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Nakai conjecture
In mathematics, the Nakai conjecture is an unproven characterization of smooth algebraic varieties, conjectured by Japanese mathematician Yoshikazu Nakai in 1961.[1] It states that if V is a complex algebraic variety, such that its ring of differential operators is generated by the derivations it contains, then V is a smooth variety. The converse statement, that smooth algebraic varieties have rings of differential operators that are generated by their derivations, is a result of Alexander Grothendieck.[2]
The Nakai conjecture is known to be true for algebraic curves[3] and Stanley–Reisner rings.[4] A proof of the conjecture would also establish the Zariski–Lipman conjecture, for a complex variety V with coordinate ring R. This conjecture states that if the derivations of R are a free module over R, then V is smooth.[5]
References
1. Nakai, Yoshikazu (1961), "On the theory of differentials in commutative rings", Journal of the Mathematical Society of Japan, 13: 63–84, doi:10.2969/jmsj/01310063, MR 0125131.
2. Schreiner, Achim (1994), "On a conjecture of Nakai", Archiv der Mathematik, 62 (6): 506–512, doi:10.1007/BF01193737, MR 1274105. Schreiner cites this converse to EGA 16.11.2.
3. Mount, Kenneth R.; Villamayor, O. E. (1973), "On a conjecture of Y. Nakai", Osaka Journal of Mathematics, 10: 325–327, MR 0327731.
4. Schreiner, Achim (1994), "On a conjecture of Nakai", Archiv der Mathematik, 62 (6): 506–512, doi:10.1007/BF01193737, MR 1274105.
5. Becker, Joseph (1977), "Higher derivations and the Zariski-Lipman conjecture", Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Part 1, Williams Coll., Williamstown, Mass., 1975), Providence, R. I.: American Mathematical Society, pp. 3–10, MR 0444654.
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Zassenhaus group
In mathematics, a Zassenhaus group, named after Hans Zassenhaus, is a certain sort of doubly transitive permutation group very closely related to rank-1 groups of Lie type.
Definition
A Zassenhaus group is a permutation group G on a finite set X with the following three properties:
• G is doubly transitive.
• Non-trivial elements of G fix at most two points.
• G has no regular normal subgroup. ("Regular" means that non-trivial elements do not fix any points of X; compare free action.)
The degree of a Zassenhaus group is the number of elements of X.
Some authors omit the third condition that G has no regular normal subgroup. This condition is put in to eliminate some "degenerate" cases. The extra examples one gets by omitting it are either Frobenius groups or certain groups of degree 2p and order 2p(2p − 1)p for a prime p, that are generated by all semilinear mappings and Galois automorphisms of a field of order 2p.
Examples
We let q = pf be a power of a prime p, and write Fq for the finite field of order q. Suzuki proved that any Zassenhaus group is of one of the following four types:
• The projective special linear group PSL2(Fq) for q > 3 odd, acting on the q + 1 points of the projective line. It has order (q + 1)q(q − 1)/2.
• The projective general linear group PGL2(Fq) for q > 3. It has order (q + 1)q(q − 1).
• A certain group containing PSL2(Fq) with index 2, for q an odd square. It has order (q + 1)q(q − 1).
• The Suzuki group Suz(Fq) for q a power of 2 that is at least 8 and not a square. The order is (q2 + 1)q2(q − 1)
The degree of these groups is q + 1 in the first three cases, q2 + 1 in the last case.
Further reading
• Finite Groups III (Grundlehren Der Mathematischen Wissenschaften Series, Vol 243) by B. Huppert, N. Blackburn, ISBN 0-387-10633-2
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Zdeněk Dvořák
Zdeněk Dvořák (born April 26, 1981) is a Czech mathematician specializing in graph theory.
Dvořák was born in Nové Město na Moravě.[1] He competed on the Czech national team in the 1999 International Mathematical Olympiad,[2] and in the same year in the International Olympiad in Informatics, where he won a gold medal.[3] He earned his Ph.D. in 2007 from Charles University in Prague, under the supervision of Jaroslav Nešetřil. He remained as a research fellow at Charles University until 2010, and then did postdoctoral studies at the Georgia Institute of Technology and Simon Fraser University. He then returned to the Computer Science Institute (IUUK) of Charles University, obtained his habilitation in 2012, and has been a full professor there since 2022.[1]
He was one of three winners of the 2015 European Prize in Combinatorics, "for his fundamental contributions to graph theory, in particular for his work on structural aspects of graph theory, including solutions to Havel's 1969 problem and the Heckman–Thomas 14/5 problem on fractional colourings of cubic triangle-free graphs.[4] This refers to two different results of Dvořák:
• Havel's conjecture is a strengthening of Grötzsch's theorem. It states that there exists a constant d such that, if a planar graph has no two triangles within distance d of each other, then it can be colored with three colors. A proof of this conjecture of Havel was announced by Dvořák and his co-authors in 2009.[5]
• C. C. Heckman and Robin Thomas conjectured in 2001 that triangle-free graphs of maximum degree three have fractional chromatic number at most 14/5.[6] A proof was announced by Dvořák and his co-authors in 2013 and published by them in 2014.[7]
References
1. Curriculum vitae: Zdeněk Dvořák (PDF), retrieved 2023-02-10.
2. Czech Republic, 40th IMO 1999, International Mathematical Olympiad, retrieved 2015-09-16.
3. IOI 1999 Results, International Olympiad in Informatics, retrieved 2015-09-16.
4. "The European Prize in Combinatorics", EuroComb 2015, University of Bergen, September 2015, retrieved 2015-09-16.
5. Dvořák, Zdeněk; Kráľ, Daniel; Thomas, Robin (2009), Three-coloring triangle-free graphs on surfaces V. Coloring planar graphs with distant anomalies, arXiv:0911.0885, Bibcode:2009arXiv0911.0885D.
6. Heckman, Christopher Carl; Thomas, Robin (2001), "A new proof of the independence ratio of triangle-free cubic graphs", Discrete Mathematics, 233 (1–3): 233–237, doi:10.1016/S0012-365X(00)00242-9, MR 1825617.
7. Dvořák, Z.; Sereni, J.-S.; Volec, J. (2014), "Subcubic triangle-free graphs have fractional chromatic number at most 14/5", Journal of the London Mathematical Society, Second Series, 89 (3): 641–662, arXiv:1301.5296, doi:10.1112/jlms/jdt085, MR 3217642, S2CID 3188176.
External links
• Home page
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Zdeněk Frolík
Zdeněk Frolík (March 10, 1933 – May 3, 1989) was a Czech mathematician. His research interests included topology and functional analysis. In particular, his work concerned covering properties of topological spaces, ultrafilters, homogeneity, measures, uniform spaces. He was one of the founders of modern descriptive theory of sets and spaces.[1]
Two classes of topological spaces are given Frolík's name: the class P of all spaces $X$ such that $X\times Y$ is pseudocompact for every pseudocompact space $Y$,[2] and the class C of all spaces $X$ such that $X\times Y$ is countably compact for every countably compact space $Y$.[3]
Frolík prepared his Ph.D. thesis under the supervision of Miroslav Katetov and Eduard Čech.[4]
Selected publications
• Generalizations of compact and Lindelöf spaces - Czechoslovak Math. J., 9 (1959), pp. 172–217 (in Russian, English summary)
• The topological product of countably compact spaces - Czechoslovak Math. J., 10 (1960), pp. 329–338
• The topological product of two pseudocompact spaces - Czechoslovak Math. J., 10 (1960), pp. 339–349
• Generalizations of the Gδ-property of complete metric spaces - Czechoslovak Math. J., 10 (1960), pp. 359–379
• On the topological product of paracompact spaces - Bull. Acad. Polon., 8 (1960), pp. 747–750
• Locally complete topological spaces - Dokl. Akad. Nauk SSSR, 137 (1961), pp. 790–792 (in Russian)
• Applications of complete families of continuous functions to the theory of Q-spaces - Czechoslovak Math. J., 11 (1961), pp. 115–133
• Invariance of Gδ-spaces under mappings - Czechoslovak Math. J., 11 (1961), pp. 258–260
• On almost real compact spaces - Bull. Acad. Polon., 9 (1961), pp. 247–250
• On two problems of W.W. Comfort - Comment. Math. Univ. Carolin., 7 (1966), pp. 139–144
• Non-homogeneity of βP- P - Comment. Math. Univ. Carolin., 7 (1966), pp. 705–710
• Sums of ultrafilters - Bull. Amer. Math. Soc., 73 (1967), pp. 87–91
• Homogeneity problems for extremally disconnected spaces - Comment. Math. Univ. Carolin., 8 (1967), pp. 757–763
• Baire sets that are Borelian subspaces - Proc. Roy. Soc. A, 299 (1967), pp. 287–290
• On the Suslin-graph theorem - Comment Math. Univ. Carolin., 9 (1968), pp. 243–249
• A survey of separable descriptive theory of sets and spaces - Czechoslovak Math. J., 20 (1970), pp. 406–467
• A measurable map with analytic domain and metrizable range is quotient - Bull. Amer. Math. Soc., 76 (1970), pp. 1112–1117
• Luzin sets are additive - Comment Math. Univ. Carolin., 21 (1980), pp. 527–534
• Refinements of perfect maps onto metrizable spaces and an application to Čech-analytic spaces - Topology Appl., 33 (1989), pp. 77–84
• Decomposability of completely Suslin additive families - Proc. Amer. Math. Soc., 82 (1981), pp. 359–365
• Applications of Luzinian separation principles (non-separable case) - Fund. Math., 117 (1983), pp. 165–185
• Analytic and Luzin spaces (non-separable case) - Topology Appl., 19 (1985), pp. 129–156
See also
• Wijsman convergence
References
1. Zdeněk Frolík 1933–1989, Mirek Husek, Jan Pelant, Topology and its Applications, Volume 44, issues 1–3, 22 May 1992, pages 11–17,(access on subscription).
2. Vaughan, Jerry E., On Frolík's characterization of class P. Czechoslovak Mathematical Journal, vol. 44 (1994), issue 1, pp. 1-6, freely available.
3. J.E. Vaughan, Countably compact and sequentially compact spaces. Handbook of Set-theoretic Topology, K. Kunen and J. Vaughan (ed.), North-Holland, Amsterdam, 1984.
4. Zdeněk Frolík on the Mathematics Genealogy Project.
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Zdeněk Hedrlín
Zdeněk Hedrlín (1933 – April 22, 2018) was a Czech mathematician, specializing in universal algebra and combinatorial theory, both in pure and applied mathematics.
Zdeněk Hedrlín received his PhD from Prague's Charles University in 1963. His thesis on commutative semigroups was supervised by Miroslav Katětov.[1] Hedrlín held the title of Docent (associated professor) at Charles University. There he worked at the Faculty of Mathematics and Physics for over 60 years until he died at age 85. He was among the first Czech mathematicians to do research on category theory.[2]
Already in the mid-1960s, the Prague school of Zdeněk Hedrlín, Aleš Pultr and Věra Trnková had devised a particularly nice notion of concrete categories over Set, the so-called functor-structured categories ...[3]
In 1970 Hedrlín was an Invited Speaker at the International Congress of Mathematicians in Nice.[4] In the later part of his career, he focused on applications of relational structures and led very successful special and interdisciplinary seminars. Applications to biological cell behavior earned him and his students a European grant.[2] (He and his students worked on computational cell models of cancer.)
Hedrlín was a member of the editorial board of the Journal of Pure and Applied Algebra.[5] His Erdős number is 1.[6] His doctoral students include Vojtěch Rödl.[1]
Selected publication
• Hedrlín, Z. (1961). "On common fixed points of commutative mappings" (PDF). Commentationes Mathematicae Universitatis Carolinae. 2 (4): 25–28.
• Hedrlín, Zdeněk (1962). "On number of commutative mappings from finite set into itself (Preliminary communication)" (PDF). Commentationes Mathematicae Universitatis Carolinae. 3 (1): 32.
• Hedrlín, Z.; Pultr, A. (1963). "Remark on topological spaces with given semigroups" (PDF). Commentationes Mathematicae Universitatis Carolinae. 4 (4): 161–163.
• Hedrlín, Z.; Pultr, A. (1964). "Relations (graphs) with given finitely generated semigroups". Monatshefte für Mathematik. 68 (3): 213–217. doi:10.1007/BF01298508. S2CID 120856684.
• Pultr, A.; Hedrlín, Z. (1964). "Relations (graphs) with given infinite semigroups". Monatshefte für Mathematik. 68 (5): 421–425. doi:10.1007/BF01304185. S2CID 122610862.
• Baayen, P. C.; Hedrlin, Z. (1964). "On the existence of well distributed sequences in compact spaces" (PDF). Stichting Mathematisch Centrum. Zuivere Wiskunde.
• Hedrlín, Z.; Pultr, A. (1965). "Symmetric relations (undirected graphs) with given semigroups". Monatshefte für Mathematik. 69 (4): 318–322. doi:10.1007/BF01297617. S2CID 120384797.
• Vopěnka, P.; Pultr, A.; Hedrlín, Z. (1965). "A rigid relation exists on any set" (PDF). Commentationes Mathematicae Universitatis Carolinae. 6 (2): 149–155.
• Hedrlín, Zdeněk; Pultr, Aleš (1966). "On full embeddings of categories of algebras". Illinois Journal of Mathematics. 10 (3): 392–406. doi:10.1215/ijm/1256054991. (over 160 citations)
• Hedrlín, Z.; Pultr, A. (1966). "On Rigid Undirected Graphs". Canadian Journal of Mathematics. 18: 1237–1242. doi:10.4153/CJM-1966-121-7. S2CID 124453196.
• Hedrlín, Z.; Vopěnka, P. (1966). "An undecidable theorem concerning full embeddings into categories of algebras" (PDF). Commentationes Mathematicae Universitatis Carolinae. 7 (3): 401–409.
• Hedrlín, Z.; Pultr, A.; Trnková, V. (1967). "Concerning a categorial approach to topological and algebraic theories" (PDF). In: (ed.): General Topology and its Relations to Modern Analysis and Algebra, Proceedings of the second Prague topological symposium, 1966. Academia Publishing House of the Czechoslovak Academy of Sciences, Praha. pp. 176–181.
• Hedrlín, Zdeněk; Lambek, Joachim (1969). "How comprehensive is the category of semigroups?". Journal of Algebra. 11 (2): 195–212. doi:10.1016/0021-8693(69)90054-4.
• Hedrlín, Zdeněk (1969). "On universal partly ordered sets and classes" (PDF). Journal of Algebra. 11 (4): 503–509. doi:10.1016/0021-8693(69)90089-1.
• Hedrlín, Z.; Mendelsohn, E. (1969). "The Category of Graphs with a Given Subgraph-with Applications to Topology and Algebra". Canadian Journal of Mathematics. 21: 1506–1517. doi:10.4153/CJM-1969-165-5. S2CID 124324655.
• Goralčík, Pavel; Hedrlín, Zdeněk (1971). "On reconstruction of monoids from their table fragments". Mathematische Zeitschrift. 122: 82–92. doi:10.1007/BF01113568. S2CID 120230682.
• Chvatal, V.; Erdös, P.; Hedrlín, Z. (1972). "Ramsey's theorem and self-complementary graphs". Discrete Mathematics. 3 (4): 301–304. doi:10.1016/0012-365X(72)90087-8.
• Goralčík, P.; Hedrlín, Z.; Koubek, V.; Ryšunková, J. (1982). "A game of composing binary relations" (PDF). R.A.I.R.O.: Informatique Théorique. 16 (4): 365–369. doi:10.1051/ita/1982160403651.
• Hedrlín, Z.; Hell, P.; Ko, C.S. (1982). "Homomorphism Interpolation and Approximation". Algebraic and Geometric Combinatorics. North-Holland Mathematics Studies. Vol. 65. pp. 213–227. doi:10.1016/S0304-0208(08)73267-5. ISBN 9780444863652.
References
1. Zdeněk Hedrlín at the Mathematics Genealogy Project
2. Kratochvíl, Jan (April 26, 2018). "Zemřel doc. Zdeněk Hedrlín (deceased docent Zdeněk Hedrlín)". Matematicko-fyzikální fakulty Univerzity Karlovy (Mathematics and Physics Faculty of Charles University).
3. Koslowski, Jürgen; Melton, Austin, eds. (6 December 2012). "Chapter. Contributions and importance of Professor George E. Strecker's Research by Jürgen Koslowski". Categorical Perspectives. Springer Science & Business Media. pp. 63–90. ISBN 978-1-4612-1370-3. (quote from p. 73)
4. Hedrlín, Z. (1970). "Extensions of structures and full embeddings of categories". In: Actes du Congrès international des mathématiciens, 1–10 Septembre 1970, Nice. Vol. 1. pp. 319–321.
5. "Managing Editors; Editors" (PDF). Journal of Pure and Applied Algebra.
6. Chvatal, V.; Erdös, P.; Hedrlín, Z. (1972). "Ramsey's theorem and self-complementary graphs". Discrete Mathematics. 3 (4): 301–304. doi:10.1016/0012-365X(72)90087-8.
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Zech's logarithm
Zech logarithms are used to implement addition in finite fields when elements are represented as powers of a generator $\alpha $.
Zech logarithms are named after Julius Zech,[1][2][3][4] and are also called Jacobi logarithms,[5] after Carl G. J. Jacobi who used them for number theoretic investigations.[6]
Definition
Given a primitive element $\alpha $ of a finite field, the Zech logarithm relative to the base $\alpha $ is defined by the equation
$\alpha ^{Z_{\alpha }(n)}=1+\alpha ^{n},$
which is often rewritten as
$Z_{\alpha }(n)=\log _{\alpha }(1+\alpha ^{n}).$
The choice of base $\alpha $ is usually dropped from the notation when it is clear from the context.
To be more precise, $Z_{\alpha }$ is a function on the integers modulo the multiplicative order of $\alpha $, and takes values in the same set. In order to describe every element, it is convenient to formally add a new symbol $-\infty $, along with the definitions
$\alpha ^{-\infty }=0$
$n+(-\infty )=-\infty $
$Z_{\alpha }(-\infty )=0$
$Z_{\alpha }(e)=-\infty $
where $e$ is an integer satisfying $\alpha ^{e}=-1$, that is $e=0$ for a field of characteristic 2, and $e={\frac {q-1}{2}}$ for a field of odd characteristic with $q$ elements.
Using the Zech logarithm, finite field arithmetic can be done in the exponential representation:
$\alpha ^{m}+\alpha ^{n}=\alpha ^{m}\cdot (1+\alpha ^{n-m})=\alpha ^{m}\cdot \alpha ^{Z(n-m)}=\alpha ^{m+Z(n-m)}$
$-\alpha ^{n}=(-1)\cdot \alpha ^{n}=\alpha ^{e}\cdot \alpha ^{n}=\alpha ^{e+n}$
$\alpha ^{m}-\alpha ^{n}=\alpha ^{m}+(-\alpha ^{n})=\alpha ^{m+Z(e+n-m)}$
$\alpha ^{m}\cdot \alpha ^{n}=\alpha ^{m+n}$
$\left(\alpha ^{m}\right)^{-1}=\alpha ^{-m}$
$\alpha ^{m}/\alpha ^{n}=\alpha ^{m}\cdot \left(\alpha ^{n}\right)^{-1}=\alpha ^{m-n}$
These formulas remain true with our conventions with the symbol $-\infty $, with the caveat that subtraction of $-\infty $ is undefined. In particular, the addition and subtraction formulas need to treat $m=-\infty $ as a special case.
This can be extended to arithmetic of the projective line by introducing another symbol $+\infty $ satisfying $\alpha ^{+\infty }=\infty $ and other rules as appropriate.
For fields of characteristic two,
$Z_{\alpha }(n)=m\iff Z_{\alpha }(m)=n$.
Uses
For sufficiently small finite fields, a table of Zech logarithms allows an especially efficient implementation of all finite field arithmetic in terms of a small number of integer addition/subtractions and table look-ups.
The utility of this method diminishes for large fields where one cannot efficiently store the table. This method is also inefficient when doing very few operations in the finite field, because one spends more time computing the table than one does in actual calculation.
Examples
Let α ∈ GF(23) be a root of the primitive polynomial x3 + x2 + 1. The traditional representation of elements of this field is as polynomials in α of degree 2 or less.
A table of Zech logarithms for this field are Z(−∞) = 0, Z(0) = −∞, Z(1) = 5, Z(2) = 3, Z(3) = 2, Z(4) = 6, Z(5) = 1, and Z(6) = 4. The multiplicative order of α is 7, so the exponential representation works with integers modulo 7.
Since α is a root of x3 + x2 + 1 then that means α3 + α2 + 1 = 0, or if we recall that since all coefficients are in GF(2), subtraction is the same as addition, we obtain α3 = α2 + 1.
The conversion from exponential to polynomial representations is given by
$\alpha ^{3}=\alpha ^{2}+1$ (as shown above)
$\alpha ^{4}=\alpha ^{3}\alpha =(\alpha ^{2}+1)\alpha =\alpha ^{3}+\alpha =\alpha ^{2}+\alpha +1$
$\alpha ^{5}=\alpha ^{4}\alpha =(\alpha ^{2}+\alpha +1)\alpha =\alpha ^{3}+\alpha ^{2}+\alpha =\alpha ^{2}+1+\alpha ^{2}+\alpha =\alpha +1$
$\alpha ^{6}=\alpha ^{5}\alpha =(\alpha +1)\alpha =\alpha ^{2}+\alpha $
Using Zech logarithms to compute α6 + α3:
$\alpha ^{6}+\alpha ^{3}=\alpha ^{6+Z(-3)}=\alpha ^{6+Z(4)}=\alpha ^{6+6}=\alpha ^{12}=\alpha ^{5}$,
or, more efficiently,
$\alpha ^{6}+\alpha ^{3}=\alpha ^{3+Z(3)}=\alpha ^{3+2}=\alpha ^{5}$,
and verifying it in the polynomial representation:
$\alpha ^{6}+\alpha ^{3}=(\alpha ^{2}+\alpha )+(\alpha ^{2}+1)=\alpha +1=\alpha ^{5}$.
See also
• Gaussian logarithm
• Irish logarithm, a similar technique derived empirically by Percy Ludgate
• Finite field arithmetic
• Logarithm table
References
1. Zech, Julius August Christoph (1849). Tafeln der Additions- und Subtractions-Logarithmen für sieben Stellen (in German) (Specially reprinted (from Vega–Hülße collection) 1st ed.). Leipzig: Weidmann'sche Buchhandlung. Archived from the original on 2018-07-14. Retrieved 2018-07-14. Also part of: Freiherr von Vega, Georg (1849). Hülße, Julius Ambrosius [in German]; Zech, Julius August Christoph (eds.). Sammlung mathematischer Tafeln (in German) (Completely reworked ed.). Leipzig: Weidmann'sche Buchhandlung. Bibcode:1849smt..book.....V. Archived from the original on 2018-07-14. Retrieved 2018-07-14.
2. Zech, Julius August Christoph (1863) [1849]. Tafeln der Additions- und Subtractions-Logarithmen für sieben Stellen (in German) (Specially reprinted (from Vega–Hülße collection) 2nd ed.). Berlin: Weidmann'sche Buchhandlung. Archived from the original on 2018-07-14. Retrieved 2018-07-13.
3. Zech, Julius August Christoph (1892) [1849]. Tafeln der Additions- und Subtractions-Logarithmen für sieben Stellen (in German) (Specially reprinted (from Vega–Hülße collection) 3rd ed.). Berlin: Weidmann'sche Buchhandlung. Archived from the original on 2018-07-14. Retrieved 2018-07-13.
4. Zech, Julius August Christoph (1910) [1849]. Tafeln der Additions- und Subtractions-Logarithmen für sieben Stellen (in German) (Specially reprinted (from Vega–Hülße collection) 4th ed.). Berlin: Weidmann'sche Buchhandlung. Archived from the original on 2018-07-14. Retrieved 2018-07-13.
5. Lidl, Rudolf; Niederreiter, Harald (1997). Finite Fields (2nd ed.). Cambridge University Press. ISBN 978-0-521-39231-0.
6. Jacoby, Carl Gustav Jacob (1846). "Über die Kreistheilung und ihre Anwendung auf die Zahlentheorie". Journal für die reine und angewandte Mathematik (in German). 1846 (30): 166–182. doi:10.1515/crll.1846.30.166. ISSN 0075-4102. S2CID 120615565. (NB. Also part of "Gesammelte Werke", Volume 6, pages 254–274.)
Further reading
• Fletcher, Alan; Miller, Jeffrey Charles Percy; Rosenhead, Louis (1946) [1943]. An Index of Mathematical Tables (1 ed.). Blackwell Scientific Publications Ltd., Oxford / McGraw-Hill, New York.
• Conway, John Horton (1968). Churchhouse, Robert F.; Herz, J.-C. (eds.). "A tabulation of some information concerning finite fields". Computers in Mathematical Research. Amsterdam: North-Holland Publishing Company: 37–50. MR 0237467.
• Lam, Clement Wing Hong; McKay, John K. S. (1973-11-01). "Algorithm 469: Arithmetic over a finite field [A1]". Communications of the ACM. Collected Algorithms of the ACM (CALGO). Association for Computing Machinery (ACM). 16 (11): 699. doi:10.1145/355611.362544. ISSN 0001-0782. S2CID 62794130. toms/469.
• Kühn, Klaus (2008). "C. F. Gauß und die Logarithmen" (PDF) (in German). Alling-Biburg, Germany. Archived (PDF) from the original on 2018-07-14. Retrieved 2018-07-14.
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Zeckendorf's theorem
In mathematics, Zeckendorf's theorem, named after Belgian amateur mathematician Edouard Zeckendorf, is a theorem about the representation of integers as sums of Fibonacci numbers.
Zeckendorf's theorem states that every positive integer can be represented uniquely as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. More precisely, if N is any positive integer, there exist positive integers ci ≥ 2, with ci + 1 > ci + 1, such that
$N=\sum _{i=0}^{k}F_{c_{i}},$
where Fn is the nth Fibonacci number. Such a sum is called the Zeckendorf representation of N. The Fibonacci coding of N can be derived from its Zeckendorf representation.
For example, the Zeckendorf representation of 64 is
64 = 55 + 8 + 1.
There are other ways of representing 64 as the sum of Fibonacci numbers
64 = 55 + 5 + 3 + 1
64 = 34 + 21 + 8 + 1
64 = 34 + 21 + 5 + 3 + 1
64 = 34 + 13 + 8 + 5 + 3 + 1
but these are not Zeckendorf representations because 34 and 21 are consecutive Fibonacci numbers, as are 5 and 3.
For any given positive integer, its Zeckendorf representation can be found by using a greedy algorithm, choosing the largest possible Fibonacci number at each stage.
History
While the theorem is named after the eponymous author who published his paper in 1972, the same result had been published 20 years earlier by Gerrit Lekkerkerker.[1] As such, the theorem is an example of Stigler's Law of Eponymy.
Proof
Zeckendorf's theorem has two parts:
1. Existence: every positive integer n has a Zeckendorf representation.
2. Uniqueness: no positive integer n has two different Zeckendorf representations.
The first part of Zeckendorf's theorem (existence) can be proven by induction. For n = 1, 2, 3 it is clearly true (as these are Fibonacci numbers), for n = 4 we have 4 = 3 + 1. If n is a Fibonacci number then there is nothing to prove. Otherwise there exists j such that Fj < n < Fj + 1 . Now suppose each positive integer a < n has a Zeckendorf representation (induction hypothesis) and consider a = n − Fj . Since a < n, a has a Zeckendorf representation by the induction hypothesis. At the same time, a = n − Fj < Fj + 1 − Fj = Fj − 1 (we apply the definition of Fibonacci number in the last equality), so the Zeckendorf representation of a does not contain Fj − 1 , and hence also does not contain Fj . As a result, n can be represented as the sum of Fj and the Zeckendorf representation of a, such that the Fibonacci numbers involved in the sum are distinct.
The second part of Zeckendorf's theorem (uniqueness) requires the following lemma:
Lemma: The sum of any non-empty set of distinct, non-consecutive Fibonacci numbers whose largest member is Fj is strictly less than the next larger Fibonacci number Fj + 1 .
The lemma can be proven by induction on j.
Now take two non-empty sets $S$ and $T$ of distinct non-consecutive Fibonacci numbers which have the same sum, $ \sum _{x\in S}x=\sum _{x\in T}x$. Consider sets $S'$ and $T'$ which are equal to $S$ and $T$ from which the common elements have been removed (i. e. $S'=S\setminus T$ and $T'=T\setminus S$). Since $S$ and $T$ had equal sum, and we have removed exactly the elements from $S\cap T$ from both sets, $S'$ and $T'$ must have the same sum as well, $ \sum _{x\in S'}x=\sum _{x\in T'}x$.
Now we will show by contradiction that at least one of $S'$ and $T'$ is empty. Assume the contrary, i. e. that $S'$ and $T'$ are both non-empty and let the largest member of $S'$ be Fs and the largest member of $T'$ be Ft. Because $S'$ and $T'$ contain no common elements, Fs ≠ Ft. Without loss of generality, suppose Fs < Ft. Then by the lemma, $ \sum _{x\in S'}x<F_{s+1}$, and, by the fact that $ F_{s}<F_{s+1}\leq F_{t}$, $ \sum _{x\in S'}x<F_{t}$, whereas clearly $ \sum _{x\in T'}x\geq F_{t}$. This contradicts the fact that $S'$ and $T'$ have the same sum, and we can conclude that either $S'$ or $T'$ must be empty.
Now assume (again without loss of generality) that $S'$ is empty. Then $S'$ has sum 0, and so must $T'$. But since $T'$ can only contain positive integers, it must be empty too. To conclude: $S'=T'=\emptyset $ which implies $S=T$, proving that each Zeckendorf representation is unique.
Fibonacci multiplication
One can define the following operation $a\circ b$ on natural numbers a, b: given the Zeckendorf representations $a=\sum _{i=0}^{k}F_{c_{i}}\;(c_{i}\geq 2)$ and $b=\sum _{j=0}^{l}F_{d_{j}}\;(d_{j}\geq 2)$ we define the Fibonacci product $a\circ b=\sum _{i=0}^{k}\sum _{j=0}^{l}F_{c_{i}+d_{j}}.$
For example, the Zeckendorf representation of 2 is $F_{3}$, and the Zeckendorf representation of 4 is $F_{4}+F_{2}$ ($F_{1}$ is disallowed from representations), so $2\circ 4=F_{3+4}+F_{3+2}=13+5=18.$
(The product is not always in Zeckendorf form. For example, $4\circ 4=(F_{4}+F_{2})\circ (F_{4}+F_{2})=F_{4+4}+2F_{4+2}+F_{2+2}=21+2\cdot 8+3=40=F_{9}+F_{5}+F_{2}.$)
A simple rearrangement of sums shows that this is a commutative operation; however, Donald Knuth proved the surprising fact that this operation is also associative.[2]
Representation with negafibonacci numbers
The Fibonacci sequence can be extended to negative index n using the rearranged recurrence relation
$F_{n-2}=F_{n}-F_{n-1},$
which yields the sequence of "negafibonacci" numbers satisfying
$F_{-n}=(-1)^{n+1}F_{n}.$
Any integer can be uniquely represented[3] as a sum of negafibonacci numbers in which no two consecutive negafibonacci numbers are used. For example:
• −11 = F−4 + F−6 = (−3) + (−8)
• 12 = F−2 + F−7 = (−1) + 13
• 24 = F−1 + F−4 + F−6 + F−9 = 1 + (−3) + (−8) + 34
• −43 = F−2 + F−7 + F−10 = (−1) + 13 + (−55)
• 0 is represented by the empty sum.
0 = F−1 + F−2 , for example, so the uniqueness of the representation does depend on the condition that no two consecutive negafibonacci numbers are used.
This gives a system of coding integers, similar to the representation of Zeckendorf's theorem. In the string representing the integer x, the nth digit is 1 if F−n appears in the sum that represents x; that digit is 0 otherwise. For example, 24 may be represented by the string 100101001, which has the digit 1 in places 9, 6, 4, and 1, because 24 = F−1 + F−4 + F−6 + F−9 . The integer x is represented by a string of odd length if and only if x > 0.
See also
• Complete sequence
• Fibonacci coding
• Fibonacci nim
• Ostrowski numeration
References
1. Historical note on the name Zeckendorf Representation by R Knott, University of Surrey
2. Knuth, Donald E. (1988). "Fibonacci multiplication" (PDF). Applied Mathematics Letters. 1 (1): 57–60. doi:10.1016/0893-9659(88)90176-0. ISSN 0893-9659. Zbl 0633.10011.
3. Knuth, Donald (2008-12-11). Negafibonacci Numbers and the Hyperbolic Plane. Annual meeting, Mathematical Association of America. The Fairmont Hotel, San Jose, CA.
• Zeckendorf, E. (1972). "Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas". Bull. Soc. R. Sci. Liège (in French). 41: 179–182. ISSN 0037-9565. Zbl 0252.10011.
External links
• Weisstein, Eric W. "Zeckendorf's Theorem". MathWorld.
• Weisstein, Eric W. "Zeckendorf Representation". MathWorld.
• Zeckendorf's theorem at cut-the-knot
• G.M. Phillips (2001) [1994], "Zeckendorf representation", Encyclopedia of Mathematics, EMS Press
• OEIS sequence A101330 (Knuth's Fibonacci (or circle) product)
This article incorporates material from proof that the Zeckendorf representation of a positive integer is unique on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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Christopher Zeeman
Sir Erik Christopher Zeeman FRS[1] (4 February 1925 – 13 February 2016), was a British mathematician,[2] known for his work in geometric topology and singularity theory.
Sir
Christopher Zeeman
Zeeman in 1980
Born
Erik Christopher Zeeman
(1925-02-04)4 February 1925
Japan
Died13 February 2016(2016-02-13) (aged 91)
Woodstock, England
CitizenshipBritish
Alma materChrist's College, Cambridge
Known forCatastrophe theory
Geometric topology
Singularity theory
Zeeman conjecture
Zeeman's comparison theorem
Stallings–Zeeman theorem
AwardsSenior Whitehead Prize (1982)
Faraday Medal (1988)
David Crighton Medal (2006)
Scientific career
FieldsMathematics
InstitutionsUniversity of Cambridge
University of Warwick
University of Oxford
Gresham College
ThesisDihomology (1955)
Doctoral advisorShaun Wylie
Doctoral studentsPeter Buneman
David Epstein
Ray Lickorish
Tim Poston
Colin Rourke
David Trotman
Terry Wall
Jenny Harrison
Notes
Fellow of the Royal Society
Overview
Zeeman's main contributions to mathematics were in topology, particularly in knot theory, the piecewise linear category, and dynamical systems.
His 1955 thesis at the University of Cambridge described a new theory termed "dihomology", an algebraic structure associated to a topological space, containing both homology and cohomology, introducing what is now known as the Zeeman spectral sequence. This was studied by Clint McCrory in his 1972 Brandeis thesis following a suggestion of Dennis Sullivan that one make "a general study of the Zeeman spectral sequence to see how singularities in a space perturb Poincaré duality". This in turn led to the discovery of intersection homology by Robert MacPherson and Mark Goresky at Brown University where McCrory was appointed in 1974.
From 1976 to 1977 he was the Donegall Lecturer in Mathematics at Trinity College Dublin.
Zeeman is known among the wider scientific public for his contribution to, and spreading awareness of catastrophe theory, which was due initially to another topologist, René Thom, and for his Christmas lectures about mathematics on television in 1978. He was especially active encouraging the application of mathematics, and catastrophe theory in particular, to biology and behavioral sciences.
Early life
Zeeman was born in Japan to a Danish father, Christian Zeeman, and a British mother. They moved to England one year after his birth. After being educated at Christ's Hospital in Horsham, West Sussex, he served as a Flying Officer with the Royal Air Force from 1943 to 1947.[2] He studied mathematics at Christ's College, Cambridge, but had forgotten much of his school mathematics while serving for the air force. He received an MA and PhD (the latter under the supervision of Shaun Wylie) from the University of Cambridge, and became a Fellow of Gonville and Caius College where he tutored David Fowler and John Horton Conway.[3]
Academic career
Zeeman is one of the founders of engulfing theory in piecewise linear topology and is credited with working out the engulfing theorem (independently also worked out by John Stallings), which can be used to prove the piecewise linear version of the Poincaré conjecture for all dimensions above four.[4][5]
After working at Cambridge (during which he spent a year abroad at University of Chicago and Princeton as a Harkness Fellow) and the Institut des Hautes Études Scientifiques, he founded the Mathematics Department and Mathematics Research Centre at the new University of Warwick in 1964. In his own words
I was 38 and had developed some fairly strong ideas on how to run a department and create a Mathematics Institute: I wanted to combine the flexibility of options that are common in most American universities, with the kind of tutorial care to be found in Oxford and Cambridge.[6]
Zeeman's style of leadership was informal, but inspirational, and he rapidly took Warwick to international recognition for the quality of its mathematical research. The first six appointments he made were all in topology, enabling the department to immediately become internationally competitive, followed by six in algebra, and finally six in analysis and six in applied mathematics. He was able to trade four academic appointments for funding that enabled PhD students to give undergraduate supervisions in groups of two for the first two years, in a manner similar to the tutorial system at Oxford and Cambridge. He remained at Warwick until 1988, but from 1966 to 1967 he was a visiting professor at the University of California at Berkeley, after which his research turned to dynamical systems, inspired by many of the world leaders in this field, including Stephen Smale and René Thom, who both spent time at Warwick. In 1963, Zeeman showed that that causality in special relativity expressed by preservation of partial ordering is given exactly and only by the Lorentz transforms.[7] Zeeman subsequently spent a sabbatical with Thom at the Institut des Hautes Études Scientifiques in Paris, where he became interested in catastrophe theory. On his return to Warwick, he taught an undergraduate course in Catastrophe Theory that became immensely popular with students; his lectures generally were "standing room only". In 1973 he gave an MSc course at Warwick giving a complete detailed proof of Thom's classification of elementary catastrophes, mainly following an unpublished manuscript, "Right-equivalence" written by John Mather at Warwick in 1969. David Trotman wrote up his notes of the course as an MSc thesis. These were then distributed in thousands of copies throughout the world and published both in the proceedings of a 1975 Seattle conference on catastrophe theory and its applications,[8] and in a 1977 collection of papers on catastrophe theory by Zeeman.[9] In 1974 Zeeman gave an invited address at the International Congress of Mathematicians in Vancouver, about applications of catastrophe theory.
Zeeman was elected as a Fellow of the Royal Society in 1975, and was awarded the Society's Faraday Medal in 1988. He was the 63rd President of the London Mathematical Society in 1986–88 giving his Presidential Address on 18 November 1988 On the classification of dynamical systems. He was awarded the Senior Whitehead Prize of the Society in 1982. He was the Society's first Forder lecturer, involving a lecture tour in New Zealand, in 1987. Between 1988 and 1994 he was the Professor of Geometry at Gresham College.[10]
In 1978, Zeeman gave the televised series of Christmas Lectures at the Royal Institution.[11] From these grew the Mathematics and Engineering Masterclasses for both primary and secondary school children that now flourish in forty centers in the United Kingdom.[12]
In 1988, Zeeman became Principal of Hertford College, Oxford. The following year he was appointed an honorary fellow of Christ's College, Cambridge. He received a knighthood in the 1991 Birthday Honours for "mathematical excellence and service to British mathematics and mathematics education".[13][14] He was invited to become President of The Mathematical Association in 2003 and based his book Three-dimensional Theorems for Schools on his 2004 Presidential Address. On Friday 6 May 2005, the University of Warwick's new Mathematics and Statistics building was named the Zeeman building in his honour. He became an Honorary Member of The Mathematical Association in 2006. In September 2006, the London Mathematical Society and the Institute of Mathematics and its Applications awarded him the David Crighton medal in recognition of his long and distinguished service to mathematics and the mathematical community.[15] The medal is awarded triennially, and Zeeman was the second ever recipient of the award.[16] He died on 13 February 2016.[17]
The Zeeman Medal
The Christopher Zeeman Medal for Communication of Mathematics[18] of the London Mathematical Society and the Institute of Mathematics and its Applications is named in Zeeman's honour. The award aims "to honour mathematicians who have excelled in promoting mathematics and engaging with the general public. They may be academic mathematicians based in universities, mathematics school teachers, industrial mathematicians, those working in the financial sector or indeed mathematicians from any number of other fields".
See also
• Mary Lou Zeeman, Zeeman's daughter, also a mathematician
• Nicolette Zeeman, Zeeman's daughter, a literary scholar
• Samuel C. Zeeman, Zeeman's son, a plant biologist
References
1. Rand, David A. (2022). "Sir Erik Christopher Zeeman. 4 February 1925—13 February 2016". Biographical Memoirs of Fellows of the Royal Society. 73: 521–547. doi:10.1098/rsbm.2022.0012. S2CID 251447255.
2. Archer, Megan (25 February 2016). "Obituary: 'Remarkable' maths professor Sir Christopher Zeeman remembered [Sir Christopher Zeeman: Mathematics professor and former college principal]". Oxford Times. p. 86. Retrieved 8 March 2016.
3. The Guardian, Obituary: David Fowler, 3 May 2004
4. 'The generalised Poincaré conjecture', Bull. Amer. Math. Soc. 67:270 (1961)
5. 'The Poincaré conjecture for n greater than or equal to 5', Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), 198–204, Prentice–Hall, 1962
6. E.C.Zeeman, Mathematical Association President's report, 2004, M-A.org.uk
7. Zeeman, E. C. (1 April 1964). "Causality Implies the Lorentz Group". Journal of Mathematical Physics. 5 (4): 490–493. Bibcode:1964JMP.....5..490Z. doi:10.1063/1.1704140. ISSN 0022-2488.
8. D.J.A.Trotman and E.C.Zeeman, The classification of elementary catastrophes of codimension less than or equal to 5, in Structural stability, the theory of catastrophes, and applications in the sciences. Proceedings of the Conference held at Battelle Seattle Research Center, Seattle, Wash., 21–25 April 1975. Edited by P. Hilton. Lecture Notes in Mathematics, Vol. 525. Springer-Verlag, Berlin-New York, 1976
9. E. C. Zeeman, Catastrophe theory. Selected papers, 1972–1977. Addison–Wesley Publishing Co., Reading, Mass.–London–Amsterdam, 1977
10. "Sir Christopher Zeeman". London, UK: Gresham College. Retrieved 8 March 2015.
11. "Mathematics into Pictures, 1978 Royal Institution Christmas Lectures". Retrieved 7 February 2022.
12. "Royal Institution Maths and Engineering Masterclasses". Retrieved 22 August 2012.
13. "No. 52563". The London Gazette (Supplement). 15 June 1991. p. 2.
14. "No. 52858". The London Gazette. 10 March 1992. p. 4257.
15. London Mathematical Society. "Honours and Awards Newsletter". Archived from the original on 12 October 2007. Retrieved 8 July 2007.
16. London Mathematical Society. "List of IMA-LMS Prizewinners". Retrieved 10 December 2014.
17. "Sir Christopher Zeeman FRS (1925–2016)". Mathematics Institute, University of Warwick. 16 February 2016. Retrieved 16 February 2016.
18. "Christopher Zeeman Medal".
External links
• O'Connor, John J.; Robertson, Edmund F., "Christopher Zeeman", MacTutor History of Mathematics Archive, University of St Andrews
• Interview in CIM Bulletin 2001
• Three references for further reading
• Bibliography
• Zeeman's Catastrophe Machine
• Zeeman's Catastrophe Machine in Flash
• AMS — The Catastrophe Machine
• Doctor Zeeman's Original Catastrophe Machine
• Video illustrating Zeeman's Catastrophe Machine
• "The Cusp of Catastrophe: René Thom, Christopher Zeeman and Denis Postle" in Maps of the Mind Charles Hampden-Turner. Collier Books, 1981. ISBN 978-0-85533-293-8
• Christopher Zeeman at the Mathematics Genealogy Project
• Mathematics into pictures, Christopher Zeeman's 1978 Royal Institution Christmas Lectures
• Zeeman building, University of Warwick
Principals of Hertford College, Oxford
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Gossard perspector
In geometry the Gossard perspector[1] (also called the Zeeman–Gossard perspector[2]) is a special point associated with a plane triangle. It is a triangle center and it is designated as X(402) in Clark Kimberling's Encyclopedia of Triangle Centers. The point was named Gossard perspector by John Conway in 1998 in honour of Harry Clinton Gossard who discovered its existence in 1916. Later it was learned that the point had appeared in an article by Christopher Zeeman published during 1899 – 1902. From 2003 onwards the Encyclopedia of Triangle Centers has been referring to this point as Zeeman–Gossard perspector.[2]
Definition
Gossard triangle
Let ABC be any triangle. Let the Euler line of triangle ABC meet the sidelines BC, CA and AB of triangle ABC at D, E and F respectively. Let AgBgCg be the triangle formed by the Euler lines of the triangles AEF, BFD and CDE, the vertex Ag being the intersection of the Euler lines of the triangles BFD and CDE, and similarly for the other two vertices. The triangle AgBgCg is called the Gossard triangle of triangle ABC.[3]
Gossard perspector
Let ABC be any triangle and let AgBgCg be its Gossard triangle. Then the lines AAg, BBg and CCg are concurrent. The point of concurrence is called the Gossard perspector of triangle ABC.
Properties
• Let AgBgCg be the Gossard triangle of triangle ABC. The lines BgCg, CgAg and AgBg are respectively parallel to the lines BC, CA and AB.[4]
• Any triangle and its Gossard triangle are congruent.
• Any triangle and its Gossard triangle have the same Euler line.
• The Gossard triangle of triangle ABC is the reflection of triangle ABC in the Gossard perspector.
Trilinear coordinates
The trilinear coordinates of the Gossard perspector of triangle ABC are
( f ( a, b, c ) : f ( b, c, a ) : f ( c, a, b ) )
where
f ( a, b, c ) = p ( a, b, c ) y ( a, b, c ) / a
where
p ( a, b, c ) = 2a4 − a2b2 − a2c2 − ( b2 − c2 )2
and
y ( a, b, c ) = a8 − a6 ( b2 + c2 ) + a4 ( 2b2 − c2 ) ( 2c2 − b2 ) + ( b2 − c2 )2 [ 3a2 ( b2 + c2 ) − b4 − c4 − 3b2c2 ]
Generalisations
The construction yielding the Gossard triangle of a triangle ABC can be generalised to produce triangles A'B'C' which are congruent to triangle ABC and whose sidelines are parallel to the sidelines of triangle ABC.
Generalisation 1
This result is due to Christopher Zeeman.[4]
Let l be any line parallel to the Euler line of triangle ABC. Let l intersect the sidelines BC, CA, AB of triangle ABC at X, Y, Z respectively. Let A'B'C' be the triangle formed by the Euler lines of the triangles AYZ, BZX and CXY. Then triangle A'B'C' is congruent to triangle ABC and its sidelines are parallel to the sidelines of triangle ABC.[4]
Generalisation 2
This generalisation is due to Paul Yiu.[1][5]
Let P be any point in the plane of the triangle ABC different from its centroid G.
Let the line PG meet the sidelines BC, CA and AB at X, Y and Z respectively.
Let the centroids of the triangles AYZ, BZX and CXY be Ga, Gb and Gc respectively.
Let Pa be a point such that YPa is parallel to CP and ZPa is parallel to BP.
Let Pb be a point such that ZPb is parallel to AP and XPb is parallel to CP.
Let Pc be a point such that XPc is parallel to BP and YPc is parallel to AP.
Let A'B'C' be the triangle formed by the lines GaPa, GbPb and GcPc.
Then the triangle A'B'C' is congruent to triangle ABC and its sides are parallel to the sides of triangle ABC.
When P coincides with the orthocenter H of triangle ABC then the line PG coincides with the Euler line of triangle ABC. The triangle A'B'C' coincides with the Gossard triangle AgBgCg of triangle ABC.
Generalisation 3
Let ABC be a triangle. Let H and O be two points, and let the line HO meets BC, CA, AB at A0, B0, C0 respectively. Let AH and AO be two points such that C0AH parallel to BH, B0AH parallel to CH and C0AO parallel to BO, B0AO parallel to CO. Define BH, BO, CH, CO cyclically. Then the triangle formed by the lines AHAO, BHBO, CHCO and triangle ABC are homothetic and congruent, and the homothetic center lies on the line OH. [6] If OH is any line through the centroid of triangle ABC, this problem is the Yiu's generalization of the Gossard perspector theorem.[6]
References
1. Kimberling, Clark. "Gossard Perspector". Archived from the original on 10 May 2012. Retrieved 17 June 2012.
2. Kimberling, Clark. "X(402) = Zeemann--Gossard perspector". Encyclopedia of Triangle Centers. Archived from the original on 19 April 2012. Retrieved 17 June 2012.
3. Kimberling, Clark. "Harry Clinton Gossard". Archived from the original on 22 May 2013. Retrieved 17 June 2012.
4. Hatzipolakis, Antreas P. "Hyacinthos Message #7564". Retrieved 17 June 2012.
5. Grinberg, Darij. "Hyacithos Message #9666". Retrieved 18 June 2012.
6. Dao Thanh Oai, A generalization of the Zeeman-Gossard perspector theorem, International Journal of Computer Discovered Mathematics, Vol.1, (2016), Issue 3, page 76-79, ISSN 2367-7775
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Zeeman's comparison theorem
In homological algebra, Zeeman's comparison theorem, introduced by Christopher Zeeman (Zeeman (1957)), gives conditions for a morphism of spectral sequences to be an isomorphism.
Statement
Comparison theorem — Let $E_{p,q}^{r},{}^{\prime }E_{p,q}^{r}$ be first quadrant spectral sequences of flat modules over a commutative ring and $f:E^{r}\to {}^{\prime }E^{r}$ a morphism between them. Then any two of the following statements implies the third:
1. $f:E_{2}^{p,0}\to {}^{\prime }E_{2}^{p,0}$ is an isomorphism for every p.
2. $f:E_{2}^{0,q}\to {}^{\prime }E_{2}^{0,q}$ is an isomorphism for every q.
3. $f:E_{\infty }^{p,q}\to {}^{\prime }E_{\infty }^{p,q}$ is an isomorphism for every p, q.
Illustrative example
As an illustration, we sketch the proof of Borel's theorem, which says the cohomology ring of a classifying space is a polynomial ring.[1]
First of all, with G as a Lie group and with $\mathbb {Q} $ as coefficient ring, we have the Serre spectral sequence $E_{2}^{p,q}$ for the fibration $G\to EG\to BG$. We have: $E_{\infty }\simeq \mathbb {Q} $ since EG is contractible. We also have a theorem of Hopf stating that $H^{*}(G;\mathbb {Q} )\simeq \Lambda (u_{1},\dots ,u_{n})$, an exterior algebra generated by finitely many homogeneous elements.
Next, we let $E(i)$ be the spectral sequence whose second page is $E(i)_{2}=\Lambda (x_{i})\otimes \mathbb {Q} [y_{i}]$ and whose nontrivial differentials on the r-th page are given by $d(x_{i})=y_{i}$ and the graded Leibniz rule. Let ${}^{\prime }E_{r}=\otimes _{i}E_{r}(i)$. Since the cohomology commutes with tensor products as we are working over a field, ${}^{\prime }E_{r}$ is again a spectral sequence such that ${}^{\prime }E_{\infty }\simeq \mathbb {Q} \otimes \dots \otimes \mathbb {Q} \simeq \mathbb {Q} $. Then we let
$f:{}^{\prime }E_{r}\to E_{r},\,x_{i}\mapsto u_{i}.$
Note, by definition, f gives the isomorphism ${}^{\prime }E_{r}^{0,q}\simeq E_{r}^{0,q}=H^{q}(G;\mathbb {Q} ).$ A crucial point is that f is a "ring homomorphism"; this rests on the technical conditions that $u_{i}$ are "transgressive" (cf. Hatcher for detailed discussion on this matter.) After this technical point is taken care, we conclude: $E_{2}^{p,0}\simeq {}^{\prime }E_{2}^{p,0}$ as ring by the comparison theorem; that is, $E_{2}^{p,0}=H^{p}(BG;\mathbb {Q} )\simeq \mathbb {Q} [y_{1},\dots ,y_{n}].$
References
1. Hatcher, Theorem 1.34 harvnb error: no target: CITEREFHatcher (help)
• McCleary, John (2001), A User's Guide to Spectral Sequences, Cambridge Studies in Advanced Mathematics, vol. 58 (2nd ed.), Cambridge University Press, ISBN 978-0-521-56759-6, MR 1793722
• Roitberg, Joseph; Hilton, Peter (1976), "On the Zeeman comparison theorem for the homology of quasi-nilpotent fibrations" (PDF), The Quarterly Journal of Mathematics, Second Series, 27 (108): 433–444, doi:10.1093/qmath/27.4.433, ISSN 0033-5606, MR 0431151
• Zeeman, Erik Christopher (1957), "A proof of the comparison theorem for spectral sequences", Proc. Cambridge Philos. Soc., 53: 57–62, doi:10.1017/S0305004100031984, MR 0084769
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Zeeman conjecture
In mathematics, the Zeeman conjecture or Zeeman's collapsibility conjecture asks whether given a finite contractible 2-dimensional CW complex $K$, the space $K\times [0,1]$ is collapsible.
The conjecture, due to Christopher Zeeman, implies the Poincaré conjecture and the Andrews–Curtis conjecture.
References
• Matveev, Sergei (2007), "1.3.4 Zeeman's Collapsing Conjecture", Algorithmic Topology and Classification of 3-Manifolds, Algorithms and Computation in Mathematics, vol. 9, Springer, pp. 46–58, ISBN 9783540458999
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Zeev Nehari
Zeev Nehari (born Willi Weissbach; 2 February 1915 – 1978) was a mathematician who worked on Complex Analysis, Univalent Functions Theory and Differential and Integral Equations. He was a student of Michael (Mihály) Fekete. The Nehari manifold is named after him.
Selected publications
• Weissbach, Willi (1941), On certain classes of analytic functions and the corresponding conformal representations, Summary of a thesis, Hebrew University, Jerusalem, MR 0017371
• Nehari, Zeev (1949), "The Schwarzian derivative and schlicht functions", Bulletin of the American Mathematical Society, 55 (6): 545–551, doi:10.1090/S0002-9904-1949-09241-8, ISSN 0002-9904, MR 0029999
• Nehari, Zeev (1952), "Some inequalities in the theory of functions", Transactions of the American Mathematical Society, 1953, Vol.75, pp. 256–286.
• Nehari, Zeev (1968), Introduction to complex analysis, Revised edition, Boston, Mass.: Allyn and Bacon Inc., MR 0224780
• Nehari, Zeev (1975) [1952], Conformal mapping, New York: Dover Publications, ISBN 978-0-486-61137-2, MR 0377031
References
• "In memoriam Zeev Nehari 1915—1978", Journal d'Analyse Mathématique, 36: vi–vii, 1979, doi:10.1007/BF02798762, ISSN 0021-7670, MR 0581795, S2CID 189782091
External links
• Zeev Nehari at the Mathematics Genealogy Project
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Selig Brodetsky
Selig Brodetsky, זליג ברודצק (10 February 1888 – 18 May 1954)[1] was a Russian-born English mathematician, a member of the World Zionist Executive, the president of the Board of Deputies of British Jews, and the second president of the Hebrew University of Jerusalem.
Selig Brodestsky
Born10 February 1888
Olviopol, Kherson Governorate, Russian Empire
Died18 May 1954(1954-05-18) (aged 66)
London, UK
EducationJews' Free School
Alma materTrinity College, Cambridge
OccupationMathematician
SpouseManya Berenblum
Children1 son, 1 daughter
Parent(s)Akiva Brodetsky
Adel Prober
RelativesSolomon Mestel (brother-in-law)
Leon Mestel (nephew)
Background
Brodetsky was born in Olviopol (now Pervomaisk) in the Kherson Governorate of the Russian Empire (present-day Ukraine), the second of 13 children born to Akiva Brodetsky (the beadle of the local synagogue) and Adel (Prober). As a child, he witnessed the murder of his uncle in a pogrom. In 1894, the family followed Akiva to the East End of London, to where he had migrated a year earlier. Brodetsky attended the Jews' Free School, where he excelled at his studies. He was awarded a scholarship, which enabled him to attend the Central Foundation Boys' School of London[2] and subsequently, in 1905, Trinity College, Cambridge.
In 1908, he completed his studies with highest honours being Senior Wrangler, to the distress of the conservative press, which was forced to recognise that a son of immigrants surpassed all the local students. The Newton scholarship enabled him to study at Leipzig University where he was awarded a doctorate in 1913. His dissertation dealt with the gravitational field.
In 1919, he married Manya Berenblum, whose family had recently emigrated from Belgium, where her father had been a diamond merchant in Antwerp. They had two children, Paul and Adele, in 1924 and 1927.
Academic career
In 1914, Brodetsky was appointed a lecturer in applied mathematics at the University of Bristol.[3][4][5] During the First World War he was employed as an advisor to the British company developing periscopes for submarines.
In 1919, Brodetsky became a lecturer at the University of Leeds. Five years later he was appointed professor of applied mathematics at Leeds where he remained until 1948. Much of his work concerned aeronautics and mechanics of aeroplanes. He was the head of the mathematics department of the University of Leeds from 1946 to 1948. He was active in the Association of University Teachers, serving as president in 1935–1936.
Brodetsky became the second president of the Hebrew University of Jerusalem in 1949, preceded by Sir Leon Simon, serving until 1952, and followed by Benjamin Mazar (1953 to 1961), at a time when the university was going through a rocky period, eventually having to abandon its campus on Mount Scopus.[6] He attempted to overhaul the structure of the university but he soon became embroiled in bitter struggles with the University Senate, which interfered in his academic and bureaucratic work. Apparently, Brodetsky thought that he was going to take up a position similar to that of Vice-Chancellor of an English university but many in Jerusalem saw the position as essentially an honorary one, like the Chancellor of an English university. This struggle affected his health and in 1952 he decided to resign his post and return to England.
Education
• Jews' Free School (JFS), London (where there is now a Brodetsky House in his honour)
• Central Foundation Boys' School, London
• Trinity College, Cambridge (senior wrangler, 1908)
• Leipzig University (PhD)
Career
• Lecturer in Applied Mathematics, University of Bristol, 1914–1919
• Reader, 1920–1924; Professor, 1924-1948 then Emeritus Professor of Applied Mathematics, University of Leeds
• President of the Hebrew University of Jerusalem and Chairman of its Executive Council, 1949–1951
Other posts
• Member of the Executive, World Zionist Organisation and Jewish Agency for Palestine
• Honorary President, Zionist Federation of Great Britain and Ireland
• Honorary President, Maccabi World Union
• President, Board of Deputies of British Jews (1940–49)[7])
He was a Fellow of the Royal Astronomical Society, Royal Aeronautical Society and Institute of Physics.
His sister Rachel married Rabbi Solomon Mestel; their son is astronomer and astrophysicist Leon Mestel.
References
1. "Dr. Selig Brodetsky". The Times. No. 52935. 19 May 1954. p. 8.
2. "Alumni". Central Foundation Boys' School. 2013. Retrieved 1 October 2015.
3. Aubin, David; Goldstein, Catherine (7 October 2014). The War of Guns and Mathematics: Mathematical Practices and Communities in ... – Google Books. American Mathematical Society. ISBN 9781470414696. Retrieved 16 February 2020.
4. Matthäus, Jürgen (18 April 2013). Jewish Responses to Persecution: 1941–1942 – Jürgen Matthäus – Google Books. AltaMira Press. ISBN 9780759122598. Retrieved 16 February 2020.
5. Kol, Moshe (22 June 2006). Mentors and friends – Moshe Kol – Google Books. Cornwall Books. ISBN 9780845347416. Retrieved 16 February 2020.
6. "Office of the President | האוניברסיטה העברית בירושלים | The Hebrew University of Jerusalem". New.huji.ac.il. 1 September 2017. Retrieved 18 February 2020.
7. www-history.mcs.st-andrews.ac.uk Selig Brodetsky
• O'Connor, John J.; Robertson, Edmund F., "Selig Brodetsky", MacTutor History of Mathematics Archive, University of St Andrews
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External links
• The personal papers of Selig Brodetsky are kept at the Central Zionist Archives in Jerusalem. The notation of the record group is A82.
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Zeller's congruence
Zeller's congruence is an algorithm devised by Christian Zeller in the 19th century to calculate the day of the week for any Julian or Gregorian calendar date. It can be considered to be based on the conversion between Julian day and the calendar date.
Formula
For the Gregorian calendar, Zeller's congruence is
$h=\left(q+\left\lfloor {\frac {13(m+1)}{5}}\right\rfloor +K+\left\lfloor {\frac {K}{4}}\right\rfloor +\left\lfloor {\frac {J}{4}}\right\rfloor -2J\right){\bmod {7}},$
for the Julian calendar it is
$h=\left(q+\left\lfloor {\frac {13(m+1)}{5}}\right\rfloor +K+\left\lfloor {\frac {K}{4}}\right\rfloor +5-J\right){\bmod {7}},$
where
• h is the day of the week (0 = Saturday, 1 = Sunday, 2 = Monday, ..., 6 = Friday)
• q is the day of the month
• m is the month (3 = March, 4 = April, 5 = May, ..., 14 = February)
• K the year of the century ($year{\bmod {1}}00$).
• J is the zero-based century (actually $\lfloor year/100\rfloor $) For example, the zero-based centuries for 1995 and 2000 are 19 and 20 respectively (not to be confused with the common ordinal century enumeration which indicates 20th for both cases).
• $\lfloor ...\rfloor $ is the floor function or integer part
• mod is the modulo operation or remainder after division
In this algorithm January and February are counted as months 13 and 14 of the previous year. E.g. if it is 2 February 2010, the algorithm counts the date as the second day of the fourteenth month of 2009 (02/14/2009 in DD/MM/YYYY format)
For an ISO week date Day-of-Week d (1 = Monday to 7 = Sunday), use
$d=((h+5){\bmod {7}})+1$
Analysis
These formulas are based on the observation that the day of the week progresses in a predictable manner based upon each subpart of that date. Each term within the formula is used to calculate the offset needed to obtain the correct day of the week.
For the Gregorian calendar, the various parts of this formula can therefore be understood as follows:
• $q$ represents the progression of the day of the week based on the day of the month, since each successive day results in an additional offset of 1 in the day of the week.
• $K$ represents the progression of the day of the week based on the year. Assuming that each year is 365 days long, the same date on each succeeding year will be offset by a value of $365{\bmod {7}}=1$.
• Since there are 366 days in each leap year, this needs to be accounted for by adding another day to the day of the week offset value. This is accomplished by adding $\left\lfloor {\frac {K}{4}}\right\rfloor $ to the offset. This term is calculated as an integer result. Any remainder is discarded.
• Using similar logic, the progression of the day of the week for each century may be calculated by observing that there are 36,524 days in a normal century and 36,525 days in each century divisible by 400. Since $36525{\bmod {7}}=6$ and $36524{\bmod {7}}=5$, the term $\left\lfloor {\frac {J}{4}}\right\rfloor -2J$ accounts for this.
• The term $\left\lfloor {\frac {13(m+1)}{5}}\right\rfloor $ adjusts for the variation in the days of the month. Starting from January, the days in a month are {31, 28/29, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31}. February's 28 or 29 days is a problem, so the formula rolls January and February around to the end so February's short count will not cause a problem. The formula is interested in days of the week, so the numbers in the sequence can be taken modulo 7. Then the number of days in a month modulo 7 (still starting with January) would be {3, 0/1, 3, 2, 3, 2, 3, 3, 2, 3, 2, 3}. Starting in March, the sequence basically alternates 3, 2, 3, 2, 3, but every five months there are two 31-day months in a row (July–August and December–January).[1] The fraction 13/5 = 2.6 and the floor function have that effect; the denominator of 5 sets a period of 5 months.
• The overall function, $\operatorname {mod} \,7$, normalizes the result to reside in the range of 0 to 6, which yields the index of the correct day of the week for the date being analyzed.
The reason that the formula differs for the Julian calendar is that this calendar does not have a separate rule for leap centuries and is offset from the Gregorian calendar by a fixed number of days each century.
Since the Gregorian calendar was adopted at different times in different regions of the world, the location of an event is significant in determining the correct day of the week for a date that occurred during this transition period. This is only required through 1929, as this was the last year that the Julian calendar was still in use by any country on earth, and thus is not required for 1930 or later.
The formulae can be used proleptically, but "Year 0" is in fact year 1 BC (see astronomical year numbering). The Julian calendar is in fact proleptic right up to 1 March AD 4 owing to mismanagement in Rome (but not Egypt) in the period since the calendar was put into effect on 1 January 45 BC (which was not a leap year). In addition, the modulo operator might truncate integers to the wrong direction (ceiling instead of floor). To accommodate this, one can add a sufficient multiple of 400 Gregorian or 700 Julian years.
Examples
For 1 January 2000, the date would be treated as the 13th month of 1999, so the values would be:
$q=1$
$m=13$
$K=99$
$J=19$
So the formula evaluates as $(1+36+99+24+4-38){\bmod {7}}=126{\bmod {7}}=0={\text{Saturday}}$.
(The 36 comes from $(13+1)\times 13/5=182/5$, truncated to an integer.)
However, for 1 March 2000, the date is treated as the 3rd month of 2000, so the values become
$q=1$
$m=3$
$K=0$
$J=20$
so the formula evaluates as $(1+10+0+0+5-40){\bmod {7}}=-24{\bmod {7}}=4={\text{Wednesday}}$.
Implementations in software
Basic modification
Further information: Modulo operation § Variants of the definition
The formulas rely on the mathematician's definition of modulo division, which means that −2 mod 7 is equal to positive 5. Unfortunately, in the truncating way most computer languages implement the remainder function, −2 mod 7 returns a result of −2. So, to implement Zeller's congruence on a computer, the formulas should be altered slightly to ensure a positive numerator. The simplest way to do this is to replace − 2J by + 5J and − J by + 6J. So the formulas become:
$h=\left(q+\left\lfloor {\frac {13(m+1)}{5}}\right\rfloor +K+\left\lfloor {\frac {K}{4}}\right\rfloor +\left\lfloor {\frac {J}{4}}\right\rfloor +5J\right){\bmod {7}},$
for the Gregorian calendar, and
$h=\left(q+\left\lfloor {\frac {13(m+1)}{5}}\right\rfloor +K+\left\lfloor {\frac {K}{4}}\right\rfloor +5+6J\right){\bmod {7}},$
for the Julian calendar.
One can readily see that, in a given year, March 1 (if that is a Saturday, then March 2) is a good test date, and that in any given century, the best test year is that which is a multiple of 100.
Common simplification
Zeller used decimal arithmetic, and found it convenient to use J and K in representing the year. But when using a computer, it is simpler to handle the modified year Y and month m, which are Y - 1 and m + 12 during January and February:
$h=\left(q+\left\lfloor {\frac {13(m+1)}{5}}\right\rfloor +Y+\left\lfloor {\frac {Y}{4}}\right\rfloor -\left\lfloor {\frac {Y}{100}}\right\rfloor +\left\lfloor {\frac {Y}{400}}\right\rfloor \right){\bmod {7}},$
for the Gregorian calendar (in this case there is no possibility of overflow because $\left\lfloor Y/4\right\rfloor \geq \left\lfloor Y/100\right\rfloor $), and
$h=\left(q+\left\lfloor {\frac {13(m+1)}{5}}\right\rfloor +Y+\left\lfloor {\frac {Y}{4}}\right\rfloor +5\right){\bmod {7}},$
for the Julian calendar.
The algorithm above is mentioned for the Gregorian case in RFC 3339, Appendix B, albeit in an abridged form that returns 0 for Sunday.
Other variations
At least three other algorithms share the overall structure of Zeller's congruence in its "common simplification" type, also using an m ∈ [3, 14] ∩ Z and the "modified year" construct.
• Michael Keith published a piece of very short C code in 1990 for Gregorian dates. The month-length component ($\left\lfloor {\frac {13(m+1)}{5}}\right\rfloor $) is replaced by $\left\lfloor {\frac {23m}{9}}\right\rfloor +4$.[2]
• J R Stockton provides a Sunday-is-0 version with $\left\lfloor {\frac {13(m-2)}{5}}\right\rfloor +2$, calling it a variation of Zeller.[2]
• Claus Tøndering describes $\left\lfloor {\frac {31(m-2)}{12}}\right\rfloor $ as a replacement.[3]
Both expressions can be shown to progress in a way that is off by one compared to the original month-length component over the required range of m, resulting in a starting value of 0 for Sunday.
See also
• Determination of the day of the week
• Doomsday rule
• ISO week date
• Julian day
References
1. The every five months rule only applies to the twelve months of a year commencing on 1 March and ending on the last day of the following February.
2. Stockton, J R. "Material Related to Zeller's Congruence". "Merlyn", archived at NCTU Taiwan.
3. Tøndering, Claus. "Week-related questions". www.tondering.dk.
Bibliography
Each of these four similar imaged papers deals firstly with the day of the week and secondly with the date of Easter Sunday, for the Julian and Gregorian calendars. The pages link to translations into English.
• Zeller, Christian (1882). "Die Grundaufgaben der Kalenderrechnung auf neue und vereinfachte Weise gelöst". Württembergische Vierteljahrshefte für Landesgeschichte (in German). V: 313–314. Archived from the original on January 11, 2015.
• Zeller, Christian (1883). "Problema duplex Calendarii fundamentale". Bulletin de la Société Mathématique de France (in Latin). 11: 59–61. Archived from the original on January 11, 2015.
• Zeller, Christian (1885). "Kalender-Formeln". Mathematisch-naturwissenschaftliche Mitteilungen des mathematisch-naturwissenschaftlichen Vereins in Württemberg (in German). 1 (1): 54–58. Archived from the original on January 11, 2015.
• Zeller, Christian (1886). "Kalender-Formeln". Acta Mathematica (in German). 9: 131–136. doi:10.1007/BF02406733.
External links
• The Calendrical Works of Rektor Chr. Zeller: The Day-of-Week and Easter Formulae by J R Stockton, near London, UK. The site includes images and translations of the above four papers, and of Zeller's reference card "Das Ganze der Kalender-Rechnung".
• This article incorporates public domain material from Paul E. Black. "Zeller's congruence". Dictionary of Algorithms and Data Structures. NIST.
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Zenzizenzizenzic
Zenzizenzizenzic is an obsolete form of mathematical notation representing the eighth power of a number (that is, the zenzizenzizenzic of x is x8), dating from a time when powers were written out in words rather than as superscript numbers. This term was suggested by Robert Recorde, a 16th-century Welsh physician, mathematician and writer of popular mathematics textbooks, in his 1557 work The Whetstone of Witte (although his spelling was zenzizenzizenzike); he wrote that it "doeth represent the square of squares squaredly".
History
At the time Recorde proposed this notation, there was no easy way of denoting the powers of numbers other than squares and cubes. The root word for Recorde's notation is zenzic, which is a German spelling of the medieval Italian word censo, meaning 'squared'.[1] Since the square of a square of a number is its fourth power, Recorde used the word zenzizenzic (spelled by him as zenzizenzike) to express it. Some of the terms had prior use in Latin zenzicubicus, zensizensicus and zensizenzum.[2] Similarly, as the sixth power of a number is equal to the square of its cube, Recorde used the word zenzicubike to express it; a more modern spelling, zenzicube, is found in Samuel Jeake's Arithmetick Surveighed and Reviewed. Finally, the word zenzizenzizenzic denotes the square of the square of a number's square, which is its eighth power: in modern notation,
$x^{8}=\left(\left(x^{2}\right)^{2}\right)^{2}.$
Samuel Jeake gives zenzizenzizenzizenzike (the square of the square of the square of the square, or 16th power) in a table in A Compleat Body of Arithmetick (1701):[3]
Indices Characters Signification of the characters
0 NAn absolute number, as if it had no mark
... ......
16 ℨℨℨℨA Zenzizenzizenzizenzike or square of squares squaredly squared
... ......
The word, as well as the system, is obsolete except as a curiosity; the Oxford English Dictionary (OED) has only one citation for it.[4][5] As well as being a mathematical oddity, it survives as a linguistic oddity: zenzizenzizenzic has more Zs than any other word in the OED.[6][7]
Notation for other powers
Recorde proposed three mathematical terms by which any power (that is, index or exponent) greater than 1 could be expressed: zenzic, i.e. squared; cubic; and sursolid, i.e. raised to a prime number greater than three, the smallest of which is five. Sursolids were as follows: 5 was the first; 7, the second; 11, the third; 13, the fourth; etc.
Therefore, a number raised to the power of six would be zenzicubic, a number raised to the power of seven would be the second sursolid, hence bissursolid (not a multiple of two and three), a number raised to the twelfth power would be the "zenzizenzicubic" and a number raised to the power of ten would be the square of the (first) sursolid. The fourteenth power was the square of the second sursolid, and the twenty-second was the square of the third sursolid.
Jeake's text appears to designate a written exponent of 0 as being equal to an "absolute number, as if it had no Mark", thus using the notation x0 to refer to an independent term of a polynomial, while a written exponent of 1, in his text, denotes "the Root of any number" (using root with the meaning of the base number, i.e. its first power x1, as demonstrated in the examples provided in the book).
Citations
1. Quinion, Michael, "Zenzizenzizenzic - the eighth power of a number", World Wide Words, retrieved 19 March 2010.
2. Michael Stifel. Arithmetica Integra (in Latin). Nuremberg. p. 61.
3. Samuel Jeake (1701). Samuel Jeake the Younger (ed.). A Compleat Body of Arithmetick. London: T. Newborough. p. 272.
4. Knight (1868).
5. Reilly (2003).
6. "Recorde also coined zenzizenzizenzic, OED with more Zs than any other" (Reilly 2003).
7. Uniquely contains six Zs. Thus, it's the only hexazetic word in the English language."Numerical Adjectives, Greek and Latin Number Prefixes". phrontistery.info. Retrieved 19 March 2010.
References
• Hebra, Alexius J. (2003), Measure for Measure: The Story of Imperial, Metric, and Other Units, The Johns Hopkins University Press, ISBN 978-0-8018-7072-9.
• Knight, Charles (1868), The English Cyclopaedia, Bradbury, Evans, p. 1045.
• Reilly, Edwin D. (2003), Milestones in Computer Science and Information Technology, Greenwood Publishing Group, p. 3, ISBN 978-1-57356-521-9.
• Todd, Richard Watson (2006), Much Ado About English, Nicholas Brealey Publishing, ISBN 978-1-85788-372-5.
• Uldrich, Jack (2008), "Chapter 2. The Power of Zenzizenzizenzic", Jump the Curve: 50 Essential Strategies to Help Your Company Stay Ahead of Emerging Technologies, Adams Media, ISBN 978-1-59869-420-8.
See also
• Prime factor exponent notation
External links
Look up zenzizenzizenzic in Wiktionary, the free dictionary.
• Entry at World Wide Words
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Zerah Colburn (mental calculator)
Zerah Colburn (September 1, 1804 – March 2, 1839)[1][2][3] was an American child prodigy of the 19th century who gained fame as a mental calculator.[4]
The Reverend
Zerah Colburn
BornSeptember 1, 1804
Cabot, Vermont, United States
DiedMarch 2, 1839(1839-03-02) (aged 34)
Norwich, Vermont, United States
Occupation(s)Schoolteacher, academic, Methodist minister
Known forMental calculator; child prodigy
Spouse
Mary Hoyt
(m. 1829)
Children6
Parent(s)Abia Colburn
Elizabeth "Betsey" Hill
RelativesZerah Colburn (nephew)
Biography
Colburn was born in Cabot, Vermont, in 1804. He was thought to be intellectually disabled until the age of six.[5] However, after six weeks of schooling, his father overheard him repeating his multiplication tables. His father was not sure whether or not he learned the tables from his older brothers and sisters, but he decided to test him further on his mathematical abilities and discovered that there was something special about his son when Zerah correctly multiplied 13 and 97.
Colburn's abilities developed rapidly and he was soon able to solve such problems as the number of seconds in 2,000 years, the product of 12,225 and 1,223, or the square root of 1,449. When he was seven years old he took six seconds to give the numbers of hours in thirty-eight years, two months, and seven days.
Zerah is reported to have been able to solve fairly complex problems. For example, the sixth Fermat number is 225+1 (or 232+1). The question is whether this number, 4,294,967,297, is prime or not. Zerah calculated in his head that it was not and had a divisor of 641. (Its other proper divisor is 6,700,417.)
His father capitalized on his boy's talents by taking Zerah around the country and eventually abroad, demonstrating the boy's exceptional abilities. The two left Vermont in the winter of 1810–11. Passing through Hanover, New Hampshire, John Wheelock, then president of Dartmouth College, offered to take upon himself the whole care and expense of his education, but his father rejected the offer. At Boston, the boy's performances attracted much attention. He was visited by Harvard College professors and eminent people from all professions, and the newspapers ran numerous articles concerning his powers of computation.[6]
After leaving Boston, his father exhibited Zerah for money throughout the middle and part of the southern states and, in January 1812, sailed with him for England. In September 1813 Colburn was being exhibited in Dublin. Colburn was pitted against the eight-year-old William Rowan Hamilton in a mental arithmetic contest, with Colburn emerging the clear victor. In reaction to his defeat, Hamilton dedicated less time to studying languages and more time to studying mathematics. After traveling over England, Scotland, and Ireland, they spent 18 months in Paris. Here Zerah was placed in the Lycée Napoléon but was soon removed by his father, who at length in 1816 returned to England in deep poverty.[6]
The Earl of Bristol soon became interested in the boy, and placed him in Westminster School, where he remained until 1819. In consequence of his father's refusal to comply with certain arrangements proposed by the earl, Zerah was removed from Westminster, and his father then proposed to Zerah that he should study to become an actor. Accordingly, he studied for this profession and was for a few months under the tuition of Charles Kemble. His first appearance, however, dissatisfied both his instructor and himself so much that he was not accepted for the stage, so he accepted a position as an assistant in a school, and soon afterward commenced a school of his own. To this he added the performing of some astronomical calculations for Thomas Young, then secretary of the Board of Longitude.[6]
In 1824 when his father died, he was enabled by the Earl of Bristol and other friends to return to the United States. Though Zerah's schooling was rather irregular, he showed talent in languages. He went to Fairfield, New York, as assistant teacher of an academy; not being pleased with his situation, he moved in March following to Burlington, Vermont, where he taught French, pursuing his studies at the same time at the University of Vermont. Toward the end of 1825 he connected himself with the Methodist Church and, after nine years of service as an itinerant preacher, settled in Norwich, Vermont, in 1835, where he was soon after appointed professor of languages at Dartmouth College in Hanover, New Hampshire.
In 1833 Colburn published his autobiography. From this it appears that his faculty of computation left him about the time he reached adulthood.[6] He died of tuberculosis at the age of 34 and was buried in Norwich's Old Meeting House Cemetery.[7]
Family
His nephew, also named Zerah Colburn, was a noted locomotive engineer and technical journalist.
See also
• Ainan Celeste Cawley
Notes
1. "Colburn, Zerah". Dictionary of American Biography. Vol. Comprehensive Index. New York: Charles Scribner's Sons. 1990.
2. Wilson, J. G.; Fiske, J., eds. (1891). "Colburn, Zerah". Appletons' Cyclopædia of American Biography. New York: D. Appleton.
3. Chisholm, Hugh, ed. (1911). "Colburn, Zerah" . Encyclopædia Britannica (11th ed.). Cambridge University Press.
4. W. W. Rouse Ball (1960) Calculating Prodigies, in Mathematical Recreations and Essays, Macmillan, New York, chapter 13.
5. "The Nineteenth Century in Print, 1833". stepanov.lk.net. Retrieved May 25, 2017.
6. One or more of the preceding sentences incorporates text from a publication now in the public domain: Ripley, George; Dana, Charles A., eds. (1900). "Colburn, Zerah" . The American Cyclopædia.
7. Brown, Jane. "The Story of Zerah Colburn, Child Math Wizard". Retrieved June 14, 2012.
Further reading
• Collins, Paul (April 7, 2007). "Have prodigy, will travel". New Scientist. 194 (2598): 50–51. doi:10.1016/S0262-4079(07)60874-4. ISSN 0262-4079.
• Colburn, Zerah (1833). A memoir of Zerah Colburn. G. & C. Merriam Company. OCLC 3394328.
External links
Wikimedia Commons has media related to Zerah Colburn (mental calculator).
• Picture with information implying he was polydactyl
• Strongly unsympathetic review of his memoir
• "Colburn, Zerah" . Appletons' Cyclopædia of American Biography. 1900.
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Axiom of limitation of size
In set theory, the axiom of limitation of size was proposed by John von Neumann in his 1925 axiom system for sets and classes.[1] It formalizes the limitation of size principle, which avoids the paradoxes encountered in earlier formulations of set theory by recognizing that some classes are too big to be sets. Von Neumann realized that the paradoxes are caused by permitting these big classes to be members of a class.[2] A class that is a member of a class is a set; a class that is not a set is a proper class. Every class is a subclass of V, the class of all sets.[lower-alpha 1] The axiom of limitation of size says that a class is a set if and only if it is smaller than V—that is, there is no function mapping it onto V. Usually, this axiom is stated in the equivalent form: A class is a proper class if and only if there is a function that maps it onto V.
Von Neumann's axiom implies the axioms of replacement, separation, union, and global choice. It is equivalent to the combination of replacement, union, and global choice in Von Neumann–Bernays–Gödel set theory (NBG) and Morse–Kelley set theory. Later expositions of class theories—such as those of Paul Bernays, Kurt Gödel, and John L. Kelley—use replacement, union, and a choice axiom equivalent to global choice rather than von Neumann's axiom.[3] In 1930, Ernst Zermelo defined models of set theory satisfying the axiom of limitation of size.[4]
Abraham Fraenkel and Azriel Lévy have stated that the axiom of limitation of size does not capture all of the "limitation of size doctrine" because it does not imply the power set axiom.[5] Michael Hallett has argued that the limitation of size doctrine does not justify the power set axiom and that "von Neumann's explicit assumption [of the smallness of power-sets] seems preferable to Zermelo's, Fraenkel's, and Lévy's obscurely hidden implicit assumption of the smallness of power-sets."[6]
Formal statement
The usual version of the axiom of limitation of size—a class is a proper class if and only if there is a function that maps it onto V —is expressed in the formal language of set theory as:
${\begin{aligned}\forall C{\Bigl [}\lnot \exists D\left(C\in D\right)\iff \exists F{\bigl [}&\,\forall y{\bigl (}\exists D(y\in D)\implies \exists x[\,x\in C\land (x,y)\in F\,]{\bigr )}\\&\,\land \,\forall x\forall y\forall z{\bigl (}\,[\,(x,y)\in F\land (x,z)\in F\,]\implies y=z{\bigr )}\,{\bigr ]}\,{\Bigr ]}\end{aligned}}$
Gödel introduced the convention that uppercase variables range over all the classes, while lowercase variables range over all the sets.[7] This convention allows us to write:
• $\exists y\,\varphi (y)$ instead of $\exists y{\bigl (}\exists D(y\in D)\land \varphi (y){\bigr )}$
• $\forall y\,\varphi (y)$ instead of $\forall y{\bigl (}\exists D(y\in D)\implies \varphi (y){\bigr )}$
With Gödel's convention, the axiom of limitation of size can be written:
${\begin{aligned}\forall C{\Bigl [}\lnot \exists D\left(C\in D\right)\iff \exists F{\bigl [}&\,\forall y\exists x{\bigl (}x\in C\land (x,y)\in F{\bigr )}\\&\,\land \,\forall x\forall y\forall z{\bigl (}\,[\,(x,y)\in F\land (x,z)\in F\,]\implies y=z{\bigr )}\,{\bigr ]}\,{\Bigr ]}\end{aligned}}$
Implications of the axiom
Von Neumann proved that the axiom of limitation of size implies the axiom of replacement, which can be expressed as: If F is a function and A is a set, then F(A) is a set. This is proved by contradiction. Let F be a function and A be a set. Assume that F(A) is a proper class. Then there is a function G that maps F(A) onto V. Since the composite function G ∘ F maps A onto V, the axiom of limitation of size implies that A is a proper class, which contradicts A being a set. Therefore, F(A) is a set. Since the axiom of replacement implies the axiom of separation, the axiom of limitation of size implies the axiom of separation.[lower-alpha 2]
Von Neumann also proved that his axiom implies that V can be well-ordered. The proof starts by proving by contradiction that Ord, the class of all ordinals, is a proper class. Assume that Ord is a set. Since it is a transitive set that is strictly well-ordered by ∈, it is an ordinal. So Ord ∈ Ord, which contradicts Ord being strictly well-ordered by ∈. Therefore, Ord is a proper class. So von Neumann's axiom implies that there is a function F that maps Ord onto V. To define a well-ordering of V, let G be the subclass of F consisting of the ordered pairs (α, x) where α is the least β such that (β, x) ∈ F; that is, G = {(α, x) ∈ F : ∀β((β, x) ∈ F ⇒ α ≤ β)}. The function G is a one-to-one correspondence between a subset of Ord and V. Therefore, x < y if G−1(x) < G−1(y) defines a well-ordering of V. This well-ordering defines a global choice function: Let Inf (x) be the least element of a non-empty set x. Since Inf (x) ∈ x, this function chooses an element of x for every non-empty set x. Therefore, Inf (x) is a global choice function, so Von Neumann's axiom implies the axiom of global choice.
In 1968, Azriel Lévy proved that von Neumann's axiom implies the axiom of union. First, he proved without using the axiom of union that every set of ordinals has an upper bound. Then he used a function that maps Ord onto V to prove that if A is a set, then ∪ A is a set.[8]
The axioms of replacement, global choice, and union (with the other axioms of NBG) imply the axiom of limitation of size.[lower-alpha 3] Therefore, this axiom is equivalent to the combination of replacement, global choice, and union in NBG or Morse–Kelley set theory. These set theories only substituted the axiom of replacement and a form of the axiom of choice for the axiom of limitation of size because von Neumann's axiom system contains the axiom of union. Lévy's proof that this axiom is redundant came many years later.[9]
The axioms of NBG with the axiom of global choice replaced by the usual axiom of choice do not imply the axiom of limitation of size. In 1964, William B. Easton used forcing to build a model of NBG with global choice replaced by the axiom of choice.[10] In Easton's model, V cannot be linearly ordered, so it cannot be well-ordered. Therefore, the axiom of limitation of size fails in this model. Ord is an example of a proper class that cannot be mapped onto V because (as proved above) if there is a function mapping Ord onto V, then V can be well-ordered.
The axioms of NBG with the axiom of replacement replaced by the weaker axiom of separation do not imply the axiom of limitation of size. Define $\omega _{\alpha }$ as the $\alpha $-th infinite initial ordinal, which is also the cardinal $\aleph _{\alpha }$; numbering starts at $0$, so $\omega _{0}=\omega .$ In 1939, Gödel pointed out that Lωω, a subset of the constructible universe, is a model of ZFC with replacement replaced by separation.[11] To expand it into a model of NBG with replacement replaced by separation, let its classes be the sets of Lωω+1, which are the constructible subsets of Lωω. This model satisfies NBG's class existence axioms because restricting the set variables of these axioms to Lωω produces instances of the axiom of separation, which holds in L.[lower-alpha 4] It satisfies the axiom of global choice because there is a function belonging to Lωω+1 that maps ωω onto Lωω, which implies that Lωω is well-ordered.[lower-alpha 5] The axiom of limitation of size fails because the proper class {ωn : n ∈ ω} has cardinality $\aleph _{0}$, so it cannot be mapped onto Lωω, which has cardinality $\aleph _{\omega }$.[lower-alpha 6]
In a 1923 letter to Zermelo, von Neumann stated the first version of his axiom: A class is a proper class if and only if there is a one-to-one correspondence between it and V.[2] The axiom of limitation of size implies von Neumann's 1923 axiom. Therefore, it also implies that all proper classes are equinumerous with V.
Proof that the axiom of limitation of size implies von Neumann's 1923 axiom
To prove the $\Longleftarrow $ direction, let $A$ be a class and $F$ be a one-to-one correspondence from $A$ to $V.$ Since $F$ maps $A$ onto $V,$ the axiom of limitation of size implies that $A$ is a proper class.
To prove the $\Longrightarrow $ direction, let $A$ be a proper class. We will define well-ordered classes $(A,<)$ and $(V,<),$ and construct order isomorphisms between $(Ord,<),(A,<),$ and $(V,<).$ Then the order isomorphism from $(A,<)$ to $(V,<)$ is a one-to-one correspondence between $A$ and $V.$
It was proved above that the axiom of limitation of size implies that there is a function $F$ that maps $Ord$ onto $V.$ Also, $G$ was defined as a subclass of $F$ that is a one-to-one correspondence between $Dom(G)$ and $V.$ It defines a well-ordering on $V\colon \,x<y\,$ if $G^{-1}(x)<G^{-1}(y).$ Therefore, $G$ is an order isomorphism from $(Dom(G),<)$ to $(V,<).$
If $(C,<)$ is well-ordered class, its proper initial segments are the classes $\{x\in C:x<y\}$ where $y\in C.$ Now $(Ord,<)$ has the property that all of its proper initial segments are sets. Since $Dom(G)\subseteq Ord,$ this property holds for $(Dom(G),<).$ The order isomorphism $G$ implies that this property holds for $(V,<).$ Since $A\subseteq V,$ this property holds for $(A,<).$
To obtain an order isomorphism from $(A,<)$ to $(V,<),$ the following theorem is used: If $P$ is a proper class and the proper initial segments of $(P,<)$ are sets, then there is an order isomorphism from $(Ord,<)$ to $(P,<).$[lower-alpha 7] Since $(A,<)$ and $(V,<)$ satisfy the theorem's hypothesis, there are order isomorphisms $I_{A}\colon (Ord,<)\rightarrow (A,<)$ and $I_{V}\colon (Ord,<)\rightarrow (V,<).$ Therefore, the order isomorphism $I_{V}\circ I_{A}^{-1}\colon (A,<)\rightarrow (V,<)$ is a one-to-one correspondence between $A$ and $V.$
Zermelo's models and the axiom of limitation of size
In 1930, Zermelo published an article on models of set theory, in which he proved that some of his models satisfy the axiom of limitation of size.[4] These models are built in ZFC by using the cumulative hierarchy Vα, which is defined by transfinite recursion:
1. V0 = ∅.[lower-alpha 8]
2. Vα+1 = Vα ∪ P(Vα). That is, the union of Vα and its power set.[lower-alpha 9]
3. For limit β: Vβ = ∪α < β Vα. That is, Vβ is the union of the preceding Vα.
Zermelo worked with models of the form Vκ where κ is a cardinal. The classes of the model are the subsets of Vκ, and the model's ∈-relation is the standard ∈-relation. The sets of the model are the classes X such that X ∈ Vκ.[lower-alpha 10] Zermelo identified cardinals κ such that Vκ satisfies:[12]
Theorem 1. A class X is a set if and only if |X| < κ.
Theorem 2. |Vκ| = κ.
Since every class is a subset of Vκ, Theorem 2 implies that every class X has cardinality ≤ κ. Combining this with Theorem 1 proves: every proper class has cardinality κ. Hence, every proper class can be put into one-to-one correspondence with Vκ. This correspondence is a subset of Vκ, so it is a class of the model. Therefore, the axiom of limitation of size holds for the model Vκ.
The theorem stating that Vκ has a well-ordering can be proved directly. Since κ is an ordinal of cardinality κ and |Vκ| = κ, there is a one-to-one correspondence between κ and Vκ. This correspondence produces a well-ordering of Vκ. Von Neumann's proof is indirect. It uses the Burali-Forti paradox to prove by contradiction that the class of all ordinals is a proper class. Hence, the axiom of limitation of size implies that there is a function that maps the class of all ordinals onto the class of all sets. This function produces a well-ordering of Vκ.[13]
The model Vω
To demonstrate that Theorems 1 and 2 hold for some Vκ, we first prove that if a set belongs to Vα then it belongs to all subsequent Vβ, or equivalently: Vα ⊆ Vβ for α ≤ β. This is proved by transfinite induction on β:
1. β = 0: V0 ⊆ V0.
2. For β+1: By inductive hypothesis, Vα ⊆ Vβ. Hence, Vα ⊆ Vβ ⊆ Vβ ∪ P(Vβ) = Vβ+1.
3. For limit β: If α < β, then Vα ⊆ ∪ξ < β Vξ = Vβ. If α = β, then Vα ⊆ Vβ.
Sets enter the cumulative hierarchy through the power set P(Vβ) at step β+1. The following definitions will be needed:
If x is a set, rank(x) is the least ordinal β such that x ∈ Vβ+1.[14]
The supremum of a set of ordinals A, denoted by sup A, is the least ordinal β such that α ≤ β for all α ∈ A.
Zermelo's smallest model is Vω. Mathematical induction proves that Vn is finite for all n < ω:
1. |V0| = 0.
2. |Vn+1| = |Vn ∪ P(Vn)| ≤ |Vn| + 2 |Vn|, which is finite since Vn is finite by inductive hypothesis.
Proof of Theorem 1: A set X enters Vω through P(Vn) for some n < ω, so X ⊆ Vn. Since Vn is finite, X is finite. Conversely: If a class X is finite, let N = sup {rank(x): x ∈ X}. Since rank(x) ≤ N for all x ∈ X, we have X ⊆ VN+1, so X ∈ VN+2 ⊆ Vω. Therefore, X ∈ Vω.
Proof of Theorem 2: Vω is the union of countably infinitely many finite sets of increasing size. Hence, it has cardinality $\aleph _{0}$, which equals ω by von Neumann cardinal assignment.
The sets and classes of Vω satisfy all the axioms of NBG except the axiom of infinity.[lower-alpha 11]
The models Vκ where κ is a strongly inaccessible cardinal
Two properties of finiteness were used to prove Theorems 1 and 2 for Vω:
1. If λ is a finite cardinal, then 2λ is finite.
2. If A is a set of ordinals such that |A| is finite, and α is finite for all α ∈ A, then sup A is finite.
To find models satisfying the axiom of infinity, replace "finite" by "< κ" to produce the properties that define strongly inaccessible cardinals. A cardinal κ is strongly inaccessible if κ > ω and:
1. If λ is a cardinal such that λ < κ, then 2λ < κ.
2. If A is a set of ordinals such that |A| < κ, and α < κ for all α ∈ A, then sup A < κ.
These properties assert that κ cannot be reached from below. The first property says κ cannot be reached by power sets; the second says κ cannot be reached by the axiom of replacement.[lower-alpha 12] Just as the axiom of infinity is required to obtain ω, an axiom is needed to obtain strongly inaccessible cardinals. Zermelo postulated the existence of an unbounded sequence of strongly inaccessible cardinals.[lower-alpha 13]
If κ is a strongly inaccessible cardinal, then transfinite induction proves |Vα| < κ for all α < κ:
1. α = 0: |V0| = 0.
2. For α+1: |Vα+1| = |Vα ∪ P(Vα)| ≤ |Vα| + 2 |Vα| = 2 |Vα| < κ. Last inequality uses inductive hypothesis and κ being strongly inaccessible.
3. For limit α: |Vα| = |∪ξ < α Vξ| ≤ sup {|Vξ| : ξ < α} < κ. Last inequality uses inductive hypothesis and κ being strongly inaccessible.
Proof of Theorem 1: A set X enters Vκ through P(Vα) for some α < κ, so X ⊆ Vα. Since |Vα| < κ, we obtain |X| < κ. Conversely: If a class X has |X| < κ, let β = sup {rank(x): x ∈ X}. Because κ is strongly inaccessible, |X| < κ and rank(x) < κ for all x ∈ X imply β = sup {rank(x): x ∈ X} < κ. Since rank(x) ≤ β for all x ∈ X, we have X ⊆ Vβ+1, so X ∈ Vβ+2 ⊆ Vκ. Therefore, X ∈ Vκ.
Proof of Theorem 2: |Vκ| = |∪α < κ Vα| ≤ sup {|Vα| : α < κ}. Let β be this supremum. Since each ordinal in the supremum is less than κ, we have β ≤ κ. Assume β < κ. Then there is a cardinal λ such that β < λ < κ; for example, let λ = 2|β|. Since λ ⊆ Vλ and |Vλ| is in the supremum, we have λ ≤ |Vλ| ≤ β. This contradicts β < λ. Therefore, |Vκ| = β = κ.
The sets and classes of Vκ satisfy all the axioms of NBG.[lower-alpha 14]
Limitation of size doctrine
The limitation of size doctrine is a heuristic principle that is used to justify axioms of set theory. It avoids the set theoretical paradoxes by restricting the full (contradictory) comprehension axiom schema:
$\forall w_{1},\ldots ,w_{n}\,\exists x\,\forall u\,(u\in x\iff \varphi (u,w_{1},\ldots ,w_{n}))$
to instances "that do not give sets 'too much bigger' than the ones they use."[15]
If "bigger" means "bigger in cardinal size," then most of the axioms can be justified: The axiom of separation produces a subset of x that is not bigger than x. The axiom of replacement produces an image set f(x) that is not bigger than x. The axiom of union produces a union whose size is not bigger than the size of the biggest set in the union times the number of sets in the union.[16] The axiom of choice produces a choice set whose size is not bigger than the size of the given set of nonempty sets.
The limitation of size doctrine does not justify the axiom of infinity:
$\exists y\,[\emptyset \in y\,\land \,\forall x(x\in y\implies x\cup \{x\}\in y)],$
which uses the empty set and sets obtained from the empty set by iterating the ordinal successor operation. Since these sets are finite, any set satisfying this axiom, such as ω, is much bigger than these sets. Fraenkel and Lévy regard the empty set and the infinite set of natural numbers, whose existence is implied by the axioms of infinity and separation, as the starting point for generating sets.[17]
Von Neumann's approach to limitation of size uses the axiom of limitation of size. As mentioned in § Implications of the axiom, von Neumann's axiom implies the axioms of separation, replacement, union, and choice. Like Fraenkel and Lévy, von Neumann had to add the axiom of infinity to his system since it cannot be proved from his other axioms.[lower-alpha 15] The differences between von Neumann's approach to limitation of size and Fraenkel and Lévy's approach are:
• Von Neumann's axiom puts limitation of size into an axiom system, making it possible to prove most set existence axioms. The limitation of size doctrine justifies axioms using informal arguments that are more open to disagreement than a proof.
• Von Neumann assumed the power set axiom since it cannot be proved from his other axioms.[lower-alpha 16] Fraenkel and Lévy state that the limitation of size doctrine justifies the power set axiom.[18]
There is disagreement on whether the limitation of size doctrine justifies the power set axiom. Michael Hallett has analyzed the arguments given by Fraenkel and Lévy. Some of their arguments measure size by criteria other than cardinal size—for example, Fraenkel introduces "comprehensiveness" and "extendability." Hallett points out what he considers to be flaws in their arguments.[19]
Hallett then argues that results in set theory seem to imply that there is no link between the size of an infinite set and the size of its power set. This would imply that the limitation of size doctrine is incapable of justifying the power set axiom because it requires that the power set of x is not "too much bigger" than x. For the case where size is measured by cardinal size, Hallett mentions Paul Cohen's work.[20] Starting with a model of ZFC and $\aleph _{\alpha }$, Cohen built a model in which the cardinality of the power set of ω is $\aleph _{\alpha }$ if the cofinality of $\aleph _{\alpha }$ is not ω; otherwise, its cardinality is $\aleph _{\alpha +1}$.[21] Since the cardinality of the power set of ω has no bound, there is no link between the cardinal size of ω and the cardinal size of P(ω).[22]
Hallett also discusses the case where size is measured by "comprehensiveness," which considers a collection "too big" if it is of "unbounded comprehension" or "unlimited extent."[23] He points out that for an infinite set, we cannot be sure that we have all its subsets without going through the unlimited extent of the universe. He also quotes John L. Bell and Moshé Machover: "... the power set P(u) of a given [infinite] set u is proportional not only to the size of u but also to the 'richness' of the entire universe ..."[24] After making these observations, Hallett states: "One is led to suspect that there is simply no link between the size (comprehensiveness) of an infinite a and the size of P(a)."[20]
Hallett considers the limitation of size doctrine valuable for justifying most of the axioms of set theory. His arguments only indicate that it cannot justify the axioms of infinity and power set.[25] He concludes that "von Neumann's explicit assumption [of the smallness of power-sets] seems preferable to Zermelo's, Fraenkel's, and Lévy's obscurely hidden implicit assumption of the smallness of power-sets."[6]
History
Von Neumann developed the axiom of limitation of size as a new method of identifying sets. ZFC identifies sets via its set building axioms. However, as Abraham Fraenkel pointed out: "The rather arbitrary character of the processes which are chosen in the axioms of Z [ZFC] as the basis of the theory, is justified by the historical development of set-theory rather than by logical arguments."[26]
The historical development of the ZFC axioms began in 1908 when Zermelo chose axioms to eliminate the paradoxes and to support his proof of the well-ordering theorem.[lower-alpha 17] In 1922, Abraham Fraenkel and Thoralf Skolem pointed out that Zermelo's axioms cannot prove the existence of the set {Z0, Z1, Z2, ...} where Z0 is the set of natural numbers, and Zn+1 is the power set of Zn.[27] They also introduced the axiom of replacement, which guarantees the existence of this set.[28] However, adding axioms as they are needed neither guarantees the existence of all reasonable sets nor clarifies the difference between sets that are safe to use and collections that lead to contradictions.
In a 1923 letter to Zermelo, von Neumann outlined an approach to set theory that identifies sets that are "too big" and might lead to contradictions.[lower-alpha 18] Von Neumann identified these sets using the criterion: "A set is 'too big' if and only if it is equivalent with the set of all things." He then restricted how these sets may be used: "... in order to avoid the paradoxes those [sets] which are 'too big' are declared to be impermissible as elements."[29] By combining this restriction with his criterion, von Neumann obtained his first version of the axiom of limitation of size, which in the language of classes states: A class is a proper class if and only if it is equinumerous with V.[2] By 1925, Von Neumann modified his axiom by changing "it is equinumerous with V " to "it can be mapped onto V ", which produces the axiom of limitation of size. This modification allowed von Neumann to give a simple proof of the axiom of replacement.[1] Von Neumann's axiom identifies sets as classes that cannot be mapped onto V. Von Neumann realized that, even with this axiom, his set theory does not fully characterize sets.[lower-alpha 19]
Gödel found von Neumann's axiom to be "of great interest":
"In particular I believe that his [von Neumann's] necessary and sufficient condition which a property must satisfy, in order to define a set, is of great interest, because it clarifies the relationship of axiomatic set theory to the paradoxes. That this condition really gets at the essence of things is seen from the fact that it implies the axiom of choice, which formerly stood quite apart from other existential principles. The inferences, bordering on the paradoxes, which are made possible by this way of looking at things, seem to me, not only very elegant, but also very interesting from the logical point of view.[lower-alpha 20] Moreover I believe that only by going farther in this direction, i.e., in the direction opposite to constructivism, will the basic problems of abstract set theory be solved."[30]
Notes
1. Proof: Let A be a class and X ∈ A. Then X is a set, so X ∈ V. Therefore, A ⊆ V.
2. Proof that uses von Neumann's axiom: Let A be a set and B be the subclass produced by the axiom of separation. Using proof by contradiction, assume B is a proper class. Then there is a function F mapping B onto V. Define the function G mapping A to V: if x ∈ B then G(x) = F(x); if x ∈ A \ B then G(x) = ∅. Since F maps A onto V, G maps A onto V. So the axiom of limitation of size implies that A is a proper class, which contradicts A being a set. Therefore, B is a set.
3. This can be rephrased as: NBG implies the axiom of limitation of size. In 1929, von Neumann proved that the axiom system that later evolved into NBG implies the axiom of limitation of size. (Ferreirós 2007, p. 380.)
4. An axiom's set variable is restricted on the right side of the "if and only if." Also, an axiom's class variables are converted to set variables. For example, the class existence axiom $\forall A\,\exists B\,\forall u\,[u\in B\Leftrightarrow u\notin A)]$ becomes $\forall a\,\exists b\,\forall u\,[u\in b\Leftrightarrow (u\in L_{\omega _{\omega }}\land u\notin a)].$ The class existence axioms are in Gödel 1940, p. 5.
5. Gödel defined a function $F$ that maps the class of ordinals onto $L$. The function ${F|}_{\omega _{\omega }}$ (which is the restriction of $F$ to $\omega _{\omega }$) maps $\omega _{\omega }$ onto $L_{\omega _{\omega }}$, and it belongs to $L_{\omega _{\omega +1}}$ because it is a constructible subset of $L_{\omega _{\omega }}$. Gödel uses the notation $F''\omega _{\alpha }$ for $L_{\omega _{\alpha }}$. (Gödel 1940, pp. 37–38, 54.)
6. Proof by contradiction that $\{\omega _{n}:n\in \omega \}$ is a proper class: Assume that it is a set. By the axiom of union, $\cup \,\{\omega _{n}:n\in \omega \}$ is a set. This union equals $\omega _{\omega }$, the model's proper class of all ordinals, which contradicts the union being a set. Therefore, $\{\omega _{n}:n\in \omega \}$ is a proper class.
Proof that $|L_{\omega _{\omega }}|=\aleph _{\omega }\!:$ The function ${F|}_{\omega _{\omega }}$ maps $\omega _{\omega }$ onto $L_{\omega _{\omega }}$, so $|L_{\omega _{\omega }}|\leq |\omega _{\omega }|.$ Also, $\omega _{\omega }\subseteq L_{\omega _{\omega }}$ implies $|\omega _{\omega }|\leq |L_{\omega _{\omega }}|.$ Therefore, $|L_{\omega _{\omega }}|=|\omega _{\omega }|=\aleph _{\omega }.$
7. This is the first half of theorem 7.7 in Gödel 1940, p. 27. Gödel defines the order isomorphism $F:(Ord,<)\rightarrow (A,<)$ by transfinite recursion: $F(\alpha )=Inf(A\setminus \{F(\beta ):\beta \in \alpha \}).$
8. This is the standard definition of V0. Zermelo let V0 be a set of urelements and proved that if this set contains a single element, the resulting model satisfies the axiom of limitation of size (his proof also works for V0 = ∅). Zermelo stated that the axiom is not true for all models built from a set of urelements. (Zermelo 1930, p. 38; English translation: Ewald 1996, p. 1227.)
9. This is Zermelo's definition (Zermelo 1930, p. 36; English translation: Ewald 1996, p. 1225.). If V0 = ∅, this definition is equivalent to the standard definition Vα+1 = P(Vα) since Vα ⊆ P(Vα) (Kunen 1980, p. 95; Kunen uses the notation R(α) instead of Vα). If V0 is a set of urelements, the standard definition eliminates the urelements at V1.
10. If X is a set, then there is a class Y such that X ∈ Y. Since Y ⊆ Vκ, we have X ∈ Vκ. Conversely: if X ∈ Vκ, then X belongs to a class, so X is a set.
11. Zermelo proved that Vω satisfies ZFC without the axiom of infinity. The class existence axioms of NBG (Gödel 1940, p. 5) are true because Vω is a set when viewed from the set theory that constructs it (namely, ZFC). Therefore, the axiom of separation produces subsets of Vω that satisfy the class existence axioms.
12. Zermelo introduced strongly inaccessible cardinals κ so that Vκ would satisfy ZFC. The axioms of power set and replacement led him to the properties of strongly inaccessible cardinals. (Zermelo 1930, pp. 31–35; English translation: Ewald 1996, pp. 1221–1224.) Independently, Wacław Sierpiński and Alfred Tarski introduced these cardinals in 1930. (Sierpiński & Tarski 1930.)
13. Zermelo used this sequence of cardinals to obtain a sequence of models that explains the paradoxes of set theory — such as, the Burali-Forti paradox and Russell's paradox. He stated that the paradoxes "depend solely on confusing set theory itself ... with individual models representing it. What appears as an 'ultrafinite non- or super-set' in one model is, in the succeeding model, a perfectly good, valid set with both a cardinal number and an ordinal type, and is itself a foundation stone for the construction of a new domain [model]." (Zermelo 1930, pp. 46–47; English translation: Ewald 1996, p. 1223.)
14. Zermelo proved that Vκ satisfies ZFC if κ is a strongly inaccessible cardinal. The class existence axioms of NBG (Gödel 1940, p. 5) are true because Vκ is a set when viewed from the set theory that constructs it (namely, ZFC + there exist infinitely many strongly inaccessible cardinals). Therefore, the axiom of separation produces subsets of Vκ that satisfy the class existence axioms.
15. The model whose sets are the elements of $V_{\omega }$ and whose classes are the subsets of $V_{\omega }$ satisfies all of his axioms except for the axiom of infinity, which fails because all sets are finite.
16. The model whose sets are the elements of $L_{\omega _{1}}$ and whose classes are the elements of $L_{\omega _{2}}$ satisfies all of his axioms except for the power set axiom. This axiom fails because all sets are countable.
17. "... we must, on the one hand, restrict these principles [axioms] sufficiently to exclude all contradictions and, on the other hand, take them sufficiently wide to retain all that is valuable in this theory." (Zermelo 1908, p. 261; English translation: van Heijenoort 1967a, p. 200). Gregory Moore argues that Zermelo's "axiomatization was primarily motivated by a desire to secure his demonstration of the Well-Ordering Theorem ..." (Moore 1982, pp. 158–160).
18. Von Neumann published an introductory article on his axiom system in 1925 (von Neumann 1925; English translation: van Heijenoort 1967c). In 1928, he provided a detailed treatment of his system (von Neumann 1928).
19. Von Neumann investigated whether his set theory is categorical; that is, whether it uniquely determines sets in the sense that any two of its models are isomorphic. He showed that it is not categorical because of a weakness in the axiom of regularity: this axiom only excludes descending ∈-sequences from existing in the model; descending sequences may still exist outside the model. A model having "external" descending sequences is not isomorphic to a model having no such sequences since this latter model lacks isomorphic images for the sets belonging to external descending sequences. This led von Neumann to conclude "that no categorical axiomatization of set theory seems to exist at all" (von Neumann 1925, p. 239; English translation: van Heijenoort 1967c, p. 412).
20. For example, von Neumann's proof that his axiom implies the well-ordering theorem uses the Burali-Forte paradox (von Neumann 1925, p. 223; English translation: van Heijenoort 1967c, p. 398).
References
1. von Neumann 1925, p. 223; English translation: van Heijenoort 1967c, pp. 397–398.
2. Hallett 1984, p. 290.
3. Bernays 1937, pp. 66–70; Bernays 1941, pp. 1–6. Gödel 1940, pp. 3–7. Kelley 1955, pp. 251–273.
4. Zermelo 1930; English translation: Ewald 1996.
5. Fraenkel, Bar-Hillel & Levy 1973, p. 137.
6. Hallett 1984, p. 295.
7. Gödel 1940, p. 3.
8. Levy 1968.
9. It came 43 years later: von Neumann stated his axioms in 1925 and Lévy's proof appeared in 1968. (von Neumann 1925, Levy 1968.)
10. Easton 1964, pp. 56a–64.
11. Gödel 1939, p. 223.
12. These theorems are part of Zermelo's Second Development Theorem. (Zermelo 1930, p. 37; English translation: Ewald 1996, p. 1226.)
13. von Neumann 1925, p. 223; English translation: van Heijenoort 1967c, p. 398. Von Neumann's proof, which only uses axioms, has the advantage of applying to all models rather than just to Vκ.
14. Kunen 1980, p. 95.
15. Fraenkel, Bar-Hillel & Levy 1973, pp. 32, 137.
16. Hallett 1984, p. 205.
17. Fraenkel, Bar-Hillel & Levy 1973, p. 95.
18. Hallett 1984, pp. 200, 202.
19. Hallett 1984, pp. 200–207.
20. Hallett 1984, pp. 206–207.
21. Cohen 1966, p. 134.
22. Hallett 1984, p. 207.
23. Hallett 1984, p. 200.
24. Bell & Machover 2007, p. 509.
25. Hallett 1984, pp. 209–210.
26. Historical Introduction in Bernays 1991, p. 31.
27. Fraenkel 1922, pp. 230–231. Skolem 1922; English translation: van Heijenoort 1967b, pp. 296–297).
28. Ferreirós 2007, p. 369. In 1917, Dmitry Mirimanoff published a form of replacement based on cardinal equivalence (Mirimanoff 1917, p. 49).
29. Hallett 1984, pp. 288, 290.
30. From a Nov. 8, 1957 letter Gödel wrote to Stanislaw Ulam (Kanamori 2003, p. 295).
Bibliography
• Bell, John L.; Machover, Moshé (2007), A Course in Mathematical Logic, Elsevier Science Ltd, ISBN 978-0-7204-2844-5.
• Bernays, Paul (1937), "A System of Axiomatic Set Theory—Part I", The Journal of Symbolic Logic, 2 (1): 65–77, doi:10.2307/2268862, JSTOR 2268862.
• Bernays, Paul (1941), "A System of Axiomatic Set Theory—Part II", The Journal of Symbolic Logic, 6 (1): 1–17, doi:10.2307/2267281, JSTOR 2267281, S2CID 250344277.
• Bernays, Paul (1991), Axiomatic Set Theory, Dover Publications, ISBN 0-486-66637-9.
• Cohen, Paul (1966), Set Theory and the Continuum Hypothesis, W. A. Benjamin, ISBN 978-0-486-46921-8.
• Easton, William B. (1964), Powers of Regular Cardinals (Ph.D. thesis), Princeton University.
• Ferreirós, José (2007), Labyrinth of Thought: A History of Set Theory and Its Role in Mathematical Thought (2nd revised ed.), Birkhäuser, ISBN 978-3-7643-8349-7.
• Fraenkel, Abraham (1922), "Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre", Mathematische Annalen, 86 (3–4): 230–237, doi:10.1007/bf01457986, S2CID 122212740.
• Fraenkel, Abraham; Bar-Hillel, Yehoshua; Levy, Azriel (1973), Foundations of Set Theory (2nd revised ed.), Basel, Switzerland: Elsevier, ISBN 0-7204-2270-1.
• Gödel, Kurt (1939), "Consistency Proof for the Generalized Continuum Hypothesis" (PDF), Proceedings of the National Academy of Sciences of the United States of America, 25 (4): 220–224, doi:10.1073/pnas.25.4.220, PMC 1077751, PMID 16588293.
• Gödel, Kurt (1940), The Consistency of the Continuum Hypothesis, Princeton University Press.
• Hallett, Michael (1984), Cantorian Set Theory and Limitation of Size, Oxford: Clarendon Press, ISBN 0-19-853179-6.
• Kanamori, Akihiro (2003), "Stanislaw Ulam" (PDF), in Solomon Feferman and John W. Dawson, Jr. (ed.), Kurt Gödel Collected Works, Volume V: Correspondence H-Z, Clarendon Press, pp. 280–300, ISBN 0-19-850075-0.
• Kelley, John L. (1955), General Topology, Van Nostrand, ISBN 978-0-387-90125-1.
• Kunen, Kenneth (1980), Set Theory: An Introduction to Independence Proofs, North-Holland, ISBN 0-444-86839-9.
• Levy, Azriel (1968), "On Von Neumann's Axiom System for Set Theory", American Mathematical Monthly, 75 (7): 762–763, doi:10.2307/2315201, JSTOR 2315201.
• Mirimanoff, Dmitry (1917), "Les antinomies de Russell et de Burali-Forti et le probleme fondamental de la theorie des ensembles", L'Enseignement Mathématique, 19: 37–52.
• Moore, Gregory H. (1982), Zermelo's Axiom of Choice: Its Origins, Development, and Influence, Springer, ISBN 0-387-90670-3.
• Sierpiński, Wacław; Tarski, Alfred (1930), "Sur une propriété caractéristique des nombres inaccessibles" (PDF), Fundamenta Mathematicae, 15: 292–300, doi:10.4064/fm-15-1-292-300, ISSN 0016-2736.
• Skolem, Thoralf (1922), "Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre", Matematikerkongressen i Helsingfors den 4-7 Juli, 1922, pp. 217–232.
• English translation: van Heijenoort, Jean (1967b), "Some remarks on axiomatized set theory", From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, pp. 290–301, ISBN 978-0-674-32449-7.
• von Neumann, John (1925), "Eine Axiomatisierung der Mengenlehre", Journal für die Reine und Angewandte Mathematik, 154: 219–240.
• English translation: van Heijenoort, Jean (1967c), "An axiomatization of set theory", From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, pp. 393–413, ISBN 978-0-674-32449-7.
• von Neumann, John (1928), "Die Axiomatisierung der Mengenlehre", Mathematische Zeitschrift, 27: 669–752, doi:10.1007/bf01171122, S2CID 123492324.
• Zermelo, Ernst (1908), "Untersuchungen über die Grundlagen der Mengenlehre", Mathematische Annalen, 65 (2): 261–281, doi:10.1007/bf01449999, S2CID 120085563.
• English translation: van Heijenoort, Jean (1967a), "Investigations in the foundations of set theory", From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, pp. 199–215, ISBN 978-0-674-32449-7.
• Zermelo, Ernst (1930), "Über Grenzzahlen und Mengenbereiche: neue Untersuchungen über die Grundlagen der Mengenlehre" (PDF), Fundamenta Mathematicae, 16: 29–47, doi:10.4064/fm-16-1-29-47.
• English translation: Ewald, William B. (1996), "On boundary numbers and domains of sets: new investigations in the foundations of set theory", From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, Oxford University Press, pp. 1208–1233, ISBN 978-0-19-853271-2.
Set theory
Overview
• Set (mathematics)
Axioms
• Adjunction
• Choice
• countable
• dependent
• global
• Constructibility (V=L)
• Determinacy
• Extensionality
• Infinity
• Limitation of size
• Pairing
• Power set
• Regularity
• Union
• Martin's axiom
• Axiom schema
• replacement
• specification
Operations
• Cartesian product
• Complement (i.e. set difference)
• De Morgan's laws
• Disjoint union
• Identities
• Intersection
• Power set
• Symmetric difference
• Union
• Concepts
• Methods
• Almost
• Cardinality
• Cardinal number (large)
• Class
• Constructible universe
• Continuum hypothesis
• Diagonal argument
• Element
• ordered pair
• tuple
• Family
• Forcing
• One-to-one correspondence
• Ordinal number
• Set-builder notation
• Transfinite induction
• Venn diagram
Set types
• Amorphous
• Countable
• Empty
• Finite (hereditarily)
• Filter
• base
• subbase
• Ultrafilter
• Fuzzy
• Infinite (Dedekind-infinite)
• Recursive
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• Subset · Superset
• Transitive
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Theories
• Alternative
• Axiomatic
• Naive
• Cantor's theorem
• Zermelo
• General
• Principia Mathematica
• New Foundations
• Zermelo–Fraenkel
• von Neumann–Bernays–Gödel
• Morse–Kelley
• Kripke–Platek
• Tarski–Grothendieck
• Paradoxes
• Problems
• Russell's paradox
• Suslin's problem
• Burali-Forti paradox
Set theorists
• Paul Bernays
• Georg Cantor
• Paul Cohen
• Richard Dedekind
• Abraham Fraenkel
• Kurt Gödel
• Thomas Jech
• John von Neumann
• Willard Quine
• Bertrand Russell
• Thoralf Skolem
• Ernst Zermelo
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Zermelo's navigation problem
In mathematical optimization, Zermelo's navigation problem, proposed in 1931 by Ernst Zermelo, is a classic optimal control problem that deals with a boat navigating on a body of water, originating from a point $A$ to a destination point $B$. The boat is capable of a certain maximum speed, and the goal is to derive the best possible control to reach $B$ in the least possible time.
Without considering external forces such as current and wind, the optimal control is for the boat to always head towards $B$. Its path then is a line segment from $A$ to $B$, which is trivially optimal. With consideration of current and wind, if the combined force applied to the boat is non-zero the control for no current and wind does not yield the optimal path.
History
In his 1931 article,[1] Ernst Zermelo formulates the following problem:
In an unbounded plane where the wind distribution is given by a vector field as a function of position and time, a ship moves with constant velocity relative to the surrounding air mass. How must the ship be steered in order to come from a starting point to a given goal in the shortest time?
This is an extension of the classical optimisation problem for geodesics – minimising the length of a curve $I[c]=\int _{a}^{b}{\sqrt {1+y'^{2}}}\,dx$ connecting points $A$ and $B$ , with the added complexity of considering some wind velocity. Although it is usually impossible to find an exact solution in most cases, the general case was solved by Zermelo himself in the form of a partial differential equation, known as Zermelo's equation, which can be numerically solved.
The problem of navigating an airship which is surrounded by air, was presented first in 1929 at a conference by Ernst Zermelo. Other mathematicians have answered the challenge over the following years. The dominant technique for solving the equations is the calculus of variations.[2]
Constant-wind case
The case of constant wind is easy to solve exactly.[3] Let $\mathbf {d} ={\vec {AB}}$, and suppose that to minimise the travel time the ship travels at a constant maximum speed $V$. Thus the position of the ship at time $t$ is $\mathbf {x} =t(\mathbf {v} +\mathbf {w} )$. Let $T$ be the time of arrival at $B$, so that $\mathbf {d} =T(\mathbf {v} +\mathbf {w} )$. Taking the dot product of this with $\mathbf {w} $ and $\mathbf {d} $ respectively results in ${\vec {d}}\cdot {\vec {w}}=T(\mathbf {v} \cdot {\vec {w}}+\mathbf {w} ^{2})$ and $d^{2}=T^{2}(v^{2}+2{\vec {v}}\cdot \mathbf {w} +\mathbf {w} ^{2})$. Eliminating ${\vec {v}}\cdot {\vec {w}}$ and writing this system as a quadratic in $T$ results in $({\vec {v}}^{2}-{\vec {w}}^{2})T^{2}+2(\mathbf {d} \cdot \mathbf {w} )T-\mathbf {d} ^{2}=0$. Upon solving this, taking the positive square-root since $T$ is positive, we obtain
${\begin{aligned}T[\mathbf {d} ]&={\frac {-2(\mathbf {d} \cdot \mathbf {w} )\pm {\sqrt {4(\mathbf {d} \cdot \mathbf {w} )^{2}+4\mathbf {d} ^{2}(\mathbf {v} ^{2}-\mathbf {w} ^{2})}}}{2(\mathbf {v} ^{2}-\mathbf {w} ^{2})}}\\[8pt]&={\sqrt {{\frac {\mathbf {d} ^{2}}{\mathbf {v} ^{2}-{\vec {w}}^{2}}}+{\frac {(\mathbf {d} \cdot \mathbf {w} )^{2}}{({\vec {v}}^{2}-{\vec {w}}^{2})^{2}}}}}-{\frac {\mathbf {d} \cdot \mathbf {w} }{\mathbf {v} ^{2}-\mathbf {w} ^{2}}}\end{aligned}}$
Claim: This defines a metric on $\mathbb {R} ^{2}$, provided $|\mathbf {v} |>|\mathbf {w} |$.
Proof
By our assumption, clearly $T[\mathbf {d} ]\geq 0$ with equality if and only if $\mathbf {d} =0$. Trivially if ${\tilde {\mathbf {d} }}={\vec {BA}}$, we have $T[\mathbf {d} ]=T[{\tilde {\mathbf {d} }}]$. It remains to show $T$ satisfies a triangle inequality $T[\mathbf {d} _{1}+\mathbf {d} _{2}]\leq T[\mathbf {d} _{1}]+T[\mathbf {d} _{2}].$
Indeed, letting $c^{2}:=\mathbf {v} ^{2}-\mathbf {w} ^{2}$, we note that this is true if and only if
${\begin{aligned}&{\sqrt {{\frac {(\mathbf {d} _{1}+\mathbf {d} _{2})^{2}}{c^{2}}}+{\frac {(({\vec {d}}_{1}+{\vec {d}}_{2})\cdot {\vec {w}})^{2}}{c^{4}}}}}-{\frac {(\mathbf {d} _{1}+\mathbf {d} _{2})\cdot \mathbf {w} }{c^{2}}}\\[8pt]\leq {}&{\sqrt {{\frac {\mathbf {d} _{1}^{2}}{c^{2}}}+{\frac {(\mathbf {d} _{1}\cdot \mathbf {w} )^{2}}{c^{4}}}-{\frac {\mathbf {d} _{2}\cdot \mathbf {w} }{c^{2}}}}}+{\sqrt {{\frac {\mathbf {d} _{2}^{2}}{c^{2}}}+{\frac {(\mathbf {d} _{2}\cdot \mathbf {w} )^{2}}{c^{4}}}}}-{\frac {\mathbf {d} _{2}\cdot \mathbf {w} }{c^{2}}}\end{aligned}}$
if and only if
${\frac {\mathbf {d} _{1}\cdot \mathbf {d} _{2}}{c^{2}}}+{\frac {(\mathbf {d} _{1}\cdot \mathbf {w} )(\mathbf {d} _{2}\cdot \mathbf {w} )}{c^{4}}}\leq \left[{\frac {{\vec {d}}_{1}^{2}}{c^{2}}}+{\frac {(\mathbf {d} _{1}\cdot \mathbf {w} )^{2}}{c^{4}}}\right]^{1/2}\left[{\frac {{\vec {d}}_{2}^{2}}{c^{2}}}+{\frac {(\mathbf {d} _{2}\cdot \mathbf {w} )^{2}}{c^{4}}}\right]^{1/2},$
which is true if and only if
${\frac {(\mathbf {d} _{1}\cdot \mathbf {d} _{2})^{2}}{c^{4}}}+{\frac {2(\mathbf {d} _{1}\cdot \mathbf {d} _{2})(\mathbf {d} _{1}\cdot \mathbf {w} )(\mathbf {d} _{2}\cdot \mathbf {w} )}{c^{6}}}\leq {\frac {\mathbf {d} _{1}^{2}\cdot \mathbf {d} _{2}^{2}}{c^{4}}}+{\frac {\mathbf {d} _{1}^{2}(\mathbf {d} _{2}\cdot \mathbf {w} )^{2}+\mathbf {d} _{2}^{2}(\mathbf {d} _{1}\cdot \mathbf {w} )^{2}}{c^{6}}}$
Using the Cauchy–Schwarz inequality, we obtain $(\mathbf {d} _{1}\cdot \mathbf {d} _{2})^{2}\leq \mathbf {d} _{1}^{2}\cdot \mathbf {d} _{2}^{2}$ with equality if and only if $\mathbf {d} _{1}$ and $\mathbf {d} _{2}$ are linearly dependent, and so the inequality is indeed true. $\blacksquare $
Note: Since this is a strict inequality if $\mathbf {d} _{1}$ and $\mathbf {d} _{2}$ are not linearly dependent, it immediately follows that a straight line from $A$ to $B$ is always a faster path than any other path made up of straight line segments. We use a limiting argument to prove this is true for any curve.
General solution
Consider the general example of a ship moving against a variable wind ${\vec {w}}(x,y)$. Writing this component-wise, we have the drift in the $x$-axis as $u(x,y)$ and the drift in the $y$-axis as $v(x,y)$. Then for a ship moving at maximum velocity $V$ at variable heading $\theta $, we have
${\begin{aligned}{\dot {x}}&=V\cos \theta +u(x,y)\\{\dot {y}}&=V\sin \theta +v(x,y)\end{aligned}}$
The Hamiltonian of the system is thus
$H=\lambda _{x}(V\cos \theta +u)+\lambda _{y}(V\sin \theta +v)+1$
Using the Euler–Lagrange equation, we obtain
${\begin{aligned}{\dot {\lambda }}_{x}&=-{\frac {\partial H}{\partial x}}=-\lambda _{x}{\frac {\partial u}{\partial x}}-\lambda _{y}{\frac {\partial v}{\partial x}}\\{\dot {\lambda }}_{y}&=-{\frac {\partial H}{\partial y}}=-\lambda _{x}{\frac {\partial u}{\partial y}}-\lambda _{y}{\frac {\partial v}{\partial y}}\\0&={\frac {\partial H}{\partial \theta }}=V(-\lambda _{x}\sin \theta +\lambda _{y}\cos \theta )\end{aligned}}$
The last equation implies that $\tan \theta =\lambda _{y}/\lambda _{x}$. We note that the system is autonomous; the Hamiltonian does not depend on time $t$, thus $H$ = constant, but since we are minimising time, the constant is equal to 0. Thus we can solve the simultaneous equations above to get[4]
${\begin{aligned}\lambda _{x}&={\frac {-\cos \theta }{V+u\cos \theta +v\sin \theta }}\\[6pt]\lambda _{y}&={\frac {-\sin \theta }{V+u\cos \theta +v\sin \theta }}\end{aligned}}$
Substituting these values into our EL-equations results in the differential equation
${\frac {d\theta }{dt}}=\sin ^{2}\theta {\frac {\partial v}{\partial x}}+\sin \theta \cos \theta \left({\frac {\partial u}{\partial x}}-{\frac {\partial v}{\partial y}}\right)-\cos ^{2}\theta {\frac {\partial u}{\partial y}}$
This result is known as Zermelo's equation. Solving this with our system allows us to find the general optimum path.
Constant-wind revisited example
If we go back to the constant wind problem $\mathbf {w} $ for all time, we have
${\frac {\partial v}{\partial y}}={\frac {\partial v}{\partial x}}={\frac {\partial u}{\partial x}}={\frac {\partial u}{\partial y}}=0$
so our general solution implies ${\frac {d\theta }{dt}}=0$, thus $\theta $ is constant, i.e. the optimum path is a straight line, as we had obtained before with an algebraic argument.
References
1. Zermelo, Ernst (1931). "Über das Navigationsproblem bei ruhender oder veränderlicher Windverteilung". Zeitschrift für Angewandte Mathematik und Mechanik. 11 (2): 114–124. Bibcode:1931ZaMM...11..114Z. doi:10.1002/zamm.19310110205.
2. Heinz-Dieter Ebbinghaus (2 June 2007). Ernst Zermelo: An Approach to His Life and Work. Springer Science & Business Media. pp. 150–. ISBN 978-3-540-49553-6.
3. Warnick, Claude (2011). "The geometry of sound rays in a wind". Contemporary Physics. 52 (3): 197–209. arXiv:1102.2409. Bibcode:2011ConPh..52..197G. doi:10.1080/00107514.2011.563515. S2CID 119728138.
4. Bryson, A.E. (1975). Applied Optimal Control: Optimization, Estimation and Control. Taylor & Francis. ISBN 9780891162285.
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Natural number
In mathematics, the natural numbers are the numbers 1, 2, 3, etc., possibly including 0 as well. Some definitions, including the standard ISO 80000-2,[1] begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, ..., whereas others start with 1, corresponding to the positive integers 1, 2, 3, ...[2][lower-alpha 1] Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers).[4] In common language, particularly in primary school education, natural numbers may be called counting numbers[5] to intuitively exclude the negative integers and zero, and also to contrast the discreteness of counting to the continuity of measurement—a hallmark characteristic of real numbers.
This article is about "positive integers" and "non-negative integers". For all the numbers ..., −2, −1, 0, 1, 2, ..., see Integer.
The natural numbers can be used for counting (as in "there are six coins on the table"), in which case they serve as cardinal numbers. They may also be used for ordering (as in "this is the third largest city in the country"), in which case they serve as ordinal numbers. Natural numbers are sometimes used as labels, known as nominal numbers, having none of the properties of numbers in a mathematical sense (e.g. sports jersey numbers).[3][6]
The natural numbers form a set, often symbolized as $ \mathbb {N} $. Many other number sets are built by successively extending the set of natural numbers: the integers, by including an additive identity 0 (if not yet in) and an additive inverse −n for each nonzero natural number n; the rational numbers, by including a multiplicative inverse $1/n$ for each nonzero integer n (and also the product of these inverses by integers); the real numbers by including the limits of Cauchy sequences[lower-alpha 2] of rationals; the complex numbers, by adjoining to the real numbers a square root of −1 (and also the sums and products thereof); and so on.[lower-alpha 3][lower-alpha 4] This chain of extensions canonically embeds the natural numbers in the other number systems.
Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics.
History
Ancient roots
The most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage—by striking out a mark and removing an object from the set.
The first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers. The ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, and all powers of 10 up to over 1 million. A stone carving from Karnak, dating back from around 1500 BCE and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622. The Babylonians had a place-value system based essentially on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one—its value being determined from context.[10]
A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation (within other numbers) dates back as early as 700 BCE by the Babylonians, who omitted such a digit when it would have been the last symbol in the number.[lower-alpha 5] The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BCE, but this usage did not spread beyond Mesoamerica.[12][13] The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628 CE. However, 0 had been used as a number in the medieval computus (the calculation of the date of Easter), beginning with Dionysius Exiguus in 525 CE, without being denoted by a numeral. Standard Roman numerals do not have a symbol for 0; instead, nulla (or the genitive form nullae) from nullus, the Latin word for "none", was employed to denote a 0 value.[14]
The first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes even not as a number at all.[lower-alpha 6] Euclid, for example, defined a unit first and then a number as a multitude of units, thus by his definition, a unit is not a number and there are no unique numbers (e.g., any two units from indefinitely many units is a 2).[16]
Independent studies on numbers also occurred at around the same time in India, China, and Mesoamerica.[17]
Modern definitions
In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. Henri Poincaré stated that axioms can only be demonstrated in their finite application, and concluded that it is "the power of the mind" which allows conceiving of the indefinite repetition of the same act.[18] Leopold Kronecker summarized his belief as "God made the integers, all else is the work of man".[lower-alpha 7]
The constructivists saw a need to improve upon the logical rigor in the foundations of mathematics.[lower-alpha 8] In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers, thus stating they were not really natural—but a consequence of definitions. Later, two classes of such formal definitions were constructed; later still, they were shown to be equivalent in most practical applications.
Set-theoretical definitions of natural numbers were initiated by Frege. He initially defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set. However, this definition turned out to lead to paradoxes, including Russell's paradox. To avoid such paradoxes, the formalism was modified so that a natural number is defined as a particular set, and any set that can be put into one-to-one correspondence with that set is said to have that number of elements.[21]
The second class of definitions was introduced by Charles Sanders Peirce, refined by Richard Dedekind, and further explored by Giuseppe Peano; this approach is now called Peano arithmetic. It is based on an axiomatization of the properties of ordinal numbers: each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several weak systems of set theory. One such system is ZFC with the axiom of infinity replaced by its negation. Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include Goodstein's theorem.[22]
With all these definitions, it is convenient to include 0 (corresponding to the empty set) as a natural number. Including 0 is now the common convention among set theorists[23] and logicians.[24] Other mathematicians also include 0,[lower-alpha 9] and computer languages often start from zero when enumerating items like loop counters and string- or array-elements.[25][26] On the other hand, many mathematicians have kept the older tradition to take 1 to be the first natural number.[27]
Notation
The set of all natural numbers is standardly denoted N or $\mathbb {N} .$[3][28] Older texts have occasionally employed J as the symbol for this set.[29]
Since natural numbers may contain 0 or not, it may be important to know which version is referred to. This is often specified by the context, but may also be done by using a subscript or a superscript in the notation, such as:[1][30]
• Naturals without zero: $\{1,2,...\}=\mathbb {N} ^{*}=\mathbb {N} ^{+}=\mathbb {N} _{0}\smallsetminus \{0\}=\mathbb {N} _{1}$
• Naturals with zero: $\;\{0,1,2,...\}=\mathbb {N} _{0}=\mathbb {N} ^{0}=\mathbb {N} ^{*}\cup \{0\}$
Alternatively, since the natural numbers naturally form a subset of the integers (often denoted $\mathbb {Z} $), they may be referred to as the positive, or the non-negative integers, respectively.[31] To be unambiguous about whether 0 is included or not, sometimes a subscript (or superscript) "0" is added in the former case, and a superscript "$*$" or "+" is added in the latter case:[1]
$\{1,2,3,\dots \}=\{x\in \mathbb {Z} :x>0\}=\mathbb {Z} ^{+}=\mathbb {Z} _{>0}$
$\{0,1,2,\dots \}=\{x\in \mathbb {Z} :x\geq 0\}=\mathbb {Z} _{0}^{+}=\mathbb {Z} _{\geq 0}$
Properties
Addition
Given the set $\mathbb {N} $ of natural numbers and the successor function $S\colon \mathbb {N} \to \mathbb {N} $ sending each natural number to the next one, one can define addition of natural numbers recursively by setting a + 0 = a and a + S(b) = S(a + b) for all a, b. Then $(\mathbb {N} ,+)$ is a commutative monoid with identity element 0. It is a free monoid on one generator. This commutative monoid satisfies the cancellation property, so it can be embedded in a group. The smallest group containing the natural numbers is the integers.
If 1 is defined as S(0), then b + 1 = b + S(0) = S(b + 0) = S(b). That is, b + 1 is simply the successor of b.
Multiplication
Analogously, given that addition has been defined, a multiplication operator $\times $ can be defined via a × 0 = 0 and a × S(b) = (a × b) + a. This turns $(\mathbb {N} ^{*},\times )$ into a free commutative monoid with identity element 1; a generator set for this monoid is the set of prime numbers.
Relationship between addition and multiplication
Addition and multiplication are compatible, which is expressed in the distribution law: a × (b + c) = (a × b) + (a × c). These properties of addition and multiplication make the natural numbers an instance of a commutative semiring. Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative. The lack of additive inverses, which is equivalent to the fact that $\mathbb {N} $ is not closed under subtraction (that is, subtracting one natural from another does not always result in another natural), means that $\mathbb {N} $ is not a ring; instead it is a semiring (also known as a rig).
If the natural numbers are taken as "excluding 0", and "starting at 1", the definitions of + and × are as above, except that they begin with a + 1 = S(a) and a × 1 = a. Furthermore, $(\mathbb {N^{*}} ,+)$ has no identity element.
Order
In this section, juxtaposed variables such as ab indicate the product a × b,[32] and the standard order of operations is assumed.
A total order on the natural numbers is defined by letting a ≤ b if and only if there exists another natural number c where a + c = b. This order is compatible with the arithmetical operations in the following sense: if a, b and c are natural numbers and a ≤ b, then a + c ≤ b + c and ac ≤ bc.
An important property of the natural numbers is that they are well-ordered: every non-empty set of natural numbers has a least element. The rank among well-ordered sets is expressed by an ordinal number; for the natural numbers, this is denoted as ω (omega).
Division
In this section, juxtaposed variables such as ab indicate the product a × b, and the standard order of operations is assumed.
While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure of division with remainder or Euclidean division is available as a substitute: for any two natural numbers a and b with b ≠ 0 there are natural numbers q and r such that
$a=bq+r{\text{ and }}r<b.$
The number q is called the quotient and r is called the remainder of the division of a by b. The numbers q and r are uniquely determined by a and b. This Euclidean division is key to the several other properties (divisibility), algorithms (such as the Euclidean algorithm), and ideas in number theory.
Algebraic properties satisfied by the natural numbers
The addition (+) and multiplication (×) operations on natural numbers as defined above have several algebraic properties:
• Closure under addition and multiplication: for all natural numbers a and b, both a + b and a × b are natural numbers.[33]
• Associativity: for all natural numbers a, b, and c, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c.[34]
• Commutativity: for all natural numbers a and b, a + b = b + a and a × b = b × a.[35]
• Existence of identity elements: for every natural number a, a + 0 = a and a × 1 = a.
• If the natural numbers are taken as "excluding 0", and "starting at 1", then for every natural number a, a × 1 = a. However, the "existence of additive identity element" property is not satisfied
• Distributivity of multiplication over addition for all natural numbers a, b, and c, a × (b + c) = (a × b) + (a × c).
• No nonzero zero divisors: if a and b are natural numbers such that a × b = 0, then a = 0 or b = 0 (or both).
• If the natural numbers are taken as "excluding 0", and "starting at 1", the "no nonzero zero divisors" property is not satisfied.
Generalizations
Two important generalizations of natural numbers arise from the two uses of counting and ordering: cardinal numbers and ordinal numbers.
• A natural number can be used to express the size of a finite set; more precisely, a cardinal number is a measure for the size of a set, which is even suitable for infinite sets. This concept of "size" relies on maps between sets, such that two sets have the same size, exactly if there exists a bijection between them. The set of natural numbers itself, and any bijective image of it, is said to be countably infinite and to have cardinality aleph-null (ℵ0).
• Natural numbers are also used as linguistic ordinal numbers: "first", "second", "third", and so forth. This way they can be assigned to the elements of a totally ordered finite set, and also to the elements of any well-ordered countably infinite set. This assignment can be generalized to general well-orderings with a cardinality beyond countability, to yield the ordinal numbers. An ordinal number may also be used to describe the notion of "size" for a well-ordered set, in a sense different from cardinality: if there is an order isomorphism (more than a bijection!) between two well-ordered sets, they have the same ordinal number. The first ordinal number that is not a natural number is expressed as ω; this is also the ordinal number of the set of natural numbers itself.
The least ordinal of cardinality ℵ0 (that is, the initial ordinal of ℵ0) is ω but many well-ordered sets with cardinal number ℵ0 have an ordinal number greater than ω.
For finite well-ordered sets, there is a one-to-one correspondence between ordinal and cardinal numbers; therefore they can both be expressed by the same natural number, the number of elements of the set. This number can also be used to describe the position of an element in a larger finite, or an infinite, sequence.
A countable non-standard model of arithmetic satisfying the Peano Arithmetic (that is, the first-order Peano axioms) was developed by Skolem in 1933. The hypernatural numbers are an uncountable model that can be constructed from the ordinary natural numbers via the ultrapower construction.
Georges Reeb used to claim provocatively that "The naïve integers don't fill up" $\mathbb {N} $. Other generalizations are discussed in the article on numbers.
Formal definitions
There are two standard methods for formally defining natural numbers. The first one, named for Giuseppe Peano, consists of an autonomous axiomatic theory called Peano arithmetic, based on few axioms called Peano axioms.
The second definition is based on set theory. It defines the natural numbers as specific sets. More precisely, each natural number n is defined as an explicitly defined set, whose elements allow counting the elements of other sets, in the sense that the sentence "a set S has n elements" means that there exists a one to one correspondence between the two sets n and S.
The sets used to define natural numbers satisfy Peano axioms. It follows that every theorem that can be stated and proved in Peano arithmetic can also be proved in set theory. However, the two definitions are not equivalent, as there are theorems that can be stated in terms of Peano arithmetic and proved in set theory, which are not provable inside Peano arithmetic. A probable example is Fermat's Last Theorem.
The definition of the integers as sets satisfying Peano axioms provide a model of Peano arithmetic inside set theory. An important consequence is that, if set theory is consistent (as it is usually guessed), then Peano arithmetic is consistent. In other words, if a contradiction could be proved in Peano arithmetic, then set theory would be contradictory, and every theorem of set theory would be both true and wrong.
Peano axioms
Main article: Peano axioms
The five Peano axioms are the following:[36][lower-alpha 10]
1. 0 is a natural number.
2. Every natural number has a successor which is also a natural number.
3. 0 is not the successor of any natural number.
4. If the successor of $x$ equals the successor of $y$, then $x$ equals $y$.
5. The axiom of induction: If a statement is true of 0, and if the truth of that statement for a number implies its truth for the successor of that number, then the statement is true for every natural number.
These are not the original axioms published by Peano, but are named in his honor. Some forms of the Peano axioms have 1 in place of 0. In ordinary arithmetic, the successor of $x$ is $x+1$.
Set-theoretic definition
Main article: Set-theoretic definition of natural numbers
Intuitively, the natural number n is the common property of all sets that have n elements. So, it seems natural to define n as an equivalence class under the relation "can be made in one to one correspondence". Unfortunately, this does not work in set theory, as such an equivalence class would not be a set (because of Russell's paradox). The standard solution is to define a particular set with n elements that will be called the natural number n.
The following definition was first published by John von Neumann,[37] although Levy attributes the idea to unpublished work of Zermelo in 1916.[38] As this definition extends to infinite set as a definition of ordinal number, the sets considered below are sometimes called von Neumann ordinals.
The definition proceeds as follows:
• Call 0 = { }, the empty set.
• Define the successor S(a) of any set a by S(a) = a ∪ {a}.
• By the axiom of infinity, there exist sets which contain 0 and are closed under the successor function. Such sets are said to be inductive. The intersection of all inductive sets is still an inductive set.
• This intersection is the set of the natural numbers.
It follows that the natural numbers are defined iteratively as follows:
• 0 = { },
• 1 = 0 ∪ {0} = {0} = {{ }},
• 2 = 1 ∪ {1} = {0, 1} = {{ }, {{ }}},
• 3 = 2 ∪ {2} = {0, 1, 2} = {{ }, {{ }}, {{ }, {{ }}}},
• n = n−1 ∪ {n−1} = {0, 1, ..., n−1} = {{ }, {{ }}, ..., {{ }, {{ }}, ...}},
• etc.
It can be checked that the natural numbers satisfies the Peano axioms.
With this definition, given a natural number n, the sentence "a set S has n elements" can be formally defined as "there exists a bijection from n to S. This formalizes the operation of counting the elements of S. Also, n ≤ m if and only if n is a subset of m. In other words, the set inclusion defines the usual total order on the natural numbers. This order is a well-order.
It follows from the definition that each natural number is equal to the set of all natural numbers less than it. This definition, can be extended to the von Neumann definition of ordinals for defining all ordinal numbers, including the infinite ones: "each ordinal is the well-ordered set of all smaller ordinals."
If one does not accept the axiom of infinity, the natural numbers may not form a set. Nevertheless, the natural numbers can still be individually defined as above, and they still satisfy the Peano axioms.
There are other set theoretical constructions. In particular, Ernst Zermelo provided a construction that is nowadays only of historical interest, and is sometimes referred to as Zermelo ordinals.[38] It consists in defining 0 as the empty set, and S(a) = {a}.
With this definition each natural number is a singleton set. So, the property of the natural numbers to represent cardinalities is not directly accessible; only the ordinal property (being the nth element of a sequence) is immediate. Unlike von Neumann's construction, the Zermelo ordinals do not extend to infinite ordinals.
See also
• Canonical representation of a positive integer – Representation of a number as a product of primes
• Countable set – Mathematical set that can be enumerated
• Sequence – Function of the natural numbers in another set
• Ordinal number – Generalization of "n-th" to infinite cases
• Cardinal number – Size of a possibly infinite set
• Set-theoretic definition of natural numbers – constructions of the whole numbers from setsPages displaying wikidata descriptions as a fallback
Number systems
Complex $:\;\mathbb {C} $ :\;\mathbb {C} }
Real $:\;\mathbb {R} $ :\;\mathbb {R} }
Rational $:\;\mathbb {Q} $ :\;\mathbb {Q} }
Integer $:\;\mathbb {Z} $ :\;\mathbb {Z} }
Natural $:\;\mathbb {N} $ :\;\mathbb {N} }
Zero: 0
One: 1
Prime numbers
Composite numbers
Negative integers
Fraction
Finite decimal
Dyadic (finite binary)
Repeating decimal
Irrational
Algebraic irrational
Transcendental
Imaginary
Notes
1. Carothers (2000, p. 3) says, "$\mathbb {N} $ is the set of natural numbers (positive integers)." Both definitions are acknowledged whenever convenient, and there is no general consensus on whether zero should be included in the natural numbers.[3]
2. Any Cauchy sequence in the Reals converges,
3. Mendelson (2008, p. x) says: "The whole fantastic hierarchy of number systems is built up by purely set-theoretic means from a few simple assumptions about natural numbers."
4. Bluman (2010, p. 1): "Numbers make up the foundation of mathematics."
5. A tablet found at Kish ... thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place.[11]
6. This convention is used, for example, in Euclid's Elements, see D. Joyce's web edition of Book VII.[15]
7. The English translation is from Gray. In a footnote, Gray attributes the German quote to: "Weber 1891–1892, 19, quoting from a lecture of Kronecker's of 1886."[19][20]
8. "Much of the mathematical work of the twentieth century has been devoted to examining the logical foundations and structure of the subject." (Eves 1990, p. 606)
9. Mac Lane & Birkhoff (1999, p. 15) include zero in the natural numbers: 'Intuitively, the set $\mathbb {N} =\{0,1,2,\ldots \}$ of all natural numbers may be described as follows: $\mathbb {N} $ contains an "initial" number 0; ...'. They follow that with their version of the Peano's axioms.
10. Hamilton (1988, pp. 117 ff) calls them "Peano's Postulates" and begins with "1. 0 is a natural number."
Halmos (1960, p. 46) uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with "(I) 0 ∈ ω (where, of course, 0 = ∅" (ω is the set of all natural numbers).
Morash (1991) gives "a two-part axiom" in which the natural numbers begin with 1. (Section 10.1: An Axiomatization for the System of Positive Integers)
References
1. "Standard number sets and intervals" (PDF). ISO 80000-2:2019. International Organization for Standardization. 19 May 2020. p. 4.
2. "natural number". Merriam-Webster.com. Merriam-Webster. Archived from the original on 13 December 2019. Retrieved 4 October 2014.
3. Weisstein, Eric W. "Natural Number". mathworld.wolfram.com. Retrieved 11 August 2020.
4. Ganssle, Jack G. & Barr, Michael (2003). "integer". Embedded Systems Dictionary. Taylor & Francis. pp. 138 (integer), 247 (signed integer), & 276 (unsigned integer). ISBN 978-1-57820-120-4. Archived from the original on 29 March 2017. Retrieved 28 March 2017 – via Google Books.
5. Weisstein, Eric W. "Counting Number". MathWorld.
6. "Natural Numbers". Brilliant Math & Science Wiki. Retrieved 11 August 2020.
7. "Introduction". Ishango bone. Brussels, Belgium: Royal Belgian Institute of Natural Sciences. Archived from the original on 4 March 2016.
8. "Flash presentation". Ishango bone. Brussels, Belgium: Royal Belgian Institute of Natural Sciences. Archived from the original on 27 May 2016.
9. "The Ishango Bone, Democratic Republic of the Congo". UNESCO's Portal to the Heritage of Astronomy. Archived from the original on 10 November 2014., on permanent display at the Royal Belgian Institute of Natural Sciences, Brussels, Belgium.
10. Ifrah, Georges (2000). The Universal History of Numbers. Wiley. ISBN 0-471-37568-3.
11. "A history of Zero". MacTutor History of Mathematics. Archived from the original on 19 January 2013. Retrieved 23 January 2013.
12. Mann, Charles C. (2005). 1491: New Revelations of the Americas before Columbus. Knopf. p. 19. ISBN 978-1-4000-4006-3. Archived from the original on 14 May 2015. Retrieved 3 February 2015 – via Google Books.
13. Evans, Brian (2014). "Chapter 10. Pre-Columbian Mathematics: The Olmec, Maya, and Inca Civilizations". The Development of Mathematics Throughout the Centuries: A brief history in a cultural context. John Wiley & Sons. ISBN 978-1-118-85397-9 – via Google Books.
14. Deckers, Michael (25 August 2003). "Cyclus Decemnovennalis Dionysii – Nineteen year cycle of Dionysius". Hbar.phys.msu.ru. Archived from the original on 15 January 2019. Retrieved 13 February 2012.
15. Euclid. "Book VII, definitions 1 and 2". In Joyce, D. (ed.). Elements. Clark University. Archived from the original on 5 August 2011.
16. Mueller, Ian (2006). Philosophy of mathematics and deductive structure in Euclid's Elements. Mineola, New York: Dover Publications. p. 58. ISBN 978-0-486-45300-2. OCLC 69792712.
17. Kline, Morris (1990) [1972]. Mathematical Thought from Ancient to Modern Times. Oxford University Press. ISBN 0-19-506135-7.
18. Poincaré, Henri (1905) [1902]. "On the nature of mathematical reasoning". La Science et l'hypothèse [Science and Hypothesis]. Translated by Greenstreet, William John. VI.
19. Gray, Jeremy (2008). Plato's Ghost: The modernist transformation of mathematics. Princeton University Press. p. 153. ISBN 978-1-4008-2904-0. Archived from the original on 29 March 2017 – via Google Books.
20. Weber, Heinrich L. (1891–1892). "Kronecker". Jahresbericht der Deutschen Mathematiker-Vereinigung [Annual report of the German Mathematicians Association]. pp. 2:5–23. (The quote is on p. 19). Archived from the original on 9 August 2018; "access to Jahresbericht der Deutschen Mathematiker-Vereinigung". Archived from the original on 20 August 2017.
21. Eves 1990, Chapter 15
22. Kirby, Laurie; Paris, Jeff (1982). "Accessible Independence Results for Peano Arithmetic". Bulletin of the London Mathematical Society. Wiley. 14 (4): 285–293. doi:10.1112/blms/14.4.285. ISSN 0024-6093.
23. Bagaria, Joan (2017). Set Theory (Winter 2014 ed.). The Stanford Encyclopedia of Philosophy. Archived from the original on 14 March 2015. Retrieved 13 February 2015.
24. Goldrei, Derek (1998). "3". Classic Set Theory: A guided independent study (1. ed., 1. print ed.). Boca Raton, Fla. [u.a.]: Chapman & Hall/CRC. p. 33. ISBN 978-0-412-60610-6.
25. Brown, Jim (1978). "In defense of index origin 0". ACM SIGAPL APL Quote Quad. 9 (2): 7. doi:10.1145/586050.586053. S2CID 40187000.
26. Hui, Roger. "Is index origin 0 a hindrance?". jsoftware.com. Archived from the original on 20 October 2015. Retrieved 19 January 2015.
27. This is common in texts about Real analysis. See, for example, Carothers (2000, p. 3) or Thomson, Bruckner & Bruckner (2008, p. 2).
28. "Listing of the Mathematical Notations used in the Mathematical Functions Website: Numbers, variables, and functions". functions.wolfram.com. Retrieved 27 July 2020.
29. Rudin, W. (1976). Principles of Mathematical Analysis. New York: McGraw-Hill. p. 25. ISBN 978-0-07-054235-8.
30. Grimaldi, Ralph P. (2004). Discrete and Combinatorial Mathematics: An applied introduction (5th ed.). Pearson Addison Wesley. ISBN 978-0-201-72634-3.
31. Grimaldi, Ralph P. (2003). A review of discrete and combinatorial mathematics (5th ed.). Boston: Addison-Wesley. p. 133. ISBN 978-0-201-72634-3.
32. Weisstein, Eric W. "Multiplication". mathworld.wolfram.com. Retrieved 27 July 2020.
33. Fletcher, Harold; Howell, Arnold A. (9 May 2014). Mathematics with Understanding. Elsevier. p. 116. ISBN 978-1-4832-8079-0. ...the set of natural numbers is closed under addition... set of natural numbers is closed under multiplication
34. Davisson, Schuyler Colfax (1910). College Algebra. Macmillian Company. p. 2. Addition of natural numbers is associative.
35. Brandon, Bertha (M.); Brown, Kenneth E.; Gundlach, Bernard H.; Cooke, Ralph J. (1962). Laidlaw mathematics series. Vol. 8. Laidlaw Bros. p. 25.
36. Mints, G.E. (ed.). "Peano axioms". Encyclopedia of Mathematics. Springer, in cooperation with the European Mathematical Society. Archived from the original on 13 October 2014. Retrieved 8 October 2014.
37. von Neumann (1923)
38. Levy (1979), p. 52
Bibliography
• Bluman, Allan (2010). Pre-Algebra DeMYSTiFieD (Second ed.). McGraw-Hill Professional. ISBN 978-0-07-174251-1 – via Google Books.
• Carothers, N.L. (2000). Real Analysis. Cambridge University Press. ISBN 978-0-521-49756-5 – via Google Books.
• Clapham, Christopher; Nicholson, James (2014). The Concise Oxford Dictionary of Mathematics (Fifth ed.). Oxford University Press. ISBN 978-0-19-967959-1 – via Google Books.
• Dedekind, Richard (1963) [1901]. Essays on the Theory of Numbers. Translated by Beman, Wooster Woodruff (reprint ed.). Dover Books. ISBN 978-0-486-21010-0 – via Archive.org.
• Dedekind, Richard (1901). Essays on the Theory of Numbers. Translated by Beman, Wooster Woodruff. Chicago, IL: Open Court Publishing Company. Retrieved 13 August 2020 – via Project Gutenberg.
• Dedekind, Richard (2007) [1901]. Essays on the Theory of Numbers. Kessinger Publishing, LLC. ISBN 978-0-548-08985-9.
• Eves, Howard (1990). An Introduction to the History of Mathematics (6th ed.). Thomson. ISBN 978-0-03-029558-4 – via Google Books.
• Halmos, Paul (1960). Naive Set Theory. Springer Science & Business Media. ISBN 978-0-387-90092-6 – via Google Books.
• Hamilton, A.G. (1988). Logic for Mathematicians (Revised ed.). Cambridge University Press. ISBN 978-0-521-36865-0 – via Google Books.
• James, Robert C.; James, Glenn (1992). Mathematics Dictionary (Fifth ed.). Chapman & Hall. ISBN 978-0-412-99041-0 – via Google Books.
• Landau, Edmund (1966). Foundations of Analysis (Third ed.). Chelsea Publishing. ISBN 978-0-8218-2693-5 – via Google Books.
• Levy, Azriel (1979). Basic Set Theory. Springer-Verlag Berlin Heidelberg. ISBN 978-3-662-02310-5.
• Mac Lane, Saunders; Birkhoff, Garrett (1999). Algebra (3rd ed.). American Mathematical Society. ISBN 978-0-8218-1646-2 – via Google Books.
• Mendelson, Elliott (2008) [1973]. Number Systems and the Foundations of Analysis. Dover Publications. ISBN 978-0-486-45792-5 – via Google Books.
• Morash, Ronald P. (1991). Bridge to Abstract Mathematics: Mathematical proof and structures (Second ed.). Mcgraw-Hill College. ISBN 978-0-07-043043-3 – via Google Books.
• Musser, Gary L.; Peterson, Blake E.; Burger, William F. (2013). Mathematics for Elementary Teachers: A contemporary approach (10th ed.). Wiley Global Education. ISBN 978-1-118-45744-3 – via Google Books.
• Szczepanski, Amy F.; Kositsky, Andrew P. (2008). The Complete Idiot's Guide to Pre-algebra. Penguin Group. ISBN 978-1-59257-772-9 – via Google Books.
• Thomson, Brian S.; Bruckner, Judith B.; Bruckner, Andrew M. (2008). Elementary Real Analysis (Second ed.). ClassicalRealAnalysis.com. ISBN 978-1-4348-4367-8 – via Google Books.
• von Neumann, John (1923). "Zur Einführung der transfiniten Zahlen" [On the Introduction of the Transfinite Numbers]. Acta Litterarum AC Scientiarum Ragiae Universitatis Hungaricae Francisco-Josephinae, Sectio Scientiarum Mathematicarum. 1: 199–208. Archived from the original on 18 December 2014. Retrieved 15 September 2013.
• von Neumann, John (January 2002) [1923]. "On the introduction of transfinite numbers". In van Heijenoort, Jean (ed.). From Frege to Gödel: A source book in mathematical logic, 1879–1931 (3rd ed.). Harvard University Press. pp. 346–354. ISBN 978-0-674-32449-7. – English translation of von Neumann 1923.
External links
Wikimedia Commons has media related to Natural numbers.
• "Natural number", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• "Axioms and construction of natural numbers". apronus.com.
Number systems
Sets of definable numbers
• Natural numbers ($\mathbb {N} $)
• Integers ($\mathbb {Z} $)
• Rational numbers ($\mathbb {Q} $)
• Constructible numbers
• Algebraic numbers ($\mathbb {A} $)
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Split
types
• Over $\mathbb {R} $:
• Split-complex numbers
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Over $\mathbb {C} $:
• Bicomplex numbers
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Other hypercomplex
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• Sedenions ($\mathbb {S} $)
• Split-biquaternions
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• Geometric algebra/Clifford algebra
• Algebra of physical space
• Spacetime algebra
Other types
• Cardinal numbers
• Extended natural numbers
• Irrational numbers
• Fuzzy numbers
• Hyperreal numbers
• Levi-Civita field
• Surreal numbers
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• Ordinal numbers
• p-adic numbers (p-adic solenoids)
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• Superreal numbers
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• Classification
• List
Classes of natural numbers
Powers and related numbers
• Achilles
• Power of 2
• Power of 3
• Power of 10
• Square
• Cube
• Fourth power
• Fifth power
• Sixth power
• Seventh power
• Eighth power
• Perfect power
• Powerful
• Prime power
Of the form a × 2b ± 1
• Cullen
• Double Mersenne
• Fermat
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Other polynomial numbers
• Hilbert
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Recursively defined numbers
• Fibonacci
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Possessing a specific set of other numbers
• Amenable
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Expressible via specific sums
• Nonhypotenuse
• Polite
• Practical
• Primary pseudoperfect
• Ulam
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Figurate numbers
2-dimensional
centered
• Centered triangular
• Centered square
• Centered pentagonal
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• Star
non-centered
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3-dimensional
centered
• Centered tetrahedral
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Zermelo set theory
Zermelo set theory (sometimes denoted by Z-), as set out in a seminal paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo–Fraenkel set theory (ZF) and its extensions, such as von Neumann–Bernays–Gödel set theory (NBG). It bears certain differences from its descendants, which are not always understood, and are frequently misquoted. This article sets out the original axioms, with the original text (translated into English) and original numbering.
The axioms of Zermelo set theory
The axioms of Zermelo set theory are stated for objects, some of which (but not necessarily all) are sets, and the remaining objects are urelements and not sets. Zermelo's language implicitly includes a membership relation ∈, an equality relation = (if it is not included in the underlying logic), and a unary predicate saying whether an object is a set. Later versions of set theory often assume that all objects are sets so there are no urelements and there is no need for the unary predicate.
1. AXIOM I. Axiom of extensionality (Axiom der Bestimmtheit) "If every element of a set M is also an element of N and vice versa ... then M $\equiv $ N. Briefly, every set is determined by its elements."
2. AXIOM II. Axiom of elementary sets (Axiom der Elementarmengen) "There exists a set, the null set, ∅, that contains no element at all. If a is any object of the domain, there exists a set {a} containing a and only a as an element. If a and b are any two objects of the domain, there always exists a set {a, b} containing as elements a and b but no object x distinct from them both." See Axiom of pairs.
3. AXIOM III. Axiom of separation (Axiom der Aussonderung) "Whenever the propositional function –(x) is defined for all elements of a set M, M possesses a subset M' containing as elements precisely those elements x of M for which –(x) is true."
4. AXIOM IV. Axiom of the power set (Axiom der Potenzmenge) "To every set T there corresponds a set T' , the power set of T, that contains as elements precisely all subsets of T ."
5. AXIOM V. Axiom of the union (Axiom der Vereinigung) "To every set T there corresponds a set ∪T, the union of T, that contains as elements precisely all elements of the elements of T ."
6. AXIOM VI. Axiom of choice (Axiom der Auswahl) "If T is a set whose elements all are sets that are different from ∅ and mutually disjoint, its union ∪T includes at least one subset S1 having one and only one element in common with each element of T ."
7. AXIOM VII. Axiom of infinity (Axiom des Unendlichen) "There exists in the domain at least one set Z that contains the null set as an element and is so constituted that to each of its elements a there corresponds a further element of the form {a}, in other words, that with each of its elements a it also contains the corresponding set {a} as element."
Connection with standard set theory
The most widely used and accepted set theory is known as ZFC, which consists of Zermelo–Fraenkel set theory including the axiom of choice (AC). The links show where the axioms of Zermelo's theory correspond. There is no exact match for "elementary sets". (It was later shown that the singleton set could be derived from what is now called the "Axiom of pairs". If a exists, a and a exist, thus {a,a} exists, and so by extensionality {a,a} = {a}.) The empty set axiom is already assumed by axiom of infinity, and is now included as part of it.
Zermelo set theory does not include the axioms of replacement and regularity. The axiom of replacement was first published in 1922 by Abraham Fraenkel and Thoralf Skolem, who had independently discovered that Zermelo's axioms cannot prove the existence of the set {Z0, Z1, Z2, ...} where Z0 is the set of natural numbers and Zn+1 is the power set of Zn. They both realized that the axiom of replacement is needed to prove this. The following year, John von Neumann pointed out that the axiom of regularity is necessary to build his theory of ordinals. The axiom of regularity was stated by von Neumann in 1925.[1]
In the modern ZFC system, the "propositional function" referred to in the axiom of separation is interpreted as "any property definable by a first-order formula with parameters", so the separation axiom is replaced by an axiom schema. The notion of "first order formula" was not known in 1908 when Zermelo published his axiom system, and he later rejected this interpretation as being too restrictive. Zermelo set theory is usually taken to be a first-order theory with the separation axiom replaced by an axiom scheme with an axiom for each first-order formula. It can also be considered as a theory in second-order logic, where now the separation axiom is just a single axiom. The second-order interpretation of Zermelo set theory is probably closer to Zermelo's own conception of it, and is stronger than the first-order interpretation.
In the usual cumulative hierarchy Vα of ZFC set theory (for ordinals α), any one of the sets Vα for α a limit ordinal larger than the first infinite ordinal ω (such as Vω·2) forms a model of Zermelo set theory. So the consistency of Zermelo set theory is a theorem of ZFC set theory. As $V_{\omega \cdot 2}$ models Zermelo's axioms while not containing $\aleph _{\omega }$ and larger infinite cardinals, by Gödel's completeness theorem Zermelo's axioms do not prove the existence of these cardinals. (Cardinals have to be defined differently in Zermelo set theory, as the usual definition of cardinals and ordinals does not work very well: with the usual definition it is not even possible to prove the existence of the ordinal ω2.)
The axiom of infinity is usually now modified to assert the existence of the first infinite von Neumann ordinal $\omega $; the original Zermelo axioms cannot prove the existence of this set, nor can the modified Zermelo axioms prove Zermelo's axiom of infinity. Zermelo's axioms (original or modified) cannot prove the existence of $V_{\omega }$ as a set nor of any rank of the cumulative hierarchy of sets with infinite index.
Zermelo allowed for the existence of urelements that are not sets and contain no elements; these are now usually omitted from set theories.
Mac Lane set theory
Mac Lane set theory, introduced by Mac Lane (1986), is Zermelo set theory with the axiom of separation restricted to first-order formulas in which every quantifier is bounded. Mac Lane set theory is similar in strength to topos theory with a natural number object, or to the system in Principia mathematica. It is strong enough to carry out almost all ordinary mathematics not directly connected with set theory or logic.
The aim of Zermelo's paper
The introduction states that the very existence of the discipline of set theory "seems to be threatened by certain contradictions or "antinomies", that can be derived from its principles – principles necessarily governing our thinking, it seems – and to which no entirely satisfactory solution has yet been found". Zermelo is of course referring to the "Russell antinomy".
He says he wants to show how the original theory of Georg Cantor and Richard Dedekind can be reduced to a few definitions and seven principles or axioms. He says he has not been able to prove that the axioms are consistent.
A non-constructivist argument for their consistency goes as follows. Define Vα for α one of the ordinals 0, 1, 2, ...,ω, ω+1, ω+2,..., ω·2 as follows:
• V0 is the empty set.
• For α a successor of the form β+1, Vα is defined to be the collection of all subsets of Vβ.
• For α a limit (e.g. ω, ω·2) then Vα is defined to be the union of Vβ for β<α.
Then the axioms of Zermelo set theory are consistent because they are true in the model Vω·2. While a non-constructivist might regard this as a valid argument, a constructivist would probably not: while there are no problems with the construction of the sets up to Vω, the construction of Vω+1 is less clear because one cannot constructively define every subset of Vω. This argument can be turned into a valid proof with the addition of a single new axiom of infinity to Zermelo set theory, simply that Vω·2 exists. This is presumably not convincing for a constructivist, but it shows that the consistency of Zermelo set theory can be proved with a theory which is not very different from Zermelo theory itself, only a little more powerful.
The axiom of separation
Zermelo comments that Axiom III of his system is the one responsible for eliminating the antinomies. It differs from the original definition by Cantor, as follows.
Sets cannot be independently defined by any arbitrary logically definable notion. They must be constructed in some way from previously constructed sets. For example, they can be constructed by taking powersets, or they can be separated as subsets of sets already "given". This, he says, eliminates contradictory ideas like "the set of all sets" or "the set of all ordinal numbers".
He disposes of the Russell paradox by means of this Theorem: "Every set $M$ possesses at least one subset $M_{0}$ that is not an element of $M$ ". Let $M_{0}$ be the subset of $M$ for which, by AXIOM III, is separated out by the notion "$x\notin x$". Then $M_{0}$ cannot be in $M$. For
1. If $M_{0}$ is in $M_{0}$, then $M_{0}$ contains an element x for which x is in x (i.e. $M_{0}$ itself), which would contradict the definition of $M_{0}$.
2. If $M_{0}$ is not in $M_{0}$, and assuming $M_{0}$ is an element of M, then $M_{0}$ is an element of M that satisfies the definition "$x\notin x$", and so is in $M_{0}$ which is a contradiction.
Therefore, the assumption that $M_{0}$ is in $M$ is wrong, proving the theorem. Hence not all objects of the universal domain B can be elements of one and the same set. "This disposes of the Russell antinomy as far as we are concerned".
This left the problem of "the domain B" which seems to refer to something. This led to the idea of a proper class.
Cantor's theorem
Zermelo's paper may be the first to mention the name "Cantor's theorem". Cantor's theorem: "If M is an arbitrary set, then always M < P(M) [the power set of M]. Every set is of lower cardinality than the set of its subsets".
Zermelo proves this by considering a function φ: M → P(M). By Axiom III this defines the following set M' :
M' = {m: m ∉ φ(m)}.
But no element m' of M could correspond to M' , i.e. such that φ(m' ) = M' . Otherwise we can construct a contradiction:
1) If m' is in M' then by definition m' ∉ φ(m' ) = M' , which is the first part of the contradiction
2) If m' is not in M' but in M then by definition m' ∉ M' = φ(m' ) which by definition implies that m' is in M' , which is the second part of the contradiction.
so by contradiction m' does not exist. Note the close resemblance of this proof to the way Zermelo disposes of Russell's paradox.
See also
• S (set theory)
References
1. Ferreirós 2007, pp. 369, 371.
Works cited
• Ferreirós, José (2007), Labyrinth of Thought: A History of Set Theory and Its Role in Mathematical Thought, Birkhäuser, ISBN 978-3-7643-8349-7.
General references
• Mac Lane, Saunders (1986), Mathematics, form and function, New York: Springer-Verlag, doi:10.1007/978-1-4612-4872-9, ISBN 0-387-96217-4, MR 0816347.
• Zermelo, Ernst (1908), "Untersuchungen über die Grundlagen der Mengenlehre I", Mathematische Annalen, 65 (2): 261–281, doi:10.1007/bf01449999, S2CID 120085563. English translation: Heijenoort, Jean van (1967), "Investigations in the foundations of set theory", From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, Source Books in the History of the Sciences, Harvard Univ. Press, pp. 199–215, ISBN 978-0-674-32449-7.
Set theory
Overview
• Set (mathematics)
Axioms
• Adjunction
• Choice
• countable
• dependent
• global
• Constructibility (V=L)
• Determinacy
• Extensionality
• Infinity
• Limitation of size
• Pairing
• Power set
• Regularity
• Union
• Martin's axiom
• Axiom schema
• replacement
• specification
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• Ernst Zermelo
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Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice",[1] and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded.
Informally,[2] Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that of a hereditary well-founded set, so that all entities in the universe of discourse are such sets. Thus the axioms of Zermelo–Fraenkel set theory refer only to pure sets and prevent its models from containing urelements (elements of sets that are not themselves sets). Furthermore, proper classes (collections of mathematical objects defined by a property shared by their members where the collections are too big to be sets) can only be treated indirectly. Specifically, Zermelo–Fraenkel set theory does not allow for the existence of a universal set (a set containing all sets) nor for unrestricted comprehension, thereby avoiding Russell's paradox. Von Neumann–Bernays–Gödel set theory (NBG) is a commonly used conservative extension of Zermelo–Fraenkel set theory that does allow explicit treatment of proper classes.
There are many equivalent formulations of the axioms of Zermelo–Fraenkel set theory. Most of the axioms state the existence of particular sets defined from other sets. For example, the axiom of pairing says that given any two sets $a$ and $b$ there is a new set $\{a,b\}$ containing exactly $a$ and $b$. Other axioms describe properties of set membership. A goal of the axioms is that each axiom should be true if interpreted as a statement about the collection of all sets in the von Neumann universe (also known as the cumulative hierarchy). Formally, ZFC is a one-sorted theory in first-order logic. The signature has equality and a single primitive binary relation, intended to formalize set membership, which is usually denoted $\in $. The formula $a\in b$ means that the set $a$ is a member of the set $b$ (which is also read, "$a$ is an element of $b$" or "$a$ is in $b$").
The metamathematics of Zermelo–Fraenkel set theory has been extensively studied. Landmark results in this area established the logical independence of the axiom of choice from the remaining Zermelo-Fraenkel axioms (see Axiom of choice § Independence) and of the continuum hypothesis from ZFC. The consistency of a theory such as ZFC cannot be proved within the theory itself, as shown by Gödel's second incompleteness theorem.
History
Main article: History of set theory
The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. However, the discovery of paradoxes in naive set theory, such as Russell's paradox, led to the desire for a more rigorous form of set theory that was free of these paradoxes.
In 1908, Ernst Zermelo proposed the first axiomatic set theory, Zermelo set theory. However, as first pointed out by Abraham Fraenkel in a 1921 letter to Zermelo, this theory was incapable of proving the existence of certain sets and cardinal numbers whose existence was taken for granted by most set theorists of the time, notably the cardinal number $\aleph _{\omega }$ and the set $\{Z_{0},{\mathcal {P}}(Z_{0}),{\mathcal {P}}({\mathcal {P}}(Z_{0})),{\mathcal {P}}({\mathcal {P}}({\mathcal {P}}(Z_{0}))),...\},$ where $Z_{0}$ is any infinite set and ${\mathcal {P}}$ is the power set operation.[3] Moreover, one of Zermelo's axioms invoked a concept, that of a "definite" property, whose operational meaning was not clear. In 1922, Fraenkel and Thoralf Skolem independently proposed operationalizing a "definite" property as one that could be formulated as a well-formed formula in a first-order logic whose atomic formulas were limited to set membership and identity. They also independently proposed replacing the axiom schema of specification with the axiom schema of replacement. Appending this schema, as well as the axiom of regularity (first proposed by John von Neumann),[4] to Zermelo set theory yields the theory denoted by ZF. Adding to ZF either the axiom of choice (AC) or a statement that is equivalent to it yields ZFC.
Axioms
Main article: Axiom
There are many equivalent formulations of the ZFC axioms; for a discussion of this, see Fraenkel, Bar-Hillel & Lévy 1973. The following particular axiom set is from Kunen (1980). The axioms per se are expressed in the symbolism of first order logic. The associated English prose is only intended to aid the intuition.
All formulations of ZFC imply that at least one set exists. Kunen includes an axiom that directly asserts the existence of a set, in addition to the axioms given below (although he notes that he does so only "for emphasis").[5] Its omission here can be justified in two ways. First, in the standard semantics of first-order logic in which ZFC is typically formalized, the domain of discourse must be nonempty. Hence, it is a logical theorem of first-order logic that something exists — usually expressed as the assertion that something is identical to itself, $\exists x(x=x)$. Consequently, it is a theorem of every first-order theory that something exists. However, as noted above, because in the intended semantics of ZFC, there are only sets, the interpretation of this logical theorem in the context of ZFC is that some set exists. Hence, there is no need for a separate axiom asserting that a set exists. Second, however, even if ZFC is formulated in so-called free logic, in which it is not provable from logic alone that something exists, the axiom of infinity (below) asserts that an infinite set exists. This implies that a set exists, and so, once again, it is superfluous to include an axiom asserting as much.
1. Axiom of extensionality
Main article: Axiom of extensionality
Two sets are equal (are the same set) if they have the same elements.
$\forall x\forall y[\forall z(z\in x\Leftrightarrow z\in y)\Rightarrow x=y].$
The converse of this axiom follows from the substitution property of equality. ZFC is constructed in first-order logic. Some formulations of first-order logic include identity; others do not. If the variety of first-order logic in which you are constructing set theory does not include equality "$=$", $x=y$ may be defined as an abbreviation for the following formula:[6] $\forall z[z\in x\Leftrightarrow z\in y]\land \forall w[x\in w\Leftrightarrow y\in w].$
In this case, the axiom of extensionality can be reformulated as
$\forall x\forall y[\forall z(z\in x\Leftrightarrow z\in y)\Rightarrow \forall w(x\in w\Leftrightarrow y\in w)],$
which says that if $x$ and $y$ have the same elements, then they belong to the same sets.[7]
2. Axiom of regularity (also called the axiom of foundation)
Main article: Axiom of regularity
Every non-empty set $x$ contains a member $y$ such that $x$ and $y$ are disjoint sets.
$\forall x[\exists a(a\in x)\Rightarrow \exists y(y\in x\land \lnot \exists z(z\in y\land z\in x))].$[8]
or in modern notation: $\forall x\,(x\neq \varnothing \Rightarrow \exists y(y\in x\land y\cap x=\varnothing )).$
This (along with the Axiom of Pairing) implies, for example, that no set is an element of itself and that every set has an ordinal rank.
3. Axiom schema of specification (or of separation, or of restricted comprehension)
Main article: Axiom schema of specification
Subsets are commonly constructed using set builder notation. For example, the even integers can be constructed as the subset of the integers $\mathbb {Z} $ satisfying the congruence modulo predicate $x\equiv 0{\pmod {2}}$:
$\{x\in \mathbb {Z} :x\equiv 0{\pmod {2}}\}.$
In general, the subset of a set $z$ obeying a formula $\varphi (x)$ with one free variable $x$ may be written as:
$\{x\in z:\varphi (x)\}.$
The axiom schema of specification states that this subset always exists (it is an axiom schema because there is one axiom for each $\varphi $). Formally, let $\varphi $ be any formula in the language of ZFC with all free variables among $x,z,w_{1},\ldots ,w_{n}$ ($y$ is not free in $\varphi $). Then:
$\forall z\forall w_{1}\forall w_{2}\ldots \forall w_{n}\exists y\forall x[x\in y\Leftrightarrow ((x\in z)\land \varphi (x,w_{1},w_{2},...,w_{n},z))].$
Note that the axiom schema of specification can only construct subsets and does not allow the construction of entities of the more general form:
$\{x:\varphi (x)\}.$
This restriction is necessary to avoid Russell's paradox (let $y=\{x:x\notin x\}$ then $y\in y\Leftrightarrow y\notin y$) and its variants that accompany naive set theory with unrestricted comprehension.
In some other axiomatizations of ZF, this axiom is redundant in that it follows from the axiom schema of replacement and the axiom of the empty set.
On the other hand, the axiom schema of specification can be used to prove the existence of the empty set, denoted $\varnothing $, once at least one set is known to exist (see above). One way to do this is to use a property $\varphi $ which no set has. For example, if $w$ is any existing set, the empty set can be constructed as
$\varnothing =\{u\in w\mid (u\in u)\land \lnot (u\in u)\}.$
Thus, the axiom of the empty set is implied by the nine axioms presented here. The axiom of extensionality implies the empty set is unique (does not depend on $w$). It is common to make a definitional extension that adds the symbol "$\varnothing $" to the language of ZFC.
4. Axiom of pairing
Main article: Axiom of pairing
If $x$ and $y$ are sets, then there exists a set which contains $x$ and $y$ as elements, for example if x = {1,2} and y = {2,3} then z will be {{1,2},{2,3}}
$\forall x\forall y\exists z((x\in z)\land (y\in z)).$
The axiom schema of specification must be used to reduce this to a set with exactly these two elements. The axiom of pairing is part of Z, but is redundant in ZF because it follows from the axiom schema of replacement if we are given a set with at least two elements. The existence of a set with at least two elements is assured by either the axiom of infinity, or by the axiom schema of specification and the axiom of the power set applied twice to any set.
5. Axiom of union
Main article: Axiom of union
The union over the elements of a set exists. For example, the union over the elements of the set $\{\{1,2\},\{2,3\}\}$ is $\{1,2,3\}.$
The axiom of union states that for any set of sets ${\mathcal {F}}$, there is a set $A$ containing every element that is a member of some member of ${\mathcal {F}}$:
$\forall {\mathcal {F}}\,\exists A\,\forall Y\,\forall x[(x\in Y\land Y\in {\mathcal {F}})\Rightarrow x\in A].$
Although this formula doesn't directly assert the existence of $\cup {\mathcal {F}}$, the set $\cup {\mathcal {F}}$ can be constructed from $A$ in the above using the axiom schema of specification:
$\cup {\mathcal {F}}=\{x\in A:\exists Y(x\in Y\land Y\in {\mathcal {F}})\}.$
6. Axiom schema of replacement
Main article: Axiom schema of replacement
The axiom schema of replacement asserts that the image of a set under any definable function will also fall inside a set.
Formally, let $\varphi $ be any formula in the language of ZFC whose free variables are among $x,y,A,w_{1},\dotsc ,w_{n},$ so that in particular $B$ is not free in $\varphi $. Then:
$\forall A\forall w_{1}\forall w_{2}\ldots \forall w_{n}{\bigl [}\forall x(x\in A\Rightarrow \exists !y\,\varphi )\Rightarrow \exists B\ \forall x{\bigl (}x\in A\Rightarrow \exists y(y\in B\land \varphi ){\bigr )}{\bigr ]}.$
(The unique existential quantifier $\exists !$ !} denotes the existence of exactly one element such that it follows a given statement. For more, see uniqueness quantification.)
In other words, if the relation $\varphi $ represents a definable function $f$, $A$ represents its domain, and $f(x)$ is a set for every $x\in A,$ then the range of $f$ is a subset of some set $B$. The form stated here, in which $B$ may be larger than strictly necessary, is sometimes called the axiom schema of collection.
7. Axiom of infinity
Main article: Axiom of infinity
First few von Neumann ordinals
0 = { } = ∅
1 = { 0} = {∅}
2 = { 0, 1} = { ∅, {∅} }
3 = { 0, 1, 2} = { ∅, {∅}, {∅, {∅}} }
4 = { 0, 1, 2, 3} = { ∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }
Let $S(w)$ abbreviate $w\cup \{w\},$ where $w$ is some set. (We can see that $\{w\}$ is a valid set by applying the Axiom of Pairing with $x=y=w$ so that the set z is $\{w\}$). Then there exists a set X such that the empty set $\varnothing $, defined axiomatically, is a member of X and, whenever a set y is a member of X then $S(y)$ is also a member of X.
$\exists X\left[\exists e(\forall z\,\neg (z\in e)\land e\in X)\land \forall y(y\in X\Rightarrow S(y)\in X)\right].$
More colloquially, there exists a set X having infinitely many members. (It must be established, however, that these members are all different because if two elements are the same, the sequence will loop around in a finite cycle of sets. The axiom of regularity prevents this from happening.) The minimal set X satisfying the axiom of infinity is the von Neumann ordinal ω which can also be thought of as the set of natural numbers $\mathbb {N} .$
8. Axiom of power set
Main article: Axiom of power set
By definition, a set $z$ is a subset of a set $x$ if and only if every element of $z$ is also an element of $x$:
$(z\subseteq x)\Leftrightarrow (\forall q(q\in z\Rightarrow q\in x)).$
The Axiom of Power Set states that for any set $x$, there is a set $y$ that contains every subset of $x$:
$\forall x\exists y\forall z[z\subseteq x\Rightarrow z\in y].$
The axiom schema of specification is then used to define the power set ${\mathcal {P}}(x)$ as the subset of such a $y$ containing the subsets of $x$ exactly:
${\mathcal {P}}(x)=\{z\in y:z\subseteq x\}.$
Axioms 1–8 define ZF. Alternative forms of these axioms are often encountered, some of which are listed in Jech (2003). Some ZF axiomatizations include an axiom asserting that the empty set exists. The axioms of pairing, union, replacement, and power set are often stated so that the members of the set $x$ whose existence is being asserted are just those sets which the axiom asserts $x$ must contain.
The following axiom is added to turn ZF into ZFC:
9. Well-ordering axiom
Main article: Well-ordering theorem
For any set $X$, there is a binary relation $R$ which well-orders $X$. This means $R$ is a linear order on $X$ such that every nonempty subset of $X$ has a member which is minimal under $R$.
$\forall X\exists R(R\;{\mbox{well-orders}}\;X).$
Given axioms 1 – 8, there are many statements provably equivalent to axiom 9, the best known of which is the axiom of choice (AC), which goes as follows. Let $X$ be a set whose members are all nonempty. Then there exists a function $f$ from $X$ to the union of the members of $X$, called a "choice function", such that for all $Y\in X$ one has $f(Y)\in Y$. Since the existence of a choice function when $X$ is a finite set is easily proved from axioms 1–8, AC only matters for certain infinite sets. AC is characterized as nonconstructive because it asserts the existence of a choice function but says nothing about how this choice function is to be "constructed."
Zorn's lemma
Main article: Zorn's lemma
The Well-ordering axiom, as well as the Axiom of choice, are each individually (logically) equivalent to Zorn's lemma.
Motivation via the cumulative hierarchy
Further information: Von Neumann universe
One motivation for the ZFC axioms is the cumulative hierarchy of sets introduced by John von Neumann.[9] In this viewpoint, the universe of set theory is built up in stages, with one stage for each ordinal number. At stage 0, there are no sets yet. At each following stage, a set is added to the universe if all of its elements have been added at previous stages. Thus the empty set is added at stage 1, and the set containing the empty set is added at stage 2.[10] The collection of all sets that are obtained in this way, over all the stages, is known as V. The sets in V can be arranged into a hierarchy by assigning to each set the first stage at which that set was added to V.
It is provable that a set is in V if and only if the set is pure and well-founded. And V satisfies all the axioms of ZFC if the class of ordinals has appropriate reflection properties. For example, suppose that a set x is added at stage α, which means that every element of x was added at a stage earlier than α. Then, every subset of x is also added at (or before) stage α, because all elements of any subset of x were also added before stage α. This means that any subset of x which the axiom of separation can construct is added at (or before) stage α, and that the powerset of x will be added at the next stage after α. For a complete argument that V satisfies ZFC see Shoenfield (1977).
The picture of the universe of sets stratified into the cumulative hierarchy is characteristic of ZFC and related axiomatic set theories such as Von Neumann–Bernays–Gödel set theory (often called NBG) and Morse–Kelley set theory. The cumulative hierarchy is not compatible with other set theories such as New Foundations.
It is possible to change the definition of V so that at each stage, instead of adding all the subsets of the union of the previous stages, subsets are only added if they are definable in a certain sense. This results in a more "narrow" hierarchy, which gives the constructible universe L, which also satisfies all the axioms of ZFC, including the axiom of choice. It is independent from the ZFC axioms whether V = L. Although the structure of L is more regular and well behaved than that of V, few mathematicians argue that V = L should be added to ZFC as an additional "axiom of constructibility".
Metamathematics
Virtual classes
As noted earlier, proper classes (collections of mathematical objects defined by a property shared by their members which are too big to be sets) can only be treated indirectly in ZF (and thus ZFC). An alternative to proper classes while staying within ZF and ZFC is the virtual class notational construct introduced by Quine (1969), where the entire construct y ∈ { x | Fx } is simply defined as Fy.[11] This provides a simple notation for classes that can contain sets but need not themselves be sets, while not committing to the ontology of classes (because the notation can be syntactically converted to one that only uses sets). Quine's approach built on the earlier approach of Bernays & Fraenkel (1958). Virtual classes are also used in Levy (2002), Takeuti & Zaring (1982), and in the Metamath implementation of ZFC.
Von Neumann–Bernays–Gödel set theory
Main article: Von Neumann–Bernays–Gödel set theory
The axiom schemata of replacement and separation each contain infinitely many instances. Montague (1961) included a result first proved in his 1957 Ph.D. thesis: if ZFC is consistent, it is impossible to axiomatize ZFC using only finitely many axioms. On the other hand, von Neumann–Bernays–Gödel set theory (NBG) can be finitely axiomatized. The ontology of NBG includes proper classes as well as sets; a set is any class that can be a member of another class. NBG and ZFC are equivalent set theories in the sense that any theorem not mentioning classes and provable in one theory can be proved in the other.
Consistency
Gödel's second incompleteness theorem says that a recursively axiomatizable system that can interpret Robinson arithmetic can prove its own consistency only if it is inconsistent. Moreover, Robinson arithmetic can be interpreted in general set theory, a small fragment of ZFC. Hence the consistency of ZFC cannot be proved within ZFC itself (unless it is actually inconsistent). Thus, to the extent that ZFC is identified with ordinary mathematics, the consistency of ZFC cannot be demonstrated in ordinary mathematics. The consistency of ZFC does follow from the existence of a weakly inaccessible cardinal, which is unprovable in ZFC if ZFC is consistent. Nevertheless, it is deemed unlikely that ZFC harbors an unsuspected contradiction; it is widely believed that if ZFC were inconsistent, that fact would have been uncovered by now. This much is certain — ZFC is immune to the classic paradoxes of naive set theory: Russell's paradox, the Burali-Forti paradox, and Cantor's paradox.
Abian & LaMacchia (1978) studied a subtheory of ZFC consisting of the axioms of extensionality, union, powerset, replacement, and choice. Using models, they proved this subtheory consistent, and proved that each of the axioms of extensionality, replacement, and power set is independent of the four remaining axioms of this subtheory. If this subtheory is augmented with the axiom of infinity, each of the axioms of union, choice, and infinity is independent of the five remaining axioms. Because there are non-well-founded models that satisfy each axiom of ZFC except the axiom of regularity, that axiom is independent of the other ZFC axioms.
If consistent, ZFC cannot prove the existence of the inaccessible cardinals that category theory requires. Huge sets of this nature are possible if ZF is augmented with Tarski's axiom.[12] Assuming that axiom turns the axioms of infinity, power set, and choice (7 – 9 above) into theorems.
Independence
Many important statements are independent of ZFC (see list of statements independent of ZFC). The independence is usually proved by forcing, whereby it is shown that every countable transitive model of ZFC (sometimes augmented with large cardinal axioms) can be expanded to satisfy the statement in question. A different expansion is then shown to satisfy the negation of the statement. An independence proof by forcing automatically proves independence from arithmetical statements, other concrete statements, and large cardinal axioms. Some statements independent of ZFC can be proven to hold in particular inner models, such as in the constructible universe. However, some statements that are true about constructible sets are not consistent with hypothesized large cardinal axioms.
Forcing proves that the following statements are independent of ZFC:
• Continuum hypothesis
• Diamond principle
• Suslin hypothesis
• Martin's axiom (which is not a ZFC axiom)
• Axiom of Constructibility (V=L) (which is also not a ZFC axiom).
Remarks:
• The consistency of V=L is provable by inner models but not forcing: every model of ZF can be trimmed to become a model of ZFC + V=L.
• The Diamond Principle implies the Continuum Hypothesis and the negation of the Suslin Hypothesis.
• Martin's axiom plus the negation of the Continuum Hypothesis implies the Suslin Hypothesis.
• The constructible universe satisfies the Generalized Continuum Hypothesis, the Diamond Principle, Martin's Axiom and the Kurepa Hypothesis.
• The failure of the Kurepa hypothesis is equiconsistent with the existence of a strongly inaccessible cardinal.
A variation on the method of forcing can also be used to demonstrate the consistency and unprovability of the axiom of choice, i.e., that the axiom of choice is independent of ZF. The consistency of choice can be (relatively) easily verified by proving that the inner model L satisfies choice. (Thus every model of ZF contains a submodel of ZFC, so that Con(ZF) implies Con(ZFC).) Since forcing preserves choice, we cannot directly produce a model contradicting choice from a model satisfying choice. However, we can use forcing to create a model which contains a suitable submodel, namely one satisfying ZF but not C.
Another method of proving independence results, one owing nothing to forcing, is based on Gödel's second incompleteness theorem. This approach employs the statement whose independence is being examined, to prove the existence of a set model of ZFC, in which case Con(ZFC) is true. Since ZFC satisfies the conditions of Gödel's second theorem, the consistency of ZFC is unprovable in ZFC (provided that ZFC is, in fact, consistent). Hence no statement allowing such a proof can be proved in ZFC. This method can prove that the existence of large cardinals is not provable in ZFC, but cannot prove that assuming such cardinals, given ZFC, is free of contradiction.
Proposed additions
The project to unify set theorists behind additional axioms to resolve the Continuum Hypothesis or other meta-mathematical ambiguities is sometimes known as "Gödel's program".[13] Mathematicians currently debate which axioms are the most plausible or "self-evident", which axioms are the most useful in various domains, and about to what degree usefulness should be traded off with plausibility; some "multiverse" set theorists argue that usefulness should be the sole ultimate criterion in which axioms to customarily adopt. One school of thought leans on expanding the "iterative" concept of a set to produce a set-theoretic universe with an interesting and complex but reasonably tractable structure by adopting forcing axioms; another school advocates for a tidier, less cluttered universe, perhaps focused on a "core" inner model.[14]
Criticisms
For criticism of set theory in general, see Objections to set theory
ZFC has been criticized both for being excessively strong and for being excessively weak, as well as for its failure to capture objects such as proper classes and the universal set.
Many mathematical theorems can be proven in much weaker systems than ZFC, such as Peano arithmetic and second-order arithmetic (as explored by the program of reverse mathematics). Saunders Mac Lane and Solomon Feferman have both made this point. Some of "mainstream mathematics" (mathematics not directly connected with axiomatic set theory) is beyond Peano arithmetic and second-order arithmetic, but still, all such mathematics can be carried out in ZC (Zermelo set theory with choice), another theory weaker than ZFC. Much of the power of ZFC, including the axiom of regularity and the axiom schema of replacement, is included primarily to facilitate the study of the set theory itself.
On the other hand, among axiomatic set theories, ZFC is comparatively weak. Unlike New Foundations, ZFC does not admit the existence of a universal set. Hence the universe of sets under ZFC is not closed under the elementary operations of the algebra of sets. Unlike von Neumann–Bernays–Gödel set theory (NBG) and Morse–Kelley set theory (MK), ZFC does not admit the existence of proper classes. A further comparative weakness of ZFC is that the axiom of choice included in ZFC is weaker than the axiom of global choice included in NBG and MK.
There are numerous mathematical statements independent of ZFC. These include the continuum hypothesis, the Whitehead problem, and the normal Moore space conjecture. Some of these conjectures are provable with the addition of axioms such as Martin's axiom or large cardinal axioms to ZFC. Some others are decided in ZF+AD where AD is the axiom of determinacy, a strong supposition incompatible with choice. One attraction of large cardinal axioms is that they enable many results from ZF+AD to be established in ZFC adjoined by some large cardinal axiom (see projective determinacy). The Mizar system and Metamath have adopted Tarski–Grothendieck set theory, an extension of ZFC, so that proofs involving Grothendieck universes (encountered in category theory and algebraic geometry) can be formalized.
See also
• Foundations of mathematics
• Inner model
• Large cardinal axiom
Related axiomatic set theories:
• Morse–Kelley set theory
• Von Neumann–Bernays–Gödel set theory
• Tarski–Grothendieck set theory
• Constructive set theory
• Internal set theory
Notes
1. Ciesielski 1997. "Zermelo-Fraenkel axioms (abbreviated as ZFC where C stands for the axiom of Choice"
2. K. Kunen, The Foundations of Mathematics (p.10). Accessed 2022-04-26.
3. Ebbinghaus 2007, p. 136.
4. Halbeisen 2011, pp. 62–63.
5. Kunen (1980, p. 10).
6. Hatcher 1982, p. 138, def. 1.
7. Fraenkel, Bar-Hillel & Lévy 1973.
8. Shoenfield 2001, p. 239.
9. Shoenfield 1977, section 2.
10. Hinman 2005, p. 467.
11. (Link 2014)
12. Tarski 1939.
13. Feferman 1996.
14. Wolchover 2013.
Works cited
• Abian, Alexander (1965). The Theory of Sets and Transfinite Arithmetic. W B Saunders.
• ———; LaMacchia, Samuel (1978). "On the Consistency and Independence of Some Set-Theoretical Axioms". Notre Dame Journal of Formal Logic. 19: 155–58. doi:10.1305/ndjfl/1093888220.
• Bernays, Paul; Fraenkel, A.A. (1958). Axiomatic Set Theory. Amsterdam: North Holland.
• Ciesielski, Krzysztof (1997). Set Theory for the Working Mathematician. Cambridge University Press. p. 4. ISBN 0-521-59441-3.
• Devlin, Keith (1996) [First published 1984]. The Joy of Sets. Springer.
• Ebbinghaus, Heinz-Dieter (2007). Ernst Zermelo: An Approach to His Life and Work. Springer. ISBN 978-3-540-49551-2.
• Feferman, Solomon (1996). "Gödel's program for new axioms: why, where, how and what?". In Hájek, Petr (ed.). Gödel '96: Logical foundations of mathematics, computer science and physics–Kurt Gödel's legacy. Springer-Verlag. pp. 3–22. ISBN 3-540-61434-6..
• Fraenkel, Abraham; Bar-Hillel, Yehoshua; Lévy, Azriel (1973) [First published 1958]. Foundations of Set Theory. North-Holland. Fraenkel's final word on ZF and ZFC.
• Halbeisen, Lorenz J. (2011). Combinatorial Set Theory: With a Gentle Introduction to Forcing. Springer. pp. 62–63. ISBN 978-1-4471-2172-5.
• Hatcher, William (1982) [First published 1968]. The Logical Foundations of Mathematics. Pergamon Press.
• van Heijenoort, Jean (1967). From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Harvard University Press. Includes annotated English translations of the classic articles by Zermelo, Fraenkel, and Skolem bearing on ZFC.
• Hinman, Peter (2005). Fundamentals of Mathematical Logic. A K Peters. ISBN 978-1-56881-262-5.
• Jech, Thomas (2003). Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.
• Kunen, Kenneth (1980). Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.
• Levy, Azriel (2002). Basic Set Theory. Dover Publications. ISBN 048642079-5.
• Link, Godehard (2014). Formalism and Beyond: On the Nature of Mathematical Discourse. Walter de Gruyter GmbH & Co KG. ISBN 978-1-61451-829-7.
• Montague, Richard (1961). "Semantical closure and non-finite axiomatizability". Infinistic Methods. London: Pergamon Press. pp. 45–69.
• Quine, Willard van Orman (1969). Set Theory and Its Logic (Revised ed.). Cambridge, Massachusetts and London, England: The Belknap Press of Harvard University Press. ISBN 0-674-80207-1.
• Shoenfield, Joseph R. (1977). "Axioms of set theory". In Barwise, K. J. (ed.). Handbook of Mathematical Logic. North-Holland Publishing Company. ISBN 0-7204-2285-X.
• Shoenfield, Joseph R. (2001) [First published 1967]. Mathematical Logic (2nd ed.). A K Peters. ISBN 978-1-56881-135-2.
• Suppes, Patrick (1972) [First published 1960]. Axiomatic Set Theory. Dover reprint.Perhaps the best exposition of ZFC before the independence of AC and the Continuum hypothesis, and the emergence of large cardinals. Includes many theorems.
• Takeuti, Gaisi; Zaring, W M (1971). Introduction to Axiomatic Set Theory. Springer-Verlag.
• Takeuti, Gaisi; Zaring, W M (1982). Introduction to Axiomatic Set Theory. Springer. ISBN 9780387906836.
• Tarski, Alfred (1939). "On well-ordered subsets of any set". Fundamenta Mathematicae. 32: 176–83. doi:10.4064/fm-32-1-176-783.
• Tiles, Mary (1989). The Philosophy of Set Theory. Dover reprint.
• Tourlakis, George (2003). Lectures in Logic and Set Theory, Vol. 2. Cambridge University Press.
• Wolchover, Natalie (2013). "To Settle Infinity Dispute, a New Law of Logic". Quanta Magazine..
• Zermelo, Ernst (1908). "Untersuchungen über die Grundlagen der Mengenlehre I". Mathematische Annalen. 65 (2): 261–281. doi:10.1007/BF01449999. S2CID 120085563. English translation in Heijenoort, Jean van (1967). "Investigations in the foundations of set theory". From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Source Books in the History of the Sciences. Harvard University Press. pp. 199–215. ISBN 978-0-674-32449-7.
• Zermelo, Ernst (1930). "Über Grenzzahlen und Mengenbereiche". Fundamenta Mathematicae. 16: 29–47. doi:10.4064/fm-16-1-29-47. ISSN 0016-2736.
External links
• "ZFC", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Stanford Encyclopedia of Philosophy articles by Thomas Jech:
• Set Theory;
• Axioms of Zermelo–Fraenkel Set Theory.
• Metamath version of the ZFC axioms — A concise and nonredundant axiomatization. The background first order logic is defined especially to facilitate machine verification of proofs.
• A derivation in Metamath of a version of the separation schema from a version of the replacement schema.
• Weisstein, Eric W. "Zermelo-Fraenkel Set Theory". MathWorld.
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Turing jump
In computability theory, the Turing jump or Turing jump operator, named for Alan Turing, is an operation that assigns to each decision problem X a successively harder decision problem X′ with the property that X′ is not decidable by an oracle machine with an oracle for X.
The operator is called a jump operator because it increases the Turing degree of the problem X. That is, the problem X′ is not Turing-reducible to X. Post's theorem establishes a relationship between the Turing jump operator and the arithmetical hierarchy of sets of natural numbers.[1] Informally, given a problem, the Turing jump returns the set of Turing machines that halt when given access to an oracle that solves that problem.
Definition
The Turing jump of X can be thought of as an oracle to the halting problem for oracle machines with an oracle for X.[1]
Formally, given a set X and a Gödel numbering φiX of the X-computable functions, the Turing jump X′ of X is defined as
$X'=\{x\mid \varphi _{x}^{X}(x)\ {\mbox{is defined}}\}.$
The nth Turing jump X(n) is defined inductively by
$X^{(0)}=X,$
$X^{(n+1)}=(X^{(n)})'.$
The ω jump X(ω) of X is the effective join of the sequence of sets X(n) for n ∈ N:
$X^{(\omega )}=\{p_{i}^{k}\mid i\in \mathbb {N} {\text{ and }}k\in X^{(i)}\},$
where pi denotes the ith prime.
The notation 0′ or ∅′ is often used for the Turing jump of the empty set. It is read zero-jump or sometimes zero-prime.
Similarly, 0(n) is the nth jump of the empty set. For finite n, these sets are closely related to the arithmetic hierarchy,[2] and is in particular connected to Post's theorem.
The jump can be iterated into transfinite ordinals: there are jump operators $j^{\delta }$ for sets of natural numbers when $\delta $ is an ordinal that has a code in Kleene's ${\mathcal {O}}$ (regardless of code, the resulting jumps are the same by a theorem of Spector),[2] in particular the sets 0(α) for α < ω1CK, where ω1CK is the Church–Kleene ordinal, are closely related to the hyperarithmetic hierarchy.[1] Beyond ω1CK, the process can be continued through the countable ordinals of the constructible universe, using Jensen's work on fine structure theory of Godel's L.[3][2] The concept has also been generalized to extend to uncountable regular cardinals.[4]
Examples
• The Turing jump 0′ of the empty set is Turing equivalent to the halting problem.[5]
• For each n, the set 0(n) is m-complete at level $\Sigma _{n}^{0}$ in the arithmetical hierarchy (by Post's theorem).
• The set of Gödel numbers of true formulas in the language of Peano arithmetic with a predicate for X is computable from X(ω).[6]
Properties
• X′ is X-computably enumerable but not X-computable.
• If A is Turing-equivalent to B, then A′ is Turing-equivalent to B′. The converse of this implication is not true.
• (Shore and Slaman, 1999) The function mapping X to X′ is definable in the partial order of the Turing degrees.[5]
Many properties of the Turing jump operator are discussed in the article on Turing degrees.
References
1. Ambos-Spies, Klaus; Fejer, Peter A. (2014), "Degrees of Unsolvability", Handbook of the History of Logic, Elsevier, vol. 9, pp. 443–494, doi:10.1016/b978-0-444-51624-4.50010-1, ISBN 9780444516244.
2. S. G. Simpson, The Hierarchy Based on the Jump Operator, p.269. The Kleene Symposium (North-Holland, 1980)
3. Hodes, Harold T. (June 1980). "Jumping Through the Transfinite: The Master Code Hierarchy of Turing Degrees". Journal of Symbolic Logic. Association for Symbolic Logic. 45 (2): 204–220. doi:10.2307/2273183. JSTOR 2273183. S2CID 41245500.
4. Lubarsky, Robert S. (December 1987). "Uncountable master codes and the jump hierarchy". The Journal of Symbolic Logic. 52 (4): 952–958. doi:10.2307/2273829. ISSN 0022-4812. JSTOR 2273829. S2CID 46113113.
5. Shore, Richard A.; Slaman, Theodore A. (1999). "Defining the Turing Jump". Mathematical Research Letters. 6 (6): 711–722. doi:10.4310/MRL.1999.v6.n6.a10.
6. Hodes, Harold T. (June 1980). "Jumping through the transfinite: the master code hierarchy of Turing degrees". The Journal of Symbolic Logic. 45 (2): 204–220. doi:10.2307/2273183. ISSN 0022-4812. JSTOR 2273183. S2CID 41245500.
• Ambos-Spies, K. and Fejer, P. Degrees of Unsolvability. Unpublished. http://www.cs.umb.edu/~fejer/articles/History_of_Degrees.pdf
• Lerman, M. (1983). Degrees of unsolvability: local and global theory. Berlin; New York: Springer-Verlag. ISBN 3-540-12155-2.
• Lubarsky, Robert S. (Dec 1987). "Uncountable Master Codes and the Jump Hierarchy". Journal of Symbolic Logic. Vol. 52, no. 4. pp. 952–958. JSTOR 2273829.
• Rogers Jr, H. (1987). Theory of recursive functions and effective computability. MIT Press, Cambridge, MA, USA. ISBN 0-07-053522-1.
• Shore, R.A.; Slaman, T.A. (1999). "Defining the Turing jump" (PDF). Mathematical Research Letters. 6 (5–6): 711–722. doi:10.4310/mrl.1999.v6.n6.a10. Retrieved 2008-07-13.
• Soare, R.I. (1987). Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets. Springer. ISBN 3-540-15299-7.
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Zero-product property
In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, ${\text{if }}ab=0,{\text{ then }}a=0{\text{ or }}b=0.$
For the product of zero factors, see empty product.
This property is also known as the rule of zero product, the null factor law, the multiplication property of zero, the nonexistence of nontrivial zero divisors, or one of the two zero-factor properties.[1] All of the number systems studied in elementary mathematics — the integers $\mathbb {Z} $, the rational numbers $\mathbb {Q} $, the real numbers $\mathbb {R} $, and the complex numbers $\mathbb {C} $ — satisfy the zero-product property. In general, a ring which satisfies the zero-product property is called a domain.
Algebraic context
Suppose $A$ is an algebraic structure. We might ask, does $A$ have the zero-product property? In order for this question to have meaning, $A$ must have both additive structure and multiplicative structure.[2] Usually one assumes that $A$ is a ring, though it could be something else, e.g. the set of nonnegative integers $\{0,1,2,\ldots \}$ with ordinary addition and multiplication, which is only a (commutative) semiring.
Note that if $A$ satisfies the zero-product property, and if $B$ is a subset of $A$, then $B$ also satisfies the zero product property: if $a$ and $b$ are elements of $B$ such that $ab=0$, then either $a=0$ or $b=0$ because $a$ and $b$ can also be considered as elements of $A$.
Examples
• A ring in which the zero-product property holds is called a domain. A commutative domain with a multiplicative identity element is called an integral domain. Any field is an integral domain; in fact, any subring of a field is an integral domain (as long as it contains 1). Similarly, any subring of a skew field is a domain. Thus, the zero-product property holds for any subring of a skew field.
• If $p$ is a prime number, then the ring of integers modulo $p$ has the zero-product property (in fact, it is a field).
• The Gaussian integers are an integral domain because they are a subring of the complex numbers.
• In the strictly skew field of quaternions, the zero-product property holds. This ring is not an integral domain, because the multiplication is not commutative.
• The set of nonnegative integers $\{0,1,2,\ldots \}$ is not a ring (being instead a semiring), but it does satisfy the zero-product property.
Non-examples
• Let $\mathbb {Z} _{n}$ denote the ring of integers modulo $n$. Then $\mathbb {Z} _{6}$ does not satisfy the zero product property: 2 and 3 are nonzero elements, yet $2\cdot 3\equiv 0{\pmod {6}}$.
• In general, if $n$ is a composite number, then $\mathbb {Z} _{n}$ does not satisfy the zero-product property. Namely, if $n=qm$ where $0<q,m<n$, then $m$ and $q$ are nonzero modulo $n$, yet $qm\equiv 0{\pmod {n}}$.
• The ring $\mathbb {Z} ^{2\times 2}$ of 2×2 matrices with integer entries does not satisfy the zero-product property: if
$M={\begin{pmatrix}1&-1\\0&0\end{pmatrix}}$
and $N={\begin{pmatrix}0&1\\0&1\end{pmatrix}},$ then
$MN={\begin{pmatrix}1&-1\\0&0\end{pmatrix}}{\begin{pmatrix}0&1\\0&1\end{pmatrix}}={\begin{pmatrix}0&0\\0&0\end{pmatrix}}=0,$
yet neither $M$ nor $N$ is zero.
• The ring of all functions $f:[0,1]\to \mathbb {R} $, from the unit interval to the real numbers, has nontrivial zero divisors: there are pairs of functions which are not identically equal to zero yet whose product is the zero function. In fact, it is not hard to construct, for any n ≥ 2, functions $f_{1},\ldots ,f_{n}$, none of which is identically zero, such that $f_{i}\,f_{j}$ is identically zero whenever $i\neq j$.
• The same is true even if we consider only continuous functions, or only even infinitely smooth functions. On the other hand, analytic functions have the zero-product property.
Application to finding roots of polynomials
Suppose $P$ and $Q$ are univariate polynomials with real coefficients, and $x$ is a real number such that $P(x)Q(x)=0$. (Actually, we may allow the coefficients and $x$ to come from any integral domain.) By the zero-product property, it follows that either $P(x)=0$ or $Q(x)=0$. In other words, the roots of $PQ$ are precisely the roots of $P$ together with the roots of $Q$.
Thus, one can use factorization to find the roots of a polynomial. For example, the polynomial $x^{3}-2x^{2}-5x+6$ factorizes as $(x-3)(x-1)(x+2)$; hence, its roots are precisely 3, 1, and −2.
In general, suppose $R$ is an integral domain and $f$ is a monic univariate polynomial of degree $d\geq 1$ with coefficients in $R$. Suppose also that $f$ has $d$ distinct roots $r_{1},\ldots ,r_{d}\in R$. It follows (but we do not prove here) that $f$ factorizes as $f(x)=(x-r_{1})\cdots (x-r_{d})$. By the zero-product property, it follows that $r_{1},\ldots ,r_{d}$ are the only roots of $f$: any root of $f$ must be a root of $(x-r_{i})$ for some $i$. In particular, $f$ has at most $d$ distinct roots.
If however $R$ is not an integral domain, then the conclusion need not hold. For example, the cubic polynomial $x^{3}+3x^{2}+2x$ has six roots in $\mathbb {Z} _{6}$ (though it has only three roots in $\mathbb {Z} $).
See also
• Fundamental theorem of algebra
• Integral domain and domain
• Prime ideal
• Zero divisor
Notes
1. The other being a⋅0 = 0⋅a = 0. Mustafa A. Munem and David J. Foulis, Algebra and Trigonometry with Applications (New York: Worth Publishers, 1982), p. 4.
2. There must be a notion of zero (the additive identity) and a notion of products, i.e., multiplication.
References
• David S. Dummit and Richard M. Foote, Abstract Algebra (3d ed.), Wiley, 2003, ISBN 0-471-43334-9.
External links
• PlanetMath: Zero rule of product
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Zero-sum Ramsey theory
In mathematics, zero-sum Ramsey theory or zero-sum theory is a branch of combinatorics. It deals with problems of the following kind: given a combinatorial structure whose elements are assigned different weights (usually elements from an Abelian group $A$), one seeks for conditions that guarantee the existence of certain substructure whose weights of its elements sum up to zero (in $A$). It combines tools from number theory, algebra, linear algebra, graph theory, discrete analysis, and other branches of mathematics.
The classic result in this area is the 1961 theorem of Paul Erdős, Abraham Ginzburg, and Abraham Ziv:[1] for any $2m-1$ elements of $\mathbb {Z} _{m}$, there is a subset of size $m$ that sums to zero.[2] (This bound is tight, as a sequence of $m-1$ zeroes and $m-1$ ones cannot have any subset of size $m$ summing to zero.[2]) There are known proofs of this result using the Cauchy-Davenport theorem, Fermat's little theorem, or the Chevalley–Warning theorem.[2]
Generalizing this result, one can define for any abelian group G the minimum quantity $EGZ(G)$ of elements of G such that there must be a subsequence of $o(G)$ elements (where $o(G)$ is the order of the group) which adds to zero. It is known that $EGZ(G)\leq 2o(G)-1$, and that this bound is strict if and only if $G=\mathbb {Z} _{m}$.[2]
See also
• Zero-sum problem
References
1. Erdős, Paul; Ginzburg, A.; Ziv, A. (1961). "Theorem in the additive number theory". Bull. Res. Council Israel. 10F: 41–43. Zbl 0063.00009.
2. "Erdös-Ginzburg-Ziv theorem - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2023-05-22.
Further reading
• Zero-sum problems - A survey (open-access journal article)
• Zero-Sum Ramsey Theory: Graphs, Sequences and More (workshop homepage)
• A. Bialostocki, "Zero-sum trees: a survey of results and open problems" N.W. Sauer (ed.) R.E. Woodrow (ed.) B. Sands (ed.), Finite and Infinite Combinatorics in Sets and Logic, Nato ASI Ser., Kluwer Acad. Publ. (1993) pp. 19–29
• Y. Caro, "Zero-sum problems: a survey" Discrete Math., 152 (1996) pp. 93–113
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Zero-symmetric graph
In the mathematical field of graph theory, a zero-symmetric graph is a connected graph in which each vertex has exactly three incident edges and, for each two vertices, there is a unique symmetry taking one vertex to the other. Such a graph is a vertex-transitive graph but cannot be an edge-transitive graph: the number of symmetries equals the number of vertices, too few to take every edge to every other edge.[1]
The smallest zero-symmetric graph, with 18 vertices and 27 edges
The truncated cuboctahedron, a zero-symmetric polyhedron
Graph families defined by their automorphisms
distance-transitive → distance-regular ← strongly regular
↓
symmetric (arc-transitive) ← t-transitive, t ≥ 2 skew-symmetric
↓
(if connected)
vertex- and edge-transitive
→ edge-transitive and regular → edge-transitive
↓ ↓ ↓
vertex-transitive → regular → (if bipartite)
biregular
↑
Cayley graph ← zero-symmetric asymmetric
The name for this class of graphs was coined by R. M. Foster in a 1966 letter to H. S. M. Coxeter.[2] In the context of group theory, zero-symmetric graphs are also called graphical regular representations of their symmetry groups.[3]
Examples
The smallest zero-symmetric graph is a nonplanar graph with 18 vertices.[4] Its LCF notation is [5,−5]9.
Among planar graphs, the truncated cuboctahedral and truncated icosidodecahedral graphs are also zero-symmetric.[5]
These examples are all bipartite graphs. However, there exist larger examples of zero-symmetric graphs that are not bipartite.[6]
These examples also have three different symmetry classes (orbits) of edges. However, there exist zero-symmetric graphs with only two orbits of edges. The smallest such graph has 20 vertices, with LCF notation [6,6,-6,-6]5.[7]
Properties
Every finite zero-symmetric graph is a Cayley graph, a property that does not always hold for cubic vertex-transitive graphs more generally and that helps in the solution of combinatorial enumeration tasks concerning zero-symmetric graphs. There are 97687 zero-symmetric graphs on up to 1280 vertices. These graphs form 89% of the cubic Cayley graphs and 88% of all connected vertex-transitive cubic graphs on the same number of vertices.[8]
Unsolved problem in mathematics:
Does every finite zero-symmetric graph contain a Hamiltonian cycle?
(more unsolved problems in mathematics)
All known finite connected zero-symmetric graphs contain a Hamiltonian cycle, but it is unknown whether every finite connected zero-symmetric graph is necessarily Hamiltonian.[9] This is a special case of the Lovász conjecture that (with five known exceptions, none of which is zero-symmetric) every finite connected vertex-transitive graph and every finite Cayley graph is Hamiltonian.
See also
• Semi-symmetric graph, graphs that have symmetries between every two edges but not between every two vertices (reversing the roles of edges and vertices in the definition of zero-symmetric graphs)
References
1. Coxeter, Harold Scott MacDonald; Frucht, Roberto; Powers, David L. (1981), Zero-symmetric graphs, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, ISBN 0-12-194580-4, MR 0658666
2. Coxeter, Frucht & Powers (1981), p. ix.
3. Lauri, Josef; Scapellato, Raffaele (2003), Topics in Graph Automorphisms and Reconstruction, London Mathematical Society Student Texts, Cambridge University Press, p. 66, ISBN 9780521529037.
4. Coxeter, Frucht & Powers (1981), Figure 1.1, p. 5.
5. Coxeter, Frucht & Powers (1981), pp. 75 and 80.
6. Coxeter, Frucht & Powers (1981), p. 55.
7. Conder, Marston D. E.; Pisanski, Tomaž; Žitnik, Arjana (2017), "Vertex-transitive graphs and their arc-types", Ars Mathematica Contemporanea, 12 (2): 383–413, arXiv:1505.02029, doi:10.26493/1855-3974.1146.f96, MR 3646702
8. Potočnik, Primož; Spiga, Pablo; Verret, Gabriel (2013), "Cubic vertex-transitive graphs on up to 1280 vertices", Journal of Symbolic Computation, 50: 465–477, arXiv:1201.5317, doi:10.1016/j.jsc.2012.09.002, MR 2996891.
9. Coxeter, Frucht & Powers (1981), p. 10.
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Zero sharp
In the mathematical discipline of set theory, 0# (zero sharp, also 0#) is the set of true formulae about indiscernibles and order-indiscernibles in the Gödel constructible universe. It is often encoded as a subset of the integers (using Gödel numbering), or as a subset of the hereditarily finite sets, or as a real number. Its existence is unprovable in ZFC, the standard form of axiomatic set theory, but follows from a suitable large cardinal axiom. It was first introduced as a set of formulae in Silver's 1966 thesis, later published as Silver (1971), where it was denoted by Σ, and rediscovered by Solovay (1967, p.52), who considered it as a subset of the natural numbers and introduced the notation O# (with a capital letter O; this later changed to the numeral '0').
Roughly speaking, if 0# exists then the universe V of sets is much larger than the universe L of constructible sets, while if it does not exist then the universe of all sets is closely approximated by the constructible sets.
Definition
Zero sharp was defined by Silver and Solovay as follows. Consider the language of set theory with extra constant symbols c1, c2, ... for each positive integer. Then 0# is defined to be the set of Gödel numbers of the true sentences about the constructible universe, with ci interpreted as the uncountable cardinal $\aleph _{i}$. (Here $\aleph _{i}$ means $\aleph _{i}$ in the full universe, not the constructible universe.)
If there is in V an uncountable set of Silver order-indiscernibles in the constructible universe L, then 0# is the set of Gödel numbers of formulas θ of set theory such that
$L_{\omega _{\omega }}\models \theta (\omega _{1},\omega _{2},...\omega _{n})$
where ω1, ... ωω are the "small" uncountable initial ordinals in V, but have all large cardinal properties consistent with V=L relative to L.
There is a subtlety about this definition: by Tarski's undefinability theorem it is not, in general, possible to define the truth of a formula of set theory in the language of set theory. To solve this, Silver and Solovay assumed the existence of a suitable large cardinal, such as a Ramsey cardinal, and showed that with this extra assumption it is possible to define the truth of statements about the constructible universe. More generally, the definition of 0# works provided that there is an uncountable set of indiscernibles for some Lα, and the phrase "0# exists" is used as a shorthand way of saying this.
There are several minor variations of the definition of 0#, which make no significant difference to its properties. There are many different choices of Gödel numbering, and 0# depends on this choice. Instead of being considered as a subset of the natural numbers, it is also possible to encode 0# as a subset of formulae of a language, or as a subset of the hereditarily finite sets, or as a real number.
Statements implying existence
The condition about the existence of a Ramsey cardinal implying that 0# exists can be weakened. The existence of ω1-Erdős cardinals implies the existence of 0#. This is close to being best possible, because the existence of 0# implies that in the constructible universe there is an α-Erdős cardinal for all countable α, so such cardinals cannot be used to prove the existence of 0#.
Chang's conjecture implies the existence of 0#.
Statements equivalent to existence
Kunen showed that 0# exists if and only if there exists a non-trivial elementary embedding for the Gödel constructible universe L into itself.
Donald A. Martin and Leo Harrington have shown that the existence of 0# is equivalent to the determinacy of lightface analytic games. In fact, the strategy for a universal lightface analytic game has the same Turing degree as 0#.
It follows from Jensen's covering theorem that the existence of 0# is equivalent to ωω being a regular cardinal in the constructible universe L.
Silver showed that the existence of an uncountable set of indiscernibles in the constructible universe is equivalent to the existence of 0#.
Consequences of existence and non-existence
Its existence implies that every uncountable cardinal in the set-theoretic universe V is an indiscernible in L and satisfies all large cardinal axioms that are realized in L (such as being totally ineffable). It follows that the existence of 0# contradicts the axiom of constructibility: V = L.
If 0# exists, then it is an example of a non-constructible Δ1
3
set of integers. This is in some sense the simplest possibility for a non-constructible set, since all Σ1
2
and Π1
2
sets of integers are constructible.
On the other hand, if 0# does not exist, then the constructible universe L is the core model—that is, the canonical inner model that approximates the large cardinal structure of the universe considered. In that case, Jensen's covering lemma holds:
For every uncountable set x of ordinals there is a constructible y such that x ⊂ y and y has the same cardinality as x.
This deep result is due to Ronald Jensen. Using forcing it is easy to see that the condition that x is uncountable cannot be removed. For example, consider Namba forcing, that preserves $\omega _{1}$ and collapses $\omega _{2}$ to an ordinal of cofinality $\omega $. Let $G$ be an $\omega $-sequence cofinal on $\omega _{2}^{L}$ and generic over L. Then no set in L of L-size smaller than $\omega _{2}^{L}$ (which is uncountable in V, since $\omega _{1}$ is preserved) can cover $G$, since $\omega _{2}$ is a regular cardinal.
Other sharps
If x is any set, then x# is defined analogously to 0# except that one uses L[x] instead of L. See the section on relative constructibility in constructible universe.
See also
• 0†, a set similar to 0# where the constructible universe is replaced by a larger inner model with a measurable cardinal.
References
• Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics ; V. 76). Elsevier Science Ltd. ISBN 0-444-10535-2.
• Harrington, Leo (1978), "Analytic determinacy and 0#", The Journal of Symbolic Logic, 43 (4): 685–693, doi:10.2307/2273508, ISSN 0022-4812, JSTOR 2273508, MR 0518675, S2CID 46061318
• Jech, Thomas (2003). Set Theory. Springer Monographs in Mathematics (Third Millennium ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-44085-7. Zbl 1007.03002.
• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. ISBN 3-540-00384-3.
• Martin, Donald A. (1970), "Measurable cardinals and analytic games", Polska Akademia Nauk. Fundamenta Mathematicae, 66 (3): 287–291, doi:10.4064/fm-66-3-287-291, ISSN 0016-2736, MR 0258637
• Silver, Jack H. (1971) [1966], "Some applications of model theory in set theory", Annals of Pure and Applied Logic, 3 (1): 45–110, doi:10.1016/0003-4843(71)90010-6, ISSN 0168-0072, MR 0409188
• Solovay, Robert M. (1967), "A nonconstructible Δ1
3
set of integers", Transactions of the American Mathematical Society, 127: 50–75, doi:10.2307/1994631, ISSN 0002-9947, JSTOR 1994631, MR 0211873
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Zero bias transform
The zero-bias transform is a transform from one probability distribution to another. The transform arises in applications of Stein's method in probability and statistics.
Formal definition
The zero bias transform may be applied to both discrete and continuous random variables. The zero bias transform of a density function f(t), defined for all real numbers t ≥ 0, is the function g(s), defined by
$g(s)=\int _{s}^{\infty }tf(t)1(t>s)\,dt$
where s and t are real numbers and f(t) is the density or mass function of the random variable T.[1]
An equivalent but alternative approach is to deduce the nature of the transformed random variable by evaluating the expected value
$\operatorname {E} (TH(T))=\sigma ^{2}E(h(T^{z}))$
where the right-side superscript denotes a zero biased random variable whereas the left hand side expectation represents the original random variable. An example from each approach is given in the examples section beneath.
If the random variable is discrete the integral becomes a sum from positive infinity to s. The zero bias transform is taken for a mean zero, variance 1 random variable which may require a location-scale transform to the random variable.
Applications
The zero bias transformation arises in applications where a normal approximation is desired. Similar to Stein's method the zero bias transform is often applied to sums of random variables with each summand having finite variance an mean zero.
The zero bias transform has been applied to CDO tranche pricing.[2]
Examples
1. Consider a Bernoulli(p) random variable B with Pr(B = 0) = 1 − p. The zero bias transform of T = (B − p) is:
${\begin{aligned}\operatorname {E} (TH(T))&=-p(1-p)H(-p)+(1-p)pH(1-p)\\&=p(1-p)[H(1-p)-H(-p)]\\&=p(1-p)\int _{-p}^{1-p}h(s)\,ds\end{aligned}}$
where h is the derivative of H. From there it follows that the random variable S is a continuous uniform random variable on the support (−p, 1 − p). This example shows how the zero bias transform smooths a discrete distribution into a continuous distribution.
2. Consider the continuous uniform on the support $(-{\sqrt {3}},{\sqrt {3}})$.
$\int _{s}^{\sqrt {3}}t1(t>s)f(t)\,dt=\int _{s}^{\sqrt {3}}{\frac {t}{2{\sqrt {3}}}}\,dt={\frac {\sqrt {3}}{4}}-{\frac {s^{2}}{{\sqrt {3}}\,4}}{\text{ where }}-{\sqrt {3}}<s<{\sqrt {3}}$
This example shows that the zero bias transform takes continuous symmetric distributions and makes them unimodular.
References
1. Goldstein, Larry; Reinert, Gesine (1997), "Stein's Method and the Zero Bias Transformation with Application to Simple Random Sampling" (PDF), The Annals of Applied Probability, 7 (4): 935–952
2. Karoui, N. El; Jiao, Y. (2009). "Stein's method and zero bias transformation for CDO tranche pricing". Finance and Stochastics. 13 (2): 151–180. doi:10.1007/s00780-008-0084-6.
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Zero crossing
A zero-crossing is a point where the sign of a mathematical function changes (e.g. from positive to negative), represented by an intercept of the axis (zero value) in the graph of the function. It is a commonly used term in electronics, mathematics, acoustics, and image processing.
In electronics
In alternating current, the zero-crossing is the instantaneous point at which there is no voltage present. In a sine wave or other simple waveform, this normally occurs twice during each cycle. It is a device for detecting the point where the voltage crosses zero in either direction.
The zero-crossing is important for systems that send digital data over AC circuits, such as modems, X10 home automation control systems, and Digital Command Control type systems for Lionel and other AC model trains.
Counting zero-crossings is also a method used in speech processing to estimate the fundamental frequency of speech.
In a system where an amplifier with digitally controlled gain is applied to an input signal, artifacts in the non-zero output signal occur when the gain of the amplifier is abruptly switched between its discrete gain settings. At audio frequencies, such as in modern consumer electronics like digital audio players, these effects are clearly audible, resulting in a 'zipping' sound when rapidly ramping the gain or a soft 'click' when a single gain change is made. Artifacts are disconcerting and clearly not desirable. If changes are made only at zero-crossings of the input signal, then no matter how the amplifier gain setting changes, the output also remains at zero, thereby minimizing the change. (The instantaneous change in gain will still produce distortion, but it will not produce a click.)
If electrical power is to be switched, no electrical interference is generated if switched at an instant when there is no current—a zero crossing. Early light dimmers and similar devices generated interference; later versions were designed to switch at the zero crossing.
In image processing
In the field of digital image Processing, great emphasis is placed on operators that seek out edges within an image. They are called edge detection or gradient filters. A gradient filter is a filter that seeks out areas of rapid change in pixel value. These points usually mark an edge or a boundary. A Laplace filter is a filter that fits in this family, though it sets about the task in a different way. It seeks out points in the signal stream where the digital signal of an image passes through a pre-set '0' value, and marks this out as a potential edge point. Because the signal has crossed through the point of zero, it is called a zero-crossing. An example can be found here, including the source in Java.
In the field of industrial radiography, it is used as a simple method for the segmentation of potential defects.[1]
See also
• Reconstruction from zero crossings
• Zero crossing control
• Zero-crossing rate
• Zero of a function (a root)
• Sign function
References
1. Mery, Domingo (2015). Computer Vision for X-Ray Testing. Switzerland: Springer International Publishing. p. 271. ISBN 978-3319207469.
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Zero dagger
In set theory, 0† (zero dagger) is a particular subset of the natural numbers, first defined by Robert M. Solovay in unpublished work in the 1960s. (The superscript † should be a dagger, but it appears as a plus sign on some browsers.) The definition is a bit awkward, because there might be no set of natural numbers satisfying the conditions. Specifically, if ZFC is consistent, then ZFC + "0† does not exist" is consistent. ZFC + "0† exists" is not known to be inconsistent (and most set theorists believe that it is consistent). In other words, it is believed to be independent (see large cardinal for a discussion). It is usually formulated as follows:
0† exists if and only if there exists a non-trivial elementary embedding j : L[U] → L[U] for the relativized Gödel constructible universe L[U], where U is an ultrafilter witnessing that some cardinal κ is measurable.
If 0† exists, then a careful analysis of the embeddings of L[U] into itself reveals that there is a closed unbounded subset of κ, and a closed unbounded proper class of ordinals greater than κ, which together are indiscernible for the structure $(L,\in ,U)$, and 0† is defined to be the set of Gödel numbers of the true formulas about the indiscernibles in L[U].
Solovay showed that the existence of 0† follows from the existence of two measurable cardinals. It is traditionally considered a large cardinal axiom, although it is not a large cardinal, nor indeed a cardinal at all.
See also
• 0#: a set of formulas (or subset of the integers) defined in a similar fashion, but simpler.
References
• Kanamori, Akihiro; Awerbuch-Friedlander, Tamara (1990). "The compleat 0†". Zeitschrift für Mathematische Logik und Grundlagen der Mathematik. 36 (2): 133–141. doi:10.1002/malq.19900360206. ISSN 0044-3050. MR 1068949.
• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. ISBN 3-540-00384-3.
External links
• Definition by "Zentralblatt math database" (PDF)
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Zero dynamics
In mathematics, zero dynamics is known as the concept of evaluating the effect of zero on systems.[1]
History
The idea was introduced thirty years ago as the nonlinear approach to the concept of transmission of zeros. The original purpose of introducing the concept was to develop an asymptotic stabilization with a set of guaranteed regions of attraction (semi-global stabilizability), to make the overall system stable.[2]
Initial working
Given the internal dynamics of any system, zero dynamics refers to the control action chosen in which the output variables of the system are kept identically zero.[3] While, various systems have an equally distinctive set of zeros, such as decoupling zeros, invariant zeros, and transmission zeros. Thus, the reason for developing this concept was to control the non-minimum phase and nonlinear systems effectively.[4]
Applications
The concept is widely utilized in SISO mechanical systems, whereby applying a few heuristic approaches, zeros can be identified for various linear systems.[5] Zero dynamics adds an essential feature to the overall system’s analysis and the design of the controllers. Mainly its behavior plays a significant role in measuring the performance limitations of specific feedback systems. In a Single Input Single Output system, the zero dynamics can be identified by using junction structure patterns. In other words, using concepts like bond graph models can help to point out the potential direction of the SISO systems.[6]
Apart from its application in nonlinear standardized systems, similar controlled results can be obtained by using zero dynamics on nonlinear discrete-time systems. In this scenario, the application of zero dynamics can be an interesting tool to measure the performance of nonlinear digital design systems (nonlinear discrete-time systems).[7]
Before the advent of zero dynamics, the problem of acquiring non-interacting control systems by using internal stability was not specifically discussed. However, with the asymptotic stability present within the zero dynamics of a system, static feedback can be ensured. Such results make zero dynamics an interesting tool to guarantee the internal stability of non-interacting control systems.[8]
References
1. Van de Straete, H.J.; Youcef-Toumi, K. (June 1996). "Physical Meaning of Zeros and Transmission Zeros from Bond Graph Models". IFAC Proceedings Volumes. 29 (1): 4422–4427. doi:10.1016/s1474-6670(17)58377-9. hdl:1721.1/11140. ISSN 1474-6670.
2. Isidori, Alberto (September 2013). "The zero dynamics of a nonlinear system: From the origin to the latest progresses of a long successful story". European Journal of Control. 19 (5): 369–378. doi:10.1016/j.ejcon.2013.05.014. ISSN 0947-3580. S2CID 15277067.
3. Youcef-Toumi, K.; Wu, S-T (June 1991). "Input/Output Linearization using Time Delay Control". 1991 American Control Conference. IEEE: 2601–2606. doi:10.23919/acc.1991.4791872. ISBN 0-87942-565-2. S2CID 20562917.
4. "Control Theory", Analytic and Geometric Study of Stratified Spaces, Lecture Notes in Mathematics, vol. 1768, Springer Berlin Heidelberg, 2001, pp. 91–149, doi:10.1007/3-540-45436-5_5, ISBN 978-3-540-42626-4
5. Miu, D. K. (1991-09-01). "Physical Interpretation of Transfer Function Zeros for Simple Control Systems With Mechanical Flexibilities". Journal of Dynamic Systems, Measurement, and Control. 113 (3): 419–424. doi:10.1115/1.2896426. ISSN 0022-0434.
6. Huang, S.Y.; Youcef-Toumi, K. (June 1996). "Zero Dynamics of Nonlinear MIMO Systems from System Configurations - A Bond Graph Approach". IFAC Proceedings Volumes. 29 (1): 4392–4397. doi:10.1016/s1474-6670(17)58372-x. ISSN 1474-6670.
7. Monaco, S.; Normand-Cyrot, D. (September 1988). "Zero dynamics of sampled nonlinear systems". Systems & Control Letters. 11 (3): 229–234. doi:10.1016/0167-6911(88)90063-1. ISSN 0167-6911.
8. Isidori, A.; Grizzle, J.W. (October 1988). "Fixed modes and nonlinear noninteracting control with stability". IEEE Transactions on Automatic Control. 33 (10): 907–914. doi:10.1109/9.7244. ISSN 0018-9286.
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ZPP (complexity)
In complexity theory, ZPP (zero-error probabilistic polynomial time) is the complexity class of problems for which a probabilistic Turing machine exists with these properties:
• It always returns the correct YES or NO answer.
• The running time is polynomial in expectation for every input.
In other words, if the algorithm is allowed to flip a truly-random coin while it is running, it will always return the correct answer and, for a problem of size n, there is some polynomial p(n) such that the average running time will be less than p(n), even though it might occasionally be much longer. Such an algorithm is called a Las Vegas algorithm.
Alternatively, ZPP can be defined as the class of problems for which a probabilistic Turing machine exists with these properties:
• It always runs in polynomial time.
• It returns an answer YES, NO or DO NOT KNOW.
• The answer is always either DO NOT KNOW or the correct answer.
• It returns DO NOT KNOW with probability at most 1/2 for every input (and the correct answer otherwise).
The two definitions are equivalent.
The definition of ZPP is based on probabilistic Turing machines, but, for clarity, note that other complexity classes based on them include BPP and RP. The class BQP is based on another machine with randomness: the quantum computer.
Intersection definition
The class ZPP is exactly equal to the intersection of the classes RP and co-RP. This is often taken to be the definition of ZPP. To show this, first note that every problem which is in both RP and co-RP has a Las Vegas algorithm as follows:
• Suppose we have a language L recognized by both the RP algorithm A and the (possibly completely different) co-RP algorithm B.
• Given an input, run A on the input for one step. If it returns YES, the answer must be YES. Otherwise, run B on the input for one step. If it returns NO, the answer must be NO. If neither occurs, repeat this step.
Note that only one machine can ever give a wrong answer, and the chance of that machine giving the wrong answer during each repetition is at most 50%. This means that the chance of reaching the kth round shrinks exponentially in k, showing that the expected running time is polynomial. This shows that RP intersect co-RP is contained in ZPP.
To show that ZPP is contained in RP intersect co-RP, suppose we have a Las Vegas algorithm C to solve a problem. We can then construct the following RP algorithm:
• Run C for at least double its expected running time. If it gives an answer, give that answer. If it doesn't give any answer before we stop it, give NO.
By Markov's Inequality, the chance that it will yield an answer before we stop it is at least 1/2. This means the chance we'll give the wrong answer on a YES instance, by stopping and yielding NO, is at most 1/2, fitting the definition of an RP algorithm. The co-RP algorithm is identical, except that it gives YES if C "times out".
Witness and proof
The classes NP, RP and ZPP can be thought of in terms of proof of membership in a set.
Definition: A verifier V for a set X is a Turing machine such that:
• if x is in X then there exists a string w such that V(x,w) accepts;
• if x is not in X, then for all strings w, V(x,w) rejects.
The string w can be thought of as the proof of membership. In the case of short proofs (of length bounded by a polynomial in the size of the input) which can be efficiently verified (V is a polynomial-time deterministic Turing machine), the string w is called a witness.
Notes:
• The definition is very asymmetric. The proof of x being in X is a single string. The proof of x not being in X is the collection of all strings, none of which is a proof of membership.
• For all x in X there must be a witness to its membership in X.
• The witness need not be a traditionally construed proof. If V is a probabilistic Turing machine which could possibly accept x if x is in X, then the proof is the string of coin flips which leads the machine to accept x (provided all members in X have some witness and the machine never accepts a non-member).
• The co-concept is a proof of non-membership, or membership in the complement set.
The classes NP, RP and ZPP are sets which have witnesses for membership. The class NP requires only that witnesses exist. They may be very rare. Of the 2f(|x|) possible strings, with f a polynomial, only one need cause the verifier to accept (if x is in X. If x is not in X, no string will cause the verifier to accept).
For the classes RP and ZPP any string chosen at random will likely be a witness.
The corresponding co-classes have witness for non-membership. In particular, co-RP is the class of sets for which, if x is not in X, any randomly chosen string is likely to be a witness for non-membership. ZPP is the class of sets for which any random string is likely to be a witness of x in X, or x not in X, which ever the case may be.
Connecting this definition with other definitions of RP, co-RP and ZPP is easy. The probabilistic polynomial-time Turing Machine V*w(x) corresponds to the deterministic polynomial-time Turing Machine V(x, w) by replacing the random tape of V* with a second input tape for V on which is written the sequence of coin flips. By selecting the witness as a random string, the verifier is a probabilistic polynomial-time Turing Machine whose probability of accepting x when x is in X is large (greater than 1/2, say), but zero if x ∉ X (for RP); of rejecting x when x is not in X is large but zero if x ∈ X (for co-RP); and of correctly accepting or rejecting x as a member of X is large, but zero of incorrectly accepting or rejecting x (for ZPP).
By repeated random selection of a possible witness, the large probability that a random string is a witness gives an expected polynomial time algorithm for accepting or rejecting an input. Conversely, if the Turing Machine is expected polynomial-time (for any given x), then a considerable fraction of the runs must be polynomial-time bounded, and the coin sequence used in such a run will be a witness.
ZPP should be contrasted with BPP. The class BPP does not require witnesses, although witnesses are sufficient (hence BPP contains RP, co-RP and ZPP). A BPP language has V(x,w) accept on a (clear) majority of strings w if x is in X, and conversely reject on a (clear) majority of strings w if x is not in X. No single string w need be definitive, and therefore they cannot in general be considered proofs or witnesses.
Complexity-theoretic properties
It is known that ZPP is closed under complement; that is, ZPP = co-ZPP.
ZPP is low for itself, meaning that a ZPP machine with the power to solve ZPP problems instantly (a ZPP oracle machine) is not any more powerful than the machine without this extra power. In symbols, ZPPZPP = ZPP.
ZPPNPBPP = ZPPNP.
NPBPP is contained in ZPPNP.
Connection to other classes
Since ZPP = RP ∩ coRP, ZPP is obviously contained in both RP and coRP.
The class P is contained in ZPP, and some computer scientists have conjectured that P = ZPP, i.e., every Las Vegas algorithm has a deterministic polynomial-time equivalent.
There exists an oracle relative to which ZPP = EXPTIME. A proof for ZPP = EXPTIME would imply that P ≠ ZPP, as P ≠ EXPTIME (see time hierarchy theorem).
See also
• BPP
• RP
External links
• Complexity Zoo: ZPP Class ZPP
Important complexity classes
Considered feasible
• DLOGTIME
• AC0
• ACC0
• TC0
• L
• SL
• RL
• NL
• NL-complete
• NC
• SC
• CC
• P
• P-complete
• ZPP
• RP
• BPP
• BQP
• APX
• FP
Suspected infeasible
• UP
• NP
• NP-complete
• NP-hard
• co-NP
• co-NP-complete
• AM
• QMA
• PH
• ⊕P
• PP
• #P
• #P-complete
• IP
• PSPACE
• PSPACE-complete
Considered infeasible
• EXPTIME
• NEXPTIME
• EXPSPACE
• 2-EXPTIME
• ELEMENTARY
• PR
• R
• RE
• ALL
Class hierarchies
• Polynomial hierarchy
• Exponential hierarchy
• Grzegorczyk hierarchy
• Arithmetical hierarchy
• Boolean hierarchy
Families of classes
• DTIME
• NTIME
• DSPACE
• NSPACE
• Probabilistically checkable proof
• Interactive proof system
List of complexity classes
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Trivial group
In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually denoted as such: $0,1,$ or $e$ depending on the context. If the group operation is denoted $\,\cdot \,$ then it is defined by $e\cdot e=e.$
The similarly defined trivial monoid is also a group since its only element is its own inverse, and is hence the same as the trivial group.
The trivial group is distinct from the empty set, which has no elements, hence lacks an identity element, and so cannot be a group.
Definitions
Given any group $G,$ the group consisting of only the identity element is a subgroup of $G,$ and, being the trivial group, is called the trivial subgroup of $G.$
The term, when referred to "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/":): G has no nontrivial proper subgroups" refers to the only subgroups of $G$ being the trivial group $\{e\}$ and the group $G$ itself.
Properties
The trivial group is cyclic of order $1$; as such it may be denoted $\mathrm {Z} _{1}$ or $\mathrm {C} _{1}.$ If the group operation is called addition, the trivial group is usually denoted by $0.$ If the group operation is called multiplication then 1 can be a notation for the trivial group. Combining these leads to the trivial ring in which the addition and multiplication operations are identical and $0=1.$
The trivial group serves as the zero object in the category of groups, meaning it is both an initial object and a terminal object.
The trivial group can be made a (bi-)ordered group by equipping it with the trivial non-strict order $\,\leq .$
See also
• Zero object (algebra) – Algebraic structure with only one element
• List of small groups
References
• Rowland, Todd & Weisstein, Eric W. "Trivial Group". MathWorld.
Groups
Basic notions
• Subgroup
• Normal subgroup
• Commutator subgroup
• Quotient group
• Group homomorphism
• (Semi-) direct product
• direct sum
Types of groups
• Finite groups
• Abelian groups
• Cyclic groups
• Infinite group
• Simple groups
• Solvable groups
• Symmetry group
• Space group
• Point group
• Wallpaper group
• Trivial group
Discrete groups
Classification of finite simple groups
Cyclic group Zn
Alternating group An
Sporadic groups
Mathieu group M11..12,M22..24
Conway group Co1..3
Janko groups J1, J2, J3, J4
Fischer group F22..24
Baby monster group B
Monster group M
Other finite groups
Symmetric group Sn
Dihedral group Dn
Rubik's Cube group
Lie groups
• General linear group GL(n)
• Special linear group SL(n)
• Orthogonal group O(n)
• Special orthogonal group SO(n)
• Unitary group U(n)
• Special unitary group SU(n)
• Symplectic group Sp(n)
Exceptional Lie groups
G2
F4
E6
E7
E8
• Circle group
• Lorentz group
• Poincaré group
• Quaternion group
Infinite dimensional groups
• Conformal group
• Diffeomorphism group
• Loop group
• Quantum group
• O(∞)
• SU(∞)
• Sp(∞)
• History
• Applications
• Abstract algebra
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Locus (mathematics)
In geometry, a locus (plural: loci) (Latin word for "place", "location") is a set of all points (commonly, a line, a line segment, a curve or a surface), whose location satisfies or is determined by one or more specified conditions.[1][2]
The set of the points that satisfy some property is often called the locus of a point satisfying this property. The use of the singular in this formulation is a witness that, until the end of the 19th century, mathematicians did not consider infinite sets. Instead of viewing lines and curves as sets of points, they viewed them as places where a point may be located or may move.
History and philosophy
Until the beginning of the 20th century, a geometrical shape (for example a curve) was not considered as an infinite set of points; rather, it was considered as an entity on which a point may be located or on which it moves. Thus a circle in the Euclidean plane was defined as the locus of a point that is at a given distance of a fixed point, the center of the circle. In modern mathematics, similar concepts are more frequently reformulated by describing shapes as sets; for instance, one says that the circle is the set of points that are at a given distance from the center.[3]
In contrast to the set-theoretic view, the old formulation avoids considering infinite collections, as avoiding the actual infinite was an important philosophical position of earlier mathematicians.[4][5]
Once set theory became the universal basis over which the whole mathematics is built,[6] the term of locus became rather old-fashioned.[7] Nevertheless, the word is still widely used, mainly for a concise formulation, for example:
• Critical locus, the set of the critical points of a differentiable function.
• Zero locus or vanishing locus, the set of points where a function vanishes, in that it takes the value zero.
• Singular locus, the set of the singular points of an algebraic variety.
• Connectedness locus, the subset of the parameter set of a family of rational functions for which the Julia set of the function is connected.
More recently, techniques such as the theory of schemes, and the use of category theory instead of set theory to give a foundation to mathematics, have returned to notions more like the original definition of a locus as an object in itself rather than as a set of points.[5]
Examples in plane geometry
Examples from plane geometry include:
• The set of points equidistant from two points is a perpendicular bisector to the line segment connecting the two points.[8]
• The set of points equidistant from two intersecting lines is the union of their two angle bisectors.
• All conic sections are loci:[9]
• Circle: the set of points at constant distance (the radius) from a fixed point (the center).
• Parabola: the set of points equidistant from a fixed point (the focus) and a line (the directrix).
• Hyperbola: the set of points for each of which the absolute value of the difference between the distances to two given foci is a constant.
• Ellipse: the set of points for each of which the sum of the distances to two given foci is a constant
Other examples of loci appear in various areas of mathematics. For example, in complex dynamics, the Mandelbrot set is a subset of the complex plane that may be characterized as the connectedness locus of a family of polynomial maps.
Proof of a locus
To prove a geometric shape is the correct locus for a given set of conditions, one generally divides the proof into two stages: the proof that all the points that satisfy the conditions are on the given shape, and the proof that all the points on the given shape satisfy the conditions.[10]
Examples
First example
Find the locus of a point P that has a given ratio of distances k = d1/d2 to two given points.
In this example k = 3, A(−1, 0) and B(0, 2) are chosen as the fixed points.
P(x, y) is a point of the locus
$\Leftrightarrow |PA|=3|PB|$
$\Leftrightarrow |PA|^{2}=9|PB|^{2}$
$\Leftrightarrow (x+1)^{2}+(y-0)^{2}=9(x-0)^{2}+9(y-2)^{2}$
$\Leftrightarrow 8(x^{2}+y^{2})-2x-36y+35=0$
$\Leftrightarrow \left(x-{\frac {1}{8}}\right)^{2}+\left(y-{\frac {9}{4}}\right)^{2}={\frac {45}{64}}.$
This equation represents a circle with center (1/8, 9/4) and radius ${\tfrac {3}{8}}{\sqrt {5}}$. It is the circle of Apollonius defined by these values of k, A, and B.
Second example
A triangle ABC has a fixed side [AB] with length c. Determine the locus of the third vertex C such that the medians from A and C are orthogonal.
Choose an orthonormal coordinate system such that A(−c/2, 0), B(c/2, 0). C(x, y) is the variable third vertex. The center of [BC] is M((2x + c)/4, y/2). The median from C has a slope y/x. The median AM has slope 2y/(2x + 3c).
C(x, y) is a point of the locus
$\Leftrightarrow $ the medians from A and C are orthogonal
$\Leftrightarrow {\frac {y}{x}}\cdot {\frac {2y}{2x+3c}}=-1$
$\Leftrightarrow 2y^{2}+2x^{2}+3cx=0$
$\Leftrightarrow x^{2}+y^{2}+(3c/2)x=0$
$\Leftrightarrow (x+3c/4)^{2}+y^{2}=9c^{2}/16.$
The locus of the vertex C is a circle with center (−3c/4, 0) and radius 3c/4.
Third example
A locus can also be defined by two associated curves depending on one common parameter. If the parameter varies, the intersection points of the associated curves describe the locus.
In the figure, the points K and L are fixed points on a given line m. The line k is a variable line through K. The line l through L is perpendicular to k. The angle $\alpha $ between k and m is the parameter. k and l are associated lines depending on the common parameter. The variable intersection point S of k and l describes a circle. This circle is the locus of the intersection point of the two associated lines.
Fourth example
A locus of points need not be one-dimensional (as a circle, line, etc.). For example,[1] the locus of the inequality 2x + 3y – 6 < 0 is the portion of the plane that is below the line of equation 2x + 3y – 6 = 0.
See also
• Algebraic variety
• Curve
• Line (geometry)
• Set-builder notation
• Shape (geometry)
References
1. James, Robert Clarke; James, Glenn (1992), Mathematics Dictionary, Springer, p. 255, ISBN 978-0-412-99041-0.
2. Whitehead, Alfred North (1911), An Introduction to Mathematics, H. Holt, p. 121, ISBN 978-1-103-19784-2.
3. Cooke, Roger L. (2012), "38.3 Topology", The History of Mathematics: A Brief Course (3rd ed.), John Wiley & Sons, ISBN 9781118460290, The word locus is one that we still use today to denote the path followed by a point moving subject to stated constraints, although, since the introduction of set theory, a locus is more often thought of statically as the set of points satisfying a given collection.
4. Bourbaki, N. (2013), Elements of the History of Mathematics, translated by J. Meldrum, Springer, p. 26, ISBN 9783642616938, the classical mathematicians carefully avoided introducing into their reasoning the 'actual infinity'.
5. Borovik, Alexandre (2010), "6.2.4 Can one live without actual infinity?", Mathematics Under the Microscope: Notes on Cognitive Aspects of Mathematical Practice, American Mathematical Society, p. 124, ISBN 9780821847619.
6. Mayberry, John P. (2000), The Foundations of Mathematics in the Theory of Sets, Encyclopedia of Mathematics and its Applications, vol. 82, Cambridge University Press, p. 7, ISBN 9780521770347, set theory provides the foundations for all mathematics.
7. Ledermann, Walter; Vajda, S. (1985), Combinatorics and Geometry, Part 1, Handbook of Applicable Mathematics, vol. 5, Wiley, p. 32, ISBN 9780471900238, We begin by explaining a slightly old-fashioned term.
8. George E. Martin, The Foundations of Geometry and the Non-Euclidean Plane, Springer-Verlag, 1975.
9. Hamilton, Henry Parr (1834), An Analytical System of Conic Sections: Designed for the Use of Students, Springer.
10. G. P. West, The new geometry: form 1.
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Zero matrix
In mathematics, particularly linear algebra, a zero matrix or null matrix is a matrix all of whose entries are zero. It also serves as the additive identity of the additive group of $m\times n$ matrices, and is denoted by the symbol $O$ or $0$ followed by subscripts corresponding to the dimension of the matrix as the context sees fit.[1][2][3] Some examples of zero matrices are
$0_{1,1}={\begin{bmatrix}0\end{bmatrix}},\ 0_{2,2}={\begin{bmatrix}0&0\\0&0\end{bmatrix}},\ 0_{2,3}={\begin{bmatrix}0&0&0\\0&0&0\end{bmatrix}}.\ $
Properties
The set of $m\times n$ matrices with entries in a ring K forms a ring $K_{m,n}$. The zero matrix $0_{K_{m,n}}\,$ in $K_{m,n}\,$ is the matrix with all entries equal to $0_{K}\,$, where $0_{K}$ is the additive identity in K.
$0_{K_{m,n}}={\begin{bmatrix}0_{K}&0_{K}&\cdots &0_{K}\\0_{K}&0_{K}&\cdots &0_{K}\\\vdots &\vdots &\ddots &\vdots \\0_{K}&0_{K}&\cdots &0_{K}\end{bmatrix}}_{m\times n}$
The zero matrix is the additive identity in $K_{m,n}\,$.[4] That is, for all $A\in K_{m,n}\,$ it satisfies the equation
$0_{K_{m,n}}+A=A+0_{K_{m,n}}=A.$
There is exactly one zero matrix of any given dimension m×n (with entries from a given ring), so when the context is clear, one often refers to the zero matrix. In general, the zero element of a ring is unique, and is typically denoted by 0 without any subscript indicating the parent ring. Hence the examples above represent zero matrices over any ring.
The zero matrix also represents the linear transformation which sends all the vectors to the zero vector.[5] It is idempotent, meaning that when it is multiplied by itself, the result is itself.
The zero matrix is the only matrix whose rank is 0.
Occurrences
The mortal matrix problem is the problem of determining, given a finite set of n × n matrices with integer entries, whether they can be multiplied in some order, possibly with repetition, to yield the zero matrix. This is known to be undecidable for a set of six or more 3 × 3 matrices, or a set of two 15 × 15 matrices.[6]
In ordinary least squares regression, if there is a perfect fit to the data, the annihilator matrix is the zero matrix.
See also
• Identity matrix, the multiplicative identity for matrices
• Matrix of ones, a matrix where all elements are one
• Nilpotent matrix
• Single-entry matrix, a matrix where all but one element is zero
References
1. Lang, Serge (1987), Linear Algebra, Undergraduate Texts in Mathematics, Springer, p. 25, ISBN 9780387964126, We have a zero matrix in which aij = 0 for all i, j. ... We shall write it O.
2. "Intro to zero matrices (article) | Matrices". Khan Academy. Retrieved 2020-08-13.
3. Weisstein, Eric W. "Zero Matrix". mathworld.wolfram.com. Retrieved 2020-08-13.
4. Warner, Seth (1990), Modern Algebra, Courier Dover Publications, p. 291, ISBN 9780486663418, The neutral element for addition is called the zero matrix, for all of its entries are zero.
5. Bronson, Richard; Costa, Gabriel B. (2007), Linear Algebra: An Introduction, Academic Press, p. 377, ISBN 9780120887842, The zero matrix represents the zero transformation 0, having the property 0(v) = 0 for every vector v ∈ V.
6. Cassaigne, Julien; Halava, Vesa; Harju, Tero; Nicolas, Francois (2014). "Tighter Undecidability Bounds for Matrix Mortality, Zero-in-the-Corner Problems, and More". arXiv:1404.0644 [cs.DM].
Matrix classes
Explicitly constrained entries
• Alternant
• Anti-diagonal
• Anti-Hermitian
• Anti-symmetric
• Arrowhead
• Band
• Bidiagonal
• Bisymmetric
• Block-diagonal
• Block
• Block tridiagonal
• Boolean
• Cauchy
• Centrosymmetric
• Conference
• Complex Hadamard
• Copositive
• Diagonally dominant
• Diagonal
• Discrete Fourier Transform
• Elementary
• Equivalent
• Frobenius
• Generalized permutation
• Hadamard
• Hankel
• Hermitian
• Hessenberg
• Hollow
• Integer
• Logical
• Matrix unit
• Metzler
• Moore
• Nonnegative
• Pentadiagonal
• Permutation
• Persymmetric
• Polynomial
• Quaternionic
• Signature
• Skew-Hermitian
• Skew-symmetric
• Skyline
• Sparse
• Sylvester
• Symmetric
• Toeplitz
• Triangular
• Tridiagonal
• Vandermonde
• Walsh
• Z
Constant
• Exchange
• Hilbert
• Identity
• Lehmer
• Of ones
• Pascal
• Pauli
• Redheffer
• Shift
• Zero
Conditions on eigenvalues or eigenvectors
• Companion
• Convergent
• Defective
• Definite
• Diagonalizable
• Hurwitz
• Positive-definite
• Stieltjes
Satisfying conditions on products or inverses
• Congruent
• Idempotent or Projection
• Invertible
• Involutory
• Nilpotent
• Normal
• Orthogonal
• Unimodular
• Unipotent
• Unitary
• Totally unimodular
• Weighing
With specific applications
• Adjugate
• Alternating sign
• Augmented
• Bézout
• Carleman
• Cartan
• Circulant
• Cofactor
• Commutation
• Confusion
• Coxeter
• Distance
• Duplication and elimination
• Euclidean distance
• Fundamental (linear differential equation)
• Generator
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Null set
In mathematical analysis, a null set is a Lebesgue measurable set of real numbers that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length.
For the set with no elements, see Empty set.
For the set of zeros of a function, see Zero set.
The notion of null set should not be confused with the empty set as defined in set theory. Although the empty set has Lebesgue measure zero, there are also non-empty sets which are null. For example, any non-empty countable set of real numbers has Lebesgue measure zero and therefore is null.
More generally, on a given measure space $M=(X,\Sigma ,\mu )$ a null set is a set $S\in \Sigma $ such that $\mu (S)=0.$
Examples
Every finite or countably infinite subset of the real numbers is a null set. For example, the set of natural numbers and the set of rational numbers are both countably infinite and therefore are null sets when considered as subsets of the real numbers.
The Cantor set is an example of an uncountable null set.
Definition
Suppose $A$ is a subset of the real line $\mathbb {R} $ such that for every $\varepsilon >0,$ there exists a sequence $U_{1},U_{2},\ldots $ of open intervals (where interval $U_{n}=(a_{n},b_{n})\subseteq \mathbb {R} $ has length $\operatorname {length} (U_{n})=b_{n}-a_{n}$) such that
$A\subseteq \bigcup _{n=1}^{\infty }U_{n}\ ~{\textrm {and}}~\ \sum _{n=1}^{\infty }\operatorname {length} (U_{n})<\varepsilon \,,$
then $A$ is a null set,[1] also known as a set of zero-content.
In terminology of mathematical analysis, this definition requires that there be a sequence of open covers of $A$ for which the limit of the lengths of the covers is zero.
Properties
The empty set is always a null set. More generally, any countable union of null sets is null. Any subset of a null set is itself a null set. Together, these facts show that the 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/":): m -null sets of $X$ form a sigma-ideal on $X.$ Similarly, the measurable $m$-null sets form a sigma-ideal of the sigma-algebra of measurable sets. Thus, null sets may be interpreted as negligible sets, defining a notion of almost everywhere.
Lebesgue measure
The Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space.
A subset $N$ of $\mathbb {R} $ has null Lebesgue measure and is considered to be a null set in $\mathbb {R} $ if and only if:
Given any positive number $\varepsilon ,$ there is a sequence $I_{1},I_{2},\ldots $ of intervals in $\mathbb {R} $ such that $N$ is contained in the union of the $I_{1},I_{2},\ldots $ and the total length of the union is less than $\varepsilon .$
This condition can be generalised to $\mathbb {R} ^{n},$ using $n$-cubes instead of intervals. In fact, the idea can be made to make sense on any manifold, even if there is no Lebesgue measure there.
For instance:
• With respect to $\mathbb {R} ^{n},$ all singleton sets are null, and therefore all countable sets are null. In particular, the set $\mathbb {Q} $ of rational numbers is a null set, despite being dense in $\mathbb {R} .$
• The standard construction of the Cantor set is an example of a null uncountable set in $\mathbb {R} ;$ ;} however other constructions are possible which assign the Cantor set any measure whatsoever.
• All the subsets of $\mathbb {R} ^{n}$ whose dimension is smaller than $n$ have null Lebesgue measure in $\mathbb {R} ^{n}.$ For instance straight lines or circles are null sets in $\mathbb {R} ^{2}.$
• Sard's lemma: the set of critical values of a smooth function has measure zero.
If $\lambda $ is Lebesgue measure for $\mathbb {R} $ and π is Lebesgue measure for $\mathbb {R} ^{2}$, then the product measure $\lambda \times \lambda =\pi .$ In terms of null sets, the following equivalence has been styled a Fubini's theorem:[2]
• For $A\subset \mathbb {R} ^{2}$ and $A_{x}=\{y:(x,y)\in A\},$
$\pi (A)=0\iff \lambda \left(\left\{x:\lambda \left(A_{x}\right)>0\right\}\right)=0.$
Uses
Null sets play a key role in the definition of the Lebesgue integral: if functions $f$ and $g$ are equal except on a null set, then $f$ is integrable if and only if $g$ is, and their integrals are equal. This motivates the formal definition of $L^{p}$ spaces as sets of equivalence classes of functions which differ only on null sets.
A measure in which all subsets of null sets are measurable is complete. Any non-complete measure can be completed to form a complete measure by asserting that subsets of null sets have measure zero. Lebesgue measure is an example of a complete measure; in some constructions, it is defined as the completion of a non-complete Borel measure.
A subset of the Cantor set which is not Borel measurable
The Borel measure is not complete. One simple construction is to start with the standard Cantor set $K,$ which is closed hence Borel measurable, and which has measure zero, and to find a subset $F$ of $K$ which is not Borel measurable. (Since the Lebesgue measure is complete, this $F$ is of course Lebesgue measurable.)
First, we have to know that every set of positive measure contains a nonmeasurable subset. Let $f$ be the Cantor function, a continuous function which is locally constant on $K^{c},$ and monotonically increasing on $[0,1],$ with $f(0)=0$ and $f(1)=1.$ Obviously, $f(K^{c})$ is countable, since it contains one point per component of $K^{c}.$ Hence $f(K^{c})$ has measure zero, so $f(K)$ has measure one. We need a strictly monotonic function, so consider $g(x)=f(x)+x.$ Since $f(x)$ is strictly monotonic and continuous, it is a homeomorphism. Furthermore, $g(K)$ has measure one. Let $E\subseteq g(K)$ be non-measurable, and let $F=g^{-1}(E).$ Because $g$ is injective, we have that $F\subseteq K,$ and so $F$ is a null set. However, if it were Borel measurable, then $f(F)$ would also be Borel measurable (here we use the fact that the preimage of a Borel set by a continuous function is measurable; $g(F)=(g^{-1})^{-1}(F)$ is the preimage of $F$ through the continuous function $h=g^{-1}.$) Therefore, $F$ is a null, but non-Borel measurable set.
Haar null
In a separable Banach space $(X,+),$ the group operation moves any subset $A\subseteq X$ to the translates $A+x$ for any $x\in X.$ When there is a probability measure μ on the σ-algebra of Borel subsets of $X,$ such that for all $x,$ $\mu (A+x)=0,$ then $A$ is a Haar null set.[3]
The term refers to the null invariance of the measures of translates, associating it with the complete invariance found with Haar measure.
Some algebraic properties of topological groups have been related to the size of subsets and Haar null sets.[4] Haar null sets have been used in Polish groups to show that when A is not a meagre set then $A^{-1}A$ contains an open neighborhood of the identity element.[5] This property is named for Hugo Steinhaus since it is the conclusion of the Steinhaus theorem.
See also
• Cantor function – Continuous function that is not absolutely continuous
• Empty set – Mathematical set containing no elements
• Measure (mathematics) – Generalization of mass, length, area and volume
• Nothing – Complete absence of anything; the opposite of everything
References
1. Franks, John (2009). A (Terse) Introduction to Lebesgue Integration. The Student Mathematical Library. Vol. 48. American Mathematical Society. p. 28. doi:10.1090/stml/048. ISBN 978-0-8218-4862-3.
2. van Douwen, Eric K. (1989). "Fubini's theorem for null sets". American Mathematical Monthly. 96 (8): 718–21. doi:10.1080/00029890.1989.11972270. JSTOR 2324722. MR 1019152.
3. Matouskova, Eva (1997). "Convexity and Haar Null Sets" (PDF). Proceedings of the American Mathematical Society. 125 (6): 1793–1799. doi:10.1090/S0002-9939-97-03776-3. JSTOR 2162223.
4. Solecki, S. (2005). "Sizes of subsets of groups and Haar null sets". Geometric and Functional Analysis. 15: 246–73. CiteSeerX 10.1.1.133.7074. doi:10.1007/s00039-005-0505-z. MR 2140632. S2CID 11511821.
5. Dodos, Pandelis (2009). "The Steinhaus property and Haar-null sets". Bulletin of the London Mathematical Society. 41 (2): 377–44. arXiv:1006.2675. Bibcode:2010arXiv1006.2675D. doi:10.1112/blms/bdp014. MR 4296513. S2CID 119174196.
Further reading
• Capinski, Marek; Kopp, Ekkehard (2005). Measure, Integral and Probability. Springer. p. 16. ISBN 978-1-85233-781-0.
• Jones, Frank (1993). Lebesgue Integration on Euclidean Spaces. Jones & Bartlett. p. 107. ISBN 978-0-86720-203-8.
• Oxtoby, John C. (1971). Measure and Category. Springer-Verlag. p. 3. ISBN 978-0-387-05349-3.
Measure theory
Basic concepts
• Absolute continuity of measures
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• Lp spaces
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Sets
• Almost everywhere
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• π-system
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• Support
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Types of Measures
• Atomic
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• Content
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Particular measures
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Maps
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• Convergence: almost everywhere
• of measures
• in measure
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• in distribution
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• Random: compact set
• element
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Main results
• Carathéodory's extension theorem
• Convergence theorems
• Dominated
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• Hahn
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• Egorov's
• Fatou's lemma
• Fubini's
• Fubini–Tonelli
• Hölder's inequality
• Minkowski inequality
• Radon–Nikodym
• Riesz–Markov–Kakutani representation theorem
Other results
• Disintegration theorem
• Lifting theory
• Lebesgue's density theorem
• Lebesgue differentiation theorem
• Sard's theorem
For Lebesgue measure
• Isoperimetric inequality
• Brunn–Minkowski theorem
• Milman's reverse
• Minkowski–Steiner formula
• Prékopa–Leindler inequality
• Vitale's random Brunn–Minkowski inequality
Applications & related
• Convex analysis
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Zero mode
In physics, a zero mode is an eigenvector with a vanishing eigenvalue.
In various subfields of physics zero modes appear whenever a physical system possesses a certain symmetry. For example, normal modes of multidimensional harmonic oscillator (e.g. a system of beads arranged around the circle, connected with springs) corresponds to elementary vibrational modes of the system. In such a system zero modes typically occur and are related with a rigid rotation around the circle.
The kernel of an operator consists of left zero modes, and the cokernel consists of the right zero modes.
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Constant problem
In mathematics, the constant problem is the problem of deciding whether a given expression is equal to zero.
The problem
This problem is also referred to as the identity problem[1] or the method of zero estimates. It has no formal statement as such but refers to a general problem prevalent in transcendental number theory. Often proofs in transcendence theory are proofs by contradiction. Specifically, they use some auxiliary function to create an integer n ≥ 0, which is shown to satisfy n < 1. Clearly, this means that n must have the value zero, and so a contradiction arises if one can show that in fact n is not zero.
In many transcendence proofs, proving that n ≠ 0 is very difficult, and hence a lot of work has been done to develop methods that can be used to prove the non-vanishing of certain expressions. The sheer generality of the problem is what makes it difficult to prove general results or come up with general methods for attacking it. The number n that arises may involve integrals, limits, polynomials, other functions, and determinants of matrices.
Results
In certain cases, algorithms or other methods exist for proving that a given expression is non-zero, or of showing that the problem is undecidable. For example, if x1, ..., xn are real numbers, then there is an algorithm[2] for deciding whether there are integers a1, ..., an such that
$a_{1}x_{1}+\cdots +a_{n}x_{n}=0\,.$
If the expression we are interested in contains an oscillating function, such as the sine or cosine function, then it has been shown that the problem is undecidable, a result known as Richardson's theorem. In general, methods specific to the expression being studied are required to prove that it cannot be zero.
See also
• Integer relation algorithm
References
1. Richardson, Daniel (1968). "Some Unsolvable Problems Involving Elementary Functions of a Real Variable". Journal of Symbolic Logic. 33: 514–520. doi:10.2307/2271358. JSTOR 2271358.
2. Bailey, David H. (January 1988). "Numerical Results on the Transcendence of Constants Involving π, e, and Euler's Constant" (PDF). Mathematics of Computation. 50 (20): 275–281. doi:10.1090/S0025-5718-1988-0917835-1.
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Null semigroup
In mathematics, a null semigroup (also called a zero semigroup) is a semigroup with an absorbing element, called zero, in which the product of any two elements is zero.[1] If every element of a semigroup is a left zero then the semigroup is called a left zero semigroup; a right zero semigroup is defined analogously.[2] According to Clifford and Preston, "In spite of their triviality, these semigroups arise naturally in a number of investigations."[1]
Null semigroup
Let S be a semigroup with zero element 0. Then S is called a null semigroup if xy = 0 for all x and y in S.
Cayley table for a null semigroup
Let S = {0, a, b, c} be (the underlying set of) a null semigroup. Then the Cayley table for S is as given below:
Cayley table for a null semigroup
0 a b c
0 0 0 0 0
a 0 0 0 0
b 0 0 0 0
c 0 0 0 0
Left zero semigroup
A semigroup in which every element is a left zero element is called a left zero semigroup. Thus a semigroup S is a left zero semigroup if xy = x for all x and y in S.
Cayley table for a left zero semigroup
Let S = {a, b, c} be a left zero semigroup. Then the Cayley table for S is as given below:
Cayley table for a left zero semigroup
a b c
a a a a
b b b b
c c c c
Right zero semigroup
A semigroup in which every element is a right zero element is called a right zero semigroup. Thus a semigroup S is a right zero semigroup if xy = y for all x and y in S.
Cayley table for a right zero semigroup
Let S = {a, b, c} be a right zero semigroup. Then the Cayley table for S is as given below:
Cayley table for a right zero semigroup
a b c
a a b c
b a b c
c a b c
Properties
A non-trivial null (left/right zero) semigroup does not contain an identity element. It follows that the only null (left/right zero) monoid is the trivial monoid.
The class of null semigroups is:
• closed under taking subsemigroups
• closed under taking quotient of subsemigroup
• closed under arbitrary direct products.
It follows that the class of null (left/right zero) semigroups is a variety of universal algebra, and thus a variety of finite semigroups. The variety of finite null semigroups is defined by the identity ab = cd.
See also
• Right group
References
1. A H Clifford; G B Preston (1964). The algebraic theory of semigroups Vol I. mathematical Surveys. Vol. 1 (2 ed.). American Mathematical Society. pp. 3–4. ISBN 978-0-8218-0272-4.
2. M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories with Applications to Wreath Products and Graphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7, p. 19
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Zero object (algebra)
In algebra, the zero object of a given algebraic structure is, in the sense explained below, the simplest object of such structure. As a set it is a singleton, and as a magma has a trivial structure, which is also an abelian group. The aforementioned abelian group structure is usually identified as addition, and the only element is called zero, so the object itself is typically denoted as {0}. One often refers to the trivial object (of a specified category) since every trivial object is isomorphic to any other (under a unique isomorphism).
This article is about trivial or zero algebraic structures. For zero elements in algebraic structures, see Zero element. For the zero object in a category, see Initial and terminal objects.
Instances of the zero object include, but are not limited to the following:
• As a group, the zero group or trivial group.
• As a ring, the zero ring or trivial ring.
• As an algebra over a field or algebra over a ring, the trivial algebra.
• As a module (over a ring R), the zero module. The term trivial module is also used, although it may be ambiguous, as a trivial G-module is a G-module with a trivial action.
• As a vector space (over a field R), the zero vector space, zero-dimensional vector space or just zero space.
These objects are described jointly not only based on the common singleton and trivial group structure, but also because of shared category-theoretical properties.
In the last three cases the scalar multiplication by an element of the base ring (or field) is defined as:
κ0 = 0 , where κ ∈ R.
The most general of them, the zero module, is a finitely-generated module with an empty generating set.
For structures requiring the multiplication structure inside the zero object, such as the trivial ring, there is only one possible, 0 × 0 = 0, because there are no non-zero elements. This structure is associative and commutative. A ring R which has both an additive and multiplicative identity is trivial if and only if 1 = 0, since this equality implies that for all r within R,
$r=r\times 1=r\times 0=0.$
In this case it is possible to define division by zero, since the single element is its own multiplicative inverse. Some properties of {0} depend on exact definition of the multiplicative identity; see § Unital structures below.
Any trivial algebra is also a trivial ring. A trivial algebra over a field is simultaneously a zero vector space considered below. Over a commutative ring, a trivial algebra is simultaneously a zero module.
The trivial ring is an example of a rng of square zero. A trivial algebra is an example of a zero algebra.
The zero-dimensional vector space is an especially ubiquitous example of a zero object, a vector space over a field with an empty basis. It therefore has dimension zero. It is also a trivial group over addition, and a trivial module mentioned above.
Properties
2↕ ${\begin{bmatrix}0\\0\end{bmatrix}}$ = ${\begin{bmatrix}\,\\\,\end{bmatrix}}$ [ ] ‹0
↔
1
^
0
↔
1
Element of the zero space, written as empty column vector (rightmost one), is multiplied by 2×0 empty matrix to obtain 2-dimensional zero vector (leftmost). Rules of matrix multiplication are respected.
The zero ring, zero module and zero vector space are the zero objects of, respectively, the category of pseudo-rings, the category of modules and the category of vector spaces. However, the zero ring is not a zero object in the category of rings, since there is no ring homomorphism of the zero ring in any other ring.
The zero object, by definition, must be a terminal object, which means that a morphism A → {0} must exist and be unique for an arbitrary object A. This morphism maps any element of A to 0.
The zero object, also by definition, must be an initial object, which means that a morphism {0} → A must exist and be unique for an arbitrary object A. This morphism maps 0, the only element of {0}, to the zero element 0 ∈ A, called the zero vector in vector spaces. This map is a monomorphism, and hence its image is isomorphic to {0}. For modules and vector spaces, this subset {0} ⊂ A is the only empty-generated submodule (or 0-dimensional linear subspace) in each module (or vector space) A.
Unital structures
The {0} object is a terminal object of any algebraic structure where it exists, like it was described for examples above. But its existence and, if it exists, the property to be an initial object (and hence, a zero object in the category-theoretical sense) depend on exact definition of the multiplicative identity 1 in a specified structure.
If the definition of 1 requires that 1 ≠ 0, then the {0} object cannot exist because it may contain only one element. In particular, the zero ring is not a field. If mathematicians sometimes talk about a field with one element, this abstract and somewhat mysterious mathematical object is not a field.
In categories where the multiplicative identity must be preserved by morphisms, but can equal to zero, the {0} object can exist. But not as initial object because identity-preserving morphisms from {0} to any object where 1 ≠ 0 do not exist. For example, in the category of rings Ring the ring of integers Z is the initial object, not {0}.
If an algebraic structure requires the multiplicative identity, but neither its preservation by morphisms nor 1 ≠ 0, then zero morphisms exist and the situation is not different from non-unital structures considered in the previous section.
Notation
Zero vector spaces and zero modules are usually denoted by 0 (instead of {0}). This is always the case when they occur in an exact sequence.
See also
• Nildimensional space
• Triviality (mathematics)
• Examples of vector spaces
• Field with one element
• Empty semigroup
• Zero element
• List of zero terms
External links
• David Sharpe (1987). Rings and factorization. Cambridge University Press. p. 10 : trivial ring. ISBN 0-521-33718-6.
• Barile, Margherita. "Trivial Module". MathWorld.
• Barile, Margherita. "Zero Module". MathWorld.
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Zero-sum game
Zero-sum game is a mathematical representation in game theory and economic theory of a situation that involves two sides, where the result is an advantage for one side and an equivalent loss for the other.[1] In other words, player one's gain is equivalent to player two's loss, with the result that the net improvement in benefit of the game is zero.[2]
Not to be confused with Empty sum or Zero game.
If the total gains of the participants are added up, and the total losses are subtracted, they will sum to zero. Thus, cutting a cake, where taking a more significant piece reduces the amount of cake available for others as much as it increases the amount available for that taker, is a zero-sum game if all participants value each unit of cake equally. Other examples of zero-sum games in daily life include games like poker, chess, and bridge where one person gains and another person loses, which results in a zero-net benefit for every player.[3] In the markets and financial instruments, futures contracts and options are zero-sum games as well.[4]
In contrast, non-zero-sum describes a situation in which the interacting parties' aggregate gains and losses can be less than or more than zero. A zero-sum game is also called a strictly competitive game, while non-zero-sum games can be either competitive or non-competitive. Zero-sum games are most often solved with the minimax theorem which is closely related to linear programming duality,[5] or with Nash equilibrium. Prisoner's Dilemma is a classic non-zero-sum game.[6]
Definition
Choice 1 Choice 2
Choice 1 −A, A B, −B
Choice 2 C, −C −D, D
Generic zero-sum game
Option 1 Option 2
Option 1 2, −2 −2, 2
Option 2 −2, 2 2, −2
Another example of the classic zero-sum game
The zero-sum property (if one gains, another loses) means that any result of a zero-sum situation is Pareto optimal. Generally, any game where all strategies are Pareto optimal is called a conflict game.[7][8]
Zero-sum games are a specific example of constant sum games where the sum of each outcome is always zero.[9] Such games are distributive, not integrative; the pie cannot be enlarged by good negotiation.
In situation where one decision maker's gain (or loss) does not necessarily result in the other decision makers' loss (or gain), they are referred to as non-zero-sum.[10] Thus, a country with an excess of bananas trading with another country for their excess of apples, where both benefit from the transaction, is in a non-zero-sum situation. Other non-zero-sum games are games in which the sum of gains and losses by the players is sometimes more or less than what they began with.
The idea of Pareto optimal payoff in a zero-sum game gives rise to a generalized relative selfish rationality standard, the punishing-the-opponent standard, where both players always seek to minimize the opponent's payoff at a favourable cost to themselves rather than prefer more over less. The punishing-the-opponent standard can be used in both zero-sum games (e.g. warfare game, chess) and non-zero-sum games (e.g. pooling selection games).[11] The player in the game has a simple enough desire to maximise the profit for them, and the opponent wishes to minimise it.[12]
Solution
For two-player finite zero-sum games, the different game theoretic solution concepts of Nash equilibrium, minimax, and maximin all give the same solution. If the players are allowed to play a mixed strategy, the game always has an equilibrium.
Example
A zero-sum game (Two person)
Blue
Red
A B C
1
−30
30
10
−10
−20
20
2
10
−10
−20
20
20
−20
A game's payoff matrix is a convenient representation. Consider these situations as an example, the two-player zero-sum game pictured at right or above.
The order of play proceeds as follows: The first player (red) chooses in secret one of the two actions 1 or 2; the second player (blue), unaware of the first player's choice, chooses in secret one of the three actions A, B or C. Then, the choices are revealed and each player's points total is affected according to the payoff for those choices.
Example: Red chooses action 2 and Blue chooses action B. When the payoff is allocated, Red gains 20 points and Blue loses 20 points.
In this example game, both players know the payoff matrix and attempt to maximize the number of their points. Red could reason as follows: "With action 2, I could lose up to 20 points and can win only 20, and with action 1 I can lose only 10 but can win up to 30, so action 1 looks a lot better." With similar reasoning, Blue would choose action C. If both players take these actions, Red will win 20 points. If Blue anticipates Red's reasoning and choice of action 1, Blue may choose action B, so as to win 10 points. If Red, in turn, anticipates this trick and goes for action 2, this wins Red 20 points.
Émile Borel and John von Neumann had the fundamental insight that probability provides a way out of this conundrum. Instead of deciding on a definite action to take, the two players assign probabilities to their respective actions, and then use a random device which, according to these probabilities, chooses an action for them. Each player computes the probabilities so as to minimize the maximum expected point-loss independent of the opponent's strategy. This leads to a linear programming problem with the optimal strategies for each player. This minimax method can compute probably optimal strategies for all two-player zero-sum games.
For the example given above, it turns out that Red should choose action 1 with probability 4/7 and action 2 with probability 3/7, and Blue should assign the probabilities 0, 4/7, and 3/7 to the three actions A, B, and C. Red will then win 20/7 points on average per game.
Solving
The Nash equilibrium for a two-player, zero-sum game can be found by solving a linear programming problem. Suppose a zero-sum game has a payoff matrix M where element Mi,j is the payoff obtained when the minimizing player chooses pure strategy i and the maximizing player chooses pure strategy j (i.e. the player trying to minimize the payoff chooses the row and the player trying to maximize the payoff chooses the column). Assume every element of M is positive. The game will have at least one Nash equilibrium. The Nash equilibrium can be found (Raghavan 1994, p. 740) by solving the following linear program to find a vector u:
Minimize:
$\sum _{i}u_{i}$
Subject to the constraints:
u ≥ 0
M u ≥ 1.
The first constraint says each element of the u vector must be nonnegative, and the second constraint says each element of the M u vector must be at least 1. For the resulting u vector, the inverse of the sum of its elements is the value of the game. Multiplying u by that value gives a probability vector, giving the probability that the maximizing player will choose each possible pure strategy.
If the game matrix does not have all positive elements, add a constant to every element that is large enough to make them all positive. That will increase the value of the game by that constant, and will not affect the equilibrium mixed strategies for the equilibrium.
The equilibrium mixed strategy for the minimizing player can be found by solving the dual of the given linear program. Alternatively, it can be found by using the above procedure to solve a modified payoff matrix which is the transpose and negation of M (adding a constant so it is positive), then solving the resulting game.
If all the solutions to the linear program are found, they will constitute all the Nash equilibria for the game. Conversely, any linear program can be converted into a two-player, zero-sum game by using a change of variables that puts it in the form of the above equations and thus such games are equivalent to linear programs, in general.[13]
Universal solution
If avoiding a zero-sum game is an action choice with some probability for players, avoiding is always an equilibrium strategy for at least one player at a zero-sum game. For any two players zero-sum game where a zero-zero draw is impossible or non-credible after the play is started, such as poker, there is no Nash equilibrium strategy other than avoiding the play. Even if there is a credible zero-zero draw after a zero-sum game is started, it is not better than the avoiding strategy. In this sense, it's interesting to find reward-as-you-go in optimal choice computation shall prevail over all two players zero-sum games concerning starting the game or not.[14]
The most common or simple example from the subfield of social psychology is the concept of "social traps". In some cases pursuing individual personal interest can enhance the collective well-being of the group, but in other situations, all parties pursuing personal interest results in mutually destructive behaviour.
Copeland's review notes that an n-player non-zero-sum game can be converted into an (n+1)-player zero-sum game, where the n+1st player, denoted the fictitious player, receives the negative of the sum of the gains of the other n-players (the global gain / loss).[15]
Zero-sum three-person games
It is clear that there are manifold relationships between players in a zero-sum three-person game, in a zero-sum two-person game, anything one player wins is necessarily lost by the other and vice versa; therefore, there is always an absolute antagonism of interests, and that is similar in the three-person game.[16] A particular move of a player in a zero-sum three-person game would be assumed to be clearly beneficial to him and may disbenefits to both other players, or benefits to one and disbenefits to the other opponent.[16] Particularly, parallelism of interests between two players makes a cooperation desirable; it may happen that a player has a choice among various policies: Get into a parallelism interest with another player by adjusting his conduct, or the opposite; that he can choose with which of other two players he prefers to build such parallelism, and to what extent.[16] The picture on the left shows that a typical example of a zero-sum three-person game. If Player 1 chooses to defence, but Player 2 & 3 chooses to offence, both of them will gain one point. At the same time, Player 2 will lose two-point because points are taken away by other players, and it is evident that Player 2 & 3 has parallelism of interests.
Economic benefits of low-cost airlines in saturated markets - net benefits or a zero-sum game [17]
Studies show that the entry of low-cost airlines into the Hong Kong market brought in $671 million in revenue and resulted in an outflow of $294 million.
Therefore, the replacement effect should be considered when introducing a new model, which will lead to economic leakage and injection. Thus introducing new models requires caution. For example, if the number of new airlines departing from and arriving at the airport is the same, the economic contribution to the host city may be a zero-sum game. Because for Hong Kong, the consumption of overseas tourists in Hong Kong is income, while the consumption of Hong Kong residents in opposite cities is outflow. In addition, the introduction of new airlines can also have a negative impact on existing airlines.
Consequently, when a new aviation model is introduced, feasibility tests need to be carried out in all aspects, taking into account the economic inflow and outflow and displacement effects caused by the model.
Zero-sum Games in Financial Markets
Derivatives trading may be considered a zero-sum game, as each dollar gained by one party in a transaction must be lost by the other, hence yielding a net transfer of wealth of zero.[18]
An options contract - whereby a buyer purchases a derivative contract which provides them with the right to buy an underlying asset from a seller at a specified strike price before a specified expiration date – is an example of a zero-sum game. A futures contract – whereby a buyer purchases a derivative contract to buy an underlying asset from the seller for a specified price on a specified date – is also an example of a zero-sum game.[19] This is because the fundamental principle of these contracts is that they are agreements between two parties, and any gain made by one party must be matched by a loss sustained by the other.
If the price of the underlying asset increases before the expiration date the buyer may exercise/ close the options/ futures contract. The buyers gain and corresponding sellers loss will be the difference between the strike price and value of the underlying asset at that time. Hence, the net transfer of wealth is zero.
Swaps, which involve the exchange of cash flows from two different financial instruments, are also considered a zero-sum game.[20] Consider a standard interest rate swap whereby Firm A pays a fixed rate and receives a floating rate; correspondingly Firm B pays a floating rate and receives a fixed rate. If rates increase, then Firm A will gain, and Firm B will lose by the rate differential (floating rate – fixed rate). If rates decrease, then Firm A will lose, and Firm B will gain by the rate differential (fixed rate – floating rate).
Whilst derivatives trading may be considered a zero-sum game, it is important to remember that this is not an absolute truth. The financial markets are complex and multifaceted, with a range of participants engaging in a variety of activities. While some trades may result in a simple transfer of wealth from one party to another, the market as a whole is not purely competitive, and many transactions serve important economic functions.
The stock market is an excellent example of a positive-sum game, often erroneously labelled as a zero-sum game. This is a zero-sum fallacy: the perception that one trader in the stock market may only increase the value of their holdings if another trader decreases their holdings.[21]
The primary goal of the stock market is to match buyers and sellers, but the prevailing price is the one which equilibrates supply and demand. Stock prices generally move according to changes in future expectations, such as acquisition announcements, upside earnings surprises, or improved guidance.[22]
For instance, if Company C announces a deal to acquire Company D, and investors believe that the acquisition will result in synergies and hence increased profitability for Company C, there will be an increased demand for Company C stock. In this scenario, all existing holders of Company C stock will enjoy gains without incurring any corresponding measurable losses to other players.
Furthermore, in the long run, the stock market is a positive-sum game. As economic growth occurs, demand increases, output increases, companies grow, and company valuations increase, leading to value creation and wealth addition in the market.
Complexity
It has been theorized by Robert Wright in his book Nonzero: The Logic of Human Destiny, that society becomes increasingly non-zero-sum as it becomes more complex, specialized, and interdependent.
Extensions
In 1944, John von Neumann and Oskar Morgenstern proved that any non-zero-sum game for n players is equivalent to a zero-sum game with n + 1 players; the (n + 1)th player representing the global profit or loss.[23]
Misunderstandings
Zero-sum games and particularly their solutions are commonly misunderstood by critics of game theory, usually with respect to the independence and rationality of the players, as well as to the interpretation of utility functions. Furthermore, the word "game" does not imply the model is valid only for recreational games.[5]
Politics is sometimes called zero sum[24][25][26] because in common usage the idea of a stalemate is perceived to be "zero sum"; politics and macroeconomics are not zero sum games, however, because they do not constitute conserved systems.
Zero-sum thinking
In psychology, zero-sum thinking refers to the perception that a given situation is like a zero-sum game, where one person's gain is equal to another person's loss.
See also
• Bimatrix game
• Comparative advantage
• Dutch disease
• Gains from trade
• Lump of labour fallacy
• Positive-sum game
• No-win situation
References
1. Cambridge business English dictionary. Cambridge: Cambridge University Press. 2011. ISBN 978-0-521-12250-4. OCLC 741548935.
2. Blakely, Sara. "Zero-Sum Game Meaning: Examples of Zero-Sum Games". Master Class. Master Class. Retrieved 2022-04-28.
3. Von Neumann, John; Oskar Morgenstern (2007). Theory of games and economic behavior (60th anniversary ed.). Princeton: Princeton University Press. ISBN 978-1-4008-2946-0. OCLC 830323721.
4. Kenton, Will. "Zero-Sum Game". Investopedia. Retrieved 2021-04-25.
5. Ken Binmore (2007). Playing for real: a text on game theory. Oxford University Press US. ISBN 978-0-19-530057-4., chapters 1 & 7
6. Chiong, Raymond; Jankovic, Lubo (2008). "Learning game strategy design through iterated Prisoner's Dilemma". International Journal of Computer Applications in Technology. 32 (3): 216. doi:10.1504/ijcat.2008.020957. ISSN 0952-8091.
7. Bowles, Samuel (2004). Microeconomics: Behavior, Institutions, and Evolution. Princeton University Press. pp. 33–36. ISBN 0-691-09163-3.
8. "Two-Person Zero-Sum Games: Basic Concepts". Neos Guide. Neos Guide. Retrieved 2022-04-28.
9. Washburn, Alan (2014). Two-Person Zero-Sum Games. International Series in Operations Research & Management Science. Vol. 201. Boston, MA: Springer US. doi:10.1007/978-1-4614-9050-0. ISBN 978-1-4614-9049-4.
10. "Non Zero Sum Game". Monash Business School. Retrieved 2021-04-25.
11. Wenliang Wang (2015). Pooling Game Theory and Public Pension Plan. ISBN 978-1507658246. Chapter 1 and Chapter 4.
12. Von Neumann, John; Oskar Morgenstern (2007). Theory of games and economic behavior (60th anniversary ed.). Princeton: Princeton University Press. p. 98. ISBN 978-1-4008-2946-0. OCLC 830323721.
13. Ilan Adler (2012) The equivalence of linear programs and zero-sum games. Springer
14. Wenliang Wang (2015). Pooling Game Theory and Public Pension Plan. ISBN 978-1507658246. Chapter 4.
15. Arthur H. Copeland (July 1945) Book review, Theory of games and economic behavior. By John von Neumann and Oskar Morgenstern (1944). Review published in the Bulletin of the American Mathematical Society 51(7) pp 498-504 (July 1945)
16. Von Neumann, John; Oskar Morgenstern (2007). Theory of games and economic behavior (60th anniversary ed.). Princeton: Princeton University Press. pp. 220–223. ISBN 978-1-4008-2946-0. OCLC 830323721.
17. Pratt, Stephen; Schucker, Markus (March 2018). "Economic impact of low-cost carrier in a saturated transport market: Net benefits or zero-sum game?". Tourism Economics: The Business and Finance of Tourism and Recreation. 25 (2): 149–170.
18. Levitt, Steven D. (February 2004). "Why are Gambling Markets Organized so Differently from Financial Markets?". The Economic Journal. 114 (10): 223–246. doi:10.1111/j.1468-0297.2004.00207.x. S2CID 2289856 – via RePEc.
19. "Options vs. Futures: What's the Difference?". Investopedia. Retrieved 2023-04-24.
20. Turnbull, Stuart M. (1987). "Swaps: A Zero Sum Game?". Financial Management. 16 (1): 15–21. doi:10.2307/3665544. ISSN 0046-3892. JSTOR 3665544.
21. Engle, Eric (September 2008). "The Stock Market as a Game: An Agent Based Approach to Trading in Stocks". Quantitative Finance Papers – via RePEc.
22. Olson, Erika S. (2010-10-26). Zero-Sum Game: The Rise of the World's Largest Derivatives Exchange. John Wiley & Sons. ISBN 978-0-470-62420-3.
23. Theory of Games and Economic Behavior. Princeton University Press (1953). June 25, 2005. ISBN 9780691130613. Retrieved 2018-02-25.
24. Rubin, Jennifer (2013-10-04). "The flaw in zero sum politics". The Washington Post. Retrieved 2017-03-08.
25. "Lexington: Zero-sum politics". The Economist. 2014-02-08. Retrieved 2017-03-08.
26. "Zero-sum game | Define Zero-sum game at". Dictionary.com. Retrieved 2017-03-08.
Further reading
• Misstating the Concept of Zero-Sum Games within the Context of Professional Sports Trading Strategies, series Pardon the Interruption (2010-09-23) ESPN, created by Tony Kornheiser and Michael Wilbon, performance by Bill Simmons
• Handbook of Game Theory – volume 2, chapter Zero-sum two-person games, (1994) Elsevier Amsterdam, by Raghavan, T. E. S., Edited by Aumann and Hart, pp. 735–759, ISBN 0-444-89427-6
• Power: Its Forms, Bases and Uses (1997) Transaction Publishers, by Dennis Wrong
External links
• Play zero-sum games online by Elmer G. Wiens.
• Game Theory & its Applications – comprehensive text on psychology and game theory. (Contents and Preface to Second Edition.)
• A playable zero-sum game and its mixed strategy Nash equilibrium.
Topics in game theory
Definitions
• Congestion game
• Cooperative game
• Determinacy
• Escalation of commitment
• Extensive-form game
• First-player and second-player win
• Game complexity
• Graphical game
• Hierarchy of beliefs
• Information set
• Normal-form game
• Preference
• Sequential game
• Simultaneous game
• Simultaneous action selection
• Solved game
• Succinct game
Equilibrium
concepts
• Bayesian Nash equilibrium
• Berge equilibrium
• Core
• Correlated equilibrium
• Epsilon-equilibrium
• Evolutionarily stable strategy
• Gibbs equilibrium
• Mertens-stable equilibrium
• Markov perfect equilibrium
• Nash equilibrium
• Pareto efficiency
• Perfect Bayesian equilibrium
• Proper equilibrium
• Quantal response equilibrium
• Quasi-perfect equilibrium
• Risk dominance
• Satisfaction equilibrium
• Self-confirming equilibrium
• Sequential equilibrium
• Shapley value
• Strong Nash equilibrium
• Subgame perfection
• Trembling hand
Strategies
• Backward induction
• Bid shading
• Collusion
• Forward induction
• Grim trigger
• Markov strategy
• Dominant strategies
• Pure strategy
• Mixed strategy
• Strategy-stealing argument
• Tit for tat
Classes
of games
• Bargaining problem
• Cheap talk
• Global game
• Intransitive game
• Mean-field game
• Mechanism design
• n-player game
• Perfect information
• Large Poisson game
• Potential game
• Repeated game
• Screening game
• Signaling game
• Stackelberg competition
• Strictly determined game
• Stochastic game
• Symmetric game
• Zero-sum game
Games
• Go
• Chess
• Infinite chess
• Checkers
• Tic-tac-toe
• Prisoner's dilemma
• Gift-exchange game
• Optional prisoner's dilemma
• Traveler's dilemma
• Coordination game
• Chicken
• Centipede game
• Lewis signaling game
• Volunteer's dilemma
• Dollar auction
• Battle of the sexes
• Stag hunt
• Matching pennies
• Ultimatum game
• Rock paper scissors
• Pirate game
• Dictator game
• Public goods game
• Blotto game
• War of attrition
• El Farol Bar problem
• Fair division
• Fair cake-cutting
• Cournot game
• Deadlock
• Diner's dilemma
• Guess 2/3 of the average
• Kuhn poker
• Nash bargaining game
• Induction puzzles
• Trust game
• Princess and monster game
• Rendezvous problem
Theorems
• Arrow's impossibility theorem
• Aumann's agreement theorem
• Folk theorem
• Minimax theorem
• Nash's theorem
• Negamax theorem
• Purification theorem
• Revelation principle
• Sprague–Grundy theorem
• Zermelo's theorem
Key
figures
• Albert W. Tucker
• Amos Tversky
• Antoine Augustin Cournot
• Ariel Rubinstein
• Claude Shannon
• Daniel Kahneman
• David K. Levine
• David M. Kreps
• Donald B. Gillies
• Drew Fudenberg
• Eric Maskin
• Harold W. Kuhn
• Herbert Simon
• Hervé Moulin
• John Conway
• Jean Tirole
• Jean-François Mertens
• Jennifer Tour Chayes
• John Harsanyi
• John Maynard Smith
• John Nash
• John von Neumann
• Kenneth Arrow
• Kenneth Binmore
• Leonid Hurwicz
• Lloyd Shapley
• Melvin Dresher
• Merrill M. Flood
• Olga Bondareva
• Oskar Morgenstern
• Paul Milgrom
• Peyton Young
• Reinhard Selten
• Robert Axelrod
• Robert Aumann
• Robert B. Wilson
• Roger Myerson
• Samuel Bowles
• Suzanne Scotchmer
• Thomas Schelling
• William Vickrey
Miscellaneous
• All-pay auction
• Alpha–beta pruning
• Bertrand paradox
• Bounded rationality
• Combinatorial game theory
• Confrontation analysis
• Coopetition
• Evolutionary game theory
• First-move advantage in chess
• Game Description Language
• Game mechanics
• Glossary of game theory
• List of game theorists
• List of games in game theory
• No-win situation
• Solving chess
• Topological game
• Tragedy of the commons
• Tyranny of small decisions
Authority control: National
• Germany
• Israel
• United States
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Zero suppression
Zero suppression is the removal of redundant zeroes from a number. This can be done for storage, page or display space constraints or formatting reasons, such as making a letter more legible.[1][2][3]
Examples
• 00049823 → 49823
• 7.678600000 → 7.6786
• 0032.3231000 → 32.3231
• 2.45000×1010 → 2.45×1010
• 0.0045×1010 → 4.5×107
One must be careful; in physics and related disciplines, trailing zeros are used to indicate the precision of the number, as an error of ±1 in the last place is assumed. Examples:
• 4.5981 is 4.5981 ± 0.0001
• 4.59810 is 4.5981 ± 0.00001
• 4.598100 is 4.5981 ± 0.000001
Data compression
It is also a way to store a large array of numbers, where many of the entries are zero. By omitting the zeroes, and instead storing the indices along with the values of the non-zero items, less space may be used in total. It only makes sense if the extra space used for storing the indices (on average) is smaller than the space saved by not storing the zeroes. This is sometimes used in a sparse array.
Example:
• Original array: 0, 1, 0, 0, 2, 5, 0, 0, 0, 4, 0, 0, 0, 0, 0
• Pairs of index and data: {2,1}, {5,2}, {6,5}, {10,4}
See also
• Run-length encoding – Form of lossless data compression
• Zero code suppression – Digital telecommunications techniquePages displaying short descriptions of redirect targets
• Zero-suppressed decision diagram – Kind of binary decision diagram
References
1. "Telecom Glossary 2000: Zero Suppression". U.S.: Institute for Telecommunication Sciences, NTIA. Archived from the original on 2008-09-25.
2. Parr, E. A. (1999). Industrial Control Handbook (3 ed.). Industrial Press, Inc. p. 582. ISBN 978-0-8311-3085-5.
3. Grabowski, Ralph (2010). Using AutoCAD 2011. Autodesk Press. p. 648. ISBN 978-1-111-12514-1.
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Symbols for zero
The modern numerical digit 0 is usually written as a circle, an ellipse or a rounded square or rectangle.
Glyphs
In most modern typefaces, the height of the 0 character is the same as the other digits. However, in typefaces with text figures, the character is often shorter (x-height).
Traditionally, many print typefaces made the capital letter O more rounded than the narrower, elliptical digit 0.[1] Typewriters originally made no distinction in shape between O and 0; some models did not even have a separate key for the digit 0. The distinction came into prominence on modern character displays.[1]
The digit 0 with a dot in the centre seems to have originated as an option on IBM 3270 displays. Its appearance has continued with Taligent's command line typeface Andalé Mono. One variation used a short vertical bar instead of the dot. This could be confused with the Greek letter Theta on a badly focused display, but in practice there was no confusion because theta was not (then) a displayable character and very little used anyway.
An alternative, the slashed zero (looking similar to the letter O except for the slash), was primarily used in hand-written coding sheets before transcription to punched cards or tape, and is also used in old-style ASCII graphic sets descended from the default typewheel on the Teletype Model 33 ASR. This form is similar to the symbol $\emptyset $, or "∅" (Unicode character U+2205), representing the empty set, as well as to the letter Ø used in several Scandinavian languages. Some Burroughs/Unisys equipment displays a digit 0 with a reversed slash.
The opposing convention that has the letter O with a slash and the digit 0 without was advocated by SHARE, a prominent IBM user group,[1] and recommended by IBM for writing FORTRAN programs,[2] and by a few other early mainframe makers; this is even more problematic for Scandinavians because it means two of their letters collide. Others advocated the opposite convention,[1] including IBM for writing Algol programs.[2] Another convention used on some early line printers left digit 0 unornamented but added a tail or hook to the capital O so that it resembled an inverted Q (like U+213A ℺) or cursive capital letter-O (${\mathcal {O}}$).[1]
Some fonts designed for use with computers made one of the capital-O–digit-0 pair more rounded and the other more angular (closer to a rectangle). The TI-99/4A computer has a more angular capital O and a more rounded digit 0, whereas others made the choice the other way around.
The typeface used on most European vehicle registration plates distinguishes the two symbols partially in this manner (having a more rectangular or wider shape for the capital O than the digit 0), but in several countries a further distinction is made by slitting open the digit 0 on the upper right side (as in German plates using the fälschungserschwerende Schrift, "forgery-impeding typeface").
Sometimes the digit 0 is used either exclusively, or not at all, to avoid confusion altogether. For example, confirmation numbers[3] used by Southwest Airlines use only the capital letters O and I instead of the digits 0 and 1, while Canadian postal codes use only the digits 1 and 0 and never the capital letters O and I, although letters and numbers always alternate.
Other
Other representations of zero
• Usual appearance of zero on seven-segment displays
• Unusual smaller appearance of zero on seven-segment displays
• International maritime signal flag for 0
On the seven-segment displays of calculators, watches, and household appliances, 0 is usually written with six line segments, though on some historical calculator models it was written with four line segments.
The international maritime signal flag has five plus signs in an X arrangement.
Zero symbols in Unicode
• U+0030 0 DIGIT ZERO
• U+0660 ٠ ARABIC-INDIC DIGIT ZERO
• U+06DF ۟ ARABIC SMALL HIGH ROUNDED ZERO
• U+06E0 ۠ ARABIC SMALL HIGH UPRIGHT RECTANGULAR ZERO
• U+06F0 ۰ EXTENDED ARABIC-INDIC DIGIT ZERO
• U+07C0 ߀ NKO DIGIT ZERO
• U+0966 ० DEVANAGARI DIGIT ZERO
• U+09E6 ০ BENGALI DIGIT ZERO
• U+0A66 ੦ GURMUKHI DIGIT ZERO
• U+0AE6 ૦ GUJARATI DIGIT ZERO
• U+0B66 ୦ ORIYA DIGIT ZERO
• U+0BE6 ௦ TAMIL DIGIT ZERO
• U+0C66 ౦ TELUGU DIGIT ZERO
• U+0C78 ౸ TELUGU FRACTION DIGIT ZERO FOR ODD POWERS OF FOUR
• U+0CE6 ೦ KANNADA DIGIT ZERO
• U+0D66 ൦ MALAYALAM DIGIT ZERO
• U+0DE6 ෦ SINHALA LITH DIGIT ZERO
• U+0E50 ๐ THAI DIGIT ZERO
• U+0ED0 ໐ LAO DIGIT ZERO
• U+0F20 ༠ TIBETAN DIGIT ZERO
• U+0F33 ༳ TIBETAN DIGIT HALF ZERO
• U+1040 ၀ MYANMAR DIGIT ZERO
• U+1090 ႐ MYANMAR SHAN DIGIT ZERO
• U+17E0 ០ KHMER DIGIT ZERO
• U+1810 ᠐ MONGOLIAN DIGIT ZERO
• U+1946 ᥆ LIMBU DIGIT ZERO
• U+19D0 ᧐ NEW TAI LUE DIGIT ZERO
• U+1A80 ᪀ TAI THAM HORA DIGIT ZERO
• U+1A90 ᪐ TAI THAM THAM DIGIT ZERO
• U+1B50 ᭐ BALINESE DIGIT ZERO
• U+1BB0 ᮰ SUNDANESE DIGIT ZERO
• U+1C40 ᱀ LEPCHA DIGIT ZERO
• U+1C50 ᱐ OL CHIKI DIGIT ZERO
• U+2070 ⁰ SUPERSCRIPT ZERO
• U+2080 ₀ SUBSCRIPT ZERO
• U+2189 ↉ VULGAR FRACTION ZERO THIRDS
• U+24EA ⓪ CIRCLED DIGIT ZERO
• U+24FF ⓿ NEGATIVE CIRCLED DIGIT ZERO
• U+3007 〇 IDEOGRAPHIC NUMBER ZERO
• U+3358 ㍘ IDEOGRAPHIC TELEGRAPH SYMBOL FOR HOUR ZERO
• U+A620 ꘠ VAI DIGIT ZERO
• U+A8D0 ꣐ SAURASHTRA DIGIT ZERO
• U+A8E0 ꣠ COMBINING DEVANAGARI DIGIT ZERO
• U+A900 ꤀ KAYAH LI DIGIT ZERO
• U+A9D0 ꧐ JAVANESE DIGIT ZERO
• U+A9F0 ꧰ MYANMAR TAI LAING DIGIT ZERO
• U+AA50 ꩐ CHAM DIGIT ZERO
• U+ABF0 ꯰ MEETEI MAYEK DIGIT ZERO
• U+FF10 ０ FULLWIDTH DIGIT ZERO
• U+1018A 𐆊 GREEK ZERO SIGN
• U+104A0 𐒠 OSMANYA DIGIT ZERO
• U+10D30 𐴰 HANIFI ROHINGYA DIGIT ZERO
• U+11066 𑁦 BRAHMI DIGIT ZERO
• U+110F0 𑃰 SORA SOMPENG DIGIT ZERO
• U+11136 𑄶 CHAKMA DIGIT ZERO
• U+111D0 𑇐 SHARADA DIGIT ZERO
• U+112F0 𑋰 KHUDAWADI DIGIT ZERO
• U+11366 𑍦 COMBINING GRANTHA DIGIT ZERO
• U+11450 𑑐 NEWA DIGIT ZERO
• U+114D0 𑓐 TIRHUTA DIGIT ZERO
• U+11650 𑙐 MODI DIGIT ZERO
• U+116C0 𑛀 TAKRI DIGIT ZERO
• U+11730 𑜰 AHOM DIGIT ZERO
• U+118E0 𑣠 WARANG CITI DIGIT ZERO
• U+11950 𑥐 DIVES AKURU DIGIT ZERO
• U+11C50 𑱐 BHAIKSUKI DIGIT ZERO
• U+11D50 𑵐 MASARAM GONDI DIGIT ZERO
• U+11DA0 𑶠 GUNJALA GONDI DIGIT ZERO
• U+16A60 𖩠 MRO DIGIT ZERO
• U+16B50 𖭐 PAHAWH HMONG DIGIT ZERO
• U+16E80 𖺀 MEDEFAIDRIN DIGIT ZERO
• U+1D2E0 𝋠 MAYAN NUMERAL ZERO
• U+1D7CE 𝟎 MATHEMATICAL BOLD DIGIT ZERO
• U+1D7D8 𝟘 MATHEMATICAL DOUBLE-STRUCK DIGIT ZERO
• U+1D7E2 𝟢 MATHEMATICAL SANS-SERIF DIGIT ZERO
• U+1D7EC 𝟬 MATHEMATICAL SANS-SERIF BOLD DIGIT ZERO
• U+1D7F6 𝟶 MATHEMATICAL MONOSPACE DIGIT ZERO
• U+1E140 𞅀 NYIAKENG PUACHUE HMONG DIGIT ZERO
• U+1E2F0 𞋰 WANCHO DIGIT ZERO
• U+1E950 𞥐 ADLAM DIGIT ZERO
• U+1F100 🄀 DIGIT ZERO FULL STOP
• U+1F101 🄁 DIGIT ZERO COMMA
• U+1F10B 🄋 DINGBAT CIRCLED SANS-SERIF DIGIT ZERO
• U+1F10C 🄌 DINGBAT NEGATIVE CIRCLED SANS-SERIF DIGIT ZERO
• U+1F10D 🄍 CIRCLED ZERO WITH SLASH
• U+1FBF0 🯰 SEGMENTED DIGIT ZERO
• U+E0030 TAG DIGIT ZERO
See also
• Arabic numeral variations § Slashed zero
• Regional handwriting variation § Arabic numerals
• Ø (disambiguation)
References
1. Bemer, Robert William (August 1967). "Towards standards for handwritten zero and oh: much ado about nothing (and a letter), or a partial dossier on distinguishing between handwritten zero and oh". Communications of the ACM. 10 (8): 513–518. doi:10.1145/363534.363563. S2CID 294510.
2. Einarsson, Bo; Shokin, Yurij (2007-05-24). "Fortran 90 for the Fortran 77 Programmer". Appendix 7: "The historical development of Fortran. Archived from the original on 2017-02-28.
3. "Check in for your Flight Reservation".
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Zero of a function
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function $f$, is a member $x$ of the domain of $f$ such that $f(x)$ vanishes at $x$; that is, the function $f$ attains the value of 0 at $x$, or equivalently, $x$ is the solution to the equation $f(x)=0$.[1] A "zero" of a function is thus an input value that produces an output of 0.[2]
"Root of a function" redirects here. For a half iterate of a function, see Functional square root.
A graph of the function $\cos(x)$ for $x$ in $\left[-2\pi ,2\pi \right]$, with zeros at $-{\tfrac {3\pi }{2}},\;-{\tfrac {\pi }{2}},\;{\tfrac {\pi }{2}}$, and ${\tfrac {3\pi }{2}},$ marked in red.
A root of a polynomial is a zero of the corresponding polynomial function.[1] The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities.[3] For example, the polynomial $f$ of degree two, defined by $f(x)=x^{2}-5x+6$ has the two roots (or zeros) that are 2 and 3.
$f(2)=2^{2}-5\times 2+6=0{\text{ and }}f(3)=3^{2}-5\times 3+6=0.$
If the function maps real numbers to real numbers, then its zeros are the $x$-coordinates of the points where its graph meets the x-axis. An alternative name for such a point $(x,0)$ in this context is an $x$-intercept.
Solution of an equation
Every equation in the unknown $x$ may be rewritten as
$f(x)=0$
by regrouping all the terms in the left-hand side. It follows that the solutions of such an equation are exactly the zeros of the function $f$. In other words, a "zero of a function" is precisely a "solution of the equation obtained by equating the function to 0", and the study of zeros of functions is exactly the same as the study of solutions of equations.
Polynomial roots
Main article: Properties of polynomial roots
Every real polynomial of odd degree has an odd number of real roots (counting multiplicities); likewise, a real polynomial of even degree must have an even number of real roots. Consequently, real odd polynomials must have at least one real root (because the smallest odd whole number is 1), whereas even polynomials may have none. This principle can be proven by reference to the intermediate value theorem: since polynomial functions are continuous, the function value must cross zero, in the process of changing from negative to positive or vice versa (which always happens for odd functions).
Fundamental theorem of algebra
Main article: Fundamental theorem of algebra
The fundamental theorem of algebra states that every polynomial of degree $n$ has $n$ complex roots, counted with their multiplicities. The non-real roots of polynomials with real coefficients come in conjugate pairs.[2] Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots.
Computing roots
Main articles: Root-finding algorithm, Real-root isolation, and Equation solving
Computing roots of functions, for example polynomial functions, frequently requires the use of specialised or approximation techniques (e.g., Newton's method). However, some polynomial functions, including all those of degree no greater than 4, can have all their roots expressed algebraically in terms of their coefficients (for more, see algebraic solution).
Zero set
In various areas of mathematics, the zero set of a function is the set of all its zeros. More precisely, if $f:X\to \mathbb {R} $ is a real-valued function (or, more generally, a function taking values in some additive group), its zero set is $f^{-1}(0)$, the inverse image of $\{0\}$ in $X$.
Under the same hypothesis on the codomain of the function, a level set of a function $f$ is the zero set of the function $f-c$ for some $c$ in the codomain of $f.$
The zero set of a linear map is also known as its kernel.
The cozero set of the function $f:X\to \mathbb {R} $ is the complement of the zero set of $f$ (i.e., the subset of $X$ on which $f$ is nonzero).
Applications
In algebraic geometry, the first definition of an algebraic variety is through zero sets. Specifically, an affine algebraic set is the intersection of the zero sets of several polynomials, in a polynomial ring $k\left[x_{1},\ldots ,x_{n}\right]$ over a field. In this context, a zero set is sometimes called a zero locus.
In analysis and geometry, any closed subset of $\mathbb {R} ^{n}$ is the zero set of a smooth function defined on all of $\mathbb {R} ^{n}$. This extends to any smooth manifold as a corollary of paracompactness.
In differential geometry, zero sets are frequently used to define manifolds. An important special case is the case that $f$ is a smooth function from $\mathbb {R} ^{p}$ to $\mathbb {R} ^{n}$. If zero is a regular value of $f$, then the zero set of $f$ is a smooth manifold of dimension $m=p-n$ by the regular value theorem.
For example, the unit $m$-sphere in $\mathbb {R} ^{m+1}$ is the zero set of the real-valued function $f(x)=\Vert x\Vert ^{2}-1$.
See also
• Marden's theorem
• Root-finding algorithm
• Sendov's conjecture
• Vanish at infinity
• Zero crossing
• Zeros and poles
References
1. "Algebra - Zeroes/Roots of Polynomials". tutorial.math.lamar.edu. Retrieved 2019-12-15.
2. Foerster, Paul A. (2006). Algebra and Trigonometry: Functions and Applications, Teacher's Edition (Classics ed.). Upper Saddle River, NJ: Prentice Hall. p. 535. ISBN 0-13-165711-9.
3. "Roots and zeros (Algebra 2, Polynomial functions)". Mathplanet. Retrieved 2019-12-15.
Further reading
• Weisstein, Eric W. "Root". MathWorld.
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Zerosumfree monoid
In abstract algebra, an additive monoid $(M,0,+)$ is said to be zerosumfree, conical, centerless or positive if nonzero elements do not sum to zero. Formally:
$(\forall a,b\in M)\ a+b=0\implies a=b=0\!$
This means that the only way zero can be expressed as a sum is as $0+0$.
References
• Wehrung, Friedrich (1996). "Tensor products of structures with interpolation". Pacific Journal of Mathematics. 176 (1): 267–285. doi:10.2140/pjm.1996.176.267. Zbl 0865.06010.
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Zeroth-order logic
Zeroth-order logic is a branch of logic without variables or quantifiers. Some authors use the phrase "zeroth-order logic" as a synonym for the propositional calculus,[1] but an alternative definition extends propositional logic by adding constants, operations, and relations on non-Boolean values.[2] Every zeroth-order language in this broader sense is complete and compact.[2]
References
1. Andrews, Peter B. (2002), An introduction to mathematical logic and type theory: to truth through proof, Applied Logic Series, vol. 27 (Second ed.), Kluwer Academic Publishers, Dordrecht, p. 201, doi:10.1007/978-94-015-9934-4, ISBN 1-4020-0763-9, MR 1932484.
2. Tao, Terence (2010), "1.4.2 Zeroth-order logic", An epsilon of room, II, American Mathematical Society, Providence, RI, pp. 27–31, doi:10.1090/gsm/117, ISBN 978-0-8218-5280-4, MR 2780010.
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Zero-dimensional space
In mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space.[1] A graphical illustration of a nildimensional space is a point.[2]
This article is about zero dimension in topology. For several kinds of zero space in algebra, see zero object (algebra).
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Definition
Specifically:
• A topological space is zero-dimensional with respect to the Lebesgue covering dimension if every open cover of the space has a refinement which is a cover by disjoint open sets.
• A topological space is zero-dimensional with respect to the finite-to-finite covering dimension if every finite open cover of the space has a refinement that is a finite open cover such that any point in the space is contained in exactly one open set of this refinement.
• A topological space is zero-dimensional with respect to the small inductive dimension if it has a base consisting of clopen sets.
The three notions above agree for separable, metrisable spaces.
Properties of spaces with small inductive dimension zero
• A zero-dimensional Hausdorff space is necessarily totally disconnected, but the converse fails. However, a locally compact Hausdorff space is zero-dimensional if and only if it is totally disconnected. (See (Arhangel'skii & Tkachenko 2008, Proposition 3.1.7, p.136) for the non-trivial direction.)
• Zero-dimensional Polish spaces are a particularly convenient setting for descriptive set theory. Examples of such spaces include the Cantor space and Baire space.
• Hausdorff zero-dimensional spaces are precisely the subspaces of topological powers $2^{I}$ where $2=\{0,1\}$ is given the discrete topology. Such a space is sometimes called a Cantor cube. If I is countably infinite, $2^{I}$ is the Cantor space.
Manifolds
All points of a zero-dimensional manifold are isolated.
In particular, the zero-dimensional hypersphere is a pair of points, and the zero-dimensional ball is a single point.
Notes
• Arhangel'skii, Alexander; Tkachenko, Mikhail (2008). Topological Groups and Related Structures. Atlantis Studies in Mathematics. Vol. 1. Atlantis Press. ISBN 978-90-78677-06-2.
• Engelking, Ryszard (1977). General Topology. PWN, Warsaw.
• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.
References
1. Hazewinkel, Michiel (1989). Encyclopaedia of Mathematics, Volume 3. Kluwer Academic Publishers. p. 190. ISBN 9789400959941.
2. Wolcott, Luke; McTernan, Elizabeth (2012). "Imagining Negative-Dimensional Space" (PDF). In Bosch, Robert; McKenna, Douglas; Sarhangi, Reza (eds.). Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture. Phoenix, Arizona, USA: Tessellations Publishing. pp. 637–642. ISBN 978-1-938664-00-7. ISSN 1099-6702. Retrieved 10 July 2015.
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1 + 1 + 1 + 1 + ⋯
In mathematics, 1 + 1 + 1 + 1 + ⋯, also written $\sum _{n=1}^{\infty }n^{0}$, $\sum _{n=1}^{\infty }1^{n}$, or simply $\sum _{n=1}^{\infty }1$, is a divergent series, meaning that its sequence of partial sums does not converge to a limit in the real numbers. The sequence 1n can be thought of as a geometric series with the common ratio 1. Unlike other geometric series with rational ratio (except −1), it converges in neither the real numbers nor in the p-adic numbers for some p. In the context of the extended real number line
$\sum _{n=1}^{\infty }1=+\infty \,,$
The series 1 + 1 + 1 + 1 + ⋯
After smoothing
since its sequence of partial sums increases monotonically without bound.
Where the sum of n0 occurs in physical applications, it may sometimes be interpreted by zeta function regularization, as the value at s = 0 of the Riemann zeta function:
$\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1-2^{1-s}}}\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{s}}}\,.$
The two formulas given above are not valid at zero however, but the analytic continuation is.
$\zeta (s)=2^{s}\pi ^{s-1}\ \sin \left({\frac {\pi s}{2}}\right)\ \Gamma (1-s)\ \zeta (1-s)\!,$
Using this one gets (given that Γ(1) = 1),
$\zeta (0)={\frac {1}{\pi }}\lim _{s\rightarrow 0}\ \sin \left({\frac {\pi s}{2}}\right)\ \zeta (1-s)={\frac {1}{\pi }}\lim _{s\rightarrow 0}\ \left({\frac {\pi s}{2}}-{\frac {\pi ^{3}s^{3}}{48}}+...\right)\ \left(-{\frac {1}{s}}+...\right)=-{\frac {1}{2}}$
where the power series expansion for ζ(s) about s = 1 follows because ζ(s) has a simple pole of residue one there. In this sense 1 + 1 + 1 + 1 + ⋯ = ζ(0) = −1/2.
Emilio Elizalde presents a comment from others about the series:
In a short period of less than a year, two distinguished physicists, A. Slavnov and F. Yndurain, gave seminars in Barcelona, about different subjects. It was remarkable that, in both presentations, at some point the speaker addressed the audience with these words: 'As everybody knows, 1 + 1 + 1 + ⋯ = −1/2.' Implying maybe: If you do not know this, it is no use to continue listening.[2]
See also
• Grandi's series
• 1 − 2 + 3 − 4 + · · ·
• 1 + 2 + 3 + 4 + · · ·
• 1 + 2 + 4 + 8 + · · ·
• 1 − 2 + 4 − 8 + ⋯
• 1 − 1 + 2 − 6 + 24 − 120 + · · ·
• Harmonic series
Notes
1. Tao, Terence (April 10, 2010), The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation, retrieved January 30, 2014
2. Elizalde, Emilio (2004). "Cosmology: Techniques and Applications". Proceedings of the II International Conference on Fundamental Interactions. arXiv:gr-qc/0409076. Bibcode:2004gr.qc.....9076E.
External links
• OEIS sequence A000012 (The simplest sequence of positive numbers: the all 1's sequence)
Sequences and series
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• 1/2 − 1/4 + 1/8 − 1/16 + ⋯
• 1/2 + 1/4 + 1/8 + 1/16 + ⋯
• 1/4 + 1/16 + 1/64 + 1/256 + ⋯
• 1 + 1/2s + 1/3s + ... (Riemann zeta function)
Divergent
• 1 + 1 + 1 + 1 + ⋯
• 1 − 1 + 1 − 1 + ⋯ (Grandi's series)
• 1 + 2 + 3 + 4 + ⋯
• 1 − 2 + 3 − 4 + ⋯
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• 1 − 2 + 4 − 8 + ⋯
• Infinite arithmetic series
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List of zeta functions
In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function
$\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}.$
Zeta functions include:
• Airy zeta function, related to the zeros of the Airy function
• Arakawa–Kaneko zeta function
• Arithmetic zeta function
• Artin–Mazur zeta function of a dynamical system
• Barnes zeta function or double zeta function
• Beurling zeta function of Beurling generalized primes
• Dedekind zeta function of a number field
• Duursma zeta function of error-correcting codes
• Epstein zeta function of a quadratic form
• Goss zeta function of a function field
• Hasse–Weil zeta function of a variety
• Height zeta function of a variety
• Hurwitz zeta function, a generalization of the Riemann zeta function
• Igusa zeta function
• Ihara zeta function of a graph
• L-function, a "twisted" zeta function
• Lefschetz zeta function of a morphism
• Lerch zeta function, a generalization of the Riemann zeta function
• Local zeta function of a characteristic-p variety
• Matsumoto zeta function
• Minakshisundaram–Pleijel zeta function of a Laplacian
• Motivic zeta function of a motive
• Multiple zeta function, or Mordell–Tornheim zeta function of several variables
• p-adic zeta function of a p-adic number
• Prime zeta function, like the Riemann zeta function, but only summed over primes
• Riemann zeta function, the archetypal example
• Ruelle zeta function
• Selberg zeta function of a Riemann surface
• Shimizu L-function
• Shintani zeta function
• Subgroup zeta function
• Witten zeta function of a Lie group
• Zeta function of an incidence algebra, a function that maps every interval of a poset to the constant value 1. Despite not resembling a holomorphic function, the special case for the poset of integer divisibility is related as a formal Dirichlet series to the Riemann zeta function.
• Zeta function of an operator or spectral zeta function
See also
Other functions called zeta functions, but not analogous to the Riemann zeta function
• Jacobi zeta function
• Weierstrass zeta function
Topics related to zeta functions
• Artin conjecture
• Birch and Swinnerton-Dyer conjecture
• Riemann hypothesis and the generalized Riemann hypothesis.
• Selberg class S
• Explicit formulae for L-functions
• Trace formula
External links
• A directory of all known zeta functions
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Zeta function regularization
In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent sums or products, and in particular can be used to define determinants and traces of some self-adjoint operators. The technique is now commonly applied to problems in physics, but has its origins in attempts to give precise meanings to ill-conditioned sums appearing in number theory.
Renormalization and regularization
Renormalization
Renormalization group
On-shell scheme
Minimal subtraction scheme
Regularization
Dimensional regularization
Pauli–Villars regularization
Lattice regularization
Zeta function regularization
Causal perturbation theory
Hadamard regularization
Point-splitting regularization
Definition
There are several different summation methods called zeta function regularization for defining the sum of a possibly divergent series a1 + a2 + ....
One method is to define its zeta regularized sum to be ζA(−1) if this is defined, where the zeta function is defined for large Re(s) by
$\zeta _{A}(s)={\frac {1}{a_{1}^{s}}}+{\frac {1}{a_{2}^{s}}}+\cdots $
if this sum converges, and by analytic continuation elsewhere.
In the case when an = n, the zeta function is the ordinary Riemann zeta function. This method was used by Euler to "sum" the series 1 + 2 + 3 + 4 + ... to ζ(−1) = −1/12.
Hawking (1977) showed that in flat space, in which the eigenvalues of Laplacians are known, the zeta function corresponding to the partition function can be computed explicitly. Consider a scalar field φ contained in a large box of volume V in flat spacetime at the temperature T = β−1. The partition function is defined by a path integral over all fields φ on the Euclidean space obtained by putting τ = it which are zero on the walls of the box and which are periodic in τ with period β. In this situation from the partition function he computes energy, entropy and pressure of the radiation of the field φ. In case of flat spaces the eigenvalues appearing in the physical quantities are generally known, while in case of curved space they are not known: in this case asymptotic methods are needed.
Another method defines the possibly divergent infinite product a1a2.... to be exp(−ζ′A(0)). Ray & Singer (1971) used this to define the determinant of a positive self-adjoint operator A (the Laplacian of a Riemannian manifold in their application) with eigenvalues a1, a2, ...., and in this case the zeta function is formally the trace of A−s. Minakshisundaram & Pleijel (1949) showed that if A is the Laplacian of a compact Riemannian manifold then the Minakshisundaram–Pleijel zeta function converges and has an analytic continuation as a meromorphic function to all complex numbers, and Seeley (1967) extended this to elliptic pseudo-differential operators A on compact Riemannian manifolds. So for such operators one can define the determinant using zeta function regularization. See "analytic torsion."
Hawking (1977) suggested using this idea to evaluate path integrals in curved spacetimes. He studied zeta function regularization in order to calculate the partition functions for thermal graviton and matter's quanta in curved background such as on the horizon of black holes and on de Sitter background using the relation by the inverse Mellin transformation to the trace of the kernel of heat equations.
Example
The first example in which zeta function regularization is available appears in the Casimir effect, which is in a flat space with the bulk contributions of the quantum field in three space dimensions. In this case we must calculate the value of Riemann zeta function at –3, which diverges explicitly. However, it can be analytically continued to s = –3 where hopefully there is no pole, thus giving a finite value to the expression. A detailed example of this regularization at work is given in the article on the detail example of the Casimir effect, where the resulting sum is very explicitly the Riemann zeta-function (and where the seemingly legerdemain analytic continuation removes an additive infinity, leaving a physically significant finite number).
An example of zeta-function regularization is the calculation of the vacuum expectation value of the energy of a particle field in quantum field theory. More generally, the zeta-function approach can be used to regularize the whole energy–momentum tensor both in flat and in curved spacetime.
The unregulated value of the energy is given by a summation over the zero-point energy of all of the excitation modes of the vacuum:
$\langle 0|T_{00}|0\rangle =\sum _{n}{\frac {\hbar |\omega _{n}|}{2}}$
Here, $T_{00}$ is the zeroth component of the energy–momentum tensor and the sum (which may be an integral) is understood to extend over all (positive and negative) energy modes $\omega _{n}$; the absolute value reminding us that the energy is taken to be positive. This sum, as written, is usually infinite ($\omega _{n}$ is typically linear in n). The sum may be regularized by writing it as
$\langle 0|T_{00}(s)|0\rangle =\sum _{n}{\frac {\hbar |\omega _{n}|}{2}}|\omega _{n}|^{-s}$
where s is some parameter, taken to be a complex number. For large, real s greater than 4 (for three-dimensional space), the sum is manifestly finite, and thus may often be evaluated theoretically.
The zeta-regularization is useful as it can often be used in a way such that the various symmetries of the physical system are preserved. Zeta-function regularization is used in conformal field theory, renormalization and in fixing the critical spacetime dimension of string theory.
Relation to other regularizations
Zeta function regularization is equivalent to dimensional regularization, see. However, the main advantage of the zeta regularization is that it can be used whenever the dimensional regularization fails, for example if there are matrices or tensors inside the calculations $\epsilon _{i,j,k}$
Relation to Dirichlet series
Zeta-function regularization gives an analytic structure to any sums over an arithmetic function f(n). Such sums are known as Dirichlet series. The regularized form
${\tilde {f}}(s)=\sum _{n=1}^{\infty }f(n)n^{-s}$
converts divergences of the sum into simple poles on the complex s-plane. In numerical calculations, the zeta-function regularization is inappropriate, as it is extremely slow to converge. For numerical purposes, a more rapidly converging sum is the exponential regularization, given by
$F(t)=\sum _{n=1}^{\infty }f(n)e^{-tn}.$
This is sometimes called the Z-transform of f, where z = exp(−t). The analytic structure of the exponential and zeta-regularizations are related. By expanding the exponential sum as a Laurent series
$F(t)={\frac {a_{N}}{t^{N}}}+{\frac {a_{N-1}}{t^{N-1}}}+\cdots $
one finds that the zeta-series has the structure
${\tilde {f}}(s)={\frac {a_{N}}{s-N}}+\cdots .$
The structure of the exponential and zeta-regulators are related by means of the Mellin transform. The one may be converted to the other by making use of the integral representation of the Gamma function:
$\Gamma (s)=\int _{0}^{\infty }t^{s-1}e^{-t}\,dt$
which leads to the identity
$\Gamma (s){\tilde {f}}(s)=\int _{0}^{\infty }t^{s-1}F(t)\,dt$
relating the exponential and zeta-regulators, and converting poles in the s-plane to divergent terms in the Laurent series.
Heat kernel regularization
The sum
$f(s)=\sum _{n}a_{n}e^{-s|\omega _{n}|}$
is sometimes called a heat kernel or a heat-kernel regularized sum; this name stems from the idea that the $\omega _{n}$ can sometimes be understood as eigenvalues of the heat kernel. In mathematics, such a sum is known as a generalized Dirichlet series; its use for averaging is known as an Abelian mean. It is closely related to the Laplace–Stieltjes transform, in that
$f(s)=\int _{0}^{\infty }e^{-st}\,d\alpha (t)$
where $\alpha (t)$ is a step function, with steps of $a_{n}$ at $t=|\omega _{n}|$. A number of theorems for the convergence of such a series exist. For example, by the Hardy-Littlewood Tauberian theorem, if
$L=\limsup _{n\to \infty }{\frac {\log \vert \sum _{k=1}^{n}a_{k}\vert }{|\omega _{n}|}}$
then the series for $f(s)$ converges in the half-plane $\Re (s)>L$ and is uniformly convergent on every compact subset of the half-plane $\Re (s)>L$. In almost all applications to physics, one has $L=0$
History
Much of the early work establishing the convergence and equivalence of series regularized with the heat kernel and zeta function regularization methods was done by G. H. Hardy and J. E. Littlewood in 1916 and is based on the application of the Cahen–Mellin integral. The effort was made in order to obtain values for various ill-defined, conditionally convergent sums appearing in number theory.
In terms of application as the regulator in physical problems, before Hawking (1977), J. Stuart Dowker and Raymond Critchley in 1976 proposed a zeta-function regularization method for quantum physical problems. Emilio Elizalde and others have also proposed a method based on the zeta regularization for the integrals $\int _{a}^{\infty }x^{m-s}dx$, here $x^{-s}$ is a regulator and the divergent integral depends on the numbers $\zeta (s-m)$ in the limit $s\to 0$ see renormalization. Also unlike other regularizations such as dimensional regularization and analytic regularization, zeta regularization has no counterterms and gives only finite results.
See also
• Generating function – Formal power series; coefficients encode information about a sequence indexed by natural numbers
• Perron's formula – Formula to calculate the sum of an arithmetic function in analytic number theory
• Renormalization – Method in physics used to deal with infinities
• 1 + 1 + 1 + 1 + ⋯ – Divergent series
• 1 + 2 + 3 + 4 + ⋯ – Divergent series
• Analytic torsion – Topological invariant of manifolds that can distinguish homotopy-equivalent manifolds
• Ramanujan summation – Mathematical techniques for summing divergent infinite series
• Minakshisundaram–Pleijel zeta function
• Zeta function (operator)
References
• ^ Tom M. Apostol, "Modular Functions and Dirichlet Series in Number Theory", "Springer-Verlag New York. (See Chapter 8.)"
• ^ A. Bytsenko, G. Cognola, E. Elizalde, V. Moretti and S. Zerbini, "Analytic Aspects of Quantum Fields", World Scientific Publishing, 2003, ISBN 981-238-364-6
• ^ G.H. Hardy and J.E. Littlewood, "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes", Acta Mathematica, 41(1916) pp. 119–196. (See, for example, theorem 2.12)
• Hawking, S. W. (1977), "Zeta function regularization of path integrals in curved spacetime", Communications in Mathematical Physics, 55 (2): 133–148, Bibcode:1977CMaPh..55..133H, doi:10.1007/BF01626516, ISSN 0010-3616, MR 0524257, S2CID 121650064
• ^ V. Moretti, "Direct z-function approach and renormalization of one-loop stress tensor in curved spacetimes, Phys. Rev.D 56, 7797 (1997).
• Minakshisundaram, S.; Pleijel, Å. (1949), "Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds", Canadian Journal of Mathematics, 1 (3): 242–256, doi:10.4153/CJM-1949-021-5, ISSN 0008-414X, MR 0031145
• Ray, D. B.; Singer, I. M. (1971), "R-torsion and the Laplacian on Riemannian manifolds", Advances in Mathematics, 7 (2): 145–210, doi:10.1016/0001-8708(71)90045-4, MR 0295381
• "Zeta-function method for regularization", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Seeley, R. T. (1967), "Complex powers of an elliptic operator", in Calderón, Alberto P. (ed.), Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), Proceedings of Symposia in Pure Mathematics, vol. 10, Providence, R.I.: Amer. Math. Soc., pp. 288–307, ISBN 978-0-8218-1410-9, MR 0237943
• ^ Dowker, J. S.; Critchley, R. (1976), "Effective Lagrangian and energy–momentum tensor in de Sitter space", Physical Review D, 13 (12): 3224–3232, Bibcode:1976PhRvD..13.3224D, doi:10.1103/PhysRevD.13.3224
• ^ D. Fermi, L. Pizzocchero, "Local zeta regularization and the scalar Casimir effect. A general approach based on integral kernels", World Scientific Publishing, ISBN: 978-981-3224-99-5 (hardcover), ISBN: 978-981-3225-01-5 (ebook). DOI: 10.1142/10570 (2017).
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ZetaGrid
ZetaGrid was at one time the largest distributed computing project, designed to explore the non-trivial roots of the Riemann zeta function, checking over one billion roots a day.
Roots of the zeta function are of particular interest in mathematics; a single root out of alignment would disprove the Riemann hypothesis, with far-reaching consequences for all of mathematics. As of June, 2023 no counterexample to the Riemann hypothesis has been found.
The project ended in November 2005 due to instability of the hosting provider.[1] The first more than 1013 zeroes were checked.[2] The project administrator stated that after the results were analyzed, they would be posted on the American Mathematical Society website.[3] The official status remains unclear, however, as it was never published nor independently verified. This is likely because there was no evidence that each zero was actually computed, as there was no process implemented to check each one as it was calculated.[4][5]
References
1. Zeta Finished – Free-DC Forum
2. Ed Pegg Jr. «Ten Trillion Zeta Zeros»
3. "ZetaGrid - News". 2010-11-18. Archived from the original on 2010-11-18. Retrieved 2023-06-04.
4. Yannick Saouter, Xavier Gourdon and Patrick Demichel. An improved lower bound for the de Bruijn-Newman constant. Math. Comp. 80 (2011) 2283. MR 2813360.
5. Yannick Saouter and Patrick Demichel. A sharp region where π(x)−li(x) is positive. Math. Comp. 79 (2010) 2398. MR 2684372.
External links
• Home page (Web archive)
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Particular values of the Riemann zeta function
In mathematics, the Riemann zeta function is a function in complex analysis, which is also important in number theory. It is often denoted ζ(s) and is named after the mathematician Bernhard Riemann. When the argument s is a real number greater than one, the zeta function satisfies the equation
$\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}\,.$
It can therefore provide the sum of various convergent infinite series, such as $ \zeta (2)={\frac {1}{1^{2}}}+$$ {\frac {1}{2^{2}}}+$$ {\frac {1}{3^{2}}}+\ldots \,.$ Explicit or numerically efficient formulae exist for ζ(s) at integer arguments, all of which have real values, including this example. This article lists these formulae, together with tables of values. It also includes derivatives and some series composed of the zeta function at integer arguments.
The same equation in s above also holds when s is a complex number whose real part is greater than one, ensuring that the infinite sum still converges. The zeta function can then be extended to the whole of the complex plane by analytic continuation, except for a simple pole at s = 1. The complex derivative exists in this more general region, making the zeta function a meromorphic function. The above equation no longer applies for these extended values of s, for which the corresponding summation would diverge. For example, the full zeta function exists at s = −1 (and is therefore finite there), but the corresponding series would be $ 1+2+3+\ldots \,,$ whose partial sums would grow indefinitely large.
The zeta function values listed below include function values at the negative even numbers (s = −2, −4, etc.), for which ζ(s) = 0 and which make up the so-called trivial zeros. The Riemann zeta function article includes a colour plot illustrating how the function varies over a continuous rectangular region of the complex plane. The successful characterisation of its non-trivial zeros in the wider plane is important in number theory, because of the Riemann hypothesis.
The Riemann zeta function at 0 and 1
At zero, one has
$\zeta (0)={B_{1}^{-}}=-{B_{1}^{+}}=-{\tfrac {1}{2}}\!$
At 1 there is a pole, so ζ(1) is not finite but the left and right limits are:
$\lim _{\varepsilon \to 0^{\pm }}\zeta (1+\varepsilon )=\pm \infty $
Since it is a pole of first order, it has a complex residue
$\lim _{\varepsilon \to 0}\varepsilon \zeta (1+\varepsilon )=1\,.$
Positive integers
Even positive integers
For the even positive integers $n$, one has the relationship to the Bernoulli numbers:
$\zeta (n)=(-1)^{{\tfrac {n}{2}}+1}{\frac {(2\pi )^{n}B_{n}}{2(n!)}}\,.$
The computation of ζ(2) is known as the Basel problem. The value of ζ(4) is related to the Stefan–Boltzmann law and Wien approximation in physics. The first few values are given by:
${\begin{aligned}\zeta (2)&=1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}\\[4pt]\zeta (4)&=1+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+\cdots ={\frac {\pi ^{4}}{90}}\\[4pt]\zeta (6)&=1+{\frac {1}{2^{6}}}+{\frac {1}{3^{6}}}+\cdots ={\frac {\pi ^{6}}{945}}\\[4pt]\zeta (8)&=1+{\frac {1}{2^{8}}}+{\frac {1}{3^{8}}}+\cdots ={\frac {\pi ^{8}}{9450}}\\[4pt]\zeta (10)&=1+{\frac {1}{2^{10}}}+{\frac {1}{3^{10}}}+\cdots ={\frac {\pi ^{10}}{93555}}\\[4pt]\zeta (12)&=1+{\frac {1}{2^{12}}}+{\frac {1}{3^{12}}}+\cdots ={\frac {691\pi ^{12}}{638512875}}\\[4pt]\zeta (14)&=1+{\frac {1}{2^{14}}}+{\frac {1}{3^{14}}}+\cdots ={\frac {2\pi ^{14}}{18243225}}\\[4pt]\zeta (16)&=1+{\frac {1}{2^{16}}}+{\frac {1}{3^{16}}}+\cdots ={\frac {3617\pi ^{16}}{325641566250}}\,.\end{aligned}}$
Taking the limit $n\rightarrow \infty $, one obtains $\zeta (\infty )=1$.
Selected values for even integers
Value Decimal expansion Source
$\zeta (2)$ 1.6449340668482264364... OEIS: A013661
$\zeta (4)$ 1.0823232337111381915... OEIS: A013662
$\zeta (6)$ 1.0173430619844491397... OEIS: A013664
$\zeta (8)$ 1.0040773561979443393... OEIS: A013666
$\zeta (10)$ 1.0009945751278180853... OEIS: A013668
$\zeta (12)$ 1.0002460865533080482... OEIS: A013670
$\zeta (14)$ 1.0000612481350587048... OEIS: A013672
$\zeta (16)$ 1.0000152822594086518... OEIS: A013674
The relationship between zeta at the positive even integers and the Bernoulli numbers may be written as
$A_{n}\zeta (2n)=\pi ^{2n}B_{n}$
where $A_{n}$ and $B_{n}$ are integers for all even $n$. These are given by the integer sequences OEIS: A002432 and OEIS: A046988, respectively, in OEIS. Some of these values are reproduced below:
coefficients
n A B
1 6 1
2 90 1
3 945 1
4 9450 1
5 93555 1
6 638512875 691
7 18243225 2
8 325641566250 3617
9 38979295480125 43867
10 1531329465290625 174611
11 13447856940643125 155366
12 201919571963756521875 236364091
13 11094481976030578125 1315862
14 564653660170076273671875 6785560294
15 5660878804669082674070015625 6892673020804
16 62490220571022341207266406250 7709321041217
17 12130454581433748587292890625 151628697551
If we let $\eta _{n}=B_{n}/A_{n}$ be the coefficient of $\pi ^{2n}$ as above,
$\zeta (2n)=\sum _{\ell =1}^{\infty }{\frac {1}{\ell ^{2n}}}=\eta _{n}\pi ^{2n}$
then we find recursively,
${\begin{aligned}\eta _{1}&=1/6\\\eta _{n}&=\sum _{\ell =1}^{n-1}(-1)^{\ell -1}{\frac {\eta _{n-\ell }}{(2\ell +1)!}}+(-1)^{n+1}{\frac {n}{(2n+1)!}}\end{aligned}}$
This recurrence relation may be derived from that for the Bernoulli numbers.
Also, there is another recurrence:
$\zeta (2n)={\frac {1}{n+{\frac {1}{2}}}}\sum _{k=1}^{n-1}\zeta (2k)\zeta (2n-2k)\quad {\text{ for }}\quad n>1$
which can be proved, using that ${\frac {d}{dx}}\cot(x)=-1-\cot ^{2}(x)$
The values of the zeta function at non-negative even integers have the generating function:
$\sum _{n=0}^{\infty }\zeta (2n)x^{2n}=-{\frac {\pi x}{2}}\cot(\pi x)=-{\frac {1}{2}}+{\frac {\pi ^{2}}{6}}x^{2}+{\frac {\pi ^{4}}{90}}x^{4}+{\frac {\pi ^{6}}{945}}x^{6}+\cdots $
Since
$\lim _{n\rightarrow \infty }\zeta (2n)=1$
The formula also shows that for $n\in \mathbb {N} ,n\rightarrow \infty $,
$\left|B_{2n}\right|\sim {\frac {(2n)!\,2}{\;~(2\pi )^{2n}\,}}$
Odd positive integers
The sum of the harmonic series is infinite.
$\zeta (1)=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots =\infty \!$
The value ζ(3) is also known as Apéry's constant and has a role in the electron's gyromagnetic ratio. The value ζ(3) also appears in Planck's law. These and additional values are:
Selected values for odd integers
Value Decimal expansion Source
$\zeta (3)$ 1.2020569031595942853... OEIS: A02117
$\zeta (5)$ 1.0369277551433699263... OEIS: A013663
$\zeta (7)$ 1.0083492773819228268... OEIS: A013665
$\zeta (9)$ 1.0020083928260822144... OEIS: A013667
$\zeta (11)$ 1.0004941886041194645... OEIS: A013669
$\zeta (13)$ 1.0001227133475784891... OEIS: A013671
$\zeta (15)$ 1.0000305882363070204... OEIS: A013673
It is known that ζ(3) is irrational (Apéry's theorem) and that infinitely many of the numbers ζ(2n + 1) : n ∈ $\mathbb {N} $ , are irrational.[1] There are also results on the irrationality of values of the Riemann zeta function at the elements of certain subsets of the positive odd integers; for example, at least one of ζ(5), ζ(7), ζ(9), or ζ(11) is irrational.[2]
The positive odd integers of the zeta function appear in physics, specifically correlation functions of antiferromagnetic XXX spin chain.[3]
Most of the identities following below are provided by Simon Plouffe. They are notable in that they converge quite rapidly, giving almost three digits of precision per iteration, and are thus useful for high-precision calculations.
Plouffe stated the following identities without proof.[4] Proofs were later given by other authors.[5]
ζ(5)
${\begin{aligned}\zeta (5)&={\frac {1}{294}}\pi ^{5}-{\frac {72}{35}}\sum _{n=1}^{\infty }{\frac {1}{n^{5}(e^{2\pi n}-1)}}-{\frac {2}{35}}\sum _{n=1}^{\infty }{\frac {1}{n^{5}(e^{2\pi n}+1)}}\\\zeta (5)&=12\sum _{n=1}^{\infty }{\frac {1}{n^{5}\sinh(\pi n)}}-{\frac {39}{20}}\sum _{n=1}^{\infty }{\frac {1}{n^{5}(e^{2\pi n}-1)}}+{\frac {1}{20}}\sum _{n=1}^{\infty }{\frac {1}{n^{5}(e^{2\pi n}+1)}}\end{aligned}}$
ζ(7)
$\zeta (7)={\frac {19}{56700}}\pi ^{7}-2\sum _{n=1}^{\infty }{\frac {1}{n^{7}(e^{2\pi n}-1)}}\!$
Note that the sum is in the form of a Lambert series.
ζ(2n + 1)
By defining the quantities
$S_{\pm }(s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}(e^{2\pi n}\pm 1)}}$
a series of relationships can be given in the form
$0=A_{n}\zeta (n)-B_{n}\pi ^{n}+C_{n}S_{-}(n)+D_{n}S_{+}(n)$
where An, Bn, Cn and Dn are positive integers. Plouffe gives a table of values:
coefficients
n A B C D
3 180 7 360 0
5 1470 5 3024 84
7 56700 19 113400 0
9 18523890 625 37122624 74844
11 425675250 1453 851350500 0
13 257432175 89 514926720 62370
15 390769879500 13687 781539759000 0
17 1904417007743250 6758333 3808863131673600 29116187100
19 21438612514068750 7708537 42877225028137500 0
21 1881063815762259253125 68529640373 3762129424572110592000 1793047592085750
These integer constants may be expressed as sums over Bernoulli numbers, as given in (Vepstas, 2006) below.
A fast algorithm for the calculation of Riemann's zeta function for any integer argument is given by E. A. Karatsuba.[6][7][8]
Negative integers
In general, for negative integers (and also zero), one has
$\zeta (-n)=(-1)^{n}{\frac {B_{n+1}}{n+1}}$
The so-called "trivial zeros" occur at the negative even integers:
$\zeta (-2n)=0$
(Ramanujan summation)
The first few values for negative odd integers are
${\begin{aligned}\zeta (-1)&=-{\frac {1}{12}}\\[4pt]\zeta (-3)&={\frac {1}{120}}\\[4pt]\zeta (-5)&=-{\frac {1}{252}}\\[4pt]\zeta (-7)&={\frac {1}{240}}\\[4pt]\zeta (-9)&=-{\frac {1}{132}}\\[4pt]\zeta (-11)&={\frac {691}{32760}}\\[4pt]\zeta (-13)&=-{\frac {1}{12}}\end{aligned}}$
However, just like the Bernoulli numbers, these do not stay small for increasingly negative odd values. For details on the first value, see 1 + 2 + 3 + 4 + · · ·.
So ζ(m) can be used as the definition of all (including those for index 0 and 1) Bernoulli numbers.
Derivatives
The derivative of the zeta function at the negative even integers is given by
$\zeta ^{\prime }(-2n)=(-1)^{n}{\frac {(2n)!}{2(2\pi )^{2n}}}\zeta (2n+1)\,.$
The first few values of which are
${\begin{aligned}\zeta ^{\prime }(-2)&=-{\frac {\zeta (3)}{4\pi ^{2}}}\\[4pt]\zeta ^{\prime }(-4)&={\frac {3}{4\pi ^{4}}}\zeta (5)\\[4pt]\zeta ^{\prime }(-6)&=-{\frac {45}{8\pi ^{6}}}\zeta (7)\\[4pt]\zeta ^{\prime }(-8)&={\frac {315}{4\pi ^{8}}}\zeta (9)\,.\end{aligned}}$
One also has
${\begin{aligned}\zeta ^{\prime }(0)&=-{\frac {1}{2}}\ln(2\pi )\\[4pt]\zeta ^{\prime }(-1)&={\frac {1}{12}}-\ln A\\[4pt]\zeta ^{\prime }(2)&={\frac {1}{6}}\pi ^{2}(\gamma +\ln 2-12\ln A+\ln \pi )\end{aligned}}$
where A is the Glaisher–Kinkelin constant. The first of these identities implies that the regularized product of the reciprocals of the positive integers is $1/{\sqrt {2\pi }}$, thus the amusing "equation" $\infty !={\sqrt {2\pi }}$ !={\sqrt {2\pi }}} .[9]
From the logarithmic derivative of the functional equation,
$2{\frac {\zeta '(1/2)}{\zeta (1/2)}}=\log(2\pi )+{\frac {\pi \cos(\pi /4)}{2\sin(\pi /4)}}-{\frac {\Gamma '(1/2)}{\Gamma (1/2)}}=\log(2\pi )+{\frac {\pi }{2}}+2\log 2+\gamma \,.$
Selected derivatives
Value Decimal expansion Source
$\zeta '(3)$ −0.19812624288563685333... OEIS: A244115
$\zeta '(2)$ −0.93754825431584375370... OEIS: A073002
$\zeta '(0)$ −0.91893853320467274178... OEIS: A075700
$\zeta '(-{\tfrac {1}{2}})$ −0.36085433959994760734... OEIS: A271854
$\zeta '(-1)$ −0.16542114370045092921... OEIS: A084448
$\zeta '(-2)$ −0.030448457058393270780... OEIS: A240966
$\zeta '(-3)$ +0.0053785763577743011444... OEIS: A259068
$\zeta '(-4)$ +0.0079838114502686242806... OEIS: A259069
$\zeta '(-5)$ −0.00057298598019863520499... OEIS: A259070
$\zeta '(-6)$ −0.0058997591435159374506... OEIS: A259071
$\zeta '(-7)$ −0.00072864268015924065246... OEIS: A259072
$\zeta '(-8)$ +0.0083161619856022473595... OEIS: A259073
Series involving ζ(n)
The following sums can be derived from the generating function:
$\sum _{k=2}^{\infty }\zeta (k)x^{k-1}=-\psi _{0}(1-x)-\gamma $
where ψ0 is the digamma function.
${\begin{aligned}\sum _{k=2}^{\infty }(\zeta (k)-1)&=1\\[4pt]\sum _{k=1}^{\infty }(\zeta (2k)-1)&={\frac {3}{4}}\\[4pt]\sum _{k=1}^{\infty }(\zeta (2k+1)-1)&={\frac {1}{4}}\\[4pt]\sum _{k=2}^{\infty }(-1)^{k}(\zeta (k)-1)&={\frac {1}{2}}\end{aligned}}$
Series related to the Euler–Mascheroni constant (denoted by γ) are
${\begin{aligned}\sum _{k=2}^{\infty }(-1)^{k}{\frac {\zeta (k)}{k}}&=\gamma \\[4pt]\sum _{k=2}^{\infty }{\frac {\zeta (k)-1}{k}}&=1-\gamma \\[4pt]\sum _{k=2}^{\infty }(-1)^{k}{\frac {\zeta (k)-1}{k}}&=\ln 2+\gamma -1\end{aligned}}$
and using the principal value
$\zeta (k)=\lim _{\varepsilon \to 0}{\frac {\zeta (k+\varepsilon )+\zeta (k-\varepsilon )}{2}}$
which of course affects only the value at 1, these formulae can be stated as
${\begin{aligned}\sum _{k=1}^{\infty }(-1)^{k}{\frac {\zeta (k)}{k}}&=0\\[4pt]\sum _{k=1}^{\infty }{\frac {\zeta (k)-1}{k}}&=0\\[4pt]\sum _{k=1}^{\infty }(-1)^{k}{\frac {\zeta (k)-1}{k}}&=\ln 2\end{aligned}}$
and show that they depend on the principal value of ζ(1) = γ .
Nontrivial zeros
Main article: Riemann hypothesis
Zeros of the Riemann zeta except negative even integers are called "nontrivial zeros". The Riemann hypothesis states that the real part of every nontrivial zero must be 1/2. In other words, all known nontrivial zeros of the Riemann zeta are of the form z = 1/2 + yi where y is a real number. The following table contains the decimal expansion of Im(z) for the first few nontrivial zeros:
Selected nontrivial zeros
Decimal expansion of Im(z) Source
14.134725141734693790... OEIS: A058303
21.022039638771554992... OEIS: A065434
25.010857580145688763... OEIS: A065452
30.424876125859513210... OEIS: A065453
32.935061587739189690... OEIS: A192492
37.586178158825671257... OEIS: A305741
40.918719012147495187... OEIS: A305742
43.327073280914999519... OEIS: A305743
48.005150881167159727... OEIS: A305744
49.773832477672302181... OEIS: A306004
Andrew Odlyzko computed the first 2 million nontrivial zeros accurate to within 4×10−9, and the first 100 zeros accurate within 1000 decimal places. See their website for the tables and bibliographies.[10][11]
Ratios
Although evaluating particular values of the zeta function is difficult, often certain ratios can be found by inserting particular values of the gamma function into the functional equation
$\zeta (s)=2^{s}\pi ^{s-1}\sin \left({\frac {\pi s}{2}}\right)\Gamma (1-s)\zeta (1-s)$
We have simple relations for half-integer arguments
${\begin{aligned}{\frac {\zeta (3/2)}{\zeta (-1/2)}}&=-4\pi \\{\frac {\zeta (5/2)}{\zeta (-3/2)}}&=-{\frac {16\pi ^{2}}{3}}\\{\frac {\zeta (7/2)}{\zeta (-5/2)}}&={\frac {64\pi ^{3}}{15}}\\{\frac {\zeta (9/2)}{\zeta (-7/2)}}&={\frac {256\pi ^{4}}{105}}\end{aligned}}$
Other examples follow for more complicated evaluations and relations of the gamma function. For example a consequence of the relation
$\Gamma \left({\tfrac {3}{4}}\right)=\left({\tfrac {\pi }{2}}\right)^{\tfrac {1}{4}}{\operatorname {AGM} \left({\sqrt {2}},1\right)}^{\tfrac {1}{2}}$
is the zeta ratio relation
${\frac {\zeta (3/4)}{\zeta (1/4)}}=2{\sqrt {\frac {\pi }{(2-{\sqrt {2}})\operatorname {AGM} \left({\sqrt {2}},1\right)}}}$
where AGM is the arithmetic–geometric mean. In a similar vein, it is possible to form radical relations, such as from
${\frac {\Gamma \left({\frac {1}{5}}\right)^{2}}{\Gamma \left({\frac {1}{10}}\right)\Gamma \left({\frac {3}{10}}\right)}}={\frac {\sqrt {1+{\sqrt {5}}}}{2^{\tfrac {7}{10}}{\sqrt[{4}]{5}}}}$
the analogous zeta relation is
${\frac {\zeta (1/5)^{2}\zeta (7/10)\zeta (9/10)}{\zeta (1/10)\zeta (3/10)\zeta (4/5)^{2}}}={\frac {(5-{\sqrt {5}})\left({\sqrt {10}}+{\sqrt {5+{\sqrt {5}}}}\right)}{10\cdot 2^{\tfrac {3}{10}}}}$
References
1. Rivoal, T. (2000). "La fonction zeta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs". Comptes Rendus de l'Académie des Sciences, Série I. 331 (4): 267–270. arXiv:math/0008051. Bibcode:2000CRASM.331..267R. doi:10.1016/S0764-4442(00)01624-4. S2CID 119678120.
2. W. Zudilin (2001). "One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational". Russ. Math. Surv. 56 (4): 774–776. Bibcode:2001RuMaS..56..774Z. doi:10.1070/rm2001v056n04abeh000427. S2CID 250734661.
3. Boos, H.E.; Korepin, V.E.; Nishiyama, Y.; Shiroishi, M. (2002). "Quantum correlations and number theory". J. Phys. A. 35 (20): 4443–4452. arXiv:cond-mat/0202346. Bibcode:2002JPhA...35.4443B. doi:10.1088/0305-4470/35/20/305. S2CID 119143600..
4. "Identities for Zeta(2*n+1)".
5. "Formulas for Odd Zeta Values and Powers of Pi".
6. Karatsuba, E. A. (1995). "Fast calculation of the Riemann zeta function ζ(s) for integer values of the argument s". Probl. Perdachi Inf. 31 (4): 69–80. MR 1367927.
7. E. A. Karatsuba: Fast computation of the Riemann zeta function for integer argument. Dokl. Math. Vol.54, No.1, p. 626 (1996).
8. E. A. Karatsuba: Fast evaluation of ζ(3). Probl. Inf. Transm. Vol.29, No.1, pp. 58–62 (1993).
9. Muñoz García, E.; Pérez Marco, R. (2008), "The Product Over All Primes is $4\pi ^{2}$", Commun. Math. Phys. (277): 69–81.
10. Odlyzko, Andrew. "Tables of zeros of the Riemann zeta function". Retrieved 7 September 2022.
11. Odlyzko, Andrew. "Papers on Zeros of the Riemann Zeta Function and Related Topics". Retrieved 7 September 2022.
Further reading
• Ciaurri, Óscar; Navas, Luis M.; Ruiz, Francisco J.; Varona, Juan L. (May 2015). "A Simple Computation of ζ(2k)". The American Mathematical Monthly. 122 (5): 444–451. doi:10.4169/amer.math.monthly.122.5.444. JSTOR 10.4169/amer.math.monthly.122.5.444. S2CID 207521195.
• Simon Plouffe, "Identities inspired from Ramanujan Notebooks Archived 2009-01-30 at the Wayback Machine", (1998).
• Simon Plouffe, "Identities inspired by Ramanujan Notebooks part 2 PDF Archived 2011-09-26 at the Wayback Machine" (2006).
• Vepstas, Linas (2006). "On Plouffe's Ramanujan identities" (PDF). The Ramanujan Journal. 27 (3): 387–408. arXiv:math.NT/0609775. doi:10.1007/s11139-011-9335-9. S2CID 8789411.
• Zudilin, Wadim (2001). "One of the Numbers ζ(5), ζ(7), ζ(9), ζ(11) Is Irrational". Russian Mathematical Surveys. 56 (4): 774–776. Bibcode:2001RuMaS..56..774Z. doi:10.1070/RM2001v056n04ABEH000427. MR 1861452. S2CID 250734661. PDF PDF Russian PS Russian
• Nontrival zeros reference by Andrew Odlyzko:
• Bibliography
• Tables
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Zeta distribution
In probability theory and statistics, the zeta distribution is a discrete probability distribution. If X is a zeta-distributed random variable with parameter s, then the probability that X takes the integer value k is given by the probability mass function
$f_{s}(k)=k^{-s}/\zeta (s)\,$
zeta
Probability mass function
Plot of the Zeta PMF on a log-log scale. (The function is only defined at integer values of k. The connecting lines do not indicate continuity.)
Cumulative distribution function
Parameters $s\in (1,\infty )$
Support $k\in \{1,2,\ldots \}$
PMF ${\frac {1/k^{s}}{\zeta (s)}}$
CDF ${\frac {H_{k,s}}{\zeta (s)}}$
Mean ${\frac {\zeta (s-1)}{\zeta (s)}}~{\textrm {for}}~s>2$
Mode $1\,$
Variance ${\frac {\zeta (s)\zeta (s-2)-\zeta (s-1)^{2}}{\zeta (s)^{2}}}~{\textrm {for}}~s>3$
Entropy $\sum _{k=1}^{\infty }{\frac {1/k^{s}}{\zeta (s)}}\log(k^{s}\zeta (s)).\,\!$
MGF does not exist
CF ${\frac {\operatorname {Li} _{s}(e^{it})}{\zeta (s)}}$
where ζ(s) is the Riemann zeta function (which is undefined for s = 1).
The multiplicities of distinct prime factors of X are independent random variables.
The Riemann zeta function being the sum of all terms $k^{-s}$ for positive integer k, it appears thus as the normalization of the Zipf distribution. The terms "Zipf distribution" and the "zeta distribution" are often used interchangeably. But while the Zeta distribution is a probability distribution by itself, it is not associated to the Zipf's law with same exponent. See also Yule–Simon distribution
Definition
The Zeta distribution is defined for positive integers $k\geq 1$, and its probability mass function is given by
$P(x=k)={\frac {1}{\zeta (s)}}k^{-s}$,
where $s>1$ is the parameter, and $\zeta (s)$ is the Riemann zeta function.
The cumulative distribution function is given by
$P(x\leq k)={\frac {H_{k,s}}{\zeta (s)}},$
where $H_{k,s}$ is the generalized harmonic number
$H_{k,s}=\sum _{i=1}^{k}{\frac {1}{i^{s}}}.$
Moments
The nth raw moment is defined as the expected value of Xn:
$m_{n}=E(X^{n})={\frac {1}{\zeta (s)}}\sum _{k=1}^{\infty }{\frac {1}{k^{s-n}}}$
The series on the right is just a series representation of the Riemann zeta function, but it only converges for values of $s-n$ that are greater than unity. Thus:
$m_{n}=\left\{{\begin{matrix}\zeta (s-n)/\zeta (s)&{\textrm {for}}~n<s-1\\\infty &{\textrm {for}}~n\geq s-1\end{matrix}}\right.$
The ratio of the zeta functions is well-defined, even for n > s − 1 because the series representation of the zeta function can be analytically continued. This does not change the fact that the moments are specified by the series itself, and are therefore undefined for large n.
Moment generating function
The moment generating function is defined as
$M(t;s)=E(e^{tX})={\frac {1}{\zeta (s)}}\sum _{k=1}^{\infty }{\frac {e^{tk}}{k^{s}}}.$
The series is just the definition of the polylogarithm, valid for $e^{t}<1$ so that
$M(t;s)={\frac {\operatorname {Li} _{s}(e^{t})}{\zeta (s)}}{\text{ for }}t<0.$
Since this does not converge on an open interval containing $t=0$, the moment generating function does not exist.
The case s = 1
ζ(1) is infinite as the harmonic series, and so the case when s = 1 is not meaningful. However, if A is any set of positive integers that has a density, i.e. if
$\lim _{n\to \infty }{\frac {N(A,n)}{n}}$
exists where N(A, n) is the number of members of A less than or equal to n, then
$\lim _{s\to 1^{+}}P(X\in A)\,$
is equal to that density.
The latter limit can also exist in some cases in which A does not have a density. For example, if A is the set of all positive integers whose first digit is d, then A has no density, but nonetheless the second limit given above exists and is proportional to
$\log(d+1)-\log(d)=\log \left(1+{\frac {1}{d}}\right),\,$
which is Benford's law.
Infinite divisibility
The Zeta distribution can be constructed with a sequence of independent random variables with a Geometric distribution. Let $p$ be a prime number and $X(p^{-s})$ be a random variable with a Geometric distribution of parameter $p^{-s}$, namely
$\quad \quad \quad \mathbb {P} \left(X(p^{-s})=k\right)=p^{-ks}(1-p^{-s})$
If the random variables $(X(p^{-s}))_{p\in {\mathcal {P}}}$ are independent, then, the random variable $Z_{s}$ defined by
$\quad \quad \quad Z_{s}=\prod _{p\in {\mathcal {P}}}p^{X(p^{-s})}$
has the Zeta distribution : $\mathbb {P} \left(Z_{s}=n\right)={\frac {1}{n^{s}\zeta (s)}}$.
Stated differently, the random variable $\log(Z_{s})=\sum _{p\in {\mathcal {P}}}X(p^{-s})\,\log(p)$ is infinitely divisible with Lévy measure given by the following sum of Dirac masses :
$\quad \quad \quad \Pi _{s}(dx)=\sum _{p\in {\mathcal {P}}}\sum _{k\geqslant 1}{\frac {p^{-ks}}{k}}\delta _{k\log(p)}(dx)$
See also
Other "power-law" distributions
• Cauchy distribution
• Lévy distribution
• Lévy skew alpha-stable distribution
• Pareto distribution
• Zipf's law
• Zipf–Mandelbrot law
• Infinitely divisible distribution
External links
• Gut, Allan. "Some remarks on the Riemann zeta distribution". CiteSeerX 10.1.1.66.3284. What Gut calls the "Riemann zeta distribution" is actually the probability distribution of −log X, where X is a random variable with what this article calls the zeta distribution.
• Weisstein, Eric W. "Zipf Distribution". MathWorld.
Probability distributions (list)
Discrete
univariate
with finite
support
• Benford
• Bernoulli
• beta-binomial
• binomial
• categorical
• hypergeometric
• negative
• Poisson binomial
• Rademacher
• soliton
• discrete uniform
• Zipf
• Zipf–Mandelbrot
with infinite
support
• beta negative binomial
• Borel
• Conway–Maxwell–Poisson
• discrete phase-type
• Delaporte
• extended negative binomial
• Flory–Schulz
• Gauss–Kuzmin
• geometric
• logarithmic
• mixed Poisson
• negative binomial
• Panjer
• parabolic fractal
• Poisson
• Skellam
• Yule–Simon
• zeta
Continuous
univariate
supported on a
bounded interval
• arcsine
• ARGUS
• Balding–Nichols
• Bates
• beta
• beta rectangular
• continuous Bernoulli
• Irwin–Hall
• Kumaraswamy
• logit-normal
• noncentral beta
• PERT
• raised cosine
• reciprocal
• triangular
• U-quadratic
• uniform
• Wigner semicircle
supported on a
semi-infinite
interval
• Benini
• Benktander 1st kind
• Benktander 2nd kind
• beta prime
• Burr
• chi
• chi-squared
• noncentral
• inverse
• scaled
• Dagum
• Davis
• Erlang
• hyper
• exponential
• hyperexponential
• hypoexponential
• logarithmic
• F
• noncentral
• folded normal
• Fréchet
• gamma
• generalized
• inverse
• gamma/Gompertz
• Gompertz
• shifted
• half-logistic
• half-normal
• Hotelling's T-squared
• inverse Gaussian
• generalized
• Kolmogorov
• Lévy
• log-Cauchy
• log-Laplace
• log-logistic
• log-normal
• log-t
• Lomax
• matrix-exponential
• Maxwell–Boltzmann
• Maxwell–Jüttner
• Mittag-Leffler
• Nakagami
• Pareto
• phase-type
• Poly-Weibull
• Rayleigh
• relativistic Breit–Wigner
• Rice
• truncated normal
• type-2 Gumbel
• Weibull
• discrete
• Wilks's lambda
supported
on the whole
real line
• Cauchy
• exponential power
• Fisher's z
• Kaniadakis κ-Gaussian
• Gaussian q
• generalized normal
• generalized hyperbolic
• geometric stable
• Gumbel
• Holtsmark
• hyperbolic secant
• Johnson's SU
• Landau
• Laplace
• asymmetric
• logistic
• noncentral t
• normal (Gaussian)
• normal-inverse Gaussian
• skew normal
• slash
• stable
• Student's t
• Tracy–Widom
• variance-gamma
• Voigt
with support
whose type varies
• generalized chi-squared
• generalized extreme value
• generalized Pareto
• Marchenko–Pastur
• Kaniadakis κ-exponential
• Kaniadakis κ-Gamma
• Kaniadakis κ-Weibull
• Kaniadakis κ-Logistic
• Kaniadakis κ-Erlang
• q-exponential
• q-Gaussian
• q-Weibull
• shifted log-logistic
• Tukey lambda
Mixed
univariate
continuous-
discrete
• Rectified Gaussian
Multivariate
(joint)
• Discrete:
• Ewens
• multinomial
• Dirichlet
• negative
• Continuous:
• Dirichlet
• generalized
• multivariate Laplace
• multivariate normal
• multivariate stable
• multivariate t
• normal-gamma
• inverse
• Matrix-valued:
• LKJ
• matrix normal
• matrix t
• matrix gamma
• inverse
• Wishart
• normal
• inverse
• normal-inverse
• complex
Directional
Univariate (circular) directional
Circular uniform
univariate von Mises
wrapped normal
wrapped Cauchy
wrapped exponential
wrapped asymmetric Laplace
wrapped Lévy
Bivariate (spherical)
Kent
Bivariate (toroidal)
bivariate von Mises
Multivariate
von Mises–Fisher
Bingham
Degenerate
and singular
Degenerate
Dirac delta function
Singular
Cantor
Families
• Circular
• compound Poisson
• elliptical
• exponential
• natural exponential
• location–scale
• maximum entropy
• mixture
• Pearson
• Tweedie
• wrapped
• Category
• Commons
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Zeta function (operator)
The zeta function of a mathematical operator ${\mathcal {O}}$ is a function defined as
$\zeta _{\mathcal {O}}(s)=\operatorname {tr} \;{\mathcal {O}}^{-s}$
for those values of s where this expression exists, and as an analytic continuation of this function for other values of s. Here "tr" denotes a functional trace.
The zeta function may also be expressible as a spectral zeta function[1] in terms of the eigenvalues $\lambda _{i}$ of the operator ${\mathcal {O}}$ by
$\zeta _{\mathcal {O}}(s)=\sum _{i}\lambda _{i}^{-s}$.
It is used in giving a rigorous definition to the functional determinant of an operator, which is given by
$\det {\mathcal {O}}:=e^{-\zeta '_{\mathcal {O}}(0)}\;.$
The Minakshisundaram–Pleijel zeta function is an example, when the operator is the Laplacian of a compact Riemannian manifold.
One of the most important motivations for Arakelov theory is the zeta functions for operators with the method of heat kernels generalized algebro-geometrically.[2]
See also
• Quillen metric
References
1. Lapidus & van Frankenhuijsen (2006) p.23
2. Soulé, C.; with the collaboration of D. Abramovich, J.-F. Burnol and J. Kramer (1992), Lectures on Arakelov geometry, Cambridge Studies in Advanced Mathematics, vol. 33, Cambridge: Cambridge University Press, pp. viii+177, ISBN 0-521-41669-8, MR 1208731
• Lapidus, Michel L.; van Frankenhuijsen, Machiel (2006), Fractal geometry, complex dimensions and zeta functions. Geometry and spectra of fractal strings, Springer Monographs in Mathematics, New York, NY: Springer-Verlag, ISBN 0-387-33285-5, Zbl 1119.28005
• Fursaev, Dmitri; Vassilevich, Dmitri (2011), Operators, Geometry and Quanta: Methods of Spectral Geometry in Quantum Field Theory, Theoretical and Mathematical Physics, Springer-Verlag, p. 98, ISBN 978-94-007-0204-2
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Riemann–Hurwitz formula
In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other. It therefore connects ramification with algebraic topology, in this case. It is a prototype result for many others, and is often applied in the theory of Riemann surfaces (which is its origin) and algebraic curves.
Statement
For a compact, connected, orientable surface $S$, the Euler characteristic $\chi (S)$ is
$\chi (S)=2-2g$,
where g is the genus (the number of handles), since the Betti numbers are $1,2g,1,0,0,\dots $. In the case of an (unramified) covering map of surfaces
$\pi \colon S'\to S$
that is surjective and of degree $N$, we have the formula
$\chi (S')=N\cdot \chi (S).$
That is because each simplex of $S$ should be covered by exactly $N$ in $S'$, at least if we use a fine enough triangulation of $S$, as we are entitled to do since the Euler characteristic is a topological invariant. What the Riemann–Hurwitz formula does is to add in a correction to allow for ramification (sheets coming together).
Now assume that $S$ and $S'$ are Riemann surfaces, and that the map $\pi $ is complex analytic. The map $\pi $ is said to be ramified at a point P in S′ if there exist analytic coordinates near P and π(P) such that π takes the form π(z) = zn, and n > 1. An equivalent way of thinking about this is that there exists a small neighborhood U of P such that π(P) has exactly one preimage in U, but the image of any other point in U has exactly n preimages in U. The number n is called the ramification index at P and also denoted by eP. In calculating the Euler characteristic of S′ we notice the loss of eP − 1 copies of P above π(P) (that is, in the inverse image of π(P)). Now let us choose triangulations of S and S′ with vertices at the branch and ramification points, respectively, and use these to compute the Euler characteristics. Then S′ will have the same number of d-dimensional faces for d different from zero, but fewer than expected vertices. Therefore, we find a "corrected" formula
$\chi (S')=N\cdot \chi (S)-\sum _{P\in S'}(e_{P}-1)$
or as it is also commonly written, using that $\chi (X)=2-2g(X)$ and multiplying through by -1:
$2g(S')-2=N\cdot (2g(S)-2)+\sum _{P\in S'}(e_{P}-1)$
(all but finitely many P have eP = 1, so this is quite safe). This formula is known as the Riemann–Hurwitz formula and also as Hurwitz's theorem.
Another useful form of the formula is:
$\chi (S')-r=N\cdot (\chi (S)-b)$
where r is the number points in S' at which the cover has nontrivial ramification (ramification points) and b is the number of points in S that are images of such points (branch points). Indeed, to obtain this formula, remove disjoint disc neighborhoods of the branch points from S and disjoint disc neighborhoods of the ramification points in S' so that the restriction of $\pi $ is a covering. Then apply the general degree formula to the restriction, use the fact that the Euler characteristic of the disc equals 1, and use the additivity of the Euler characteristic under connected sums.
Examples
The Weierstrass $\wp $-function, considered as a meromorphic function with values in the Riemann sphere, yields a map from an elliptic curve (genus 1) to the projective line (genus 0). It is a double cover (N = 2), with ramification at four points only, at which e = 2. The Riemann–Hurwitz formula then reads
$0=2\cdot 2-4\cdot (2-1)$
with the summation taken over four ramification points.
The formula may also be used to calculate the genus of hyperelliptic curves.
As another example, the Riemann sphere maps to itself by the function zn, which has ramification index n at 0, for any integer n > 1. There can only be other ramification at the point at infinity. In order to balance the equation
$2=n\cdot 2-(n-1)-(e_{\infty }-1)$
we must have ramification index n at infinity, also.
Consequences
Several results in algebraic topology and complex analysis follow.
Firstly, there are no ramified covering maps from a curve of lower genus to a curve of higher genus – and thus, since non-constant meromorphic maps of curves are ramified covering spaces, there are no non-constant meromorphic maps from a curve of lower genus to a curve of higher genus.
As another example, it shows immediately that a curve of genus 0 has no cover with N > 1 that is unramified everywhere: because that would give rise to an Euler characteristic > 2.
Generalizations
For a correspondence of curves, there is a more general formula, Zeuthen's theorem, which gives the ramification correction to the first approximation that the Euler characteristics are in the inverse ratio to the degrees of the correspondence.
An orbifold covering of degree N between orbifold surfaces S' and S is a branched covering, so the Riemann–Hurwitz formula implies the usual formula for coverings
$\chi (S')=N\cdot \chi (S)\,$
denoting with $\chi \,$ the orbifold Euler characteristic.
References
• Hartshorne, Robin (1977), Algebraic Geometry, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157, OCLC 13348052, section IV.2.
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
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Zeuthen–Segre invariant
In algebraic geometry, the Zeuthen–Segre invariant I is an invariant of a projective surface found in a complex projective space which was introduced by Zeuthen (1871) and rediscovered by Corrado Segre (1896).
The invariant I is defined to be d – 4g – b if the surface has a pencil of curves, non-singular of genus g except for d curves with 1 ordinary node, and with b base points where the curves are non-singular and transverse.
Alexander (1914) showed that the Zeuthen–Segre invariant I is χ–4, where χ is the topological Euler–Poincaré characteristic introduced by Poincaré (1895), which is equal to the Chern number c2 of the surface.
References
• Alexander, J. W. (1914), "Sur les cycles des surfaces algébriques et sur une définition topologique de l'invariant de Zeuthen-Segre", Atti della Accademia Nazionale dei Lincei. Rend. V (2), 23: 55–62
• Baker, Henry Frederick (1933), Principles of geometry. Volume 6. Introduction to the theory of algebraic surfaces and higher loci., Cambridge Library Collection, Cambridge University Press, ISBN 978-1-108-01782-4, MR 2850141 Reprinted 2010
• Fulton, William (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323
• Poincaré, Henri (1895), "Analysis Situs", Journal de l'École Polytechnique, 1: 1–123
• Segre, C. (1896), "Intorno ad un carattere delle superficie e delle varietà superiori algebriche.", Atti della Accademia delle Scienze di Torino (in Italian), 31: 485–501
• Zeuthen, H. G. (1871), "Études géométriques de quelques-unes des propriétés de deux surfaces dont les points se correspondent un-à-un", Mathematische Annalen, Springer Berlin / Heidelberg, 4: 21–49, doi:10.1007/BF01443296, ISSN 0025-5831, S2CID 121840169
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Sign extension
Sign extension (abbreviated as sext) is the operation, in computer arithmetic, of increasing the number of bits of a binary number while preserving the number's sign (positive/negative) and value. This is done by appending digits to the most significant side of the number, following a procedure dependent on the particular signed number representation used.
For example, if six bits are used to represent the number "00 1010" (decimal positive 10) and the sign extend operation increases the word length to 16 bits, then the new representation is simply "0000 0000 0000 1010". Thus, both the value and the fact that the value was positive are maintained.
If ten bits are used to represent the value "11 1111 0001" (decimal negative 15) using two's complement, and this is sign extended to 16 bits, the new representation is "1111 1111 1111 0001". Thus, by padding the left side with ones, the negative sign and the value of the original number are maintained.
In the Intel x86 instruction set, for example, there are two ways of doing sign extension:
• using the instructions cbw, cwd, cwde, and cdq: convert byte to word, word to doubleword, word to extended doubleword, and doubleword to quadword, respectively (in the x86 context a byte has 8 bits, a word 16 bits, a doubleword and extended doubleword 32 bits, and a quadword 64 bits);
• using one of the sign extended moves, accomplished by the movsx ("move with sign extension") family of instructions.
Zero extension
A similar concept is zero extension (abbreviated as zext). In a move or convert operation, zero extension refers to setting the high bits of the destination to zero, rather than setting them to a copy of the most significant bit of the source. If the source of the operation is an unsigned number, then zero extension is usually the correct way to move it to a larger field while preserving its numeric value, while sign extension is correct for signed numbers.
In the x86 and x64 instruction sets, the movzx instruction ("move with zero extension") performs this function. For example, movzx ebx, al copies a byte from the al register to the low-order byte of ebx and then fills the remaining bytes of ebx with zeroes.
On x64, most instructions that write to the entirety of lower 32 bits of any of the general-purpose registers will zero the upper half of the destination register. For example, the instruction mov eax, 1234 will clear the upper 32 bits of the rax[lower-alpha 1] register.
See also
• Arithmetic shift and logical shift
References
• Mano, Morris M.; Kime, Charles R. (2004). Logic and Computer Design Fundamentals (3rd ed.), pp 453. Pearson Prentice Hall. ISBN 0-13-140539-X.
Notes
1. RAX - 64 bit accumulator
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Yitang Zhang
Yitang Zhang (Chinese: 张益唐; born February 5, 1955)[3] is a Chinese-American mathematician primarily working on number theory and a professor of mathematics at the University of California, Santa Barbara since 2015.[4]
Yitang Zhang
Zhang in 2014
Born (1955-02-05) February 5, 1955
Shanghai, China
CitizenshipUnited States
Alma materPeking University (BS, MA)
Purdue University (PhD)
Known forEstablishing the existence of an infinitely repeatable prime 2-tuple[1]
AwardsOstrowski Prize (2013)
Cole Prize (2014)
Rolf Schock Prize (2014)
MacArthur Fellowship (2014)
Scientific career
FieldsNumber theory
InstitutionsUniversity of New Hampshire
University of California, Santa Barbara
ThesisThe Jacobian conjecture and the degree of field extension (1992)
Doctoral advisorTzuong-Tsieng Moh (莫宗堅)[2]
Previously working at the University of New Hampshire as a lecturer, Zhang submitted a paper to the Annals of Mathematics in 2013 which established the first finite bound on the least gap between consecutive primes that is attained infinitely often. This work led to a 2013 Ostrowski Prize, a 2014 Cole Prize, a 2014 Rolf Schock Prize, and a 2014 MacArthur Fellowship. Zhang became a professor of mathematics at the University of California, Santa Barbara in fall 2015.[5][6][7][8]
Early life and education
Zhang was born in Shanghai, China, with his ancestral home in Pinghu, Zhejiang. He lived in Shanghai with his grandmother until he went to Peking University. At around the age of nine, he found a proof of the Pythagorean theorem.[9] He first learned about Fermat's Last Theorem and the Goldbach conjecture when he was 10.[9] During the Cultural Revolution, he and his mother were sent to the countryside to work in the fields. He worked as a laborer for 10 years and was unable to attend high school.[9] After the Cultural Revolution ended, Zhang entered Peking University in 1978 as an undergraduate student and received a bachelor of science in mathematics in 1982. He became a graduate student of Professor Pan Chengbiao, a number theorist at Peking University, and obtained a master of science in mathematics in 1984.[10]
After receiving his master's degree in mathematics, with recommendations from Professor Ding Shisun, the President of Peking University, and Professor Deng Donggao, Chair of the university's Math Department,[11] Zhang was granted a full scholarship at Purdue University. Zhang arrived at Purdue in January 1985, studied there for six and a half years, and obtained his PhD in mathematics in December 1991.
Career
Zhang's PhD work was on the Jacobian conjecture. After graduation, Zhang had trouble finding an academic position. In a 2013 interview with Nautilus magazine, Zhang said he did not get a job after graduation. "During that period it was difficult to find a job in academics. That was a job market problem. Also, my advisor [Tzuong-Tsieng Moh] did not write me letters of recommendation."[12] Zhang made this claim again in George Csicsery's documentary film "Counting from Infinity: Yitang Zhang and the Twin Prime Conjecture"[13] while discussing his difficulties at Purdue and in the years that followed.[9] Moh claimed that Zhang never came back to him requesting recommendation letters.[11] In a detailed profile published in The New Yorker magazine in February 2015, Alec Wilkinson wrote Zhang "parted unhappily" with Moh, and that Zhang "left Purdue without Moh's support, and, having published no papers, was unable to find an academic job".[7] In 2018, responding to reports of his treatment of Zhang, Moh posted an update on his website. Moh wrote that Zhang "failed miserably" in proving the Jacobian conjecture, "never published any paper on algebraic geometry" after leaving Purdue, and "wasted seven years of his own life and my time".[14]
After some years, Zhang managed to find a position as a lecturer at the University of New Hampshire, where he was hired by Kenneth Appel in 1999. Prior to getting back to academia, he worked for several years as an accountant and a delivery worker for a New York City restaurant. He also worked in a motel in Kentucky and in a Subway sandwich shop.[1] A profile published in the Quanta Magazine reports that Zhang used to live in his car during the initial job-hunting days.[9] He served as lecturer at UNH from 1999[15] until around January 2014, when UNH appointed him to a full professorship as a result of his breakthrough on prime numbers.[16] Zhang stayed for a semester at The Institute For Advanced Study in Princeton, NJ, in 2014, and he joined the University of California, Santa Barbara in fall 2015.[17]
Research
On April 17, 2013, Zhang announced a proof that there are infinitely many pairs of prime numbers that differ by less than 70 million. This result implies the existence of an infinitely repeatable prime 2-tuple,[1] thus establishing a theorem akin to the twin prime conjecture. Zhang's paper was accepted by Annals of Mathematics in early May 2013,[6] his first publication since his last paper in 2001.[18] The proof was refereed by leading experts in analytic number theory.[7] Zhang's result set off a flurry of activity in the field, such as the Polymath8 project.
If P(N) stands for the proposition that there is an infinitude of pairs of prime numbers (not necessarily consecutive primes) that differ by exactly N, then Zhang's result is equivalent to the statement that there exists at least one even integer k < 70,000,000 such that P(k) is true. The classical form of the twin prime conjecture is equivalent to P(2); and in fact it has been conjectured that P(k) holds for all even integers k.[19][20] While these stronger conjectures remain unproven, a result due to James Maynard in November 2013, employing a different technique, showed that P(k) holds for some k ≤ 600.[21] Subsequently, in April 2014, the Polymath project 8 lowered the bound to k ≤ 246.[22] With current methods k ≤ 6 is the best attainable, and in fact k ≤ 12 and k ≤ 6 follow using current methods if the Elliott–Halberstam conjecture and its generalization, respectively, hold.[7][22]
Honors and awards
Zhang was awarded the 2013 Morningside Special Achievement Award in Mathematics,[23] the 2013 Ostrowski Prize,[24] the 2014 Frank Nelson Cole Prize in Number Theory,[16][25] and the 2014 Rolf Schock Prize[26] in Mathematics.
He is a recipient of the 2014 MacArthur award,[27] and was elected as an Academia Sinica Fellow during the same year.[10] He was an invited speaker at the 2014 International Congress of Mathematicians.
Political views
In 1989 Zhang joined a group interested in Chinese democracy (中国民联). In a 2013 interview, he affirmed that his political views on the subject had not changed since.[7][28]
Publications
• Zhang, Yitang (2007). "On the Landau-Siegel Zeros Conjecture". arXiv:0705.4306 [math.NT].
• Zhang, Yitang (2014). "Bounded gaps between primes". Annals of Mathematics. 179 (3): 1121–1174. doi:10.4007/annals.2014.179.3.7.
• Zhang, Yitang (2022). "Discrete mean estimates and the Landau-Siegel zero". arXiv:2211.02515 [math.NT].
References
1. Klarreich, Erica (May 19, 2013). "Unheralded Mathematician Bridges the Prime Gap". Quanta Magazine. Retrieved May 19, 2013.
2. Yitang Zhang at the Mathematics Genealogy Project
3. Zhang, Yitang (1991). The Jacobian conjecture and the degree of field extension. Purdue University. pp. 1–24. Retrieved March 4, 2021.
4. "Yitang (Tom) Zhang | Department of Mathematics – UC Santa Barbara". math.ucsb.edu. Retrieved October 19, 2022.
5. Yitang Zhang, Mathematician, MacArthur Fellows Program, MacArthur Foundation, September 17, 2014
6. Zhang, Yitang (2014). "Bounded gaps between primes". Annals of Mathematics. 179 (3): 1121–1174. doi:10.4007/annals.2014.179.3.7. MR 3171761. Zbl 1290.11128. (subscription required)
7. Wilkinson, Alec. "The Pursuit of Beauty". The New Yorker. No. February 2, 2015.
8. "Yitang (Tom) Zhang | Department of Mathematics – UC Santa Barbara". math.ucsb.edu. Retrieved February 15, 2018.
9. Thomas Lin (April 2, 2015). "After Prime Proof, an Unlikely Star Rises". Quanta Magazine.
10. "Mathematics and Physical Sciences Yitang Zhang". sinica.edu.tw. 2014.
11. Moh, Tzuong-Tsieng. "Zhang, Yitang's life at Purdue (Jan. 1985-Dec, 1991)" (PDF). Retrieved May 24, 2013.
12. "The Twin Prime Hero". Nautilus.
13. Counting from Infinity: Yitang Zhang and the Twin Prime Conjecture on IMdB
14. "Bio" (PDF). math.purdue.edu. 2013. Retrieved August 9, 2021.
15. Macalaster, Gretyl (December 14, 2013). "Math world stunned by UNH lecturer's find". New Hampshire Union Leader.
16. "January 2014 AMS-MAA Prize booklet" (PDF). p. 7.
17. "Celebrity Mathematician Joins UCSB Faculty | The Daily Nexus". September 17, 2015.
18. Jordan Ellenberg (May 22, 2013). "The Beauty of Bounded Gaps". Slate. Retrieved January 23, 2017.
19. McKee, Maggie (May 14, 2013). "First proof that infinitely many prime numbers come in pairs". Nature. Retrieved May 21, 2013.
20. Chang, Kenneth (May 20, 2013). "Solving a Riddle of Primes". The New York Times. Retrieved May 21, 2013.
21. Klarreich, Erica (November 20, 2013). "Together and Alone, Closing the Prime Gap". Retrieved November 20, 2013.
22. "Bounded gaps between primes". Polymath.
23. "ICCM 2013: Morningside Awards".
24. "The 2013 Ostrowski Prize".
25. "Yitang Zhang Receives 2014 AMS Cole Prize in Number Theory".
26. "The 2014 Rolf Schock Prize".
27. Lee, Felicia R. (September 17, 2014). "MacArthur Awards Go to 21 Diverse Fellows". The New York Times.
28. "张益唐问答录" (in Chinese). July 1, 2013. Retrieved June 30, 2015.
External links
• Alec Wilkinson, The Pursuit of Beauty, Yitang Zhang solves a pure-math mystery, The New Yorker, Profiles, February 2, 2015, issue
• Discover Magazine article by Steve Nadis, "Prime Solver"
• Gaps between Primes – Numberphile – University of Nottingham video (shorter version)
• Gaps between Primes (extra footage) – Numberphile (longer version)
Rolf Schock Prize laureates
Logic and philosophy
• Willard Van Orman Quine (1993)
• Michael Dummett (1995)
• Dana Scott (1997)
• John Rawls (1999)
• Saul Kripke (2001)
• Solomon Feferman (2003)
• Jaakko Hintikka (2005)
• Thomas Nagel (2008)
• Hilary Putnam (2011)
• Derek Parfit (2014)
• Ruth Millikan (2017)
• Saharon Shelah (2018)
• Dag Prawitz / Per Martin-Löf (2020)
• David Kaplan (2022)
Mathematics
• Elias M. Stein (1993)
• Andrew Wiles (1995)
• Mikio Sato (1997)
• Yuri I. Manin (1999)
• Elliott H. Lieb (2001)
• Richard P. Stanley (2003)
• Luis Caffarelli (2005)
• Endre Szemerédi (2008)
• Michael Aschbacher (2011)
• Yitang Zhang (2014)
• Richard Schoen (2017)
• Ronald Coifman (2018)
• Nikolai G. Makarov (2020)
• Jonathan Pila (2022)
Visual arts
• Rafael Moneo (1993)
• Claes Oldenburg (1995)
• Torsten Andersson (1997)
• Herzog & de Meuron (1999)
• Giuseppe Penone (2001)
• Susan Rothenberg (2003)
• SANAA / Kazuyo Sejima + Ryue Nishizawa (2005)
• Mona Hatoum (2008)
• Marlene Dumas (2011)
• Anne Lacaton / Jean-Philippe Vassal (2014)
• Doris Salcedo (2017)
• Andrea Branzi (2018)
• Francis Alÿs (2020)
• Rem Koolhaas (2022)
Musical arts
• Ingvar Lidholm (1993)
• György Ligeti (1995)
• Jorma Panula (1997)
• Kronos Quartet (1999)
• Kaija Saariaho (2001)
• Anne Sofie von Otter (2003)
• Mauricio Kagel (2005)
• Gidon Kremer (2008)
• Andrew Manze (2011)
• Herbert Blomstedt (2014)
• Wayne Shorter (2017)
• Barbara Hannigan (2018)
• György Kurtág (2020)
• Víkingur Ólafsson (2022)
Authority control
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Zhang Luping
Zhang Luping (simplified Chinese: 张鹭平; traditional Chinese: 張鹭平; 1945-1998) was a Chinese martial artist and mathematician born in Jiaxing, Zhejiang province. He was best known in China for his exceptional skill at tai chi's push hands, and for an incident in his hometown in which he accidentally broke a weightlifting champion’s forearm during an arm wrestling match. He was a student of Cai Hong Xiang (蔡鸿祥),[1]: 2–7 Wang Zi-Ping (王子平),[1]: 6 and Fu Zhong Wen (傅鍾文) .[1] He was also a descendant of Zhang Jun (traditional Chinese: 張浚; simplified Chinese: 张浚; pinyin: Zhāng Jùn, 1097–1164) and of Zhang Jiugao (simplified Chinese: 張九皋), who was the brother of Zhang Jiuling (simplified Chinese: 张九龄). He was noted for his deep knowledge of the five styles of tai chi, his superb application of the principles, and his highly developed internal power. In an age when many great martial arts teachers remained reluctant to share their highest insights and techniques, Zhang championed in his teaching an attitude of openness and a strong desire to ensure the continuation of Chinese martial traditions.
Zhang Luping
Master Zhang Lu Ping demonstrating tai chi at the Tai Chi Farm (Zhang San Feng Festival)
Martial arts career
Zhang started learning Shaolin Kung Fu when he was 13 years old from Shaolin and Jin Woo grandmaster Fang Nan Tang (方南堂). Zhang was captain of the Wushu team [2] at East Normal University in Shanghai, where he studied under three-time Chinese Wushu national champion (1953-1960) and five-time Chinese national martial arts competition gold medalist Cai Hong Xiang. Because of his excellent performance and dedication to learning the arts, Cai Hong Xiang arranged for Zhang to study under the famous grandmaster Wang Zi-Ping. Zhang also had the privilege at that time to study alongside The Magic Fist Dragon, Cai Long Yun.
Eventually, Zhang developed an interest in tai chi. He learned Chen-style taijiquan from many Chen lineage holders, including Dong Xiang Gen (董祥根)[1]: 3 and Du Wen Cai (都文才).[1]: 2 whom was the last student of Chen Zhao Kui(陈照奎). His form was also corrected by grandmaster Gu Liu Xin (顾留馨) (a former student of Chen Fake, Sun Lutang, and Chen Weiming, who corrected his style with a level of detail that would set him apart from other Chen style practitioners). Zhang also studied Wu style tai chi with master Sun Ren Zhi (孙润志)[1]: 2 and Xin Yi, another internal style similar to tai chi, with the well-known Shang Hai-based master, "Little Tiger" Zhang Hai Sheng (小老虎章海深),[1]: 5 who was highly respected for his skill in combat.
Scholarship and emigration to the United States
Zhang earned his master's degree in mathematics at East Normal University in Shang Hai under Prof. Cheng Chang Ping (陈昌平教授), who had done extensive mathematics work with Prof. Wolf Von Wahl at the University of Bayreuth in West Germany. Zhang also studied under professor Wang Guang Yin (王光寅) at the math research department of Chinese Academy of Sciences in Bei Jing. In 1983, Zhang published an article in The Mathematical Journal (数学学报) entitled “Hλ solutions of the 1st class of Fuchs type equations with operator coefficients” (一类具算子系数的Fuchs型方程的Hλ解) with his colleague Wang Ju Yan (王继延).
Zhang Luping was known in the academic community for his ground-breaking work in differential equations. He came to the United States in 1985 for a master's degree in mathematics at Carnegie-Mellon University in Pittsburgh, PA, following which he completed a doctorate degree and a post doctorate degree at the University of Massachusetts, Amherst under Prof. M. S. Berger. In November of 1994, he co-authored with professor Berger a paper published by International Publications for the PanAmerican Mathematical Journal entitled “A New Method for Large Quasiperiodic Nonlinear Oscillations with Fixed Frequencies for the Non-dissipative Second Order Conservative System of the Second Type” about the communication of applied nonlinear analysis. After completing his post-doctorate work, he taught mathematics at the University of California, Irvine and the University of Massachusetts, Amherst.
Legacy
In 1975, Zhang became the Zhejiang Province chen style tai chi champion.[2] MA. In 1998, Zhang defeated several local martial artists in Pittsburgh, PA and was invited to teach seminars at the Zhang San Feng Festival at the Tai Chi Farm owned by late master Jou Tsung Hwa. He held seminars all across the U.S. and judged many U.S. competitions, including the Houston 1990 United States National Chinese Martial Arts Competition. He was a Special Master for Taste of China and many similar martial arts events. He was twice pictured on the cover of Tai Chi International Magazine, as well as Inside Kung Fu magazine and the Pa Kua Zhang Newsletter. The Australian magazine "Tai Chi Combat and Health" called Master Zhang the "Real Thing"[3] In Tai Chi for Dummies, author Therese Ikoian wrote, "The late tai chi Master Zhang Lu Ping knew the spiraling technique well. Manny Fuentes had the 'privilege of being thrown around by him'. No matter how well Manny thought that he'd prepared a pending movement, he said that he was moved as easily as if he were a leaf, not a 175-pound man! By the time Zhang manifested the spiraling force up into his arms and hands, it contained an irresistible momentum."[4]
Zhang died in 1998 in Amherst, Massachusetts. His son Huan Zhang, a practitioner of tai chi and a scholar like his father, wrote a biographical article for China’s Premier Tai Chi - The spirit of Kung Fu Magazine（太极武魂杂志)[5] in his father's memory.
References
• Huan Zhang, (June 2018), "In memory of my father, Zhang Lu Ping", Tai Chi - The spirit of Kung Fu Magazine, Beijing Physical Education Press , China
1. B. Jones, (April 1990). "A LOOK at T'ai Chi Teachers in China", T'ai Chi Magazine, The Leading International Magazine of Tai Chi Chuan
2. Teri Lynn Breier, (May 1989).P-34,"China's Lu Ping Zhang: Teaching While Learning", Inside Kung Fu Magazine
3. Leroy Clark, (June, 1991). P7-8, "Tai Chi Combat and Health", Taiji Publications, Murwillumbah, Australia
4. Therese Ikoian, Manny Fuentes, (2001), "Tai Chi for Dummies", Wiley Publishing Inc, ISBN 0-7645-5351-8 p 71.
5. William Phillips, (2019), "In the presence of Cheng Man-Ching", My life and lessons with the Master of Five Excellences, Floating World Press, ISBN 978-0648283126 p ii
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Sun-Yung Alice Chang
Sun-Yung Alice Chang (Chinese: 張聖容; pinyin: Zhāng Shèngróng, Hakka: Chông Sṳn-yùng, [t͡soŋ sɨn juŋ]; born 1948) is a Taiwanese American mathematician specializing in aspects of mathematical analysis ranging from harmonic analysis and partial differential equations to differential geometry. She is the Eugene Higgins Professor of Mathematics at Princeton University.[1]
Sun-Yung Alice Chang
Sun-Yung Alice Chang, 2007
Born1948
Xian, China
NationalityAmerican
Other namesAlice Chang
Alma materNational Taiwan University
University of California, Berkeley
SpousePaul C. Yang
Scientific career
FieldsMathematics
InstitutionsUniversity of California, Los Angeles
Princeton University
Doctoral advisorDonald Sarason
Life
Chang was born in Xian, China in 1948 and grew up in Taiwan. She received her Bachelor of Science degree in 1970 from National Taiwan University, and her doctorate in 1974 from the University of California, Berkeley.[2] At Berkeley, Chang wrote her thesis on the study of bounded analytic functions. Chang became a full professor at UCLA in 1980 before moving to Princeton in 1998.[3]
Career and research
Chang's research interests include the study of geometric types of nonlinear partial differential equations and problems in isospectral geometry. Working with her husband Paul Yang and others, she produced contributions to differential equations in relation to geometry and topology.[3]
She teaches at Princeton University as of 1998. Before that, she held visiting positions at University of California-Berkeley; Institute for Advanced Study, Princeton, N.J.; and Swiss Federal Institute of Technology, Zurich, Switzerland.[3] She served at Swiss Federal Institute of Technology as a visiting professor in 2015.[4]
In 2004,[5] she was interviewed by Yu Kiang Leong for Creative Minds, Charmed Lives: Interviews at Institute for Mathematical Sciences, National University of Singapore, and she declared:
«In the mathematical community, we should leave room for people who want to do work in their own way. Mathematical research is not just a scientific approach; the nature of mathematics is sometimes close to that of art. Some people want individual character and an individual way of working things out. They should be appreciated too. There should be room for single research and collaborative research».[6]
Chang's life was profiled in the 2017 documentary film Girls who fell in love with Math.[7]
Service and honors
• Sloan Foundation Research Fellowship, 1979–1981[8]
• Invited speaker at the International Congress of Mathematicians in Berkeley, 1986 [8]
• Vice president of the American Mathematical Society, 1989-1991[8]
• Ruth Lyttle Satter Prize in Mathematics of the American Mathematical Society, 1995 [8]
• Guggenheim Fellowship, 1998 [9]
• Plenary Speaker at the International Congress of Mathematicians in Beijing, 2002[10]
• Member, American Academy of Arts and Sciences, 2008 [11]
• Honorary Degree, UPMC, 2013 [12]
• Fellow, National Academy of Sciences, 2009 [13]
• Fellow, Academia Sinica, 2012[14]
• Fellow, American Mathematical Society, 2015[15]
• Fellow, Association for Women in Mathematics, 2019[16]
• MSRI Simons Professor, 2015-2016[17]
Publications
• Chang, Sun-Yung A.; Yang, Paul C. Conformal deformation of metrics on $S^{2}$. J. Differential Geom. 27 (1988), no. 2, 259–296.
• Chang, Sun-Yung Alice; Yang, Paul C. Prescribing Gaussian curvature on $S^{2}$. Acta Math. 159 (1987), no. 3–4, 215–259.
• Chang, Sun-Yung A.; Yang, Paul C. Extremal metrics of zeta function determinants on 4-manifolds. Ann. of Math. (2) 142 (1995), no. 1, 171–212.
• Chang, Sun-Yung A.; Gursky, Matthew J.; Yang, Paul C. The scalar curvature equation on 2- and 3-spheres. Calc. Var. Partial Differential Equations 1 (1993), no. 2, 205–229.
• Chang, Sun-Yung A.; Gursky, Matthew J.; Yang, Paul C. An equation of Monge-Ampère type in conformal geometry, and four-manifolds of positive Ricci curvature. Ann. of Math. (2) 155 (2002), no. 3, 709–787.
• Chang, S.-Y. A.; Wilson, J. M.; Wolff, T. H. Some weighted norm inequalities concerning the Schrödinger operators. Comment. Math. Helv. 60 (1985), no. 2, 217–246.
• Carleson, Lennart; Chang, Sun-Yung A. On the existence of an extremal function for an inequality of J. Moser. Bull. Sci. Math. (2) 110 (1986), no. 2, 113–127.
• Chang, Sun-Yung A.; Fefferman, Robert Some recent developments in Fourier analysis and $H^{p}$-theory on product domains. Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 1, 1–43.
• Chang, Sun-Yung A.; Fefferman, Robert A continuous version of duality of $H^{1}$ with BMO on the bidisc. Ann. of Math. (2) 112 (1980), no. 1, 179–201.
References
• O'Connor, John J.; Robertson, Edmund F., "Sun-Yung Alice Chang", MacTutor History of Mathematics Archive, University of St Andrews
1. "Sun-Yung Alice Chang". web.math.princeton.edu. Retrieved 2020-11-12.
2. Gursky, Matthew J.; Wang, Yi (March 2020). "Sun-Yung Alice Chang and Geometric Analysis" (PDF). Notices of the American Mathematical Society. 67 (3): 318. doi:10.1090/noti2037.
3. "Sun-Yung Alice Chang". Faculty Profiles. Princeton University. Retrieved 23 February 2014.
4. "List of guests". Swiss Federal Institute of Technology. Retrieved 23 February 2014.
5. Leong, Y K (Summer 2012). "An Interview with Sun-Yung Alice Chang" (PDF). Asia Pacific Mathematics Newsletter. 2: 25–29.
6. Leong, Yu Kiang (2010). Creative Minds, Charmed Lives: Interviews at Institute for Mathematical Sciences, National University of Singapore. World Scientific. ISBN 9789814317580. Retrieved 2017-09-10.
7. "Girls who fell in love with Math". Taiwan Film Institute. 31 August 2017. Retrieved 2018-02-04.
8. Oakes, Elizabeth H. (2002). International encyclopedia of women scientists. New York, NY: Facts on File. p. 58. ISBN 0816043817.
9. "Sun-Yung Alice Chang". Guggenheim Foundation. Archived from the original on 27 February 2014. Retrieved 23 February 2014.
10. "Plenary Speakers". International Congress of Mathematics. Archived from the original on 23 February 2014. Retrieved 23 February 2014.
11. "Members of the American Academy of Arts & Sciences: 1780-2013" (PDF). American Academy of Arts and Sciences. Retrieved 23 February 2014.
12. "Professor Alice Chang awarded Doctor Honoris Causa of Pierre and Marie Curie University". Princeton University. Retrieved 26 January 2019.
13. "Sun-Yung Alice Chang". National Academy of Sciences. Retrieved 23 February 2014.
14. 2012 Academicians Announced
15. 2016 Class of the Fellows of the AMS, American Mathematical Society, retrieved 2015-11-16.
16. 2019 Class of AWM Fellows, Association for Women in Mathematics, retrieved 2019-01-19
17. MSRI. "Mathematical Sciences Research Institute". www.msri.org. Retrieved 2021-06-07.
External links
• Leong, Y K (July 2012). "An Interview with Sun-Yung Alice Chang" (PDF). Asia Pacific Mathematics Newsletter. pp. 25–29.
• AWM Fellows List 2019
• Gursky, Matthew J.; Wang, Yi (March 2020). "Sun-Yung Alice Chang and Geometric Analysis" (PDF). Notices of the American Mathematical Society. 67 (3): 318–326. doi:10.1090/noti2037.
Ruth Lyttle Satter Prize in Mathematics recipients
• 1991 Dusa McDuff
• 1993 Lai-Sang Young
• 1995 Sun-Yung Alice Chang
• 1997 Ingrid Daubechies
• 1999 Bernadette Perrin-Riou
• 2001 Karen E. Smith & Sijue Wu
• 2003 Abigail Thompson
• 2005 Svetlana Jitomirskaya
• 2007 Claire Voisin
• 2009 Laure Saint-Raymond
• 2011 Amie Wilkinson
• 2013 Maryam Mirzakhani
• 2015 Hee Oh
• 2017 Laura DeMarco
• 2019 Maryna Viazovska
• 2021 Kaisa Matomäki
• 2023 Panagiota Daskalopoulos & Nataša Šešum
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Shou-Wu Zhang
Shou-Wu Zhang (Chinese: 张寿武; pinyin: Zhāng Shòuwǔ; born October 9, 1962) is a Chinese-American mathematician known for his work in number theory and arithmetic geometry. He is currently a Professor of Mathematics at Princeton University.
Shou-Wu Zhang
Shou-Wu Zhang in 2014
Born (1962-10-09) October 9, 1962
Hexian, Anhui, China
NationalityAmerican
Alma mater
• Columbia University
• Chinese Academy of Sciences
• Sun Yat-sen University
Known for
• Arakelov theory
• Arithmetic dynamics
• Bogomolov conjecture
• Gross–Zagier theorem
Awards
List of Awards
• Sloan Fellowship (1997)
• Morningside Medal (1998)
• Clay Prize Fellow (2003)
• Guggenheim Fellow (2009)
• American Academy of Arts and Sciences Fellow (2011)
• American Mathematical Society Fellow (2016)
Scientific career
FieldsMathematics
Institutions
• Princeton University
• Columbia University
• Institute for Advanced Study
ThesisPositive Line Bundles on Arithmetic Surfaces (1991)
Doctoral advisorLucien Szpiro
Other academic advisorsWang Yuan
Doctoral students
• Wei Zhang
• Xinyi Yuan
• Tian Ye
• Yifeng Liu
Other notable students
• Bhargav Bhatt
Influences
• Gerd Faltings
• Dorian M. Goldfeld
• Benedict Gross
Biography
Early life
Shou-Wu Zhang was born in Hexian, Ma'anshan, Anhui, China on October 9, 1962.[1][2][3] Zhang grew up in a poor farming household and could not attend school until eighth grade due to the Cultural Revolution.[1] He spent most of his childhood raising ducks in the countryside and self-studying mathematics textbooks that he acquired from sent-down youth in trades for frogs.[1][2] By the time he entered junior high school at the age of fourteen, he had self-learned calculus and had become interested in number theory after reading about Chen Jingrun's proof of Chen's theorem which made substantial progress on Goldbach's conjecture.[1][2][4]
Education
Zhang was admitted to the Sun Yat-sen University chemistry department in 1980 after scoring poorly on his mathematics entrance examinations, but he later transferred to the mathematics department after feigning color blindness and received his bachelor's degree in mathematics in 1983.[5][1][2][3][4][6] He then studied under analytic number theorist Wang Yuan at the Chinese Academy of Sciences where he received his master's degree in 1986.[1][4][3][6] In 1986, Zhang was brought to the United States to pursue his doctoral studies at Columbia University by Dorian M. Goldfeld.[1][2] He then studied under Goldfeld, Hervé Jacquet, Lucien Szpiro, and Gerd Faltings, and then completed his PhD at Columbia University under Szpiro in 1991.[7][1][2][4][3][6]
Career
Zhang was a member of the Institute for Advanced Study and an assistant professor at Princeton University from 1991 to 1996.[3][6] In 1996, Zhang moved back to Columbia University where he was a tenured professor until 2013.[1][5][3][6] He has been a professor at Princeton University since 2011[5][6] and is an Eugene Higgins Professor since 2021. [8]
Zhang is on the editorial boards of: Acta Mathematica Sinica, Algebra & Number Theory, Forum of Mathematics, Journal of Differential Geometry, National Science Review, Pure and Applied Mathematics Quarterly, Science in China, and Research in Number Theory.[5] He has previously served on the editorial boards of: Journal of Number Theory, Journal of the American Mathematical Society, Journal of Algebraic Geometry, and International Journal of Number Theory.[5]
Research
Zhang's doctoral thesis Positive line bundles on Arithmetic Surfaces (Zhang 1992) proved a Nakai–Moishezon type theorem in intersection theory using a result from differential geometry already proved in Tian Gang's doctoral thesis.[5] In a series of subsequent papers (Zhang 1993, 1995a, 1995b, Szpiro, Ullmo & Zhang 1997), he further developed his theory of 'positive line bundles' in Arakelov theory which culminated in a proof (with Emmanuel Ullmo) of the Bogomolov conjecture (Zhang 1998).[5]
In a series of works in the 2000s (Zhang 2001b, 2004, Yuan, Zhang & W. Zhang 2009), Zhang proved a generalization of the Gross–Zagier theorem from elliptic curves over rationals to modular abelian varieties of GL(2) type over totally real fields.[5] In particular, the latter result led him to a proof of the rank one Birch-Swinnerton-Dyer conjecture for modular abelian varieties of GL(2) type over totally real fields through his work relating the Néron–Tate height of Heegner points to special values of L-functions in (Zhang 1997, 2001a).[5][9] Eventually, Yuan, Zhang, and W. Zhang (2013) established a full generalization of the Gross–Zagier theorem to all Shimura curves.
In arithmetic dynamics, Zhang (1995a, 2006) posed conjectures on the Zariski density of non-fibered endomorphisms of quasi-projective varieties and Ghioca, Tucker, and Zhang (2011) proposed a dynamical analogue of the Manin–Mumford conjecture.[10][5]
In 2018, Yuan and Zhang (2018) proved the averaged Colmez conjecture which was shown to imply the André–Oort conjecture for Siegel modular varieties by Jacob Tsimerman.[11]
Awards
Zhang has received a Sloan Foundation Research Fellowship (1997) and a Morningside Gold Medal of Mathematics (1998). He is also a Clay Foundation Prize Fellow (2003), Guggenheim Foundation Fellow (2009), Fellow of the American Academy of Arts and Sciences (2011), and Fellow of the American Mathematical Society (2016).[12][13][5] He was also an invited speaker at the International Congress of Mathematicians in 1998.[14][5][6][15]
Selected publications
Arakelov theory
• Zhang, Shou-Wu (1993), "Admissible pairing on a curve", Inventiones Mathematicae, 112 (1): 421–432, Bibcode:1993InMat.112..171Z, doi:10.1007/BF01232429, S2CID 120229374.
• Zhang, Shou-Wu (1995a), "Small points and adelic metrics", Journal of Algebraic Geometry, 8 (1): 281–300.
• Zhang, Shou-Wu (1995b), "Positive line bundles on arithmetic varieties", Journal of the American Mathematical Society, 136 (3): 187–221, doi:10.1090/S0894-0347-1995-1254133-7.
• Zhang, Shou-Wu (1996), "Heights and reductions of semi-stable varieties", Compositio Mathematica, 104 (1): 77–105.
• Zhang, Shou-Wu (2010), "Gross–Schoen cycles and Dualising sheaves", Invent. Math., 179 (1): 1–73, Bibcode:2010InMat.179....1Z, doi:10.1007/s00222-009-0209-3, S2CID 5698835.
• Yuan, Xinyi; Zhang, Shou-Wu (2017), "The arithmetic Hodge index theorem for adelic line bundles", Math. Ann., 367 (3–4): 1123–1171, doi:10.1007/s00208-016-1414-1, S2CID 2813125.
Bogomolov Conjecture
• Szpiro, Lucien; Ullmo, Emmanuel; Zhang, Shou-Wu (1997), "Equirépartition des petits points", Inventiones Mathematicae, 127 (2): 337–347, Bibcode:1997InMat.127..337S, doi:10.1007/s002220050123, S2CID 119668209.
• Zhang, Shou-Wu (1998), "Equidistribution of small points on abelian varieties", Annals of Mathematics, 147 (1): 159–165, doi:10.2307/120986, JSTOR 120986.
Gross--Zagier formulae
• Zhang, Shou-Wu (1997), "Heights of Heegner cycles and derivatives of L-series", Inventiones Mathematicae, 130 (1): 99–152, Bibcode:1997InMat.130...99Z, doi:10.1007/s002220050179, S2CID 10537873.
• Zhang, Shou-Wu (2001), "Heights of Heegner points on Shimura curves", Annals of Mathematics, 153 (1): 27–147, arXiv:math/0101269, doi:10.2307/2661372, JSTOR 2661372, S2CID 119624920.
• Yuan, Xinyi; Zhang, Shou-Wu; Zhang, Wei (2009), "The Gross–Kohnen–Zagier Theorem over Totally Real Fields", Compositio Mathematica, 145 (5): 1147–1162, doi:10.1112/S0010437X08003734.
• Yuan, Xinyi; Zhang, Shou-Wu; Zhang, Wei (2013), The Gross–Zagier formula on Shimura curves, Annals of Mathematics Studies, vol. 184.
• Liu, Yifeng; Zhang, Shou-Wu; Zhang, Wei (2018a), "A p-adic Waldspurger formula", Duke Mathematical Journal, 167 (4): 743–833, arXiv:1511.08172, doi:10.1215/00127094-2017-0045, S2CID 4867572.
• Yuan, Xinyi; Zhang, Shou-Wu (2018b), "On the averaged Colmez conjecture", Annals of Mathematics, 187 (2): 553–638, arXiv:1507.06903, doi:10.4007/annals.2018.187.2.4, S2CID 118916754.
Arithmetic dynamics
• Zhang, Shou-Wu (2006), "Distributions in algebraic dynamics", in Yau, Shing-Tung (ed.), Essays in geometry in memory of S.S. Chern, Surveys in Differential Geometry, vol. 10, Somerville, MA: International Press, pp. 381–430, doi:10.4310/SDG.2005.v10.n1.a9, MR 2408228.
• Ghioca, Dragos; Tucker, Thomas J.; Zhang, Shou-Wu (2011), "Towards a dynamical Manin-Mumford conjecture", International Mathematics Research Notices, 22: 5109–5122, doi:10.1093/imrn/rnq283, MR 2854724.
References
1. "从放鸭娃到数学大师" [From ducklings to mathematics master] (in Chinese). Academy of Mathematics and Systems Science. 11 November 2011. Retrieved 5 May 2019.
2. "專訪張壽武：在數學殿堂里，依然懷抱小學四年級的夢想" [Interview with Zhang Shou-Wu: In the mathematics department, he still has his dream from fourth grade of elementary school] (in Chinese). Beijing Sina Net. 3 May 2019. Retrieved 5 May 2019.
3. "旅美青年数学家张寿武" [Zhang Shouwu, a young mathematician in the United States] (in Chinese). He County Government. 2 November 2017. Retrieved 5 May 2019.
4. "专访数学家张寿武：要让别人解中国人出的数学题" [Interview with mathematician Zhang Shouwu: Let others solve the math problems of Chinese people] (in Chinese). Sina Education. 4 May 2019. Retrieved 5 May 2019.
5. Leong, Y. K. (July–December 2018). "Shou-Wu Zhang: Number Theory and Arithmetic Algebraic Geometry" (PDF). Imprints. No. 32. The Institute for Mathematical Sciences, National University of Singapore. pp. 32–36. Retrieved 5 May 2019.
6. "专访数学家张寿武：数学苍穹闪烁中国新星" [Interview with mathematician Zhang Shouwu: A new Chinese star flashing in the mathematical sky] (in Chinese). Zhishi Fenzi. 4 December 2017. Retrieved 5 May 2019.
7. Shou-Wu Zhang at the Mathematics Genealogy Project
8. Office of Communications. "Faculty members named to endowed professorships". Princeton University. Retrieved May 26, 2021.
9. Zhang, Wei (2013). "The Birch–Swinnerton-Dyer conjecture and Heegner points: a survey". Current Developments in Mathematics. 2013: 169–203. doi:10.4310/CDM.2013.v2013.n1.a3..
10. Benedetto, Robert; Ingram, Patrick; Jones, Rafe; Manes, Michelle; Silverman, Joseph H.; Tucker, Thomas J. (2019). "Current trends and open problems in arithmetic dynamics". Bulletin of the American Mathematical Society. 56 (4): 611–685. arXiv:1806.04980. doi:10.1090/bull/1665. S2CID 53550119.
11. "February 2018". Notices of the American Mathematical Society. 65 (2): 191. 2018. ISSN 1088-9477.
12. 2016 Class of the Fellows of the AMS, American Mathematical Society, retrieved 2015-11-16
13. "Shou-Wu Zhang". John Simon Guggenheim Foundation. Retrieved 31 January 2019.
14. "ICM Plenary and Invited Speakers". Retrieved 31 January 2019.
15. Zhang, Shou-Wu (1998). "Small points and Arakelov theory". Doc. Math. (Bielefeld) Extra Vol. ICM Berlin, 1998, vol. II. pp. 217–225.
External links
• Princeton home page
• Shou-Wu Zhang at the Mathematics Genealogy Project
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Wei Zhang (mathematician)
Wei Zhang (Chinese: 张伟; born 1981) is a Chinese mathematician specializing in number theory. He is currently a Professor of Mathematics at the Massachusetts Institute of Technology.[1]
Wei Zhang
Zhang at the Mathematical Research Institute of Oberwolfach in 2017
Born1981 (age 41–42)
Alma materColumbia University
Peking University
Awards
• SASTRA Ramanujan Prize (2010)
• Sloan Research Fellowship (2013)
• Morningside Gold Medal (2016)
• New Horizons In Mathematics Prize (2018)
• Clay Research Award (2019)
• Fellow of the American Mathematical Society (2019)
Scientific career
FieldsMathematics
InstitutionsMassachusetts Institute of Technology
Columbia University
ThesisModularity of Generating Functions of Special Cycles on Shimura Varieties (2009)
Doctoral advisorShou-Wu Zhang
Education
Zhang grew up in Sichuan province in China and attended Chengdu No.7 High School. [2] He earned his B.S. in Mathematics from Peking University in 2004 and his Ph.D. from Columbia University in 2009 under the supervision of Shou-Wu Zhang.[3][4]
Career
Zhang was a postdoctoral researcher and Benjamin Peirce Fellow at Harvard University from 2009 to 2011. He was a member of the mathematics faculty at Columbia University from 2011 to 2017, initially as an assistant professor before becoming a full professor in 2015. He has been a full professor at the Massachusetts Institute of Technology since 2017.[4][5]
Work
His collaborations with Zhiwei Yun, Xinyi Yuan and Xinwen Zhu have received attention in publications such as Quanta Magazine and Business Insider.[6][7] In particular, his work with Zhiwei Yun on the Taylor expansion of L-functions is "already being hailed as one of the most exciting breakthroughs in an important area of number theory in the last 30 years."[6]
Zhang has also made substantial contributions to the global Gan–Gross–Prasad conjecture.
Awards
He was a recipient of the SASTRA Ramanujan Prize in 2010, for "far-reaching contributions by himself and in collaboration with others to a broad range of areas in mathematics, including number theory, automorphic forms, L-functions, trace formulas, representation theory, and algebraic geometry.”[8] In 2013, Zhang received a Sloan Research Fellowship; in 2016 Zhang was awarded the Morningside Gold Medal of Mathematics.[4][9] In December 2017 he was awarded 2018 New Horizons In Mathematics Prize together with Zhiwei Yun, Aaron Naber and Maryna Viazovska. In 2019 he received the Clay Research Award.[10]
He was included in the 2019 class of fellows of the American Mathematical Society "for contributions to number theory, algebraic geometry and geometric representation theory".[11] He was elected to the American Academy of Arts and Sciences in 2023.[12]
Publications (selected)
• "Automorphic period and the central value of Rankin-Selberg L-function", J. Amer. Math. Soc. 27 (2014), 541–612.
• "On arithmetic fundamental lemmas", Invent. Math., 188 (2012), No. 1, 197–252.
• "Fourier transform and the global Gan–Gross–Prasad conjecture for unitary groups", Annals of Mathematics 180 (2014), No. 3, 971–1049.
• "Selmer groups and the indivisibility of Heegner points", Cambridge Journal of Mathematics 2 (2014), no. 2, 191–253.
• (with Michael Rapoport, Ulrich Terstiege) "On the Arithmetic Fundamental Lemma in the minuscule case", Compositio Mathematica 149 (2013), no. 10, 1631–1666.
• (with Xinyi Yuan, Shou-Wu Zhang) "The Gross–Kohnen–Zagier theorem over totally real fields", Compositio Mathematica 145 (2009), no. 5, 1147–1162.
• (with Xinyi Yuan, Shou-Wu Zhang) "The Gross–Zagier formula on Shimura curves", Annals of Mathematics Studies vol. 184, Princeton University Press, 2012.
• (with Manjul Bhargava, Christopher Skinner) "A majority of elliptic curves over Q satisfy the Birch and Swinnerton-Dyer conjecture", preprint.
• (with Zhiwei Yun) "Shtukas and the Taylor expansion of L-functions", Annals of Mathematics 186 (2017), No. 3, 767–911.
• (with Xinyi Yuan, Shou-Wu Zhang) "Triple product L-series and Gross–Kudla–Schoen cycles", preprint.
• (with Yifeng Liu, Shou-Wu Zhang) "On p-adic Waldspurger formula", preprint.
References
1. "Wei Zhang | MIT Mathematics".
2. "成都七中2000届校友张伟获2019年克雷研究奖，首位华人数学家". 15 September 2019.
3. "Wei Zhang", Mathematics Genealogy Project.
4. "Curriculum Vitae" (PDF). Wei Zhang. Retrieved September 4, 2020.
5. "Wei Zhang". Massachusetts Institute of Technology Department of Mathematics. Retrieved September 4, 2020.
6. "Math Quartet Joins Forces on Unified Theory", Quanta Magazine. Retrieved on 4 December 2016.
7. "Math Quartet Joins Forces on Unified Theory", Business Insider. Retrieved on 4 December 2016.
8. Notices of the AMS, January 2011, American Mathematical Society.
9. "Wei Zhang awarded the 2016 ICCM Morningside Gold Medal", Columbia University. Published 18 August 2016.
10. Clay Research Award 2019
11. 2019 Class of the Fellows of the AMS, American Mathematical Society, retrieved 2018-11-07
12. New members, American Academy of Arts and Sciences, 2023, retrieved 2023-04-21
Recipients of SASTRA Ramanujan Prize
• Manjul Bhargava (2005)
• Kannan Soundararajan (2005)
• Terence Tao (2006)
• Ben Green (2007)
• Akshay Venkatesh (2008)
• Kathrin Bringmann (2009)
• Wei Zhang (2010)
• Roman Holowinsky (2011)
• Zhiwei Yun (2012)
• Peter Scholze (2013)
• James Maynard (2014)
• Jacob Tsimerman (2015)
• Kaisa Matomäki (2016)
• Maksym Radziwill (2016)
• Maryna Viazovska (2017)
• Yifeng Liu (2018)
• Jack Thorne (2018)
• Adam Harper (2019)
• Shai Evra (2020)
• Will Sawin (2021)
• Yunqing Tang (2022)
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Hongkai Zhao
Hongkai Zhao is a Chinese mathematician and Ruth F. DeVarney Distinguished Professor of Mathematics at Duke University. He was formerly the Chancellor's Professor in the Department of Mathematics at the University of California, Irvine. He is known for his work in scientific computing, imaging and numerical analysis, such as the fast sweeping method for Hamilton-Jacobi equation[1] and numerical methods for moving interface problems.[2]
Zhao had obtained his Bachelor of Science degree in the applied mathematics from the Peking University in 1990 and two years later got his Master's in the same field from the University of Southern California. From 1992 to 1996 he attended University of California, Los Angeles where he got his Ph.D. in mathematics. From 1996 to 1998 Zhao was a Gábor Szegő Assistant Professor at the Department of Mathematics of Stanford University and then got promoted to Research Associate which he kept till 1999. He has been at the University of California, Irvine since. At the same time he is also a member of the Institute for Mathematical Behavioral Sciences and the Department of Computer Science of UCI. From 2010 to 2013 and 2016 to 2019, Zhao was the chairman of the Department of Mathematics and since 2016 serves as Chancellor's Professor of mathematics.[3]
Hongkai Zhao received Alfred P. Sloan Fellowship in 2002 and the Feng Kang Prize[4] in Scientific Computing in 2007. He was elected as a Fellow of the Society for Industrial and Applied Mathematics, in the 2022 Class of SIAM Fellows, "for seminal contributions to scientific computation, numerical analysis, and applications in science and engineering".[5]
When it comes to free time he likes to watch and play sports games.[6]
References
1. Zhao, Hongkai (2005). "A fast sweeping method for Eikonal equations". Mathematics of Computation. 74 (250): 603–627. doi:10.1090/S0025-5718-04-01678-3. ISSN 0025-5718.
2. Zhao, Hong-Kai; Chan, T.; Merriman, B.; Osher, S. (1996-08-01). "A Variational Level Set Approach to Multiphase Motion". Journal of Computational Physics. 127 (1): 179–195. Bibcode:1996JCoPh.127..179Z. doi:10.1006/jcph.1996.0167. ISSN 0021-9991.
3. "Hongkai Zhao CV" (PDF). Retrieved 2019-01-29.
4. "Hongkai Zhao awarded the 2007 Feng Kang Prize". UCI. 2007-10-05. Retrieved 2019-01-29.
5. "SIAM Announces Class of 2022 Fellows". SIAM News. March 31, 2022. Retrieved 2022-03-31.
6. Jennifer Fitzenberger (2009-01-22). "Mathematics in the real world". UCI. Retrieved 2019-01-29.
External links
• Hongkai Zhao publications indexed by Google Scholar
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Zhao Youqin's π algorithm
Zhao Youqin's π algorithm was an algorithm devised by Yuan dynasty Chinese astronomer and mathematician Zhao Youqin (赵友钦, ? – 1330) to calculate the value of π in his book Ge Xiang Xin Shu (革象新书).
Algorithm
Zhao Youqin started with an inscribed square in a circle with radius r.[1]
If $\ell $ denotes the length of a side of the square, draw a perpendicular line d from the center of the circle to side l. Let e denotes r − d. Then from the diagram:
$d={\sqrt {r^{2}-\left({\frac {\ell }{2}}\right)^{2}}}$
$e=r-d=r-{\sqrt {r^{2}-\left({\frac {\ell }{2}}\right)^{2}}}.$
Extend the perpendicular line d to dissect the circle into an octagon; $\ell _{2}$ denotes the length of one side of octagon.
$\ell _{2}={\sqrt {\left({\frac {\ell }{2}}\right)^{2}+e^{2}}}$
$\ell _{2}={\frac {1}{2}}{\sqrt {\ell ^{2}+4\left(r-{\frac {1}{2}}{\sqrt {4r^{2}-\ell ^{2}}}\right)^{2}}}$
Let $l_{3}$ denotes the length of a side of hexadecagon
$\ell _{3}={\frac {1}{2}}{\sqrt {\ell _{2}^{2}+4\left(r-{\frac {1}{2}}{\sqrt {4r^{2}-\ell _{2}^{2}}}\right)^{2}}}$
similarly
$\ell _{n+1}={\frac {1}{2}}{\sqrt {\ell _{n}^{2}+4\left(r-{\frac {1}{2}}{\sqrt {4r^{2}-\ell _{n}^{2}}}\right)^{2}}}$
Proceeding in this way, he at last calculated the side of a 16384-gon, multiplying it by 16384 to obtain 3141.592 for a circle with diameter = 1000 units, or
$\pi =3.141592.\,$
He multiplied this number by 113 and obtained 355. From this he deduced that of the traditional values of π, that is 3, 3.14, 22/7 and 355/113, the last is the most exact.[2]
See also
• Liu Hui's π algorithm
References
1. Yoshio Mikami, Development of Mathematics in China and Japan, Chapter 20, The Studies about the Value of π etc., pp 135–138
2. Yoshio Mikami, p136
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Zhen Luan
Zhen Luan (甄鸾) (535 – 566) was a Chinese mathematician, astronomer and daoist who was active during the Northern Zhou (557-581) of the Southern and Northern Dynasties period.
Born in the Wuji County of the present day Hubei Province, he is primarily known for the comments on the ancient mathematical treatises. Proceeding from them, he paid special attention to the "Nine Palaces" calculation technique; his description of the Luo shu represents an early example of textual comment on this scheme.
Zhen Luan developed the Tianhe calendar which was implemented in 566 and was current for the next 18 years.
Zhen trained in a Daoist congregation,[1] but converted to Buddhism out of disgust with Daoist sexual practices.[2][3] He wrote the anti-Daoist text Xiaodao Lun in 570 for Emperor Wu of Northern Zhou.[4][5]
His solid scholarship was commended by Yan Yuan (1635 - 1704).
Literature
• Schuyler Cammann, "The Magic Square of Three in Old Chinese Philosophy and Religion" in History of Religions, Vol. 1, No. 1 (Summer, 1961), pp. 37–80.
References
1. Wile, Douglas (1992). "Sexual Practices and Taoism". Art of the Bedchamber: The Chinese Sexual Yoga Classics Including Women's Solo Meditation Texts. SUNY Press. pp. 25–26.
2. Gulik, Robert H Van (1974). Sexual Life in Ancient China. Brill Archive. p. 89.
3. Farzeen Baldrian-Hussein, Farzeen (October 1996). "Laughing at the Tao: Debates among Buddhists and Taoists in Medieval China by Livia Kohn (review)". Asian Folklore Studies. 55 (2): 361–363. doi:10.2307/1178836. JSTOR 1178836.
4. Bokenkamp, Stephen R (1990). "Stages of Transcendence". In Buswell, Robert E (ed.). Chinese Buddhist Apocrypha. University of Hawaii Press.
5. Komjathy, Louis (2012). "The Daoist Tradition in China". In Nadeau, Randall L (ed.). The Wiley-Blackwell Companion to Chinese Religions. John Wiley & Sons. pp. 179–180.
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Zhi-Ming Ma
Zhi-Ming Ma. (Chinese: 馬志明; pinyin: Zhi-Ming Ma) is a Chinese mathematics professor of Chinese Academy of Sciences.[1] Ma is a former Vice Chairman of the Executive Committee for International Mathematical Union.[2], a two times president of Chinese Mathematical Society, an elected member of World Academy of Sciences and the Chairman of Graduate Degree Committee of Academy of Math and Systems Science, Chinese Academy of Sciences.[3][1]
Zhi-Ming Ma
馬志明
Born1948
Chengdu, Sichuan
Citizenship China
Alma materChongqing Normal University,
China Science and Technology University
Beijing Municipality,
Chinese Academy of Sciences
Known forHe discovered a new framework of quasi-regular Dirichlet forms which correspond to right processes in one-to-one manner.
Scientific career
FieldsMathematics
InstitutionsChinese Academy of Sciences
International Mathematical Union]]
Biography
Ma was born in January, 1948 at Chengdu, Sichuan while Jiaocheng County in Shanxi Province was his native origin. He obtained his first degree in Mathematics from Chongqing Normal University in 1978.[1] In 1981, he received his Master's degree from China Science and Technology University, Graduate School of Science Beijing Municipality. In 1984 he received his doctorate degree in Applied Mathematics from Chinese Academy of Sciences.[4]
Contributions
Ma's contribution in the theory of Dirichlet forms and Markov processes brought an end to a twenty years puzzle in the field. Ma and his team discovered a new framework of quasi-regular Dirichlet forms which correspond to right processes in one-to-one manner. His book, written in collaboration with Michael Rockner, An Introduction to the Theory of (Non-symmetric) Dirichlet Forms, has become a notable text in this field. His work on the proof of the Feynman-Kac probabilistic representation of mixed boundary problems of Schrodinger operators with measure-valued potentials is an important contribution to the theory of the Schrodinger equation.[1]
Selected publications
• Zhi-Ming Ma, M. Röckner Introduction to the theory of (non-symmetric) Dirichlet forms.1 November, 1992[5]
• Guan, QY., Ma, ZM. Reflected Symmetric α-Stable Processes and Regional Fractional Laplacian. Probab. Theory Relat. Fields 134, 649–694 (2006). https://doi.org/10.1007/s00440-005-0438-3[6]
• Albeverio, Sergio, Ma, Zhiming Additive functionals, nowhere Radon and Kato class smooth measures associated with Dirichlet forms. https://doi.org/10.18910/8386[7]
Awards
Ma's contributions to science have led to his recognition with various awards including; the First Class Prize for Natural Sciences by the Chinese Academy of Sciences, the Max-Planck Research Award by the Max-Planck Society and Alexander von Humboldt Foundation, the Chinese National Natural Sciences Prize, the Shiing Shen. Chern Mathematics Prize, the Qiu-Shi Outstanding Young Scholars Prize, the He-Liang-He-Li Sciences and Technology Progress Prize，and the Hua Loo-Keng Mathematics Prize.[1]
Memberships
Ma is a member of different recognised scientific organisations. He was elected as an Academician of the Chinese Academy of Sciences in 1995 and he became a fellow of the Third World Academy of Sciences in 1999.[8] He was the Chairman of the organizing committee for the International Congress of Mathematicians that was held in Beijing (2002). He was elected as a member of the Executive Committee for International Mathematical Union in 2003 and he became the Vice president in 2007. He was elected as a member of Chinese Mathematical Society and he became the president in 2003.[1][8]
References
1. "Ma Zhiming". en.nankai.edu.cn. Retrieved 2022-05-28.
2. "IMU Executive Committee | International Mathematical Union (IMU)". www.mathunion.org. Retrieved 2022-05-28.
3. "Ma, Zhi-Ming". TWAS. Retrieved 2022-05-28.
4. "China Vitae : Biography of Ma Zhiming". www.chinavitae.com. Retrieved 2022-05-28.
5. Ma, Zhi-Ming; Röckner, M. (1992). Introduction to the theory of (non-symmetric) Dirichlet forms. Universitext. doi:10.1007/978-3-642-77739-4. ISBN 978-3-540-55848-4. S2CID 122441749.
6. Guan, Qing-Yang; Ma, Zhi-Ming (2006-04-01). "Reflected Symmetric α-Stable Processes and Regional Fractional Laplacian". Probability Theory and Related Fields. 134 (4): 649–694. doi:10.1007/s00440-005-0438-3. ISSN 1432-2064. S2CID 18733207.
7. Albeverio, Sergio; Ma, Zhiming (1992). "Additive functionals, nowhere Radon and Kato class smooth measures associated with Dirichlet forms". Osaka Journal of Mathematics. 29 (2): 247–265. doi:10.18910/8386. hdl:11094/8386.
8. "Zhi-Ming Ma curriculum vitae" (PDF).
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Zhihong Xia
Zhihong "Jeff" Xia (Chinese: 夏志宏; pinyin: Xià Zhìhóng; born 20 September 1962, in Dongtai, Jiangsu, China) is a Chinese-American mathematician.
Education and career
Xia received, in 1982, from Nanjing University a bachelor's degree in astronomy and in 1988, a PhD in mathematics from Northwestern University with thesis advisor Donald G. Saari, for his thesis, The Existence of the Non-Collision Singularities.[1] From 1988 to 1990, Xia was an assistant professor at Harvard University and from 1990 to 1994, an associate professor at Georgia Institute of Technology (and Institute Fellow). In 1994, he became a full professor at Northwestern University and since 2000, he has been the Arthur and Gladys Pancoe Professor of Mathematics.[2]
His research deals with celestial mechanics, dynamical systems, Hamiltonian dynamics, and ergodic theory. In his dissertation, he solved the Painlevé conjecture, a long-standing problem posed in 1895 by Paul Painlevé. The problem concerns the existence of singularities of non-collision character in the $N$-body problem in three-dimensional space; Xia proved the existence for $N\geq 5$. For the existence proof, he constructed an example of five masses, of which four are separated into two pairs which revolve around each other in eccentric elliptical orbits about the z-axis of symmetry, and a fifth mass moves along the z-axis. For selected initial conditions, the fifth mass can be accelerated to an infinite velocity in a finite time interval (without any collision between the bodies involved in the example).[3] The case $N=4$ was open until 2014,[4] when it was solved by Jinxin Xue.[5][6] For $N=3$, Painlevé had proven that the singularities (points of the orbit in which accelerations become infinite in a finite time interval) must be of the collision type. However, Painlevé's proof did not extend to the case $N>3$.
In 1993, Xia was the inaugural winner of the Blumenthal Award of the American Mathematical Society. From 1989 to 1991, he was a Sloan Fellow. From 1993 to 1998, he received the National Young Investigator Award from the National Science Foundation. In 1995, he received the Monroe H. Martin Prize in Applied Mathematics from the University of Maryland.[7] In 1998, he was an Invited Speaker of the International Congress of Mathematicians in Berlin.[8]
Selected publications
• Xia, Zhihong (1992). "The Existence of Noncollision Singularities in Newtonian Systems". Annals of Mathematics. Series 2. 135 (3): 411–468. doi:10.2307/2946572. JSTOR 2946572.
• Xia, Zhihong (1992). "Existence of invariant tori in volume-preserving diffeomorphisms". Ergodic Theory and Dynamical Systems. 12 (3): 621–631. doi:10.1017/S0143385700006969. S2CID 122761956.
• Xia, Zhihong (1992). "Melnikov method and transversal homoclinic points in the restricted three-body problem" (PDF). Journal of Differential Equations. 96 (1): 170–184. Bibcode:1992JDE....96..170X. doi:10.1016/0022-0396(92)90149-H.
• Saari, Donald G.; Xia, Zhihong (1993). "Off to Infinity in Finite Time" (PDF). Notices of the AMS. 42: 538–546.
• Xia, Z (1994). "Arnold diffusion and oscillatory solutions in the planar three-body problem". Journal of Differential Equations. 110 (2): 289–321. Bibcode:1994JDE...110..289X. doi:10.1006/jdeq.1994.1069.
• Saari, Donald G; Xia, Zhihong (1996). "Singularities in the Newtonian 𝑛-body problem". Hamiltonian Dynamics and Celestial Mechanics. Contemporary Mathematics. Vol. 198. American Mathematical Society. pp. 21–30. CiteSeerX 10.1.1.24.1325. doi:10.1090/conm/198/02493. ISBN 978-0-8218-0566-4.
• Xia, Zhihong (1996). "Homoclinic points in symplectic and volume-preserving diffeomorphisms". Communications in Mathematical Physics. 177 (2): 435–449. Bibcode:1996CMaPh.177..435X. doi:10.1007/BF02101901. S2CID 17732615.
• Zhu, Deming; Xia, Zhihong (1998). "Bifurcations of heteroclinic loops". Science in China Series A: Mathematics. 41 (8): 837–848. Bibcode:1998ScChA..41..837Z. doi:10.1007/BF02871667. S2CID 120519869.
• Xia, Zhihong (2004). "Convex central configurations for the n-body problem". Journal of Differential Equations. 200 (2): 185–190. Bibcode:2004JDE...200..185X. doi:10.1016/j.jde.2003.10.001.
• Xia, Zhihong (2006). "Area-preserving surface diffeomorphisms". Communications in Mathematical Physics. 263 (3): 723–735. arXiv:math/0503223. Bibcode:2006CMaPh.263..723X. CiteSeerX 10.1.1.235.4920. doi:10.1007/s00220-005-1514-3. S2CID 14540760.
• Saghin, Radu; Xia, Zhihong (2009). "Geometric expansion, Lyapunov exponents and foliations" (PDF). Annales de l'Institut Henri Poincaré C. 26 (2): 689–704. Bibcode:2009AIHPC..26..689S. doi:10.1016/j.anihpc.2008.07.001. S2CID 119147899.
• Xia, Zhihong; Zhang, Pengfei (2014). "Homoclinic points for convex billiards". Nonlinearity. 27 (6): 1181–1192. arXiv:1310.5279. Bibcode:2014Nonli..27.1181X. doi:10.1088/0951-7715/27/6/1181. S2CID 119627854.
• Xia, Zhihong; Zhang, Pengfei (2017). "Homoclinic intersections for geodesic flows on convex spheres". Dynamical Systems, Ergodic Theory, and Probability: in Memory of Kolya Chernov. Contemporary Mathematics. Vol. 698. American Mathematical Society. pp. 221–238.
References
1. Zhi-Hong Xia at the Mathematics Genealogy Project
2. "Zhihong Jeff Xia". Northwestern University.
3. In 1908 Edvard Hugo von Zeipel proved the surprising fact that the existence of a non-collision singularity in the $N$-body problem necessarily causes the velocity of at least one particle to become unbounded.
4. Already in 2003, Joseph L. Gerver gave arguments (a heuristic model) for the existence of a non-collision singularity for the planar Newtonian 4-body problem — however, at the time there was still no rigorous proof. See Gerver, Joseph L. (2003). "Noncollision Singularities: Do Four Bodies Suffice?". Exp. Math. 12 (2): 187–198. doi:10.1080/10586458.2003.10504491. S2CID 23816314.
5. Xue, Jinxin (2014). "Noncollision Singularities in a Planar Four-body Problem". arXiv:1409.0048 [math.DS].
6. Xue, Jinxin (2020). "Non-collision singularities in a planar 4-body problem". Acta Mathematica. 224 (2): 253–388. doi:10.4310/ACTA.2020.v224.n2.a2.
7. "Monroe H. Martin Prize". Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park.
8. Xia, Zhihong (1998). "Arnold diffusion: a variational construction". Doc. Math. (Bielefeld) Extra Vol. ICM Berlin, 1998, vol. II. pp. 867–877.
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Zhiwei Yun
Zhiwei Yun (Chinese: 恽之玮; pinyin: Yùn Zhīwěi; born September 1982) is a Professor of Mathematics at MIT specializing in number theory, algebraic geometry and representation theory, with a particular focus on the Langlands program.
Zhiwei Yun
Born
Yun Zhiwei (恽之玮)
September 1982 (age 40)
Changzhou, China[1]
Alma materPeking University
Princeton University
Known forcontributions to number theory, representation theory and algebraic geometry
AwardsGold Medal, IMO (2000)[2]
SASTRA Ramanujan Prize (2012)[3]
2018 New Horizons In Mathematics Prize (2018)
Morningside Medal (2019)
Scientific career
FieldsMathematics
InstitutionsMassachusetts Institute of Technology
Stanford University
Yale University
Doctoral advisorRobert MacPherson
He was previously a C. L. E. Moore instructor at Massachusetts Institute of Technology from 2010 to 2012, assistant professor then associate professor at Stanford University from 2012 to 2016, and professor at Yale University from 2016 to 2017.
Education
Yun was born in Changzhou, China.[1] As a high schooler, he participated in the International Mathematical Olympiad in 2000; he received a gold medal with a perfect score.[2] Yun received his bachelor's degree from Peking University in 2004. In 2009, he received his Ph.D. from Princeton University, under the direction of Robert MacPherson.[4][5]
Work
His collaborations with Wei Zhang, Xinyi Yuan and Xinwen Zhu have received attention in publications such as Quanta Magazine and Business Insider.[6][7] In particular, his work with Wei Zhang on the Taylor expansion of L-functions is "already being hailed as one of the most exciting breakthroughs in an important area of number theory in the last 30 years."
Yun also made substantial contributions towards the global Gan–Gross–Prasad conjecture.
Awards
Yun was awarded the SASTRA Ramanujan Prize in 2012 for his "fundamental contributions to several areas that lie at the interface of representation theory, algebraic geometry and number theory."[3]
In December 2017, he was awarded 2018 New Horizons In Mathematics Prize together with Wei Zhang, Aaron Naber and Maryna Viazovska.
He was included in the 2019 class of fellows of the American Mathematical Society "for contributions to geometry, number theory, and representation theory, including his construction of motives with exceptional Galois groups".[8] In 2019 he received the Morningside Medal jointly with Xinwen Zhu.[9]
Selected publications
• Yun, Zhiwei; Vincent, Christelle (2015). "Galois representations attached to moments of Kloosterman sums and conjectures of Evans". Compositio Mathematica. 151 (1): 68–120. arXiv:1308.3920. doi:10.1112/S0010437X14007593.
• (with Davesh Maulik) Maulik, Davesh; Yun, Zhiwei (2014). "Macdonald formula for curves with planar singularities". Journal für die reine und angewandte Mathematik. 2014 (694): 27–48. arXiv:1107.2175. doi:10.1515/crelle-2012-0093. S2CID 5317997.
• Yun, Zhiwei (2014). "Motives with exceptional Galois groups and the inverse Galois problem". Inventiones Mathematicae. 196 (2): 267–337. arXiv:1112.2434. Bibcode:2014InMat.196..267Y. doi:10.1007/s00222-013-0469-9.
• (with Roman Bezrukavnikov) Bezrukavnikov, Roman; Yun, Zhiwei (2013). "On Koszul duality for Kac–Moody groups". Representation Theory. 17: 1–98. arXiv:1101.1253. doi:10.1090/S1088-4165-2013-00421-1.
• (with Ngô Bảo Châu and Jochen Heinloth) Heinloth, Jochen; Ngô, Bao-Châu; Yun, Zhiwei (2013). "Kloosterman sheaves for reductive groups". Annals of Mathematics. 177 (1): 241–310. doi:10.4007/annals.2013.177.1.5.
• Yun, Zhiwei (2012). "Langlands duality and global Springer theory". Compositio Mathematica. 148 (3): 835–867. doi:10.1112/S0010437X11007433.
• Yun, Zhiwei (2011). "Global Springer theory". Advances in Mathematics. 228 (1): 266–328. doi:10.1016/j.aim.2011.05.012.
• Gordon, Julia; Yun, Zhiwei (2011). "The fundamental lemma of Jacquet and Rallis". Duke Mathematical Journal. 156 (2): 167–227. arXiv:0901.0900. doi:10.1215/00127094-2010-210. S2CID 14295843.
• Yun, Zhiwei (2009). "Weights of mixed tilting sheaves and geometric Ringel duality". Selecta Mathematica. New Series. 14 (2): 299–320. arXiv:0805.1495. doi:10.1007/s00029-008-0066-8.
• (with Alexei Oblomkov) Oblomkov, Alexei; Yun, Zhiwei (2016). "Geometric representations of graded and rational Cherednik algebras". Advances in Mathematics. 292: 601–706. arXiv:1407.5685. doi:10.1016/j.aim.2016.01.015.
• (with Wei Zhang) Yun, Zhiwei; Zhang, Wei (2017). "Shtukas and the Taylor expansion of L-functions". Annals of Mathematics. 186 (3): 767–911. arXiv:1512.02683. doi:10.4007/annals.2017.186.3.2.
References
1. "北京大学校友恽之玮获2012年"拉马努金"奖". 30 August 2012.
2. "International Mathematical Olympiad".
3. "ZHIWEI YUN TO RECEIVE 2012 SASTRA RAMANUJAN PRIZE". Sastra University. Retrieved 2 September 2016.
4. "Zhiwei Yun" (PDF). Stanford University. Retrieved 2 September 2016.
5. "Zhiwei Yun", Mathematics Genealogy Project. Retrieved on 4 December 2016.
6. "Math Quartet Joins Forces on Unified Theory", Quanta Magazine. Retrieved on 4 December 2016.
7. "Math Quartet Joins Forces on Unified Theory", Business Insider. Retrieved on 4 December 2016.
8. 2019 Class of the Fellows of the AMS, American Mathematical Society, retrieved 2018-11-07
9. Morningside Medal 2019
Recipients of SASTRA Ramanujan Prize
• Manjul Bhargava (2005)
• Kannan Soundararajan (2005)
• Terence Tao (2006)
• Ben Green (2007)
• Akshay Venkatesh (2008)
• Kathrin Bringmann (2009)
• Wei Zhang (2010)
• Roman Holowinsky (2011)
• Zhiwei Yun (2012)
• Peter Scholze (2013)
• James Maynard (2014)
• Jacob Tsimerman (2015)
• Kaisa Matomäki (2016)
• Maksym Radziwill (2016)
• Maryna Viazovska (2017)
• Yifeng Liu (2018)
• Jack Thorne (2018)
• Adam Harper (2019)
• Shai Evra (2020)
• Will Sawin (2021)
• Yunqing Tang (2022)
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Zhoubi Suanjing
The Zhoubi Suanjing, also known by many other names, is an ancient Chinese astronomical and mathematical work. The Zhoubi is most famous for its presentation of Chinese cosmology and a form of the Pythagorean theorem. It claims to present 246 problems worked out by the early Zhou culture hero Ji Dan and members of his court, placing its contents in the 11th century BC. However, the present form of the book does not seem to be earlier than the 2nd century Eastern Han, with some additions and commentaries continuing to be added for several more centuries.
The Gougu Theorem diagram added to the Zhoubi by Zhao Shuang
Zhoubi Suanjing
Traditional Chinese《周髀算經》
Simplified Chinese《周髀算经》
Transcriptions
Standard Mandarin
Hanyu PinyinZhōubì suànjīng
Wade–GilesChou-pi Suan-ching
Zhoubi
Chinese《周髀》
Literal meaningThe Zhou Gnomon
On Gnomons and Circular Paths
Transcriptions
Standard Mandarin
Hanyu PinyinZhōubì
Wade–GilesChou-pi
Zhoubi
Traditional Chinese《算經》
Simplified Chinese《算经》
Literal meaningThe Classic of Computation
The Arithmetic Classic
Transcriptions
Standard Mandarin
Hanyu PinyinSuànjīng
Wade–GilesSuan-ching
Names
Zhoubi Suanjing is the atonal pinyin romanization of the modern standard Mandarin pronunciation of the work's Classical Chinese name, 《周髀算經》. The same name has been variously romanized as the Chou Pei Suan Ching,[1] the Tcheou-pi Souane,[2] &c. Its original title was simply the Zhoubi. The character 髀 is a literary term for the femur or thighbone but in context only refers to one or more gnomons, large sticks whose shadows were used for Chinese calendrical and astronomical calculations.[3] Because of the ambiguous nature of the character 周, it has been alternately understood and translated as "On the Gnomon and the Circular Paths of Heaven",[3] the "Zhou Shadow Gauge Manual",[4] "The Gnomon of the Zhou Sundial",[5] and "Gnomon of the Zhou Dynasty".[6] The honorific Suanjing—"Arithmetical Classic",[1] "Sacred Book of Arithmetic",[7] "Mathematical Canon",[6] "Classic of Computations",[8] &c.—was added later.
Dating
Examples of the gnomon described in the work have been found from as early as 2300 BC and Ji Dan, better known as the Duke of Zhou, was an 11th century BC regent and noble during the first generation of the Zhou dynasty. The Zhoubi was traditionally dated to Ji Dan's own life[9] and considered to be the oldest Chinese mathematical treatise.[3] However, although some passages seem to come from the Warring States Period or earlier,[9] the current text of the work mentions Lü Bowei and is believed to have received its current form no earlier than the Eastern Han, during the 1st or 2nd century. It does not appear at all in the Book of Han's account of calendrical, astronomical, and mathematical works, although Joseph Needham allows that this may have been from its current contents having previously been provided in several different works listed in the Han history which are otherwise unknown.[3]
Contents
The Zhoubi is an anonymous collection of 246 problems encountered by the Duke of Zhou and figures in his court, including the astrologer Shang Gao. Each problem includes an answer and a corresponding arithmetic algorithm.
It is an important source on early Chinese cosmology, glossing the ancient idea of a round heaven over a square earth (天圆地方, tiānyuán dìfāng) as similar to the round parasol suspended over some ancient Chinese chariots[10] or a Chinese chessboard.[11] All things measurable were considered variants of the square, while the expansion of a polygon to infinite sides approaches the immeasurable circle.[4] This concept of a "canopy heaven" (蓋天, gàitiān) had earlier produced the jade bì (璧) and cóng (琮) objects and myths about Gonggong, Mount Buzhou, Nüwa, and repairing the sky. Although this eventually developed into an idea of a "spherical heaven" (渾天, hùntiān),[12] the Zhoubi offers numerous explorations of the geometric relationships of simple circles circumscribed by squares and squares circumscribed by circles.[13] A large part of this involves analysis of solar declination in the Northern Hemisphere at various points throughout the year.[3]
At one point during its discussion of the shadows cast by gnomons, the work presents a form of the Pythagorean theorem known as the gougu theorem (勾股定理, gōugǔ dìnglǐ)[14] from the Chinese names—lit. "hook" and "thigh"—of the two sides of the carpenter or try square.[15] In the 3rd century, Zhao Shuang's commentary on the Zhoubi included a diagram effectively proving the theorem[16] for the case of a 3-4-5 triangle,[17] whence it can be generalized to all right triangles. The original text being ambiguous on its own, there is disagreement as to whether this proof was established by Zhao or merely represented an illustration of a previously understood concept earlier than Pythagoras.[18][14] Shang Gao concludes the gougu problem saying "He who understands the earth is a wise man, and he who understands the heavens is a sage. Knowledge is derived from the shadow [straight line], and the shadow is derived from the gnomon [right angle]. The combination of the gnomon with numbers is what guides and rules the ten thousand things."[19]
Commentaries
The Zhoubi has had a prominent place in Chinese mathematics and was the subject of specific commentaries by Zhao Shuang in the 3rd century, Liu Hui in 263, by Zu Gengzhi in the early 6th century, Li Chunfeng in the 7th century, and Yang Hui in 1270.
See also
• Tsinghua Bamboo Slips
• Dunhuang Star Chart
References
Citations
1. Needham & al. (1959), p. 815.
2. EB, 1st ed. (1771), p. 188.
3. Needham & al. (1959), p. 19.
4. Zou (2011), p. 104.
5. Pang-White (2018), p. 464.
6. Cullen (2018), p. 758.
7. Davis & al. (1995), p. 28.
8. Elman (2015), p. 240.
9. Needham & al. (1959), p. 20.
10. Tseng (2011), pp. 45–49.
11. Ding (2020), p. 172.
12. Tseng (2011), p. 50.
13. Tseng (2011), p. 51.
14. Cullen (1996), p. 82.
15. Gamwell (2016), p. 39.
16. Cullen (1996), p. 208.
17. Chemla (2005), p. .
18. Chemla (2005).
19. Gamwell (2016), p. 41.
Works cited
• "Chinese", Encyclopaedia Britannica, vol. II (1st ed.), Edinburgh: Colin Macfarquhar, 1771, pp. 184–192.
• Chemla, Karine (2005), Geometrical Figures and Generality in Ancient China and Beyond, Science in Context, ISBN 0-521-55089-0.
• Cullen, Christopher (1996), Astronomy and Mathematics in Ancient China, Cambridge University Press, ISBN 0-521-55089-0.
• Cullen, Christopher (2018), "Chinese Astronomy in the Early Imperial Age", The Cambridge History of Science, Vol. I: Ancient Science, Cambridge University Press, ISBN 978-110868262-6.
• Davis, Philip J.; et al., eds. (1995), "Brief Chronological Table to 1910", The Mathematical Experience, Modern Birkhäuser Classics, Boston: Birkhäuser, pp. 26–29, ISBN 978-081768294-1.
• Ding, D.X. Daniel (2020), The Historical Roots of Technical Communication in the Chinese Tradition, Newcastle-upon-Tyne: Cambridge Scholars, ISBN 978-152755989-9.
• Elman, Benjamin (2015), "Early Modern or Late Imperial? The Crisis of Classical Philology in Eighteenth-Century China", World Philology, Cambridge: Harvard University Press, pp. 225–244.
• Gamwell, Lynn (2016), Mathematics + Art: A Cultural History, Princeton University Press, ISBN 978-069116528-8.
• Needham, Joseph; et al. (1959), Science & Civilisation in China, Vol. III: Mathematics and the Sciences of the Heavens and the Earth, Cambridge University Press, ISBN 978-052105801-8.
• Pang-White, A. Ann (2018), The Confucian Four Books for Women, Oxford University Press, ISBN 978-0-19-046091-4.
• Tseng, L.Y. Lillian (2011), Picturing Heaven in Early China, East Asian Monographs, Cambridge: Harvard University Asia Center, ISBN 978-0-674-06069-2.
• Zou Hui (2011), A Jesuit Garden in Beijing and Early Modern Chinese Culture, West Lafayette: Purdue University Press, ISBN 978-155753583-2.
Further reading
• 《周髀算經》 (in Chinese), Chinese Text Project.
• 《周髀算經》 (in Chinese), Project Gutenberg.
• Boyer, Carl B. (1991), A History of Mathematics, John Wiley & Sons, ISBN 0-471-54397-7.
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M. D. Chow
M. D. Chow (1878 – February 13, 1949 Shanghai), also known by the Chinese names Zhou Jinjue (Chinese: 周今覺) and Zhou Mingda (Chinese: 周明达), was a Chinese philatelist and mathematician. He was nicknamed the "king of Chinese philately".[1]
Names
Having multiple names was the custom. He was also known as Zhou Meiquan (Chinese: 周美权 or 周梅泉), Zhou Jinjue (Chinese: 周今觉; formerly romanised Chow Chin Tso).
Early life
He was born into a salt merchant family in Yangzhou and moved to Shanghai in 1912. He was home schooled.
Philately
He was most noted as the founding father of Chinese philately and was crowned the King of Chinese Stamps after his acquisition of the rarest stamp, the block of four Red Revenue stamps from the original owner R. A. de Villard in 1927.[2][3][4] He championed the study of the Red Revenues.[5]
To entertain his sick son Wei-Liang Chow in 1923, he brought home many colorful foreign stamps during his recuperation. Soon they both caught the bug and began learning and collecting stamps.[6] He found Chinese Philatelic Society on November 15, 1925.[7] His bi-lingual Philatelic Bulletin won a Special Bronze Medal at the International Philatelic Exhibition in New York in 1926.[8] Chinese stamps eventually became a gold medal contender in 1927 at the Strasbourg International Exhibition in France.[3][4] He's the first Chinese to be granted a fellow of F.R.P.S.L., the Royal Philatelic Society London.[3][4]
Math clubs
In Yangzhou in 1900, he created Zhixin Math Club (Chinese: 知新算社) with Bao Mofen (Chinese: 包墨芬) and Yu Yudong (Chinese: 余雨东). As one of the finest mathematicians in China, he was highly praised by Japanese scholars.[9] In the 1920s, he created Science Society of China (Chinese: 中国科学社) with Ren Hongjuan (Chinese: 任鸿隽) and Hu Mingxia (Chinese: 胡明夏), and was named co-honorary president with Zhang Jian.
Awards and honors
His bi-lingual Philatelic Bulletin won a Special Bronze Medal at the International Philatelic Exhibition in New York in 1926.
References
1. Chen Tse-chuan (1 January 1962). "The Story of Chinese Stamps". Taiwan Today.
2. Ma, Runsheng (1947.) Shanghai: Ma’s Illustrated Catalogue of the Stamps of China
3. Woo, L.Y. (Chinese: 吳樂園) (1983). Taipei: Red Revenue Surcharges Stamp Collection (Chinese: 紅印花加蓋郵票專集)
4. Ministry of Transportation Post Office (Chinese: 交通部郵政總局) (1984). Taipei: Red Revenue Surcharge, Part I of 2 (Chinese: 紅印花郵票上編)
5. Matthew Bennett, Inc. (2001) The "Sun" Collection of the 1897 Red Revenue Surcharge of China. New York: Matthew Bennett, Inc.
6. Philatelic Bulletin, 1923, 3:3, page 27
7. Illustriertes Briefmarken Journal, No. 2, Jan 16, 1926
8. New York Times, World Stamp Show to Give 582 Awards 1926.04.18 retrieved 2015.09.10
9. Sotheby’s catalog titled Postage Stamps of the Far East, April 29, 1996
External links
• (in Chinese) Zhou Jinjue on CNKI
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Xinwen Zhu
Xinwen Zhu (Chinese: 朱歆文; born 1982 in Sichuan) is a Chinese mathematician and professor at Stanford University. His work deals primarily with geometric representation theory and in particular the Langlands program, tying number theory to algebraic geometry and quantum physics.[1][2]
Biography
Zhu obtained his A.B. in mathematics from Peking University in 2004 and his Ph.D. in mathematics from the University of California, Berkeley in 2009 under the direction of Edward Frenkel.[1] He taught at Harvard University as a Benjamin Peirce Lecturer and at Northwestern University as an assistant professor before joining the Caltech faculty in 2014. According to the American Mathematical Society, "[Zhu] studies the geometry and topology of flag varieties of loop groups and applies techniques from the geometric Langlands program to arithmetic geometry."[3]
The awards Zhu has received include an AMS Centennial Fellowship in 2013 and a Sloan Fellowship in 2015.[4] His research has been published in Annals of Mathematics and Inventiones mathematicae, among other mathematics journals. Zhu, Wei Zhang, Xinyi Yuan and Zhiwei Yun are frequent collaborators.[5] In 2019 he received the Morningside Medal jointly with Zhiwei Yun.[6] Zhu won the 2020 New Horizons in Mathematics Breakthrough Prize "For work in arithmetic algebraic geometry including applications to the theory of Shimura varieties and the Riemann-Hilbert problem for p-adic varieties."
Publications (selected)
• (with Edward Frenkel) "Gerbal Representations of Double Loop Groups", International Mathematics Research Notices 2012 (2012), No. 17, 3929–4013.
• (with George Pappas]) "Local models of Shimura varieties and a conjecture of Kottwitz", Inventiones mathematicae 194 (2013), No. 1, 147–254.
• "On the coherence conjecture of Pappas and Rapoport", Annals of Mathematics 180 (2014), No. 1, 1–85.
• (with Denis Osipov) "A categorical proof of the Parshin reciprocity laws on algebraic surfaces", Algebra & Number Theory 5 (2011), No. 3, 289–337.
• "Affine Demazure modules and T-fixed point subschemes in the affine Grassmannian", Advances in Mathematics 221 (2009), No. 2, 570–600.
• "Affine Grassmannians and the geometric Satake in mixed characteristic", Annals of Mathematics 185 (2017), No. 2, 403–492.
• (with Edward Frenkel) "Any flat bundle on a punctured disc has an oper structure", Mathematical Research Letters 17 (2010), no. 1, 27–37.
• "The geometric Satake correspondence for ramified groups", Annales Scientifiques de l'École Normale Supérieure 48 (2015), no. 2, 409–451.
• (with Zhiwei Yun) "Integral homology of loop groups via Langlands dual groups", Representation Theory 15 (2011), 347–369.
• (with An Huang, Bong H. Lian) "Period integrals and the Riemann–Hilbert correspondence", Journal of Differential Geometry 104 (2016), No. 2, 325–369.
• (with Tsao-Hsien Chen) "Geometric Langlands in prime characteristic", Compositio Mathematica 153 (2017), No. 2, 395–452.
References
1. "Prime Numbers, Quantum Fields, and Donuts: An Interview with Xinwen Zhu", Caltech. Retrieved on 3 December 2016.
2. 北大数学校友创新合作: 统一数论与几何 [New collaboration among Peking University mathematics alumni: unifying number theory and geometry]. Peking University. 15 December 2015. Retrieved 7 August 2017.
3. "Mathematics People", Notices of the AMS. Retrieved on 3 December 2016.
4. "Caltech Professors Awarded 2015 Sloan Fellowships", Caltech. Retrieved on 3 December 2016.
5. "Math Quartet Joins Forces on Unified Theory", Quanta Magazine. Retrieved on 3 December 2016.
6. Morningside Medal 2019
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Zhuo Qun Song
Zhuo Qun Song (Chinese: 宋卓群; pinyin: Sòng Zhuōqún; born 1997), also called Alex Song, is a Chinese-Canadian who is currently the most highly decorated International Mathematical Olympiad (IMO) contestant, with five gold medals and one bronze medal.
Zhuo Qun Song
宋卓群
Song in 2015
Born1997 (age 25–26)
Tianjin, People's Republic of China
NationalityCanadian
Other namesAlex
Alma materPrinceton University
Phillips Exeter Academy
Known forMost highly decorated IMO contestant with 5 golds and 1 bronze medal
Zhuo Qun Song
Chinese宋卓群
Transcriptions
Standard Mandarin
Hanyu PinyinSòng Zhuōqún
IPA[sʊ̂ŋ ʈʂwó.tɕʰy̌n]
Early life
Song was born in Tianjin, China in 1997.[1] He and his parents moved to Canada in 2002.[1] Song was brought up in Waterloo, Ontario.[2][3]
Song was interested in mathematics at a very young age where he started participating in competitions in first grade. By fourth grade, Song was participating in competitions such as the Canadian Open Mathematics Challenge and the American Mathematics Competitions. In fifth grade, Song became interested in solving Olympiad type questions and started training to solve them.[1]
In 2011, Song moved to the United States to attend Phillips Exeter Academy.[3]
International Mathematical Olympiad
In 2010, when Song was in the seventh grade, he represented Vincent Massey Secondary School in the Canadian Mathematical Olympiad where he finished first place.[1][4]
In the same year, Song represented Canada in the 2010 IMO where he won a Bronze Medal.[4] He would continue to represent Canada for 5 subsequent IMOs where he obtained a gold medal each time. He obtained a perfect score on his final run in 2015, the only contestant to do so that year.[2][3][5] The performances made Song the most decorated contestant of all time.[2][3][6] In 2015, Song was also one of the twelve top scorers of the United States of America Mathematical Olympiad, representing Phillips Exeter Academy.[7]
Results
Year Venue Result
2015 Chiang Mai Gold medal (P)[8]
2014 Cape Town Gold medal[9]
2013 Santa Marta Gold medal[10]
2012 Mar del Plata Gold medal[11]
2011 Amsterdam Gold medal[12]
2010 Astana Bronze medal[13]
Post-IMO
Song graduated from Phillips Exeter Academy in 2015.[2][3]
Song attended Princeton University where he graduated in 2019 with a Bachelor of Arts in Mathematics.[14]
During his time at Princeton, Song was part of the team that participated in the Putnam Competition. His team won second place in 2016[15] and third place in 2017.[16]
Song was previously a Quantitative Researcher at Citadel LLC.[17] He is currently a PhD student at the University of Illinois Urbana–Champaign.[18] He also has been lead coach for the Canadian IMO team since 2020. [19]
Publications
• Kaushansky, Vadim; Reisinger, Christoph; Shkolnikov, Mykhaylo; Song, Zhuo Qun (11 October 2020). "Convergence of a time-stepping scheme to the free boundary in the supercooled Stefan problem". arXiv:2010.05281 [math.PR].
• Song, Zhuo Qun (26 July 2019). "The Convergence of a Time-Stepping Scheme for a McKean-Vlasov Equation with Blow-Ups". {{cite journal}}: Cite journal requires |journal= (help)
• Liu, Yang; Park, Peter S.; Song, Zhuo Qun (1 December 2017). "Bounded gaps between products of distinct primes". Research in Number Theory. 3 (1): 26. doi:10.1007/s40993-017-0089-3. ISSN 2363-9555. S2CID 37218431.
• Liu, Yang; Park, Peter S.; Song, Zhuo Qun (11 December 2016). "The "Riemann Hypothesis" is true for period polynomials of almost all newforms". Research in the Mathematical Sciences. 3 (1): 31. arXiv:1607.04699. doi:10.1186/s40687-016-0081-x. ISSN 2197-9847. S2CID 44531385.
See also
• List of International Mathematical Olympiad participants
References
1. "Team Biographies" (PDF). Canadian Mathematical Society. 2012.{{cite web}}: CS1 maint: url-status (link)
2. Casey, Liam (27 July 2015). "Canadian math whiz wins international competition". CTVNews. Retrieved 25 November 2021.
3. International, Radio Canada (27 July 2015). "Alex Song tops International Math Olympiad". RCI | English. Retrieved 25 November 2021.
4. "Calgary mathlete brings home gold". CBC News. 15 July 2010.{{cite web}}: CS1 maint: url-status (link)
5. Kilkenny, Carmel (29 July 2015). "Alex Song and the Canadian Math Team". Radio Canada International. Retrieved 27 November 2021.
6. "Hall of Fame". International Mathematics Olympiad. Retrieved 27 November 2021.{{cite web}}: CS1 maint: url-status (link)
7. "Winners of the 2015 USA Mathematical Olympiad Announced". Mathematics Association of America. 27 May 2015. Retrieved 27 November 2021.
8. "Alex Song '15 Breaks IMO Record with Five Golds". Phillips Exeter Academy. 21 July 2015. Retrieved 28 November 2021.
9. Hammer, Kate (16 July 2014). "Canada's mathletes ninth in the world at math Olympiad in South Africa". The Globe and Mail. Retrieved 28 November 2021.
10. Hickey, Walt (14 August 2013). "Three American High Schoolers Swept An International Competition By Crushing These Math Problems". Business Insider. Retrieved 28 November 2021.
11. "53rd International Mathematical Olympiad Mar del Plata, Argentina — July 4 – 16, 2012". Canadian Mathematical Society. Retrieved 28 November 2021.
12. "52nd International Mathematical Olympiad Amsterdam, Netherlands — July 16 – 24, 2011". Canadian Mathematical Society. Retrieved 28 November 2021.
13. "51st International Mathematical Olympiad Astana, Kazakhstan — July 5 – 14, 2010". Canadian Mathematical Society. Retrieved 28 November 2021.
14. "Congratulations Class of 2019! | Math". www.math.princeton.edu. Retrieved 25 November 2021.
15. "2016 results".{{cite web}}: CS1 maint: url-status (link)
16. "2017 results".{{cite web}}: CS1 maint: url-status (link)
17. "Six Top Mathletes Selected for Math Team Canada 2021". CMS-SMC. Retrieved 27 November 2021.
18. "IDEA MATH". ideamath.education. Retrieved 19 November 2022.
19. "International Mathematical Olympiad". www.imo-official.org. Retrieved 19 November 2022.
External links
• Zhuo Qun Song's results at International Mathematical Olympiad
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Chinese Zhusuan
Zhusuan (Chinese: 珠算; pinyin: zhūsuàn; literally: "bead calculation") is the knowledge and practices of arithmetic calculation through the suanpan or Chinese abacus. In the year 2013, it has been inscribed on the UNESCO Representative List of the Intangible Cultural Heritage of Humanity.[1] While deciding on the inscription, the Intergovernmental Committee noted that "Zhusuan is considered by Chinese people as a cultural symbol of their identity as well as a practical tool; transmitted from generation to generation, it is a calculating technique adapted to multiple aspects of daily life, serving multiform socio-cultural functions and offering the world an alternative knowledge system."[2] The movement to get Chinese Zhusuan inscribed in the list was spearheaded by Chinese Abacus and Mental Arithmetic Association.
History
Zhusuan was a abacus invented in China at the end of the 2nd century CE and reached its peak during the period from the 13th to the 16th century CE. In the 13th century, Guo Shoujing (郭守敬) used Zhusuan to calculate the length of each orbital year and found it to be 365.2425 days. In the 16th century, Zhu Zaiyu (朱載堉) calculated the musical Twelve-interval Equal Temperament using Zhusuan. And again in the 16th century, Wang Wensu (王文素) and Cheng Dawei (程大位) wrote respectively Principles of Algorithms and General Rules of Calculation, summarizing and refining the mathematical algorithms of Zhusuan, thus further boosting the popularity and promotion of Zhusuan. At the end of the 16th century, Zhusuan was introduced to neighboring countries and regions.[3]
In culture
Zhusuan is an important part of the traditional Chinese culture. Zhusuan has a far-reaching effect on various fields of Chinese society, like Chinese folk custom, language, literature, sculpture, architecture, etc., creating a Zhusuan-related cultural phenomenon. For example, ‘Iron Abacus’ (鐵算盤) refers to someone good at calculating; ‘Plus three equals plus five and minus two’ (三下五除二; +3 = +5 − 2) means quick and decisive; ‘3 times 7 equals 21’ indicates quick and rash; and in some places of China, there is a custom of telling children's fortune by placing various daily necessities before them on their first birthday and letting them choose one to predict their future lives. Among the items is an abacus, which symbolizes wisdom and wealth.[3]
References
1. "Chinese Zhusuan, knowledge and practices of mathematical calculation through the abacus". www.unesco.org. UNESCO. Retrieved 29 November 2016.
2. "Decision of the Intergovernmental Committee: 8.COM 8.8". www.unesco.org. UNESCO. Retrieved 29 November 2016.
3. "Nomination File No. 00426". www.unesco.org. UNESCO. Retrieved 29 November 2016.
External links
Look up chinese zhusuan in Wiktionary, the free dictionary.
• UNESCO video on Chinese Zhusuan on YouTube (Published on Dec 4, 2013): Zhusuan
Authority control: National
• Japan
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Ziegler spectrum
In mathematics, the (right) Ziegler spectrum of a ring R is a topological space whose points are (isomorphism classes of) indecomposable pure-injective right R-modules. Its closed subsets correspond to theories of modules closed under arbitrary products and direct summands. Ziegler spectra are named after Martin Ziegler, who first defined and studied them in 1984.[1]
Definition
Let R be a ring (associative, with 1, not necessarily commutative). A (right) pp-n-formula is a formula in the language of (right) R-modules of the form
$\exists {\overline {y}}\ ({\overline {y}},{\overline {x}})A=0$
where $\ell ,n,m$ are natural numbers, $A$ is an $(\ell +n)\times m$ matrix with entries from R, and ${\overline {y}}$ is an $\ell $-tuple of variables and ${\overline {x}}$ is an $n$-tuple of variables.
The (right) Ziegler spectrum, $\operatorname {Zg} _{R}$, of R is the topological space whose points are isomorphism classes of indecomposable pure-injective right modules, denoted by $\operatorname {pinj} _{R}$, and the topology has the sets
$(\varphi /\psi )=\{N\in \operatorname {pinj} _{R}\mid \varphi (N)\supsetneq \psi (N)\cap \varphi (N)\}$
as subbasis of open sets, where $\varphi ,\psi $ range over (right) pp-1-formulae and $\varphi (N)$ denotes the subgroup of $N$ consisting of all elements that satisfy the one-variable formula $\varphi $. One can show that these sets form a basis.
Properties
Ziegler spectra are rarely Hausdorff and often fail to have the $T_{0}$-property. However they are always compact and have a basis of compact open sets given by the sets $(\varphi /\psi )$ where $\varphi ,\psi $ are pp-1-formulae.
When the ring R is countable $\operatorname {Zg} _{R}$ is sober.[2] It is not currently known if all Ziegler spectra are sober.
Generalization
Ivo Herzog showed in 1997 how to define the Ziegler spectrum of a locally coherent Grothendieck category, which generalizes the construction above.[3]
References
1. Ziegler, Martin (1984-04-01). "Model theory of modules" (PDF). Annals of Pure and Applied Logic. SPECIAL ISSUE. 26 (2): 149–213. doi:10.1016/0168-0072(84)90014-9.
2. Ivo Herzog (1993). Elementary duality of modules. Trans. Amer. Math. Soc., 340:1 37–69
3. Herzog, I. (1997). "The Ziegler Spectrum of a Locally Coherent Grothendieck Category". Proceedings of the London Mathematical Society. 74 (3): 503–558. doi:10.1112/S002461159700018X.
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Zig-zag product
In graph theory, the zig-zag product of regular graphs $G,H$, denoted by $G\circ H$, is a binary operation which takes a large graph ($G$) and a small graph ($H$) and produces a graph that approximately inherits the size of the large one but the degree of the small one. An important property of the zig-zag product is that if $H$ is a good expander, then the expansion of the resulting graph is only slightly worse than the expansion of $G$.
Roughly speaking, the zig-zag product $G\circ H$ replaces each vertex of $G$ with a copy (cloud) of $H$, and connects the vertices by moving a small step (zig) inside a cloud, followed by a big step (zag) between two clouds, and finally performs another small step inside the destination cloud.
The zigzag product was introduced by Reingold, Vadhan & Wigderson (2000). When the zig-zag product was first introduced, it was used for the explicit construction of constant degree expanders and extractors. Later on, the zig-zag product was used in computational complexity theory to prove that symmetric logspace and logspace are equal (Reingold 2008).
Definition
Let $G$ be a $D$-regular graph on $[N]$ with rotation map $\mathrm {Rot} _{G}$ and let $H$ be a $d$-regular graph on $[D]$ with rotation map $\mathrm {Rot} _{H}$. The zig-zag product $G\circ H$ is defined to be the $d^{2}$-regular graph on $[N]\times [D]$ whose rotation map $\mathrm {Rot} _{G\circ H}$ is as follows:
$\mathrm {Rot} _{G\circ H}((v,a),(i,j))$:
1. Let $(a',i')=\mathrm {Rot} _{H}(a,i)$.
2. Let $(w,b')=\mathrm {Rot} _{G}(v,a')$.
3. Let $(b,j')=\mathrm {Rot} _{H}(b',j)$.
4. Output $((w,b),(j',i'))$.
Properties
Reduction of the degree
It is immediate from the definition of the zigzag product that it transforms a graph $G$ to a new graph which is $d^{2}$-regular. Thus if $G$ is a significantly larger than $H$, the zigzag product will reduce the degree of $G$. Roughly speaking, by amplifying each vertex of $G$ into a cloud of the size of $H$ the product in fact splits the edges of each original vertex between the vertices of the cloud that replace it.
Spectral gap preservation
The expansion of a graph can be measured by its spectral gap, with an important property of the zigzag product the preservation of the spectral gap. That is, if $H$ is a “good enough” expander (has a large spectral gap) then the expansion of the zigzag product is close to the original expansion of $G$.
Formally: Define a $(N,D,\lambda )$-graph as any $D$-regular graph on $N$ vertices, whose second largest eigenvalue (of the associated random walk) has absolute value at most $\lambda $.
Let $G_{1}$ be a $(N_{1},D_{1},\lambda _{1})$-graph and $G_{2}$ be a $(D_{1},D_{2},\lambda _{2})$-graph, then $G_{1}\circ G_{2}$ is a $(N_{1}\cdot D_{1},D_{2}^{2},f(\lambda _{1},\lambda _{2}))$-graph, where $f(\lambda _{1},\lambda _{2})<\lambda _{1}+\lambda _{2}+\lambda _{2}^{2}$.
Connectivity preservation
The zigzag product $G\circ H$ operates separately on each connected component of $G$.
Formally speaking, given two graphs: $G$, a $D$-regular graph on $[N]$ and $H$, a $d$-regular graph on $[D]$ - if $S\subseteq [N]$ is a connected component of $G$ then $G|_{S}\circ H=G\circ H|_{S\times D}$, where $G|_{S}$ is the subgraph of $G$ induced by $S$ (i.e., the graph on $S$ which contains all of the edges in $G$ between vertices in $S$).
Applications
Construction of constant degree expanders
In 2002 Omer Reingold, Salil Vadhan, and Avi Wigderson gave a simple, explicit combinatorial construction of constant-degree expander graphs. The construction is iterative, and needs as a basic building block a single, expander of constant size. In each iteration the zigzag product is used in order to generate another graph whose size is increased but its degree and expansion remains unchanged. This process continues, yielding arbitrarily large expanders.
From the properties of the zigzag product mentioned above, we see that the product of a large graph with a small graph, inherits a size similar to the large graph, and degree similar to the small graph, while preserving its expansion properties from both, thus enabling to increase the size of the expander without deleterious effects.
Solving the undirected s-t connectivity problem in logarithmic space
In 2005 Omer Reingold introduced an algorithm that solves the undirected st-connectivity problem, the problem of testing whether there is a path between two given vertices in an undirected graph, using only logarithmic space. The algorithm relies heavily on the zigzag product.
Roughly speaking, in order to solve the undirected s-t connectivity problem in logarithmic space, the input graph is transformed, using a combination of powering and the zigzag product, into a constant-degree regular graph with a logarithmic diameter. The power product increases the expansion (hence reduces the diameter) at the price of increasing the degree, and the zigzag product is used to reduce the degree while preserving the expansion.
See also
• Graph operations
References
• Reingold, O.; Vadhan, S.; Wigderson, A. (2000), "Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors", Proc. 41st IEEE Symposium on Foundations of Computer Science (FOCS), pp. 3–13, arXiv:math/0406038, doi:10.1109/SFCS.2000.892006.
• Reingold, O (2008), "Undirected connectivity in log-space", Journal of the ACM, 55 (4): Article 17, 24 pages, doi:10.1145/1391289.1391291.
• Reingold, O.; Trevisan, L.; Vadhan, S. (2006), "Pseudorandom walks on regular digraphs and the RL vs. L problem", Proc. 38th ACM Symposium on Theory of Computing (STOC), pp. 457–466, doi:10.1145/1132516.1132583.
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Ziggurat algorithm
The ziggurat algorithm is an algorithm for pseudo-random number sampling. Belonging to the class of rejection sampling algorithms, it relies on an underlying source of uniformly-distributed random numbers, typically from a pseudo-random number generator, as well as precomputed tables. The algorithm is used to generate values from a monotonically decreasing probability distribution. It can also be applied to symmetric unimodal distributions, such as the normal distribution, by choosing a value from one half of the distribution and then randomly choosing which half the value is considered to have been drawn from. It was developed by George Marsaglia and others in the 1960s.
A typical value produced by the algorithm only requires the generation of one random floating-point value and one random table index, followed by one table lookup, one multiply operation and one comparison. Sometimes (2.5% of the time, in the case of a normal or exponential distribution when using typical table sizes) more computations are required. Nevertheless, the algorithm is computationally much faster than the two most commonly used methods of generating normally distributed random numbers, the Marsaglia polar method and the Box–Muller transform, which require at least one logarithm and one square root calculation for each pair of generated values. However, since the ziggurat algorithm is more complex to implement it is best used when large quantities of random numbers are required.
The term ziggurat algorithm dates from Marsaglia's paper with Wai Wan Tsang in 2000; it is so named because it is conceptually based on covering the probability distribution with rectangular segments stacked in decreasing order of size, resulting in a figure that resembles a ziggurat.
Theory of operation
The ziggurat algorithm is a rejection sampling algorithm; it randomly generates a point in a distribution slightly larger than the desired distribution, then tests whether the generated point is inside the desired distribution. If not, it tries again. Given a random point underneath a probability density curve, its x coordinate is a random number with the desired distribution.
The distribution the ziggurat algorithm chooses from is made up of n equal-area regions; n − 1 rectangles that cover the bulk of the desired distribution, on top of a non-rectangular base that includes the tail of the distribution.
Given a monotone decreasing probability density function f(x), defined for all x ≥ 0, the base of the ziggurat is defined as all points inside the distribution and below y1 = f(x1). This consists of a rectangular region from (0, 0) to (x1, y1), and the (typically infinite) tail of the distribution, where x > x1 (and y < y1).
This layer (call it layer 0) has area A. On top of this, add a rectangular layer of width x1 and height A/x1, so it also has area A. The top of this layer is at height y2 = y1 + A/x1, and intersects the density function at a point (x2, y2), where y2 = f(x2). This layer includes every point in the density function between y1 and y2, but (unlike the base layer) also includes points such as (x1, y2) which are not in the desired distribution.
Further layers are then stacked on top. To use a precomputed table of size n (n = 256 is typical), one chooses x1 such that xn = 0, meaning that the top box, layer n − 1, reaches the distribution's peak at (0, f(0)) exactly.
Layer i extends vertically from yi to yi+1, and can be divided into two regions horizontally: the (generally larger) portion from 0 to xi+1 which is entirely contained within the desired distribution, and the (small) portion from xi+1 to xi, which is only partially contained.
Ignoring for a moment the problem of layer 0, and given uniform random variables U0 and U1 ∈ [0,1), the ziggurat algorithm can be described as:
1. Choose a random layer 0 ≤ i < n.
2. Let x = U0xi.
3. If x < xi+1, return x.
4. Let y = yi + U1(yi+1 − yi).
5. Compute f(x). If y < f(x), return x.
6. Otherwise, choose new random numbers and go back to step 1.
Step 1 amounts to choosing a low-resolution y coordinate. Step 3 tests if the x coordinate is clearly within the desired density function without knowing more about the y coordinate. If it is not, step 4 chooses a high-resolution y coordinate, and step 5 does the rejection test.
With closely spaced layers, the algorithm terminates at step 3 a very large fraction of the time. For the top layer n − 1, however, this test always fails, because xn = 0.
Layer 0 can also be divided into a central region and an edge, but the edge is an infinite tail. To use the same algorithm to check if the point is in the central region, generate a fictitious x0 = A/y1. This will generate points with x < x1 with the correct frequency, and in the rare case that layer 0 is selected and x ≥ x1, use a special fallback algorithm to select a point at random from the tail. Because the fallback algorithm is used less than one time in a thousand, speed is not essential.
Thus, the full ziggurat algorithm for one-sided distributions is:
1. Choose a random layer 0 ≤ i < n.
2. Let x = U0xi
3. If x < xi+1, return x.
4. If i = 0, generate a point from the tail using the fallback algorithm.
5. Let y = yi + U1(yi+1 − yi).
6. Compute f(x). If y < f(x), return x.
7. Otherwise, choose new random numbers and go back to step 1.
For a two-sided distribution, the result must be negated 50% of the time. This can often be done conveniently by choosing U0 ∈ (−1,1) and, in step 3, testing if |x| < xi+1.
Fallback algorithms for the tail
Because the ziggurat algorithm only generates most outputs very rapidly, and requires a fallback algorithm whenever x > x1, it is always more complex than a more direct implementation. The specific fallback algorithm depends on the distribution.
For an exponential distribution, the tail looks just like the body of the distribution. One way is to fall back to the most elementary algorithm E = −ln(U1) and let x = x1 − ln(U1). Another is to call the ziggurat algorithm recursively and add x1 to the result.
For a normal distribution, Marsaglia suggests a compact algorithm:
1. Let x = −ln(U1)/x1.
2. Let y = −ln(U2).
3. If 2y > x2, return x + x1.
4. Otherwise, go back to step 1.
Since x1 ≈ 3.5 for typical table sizes, the test in step 3 is almost always successful.
Optimizations
The algorithm can be performed efficiently with precomputed tables of xi and yi = f(xi), but there are some modifications to make it even faster:
• Nothing in the ziggurat algorithm depends on the probability distribution function being normalized (integral under the curve equal to 1), removing normalizing constants can speed up the computation of f(x).
• Most uniform random number generators are based on integer random number generators which return an integer in the range [0, 232 − 1]. A table of 2−32xi lets you use such numbers directly for U0.
• When computing two-sided distributions using a two-sided U0 as described earlier, the random integer can be interpreted as a signed number in the range [−231, 231 − 1], and a scale factor of 2−31 can be used.
• Rather than comparing U0xi to xi+1 in step 3, it is possible to precompute xi+1/xi and compare U0 with that directly. If U0 is an integer random number generator, these limits may be premultiplied by 232 (or 231, as appropriate) so an integer comparison can be used.
• With the above two changes, the table of unmodified xi values is no longer needed and may be deleted.
• When generating IEEE 754 single-precision floating point values, which only have a 24-bit mantissa (including the implicit leading 1), the least-significant bits of a 32-bit integer random number are not used. These bits may be used to select the layer number. (See the references below for a detailed discussion of this.)
• The first three steps may be put into an inline function, which can call an out-of-line implementation of the less frequently needed steps.
Generating the tables
It is possible to store the entire table precomputed, or just include the values n, y1, A, and an implementation of f −1(y) in the source code, and compute the remaining values when initializing the random number generator.
As previously described, you can find xi = f −1(yi) and yi+1 = yi + A/xi. Repeat n − 1 times for the layers of the ziggurat. At the end, you should have yn = f(0). There will be some round-off error, but it is a useful sanity test to see that it is acceptably small.
When actually filling in the table values, just assume that xn = 0 and yn = f(0), and accept the slight difference in layer n − 1's area as rounding error.
Finding x1 and A
Given an initial (guess at) x1, you need a way to compute the area t of the tail for which x > x1. For the exponential distribution, this is just e−x1, while for the normal distribution, assuming you are using the unnormalized f(x) = e−x2/2, this is √π/2 erfc(x/√2). For more awkward distributions, numerical integration may be required.
With this in hand, from x1, you can find y1 = f(x1), the area t in the tail, and the area of the base layer A = x1y1 + t.
Then compute the series yi and xi as above. If yi > f(0) for any i < n, then the initial estimate x1 was too low, leading to too large an area A. If yn < f(0), then the initial estimate x1 was too high.
Given this, use a root-finding algorithm (such as the bisection method) to find the value x1 which produces yn−1 as close to f(0) as possible. Alternatively, look for the value which makes the area of the topmost layer, xn−1(f(0) − yn−1), as close to the desired value A as possible. This saves one evaluation of f −1(x) and is actually the condition of greatest interest.
References
• George Marsaglia; Wai Wan Tsang (2000). "The Ziggurat Method for Generating Random Variables". Journal of Statistical Software. 5 (8). Retrieved 2007-06-20. This paper numbers the layers from 1 starting at the top, and makes layer 0 at the bottom a special case, while the explanation above numbers layers from 0 at the bottom.
• C implementation of the ziggurat method for the normal density function and the exponential density function, that is essentially a copy of the code in the paper. (Potential users should be aware that this C code assumes 32-bit integers.)
• A C# implementation of the ziggurat algorithm and overview of the method.
• Jurgen A. Doornik (2005). "An Improved Ziggurat Method to Generate Normal Random Samples" (Document). Nuffield College, Oxford. {{cite document}}: Unknown parameter |access-date= ignored (help); Unknown parameter |url= ignored (help) Describes the hazards of using the least-significant bits of the integer random number generator to choose the layer number.
• Normal Behavior By Cleve Moler, MathWorks, describing the ziggurat algorithm introduced in MATLAB version 5, 2001.
• The Ziggurat Random Normal Generator Blogs of MathWorks, posted by Cleve Moler, May 18, 2015.
• David B. Thomas; Philip H.W. Leong; Wayne Luk; John D. Villasenor (October 2007). "Gaussian Random Number Generators" (PDF). ACM Computing Surveys. 39 (4): 11:1–38. doi:10.1145/1287620.1287622. ISSN 0360-0300. S2CID 10948255. Retrieved 2009-07-27. [W]hen maintaining extremely high statistical quality is the first priority, and subject to that constraint, speed is also desired, the Ziggurat method will often be the most appropriate choice. Comparison of several algorithms for generating Gaussian random numbers.
• Nadler, Boaz (2006). "Design Flaws in the Implementation of the Ziggurat and Monty Python methods (And some remarks on Matlab randn)". arXiv:math/0603058.. Illustrates problems with underlying uniform pseudo-random number generators and how those problems affect the ziggurat algorithm's output.
• Edrees, Hassan M.; Cheung, Brian; Sandora, McCullen; Nummey, David; Stefan, Deian (13–16 July 2009). Hardware-Optimized Ziggurat Algorithm for High-Speed Gaussian Random Number Generators (PDF). 2009 International Conference on Engineering of Reconfigurable Systems & Algorithms. Las Vegas.
• Marsaglia, George (September 1963). Generating a Variable from the Tail of the Normal Distribution (Technical report). Boeing Scientific Research Labs. Mathematical Note No. 322, DTIC accession number AD0423993. Archived from the original on September 10, 2014 – via Defense Technical Information Center.
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Zig-zag lemma
In mathematics, particularly homological algebra, the zig-zag lemma asserts the existence of a particular long exact sequence in the homology groups of certain chain complexes. The result is valid in every abelian category.
Statement
In an abelian category (such as the category of abelian groups or the category of vector spaces over a given field), let $({\mathcal {A}},\partial _{\bullet }),({\mathcal {B}},\partial _{\bullet }')$ and $({\mathcal {C}},\partial _{\bullet }'')$ be chain complexes that fit into the following short exact sequence:
$0\longrightarrow {\mathcal {A}}\mathrel {\stackrel {\alpha }{\longrightarrow }} {\mathcal {B}}\mathrel {\stackrel {\beta }{\longrightarrow }} {\mathcal {C}}\longrightarrow 0$
Such a sequence is shorthand for the following commutative diagram:
where the rows are exact sequences and each column is a chain complex.
The zig-zag lemma asserts that there is a collection of boundary maps
$\delta _{n}:H_{n}({\mathcal {C}})\longrightarrow H_{n-1}({\mathcal {A}}),$
that makes the following sequence exact:
The maps $\alpha _{*}^{}$ and $\beta _{*}^{}$ are the usual maps induced by homology. The boundary maps $\delta _{n}^{}$ are explained below. The name of the lemma arises from the "zig-zag" behavior of the maps in the sequence. A variant version of the zig-zag lemma is commonly known as the "snake lemma" (it extracts the essence of the proof of the zig-zag lemma given below).
Construction of the boundary maps
The maps $\delta _{n}^{}$ are defined using a standard diagram chasing argument. Let $c\in C_{n}$ represent a class in $H_{n}({\mathcal {C}})$, so $\partial _{n}''(c)=0$. Exactness of the row implies that $\beta _{n}^{}$ is surjective, so there must be some $b\in B_{n}$ with $\beta _{n}^{}(b)=c$. By commutativity of the diagram,
$\beta _{n-1}\partial _{n}'(b)=\partial _{n}''\beta _{n}(b)=\partial _{n}''(c)=0.$
By exactness,
$\partial _{n}'(b)\in \ker \beta _{n-1}=\mathrm {im} \;\alpha _{n-1}.$
Thus, since $\alpha _{n-1}^{}$ is injective, there is a unique element $a\in A_{n-1}$ such that $\alpha _{n-1}(a)=\partial _{n}'(b)$. This is a cycle, since $\alpha _{n-2}^{}$ is injective and
$\alpha _{n-2}\partial _{n-1}(a)=\partial _{n-1}'\alpha _{n-1}(a)=\partial _{n-1}'\partial _{n}'(b)=0,$
since $\partial ^{2}=0$. That is, $\partial _{n-1}(a)\in \ker \alpha _{n-2}=\{0\}$. This means $a$ is a cycle, so it represents a class in $H_{n-1}({\mathcal {A}})$. We can now define
$\delta _{}^{}[c]=[a].$
With the boundary maps defined, one can show that they are well-defined (that is, independent of the choices of c and b). The proof uses diagram chasing arguments similar to that above. Such arguments are also used to show that the sequence in homology is exact at each group.
See also
• Mayer–Vietoris sequence
References
• Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. ISBN 0-521-79540-0.
• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556
• Munkres, James R. (1993). Elements of Algebraic Topology. New York: Westview Press. ISBN 0-201-62728-0.
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Fence (mathematics)
In mathematics, a fence, also called a zigzag poset, is a partially ordered set (poset) in which the order relations form a path with alternating orientations:
$a<b>c<d>e<f>h<i\cdots $
or
$a>b<c>d<e>f<h>i\cdots $
A fence may be finite, or it may be formed by an infinite alternating sequence extending in both directions. The incidence posets of path graphs form examples of fences.
A linear extension of a fence is called an alternating permutation; André's problem of counting the number of different linear extensions has been studied since the 19th century.[1] The solutions to this counting problem, the so-called Euler zigzag numbers or up/down numbers, are:
$1,1,2,4,10,32,122,544,2770,15872,101042.$
(sequence A001250 in the OEIS).
The number of antichains in a fence is a Fibonacci number; the distributive lattice with this many elements, generated from a fence via Birkhoff's representation theorem, has as its graph the Fibonacci cube.[2]
A partially ordered set is series-parallel if and only if it does not have four elements forming a fence.[3]
Several authors have also investigated the number of order-preserving maps from fences to themselves, or to fences of other sizes.[4]
An up-down poset Q(a,b) is a generalization of a zigzag poset in which there are a downward orientations for every upward one and b total elements.[5] For instance, Q(2,9) has the elements and relations
$a>b>c<d>e>f<g>h>i.$
In this notation, a fence is a partially ordered set of the form Q(1,n).
Equivalent conditions
The following conditions are equivalent for a poset P:
1. P is a disjoint union of zigzag posets.
2. If a ≤ b ≤ c in P, either a = b or b = c.
3. $<\circ <\;=\emptyset $, i.e. it is never the case that a < b and b < c, so that < is vacuously transitive.
4. P has dimension at most one (defined analogously to the Krull dimension of a commutative ring).
5. Every element of P is either maximal or minimal.
6. The slice category Pos/P is cartesian closed.[lower-alpha 1]
The prime ideals of a commutative ring R, ordered by inclusion, satisfy the equivalent conditions above if and only if R has Krull dimension at most one.
Notes
1. Here, Pos denotes the category of partially ordered sets.
References
1. André (1881).
2. Gansner (1982) calls the fact that this lattice has a Fibonacci number of elements a “well known fact,” while Stanley (1986) asks for a description of it in an exercise. See also Höft & Höft (1985), Beck (1990), and Salvi & Salvi (2008).
3. Valdes, Tarjan & Lawler (1982).
4. Currie & Visentin (1991); Duffus et al. (1992); Rutkowski (1992a); Rutkowski (1992b); Farley (1995).
5. Gansner (1982).
• André, Désiré (1881), "Sur les permutations alternées", J. Math. Pures Appl., (Ser. 3), 7: 167–184.
• Beck, István (1990), "Partial orders and the Fibonacci numbers", Fibonacci Quarterly, 28 (2): 172–174, MR 1051291.
• Currie, J. D.; Visentin, T. I. (1991), "The number of order-preserving maps of fences and crowns", Order, 8 (2): 133–142, doi:10.1007/BF00383399, hdl:10680/1724, MR 1137906, S2CID 122356472.
• Duffus, Dwight; Rödl, Vojtěch; Sands, Bill; Woodrow, Robert (1992), "Enumeration of order preserving maps", Order, 9 (1): 15–29, doi:10.1007/BF00419036, MR 1194849, S2CID 84180809.
• Farley, Jonathan David (1995), "The number of order-preserving maps between fences and crowns", Order, 12 (1): 5–44, doi:10.1007/BF01108588, MR 1336535, S2CID 120372679.
• Gansner, Emden R. (1982), "On the lattice of order ideals of an up-down poset", Discrete Mathematics, 39 (2): 113–122, doi:10.1016/0012-365X(82)90134-0, MR 0675856.
• Höft, Hartmut; Höft, Margret (1985), "A Fibonacci sequence of distributive lattices", Fibonacci Quarterly, 23 (3): 232–237, MR 0806293.
• Kelly, David; Rival, Ivan (1974), "Crowns, fences, and dismantlable lattices", Canadian Journal of Mathematics, 26 (5): 1257–1271, doi:10.4153/cjm-1974-120-2, MR 0417003.
• Rutkowski, Aleksander (1992a), "The number of strictly increasing mappings of fences", Order, 9 (1): 31–42, doi:10.1007/BF00419037, MR 1194850, S2CID 120965362.
• Rutkowski, Aleksander (1992b), "The formula for the number of order-preserving self-mappings of a fence", Order, 9 (2): 127–137, doi:10.1007/BF00814405, MR 1199291, S2CID 121879635.
• Salvi, Rodolfo; Salvi, Norma Zagaglia (2008), "Alternating unimodal sequences of Whitney numbers", Ars Combinatoria, 87: 105–117, MR 2414008.
• Stanley, Richard P. (1986), Enumerative Combinatorics, Wadsworth, Inc. Exercise 3.23a, page 157.
• Valdes, Jacobo; Tarjan, Robert E.; Lawler, Eugene L. (1982), "The Recognition of Series Parallel Digraphs", SIAM Journal on Computing, 11 (2): 298–313, doi:10.1137/0211023.
External links
• Weisstein, Eric W. "Fence Poset". MathWorld.
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Zilber–Pink conjecture
In mathematics, the Zilber–Pink conjecture is a far-reaching generalisation of many famous Diophantine conjectures and statements, such as André–Oort, Manin–Mumford, and Mordell–Lang. For algebraic tori and semiabelian varieties it was proposed by Boris Zilber[1] and independently by Enrico Bombieri, David Masser, Umberto Zannier[2] in the early 2000's. For semiabelian varieties the conjecture implies the Mordell–Lang and Manin–Mumford conjectures. Richard Pink proposed (again independently) a more general conjecture for Shimura varieties which also implies the André–Oort conjecture.[3] In the case of algebraic tori, Zilber called it the Conjecture on Intersection with Tori (CIT). The general version is now known as the Zilber–Pink conjecture. It states roughly that atypical or unlikely intersections of an algebraic variety with certain special varieties are accounted for by finitely many special varieties.
Statement
Atypical and unlikely intersections
The intersection of two algebraic varieties is called atypical if its dimension is larger than expected. More precisely, given three varieties $X,Y\subseteq U$, a component $Z$ of the intersection $X\cap Y$ is said to be atypical in $U$ if $\dim Z>\dim X+\dim Y-\dim U$. Since the expected dimension of $X\cap Y$ is $\dim X+\dim Y-\dim U$, atypical intersections are "atypically large" and are not expected to occur. When $\dim X+\dim Y-\dim U<0$, the varieties $X$ and $Y$ are not expected to intersect at all, so when they do, the intersection is said to be unlikely. For example, if in a 3-dimensional space two lines intersect, then it is an unlikely intersection, for two randomly chosen lines would almost never intersect.
Special varieties
Special varieties of a Shimura variety are certain arithmetically defined subvarieties. They are higher dimensional versions of special points. For example, in semiabelian varieties special points are torsion points and special varieties are translates of irreducible algebraic subgroups by torsion points. In the modular setting special points are the singular moduli and special varieties are irreducible components of varieties defined by modular equations.
Given a mixed Shimura variety $X$ and a subvariety $V\subseteq X$, an atypical subvariety of $V$ is an atypical component of an intersection $V\cap T$ where $T\subseteq X$ is a special subvariety.
The Zilber–Pink conjecture
Let $X$ be a mixed Shimura variety or a semiabelian variety defined over $\mathbb {C} $, and let $V\subseteq X$ be a subvariety. Then $V$ contains only finitely many maximal atypical subvarieties.[4]
The abelian and modular versions of the Zilber–Pink conjecture are special cases of the conjecture for Shimura varieties, while in general the semiabelian case is not. However, special subvarieties of semiabelian and Shimura varieties share many formal properties which makes the same formulation valid in both settings.
Partial results and special cases
While the Zilber–Pink conjecture is wide open, many special cases and weak versions have been proven.
If a variety $V\subseteq X$ contains a special variety $T$ then by definition $T$ is an atypical subvariety of $V$. Hence, the Zilber–Pink conjecture implies that $V$ contains only finitely many maximal special subvarieties. This is the Manin–Mumford conjecture in the semiabelian setting and the André–Oort conjecture in the Shimura setting. Both are now theorems; the former has been known for several decades,[5] while the latter was proven in full generality only recently.[6]
Many partial results have been proven on the Zilber–Pink conjecture.[7][8][9] An example in the modular setting is the result that any variety contains only finitely many maximal strongly atypical subvarieties, where a strongly atypical subvariety is an atypical subvariety with no constant coordinate.[10][11]
References
1. Zilber, Boris (2002), "Exponential sums equations and the Schanuel conjecture", J. London Math. Soc., 65 (2): 27–44, doi:10.1112/S0024610701002861.
2. Bombieri, Enrico; Masser, David; Zannier, Umberto (2007), Anomalous Subvarieties—Structure Theorems and Applications, International Mathematics Research Notices, vol. 2007.
3. Pink, Richard (2005). "A Combination of the Conjectures of Mordell–Lang and André–Oort". Geometric Methods in Algebra and Number Theory. Progress in Mathematics. Vol. 235. pp. 251–282. CiteSeerX 10.1.1.499.3023. doi:10.1007/0-8176-4417-2_11. ISBN 0-8176-4349-4.
4. Habegger, Philipp; Pila, Jonathan (2016), o-minimality and certain atypical intersections, Ann. Sci. Éc. Norm. Supér, vol. 49, pp. 813–858.
5. Raynaud, Michel (1983). "Sous-variétés d'une variété abélienne et points de torsion". In Artin, Michael; Tate, John (eds.). Arithmetic and geometry. Papers dedicated to I. R. Shafarevich on the occasion of his sixtieth birthday. Vol. I: Arithmetic. Progress in Mathematics (in French). Vol. 35. Birkhäuser-Boston. pp. 327–352. MR 0717600. Zbl 0581.14031.
6. Pila, Jonathan; Shankar, Ananth; Tsimerman, Jacob; Esnault, Hélène; Groechenig, Michael (2021-09-17). "Canonical Heights on Shimura Varieties and the André-Oort Conjecture". arXiv:2109.08788 [math.NT].
7. Habegger, Philipp; Pila, Jonathan (2012), Some unlikely intersections beyond André-Oort, Compositio Math., vol. 148, pp. 1–27.
8. Daw, Christopher; Orr, Martin (2021), Unlikely intersections with $E\times $CM curves in ${\mathcal {A}}_{2}$, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), vol. 22, pp. 1705–1745.
9. Daw, Christopher; Orr, Martin (2022), Quantitative Reduction Theory and Unlikely Intersections, IMRN, vol. 2022, pp. 16138–16195.
10. Pila, Jonathan; Tsimerman, Jacob (2016), "Ax-Schanuel for the j-function", Duke Math. J., 165 (13): 2587–2605, arXiv:1412.8255, doi:10.1215/00127094-3620005, S2CID 118973278
11. Aslanyan, Vahagn (2021), "Weak Modular Zilber–Pink with Derivatives", Math. Ann., arXiv:1803.05895, doi:10.1007/s00208-021-02213-7, S2CID 119654268
Further reading
• Pila, Jonathan (2022). Point-Counting and the Zilber–Pink Conjecture. Cambridge University Press. ISBN 9781009170321.
• Zannier, Umberto (2012). Some Problems of Unlikely Intersections in Arithmetic and Geometry. Princeton: Princeton University Press. ISBN 978-0-691-15370-4.
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Zimmer's conjecture
Zimmer's conjecture is a statement in mathematics "which has to do with the circumstances under which geometric spaces exhibit certain kinds of symmetries."[1] It was named after the mathematician Robert Zimmer. The conjecture states that there can exist symmetries (specifically higher-rank lattices) in a higher dimension that cannot exist in lower dimensions.
In 2017, the conjecture was proven by Aaron Brown and Sebastián Hurtado-Salazar of the University of Chicago and David Fisher of Indiana University.[1][2][3]
References
1. Hartnett, Kevin (2018-10-23). "A Proof About Where Symmetries Can't Exist". Quanta Magazine. Retrieved 2018-11-02.
2. Brown, Aaron; Fisher, David; Hurtado, Sebastian (2017-10-07). "Zimmer's conjecture for actions of SL(𝑚,ℤ)". arXiv:1710.02735 [math.DS].
3. "New Methods for Zimmer's Conjecture". IPAM. Retrieved 2018-11-02.
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Zimmert set
In mathematics, a Zimmert set is a set of positive integers associated with the structure of quotients of hyperbolic three-space by a Bianchi group.
Definition
Fix an integer d and let D be the discriminant of the imaginary quadratic field Q(√-d). The Zimmert set Z(d) is the set of positive integers n such that 4n2 < -D-3 and n ≠ 2; D is a quadratic non-residue of all odd primes in d; n is odd if D is not congruent to 5 modulo 8. The cardinality of Z(d) may be denoted by z(d).
Property
For all but a finite number of d we have z(d) > 1: indeed this is true for all d > 10476.[1]
Application
Let Γd denote the Bianchi group PSL(2,Od), where Od is the ring of integers of. As a subgroup of PSL(2,C), there is an action of Γd on hyperbolic 3-space H3, with a fundamental domain. It is a theorem that there are only finitely many values of d for which Γd can contain an arithmetic subgroup G for which the quotient H3/G is a link complement. Zimmert sets are used to obtain results in this direction: z(d) is a lower bound for the rank of the largest free quotient of Γd[2] and so the result above implies that almost all Bianchi groups have non-cyclic free quotients.[1]
References
1. Mason, A.W.; Odoni, R.W.K.; Stothers, W.W. (1992). "Almost all Bianchi groups have free, non-cyclic quotients". Math. Proc. Camb. Philos. Soc. 111 (1): 1–6. Bibcode:1992MPCPS.111....1M. doi:10.1017/S0305004100075101. S2CID 122325132. Zbl 0758.20009.
2. Zimmert, R. (1973). "Zur SL2 der ganzen Zahlen eines imaginär-quadratischen Zahlkörpers". Inventiones Mathematicae. 19: 73–81. Bibcode:1973InMat..19...73Z. doi:10.1007/BF01418852. S2CID 121281237. Zbl 0254.10019.
• Maclachlan, Colin; Reid, Alan W. (2003). The Arithmetic of Hyperbolic 3-Manifolds. Graduate Texts in Mathematics. Vol. 219. Springer-Verlag. ISBN 0-387-98386-4. Zbl 1025.57001.
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Zinovy Reichstein
Zinovy Reichstein (born 1961) is a Russian-born American mathematician. He is a professor at the University of British Columbia in Vancouver. He studies mainly algebra, algebraic geometry and algebraic groups. He introduced (with Joe P. Buhler) the concept of essential dimension.[1]
Zinovy Reichstein
Born1961
Alma materHarvard University
Known forEssential dimension
Scientific career
FieldsMathematics
InstitutionsUniversity of British Columbia
Doctoral advisorMichael Artin
Early life and education
In high school, Reichstein participated in the national mathematics olympiad in Russia and was the third highest scorer in 1977 and second highest scorer in 1978.
Because of the Antisemitism in the Soviet Union at the time, Reichstein was not accepted to Moscow University, even though he had passed the special math entrance exams. He attended a semester of college at Russian University of Transport instead.
His family then decided to emigrate, arriving in Vienna, Austria, in August 1979 and New York, United States in the fall of 1980. Reichstein worked as a delivery boy for a short period of time in New York. He was then accepted to and attended California Institute of Technology for his undergraduate studies.[2]
Reichstein received his PhD degree in 1988 from Harvard University under the supervision of Michael Artin. Parts of his thesis entitled "The Behavior of Stability under Equivariant Maps" were published in the journal Inventiones Mathematicae.[3]
Career
As of 2011, he is on the editorial board of the mathematics journal Transformation groups.[4]
Awards
• Winner of the 2013 Jeffery-Williams Prize awarded by the Canadian Mathematical Society[5]
• Fellow of the American Mathematical Society, 2012[6]
• Invited Speaker to the International Congress of Mathematicians (Hyderabad, India 2010)[7]
References
1. J. Buhler, Z. Reichstein (1997). "On the Essential Dimension of a Finite Group". Compositio Mathematica. 106 (2): 159–179. doi:10.1023/A:1000144403695.
2. Dietrich, JS. "To Do Mathematics: The Odyssey of a Soviet Emigre" (PDF). calteches.library.caltech.edu. Archived from the original (PDF) on 2010-08-06.
3. Reichstein, Zinovy (1989), "Stability and equivariant maps", Inventiones Mathematicae, 96 (2): 349–383, Bibcode:1989InMat..96..349R, doi:10.1007/BF01393967, S2CID 120929091
4. "Transformation groups (editorial board)". Springer.
5. UBC PROFESSOR GARNERS PRESTIGIOUS NATIONAL AWARD
6. List of Fellows of the American Mathematical Society, retrieved 2013-06-09.
7. Speakers of the International Congress of Mathematicians, retrieved 2011-05-24
External links
• Official website
• Zinovy Reichstein at the Mathematics Genealogy Project
Authority control
International
• VIAF
Academics
• CiNii
• DBLP
• MathSciNet
• Mathematics Genealogy Project
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Zipf–Mandelbrot law
In probability theory and statistics, the Zipf–Mandelbrot law is a discrete probability distribution. Also known as the Pareto–Zipf law, it is a power-law distribution on ranked data, named after the linguist George Kingsley Zipf who suggested a simpler distribution called Zipf's law, and the mathematician Benoit Mandelbrot, who subsequently generalized it.
Zipf–Mandelbrot
Parameters $N\in \{1,2,3\ldots \}$ (integer)
$q\in [0;\infty )$ (real)
$s>0\,$ (real)
Support $k\in \{1,2,\ldots ,N\}$
PMF ${\frac {1/(k+q)^{s}}{H_{N,q,s}}}$
CDF ${\frac {H_{k,q,s}}{H_{N,q,s}}}$
Mean ${\frac {H_{N,q,s-1}}{H_{N,q,s}}}-q$
Mode $1\,$
Entropy ${\frac {s}{H_{N,q,s}}}\sum _{k=1}^{N}{\frac {\ln(k+q)}{(k+q)^{s}}}+\ln(H_{N,q,s})$
The probability mass function is given by:
$f(k;N,q,s)={\frac {1/(k+q)^{s}}{H_{N,q,s}}}$
where $H_{N,q,s}$ is given by:
$H_{N,q,s}=\sum _{i=1}^{N}{\frac {1}{(i+q)^{s}}}$
which may be thought of as a generalization of a harmonic number. In the formula, $k$ is the rank of the data, and $q$ and $s$ are parameters of the distribution. In the limit as $N$ approaches infinity, this becomes the Hurwitz zeta function $\zeta (s,q)$. For finite $N$ and $q=0$ the Zipf–Mandelbrot law becomes Zipf's law. For infinite $N$ and $q=0$ it becomes a Zeta distribution.
Applications
The distribution of words ranked by their frequency in a random text corpus is approximated by a power-law distribution, known as Zipf's law.
If one plots the frequency rank of words contained in a moderately sized corpus of text data versus the number of occurrences or actual frequencies, one obtains a power-law distribution, with exponent close to one (but see Powers, 1998 and Gelbukh & Sidorov, 2001). Zipf's law implicitly assumes a fixed vocabulary size, but the Harmonic series with s=1 does not converge, while the Zipf–Mandelbrot generalization with s>1 does. Furthermore, there is evidence that the closed class of functional words that define a language obeys a Zipf–Mandelbrot distribution with different parameters from the open classes of contentive words that vary by topic, field and register.[1]
In ecological field studies, the relative abundance distribution (i.e. the graph of the number of species observed as a function of their abundance) is often found to conform to a Zipf–Mandelbrot law.[2]
Within music, many metrics of measuring "pleasing" music conform to Zipf–Mandelbrot distributions.[3]
Notes
1. Powers, David M W (1998). "Applications and explanations of Zipf's law". New methods in language processing and computational natural language learning. Joint conference on new methods in language processing and computational natural language learning. Association for Computational Linguistics. pp. 151–160.
2. Mouillot, D; Lepretre, A (2000). "Introduction of relative abundance distribution (RAD) indices, estimated from the rank-frequency diagrams (RFD), to assess changes in community diversity". Environmental Monitoring and Assessment. Springer. 63 (2): 279–295. doi:10.1023/A:1006297211561. S2CID 102285701. Retrieved 24 Dec 2008.
3. Manaris, B; Vaughan, D; Wagner, CS; Romero, J; Davis, RB. "Evolutionary Music and the Zipf–Mandelbrot Law: Developing Fitness Functions for Pleasant Music". Proceedings of 1st European Workshop on Evolutionary Music and Art (EvoMUSART2003). 611.
References
• Mandelbrot, Benoît (1965). "Information Theory and Psycholinguistics". In B.B. Wolman and E. Nagel (ed.). Scientific psychology. Basic Books. Reprinted as
• Mandelbrot, Benoît (1968) [1965]. "Information Theory and Psycholinguistics". In R.C. Oldfield and J.C. Marchall (ed.). Language. Penguin Books.
• Powers, David M W (1998). "Applications and explanations of Zipf's law". New methods in language processing and computational natural language learning. Joint conference on new methods in language processing and computational natural language learning. Association for Computational Linguistics. pp. 151–160.
• Zipf, George Kingsley (1932). Selected Studies of the Principle of Relative Frequency in Language. Cambridge, MA: Harvard University Press.
• Van Droogenbroeck F.J., 'An essential rephrasing of the Zipf–Mandelbrot law to solve authorship attribution applications by Gaussian statistics' (2019)
External links
• Z. K. Silagadze: Citations and the Zipf–Mandelbrot's law
• NIST: Zipf's law
• W. Li's References on Zipf's law
• Gelbukh & Sidorov, 2001: Zipf and Heaps Laws’ Coefficients Depend on Language
• C++ Library for generating random Zipf–Mandelbrot deviates.
Probability distributions (list)
Discrete
univariate
with finite
support
• Benford
• Bernoulli
• beta-binomial
• binomial
• categorical
• hypergeometric
• negative
• Poisson binomial
• Rademacher
• soliton
• discrete uniform
• Zipf
• Zipf–Mandelbrot
with infinite
support
• beta negative binomial
• Borel
• Conway–Maxwell–Poisson
• discrete phase-type
• Delaporte
• extended negative binomial
• Flory–Schulz
• Gauss–Kuzmin
• geometric
• logarithmic
• mixed Poisson
• negative binomial
• Panjer
• parabolic fractal
• Poisson
• Skellam
• Yule–Simon
• zeta
Continuous
univariate
supported on a
bounded interval
• arcsine
• ARGUS
• Balding–Nichols
• Bates
• beta
• beta rectangular
• continuous Bernoulli
• Irwin–Hall
• Kumaraswamy
• logit-normal
• noncentral beta
• PERT
• raised cosine
• reciprocal
• triangular
• U-quadratic
• uniform
• Wigner semicircle
supported on a
semi-infinite
interval
• Benini
• Benktander 1st kind
• Benktander 2nd kind
• beta prime
• Burr
• chi
• chi-squared
• noncentral
• inverse
• scaled
• Dagum
• Davis
• Erlang
• hyper
• exponential
• hyperexponential
• hypoexponential
• logarithmic
• F
• noncentral
• folded normal
• Fréchet
• gamma
• generalized
• inverse
• gamma/Gompertz
• Gompertz
• shifted
• half-logistic
• half-normal
• Hotelling's T-squared
• inverse Gaussian
• generalized
• Kolmogorov
• Lévy
• log-Cauchy
• log-Laplace
• log-logistic
• log-normal
• log-t
• Lomax
• matrix-exponential
• Maxwell–Boltzmann
• Maxwell–Jüttner
• Mittag-Leffler
• Nakagami
• Pareto
• phase-type
• Poly-Weibull
• Rayleigh
• relativistic Breit–Wigner
• Rice
• truncated normal
• type-2 Gumbel
• Weibull
• discrete
• Wilks's lambda
supported
on the whole
real line
• Cauchy
• exponential power
• Fisher's z
• Kaniadakis κ-Gaussian
• Gaussian q
• generalized normal
• generalized hyperbolic
• geometric stable
• Gumbel
• Holtsmark
• hyperbolic secant
• Johnson's SU
• Landau
• Laplace
• asymmetric
• logistic
• noncentral t
• normal (Gaussian)
• normal-inverse Gaussian
• skew normal
• slash
• stable
• Student's t
• Tracy–Widom
• variance-gamma
• Voigt
with support
whose type varies
• generalized chi-squared
• generalized extreme value
• generalized Pareto
• Marchenko–Pastur
• Kaniadakis κ-exponential
• Kaniadakis κ-Gamma
• Kaniadakis κ-Weibull
• Kaniadakis κ-Logistic
• Kaniadakis κ-Erlang
• q-exponential
• q-Gaussian
• q-Weibull
• shifted log-logistic
• Tukey lambda
Mixed
univariate
continuous-
discrete
• Rectified Gaussian
Multivariate
(joint)
• Discrete:
• Ewens
• multinomial
• Dirichlet
• negative
• Continuous:
• Dirichlet
• generalized
• multivariate Laplace
• multivariate normal
• multivariate stable
• multivariate t
• normal-gamma
• inverse
• Matrix-valued:
• LKJ
• matrix normal
• matrix t
• matrix gamma
• inverse
• Wishart
• normal
• inverse
• normal-inverse
• complex
Directional
Univariate (circular) directional
Circular uniform
univariate von Mises
wrapped normal
wrapped Cauchy
wrapped exponential
wrapped asymmetric Laplace
wrapped Lévy
Bivariate (spherical)
Kent
Bivariate (toroidal)
bivariate von Mises
Multivariate
von Mises–Fisher
Bingham
Degenerate
and singular
Degenerate
Dirac delta function
Singular
Cantor
Families
• Circular
• compound Poisson
• elliptical
• exponential
• natural exponential
• location–scale
• maximum entropy
• mixture
• Pearson
• Tweedie
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Zipf's law
Zipf's law (/zɪf/, German: [ts͡ɪpf]) is an empirical law that often holds, approximately, when a list of measured values is sorted in decreasing order. It states that the value of the nth entry is inversely proportional to n.
The best known instance of Zipf's law applies to the frequency table of words in a text or corpus of natural language:
${\text{word frequency}}\propto {\frac {1}{\text{word rank}}}$
Namely, it is usually found that the most common word occurs approximately twice as often as the next common one, three times as often as the third most common, and so on. For example, in the Brown Corpus of American English text, the word "the" is the most frequently occurring word, and by itself accounts for nearly 7% of all word occurrences (69,971 out of slightly over 1 million). True to Zipf's Law, the second-place word "of" accounts for slightly over 3.5% of words (36,411 occurrences), followed by "and" (28,852).[2] It is often used in the following form, called Zipf-Mandelbrot law:
${\text{frequency}}\propto {\frac {1}{({\text{rank}}+b)^{a}}}$
where $a,b$ are fitted parameters, with $a\approx 1$, and $b\approx 2.7$.[1]
This "law" is named after the American linguist George Kingsley Zipf,[3][4][5] and is still an important concept in quantitative linguistics. It has been found to apply to many other types of data studied in the physical and social sciences.
In mathematical statistics, the concept has been formalized as the Zipfian distribution: a family of related discrete probability distributions whose rank-frequency distribution is an inverse power law relation. They are related to Benford's law and the Pareto distribution.
Some sets of time-dependent empirical data deviate somewhat from Zipf's law. Such empirical distributions are said to be quasi-Zipfian.
History
In 1913, the German physicist Felix Auerbach observed an inverse proportionality between the population sizes of cities, and their ranks when sorted by decreasing order of that variable.[6]
Zipf's law has been discovered before Zipf,[lower-alpha 1] by the French stenographer Jean-Baptiste Estoup' Gammes Stenographiques (4th ed) in 1916,[7] with G. Dewey in 1923,[8] and with E. Condon in 1928.[9]
The same relation for frequencies of words in natural language texts was observed by George Zipf in 1932,[4] but he never claimed to have originated it. In fact, Zipf didn't like mathematics. In his 1932 publication,[10] the author speaks with disdain about mathematical involvement in linguistics, a. o. ibidem, p. 21: (…) let me say here for the sake of any mathematician who may plan to formulate the ensuing data more exactly, the ability of the highly intense positive to become the highly intense negative, in my opinion, introduces the devil into the formula in the form of √(-i). The only mathematical expression Zipf used looks like a.b2 = constant, which he "borrowed" from Alfred J. Lotka's 1926 publication.[11]
The same relationship was found to occur in many other contexts, and for other variables besides frequency.[1] For example, when corporations are ranked by decreasing size, their sizes are found to be inversely proportional to the rank.[12] The same relation is found for personal incomes (where it is called Pareto principle[13]), number of people watching the same TV channel,[14] notes in music,[15] cells transcriptomes[16][17] and more.
Formal definition
Zipf's law
Probability mass function
Zipf PMF for N = 10 on a log–log scale. The horizontal axis is the index k . (Note that the function is only defined at integer values of k. The connecting lines do not indicate continuity.)
Cumulative distribution function
Zipf CDF for N = 10. The horizontal axis is the index k . (Note that the function is only defined at integer values of k. The connecting lines do not indicate continuity.)
Parameters $s\geq 0\,$ (real)
$N\in \{1,2,3\ldots \}$ (integer)
Support $k\in \{1,2,\ldots ,N\}$
PMF ${\frac {1/k^{s}}{H_{N,s}}}$ where HN,s is the Nth generalized harmonic number
CDF ${\frac {H_{k,s}}{H_{N,s}}}$
Mean ${\frac {H_{N,s-1}}{H_{N,s}}}$
Mode $1\,$
Variance ${\frac {H_{N,s-2}}{H_{N,s}}}-{\frac {H_{N,s-1}^{2}}{H_{N,s}^{2}}}$
Entropy ${\frac {s}{H_{N,s}}}\sum \limits _{k=1}^{N}{\frac {\ln(k)}{k^{s}}}+\ln(H_{N,s})$
MGF ${\frac {1}{H_{N,s}}}\sum \limits _{n=1}^{N}{\frac {e^{nt}}{n^{s}}}$
CF ${\frac {1}{H_{N,s}}}\sum \limits _{n=1}^{N}{\frac {e^{int}}{n^{s}}}$
Formally, the Zipf distribution on N elements assigns to the element of rank k (counting from 1) the probability
$f(k;N)={\frac {1}{H_{N}}}\,{\frac {1}{k}}$
where HN is a normalization constant, the Nth harmonic number:
$H_{N}=\sum _{k=1}^{N}{\frac {1}{k}}\ .$
The distribution is sometimes generalized to an inverse power law with exponent s instead of 1.[18] Namely,
$f(k;s,N)={\frac {1}{H_{s,N}}}\,{\frac {1}{k^{s}}}$
where Hs,N is a generalized harmonic number
$H_{s,N}=\sum _{k=1}^{N}{\frac {1}{k^{s}}}\ .$
The generalized Zipf distribution can be extended to infinitely many items (N = ∞) only if the exponent s exceeds 1. In that case, the normalization constant Hs,N becomes Riemann's zeta function,
$\zeta (s)=\sum _{k=1}^{\infty }{\frac {1}{k^{s}}}<\infty \ .$
If the exponent s is 1 or less, the normalization constant Hs,N diverges as N tends to infinity.
Empirical testing
Empirically, a data set can be tested to see whether Zipf's law applies by checking the goodness of fit of an empirical distribution to the hypothesized power law distribution with a Kolmogorov–Smirnov test, and then comparing the (log) likelihood ratio of the power law distribution to alternative distributions like an exponential distribution or lognormal distribution.[19]
Zipf's law can be visuallized by plotting the item frequency data on a log-log graph, with the axes being the logarithm of rank order, and logarithm of frequency. The data conform to Zipf's law with exponent s to the extent that the plot approximates a linear (more precisely, affine) function with slope −s. For exponent s = 1, one can also plot the reciprocal of the frequency (mean interword interval) against rank, or the reciprocal of rank against frequency, and compare the result with the line through the origin with slope 1.[3]
Statistical explanations
Although Zipf's Law holds for most natural languages, even some non-natural ones like Esperanto,[20] the reason is still not well understood.[21] Recent reviews of generative processes for Zipf's law include.[22][23]
However, it may be partially explained by the statistical analysis of randomly generated texts. Wentian Li has shown that in a document in which each character has been chosen randomly from a uniform distribution of all letters (plus a space character), the "words" with different lengths follow the macro-trend of the Zipf's law (the more probable words are the shortest with equal probability).[24] In 1959, Vitold Belevitch observed that if any of a large class of well-behaved statistical distributions (not only the normal distribution) is expressed in terms of rank and expanded into a Taylor series, the first-order truncation of the series results in Zipf's law. Further, a second-order truncation of the Taylor series resulted in Mandelbrot's law.[25][26]
The principle of least effort is another possible explanation: Zipf himself proposed that neither speakers nor hearers using a given language want to work any harder than necessary to reach understanding, and the process that results in approximately equal distribution of effort leads to the observed Zipf distribution.[5][27]
A minimal explanation assumes that words are generated by monkeys typing randomly. If language is generated by a single monkey typing randomly, with fixed and nonzero probability of hitting each letter key or white space, then the words (letter strings separated by white spaces) produced by the monkey follows Zipf's law.[28]
Another possible cause for the Zipf distribution is a preferential attachment process, in which the value x of an item tends to grow at a rate proportional to x (intuitively, "the rich get richer" or "success breeds success"). Such a growth process results in the Yule–Simon distribution, which has been shown to fit word frequency versus rank in language[29] and population versus city rank[30] better than Zipf's law. It was originally derived to explain population versus rank in species by Yule, and applied to cities by Simon.
A similar explanation is based on atlas models, systems of exchangeable positive-valued diffusion processes with drift and variance parameters that depend only on the rank of the process. It has been shown mathematically that Zipf's law holds for Atlas models that satisfy certain natural regularity conditions.[31][32] Quasi-Zipfian distributions can result from quasi-Atlas models.
Related laws
A generalization of Zipf's law is the Zipf–Mandelbrot law, proposed by Benoit Mandelbrot, whose frequencies are:
$f(k;N,q,s)={\frac {1}{C}}\,{\frac {(k+q)^{s}}{.}}\,$
The constant C is the Hurwitz zeta function evaluated at s.
Zipfian distributions can be obtained from Pareto distributions by an exchange of variables.[18]
The Zipf distribution is sometimes called the discrete Pareto distribution[33] because it is analogous to the continuous Pareto distribution in the same way that the discrete uniform distribution is analogous to the continuous uniform distribution.
The tail frequencies of the Yule–Simon distribution are approximately
$f(k;\rho )\approx {\frac {[{\text{constant}}]}{k^{\rho +1}}}$
for any choice of ρ > 0.
In the parabolic fractal distribution, the logarithm of the frequency is a quadratic polynomial of the logarithm of the rank. This can markedly improve the fit over a simple power-law relationship.[34] Like fractal dimension, it is possible to calculate Zipf dimension, which is a useful parameter in the analysis of texts.[35]
It has been argued that Benford's law is a special bounded case of Zipf's law,[34] with the connection between these two laws being explained by their both originating from scale invariant functional relations from statistical physics and critical phenomena.[36] The ratios of probabilities in Benford's law are not constant. The leading digits of data satisfying Zipf's law with s = 1 satisfy Benford's law.
$n$ Benford's law: $P(n)=$
$\log _{10}(n+1)-\log _{10}(n)$
${\frac {\log(P(n)/P(n-1))}{\log(n/(n-1))}}$
1 0.30103000
2 0.17609126 −0.7735840
3 0.12493874 −0.8463832
4 0.09691001 −0.8830605
5 0.07918125 −0.9054412
6 0.06694679 −0.9205788
7 0.05799195 −0.9315169
8 0.05115252 −0.9397966
9 0.04575749 −0.9462848
Occurrences
City sizes
Following Auerbach's 1913 observation, there has been substantial examination of Zipf's law for city sizes.[37] However, more recent empirical[38][39] and theoretical[40] studies have challenged the relevance of Zipf's law for cities.
Word frequencies in natural languages
In many texts in human languages, word frequencies approximately follow a Zipf distribution with exponent s close to 1: that is, the most common word occurs about n times the nth most common one.
The actual rank-frequency plot of a natural language text deviates in some extent from the ideal Zipf distribution, especially at the two ends of the range. The deviations may depend on the language, on the topic of the text, on the author, on whether the text was translated from another language, and on the spelling rules used. Some deviation is inevitable because of sampling error.
At the low-frequency end, where the rank approaches N, the plot takes a staircase shape, because each word can occur only an integer number of times.
• Zipf's law plots for several languages
• Texts in German (1669), Russian (1972), French (1865), Italian (1840), and Medieval English (1460).
• Cervantes' Don Quixote Part I (Spanish, 1605) and Assis's Dom Casmurro (Portuguese, 1899).
• Ge'ez (14th century), Arabic (~650 CE), Hebrew (500-800 CE), all with vowels.
• Lhasa Tibetan, Chinese, Vietnamese, all with separated syllables.
• Biblical texts: Pentateuch from the Latin Vulgate and Russian Synodal Bible, the four Gospels from the Byzantine Greek Majority version
• Cervantes's Don Quixote, Part I (1605) and Part II (1615).
• First five books of the Old Testament (the Torah) in Hebrew, with vowels.
• First five books of the Old Testament (the Pentateuch) in the Latin Vulgate version.
• First four books of the New Testament (the Gospels) in the Latin Vulgate version.
In some Romance languages, the frequencies of the dozen or so most frequent words deviate significantly from the ideal Zipf distribution, because of those words include articles inflected for grammatical gender and number.
In many East Asian languages, such as Chinese, Lhasa Tibetan, and Vietnamese, each "word" consists of a single syllable; a word of English being often translated to a compound of two such syllables. The rank-frequency table for those "words" deviates significantly from the ideal Zipf law, at both ends of the range.
Even in English, the deviations from the ideal Zipf's law become more apparent as one examines large collections of texts. Analysis of a corpus of 30,000 English texts showed that only about 15% of the texts in have a good fit to Zipf's law. Slight changes in the definition of Zipf's law can increase this percentage up to close to 50%.[41]
In these cases, the observed frequency-rank relation can be modeled more accurately as by separate Zipf–Mandelbrot laws distributions for different subsets or subtypes of words. This is the case for the frequency-rank plot of the first 10 million words of the English Wikipedia. In particular, the frequencies of the closed class of function words in English is better described with s lower than 1, while open-ended vocabulary growth with document size and corpus size require s greater than 1 for convergence of the Generalized Harmonic Series.[3]
When a text is encrypted in such a way that every occurrence of each distinct plaintext word is always mapped to the same encrypted word (as in the case of simple substitution ciphers, like the Caesar ciphers, or simple codebook ciphers), the frequency-rank distribution is not affected. On the other hand, if separate occurrences of the same word may be mapped to two or more different words (as happens with the Vigenère cipher), the Zipf distribution will typically have a flat part at the high-frequency end.
Applications
Zipf's law has been used for extraction of parallel fragments of texts out of comparable corpora.[42] Zipf's law has also been used in the search for extraterrestrial intelligence.[43][44]
The frequency-rank word distribution is often characteristic of the author and changes little over time. This feature has been used in the analysis of texts for authorship attribution.[45][46]
The word-like sign groups of the 15th-century codex Voynich Manuscript have been found to satisfy Zipf's law, suggesting that text is most likely not a hoax but rather written in an obscure language or cipher.[47][48]
See also
• 1% rule (Internet culture) – Hypothesis that more people will lurk in a virtual community than will participatePages displaying short descriptions of redirect targets
• Benford's law – Observation that in many real-life datasets, the leading digit is likely to be small
• Bradford's law – Pattern of references in science journals
• Brevity law – Linguistics law
• Demographic gravitation
• Frequency list – Bare list of a language's words in corpus linguisticsPages displaying short descriptions of redirect targets
• Gibrat's law – Economic principle
• Hapax legomenon – Word that only appears once in a given text or record
• Heaps' law – Heuristic for distinct words in a document
• King effect – Phenomenon in statistics where highest-ranked data points are outliers
• Long tail – Feature of some statistical distributions
• Lorenz curve – Graphical representation of the distribution of income or of wealth
• Lotka's law – An application of Zipf's law describing the frequency of publication by authors in any given field
• Menzerath's law – Linguistic law
• Pareto distribution – Probability distribution
• Pareto principle – Statistical principle about ratio of effects to causes, a.k.a. the "80–20 rule"
• Price's law – Historian of SciencePages displaying short descriptions of redirect targets
• Principle of least effort – Idea that agents prefer to do what's easiest
• Rank-size distribution – distribution of size by rankPages displaying wikidata descriptions as a fallback
• Stigler's law of eponymy – Observation that no scientific discovery is named after its discoverer
Notes
1. as Zipf acknowledged[5]: 546
References
1. Piantadosi, Steven (March 25, 2014). "Zipf's word frequency law in natural language: A critical review and future directions". Psychon Bull Rev. 21 (5): 1112–1130. doi:10.3758/s13423-014-0585-6. PMC 4176592. PMID 24664880.
2. Fagan, Stephen; Gençay, Ramazan (2010), "An introduction to textual econometrics", in Ullah, Aman; Giles, David E. A. (eds.), Handbook of Empirical Economics and Finance, CRC Press, pp. 133–153, ISBN 9781420070361. P. 139: "For example, in the Brown Corpus, consisting of over one million words, half of the word volume consists of repeated uses of only 135 words."
3. Powers, David M W (1998). Applications and explanations of Zipf's law. Joint conference on new methods in language processing and computational natural language learning. Association for Computational Linguistics. pp. 151–160.
4. George K. Zipf (1935): The Psychobiology of Language. Houghton-Mifflin.
5. George K. Zipf (1949). Human Behavior and the Principle of Least Effort. Cambridge, Massachusetts: Addison-Wesley. p. 1.
6. Auerbach F. (1913) Das Gesetz der Bevölkerungskonzentration. Petermann’s Geographische Mitteilungen 59, 74–76
7. Christopher D. Manning, Hinrich Schütze Foundations of Statistical Natural Language Processing, MIT Press (1999), ISBN 978-0-262-13360-9, p. 24
8. Dewey, Godfrey. Relativ frequency of English speech sounds. Harvard University Press, 1923.
9. Condon, EDWARD U. "Statistics of vocabulary." Science 67.1733 (1928): 300-300.
10. George K. Zipf (1932): Selected Studies on the Principle of Relative Frequency in Language. Harvard, MA: Harvard University Press.
11. Zipf, George Kingsley (1942). "The Unity of Nature, Least-Action, and Natural Social Science". Sociometry. 5 (1): 48–62. doi:10.2307/2784953. ISSN 0038-0431. JSTOR 2784953.
12. Axtell, Robert L (2001): Zipf distribution of US firm sizes, Science, 293, 5536, 1818, American Association for the Advancement of Science.
13. Sandmo, Agnar (2015-01-01), Atkinson, Anthony B.; Bourguignon, François (eds.), Chapter 1 - The Principal Problem in Political Economy: Income Distribution in the History of Economic Thought, Handbook of Income Distribution, vol. 2, Elsevier, pp. 3–65, doi:10.1016/B978-0-444-59428-0.00002-3, retrieved 2023-07-11
14. M. Eriksson, S.M. Hasibur Rahman, F. Fraille, M. Sjöström, Efficient Interactive Multicast over DVB-T2 - Utilizing Dynamic SFNs and PARPS Archived 2014-05-02 at the Wayback Machine, 2013 IEEE International Conference on Computer and Information Technology (BMSB'13), London, UK, June 2013. Suggests a heterogeneous Zipf-law TV channel-selection model
15. Zanette, Damián H. (June 7, 2004). "Zipf's law and the creation of musical context". arXiv:cs/0406015.
16. Lazzardi, Silvia; Valle, Filippo; Mazzolini, Andrea; Scialdone, Antonio; Caselle, Michele; Osella, Matteo (2021-06-17). "Emergent Statistical Laws in Single-Cell Transcriptomic Data". bioRxiv: 2021–06.16.448706. doi:10.1101/2021.06.16.448706. S2CID 235482777. Retrieved 2021-06-18.
17. Ramu Chenna, Toby Gibson; Evaluation of the Suitability of a Zipfian Gap Model for Pairwise Sequence Alignment, International Conference on Bioinformatics Computational Biology: 2011.
18. Adamic, Lada A. (2000). Zipf, power-laws, and Pareto - a ranking tutorial (Report). Hewlett-Packard Company. Archived from the original on 2007-10-26. "originally published". www.parc.xerox.com. Xerox Corporation.
19. Clauset, A., Shalizi, C. R., & Newman, M. E. J. (2009). Power-Law Distributions in Empirical Data. SIAM Review, 51(4), 661–703. doi:10.1137/070710111
20. Bill Manaris; Luca Pellicoro; George Pothering; Harland Hodges (13 February 2006). Investigating Esperanto's statistical proportions relative to other languages using neural networks and Zipf's law (PDF). Artificial Intelligence and Applications. Innsbruck, Austria. pp. 102–108. Archived from the original (PDF) on 5 March 2016.
21. Léon Brillouin, La science et la théorie de l'information, 1959, réédité en 1988, traduction anglaise rééditée en 2004
22. Mitzenmacher, Michael (January 2004). "A Brief History of Generative Models for Power Law and Lognormal Distributions". Internet Mathematics. 1 (2): 226–251. doi:10.1080/15427951.2004.10129088. ISSN 1542-7951. S2CID 1671059.
23. Simkin, M. V.; Roychowdhury, V. P. (2011-05-01). "Re-inventing Willis". Physics Reports. 502 (1): 1–35. arXiv:physics/0601192. doi:10.1016/j.physrep.2010.12.004. ISSN 0370-1573. S2CID 88517297.
24. Wentian Li (1992). "Random Texts Exhibit Zipf's-Law-Like Word Frequency Distribution". IEEE Transactions on Information Theory. 38 (6): 1842–1845. CiteSeerX 10.1.1.164.8422. doi:10.1109/18.165464.
25. Belevitch V (18 December 1959). "On the statistical laws of linguistic distributions" (PDF). Annales de la Société Scientifique de Bruxelles. I. 73: 310–326.
26. Neumann, Peter G. "Statistical metalinguistics and Zipf/Pareto/Mandelbrot", SRI International Computer Science Laboratory, accessed and archived 29 May 2011.
27. Ramon Ferrer i Cancho & Ricard V. Sole (2003). "Least effort and the origins of scaling in human language". Proceedings of the National Academy of Sciences of the United States of America. 100 (3): 788–791. Bibcode:2003PNAS..100..788C. doi:10.1073/pnas.0335980100. PMC 298679. PMID 12540826.
28. Conrad, B.; Mitzenmacher, M. (July 2004). "Power laws for monkeys typing randomly: the case of unequal probabilities". IEEE Transactions on Information Theory. 50 (7): 1403–1414. doi:10.1109/TIT.2004.830752. ISSN 1557-9654. S2CID 8913575.
29. Lin, Ruokuang; Ma, Qianli D. Y.; Bian, Chunhua (2014). "Scaling laws in human speech, decreasing emergence of new words and a generalized model". arXiv:1412.4846 [cs.CL].
30. Vitanov, Nikolay K.; Ausloos, Marcel; Bian, Chunhua (2015). "Test of two hypotheses explaining the size of populations in a system of cities". Journal of Applied Statistics. 42 (12): 2686–2693. arXiv:1506.08535. Bibcode:2015JApSt..42.2686V. doi:10.1080/02664763.2015.1047744. S2CID 10599428.
31. Ricardo T. Fernholz; Robert Fernholz (December 2020). "Zipf's law for atlas models". Journal of Applied Probability. 57 (4): 1276–1297. doi:10.1017/jpr.2020.64. S2CID 146808080.
32. Terence Tao (2012). "E Pluribus Unum: From Complexity, Universality". Daedalus. 141 (3): 23–34. doi:10.1162/DAED_a_00158. S2CID 14535989.
33. N. L. Johnson; S. Kotz & A. W. Kemp (1992). Univariate Discrete Distributions (second ed.). New York: John Wiley & Sons, Inc. ISBN 978-0-471-54897-3., p. 466.
34. Johan Gerard van der Galien (2003-11-08). "Factorial randomness: the Laws of Benford and Zipf with respect to the first digit distribution of the factor sequence from the natural numbers". Archived from the original on 2007-03-05. Retrieved 8 July 2016.
35. Eftekhari, Ali (2006). "Fractal geometry of texts: An initial application to the works of Shakespeare". Journal of Quantitative Linguistic. 13 (2–3): 177–193. doi:10.1080/09296170600850106. S2CID 17657731.
36. Pietronero, L.; Tosatti, E.; Tosatti, V.; Vespignani, A. (2001). "Explaining the uneven distribution of numbers in nature: The laws of Benford and Zipf". Physica A. 293 (1–2): 297–304. Bibcode:2001PhyA..293..297P. doi:10.1016/S0378-4371(00)00633-6.
37. Gabaix, Xavier (1999). "Zipf's Law for Cities: An Explanation". The Quarterly Journal of Economics. 114 (3): 739–767. doi:10.1162/003355399556133. ISSN 0033-5533. JSTOR 2586883.
38. Arshad, Sidra; Hu, Shougeng; Ashraf, Badar Nadeem (2018-02-15). "Zipf's law and city size distribution: A survey of the literature and future research agenda". Physica A: Statistical Mechanics and Its Applications. 492: 75–92. Bibcode:2018PhyA..492...75A. doi:10.1016/j.physa.2017.10.005. ISSN 0378-4371.
39. Gan, Li; Li, Dong; Song, Shunfeng (2006-08-01). "Is the Zipf law spurious in explaining city-size distributions?". Economics Letters. 92 (2): 256–262. doi:10.1016/j.econlet.2006.03.004. ISSN 0165-1765.
40. Verbavatz, Vincent; Barthelemy, Marc (November 2020). "The growth equation of cities". Nature. 587 (7834): 397–401. arXiv:2011.09403. Bibcode:2020Natur.587..397V. doi:10.1038/s41586-020-2900-x. ISSN 1476-4687. PMID 33208958. S2CID 227012701.
41. Moreno-Sánchez, I.; Font-Clos, F.; Corral, A. (2016). "Large-scale analysis of Zipf's Law in English texts". PLOS ONE. 11 (1): e0147073. arXiv:1509.04486. Bibcode:2016PLoSO..1147073M. doi:10.1371/journal.pone.0147073. PMC 4723055. PMID 26800025.
42. Mohammadi, Mehdi (2016). "Parallel Document Identification using Zipf's Law" (PDF). Proceedings of the Ninth Workshop on Building and Using Comparable Corpora. LREC 2016. Portorož, Slovenia. pp. 21–25. Archived (PDF) from the original on 2018-03-23.
43. Doyle, Laurance R.; Mao, Tianhua (2016-11-18). "Why Alien Language Would Stand Out Among All the Noise of the Universe". Nautilus Quarterly.
44. Kershenbaum, Arik (2021-03-16). The Zoologist's Guide to the Galaxy: What Animals on Earth Reveal About Aliens--and Ourselves. Penguin. pp. 251–256. ISBN 978-1-9848-8197-7. OCLC 1242873084.
45. Frans J. Van Droogenbroeck (2016): Handling the Zipf distribution in computerized authorship attribution
46. Frans J. Van Droogenbroeck (2019): An essential rephrasing of the Zipf-Mandelbrot law to solve authorship attribution applications by Gaussian statistics
47. Boyle, Rebecca. "Mystery text's language-like patterns may be an elaborate hoax". New Scientist. Retrieved 2022-02-25.
48. Montemurro, Marcelo A.; Zanette, Damián H. (2013-06-21). "Keywords and Co-Occurrence Patterns in the Voynich Manuscript: An Information-Theoretic Analysis". PLOS ONE. 8 (6): e66344. Bibcode:2013PLoSO...866344M. doi:10.1371/journal.pone.0066344. ISSN 1932-6203. PMC 3689824. PMID 23805215.
Further reading
• Alexander Gelbukh and Grigori Sidorov (2001) "Zipf and Heaps Laws’ Coefficients Depend on Language". Proc. CICLing-2001, Conference on Intelligent Text Processing and Computational Linguistics, February 18–24, 2001, Mexico City. Lecture Notes in Computer Science N 2004, ISSN 0302-9743, ISBN 3-540-41687-0, Springer-Verlag: 332–335.
• Kali R. (2003) "The city as a giant component: a random graph approach to Zipf's law," Applied Economics Letters 10: 717–720(4)
• Shyklo A. (2017); Simple Explanation of Zipf's Mystery via New Rank-Share Distribution, Derived from Combinatorics of the Ranking Process, Available at SSRN: https://ssrn.com/abstract=2918642.
External links
Library resources about
Zipf's law
• Resources in your library
• Resources in other libraries
Wikimedia Commons has media related to Zipf's law.
• Strogatz, Steven (2009-05-29). "Guest Column: Math and the City". The New York Times. Archived from the original on 2015-09-27. Retrieved 2009-05-29.—An article on Zipf's law applied to city populations
• Seeing Around Corners (Artificial societies turn up Zipf's law)
• PlanetMath article on Zipf's law
• Distributions de type "fractal parabolique" dans la Nature (French, with English summary) Archived 2004-10-24 at the Wayback Machine
• An analysis of income distribution
• Zipf List of French words Archived 2007-06-23 at the Wayback Machine
• Zipf list for English, French, Spanish, Italian, Swedish, Icelandic, Latin, Portuguese and Finnish from Gutenberg Project and online calculator to rank words in texts Archived 2011-04-08 at the Wayback Machine
• Citations and the Zipf–Mandelbrot's law
• Zipf's Law examples and modelling (1985)
• Complex systems: Unzipping Zipf's law (2011)
• Benford’s law, Zipf’s law, and the Pareto distribution by Terence Tao.
• "Zipf law", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Probability distributions (list)
Discrete
univariate
with finite
support
• Benford
• Bernoulli
• beta-binomial
• binomial
• categorical
• hypergeometric
• negative
• Poisson binomial
• Rademacher
• soliton
• discrete uniform
• Zipf
• Zipf–Mandelbrot
with infinite
support
• beta negative binomial
• Borel
• Conway–Maxwell–Poisson
• discrete phase-type
• Delaporte
• extended negative binomial
• Flory–Schulz
• Gauss–Kuzmin
• geometric
• logarithmic
• mixed Poisson
• negative binomial
• Panjer
• parabolic fractal
• Poisson
• Skellam
• Yule–Simon
• zeta
Continuous
univariate
supported on a
bounded interval
• arcsine
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Zlil Sela
Zlil Sela is an Israeli mathematician working in the area of geometric group theory. He is a Professor of Mathematics at the Hebrew University of Jerusalem. Sela is known for the solution[1] of the isomorphism problem for torsion-free word-hyperbolic groups and for the solution of the Tarski conjecture about equivalence of first-order theories of finitely generated non-abelian free groups.[2]
Biographical data
Sela received his Ph.D. in 1991 from the Hebrew University of Jerusalem, where his doctoral advisor was Eliyahu Rips. Prior to his current appointment at the Hebrew University, he held an Associate Professor position at Columbia University in New York.[3] While at Columbia, Sela won the Sloan Fellowship from the Sloan Foundation.[3][4]
Sela gave an Invited Address at the 2002 International Congress of Mathematicians in Beijing.[2][5] He gave a plenary talk at the 2002 annual meeting of the Association for Symbolic Logic,[6] and he delivered an AMS Invited Address at the October 2003 meeting of the American Mathematical Society[7] and the 2005 Tarski Lectures at the University of California at Berkeley.[8] He was also awarded the 2003 Erdős Prize from the Israel Mathematical Union.[9] Sela also received the 2008 Carol Karp Prize from the Association for Symbolic Logic for his work on the Tarski conjecture and on discovering and developing new connections between model theory and geometric group theory.[10][11]
Mathematical contributions
Sela's early important work was his solution[1] in mid-1990s of the isomorphism problem for torsion-free word-hyperbolic groups. The machinery of group actions on real trees, developed by Eliyahu Rips, played a key role in Sela's approach. The solution of the isomorphism problem also relied on the notion of canonical representatives for elements of hyperbolic groups, introduced by Rips and Sela in a joint 1995 paper.[12] The machinery of the canonical representatives allowed Rips and Sela to prove[12] algorithmic solvability of finite systems of equations in torsion-free hyperbolic groups, by reducing the problem to solving equations in free groups, where the Makanin–Razborov algorithm can be applied. The technique of canonical representatives was later generalized by Dahmani[13] to the case of relatively hyperbolic groups and played a key role in the solution of the isomorphism problem for toral relatively hyperbolic groups.[14]
In his work on the isomorphism problem Sela also introduced and developed the notion of a JSJ-decomposition for word-hyperbolic groups,[15] motivated by the notion of a JSJ decomposition for 3-manifolds. A JSJ-decomposition is a representation of a word-hyperbolic group as the fundamental group of a graph of groups which encodes in a canonical way all possible splittings over infinite cyclic subgroups. The idea of JSJ-decomposition was later extended by Rips and Sela to torsion-free finitely presented groups[16] and this work gave rise a systematic development of the JSJ-decomposition theory with many further extensions and generalizations by other mathematicians.[17][18][19][20] Sela applied a combination of his JSJ-decomposition and real tree techniques to prove that torsion-free word-hyperbolic groups are Hopfian.[21] This result and Sela's approach were later generalized by others to finitely generated subgroups of hyperbolic groups[22] and to the setting of relatively hyperbolic groups.
Sela's most important work came in early 2000s when he produced a solution to a famous Tarski conjecture. Namely, in a long series of papers,[23][24][25][26][27][28][29] he proved that any two non-abelian finitely generated free groups have the same first-order theory. Sela's work relied on applying his earlier JSJ-decomposition and real tree techniques as well as developing new ideas and machinery of "algebraic geometry" over free groups.
Sela pushed this work further to study first-order theory of arbitrary torsion-free word-hyperbolic groups and to characterize all groups that are elementarily equivalent to (that is, have the same first-order theory as) a given torsion-free word-hyperbolic group. In particular, his work implies that if a finitely generated group G is elementarily equivalent to a word-hyperbolic group then G is word-hyperbolic as well.
Sela also proved that the first-order theory of a finitely generated free group is stable in the model-theoretic sense, providing a brand-new and qualitatively different source of examples for the stability theory.
An alternative solution for the Tarski conjecture has been presented by Olga Kharlampovich and Alexei Myasnikov.[30][31][32][33]
The work of Sela on first-order theory of free and word-hyperbolic groups substantially influenced the development of geometric group theory, in particular by stimulating the development and the study of the notion of limit groups and of relatively hyperbolic groups.[34]
Sela's classification theorem
Theorem. Two non-abelian torsion-free hyperbolic groups are elementarily equivalent if and only if their cores are isomorphic.[35]
Published work
• Sela, Zlil; Rips, Eliyahu (1995), "Canonical representatives and equations in hyperbolic groups", Inventiones Mathematicae, 120 (3): 489–512, Bibcode:1995InMat.120..489R, doi:10.1007/BF01241140, MR 1334482, S2CID 121404710
• Sela, Zlil (1995), "The isomorphism problem for hyperbolic groups", Annals of Mathematics, Second Series, 141 (2): 217–283, doi:10.2307/2118520, JSTOR 2118520, MR 1324134
• Sela, Zlil (1997), "Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups. II.", Geometric and Functional Analysis, 7 (3): 561–593, doi:10.1007/s000390050019, MR 1466338, S2CID 120486267
• Sela, Zlil; Rips, Eliyahu (1997), "Cyclic splittings of finitely presented groups and the canonical JSJ decomposition", Annals of Mathematics, Second Series, 146 (1): 53–109, doi:10.2307/2951832, JSTOR 2951832, MR 1469317
• Sela, Zlil (1997), "Acylindrical accessibility for groups", Inventiones Mathematicae, 129 (3): 527–565, Bibcode:1997InMat.129..527S, doi:10.1007/s002220050172, S2CID 122548154 (Sela's theorem on acylindrical accessibility for groups)[36]
• Sela, Zlil (2001), "Diophantine geometry over groups. I. Makanin-Razborov diagrams" (PDF), Publications Mathématiques de l'IHÉS, 93 (1): 31–105, doi:10.1007/s10240-001-8188-y, MR 1863735, S2CID 51799226
• Sela, Zlil (2003), "Diophantine geometry over groups. II. Completions, closures and formal solutions", Israel Journal of Mathematics, 134 (1): 173–254, doi:10.1007/BF02787407, MR 1972179
• Sela, Zlil (2006), "Diophantine geometry over groups. VI. The elementary theory of a free group", Geometric and Functional Analysis, 16 (3): 707–730, doi:10.1007/s00039-006-0565-8, MR 2238945, S2CID 123197664
See also
• Geometric group theory
• Stable theory
• Free group
• Word-hyperbolic group
• Group isomorphism problem
• Real trees
• JSJ decomposition
References
1. Z. Sela. "The isomorphism problem for hyperbolic groups. I." Annals of Mathematics (2), vol. 141 (1995), no. 2, pp. 217–283.
2. Z. Sela. Diophantine geometry over groups and the elementary theory of free and hyperbolic groups. Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pp. 87 92, Higher Ed. Press, Beijing, 2002. ISBN 7-04-008690-5
3. Faculty Members Win Fellowships Columbia University Record, May 15, 1996, Vol. 21, No. 27.
4. Sloan Fellowships Awarded Notices of the American Mathematical Society, vol. 43 (1996), no. 7, pp. 781–782
5. Invited Speakers for ICM2002. Notices of the American Mathematical Society, vol. 48, no. 11, December 2001; pp. 1343 1345
6. The 2002 annual meeting of the Association for Symbolic Logic. Bulletin of Symbolic Logic, vol. 9 (2003), pp. 51–70
7. AMS Meeting at Binghamton, New York. Notices of the American Mathematical Society, vol. 50 (2003), no. 9, p. 1174
8. 2005 Tarski Lectures. Department of Mathematics, University of California at Berkeley. Accessed September 14, 2008.
9. Erdős Prize. Israel Mathematical Union. Accessed September 14, 2008
10. Karp Prize Recipients. Archived 2008-05-13 at the Wayback Machine Association for Symbolic Logic. Accessed September 13, 2008
11. ASL Karp and Sacks Prizes Awarded, Notices of the American Mathematical Society, vol. 56 (2009), no. 5, p. 638
12. Z. Sela, and E. Rips. Canonical representatives and equations in hyperbolic groups, Inventiones Mathematicae vol. 120 (1995), no. 3, pp. 489–512
13. François Dahmani. "Accidental parabolics and relatively hyperbolic groups." Israel Journal of Mathematics, vol. 153 (2006), pp. 93–127
14. François Dahmani, and Daniel Groves, "The isomorphism problem for toral relatively hyperbolic groups". Publications Mathématiques de l'IHÉS, vol. 107 (2008), pp. 211–290
15. Z. Sela. "Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups. II." Geometric and Functional Analysis, vol. 7 (1997), no. 3, pp. 561–593
16. E. Rips, and Z. Sela. "Cyclic splittings of finitely presented groups and the canonical JSJ decomposition." Annals of Mathematics (2), vol. 146 (1997), no. 1, pp. 53–109
17. M. J. Dunwoody, and M. E. Sageev. "JSJ-splittings for finitely presented groups over slender groups." Inventiones Mathematicae, vol. 135 (1999), no. 1, pp. 25 44
18. P. Scott and G. A. Swarup. "Regular neighbourhoods and canonical decompositions for groups." Electronic Research Announcements of the American Mathematical Society, vol. 8 (2002), pp. 20–28
19. B. H. Bowditch. "Cut points and canonical splittings of hyperbolic groups." Acta Mathematica, vol. 180 (1998), no. 2, pp. 145–186
20. K. Fujiwara, and P. Papasoglu, "JSJ-decompositions of finitely presented groups and complexes of groups." Geometric and Functional Analysis, vol. 16 (2006), no. 1, pp. 70–125
21. Zlil Sela, "Endomorphisms of hyperbolic groups. I. The Hopf property." Topology, vol. 38 (1999), no. 2, pp. 301–321
22. Inna Bumagina, "The Hopf property for subgroups of hyperbolic groups." Geometriae Dedicata, vol. 106 (2004), pp. 211–230
23. Z. Sela. "Diophantine geometry over groups. I. Makanin-Razborov diagrams." Publications Mathématiques. Institut de Hautes Études Scientifiques, vol. 93 (2001), pp. 31–105
24. Z. Sela. Diophantine geometry over groups. II. Completions, closures and formal solutions. Israel Journal of Mathematics, vol. 134 (2003), pp. 173–254
25. Z. Sela. "Diophantine geometry over groups. III. Rigid and solid solutions." Israel Journal of Mathematics, vol. 147 (2005), pp. 1–73
26. Z. Sela. "Diophantine geometry over groups. IV. An iterative procedure for validation of a sentence." Israel Journal of Mathematics, vol. 143 (2004), pp. 1–130
27. Z. Sela. "Diophantine geometry over groups. V1. Quantifier elimination. I." Israel Journal of Mathematics, vol. 150 (2005), pp. 1–197
28. Z. Sela. "Diophantine geometry over groups. V2. Quantifier elimination. II." Geometric and Functional Analysis, vol. 16 (2006), no. 3, pp. 537–706
29. Z. Sela. "Diophantine geometry over groups. VI. The elementary theory of a free group." Geometric and Functional Analysis, vol. 16 (2006), no. 3, pp. 707–730
30. O. Kharlampovich, and A. Myasnikov. "Tarski's problem about the elementary theory of free groups has a positive solution." Electronic Research Announcements of the American Mathematical Society, vol. 4 (1998), pp. 101–108
31. O. Kharlampovich, and A. Myasnikov. Implicit function theorem over free groups. Journal of Algebra, vol. 290 (2005), no. 1, pp. 1–203
32. O. Kharlampovich, and A. Myasnikov. "Algebraic geometry over free groups: lifting solutions into generic points." Groups, languages, algorithms, pp. 213–318, Contemporary Mathematics, vol. 378, American Mathematical Society, Providence, RI, 2005
33. O. Kharlampovich, and A. Myasnikov. "Elementary theory of free non-abelian groups." Journal of Algebra, vol. 302 (2006), no. 2, pp. 451–552
34. Frédéric Paulin. Sur la théorie élémentaire des groupes libres (d'après Sela). Astérisque No. 294 (2004), pp. 63–402
35. Guirardel, Vincent; Levitt, Gilbert; Salinos, Rizos (2020). "Towers and the first-order theory of hyperbolic groups". arXiv:2007.14148 [math.GR]. (See p. 8.)
36. Kapovich, Ilya; Weidmann, Richard (2002). "Acylindrical accessibility for groups acting on R-tree". arXiv:math/0210308.
External links
• Zlil Sela's webpage at the Hebrew University
• Zlil Sela at the Mathematics Genealogy Project
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Multiplicative group of integers modulo n
In modular arithmetic, the integers coprime (relatively prime) to n from the set $\{0,1,\dots ,n-1\}$ of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n. Equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n. Hence another name is the group of primitive residue classes modulo n. In the theory of rings, a branch of abstract algebra, it is described as the group of units of the ring of integers modulo n. Here units refers to elements with a multiplicative inverse, which, in this ring, are exactly those coprime to n.
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
This quotient group, usually denoted $(\mathbb {Z} /n\mathbb {Z} )^{\times }$, is fundamental in number theory. It is used in cryptography, integer factorization, and primality testing. It is an abelian, finite group whose order is given by Euler's totient function: $|(\mathbb {Z} /n\mathbb {Z} )^{\times }|=\varphi (n).$ For prime n the group is cyclic, and in general the structure is easy to describe, but no simple general formula for finding generators is known.
Group axioms
It is a straightforward exercise to show that, under multiplication, the set of congruence classes modulo n that are coprime to n satisfy the axioms for an abelian group.
Indeed, a is coprime to n if and only if gcd(a, n) = 1. Integers in the same congruence class a ≡ b (mod n) satisfy gcd(a, n) = gcd(b, n), hence one is coprime to n if and only if the other is. Thus the notion of congruence classes modulo n that are coprime to n is well-defined.
Since gcd(a, n) = 1 and gcd(b, n) = 1 implies gcd(ab, n) = 1, the set of classes coprime to n is closed under multiplication.
Integer multiplication respects the congruence classes, that is, a ≡ a' and b ≡ b' (mod n) implies ab ≡ a'b' (mod n). This implies that the multiplication is associative, commutative, and that the class of 1 is the unique multiplicative identity.
Finally, given a, the multiplicative inverse of a modulo n is an integer x satisfying ax ≡ 1 (mod n). It exists precisely when a is coprime to n, because in that case gcd(a, n) = 1 and by Bézout's lemma there are integers x and y satisfying ax + ny = 1. Notice that the equation ax + ny = 1 implies that x is coprime to n, so the multiplicative inverse belongs to the group.
Notation
The set of (congruence classes of) integers modulo n with the operations of addition and multiplication is a ring. It is denoted $\mathbb {Z} /n\mathbb {Z} $ or $\mathbb {Z} /(n)$ (the notation refers to taking the quotient of integers modulo the ideal $n\mathbb {Z} $ or $(n)$ consisting of the multiples of n). Outside of number theory the simpler notation $\mathbb {Z} _{n}$ is often used, though it can be confused with the p-adic integers when n is a prime number.
The multiplicative group of integers modulo n, which is the group of units in this ring, may be written as (depending on the author) $(\mathbb {Z} /n\mathbb {Z} )^{\times },$ $(\mathbb {Z} /n\mathbb {Z} )^{*},$ $\mathrm {U} (\mathbb {Z} /n\mathbb {Z} ),$ $\mathrm {E} (\mathbb {Z} /n\mathbb {Z} )$ (for German Einheit, which translates as unit), $\mathbb {Z} _{n}^{*}$, or similar notations. This article uses $(\mathbb {Z} /n\mathbb {Z} )^{\times }.$
The notation $\mathrm {C} _{n}$ refers to the cyclic group of order n. It is isomorphic to the group of integers modulo n under addition. Note that $\mathbb {Z} /n\mathbb {Z} $ or $\mathbb {Z} _{n}$ may also refer to the group under addition. For example, the multiplicative group $(\mathbb {Z} /p\mathbb {Z} )^{\times }$ for a prime p is cyclic and hence isomorphic to the additive group $\mathbb {Z} /(p-1)\mathbb {Z} $, but the isomorphism is not obvious.
Structure
The order of the multiplicative group of integers modulo n is the number of integers in $\{0,1,\dots ,n-1\}$ coprime to n. It is given by Euler's totient function: $|(\mathbb {Z} /n\mathbb {Z} )^{\times }|=\varphi (n)$ (sequence A000010 in the OEIS). For prime p, $\varphi (p)=p-1$.
Cyclic case
Main article: primitive root modulo n
The group $(\mathbb {Z} /n\mathbb {Z} )^{\times }$ is cyclic if and only if n is 1, 2, 4, pk or 2pk, where p is an odd prime and k > 0. For all other values of n the group is not cyclic.[1][2][3] This was first proved by Gauss.[4]
This means that for these n:
$(\mathbb {Z} /n\mathbb {Z} )^{\times }\cong \mathrm {C} _{\varphi (n)},$ where $\varphi (p^{k})=\varphi (2p^{k})=p^{k}-p^{k-1}.$
By definition, the group is cyclic if and only if it has a generator g (a generating set {g} of size one), that is, the powers $g^{0},g^{1},g^{2},\dots ,$ give all possible residues modulo n coprime to n (the first $\varphi (n)$ powers $g^{0},\dots ,g^{\varphi (n)-1}$ give each exactly once). A generator of $(\mathbb {Z} /n\mathbb {Z} )^{\times }$ is called a primitive root modulo n.[5] If there is any generator, then there are $\varphi (\varphi (n))$ of them.
Powers of 2
Modulo 1 any two integers are congruent, i.e., there is only one congruence class, [0], coprime to 1. Therefore, $(\mathbb {Z} /1\,\mathbb {Z} )^{\times }\cong \mathrm {C} _{1}$ is the trivial group with φ(1) = 1 element. Because of its trivial nature, the case of congruences modulo 1 is generally ignored and some authors choose not to include the case of n = 1 in theorem statements.
Modulo 2 there is only one coprime congruence class, [1], so $(\mathbb {Z} /2\mathbb {Z} )^{\times }\cong \mathrm {C} _{1}$ is the trivial group.
Modulo 4 there are two coprime congruence classes, [1] and [3], so $(\mathbb {Z} /4\mathbb {Z} )^{\times }\cong \mathrm {C} _{2},$ the cyclic group with two elements.
Modulo 8 there are four coprime congruence classes, [1], [3], [5] and [7]. The square of each of these is 1, so $(\mathbb {Z} /8\mathbb {Z} )^{\times }\cong \mathrm {C} _{2}\times \mathrm {C} _{2},$ the Klein four-group.
Modulo 16 there are eight coprime congruence classes [1], [3], [5], [7], [9], [11], [13] and [15]. $\{\pm 1,\pm 7\}\cong \mathrm {C} _{2}\times \mathrm {C} _{2},$ is the 2-torsion subgroup (i.e., the square of each element is 1), so $(\mathbb {Z} /16\mathbb {Z} )^{\times }$ is not cyclic. The powers of 3, $\{1,3,9,11\}$ are a subgroup of order 4, as are the powers of 5, $\{1,5,9,13\}.$ Thus $(\mathbb {Z} /16\mathbb {Z} )^{\times }\cong \mathrm {C} _{2}\times \mathrm {C} _{4}.$
The pattern shown by 8 and 16 holds[6] for higher powers 2k, k > 2: $\{\pm 1,2^{k-1}\pm 1\}\cong \mathrm {C} _{2}\times \mathrm {C} _{2},$ is the 2-torsion subgroup (so $(\mathbb {Z} /2^{k}\mathbb {Z} )^{\times }$ is not cyclic) and the powers of 3 are a cyclic subgroup of order 2k − 2, so $(\mathbb {Z} /2^{k}\mathbb {Z} )^{\times }\cong \mathrm {C} _{2}\times \mathrm {C} _{2^{k-2}}.$
General composite numbers
By the fundamental theorem of finite abelian groups, the group $(\mathbb {Z} /n\mathbb {Z} )^{\times }$ is isomorphic to a direct product of cyclic groups of prime power orders.
More specifically, the Chinese remainder theorem[7] says that if $\;\;n=p_{1}^{k_{1}}p_{2}^{k_{2}}p_{3}^{k_{3}}\dots ,\;$ then the ring $\mathbb {Z} /n\mathbb {Z} $ is the direct product of the rings corresponding to each of its prime power factors:
$\mathbb {Z} /n\mathbb {Z} \cong \mathbb {Z} /{p_{1}^{k_{1}}}\mathbb {Z} \;\times \;\mathbb {Z} /{p_{2}^{k_{2}}}\mathbb {Z} \;\times \;\mathbb {Z} /{p_{3}^{k_{3}}}\mathbb {Z} \dots \;\;$
Similarly, the group of units $(\mathbb {Z} /n\mathbb {Z} )^{\times }$ is the direct product of the groups corresponding to each of the prime power factors:
$(\mathbb {Z} /n\mathbb {Z} )^{\times }\cong (\mathbb {Z} /{p_{1}^{k_{1}}}\mathbb {Z} )^{\times }\times (\mathbb {Z} /{p_{2}^{k_{2}}}\mathbb {Z} )^{\times }\times (\mathbb {Z} /{p_{3}^{k_{3}}}\mathbb {Z} )^{\times }\dots \;.$
For each odd prime power $p^{k}$ the corresponding factor $(\mathbb {Z} /{p^{k}}\mathbb {Z} )^{\times }$ is the cyclic group of order $\varphi (p^{k})=p^{k}-p^{k-1}$, which may further factor into cyclic groups of prime-power orders. For powers of 2 the factor $(\mathbb {Z} /{2^{k}}\mathbb {Z} )^{\times }$ is not cyclic unless k = 0, 1, 2, but factors into cyclic groups as described above.
The order of the group $\varphi (n)$ is the product of the orders of the cyclic groups in the direct product. The exponent of the group, that is, the least common multiple of the orders in the cyclic groups, is given by the Carmichael function $\lambda (n)$ (sequence A002322 in the OEIS). In other words, $\lambda (n)$ is the smallest number such that for each a coprime to n, $a^{\lambda (n)}\equiv 1{\pmod {n}}$ holds. It divides $\varphi (n)$ and is equal to it if and only if the group is cyclic.
Subgroup of false witnesses
If n is composite, there exists a subgroup of the multiplicative group, called the "group of false witnesses", in which the elements, when raised to the power n − 1, are congruent to 1 modulo n. (Because the residue 1 when raised to any power is congruent to 1 modulo n, the set of such elements is nonempty.)[8] One could say, because of Fermat's Little Theorem, that such residues are "false positives" or "false witnesses" for the primality of n. The number 2 is the residue most often used in this basic primality check, hence 341 = 11 × 31 is famous since 2340 is congruent to 1 modulo 341, and 341 is the smallest such composite number (with respect to 2). For 341, the false witnesses subgroup contains 100 residues and so is of index 3 inside the 300 element multiplicative group mod 341.
n = 9
The smallest example with a nontrivial subgroup of false witnesses is 9 = 3 × 3. There are 6 residues coprime to 9: 1, 2, 4, 5, 7, 8. Since 8 is congruent to −1 modulo 9, it follows that 88 is congruent to 1 modulo 9. So 1 and 8 are false positives for the "primality" of 9 (since 9 is not actually prime). These are in fact the only ones, so the subgroup {1,8} is the subgroup of false witnesses. The same argument shows that n − 1 is a "false witness" for any odd composite n.
n = 91
For n = 91 (= 7 × 13), there are $\varphi (91)=72$ residues coprime to 91, half of them (i.e., 36 of them) are false witnesses of 91, namely 1, 3, 4, 9, 10, 12, 16, 17, 22, 23, 25, 27, 29, 30, 36, 38, 40, 43, 48, 51, 53, 55, 61, 62, 64, 66, 68, 69, 74, 75, 79, 81, 82, 87, 88, and 90, since for these values of x, x90 is congruent to 1 mod 91.
n = 561
n = 561 (= 3 × 11 × 17) is a Carmichael number, thus s560 is congruent to 1 modulo 561 for any integer s coprime to 561. The subgroup of false witnesses is, in this case, not proper; it is the entire group of multiplicative units modulo 561, which consists of 320 residues.
Examples
This table shows the cyclic decomposition of $(\mathbb {Z} /n\mathbb {Z} )^{\times }$ and a generating set for n ≤ 128. The decomposition and generating sets are not unique; for example,
$\displaystyle {\begin{aligned}(\mathbb {Z} /35\mathbb {Z} )^{\times }&\cong (\mathbb {Z} /5\mathbb {Z} )^{\times }\times (\mathbb {Z} /7\mathbb {Z} )^{\times }\cong \mathrm {C} _{4}\times \mathrm {C} _{6}\cong \mathrm {C} _{4}\times \mathrm {C} _{2}\times \mathrm {C} _{3}\cong \mathrm {C} _{2}\times \mathrm {C} _{12}\cong (\mathbb {Z} /4\mathbb {Z} )^{\times }\times (\mathbb {Z} /13\mathbb {Z} )^{\times }\\&\cong (\mathbb {Z} /52\mathbb {Z} )^{\times }\end{aligned}}$
(but $\not \cong \mathrm {C} _{24}\cong \mathrm {C} _{8}\times \mathrm {C} _{3}$). The table below lists the shortest decomposition (among those, the lexicographically first is chosen – this guarantees isomorphic groups are listed with the same decompositions). The generating set is also chosen to be as short as possible, and for n with primitive root, the smallest primitive root modulo n is listed.
For example, take $(\mathbb {Z} /20\mathbb {Z} )^{\times }$. Then $\varphi (20)=8$ means that the order of the group is 8 (i.e., there are 8 numbers less than 20 and coprime to it); $\lambda (20)=4$ means the order of each element divides 4, that is, the fourth power of any number coprime to 20 is congruent to 1 (mod 20). The set {3,19} generates the group, which means that every element of $(\mathbb {Z} /20\mathbb {Z} )^{\times }$ is of the form 3a × 19b (where a is 0, 1, 2, or 3, because the element 3 has order 4, and similarly b is 0 or 1, because the element 19 has order 2).
Smallest primitive root mod n are (0 if no root exists)
0, 1, 2, 3, 2, 5, 3, 0, 2, 3, 2, 0, 2, 3, 0, 0, 3, 5, 2, 0, 0, 7, 5, 0, 2, 7, 2, 0, 2, 0, 3, 0, 0, 3, 0, 0, 2, 3, 0, 0, 6, 0, 3, 0, 0, 5, 5, 0, 3, 3, 0, 0, 2, 5, 0, 0, 0, 3, 2, 0, 2, 3, 0, 0, 0, 0, 2, 0, 0, 0, 7, 0, 5, 5, 0, 0, 0, 0, 3, 0, 2, 7, 2, 0, 0, 3, 0, 0, 3, 0, ... (sequence A046145 in the OEIS)
Numbers of the elements in a minimal generating set of mod n are
0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 3, 1, 2, 1, 2, 2, 1, 1, 3, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 3, 1, 1, 1, 3, 2, 1, 2, 3, 1, 2, ... (sequence A046072 in the OEIS)
Group structure of (Z/nZ)×
$n\;$$(\mathbb {Z} /n\mathbb {Z} )^{\times }$$\varphi (n)$$\lambda (n)\;$Generating set $n\;$$(\mathbb {Z} /n\mathbb {Z} )^{\times }$$\varphi (n)$$\lambda (n)\;$Generating set $n\;$$(\mathbb {Z} /n\mathbb {Z} )^{\times }$$\varphi (n)$$\lambda (n)\;$Generating set $n\;$$(\mathbb {Z} /n\mathbb {Z} )^{\times }$$\varphi (n)$$\lambda (n)\;$Generating set
1 C1110 33 C2×C1020102, 10 65 C4×C1248122, 12 97 C9696965
2 C1111 34 C1616163 66 C2×C1020105, 7 98 C4242423
3 C2222 35 C2×C1224122, 6 67 C6666662 99 C2×C3060302, 5
4 C2223 36 C2×C61265, 19 68 C2×C1632163, 67 100 C2×C2040203, 99
5 C4442 37 C3636362 69 C2×C2244222, 68 101 C1001001002
6 C2225 38 C1818183 70 C2×C1224123, 69 102 C2×C1632165, 101
7 C6663 39 C2×C1224122, 38 71 C7070707 103 C1021021025
8 C2×C2423, 5 40 C2×C2×C41643, 11, 39 72 C2×C2×C62465, 17, 19 104 C2×C2×C1248123, 5, 103
9 C6662 41 C4040406 73 C7272725 105 C2×C2×C1248122, 29, 41
10 C4443 42 C2×C61265, 13 74 C3636365 106 C5252523
11 C1010102 43 C4242423 75 C2×C2040202, 74 107 C1061061062
12 C2×C2425, 7 44 C2×C1020103, 43 76 C2×C1836183, 37 108 C2×C1836185, 107
13 C1212122 45 C2×C1224122, 44 77 C2×C3060302, 76 109 C1081081086
14 C6663 46 C2222225 78 C2×C1224125, 7 110 C2×C2040203, 109
15 C2×C4842, 14 47 C4646465 79 C7878783 111 C2×C3672362, 110
16 C2×C4843, 15 48 C2×C2×C41645, 7, 47 80 C2×C4×C43243, 7, 79 112 C2×C2×C1248123, 5, 111
17 C1616163 49 C4242423 81 C5454542 113 C1121121123
18 C6665 50 C2020203 82 C4040407 114 C2×C1836185, 37
19 C1818182 51 C2×C1632165, 50 83 C8282822 115 C2×C4488442, 114
20 C2×C4843, 19 52 C2×C1224127, 51 84 C2×C2×C62465, 11, 13 116 C2×C2856283, 115
21 C2×C61262, 20 53 C5252522 85 C4×C1664162, 3 117 C6×C1272122, 17
22 C1010107 54 C1818185 86 C4242423 118 C58585811
23 C2222225 55 C2×C2040202, 21 87 C2×C2856282, 86 119 C2×C4896483, 118
24 C2×C2×C2825, 7, 13 56 C2×C2×C62463, 13, 29 88 C2×C2×C1040103, 5, 7 120 C2×C2×C2×C43247, 11, 19, 29
25 C2020202 57 C2×C1836182, 20 89 C8888883 121 C1101101102
26 C1212127 58 C2828283 90 C2×C1224127, 11 122 C6060607
27 C1818182 59 C5858582 91 C6×C1272122, 3 123 C2×C4080407, 83
28 C2×C61263, 13 60 C2×C2×C41647, 11, 19 92 C2×C2244223, 91 124 C2×C3060303, 61
29 C2828282 61 C6060602 93 C2×C30603011, 61 125 C1001001002
30 C2×C4847, 11 62 C3030303 94 C4646465 126 C6×C63665, 13
31 C3030303 63 C6×C63662, 5 95 C2×C3672362, 94 127 C1261261263
32 C2×C81683, 31 64 C2×C1632163, 63 96 C2×C2×C83285, 17, 31 128 C2×C3264323, 127
See also
• Lenstra elliptic curve factorization
Notes
1. Weisstein, Eric W. "Modulo Multiplication Group". MathWorld.
2. Primitive root, Encyclopedia of Mathematics
3. (Vinogradov 2003, pp. 105–121, § VI PRIMITIVE ROOTS AND INDICES)
4. (Gauss & Clarke 1986, arts. 52–56, 82–891) harv error: no target: CITEREFGaussClarke1986 (help)
5. (Vinogradov 2003, p. 106)
6. (Gauss & Clarke 1986, arts. 90–91) harv error: no target: CITEREFGaussClarke1986 (help)
7. Riesel covers all of this. (Riesel 1994, pp. 267–275)
8. Erdős, Paul; Pomerance, Carl (1986). "On the number of false witnesses for a composite number". Math. Comput. 46 (173): 259–279. doi:10.1090/s0025-5718-1986-0815848-x. Zbl 0586.10003.
References
The Disquisitiones Arithmeticae has been translated from Gauss's Ciceronian Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.
• Gauss, Carl Friedrich (1986), Disquisitiones Arithmeticae (English translation, Second, corrected edition), translated by Clarke, Arthur A., New York: Springer, ISBN 978-0-387-96254-2
• Gauss, Carl Friedrich (1965), Untersuchungen uber hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (German translation, Second edition), translated by Maser, H., New York: Chelsea, ISBN 978-0-8284-0191-3
• Riesel, Hans (1994), Prime Numbers and Computer Methods for Factorization (second edition), Boston: Birkhäuser, ISBN 978-0-8176-3743-9
• Vinogradov, I. M. (2003), "§ VI PRIMITIVE ROOTS AND INDICES", Elements of Number Theory, Mineola, NY: Dover Publications, pp. 105–121, ISBN 978-0-486-49530-9
External links
• Weisstein, Eric W. "Modulo Multiplication Group". MathWorld.
• Weisstein, Eric W. "Primitive Root". MathWorld.
• Web-based tool to interactively compute group tables by John Jones
• OEIS sequence A033948 (Numbers that have a primitive root (the multiplicative group modulo n is cyclic))
• Numbers n such that the multiplicative group modulo n is the direct product of k cyclic groups:
• k = 2 OEIS sequence A272592 (2 cyclic groups)
• k = 3 OEIS sequence A272593 (3 cyclic groups)
• k = 4 OEIS sequence A272594 (4 cyclic groups)
• OEIS sequence A272590 (The smallest number m such that the multiplicative group modulo m is the direct product of n cyclic groups)
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Znám's problem
In number theory, Znám's problem asks which sets of integers have the property that each integer in the set is a proper divisor of the product of the other integers in the set, plus 1. Znám's problem is named after the Slovak mathematician Štefan Znám, who suggested it in 1972, although other mathematicians had considered similar problems around the same time.
The initial terms of Sylvester's sequence almost solve this problem, except that the last chosen term equals one plus the product of the others, rather than being a proper divisor. Sun (1983) showed that there is at least one solution to the (proper) Znám problem for each $k\geq 5$. Sun's solution is based on a recurrence similar to that for Sylvester's sequence, but with a different set of initial values.
The Znám problem is closely related to Egyptian fractions. It is known that there are only finitely many solutions for any fixed $k$. It is unknown whether there are any solutions to Znám's problem using only odd numbers, and there remain several other open questions.
The problem
Znám's problem asks which sets of integers have the property that each integer in the set is a proper divisor of the product of the other integers in the set, plus 1. That is, given $k$, what sets of integers
$\{n_{1},\ldots ,n_{k}\}$
are there such that, for each $i$, $n_{i}$ divides but is not equal to
${\Bigl (}\prod _{j\neq i}^{n}n_{j}{\Bigr )}+1?$
A closely related problem concerns sets of integers in which each integer in the set is a divisor, but not necessarily a proper divisor, of one plus the product of the other integers in the set. This problem does not seem to have been named in the literature, and will be referred to as the improper Znám problem. Any solution to Znám's problem is also a solution to the improper Znám problem, but not necessarily vice versa.
History
Znám's problem is named after the Slovak mathematician Štefan Znám, who suggested it in 1972. Barbeau (1971) had posed the improper Znám problem for $k=3$, and Mordell (1973), independently of Znám, found all solutions to the improper problem for $k\leq 5$. Skula (1975) showed that Znám's problem is unsolvable for $k<5$, and credited J. Janák with finding the solution $\{2,3,11,23,315\}$ for $k=5$.[1]
Examples
Sylvester's sequence is an integer sequence in which each term is one plus the product of the previous terms. The first few terms of the sequence are
2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443 (sequence A000058 in the OEIS).
Stopping the sequence early produces a set like $\{2,3,7,43\}$ that almost meets the conditions of Znám's problem, except that the largest value equals one plus the product of the other terms, rather than being a proper divisor.[2] Thus, it is a solution to the improper Znám problem, but not a solution to Znám's problem as it is usually defined.
One solution to the proper Znám problem, for $k=5$, is $\{2,3,7,47,395\}$. A few calculations will show that
3 × 7 × 47 × 395+ 1 =389866, which is divisible by but unequal to 2,
2 × 7 × 47 × 395+ 1 =259911, which is divisible by but unequal to 3,
2 × 3 × 47 × 395+ 1 =111391, which is divisible by but unequal to 7,
2 × 3 × 7 × 395+ 1 =16591, which is divisible by but unequal to 47, and
2 × 3 × 7 × 47+ 1 =1975, which is divisible by but unequal to 395.
Connection to Egyptian fractions
Any solution to the improper Znám problem is equivalent (via division by the product of the values $x_{i}$) to a solution to the equation
$\sum {\frac {1}{x_{i}}}+\prod {\frac {1}{x_{i}}}=y,$
where $y$ as well as each $x_{i}$ must be an integer, and conversely any such solution corresponds to a solution to the improper Znám problem. However, all known solutions have $y=1$, so they satisfy the equation
$\sum {\frac {1}{x_{i}}}+\prod {\frac {1}{x_{i}}}=1.$
That is, they lead to an Egyptian fraction representation of the number one as a sum of unit fractions. Several of the cited papers on Znám's problem study also the solutions to this equation. Brenton & Hill (1988) describe an application of the equation in topology, to the classification of singularities on surfaces,[2] and Domaratzki et al. (2005) describe an application to the theory of nondeterministic finite automata.[3]
Number of solutions
The number of solutions to Znám's problem for any $k$ is finite, so it makes sense to count the total number of solutions for each $k$.[4] Sun (1983) showed that there is at least one solution to the (proper) Znám problem for each $k\geq 5$. Sun's solution is based on a recurrence similar to that for Sylvester's sequence, but with a different set of initial values.[5] The number of solutions for small values of $k$, starting with $k=5$, forms the sequence[6]
2, 5, 18, 96 (sequence A075441 in the OEIS).
Presently, a few solutions are known for $k=9$ and $k=10$, but it is unclear how many solutions remain undiscovered for those values of $k$. However, there are infinitely many solutions if $k$ is not fixed: Cao & Jing (1998) showed that there are at least 39 solutions for each $k\geq 12$, improving earlier results proving the existence of fewer solutions;[7] Sun & Cao (1988) conjecture that the number of solutions for each value of $k$ grows monotonically with $k$.[8]
It is unknown whether there are any solutions to Znám's problem using only odd numbers. With one exception, all known solutions start with 2. If all numbers in a solution to Znám's problem or the improper Znám problem are prime, their product is a primary pseudoperfect number;[9] it is unknown whether infinitely many solutions of this type exist.
References
Notes
1. Barbeau 1971; Mordell 1973; Skula 1975
2. Brenton & Hill 1988.
3. Domaratzki et al. 2005.
4. Janák & Skula 1978.
5. Sun 1983.
6. Brenton & Vasiliu 2002.
7. Cao, Liu & Zhang 1987 Sun & Cao 1988
8. Sun & Cao 1988.
9. Butske, Jaje & Mayernik 2000.
Sources
• Barbeau, G. E. J. (1971), "Problem 179", Canadian Mathematical Bulletin, 14 (1): 129.
• Brenton, Lawrence; Hill, Richard (1988), "On the Diophantine equation $ 1=\sum 1/n_{i}+1/\prod n_{i}$ and a class of homologically trivial complex surface singularities", Pacific Journal of Mathematics, 133 (1): 41–67, doi:10.2140/pjm.1988.133.41, MR 0936356.
• Brenton, Lawrence; Vasiliu, Ana (2002), "Znám's problem", Mathematics Magazine, 75 (1): 3–11, doi:10.2307/3219178, JSTOR 3219178.
• Butske, William; Jaje, Lynda M.; Mayernik, Daniel R. (2000), "On the equation $ \sum _{p|N}{\frac {1}{p}}+{\frac {1}{N}}=1$, pseudoperfect numbers, and perfectly weighted graphs", Mathematics of Computation, 69: 407–420, doi:10.1090/S0025-5718-99-01088-1, MR 1648363.
• Cao, Zhen Fu; Jing, Cheng Ming (1998), "On the number of solutions of Znám's problem", J. Harbin Inst. Tech., 30 (1): 46–49, MR 1651784.
• Cao, Zhen Fu; Liu, Rui; Zhang, Liang Rui (1987), "On the equation $ \sum _{j=1}^{s}(1/x_{j})+(1/(x_{1}\cdots x_{s}))=1$ and Znám's problem", Journal of Number Theory, 27 (2): 206–211, doi:10.1016/0022-314X(87)90062-X, MR 0909837.
• Domaratzki, Michael; Ellul, Keith; Shallit, Jeffrey; Wang, Ming-Wei (2005), "Non-uniqueness and radius of cyclic unary NFAs", International Journal of Foundations of Computer Science, 16 (5): 883–896, doi:10.1142/S0129054105003352, MR 2174328.
• Janák, Jaroslav; Skula, Ladislav (1978), "On the integers $ x_{i}$ for which $ x_{i}|x_{1}\cdots x_{i-1}x_{i+1}\cdots x_{n}+1$", Math. Slovaca, 28 (3): 305–310, MR 0534998.
• Mordell, L. J. (1973), "Systems of congruences", Canadian Mathematical Bulletin, 16 (3): 457–462, doi:10.4153/CMB-1973-077-3, MR 0332650.
• Skula, Ladislav (1975), "On a problem of Znám", Acta Fac. Rerum Natur. Univ. Comenian. Math. (Russian, Slovak summary), 32: 87–90, MR 0539862.
• Sun, Qi (1983), "On a problem of Š. Znám", Sichuan Daxue Xuebao (4): 9–12, MR 0750288.
• Sun, Qi; Cao, Zhen Fu (1988), "On the equation $ \sum _{j=1}^{s}1/x_{j}+1/x_{1}\cdots x_{s}=n$ and the number of solutions of Znám's problem", Northeastern Mathematics Journal, 4 (1): 43–48, MR 0970644.
External links
• Primefan, Solutions to Znám's Problem
• Weisstein, Eric W., "Znám's Problem", MathWorld
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Zoel García de Galdeano
Zoel García de Galdeano y Yanguas (5 July 1846 – 28 March 1924) was a Spanish mathematician. He was considered by Julio Rey Pastor as "The apostle of modern mathematics".[1]
Zoel García de Galdeano
Born(1846-07-05)5 July 1846
Pamplona, Spain
Died28 March 1924(1924-03-28) (aged 77)
Zaragoza, Spain
Nationality Spanish
Alma materUniversity of Zaragoza
Scientific career
FieldsMathematics
Biography
His father was a military man, and was killed in war action, so his maternal grandfather, the historian José Yanguas y Miranda (1782-1863), took care of Zoel. To continue his studies, in 1863, Zoel moved to Zaragoza, where he received the title of professor and expert surveyor. In 1869 he graduated as Bachelor. Later he began his studies of Philosophy and Letters, and Sciences at the University of Zaragoza. In 1871, he graduated from these two specialties.[2]
Between 1872 and 1879, Zoel served as professor of mathematics at various schools and institutes in Spain. While he worked in the city of Toledo, he began to write mathematical works that introduced the modern concepts of the European Mathematical in Spain.
In 1889 he obtained the professorship of Analytic geometry at the University of Zaragoza, and in 1896, he was appointed to the professorship of Infinitesimal calculus. He worked at this university until his retirement in 1918.
In 1891, Zoel created El Progreso Matemático, the first strictly mathematical journal published in Spain.[3] He was the principal editor in the two periods in which the journal was published (1891 – 1895 and 1899 – 1900). He was also the first contemporary Spanish mathematician to regularly participate in international congresses of mathematics.
He died in Zaragoza on 28 March 1924.
Notes
1. "García de Galdeano, Zoel". Gran Enciclopedia Aragonesa (in Spanish).
2. Ausejo Martínez, Elena (2010). "Zoel García de Galdeano y Yanguas (Pamplona, 1846 - Zaragoza, 1924)". Números (in Spanish). 73: 5–22. Retrieved 23 June 2016.
3. Hormigón, Mariano (1981). "El Progreso Matemático (1891-1900): Un estudio sobre la primera revista matemática española". Llull: Revista de la Sociedad Española de Historia de las Ciencias y de las Técnicas (in Spanish). Llull. 4 (6): 87–115. Retrieved 23 June 2016.
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Zofia Szmydt
Zofia Szmydt (29 July 1923 – 26 November 2010) was a Polish mathematician working in the areas of differential equations, potential theory and the theory of distributions. She was a winner of the Stefan Banach Prize for mathematics in 1956.
Zofia Szmydt
Born(1923-07-29)July 29, 1923
Warsaw
DiedNovember 26, 2010(2010-11-26) (aged 87)
CitizenshipPolish
Alma materJagiellonian University
AwardsStefan Banach Prize
Scientific career
FieldsDifferential equations
InstitutionsJagiellonian University,
University of Warsaw
ThesisO całkach pierwszych równania różniczkowego
Doctoral advisorTadeusz Ważewski
Life
Zofia Szmydt was born in Warsaw on 29 July 1923. Her mother, Zofia Szmydtowa (née Gąsiorowska), was a historian and philologist.[1]
Szmydt studied at the University of Warsaw in clandestine classes during the Second World War. Following the Warsaw Uprising, she and her family were deported to Krakow.[1]
In 1946, Szmydt graduated in mathematics from the Jagiellonian University. She defended her doctoral thesis in 1949, written under the direction of Tadeusz Ważewski.[2]
Szmydt died on 27 November 2010.[3]
Career
Until 1952, Szmydt worked at the Jagiellonian University. She was a member of the Mathematical Institute of the Polish Academy of Sciences between 1949 and 1971. In 1971, she joined the University of Warsaw where she became a professor in 1984. She retired in 1993.
Contributions
In her 1951 paper, Sur l’allure asymptotique des intégrales des équations différentielles ordinaires, Szmydt applied the topological method by Ważewski to generalizations of Perron's classic results on the asymptotics of systems of solutions of ordinary differential equations.[4]
Szmydt's work on hyperbolic differential equations Sur un problème concernant un systèmes d’équations différentielles hyperboliques d’ordre arbitraire à deux variables indépendantes (1957) proposed a generalised solution for the functional differential equation, which subsumed the Darboux, Cauchy, Picard and Goursat problems as special cases.[5] This was in later literature referred to as the Szmydt problem.[6]
Szmydt's textbook Fourier Transformation and Linear Differential Equations (1971) was the first on the topic to be published in the Polish language.[7] Her motivation was to present the basics of the theory of partial differential equations with a particular emphasis on distributions in limit problems of the classical equations (the heat equation, Schrödinger equation, and the Laplace and Poisson equations).[8]
In Paley–Wiener theorems for the Mellin transformations (1990), Szmydt gave a full characterization of the space of multipliers for Mellin's distribution in terms of the Mellin transform (equivalent to the Paley–Wiener theorem) and established relationships between Schwartz and Mellin distribution spaces.[9]
Honours
In 1956, Szmydt won the Stefan Banach prize of the Polish Academy of Sciences for her research into topological methods in nonlinear ordinary differential equations.[2] In 1973, she awarded the Commander's Cross of the Order of Polonia Restituta for her services to mathematical education.[7]
Selected works
Books
• Topological Imbedding of Laplace Distributions in Laplace Hyperfunctions. Polish Academy of Sciences. 1998. (with Bogdan Ziemian)
• The Mellin Transformation and Fuchsian Type Partial Differential Equations. Springer. 1992. ISBN 978-0792316831. (with Bogdan Ziemian)
• Fourier Transformation and Linear Differential Equations. Springer. 1977. ISBN 978-90-277-0622-5.
Articles
• "Paley–Wiener theorems for the Mellin transformations". Ann. Polon. Math. 51. 1990.
• "Sur un problème concernant un systèmes d'équations différentielles hyperboliques d'ordre arbitraire à deux variables indépendantes". Bull. Acad. Polon. Sci. III (5). 1957.
• "Sur l'allure asymptotique des intégrales des équations différentielles ordinaires". Ann. Soc. Polon. Math. 24 (2). 1951.
References
1. Łysik 2015, p. 283.
2. Kenney 2017, p. 76.
3. Łysik 2015, p. 285.
4. Łysik 2015, p. 287.
5. Karpowicz 2014, p. 866.
6. Łysik 2015, p. 288.
7. Łysik 2015, p. 284.
8. Łysik 2015, p. 290.
9. Łysik 2015, p. 291.
Bibliography
• Łysik, Grzegorz (2015). "Zofia Szmydt (1923–2010)". Wiadomości Matematyczne. 51 (2).
• Karpowicz, Adrian (2014). "The Existence of a Unique Solution of the Hyperbolic Functional Differential Equation" (PDF). Demonstratio Mathematica. XLVII (4).
• Kenney, Emelie Agnes (2017). "Making Her Mark on a Century of Turmoil and Triumph: A Tribute to Polish Women in Mathematics". In Janet L. Beery; Sarah J. Greenwald; Jacqueline A. Jensen-Vallin; Maura B. Mast (eds.). Women in Mathematics: Celebrating the Centennial of the Mathematical Association of America. Springer. ISBN 978-3-319-66694-5.
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Zoll surface
In mathematics, particularly in differential geometry, a Zoll surface, named after Otto Zoll, is a surface homeomorphic to the 2-sphere, equipped with a Riemannian metric all of whose geodesics are closed and of equal length. While the usual unit-sphere metric on S2 obviously has this property, it also has an infinite-dimensional family of geometrically distinct deformations that are still Zoll surfaces. In particular, most Zoll surfaces do not have constant curvature.
Zoll, a student of David Hilbert, discovered the first non-trivial examples.
See also
• Funk transform: The original motivation for studying the Funk transform was to describe Zoll metrics on the sphere.
References
• Besse, Arthur L. (1978), Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 93, Springer, Berlin, doi:10.1007/978-3-642-61876-5
• Funk, Paul (1913), "Über Flächen mit lauter geschlossenen geodätischen Linien", Mathematische Annalen, 74: 278–300, doi:10.1007/BF01456044
• Guillemin, Victor (1976), "The Radon transform on Zoll surfaces", Advances in Mathematics, 22 (1): 85–119, doi:10.1016/0001-8708(76)90139-0
• LeBrun, Claude; Mason, L.J. (July 2002), "Zoll manifolds and complex surfaces", Journal of Differential Geometry, 61 (3): 453–535, doi:10.4310/jdg/1090351530
• Zoll, Otto (March 1903). "Über Flächen mit Scharen geschlossener geodätischer Linien". Mathematische Annalen (in German). 57 (1): 108–133. doi:10.1007/bf01449019.
External links
• Tannery's pear, an example of Zoll surface where all closed geodesics (up to the meridians) are shaped like a curved-figure eight.
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Zolotarev's lemma
In number theory, Zolotarev's lemma states that the Legendre symbol
$\left({\frac {a}{p}}\right)$
for an integer a modulo an odd prime number p, where p does not divide a, can be computed as the sign of a permutation:
$\left({\frac {a}{p}}\right)=\varepsilon (\pi _{a})$
where ε denotes the signature of a permutation and πa is the permutation of the nonzero residue classes mod p induced by multiplication by a.
For example, take a = 2 and p = 7. The nonzero squares mod 7 are 1, 2, and 4, so (2|7) = 1 and (6|7) = −1. Multiplication by 2 on the nonzero numbers mod 7 has the cycle decomposition (1,2,4)(3,6,5), so the sign of this permutation is 1, which is (2|7). Multiplication by 6 on the nonzero numbers mod 7 has cycle decomposition (1,6)(2,5)(3,4), whose sign is −1, which is (6|7).
Proof
In general, for any finite group G of order n, it is straightforward to determine the signature of the permutation πg made by left-multiplication by the element g of G. The permutation πg will be even, unless there are an odd number of orbits of even size. Assuming n even, therefore, the condition for πg to be an odd permutation, when g has order k, is that n/k should be odd, or that the subgroup <g> generated by g should have odd index.
We will apply this to the group of nonzero numbers mod p, which is a cyclic group of order p − 1. The jth power of a primitive root modulo p will have index the greatest common divisor
i = (j, p − 1).
The condition for a nonzero number mod p to be a quadratic non-residue is to be an odd power of a primitive root. The lemma therefore comes down to saying that i is odd when j is odd, which is true a fortiori, and j is odd when i is odd, which is true because p − 1 is even (p is odd).
Another proof
Zolotarev's lemma can be deduced easily from Gauss's lemma and vice versa. The example
$\left({\frac {3}{11}}\right)$,
i.e. the Legendre symbol (a/p) with a = 3 and p = 11, will illustrate how the proof goes. Start with the set {1, 2, . . . , p − 1} arranged as a matrix of two rows such that the sum of the two elements in any column is zero mod p, say:
1 2 3 4 5
10 9 8 7 6
Apply the permutation $U:x\mapsto ax{\pmod {p}}$:
3 6 9 1 4
8 5 2 10 7
The columns still have the property that the sum of two elements in one column is zero mod p. Now apply a permutation V which swaps any pairs in which the upper member was originally a lower member:
3 5 2 1 4
8 6 9 10 7
Finally, apply a permutation W which gets back the original matrix:
1 2 3 4 5
10 9 8 7 6
We have W−1 = VU. Zolotarev's lemma says (a/p) = 1 if and only if the permutation U is even. Gauss's lemma says (a/p) = 1 iff V is even. But W is even, so the two lemmas are equivalent for the given (but arbitrary) a and p.
Jacobi symbol
This interpretation of the Legendre symbol as the sign of a permutation can be extended to the Jacobi symbol
$\left({\frac {a}{n}}\right),$
where a and n are relatively prime integers with odd n > 0: a is invertible mod n, so multiplication by a on Z/nZ is a permutation and a generalization of Zolotarev's lemma is that the Jacobi symbol above is the sign of this permutation.
For example, multiplication by 2 on Z/21Z has cycle decomposition (0)(1,2,4,8,16,11)(3,6,12)(5,10,20,19,17,13)(7,14)(9,18,15), so the sign of this permutation is (1)(−1)(1)(−1)(−1)(1) = −1 and the Jacobi symbol (2|21) is −1. (Note that multiplication by 2 on the units mod 21 is a product of two 6-cycles, so its sign is 1. Thus it's important to use all integers mod n and not just the units mod n to define the right permutation.)
When n = p is an odd prime and a is not divisible by p, multiplication by a fixes 0 mod p, so the sign of multiplication by a on all numbers mod p and on the units mod p have the same sign. But for composite n that is not the case, as we see in the example above.
History
This lemma was introduced by Yegor Ivanovich Zolotarev in an 1872 proof of quadratic reciprocity.
See also: Gauss's lemma
References
• Zolotareff G. (1872). "Nouvelle démonstration de la loi de réciprocité de Legendre" (PDF). Nouvelles Annales de Mathématiques. 2e série. 11: 354–362.
External links
• PlanetMath article on Zolotarev's lemma; includes his proof of quadratic reciprocity
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Zonal polynomial
In mathematics, a zonal polynomial is a multivariate symmetric homogeneous polynomial. The zonal polynomials form a basis of the space of symmetric polynomials.
They appear as zonal spherical functions of the Gelfand pairs $(S_{2n},H_{n})$ (here, $H_{n}$ is the hyperoctahedral group) and $(Gl_{n}(\mathbb {R} ),O_{n})$, which means that they describe canonical basis of the double class algebras $\mathbb {C} [H_{n}\backslash S_{2n}/H_{n}]$ and $\mathbb {C} [O_{d}(\mathbb {R} )\backslash M_{d}(\mathbb {R} )/O_{d}(\mathbb {R} )]$.
They are applied in multivariate statistics.
The zonal polynomials are the $\alpha =2$ case of the C normalization of the Jack function.
References
• Robb Muirhead, Aspects of Multivariate Statistical Theory, John Wiley & Sons, Inc., New York, 1984.
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Zonal spherical function
In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K (often a maximal compact subgroup) that arises as the matrix coefficient of a K-invariant vector in an irreducible representation of G. The key examples are the matrix coefficients of the spherical principal series, the irreducible representations appearing in the decomposition of the unitary representation of G on L2(G/K). In this case the commutant of G is generated by the algebra of biinvariant functions on G with respect to K acting by right convolution. It is commutative if in addition G/K is a symmetric space, for example when G is a connected semisimple Lie group with finite centre and K is a maximal compact subgroup. The matrix coefficients of the spherical principal series describe precisely the spectrum of the corresponding C* algebra generated by the biinvariant functions of compact support, often called a Hecke algebra. The spectrum of the commutative Banach *-algebra of biinvariant L1 functions is larger; when G is a semisimple Lie group with maximal compact subgroup K, additional characters come from matrix coefficients of the complementary series, obtained by analytic continuation of the spherical principal series.
Zonal spherical functions have been explicitly determined for real semisimple groups by Harish-Chandra. For special linear groups, they were independently discovered by Israel Gelfand and Mark Naimark. For complex groups, the theory simplifies significantly, because G is the complexification of K, and the formulas are related to analytic continuations of the Weyl character formula on K. The abstract functional analytic theory of zonal spherical functions was first developed by Roger Godement. Apart from their group theoretic interpretation, the zonal spherical functions for a semisimple Lie group G also provide a set of simultaneous eigenfunctions for the natural action of the centre of the universal enveloping algebra of G on L2(G/K), as differential operators on the symmetric space G/K. For semisimple p-adic Lie groups, the theory of zonal spherical functions and Hecke algebras was first developed by Satake and Ian G. Macdonald. The analogues of the Plancherel theorem and Fourier inversion formula in this setting generalise the eigenfunction expansions of Mehler, Weyl and Fock for singular ordinary differential equations: they were obtained in full generality in the 1960s in terms of Harish-Chandra's c-function.
The name "zonal spherical function" comes from the case when G is SO(3,R) acting on a 2-sphere and K is the subgroup fixing a point: in this case the zonal spherical functions can be regarded as certain functions on the sphere invariant under rotation about a fixed axis.
Definitions
See also: Hecke algebra of a locally compact group
Let G be a locally compact unimodular topological group and K a compact subgroup and let H1 = L2(G/K). Thus, H1 admits a unitary representation π of G by left translation. This is a subrepresentation of the regular representation, since if H= L2(G) with left and right regular representations λ and ρ of G and P is the orthogonal projection
$P=\int _{K}\rho (k)\,dk$
from H to H1 then H1 can naturally be identified with PH with the action of G given by the restriction of λ.
On the other hand, by von Neumann's commutation theorem[1]
$\lambda (G)^{\prime }=\rho (G)^{\prime \prime },$
where S' denotes the commutant of a set of operators S, so that
$\pi (G)^{\prime }=P\rho (G)^{\prime \prime }P.$
Thus the commutant of π is generated as a von Neumann algebra by operators
$P\rho (f)P=\int _{G}f(g)(P\rho (g)P)\,dg$
where f is a continuous function of compact support on G.[lower-alpha 1]
However Pρ(f) P is just the restriction of ρ(F) to H1, where
$F(g)=\int _{K}\int _{K}f(kgk^{\prime })\,dk\,dk^{\prime }$
is the K-biinvariant continuous function of compact support obtained by averaging f by K on both sides.
Thus the commutant of π is generated by the restriction of the operators ρ(F) with F in Cc(K\G/K), the K-biinvariant continuous functions of compact support on G.
These functions form a * algebra under convolution with involution
$F^{*}(g)={\overline {F(g^{-1})}},$
often called the Hecke algebra for the pair (G, K).
Let A(K\G/K) denote the C* algebra generated by the operators ρ(F) on H1.
The pair (G, K) is said to be a Gelfand pair[2] if one, and hence all, of the following algebras are commutative:
• $\pi (G)^{\prime }$
• $C_{c}(K\backslash G/K)$
• $A(K\backslash G/K).$
Since A(K\G/K) is a commutative C* algebra, by the Gelfand–Naimark theorem it has the form C0(X), where X is the locally compact space of norm continuous * homomorphisms of A(K\G/K) into C.
A concrete realization of the * homomorphisms in X as K-biinvariant uniformly bounded functions on G is obtained as follows.[2][3][4][5][6]
Because of the estimate
$\|\pi (F)\|\leq \int _{G}|F(g)|\,dg\equiv \|F\|_{1},$
the representation π of Cc(K\G/K) in A(K\G/K) extends by continuity to L1(K\G/K), the * algebra of K-biinvariant integrable functions. The image forms a dense * subalgebra of A(K\G/K). The restriction of a * homomorphism χ continuous for the operator norm is also continuous for the norm ||·||1. Since the Banach space dual of L1 is L∞, it follows that
$\chi (\pi (F))=\int _{G}F(g)h(g)\,dg,$
for some unique uniformly bounded K-biinvariant function h on G. These functions h are exactly the zonal spherical functions for the pair (G, K).
Properties
A zonal spherical function h has the following properties:[2]
1. h is uniformly continuous on G
2. $h(x)h(y)=\int _{K}h(xky)\,dk\,\,(x,y\in G).$
3. h(1) =1 (normalisation)
4. h is a positive definite function on G
5. f * h is proportional to h for all f in Cc(K\G/K).
These are easy consequences of the fact that the bounded linear functional χ defined by h is a homomorphism. Properties 2, 3 and 4 or properties 3, 4 and 5 characterize zonal spherical functions. A more general class of zonal spherical functions can be obtained by dropping positive definiteness from the conditions, but for these functions there is no longer any connection with unitary representations. For semisimple Lie groups, there is a further characterization as eigenfunctions of invariant differential operators on G/K (see below).
In fact, as a special case of the Gelfand–Naimark–Segal construction, there is one-one correspondence between irreducible representations σ of G having a unit vector v fixed by K and zonal spherical functions h given by
$h(g)=(\sigma (g)v,v).$
Such irreducible representations are often described as having class one. They are precisely the irreducible representations required to decompose the induced representation π on H1. Each representation σ extends uniquely by continuity to A(K\G/K), so that each zonal spherical function satisfies
$\left|\int _{G}f(g)h(g)\,dg\right|\leq \|\pi (f)\|$
for f in A(K\G/K). Moreover, since the commutant π(G)' is commutative, there is a unique probability measure μ on the space of * homomorphisms X such that
$\int _{G}|f(g)|^{2}\,dg=\int _{X}|\chi (\pi (f))|^{2}\,d\mu (\chi ).$
μ is called the Plancherel measure. Since π(G)' is the centre of the von Neumann algebra generated by G, it also gives the measure associated with the direct integral decomposition of H1 in terms of the irreducible representations σχ.
Gelfand pairs
See also: Gelfand pair
If G is a connected Lie group, then, thanks to the work of Cartan, Malcev, Iwasawa and Chevalley, G has a maximal compact subgroup, unique up to conjugation.[7][8] In this case K is connected and the quotient G/K is diffeomorphic to a Euclidean space. When G is in addition semisimple, this can be seen directly using the Cartan decomposition associated to the symmetric space G/K, a generalisation of the polar decomposition of invertible matrices. Indeed, if τ is the associated period two automorphism of G with fixed point subgroup K, then
$G=P\cdot K,$
where
$P=\{g\in G|\tau (g)=g^{-1}\}.$
Under the exponential map, P is diffeomorphic to the -1 eigenspace of τ in the Lie algebra of G. Since τ preserves K, it induces an automorphism of the Hecke algebra Cc(K\G/K). On the other hand, if F lies in Cc(K\G/K), then
F(τg) = F(g−1),
so that τ induces an anti-automorphism, because inversion does. Hence, when G is semisimple,
• the Hecke algebra is commutative
• (G,K) is a Gelfand pair.
More generally the same argument gives the following criterion of Gelfand for (G,K) to be a Gelfand pair:[9]
• G is a unimodular locally compact group;
• K is a compact subgroup arising as the fixed points of a period two automorphism τ of G;
• G =K·P (not necessarily a direct product), where P is defined as above.
The two most important examples covered by this are when:
• G is a compact connected semisimple Lie group with τ a period two automorphism;[10][11]
• G is a semidirect product $A\rtimes K$, with A a locally compact Abelian group without 2-torsion and τ(a· k)= k·a−1 for a in A and k in K.
The three cases cover the three types of symmetric spaces G/K:[5]
1. Non-compact type, when K is a maximal compact subgroup of a non-compact real semisimple Lie group G;
2. Compact type, when K is the fixed point subgroup of a period two automorphism of a compact semisimple Lie group G;
3. Euclidean type, when A is a finite-dimensional Euclidean space with an orthogonal action of K.
Cartan–Helgason theorem
Let G be a compact semisimple connected and simply connected Lie group and τ a period two automorphism of a G with fixed point subgroup K = Gτ. In this case K is a connected compact Lie group.[5] In addition let T be a maximal torus of G invariant under τ, such that T $\cap $ P is a maximal torus in P, and set[12]
$S=K\cap T=T^{\tau }.$
S is the direct product of a torus and an elementary abelian 2-group.
In 1929 Élie Cartan found a rule to determine the decomposition of L2(G/K) into the direct sum of finite-dimensional irreducible representations of G, which was proved rigorously only in 1970 by Sigurdur Helgason. Because the commutant of G on L2(G/K) is commutative, each irreducible representation appears with multiplicity one. By Frobenius reciprocity for compact groups, the irreducible representations V that occur are precisely those admitting a non-zero vector fixed by K.
From the representation theory of compact semisimple groups, irreducible representations of G are classified by their highest weight. This is specified by a homomorphism of the maximal torus T into T.
The Cartan–Helgason theorem[13][14] states that
the irreducible representations of G admitting a non-zero vector fixed by K are precisely those with highest weights corresponding to homomorphisms trivial on S.
The corresponding irreducible representations are called spherical representations.
The theorem can be proved[5] using the Iwasawa decomposition:
${\mathfrak {g}}={\mathfrak {k}}\oplus {\mathfrak {a}}\oplus {\mathfrak {n}},$
where ${\mathfrak {g}}$, ${\mathfrak {k}}$, ${\mathfrak {a}}$ are the complexifications of the Lie algebras of G, K, A = T $\cap $ P and
${\mathfrak {n}}=\bigoplus {\mathfrak {g}}_{\alpha },$
summed over all eigenspaces for T in ${\mathfrak {g}}$ corresponding to positive roots α not fixed by τ.
Let V be a spherical representation with highest weight vector v0 and K-fixed vector vK. Since v0 is an eigenvector of the solvable Lie algebra ${\mathfrak {a}}\oplus {\mathfrak {n}}$, the Poincaré–Birkhoff–Witt theorem implies that the K-module generated by v0 is the whole of V. If Q is the orthogonal projection onto the fixed points of K in V obtained by averaging over G with respect to Haar measure, it follows that
$\displaystyle {v_{K}=cQv_{0}}$
for some non-zero constant c. Because vK is fixed by S and v0 is an eigenvector for S, the subgroup S must actually fix v0, an equivalent form of the triviality condition on S.
Conversely if v0 is fixed by S, then it can be shown[15] that the matrix coefficient
$\displaystyle {f(g)=(gv_{0},v_{0})}$
is non-negative on K. Since f(1) > 0, it follows that (Qv0, v0) > 0 and hence that Qv0 is a non-zero vector fixed by K.
Harish-Chandra's formula
If G is a non-compact semisimple Lie group, its maximal compact subgroup K acts by conjugation on the component P in the Cartan decomposition. If A is a maximal Abelian subgroup of G contained in P, then A is diffeomorphic to its Lie algebra under the exponential map and, as a further generalisation of the polar decomposition of matrices, every element of P is conjugate under K to an element of A, so that[16]
G =KAK.
There is also an associated Iwasawa decomposition
G =KAN,
where N is a closed nilpotent subgroup, diffeomorphic to its Lie algebra under the exponential map and normalised by A. Thus S=AN is a closed solvable subgroup of G, the semidirect product of N by A, and G = KS.
If α in Hom(A,T) is a character of A, then α extends to a character of S, by defining it to be trivial on N. There is a corresponding unitary induced representation σ of G on L2(G/S) = L2(K),[17] a so-called (spherical) principal series representation.
This representation can be described explicitly as follows. Unlike G and K, the solvable Lie group S is not unimodular. Let dx denote left invariant Haar measure on S and ΔS the modular function of S. Then[5]
$\int _{G}f(g)\,dg=\int _{S}\int _{K}f(x\cdot k)\,dx\,dk=\int _{S}\int _{K}f(k\cdot x)\Delta _{S}(x)\,dx\,dk.$
The principal series representation σ is realised on L2(K) as[18]
$(\sigma (g)\xi )(k)=\alpha ^{\prime }(g^{-1}k)^{-1}\,\xi (U(g^{-1}k)),$
where
$g=U(g)\cdot X(g)$
is the Iwasawa decomposition of g with U(g) in K and X(g) in S and
$\alpha ^{\prime }(kx)=\Delta _{S}(x)^{1/2}\alpha (x)$
for k in K and x in S.
The representation σ is irreducible, so that if v denotes the constant function 1 on K, fixed by K,
$\varphi _{\alpha }(g)=(\sigma (g)v,v)$
defines a zonal spherical function of G.
Computing the inner product above leads to Harish-Chandra's formula for the zonal spherical function
$\varphi _{\alpha }(g)=\int _{K}\alpha ^{\prime }(gk)^{-1}\,dk$
as an integral over K.
Harish-Chandra proved that these zonal spherical functions exhaust the characters of the C* algebra generated by the Cc(K \ G / K) acting by right convolution on L2(G / K). He also showed that two different characters α and β give the same zonal spherical function if and only if α = β·s, where s is in the Weyl group of A
$W(A)=N_{K}(A)/C_{K}(A),$
the quotient of the normaliser of A in K by its centraliser, a finite reflection group.
It can also be verified directly[2] that this formula defines a zonal spherical function, without using representation theory. The proof for general semisimple Lie groups that every zonal spherical formula arises in this way requires the detailed study of G-invariant differential operators on G/K and their simultaneous eigenfunctions (see below).[4][5] In the case of complex semisimple groups, Harish-Chandra and Felix Berezin realised independently that the formula simplified considerably and could be proved more directly.[5][19][20][21][22]
The remaining positive-definite zonal spherical functions are given by Harish-Chandra's formula with α in Hom(A,C*) instead of Hom(A,T). Only certain α are permitted and the corresponding irreducible representations arise as analytic continuations of the spherical principal series. This so-called "complementary series" was first studied by Bargmann (1947) for G = SL(2,R) and by Harish-Chandra (1947) and Gelfand & Naimark (1947) for G = SL(2,C). Subsequently in the 1960s, the construction of a complementary series by analytic continuation of the spherical principal series was systematically developed for general semisimple Lie groups by Ray Kunze, Elias Stein and Bertram Kostant.[23][24][25] Since these irreducible representations are not tempered, they are not usually required for harmonic analysis on G (or G / K).
Eigenfunctions
Harish-Chandra proved[4][5] that zonal spherical functions can be characterised as those normalised positive definite K-invariant functions on G/K that are eigenfunctions of D(G/K), the algebra of invariant differential operators on G. This algebra acts on G/K and commutes with the natural action of G by left translation. It can be identified with the subalgebra of the universal enveloping algebra of G fixed under the adjoint action of K. As for the commutant of G on L2(G/K) and the corresponding Hecke algebra, this algebra of operators is commutative; indeed it is a subalgebra of the algebra of mesurable operators affiliated with the commutant π(G)', an Abelian von Neumann algebra. As Harish-Chandra proved, it is isomorphic to the algebra of W(A)-invariant polynomials on the Lie algebra of A, which itself is a polynomial ring by the Chevalley–Shephard–Todd theorem on polynomial invariants of finite reflection groups. The simplest invariant differential operator on G/K is the Laplacian operator; up to a sign this operator is just the image under π of the Casimir operator in the centre of the universal enveloping algebra of G.
Thus a normalised positive definite K-biinvariant function f on G is a zonal spherical function if and only if for each D in D(G/K) there is a constant λD such that
$\displaystyle \pi (D)f=\lambda _{D}f,$
i.e. f is a simultaneous eigenfunction of the operators π(D).
If ψ is a zonal spherical function, then, regarded as a function on G/K, it is an eigenfunction of the Laplacian there, an elliptic differential operator with real analytic coefficients. By analytic elliptic regularity, ψ is a real analytic function on G/K, and hence G.
Harish-Chandra used these facts about the structure of the invariant operators to prove that his formula gave all zonal spherical functions for real semisimple Lie groups.[26][27][28] Indeed, the commutativity of the commutant implies that the simultaneous eigenspaces of the algebra of invariant differential operators all have dimension one; and the polynomial structure of this algebra forces the simultaneous eigenvalues to be precisely those already associated with Harish-Chandra's formula.
Example: SL(2,C)
See also: SL(2,C); Representations of the Lorentz group; and Spectral theory of ordinary differential equations
The group G = SL(2,C) is the complexification of the compact Lie group K = SU(2) and the double cover of the Lorentz group. The infinite-dimensional representations of the Lorentz group were first studied by Dirac in 1945, who considered the discrete series representations, which he termed expansors. A systematic study was taken up shortly afterwards by Harish-Chandra, Gelfand–Naimark and Bargmann. The irreducible representations of class one, corresponding to the zonal spherical functions, can be determined easily using the radial component of the Laplacian operator.[5]
Indeed, any unimodular complex 2×2 matrix g admits a unique polar decomposition g = pv with v unitary and p positive. In turn p = uau*, with u unitary and a a diagonal matrix with positive entries. Thus g = uaw with w = u* v, so that any K-biinvariant function on G corresponds to a function of the diagonal matrix
$a={\begin{pmatrix}e^{r/2}&0\\0&e^{-r/2}\end{pmatrix}},$
invariant under the Weyl group. Identifying G/K with hyperbolic 3-space, the zonal hyperbolic functions ψ correspond to radial functions that are eigenfunctions of the Laplacian. But in terms of the radial coordinate r, the Laplacian is given by[29]
$L=-\partial _{r}^{2}-2\coth r\partial _{r}.$
Setting f(r) = sinh (r)·ψ(r), it follows that f is an odd function of r and an eigenfunction of $\partial _{r}^{2}$.
Hence
$\varphi (r)={\sin(\ell r) \over \ell \sinh r}$
where $\ell $ is real.
There is a similar elementary treatment for the generalized Lorentz groups SO(N,1) in Takahashi (1963) and Faraut & Korányi (1994) (recall that SO0(3,1) = SL(2,C) / ±I).
Complex case
If G is a complex semisimple Lie group, it is the complexification of its maximal compact subgroup K. If ${\mathfrak {g}}$ and ${\mathfrak {k}}$ are their Lie algebras, then
${\mathfrak {g}}={\mathfrak {k}}\oplus i{\mathfrak {k}}.$
Let T be a maximal torus in K with Lie algebra ${\mathfrak {t}}$. Then
$A=\exp i{\mathfrak {t}},\,\,P=\exp i{\mathfrak {k}}.$
Let
$W=N_{K}(T)/T$
be the Weyl group of T in K. Recall characters in Hom(T,T) are called weights and can be identified with elements of the weight lattice Λ in Hom(${\mathfrak {t}}$, R) = ${\mathfrak {t}}^{*}$. There is a natural ordering on weights and every finite-dimensional irreducible representation (π, V) of K has a unique highest weight λ. The weights of the adjoint representation of K on ${\mathfrak {k}}\ominus {\mathfrak {t}}$ are called roots and ρ is used to denote half the sum of the positive roots α, Weyl's character formula asserts that for z = exp X in T
$\displaystyle \chi _{\lambda }(e^{X})\equiv {\rm {Tr}}\,\pi (z)=A_{\lambda +\rho }(e^{X})/A_{\rho }(e^{X}),$
where, for μ in ${\mathfrak {t}}^{*}$, Aμ denotes the antisymmetrisation
$\displaystyle A_{\mu }(e^{X})=\sum _{s\in W}\varepsilon (s)e^{i\mu (sX)},$
and ε denotes the sign character of the finite reflection group W.
Weyl's denominator formula expresses the denominator Aρ as a product:
$\displaystyle A_{\rho }(e^{X})=e^{i\rho (X)}\prod _{\alpha >0}(1-e^{-i\alpha (X)}),$
where the product is over the positive roots.
Weyl's dimension formula asserts that
$\displaystyle \chi _{\lambda }(1)\equiv {\rm {dim}}\,V={\prod _{\alpha >0}(\lambda +\rho ,\alpha ) \over \prod _{\alpha >0}(\rho ,\alpha )}.$
where the inner product on ${\mathfrak {t}}^{*}$ is that associated with the Killing form on ${\mathfrak {k}}$.
Now
• every irreducible representation of K extends holomorphically to the complexification G
• every irreducible character χλ(k) of K extends holomorphically to the complexification of K and ${\mathfrak {t}}^{*}$.
• for every λ in Hom(A,T) = $i{\mathfrak {t}}^{*}$, there is a zonal spherical function φλ.
The Berezin–Harish–Chandra formula[5] asserts that for X in $i{\mathfrak {t}}$
$\varphi _{\lambda }(e^{X})={\chi _{\lambda }(e^{X}) \over \chi _{\lambda }(1)}.$
In other words:
• the zonal spherical functions on a complex semisimple Lie group are given by analytic continuation of the formula for the normalised characters.
One of the simplest proofs[30] of this formula involves the radial component on A of the Laplacian on G, a proof formally parallel to Helgason's reworking of Freudenthal's classical proof of the Weyl character formula, using the radial component on T of the Laplacian on K.[31]
In the latter case the class functions on K can be identified with W-invariant functions on T. The radial component of ΔK on T is just the expression for the restriction of ΔK to W-invariant functions on T, where it is given by the formula
$\displaystyle \Delta _{K}=h^{-1}\circ \Delta _{T}\circ h+\|\rho \|^{2},$
where
$\displaystyle h(e^{X})=A_{\rho }(e^{X})$
for X in ${\mathfrak {t}}$. If χ is a character with highest weight λ, it follows that φ = h·χ satisfies
$\Delta _{T}\varphi =(\|\lambda +\rho \|^{2}-\|\rho \|^{2})\varphi .$
Thus for every weight μ with non-zero Fourier coefficient in φ,
$\displaystyle \|\lambda +\rho \|^{2}=\|\mu +\rho \|^{2}.$
The classical argument of Freudenthal shows that μ + ρ must have the form s(λ + ρ) for some s in W, so the character formula follows from the antisymmetry of φ.
Similarly K-biinvariant functions on G can be identified with W(A)-invariant functions on A. The radial component of ΔG on A is just the expression for the restriction of ΔG to W(A)-invariant functions on A. It is given by the formula
$\displaystyle \Delta _{G}=H^{-1}\circ \Delta _{A}\circ H-\|\rho \|^{2},$
where
$\displaystyle H(e^{X})=A_{\rho }(e^{X})$
for X in $i{\mathfrak {t}}$.
The Berezin–Harish–Chandra formula for a zonal spherical function φ can be established by introducing the antisymmetric function
$\displaystyle f=H\cdot \varphi ,$
which is an eigenfunction of the Laplacian ΔA. Since K is generated by copies of subgroups that are homomorphic images of SU(2) corresponding to simple roots, its complexification G is generated by the corresponding homomorphic images of SL(2,C). The formula for zonal spherical functions of SL(2,C) implies that f is a periodic function on $i{\mathfrak {t}}$ with respect to some sublattice. Antisymmetry under the Weyl group and the argument of Freudenthal again imply that ψ must have the stated form up to a multiplicative constant, which can be determined using the Weyl dimension formula.
Example: SL(2,R)
See also: SL(2,R); Representation theory of SL2(R); and Spectral theory of ordinary differential equations
The theory of zonal spherical functions for SL(2,R) originated in the work of Mehler in 1881 on hyperbolic geometry. He discovered the analogue of the Plancherel theorem, which was rediscovered by Fock in 1943. The corresponding eigenfunction expansion is termed the Mehler–Fock transform. It was already put on a firm footing in 1910 by Hermann Weyl's important work on the spectral theory of ordinary differential equations. The radial part of the Laplacian in this case leads to a hypergeometric differential equation, the theory of which was treated in detail by Weyl. Weyl's approach was subsequently generalised by Harish-Chandra to study zonal spherical functions and the corresponding Plancherel theorem for more general semisimple Lie groups. Following the work of Dirac on the discrete series representations of SL(2,R), the general theory of unitary irreducible representations of SL(2,R) was developed independently by Bargmann, Harish-Chandra and Gelfand–Naimark. The irreducible representations of class one, or equivalently the theory of zonal spherical functions, form an important special case of this theory.
The group G = SL(2,R) is a double cover of the 3-dimensional Lorentz group SO(2,1), the symmetry group of the hyperbolic plane with its Poincaré metric. It acts by Möbius transformations. The upper half-plane can be identified with the unit disc by the Cayley transform. Under this identification G becomes identified with the group SU(1,1), also acting by Möbius transformations. Because the action is transitive, both spaces can be identified with G/K, where K = SO(2). The metric is invariant under G and the associated Laplacian is G-invariant, coinciding with the image of the Casimir operator. In the upper half-plane model the Laplacian is given by the formula[5][6]
$\displaystyle \Delta =-4y^{2}(\partial _{x}^{2}+\partial _{y}^{2}).$
If s is a complex number and z = x + i y with y > 0, the function
$\displaystyle f_{s}(z)=y^{s}=\exp({s}\cdot \log y),$
is an eigenfunction of Δ:
$\displaystyle \Delta f_{s}=4s(1-s)f_{s}.$
Since Δ commutes with G, any left translate of fs is also an eigenfunction with the same eigenvalue. In particular, averaging over K, the function
$\varphi _{s}(z)=\int _{K}f_{s}(k\cdot z)\,dk$
is a K-invariant eigenfunction of Δ on G/K. When
$\displaystyle s={1 \over 2}+i\tau ,$
with τ real, these functions give all the zonal spherical functions on G. As with Harish-Chandra's more general formula for semisimple Lie groups, φs is a zonal spherical function because it is the matrix coefficient corresponding to a vector fixed by K in the principal series. Various arguments are available to prove that there are no others. One of the simplest classical Lie algebraic arguments[5][6][32][33][34] is to note that, since Δ is an elliptic operator with analytic coefficients, by analytic elliptic regularity any eigenfunction is necessarily real analytic. Hence, if the zonal spherical function corresponds to the matrix coefficient for a vector v and representation σ, the vector v is an analytic vector for G and
$\displaystyle (\sigma (e^{X})v,v)=\sum _{n=0}^{\infty }(\sigma (X)^{n}v,v)/n!$
for X in $i{\mathfrak {t}}$. The infinitesimal form of the irreducible unitary representations with a vector fixed by K were worked out classically by Bargmann.[32][33] They correspond precisely to the principal series of SL(2,R). It follows that the zonal spherical function corresponds to a principal series representation.
Another classical argument[35] proceeds by showing that on radial functions the Laplacian has the form
$\displaystyle \Delta =-\partial _{r}^{2}-\coth(r)\cdot \partial _{r},$
so that, as a function of r, the zonal spherical function φ(r) must satisfy the ordinary differential equation
$\displaystyle \varphi ^{\prime \prime }+\coth r\,\varphi ^{\prime }=\alpha \,\varphi $
for some constant α. The change of variables t = sinh r transforms this equation into the hypergeometric differential equation. The general solution in terms of Legendre functions of complex index is given by[2][36]
$\varphi (r)=P_{\rho }(\cosh r)={1 \over 2\pi }\int _{0}^{2\pi }(\cosh r+\sinh r\,\cos \theta )^{\rho }\,d\theta ,$
where α = ρ(ρ+1). Further restrictions on ρ are imposed by boundedness and positive-definiteness of the zonal spherical function on G.
There is yet another approach, due to Mogens Flensted-Jensen, which derives the properties of the zonal spherical functions on SL(2,R), including the Plancherel formula, from the corresponding results for SL(2,C), which are simple consequences of the Plancherel formula and Fourier inversion formula for R. This "method of descent" works more generally, allowing results for a real semisimple Lie group to be derived by descent from the corresponding results for its complexification.[37][38]
Further directions
• The theory of zonal functions that are not necessarily positive-definite. These are given by the same formulas as above, but without restrictions on the complex parameter s or ρ. They correspond to non-unitary representations.[5]
• Harish-Chandra's eigenfunction expansion and inversion formula for spherical functions.[39] This is an important special case of his Plancherel theorem for real semisimple Lie groups.
• The structure of the Hecke algebra. Harish-Chandra and Godement proved that, as convolution algebras, there are natural isomorphisms between Cc∞(K \ G / K ) and Cc∞(A)W, the subalgebra invariant under the Weyl group.[3] This is straightforward to establish for SL(2,R).[6]
• Spherical functions for Euclidean motion groups and compact Lie groups.[5]
• Spherical functions for p-adic Lie groups. These were studied in depth by Satake and Macdonald.[40][41] Their study, and that of the associated Hecke algebras, was one of the first steps in the extensive representation theory of semisimple p-adic Lie groups, a key element in the Langlands program.
See also
• Plancherel theorem for spherical functions
• Hecke algebra of a locally compact group
• Representations of Lie groups
• Non-commutative harmonic analysis
• Tempered representation
• Positive definite function on a group
• Symmetric space
• Gelfand pair
Notes
1. If σ is a unitary representation of G, then $\sigma (f)=\int _{G}f(g)\sigma (g)\,dg$.
Citations
1. Dixmier 1996, Algèbres hilbertiennes.
2. Dieudonné 1978.
3. Godement 1952.
4. Helgason 2001.
5. Helgason 1984.
6. Lang 1985.
7. Cartier 1954–1955.
8. Hochschild 1965.
9. Dieudonné 1978, pp. 55–57.
10. Dieudonné 1977.
11. Helgason 1978, p. 249.
12. Helgason 1978, pp. 257–264.
13. Helgason 1984, pp. 534–538.
14. Goodman & Wallach 1998, pp. 549–550.
15. Goodman & Wallach 1998, p. 550.
16. Helgason 1978, Chapter IX.
17. Harish-Chandra 1954a, p. 251.
18. Wallach 1973.
19. Berezin 1956a.
20. Berezin 1956b.
21. Harish-Chandra 1954b.
22. Harish-Chandra 1954c.
23. Kunze & Stein 1961.
24. Stein 1970.
25. Kostant 1969.
26. Harish-Chandra 1958.
27. Helgason 2001, pages 418–422, 427-434
28. Helgason 1984, p. 418.
29. Davies 1990.
30. Helgason 1984, pp. 432–433.
31. Helgason 1984, pp. 501–502.
32. Bargmann 1947.
33. Howe & Tan 1992.
34. Wallach 1988.
35. Helgason 2001, p. 405.
36. Bateman & Erdélyi 1953, p. 156.
37. Flensted-Jensen 1978.
38. Helgason 1984, pp. 489–491.
39. Helgason 1984, pp. 434–458.
40. Satake 1963.
41. Macdonald 1971.
Sources
• Bargmann, V. (1947), "Irreducible Unitary Representations of the Lorentz Group", Annals of Mathematics, 48 (3): 568–640, doi:10.2307/1969129, JSTOR 1969129
• Barnett, Adam; Smart, Nigel P. (2003), "Mental Poker Revisited", Cryptography and Coding, Lecture Notes in Computer Science, vol. 2898, pp. 370–383, doi:10.1007/978-3-540-40974-8_29, ISBN 978-3-540-20663-7
• Bateman, Harry; Erdélyi, Arthur (1953), Higher transcendental functions, Vol. I (PDF), McGraw–Hill, ISBN 0-07-019546-3
• Berezin, F. A. (1956a), "Операторы Лапласа на полупростых группах" [Laplace operators on semisimple groups], Doklady Akademii Nauk SSSR, 107: 9–12.
• Berezin, F. A. (1956b), "Representation of complex semisimple Lie groups in Banach space", Doklady Akademii Nauk SSSR, 110: 897–900
• Brychkov, Yu A.; Prudnikov, A.P. (2001) [1994], "Spherical functions", Encyclopedia of Mathematics, EMS Press
• Cartier, Pierre (1954–1955), Structure topologique des groupes de Lie généraux, Exposé No. 22 (PDF), Séminaire "Sophus Lie", vol. 1.
• Davies, E. B. (1990), Heat Kernels and Spectral Theory, Cambridge University Press, ISBN 0-521-40997-7
• Dieudonné, Jean (1977), Treatise on Analysis, Vol. V, Academic Press
• Dieudonné, Jean (1978), Treatise on Analysis, Vol. VI, Academic Press, ISBN 0-12-215506-8
• Dirac, P. A. M. (1945), "Unitary representations of the Lorentz group", Proceedings of the Royal Society A, 183 (994): 284–295, Bibcode:1945RSPSA.183..284D, doi:10.1098/rspa.1945.0003
• Dixmier, Jacques (1996), Les algèbres d'opérateurs dans l'espace hilbertien (algèbres de von Neumann), Les Grands Classiques Gauthier-Villars., Éditions Jacques Gabay, ISBN 2-87647-012-8
• Faraut, Jacques; Korányi, Adam (1994), Analysis on symmetric cones, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, ISBN 0-19-853477-9, MR 1446489, Chapter XIV.
• Flensted-Jensen, Mogens (1978), "Spherical functions of a real semisimple Lie group. A method of reduction to the complex case", J. Funct. Anal., 30: 106–146, doi:10.1016/0022-1236(78)90058-7
• Gelfand, I.M.; Naimark, M.A. (1947), "Unitary representations of the Lorentz group", Izv. Akad. Nauk SSSR, Ser. Mat., 37: 411–504
• Gelfand, I.M.; Naimark, M.A. (1948), "An analogue of Plancherel's theorem for the complex unimodular group", Doklady Akademii Nauk SSSR, 63: 609–612
• Gelfand, I.M.; Naimark, M.A. (1952), "Unitary representations of the unimodular group containing the identity representation of the unitary subgroup", Trudy Moscov. Mat. Obšč., 1: 423–475
• Godement, Roger (1952), "A theory of spherical functions. I", Transactions of the American Mathematical Society, 73 (3): 496–556, doi:10.2307/1990805, JSTOR 1990805
• Goodman, Roe; Wallach, Nolan (1998), Representations and Invariants of the Classical Groups, Cambridge University Press, ISBN 0-521-66348-2
• Harish-Chandra (1947), "Infinite irreducible representations of the Lorentz group", Proceedings of the Royal Society A, 189 (1018): 372–401, Bibcode:1947RSPSA.189..372H, doi:10.1098/rspa.1947.0047
• Harish-Chandra (1954a), "Representations of Semisimple Lie Groups. II", Trans. Amer. Math. Soc., 76 (1): 26–65, doi:10.2307/1990743, JSTOR 1990743, PMC 1063373, PMID 16578367
• Harish-Chandra (1954b), "Representations of Semisimple Lie Groups. III", Trans. Amer. Math. Soc., 76 (2): 234–253, doi:10.2307/1990767, JSTOR 1990767 (Simplification of formula for complex semisimple Lie groups)
• Harish-Chandra (1954c), "The Plancherel formula for complex semisimple Lie groups", Trans. Amer. Math. Soc., 76 (3): 485–528, doi:10.2307/1990793, JSTOR 1990793, PMC 1063477, PMID 16589034 (Second proof of formula for complex semisimple Lie groups)
• Harish-Chandra (1958), "Spherical functions on a semisimple Lie group I, II", Amer. J. Math., 80 (2): 241–310, 553–613, doi:10.2307/2372786, JSTOR 2372786, PMC 528464, PMID 16590028 (Determination of Plancherel measure)
• Helgason, Sigurdur (1978), Differential geometry, Lie groups and symmetric spaces, Academic Press, ISBN 0-12-338460-5
• Helgason, Sigurdur (1984), Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, Academic Press, ISBN 0-12-338301-3 – via Internet Archive
• Helgason, Sigurdur (2001), Differential geometry and symmetric spaces (reprint of 1962 edition), American Mathematical Society, ISBN 0-8218-2735-9
• Hochschild, Gerhard P. (1965), The structure of Lie groups, Holden–Day
• Howe, Roger; Tan, Eng-chye (1992), Non-abelian Harmonic Analysis: Applications of SL(2,R), Universitext, Springer-Verlag, ISBN 0-387-97768-6
• Kostant, Bertram (1969), "On the existence and irreducibility of certain series of representations", Bull. Amer. Math. Soc., 75 (4): 627–642, doi:10.1090/S0002-9904-1969-12235-4
• Kunze, Raymond A.; Stein, Elias M. (1961), "Analytic continuation of the principal series", Bull. Amer. Math. Soc., 67 (6): 593–596, doi:10.1090/S0002-9904-1961-10705-2
• Lang, Serge (1985), SL(2,R), Graduate Texts in Mathematics, vol. 105, Springer-Verlag, ISBN 0-387-96198-4
• Macdonald, Ian G. (1971), Spherical Functions on a Group of p-adic Type, Publ. Ramanujan Institute, vol. 2, University of Madras
• Satake, I. (1963), "Theory of spherical functions on reductive algebraic groups over p-adic fields", Publ. Math. IHÉS, 18: 5–70, doi:10.1007/bf02684781, S2CID 4666554
• Stein, Elias M. (1970), "Analytic continuation of group representations", Advances in Mathematics, 4 (2): 172–207, doi:10.1016/0001-8708(70)90022-8
• Takahashi, R. (1963), "Sur les représentations unitaires des groupes de Lorentz généralisés", Bull. Soc. Math. France, 91: 289–433, doi:10.24033/bsmf.1598
• Wallach, Nolan (1973), Harmonic Analysis on Homogeneous Spaces, Marcel Decker, ISBN 0-8247-6010-7
• Wallach, Nolan (1988), Real Reductive Groups I, Academic Press, ISBN 0-12-732960-9 – via Internet Archive
External links
• Casselman, William, Notes on spherical functions (PDF)
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Zonal spherical harmonics
In the mathematical study of rotational symmetry, the zonal spherical harmonics are special spherical harmonics that are invariant under the rotation through a particular fixed axis. The zonal spherical functions are a broad extension of the notion of zonal spherical harmonics to allow for a more general symmetry group.
On the two-dimensional sphere, the unique zonal spherical harmonic of degree ℓ invariant under rotations fixing the north pole is represented in spherical coordinates by
$Z^{(\ell )}(\theta ,\phi )=P_{\ell }(\cos \theta )$
where Pℓ is a Legendre polynomial of degree ℓ. The general zonal spherical harmonic of degree ℓ is denoted by $Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )$, where x is a point on the sphere representing the fixed axis, and y is the variable of the function. This can be obtained by rotation of the basic zonal harmonic $Z^{(\ell )}(\theta ,\phi ).$
In n-dimensional Euclidean space, zonal spherical harmonics are defined as follows. Let x be a point on the (n−1)-sphere. Define $Z_{\mathbf {x} }^{(\ell )}$ to be the dual representation of the linear functional
$P\mapsto P(\mathbf {x} )$
in the finite-dimensional Hilbert space Hℓ of spherical harmonics of degree ℓ. In other words, the following reproducing property holds:
$Y(\mathbf {x} )=\int _{S^{n-1}}Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )Y(\mathbf {y} )\,d\Omega (y)$
for all Y ∈ Hℓ. The integral is taken with respect to the invariant probability measure.
Relationship with harmonic potentials
The zonal harmonics appear naturally as coefficients of the Poisson kernel for the unit ball in Rn: for x and y unit vectors,
${\frac {1}{\omega _{n-1}}}{\frac {1-r^{2}}{|\mathbf {x} -r\mathbf {y} |^{n}}}=\sum _{k=0}^{\infty }r^{k}Z_{\mathbf {x} }^{(k)}(\mathbf {y} ),$
where $\omega _{n-1}$ is the surface area of the (n-1)-dimensional sphere. They are also related to the Newton kernel via
${\frac {1}{|\mathbf {x} -\mathbf {y} |^{n-2}}}=\sum _{k=0}^{\infty }c_{n,k}{\frac {|\mathbf {x} |^{k}}{|\mathbf {y} |^{n+k-2}}}Z_{\mathbf {x} /|\mathbf {x} |}^{(k)}(\mathbf {y} /|\mathbf {y} |)$
where x,y ∈ Rn and the constants cn,k are given by
$c_{n,k}={\frac {1}{\omega _{n-1}}}{\frac {2k+n-2}{(n-2)}}.$
The coefficients of the Taylor series of the Newton kernel (with suitable normalization) are precisely the ultraspherical polynomials. Thus, the zonal spherical harmonics can be expressed as follows. If α = (n−2)/2, then
$Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )={\frac {n+2\ell -2}{n-2}}C_{\ell }^{(\alpha )}(\mathbf {x} \cdot \mathbf {y} )$
where cn,ℓ are the constants above and $C_{\ell }^{(\alpha )}$ is the ultraspherical polynomial of degree ℓ.
Properties
• The zonal spherical harmonics are rotationally invariant, meaning that
$Z_{R\mathbf {x} }^{(\ell )}(R\mathbf {y} )=Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )$
for every orthogonal transformation R. Conversely, any function f(x,y) on Sn−1×Sn−1 that is a spherical harmonic in y for each fixed x, and that satisfies this invariance property, is a constant multiple of the degree ℓ zonal harmonic.
• If Y1, ..., Yd is an orthonormal basis of Hℓ, then
$Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )=\sum _{k=1}^{d}Y_{k}(\mathbf {x} ){\overline {Y_{k}(\mathbf {y} )}}.$
• Evaluating at x = y gives
$Z_{\mathbf {x} }^{(\ell )}(\mathbf {x} )=\omega _{n-1}^{-1}\dim \mathbf {H} _{\ell }.$
References
• Stein, Elias; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton, N.J.: Princeton University Press, ISBN 978-0-691-08078-9.
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Zone theorem
In geometry, the zone theorem is a result that establishes the complexity of the zone of a line in an arrangement of lines.
Definition
A line arrangement, denoted as $A(L)$, is a subdivision of the plane, induced by a set of lines $L$, into cells ($2$-dimensional faces), edges ($1$-dimensional faces) and vertices ($0$-dimensional faces). Given a set of $n$ lines $L$, the line arrangement $A(L)$, and a line $l$ (not belonging to $L$), the zone of $l$ is the set of faces intersected by $l$. The complexity of a zone is the total number of edges in its boundary, expressed as a function of $n$. The zone theorem states that said complexity is $O(n)$.
History
This result was published for the first time in 1985;[1] Chazelle et al. gave the upper bound of $10n+2$ for the complexity of the zone of a line in an arrangement. In 1991,[2] this bound was improved to $\lfloor 9.5n\rfloor -1$, and it was also shown that this is the best possible upper bound up to a small additive factor. Then, in 2011,[3] Rom Pinchasi proved that the complexity of the zone of a line in an arrangement is at most $\lfloor 9.5n\rfloor -3$, and this is a tight bound.
Some paradigms used in the different proofs of the theorem are induction,[1] sweep technique,[2][4] tree construction,[5] and Davenport-Schinzel sequences.[6][7]
Generalizations
Although the most popular version is for arrangements of lines in the plane, there exist some generalizations of the zone theorem. For instance, in dimension $d$, considering arrangements of hyperplanes, the complexity of the zone of a hyperplane $h$ is the number of facets ($d-1$ - dimensional faces) bounding the set of cells ($d$-dimensional faces) intersected by $h$. Analogously, the $d$-dimensional zone theorem states that the complexity of the zone of a hyperplane is $O(n^{d-1})$.[7] There are considerably fewer proofs for the theorem for dimension $d\geq 3$. For the $3$-dimensional case, there are proofs based on sweep techniques and for higher dimensions is used Euler’s relation:[8]
$\sum _{i=0}^{d}(-1)^{i}F_{i}\geq 0.$
Another generalization is considering arrangements of pseudolines (and pseudohyperplanes in dimension $d$) instead of lines (and hyperplanes). Some proofs for the theorem work well in this case since they do not use the straightness of the lines substantially through their arguments.[7]
Motivation
The primary motivation to study the zone complexity in arrangements arises from looking for efficient algorithms to construct arrangements. A classical algorithm is the incremental construction, which can be roughly described as adding the lines one after the other and storing all faces generated by each in an appropriate data structure (the usual structure for arrangements is the doubly connected edge list (DCEL)). Here, the consequence of the zone theorem is that the entire construction of any arrangement of $n$ lines can be done in time $O(n^{2})$, since the insertion of each line takes time $O(n)$.
Notes
1. Chazelle, Guibas & Lee (1985)
2. Bern et al. (1991)
3. Pinchasi (2011)
4. Edelsbrunner, O'Rourke & Seidel (1986)
5. Edelsbrunner & Guibas (1989)
6. Edelsbrunner et al. (1992)
7. Edelsbrunner, Seidel & Sharir (1991)
8. Saxena (2021)
References
• Agarwal, P. K.; Sharir, M. (2000), "Arrangements and their applications" (PDF), in Sack, J.-R.; Urrutia, J. (eds.), Handbook of Computational Geometry, Elsevier, pp. 49–119.
• Agarwal, P. K.; Sharir, M. (2002), "Pseudo-line arrangements: duality, algorithms, and applications", Proc. 13th ACM-SIAM Symposium on Discrete Algorithms (SODA '02), San Francisco: Society for Industrial and Applied Mathematics, pp. 800–809.
• Aharoni, Y.; Halperin, D.; Hanniel, I.; Har-Peled, S.; Linhart, C. (1999), "On-line zone construction in arrangements of lines in the plane", in Vitter, Jeffrey S.; Zaroliagis, Christos D. (eds.), Algorithm Engineering: 3rd International Workshop, WAE'99, London, UK, July 19–21, 1999, Proceedings, Lecture Notes in Computer Science, vol. 1668, Springer-Verlag, pp. 139–153, CiteSeerX 10.1.1.35.7681, doi:10.1007/3-540-48318-7_13, ISBN 978-3-540-66427-7.
• Bern, M. W.; Eppstein, D.; Plassman, P. E.; Yao, F. F. (1991), "Horizon theorems for lines and polygons", in Goodman, J. E.; Pollack, R.; Steiger, W. (eds.), Discrete and Computational Geometry: Papers from the DIMACS Special Year, DIMACS Ser. Discrete Math. and Theoretical Computer Science (6 ed.), Amer. Math. Soc., pp. 45–66, MR 1143288.
• Chazelle, B.; Guibas, L. J.; Lee, D. T. (1985), "The power of geometric duality", BIT Numerical Mathematics, 25 (1): 76–90, doi:10.1007/BF01934990, S2CID 122411548.
• Edelsbrunner, H. (1987), Algorithms in Combinatorial Geometry, EATCS Monographs in Theoretical Computer Science, Springer-Verlag, ISBN 978-3-540-13722-1.
• Edelsbrunner, H.; O'Rourke, J.; Seidel, R. (1986), "Constructing arrangements of lines and hyperplanes with applications", SIAM Journal on Computing, 15 (2): 341–363, doi:10.1137/0215024.
• Edelsbrunner, H.; Guibas, L. J. (1989), "Topologically sweeping an arrangement", Journal of Computer and System Sciences, 38 (1): 165–194, doi:10.1016/0022-0000(89)90038-X.
• Edelsbrunner, H.; Guibas, L. J.; Pach, J.; Pollack, R.; Seidel, R.; Sharir, M. (1992), "Arrangements of curves in the plane—topology, combinatorics, and algorithms", Theoretical Computer Science, 92 (2): 319–336, doi:10.1016/0304-3975(92)90319-B.
• Edelsbrunner, H.; Seidel, R.; Sharir, M. (1991), "On the zone theorem for hyperplane arrangements", New Results and New Trends in Computer Science, Graz, Austria: Springer Science & Business Media, 555: 108, doi:10.1016/0304-3975(92)90319-B
• Grünbaum, B. (1972), Arrangements and Spreads, Regional Conference Series in Mathematics, vol. 10, Providence, R.I.: American Mathematical Society.
• Pinchasi, R. (2011), "The zone theorem revisited", Manuscript
• Saxena, S. (2021), "Zone theorem for arrangements in dimension three", Information Processing Letters, 172: 106161, arXiv:2006.01428, doi:10.1016/j.ipl.2021.106161, S2CID 219179345
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Zonohedron
In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in three-dimensional space, or as a three-dimensional projection of a hypercube. Zonohedra were originally defined and studied by E. S. Fedorov, a Russian crystallographer. More generally, in any dimension, the Minkowski sum of line segments forms a polytope known as a zonotope.
Zonohedra that tile space
The original motivation for studying zonohedra is that the Voronoi diagram of any lattice forms a convex uniform honeycomb in which the cells are zonohedra. Any zonohedron formed in this way can tessellate 3-dimensional space and is called a primary parallelohedron. Each primary parallelohedron is combinatorially equivalent to one of five types: the rhombohedron (including the cube), hexagonal prism, truncated octahedron, rhombic dodecahedron, and the rhombo-hexagonal dodecahedron.
Zonohedra from Minkowski sums
Let $\{v_{0},v_{1},\dots \}$ be a collection of three-dimensional vectors. With each vector $v_{i}$ we may associate a line segment $ \{x_{i}v_{i}\mid 0\leq x_{i}\leq 1\}$. The Minkowski sum $ \ \sum _{i}x_{i}v_{i}\mid 0\leq x_{i}\leq 1\}$ forms a zonohedron, and all zonohedra that contain the origin have this form. The vectors from which the zonohedron is formed are called its generators. This characterization allows the definition of zonohedra to be generalized to higher dimensions, giving zonotopes.
Each edge in a zonohedron is parallel to at least one of the generators, and has length equal to the sum of the lengths of the generators to which it is parallel. Therefore, by choosing a set of generators with no parallel pairs of vectors, and by setting all vector lengths equal, we may form an equilateral version of any combinatorial type of zonohedron.
By choosing sets of vectors with high degrees of symmetry, we can form in this way, zonohedra with at least as much symmetry. For instance, generators equally spaced around the equator of a sphere, together with another pair of generators through the poles of the sphere, form zonohedra in the form of prism over regular $2k$-gons: the cube, hexagonal prism, octagonal prism, decagonal prism, dodecagonal prism, etc. Generators parallel to the edges of an octahedron form a truncated octahedron, and generators parallel to the long diagonals of a cube form a rhombic dodecahedron.[1]
The Minkowski sum of any two zonohedra is another zonohedron, generated by the union of the generators of the two given zonohedra. Thus, the Minkowski sum of a cube and a truncated octahedron forms the truncated cuboctahedron, while the Minkowski sum of the cube and the rhombic dodecahedron forms the truncated rhombic dodecahedron. Both of these zonohedra are simple (three faces meet at each vertex), as is the truncated small rhombicuboctahedron formed from the Minkowski sum of the cube, truncated octahedron, and rhombic dodecahedron.[1]
Zonohedra from arrangements
The Gauss map of any convex polyhedron maps each face of the polygon to a point on the unit sphere, and maps each edge of the polygon separating a pair of faces to a great circle arc connecting the corresponding two points. In the case of a zonohedron, the edges surrounding each face can be grouped into pairs of parallel edges, and when translated via the Gauss map any such pair becomes a pair of contiguous segments on the same great circle. Thus, the edges of the zonohedron can be grouped into zones of parallel edges, which correspond to the segments of a common great circle on the Gauss map, and the 1-skeleton of the zonohedron can be viewed as the planar dual graph to an arrangement of great circles on the sphere. Conversely any arrangement of great circles may be formed from the Gauss map of a zonohedron generated by vectors perpendicular to the planes through the circles.
Any simple zonohedron corresponds in this way to a simplicial arrangement, one in which each face is a triangle. Simplicial arrangements of great circles correspond via central projection to simplicial arrangements of lines in the projective plane. There are three known infinite families of simplicial arrangements, one of which leads to the prisms when converted to zonohedra, and the other two of which correspond to additional infinite families of simple zonohedra. There are also many sporadic examples that do not fit into these three families.[2]
It follows from the correspondence between zonohedra and arrangements, and from the Sylvester–Gallai theorem which (in its projective dual form) proves the existence of crossings of only two lines in any arrangement, that every zonohedron has at least one pair of opposite parallelogram faces. (Squares, rectangles, and rhombuses count for this purpose as special cases of parallelograms.) More strongly, every zonohedron has at least six parallelogram faces, and every zonohedron has a number of parallelogram faces that is linear in its number of generators.[3]
Types of zonohedra
Any prism over a regular polygon with an even number of sides forms a zonohedron. These prisms can be formed so that all faces are regular: two opposite faces are equal to the regular polygon from which the prism was formed, and these are connected by a sequence of square faces. Zonohedra of this type are the cube, hexagonal prism, octagonal prism, decagonal prism, dodecagonal prism, etc.
In addition to this infinite family of regular-faced zonohedra, there are three Archimedean solids, all omnitruncations of the regular forms:
• The truncated octahedron, with 6 square and 8 hexagonal faces. (Omnitruncated tetrahedron)
• The truncated cuboctahedron, with 12 squares, 8 hexagons, and 6 octagons. (Omnitruncated cube)
• The truncated icosidodecahedron, with 30 squares, 20 hexagons and 12 decagons. (Omnitruncated dodecahedron)
In addition, certain Catalan solids (duals of Archimedean solids) are again zonohedra:
• Kepler's rhombic dodecahedron is the dual of the cuboctahedron.
• The rhombic triacontahedron is the dual of the icosidodecahedron.
Others with congruent rhombic faces:
• Bilinski's rhombic dodecahedron.
• Rhombic icosahedron
• Rhombohedron
There are infinitely many zonohedra with rhombic faces that are not all congruent to each other. They include:
• Rhombic enneacontahedron
zonohedron image number of
generators
regular face face
transitive
edge
transitive
vertex
transitive
Parallelohedron
(space-filling)
simple
Cube
4.4.4
3 Yes Yes Yes Yes Yes Yes
Hexagonal prism
4.4.6
4 Yes No No Yes Yes Yes
2n-prism (n > 3)
4.4.2n
n + 1 Yes No No Yes No Yes
Truncated octahedron
4.6.6
6 Yes No No Yes Yes Yes
Truncated cuboctahedron
4.6.8
9 Yes No No Yes No Yes
Truncated icosidodecahedron
4.6.10
15 Yes No No Yes No Yes
Parallelepiped 3 No Yes No No Yes Yes
Rhombic dodecahedron
V3.4.3.4
4 No Yes Yes No Yes No
Bilinski dodecahedron 4 No No No No Yes No
Rhombic icosahedron 5 No No No No No No
Rhombic triacontahedron
V3.5.3.5
6 No Yes Yes No No No
Rhombo-hexagonal dodecahedron 5 No No No No Yes No
Truncated rhombic dodecahedron 7 No No No No No Yes
Dissection of zonohedra
Although it is not generally true that any polyhedron has a dissection into any other polyhedron of the same volume (see Hilbert's third problem), it is known that any two zonohedra of equal volumes can be dissected into each other.
Zonohedrification
Zonohedrification is a process defined by George W. Hart for creating a zonohedron from another polyhedron.[4][5]
First the vertices of any seed polyhedron are considered vectors from the polyhedron center. These vectors create the zonohedron which we call the zonohedrification of the original polyhedron. If the seed polyhedron has central symmetry, opposite points define the same direction, so the number of zones in the zonohedron is half the number of vertices of the seed. For any two vertices of the original polyhedron, there are two opposite planes of the zonohedrification which each have two edges parallel to the vertex vectors.
Examples
SymmetryDihedralOctahedralicosahedral
Seed
8 vertex
V4.4.6
6 vertex
{3,4}
8 vertex
{4,3}
12 vertex
3.4.3.4
14 vertex
V3.4.3.4
12 vertex
{3,5}
20 vertex
{5,3}
30 vertex
3.5.3.5
32 vertex
V3.5.3.5
Zonohedron
4 zone
4.4.6
3 zone
{4,3}
4 zone
Rhomb.12
6 zone
4.6.6
7 zone
Ch.cube
6 zone
Rhomb.30
10 zone
Rhomb.90
15 zone
4.6.10
16 zone
Rhomb.90
Zonotopes
The Minkowski sum of line segments in any dimension forms a type of polytope called a zonotope. Equivalently, a zonotope $Z$ generated by vectors $v_{1},...,v_{k}\in \mathbb {R} ^{n}$ is given by $Z=\{a_{1}v_{1}+\cdots +a_{k}v_{k}|\;\forall (j)a_{j}\in [0,1]\}$. Note that in the special case where $k\leq n$, the zonotope $Z$ is a (possibly degenerate) parallelotope.
The facets of any zonotope are themselves zonotopes of one lower dimension; for instance, the faces of zonohedra are zonogons. Examples of four-dimensional zonotopes include the tesseract (Minkowski sums of d mutually perpendicular equal length line segments), the omnitruncated 5-cell, and the truncated 24-cell. Every permutohedron is a zonotope.
Zonotopes and Matroids
Fix a zonotope $Z$ defined from the set of vectors $V=\{v_{1},\dots ,v_{n}\}\subset \mathbb {R} ^{d}$ and let $M$ be the $d\times n$ matrix whose columns are the $v_{i}$. Then the vector matroid ${\underline {\mathcal {M}}}$ on the columns of $M$ encodes a wealth of information about $Z$, that is, many properties of $Z$ are purely combinatorial in nature.
For example, pairs of opposite facets of $Z$ are naturally indexed by the cocircuits of ${\mathcal {M}}$ and if we consider the oriented matroid ${\mathcal {M}}$ represented by ${M}$, then we obtain a bijection between facets of $Z$ and signed cocircuits of ${\mathcal {M}}$ which extends to a poset anti-isomorphism between the face lattice of $Z$ and the covectors of ${\mathcal {M}}$ ordered by component-wise extension of $0\prec +,-$. In particular, if $M$ and $N$ are two matrices that differ by a projective transformation then their respective zonotopes are combinatorially equivalent. The converse of the previous statement does not hold: the segment $[0,2]\subset \mathbb {R} $ is a zonotope and is generated by both $\{2\mathbf {e} _{1}\}$ and by $\{\mathbf {e} _{1},\mathbf {e} _{1}\}$ whose corresponding matrices, $[2]$ and $[1~1]$, do not differ by a projective transformation.
Tilings
Tiling properties of the zonotope $Z$ are also closely related to the oriented matroid ${\mathcal {M}}$ associated to it. First we consider the space-tiling property. The zonotope $Z$ is said to tile $\mathbb {R} ^{d}$ if there is a set of vectors $\Lambda \subset \mathbb {R} ^{d}$ such that the union of all translates $Z+\lambda $ ($\lambda \in \Lambda $) is $\mathbb {R} ^{d}$ and any two translates intersect in a (possibly empty) face of each. Such a zonotope is called a space-tiling zonotope. The following classification of space-tiling zonotopes is due to McMullen:[6] The zonotope $Z$ generated by the vectors $V$ tiles space if and only if the corresponding oriented matroid is regular. So the seemingly geometric condition of being a space-tiling zonotope actually depends only on the combinatorial structure of the generating vectors.
Another family of tilings associated to the zonotope $Z$ are the zonotopal tilings of $Z$. A collection of zonotopes is a zonotopal tiling of $Z$ if it a polyhedral complex with support $Z$, that is, if the union of all zonotopes in the collection is $Z$ and any two intersect in a common (possibly empty) face of each. Many of the images of zonohedra on this page can be viewed as zonotopal tilings of a 2-dimensional zonotope by simply considering them as planar objects (as opposed to planar representations of three dimensional objects). The Bohne-Dress Theorem states that there is a bijection between zonotopal tilings of the zonotope $Z$ and single-element lifts of the oriented matroid ${\mathcal {M}}$ associated to $Z$.[7][8]
Volume
Zonohedra, and n-dimensional zonotopes in general, are noteworthy for admitting a simple analytic formula for their volume.[9]
Let $Z(S)$ be the zonotope $Z=\{a_{1}v_{1}+\cdots +a_{k}v_{k}|\;\forall (j)a_{j}\in [0,1]\}$ generated by a set of vectors $S=\{v_{1},\dots ,v_{k}\in \mathbb {R} ^{n}\}$. Then the n-dimensional volume of $Z(S)$ is given by $\sum _{T\subset S\;:\;|T|=n}|\det(Z(T))|$.
The determinant in this formula makes sense because (as noted above) when the set $T$ has cardinality equal to the dimension $n$ of the ambient space, the zonotope is a parallelotope.
Note that when $k<n$, this formula simply states that the zonotope has n-volume zero.
References
1. Eppstein, David (1996). "Zonohedra and zonotopes". Mathematica in Education and Research. 5 (4): 15–21.
2. Grünbaum, Branko (2009). "A catalogue of simplicial arrangements in the real projective plane". Ars Mathematica Contemporanea. 2 (1): 1–25. doi:10.26493/1855-3974.88.e12. hdl:1773/2269. MR 2485643.
3. Shephard, G. C. (1968). "Twenty problems on convex polyhedra, part I". The Mathematical Gazette. 52 (380): 136–156. doi:10.2307/3612678. JSTOR 3612678. MR 0231278. S2CID 250442107.
4. "Zonohedrification".
5. Zonohedrification, George W. Hart, The Mathematica Journal, 1999, Volume: 7, Issue: 3, pp. 374-389
6. McMullen, Peter (1975). "Space tiling zonotopes". Mathematika. 22 (2): 202–211. doi:10.1112/S0025579300006082.
7. J. Bohne, Eine kombinatorische Analyse zonotopaler Raumaufteilungen, Dissertation, Bielefeld 1992; Preprint 92-041, SFB 343, Universität Bielefeld 1992, 100 pages.
8. Richter-Gebert, J., & Ziegler, G. M. (1994). Zonotopal tilings and the Bohne-Dress theorem. Contemporary Mathematics, 178, 211-211.
9. McMullen, Peter (1984-05-01). "Volumes of Projections of unit Cubes". Bulletin of the London Mathematical Society. 16 (3): 278–280. doi:10.1112/blms/16.3.278. ISSN 0024-6093.
• Coxeter, H. S. M (1962). "The Classification of Zonohedra by Means of Projective Diagrams". J. Math. Pures Appl. 41: 137–156. Reprinted in Coxeter, H. S. M (1999). The Beauty of Geometry. Mineola, NY: Dover. pp. 54–74. ISBN 0-486-40919-8.
• Fedorov, E. S. (1893). "Elemente der Gestaltenlehre". Zeitschrift für Krystallographie und Mineralogie. 21: 671–694.
• Rolf Schneider, Chapter 3.5 "Zonoids and other classes of convex bodies" in Convex bodies: the Brunn-Minkowski theory, Cambridge University Press, Cambridge, 1993.
• Shephard, G. C. (1974). "Space-filling zonotopes". Mathematika. 21 (2): 261–269. doi:10.1112/S0025579300008652.
• Taylor, Jean E. (1992). "Zonohedra and generalized zonohedra". American Mathematical Monthly. 99 (2): 108–111. doi:10.2307/2324178. JSTOR 2324178.
• Beck, M.; Robins, S. (2007). Computing the continuous discretely. Springer Science+ Business Media, LLC.
External links
• Weisstein, Eric W. "Zonohedron". MathWorld.
• Eppstein, David. "The Geometry Junkyard: Zonohedra and Zonotopes".
• Hart, George W. "Virtual Polyhedra: Zonohedra".
• Weisstein, Eric W. "Primary Parallelohedron". MathWorld.
• Bulatov, Vladimir. "Zonohedral Polyhedra Completion".
• Centore, Paul. "Chap. 2 of The Geometry of Colour" (PDF).
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Zonogon
In geometry, a zonogon is a centrally-symmetric, convex polygon.[1] Equivalently, it is a convex polygon whose sides can be grouped into parallel pairs with equal lengths and opposite orientations.
Examples
A regular polygon is a zonogon if and only if it has an even number of sides.[2] Thus, the square, regular hexagon, and regular octagon are all zonogons. The four-sided zonogons are the square, the rectangles, the rhombi, and the parallelograms.
Tiling and equidissection
The four-sided and six-sided zonogons are parallelogons, able to tile the plane by translated copies of themselves, and all convex parallelogons have this form.[3]
Every $2n$-sided zonogon can be tiled by ${\tbinom {n}{2}}$ parallelograms.[4] (For equilateral zonogons, a $2n$-sided one can be tiled by ${\tbinom {n}{2}}$ rhombi.) In this tiling, there is parallelogram for each pair of slopes of sides in the $2n$-sided zonogon. At least three of the zonogon's vertices must be vertices of only one of the parallelograms in any such tiling.[5] For instance, the regular octagon can be tiled by two squares and four 45° rhombi.[6]
In a generalization of Monsky's theorem, Paul Monsky (1990) proved that no zonogon has an equidissection into an odd number of equal-area triangles.[7][8]
Other properties
In an $n$-sided zonogon, at most $2n-3$ pairs of vertices can be at unit distance from each other. There exist $n$-sided zonogons with $2n-O({\sqrt {n}})$ unit-distance pairs.[9]
Related shapes
Zonogons are the two-dimensional analogues of three-dimensional zonohedra and higher-dimensional zonotopes. As such, each zonogon can be generated as the Minkowski sum of a collection of line segments in the plane.[1] If no two of the generating line segments are parallel, there will be one pair of parallel edges for each line segment. Every face of a zonohedron is a zonogon, and every zonogon is the face of at least one zonohedron, the prism over that zonogon. Additionally, every planar cross-section through the center of a centrally-symmetric polyhedron (such as a zonohedron) is a zonogon.
References
1. Boltyanski, Vladimir; Martini, Horst; Soltan, P. S. (2012), Excursions into Combinatorial Geometry, Springer, p. 319, ISBN 9783642592379
2. Young, John Wesley; Schwartz, Albert John (1915), Plane Geometry, H. Holt, p. 121, If a regular polygon has an even number of sides, its center is a center of symmetry of the polygon
3. Alexandrov, A. D. (2005), Convex Polyhedra, Springer, p. 351, ISBN 9783540231585
4. Beck, József (2014), Probabilistic Diophantine Approximation: Randomness in Lattice Point Counting, Springer, p. 28, ISBN 9783319107417
5. Andreescu, Titu; Feng, Zuming (2000), Mathematical Olympiads 1998-1999: Problems and Solutions from Around the World, Cambridge University Press, p. 125, ISBN 9780883858035
6. Frederickson, Greg N. (1997), Dissections: Plane and Fancy, Cambridge University Press, Cambridge, p. 10, doi:10.1017/CBO9780511574917, ISBN 978-0-521-57197-5, MR 1735254
7. Monsky, Paul (1990), "A conjecture of Stein on plane dissections", Mathematische Zeitschrift, 205 (4): 583–592, doi:10.1007/BF02571264, MR 1082876, S2CID 122009844
8. Stein, Sherman; Szabó, Sandor (1994), Algebra and Tiling: Homomorphisms in the Service of Geometry, Carus Mathematical Monographs, vol. 25, Cambridge University Press, p. 130, ISBN 9780883850282
9. Ábrego, Bernardo M.; Fernández-Merchant, Silvia (2002), "The unit distance problem for centrally symmetric convex polygons", Discrete & Computational Geometry, 28 (4): 467–473, doi:10.1007/s00454-002-2882-5, MR 1949894
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Zorn's lemma
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least one maximal element.
The lemma was proved (assuming the axiom of choice) by Kazimierz Kuratowski in 1922 and independently by Max Zorn in 1935.[2] It occurs in the proofs of several theorems of crucial importance, for instance the Hahn–Banach theorem in functional analysis, the theorem that every vector space has a basis,[3] Tychonoff's theorem in topology stating that every product of compact spaces is compact, and the theorems in abstract algebra that in a ring with identity every proper ideal is contained in a maximal ideal and that every field has an algebraic closure.[4]
Zorn's lemma is equivalent to the well-ordering theorem and also to the axiom of choice, in the sense that within ZF (Zermelo–Fraenkel set theory without the axiom of choice) any one of the three is sufficient to prove the other two.[5] An earlier formulation of Zorn's lemma is Hausdorff's maximum principle which states that every totally ordered subset of a given partially ordered set is contained in a maximal totally ordered subset of that partially ordered set.[6]
Motivation
To prove the existence of a mathematical object that can be viewed as a maximal element in some partially ordered set in some way, one can try proving the existence of such an object by assuming there is no maximal element and using transfinite induction and the assumptions of the situation to get a contradiction. Zorn's lemma tidies up the conditions a situation needs to satisfy in order for such an argument to work and enables mathematicians to not have to repeat the transfinite induction argument by hand each time, but just check the conditions of Zorn's lemma.
If you are building a mathematical object in stages and find that (i) you have not finished even after infinitely many stages, and (ii) there seems to be nothing to stop you continuing to build, then Zorn’s lemma may well be able to help you.
— William Timothy Gowers, "How to use Zorn’s lemma"[7]
Statement of the lemma
Preliminary notions:
• A set P equipped with a binary relation ≤ that is reflexive (x ≤ x for every x), antisymmetric (if both x ≤ y and y ≤ x hold, then x = y), and transitive (if x ≤ y and y ≤ z then x ≤ z) is said to be (partially) ordered by ≤. Given two elements x and y of P with x ≤ y, y is said to be greater than or equal to x. The word "partial" is meant to indicate that not every pair of elements of a partially ordered set is required to be comparable under the order relation, that is, in a partially ordered set P with order relation ≤ there may be elements x and y with neither x ≤ y nor y ≤ x. An ordered set in which every pair of elements is comparable is called totally ordered.
• Every subset S of a partially ordered set P can itself be seen as partially ordered by restricting the order relation inherited from P to S. A subset S of a partially ordered set P is called a chain (in P) if it is totally ordered in the inherited order.
• An element m of a partially ordered set P with order relation ≤ is maximal (with respect to ≤) if there is no other element of P greater than m, that is, if there is no s in P with s ≠ m and m ≤ s. Depending on the order relation, a partially ordered set may have any number of maximal elements. However, a totally ordered set can have at most one maximal element.
• Given a subset S of a partially ordered set P, an element u of P is an upper bound of S if it is greater than or equal to every element of S. Here, S is not required to be a chain, and u is required to be comparable to every element of S but need not itself be an element of S.
Zorn's lemma can then be stated as:
Zorn's lemma — Suppose a partially ordered set P has the property that every chain in P has an upper bound in P. Then the set P contains at least one maximal element.
Variants of this formulation are sometimes used, such as requiring that the set P and the chains be non-empty.[8]
Zorn's lemma (for non-empty sets) — Suppose a non-empty partially ordered set P has the property that every non-empty chain has an upper bound in P. Then the set P contains at least one maximal element.
Although this formulation appears to be formally weaker (since it places on P the additional condition of being non-empty, but obtains the same conclusion about P), in fact the two formulations are equivalent. To verify this, suppose first that P satisfies the condition that every chain in P has an upper bound in P. Then the empty subset of P is a chain, as it satisfies the definition vacuously; so the hypothesis implies that this subset must have an upper bound in P, and this upper bound shows that P is in fact non-empty. Conversely, if P is assumed to be non-empty and satisfies the hypothesis that every non-empty chain has an upper bound in P, then P also satisfies the condition that every chain has an upper bound, as an arbitrary element of P serves as an upper bound for the empty chain (that is, the empty subset viewed as a chain).
The difference may seem subtle, but in many proofs that invoke Zorn's lemma one takes unions of some sort to produce an upper bound, and so the case of the empty chain may be overlooked; that is, the verification that all chains have upper bounds may have to deal with empty and non-empty chains separately. So many authors prefer to verify the non-emptiness of the set P rather than deal with the empty chain in the general argument.[9]
Example applications
Every vector space has a basis
Zorn's lemma can be used to show that every vector space V has a basis.[10]
If V = {0}, then the empty set is a basis for V. Now, suppose that V ≠ {0}. Let P be the set consisting of all linearly independent subsets of V. Since V is not the zero vector space, there exists a nonzero element v of V, so P contains the linearly independent subset {v}. Furthermore, P is partially ordered by set inclusion (see inclusion order). Finding a maximal linearly independent subset of V is the same as finding a maximal element in P.
To apply Zorn's lemma, take a chain T in P (that is, T is a subset of P that is totally ordered). If T is the empty set, then {v} is an upper bound for T in P. Suppose then that T is non-empty. We need to show that T has an upper bound, that is, there exists a linearly independent subset B of V containing all the members of T.
Take B to be the union of all the sets in T. We wish to show that B is an upper bound for T in P. To do this, it suffices to show that B is a linearly independent subset of V.
Suppose otherwise, that B is not linearly independent. Then there exists vectors v1, v2, ..., vk ∈ B and scalars a1, a2, ..., ak, not all zero, such that
$a_{1}\mathbf {v} _{1}+a_{2}\mathbf {v} _{2}+\cdots +a_{k}\mathbf {v} _{k}=\mathbf {0} .$
Since B is the union of all the sets in T, there are some sets S1, S2, ..., Sk ∈ T such that vi ∈ Si for every i = 1, 2, ..., k. As T is totally ordered, one of the sets S1, S2, ..., Sk must contain the others, so there is some set Si that contains all of v1, v2, ..., vk. This tells us there is a linearly dependent set of vectors in Si, contradicting that Si is linearly independent (because it is a member of P).
The hypothesis of Zorn's lemma has been checked, and thus there is a maximal element in P, in other words a maximal linearly independent subset B of V.
Finally, we show that B is indeed a basis of V. It suffices to show that B is a spanning set of V. Suppose for the sake of contradiction that B is not spanning. Then there exists some v ∈ V not covered by the span of B. This says that B ∪ {v} is a linearly independent subset of V that is larger than B, contradicting the maximality of B. Therefore, B is a spanning set of V, and thus, a basis of V.
Every nontrivial ring with unity contains a maximal ideal
Zorn's lemma can be used to show that every nontrivial ring R with unity contains a maximal ideal.
Let P be the set consisting of all proper ideals in R (that is, all ideals in R except R itself). Since R is non-trivial, the set P contains the trivial ideal {0}. Furthermore, P is partially ordered by set inclusion. Finding a maximal ideal in R is the same as finding a maximal element in P.
To apply Zorn's lemma, take a chain T in P. If T is empty, then the trivial ideal {0} is an upper bound for T in P. Assume then that T is non-empty. It is necessary to show that T has an upper bound, that is, there exists an ideal I ⊆ R containing all the members of T but still smaller than R (otherwise it would not be a proper ideal, so it is not in P).
Take I to be the union of all the ideals in T. We wish to show that I is an upper bound for T in P. We will first show that I is an ideal of R. For I to be an ideal, it must satisfy three conditions:
1. I is a nonempty subset of R,
2. For every x, y ∈ I, the sum x + y is in I,
3. For every r ∈ R and every x ∈ I, the product rx is in I.
#1 - I is a nonempty subset of R.
Because T contains at least one element, and that element contains at least 0, the union I contains at least 0 and is not empty. Every element of T is a subset of R, so the union I only consists of elements in R.
#2 - For every x, y ∈ I, the sum x + y is in I.
Suppose x and y are elements of I. Then there exist two ideals J, K ∈ T such that x is an element of J and y is an element of K. Since T is totally ordered, we know that J ⊆ K or K ⊆ J. Without loss of generality, assume the first case. Both x and y are members of the ideal K, therefore their sum x + y is a member of K, which shows that x + y is a member of I.
#3 - For every r ∈ R and every x ∈ I, the product rx is in I.
Suppose x is an element of I. Then there exists an ideal J ∈ T such that x is in J. If r ∈ R, then rx is an element of J and hence an element of I. Thus, I is an ideal in R.
Now, we show that I is a proper ideal. An ideal is equal to R if and only if it contains 1. (It is clear that if it is R then it contains 1; on the other hand, if it contains 1 and r is an arbitrary element of R, then r1 = r is an element of the ideal, and so the ideal is equal to R.) So, if I were equal to R, then it would contain 1, and that means one of the members of T would contain 1 and would thus be equal to R – but R is explicitly excluded from P.
The hypothesis of Zorn's lemma has been checked, and thus there is a maximal element in P, in other words a maximal ideal in R.
Proof sketch
A sketch of the proof of Zorn's lemma follows, assuming the axiom of choice. Suppose the lemma is false. Then there exists a partially ordered set, or poset, P such that every totally ordered subset has an upper bound, and that for every element in P there is another element bigger than it. For every totally ordered subset T we may then define a bigger element b(T), because T has an upper bound, and that upper bound has a bigger element. To actually define the function b, we need to employ the axiom of choice.
Using the function b, we are going to define elements a0 < a1 < a2 < a3 < ... < aω < aω+1 <…, in P. This uncountably infinite sequence is really long: the indices are not just the natural numbers, but all ordinals. In fact, the sequence is too long for the set P; there are too many ordinals (a proper class), more than there are elements in any set (in other words, given any set of ordinals, there exists a larger ordinal), and the set P will be exhausted before long and then we will run into the desired contradiction.
The ai are defined by transfinite recursion: we pick a0 in P arbitrary (this is possible, since P contains an upper bound for the empty set and is thus not empty) and for any other ordinal w we set aw = b({av : v < w}). Because the av are totally ordered, this is a well-founded definition.
The above proof can be formulated without explicitly referring to ordinals by considering the initial segments {av : v < w} as subsets of P. Such sets can be easily characterized as well-ordered chains S ⊆ P where each x ∈ S satisfies x = b({y ∈ S : y < x}). Contradiction is reached by noting that we can always find a "next" initial segment either by taking the union of all such S (corresponding to the limit ordinal case) or by appending b(S) to the "last" S (corresponding to the successor ordinal case).[11]
This proof shows that actually a slightly stronger version of Zorn's lemma is true:
Lemma — If P is a poset in which every well-ordered subset has an upper bound, and if x is any element of P, then P has a maximal element greater than or equal to x. That is, there is a maximal element which is comparable to x.
History
The Hausdorff maximal principle is an early statement similar to Zorn's lemma.
Kazimierz Kuratowski proved in 1922[12] a version of the lemma close to its modern formulation (it applies to sets ordered by inclusion and closed under unions of well-ordered chains). Essentially the same formulation (weakened by using arbitrary chains, not just well-ordered) was independently given by Max Zorn in 1935,[13] who proposed it as a new axiom of set theory replacing the well-ordering theorem, exhibited some of its applications in algebra, and promised to show its equivalence with the axiom of choice in another paper, which never appeared.
The name "Zorn's lemma" appears to be due to John Tukey, who used it in his book Convergence and Uniformity in Topology in 1940. Bourbaki's Théorie des Ensembles of 1939 refers to a similar maximal principle as "le théorème de Zorn".[14] The name "Kuratowski–Zorn lemma" prevails in Poland and Russia.
Equivalent forms of Zorn's lemma
See also: Axiom of choice § Equivalents
Zorn's lemma is equivalent (in ZF) to three main results:
1. Hausdorff maximal principle
2. Axiom of choice
3. Well-ordering theorem.
A well-known joke alluding to this equivalency (which may defy human intuition) is attributed to Jerry Bona: "The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"[15]
Zorn's lemma is also equivalent to the strong completeness theorem of first-order logic.[16]
Moreover, Zorn's lemma (or one of its equivalent forms) implies some major results in other mathematical areas. For example,
1. Banach's extension theorem which is used to prove one of the most fundamental results in functional analysis, the Hahn–Banach theorem
2. Every vector space has a basis, a result from linear algebra (to which it is equivalent[17]). In particular, the real numbers, as a vector space over the rational numbers, possess a Hamel basis.
3. Every commutative unital ring has a maximal ideal, a result from ring theory known as Krull's theorem, to which Zorn's lemma is equivalent[18]
4. Tychonoff's theorem in topology (to which it is also equivalent[19])
5. Every proper filter is contained in an ultrafilter, a result that yields completeness theorem of first-order logic[20]
In this sense, we see how Zorn's lemma can be seen as a powerful tool, applicable to many areas of mathematics.
Analogs under weakenings of the axiom of choice
See also: Axiom of dependent choice
A weakened form of Zorn's lemma can be proven from ZF + DC (Zermelo–Fraenkel set theory with the axiom of choice replaced by the axiom of dependent choice). Zorn's lemma can be expressed straightforwardly by observing that the set having no maximal element would be equivalent to stating that the set's ordering relation would be entire, which would allow us to apply the axiom of dependent choice to construct a countable chain. As a result, any partially ordered set with exclusively finite chains must have a maximal element.[21]
More generally, strengthening the axiom of dependent choice to higher ordinals allows us to generalize the statement in the previous paragraph to higher cardinalities.[21] In the limit where we allow arbitrarily large ordinals, we recover the proof of the full Zorn's lemma using the axiom of choice in the preceding section.
In popular culture
The 1970 film Zorns Lemma is named after the lemma.
The lemma was referenced on The Simpsons in the episode "Bart's New Friend".[22]
See also
• Antichain – Subset of incomparable elements
• Bourbaki–Witt theorem
• Chain-complete partial order – a partially ordered set in which every chain has a least upper bound
• Szpilrajn extension theorem – Mathematical result on order relations
• Tarski finiteness – Mathematical set containing a finite number of elementsPages displaying short descriptions of redirect targets
• Teichmüller–Tukey lemma (sometimes named Tukey's lemma)
Notes
1. Serre, Jean-Pierre (2003), Trees, Springer Monographs in Mathematics, Springer, p. 23
2. Moore 2013, p. 168
3. Wilansky, Albert (1964). Functional Analysis. New York: Blaisdell. pp. 16–17.
4. Jech 2008, ch. 2, §2 Some applications of the Axiom of Choice in mathematics
5. Jech 2008, p. 9
6. Moore 2013, p. 168
7. William Timothy Gowers (12 August 2008). "How to use Zorn's lemma".
8. For example, Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Vol. 211 (Revised 3rd ed.). Springer-Verlag. p. 880. ISBN 978-0-387-95385-4., Dummit, David S.; Foote, Richard M. (1998). Abstract Algebra (2nd ed.). Prentice Hall. p. 875. ISBN 978-0-13-569302-5., and Bergman, George M (2015). An Invitation to General Algebra and Universal Constructions. Universitext (2nd ed.). Springer-Verlag. p. 162. ISBN 978-3-319-11477-4..
9. Bergman, George M (2015). An Invitation to General Algebra and Universal Constructions. Universitext (Second ed.). Springer-Verlag. p. 164. ISBN 978-3-319-11477-4.
10. Smits, Tim. "A Proof that every Vector Space has a Basis" (PDF). Retrieved 14 August 2022.
11. Lewin, Jonathan W. (1991). "A simple proof of Zorn's lemma". The American Mathematical Monthly. 98 (4): 353–354. doi:10.1080/00029890.1991.12000768.
12. Kuratowski, Casimir (1922). "Une méthode d'élimination des nombres transfinis des raisonnements mathématiques" [A method of disposing of transfinite numbers of mathematical reasoning] (PDF). Fundamenta Mathematicae (in French). 3: 76–108. doi:10.4064/fm-3-1-76-108. Retrieved 24 April 2013.
13. Zorn, Max (1935). "A remark on method in transfinite algebra". Bulletin of the American Mathematical Society. 41 (10): 667–670. doi:10.1090/S0002-9904-1935-06166-X.
14. Campbell 1978, p. 82.
15. Krantz, Steven G. (2002), "The Axiom of Choice", Handbook of Logic and Proof Techniques for Computer Science, Springer, pp. 121–126, doi:10.1007/978-1-4612-0115-1_9, ISBN 978-1-4612-6619-8.
16. J.L. Bell & A.B. Slomson (1969). Models and Ultraproducts. North Holland Publishing Company. Chapter 5, Theorem 4.3, page 103.
17. Blass, Andreas (1984). "Existence of bases implies the Axiom of Choice". Axiomatic Set Theory. pp. 31–33. doi:10.1090/conm/031/763890. ISBN 9780821850268. {{cite book}}: |journal= ignored (help)
18. Hodges, W. (1979). "Krull implies Zorn". Journal of the London Mathematical Society. s2-19 (2): 285–287. doi:10.1112/jlms/s2-19.2.285.
19. Kelley, John L. (1950). "The Tychonoff product theorem implies the axiom of choice". Fundamenta Mathematicae. 37: 75–76. doi:10.4064/fm-37-1-75-76.
20. J.L. Bell & A.B. Slomson (1969). Models and Ultraproducts. North Holland Publishing Company.
21. Wolk, Elliot S. (1983), "On the principle of dependent choices and some forms of Zorn's lemma", Canadian Mathematical Bulletin, 26 (3): 365–367, doi:10.4153/CMB-1983-062-5
22. "Zorn's Lemma | The Simpsons and their Mathematical Secrets".
References
• Campbell, Paul J. (February 1978). "The Origin of 'Zorn's Lemma'". Historia Mathematica. 5 (1): 77–89. doi:10.1016/0315-0860(78)90136-2.
• Ciesielski, Krzysztof (1997). Set Theory for the Working Mathematician. Cambridge University Press. ISBN 978-0-521-59465-3.
• Jech, Thomas (2008) [1973]. The Axiom of Choice. Mineola, New York: Dover Publications. ISBN 978-0-486-46624-8.
• Moore, Gregory H. (2013) [1982]. Zermelo's axiom of choice: Its origins, development & influence. Dover Publications. ISBN 978-0-486-48841-7.
External links
• "Zorn lemma", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Zorn's Lemma at ProvenMath contains a formal proof down to the finest detail of the equivalence of the axiom of choice and Zorn's Lemma.
• Zorn's Lemma at Metamath is another formal proof. (Unicode version for recent browsers.)
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Zorn ring
In mathematics, a Zorn ring is an alternative ring in which for every non-nilpotent x there exists an element y such that xy is a non-zero idempotent (Kaplansky 1968, pages 19, 25). Kaplansky (1951) named them after Max August Zorn, who studied a similar condition in (Zorn 1941).
For associative rings, the definition of Zorn ring can be restated as follows: the Jacobson radical J(R) is a nil ideal and every right ideal of R which is not contained in J(R) contains a nonzero idempotent. Replacing "right ideal" with "left ideal" yields an equivalent definition. Left or right Artinian rings, left or right perfect rings, semiprimary rings and von Neumann regular rings are all examples of associative Zorn rings.
References
• Kaplansky, Irving (1951), "Semi-simple alternative rings", Portugaliae Mathematica, 10 (1): 37–50, MR 0041835
• Kaplansky, I. (1968), Rings of Operators, New York: W. A. Benjamin, Inc.
• Tuganbaev, A. A. (2002), "Semiregular, weakly regular, and π-regular rings", J. Math. Sci. (New York), 109 (3): 1509–1588, doi:10.1023/A:1013929008743, MR 1871186, S2CID 189870092
• Zorn, Max (1941), "Alternative rings and related questions I: existence of the radical", Annals of Mathematics, Second Series, 42 (3): 676–686, doi:10.2307/1969256, JSTOR 1969256, MR 0005098
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Zorya Shapiro
Zorya Yakovlevna Shapiro (Russian: Зоря Яковлевна Шапиро; 7 December 1914 – 4 July 2013) was a Soviet mathematician, educator and translator. She is known for her contributions to representation theory and functional analysis in her collaboration with Israel Gelfand, and the Shapiro-Lobatinski condition in elliptical boundary value problems.
Zorya Shapiro
Born(1914-12-07)December 7, 1914
DiedJuly 4, 2013(2013-07-04) (aged 98)
River Forest, Illinois
CitizenshipSoviet
Alma materMSU Faculty of Mechanics and Mathematics
Known forShapiro-Lopatinski condition in elliptic boundary value problems
SpouseIsrael Gelfand
Scientific career
Fieldsrepresentation theory
Thesis (1938)
Life
Zorya Shapiro attended the Moscow State University Faculty of Mechanics and Mathematics from where she received her undergraduate and doctoral degrees by 1938.[1] She was active in the military department of the university, especially in aviation, learning to fly and land aeroplanes.[2]
She started her teaching career at the Faculty, shortly after Zoya Kishkina (1917–1989) and Natalya Eisenstadt (1912–1985), and very quickly became recognized for her courses in analysis.[1]
Shapiro married Israel Gelfand in 1942. They had 3 sons, one of whom died in childhood.[3] Shapiro and Gelfand later divorced.[4]
In the 1980s, Shapiro lived in the same house as Akiva Yaglom.[5] In 1991 Shapiro moved to River Forest, Illinois to live with her younger son. She died there on 4 July 2013.
Career
Shapiro published several works on representation theory. A contribution (with Gelfand) in integral geometry was to find inversion formulae for the reconstruction of the value of a function on a manifold in terms of integrals over a family of submanifolds, a result with applicability in non-linear differential equations, tomography, multi-dimensional complex analysis and other domains.[6] Another work was on the representations of rotation groups of 3-dimensional spaces.[7]
Shapiro is best known for her elucidation of the conditions for well-defined solutions to the elliptical boundary value problem on Sobolev spaces.[8]
Selected publications
Articles
• "О существовании квазиконформных отображений". Доклады АН СССР. 30 (8). 1941.
• "Об эллиптических системах уравнений с частными производными". Доклады АН СССР. XLVI (4): 146–149. 1945.
• "Первая краевая задача для эллиптической системы дифференциальных уравнений" (PDF). Математический сборник. 28(70) (1): 55–78. 1951.
• "Представления группы вращений трёхмерного пространства и их применения". УМН. 7 (1(47)): 3–117. 1952. (with I.M. Gelfand)
• "Об общих краевых задачах для уравнений эллиптического типа" (PDF). Известия АН СССР. 17 (6): 539–565. 1953.
• "Однородные функции и их приложения" (PDF). Успехи математических наук. 10 (3(65)): 3–70. 1955. (with I.M. Gelfand)
• "Об одном классе обобщённых функций" (PDF). Успехи математических наук. 13 (3(81)): 205–212. 1958.
• "Интегральная геометрия на многообразии k-мерных плоскостей". Доклады АН СССР. 168 (6): 1236–1238. 1966. (with I.M. Gelfand, M.I. Graev)
• "Интегральная геометрия на k-мерных плоскостях" (PDF). Функциональный анализ и его приложения. 1 (1): 15–31. 1967. (with I.M. Gelfand, M.I. Graev)
• "Дифференциальные формы и интегральная геометрия" (PDF). Функциональный анализ и его приложения. 3 (2): 24–40. 1969. (with I.M. Gelfand, M.I. Graev)
• "Интегральная геометрия в проективном пространстве". Функциональный анализ и его приложения. 4 (1): 14–32. 1970. (with I.M. Gelfand, M.I. Graev)
• "Локальная задача интегральной геометрии в пространстве кривых" (PDF). Функциональный анализ и его приложения. 13 (2): 11–31. 1979. (with I.M. Gelfand, S.G. Gindikin)
Books
• Representations of the rotation and Lorentz groups and their applications. Macmillan. 1963. (with I.M. Gelfand, R.A. Minlos)
From French
• Jean Leray (1961). Дифференциальное и интегральное исчисления на комплексном аналитическом многообразии. Moscow: Foreign Literature.
From English
• Stanislaw Ulam (1964). Collection of Mathematical Problems [Нерешённые математические задачи]. Moscow: Nauka.
• Robert Finn (1989). Equilibrium Capillary Surfaces [Равновесные капиллярные поверхности: Математическая теория]. Moscow: Mir.
References
1. Vladimir Tikhomirov (2010). "Прогулки с И.М. Гельфандом". Семь Исскуств. 11 (12).
2. Vladimir Tikhomirov (2011). Ровесники Октября. p. 6. ISBN 9785040175239.
3. "Israel Gelfand". The Daily Telegraph. 26 October 2009. Retrieved 28 October 2018.
4. Thomas Maugh (2 November 2009). "Mathematics genius had it all figured out". The Sydney Morning Herald. Retrieved 28 October 2018.
5. "Akiva M. Yaglom, Dec. 2, 1988; Part 2". Retrieved 28 October 2018.
6. David Kazhdan (2003). "Works of I. Gelfand on the theory of representations" (PDF). An International Conference on "The Unity of Mathematics": 5.
7. A.A. Yushkevich (5 September 2017). "Колмогоров на моем пути в математику". Колмогоров в воспоминаниях учеников. p. 425. ISBN 978-5-457-91890-0.
8. Katsiaryna Krupchyk; Jukka Tuomela (2006). "The Shapiro–Lopatinskij Condition for Elliptic Boundary Value Problems". LMS Journal of Computation and Mathematics. 9: 287–329. doi:10.1112/S1461157000001285.
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Zoé Chatzidakis
Zoé Maria Chatzidakis is a mathematician who works as a director of research at the École Normale Supérieure in Paris, France.[1] Her research concerns model theory and difference algebra. She was invited to give the Tarski Lectures in 2020, though the lectures were postponed due to the COVID-19 pandemic.[2]
Zoé Chatzidakis
Alma materYale university
AwardsLeconte Prize (2013) Tarski Lectures (2020)
Scientific career
FieldsMathematics, Model theory, Algebra
InstitutionsÉcole normale supérieure (Paris)
ThesisModel Theory of Profinite Groups (1984)
Doctoral advisorAngus John Macintyre
Education and employment
Chatzidakis earned her Ph.D. in 1984 from Yale University, under the supervision of Angus Macintyre, with a dissertation on the model theory of profinite groups.[3] She is Senior researcher and team director in Algebra and Geometry in the Département de mathématiques et applications de l'École Normale Supérieure.[4][5]
Honors and awards
She was the 2013 winner of the Leconte Prize,[6] and was an invited speaker at the International Congress of Mathematicians in 2014.[7] She was named MSRI Chern Professor for Fall 2020.[8]
References
1. Member directory, ENS/DMA, retrieved 2016-07-02.
2. "The Tarski Lectures | Department of Mathematics at University of California Berkeley". math.berkeley.edu. Retrieved 2021-11-02. Update on March 10th 2020: The event has been postponed to next year
3. Zoé Chatzidakis at the Mathematics Genealogy Project.
4. "Mathematics at Ecole Normale Supérieure - Algebra and Geometry". www.math.ens.fr. Retrieved 2021-06-07.
5. "Gestion membre". www.math.ens.fr. Retrieved 2021-06-07.
6. Leconte Prize citation, French Academy of Sciences, retrieved 2016-07-02.
7. ICM Plenary and Invited Speakers since 1897, International Mathematical Union, retrieved 2016-07-02.
8. MSRI. "Mathematical Sciences Research Institute". www.msri.org. Retrieved 2021-06-07.
External links
• Home page
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• ISNI
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• Israel
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