of measure zero to make the result work.
In the above theorem, the isomorphism is required to preserve the weak operator topology. As it turns out (and follows easily from the definitions), for algebras L∞(X, μ), the following topologies agree on norm bounded sets:
1. The weak operator topology on L∞(X, μ);
2. The ultraweak operator topology on L∞(X, μ);
3. The topology of weak* convergence on L∞(X, μ) considered as the dual space of L1(X, μ).
However, for an abelian von Neumann algebra A the realization of A as an algebra of operators on a separable Hilbert space is highly non-unique. The complete classification of the operator algebra realizations of A is given by spectral multiplicity theory and requires the use of direct integrals.
Spatial isomorphism
Using direct integral theory, it can be shown that the abelian von Neumann algebras of the form L∞(X, μ) acting as operators on L2(X, μ) are all maximal abelian. This means that they cannot be extended to properly larger abelian algebras. They are also referred to as Maximal abelian self-adjoint algebras (or M.A.S.A.s). Another phrase used to describe them is abelian von Neumann algebras of uniform multiplicity 1; this description makes sense only in relation to multiplicity theory described below.
Von Neumann algebras A on H, B on K are spatially isomorphic (or unitarily isomorphic) if and only if there is a unitary operator U: H → K such that
$UAU^{*}=B.$
In particular spatially isomorphic von Neumann algebras are algebraically isomorphic.
To describe the most general abelian von Neumann algebra on a separable Hilbert space H up to spatial isomorphism, we need to refer the direct integral decomposition of H. The details of this decomposition are discussed in decomposition of abelian von Neumann algebras. In particular:
Theorem Any abelian von Neumann algebra on a separable Hilbert space H is spatially isomorphic to L∞(X, μ) acting on
$\int _{X}^{\oplus }H(x)\,d\mu (x)$
for some measurable family of Hilbert spaces {Hx}x ∈ X.
Note that for abelian von Neumann algebras acting on such direct integral spaces, the equivalence of the weak operator topology, the ultraweak topology and the weak* topology on norm bounded sets still hold.
Point and spatial realization of automorphisms
Many problems in ergodic theory reduce to problems about automorphisms of abelian von Neumann algebras. In that regard, the following results are useful:
Theorem.[2] Suppose μ, ν are standard measures on X, Y respectively. Then any involutive isomorphism
$\Phi :L^{\infty }(X,\mu )\rightarrow L^{\infty }(Y,\nu )$
which is weak*-bicontinuous corresponds to a point transformation in the following sense: There are Borel null subsets M of X and N of Y and a Borel isomorphism
$\eta :X\setminus M\rightarrow Y\setminus N$
such that
1. η carries the measure μ into a measure μ' on Y which is equivalent to ν in the sense that μ' and ν have the same sets of measure zero;
2. η realizes the transformation Φ, that is
$\Phi (f)=f\circ \eta ^{-1}.$
Note that in general we cannot expect η to carry μ into ν.
The next result concerns unitary transformations which induce a weak*-bicontinuous isomorphism between abelian von Neumann algebras.
Theorem.[3] Suppose μ, ν are standard measures on X, Y and
$H=\int _{X}^{\oplus }H_{x}d\mu (x),\quad K=\int _{Y}^{\oplus }K_{y}d\nu (y)$
for measurable families of Hilbert spaces {Hx}x ∈ X, {Ky}y ∈ Y. If U : H → K is a unitary such that
$U\,L^{\infty }(X,\mu )\,U^{*}=L^{\infty }(Y,\nu )$
then there is an almost everywhere defined Borel point transformation η : X → Y as in the previous theorem and a measurable family {Ux}x ∈ X of unitary operators
$U_{x}:H_{x}\rightarrow K_{\eta (x)}$
such that
$U{\bigg (}\int _{X}^{\oplus }\psi _{x}d\mu (x){\bigg )}=\int _{Y}^{\oplus }{\sqrt {{\frac {d(\mu \circ \eta ^{-1})}{d\nu }}(y)}}\ U_{\eta ^{-1}(y)}{\bigg (}\psi _{\eta ^{-1}(y)}{\bigg )}d\nu (y),$
where the expression in square root sign is the Radon–Nikodym derivative of μ η−1 with respect to ν. The statement follows combining the theorem on point realization of automorphisms stated above with the theorem characterizing the algebra of diagonalizable operators stated in the article on direct integrals.
Notes
1. Bogachev, V.I. (2007). Measure theory. Vol. II. Springer-Verlag. p. 275. ISBN 978-3-540-34513-8.
2. Takesaki, Masamichi (2001), Theory of Operator Algebras I, Springer-Verlag, ISBN 3-540-42248-X, Chapter IV, Lemma 8.22, p. 275
3. Takesaki, Masamichi (2001), Theory of Operator Algebras I, Springer-Verlag, ISBN 3-540-42248-X, Chapter IV, Theorem 8.23, p. 277
References
• J. Dixmier, Les algèbres d'opérateurs dans l'espace Hilbertien, Gauthier-Villars, 1969. See chapter I, section 6.
• Masamichi Takesaki Theory of Operator Algebras I,II,III", encyclopedia of mathematical sciences, Springer-Verlag, 2001–2003 (the first volume was published 1979 in 1. Edition) ISBN 3-540-42248-X
| Wikipedia |
Abel–Ruffini theorem
In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, general means that the coefficients of the equation are viewed and manipulated as indeterminates.
Not to be confused with Abel's theorem.
The theorem is named after Paolo Ruffini, who made an incomplete proof in 1799,[1] (which was refined and completed in 1813[2] and accepted by Cauchy) and Niels Henrik Abel, who provided a proof in 1824.[3][4]
Abel–Ruffini theorem refers also to the slightly stronger result that there are equations of degree five and higher that cannot be solved by radicals. This does not follow from Abel's statement of the theorem, but is a corollary of his proof, as his proof is based on the fact that some polynomials in the coefficients of the equation are not the zero polynomial. This improved statement follows directly from Galois theory § A non-solvable quintic example. Galois theory implies also that
$x^{5}-x-1=0$
is the simplest equation that cannot be solved in radicals, and that almost all polynomials of degree five or higher cannot be solved in radicals.
The impossibility of solving in degree five or higher contrasts with the case of lower degree: one has the quadratic formula, the cubic formula, and the quartic formula for degrees two, three, and four, respectively.
Context
Polynomial equations of degree two can be solved with the quadratic formula, which has been known since antiquity. Similarly the cubic formula for degree three, and the quartic formula for degree four, were found during the 16th century. At that time a fundamental problem was whether equations of higher degree could be solved in a similar way.
The fact that every polynomial equation of positive degree has solutions, possibly non-real, was asserted during the 17th century, but completely proved only at the beginning of the 19th century. This is the fundamental theorem of algebra, which does not provide any tool for computing exactly the solutions, although Newton's method allows approximating the solutions to any desired accuracy.
From the 16th century to beginning of the 19th century, the main problem of algebra was to search for a formula for the solutions of polynomial equations of degree five and higher, hence the name the "fundamental theorem of algebra". This meant a solution in radicals, that is, an expression involving only the coefficients of the equation, and the operations of addition, subtraction, multiplication, division, and nth root extraction.
The Abel–Ruffini theorem proves that this is impossible. However, this impossibility does not imply that a specific equation of any degree cannot be solved in radicals. On the contrary, there are equations of any degree that can be solved in radicals. This is the case of the equation $x^{n}-1=0$ for any n, and the equations defined by cyclotomic polynomials, all of whose solutions can be expressed in radicals.
Abel's proof of the theorem does not explicitly contain the assertion that there are specific equations that cannot be solved by radicals. Such an assertion is not a consequence of Abel's statement of the theorem, as the statement does not exclude the possibility that "every particular quintic equation might be soluble, with a special formula for each equation."[5] However, the existence of specific equations that cannot be solved in radicals seems to be a consequence of Abel's proof, as the proof uses the fact that some polynomials in the coefficients are not the zero polynomial, and, given a finite number of polynomials, there are values of the variables at which none of the polynomials takes the value zero.
Soon after Abel's publication of its proof, Évariste Galois introduced a theory, now called Galois theory that allows deciding, for any given equation, whether it is solvable in radicals. This was purely theoretical before the rise of electronic computers. With modern computers and programs, deciding whether a polynomial is solvable by radicals can be done for polynomials of degree up to 31. Computing the solutions in radicals of solvable polynomials requires huge computations and, as of 2023, no implemented algorithm has been published for polynomials of degree higher than seven. Even for the degree five, the expression of the solutions is so huge that it has no practical interest.
Proof
The proof of the Abel–Ruffini theorem predates Galois theory. However, Galois theory allows a better understanding of the subject, and modern proofs are generally based on it, while the original proofs of the Abel–Ruffini theorem are still presented for historical purposes.[1][6][7][8]
The proofs based on Galois theory comprise four main steps: the characterization of solvable equations in terms of field theory; the use of the Galois correspondence between subfields of a given field and the subgroups of its Galois group for expressing this characterization in terms of solvable groups; the proof that the symmetric group is not solvable if its degree is five or higher; and the existence of polynomials with a symmetric Galois group.
Algebraic solutions and field theory
An algebraic solution of a polynomial equation is an expression involving the four basic arithmetic operations (addition, subtraction, multiplication, and division), and root extractions. Such an expression may be viewed as the description of a computation that starts from the coefficients of the equation to be solved and proceeds by computing some numbers, one after the other.
At each step of the computation, one may consider the smallest field that contains all numbers that have been computed so far. This field is changed only for the steps involving the computation of an nth root.
So, an algebraic solution produces a sequence
$F_{0}\subseteq F_{1}\subseteq \cdots \subseteq F_{k}$
of fields, and elements $x_{i}\in F_{i}$ such that $F_{i}=F_{i-1}(x_{i})$ for $i=1,\ldots ,k,$ with $x_{i}^{n_{i}}\in F_{i-1}$ for some integer $n_{i}>1.$ An algebraic solution of the initial polynomial equation exists if and only if there exists such a sequence of fields such that $F_{k}$ contains a solution.
For having normal extensions, which are fundamental for the theory, one must refine the sequence of fields as follows. If $F_{i-1}$ does not contain all $n_{i}$-th roots of unity, one introduces the field $K_{i}$ that extends $F_{i-1}$ by a primitive root of unity, and one redefines $F_{i}$ as $K_{i}(x_{i}).$
So, if one starts from a solution in terms of radicals, one gets an increasing sequence of fields such that the last one contains the solution, and each is a normal extension of the preceding one with a Galois group that is cyclic.
Conversely, if one has such a sequence of fields, the equation is solvable in terms of radicals. For proving this, it suffices to prove that a normal extension with a cyclic Galois group can be built from a succession of radical extensions.
Galois correspondence
The Galois correspondence establishes a one to one correspondence between the subextensions of a normal field extension $F/E$ and the subgroups of the Galois group of the extension. This correspondence maps a field K such $E\subseteq K\subseteq F$ to the Galois group $\operatorname {Gal} (F/K)$ of the automorphisms of F that leave K fixed, and, conversely, maps a subgroup H of $\operatorname {Gal} (F/E)$ to the field of the elements of F that are fixed by H.
The preceding section shows that an equation is solvable in terms of radicals if and only if the Galois group of its splitting field (the smallest field that contains all the roots) is solvable, that is, it contains a sequence of subgroups such that each is normal in the preceding one, with a quotient group that is cyclic. (Solvable groups are commonly defined with abelian instead of cyclic quotient groups, but the fundamental theorem of finite abelian groups shows that the two definitions are equivalent).
So, for proving Abel–Ruffini theorem, it remains to prove that the symmetric group $S_{5}$ is not solvable, and that there are polynomials with symmetric Galois group.
Solvable symmetric groups
For n > 4, the symmetric group ${\mathcal {S}}_{n}$ of degree n has only the alternating group ${\mathcal {A}}_{n}$ as a nontrivial normal subgroup (see Symmetric group § Normal subgroups). For n > 4, the alternating group ${\mathcal {A}}_{n}$ is not abelian and simple (that is, it does not have any nontrivial normal subgroup). This implies that both ${\mathcal {A}}_{n}$ and ${\mathcal {S}}_{n}$ are not solvable for n > 4. Thus, the Abel–Ruffini theorem results from the existence of polynomials with a symmetric Galois group; this will be shown in the next section.
On the other hand, for n ≤ 4, the symmetric group and all its subgroups are solvable. This explains the existence of the quadratic, cubic, and quartic formulas, since a major result of Galois theory is that a polynomial equation has a solution in radicals if and only if its Galois group is solvable (the term "solvable group" takes its origin from this theorem).
General equation
The general or generic polynomial equation of degree n is the equation
$x^{n}+a_{1}x^{n-1}+\cdots +a_{n-1}x+a_{n}=0,$
where $a_{1},\ldots ,a_{n}$ are distinct indeterminates. This is an equation defined over the field $F=\mathbb {Q} (a_{1},\ldots ,a_{n})$ of the rational fractions in $a_{1},\ldots ,a_{n}$ with rational number coefficients. The original Abel–Ruffini theorem asserts that, for n > 4, this equation is not solvable in radicals. In view of the preceding sections, this results from the fact that the Galois group over F of the equation is the symmetric group ${\mathcal {S}}_{n}$ (this Galois group is the group of the field automorphisms of the splitting field of the equation that fix the elements of F, where the splitting field is the smallest field containing all the roots of the equation).
For proving that the Galois group is ${\mathcal {S}}_{n},$ it is simpler to start from the roots. Let $x_{1},\ldots ,x_{n}$ be new indeterminates, aimed to be the roots, and consider the polynomial
$P(x)=x^{n}+b_{1}x^{n-1}+\cdots +b_{n-1}x+b_{n}=(x-x_{1})\cdots (x-x_{n}).$
Let $H=\mathbb {Q} (x_{1},\ldots ,x_{n})$ be the field of the rational fractions in $x_{1},\ldots ,x_{n},$ and $K=\mathbb {Q} (b_{1},\ldots ,b_{n})$ be its subfield generated by the coefficients of $P(x).$ The permutations of the $x_{i}$ induce automorphisms of H. Vieta's formulas imply that every element of K is a symmetric function of the $x_{i},$ and is thus fixed by all these automorphisms. It follows that the Galois group $\operatorname {Gal} (H/K)$ is the symmetric group ${\mathcal {S}}_{n}.$
The fundamental theorem of symmetric polynomials implies that the $b_{i}$ are algebraic independent, and thus that the map that sends each $a_{i}$ to the corresponding $b_{i}$ is a field isomorphism from F to K. This means that one may consider $P(x)=0$ as a generic equation. This finishes the proof that the Galois group of a general equation is the symmetric group, and thus proves the original Abel–Ruffini theorem, which asserts that the general polynomial equation of degree n cannot be solved in radicals for n > 4.
Explicit example
See also: Galois theory § A non-solvable quintic example
The equation $x^{5}-x-1=0$ is not solvable in radicals, as will be explained below.
Let q be $x^{5}-x-1$. Let G be its Galois group, which acts faithfully on the set of complex roots of q. Numbering the roots lets one identify G with a subgroup of the symmetric group ${\mathcal {S}}_{5}$. Since $q{\bmod {2}}$ factors as $(x^{2}+x+1)(x^{3}+x^{2}+1)$ in $\mathbb {F} _{2}[x]$, the group G contains a permutation g that is a product of disjoint cycles of lengths 2 and 3 (in general, when a monic integer polynomial reduces modulo a prime to a product of distinct monic irreducible polynomials, the degrees of the factors give the lengths of the disjoint cycles in some permutation belonging to the Galois group); then G also contains $g^{3}$, which is a transposition. Since $q{\bmod {3}}$ is irreducible in $\mathbb {F} _{3}[x]$, the same principle shows that G contains a 5-cycle. Because 5 is prime, any transposition and 5-cycle in ${\mathcal {S}}_{5}$ generate the whole group; see Symmetric group § Generators and relations. Thus $G={\mathcal {S}}_{5}$. Since the group ${\mathcal {S}}_{5}$ is not solvable, the equation $x^{5}-x-1=0$ is not solvable in radicals.
Cayley's resolvent
Testing whether a specific quintic is solvable in radicals can be done by using Cayley's resolvent. This is a univariate polynomial of degree six whose coefficients are polynomials in the coefficients of a generic quintic. A specific irreducible quintic is solvable in radicals if and only, when its coefficients are substituted in Cayley's resolvent, the resulting sextic polynomial has a rational root.
History
Around 1770, Joseph Louis Lagrange began the groundwork that unified the many different tricks that had been used up to that point to solve equations, relating them to the theory of groups of permutations, in the form of Lagrange resolvents.[9] This innovative work by Lagrange was a precursor to Galois theory, and its failure to develop solutions for equations of fifth and higher degrees hinted that such solutions might be impossible, but it did not provide conclusive proof. The first person who conjectured that the problem of solving quintics by radicals might be impossible to solve was Carl Friedrich Gauss, who wrote in 1798 in section 359 of his book Disquisitiones Arithmeticae (which would be published only in 1801) that "there is little doubt that this problem does not so much defy modern methods of analysis as that it proposes the impossible". The next year, in his thesis, he wrote "After the labors of many geometers left little hope of ever arriving at the resolution of the general equation algebraically, it appears more and more likely that this resolution is impossible and contradictory." And he added "Perhaps it will not be so difficult to prove, with all rigor, the impossibility for the fifth degree. I shall set forth my investigations of this at greater length in another place." Actually, Gauss published nothing else on this subject.[1]
The theorem was first nearly proved by Paolo Ruffini in 1799.[10] He sent his proof to several mathematicians to get it acknowledged, amongst them Lagrange (who did not reply) and Augustin-Louis Cauchy, who sent him a letter saying: "Your memoir on the general solution of equations is a work which I have always believed should be kept in mind by mathematicians and which, in my opinion, proves conclusively the algebraic unsolvability of general equations of higher than fourth degree."[11] However, in general, Ruffini's proof was not considered convincing. Abel wrote: "The first and, if I am not mistaken, the only one who, before me, has sought to prove the impossibility of the algebraic solution of general equations is the mathematician Ruffini. But his memoir is so complicated that it is very difficult to determine the validity of his argument. It seems to me that his argument is not completely satisfying."[11][12]
The proof also, as it was discovered later, was incomplete. Ruffini assumed that all radicals that he was dealing with could be expressed from the roots of the polynomial using field operations alone; in modern terms, he assumed that the radicals belonged to the splitting field of the polynomial. To see why this is really an extra assumption, consider, for instance, the polynomial $P(x)=x^{3}-15x-20$. According to Cardano's formula, one of its roots (all of them, actually) can be expressed as the sum of a cube root of $10+5i$ with a cube root of $10-5i$. On the other hand, since $P(-3)<0$, $P(-2)>0$, $P(-1)<0$, and $P(5)>0$, the roots $r_{1}$, $r_{2}$, and $r_{3}$ of $P(x)$ are all real and therefore the field $\mathbf {Q} (r_{1},r_{2},r_{3})$ is a subfield of $\mathbf {R} $. But then the numbers $10\pm 5i$ cannot belong to $\mathbf {Q} (r_{1},r_{2},r_{3})$. While Cauchy either did not notice Ruffini's assumption or felt that it was a minor one, most historians believe that the proof was not complete until Abel proved the theorem on natural irrationalities, which asserts that the assumption holds in the case of general polynomials.[7][13] The Abel–Ruffini theorem is thus generally credited to Abel, who published a proof compressed into just six pages in 1824.[3] (Abel adopted a very terse style to save paper and money: the proof was printed at his own expense.[8]) A more elaborated version of the proof would be published in 1826.[4]
Proving that the general quintic (and higher) equations were unsolvable by radicals did not completely settle the matter, because the Abel–Ruffini theorem does not provide necessary and sufficient conditions for saying precisely which quintic (and higher) equations are unsolvable by radicals. Abel was working on a complete characterization when he died in 1829.[14]
According to Nathan Jacobson, "The proofs of Ruffini and of Abel [...] were soon superseded by the crowning achievement of this line of research: Galois' discoveries in the theory of equations."[15] In 1830, Galois (at the age of 18) submitted to the Paris Academy of Sciences a memoir on his theory of solvability by radicals, which was ultimately rejected in 1831 as being too sketchy and for giving a condition in terms of the roots of the equation instead of its coefficients. Galois was aware of the contributions of Ruffini and Abel, since he wrote "It is a common truth, today, that the general equation of degree greater than 4 cannot be solved by radicals... this truth has become common (by hearsay) despite the fact that geometers have ignored the proofs of Abel and Ruffini..."[1] Galois then died in 1832 and his paper Mémoire sur les conditions de resolubilité des équations par radicaux[16] remained unpublished until 1846, when it was published by Joseph Liouville accompanied by some of his own explanations.[14] Prior to this publication, Liouville announced Galois' result to the academy in a speech he gave on 4 July 1843.[5] A simplification of Abel's proof was published by Pierre Wantzel in 1845.[17] When Wantzel published it, he was already aware of the contributions by Galois and he mentions that, whereas Abel's proof is valid only for general polynomials, Galois' approach can be used to provide a concrete polynomial of degree 5 whose roots cannot be expressed in radicals from its coefficients.
In 1963, Vladimir Arnold discovered a topological proof of the Abel–Ruffini theorem,[18][19][20] which served as a starting point for topological Galois theory.[21]
References
1. Ayoub, Raymond G. (1980), "Paolo Ruffini's Contributions to the Quintic", Archive for History of Exact Sciences, 22 (3): 253–277, doi:10.1007/BF00357046, JSTOR 41133596, MR 0606270, S2CID 123447349, Zbl 0471.01008
2. Ruffini, Paolo (1813). Riflessioni intorno alla soluzione delle equazioni algebraiche generali opuscolo del cav. dott. Paolo Ruffini ... (in Italian). presso la Societa Tipografica.
3. Abel, Niels Henrik (1881) [1824], "Mémoire sur les équations algébriques, ou l'on démontre l'impossibilité de la résolution de l'équation générale du cinquième degré" (PDF), in Sylow, Ludwig; Lie, Sophus (eds.), Œuvres Complètes de Niels Henrik Abel (in French), vol. I (2nd ed.), Grøndahl & Søn, pp. 28–33
4. Abel, Niels Henrik (1881) [1826], "Démonstration de l'impossibilité de la résolution algébrique des équations générales qui passent le quatrième degré" (PDF), in Sylow, Ludwig; Lie, Sophus (eds.), Œuvres Complètes de Niels Henrik Abel (in French), vol. I (2nd ed.), Grøndahl & Søn, pp. 66–87
5. Stewart, Ian (2015), "Historical Introduction", Galois Theory (4th ed.), CRC Press, ISBN 978-1-4822-4582-0
6. Rosen, Michael I. (1995), "Niels Hendrik Abel and Equations of the Fifth Degree", American Mathematical Monthly, 102 (6): 495–505, doi:10.2307/2974763, JSTOR 2974763, MR 1336636, Zbl 0836.01015
7. Tignol, Jean-Pierre (2016), "Ruffini and Abel on General Equations", Galois' Theory of Algebraic Equations (2nd ed.), World Scientific, ISBN 978-981-4704-69-4, Zbl 1333.12001
8. Pesic, Peter (2004), Abel's Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability, Cambridge: MIT Press, ISBN 0-262-66182-9, Zbl 1166.01010
9. Lagrange, Joseph-Louis (1869) [1771], "Réflexions sur la résolution algébrique des équations", in Serret, Joseph-Alfred (ed.), Œuvres de Lagrange, vol. III, Gauthier-Villars, pp. 205–421
10. Ruffini, Paolo (1799), Teoria generale delle equazioni, in cui si dimostra impossibile la soluzione algebraica delle equazioni generali di grado superiore al quarto (in Italian), Stamperia di S. Tommaso d'Aquino
11. Kiernan, B. Melvin (1971), "The Development of Galois Theory from Lagrange to Artin", Archive for History of Exact Sciences, 8 (1/2): 40–154, doi:10.1007/BF00327219, JSTOR 41133337, S2CID 121442989
12. Abel, Niels Henrik (1881) [1828], "Sur la resolution algébrique des équations" (PDF), in Sylow, Ludwig; Lie, Sophus (eds.), Œuvres Complètes de Niels Henrik Abel (in French), vol. II (2nd ed.), Grøndahl & Søn, pp. 217–243
13. Stewart, Ian (2015), "The Idea Behind Galois Theory", Galois Theory (4th ed.), CRC Press, ISBN 978-1-4822-4582-0
14. Tignol, Jean-Pierre (2016), "Galois", Galois' Theory of Algebraic Equations (2nd ed.), World Scientific, ISBN 978-981-4704-69-4, Zbl 1333.12001
15. Jacobson, Nathan (2009), "Galois Theory of Equations", Basic Algebra, vol. 1 (2nd ed.), Dover, ISBN 978-0-486-47189-1
16. Galois, Évariste (1846), "Mémoire sur les conditions de resolubilité des équations par radicaux" (PDF), Journal de Mathématiques Pures et Appliquées (in French), XI: 417–433
17. Wantzel, Pierre (1845), "Démonstration de l'impossibilité de résoudre toutes les équations algébriques avec des radicaux", Nouvelles Annales de Mathématiques (in French), 4: 57–65
18. Alekseev, V. B. (2004), Abel's Theorem in Problems and Solutions: Based on the Lectures of Professor V. I. Arnold, Kluwer Academic Publishers, ISBN 1-4020-2186-0, Zbl 1065.12001
19. "Short Proof of Abel's Theorem that 5th Degree Polynomial Equations Cannot be Solved" on YouTube
20. Goldmakher, Leo, Arnold's Elementary Proof of the Insolvability of the Quintic (PDF)
21. Khovanskii, Askold (2014), Topological Galois Theory: Solvability and Unsolvability of Equations in Finite Terms, Springer Monographs in Mathematics, Springer-Verlag, doi:10.1007/978-3-642-38871-2, ISBN 978-3-642-38870-5
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